LMFDB - Embedded newform 912.2.d.a.191.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(191,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("912.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.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:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.2593100598870016.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + x^8 + 4x^6 + 4x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.9
Root			$-0.430469 + 1.34711i$ of defining polynomial
Character	$\chi$	$=$	912.191
Dual form			912.2.d.a.191.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.916638 - 1.46962i) q^{3} +0.860938i q^{5} +2.31955i q^{7} +(-1.31955 - 2.69421i) q^{9} +O(q^{10})$	\(q+(0.916638 - 1.46962i) q^{3} +0.860938i q^{5} +2.31955i q^{7} +(-1.31955 - 2.69421i) q^{9} +0.512326 q^{11} +2.93923 q^{13} +(1.26525 + 0.789168i) q^{15} -5.22471i q^{17} +1.00000i q^{19} +(3.40885 + 2.12619i) q^{21} +9.29219 q^{23} +4.25879 q^{25} +(-5.16901 - 0.530383i) q^{27} -6.87309i q^{29} +4.93923i q^{31} +(0.469617 - 0.752923i) q^{33} -1.99699 q^{35} +9.45681 q^{37} +(2.69421 - 4.31955i) q^{39} +6.43426i q^{41} +1.68045i q^{43} +(2.31955 - 1.13605i) q^{45} -6.59798 q^{47} +1.61968 q^{49} +(-7.67833 - 4.78917i) q^{51} -3.31794i q^{53} +0.441081i q^{55} +(1.46962 + 0.916638i) q^{57} -1.13605 q^{59} -2.01942 q^{61} +(6.24936 - 3.06077i) q^{63} +2.53050i q^{65} -7.15667i q^{67} +(8.51757 - 13.6560i) q^{69} -10.3380 q^{71} -4.49815 q^{73} +(3.90376 - 6.25879i) q^{75} +1.18837i q^{77} +5.36090i q^{79} +(-5.51757 + 7.11030i) q^{81} -8.96476 q^{83} +4.49815 q^{85} +(-10.1008 - 6.30013i) q^{87} -1.04584i q^{89} +6.81770i q^{91} +(7.25879 + 4.52749i) q^{93} -0.860938 q^{95} -8.21744 q^{97} +(-0.676040 - 1.38032i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9}+O(q^{10}) $ 12 * q - 4 * q^9	$ 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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			0
$2$		0			0
$3$	0.916638	−	1.46962i	0.529221	−	0.848484i
$3$	0.916638	−	1.46962i	0.529221	−	0.848484i
$4$		0			0
$5$			0.860938i			0.385023i	0.981295	+	0.192512i	$0.0616633\pi$
$5$			0.860938i			0.385023i	−0.981295	+	0.192512i	$0.938337\pi$
$6$		0			0
$7$			2.31955i			0.876708i	0.898802	+	0.438354i	$0.144438\pi$
$7$			2.31955i			0.876708i	−0.898802	+	0.438354i	$0.855562\pi$
$8$		0			0
$9$	−1.31955	−	2.69421i	−0.439850	−	0.898071i
$10$		0			0
$11$	0.512326			0.154472			0.0772361	−	0.997013i	$-0.475390\pi$
$11$	0.512326			0.154472			0.0772361	+	0.997013i	$0.475390\pi$
$12$		0			0
$13$	2.93923			0.815197			0.407599	−	0.913161i	$-0.366366\pi$
$13$	2.93923			0.815197			0.407599	+	0.913161i	$0.366366\pi$
$14$		0			0
$15$	1.26525	+	0.789168i	0.326686	+	0.203762i
$16$		0			0
$17$		−	5.22471i		−	1.26718i	−0.773669	−	0.633589i	$-0.781581\pi$
$17$		−	5.22471i		−	1.26718i	0.773669	−	0.633589i	$-0.218419\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$		0			0
$21$	3.40885	+	2.12619i	0.743873	+	0.463972i
$22$		0			0
$23$	9.29219			1.93756			0.968778	−	0.247931i	$-0.0797505\pi$
$23$	9.29219			1.93756			0.968778	+	0.247931i	$0.0797505\pi$
$24$		0			0
$25$	4.25879			0.851757
$26$		0			0
$27$	−5.16901	−	0.530383i	−0.994777	−	0.102072i
$28$		0			0
$29$		−	6.87309i		−	1.27630i	−0.769912	−	0.638150i	$-0.779700\pi$
$29$		−	6.87309i		−	1.27630i	0.769912	−	0.638150i	$-0.220300\pi$
$30$		0			0
$31$			4.93923i			0.887113i	0.896246	+	0.443556i	$0.146283\pi$
$31$			4.93923i			0.887113i	−0.896246	+	0.443556i	$0.853717\pi$
$32$		0			0
$33$	0.469617	−	0.752923i	0.0817499	−	0.131067i
$34$		0			0
$35$	−1.99699			−0.337553
$36$		0			0
$37$	9.45681			1.55469			0.777345	−	0.629075i	$-0.216566\pi$
$37$	9.45681			1.55469			0.777345	+	0.629075i	$0.216566\pi$
$38$		0			0
$39$	2.69421	−	4.31955i	0.431419	−	0.691682i
$40$		0			0
$41$			6.43426i			1.00486i	0.864617	+	0.502431i	$0.167561\pi$
$41$			6.43426i			1.00486i	−0.864617	+	0.502431i	$0.832439\pi$
$42$		0			0
$43$			1.68045i			0.256266i	0.991757	+	0.128133i	$0.0408985\pi$
$43$			1.68045i			0.256266i	−0.991757	+	0.128133i	$0.959102\pi$
$44$		0			0
$45$	2.31955	−	1.13605i	0.345778	−	0.169353i
$46$		0			0
$47$	−6.59798			−0.962414			−0.481207	−	0.876607i	$-0.659801\pi$
$47$	−6.59798			−0.962414			−0.481207	+	0.876607i	$0.659801\pi$
$48$		0			0
$49$	1.61968			0.231383
$50$		0			0
$51$	−7.67833	−	4.78917i	−1.07518	−	0.670618i
$52$		0			0
$53$		−	3.31794i		−	0.455754i	−0.973690	−	0.227877i	$-0.926822\pi$
$53$		−	3.31794i		−	0.455754i	0.973690	−	0.227877i	$-0.0731784\pi$
$54$		0			0
$55$			0.441081i			0.0594754i
$56$		0			0
$57$	1.46962	+	0.916638i	0.194656	+	0.121412i
$58$		0			0
$59$	−1.13605			−0.147901			−0.0739507	−	0.997262i	$-0.523561\pi$
$59$	−1.13605			−0.147901			−0.0739507	+	0.997262i	$0.523561\pi$
$60$		0			0
$61$	−2.01942			−0.258560			−0.129280	−	0.991608i	$-0.541267\pi$
$61$	−2.01942			−0.258560			−0.129280	+	0.991608i	$0.541267\pi$
$62$		0			0
$63$	6.24936	−	3.06077i	0.787346	−	0.385620i
$64$		0			0
$65$			2.53050i			0.313870i
$66$		0			0
$67$		−	7.15667i		−	0.874327i	−0.899382	−	0.437163i	$-0.855983\pi$
$67$		−	7.15667i		−	0.874327i	0.899382	−	0.437163i	$-0.144017\pi$
$68$		0			0
$69$	8.51757	−	13.6560i	1.02540	−	1.64398i
$70$		0			0
$71$	−10.3380			−1.22690			−0.613449	−	0.789734i	$-0.710218\pi$
$71$	−10.3380			−1.22690			−0.613449	+	0.789734i	$0.710218\pi$
$72$		0			0
$73$	−4.49815			−0.526469			−0.263235	−	0.964732i	$-0.584789\pi$
$73$	−4.49815			−0.526469			−0.263235	+	0.964732i	$0.584789\pi$
$74$		0			0
$75$	3.90376	−	6.25879i	0.450768	−	0.722702i
$76$		0			0
$77$			1.18837i			0.135427i
$78$		0			0
