LMFDB - Embedded newform 177.5.c.a.58.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(58,177)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("177.58");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.7
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.34

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.77154i q^{2} -5.19615 q^{3} -17.3107 q^{4} +34.3452 q^{5} +29.9898i q^{6} -39.2727 q^{7} +7.56484i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-5.77154i q^{2} -5.19615 q^{3} -17.3107 q^{4} +34.3452 q^{5} +29.9898i q^{6} -39.2727 q^{7} +7.56484i q^{8} +27.0000 q^{9} -198.225i q^{10} +56.8541i q^{11} +89.9491 q^{12} -232.684i q^{13} +226.664i q^{14} -178.463 q^{15} -233.311 q^{16} -391.037 q^{17} -155.832i q^{18} -352.390 q^{19} -594.541 q^{20} +204.067 q^{21} +328.136 q^{22} -173.061i q^{23} -39.3081i q^{24} +554.595 q^{25} -1342.94 q^{26} -140.296 q^{27} +679.838 q^{28} +90.7462 q^{29} +1030.01i q^{30} +128.035i q^{31} +1467.60i q^{32} -295.422i q^{33} +2256.89i q^{34} -1348.83 q^{35} -467.389 q^{36} -413.192i q^{37} +2033.83i q^{38} +1209.06i q^{39} +259.816i q^{40} -761.796 q^{41} -1177.78i q^{42} +874.013i q^{43} -984.184i q^{44} +927.321 q^{45} -998.831 q^{46} +895.574i q^{47} +1212.32 q^{48} -858.658 q^{49} -3200.87i q^{50} +2031.89 q^{51} +4027.92i q^{52} +3689.46 q^{53} +809.725i q^{54} +1952.67i q^{55} -297.092i q^{56} +1831.07 q^{57} -523.746i q^{58} +(-1547.97 - 3117.88i) q^{59} +3089.32 q^{60} +1498.06i q^{61} +738.962 q^{62} -1060.36 q^{63} +4737.35 q^{64} -7991.57i q^{65} -1705.04 q^{66} -3683.48i q^{67} +6769.13 q^{68} +899.253i q^{69} +7784.82i q^{70} -9230.55 q^{71} +204.251i q^{72} +4308.55i q^{73} -2384.76 q^{74} -2881.76 q^{75} +6100.12 q^{76} -2232.81i q^{77} +6978.14 q^{78} -4707.88 q^{79} -8013.11 q^{80} +729.000 q^{81} +4396.74i q^{82} -398.453i q^{83} -3532.54 q^{84} -13430.3 q^{85} +5044.40 q^{86} -471.531 q^{87} -430.092 q^{88} -12419.6i q^{89} -5352.08i q^{90} +9138.11i q^{91} +2995.82i q^{92} -665.292i q^{93} +5168.84 q^{94} -12102.9 q^{95} -7625.87i q^{96} +397.504i q^{97} +4955.78i q^{98} +1535.06i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	5.77154i		−	1.44289i	−0.692474	−	0.721443i	$-0.743479\pi$
$2$		−	5.77154i		−	1.44289i	0.692474	−	0.721443i	$-0.256521\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	−17.3107			−1.08192
$5$	34.3452			1.37381			0.686905	−	0.726748i	$-0.258969\pi$
$5$	34.3452			1.37381			0.686905	+	0.726748i	$0.258969\pi$
$6$			29.9898i			0.833051i
$7$	−39.2727			−0.801483			−0.400741	−	0.916191i	$-0.631247\pi$
$7$	−39.2727			−0.801483			−0.400741	+	0.916191i	$0.631247\pi$
$8$			7.56484i			0.118201i
$9$	27.0000			0.333333
$10$		−	198.225i		−	1.98225i
$11$			56.8541i			0.469868i	0.972011	+	0.234934i	$0.0754874\pi$
$11$			56.8541i			0.469868i	−0.972011	+	0.234934i	$0.924513\pi$
$12$	89.9491			0.624647
$13$		−	232.684i		−	1.37683i	−0.725319	−	0.688413i	$-0.758308\pi$
$13$		−	232.684i		−	1.37683i	0.725319	−	0.688413i	$-0.241692\pi$
$14$			226.664i			1.15645i
$15$	−178.463			−0.793169
$16$	−233.311			−0.911370
$17$	−391.037			−1.35307			−0.676535	−	0.736411i	$-0.736519\pi$
$17$	−391.037			−1.35307			−0.676535	+	0.736411i	$0.736519\pi$
$18$		−	155.832i		−	0.480962i
$19$	−352.390			−0.976149			−0.488075	−	0.872802i	$-0.662301\pi$
$19$	−352.390			−0.976149			−0.488075	+	0.872802i	$0.662301\pi$
$20$	−594.541			−1.48635
$21$	204.067			0.462736
$22$	328.136			0.677966
$23$		−	173.061i		−	0.327148i	−0.986531	−	0.163574i	$-0.947698\pi$
$23$		−	173.061i		−	0.327148i	0.986531	−	0.163574i	$-0.0523023\pi$
$24$		−	39.3081i		−	0.0682432i
$25$	554.595			0.887352
$26$	−1342.94			−1.98660
$27$	−140.296			−0.192450
$28$	679.838			0.867140
$29$	90.7462			0.107903			0.0539514	−	0.998544i	$-0.482818\pi$
$29$	90.7462			0.107903			0.0539514	+	0.998544i	$0.482818\pi$
$30$			1030.01i			1.14445i
$31$			128.035i			0.133232i	0.997779	+	0.0666158i	$0.0212202\pi$
$31$			128.035i			0.133232i	−0.997779	+	0.0666158i	$0.978780\pi$
$32$			1467.60i			1.43320i
$33$		−	295.422i		−	0.271279i
$34$			2256.89i			1.95233i
$35$	−1348.83			−1.10108
$36$	−467.389			−0.360640
$37$		−	413.192i		−	0.301821i	−0.988547	−	0.150910i	$-0.951780\pi$
$37$		−	413.192i		−	0.301821i	0.988547	−	0.150910i	$-0.0482204\pi$
$38$			2033.83i			1.40847i
$39$			1209.06i			0.794911i
$40$			259.816i			0.162385i
$41$	−761.796			−0.453180			−0.226590	−	0.973990i	$-0.572758\pi$
$41$	−761.796			−0.453180			−0.226590	+	0.973990i	$0.572758\pi$
$42$		−	1177.78i		−	0.667676i
$43$			874.013i			0.472695i	0.971669	+	0.236347i	$0.0759504\pi$
$43$			874.013i			0.472695i	−0.971669	+	0.236347i	$0.924050\pi$
$44$		−	984.184i		−	0.508360i
$45$	927.321			0.457936
$46$	−998.831			−0.472038
$47$			895.574i			0.405420i	0.979239	+	0.202710i	$0.0649749\pi$
$47$			895.574i			0.405420i	−0.979239	+	0.202710i	$0.935025\pi$
$48$	1212.32			0.526179
$49$	−858.658			−0.357625
$50$		−	3200.87i		−	1.28035i
$51$	2031.89			0.781195
$52$			4027.92i			1.48962i
$53$	3689.46			1.31344			0.656721	−	0.754134i	$-0.271943\pi$
$53$	3689.46			1.31344			0.656721	+	0.754134i	$0.271943\pi$
$54$			809.725i			0.277684i
$55$			1952.67i			0.645509i
$56$		−	297.092i		−	0.0947358i
$57$	1831.07			0.563580
$58$		−	523.746i		−	0.155691i
$59$	−1547.97	−	3117.88i	−0.444691	−	0.895684i
$59$	−1547.97	−	3117.88i	−0.444691	−	0.895684i
$60$	3089.32			0.858145
$61$			1498.06i			0.402596i	0.979530	+	0.201298i	$0.0645159\pi$
$61$			1498.06i			0.402596i	−0.979530	+	0.201298i	$0.935484\pi$
$62$	738.962			0.192238
$63$	−1060.36			−0.267161
$64$	4737.35			1.15658
$65$		−	7991.57i		−	1.89150i
$66$	−1705.04			−0.391424
$67$		−	3683.48i		−	0.820558i	−0.911960	−	0.410279i	$-0.865431\pi$
$67$		−	3683.48i		−	0.820558i	0.911960	−	0.410279i	$-0.134569\pi$
$68$	6769.13			1.46391
$69$			899.253i			0.188879i
$70$			7784.82i			1.58874i
$71$	−9230.55			−1.83109			−0.915547	−	0.402211i	$-0.868242\pi$
$71$	−9230.55			−1.83109			−0.915547	+	0.402211i	$0.868242\pi$
$72$			204.251i			0.0394002i
$73$			4308.55i			0.808510i	0.914646	+	0.404255i	$0.132469\pi$
$73$			4308.55i			0.808510i	−0.914646	+	0.404255i	$0.867531\pi$
$74$	−2384.76			−0.435493
$75$	−2881.76			−0.512313
$76$	6100.12			1.05612
$77$		−	2232.81i		−	0.376591i
$78$	6978.14			1.14697
