LMFDB - Embedded newform 2880.2.h.e.1151.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1151,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.h (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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.8
Root			$0.500000 - 1.44392i$ of defining polynomial
Character	$\chi$	$=$	2880.1151
Dual form			2880.2.h.e.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} +5.03127i q^{7} +O(q^{10})$	$q+1.00000i q^{5} +5.03127i q^{7} -2.08402 q^{11} -3.41421 q^{13} -4.00000i q^{17} +4.16804i q^{19} -2.94725 q^{23} -1.00000 q^{25} +3.65685i q^{29} -2.94725i q^{31} -5.03127 q^{35} -5.07107 q^{37} +1.41421i q^{41} -4.16804i q^{43} +10.0625 q^{47} -18.3137 q^{49} -10.8284i q^{53} -2.08402i q^{55} +6.25206 q^{59} -4.82843 q^{61} -3.41421i q^{65} +5.89450i q^{67} -14.2306 q^{71} +3.17157 q^{73} -10.4853i q^{77} -11.2833i q^{79} +10.0625 q^{83} +4.00000 q^{85} -2.58579i q^{89} -17.1778i q^{91} -4.16804 q^{95} -2.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} - 56 q^{49} - 16 q^{61} + 48 q^{73} + 32 q^{85} + 48 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 - 56 * q^49 - 16 * q^61 + 48 * q^73 + 32 * q^85 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$7$	5.03127i	1.90164i	0.309740	+	0.950821i	$0.399758\pi$
$7$	5.03127i	1.90164i	−0.309740	+	0.950821i	$0.600242\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.08402	−0.628356	−0.314178	−	0.949364i	$-0.601729\pi$
$11$	−2.08402	−0.628356	−0.314178	+	0.949364i	$0.601729\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	4.16804i	0.956215i	0.878301	+	0.478107i	$0.158677\pi$
$19$	4.16804i	0.956215i	−0.878301	+	0.478107i	$0.841323\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.94725	−0.614544	−0.307272	−	0.951622i	$-0.599416\pi$
$23$	−2.94725	−0.614544	−0.307272	+	0.951622i	$0.599416\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.65685i	0.679061i	0.940595	+	0.339530i	$0.110268\pi$
$29$	3.65685i	0.679061i	−0.940595	+	0.339530i	$0.889732\pi$
$30$	0	0
$31$	− 2.94725i	− 0.529342i	−0.964339	−	0.264671i	$-0.914737\pi$
$31$	− 2.94725i	− 0.529342i	0.964339	−	0.264671i	$-0.0852634\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.03127	−0.850440
$36$	0	0
$37$	−5.07107	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$37$	−5.07107	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.41421i	0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$	1.41421i	0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$	0	0
$43$	− 4.16804i	− 0.635621i	−0.948154	−	0.317810i	$-0.897052\pi$
$43$	− 4.16804i	− 0.635621i	0.948154	−	0.317810i	$-0.102948\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0625	1.46777	0.733887	−	0.679272i	$-0.237704\pi$
$47$	10.0625	1.46777	0.733887	+	0.679272i	$0.237704\pi$
$48$	0	0
$49$	−18.3137	−2.61624
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.8284i	− 1.48740i	−0.668514	−	0.743699i	$-0.733069\pi$
$53$	− 10.8284i	− 1.48740i	0.668514	−	0.743699i	$-0.266931\pi$
$54$	0	0
$55$	− 2.08402i	− 0.281009i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.25206	0.813949	0.406975	−	0.913439i	$-0.366584\pi$
$59$	6.25206	0.813949	0.406975	+	0.913439i	$0.366584\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.41421i	− 0.423481i
$66$	0	0
$67$	5.89450i	0.720128i	0.932928	+	0.360064i	$0.117245\pi$
$67$	5.89450i	0.720128i	−0.932928	+	0.360064i	$0.882755\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.2306	−1.68886	−0.844430	−	0.535666i	$-0.820061\pi$
$71$	−14.2306	−1.68886	−0.844430	+	0.535666i	$0.820061\pi$
$72$	0	0
$73$	3.17157	0.371205	0.185602	−	0.982625i	$-0.440576\pi$
$73$	3.17157	0.371205	0.185602	+	0.982625i	$0.440576\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.4853i	− 1.19491i
$78$	0	0
$79$	− 11.2833i	− 1.26947i	−0.772728	−	0.634737i	$-0.781108\pi$
$79$	− 11.2833i	− 1.26947i	0.772728	−	0.634737i	$-0.218892\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.0625	1.10451	0.552254	−	0.833676i	$-0.313768\pi$
$83$	10.0625	1.10451	0.552254	+	0.833676i	$0.313768\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.58579i	− 0.274093i	−0.990565	−	0.137046i	$-0.956239\pi$
$89$	− 2.58579i	− 0.274093i	0.990565	−	0.137046i	$-0.0437609\pi$
$90$	0	0
$91$	− 17.1778i	− 1.80073i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.16804	−0.427632
$96$	0	0
$97$	−2.48528	−0.252342	−0.126171	−	0.992009i	$-0.540269\pi$
$97$	−2.48528	−0.252342	−0.126171	+	0.992009i	$0.540269\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 12.8284i	− 1.27648i	−0.769839	−	0.638238i	$-0.779664\pi$
