LMFDB - Embedded newform 3584.2.b.i.1793.3

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			1793.3
Root			$-0.232297 - 0.560814i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.i.1793.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53584i q^{3} -1.47134i q^{5} -1.00000 q^{7} -3.43049 q^{9} +O(q^{10})$	$q-2.53584i q^{3} -1.47134i q^{5} -1.00000 q^{7} -3.43049 q^{9} -5.35014i q^{11} -3.55212i q^{13} -3.73108 q^{15} -2.73782 q^{17} +8.61663i q^{19} +2.53584i q^{21} -6.17919 q^{23} +2.83517 q^{25} +1.09166i q^{27} +8.49543i q^{29} -10.0009 q^{31} -13.5671 q^{33} +1.47134i q^{35} +0.333865i q^{37} -9.00761 q^{39} +1.30058 q^{41} -5.82230i q^{43} +5.04741i q^{45} +9.00348 q^{47} +1.00000 q^{49} +6.94267i q^{51} -1.20997i q^{53} -7.87186 q^{55} +21.8504 q^{57} +2.71738i q^{59} -4.48221i q^{61} +3.43049 q^{63} -5.22636 q^{65} -8.45438i q^{67} +15.6694i q^{69} -2.46718 q^{71} +6.31997 q^{73} -7.18954i q^{75} +5.35014i q^{77} -11.3091 q^{79} -7.52320 q^{81} -5.41031i q^{83} +4.02825i q^{85} +21.5431 q^{87} -6.44811 q^{89} +3.55212i q^{91} +25.3606i q^{93} +12.6780 q^{95} +4.95282 q^{97} +18.3536i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 2.53584i	− 1.46407i	−0.681268	−	0.732034i	$-0.738571\pi$
$3$	− 2.53584i	− 1.46407i	0.681268	−	0.732034i	$-0.261429\pi$
$4$	0	0
$5$	− 1.47134i	− 0.658002i	−0.944330	−	0.329001i	$-0.893288\pi$
$5$	− 1.47134i	− 0.658002i	0.944330	−	0.329001i	$-0.106712\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−3.43049	−1.14350
$10$	0	0
$11$	− 5.35014i	− 1.61313i	−0.591146	−	0.806564i	$-0.701324\pi$
$11$	− 5.35014i	− 1.61313i	0.591146	−	0.806564i	$-0.298676\pi$
$12$	0	0
$13$	− 3.55212i	− 0.985181i	−0.870261	−	0.492590i	$-0.836050\pi$
$13$	− 3.55212i	− 0.985181i	0.870261	−	0.492590i	$-0.163950\pi$
$14$	0	0
$15$	−3.73108	−0.963359
$16$	0	0
$17$	−2.73782	−0.664018	−0.332009	−	0.943276i	$-0.607727\pi$
$17$	−2.73782	−0.664018	−0.332009	+	0.943276i	$0.607727\pi$
$18$	0	0
$19$	8.61663i	1.97679i	0.151908	+	0.988395i	$0.451458\pi$
$19$	8.61663i	1.97679i	−0.151908	+	0.988395i	$0.548542\pi$
$20$	0	0
$21$	2.53584i	0.553366i
$22$	0	0
$23$	−6.17919	−1.28845	−0.644225	−	0.764836i	$-0.722820\pi$
$23$	−6.17919	−1.28845	−0.644225	+	0.764836i	$0.722820\pi$
$24$	0	0
$25$	2.83517	0.567034
$26$	0	0
$27$	1.09166i	0.210090i
$28$	0	0
$29$	8.49543i	1.57756i	0.614674	+	0.788781i	$0.289287\pi$
$29$	8.49543i	1.57756i	−0.614674	+	0.788781i	$0.710713\pi$
$30$	0	0
$31$	−10.0009	−1.79621	−0.898105	−	0.439782i	$-0.855056\pi$
$31$	−10.0009	−1.79621	−0.898105	+	0.439782i	$0.855056\pi$
$32$	0	0
$33$	−13.5671	−2.36173
$34$	0	0
$35$	1.47134i	0.248701i
$36$	0	0
$37$	0.333865i	0.0548871i	0.999623	+	0.0274435i	$0.00873664\pi$
$37$	0.333865i	0.0548871i	−0.999623	+	0.0274435i	$0.991263\pi$
$38$	0	0
$39$	−9.00761	−1.44237
$40$	0	0
$41$	1.30058	0.203117	0.101559	−	0.994830i	$-0.467617\pi$
$41$	1.30058	0.203117	0.101559	+	0.994830i	$0.467617\pi$
$42$	0	0
$43$	− 5.82230i	− 0.887892i	−0.896053	−	0.443946i	$-0.853578\pi$
$43$	− 5.82230i	− 0.887892i	0.896053	−	0.443946i	$-0.146422\pi$
$44$	0	0
$45$	5.04741i	0.752423i
$46$	0	0
$47$	9.00348	1.31329	0.656646	−	0.754199i	$-0.271974\pi$
$47$	9.00348	1.31329	0.656646	+	0.754199i	$0.271974\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.94267i	0.972169i
$52$	0	0
$53$	− 1.20997i	− 0.166203i	−0.996541	−	0.0831014i	$-0.973517\pi$
$53$	− 1.20997i	− 0.166203i	0.996541	−	0.0831014i	$-0.0264825\pi$
$54$	0	0
$55$	−7.87186	−1.06144
$56$	0	0
$57$	21.8504	2.89416
$58$	0	0
$59$	2.71738i	0.353773i	0.984231	+	0.176887i	$0.0566026\pi$
$59$	2.71738i	0.353773i	−0.984231	+	0.176887i	$0.943397\pi$
$60$	0	0
$61$	− 4.48221i	− 0.573888i	−0.957947	−	0.286944i	$-0.907361\pi$
$61$	− 4.48221i	− 0.573888i	0.957947	−	0.286944i	$-0.0926394\pi$
$62$	0	0
$63$	3.43049	0.432201
$64$	0	0
$65$	−5.22636	−0.648250
$66$	0	0
$67$	− 8.45438i	− 1.03287i	−0.856327	−	0.516434i	$-0.827259\pi$
$67$	− 8.45438i	− 1.03287i	0.856327	−	0.516434i	$-0.172741\pi$
$68$	0	0
$69$	15.6694i	1.88638i
$70$	0	0
$71$	−2.46718	−0.292801	−0.146400	−	0.989225i	$-0.546769\pi$
$71$	−2.46718	−0.292801	−0.146400	+	0.989225i	$0.546769\pi$
$72$	0	0
$73$	6.31997	0.739696	0.369848	−	0.929092i	$-0.379410\pi$
$73$	6.31997	0.739696	0.369848	+	0.929092i	$0.379410\pi$
$74$	0	0
$75$	− 7.18954i	− 0.830177i
$76$	0	0
$77$	5.35014i	0.609705i
$78$	0	0
$79$	−11.3091	−1.27237	−0.636186	−	0.771536i	$-0.719489\pi$
$79$	−11.3091	−1.27237	−0.636186	+	0.771536i	$0.719489\pi$
$80$	0	0
$81$	−7.52320	−0.835911
$82$	0	0
$83$	− 5.41031i	− 0.593859i	−0.954899	−	0.296929i	$-0.904037\pi$
$83$	− 5.41031i	− 0.593859i	0.954899	−	0.296929i	$-0.0959626\pi$
