LMFDB - Embedded newform 315.2.g.a.314.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(314,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("315.314");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.g (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:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 32x^{14} + 324x^{12} + 1328x^{10} + 2314x^{8} + 1920x^{6} + 780x^{4} + 144x^{2} + 9 $ x^16 + 32x^14 + 324x^12 + 1328x^10 + 2314x^8 + 1920x^6 + 780x^4 + 144*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			314.10
Root			$1.06294i$ of defining polynomial
Character	$\chi$	$=$	315.314
Dual form			315.2.g.a.314.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.741964 q^{2} -1.44949 q^{4} +(-2.05542 + 0.880486i) q^{5} +(-1.24519 - 2.33441i) q^{7} -2.55940 q^{8} +(-1.52505 + 0.653289i) q^{10} -1.41421i q^{11} -5.54048 q^{13} +(-0.923889 - 1.73205i) q^{14} +1.00000 q^{16} +6.07445i q^{17} -7.12018i q^{19} +(2.97931 - 1.27626i) q^{20} -1.04930i q^{22} -4.78529 q^{23} +(3.44949 - 3.61953i) q^{25} -4.11084 q^{26} +(1.80490 + 3.38371i) q^{28} +5.51399i q^{29} +1.30658i q^{31} +5.86076 q^{32} +4.50702i q^{34} +(4.61481 + 3.70182i) q^{35} -2.57024i q^{37} -5.28291i q^{38} +(5.26063 - 2.25351i) q^{40} +5.95862 q^{41} +6.76742i q^{43} +2.04989i q^{44} -3.55051 q^{46} -7.83542i q^{47} +(-3.89898 + 5.81360i) q^{49} +(2.55940 - 2.68556i) q^{50} +8.03087 q^{52} -9.90408 q^{53} +(1.24519 + 2.90680i) q^{55} +(3.18695 + 5.97469i) q^{56} +4.09118i q^{58} -1.84778 q^{59} -11.6272i q^{61} +0.969433i q^{62} +2.34847 q^{64} +(11.3880 - 4.87832i) q^{65} +7.23907i q^{67} -8.80486i q^{68} +(3.42403 + 2.74662i) q^{70} -8.34242i q^{71} +0.559702 q^{73} -1.90702i q^{74} +10.3206i q^{76} +(-3.30136 + 1.76097i) q^{77} -4.00000 q^{79} +(-2.05542 + 0.880486i) q^{80} +4.42108 q^{82} -7.83542i q^{83} +(-5.34847 - 12.4855i) q^{85} +5.02118i q^{86} +3.61953i q^{88} -12.3325 q^{89} +(6.89898 + 12.9338i) q^{91} +6.93623 q^{92} -5.81360i q^{94} +(6.26922 + 14.6349i) q^{95} -5.54048 q^{97} +(-2.89290 + 4.31348i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4} + 16 q^{16} + 16 q^{25} - 96 q^{46} + 16 q^{49} - 80 q^{64} - 48 q^{70} - 64 q^{79} + 32 q^{85} + 32 q^{91}+O(q^{100}) $ 16 * q + 16 * q^4 + 16 * q^16 + 16 * q^25 - 96 * q^46 + 16 * q^49 - 80 * q^64 - 48 * q^70 - 64 * q^79 + 32 * q^85 + 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0.741964			0.524648			0.262324	−	0.964980i	$-0.415511\pi$
$2$	0.741964			0.524648			0.262324	+	0.964980i	$0.415511\pi$
$3$		0			0
$3$		0			0
$4$	−1.44949			−0.724745
$5$	−2.05542	+	0.880486i	−0.919211	+	0.393765i
$5$	−2.05542	+	0.880486i	−0.919211	+	0.393765i
$6$		0			0
$7$	−1.24519	−	2.33441i	−0.470639	−	0.882326i
$7$	−1.24519	−	2.33441i	−0.470639	−	0.882326i
$8$	−2.55940			−0.904883
$9$		0			0
$10$	−1.52505	+	0.653289i	−0.482262	+	0.206588i
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$	−5.54048			−1.53665			−0.768327	−	0.640058i	$-0.778910\pi$
$13$	−5.54048			−1.53665			−0.768327	+	0.640058i	$0.778910\pi$
$14$	−0.923889	−	1.73205i	−0.246920	−	0.462910i
$15$		0			0
$16$	1.00000			0.250000
$17$			6.07445i			1.47327i	0.676290	+	0.736636i	$0.263587\pi$
$17$			6.07445i			1.47327i	−0.676290	+	0.736636i	$0.736413\pi$
$18$		0			0
$19$		−	7.12018i		−	1.63348i	−0.577005	−	0.816740i	$-0.695779\pi$
$19$		−	7.12018i		−	1.63348i	0.577005	−	0.816740i	$-0.304221\pi$
$20$	2.97931	−	1.27626i	0.666194	−	0.285379i
$21$		0			0
$22$		−	1.04930i		−	0.223710i
$23$	−4.78529			−0.997801			−0.498901	−	0.866659i	$-0.666263\pi$
$23$	−4.78529			−0.997801			−0.498901	+	0.866659i	$0.666263\pi$
$24$		0			0
$25$	3.44949	−	3.61953i	0.689898	−	0.723907i
$26$	−4.11084			−0.806201
$27$		0			0
$28$	1.80490	+	3.38371i	0.341094	+	0.639461i
$29$			5.51399i			1.02392i	0.859009	+	0.511961i	$0.171081\pi$
$29$			5.51399i			1.02392i	−0.859009	+	0.511961i	$0.828919\pi$
$30$		0			0
$31$			1.30658i			0.234668i	0.993092	+	0.117334i	$0.0374348\pi$
$31$			1.30658i			0.234668i	−0.993092	+	0.117334i	$0.962565\pi$
$32$	5.86076			1.03605
$33$		0			0
$34$			4.50702i			0.772948i
$35$	4.61481	+	3.70182i	0.780046	+	0.625722i
$36$		0			0
$37$		−	2.57024i		−	0.422545i	−0.977427	−	0.211272i	$-0.932239\pi$
$37$		−	2.57024i		−	0.422545i	0.977427	−	0.211272i	$-0.0677607\pi$
$38$		−	5.28291i		−	0.857002i
$39$		0			0
$40$	5.26063	−	2.25351i	0.831779	−	0.356312i
$41$	5.95862			0.930579			0.465290	−	0.885158i	$-0.345950\pi$
$41$	5.95862			0.930579			0.465290	+	0.885158i	$0.345950\pi$
$42$		0			0
$43$			6.76742i			1.03202i	0.856582	+	0.516011i	$0.172584\pi$
$43$			6.76742i			1.03202i	−0.856582	+	0.516011i	$0.827416\pi$
$44$			2.04989i			0.309032i
$45$		0			0
$46$	−3.55051			−0.523494
$47$		−	7.83542i		−	1.14291i	−0.820632	−	0.571457i	$-0.806378\pi$
$47$		−	7.83542i		−	1.14291i	0.820632	−	0.571457i	$-0.193622\pi$
$48$		0			0
$49$	−3.89898	+	5.81360i	−0.556997	+	0.830514i
$50$	2.55940	−	2.68556i	0.361953	−	0.379796i
$51$		0			0
$52$	8.03087			1.11368
$53$	−9.90408			−1.36043			−0.680215	−	0.733013i	$-0.738114\pi$
$53$	−9.90408			−1.36043			−0.680215	+	0.733013i	$0.738114\pi$
$54$		0			0
$55$	1.24519	+	2.90680i	0.167902	+	0.391953i
$56$	3.18695	+	5.97469i	0.425874	+	0.798402i
$57$		0			0
$58$			4.09118i			0.537198i
$59$	−1.84778			−0.240560			−0.120280	−	0.992740i	$-0.538379\pi$
$59$	−1.84778			−0.240560			−0.120280	+	0.992740i	$0.538379\pi$
$60$		0			0
$61$		−	11.6272i		−	1.48871i	−0.667784	−	0.744355i	$-0.732757\pi$
$61$		−	11.6272i		−	1.48871i	0.667784	−	0.744355i	$-0.267243\pi$
$62$			0.969433i			0.123118i
$63$		0			0
$64$	2.34847			0.293559
$65$	11.3880	−	4.87832i	1.41251	−	0.605081i
$66$		0			0
$67$			7.23907i			0.884393i	0.896918	+	0.442196i	$0.145801\pi$
$67$			7.23907i			0.884393i	−0.896918	+	0.442196i	$0.854199\pi$
$68$		−	8.80486i		−	1.06775i
$69$		0			0
$70$	3.42403	+	2.74662i	0.409249	+	0.328284i
$71$		−	8.34242i		−	0.990063i	−0.868875	−	0.495031i	$-0.835157\pi$
$71$		−	8.34242i		−	0.990063i	0.868875	−	0.495031i	$-0.164843\pi$
$72$		0			0
$73$	0.559702			0.0655082			0.0327541	−	0.999463i	$-0.489572\pi$
$73$	0.559702			0.0655082			0.0327541	+	0.999463i	$0.489572\pi$
$74$		−	1.90702i		−	0.221687i
$75$		0			0
$76$			10.3206i			1.18386i