$101$	− 12.8284i	− 1.27648i	0.769839	−	0.638238i	$-0.220336\pi$
$102$	0	0
$103$	3.30481i	0.325633i	0.986656	+	0.162816i	$0.0520578\pi$
$103$	3.30481i	0.325633i	−0.986656	+	0.162816i	$0.947942\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.0625	0.972783	0.486392	−	0.873741i	$-0.338313\pi$
$107$	10.0625	0.972783	0.486392	+	0.873741i	$0.338313\pi$
$108$	0	0
$109$	−0.828427	−0.0793489	−0.0396745	−	0.999213i	$-0.512632\pi$
$109$	−0.828427	−0.0793489	−0.0396745	+	0.999213i	$0.512632\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.48528i	0.421940i	0.977493	+	0.210970i	$0.0676622\pi$
$113$	4.48528i	0.421940i	−0.977493	+	0.210970i	$0.932338\pi$
$114$	0	0
$115$	− 2.94725i	− 0.274833i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.1251	1.84486
$120$	0	0
$121$	−6.65685	−0.605169
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	5.03127i	0.446453i	0.974767	+	0.223227i	$0.0716590\pi$
$127$	5.03127i	0.446453i	−0.974767	+	0.223227i	$0.928341\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.1466	−1.06125	−0.530625	−	0.847607i	$-0.678043\pi$
$131$	−12.1466	−1.06125	−0.530625	+	0.847607i	$0.678043\pi$
$132$	0	0
$133$	−20.9706	−1.81838
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.1421i	− 1.20824i	−0.796892	−	0.604122i	$-0.793524\pi$
$137$	− 14.1421i	− 1.20824i	0.796892	−	0.604122i	$-0.206476\pi$
$138$	0	0
$139$	7.11529i	0.603511i	0.953385	+	0.301756i	$0.0975727\pi$
$139$	7.11529i	0.603511i	−0.953385	+	0.301756i	$0.902427\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.11529	0.595011
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.4853i	0.858988i	0.903070	+	0.429494i	$0.141308\pi$
$149$	10.4853i	0.858988i	−0.903070	+	0.429494i	$0.858692\pi$
$150$	0	0
$151$	− 7.11529i	− 0.579034i	−0.957173	−	0.289517i	$-0.906505\pi$
$151$	− 7.11529i	− 0.579034i	0.957173	−	0.289517i	$-0.0934948\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.94725	0.236729
$156$	0	0
$157$	10.2426	0.817452	0.408726	−	0.912657i	$-0.365973\pi$
$157$	10.2426	0.817452	0.408726	+	0.912657i	$0.365973\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 14.8284i	− 1.16864i
$162$	0	0
$163$	4.16804i	0.326466i	0.986588	+	0.163233i	$0.0521923\pi$
$163$	4.16804i	0.326466i	−0.986588	+	0.163233i	$0.947808\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.84175	−0.684196	−0.342098	−	0.939664i	$-0.611137\pi$
$167$	−8.84175	−0.684196	−0.342098	+	0.939664i	$0.611137\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.4853i	0.949238i	0.880191	+	0.474619i	$0.157414\pi$
$173$	12.4853i	0.949238i	−0.880191	+	0.474619i	$0.842586\pi$
$174$	0	0
$175$	− 5.03127i	− 0.380328i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.3146	−1.21941	−0.609706	−	0.792628i	$-0.708712\pi$
$179$	−16.3146	−1.21941	−0.609706	+	0.792628i	$0.708712\pi$
$180$	0	0
$181$	−21.7990	−1.62031	−0.810153	−	0.586218i	$-0.800616\pi$
$181$	−21.7990	−1.62031	−0.810153	+	0.586218i	$0.800616\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.07107i	− 0.372832i
$186$	0	0
$187$	8.33609i	0.609595i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.72646	−0.124922	−0.0624611	−	0.998047i	$-0.519895\pi$
$191$	−1.72646	−0.124922	−0.0624611	+	0.998047i	$0.519895\pi$
$192$	0	0
$193$	−15.6569	−1.12701	−0.563503	−	0.826114i	$-0.690546\pi$
$193$	−15.6569	−1.12701	−0.563503	+	0.826114i	$0.690546\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.6569i	− 1.40049i	−0.713901	−	0.700246i	$-0.753073\pi$
$197$	− 19.6569i	− 1.40049i	0.713901	−	0.700246i	$-0.246927\pi$
$198$	0	0
$199$	27.2404i	1.93102i	0.260366	+	0.965510i	$0.416157\pi$
$199$	27.2404i	1.93102i	−0.260366	+	0.965510i	$0.583843\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−18.3986	−1.29133
$204$	0	0
$205$	−1.41421	−0.0987730
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 8.68629i	− 0.600843i
$210$	0	0
$211$	13.0098i	0.895631i	0.894126	+	0.447816i	$0.147798\pi$
$211$	13.0098i	0.895631i	−0.894126	+	0.447816i	$0.852202\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.16804	0.284258
$216$	0	0
$217$	14.8284	1.00662
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.6569i	0.918659i
$222$	0	0
$223$	20.9883i	1.40548i	0.711446	+	0.702741i	$0.248041\pi$
$223$	20.9883i	1.40548i	−0.711446	+	0.702741i	$0.751959\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.33609	0.553285	0.276643	−	0.960973i	$-0.410778\pi$
$227$	8.33609	0.553285	0.276643	+	0.960973i	$0.410778\pi$
$228$	0	0
