LMFDB - Embedded newform 4800.2.f.q.3649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(3649,4800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.f (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:	$38.3281929702$
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 2400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4800.3649
Dual form			4800.2.f.q.3649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -1.00000i q^{13} -3.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} -1.00000i q^{27} +4.00000 q^{29} +7.00000 q^{31} -6.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} +9.00000i q^{43} +6.00000i q^{47} +6.00000 q^{49} -2.00000i q^{53} -3.00000i q^{57} -10.0000 q^{59} +1.00000 q^{61} -1.00000i q^{63} +3.00000i q^{67} +4.00000 q^{69} +14.0000 q^{71} +10.0000i q^{73} +8.00000 q^{79} +1.00000 q^{81} +18.0000i q^{83} +4.00000i q^{87} +1.00000 q^{91} +7.00000i q^{93} +3.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 6 q^{19} - 2 q^{21} + 8 q^{29} + 14 q^{31} + 2 q^{39} + 12 q^{41} + 12 q^{49} - 20 q^{59} + 2 q^{61} + 8 q^{69} + 28 q^{71} + 16 q^{79} + 2 q^{81} + 2 q^{91}+O(q^{100}) $ 2 * q - 2 * q^9 - 6 * q^19 - 2 * q^21 + 8 * q^29 + 14 * q^31 + 2 * q^39 + 12 * q^41 + 12 * q^49 - 20 * q^59 + 2 * q^61 + 8 * q^69 + 28 * q^71 + 16 * q^79 + 2 * q^81 + 2 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	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
$43$	9.00000i	1.37249i	0.727372	+	0.686244i	$0.240742\pi$
$43$	9.00000i	1.37249i	−0.727372	+	0.686244i	$0.759258\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	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
$57$	− 3.00000i	− 0.397360i
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	− 1.00000i	− 0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.00000i	0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$	3.00000i	0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	18.0000i	1.97576i	0.155230	+	0.987878i	$0.450388\pi$
$83$	18.0000i	1.97576i	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000i	0.428845i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	7.00000i	0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.00000i	0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$	3.00000i	0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	0	0
$129$	−9.00000	−0.792406
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	− 3.00000i	− 0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.00000i	0.494872i
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 9.00000i	− 0.718278i	−0.933284	−	0.359139i	$-0.883070\pi$
$157$	− 9.00000i	− 0.718278i	0.933284	−	0.359139i	$-0.116930\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.0000i	− 0.751646i
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	−3.00000	−0.211604
$202$	0	0
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.00000i	0.278019i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.00000	0.481900	0.240950	−	0.970538i	$-0.422541\pi$
$211$	7.00000	0.481900	0.240950	+	0.970538i	$0.422541\pi$
$212$	0	0
$213$	14.0000i	0.959264i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.00000i	0.475191i
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.0000i	0.736614i	0.929704	+	0.368307i	$0.120063\pi$
$223$	11.0000i	0.736614i	−0.929704	+	0.368307i	$0.879937\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	0	0
$229$	−23.0000	−1.51988	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−23.0000	−1.51988	−0.759941	+	0.649992i	$0.774772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.00000i	0.190885i
$248$	0	0
$249$	−18.0000	−1.14070
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	16.0000i	0.998053i	0.866587	+	0.499026i	$0.166309\pi$
$257$	16.0000i	0.998053i	−0.866587	+	0.499026i	$0.833691\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	2.00000i	0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$	2.00000i	0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	1.00000i	0.0605228i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 11.0000i	− 0.660926i	−0.943819	−	0.330463i	$-0.892795\pi$