$77$	−3.30136	+	1.76097i	−0.376225	+	0.200681i
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.05542	+	0.880486i	−0.229803	+	0.0984413i
$81$		0			0
$82$	4.42108			0.488226
$83$		−	7.83542i		−	0.860050i	−0.902817	−	0.430025i	$-0.858505\pi$
$83$		−	7.83542i		−	0.860050i	0.902817	−	0.430025i	$-0.141495\pi$
$84$		0			0
$85$	−5.34847	−	12.4855i	−0.580123	−	1.35425i
$86$			5.02118i			0.541448i
$87$		0			0
$88$			3.61953i			0.385844i
$89$	−12.3325			−1.30724			−0.653622	−	0.756821i	$-0.726751\pi$
$89$	−12.3325			−1.30724			−0.653622	+	0.756821i	$0.726751\pi$
$90$		0			0
$91$	6.89898	+	12.9338i	0.723210	+	1.35583i
$92$	6.93623			0.723152
$93$		0			0
$94$		−	5.81360i		−	0.599627i
$95$	6.26922	+	14.6349i	0.643208	+	1.50151i
$96$		0			0
$97$	−5.54048			−0.562551			−0.281275	−	0.959627i	$-0.590757\pi$
$97$	−5.54048			−0.562551			−0.281275	+	0.959627i	$0.590757\pi$
$98$	−2.89290	+	4.31348i	−0.292227	+	0.435727i
$99$		0			0
$100$	−5.00000	+	5.24648i	−0.500000	+	0.524648i
$101$	−5.95862			−0.592904			−0.296452	−	0.955048i	$-0.595804\pi$
$101$	−5.95862			−0.592904			−0.296452	+	0.955048i	$0.595804\pi$
$102$		0			0
$103$	4.98078			0.490771			0.245385	−	0.969426i	$-0.421085\pi$
$103$	4.98078			0.490771			0.245385	+	0.969426i	$0.421085\pi$
$104$	14.1803			1.39049
$105$		0			0
$106$	−7.34847			−0.713746
$107$	−1.81743			−0.175698			−0.0878489	−	0.996134i	$-0.527999\pi$
$107$	−1.81743			−0.175698			−0.0878489	+	0.996134i	$0.527999\pi$
$108$		0			0
$109$	3.10102			0.297024			0.148512	−	0.988911i	$-0.452552\pi$
$109$	3.10102			0.297024			0.148512	+	0.988911i	$0.452552\pi$
$110$	0.923889	+	2.15674i	0.0880894	+	0.205637i
$111$		0			0
$112$	−1.24519	−	2.33441i	−0.117660	−	0.220581i
$113$	−0.333505			−0.0313735			−0.0156868	−	0.999877i	$-0.504993\pi$
$113$	−0.333505			−0.0313735			−0.0156868	+	0.999877i	$0.504993\pi$
$114$		0			0
$115$	9.83577	−	4.21338i	0.917190	−	0.392899i
$116$		−	7.99247i		−	0.742082i
$117$		0			0
$118$	−1.37099			−0.126209
$119$	14.1803	−	7.56388i	1.29991	−	0.693380i
$120$		0			0
$121$	9.00000			0.818182
$122$		−	8.62696i		−	0.781049i
$123$		0			0
$124$		−	1.89387i		−	0.170075i
$125$	−3.90320	+	10.4769i	−0.349113	+	0.937081i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−9.97903			−0.882030
$129$		0			0
$130$	8.44949	−	3.61953i	0.741069	−	0.317454i
$131$	−6.37389			−0.556890			−0.278445	−	0.960452i	$-0.589819\pi$
$131$	−6.37389			−0.556890			−0.278445	+	0.960452i	$0.589819\pi$
$132$		0			0
$133$	−16.6214	+	8.86601i	−1.44126	+	0.768781i
$134$			5.37113i			0.463995i
$135$		0			0
$136$		−	15.5469i		−	1.33314i
$137$	−3.30136			−0.282054			−0.141027	−	0.990006i	$-0.545040\pi$
$137$	−3.30136			−0.282054			−0.141027	+	0.990006i	$0.545040\pi$
$138$		0			0
$139$			1.30658i			0.110822i	0.998464	+	0.0554112i	$0.0176470\pi$
$139$			1.30658i			0.110822i	−0.998464	+	0.0554112i	$0.982353\pi$
$140$	−6.68913	−	5.36575i	−0.565334	−	0.453489i
$141$		0			0
$142$		−	6.18977i		−	0.519434i
$143$			7.83542i			0.655231i
$144$		0			0
$145$	−4.85499	−	11.3336i	−0.403185	−	0.941201i
$146$	0.415279			0.0343687
$147$		0			0
$148$			3.72553i			0.306237i
$149$		−	9.89949i		−	0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$		−	9.89949i		−	0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$		0			0
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$			18.2234i			1.47811i
$153$		0			0
$154$	−2.44949	+	1.30658i	−0.197386	+	0.105287i
$155$	−1.15042	−	2.68556i	−0.0924042	−	0.215710i
$156$		0			0
$157$	−13.0117			−1.03844			−0.519221	−	0.854640i	$-0.673778\pi$
$157$	−13.0117			−1.03844			−0.519221	+	0.854640i	$0.673778\pi$
$158$	−2.96786			−0.236110
$159$		0			0
$160$	−12.0463	+	5.16031i	−0.952344	+	0.407959i
$161$	5.95862	+	11.1708i	0.469605	+	0.880386i
$162$		0			0
$163$		−	17.7320i		−	1.38888i	−0.719551	−	0.694439i	$-0.755652\pi$
$163$		−	17.7320i		−	1.38888i	0.719551	−	0.694439i	$-0.244348\pi$
$164$	−8.63695			−0.674433
$165$		0			0
$166$		−	5.81360i		−	0.451223i
$167$			5.10502i			0.395038i	0.980299	+	0.197519i	$0.0632885\pi$
$167$			5.10502i			0.395038i	−0.980299	+	0.197519i	$0.936712\pi$
$168$		0			0
$169$	17.6969			1.36130
$170$	−3.96837	−	9.26382i	−0.304360	−	0.710503i
$171$		0			0
$172$		−	9.80930i		−	0.747952i
$173$			1.76097i			0.133884i	0.997757	+	0.0669421i	$0.0213243\pi$
$173$			1.76097i			0.133884i	−0.997757	+	0.0669421i	$0.978676\pi$
$174$		0			0
$175$	−12.7448	−	3.54551i	−0.963415	−	0.268016i
$176$		−	1.41421i		−	0.106600i
$177$		0			0
$178$	−9.15028			−0.685842
$179$			0.142865i			0.0106782i	0.999986	+	0.00533910i	$0.00169950\pi$
$179$			0.142865i			0.0106782i	−0.999986	+	0.00533910i	$0.998301\pi$
$180$		0			0
$181$		−	3.20045i		−	0.237887i	−0.992901	−	0.118944i	$-0.962049\pi$
$181$		−	3.20045i		−	0.237887i	0.992901	−	0.118944i	$-0.0379508\pi$
$182$	5.11879	+	9.59640i	0.379430	+	0.711332i
$183$		0			0
$184$	12.2474			0.902894
$185$	2.26306	+	5.28291i	0.166383	+	0.388408i
$186$		0			0
$187$	8.59057			0.628205
$188$			11.3574i			0.828321i
$189$		0			0
$190$	4.65153	+	10.8586i	0.337458	+	0.787766i
$191$			22.4846i			1.62693i	0.581617	+	0.813463i	$0.302420\pi$
$191$			22.4846i			1.62693i	−0.581617	+	0.813463i	$0.697580\pi$
$192$		0			0
$193$		−	6.76742i		−	0.487129i	−0.969885	−	0.243565i	$-0.921683\pi$
$193$		−	6.76742i		−	0.487129i	0.969885	−	0.243565i	$-0.0783168\pi$
$194$	−4.11084			−0.295141
$195$		0			0
$196$	5.65153	−	8.42676i	0.403681	−	0.601911i
$197$	−13.5389			−0.964610			−0.482305	−	0.876003i	$-0.660200\pi$
$197$	−13.5389			−0.964610			−0.482305	+	0.876003i	$0.660200\pi$
$198$		0			0
$199$			7.12018i			0.504736i	0.967631	+	0.252368i	$0.0812094\pi$
$199$			7.12018i			0.504736i	−0.967631	+	0.252368i	$0.918791\pi$
$200$	−8.82861	+	9.26382i	−0.624277	+	0.655051i
$201$		0			0
$202$	−4.42108			−0.311066
$203$	12.8719	−	6.86599i	0.903433	−	0.481898i
$204$		0			0
$205$	−12.2474	+	5.24648i	−0.855399	+	0.366430i
$206$	3.69556			0.257482
$207$		0			0
$208$	−5.54048			−0.384163
$209$	−10.0695			−0.696519
$210$		0			0
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$	14.3559			0.985965
$213$		0			0
$214$	−1.34847			−0.0921795
$215$	−5.95862	−	13.9099i	−0.406374	−	0.948646i
$216$		0			0
$217$	3.05009	−	1.62694i	0.207054	−	0.110444i
