LMFDB - Embedded newform 1300.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1,1300)]

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, 0, 0]))

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

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

chi := DirichletCharacter("1300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.a (trivial)

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:	yes
Analytic conductor:	$10.3805522628$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	1300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67513 q^{3} -1.19394 q^{7} -0.193937 q^{9} +O(q^{10})$	$q+1.67513 q^{3} -1.19394 q^{7} -0.193937 q^{9} -5.44358 q^{11} -1.00000 q^{13} -6.96239 q^{17} -0.869067 q^{19} -2.00000 q^{21} -0.324869 q^{23} -5.35026 q^{27} -4.15633 q^{29} +2.09332 q^{31} -9.11871 q^{33} +1.50659 q^{37} -1.67513 q^{39} +9.92478 q^{41} +7.86177 q^{43} -6.80606 q^{47} -5.57452 q^{49} -11.6629 q^{51} -11.3503 q^{53} -1.45580 q^{57} +9.18172 q^{59} +9.43136 q^{61} +0.231548 q^{63} +7.11871 q^{67} -0.544198 q^{69} +8.21933 q^{71} -3.58181 q^{73} +6.49929 q^{77} -2.57452 q^{79} -8.38058 q^{81} -2.99271 q^{83} -6.96239 q^{87} +9.47627 q^{89} +1.19394 q^{91} +3.50659 q^{93} -14.4993 q^{97} +1.05571 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 4 * q^7 - q^9	$ 3 q - 4 q^{7} - q^{9} - 3 q^{13} - 10 q^{17} + 2 q^{19} - 6 q^{21} - 6 q^{23} - 6 q^{27} - 2 q^{29} - 6 q^{33} - 16 q^{37} + 8 q^{41} + 6 q^{43} - 20 q^{47} - 5 q^{49} - 4 q^{51} - 24 q^{53} - 14 q^{57} + 2 q^{59} - 14 q^{61} + 12 q^{63} + 8 q^{69} + 10 q^{71} - 12 q^{73} - 14 q^{77} + 4 q^{79} - 13 q^{81} + 4 q^{83} - 10 q^{87} + 10 q^{89} + 4 q^{91} - 10 q^{93} - 10 q^{97} - 14 q^{99}+O(q^{100}) $ 3 * q - 4 * q^7 - q^9 - 3 * q^13 - 10 * q^17 + 2 * q^19 - 6 * q^21 - 6 * q^23 - 6 * q^27 - 2 * q^29 - 6 * q^33 - 16 * q^37 + 8 * q^41 + 6 * q^43 - 20 * q^47 - 5 * q^49 - 4 * q^51 - 24 * q^53 - 14 * q^57 + 2 * q^59 - 14 * q^61 + 12 * q^63 + 8 * q^69 + 10 * q^71 - 12 * q^73 - 14 * q^77 + 4 * q^79 - 13 * q^81 + 4 * q^83 - 10 * q^87 + 10 * q^89 + 4 * q^91 - 10 * q^93 - 10 * q^97 - 14 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.67513	0.967137	0.483569	−	0.875306i	$-0.339340\pi$
$3$	1.67513	0.967137	0.483569	+	0.875306i	$0.339340\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.19394	−0.451266	−0.225633	−	0.974212i	$-0.572445\pi$
$7$	−1.19394	−0.451266	−0.225633	+	0.974212i	$0.572445\pi$
$8$	0	0
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−5.44358	−1.64130	−0.820651	−	0.571430i	$-0.806389\pi$
$11$	−5.44358	−1.64130	−0.820651	+	0.571430i	$0.806389\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.96239	−1.68863	−0.844314	−	0.535849i	$-0.819992\pi$
$17$	−6.96239	−1.68863	−0.844314	+	0.535849i	$0.819992\pi$
$18$	0	0
$19$	−0.869067	−0.199378	−0.0996889	−	0.995019i	$-0.531785\pi$
$19$	−0.869067	−0.199378	−0.0996889	+	0.995019i	$0.531785\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−0.324869	−0.0677399	−0.0338699	−	0.999426i	$-0.510783\pi$