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.00000i	− 0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$	− 4.00000i	− 0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$	0	0
$235$	10.0625i	0.656408i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.2931	−1.57139	−0.785696	−	0.618613i	$-0.787695\pi$
$239$	−24.2931	−1.57139	−0.785696	+	0.618613i	$0.787695\pi$
$240$	0	0
$241$	−12.9706	−0.835507	−0.417754	−	0.908560i	$-0.637183\pi$
$241$	−12.9706	−0.835507	−0.417754	+	0.908560i	$0.637183\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 18.3137i	− 1.17002i
$246$	0	0
$247$	− 14.2306i	− 0.905471i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.97852	−0.503600	−0.251800	−	0.967779i	$-0.581023\pi$
$251$	−7.97852	−0.503600	−0.251800	+	0.967779i	$0.581023\pi$
$252$	0	0
$253$	6.14214	0.386153
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.14214i	− 0.383136i	−0.981479	−	0.191568i	$-0.938643\pi$
$257$	− 6.14214i	− 0.383136i	0.981479	−	0.191568i	$-0.0613572\pi$
$258$	0	0
$259$	− 25.5139i	− 1.58536i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.38883	−0.332290	−0.166145	−	0.986101i	$-0.553132\pi$
$263$	−5.38883	−0.332290	−0.166145	+	0.986101i	$0.553132\pi$
$264$	0	0
$265$	10.8284	0.665185
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 2.00000i	− 0.121942i	−0.998140	−	0.0609711i	$-0.980580\pi$
$269$	− 2.00000i	− 0.121942i	0.998140	−	0.0609711i	$-0.0194197\pi$
$270$	0	0
$271$	7.11529i	0.432223i	0.976369	+	0.216112i	$0.0693375\pi$
$271$	7.11529i	0.432223i	−0.976369	+	0.216112i	$0.930662\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.08402	0.125671
$276$	0	0
$277$	5.75736	0.345926	0.172963	−	0.984928i	$-0.444666\pi$
$277$	5.75736	0.345926	0.172963	+	0.984928i	$0.444666\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.3848i	1.57398i	0.616963	+	0.786992i	$0.288363\pi$
$281$	26.3848i	1.57398i	−0.616963	+	0.786992i	$0.711637\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.11529	−0.420003
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.65685i	0.447318i	0.974667	+	0.223659i	$0.0718002\pi$
$293$	7.65685i	0.447318i	−0.974667	+	0.223659i	$0.928200\pi$
$294$	0	0
$295$	6.25206i	0.364009i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.0625	0.581932
$300$	0	0
$301$	20.9706	1.20872
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.82843i	− 0.276475i
$306$	0	0
$307$	− 20.1251i	− 1.14860i	−0.818645	−	0.574300i	$-0.805274\pi$
$307$	− 20.1251i	− 1.14860i	0.818645	−	0.574300i	$-0.194726\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.8516	1.23909	0.619544	−	0.784962i	$-0.287318\pi$
$311$	21.8516	1.23909	0.619544	+	0.784962i	$0.287318\pi$
$312$	0	0
$313$	−12.3431	−0.697676	−0.348838	−	0.937183i	$-0.613424\pi$
$313$	−12.3431	−0.697676	−0.348838	+	0.937183i	$0.613424\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.97056i	− 0.391506i	−0.980653	−	0.195753i	$-0.937285\pi$
$317$	− 6.97056i	− 0.391506i	0.980653	−	0.195753i	$-0.0627151\pi$
$318$	0	0
$319$	− 7.62096i	− 0.426692i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.6722	0.927664
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	0	0
$328$	0	0
$329$	50.6274i	2.79118i
$330$	0	0
$331$	18.3986i	1.01128i	0.862744	+	0.505640i	$0.168744\pi$
$331$	18.3986i	1.01128i	−0.862744	+	0.505640i	$0.831256\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.89450	−0.322051
$336$	0	0
$337$	−8.14214	−0.443530	−0.221765	−	0.975100i	$-0.571182\pi$
$337$	−8.14214	−0.443530	−0.221765	+	0.975100i	$0.571182\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.14214i	0.332615i
$342$	0	0
$343$	− 56.9224i	− 3.07352i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.2306	−0.763938	−0.381969	−	0.924175i	$-0.624754\pi$
$347$	−14.2306	−0.763938	−0.381969	+	0.924175i	$0.624754\pi$
$348$	0	0
$349$	−13.5147	−0.723426	−0.361713	−	0.932289i	$-0.617808\pi$
$349$	−13.5147	−0.723426	−0.361713	+	0.932289i	$0.617808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.3137i	0.602168i	0.953598	+	0.301084i	$0.0973484\pi$
$353$	11.3137i	0.602168i	−0.953598	+	0.301084i	$0.902652\pi$
$354$	0	0
$355$	− 14.2306i	− 0.755281i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.72646	−0.0911191	−0.0455595	−	0.998962i	$-0.514507\pi$
$359$	−1.72646	−0.0911191	−0.0455595	+	0.998962i	$0.514507\pi$
$360$	0	0
$361$	1.62742	0.0856535
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.17157i	0.166008i
$366$	0	0
$367$	− 12.6522i	− 0.660441i	−0.943904	−	0.330221i	$-0.892877\pi$
$367$	− 12.6522i	− 0.660441i	0.943904	−	0.330221i	$-0.107123\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	54.4808	2.82850
