LMFDB - Embedded newform 1300.2.c.f.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1249,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1300.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.c (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:	$10.3805522628$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$0.675970 + 0.675970i$ of defining polynomial
Character	$\chi$	$=$	1300.1249
Dual form			1300.2.c.f.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35194i q^{3} +4.17226i q^{7} +1.17226 q^{9} +O(q^{10})$	$q-1.35194i q^{3} +4.17226i q^{7} +1.17226 q^{9} -5.52420 q^{11} -1.00000i q^{13} -0.703878i q^{17} -6.82032 q^{19} +5.64064 q^{21} +2.64806i q^{23} -5.64064i q^{27} -8.17226 q^{29} -9.52420 q^{31} +7.46838i q^{33} -6.87614i q^{37} -1.35194 q^{39} +0.703878 q^{41} +1.35194i q^{43} -8.17226i q^{47} -10.4078 q^{49} -0.951601 q^{51} -5.04840i q^{53} +9.22066i q^{57} +12.2281 q^{59} -0.172260 q^{61} +4.89098i q^{63} +10.8761i q^{67} +3.58002 q^{69} -5.52420 q^{71} +11.2207i q^{73} -23.0484i q^{77} -1.29612 q^{79} -4.10902 q^{81} +9.58002i q^{83} +11.0484i q^{87} -11.7523 q^{89} +4.17226 q^{91} +12.8761i q^{93} +18.3445i q^{97} -6.47580 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{9}+O(q^{10}) $ 6 * q - 22 * q^9	$ 6 q - 22 q^{9} - 16 q^{19} - 16 q^{21} - 20 q^{29} - 24 q^{31} - 4 q^{39} - 4 q^{41} - 46 q^{49} - 72 q^{51} + 32 q^{59} + 28 q^{61} - 24 q^{69} - 16 q^{79} + 46 q^{81} + 4 q^{89} - 4 q^{91} - 72 q^{99}+O(q^{100}) $ 6 * q - 22 * q^9 - 16 * q^19 - 16 * q^21 - 20 * q^29 - 24 * q^31 - 4 * q^39 - 4 * q^41 - 46 * q^49 - 72 * q^51 + 32 * q^59 + 28 * q^61 - 24 * q^69 - 16 * q^79 + 46 * q^81 + 4 * q^89 - 4 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.35194i	− 0.780542i	−0.920700	−	0.390271i	$-0.872381\pi$
$3$	− 1.35194i	− 0.780542i	0.920700	−	0.390271i	$-0.127619\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.17226i	1.57697i	0.615056	+	0.788483i	$0.289133\pi$
$7$	4.17226i	1.57697i	−0.615056	+	0.788483i	$0.710867\pi$
$8$	0	0
$9$	1.17226	0.390753
$10$	0	0
$11$	−5.52420	−1.66561	−0.832804	−	0.553567i	$-0.813266\pi$
$11$	−5.52420	−1.66561	−0.832804	+	0.553567i	$0.813266\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.703878i	− 0.170716i	−0.996350	−	0.0853578i	$-0.972797\pi$
$17$	− 0.703878i	− 0.170716i	0.996350	−	0.0853578i	$-0.0272033\pi$
$18$	0	0
$19$	−6.82032	−1.56469	−0.782344	−	0.622846i	$-0.785976\pi$
$19$	−6.82032	−1.56469	−0.782344	+	0.622846i	$0.785976\pi$
$20$	0	0
$21$	5.64064	1.23089
$22$	0	0
$23$	2.64806i	0.552159i	0.961135	+	0.276079i	$0.0890353\pi$
$23$	2.64806i	0.552159i	−0.961135	+	0.276079i	$0.910965\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.64064i	− 1.08554i
$28$	0	0
$29$	−8.17226	−1.51755	−0.758775	−	0.651352i	$-0.774202\pi$
$29$	−8.17226	−1.51755	−0.758775	+	0.651352i	$0.774202\pi$
$30$	0	0
$31$	−9.52420	−1.71060	−0.855298	−	0.518136i	$-0.826626\pi$
$31$	−9.52420	−1.71060	−0.855298	+	0.518136i	$0.826626\pi$
$32$	0	0
$33$	7.46838i	1.30008i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.87614i	− 1.13043i	−0.824944	−	0.565215i	$-0.808793\pi$
$37$	− 6.87614i	− 1.13043i	0.824944	−	0.565215i	$-0.191207\pi$
$38$	0	0
$39$	−1.35194	−0.216484
$40$	0	0
$41$	0.703878	0.109927	0.0549637	−	0.998488i	$-0.482496\pi$
$41$	0.703878	0.109927	0.0549637	+	0.998488i	$0.482496\pi$
$42$	0	0
$43$	1.35194i	0.206169i	0.994673	+	0.103084i	$0.0328712\pi$
$43$	1.35194i	0.206169i	−0.994673	+	0.103084i	$0.967129\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.17226i	− 1.19205i	−0.802967	−	0.596023i	$-0.796747\pi$
$47$	− 8.17226i	− 1.19205i	0.802967	−	0.596023i	$-0.203253\pi$
$48$	0	0
$49$	−10.4078	−1.48682
$50$	0	0
$51$	−0.951601	−0.133251
$52$	0	0
$53$	− 5.04840i	− 0.693451i	−0.937967	−	0.346725i	$-0.887294\pi$
$53$	− 5.04840i	− 0.693451i	0.937967	−	0.346725i	$-0.112706\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.22066i	1.22131i
$58$	0	0
$59$	12.2281	1.59196	0.795980	−	0.605323i	$-0.206956\pi$
$59$	12.2281	1.59196	0.795980	+	0.605323i	$0.206956\pi$
$60$	0	0
$61$	−0.172260	−0.0220557	−0.0110278	−	0.999939i	$-0.503510\pi$
$61$	−0.172260	−0.0220557	−0.0110278	+	0.999939i	$0.503510\pi$
$62$	0	0
$63$	4.89098i	0.616205i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8761i	1.32873i	0.747407	+	0.664366i	$0.231298\pi$
$67$	10.8761i	1.32873i	−0.747407	+	0.664366i	$0.768702\pi$
$68$	0	0
$69$	3.58002	0.430983
$70$	0	0
$71$	−5.52420	−0.655602	−0.327801	−	0.944747i	$-0.606308\pi$
$71$	−5.52420	−0.655602	−0.327801	+	0.944747i	$0.606308\pi$
$72$	0	0
$73$	11.2207i	1.31328i	0.754205	+	0.656639i	$0.228023\pi$