$84$	0	0
$85$	4.02825i	0.436925i
$86$	0	0
$87$	21.5431	2.30966
$88$	0	0
$89$	−6.44811	−0.683498	−0.341749	−	0.939791i	$-0.611019\pi$
$89$	−6.44811	−0.683498	−0.341749	+	0.939791i	$0.611019\pi$
$90$	0	0
$91$	3.55212i	0.372363i
$92$	0	0
$93$	25.3606i	2.62977i
$94$	0	0
$95$	12.6780	1.30073
$96$	0	0
$97$	4.95282	0.502883	0.251441	−	0.967872i	$-0.419095\pi$
$97$	4.95282	0.502883	0.251441	+	0.967872i	$0.419095\pi$
$98$	0	0
$99$	18.3536i	1.84461i
$100$	0	0
$101$	9.20897i	0.916327i	0.888868	+	0.458164i	$0.151493\pi$
$101$	9.20897i	0.916327i	−0.888868	+	0.458164i	$0.848507\pi$
$102$	0	0
$103$	−1.26803	−0.124942	−0.0624712	−	0.998047i	$-0.519898\pi$
$103$	−1.26803	−0.124942	−0.0624712	+	0.998047i	$0.519898\pi$
$104$	0	0
$105$	3.73108	0.364116
$106$	0	0
$107$	− 1.25678i	− 0.121497i	−0.998153	−	0.0607487i	$-0.980651\pi$
$107$	− 1.25678i	− 0.121497i	0.998153	−	0.0607487i	$-0.0193488\pi$
$108$	0	0
$109$	11.3048i	1.08280i	0.840764	+	0.541401i	$0.182106\pi$
$109$	11.3048i	1.08280i	−0.840764	+	0.541401i	$0.817894\pi$
$110$	0	0
$111$	0.846628	0.0803584
$112$	0	0
$113$	14.9407	1.40550	0.702751	−	0.711436i	$-0.251955\pi$
$113$	14.9407	1.40550	0.702751	+	0.711436i	$0.251955\pi$
$114$	0	0
$115$	9.09166i	0.847802i
$116$	0	0
$117$	12.1855i	1.12655i
$118$	0	0
$119$	2.73782	0.250975
$120$	0	0
$121$	−17.6240	−1.60218
$122$	0	0
$123$	− 3.29807i	− 0.297377i
$124$	0	0
$125$	− 11.5282i	− 1.03111i
$126$	0	0
$127$	−15.3442	−1.36158	−0.680790	−	0.732478i	$-0.738363\pi$
$127$	−15.3442	−1.36158	−0.680790	+	0.732478i	$0.738363\pi$
$128$	0	0
$129$	−14.7644	−1.29994
$130$	0	0
$131$	6.07758i	0.531001i	0.964111	+	0.265500i	$0.0855371\pi$
$131$	6.07758i	0.531001i	−0.964111	+	0.265500i	$0.914463\pi$
$132$	0	0
$133$	− 8.61663i	− 0.747156i
$134$	0	0
$135$	1.60620	0.138239
$136$	0	0
$137$	−8.27393	−0.706889	−0.353445	−	0.935455i	$-0.614990\pi$
$137$	−8.27393	−0.706889	−0.353445	+	0.935455i	$0.614990\pi$
$138$	0	0
$139$	− 22.5020i	− 1.90859i	−0.298863	−	0.954296i	$-0.596607\pi$
$139$	− 22.5020i	− 1.90859i	0.298863	−	0.954296i	$-0.403393\pi$
$140$	0	0
$141$	− 22.8314i	− 1.92275i
$142$	0	0
$143$	−19.0043	−1.58922
$144$	0	0
$145$	12.4996	1.03804
$146$	0	0
$147$	− 2.53584i	− 0.209153i
$148$	0	0
$149$	4.36212i	0.357358i	0.983907	+	0.178679i	$0.0571824\pi$
$149$	4.36212i	0.357358i	−0.983907	+	0.178679i	$0.942818\pi$
$150$	0	0
$151$	−16.0941	−1.30972	−0.654858	−	0.755752i	$-0.727272\pi$
$151$	−16.0941	−1.30972	−0.654858	+	0.755752i	$0.727272\pi$
$152$	0	0
$153$	9.39206	0.759303
$154$	0	0
$155$	14.7146i	1.18191i
$156$	0	0
$157$	0.677651i	0.0540824i	0.999634	+	0.0270412i	$0.00860853\pi$
$157$	0.677651i	0.0540824i	−0.999634	+	0.0270412i	$0.991391\pi$
$158$	0	0
$159$	−3.06830	−0.243332
$160$	0	0
$161$	6.17919	0.486988
$162$	0	0
$163$	− 2.49003i	− 0.195034i	−0.995234	−	0.0975171i	$-0.968910\pi$
$163$	− 2.49003i	− 0.195034i	0.995234	−	0.0975171i	$-0.0310901\pi$
$164$	0	0
$165$	19.9618i	1.55402i
$166$	0	0
$167$	21.0052	1.62543	0.812716	−	0.582660i	$-0.197988\pi$
$167$	21.0052	1.62543	0.812716	+	0.582660i	$0.197988\pi$
$168$	0	0
$169$	0.382445	0.0294189
$170$	0	0
$171$	− 29.5593i	− 2.26045i
$172$	0	0
$173$	21.2107i	1.61262i	0.591493	+	0.806310i	$0.298539\pi$
$173$	21.2107i	1.61262i	−0.591493	+	0.806310i	$0.701461\pi$
$174$	0	0
$175$	−2.83517	−0.214319
$176$	0	0
$177$	6.89086	0.517949
$178$	0	0
$179$	5.14067i	0.384232i	0.981372	+	0.192116i	$0.0615350\pi$
$179$	5.14067i	0.384232i	−0.981372	+	0.192116i	$0.938465\pi$
$180$	0	0
$181$	− 6.81366i	− 0.506455i	−0.967407	−	0.253228i	$-0.918508\pi$
$181$	− 6.81366i	− 0.506455i	0.967407	−	0.253228i	$-0.0814921\pi$
$182$	0	0
$183$	−11.3662	−0.840212
$184$	0	0
$185$	0.491228	0.0361158
$186$	0	0
$187$	14.6477i	1.07115i
$188$	0	0
$189$	− 1.09166i	− 0.0794065i
$190$	0	0
$191$	17.1650	1.24202	0.621010	−	0.783803i	$-0.286723\pi$
$191$	17.1650	1.24202	0.621010	+	0.783803i	$0.286723\pi$
$192$	0	0
$193$	−18.9566	−1.36453	−0.682263	−	0.731107i	$-0.739004\pi$
$193$	−18.9566	−1.36453	−0.682263	+	0.731107i	$0.739004\pi$
$194$	0	0
$195$	13.2532i	0.949083i
$196$	0	0
$197$	3.08800i	0.220011i	0.993931	+	0.110005i	$0.0350868\pi$
$197$	3.08800i	0.220011i	−0.993931	+	0.110005i	$0.964913\pi$
$198$	0	0
$199$	1.49652	0.106085	0.0530426	−	0.998592i	$-0.483108\pi$
$199$	1.49652	0.106085	0.0530426	+	0.998592i	$0.483108\pi$
$200$	0	0
$201$	−21.4390	−1.51219
$202$	0	0
$203$	− 8.49543i	− 0.596262i
$204$	0	0
$205$	− 1.91360i	− 0.133651i
$206$	0	0
$207$	21.1976	1.47334
$208$	0	0
$209$	46.1002	3.18882
$210$	0	0
$211$	20.0643i	1.38129i	0.723196	+	0.690643i	$0.242672\pi$
$211$	20.0643i	1.38129i	−0.723196	+	0.690643i	$0.757328\pi$
$212$	0	0
$213$	6.25638i	0.428680i
