LMFDB - Embedded newform 4320.2.b.d.431.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4320,2,Mod(431,4320)] 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("4320.431"); 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, 1, 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.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == 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} - 2 x^{15} + x^{14} + 2 x^{13} - 6 x^{12} + 8 x^{11} - 6 x^{10} - 8 x^{9} + 32 x^{8} - 16 x^{7} + \cdots + 256 $ x^16 - 2x^15 + x^14 + 2x^13 - 6x^12 + 8x^11 - 6x^10 - 8x^9 + 32x^8 - 16x^7 - 24x^6 + 64x^5 - 96x^4 + 64x^3 + 64x^2 - 256x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.7
Root			$-1.23596 - 0.687323i$ of defining polynomial
Character	$\chi$	$=$	4320.431
Dual form			4320.2.b.d.431.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.920206i q^{7} -0.482221i q^{11} +2.01753i q^{13} +0.701532i q^{17} -0.782948 q^{19} -1.44101 q^{23} +1.00000 q^{25} +2.14505 q^{29} -6.59164i q^{31} -0.920206i q^{35} -2.33837i q^{37} -7.36833i q^{41} +7.06978 q^{43} -0.294304 q^{47} +6.15322 q^{49} +10.5190 q^{53} -0.482221i q^{55} +10.8042i q^{59} +8.65852i q^{61} +2.01753i q^{65} +11.2157 q^{67} -5.57514 q^{71} +1.01962 q^{73} -0.443743 q^{77} -1.86662i q^{79} +1.75905i q^{83} +0.701532i q^{85} -11.9152i q^{89} +1.85654 q^{91} -0.782948 q^{95} +3.93818 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} + 8 q^{19} - 8 q^{23} + 16 q^{25} - 16 q^{47} - 8 q^{49} - 32 q^{53} - 32 q^{67} + 24 q^{71} + 8 q^{73} - 48 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100}) $ 16 * q + 16 * q^5 + 8 * q^19 - 8 * q^23 + 16 * q^25 - 16 * q^47 - 8 * q^49 - 32 * q^53 - 32 * q^67 + 24 * q^71 + 8 * q^73 - 48 * q^91 + 8 * q^95 - 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	− 0.920206i	− 0.347805i	−0.984763	−	0.173903i	$-0.944362\pi$
$7$	− 0.920206i	− 0.347805i	0.984763	−	0.173903i	$-0.0556378\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.482221i	− 0.145395i	−0.997354	−	0.0726976i	$-0.976839\pi$
$11$	− 0.482221i	− 0.145395i	0.997354	−	0.0726976i	$-0.0231608\pi$
$12$	0	0
$13$	2.01753i	0.559561i	0.960064	+	0.279781i	$0.0902617\pi$
$13$	2.01753i	0.559561i	−0.960064	+	0.279781i	$0.909738\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.701532i	0.170146i	0.996375	+	0.0850732i	$0.0271124\pi$
$17$	0.701532i	0.170146i	−0.996375	+	0.0850732i	$0.972888\pi$
$18$	0	0
$19$	−0.782948	−0.179621	−0.0898103	−	0.995959i	$-0.528626\pi$
$19$	−0.782948	−0.179621	−0.0898103	+	0.995959i	$0.528626\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.44101	−0.300472	−0.150236	−	0.988650i	$-0.548003\pi$
$23$	−1.44101	−0.300472	−0.150236	+	0.988650i	$0.548003\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.14505	0.398326	0.199163	−	0.979966i	$-0.436178\pi$
$29$	2.14505	0.398326	0.199163	+	0.979966i	$0.436178\pi$
$30$	0	0
$31$	− 6.59164i	− 1.18389i	−0.805977	−	0.591946i	$-0.798360\pi$
$31$	− 6.59164i	− 1.18389i	0.805977	−	0.591946i	$-0.201640\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.920206i	− 0.155543i
$36$	0	0
$37$	− 2.33837i	− 0.384426i	−0.981353	−	0.192213i	$-0.938433\pi$
$37$	− 2.33837i	− 0.384426i	0.981353	−	0.192213i	$-0.0615665\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.36833i	− 1.15074i	−0.817893	−	0.575370i	$-0.804858\pi$
$41$	− 7.36833i	− 1.15074i	0.817893	−	0.575370i	$-0.195142\pi$
$42$	0	0
$43$	7.06978	1.07813	0.539066	−	0.842264i	$-0.318777\pi$
$43$	7.06978	1.07813	0.539066	+	0.842264i	$0.318777\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.294304	−0.0429287	−0.0214643	−	0.999770i	$-0.506833\pi$
$47$	−0.294304	−0.0429287	−0.0214643	+	0.999770i	$0.506833\pi$
$48$	0	0
$49$	6.15322	0.879031
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5190	1.44489	0.722445	−	0.691428i	$-0.243018\pi$
$53$	10.5190	1.44489	0.722445	+	0.691428i	$0.243018\pi$
$54$	0	0
$55$	− 0.482221i	− 0.0650227i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.8042i	1.40659i	0.710899	+	0.703295i	$0.248288\pi$
$59$	10.8042i	1.40659i	−0.710899	+	0.703295i	$0.751712\pi$
$60$	0	0
$61$	8.65852i	1.10861i	0.832313	+	0.554305i	$0.187016\pi$
$61$	8.65852i	1.10861i	−0.832313	+	0.554305i	$0.812984\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.01753i	0.250243i
$66$	0	0
$67$	11.2157	1.37022	0.685110	−	0.728439i	$-0.259754\pi$
$67$	11.2157	1.37022	0.685110	+	0.728439i	$0.259754\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.57514	−0.661647	−0.330824	−	0.943693i	$-0.607327\pi$
$71$	−5.57514	−0.661647	−0.330824	+	0.943693i	$0.607327\pi$
$72$	0	0
$73$	1.01962	0.119337	0.0596686	−	0.998218i	$-0.480996\pi$