$79$			5.36090i			0.603148i	0.953443	+	0.301574i	$0.0975121\pi$
$79$			5.36090i			0.603148i	−0.953443	+	0.301574i	$0.902488\pi$
$80$		0			0
$81$	−5.51757	+	7.11030i	−0.613063	+	0.790034i
$82$		0			0
$83$	−8.96476			−0.984010			−0.492005	−	0.870592i	$-0.663736\pi$
$83$	−8.96476			−0.984010			−0.492005	+	0.870592i	$0.663736\pi$
$84$		0			0
$85$	4.49815			0.487893
$86$		0			0
$87$	−10.1008	−	6.30013i	−1.08292	−	0.675445i
$88$		0			0
$89$		−	1.04584i		−	0.110858i	−0.998463	−	0.0554292i	$-0.982347\pi$
$89$		−	1.04584i		−	0.110858i	0.998463	−	0.0554292i	$-0.0176527\pi$
$90$		0			0
$91$			6.81770i			0.714690i
$92$		0			0
$93$	7.25879	+	4.52749i	0.752701	+	0.469479i
$94$		0			0
$95$	−0.860938			−0.0883304
$96$		0			0
$97$	−8.21744			−0.834354			−0.417177	−	0.908825i	$-0.636981\pi$
$97$	−8.21744			−0.834354			−0.417177	+	0.908825i	$0.636981\pi$
$98$		0			0
$99$	−0.676040	−	1.38032i	−0.0679446	−	0.138727i
$100$		0			0
$101$		−	11.4741i		−	1.14171i	−0.821050	−	0.570857i	$-0.806611\pi$
$101$		−	11.4741i		−	1.14171i	0.821050	−	0.570857i	$-0.193389\pi$
$102$		0			0
$103$			13.1567i			1.29637i	0.761485	+	0.648183i	$0.224471\pi$
$103$			13.1567i			1.29637i	−0.761485	+	0.648183i	$0.775529\pi$
$104$		0			0
$105$	−1.83052	+	2.93481i	−0.178640	+	0.286408i
$106$		0			0
$107$	14.1160			1.36464			0.682321	−	0.731052i	$-0.260970\pi$
$107$	14.1160			1.36464			0.682321	+	0.731052i	$0.260970\pi$
$108$		0			0
$109$	1.36090			0.130350			0.0651752	−	0.997874i	$-0.479239\pi$
$109$	1.36090			0.130350			0.0651752	+	0.997874i	$0.479239\pi$
$110$		0			0
$111$	8.66846	−	13.8979i	0.822774	−	1.31913i
$112$		0			0
$113$		−	18.1244i		−	1.70500i	−0.522729	−	0.852499i	$-0.675086\pi$
$113$		−	18.1244i		−	1.70500i	0.522729	−	0.852499i	$-0.324914\pi$
$114$		0			0
$115$			8.00000i			0.746004i
$116$		0			0
$117$	−3.87847	−	7.91893i	−0.358565	−	0.732105i
$118$		0			0
$119$	12.1190			1.11095
$120$		0			0
$121$	−10.7375			−0.976138
$122$		0			0
$123$	9.45590	+	5.89789i	0.852610	+	0.531795i
$124$		0			0
$125$			7.97124i			0.712969i
$126$		0			0
$127$			1.48243i			0.131544i	0.997835	+	0.0657721i	$0.0209510\pi$
$127$			1.48243i			0.131544i	−0.997835	+	0.0657721i	$0.979049\pi$
$128$		0			0
$129$	2.46962	+	1.54036i	0.217438	+	0.135621i
$130$		0			0
$131$	−6.82078			−0.595934			−0.297967	−	0.954576i	$-0.596309\pi$
$131$	−6.82078			−0.595934			−0.297967	+	0.954576i	$0.596309\pi$
$132$		0			0
$133$	−2.31955			−0.201131
$134$		0			0
$135$	0.456627	−	4.45020i	0.0393002	−	0.383012i
$136$		0			0
$137$			19.6681i			1.68036i	0.542307	+	0.840180i	$0.317551\pi$
$137$			19.6681i			1.68036i	−0.542307	+	0.840180i	$0.682449\pi$
$138$		0			0
$139$			17.2332i			1.46170i	0.682539	+	0.730849i	$0.260876\pi$
$139$			17.2332i			1.46170i	−0.682539	+	0.730849i	$0.739124\pi$
$140$		0			0
$141$	−6.04795	+	9.69650i	−0.509330	+	0.816593i
$142$		0			0
$143$	1.50585			0.125925
$144$		0			0
$145$	5.91730			0.491405
$146$		0			0
$147$	1.48466	−	2.38032i	0.122453	−	0.196325i
$148$		0			0
$149$			13.7295i			1.12476i	0.826878	+	0.562381i	$0.190114\pi$
$149$			13.7295i			1.12476i	−0.826878	+	0.562381i	$0.809886\pi$
$150$		0			0
$151$			3.18230i			0.258972i	0.991581	+	0.129486i	$0.0413327\pi$
$151$			3.18230i			0.258972i	−0.991581	+	0.129486i	$0.958667\pi$
$152$		0			0
$153$	−14.0765	+	6.89427i	−1.13802	+	0.557369i
$154$		0			0
$155$	−4.25238			−0.341559
$156$		0			0
$157$	−17.0351			−1.35955			−0.679776	−	0.733420i	$-0.737923\pi$
$157$	−17.0351			−1.35955			−0.679776	+	0.733420i	$0.737923\pi$
$158$		0			0
$159$	−4.87610	−	3.04135i	−0.386700	−	0.241195i
$160$		0			0
$161$			21.5537i			1.69867i
$162$		0			0
$163$		−	4.60027i		−	0.360321i	−0.983637	−	0.180160i	$-0.942338\pi$
$163$		−	4.60027i		−	0.360321i	0.983637	−	0.180160i	$-0.0576617\pi$
$164$		0			0
$165$	0.648220	+	0.404312i	0.0504639	+	0.0314756i
$166$		0			0
$167$	11.1043			0.859275			0.429638	−	0.903001i	$-0.358641\pi$
$167$	11.1043			0.859275			0.429638	+	0.903001i	$0.358641\pi$
$168$		0			0
$169$	−4.36090			−0.335454
$170$		0			0
$171$	2.69421	−	1.31955i	0.206032	−	0.100909i
$172$		0			0
$173$			2.73211i			0.207719i	0.994592	+	0.103859i	$0.0331192\pi$
$173$			2.73211i			0.207719i	−0.994592	+	0.103859i	$0.966881\pi$
$174$		0			0
$175$			9.87847i			0.746742i
$176$		0			0
$177$	−1.04135	+	1.66956i	−0.0782725	+	0.125492i
$178$		0			0
$179$	7.66053			0.572575			0.286287	−	0.958144i	$-0.407579\pi$
$179$	7.66053			0.572575			0.286287	+	0.958144i	$0.407579\pi$
$180$		0			0
$181$	16.5564			1.23063			0.615314	−	0.788282i	$-0.289029\pi$
$181$	16.5564			1.23063			0.615314	+	0.788282i	$0.289029\pi$
$182$		0			0
$183$	−1.85107	+	2.96777i	−0.136835	+	0.219384i
$184$		0			0
$185$			8.14172i			0.598591i
$186$		0			0
$187$		−	2.67676i		−	0.195744i
$188$		0			0
$189$	1.23025	−	11.9898i	0.0894875	−	0.872129i
$190$		0			0
$191$	−2.23420			−0.161661			−0.0808306	−	0.996728i	$-0.525757\pi$
$191$	−2.23420			−0.161661			−0.0808306	+	0.996728i	$0.525757\pi$
$192$		0			0
$193$	−14.3960			−1.03625			−0.518125	−	0.855305i	$-0.673370\pi$
$193$	−14.3960			−1.03625			−0.518125	+	0.855305i	$0.673370\pi$
$194$		0			0
$195$	3.71887	+	2.31955i	0.266314	+	0.166106i
$196$		0			0
$197$		−	17.8872i		−	1.27441i	−0.770696	−	0.637204i	$-0.780091\pi$
$197$		−	17.8872i		−	1.27441i	0.770696	−	0.637204i	$-0.219909\pi$
$198$		0			0
$199$		−	2.95865i		−	0.209733i	−0.994486	−	0.104867i	$-0.966558\pi$
$199$		−	2.95865i		−	0.209733i	0.994486	−	0.104867i	$-0.0334416\pi$
$200$		0			0
$201$	−10.5176	−	6.56008i	−0.741852	−	0.462712i
$202$		0			0
$203$	15.9425			1.11894
$204$		0			0
$205$	−5.53950			−0.386896
$206$		0			0
$207$	−12.2615	−	25.0351i	−0.852234	−	1.74006i
