LMFDB - Embedded newform 3840.2.k.s.1921.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1921,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, 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("3840.1921");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1921.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.1921
Dual form			3840.2.k.s.1921.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -2.00000i q^{13} +1.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -4.00000 q^{31} +10.0000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} -7.00000 q^{49} -6.00000i q^{51} -2.00000i q^{53} -4.00000 q^{57} -8.00000i q^{59} +2.00000i q^{61} +2.00000 q^{65} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +14.0000 q^{73} +1.00000i q^{75} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +6.00000i q^{85} +2.00000 q^{87} +14.0000 q^{89} +4.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 2 q^{15} + 12 q^{17} + 16 q^{23} - 2 q^{25} - 8 q^{31} - 4 q^{39} - 4 q^{41} - 16 q^{47} - 14 q^{49} - 8 q^{57} + 4 q^{65} + 16 q^{71} + 28 q^{73} + 24 q^{79} + 2 q^{81} + 4 q^{87} + 28 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^9 + 2 * q^15 + 12 * q^17 + 16 * q^23 - 2 * q^25 - 8 * q^31 - 4 * q^39 - 4 * q^41 - 16 * q^47 - 14 * q^49 - 8 * q^57 + 4 * q^65 + 16 * q^71 + 28 * q^73 + 24 * q^79 + 2 * q^81 + 4 * q^87 + 28 * q^89 + 8 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$.

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 6.00000i	− 0.840168i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	0	0
$61$	2.00000i	0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$	2.00000i	0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	2.00000i	0.180334i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	− 8.00000i	− 0.698963i	−0.936943	−	0.349482i	$-0.886358\pi$
$131$	− 8.00000i	− 0.698963i	0.936943	−	0.349482i	$-0.113642\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	− 20.0000i	− 1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	− 20.0000i	− 1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$	0	0
$141$	8.00000i	0.673722i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	− 10.0000i	− 0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$	− 10.0000i	− 0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 4.00000i	− 0.321288i
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000i	0.305888i
$172$	0	0
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	16.0000i	1.19590i	0.801535	+	0.597948i	$0.204017\pi$
$179$	16.0000i	1.19590i	−0.801535	+	0.597948i	$0.795983\pi$
$180$	0	0
$181$	14.0000i	1.04061i	0.853980	+	0.520306i	$0.174182\pi$
$181$	14.0000i	1.04061i	−0.853980	+	0.520306i	$0.825818\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	− 2.00000i	− 0.143223i
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 2.00000i	− 0.139686i
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.0000i	1.92760i	0.266627	+	0.963800i	$0.414091\pi$
$211$	28.0000i	1.92760i	−0.266627	+	0.963800i	$0.585909\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 14.0000i	− 0.946032i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	6.00000i	0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$	6.00000i	0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	− 12.0000i	− 0.779484i
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 7.00000i	− 0.447214i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	24.0000i	1.51487i	0.652913	+	0.757433i	$0.273547\pi$
$251$	24.0000i	1.51487i	−0.652913	+	0.757433i	$0.726453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	− 30.0000i	− 1.82913i	−0.404436	−	0.914566i	$-0.632532\pi$
$269$	− 30.0000i	− 1.82913i	0.404436	−	0.914566i	$-0.367468\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	0	0
$285$	− 4.00000i	− 0.236940i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16.0000i	− 0.925304i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	− 8.00000i	− 0.455104i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000i	1.46031i	0.683284	+	0.730153i	$0.260551\pi$
$317$	26.0000i	1.46031i	−0.683284	+	0.730153i	$0.739449\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	2.00000i	0.110940i
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	0	0
$333$	− 10.0000i	− 0.547997i
$334$	0	0
$335$	12.0000	0.655630
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	10.0000i	0.543125i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	2.00000i	0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$	2.00000i	0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	8.00000i	0.424596i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	14.0000i	0.732793i
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	− 16.0000i	− 0.819705i
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	12.0000i	0.603786i
