LMFDB - Embedded newform 177.9.c.a.58.9

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	177.c (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$72.1060139808$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.9
Character	$\chi$	$=$	177.58
Dual form			177.9.c.a.58.72

$q$-expansion

$f(q)$	$=$	$q-25.6810i q^{2} +46.7654 q^{3} -403.515 q^{4} +951.178 q^{5} -1200.98i q^{6} +4182.78 q^{7} +3788.34i q^{8} +2187.00 q^{9} +O(q^{10})$	\(q-25.6810i q^{2} +46.7654 q^{3} -403.515 q^{4} +951.178 q^{5} -1200.98i q^{6} +4182.78 q^{7} +3788.34i q^{8} +2187.00 q^{9} -24427.2i q^{10} +17992.5i q^{11} -18870.5 q^{12} +12240.6i q^{13} -107418. i q^{14} +44482.2 q^{15} -6011.37 q^{16} +21398.1 q^{17} -56164.4i q^{18} -176067. q^{19} -383815. q^{20} +195609. q^{21} +462066. q^{22} -346513. i q^{23} +177163. i q^{24} +514114. q^{25} +314351. q^{26} +102276. q^{27} -1.68781e6 q^{28} +1.08118e6 q^{29} -1.14235e6i q^{30} +659909. i q^{31} +1.12419e6i q^{32} +841425. i q^{33} -549526. i q^{34} +3.97856e6 q^{35} -882488. q^{36} +809547. i q^{37} +4.52157e6i q^{38} +572435. i q^{39} +3.60339e6i q^{40} +2.01892e6 q^{41} -5.02344e6i q^{42} -2.82734e6i q^{43} -7.26024e6i q^{44} +2.08023e6 q^{45} -8.89881e6 q^{46} -3.68905e6i q^{47} -281124. q^{48} +1.17308e7 q^{49} -1.32030e7i q^{50} +1.00069e6 q^{51} -4.93926e6i q^{52} +1.08310e7 q^{53} -2.62655e6i q^{54} +1.71140e7i q^{55} +1.58458e7i q^{56} -8.23382e6 q^{57} -2.77658e7i q^{58} +(3.87060e6 - 1.14825e7i) q^{59} -1.79492e7 q^{60} +1.10547e7i q^{61} +1.69471e7 q^{62} +9.14773e6 q^{63} +2.73315e7 q^{64} +1.16430e7i q^{65} +2.16087e7 q^{66} -3.56627e7i q^{67} -8.63447e6 q^{68} -1.62048e7i q^{69} -1.02174e8i q^{70} -4.05386e7 q^{71} +8.28510e6i q^{72} -1.60542e7i q^{73} +2.07900e7 q^{74} +2.40427e7 q^{75} +7.10456e7 q^{76} +7.52586e7i q^{77} +1.47007e7 q^{78} +1.42083e7 q^{79} -5.71788e6 q^{80} +4.78297e6 q^{81} -5.18479e7i q^{82} +9.26581e7i q^{83} -7.89313e7 q^{84} +2.03534e7 q^{85} -7.26089e7 q^{86} +5.05618e7 q^{87} -6.81617e7 q^{88} -2.50440e7i q^{89} -5.34223e7i q^{90} +5.11996e7i q^{91} +1.39823e8i q^{92} +3.08609e7i q^{93} -9.47386e7 q^{94} -1.67471e8 q^{95} +5.25733e7i q^{96} +8.23795e7i q^{97} -3.01260e8i q^{98} +3.93496e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) $	$ 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) $

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 $

$2$		−	25.6810i		−	1.60506i	−0.596609	−	0.802532i	$-0.703486\pi$
$2$		−	25.6810i		−	1.60506i	0.596609	−	0.802532i	$-0.296514\pi$
$3$	46.7654			0.577350
$3$	46.7654			0.577350
$4$	−403.515			−1.57623
$5$	951.178			1.52188			0.760942	−	0.648820i	$-0.224737\pi$
$5$	951.178			1.52188			0.760942	+	0.648820i	$0.224737\pi$
$6$		−	1200.98i		−	0.926684i
$7$	4182.78			1.74210			0.871049	−	0.491196i	$-0.163440\pi$
$7$	4182.78			1.74210			0.871049	+	0.491196i	$0.163440\pi$
$8$			3788.34i			0.924888i
$9$	2187.00			0.333333
$10$		−	24427.2i		−	2.44272i
$11$			17992.5i			1.22891i	0.788951	+	0.614456i	$0.210624\pi$
$11$			17992.5i			1.22891i	−0.788951	+	0.614456i	$0.789376\pi$
$12$	−18870.5			−0.910038
$13$			12240.6i			0.428577i	0.976770	+	0.214288i	$0.0687432\pi$
$13$			12240.6i			0.428577i	−0.976770	+	0.214288i	$0.931257\pi$
$14$		−	107418.i		−	2.79618i
$15$	44482.2			0.878660
$16$	−6011.37			−0.0917262
$17$	21398.1			0.256201			0.128100	−	0.991761i	$-0.459112\pi$
$17$	21398.1			0.256201			0.128100	+	0.991761i	$0.459112\pi$
$18$		−	56164.4i		−	0.535021i
$19$	−176067.			−1.35102			−0.675511	−	0.737350i	$-0.736077\pi$
$19$	−176067.			−1.35102			−0.675511	+	0.737350i	$0.736077\pi$
$20$	−383815.			−2.39884
$21$	195609.			1.00580
$22$	462066.			1.97248
$23$		−	346513.i		−	1.23825i	−0.785293	−	0.619125i	$-0.787488\pi$
$23$		−	346513.i		−	1.23825i	0.785293	−	0.619125i	$-0.212512\pi$
$24$			177163.i			0.533984i
$25$	514114.			1.31613
$26$	314351.			0.687893
$27$	102276.			0.192450
$28$	−1.68781e6			−2.74595
$29$	1.08118e6			1.52864			0.764322	−	0.644835i	$-0.223074\pi$
$29$	1.08118e6			1.52864			0.764322	+	0.644835i	$0.223074\pi$
$30$		−	1.14235e6i		−	1.41031i
$31$			659909.i			0.714557i	0.933998	+	0.357279i	$0.116295\pi$
$31$			659909.i			0.714557i	−0.933998	+	0.357279i	$0.883705\pi$
$32$			1.12419e6i			1.07211i
$33$			841425.i			0.709512i
$34$		−	549526.i		−	0.411218i
$35$	3.97856e6			2.65127
$36$	−882488.			−0.525410
$37$			809547.i			0.431951i	0.976399	+	0.215976i	$0.0692932\pi$
$37$			809547.i			0.431951i	−0.976399	+	0.215976i	$0.930707\pi$
$38$			4.52157e6i			2.16848i
$39$			572435.i			0.247439i
$40$			3.60339e6i			1.40757i
$41$	2.01892e6			0.714468			0.357234	−	0.934015i	$-0.383720\pi$
$41$	2.01892e6			0.714468			0.357234	+	0.934015i	$0.383720\pi$
$42$		−	5.02344e6i		−	1.61437i
$43$		−	2.82734e6i		−	0.826997i	−0.910505	−	0.413498i	$-0.864307\pi$
$43$		−	2.82734e6i		−	0.826997i	0.910505	−	0.413498i	$-0.135693\pi$
$44$		−	7.26024e6i		−	1.93705i
$45$	2.08023e6			0.507295
$46$	−8.89881e6			−1.98747
$47$		−	3.68905e6i		−	0.756002i	−0.925805	−	0.378001i	$-0.876612\pi$
$47$		−	3.68905e6i		−	0.756002i	0.925805	−	0.378001i	$-0.123388\pi$
$48$	−281124.			−0.0529582
$49$	1.17308e7			2.03490
$50$		−	1.32030e7i		−	2.11248i
$51$	1.00069e6			0.147917
$52$		−	4.93926e6i		−	0.675536i
$53$	1.08310e7			1.37266			0.686332	−	0.727289i	$-0.259220\pi$
$53$	1.08310e7			1.37266			0.686332	+	0.727289i	$0.259220\pi$
$54$		−	2.62655e6i		−	0.308895i
$55$			1.71140e7i			1.87026i
$56$			1.58458e7i			1.61125i
