LMFDB - Embedded newform 4320.2.h.b.2591.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(2591,4320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4320.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 523 x^{12} - 1120 x^{11} + 2214 x^{10} - 3524 x^{9} + \cdots + 2116 $ x^16 - 8x^15 + 44x^14 - 168x^13 + 523x^12 - 1120x^11 + 2214x^10 - 3524x^9 + 6063x^8 - 18332x^7 + 43638x^6 - 65248x^5 + 6502x^4 - 2712x^3 + 160676x^2 + 36616*x + 2116
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.4
Root			$1.43545 - 1.87072i$ of defining polynomial
Character	$\chi$	$=$	4320.2591
Dual form			4320.2.h.b.2591.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -0.236402i q^{7} +4.19327 q^{11} -6.12645 q^{13} +0.219916i q^{17} +3.97822i q^{19} +3.96845 q^{23} -1.00000 q^{25} -6.25975i q^{29} -3.53413i q^{31} -0.236402 q^{35} +11.5231 q^{37} +12.1049i q^{41} -5.59917i q^{43} -12.4661 q^{47} +6.94411 q^{49} -2.55071i q^{53} -4.19327i q^{55} -1.28245 q^{59} +7.97709 q^{61} +6.12645i q^{65} -5.49100i q^{67} +13.4061 q^{71} +12.3280 q^{73} -0.991298i q^{77} -11.7061i q^{79} -7.81237 q^{83} +0.219916 q^{85} +2.39093i q^{89} +1.44830i q^{91} +3.97822 q^{95} +1.29917 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{11} - 8 q^{13} + 32 q^{23} - 16 q^{25} + 8 q^{37} + 16 q^{47} - 8 q^{49} - 32 q^{59} + 8 q^{61} - 32 q^{71} - 8 q^{73} - 32 q^{83} - 8 q^{97}+O(q^{100}) $ 16 * q - 8 * q^11 - 8 * q^13 + 32 * q^23 - 16 * q^25 + 8 * q^37 + 16 * q^47 - 8 * q^49 - 32 * q^59 + 8 * q^61 - 32 * q^71 - 8 * q^73 - 32 * q^83 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4320\mathbb{Z}\right)^\times$.

$n$	$2081$	$2431$	$3457$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	− 0.236402i	− 0.0893515i	−0.999002	−	0.0446758i	$-0.985775\pi$
$7$	− 0.236402i	− 0.0893515i	0.999002	−	0.0446758i	$-0.0142255\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.19327	1.26432	0.632160	−	0.774838i	$-0.282169\pi$
$11$	4.19327	1.26432	0.632160	+	0.774838i	$0.282169\pi$
$12$	0	0
$13$	−6.12645	−1.69917	−0.849586	−	0.527451i	$-0.823148\pi$
$13$	−6.12645	−1.69917	−0.849586	+	0.527451i	$0.823148\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.219916i	0.0533376i	0.999644	+	0.0266688i	$0.00848994\pi$
$17$	0.219916i	0.0533376i	−0.999644	+	0.0266688i	$0.991510\pi$
$18$	0	0
$19$	3.97822i	0.912665i	0.889809	+	0.456333i	$0.150837\pi$
$19$	3.97822i	0.912665i	−0.889809	+	0.456333i	$0.849163\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.96845	0.827480	0.413740	−	0.910395i	$-0.364222\pi$
$23$	3.96845	0.827480	0.413740	+	0.910395i	$0.364222\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.25975i	− 1.16241i	−0.813758	−	0.581203i	$-0.802582\pi$
$29$	− 6.25975i	− 1.16241i	0.813758	−	0.581203i	$-0.197418\pi$
$30$	0	0
$31$	− 3.53413i	− 0.634749i	−0.948300	−	0.317375i	$-0.897199\pi$
$31$	− 3.53413i	− 0.634749i	0.948300	−	0.317375i	$-0.102801\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.236402	−0.0399592
$36$	0	0
$37$	11.5231	1.89438	0.947189	−	0.320674i	$-0.103910\pi$
$37$	11.5231	1.89438	0.947189	+	0.320674i	$0.103910\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.1049i	1.89047i	0.326395	+	0.945234i	$0.394166\pi$
$41$	12.1049i	1.89047i	−0.326395	+	0.945234i	$0.605834\pi$
$42$	0	0
$43$	− 5.59917i	− 0.853866i	−0.904283	−	0.426933i	$-0.859594\pi$
$43$	− 5.59917i	− 0.853866i	0.904283	−	0.426933i	$-0.140406\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.4661	−1.81837	−0.909183	−	0.416396i	$-0.863293\pi$
$47$	−12.4661	−1.81837	−0.909183	+	0.416396i	$0.863293\pi$
$48$	0	0
$49$	6.94411	0.992016
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.55071i	− 0.350368i	−0.984536	−	0.175184i	$-0.943948\pi$
$53$	− 2.55071i	− 0.350368i	0.984536	−	0.175184i	$-0.0560520\pi$
$54$	0	0
$55$	− 4.19327i	− 0.565421i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.28245	−0.166961	−0.0834807	−	0.996509i	$-0.526604\pi$
$59$	−1.28245	−0.166961	−0.0834807	+	0.996509i	$0.526604\pi$
$60$	0	0
$61$	7.97709	1.02136	0.510681	−	0.859771i	$-0.329393\pi$
$61$	7.97709	1.02136	0.510681	+	0.859771i	$0.329393\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.12645i	0.759893i
$66$	0	0
$67$	− 5.49100i	− 0.670833i	−0.942070	−	0.335416i	$-0.891123\pi$
$67$	− 5.49100i	− 0.670833i	0.942070	−	0.335416i	$-0.108877\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.4061	1.59101	0.795507	−	0.605944i	$-0.207204\pi$
$71$	13.4061	1.59101	0.795507	+	0.605944i	$0.207204\pi$
$72$	0	0
$73$	12.3280	1.44289	0.721443	−	0.692474i	$-0.243479\pi$