$214$	0	0
$215$	−8.56656	−0.584235
$216$	0	0
$217$	10.0009	0.678903
$218$	0	0
$219$	− 16.0264i	− 1.08297i
$220$	0	0
$221$	9.72506i	0.654178i
$222$	0	0
$223$	−16.1569	−1.08194	−0.540971	−	0.841041i	$-0.681943\pi$
$223$	−16.1569	−1.08194	−0.540971	+	0.841041i	$0.681943\pi$
$224$	0	0
$225$	−9.72603	−0.648402
$226$	0	0
$227$	6.81256i	0.452165i	0.974108	+	0.226083i	$0.0725920\pi$
$227$	6.81256i	0.452165i	−0.974108	+	0.226083i	$0.927408\pi$
$228$	0	0
$229$	− 6.88613i	− 0.455048i	−0.973772	−	0.227524i	$-0.926937\pi$
$229$	− 6.88613i	− 0.455048i	0.973772	−	0.227524i	$-0.0730631\pi$
$230$	0	0
$231$	13.5671	0.892651
$232$	0	0
$233$	−0.00584477	−0.000382903 0	−0.000191452	−	1.00000i	$-0.500061\pi$
$233$	−0.00584477	−0.000382903 0	−0.000191452		1.00000i	$0.500061\pi$
$234$	0	0
$235$	− 13.2471i	− 0.864148i
$236$	0	0
$237$	28.6781i	1.86284i
$238$	0	0
$239$	12.7496	0.824703	0.412351	−	0.911025i	$-0.364708\pi$
$239$	12.7496	0.824703	0.412351	+	0.911025i	$0.364708\pi$
$240$	0	0
$241$	25.1923	1.62278	0.811389	−	0.584507i	$-0.198712\pi$
$241$	25.1923	1.62278	0.811389	+	0.584507i	$0.198712\pi$
$242$	0	0
$243$	22.3526i	1.43392i
$244$	0	0
$245$	− 1.47134i	− 0.0940002i
$246$	0	0
$247$	30.6073	1.94749
$248$	0	0
$249$	−13.7197	−0.869450
$250$	0	0
$251$	− 19.7552i	− 1.24694i	−0.781849	−	0.623468i	$-0.785723\pi$
$251$	− 19.7552i	− 1.24694i	0.781849	−	0.623468i	$-0.214277\pi$
$252$	0	0
$253$	33.0595i	2.07843i
$254$	0	0
$255$	10.2150	0.639688
$256$	0	0
$257$	3.70413	0.231057	0.115529	−	0.993304i	$-0.463144\pi$
$257$	3.70413	0.231057	0.115529	+	0.993304i	$0.463144\pi$
$258$	0	0
$259$	− 0.333865i	− 0.0207454i
$260$	0	0
$261$	− 29.1435i	− 1.80394i
$262$	0	0
$263$	−8.22781	−0.507348	−0.253674	−	0.967290i	$-0.581639\pi$
$263$	−8.22781	−0.507348	−0.253674	+	0.967290i	$0.581639\pi$
$264$	0	0
$265$	−1.78028	−0.109362
$266$	0	0
$267$	16.3514i	1.00069i
$268$	0	0
$269$	− 1.57387i	− 0.0959606i	−0.998848	−	0.0479803i	$-0.984722\pi$
$269$	− 1.57387i	− 0.0959606i	0.998848	−	0.0479803i	$-0.0152785\pi$
$270$	0	0
$271$	23.5357	1.42969	0.714845	−	0.699283i	$-0.246497\pi$
$271$	23.5357	1.42969	0.714845	+	0.699283i	$0.246497\pi$
$272$	0	0
$273$	9.00761	0.545166
$274$	0	0
$275$	− 15.1686i	− 0.914699i
$276$	0	0
$277$	− 6.62846i	− 0.398265i	−0.979973	−	0.199133i	$-0.936188\pi$
$277$	− 6.62846i	− 0.398265i	0.979973	−	0.199133i	$-0.0638125\pi$
$278$	0	0
$279$	34.3079	2.05396
$280$	0	0
$281$	−19.3905	−1.15674	−0.578370	−	0.815774i	$-0.696311\pi$
$281$	−19.3905	−1.15674	−0.578370	+	0.815774i	$0.696311\pi$
$282$	0	0
$283$	− 4.65625i	− 0.276785i	−0.990377	−	0.138393i	$-0.955806\pi$
$283$	− 4.65625i	− 0.276785i	0.990377	−	0.138393i	$-0.0441936\pi$
$284$	0	0
$285$	− 32.1493i	− 1.90436i
$286$	0	0
$287$	−1.30058	−0.0767710
$288$	0	0
$289$	−9.50435	−0.559080
$290$	0	0
$291$	− 12.5596i	− 0.736255i
$292$	0	0
$293$	27.3806i	1.59959i	0.600272	+	0.799796i	$0.295059\pi$
$293$	27.3806i	1.59959i	−0.600272	+	0.799796i	$0.704941\pi$
$294$	0	0
$295$	3.99819	0.232783
$296$	0	0
$297$	5.84053	0.338902
$298$	0	0
$299$	21.9492i	1.26936i
$300$	0	0
$301$	5.82230i	0.335592i
$302$	0	0
$303$	23.3525	1.34157
$304$	0	0
$305$	−6.59484	−0.377619
$306$	0	0
$307$	9.34520i	0.533359i	0.963785	+	0.266679i	$0.0859264\pi$
$307$	9.34520i	0.533359i	−0.963785	+	0.266679i	$0.914074\pi$
$308$	0	0
$309$	3.21551i	0.182924i
$310$	0	0
$311$	0.237720	0.0134798	0.00673992	−	0.999977i	$-0.497855\pi$
$311$	0.237720	0.0134798	0.00673992	+	0.999977i	$0.497855\pi$
$312$	0	0
$313$	−4.86308	−0.274878	−0.137439	−	0.990510i	$-0.543887\pi$
$313$	−4.86308	−0.274878	−0.137439	+	0.990510i	$0.543887\pi$
$314$	0	0
$315$	− 5.04741i	− 0.284389i
$316$	0	0
$317$	− 23.8299i	− 1.33842i	−0.743073	−	0.669210i	$-0.766633\pi$
$317$	− 23.8299i	− 1.33842i	0.743073	−	0.669210i	$-0.233367\pi$
$318$	0	0
$319$	45.4518	2.54481
$320$	0	0
$321$	−3.18699	−0.177881
$322$	0	0
$323$	− 23.5908i	− 1.31262i
$324$	0	0
$325$	− 10.0709i	− 0.558631i
$326$	0	0
$327$	28.6671	1.58530
$328$	0	0
$329$	−9.00348	−0.496378
$330$	0	0
$331$	8.23230i	0.452488i	0.974071	+	0.226244i	$0.0726447\pi$
$331$	8.23230i	0.452488i	−0.974071	+	0.226244i	$0.927355\pi$
$332$	0	0
$333$	− 1.14532i	− 0.0627632i
$334$	0	0
$335$	−12.4392	−0.679628
$336$	0	0
$337$	−20.2587	−1.10356	−0.551780	−	0.833990i	$-0.686051\pi$
$337$	−20.2587	−1.10356	−0.551780	+	0.833990i	$0.686051\pi$
$338$	0	0
$339$	− 37.8872i	− 2.05775i
$340$	0	0
$341$	53.5061i	2.89752i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	23.0550	1.24124
$346$	0	0
$347$	− 21.7190i	− 1.16594i	−0.812495	−	0.582968i	$-0.801891\pi$
$347$	− 21.7190i	− 1.16594i	0.812495	−	0.582968i	$-0.198109\pi$
$348$	0	0
