LMFDB - Embedded newform 177.5.c.a.58.13

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			58.13
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.44422i q^{2} +5.19615 q^{3} +4.13738 q^{4} -16.2205 q^{5} -17.8967i q^{6} +92.6602 q^{7} -69.3575i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-3.44422i q^{2} +5.19615 q^{3} +4.13738 q^{4} -16.2205 q^{5} -17.8967i q^{6} +92.6602 q^{7} -69.3575i q^{8} +27.0000 q^{9} +55.8670i q^{10} +48.6771i q^{11} +21.4985 q^{12} +216.742i q^{13} -319.142i q^{14} -84.2844 q^{15} -172.684 q^{16} +171.979 q^{17} -92.9938i q^{18} +267.365 q^{19} -67.1105 q^{20} +481.476 q^{21} +167.654 q^{22} -734.072i q^{23} -360.392i q^{24} -361.894 q^{25} +746.506 q^{26} +140.296 q^{27} +383.370 q^{28} +999.352 q^{29} +290.294i q^{30} +568.212i q^{31} -514.959i q^{32} +252.934i q^{33} -592.334i q^{34} -1503.00 q^{35} +111.709 q^{36} -1003.07i q^{37} -920.864i q^{38} +1126.22i q^{39} +1125.02i q^{40} -2301.23 q^{41} -1658.31i q^{42} -2157.18i q^{43} +201.396i q^{44} -437.954 q^{45} -2528.30 q^{46} +1199.13i q^{47} -897.293 q^{48} +6184.90 q^{49} +1246.44i q^{50} +893.631 q^{51} +896.743i q^{52} -1101.00 q^{53} -483.210i q^{54} -789.569i q^{55} -6426.67i q^{56} +1389.27 q^{57} -3441.98i q^{58} +(-1324.95 - 3218.99i) q^{59} -348.716 q^{60} -4030.40i q^{61} +1957.05 q^{62} +2501.82 q^{63} -4536.57 q^{64} -3515.67i q^{65} +871.158 q^{66} +8142.38i q^{67} +711.544 q^{68} -3814.35i q^{69} +5176.65i q^{70} -2537.07 q^{71} -1872.65i q^{72} +792.331i q^{73} -3454.80 q^{74} -1880.46 q^{75} +1106.19 q^{76} +4510.43i q^{77} +3878.96 q^{78} -3214.46 q^{79} +2801.03 q^{80} +729.000 q^{81} +7925.93i q^{82} +10537.1i q^{83} +1992.05 q^{84} -2789.60 q^{85} -7429.78 q^{86} +5192.79 q^{87} +3376.12 q^{88} +226.497i q^{89} +1508.41i q^{90} +20083.3i q^{91} -3037.13i q^{92} +2952.52i q^{93} +4130.07 q^{94} -4336.81 q^{95} -2675.80i q^{96} +7280.88i q^{97} -21302.1i q^{98} +1314.28i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.44422i		−	0.861054i	−0.902578	−	0.430527i	$-0.858328\pi$
$2$		−	3.44422i		−	0.861054i	0.902578	−	0.430527i	$-0.141672\pi$
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	4.13738			0.258586
$5$	−16.2205			−0.648821			−0.324411	−	0.945916i	$-0.605166\pi$
$5$	−16.2205			−0.648821			−0.324411	+	0.945916i	$0.605166\pi$
$6$		−	17.8967i		−	0.497130i
$7$	92.6602			1.89102			0.945512	−	0.325588i	$-0.105562\pi$
$7$	92.6602			1.89102			0.945512	+	0.325588i	$0.105562\pi$
$8$		−	69.3575i		−	1.08371i
$9$	27.0000			0.333333
$10$			55.8670i			0.558670i
$11$			48.6771i			0.402290i	0.979561	+	0.201145i	$0.0644663\pi$
$11$			48.6771i			0.402290i	−0.979561	+	0.201145i	$0.935534\pi$
$12$	21.4985			0.149295
$13$			216.742i			1.28250i	0.767334	+	0.641248i	$0.221583\pi$
$13$			216.742i			1.28250i	−0.767334	+	0.641248i	$0.778417\pi$
$14$		−	319.142i		−	1.62827i
$15$	−84.2844			−0.374597
$16$	−172.684			−0.674547
$17$	171.979			0.595084			0.297542	−	0.954709i	$-0.403833\pi$
$17$	171.979			0.595084			0.297542	+	0.954709i	$0.403833\pi$
$18$		−	92.9938i		−	0.287018i
$19$	267.365			0.740624			0.370312	−	0.928907i	$-0.379251\pi$
$19$	267.365			0.740624			0.370312	+	0.928907i	$0.379251\pi$
$20$	−67.1105			−0.167776
$21$	481.476			1.09178
$22$	167.654			0.346394
$23$		−	734.072i		−	1.38766i	−0.720139	−	0.693830i	$-0.755922\pi$
$23$		−	734.072i		−	1.38766i	0.720139	−	0.693830i	$-0.244078\pi$
$24$		−	360.392i		−	0.625681i
$25$	−361.894			−0.579031
$26$	746.506			1.10430
$27$	140.296			0.192450
$28$	383.370			0.488993
$29$	999.352			1.18829			0.594145	−	0.804358i	$-0.297490\pi$
$29$	999.352			1.18829			0.594145	+	0.804358i	$0.297490\pi$
$30$			290.294i			0.322548i
$31$			568.212i			0.591272i	0.955301	+	0.295636i	$0.0955315\pi$
$31$			568.212i			0.591272i	−0.955301	+	0.295636i	$0.904468\pi$
$32$		−	514.959i		−	0.502889i
$33$			252.934i			0.232262i
$34$		−	592.334i		−	0.512400i
$35$	−1503.00			−1.22694
$36$	111.709			0.0861954
$37$		−	1003.07i		−	0.732704i	−0.930476	−	0.366352i	$-0.880607\pi$
$37$		−	1003.07i		−	0.732704i	0.930476	−	0.366352i	$-0.119393\pi$
$38$		−	920.864i		−	0.637717i
$39$			1126.22i			0.740449i
$40$			1125.02i			0.703135i
$41$	−2301.23			−1.36897			−0.684483	−	0.729029i	$-0.739972\pi$
$41$	−2301.23			−1.36897			−0.684483	+	0.729029i	$0.739972\pi$
$42$		−	1658.31i		−	0.940084i
$43$		−	2157.18i		−	1.16667i	−0.812231	−	0.583336i	$-0.801747\pi$
$43$		−	2157.18i		−	1.16667i	0.812231	−	0.583336i	$-0.198253\pi$
$44$			201.396i			0.104027i
$45$	−437.954			−0.216274
$46$	−2528.30			−1.19485
$47$			1199.13i			0.542840i	0.962461	+	0.271420i	$0.0874932\pi$
$47$			1199.13i			0.542840i	−0.962461	+	0.271420i	$0.912507\pi$
$48$	−897.293			−0.389450
$49$	6184.90			2.57597
$50$			1246.44i			0.498577i
$51$	893.631			0.343572
$52$			896.743i			0.331636i
$53$	−1101.00			−0.391956			−0.195978	−	0.980608i	$-0.562788\pi$
$53$	−1101.00			−0.391956			−0.195978	+	0.980608i	$0.562788\pi$
$54$		−	483.210i		−	0.165710i
$55$		−	789.569i		−	0.261015i
$56$		−	6426.67i		−	2.04932i
$57$	1389.27			0.427599
$58$		−	3441.98i		−	1.02318i
$59$	−1324.95	−	3218.99i	−0.380623	−	0.924730i
$59$	−1324.95	−	3218.99i	−0.380623	−	0.924730i
$60$	−348.716			−0.0968657
$61$		−	4030.40i		−	1.08315i	−0.840653	−	0.541574i	$-0.817828\pi$
$61$		−	4030.40i		−	1.08315i	0.840653	−	0.541574i	$-0.182172\pi$
$62$	1957.05			0.509117
$63$	2501.82			0.630341
$64$	−4536.57			−1.10756
$65$		−	3515.67i		−	0.832111i
$66$	871.158			0.199990
$67$			8142.38i			1.81385i	0.421290	+	0.906926i	$0.361577\pi$
$67$			8142.38i			1.81385i	−0.421290	+	0.906926i	$0.638423\pi$
$68$	711.544			0.153881
$69$		−	3814.35i		−	0.801165i
$70$			5176.65i			1.05646i
$71$	−2537.07			−0.503286			−0.251643	−	0.967820i	$-0.580971\pi$
$71$	−2537.07			−0.503286			−0.251643	+	0.967820i	$0.580971\pi$
$72$		−	1872.65i		−	0.361237i
$73$			792.331i			0.148683i	0.997233	+	0.0743414i	$0.0236855\pi$
$73$			792.331i			0.148683i	−0.997233	+	0.0743414i	$0.976315\pi$
$74$	−3454.80			−0.630898
$75$	−1880.46			−0.334304
$76$	1106.19			0.191515
$77$			4510.43i			0.760740i
$78$	3878.96			0.637567
$79$	−3214.46			−0.515056			−0.257528	−	0.966271i	$-0.582908\pi$
$79$	−3214.46			−0.515056			−0.257528	+	0.966271i	$0.582908\pi$
$80$	2801.03			0.437661
