LMFDB - Embedded newform 177.4.d.b.176.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(176,177)]

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([1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("177.176");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.d (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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-59})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} - 15x + 225 $ x^4 - x^3 - 14x^2 - 15x + 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			176.1
Root			$3.57603 - 1.48727i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.4.d.b.176.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.555050\pi$
$5$		−	22.0271i		−	1.97016i	0.172085	−	0.985082i	$-0.444950\pi$
$6$		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	$-0.855566\pi$
$17$		−	61.4492i		−	0.876683i	0.898808	−	0.438342i	$-0.144434\pi$
$18$		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	$0.336674\pi$
$29$			272.118i			1.74245i	−0.490884	+	0.871225i	$0.663326\pi$
$30$		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	$-0.662010\pi$
$41$		−	458.504i		−	1.74649i	0.487277	−	0.873247i	$-0.337990\pi$
$42$		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	$-0.694049\pi$
$53$		−	632.682i		−	1.63973i	0.572558	−	0.819864i	$-0.305951\pi$
$54$		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	$0.451932\pi$
$71$			1182.90i			1.97724i	−0.150437	+	0.988620i	$0.548068\pi$
$72$		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	$0.137724\pi$
$107$			928.173i			0.838597i	−0.907848	+	0.419299i	$0.862276\pi$
$108$	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	$0.119749\pi$
$137$			1178.26i			0.734784i	−0.930066	+	0.367392i	$0.880251\pi$
$138$		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	$0.0533076\pi$
$167$			719.468i			0.333378i	−0.986010	+	0.166689i	$0.946692\pi$
$168$		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	$0.264673\pi$
$197$			4086.37i			1.47788i	−0.673773	+	0.738939i	$0.735327\pi$
$198$		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	$0.0688628\pi$
$239$			1586.24i			0.429311i	−0.976690	+	0.214655i	$0.931137\pi$
$240$	−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	$-0.761278\pi$
$251$		−	5420.99i		−	1.36323i	0.731712	−	0.681614i	$-0.238722\pi$
$252$	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	$0.201081\pi$
$257$			4866.00i			1.18106i	−0.807015	+	0.590530i	$0.798919\pi$
$258$		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	$-0.787388\pi$
$263$		−	5283.40i		−	1.23874i	0.785099	−	0.619370i	$-0.212612\pi$
$264$		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	$-0.675751\pi$
$281$		−	8020.94i		−	1.70281i	0.524510	−	0.851404i	$-0.324249\pi$
$282$		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	$-0.847034\pi$
$293$		−	4636.93i		−	0.924548i	0.886737	−	0.462274i	$-0.152966\pi$
$294$		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	$0.0409154\pi$
$311$			1406.08i			0.256371i	−0.991750	+	0.128186i	$0.959085\pi$
$312$		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	$0.00606515\pi$
$317$			215.072i			0.0381062i	−0.999818	+	0.0190531i	$0.993935\pi$
$318$		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	$0.466229\pi$
$359$			13527.7i			1.98875i	−0.105895	+	0.994377i	$0.533771\pi$
$360$		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	$-0.614843\pi$
$383$		−	14025.8i		−	1.87124i	0.353014	−	0.935618i	$-0.385157\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			11829.0i			1.54178i	0.636968	+	0.770890i	$0.280188\pi$
$389$			11829.0i			1.54178i	−0.636968	+	0.770890i	$0.719812\pi$