$73$	12.3280	1.44289	0.721443	+	0.692474i	$0.243479\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.991298i	− 0.112969i
$78$	0	0
$79$	− 11.7061i	− 1.31704i	−0.752561	−	0.658522i	$-0.771182\pi$
$79$	− 11.7061i	− 1.31704i	0.752561	−	0.658522i	$-0.228818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.81237	−0.857519	−0.428759	−	0.903419i	$-0.641049\pi$
$83$	−7.81237	−0.857519	−0.428759	+	0.903419i	$0.641049\pi$
$84$	0	0
$85$	0.219916	0.0238533
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.39093i	0.253438i	0.991939	+	0.126719i	$0.0404447\pi$
$89$	2.39093i	0.253438i	−0.991939	+	0.126719i	$0.959555\pi$
$90$	0	0
$91$	1.44830i	0.151824i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.97822	0.408156
$96$	0	0
$97$	1.29917	0.131911	0.0659554	−	0.997823i	$-0.478990\pi$
$97$	1.29917	0.131911	0.0659554	+	0.997823i	$0.478990\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.33190i	0.630048i	0.949084	+	0.315024i	$0.102013\pi$
$101$	6.33190i	0.630048i	−0.949084	+	0.315024i	$0.897987\pi$
$102$	0	0
$103$	− 14.7827i	− 1.45659i	−0.685266	−	0.728293i	$-0.740314\pi$
$103$	− 14.7827i	− 1.45659i	0.685266	−	0.728293i	$-0.259686\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.63259	0.641196	0.320598	−	0.947215i	$-0.396116\pi$
$107$	6.63259	0.641196	0.320598	+	0.947215i	$0.396116\pi$
$108$	0	0
$109$	−6.13296	−0.587431	−0.293715	−	0.955893i	$-0.594892\pi$
$109$	−6.13296	−0.587431	−0.293715	+	0.955893i	$0.594892\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.97294i	− 0.750031i	−0.927019	−	0.375015i	$-0.877637\pi$
$113$	− 7.97294i	− 0.750031i	0.927019	−	0.375015i	$-0.122363\pi$
$114$	0	0
$115$	− 3.96845i	− 0.370060i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.0519887	0.00476579
$120$	0	0
$121$	6.58354	0.598503
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	4.71824i	0.418677i	0.977843	+	0.209338i	$0.0671310\pi$
$127$	4.71824i	0.418677i	−0.977843	+	0.209338i	$0.932869\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.5277	−1.00718	−0.503589	−	0.863943i	$-0.667988\pi$
$131$	−11.5277	−1.00718	−0.503589	+	0.863943i	$0.667988\pi$
$132$	0	0
$133$	0.940458	0.0815480
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.82762i	− 0.156144i	−0.996948	−	0.0780722i	$-0.975124\pi$
$137$	− 1.82762i	− 0.156144i	0.996948	−	0.0780722i	$-0.0248765\pi$
$138$	0	0
$139$	− 14.3382i	− 1.21615i	−0.793879	−	0.608075i	$-0.791942\pi$
$139$	− 14.3382i	− 1.21615i	0.793879	−	0.608075i	$-0.208058\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.6899	−2.14830
$144$	0	0
$145$	−6.25975	−0.519844
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1.10903i	− 0.0908556i	−0.998968	−	0.0454278i	$-0.985535\pi$
$149$	− 1.10903i	− 0.0908556i	0.998968	−	0.0454278i	$-0.0144651\pi$
$150$	0	0
$151$	14.8568i	1.20903i	0.796595	+	0.604513i	$0.206632\pi$
$151$	14.8568i	1.20903i	−0.796595	+	0.604513i	$0.793368\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.53413	−0.283868
$156$	0	0
$157$	−3.74387	−0.298794	−0.149397	−	0.988777i	$-0.547733\pi$
$157$	−3.74387	−0.298794	−0.149397	+	0.988777i	$0.547733\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 0.938150i	− 0.0739366i
$162$	0	0
$163$	− 21.7665i	− 1.70488i	−0.522822	−	0.852442i	$-0.675121\pi$
$163$	− 21.7665i	− 1.70488i	0.522822	−	0.852442i	$-0.324879\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.38143	0.184280	0.0921402	−	0.995746i	$-0.470629\pi$
$167$	2.38143	0.184280	0.0921402	+	0.995746i	$0.470629\pi$
$168$	0	0
$169$	24.5334	1.88718
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.1187i	− 1.14945i	−0.818346	−	0.574727i	$-0.805108\pi$
$173$	− 15.1187i	− 1.14945i	0.818346	−	0.574727i	$-0.194892\pi$
$174$	0	0
$175$	0.236402i	0.0178703i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.2389	0.989521	0.494760	−	0.869029i	$-0.335256\pi$
$179$	13.2389	0.989521	0.494760	+	0.869029i	$0.335256\pi$
$180$	0	0
$181$	16.4059	1.21944	0.609722	−	0.792616i	$-0.291281\pi$
$181$	16.4059	1.21944	0.609722	+	0.792616i	$0.291281\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 11.5231i	− 0.847192i
$186$	0	0
$187$	0.922169i	0.0674357i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.7066	1.57063	0.785316	−	0.619096i	$-0.212501\pi$
$191$	21.7066	1.57063	0.785316	+	0.619096i	$0.212501\pi$
$192$	0	0
$193$	−9.74708	−0.701610	−0.350805	−	0.936448i	$-0.614092\pi$
$193$	−9.74708	−0.701610	−0.350805	+	0.936448i	$0.614092\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 17.7484i	− 1.26452i	−0.774757	−	0.632260i	$-0.782128\pi$
$197$	− 17.7484i	− 1.26452i	0.774757	−	0.632260i	$-0.217872\pi$
$198$	0	0
$199$	9.39842i	0.666236i	0.942885	+	0.333118i	$0.108101\pi$