$57$	−8.23382e6			−0.780013
$58$		−	2.77658e7i		−	2.45357i
$59$	3.87060e6	−	1.14825e7i	0.319426	−	0.947611i
$59$	3.87060e6	−	1.14825e7i	0.319426	−	0.947611i
$60$	−1.79492e7			−1.38497
$61$			1.10547e7i			0.798414i	0.916861	+	0.399207i	$0.130715\pi$
$61$			1.10547e7i			0.798414i	−0.916861	+	0.399207i	$0.869285\pi$
$62$	1.69471e7			1.14691
$63$	9.14773e6			0.580699
$64$	2.73315e7			1.62909
$65$			1.16430e7i			0.652244i
$66$	2.16087e7			1.13881
$67$		−	3.56627e7i		−	1.76976i	−0.465817	−	0.884881i	$-0.654240\pi$
$67$		−	3.56627e7i		−	1.76976i	0.465817	−	0.884881i	$-0.345760\pi$
$68$	−8.63447e6			−0.403831
$69$		−	1.62048e7i		−	0.714904i
$70$		−	1.02174e8i		−	4.25546i
$71$	−4.05386e7			−1.59527			−0.797637	−	0.603138i	$-0.793917\pi$
$71$	−4.05386e7			−1.59527			−0.797637	+	0.603138i	$0.793917\pi$
$72$			8.28510e6i			0.308296i
$73$		−	1.60542e7i		−	0.565322i	−0.959220	−	0.282661i	$-0.908783\pi$
$73$		−	1.60542e7i		−	0.565322i	0.959220	−	0.282661i	$-0.0912172\pi$
$74$	2.07900e7			0.693310
$75$	2.40427e7			0.759869
$76$	7.10456e7			2.12952
$77$			7.52586e7i			2.14088i
$78$	1.47007e7			0.397155
$79$	1.42083e7			0.364781			0.182391	−	0.983226i	$-0.441616\pi$
$79$	1.42083e7			0.364781			0.182391	+	0.983226i	$0.441616\pi$
$80$	−5.71788e6			−0.139597
$81$	4.78297e6			0.111111
$82$		−	5.18479e7i		−	1.14677i
$83$			9.26581e7i			1.95241i	0.216852	+	0.976205i	$0.430421\pi$
$83$			9.26581e7i			1.95241i	−0.216852	+	0.976205i	$0.569579\pi$
$84$	−7.89313e7			−1.58537
$85$	2.03534e7			0.389908
$86$	−7.26089e7			−1.32738
$87$	5.05618e7			0.882563
$88$	−6.81617e7			−1.13661
$89$		−	2.50440e7i		−	0.399157i	−0.979882	−	0.199579i	$-0.936043\pi$
$89$		−	2.50440e7i		−	0.399157i	0.979882	−	0.199579i	$-0.0639573\pi$
$90$		−	5.34223e7i		−	0.814241i
$91$			5.11996e7i			0.746623i
$92$			1.39823e8i			1.95177i
$93$			3.08609e7i			0.412550i
$94$	−9.47386e7			−1.21343
$95$	−1.67471e8			−2.05610
$96$			5.25733e7i			0.618986i
$97$			8.23795e7i			0.930534i	0.885170	+	0.465267i	$0.154042\pi$
$97$			8.23795e7i			0.930534i	−0.885170	+	0.465267i	$0.845958\pi$
$98$		−	3.01260e8i		−	3.26615i
$99$			3.93496e7i			0.409637i
$100$	−2.07453e8			−2.07453
$101$			1.45553e8i			1.39873i	0.714763	+	0.699367i	$0.246535\pi$
$101$			1.45553e8i			1.39873i	−0.714763	+	0.699367i	$0.753465\pi$
$102$		−	2.56988e7i		−	0.237417i
$103$		−	1.57948e8i		−	1.40335i	−0.712497	−	0.701675i	$-0.752436\pi$
$103$		−	1.57948e8i		−	1.40335i	0.712497	−	0.701675i	$-0.247564\pi$
$104$	−4.63715e7			−0.396386
$105$	1.86059e8			1.53071
$106$		−	2.78150e8i		−	2.20321i
$107$	3.86465e7			0.294833			0.147416	−	0.989075i	$-0.452904\pi$
$107$	3.86465e7			0.294833			0.147416	+	0.989075i	$0.452904\pi$
$108$	−4.12699e7			−0.303346
$109$			1.37706e8i			0.975543i	0.872971	+	0.487771i	$0.162190\pi$
$109$			1.37706e8i			0.975543i	−0.872971	+	0.487771i	$0.837810\pi$
$110$	4.39506e8			3.00189
$111$			3.78587e7i			0.249387i
$112$	−2.51442e7			−0.159796
$113$		−	1.35512e8i		−	0.831120i	−0.909566	−	0.415560i	$-0.863586\pi$
$113$		−	1.35512e8i		−	0.831120i	0.909566	−	0.415560i	$-0.136414\pi$
$114$			2.11453e8i			1.25197i
$115$		−	3.29595e8i		−	1.88447i
$116$	−4.36273e8			−2.40950
$117$			2.67701e7i			0.142859i
$118$	−2.94884e8	−	9.94011e7i	−1.52098	−	0.512700i
$119$	8.95036e7			0.446326
$120$			1.68514e8i			0.812663i
$121$	−1.09371e8			−0.510222
$122$	2.83897e8			1.28151
$123$	9.44154e7			0.412499
$124$		−	2.66283e8i		−	1.12631i
$125$	1.17460e8			0.481115
$126$		−	2.34923e8i		−	0.932060i
$127$	9.80202e6			0.0376791			0.0188396	−	0.999823i	$-0.494003\pi$
$127$	9.80202e6			0.0376791			0.0188396	+	0.999823i	$0.494003\pi$
$128$		−	4.14108e8i		−	1.54267i
$129$		−	1.32222e8i		−	0.477467i
$130$	2.99003e8			1.04689
$131$		−	7.85281e7i		−	0.266649i	−0.991072	−	0.133324i	$-0.957435\pi$
$131$		−	7.85281e7i		−	0.266649i	0.991072	−	0.133324i	$-0.0425652\pi$
$132$		−	3.39528e8i		−	1.11836i
$133$	−7.36447e8			−2.35361
$134$	−9.15855e8			−2.84058
$135$	9.72825e7			0.292887
$136$			8.10634e7i			0.236957i
$137$	−6.23729e8			−1.77057			−0.885287	−	0.465045i	$-0.846038\pi$
$137$	−6.23729e8			−1.77057			−0.885287	+	0.465045i	$0.846038\pi$
$138$	−4.16156e8			−1.14747
$139$	−1.27501e8			−0.341551			−0.170776	−	0.985310i	$-0.554627\pi$
$139$	−1.27501e8			−0.341551			−0.170776	+	0.985310i	$0.554627\pi$
$140$	−1.60541e9			−4.17902
$141$		−	1.72520e8i		−	0.436478i
$142$			1.04107e9i			2.56052i
$143$	−2.20238e8			−0.526683
$144$	−1.31469e7			−0.0305754
$145$	1.02839e9			2.32642
$146$	−4.12287e8			−0.907378
$147$	5.48596e8			1.17485
$148$		−	3.26664e8i		−	0.680855i
$149$		−	7.13072e8i		−	1.44673i	−0.690465	−	0.723366i	$-0.742594\pi$
$149$		−	7.13072e8i		−	1.44673i	0.690465	−	0.723366i	$-0.257406\pi$
$150$		−	6.17442e8i		−	1.21964i
$151$			3.74058e8i			0.719501i	0.933049	+	0.359751i	$0.117138\pi$
$151$			3.74058e8i			0.719501i	−0.933049	+	0.359751i	$0.882862\pi$
$152$		−	6.67001e8i		−	1.24954i
$153$	4.67977e7			0.0854002
$154$	1.93272e9			3.43625
$155$			6.27690e8i			1.08747i
$156$		−	2.30986e8i		−	0.390021i
$157$		−	7.36816e8i		−	1.21272i	−0.795190	−	0.606360i	$-0.792629\pi$
$157$		−	7.36816e8i		−	1.21272i	0.795190	−	0.606360i	$-0.207371\pi$
$158$		−	3.64883e8i		−	0.585498i
$159$	5.06514e8			0.792507
$160$			1.06931e9i			1.63163i
$161$		−	1.44939e9i		−	2.15715i
$162$		−	1.22832e8i		−	0.178340i
$163$	−4.00064e8			−0.566733			−0.283367	−	0.959012i	$-0.591451\pi$
$163$	−4.00064e8			−0.566733			−0.283367	+	0.959012i	$0.591451\pi$