$390$		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	$0.168686\pi$
$449$			9618.49i			1.01097i	−0.862836	+	0.505484i	$0.831314\pi$
$450$		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	$0.238606\pi$
$461$			13488.1i			1.36270i	−0.731959	+	0.681348i	$0.761394\pi$
$462$		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	$0.0221742\pi$
$479$			1459.42i			0.139212i	−0.997575	+	0.0696059i	$0.977826\pi$
$480$		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	$-0.817791\pi$
$491$		−	11786.6i		−	1.08335i	0.840589	−	0.541674i	$-0.182209\pi$
$492$	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	$0.161443\pi$
$521$			11552.4i			0.971443i	−0.874114	+	0.485721i	$0.838557\pi$
$522$		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	$-0.765229\pi$
$557$		−	17680.4i		−	1.34496i	0.740115	−	0.672481i	$-0.234771\pi$
$558$		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	$0.408616\pi$
$593$			27698.9i			1.91814i	−0.283163	+	0.959072i	$0.591384\pi$
$594$		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	$0.352491\pi$
$599$			26228.1i			1.78906i	−0.447004	+	0.894532i	$0.647509\pi$
$600$		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	$-0.510391\pi$
$617$		−	30635.6i		−	1.99893i	0.0326376	−	0.999467i	$-0.489609\pi$
$618$		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	$-0.822155\pi$
$641$		−	17205.8i		−	1.06020i	0.847936	−	0.530099i	$-0.177845\pi$
$642$		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	$0.156714\pi$
$647$			15558.0i			0.945362i	−0.881234	+	0.472681i	$0.843286\pi$
$648$		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	$0.0653328\pi$
$653$			6801.85i			0.407622i	−0.979010	+	0.203811i	$0.934667\pi$
$654$		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	$0.396885\pi$
$677$			33397.6i			1.89597i	−0.318308	+	0.947987i	$0.603115\pi$
$678$		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	$-0.658373\pi$
$743$		−	35594.4i		−	1.75751i	0.477269	−	0.878757i	$-0.341627\pi$
$744$		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	$0.360766\pi$
$761$			38033.2i			1.81170i	−0.423601	+	0.905849i	$0.639234\pi$
$762$		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			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$			75009.2i			3.24176i
$813$	−32602.4	−	7134.48i	−1.40642	−	0.307770i
$814$		0			0
$815$			89165.7i			3.83232i
$816$	−4368.50	+	19962.8i	−0.187412	+	0.856417i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	80796.1			3.44088
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18603.7i			0.782243i	0.920339	+	0.391122i	$0.127913\pi$
$827$			18603.7i			0.782243i	−0.920339	+	0.391122i	$0.872087\pi$
$828$		0			0
$829$	−47472.1			−1.98887			−0.994436	−	0.105339i	$-0.966407\pi$
$829$	−47472.1			−1.98887			−0.994436	+	0.105339i	$0.966407\pi$
$830$		0			0
$831$	−18261.8	−	3996.27i	−0.762327	−	0.166822i
$832$		0			0
$833$		−	51877.2i		−	2.15779i
$834$		0			0
$835$	15847.8			0.656809
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−49659.3			−2.03613
$842$		0			0
$843$	−8909.68	+	40714.6i	−0.364016	+	1.66345i
$844$		0			0
$845$		−	48393.6i		−	1.97016i
$846$		0			0
$847$	45861.2			1.86046
$848$		−	40491.7i		−	1.63973i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−10511.7	+	48035.4i	−0.422682	+	1.93153i
$853$	−48375.5			−1.94179			−0.970895	−	0.239506i	$-0.923015\pi$
$853$	−48375.5			−1.94179			−0.970895	+	0.239506i	$0.923015\pi$
$854$		0			0
$855$	10005.7	−	21766.8i	0.400221	−	0.870654i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$	17548.8	−	80192.7i	0.694612	−	3.17417i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5771.45	−	1262.98i	−0.226077	−	0.0494730i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	178505.i		−	6.89665i
