LMFDB - Embedded newform 3584.2.b.i.1793.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (of order $2$, degree $1$, not 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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 4 x^{10} + 8 x^{9} - 24 x^{8} + 24 x^{7} + 176 x^{6} + 160 x^{5} + 145 x^{4} + \cdots + 8 $ x^12 - 4x^11 + 4x^10 + 8x^9 - 24x^8 + 24x^7 + 176x^6 + 160x^5 + 145x^4 + 108x^3 + 68x^2 + 32*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.9
Root			$-0.484138 + 0.200537i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81529i q^{3} -2.66965i q^{5} -1.00000 q^{7} -0.295267 q^{9} +O(q^{10})$	$q+1.81529i q^{3} -2.66965i q^{5} -1.00000 q^{7} -0.295267 q^{9} -4.22372i q^{11} +1.10581i q^{13} +4.84619 q^{15} +5.14481 q^{17} -1.59075i q^{19} -1.81529i q^{21} -4.83578 q^{23} -2.12705 q^{25} +4.90987i q^{27} -4.03668i q^{29} -2.30598 q^{31} +7.66726 q^{33} +2.66965i q^{35} -0.485763i q^{37} -2.00736 q^{39} -4.14145 q^{41} -4.91069i q^{43} +0.788260i q^{45} -6.97659 q^{47} +1.00000 q^{49} +9.33931i q^{51} +6.45784i q^{53} -11.2759 q^{55} +2.88766 q^{57} -0.825767i q^{59} -15.3550i q^{61} +0.295267 q^{63} +2.95212 q^{65} +1.98790i q^{67} -8.77833i q^{69} -7.69818 q^{71} +16.9578 q^{73} -3.86121i q^{75} +4.22372i q^{77} -12.2725 q^{79} -9.79862 q^{81} +3.82200i q^{83} -13.7349i q^{85} +7.32774 q^{87} -13.6820 q^{89} -1.10581i q^{91} -4.18601i q^{93} -4.24674 q^{95} +11.7879 q^{97} +1.24712i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 20 * q^9	$ 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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	0
$2$	0	0
$3$	1.81529i	1.04806i	0.851701	+	0.524028i	$0.175571\pi$
$3$	1.81529i	1.04806i	−0.851701	+	0.524028i	$0.824429\pi$
$4$	0	0
$5$	− 2.66965i	− 1.19391i	−0.802276	−	0.596953i	$-0.796378\pi$
$5$	− 2.66965i	− 1.19391i	0.802276	−	0.596953i	$-0.203622\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.295267	−0.0984222
$10$	0	0
$11$	− 4.22372i	− 1.27350i	−0.771071	−	0.636749i	$-0.780279\pi$
$11$	− 4.22372i	− 1.27350i	0.771071	−	0.636749i	$-0.219721\pi$
$12$	0	0
$13$	1.10581i	0.306696i	0.988172	+	0.153348i	$0.0490055\pi$
$13$	1.10581i	0.306696i	−0.988172	+	0.153348i	$0.950995\pi$
$14$	0	0
$15$	4.84619	1.25128
$16$	0	0
$17$	5.14481	1.24780	0.623900	−	0.781504i	$-0.285547\pi$
$17$	5.14481	1.24780	0.623900	+	0.781504i	$0.285547\pi$
$18$	0	0
$19$	− 1.59075i	− 0.364942i	−0.983211	−	0.182471i	$-0.941590\pi$
$19$	− 1.59075i	− 0.364942i	0.983211	−	0.182471i	$-0.0584096\pi$
$20$	0	0
$21$	− 1.81529i	− 0.396128i
$22$	0	0
$23$	−4.83578	−1.00833	−0.504165	−	0.863607i	$-0.668200\pi$
$23$	−4.83578	−1.00833	−0.504165	+	0.863607i	$0.668200\pi$
$24$	0	0
$25$	−2.12705	−0.425410
$26$	0	0
$27$	4.90987i	0.944904i
$28$	0	0
$29$	− 4.03668i	− 0.749593i	−0.927107	−	0.374797i	$-0.877713\pi$
$29$	− 4.03668i	− 0.749593i	0.927107	−	0.374797i	$-0.122287\pi$
$30$	0	0
$31$	−2.30598	−0.414166	−0.207083	−	0.978323i	$-0.566397\pi$
$31$	−2.30598	−0.414166	−0.207083	+	0.978323i	$0.566397\pi$
$32$	0	0
$33$	7.66726	1.33470
$34$	0	0
$35$	2.66965i	0.451254i
$36$	0	0
$37$	− 0.485763i	− 0.0798590i	−0.999202	−	0.0399295i	$-0.987287\pi$
$37$	− 0.485763i	− 0.0798590i	0.999202	−	0.0399295i	$-0.0127133\pi$
$38$	0	0
$39$	−2.00736	−0.321434
$40$	0	0
$41$	−4.14145	−0.646786	−0.323393	−	0.946265i	$-0.604824\pi$
$41$	−4.14145	−0.646786	−0.323393	+	0.946265i	$0.604824\pi$
$42$	0	0
$43$	− 4.91069i	− 0.748873i	−0.927253	−	0.374437i	$-0.877836\pi$
$43$	− 4.91069i	− 0.748873i	0.927253	−	0.374437i	$-0.122164\pi$
$44$	0	0
$45$	0.788260i	0.117507i
$46$	0	0
$47$	−6.97659	−1.01764	−0.508820	−	0.860873i	$-0.669918\pi$
$47$	−6.97659	−1.01764	−0.508820	+	0.860873i	$0.669918\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.33931i	1.30776i
$52$	0	0
$53$	6.45784i	0.887052i	0.896261	+	0.443526i	$0.146273\pi$
$53$	6.45784i	0.887052i	−0.896261	+	0.443526i	$0.853727\pi$
$54$	0	0
$55$	−11.2759	−1.52044
$56$	0	0
$57$	2.88766	0.382480
$58$	0	0
$59$	− 0.825767i	− 0.107506i	−0.998554	−	0.0537528i	$-0.982882\pi$
$59$	− 0.825767i	− 0.107506i	0.998554	−	0.0537528i	$-0.0171183\pi$
$60$	0	0
$61$	− 15.3550i	− 1.96600i	−0.183595	−	0.983002i	$-0.558774\pi$
$61$	− 15.3550i	− 1.96600i	0.183595	−	0.983002i	$-0.441226\pi$
$62$	0	0
$63$	0.295267	0.0372001
$64$	0	0
$65$	2.95212	0.366166
$66$	0	0
$67$	1.98790i	0.242860i	0.992600	+	0.121430i	$0.0387480\pi$
$67$	1.98790i	0.242860i	−0.992600	+	0.121430i	$0.961252\pi$
$68$	0	0
$69$	− 8.77833i	− 1.05679i
$70$	0	0
$71$	−7.69818	−0.913606	−0.456803	−	0.889568i	$-0.651005\pi$
$71$	−7.69818	−0.913606	−0.456803	+	0.889568i	$0.651005\pi$
$72$	0	0
$73$	16.9578	1.98476	0.992382	−	0.123199i	$-0.0393152\pi$
$73$	16.9578	1.98476	0.992382	+	0.123199i	$0.0393152\pi$
$74$	0	0