$349$	− 15.6559i	− 0.838043i	−0.907976	−	0.419021i	$-0.862373\pi$
$349$	− 15.6559i	− 0.838043i	0.907976	−	0.419021i	$-0.137627\pi$
$350$	0	0
$351$	3.87770	0.206977
$352$	0	0
$353$	15.1663	0.807219	0.403610	−	0.914931i	$-0.367755\pi$
$353$	15.1663	0.807219	0.403610	+	0.914931i	$0.367755\pi$
$354$	0	0
$355$	3.63005i	0.192663i
$356$	0	0
$357$	− 6.94267i	− 0.367445i
$358$	0	0
$359$	−35.1850	−1.85699	−0.928496	−	0.371342i	$-0.878898\pi$
$359$	−35.1850	−1.85699	−0.928496	+	0.371342i	$0.878898\pi$
$360$	0	0
$361$	−55.2462	−2.90770
$362$	0	0
$363$	44.6918i	2.34571i
$364$	0	0
$365$	− 9.29880i	− 0.486721i
$366$	0	0
$367$	10.1677	0.530751	0.265376	−	0.964145i	$-0.414504\pi$
$367$	10.1677	0.530751	0.265376	+	0.964145i	$0.414504\pi$
$368$	0	0
$369$	−4.46164	−0.232264
$370$	0	0
$371$	1.20997i	0.0628187i
$372$	0	0
$373$	11.0935i	0.574400i	0.957871	+	0.287200i	$0.0927245\pi$
$373$	11.0935i	0.574400i	−0.957871	+	0.287200i	$0.907276\pi$
$374$	0	0
$375$	−29.2336	−1.50962
$376$	0	0
$377$	30.1768	1.55418
$378$	0	0
$379$	− 36.0994i	− 1.85430i	−0.374686	−	0.927152i	$-0.622249\pi$
$379$	− 36.0994i	− 1.85430i	0.374686	−	0.927152i	$-0.377751\pi$
$380$	0	0
$381$	38.9105i	1.99345i
$382$	0	0
$383$	−7.18325	−0.367047	−0.183523	−	0.983015i	$-0.558750\pi$
$383$	−7.18325	−0.367047	−0.183523	+	0.983015i	$0.558750\pi$
$384$	0	0
$385$	7.87186	0.401187
$386$	0	0
$387$	19.9734i	1.01530i
$388$	0	0
$389$	− 25.6497i	− 1.30049i	−0.759725	−	0.650245i	$-0.774666\pi$
$389$	− 25.6497i	− 1.30049i	0.759725	−	0.650245i	$-0.225334\pi$
$390$	0	0
$391$	16.9175	0.855554
$392$	0	0
$393$	15.4118	0.777422
$394$	0	0
$395$	16.6395i	0.837223i
$396$	0	0
$397$	− 28.1738i	− 1.41400i	−0.707212	−	0.707002i	$-0.750047\pi$
$397$	− 28.1738i	− 1.41400i	0.707212	−	0.707002i	$-0.249953\pi$
$398$	0	0
$399$	−21.8504	−1.09389
$400$	0	0
$401$	10.2803	0.513371	0.256686	−	0.966495i	$-0.417369\pi$
$401$	10.2803	0.513371	0.256686	+	0.966495i	$0.417369\pi$
$402$	0	0
$403$	35.5243i	1.76959i
$404$	0	0
$405$	11.0692i	0.550031i
$406$	0	0
$407$	1.78623	0.0885399
$408$	0	0
$409$	19.0375	0.941343	0.470672	−	0.882309i	$-0.344012\pi$
$409$	19.0375	0.941343	0.470672	+	0.882309i	$0.344012\pi$
$410$	0	0
$411$	20.9814i	1.03493i
$412$	0	0
$413$	− 2.71738i	− 0.133714i
$414$	0	0
$415$	−7.96039	−0.390760
$416$	0	0
$417$	−57.0614	−2.79431
$418$	0	0
$419$	0.952050i	0.0465107i	0.999730	+	0.0232553i	$0.00740307\pi$
$419$	0.952050i	0.0465107i	−0.999730	+	0.0232553i	$0.992597\pi$
$420$	0	0
$421$	27.1640i	1.32389i	0.749552	+	0.661946i	$0.230269\pi$
$421$	27.1640i	1.32389i	−0.749552	+	0.661946i	$0.769731\pi$
$422$	0	0
$423$	−30.8864	−1.50175
$424$	0	0
$425$	−7.76218	−0.376521
$426$	0	0
$427$	4.48221i	0.216909i
$428$	0	0
$429$	48.1920i	2.32673i
$430$	0	0
$431$	−36.4598	−1.75621	−0.878104	−	0.478470i	$-0.841191\pi$
$431$	−36.4598	−1.75621	−0.878104	+	0.478470i	$0.841191\pi$
$432$	0	0
$433$	−18.9245	−0.909456	−0.454728	−	0.890630i	$-0.650263\pi$
$433$	−18.9245	−0.909456	−0.454728	+	0.890630i	$0.650263\pi$
$434$	0	0
$435$	− 31.6971i	− 1.51976i
$436$	0	0
$437$	− 53.2437i	− 2.54699i
$438$	0	0
$439$	−20.9378	−0.999306	−0.499653	−	0.866226i	$-0.666539\pi$
$439$	−20.9378	−0.999306	−0.499653	+	0.866226i	$0.666539\pi$
$440$	0	0
$441$	−3.43049	−0.163357
$442$	0	0
$443$	− 28.7793i	− 1.36734i	−0.729790	−	0.683672i	$-0.760382\pi$
$443$	− 28.7793i	− 1.36734i	0.729790	−	0.683672i	$-0.239618\pi$
$444$	0	0
$445$	9.48734i	0.449743i
$446$	0	0
$447$	11.0616	0.523197
$448$	0	0
$449$	−22.1940	−1.04740	−0.523700	−	0.851903i	$-0.675449\pi$
$449$	−22.1940	−1.04740	−0.523700	+	0.851903i	$0.675449\pi$
$450$	0	0
$451$	− 6.95831i	− 0.327654i
$452$	0	0
$453$	40.8120i	1.91751i
$454$	0	0
$455$	5.22636	0.245016
$456$	0	0
$457$	−24.6211	−1.15173	−0.575863	−	0.817546i	$-0.695334\pi$
$457$	−24.6211	−1.15173	−0.575863	+	0.817546i	$0.695334\pi$
$458$	0	0
$459$	− 2.98876i	− 0.139504i
$460$	0	0
$461$	18.4057i	0.857239i	0.903485	+	0.428619i	$0.141000\pi$
$461$	18.4057i	0.857239i	−0.903485	+	0.428619i	$0.859000\pi$
$462$	0	0
$463$	−40.8454	−1.89825	−0.949123	−	0.314905i	$-0.898027\pi$
$463$	−40.8454	−1.89825	−0.949123	+	0.314905i	$0.898027\pi$
$464$	0	0
$465$	37.3140	1.73040
$466$	0	0
$467$	− 7.56396i	− 0.350018i	−0.984567	−	0.175009i	$-0.944004\pi$
$467$	− 7.56396i	− 0.350018i	0.984567	−	0.175009i	$-0.0559955\pi$
$468$	0	0
$469$	8.45438i	0.390387i
$470$	0	0
$471$	1.71841	0.0791804
$472$	0	0
$473$	−31.1501	−1.43228
$474$	0	0
$475$	24.4296i	1.12091i
$476$	0	0
$477$	4.15081i	0.190052i
$478$	0	0
$479$	−37.8779	−1.73069	−0.865343	−	0.501181i	$-0.832899\pi$
$479$	−37.8779	−1.73069	−0.865343	+	0.501181i	$0.832899\pi$
$480$	0	0
$481$	1.18593	0.0540737
$482$	0	0