$876$		0			0
$877$	−35493.1			−1.36661			−0.683305	−	0.730133i	$-0.739458\pi$
$877$	−35493.1			−1.36661			−0.683305	+	0.730133i	$0.739458\pi$
$878$		0			0
$879$	−5150.71	+	23537.2i	−0.197644	+	0.903175i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	26098.8			0.994671			0.497336	−	0.867558i	$-0.334312\pi$
$883$	26098.8			0.994671			0.497336	+	0.867558i	$0.334312\pi$
$884$		0			0
$885$	−50671.1	−	11088.5i	−1.92462	−	0.421170i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−67681.9			−2.55341
$890$		0			0
$891$		0			0
$892$	−46816.0			−1.75730
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	70690.8	+	32495.0i	2.61818	+	1.20352i
$901$	−38877.8			−1.43752
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	36545.6i		−	1.34234i
$906$		0			0
$907$	−33289.5			−1.21870			−0.609350	−	0.792901i	$-0.708570\pi$
$907$	−33289.5			−1.21870			−0.609350	+	0.792901i	$0.708570\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	42496.6i		−	1.54553i	−0.634695	−	0.772763i	$-0.718874\pi$
$911$		−	42496.6i		−	1.54553i	0.634695	−	0.772763i	$-0.281126\pi$
$912$	−13085.9	−	2863.63i	−0.475130	−	0.103974i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	35178.3	+	7698.16i	1.25859	+	0.275421i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	34006.4			1.19712
$932$		0			0
$933$	1561.88	−	7137.31i	0.0548056	−	0.250445i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			29004.0i			1.00000i
$945$	64127.4	−	85004.7i	2.20748	−	2.92614i
$946$		0			0
$947$		−	10490.9i		−	0.359989i	−0.983668	−	0.179994i	$-0.942392\pi$
$947$		−	10490.9i		−	0.359989i	0.983668	−	0.179994i	$-0.0576080\pi$
$948$	−22755.0	−	4979.54i	−0.779587	−	0.170599i
$949$		0			0
$950$		0			0
$951$	238.903	−	1091.71i	0.00814611	−	0.0372253i
$952$		0			0
$953$			56348.9i			1.91534i	0.287868	+	0.957670i	$0.407054\pi$
$953$			56348.9i			1.91534i	−0.287868	+	0.957670i	$0.592946\pi$
$954$		0			0
$955$		0			0
$956$		−	12689.9i		−	0.429311i
$957$		0			0
$958$		0			0
$959$		−	40598.3i		−	1.36704i
$960$	12527.5	−	57246.9i	0.421170	−	1.92462i
$961$	29791.0			1.00000
$962$		0			0
$963$	−10467.0	+	22770.2i	−0.350252	+	0.761951i
$964$	9516.17			0.317941
$965$			116722.i			3.89368i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−2749.50	+	12564.4i	−0.0911523	+	0.416539i
$970$		0			0
$971$		−	11222.9i		−	0.370916i	−0.982652	−	0.185458i	$-0.940623\pi$
$971$		−	11222.9i		−	0.370916i	0.982652	−	0.185458i	$-0.0593769\pi$
$972$	14358.3	+	26686.5i	0.473811	+	0.880627i
$973$	67534.2			2.22512
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$			148768.i			4.84919i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	90010.9			2.91166
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	26953.9	+	5898.40i	0.861386	+	0.188499i
$994$		0			0
$995$			31021.8i			0.988399i
$996$		0			0
$997$	−20771.8			−0.659829			−0.329914	−	0.944011i	$-0.607020\pi$
$997$	−20771.8			−0.659829			−0.329914	+	0.944011i	$0.607020\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.d.b.176.1	✓	4
3.2	odd	2	inner	177.4.d.b.176.2	yes	4
59.58	odd	2	CM	177.4.d.b.176.1	✓	4
177.176	even	2	inner	177.4.d.b.176.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.d.b.176.1	✓	4	1.1	even	1	trivial
177.4.d.b.176.1	✓	4	59.58	odd	2	CM
177.4.d.b.176.2	yes	4	3.2	odd	2	inner
177.4.d.b.176.2	yes	4	177.176	even	2	inner

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)

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

Introduction

L-functions

Modular forms

Varieties

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Database

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