LMFDB - Embedded newform 177.4.d.b.176.2

Newspace parameters

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-59})$
Defining polynomial:	$x^{4} - x^{3} - 14 x^{2} - 15 x + 225$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			176.2
Root			$3.57603 + 1.48727i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.4.d.b.176.1

$q$-expansion

$f(q)$	$=$	$q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +O(q^{10})$	$q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +(40.6083 - 8.88642i) q^{12} +(-24.4678 - 111.810i) q^{15} +64.0000 q^{16} +61.4492i q^{17} +40.2810 q^{19} -176.217i q^{20} +(174.901 - 38.2740i) q^{21} -360.193 q^{25} +(-112.000 + 84.4926i) q^{27} +275.650 q^{28} -272.118i q^{29} -758.970i q^{35} +(-196.258 + 90.2155i) q^{36} +458.504i q^{41} +(248.398 + 540.374i) q^{45} +(-324.866 + 71.0914i) q^{48} +844.230 q^{49} +(-68.2579 - 311.918i) q^{51} +632.682i q^{53} +(-204.468 + 44.7442i) q^{57} -453.188i q^{59} +(195.742 + 894.483i) q^{60} +(-845.288 + 388.561i) q^{63} -512.000 q^{64} -491.593i q^{68} -1182.90i q^{71} +(1828.35 - 400.104i) q^{75} -322.248 q^{76} -560.354 q^{79} +1409.73i q^{80} +(474.661 - 553.297i) q^{81} +(-1399.21 + 306.192i) q^{84} -1353.55 q^{85} +(302.269 + 1381.28i) q^{87} +887.274i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10}) $	$ 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + 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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	−5.07603	+	1.11080i	−0.976883	+	0.213774i
$3$	−5.07603	+	1.11080i	−0.976883	+	0.213774i
$4$	−8.00000			−1.00000
$5$			22.0271i			1.97016i	0.172085	+	0.985082i	$0.444950\pi$
$5$			22.0271i			1.97016i	−0.172085	+	0.985082i	$0.555050\pi$
$6$		0			0
$7$	−34.4562			−1.86046			−0.930230	−	0.366977i	$-0.880393\pi$
$7$	−34.4562			−1.86046			−0.930230	+	0.366977i	$0.880393\pi$
$8$		0			0
$9$	24.5322	−	11.2769i	0.908601	−	0.417665i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	40.6083	−	8.88642i	0.976883	−	0.213774i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−24.4678	−	111.810i	−0.421170	−	1.92462i
$16$	64.0000			1.00000
$17$			61.4492i			0.876683i	0.898808	+	0.438342i	$0.144434\pi$
$17$			61.4492i			0.876683i	−0.898808	+	0.438342i	$0.855566\pi$
$18$		0			0
$19$	40.2810			0.486374			0.243187	−	0.969979i	$-0.421807\pi$
$19$	40.2810			0.486374			0.243187	+	0.969979i	$0.421807\pi$
$20$		−	176.217i		−	1.97016i
$21$	174.901	−	38.2740i	1.81745	−	0.397718i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−360.193			−2.88155
$26$		0			0
$27$	−112.000	+	84.4926i	−0.798311	+	0.602245i
$28$	275.650			1.86046
$29$		−	272.118i		−	1.74245i	−0.490884	−	0.871225i	$-0.663326\pi$
$29$		−	272.118i		−	1.74245i	0.490884	−	0.871225i	$-0.336674\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	758.970i		−	3.66541i
$36$	−196.258	+	90.2155i	−0.908601	+	0.417665i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			458.504i			1.74649i	0.487277	+	0.873247i	$0.337990\pi$
$41$			458.504i			1.74649i	−0.487277	+	0.873247i	$0.662010\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	248.398	+	540.374i	0.822868	+	1.79009i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−324.866	+	71.0914i	−0.976883	+	0.213774i
