LMFDB - Embedded newform 531.5.c.d.235.34

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.34
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} -7.56484i q^{8} +O(q^{10})$	\(q+5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} -7.56484i q^{8} -198.225i q^{10} -56.8541i q^{11} -232.684i q^{13} -226.664i q^{14} -233.311 q^{16} +391.037 q^{17} -352.390 q^{19} +594.541 q^{20} +328.136 q^{22} +173.061i q^{23} +554.595 q^{25} +1342.94 q^{26} +679.838 q^{28} -90.7462 q^{29} +128.035i q^{31} -1467.60i q^{32} +2256.89i q^{34} +1348.83 q^{35} -413.192i q^{37} -2033.83i q^{38} +259.816i q^{40} +761.796 q^{41} +874.013i q^{43} +984.184i q^{44} -998.831 q^{46} -895.574i q^{47} -858.658 q^{49} +3200.87i q^{50} +4027.92i q^{52} -3689.46 q^{53} +1952.67i q^{55} +297.092i q^{56} -523.746i q^{58} +(1547.97 + 3117.88i) q^{59} +1498.06i q^{61} -738.962 q^{62} +4737.35 q^{64} +7991.57i q^{65} -3683.48i q^{67} -6769.13 q^{68} +7784.82i q^{70} +9230.55 q^{71} +4308.55i q^{73} +2384.76 q^{74} +6100.12 q^{76} +2232.81i q^{77} -4707.88 q^{79} +8013.11 q^{80} +4396.74i q^{82} +398.453i q^{83} -13430.3 q^{85} -5044.40 q^{86} -430.092 q^{88} +12419.6i q^{89} +9138.11i q^{91} -2995.82i q^{92} +5168.84 q^{94} +12102.9 q^{95} +397.504i q^{97} -4955.78i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 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} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 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 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 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 $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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	$0.256521\pi$
$2$			5.77154i			1.44289i	−0.692474	+	0.721443i	$0.743479\pi$
$3$		0			0
$3$		0			0
$4$	−17.3107			−1.08192
$5$	−34.3452			−1.37381			−0.686905	−	0.726748i	$-0.741031\pi$
$5$	−34.3452			−1.37381			−0.686905	+	0.726748i	$0.741031\pi$
$6$		0			0
$7$	−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
$9$		0			0
$10$		−	198.225i		−	1.98225i
$11$		−	56.8541i		−	0.469868i	−0.972011	−	0.234934i	$-0.924513\pi$
$11$		−	56.8541i		−	0.469868i	0.972011	−	0.234934i	$-0.0754874\pi$
$12$		0			0
$13$		−	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
$15$		0			0
$16$	−233.311			−0.911370
$17$	391.037			1.35307			0.676535	−	0.736411i	$-0.263481\pi$
$17$	391.037			1.35307			0.676535	+	0.736411i	$0.263481\pi$
$18$		0			0
$19$	−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
$21$		0			0
$22$	328.136			0.677966
$23$			173.061i			0.327148i	0.986531	+	0.163574i	$0.0523023\pi$
$23$			173.061i			0.327148i	−0.986531	+	0.163574i	$0.947698\pi$
$24$		0			0
$25$	554.595			0.887352
$26$	1342.94			1.98660
$27$		0			0
$28$	679.838			0.867140
$29$	−90.7462			−0.107903			−0.0539514	−	0.998544i	$-0.517182\pi$
$29$	−90.7462			−0.107903			−0.0539514	+	0.998544i	$0.517182\pi$
$30$		0			0
$31$			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
$33$		0			0
$34$			2256.89i			1.95233i
$35$	1348.83			1.10108
$36$		0			0
$37$		−	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
$39$		0			0
$40$			259.816i			0.162385i
$41$	761.796			0.453180			0.226590	−	0.973990i	$-0.427242\pi$
$41$	761.796			0.453180			0.226590	+	0.973990i	$0.427242\pi$
$42$		0			0
$43$			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
$45$		0			0
$46$	−998.831			−0.472038
$47$		−	895.574i		−	0.405420i	−0.979239	−	0.202710i	$-0.935025\pi$
$47$		−	895.574i		−	0.405420i	0.979239	−	0.202710i	$-0.0649749\pi$
$48$		0			0
$49$	−858.658			−0.357625
$50$			3200.87i			1.28035i
$51$		0			0
$52$			4027.92i			1.48962i
$53$	−3689.46			−1.31344			−0.656721	−	0.754134i	$-0.728057\pi$
$53$	−3689.46			−1.31344			−0.656721	+	0.754134i	$0.728057\pi$
$54$		0			0
$55$			1952.67i			0.645509i
$56$			297.092i			0.0947358i
$57$		0			0
$58$		−	523.746i		−	0.155691i
$59$	1547.97	+	3117.88i	0.444691	+	0.895684i
$59$	1547.97	+	3117.88i	0.444691	+	0.895684i
$60$		0			0
$61$			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
$63$		0			0
$64$	4737.35			1.15658
$65$			7991.57i			1.89150i
$66$		0			0
$67$		−	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
$69$		0			0
$70$			7784.82i			1.58874i
$71$	9230.55			1.83109			0.915547	−	0.402211i	$-0.131758\pi$
$71$	9230.55			1.83109			0.915547	+	0.402211i	$0.131758\pi$
$72$		0			0
$73$			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
$75$		0			0
$76$	6100.12			1.05612
$77$			2232.81i			0.376591i
