LMFDB - Embedded newform 4320.2.k.d.2161.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(2161,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4320.2161");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.k (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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 $ x^20 - x^18 + 5x^16 + 28x^12 - 28x^10 + 112x^8 + 320x^4 - 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.14
Root			$1.17425 + 0.788128i$ of defining polynomial
Character	$\chi$	$=$	4320.2161
Dual form			4320.2.k.d.2161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{5} -2.12726 q^{7} +O(q^{10})$	$q+1.00000i q^{5} -2.12726 q^{7} +5.48036i q^{11} +4.91228i q^{13} -0.235541 q^{17} +5.23139i q^{19} -7.42390 q^{23} -1.00000 q^{25} -4.24894i q^{29} -5.05609 q^{31} -2.12726i q^{35} +2.27608i q^{37} +3.26132 q^{41} -9.90812i q^{43} +8.17359 q^{47} -2.47478 q^{49} -10.6637i q^{53} -5.48036 q^{55} -0.702988i q^{59} -0.319109i q^{61} -4.91228 q^{65} -2.70667i q^{67} -7.18836 q^{71} +9.66370 q^{73} -11.6581i q^{77} +10.8565 q^{79} -8.53645i q^{83} -0.235541i q^{85} -9.90812 q^{89} -10.4497i q^{91} -5.23139 q^{95} +14.8895 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q - 20 q^{25} - 20 q^{31} + 36 q^{49} - 40 q^{73} + 4 q^{79} + 40 q^{97}+O(q^{100}) $ 20 * q - 20 * q^25 - 20 * q^31 + 36 * q^49 - 40 * q^73 + 4 * q^79 + 40 * q^97

Display 100 coefficients 10 coefficients

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
$7$	−2.12726	−0.804027	−0.402014	−	0.915634i	$-0.631690\pi$
$7$	−2.12726	−0.804027	−0.402014	+	0.915634i	$0.631690\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.48036i	1.65239i	0.563384	+	0.826195i	$0.309499\pi$
$11$	5.48036i	1.65239i	−0.563384	+	0.826195i	$0.690501\pi$
$12$	0	0
$13$	4.91228i	1.36242i	0.732088	+	0.681210i	$0.238546\pi$
$13$	4.91228i	1.36242i	−0.732088	+	0.681210i	$0.761454\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.235541	−0.0571270	−0.0285635	−	0.999592i	$-0.509093\pi$
$17$	−0.235541	−0.0571270	−0.0285635	+	0.999592i	$0.509093\pi$
$18$	0	0
$19$	5.23139i	1.20016i	0.799939	+	0.600081i	$0.204865\pi$
$19$	5.23139i	1.20016i	−0.799939	+	0.600081i	$0.795135\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.42390	−1.54799	−0.773995	−	0.633192i	$-0.781744\pi$
$23$	−7.42390	−1.54799	−0.773995	+	0.633192i	$0.781744\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 4.24894i	− 0.789008i	−0.918894	−	0.394504i	$-0.870917\pi$
$29$	− 4.24894i	− 0.789008i	0.918894	−	0.394504i	$-0.129083\pi$
$30$	0	0
$31$	−5.05609	−0.908100	−0.454050	−	0.890976i	$-0.650021\pi$
$31$	−5.05609	−0.908100	−0.454050	+	0.890976i	$0.650021\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.12726i	− 0.359572i
$36$	0	0
$37$	2.27608i	0.374185i	0.982342	+	0.187093i	$0.0599065\pi$
$37$	2.27608i	0.374185i	−0.982342	+	0.187093i	$0.940094\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.26132	0.509332	0.254666	−	0.967029i	$-0.418034\pi$
$41$	3.26132	0.509332	0.254666	+	0.967029i	$0.418034\pi$
$42$	0	0
$43$	− 9.90812i	− 1.51097i	−0.655163	−	0.755487i	$-0.727400\pi$
$43$	− 9.90812i	− 1.51097i	0.655163	−	0.755487i	$-0.272600\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.17359	1.19224	0.596121	−	0.802895i	$-0.296708\pi$
$47$	8.17359	1.19224	0.596121	+	0.802895i	$0.296708\pi$
$48$	0	0
$49$	−2.47478	−0.353540
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.6637i	− 1.46477i	−0.680890	−	0.732386i	$-0.738407\pi$
$53$	− 10.6637i	− 1.46477i	0.680890	−	0.732386i	$-0.261593\pi$
$54$	0	0
$55$	−5.48036	−0.738971
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 0.702988i	− 0.0915212i	−0.998952	−	0.0457606i	$-0.985429\pi$
$59$	− 0.702988i	− 0.0915212i	0.998952	−	0.0457606i	$-0.0145711\pi$
$60$	0	0
$61$	− 0.319109i	− 0.0408577i	−0.999791	−	0.0204288i	$-0.993497\pi$
$61$	− 0.319109i	− 0.0408577i	0.999791	−	0.0204288i	$-0.00650315\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.91228	−0.609293
$66$	0	0
$67$	− 2.70667i	− 0.330672i	−0.986237	−	0.165336i	$-0.947129\pi$
$67$	− 2.70667i	− 0.330672i	0.986237	−	0.165336i	$-0.0528708\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.18836	−0.853101	−0.426551	−	0.904464i	$-0.640271\pi$
$71$	−7.18836	−0.853101	−0.426551	+	0.904464i	$0.640271\pi$
$72$	0	0
$73$	9.66370	1.13105	0.565525	−	0.824731i	$-0.308673\pi$
$73$	9.66370	1.13105	0.565525	+	0.824731i	$0.308673\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.6581i	− 1.32857i
$78$	0	0
$79$	10.8565	1.22146	0.610729	−	0.791840i	$-0.290877\pi$