$73$	11.2207i	1.31328i	−0.754205	+	0.656639i	$0.771977\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 23.0484i	− 2.62661i
$78$	0	0
$79$	−1.29612	−0.145825	−0.0729125	−	0.997338i	$-0.523229\pi$
$79$	−1.29612	−0.145825	−0.0729125	+	0.997338i	$0.523229\pi$
$80$	0	0
$81$	−4.10902	−0.456558
$82$	0	0
$83$	9.58002i	1.05154i	0.850626	+	0.525772i	$0.176223\pi$
$83$	9.58002i	1.05154i	−0.850626	+	0.525772i	$0.823777\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	11.0484i	1.18451i
$88$	0	0
$89$	−11.7523	−1.24574	−0.622869	−	0.782326i	$-0.714033\pi$
$89$	−11.7523	−1.24574	−0.622869	+	0.782326i	$0.714033\pi$
$90$	0	0
$91$	4.17226	0.437372
$92$	0	0
$93$	12.8761i	1.33519i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.3445i	1.86260i	0.364248	+	0.931302i	$0.381326\pi$
$97$	18.3445i	1.86260i	−0.364248	+	0.931302i	$0.618674\pi$
$98$	0	0
$99$	−6.47580	−0.650842
$100$	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
$103$	10.9926i	1.08313i	0.840658	+	0.541566i	$0.182168\pi$
$103$	10.9926i	1.08313i	−0.840658	+	0.541566i	$0.817832\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.99258i	0.675998i	0.941146	+	0.337999i	$0.109750\pi$
$107$	6.99258i	0.675998i	−0.941146	+	0.337999i	$0.890250\pi$
$108$	0	0
$109$	−3.40776	−0.326404	−0.163202	−	0.986593i	$-0.552182\pi$
$109$	−3.40776	−0.326404	−0.163202	+	0.986593i	$0.552182\pi$
$110$	0	0
$111$	−9.29612	−0.882349
$112$	0	0
$113$	− 3.64064i	− 0.342483i	−0.985229	−	0.171241i	$-0.945222\pi$
$113$	− 3.64064i	− 0.342483i	0.985229	−	0.171241i	$-0.0547778\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.17226i	− 0.108376i
$118$	0	0
$119$	2.93676	0.269213
$120$	0	0
$121$	19.5168	1.77425
$122$	0	0
$123$	− 0.951601i	− 0.0858030i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.69646i	− 0.505479i	−0.967534	−	0.252740i	$-0.918668\pi$
$127$	− 5.69646i	− 0.505479i	0.967534	−	0.252740i	$-0.0813316\pi$
$128$	0	0
$129$	1.82774	0.160923
$130$	0	0
$131$	17.7523	1.55102	0.775512	−	0.631333i	$-0.217492\pi$
$131$	17.7523	1.55102	0.775512	+	0.631333i	$0.217492\pi$
$132$	0	0
$133$	− 28.4562i	− 2.46746i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.93676i	− 0.763519i	−0.924262	−	0.381760i	$-0.875318\pi$
$137$	− 8.93676i	− 0.763519i	0.924262	−	0.381760i	$-0.124682\pi$
$138$	0	0
$139$	−13.2961	−1.12776	−0.563881	−	0.825856i	$-0.690692\pi$
$139$	−13.2961	−1.12776	−0.563881	+	0.825856i	$0.690692\pi$
$140$	0	0
$141$	−11.0484	−0.930443
$142$	0	0
$143$	5.52420i	0.461957i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	14.0707i	1.16053i
$148$	0	0
$149$	8.93676	0.732128	0.366064	−	0.930590i	$-0.380705\pi$
$149$	8.93676	0.732128	0.366064	+	0.930590i	$0.380705\pi$
$150$	0	0
$151$	2.93196	0.238599	0.119300	−	0.992858i	$-0.461935\pi$
$151$	2.93196	0.238599	0.119300	+	0.992858i	$0.461935\pi$
$152$	0	0
$153$	− 0.825129i	− 0.0667077i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.592243i	0.0472662i	0.999721	+	0.0236331i	$0.00752334\pi$
$157$	0.592243i	0.0472662i	−0.999721	+	0.0236331i	$0.992477\pi$
$158$	0	0
$159$	−6.82513	−0.541268
$160$	0	0
$161$	−11.0484	−0.870736
$162$	0	0
$163$	− 16.6284i	− 1.30244i	−0.758890	−	0.651219i	$-0.774258\pi$
$163$	− 16.6284i	− 1.30244i	0.758890	−	0.651219i	$-0.225742\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.82774i	− 0.296199i	−0.988972	−	0.148100i	$-0.952684\pi$
$167$	− 3.82774i	− 0.296199i	0.988972	−	0.148100i	$-0.0473156\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−7.99519	−0.611408
$172$	0	0
$173$	− 14.1116i	− 1.07289i	−0.843936	−	0.536444i	$-0.819767\pi$
$173$	− 14.1116i	− 1.07289i	0.843936	−	0.536444i	$-0.180233\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 16.5316i	− 1.24259i
$178$	0	0
$179$	−5.75228	−0.429945	−0.214973	−	0.976620i	$-0.568966\pi$
$179$	−5.75228	−0.429945	−0.214973	+	0.976620i	$0.568966\pi$
$180$	0	0
$181$	2.76450	0.205484	0.102742	−	0.994708i	$-0.467238\pi$
$181$	2.76450	0.205484	0.102742	+	0.994708i	$0.467238\pi$
$182$	0	0
$183$	0.232886i	0.0172154i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.88836i	0.284345i
$188$	0	0
$189$	23.5342	1.71186
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	14.3445i	1.03254i	0.856426	+	0.516271i	$0.172680\pi$
$193$	14.3445i	1.03254i	−0.856426	+	0.516271i	$0.827320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 7.40776i	− 0.527781i	−0.964553	−	0.263890i	$-0.914994\pi$
$197$	− 7.40776i	− 0.527781i	0.964553	−	0.263890i	$-0.0850057\pi$
$198$	0	0
$199$	−1.75228	−0.124216	−0.0621078	−	0.998069i	$-0.519782\pi$
$199$	−1.75228	−0.124216	−0.0621078	+	0.998069i	$0.519782\pi$
$200$	0	0
$201$	14.7039	1.03713
$202$	0	0
$203$	− 34.0968i	− 2.39313i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.10422i	0.215758i
$208$	0	0
$209$	37.6768	2.60616