$277$	− 11.0000i	− 0.660926i	0.943819	−	0.330463i	$-0.107205\pi$
$278$	0	0
$279$	−7.00000	−0.419079
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	− 5.00000i	− 0.297219i	−0.988896	−	0.148610i	$-0.952520\pi$
$283$	− 5.00000i	− 0.297219i	0.988896	−	0.148610i	$-0.0474798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	−3.00000	−0.175863
$292$	0	0
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−9.00000	−0.518751
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.0000i	0.741949i	0.928643	+	0.370975i	$0.120976\pi$
$307$	13.0000i	0.741949i	−0.928643	+	0.370975i	$0.879024\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	0	0
$313$	23.0000i	1.30004i	0.759918	+	0.650018i	$0.225239\pi$
$313$	23.0000i	1.30004i	−0.759918	+	0.650018i	$0.774761\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 15.0000i	− 0.829502i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.0000i	1.03500i	0.855684	+	0.517498i	$0.173136\pi$
$337$	19.0000i	1.03500i	−0.855684	+	0.517498i	$0.826864\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	0	0
$342$	0	0
$343$	13.0000i	0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00000i	0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$	2.00000i	0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	− 36.0000i	− 1.91609i	−0.286623	−	0.958043i	$-0.592533\pi$
$353$	− 36.0000i	− 1.91609i	0.286623	−	0.958043i	$-0.407467\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.00000i	0.156599i	0.996930	+	0.0782994i	$0.0249490\pi$
$367$	3.00000i	0.156599i	−0.996930	+	0.0782994i	$0.975051\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	− 15.0000i	− 0.776671i	−0.921518	−	0.388335i	$-0.873050\pi$
$373$	− 15.0000i	− 0.776671i	0.921518	−	0.388335i	$-0.126950\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	−13.0000	−0.667765	−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000	−0.667765	−0.333883	+	0.942615i	$0.608359\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	32.0000i	1.63512i	0.575841	+	0.817562i	$0.304675\pi$
$383$	32.0000i	1.63512i	−0.575841	+	0.817562i	$0.695325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.00000i	− 0.457496i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6.00000i	0.302660i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13.0000i	− 0.652451i	−0.945292	−	0.326226i	$-0.894223\pi$
$397$	− 13.0000i	− 0.652451i	0.945292	−	0.326226i	$-0.105777\pi$
$398$	0	0
$399$	3.00000	0.150188
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 7.00000i	− 0.348695i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	0	0
$411$	10.0000	0.493264
$412$	0	0
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	26.0000	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$419$	26.0000	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.00000i	0.0483934i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	15.0000i	0.720854i	0.932787	+	0.360427i	$0.117369\pi$
$433$	15.0000i	0.720854i	−0.932787	+	0.360427i	$0.882631\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−23.0000	−1.09773	−0.548865	−	0.835911i	$-0.684940\pi$
$439$	−23.0000	−1.09773	−0.548865	+	0.835911i	$0.684940\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	14.0000i	0.662177i
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	17.0000i	0.798730i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	−3.00000	−0.138527
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	4.00000i	0.182006i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 29.0000i	− 1.31412i	−0.753840	−	0.657058i	$-0.771801\pi$
$487$	− 29.0000i	− 1.31412i	0.753840	−	0.657058i	$-0.228199\pi$
$488$	0	0
$489$	−19.0000	−0.859210
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.0000i	0.627986i
$498$	0	0
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	18.0000i	0.802580i	0.915951	+	0.401290i	$0.131438\pi$