$81$	729.000			0.111111
$82$			7925.93i			1.17875i
$83$			10537.1i			1.52956i	0.644292	+	0.764779i	$0.277152\pi$
$83$			10537.1i			1.52956i	−0.644292	+	0.764779i	$0.722848\pi$
$84$	1992.05			0.282320
$85$	−2789.60			−0.386103
$86$	−7429.78			−1.00457
$87$	5192.79			0.686060
$88$	3376.12			0.435966
$89$			226.497i			0.0285945i	0.999898	+	0.0142972i	$0.00455111\pi$
$89$			226.497i			0.0285945i	−0.999898	+	0.0142972i	$0.995449\pi$
$90$			1508.41i			0.186223i
$91$			20083.3i			2.42523i
$92$		−	3037.13i		−	0.358829i
$93$			2952.52i			0.341371i
$94$	4130.07			0.467414
$95$	−4336.81			−0.480533
$96$		−	2675.80i		−	0.290343i
$97$			7280.88i			0.773821i	0.922117	+	0.386911i	$0.126458\pi$
$97$			7280.88i			0.773821i	−0.922117	+	0.386911i	$0.873542\pi$
$98$		−	21302.1i		−	2.21805i
$99$			1314.28i			0.134097i
$100$	−1497.29			−0.149729
$101$			11745.9i			1.15145i	0.817643	+	0.575725i	$0.195280\pi$
$101$			11745.9i			1.15145i	−0.817643	+	0.575725i	$0.804720\pi$
$102$		−	3077.86i		−	0.295834i
$103$			19029.4i			1.79370i	0.442333	+	0.896851i	$0.354151\pi$
$103$			19029.4i			1.79370i	−0.442333	+	0.896851i	$0.645849\pi$
$104$	15032.7			1.38985
$105$	−7809.80			−0.708372
$106$			3792.10i			0.337495i
$107$	6751.89			0.589737			0.294868	−	0.955538i	$-0.404724\pi$
$107$	6751.89			0.589737			0.294868	+	0.955538i	$0.404724\pi$
$108$	580.458			0.0497649
$109$		−	22919.8i		−	1.92911i	−0.263876	−	0.964556i	$-0.585001\pi$
$109$		−	22919.8i		−	1.92911i	0.263876	−	0.964556i	$-0.414999\pi$
$110$	−2719.45			−0.224748
$111$		−	5212.12i		−	0.423027i
$112$	−16000.9			−1.27558
$113$		−	1167.57i		−	0.0914377i	−0.998954	−	0.0457188i	$-0.985442\pi$
$113$		−	1167.57i		−	0.0914377i	0.998954	−	0.0457188i	$-0.0145578\pi$
$114$		−	4784.95i		−	0.368186i
$115$			11907.0i			0.900343i
$116$	4134.70			0.307276
$117$			5852.03i			0.427499i
$118$	−11086.9	+	4563.41i	−0.796243	+	0.327737i
$119$	15935.6			1.12532
$120$			5845.75i			0.405955i
$121$	12271.5			0.838163
$122$	−13881.6			−0.932649
$123$	−11957.5			−0.790372
$124$			2350.91i			0.152895i
$125$	16008.0			1.02451
$126$		−	8616.82i		−	0.542758i
$127$	−14064.3			−0.871990			−0.435995	−	0.899949i	$-0.643603\pi$
$127$	−14064.3			−0.871990			−0.435995	+	0.899949i	$0.643603\pi$
$128$			7385.60i			0.450781i
$129$		−	11209.0i		−	0.673578i
$130$	−12108.7			−0.716492
$131$			15833.5i			0.922644i	0.887233	+	0.461322i	$0.152625\pi$
$131$			15833.5i			0.922644i	−0.887233	+	0.461322i	$0.847375\pi$
$132$			1046.48i			0.0600598i
$133$	24774.1			1.40054
$134$	28044.1			1.56182
$135$	−2275.68			−0.124866
$136$		−	11928.1i		−	0.644899i
$137$	−21940.7			−1.16899			−0.584493	−	0.811399i	$-0.698707\pi$
$137$	−21940.7			−1.16899			−0.584493	+	0.811399i	$0.698707\pi$
$138$	−13137.4			−0.689847
$139$	−33025.6			−1.70931			−0.854655	−	0.519196i	$-0.826231\pi$
$139$	−33025.6			−1.70931			−0.854655	+	0.519196i	$0.826231\pi$
$140$	−6218.47			−0.317269
$141$			6230.88i			0.313409i
$142$			8738.20i			0.433357i
$143$	−10550.4			−0.515936
$144$	−4662.47			−0.224849
$145$	−16210.0			−0.770988
$146$	2728.96			0.128024
$147$	32137.7			1.48724
$148$		−	4150.09i		−	0.189467i
$149$			20821.5i			0.937863i	0.883235	+	0.468932i	$0.155361\pi$
$149$			20821.5i			0.937863i	−0.883235	+	0.468932i	$0.844639\pi$
$150$			6476.70i			0.287853i
$151$		−	9654.62i		−	0.423430i	−0.977331	−	0.211715i	$-0.932095\pi$
$151$		−	9654.62i		−	0.423430i	0.977331	−	0.211715i	$-0.0679048\pi$
$152$		−	18543.8i		−	0.802622i
$153$	4643.44			0.198361
$154$	15534.9			0.655038
$155$		−	9216.71i		−	0.383630i
$156$			4659.61i			0.191470i
$157$			23483.9i			0.952731i	0.879247	+	0.476366i	$0.158046\pi$
$157$			23483.9i			0.952731i	−0.879247	+	0.476366i	$0.841954\pi$
$158$			11071.3i			0.443491i
$159$	−5720.99			−0.226296
$160$			8352.90i			0.326285i
$161$		−	68019.2i		−	2.62410i
$162$		−	2510.83i		−	0.0956727i
$163$	12805.6			0.481977			0.240988	−	0.970528i	$-0.422528\pi$
$163$	12805.6			0.481977			0.240988	+	0.970528i	$0.422528\pi$
$164$	−9521.06			−0.353995
$165$		−	4102.72i		−	0.150697i
$166$	36292.2			1.31703
$167$	−29956.3			−1.07413			−0.537063	−	0.843542i	$-0.680466\pi$
$167$	−29956.3			−1.07413			−0.537063	+	0.843542i	$0.680466\pi$
$168$		−	33394.0i		−	1.18318i
$169$	−18416.0			−0.644796
$170$			9607.98i			0.332456i
$171$	7218.86			0.246875
$172$		−	8925.05i		−	0.301685i
$173$		−	51496.1i		−	1.72061i	−0.509781	−	0.860304i	$-0.670274\pi$
$173$		−	51496.1i		−	1.72061i	0.509781	−	0.860304i	$-0.329726\pi$
$174$		−	17885.1i		−	0.590735i
$175$	−33533.2			−1.09496
$176$		−	8405.76i		−	0.271364i
$177$	−6884.64	−	16726.3i	−0.219753	−	0.533893i
$178$	780.104			0.0246214
$179$		−	17057.4i		−	0.532363i	−0.963923	−	0.266181i	$-0.914238\pi$
$179$		−	17057.4i		−	0.532363i	0.963923	−	0.266181i	$-0.0857620\pi$
$180$	−1811.98			−0.0559254
$181$	7526.56			0.229741			0.114871	−	0.993380i	$-0.463355\pi$
$181$	7526.56			0.229741			0.114871	+	0.993380i	$0.463355\pi$
$182$	69171.3			2.08825
$183$		−	20942.6i		−	0.625356i
$184$	−50913.4			−1.50382
$185$			16270.4i			0.475394i
$186$	10169.1			0.293939
$187$			8371.46i			0.239397i
$188$			4961.27i			0.140371i
$189$	12999.9			0.363928
$190$			14936.9i			0.413765i
$191$			34367.0i			0.942053i	0.882119	+	0.471026i	$0.156116\pi$
$191$			34367.0i			0.942053i	−0.882119	+	0.471026i	$0.843884\pi$
$192$	−23572.7			−0.639451
$193$	−35591.7			−0.955507			−0.477753	−	0.878494i	$-0.658549\pi$
$193$	−35591.7			−0.955507			−0.477753	+	0.878494i	$0.658549\pi$
$194$	25076.9			0.666302
$195$		−	18267.9i		−	0.480419i
$196$	25589.3			0.666110
$197$	−25127.2			−0.647458			−0.323729	−	0.946150i	$-0.604937\pi$
$197$	−25127.2			−0.647458			−0.323729	+	0.946150i	$0.604937\pi$
$198$	4526.67			0.115465
$199$	19989.2			0.504765			0.252382	−	0.967628i	$-0.418786\pi$
$199$	19989.2			0.504765			0.252382	+	0.967628i	$0.418786\pi$
$200$			25100.1i			0.627502i
$201$			42309.1i			1.04723i
$202$	40455.6			0.991460
$203$	92600.1			2.24709
$204$	3697.29			0.0888430
$205$	37327.2			0.888214
$206$	65541.3			1.54447
$207$		−	19819.9i		−	0.462553i
$208$		−	37427.9i		−	0.865104i
$209$			13014.6i			0.297946i
$210$			26898.6i			0.609947i
$211$		−	32244.7i		−	0.724257i	−0.932128	−	0.362129i	$-0.882050\pi$
$211$		−	32244.7i		−	0.724257i	0.932128	−	0.362129i	$-0.117950\pi$