$164$	−8.14664e8			−1.12617
$165$			8.00345e8i			1.07980i
$166$	2.37955e9			3.13374
$167$	6.04579e8			0.777297			0.388648	−	0.921386i	$-0.372942\pi$
$167$	6.04579e8			0.777297			0.388648	+	0.921386i	$0.372942\pi$
$168$			7.41034e8i			0.930253i
$169$	6.65899e8			0.816322
$170$		−	5.22697e8i		−	0.625827i
$171$	−3.85058e8			−0.450341
$172$			1.14087e9i			1.30354i
$173$			1.13601e9i			1.26822i	0.773241	+	0.634112i	$0.218634\pi$
$173$			1.13601e9i			1.26822i	−0.773241	+	0.634112i	$0.781366\pi$
$174$		−	1.29848e9i		−	1.41657i
$175$	2.15042e9			2.29283
$176$		−	1.08160e8i		−	0.112723i
$177$	1.81010e8	−	5.36986e8i	0.184421	−	0.547104i
$178$	−6.43156e8			−0.640673
$179$			1.64528e9i			1.60261i	0.598257	+	0.801304i	$0.295860\pi$
$179$			1.64528e9i			1.60261i	−0.598257	+	0.801304i	$0.704140\pi$
$180$	−8.39403e8			−0.799614
$181$	−1.27022e9			−1.18349			−0.591747	−	0.806124i	$-0.701562\pi$
$181$	−1.27022e9			−1.18349			−0.591747	+	0.806124i	$0.701562\pi$
$182$	1.31486e9			1.19838
$183$			5.16978e8i			0.460965i
$184$	1.31271e9			1.14524
$185$			7.70023e8i			0.657380i
$186$	7.92539e8			0.662169
$187$			3.85005e8i			0.314848i
$188$			1.48859e9i			1.19163i
$189$	4.27797e8			0.335267
$190$			4.30082e9i			3.30017i
$191$			5.84587e8i			0.439254i	0.975584	+	0.219627i	$0.0704840\pi$
$191$			5.84587e8i			0.439254i	−0.975584	+	0.219627i	$0.929516\pi$
$192$	1.27817e9			0.940554
$193$	−5.03544e8			−0.362918			−0.181459	−	0.983399i	$-0.558082\pi$
$193$	−5.03544e8			−0.362918			−0.181459	+	0.983399i	$0.558082\pi$
$194$	2.11559e9			1.49357
$195$			5.44488e8i			0.376573i
$196$	−4.73356e9			−3.20748
$197$	−9.45241e8			−0.627593			−0.313796	−	0.949490i	$-0.601601\pi$
$197$	−9.45241e8			−0.627593			−0.313796	+	0.949490i	$0.601601\pi$
$198$	1.01054e9			0.657494
$199$	−3.11939e8			−0.198910			−0.0994550	−	0.995042i	$-0.531710\pi$
$199$	−3.11939e8			−0.198910			−0.0994550	+	0.995042i	$0.531710\pi$
$200$			1.94764e9i			1.21727i
$201$		−	1.66778e9i		−	1.02177i
$202$	3.73794e9			2.24506
$203$	4.52234e9			2.66305
$204$	−4.03794e8			−0.233152
$205$	1.92035e9			1.08734
$206$	−4.05627e9			−2.25247
$207$		−	7.57824e8i		−	0.412750i
$208$		−	7.35827e7i		−	0.0393117i
$209$		−	3.16788e9i		−	1.66029i
$210$		−	4.77819e9i		−	2.45689i
$211$		−	4.38644e8i		−	0.221300i	−0.993859	−	0.110650i	$-0.964707\pi$
$211$		−	4.38644e8i		−	0.221300i	0.993859	−	0.110650i	$-0.0352933\pi$
$212$	−4.37046e9			−2.16363
$213$	−1.89580e9			−0.921031
$214$		−	9.92483e8i		−	0.473225i
$215$		−	2.68930e9i		−	1.25859i
$216$			3.87456e8i			0.177995i
$217$			2.76025e9i			1.24483i
$218$	3.53643e9			1.56581
$219$		−	7.50779e8i		−	0.326389i
$220$		−	6.90578e9i		−	2.94796i
$221$			2.61925e8i			0.109802i
$222$	9.72251e8			0.400283
$223$	2.71647e9			1.09846			0.549232	−	0.835670i	$-0.314920\pi$
$223$	2.71647e9			1.09846			0.549232	+	0.835670i	$0.314920\pi$
$224$			4.70225e9i			1.86773i
$225$	1.12437e9			0.438710
$226$	−3.48009e9			−1.33400
$227$		−	4.74002e9i		−	1.78516i	−0.450892	−	0.892578i	$-0.648894\pi$
$227$		−	4.74002e9i		−	1.78516i	0.450892	−	0.892578i	$-0.351106\pi$
$228$	3.32247e9			1.22948
$229$			1.98073e9i			0.720252i	0.932904	+	0.360126i	$0.117266\pi$
$229$			1.98073e9i			0.720252i	−0.932904	+	0.360126i	$0.882734\pi$
$230$	−8.46435e9			−3.02470
$231$			3.51949e9i			1.23604i
$232$			4.09588e9i			1.41382i
$233$			2.97503e9i			1.00941i	0.863291	+	0.504706i	$0.168399\pi$
$233$			2.97503e9i			1.00941i	−0.863291	+	0.504706i	$0.831601\pi$
$234$	6.87485e8			0.229298
$235$		−	3.50894e9i		−	1.15055i
$236$	−1.56185e9	+	4.63338e9i	−0.503490	+	1.49365i
$237$	6.64455e8			0.210607
$238$		−	2.29854e9i		−	0.716383i
$239$	−4.45303e9			−1.36478			−0.682392	−	0.730987i	$-0.739060\pi$
$239$	−4.45303e9			−1.36478			−0.682392	+	0.730987i	$0.739060\pi$
$240$	−2.67399e8			−0.0805962
$241$	−6.45336e9			−1.91301			−0.956507	−	0.291711i	$-0.905776\pi$
$241$	−6.45336e9			−1.91301			−0.956507	+	0.291711i	$0.905776\pi$
$242$			2.80875e9i			0.818940i
$243$	2.23677e8			0.0641500
$244$		−	4.46075e9i		−	1.25849i
$245$	1.11581e10			3.09689
$246$		−	2.42468e9i		−	0.662087i
$247$		−	2.15516e9i		−	0.579017i
$248$	−2.49996e9			−0.660886
$249$			4.33319e9i			1.12722i
$250$		−	3.01649e9i		−	0.772221i
$251$	−3.67346e9			−0.925508			−0.462754	−	0.886487i	$-0.653139\pi$
$251$	−3.67346e9			−0.925508			−0.462754	+	0.886487i	$0.653139\pi$
$252$	−3.69125e9			−0.915316
$253$	6.23463e9			1.52170
$254$		−	2.51726e8i		−	0.0604774i
$255$	9.51835e8			0.225113
$256$	−3.63786e9			−0.847005
$257$	−6.27039e9			−1.43735			−0.718674	−	0.695347i	$-0.755251\pi$
$257$	−6.27039e9			−1.43735			−0.718674	+	0.695347i	$0.755251\pi$
$258$	−3.39558e9			−0.766365
$259$			3.38615e9i			0.752502i
$260$		−	4.69811e9i		−	1.02809i
$261$	2.36454e9			0.509548
$262$	−2.01668e9			−0.427989
$263$	−2.37803e9			−0.497044			−0.248522	−	0.968626i	$-0.579945\pi$
$263$	−2.37803e9			−0.497044			−0.248522	+	0.968626i	$0.579945\pi$
$264$	−3.18761e9			−0.656219
$265$	1.03022e10			2.08903
$266$			1.89127e10i			3.77770i
$267$		−	1.17119e9i		−	0.230453i
$268$			1.43904e10i			2.78955i
$269$		−	2.51448e9i		−	0.480218i	−0.970746	−	0.240109i	$-0.922817\pi$
$269$		−	2.51448e9i		−	0.480218i	0.970746	−	0.240109i	$-0.0771832\pi$
$270$		−	2.49832e9i		−	0.470102i
$271$	−2.01336e9			−0.373288			−0.186644	−	0.982428i	$-0.559761\pi$
$271$	−2.01336e9			−0.373288			−0.186644	+	0.982428i	$0.559761\pi$
$272$	−1.28632e8			−0.0235003