$79$	10.8565	1.22146	0.610729	+	0.791840i	$0.290877\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.53645i	− 0.936997i	−0.883464	−	0.468498i	$-0.844795\pi$
$83$	− 8.53645i	− 0.936997i	0.883464	−	0.468498i	$-0.155205\pi$
$84$	0	0
$85$	− 0.235541i	− 0.0255480i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.90812	−1.05026	−0.525129	−	0.851022i	$-0.675983\pi$
$89$	−9.90812	−1.05026	−0.525129	+	0.851022i	$0.675983\pi$
$90$	0	0
$91$	− 10.4497i	− 1.09542i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.23139	−0.536729
$96$	0	0
$97$	14.8895	1.51180	0.755902	−	0.654685i	$-0.227199\pi$
$97$	14.8895	1.51180	0.755902	+	0.654685i	$0.227199\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.13283i	0.610240i	0.952314	+	0.305120i	$0.0986965\pi$
$101$	6.13283i	0.610240i	−0.952314	+	0.305120i	$0.901303\pi$
$102$	0	0
$103$	8.82788	0.869837	0.434918	−	0.900470i	$-0.356777\pi$
$103$	8.82788	0.869837	0.434918	+	0.900470i	$0.356777\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.48593i	0.723693i	0.932238	+	0.361846i	$0.117853\pi$
$107$	7.48593i	0.723693i	−0.932238	+	0.361846i	$0.882147\pi$
$108$	0	0
$109$	17.8462i	1.70935i	0.519160	+	0.854677i	$0.326245\pi$
$109$	17.8462i	1.70935i	−0.519160	+	0.854677i	$0.673755\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.8609	−1.39800	−0.698998	−	0.715124i	$-0.746370\pi$
$113$	−14.8609	−1.39800	−0.698998	+	0.715124i	$0.746370\pi$
$114$	0	0
$115$	− 7.42390i	− 0.692282i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.501055	0.0459316
$120$	0	0
$121$	−19.0343	−1.73039
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−6.82788	−0.605876	−0.302938	−	0.953010i	$-0.597968\pi$
$127$	−6.82788	−0.605876	−0.302938	+	0.953010i	$0.597968\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 21.3874i	− 1.86863i	−0.356452	−	0.934314i	$-0.616014\pi$
$131$	− 21.3874i	− 1.86863i	0.356452	−	0.934314i	$-0.383986\pi$
$132$	0	0
$133$	− 11.1285i	− 0.964963i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−19.6081	−1.67523	−0.837617	−	0.546259i	$-0.816052\pi$
$137$	−19.6081	−1.67523	−0.837617	+	0.546259i	$0.816052\pi$
$138$	0	0
$139$	− 0.901666i	− 0.0764784i	−0.999269	−	0.0382392i	$-0.987825\pi$
$139$	− 0.901666i	− 0.0764784i	0.999269	−	0.0382392i	$-0.0121749\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.9210	−2.25125
$144$	0	0
$145$	4.24894	0.352855
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.75106i	0.143453i	0.997424	+	0.0717264i	$0.0228508\pi$
$149$	1.75106i	0.143453i	−0.997424	+	0.0717264i	$0.977149\pi$
$150$	0	0
$151$	−2.90141	−0.236114	−0.118057	−	0.993007i	$-0.537666\pi$
$151$	−2.90141	−0.236114	−0.118057	+	0.993007i	$0.537666\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5.05609i	− 0.406115i
$156$	0	0
$157$	19.7601i	1.57703i	0.615018	+	0.788513i	$0.289149\pi$
$157$	19.7601i	1.57703i	−0.615018	+	0.788513i	$0.710851\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.7925	1.24463
$162$	0	0
$163$	− 21.4946i	− 1.68359i	−0.539799	−	0.841794i	$-0.681500\pi$
$163$	− 21.4946i	− 1.68359i	0.539799	−	0.841794i	$-0.318500\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.0666	−1.47541	−0.737707	−	0.675121i	$-0.764091\pi$
$167$	−19.0666	−1.47541	−0.737707	+	0.675121i	$0.764091\pi$
$168$	0	0
$169$	−11.1305	−0.856190
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.31060i	− 0.707872i	−0.935270	−	0.353936i	$-0.884843\pi$
$173$	− 9.31060i	− 0.707872i	0.935270	−	0.353936i	$-0.115157\pi$
$174$	0	0
$175$	2.12726	0.160805
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 2.70620i	− 0.202271i	−0.994873	−	0.101136i	$-0.967752\pi$
$179$	− 2.70620i	− 0.202271i	0.994873	−	0.101136i	$-0.0322476\pi$
$180$	0	0
$181$	7.42390i	0.551814i	0.961184	+	0.275907i	$0.0889782\pi$
$181$	7.42390i	0.551814i	−0.961184	+	0.275907i	$0.911022\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.27608	−0.167341
$186$	0	0
$187$	− 1.29085i	− 0.0943960i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.2099	−0.955837	−0.477919	−	0.878404i	$-0.658609\pi$
$191$	−13.2099	−0.955837	−0.477919	+	0.878404i	$0.658609\pi$
$192$	0	0
$193$	−3.73487	−0.268842	−0.134421	−	0.990924i	$-0.542917\pi$
$193$	−3.73487	−0.268842	−0.134421	+	0.990924i	$0.542917\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.00321i	0.498958i	0.968380	+	0.249479i	$0.0802594\pi$
$197$	7.00321i	0.498958i	−0.968380	+	0.249479i	$0.919741\pi$
$198$	0	0
$199$	−21.2769	−1.50828	−0.754140	−	0.656714i	$-0.771946\pi$
$199$	−21.2769	−1.50828	−0.754140	+	0.656714i	$0.771946\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.03858i	0.634384i
$204$	0	0