$79$	−4707.88			−0.754346			−0.377173	−	0.926143i	$-0.623104\pi$
$79$	−4707.88			−0.754346			−0.377173	+	0.926143i	$0.623104\pi$
$80$	−8013.11			−1.25205
$81$	729.000			0.111111
$82$			4396.74i			0.653887i
$83$		−	398.453i		−	0.0578390i	−0.999582	−	0.0289195i	$-0.990793\pi$
$83$		−	398.453i		−	0.0578390i	0.999582	−	0.0289195i	$-0.00920665\pi$
$84$	−3532.54			−0.500644
$85$	−13430.3			−1.85886
$86$	5044.40			0.682045
$87$	−471.531			−0.0622977
$88$	−430.092			−0.0555387
$89$		−	12419.6i		−	1.56794i	−0.620802	−	0.783968i	$-0.713193\pi$
$89$		−	12419.6i		−	1.56794i	0.620802	−	0.783968i	$-0.286807\pi$
$90$		−	5352.08i		−	0.660750i
$91$			9138.11i			1.10350i
$92$			2995.82i			0.353948i
$93$		−	665.292i		−	0.0769212i
$94$	5168.84			0.584975
$95$	−12102.9			−1.34104
$96$		−	7625.87i		−	0.827460i
$97$			397.504i			0.0422472i	0.999777	+	0.0211236i	$0.00672435\pi$
$97$			397.504i			0.0422472i	−0.999777	+	0.0211236i	$0.993276\pi$
$98$			4955.78i			0.516012i
$99$			1535.06i			0.156623i
$100$	−9600.44			−0.960044
$101$		−	7090.10i		−	0.695040i	−0.937673	−	0.347520i	$-0.887024\pi$
$101$		−	7090.10i		−	0.695040i	0.937673	−	0.347520i	$-0.112976\pi$
$102$		−	11727.1i		−	1.12718i
$103$			6266.89i			0.590714i	0.955387	+	0.295357i	$0.0954387\pi$
$103$			6266.89i			0.590714i	−0.955387	+	0.295357i	$0.904561\pi$
$104$	1760.22			0.162742
$105$	7008.72			0.635712
$106$		−	21293.9i		−	1.89515i
$107$	1508.55			0.131763			0.0658814	−	0.997827i	$-0.479014\pi$
$107$	1508.55			0.131763			0.0658814	+	0.997827i	$0.479014\pi$
$108$	2428.63			0.208216
$109$		−	16261.8i		−	1.36872i	−0.729144	−	0.684361i	$-0.760081\pi$
$109$		−	16261.8i		−	1.36872i	0.729144	−	0.684361i	$-0.239919\pi$
$110$	11269.9			0.931396
$111$			2147.01i			0.174256i
$112$	9162.73			0.730447
$113$		−	22671.8i		−	1.77553i	−0.460293	−	0.887767i	$-0.652256\pi$
$113$		−	22671.8i		−	1.77553i	0.460293	−	0.887767i	$-0.347744\pi$
$114$		−	10568.1i		−	0.813182i
$115$		−	5943.83i		−	0.449439i
$116$	−1570.88			−0.116742
$117$		−	6282.46i		−	0.458942i
$118$	−17995.0	+	8934.17i	−1.29237	+	0.641638i
$119$	15357.1			1.08446
$120$		−	1350.05i		−	0.0937531i
$121$	11408.6			0.779224
$122$	8646.11			0.580900
$123$	3958.41			0.261644
$124$		−	2216.39i		−	0.144146i
$125$	−2418.07			−0.154757
$126$			6119.93i			0.385483i
$127$	7430.12			0.460668			0.230334	−	0.973112i	$-0.426018\pi$
$127$	7430.12			0.460668			0.230334	+	0.973112i	$0.426018\pi$
$128$		−	3860.20i		−	0.235608i
$129$		−	4541.50i		−	0.272911i
$130$	−46123.7			−2.72921
$131$			30063.5i			1.75185i	0.482446	+	0.875926i	$0.339748\pi$
$131$			30063.5i			1.75185i	−0.482446	+	0.875926i	$0.660252\pi$
$132$			5113.97i			0.293502i
$133$	13839.3			0.782367
$134$	−21259.4			−1.18397
$135$	−4818.50			−0.264390
$136$		−	2958.14i		−	0.159934i
$137$	20907.4			1.11393			0.556967	−	0.830535i	$-0.311965\pi$
$137$	20907.4			1.11393			0.556967	+	0.830535i	$0.311965\pi$
$138$	5190.08			0.272531
$139$	22666.8			1.17317			0.586583	−	0.809889i	$-0.300473\pi$
$139$	22666.8			1.17317			0.586583	+	0.809889i	$0.300473\pi$
$140$	23349.2			1.19129
$141$		−	4653.54i		−	0.234070i
$142$			53274.5i			2.64206i
$143$	13229.0			0.646927
$144$	−6299.39			−0.303790
$145$	3116.70			0.148238
$146$	24867.0			1.16659
$147$	4461.72			0.206475
$148$			7152.65i			0.326546i
$149$			14748.1i			0.664299i	0.943227	+	0.332150i	$0.107774\pi$
$149$			14748.1i			0.664299i	−0.943227	+	0.332150i	$0.892226\pi$
$150$			16632.2i			0.739209i
$151$		−	1352.46i		−	0.0593158i	−0.999560	−	0.0296579i	$-0.990558\pi$
$151$		−	1352.46i		−	0.0593158i	0.999560	−	0.0296579i	$-0.00944179\pi$
$152$		−	2665.77i		−	0.115382i
$153$	−10558.0			−0.451023
$154$	−12886.8			−0.543378
$155$			4397.41i			0.183035i
$156$		−	20929.7i		−	0.860030i
$157$		−	43867.4i		−	1.77968i	−0.456270	−	0.889841i	$-0.650815\pi$
$157$		−	43867.4i		−	1.77968i	0.456270	−	0.889841i	$-0.349185\pi$
$158$			27171.7i			1.08844i
$159$	−19171.0			−0.758316
$160$			50405.1i			1.96895i
$161$			6796.58i			0.262204i
$162$		−	4207.46i		−	0.160321i
$163$	24914.8			0.937741			0.468870	−	0.883267i	$-0.344661\pi$
$163$	24914.8			0.937741			0.468870	+	0.883267i	$0.344661\pi$
$164$	13187.2			0.490304
$165$		−	10146.3i		−	0.372685i
$166$	−2299.69			−0.0834551
$167$	−15384.8			−0.551644			−0.275822	−	0.961209i	$-0.588950\pi$
$167$	−15384.8			−0.551644			−0.275822	+	0.961209i	$0.588950\pi$
$168$			1543.73i			0.0546958i
$169$	−25580.7			−0.895651
$170$			77513.4i			2.68212i
$171$	−9514.53			−0.325383
$172$		−	15129.8i		−	0.511418i
$173$		−	54918.5i		−	1.83496i	−0.397784	−	0.917479i	$-0.630221\pi$
$173$		−	54918.5i		−	1.83496i	0.397784	−	0.917479i	$-0.369779\pi$
$174$			2721.46i			0.0898884i
$175$	−21780.4			−0.711198
$176$		−	13264.7i		−	0.428224i
$177$	8043.48	+	16201.0i	0.256742	+	0.517123i
$178$	−71680.3			−2.26235
$179$		−	50544.7i		−	1.57750i	−0.614713	−	0.788751i	$-0.710728\pi$
$179$		−	50544.7i		−	1.57750i	0.614713	−	0.788751i	$-0.289272\pi$
$180$	−16052.6			−0.495450
$181$	34588.1			1.05577			0.527885	−	0.849316i	$-0.322985\pi$
$181$	34588.1			1.05577			0.527885	+	0.849316i	$0.322985\pi$
$182$	52741.0			1.59223
$183$		−	7784.14i		−	0.232439i
$184$	1309.18			0.0386691
$185$		−	14191.2i		−	0.414644i
$186$	−3839.76			−0.110989
$187$		−	22232.0i		−	0.635765i
$188$		−	15503.0i		−	0.438632i
$189$	5509.80			0.154245
$190$			69852.5i			1.93497i
$191$			29175.3i			0.799739i	0.916572	+	0.399869i	$0.130945\pi$
$191$			29175.3i			0.799739i	−0.916572	+	0.399869i	$0.869055\pi$
$192$	−24616.0			−0.667751
$193$	31941.0			0.857500			0.428750	−	0.903423i	$-0.358954\pi$
$193$	31941.0			0.857500			0.428750	+	0.903423i	$0.358954\pi$
$194$	2294.21			0.0609579
$195$			41525.4i			1.09206i
$196$	14864.0			0.386922
$197$	−4250.44			−0.109522			−0.0547610	−	0.998499i	$-0.517440\pi$
$197$	−4250.44			−0.109522			−0.0547610	+	0.998499i	$0.517440\pi$
$198$	8859.66			0.225989
$199$	28562.3			0.721251			0.360625	−	0.932711i	$-0.382563\pi$
$199$	28562.3			0.721251			0.360625	+	0.932711i	$0.382563\pi$
$200$			4195.43i			0.104886i
$201$			19139.9i			0.473749i
$202$	−40920.8			−1.00286
$203$	−3563.84			−0.0864822
$204$	−35173.4			−0.845190
$205$	−26164.1			−0.622583
$206$	36169.6			0.852334
$207$		−	4672.66i		−	0.109049i
$208$			54287.6i			1.25480i
$209$		−	20034.8i		−	0.458662i