$212$	−4555.27			−0.101354
$213$	−13183.0			−0.290573
$214$		−	23255.0i		−	0.507795i
$215$			34990.5i			0.756962i
$216$		−	9730.58i		−	0.208560i
$217$			52650.6i			1.11811i
$218$	−78940.7			−1.66107
$219$			4117.07i			0.0858421i
$220$		−	3266.75i		−	0.0674948i
$221$			37275.1i			0.763193i
$222$	−17951.7			−0.364249
$223$	−28806.5			−0.579270			−0.289635	−	0.957137i	$-0.593534\pi$
$223$	−28806.5			−0.579270			−0.289635	+	0.957137i	$0.593534\pi$
$224$		−	47716.1i		−	0.950975i
$225$	−9771.14			−0.193010
$226$	−4021.36			−0.0787328
$227$		−	57367.6i		−	1.11331i	−0.830745	−	0.556654i	$-0.812085\pi$
$227$		−	57367.6i		−	1.11331i	0.830745	−	0.556654i	$-0.187915\pi$
$228$	5747.94			0.110571
$229$			8038.50i			0.153287i	0.997059	+	0.0766433i	$0.0244203\pi$
$229$			8038.50i			0.153287i	−0.997059	+	0.0766433i	$0.975580\pi$
$230$	41010.4			0.775244
$231$			23436.9i			0.439214i
$232$		−	69312.6i		−	1.28776i
$233$			55821.9i			1.02824i	0.857719	+	0.514118i	$0.171881\pi$
$233$			55821.9i			1.02824i	−0.857719	+	0.514118i	$0.828119\pi$
$234$	20155.7			0.368099
$235$		−	19450.6i		−	0.352206i
$236$	−5481.82	−	13318.2i	−0.0984239	−	0.239122i
$237$	−16702.8			−0.297368
$238$		−	54885.8i		−	0.968960i
$239$	−35079.1			−0.614120			−0.307060	−	0.951690i	$-0.599345\pi$
$239$	−35079.1			−0.614120			−0.307060	+	0.951690i	$0.599345\pi$
$240$	14554.6			0.252683
$241$	84946.1			1.46255			0.731273	−	0.682085i	$-0.238927\pi$
$241$	84946.1			1.46255			0.731273	+	0.682085i	$0.238927\pi$
$242$		−	42265.8i		−	0.721703i
$243$	3788.00			0.0641500
$244$		−	16675.3i		−	0.280087i
$245$	−100322.			−1.67134
$246$			41184.4i			0.680553i
$247$			57949.2i			0.949847i
$248$	39409.8			0.640768
$249$			54752.5i			0.883091i
$250$		−	55134.8i		−	0.882157i
$251$	−62187.4			−0.987086			−0.493543	−	0.869721i	$-0.664298\pi$
$251$	−62187.4			−0.987086			−0.493543	+	0.869721i	$0.664298\pi$
$252$	10351.0			0.162998
$253$	35732.5			0.558242
$254$			48440.5i			0.750830i
$255$	−14495.2			−0.222917
$256$	−47147.6			−0.719415
$257$	−11454.6			−0.173426			−0.0867129	−	0.996233i	$-0.527636\pi$
$257$	−11454.6			−0.173426			−0.0867129	+	0.996233i	$0.527636\pi$
$258$	−38606.3			−0.579987
$259$		−	92944.8i		−	1.38556i
$260$		−	14545.7i		−	0.215172i
$261$	26982.5			0.396097
$262$	54533.9			0.794446
$263$	39389.7			0.569471			0.284735	−	0.958606i	$-0.408094\pi$
$263$	39389.7			0.569471			0.284735	+	0.958606i	$0.408094\pi$
$264$	17542.8			0.251705
$265$	17858.9			0.254310
$266$		−	85327.4i		−	1.20594i
$267$			1176.91i			0.0165090i
$268$			33688.1i			0.469037i
$269$			13109.8i			0.181172i	0.995889	+	0.0905861i	$0.0288740\pi$
$269$			13109.8i			0.181172i	−0.995889	+	0.0905861i	$0.971126\pi$
$270$			7837.93i			0.107516i
$271$	63942.9			0.870670			0.435335	−	0.900268i	$-0.356630\pi$
$271$	63942.9			0.870670			0.435335	+	0.900268i	$0.356630\pi$
$272$	−29698.1			−0.401412
$273$			104356.i			1.40021i
$274$			75568.5i			1.00656i
$275$		−	17616.0i		−	0.232938i
$276$		−	15781.4i		−	0.207170i
$277$	−26097.8			−0.340130			−0.170065	−	0.985433i	$-0.554398\pi$
$277$	−26097.8			−0.340130			−0.170065	+	0.985433i	$0.554398\pi$
$278$			113747.i			1.47181i
$279$			15341.7i			0.197091i
$280$			104244.i			1.32964i
$281$	−33934.6			−0.429765			−0.214882	−	0.976640i	$-0.568937\pi$
$281$	−33934.6			−0.429765			−0.214882	+	0.976640i	$0.568937\pi$
$282$	21460.5			0.269862
$283$		−	50184.5i		−	0.626609i	−0.949653	−	0.313305i	$-0.898564\pi$
$283$		−	50184.5i		−	0.626609i	0.949653	−	0.313305i	$-0.101436\pi$
$284$	−10496.8			−0.130143
$285$	−22534.7			−0.277436
$286$			36337.7i			0.444248i
$287$	−213232.			−2.58875
$288$		−	13903.9i		−	0.167630i
$289$	−53944.1			−0.645875
$290$			55830.8i			0.663863i
$291$			37832.6i			0.446766i
$292$			3278.17i			0.0384473i
$293$	−145740.			−1.69764			−0.848819	−	0.528684i	$-0.822686\pi$
$293$	−145740.			−1.69764			−0.848819	+	0.528684i	$0.822686\pi$
$294$		−	110689.i		−	1.28059i
$295$	21491.4	+	52213.7i	0.246957	+	0.599985i
$296$	−69570.6			−0.794040
$297$			6829.21i			0.0774208i
$298$	71713.7			0.807551
$299$	159104.			1.77967
$300$	−7780.17			−0.0864463
$301$		−	199884.i		−	2.20620i
$302$	−33252.6			−0.364596
$303$			61033.7i			0.664790i
$304$	−46169.7			−0.499586
$305$			65375.2i			0.702770i
$306$		−	15993.0i		−	0.170800i
$307$	−37666.1			−0.399645			−0.199822	−	0.979832i	$-0.564037\pi$
$307$	−37666.1			−0.399645			−0.199822	+	0.979832i	$0.564037\pi$
$308$			18661.4i			0.196717i
$309$			98879.5i			1.03559i
$310$	−31744.3			−0.330326
$311$	−145763.			−1.50704			−0.753522	−	0.657423i	$-0.771647\pi$
$311$	−145763.			−1.50704			−0.753522	+	0.657423i	$0.771647\pi$
$312$	78112.0			0.802433
$313$		−	28891.9i		−	0.294908i	−0.989069	−	0.147454i	$-0.952892\pi$
$313$		−	28891.9i		−	0.294908i	0.989069	−	0.147454i	$-0.0471079\pi$
$314$	80883.5			0.820353
$315$	−40580.9			−0.408979
$316$	−13299.5			−0.133186
$317$	149119.			1.48393			0.741967	−	0.670437i	$-0.233893\pi$
$317$	149119.			1.48393			0.741967	+	0.670437i	$0.233893\pi$
$318$			19704.3i			0.194853i
$319$			48645.6i			0.478038i
$320$	73585.6			0.718610
$321$	35083.9			0.340485
$322$	−234273.			−2.25949
$323$	45981.3			0.440734
$324$	3016.15			0.0287318
$325$		−	78437.6i		−	0.742605i
$326$		−	44105.4i		−	0.415008i
$327$		−	119095.i		−	1.11377i
$328$			159608.i			1.48356i
$329$			111112.i			1.02652i
$330$	−14130.7			−0.129758
$331$	20162.8			0.184033			0.0920164	−	0.995757i	$-0.470669\pi$
$331$	20162.8			0.184033			0.0920164	+	0.995757i	$0.470669\pi$
$332$			43596.1i			0.395523i
$333$		−	27083.0i		−	0.244235i
$334$			103176.i			0.924880i
$335$		−	132074.i		−	1.17687i
$336$	−83143.3			−0.736459
$337$		−	56644.5i		−	0.498767i	−0.968405	−	0.249384i	$-0.919772\pi$
$337$		−	56644.5i		−	0.498767i	0.968405	−	0.249384i	$-0.0802280\pi$
$338$			63428.7i			0.555204i
$339$		−	6066.86i		−	0.0527916i
$340$	−11541.6			−0.0998410
$341$	−27658.9			−0.237863
$342$		−	24863.3i		−	0.212572i
$343$	350617.			2.98020
$344$	−149616.			−1.26433
$345$			61870.8i			0.519813i
$346$	−177364.			−1.48154
$347$		−	98468.3i		−	0.817782i	−0.912583	−	0.408891i	$-0.865916\pi$
$347$		−	98468.3i		−	0.817782i	0.912583	−	0.408891i	$-0.134084\pi$
$348$	21484.5			0.177406