$75$	− 3.86121i	− 0.445854i
$76$	0	0
$77$	4.22372i	0.481337i
$78$	0	0
$79$	−12.2725	−1.38076	−0.690382	−	0.723445i	$-0.742558\pi$
$79$	−12.2725	−1.38076	−0.690382	+	0.723445i	$0.742558\pi$
$80$	0	0
$81$	−9.79862	−1.08874
$82$	0	0
$83$	3.82200i	0.419519i	0.977753	+	0.209760i	$0.0672681\pi$
$83$	3.82200i	0.419519i	−0.977753	+	0.209760i	$0.932732\pi$
$84$	0	0
$85$	− 13.7349i	− 1.48975i
$86$	0	0
$87$	7.32774	0.785616
$88$	0	0
$89$	−13.6820	−1.45029	−0.725143	−	0.688598i	$-0.758226\pi$
$89$	−13.6820	−1.45029	−0.725143	+	0.688598i	$0.758226\pi$
$90$	0	0
$91$	− 1.10581i	− 0.115920i
$92$	0	0
$93$	− 4.18601i	− 0.434070i
$94$	0	0
$95$	−4.24674	−0.435707
$96$	0	0
$97$	11.7879	1.19688	0.598440	−	0.801167i	$-0.295787\pi$
$97$	11.7879	1.19688	0.598440	+	0.801167i	$0.295787\pi$
$98$	0	0
$99$	1.24712i	0.125341i
$100$	0	0
$101$	− 6.76266i	− 0.672910i	−0.941700	−	0.336455i	$-0.890772\pi$
$101$	− 6.76266i	− 0.672910i	0.941700	−	0.336455i	$-0.109228\pi$
$102$	0	0
$103$	3.56041	0.350818	0.175409	−	0.984496i	$-0.443875\pi$
$103$	3.56041	0.350818	0.175409	+	0.984496i	$0.443875\pi$
$104$	0	0
$105$	−4.84619	−0.472939
$106$	0	0
$107$	− 19.1207i	− 1.84846i	−0.381832	−	0.924232i	$-0.624707\pi$
$107$	− 19.1207i	− 1.84846i	0.381832	−	0.924232i	$-0.375293\pi$
$108$	0	0
$109$	− 4.88132i	− 0.467545i	−0.972291	−	0.233773i	$-0.924893\pi$
$109$	− 4.88132i	− 0.467545i	0.972291	−	0.233773i	$-0.0751072\pi$
$110$	0	0
$111$	0.881800	0.0836967
$112$	0	0
$113$	−19.8790	−1.87006	−0.935031	−	0.354566i	$-0.884629\pi$
$113$	−19.8790	−1.87006	−0.935031	+	0.354566i	$0.884629\pi$
$114$	0	0
$115$	12.9099i	1.20385i
$116$	0	0
$117$	− 0.326508i	− 0.0301857i
$118$	0	0
$119$	−5.14481	−0.471624
$120$	0	0
$121$	−6.83978	−0.621799
$122$	0	0
$123$	− 7.51793i	− 0.677869i
$124$	0	0
$125$	− 7.66978i	− 0.686006i
$126$	0	0
$127$	13.6917	1.21494	0.607472	−	0.794341i	$-0.292184\pi$
$127$	13.6917	1.21494	0.607472	+	0.794341i	$0.292184\pi$
$128$	0	0
$129$	8.91431	0.784861
$130$	0	0
$131$	− 9.94586i	− 0.868973i	−0.900678	−	0.434487i	$-0.856930\pi$
$131$	− 9.94586i	− 0.868973i	0.900678	−	0.434487i	$-0.143070\pi$
$132$	0	0
$133$	1.59075i	0.137935i
$134$	0	0
$135$	13.1076	1.12813
$136$	0	0
$137$	−18.1361	−1.54947	−0.774737	−	0.632284i	$-0.782118\pi$
$137$	−18.1361	−1.54947	−0.774737	+	0.632284i	$0.782118\pi$
$138$	0	0
$139$	− 17.0879i	− 1.44937i	−0.689078	−	0.724687i	$-0.741984\pi$
$139$	− 17.0879i	− 1.44937i	0.689078	−	0.724687i	$-0.258016\pi$
$140$	0	0
$141$	− 12.6645i	− 1.06654i
$142$	0	0
$143$	4.67061	0.390576
$144$	0	0
$145$	−10.7765	−0.894944
$146$	0	0
$147$	1.81529i	0.149722i
$148$	0	0
$149$	− 14.2206i	− 1.16500i	−0.812831	−	0.582500i	$-0.802075\pi$
$149$	− 14.2206i	− 1.16500i	0.812831	−	0.582500i	$-0.197925\pi$
$150$	0	0
$151$	18.7608	1.52673	0.763367	−	0.645964i	$-0.223545\pi$
$151$	18.7608	1.52673	0.763367	+	0.645964i	$0.223545\pi$
$152$	0	0
$153$	−1.51909	−0.122811
$154$	0	0
$155$	6.15617i	0.494475i
$156$	0	0
$157$	0.900906i	0.0719002i	0.999354	+	0.0359501i	$0.0114457\pi$
$157$	0.900906i	0.0719002i	−0.999354	+	0.0359501i	$0.988554\pi$
$158$	0	0
$159$	−11.7228	−0.929681
$160$	0	0
$161$	4.83578	0.381113
$162$	0	0
$163$	0.0608368i	0.00476511i	0.999997	+	0.00238255i	$0.000758391\pi$
$163$	0.0608368i	0.00476511i	−0.999997	+	0.00238255i	$0.999242\pi$
$164$	0	0
$165$	− 20.4689i	− 1.59350i
$166$	0	0
$167$	−10.3646	−0.802040	−0.401020	−	0.916069i	$-0.631344\pi$
$167$	−10.3646	−0.802040	−0.401020	+	0.916069i	$0.631344\pi$
$168$	0	0
$169$	11.7772	0.905938
$170$	0	0
$171$	0.469695i	0.0359184i
$172$	0	0
$173$	− 10.1507i	− 0.771744i	−0.922552	−	0.385872i	$-0.873901\pi$
$173$	− 10.1507i	− 0.771744i	0.922552	−	0.385872i	$-0.126099\pi$
$174$	0	0
$175$	2.12705	0.160790
$176$	0	0
$177$	1.49900	0.112672
$178$	0	0
$179$	17.3258i	1.29499i	0.762069	+	0.647496i	$0.224184\pi$
$179$	17.3258i	1.29499i	−0.762069	+	0.647496i	$0.775816\pi$
$180$	0	0
$181$	13.8039i	1.02604i	0.858378	+	0.513018i	$0.171473\pi$
$181$	13.8039i	1.02604i	−0.858378	+	0.513018i	$0.828527\pi$
$182$	0	0
$183$	27.8737	2.06048
$184$	0	0
$185$	−1.29682	−0.0953441
$186$	0	0
$187$	− 21.7302i	− 1.58907i
$188$	0	0
$189$	− 4.90987i	− 0.357140i
$190$	0	0
$191$	−10.5275	−0.761744	−0.380872	−	0.924628i	$-0.624376\pi$
$191$	−10.5275	−0.761744	−0.380872	+	0.924628i	$0.624376\pi$
$192$	0	0
$193$	−16.1968	−1.16587	−0.582937	−	0.812517i	$-0.698097\pi$
$193$	−16.1968	−1.16587	−0.582937	+	0.812517i	$0.698097\pi$
$194$	0	0
$195$	5.35895i	0.383762i
$196$	0	0
$197$	13.5990i	0.968886i	0.874823	+	0.484443i	$0.160978\pi$
$197$	13.5990i	0.968886i	−0.874823	+	0.484443i	$0.839022\pi$
$198$	0	0
$199$	12.7751	0.905600	0.452800	−	0.891612i	$-0.350425\pi$
$199$	12.7751	0.905600	0.452800	+	0.891612i	$0.350425\pi$
$200$	0	0
$201$	−3.60860	−0.254531
$202$	0	0