$205$	3.26132i	0.227780i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−28.6699	−1.98314
$210$	0	0
$211$	24.4773i	1.68509i	0.538628	+	0.842544i	$0.318943\pi$
$211$	24.4773i	1.68509i	−0.538628	+	0.842544i	$0.681057\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.90812	0.675728
$216$	0	0
$217$	10.7556	0.730138
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.15704i	− 0.0778310i
$222$	0	0
$223$	16.9862	1.13748	0.568741	−	0.822516i	$-0.307431\pi$
$223$	16.9862	1.13748	0.568741	+	0.822516i	$0.307431\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 20.5203i	− 1.36198i	−0.732294	−	0.680988i	$-0.761550\pi$
$227$	− 20.5203i	− 1.36198i	0.732294	−	0.680988i	$-0.238450\pi$
$228$	0	0
$229$	− 3.01268i	− 0.199083i	−0.995033	−	0.0995416i	$-0.968262\pi$
$229$	− 3.01268i	− 0.199083i	0.995033	−	0.0995416i	$-0.0317376\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.2379	0.998267	0.499134	−	0.866525i	$-0.333652\pi$
$233$	15.2379	0.998267	0.499134	+	0.866525i	$0.333652\pi$
$234$	0	0
$235$	8.17359i	0.533186i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8761	1.02694	0.513470	−	0.858108i	$-0.328360\pi$
$239$	15.8761	1.02694	0.513470	+	0.858108i	$0.328360\pi$
$240$	0	0
$241$	20.9838	1.35169	0.675843	−	0.737046i	$-0.263780\pi$
$241$	20.9838	1.35169	0.675843	+	0.737046i	$0.263780\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.47478i	− 0.158108i
$246$	0	0
$247$	−25.6980	−1.63513
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 8.49787i	− 0.536381i	−0.963366	−	0.268190i	$-0.913574\pi$
$251$	− 8.49787i	− 0.536381i	0.963366	−	0.268190i	$-0.0864256\pi$
$252$	0	0
$253$	− 40.6856i	− 2.55788i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.222442	0.0138756	0.00693778	−	0.999976i	$-0.497792\pi$
$257$	0.222442	0.0138756	0.00693778	+	0.999976i	$0.497792\pi$
$258$	0	0
$259$	− 4.84181i	− 0.300855i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.74301	0.477454	0.238727	−	0.971087i	$-0.423270\pi$
$263$	7.74301	0.477454	0.238727	+	0.971087i	$0.423270\pi$
$264$	0	0
$265$	10.6637	0.655066
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.50903i	0.152978i	0.997070	+	0.0764890i	$0.0243710\pi$
$269$	2.50903i	0.152978i	−0.997070	+	0.0764890i	$0.975629\pi$
$270$	0	0
$271$	7.00558	0.425559	0.212779	−	0.977100i	$-0.431748\pi$
$271$	7.00558	0.425559	0.212779	+	0.977100i	$0.431748\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.48036i	− 0.330478i
$276$	0	0
$277$	− 8.13052i	− 0.488516i	−0.969710	−	0.244258i	$-0.921456\pi$
$277$	− 8.13052i	− 0.488516i	0.969710	−	0.244258i	$-0.0785443\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.2038	−1.20526	−0.602628	−	0.798023i	$-0.705880\pi$
$281$	−20.2038	−1.20526	−0.602628	+	0.798023i	$0.705880\pi$
$282$	0	0
$283$	27.4626i	1.63248i	0.577712	+	0.816241i	$0.303946\pi$
$283$	27.4626i	1.63248i	−0.577712	+	0.816241i	$0.696054\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.93765	−0.409517
$288$	0	0
$289$	−16.9445	−0.996737
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.00321i	0.292291i	0.989263	+	0.146145i	$0.0466867\pi$
$293$	5.00321i	0.292291i	−0.989263	+	0.146145i	$0.953313\pi$
$294$	0	0
$295$	0.702988	0.0409295
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 36.4683i	− 2.10901i
$300$	0	0
$301$	21.0771i	1.21486i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.319109	0.0182721
$306$	0	0
$307$	9.18634i	0.524292i	0.965028	+	0.262146i	$0.0844302\pi$
$307$	9.18634i	0.524292i	−0.965028	+	0.262146i	$0.915570\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.99798	−0.113295	−0.0566475	−	0.998394i	$-0.518041\pi$
$311$	−1.99798	−0.113295	−0.0566475	+	0.998394i	$0.518041\pi$
$312$	0	0
$313$	−10.6052	−0.599440	−0.299720	−	0.954027i	$-0.596893\pi$
$313$	−10.6052	−0.599440	−0.299720	+	0.954027i	$0.596893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.76229i	0.548305i	0.961686	+	0.274152i	$0.0883973\pi$
$317$	9.76229i	0.548305i	−0.961686	+	0.274152i	$0.911603\pi$
$318$	0	0
$319$	23.2857	1.30375
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.23220i	− 0.0685616i
$324$	0	0
$325$	− 4.91228i	− 0.272484i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.3873	−0.958594
$330$	0	0
$331$	5.62096i	0.308956i	0.987996	+	0.154478i	$0.0493696\pi$
$331$	5.62096i	0.308956i	−0.987996	+	0.154478i	$0.950630\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.70667	0.147881
$336$	0	0
$337$	−0.804011	−0.0437973	−0.0218986	−	0.999760i	$-0.506971\pi$
$337$	−0.804011	−0.0437973	−0.0218986	+	0.999760i	$0.506971\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 27.7092i	− 1.50054i
$342$	0	0
$343$	20.1553	1.08828