$210$	0	0
$211$	−9.75228	−0.671374	−0.335687	−	0.941974i	$-0.608969\pi$
$211$	−9.75228	−0.671374	−0.335687	+	0.941974i	$0.608969\pi$
$212$	0	0
$213$	7.46838i	0.511725i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 39.7374i	− 2.69755i
$218$	0	0
$219$	15.1696	1.02507
$220$	0	0
$221$	−0.703878	−0.0473480
$222$	0	0
$223$	− 2.76450i	− 0.185125i	−0.995707	−	0.0925624i	$-0.970494\pi$
$223$	− 2.76450i	− 0.185125i	0.995707	−	0.0925624i	$-0.0295058\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8129i	1.18228i	0.806568	+	0.591142i	$0.201323\pi$
$227$	17.8129i	1.18228i	−0.806568	+	0.591142i	$0.798677\pi$
$228$	0	0
$229$	−10.1116	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$229$	−10.1116	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$230$	0	0
$231$	−31.1600	−2.05018
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.75228i	0.113823i
$238$	0	0
$239$	−9.86872	−0.638354	−0.319177	−	0.947695i	$-0.603407\pi$
$239$	−9.86872	−0.638354	−0.319177	+	0.947695i	$0.603407\pi$
$240$	0	0
$241$	−1.88836	−0.121640	−0.0608201	−	0.998149i	$-0.519372\pi$
$241$	−1.88836	−0.121640	−0.0608201	+	0.998149i	$0.519372\pi$
$242$	0	0
$243$	− 11.3668i	− 0.729179i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.82032i	0.433967i
$248$	0	0
$249$	12.9516	0.820774
$250$	0	0
$251$	2.35936	0.148921	0.0744607	−	0.997224i	$-0.476276\pi$
$251$	2.35936	0.148921	0.0744607	+	0.997224i	$0.476276\pi$
$252$	0	0
$253$	− 14.6284i	− 0.919681i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.65548i	− 0.103266i	−0.998666	−	0.0516330i	$-0.983557\pi$
$257$	− 1.65548i	− 0.103266i	0.998666	−	0.0516330i	$-0.0164426\pi$
$258$	0	0
$259$	28.6890	1.78265
$260$	0	0
$261$	−9.58002	−0.592988
$262$	0	0
$263$	16.6332i	1.02565i	0.858494	+	0.512824i	$0.171401\pi$
$263$	16.6332i	1.02565i	−0.858494	+	0.512824i	$0.828599\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.8884i	0.972352i
$268$	0	0
$269$	−16.0968	−0.981439	−0.490720	−	0.871318i	$-0.663266\pi$
$269$	−16.0968	−0.981439	−0.490720	+	0.871318i	$0.663266\pi$
$270$	0	0
$271$	−10.4758	−0.636360	−0.318180	−	0.948030i	$-0.603072\pi$
$271$	−10.4758	−0.636360	−0.318180	+	0.948030i	$0.603072\pi$
$272$	0	0
$273$	− 5.64064i	− 0.341387i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.3929i	1.04504i	0.852627	+	0.522520i	$0.175008\pi$
$277$	17.3929i	1.04504i	−0.852627	+	0.522520i	$0.824992\pi$
$278$	0	0
$279$	−11.1648	−0.668422
$280$	0	0
$281$	11.2961	0.673870	0.336935	−	0.941528i	$-0.390610\pi$
$281$	11.2961	0.673870	0.336935	+	0.941528i	$0.390610\pi$
$282$	0	0
$283$	− 9.24030i	− 0.549279i	−0.961547	−	0.274640i	$-0.911441\pi$
$283$	− 9.24030i	− 0.549279i	0.961547	−	0.274640i	$-0.0885585\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.93676i	0.173352i
$288$	0	0
$289$	16.5046	0.970856
$290$	0	0
$291$	24.8007	1.45384
$292$	0	0
$293$	16.4051i	0.958399i	0.877706	+	0.479199i	$0.159073\pi$
$293$	16.4051i	0.958399i	−0.877706	+	0.479199i	$0.840927\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	31.1600i	1.80809i
$298$	0	0
$299$	2.64806	0.153141
$300$	0	0
$301$	−5.64064	−0.325121
$302$	0	0
$303$	− 8.11164i	− 0.466001i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.1090i	− 0.862318i	−0.902276	−	0.431159i	$-0.858105\pi$
$307$	− 15.1090i	− 0.862318i	0.902276	−	0.431159i	$-0.141895\pi$
$308$	0	0
$309$	14.8613	0.845430
$310$	0	0
$311$	−8.11164	−0.459969	−0.229984	−	0.973194i	$-0.573867\pi$
$311$	−8.11164	−0.459969	−0.229984	+	0.973194i	$0.573867\pi$
$312$	0	0
$313$	1.39292i	0.0787325i	0.999225	+	0.0393662i	$0.0125339\pi$
$313$	1.39292i	0.0787325i	−0.999225	+	0.0393662i	$0.987466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 29.8129i	− 1.67446i	−0.546851	−	0.837230i	$-0.684174\pi$
$317$	− 29.8129i	− 1.67446i	0.546851	−	0.837230i	$-0.315826\pi$
$318$	0	0
$319$	45.1452	2.52765
$320$	0	0
$321$	9.45355	0.527645
$322$	0	0
$323$	4.80068i	0.267117i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.60708i	0.254772i
$328$	0	0
$329$	34.0968	1.87982
$330$	0	0
$331$	−19.1648	−1.05339	−0.526697	−	0.850053i	$-0.676570\pi$
$331$	−19.1648	−1.05339	−0.526697	+	0.850053i	$0.676570\pi$
$332$	0	0
$333$	− 8.06063i	− 0.441720i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.6406i	0.634106i	0.948408	+	0.317053i	$0.102693\pi$
$337$	11.6406i	0.634106i	−0.948408	+	0.317053i	$0.897307\pi$
$338$	0	0
$339$	−4.92193	−0.267322
$340$	0	0
$341$	52.6136	2.84919
$342$	0	0
$343$	− 14.2180i	− 0.767702i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.7597i	− 1.22180i	−0.791706	−	0.610902i	$-0.790807\pi$
$347$	− 22.7597i	− 1.22180i	0.791706	−	0.610902i	$-0.209193\pi$
$348$	0	0
$349$	−1.04840	−0.0561195	−0.0280598	−	0.999606i	$-0.508933\pi$