$349$			11046.8i			0.0906953i	0.998971	+	0.0453476i	$0.0144396\pi$
$349$			11046.8i			0.0906953i	−0.998971	+	0.0453476i	$0.985560\pi$
$350$			115495.i			0.942820i
$351$			30408.0i			0.246816i
$352$	25066.7			0.202307
$353$			82041.3i			0.658390i	0.944262	+	0.329195i	$0.106777\pi$
$353$			82041.3i			0.658390i	−0.944262	+	0.329195i	$0.893223\pi$
$354$	−57609.1	+	23712.2i	−0.459711	+	0.189219i
$355$	41152.6			0.326543
$356$			937.104i			0.00739414i
$357$	82804.0			0.649703
$358$	−58749.5			−0.458393
$359$	−48438.3			−0.375837			−0.187919	−	0.982185i	$-0.560174\pi$
$359$	−48438.3			−0.375837			−0.187919	+	0.982185i	$0.560174\pi$
$360$			30375.4i			0.234378i
$361$	−58836.8			−0.451476
$362$		−	25923.1i		−	0.197820i
$363$	63764.8			0.483913
$364$			83092.3i			0.627131i
$365$		−	12852.0i		−	0.0964686i
$366$	−72130.7			−0.538465
$367$		−	206944.i		−	1.53646i	−0.640175	−	0.768229i	$-0.721138\pi$
$367$		−	206944.i		−	1.53646i	0.640175	−	0.768229i	$-0.278862\pi$
$368$			126762.i			0.936041i
$369$	−62133.2			−0.456322
$370$	56038.7			0.409340
$371$	−102019.			−0.741198
$372$			12215.7i			0.0882738i
$373$	96391.4			0.692821			0.346410	−	0.938083i	$-0.387401\pi$
$373$	96391.4			0.692821			0.346410	+	0.938083i	$0.387401\pi$
$374$	28833.1			0.206133
$375$	83179.8			0.591501
$376$	83168.8			0.588281
$377$			216601.i			1.52398i
$378$		−	44774.3i		−	0.313361i
$379$	222227.			1.54710			0.773549	−	0.633737i	$-0.218480\pi$
$379$	222227.			1.54710			0.773549	+	0.633737i	$0.218480\pi$
$380$	−17943.0			−0.124259
$381$	−73080.3			−0.503443
$382$	118367.			0.811158
$383$	251261.			1.71288			0.856440	−	0.516246i	$-0.172671\pi$
$383$	251261.			1.71288			0.856440	+	0.516246i	$0.172671\pi$
$384$			38376.7i			0.260259i
$385$		−	73161.6i		−	0.493585i
$386$			122585.i			0.822743i
$387$		−	58243.7i		−	0.388891i
$388$			30123.8i			0.200100i
$389$	247468.			1.63538			0.817691	−	0.575657i	$-0.195254\pi$
$389$	247468.			1.63538			0.817691	+	0.575657i	$0.195254\pi$
$390$	−62918.8			−0.413667
$391$		−	126245.i		−	0.825774i
$392$		−	428969.i		−	2.79161i
$393$			82273.2i			0.532689i
$394$			86543.4i			0.557496i
$395$	52140.3			0.334179
$396$			5437.68i			0.0346756i
$397$		−	15188.5i		−	0.0963684i	−0.998838	−	0.0481842i	$-0.984657\pi$
$397$		−	15188.5i		−	0.0963684i	0.998838	−	0.0481842i	$-0.0153435\pi$
$398$		−	68847.1i		−	0.434630i
$399$	128730.			0.808601
$400$	62493.4			0.390583
$401$			41102.4i			0.255611i	0.991799	+	0.127805i	$0.0407933\pi$
$401$			41102.4i			0.255611i	−0.991799	+	0.127805i	$0.959207\pi$
$402$	145722.			0.901720
$403$	−123155.			−0.758304
$404$			48597.4i			0.297749i
$405$	−11824.8			−0.0720913
$406$		−	318935.i		−	1.93486i
$407$	48826.7			0.294760
$408$		−	61980.0i		−	0.372333i
$409$		−	119091.i		−	0.711925i	−0.934500	−	0.355962i	$-0.884153\pi$
$409$		−	119091.i		−	0.711925i	0.934500	−	0.355962i	$-0.115847\pi$
$410$		−	128563.i		−	0.764800i
$411$	−114007.			−0.674915
$412$			78731.7i			0.463826i
$413$	−122770.	−	298272.i	−0.719768	−	1.74869i
$414$	−68264.1			−0.398283
$415$		−	170918.i		−	0.992411i
$416$	111613.			0.644953
$417$	−171606.			−0.986871
$418$	44825.0			0.256547
$419$			228690.i			1.30263i	0.758809	+	0.651313i	$0.225782\pi$
$419$			228690.i			1.30263i	−0.758809	+	0.651313i	$0.774218\pi$
$420$	−32312.1			−0.183175
$421$		−	282680.i		−	1.59489i	−0.603390	−	0.797447i	$-0.706184\pi$
$421$		−	282680.i		−	1.59489i	0.603390	−	0.797447i	$-0.293816\pi$
$422$	−111058.			−0.623624
$423$			32376.6i			0.180947i
$424$			76362.9i			0.424767i
$425$	−62238.3			−0.344572
$426$			45405.0i			0.250199i
$427$		−	373457.i		−	2.04826i
$428$	27935.1			0.152498
$429$	−54821.3			−0.297876
$430$	120515.			0.651785
$431$			270326.i			1.45524i	0.685981	+	0.727619i	$0.259373\pi$
$431$			270326.i			1.45524i	−0.685981	+	0.727619i	$0.740627\pi$
$432$	−24226.9			−0.129817
$433$	−201414.			−1.07427			−0.537136	−	0.843496i	$-0.680494\pi$
$433$	−201414.			−1.07427			−0.537136	+	0.843496i	$0.680494\pi$
$434$	181340.			0.962752
$435$	−84229.8			−0.445130
$436$		−	94827.9i		−	0.498842i
$437$		−	196265.i		−	1.02773i
$438$	14180.1			0.0739147
$439$	−10876.9			−0.0564388			−0.0282194	−	0.999602i	$-0.508984\pi$
$439$	−10876.9			−0.0564388			−0.0282194	+	0.999602i	$0.508984\pi$
$440$	−54762.5			−0.282864
$441$	166992.			0.858657
$442$	128384.			0.657151
$443$		−	28282.8i		−	0.144117i	−0.997400	−	0.0720584i	$-0.977043\pi$
$443$		−	28282.8i		−	0.144117i	0.997400	−	0.0720584i	$-0.0229568\pi$
$444$		−	21564.5i		−	0.109389i
$445$		−	3673.90i		−	0.0185527i
$446$			99215.8i			0.498782i
$447$			108192.i			0.541475i
$448$	−420360.			−2.09443
$449$	158321.			0.785320			0.392660	−	0.919684i	$-0.371555\pi$
$449$	158321.			0.785320			0.392660	+	0.919684i	$0.371555\pi$
$450$			33653.9i			0.166192i
$451$		−	112017.i		−	0.550721i
$452$		−	4830.67i		−	0.0236445i
$453$		−	50166.9i		−	0.244467i
$454$	−197586.			−0.958618
$455$		−	325762.i		−	1.57354i
$456$		−	96356.3i		−	0.463394i
$457$			35833.3i			0.171575i	0.996313	+	0.0857876i	$0.0273406\pi$
$457$			35833.3i			0.171575i	−0.996313	+	0.0857876i	$0.972659\pi$
$458$	27686.3			0.131988
$459$	24128.0			0.114524
$460$			49263.9i			0.232816i
$461$	307571.			1.44725			0.723625	−	0.690193i	$-0.242474\pi$
$461$	307571.			1.44725			0.723625	+	0.690193i	$0.242474\pi$
$462$	80721.7			0.378187
$463$			272379.i			1.27061i	0.772262	+	0.635304i	$0.219125\pi$
$463$			272379.i			1.27061i	−0.772262	+	0.635304i	$0.780875\pi$
$464$	−172572.			−0.801558
$465$		−	47891.4i		−	0.221489i
$466$	192263.			0.885367
$467$			217459.i			0.997111i	0.866858	+	0.498556i	$0.166136\pi$
$467$			217459.i			0.997111i	−0.866858	+	0.498556i	$0.833864\pi$
$468$			24212.1i			0.110545i
$469$			754474.i			3.43004i
$470$	−66992.0			−0.303268
$471$			122026.i			0.550060i
$472$	−223261.	+	91895.2i	−1.00214	+	0.412485i
$473$	105005.			0.469341
$474$			57528.2i			0.256050i
$475$	−96757.9			−0.428844
$476$	65931.8			0.290992
$477$	−29727.1			−0.130652
$478$			120820.i			0.528790i
$479$	−56488.8			−0.246202			−0.123101	−	0.992394i	$-0.539284\pi$
$479$	−56488.8			−0.246202			−0.123101	+	0.992394i	$0.539284\pi$
$480$			43403.0i			0.188381i
$481$	217408.			0.939691
$482$		−	292573.i		−	1.25933i
$483$		−	353438.i		−	1.51502i
$484$	50772.0			0.216737