$218$	2.30084			0.155833
$219$		0			0
$220$	−1.80490	−	4.21338i	−0.121686	−	0.284066i
$221$		−	33.6554i		−	2.26391i
$222$		0			0
$223$	18.5521			1.24234			0.621171	−	0.783675i	$-0.286657\pi$
$223$	18.5521			1.24234			0.621171	+	0.783675i	$0.286657\pi$
$224$	−7.29778	−	13.6814i	−0.487604	−	0.914129i
$225$		0			0
$226$	−0.247449			−0.0164600
$227$		−	18.4013i		−	1.22133i	−0.791887	−	0.610667i	$-0.790901\pi$
$227$		−	18.4013i		−	1.22133i	0.791887	−	0.610667i	$-0.209099\pi$
$228$		0			0
$229$		−	5.81360i		−	0.384174i	−0.981378	−	0.192087i	$-0.938474\pi$
$229$		−	5.81360i		−	0.384174i	0.981378	−	0.192087i	$-0.0615255\pi$
$230$	7.29778	−	3.12617i	0.481202	−	0.206134i
$231$		0			0
$232$		−	14.1125i		−	0.926530i
$233$	−2.63435			−0.172582			−0.0862910	−	0.996270i	$-0.527501\pi$
$233$	−2.63435			−0.172582			−0.0862910	+	0.996270i	$0.527501\pi$
$234$		0			0
$235$	6.89898	+	16.1051i	0.450040	+	1.05058i
$236$	2.67834			0.174345
$237$		0			0
$238$	10.5213	−	5.61212i	0.681992	−	0.363780i
$239$			13.9993i			0.905538i	0.891628	+	0.452769i	$0.149564\pi$
$239$			13.9993i			0.905538i	−0.891628	+	0.452769i	$0.850436\pi$
$240$		0			0
$241$			20.0540i			1.29179i	0.763427	+	0.645894i	$0.223515\pi$
$241$			20.0540i			1.29179i	−0.763427	+	0.645894i	$0.776485\pi$
$242$	6.67767			0.429257
$243$		0			0
$244$			16.8535i			1.07894i
$245$	2.89524	−	15.3824i	0.184970	−	0.982744i
$246$		0			0
$247$			39.4492i			2.51009i
$248$		−	3.34405i		−	0.212347i
$249$		0			0
$250$	−2.89603	+	7.77347i	−0.183161	+	0.491637i
$251$	−10.0695			−0.635578			−0.317789	−	0.948161i	$-0.602940\pi$
$251$	−10.0695			−0.635578			−0.317789	+	0.948161i	$0.602940\pi$
$252$		0			0
$253$			6.76742i			0.425464i
$254$		0			0
$255$		0			0
$256$	−12.1010			−0.756314
$257$		−	13.1183i		−	0.818299i	−0.912467	−	0.409150i	$-0.865825\pi$
$257$		−	13.1183i		−	0.818299i	0.912467	−	0.409150i	$-0.134175\pi$
$258$		0			0
$259$	−6.00000	+	3.20045i	−0.372822	+	0.198866i
$260$	−16.5068	+	7.07107i	−1.02371	+	0.438529i
$261$		0			0
$262$	−4.72920			−0.292171
$263$	28.2283			1.74063			0.870316	−	0.492493i	$-0.163914\pi$
$263$	28.2283			1.74063			0.870316	+	0.492493i	$0.163914\pi$
$264$		0			0
$265$	20.3570	−	8.72040i	1.25052	−	0.535690i
$266$	−12.3325	+	6.57826i	−0.756155	+	0.403339i
$267$		0			0
$268$		−	10.4930i		−	0.640959i
$269$	22.4020			1.36587			0.682936	−	0.730478i	$-0.260703\pi$
$269$	22.4020			1.36587			0.682936	+	0.730478i	$0.260703\pi$
$270$		0			0
$271$			10.3206i			0.626933i	0.949599	+	0.313467i	$0.101490\pi$
$271$			10.3206i			0.626933i	−0.949599	+	0.313467i	$0.898510\pi$
$272$			6.07445i			0.368318i
$273$		0			0
$274$	−2.44949			−0.147979
$275$	−5.11879	−	4.87832i	−0.308675	−	0.294173i
$276$		0			0
$277$		−	7.23907i		−	0.434953i	−0.976066	−	0.217477i	$-0.930217\pi$
$277$		−	7.23907i		−	0.434953i	0.976066	−	0.217477i	$-0.0697826\pi$
$278$			0.969433i			0.0581427i
$279$		0			0
$280$	−11.8111	−	9.47443i	−0.705851	−	0.566205i
$281$		−	1.41421i		−	0.0843649i	−0.999110	−	0.0421825i	$-0.986569\pi$
$281$		−	1.41421i		−	0.0843649i	0.999110	−	0.0421825i	$-0.0134311\pi$
$282$		0			0
$283$	18.5521			1.10281			0.551405	−	0.834238i	$-0.314092\pi$
$283$	18.5521			1.10281			0.551405	+	0.834238i	$0.314092\pi$
$284$			12.0922i			0.717543i
$285$		0			0
$286$			5.81360i			0.343765i
$287$	−7.41964	−	13.9099i	−0.437967	−	0.821074i
$288$		0			0
$289$	−19.8990			−1.17053
$290$	−3.60223	−	8.40909i	−0.211530	−	0.493799i
$291$		0			0
$292$	−0.811283			−0.0474767
$293$			14.7014i			0.858866i	0.903099	+	0.429433i	$0.141287\pi$
$293$			14.7014i			0.858866i	−0.903099	+	0.429433i	$0.858713\pi$
$294$		0			0
$295$	3.79796	−	1.62694i	0.221126	−	0.0947243i
$296$			6.57826i			0.382353i
$297$		0			0
$298$		−	7.34507i		−	0.425488i
$299$	26.5128			1.53327
$300$		0			0
$301$	15.7980	−	8.42676i	0.910579	−	0.485710i
$302$	−7.41964			−0.426952
$303$		0			0
$304$		−	7.12018i		−	0.408370i
$305$	10.2376	+	23.8988i	0.586202	+	1.36844i
$306$		0			0
$307$	24.6523			1.40698			0.703491	−	0.710704i	$-0.251624\pi$
$307$	24.6523			1.40698			0.703491	+	0.710704i	$0.251624\pi$
$308$	4.78529	−	2.55251i	0.272667	−	0.145443i
$309$		0			0
$310$	−0.853572	−	1.99259i	−0.0484796	−	0.113172i
$311$	18.2911			1.03719			0.518597	−	0.855019i	$-0.326454\pi$
$311$	18.2911			1.03719			0.518597	+	0.855019i	$0.326454\pi$
$312$		0			0
$313$	14.1311			0.798734			0.399367	−	0.916791i	$-0.369230\pi$
$313$	14.1311			0.798734			0.399367	+	0.916791i	$0.369230\pi$
$314$	−9.65417			−0.544817
$315$		0			0
$316$	5.79796			0.326161
$317$	−3.30136			−0.185423			−0.0927114	−	0.995693i	$-0.529553\pi$
$317$	−3.30136			−0.185423			−0.0927114	+	0.995693i	$0.529553\pi$
$318$		0			0
$319$	7.79796			0.436602
$320$	−4.82709	+	2.06779i	−0.269842	+	0.115593i
$321$		0			0
$322$	4.42108	+	8.28836i	0.246377	+	0.461892i
$323$	43.2512			2.40656
$324$		0			0
$325$	−19.1118	+	20.0540i	−1.06013	+	1.11239i
$326$		−	13.1565i		−	0.728672i
$327$		0			0
$328$	−15.2505			−0.842066
$329$	−18.2911	+	9.75663i	−1.00842	+	0.537900i
$330$		0			0
$331$	31.3939			1.72556			0.862782	−	0.505577i	$-0.168720\pi$
$331$	31.3939			1.72556			0.862782	+	0.505577i	$0.168720\pi$
$332$			11.3574i			0.623317i
$333$		0			0
$334$			3.78774i			0.207256i
$335$	−6.37389	−	14.8793i	−0.348243	−	0.812944i
$336$		0			0
$337$			22.4008i			1.22025i	0.792304	+	0.610126i	$0.208881\pi$
$337$			22.4008i			1.22025i	−0.792304	+	0.610126i	$0.791119\pi$
$338$	13.1305			0.714204
$339$		0			0
$340$	7.75255	+	18.0977i	0.420441	+	0.981484i
$341$	1.84778			0.100063
$342$		0			0
$343$	18.4263	+	1.86277i	0.994929	+	0.100580i
$344$		−	17.3205i		−	0.933859i
$345$		0			0
$346$			1.30658i			0.0702420i
$347$	−37.1319			−1.99334			−0.996672	−	0.0815179i	$-0.974023\pi$
$347$	−37.1319			−1.99334			−0.996672	+	0.0815179i	$0.974023\pi$
$348$		0			0
$349$		−	25.8676i		−	1.38466i	−0.721582	−	0.692329i	$-0.756585\pi$
$349$		−	25.8676i		−	1.38466i	0.721582	−	0.692329i	$-0.243415\pi$
$350$	−9.45616	−	2.63064i	−0.505453	−	0.140614i
$351$		0			0
$352$		−	8.28836i		−	0.441771i
$353$			1.76097i			0.0937271i	0.998901	+	0.0468635i	$0.0149226\pi$
$353$			1.76097i			0.0937271i	−0.998901	+	0.0468635i	$0.985077\pi$
$354$		0			0
$355$	7.34538	+	17.1472i	0.389852	+	0.910077i