$208$		0			0
$209$			0.512326i			0.0354383i
$210$		0			0
$211$		−	26.6135i		−	1.83215i	−0.401010	−	0.916074i	$-0.631341\pi$
$211$		−	26.6135i		−	1.83215i	0.401010	−	0.916074i	$-0.368659\pi$
$212$		0			0
$213$	−9.47622	+	15.1929i	−0.649300	+	1.04100i
$214$		0			0
$215$	−1.44676			−0.0986684
$216$		0			0
$217$	−11.4568			−0.777739
$218$		0			0
$219$	−4.12318	+	6.61057i	−0.278618	+	0.446701i
$220$		0			0
$221$		−	15.3567i		−	1.03300i
$222$		0			0
$223$			22.0959i			1.47965i	0.672799	+	0.739826i	$0.265092\pi$
$223$			22.0959i			1.47965i	−0.672799	+	0.739826i	$0.734908\pi$
$224$		0			0
$225$	−5.61968	−	11.4741i	−0.374646	−	0.764938i
$226$		0			0
$227$	−9.49381			−0.630126			−0.315063	−	0.949071i	$-0.602026\pi$
$227$	−9.49381			−0.630126			−0.315063	+	0.949071i	$0.602026\pi$
$228$		0			0
$229$	5.85905			0.387177			0.193589	−	0.981083i	$-0.437987\pi$
$229$	5.85905			0.387177			0.193589	+	0.981083i	$0.437987\pi$
$230$		0			0
$231$	1.74644	+	1.08930i	0.114908	+	0.0716708i
$232$		0			0
$233$			15.5271i			1.01722i	0.860998	+	0.508608i	$0.169840\pi$
$233$			15.5271i			1.01722i	−0.860998	+	0.508608i	$0.830160\pi$
$234$		0			0
$235$		−	5.68045i		−	0.370552i
$236$		0			0
$237$	7.87847	+	4.91400i	0.511762	+	0.319199i
$238$		0			0
$239$	−26.6526			−1.72401			−0.862007	−	0.506897i	$-0.830793\pi$
$239$	−26.6526			−1.72401			−0.862007	+	0.506897i	$0.830793\pi$
$240$		0			0
$241$	−7.33528			−0.472507			−0.236253	−	0.971691i	$-0.575920\pi$
$241$	−7.33528			−0.472507			−0.236253	+	0.971691i	$0.575920\pi$
$242$		0			0
$243$	5.39181	+	14.6263i	0.345885	+	0.938277i
$244$		0			0
$245$			1.39445i			0.0890880i
$246$		0			0
$247$			2.93923i			0.187019i
$248$		0			0
$249$	−8.21744	+	13.1748i	−0.520759	+	0.834917i
$250$		0			0
$251$	−8.31985			−0.525144			−0.262572	−	0.964912i	$-0.584571\pi$
$251$	−8.31985			−0.525144			−0.262572	+	0.964912i	$0.584571\pi$
$252$		0			0
$253$	4.76063			0.299298
$254$		0			0
$255$	4.12318	−	6.61057i	0.258203	−	0.413970i
$256$		0			0
$257$		−	15.0504i		−	0.938819i	−0.882981	−	0.469409i	$-0.844467\pi$
$257$		−	15.0504i		−	0.938819i	0.882981	−	0.469409i	$-0.155533\pi$
$258$		0			0
$259$			21.9355i			1.36301i
$260$		0			0
$261$	−18.5176	+	9.06939i	−1.14621	+	0.561381i
$262$		0			0
$263$	−6.12355			−0.377594			−0.188797	−	0.982016i	$-0.560459\pi$
$263$	−6.12355			−0.377594			−0.188797	+	0.982016i	$0.560459\pi$
$264$		0			0
$265$	2.85654			0.175476
$266$		0			0
$267$	−1.53698	−	0.958652i	−0.0940616	−	0.0586686i
$268$		0			0
$269$			10.4282i			0.635821i	0.948121	+	0.317911i	$0.102981\pi$
$269$			10.4282i			0.635821i	−0.948121	+	0.317911i	$0.897019\pi$
$270$		0			0
$271$		−	0.677937i		−	0.0411817i	−0.999788	−	0.0205909i	$-0.993445\pi$
$271$		−	0.677937i		−	0.0411817i	0.999788	−	0.0205909i	$-0.00655474\pi$
$272$		0			0
$273$	10.0194	+	6.24936i	0.606403	+	0.378229i
$274$		0			0
$275$	2.18189			0.131573
$276$		0			0
$277$	−0.0194175			−0.00116669			−0.000583344	−	1.00000i	$-0.500186\pi$
$277$	−0.0194175			−0.00116669			−0.000583344		1.00000i	$0.500186\pi$
$278$		0			0
$279$	13.3074	−	6.51757i	0.796690	−	0.390197i
$280$		0			0
$281$			1.81209i			0.108100i	0.998538	+	0.0540502i	$0.0172131\pi$
$281$			1.81209i			0.108100i	−0.998538	+	0.0540502i	$0.982787\pi$
$282$		0			0
$283$		−	29.1943i		−	1.73542i	−0.497068	−	0.867711i	$-0.665590\pi$
$283$		−	29.1943i		−	1.73542i	0.497068	−	0.867711i	$-0.334410\pi$
$284$		0			0
$285$	−0.789168	+	1.26525i	−0.0467463	+	0.0749469i
$286$		0			0
$287$	−14.9246			−0.880971
$288$		0			0
$289$	−10.2976			−0.605742
$290$		0			0
$291$	−7.53241	+	12.0765i	−0.441558	+	0.707936i
$292$		0			0
$293$			3.53397i			0.206457i	0.994658	+	0.103228i	$0.0329172\pi$
$293$			3.53397i			0.206457i	−0.994658	+	0.103228i	$0.967083\pi$
$294$		0			0
$295$		−	0.978070i		−	0.0569454i
$296$		0			0
$297$	−2.64822	−	0.271729i	−0.153665	−	0.0157673i
$298$		0			0
$299$	27.3119			1.57949
$300$		0			0
$301$	−3.89789			−0.224670
$302$		0			0
$303$	−16.8625	−	10.5176i	−0.968726	−	0.604219i
$304$		0			0
$305$		−	1.73859i		−	0.0995516i
$306$		0			0
$307$		−	10.6391i		−	0.607206i	−0.952799	−	0.303603i	$-0.901810\pi$
$307$		−	10.6391i		−	0.607206i	0.952799	−	0.303603i	$-0.0981896\pi$
$308$		0			0
$309$	19.3353	+	12.0599i	1.09995	+	0.686064i
$310$		0			0
$311$	7.76963			0.440575			0.220288	−	0.975435i	$-0.429300\pi$
$311$	7.76963			0.440575			0.220288	+	0.975435i	$0.429300\pi$
$312$		0			0
$313$	−30.0703			−1.69967			−0.849837	−	0.527046i	$-0.823299\pi$
$313$	−30.0703			−1.69967			−0.849837	+	0.527046i	$0.823299\pi$
$314$		0			0
$315$	2.63513	+	5.38032i	0.148473	+	0.303146i
$316$		0			0
$317$			7.13149i			0.400544i	0.979740	+	0.200272i	$0.0641826\pi$
$317$			7.13149i			0.400544i	−0.979740	+	0.200272i	$0.935817\pi$
$318$		0			0
$319$		−	3.52126i		−	0.197153i
$320$		0			0
$321$	12.9392	−	20.7451i	0.722198	−	1.15788i
$322$		0			0
$323$	5.22471			0.290711
$324$		0			0
$325$	12.5176			0.694350
$326$		0			0
$327$	1.24745	−	2.00000i	0.0689842	−	0.110600i
$328$		0			0
$329$		−	15.3043i		−	0.843756i
$330$		0			0
$331$			22.1530i			1.21764i	0.793309	+	0.608819i	$0.208356\pi$
$331$			22.1530i			1.21764i	−0.793309	+	0.608819i	$0.791644\pi$
$332$		0			0
$333$	−12.4787	−	25.4787i	−0.683831	−	1.39622i
$334$		0			0
$335$	6.16145			0.336636
$336$		0			0
$337$	12.8177			0.698225			0.349112	−	0.937081i	$-0.386483\pi$
$337$	12.8177			0.698225			0.349112	+	0.937081i	$0.386483\pi$
$338$		0			0
$339$	−26.6359	−	16.6135i	−1.44666	−	0.902320i
$340$		0			0
$341$			2.53050i			0.137034i
$342$		0			0
$343$			19.9938i			1.07956i
$344$		0			0
$345$	11.7569	+	7.33310i	0.632972	+	0.394801i
$346$		0			0