$49$	844.230			2.46131
$50$		0			0
$51$	−68.2579	−	311.918i	−0.187412	−	0.856417i
$52$		0			0
$53$			632.682i			1.63973i	0.572558	+	0.819864i	$0.305951\pi$
$53$			632.682i			1.63973i	−0.572558	+	0.819864i	$0.694049\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−204.468	+	44.7442i	−0.475130	+	0.103974i
$58$		0			0
$59$		−	453.188i		−	1.00000i
$59$		−	453.188i		−	1.00000i
$60$	195.742	+	894.483i	0.421170	+	1.92462i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−845.288	+	388.561i	−1.69042	+	0.777048i
$64$	−512.000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		−	491.593i		−	0.876683i
$69$		0			0
$70$		0			0
$71$		−	1182.90i		−	1.97724i	−0.150437	−	0.988620i	$-0.548068\pi$
$71$		−	1182.90i		−	1.97724i	0.150437	−	0.988620i	$-0.451932\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1828.35	−	400.104i	2.81493	−	0.616000i
$76$	−322.248			−0.486374
$77$		0			0
$78$		0			0
$79$	−560.354			−0.798035			−0.399017	−	0.916943i	$-0.630649\pi$
$79$	−560.354			−0.798035			−0.399017	+	0.916943i	$0.630649\pi$
$80$			1409.73i			1.97016i
$81$	474.661	−	553.297i	0.651113	−	0.758981i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−1399.21	+	306.192i	−1.81745	+	0.397718i
$85$	−1353.55			−1.72721
$86$		0			0
$87$	302.269	+	1381.28i	0.372491	+	1.70217i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			887.274i			0.958236i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	2881.55			2.88155
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	843.066	+	3852.56i	0.783570	+	3.58068i
$106$		0			0
$107$		−	928.173i		−	0.838597i	−0.907848	−	0.419299i	$-0.862276\pi$
$107$		−	928.173i		−	0.838597i	0.907848	−	0.419299i	$-0.137724\pi$
$108$	896.000	−	675.941i	0.798311	−	0.602245i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−2205.20			−1.86046
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$			2176.94i			1.74245i
$117$		0			0
$118$		0			0
$119$		−	2117.30i		−	1.63103i
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$	−509.307	−	2327.38i	−0.373355	−	1.70612i
$124$		0			0
$125$		−	5180.63i		−	3.70696i
$126$		0			0
$127$	1964.29			1.37246			0.686230	−	0.727385i	$-0.259264\pi$
$127$	1964.29			1.37246			0.686230	+	0.727385i	$0.259264\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−1387.93			−0.904879
$134$		0			0
$135$	−1861.13	−	2467.04i	−1.18652	−	1.57280i
$136$		0			0
$137$		−	1178.26i		−	0.734784i	−0.930066	−	0.367392i	$-0.880251\pi$
$137$		−	1178.26i		−	0.734784i	0.930066	−	0.367392i	$-0.119749\pi$
$138$		0			0
$139$	−1960.00			−1.19601			−0.598004	−	0.801493i	$-0.704039\pi$
$139$	−1960.00			−1.19601			−0.598004	+	0.801493i	$0.704039\pi$
$140$			6071.76i			3.66541i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1570.06	−	721.724i	0.908601	−	0.417665i
$145$	5993.97			3.43291
$146$		0			0
$147$	−4285.34	+	937.773i	−2.40441	+	0.526165i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	692.959	+	1507.49i	0.366159	+	0.796555i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−702.785	−	3211.52i	−0.350531	−	1.60182i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4048.00			−1.94518			−0.972588	−	0.232533i	$-0.925299\pi$