$23$	−0.324869	−0.0677399	−0.0338699	+	0.999426i	$0.510783\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.35026	−1.02966
$28$	0	0
$29$	−4.15633	−0.771810	−0.385905	−	0.922538i	$-0.626111\pi$
$29$	−4.15633	−0.771810	−0.385905	+	0.922538i	$0.626111\pi$
$30$	0	0
$31$	2.09332	0.375972	0.187986	−	0.982172i	$-0.439804\pi$
$31$	2.09332	0.375972	0.187986	+	0.982172i	$0.439804\pi$
$32$	0	0
$33$	−9.11871	−1.58736
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.50659	0.247681	0.123841	−	0.992302i	$-0.460479\pi$
$37$	1.50659	0.247681	0.123841	+	0.992302i	$0.460479\pi$
$38$	0	0
$39$	−1.67513	−0.268236
$40$	0	0
$41$	9.92478	1.54999	0.774995	−	0.631967i	$-0.217752\pi$
$41$	9.92478	1.54999	0.774995	+	0.631967i	$0.217752\pi$
$42$	0	0
$43$	7.86177	1.19891	0.599455	−	0.800409i	$-0.295384\pi$
$43$	7.86177	1.19891	0.599455	+	0.800409i	$0.295384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.80606	−0.992766	−0.496383	−	0.868104i	$-0.665339\pi$
$47$	−6.80606	−0.992766	−0.496383	+	0.868104i	$0.665339\pi$
$48$	0	0
$49$	−5.57452	−0.796359
$50$	0	0
$51$	−11.6629	−1.63313
$52$	0	0
$53$	−11.3503	−1.55908	−0.779539	−	0.626353i	$-0.784547\pi$
$53$	−11.3503	−1.55908	−0.779539	+	0.626353i	$0.784547\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.45580	−0.192826
$58$	0	0
$59$	9.18172	1.19536	0.597679	−	0.801736i	$-0.296090\pi$
$59$	9.18172	1.19536	0.597679	+	0.801736i	$0.296090\pi$
$60$	0	0
$61$	9.43136	1.20756	0.603781	−	0.797150i	$-0.293660\pi$
$61$	9.43136	1.20756	0.603781	+	0.797150i	$0.293660\pi$
$62$	0	0
$63$	0.231548	0.0291723
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.11871	0.869689	0.434845	−	0.900505i	$-0.356803\pi$
$67$	7.11871	0.869689	0.434845	+	0.900505i	$0.356803\pi$
$68$	0	0
$69$	−0.544198	−0.0655138
$70$	0	0
$71$	8.21933	0.975455	0.487727	−	0.872996i	$-0.337826\pi$
$71$	8.21933	0.975455	0.487727	+	0.872996i	$0.337826\pi$
$72$	0	0
$73$	−3.58181	−0.419219	−0.209610	−	0.977785i	$-0.567219\pi$
$73$	−3.58181	−0.419219	−0.209610	+	0.977785i	$0.567219\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.49929	0.740663
$78$	0	0
$79$	−2.57452	−0.289656	−0.144828	−	0.989457i	$-0.546263\pi$
$79$	−2.57452	−0.289656	−0.144828	+	0.989457i	$0.546263\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	−2.99271	−0.328492	−0.164246	−	0.986419i	$-0.552519\pi$
$83$	−2.99271	−0.328492	−0.164246	+	0.986419i	$0.552519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.96239	−0.746446
$88$	0	0
$89$	9.47627	1.00448	0.502241	−	0.864728i	$-0.332509\pi$
$89$	9.47627	1.00448	0.502241	+	0.864728i	$0.332509\pi$
$90$	0	0
$91$	1.19394	0.125159
$92$	0	0
$93$	3.50659	0.363616
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.4993	−1.47218	−0.736090	−	0.676884i	$-0.763330\pi$
$97$	−14.4993	−1.47218	−0.736090	+	0.676884i	$0.763330\pi$
$98$	0	0
$99$	1.05571	0.106103
$100$	0	0
$101$	−4.57452	−0.455181	−0.227591	−	0.973757i	$-0.573085\pi$
