LMFDB - Embedded newform 4800.2.f.e.3649.1

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:	$1$
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 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} +6.00000i q^{17} -4.00000 q^{19} +4.00000 q^{21} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +4.00000i q^{33} -2.00000i q^{37} +2.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} -8.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} -10.0000i q^{53} +4.00000i q^{57} -4.00000 q^{59} -6.00000 q^{61} -4.00000i q^{63} +4.00000i q^{67} -16.0000 q^{71} -6.00000i q^{73} -16.0000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{87} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 8 q^{11} - 8 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} + 4 q^{39} + 4 q^{41} - 18 q^{49} + 12 q^{51} - 8 q^{59} - 12 q^{61} - 32 q^{71} - 8 q^{79} + 2 q^{81} - 20 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 8 * q^11 - 8 * q^19 + 8 * q^21 + 4 * q^29 + 8 * q^31 + 4 * q^39 + 4 * q^41 - 18 * q^49 + 12 * q^51 - 8 * q^59 - 12 * q^61 - 32 * q^71 - 8 * q^79 + 2 * q^81 - 20 * q^89 - 16 * q^91 + 8 * q^99

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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	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
$57$	4.00000i	0.529813i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	− 4.00000i	− 0.503953i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 16.0000i	− 1.82337i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	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
$85$	0	0
$86$	0	0
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	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
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	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
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000i	0.300658i
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	8.00000i	0.561490i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	16.0000i	1.09630i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	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
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	10.0000i	0.611990i
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	8.00000i	0.484182i
$274$	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
$279$	−4.00000	−0.239474
$280$	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
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	14.0000	0.820695
$292$	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
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 14.0000i	− 0.774202i
$328$	0	0
$329$	32.0000	1.76422
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 8.00000i	− 0.431959i
$344$	0	0
$345$	0	0
$346$	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
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.0000i	1.27021i
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	40.0000	2.07670
$372$	0	0
$373$	34.0000i	1.76045i	0.474554	+	0.880227i	$0.342610\pi$
$373$	34.0000i	1.76045i	−0.474554	+	0.880227i	$0.657390\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000i	0.206010i
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	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
$387$	4.00000i	0.203331i
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000i	0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	− 16.0000i	− 0.787309i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000i	0.979404i
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	0	0
$423$	8.00000i	0.388973i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 18.0000i	− 0.851371i
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	− 12.0000i	− 0.563809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 26.0000i	− 1.21623i	−0.793849	−	0.608114i	$-0.791926\pi$
$457$	− 26.0000i	− 1.21623i	0.793849	−	0.608114i	$-0.208074\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	16.0000i	0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 64.0000i	− 2.87079i
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0000i	1.04546i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	4.00000i	0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000i	0.517838i
