LMFDB - Embedded newform 3840.2.k.e.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:	$1$
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 1920)
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.e.1921.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{11} +4.00000i q^{13} +1.00000 q^{15} +4.00000 q^{17} -8.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} +2.00000 q^{31} -6.00000 q^{33} -4.00000i q^{35} +4.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} +9.00000 q^{49} -4.00000i q^{51} -14.0000i q^{53} +6.00000 q^{55} -8.00000 q^{57} +6.00000i q^{59} +6.00000i q^{61} +4.00000 q^{63} -4.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +2.00000 q^{73} +1.00000i q^{75} +24.0000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} -16.0000i q^{91} -2.00000i q^{93} +8.00000 q^{95} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 2 q^{9} + 2 q^{15} + 8 q^{17} - 2 q^{25} + 4 q^{31} - 12 q^{33} + 8 q^{39} - 12 q^{41} + 18 q^{49} + 12 q^{55} - 16 q^{57} + 8 q^{63} - 8 q^{65} - 16 q^{71} + 4 q^{73} - 12 q^{79} + 2 q^{81}+ \cdots - 28 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 2 * q^9 + 2 * q^15 + 8 * q^17 - 2 * q^25 + 4 * q^31 - 12 * q^33 + 8 * q^39 - 12 * q^41 + 18 * q^49 + 12 * q^55 - 16 * q^57 + 8 * q^63 - 8 * q^65 - 16 * q^71 + 4 * q^73 - 12 * q^79 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 16 * q^95 - 28 * q^97

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	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	− 6.00000i	− 1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	4.00000i	0.872872i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	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	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	− 4.00000i	− 0.676123i
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	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
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	− 4.00000i	− 0.560112i
$52$	0	0
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	6.00000i	0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$	6.00000i	0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	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
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	24.0000i	2.73505i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	4.00000i	0.433861i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	− 16.0000i	− 1.67726i
$92$	0	0
$93$	− 2.00000i	− 0.207390i
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	6.00000i	0.603023i
$100$	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
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	18.0000i	1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$	18.0000i	1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	−25.0000	−2.27273
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	− 2.00000i	− 0.174741i	−0.996176	−	0.0873704i	$-0.972154\pi$
$131$	− 2.00000i	− 0.174741i	0.996176	−	0.0873704i	$-0.0278464\pi$
$132$	0	0
$133$	32.0000i	2.77475i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	8.00000i	0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$	8.00000i	0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	2.00000i	0.160644i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−14.0000	−1.11027
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	− 6.00000i	− 0.467099i
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	8.00000i	0.611775i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	0	0
$181$	− 18.0000i	− 1.33793i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	− 18.0000i	− 1.33793i	0.743294	−	0.668965i	$-0.233262\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	4.00000i	0.286446i
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	− 6.00000i	− 0.419058i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−48.0000	−3.32023
$210$	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
$217$	−8.00000	−0.543075
$218$	0	0
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	16.0000i	1.07628i
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.00000i	0.389742i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	9.00000i	0.574989i
$246$	0	0
$247$	32.0000	2.03611
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	− 18.0000i	− 1.13615i	−0.822977	−	0.568075i	$-0.807688\pi$
$251$	− 18.0000i	− 1.13615i	0.822977	−	0.568075i	$-0.192312\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	4.00000	0.249513	0.124757	−	0.992187i	$-0.460185\pi$
$257$	4.00000	0.249513	0.124757	+	0.992187i	$0.460185\pi$
$258$	0	0
$259$	− 16.0000i	− 0.994192i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	14.0000	0.860013
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	30.0000i	1.82913i	0.404436	+	0.914566i	$0.367468\pi$
$269$	30.0000i	1.82913i	−0.404436	+	0.914566i	$0.632532\pi$
$270$	0	0
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	−16.0000	−0.968364
$274$	0	0