$273$			2.39437e9i			0.431063i
$274$			1.60180e10i			2.84188i
$275$			9.25019e9i			1.61741i
$276$			6.53889e9i			1.12685i
$277$	7.95593e9			1.35136			0.675681	−	0.737194i	$-0.263850\pi$
$277$	7.95593e9			1.35136			0.675681	+	0.737194i	$0.263850\pi$
$278$			3.27437e9i			0.548212i
$279$			1.44322e9i			0.238186i
$280$			1.50722e10i			2.45213i
$281$	8.46663e8			0.135795			0.0678977	−	0.997692i	$-0.478371\pi$
$281$	8.46663e8			0.135795			0.0678977	+	0.997692i	$0.478371\pi$
$282$	−4.43048e9			−0.700575
$283$			3.53661e8i			0.0551367i	0.999620	+	0.0275684i	$0.00877639\pi$
$283$			3.53661e8i			0.0551367i	−0.999620	+	0.0275684i	$0.991224\pi$
$284$	1.63579e10			2.51452
$285$	−7.83183e9			−1.18709
$286$			5.65595e9i			0.845359i
$287$	8.44468e9			1.24467
$288$			2.45861e9i			0.357372i
$289$	−6.51788e9			−0.934361
$290$		−	2.64102e10i		−	3.73405i
$291$			3.85251e9i			0.537244i
$292$			6.47810e9i			0.891078i
$293$	1.97817e9			0.268407			0.134203	−	0.990954i	$-0.457152\pi$
$293$	1.97817e9			0.268407			0.134203	+	0.990954i	$0.457152\pi$
$294$		−	1.40885e10i		−	1.88571i
$295$	3.68163e9	−	1.09219e10i	0.486130	−	1.44215i
$296$	−3.06684e9			−0.399507
$297$			1.84020e9i			0.236504i
$298$	−1.83124e10			−2.32210
$299$	4.24152e9			0.530685
$300$	−9.70161e9			−1.19773
$301$		−	1.18261e10i		−	1.44071i
$302$	9.60620e9			1.15485
$303$			6.80683e9i			0.807559i
$304$	1.05840e9			0.123924
$305$			1.05150e10i			1.21509i
$306$		−	1.20181e9i		−	0.137073i
$307$	1.04677e10			1.17841			0.589205	−	0.807984i	$-0.299441\pi$
$307$	1.04677e10			1.17841			0.589205	+	0.807984i	$0.299441\pi$
$308$		−	3.03680e10i		−	3.37453i
$309$		−	7.38651e9i		−	0.810225i
$310$	1.61197e10			1.74546
$311$	−7.77197e9			−0.830787			−0.415393	−	0.909642i	$-0.636356\pi$
$311$	−7.77197e9			−0.830787			−0.415393	+	0.909642i	$0.636356\pi$
$312$	−2.16858e9			−0.228853
$313$		−	3.77331e9i		−	0.393138i	−0.980490	−	0.196569i	$-0.937020\pi$
$313$		−	3.77331e9i		−	0.393138i	0.980490	−	0.196569i	$-0.0629799\pi$
$314$	−1.89222e10			−1.94649
$315$	8.70112e9			0.883757
$316$	−5.73325e9			−0.574980
$317$	2.22856e9			0.220692			0.110346	−	0.993893i	$-0.464804\pi$
$317$	2.22856e9			0.220692			0.110346	+	0.993893i	$0.464804\pi$
$318$		−	1.30078e10i		−	1.27203i
$319$			1.94531e10i			1.87857i
$320$	2.59972e10			2.47928
$321$	1.80732e9			0.170222
$322$	−3.72217e10			−3.46237
$323$	−3.76750e9			−0.346133
$324$	−1.93000e9			−0.175137
$325$			6.29305e9i			0.564063i
$326$			1.02740e10i			0.909643i
$327$			6.43986e9i			0.563230i
$328$			7.64835e9i			0.660803i
$329$		−	1.54305e10i		−	1.31703i
$330$	2.05537e10			1.73314
$331$	−9.97024e9			−0.830603			−0.415302	−	0.909684i	$-0.636324\pi$
$331$	−9.97024e9			−0.830603			−0.415302	+	0.909684i	$0.636324\pi$
$332$		−	3.73889e10i		−	3.07745i
$333$			1.77048e9i			0.143984i
$334$		−	1.55262e10i		−	1.24761i
$335$		−	3.39216e10i		−	2.69337i
$336$	−1.17588e9			−0.0922583
$337$			3.14033e9i			0.243476i	0.992562	+	0.121738i	$0.0388467\pi$
$337$			3.14033e9i			0.243476i	−0.992562	+	0.121738i	$0.961153\pi$
$338$		−	1.71010e10i		−	1.31025i
$339$		−	6.33727e9i		−	0.479848i
$340$	−8.21291e9			−0.614584
$341$	−1.18734e10			−0.878127
$342$			9.88868e9i			0.722826i
$343$	2.49545e10			1.80291
$344$	1.07109e10			0.764880
$345$		−	1.54136e10i		−	1.08800i
$346$	2.91738e10			2.03558
$347$			1.41172e10i			0.973709i	0.873483	+	0.486855i	$0.161856\pi$
$347$			1.41172e10i			0.973709i	−0.873483	+	0.486855i	$0.838144\pi$
$348$	−2.04025e10			−1.39112
$349$			1.18938e10i			0.801712i	0.916141	+	0.400856i	$0.131287\pi$
$349$			1.18938e10i			0.801712i	−0.916141	+	0.400856i	$0.868713\pi$
$350$		−	5.52251e10i		−	3.68014i
$351$			1.25192e9i			0.0824796i
$352$	−2.02270e10			−1.31753
$353$			1.21056e10i			0.779629i	0.920893	+	0.389814i	$0.127461\pi$
$353$			1.21056e10i			0.779629i	−0.920893	+	0.389814i	$0.872539\pi$
$354$	−1.37903e10	−	4.64853e9i	−0.878136	−	0.296007i
$355$	−3.85594e10			−2.42782
$356$			1.01056e10i			0.629164i
$357$	4.18567e9			0.257687
$358$	4.22524e10			2.57229
$359$	−2.50527e10			−1.50826			−0.754132	−	0.656723i	$-0.771942\pi$
$359$	−2.50527e10			−1.50826			−0.754132	+	0.656723i	$0.771942\pi$
$360$			7.88061e9i			0.469191i
$361$	1.40159e10			0.825262
$362$			3.26207e10i			1.89958i
$363$	−5.11476e9			−0.294577
$364$		−	2.06598e10i		−	1.17685i
$365$		−	1.52704e10i		−	0.860355i
$366$	1.32765e10			0.739878
$367$		−	9.78687e9i		−	0.539485i	−0.962932	−	0.269743i	$-0.913061\pi$
$367$		−	9.78687e9i		−	0.539485i	0.962932	−	0.269743i	$-0.0869386\pi$
$368$			2.08302e9i			0.113580i
$369$	4.41537e9			0.238156
$370$	1.97750e10			1.05514
$371$	4.53035e10			2.39131
$372$		−	1.24528e10i		−	0.650274i
$373$	−4.70700e9			−0.243169			−0.121585	−	0.992581i	$-0.538798\pi$
$373$	−4.70700e9			−0.243169			−0.121585	+	0.992581i	$0.538798\pi$
$374$	9.88734e9			0.505351
$375$	5.49305e9			0.277772
$376$	1.39754e10			0.699217
$377$			1.32343e10i			0.655141i
$378$		−	1.09863e10i		−	0.538125i
$379$	−7.32437e8			−0.0354988			−0.0177494	−	0.999842i	$-0.505650\pi$
$379$	−7.32437e8			−0.0354988			−0.0177494	+	0.999842i	$0.505650\pi$
$380$	6.75769e10			3.24089
$381$	4.58395e8			0.0217541
$382$	1.50128e10			0.705030
$383$	−1.41091e10			−0.655699			−0.327849	−	0.944730i	$-0.606324\pi$
$383$	−1.41091e10			−0.655699			−0.327849	+	0.944730i	$0.606324\pi$
$384$		−	1.93659e10i		−	0.890663i
$385$			7.15843e10i			3.25818i
$386$			1.29315e10i			0.582506i
$387$		−	6.18339e9i		−	0.275666i