$485$		−	118100.i		−	0.502072i
$486$		−	13046.7i		−	0.0552366i
$487$	−374783.			−1.58024			−0.790118	−	0.612954i	$-0.789981\pi$
$487$	−374783.			−1.58024			−0.790118	+	0.612954i	$0.789981\pi$
$488$	−279538.			−1.17382
$489$	66540.1			0.278270
$490$			345532.i			1.43912i
$491$	−61744.1			−0.256114			−0.128057	−	0.991767i	$-0.540874\pi$
$491$	−61744.1			−0.256114			−0.128057	+	0.991767i	$0.540874\pi$
$492$	−49472.9			−0.204379
$493$	171868.			0.707133
$494$	199590.			0.817870
$495$		−	21318.4i		−	0.0870048i
$496$		−	98121.2i		−	0.398841i
$497$	−235085.			−0.951726
$498$	188580.			0.760389
$499$	−212124.			−0.851901			−0.425950	−	0.904747i	$-0.640060\pi$
$499$	−212124.			−0.851901			−0.425950	+	0.904747i	$0.640060\pi$
$500$	66231.0			0.264924
$501$	−155657.			−0.620147
$502$			214187.i			0.849934i
$503$		−	268106.i		−	1.05967i	−0.848101	−	0.529834i	$-0.822254\pi$
$503$		−	268106.i		−	1.05967i	0.848101	−	0.529834i	$-0.177746\pi$
$504$		−	173520.i		−	0.683107i
$505$		−	190525.i		−	0.747085i
$506$		−	123070.i		−	0.480676i
$507$	−95692.4			−0.372273
$508$	−58189.4			−0.225484
$509$			241532.i			0.932265i	0.884715	+	0.466133i	$0.154353\pi$
$509$			241532.i			0.932265i	−0.884715	+	0.466133i	$0.845647\pi$
$510$			49924.5i			0.191944i
$511$			73417.5i			0.281163i
$512$			280556.i			1.07024i
$513$	37510.3			0.142533
$514$			39452.1i			0.149329i
$515$		−	308667.i		−	1.16379i
$516$		−	46375.9i		−	0.174178i
$517$	−58370.3			−0.218379
$518$	−320122.			−1.19304
$519$		−	267581.i		−	0.993394i
$520$	−243838.			−0.901767
$521$	81137.0			0.298912			0.149456	−	0.988768i	$-0.452248\pi$
$521$	81137.0			0.298912			0.149456	+	0.988768i	$0.452248\pi$
$522$		−	92933.6i		−	0.341061i
$523$	507362.			1.85488			0.927438	−	0.373977i	$-0.122006\pi$
$523$	507362.			1.85488			0.927438	+	0.373977i	$0.122006\pi$
$524$			65509.1i			0.238583i
$525$	−174243.			−0.632176
$526$		−	135667.i		−	0.490345i
$527$			97720.8i			0.351857i
$528$		−	43677.6i		−	0.156672i
$529$	−259020.			−0.925598
$530$		−	61509.9i		−	0.218974i
$531$	−35773.6	−	86912.6i	−0.126874	−	0.308243i
$532$	102500.			0.362160
$533$		−	498773.i		−	1.75569i
$534$	4053.54			0.0142152
$535$	−109519.			−0.382634
$536$	564735.			1.96569
$537$		−	88633.0i		−	0.307360i
$538$	45153.0			0.155999
$539$			301063.i			1.03629i
$540$	−9415.34			−0.0322886
$541$			303310.i			1.03632i	0.855285	+	0.518158i	$0.173382\pi$
$541$			303310.i			1.03632i	−0.855285	+	0.518158i	$0.826618\pi$
$542$		−	220233.i		−	0.749694i
$543$	39109.2			0.132641
$544$		−	88562.2i		−	0.299261i
$545$			371771.i			1.25165i
$546$	359425.			1.20565
$547$	185234.			0.619077			0.309539	−	0.950887i	$-0.399825\pi$
$547$	185234.			0.619077			0.309539	+	0.950887i	$0.399825\pi$
$548$	−90777.0			−0.302284
$549$		−	108821.i		−	0.361050i
$550$	−60673.2			−0.200573
$551$	267192.			0.880076
$552$	−264554.			−0.868231
$553$	−297853.			−0.973983
$554$			89886.5i			0.292870i
$555$			84543.3i			0.274469i
$556$	−136639.			−0.442004
$557$	300545.			0.968723			0.484361	−	0.874868i	$-0.339052\pi$
$557$	300545.			0.968723			0.484361	+	0.874868i	$0.339052\pi$
$558$	52840.2			0.169706
$559$	467550.			1.49625
$560$	259544.			0.827626
$561$			43499.4i			0.138216i
$562$			116878.i			0.370050i
$563$		−	110761.i		−	0.349437i	−0.984618	−	0.174719i	$-0.944098\pi$
$563$		−	110761.i		−	0.349437i	0.984618	−	0.174719i	$-0.0559016\pi$
$564$			25779.5i			0.0810432i
$565$			18938.6i			0.0593267i
$566$	−172846.			−0.539544
$567$	67549.3			0.210114
$568$			175965.i			0.545417i
$569$			85112.7i			0.262887i	0.991324	+	0.131444i	$0.0419612\pi$
$569$			85112.7i			0.262887i	−0.991324	+	0.131444i	$0.958039\pi$
$570$			77614.4i			0.238887i
$571$		−	419082.i		−	1.28537i	−0.766132	−	0.642683i	$-0.777821\pi$
$571$		−	419082.i		−	1.28537i	0.766132	−	0.642683i	$-0.222179\pi$
$572$	−43650.9			−0.133414
$573$			178576.i			0.543894i
$574$			734418.i			2.22905i
$575$			265656.i			0.803497i
$576$	−122487.			−0.369187
$577$	−210330.			−0.631757			−0.315878	−	0.948800i	$-0.602299\pi$
$577$	−210330.			−0.631757			−0.315878	+	0.948800i	$0.602299\pi$
$578$			185795.i			0.556133i
$579$	−184940.			−0.551662
$580$	−67067.0			−0.199367
$581$			976372.i			2.89243i
$582$	130304.			0.384690
$583$		−	53593.7i		−	0.157680i
$584$	54954.1			0.161129
$585$		−	94923.1i		−	0.277370i
$586$			501962.i			1.46176i
$587$		−	645237.i		−	1.87259i	−0.351213	−	0.936296i	$-0.614231\pi$
$587$		−	645237.i		−	1.87259i	0.351213	−	0.936296i	$-0.385769\pi$
$588$	132966.			0.384579
$589$			151920.i			0.437910i
$590$	179835.	−	74021.0i	0.516619	−	0.212643i
$591$	−130565.			−0.373810
$592$			173215.i			0.494244i
$593$	−386221.			−1.09831			−0.549157	−	0.835719i	$-0.685051\pi$
$593$	−386221.			−1.09831			−0.549157	+	0.835719i	$0.685051\pi$
$594$	23521.3			0.0666635
$595$	−258485.			−0.730131
$596$			86146.4i			0.242518i
$597$	103867.			0.291426
$598$		−	547989.i		−	1.53239i
$599$	−388204.			−1.08195			−0.540974	−	0.841039i	$-0.681944\pi$
$599$	−388204.			−1.08195			−0.540974	+	0.841039i	$0.681944\pi$
$600$			130424.i			0.362288i
$601$			510006.i			1.41197i	0.708225	+	0.705987i	$0.249496\pi$
$601$			510006.i			1.41197i	−0.708225	+	0.705987i	$0.750504\pi$
$602$	−688444.			−1.89966
$603$			219844.i			0.604617i
$604$		−	39944.8i		−	0.109493i
$605$	−199051.			−0.543818
$606$	210213.			0.572420
$607$	−560179.			−1.52037			−0.760185	−	0.649707i	$-0.774892\pi$
$607$	−560179.			−1.52037			−0.760185	+	0.649707i	$0.774892\pi$
$608$		−	137682.i		−	0.372452i
$609$	481164.			1.29736
$610$	225166.			0.605123
$611$	−259902.			−0.696190
$612$	19211.7			0.0512935
$613$		−	306896.i		−	0.816715i	−0.912822	−	0.408358i	$-0.866102\pi$
$613$		−	306896.i		−	0.816715i	0.912822	−	0.408358i	$-0.133898\pi$
$614$			129730.i			0.344116i
$615$	193958.			0.512811
$616$	312832.			0.824422
$617$	465889.			1.22380			0.611902	−	0.790933i	$-0.290405\pi$
$617$	465889.			1.22380			0.611902	+	0.790933i	$0.290405\pi$
$618$	340562.			0.891702
$619$	608265.			1.58749			0.793746	−	0.608250i	$-0.208128\pi$
$619$	608265.			1.58749			0.793746	+	0.608250i	$0.208128\pi$
$620$		−	38133.0i		−	0.0992014i
$621$		−	102987.i		−	0.267055i
$622$			502039.i			1.29765i
$623$			20987.2i			0.0540728i
$624$		−	194481.i		−	0.499468i
$625$	−33473.7			−0.0856927