$503$	18.0000i	0.802580i	−0.915951	+	0.401290i	$0.868562\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.0000i	0.532939i
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	3.00000i	0.132453i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	− 37.0000i	− 1.61790i	−0.587879	−	0.808949i	$-0.700037\pi$
$523$	− 37.0000i	− 1.61790i	0.587879	−	0.808949i	$-0.299963\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.00000	−0.214967	−0.107483	−	0.994207i	$-0.534279\pi$
$541$	−5.00000	−0.214967	−0.107483	+	0.994207i	$0.534279\pi$
$542$	0	0
$543$	11.0000i	0.472055i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	0	0
$549$	−1.00000	−0.0426790
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	8.00000i	0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 6.00000i	− 0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$	− 6.00000i	− 0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$	0	0
$559$	9.00000	0.380659
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 8.00000i	− 0.337160i	−0.985688	−	0.168580i	$-0.946082\pi$
$563$	− 8.00000i	− 0.337160i	0.985688	−	0.168580i	$-0.0539181\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000i	0.0419961i
$568$	0	0
$569$	20.0000	0.838444	0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000	0.838444	0.419222	+	0.907884i	$0.362303\pi$
$570$	0	0
$571$	−5.00000	−0.209243	−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000	−0.209243	−0.104622	+	0.994512i	$0.533363\pi$
$572$	0	0
$573$	16.0000i	0.668410i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 31.0000i	− 1.29055i	−0.763952	−	0.645273i	$-0.776743\pi$
$577$	− 31.0000i	− 1.29055i	0.763952	−	0.645273i	$-0.223257\pi$
$578$	0	0
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−18.0000	−0.746766
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	−21.0000	−0.865290
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13.0000i	− 0.532055i
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	−21.0000	−0.856608	−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000	−0.856608	−0.428304	+	0.903635i	$0.640889\pi$
$602$	0	0
$603$	− 3.00000i	− 0.122169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	6.00000	0.242734
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	0	0
$619$	−19.0000	−0.763674	−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000	−0.763674	−0.381837	+	0.924230i	$0.624709\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	0	0
$633$	7.00000i	0.278225i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	−14.0000	−0.553831
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−7.00000	−0.274352
$652$	0	0
$653$	− 10.0000i	− 0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$	− 10.0000i	− 0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.0000i	− 0.390137i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	−11.0000	−0.425285
$670$	0	0
$671$	0	0
$672$	0	0
$673$	22.0000i	0.848038i	0.905653	+	0.424019i	$0.139381\pi$
$673$	22.0000i	0.848038i	−0.905653	+	0.424019i	$0.860619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 23.0000i	− 0.877505i
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	0	0
$703$	18.0000i	0.678883i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	−3.00000	−0.112667	−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000	−0.112667	−0.0563337	+	0.998412i	$0.517941\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	− 28.0000i	− 1.04861i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000i	0.224074i
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	− 11.0000i	− 0.409094i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.00000i	− 0.259616i	−0.991539	−	0.129808i	$-0.958564\pi$
$727$	− 7.00000i	− 0.259616i	0.991539	−	0.129808i	$-0.0414360\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	0	0
$743$	− 28.0000i	− 1.02722i	−0.858024	−	0.513610i	$-0.828308\pi$
$743$	− 28.0000i	− 1.02722i	0.858024	−	0.513610i	$-0.171692\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 18.0000i	− 0.658586i