$78$		0			0
$79$	−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
$81$		0			0
$82$			4396.74i			0.653887i
$83$			398.453i			0.0578390i	0.999582	+	0.0289195i	$0.00920665\pi$
$83$			398.453i			0.0578390i	−0.999582	+	0.0289195i	$0.990793\pi$
$84$		0			0
$85$	−13430.3			−1.85886
$86$	−5044.40			−0.682045
$87$		0			0
$88$	−430.092			−0.0555387
$89$			12419.6i			1.56794i	0.620802	+	0.783968i	$0.286807\pi$
$89$			12419.6i			1.56794i	−0.620802	+	0.783968i	$0.713193\pi$
$90$		0			0
$91$			9138.11i			1.10350i
$92$		−	2995.82i		−	0.353948i
$93$		0			0
$94$	5168.84			0.584975
$95$	12102.9			1.34104
$96$		0			0
$97$			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
$99$		0			0
$100$	−9600.44			−0.960044
$101$			7090.10i			0.695040i	0.937673	+	0.347520i	$0.112976\pi$
$101$			7090.10i			0.695040i	−0.937673	+	0.347520i	$0.887024\pi$
$102$		0			0
$103$			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
$105$		0			0
$106$		−	21293.9i		−	1.89515i
$107$	−1508.55			−0.131763			−0.0658814	−	0.997827i	$-0.520986\pi$
$107$	−1508.55			−0.131763			−0.0658814	+	0.997827i	$0.520986\pi$
$108$		0			0
$109$		−	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
$111$		0			0
$112$	9162.73			0.730447
$113$			22671.8i			1.77553i	0.460293	+	0.887767i	$0.347744\pi$
$113$			22671.8i			1.77553i	−0.460293	+	0.887767i	$0.652256\pi$
$114$		0			0
$115$		−	5943.83i		−	0.449439i
$116$	1570.88			0.116742
$117$		0			0
$118$	−17995.0	+	8934.17i	−1.29237	+	0.641638i
$119$	−15357.1			−1.08446
$120$		0			0
$121$	11408.6			0.779224
$122$	−8646.11			−0.580900
$123$		0			0
$124$		−	2216.39i		−	0.144146i
$125$	2418.07			0.154757
$126$		0			0
$127$	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
$129$		0			0
$130$	−46123.7			−2.72921
$131$		−	30063.5i		−	1.75185i	−0.482446	−	0.875926i	$-0.660252\pi$
$131$		−	30063.5i		−	1.75185i	0.482446	−	0.875926i	$-0.339748\pi$
$132$		0			0
$133$	13839.3			0.782367
$134$	21259.4			1.18397
$135$		0			0
$136$		−	2958.14i		−	0.159934i
$137$	−20907.4			−1.11393			−0.556967	−	0.830535i	$-0.688035\pi$
$137$	−20907.4			−1.11393			−0.556967	+	0.830535i	$0.688035\pi$
$138$		0			0
$139$	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
$141$		0			0
$142$			53274.5i			2.64206i
$143$	−13229.0			−0.646927
$144$		0			0
$145$	3116.70			0.148238
$146$	−24867.0			−1.16659
$147$		0			0
$148$			7152.65i			0.326546i
$149$		−	14748.1i		−	0.664299i	−0.943227	−	0.332150i	$-0.892226\pi$
$149$		−	14748.1i		−	0.664299i	0.943227	−	0.332150i	$-0.107774\pi$
$150$		0			0
$151$		−	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
$153$		0			0
$154$	−12886.8			−0.543378
$155$		−	4397.41i		−	0.183035i
$156$		0			0
$157$		−	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
$159$		0			0
$160$			50405.1i			1.96895i
$161$		−	6796.58i		−	0.262204i
$162$		0			0
$163$	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
$165$		0			0
$166$	−2299.69			−0.0834551
$167$	15384.8			0.551644			0.275822	−	0.961209i	$-0.411050\pi$
$167$	15384.8			0.551644			0.275822	+	0.961209i	$0.411050\pi$
$168$		0			0
$169$	−25580.7			−0.895651
$170$		−	77513.4i		−	2.68212i
$171$		0			0
$172$		−	15129.8i		−	0.511418i
$173$			54918.5i			1.83496i	0.397784	+	0.917479i	$0.369779\pi$
$173$			54918.5i			1.83496i	−0.397784	+	0.917479i	$0.630221\pi$
$174$		0			0
$175$	−21780.4			−0.711198
$176$			13264.7i			0.428224i
$177$		0			0
$178$	−71680.3			−2.26235
$179$			50544.7i			1.57750i	0.614713	+	0.788751i	$0.289272\pi$
$179$			50544.7i			1.57750i	−0.614713	+	0.788751i	$0.710728\pi$
$180$		0			0
$181$	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
$183$		0			0
$184$	1309.18			0.0386691
$185$			14191.2i			0.414644i
$186$		0			0
$187$		−	22232.0i		−	0.635765i
$188$			15503.0i			0.438632i
$189$		0			0
$190$			69852.5i			1.93497i
$191$		−	29175.3i		−	0.799739i	−0.916572	−	0.399869i	$-0.869055\pi$
$191$		−	29175.3i		−	0.799739i	0.916572	−	0.399869i	$-0.130945\pi$
$192$		0			0
$193$	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
$195$		0			0
$196$	14864.0			0.386922
$197$	4250.44			0.109522			0.0547610	−	0.998499i	$-0.482560\pi$
$197$	4250.44			0.109522			0.0547610	+	0.998499i	$0.482560\pi$
$198$		0			0
$199$	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
$201$		0			0
$202$	−40920.8			−1.00286
$203$	3563.84			0.0864822
$204$		0			0
$205$	−26164.1			−0.622583
$206$	−36169.6			−0.852334
$207$		0			0
$208$			54287.6i			1.25480i
$209$			20034.8i			0.458662i
$210$		0			0