$349$	−1.04840	−0.0561195	−0.0280598	+	0.999606i	$0.508933\pi$
$350$	0	0
$351$	−5.64064	−0.301075
$352$	0	0
$353$	− 4.40515i	− 0.234462i	−0.993105	−	0.117231i	$-0.962598\pi$
$353$	− 4.40515i	− 0.234462i	0.993105	−	0.117231i	$-0.0374018\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.97033i	− 0.210132i
$358$	0	0
$359$	−24.2281	−1.27871	−0.639355	−	0.768912i	$-0.720798\pi$
$359$	−24.2281	−1.27871	−0.639355	+	0.768912i	$0.720798\pi$
$360$	0	0
$361$	27.5168	1.44825
$362$	0	0
$363$	− 26.3855i	− 1.38488i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 27.3371i	− 1.42699i	−0.700663	−	0.713493i	$-0.747112\pi$
$367$	− 27.3371i	− 1.42699i	0.700663	−	0.713493i	$-0.252888\pi$
$368$	0	0
$369$	0.825129	0.0429545
$370$	0	0
$371$	21.0632	1.09355
$372$	0	0
$373$	− 22.6890i	− 1.17479i	−0.809299	−	0.587397i	$-0.800153\pi$
$373$	− 22.6890i	− 1.17479i	0.809299	−	0.587397i	$-0.199847\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.17226i	0.420893i
$378$	0	0
$379$	−30.3249	−1.55768	−0.778842	−	0.627220i	$-0.784193\pi$
$379$	−30.3249	−1.55768	−0.778842	+	0.627220i	$0.784193\pi$
$380$	0	0
$381$	−7.70127	−0.394548
$382$	0	0
$383$	5.23550i	0.267521i	0.991014	+	0.133761i	$0.0427053\pi$
$383$	5.23550i	0.267521i	−0.991014	+	0.133761i	$0.957295\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.58482i	0.0805612i
$388$	0	0
$389$	−23.8735	−1.21044	−0.605218	−	0.796060i	$-0.706914\pi$
$389$	−23.8735	−1.21044	−0.605218	+	0.796060i	$0.706914\pi$
$390$	0	0
$391$	1.86391	0.0942621
$392$	0	0
$393$	− 24.0000i	− 1.21064i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	39.2207i	1.96843i	0.176981	+	0.984214i	$0.443367\pi$
$397$	39.2207i	1.96843i	−0.176981	+	0.984214i	$0.556633\pi$
$398$	0	0
$399$	−38.4710	−1.92596
$400$	0	0
$401$	31.0336	1.54974	0.774871	−	0.632119i	$-0.217815\pi$
$401$	31.0336	1.54974	0.774871	+	0.632119i	$0.217815\pi$
$402$	0	0
$403$	9.52420i	0.474434i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.9852i	1.88285i
$408$	0	0
$409$	18.1116	0.895563	0.447781	−	0.894143i	$-0.352214\pi$
$409$	18.1116	0.895563	0.447781	+	0.894143i	$0.352214\pi$
$410$	0	0
$411$	−12.0820	−0.595959
$412$	0	0
$413$	51.0187i	2.51047i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.9755i	0.880266i
$418$	0	0
$419$	23.0484	1.12599	0.562994	−	0.826461i	$-0.309649\pi$
$419$	23.0484	1.12599	0.562994	+	0.826461i	$0.309649\pi$
$420$	0	0
$421$	−34.9123	−1.70152	−0.850761	−	0.525553i	$-0.823859\pi$
$421$	−34.9123	−1.70152	−0.850761	+	0.525553i	$0.823859\pi$
$422$	0	0
$423$	− 9.58002i	− 0.465796i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.718715i	− 0.0347811i
$428$	0	0
$429$	7.46838	0.360577
$430$	0	0
$431$	6.47580	0.311928	0.155964	−	0.987763i	$-0.450152\pi$
$431$	6.47580	0.311928	0.155964	+	0.987763i	$0.450152\pi$
$432$	0	0
$433$	24.8155i	1.19256i	0.802777	+	0.596279i	$0.203355\pi$
$433$	24.8155i	1.19256i	−0.802777	+	0.596279i	$0.796645\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.0606i	− 0.863957i
$438$	0	0
$439$	−10.8155	−0.516196	−0.258098	−	0.966119i	$-0.583096\pi$
$439$	−10.8155	−0.516196	−0.258098	+	0.966119i	$0.583096\pi$
$440$	0	0
$441$	−12.2006	−0.580981
$442$	0	0
$443$	− 18.0410i	− 0.857153i	−0.903506	−	0.428576i	$-0.859015\pi$
$443$	− 18.0410i	− 0.857153i	0.903506	−	0.428576i	$-0.140985\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 12.0820i	− 0.571457i
$448$	0	0
$449$	−27.1452	−1.28106	−0.640531	−	0.767933i	$-0.721286\pi$
$449$	−27.1452	−1.28106	−0.640531	+	0.767933i	$0.721286\pi$
$450$	0	0
$451$	−3.88836	−0.183096
$452$	0	0
$453$	− 3.96383i	− 0.186237i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.75228i	− 0.175524i	−0.996141	−	0.0877621i	$-0.972028\pi$
$457$	− 3.75228i	− 0.175524i	0.996141	−	0.0877621i	$-0.0279715\pi$
$458$	0	0
$459$	−3.97033	−0.185319
$460$	0	0
$461$	36.3297	1.69204	0.846021	−	0.533150i	$-0.178992\pi$
$461$	36.3297	1.69204	0.846021	+	0.533150i	$0.178992\pi$
$462$	0	0
$463$	29.4684i	1.36951i	0.728772	+	0.684756i	$0.240091\pi$
$463$	29.4684i	1.36951i	−0.728772	+	0.684756i	$0.759909\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.5848i	− 0.813729i	−0.913489	−	0.406864i	$-0.866622\pi$
$467$	− 17.5848i	− 0.813729i	0.913489	−	0.406864i	$-0.133378\pi$
$468$	0	0
$469$	−45.3781	−2.09537
$470$	0	0
$471$	0.800677	0.0368932
$472$	0	0
$473$	− 7.46838i	− 0.343397i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5.91804i	− 0.270968i
$478$	0	0
$479$	11.2765	0.515235	0.257618	−	0.966247i	$-0.417062\pi$
$479$	11.2765	0.515235	0.257618	+	0.966247i	$0.417062\pi$
$480$	0	0
$481$	−6.87614	−0.313525
$482$	0	0
$483$	14.9368i	0.679646i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 26.1574i	− 1.18531i	−0.805458	−	0.592653i	$-0.798081\pi$