$73$	1.01962	0.119337	0.0596686	+	0.998218i	$0.480996\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.443743	−0.0505692
$78$	0	0
$79$	− 1.86662i	− 0.210012i	−0.994472	−	0.105006i	$-0.966514\pi$
$79$	− 1.86662i	− 0.210012i	0.994472	−	0.105006i	$-0.0334861\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.75905i	0.193081i	0.995329	+	0.0965404i	$0.0307777\pi$
$83$	1.75905i	0.193081i	−0.995329	+	0.0965404i	$0.969222\pi$
$84$	0	0
$85$	0.701532i	0.0760918i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 11.9152i	− 1.26301i	−0.775373	−	0.631503i	$-0.782438\pi$
$89$	− 11.9152i	− 1.26301i	0.775373	−	0.631503i	$-0.217562\pi$
$90$	0	0
$91$	1.85654	0.194618
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.782948	−0.0803287
$96$	0	0
$97$	3.93818	0.399862	0.199931	−	0.979810i	$-0.435928\pi$
$97$	3.93818	0.399862	0.199931	+	0.979810i	$0.435928\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.11015	−0.806990	−0.403495	−	0.914982i	$-0.632205\pi$
$101$	−8.11015	−0.806990	−0.403495	+	0.914982i	$0.632205\pi$
$102$	0	0
$103$	− 14.0930i	− 1.38863i	−0.719672	−	0.694314i	$-0.755708\pi$
$103$	− 14.0930i	− 1.38863i	0.719672	−	0.694314i	$-0.244292\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.23739i	0.216297i	0.994135	+	0.108149i	$0.0344922\pi$
$107$	2.23739i	0.216297i	−0.994135	+	0.108149i	$0.965508\pi$
$108$	0	0
$109$	− 14.7481i	− 1.41261i	−0.707907	−	0.706306i	$-0.750360\pi$
$109$	− 14.7481i	− 1.41261i	0.707907	−	0.706306i	$-0.249640\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 17.3371i	− 1.63094i	−0.578802	−	0.815468i	$-0.696480\pi$
$113$	− 17.3371i	− 1.63094i	0.578802	−	0.815468i	$-0.303520\pi$
$114$	0	0
$115$	−1.44101	−0.134375
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.645554	0.0591778
$120$	0	0
$121$	10.7675	0.978860
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.45636i	0.306702i	0.988172	+	0.153351i	$0.0490066\pi$
$127$	3.45636i	0.306702i	−0.988172	+	0.153351i	$0.950993\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.4536i	1.52493i	0.647032	+	0.762463i	$0.276010\pi$
$131$	17.4536i	1.52493i	−0.647032	+	0.762463i	$0.723990\pi$
$132$	0	0
$133$	0.720473i	0.0624730i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.15998i	0.782590i	0.920265	+	0.391295i	$0.127973\pi$
$137$	9.15998i	0.782590i	−0.920265	+	0.391295i	$0.872027\pi$
$138$	0	0
$139$	8.97927	0.761611	0.380806	−	0.924655i	$-0.375647\pi$
$139$	8.97927	0.761611	0.380806	+	0.924655i	$0.375647\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.972894	0.0813575
$144$	0	0
$145$	2.14505	0.178137
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.27614	0.432238	0.216119	−	0.976367i	$-0.430660\pi$
$149$	5.27614	0.432238	0.216119	+	0.976367i	$0.430660\pi$
$150$	0	0
$151$	− 16.8834i	− 1.37395i	−0.726681	−	0.686975i	$-0.758938\pi$
$151$	− 16.8834i	− 1.37395i	0.726681	−	0.686975i	$-0.241062\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 6.59164i	− 0.529453i
$156$	0	0
$157$	− 14.2642i	− 1.13841i	−0.822196	−	0.569204i	$-0.807251\pi$
$157$	− 14.2642i	− 1.13841i	0.822196	−	0.569204i	$-0.192749\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.32603i	0.104506i
$162$	0	0
$163$	2.68674	0.210442	0.105221	−	0.994449i	$-0.466445\pi$
$163$	2.68674	0.210442	0.105221	+	0.994449i	$0.466445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.28504	−0.563733	−0.281867	−	0.959454i	$-0.590954\pi$
$167$	−7.28504	−0.563733	−0.281867	+	0.959454i	$0.590954\pi$
$168$	0	0
$169$	8.92959	0.686891
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.06567	−0.461164	−0.230582	−	0.973053i	$-0.574063\pi$
$173$	−6.06567	−0.461164	−0.230582	+	0.973053i	$0.574063\pi$
$174$	0	0
$175$	− 0.920206i	− 0.0695611i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.1969i	1.36010i	0.733165	+	0.680050i	$0.238042\pi$
$179$	18.1969i	1.36010i	−0.733165	+	0.680050i	$0.761958\pi$
$180$	0	0
$181$	− 5.60765i	− 0.416813i	−0.978042	−	0.208407i	$-0.933172\pi$
$181$	− 5.60765i	− 0.416813i	0.978042	−	0.208407i	$-0.0668278\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.33837i	− 0.171921i
$186$	0	0
$187$	0.338293	0.0247385
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.21210	−0.594206	−0.297103	−	0.954845i	$-0.596021\pi$
$191$	−8.21210	−0.594206	−0.297103	+	0.954845i	$0.596021\pi$
$192$	0	0
$193$	14.2696	1.02715	0.513573	−	0.858046i	$-0.328322\pi$
$193$	14.2696	1.02715	0.513573	+	0.858046i	$0.328322\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.98477	0.283903	0.141952	−	0.989874i	$-0.454662\pi$
$197$	3.98477	0.283903	0.141952	+	0.989874i	$0.454662\pi$
$198$	0	0
$199$	27.7090i	1.96424i	0.188264	+	0.982118i	$0.439714\pi$