$199$	9.39842i	0.666236i	−0.942885	+	0.333118i	$0.891899\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.47982	−0.103863
$204$	0	0
$205$	12.1049	0.845443
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.6817i	1.15390i
$210$	0	0
$211$	− 15.7220i	− 1.08235i	−0.840910	−	0.541175i	$-0.817980\pi$
$211$	− 15.7220i	− 1.08235i	0.840910	−	0.541175i	$-0.182020\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.59917	−0.381860
$216$	0	0
$217$	−0.835476	−0.0567158
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.34731i	− 0.0906296i
$222$	0	0
$223$	11.5068i	0.770552i	0.922801	+	0.385276i	$0.125894\pi$
$223$	11.5068i	0.770552i	−0.922801	+	0.385276i	$0.874106\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.1067	1.26816	0.634078	−	0.773269i	$-0.281380\pi$
$227$	19.1067	1.26816	0.634078	+	0.773269i	$0.281380\pi$
$228$	0	0
$229$	−3.91289	−0.258571	−0.129286	−	0.991607i	$-0.541268\pi$
$229$	−3.91289	−0.258571	−0.129286	+	0.991607i	$0.541268\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.4964i	0.884178i	0.896971	+	0.442089i	$0.145762\pi$
$233$	13.4964i	0.884178i	−0.896971	+	0.442089i	$0.854238\pi$
$234$	0	0
$235$	12.4661i	0.813198i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.39860	−0.284522	−0.142261	−	0.989829i	$-0.545437\pi$
$239$	−4.39860	−0.284522	−0.142261	+	0.989829i	$0.545437\pi$
$240$	0	0
$241$	1.95898	0.126189	0.0630947	−	0.998008i	$-0.479903\pi$
$241$	1.95898	0.126189	0.0630947	+	0.998008i	$0.479903\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 6.94411i	− 0.443643i
$246$	0	0
$247$	− 24.3723i	− 1.55077i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.30208	0.145306	0.0726531	−	0.997357i	$-0.476853\pi$
$251$	2.30208	0.145306	0.0726531	+	0.997357i	$0.476853\pi$
$252$	0	0
$253$	16.6408	1.04620
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.1997i	1.82143i	0.413036	+	0.910714i	$0.364468\pi$
$257$	29.1997i	1.82143i	−0.413036	+	0.910714i	$0.635532\pi$
$258$	0	0
$259$	− 2.72407i	− 0.169266i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.85019	0.175750	0.0878752	−	0.996131i	$-0.471992\pi$
$263$	2.85019	0.175750	0.0878752	+	0.996131i	$0.471992\pi$
$264$	0	0
$265$	−2.55071	−0.156689
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 16.6886i	− 1.01752i	−0.860908	−	0.508761i	$-0.830104\pi$
$269$	− 16.6886i	− 1.01752i	0.860908	−	0.508761i	$-0.169896\pi$
$270$	0	0
$271$	− 28.5186i	− 1.73238i	−0.499712	−	0.866192i	$-0.666561\pi$
$271$	− 28.5186i	− 1.73238i	0.499712	−	0.866192i	$-0.333439\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.19327	−0.252864
$276$	0	0
$277$	15.1092	0.907824	0.453912	−	0.891046i	$-0.350028\pi$
$277$	15.1092	0.907824	0.453912	+	0.891046i	$0.350028\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.98895i	0.536236i	0.963386	+	0.268118i	$0.0864017\pi$
$281$	8.98895i	0.536236i	−0.963386	+	0.268118i	$0.913598\pi$
$282$	0	0
$283$	14.2175i	0.845141i	0.906330	+	0.422570i	$0.138872\pi$
$283$	14.2175i	0.845141i	−0.906330	+	0.422570i	$0.861128\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.86162	0.168916
$288$	0	0
$289$	16.9516	0.997155
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 31.7676i	− 1.85588i	−0.372725	−	0.927942i	$-0.621577\pi$
$293$	− 31.7676i	− 1.85588i	0.372725	−	0.927942i	$-0.378423\pi$
$294$	0	0
$295$	1.28245i	0.0746674i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.3125	−1.40603
$300$	0	0
$301$	−1.32366	−0.0762942
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 7.97709i	− 0.456767i
$306$	0	0
$307$	− 12.9510i	− 0.739155i	−0.929200	−	0.369578i	$-0.879502\pi$
$307$	− 12.9510i	− 0.739155i	0.929200	−	0.369578i	$-0.120498\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.36337	−0.247424	−0.123712	−	0.992318i	$-0.539480\pi$
$311$	−4.36337	−0.247424	−0.123712	+	0.992318i	$0.539480\pi$
$312$	0	0
$313$	4.18119	0.236335	0.118167	−	0.992994i	$-0.462298\pi$
$313$	4.18119	0.236335	0.118167	+	0.992994i	$0.462298\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.270142i	0.0151727i	0.999971	+	0.00758633i	$0.00241483\pi$
$317$	0.270142i	0.0151727i	−0.999971	+	0.00758633i	$0.997585\pi$
$318$	0	0
$319$	− 26.2488i	− 1.46965i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.874875	−0.0486793
$324$	0	0
$325$	6.12645	0.339834
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.94701i	0.162474i
$330$	0	0
$331$	− 23.5507i	− 1.29446i	−0.762293	−	0.647232i	$-0.775927\pi$
$331$	− 23.5507i	− 1.29446i	0.762293	−	0.647232i	$-0.224073\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.49100	−0.300006
$336$	0	0