$275$	6.00000i	0.361814i
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 8.00000i	− 0.473879i
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 48.0000i	− 2.76667i
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−6.00000	−0.343559
$306$	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
$309$	16.0000i	0.910208i
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	4.00000i	0.225374i
$316$	0	0
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	− 32.0000i	− 1.78053i
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	18.0000	0.995402
$328$	0	0
$329$	0	0
$330$	0	0
$331$	12.0000i	0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$	12.0000i	0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	0	0
$339$	12.0000i	0.651751i
$340$	0	0
$341$	− 12.0000i	− 0.649836i
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	− 2.00000i	− 0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$	− 2.00000i	− 0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−16.0000	−0.851594	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000	−0.851594	−0.425797	+	0.904819i	$0.640006\pi$
$354$	0	0
$355$	− 8.00000i	− 0.424596i
$356$	0	0
$357$	16.0000i	0.846810i
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	25.0000i	1.31216i
$364$	0	0
$365$	2.00000i	0.104685i
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	56.0000i	2.90738i
$372$	0	0
$373$	32.0000i	1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$	32.0000i	1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	− 12.0000i	− 0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$	− 12.0000i	− 0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	0	0
$381$	16.0000i	0.819705i
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	− 12.0000i	− 0.609994i
$388$	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
$391$	0	0
$392$	0	0
$393$	−2.00000	−0.100887
$394$	0	0
$395$	− 6.00000i	− 0.301893i
$396$	0	0
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	32.0000	1.60200
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	− 8.00000i	− 0.394611i
$412$	0	0
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	34.0000i	1.66101i	0.557012	+	0.830504i	$0.311948\pi$
$419$	34.0000i	1.66101i	−0.557012	+	0.830504i	$0.688052\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	0	0
$429$	− 24.0000i	− 1.15873i
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 6.00000i	− 0.284427i
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	0	0
$451$	36.0000i	1.69517i
$452$	0	0
$453$	6.00000i	0.281905i
$454$	0	0
$455$	16.0000	0.750092
$456$	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
$459$	4.00000i	0.186704i
$460$	0	0
$461$	− 10.0000i	− 0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$	− 10.0000i	− 0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	16.0000i	0.738811i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	72.0000	3.31056
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	14.0000i	0.641016i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.0000i	− 0.635707i
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	26.0000i	1.17336i	0.809818	+	0.586682i	$0.199566\pi$
$491$	26.0000i	1.17336i	−0.809818	+	0.586682i	$0.800434\pi$
$492$	0	0
$493$	− 8.00000i	− 0.360302i
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	26.0000i	1.15243i	0.817298	+	0.576215i	$0.195471\pi$
$509$	26.0000i	1.15243i	−0.817298	+	0.576215i	$0.804529\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 6.00000i	− 0.260378i
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	− 54.0000i	− 2.32594i
$540$	0	0
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	− 6.00000i	− 0.256074i
$550$	0	0
$551$	−16.0000	−0.681623
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	4.00000i	0.169791i
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	− 28.0000i	− 1.17176i	−0.810397	−	0.585882i	$-0.800748\pi$
$571$	− 28.0000i	− 1.17176i	0.810397	−	0.585882i	$-0.199252\pi$
$572$	0	0
$573$	12.0000i	0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	− 14.0000i	− 0.581820i
$580$	0	0
$581$	− 48.0000i	− 1.99138i
$582$	0	0
$583$	−84.0000	−3.47892
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	16.0000i	0.660391i	0.943913	+	0.330195i	$0.107115\pi$
$587$	16.0000i	0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$	0	0
$589$	− 16.0000i	− 0.659269i
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−4.00000	−0.164260	−0.0821302	−	0.996622i	$-0.526172\pi$
$593$	−4.00000	−0.164260	−0.0821302	+	0.996622i	$0.526172\pi$
$594$	0	0
$595$	− 16.0000i	− 0.655936i
$596$	0	0
$597$	2.00000i	0.0818546i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 25.0000i	− 1.01639i
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 4.00000i	− 0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$	− 4.00000i	− 0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	24.0000i	0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$	24.0000i	0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	48.0000i	1.91694i