$483$	− 15.6694i	− 0.712984i
$484$	0	0
$485$	− 7.28727i	− 0.330898i
$486$	0	0
$487$	−18.7938	−0.851630	−0.425815	−	0.904810i	$-0.640013\pi$
$487$	−18.7938	−0.851630	−0.425815	+	0.904810i	$0.640013\pi$
$488$	0	0
$489$	−6.31432	−0.285543
$490$	0	0
$491$	11.9445i	0.539045i	0.962994	+	0.269523i	$0.0868659\pi$
$491$	11.9445i	0.539045i	−0.962994	+	0.269523i	$0.913134\pi$
$492$	0	0
$493$	− 23.2589i	− 1.04753i
$494$	0	0
$495$	27.0043	1.21376
$496$	0	0
$497$	2.46718	0.110668
$498$	0	0
$499$	− 16.0387i	− 0.717992i	−0.933339	−	0.358996i	$-0.883119\pi$
$499$	− 16.0387i	− 0.717992i	0.933339	−	0.358996i	$-0.116881\pi$
$500$	0	0
$501$	− 53.2659i	− 2.37975i
$502$	0	0
$503$	2.25982	0.100760	0.0503801	−	0.998730i	$-0.483957\pi$
$503$	2.25982	0.100760	0.0503801	+	0.998730i	$0.483957\pi$
$504$	0	0
$505$	13.5495	0.602945
$506$	0	0
$507$	− 0.969820i	− 0.0430712i
$508$	0	0
$509$	− 19.8827i	− 0.881285i	−0.897683	−	0.440642i	$-0.854751\pi$
$509$	− 19.8827i	− 0.881285i	0.897683	−	0.440642i	$-0.145249\pi$
$510$	0	0
$511$	−6.31997	−0.279579
$512$	0	0
$513$	−9.40642	−0.415303
$514$	0	0
$515$	1.86569i	0.0822123i
$516$	0	0
$517$	− 48.1699i	− 2.11851i
$518$	0	0
$519$	53.7870	2.36099
$520$	0	0
$521$	14.9222	0.653754	0.326877	−	0.945067i	$-0.394004\pi$
$521$	14.9222	0.653754	0.326877	+	0.945067i	$0.394004\pi$
$522$	0	0
$523$	16.2480i	0.710474i	0.934776	+	0.355237i	$0.115600\pi$
$523$	16.2480i	0.710474i	−0.934776	+	0.355237i	$0.884400\pi$
$524$	0	0
$525$	7.18954i	0.313777i
$526$	0	0
$527$	27.3806	1.19272
$528$	0	0
$529$	15.1823	0.660101
$530$	0	0
$531$	− 9.32197i	− 0.404539i
$532$	0	0
$533$	− 4.61983i	− 0.200107i
$534$	0	0
$535$	−1.84914	−0.0799455
$536$	0	0
$537$	13.0359	0.562542
$538$	0	0
$539$	− 5.35014i	− 0.230447i
$540$	0	0
$541$	− 39.7623i	− 1.70952i	−0.519027	−	0.854758i	$-0.673705\pi$
$541$	− 39.7623i	− 1.70952i	0.519027	−	0.854758i	$-0.326295\pi$
$542$	0	0
$543$	−17.2784	−0.741485
$544$	0	0
$545$	16.6331	0.712486
$546$	0	0
$547$	36.7280i	1.57038i	0.619256	+	0.785189i	$0.287434\pi$
$547$	36.7280i	1.57038i	−0.619256	+	0.785189i	$0.712566\pi$
$548$	0	0
$549$	15.3762i	0.656240i
$550$	0	0
$551$	−73.2020	−3.11851
$552$	0	0
$553$	11.3091	0.480912
$554$	0	0
$555$	− 1.24568i	− 0.0528760i
$556$	0	0
$557$	− 25.0967i	− 1.06338i	−0.846939	−	0.531691i	$-0.821557\pi$
$557$	− 25.0967i	− 1.06338i	0.846939	−	0.531691i	$-0.178443\pi$
$558$	0	0
$559$	−20.6815	−0.874734
$560$	0	0
$561$	37.1443	1.56823
$562$	0	0
$563$	27.5781i	1.16228i	0.813805	+	0.581138i	$0.197392\pi$
$563$	27.5781i	1.16228i	−0.813805	+	0.581138i	$0.802608\pi$
$564$	0	0
$565$	− 21.9828i	− 0.924822i
$566$	0	0
$567$	7.52320	0.315945
$568$	0	0
$569$	25.4129	1.06536	0.532682	−	0.846315i	$-0.321184\pi$
$569$	25.4129	1.06536	0.532682	+	0.846315i	$0.321184\pi$
$570$	0	0
$571$	6.29583i	0.263472i	0.991285	+	0.131736i	$0.0420552\pi$
$571$	6.29583i	0.263472i	−0.991285	+	0.131736i	$0.957945\pi$
$572$	0	0
$573$	− 43.5278i	− 1.81840i
$574$	0	0
$575$	−17.5190	−0.730594
$576$	0	0
$577$	8.39622	0.349539	0.174770	−	0.984609i	$-0.444082\pi$
$577$	8.39622	0.349539	0.174770	+	0.984609i	$0.444082\pi$
$578$	0	0
$579$	48.0709i	1.99776i
$580$	0	0
$581$	5.41031i	0.224458i
$582$	0	0
$583$	−6.47354	−0.268106
$584$	0	0
$585$	17.9290	0.741273
$586$	0	0
$587$	36.7104i	1.51520i	0.652718	+	0.757601i	$0.273629\pi$
$587$	36.7104i	1.51520i	−0.652718	+	0.757601i	$0.726371\pi$
$588$	0	0
$589$	− 86.1737i	− 3.55073i
$590$	0	0
$591$	7.83068	0.322111
$592$	0	0
$593$	−37.3627	−1.53430	−0.767152	−	0.641466i	$-0.778327\pi$
$593$	−37.3627	−1.53430	−0.767152	+	0.641466i	$0.778327\pi$
$594$	0	0
$595$	− 4.02825i	− 0.165142i
$596$	0	0
$597$	− 3.79493i	− 0.155316i
$598$	0	0
$599$	−18.2380	−0.745185	−0.372592	−	0.927995i	$-0.621531\pi$
$599$	−18.2380	−0.745185	−0.372592	+	0.927995i	$0.621531\pi$
$600$	0	0
$601$	−11.2357	−0.458314	−0.229157	−	0.973389i	$-0.573597\pi$
$601$	−11.2357	−0.458314	−0.229157	+	0.973389i	$0.573597\pi$
$602$	0	0
$603$	29.0027i	1.18108i
$604$	0	0
$605$	25.9309i	1.05424i
$606$	0	0
$607$	16.1984	0.657473	0.328737	−	0.944422i	$-0.393377\pi$
$607$	16.1984	0.657473	0.328737	+	0.944422i	$0.393377\pi$
$608$	0	0
$609$	−21.5431	−0.872969
$610$	0	0
$611$	− 31.9814i	− 1.29383i
$612$	0	0
$613$	− 23.5593i	− 0.951550i	−0.879567	−	0.475775i	$-0.842168\pi$
$613$	− 23.5593i	− 0.951550i	0.879567	−	0.475775i	$-0.157832\pi$
$614$	0	0
$615$	−4.85257	−0.195675
$616$	0	0
$617$	17.5014	0.704580	0.352290	−	0.935891i	$-0.385403\pi$
$617$	17.5014	0.704580	0.352290	+	0.935891i	$0.385403\pi$
$618$	0	0
$619$	− 22.7093i	− 0.912764i	−0.889784	−	0.456382i	$-0.849145\pi$
$619$	− 22.7093i	− 0.912764i	0.889784	−	0.456382i	$-0.150855\pi$