$487$	− 26.1574i	− 1.18531i	0.805458	−	0.592653i	$-0.201919\pi$
$488$	0	0
$489$	−22.4806	−1.01661
$490$	0	0
$491$	0.456156	0.0205860	0.0102930	−	0.999947i	$-0.496724\pi$
$491$	0.456156	0.0205860	0.0102930	+	0.999947i	$0.496724\pi$
$492$	0	0
$493$	5.75228i	0.259070i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 23.0484i	− 1.03386i
$498$	0	0
$499$	12.4610	0.557829	0.278915	−	0.960316i	$-0.410025\pi$
$499$	12.4610	0.557829	0.278915	+	0.960316i	$0.410025\pi$
$500$	0	0
$501$	−5.17487	−0.231196
$502$	0	0
$503$	11.7933i	0.525835i	0.964818	+	0.262918i	$0.0846848\pi$
$503$	11.7933i	0.525835i	−0.964818	+	0.262918i	$0.915315\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.35194i	0.0600417i
$508$	0	0
$509$	29.0484	1.28755	0.643774	−	0.765216i	$-0.277368\pi$
$509$	29.0484	1.28755	0.643774	+	0.765216i	$0.277368\pi$
$510$	0	0
$511$	−46.8155	−2.07100
$512$	0	0
$513$	38.4710i	1.69854i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	45.1452i	1.98548i
$518$	0	0
$519$	−19.0781	−0.837434
$520$	0	0
$521$	−15.3323	−0.671720	−0.335860	−	0.941912i	$-0.609027\pi$
$521$	−15.3323	−0.671720	−0.335860	+	0.941912i	$0.609027\pi$
$522$	0	0
$523$	− 15.4487i	− 0.675526i	−0.941231	−	0.337763i	$-0.890330\pi$
$523$	− 15.4487i	− 0.675526i	0.941231	−	0.337763i	$-0.109670\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.70388i	0.292026i
$528$	0	0
$529$	15.9878	0.695121
$530$	0	0
$531$	14.3345	0.622064
$532$	0	0
$533$	− 0.703878i	− 0.0304884i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.77673i	0.335591i
$538$	0	0
$539$	57.4945	2.47646
$540$	0	0
$541$	−38.8007	−1.66817	−0.834086	−	0.551635i	$-0.814004\pi$
$541$	−38.8007	−1.66817	−0.834086	+	0.551635i	$0.814004\pi$
$542$	0	0
$543$	− 3.73744i	− 0.160389i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.9926i	0.641036i	0.947242	+	0.320518i	$0.103857\pi$
$547$	14.9926i	0.641036i	−0.947242	+	0.320518i	$0.896143\pi$
$548$	0	0
$549$	−0.201934	−0.00861833
$550$	0	0
$551$	55.7374	2.37449
$552$	0	0
$553$	− 5.40776i	− 0.229961i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26.3807i	− 1.11779i	−0.829240	−	0.558893i	$-0.811226\pi$
$557$	− 26.3807i	− 1.11779i	0.829240	−	0.558893i	$-0.188774\pi$
$558$	0	0
$559$	1.35194	0.0571809
$560$	0	0
$561$	5.25683	0.221944
$562$	0	0
$563$	42.0410i	1.77182i	0.463861	+	0.885908i	$0.346464\pi$
$563$	42.0410i	1.77182i	−0.463861	+	0.885908i	$0.653536\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17.1439i	− 0.719977i
$568$	0	0
$569$	−30.2691	−1.26894	−0.634472	−	0.772945i	$-0.718783\pi$
$569$	−30.2691	−1.26894	−0.634472	+	0.772945i	$0.718783\pi$
$570$	0	0
$571$	−12.1116	−0.506856	−0.253428	−	0.967354i	$-0.581558\pi$
$571$	−12.1116	−0.506856	−0.253428	+	0.967354i	$0.581558\pi$
$572$	0	0
$573$	− 16.2233i	− 0.677737i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8.40515i	− 0.349911i	−0.984576	−	0.174955i	$-0.944022\pi$
$577$	− 8.40515i	− 0.349911i	0.984576	−	0.174955i	$-0.0559781\pi$
$578$	0	0
$579$	19.3929	0.805942
$580$	0	0
$581$	−39.9703	−1.65825
$582$	0	0
$583$	27.8884i	1.15502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.4439i	1.95822i	0.203330	+	0.979110i	$0.434824\pi$
$587$	47.4439i	1.95822i	−0.203330	+	0.979110i	$0.565176\pi$
$588$	0	0
$589$	64.9581	2.67655
$590$	0	0
$591$	−10.0148	−0.411955
$592$	0	0
$593$	− 32.4413i	− 1.33221i	−0.745860	−	0.666103i	$-0.767961\pi$
$593$	− 32.4413i	− 1.33221i	0.745860	−	0.666103i	$-0.232039\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.36897i	0.0969556i
$598$	0	0
$599$	34.6742	1.41675	0.708375	−	0.705836i	$-0.249429\pi$
$599$	34.6742	1.41675	0.708375	+	0.705836i	$0.249429\pi$
$600$	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
$603$	12.7497i	0.519207i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.30354i	− 0.0934978i	−0.998907	−	0.0467489i	$-0.985114\pi$
$607$	− 2.30354i	− 0.0934978i	0.998907	−	0.0467489i	$-0.0148861\pi$
$608$	0	0
$609$	−46.0968	−1.86794
$610$	0	0
$611$	−8.17226	−0.330614
$612$	0	0
$613$	15.8735i	0.641126i	0.947227	+	0.320563i	$0.103872\pi$
$613$	15.8735i	0.641126i	−0.947227	+	0.320563i	$0.896128\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.3445i	0.416455i	0.978080	+	0.208227i	$0.0667694\pi$
$617$	10.3445i	0.416455i	−0.978080	+	0.208227i	$0.933231\pi$
$618$	0	0
$619$	−37.1500	−1.49318	−0.746592	−	0.665282i	$-0.768311\pi$
$619$	−37.1500	−1.49318	−0.746592	+	0.665282i	$0.768311\pi$
$620$	0	0
$621$	14.9368	0.599392
$622$	0	0
$623$	− 49.0336i	− 1.96449i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 50.9368i	− 2.03422i
$628$	0	0
$629$	−4.83997	−0.192982
$630$	0	0
$631$	10.2132	0.406583	0.203291	−	0.979118i	$-0.434836\pi$