$210$		−	40451.1i		−	0.917259i
$211$			26481.8i			0.594816i	0.954750	+	0.297408i	$0.0961222\pi$
$211$			26481.8i			0.594816i	−0.954750	+	0.297408i	$0.903878\pi$
$212$	−63867.2			−1.42104
$213$	47963.3			1.05718
$214$		−	8706.68i		−	0.190119i
$215$			30018.2i			0.649393i
$216$		−	1061.32i		−	0.0227477i
$217$		−	5028.29i		−	0.106783i
$218$	−93855.5			−1.97491
$219$		−	22387.9i		−	0.466793i
$220$		−	33802.0i		−	0.698389i
$221$			90988.0i			1.86294i
$222$	12391.6			0.251432
$223$	−20499.0			−0.412215			−0.206108	−	0.978529i	$-0.566080\pi$
$223$	−20499.0			−0.412215			−0.206108	+	0.978529i	$0.566080\pi$
$224$		−	57636.6i		−	1.14869i
$225$	14974.1			0.295784
$226$	−130851.			−2.56189
$227$		−	78399.4i		−	1.52146i	−0.649067	−	0.760731i	$-0.724841\pi$
$227$		−	78399.4i		−	1.52146i	0.649067	−	0.760731i	$-0.275159\pi$
$228$	−31697.2			−0.609748
$229$		−	31210.2i		−	0.595149i	−0.954699	−	0.297574i	$-0.903822\pi$
$229$		−	31210.2i		−	0.595149i	0.954699	−	0.297574i	$-0.0961776\pi$
$230$	−34305.1			−0.648490
$231$			11602.0i			0.217425i
$232$			686.481i			0.0127542i
$233$			91009.4i			1.67639i	0.545372	+	0.838194i	$0.316388\pi$
$233$			91009.4i			1.67639i	−0.545372	+	0.838194i	$0.683612\pi$
$234$	−36259.5			−0.662201
$235$			30758.7i			0.556970i
$236$	26796.5	+	53972.7i	0.481120	+	0.969058i
$237$	24462.8			0.435522
$238$		−	88634.0i		−	1.56476i
$239$	−96173.1			−1.68367			−0.841837	−	0.539732i	$-0.818526\pi$
$239$	−96173.1			−1.68367			−0.841837	+	0.539732i	$0.818526\pi$
$240$	41637.3			0.722870
$241$	−69208.9			−1.19159			−0.595796	−	0.803136i	$-0.703163\pi$
$241$	−69208.9			−1.19159			−0.595796	+	0.803136i	$0.703163\pi$
$242$		−	65845.3i		−	1.12433i
$243$	−3788.00			−0.0641500
$244$		−	25932.5i		−	0.435576i
$245$	−29490.8			−0.491309
$246$		−	22846.1i		−	0.377522i
$247$			81995.4i			1.34399i
$248$	−968.568			−0.0157481
$249$			2070.42i			0.0333934i
$250$			13956.0i			0.223296i
$251$	94134.4			1.49417			0.747087	−	0.664726i	$-0.231452\pi$
$251$	94134.4			1.49417			0.747087	+	0.664726i	$0.231452\pi$
$252$	18355.6			0.289047
$253$	9839.24			0.153717
$254$		−	42883.3i		−	0.664692i
$255$	69785.7			1.07321
$256$	53518.2			0.816623
$257$	11680.1			0.176841			0.0884203	−	0.996083i	$-0.471818\pi$
$257$	11680.1			0.176841			0.0884203	+	0.996083i	$0.471818\pi$
$258$	−26211.5			−0.393779
$259$			16227.2i			0.241904i
$260$			138340.i			2.04645i
$261$	2450.15			0.0359676
$262$	173513.			2.52772
$263$	−67551.5			−0.976616			−0.488308	−	0.872671i	$-0.662386\pi$
$263$	−67551.5			−0.976616			−0.488308	+	0.872671i	$0.662386\pi$
$264$	2234.82			0.0320653
$265$	126715.			1.80442
$266$		−	79874.1i		−	1.12887i
$267$			64534.2i			0.905248i
$268$			63763.7i			0.887778i
$269$		−	8085.17i		−	0.111734i	−0.998438	−	0.0558669i	$-0.982208\pi$
$269$		−	8085.17i		−	0.111734i	0.998438	−	0.0558669i	$-0.0177923\pi$
$270$			27810.2i			0.381484i
$271$	−3043.53			−0.0414419			−0.0207209	−	0.999785i	$-0.506596\pi$
$271$	−3043.53			−0.0414419			−0.0207209	+	0.999785i	$0.506596\pi$
$272$	91233.1			1.23315
$273$		−	47483.0i		−	0.637108i
$274$		−	120668.i		−	1.60728i
$275$			31531.0i			0.416939i
$276$		−	15566.7i		−	0.204352i
$277$	110726.			1.44308			0.721540	−	0.692373i	$-0.243435\pi$
$277$	110726.			1.44308			0.721540	+	0.692373i	$0.243435\pi$
$278$		−	130822.i		−	1.69275i
$279$			3456.96i			0.0444105i
$280$		−	10203.7i		−	0.130149i
$281$	−132826.			−1.68217			−0.841085	−	0.540903i	$-0.818083\pi$
$281$	−132826.			−1.68217			−0.841085	+	0.540903i	$0.818083\pi$
$282$	−26858.1			−0.337736
$283$		−	136650.i		−	1.70622i	−0.521728	−	0.853112i	$-0.674713\pi$
$283$		−	136650.i		−	1.70622i	0.521728	−	0.853112i	$-0.325287\pi$
$284$	159787.			1.98110
$285$	62888.6			0.774252
$286$		−	76351.8i		−	0.933442i
$287$	29917.7			0.363216
$288$			39625.2i			0.477734i
$289$	69389.1			0.830798
$290$		−	17988.2i		−	0.213890i
$291$		−	2065.49i		−	0.0243914i
$292$		−	74584.1i		−	0.874743i
$293$	−138078.			−1.60838			−0.804189	−	0.594374i	$-0.797400\pi$
$293$	−138078.			−1.60838			−0.804189	+	0.594374i	$0.797400\pi$
$294$		−	25751.0i		−	0.297920i
$295$	−53165.4	−	107084.i	−0.610921	−	1.23050i
$296$	3125.74			0.0356754
$297$		−	7976.40i		−	0.0904262i
$298$	85119.4			0.958508
$299$	−40268.6			−0.450426
$300$	49885.3			0.554282
$301$		−	34324.8i		−	0.378857i
$302$	−7805.78			−0.0855859
$303$			36841.3i			0.401281i
$304$	82216.3			0.889633
$305$			51451.2i			0.553090i
$306$			60936.0i			0.650775i
$307$	88705.4			0.941181			0.470591	−	0.882352i	$-0.344041\pi$
$307$	88705.4			0.941181			0.470591	+	0.882352i	$0.344041\pi$
$308$			38651.5i			0.407442i
$309$		−	32563.7i		−	0.341049i
$310$	25379.8			0.264098
$311$	−120340.			−1.24420			−0.622101	−	0.782937i	$-0.713721\pi$
$311$	−120340.			−1.24420			−0.622101	+	0.782937i	$0.713721\pi$
$312$	−9146.35			−0.0939590
$313$			153842.i			1.57031i	0.619299	+	0.785155i	$0.287417\pi$
$313$			153842.i			1.57031i	−0.619299	+	0.785155i	$0.712583\pi$
$314$	−253183.			−2.56788
$315$	−36418.4			−0.367028
$316$	81496.7			0.816142
$317$	13221.7			0.131574			0.0657868	−	0.997834i	$-0.479044\pi$
$317$	13221.7			0.131574			0.0657868	+	0.997834i	$0.479044\pi$
$318$			110646.i			1.09416i
$319$			5159.29i			0.0507001i
$320$	162705.			1.58892
$321$	−7838.67			−0.0760733
$322$	39226.8			0.378330
$323$	137798.			1.32080
$324$	−12619.5			−0.120213
$325$		−	129045.i		−	1.22173i
$326$		−	143797.i		−	1.35305i
$327$			84498.7i			0.790232i
$328$		−	5762.87i		−	0.0535662i
$329$		−	35171.6i		−	0.324938i
$330$	−58560.1			−0.537742
$331$	56327.8			0.514123			0.257061	−	0.966395i	$-0.417246\pi$
$331$	56327.8			0.514123			0.257061	+	0.966395i	$0.417246\pi$
$332$			6897.50i			0.0625771i
$333$		−	11156.2i		−	0.100607i
$334$			88794.1i			0.795960i
$335$		−	126510.i		−	1.12729i
$336$	−47610.9			−0.421724
$337$		−	115339.i		−	1.01558i	−0.861481	−	0.507791i	$-0.830462\pi$
$337$		−	115339.i		−	1.01558i	0.861481	−	0.507791i	$-0.169538\pi$
$338$			147640.i			1.29232i
$339$			117806.i			1.02511i
$340$	232487.			2.01114
$341$	−7279.34			−0.0626013
$342$			54913.5i			0.469491i
$343$	128015.			1.08811
$344$	−6611.77			−0.0558729
$345$			30885.1i			0.259484i
$346$	−316964.			−2.64763
$347$			9584.04i			0.0795957i	0.999208	+	0.0397979i	$0.0126714\pi$