$388$		−	3.32414e10i		−	1.46674i
$389$	−2.85066e10			−1.24494			−0.622469	−	0.782645i	$-0.713870\pi$
$389$	−2.85066e10			−1.24494			−0.622469	+	0.782645i	$0.713870\pi$
$390$	1.39830e10			0.604424
$391$		−	7.41473e9i		−	0.317240i
$392$			4.44404e10i			1.88206i
$393$		−	3.67239e9i		−	0.153950i
$394$			2.42748e10i			1.00733i
$395$	1.35146e10			0.555155
$396$		−	1.58781e10i		−	0.645683i
$397$			4.82585e7i			0.00194273i	1.00000		0.000971363i	$0.000309195\pi$
$397$			4.82585e7i			0.00194273i	−1.00000		0.000971363i	$0.999691\pi$
$398$			8.01090e9i			0.319263i
$399$	−3.44402e10			−1.35886
$400$	−3.09053e9			−0.120724
$401$			3.13170e10i			1.21116i	0.795783	+	0.605581i	$0.207059\pi$
$401$			3.13170e10i			1.21116i	−0.795783	+	0.605581i	$0.792941\pi$
$402$	−4.28303e10			−1.64001
$403$	−8.07766e9			−0.306243
$404$		−	5.87328e10i		−	2.20473i
$405$	4.54945e9			0.169098
$406$		−	1.16138e11i		−	4.27436i
$407$	−1.45658e10			−0.530830
$408$			3.79096e9i			0.136807i
$409$			3.47833e10i			1.24302i	0.783406	+	0.621510i	$0.213480\pi$
$409$			3.47833e10i			1.24302i	−0.783406	+	0.621510i	$0.786520\pi$
$410$		−	4.93165e10i		−	1.74525i
$411$	−2.91689e10			−1.02224
$412$			6.37345e10i			2.21200i
$413$	1.61899e10	−	4.80289e10i	0.556472	−	1.65083i
$414$	−1.94617e10			−0.662490
$415$			8.81343e10i			2.97134i
$416$	−1.37608e10			−0.459483
$417$	−5.96265e9			−0.197195
$418$	−8.13543e10			−2.66487
$419$			2.76733e9i			0.0897851i	0.998992	+	0.0448926i	$0.0142946\pi$
$419$			2.76733e9i			0.0897851i	−0.998992	+	0.0448926i	$0.985705\pi$
$420$	−7.50776e10			−2.41276
$421$		−	5.62314e10i		−	1.78999i	−0.446078	−	0.894994i	$-0.647180\pi$
$421$		−	5.62314e10i		−	1.78999i	0.446078	−	0.894994i	$-0.352820\pi$
$422$	−1.12648e10			−0.355201
$423$		−	8.06795e9i		−	0.252001i
$424$			4.10314e10i			1.26956i
$425$	1.10011e10			0.337194
$426$			4.86861e10i			1.47831i
$427$			4.62394e10i			1.39092i
$428$	−1.55945e10			−0.464724
$429$	−1.02995e10			−0.304080
$430$	−6.90640e10			−2.02012
$431$		−	4.15179e10i		−	1.20317i	−0.798810	−	0.601584i	$-0.794537\pi$
$431$		−	4.15179e10i		−	1.20317i	0.798810	−	0.601584i	$-0.205463\pi$
$432$	−6.14818e8			−0.0176527
$433$	2.35994e10			0.671351			0.335676	−	0.941978i	$-0.391035\pi$
$433$	2.35994e10			0.671351			0.335676	+	0.941978i	$0.391035\pi$
$434$	7.08861e10			1.99803
$435$	4.80933e10			1.34316
$436$		−	5.55664e10i		−	1.53768i
$437$			6.10094e10i			1.67290i
$438$	−1.92808e10			−0.523875
$439$	2.64779e9			0.0712895			0.0356447	−	0.999365i	$-0.488652\pi$
$439$	2.64779e9			0.0712895			0.0356447	+	0.999365i	$0.488652\pi$
$440$	−6.48339e10			−1.72978
$441$	2.56553e10			0.678302
$442$	6.72651e9			0.176239
$443$		−	2.31302e10i		−	0.600572i	−0.953849	−	0.300286i	$-0.902918\pi$
$443$		−	2.31302e10i		−	0.600572i	0.953849	−	0.300286i	$-0.0970821\pi$
$444$		−	1.52766e10i		−	0.393092i
$445$		−	2.38213e10i		−	0.607471i
$446$		−	6.97618e10i		−	1.76311i
$447$		−	3.33471e10i		−	0.835271i
$448$	1.14322e11			2.83803
$449$	−2.70386e10			−0.665272			−0.332636	−	0.943055i	$-0.607938\pi$
$449$	−2.70386e10			−0.665272			−0.332636	+	0.943055i	$0.607938\pi$
$450$		−	2.88749e10i		−	0.704159i
$451$			3.63253e10i			0.878018i
$452$			5.46811e10i			1.31004i
$453$			1.74930e10i			0.415404i
$454$	−1.21728e11			−2.86529
$455$			4.86999e10i			1.13627i
$456$		−	3.11925e10i		−	0.721425i
$457$			5.14489e10i			1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$457$			5.14489e10i			1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$458$	5.08673e10			1.15605
$459$	2.18851e9			0.0493058
$460$			1.32997e11i			2.97036i
$461$	−3.76966e10			−0.834639			−0.417319	−	0.908760i	$-0.637030\pi$
$461$	−3.76966e10			−0.834639			−0.417319	+	0.908760i	$0.637030\pi$
$462$	9.03842e10			1.98392
$463$			3.52785e9i			0.0767691i	0.999263	+	0.0383846i	$0.0122212\pi$
$463$			3.52785e9i			0.0767691i	−0.999263	+	0.0383846i	$0.987779\pi$
$464$	−6.49938e9			−0.140217
$465$			2.93542e10i			0.627853i
$466$	7.64019e10			1.62017
$467$		−	1.82138e10i		−	0.382941i	−0.981498	−	0.191471i	$-0.938674\pi$
$467$		−	1.82138e10i		−	0.382941i	0.981498	−	0.191471i	$-0.0613257\pi$
$468$		−	1.08022e10i		−	0.225179i
$469$		−	1.49169e11i		−	3.08310i
$470$	−9.01132e10			−1.84670
$471$		−	3.44575e10i		−	0.700164i
$472$	4.34998e10	+	1.46632e10i	0.876434	+	0.295434i
$473$	5.08708e10			1.01631
$474$		−	1.70639e10i		−	0.338037i
$475$	−9.05183e10			−1.77812
$476$	−3.61161e10			−0.703514
$477$	2.36873e10			0.457554
$478$			1.14358e11i			2.19057i
$479$	5.83560e10			1.10852			0.554260	−	0.832343i	$-0.313001\pi$
$479$	5.83560e10			1.10852			0.554260	+	0.832343i	$0.313001\pi$
$480$			5.00066e10i			0.942025i
$481$	−9.90932e9			−0.185124
$482$			1.65729e11i			3.07051i
$483$		−	6.77811e10i		−	1.24543i
$484$	4.41327e10			0.804228
$485$			7.83576e10i			1.41617i
$486$		−	5.74426e9i		−	0.102965i
$487$	−4.42685e9			−0.0787007			−0.0393504	−	0.999225i	$-0.512529\pi$
$487$	−4.42685e9			−0.0787007			−0.0393504	+	0.999225i	$0.512529\pi$
$488$	−4.18791e10			−0.738444
$489$	−1.87091e10			−0.327204
$490$		−	2.86551e11i		−	4.97071i
$491$	1.59957e10			0.275219			0.137610	−	0.990487i	$-0.456058\pi$
$491$	1.59957e10			0.275219			0.137610	+	0.990487i	$0.456058\pi$
$492$	−3.80981e10			−0.650193
$493$	2.31352e10			0.391639
$494$	−5.53467e10			−0.929359
$495$			3.74284e10i			0.623420i
$496$		−	3.96695e9i		−	0.0655436i
$497$	−1.69564e11			−2.77912
$498$	1.11281e11			1.80927
$499$	−6.22322e10			−1.00372			−0.501860	−	0.864949i	$-0.667351\pi$