$337$	10.2751	0.559718	0.279859	−	0.960041i	$-0.409712\pi$
$337$	10.2751	0.559718	0.279859	+	0.960041i	$0.409712\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 14.8196i	− 0.802526i
$342$	0	0
$343$	− 3.29642i	− 0.177990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.44554	−0.292332	−0.146166	−	0.989260i	$-0.546693\pi$
$347$	−5.44554	−0.292332	−0.146166	+	0.989260i	$0.546693\pi$
$348$	0	0
$349$	2.96450	0.158686	0.0793429	−	0.996847i	$-0.474718\pi$
$349$	2.96450	0.158686	0.0793429	+	0.996847i	$0.474718\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5030i	0.825143i	0.910925	+	0.412571i	$0.135369\pi$
$353$	15.5030i	0.825143i	−0.910925	+	0.412571i	$0.864631\pi$
$354$	0	0
$355$	− 13.4061i	− 0.711523i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.96827	−0.367771	−0.183886	−	0.982948i	$-0.558868\pi$
$359$	−6.96827	−0.367771	−0.183886	+	0.982948i	$0.558868\pi$
$360$	0	0
$361$	3.17380	0.167042
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12.3280i	− 0.645278i
$366$	0	0
$367$	− 16.5599i	− 0.864417i	−0.901774	−	0.432209i	$-0.857734\pi$
$367$	− 16.5599i	− 0.864417i	0.901774	−	0.432209i	$-0.142266\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.602994	−0.0313059
$372$	0	0
$373$	9.82130	0.508528	0.254264	−	0.967135i	$-0.418167\pi$
$373$	9.82130	0.508528	0.254264	+	0.967135i	$0.418167\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	38.3501i	1.97513i
$378$	0	0
$379$	10.7763i	0.553544i	0.960936	+	0.276772i	$0.0892646\pi$
$379$	10.7763i	0.553544i	−0.960936	+	0.276772i	$0.910735\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.51537	−0.486213	−0.243106	−	0.970000i	$-0.578166\pi$
$383$	−9.51537	−0.486213	−0.243106	+	0.970000i	$0.578166\pi$
$384$	0	0
$385$	−0.991298	−0.0505212
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.1237i	1.42593i	0.701199	+	0.712965i	$0.252648\pi$
$389$	28.1237i	1.42593i	−0.701199	+	0.712965i	$0.747352\pi$
$390$	0	0
$391$	0.872728i	0.0441357i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.7061	−0.589000
$396$	0	0
$397$	−3.49390	−0.175354	−0.0876770	−	0.996149i	$-0.527944\pi$
$397$	−3.49390	−0.175354	−0.0876770	+	0.996149i	$0.527944\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.6721i	1.63157i	0.578357	+	0.815784i	$0.303694\pi$
$401$	32.6721i	1.63157i	−0.578357	+	0.815784i	$0.696306\pi$
$402$	0	0
$403$	21.6517i	1.07855i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.3193	2.39510
$408$	0	0
$409$	−38.5509	−1.90622	−0.953109	−	0.302627i	$-0.902136\pi$
$409$	−38.5509	−1.90622	−0.953109	+	0.302627i	$0.902136\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.303175i	0.0149183i
$414$	0	0
$415$	7.81237i	0.383494i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.8172	1.70093	0.850467	−	0.526029i	$-0.176320\pi$
$419$	34.8172	1.70093	0.850467	+	0.526029i	$0.176320\pi$
$420$	0	0
$421$	−6.33400	−0.308700	−0.154350	−	0.988016i	$-0.549328\pi$
$421$	−6.33400	−0.308700	−0.154350	+	0.988016i	$0.549328\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 0.219916i	− 0.0106675i
$426$	0	0
$427$	− 1.88580i	− 0.0912602i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.0070	−1.20455	−0.602273	−	0.798290i	$-0.705738\pi$
$431$	−25.0070	−1.20455	−0.602273	+	0.798290i	$0.705738\pi$
$432$	0	0
$433$	30.3390	1.45800	0.728999	−	0.684514i	$-0.239986\pi$
$433$	30.3390	1.45800	0.728999	+	0.684514i	$0.239986\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.7874i	0.755212i
$438$	0	0
$439$	13.4814i	0.643431i	0.946836	+	0.321715i	$0.104259\pi$
$439$	13.4814i	0.643431i	−0.946836	+	0.321715i	$0.895741\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.2417	−1.43683	−0.718413	−	0.695617i	$-0.755131\pi$
$443$	−30.2417	−1.43683	−0.718413	+	0.695617i	$0.755131\pi$
$444$	0	0
$445$	2.39093	0.113341
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.8055i	0.745907i	0.927850	+	0.372953i	$0.121655\pi$
$449$	15.8055i	0.745907i	−0.927850	+	0.372953i	$0.878345\pi$
$450$	0	0
$451$	50.7591i	2.39015i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.44830	0.0678976
$456$	0	0
$457$	16.0036	0.748616	0.374308	−	0.927304i	$-0.377880\pi$
$457$	16.0036	0.748616	0.374308	+	0.927304i	$0.377880\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.9604i	− 0.603625i	−0.953367	−	0.301813i	$-0.902408\pi$
$461$	− 12.9604i	− 0.603625i	0.953367	−	0.301813i	$-0.0975917\pi$
$462$	0	0
$463$	− 8.44368i	− 0.392411i	−0.980563	−	0.196206i	$-0.937138\pi$
$463$	− 8.44368i	− 0.392411i	0.980563	−	0.196206i	$-0.0628620\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.6404	1.18650	0.593249	−	0.805019i	$-0.297845\pi$