$347$			9584.04i			0.0795957i	−0.999208	+	0.0397979i	$0.987329\pi$
$348$	8162.54			0.0674011
$349$		−	148678.i		−	1.22066i	−0.792147	−	0.610330i	$-0.791037\pi$
$349$		−	148678.i		−	1.22066i	0.792147	−	0.610330i	$-0.208963\pi$
$350$			125707.i			1.02618i
$351$			32644.6i			0.264970i
$352$	−83439.0			−0.673416
$353$		−	12045.3i		−	0.0966652i	−0.998831	−	0.0483326i	$-0.984609\pi$
$353$		−	12045.3i		−	0.0966652i	0.998831	−	0.0483326i	$-0.0153907\pi$
$354$	93504.5	−	46423.3i	0.746150	−	0.370450i
$355$	−317025.			−2.51557
$356$			214992.i			1.69638i
$357$	−79797.7			−0.626115
$358$	−291721.			−2.27615
$359$	−11159.4			−0.0865872			−0.0432936	−	0.999062i	$-0.513785\pi$
$359$	−11159.4			−0.0865872			−0.0432936	+	0.999062i	$0.513785\pi$
$360$			7015.04i			0.0541284i
$361$	−6142.36			−0.0471326
$362$		−	199627.i		−	1.52336i
$363$	−59280.9			−0.449885
$364$		−	158187.i		−	1.19390i
$365$			147978.i			1.11074i
$366$	−44926.5			−0.335383
$367$			24075.5i			0.178749i	0.995998	+	0.0893744i	$0.0284868\pi$
$367$			24075.5i			0.178749i	−0.995998	+	0.0893744i	$0.971513\pi$
$368$			40377.1i			0.298153i
$369$	−20568.5			−0.151060
$370$	−81905.0			−0.598284
$371$	−144895.			−1.05270
$372$			11516.7i			0.0832226i
$373$	−87079.2			−0.625888			−0.312944	−	0.949772i	$-0.601315\pi$
$373$	−87079.2			−0.625888			−0.312944	+	0.949772i	$0.601315\pi$
$374$	−128313.			−0.917336
$375$	12564.7			0.0893487
$376$	−6774.87			−0.0479210
$377$		−	21115.2i		−	0.148563i
$378$		−	31800.1i		−	0.222559i
$379$	201608.			1.40355			0.701776	−	0.712398i	$-0.252391\pi$
$379$	201608.			1.40355			0.701776	+	0.712398i	$0.252391\pi$
$380$	209510.			1.45090
$381$	−38608.0			−0.265967
$382$	168386.			1.15393
$383$	181721.			1.23882			0.619408	−	0.785069i	$-0.287373\pi$
$383$	181721.			1.23882			0.619408	+	0.785069i	$0.287373\pi$
$384$			20058.2i			0.136028i
$385$		−	76686.4i		−	0.517365i
$386$		−	184349.i		−	1.23727i
$387$			23598.4i			0.157565i
$388$		−	6881.08i		−	0.0457081i
$389$	−133518.			−0.882350			−0.441175	−	0.897421i	$-0.645438\pi$
$389$	−133518.			−0.882350			−0.441175	+	0.897421i	$0.645438\pi$
$390$	239666.			1.57571
$391$			67673.4i			0.442654i
$392$		−	6495.61i		−	0.0422715i
$393$		−	156215.i		−	1.01143i
$394$			24531.6i			0.158028i
$395$	−161693.			−1.03633
$396$		−	26573.0i		−	0.169453i
$397$		−	27629.4i		−	0.175304i	−0.996151	−	0.0876518i	$-0.972064\pi$
$397$		−	27629.4i		−	0.175304i	0.996151	−	0.0876518i	$-0.0279363\pi$
$398$		−	164848.i		−	1.04068i
$399$	−71911.1			−0.451700
$400$	−129393.			−0.808706
$401$		−	208735.i		−	1.29809i	−0.760748	−	0.649047i	$-0.775168\pi$
$401$		−	208735.i		−	1.29809i	0.760748	−	0.649047i	$-0.224832\pi$
$402$	110467.			0.683566
$403$	29791.8			0.183437
$404$			122735.i			0.751977i
$405$	25037.7			0.152645
$406$			20568.9i			0.124784i
$407$	23491.7			0.141816
$408$			15370.9i			0.0923378i
$409$			278718.i			1.66616i	0.553150	+	0.833082i	$0.313426\pi$
$409$			278718.i			1.66616i	−0.553150	+	0.833082i	$0.686574\pi$
$410$			151007.i			0.898316i
$411$	−108638.			−0.643130
$412$		−	108484.i		−	0.639106i
$413$	60792.9	+	122447.i	0.356412	+	0.717875i
$414$	−26968.4			−0.157346
$415$		−	13685.0i		−	0.0794598i
$416$	341486.			1.97327
$417$	−117780.			−0.677328
$418$	−115632.			−0.661796
$419$		−	15318.9i		−	0.0872570i	−0.999048	−	0.0436285i	$-0.986108\pi$
$419$		−	15318.9i		−	0.0872570i	0.999048	−	0.0436285i	$-0.0138918\pi$
$420$	−121326.			−0.687789
$421$			278756.i			1.57275i	0.617747	+	0.786377i	$0.288046\pi$
$421$			278756.i			1.57275i	−0.617747	+	0.786377i	$0.711954\pi$
$422$	152841.			0.858252
$423$			24180.5i			0.135140i
$424$			27910.2i			0.155250i
$425$	−216867.			−1.20065
$426$		−	276822.i		−	1.52539i
$427$		−	58832.8i		−	0.322674i
$428$	−26114.1			−0.142557
$429$	−68739.9			−0.373503
$430$	173251.			0.937000
$431$		−	202239.i		−	1.08870i	−0.838857	−	0.544352i	$-0.816776\pi$
$431$		−	202239.i		−	1.08870i	0.838857	−	0.544352i	$-0.183224\pi$
$432$	32732.6			0.175393
$433$	−88674.2			−0.472957			−0.236478	−	0.971637i	$-0.575993\pi$
$433$	−88674.2			−0.472957			−0.236478	+	0.971637i	$0.575993\pi$
$434$	−29021.0			−0.154075
$435$	−16194.8			−0.0855851
$436$			281503.i			1.48085i
$437$			60985.1i			0.319345i
$438$	−129213.			−0.673530
$439$	−374655.			−1.94403			−0.972013	−	0.234928i	$-0.924515\pi$
$439$	−374655.			−1.94403			−0.972013	+	0.234928i	$0.924515\pi$
$440$	−14771.6			−0.0762997
$441$	−23183.8			−0.119208
$442$	525141.			2.68801
$443$			19857.5i			0.101185i	0.998719	+	0.0505925i	$0.0161110\pi$
$443$			19857.5i			0.101185i	−0.998719	+	0.0505925i	$0.983889\pi$
$444$		−	37166.3i		−	0.188531i
$445$		−	426555.i		−	2.15404i
$446$			118311.i			0.594779i
$447$		−	76633.4i		−	0.383533i
$448$	−186048.			−0.926978
$449$	71307.0			0.353703			0.176852	−	0.984238i	$-0.443409\pi$
$449$	71307.0			0.353703			0.176852	+	0.984238i	$0.443409\pi$
$450$		−	86423.5i		−	0.426783i
$451$		−	43311.2i		−	0.212935i
$452$			392465.i			1.92099i
$453$			7027.59i			0.0342460i
$454$	−452486.			−2.19530
$455$			313850.i			1.51600i
$456$			13851.8i			0.0666156i
$457$			134049.i			0.641846i	0.947105	+	0.320923i	$0.103993\pi$
$457$			134049.i			0.641846i	−0.947105	+	0.320923i	$0.896007\pi$
$458$	−180131.			−0.858732
$459$	54861.0			0.260398
$460$			102892.i			0.486257i
$461$	281430.			1.32425			0.662123	−	0.749395i	$-0.269656\pi$
$461$	281430.			1.32425			0.662123	+	0.749395i	$0.269656\pi$
$462$	66961.6			0.313720
$463$			323834.i			1.51064i	0.655356	+	0.755320i	$0.272518\pi$
$463$			323834.i			1.51064i	−0.655356	+	0.755320i	$0.727482\pi$
$464$	−21172.0			−0.0983393
$465$		−	22849.6i		−	0.105675i
$466$	525265.			2.41884
$467$			411702.i			1.88777i	0.330272	+	0.943886i	$0.392859\pi$
$467$			411702.i			1.88777i	−0.330272	+	0.943886i	$0.607141\pi$
$468$			108754.i			0.496538i
$469$			144660.i			0.657663i
$470$	177525.			0.803645
$471$			227942.i			1.02750i
$472$	23586.2	−	11710.1i	0.105870	−	0.0525628i
$473$	−49691.2			−0.222104
$474$		−	141188.i		−	0.628409i
$475$	−195434.			−0.866188
$476$	−265842.			−1.17330
$477$	99615.4			0.437814
$478$			555067.i			2.42935i
$479$	84811.3			0.369643			0.184822	−	0.982772i	$-0.440829\pi$
$479$	84811.3			0.369643			0.184822	+	0.982772i	$0.440829\pi$
$480$		−	261912.i		−	1.13677i
$481$	−96143.1			−0.415554