$356$	17.8758			0.947418
$357$		0			0
$358$			0.106000i			0.00560229i
$359$		−	9.89949i		−	0.522475i	−0.965275	−	0.261238i	$-0.915869\pi$
$359$		−	9.89949i		−	0.522475i	0.965275	−	0.261238i	$-0.0841306\pi$
$360$		0			0
$361$	−31.6969			−1.66826
$362$		−	2.37462i		−	0.124807i
$363$		0			0
$364$	−10.0000	−	18.7474i	−0.524142	−	0.982630i
$365$	−1.15042	+	0.492810i	−0.0602159	+	0.0257948i
$366$		0			0
$367$	3.60979			0.188430			0.0942149	−	0.995552i	$-0.469966\pi$
$367$	3.60979			0.188430			0.0942149	+	0.995552i	$0.469966\pi$
$368$	−4.78529			−0.249450
$369$		0			0
$370$	1.67911	+	3.91973i	0.0872926	+	0.203777i
$371$	12.3325	+	23.1202i	0.640272	+	1.20034i
$372$		0			0
$373$		−	32.2102i		−	1.66778i	−0.551932	−	0.833889i	$-0.686109\pi$
$373$		−	32.2102i		−	1.66778i	0.551932	−	0.833889i	$-0.313891\pi$
$374$	6.37389			0.329586
$375$		0			0
$376$			20.0540i			1.03420i
$377$		−	30.5502i		−	1.57341i
$378$		0			0
$379$	−17.5959			−0.903842			−0.451921	−	0.892058i	$-0.649261\pi$
$379$	−17.5959			−0.903842			−0.451921	+	0.892058i	$0.649261\pi$
$380$	−9.08716	−	21.2132i	−0.466162	−	1.08821i
$381$		0			0
$382$			16.6827i			0.853562i
$383$			26.2367i			1.34063i	0.742076	+	0.670316i	$0.233841\pi$
$383$			26.2367i			1.34063i	−0.742076	+	0.670316i	$0.766159\pi$
$384$		0			0
$385$	5.23517	−	6.52633i	0.266809	−	0.332613i
$386$		−	5.02118i		−	0.255571i
$387$		0			0
$388$	8.03087			0.407706
$389$			19.3704i			0.982118i	0.871126	+	0.491059i	$0.163390\pi$
$389$			19.3704i			0.982118i	−0.871126	+	0.491059i	$0.836610\pi$
$390$		0			0
$391$		−	29.0680i		−	1.47003i
$392$	9.97903	−	14.8793i	0.504017	−	0.751519i
$393$		0			0
$394$	−10.0454			−0.506080
$395$	8.22167	−	3.52194i	0.413677	−	0.177208i
$396$		0			0
$397$	−31.3122			−1.57151			−0.785757	−	0.618535i	$-0.787726\pi$
$397$	−31.3122			−1.57151			−0.785757	+	0.618535i	$0.787726\pi$
$398$			5.28291i			0.264809i
$399$		0			0
$400$	3.44949	−	3.61953i	0.172474	−	0.180977i
$401$			22.4846i			1.12282i	0.827536	+	0.561412i	$0.189742\pi$
$401$			22.4846i			1.12282i	−0.827536	+	0.561412i	$0.810258\pi$
$402$		0			0
$403$		−	7.23907i		−	0.360604i
$404$	8.63695			0.429704
$405$		0			0
$406$	9.55051	−	5.09432i	0.473984	−	0.252827i
$407$	−3.63487			−0.180174
$408$		0			0
$409$			14.8276i			0.733180i	0.930383	+	0.366590i	$0.119475\pi$
$409$			14.8276i			0.733180i	−0.930383	+	0.366590i	$0.880525\pi$
$410$	−9.08716	+	3.89270i	−0.448783	+	0.192247i
$411$		0			0
$412$	−7.21959			−0.355684
$413$	2.30084	+	4.31348i	0.113217	+	0.212253i
$414$		0			0
$415$	6.89898	+	16.1051i	0.338658	+	0.790567i
$416$	−32.4714			−1.59204
$417$		0			0
$418$	−7.47117			−0.365427
$419$	−16.4433			−0.803310			−0.401655	−	0.915791i	$-0.631565\pi$
$419$	−16.4433			−0.803310			−0.401655	+	0.915791i	$0.631565\pi$
$420$		0			0
$421$	−12.6969			−0.618811			−0.309405	−	0.950930i	$-0.600130\pi$
$421$	−12.6969			−0.618811			−0.309405	+	0.950930i	$0.600130\pi$
$422$	−7.41964			−0.361182
$423$		0			0
$424$	25.3485			1.23103
$425$	21.9867	+	20.9538i	1.06651	+	1.01641i
$426$		0			0
$427$	−27.1427	+	14.4781i	−1.31353	+	0.700646i
$428$	2.63435			0.127336
$429$		0			0
$430$	−4.42108	−	10.3206i	−0.213203	−	0.497705i
$431$		−	37.6123i		−	1.81172i	−0.423576	−	0.905861i	$-0.639225\pi$
$431$		−	37.6123i		−	1.81172i	0.423576	−	0.905861i	$-0.360775\pi$
$432$		0			0
$433$	1.93069			0.0927829			0.0463915	−	0.998923i	$-0.485228\pi$
$433$	1.93069			0.0927829			0.0463915	+	0.998923i	$0.485228\pi$
$434$	2.26306	−	1.20713i	0.108630	−	0.0579442i
$435$		0			0
$436$	−4.49490			−0.215267
$437$			34.0721i			1.62989i
$438$		0			0
$439$			30.3746i			1.44970i	0.688907	+	0.724850i	$0.258091\pi$
$439$			30.3746i			1.44970i	−0.688907	+	0.724850i	$0.741909\pi$
$440$	−3.18695	−	7.43966i	−0.151932	−	0.354672i
$441$		0			0
$442$		−	24.9711i		−	1.18775i
$443$	4.78529			0.227356			0.113678	−	0.993518i	$-0.463737\pi$
$443$	4.78529			0.227356			0.113678	+	0.993518i	$0.463737\pi$
$444$		0			0
$445$	25.3485	−	10.8586i	1.20163	−	0.514747i
$446$	13.7650			0.651792
$447$		0			0
$448$	−2.92430	−	5.48230i	−0.138160	−	0.259014i
$449$		−	5.22826i		−	0.246737i	−0.992361	−	0.123368i	$-0.960630\pi$
$449$		−	5.22826i		−	0.246737i	0.992361	−	0.123368i	$-0.0393697\pi$
$450$		0			0
$451$		−	8.42676i		−	0.396800i
$452$	0.483412			0.0227378
$453$		0			0
$454$		−	13.6531i		−	0.640770i
$455$	−25.5683	−	20.5099i	−1.19866	−	0.961518i
$456$		0			0
$457$		−	4.66883i		−	0.218399i	−0.994020	−	0.109199i	$-0.965171\pi$
$457$		−	4.66883i		−	0.218399i	0.994020	−	0.109199i	$-0.0348287\pi$
$458$		−	4.31348i		−	0.201556i
$459$		0			0
$460$	−14.2568	+	6.10725i	−0.664729	+	0.284752i
$461$	−40.6931			−1.89527			−0.947633	−	0.319361i	$-0.896532\pi$
$461$	−40.6931			−1.89527			−0.947633	+	0.319361i	$0.896532\pi$
$462$		0			0
$463$		−	32.2102i		−	1.49693i	−0.663173	−	0.748466i	$-0.730791\pi$
$463$		−	32.2102i		−	1.49693i	0.663173	−	0.748466i	$-0.269209\pi$
$464$			5.51399i			0.255981i
$465$		0			0
$466$	−1.95459			−0.0905447
$467$			0.791539i			0.0366280i	0.999832	+	0.0183140i	$0.00582986\pi$
$467$			0.791539i			0.0366280i	−0.999832	+	0.0183140i	$0.994170\pi$
$468$		0			0
$469$	16.8990	−	9.01405i	0.780322	−	0.416230i
$470$	5.11879	+	11.9494i	0.236112	+	0.551184i
$471$		0			0
$472$	4.72920			0.217679
$473$	9.57058			0.440056
$474$		0			0
$475$	−25.7717	−	24.5610i	−1.18249	−	1.12694i
$476$	−20.5542	+	10.9638i	−0.942099	+	0.502523i
$477$		0			0
$478$			10.3870i			0.475088i
$479$	−38.4300			−1.75591			−0.877956	−	0.478740i	$-0.841093\pi$
$479$	−38.4300			−1.75591			−0.877956	+	0.478740i	$0.841093\pi$
$480$		0			0
$481$			14.2404i			0.649304i
$482$			14.8793i			0.677734i
$483$		0			0
$484$	−13.0454			−0.592973
$485$	11.3880	−	4.87832i	0.517103	−	0.221513i
$486$		0			0
$487$		−	15.1618i		−	0.687046i	−0.939144	−	0.343523i	$-0.888380\pi$
$487$		−	15.1618i		−	0.687046i	0.939144	−	0.343523i	$-0.111620\pi$
$488$			29.7586i			1.34711i
$489$		0			0
$490$	2.14816	−	11.4132i	0.0970442	−	0.515594i
$491$		−	26.8701i		−	1.21263i	−0.795225	−	0.606314i	$-0.792647\pi$
$491$		−	26.8701i		−	1.21263i	0.795225	−	0.606314i	$-0.207353\pi$
$492$		0			0
$493$	−33.4945			−1.50852
$494$			29.2699i			1.31691i
$495$		0			0
$496$			1.30658i			0.0586670i