$199$	27.7090i	1.96424i	−0.188264	+	0.982118i	$0.560286\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.97389i	− 0.138540i
$204$	0	0
$205$	− 7.36833i	− 0.514626i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.377554i	0.0261160i
$210$	0	0
$211$	17.5428	1.20769	0.603847	−	0.797100i	$-0.293634\pi$
$211$	17.5428	1.20769	0.603847	+	0.797100i	$0.293634\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.06978	0.482155
$216$	0	0
$217$	−6.06567	−0.411764
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.41536	−0.0952073
$222$	0	0
$223$	− 0.406159i	− 0.0271984i	−0.999908	−	0.0135992i	$-0.995671\pi$
$223$	− 0.406159i	− 0.0271984i	0.999908	−	0.0135992i	$-0.00432890\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 17.6015i	− 1.16825i	−0.811663	−	0.584126i	$-0.801437\pi$
$227$	− 17.6015i	− 1.16825i	0.811663	−	0.584126i	$-0.198563\pi$
$228$	0	0
$229$	− 25.2194i	− 1.66655i	−0.552860	−	0.833274i	$-0.686464\pi$
$229$	− 25.2194i	− 1.66655i	0.552860	−	0.833274i	$-0.313536\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.9599i	− 0.849034i	−0.905420	−	0.424517i	$-0.860444\pi$
$233$	− 12.9599i	− 0.849034i	0.905420	−	0.424517i	$-0.139556\pi$
$234$	0	0
$235$	−0.294304	−0.0191983
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.1176	−0.783821	−0.391911	−	0.920003i	$-0.628186\pi$
$239$	−12.1176	−0.783821	−0.391911	+	0.920003i	$0.628186\pi$
$240$	0	0
$241$	6.52697	0.420439	0.210220	−	0.977654i	$-0.432582\pi$
$241$	6.52697	0.420439	0.210220	+	0.977654i	$0.432582\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.15322	0.393115
$246$	0	0
$247$	− 1.57962i	− 0.100509i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 8.66260i	− 0.546779i	−0.961903	−	0.273389i	$-0.911855\pi$
$251$	− 8.66260i	− 0.546779i	0.961903	−	0.273389i	$-0.0881447\pi$
$252$	0	0
$253$	0.694887i	0.0436872i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.7615i	1.41982i	0.704291	+	0.709912i	$0.251265\pi$
$257$	22.7615i	1.41982i	−0.704291	+	0.709912i	$0.748735\pi$
$258$	0	0
$259$	−2.15179	−0.133706
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.7096	1.27701	0.638506	−	0.769617i	$-0.279553\pi$
$263$	20.7096	1.27701	0.638506	+	0.769617i	$0.279553\pi$
$264$	0	0
$265$	10.5190	0.646175
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.2212	−0.806112	−0.403056	−	0.915175i	$-0.632052\pi$
$269$	−13.2212	−0.806112	−0.403056	+	0.915175i	$0.632052\pi$
$270$	0	0
$271$	4.05018i	0.246031i	0.992405	+	0.123016i	$0.0392565\pi$
$271$	4.05018i	0.246031i	−0.992405	+	0.123016i	$0.960743\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.482221i	− 0.0290790i
$276$	0	0
$277$	− 4.33623i	− 0.260539i	−0.991479	−	0.130269i	$-0.958416\pi$
$277$	− 4.33623i	− 0.260539i	0.991479	−	0.130269i	$-0.0415842\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.2605i	1.44726i	0.690188	+	0.723630i	$0.257528\pi$
$281$	24.2605i	1.44726i	−0.690188	+	0.723630i	$0.742472\pi$
$282$	0	0
$283$	27.0761	1.60951	0.804754	−	0.593609i	$-0.202297\pi$
$283$	27.0761	1.60951	0.804754	+	0.593609i	$0.202297\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.78038	−0.400233
$288$	0	0
$289$	16.5079	0.971050
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.2078	1.18055	0.590275	−	0.807202i	$-0.299019\pi$
$293$	20.2078	1.18055	0.590275	+	0.807202i	$0.299019\pi$
$294$	0	0
$295$	10.8042i	0.629046i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.90728i	− 0.168132i
$300$	0	0
$301$	− 6.50566i	− 0.374980i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.65852i	0.495786i
$306$	0	0
$307$	9.07799	0.518108	0.259054	−	0.965863i	$-0.416589\pi$
$307$	9.07799	0.518108	0.259054	+	0.965863i	$0.416589\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.9031	−0.901782	−0.450891	−	0.892579i	$-0.648894\pi$
$311$	−15.9031	−0.901782	−0.450891	+	0.892579i	$0.648894\pi$
$312$	0	0
$313$	−22.2979	−1.26035	−0.630176	−	0.776452i	$-0.717017\pi$
$313$	−22.2979	−1.26035	−0.630176	+	0.776452i	$0.717017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.9636	−1.40210	−0.701048	−	0.713114i	$-0.747284\pi$
$317$	−24.9636	−1.40210	−0.701048	+	0.713114i	$0.747284\pi$
$318$	0	0
$319$	− 1.03439i	− 0.0579146i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 0.549262i	− 0.0305618i
$324$	0	0
$325$	2.01753i	0.111912i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.270821i	0.0149308i
$330$	0	0
$331$	1.81036	0.0995065	0.0497532	−	0.998762i	$-0.484157\pi$
$331$	1.81036	0.0995065	0.0497532	+	0.998762i	$0.484157\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.2157	0.612781
$336$	0	0
$337$	27.1410	1.47847	0.739233	−	0.673449i	$-0.235188\pi$
$337$	27.1410	1.47847	0.739233	+	0.673449i	$0.235188\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.17863	−0.172132