$203$	4.03668i	0.283320i
$204$	0	0
$205$	11.0562i	0.772202i
$206$	0	0
$207$	1.42785	0.0992421
$208$	0	0
$209$	−6.71887	−0.464754
$210$	0	0
$211$	5.22756i	0.359880i	0.983678	+	0.179940i	$0.0575903\pi$
$211$	5.22756i	0.359880i	−0.983678	+	0.179940i	$0.942410\pi$
$212$	0	0
$213$	− 13.9744i	− 0.957510i
$214$	0	0
$215$	−13.1098	−0.894084
$216$	0	0
$217$	2.30598	0.156540
$218$	0	0
$219$	30.7833i	2.08014i
$220$	0	0
$221$	5.68917i	0.382695i
$222$	0	0
$223$	−0.141606	−0.00948263	−0.00474132	−	0.999989i	$-0.501509\pi$
$223$	−0.141606	−0.00948263	−0.00474132	+	0.999989i	$0.501509\pi$
$224$	0	0
$225$	0.628047	0.0418698
$226$	0	0
$227$	13.3258i	0.884462i	0.896901	+	0.442231i	$0.145813\pi$
$227$	13.3258i	0.884462i	−0.896901	+	0.442231i	$0.854187\pi$
$228$	0	0
$229$	11.5223i	0.761416i	0.924695	+	0.380708i	$0.124320\pi$
$229$	11.5223i	0.761416i	−0.924695	+	0.380708i	$0.875680\pi$
$230$	0	0
$231$	−7.66726	−0.504469
$232$	0	0
$233$	12.7052	0.832347	0.416173	−	0.909285i	$-0.363371\pi$
$233$	12.7052	0.832347	0.416173	+	0.909285i	$0.363371\pi$
$234$	0	0
$235$	18.6251i	1.21497i
$236$	0	0
$237$	− 22.2781i	− 1.44712i
$238$	0	0
$239$	27.9471	1.80775	0.903874	−	0.427798i	$-0.140711\pi$
$239$	27.9471	1.80775	0.903874	+	0.427798i	$0.140711\pi$
$240$	0	0
$241$	−14.4371	−0.929975	−0.464987	−	0.885317i	$-0.653941\pi$
$241$	−14.4371	−0.929975	−0.464987	+	0.885317i	$0.653941\pi$
$242$	0	0
$243$	− 3.05770i	− 0.196152i
$244$	0	0
$245$	− 2.66965i	− 0.170558i
$246$	0	0
$247$	1.75906	0.111926
$248$	0	0
$249$	−6.93803	−0.439680
$250$	0	0
$251$	− 6.44681i	− 0.406919i	−0.979083	−	0.203460i	$-0.934781\pi$
$251$	− 6.44681i	− 0.406919i	0.979083	−	0.203460i	$-0.0652186\pi$
$252$	0	0
$253$	20.4250i	1.28411i
$254$	0	0
$255$	24.9327	1.56135
$256$	0	0
$257$	4.04585	0.252373	0.126187	−	0.992007i	$-0.459726\pi$
$257$	4.04585	0.252373	0.126187	+	0.992007i	$0.459726\pi$
$258$	0	0
$259$	0.485763i	0.0301839i
$260$	0	0
$261$	1.19190i	0.0737766i
$262$	0	0
$263$	21.7257	1.33966	0.669832	−	0.742513i	$-0.266366\pi$
$263$	21.7257	1.33966	0.669832	+	0.742513i	$0.266366\pi$
$264$	0	0
$265$	17.2402	1.05906
$266$	0	0
$267$	− 24.8367i	− 1.51998i
$268$	0	0
$269$	− 16.2648i	− 0.991685i	−0.868412	−	0.495843i	$-0.834859\pi$
$269$	− 16.2648i	− 0.991685i	0.868412	−	0.495843i	$-0.165141\pi$
$270$	0	0
$271$	−16.3342	−0.992231	−0.496115	−	0.868257i	$-0.665241\pi$
$271$	−16.3342	−0.992231	−0.496115	+	0.868257i	$0.665241\pi$
$272$	0	0
$273$	2.00736	0.121491
$274$	0	0
$275$	8.98406i	0.541759i
$276$	0	0
$277$	− 26.0088i	− 1.56271i	−0.624084	−	0.781357i	$-0.714528\pi$
$277$	− 26.0088i	− 1.56271i	0.624084	−	0.781357i	$-0.285472\pi$
$278$	0	0
$279$	0.680879	0.0407632
$280$	0	0
$281$	29.8862	1.78286	0.891432	−	0.453155i	$-0.149702\pi$
$281$	29.8862	1.78286	0.891432	+	0.453155i	$0.149702\pi$
$282$	0	0
$283$	− 23.1334i	− 1.37514i	−0.726120	−	0.687568i	$-0.758678\pi$
$283$	− 23.1334i	− 1.37514i	0.726120	−	0.687568i	$-0.241322\pi$
$284$	0	0
$285$	− 7.70906i	− 0.456645i
$286$	0	0
$287$	4.14145	0.244462
$288$	0	0
$289$	9.46907	0.557004
$290$	0	0
$291$	21.3984i	1.25440i
$292$	0	0
$293$	− 23.7587i	− 1.38800i	−0.719977	−	0.693998i	$-0.755848\pi$
$293$	− 23.7587i	− 1.38800i	0.719977	−	0.693998i	$-0.244152\pi$
$294$	0	0
$295$	−2.20451	−0.128352
$296$	0	0
$297$	20.7379	1.20333
$298$	0	0
$299$	− 5.34744i	− 0.309251i
$300$	0	0
$301$	4.91069i	0.283047i
$302$	0	0
$303$	12.2762	0.705248
$304$	0	0
$305$	−40.9925	−2.34722
$306$	0	0
$307$	1.34008i	0.0764824i	0.999269	+	0.0382412i	$0.0121755\pi$
$307$	1.34008i	0.0764824i	−0.999269	+	0.0382412i	$0.987824\pi$
$308$	0	0
$309$	6.46317i	0.367677i
$310$	0	0
$311$	−30.8370	−1.74860	−0.874302	−	0.485383i	$-0.838680\pi$
$311$	−30.8370	−1.74860	−0.874302	+	0.485383i	$0.838680\pi$
$312$	0	0
$313$	−1.77750	−0.100470	−0.0502351	−	0.998737i	$-0.515997\pi$
$313$	−1.77750	−0.100470	−0.0502351	+	0.998737i	$0.515997\pi$
$314$	0	0
$315$	− 0.788260i	− 0.0444134i
$316$	0	0
$317$	− 2.69913i	− 0.151598i	−0.997123	−	0.0757991i	$-0.975849\pi$
$317$	− 2.69913i	− 0.151598i	0.997123	−	0.0757991i	$-0.0241508\pi$
$318$	0	0
$319$	−17.0498	−0.954606
$320$	0	0
$321$	34.7095	1.93729
$322$	0	0
$323$	− 8.18409i	− 0.455375i
$324$	0	0
$325$	− 2.35211i	− 0.130471i
$326$	0	0
$327$	8.86099	0.490014
$328$	0	0
$329$	6.97659	0.384632
$330$	0	0
$331$	2.01742i	0.110888i	0.998462	+	0.0554438i	$0.0176574\pi$
$331$	2.01742i	0.110888i	−0.998462	+	0.0554438i	$0.982343\pi$
$332$	0	0
$333$	0.143430i	0.00785990i
$334$	0	0
$335$	5.30700	0.289952
$336$	0	0
$337$	−11.3824	−0.620041	−0.310020	−	0.950730i	$-0.600336\pi$
$337$	−11.3824	−0.620041	−0.310020	+	0.950730i	$0.600336\pi$
$338$	0	0
$339$	− 36.0861i	− 1.95993i
$340$	0	0
$341$	9.73981i	0.527440i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−23.4351	−1.26170