$482$			399442.i			1.71933i
$483$		−	35316.1i		−	0.151383i
$484$	−197491.			−0.843058
$485$			13652.4i			0.0580396i
$486$			21862.6i			0.0925612i
$487$	249337.			1.05131			0.525653	−	0.850699i	$-0.323821\pi$
$487$	249337.			1.05131			0.525653	+	0.850699i	$0.323821\pi$
$488$	−11332.6			−0.0475871
$489$	−129461.			−0.541405
$490$			170207.i			0.708902i
$491$	50328.2			0.208760			0.104380	−	0.994537i	$-0.466714\pi$
$491$	50328.2			0.208760			0.104380	+	0.994537i	$0.466714\pi$
$492$	−68522.8			−0.283077
$493$	−35485.1			−0.146000
$494$	473240.			1.93922
$495$			52722.0i			0.215170i
$496$		−	29872.0i		−	0.121423i
$497$	362508.			1.46759
$498$	11949.5			0.0481828
$499$	−23067.0			−0.0926380			−0.0463190	−	0.998927i	$-0.514749\pi$
$499$	−23067.0			−0.0926380			−0.0463190	+	0.998927i	$0.514749\pi$
$500$	41858.5			0.167434
$501$	79941.8			0.318492
$502$		−	543301.i		−	2.15592i
$503$			410578.i			1.62278i	0.584505	+	0.811390i	$0.301289\pi$
$503$			410578.i			1.62278i	−0.584505	+	0.811390i	$0.698711\pi$
$504$		−	8021.47i		−	0.0315786i
$505$		−	243511.i		−	0.954852i
$506$		−	56787.6i		−	0.221795i
$507$	132921.			0.517104
$508$	−128621.			−0.498406
$509$		−	19719.0i		−	0.0761115i	−0.999276	−	0.0380557i	$-0.987884\pi$
$509$		−	19719.0i		−	0.0761115i	0.999276	−	0.0380557i	$-0.0121164\pi$
$510$		−	402771.i		−	1.54852i
$511$		−	169208.i		−	0.648007i
$512$		−	370646.i		−	1.41390i
$513$	49438.9			0.187860
$514$		−	67412.5i		−	0.255161i
$515$			215238.i			0.811529i
$516$			78616.7i			0.295267i
$517$	−50917.0			−0.190494
$518$	93655.8			0.349040
$519$			285365.i			1.05941i
$520$	60455.0			0.223576
$521$	420195.			1.54802			0.774008	−	0.633175i	$-0.218249\pi$
$521$	420195.			1.54802			0.774008	+	0.633175i	$0.218249\pi$
$522$		−	14141.1i		−	0.0518971i
$523$	−523604.			−1.91425			−0.957126	−	0.289670i	$-0.906454\pi$
$523$	−523604.			−1.91425			−0.957126	+	0.289670i	$0.906454\pi$
$524$		−	520421.i		−	1.89536i
$525$	113174.			0.410610
$526$			389877.i			1.40915i
$527$		−	50066.6i		−	0.180272i
$528$			68925.2i			0.247235i
$529$	249891.			0.892974
$530$		−	731343.i		−	2.60357i
$531$	−41795.2	−	84182.7i	−0.148230	−	0.298561i
$532$	−239568.			−0.846458
$533$			177257.i			0.623950i
$534$	372462.			1.30617
$535$	51811.6			0.181017
$536$	27865.0			0.0969905
$537$			262638.i			0.910771i
$538$	−46663.9			−0.161219
$539$		−	48818.2i		−	0.168037i
$540$	83411.7			0.286048
$541$			62563.9i			0.213761i	0.994272	+	0.106881i	$0.0340863\pi$
$541$			62563.9i			0.213761i	−0.994272	+	0.106881i	$0.965914\pi$
$542$			17565.9i			0.0597959i
$543$	−179725.			−0.609549
$544$		−	573886.i		−	1.93922i
$545$		−	558515.i		−	1.88036i
$546$	−274050.			−0.919274
$547$	28370.0			0.0948166			0.0474083	−	0.998876i	$-0.484904\pi$
$547$	28370.0			0.0948166			0.0474083	+	0.998876i	$0.484904\pi$
$548$	−361923.			−1.20519
$549$			40447.6i			0.134199i
$550$	181982.			0.601595
$551$	−31978.0			−0.105329
$552$	−6802.71			−0.0223256
$553$	184891.			0.604596
$554$		−	639060.i		−	2.08220i
$555$			73739.6i			0.239395i
$556$	−392378.			−1.26927
$557$	−85305.1			−0.274957			−0.137478	−	0.990505i	$-0.543900\pi$
$557$	−85305.1			−0.274957			−0.137478	+	0.990505i	$0.543900\pi$
$558$	19952.0			0.0640793
$559$	203369.			0.650819
$560$	314696.			1.00350
$561$			115521.i			0.367059i
$562$			766610.i			2.42718i
$563$		−	625430.i		−	1.97316i	−0.163282	−	0.986580i	$-0.552208\pi$
$563$		−	625430.i		−	1.97316i	0.163282	−	0.986580i	$-0.447792\pi$
$564$			80556.0i			0.253244i
$565$		−	778668.i		−	2.43925i
$566$	−788680.			−2.46189
$567$	−28629.8			−0.0890537
$568$		−	69827.6i		−	0.216437i
$569$			182688.i			0.564269i	0.959375	+	0.282134i	$0.0910424\pi$
$569$			182688.i			0.564269i	−0.959375	+	0.282134i	$0.908958\pi$
$570$		−	362964.i		−	1.11716i
$571$		−	366777.i		−	1.12494i	−0.826817	−	0.562471i	$-0.809851\pi$
$571$		−	366777.i		−	1.12494i	0.826817	−	0.562471i	$-0.190149\pi$
$572$	−229004.			−0.699923
$573$		−	151599.i		−	0.461729i
$574$		−	172672.i		−	0.524079i
$575$		−	95979.0i		−	0.290296i
$576$	127908.			0.385526
$577$	485860.			1.45935			0.729675	−	0.683794i	$-0.239671\pi$
$577$	485860.			1.45935			0.729675	+	0.683794i	$0.239671\pi$
$578$		−	400482.i		−	1.19875i
$579$	−165970.			−0.495078
$580$	−53952.3			−0.160381
$581$			15648.3i			0.0463570i
$582$	−11921.1			−0.0351940
$583$			209761.i			0.617145i
$584$	−32593.5			−0.0955664
$585$		−	215773.i		−	0.630499i
$586$			796921.i			2.32071i
$587$			466181.i			1.35294i	0.736470	+	0.676470i	$0.236491\pi$
$587$			466181.i			1.35294i	−0.736470	+	0.676470i	$0.763509\pi$
$588$	−77235.5			−0.223389
$589$		−	45118.4i		−	0.130054i
$590$	−618041.	+	306846.i	−1.77547	+	0.881489i
$591$	22085.9			0.0632326
$592$			96402.1i			0.275070i
$593$	−263489.			−0.749294			−0.374647	−	0.927167i	$-0.622236\pi$
$593$	−263489.			−0.749294			−0.374647	+	0.927167i	$0.622236\pi$
$594$	−46036.2			−0.130475
$595$	527442.			1.48984
$596$		−	255300.i		−	0.718719i
$597$	−148414.			−0.416414
$598$			232412.i			0.649914i
$599$	−182847.			−0.509606			−0.254803	−	0.966993i	$-0.582011\pi$
$599$	−182847.			−0.509606			−0.254803	+	0.966993i	$0.582011\pi$
$600$		−	21800.1i		−	0.0605558i
$601$		−	529477.i		−	1.46588i	−0.680294	−	0.732939i	$-0.738148\pi$
$601$		−	529477.i		−	1.46588i	0.680294	−	0.732939i	$-0.261852\pi$
$602$	−198107.			−0.546647
$603$		−	99454.1i		−	0.273519i
$604$			23412.0i			0.0641749i
$605$	391832.			1.07051
$606$	212631.			0.579003
$607$	−161986.			−0.439642			−0.219821	−	0.975540i	$-0.570547\pi$
$607$	−161986.			−0.439642			−0.219821	+	0.975540i	$0.570547\pi$
$608$		−	517167.i		−	1.39902i
$609$	18518.3			0.0499305
$610$	296953.			0.798046
$611$	208385.			0.558193
$612$	182767.			0.487971
$613$			97048.9i			0.258268i	0.991627	+	0.129134i	$0.0412197\pi$
$613$			97048.9i			0.258268i	−0.991627	+	0.129134i	$0.958780\pi$
$614$		−	511967.i		−	1.35802i
$615$	135952.			0.359448
$616$	16890.9			0.0445134
$617$	−17414.4			−0.0457444			−0.0228722	−	0.999738i	$-0.507281\pi$
$617$	−17414.4			−0.0457444			−0.0228722	+	0.999738i	$0.507281\pi$
$618$	−187943.			−0.492095
$619$	−216510.			−0.565061			−0.282531	−	0.959258i	$-0.591174\pi$
$619$	−216510.			−0.565061			−0.282531	+	0.959258i	$0.591174\pi$
$620$		−	76122.3i		−	0.198029i
$621$			24279.8i			0.0629597i
$622$			694550.i			1.79524i