$497$	−19.4747	+	10.3879i	−0.873558	+	0.465963i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$	5.65764	−	15.1861i	0.253018	−	0.679144i
$501$		0			0
$502$	−7.47117			−0.333455
$503$		−	20.7759i		−	0.926350i	−0.886267	−	0.463175i	$-0.846710\pi$
$503$		−	20.7759i		−	0.926350i	0.886267	−	0.463175i	$-0.153290\pi$
$504$		0			0
$505$	12.2474	−	5.24648i	0.545004	−	0.233465i
$506$			5.02118i			0.223219i
$507$		0			0
$508$		0			0
$509$	−14.1803			−0.628530			−0.314265	−	0.949335i	$-0.601758\pi$
$509$	−14.1803			−0.628530			−0.314265	+	0.949335i	$0.601758\pi$
$510$		0			0
$511$	−0.696938	−	1.30658i	−0.0308307	−	0.0577996i
$512$	10.9795			0.485232
$513$		0			0
$514$		−	9.73333i		−	0.429319i
$515$	−10.2376	+	4.38551i	−0.451122	+	0.193248i
$516$		0			0
$517$	−11.0810			−0.487340
$518$	−4.45178	+	2.37462i	−0.195600	+	0.104335i
$519$		0			0
$520$	−29.1464	+	12.4855i	−1.27816	+	0.547527i
$521$	26.9281			1.17974			0.589870	−	0.807498i	$-0.299179\pi$
$521$	26.9281			1.17974			0.589870	+	0.807498i	$0.299179\pi$
$522$		0			0
$523$	−34.3623			−1.50256			−0.751279	−	0.659985i	$-0.770563\pi$
$523$	−34.3623			−1.50256			−0.751279	+	0.659985i	$0.770563\pi$
$524$	9.23889			0.403603
$525$		0			0
$526$	20.9444			0.913219
$527$	−7.93674			−0.345730
$528$		0			0
$529$	−0.101021			−0.00439220
$530$	15.1042	−	6.47022i	0.656084	−	0.281048i
$531$		0			0
$532$	24.0926	−	12.8512i	1.04455	−	0.557170i
$533$	−33.0136			−1.42998
$534$		0			0
$535$	3.73558	−	1.60022i	0.161503	−	0.0691837i
$536$		−	18.5276i		−	0.800272i
$537$		0			0
$538$	16.6214			0.716601
$539$	8.22167	+	5.51399i	0.354133	+	0.237504i
$540$		0			0
$541$	21.1010			0.907204			0.453602	−	0.891204i	$-0.350139\pi$
$541$	21.1010			0.907204			0.453602	+	0.891204i	$0.350139\pi$
$542$			7.65753i			0.328919i
$543$		0			0
$544$			35.6009i			1.52638i
$545$	−6.37389	+	2.73040i	−0.273028	+	0.116958i
$546$		0			0
$547$			10.4930i			0.448646i	0.974515	+	0.224323i	$0.0720171\pi$
$547$			10.4930i			0.448646i	−0.974515	+	0.224323i	$0.927983\pi$
$548$	4.78529			0.204417
$549$		0			0
$550$	−3.79796	−	3.61953i	−0.161946	−	0.154337i
$551$	39.2606			1.67256
$552$		0			0
$553$	4.98078	+	9.33766i	0.211804	+	0.397078i
$554$		−	5.37113i		−	0.228197i
$555$		0			0
$556$		−	1.89387i		−	0.0803180i
$557$	29.7122			1.25895			0.629474	−	0.777022i	$-0.283270\pi$
$557$	29.7122			1.25895			0.629474	+	0.777022i	$0.283270\pi$
$558$		0			0
$559$		−	37.4948i		−	1.58586i
$560$	4.61481	+	3.70182i	0.195012	+	0.156431i
$561$		0			0
$562$		−	1.04930i		−	0.0442618i
$563$			26.2367i			1.10574i	0.833266	+	0.552872i	$0.186468\pi$
$563$			26.2367i			1.10574i	−0.833266	+	0.552872i	$0.813532\pi$
$564$		0			0
$565$	0.685493	−	0.293646i	0.0288389	−	0.0123538i
$566$	13.7650			0.578587
$567$		0			0
$568$			21.3516i			0.895891i
$569$		−	39.1694i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$569$		−	39.1694i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$570$		0			0
$571$	−33.3939			−1.39749			−0.698745	−	0.715371i	$-0.746258\pi$
$571$	−33.3939			−1.39749			−0.698745	+	0.715371i	$0.746258\pi$
$572$		−	11.3574i		−	0.474875i
$573$		0			0
$574$	−5.50510	−	10.3206i	−0.229779	−	0.430775i
$575$	−16.5068	+	17.3205i	−0.688381	+	0.722315i
$576$		0			0
$577$	26.3314			1.09619			0.548096	−	0.836416i	$-0.315353\pi$
$577$	26.3314			1.09619			0.548096	+	0.836416i	$0.315353\pi$
$578$	−14.7643			−0.614115
$579$		0			0
$580$	7.03726	+	16.4279i	0.292206	+	0.682130i
$581$	−18.2911	+	9.75663i	−0.758844	+	0.404773i
$582$		0			0
$583$			14.0065i			0.580089i
$584$	−1.43250			−0.0592773
$585$		0			0
$586$			10.9079i			0.450602i
$587$		−	3.52194i		−	0.145366i	−0.997355	−	0.0726831i	$-0.976844\pi$
$587$		−	3.52194i		−	0.145366i	0.997355	−	0.0726831i	$-0.0231562\pi$
$588$		0			0
$589$	9.30306			0.383326
$590$	2.81795	−	1.20713i	0.116013	−	0.0496969i
$591$		0			0
$592$		−	2.57024i		−	0.105636i
$593$		−	47.1904i		−	1.93788i	−0.247298	−	0.968940i	$-0.579543\pi$
$593$		−	47.1904i		−	1.93788i	0.247298	−	0.968940i	$-0.420457\pi$
$594$		0			0
$595$	−22.4865	+	28.0325i	−0.921858	+	1.14922i
$596$			14.3492i			0.587767i
$597$		0			0
$598$	19.6715			0.804429
$599$			10.8851i			0.444754i	0.974961	+	0.222377i	$0.0713815\pi$
$599$			10.8851i			0.444754i	−0.974961	+	0.222377i	$0.928618\pi$
$600$		0			0
$601$		−	19.4667i		−	0.794062i	−0.917805	−	0.397031i	$-0.870041\pi$
$601$		−	19.4667i		−	0.794062i	0.917805	−	0.397031i	$-0.129959\pi$
$602$	11.7215	−	6.25235i	0.477733	−	0.254827i
$603$		0			0
$604$	14.4949			0.589789
$605$	−18.4988	+	7.92437i	−0.752082	+	0.322172i
$606$		0			0
$607$	−17.4327			−0.707573			−0.353786	−	0.935326i	$-0.615106\pi$
$607$	−17.4327			−0.707573			−0.353786	+	0.935326i	$0.615106\pi$
$608$		−	41.7296i		−	1.69236i
$609$		0			0
$610$	7.59592	+	17.7320i	0.307550	+	0.717948i
$611$			43.4120i			1.75626i
$612$		0			0
$613$			17.7320i			0.716189i	0.933685	+	0.358095i	$0.116574\pi$
$613$			17.7320i			0.716189i	−0.933685	+	0.358095i	$0.883426\pi$
$614$	18.2911			0.738170
$615$		0			0
$616$	8.44949	−	4.50702i	0.340440	−	0.181593i
$617$	15.8398			0.637686			0.318843	−	0.947808i	$-0.396706\pi$
$617$	15.8398			0.637686			0.318843	+	0.947808i	$0.396706\pi$
$618$		0			0
$619$			6.53289i			0.262579i	0.991344	+	0.131289i	$0.0419117\pi$
$619$			6.53289i			0.262579i	−0.991344	+	0.131289i	$0.958088\pi$
$620$	1.66753	+	3.89270i	0.0669694	+	0.156334i
$621$		0			0
$622$	13.5714			0.544162
$623$	15.3564	+	28.7892i	0.615240	+	1.15341i
$624$		0			0
$625$	−1.20204	−	24.9711i	−0.0480816	−	0.998843i
$626$	10.4847			0.419054
$627$		0			0
$628$	18.8603			0.752606
$629$	15.6128			0.622523
$630$		0			0
$631$	5.79796			0.230813			0.115407	−	0.993318i	$-0.463183\pi$
$631$	5.79796			0.230813			0.115407	+	0.993318i	$0.463183\pi$
$632$	10.2376			0.407229
$633$		0			0
$634$	−2.44949			−0.0972817
$635$		0			0
$636$		0			0
$637$	21.6022	−	32.2102i	0.855911	−	1.27621i
$638$	5.78580			0.229062
$639$		0			0
$640$	20.5111	−	8.78640i	0.810772	−	0.347313i
$641$		−	32.2412i		−	1.27345i	−0.771091	−	0.636725i	$-0.780289\pi$
$641$		−	32.2412i		−	1.27345i	0.771091	−	0.636725i	$-0.219711\pi$
$642$		0			0
$643$	18.5521			0.731625			0.365812	−	0.930689i	$-0.380791\pi$