$163$	−4048.00			−1.94518			−0.972588	+	0.232533i	$0.925299\pi$
$164$		−	3668.03i		−	1.74649i
$165$		0			0
$166$		0			0
$167$		−	719.468i		−	0.333378i	−0.986010	−	0.166689i	$-0.946692\pi$
$167$		−	719.468i		−	0.333378i	0.986010	−	0.166689i	$-0.0533076\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$	988.183	−	454.247i	0.441920	−	0.203141i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	12410.9			5.36100
$176$		0			0
$177$	503.402	+	2300.40i	0.213774	+	0.976883i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−1987.19	−	4322.99i	−0.822868	−	1.79009i
$181$	1659.12			0.681334			0.340667	−	0.940184i	$-0.389347\pi$
$181$	1659.12			0.681334			0.340667	+	0.940184i	$0.389347\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3859.09	−	2911.29i	1.48523	−	1.12045i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	2598.93	−	568.731i	0.976883	−	0.213774i
$193$	−5299.00			−1.97632			−0.988161	−	0.153422i	$-0.950971\pi$
$193$	−5299.00			−1.97632			−0.988161	+	0.153422i	$0.950971\pi$
$194$		0			0
$195$		0			0
$196$	−6753.84			−2.46131
$197$		−	4086.37i		−	1.47788i	−0.673773	−	0.738939i	$-0.735327\pi$
$197$		−	4086.37i		−	1.47788i	0.673773	−	0.738939i	$-0.264673\pi$
$198$		0			0
$199$	−1408.35			−0.501684			−0.250842	−	0.968028i	$-0.580707\pi$
$199$	−1408.35			−0.501684			−0.250842	+	0.968028i	$0.580707\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9376.16i			3.24176i
$204$	546.063	+	2495.34i	0.187412	+	0.856417i
$205$	−10099.5			−3.44088
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	5061.46i		−	1.63973i
$213$	1313.96	+	6004.42i	0.422682	+	1.93153i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	5852.00			1.75730			0.878652	−	0.477462i	$-0.158443\pi$
$223$	5852.00			1.75730			0.878652	+	0.477462i	$0.158443\pi$
$224$		0			0
$225$	−8836.35	+	4061.88i	−2.61818	+	1.20352i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	1635.74	−	357.954i	0.475130	−	0.103974i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$			3625.50i			1.00000i
$237$	2844.38	−	622.442i	0.779587	−	0.170599i
$238$		0			0
$239$		−	1586.24i		−	0.429311i	−0.976690	−	0.214655i	$-0.931137\pi$
$239$		−	1586.24i		−	0.429311i	0.976690	−	0.214655i	$-0.0688628\pi$
$240$	−1565.94	−	7155.86i	−0.421170	−	1.92462i
$241$	−1189.52			−0.317941			−0.158971	−	0.987283i	$-0.550817\pi$
$241$	−1189.52			−0.317941			−0.158971	+	0.987283i	$0.550817\pi$
$242$		0			0
$243$	−1794.79	+	3335.81i	−0.473811	+	0.880627i
$244$		0			0
$245$			18595.9i			4.84919i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			5420.99i			1.36323i	0.731712	+	0.681614i	$0.238722\pi$
$251$			5420.99i			1.36323i	−0.731712	+	0.681614i	$0.761278\pi$
$252$	6762.30	−	3108.48i	1.69042	−	0.777048i
$253$		0			0
$254$		0			0
$255$	6870.65	−	1503.52i	1.68728	−	0.369233i
$256$	4096.00			1.00000
$257$		−	4866.00i		−	1.18106i	−0.807015	−	0.590530i	$-0.798919\pi$
$257$		−	4866.00i		−	1.18106i	0.807015	−	0.590530i	$-0.201081\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3068.66	−	6675.67i	−0.727760	−	1.58319i
$262$		0			0
$263$			5283.40i			1.23874i	0.785099	+	0.619370i	$0.212612\pi$
$263$			5283.40i			1.23874i	−0.785099	+	0.619370i	$0.787388\pi$