$620$	0	0
$621$	− 6.74556i	− 0.270690i
$622$	0	0
$623$	6.44811	0.258338
$624$	0	0
$625$	−2.78596	−0.111439
$626$	0	0
$627$	− 116.903i	− 4.66865i
$628$	0	0
$629$	− 0.914061i	− 0.0364460i
$630$	0	0
$631$	−1.05047	−0.0418185	−0.0209092	−	0.999781i	$-0.506656\pi$
$631$	−1.05047	−0.0418185	−0.0209092	+	0.999781i	$0.506656\pi$
$632$	0	0
$633$	50.8799	2.02230
$634$	0	0
$635$	22.5765i	0.895922i
$636$	0	0
$637$	− 3.55212i	− 0.140740i
$638$	0	0
$639$	8.46365	0.334817
$640$	0	0
$641$	−5.48096	−0.216485	−0.108242	−	0.994125i	$-0.534522\pi$
$641$	−5.48096	−0.216485	−0.108242	+	0.994125i	$0.534522\pi$
$642$	0	0
$643$	− 1.22391i	− 0.0482662i	−0.999709	−	0.0241331i	$-0.992317\pi$
$643$	− 1.22391i	− 0.0482662i	0.999709	−	0.0241331i	$-0.00768255\pi$
$644$	0	0
$645$	21.7234i	0.855360i
$646$	0	0
$647$	−17.7538	−0.697975	−0.348987	−	0.937127i	$-0.613474\pi$
$647$	−17.7538	−0.697975	−0.348987	+	0.937127i	$0.613474\pi$
$648$	0	0
$649$	14.5384	0.570682
$650$	0	0
$651$	− 25.3606i	− 0.993961i
$652$	0	0
$653$	27.0233i	1.05751i	0.848776	+	0.528753i	$0.177340\pi$
$653$	27.0233i	1.05751i	−0.848776	+	0.528753i	$0.822660\pi$
$654$	0	0
$655$	8.94216	0.349399
$656$	0	0
$657$	−21.6806	−0.845841
$658$	0	0
$659$	29.2572i	1.13970i	0.821750	+	0.569849i	$0.192998\pi$
$659$	29.2572i	1.13970i	−0.821750	+	0.569849i	$0.807002\pi$
$660$	0	0
$661$	− 1.24828i	− 0.0485523i	−0.999705	−	0.0242762i	$-0.992272\pi$
$661$	− 1.24828i	− 0.0485523i	0.999705	−	0.0242762i	$-0.00772810\pi$
$662$	0	0
$663$	24.6612	0.957762
$664$	0	0
$665$	−12.6780	−0.491630
$666$	0	0
$667$	− 52.4949i	− 2.03261i
$668$	0	0
$669$	40.9712i	1.58404i
$670$	0	0
$671$	−23.9805	−0.925756
$672$	0	0
$673$	16.0287	0.617861	0.308931	−	0.951085i	$-0.400029\pi$
$673$	16.0287	0.617861	0.308931	+	0.951085i	$0.400029\pi$
$674$	0	0
$675$	3.09504i	0.119128i
$676$	0	0
$677$	− 3.28162i	− 0.126123i	−0.998010	−	0.0630614i	$-0.979914\pi$
$677$	− 3.28162i	− 0.126123i	0.998010	−	0.0630614i	$-0.0200864\pi$
$678$	0	0
$679$	−4.95282	−0.190072
$680$	0	0
$681$	17.2756	0.662001
$682$	0	0
$683$	18.1046i	0.692755i	0.938095	+	0.346377i	$0.112588\pi$
$683$	18.1046i	0.692755i	−0.938095	+	0.346377i	$0.887412\pi$
$684$	0	0
$685$	12.1737i	0.465134i
$686$	0	0
$687$	−17.4621	−0.666222
$688$	0	0
$689$	−4.29797	−0.163740
$690$	0	0
$691$	16.3412i	0.621648i	0.950467	+	0.310824i	$0.100605\pi$
$691$	16.3412i	0.621648i	−0.950467	+	0.310824i	$0.899395\pi$
$692$	0	0
$693$	− 18.3536i	− 0.697196i
$694$	0	0
$695$	−33.1080	−1.25586
$696$	0	0
$697$	−3.56076	−0.134873
$698$	0	0
$699$	0.0148214i	0 0.000560597i
$700$	0	0
$701$	5.84196i	0.220648i	0.993896	+	0.110324i	$0.0351888\pi$
$701$	5.84196i	0.220648i	−0.993896	+	0.110324i	$0.964811\pi$
$702$	0	0
$703$	−2.87679	−0.108500
$704$	0	0
$705$	−33.5927	−1.26517
$706$	0	0
$707$	− 9.20897i	− 0.346339i
$708$	0	0
$709$	− 13.7622i	− 0.516851i	−0.966031	−	0.258426i	$-0.916796\pi$
$709$	− 13.7622i	− 0.516851i	0.966031	−	0.258426i	$-0.0832037\pi$
$710$	0	0
$711$	38.7958	1.45495
$712$	0	0
$713$	61.7972	2.31432
$714$	0	0
$715$	27.9618i	1.04571i
$716$	0	0
$717$	− 32.3309i	− 1.20742i
$718$	0	0
$719$	−25.5551	−0.953045	−0.476522	−	0.879162i	$-0.658103\pi$
$719$	−25.5551	−0.953045	−0.476522	+	0.879162i	$0.658103\pi$
$720$	0	0
$721$	1.26803	0.0472238
$722$	0	0
$723$	− 63.8836i	− 2.37586i
$724$	0	0
$725$	24.0860i	0.894531i
$726$	0	0
$727$	−29.7659	−1.10395	−0.551977	−	0.833859i	$-0.686126\pi$
$727$	−29.7659	−1.10395	−0.551977	+	0.833859i	$0.686126\pi$
$728$	0	0
$729$	34.1131	1.26345
$730$	0	0
$731$	15.9404i	0.589577i
$732$	0	0
$733$	− 32.2469i	− 1.19107i	−0.803331	−	0.595533i	$-0.796941\pi$
$733$	− 32.2469i	− 1.19107i	0.803331	−	0.595533i	$-0.203059\pi$
$734$	0	0
$735$	−3.73108	−0.137623
$736$	0	0
$737$	−45.2322	−1.66615
$738$	0	0
$739$	− 8.37793i	− 0.308187i	−0.988056	−	0.154094i	$-0.950754\pi$
$739$	− 8.37793i	− 0.308187i	0.988056	−	0.154094i	$-0.0492457\pi$
$740$	0	0
$741$	− 77.6152i	− 2.85127i
$742$	0	0
$743$	25.4881	0.935068	0.467534	−	0.883975i	$-0.345143\pi$
$743$	25.4881	0.935068	0.467534	+	0.883975i	$0.345143\pi$
$744$	0	0
$745$	6.41814	0.235142
$746$	0	0
$747$	18.5600i	0.679076i
$748$	0	0
$749$	1.25678i	0.0459217i
$750$	0	0
$751$	−27.7490	−1.01258	−0.506288	−	0.862365i	$-0.668983\pi$
$751$	−27.7490	−1.01258	−0.506288	+	0.862365i	$0.668983\pi$
$752$	0	0
$753$	−50.0960	−1.82560
$754$	0	0
$755$	23.6798i	0.861795i
$756$	0	0
$757$	− 12.5446i	− 0.455942i	−0.973668	−	0.227971i	$-0.926791\pi$
$757$	− 12.5446i	− 0.455942i	0.973668	−	0.227971i	$-0.0732091\pi$
$758$	0	0
$759$	83.8337	3.04297
$760$	0	0
$761$	−36.9458	−1.33929	−0.669643	−	0.742683i	$-0.733553\pi$
$761$	−36.9458	−1.33929	−0.669643	+	0.742683i	$0.733553\pi$