$631$	10.2132	0.406583	0.203291	+	0.979118i	$0.434836\pi$
$632$	0	0
$633$	13.1845i	0.524036i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.4078i	0.412370i
$638$	0	0
$639$	−6.47580	−0.256179
$640$	0	0
$641$	−16.0968	−0.635785	−0.317893	−	0.948127i	$-0.602975\pi$
$641$	−16.0968	−0.635785	−0.317893	+	0.948127i	$0.602975\pi$
$642$	0	0
$643$	18.0458i	0.711656i	0.934551	+	0.355828i	$0.115801\pi$
$643$	18.0458i	0.711656i	−0.934551	+	0.355828i	$0.884199\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 43.5700i	− 1.71291i	−0.516219	−	0.856456i	$-0.672661\pi$
$647$	− 43.5700i	− 1.71291i	0.516219	−	0.856456i	$-0.327339\pi$
$648$	0	0
$649$	−67.5503	−2.65158
$650$	0	0
$651$	−53.7226	−2.10555
$652$	0	0
$653$	22.3445i	0.874409i	0.899362	+	0.437204i	$0.144031\pi$
$653$	22.3445i	0.874409i	−0.899362	+	0.437204i	$0.855969\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.1535i	0.513168i
$658$	0	0
$659$	−28.9219	−1.12664	−0.563319	−	0.826239i	$-0.690476\pi$
$659$	−28.9219	−1.12664	−0.563319	+	0.826239i	$0.690476\pi$
$660$	0	0
$661$	−18.6890	−0.726919	−0.363460	−	0.931610i	$-0.618405\pi$
$661$	−18.6890	−0.726919	−0.363460	+	0.931610i	$0.618405\pi$
$662$	0	0
$663$	0.951601i	0.0369571i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 21.6406i	− 0.837929i
$668$	0	0
$669$	−3.73744	−0.144498
$670$	0	0
$671$	0.951601	0.0367361
$672$	0	0
$673$	− 23.5194i	− 0.906606i	−0.891357	−	0.453303i	$-0.850246\pi$
$673$	− 23.5194i	− 0.906606i	0.891357	−	0.453303i	$-0.149754\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.35936i	− 0.321276i	−0.987013	−	0.160638i	$-0.948645\pi$
$677$	− 8.35936i	− 0.321276i	0.987013	−	0.160638i	$-0.0513552\pi$
$678$	0	0
$679$	−76.5381	−2.93726
$680$	0	0
$681$	24.0820	0.922823
$682$	0	0
$683$	11.9394i	0.456847i	0.973562	+	0.228424i	$0.0733572\pi$
$683$	11.9394i	0.456847i	−0.973562	+	0.228424i	$0.926643\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.6703i	0.521555i
$688$	0	0
$689$	−5.04840	−0.192329
$690$	0	0
$691$	15.4274	0.586886	0.293443	−	0.955977i	$-0.405199\pi$
$691$	15.4274	0.586886	0.293443	+	0.955977i	$0.405199\pi$
$692$	0	0
$693$	− 27.0187i	− 1.02636i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 0.495445i	− 0.0187663i
$698$	0	0
$699$	8.11164	0.306810
$700$	0	0
$701$	37.2813	1.40809	0.704047	−	0.710153i	$-0.251374\pi$
$701$	37.2813	1.40809	0.704047	+	0.710153i	$0.251374\pi$
$702$	0	0
$703$	46.8975i	1.76877i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	25.0336i	0.941484i
$708$	0	0
$709$	11.9852	0.450112	0.225056	−	0.974346i	$-0.427743\pi$
$709$	11.9852	0.450112	0.225056	+	0.974346i	$0.427743\pi$
$710$	0	0
$711$	−1.51939	−0.0569817
$712$	0	0
$713$	− 25.2207i	− 0.944521i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.3419i	0.498263i
$718$	0	0
$719$	−23.6258	−0.881094	−0.440547	−	0.897730i	$-0.645215\pi$
$719$	−23.6258	−0.881094	−0.440547	+	0.897730i	$0.645215\pi$
$720$	0	0
$721$	−45.8639	−1.70806
$722$	0	0
$723$	2.55295i	0.0949454i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	22.1526i	0.821595i	0.911727	+	0.410798i	$0.134750\pi$
$727$	22.1526i	0.821595i	−0.911727	+	0.410798i	$0.865250\pi$
$728$	0	0
$729$	−27.6943	−1.02571
$730$	0	0
$731$	0.951601	0.0351962
$732$	0	0
$733$	10.0000i	0.369358i	0.982799	+	0.184679i	$0.0591246\pi$
$733$	10.0000i	0.369358i	−0.982799	+	0.184679i	$0.940875\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 60.0820i	− 2.21315i
$738$	0	0
$739$	19.0436	0.700530	0.350265	−	0.936651i	$-0.386092\pi$
$739$	19.0436	0.700530	0.350265	+	0.936651i	$0.386092\pi$
$740$	0	0
$741$	9.22066	0.338729
$742$	0	0
$743$	− 6.26906i	− 0.229989i	−0.993366	−	0.114995i	$-0.963315\pi$
$743$	− 6.26906i	− 0.229989i	0.993366	−	0.114995i	$-0.0366851\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.2303i	0.410894i
$748$	0	0
$749$	−29.1749	−1.06603
$750$	0	0
$751$	−20.9219	−0.763452	−0.381726	−	0.924276i	$-0.624670\pi$
$751$	−20.9219	−0.763452	−0.381726	+	0.924276i	$0.624670\pi$
$752$	0	0
$753$	− 3.18971i	− 0.116239i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 48.8975i	− 1.77721i	−0.458674	−	0.888604i	$-0.651676\pi$
$757$	− 48.8975i	− 1.77721i	0.458674	−	0.888604i	$-0.348324\pi$
$758$	0	0
$759$	−19.7767	−0.717850
$760$	0	0
$761$	−4.09680	−0.148509	−0.0742544	−	0.997239i	$-0.523658\pi$
$761$	−4.09680	−0.148509	−0.0742544	+	0.997239i	$0.523658\pi$
$762$	0	0
$763$	− 14.2180i	− 0.514728i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 12.2281i	− 0.441530i
$768$	0	0
$769$	28.7858	1.03804	0.519022	−	0.854761i	$-0.326296\pi$
$769$	28.7858	1.03804	0.519022	+	0.854761i	$0.326296\pi$
$770$	0	0
$771$	−2.23811	−0.0806035
$772$	0	0
$773$	− 7.22066i	− 0.259709i	−0.991533	−	0.129855i	$-0.958549\pi$