$211$			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
$213$		0			0
$214$		−	8706.68i		−	0.190119i
$215$		−	30018.2i		−	0.649393i
$216$		0			0
$217$		−	5028.29i		−	0.106783i
$218$	93855.5			1.97491
$219$		0			0
$220$		−	33802.0i		−	0.698389i
$221$		−	90988.0i		−	1.86294i
$222$		0			0
$223$	−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
$225$		0			0
$226$	−130851.			−2.56189
$227$			78399.4i			1.52146i	0.649067	+	0.760731i	$0.275159\pi$
$227$			78399.4i			1.52146i	−0.649067	+	0.760731i	$0.724841\pi$
$228$		0			0
$229$		−	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
$231$		0			0
$232$			686.481i			0.0127542i
$233$		−	91009.4i		−	1.67639i	−0.545372	−	0.838194i	$-0.683612\pi$
$233$		−	91009.4i		−	1.67639i	0.545372	−	0.838194i	$-0.316388\pi$
$234$		0			0
$235$			30758.7i			0.556970i
$236$	−26796.5	−	53972.7i	−0.481120	−	0.969058i
$237$		0			0
$238$		−	88634.0i		−	1.56476i
$239$	96173.1			1.68367			0.841837	−	0.539732i	$-0.181474\pi$
$239$	96173.1			1.68367			0.841837	+	0.539732i	$0.181474\pi$
$240$		0			0
$241$	−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
$243$		0			0
$244$		−	25932.5i		−	0.435576i
$245$	29490.8			0.491309
$246$		0			0
$247$			81995.4i			1.34399i
$248$	968.568			0.0157481
$249$		0			0
$250$			13956.0i			0.223296i
$251$	−94134.4			−1.49417			−0.747087	−	0.664726i	$-0.768548\pi$
$251$	−94134.4			−1.49417			−0.747087	+	0.664726i	$0.768548\pi$
$252$		0			0
$253$	9839.24			0.153717
$254$			42883.3i			0.664692i
$255$		0			0
$256$	53518.2			0.816623
$257$	−11680.1			−0.176841			−0.0884203	−	0.996083i	$-0.528182\pi$
$257$	−11680.1			−0.176841			−0.0884203	+	0.996083i	$0.528182\pi$
$258$		0			0
$259$			16227.2i			0.241904i
$260$		−	138340.i		−	2.04645i
$261$		0			0
$262$	173513.			2.52772
$263$	67551.5			0.976616			0.488308	−	0.872671i	$-0.337614\pi$
$263$	67551.5			0.976616			0.488308	+	0.872671i	$0.337614\pi$
$264$		0			0
$265$	126715.			1.80442
$266$			79874.1i			1.12887i
$267$		0			0
$268$			63763.7i			0.887778i
$269$			8085.17i			0.111734i	0.998438	+	0.0558669i	$0.0177923\pi$
$269$			8085.17i			0.111734i	−0.998438	+	0.0558669i	$0.982208\pi$
$270$		0			0
$271$	−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
$273$		0			0
$274$		−	120668.i		−	1.60728i
$275$		−	31531.0i		−	0.416939i
$276$		0			0
$277$	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
$279$		0			0
$280$		−	10203.7i		−	0.130149i
$281$	132826.			1.68217			0.841085	−	0.540903i	$-0.181917\pi$
$281$	132826.			1.68217			0.841085	+	0.540903i	$0.181917\pi$
$282$		0			0
$283$		−	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
$285$		0			0
$286$		−	76351.8i		−	0.933442i
$287$	−29917.7			−0.363216
$288$		0			0
$289$	69389.1			0.830798
$290$			17988.2i			0.213890i
$291$		0			0
$292$		−	74584.1i		−	0.874743i
$293$	138078.			1.60838			0.804189	−	0.594374i	$-0.202600\pi$
$293$	138078.			1.60838			0.804189	+	0.594374i	$0.202600\pi$
$294$		0			0
$295$	−53165.4	−	107084.i	−0.610921	−	1.23050i
$296$	−3125.74			−0.0356754
$297$		0			0
$298$	85119.4			0.958508
$299$	40268.6			0.450426
$300$		0			0
$301$		−	34324.8i		−	0.378857i
$302$	7805.78			0.0855859
$303$		0			0
$304$	82216.3			0.889633
$305$		−	51451.2i		−	0.553090i
$306$		0			0
$307$	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
$309$		0			0
$310$	25379.8			0.264098
$311$	120340.			1.24420			0.622101	−	0.782937i	$-0.286279\pi$
$311$	120340.			1.24420			0.622101	+	0.782937i	$0.286279\pi$
$312$		0			0
$313$			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
$315$		0			0
$316$	81496.7			0.816142
$317$	−13221.7			−0.131574			−0.0657868	−	0.997834i	$-0.520956\pi$
$317$	−13221.7			−0.131574			−0.0657868	+	0.997834i	$0.520956\pi$
$318$		0			0
$319$			5159.29i			0.0507001i
$320$	−162705.			−1.58892
$321$		0			0
$322$	39226.8			0.378330
$323$	−137798.			−1.32080
$324$		0			0
$325$		−	129045.i		−	1.22173i
$326$			143797.i			1.35305i
$327$		0			0
$328$		−	5762.87i		−	0.0535662i
$329$			35171.6i			0.324938i
$330$		0			0
$331$	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
$333$		0			0
$334$			88794.1i			0.795960i
$335$			126510.i			1.12729i
$336$		0			0
$337$		−	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
$339$		0			0
$340$	232487.			2.01114
$341$	7279.34			0.0626013
$342$		0			0
$343$	128015.			1.08811
$344$	6611.77			0.0558729
$345$		0			0
$346$	−316964.			−2.64763
$347$		−	9584.04i		−	0.0795957i	−0.999208	−	0.0397979i	$-0.987329\pi$
$347$		−	9584.04i		−	0.0795957i	0.999208	−	0.0397979i	$-0.0126714\pi$