$626$	−99509.8			−0.253932
$627$			67625.7i			0.172019i
$628$			97161.7i			0.246363i
$629$		−	172508.i		−	0.436021i
$630$			139769.i			0.352153i
$631$	−51223.1			−0.128649			−0.0643245	−	0.997929i	$-0.520489\pi$
$631$	−51223.1			−0.128649			−0.0643245	+	0.997929i	$0.520489\pi$
$632$			222947.i			0.558171i
$633$		−	167548.i		−	0.418150i
$634$		−	513598.i		−	1.27775i
$635$	228131.			0.565766
$636$	−23669.9			−0.0585170
$637$			1.34053e6i			3.30367i
$638$	167546.			0.411616
$639$	−68500.8			−0.167762
$640$		−	119798.i		−	0.292476i
$641$	−194528.			−0.473440			−0.236720	−	0.971578i	$-0.576072\pi$
$641$	−194528.			−0.473440			−0.236720	+	0.971578i	$0.576072\pi$
$642$		−	120836.i		−	0.293176i
$643$	−127058.			−0.307312			−0.153656	−	0.988124i	$-0.549105\pi$
$643$	−127058.			−0.307312			−0.153656	+	0.988124i	$0.549105\pi$
$644$		−	281421.i		−	0.678555i
$645$			181816.i			0.437032i
$646$		−	158370.i		−	0.379495i
$647$	298850.			0.713912			0.356956	−	0.934121i	$-0.383815\pi$
$647$	298850.			0.713912			0.356956	+	0.934121i	$0.383815\pi$
$648$		−	50561.6i		−	0.120412i
$649$	156691.	−	64494.7i	0.372010	−	0.153121i
$650$	−270156.			−0.639423
$651$			273581.i			0.645540i
$652$	52981.8			0.124633
$653$	−808235.			−1.89545			−0.947723	−	0.319094i	$-0.896621\pi$
$653$	−808235.			−1.89545			−0.947723	+	0.319094i	$0.896621\pi$
$654$	−410188.			−0.959019
$655$		−	256828.i		−	0.598631i
$656$	397386.			0.923431
$657$			21392.9i			0.0495609i
$658$	382693.			0.883891
$659$		−	205299.i		−	0.472732i	−0.971664	−	0.236366i	$-0.924044\pi$
$659$		−	205299.i		−	0.472732i	0.971664	−	0.236366i	$-0.0759565\pi$
$660$		−	16974.5i		−	0.0389681i
$661$	−218321.			−0.499680			−0.249840	−	0.968287i	$-0.580378\pi$
$661$	−218321.			−0.499680			−0.249840	+	0.968287i	$0.580378\pi$
$662$		−	69445.1i		−	0.158462i
$663$			193687.i			0.440630i
$664$	730829.			1.65760
$665$	−401849.			−0.908699
$666$	−93279.5			−0.210299
$667$		−	733596.i		−	1.64894i
$668$	−123940.			−0.277754
$669$	−149683.			−0.334441
$670$	−454891.			−1.01335
$671$	196188.			0.435740
$672$		−	247940.i		−	0.549046i
$673$		−	308870.i		−	0.681939i	−0.940074	−	0.340970i	$-0.889245\pi$
$673$		−	308870.i		−	0.681939i	0.940074	−	0.340970i	$-0.110755\pi$
$674$	−195096.			−0.429466
$675$	−50772.4			−0.111435
$676$	−76194.0			−0.166735
$677$	−765077.			−1.66927			−0.834637	−	0.550800i	$-0.814323\pi$
$677$	−765077.			−1.66927			−0.834637	+	0.550800i	$0.814323\pi$
$678$	−20895.6			−0.0454564
$679$			674648.i			1.46331i
$680$			193479.i			0.418424i
$681$		−	298091.i		−	0.642768i
$682$			95263.3i			0.204813i
$683$		−	32889.7i		−	0.0705047i	−0.999378	−	0.0352523i	$-0.988777\pi$
$683$		−	32889.7i		−	0.0705047i	0.999378	−	0.0352523i	$-0.0112235\pi$
$684$	29867.2			0.0638384
$685$	355890.			0.758463
$686$		−	1.20760e6i		−	2.56611i
$687$			41769.3i			0.0885000i
$688$			372510.i			0.786975i
$689$		−	238634.i		−	0.502682i
$690$	213096.			0.447587
$691$		−	498934.i		−	1.04493i	−0.852661	−	0.522465i	$-0.825013\pi$
$691$		−	498934.i		−	1.04493i	0.852661	−	0.522465i	$-0.174987\pi$
$692$		−	213059.i		−	0.444925i
$693$			121782.i			0.253580i
$694$	−339146.			−0.704154
$695$	535693.			1.10904
$696$		−	360159.i		−	0.743490i
$697$	−395764.			−0.814650
$698$	38047.5			0.0780935
$699$			290059.i			0.593653i
$700$	−138739.			−0.283142
$701$		−	70692.6i		−	0.143859i	−0.997410	−	0.0719297i	$-0.977084\pi$
$701$		−	70692.6i		−	0.143859i	0.997410	−	0.0719297i	$-0.0229157\pi$
$702$	104732.			0.212522
$703$		−	268187.i		−	0.542658i
$704$		−	220827.i		−	0.445561i
$705$		−	101068.i		−	0.203346i
$706$	282568.			0.566909
$707$			1.08838e6i			2.17742i
$708$	−28484.4	−	69203.2i	−0.0568251	−	0.138057i
$709$	−845715.			−1.68241			−0.841204	−	0.540717i	$-0.818153\pi$
$709$	−845715.			−1.68241			−0.841204	+	0.540717i	$0.818153\pi$
$710$		−	141738.i		−	0.281171i
$711$	−86790.5			−0.171685
$712$	15709.3			0.0309881
$713$	417108.			0.820484
$714$		−	285195.i		−	0.559429i
$715$	171133.			0.334750
$716$		−	70573.0i		−	0.137662i
$717$	−182277.			−0.354562
$718$			166832.i			0.323616i
$719$		−	541985.i		−	1.04841i	−0.851593	−	0.524203i	$-0.824363\pi$
$719$		−	541985.i		−	1.04841i	0.851593	−	0.524203i	$-0.175637\pi$
$720$	75627.7			0.145887
$721$			1.76327e6i			3.39193i
$722$			202647.i			0.388745i
$723$	441393.			0.844401
$724$	31140.2			0.0594080
$725$	−361660.			−0.688057
$726$		−	219620.i		−	0.416676i
$727$	324956.			0.614832			0.307416	−	0.951575i	$-0.400536\pi$
$727$	324956.			0.614832			0.307416	+	0.951575i	$0.400536\pi$
$728$	1.39293e6			2.62825
$729$	19683.0			0.0370370
$730$	−44265.2			−0.0830647
$731$		−	370990.i		−	0.694268i
$732$		−	86647.3i		−	0.161708i
$733$	−228228.			−0.424777			−0.212389	−	0.977185i	$-0.568124\pi$
$733$	−228228.			−0.424777			−0.212389	+	0.977185i	$0.568124\pi$
$734$	−712760.			−1.32297
$735$	−521291.			−0.964951
$736$	−378016.			−0.697839
$737$	−396348.			−0.729695
$738$			214000.i			0.392918i
$739$			153824.i			0.281667i	0.990033	+	0.140834i	$0.0449782\pi$
$739$			153824.i			0.281667i	−0.990033	+	0.140834i	$0.955022\pi$
$740$			67316.7i			0.122930i
$741$			301113.i			0.548395i
$742$			351376.i			0.638212i
$743$	40762.4			0.0738384			0.0369192	−	0.999318i	$-0.488246\pi$
$743$	40762.4			0.0738384			0.0369192	+	0.999318i	$0.488246\pi$
$744$	204779.			0.369947
$745$		−	337736.i		−	0.608506i
$746$		−	331993.i		−	0.596556i
$747$			284503.i			0.509853i
$748$			34635.9i			0.0619047i
$749$	625632.			1.11521
$750$		−	286489.i		−	0.509314i
$751$		−	947341.i		−	1.67968i	−0.542834	−	0.839840i	$-0.682649\pi$
$751$		−	947341.i		−	1.67968i	0.542834	−	0.839840i	$-0.317351\pi$
$752$		−	207071.i		−	0.366171i
$753$	−323135.			−0.569894
$754$	746022.			1.31223
$755$			156603.i			0.274730i
$756$	53785.3			0.0941067
$757$	−384807.			−0.671508			−0.335754	−	0.941950i	$-0.608991\pi$
$757$	−384807.			−0.671508			−0.335754	+	0.941950i	$0.608991\pi$
$758$		−	765397.i		−	1.33213i
$759$	185671.			0.322301
$760$			300790.i			0.520758i
$761$	621369.			1.07295			0.536476	−	0.843916i	$-0.319755\pi$
$761$	621369.			1.07295			0.536476	+	0.843916i	$0.319755\pi$
$762$			251704.i			0.433492i
$763$		−	2.12375e6i		−	3.64800i
$764$			142189.i			0.243602i
$765$	−75319.1			−0.128701