$347$	−27.6015			−1.48172			−0.740862	−	0.671657i	$-0.765583\pi$
$347$	−27.6015			−1.48172			−0.740862	+	0.671657i	$0.765583\pi$
$348$		0			0
$349$	22.6900			1.21457			0.607283	−	0.794486i	$-0.292259\pi$
$349$	22.6900			1.21457			0.607283	+	0.794486i	$0.292259\pi$
$350$		0			0
$351$	−15.1929	−	1.55892i	−0.810939	−	0.0832089i
$352$		0			0
$353$			27.0891i			1.44181i	0.693034	+	0.720904i	$0.256273\pi$
$353$			27.0891i			1.44181i	−0.693034	+	0.720904i	$0.743727\pi$
$354$		0			0
$355$		−	8.90040i		−	0.472384i
$356$		0			0
$357$	11.1087	−	17.8103i	0.587936	−	0.942620i
$358$		0			0
$359$	−12.4320			−0.656136			−0.328068	−	0.944654i	$-0.606397\pi$
$359$	−12.4320			−0.656136			−0.328068	+	0.944654i	$0.606397\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	−9.84242	+	15.7800i	−0.516593	+	0.828238i
$364$		0			0
$365$		−	3.87263i		−	0.202703i
$366$		0			0
$367$			20.0000i			1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$			20.0000i			1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$		0			0
$369$	17.3353	−	8.49034i	0.902438	−	0.441989i
$370$		0			0
$371$	7.69613			0.399563
$372$		0			0
$373$	5.79577			0.300094			0.150047	−	0.988679i	$-0.452058\pi$
$373$	5.79577			0.300094			0.150047	+	0.988679i	$0.452058\pi$
$374$		0			0
$375$	11.7147	+	7.30674i	0.604943	+	0.377318i
$376$		0			0
$377$		−	20.2016i		−	1.04044i
$378$		0			0
$379$		−	35.6354i		−	1.83047i	−0.402923	−	0.915234i	$-0.632006\pi$
$379$		−	35.6354i		−	1.83047i	0.402923	−	0.915234i	$-0.367994\pi$
$380$		0			0
$381$	2.17860	+	1.35885i	0.111613	+	0.0696160i
$382$		0			0
$383$	37.2802			1.90493			0.952464	−	0.304652i	$-0.0985403\pi$
$383$	37.2802			1.90493			0.952464	+	0.304652i	$0.0985403\pi$
$384$		0			0
$385$	−1.02311			−0.0521425
$386$		0			0
$387$	4.52749	−	2.21744i	0.230145	−	0.112719i
$388$		0			0
$389$		−	24.2547i		−	1.22976i	−0.788620	−	0.614881i	$-0.789204\pi$
$389$		−	24.2547i		−	1.22976i	0.788620	−	0.614881i	$-0.210796\pi$
$390$		0			0
$391$		−	48.5490i		−	2.45523i
$392$		0			0
$393$	−6.25218	+	10.0239i	−0.315381	+	0.505640i
$394$		0			0
$395$	−4.61540			−0.232226
$396$		0			0
$397$	−23.1373			−1.16123			−0.580613	−	0.814180i	$-0.697187\pi$
$397$	−23.1373			−1.16123			−0.580613	+	0.814180i	$0.697187\pi$
$398$		0			0
$399$	−2.12619	+	3.40885i	−0.106443	+	0.170656i
$400$		0			0
$401$			25.1300i			1.25493i	0.778643	+	0.627467i	$0.215908\pi$
$401$			25.1300i			1.25493i	−0.778643	+	0.627467i	$0.784092\pi$
$402$		0			0
$403$			14.5176i			0.723172i
$404$		0			0
$405$	−6.12153	−	4.75029i	−0.304181	−	0.236044i
$406$		0			0
$407$	4.84497			0.240156
$408$		0			0
$409$	20.8177			1.02937			0.514685	−	0.857380i	$-0.327909\pi$
$409$	20.8177			1.02937			0.514685	+	0.857380i	$0.327909\pi$
$410$		0			0
$411$	28.9046	+	18.0285i	1.42576	+	0.889282i
$412$		0			0
$413$		−	2.63513i		−	0.129666i
$414$		0			0
$415$		−	7.71810i		−	0.378867i
$416$		0			0
$417$	25.3262	+	15.7966i	1.24023	+	0.773561i
$418$		0			0
$419$	19.3718			0.946375			0.473188	−	0.880962i	$-0.343103\pi$
$419$	19.3718			0.946375			0.473188	+	0.880962i	$0.343103\pi$
$420$		0			0
$421$	27.7313			1.35154			0.675771	−	0.737112i	$-0.263811\pi$
$421$	27.7313			1.35154			0.675771	+	0.737112i	$0.263811\pi$
$422$		0			0
$423$	8.70637	+	17.7764i	0.423318	+	0.864316i
$424$		0			0
$425$		−	22.2509i		−	1.07933i
$426$		0			0
$427$		−	4.68414i		−	0.226681i
$428$		0			0
$429$	1.38032	−	2.21302i	0.0666423	−	0.106846i
$430$		0			0
$431$	−39.3295			−1.89443			−0.947217	−	0.320594i	$-0.896118\pi$
$431$	−39.3295			−1.89443			−0.947217	+	0.320594i	$0.896118\pi$
$432$		0			0
$433$	6.35721			0.305508			0.152754	−	0.988264i	$-0.451186\pi$
$433$	6.35721			0.305508			0.152754	+	0.988264i	$0.451186\pi$
$434$		0			0
$435$	5.42402	−	8.69617i	0.260062	−	0.416950i
$436$		0			0
$437$			9.29219i			0.444506i
$438$		0			0
$439$		−	16.2357i		−	0.774887i	−0.921894	−	0.387443i	$-0.873358\pi$
$439$		−	16.2357i		−	0.774887i	0.921894	−	0.387443i	$-0.126642\pi$
$440$		0			0
$441$	−2.13726	−	4.36377i	−0.101774	−	0.207799i
$442$		0			0
$443$	29.9494			1.42294			0.711469	−	0.702718i	$-0.248030\pi$
$443$	29.9494			1.42294			0.711469	+	0.702718i	$0.248030\pi$
$444$		0			0
$445$	0.900400			0.0426830
$446$		0			0
$447$	20.1771	+	12.5849i	0.954342	+	0.595247i
$448$		0			0
$449$		−	31.2915i		−	1.47674i	−0.674398	−	0.738368i	$-0.735597\pi$
$449$		−	31.2915i		−	1.47674i	0.674398	−	0.738368i	$-0.264403\pi$
$450$		0			0
$451$			3.29644i			0.155223i
$452$		0			0
$453$	4.67676	+	2.91701i	0.219733	+	0.137053i
$454$		0			0
$455$	−5.86962			−0.275172
$456$		0			0
$457$	7.89789			0.369448			0.184724	−	0.982790i	$-0.440861\pi$
$457$	7.89789			0.369448			0.184724	+	0.982790i	$0.440861\pi$
$458$		0			0
$459$	−2.77110	+	27.0066i	−0.129344	+	1.26056i
$460$		0			0
$461$		−	10.7178i		−	0.499176i	−0.968352	−	0.249588i	$-0.919705\pi$
$461$		−	10.7178i		−	0.499176i	0.968352	−	0.249588i	$-0.0802952\pi$
$462$		0			0
$463$		−	22.7156i		−	1.05568i	−0.849343	−	0.527842i	$-0.823001\pi$
$463$		−	22.7156i		−	1.05568i	0.849343	−	0.527842i	$-0.176999\pi$
$464$		0			0
$465$	−3.89789	+	6.24936i	−0.180760	+	0.289807i
$466$		0			0
$467$	−0.365329			−0.0169054			−0.00845271	−	0.999964i	$-0.502691\pi$
$467$	−0.365329			−0.0169054			−0.00845271	+	0.999964i	$0.502691\pi$
$468$		0			0
$469$	16.6003			0.766529
$470$		0			0
$471$	−15.6151	+	25.0351i	−0.719504	+	1.15356i
$472$		0			0
$473$			0.860938i			0.0395860i
$474$		0			0
$475$			4.25879i			0.195406i
$476$		0			0
$477$	−8.93923	+	4.37819i	−0.409299	+	0.200463i
$478$		0			0
$479$	−9.11176			−0.416327			−0.208163	−	0.978094i	$-0.566749\pi$
$479$	−9.11176			−0.416327			−0.208163	+	0.978094i	$0.566749\pi$