$348$		0			0
$349$		−	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
$351$		0			0
$352$	−83439.0			−0.673416
$353$			12045.3i			0.0966652i	0.998831	+	0.0483326i	$0.0153907\pi$
$353$			12045.3i			0.0966652i	−0.998831	+	0.0483326i	$0.984609\pi$
$354$		0			0
$355$	−317025.			−2.51557
$356$		−	214992.i		−	1.69638i
$357$		0			0
$358$	−291721.			−2.27615
$359$	11159.4			0.0865872			0.0432936	−	0.999062i	$-0.486215\pi$
$359$	11159.4			0.0865872			0.0432936	+	0.999062i	$0.486215\pi$
$360$		0			0
$361$	−6142.36			−0.0471326
$362$			199627.i			1.52336i
$363$		0			0
$364$		−	158187.i		−	1.19390i
$365$		−	147978.i		−	1.11074i
$366$		0			0
$367$			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
$369$		0			0
$370$	−81905.0			−0.598284
$371$	144895.			1.05270
$372$		0			0
$373$	−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
$375$		0			0
$376$	−6774.87			−0.0479210
$377$			21115.2i			0.148563i
$378$		0			0
$379$	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
$381$		0			0
$382$	168386.			1.15393
$383$	−181721.			−1.23882			−0.619408	−	0.785069i	$-0.712627\pi$
$383$	−181721.			−1.23882			−0.619408	+	0.785069i	$0.712627\pi$
$384$		0			0
$385$		−	76686.4i		−	0.517365i
$386$			184349.i			1.23727i
$387$		0			0
$388$		−	6881.08i		−	0.0457081i
$389$	133518.			0.882350			0.441175	−	0.897421i	$-0.354562\pi$
$389$	133518.			0.882350			0.441175	+	0.897421i	$0.354562\pi$
$390$		0			0
$391$			67673.4i			0.442654i
$392$			6495.61i			0.0422715i
$393$		0			0
$394$			24531.6i			0.158028i
$395$	161693.			1.03633
$396$		0			0
$397$		−	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
$399$		0			0
$400$	−129393.			−0.808706
$401$			208735.i			1.29809i	0.760748	+	0.649047i	$0.224832\pi$
$401$			208735.i			1.29809i	−0.760748	+	0.649047i	$0.775168\pi$
$402$		0			0
$403$	29791.8			0.183437
$404$		−	122735.i		−	0.751977i
$405$		0			0
$406$			20568.9i			0.124784i
$407$	−23491.7			−0.141816
$408$		0			0
$409$			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
$411$		0			0
$412$		−	108484.i		−	0.639106i
$413$	−60792.9	−	122447.i	−0.356412	−	0.717875i
$414$		0			0
$415$		−	13685.0i		−	0.0794598i
$416$	−341486.			−1.97327
$417$		0			0
$418$	−115632.			−0.661796
$419$			15318.9i			0.0872570i	0.999048	+	0.0436285i	$0.0138918\pi$
$419$			15318.9i			0.0872570i	−0.999048	+	0.0436285i	$0.986108\pi$
$420$		0			0
$421$			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
$423$		0			0
$424$			27910.2i			0.155250i
$425$	216867.			1.20065
$426$		0			0
$427$		−	58832.8i		−	0.322674i
$428$	26114.1			0.142557
$429$		0			0
$430$	173251.			0.937000
$431$			202239.i			1.08870i	0.838857	+	0.544352i	$0.183224\pi$
$431$			202239.i			1.08870i	−0.838857	+	0.544352i	$0.816776\pi$
$432$		0			0
$433$	−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
$435$		0			0
$436$			281503.i			1.48085i
$437$		−	60985.1i		−	0.319345i
$438$		0			0
$439$	−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
$441$		0			0
$442$	525141.			2.68801
$443$		−	19857.5i		−	0.101185i	−0.998719	−	0.0505925i	$-0.983889\pi$
$443$		−	19857.5i		−	0.101185i	0.998719	−	0.0505925i	$-0.0161110\pi$
$444$		0			0
$445$		−	426555.i		−	2.15404i
$446$		−	118311.i		−	0.594779i
$447$		0			0
$448$	−186048.			−0.926978
$449$	−71307.0			−0.353703			−0.176852	−	0.984238i	$-0.556591\pi$
$449$	−71307.0			−0.353703			−0.176852	+	0.984238i	$0.556591\pi$
$450$		0			0
$451$		−	43311.2i		−	0.212935i
$452$		−	392465.i		−	1.92099i
$453$		0			0
$454$	−452486.			−2.19530
$455$		−	313850.i		−	1.51600i
$456$		0			0
$457$			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
$459$		0			0
$460$			102892.i			0.486257i
$461$	−281430.			−1.32425			−0.662123	−	0.749395i	$-0.730344\pi$
$461$	−281430.			−1.32425			−0.662123	+	0.749395i	$0.730344\pi$
$462$		0			0
$463$			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
$465$		0			0
$466$	525265.			2.41884
$467$		−	411702.i		−	1.88777i	−0.330272	−	0.943886i	$-0.607141\pi$
$467$		−	411702.i		−	1.88777i	0.330272	−	0.943886i	$-0.392859\pi$
$468$		0			0
$469$			144660.i			0.657663i
$470$	−177525.			−0.803645
$471$		0			0
$472$	23586.2	−	11710.1i	0.105870	−	0.0525628i
$473$	49691.2			0.222104
$474$		0			0
$475$	−195434.			−0.866188
$476$	265842.			1.17330
$477$		0			0
$478$			555067.i			2.42935i
$479$	−84811.3			−0.369643			−0.184822	−	0.982772i	$-0.559171\pi$
$479$	−84811.3			−0.369643			−0.184822	+	0.982772i	$0.559171\pi$
$480$		0			0
$481$	−96143.1			−0.415554