$346$	0	0
$347$	18.3962i	0.987557i	0.869588	+	0.493779i	$0.164385\pi$
$347$	18.3962i	0.987557i	−0.869588	+	0.493779i	$0.835615\pi$
$348$	0	0
$349$	− 3.94902i	− 0.211386i	−0.994399	−	0.105693i	$-0.966294\pi$
$349$	− 3.94902i	− 0.211386i	0.994399	−	0.105693i	$-0.0337061\pi$
$350$	0	0
$351$	−5.42936	−0.289798
$352$	0	0
$353$	−1.64653	−0.0876358	−0.0438179	−	0.999040i	$-0.513952\pi$
$353$	−1.64653	−0.0876358	−0.0438179	+	0.999040i	$0.513952\pi$
$354$	0	0
$355$	20.5515i	1.09076i
$356$	0	0
$357$	− 9.33931i	− 0.494289i
$358$	0	0
$359$	−20.6386	−1.08927	−0.544633	−	0.838675i	$-0.683331\pi$
$359$	−20.6386	−1.08927	−0.544633	+	0.838675i	$0.683331\pi$
$360$	0	0
$361$	16.4695	0.866817
$362$	0	0
$363$	− 12.4162i	− 0.651680i
$364$	0	0
$365$	− 45.2715i	− 2.36962i
$366$	0	0
$367$	13.2300	0.690601	0.345301	−	0.938492i	$-0.387777\pi$
$367$	13.2300	0.690601	0.345301	+	0.938492i	$0.387777\pi$
$368$	0	0
$369$	1.22283	0.0636582
$370$	0	0
$371$	− 6.45784i	− 0.335274i
$372$	0	0
$373$	− 20.8277i	− 1.07842i	−0.842172	−	0.539208i	$-0.818723\pi$
$373$	− 20.8277i	− 1.07842i	0.842172	−	0.539208i	$-0.181277\pi$
$374$	0	0
$375$	13.9229	0.718973
$376$	0	0
$377$	4.46379	0.229897
$378$	0	0
$379$	− 28.9858i	− 1.48890i	−0.667678	−	0.744451i	$-0.732712\pi$
$379$	− 28.9858i	− 1.48890i	0.667678	−	0.744451i	$-0.267288\pi$
$380$	0	0
$381$	24.8544i	1.27333i
$382$	0	0
$383$	14.8173	0.757128	0.378564	−	0.925575i	$-0.376418\pi$
$383$	14.8173	0.757128	0.378564	+	0.925575i	$0.376418\pi$
$384$	0	0
$385$	11.2759	0.574671
$386$	0	0
$387$	1.44996i	0.0737058i
$388$	0	0
$389$	− 5.38750i	− 0.273157i	−0.990629	−	0.136578i	$-0.956389\pi$
$389$	− 5.38750i	− 0.273157i	0.990629	−	0.136578i	$-0.0436106\pi$
$390$	0	0
$391$	−24.8792	−1.25819
$392$	0	0
$393$	18.0546	0.910733
$394$	0	0
$395$	32.7633i	1.64850i
$396$	0	0
$397$	27.1423i	1.36223i	0.732176	+	0.681115i	$0.238505\pi$
$397$	27.1423i	1.36223i	−0.732176	+	0.681115i	$0.761495\pi$
$398$	0	0
$399$	−2.88766	−0.144564
$400$	0	0
$401$	−25.1393	−1.25540	−0.627698	−	0.778457i	$-0.716003\pi$
$401$	−25.1393	−1.25540	−0.627698	+	0.778457i	$0.716003\pi$
$402$	0	0
$403$	− 2.54997i	− 0.127023i
$404$	0	0
$405$	26.1589i	1.29985i
$406$	0	0
$407$	−2.05173	−0.101700
$408$	0	0
$409$	12.9158	0.638647	0.319323	−	0.947646i	$-0.396544\pi$
$409$	12.9158	0.638647	0.319323	+	0.947646i	$0.396544\pi$
$410$	0	0
$411$	− 32.9223i	− 1.62394i
$412$	0	0
$413$	0.825767i	0.0406333i
$414$	0	0
$415$	10.2034	0.500866
$416$	0	0
$417$	31.0194	1.51903
$418$	0	0
$419$	− 19.9526i	− 0.974747i	−0.873194	−	0.487373i	$-0.837955\pi$
$419$	− 19.9526i	− 0.974747i	0.873194	−	0.487373i	$-0.162045\pi$
$420$	0	0
$421$	36.7191i	1.78958i	0.446487	+	0.894790i	$0.352675\pi$
$421$	36.7191i	1.78958i	−0.446487	+	0.894790i	$0.647325\pi$
$422$	0	0
$423$	2.05996	0.100158
$424$	0	0
$425$	−10.9433	−0.530827
$426$	0	0
$427$	15.3550i	0.743080i
$428$	0	0
$429$	8.47851i	0.409346i
$430$	0	0
$431$	−22.3374	−1.07595	−0.537977	−	0.842959i	$-0.680811\pi$
$431$	−22.3374	−1.07595	−0.537977	+	0.842959i	$0.680811\pi$
$432$	0	0
$433$	26.9387	1.29459	0.647295	−	0.762239i	$-0.275900\pi$
$433$	26.9387	1.29459	0.647295	+	0.762239i	$0.275900\pi$
$434$	0	0
$435$	− 19.5625i	− 0.937951i
$436$	0	0
$437$	7.69251i	0.367983i
$438$	0	0
$439$	11.9820	0.571869	0.285935	−	0.958249i	$-0.407696\pi$
$439$	11.9820	0.571869	0.285935	+	0.958249i	$0.407696\pi$
$440$	0	0
$441$	−0.295267	−0.0140603
$442$	0	0
$443$	20.4778i	0.972929i	0.873700	+	0.486464i	$0.161714\pi$
$443$	20.4778i	0.972929i	−0.873700	+	0.486464i	$0.838286\pi$
$444$	0	0
$445$	36.5261i	1.73150i
$446$	0	0
$447$	25.8145	1.22098
$448$	0	0
$449$	18.0256	0.850683	0.425341	−	0.905033i	$-0.360154\pi$
$449$	18.0256	0.850683	0.425341	+	0.905033i	$0.360154\pi$
$450$	0	0
$451$	17.4923i	0.823682i
$452$	0	0
$453$	34.0563i	1.60010i
$454$	0	0
$455$	−2.95212	−0.138398
$456$	0	0
$457$	−24.6480	−1.15298	−0.576492	−	0.817103i	$-0.695579\pi$
$457$	−24.6480	−1.15298	−0.576492	+	0.817103i	$0.695579\pi$
$458$	0	0
$459$	25.2603i	1.17905i
$460$	0	0
$461$	30.0660i	1.40031i	0.713989	+	0.700157i	$0.246887\pi$
$461$	30.0660i	1.40031i	−0.713989	+	0.700157i	$0.753113\pi$
$462$	0	0
$463$	−5.79825	−0.269467	−0.134734	−	0.990882i	$-0.543018\pi$
$463$	−5.79825	−0.269467	−0.134734	+	0.990882i	$0.543018\pi$
$464$	0	0
$465$	−11.1752	−0.518238
$466$	0	0
$467$	− 35.4774i	− 1.64170i	−0.571145	−	0.820849i	$-0.693501\pi$
$467$	− 35.4774i	− 1.64170i	0.571145	−	0.820849i	$-0.306499\pi$
$468$	0	0
$469$	− 1.98790i	− 0.0917925i
$470$	0	0
$471$	−1.63540	−0.0753554
$472$	0	0
$473$	−20.7414	−0.953689
$474$	0	0
$475$	3.38360i	0.155250i
$476$	0	0
$477$	− 1.90678i	− 0.0873056i
$478$	0	0
$479$	25.1846	1.15071	0.575356	−	0.817903i	$-0.304864\pi$
$479$	25.1846	1.15071	0.575356	+	0.817903i	$0.304864\pi$