$342$	0	0
$343$	− 12.1037i	− 0.653537i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.1993i	− 1.56750i	−0.621075	−	0.783751i	$-0.713304\pi$
$347$	− 29.1993i	− 1.56750i	0.621075	−	0.783751i	$-0.286696\pi$
$348$	0	0
$349$	− 4.81219i	− 0.257591i	−0.991671	−	0.128795i	$-0.958889\pi$
$349$	− 4.81219i	− 0.257591i	0.991671	−	0.128795i	$-0.0411110\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.68613i	0.142968i	0.997442	+	0.0714841i	$0.0227735\pi$
$353$	2.68613i	0.142968i	−0.997442	+	0.0714841i	$0.977226\pi$
$354$	0	0
$355$	−5.57514	−0.295898
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.20997	−0.0638600	−0.0319300	−	0.999490i	$-0.510165\pi$
$359$	−1.20997	−0.0638600	−0.0319300	+	0.999490i	$0.510165\pi$
$360$	0	0
$361$	−18.3870	−0.967736
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.01962	0.0533692
$366$	0	0
$367$	− 2.67794i	− 0.139787i	−0.997554	−	0.0698936i	$-0.977734\pi$
$367$	− 2.67794i	− 0.139787i	0.997554	−	0.0698936i	$-0.0222660\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.67962i	− 0.502541i
$372$	0	0
$373$	17.6359i	0.913155i	0.889684	+	0.456577i	$0.150925\pi$
$373$	17.6359i	0.913155i	−0.889684	+	0.456577i	$0.849075\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.32769i	0.222888i
$378$	0	0
$379$	−26.4128	−1.35673	−0.678367	−	0.734723i	$-0.737312\pi$
$379$	−26.4128	−1.35673	−0.678367	+	0.734723i	$0.737312\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.4773	1.60841	0.804207	−	0.594349i	$-0.202590\pi$
$383$	31.4773	1.60841	0.804207	+	0.594349i	$0.202590\pi$
$384$	0	0
$385$	−0.443743	−0.0226152
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.9035	0.654233	0.327116	−	0.944984i	$-0.393923\pi$
$389$	12.9035	0.654233	0.327116	+	0.944984i	$0.393923\pi$
$390$	0	0
$391$	− 1.01092i	− 0.0511242i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.86662i	− 0.0939200i
$396$	0	0
$397$	29.0747i	1.45922i	0.683866	+	0.729608i	$0.260297\pi$
$397$	29.0747i	1.45922i	−0.683866	+	0.729608i	$0.739703\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.03847i	− 0.201672i	−0.994903	−	0.100836i	$-0.967848\pi$
$401$	− 4.03847i	− 0.201672i	0.994903	−	0.100836i	$-0.0321517\pi$
$402$	0	0
$403$	13.2988	0.662461
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.12761	−0.0558937
$408$	0	0
$409$	23.4511	1.15958	0.579792	−	0.814764i	$-0.303134\pi$
$409$	23.4511	1.15958	0.579792	+	0.814764i	$0.303134\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.94211	0.489219
$414$	0	0
$415$	1.75905i	0.0863484i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10.7206i	− 0.523734i	−0.965104	−	0.261867i	$-0.915662\pi$
$419$	− 10.7206i	− 0.523734i	0.965104	−	0.261867i	$-0.0843381\pi$
$420$	0	0
$421$	14.6900i	0.715945i	0.933732	+	0.357972i	$0.116532\pi$
$421$	14.6900i	0.715945i	−0.933732	+	0.357972i	$0.883468\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.701532i	0.0340293i
$426$	0	0
$427$	7.96763	0.385581
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.1052	−0.534921	−0.267460	−	0.963569i	$-0.586184\pi$
$431$	−11.1052	−0.534921	−0.267460	+	0.963569i	$0.586184\pi$
$432$	0	0
$433$	−13.1055	−0.629812	−0.314906	−	0.949123i	$-0.601973\pi$
$433$	−13.1055	−0.629812	−0.314906	+	0.949123i	$0.601973\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.12824	0.0539709
$438$	0	0
$439$	11.5099i	0.549338i	0.961539	+	0.274669i	$0.0885682\pi$
$439$	11.5099i	0.549338i	−0.961539	+	0.274669i	$0.911432\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 14.5661i	− 0.692055i	−0.938224	−	0.346028i	$-0.887530\pi$
$443$	− 14.5661i	− 0.692055i	0.938224	−	0.346028i	$-0.112470\pi$
$444$	0	0
$445$	− 11.9152i	− 0.564834i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.5436i	− 0.591967i	−0.955193	−	0.295984i	$-0.904353\pi$
$449$	− 12.5436i	− 0.591967i	0.955193	−	0.295984i	$-0.0956474\pi$
$450$	0	0
$451$	−3.55317	−0.167312
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.85654	0.0870360
$456$	0	0
$457$	5.52282	0.258347	0.129173	−	0.991622i	$-0.458768\pi$
$457$	5.52282	0.258347	0.129173	+	0.991622i	$0.458768\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−37.7632	−1.75881	−0.879405	−	0.476075i	$-0.842059\pi$
$461$	−37.7632	−1.75881	−0.879405	+	0.476075i	$0.842059\pi$
$462$	0	0
$463$	− 34.1260i	− 1.58597i	−0.609241	−	0.792985i	$-0.708526\pi$
$463$	− 34.1260i	− 1.58597i	0.609241	−	0.792985i	$-0.291474\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.9667i	0.600026i	0.953935	+	0.300013i	$0.0969910\pi$
$467$	12.9667i	0.600026i	−0.953935	+	0.300013i	$0.903009\pi$
$468$	0	0
$469$	− 10.3208i	− 0.476570i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.40920i	− 0.156755i