$467$	25.6404	1.18650	0.593249	+	0.805019i	$0.297845\pi$
$468$	0	0
$469$	−1.29808	−0.0599400
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 23.4789i	− 1.07956i
$474$	0	0
$475$	− 3.97822i	− 0.182533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.44511	0.157411	0.0787056	−	0.996898i	$-0.474921\pi$
$479$	3.44511	0.157411	0.0787056	+	0.996898i	$0.474921\pi$
$480$	0	0
$481$	−70.5954	−3.21887
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.29917i	− 0.0589923i
$486$	0	0
$487$	32.4887i	1.47221i	0.676870	+	0.736103i	$0.263336\pi$
$487$	32.4887i	1.47221i	−0.676870	+	0.736103i	$0.736664\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.93386	0.222662	0.111331	−	0.993783i	$-0.464489\pi$
$491$	4.93386	0.222662	0.111331	+	0.993783i	$0.464489\pi$
$492$	0	0
$493$	1.37662	0.0619999
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.16923i	− 0.142160i
$498$	0	0
$499$	− 16.2685i	− 0.728277i	−0.931345	−	0.364138i	$-0.881364\pi$
$499$	− 16.2685i	− 0.728277i	0.931345	−	0.364138i	$-0.118636\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.2343	−0.679264	−0.339632	−	0.940558i	$-0.610303\pi$
$503$	−15.2343	−0.679264	−0.339632	+	0.940558i	$0.610303\pi$
$504$	0	0
$505$	6.33190	0.281766
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 23.2521i	− 1.03063i	−0.857000	−	0.515316i	$-0.827674\pi$
$509$	− 23.2521i	− 1.03063i	0.857000	−	0.515316i	$-0.172326\pi$
$510$	0	0
$511$	− 2.91437i	− 0.128924i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.7827	−0.651405
$516$	0	0
$517$	−52.2737	−2.29900
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.5030i	1.51160i	0.654801	+	0.755801i	$0.272752\pi$
$521$	34.5030i	1.51160i	−0.654801	+	0.755801i	$0.727248\pi$
$522$	0	0
$523$	34.4523i	1.50649i	0.657738	+	0.753247i	$0.271513\pi$
$523$	34.4523i	1.50649i	−0.657738	+	0.753247i	$0.728487\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.777214	0.0338560
$528$	0	0
$529$	−7.25138	−0.315278
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 74.1600i	− 3.21223i
$534$	0	0
$535$	− 6.63259i	− 0.286752i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	29.1186	1.25423
$540$	0	0
$541$	28.8403	1.23994	0.619971	−	0.784625i	$-0.287144\pi$
$541$	28.8403	1.23994	0.619971	+	0.784625i	$0.287144\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.13296i	0.262707i
$546$	0	0
$547$	29.5474i	1.26336i	0.775230	+	0.631679i	$0.217634\pi$
$547$	29.5474i	1.26336i	−0.775230	+	0.631679i	$0.782366\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	24.9026	1.06089
$552$	0	0
$553$	−2.76735	−0.117680
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.4359i	0.611668i	0.952085	+	0.305834i	$0.0989352\pi$
$557$	14.4359i	0.611668i	−0.952085	+	0.305834i	$0.901065\pi$
$558$	0	0
$559$	34.3030i	1.45086i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.8884	−0.922485	−0.461242	−	0.887274i	$-0.652596\pi$
$563$	−21.8884	−0.922485	−0.461242	+	0.887274i	$0.652596\pi$
$564$	0	0
$565$	−7.97294	−0.335424
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5.36768i	− 0.225025i	−0.993650	−	0.112512i	$-0.964110\pi$
$569$	− 5.36768i	− 0.225025i	0.993650	−	0.112512i	$-0.0358898\pi$
$570$	0	0
$571$	− 44.3735i	− 1.85697i	−0.371368	−	0.928486i	$-0.621111\pi$
$571$	− 44.3735i	− 1.85697i	0.371368	−	0.928486i	$-0.378889\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.96845	−0.165496
$576$	0	0
$577$	37.6815	1.56870	0.784351	−	0.620318i	$-0.212996\pi$
$577$	37.6815	1.56870	0.784351	+	0.620318i	$0.212996\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.84686i	0.0766206i
$582$	0	0
$583$	− 10.6958i	− 0.442977i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.7352	−1.26858	−0.634289	−	0.773096i	$-0.718707\pi$
$587$	−30.7352	−1.26858	−0.634289	+	0.773096i	$0.718707\pi$
$588$	0	0
$589$	14.0595	0.579314
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 44.2600i	− 1.81754i	−0.417295	−	0.908771i	$-0.637022\pi$
$593$	− 44.2600i	− 1.81754i	0.417295	−	0.908771i	$-0.362978\pi$
$594$	0	0
$595$	− 0.0519887i	− 0.00213133i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.7660	1.82909	0.914545	−	0.404484i	$-0.132549\pi$
$599$	44.7660	1.82909	0.914545	+	0.404484i	$0.132549\pi$
$600$	0	0
$601$	−37.5308	−1.53091	−0.765457	−	0.643487i	$-0.777487\pi$
$601$	−37.5308	−1.53091	−0.765457	+	0.643487i	$0.777487\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.58354i	− 0.267659i
$606$	0	0
$607$	0.939378i	0.0381282i	0.999818	+	0.0190641i	$0.00606866\pi$
$607$	0.939378i	0.0381282i	−0.999818	+	0.0190641i	$0.993931\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	76.3729	3.08972
$612$	0	0