$766$		−	865396.i		−	1.47488i
$767$	697689.	−	287172.i	1.18596	−	0.488148i
$768$	−244986.			−0.415354
$769$			140425.i			0.237460i	0.992927	+	0.118730i	$0.0378823\pi$
$769$			140425.i			0.237460i	−0.992927	+	0.118730i	$0.962118\pi$
$770$	−251984.			−0.425003
$771$	−59519.9			−0.100127
$772$	−147256.			−0.247081
$773$			233110.i			0.390124i	0.980791	+	0.195062i	$0.0624908\pi$
$773$			233110.i			0.390124i	−0.980791	+	0.195062i	$0.937509\pi$
$774$	−200604.			−0.334856
$775$		−	205633.i		−	0.342365i
$776$	504984.			0.838598
$777$		−	482956.i		−	0.799954i
$778$		−	852332.i		−	1.40815i
$779$	−615269.			−1.01389
$780$		−	75581.4i		−	0.124230i
$781$		−	123497.i		−	0.202467i
$782$	−434816.			−0.711036
$783$	140205.			0.228687
$784$	−1.06803e6			−1.73761
$785$		−	380921.i		−	0.618153i
$786$	283367.			0.458674
$787$	95032.6			0.153435			0.0767173	−	0.997053i	$-0.475556\pi$
$787$	95032.6			0.153435			0.0767173	+	0.997053i	$0.475556\pi$
$788$	−103961.			−0.167424
$789$	204675.			0.328784
$790$		−	179583.i		−	0.287746i
$791$		−	108187.i		−	0.172911i
$792$	91155.3			0.145322
$793$	873555.			1.38913
$794$	−52312.6			−0.0829784
$795$	92797.5			0.146826
$796$	82702.9			0.130525
$797$			1.06273e6i			1.67304i	0.547936	+	0.836520i	$0.315414\pi$
$797$			1.06273e6i			1.67304i	−0.547936	+	0.836520i	$0.684586\pi$
$798$		−	443374.i		−	0.696249i
$799$			206226.i			0.323035i
$800$			186361.i			0.291188i
$801$			6115.42i			0.00953149i
$802$	141566.			0.220094
$803$	−38568.4			−0.0598137
$804$			175049.i			0.270799i
$805$			1.10331e6i			1.70257i
$806$			424174.i			0.652940i
$807$			68120.6i			0.104600i
$808$	814669.			1.24784
$809$			961800.i			1.46956i	0.678305	+	0.734780i	$0.262715\pi$
$809$			961800.i			1.46956i	−0.678305	+	0.734780i	$0.737285\pi$
$810$			40727.1i			0.0620745i
$811$		−	957274.i		−	1.45544i	−0.685874	−	0.727721i	$-0.740580\pi$
$811$		−	957274.i		−	1.45544i	0.685874	−	0.727721i	$-0.259420\pi$
$812$	383122.			0.581065
$813$	332257.			0.502682
$814$		−	168170.i		−	0.253804i
$815$	−207714.			−0.312717
$816$	−154316.			−0.231756
$817$		−	576754.i		−	0.864065i
$818$	−410177.			−0.613006
$819$			542250.i			0.808410i
$820$	154437.			0.229680
$821$		−	698787.i		−	1.03671i	−0.855165	−	0.518357i	$-0.826544\pi$
$821$		−	698787.i		−	1.03671i	0.855165	−	0.518357i	$-0.173456\pi$
$822$			392666.i			0.581138i
$823$		−	551034.i		−	0.813540i	−0.913531	−	0.406770i	$-0.866655\pi$
$823$		−	551034.i		−	0.813540i	0.913531	−	0.406770i	$-0.133345\pi$
$824$	1.31983e6			1.94385
$825$		−	91535.3i		−	0.134487i
$826$	−1.02731e6	+	422846.i	−1.50571	+	0.619759i
$827$	833437.			1.21860			0.609301	−	0.792939i	$-0.291450\pi$
$827$	833437.			1.21860			0.609301	+	0.792939i	$0.291450\pi$
$828$		−	82002.6i		−	0.119610i
$829$	882949.			1.28477			0.642387	−	0.766381i	$-0.277944\pi$
$829$	882949.			1.28477			0.642387	+	0.766381i	$0.277944\pi$
$830$	−588678.			−0.854519
$831$	−135608.			−0.196374
$832$		−	983265.i		−	1.42044i
$833$	1.06368e6			1.53292
$834$			591048.i			0.849749i
$835$	485907.			0.696916
$836$			53846.2i			0.0770447i
$837$			79718.0i			0.113790i
$838$	787659.			1.12163
$839$			959966.i			1.36374i	0.731473	+	0.681871i	$0.238833\pi$
$839$			959966.i			1.36374i	−0.731473	+	0.681871i	$0.761167\pi$
$840$			541668.i			0.767670i
$841$	291424.			0.412034
$842$	−973612.			−1.37329
$843$	−176330.			−0.248125
$844$		−	133408.i		−	0.187283i
$845$	298718.			0.418357
$846$	111512.			0.155805
$847$	1.13708e6			1.58499
$848$	190126.			0.264393
$849$		−	260766.i		−	0.361773i
$850$			214362.i			0.296695i
$851$	−736327.			−1.01674
$852$	−54543.0			−0.0751380
$853$	8821.74			0.0121243			0.00606214	−	0.999982i	$-0.498070\pi$
$853$	8821.74			0.0121243			0.00606214	+	0.999982i	$0.498070\pi$
$854$	−1.28627e6			−1.76366
$855$	−117094.			−0.160178
$856$		−	468294.i		−	0.639104i
$857$			74384.0i			0.101279i	0.998717	+	0.0506394i	$0.0161259\pi$
$857$			74384.0i			0.101279i	−0.998717	+	0.0506394i	$0.983874\pi$
$858$			188816.i			0.256487i
$859$			106512.i			0.144349i	0.997392	+	0.0721746i	$0.0229939\pi$
$859$			106512.i			0.144349i	−0.997392	+	0.0721746i	$0.977006\pi$
$860$			144769.i			0.195740i
$861$	−1.10799e6			−1.49461
$862$	931063.			1.25304
$863$			662734.i			0.889852i	0.895567	+	0.444926i	$0.146770\pi$
$863$			662734.i			0.889852i	−0.895567	+	0.444926i	$0.853230\pi$
$864$		−	72246.7i		−	0.0967811i
$865$			835294.i			1.11637i
$866$			693714.i			0.925006i
$867$	−280302.			−0.372896
$868$			217836.i			0.289128i
$869$		−	156471.i		−	0.207202i
$870$			290106.i			0.383281i
$871$	−1.76479e6			−2.32626
$872$	−1.58966e6			−2.09060
$873$			196584.i			0.257940i
$874$	−675980.			−0.884934
$875$	1.48330e6			1.93737
$876$			17033.9i			0.0221976i
$877$	1.48294e6			1.92808			0.964040	−	0.265757i	$-0.0856217\pi$
$877$	1.48294e6			1.92808			0.964040	+	0.265757i	$0.0856217\pi$
$878$			37462.5i			0.0485968i
$879$	−757290.			−0.980132
$880$			136346.i			0.176067i
$881$		−	734191.i		−	0.945926i	−0.881082	−	0.472963i	$-0.843184\pi$
$881$		−	734191.i		−	0.945926i	0.881082	−	0.472963i	$-0.156816\pi$
$882$		−	575158.i		−	0.739350i
$883$	824634.			1.05765			0.528823	−	0.848732i	$-0.322634\pi$
$883$	824634.			1.05765			0.528823	+	0.848732i	$0.322634\pi$
$884$			154221.i			0.197351i
$885$	111673.	+	271310.i	0.142580	+	0.346401i
$886$	−97412.0			−0.124092
$887$			786206.i			0.999284i	0.866232	+	0.499642i	$0.166535\pi$
$887$			786206.i			0.999284i	−0.866232	+	0.499642i	$0.833465\pi$
$888$	−361499.			−0.458439
$889$	−1.30320e6			−1.64895
$890$	−12653.7			−0.0159749
$891$			35485.6i			0.0446989i
$892$	−119183.			−0.149791
$893$			320606.i			0.402040i
$894$	372635.			0.466240
$895$			276681.i			0.345408i
$896$			684351.i			0.852438i
$897$	826729.			1.02749
$898$		−	545293.i		−	0.676203i
$899$			567844.i			0.702603i
$900$	−40426.9			−0.0499098
$901$	−189350.			−0.233247
$902$	−385812.			−0.474201
$903$		−	1.03863e6i		−	1.27375i
$904$	−80979.6			−0.0990920
$905$	−122085.			−0.149061
$906$	−172786.			−0.210499
$907$	−1.44869e6			−1.76101			−0.880504	−	0.474038i	$-0.842796\pi$
$907$	−1.44869e6			−1.76101			−0.880504	+	0.474038i	$0.842796\pi$
$908$		−	237352.i		−	0.287886i
$909$			317140.i			0.383817i
$910$	−1.12200e6			−1.35490
$911$	−812697.			−0.979246			−0.489623	−	0.871934i	$-0.662866\pi$