$480$	0	0
$481$	0.537160	0.0244924
$482$	0	0
$483$	8.77833i	0.399428i
$484$	0	0
$485$	− 31.4696i	− 1.42896i
$486$	0	0
$487$	−7.95563	−0.360504	−0.180252	−	0.983620i	$-0.557691\pi$
$487$	−7.95563	−0.360504	−0.180252	+	0.983620i	$0.557691\pi$
$488$	0	0
$489$	−0.110436	−0.00499410
$490$	0	0
$491$	18.4713i	0.833599i	0.908998	+	0.416799i	$0.136848\pi$
$491$	18.4713i	0.833599i	−0.908998	+	0.416799i	$0.863152\pi$
$492$	0	0
$493$	− 20.7680i	− 0.935343i
$494$	0	0
$495$	3.32939	0.149645
$496$	0	0
$497$	7.69818	0.345310
$498$	0	0
$499$	30.1446i	1.34946i	0.738066	+	0.674729i	$0.235739\pi$
$499$	30.1446i	1.34946i	−0.738066	+	0.674729i	$0.764261\pi$
$500$	0	0
$501$	− 18.8148i	− 0.840583i
$502$	0	0
$503$	6.87344	0.306472	0.153236	−	0.988190i	$-0.451031\pi$
$503$	6.87344	0.306472	0.153236	+	0.988190i	$0.451031\pi$
$504$	0	0
$505$	−18.0540	−0.803391
$506$	0	0
$507$	21.3790i	0.949474i
$508$	0	0
$509$	− 39.6725i	− 1.75845i	−0.476403	−	0.879227i	$-0.658060\pi$
$509$	− 39.6725i	− 1.75845i	0.476403	−	0.879227i	$-0.341940\pi$
$510$	0	0
$511$	−16.9578	−0.750170
$512$	0	0
$513$	7.81036	0.344836
$514$	0	0
$515$	− 9.50507i	− 0.418844i
$516$	0	0
$517$	29.4672i	1.29596i
$518$	0	0
$519$	18.4264	0.808831
$520$	0	0
$521$	−22.5717	−0.988885	−0.494442	−	0.869210i	$-0.664628\pi$
$521$	−22.5717	−0.988885	−0.494442	+	0.869210i	$0.664628\pi$
$522$	0	0
$523$	2.42035i	0.105835i	0.998599	+	0.0529173i	$0.0168520\pi$
$523$	2.42035i	0.105835i	−0.998599	+	0.0529173i	$0.983148\pi$
$524$	0	0
$525$	3.86121i	0.168517i
$526$	0	0
$527$	−11.8638	−0.516796
$528$	0	0
$529$	0.384803	0.0167306
$530$	0	0
$531$	0.243821i	0.0105809i
$532$	0	0
$533$	− 4.57965i	− 0.198367i
$534$	0	0
$535$	−51.0455	−2.20689
$536$	0	0
$537$	−31.4513	−1.35722
$538$	0	0
$539$	− 4.22372i	− 0.181928i
$540$	0	0
$541$	1.90077i	0.0817203i	0.999165	+	0.0408602i	$0.0130098\pi$
$541$	1.90077i	0.0817203i	−0.999165	+	0.0408602i	$0.986990\pi$
$542$	0	0
$543$	−25.0581	−1.07534
$544$	0	0
$545$	−13.0314	−0.558205
$546$	0	0
$547$	− 22.7593i	− 0.973117i	−0.873648	−	0.486559i	$-0.838252\pi$
$547$	− 22.7593i	− 0.973117i	0.873648	−	0.486559i	$-0.161748\pi$
$548$	0	0
$549$	4.53381i	0.193498i
$550$	0	0
$551$	−6.42134	−0.273558
$552$	0	0
$553$	12.2725	0.521880
$554$	0	0
$555$	− 2.35410i	− 0.0999260i
$556$	0	0
$557$	11.7423i	0.497536i	0.968563	+	0.248768i	$0.0800257\pi$
$557$	11.7423i	0.497536i	−0.968563	+	0.248768i	$0.919974\pi$
$558$	0	0
$559$	5.43027	0.229676
$560$	0	0
$561$	39.4466	1.66544
$562$	0	0
$563$	17.1884i	0.724406i	0.932099	+	0.362203i	$0.117975\pi$
$563$	17.1884i	0.724406i	−0.932099	+	0.362203i	$0.882025\pi$
$564$	0	0
$565$	53.0701i	2.23268i
$566$	0	0
$567$	9.79862	0.411503
$568$	0	0
$569$	8.80662	0.369193	0.184596	−	0.982814i	$-0.440902\pi$
$569$	8.80662	0.369193	0.184596	+	0.982814i	$0.440902\pi$
$570$	0	0
$571$	− 22.9690i	− 0.961222i	−0.876934	−	0.480611i	$-0.840415\pi$
$571$	− 22.9690i	− 0.961222i	0.876934	−	0.480611i	$-0.159585\pi$
$572$	0	0
$573$	− 19.1105i	− 0.798351i
$574$	0	0
$575$	10.2860	0.428954
$576$	0	0
$577$	27.5655	1.14757	0.573783	−	0.819007i	$-0.305475\pi$
$577$	27.5655	1.14757	0.573783	+	0.819007i	$0.305475\pi$
$578$	0	0
$579$	− 29.4019i	− 1.22190i
$580$	0	0
$581$	− 3.82200i	− 0.158563i
$582$	0	0
$583$	27.2761	1.12966
$584$	0	0
$585$	−0.871663	−0.0360388
$586$	0	0
$587$	12.6646i	0.522726i	0.965241	+	0.261363i	$0.0841719\pi$
$587$	12.6646i	0.522726i	−0.965241	+	0.261363i	$0.915828\pi$
$588$	0	0
$589$	3.66823i	0.151147i
$590$	0	0
$591$	−24.6860	−1.01545
$592$	0	0
$593$	−11.0010	−0.451755	−0.225877	−	0.974156i	$-0.572525\pi$
$593$	−11.0010	−0.451755	−0.225877	+	0.974156i	$0.572525\pi$
$594$	0	0
$595$	13.7349i	0.563074i
$596$	0	0
$597$	23.1904i	0.949119i
$598$	0	0
$599$	38.6985	1.58118	0.790588	−	0.612348i	$-0.209775\pi$
$599$	38.6985	1.58118	0.790588	+	0.612348i	$0.209775\pi$
$600$	0	0
$601$	4.53332	0.184918	0.0924589	−	0.995716i	$-0.470527\pi$
$601$	4.53332	0.184918	0.0924589	+	0.995716i	$0.470527\pi$
$602$	0	0
$603$	− 0.586960i	− 0.0239028i
$604$	0	0
$605$	18.2599i	0.742369i
$606$	0	0
$607$	−47.2043	−1.91596	−0.957982	−	0.286827i	$-0.907399\pi$
$607$	−47.2043	−1.91596	−0.957982	+	0.286827i	$0.907399\pi$
$608$	0	0
$609$	−7.32774	−0.296935
$610$	0	0
$611$	− 7.71476i	− 0.312106i
$612$	0	0
$613$	28.4815i	1.15036i	0.818028	+	0.575179i	$0.195067\pi$
$613$	28.4815i	1.15036i	−0.818028	+	0.575179i	$0.804933\pi$
$614$	0	0
$615$	−20.0703	−0.809311
$616$	0	0
$617$	−0.0639668	−0.00257521	−0.00128760	−	0.999999i	$-0.500410\pi$
$617$	−0.0639668	−0.00257521	−0.00128760	+	0.999999i	$0.500410\pi$
$618$	0	0
$619$	21.4195i	0.860920i	0.902610	+	0.430460i	$0.141649\pi$
$619$	21.4195i	0.860920i	−0.902610	+	0.430460i	$0.858351\pi$
$620$	0	0
$621$	− 23.7431i	− 0.952776i
$622$	0	0
$623$	13.6820	0.548157