$264$		0			0
$265$	−13936.2			−3.23053
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	6422.81			1.43970			0.719849	−	0.694131i	$-0.244211\pi$
$271$	6422.81			1.43970			0.719849	+	0.694131i	$0.244211\pi$
$272$			3932.75i			0.876683i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3597.65			0.780367			0.390183	−	0.920737i	$-0.372412\pi$
$277$	3597.65			0.780367			0.390183	+	0.920737i	$0.372412\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			8020.94i			1.70281i	0.524510	+	0.851404i	$0.324249\pi$
$281$			8020.94i			1.70281i	−0.524510	+	0.851404i	$0.675751\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			9463.17i			1.97724i
$285$	−985.586	−	4503.83i	−0.204846	−	0.936084i
$286$		0			0
$287$		−	15798.3i		−	3.24928i
$288$		0			0
$289$	1137.00			0.231427
$290$		0			0
$291$		0			0
$292$		0			0
$293$			4636.93i			0.924548i	0.886737	+	0.462274i	$0.152966\pi$
$293$			4636.93i			0.924548i	−0.886737	+	0.462274i	$0.847034\pi$
$294$		0			0
$295$	9982.41			1.97016
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−14626.8	+	3200.83i	−2.81493	+	0.616000i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	2577.98			0.486374
$305$		0			0
$306$		0			0
$307$	−6930.27			−1.28838			−0.644188	−	0.764867i	$-0.722805\pi$
$307$	−6930.27			−1.28838			−0.644188	+	0.764867i	$0.722805\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	1406.08i		−	0.256371i	−0.991750	−	0.128186i	$-0.959085\pi$
$311$		−	1406.08i		−	0.256371i	0.991750	−	0.128186i	$-0.0409154\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−8558.87	−	18619.2i	−1.53091	−	3.33040i
$316$	4482.83			0.798035
$317$		−	215.072i		−	0.0381062i	−0.999818	−	0.0190531i	$-0.993935\pi$
$317$		−	215.072i		−	0.0381062i	0.999818	−	0.0190531i	$-0.00606515\pi$
$318$		0			0
$319$		0			0
$320$		−	11277.9i		−	1.97016i
$321$	1031.02	+	4711.44i	0.179270	+	0.819211i
$322$		0			0
$323$			2475.23i			0.426395i
$324$	−3797.29	+	4426.38i	−0.651113	+	0.758981i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5310.03			−0.881770			−0.440885	−	0.897564i	$-0.645335\pi$
$331$	−5310.03			−0.881770			−0.440885	+	0.897564i	$0.645335\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	11193.7	−	2449.54i	1.81745	−	0.397718i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	10828.4			1.72721
$341$		0			0
$342$		0			0
$343$	−17270.5			−2.71871
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−2418.16	−	11050.2i	−0.372491	−	1.70217i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	26055.8			3.89549
$356$		0			0
$357$	2351.91	+	10747.5i	0.348673	+	1.59333i
$358$		0			0
$359$		−	13527.7i		−	1.98875i	−0.105895	−	0.994377i	$-0.533771\pi$
$359$		−	13527.7i		−	1.98875i	0.105895	−	0.994377i	$-0.466229\pi$
$360$		0			0
$361$	−5236.44			−0.763441
$362$		0			0
$363$	6756.20	−	1478.48i	0.976883	−	0.213774i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	5170.52	+	11248.1i	0.729449	+	1.58687i
$370$		0			0
$371$		−	21799.8i		−	3.05065i
$372$		0			0
$373$	4898.00			0.679916			0.339958	−	0.940441i	$-0.389587\pi$
$373$	4898.00			0.679916			0.339958	+	0.940441i	$0.389587\pi$
$374$		0			0