$482$		−	399442.i		−	1.71933i
$483$		0			0
$484$	−197491.			−0.843058
$485$		−	13652.4i		−	0.0580396i
$486$		0			0
$487$	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
$489$		0			0
$490$			170207.i			0.708902i
$491$	−50328.2			−0.208760			−0.104380	−	0.994537i	$-0.533286\pi$
$491$	−50328.2			−0.208760			−0.104380	+	0.994537i	$0.533286\pi$
$492$		0			0
$493$	−35485.1			−0.146000
$494$	−473240.			−1.93922
$495$		0			0
$496$		−	29872.0i		−	0.121423i
$497$	−362508.			−1.46759
$498$		0			0
$499$	−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
$501$		0			0
$502$		−	543301.i		−	2.15592i
$503$		−	410578.i		−	1.62278i	−0.584505	−	0.811390i	$-0.698711\pi$
$503$		−	410578.i		−	1.62278i	0.584505	−	0.811390i	$-0.301289\pi$
$504$		0			0
$505$		−	243511.i		−	0.954852i
$506$			56787.6i			0.221795i
$507$		0			0
$508$	−128621.			−0.498406
$509$			19719.0i			0.0761115i	0.999276	+	0.0380557i	$0.0121164\pi$
$509$			19719.0i			0.0761115i	−0.999276	+	0.0380557i	$0.987884\pi$
$510$		0			0
$511$		−	169208.i		−	0.648007i
$512$			370646.i			1.41390i
$513$		0			0
$514$		−	67412.5i		−	0.255161i
$515$		−	215238.i		−	0.811529i
$516$		0			0
$517$	−50917.0			−0.190494
$518$	−93655.8			−0.349040
$519$		0			0
$520$	60455.0			0.223576
$521$	−420195.			−1.54802			−0.774008	−	0.633175i	$-0.781751\pi$
$521$	−420195.			−1.54802			−0.774008	+	0.633175i	$0.781751\pi$
$522$		0			0
$523$	−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
$525$		0			0
$526$			389877.i			1.40915i
$527$			50066.6i			0.180272i
$528$		0			0
$529$	249891.			0.892974
$530$			731343.i			2.60357i
$531$		0			0
$532$	−239568.			−0.846458
$533$		−	177257.i		−	0.623950i
$534$		0			0
$535$	51811.6			0.181017
$536$	−27865.0			−0.0969905
$537$		0			0
$538$	−46663.9			−0.161219
$539$			48818.2i			0.168037i
$540$		0			0
$541$			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
$543$		0			0
$544$		−	573886.i		−	1.93922i
$545$			558515.i			1.88036i
$546$		0			0
$547$	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
$549$		0			0
$550$	181982.			0.601595
$551$	31978.0			0.105329
$552$		0			0
$553$	184891.			0.604596
$554$			639060.i			2.08220i
$555$		0			0
$556$	−392378.			−1.26927
$557$	85305.1			0.274957			0.137478	−	0.990505i	$-0.456100\pi$
$557$	85305.1			0.274957			0.137478	+	0.990505i	$0.456100\pi$
$558$		0			0
$559$	203369.			0.650819
$560$	−314696.			−1.00350
$561$		0			0
$562$			766610.i			2.42718i
$563$			625430.i			1.97316i	0.163282	+	0.986580i	$0.447792\pi$
$563$			625430.i			1.97316i	−0.163282	+	0.986580i	$0.552208\pi$
$564$		0			0
$565$		−	778668.i		−	2.43925i
$566$	788680.			2.46189
$567$		0			0
$568$		−	69827.6i		−	0.216437i
$569$		−	182688.i		−	0.564269i	−0.959375	−	0.282134i	$-0.908958\pi$
$569$		−	182688.i		−	0.564269i	0.959375	−	0.282134i	$-0.0910424\pi$
$570$		0			0
$571$		−	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
$573$		0			0
$574$		−	172672.i		−	0.524079i
$575$			95979.0i			0.290296i
$576$		0			0
$577$	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
$579$		0			0
$580$	−53952.3			−0.160381
$581$		−	15648.3i		−	0.0463570i
$582$		0			0
$583$			209761.i			0.617145i
$584$	32593.5			0.0955664
$585$		0			0
$586$			796921.i			2.32071i
$587$		−	466181.i		−	1.35294i	−0.736470	−	0.676470i	$-0.763509\pi$
$587$		−	466181.i		−	1.35294i	0.736470	−	0.676470i	$-0.236491\pi$
$588$		0			0
$589$		−	45118.4i		−	0.130054i
$590$	618041.	−	306846.i	1.77547	−	0.881489i
$591$		0			0
$592$			96402.1i			0.275070i
$593$	263489.			0.749294			0.374647	−	0.927167i	$-0.377764\pi$
$593$	263489.			0.749294			0.374647	+	0.927167i	$0.377764\pi$
$594$		0			0
$595$	527442.			1.48984
$596$			255300.i			0.718719i
$597$		0			0
$598$			232412.i			0.649914i
$599$	182847.			0.509606			0.254803	−	0.966993i	$-0.417989\pi$
$599$	182847.			0.509606			0.254803	+	0.966993i	$0.417989\pi$
$600$		0			0
$601$		−	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
$603$		0			0
$604$			23412.0i			0.0641749i
$605$	−391832.			−1.07051
$606$		0			0
$607$	−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
$609$		0			0
$610$	296953.			0.798046
$611$	−208385.			−0.558193
$612$		0			0
$613$			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
$615$		0			0
$616$	16890.9			0.0445134
$617$	17414.4			0.0457444			0.0228722	−	0.999738i	$-0.492719\pi$
$617$	17414.4			0.0457444			0.0228722	+	0.999738i	$0.492719\pi$
$618$		0			0
$619$	−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
$621$		0			0
$622$			694550.i			1.79524i
