LMFDB - Embedded newform 3840.2.k.w.1921.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} -8.00000 q^{31} +4.00000i q^{35} +6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} -12.0000 q^{47} +9.00000 q^{49} -6.00000i q^{51} -10.0000i q^{53} +8.00000i q^{59} +10.0000i q^{61} -4.00000 q^{63} -2.00000 q^{65} +12.0000i q^{67} +4.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -6.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} +8.00000i q^{91} -8.00000i q^{93} +18.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q + 8 * q^7 - 2 * q^9	$ 2 q + 8 q^{7} - 2 q^{9} - 2 q^{15} - 12 q^{17} + 8 q^{23} - 2 q^{25} - 16 q^{31} - 4 q^{39} + 12 q^{41} - 24 q^{47} + 18 q^{49} - 8 q^{63} - 4 q^{65} - 16 q^{71} - 20 q^{73} + 32 q^{79} + 2 q^{81} - 4 q^{87} + 12 q^{89} + 36 q^{97}+O(q^{100}) $ 2 * q + 8 * q^7 - 2 * q^9 - 2 * q^15 - 12 * q^17 + 8 * q^23 - 2 * q^25 - 16 * q^31 - 4 * q^39 + 12 * q^41 - 24 * q^47 + 18 * q^49 - 8 * q^63 - 4 * q^65 - 16 * q^71 - 20 * q^73 + 32 * q^79 + 2 * q^81 - 4 * q^87 + 12 * q^89 + 36 * 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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	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	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000i	0.676123i
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	− 6.00000i	− 0.840168i
$52$	0	0
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	4.00000i	0.481543i
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	− 6.00000i	− 0.650791i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	8.00000i	0.838628i
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	− 12.0000i	− 1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	9.00000i	0.742307i
$148$	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
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	− 8.00000i	− 0.642575i
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	24.0000i	1.79384i	0.442189	+	0.896922i	$0.354202\pi$
$179$	24.0000i	1.79384i	−0.442189	+	0.896922i	$0.645798\pi$
$180$	0	0
$181$	− 2.00000i	− 0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$	− 2.00000i	− 0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	− 2.00000i	− 0.143223i
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	6.00000i	0.419058i
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 8.00000i	− 0.550743i	−0.961338	−	0.275371i	$-0.911199\pi$
$211$	− 8.00000i	− 0.550743i	0.961338	−	0.275371i	$-0.0888008\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	−32.0000	−2.17230
$218$	0	0
$219$	− 10.0000i	− 0.675737i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	0	0
$229$	− 2.00000i	− 0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$	− 2.00000i	− 0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	− 12.0000i	− 0.782794i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	9.00000i	0.574989i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 16.0000i	− 1.00991i	−0.863145	−	0.504956i	$-0.831509\pi$
$251$	− 16.0000i	− 1.00991i	0.863145	−	0.504956i	$-0.168491\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.00000	0.375735
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	24.0000i	1.49129i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	2.00000i	0.121942i	0.998140	+	0.0609711i	$0.0194197\pi$
$269$	2.00000i	0.121942i	−0.998140	+	0.0609711i	$0.980580\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	18.0000i	1.05518i
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000i	0.462652i
$300$	0	0
$301$	48.0000i	2.76667i
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−10.0000	−0.572598
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	− 4.00000i	− 0.227552i
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 2.00000i	− 0.110940i
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	−48.0000	−2.64633
$330$	0	0
$331$	− 8.00000i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$	− 8.00000i	− 0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	0	0
$349$	− 6.00000i	− 0.321173i	−0.987022	−	0.160586i	$-0.948662\pi$
$349$	− 6.00000i	− 0.321173i	0.987022	−	0.160586i	$-0.0513385\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	− 10.0000i	− 0.523424i
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	− 40.0000i	− 2.07670i
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	− 4.00000i	− 0.204926i
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	0	0
$389$	− 26.0000i	− 1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$	− 26.0000i	− 1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.0000i	0.805047i
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	14.0000i	0.690569i
$412$	0	0
$413$	32.0000i	1.57462i
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	40.0000i	1.95413i	0.212946	+	0.977064i	$0.431694\pi$