$375$	5754.66	+	26297.1i	0.792451	+	3.62126i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12519.3			−1.69677			−0.848383	−	0.529383i	$-0.822423\pi$
$379$	−12519.3			−1.69677			−0.848383	+	0.529383i	$0.822423\pi$
$380$		−	7098.19i		−	0.958236i
$381$	−9970.79	+	2181.94i	−1.34073	+	0.293396i
$382$		0			0
$383$			14025.8i			1.87124i	0.353014	+	0.935618i	$0.385157\pi$
$383$			14025.8i			1.87124i	−0.353014	+	0.935618i	$0.614843\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	11829.0i		−	1.54178i	−0.636968	−	0.770890i	$-0.719812\pi$
$389$		−	11829.0i		−	1.54178i	0.636968	−	0.770890i	$-0.280188\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12343.0i		−	1.57226i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	7045.18	−	1541.72i	0.883961	−	0.193440i
$400$	−23052.4			−2.88155
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	12187.5	+	10455.4i	1.49532	+	1.28280i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	1308.81	+	5980.88i	0.157078	+	0.717798i
$412$		0			0
$413$			15615.1i			1.86046i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	9949.03	−	2177.17i	1.16836	−	0.255675i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−6744.53	−	30820.5i	−0.783570	−	3.58068i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	22133.6i		−	2.52620i
$426$		0			0
$427$		0			0
$428$			7425.39i			0.838597i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−7168.00	+	5407.53i	−0.798311	+	0.602245i
$433$	−1808.48			−0.200716			−0.100358	−	0.994951i	$-0.531999\pi$
$433$	−1808.48			−0.200716			−0.100358	+	0.994951i	$0.531999\pi$
$434$		0			0
$435$	−30425.6	+	6658.12i	−3.35356	+	0.733868i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−3220.00			−0.350073			−0.175037	−	0.984562i	$-0.556004\pi$
$439$	−3220.00			−0.350073			−0.175037	+	0.984562i	$0.556004\pi$
$440$		0			0
$441$	20710.8	−	9520.33i	2.23635	−	1.02800i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	17641.6			1.86046
$449$		−	9618.49i		−	1.01097i	−0.862836	−	0.505484i	$-0.831314\pi$
$449$		−	9618.49i		−	1.01097i	0.862836	−	0.505484i	$-0.168686\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	−5192.00	−	6882.31i	−0.527978	−	0.699866i
$460$		0			0
$461$		−	13488.1i		−	1.36270i	−0.731959	−	0.681348i	$-0.761394\pi$
$461$		−	13488.1i		−	1.36270i	0.731959	−	0.681348i	$-0.238606\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		−	17415.6i		−	1.74245i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−14509.0			−1.40151
$476$			16938.4i			1.63103i
$477$	7134.72	+	15521.1i	0.684856	+	1.48986i
$478$		0			0
$479$		−	1459.42i		−	0.139212i	−0.997575	−	0.0696059i	$-0.977826\pi$
$479$		−	1459.42i		−	0.139212i	0.997575	−	0.0696059i	$-0.0221742\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10648.0			1.00000
$485$		0			0
$486$		0			0
$487$	−2456.75			−0.228595			−0.114298	−	0.993447i	$-0.536462\pi$
$487$	−2456.75			−0.228595			−0.114298	+	0.993447i	$0.536462\pi$
$488$		0			0
$489$	20547.8	−	4496.53i	1.90021	−	0.415828i
$490$		0			0
$491$			11786.6i			1.08335i	0.840589	+	0.541674i	$0.182209\pi$
$491$			11786.6i			1.08335i	−0.840589	+	0.541674i	$0.817791\pi$
$492$	4074.46	+	18619.1i	0.373355	+	1.70612i
$493$	16721.4			1.52758
$494$		0			0
$495$		0			0
$496$		0			0