$623$		−	487751.i		−	1.25667i
$624$		0			0
$625$	−429671.			−1.09996
$626$	−887904.			−2.26578
$627$		0			0
$628$			759376.i			1.92547i
$629$		−	161574.i		−	0.408384i
$630$		0			0
$631$	−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
$633$		0			0
$634$		−	76309.6i		−	0.189846i
$635$	−255189.			−0.632871
$636$		0			0
$637$			199796.i			0.492388i
$638$	−29777.1			−0.0731544
$639$		0			0
$640$		−	132580.i		−	0.323681i
$641$	210985.			0.513495			0.256747	−	0.966479i	$-0.417349\pi$
$641$	210985.			0.513495			0.256747	+	0.966479i	$0.417349\pi$
$642$		0			0
$643$	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
$645$		0			0
$646$		−	795305.i		−	1.90576i
$647$	671147.			1.60328			0.801639	−	0.597808i	$-0.203962\pi$
$647$	671147.			1.60328			0.801639	+	0.597808i	$0.203962\pi$
$648$		0			0
$649$	177264.	−	88008.3i	0.420853	−	0.208946i
$650$	744790.			1.76282
$651$		0			0
$652$	−431294.			−1.01456
$653$	−330138.			−0.774227			−0.387114	−	0.922032i	$-0.626528\pi$
$653$	−330138.			−0.774227			−0.387114	+	0.922032i	$0.626528\pi$
$654$		0			0
$655$			1.03254e6i			2.40671i
$656$	−177735.			−0.413014
$657$		0			0
$658$	−202994.			−0.468848
$659$			45073.9i			0.103790i	0.998653	+	0.0518949i	$0.0165261\pi$
$659$			45073.9i			0.103790i	−0.998653	+	0.0518949i	$0.983474\pi$
$660$		0			0
$661$	−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
$663$		0			0
$664$	3014.23			0.00683661
$665$	−475314.			−1.07482
$666$		0			0
$667$		−	15704.7i		−	0.0353002i
$668$	−266322.			−0.596835
$669$		0			0
$670$	−730159.			−1.62655
$671$	85170.7			0.189167
$672$		0			0
$673$		−	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
$675$		0			0
$676$	442820.			0.969022
$677$	254006.			0.554199			0.277100	−	0.960841i	$-0.410627\pi$
$677$	254006.			0.554199			0.277100	+	0.960841i	$0.410627\pi$
$678$		0			0
$679$		−	15611.0i		−	0.0338604i
$680$			101598.i			0.219719i
$681$		0			0
$682$			42013.0i			0.0903265i
$683$			537333.i			1.15187i	0.817497	+	0.575933i	$0.195361\pi$
$683$			537333.i			1.15187i	−0.817497	+	0.575933i	$0.804639\pi$
$684$		0			0
$685$	718071.			1.53033
$686$			738847.i			1.57002i
$687$		0			0
$688$		−	203917.i		−	0.430800i
$689$			858477.i			1.80838i
$690$		0			0
$691$		−	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
$693$		0			0
$694$	55314.7			0.114848
$695$	−778495.			−1.61171
$696$		0			0
$697$	297890.			0.613184
$698$	858099.			1.76127
$699$		0			0
$700$	377035.			0.769459
$701$		−	792788.i		−	1.61332i	−0.591015	−	0.806661i	$-0.701273\pi$
$701$		−	792788.i		−	1.61332i	0.591015	−	0.806661i	$-0.298727\pi$
$702$		0			0
$703$			145605.i			0.294622i
$704$		−	269337.i		−	0.543440i
$705$		0			0
$706$	−69520.3			−0.139477
$707$		−	278447.i		−	0.557063i
$708$		0			0
$709$	−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
$711$		0			0
$712$	93952.4			0.185331
$713$	−22158.0			−0.0435864
$714$		0			0
$715$	454353.			0.888754
$716$		−	874965.i		−	1.70673i
$717$		0			0
$718$			64407.2i			0.124935i
$719$		−	267439.i		−	0.517329i	−0.965967	−	0.258665i	$-0.916718\pi$
$719$		−	267439.i		−	0.517329i	0.965967	−	0.258665i	$-0.0832825\pi$
$720$		0			0
$721$		−	246117.i		−	0.473448i
$722$		−	35450.9i		−	0.0680069i
$723$		0			0
$724$	−598745.			−1.14226
$725$	−50327.4			−0.0957477
$726$		0			0
$727$	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
$729$		0			0
$730$	854062.			1.60267
$731$			341772.i			0.639589i
$732$		0			0
$733$	−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
$735$		0			0
$736$	253985.			0.468870
$737$	−209421.			−0.385554
$738$		0			0
$739$			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
$741$		0			0
$742$			836267.i			1.51893i
$743$	266645.			0.483009			0.241505	−	0.970400i	$-0.422359\pi$
$743$	266645.			0.483009			0.241505	+	0.970400i	$0.422359\pi$
$744$		0			0
$745$			506527.i			0.912621i
$746$		−	502581.i		−	0.903085i
$747$		0			0
$748$			384853.i			0.687846i
$749$	59244.9			0.105606
$750$		0			0
$751$		−	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
$753$		0			0
$754$	−121867.			−0.214360
$755$			46450.5i			0.0814886i
$756$		0			0
$757$	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
$759$		0			0
$760$		−	91556.7i		−	0.158512i
$761$	326900.			0.564476			0.282238	−	0.959344i	$-0.408923\pi$
$761$	326900.			0.564476			0.282238	+	0.959344i	$0.408923\pi$
$762$		0			0
$763$			638643.i			1.09701i
$764$			505045.i			0.865253i
$765$		0			0
$766$		−	1.04881e6i		−	1.78747i
$767$	725479.	−	360187.i	1.23320	−	0.612262i