$372$	0	0
$373$	12.5858	0.651667	0.325834	−	0.945427i	$-0.394355\pi$
$373$	12.5858	0.651667	0.325834	+	0.945427i	$0.394355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.4853i	− 0.643025i
$378$	0	0
$379$	24.7988i	1.27383i	0.770934	+	0.636914i	$0.219790\pi$
$379$	24.7988i	1.27383i	−0.770934	+	0.636914i	$0.780210\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.0625	0.514172	0.257086	−	0.966389i	$-0.417238\pi$
$383$	10.0625	0.514172	0.257086	+	0.966389i	$0.417238\pi$
$384$	0	0
$385$	10.4853	0.534379
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5.02944i	− 0.255003i	−0.991838	−	0.127501i	$-0.959304\pi$
$389$	− 5.02944i	− 0.255003i	0.991838	−	0.127501i	$-0.0406957\pi$
$390$	0	0
$391$	11.7890i	0.596196i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.2833	0.567726
$396$	0	0
$397$	33.0711	1.65979	0.829895	−	0.557920i	$-0.188400\pi$
$397$	33.0711	1.65979	0.829895	+	0.557920i	$0.188400\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 21.2132i	− 1.05934i	−0.848205	−	0.529668i	$-0.822316\pi$
$401$	− 21.2132i	− 1.05934i	0.848205	−	0.529668i	$-0.177684\pi$
$402$	0	0
$403$	10.0625i	0.501251i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.5682	0.523847
$408$	0	0
$409$	14.9706	0.740247	0.370123	−	0.928983i	$-0.379315\pi$
$409$	14.9706	0.740247	0.370123	+	0.928983i	$0.379315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	31.4558i	1.54784i
$414$	0	0
$415$	10.0625i	0.493951i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.52560	−0.221090	−0.110545	−	0.993871i	$-0.535260\pi$
$419$	−4.52560	−0.221090	−0.110545	+	0.993871i	$0.535260\pi$
$420$	0	0
$421$	14.9706	0.729621	0.364810	−	0.931082i	$-0.381134\pi$
$421$	14.9706	0.729621	0.364810	+	0.931082i	$0.381134\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.00000i	0.194029i
$426$	0	0
$427$	− 24.2931i	− 1.17563i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0821	1.73802	0.869008	−	0.494798i	$-0.164758\pi$
$431$	36.0821	1.73802	0.869008	+	0.494798i	$0.164758\pi$
$432$	0	0
$433$	−31.4558	−1.51167	−0.755836	−	0.654761i	$-0.772769\pi$
$433$	−31.4558	−1.51167	−0.755836	+	0.654761i	$0.772769\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.2843i	− 0.587636i
$438$	0	0
$439$	− 13.0098i	− 0.620924i	−0.950586	−	0.310462i	$-0.899516\pi$
$439$	− 13.0098i	− 0.620924i	0.950586	−	0.310462i	$-0.100484\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.3557	−1.63229	−0.816144	−	0.577849i	$-0.803892\pi$
$443$	−34.3557	−1.63229	−0.816144	+	0.577849i	$0.803892\pi$
$444$	0	0
$445$	2.58579	0.122578
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.2132i	1.56743i	0.621122	+	0.783714i	$0.286677\pi$
$449$	33.2132i	1.56743i	−0.621122	+	0.783714i	$0.713323\pi$
$450$	0	0
$451$	− 2.94725i	− 0.138781i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17.1778	0.805310
$456$	0	0
$457$	39.4558	1.84567	0.922833	−	0.385200i	$-0.125867\pi$
$457$	39.4558	1.84567	0.922833	+	0.385200i	$0.125867\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.79899i	− 0.270086i	−0.990840	−	0.135043i	$-0.956883\pi$
$461$	− 5.79899i	− 0.270086i	0.990840	−	0.135043i	$-0.0431172\pi$
$462$	0	0
$463$	12.6522i	0.587999i	0.955806	+	0.294000i	$0.0949864\pi$
$463$	12.6522i	0.587999i	−0.955806	+	0.294000i	$0.905014\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−40.9653	−1.89565	−0.947824	−	0.318794i	$-0.896722\pi$
$467$	−40.9653	−1.89565	−0.947824	+	0.318794i	$0.896722\pi$
$468$	0	0
$469$	−29.6569	−1.36943
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.68629i	0.399396i
$474$	0	0
$475$	− 4.16804i	− 0.191243i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−34.3557	−1.56975	−0.784876	−	0.619653i	$-0.787273\pi$
$479$	−34.3557	−1.56975	−0.784876	+	0.619653i	$0.787273\pi$
$480$	0	0
$481$	17.3137	0.789437
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 2.48528i	− 0.112851i
$486$	0	0
$487$	− 5.03127i	− 0.227989i	−0.993481	−	0.113994i	$-0.963635\pi$
$487$	− 5.03127i	− 0.227989i	0.993481	−	0.113994i	$-0.0363646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.5882	0.658354	0.329177	−	0.944268i	$-0.393229\pi$
$491$	14.5882	0.658354	0.329177	+	0.944268i	$0.393229\pi$
$492$	0	0
$493$	14.6274	0.658786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 71.5980i	− 3.21161i
$498$	0	0
$499$	− 30.1876i	− 1.35138i	−0.737184	−	0.675692i	$-0.763845\pi$
$499$	− 30.1876i	− 1.35138i	0.737184	−	0.675692i	$-0.236155\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.8516	−0.974313	−0.487156	−	0.873315i	$-0.661966\pi$