$497$			40758.1i			3.67857i
$498$		0			0
$499$	22217.2			1.99314			0.996571	−	0.0827472i	$-0.0263694\pi$
$499$	22217.2			1.99314			0.996571	+	0.0827472i	$0.0263694\pi$
$500$			41445.0i			3.70696i
$501$	799.187	+	3652.04i	0.0712675	+	0.325671i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−11152.0	+	2440.43i	−0.976883	+	0.213774i
$508$	−15714.3			−1.37246
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−4511.47	+	3403.45i	−0.388278	+	0.292916i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	11552.4i		−	0.971443i	−0.874114	−	0.485721i	$-0.838557\pi$
$521$		−	11552.4i		−	0.971443i	0.874114	−	0.485721i	$-0.161443\pi$
$522$		0			0
$523$	−23099.6			−1.93131			−0.965654	−	0.259833i	$-0.916333\pi$
$523$	−23099.6			−1.93131			−0.965654	+	0.259833i	$0.916333\pi$
$524$		0			0
$525$	−62998.1	+	13786.1i	−5.23707	+	1.14604i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	−5110.57	−	11117.7i	−0.417665	−	0.908601i
$532$	11103.4			0.904879
$533$		0			0
$534$		0			0
$535$	20445.0			1.65217
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	14889.0	+	19736.3i	1.18652	+	1.57280i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−8421.75	+	1842.96i	−0.665584	+	0.145652i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12616.0			−0.986145			−0.493072	−	0.869988i	$-0.664126\pi$
$547$	−12616.0			−0.986145			−0.493072	+	0.869988i	$0.664126\pi$
$548$			9426.07i			0.734784i
$549$		0			0
$550$		0			0
$551$		−	10961.2i		−	0.847482i
$552$		0			0
$553$	19307.7			1.48471
$554$		0			0
$555$		0			0
$556$	15680.0			1.19601
$557$			17680.4i			1.34496i	0.740115	+	0.672481i	$0.234771\pi$
$557$			17680.4i			1.34496i	−0.740115	+	0.672481i	$0.765229\pi$
$558$		0			0
$559$		0			0
$560$		−	48574.1i		−	3.66541i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16355.0	+	19064.5i	−1.21137	+	1.41205i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−12560.5	+	5773.79i	−0.908601	+	0.417665i
$577$	−27315.1			−1.97078			−0.985390	−	0.170311i	$-0.945523\pi$
$577$	−27315.1			−1.97078			−0.985390	+	0.170311i	$0.945523\pi$
$578$		0			0
$579$	26897.9	−	5886.14i	1.93064	−	0.422486i
$580$	−47951.8			−3.43291
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	34282.7	−	7502.18i	2.40441	−	0.526165i
$589$		0			0
$590$		0			0
$591$	4539.15	+	20742.5i	0.315932	+	1.44371i
$592$		0			0
$593$		−	27698.9i		−	1.91814i	−0.283163	−	0.959072i	$-0.591384\pi$
$593$		−	27698.9i		−	1.91814i	0.283163	−	0.959072i	$-0.408616\pi$
$594$		0			0
$595$	46638.1			3.21340
$596$		0			0
$597$	7148.82	−	1564.40i	0.490086	−	0.107247i
$598$		0			0
$599$		−	26228.1i		−	1.78906i	−0.447004	−	0.894532i	$-0.647509\pi$
$599$		−	26228.1i		−	1.78906i	0.447004	−	0.894532i	$-0.352491\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	29318.1i		−	1.97016i
$606$		0			0
$607$	−26449.2			−1.76860			−0.884301	−	0.466917i	$-0.845365\pi$
$607$	−26449.2			−1.76860			−0.884301	+	0.466917i	$0.845365\pi$
$608$		0			0
$609$	−10415.1	−	47593.7i	−0.693004	−	3.16682i
$610$		0			0
$611$		0			0
$612$	−5543.67	−	12059.9i	−0.366159	−	0.796555i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	51265.5	−	11218.6i	3.36134	−	0.735571i
$616$		0			0