$101$	−4.57452	−0.455181	−0.227591	+	0.973757i	$0.573085\pi$
$102$	0	0
$103$	−10.9502	−1.07895	−0.539476	−	0.842001i	$-0.681378\pi$
$103$	−10.9502	−1.07895	−0.539476	+	0.842001i	$0.681378\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−20.6131	−1.99274	−0.996371	−	0.0851175i	$-0.972873\pi$
$107$	−20.6131	−1.99274	−0.996371	+	0.0851175i	$0.972873\pi$
$108$	0	0
$109$	10.8872	1.04280	0.521401	−	0.853312i	$-0.325410\pi$
$109$	10.8872	1.04280	0.521401	+	0.853312i	$0.325410\pi$
$110$	0	0
$111$	2.52373	0.239542
$112$	0	0
$113$	4.88717	0.459746	0.229873	−	0.973221i	$-0.426169\pi$
$113$	4.88717	0.459746	0.229873	+	0.973221i	$0.426169\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.193937	0.0179294
$118$	0	0
$119$	8.31265	0.762019
$120$	0	0
$121$	18.6326	1.69387
$122$	0	0
$123$	16.6253	1.49905
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.7259	1.04051	0.520253	−	0.854012i	$-0.325837\pi$
$127$	11.7259	1.04051	0.520253	+	0.854012i	$0.325837\pi$
$128$	0	0
$129$	13.1695	1.15951
$130$	0	0
$131$	−1.61213	−0.140852	−0.0704261	−	0.997517i	$-0.522436\pi$
$131$	−1.61213	−0.140852	−0.0704261	+	0.997517i	$0.522436\pi$
$132$	0	0
$133$	1.03761	0.0899723
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.8641	−1.01362	−0.506811	−	0.862057i	$-0.669176\pi$
$137$	−11.8641	−1.01362	−0.506811	+	0.862057i	$0.669176\pi$
$138$	0	0
$139$	−6.57452	−0.557643	−0.278822	−	0.960343i	$-0.589944\pi$
$139$	−6.57452	−0.557643	−0.278822	+	0.960343i	$0.589944\pi$
$140$	0	0
$141$	−11.4010	−0.960141
$142$	0	0
$143$	5.44358	0.455215
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−9.33804	−0.770189
$148$	0	0
$149$	−10.0508	−0.823392	−0.411696	−	0.911321i	$-0.635064\pi$
$149$	−10.0508	−0.823392	−0.411696	+	0.911321i	$0.635064\pi$
$150$	0	0
$151$	−13.5188	−1.10014	−0.550072	−	0.835117i	$-0.685400\pi$
$151$	−13.5188	−1.10014	−0.550072	+	0.835117i	$0.685400\pi$
$152$	0	0
$153$	1.35026	0.109162
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.47627	−0.437054	−0.218527	−	0.975831i	$-0.570125\pi$
$157$	−5.47627	−0.437054	−0.218527	+	0.975831i	$0.570125\pi$
$158$	0	0
$159$	−19.0132	−1.50784
$160$	0	0
$161$	0.387873	0.0305687
$162$	0	0
$163$	−0.0811024	−0.00635243	−0.00317621	−	0.999995i	$-0.501011\pi$
$163$	−0.0811024	−0.00635243	−0.00317621	+	0.999995i	$0.501011\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.1695	1.94767	0.973837	−	0.227247i	$-0.0729725\pi$
$167$	25.1695	1.94767	0.973837	+	0.227247i	$0.0729725\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.168544	0.0128889
$172$	0	0
$173$	−12.6497	−0.961742	−0.480871	−	0.876791i	$-0.659679\pi$
$173$	−12.6497	−0.961742	−0.480871	+	0.876791i	$0.659679\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.3806	1.15608
$178$	0	0
$179$	11.4109	0.852890	0.426445	−	0.904513i	$-0.359766\pi$
$179$	11.4109	0.852890	0.426445	+	0.904513i	$0.359766\pi$
$180$	0	0
$181$	−15.6326	−1.16196	−0.580981	−	0.813917i	$-0.697331\pi$
$181$	−15.6326	−1.16196	−0.580981	+	0.813917i	$0.697331\pi$
$182$	0	0