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.7019i	0.574507i	0.957855	+	0.287253i	$0.0927421\pi$
$347$	10.7019i	0.574507i	−0.957855	+	0.287253i	$0.907258\pi$
$348$	0	0
$349$	− 24.9915i	− 1.33776i	−0.743369	−	0.668881i	$-0.766774\pi$
$349$	− 24.9915i	− 1.33776i	0.743369	−	0.668881i	$-0.233226\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.4860	0.824237	0.412119	−	0.911130i	$-0.364789\pi$
$353$	15.4860	0.824237	0.412119	+	0.911130i	$0.364789\pi$
$354$	0	0
$355$	− 7.18836i	− 0.381518i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.9335	0.629826	0.314913	−	0.949121i	$-0.398025\pi$
$359$	11.9335	0.629826	0.314913	+	0.949121i	$0.398025\pi$
$360$	0	0
$361$	−8.36740	−0.440390
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.66370i	0.505821i
$366$	0	0
$367$	−6.95514	−0.363055	−0.181528	−	0.983386i	$-0.558104\pi$
$367$	−6.95514	−0.363055	−0.181528	+	0.983386i	$0.558104\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	22.6844i	1.17772i
$372$	0	0
$373$	− 11.6296i	− 0.602156i	−0.953600	−	0.301078i	$-0.902654\pi$
$373$	− 11.6296i	− 0.602156i	0.953600	−	0.301078i	$-0.0973464\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.8720	1.07496
$378$	0	0
$379$	0.235541i	0.0120989i	0.999982	+	0.00604945i	$0.00192561\pi$
$379$	0.235541i	0.0120989i	−0.999982	+	0.00604945i	$0.998074\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.23305	−0.216299	−0.108149	−	0.994135i	$-0.534493\pi$
$383$	−4.23305	−0.216299	−0.108149	+	0.994135i	$0.534493\pi$
$384$	0	0
$385$	11.6581	0.594153
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.0991i	1.37398i	0.726667	+	0.686990i	$0.241068\pi$
$389$	27.0991i	1.37398i	−0.726667	+	0.686990i	$0.758932\pi$
$390$	0	0
$391$	1.74863	0.0884320
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.8565i	0.546252i
$396$	0	0
$397$	− 21.9552i	− 1.10190i	−0.834539	−	0.550949i	$-0.814266\pi$
$397$	− 21.9552i	− 1.10190i	0.834539	−	0.550949i	$-0.185734\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.36813	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$401$	−8.36813	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$402$	0	0
$403$	− 24.8369i	− 1.23721i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.4737	−0.618300
$408$	0	0
$409$	−28.2474	−1.39675	−0.698373	−	0.715734i	$-0.746092\pi$
$409$	−28.2474	−1.39675	−0.698373	+	0.715734i	$0.746092\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.49544i	0.0735856i
$414$	0	0
$415$	8.53645	0.419038
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.68443i	0.424262i	0.977241	+	0.212131i	$0.0680404\pi$
$419$	8.68443i	0.424262i	−0.977241	+	0.212131i	$0.931960\pi$
$420$	0	0
$421$	4.73033i	0.230542i	0.993334	+	0.115271i	$0.0367737\pi$
$421$	4.73033i	0.230542i	−0.993334	+	0.115271i	$0.963226\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.235541	0.0114254
$426$	0	0
$427$	0.678826i	0.0328507i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−39.1052	−1.88363	−0.941817	−	0.336127i	$-0.890883\pi$
$431$	−39.1052	−1.88363	−0.941817	+	0.336127i	$0.890883\pi$
$432$	0	0
$433$	33.8221	1.62538	0.812692	−	0.582693i	$-0.198001\pi$
$433$	33.8221	1.62538	0.812692	+	0.582693i	$0.198001\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 38.8373i	− 1.85784i
$438$	0	0
$439$	−1.59960	−0.0763448	−0.0381724	−	0.999271i	$-0.512154\pi$
$439$	−1.59960	−0.0763448	−0.0381724	+	0.999271i	$0.512154\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.64862i	0.410909i	0.978667	+	0.205454i	$0.0658672\pi$
$443$	8.64862i	0.410909i	−0.978667	+	0.205454i	$0.934133\pi$
$444$	0	0
$445$	− 9.90812i	− 0.469690i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.9597	−0.753184	−0.376592	−	0.926379i	$-0.622904\pi$
$449$	−15.9597	−0.753184	−0.376592	+	0.926379i	$0.622904\pi$
$450$	0	0
$451$	17.8732i	0.841615i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.4497	0.489888
$456$	0	0
$457$	−22.6462	−1.05934	−0.529672	−	0.848203i	$-0.677685\pi$
$457$	−22.6462	−1.05934	−0.529672	+	0.848203i	$0.677685\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 22.4400i	− 1.04513i	−0.852599	−	0.522566i	$-0.824975\pi$
$461$	− 22.4400i	− 1.04513i	0.852599	−	0.522566i	$-0.175025\pi$
$462$	0	0
$463$	−30.8710	−1.43470	−0.717348	−	0.696715i	$-0.754644\pi$
$463$	−30.8710	−1.43470	−0.717348	+	0.696715i	$0.754644\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.05734i	− 0.141477i	−0.997495	−	0.0707383i	$-0.977464\pi$
$467$	− 3.05734i	− 0.141477i	0.997495	−	0.0707383i	$-0.0225355\pi$
$468$	0	0
$469$	5.75777i	0.265869i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	54.3001	2.49672
$474$	0	0