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	16.0000i	0.583072i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 47.0000i	− 1.70824i	−0.520073	−	0.854122i	$-0.674095\pi$
$757$	− 47.0000i	− 1.70824i	0.520073	−	0.854122i	$-0.325905\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	44.0000	1.59500	0.797499	−	0.603320i	$-0.206156\pi$
$761$	44.0000	1.59500	0.797499	+	0.603320i	$0.206156\pi$
$762$	0	0
$763$	− 15.0000i	− 0.543036i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.0000i	0.361079i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.00000i	0.215249i
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 4.00000i	− 0.142948i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.00000i	− 0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$	− 7.00000i	− 0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$	0	0
$789$	−2.00000	−0.0712019
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 1.00000i	− 0.0355110i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 48.0000i	− 1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$	− 48.0000i	− 1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.0000i	0.563227i
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	0	0
$813$	4.00000i	0.140286i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 27.0000i	− 0.944610i
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	− 41.0000i	− 1.42917i	−0.699549	−	0.714585i	$-0.746616\pi$
$823$	− 41.0000i	− 1.42917i	0.699549	−	0.714585i	$-0.253384\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 34.0000i	− 1.18230i	−0.806563	−	0.591148i	$-0.798675\pi$
$827$	− 34.0000i	− 1.18230i	0.806563	−	0.591148i	$-0.201325\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	11.0000	0.381586
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 7.00000i	− 0.241955i
$838$	0	0
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	4.00000i	0.137767i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 11.0000i	− 0.377964i
$848$	0	0
$849$	5.00000	0.171600
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	37.0000i	1.26686i	0.773802	+	0.633428i	$0.218353\pi$
$853$	37.0000i	1.26686i	−0.773802	+	0.633428i	$0.781647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 32.0000i	− 1.09310i	−0.837427	−	0.546550i	$-0.815941\pi$
$857$	− 32.0000i	− 1.09310i	0.837427	−	0.546550i	$-0.184059\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	4.00000i	0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$	4.00000i	0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.0000i	0.577350i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3.00000	0.101651
$872$	0	0
$873$	− 3.00000i	− 0.101535i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	43.0000i	1.45201i	0.687691	+	0.726003i	$0.258624\pi$
$877$	43.0000i	1.45201i	−0.687691	+	0.726003i	$0.741376\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	29.0000i	0.975928i	0.872864	+	0.487964i	$0.162260\pi$
$883$	29.0000i	0.975928i	−0.872864	+	0.487964i	$0.837740\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 38.0000i	− 1.27592i	−0.770072	−	0.637958i	$-0.779780\pi$
$887$	− 38.0000i	− 1.27592i	0.770072	−	0.637958i	$-0.220220\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 4.00000i	− 0.133556i
$898$	0	0
$899$	28.0000	0.933852
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 9.00000i	− 0.299501i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 20.0000i	− 0.664089i	−0.943264	−	0.332045i	$-0.892262\pi$
$907$	− 20.0000i	− 0.664089i	0.943264	−	0.332045i	$-0.107738\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−42.0000	−1.39152	−0.695761	−	0.718273i	$-0.744933\pi$
$911$	−42.0000	−1.39152	−0.695761	+	0.718273i	$0.744933\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000i	0.198137i
$918$	0	0
$919$	−1.00000	−0.0329870	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000	−0.0329870	−0.0164935	+	0.999864i	$0.505250\pi$
$920$	0	0
$921$	−13.0000	−0.428365
$922$	0	0