$480$		0			0
$481$	27.7958			1.26738
$482$		0			0
$483$	31.6757	+	19.7569i	1.44129	+	0.898972i
$484$		0			0
$485$		−	7.07471i		−	0.321246i
$486$		0			0
$487$			38.1274i			1.72772i	0.503736	+	0.863858i	$0.331959\pi$
$487$			38.1274i			1.72772i	−0.503736	+	0.863858i	$0.668041\pi$
$488$		0			0
$489$	−6.76063	−	4.21678i	−0.305726	−	0.190689i
$490$		0			0
$491$	−36.1297			−1.63051			−0.815255	−	0.579102i	$-0.803403\pi$
$491$	−36.1297			−1.63051			−0.815255	+	0.579102i	$0.803403\pi$
$492$		0			0
$493$	−35.9099			−1.61730
$494$		0			0
$495$	1.18837	−	0.582029i	0.0534131	−	0.0261603i
$496$		0			0
$497$		−	23.9796i		−	1.07563i
$498$		0			0
$499$		−	9.88467i		−	0.442499i	−0.975217	−	0.221249i	$-0.928987\pi$
$499$		−	9.88467i		−	0.442499i	0.975217	−	0.221249i	$-0.0710135\pi$
$500$		0			0
$501$	10.1786	−	16.3190i	0.454746	−	0.729081i
$502$		0			0
$503$	32.2403			1.43753			0.718763	−	0.695255i	$-0.244709\pi$
$503$	32.2403			1.43753			0.718763	+	0.695255i	$0.244709\pi$
$504$		0			0
$505$	9.87847			0.439586
$506$		0			0
$507$	−3.99736	+	6.40885i	−0.177529	+	0.284627i
$508$		0			0
$509$			12.9943i			0.575964i	0.957636	+	0.287982i	$0.0929843\pi$
$509$			12.9943i			0.575964i	−0.957636	+	0.287982i	$0.907016\pi$
$510$		0			0
$511$		−	10.4337i		−	0.461560i
$512$		0			0
$513$	0.530383	−	5.16901i	0.0234170	−	0.228217i
$514$		0			0
$515$	−11.3271			−0.499131
$516$		0			0
$517$	−3.38032			−0.148666
$518$		0			0
$519$	4.01516	+	2.50436i	0.176246	+	0.109929i
$520$		0			0
$521$		−	10.4638i		−	0.458429i	−0.973376	−	0.229215i	$-0.926384\pi$
$521$		−	10.4638i		−	0.458429i	0.973376	−	0.229215i	$-0.0736157\pi$
$522$		0			0
$523$		−	35.1178i		−	1.53560i	−0.640692	−	0.767798i	$-0.721352\pi$
$523$		−	35.1178i		−	1.53560i	0.640692	−	0.767798i	$-0.278648\pi$
$524$		0			0
$525$	14.5176	+	9.05498i	0.633599	+	0.395192i
$526$		0			0
$527$	25.8061			1.12413
$528$		0			0
$529$	63.3448			2.75412
$530$		0			0
$531$	1.49908	+	3.06077i	0.0650544	+	0.132826i
$532$		0			0
$533$			18.9118i			0.819161i
$534$		0			0
$535$			12.1530i			0.525419i
$536$		0			0
$537$	7.02193	−	11.2580i	0.303019	−	0.485821i
$538$		0			0
$539$	0.829806			0.0357423
$540$		0			0
$541$	0.536989			0.0230870			0.0115435	−	0.999933i	$-0.496326\pi$
$541$	0.536989			0.0230870			0.0115435	+	0.999933i	$0.496326\pi$
$542$		0			0
$543$	15.1762	−	24.3316i	0.651274	−	1.04417i
$544$		0			0
$545$			1.17165i			0.0501879i
$546$		0			0
$547$			22.6573i			0.968758i	0.874858	+	0.484379i	$0.160954\pi$
$547$			22.6573i			0.968758i	−0.874858	+	0.484379i	$0.839046\pi$
$548$		0			0
$549$	2.66472	+	5.44074i	0.113728	+	0.232205i
$550$		0			0
$551$	6.87309			0.292803
$552$		0			0
$553$	−12.4349			−0.528785
$554$		0			0
$555$	11.9652	+	7.46301i	0.507895	+	0.316787i
$556$		0			0
$557$			39.1725i			1.65979i	0.557917	+	0.829896i	$0.311601\pi$
$557$			39.1725i			1.65979i	−0.557917	+	0.829896i	$0.688399\pi$
$558$		0			0
$559$			4.93923i			0.208907i
$560$		0			0
$561$	−3.93381	−	2.45362i	−0.166086	−	0.103592i
$562$		0			0
$563$	−11.9417			−0.503284			−0.251642	−	0.967820i	$-0.580971\pi$
$563$	−11.9417			−0.503284			−0.251642	+	0.967820i	$0.580971\pi$
$564$		0			0
$565$	15.6040			0.656463
$566$		0			0
$567$	−16.4927	−	12.7983i	−0.692629	−	0.537478i
$568$		0			0
$569$		−	25.3461i		−	1.06256i	−0.847195	−	0.531281i	$-0.821711\pi$
$569$		−	25.3461i		−	1.06256i	0.847195	−	0.531281i	$-0.178289\pi$
$570$		0			0
$571$		−	27.1567i		−	1.13647i	−0.822866	−	0.568236i	$-0.807626\pi$
$571$		−	27.1567i		−	1.13647i	0.822866	−	0.568236i	$-0.192374\pi$
$572$		0			0
$573$	−2.04795	+	3.28342i	−0.0855545	+	0.137167i
$574$		0			0
$575$	39.5734			1.65033
$576$		0			0
$577$	−14.0194			−0.583636			−0.291818	−	0.956474i	$-0.594260\pi$
$577$	−14.0194			−0.583636			−0.291818	+	0.956474i	$0.594260\pi$
$578$		0			0
$579$	−13.1960	+	21.1567i	−0.548405	+	0.879241i
$580$		0			0
$581$		−	20.7942i		−	0.862690i
$582$		0			0
$583$		−	1.69987i		−	0.0704013i
$584$		0			0
$585$	6.81770	−	3.33912i	0.281877	−	0.138056i
$586$		0			0
$587$	−35.7076			−1.47381			−0.736905	−	0.675996i	$-0.763713\pi$
$587$	−35.7076			−1.47381			−0.736905	+	0.675996i	$0.763713\pi$
$588$		0			0
$589$	−4.93923			−0.203518
$590$		0			0
$591$	−26.2873	−	16.3960i	−1.08131	−	0.674443i
$592$		0			0
$593$			4.73357i			0.194384i	0.995266	+	0.0971922i	$0.0309862\pi$
$593$			4.73357i			0.194384i	−0.995266	+	0.0971922i	$0.969014\pi$
$594$		0			0
$595$			10.4337i			0.427740i
$596$		0			0
$597$	−4.34809	−	2.71201i	−0.177955	−	0.110995i
$598$		0			0
$599$	−20.6761			−0.844801			−0.422400	−	0.906409i	$-0.638812\pi$
$599$	−20.6761			−0.844801			−0.422400	+	0.906409i	$0.638812\pi$
$600$		0			0
$601$	−17.2137			−0.702163			−0.351082	−	0.936345i	$-0.614186\pi$
$601$	−17.2137			−0.702163			−0.351082	+	0.936345i	$0.614186\pi$
$602$		0			0
$603$	−19.2816	+	9.44359i	−0.785208	+	0.384573i
$604$		0			0
$605$		−	9.24434i		−	0.375836i
$606$		0			0
$607$		−	5.53950i		−	0.224841i	−0.993661	−	0.112421i	$-0.964140\pi$
$607$		−	5.53950i		−	0.224841i	0.993661	−	0.112421i	$-0.0358604\pi$
$608$		0			0
$609$	14.6135	−	23.4293i	0.592168	−	0.949405i
$610$		0			0
$611$	−19.3930			−0.784557
$612$		0			0
$613$	6.41546			0.259118			0.129559	−	0.991572i	$-0.458644\pi$
$613$	6.41546			0.259118			0.129559	+	0.991572i	$0.458644\pi$
$614$		0			0
$615$	−5.07772	+	8.14095i	−0.204753	+	0.328275i
$616$		0			0
$617$			41.5158i			1.67136i	0.549214	+	0.835682i	$0.314927\pi$
$617$			41.5158i			1.67136i	−0.549214	+	0.835682i	$0.685073\pi$
$618$		0			0
$619$		−	30.7094i		−	1.23431i	−0.786840	−	0.617157i	$-0.788284\pi$
$619$		−	30.7094i		−	1.23431i	0.786840	−	0.617157i	$-0.211716\pi$