$613$	−7.52274	−0.303840	−0.151920	−	0.988393i	$-0.548546\pi$
$613$	−7.52274	−0.303840	−0.151920	+	0.988393i	$0.548546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.94062i	− 0.400194i	−0.979776	−	0.200097i	$-0.935874\pi$
$617$	− 9.94062i	− 0.400194i	0.979776	−	0.200097i	$-0.0641257\pi$
$618$	0	0
$619$	4.21615i	0.169461i	0.996404	+	0.0847306i	$0.0270030\pi$
$619$	4.21615i	0.169461i	−0.996404	+	0.0847306i	$0.972997\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.565221	0.0226451
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.53411i	0.101042i
$630$	0	0
$631$	− 14.1230i	− 0.562228i	−0.959674	−	0.281114i	$-0.909296\pi$
$631$	− 14.1230i	− 0.562228i	0.959674	−	0.281114i	$-0.0907039\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.71824	0.187238
$636$	0	0
$637$	−42.5428	−1.68561
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 18.0411i	− 0.712581i	−0.934375	−	0.356290i	$-0.884041\pi$
$641$	− 18.0411i	− 0.712581i	0.934375	−	0.356290i	$-0.115959\pi$
$642$	0	0
$643$	− 33.5059i	− 1.32134i	−0.750675	−	0.660671i	$-0.770272\pi$
$643$	− 33.5059i	− 1.32134i	0.750675	−	0.660671i	$-0.229728\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.6664	−1.55945	−0.779723	−	0.626125i	$-0.784640\pi$
$647$	−39.6664	−1.55945	−0.779723	+	0.626125i	$0.784640\pi$
$648$	0	0
$649$	−5.37768	−0.211093
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.7726i	1.63469i	0.576149	+	0.817345i	$0.304555\pi$
$653$	41.7726i	1.63469i	−0.576149	+	0.817345i	$0.695445\pi$
$654$	0	0
$655$	11.5277i	0.450424i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.7863	−0.731811	−0.365906	−	0.930652i	$-0.619241\pi$
$659$	−18.7863	−0.731811	−0.365906	+	0.930652i	$0.619241\pi$
$660$	0	0
$661$	10.3787	0.403684	0.201842	−	0.979418i	$-0.435307\pi$
$661$	10.3787	0.403684	0.201842	+	0.979418i	$0.435307\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.940458i	− 0.0364694i
$666$	0	0
$667$	− 24.8415i	− 0.961868i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.4501	1.29133
$672$	0	0
$673$	4.61169	0.177768	0.0888839	−	0.996042i	$-0.471670\pi$
$673$	4.61169	0.177768	0.0888839	+	0.996042i	$0.471670\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.7833i	0.568169i	0.958799	+	0.284085i	$0.0916897\pi$
$677$	14.7833i	0.568169i	−0.958799	+	0.284085i	$0.908310\pi$
$678$	0	0
$679$	− 0.307126i	− 0.0117864i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−28.8535	−1.10405	−0.552023	−	0.833829i	$-0.686144\pi$
$683$	−28.8535	−1.10405	−0.552023	+	0.833829i	$0.686144\pi$
$684$	0	0
$685$	−1.82762	−0.0698299
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15.6268i	0.595335i
$690$	0	0
$691$	13.1263i	0.499346i	0.968330	+	0.249673i	$0.0803232\pi$
$691$	13.1263i	0.499346i	−0.968330	+	0.249673i	$0.919677\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.3382	−0.543879
$696$	0	0
$697$	−2.66206	−0.100833
$698$	0	0
$699$	0	0
$700$	0	0
$701$	40.5190i	1.53038i	0.643804	+	0.765191i	$0.277355\pi$
$701$	40.5190i	1.53038i	−0.643804	+	0.765191i	$0.722645\pi$
$702$	0	0
$703$	45.8412i	1.72893i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.49687	0.0562958
$708$	0	0
$709$	12.8050	0.480901	0.240450	−	0.970661i	$-0.422705\pi$
$709$	12.8050	0.480901	0.240450	+	0.970661i	$0.422705\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 14.0250i	− 0.525242i
$714$	0	0
$715$	25.6899i	0.960747i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24.5877	−0.916968	−0.458484	−	0.888703i	$-0.651607\pi$
$719$	−24.5877	−0.916968	−0.458484	+	0.888703i	$0.651607\pi$
$720$	0	0
$721$	−3.49467	−0.130148
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.25975i	0.232481i
$726$	0	0
$727$	23.5253i	0.872506i	0.899824	+	0.436253i	$0.143695\pi$
$727$	23.5253i	0.872506i	−0.899824	+	0.436253i	$0.856305\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.23135	0.0455431
$732$	0	0
$733$	−36.3062	−1.34100	−0.670500	−	0.741909i	$-0.733921\pi$
$733$	−36.3062	−1.34100	−0.670500	+	0.741909i	$0.733921\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 23.0253i	− 0.848147i
$738$	0	0
$739$	− 19.9466i	− 0.733749i	−0.930270	−	0.366875i	$-0.880428\pi$
$739$	− 19.9466i	− 0.733749i	0.930270	−	0.366875i	$-0.119572\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−46.8143	−1.71745	−0.858726	−	0.512435i	$-0.828743\pi$
$743$	−46.8143	−1.71745	−0.858726	+	0.512435i	$0.828743\pi$
$744$	0	0
$745$	−1.10903	−0.0406319
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.56796i	− 0.0572919i
$750$	0	0
$751$	20.9312i	0.763790i	0.924206	+	0.381895i	$0.124728\pi$