$762$	0	0
$763$	− 11.3048i	− 0.409261i
$764$	0	0
$765$	− 13.8189i	− 0.499623i
$766$	0	0
$767$	9.65248	0.348531
$768$	0	0
$769$	−34.5020	−1.24417	−0.622087	−	0.782948i	$-0.713715\pi$
$769$	−34.5020	−1.24417	−0.622087	+	0.782948i	$0.713715\pi$
$770$	0	0
$771$	− 9.39308i	− 0.338283i
$772$	0	0
$773$	− 12.1677i	− 0.437641i	−0.975765	−	0.218820i	$-0.929779\pi$
$773$	− 12.1677i	− 0.437641i	0.975765	−	0.218820i	$-0.0702209\pi$
$774$	0	0
$775$	−28.3542	−1.01851
$776$	0	0
$777$	−0.846628	−0.0303726
$778$	0	0
$779$	11.2066i	0.401520i
$780$	0	0
$781$	13.1998i	0.472325i
$782$	0	0
$783$	−9.27412	−0.331430
$784$	0	0
$785$	0.997052	0.0355863
$786$	0	0
$787$	− 24.2744i	− 0.865289i	−0.901565	−	0.432644i	$-0.857581\pi$
$787$	− 24.2744i	− 0.865289i	0.901565	−	0.432644i	$-0.142419\pi$
$788$	0	0
$789$	20.8644i	0.742793i
$790$	0	0
$791$	−14.9407	−0.531230
$792$	0	0
$793$	−15.9214	−0.565384
$794$	0	0
$795$	4.51451i	0.160113i
$796$	0	0
$797$	− 4.58429i	− 0.162384i	−0.996698	−	0.0811919i	$-0.974127\pi$
$797$	− 4.58429i	− 0.162384i	0.996698	−	0.0811919i	$-0.0258727\pi$
$798$	0	0
$799$	−24.6499	−0.872050
$800$	0	0
$801$	22.1202	0.781578
$802$	0	0
$803$	− 33.8127i	− 1.19323i
$804$	0	0
$805$	− 9.09166i	− 0.320439i
$806$	0	0
$807$	−3.99109	−0.140493
$808$	0	0
$809$	31.9465	1.12318	0.561589	−	0.827416i	$-0.310190\pi$
$809$	31.9465	1.12318	0.561589	+	0.827416i	$0.310190\pi$
$810$	0	0
$811$	9.10676i	0.319781i	0.987135	+	0.159891i	$0.0511142\pi$
$811$	9.10676i	0.319781i	−0.987135	+	0.159891i	$0.948886\pi$
$812$	0	0
$813$	− 59.6827i	− 2.09316i
$814$	0	0
$815$	−3.66367	−0.128333
$816$	0	0
$817$	50.1686	1.75518
$818$	0	0
$819$	− 12.1855i	− 0.425796i
$820$	0	0
$821$	− 28.5354i	− 0.995891i	−0.867208	−	0.497946i	$-0.834088\pi$
$821$	− 28.5354i	− 0.995891i	0.867208	−	0.497946i	$-0.165912\pi$
$822$	0	0
$823$	7.58038	0.264235	0.132118	−	0.991234i	$-0.457822\pi$
$823$	7.58038	0.264235	0.132118	+	0.991234i	$0.457822\pi$
$824$	0	0
$825$	−38.4651	−1.33918
$826$	0	0
$827$	17.7460i	0.617087i	0.951210	+	0.308544i	$0.0998416\pi$
$827$	17.7460i	0.617087i	−0.951210	+	0.308544i	$0.900158\pi$
$828$	0	0
$829$	10.6545i	0.370046i	0.982734	+	0.185023i	$0.0592359\pi$
$829$	10.6545i	0.370046i	−0.982734	+	0.185023i	$0.940764\pi$
$830$	0	0
$831$	−16.8087	−0.583088
$832$	0	0
$833$	−2.73782	−0.0948598
$834$	0	0
$835$	− 30.9057i	− 1.06954i
$836$	0	0
$837$	− 10.9175i	− 0.377365i
$838$	0	0
$839$	34.1821	1.18010	0.590048	−	0.807368i	$-0.299109\pi$
$839$	34.1821	1.18010	0.590048	+	0.807368i	$0.299109\pi$
$840$	0	0
$841$	−43.1724	−1.48870
$842$	0	0
$843$	49.1713i	1.69355i
$844$	0	0
$845$	− 0.562705i	− 0.0193577i
$846$	0	0
$847$	17.6240	0.605569
$848$	0	0
$849$	−11.8075	−0.405233
$850$	0	0
$851$	− 2.06301i	− 0.0707192i
$852$	0	0
$853$	28.6019i	0.979312i	0.871916	+	0.489656i	$0.162878\pi$
$853$	28.6019i	0.979312i	−0.871916	+	0.489656i	$0.837122\pi$
$854$	0	0
$855$	−43.4916	−1.48738
$856$	0	0
$857$	4.18332	0.142899	0.0714497	−	0.997444i	$-0.477237\pi$
$857$	4.18332	0.142899	0.0714497	+	0.997444i	$0.477237\pi$
$858$	0	0
$859$	− 25.9454i	− 0.885245i	−0.896708	−	0.442622i	$-0.854048\pi$
$859$	− 25.9454i	− 0.885245i	0.896708	−	0.442622i	$-0.145952\pi$
$860$	0	0
$861$	3.29807i	0.112398i
$862$	0	0
$863$	−2.24075	−0.0762759	−0.0381379	−	0.999272i	$-0.512143\pi$
$863$	−2.24075	−0.0762759	−0.0381379	+	0.999272i	$0.512143\pi$
$864$	0	0
$865$	31.2081	1.06111
$866$	0	0
$867$	24.1015i	0.818531i
$868$	0	0
$869$	60.5053i	2.05250i
$870$	0	0
$871$	−30.0310	−1.01756
$872$	0	0
$873$	−16.9906	−0.575045
$874$	0	0
$875$	11.5282i	0.389723i
$876$	0	0
$877$	30.6413i	1.03468i	0.855779	+	0.517341i	$0.173078\pi$
$877$	30.6413i	1.03468i	−0.855779	+	0.517341i	$0.826922\pi$
$878$	0	0
$879$	69.4328	2.34191
$880$	0	0
$881$	39.8271	1.34181	0.670905	−	0.741543i	$-0.265906\pi$
$881$	39.8271	1.34181	0.670905	+	0.741543i	$0.265906\pi$
$882$	0	0
$883$	2.90916i	0.0979010i	0.998801	+	0.0489505i	$0.0155876\pi$
$883$	2.90916i	0.0979010i	−0.998801	+	0.0489505i	$0.984412\pi$
$884$	0	0
$885$	− 10.1388i	− 0.340811i
$886$	0	0
$887$	12.3473	0.414580	0.207290	−	0.978279i	$-0.433536\pi$
$887$	12.3473	0.414580	0.207290	+	0.978279i	$0.433536\pi$
$888$	0	0
$889$	15.3442	0.514629
$890$	0	0
$891$	40.2502i	1.34843i
$892$	0	0
$893$	77.5796i	2.59610i
$894$	0	0
$895$	7.56366	0.252825
$896$	0	0
$897$	55.6597	1.85842
$898$	0	0
$899$	− 84.9617i	− 2.83363i
$900$	0	0
$901$	3.31269i	0.110362i
$902$	0	0
$903$	14.7644	0.491329
$904$	0	0
$905$	−10.0252	−0.333248
$906$	0	0
$907$	− 28.0705i	− 0.932067i	−0.884767	−	0.466034i	$-0.845683\pi$
$907$	− 28.0705i	− 0.932067i	0.884767	−	0.466034i	$-0.154317\pi$
$908$	0	0
$909$	− 31.5913i	− 1.04782i