$623$			487751.i			1.25667i
$624$		−	282086.i		−	0.724458i
$625$	−429671.			−1.09996
$626$	887904.			2.26578
$627$			104104.i			0.264808i
$628$			759376.i			1.92547i
$629$			161574.i			0.408384i
$630$			210190.i			0.529580i
$631$	−218714.			−0.549310			−0.274655	−	0.961543i	$-0.588564\pi$
$631$	−218714.			−0.549310			−0.274655	+	0.961543i	$0.588564\pi$
$632$		−	35614.3i		−	0.0891643i
$633$		−	137604.i		−	0.343417i
$634$		−	76309.6i		−	0.189846i
$635$	255189.			0.632871
$636$	331864.			0.820437
$637$			199796.i			0.492388i
$638$	29777.1			0.0731544
$639$	−249225.			−0.610365
$640$		−	132580.i		−	0.323681i
$641$	−210985.			−0.513495			−0.256747	−	0.966479i	$-0.582651\pi$
$641$	−210985.			−0.513495			−0.256747	+	0.966479i	$0.582651\pi$
$642$			45241.2i			0.109765i
$643$	192225.			0.464932			0.232466	−	0.972605i	$-0.425321\pi$
$643$	192225.			0.464932			0.232466	+	0.972605i	$0.425321\pi$
$644$		−	117654.i		−	0.283683i
$645$		−	155979.i		−	0.374927i
$646$		−	795305.i		−	1.90576i
$647$	−671147.			−1.60328			−0.801639	−	0.597808i	$-0.796038\pi$
$647$	−671147.			−1.60328			−0.801639	+	0.597808i	$0.796038\pi$
$648$			5514.77i			0.0131334i
$649$	177264.	−	88008.3i	0.420853	−	0.208946i
$650$	−744790.			−1.76282
$651$			26127.8i			0.0616511i
$652$	−431294.			−1.01456
$653$	330138.			0.774227			0.387114	−	0.922032i	$-0.373472\pi$
$653$	330138.			0.774227			0.387114	+	0.922032i	$0.373472\pi$
$654$	487688.			1.14021
$655$			1.03254e6i			2.40671i
$656$	177735.			0.413014
$657$			116331.i			0.269503i
$658$	−202994.			−0.468848
$659$		−	45073.9i		−	0.103790i	−0.998653	−	0.0518949i	$-0.983474\pi$
$659$		−	45073.9i		−	0.103790i	0.998653	−	0.0518949i	$-0.0165261\pi$
$660$			175641.i			0.403215i
$661$	−304683.			−0.697341			−0.348670	−	0.937245i	$-0.613367\pi$
$661$	−304683.			−0.697341			−0.348670	+	0.937245i	$0.613367\pi$
$662$		−	325098.i		−	0.741821i
$663$		−	472787.i		−	1.07557i
$664$	3014.23			0.00683661
$665$	475314.			1.07482
$666$	−64388.4			−0.145164
$667$		−	15704.7i		−	0.0353002i
$668$	266322.			0.596835
$669$	106516.			0.237992
$670$	−730159.			−1.62655
$671$	−85170.7			−0.189167
$672$			299488.i			0.663195i
$673$		−	188450.i		−	0.416070i	−0.978121	−	0.208035i	$-0.933293\pi$
$673$		−	188450.i		−	0.416070i	0.978121	−	0.208035i	$-0.0667068\pi$
$674$	−665681.			−1.46537
$675$	−77807.5			−0.170771
$676$	442820.			0.969022
$677$	−254006.			−0.554199			−0.277100	−	0.960841i	$-0.589373\pi$
$677$	−254006.			−0.554199			−0.277100	+	0.960841i	$0.589373\pi$
$678$	679923.			1.47911
$679$		−	15611.0i		−	0.0338604i
$680$		−	101598.i		−	0.219719i
$681$			407375.i			0.878417i
$682$			42013.0i			0.0903265i
$683$		−	537333.i		−	1.15187i	−0.817497	−	0.575933i	$-0.804639\pi$
$683$		−	537333.i		−	1.15187i	0.817497	−	0.575933i	$-0.195361\pi$
$684$	164703.			0.352038
$685$	718071.			1.53033
$686$		−	738847.i		−	1.57002i
$687$			162173.i			0.343609i
$688$		−	203917.i		−	0.430800i
$689$		−	858477.i		−	1.80838i
$690$	178255.			0.374406
$691$		−	70666.8i		−	0.147999i	−0.997258	−	0.0739996i	$-0.976424\pi$
$691$		−	70666.8i		−	0.147999i	0.997258	−	0.0739996i	$-0.0235763\pi$
$692$			950678.i			1.98528i
$693$		−	60285.9i		−	0.125530i
$694$	55314.7			0.114848
$695$	778495.			1.61171
$696$		−	3567.06i		−	0.00736363i
$697$	297890.			0.613184
$698$	−858099.			−1.76127
$699$		−	472899.i		−	0.967863i
$700$	377035.			0.769459
$701$			792788.i			1.61332i	0.591015	+	0.806661i	$0.298727\pi$
$701$			792788.i			1.61332i	−0.591015	+	0.806661i	$0.701273\pi$
$702$	188410.			0.382322
$703$			145605.i			0.294622i
$704$			269337.i			0.543440i
$705$		−	159827.i		−	0.321567i
$706$	−69520.3			−0.139477
$707$			278447.i			0.557063i
$708$	−139238.	−	280450.i	−0.277775	−	0.559486i
$709$	−867302.			−1.72535			−0.862677	−	0.505756i	$-0.831214\pi$
$709$	−867302.			−1.72535			−0.862677	+	0.505756i	$0.831214\pi$
$710$			1.82972e6i			3.62969i
$711$	−127113.			−0.251449
$712$	93952.4			0.185331
$713$	22158.0			0.0435864
$714$			460556.i			0.903412i
$715$	454353.			0.888754
$716$			874965.i			1.70673i
$717$	499730.			0.972070
$718$			64407.2i			0.124935i
$719$			267439.i			0.517329i	0.965967	+	0.258665i	$0.0832825\pi$
$719$			267439.i			0.517329i	−0.965967	+	0.258665i	$0.916718\pi$
$720$	−216354.			−0.417349
$721$		−	246117.i		−	0.473448i
$722$			35450.9i			0.0680069i
$723$	359620.			0.687966
$724$	−598745.			−1.14226
$725$	50327.4			0.0957477
$726$			342142.i			0.649133i
$727$	717136.			1.35685			0.678426	−	0.734668i	$-0.262662\pi$
$727$	717136.			1.35685			0.678426	+	0.734668i	$0.262662\pi$
$728$	−69128.4			−0.130435
$729$	19683.0			0.0370370
$730$	854062.			1.60267
$731$		−	341772.i		−	0.639589i
$732$			134749.i			0.251480i
$733$	−443873.			−0.826134			−0.413067	−	0.910701i	$-0.635543\pi$
$733$	−443873.			−0.826134			−0.413067	+	0.910701i	$0.635543\pi$
$734$	138953.			0.257914
$735$	153239.			0.283657
$736$	253985.			0.468870
$737$	209421.			0.385554
$738$			118712.i			0.217962i
$739$			177524.i			0.325063i	0.986703	+	0.162532i	$0.0519659\pi$
$739$			177524.i			0.325063i	−0.986703	+	0.162532i	$0.948034\pi$
$740$			245660.i			0.448611i
$741$		−	426060.i		−	0.775952i
$742$			836267.i			1.51893i
$743$	−266645.			−0.483009			−0.241505	−	0.970400i	$-0.577641\pi$
$743$	−266645.			−0.483009			−0.241505	+	0.970400i	$0.577641\pi$
$744$	5032.83			0.00909214
$745$			506527.i			0.912621i
$746$			502581.i			0.903085i
$747$		−	10758.2i		−	0.0192797i
$748$			384853.i			0.687846i
$749$	−59244.9			−0.105606
$750$		−	72517.5i		−	0.128920i
$751$		−	438213.i		−	0.776972i	−0.921455	−	0.388486i	$-0.872998\pi$
$751$		−	438213.i		−	0.776972i	0.921455	−	0.388486i	$-0.127002\pi$
$752$		−	208947.i		−	0.369488i
$753$	−489137.			−0.862662
$754$	−121867.			−0.214360
$755$		−	46450.5i		−	0.0814886i
$756$	−95378.6			−0.166881
$757$	999861.			1.74481			0.872404	−	0.488785i	$-0.162560\pi$
$757$	999861.			1.74481			0.872404	+	0.488785i	$0.162560\pi$
$758$		−	1.16359e6i		−	2.02516i
$759$	−51126.2			−0.0887483
$760$		−	91556.7i		−	0.158512i
$761$	−326900.			−0.564476			−0.282238	−	0.959344i	$-0.591077\pi$
$761$	−326900.			−0.564476			−0.282238	+	0.959344i	$0.591077\pi$
$762$			222828.i			0.383760i
$763$			638643.i			1.09701i
$764$		−	505045.i		−	0.865253i
$765$	−362617.			−0.619620