$620$		0			0
$621$	−48.0315	−	4.92842i	−1.92744	−	0.197771i
$622$		0			0
$623$	2.42587			0.0971904
$624$		0			0
$625$	14.4312			0.577247
$626$		0			0
$627$	0.752923	+	0.469617i	0.0300689	+	0.0187547i
$628$		0			0
$629$		−	49.4091i		−	1.97007i
$630$		0			0
$631$		−	2.56261i		−	0.102016i	−0.998698	−	0.0510080i	$-0.983757\pi$
$631$		−	2.56261i		−	0.102016i	0.998698	−	0.0510080i	$-0.0162434\pi$
$632$		0			0
$633$	−39.1116	−	24.3949i	−1.55455	−	0.969611i
$634$		0			0
$635$	−1.27628			−0.0506476
$636$		0			0
$637$	4.76063			0.188623
$638$		0			0
$639$	13.6416	+	27.8528i	0.539651	+	1.10184i
$640$		0			0
$641$		−	25.3884i		−	1.00278i	−0.865221	−	0.501392i	$-0.832822\pi$
$641$		−	25.3884i		−	1.00278i	0.865221	−	0.501392i	$-0.167178\pi$
$642$		0			0
$643$		−	43.0728i		−	1.69863i	−0.527890	−	0.849313i	$-0.677017\pi$
$643$		−	43.0728i		−	1.69863i	0.527890	−	0.849313i	$-0.322983\pi$
$644$		0			0
$645$	−1.32616	+	2.12619i	−0.0522174	+	0.0837186i
$646$		0			0
$647$	8.87008			0.348719			0.174359	−	0.984682i	$-0.444215\pi$
$647$	8.87008			0.348719			0.174359	+	0.984682i	$0.444215\pi$
$648$		0			0
$649$	−0.582029			−0.0228466
$650$		0			0
$651$	−10.5017	+	16.8371i	−0.411596	+	0.659899i
$652$		0			0
$653$			7.61498i			0.297997i	0.988837	+	0.148999i	$0.0476050\pi$
$653$			7.61498i			0.297997i	−0.988837	+	0.148999i	$0.952395\pi$
$654$		0			0
$655$		−	5.87226i		−	0.229448i
$656$		0			0
$657$	5.93554	+	12.1190i	0.231568	+	0.472807i
$658$		0			0
$659$	−25.8129			−1.00553			−0.502763	−	0.864424i	$-0.667683\pi$
$659$	−25.8129			−1.00553			−0.502763	+	0.864424i	$0.667683\pi$
$660$		0			0
$661$	−22.9963			−0.894453			−0.447226	−	0.894421i	$-0.647588\pi$
$661$	−22.9963			−0.894453			−0.447226	+	0.894421i	$0.647588\pi$
$662$		0			0
$663$	−22.5684	−	14.0765i	−0.876484	−	0.546686i
$664$		0			0
$665$		−	1.99699i		−	0.0774399i
$666$		0			0
$667$		−	63.8661i		−	2.47290i
$668$		0			0
$669$	32.4725	+	20.2539i	1.25546	+	0.783063i
$670$		0			0
$671$	−1.03460			−0.0399403
$672$		0			0
$673$	23.7702			0.916272			0.458136	−	0.888882i	$-0.348517\pi$
$673$	23.7702			0.916272			0.458136	+	0.888882i	$0.348517\pi$
$674$		0			0
$675$	−22.0137	−	2.25879i	−0.847308	−	0.0869407i
$676$		0			0
$677$			40.5981i			1.56031i	0.625585	+	0.780156i	$0.284860\pi$
$677$			40.5981i			1.56031i	−0.625585	+	0.780156i	$0.715140\pi$
$678$		0			0
$679$		−	19.0608i		−	0.731485i
$680$		0			0
$681$	−8.70238	+	13.9523i	−0.333476	+	0.534652i
$682$		0			0
$683$	−4.69797			−0.179763			−0.0898815	−	0.995952i	$-0.528649\pi$
$683$	−4.69797			−0.179763			−0.0898815	+	0.995952i	$0.528649\pi$
$684$		0			0
$685$	−16.9330			−0.646978
$686$		0			0
$687$	5.37063	−	8.61057i	0.204902	−	0.328514i
$688$		0			0
$689$		−	9.75220i		−	0.371529i
$690$		0			0
$691$			13.9235i			0.529675i	0.964293	+	0.264838i	$0.0853184\pi$
$691$			13.9235i			0.529675i	−0.964293	+	0.264838i	$0.914682\pi$
$692$		0			0
$693$	3.20171	−	1.56811i	0.121623	−	0.0595676i
$694$		0			0
$695$	−14.8367			−0.562788
$696$		0			0
$697$	33.6172			1.27334
$698$		0			0
$699$	22.8190	+	14.2328i	0.863092	+	0.538332i
$700$		0			0
$701$			20.8565i			0.787738i	0.919166	+	0.393869i	$0.128864\pi$
$701$			20.8565i			0.787738i	−0.919166	+	0.393869i	$0.871136\pi$
$702$		0			0
$703$			9.45681i			0.356670i
$704$		0			0
$705$	−8.34809	−	5.20691i	−0.314407	−	0.196104i
$706$		0			0
$707$	26.6147			1.00095
$708$		0			0
$709$	0.843327			0.0316718			0.0158359	−	0.999875i	$-0.494959\pi$
$709$	0.843327			0.0316718			0.0158359	+	0.999875i	$0.494959\pi$
$710$		0			0
$711$	14.4434	−	7.07398i	0.541670	−	0.265295i
$712$		0			0
$713$			45.8963i			1.71883i
$714$		0			0
$715$			1.29644i			0.0484841i
$716$		0			0
$717$	−24.4308	+	39.1691i	−0.912384	+	1.46280i
$718$		0			0
$719$	−0.630495			−0.0235135			−0.0117567	−	0.999931i	$-0.503742\pi$
$719$	−0.630495			−0.0235135			−0.0117567	+	0.999931i	$0.503742\pi$
$720$		0			0
$721$	−30.5176			−1.13653
$722$		0			0
$723$	−6.72379	+	10.7800i	−0.250060	+	0.400914i
$724$		0			0
$725$		−	29.2710i		−	1.08710i
$726$		0			0
$727$			44.9901i			1.66859i	0.551318	+	0.834295i	$0.314125\pi$
$727$			44.9901i			1.66859i	−0.551318	+	0.834295i	$0.685875\pi$
$728$		0			0
$729$	26.4374	+	5.48311i	0.979163	+	0.203078i
$730$		0			0
$731$	8.77986			0.324735
$732$		0			0
$733$	−33.9926			−1.25555			−0.627773	−	0.778397i	$-0.716033\pi$
$733$	−33.9926			−1.25555			−0.627773	+	0.778397i	$0.716033\pi$
$734$		0			0
$735$	2.04930	+	1.27820i	0.0755897	+	0.0471472i
$736$		0			0
$737$		−	3.66655i		−	0.135059i
$738$		0			0
$739$		−	42.4337i		−	1.56095i	−0.625188	−	0.780474i	$-0.714978\pi$
$739$		−	42.4337i		−	1.56095i	0.625188	−	0.780474i	$-0.285022\pi$
$740$		0			0
$741$	4.31955	+	2.69421i	0.158683	+	0.0989744i
$742$		0			0
$743$	−35.5227			−1.30320			−0.651600	−	0.758562i	$-0.725902\pi$
$743$	−35.5227			−1.30320			−0.651600	+	0.758562i	$0.725902\pi$
$744$		0			0
$745$	−11.8202			−0.433059
$746$		0			0
$747$	11.8295	+	24.1530i	0.432817	+	0.883711i
$748$		0			0
$749$			32.7427i			1.19639i
$750$		0			0
$751$			9.30965i			0.339714i	0.985469	+	0.169857i	$0.0543306\pi$
$751$			9.30965i			0.339714i	−0.985469	+	0.169857i	$0.945669\pi$
$752$		0			0
$753$	−7.62629	+	12.2270i	−0.277917	+	0.445577i
$754$		0			0
$755$	−2.73976			−0.0997100
$756$		0			0
$757$	−24.8503			−0.903201			−0.451600	−	0.892220i	$-0.649147\pi$
$757$	−24.8503			−0.903201			−0.451600	+	0.892220i	$0.649147\pi$
$758$		0			0
$759$	4.36377	−	6.99631i	0.158395	−	0.253950i
$760$		0			0
$761$			7.49682i			0.271759i	0.990725	+	0.135880i	$0.0433861\pi$
$761$			7.49682i			0.271759i	−0.990725	+	0.135880i	$0.956614\pi$
$762$		0			0
$763$			3.15667i			0.114279i