$475$	− 5.23139i	− 0.240032i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.4262	1.02468	0.512341	−	0.858782i	$-0.328779\pi$
$479$	22.4262	1.02468	0.512341	+	0.858782i	$0.328779\pi$
$480$	0	0
$481$	−11.1807	−0.509798
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.8895i	0.676100i
$486$	0	0
$487$	11.9961	0.543594	0.271797	−	0.962355i	$-0.412382\pi$
$487$	11.9961	0.543594	0.271797	+	0.962355i	$0.412382\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.261916i	0.0118201i	0.999983	+	0.00591005i	$0.00188124\pi$
$491$	0.261916i	0.0118201i	−0.999983	+	0.00591005i	$0.998119\pi$
$492$	0	0
$493$	1.00080i	0.0450736i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.2915	0.685917
$498$	0	0
$499$	43.0171i	1.92571i	0.270018	+	0.962855i	$0.412970\pi$
$499$	43.0171i	1.92571i	−0.270018	+	0.962855i	$0.587030\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.3295	−0.951037	−0.475518	−	0.879706i	$-0.657740\pi$
$503$	−21.3295	−0.951037	−0.475518	+	0.879706i	$0.657740\pi$
$504$	0	0
$505$	−6.13283	−0.272908
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 15.0919i	− 0.668936i	−0.942407	−	0.334468i	$-0.891443\pi$
$509$	− 15.0919i	− 0.668936i	0.942407	−	0.334468i	$-0.108557\pi$
$510$	0	0
$511$	−20.5572	−0.909396
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.82788i	0.389003i
$516$	0	0
$517$	44.7942i	1.97005i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−33.8443	−1.48274	−0.741372	−	0.671094i	$-0.765825\pi$
$521$	−33.8443	−1.48274	−0.741372	+	0.671094i	$0.765825\pi$
$522$	0	0
$523$	− 10.5034i	− 0.459281i	−0.973276	−	0.229640i	$-0.926245\pi$
$523$	− 10.5034i	− 0.459281i	0.973276	−	0.229640i	$-0.0737550\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.19091	0.0518770
$528$	0	0
$529$	32.1143	1.39627
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.0205i	0.693924i
$534$	0	0
$535$	−7.48593	−0.323645
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 13.5627i	− 0.584186i
$540$	0	0
$541$	18.2058i	0.782728i	0.920236	+	0.391364i	$0.127997\pi$
$541$	18.2058i	0.782728i	−0.920236	+	0.391364i	$0.872003\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.8462	−0.764446
$546$	0	0
$547$	7.81348i	0.334080i	0.985950	+	0.167040i	$0.0534209\pi$
$547$	7.81348i	0.334080i	−0.985950	+	0.167040i	$0.946579\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.2278	0.946937
$552$	0	0
$553$	−23.0947	−0.982085
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.7679i	− 0.625735i	−0.949797	−	0.312867i	$-0.898710\pi$
$557$	− 14.7679i	− 0.625735i	0.949797	−	0.312867i	$-0.101290\pi$
$558$	0	0
$559$	48.6715	2.05858
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.5041i	1.24345i	0.783236	+	0.621724i	$0.213567\pi$
$563$	29.5041i	1.24345i	−0.783236	+	0.621724i	$0.786433\pi$
$564$	0	0
$565$	− 14.8609i	− 0.625203i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−29.0862	−1.21935	−0.609677	−	0.792650i	$-0.708701\pi$
$569$	−29.0862	−1.21935	−0.609677	+	0.792650i	$0.708701\pi$
$570$	0	0
$571$	42.1417i	1.76357i	0.471649	+	0.881787i	$0.343659\pi$
$571$	42.1417i	1.76357i	−0.471649	+	0.881787i	$0.656341\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.42390	0.309598
$576$	0	0
$577$	−33.8258	−1.40819	−0.704094	−	0.710107i	$-0.748646\pi$
$577$	−33.8258	−1.40819	−0.704094	+	0.710107i	$0.748646\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	18.1592i	0.753371i
$582$	0	0
$583$	58.4409	2.42037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.36693i	− 0.0976937i	−0.998806	−	0.0488469i	$-0.984445\pi$
$587$	− 2.36693i	− 0.0976937i	0.998806	−	0.0488469i	$-0.0155546\pi$
$588$	0	0
$589$	− 26.4504i	− 1.08987i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.0133	−0.575459	−0.287729	−	0.957712i	$-0.592900\pi$
$593$	−14.0133	−0.575459	−0.287729	+	0.957712i	$0.592900\pi$
$594$	0	0
$595$	0.501055i	0.0205413i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.194646	0.00795304	0.00397652	−	0.999992i	$-0.498734\pi$
$599$	0.194646	0.00795304	0.00397652	+	0.999992i	$0.498734\pi$
$600$	0	0
$601$	−40.3962	−1.64780	−0.823898	−	0.566738i	$-0.808205\pi$
$601$	−40.3962	−1.64780	−0.823898	+	0.566738i	$0.808205\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 19.0343i	− 0.773855i
$606$	0	0
$607$	−34.9823	−1.41989	−0.709944	−	0.704258i	$-0.751280\pi$
$607$	−34.9823	−1.41989	−0.709944	+	0.704258i	$0.751280\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.1510i	1.62433i
$612$	0	0
$613$	6.46784i	0.261233i	0.991433	+	0.130617i	$0.0416957\pi$
$613$	6.46784i	0.261233i	−0.991433	+	0.130617i	$0.958304\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.9261	−0.842451	−0.421226	−	0.906956i	$-0.638400\pi$