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	− 10.0000i	− 0.429141i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.00000i	0.167984i
$568$	0	0
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	40.0000i	1.65663i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.00000i	0.163709i
$598$	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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 40.0000i	− 1.60257i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 16.0000i	− 0.638978i
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	0	0
$633$	20.0000i	0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	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
$647$	− 48.0000i	− 1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$	− 48.0000i	− 1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	16.0000	0.627089
$652$	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
$657$	6.00000i	0.234082i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	12.0000i	0.466041i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 14.0000i	− 0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$	− 14.0000i	− 0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$	0	0
$679$	−56.0000	−2.14908
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	0	0
$689$	20.0000	0.761939
$690$	0	0
$691$	52.0000	1.97817	0.989087	−	0.147335i	$-0.0470696\pi$
$691$	52.0000	1.97817	0.989087	+	0.147335i	$0.0470696\pi$
$692$	0	0
$693$	16.0000i	0.607790i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000i	0.902613i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	48.0000	1.78761
$722$	0	0
$723$	14.0000i	0.520666i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	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
$741$	−8.00000	−0.293887
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	56.0000i	2.02734i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	−8.00000	−0.286630
$780$	0	0
$781$	64.0000	2.29010
$782$	0	0
$783$	2.00000i	0.0714742i
$784$	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
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	− 12.0000i	− 0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 18.0000i	− 0.633630i
$808$	0	0
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	− 12.0000i	− 0.420858i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	0	0
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	0	0
$825$	0	0
$826$	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
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	0	0
$833$	− 54.0000i	− 1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	0	0
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.0000i	0.687208i
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	18.0000i	0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$	18.0000i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	−8.00000	−0.271070
$872$	0	0
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	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
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	−80.0000	−2.68311
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	60.0000	1.99889
$902$	0	0
$903$	− 16.0000i	− 0.532447i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.0000i	0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$	20.0000i	0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.0000i	− 0.528367i
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	0	0
$923$	− 32.0000i	− 1.05329i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.0000i	0.394132i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	24.0000i	0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000i	0.258603i
$958$	0	0
$959$	72.0000	2.32500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.0000i	0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$	12.0000i	0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	− 80.0000i	− 2.56468i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.0000i	0.703842i	0.936030	+	0.351921i	$0.114471\pi$
$977$	22.0000i	0.703842i	−0.936030	+	0.351921i	$0.885529\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 32.0000i	− 1.01857i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	28.0000i	0.888553i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18.0000i	− 0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$	− 18.0000i	− 0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.f.e.3649.1		2