$768$		0			0
$769$		−	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
$771$		0			0
$772$	−552922.			−0.927746
$773$		−	210457.i		−	0.352213i	−0.984371	−	0.176106i	$-0.943650\pi$
$773$		−	210457.i		−	0.352213i	0.984371	−	0.176106i	$-0.0563503\pi$
$774$		0			0
$775$			71007.9i			0.118223i
$776$	3007.05			0.00499365
$777$		0			0
$778$			770606.i			1.27313i
$779$	−268449.			−0.442371
$780$		0			0
$781$		−	524794.i		−	0.860373i
$782$	−390580.			−0.638700
$783$		0			0
$784$	200334.			0.325929
$785$			1.50664e6i			2.44494i
$786$		0			0
$787$	−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
$789$		0			0
$790$			933219.i			1.49530i
$791$		−	890382.i		−	1.42306i
$792$		0			0
$793$	348574.			0.554305
$794$	159464.			0.252943
$795$		0			0
$796$	−494433.			−0.780335
$797$			592344.i			0.932518i	0.884648	+	0.466259i	$0.154399\pi$
$797$			592344.i			0.932518i	−0.884648	+	0.466259i	$0.845601\pi$
$798$		0			0
$799$		−	350203.i		−	0.548562i
$800$		−	813924.i		−	1.27176i
$801$		0			0
$802$	−1.20472e6			−1.87300
$803$	244959.			0.379893
$804$		0			0
$805$			233430.i			0.360218i
$806$			171944.i			0.264678i
$807$		0			0
$808$	53635.5			0.0821542
$809$		−	340035.i		−	0.519550i	−0.965669	−	0.259775i	$-0.916352\pi$
$809$		−	340035.i		−	0.519550i	0.965669	−	0.259775i	$-0.0836484\pi$
$810$		0			0
$811$			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
$813$		0			0
$814$		−	135583.i		−	0.204624i
$815$	−855706.			−1.28828
$816$		0			0
$817$		−	307993.i		−	0.461421i
$818$	−1.60863e6			−2.40408
$819$		0			0
$820$	452918.			0.673585
$821$		−	623437.i		−	0.924924i	−0.886639	−	0.462462i	$-0.846966\pi$
$821$		−	623437.i		−	0.924924i	0.886639	−	0.462462i	$-0.153034\pi$
$822$		0			0
$823$		−	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
$825$		0			0
$826$	706710.	−	350869.i	1.03581	−	0.514262i
$827$	72588.8			0.106135			0.0530675	−	0.998591i	$-0.483100\pi$
$827$	72588.8			0.106135			0.0530675	+	0.998591i	$0.483100\pi$
$828$		0			0
$829$	−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
$831$		0			0
$832$		−	1.10230e6i		−	1.59241i
$833$	−335767.			−0.483892
$834$		0			0
$835$	−528395.			−0.757854
$836$		−	346817.i		−	0.496235i
$837$		0			0
$838$	−88413.8			−0.125902
$839$		−	129224.i		−	0.183578i	−0.995778	−	0.0917888i	$-0.970742\pi$
$839$		−	129224.i		−	0.183578i	0.995778	−	0.0917888i	$-0.0292585\pi$
$840$		0			0
$841$	−699046.			−0.988357
$842$	−1.60885e6			−2.26930
$843$		0			0
$844$		−	458419.i		−	0.643543i
$845$	878574.			1.23045
$846$		0			0
$847$	−448047.			−0.624535
$848$	860790.			1.19703
$849$		0			0
$850$			1.25166e6i			1.73240i
$851$	71507.6			0.0987400
$852$		0			0
$853$	−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
$855$		0			0
$856$			11412.0i			0.0155745i
$857$			28783.4i			0.0391905i	0.999808	+	0.0195952i	$0.00623775\pi$
$857$			28783.4i			0.0391905i	−0.999808	+	0.0195952i	$0.993762\pi$
$858$		0			0
$859$		−	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
$861$		0			0
$862$	−1.16723e6			−1.57088
$863$		−	287494.i		−	0.386018i	−0.981197	−	0.193009i	$-0.938175\pi$
$863$		−	287494.i		−	0.386018i	0.981197	−	0.193009i	$-0.0618247\pi$
$864$		0			0
$865$		−	1.88619e6i		−	2.52088i
$866$		−	511787.i		−	0.682423i
$867$		0			0
$868$			87043.4i			0.115530i
$869$			267662.i			0.354443i
$870$		0			0
$871$	−857087.			−1.12977
$872$	−123018.			−0.161784
$873$		0			0
$874$	351978.			0.460779
$875$	−94964.1			−0.124035
$876$		0			0
$877$	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
$879$		0			0
$880$		−	455578.i		−	0.588298i
$881$			389453.i			0.501769i	0.968017	+	0.250884i	$0.0807214\pi$
$881$			389453.i			0.501769i	−0.968017	+	0.250884i	$0.919279\pi$
$882$		0			0
$883$	−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
$885$		0			0
$886$	114608.			0.145999
$887$			283208.i			0.359963i	0.983670	+	0.179981i	$0.0576038\pi$
$887$			283208.i			0.359963i	−0.983670	+	0.179981i	$0.942396\pi$
$888$		0			0
$889$	−291801.			−0.369218
$890$	2.46188e6			3.10804
$891$		0			0
$892$	354853.			0.445984
$893$			315591.i			0.395751i
$894$		0			0
$895$		−	1.73597e6i		−	2.16719i
$896$		−	151600.i		−	0.188836i
$897$		0			0
$898$		−	411551.i		−	0.510354i
$899$		−	11618.7i		−	0.0143760i
$900$		0			0
$901$	−1.44272e6			−1.77718
$902$	249972.			0.307241
$903$		0			0
$904$	171509.			0.209869
$905$	−1.18794e6			−1.45043
$906$		0			0
$907$	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
$909$		0			0
$910$	1.81140e6			2.18742