$503$	−21.8516	−0.974313	−0.487156	+	0.873315i	$0.661966\pi$
$504$	0	0
$505$	12.8284	0.570858
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.1421i	1.77927i	0.456673	+	0.889634i	$0.349041\pi$
$509$	40.1421i	1.77927i	−0.456673	+	0.889634i	$0.650959\pi$
$510$	0	0
$511$	15.9570i	0.705898i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.30481	−0.145627
$516$	0	0
$517$	−20.9706	−0.922284
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.5858i	0.463772i	0.972743	+	0.231886i	$0.0744896\pi$
$521$	10.5858i	0.463772i	−0.972743	+	0.231886i	$0.925510\pi$
$522$	0	0
$523$	− 7.62096i	− 0.333241i	−0.986021	−	0.166621i	$-0.946714\pi$
$523$	− 7.62096i	− 0.333241i	0.986021	−	0.166621i	$-0.0532855\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.7890	−0.513537
$528$	0	0
$529$	−14.3137	−0.622335
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.82843i	− 0.209142i
$534$	0	0
$535$	10.0625i	0.435042i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38.1662	1.64393
$540$	0	0
$541$	20.1421	0.865978	0.432989	−	0.901399i	$-0.357459\pi$
$541$	20.1421	0.865978	0.432989	+	0.901399i	$0.357459\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 0.828427i	− 0.0354859i
$546$	0	0
$547$	18.3986i	0.786669i	0.919396	+	0.393334i	$0.128679\pi$
$547$	18.3986i	0.786669i	−0.919396	+	0.393334i	$0.871321\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−15.2419	−0.649328
$552$	0	0
$553$	56.7696	2.41409
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.1127i	− 0.979316i	−0.871914	−	0.489658i	$-0.837122\pi$
$557$	− 23.1127i	− 0.979316i	0.871914	−	0.489658i	$-0.162878\pi$
$558$	0	0
$559$	14.2306i	0.601890i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.2306	0.599748	0.299874	−	0.953979i	$-0.403055\pi$
$563$	14.2306	0.599748	0.299874	+	0.953979i	$0.403055\pi$
$564$	0	0
$565$	−4.48528	−0.188697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9.89949i	0.415008i	0.978234	+	0.207504i	$0.0665341\pi$
$569$	9.89949i	0.415008i	−0.978234	+	0.207504i	$0.933466\pi$
$570$	0	0
$571$	18.3986i	0.769959i	0.922925	+	0.384979i	$0.125791\pi$
$571$	18.3986i	0.769959i	−0.922925	+	0.384979i	$0.874209\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.94725	0.122909
$576$	0	0
$577$	37.7990	1.57359	0.786796	−	0.617213i	$-0.211738\pi$
$577$	37.7990	1.57359	0.786796	+	0.617213i	$0.211738\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	50.6274i	2.10038i
$582$	0	0
$583$	22.5667i	0.934616i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.9766	−1.73256	−0.866281	−	0.499557i	$-0.833496\pi$
$587$	−41.9766	−1.73256	−0.866281	+	0.499557i	$0.833496\pi$
$588$	0	0
$589$	12.2843	0.506165
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 9.85786i	− 0.404814i	−0.979301	−	0.202407i	$-0.935124\pi$
$593$	− 9.85786i	− 0.404814i	0.979301	−	0.202407i	$-0.0648764\pi$
$594$	0	0
$595$	20.1251i	0.825048i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.5041	0.510905	0.255452	−	0.966822i	$-0.417776\pi$
$599$	12.5041	0.510905	0.255452	+	0.966822i	$0.417776\pi$
$600$	0	0
$601$	15.3137	0.624659	0.312330	−	0.949974i	$-0.398891\pi$
$601$	15.3137	0.624659	0.312330	+	0.949974i	$0.398891\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.65685i	− 0.270640i
$606$	0	0
$607$	18.5467i	0.752789i	0.926460	+	0.376394i	$0.122836\pi$
$607$	18.5467i	0.752789i	−0.926460	+	0.376394i	$0.877164\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.3557	−1.38988
$612$	0	0
$613$	28.3848	1.14645	0.573225	−	0.819398i	$-0.305692\pi$
$613$	28.3848	1.14645	0.573225	+	0.819398i	$0.305692\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	40.0000i	1.61034i	0.593045	+	0.805170i	$0.297926\pi$
$617$	40.0000i	1.61034i	−0.593045	+	0.805170i	$0.702074\pi$
$618$	0	0
$619$	− 18.9043i	− 0.759828i	−0.925022	−	0.379914i	$-0.875954\pi$
$619$	− 18.9043i	− 0.759828i	0.925022	−	0.379914i	$-0.124046\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.0098	0.521227
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.2843i	0.808787i
$630$	0	0
$631$	− 0.505668i	− 0.0201303i	−0.999949	−	0.0100652i	$-0.996796\pi$
$631$	− 0.505668i	− 0.0201303i	0.999949	−	0.0100652i	$-0.00320390\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.03127	−0.199660
$636$	0	0
$637$	62.5269	2.47741
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.8995i	0.864978i	0.901639	+	0.432489i	$0.142365\pi$
$641$	21.8995i	0.864978i	−0.901639	+	0.432489i	$0.857635\pi$
$642$	0	0
$643$	18.3986i	0.725571i	0.931873	+	0.362786i	$0.118174\pi$