$751$	20.9312i	0.763790i	−0.924206	+	0.381895i	$0.875272\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.8568	0.540693
$756$	0	0
$757$	−33.9548	−1.23411	−0.617054	−	0.786921i	$-0.711674\pi$
$757$	−33.9548	−1.23411	−0.617054	+	0.786921i	$0.711674\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28.4808i	1.03243i	0.856460	+	0.516214i	$0.172659\pi$
$761$	28.4808i	1.03243i	−0.856460	+	0.516214i	$0.827341\pi$
$762$	0	0
$763$	1.44984i	0.0524879i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.85689	0.283696
$768$	0	0
$769$	16.8525	0.607717	0.303858	−	0.952717i	$-0.401725\pi$
$769$	16.8525	0.607717	0.303858	+	0.952717i	$0.401725\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 12.1668i	− 0.437608i	−0.975769	−	0.218804i	$-0.929784\pi$
$773$	− 12.1668i	− 0.437608i	0.975769	−	0.218804i	$-0.0702155\pi$
$774$	0	0
$775$	3.53413i	0.126950i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−48.1559	−1.72536
$780$	0	0
$781$	56.2155	2.01155
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.74387i	0.133625i
$786$	0	0
$787$	− 4.70174i	− 0.167599i	−0.996483	−	0.0837995i	$-0.973294\pi$
$787$	− 4.70174i	− 0.167599i	0.996483	−	0.0837995i	$-0.0267055\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.88482	−0.0670164
$792$	0	0
$793$	−48.8712	−1.73547
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 46.8127i	− 1.65819i	−0.559108	−	0.829095i	$-0.688856\pi$
$797$	− 46.8127i	− 1.65819i	0.559108	−	0.829095i	$-0.311144\pi$
$798$	0	0
$799$	− 2.74150i	− 0.0969872i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	51.6948	1.82427
$804$	0	0
$805$	−0.938150	−0.0330654
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 14.1000i	− 0.495728i	−0.968795	−	0.247864i	$-0.920271\pi$
$809$	− 14.1000i	− 0.495728i	0.968795	−	0.247864i	$-0.0797287\pi$
$810$	0	0
$811$	24.8786i	0.873605i	0.899557	+	0.436802i	$0.143889\pi$
$811$	24.8786i	0.873605i	−0.899557	+	0.436802i	$0.856111\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.7665	−0.762447
$816$	0	0
$817$	22.2747	0.779294
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.9066i	1.11355i	0.830664	+	0.556774i	$0.187961\pi$
$821$	31.9066i	1.11355i	−0.830664	+	0.556774i	$0.812039\pi$
$822$	0	0
$823$	− 31.6861i	− 1.10451i	−0.833675	−	0.552255i	$-0.813768\pi$
$823$	− 31.6861i	− 1.10451i	0.833675	−	0.552255i	$-0.186232\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.7538	−0.443495	−0.221747	−	0.975104i	$-0.571176\pi$
$827$	−12.7538	−0.443495	−0.221747	+	0.975104i	$0.571176\pi$
$828$	0	0
$829$	−15.9233	−0.553040	−0.276520	−	0.961008i	$-0.589181\pi$
$829$	−15.9233	−0.553040	−0.276520	+	0.961008i	$0.589181\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.52712i	0.0529117i
$834$	0	0
$835$	− 2.38143i	− 0.0824127i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.9658	−0.447630	−0.223815	−	0.974632i	$-0.571851\pi$
$839$	−12.9658	−0.447630	−0.223815	+	0.974632i	$0.571851\pi$
$840$	0	0
$841$	−10.1845	−0.351190
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 24.5334i	− 0.843974i
$846$	0	0
$847$	− 1.55636i	− 0.0534772i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	45.7287	1.56756
$852$	0	0
$853$	−8.78589	−0.300823	−0.150411	−	0.988623i	$-0.548060\pi$
$853$	−8.78589	−0.300823	−0.150411	+	0.988623i	$0.548060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.5798i	1.55698i	0.627659	+	0.778488i	$0.284013\pi$
$857$	45.5798i	1.55698i	−0.627659	+	0.778488i	$0.715987\pi$
$858$	0	0
$859$	50.3847i	1.71910i	0.511049	+	0.859552i	$0.329257\pi$
$859$	50.3847i	1.71910i	−0.511049	+	0.859552i	$0.670743\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.4555	−1.07076	−0.535378	−	0.844612i	$-0.679831\pi$
$863$	−31.4555	−1.07076	−0.535378	+	0.844612i	$0.679831\pi$
$864$	0	0
$865$	−15.1187	−0.514051
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 49.0870i	− 1.66516i
$870$	0	0
$871$	33.6404i	1.13986i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.236402	0.00799184
$876$	0	0
$877$	−27.8719	−0.941168	−0.470584	−	0.882355i	$-0.655957\pi$
$877$	−27.8719	−0.941168	−0.470584	+	0.882355i	$0.655957\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.8082i	1.50963i	0.655941	+	0.754813i	$0.272272\pi$
$881$	44.8082i	1.50963i	−0.655941	+	0.754813i	$0.727728\pi$
$882$	0	0
$883$	2.50194i	0.0841969i	0.999113	+	0.0420984i	$0.0134043\pi$
$883$	2.50194i	0.0841969i	−0.999113	+	0.0420984i	$0.986596\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.3502	1.65702	0.828509	−	0.559976i	$-0.189190\pi$
$887$	49.3502	1.65702	0.828509	+	0.559976i	$0.189190\pi$
$888$	0	0
$889$	1.11540	0.0374094
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 49.5928i	− 1.65956i