$643$	18.5521			0.731625			0.365812	+	0.930689i	$0.380791\pi$
$644$	−8.63695	−	16.1920i	−0.340344	−	0.638055i
$645$		0			0
$646$	32.0908			1.26260
$647$		−	31.3417i		−	1.23217i	−0.787680	−	0.616085i	$-0.788718\pi$
$647$		−	31.3417i		−	1.23217i	0.787680	−	0.616085i	$-0.211282\pi$
$648$		0			0
$649$			2.61315i			0.102575i
$650$	−14.1803	+	14.8793i	−0.556197	+	0.583615i
$651$		0			0
$652$			25.7024i			1.00658i
$653$	−32.6801			−1.27887			−0.639436	−	0.768845i	$-0.720832\pi$
$653$	−32.6801			−1.27887			−0.639436	+	0.768845i	$0.720832\pi$
$654$		0			0
$655$	13.1010	−	5.61212i	0.511899	−	0.219284i
$656$	5.95862			0.232645
$657$		0			0
$658$	−13.5714	+	7.23907i	−0.529066	+	0.282208i
$659$			3.25702i			0.126876i	0.997986	+	0.0634378i	$0.0202064\pi$
$659$			3.25702i			0.126876i	−0.997986	+	0.0634378i	$0.979794\pi$
$660$		0			0
$661$			18.0281i			0.701212i	0.936523	+	0.350606i	$0.114024\pi$
$661$			18.0281i			0.701212i	−0.936523	+	0.350606i	$0.885976\pi$
$662$	23.2931			0.905313
$663$		0			0
$664$			20.0540i			0.778244i
$665$	26.3576	−	32.8583i	1.02211	−	1.27419i
$666$		0			0
$667$		−	26.3860i		−	1.02167i
$668$		−	7.39967i		−	0.286302i
$669$		0			0
$670$	−4.72920	−	11.0399i	−0.182705	−	0.426509i
$671$	−16.4433			−0.634788
$672$		0			0
$673$			22.1888i			0.855317i	0.903940	+	0.427659i	$0.140661\pi$
$673$			22.1888i			0.855317i	−0.903940	+	0.427659i	$0.859339\pi$
$674$			16.6206i			0.640202i
$675$		0			0
$676$	−25.6515			−0.986597
$677$			44.4600i			1.70874i	0.519667	+	0.854369i	$0.326056\pi$
$677$			44.4600i			1.70874i	−0.519667	+	0.854369i	$0.673944\pi$
$678$		0			0
$679$	6.89898	+	12.9338i	0.264759	+	0.496353i
$680$	13.6889	+	31.9555i	0.524944	+	1.22544i
$681$		0			0
$682$	1.37099			0.0524977
$683$	−25.2605			−0.966565			−0.483282	−	0.875465i	$-0.660556\pi$
$683$	−25.2605			−0.966565			−0.483282	+	0.875465i	$0.660556\pi$
$684$		0			0
$685$	6.78568	−	2.90680i	0.259267	−	0.111063i
$686$	13.6717	+	1.38211i	0.521987	+	0.0527690i
$687$		0			0
$688$			6.76742i			0.258005i
$689$	54.8734			2.09051
$690$		0			0
$691$			12.9338i			0.492024i	0.969267	+	0.246012i	$0.0791203\pi$
$691$			12.9338i			0.492024i	−0.969267	+	0.246012i	$0.920880\pi$
$692$		−	2.55251i		−	0.0970319i
$693$		0			0
$694$	−27.5505			−1.04580
$695$	−1.15042	−	2.68556i	−0.0436380	−	0.101869i
$696$		0			0
$697$			36.1953i			1.37100i
$698$		−	19.1928i		−	0.726458i
$699$		0			0
$700$	18.4734	+	5.13919i	0.698230	+	0.194243i
$701$			20.9275i			0.790420i	0.918591	+	0.395210i	$0.129328\pi$
$701$			20.9275i			0.790420i	−0.918591	+	0.395210i	$0.870672\pi$
$702$		0			0
$703$	−18.3006			−0.690218
$704$		−	3.32124i		−	0.125174i
$705$		0			0
$706$			1.30658i			0.0491737i
$707$	7.41964	+	13.9099i	0.279044	+	0.523135i
$708$		0			0
$709$	−13.7980			−0.518193			−0.259097	−	0.965851i	$-0.583425\pi$
$709$	−13.7980			−0.518193			−0.259097	+	0.965851i	$0.583425\pi$
$710$	5.45001	+	12.7226i	0.204535	+	0.477470i
$711$		0			0
$712$	31.5638			1.18290
$713$		−	6.25235i		−	0.234152i
$714$		0			0
$715$	−6.89898	−	16.1051i	−0.258007	−	0.602296i
$716$		−	0.207081i		−	0.00773897i
$717$		0			0
$718$		−	7.34507i		−	0.274115i
$719$	5.54334			0.206732			0.103366	−	0.994643i	$-0.467039\pi$
$719$	5.54334			0.206732			0.103366	+	0.994643i	$0.467039\pi$
$720$		0			0
$721$	−6.20204	−	11.6272i	−0.230976	−	0.433020i
$722$	−23.5180			−0.875249
$723$		0			0
$724$			4.63902i			0.172408i
$725$	19.9581	+	19.0205i	0.741224	+	0.706402i
$726$		0			0
$727$	−7.21959			−0.267760			−0.133880	−	0.990998i	$-0.542744\pi$
$727$	−7.21959			−0.267760			−0.133880	+	0.990998i	$0.542744\pi$
$728$	−17.6572	−	33.1027i	−0.654420	−	1.22687i
$729$		0			0
$730$	−0.853572	+	0.365647i	−0.0315921	+	0.0135332i
$731$	−41.1084			−1.52045
$732$		0			0
$733$	6.65989			0.245989			0.122994	−	0.992407i	$-0.460750\pi$
$733$	6.65989			0.245989			0.122994	+	0.992407i	$0.460750\pi$
$734$	2.67834			0.0988592
$735$		0			0
$736$	−28.0454			−1.03377
$737$	10.2376			0.377106
$738$		0			0
$739$	37.3939			1.37556			0.687778	−	0.725921i	$-0.258586\pi$
$739$	37.3939			1.37556			0.687778	+	0.725921i	$0.258586\pi$
$740$	−3.28028	−	7.65753i	−0.120585	−	0.281496i
$741$		0			0
$742$	9.15028	+	17.1544i	0.335917	+	0.629757i
$743$	−28.2283			−1.03560			−0.517798	−	0.855503i	$-0.673248\pi$
$743$	−28.2283			−1.03560			−0.517798	+	0.855503i	$0.673248\pi$
$744$		0			0
$745$	8.71636	+	20.3476i	0.319343	+	0.745479i
$746$		−	23.8988i		−	0.874996i
$747$		0			0
$748$	−12.4519			−0.455288
$749$	2.26306	+	4.24264i	0.0826903	+	0.155023i
$750$		0			0
$751$	−25.1918			−0.919263			−0.459632	−	0.888110i	$-0.652019\pi$
$751$	−25.1918			−0.919263			−0.459632	+	0.888110i	$0.652019\pi$
$752$		−	7.83542i		−	0.285729i
$753$		0			0
$754$		−	22.6671i		−	0.825488i
$755$	20.5542	−	8.80486i	0.748043	−	0.320442i
$756$		0			0
$757$		−	24.2874i		−	0.882742i	−0.897325	−	0.441371i	$-0.854492\pi$
$757$		−	24.2874i		−	0.882742i	0.897325	−	0.441371i	$-0.145508\pi$
$758$	−13.0555			−0.474198
$759$		0			0
$760$	−16.0454	−	37.4566i	−0.582028	−	1.35869i
$761$	−44.3886			−1.60909			−0.804544	−	0.593894i	$-0.797590\pi$
$761$	−44.3886			−1.60909			−0.804544	+	0.593894i	$0.797590\pi$
$762$		0			0
$763$	−3.86137	−	7.23907i	−0.139791	−	0.262072i
$764$		−	32.5911i		−	1.17911i
$765$		0			0
$766$			19.4667i			0.703359i
$767$	10.2376			0.369658
$768$		0			0
$769$			19.4667i			0.701986i	0.936378	+	0.350993i	$0.114156\pi$
$769$			19.4667i			0.701986i	−0.936378	+	0.350993i	$0.885844\pi$
$770$	3.88430	−	4.84230i	0.139981	−	0.174504i
$771$		0			0
$772$			9.80930i			0.353045i
$773$		−	17.4318i		−	0.626979i	−0.949592	−	0.313490i	$-0.898502\pi$
$773$		−	17.4318i		−	0.626979i	0.949592	−	0.313490i	$-0.101498\pi$
$774$		0			0
$775$	4.72920	+	4.50702i	0.169878	+	0.161897i
$776$	14.1803			0.509043
$777$		0			0
$778$			14.3721i			0.515266i
$779$		−	42.4264i		−	1.52008i
$780$		0			0
$781$	−11.7980			−0.422164
$782$		−	21.5674i		−	0.771249i
$783$		0			0
$784$	−3.89898	+	5.81360i	−0.139249	+	0.207629i
$785$	26.7444	−	11.4566i	0.954548	−	0.408903i
$786$		0			0
$787$	−40.4625			−1.44233			−0.721166	−	0.692762i	$-0.756393\pi$
$787$	−40.4625			−1.44233			−0.721166	+	0.692762i	$0.756393\pi$
$788$	19.6246			0.699096
$789$		0			0
$790$	6.10018	−	2.61315i	0.217035	−	0.0929719i