$617$			30635.6i			1.99893i	0.0326376	+	0.999467i	$0.489609\pi$
$617$			30635.6i			1.99893i	−0.0326376	+	0.999467i	$0.510391\pi$
$618$		0			0
$619$	−11430.9			−0.742238			−0.371119	−	0.928585i	$-0.621026\pi$
$619$	−11430.9			−0.742238			−0.371119	+	0.928585i	$0.621026\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	69090.1			4.42177
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−29860.0			−1.88385			−0.941924	−	0.335827i	$-0.890984\pi$
$631$	−29860.0			−1.88385			−0.941924	+	0.335827i	$0.890984\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			43267.6i			2.70397i
$636$	5622.28	+	25692.1i	0.350531	+	1.60182i
$637$		0			0
$638$		0			0
$639$	−13339.5	−	29019.1i	−0.825823	−	1.79652i
$640$		0			0
$641$			17205.8i			1.06020i	0.847936	+	0.530099i	$0.177845\pi$
$641$			17205.8i			1.06020i	−0.847936	+	0.530099i	$0.822155\pi$
$642$		0			0
$643$	−13556.0			−0.831410			−0.415705	−	0.909500i	$-0.636465\pi$
$643$	−13556.0			−0.831410			−0.415705	+	0.909500i	$0.636465\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	15558.0i		−	0.945362i	−0.881234	−	0.472681i	$-0.843286\pi$
$647$		−	15558.0i		−	0.945362i	0.881234	−	0.472681i	$-0.156714\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	32384.0			1.94518
$653$		−	6801.85i		−	0.407622i	−0.979010	−	0.203811i	$-0.934667\pi$
$653$		−	6801.85i		−	0.407622i	0.979010	−	0.203811i	$-0.0653328\pi$
$654$		0			0
$655$		0			0
$656$			29344.3i			1.74649i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−5355.58			−0.315141			−0.157570	−	0.987508i	$-0.550366\pi$
$661$	−5355.58			−0.315141			−0.157570	+	0.987508i	$0.550366\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	30572.1i		−	1.78276i
$666$		0			0
$667$		0			0
$668$			5755.74i			0.333378i
$669$	−29704.9	+	6500.42i	−1.71668	+	0.375666i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	40341.7	−	30433.7i	2.30037	−	1.73540i
$676$	−17576.0			−1.00000
$677$		−	33397.6i		−	1.89597i	−0.318308	−	0.947987i	$-0.603115\pi$
$677$		−	33397.6i		−	1.89597i	0.318308	−	0.947987i	$-0.396885\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−7905.47	+	3633.97i	−0.441920	+	0.203141i
$685$	25953.6			1.44765
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	43173.1i		−	2.35633i
$696$		0			0
$697$	−28174.7			−1.53112
$698$		0			0
$699$		0			0
$700$	−99287.2			−5.36100
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−4027.22	−	18403.2i	−0.213774	−	0.976883i
$709$	37498.6			1.98630			0.993152	−	0.116833i	$-0.0372743\pi$
$709$	37498.6			1.98630			0.993152	+	0.116833i	$0.0372743\pi$
$710$		0			0
$711$	−13746.7	+	6319.08i	−0.725095	+	0.333311i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1762.00	+	8051.80i	0.0917755	+	0.419386i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	15897.5	+	34583.9i	0.822868	+	1.79009i
$721$		0			0
$722$		0			0
$723$	6038.05	−	1321.32i	0.310591	−	0.0679676i
$724$	−13273.0			−0.681334
$725$			98015.1i			5.02095i
$726$		0			0
$727$	39116.0			1.99551			0.997753	−	0.0670071i	$-0.0213450\pi$
$727$	39116.0			1.99551			0.997753	+	0.0670071i	$0.0213450\pi$
$728$		0			0
$729$	5405.00	−	18926.3i	0.274602	−	0.961558i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9898.00			−0.498760			−0.249380	−	0.968406i	$-0.580227\pi$