$764$		0			0
$765$	−5.93554	−	12.1190i	−0.214600	−	0.438163i
$766$		0			0
$767$	−3.33912			−0.120569
$768$		0			0
$769$	16.8942			0.609220			0.304610	−	0.952477i	$-0.401474\pi$
$769$	16.8942			0.609220			0.304610	+	0.952477i	$0.401474\pi$
$770$		0			0
$771$	−22.1183	−	13.7958i	−0.796573	−	0.496843i
$772$		0			0
$773$			8.88680i			0.319636i	0.987147	+	0.159818i	$0.0510907\pi$
$773$			8.88680i			0.319636i	−0.987147	+	0.159818i	$0.948909\pi$
$774$		0			0
$775$			21.0351i			0.755605i
$776$		0			0
$777$	32.2369	+	20.1069i	1.15649	+	0.721333i
$778$		0			0
$779$	−6.43426			−0.230531
$780$		0			0
$781$	−5.29644			−0.189522
$782$		0			0
$783$	−3.64537	+	35.5271i	−0.130275	+	1.26963i
$784$		0			0
$785$		−	14.6662i		−	0.523459i
$786$		0			0
$787$			16.9575i			0.604469i	0.953234	+	0.302234i	$0.0977325\pi$
$787$			16.9575i			0.604469i	−0.953234	+	0.302234i	$0.902267\pi$
$788$		0			0
$789$	−5.61308	+	8.99928i	−0.199831	+	0.320383i
$790$		0			0
$791$	42.0404			1.49478
$792$		0			0
$793$	−5.93554			−0.210777
$794$		0			0
$795$	2.61841	−	4.19802i	0.0928655	−	0.148888i
$796$		0			0
$797$			24.6556i			0.873347i	0.899620	+	0.436673i	$0.143843\pi$
$797$			24.6556i			0.873347i	−0.899620	+	0.436673i	$0.856157\pi$
$798$		0			0
$799$			34.4725i			1.21955i
$800$		0			0
$801$	−2.81770	+	1.38003i	−0.0995587	+	0.0487611i
$802$		0			0
$803$	−2.30452			−0.0813248
$804$		0			0
$805$	−18.5564			−0.654027
$806$		0			0
$807$	15.3255	+	9.55892i	0.539484	+	0.336490i
$808$		0			0
$809$		−	24.0319i		−	0.844916i	−0.906382	−	0.422458i	$-0.861167\pi$
$809$		−	24.0319i		−	0.844916i	0.906382	−	0.422458i	$-0.138833\pi$
$810$		0			0
$811$		−	26.9004i		−	0.944601i	−0.881438	−	0.472300i	$-0.843424\pi$
$811$		−	26.9004i		−	0.944601i	0.881438	−	0.472300i	$-0.156576\pi$
$812$		0			0
$813$	−0.996308	−	0.621422i	−0.0349420	−	0.0217942i
$814$		0			0
$815$	3.96054			0.138732
$816$		0			0
$817$	−1.68045			−0.0587915
$818$		0			0
$819$	18.3684	−	8.99631i	0.641842	−	0.314356i
$820$		0			0
$821$		−	0.0455458i		−	0.00158956i	−1.00000		0.000794780i	$-0.999747\pi$
$821$		−	0.0455458i		−	0.00158956i	1.00000		0.000794780i	$-0.000252986\pi$
$822$		0			0
$823$		−	13.5151i		−	0.471105i	−0.971862	−	0.235553i	$-0.924310\pi$
$823$		−	13.5151i		−	0.471105i	0.971862	−	0.235553i	$-0.0756900\pi$
$824$		0			0
$825$	2.00000	−	3.20654i	0.0696311	−	0.111637i
$826$		0			0
$827$	−29.8492			−1.03796			−0.518979	−	0.854787i	$-0.673688\pi$
$827$	−29.8492			−1.03796			−0.518979	+	0.854787i	$0.673688\pi$
$828$		0			0
$829$	−31.0740			−1.07924			−0.539622	−	0.841907i	$-0.681433\pi$
$829$	−31.0740			−1.07924			−0.539622	+	0.841907i	$0.681433\pi$
$830$		0			0
$831$	−0.0177989	+	0.0285364i	−0.000617435	+	0.000989916i
$832$		0			0
$833$		−	8.46238i		−	0.293204i
$834$		0			0
$835$			9.56010i			0.330841i
$836$		0			0
$837$	2.61968	−	25.5310i	0.0905495	−	0.882479i
$838$		0			0
$839$	−12.7504			−0.440191			−0.220096	−	0.975478i	$-0.570637\pi$
$839$	−12.7504			−0.440191			−0.220096	+	0.975478i	$0.570637\pi$
$840$		0			0
$841$	−18.2394			−0.628944
$842$		0			0
$843$	2.66308	+	1.66103i	0.0917214	+	0.0572090i
$844$		0			0
$845$		−	3.75446i		−	0.129157i
$846$		0			0
$847$		−	24.9062i		−	0.855788i
$848$		0			0
$849$	−42.9045	−	26.7606i	−1.47248	−	0.918422i
$850$		0			0
$851$	87.8744			3.01230
$852$		0			0
$853$	−49.5527			−1.69665			−0.848326	−	0.529474i	$-0.822389\pi$
$853$	−49.5527			−1.69665			−0.848326	+	0.529474i	$0.822389\pi$
$854$		0			0
$855$	1.13605	+	2.31955i	0.0388521	+	0.0793270i
$856$		0			0
$857$			30.4829i			1.04128i	0.853778	+	0.520638i	$0.174306\pi$
$857$			30.4829i			1.04128i	−0.853778	+	0.520638i	$0.825694\pi$
$858$		0			0
$859$		−	0.645307i		−	0.0220176i	−0.999939	−	0.0110088i	$-0.996496\pi$
$859$		−	0.645307i		−	0.0220176i	0.999939	−	0.0110088i	$-0.00350428\pi$
$860$		0			0
$861$	−13.6804	+	21.9335i	−0.466228	+	0.747490i
$862$		0			0
$863$	28.8800			0.983088			0.491544	−	0.870853i	$-0.336433\pi$
$863$	28.8800			0.983088			0.491544	+	0.870853i	$0.336433\pi$
$864$		0			0
$865$	−2.35218			−0.0799766
$866$		0			0
$867$	−9.43919	+	15.1336i	−0.320572	+	0.513963i
$868$		0			0
$869$			2.74653i			0.0931696i
$870$		0			0
$871$		−	21.0351i		−	0.712749i
$872$		0			0
$873$	10.8433	+	22.1395i	0.366991	+	0.749310i
$874$		0			0
$875$	−18.4897			−0.625066
$876$		0			0
$877$	−49.2708			−1.66376			−0.831879	−	0.554958i	$-0.812734\pi$
$877$	−49.2708			−1.66376			−0.831879	+	0.554958i	$0.812734\pi$
$878$		0			0
$879$	5.19358	+	3.23937i	0.175175	+	0.109261i
$880$		0			0
$881$			44.4428i			1.49732i	0.662957	+	0.748658i	$0.269301\pi$
$881$			44.4428i			1.49732i	−0.662957	+	0.748658i	$0.730699\pi$
$882$		0			0
$883$			4.23686i			0.142582i	0.997456	+	0.0712908i	$0.0227118\pi$
$883$			4.23686i			0.142582i	−0.997456	+	0.0712908i	$0.977288\pi$
$884$		0			0
$885$	−1.43739	−	0.896536i	−0.0483173	−	0.0301367i
$886$		0			0
$887$	45.6379			1.53237			0.766186	−	0.642619i	$-0.222152\pi$
$887$	45.6379			1.53237			0.766186	+	0.642619i	$0.222152\pi$
$888$		0			0
$889$	−3.43857			−0.115326
$890$		0			0
$891$	−2.82680	+	3.64279i	−0.0947012	+	0.122038i
$892$		0			0
$893$		−	6.59798i		−	0.220793i
$894$		0			0
$895$			6.59524i			0.220455i
$896$		0			0
$897$	25.0351	−	40.1381i	0.835899	−	1.34017i
$898$		0			0
$899$	33.9478			1.13222
$900$		0			0
$901$	−17.3353			−0.577522
$902$		0			0
$903$	−3.57295	+	5.72840i	−0.118900	+	0.190629i
$904$		0			0
$905$			14.2540i			0.473820i
$906$		0			0
$907$			40.3390i			1.33943i	0.742617	+	0.669717i	$0.233584\pi$
$907$			40.3390i			1.33943i	−0.742617	+	0.669717i	$0.766416\pi$
$908$		0			0
$909$	−30.9136	+	15.1406i	−1.02534	+	0.502183i