$791$	0.415279	+	0.778539i	0.0147656	+	0.0276817i
$792$		0			0
$793$			64.4203i			2.28763i
$794$	−23.2325			−0.824491
$795$		0			0
$796$		−	10.3206i		−	0.365805i
$797$		−	15.4930i		−	0.548789i	−0.961617	−	0.274394i	$-0.911523\pi$
$797$		−	15.4930i		−	0.548789i	0.961617	−	0.274394i	$-0.0884773\pi$
$798$		0			0
$799$	47.5959			1.68382
$800$	20.2166	−	21.2132i	0.714765	−	0.750000i
$801$		0			0
$802$			16.6827i			0.589087i
$803$		−	0.791539i		−	0.0279328i
$804$		0			0
$805$	−22.0832	−	17.7143i	−0.778331	−	0.624346i
$806$		−	5.37113i		−	0.189190i
$807$		0			0
$808$	15.2505			0.536509
$809$			8.62815i			0.303349i	0.988430	+	0.151675i	$0.0484666\pi$
$809$			8.62815i			0.303349i	−0.988430	+	0.151675i	$0.951533\pi$
$810$		0			0
$811$			10.3206i			0.362406i	0.983446	+	0.181203i	$0.0579991\pi$
$811$			10.3206i			0.362406i	−0.983446	+	0.181203i	$0.942001\pi$
$812$	−18.6577	+	9.95218i	−0.654758	+	0.349253i
$813$		0			0
$814$	−2.69694			−0.0945276
$815$	15.6128	+	36.4467i	0.546892	+	1.27667i
$816$		0			0
$817$	48.1852			1.68579
$818$			11.0016i			0.384661i
$819$		0			0
$820$	17.7526	−	7.60471i	0.619946	−	0.265568i
$821$			27.8557i			0.972170i	0.873912	+	0.486085i	$0.161575\pi$
$821$			27.8557i			0.972170i	−0.873912	+	0.486085i	$0.838425\pi$
$822$		0			0
$823$		−	14.4781i		−	0.504676i	−0.967639	−	0.252338i	$-0.918801\pi$
$823$		−	14.4781i		−	0.504676i	0.967639	−	0.252338i	$-0.0811995\pi$
$824$	−12.7478			−0.444090
$825$		0			0
$826$	1.70714	+	3.20045i	0.0593991	+	0.111358i
$827$	10.7210			0.372806			0.186403	−	0.982473i	$-0.440317\pi$
$827$	10.7210			0.372806			0.186403	+	0.982473i	$0.440317\pi$
$828$		0			0
$829$			17.4408i			0.605744i	0.953031	+	0.302872i	$0.0979455\pi$
$829$			17.4408i			0.605744i	−0.953031	+	0.302872i	$0.902055\pi$
$830$	5.11879	+	11.9494i	0.177676	+	0.414769i
$831$		0			0
$832$	−13.0117			−0.451098
$833$	−35.3144	−	23.6842i	−1.22357	−	0.820608i
$834$		0			0
$835$	−4.49490	−	10.4930i	−0.155552	−	0.363124i
$836$	14.5956			0.504798
$837$		0			0
$838$	−12.2004			−0.421455
$839$	−28.3606			−0.979116			−0.489558	−	0.871971i	$-0.662842\pi$
$839$	−28.3606			−0.979116			−0.489558	+	0.871971i	$0.662842\pi$
$840$		0			0
$841$	−1.40408			−0.0484166
$842$	−9.42067			−0.324658
$843$		0			0
$844$	14.4949			0.498935
$845$	−36.3746	+	15.5819i	−1.25132	+	0.536034i
$846$		0			0
$847$	−11.2068	−	21.0097i	−0.385069	−	0.721903i
$848$	−9.90408			−0.340108
$849$		0			0
$850$	16.3133	+	15.5469i	0.559542	+	0.533255i
$851$			12.2993i			0.421616i
$852$		0			0
$853$	−34.0542			−1.16599			−0.582996	−	0.812475i	$-0.698120\pi$
$853$	−34.0542			−1.16599			−0.582996	+	0.812475i	$0.698120\pi$
$854$	−20.1389	+	10.7423i	−0.689139	+	0.367592i
$855$		0			0
$856$	4.65153			0.158986
$857$		−	15.4930i		−	0.529229i	−0.964354	−	0.264615i	$-0.914755\pi$
$857$		−	15.4930i		−	0.529229i	0.964354	−	0.264615i	$-0.0852448\pi$
$858$		0			0
$859$		−	30.3746i		−	1.03637i	−0.855269	−	0.518184i	$-0.826608\pi$
$859$		−	30.3746i		−	1.03637i	0.855269	−	0.518184i	$-0.173392\pi$
$860$	8.63695	+	20.1622i	0.294518	+	0.687526i
$861$		0			0
$862$		−	27.9070i		−	0.950515i
$863$	37.7989			1.28669			0.643345	−	0.765577i	$-0.277546\pi$
$863$	37.7989			1.28669			0.643345	+	0.765577i	$0.277546\pi$
$864$		0			0
$865$	−1.55051	−	3.61953i	−0.0527189	−	0.123068i
$866$	1.43250			0.0486783
$867$		0			0
$868$	−4.42108	+	2.35824i	−0.150061	+	0.0800438i
$869$			5.65685i			0.191896i
$870$		0			0
$871$		−	40.1079i		−	1.35900i
$872$	−7.93674			−0.268772
$873$		0			0
$874$			25.2803i			0.855118i
$875$	29.3176	−	3.93408i	0.991117	−	0.132996i
$876$		0			0
$877$		−	14.4781i		−	0.488892i	−0.969663	−	0.244446i	$-0.921394\pi$
$877$		−	14.4781i		−	0.488892i	0.969663	−	0.244446i	$-0.0786061\pi$
$878$			22.5368i			0.760582i
$879$		0			0
$880$	1.24519	+	2.90680i	0.0419755	+	0.0979882i
$881$	36.9975			1.24648			0.623239	−	0.782031i	$-0.285816\pi$
$881$	36.9975			1.24648			0.623239	+	0.782031i	$0.285816\pi$
$882$		0			0
$883$		−	14.0065i		−	0.471356i	−0.971831	−	0.235678i	$-0.924269\pi$
$883$		−	14.0065i		−	0.471356i	0.971831	−	0.235678i	$-0.0757310\pi$
$884$			48.7832i			1.64075i
$885$		0			0
$886$	3.55051			0.119282
$887$			30.5502i			1.02577i	0.858456	+	0.512887i	$0.171424\pi$
$887$			30.5502i			1.02577i	−0.858456	+	0.512887i	$0.828576\pi$
$888$		0			0
$889$		0			0
$890$	18.8076	−	8.05669i	0.630434	−	0.270061i
$891$		0			0
$892$	−26.8911			−0.900382
$893$	−55.7896			−1.86693
$894$		0			0
$895$	−0.125790	−	0.293646i	−0.00420470	−	0.00981552i
$896$	12.4258	+	23.2952i	0.415118	+	0.778238i
$897$		0			0
$898$		−	3.87918i		−	0.129450i
$899$	−7.20445			−0.240282
$900$		0			0
$901$		−	60.1619i		−	2.00428i
$902$		−	6.25235i		−	0.208180i
$903$		0			0
$904$	0.853572			0.0283894
$905$	2.81795	+	6.57826i	0.0936718	+	0.218669i
$906$		0			0
$907$		−	22.4008i		−	0.743808i	−0.928271	−	0.371904i	$-0.878705\pi$
$907$		−	22.4008i		−	0.743808i	0.928271	−	0.371904i	$-0.121295\pi$
$908$			26.6724i			0.885156i
$909$		0			0
$910$	−18.9708	−	15.2176i	−0.628874	−	0.504458i
$911$		−	26.8701i		−	0.890245i	−0.895470	−	0.445122i	$-0.853160\pi$
$911$		−	26.8701i		−	0.890245i	0.895470	−	0.445122i	$-0.146840\pi$
$912$		0			0
$913$	−11.0810			−0.366726
$914$		−	3.46410i		−	0.114582i
$915$		0			0
$916$			8.42676i			0.278428i
$917$	7.93674	+	14.8793i	0.262094	+	0.491358i
$918$		0			0
$919$	−10.0000			−0.329870			−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000			−0.329870			−0.164935	+	0.986304i	$0.552741\pi$
$920$	−25.1736	+	10.7837i	−0.829950	+	0.355528i
$921$		0			0
$922$	−30.1928			−0.994347
$923$			46.2210i			1.52138i
$924$		0			0
$925$	−9.30306	−	8.86601i	−0.305883	−	0.291513i
$926$		−	23.8988i		−	0.785362i
$927$		0			0
$928$			32.3162i			1.06083i
$929$	8.63695			0.283369			0.141685	−	0.989912i	$-0.454748\pi$
$929$	8.63695			0.283369			0.141685	+	0.989912i	$0.454748\pi$
$930$		0			0
$931$	41.3939	+	27.7614i	1.35663	+	0.909844i
$932$	3.81846			0.125078
$933$		0			0
$934$			0.587293i			0.0192168i
$935$	−17.6572	+	7.56388i	−0.577453	+	0.247365i
$936$		0			0
$937$	8.03087			0.262357			0.131179	−	0.991359i	$-0.458124\pi$
$937$	8.03087			0.262357			0.131179	+	0.991359i	$0.458124\pi$
$938$	12.5384	−	6.68810i	0.409394	−	0.218374i