$419$	40.0000i	1.95413i	−0.212946	+	0.977064i	$0.568306\pi$
$420$	0	0
$421$	− 18.0000i	− 0.877266i	−0.898666	−	0.438633i	$-0.855463\pi$
$421$	− 18.0000i	− 0.877266i	0.898666	−	0.438633i	$-0.144537\pi$
$422$	0	0
$423$	12.0000	0.583460
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	40.0000i	1.93574i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	6.00000i	0.284427i
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 8.00000i	− 0.375873i
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	6.00000i	0.280056i
$460$	0	0
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	48.0000i	2.21643i
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	16.0000i	0.728025i
$484$	0	0
$485$	18.0000i	0.817338i
$486$	0	0
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	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
$497$	−32.0000	−1.43540
$498$	0	0
$499$	− 32.0000i	− 1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	− 32.0000i	− 1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$	0	0
$501$	12.0000i	0.536120i
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	9.00000i	0.399704i
$508$	0	0
$509$	2.00000i	0.0886484i	0.999017	+	0.0443242i	$0.0141135\pi$
$509$	2.00000i	0.0886484i	−0.999017	+	0.0443242i	$0.985887\pi$
$510$	0	0
$511$	−40.0000	−1.76950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.00000i	− 0.176261i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	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
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	− 8.00000i	− 0.347170i
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	−24.0000	−1.03568
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 38.0000i	− 1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$	− 38.0000i	− 1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	0	0
$549$	− 10.0000i	− 0.426790i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	64.0000	2.72156
$554$	0	0
$555$	− 6.00000i	− 0.254686i
$556$	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
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	− 6.00000i	− 0.252422i
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	− 40.0000i	− 1.67395i	−0.547243	−	0.836974i	$-0.684323\pi$
$571$	− 40.0000i	− 1.67395i	0.547243	−	0.836974i	$-0.315677\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	2.00000i	0.0831172i
$580$	0	0
$581$	− 48.0000i	− 1.99138i
$582$	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
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	− 24.0000i	− 0.983904i
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	11.0000i	0.447214i
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	− 24.0000i	− 0.970936i
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	− 16.0000i	− 0.643094i	−0.946894	−	0.321547i	$-0.895797\pi$
$619$	− 16.0000i	− 0.643094i	0.946894	−	0.321547i	$-0.104203\pi$
$620$	0	0
$621$	− 4.00000i	− 0.160514i
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 36.0000i	− 1.43541i
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	− 4.00000i	− 0.158735i
$636$	0	0
$637$	18.0000i	0.713186i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	36.0000i	1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$	36.0000i	1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$	0	0
$645$	− 12.0000i	− 0.472500i
$646$	0	0
$647$	−4.00000	−0.157256	−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000	−0.157256	−0.0786281	+	0.996904i	$0.525054\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 32.0000i	− 1.25418i
$652$	0	0
$653$	50.0000i	1.95665i	0.207072	+	0.978326i	$0.433606\pi$
$653$	50.0000i	1.95665i	−0.207072	+	0.978326i	$0.566394\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	− 24.0000i	− 0.934907i	−0.884018	−	0.467454i	$-0.845171\pi$
$659$	− 24.0000i	− 0.934907i	0.884018	−	0.467454i	$-0.154829\pi$
$660$	0	0
$661$	14.0000i	0.544537i	0.962221	+	0.272268i	$0.0877739\pi$
$661$	14.0000i	0.544537i	−0.962221	+	0.272268i	$0.912226\pi$
$662$	0	0
$663$	12.0000	0.466041
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	− 28.0000i	− 1.08254i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	72.0000	2.76311
$680$	0	0
$681$	−28.0000	−1.07296
$682$	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
$685$	14.0000i	0.534913i
$686$	0	0
$687$	2.00000	0.0763048
$688$	0	0
$689$	20.0000	0.761939
$690$	0	0
$691$	− 40.0000i	− 1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$	− 40.0000i	− 1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 18.0000i	− 0.680823i
$700$	0	0
$701$	18.0000i	0.679851i	0.940452	+	0.339925i	$0.110402\pi$
$701$	18.0000i	0.679851i	−0.940452	+	0.339925i	$0.889598\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	30.0000i	1.12667i	0.826227	+	0.563337i	$0.190483\pi$
$709$	30.0000i	1.12667i	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 72.0000i	− 2.66302i