$499$	−6.22322e10			−1.00372			−0.501860	+	0.864949i	$0.667351\pi$
$500$	−4.73968e10			−0.758349
$501$	2.82733e10			0.448773
$502$			9.43381e10i			1.48550i
$503$			1.08220e11i			1.69058i	0.534311	+	0.845288i	$0.320571\pi$
$503$			1.08220e11i			1.69058i	−0.534311	+	0.845288i	$0.679429\pi$
$504$			3.46547e10i			0.537082i
$505$			1.38447e11i			2.12871i
$506$		−	1.60112e11i		−	2.44242i
$507$	3.11410e10			0.471304
$508$	−3.95527e9			−0.0593910
$509$			5.44215e10i			0.810773i	0.914145	+	0.405386i	$0.132863\pi$
$509$			5.44215e10i			0.810773i	−0.914145	+	0.405386i	$0.867137\pi$
$510$		−	2.44441e10i		−	0.361321i
$511$		−	6.71509e10i		−	0.984846i
$512$		−	1.25879e10i		−	0.183178i
$513$	−1.80074e10			−0.260004
$514$			1.61030e11i			2.30704i
$515$		−	1.50237e11i		−	2.13574i
$516$			5.33534e10i			0.752598i
$517$	6.63752e10			0.929059
$518$	8.69599e10			1.20781
$519$			5.31257e10i			0.732209i
$520$	−4.41075e10			−0.603253
$521$	−5.92887e10			−0.804676			−0.402338	−	0.915491i	$-0.631802\pi$
$521$	−5.92887e10			−0.804676			−0.402338	+	0.915491i	$0.631802\pi$
$522$		−	6.07239e10i		−	0.817857i
$523$	−3.98393e10			−0.532482			−0.266241	−	0.963906i	$-0.585782\pi$
$523$	−3.98393e10			−0.532482			−0.266241	+	0.963906i	$0.585782\pi$
$524$			3.16873e10i			0.420300i
$525$	1.00565e11			1.32377
$526$			6.10703e10i			0.797787i
$527$			1.41208e10i			0.183070i
$528$		−	5.05812e9i		−	0.0650809i
$529$	−4.17602e10			−0.533261
$530$		−	2.64571e11i		−	3.35303i
$531$	8.46501e9	−	2.51123e10i	0.106475	−	0.315870i
$532$	2.97168e11			3.70984
$533$			2.47127e10i			0.306205i
$534$	−3.00774e10			−0.369893
$535$	3.67597e10			0.448701
$536$	1.35102e11			1.63683
$537$			7.69421e10i			0.925266i
$538$	−6.45744e10			−0.770781
$539$			2.11067e11i			2.50072i
$540$	−3.92550e10			−0.461657
$541$		−	1.29570e11i		−	1.51257i	−0.654245	−	0.756283i	$-0.727013\pi$
$541$		−	1.29570e11i		−	1.51257i	0.654245	−	0.756283i	$-0.272987\pi$
$542$			5.17051e10i			0.599151i
$543$	−5.94025e10			−0.683291
$544$			2.40556e10i			0.274676i
$545$			1.30983e11i			1.48466i
$546$	6.14899e10			0.691883
$547$	1.30171e11			1.45400			0.727000	−	0.686637i	$-0.240914\pi$
$547$	1.30171e11			1.45400			0.727000	+	0.686637i	$0.240914\pi$
$548$	2.51684e11			2.79083
$549$			2.41767e10i			0.266138i
$550$	2.37554e11			2.59604
$551$	−1.90360e11			−2.06523
$552$	6.13893e10			0.661206
$553$	5.94300e10			0.635485
$554$		−	2.04316e11i		−	2.16902i
$555$			3.60104e10i			0.379539i
$556$	5.14488e10			0.538364
$557$	−8.74828e10			−0.908870			−0.454435	−	0.890780i	$-0.650159\pi$
$557$	−8.74828e10			−0.908870			−0.454435	+	0.890780i	$0.650159\pi$
$558$	3.70634e10			0.382303
$559$	3.46083e10			0.354432
$560$	−2.39166e10			−0.243191
$561$			1.80049e10i			0.181777i
$562$		−	2.17432e10i		−	0.217960i
$563$			8.21664e10i			0.817825i	0.912574	+	0.408913i	$0.134092\pi$
$563$			8.21664e10i			0.817825i	−0.912574	+	0.408913i	$0.865908\pi$
$564$			6.96143e10i			0.687990i
$565$		−	1.28896e11i		−	1.26487i
$566$	9.08237e9			0.0884980
$567$	2.00061e10			0.193566
$568$		−	1.53574e11i		−	1.47545i
$569$			1.48678e11i			1.41840i	0.705009	+	0.709198i	$0.250943\pi$
$569$			1.48678e11i			1.41840i	−0.705009	+	0.709198i	$0.749057\pi$
$570$			2.01129e11i			1.90536i
$571$		−	8.59763e10i		−	0.808788i	−0.914585	−	0.404394i	$-0.867483\pi$
$571$		−	8.59763e10i		−	0.808788i	0.914585	−	0.404394i	$-0.132517\pi$
$572$	8.88696e10			0.830174
$573$			2.73384e10i			0.253603i
$574$		−	2.16868e11i		−	1.99778i
$575$		−	1.78147e11i		−	1.62970i
$576$	5.97741e10			0.543029
$577$	−1.13285e11			−1.02204			−0.511021	−	0.859568i	$-0.670732\pi$
$577$	−1.13285e11			−1.02204			−0.511021	+	0.859568i	$0.670732\pi$
$578$			1.67386e11i			1.49971i
$579$	−2.35484e10			−0.209531
$580$	−4.14973e11			−3.66697
$581$			3.87568e11i			3.40129i
$582$	9.89364e10			0.862311
$583$			1.94876e11i			1.68688i
$584$	6.08186e10			0.522860
$585$			2.54632e10i			0.217415i
$586$		−	5.08015e10i		−	0.430810i
$587$		−	2.73783e10i		−	0.230597i	−0.993331	−	0.115299i	$-0.963217\pi$
$587$		−	2.73783e10i		−	0.230597i	0.993331	−	0.115299i	$-0.0367825\pi$
$588$	−2.21367e11			−1.85184
$589$		−	1.16188e11i		−	0.965383i
$590$	−2.80487e11	−	9.45481e10i	−2.31475	−	0.780270i
$591$	−4.42046e10			−0.362341
$592$		−	4.86648e9i		−	0.0396213i
$593$	2.13025e11			1.72271			0.861355	−	0.508004i	$-0.169616\pi$
$593$	2.13025e11			1.72271			0.861355	+	0.508004i	$0.169616\pi$
$594$	4.72582e10			0.379604
$595$	8.51338e10			0.679257
$596$			2.87735e11i			2.28038i
$597$	−1.45879e10			−0.114841
$598$		−	1.08927e11i		−	0.851783i
$599$	−9.11473e10			−0.708005			−0.354003	−	0.935244i	$-0.615180\pi$
$599$	−9.11473e10			−0.708005			−0.354003	+	0.935244i	$0.615180\pi$
$600$			9.10821e10i			0.702794i
$601$		−	2.68033e10i		−	0.205443i	−0.994710	−	0.102721i	$-0.967245\pi$
$601$		−	2.68033e10i		−	0.205443i	0.994710	−	0.102721i	$-0.0327550\pi$
$602$	−3.03707e11			−2.31243
$603$		−	7.79943e10i		−	0.589921i
$604$		−	1.50938e11i		−	1.13410i
$605$	−1.04031e11			−0.776499
$606$	1.74806e11			1.29618
$607$	7.40206e10			0.545253			0.272626	−	0.962120i	$-0.412108\pi$
$607$	7.40206e10			0.545253			0.272626	+	0.962120i	$0.412108\pi$
$608$		−	1.97933e11i		−	1.44845i
$609$	2.11489e11			1.53751
$610$	2.70036e11			1.95030
$611$	4.51561e10			0.324005
$612$	−1.88836e10			−0.134610
$613$		−	3.05315e10i		−	0.216225i	−0.994139	−	0.108113i	$-0.965519\pi$
$613$		−	3.05315e10i		−	0.216225i	0.994139	−	0.108113i	$-0.0344807\pi$