$617$	−20.9261	−0.842451	−0.421226	+	0.906956i	$0.638400\pi$
$618$	0	0
$619$	11.1568i	0.448430i	0.974540	+	0.224215i	$0.0719818\pi$
$619$	11.1568i	0.448430i	−0.974540	+	0.224215i	$0.928018\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	21.0771	0.844437
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 0.536109i	− 0.0213761i
$630$	0	0
$631$	−4.79278	−0.190798	−0.0953989	−	0.995439i	$-0.530413\pi$
$631$	−4.79278	−0.190798	−0.0953989	+	0.995439i	$0.530413\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 6.82788i	− 0.270956i
$636$	0	0
$637$	− 12.1568i	− 0.481670i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.9892	1.69797	0.848986	−	0.528416i	$-0.177214\pi$
$641$	42.9892	1.69797	0.848986	+	0.528416i	$0.177214\pi$
$642$	0	0
$643$	− 28.0578i	− 1.10649i	−0.833017	−	0.553247i	$-0.813389\pi$
$643$	− 28.0578i	− 1.10649i	0.833017	−	0.553247i	$-0.186611\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.30186	−0.208437	−0.104219	−	0.994554i	$-0.533234\pi$
$647$	−5.30186	−0.208437	−0.104219	+	0.994554i	$0.533234\pi$
$648$	0	0
$649$	3.85263	0.151229
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 10.6725i	− 0.417647i	−0.977953	−	0.208823i	$-0.933037\pi$
$653$	− 10.6725i	− 0.417647i	0.977953	−	0.208823i	$-0.0669634\pi$
$654$	0	0
$655$	21.3874	0.835676
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.6338i	0.609008i	0.952511	+	0.304504i	$0.0984907\pi$
$659$	15.6338i	0.609008i	−0.952511	+	0.304504i	$0.901509\pi$
$660$	0	0
$661$	35.2867i	1.37249i	0.727369	+	0.686247i	$0.240743\pi$
$661$	35.2867i	1.37249i	−0.727369	+	0.686247i	$0.759257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.1285	0.431545
$666$	0	0
$667$	31.5437i	1.22138i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.74883	0.0675128
$672$	0	0
$673$	43.2233	1.66614	0.833068	−	0.553170i	$-0.186582\pi$
$673$	43.2233	1.66614	0.833068	+	0.553170i	$0.186582\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.8297i	1.53078i	0.643566	+	0.765391i	$0.277454\pi$
$677$	39.8297i	1.53078i	−0.643566	+	0.765391i	$0.722546\pi$
$678$	0	0
$679$	−31.6739	−1.21553
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.7286i	− 0.563574i	−0.959477	−	0.281787i	$-0.909073\pi$
$683$	− 14.7286i	− 0.563574i	0.959477	−	0.281787i	$-0.0909271\pi$
$684$	0	0
$685$	− 19.6081i	− 0.749187i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52.3831	1.99564
$690$	0	0
$691$	31.2491i	1.18877i	0.804180	+	0.594386i	$0.202605\pi$
$691$	31.2491i	1.18877i	−0.804180	+	0.594386i	$0.797395\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.901666	0.0342022
$696$	0	0
$697$	−0.768172	−0.0290966
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.52968i	0.171084i	0.996335	+	0.0855419i	$0.0272621\pi$
$701$	4.52968i	0.171084i	−0.996335	+	0.0855419i	$0.972738\pi$
$702$	0	0
$703$	−11.9071	−0.449083
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 13.0461i	− 0.490649i
$708$	0	0
$709$	− 15.1260i	− 0.568069i	−0.958814	−	0.284035i	$-0.908327\pi$
$709$	− 15.1260i	− 0.568069i	0.958814	−	0.284035i	$-0.0916731\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37.5359	1.40573
$714$	0	0
$715$	− 26.9210i	− 1.00679i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.8773	0.480243	0.240122	−	0.970743i	$-0.422813\pi$
$719$	12.8773	0.480243	0.240122	+	0.970743i	$0.422813\pi$
$720$	0	0
$721$	−18.7792	−0.699373
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.24894i	0.157802i
$726$	0	0
$727$	−7.50213	−0.278239	−0.139119	−	0.990276i	$-0.544427\pi$
$727$	−7.50213	−0.278239	−0.139119	+	0.990276i	$0.544427\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.33376i	0.0863174i
$732$	0	0
$733$	− 12.4046i	− 0.458174i	−0.973406	−	0.229087i	$-0.926426\pi$
$733$	− 12.4046i	− 0.458174i	0.973406	−	0.229087i	$-0.0735740\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.8335	0.546399
$738$	0	0
$739$	− 49.6664i	− 1.82701i	−0.406830	−	0.913504i	$-0.633366\pi$
$739$	− 49.6664i	− 1.82701i	0.406830	−	0.913504i	$-0.366634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.1312	−0.958662	−0.479331	−	0.877634i	$-0.659121\pi$
$743$	−26.1312	−0.958662	−0.479331	+	0.877634i	$0.659121\pi$
$744$	0	0
$745$	−1.75106	−0.0641540
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 15.9245i	− 0.581869i
$750$	0	0
$751$	−42.6818	−1.55748	−0.778741	−	0.627346i	$-0.784141\pi$
$751$	−42.6818	−1.55748	−0.778741	+	0.627346i	$0.784141\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.90141i	− 0.105593i
$756$	0	0
$757$	− 37.3850i	− 1.35878i	−0.733777	−	0.679391i	$-0.762244\pi$