$910$		0			0
$911$	−11.3271			−0.375283			−0.187641	−	0.982238i	$-0.560084\pi$
$911$	−11.3271			−0.375283			−0.187641	+	0.982238i	$0.560084\pi$
$912$		0			0
$913$	−4.59288			−0.152002
$914$		0			0
$915$	−2.55507	−	1.59366i	−0.0844679	−	0.0526848i
$916$		0			0
$917$		−	15.8211i		−	0.522460i
$918$		0			0
$919$		−	57.2270i		−	1.88774i	−0.330313	−	0.943872i	$-0.607154\pi$
$919$		−	57.2270i		−	1.88774i	0.330313	−	0.943872i	$-0.392846\pi$
$920$		0			0
$921$	−15.6354	−	9.75220i	−0.515204	−	0.321346i
$922$		0			0
$923$	−30.3859			−1.00016
$924$		0			0
$925$	40.2745			1.32422
$926$		0			0
$927$	35.4469	−	17.3609i	1.16423	−	0.570207i
$928$		0			0
$929$		−	16.0607i		−	0.526933i	−0.964669	−	0.263466i	$-0.915134\pi$
$929$		−	16.0607i		−	0.526933i	0.964669	−	0.263466i	$-0.0848658\pi$
$930$		0			0
$931$			1.61968i			0.0530830i
$932$		0			0
$933$	7.12193	−	11.4184i	0.233162	−	0.373821i
$934$		0			0
$935$	2.30452			0.0753659
$936$		0			0
$937$	17.2200			0.562551			0.281276	−	0.959627i	$-0.409242\pi$
$937$	17.2200			0.562551			0.281276	+	0.959627i	$0.409242\pi$
$938$		0			0
$939$	−27.5636	+	44.1918i	−0.899503	+	1.44215i
$940$		0			0
$941$		−	1.84092i		−	0.0600123i	−0.999550	−	0.0300061i	$-0.990447\pi$
$941$		−	1.84092i		−	0.0600123i	0.999550	−	0.0300061i	$-0.00955269\pi$
$942$		0			0
$943$			59.7884i			1.94698i
$944$		0			0
$945$	10.3225	+	1.05917i	0.335790	+	0.0344548i
$946$		0			0
$947$	17.3513			0.563843			0.281921	−	0.959437i	$-0.409028\pi$
$947$	17.3513			0.563843			0.281921	+	0.959437i	$0.409028\pi$
$948$		0			0
$949$	−13.2211			−0.429176
$950$		0			0
$951$	10.4806	+	6.53699i	0.339855	+	0.211976i
$952$		0			0
$953$			38.8762i			1.25932i	0.776869	+	0.629662i	$0.216807\pi$
$953$			38.8762i			1.25932i	−0.776869	+	0.629662i	$0.783193\pi$
$954$		0			0
$955$		−	1.92351i		−	0.0622433i
$956$		0			0
$957$	−5.17491	−	3.22772i	−0.167281	−	0.104337i
$958$		0			0
$959$	−45.6212			−1.47319
$960$		0			0
$961$	6.60396			0.213031
$962$		0			0
$963$	−18.6267	−	38.0315i	−0.600239	−	1.22555i
$964$		0			0
$965$		−	12.3941i		−	0.398980i
$966$		0			0
$967$		−	51.2708i		−	1.64876i	−0.566038	−	0.824379i	$-0.691524\pi$
$967$		−	51.2708i		−	1.64876i	0.566038	−	0.824379i	$-0.308476\pi$
$968$		0			0
$969$	4.78917	−	7.67833i	0.153850	−	0.246663i
$970$		0			0
$971$	32.3594			1.03846			0.519231	−	0.854634i	$-0.326218\pi$
$971$	32.3594			1.03846			0.519231	+	0.854634i	$0.326218\pi$
$972$		0			0
$973$	−39.9732			−1.28148
$974$		0			0
$975$	11.4741	−	18.3960i	0.367465	−	0.589145i
$976$		0			0
$977$		−	53.5357i		−	1.71276i	−0.516348	−	0.856379i	$-0.672709\pi$
$977$		−	53.5357i		−	1.71276i	0.516348	−	0.856379i	$-0.327291\pi$
$978$		0			0
$979$		−	0.535809i		−	0.0171245i
$980$		0			0
$981$	−1.79577	−	3.66655i	−0.0573347	−	0.117064i
$982$		0			0
$983$	−50.0797			−1.59729			−0.798646	−	0.601801i	$-0.794450\pi$
$983$	−50.0797			−1.59729			−0.798646	+	0.601801i	$0.794450\pi$
$984$		0			0
$985$	15.3997			0.490676
$986$		0			0
$987$	−22.4915	−	14.0285i	−0.715913	−	0.446533i
$988$		0			0
$989$			15.6151i			0.496530i
$990$		0			0
$991$			47.8917i			1.52133i	0.649145	+	0.760665i	$0.275127\pi$
$991$			47.8917i			1.52133i	−0.649145	+	0.760665i	$0.724873\pi$
$992$		0			0
$993$	32.5564	+	20.3063i	1.03315	+	0.644400i
$994$		0			0
$995$	2.54722			0.0807522
$996$		0			0
$997$	−5.94175			−0.188177			−0.0940885	−	0.995564i	$-0.529994\pi$
$997$	−5.94175			−0.188177			−0.0940885	+	0.995564i	$0.529994\pi$
$998$		0			0
$999$	−48.8824	−	5.01573i	−1.54657	−	0.158691i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.d.a.191.9	yes	12
3.2	odd	2	inner	912.2.d.a.191.3	✓	12
4.3	odd	2	inner	912.2.d.a.191.4	yes	12
12.11	even	2	inner	912.2.d.a.191.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.d.a.191.3	✓	12	3.2	odd	2	inner
912.2.d.a.191.4	yes	12	4.3	odd	2	inner
912.2.d.a.191.9	yes	12	1.1	even	1	trivial
912.2.d.a.191.10	yes	12	12.11	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.916638 - 1.46962i) q^{3} +0.860938i q^{5} +2.31955i q^{7} +(-1.31955 - 2.69421i) q^{9} +O(q^{10})\)	\(q+(0.916638 - 1.46962i) q^{3} +0.860938i q^{5} +2.31955i q^{7} +(-1.31955 - 2.69421i) q^{9} +0.512326 q^{11} +2.93923 q^{13} +(1.26525 + 0.789168i) q^{15} -5.22471i q^{17} +1.00000i q^{19} +(3.40885 + 2.12619i) q^{21} +9.29219 q^{23} +4.25879 q^{25} +(-5.16901 - 0.530383i) q^{27} -6.87309i q^{29} +4.93923i q^{31} +(0.469617 - 0.752923i) q^{33} -1.99699 q^{35} +9.45681 q^{37} +(2.69421 - 4.31955i) q^{39} +6.43426i q^{41} +1.68045i q^{43} +(2.31955 - 1.13605i) q^{45} -6.59798 q^{47} +1.61968 q^{49} +(-7.67833 - 4.78917i) q^{51} -3.31794i q^{53} +0.441081i q^{55} +(1.46962 + 0.916638i) q^{57} -1.13605 q^{59} -2.01942 q^{61} +(6.24936 - 3.06077i) q^{63} +2.53050i q^{65} -7.15667i q^{67} +(8.51757 - 13.6560i) q^{69} -10.3380 q^{71} -4.49815 q^{73} +(3.90376 - 6.25879i) q^{75} +1.18837i q^{77} +5.36090i q^{79} +(-5.51757 + 7.11030i) q^{81} -8.96476 q^{83} +4.49815 q^{85} +(-10.1008 - 6.30013i) q^{87} -1.04584i q^{89} +6.81770i q^{91} +(7.25879 + 4.52749i) q^{93} -0.860938 q^{95} -8.21744 q^{97} +(-0.676040 - 1.38032i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9}+O(q^{10}) \) 12 * q - 4 * q^9	\( 12 q - 4 q^{9} - 12 q^{21} + 4 q^{25} - 12 q^{33} - 16 q^{37} + 16 q^{45} - 4 q^{49} - 24 q^{61} + 8 q^{69} + 40 q^{73} + 28 q^{81} - 40 q^{85} + 40 q^{93} - 16 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 12 * q^21 + 4 * q^25 - 12 * q^33 - 16 * q^37 + 16 * q^45 - 4 * q^49 - 24 * q^61 + 8 * q^69 + 40 * q^73 + 28 * q^81 - 40 * q^85 + 40 * q^93 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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