4.3	odd	2		4800.2.f.bh.3649.2		2
5.2	odd	4		192.2.a.a.1.1		1
5.3	odd	4		4800.2.a.co.1.1		1
5.4	even	2	inner	4800.2.f.e.3649.2		2
8.3	odd	2		2400.2.f.a.1249.1		2
8.5	even	2		2400.2.f.r.1249.2		2
15.2	even	4		576.2.a.g.1.1		1
20.3	even	4		4800.2.a.f.1.1		1
20.7	even	4		192.2.a.c.1.1		1
20.19	odd	2		4800.2.f.bh.3649.1		2
24.5	odd	2		7200.2.f.f.6049.2		2
24.11	even	2		7200.2.f.x.6049.1		2
35.27	even	4		9408.2.a.ct.1.1		1
40.3	even	4		2400.2.a.r.1.1		1
40.13	odd	4		2400.2.a.q.1.1		1
40.19	odd	2		2400.2.f.a.1249.2		2
40.27	even	4		96.2.a.a.1.1	✓	1
40.29	even	2		2400.2.f.r.1249.1		2
40.37	odd	4		96.2.a.b.1.1	yes	1
60.47	odd	4		576.2.a.h.1.1		1
80.27	even	4		768.2.d.a.385.1		2
80.37	odd	4		768.2.d.h.385.2		2
80.67	even	4		768.2.d.a.385.2		2
80.77	odd	4		768.2.d.h.385.1		2
120.29	odd	2		7200.2.f.f.6049.1		2
120.53	even	4		7200.2.a.bx.1.1		1
120.59	even	2		7200.2.f.x.6049.2		2
120.77	even	4		288.2.a.b.1.1		1
120.83	odd	4		7200.2.a.e.1.1		1
120.107	odd	4		288.2.a.c.1.1		1
140.27	odd	4		9408.2.a.bj.1.1		1
240.77	even	4		2304.2.d.s.1153.1		2
240.107	odd	4		2304.2.d.c.1153.2		2
240.197	even	4		2304.2.d.s.1153.2		2
240.227	odd	4		2304.2.d.c.1153.1		2
280.27	odd	4		4704.2.a.t.1.1		1
280.237	even	4		4704.2.a.e.1.1		1
360.67	even	12		2592.2.i.b.865.1		2
360.77	even	12		2592.2.i.w.865.1		2
360.157	odd	12		2592.2.i.h.865.1		2
360.187	even	12		2592.2.i.b.1729.1		2
360.227	odd	12		2592.2.i.q.1729.1		2
360.277	odd	12		2592.2.i.h.1729.1		2
360.317	even	12		2592.2.i.w.1729.1		2
360.347	odd	12		2592.2.i.q.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.a.a.1.1	✓	1	40.27	even	4
96.2.a.b.1.1	yes	1	40.37	odd	4
192.2.a.a.1.1		1	5.2	odd	4
192.2.a.c.1.1		1	20.7	even	4
288.2.a.b.1.1		1	120.77	even	4
288.2.a.c.1.1		1	120.107	odd	4
576.2.a.g.1.1		1	15.2	even	4
576.2.a.h.1.1		1	60.47	odd	4
768.2.d.a.385.1		2	80.27	even	4
768.2.d.a.385.2		2	80.67	even	4
768.2.d.h.385.1		2	80.77	odd	4
768.2.d.h.385.2		2	80.37	odd	4
2304.2.d.c.1153.1		2	240.227	odd	4
2304.2.d.c.1153.2		2	240.107	odd	4
2304.2.d.s.1153.1		2	240.77	even	4
2304.2.d.s.1153.2		2	240.197	even	4
2400.2.a.q.1.1		1	40.13	odd	4
2400.2.a.r.1.1		1	40.3	even	4
2400.2.f.a.1249.1		2	8.3	odd	2
2400.2.f.a.1249.2		2	40.19	odd	2
2400.2.f.r.1249.1		2	40.29	even	2
2400.2.f.r.1249.2		2	8.5	even	2
2592.2.i.b.865.1		2	360.67	even	12
2592.2.i.b.1729.1		2	360.187	even	12
2592.2.i.h.865.1		2	360.157	odd	12
2592.2.i.h.1729.1		2	360.277	odd	12
2592.2.i.q.865.1		2	360.347	odd	12
2592.2.i.q.1729.1		2	360.227	odd	12
2592.2.i.w.865.1		2	360.77	even	12
2592.2.i.w.1729.1		2	360.317	even	12
4704.2.a.e.1.1		1	280.237	even	4
4704.2.a.t.1.1		1	280.27	odd	4
4800.2.a.f.1.1		1	20.3	even	4
4800.2.a.co.1.1		1	5.3	odd	4
4800.2.f.e.3649.1		2	1.1	even	1	trivial
4800.2.f.e.3649.2		2	5.4	even	2	inner
4800.2.f.bh.3649.1		2	20.19	odd	2
4800.2.f.bh.3649.2		2	4.3	odd	2
7200.2.a.e.1.1		1	120.83	odd	4
7200.2.a.bx.1.1		1	120.53	even	4
7200.2.f.f.6049.1		2	120.29	odd	2
7200.2.f.f.6049.2		2	24.5	odd	2
7200.2.f.x.6049.1		2	24.11	even	2
7200.2.f.x.6049.2		2	120.59	even	2
9408.2.a.bj.1.1		1	140.27	odd	4
9408.2.a.ct.1.1		1	35.27	even	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:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} +6.00000i q^{17} -4.00000 q^{19} +4.00000 q^{21} +1.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +4.00000i q^{33} -2.00000i q^{37} +2.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} -8.00000i q^{47} -9.00000 q^{49} +6.00000 q^{51} -10.0000i q^{53} +4.00000i q^{57} -4.00000 q^{59} -6.00000 q^{61} -4.00000i q^{63} +4.00000i q^{67} -16.0000 q^{71} -6.00000i q^{73} -16.0000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} -2.00000i q^{87} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 8 q^{11} - 8 q^{19} + 8 q^{21} + 4 q^{29} + 8 q^{31} + 4 q^{39} + 4 q^{41} - 18 q^{49} + 12 q^{51} - 8 q^{59} - 12 q^{61} - 32 q^{71} - 8 q^{79} + 2 q^{81} - 20 q^{89} - 16 q^{91} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 8 * q^11 - 8 * q^19 + 8 * q^21 + 4 * q^29 + 8 * q^31 + 4 * q^39 + 4 * q^41 - 18 * q^49 + 12 * q^51 - 8 * q^59 - 12 * q^61 - 32 * q^71 - 8 * q^79 + 2 * q^81 - 20 * q^89 - 16 * q^91 + 8 * q^99

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