$624$	0	0
$625$	−31.1109	−1.24444
$626$	0	0
$627$	− 12.1967i	− 0.487088i
$628$	0	0
$629$	− 2.49916i	− 0.0996481i
$630$	0	0
$631$	−17.1912	−0.684371	−0.342185	−	0.939632i	$-0.611167\pi$
$631$	−17.1912	−0.684371	−0.342185	+	0.939632i	$0.611167\pi$
$632$	0	0
$633$	−9.48951	−0.377174
$634$	0	0
$635$	− 36.5522i	− 1.45053i
$636$	0	0
$637$	1.10581i	0.0438137i
$638$	0	0
$639$	2.27302	0.0899191
$640$	0	0
$641$	−18.4865	−0.730172	−0.365086	−	0.930974i	$-0.618960\pi$
$641$	−18.4865	−0.730172	−0.365086	+	0.930974i	$0.618960\pi$
$642$	0	0
$643$	− 26.3286i	− 1.03830i	−0.854683	−	0.519150i	$-0.826249\pi$
$643$	− 26.3286i	− 1.03830i	0.854683	−	0.519150i	$-0.173751\pi$
$644$	0	0
$645$	− 23.7981i	− 0.937050i
$646$	0	0
$647$	30.3147	1.19179	0.595897	−	0.803061i	$-0.296796\pi$
$647$	30.3147	1.19179	0.595897	+	0.803061i	$0.296796\pi$
$648$	0	0
$649$	−3.48780	−0.136908
$650$	0	0
$651$	4.18601i	0.164063i
$652$	0	0
$653$	43.9134i	1.71846i	0.511587	+	0.859231i	$0.329058\pi$
$653$	43.9134i	1.71846i	−0.511587	+	0.859231i	$0.670942\pi$
$654$	0	0
$655$	−26.5520	−1.03747
$656$	0	0
$657$	−5.00708	−0.195345
$658$	0	0
$659$	7.83648i	0.305266i	0.988283	+	0.152633i	$0.0487752\pi$
$659$	7.83648i	0.305266i	−0.988283	+	0.152633i	$0.951225\pi$
$660$	0	0
$661$	− 17.6047i	− 0.684744i	−0.939564	−	0.342372i	$-0.888770\pi$
$661$	− 17.6047i	− 0.684744i	0.939564	−	0.342372i	$-0.111230\pi$
$662$	0	0
$663$	−10.3275	−0.401086
$664$	0	0
$665$	4.24674	0.164682
$666$	0	0
$667$	19.5205i	0.755838i
$668$	0	0
$669$	− 0.257055i	− 0.00993834i
$670$	0	0
$671$	−64.8551	−2.50370
$672$	0	0
$673$	3.41747	0.131734	0.0658668	−	0.997828i	$-0.479019\pi$
$673$	3.41747	0.131734	0.0658668	+	0.997828i	$0.479019\pi$
$674$	0	0
$675$	− 10.4435i	− 0.401972i
$676$	0	0
$677$	− 20.5328i	− 0.789138i	−0.918866	−	0.394569i	$-0.870894\pi$
$677$	− 20.5328i	− 0.789138i	0.918866	−	0.394569i	$-0.129106\pi$
$678$	0	0
$679$	−11.7879	−0.452378
$680$	0	0
$681$	−24.1901	−0.926966
$682$	0	0
$683$	46.8834i	1.79395i	0.442086	+	0.896973i	$0.354239\pi$
$683$	46.8834i	1.79395i	−0.442086	+	0.896973i	$0.645761\pi$
$684$	0	0
$685$	48.4172i	1.84992i
$686$	0	0
$687$	−20.9163	−0.798007
$688$	0	0
$689$	−7.14112	−0.272055
$690$	0	0
$691$	− 25.2722i	− 0.961398i	−0.876886	−	0.480699i	$-0.840383\pi$
$691$	− 25.2722i	− 0.961398i	0.876886	−	0.480699i	$-0.159617\pi$
$692$	0	0
$693$	− 1.24712i	− 0.0473743i
$694$	0	0
$695$	−45.6187	−1.73042
$696$	0	0
$697$	−21.3070	−0.807060
$698$	0	0
$699$	23.0636i	0.872347i
$700$	0	0
$701$	46.0899i	1.74079i	0.492353	+	0.870395i	$0.336137\pi$
$701$	46.0899i	1.74079i	−0.492353	+	0.870395i	$0.663863\pi$
$702$	0	0
$703$	−0.772727	−0.0291439
$704$	0	0
$705$	−33.8099	−1.27335
$706$	0	0
$707$	6.76266i	0.254336i
$708$	0	0
$709$	14.4781i	0.543736i	0.962335	+	0.271868i	$0.0876414\pi$
$709$	14.4781i	0.543736i	−0.962335	+	0.271868i	$0.912359\pi$
$710$	0	0
$711$	3.62366	0.135898
$712$	0	0
$713$	11.1512	0.417616
$714$	0	0
$715$	− 12.4689i	− 0.466311i
$716$	0	0
$717$	50.7320i	1.89462i
$718$	0	0
$719$	−2.25703	−0.0841731	−0.0420866	−	0.999114i	$-0.513401\pi$
$719$	−2.25703	−0.0841731	−0.0420866	+	0.999114i	$0.513401\pi$
$720$	0	0
$721$	−3.56041	−0.132597
$722$	0	0
$723$	− 26.2075i	− 0.974666i
$724$	0	0
$725$	8.58623i	0.318885i
$726$	0	0
$727$	27.3854	1.01567	0.507835	−	0.861455i	$-0.330446\pi$
$727$	27.3854	1.01567	0.507835	+	0.861455i	$0.330446\pi$
$728$	0	0
$729$	−23.8452	−0.883157
$730$	0	0
$731$	− 25.2646i	− 0.934444i
$732$	0	0
$733$	− 45.5500i	− 1.68243i	−0.540702	−	0.841214i	$-0.681841\pi$
$733$	− 45.5500i	− 1.68243i	0.540702	−	0.841214i	$-0.318159\pi$
$734$	0	0
$735$	4.84619	0.178754
$736$	0	0
$737$	8.39631	0.309282
$738$	0	0
$739$	30.4630i	1.12060i	0.828290	+	0.560299i	$0.189314\pi$
$739$	30.4630i	1.12060i	−0.828290	+	0.560299i	$0.810686\pi$
$740$	0	0
$741$	3.19320i	0.117305i
$742$	0	0
$743$	−47.2219	−1.73240	−0.866202	−	0.499694i	$-0.833446\pi$
$743$	−47.2219	−1.73240	−0.866202	+	0.499694i	$0.833446\pi$
$744$	0	0
$745$	−37.9641	−1.39090
$746$	0	0
$747$	− 1.12851i	− 0.0412900i
$748$	0	0
$749$	19.1207i	0.698654i
$750$	0	0
$751$	−9.98015	−0.364181	−0.182090	−	0.983282i	$-0.558286\pi$
$751$	−9.98015	−0.364181	−0.182090	+	0.983282i	$0.558286\pi$
$752$	0	0
$753$	11.7028	0.426474
$754$	0	0
$755$	− 50.0849i	− 1.82278i
$756$	0	0
$757$	22.1282i	0.804264i	0.915582	+	0.402132i	$0.131731\pi$
$757$	22.1282i	0.804264i	−0.915582	+	0.402132i	$0.868269\pi$
$758$	0	0
$759$	−37.0772	−1.34582
$760$	0	0
$761$	36.0367	1.30633	0.653165	−	0.757215i	$-0.273441\pi$
$761$	36.0367	1.30633	0.653165	+	0.757215i	$0.273441\pi$
$762$	0	0
$763$	4.88132i	0.176716i
$764$	0	0
$765$	4.05545i	0.146625i
$766$	0	0
$767$	0.913138	0.0329715
$768$	0	0
$769$	−2.85648	−0.103007	−0.0515036	−	0.998673i	$-0.516401\pi$