$766$		−	1.04881e6i		−	1.78747i
$767$	−725479.	+	360187.i	−1.23320	+	0.612262i
$768$	−278089.			−0.471478
$769$		−	235002.i		−	0.397392i	−0.980061	−	0.198696i	$-0.936329\pi$
$769$		−	235002.i		−	0.397392i	0.980061	−	0.198696i	$-0.0636707\pi$
$770$	−442599.			−0.746498
$771$	−60691.8			−0.102099
$772$	−552922.			−0.927746
$773$			210457.i			0.352213i	0.984371	+	0.176106i	$0.0563503\pi$
$773$			210457.i			0.352213i	−0.984371	+	0.176106i	$0.943650\pi$
$774$	136199.			0.227348
$775$			71007.9i			0.118223i
$776$	−3007.05			−0.00499365
$777$		−	84318.8i		−	0.139663i
$778$			770606.i			1.27313i
$779$	268449.			0.442371
$780$		−	718835.i		−	1.18152i
$781$		−	524794.i		−	0.860373i
$782$	390580.			0.638700
$783$	−12731.3			−0.0207659
$784$	200334.			0.325929
$785$		−	1.50664e6i		−	2.44494i
$786$	−901600.			−1.45938
$787$	−11734.5			−0.0189459			−0.00947293	−	0.999955i	$-0.503015\pi$
$787$	−11734.5			−0.0189459			−0.00947293	+	0.999955i	$0.503015\pi$
$788$	73578.1			0.118494
$789$	351008.			0.563849
$790$			933219.i			1.49530i
$791$			890382.i			1.42306i
$792$	−11612.5			−0.0185129
$793$	348574.			0.554305
$794$	−159464.			−0.252943
$795$	−658432.			−1.04178
$796$	−494433.			−0.780335
$797$		−	592344.i		−	0.932518i	−0.884648	−	0.466259i	$-0.845601\pi$
$797$		−	592344.i		−	0.932518i	0.884648	−	0.466259i	$-0.154399\pi$
$798$			415038.i			0.651751i
$799$		−	350203.i		−	0.548562i
$800$			813924.i			1.27176i
$801$		−	335330.i		−	0.522645i
$802$	−1.20472e6			−1.87300
$803$	−244959.			−0.379893
$804$		−	331326.i		−	0.512559i
$805$			233430.i			0.360218i
$806$		−	171944.i		−	0.264678i
$807$			42011.8i			0.0645096i
$808$	53635.5			0.0821542
$809$			340035.i			0.519550i	0.965669	+	0.259775i	$0.0836484\pi$
$809$			340035.i			0.519550i	−0.965669	+	0.259775i	$0.916352\pi$
$810$		−	144506.i		−	0.220250i
$811$			10322.5i			0.0156943i	0.999969	+	0.00784717i	$0.00249786\pi$
$811$			10322.5i			0.0156943i	−0.999969	+	0.00784717i	$0.997502\pi$
$812$	61692.7			0.0935668
$813$	15814.7			0.0239265
$814$		−	135583.i		−	0.204624i
$815$	855706.			1.28828
$816$	−474061.			−0.711958
$817$		−	307993.i		−	0.461421i
$818$	1.60863e6			2.40408
$819$			246729.i			0.367834i
$820$	452918.			0.673585
$821$			623437.i			0.924924i	0.886639	+	0.462462i	$0.153034\pi$
$821$			623437.i			0.924924i	−0.886639	+	0.462462i	$0.846966\pi$
$822$			627010.i			0.927963i
$823$		−	193091.i		−	0.285077i	−0.989789	−	0.142538i	$-0.954474\pi$
$823$		−	193091.i		−	0.285077i	0.989789	−	0.142538i	$-0.0455264\pi$
$824$	−47408.0			−0.0698229
$825$		−	163840.i		−	0.240720i
$826$	706710.	−	350869.i	1.03581	−	0.514262i
$827$	−72588.8			−0.106135			−0.0530675	−	0.998591i	$-0.516900\pi$
$827$	−72588.8			−0.106135			−0.0530675	+	0.998591i	$0.516900\pi$
$828$			80887.0i			0.117983i
$829$	−941304.			−1.36969			−0.684843	−	0.728691i	$-0.740129\pi$
$829$	−941304.			−1.36969			−0.684843	+	0.728691i	$0.740129\pi$
$830$	−78983.3			−0.114651
$831$	−575350.			−0.833163
$832$		−	1.10230e6i		−	1.59241i
$833$	335767.			0.483892
$834$			679772.i			0.977307i
$835$	−528395.			−0.757854
$836$			346817.i			0.496235i
$837$		−	17962.9i		−	0.0256404i
$838$	−88413.8			−0.125902
$839$			129224.i			0.183578i	0.995778	+	0.0917888i	$0.0292585\pi$
$839$			129224.i			0.183578i	−0.995778	+	0.0917888i	$0.970742\pi$
$840$			53019.9i			0.0751415i
$841$	−699046.			−0.988357
$842$	1.60885e6			2.26930
$843$	690183.			0.971201
$844$		−	458419.i		−	0.643543i
$845$	−878574.			−1.23045
$846$	139559.			0.194992
$847$	−448047.			−0.624535
$848$	−860790.			−1.19703
$849$			710053.i			0.985089i
$850$			1.25166e6i			1.73240i
$851$	−71507.6			−0.0987400
$852$	−830279.			−1.14379
$853$	−11458.1			−0.0157476			−0.00787381	−	0.999969i	$-0.502506\pi$
$853$	−11458.1			−0.0157476			−0.00787381	+	0.999969i	$0.502506\pi$
$854$	−339556.			−0.465581
$855$	−326779.			−0.447014
$856$			11412.0i			0.0155745i
$857$		−	28783.4i		−	0.0391905i	−0.999808	−	0.0195952i	$-0.993762\pi$
$857$		−	28783.4i		−	0.0391905i	0.999808	−	0.0195952i	$-0.00623775\pi$
$858$			396736.i			0.538923i
$859$		−	969548.i		−	1.31396i	−0.753907	−	0.656982i	$-0.771833\pi$
$859$		−	969548.i		−	1.31396i	0.753907	−	0.656982i	$-0.228167\pi$
$860$		−	519636.i		−	0.702591i
$861$	−155457.			−0.209703
$862$	−1.16723e6			−1.57088
$863$			287494.i			0.386018i	0.981197	+	0.193009i	$0.0618247\pi$
$863$			287494.i			0.386018i	−0.981197	+	0.193009i	$0.938175\pi$
$864$		−	205899.i		−	0.275820i
$865$		−	1.88619e6i		−	2.52088i
$866$			511787.i			0.682423i
$867$	−360556.			−0.479661
$868$			87043.4i			0.115530i
$869$		−	267662.i		−	0.354443i
$870$			93469.3i			0.123490i
$871$	−857087.			−1.12977
$872$	123018.			0.161784
$873$			10732.6i			0.0140824i
$874$	351978.			0.460779
$875$	94964.1			0.124035
$876$			387550.i			0.505033i
$877$	979332.			1.27330			0.636650	−	0.771153i	$-0.280320\pi$
$877$	979332.			1.27330			0.636650	+	0.771153i	$0.280320\pi$
$878$			2.16234e6i			2.80501i
$879$	717473.			0.928598
$880$		−	455578.i		−	0.588298i
$881$		−	389453.i		−	0.501769i	−0.968017	−	0.250884i	$-0.919279\pi$
$881$		−	389453.i		−	0.501769i	0.968017	−	0.250884i	$-0.0807214\pi$
$882$			133806.i			0.172004i
$883$	−883630.			−1.13331			−0.566655	−	0.823955i	$-0.691763\pi$
$883$	−883630.			−1.13331			−0.566655	+	0.823955i	$0.691763\pi$
$884$		−	1.57507e6i		−	2.01555i
$885$	276255.	+	556426.i	0.352715	+	0.710429i
$886$	114608.			0.145999
$887$		−	283208.i		−	0.359963i	−0.983670	−	0.179981i	$-0.942396\pi$
$887$		−	283208.i		−	0.359963i	0.983670	−	0.179981i	$-0.0576038\pi$
$888$	−16241.8			−0.0205972
$889$	−291801.			−0.369218
$890$	−2.46188e6			−3.10804
$891$			41446.6i			0.0522076i
$892$	354853.			0.445984
$893$		−	315591.i		−	0.395751i
$894$	−442293.			−0.553395
$895$		−	1.73597e6i		−	2.16719i
$896$			151600.i			0.188836i
$897$	209242.			0.260054
$898$		−	411551.i		−	0.510354i
$899$			11618.7i			0.0143760i
$900$	−259212.			−0.320015
$901$	−1.44272e6			−1.77718
$902$	−249972.			−0.307241
$903$			178357.i			0.218733i
$904$	171509.			0.209869
$905$	1.18794e6			1.45043
$906$	40560.0			0.0494131
$907$	312945.			0.380412			0.190206	−	0.981744i	$-0.439084\pi$
$907$	312945.			0.380412			0.190206	+	0.981744i	$0.439084\pi$
$908$			1.35715e6i			1.64610i
$909$		−	191433.i		−	0.231680i