$773$	− 7.22066i	− 0.259709i	0.991533	−	0.129855i	$-0.0414510\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 38.7858i	− 1.39143i
$778$	0	0
$779$	−4.80068	−0.172002
$780$	0	0
$781$	30.5168	1.09198
$782$	0	0
$783$	46.0968i	1.64737i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	41.5800i	1.48217i	0.671413	+	0.741084i	$0.265688\pi$
$787$	41.5800i	1.48217i	−0.671413	+	0.741084i	$0.734312\pi$
$788$	0	0
$789$	22.4871	0.800562
$790$	0	0
$791$	15.1897	0.540084
$792$	0	0
$793$	0.172260i	0.00611715i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.4078i	0.687458i	0.939069	+	0.343729i	$0.111690\pi$
$797$	19.4078i	0.687458i	−0.939069	+	0.343729i	$0.888310\pi$
$798$	0	0
$799$	−5.75228	−0.203501
$800$	0	0
$801$	−13.7767	−0.486777
$802$	0	0
$803$	− 61.9852i	− 2.18741i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.7619i	0.766055i
$808$	0	0
$809$	−43.7981	−1.53986	−0.769929	−	0.638130i	$-0.779708\pi$
$809$	−43.7981	−1.53986	−0.769929	+	0.638130i	$0.779708\pi$
$810$	0	0
$811$	−5.75709	−0.202159	−0.101079	−	0.994878i	$-0.532230\pi$
$811$	−5.75709	−0.202159	−0.101079	+	0.994878i	$0.532230\pi$
$812$	0	0
$813$	14.1626i	0.496706i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.22066i	− 0.322590i
$818$	0	0
$819$	4.89098	0.170905
$820$	0	0
$821$	−32.9368	−1.14950	−0.574750	−	0.818329i	$-0.694901\pi$
$821$	−32.9368	−1.14950	−0.574750	+	0.818329i	$0.694901\pi$
$822$	0	0
$823$	57.0894i	1.99001i	0.0998217	+	0.995005i	$0.468173\pi$
$823$	57.0894i	1.99001i	−0.0998217	+	0.995005i	$0.531827\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 27.4535i	− 0.954653i	−0.878726	−	0.477327i	$-0.841606\pi$
$827$	− 27.4535i	− 0.954653i	0.878726	−	0.477327i	$-0.158394\pi$
$828$	0	0
$829$	−10.4200	−0.361901	−0.180950	−	0.983492i	$-0.557917\pi$
$829$	−10.4200	−0.361901	−0.180950	+	0.983492i	$0.557917\pi$
$830$	0	0
$831$	23.5142	0.815698
$832$	0	0
$833$	7.32580i	0.253824i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	53.7226i	1.85692i
$838$	0	0
$839$	−40.1984	−1.38780	−0.693902	−	0.720070i	$-0.744110\pi$
$839$	−40.1984	−1.38780	−0.693902	+	0.720070i	$0.744110\pi$
$840$	0	0
$841$	37.7858	1.30296
$842$	0	0
$843$	− 15.2717i	− 0.525984i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	81.4291i	2.79794i
$848$	0	0
$849$	−12.4923	−0.428736
$850$	0	0
$851$	18.2084	0.624177
$852$	0	0
$853$	− 34.8761i	− 1.19414i	−0.802191	−	0.597068i	$-0.796332\pi$
$853$	− 34.8761i	− 1.19414i	0.802191	−	0.597068i	$-0.203668\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.1600i	0.859450i	0.902960	+	0.429725i	$0.141390\pi$
$857$	25.1600i	0.859450i	−0.902960	+	0.429725i	$0.858610\pi$
$858$	0	0
$859$	15.0484	0.513445	0.256722	−	0.966485i	$-0.417357\pi$
$859$	15.0484	0.513445	0.256722	+	0.966485i	$0.417357\pi$
$860$	0	0
$861$	3.97033	0.135308
$862$	0	0
$863$	− 2.29873i	− 0.0782498i	−0.999234	−	0.0391249i	$-0.987543\pi$
$863$	− 2.29873i	− 0.0782498i	0.999234	−	0.0391249i	$-0.0124570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 22.3132i	− 0.757794i
$868$	0	0
$869$	7.16003	0.242888
$870$	0	0
$871$	10.8761	0.368524
$872$	0	0
$873$	21.5046i	0.727819i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 10.0000i	− 0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$	− 10.0000i	− 0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$	0	0
$879$	22.1788	0.748071
$880$	0	0
$881$	−48.1426	−1.62196	−0.810982	−	0.585070i	$-0.801067\pi$
$881$	−48.1426	−1.62196	−0.810982	+	0.585070i	$0.801067\pi$
$882$	0	0
$883$	12.8565i	0.432655i	0.976321	+	0.216328i	$0.0694080\pi$
$883$	12.8565i	0.432655i	−0.976321	+	0.216328i	$0.930592\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32.5216i	− 1.09197i	−0.837796	−	0.545984i	$-0.816156\pi$
$887$	− 32.5216i	− 1.09197i	0.837796	−	0.545984i	$-0.183844\pi$
$888$	0	0
$889$	23.7671	0.797123
$890$	0	0
$891$	22.6991	0.760447
$892$	0	0
$893$	55.7374i	1.86518i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.58002i	− 0.119533i
$898$	0	0
$899$	77.8342	2.59592
$900$	0	0
$901$	−3.55346	−0.118383
$902$	0	0
$903$	7.62581i	0.253771i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 21.0894i	− 0.700261i	−0.936701	−	0.350131i	$-0.886137\pi$
$907$	− 21.0894i	− 0.700261i	0.936701	−	0.350131i	$-0.113863\pi$
$908$	0	0
$909$	7.03356	0.233289
$910$	0	0
$911$	43.1600	1.42996	0.714978	−	0.699147i	$-0.246437\pi$
$911$	43.1600	1.42996	0.714978	+	0.699147i	$0.246437\pi$
$912$	0	0
$913$	− 52.9219i	− 1.75146i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	74.0671i	2.44591i
$918$	0	0
$919$	−11.8884	−0.392161	−0.196080	−	0.980588i	$-0.562821\pi$
$919$	−11.8884	−0.392161	−0.196080	+	0.980588i	$0.562821\pi$
$920$	0	0