$628$	0	0
$629$	16.0000i	0.637962i
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	− 16.0000i	− 0.634941i
$636$	0	0
$637$	36.0000i	1.42637i
$638$	0	0
$639$	8.00000	0.316475
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	12.0000i	0.472500i
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	8.00000i	0.313545i
$652$	0	0
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	2.00000	0.0781465
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	− 10.0000i	− 0.389545i	−0.980848	−	0.194772i	$-0.937603\pi$
$659$	− 10.0000i	− 0.389545i	0.980848	−	0.194772i	$-0.0623968\pi$
$660$	0	0
$661$	− 38.0000i	− 1.47803i	−0.673690	−	0.739014i	$-0.735292\pi$
$661$	− 38.0000i	− 1.47803i	0.673690	−	0.739014i	$-0.264708\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	−32.0000	−1.24091
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 16.0000i	− 0.618596i
$670$	0	0
$671$	36.0000	1.38976
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	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
$679$	56.0000	2.14908
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	8.00000i	0.305664i
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	56.0000	2.13343
$690$	0	0
$691$	− 16.0000i	− 0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$	− 16.0000i	− 0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	0	0
$693$	− 24.0000i	− 0.911685i
$694$	0	0
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	− 8.00000i	− 0.302588i
$700$	0	0
$701$	− 38.0000i	− 1.43524i	−0.696435	−	0.717620i	$-0.745231\pi$
$701$	− 38.0000i	− 1.43524i	0.696435	−	0.717620i	$-0.254769\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.0000i	1.50435i
$708$	0	0
$709$	− 38.0000i	− 1.42712i	−0.700594	−	0.713560i	$-0.747082\pi$
$709$	− 38.0000i	− 1.42712i	0.700594	−	0.713560i	$-0.252918\pi$
$710$	0	0
$711$	6.00000	0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	24.0000i	0.897549i
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	0	0
$721$	64.0000	2.38348
$722$	0	0
$723$	18.0000i	0.669427i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	48.0000i	1.77534i
$732$	0	0
$733$	− 40.0000i	− 1.47743i	−0.674016	−	0.738717i	$-0.735432\pi$
$733$	− 40.0000i	− 1.47743i	0.674016	−	0.738717i	$-0.264568\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	− 44.0000i	− 1.61857i	−0.587419	−	0.809283i	$-0.699856\pi$
$739$	− 44.0000i	− 1.61857i	0.587419	−	0.809283i	$-0.300144\pi$
$740$	0	0
$741$	− 32.0000i	− 1.17555i
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	− 32.0000i	− 1.16925i
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	− 6.00000i	− 0.218362i
$756$	0	0
$757$	48.0000i	1.74459i	0.488980	+	0.872295i	$0.337369\pi$
$757$	48.0000i	1.74459i	−0.488980	+	0.872295i	$0.662631\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	− 72.0000i	− 2.60658i
$764$	0	0
$765$	− 4.00000i	− 0.144620i
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	− 4.00000i	− 0.144056i
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	48.0000i	1.71978i
$780$	0	0
$781$	48.0000i	1.71758i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.0000i	0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$	12.0000i	0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$	0	0
$789$	− 16.0000i	− 0.569615i
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	− 14.0000i	− 0.496529i
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	− 52.0000i	− 1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$	− 52.0000i	− 1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	0	0
$813$	22.0000i	0.771574i
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	96.0000	3.35861
$818$	0	0
$819$	16.0000i	0.559085i
$820$	0	0
$821$	− 54.0000i	− 1.88461i	−0.334751	−	0.942306i	$-0.608652\pi$
$821$	− 54.0000i	− 1.88461i	0.334751	−	0.942306i	$-0.391348\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	2.00000i	0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$	2.00000i	0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	0	0
$833$	36.0000	1.24733
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.00000i	0.0691301i
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	− 3.00000i	− 0.103203i
$846$	0	0
$847$	100.000	3.43604
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 28.0000i	− 0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$	− 28.0000i	− 0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	28.0000i	0.955348i	0.878537	+	0.477674i	$0.158520\pi$
$859$	28.0000i	0.955348i	−0.878537	+	0.477674i	$0.841480\pi$
$860$	0	0
$861$	− 24.0000i	− 0.817918i
$862$	0	0
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	36.0000i	1.22122i
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	14.0000	0.473828
$874$	0	0
$875$	4.00000i	0.135225i
$876$	0	0
$877$	32.0000i	1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$	32.0000i	1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	−58.0000	−1.95407	−0.977035	−	0.213080i	$-0.931651\pi$