$757$	− 37.3850i	− 1.35878i	0.733777	−	0.679391i	$-0.237756\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.2038	0.732386	0.366193	−	0.930539i	$-0.380661\pi$
$761$	20.2038	0.732386	0.366193	+	0.930539i	$0.380661\pi$
$762$	0	0
$763$	− 37.9634i	− 1.37437i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.45327	0.124690
$768$	0	0
$769$	28.9265	1.04312	0.521558	−	0.853216i	$-0.325351\pi$
$769$	28.9265	1.04312	0.521558	+	0.853216i	$0.325351\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 47.0349i	− 1.69173i	−0.533400	−	0.845863i	$-0.679086\pi$
$773$	− 47.0349i	− 1.69173i	0.533400	−	0.845863i	$-0.320914\pi$
$774$	0	0
$775$	5.05609	0.181620
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.0612i	0.611281i
$780$	0	0
$781$	− 39.3948i	− 1.40966i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.7601	−0.705267
$786$	0	0
$787$	27.5698i	0.982756i	0.870946	+	0.491378i	$0.163507\pi$
$787$	27.5698i	0.982756i	−0.870946	+	0.491378i	$0.836493\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	31.6129	1.12403
$792$	0	0
$793$	1.56755	0.0556653
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.4890i	1.64672i	0.567516	+	0.823362i	$0.307904\pi$
$797$	46.4890i	1.64672i	−0.567516	+	0.823362i	$0.692096\pi$
$798$	0	0
$799$	−1.92521	−0.0681091
$800$	0	0
$801$	0	0
$802$	0	0
$803$	52.9605i	1.86894i
$804$	0	0
$805$	15.7925i	0.556614i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.2189	1.66013	0.830064	−	0.557668i	$-0.188304\pi$
$809$	47.2189	1.66013	0.830064	+	0.557668i	$0.188304\pi$
$810$	0	0
$811$	9.83765i	0.345447i	0.984970	+	0.172723i	$0.0552567\pi$
$811$	9.83765i	0.345447i	−0.984970	+	0.172723i	$0.944743\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.4946	0.752923
$816$	0	0
$817$	51.8332	1.81341
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 35.6781i	− 1.24517i	−0.782551	−	0.622587i	$-0.786082\pi$
$821$	− 35.6781i	− 1.24517i	0.782551	−	0.622587i	$-0.213918\pi$
$822$	0	0
$823$	23.7092	0.826450	0.413225	−	0.910629i	$-0.364402\pi$
$823$	23.7092	0.826450	0.413225	+	0.910629i	$0.364402\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9.64454i	− 0.335373i	−0.985840	−	0.167687i	$-0.946370\pi$
$827$	− 9.64454i	− 0.335373i	0.985840	−	0.167687i	$-0.0536297\pi$
$828$	0	0
$829$	30.3639i	1.05458i	0.849685	+	0.527291i	$0.176792\pi$
$829$	30.3639i	1.05458i	−0.849685	+	0.527291i	$0.823208\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.582911	0.0201967
$834$	0	0
$835$	− 19.0666i	− 0.659825i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.58047	0.192659	0.0963296	−	0.995349i	$-0.469290\pi$
$839$	5.58047	0.192659	0.0963296	+	0.995349i	$0.469290\pi$
$840$	0	0
$841$	10.9465	0.377467
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 11.1305i	− 0.382900i
$846$	0	0
$847$	40.4909	1.39128
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 16.8974i	− 0.579235i
$852$	0	0
$853$	− 50.6473i	− 1.73413i	−0.498193	−	0.867066i	$-0.666003\pi$
$853$	− 50.6473i	− 1.73413i	0.498193	−	0.867066i	$-0.333997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.1906	0.655539	0.327769	−	0.944758i	$-0.393703\pi$
$857$	19.1906	0.655539	0.327769	+	0.944758i	$0.393703\pi$
$858$	0	0
$859$	1.31864i	0.0449915i	0.999747	+	0.0224957i	$0.00716122\pi$
$859$	1.31864i	0.0449915i	−0.999747	+	0.0224957i	$0.992839\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	12.3923	0.421840	0.210920	−	0.977503i	$-0.432354\pi$
$863$	12.3923	0.421840	0.210920	+	0.977503i	$0.432354\pi$
$864$	0	0
$865$	9.31060	0.316570
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.4978i	2.01832i
$870$	0	0
$871$	13.2959	0.450514
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.12726i	0.0719144i
$876$	0	0
$877$	36.8292i	1.24363i	0.783163	+	0.621816i	$0.213605\pi$
$877$	36.8292i	1.24363i	−0.783163	+	0.621816i	$0.786395\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	21.4110	0.721356	0.360678	−	0.932690i	$-0.382545\pi$
$881$	21.4110	0.721356	0.360678	+	0.932690i	$0.382545\pi$
$882$	0	0
$883$	− 1.59479i	− 0.0536690i	−0.999640	−	0.0268345i	$-0.991457\pi$
$883$	− 1.59479i	− 0.0536690i	0.999640	−	0.0268345i	$-0.00854271\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.6129	0.960727	0.480363	−	0.877070i	$-0.340505\pi$
$887$	28.6129	0.960727	0.480363	+	0.877070i	$0.340505\pi$
$888$	0	0
$889$	14.5247	0.487141
$890$	0	0
$891$	0	0
$892$	0	0
$893$	42.7592i	1.43088i
$894$	0	0
$895$	2.70620	0.0904584
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.4830i	0.716498i
$900$	0	0
$901$	2.51173i	0.0836780i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.42390	−0.246779
$906$	0	0