$474$	0	0
$475$	−0.782948	−0.0359241
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.0652	1.64786	0.823931	−	0.566690i	$-0.191776\pi$
$479$	36.0652	1.64786	0.823931	+	0.566690i	$0.191776\pi$
$480$	0	0
$481$	4.71773	0.215110
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.93818	0.178824
$486$	0	0
$487$	− 40.1129i	− 1.81769i	−0.417135	−	0.908845i	$-0.636966\pi$
$487$	− 40.1129i	− 1.81769i	0.417135	−	0.908845i	$-0.363034\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.8605i	1.75375i	0.480720	+	0.876874i	$0.340375\pi$
$491$	38.8605i	1.75375i	−0.480720	+	0.876874i	$0.659625\pi$
$492$	0	0
$493$	1.50482i	0.0677737i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.13028i	0.230124i
$498$	0	0
$499$	10.1400	0.453928	0.226964	−	0.973903i	$-0.427120\pi$
$499$	10.1400	0.453928	0.226964	+	0.973903i	$0.427120\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.8213	1.32967	0.664834	−	0.746991i	$-0.268502\pi$
$503$	29.8213	1.32967	0.664834	+	0.746991i	$0.268502\pi$
$504$	0	0
$505$	−8.11015	−0.360897
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.1769	−1.38189	−0.690945	−	0.722907i	$-0.742805\pi$
$509$	−31.1769	−1.38189	−0.690945	+	0.722907i	$0.742805\pi$
$510$	0	0
$511$	− 0.938258i	− 0.0415061i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14.0930i	− 0.621013i
$516$	0	0
$517$	0.141920i	0.00624162i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.7989i	1.34932i	0.738127	+	0.674662i	$0.235711\pi$
$521$	30.7989i	1.34932i	−0.738127	+	0.674662i	$0.764289\pi$
$522$	0	0
$523$	−26.0367	−1.13851	−0.569253	−	0.822162i	$-0.692767\pi$
$523$	−26.0367	−1.13851	−0.569253	+	0.822162i	$0.692767\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.62424	0.201435
$528$	0	0
$529$	−20.9235	−0.909717
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.8658	0.643909
$534$	0	0
$535$	2.23739i	0.0967310i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.96721i	− 0.127807i
$540$	0	0
$541$	17.2351i	0.740994i	0.928834	+	0.370497i	$0.120813\pi$
$541$	17.2351i	0.740994i	−0.928834	+	0.370497i	$0.879187\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.7481i	− 0.631739i
$546$	0	0
$547$	−33.1887	−1.41905	−0.709523	−	0.704682i	$-0.751090\pi$
$547$	−33.1887	−1.41905	−0.709523	+	0.704682i	$0.751090\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.67946	−0.0715475
$552$	0	0
$553$	−1.71768	−0.0730431
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.8024	−0.839053	−0.419526	−	0.907743i	$-0.637804\pi$
$557$	−19.8024	−0.839053	−0.419526	+	0.907743i	$0.637804\pi$
$558$	0	0
$559$	14.2635i	0.603280i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 23.6881i	− 0.998333i	−0.866506	−	0.499166i	$-0.833640\pi$
$563$	− 23.6881i	− 0.998333i	0.866506	−	0.499166i	$-0.166360\pi$
$564$	0	0
$565$	− 17.3371i	− 0.729377i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.15817i	− 0.0485531i	−0.999705	−	0.0242765i	$-0.992272\pi$
$569$	− 1.15817i	− 0.0485531i	0.999705	−	0.0242765i	$-0.00772822\pi$
$570$	0	0
$571$	26.1823	1.09570	0.547848	−	0.836578i	$-0.315447\pi$
$571$	26.1823	1.09570	0.547848	+	0.836578i	$0.315447\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.44101	−0.0600944
$576$	0	0
$577$	−35.9581	−1.49696	−0.748479	−	0.663159i	$-0.769215\pi$
$577$	−35.9581	−1.49696	−0.748479	+	0.663159i	$0.769215\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.61869	0.0671545
$582$	0	0
$583$	− 5.07247i	− 0.210080i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.68401i	0.317153i	0.987347	+	0.158577i	$0.0506905\pi$
$587$	7.68401i	0.317153i	−0.987347	+	0.158577i	$0.949310\pi$
$588$	0	0
$589$	5.16091i	0.212651i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 14.3657i	− 0.589930i	−0.955508	−	0.294965i	$-0.904692\pi$
$593$	− 14.3657i	− 0.589930i	0.955508	−	0.294965i	$-0.0953080\pi$
$594$	0	0
$595$	0.645554	0.0264651
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.9780	−0.734562	−0.367281	−	0.930110i	$-0.619711\pi$
$599$	−17.9780	−0.734562	−0.367281	+	0.930110i	$0.619711\pi$
$600$	0	0
$601$	−21.2552	−0.867018	−0.433509	−	0.901149i	$-0.642725\pi$
$601$	−21.2552	−0.867018	−0.433509	+	0.901149i	$0.642725\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.7675	0.437760
$606$	0	0
$607$	0.428639i	0.0173979i	0.999962	+	0.00869897i	$0.00276900\pi$
$607$	0.428639i	0.0173979i	−0.999962	+	0.00869897i	$0.997231\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 0.593766i	− 0.0240212i
$612$	0	0
$613$	40.6593i	1.64221i	0.570774	+	0.821107i	$0.306643\pi$
$613$	40.6593i	1.64221i	−0.570774	+	0.821107i	$0.693357\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16.8640i	0.678920i	0.940621	+	0.339460i	$0.110244\pi$