$921$	−20.4265	−0.673075
$922$	0	0
$923$	5.52420i	0.181831i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.8862i	0.423237i
$928$	0	0
$929$	34.4265	1.12950	0.564748	−	0.825263i	$-0.308973\pi$
$929$	34.4265	1.12950	0.564748	+	0.825263i	$0.308973\pi$
$930$	0	0
$931$	70.9842	2.32641
$932$	0	0
$933$	10.9664i	0.359025i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	0	0
$939$	1.88314	0.0614541
$940$	0	0
$941$	−38.1116	−1.24240	−0.621202	−	0.783651i	$-0.713355\pi$
$941$	−38.1116	−1.24240	−0.621202	+	0.783651i	$0.713355\pi$
$942$	0	0
$943$	1.86391i	0.0606974i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13.9245i	0.452487i	0.974071	+	0.226243i	$0.0726445\pi$
$947$	13.9245i	0.452487i	−0.974071	+	0.226243i	$0.927356\pi$
$948$	0	0
$949$	11.2207	0.364238
$950$	0	0
$951$	−40.3052	−1.30699
$952$	0	0
$953$	23.7523i	0.769412i	0.923039	+	0.384706i	$0.125697\pi$
$953$	23.7523i	0.769412i	−0.923039	+	0.384706i	$0.874303\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 61.0336i	− 1.97293i
$958$	0	0
$959$	37.2865	1.20404
$960$	0	0
$961$	59.7104	1.92614
$962$	0	0
$963$	8.19713i	0.264149i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 6.87614i	− 0.221122i	−0.993869	−	0.110561i	$-0.964735\pi$
$967$	− 6.87614i	− 0.221122i	0.993869	−	0.110561i	$-0.0352647\pi$
$968$	0	0
$969$	6.49022	0.208496
$970$	0	0
$971$	6.24772	0.200499	0.100249	−	0.994962i	$-0.468036\pi$
$971$	6.24772	0.200499	0.100249	+	0.994962i	$0.468036\pi$
$972$	0	0
$973$	− 55.4749i	− 1.77844i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 34.5316i	− 1.10476i	−0.833591	−	0.552382i	$-0.813719\pi$
$977$	− 34.5316i	− 1.10476i	0.833591	−	0.552382i	$-0.186281\pi$
$978$	0	0
$979$	64.9219	2.07491
$980$	0	0
$981$	−3.99478	−0.127543
$982$	0	0
$983$	11.5652i	0.368872i	0.982845	+	0.184436i	$0.0590458\pi$
$983$	11.5652i	0.368872i	−0.982845	+	0.184436i	$0.940954\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 46.0968i	− 1.46728i
$988$	0	0
$989$	−3.58002	−0.113838
$990$	0	0
$991$	3.77673	0.119972	0.0599859	−	0.998199i	$-0.480894\pi$
$991$	3.77673	0.119972	0.0599859	+	0.998199i	$0.480894\pi$
$992$	0	0
$993$	25.9097i	0.822220i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	54.4265i	1.72370i	0.507160	+	0.861852i	$0.330695\pi$
$997$	54.4265i	1.72370i	−0.507160	+	0.861852i	$0.669305\pi$
$998$	0	0
$999$	−38.7858	−1.22713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.c.f.1249.3		6
5.2	odd	4		1300.2.a.i.1.2		3
5.3	odd	4		260.2.a.b.1.2	✓	3
5.4	even	2	inner	1300.2.c.f.1249.4		6
15.8	even	4		2340.2.a.n.1.3		3
20.3	even	4		1040.2.a.o.1.2		3
20.7	even	4		5200.2.a.ci.1.2		3
40.3	even	4		4160.2.a.br.1.2		3
40.13	odd	4		4160.2.a.bo.1.2		3
60.23	odd	4		9360.2.a.da.1.1		3
65.8	even	4		3380.2.f.h.3041.4		6
65.18	even	4		3380.2.f.h.3041.3		6
65.38	odd	4		3380.2.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.a.b.1.2	✓	3	5.3	odd	4
1040.2.a.o.1.2		3	20.3	even	4
1300.2.a.i.1.2		3	5.2	odd	4
1300.2.c.f.1249.3		6	1.1	even	1	trivial
1300.2.c.f.1249.4		6	5.4	even	2	inner
2340.2.a.n.1.3		3	15.8	even	4
3380.2.a.o.1.2		3	65.38	odd	4
3380.2.f.h.3041.3		6	65.18	even	4
3380.2.f.h.3041.4		6	65.8	even	4
4160.2.a.bo.1.2		3	40.13	odd	4
4160.2.a.br.1.2		3	40.3	even	4
5200.2.a.ci.1.2		3	20.7	even	4
9360.2.a.da.1.1		3	60.23	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.35194i q^{3} +4.17226i q^{7} +1.17226 q^{9} +O(q^{10})\)	\(q-1.35194i q^{3} +4.17226i q^{7} +1.17226 q^{9} -5.52420 q^{11} -1.00000i q^{13} -0.703878i q^{17} -6.82032 q^{19} +5.64064 q^{21} +2.64806i q^{23} -5.64064i q^{27} -8.17226 q^{29} -9.52420 q^{31} +7.46838i q^{33} -6.87614i q^{37} -1.35194 q^{39} +0.703878 q^{41} +1.35194i q^{43} -8.17226i q^{47} -10.4078 q^{49} -0.951601 q^{51} -5.04840i q^{53} +9.22066i q^{57} +12.2281 q^{59} -0.172260 q^{61} +4.89098i q^{63} +10.8761i q^{67} +3.58002 q^{69} -5.52420 q^{71} +11.2207i q^{73} -23.0484i q^{77} -1.29612 q^{79} -4.10902 q^{81} +9.58002i q^{83} +11.0484i q^{87} -11.7523 q^{89} +4.17226 q^{91} +12.8761i q^{93} +18.3445i q^{97} -6.47580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{9}+O(q^{10}) \) 6 * q - 22 * q^9	\( 6 q - 22 q^{9} - 16 q^{19} - 16 q^{21} - 20 q^{29} - 24 q^{31} - 4 q^{39} - 4 q^{41} - 46 q^{49} - 72 q^{51} + 32 q^{59} + 28 q^{61} - 24 q^{69} - 16 q^{79} + 46 q^{81} + 4 q^{89} - 4 q^{91} - 72 q^{99}+O(q^{100}) \) 6 * q - 22 * q^9 - 16 * q^19 - 16 * q^21 - 20 * q^29 - 24 * q^31 - 4 * q^39 - 4 * q^41 - 46 * q^49 - 72 * q^51 + 32 * q^59 + 28 * q^61 - 24 * q^69 - 16 * q^79 + 46 * q^81 + 4 * q^89 - 4 * q^91 - 72 * q^99

Introduction

L-functions

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Database

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