$907$	− 39.3412i	− 1.30630i	−0.757228	−	0.653151i	$-0.773447\pi$
$907$	− 39.3412i	− 1.30630i	0.757228	−	0.653151i	$-0.226553\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.6919	1.11626	0.558131	−	0.829753i	$-0.311519\pi$
$911$	33.6919	1.11626	0.558131	+	0.829753i	$0.311519\pi$
$912$	0	0
$913$	46.7828	1.54828
$914$	0	0
$915$	0	0
$916$	0	0
$917$	45.4965i	1.50243i
$918$	0	0
$919$	−39.2196	−1.29373	−0.646867	−	0.762603i	$-0.723921\pi$
$919$	−39.2196	−1.29373	−0.646867	+	0.762603i	$0.723921\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 35.3112i	− 1.16228i
$924$	0	0
$925$	− 2.27608i	− 0.0748371i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.1242	−1.48048	−0.740239	−	0.672344i	$-0.765288\pi$
$929$	−45.1242	−1.48048	−0.740239	+	0.672344i	$0.765288\pi$
$930$	0	0
$931$	− 12.9465i	− 0.424305i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.29085	0.0422152
$936$	0	0
$937$	−56.9008	−1.85887	−0.929435	−	0.368986i	$-0.879705\pi$
$937$	−56.9008	−1.85887	−0.929435	+	0.368986i	$0.879705\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	56.6437i	1.84653i	0.384163	+	0.923265i	$0.374490\pi$
$941$	56.6437i	1.84653i	−0.384163	+	0.923265i	$0.625510\pi$
$942$	0	0
$943$	−24.2117	−0.788441
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.1789i	0.623230i	0.950208	+	0.311615i	$0.100870\pi$
$947$	19.1789i	0.623230i	−0.950208	+	0.311615i	$0.899130\pi$
$948$	0	0
$949$	47.4708i	1.54097i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.3512	−0.950777	−0.475389	−	0.879776i	$-0.657693\pi$
$953$	−29.3512	−0.950777	−0.475389	+	0.879776i	$0.657693\pi$
$954$	0	0
$955$	− 13.2099i	− 0.427463i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	41.7115	1.34693
$960$	0	0
$961$	−5.43596	−0.175354
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3.73487i	− 0.120230i
$966$	0	0
$967$	57.6440	1.85371	0.926854	−	0.375423i	$-0.122503\pi$
$967$	57.6440	1.85371	0.926854	+	0.375423i	$0.122503\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 6.42670i	− 0.206243i	−0.994669	−	0.103121i	$-0.967117\pi$
$971$	− 6.42670i	− 0.206243i	0.994669	−	0.103121i	$-0.0328830\pi$
$972$	0	0
$973$	1.91808i	0.0614907i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.6408	0.948293	0.474147	−	0.880446i	$-0.342757\pi$
$977$	29.6408	0.948293	0.474147	+	0.880446i	$0.342757\pi$
$978$	0	0
$979$	− 54.3001i	− 1.73544i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.4287	−0.651573	−0.325787	−	0.945443i	$-0.605629\pi$
$983$	−20.4287	−0.651573	−0.325787	+	0.945443i	$0.605629\pi$
$984$	0	0
$985$	−7.00321	−0.223141
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.5569i	2.33897i
$990$	0	0
$991$	−43.0135	−1.36637	−0.683184	−	0.730246i	$-0.739405\pi$
$991$	−43.0135	−1.36637	−0.683184	+	0.730246i	$0.739405\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 21.2769i	− 0.674523i
$996$	0	0
$997$	51.4261i	1.62868i	0.580387	+	0.814341i	$0.302901\pi$
$997$	51.4261i	1.62868i	−0.580387	+	0.814341i	$0.697099\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.k.d.2161.14		20
3.2	odd	2	inner	4320.2.k.d.2161.4		20
4.3	odd	2		1080.2.k.d.541.15	yes	20
8.3	odd	2		1080.2.k.d.541.16	yes	20
8.5	even	2	inner	4320.2.k.d.2161.3		20
12.11	even	2		1080.2.k.d.541.6	yes	20
24.5	odd	2	inner	4320.2.k.d.2161.13		20
24.11	even	2		1080.2.k.d.541.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.k.d.541.5	✓	20	24.11	even	2
1080.2.k.d.541.6	yes	20	12.11	even	2
1080.2.k.d.541.15	yes	20	4.3	odd	2
1080.2.k.d.541.16	yes	20	8.3	odd	2
4320.2.k.d.2161.3		20	8.5	even	2	inner
4320.2.k.d.2161.4		20	3.2	odd	2	inner
4320.2.k.d.2161.13		20	24.5	odd	2	inner
4320.2.k.d.2161.14		20	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{5} -2.12726 q^{7} +O(q^{10})\)	\(q+1.00000i q^{5} -2.12726 q^{7} +5.48036i q^{11} +4.91228i q^{13} -0.235541 q^{17} +5.23139i q^{19} -7.42390 q^{23} -1.00000 q^{25} -4.24894i q^{29} -5.05609 q^{31} -2.12726i q^{35} +2.27608i q^{37} +3.26132 q^{41} -9.90812i q^{43} +8.17359 q^{47} -2.47478 q^{49} -10.6637i q^{53} -5.48036 q^{55} -0.702988i q^{59} -0.319109i q^{61} -4.91228 q^{65} -2.70667i q^{67} -7.18836 q^{71} +9.66370 q^{73} -11.6581i q^{77} +10.8565 q^{79} -8.53645i q^{83} -0.235541i q^{85} -9.90812 q^{89} -10.4497i q^{91} -5.23139 q^{95} +14.8895 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q - 20 q^{25} - 20 q^{31} + 36 q^{49} - 40 q^{73} + 4 q^{79} + 40 q^{97}+O(q^{100}) \) 20 * q - 20 * q^25 - 20 * q^31 + 36 * q^49 - 40 * q^73 + 4 * q^79 + 40 * q^97

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Database

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