$643$	18.3986i	0.725571i	−0.931873	+	0.362786i	$0.881826\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.3986	−0.723325	−0.361662	−	0.932309i	$-0.617791\pi$
$647$	−18.3986	−0.723325	−0.361662	+	0.932309i	$0.617791\pi$
$648$	0	0
$649$	−13.0294	−0.511450
$650$	0	0
$651$	0	0
$652$	0	0
$653$	23.6569i	0.925764i	0.886420	+	0.462882i	$0.153185\pi$
$653$	23.6569i	0.925764i	−0.886420	+	0.462882i	$0.846815\pi$
$654$	0	0
$655$	− 12.1466i	− 0.474606i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−45.7871	−1.78361	−0.891807	−	0.452417i	$-0.850562\pi$
$659$	−45.7871	−1.78361	−0.891807	+	0.452417i	$0.850562\pi$
$660$	0	0
$661$	−26.4853	−1.03016	−0.515079	−	0.857143i	$-0.672237\pi$
$661$	−26.4853	−1.03016	−0.515079	+	0.857143i	$0.672237\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20.9706i	− 0.813204i
$666$	0	0
$667$	− 10.7777i	− 0.417313i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0625	0.388460
$672$	0	0
$673$	−10.2010	−0.393220	−0.196610	−	0.980482i	$-0.562993\pi$
$673$	−10.2010	−0.393220	−0.196610	+	0.980482i	$0.562993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.31371i	0.204222i	0.994773	+	0.102111i	$0.0325598\pi$
$677$	5.31371i	0.204222i	−0.994773	+	0.102111i	$0.967440\pi$
$678$	0	0
$679$	− 12.5041i	− 0.479864i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.9570	0.610580	0.305290	−	0.952260i	$-0.401247\pi$
$683$	15.9570	0.610580	0.305290	+	0.952260i	$0.401247\pi$
$684$	0	0
$685$	14.1421	0.540343
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36.9706i	1.40847i
$690$	0	0
$691$	27.7461i	1.05551i	0.849397	+	0.527755i	$0.176966\pi$
$691$	27.7461i	1.05551i	−0.849397	+	0.527755i	$0.823034\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.11529	−0.269899
$696$	0	0
$697$	5.65685	0.214269
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.1421i	− 0.911836i	−0.890022	−	0.455918i	$-0.849311\pi$
$701$	− 24.1421i	− 0.911836i	0.890022	−	0.455918i	$-0.150689\pi$
$702$	0	0
$703$	− 21.1364i	− 0.797176i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	64.5433	2.42740
$708$	0	0
$709$	−1.31371	−0.0493374	−0.0246687	−	0.999696i	$-0.507853\pi$
$709$	−1.31371	−0.0493374	−0.0246687	+	0.999696i	$0.507853\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.68629i	0.325304i
$714$	0	0
$715$	7.11529i	0.266097i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	13.5155	0.504042	0.252021	−	0.967722i	$-0.418905\pi$
$719$	13.5155	0.504042	0.252021	+	0.967722i	$0.418905\pi$
$720$	0	0
$721$	−16.6274	−0.619237
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.65685i	− 0.135812i
$726$	0	0
$727$	25.1564i	0.932998i	0.884522	+	0.466499i	$0.154485\pi$
$727$	25.1564i	0.932998i	−0.884522	+	0.466499i	$0.845515\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−16.6722	−0.616643
$732$	0	0
$733$	−33.0711	−1.22151	−0.610754	−	0.791820i	$-0.709134\pi$
$733$	−33.0711	−1.22151	−0.610754	+	0.791820i	$0.709134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.2843i	− 0.452497i
$738$	0	0
$739$	38.5237i	1.41712i	0.705652	+	0.708559i	$0.250654\pi$
$739$	38.5237i	1.41712i	−0.705652	+	0.708559i	$0.749346\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.60963	−0.242484	−0.121242	−	0.992623i	$-0.538688\pi$
$743$	−6.60963	−0.242484	−0.121242	+	0.992623i	$0.538688\pi$
$744$	0	0
$745$	−10.4853	−0.384151
$746$	0	0
$747$	0	0
$748$	0	0
$749$	50.6274i	1.84989i
$750$	0	0
$751$	− 17.1778i	− 0.626828i	−0.949617	−	0.313414i	$-0.898527\pi$
$751$	− 17.1778i	− 0.626828i	0.949617	−	0.313414i	$-0.101473\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.11529	0.258952
$756$	0	0
$757$	42.0416	1.52803	0.764015	−	0.645199i	$-0.223226\pi$
$757$	42.0416	1.52803	0.764015	+	0.645199i	$0.223226\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.3848i	0.811448i	0.913996	+	0.405724i	$0.132980\pi$
$761$	22.3848i	0.811448i	−0.913996	+	0.405724i	$0.867020\pi$
$762$	0	0
$763$	− 4.16804i	− 0.150893i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.3459	−0.770755
$768$	0	0
$769$	−1.37258	−0.0494966	−0.0247483	−	0.999694i	$-0.507878\pi$
$769$	−1.37258	−0.0494966	−0.0247483	+	0.999694i	$0.507878\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 42.2843i	− 1.52086i	−0.649420	−	0.760430i	$-0.724988\pi$
$773$	− 42.2843i	− 1.52086i	0.649420	−	0.760430i	$-0.275012\pi$
$774$	0	0
$775$	2.94725i	0.105868i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.89450	−0.211192
$780$	0	0
$781$	29.6569	1.06121
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.2426i	0.365576i
$786$	0	0
$787$	18.3986i	0.655840i	0.944705	+	0.327920i	$0.106348\pi$