$881$	−58.0000	−1.95407	−0.977035	+	0.213080i	$0.931651\pi$
$882$	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
$885$	6.00000i	0.201688i
$886$	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
$889$	64.0000	2.14649
$890$	0	0
$891$	− 6.00000i	− 0.201008i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.00000i	− 0.133407i
$900$	0	0
$901$	− 56.0000i	− 1.86563i
$902$	0	0
$903$	−48.0000	−1.59734
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	0	0
$915$	6.00000i	0.198354i
$916$	0	0
$917$	8.00000i	0.264183i
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	− 32.0000i	− 1.05329i
$924$	0	0
$925$	− 4.00000i	− 0.131519i
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	− 72.0000i	− 2.35970i
$932$	0	0
$933$	− 12.0000i	− 0.392862i
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	46.0000	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$938$	0	0
$939$	6.00000i	0.195803i
$940$	0	0
$941$	− 14.0000i	− 0.456387i	−0.973616	−	0.228193i	$-0.926718\pi$
$941$	− 14.0000i	− 0.456387i	0.973616	−	0.228193i	$-0.0732819\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	0	0
$949$	8.00000i	0.259691i
$950$	0	0
$951$	14.0000	0.453981
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	− 12.0000i	− 0.388311i
$956$	0	0
$957$	12.0000i	0.387905i
$958$	0	0
$959$	−32.0000	−1.03333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	− 8.00000i	− 0.257796i
$964$	0	0
$965$	14.0000i	0.450676i
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	−32.0000	−1.02799
$970$	0	0
$971$	30.0000i	0.962746i	0.876516	+	0.481373i	$0.159862\pi$
$971$	30.0000i	0.962746i	−0.876516	+	0.481373i	$0.840138\pi$
$972$	0	0
$973$	− 32.0000i	− 1.02587i
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	4.00000	0.127971	0.0639857	−	0.997951i	$-0.479619\pi$
$977$	4.00000	0.127971	0.0639857	+	0.997951i	$0.479619\pi$
$978$	0	0
$979$	36.0000i	1.15056i
$980$	0	0
$981$	− 18.0000i	− 0.574696i
$982$	0	0
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	30.0000	0.952981	0.476491	−	0.879180i	$-0.341909\pi$
$991$	30.0000	0.952981	0.476491	+	0.879180i	$0.341909\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	− 2.00000i	− 0.0634043i
$996$	0	0
$997$	− 24.0000i	− 0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$	− 24.0000i	− 0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$	0	0
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.k.e.1921.1		2
4.3	odd	2		3840.2.k.x.1921.2		2
8.3	odd	2		3840.2.k.x.1921.1		2
8.5	even	2	inner	3840.2.k.e.1921.2		2
16.3	odd	4		1920.2.a.g.1.1	yes	1
16.5	even	4		1920.2.a.f.1.1	✓	1
16.11	odd	4		1920.2.a.m.1.1	yes	1
16.13	even	4		1920.2.a.x.1.1	yes	1
48.5	odd	4		5760.2.a.bu.1.1		1
48.11	even	4		5760.2.a.z.1.1		1
48.29	odd	4		5760.2.a.x.1.1		1
48.35	even	4		5760.2.a.a.1.1		1
80.19	odd	4		9600.2.a.cd.1.1		1
80.29	even	4		9600.2.a.a.1.1		1
80.59	odd	4		9600.2.a.y.1.1		1
80.69	even	4		9600.2.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.f.1.1	✓	1	16.5	even	4
1920.2.a.g.1.1	yes	1	16.3	odd	4
1920.2.a.m.1.1	yes	1	16.11	odd	4
1920.2.a.x.1.1	yes	1	16.13	even	4
3840.2.k.e.1921.1		2	1.1	even	1	trivial
3840.2.k.e.1921.2		2	8.5	even	2	inner
3840.2.k.x.1921.1		2	8.3	odd	2
3840.2.k.x.1921.2		2	4.3	odd	2
5760.2.a.a.1.1		1	48.35	even	4
5760.2.a.x.1.1		1	48.29	odd	4
5760.2.a.z.1.1		1	48.11	even	4
5760.2.a.bu.1.1		1	48.5	odd	4
9600.2.a.a.1.1		1	80.29	even	4
9600.2.a.y.1.1		1	80.59	odd	4
9600.2.a.bf.1.1		1	80.69	even	4
9600.2.a.cd.1.1		1	80.19	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.00000i q^{5} -4.00000 q^{7} -1.00000 q^{9} -6.00000i q^{11} +4.00000i q^{13} +1.00000 q^{15} +4.00000 q^{17} -8.00000i q^{19} +4.00000i q^{21} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} +2.00000 q^{31} -6.00000 q^{33} -4.00000i q^{35} +4.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} +12.0000i q^{43} -1.00000i q^{45} +9.00000 q^{49} -4.00000i q^{51} -14.0000i q^{53} +6.00000 q^{55} -8.00000 q^{57} +6.00000i q^{59} +6.00000i q^{61} +4.00000 q^{63} -4.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} +2.00000 q^{73} +1.00000i q^{75} +24.0000i q^{77} -6.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +4.00000i q^{85} -2.00000 q^{87} -6.00000 q^{89} -16.0000i q^{91} -2.00000i q^{93} +8.00000 q^{95} -14.0000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 2 q^{9} + 2 q^{15} + 8 q^{17} - 2 q^{25} + 4 q^{31} - 12 q^{33} + 8 q^{39} - 12 q^{41} + 18 q^{49} + 12 q^{55} - 16 q^{57} + 8 q^{63} - 8 q^{65} - 16 q^{71} + 4 q^{73} - 12 q^{79} + 2 q^{81}+ \cdots - 28 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 2 * q^9 + 2 * q^15 + 8 * q^17 - 2 * q^25 + 4 * q^31 - 12 * q^33 + 8 * q^39 - 12 * q^41 + 18 * q^49 + 12 * q^55 - 16 * q^57 + 8 * q^63 - 8 * q^65 - 16 * q^71 + 4 * q^73 - 12 * q^79 + 2 * q^81 - 4 * q^87 - 12 * q^89 + 16 * q^95 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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