$911$	150467.			0.181303			0.0906514	−	0.995883i	$-0.471105\pi$
$911$	150467.			0.181303			0.0906514	+	0.995883i	$0.471105\pi$
$912$		0			0
$913$	22653.7			0.0271767
$914$	−773669.			−0.926111
$915$		0			0
$916$			540271.i			0.643903i
$917$			1.18067e6i			1.40408i
$918$		0			0
$919$		−	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
$921$		0			0
$922$		−	1.62429e6i		−	1.91074i
$923$		−	2.14780e6i		−	2.52110i
$924$		0			0
$925$		−	229154.i		−	0.267821i
$926$	−1.86902e6			−2.17968
$927$		0			0
$928$			133179.i			0.154647i
$929$			167906.i			0.194551i	0.995257	+	0.0972757i	$0.0310129\pi$
$929$			167906.i			0.194551i	−0.995257	+	0.0972757i	$0.968987\pi$
$930$		0			0
$931$	302582.			0.349095
$932$			1.57544e6i			1.81372i
$933$		0			0
$934$	2.37616e6			2.72384
$935$			763565.i			0.873419i
$936$		0			0
$937$			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
$939$		0			0
$940$		−	532455.i		−	0.602597i
$941$		−	1.41681e6i		−	1.60004i	−0.599973	−	0.800020i	$-0.704822\pi$
$941$		−	1.41681e6i		−	1.60004i	0.599973	−	0.800020i	$-0.295178\pi$
$942$		0			0
$943$			131837.i			0.148257i
$944$	−361158.	−	727434.i	−0.405278	−	0.816299i
$945$		0			0
$946$			286795.i			0.320471i
$947$	39435.9			0.0439735			0.0219868	−	0.999758i	$-0.493001\pi$
$947$	39435.9			0.0439735			0.0219868	+	0.999758i	$0.493001\pi$
$948$		0			0
$949$	1.00253e6			1.11318
$950$		−	1.12795e6i		−	1.24981i
$951$		0			0
$952$			116174.i			0.128184i
$953$	−1.14306e6			−1.25858			−0.629292	−	0.777169i	$-0.716655\pi$
$953$	−1.14306e6			−1.25858			−0.629292	+	0.777169i	$0.716655\pi$
$954$		0			0
$955$			1.00203e6i			1.09869i
$956$	−1.66483e6			−1.82160
$957$		0			0
$958$		−	489492.i		−	0.533353i
$959$	821090.			0.892799
$960$		0			0
$961$	907128.			0.982249
$962$		−	554894.i		−	0.599598i
$963$		0			0
$964$	1.19806e6			1.28921
$965$	−1.09702e6			−1.17804
$966$		0			0
$967$		−	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
$969$		0			0
$970$	78795.2			0.0837445
$971$	1.58897e6			1.68530			0.842651	−	0.538460i	$-0.180994\pi$
$971$	1.58897e6			1.68530			0.842651	+	0.538460i	$0.180994\pi$
$972$		0			0
$973$	−890184.			−0.940273
$974$			1.43906e6i			1.51691i
$975$		0			0
$976$		−	349513.i		−	0.366914i
$977$			1.26322e6i			1.32340i	0.749770	+	0.661699i	$0.230164\pi$
$977$			1.26322e6i			1.32340i	−0.749770	+	0.661699i	$0.769836\pi$
$978$		0			0
$979$	706105.			0.736723
$980$	−510507.			−0.531557
$981$		0			0
$982$		−	290471.i		−	0.301217i
$983$			757024.i			0.783435i	0.920086	+	0.391717i	$0.128119\pi$
$983$			757024.i			0.783435i	−0.920086	+	0.391717i	$0.871881\pi$
$984$		0			0
$985$	−145982.			−0.150462
$986$		−	204804.i		−	0.210661i
$987$		0			0
$988$		−	1.41940e6i		−	1.45409i
$989$	−151258.			−0.154641
$990$		0			0
$991$		−	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
$993$		0			0
$994$		−	2.09223e6i		−	2.11757i
$995$	−980977.			−0.990861
$996$		0			0
$997$	−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
$999$		0			0

Twists

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

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

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} -7.56484i q^{8} +O(q^{10})\)	\(q+5.77154i q^{2} -17.3107 q^{4} -34.3452 q^{5} -39.2727 q^{7} -7.56484i q^{8} -198.225i q^{10} -56.8541i q^{11} -232.684i q^{13} -226.664i q^{14} -233.311 q^{16} +391.037 q^{17} -352.390 q^{19} +594.541 q^{20} +328.136 q^{22} +173.061i q^{23} +554.595 q^{25} +1342.94 q^{26} +679.838 q^{28} -90.7462 q^{29} +128.035i q^{31} -1467.60i q^{32} +2256.89i q^{34} +1348.83 q^{35} -413.192i q^{37} -2033.83i q^{38} +259.816i q^{40} +761.796 q^{41} +874.013i q^{43} +984.184i q^{44} -998.831 q^{46} -895.574i q^{47} -858.658 q^{49} +3200.87i q^{50} +4027.92i q^{52} -3689.46 q^{53} +1952.67i q^{55} +297.092i q^{56} -523.746i q^{58} +(1547.97 + 3117.88i) q^{59} +1498.06i q^{61} -738.962 q^{62} +4737.35 q^{64} +7991.57i q^{65} -3683.48i q^{67} -6769.13 q^{68} +7784.82i q^{70} +9230.55 q^{71} +4308.55i q^{73} +2384.76 q^{74} +6100.12 q^{76} +2232.81i q^{77} -4707.88 q^{79} +8013.11 q^{80} +4396.74i q^{82} +398.453i q^{83} -13430.3 q^{85} -5044.40 q^{86} -430.092 q^{88} +12419.6i q^{89} +9138.11i q^{91} -2995.82i q^{92} +5168.84 q^{94} +12102.9 q^{95} +397.504i q^{97} -4955.78i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 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 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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