$396$	0	0
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	22.0000i	1.08518i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	− 32.0000i	− 1.56330i	−0.623716	−	0.781651i	$-0.714378\pi$
$419$	− 32.0000i	− 1.56330i	0.623716	−	0.781651i	$-0.285622\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	2.00000i	0.0958927i
$436$	0	0
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	14.0000i	0.663664i
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	6.00000i	0.280056i
$460$	0	0
$461$	− 14.0000i	− 0.652045i	−0.945362	−	0.326023i	$-0.894291\pi$
$461$	− 14.0000i	− 0.652045i	0.945362	−	0.326023i	$-0.105709\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.00000i	0.183533i
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000i	0.0908153i
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	0	0
$491$	16.0000i	0.722070i	0.932552	+	0.361035i	$0.117576\pi$
$491$	16.0000i	0.722070i	−0.932552	+	0.361035i	$0.882424\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 30.0000i	− 1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$	− 30.0000i	− 1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	8.00000i	0.352522i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.0000	−1.04546
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	8.00000i	0.347170i
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	0	0
$540$	0	0
$541$	42.0000i	1.80572i	0.429934	+	0.902861i	$0.358537\pi$
$541$	42.0000i	1.80572i	−0.429934	+	0.902861i	$0.641463\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	− 2.00000i	− 0.0853579i
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	0	0
$555$	10.0000i	0.424476i
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	− 10.0000i	− 0.420703i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	− 10.0000i	− 0.415586i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	16.0000i	0.659269i
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0000i	0.818546i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	12.0000i	0.488678i
$604$	0	0
$605$	11.0000i	0.447214i
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000i	0.647291i
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	8.00000i	0.321029i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	60.0000i	2.39236i
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	0	0
$633$	28.0000	1.11290
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	14.0000i	0.554700i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	4.00000i	0.157500i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	24.0000i	0.934907i	0.884018	+	0.467454i	$0.154829\pi$
$659$	24.0000i	0.934907i	−0.884018	+	0.467454i	$0.845171\pi$
$660$	0	0
$661$	38.0000i	1.47803i	0.673690	+	0.739014i	$0.264708\pi$
$661$	38.0000i	1.47803i	−0.673690	+	0.739014i	$0.735292\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	− 22.0000i	− 0.840577i
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	28.0000i	1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$	28.0000i	1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	6.00000i	0.226941i
$700$	0	0
$701$	− 30.0000i	− 1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$	− 30.0000i	− 1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 2.00000i	− 0.0743808i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	0	0
$735$	−7.00000	−0.258199
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 28.0000i	− 1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$	− 28.0000i	− 1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$	0	0
$741$	8.00000i	0.293887i
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	20.0000i	0.727875i
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.00000i	0.286630i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	44.0000i	1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$	44.0000i	1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	− 2.00000i	− 0.0709327i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	36.0000i	1.26413i	0.774915	+	0.632065i	$0.217793\pi$
$811$	36.0000i	1.26413i	−0.774915	+	0.632065i	$0.782207\pi$
$812$	0	0
$813$	4.00000i	0.140286i
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 42.0000i	− 1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$	− 42.0000i	− 1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	0	0
$829$	− 6.00000i	− 0.208389i	−0.994557	−	0.104194i	$-0.966774\pi$
$829$	− 6.00000i	− 0.208389i	0.994557	−	0.104194i	$-0.0332264\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	−42.0000	−1.45521
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 4.00000i	− 0.138260i
$838$	0	0
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	2.00000i	0.0688837i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	80.0000i	2.74236i