$911$	−812697.			−0.979246			−0.489623	+	0.871934i	$0.662866\pi$
$912$	−239905.			−0.288436
$913$	−512917.			−0.615327
$914$	123418.			0.147735
$915$			339699.i			0.405745i
$916$			33258.3i			0.0396378i
$917$			1.46713e6i			1.74474i
$918$		−	83102.2i		−	0.0986114i
$919$			851885.i			1.00867i	0.863507	+	0.504336i	$0.168263\pi$
$919$			851885.i			1.00867i	−0.863507	+	0.504336i	$0.831737\pi$
$920$	825842.			0.975711
$921$	−195719.			−0.230735
$922$		−	1.05934e6i		−	1.24616i
$923$		−	549888.i		−	0.645463i
$924$			96967.2i			0.113575i
$925$			363006.i			0.424258i
$926$	938132.			1.09406
$927$			513793.i			0.597900i
$928$		−	514625.i		−	0.597578i
$929$		−	227195.i		−	0.263250i	−0.991300	−	0.131625i	$-0.957981\pi$
$929$		−	227195.i		−	0.263250i	0.991300	−	0.131625i	$-0.0420194\pi$
$930$	−164948.			−0.190714
$931$	1.65363e6			1.90782
$932$			230957.i			0.265888i
$933$	−757406.			−0.870092
$934$	748976.			0.858567
$935$		−	135790.i		−	0.155326i
$936$	405882.			0.463285
$937$		−	961855.i		−	1.09554i	−0.836627	−	0.547772i	$-0.815476\pi$
$937$		−	961855.i		−	1.09554i	0.836627	−	0.547772i	$-0.184524\pi$
$938$	2.59857e6			2.95345
$939$		−	150127.i		−	0.170265i
$940$		−	80474.4i		−	0.0910756i
$941$			192956.i			0.217911i	0.994047	+	0.108955i	$0.0347506\pi$
$941$			192956.i			0.217911i	−0.994047	+	0.108955i	$0.965249\pi$
$942$	420283.			0.473631
$943$			1.68927e6i			1.89966i
$944$	228798.	+	555867.i	0.256748	+	0.623774i
$945$	−210865.			−0.236124
$946$		−	361660.i		−	0.404128i
$947$	−846036.			−0.943385			−0.471693	−	0.881763i	$-0.656357\pi$
$947$	−846036.			−0.943385			−0.471693	+	0.881763i	$0.656357\pi$
$948$	−69106.0			−0.0768952
$949$	−171731.			−0.190685
$950$			333255.i			0.369258i
$951$	774845.			0.856749
$952$		−	1.10526e6i		−	1.21952i
$953$	596807.			0.657125			0.328563	−	0.944482i	$-0.393436\pi$
$953$	596807.			0.657125			0.328563	+	0.944482i	$0.393436\pi$
$954$			102387.i			0.112498i
$955$		−	557452.i		−	0.611224i
$956$	−145136.			−0.158803
$957$			252770.i			0.275995i
$958$			194560.i			0.211993i
$959$	−2.03303e6			−2.21058
$960$	382362.			0.414890
$961$	600656.			0.650398
$962$		−	748799.i		−	0.809124i
$963$	182301.			0.196579
$964$	351454.			0.378194
$965$	577316.			0.619953
$966$	−1.21732e6			−1.30452
$967$			380274.i			0.406672i	0.979109	+	0.203336i	$0.0651784\pi$
$967$			380274.i			0.406672i	−0.979109	+	0.203336i	$0.934822\pi$
$968$		−	851123.i		−	0.908326i
$969$	238926.			0.254458
$970$	−406761.			−0.432311
$971$	1.45439e6			1.54256			0.771281	−	0.636495i	$-0.219616\pi$
$971$	1.45439e6			1.54256			0.771281	+	0.636495i	$0.219616\pi$
$972$	15672.4			0.0165883
$973$	−3.06016e6			−3.23235
$974$			1.29083e6i			1.36067i
$975$		−	407574.i		−	0.428743i
$976$			695985.i			0.730635i
$977$		−	1.71629e6i		−	1.79805i	−0.437893	−	0.899027i	$-0.644275\pi$
$977$		−	1.71629e6i		−	1.79805i	0.437893	−	0.899027i	$-0.355725\pi$
$978$		−	229178.i		−	0.239605i
$979$	−11025.2			−0.0115033
$980$	−415072.			−0.432187
$981$		−	618834.i		−	0.643038i
$982$			212660.i			0.220528i
$983$		−	1.55447e6i		−	1.60870i	−0.594155	−	0.804350i	$-0.702513\pi$
$983$		−	1.55447e6i		−	1.60870i	0.594155	−	0.804350i	$-0.297487\pi$
$984$			829345.i			0.856535i
$985$	407576.			0.420084
$986$		−	591950.i		−	0.608880i
$987$			577354.i			0.592663i
$988$			239758.i			0.245617i
$989$	−1.58352e6			−1.61894
$990$	−73425.0			−0.0749159
$991$			1.55870e6i			1.58714i	0.608482	+	0.793568i	$0.291779\pi$
$991$			1.55870e6i			1.58714i	−0.608482	+	0.793568i	$0.708221\pi$
$992$	292606.			0.297344
$993$	104769.			0.106251
$994$			809683.i			0.819488i
$995$	−324235.			−0.327502
$996$			226532.i			0.228355i
$997$	−55668.1			−0.0560037			−0.0280018	−	0.999608i	$-0.508914\pi$
$997$	−55668.1			−0.0560037			−0.0280018	+	0.999608i	$0.508914\pi$
$998$			730601.i			0.733532i
$999$		−	140727.i		−	0.141009i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.13	✓	40
3.2	odd	2		531.5.c.d.235.28		40
59.58	odd	2	inner	177.5.c.a.58.28	yes	40
177.176	even	2		531.5.c.d.235.13		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.13	✓	40	1.1	even	1	trivial
177.5.c.a.58.28	yes	40	59.58	odd	2	inner
531.5.c.d.235.13		40	177.176	even	2
531.5.c.d.235.28		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)

\(f(q)\)	\(=\)	\(q-3.44422i q^{2} +5.19615 q^{3} +4.13738 q^{4} -16.2205 q^{5} -17.8967i q^{6} +92.6602 q^{7} -69.3575i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-3.44422i q^{2} +5.19615 q^{3} +4.13738 q^{4} -16.2205 q^{5} -17.8967i q^{6} +92.6602 q^{7} -69.3575i q^{8} +27.0000 q^{9} +55.8670i q^{10} +48.6771i q^{11} +21.4985 q^{12} +216.742i q^{13} -319.142i q^{14} -84.2844 q^{15} -172.684 q^{16} +171.979 q^{17} -92.9938i q^{18} +267.365 q^{19} -67.1105 q^{20} +481.476 q^{21} +167.654 q^{22} -734.072i q^{23} -360.392i q^{24} -361.894 q^{25} +746.506 q^{26} +140.296 q^{27} +383.370 q^{28} +999.352 q^{29} +290.294i q^{30} +568.212i q^{31} -514.959i q^{32} +252.934i q^{33} -592.334i q^{34} -1503.00 q^{35} +111.709 q^{36} -1003.07i q^{37} -920.864i q^{38} +1126.22i q^{39} +1125.02i q^{40} -2301.23 q^{41} -1658.31i q^{42} -2157.18i q^{43} +201.396i q^{44} -437.954 q^{45} -2528.30 q^{46} +1199.13i q^{47} -897.293 q^{48} +6184.90 q^{49} +1246.44i q^{50} +893.631 q^{51} +896.743i q^{52} -1101.00 q^{53} -483.210i q^{54} -789.569i q^{55} -6426.67i q^{56} +1389.27 q^{57} -3441.98i q^{58} +(-1324.95 - 3218.99i) q^{59} -348.716 q^{60} -4030.40i q^{61} +1957.05 q^{62} +2501.82 q^{63} -4536.57 q^{64} -3515.67i q^{65} +871.158 q^{66} +8142.38i q^{67} +711.544 q^{68} -3814.35i q^{69} +5176.65i q^{70} -2537.07 q^{71} -1872.65i q^{72} +792.331i q^{73} -3454.80 q^{74} -1880.46 q^{75} +1106.19 q^{76} +4510.43i q^{77} +3878.96 q^{78} -3214.46 q^{79} +2801.03 q^{80} +729.000 q^{81} +7925.93i q^{82} +10537.1i q^{83} +1992.05 q^{84} -2789.60 q^{85} -7429.78 q^{86} +5192.79 q^{87} +3376.12 q^{88} +226.497i q^{89} +1508.41i q^{90} +20083.3i q^{91} -3037.13i q^{92} +2952.52i q^{93} +4130.07 q^{94} -4336.81 q^{95} -2675.80i q^{96} +7280.88i q^{97} -21302.1i q^{98} +1314.28i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 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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