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.00000	0.146746	0.0733729	−	0.997305i	$-0.476624\pi$
$743$	4.00000	0.146746	0.0733729	+	0.997305i	$0.476624\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	16.0000i	0.584627i
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	16.0000	0.583072
$754$	0	0
$755$	− 8.00000i	− 0.291150i
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	− 24.0000i	− 0.868858i
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	− 22.0000i	− 0.792311i
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−18.0000	−0.642448
$786$	0	0
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	0	0
$789$	4.00000i	0.142404i
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−20.0000	−0.710221
$794$	0	0
$795$	10.0000i	0.354663i
$796$	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
$799$	72.0000	2.54718
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	16.0000i	0.563926i
$806$	0	0
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	− 40.0000i	− 1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$	− 40.0000i	− 1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 8.00000i	− 0.279543i
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	10.0000i	0.347314i	0.984806	+	0.173657i	$0.0555585\pi$
$829$	10.0000i	0.347314i	−0.984806	+	0.173657i	$0.944442\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	−54.0000	−1.87099
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	8.00000i	0.276520i
$838$	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
$843$	− 10.0000i	− 0.344418i
$844$	0	0
$845$	9.00000i	0.309609i
$846$	0	0
$847$	44.0000	1.51186
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	24.0000i	0.822709i
$852$	0	0
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	0	0
$859$	32.0000i	1.09183i	0.837842	+	0.545913i	$0.183817\pi$
$859$	32.0000i	1.09183i	−0.837842	+	0.545913i	$0.816183\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	28.0000	0.953131	0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000	0.953131	0.476566	+	0.879139i	$0.341881\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−18.0000	−0.609208
$874$	0	0
$875$	− 4.00000i	− 0.135225i
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	0	0
$885$	− 8.00000i	− 0.268917i
$886$	0	0
$887$	28.0000	0.940148	0.470074	−	0.882627i	$-0.344227\pi$
$887$	28.0000	0.940148	0.470074	+	0.882627i	$0.344227\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	60.0000i	1.99889i
$902$	0	0
$903$	−48.0000	−1.59734
$904$	0	0
$905$	2.00000	0.0664822
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	0	0
$909$	− 6.00000i	− 0.199007i
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 10.0000i	− 0.330590i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	4.00000	0.131377
$928$	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
$931$	0	0
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	22.0000i	0.717943i
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	− 20.0000i	− 0.649227i
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56.0000	1.80833
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	2.00000i	0.0643823i
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6.00000i	0.191565i
$982$	0	0
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	− 48.0000i	− 1.52786i
$988$	0	0
$989$	48.0000i	1.52631i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	8.00000	0.253872
$994$	0	0
$995$	− 16.0000i	− 0.507234i
$996$	0	0
$997$	− 42.0000i	− 1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$	− 42.0000i	− 1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{13} -1.00000 q^{15} -6.00000 q^{17} +4.00000i q^{21} +4.00000 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} -8.00000 q^{31} +4.00000i q^{35} +6.00000i q^{37} -2.00000 q^{39} +6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} -12.0000 q^{47} +9.00000 q^{49} -6.00000i q^{51} -10.0000i q^{53} +8.00000i q^{59} +10.0000i q^{61} -4.00000 q^{63} -2.00000 q^{65} +12.0000i q^{67} +4.00000i q^{69} -8.00000 q^{71} -10.0000 q^{73} -1.00000i q^{75} +16.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -6.00000i q^{85} -2.00000 q^{87} +6.00000 q^{89} +8.00000i q^{91} -8.00000i q^{93} +18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q + 8 * q^7 - 2 * q^9	\( 2 q + 8 q^{7} - 2 q^{9} - 2 q^{15} - 12 q^{17} + 8 q^{23} - 2 q^{25} - 16 q^{31} - 4 q^{39} + 12 q^{41} - 24 q^{47} + 18 q^{49} - 8 q^{63} - 4 q^{65} - 16 q^{71} - 20 q^{73} + 32 q^{79} + 2 q^{81} - 4 q^{87} + 12 q^{89} + 36 q^{97}+O(q^{100}) \) 2 * q + 8 * q^7 - 2 * q^9 - 2 * q^15 - 12 * q^17 + 8 * q^23 - 2 * q^25 - 16 * q^31 - 4 * q^39 + 12 * q^41 - 24 * q^47 + 18 * q^49 - 8 * q^63 - 4 * q^65 - 16 * q^71 - 20 * q^73 + 32 * q^79 + 2 * q^81 - 4 * q^87 + 12 * q^89 + 36 * q^97

Introduction

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