$787$	18.3986i	0.655840i	−0.944705	+	0.327920i	$0.893652\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22.5667	−0.802379
$792$	0	0
$793$	16.4853	0.585410
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 13.4558i	− 0.476630i	−0.971188	−	0.238315i	$-0.923405\pi$
$797$	− 13.4558i	− 0.476630i	0.971188	−	0.238315i	$-0.0765951\pi$
$798$	0	0
$799$	− 40.2502i	− 1.42395i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.60963	−0.233249
$804$	0	0
$805$	14.8284	0.522633
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.87006i	0.241538i	0.992681	+	0.120769i	$0.0385361\pi$
$809$	6.87006i	0.241538i	−0.992681	+	0.120769i	$0.961464\pi$
$810$	0	0
$811$	− 43.9126i	− 1.54198i	−0.636848	−	0.770989i	$-0.719762\pi$
$811$	− 43.9126i	− 1.54198i	0.636848	−	0.770989i	$-0.280238\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.16804	−0.146000
$816$	0	0
$817$	17.3726	0.607790
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 36.3431i	− 1.26838i	−0.773175	−	0.634192i	$-0.781333\pi$
$821$	− 36.3431i	− 1.26838i	0.773175	−	0.634192i	$-0.218667\pi$
$822$	0	0
$823$	39.3870i	1.37294i	0.727157	+	0.686471i	$0.240841\pi$
$823$	39.3870i	1.37294i	−0.727157	+	0.686471i	$0.759159\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.4808	1.89448	0.947241	−	0.320522i	$-0.103858\pi$
$827$	54.4808	1.89448	0.947241	+	0.320522i	$0.103858\pi$
$828$	0	0
$829$	41.1127	1.42790	0.713952	−	0.700195i	$-0.246904\pi$
$829$	41.1127	1.42790	0.713952	+	0.700195i	$0.246904\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	73.2548i	2.53813i
$834$	0	0
$835$	− 8.84175i	− 0.305982i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.8516	0.754399	0.377200	−	0.926132i	$-0.376887\pi$
$839$	21.8516	0.754399	0.377200	+	0.926132i	$0.376887\pi$
$840$	0	0
$841$	15.6274	0.538876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1.34315i	− 0.0462056i
$846$	0	0
$847$	− 33.4925i	− 1.15081i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14.9457	0.512332
$852$	0	0
$853$	−39.8995	−1.36613	−0.683066	−	0.730356i	$-0.739354\pi$
$853$	−39.8995	−1.36613	−0.683066	+	0.730356i	$0.739354\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 0.970563i	− 0.0331538i	−0.999863	−	0.0165769i	$-0.994723\pi$
$857$	− 0.970563i	− 0.0331538i	0.999863	−	0.0165769i	$-0.00527683\pi$
$858$	0	0
$859$	− 10.5682i	− 0.360583i	−0.983613	−	0.180291i	$-0.942296\pi$
$859$	− 10.5682i	− 0.360583i	0.983613	−	0.180291i	$-0.0577041\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.7445	1.35292	0.676460	−	0.736480i	$-0.263513\pi$
$863$	39.7445	1.35292	0.676460	+	0.736480i	$0.263513\pi$
$864$	0	0
$865$	−12.4853	−0.424512
$866$	0	0
$867$	0	0
$868$	0	0
$869$	23.5147i	0.797682i
$870$	0	0
$871$	− 20.1251i	− 0.681913i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.03127	0.170088
$876$	0	0
$877$	−6.92893	−0.233973	−0.116987	−	0.993133i	$-0.537323\pi$
$877$	−6.92893	−0.233973	−0.116987	+	0.993133i	$0.537323\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 48.2426i	− 1.62534i	−0.582727	−	0.812668i	$-0.698014\pi$
$881$	− 48.2426i	− 1.62534i	0.582727	−	0.812668i	$-0.301986\pi$
$882$	0	0
$883$	− 49.3014i	− 1.65912i	−0.558415	−	0.829562i	$-0.688590\pi$
$883$	− 49.3014i	− 1.65912i	0.558415	−	0.829562i	$-0.311410\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.2833	0.378857	0.189429	−	0.981894i	$-0.439336\pi$
$887$	11.2833	0.378857	0.189429	+	0.981894i	$0.439336\pi$
$888$	0	0
$889$	−25.3137	−0.848995
$890$	0	0
$891$	0	0
$892$	0	0
$893$	41.9411i	1.40351i
$894$	0	0
$895$	− 16.3146i	− 0.545337i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.7777	0.359455
$900$	0	0
$901$	−43.3137	−1.44299
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 21.7990i	− 0.724623i
$906$	0	0
$907$	21.8516i	0.725569i	0.931873	+	0.362784i	$0.118174\pi$
$907$	21.8516i	0.725569i	−0.931873	+	0.362784i	$0.881826\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.2502	−1.33355	−0.666774	−	0.745260i	$-0.732325\pi$
$911$	−40.2502	−1.33355	−0.666774	+	0.745260i	$0.732325\pi$
$912$	0	0
$913$	−20.9706	−0.694024
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 61.1127i	− 2.01812i
$918$	0	0
$919$	43.1974i	1.42495i	0.701697	+	0.712476i	$0.252426\pi$
$919$	43.1974i	1.42495i	−0.701697	+	0.712476i	$0.747574\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	48.5863	1.59924
$924$	0	0
$925$	5.07107	0.166736
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 44.0416i	− 1.44496i	−0.691392	−	0.722480i	$-0.743002\pi$
$929$	− 44.0416i	− 1.44496i	0.691392	−	0.722480i	$-0.256998\pi$
$930$	0	0