$769$	−2.85648	−0.103007	−0.0515036	+	0.998673i	$0.516401\pi$
$770$	0	0
$771$	7.34437i	0.264501i
$772$	0	0
$773$	− 12.5383i	− 0.450970i	−0.974247	−	0.225485i	$-0.927603\pi$
$773$	− 12.5383i	− 0.450970i	0.974247	−	0.225485i	$-0.0723967\pi$
$774$	0	0
$775$	4.90493	0.176190
$776$	0	0
$777$	−0.881800	−0.0316344
$778$	0	0
$779$	6.58801i	0.236040i
$780$	0	0
$781$	32.5149i	1.16348i
$782$	0	0
$783$	19.8196	0.708294
$784$	0	0
$785$	2.40511	0.0858420
$786$	0	0
$787$	36.3630i	1.29620i	0.761555	+	0.648100i	$0.224436\pi$
$787$	36.3630i	1.29620i	−0.761555	+	0.648100i	$0.775564\pi$
$788$	0	0
$789$	39.4384i	1.40404i
$790$	0	0
$791$	19.8790	0.706817
$792$	0	0
$793$	16.9796	0.602965
$794$	0	0
$795$	31.2959i	1.10995i
$796$	0	0
$797$	− 23.7978i	− 0.842963i	−0.906837	−	0.421481i	$-0.861510\pi$
$797$	− 23.7978i	− 0.842963i	0.906837	−	0.421481i	$-0.138490\pi$
$798$	0	0
$799$	−35.8933	−1.26981
$800$	0	0
$801$	4.03983	0.142740
$802$	0	0
$803$	− 71.6251i	− 2.52759i
$804$	0	0
$805$	− 12.9099i	− 0.455013i
$806$	0	0
$807$	29.5254	1.03934
$808$	0	0
$809$	−16.0762	−0.565209	−0.282604	−	0.959237i	$-0.591198\pi$
$809$	−16.0762	−0.565209	−0.282604	+	0.959237i	$0.591198\pi$
$810$	0	0
$811$	− 4.97298i	− 0.174625i	−0.996181	−	0.0873125i	$-0.972172\pi$
$811$	− 4.97298i	− 0.174625i	0.996181	−	0.0873125i	$-0.0278279\pi$
$812$	0	0
$813$	− 29.6512i	− 1.03991i
$814$	0	0
$815$	0.162413	0.00568909
$816$	0	0
$817$	−7.81167	−0.273296
$818$	0	0
$819$	0.326508i	0.0114091i
$820$	0	0
$821$	27.1820i	0.948658i	0.880348	+	0.474329i	$0.157309\pi$
$821$	27.1820i	0.948658i	−0.880348	+	0.474329i	$0.842691\pi$
$822$	0	0
$823$	−11.8570	−0.413308	−0.206654	−	0.978414i	$-0.566257\pi$
$823$	−11.8570	−0.413308	−0.206654	+	0.978414i	$0.566257\pi$
$824$	0	0
$825$	−16.3086	−0.567794
$826$	0	0
$827$	48.6355i	1.69122i	0.533798	+	0.845612i	$0.320764\pi$
$827$	48.6355i	1.69122i	−0.533798	+	0.845612i	$0.679236\pi$
$828$	0	0
$829$	47.2987i	1.64275i	0.570388	+	0.821375i	$0.306793\pi$
$829$	47.2987i	1.64275i	−0.570388	+	0.821375i	$0.693207\pi$
$830$	0	0
$831$	47.2134	1.63781
$832$	0	0
$833$	5.14481	0.178257
$834$	0	0
$835$	27.6700i	0.957559i
$836$	0	0
$837$	− 11.3221i	− 0.391347i
$838$	0	0
$839$	30.9387	1.06812	0.534062	−	0.845445i	$-0.320665\pi$
$839$	30.9387	1.06812	0.534062	+	0.845445i	$0.320665\pi$
$840$	0	0
$841$	12.7052	0.438110
$842$	0	0
$843$	54.2521i	1.86854i
$844$	0	0
$845$	− 31.4410i	− 1.08160i
$846$	0	0
$847$	6.83978	0.235018
$848$	0	0
$849$	41.9937	1.44122
$850$	0	0
$851$	2.34905i	0.0805243i
$852$	0	0
$853$	− 14.8869i	− 0.509719i	−0.966978	−	0.254859i	$-0.917971\pi$
$853$	− 14.8869i	− 0.509719i	0.966978	−	0.254859i	$-0.0820291\pi$
$854$	0	0
$855$	1.25392	0.0428832
$856$	0	0
$857$	11.8197	0.403754	0.201877	−	0.979411i	$-0.435296\pi$
$857$	11.8197	0.403754	0.201877	+	0.979411i	$0.435296\pi$
$858$	0	0
$859$	− 1.73604i	− 0.0592329i	−0.999561	−	0.0296165i	$-0.990571\pi$
$859$	− 1.73604i	− 0.0592329i	0.999561	−	0.0296165i	$-0.00942859\pi$
$860$	0	0
$861$	7.51793i	0.256210i
$862$	0	0
$863$	−8.85723	−0.301504	−0.150752	−	0.988572i	$-0.548169\pi$
$863$	−8.85723	−0.301504	−0.150752	+	0.988572i	$0.548169\pi$
$864$	0	0
$865$	−27.0989	−0.921389
$866$	0	0
$867$	17.1891i	0.583772i
$868$	0	0
$869$	51.8356i	1.75840i
$870$	0	0
$871$	−2.19823	−0.0744842
$872$	0	0
$873$	−3.48058	−0.117800
$874$	0	0
$875$	7.66978i	0.259286i
$876$	0	0
$877$	22.4641i	0.758560i	0.925282	+	0.379280i	$0.123828\pi$
$877$	22.4641i	0.758560i	−0.925282	+	0.379280i	$0.876172\pi$
$878$	0	0
$879$	43.1288	1.45470
$880$	0	0
$881$	−28.3137	−0.953914	−0.476957	−	0.878927i	$-0.658260\pi$
$881$	−28.3137	−0.953914	−0.476957	+	0.878927i	$0.658260\pi$
$882$	0	0
$883$	− 38.7038i	− 1.30249i	−0.758868	−	0.651244i	$-0.774247\pi$
$883$	− 38.7038i	− 1.30249i	0.758868	−	0.651244i	$-0.225753\pi$
$884$	0	0
$885$	− 4.00182i	− 0.134520i
$886$	0	0
$887$	53.2495	1.78794	0.893972	−	0.448124i	$-0.147908\pi$
$887$	53.2495	1.78794	0.893972	+	0.448124i	$0.147908\pi$
$888$	0	0
$889$	−13.6917	−0.459206
$890$	0	0
$891$	41.3866i	1.38650i
$892$	0	0
$893$	11.0980i	0.371380i
$894$	0	0
$895$	46.2539	1.54610
$896$	0	0
$897$	9.70714	0.324112
$898$	0	0
$899$	9.30851i	0.310456i
$900$	0	0
$901$	33.2243i	1.10686i
$902$	0	0
$903$	−8.91431	−0.296650
$904$	0	0
$905$	36.8517	1.22499
$906$	0	0
$907$	− 46.0072i	− 1.52764i	−0.645428	−	0.763821i	$-0.723321\pi$
$907$	− 46.0072i	− 1.52764i	0.645428	−	0.763821i	$-0.276679\pi$
$908$	0	0
$909$	1.99679i	0.0662293i
$910$	0	0
$911$	−9.96023	−0.329997	−0.164999	−	0.986294i	$-0.552762\pi$
$911$	−9.96023	−0.329997	−0.164999	+	0.986294i	$0.552762\pi$
$912$	0	0
$913$	16.1430	0.534257
$914$	0	0
$915$	− 74.4131i	− 2.46002i
$916$	0	0
$917$	9.94586i	0.328441i
$918$	0	0
$919$	32.4154	1.06929	0.534643	−	0.845078i	$-0.320446\pi$