$923$	− 14.0000i	− 0.460816i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	10.0000i	0.327385i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 19.0000i	− 0.620703i	−0.950622	−	0.310351i	$-0.899553\pi$
$937$	− 19.0000i	− 0.620703i	0.950622	−	0.310351i	$-0.100447\pi$
$938$	0	0
$939$	−23.0000	−0.750577
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	− 24.0000i	− 0.781548i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.0000i	− 0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$	− 20.0000i	− 0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	− 8.00000i	− 0.259145i	−0.991570	−	0.129573i	$-0.958639\pi$
$953$	− 8.00000i	− 0.259145i	0.991570	−	0.129573i	$-0.0413606\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	2.00000i	0.0644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	− 12.0000i	− 0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.0000	0.478913
$982$	0	0
$983$	− 28.0000i	− 0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$	− 28.0000i	− 0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	0	0
$989$	36.0000	1.14473
$990$	0	0
$991$	31.0000	0.984747	0.492374	−	0.870384i	$-0.336129\pi$
$991$	31.0000	0.984747	0.492374	+	0.870384i	$0.336129\pi$
$992$	0	0
$993$	32.0000i	1.01549i
$994$	0	0
$995$	0	0
$996$	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
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.q.3649.2		2
4.3	odd	2		4800.2.f.t.3649.1		2
5.2	odd	4		4800.2.a.bw.1.1		1
5.3	odd	4		4800.2.a.x.1.1		1
5.4	even	2	inner	4800.2.f.q.3649.1		2
8.3	odd	2		2400.2.f.h.1249.2		2
8.5	even	2		2400.2.f.k.1249.1		2
20.3	even	4		4800.2.a.bx.1.1		1
20.7	even	4		4800.2.a.w.1.1		1
20.19	odd	2		4800.2.f.t.3649.2		2
24.5	odd	2		7200.2.f.r.6049.2		2
24.11	even	2		7200.2.f.l.6049.1		2
40.3	even	4		2400.2.a.f.1.1	✓	1
40.13	odd	4		2400.2.a.bb.1.1	yes	1
40.19	odd	2		2400.2.f.h.1249.1		2
40.27	even	4		2400.2.a.bc.1.1	yes	1
40.29	even	2		2400.2.f.k.1249.2		2
40.37	odd	4		2400.2.a.g.1.1	yes	1
120.29	odd	2		7200.2.f.r.6049.1		2
120.53	even	4		7200.2.a.bi.1.1		1
120.59	even	2		7200.2.f.l.6049.2		2
120.77	even	4		7200.2.a.s.1.1		1
120.83	odd	4		7200.2.a.r.1.1		1
120.107	odd	4		7200.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.2.a.f.1.1	✓	1	40.3	even	4
2400.2.a.g.1.1	yes	1	40.37	odd	4
2400.2.a.bb.1.1	yes	1	40.13	odd	4
2400.2.a.bc.1.1	yes	1	40.27	even	4
2400.2.f.h.1249.1		2	40.19	odd	2
2400.2.f.h.1249.2		2	8.3	odd	2
2400.2.f.k.1249.1		2	8.5	even	2
2400.2.f.k.1249.2		2	40.29	even	2
4800.2.a.w.1.1		1	20.7	even	4
4800.2.a.x.1.1		1	5.3	odd	4
4800.2.a.bw.1.1		1	5.2	odd	4
4800.2.a.bx.1.1		1	20.3	even	4
4800.2.f.q.3649.1		2	5.4	even	2	inner
4800.2.f.q.3649.2		2	1.1	even	1	trivial
4800.2.f.t.3649.1		2	4.3	odd	2
4800.2.f.t.3649.2		2	20.19	odd	2
7200.2.a.r.1.1		1	120.83	odd	4
7200.2.a.s.1.1		1	120.77	even	4
7200.2.a.bi.1.1		1	120.53	even	4
7200.2.a.bj.1.1		1	120.107	odd	4
7200.2.f.l.6049.1		2	24.11	even	2
7200.2.f.l.6049.2		2	120.59	even	2
7200.2.f.r.6049.1		2	120.29	odd	2
7200.2.f.r.6049.2		2	24.5	odd	2

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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{7} -1.00000 q^{9} -1.00000i q^{13} -3.00000 q^{19} -1.00000 q^{21} -4.00000i q^{23} -1.00000i q^{27} +4.00000 q^{29} +7.00000 q^{31} -6.00000i q^{37} +1.00000 q^{39} +6.00000 q^{41} +9.00000i q^{43} +6.00000i q^{47} +6.00000 q^{49} -2.00000i q^{53} -3.00000i q^{57} -10.0000 q^{59} +1.00000 q^{61} -1.00000i q^{63} +3.00000i q^{67} +4.00000 q^{69} +14.0000 q^{71} +10.0000i q^{73} +8.00000 q^{79} +1.00000 q^{81} +18.0000i q^{83} +4.00000i q^{87} +1.00000 q^{91} +7.00000i q^{93} +3.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 6 q^{19} - 2 q^{21} + 8 q^{29} + 14 q^{31} + 2 q^{39} + 12 q^{41} + 12 q^{49} - 20 q^{59} + 2 q^{61} + 8 q^{69} + 28 q^{71} + 16 q^{79} + 2 q^{81} + 2 q^{91}+O(q^{100}) \) 2 * q - 2 * q^9 - 6 * q^19 - 2 * q^21 + 8 * q^29 + 14 * q^31 + 2 * q^39 + 12 * q^41 + 12 * q^49 - 20 * q^59 + 2 * q^61 + 8 * q^69 + 28 * q^71 + 16 * q^79 + 2 * q^81 + 2 * q^91

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