$894$	0	0
$895$	− 13.2389i	− 0.442527i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−22.1228	−0.737837
$900$	0	0
$901$	0.560944	0.0186878
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 16.4059i	− 0.545352i
$906$	0	0
$907$	21.3269i	0.708149i	0.935217	+	0.354074i	$0.115204\pi$
$907$	21.3269i	0.708149i	−0.935217	+	0.354074i	$0.884796\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.02938	0.0341047	0.0170524	−	0.999855i	$-0.494572\pi$
$911$	1.02938	0.0341047	0.0170524	+	0.999855i	$0.494572\pi$
$912$	0	0
$913$	−32.7594	−1.08418
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.72517i	0.0899930i
$918$	0	0
$919$	− 2.07618i	− 0.0684869i	−0.999414	−	0.0342435i	$-0.989098\pi$
$919$	− 2.07618i	− 0.0684869i	0.999414	−	0.0342435i	$-0.0109022\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−82.1320	−2.70341
$924$	0	0
$925$	−11.5231	−0.378876
$926$	0	0
$927$	0	0
$928$	0	0
$929$	47.7155i	1.56549i	0.622340	+	0.782747i	$0.286182\pi$
$929$	47.7155i	1.56549i	−0.622340	+	0.782747i	$0.713818\pi$
$930$	0	0
$931$	27.6252i	0.905379i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.922169	0.0301582
$936$	0	0
$937$	35.7608	1.16826	0.584128	−	0.811662i	$-0.301437\pi$
$937$	35.7608	1.16826	0.584128	+	0.811662i	$0.301437\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.7639i	0.905077i	0.891745	+	0.452538i	$0.149481\pi$
$941$	27.7639i	0.905077i	−0.891745	+	0.452538i	$0.850519\pi$
$942$	0	0
$943$	48.0377i	1.56432i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.5707	−1.05841	−0.529203	−	0.848495i	$-0.677509\pi$
$947$	−32.5707	−1.05841	−0.529203	+	0.848495i	$0.677509\pi$
$948$	0	0
$949$	−75.5270	−2.45171
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0.347929i	0.0112705i	0.999984	+	0.00563526i	$0.00179377\pi$
$953$	0.347929i	0.0112705i	−0.999984	+	0.00563526i	$0.998206\pi$
$954$	0	0
$955$	− 21.7066i	− 0.702408i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.432053	−0.0139517
$960$	0	0
$961$	18.5099	0.597093
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.74708i	0.313770i
$966$	0	0
$967$	27.9086i	0.897480i	0.893662	+	0.448740i	$0.148127\pi$
$967$	27.9086i	0.897480i	−0.893662	+	0.448740i	$0.851873\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.85692	0.252141	0.126070	−	0.992021i	$-0.459764\pi$
$971$	7.85692	0.252141	0.126070	+	0.992021i	$0.459764\pi$
$972$	0	0
$973$	−3.38958	−0.108665
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.14065i	0.228449i	0.993455	+	0.114225i	$0.0364384\pi$
$977$	7.14065i	0.228449i	−0.993455	+	0.114225i	$0.963562\pi$
$978$	0	0
$979$	10.0258i	0.320427i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20.7139	0.660669	0.330335	−	0.943864i	$-0.392838\pi$
$983$	20.7139	0.660669	0.330335	+	0.943864i	$0.392838\pi$
$984$	0	0
$985$	−17.7484	−0.565510
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 22.2201i	− 0.706557i
$990$	0	0
$991$	29.8200i	0.947264i	0.880723	+	0.473632i	$0.157057\pi$
$991$	29.8200i	0.947264i	−0.880723	+	0.473632i	$0.842943\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.39842	0.297950
$996$	0	0
$997$	−22.6671	−0.717874	−0.358937	−	0.933362i	$-0.616861\pi$
$997$	−22.6671	−0.717874	−0.358937	+	0.933362i	$0.616861\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.h.b.2591.4	✓	16
3.2	odd	2		4320.2.h.c.2591.12	yes	16
4.3	odd	2		4320.2.h.c.2591.5	yes	16
12.11	even	2	inner	4320.2.h.b.2591.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.h.b.2591.4	✓	16	1.1	even	1	trivial
4320.2.h.b.2591.13	yes	16	12.11	even	2	inner
4320.2.h.c.2591.5	yes	16	4.3	odd	2
4320.2.h.c.2591.12	yes	16	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -0.236402i q^{7} +4.19327 q^{11} -6.12645 q^{13} +0.219916i q^{17} +3.97822i q^{19} +3.96845 q^{23} -1.00000 q^{25} -6.25975i q^{29} -3.53413i q^{31} -0.236402 q^{35} +11.5231 q^{37} +12.1049i q^{41} -5.59917i q^{43} -12.4661 q^{47} +6.94411 q^{49} -2.55071i q^{53} -4.19327i q^{55} -1.28245 q^{59} +7.97709 q^{61} +6.12645i q^{65} -5.49100i q^{67} +13.4061 q^{71} +12.3280 q^{73} -0.991298i q^{77} -11.7061i q^{79} -7.81237 q^{83} +0.219916 q^{85} +2.39093i q^{89} +1.44830i q^{91} +3.97822 q^{95} +1.29917 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{11} - 8 q^{13} + 32 q^{23} - 16 q^{25} + 8 q^{37} + 16 q^{47} - 8 q^{49} - 32 q^{59} + 8 q^{61} - 32 q^{71} - 8 q^{73} - 32 q^{83} - 8 q^{97}+O(q^{100}) \) 16 * q - 8 * q^11 - 8 * q^13 + 32 * q^23 - 16 * q^25 + 8 * q^37 + 16 * q^47 - 8 * q^49 - 32 * q^59 + 8 * q^61 - 32 * q^71 - 8 * q^73 - 32 * q^83 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more