$931$	− 76.3323i	− 2.50169i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.33609	−0.272619
$936$	0	0
$937$	−0.343146	−0.0112101	−0.00560504	−	0.999984i	$-0.501784\pi$
$937$	−0.343146	−0.0112101	−0.00560504	+	0.999984i	$0.501784\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	16.3431i	0.532771i	0.963867	+	0.266386i	$0.0858295\pi$
$941$	16.3431i	0.532771i	−0.963867	+	0.266386i	$0.914171\pi$
$942$	0	0
$943$	− 4.16804i	− 0.135730i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0196	−0.845523	−0.422762	−	0.906241i	$-0.638939\pi$
$947$	−26.0196	−0.845523	−0.422762	+	0.906241i	$0.638939\pi$
$948$	0	0
$949$	−10.8284	−0.351506
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.1421i	0.458109i	0.973414	+	0.229054i	$0.0735634\pi$
$953$	14.1421i	0.458109i	−0.973414	+	0.229054i	$0.926437\pi$
$954$	0	0
$955$	− 1.72646i	− 0.0558669i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	71.1529	2.29765
$960$	0	0
$961$	22.3137	0.719797
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 15.6569i	− 0.504012i
$966$	0	0
$967$	− 14.3787i	− 0.462388i	−0.972908	−	0.231194i	$-0.925737\pi$
$967$	− 14.3787i	− 0.462388i	0.972908	−	0.231194i	$-0.0742632\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.4314	0.366853	0.183426	−	0.983033i	$-0.441281\pi$
$971$	11.4314	0.366853	0.183426	+	0.983033i	$0.441281\pi$
$972$	0	0
$973$	−35.7990	−1.14766
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.3431i	0.330907i	0.986218	+	0.165453i	$0.0529087\pi$
$977$	10.3431i	0.330907i	−0.986218	+	0.165453i	$0.947091\pi$
$978$	0	0
$979$	5.38883i	0.172228i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.5155	−0.431076	−0.215538	−	0.976495i	$-0.569151\pi$
$983$	−13.5155	−0.431076	−0.215538	+	0.976495i	$0.569151\pi$
$984$	0	0
$985$	19.6569	0.626319
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.2843i	0.390617i
$990$	0	0
$991$	− 15.4514i	− 0.490829i	−0.969418	−	0.245415i	$-0.921076\pi$
$991$	− 15.4514i	− 0.490829i	0.969418	−	0.245415i	$-0.0789241\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27.2404	−0.863578
$996$	0	0
$997$	2.92893	0.0927602	0.0463801	−	0.998924i	$-0.485231\pi$
$997$	2.92893	0.0927602	0.0463801	+	0.998924i	$0.485231\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.h.e.1151.8		8
3.2	odd	2	inner	2880.2.h.e.1151.4		8
4.3	odd	2	inner	2880.2.h.e.1151.5		8
8.3	odd	2		180.2.e.a.71.4	yes	8
8.5	even	2		180.2.e.a.71.6	yes	8
12.11	even	2	inner	2880.2.h.e.1151.1		8
24.5	odd	2		180.2.e.a.71.3	✓	8
24.11	even	2		180.2.e.a.71.5	yes	8
40.3	even	4		900.2.h.c.899.7		8
40.13	odd	4		900.2.h.c.899.6		8
40.19	odd	2		900.2.e.d.251.5		8
40.27	even	4		900.2.h.b.899.2		8
40.29	even	2		900.2.e.d.251.3		8
40.37	odd	4		900.2.h.b.899.3		8
120.29	odd	2		900.2.e.d.251.6		8
120.53	even	4		900.2.h.b.899.4		8
120.59	even	2		900.2.e.d.251.4		8
120.77	even	4		900.2.h.c.899.5		8
120.83	odd	4		900.2.h.b.899.1		8
120.107	odd	4		900.2.h.c.899.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.e.a.71.3	✓	8	24.5	odd	2
180.2.e.a.71.4	yes	8	8.3	odd	2
180.2.e.a.71.5	yes	8	24.11	even	2
180.2.e.a.71.6	yes	8	8.5	even	2
900.2.e.d.251.3		8	40.29	even	2
900.2.e.d.251.4		8	120.59	even	2
900.2.e.d.251.5		8	40.19	odd	2
900.2.e.d.251.6		8	120.29	odd	2
900.2.h.b.899.1		8	120.83	odd	4
900.2.h.b.899.2		8	40.27	even	4
900.2.h.b.899.3		8	40.37	odd	4
900.2.h.b.899.4		8	120.53	even	4
900.2.h.c.899.5		8	120.77	even	4
900.2.h.c.899.6		8	40.13	odd	4
900.2.h.c.899.7		8	40.3	even	4
900.2.h.c.899.8		8	120.107	odd	4
2880.2.h.e.1151.1		8	12.11	even	2	inner
2880.2.h.e.1151.4		8	3.2	odd	2	inner
2880.2.h.e.1151.5		8	4.3	odd	2	inner
2880.2.h.e.1151.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} +5.03127i q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} +5.03127i q^{7} -2.08402 q^{11} -3.41421 q^{13} -4.00000i q^{17} +4.16804i q^{19} -2.94725 q^{23} -1.00000 q^{25} +3.65685i q^{29} -2.94725i q^{31} -5.03127 q^{35} -5.07107 q^{37} +1.41421i q^{41} -4.16804i q^{43} +10.0625 q^{47} -18.3137 q^{49} -10.8284i q^{53} -2.08402i q^{55} +6.25206 q^{59} -4.82843 q^{61} -3.41421i q^{65} +5.89450i q^{67} -14.2306 q^{71} +3.17157 q^{73} -10.4853i q^{77} -11.2833i q^{79} +10.0625 q^{83} +4.00000 q^{85} -2.58579i q^{89} -17.1778i q^{91} -4.16804 q^{95} -2.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{13} - 8 q^{25} + 16 q^{37} - 56 q^{49} - 16 q^{61} + 48 q^{73} + 32 q^{85} + 48 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 8 * q^25 + 16 * q^37 - 56 * q^49 - 16 * q^61 + 48 * q^73 + 32 * q^85 + 48 * q^97

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