$910$	0	0
$911$	1.04576	0.0346474	0.0173237	−	0.999850i	$-0.494485\pi$
$911$	1.04576	0.0346474	0.0173237	+	0.999850i	$0.494485\pi$
$912$	0	0
$913$	−28.9459	−0.957971
$914$	0	0
$915$	16.7235i	0.552861i
$916$	0	0
$917$	− 6.07758i	− 0.200699i
$918$	0	0
$919$	1.28887	0.0425159	0.0212579	−	0.999774i	$-0.493233\pi$
$919$	1.28887	0.0425159	0.0212579	+	0.999774i	$0.493233\pi$
$920$	0	0
$921$	23.6979	0.780874
$922$	0	0
$923$	8.76372i	0.288461i
$924$	0	0
$925$	0.946564i	0.0311228i
$926$	0	0
$927$	4.34996	0.142871
$928$	0	0
$929$	−5.26769	−0.172827	−0.0864137	−	0.996259i	$-0.527541\pi$
$929$	−5.26769	−0.172827	−0.0864137	+	0.996259i	$0.527541\pi$
$930$	0	0
$931$	8.61663i	0.282398i
$932$	0	0
$933$	− 0.602819i	− 0.0197354i
$934$	0	0
$935$	21.5517	0.704817
$936$	0	0
$937$	−37.7934	−1.23466	−0.617329	−	0.786705i	$-0.711785\pi$
$937$	−37.7934	−1.23466	−0.617329	+	0.786705i	$0.711785\pi$
$938$	0	0
$939$	12.3320i	0.402440i
$940$	0	0
$941$	− 19.6472i	− 0.640481i	−0.947336	−	0.320241i	$-0.896236\pi$
$941$	− 19.6472i	− 0.640481i	0.947336	−	0.320241i	$-0.103764\pi$
$942$	0	0
$943$	−8.03655	−0.261706
$944$	0	0
$945$	−1.60620	−0.0522496
$946$	0	0
$947$	9.77421i	0.317619i	0.987309	+	0.158810i	$0.0507656\pi$
$947$	9.77421i	0.317619i	−0.987309	+	0.158810i	$0.949234\pi$
$948$	0	0
$949$	− 22.4493i	− 0.728735i
$950$	0	0
$951$	−60.4289	−1.95954
$952$	0	0
$953$	−58.8098	−1.90504	−0.952519	−	0.304480i	$-0.901517\pi$
$953$	−58.8098	−1.90504	−0.952519	+	0.304480i	$0.901517\pi$
$954$	0	0
$955$	− 25.2556i	− 0.817251i
$956$	0	0
$957$	− 115.259i	− 3.72578i
$958$	0	0
$959$	8.27393	0.267179
$960$	0	0
$961$	69.0174	2.22637
$962$	0	0
$963$	4.31137i	0.138932i
$964$	0	0
$965$	27.8915i	0.897860i
$966$	0	0
$967$	45.9820	1.47868	0.739340	−	0.673332i	$-0.235138\pi$
$967$	45.9820	1.47868	0.739340	+	0.673332i	$0.235138\pi$
$968$	0	0
$969$	−59.8224	−1.92177
$970$	0	0
$971$	18.1060i	0.581048i	0.956868	+	0.290524i	$0.0938296\pi$
$971$	18.1060i	0.581048i	−0.956868	+	0.290524i	$0.906170\pi$
$972$	0	0
$973$	22.5020i	0.721380i
$974$	0	0
$975$	−25.5381	−0.817874
$976$	0	0
$977$	20.6440	0.660461	0.330231	−	0.943900i	$-0.392873\pi$
$977$	20.6440	0.660461	0.330231	+	0.943900i	$0.392873\pi$
$978$	0	0
$979$	34.4983i	1.10257i
$980$	0	0
$981$	− 38.7810i	− 1.23818i
$982$	0	0
$983$	57.7075	1.84058	0.920291	−	0.391234i	$-0.127952\pi$
$983$	57.7075	1.84058	0.920291	+	0.391234i	$0.127952\pi$
$984$	0	0
$985$	4.54349	0.144767
$986$	0	0
$987$	22.8314i	0.726731i
$988$	0	0
$989$	35.9771i	1.14400i
$990$	0	0
$991$	−15.8234	−0.502647	−0.251323	−	0.967903i	$-0.580866\pi$
$991$	−15.8234	−0.502647	−0.251323	+	0.967903i	$0.580866\pi$
$992$	0	0
$993$	20.8758	0.662474
$994$	0	0
$995$	− 2.20188i	− 0.0698043i
$996$	0	0
$997$	− 41.9024i	− 1.32706i	−0.748149	−	0.663530i	$-0.769057\pi$
$997$	− 41.9024i	− 1.32706i	0.748149	−	0.663530i	$-0.230943\pi$
$998$	0	0
$999$	−0.364467	−0.0115312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.i.1793.3		12
4.3	odd	2		3584.2.b.k.1793.10		12
8.3	odd	2		3584.2.b.k.1793.3		12
8.5	even	2	inner	3584.2.b.i.1793.10		12
16.3	odd	4		3584.2.a.k.1.1	yes	6
16.5	even	4		3584.2.a.l.1.1	yes	6
16.11	odd	4		3584.2.a.e.1.6	✓	6
16.13	even	4		3584.2.a.f.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.e.1.6	✓	6	16.11	odd	4
3584.2.a.f.1.6	yes	6	16.13	even	4
3584.2.a.k.1.1	yes	6	16.3	odd	4
3584.2.a.l.1.1	yes	6	16.5	even	4
3584.2.b.i.1793.3		12	1.1	even	1	trivial
3584.2.b.i.1793.10		12	8.5	even	2	inner
3584.2.b.k.1793.3		12	8.3	odd	2
3584.2.b.k.1793.10		12	4.3	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)

\(f(q)\)	\(=\)	\(q-2.53584i q^{3} -1.47134i q^{5} -1.00000 q^{7} -3.43049 q^{9} +O(q^{10})\)	\(q-2.53584i q^{3} -1.47134i q^{5} -1.00000 q^{7} -3.43049 q^{9} -5.35014i q^{11} -3.55212i q^{13} -3.73108 q^{15} -2.73782 q^{17} +8.61663i q^{19} +2.53584i q^{21} -6.17919 q^{23} +2.83517 q^{25} +1.09166i q^{27} +8.49543i q^{29} -10.0009 q^{31} -13.5671 q^{33} +1.47134i q^{35} +0.333865i q^{37} -9.00761 q^{39} +1.30058 q^{41} -5.82230i q^{43} +5.04741i q^{45} +9.00348 q^{47} +1.00000 q^{49} +6.94267i q^{51} -1.20997i q^{53} -7.87186 q^{55} +21.8504 q^{57} +2.71738i q^{59} -4.48221i q^{61} +3.43049 q^{63} -5.22636 q^{65} -8.45438i q^{67} +15.6694i q^{69} -2.46718 q^{71} +6.31997 q^{73} -7.18954i q^{75} +5.35014i q^{77} -11.3091 q^{79} -7.52320 q^{81} -5.41031i q^{83} +4.02825i q^{85} +21.5431 q^{87} -6.44811 q^{89} +3.55212i q^{91} +25.3606i q^{93} +12.6780 q^{95} +4.95282 q^{97} +18.3536i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) 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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