$733$	−9898.00			−0.498760			−0.249380	+	0.968406i	$0.580227\pi$
$734$		0			0
$735$	−20656.4	−	94393.6i	−1.03663	−	4.73709i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			35594.4i			1.75751i	0.477269	+	0.878757i	$0.341627\pi$
$743$			35594.4i			1.75751i	−0.477269	+	0.878757i	$0.658373\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			31981.3i			1.56018i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−6021.65	−	27517.1i	−0.291423	−	1.33171i
$754$		0			0
$755$		0			0
$756$	−30872.8	+	23290.4i	−1.48523	+	1.12045i
$757$	41640.9			1.99929			0.999646	−	0.0265962i	$-0.00846682\pi$
$757$	41640.9			1.99929			0.999646	+	0.0265962i	$0.00846682\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	38033.2i		−	1.81170i	−0.423601	−	0.905849i	$-0.639234\pi$
$761$		−	38033.2i		−	1.81170i	0.423601	−	0.905849i	$-0.360766\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−33205.5	+	15263.9i	−1.56934	+	0.721394i
$766$		0			0
$767$		0			0
$768$	−20791.4	+	4549.85i	−0.976883	+	0.213774i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	5405.16	+	24700.0i	0.252480	+	1.15376i
$772$	42392.0			1.97632
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			18469.0i			0.849449i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	22992.0	+	30477.2i	1.04938	+	1.39102i
$784$	54030.7			2.46131
$785$		0			0
$786$		0			0
$787$	−42784.0			−1.93785			−0.968923	−	0.247362i	$-0.920436\pi$
$787$	−42784.0			−1.93785			−0.968923	+	0.247362i	$0.920436\pi$
$788$			32691.0i			1.47788i
$789$	−5868.82	−	26818.7i	−0.264810	−	1.21010i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	70740.4	−	15480.3i	3.15585	−	0.690604i
$796$	11266.8			0.501684
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +O(q^{10})\)	\(q+(-5.07603 + 1.11080i) q^{3} -8.00000 q^{4} +22.0271i q^{5} -34.4562 q^{7} +(24.5322 - 11.2769i) q^{9} +(40.6083 - 8.88642i) q^{12} +(-24.4678 - 111.810i) q^{15} +64.0000 q^{16} +61.4492i q^{17} +40.2810 q^{19} -176.217i q^{20} +(174.901 - 38.2740i) q^{21} -360.193 q^{25} +(-112.000 + 84.4926i) q^{27} +275.650 q^{28} -272.118i q^{29} -758.970i q^{35} +(-196.258 + 90.2155i) q^{36} +458.504i q^{41} +(248.398 + 540.374i) q^{45} +(-324.866 + 71.0914i) q^{48} +844.230 q^{49} +(-68.2579 - 311.918i) q^{51} +632.682i q^{53} +(-204.468 + 44.7442i) q^{57} -453.188i q^{59} +(195.742 + 894.483i) q^{60} +(-845.288 + 388.561i) q^{63} -512.000 q^{64} -491.593i q^{68} -1182.90i q^{71} +(1828.35 - 400.104i) q^{75} -322.248 q^{76} -560.354 q^{79} +1409.73i q^{80} +(474.661 - 553.297i) q^{81} +(-1399.21 + 306.192i) q^{84} -1353.55 q^{85} +(302.269 + 1381.28i) q^{87} +887.274i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + O(q^{10}) \)	\( 4q - 7q^{3} - 32q^{4} - 58q^{7} + 5q^{9} + 56q^{12} - 191q^{15} + 256q^{16} - 238q^{19} + 367q^{21} - 882q^{25} - 448q^{27} + 464q^{28} - 40q^{36} + 49q^{45} - 448q^{48} + 1062q^{49} + 472q^{51} - 911q^{57} + 1528q^{60} - 1931q^{63} - 2048q^{64} + 3402q^{75} + 1904q^{76} + 1670q^{79} + 1433q^{81} - 2936q^{84} - 944q^{85} + 637q^{87} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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