$617$	16.8640i	0.678920i	−0.940621	+	0.339460i	$0.889756\pi$
$618$	0	0
$619$	−16.2055	−0.651353	−0.325677	−	0.945481i	$-0.605592\pi$
$619$	−16.2055	−0.651353	−0.325677	+	0.945481i	$0.605592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.9644	−0.439280
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.64044	0.0654088
$630$	0	0
$631$	10.7163i	0.426609i	0.976986	+	0.213304i	$0.0684226\pi$
$631$	10.7163i	0.426609i	−0.976986	+	0.213304i	$0.931577\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.45636i	0.137161i
$636$	0	0
$637$	12.4143i	0.491872i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	25.7665i	1.01772i	0.860851	+	0.508858i	$0.169932\pi$
$641$	25.7665i	1.01772i	−0.860851	+	0.508858i	$0.830068\pi$
$642$	0	0
$643$	9.51281	0.375149	0.187574	−	0.982250i	$-0.439937\pi$
$643$	9.51281	0.375149	0.187574	+	0.982250i	$0.439937\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−0.478556	−0.0188140	−0.00940699	−	0.999956i	$-0.502994\pi$
$647$	−0.478556	−0.0188140	−0.00940699	+	0.999956i	$0.502994\pi$
$648$	0	0
$649$	5.21002	0.204511
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.1846	1.37688	0.688439	−	0.725294i	$-0.258296\pi$
$653$	35.1846	1.37688	0.688439	+	0.725294i	$0.258296\pi$
$654$	0	0
$655$	17.4536i	0.681967i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 11.6312i	− 0.453088i	−0.974001	−	0.226544i	$-0.927257\pi$
$659$	− 11.6312i	− 0.453088i	0.974001	−	0.226544i	$-0.0727427\pi$
$660$	0	0
$661$	− 25.6802i	− 0.998843i	−0.866359	−	0.499421i	$-0.833546\pi$
$661$	− 25.6802i	− 0.998843i	0.866359	−	0.499421i	$-0.166454\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.720473i	0.0279388i
$666$	0	0
$667$	−3.09104	−0.119686
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.17533	0.161187
$672$	0	0
$673$	−1.60255	−0.0617738	−0.0308869	−	0.999523i	$-0.509833\pi$
$673$	−1.60255	−0.0617738	−0.0308869	+	0.999523i	$0.509833\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.62519	0.293060	0.146530	−	0.989206i	$-0.453190\pi$
$677$	7.62519	0.293060	0.146530	+	0.989206i	$0.453190\pi$
$678$	0	0
$679$	− 3.62394i	− 0.139074i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 28.3435i	− 1.08453i	−0.840207	−	0.542266i	$-0.817566\pi$
$683$	− 28.3435i	− 1.08453i	0.840207	−	0.542266i	$-0.182434\pi$
$684$	0	0
$685$	9.15998i	0.349985i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21.2223i	0.808505i
$690$	0	0
$691$	−19.3466	−0.735979	−0.367990	−	0.929830i	$-0.619954\pi$
$691$	−19.3466	−0.735979	−0.367990	+	0.929830i	$0.619954\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.97927	0.340603
$696$	0	0
$697$	5.16912	0.195794
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−49.8239	−1.88182	−0.940912	−	0.338650i	$-0.890030\pi$
$701$	−49.8239	−1.88182	−0.940912	+	0.338650i	$0.890030\pi$
$702$	0	0
$703$	1.83082i	0.0690509i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.46301i	0.280675i
$708$	0	0
$709$	2.93005i	0.110040i	0.998485	+	0.0550202i	$0.0175223\pi$
$709$	2.93005i	0.110040i	−0.998485	+	0.0550202i	$0.982478\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.49863i	0.355726i
$714$	0	0
$715$	0.972894	0.0363842
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.4103	−1.58164	−0.790818	−	0.612051i	$-0.790345\pi$
$719$	−42.4103	−1.58164	−0.790818	+	0.612051i	$0.790345\pi$
$720$	0	0
$721$	−12.9685	−0.482972
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.14505	0.0796651
$726$	0	0
$727$	33.8955i	1.25712i	0.777763	+	0.628558i	$0.216355\pi$
$727$	33.8955i	1.25712i	−0.777763	+	0.628558i	$0.783645\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.95967i	0.183440i
$732$	0	0
$733$	4.65523i	0.171945i	0.996298	+	0.0859724i	$0.0273997\pi$
$733$	4.65523i	0.171945i	−0.996298	+	0.0859724i	$0.972600\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.40847i	− 0.199223i
$738$	0	0
$739$	−42.8311	−1.57557	−0.787783	−	0.615953i	$-0.788771\pi$
$739$	−42.8311	−1.57557	−0.787783	+	0.615953i	$0.788771\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.1930	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$743$	22.1930	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$744$	0	0
$745$	5.27614	0.193303
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.05886	0.0752293
$750$	0	0
$751$	0.0193272i	0 0.000705259i	1.00000		0.000352629i	$0.000112245\pi$
$751$	0.0193272i	0 0.000705259i	−1.00000		0.000352629i	$0.999888\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 16.8834i	− 0.614450i
$756$	0	0
$757$	47.2539i	1.71747i	0.512417	+	0.858736i	$0.328750\pi$
$757$	47.2539i	1.71747i	−0.512417	+	0.858736i	$0.671250\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 40.7068i	− 1.47562i	−0.675007	−	0.737811i	$-0.735860\pi$