$919$	32.4154	1.06929	0.534643	+	0.845078i	$0.320446\pi$
$920$	0	0
$921$	−2.43263	−0.0801578
$922$	0	0
$923$	− 8.51270i	− 0.280199i
$924$	0	0
$925$	1.03324i	0.0339728i
$926$	0	0
$927$	−1.05127	−0.0345283
$928$	0	0
$929$	29.2818	0.960706	0.480353	−	0.877075i	$-0.340509\pi$
$929$	29.2818	0.960706	0.480353	+	0.877075i	$0.340509\pi$
$930$	0	0
$931$	− 1.59075i	− 0.0521346i
$932$	0	0
$933$	− 55.9779i	− 1.83263i
$934$	0	0
$935$	−58.0122	−1.89720
$936$	0	0
$937$	14.9553	0.488567	0.244284	−	0.969704i	$-0.421447\pi$
$937$	14.9553	0.488567	0.244284	+	0.969704i	$0.421447\pi$
$938$	0	0
$939$	− 3.22667i	− 0.105299i
$940$	0	0
$941$	19.3908i	0.632122i	0.948739	+	0.316061i	$0.102360\pi$
$941$	19.3908i	0.632122i	−0.948739	+	0.316061i	$0.897640\pi$
$942$	0	0
$943$	20.0272	0.652175
$944$	0	0
$945$	−13.1076	−0.426392
$946$	0	0
$947$	22.9007i	0.744172i	0.928198	+	0.372086i	$0.121357\pi$
$947$	22.9007i	0.744172i	−0.928198	+	0.372086i	$0.878643\pi$
$948$	0	0
$949$	18.7521i	0.608718i
$950$	0	0
$951$	4.89970	0.158884
$952$	0	0
$953$	−2.44851	−0.0793149	−0.0396574	−	0.999213i	$-0.512627\pi$
$953$	−2.44851	−0.0793149	−0.0396574	+	0.999213i	$0.512627\pi$
$954$	0	0
$955$	28.1048i	0.909451i
$956$	0	0
$957$	− 30.9503i	− 1.00048i
$958$	0	0
$959$	18.1361	0.585646
$960$	0	0
$961$	−25.6825	−0.828466
$962$	0	0
$963$	5.64569i	0.181930i
$964$	0	0
$965$	43.2400i	1.39194i
$966$	0	0
$967$	−25.1630	−0.809186	−0.404593	−	0.914497i	$-0.632587\pi$
$967$	−25.1630	−0.809186	−0.404593	+	0.914497i	$0.632587\pi$
$968$	0	0
$969$	14.8565	0.477259
$970$	0	0
$971$	− 6.88514i	− 0.220955i	−0.993879	−	0.110477i	$-0.964762\pi$
$971$	− 6.88514i	− 0.220955i	0.993879	−	0.110477i	$-0.0352380\pi$
$972$	0	0
$973$	17.0879i	0.547812i
$974$	0	0
$975$	4.26975	0.136741
$976$	0	0
$977$	44.5985	1.42683	0.713416	−	0.700741i	$-0.247147\pi$
$977$	44.5985	1.42683	0.713416	+	0.700741i	$0.247147\pi$
$978$	0	0
$979$	57.7888i	1.84694i
$980$	0	0
$981$	1.44129i	0.0460169i
$982$	0	0
$983$	8.28723	0.264322	0.132161	−	0.991228i	$-0.457808\pi$
$983$	8.28723	0.264322	0.132161	+	0.991228i	$0.457808\pi$
$984$	0	0
$985$	36.3045	1.15676
$986$	0	0
$987$	12.6645i	0.403116i
$988$	0	0
$989$	23.7470i	0.755112i
$990$	0	0
$991$	12.2190	0.388149	0.194074	−	0.980987i	$-0.437830\pi$
$991$	12.2190	0.388149	0.194074	+	0.980987i	$0.437830\pi$
$992$	0	0
$993$	−3.66220	−0.116217
$994$	0	0
$995$	− 34.1050i	− 1.08120i
$996$	0	0
$997$	48.2292i	1.52743i	0.645552	+	0.763717i	$0.276627\pi$
$997$	48.2292i	1.52743i	−0.645552	+	0.763717i	$0.723373\pi$
$998$	0	0
$999$	2.38503	0.0754591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.9		12
4.3	odd	2		3584.2.b.k.1793.4		12
8.3	odd	2		3584.2.b.k.1793.9		12
8.5	even	2	inner	3584.2.b.i.1793.4		12
16.3	odd	4		3584.2.a.k.1.4	yes	6
16.5	even	4		3584.2.a.l.1.4	yes	6
16.11	odd	4		3584.2.a.e.1.3	✓	6
16.13	even	4		3584.2.a.f.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.3	✓	6	16.11	odd	4
3584.2.a.f.1.3	yes	6	16.13	even	4
3584.2.a.k.1.4	yes	6	16.3	odd	4
3584.2.a.l.1.4	yes	6	16.5	even	4
3584.2.b.i.1793.4		12	8.5	even	2	inner
3584.2.b.i.1793.9		12	1.1	even	1	trivial
3584.2.b.k.1793.4		12	4.3	odd	2
3584.2.b.k.1793.9		12	8.3	odd	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.81529i q^{3} -2.66965i q^{5} -1.00000 q^{7} -0.295267 q^{9} +O(q^{10})\)	\(q+1.81529i q^{3} -2.66965i q^{5} -1.00000 q^{7} -0.295267 q^{9} -4.22372i q^{11} +1.10581i q^{13} +4.84619 q^{15} +5.14481 q^{17} -1.59075i q^{19} -1.81529i q^{21} -4.83578 q^{23} -2.12705 q^{25} +4.90987i q^{27} -4.03668i q^{29} -2.30598 q^{31} +7.66726 q^{33} +2.66965i q^{35} -0.485763i q^{37} -2.00736 q^{39} -4.14145 q^{41} -4.91069i q^{43} +0.788260i q^{45} -6.97659 q^{47} +1.00000 q^{49} +9.33931i q^{51} +6.45784i q^{53} -11.2759 q^{55} +2.88766 q^{57} -0.825767i q^{59} -15.3550i q^{61} +0.295267 q^{63} +2.95212 q^{65} +1.98790i q^{67} -8.77833i q^{69} -7.69818 q^{71} +16.9578 q^{73} -3.86121i q^{75} +4.22372i q^{77} -12.2725 q^{79} -9.79862 q^{81} +3.82200i q^{83} -13.7349i q^{85} +7.32774 q^{87} -13.6820 q^{89} -1.10581i q^{91} -4.18601i q^{93} -4.24674 q^{95} +11.7879 q^{97} +1.24712i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 20 * q^9	\( 12 q - 12 q^{7} - 20 q^{9} - 16 q^{15} - 4 q^{25} - 16 q^{31} + 8 q^{33} + 8 q^{41} + 16 q^{47} + 12 q^{49} - 32 q^{55} + 8 q^{57} + 20 q^{63} - 16 q^{65} + 16 q^{71} - 32 q^{73} - 48 q^{79} + 20 q^{81} + 16 q^{87} - 32 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 12 * q - 12 * q^7 - 20 * q^9 - 16 * q^15 - 4 * q^25 - 16 * q^31 + 8 * q^33 + 8 * q^41 + 16 * q^47 + 12 * q^49 - 32 * q^55 + 8 * q^57 + 20 * q^63 - 16 * q^65 + 16 * q^71 - 32 * q^73 - 48 * q^79 + 20 * q^81 + 16 * q^87 - 32 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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