$910$	1.81140e6			2.18742
$911$	−150467.			−0.181303			−0.0906514	−	0.995883i	$-0.528895\pi$
$911$	−150467.			−0.181303			−0.0906514	+	0.995883i	$0.528895\pi$
$912$	−427208.			−0.513630
$913$	22653.7			0.0271767
$914$	773669.			0.926111
$915$		−	267348.i		−	0.319327i
$916$			540271.i			0.643903i
$917$		−	1.18067e6i		−	1.40408i
$918$		−	316633.i		−	0.375725i
$919$		−	717926.i		−	0.850059i	−0.905180	−	0.425029i	$-0.860264\pi$
$919$		−	717926.i		−	0.850059i	0.905180	−	0.425029i	$-0.139736\pi$
$920$	44964.2			0.0531240
$921$	−460927.			−0.543391
$922$		−	1.62429e6i		−	1.91074i
$923$			2.14780e6i			2.52110i
$924$		−	200839.i		−	0.235237i
$925$		−	229154.i		−	0.267821i
$926$	1.86902e6			2.17968
$927$			169206.i			0.196905i
$928$			133179.i			0.154647i
$929$		−	167906.i		−	0.194551i	−0.995257	−	0.0972757i	$-0.968987\pi$
$929$		−	167906.i		−	0.194551i	0.995257	−	0.0972757i	$-0.0310129\pi$
$930$	−131877.			−0.152477
$931$	302582.			0.349095
$932$		−	1.57544e6i		−	1.81372i
$933$	625307.			0.718340
$934$	2.37616e6			2.72384
$935$		−	763565.i		−	0.873419i
$936$	47525.8			0.0542473
$937$			1.16725e6i			1.32949i	0.747069	+	0.664747i	$0.231461\pi$
$937$			1.16725e6i			1.32949i	−0.747069	+	0.664747i	$0.768539\pi$
$938$	834913.			0.948933
$939$		−	799385.i		−	0.906619i
$940$		−	532455.i		−	0.602597i
$941$			1.41681e6i			1.60004i	0.599973	+	0.800020i	$0.295178\pi$
$941$			1.41681e6i			1.60004i	−0.599973	+	0.800020i	$0.704822\pi$
$942$	1.31558e6			1.48257
$943$			131837.i			0.148257i
$944$	361158.	+	727434.i	0.405278	+	0.816299i
$945$	189235.			0.211904
$946$			286795.i			0.320471i
$947$	−39435.9			−0.0439735			−0.0219868	−	0.999758i	$-0.506999\pi$
$947$	−39435.9			−0.0439735			−0.0219868	+	0.999758i	$0.506999\pi$
$948$	−423469.			−0.471200
$949$	1.00253e6			1.11318
$950$			1.12795e6i			1.24981i
$951$	−68702.0			−0.0759641
$952$			116174.i			0.128184i
$953$	1.14306e6			1.25858			0.629292	−	0.777169i	$-0.283345\pi$
$953$	1.14306e6			1.25858			0.629292	+	0.777169i	$0.283345\pi$
$954$		−	574935.i		−	0.631716i
$955$			1.00203e6i			1.09869i
$956$	1.66483e6			1.82160
$957$		−	26808.5i		−	0.0292717i
$958$		−	489492.i		−	0.533353i
$959$	−821090.			−0.892799
$960$	−845441.			−0.917363
$961$	907128.			0.982249
$962$			554894.i			0.599598i
$963$	40730.9			0.0439210
$964$	1.19806e6			1.28921
$965$	1.09702e6			1.17804
$966$	−203828.			−0.218429
$967$		−	1.11507e6i		−	1.19247i	−0.802810	−	0.596235i	$-0.796662\pi$
$967$		−	1.11507e6i		−	1.19247i	0.802810	−	0.596235i	$-0.203338\pi$
$968$			86304.4i			0.0921048i
$969$	−716017.			−0.762563
$970$	78795.2			0.0837445
$971$	−1.58897e6			−1.68530			−0.842651	−	0.538460i	$-0.819006\pi$
$971$	−1.58897e6			−1.68530			−0.842651	+	0.538460i	$0.819006\pi$
$972$	65572.9			0.0694052
$973$	−890184.			−0.940273
$974$		−	1.43906e6i		−	1.51691i
$975$			670539.i			0.705366i
$976$		−	349513.i		−	0.366914i
$977$		−	1.26322e6i		−	1.32340i	−0.749770	−	0.661699i	$-0.769836\pi$
$977$		−	1.26322e6i		−	1.32340i	0.749770	−	0.661699i	$-0.230164\pi$
$978$			747191.i			0.781185i
$979$	706105.			0.736723
$980$	510507.			0.531557
$981$		−	439068.i		−	0.456240i
$982$		−	290471.i		−	0.301217i
$983$		−	757024.i		−	0.783435i	−0.920086	−	0.391717i	$-0.871881\pi$
$983$		−	757024.i		−	0.783435i	0.920086	−	0.391717i	$-0.128119\pi$
$984$			29944.7i			0.0309265i
$985$	−145982.			−0.150462
$986$			204804.i			0.210661i
$987$			182757.i			0.187603i
$988$		−	1.41940e6i		−	1.45409i
$989$	151258.			0.154641
$990$	304287.			0.310465
$991$		−	814885.i		−	0.829753i	−0.909878	−	0.414877i	$-0.863825\pi$
$991$		−	814885.i		−	0.829753i	0.909878	−	0.414877i	$-0.136175\pi$
$992$	−187905.			−0.190948
$993$	−292688.			−0.296829
$994$		−	2.09223e6i		−	2.11757i
$995$	980977.			0.990861
$996$		−	35840.5i		−	0.0361289i
$997$	−472205.			−0.475051			−0.237525	−	0.971381i	$-0.576336\pi$
$997$	−472205.			−0.475051			−0.237525	+	0.971381i	$0.576336\pi$
$998$			133132.i			0.133666i
$999$			57969.3i			0.0580854i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.7	✓	40
3.2	odd	2		531.5.c.d.235.34		40
59.58	odd	2	inner	177.5.c.a.58.34	yes	40
177.176	even	2		531.5.c.d.235.7		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.7	✓	40	1.1	even	1	trivial
177.5.c.a.58.34	yes	40	59.58	odd	2	inner
531.5.c.d.235.7		40	177.176	even	2
531.5.c.d.235.34		40	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.77154i q^{2} -5.19615 q^{3} -17.3107 q^{4} +34.3452 q^{5} +29.9898i q^{6} -39.2727 q^{7} +7.56484i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-5.77154i q^{2} -5.19615 q^{3} -17.3107 q^{4} +34.3452 q^{5} +29.9898i q^{6} -39.2727 q^{7} +7.56484i q^{8} +27.0000 q^{9} -198.225i q^{10} +56.8541i q^{11} +89.9491 q^{12} -232.684i q^{13} +226.664i q^{14} -178.463 q^{15} -233.311 q^{16} -391.037 q^{17} -155.832i q^{18} -352.390 q^{19} -594.541 q^{20} +204.067 q^{21} +328.136 q^{22} -173.061i q^{23} -39.3081i q^{24} +554.595 q^{25} -1342.94 q^{26} -140.296 q^{27} +679.838 q^{28} +90.7462 q^{29} +1030.01i q^{30} +128.035i q^{31} +1467.60i q^{32} -295.422i q^{33} +2256.89i q^{34} -1348.83 q^{35} -467.389 q^{36} -413.192i q^{37} +2033.83i q^{38} +1209.06i q^{39} +259.816i q^{40} -761.796 q^{41} -1177.78i q^{42} +874.013i q^{43} -984.184i q^{44} +927.321 q^{45} -998.831 q^{46} +895.574i q^{47} +1212.32 q^{48} -858.658 q^{49} -3200.87i q^{50} +2031.89 q^{51} +4027.92i q^{52} +3689.46 q^{53} +809.725i q^{54} +1952.67i q^{55} -297.092i q^{56} +1831.07 q^{57} -523.746i q^{58} +(-1547.97 - 3117.88i) q^{59} +3089.32 q^{60} +1498.06i q^{61} +738.962 q^{62} -1060.36 q^{63} +4737.35 q^{64} -7991.57i q^{65} -1705.04 q^{66} -3683.48i q^{67} +6769.13 q^{68} +899.253i q^{69} +7784.82i q^{70} -9230.55 q^{71} +204.251i q^{72} +4308.55i q^{73} -2384.76 q^{74} -2881.76 q^{75} +6100.12 q^{76} -2232.81i q^{77} +6978.14 q^{78} -4707.88 q^{79} -8013.11 q^{80} +729.000 q^{81} +4396.74i q^{82} -398.453i q^{83} -3532.54 q^{84} -13430.3 q^{85} +5044.40 q^{86} -471.531 q^{87} -430.092 q^{88} -12419.6i q^{89} -5352.08i q^{90} +9138.11i q^{91} +2995.82i q^{92} -665.292i q^{93} +5168.84 q^{94} -12102.9 q^{95} -7625.87i q^{96} +397.504i q^{97} +4955.78i q^{98} +1535.06i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Introduction

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