$761$	− 40.7068i	− 1.47562i	0.675007	−	0.737811i	$-0.264140\pi$
$762$	0	0
$763$	−13.5713	−0.491314
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.7978	−0.787073
$768$	0	0
$769$	1.80744	0.0651780	0.0325890	−	0.999469i	$-0.489625\pi$
$769$	1.80744	0.0651780	0.0325890	+	0.999469i	$0.489625\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.1312	1.40745	0.703726	−	0.710471i	$-0.251518\pi$
$773$	39.1312	1.40745	0.703726	+	0.710471i	$0.251518\pi$
$774$	0	0
$775$	− 6.59164i	− 0.236779i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.76902i	0.206696i
$780$	0	0
$781$	2.68845i	0.0962004i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 14.2642i	− 0.509112i
$786$	0	0
$787$	−4.89070	−0.174335	−0.0871673	−	0.996194i	$-0.527781\pi$
$787$	−4.89070	−0.174335	−0.0871673	+	0.996194i	$0.527781\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.9537	−0.567248
$792$	0	0
$793$	−17.4688	−0.620335
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.9556	−1.13193	−0.565963	−	0.824430i	$-0.691496\pi$
$797$	−31.9556	−1.13193	−0.565963	+	0.824430i	$0.691496\pi$
$798$	0	0
$799$	− 0.206464i	− 0.00730416i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 0.491681i	− 0.0173511i
$804$	0	0
$805$	1.32603i	0.0467364i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.3798i	1.31420i	0.753801	+	0.657102i	$0.228218\pi$
$809$	37.3798i	1.31420i	−0.753801	+	0.657102i	$0.771782\pi$
$810$	0	0
$811$	−17.2532	−0.605842	−0.302921	−	0.953016i	$-0.597962\pi$
$811$	−17.2532	−0.605842	−0.302921	+	0.953016i	$0.597962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.68674	0.0941125
$816$	0	0
$817$	−5.53527	−0.193655
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.86159	0.204571	0.102285	−	0.994755i	$-0.467384\pi$
$821$	5.86159	0.204571	0.102285	+	0.994755i	$0.467384\pi$
$822$	0	0
$823$	2.27164i	0.0791843i	0.999216	+	0.0395921i	$0.0126059\pi$
$823$	2.27164i	0.0791843i	−0.999216	+	0.0395921i	$0.987394\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.9531i	1.52840i	0.644981	+	0.764199i	$0.276865\pi$
$827$	43.9531i	1.52840i	−0.644981	+	0.764199i	$0.723135\pi$
$828$	0	0
$829$	− 19.7872i	− 0.687239i	−0.939109	−	0.343620i	$-0.888347\pi$
$829$	− 19.7872i	− 0.687239i	0.939109	−	0.343620i	$-0.111653\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.31668i	0.149564i
$834$	0	0
$835$	−7.28504	−0.252109
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17.6073	0.607872	0.303936	−	0.952692i	$-0.401699\pi$
$839$	17.6073	0.607872	0.303936	+	0.952692i	$0.401699\pi$
$840$	0	0
$841$	−24.3988	−0.841337
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.92959	0.307187
$846$	0	0
$847$	− 9.90829i	− 0.340453i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.36963i	0.115509i
$852$	0	0
$853$	31.9062i	1.09245i	0.837640	+	0.546223i	$0.183935\pi$
$853$	31.9062i	1.09245i	−0.837640	+	0.546223i	$0.816065\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 16.0007i	− 0.546574i	−0.961933	−	0.273287i	$-0.911889\pi$
$857$	− 16.0007i	− 0.546574i	0.961933	−	0.273287i	$-0.0881108\pi$
$858$	0	0
$859$	−5.29765	−0.180754	−0.0903768	−	0.995908i	$-0.528807\pi$
$859$	−5.29765	−0.180754	−0.0903768	+	0.995908i	$0.528807\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.1157	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$863$	30.1157	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$864$	0	0
$865$	−6.06567	−0.206239
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.900125	−0.0305347
$870$	0	0
$871$	22.6280i	0.766722i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.920206i	− 0.0311087i
$876$	0	0
$877$	− 27.0663i	− 0.913963i	−0.889476	−	0.456981i	$-0.848931\pi$
$877$	− 27.0663i	− 0.913963i	0.889476	−	0.456981i	$-0.151069\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	26.2189i	0.883338i	0.897178	+	0.441669i	$0.145613\pi$
$881$	26.2189i	0.883338i	−0.897178	+	0.441669i	$0.854387\pi$
$882$	0	0
$883$	−1.87380	−0.0630584	−0.0315292	−	0.999503i	$-0.510038\pi$
$883$	−1.87380	−0.0630584	−0.0315292	+	0.999503i	$0.510038\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.9667	−0.703992	−0.351996	−	0.936001i	$-0.614497\pi$
$887$	−20.9667	−0.703992	−0.351996	+	0.936001i	$0.614497\pi$
$888$	0	0
$889$	3.18056	0.106673
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.230425	0.00771087
$894$	0	0
$895$	18.1969i	0.608256i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 14.1394i	− 0.471575i
$900$	0	0
$901$	7.37938i	0.245843i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.60765i	− 0.186405i
$906$	0	0
$907$	−7.98889	−0.265267	−0.132633	−	0.991165i	$-0.542343\pi$