$614$		−	2.68820e11i		−	1.89142i
$615$	8.98058e10			0.627775
$616$	−2.85105e11			−1.98008
$617$	1.70239e11			1.17468			0.587338	−	0.809342i	$-0.300176\pi$
$617$	1.70239e11			1.17468			0.587338	+	0.809342i	$0.300176\pi$
$618$	−1.89693e11			−1.30046
$619$	−4.59600e10			−0.313052			−0.156526	−	0.987674i	$-0.550030\pi$
$619$	−4.59600e10			−0.313052			−0.156526	+	0.987674i	$0.550030\pi$
$620$		−	2.53283e11i		−	1.71411i
$621$		−	3.54399e10i		−	0.238301i
$622$			1.99592e11i			1.33347i
$623$		−	1.04753e11i		−	0.695371i
$624$		−	3.44112e9i		−	0.0226966i
$625$	−8.91006e10			−0.583929
$626$	−9.69024e10			−0.631011
$627$		−	1.48147e11i		−	0.958567i
$628$			2.97317e11i			1.91153i
$629$			1.73228e10i			0.110666i
$630$		−	2.23454e11i		−	1.41849i
$631$	2.58454e11			1.63029			0.815147	−	0.579254i	$-0.196656\pi$
$631$	2.58454e11			1.63029			0.815147	+	0.579254i	$0.196656\pi$
$632$			5.38258e10i			0.337382i
$633$		−	2.05133e10i		−	0.127768i
$634$		−	5.72317e10i		−	0.354225i
$635$	9.32347e9			0.0573433
$636$	−2.04386e11			−1.24917
$637$			1.43592e11i			0.872113i
$638$	4.99576e11			3.01522
$639$	−8.86579e10			−0.531758
$640$		−	3.93891e11i		−	2.34777i
$641$	−1.62707e11			−0.963770			−0.481885	−	0.876234i	$-0.660048\pi$
$641$	−1.62707e11			−0.963770			−0.481885	+	0.876234i	$0.660048\pi$
$642$		−	4.64138e10i		−	0.273217i
$643$	3.41504e11			1.99780			0.998901	−	0.0468767i	$-0.0149268\pi$
$643$	3.41504e11			1.99780			0.998901	+	0.0468767i	$0.0149268\pi$
$644$			5.84849e11i			3.40017i
$645$		−	1.25766e11i		−	0.726649i
$646$			9.67532e10i			0.555565i
$647$	1.73298e11			0.988954			0.494477	−	0.869191i	$-0.335360\pi$
$647$	1.73298e11			0.988954			0.494477	+	0.869191i	$0.335360\pi$
$648$			1.81195e10i			0.102765i
$649$	2.06600e11	+	6.96418e10i	1.16453	+	0.392546i
$650$	1.61612e11			0.905358
$651$			1.29084e11i			0.718702i
$652$	1.61432e11			0.893303
$653$	2.36255e11			1.29936			0.649679	−	0.760209i	$-0.274903\pi$
$653$	2.36255e11			1.29936			0.649679	+	0.760209i	$0.274903\pi$
$654$	1.65382e11			0.904020
$655$		−	7.46941e10i		−	0.405809i
$656$	−1.21365e10			−0.0655355
$657$		−	3.51104e10i		−	0.188441i
$658$	−3.96270e11			−2.11392
$659$		−	1.18417e11i		−	0.627871i	−0.949444	−	0.313936i	$-0.898352\pi$
$659$		−	1.18417e11i		−	0.627871i	0.949444	−	0.313936i	$-0.101648\pi$
$660$		−	3.22951e11i		−	1.70201i
$661$	7.70053e10			0.403380			0.201690	−	0.979449i	$-0.435357\pi$
$661$	7.70053e10			0.403380			0.201690	+	0.979449i	$0.435357\pi$
$662$			2.56046e11i			1.33317i
$663$			1.22490e10i			0.0633940i
$664$	−3.51020e11			−1.80576
$665$	−7.00492e11			−3.58193
$666$	4.54677e10			0.231103
$667$		−	3.74643e11i		−	1.89284i
$668$	−2.43957e11			−1.22520
$669$	1.27037e11			0.634199
$670$	−8.71140e11			−4.32304
$671$	−1.98902e11			−0.981180
$672$			2.19903e11i			1.07833i
$673$		−	2.12639e11i		−	1.03653i	−0.855220	−	0.518265i	$-0.826578\pi$
$673$		−	2.12639e11i		−	1.03653i	0.855220	−	0.518265i	$-0.173422\pi$
$674$	8.06468e10			0.390794
$675$	5.25814e10			0.253290
$676$	−2.68700e11			−1.28671
$677$	−3.86837e11			−1.84150			−0.920752	−	0.390148i	$-0.872424\pi$
$677$	−3.86837e11			−1.84150			−0.920752	+	0.390148i	$0.872424\pi$
$678$	−1.62748e11			−0.770186
$679$			3.44575e11i			1.62108i
$680$			7.71057e10i			0.360621i
$681$		−	2.21669e11i		−	1.03066i
$682$			3.04921e11i			1.40945i
$683$		−	2.43065e11i		−	1.11696i	−0.829516	−	0.558482i	$-0.811384\pi$
$683$		−	2.43065e11i		−	1.11696i	0.829516	−	0.558482i	$-0.188616\pi$
$684$	1.55377e11			0.709841
$685$	−5.93278e11			−2.69461
$686$		−	6.40858e11i		−	2.89378i
$687$			9.26298e10i			0.415838i
$688$			1.69962e10i			0.0758573i
$689$			1.32577e11i			0.588291i
$690$	−3.95838e11			−1.74631
$691$			1.98105e11i			0.868926i	0.900690	+	0.434463i	$0.143062\pi$
$691$			1.98105e11i			0.868926i	−0.900690	+	0.434463i	$0.856938\pi$
$692$		−	4.58395e11i		−	1.99901i
$693$			1.64590e11i			0.713628i
$694$	3.62543e11			1.56287
$695$	−1.21277e11			−0.519801
$696$			1.91545e11i			0.816272i
$697$	4.32010e10			0.183047
$698$	3.05444e11			1.28680
$699$			1.39129e11i			0.582784i
$700$	−8.67729e11			−3.61403
$701$		−	2.21452e11i		−	0.917079i	−0.888674	−	0.458539i	$-0.848373\pi$
$701$		−	2.21452e11i		−	0.917079i	0.888674	−	0.458539i	$-0.151627\pi$
$702$	3.21505e10			0.132385
$703$		−	1.42534e11i		−	0.583576i
$704$			4.91762e11i			2.00200i
$705$		−	1.64097e11i		−	0.664269i
$706$	3.10885e11			1.25135
$707$			6.08815e11i			2.43673i
$708$	−7.30404e10	+	2.16682e11i	−0.290690	+	0.862362i
$709$	3.27105e11			1.29450			0.647250	−	0.762278i	$-0.275919\pi$
$709$	3.27105e11			1.29450			0.647250	+	0.762278i	$0.275919\pi$
$710$			9.90245e11i			3.89681i
$711$	3.10735e10			0.121594
$712$	9.48753e10			0.369176
$713$	2.28667e11			0.884800
$714$		−	1.07492e11i		−	0.413604i
$715$	−2.09486e11			−0.801550
$716$		−	6.63895e11i		−	2.52608i
$717$	−2.08247e11			−0.787958
$718$			6.43380e11i			2.42086i
$719$		−	1.19003e11i		−	0.445291i	−0.974899	−	0.222646i	$-0.928531\pi$
$719$		−	1.19003e11i		−	0.445291i	0.974899	−	0.222646i	$-0.0714693\pi$
$720$	−1.25050e10			−0.0465322
$721$		−	6.60662e11i		−	2.44477i
$722$		−	3.59943e11i		−	1.32460i
$723$	−3.01794e11			−1.10448
$724$	5.12555e11			1.86546
$725$	5.55850e11			2.01190
$726$			1.31352e11i			0.472815i
$727$	−1.19400e9			−0.00427432			−0.00213716	−	0.999998i	$-0.500680\pi$
$727$	−1.19400e9			−0.00427432			−0.00213716	+	0.999998i	$0.500680\pi$
$728$	−1.93962e11			−0.690542