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	20.0000i	0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$	20.0000i	0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−16.0000	−0.534224
$898$	0	0
$899$	− 8.00000i	− 0.266815i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.00000i	0.0661180i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	− 10.0000i	− 0.328798i
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	28.0000i	0.917663i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	2.00000i	0.0652675i
$940$	0	0
$941$	18.0000i	0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$	18.0000i	0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000i	0.909878i	0.890523	+	0.454939i	$0.150339\pi$
$947$	28.0000i	0.909878i	−0.890523	+	0.454939i	$0.849661\pi$
$948$	0	0
$949$	− 28.0000i	− 0.908918i
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	24.0000i	0.776622i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	10.0000i	0.321911i
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	− 8.00000i	− 0.256732i	−0.991727	−	0.128366i	$-0.959027\pi$
$971$	− 8.00000i	− 0.256732i	0.991727	−	0.128366i	$-0.0409733\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 10.0000i	− 0.319275i
$982$	0	0
$983$	−56.0000	−1.78612	−0.893061	−	0.449935i	$-0.851447\pi$
$983$	−56.0000	−1.78612	−0.893061	+	0.449935i	$0.851447\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.0000i	1.01754i
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	− 20.0000i	− 0.634043i
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.s.1921.1		2
4.3	odd	2		3840.2.k.n.1921.2		2
8.3	odd	2		3840.2.k.n.1921.1		2
8.5	even	2	inner	3840.2.k.s.1921.2		2
16.3	odd	4		480.2.a.d.1.1	✓	1
16.5	even	4		960.2.a.b.1.1		1
16.11	odd	4		960.2.a.k.1.1		1
16.13	even	4		480.2.a.g.1.1	yes	1
48.5	odd	4		2880.2.a.z.1.1		1
48.11	even	4		2880.2.a.ba.1.1		1
48.29	odd	4		1440.2.a.d.1.1		1
48.35	even	4		1440.2.a.c.1.1		1
80.3	even	4		2400.2.f.l.1249.1		2
80.13	odd	4		2400.2.f.g.1249.2		2
80.19	odd	4		2400.2.a.z.1.1		1
80.27	even	4		4800.2.f.o.3649.1		2
80.29	even	4		2400.2.a.i.1.1		1
80.37	odd	4		4800.2.f.v.3649.2		2
80.43	even	4		4800.2.f.o.3649.2		2
80.53	odd	4		4800.2.f.v.3649.1		2
80.59	odd	4		4800.2.a.s.1.1		1
80.67	even	4		2400.2.f.l.1249.2		2
80.69	even	4		4800.2.a.cb.1.1		1
80.77	odd	4		2400.2.f.g.1249.1		2
240.29	odd	4		7200.2.a.ba.1.1		1
240.77	even	4		7200.2.f.k.6049.1		2
240.83	odd	4		7200.2.f.s.6049.2		2
240.173	even	4		7200.2.f.k.6049.2		2
240.179	even	4		7200.2.a.z.1.1		1
240.227	odd	4		7200.2.f.s.6049.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.a.d.1.1	✓	1	16.3	odd	4
480.2.a.g.1.1	yes	1	16.13	even	4
960.2.a.b.1.1		1	16.5	even	4
960.2.a.k.1.1		1	16.11	odd	4
1440.2.a.c.1.1		1	48.35	even	4
1440.2.a.d.1.1		1	48.29	odd	4
2400.2.a.i.1.1		1	80.29	even	4
2400.2.a.z.1.1		1	80.19	odd	4
2400.2.f.g.1249.1		2	80.77	odd	4
2400.2.f.g.1249.2		2	80.13	odd	4
2400.2.f.l.1249.1		2	80.3	even	4
2400.2.f.l.1249.2		2	80.67	even	4
2880.2.a.z.1.1		1	48.5	odd	4
2880.2.a.ba.1.1		1	48.11	even	4
3840.2.k.n.1921.1		2	8.3	odd	2
3840.2.k.n.1921.2		2	4.3	odd	2
3840.2.k.s.1921.1		2	1.1	even	1	trivial
3840.2.k.s.1921.2		2	8.5	even	2	inner
4800.2.a.s.1.1		1	80.59	odd	4
4800.2.a.cb.1.1		1	80.69	even	4
4800.2.f.o.3649.1		2	80.27	even	4
4800.2.f.o.3649.2		2	80.43	even	4
4800.2.f.v.3649.1		2	80.53	odd	4
4800.2.f.v.3649.2		2	80.37	odd	4
7200.2.a.z.1.1		1	240.179	even	4
7200.2.a.ba.1.1		1	240.29	odd	4
7200.2.f.k.6049.1		2	240.77	even	4
7200.2.f.k.6049.2		2	240.173	even	4
7200.2.f.s.6049.1		2	240.227	odd	4
7200.2.f.s.6049.2		2	240.83	odd	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +1.00000i q^{5} -1.00000 q^{9} -2.00000i q^{13} +1.00000 q^{15} +6.00000 q^{17} -4.00000i q^{19} +8.00000 q^{23} -1.00000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -4.00000 q^{31} +10.0000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} -1.00000i q^{45} -8.00000 q^{47} -7.00000 q^{49} -6.00000i q^{51} -2.00000i q^{53} -4.00000 q^{57} -8.00000i q^{59} +2.00000i q^{61} +2.00000 q^{65} -12.0000i q^{67} -8.00000i q^{69} +8.00000 q^{71} +14.0000 q^{73} +1.00000i q^{75} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +6.00000i q^{85} +2.00000 q^{87} +14.0000 q^{89} +4.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 2 q^{15} + 12 q^{17} + 16 q^{23} - 2 q^{25} - 8 q^{31} - 4 q^{39} - 4 q^{41} - 16 q^{47} - 14 q^{49} - 8 q^{57} + 4 q^{65} + 16 q^{71} + 28 q^{73} + 24 q^{79} + 2 q^{81} + 4 q^{87} + 28 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^9 + 2 * q^15 + 12 * q^17 + 16 * q^23 - 2 * q^25 - 8 * q^31 - 4 * q^39 - 4 * q^41 - 16 * q^47 - 14 * q^49 - 8 * q^57 + 4 * q^65 + 16 * q^71 + 28 * q^73 + 24 * q^79 + 2 * q^81 + 4 * q^87 + 28 * q^89 + 8 * q^95 + 4 * q^97

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