$907$	−7.98889	−0.265267	−0.132633	+	0.991165i	$0.542343\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−55.3654	−1.83434	−0.917169	−	0.398497i	$-0.869532\pi$
$911$	−55.3654	−1.83434	−0.917169	+	0.398497i	$0.869532\pi$
$912$	0	0
$913$	0.848251	0.0280730
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.0609	0.530377
$918$	0	0
$919$	39.2576i	1.29499i	0.762070	+	0.647495i	$0.224183\pi$
$919$	39.2576i	1.29499i	−0.762070	+	0.647495i	$0.775817\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 11.2480i	− 0.370232i
$924$	0	0
$925$	− 2.33837i	− 0.0768853i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 42.8360i	− 1.40540i	−0.711485	−	0.702701i	$-0.751977\pi$
$929$	− 42.8360i	− 1.40540i	0.711485	−	0.702701i	$-0.248023\pi$
$930$	0	0
$931$	−4.81765	−0.157892
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.338293	0.0110634
$936$	0	0
$937$	−30.5942	−0.999469	−0.499734	−	0.866179i	$-0.666569\pi$
$937$	−30.5942	−0.999469	−0.499734	+	0.866179i	$0.666569\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.8808	−1.52827	−0.764136	−	0.645055i	$-0.776834\pi$
$941$	−46.8808	−1.52827	−0.764136	+	0.645055i	$0.776834\pi$
$942$	0	0
$943$	10.6179i	0.345765i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.5877i	1.74137i	0.491844	+	0.870684i	$0.336323\pi$
$947$	53.5877i	1.74137i	−0.491844	+	0.870684i	$0.663677\pi$
$948$	0	0
$949$	2.05711i	0.0667765i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23.4773i	− 0.760503i	−0.924883	−	0.380252i	$-0.875837\pi$
$953$	− 23.4773i	− 0.760503i	0.924883	−	0.380252i	$-0.124163\pi$
$954$	0	0
$955$	−8.21210	−0.265737
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.42907	0.272189
$960$	0	0
$961$	−12.4497	−0.401603
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.2696	0.459354
$966$	0	0
$967$	− 0.755493i	− 0.0242950i	−0.999926	−	0.0121475i	$-0.996133\pi$
$967$	− 0.755493i	− 0.0242950i	0.999926	−	0.0121475i	$-0.00386677\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 0.229783i	− 0.00737407i	−0.999993	−	0.00368704i	$-0.998826\pi$
$971$	− 0.229783i	− 0.00737407i	0.999993	−	0.00368704i	$-0.00117362\pi$
$972$	0	0
$973$	− 8.26278i	− 0.264893i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.4218i	− 0.397410i	−0.980059	−	0.198705i	$-0.936326\pi$
$977$	− 12.4218i	− 0.397410i	0.980059	−	0.198705i	$-0.0636735\pi$
$978$	0	0
$979$	−5.74575	−0.183635
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.7013	−0.755956	−0.377978	−	0.925815i	$-0.623380\pi$
$983$	−23.7013	−0.755956	−0.377978	+	0.925815i	$0.623380\pi$
$984$	0	0
$985$	3.98477	0.126965
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.1876	−0.323948
$990$	0	0
$991$	− 4.03872i	− 0.128294i	−0.997940	−	0.0641471i	$-0.979567\pi$
$991$	− 4.03872i	− 0.128294i	0.997940	−	0.0641471i	$-0.0204327\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.7090i	0.878433i
$996$	0	0
$997$	− 49.3776i	− 1.56380i	−0.623401	−	0.781902i	$-0.714250\pi$
$997$	− 49.3776i	− 1.56380i	0.623401	−	0.781902i	$-0.285750\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.b.d.431.7		16
3.2	odd	2		4320.2.b.b.431.7		16
4.3	odd	2		1080.2.b.a.971.14	yes	16
8.3	odd	2		4320.2.b.b.431.10		16
8.5	even	2		1080.2.b.d.971.4	yes	16
12.11	even	2		1080.2.b.d.971.3	yes	16
24.5	odd	2		1080.2.b.a.971.13	✓	16
24.11	even	2	inner	4320.2.b.d.431.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.b.a.971.13	✓	16	24.5	odd	2
1080.2.b.a.971.14	yes	16	4.3	odd	2
1080.2.b.d.971.3	yes	16	12.11	even	2
1080.2.b.d.971.4	yes	16	8.5	even	2
4320.2.b.b.431.7		16	3.2	odd	2
4320.2.b.b.431.10		16	8.3	odd	2
4320.2.b.d.431.7		16	1.1	even	1	trivial
4320.2.b.d.431.10		16	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.920206i q^{7} -0.482221i q^{11} +2.01753i q^{13} +0.701532i q^{17} -0.782948 q^{19} -1.44101 q^{23} +1.00000 q^{25} +2.14505 q^{29} -6.59164i q^{31} -0.920206i q^{35} -2.33837i q^{37} -7.36833i q^{41} +7.06978 q^{43} -0.294304 q^{47} +6.15322 q^{49} +10.5190 q^{53} -0.482221i q^{55} +10.8042i q^{59} +8.65852i q^{61} +2.01753i q^{65} +11.2157 q^{67} -5.57514 q^{71} +1.01962 q^{73} -0.443743 q^{77} -1.86662i q^{79} +1.75905i q^{83} +0.701532i q^{85} -11.9152i q^{89} +1.85654 q^{91} -0.782948 q^{95} +3.93818 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} + 8 q^{19} - 8 q^{23} + 16 q^{25} - 16 q^{47} - 8 q^{49} - 32 q^{53} - 32 q^{67} + 24 q^{71} + 8 q^{73} - 48 q^{91} + 8 q^{95} - 8 q^{97}+O(q^{100}) \) 16 * q + 16 * q^5 + 8 * q^19 - 8 * q^23 + 16 * q^25 - 16 * q^47 - 8 * q^49 - 32 * q^53 - 32 * q^67 + 24 * q^71 + 8 * q^73 - 48 * q^91 + 8 * q^95 - 8 * q^97

Introduction

L-functions

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