$729$	1.04604e10			0.0370370
$730$	−3.92158e11			−1.38092
$731$		−	6.04997e10i		−	0.211877i
$732$		−	2.08609e11i		−	0.726587i
$733$	3.01012e11			1.04272			0.521361	−	0.853336i	$-0.325425\pi$
$733$	3.01012e11			1.04272			0.521361	+	0.853336i	$0.325425\pi$
$734$	−2.51337e11			−0.865908
$735$	5.21812e11			1.78799
$736$	3.89548e11			1.32755
$737$	6.41660e11			2.17488
$738$		−	1.13391e11i		−	0.382256i
$739$			4.50964e11i			1.51204i	0.654547	+	0.756022i	$0.272860\pi$
$739$			4.50964e11i			1.51204i	−0.654547	+	0.756022i	$0.727140\pi$
$740$		−	3.10716e11i		−	1.03618i
$741$		−	1.00787e11i		−	0.334296i
$742$		−	1.16344e12i		−	3.83821i
$743$	−4.32552e11			−1.41933			−0.709664	−	0.704540i	$-0.751153\pi$
$743$	−4.32552e11			−1.41933			−0.709664	+	0.704540i	$0.751153\pi$
$744$	−1.16912e11			−0.381562
$745$		−	6.78258e11i		−	2.20176i
$746$			1.20881e11i			0.390302i
$747$			2.02643e11i			0.650803i
$748$		−	1.55356e11i		−	0.496273i
$749$	1.61650e11			0.513627
$750$		−	1.41067e11i		−	0.445842i
$751$			8.66836e10i			0.272507i	0.990674	+	0.136253i	$0.0435061\pi$
$751$			8.66836e10i			0.272507i	−0.990674	+	0.136253i	$0.956494\pi$
$752$			2.21762e10i			0.0693452i
$753$	−1.71791e11			−0.534342
$754$	3.39870e11			1.05154
$755$			3.55796e11i			1.09500i
$756$	−1.72623e11			−0.528458
$757$	−2.29030e11			−0.697442			−0.348721	−	0.937227i	$-0.613384\pi$
$757$	−2.29030e11			−0.697442			−0.348721	+	0.937227i	$0.613384\pi$
$758$			1.88097e10i			0.0569779i
$759$	2.91565e11			0.878553
$760$		−	6.34436e11i		−	1.90166i
$761$	−3.30547e11			−0.985585			−0.492793	−	0.870147i	$-0.664024\pi$
$761$	−3.30547e11			−0.985585			−0.492793	+	0.870147i	$0.664024\pi$
$762$		−	1.17721e10i		−	0.0349167i
$763$			5.75993e11i			1.69949i
$764$		−	2.35890e11i		−	0.692365i
$765$	4.45129e10			0.129969
$766$			3.62336e11i			1.05244i
$767$	1.40553e11	+	4.73784e10i	0.406124	+	0.136899i
$768$	−1.70126e11			−0.489018
$769$		−	5.16985e11i		−	1.47834i	−0.673522	−	0.739168i	$-0.735219\pi$
$769$		−	5.16985e11i		−	1.47834i	0.673522	−	0.739168i	$-0.264781\pi$
$770$	1.83836e12			5.22958
$771$	−2.93237e11			−0.829854
$772$	2.03188e11			0.572042
$773$		−	3.80459e11i		−	1.06559i	−0.846245	−	0.532794i	$-0.821142\pi$
$773$		−	3.80459e11i		−	1.06559i	0.846245	−	0.532794i	$-0.178858\pi$
$774$	−1.58796e11			−0.442461
$775$			3.39268e11i			0.940451i
$776$	−3.12082e11			−0.860640
$777$			1.58355e11i			0.434457i
$778$			7.32080e11i			1.99821i
$779$	−3.55464e11			−0.965263
$780$		−	2.19709e11i		−	0.593567i
$781$		−	7.29390e11i		−	1.96045i
$782$	−1.90418e11			−0.509191

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-25.6810i q^{2} +46.7654 q^{3} -403.515 q^{4} +951.178 q^{5} -1200.98i q^{6} +4182.78 q^{7} +3788.34i q^{8} +2187.00 q^{9} +O(q^{10})\)	\(q-25.6810i q^{2} +46.7654 q^{3} -403.515 q^{4} +951.178 q^{5} -1200.98i q^{6} +4182.78 q^{7} +3788.34i q^{8} +2187.00 q^{9} -24427.2i q^{10} +17992.5i q^{11} -18870.5 q^{12} +12240.6i q^{13} -107418. i q^{14} +44482.2 q^{15} -6011.37 q^{16} +21398.1 q^{17} -56164.4i q^{18} -176067. q^{19} -383815. q^{20} +195609. q^{21} +462066. q^{22} -346513. i q^{23} +177163. i q^{24} +514114. q^{25} +314351. q^{26} +102276. q^{27} -1.68781e6 q^{28} +1.08118e6 q^{29} -1.14235e6i q^{30} +659909. i q^{31} +1.12419e6i q^{32} +841425. i q^{33} -549526. i q^{34} +3.97856e6 q^{35} -882488. q^{36} +809547. i q^{37} +4.52157e6i q^{38} +572435. i q^{39} +3.60339e6i q^{40} +2.01892e6 q^{41} -5.02344e6i q^{42} -2.82734e6i q^{43} -7.26024e6i q^{44} +2.08023e6 q^{45} -8.89881e6 q^{46} -3.68905e6i q^{47} -281124. q^{48} +1.17308e7 q^{49} -1.32030e7i q^{50} +1.00069e6 q^{51} -4.93926e6i q^{52} +1.08310e7 q^{53} -2.62655e6i q^{54} +1.71140e7i q^{55} +1.58458e7i q^{56} -8.23382e6 q^{57} -2.77658e7i q^{58} +(3.87060e6 - 1.14825e7i) q^{59} -1.79492e7 q^{60} +1.10547e7i q^{61} +1.69471e7 q^{62} +9.14773e6 q^{63} +2.73315e7 q^{64} +1.16430e7i q^{65} +2.16087e7 q^{66} -3.56627e7i q^{67} -8.63447e6 q^{68} -1.62048e7i q^{69} -1.02174e8i q^{70} -4.05386e7 q^{71} +8.28510e6i q^{72} -1.60542e7i q^{73} +2.07900e7 q^{74} +2.40427e7 q^{75} +7.10456e7 q^{76} +7.52586e7i q^{77} +1.47007e7 q^{78} +1.42083e7 q^{79} -5.71788e6 q^{80} +4.78297e6 q^{81} -5.18479e7i q^{82} +9.26581e7i q^{83} -7.89313e7 q^{84} +2.03534e7 q^{85} -7.26089e7 q^{86} +5.05618e7 q^{87} -6.81617e7 q^{88} -2.50440e7i q^{89} -5.34223e7i q^{90} +5.11996e7i q^{91} +1.39823e8i q^{92} +3.08609e7i q^{93} -9.47386e7 q^{94} -1.67471e8 q^{95} +5.25733e7i q^{96} +8.23795e7i q^{97} -3.01260e8i q^{98} +3.93496e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} + O(q^{10}) \)	\( 80q - 10240q^{4} + 160q^{7} + 174960q^{9} - 22680q^{12} - 59616q^{15} + 1199848q^{16} - 10608q^{17} - 27516q^{19} - 146436q^{20} - 974696q^{22} + 5718040q^{25} - 797484q^{26} - 3133000q^{28} + 1725924q^{29} + 4318800q^{35} - 22394880q^{36} - 732180q^{41} + 22752084q^{46} + 8703936q^{48} + 55899176q^{49} - 10373832q^{51} - 39265944q^{53} - 11408040q^{57} - 33575112q^{59} - 18034488q^{60} + 13038600q^{62} + 349920q^{63} - 241654260q^{64} - 35711928q^{66} + 36772608q^{68} - 235272660q^{71} - 63050712q^{74} + 74363184q^{75} + 9454680q^{76} - 10865988q^{78} + 17252580q^{79} + 318203976q^{80} + 382637520q^{81} - 20743128q^{84} - 27245820q^{85} + 105666984q^{86} + 29437992q^{87} + 82079788q^{88} + 121215992q^{94} - 690837276q^{95} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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