$939$		0			0
$940$	−10.0000	−	23.3441i	−0.326164	−	0.761402i
$941$	36.1670			1.17901			0.589505	−	0.807765i	$-0.299323\pi$
$941$	36.1670			1.17901			0.589505	+	0.807765i	$0.299323\pi$
$942$		0			0
$943$	−28.5137			−0.928534
$944$	−1.84778			−0.0601401
$945$		0			0
$946$	7.10102			0.230874
$947$	−13.0218			−0.423153			−0.211577	−	0.977361i	$-0.567860\pi$
$947$	−13.0218			−0.423153			−0.211577	+	0.977361i	$0.567860\pi$
$948$		0			0
$949$	−3.10102			−0.100663
$950$	−19.1217	−	18.2234i	−0.620389	−	0.591244i
$951$		0			0
$952$	−36.2930	+	19.3590i	−1.17626	+	0.627428i
$953$	−27.7112			−0.897654			−0.448827	−	0.893619i	$-0.648158\pi$
$953$	−27.7112			−0.897654			−0.448827	+	0.893619i	$0.648158\pi$
$954$		0			0
$955$	−19.7973	−	46.2152i	−0.640626	−	1.49549i
$956$		−	20.2918i		−	0.656284i
$957$		0			0
$958$	−28.5137			−0.921236
$959$	4.11084	+	7.70674i	0.132746	+	0.248864i
$960$		0			0
$961$	29.2929			0.944931
$962$			10.5658i			0.340656i
$963$		0			0
$964$		−	29.0680i		−	0.936217i
$965$	5.95862	+	13.9099i	0.191815	+	0.447775i
$966$		0			0
$967$			42.7031i			1.37324i	0.727017	+	0.686620i	$0.240906\pi$
$967$			42.7031i			1.37324i	−0.727017	+	0.686620i	$0.759094\pi$
$968$	−23.0346			−0.740359
$969$		0			0
$970$	8.44949	−	3.61953i	0.271297	−	0.116216i
$971$	−40.2778			−1.29258			−0.646288	−	0.763093i	$-0.723680\pi$
$971$	−40.2778			−1.29258			−0.646288	+	0.763093i	$0.723680\pi$
$972$		0			0
$973$	3.05009	−	1.62694i	0.0977815	−	0.0521574i
$974$		−	11.2495i		−	0.360457i
$975$		0			0
$976$		−	11.6272i		−	0.372178i
$977$	−34.3139			−1.09780			−0.548900	−	0.835888i	$-0.684953\pi$
$977$	−34.3139			−1.09780			−0.548900	+	0.835888i	$0.684953\pi$
$978$		0			0
$979$			17.4408i			0.557411i
$980$	−4.19662	+	22.2966i	−0.134056	+	0.712239i
$981$		0			0
$982$		−	19.9366i		−	0.636203i
$983$		−	54.8480i		−	1.74938i	−0.484684	−	0.874689i	$-0.661065\pi$
$983$		−	54.8480i		−	1.74938i	0.484684	−	0.874689i	$-0.338935\pi$
$984$		0			0
$985$	27.8282	−	11.9208i	0.886680	−	0.379830i
$986$	−24.8517			−0.791439
$987$		0			0
$988$		−	57.1812i		−	1.81918i
$989$		−	32.3840i		−	1.02975i
$990$		0			0
$991$	23.7980			0.755967			0.377984	−	0.925812i	$-0.376618\pi$
$991$	23.7980			0.755967			0.377984	+	0.925812i	$0.376618\pi$
$992$			7.65753i			0.243127i
$993$		0			0
$994$	−14.4495	+	7.70747i	−0.458310	+	0.244466i
$995$	−6.26922	−	14.6349i	−0.198747	−	0.463959i
$996$		0			0
$997$	−4.16950			−0.132049			−0.0660246	−	0.997818i	$-0.521032\pi$
$997$	−4.16950			−0.132049			−0.0660246	+	0.997818i	$0.521032\pi$
$998$	−7.41964			−0.234865
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.g.a.314.10	yes	16
3.2	odd	2	inner	315.2.g.a.314.7	yes	16
4.3	odd	2		5040.2.k.g.1889.4		16
5.2	odd	4		1575.2.b.h.251.11		16
5.3	odd	4		1575.2.b.h.251.6		16
5.4	even	2	inner	315.2.g.a.314.5	✓	16
7.6	odd	2	inner	315.2.g.a.314.11	yes	16
12.11	even	2		5040.2.k.g.1889.14		16
15.2	even	4		1575.2.b.h.251.7		16
15.8	even	4		1575.2.b.h.251.10		16
15.14	odd	2	inner	315.2.g.a.314.12	yes	16
20.19	odd	2		5040.2.k.g.1889.1		16
21.20	even	2	inner	315.2.g.a.314.6	yes	16
28.27	even	2		5040.2.k.g.1889.13		16
35.13	even	4		1575.2.b.h.251.5		16
35.27	even	4		1575.2.b.h.251.12		16
35.34	odd	2	inner	315.2.g.a.314.8	yes	16
60.59	even	2		5040.2.k.g.1889.15		16
84.83	odd	2		5040.2.k.g.1889.3		16
105.62	odd	4		1575.2.b.h.251.8		16
105.83	odd	4		1575.2.b.h.251.9		16
105.104	even	2	inner	315.2.g.a.314.9	yes	16
140.139	even	2		5040.2.k.g.1889.16		16
420.419	odd	2		5040.2.k.g.1889.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.g.a.314.5	✓	16	5.4	even	2	inner
315.2.g.a.314.6	yes	16	21.20	even	2	inner
315.2.g.a.314.7	yes	16	3.2	odd	2	inner
315.2.g.a.314.8	yes	16	35.34	odd	2	inner
315.2.g.a.314.9	yes	16	105.104	even	2	inner
315.2.g.a.314.10	yes	16	1.1	even	1	trivial
315.2.g.a.314.11	yes	16	7.6	odd	2	inner
315.2.g.a.314.12	yes	16	15.14	odd	2	inner
1575.2.b.h.251.5		16	35.13	even	4
1575.2.b.h.251.6		16	5.3	odd	4
1575.2.b.h.251.7		16	15.2	even	4
1575.2.b.h.251.8		16	105.62	odd	4
1575.2.b.h.251.9		16	105.83	odd	4
1575.2.b.h.251.10		16	15.8	even	4
1575.2.b.h.251.11		16	5.2	odd	4
1575.2.b.h.251.12		16	35.27	even	4
5040.2.k.g.1889.1		16	20.19	odd	2
5040.2.k.g.1889.2		16	420.419	odd	2
5040.2.k.g.1889.3		16	84.83	odd	2
5040.2.k.g.1889.4		16	4.3	odd	2
5040.2.k.g.1889.13		16	28.27	even	2
5040.2.k.g.1889.14		16	12.11	even	2
5040.2.k.g.1889.15		16	60.59	even	2
5040.2.k.g.1889.16		16	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.741964 q^{2} -1.44949 q^{4} +(-2.05542 + 0.880486i) q^{5} +(-1.24519 - 2.33441i) q^{7} -2.55940 q^{8} +(-1.52505 + 0.653289i) q^{10} -1.41421i q^{11} -5.54048 q^{13} +(-0.923889 - 1.73205i) q^{14} +1.00000 q^{16} +6.07445i q^{17} -7.12018i q^{19} +(2.97931 - 1.27626i) q^{20} -1.04930i q^{22} -4.78529 q^{23} +(3.44949 - 3.61953i) q^{25} -4.11084 q^{26} +(1.80490 + 3.38371i) q^{28} +5.51399i q^{29} +1.30658i q^{31} +5.86076 q^{32} +4.50702i q^{34} +(4.61481 + 3.70182i) q^{35} -2.57024i q^{37} -5.28291i q^{38} +(5.26063 - 2.25351i) q^{40} +5.95862 q^{41} +6.76742i q^{43} +2.04989i q^{44} -3.55051 q^{46} -7.83542i q^{47} +(-3.89898 + 5.81360i) q^{49} +(2.55940 - 2.68556i) q^{50} +8.03087 q^{52} -9.90408 q^{53} +(1.24519 + 2.90680i) q^{55} +(3.18695 + 5.97469i) q^{56} +4.09118i q^{58} -1.84778 q^{59} -11.6272i q^{61} +0.969433i q^{62} +2.34847 q^{64} +(11.3880 - 4.87832i) q^{65} +7.23907i q^{67} -8.80486i q^{68} +(3.42403 + 2.74662i) q^{70} -8.34242i q^{71} +0.559702 q^{73} -1.90702i q^{74} +10.3206i q^{76} +(-3.30136 + 1.76097i) q^{77} -4.00000 q^{79} +(-2.05542 + 0.880486i) q^{80} +4.42108 q^{82} -7.83542i q^{83} +(-5.34847 - 12.4855i) q^{85} +5.02118i q^{86} +3.61953i q^{88} -12.3325 q^{89} +(6.89898 + 12.9338i) q^{91} +6.93623 q^{92} -5.81360i q^{94} +(6.26922 + 14.6349i) q^{95} -5.54048 q^{97} +(-2.89290 + 4.31348i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4} + 16 q^{16} + 16 q^{25} - 96 q^{46} + 16 q^{49} - 80 q^{64} - 48 q^{70} - 64 q^{79} + 32 q^{85} + 32 q^{91}+O(q^{100}) \) 16 * q + 16 * q^4 + 16 * q^16 + 16 * q^25 - 96 * q^46 + 16 * q^49 - 80 * q^64 - 48 * q^70 - 64 * q^79 + 32 * q^85 + 32 * q^91

Introduction

L-functions

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