$183$	15.7988	1.16788
$184$	0	0
$185$	0	0
$186$	0	0
$187$	37.9003	2.77155
$188$	0	0
$189$	6.38787	0.464649
$190$	0	0
$191$	−17.9248	−1.29699	−0.648496	−	0.761218i	$-0.724602\pi$
$191$	−17.9248	−1.29699	−0.648496	+	0.761218i	$0.724602\pi$
$192$	0	0
$193$	−2.83638	−0.204167	−0.102084	−	0.994776i	$-0.532551\pi$
$193$	−2.83638	−0.204167	−0.102084	+	0.994776i	$0.532551\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.44851	−0.316943	−0.158472	−	0.987364i	$-0.550657\pi$
$197$	−4.44851	−0.316943	−0.158472	+	0.987364i	$0.550657\pi$
$198$	0	0
$199$	−14.3634	−1.01820	−0.509098	−	0.860708i	$-0.670021\pi$
$199$	−14.3634	−1.01820	−0.509098	+	0.860708i	$0.670021\pi$
$200$	0	0
$201$	11.9248	0.841109
$202$	0	0
$203$	4.96239	0.348291
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.0630040	0.00437908
$208$	0	0
$209$	4.73084	0.327239
$210$	0	0
$211$	5.58769	0.384672	0.192336	−	0.981329i	$-0.438394\pi$
$211$	5.58769	0.384672	0.192336	+	0.981329i	$0.438394\pi$
$212$	0	0
$213$	13.7685	0.943399
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.49929	−0.169663
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	6.96239	0.468341
$222$	0	0
$223$	−11.7440	−0.786437	−0.393219	−	0.919445i	$-0.628638\pi$
$223$	−11.7440	−0.786437	−0.393219	+	0.919445i	$0.628638\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.8061	0.982713	0.491356	−	0.870959i	$-0.336501\pi$
$227$	14.8061	0.982713	0.491356	+	0.870959i	$0.336501\pi$
$228$	0	0
$229$	−29.5125	−1.95024	−0.975119	−	0.221681i	$-0.928846\pi$
$229$	−29.5125	−1.95024	−0.975119	+	0.221681i	$0.928846\pi$
$230$	0	0
$231$	10.8872	0.716323
$232$	0	0
$233$	−25.4617	−1.66805	−0.834025	−	0.551726i	$-0.813969\pi$
$233$	−25.4617	−1.66805	−0.834025	+	0.551726i	$0.813969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.31265	−0.280137
$238$	0	0
$239$	−11.2424	−0.727207	−0.363604	−	0.931554i	$-0.618454\pi$
$239$	−11.2424	−0.727207	−0.363604	+	0.931554i	$0.618454\pi$
$240$	0	0
$241$	1.76257	0.113537	0.0567686	−	0.998387i	$-0.481920\pi$
$241$	1.76257	0.113537	0.0567686	+	0.998387i	$0.481920\pi$
$242$	0	0
$243$	2.01222	0.129084
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.869067	0.0552974
$248$	0	0
$249$	−5.01317	−0.317697
$250$	0	0
$251$	23.5369	1.48564	0.742818	−	0.669493i	$-0.233489\pi$
$251$	23.5369	1.48564	0.742818	+	0.669493i	$0.233489\pi$
$252$	0	0
$253$	1.76845	0.111182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.78892	0.485859	0.242930	−	0.970044i	$-0.421892\pi$
$257$	7.78892	0.485859	0.242930	+	0.970044i	$0.421892\pi$
$258$	0	0
$259$	−1.79877	−0.111770
$260$	0	0
$261$	0.806063	0.0498941
$262$	0	0
$263$	3.47390	0.214210	0.107105	−	0.994248i	$-0.465842\pi$
$263$	3.47390	0.214210	0.107105	+	0.994248i	$0.465842\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	15.8740	0.971473
$268$	0	0
$269$	−20.8265	−1.26982	−0.634908	−	0.772588i	$-0.718962\pi$
$269$	−20.8265	−1.26982	−0.634908	+	0.772588i	$0.718962\pi$
$270$	0	0
$271$	27.4191	1.66559	0.832797	−	0.553578i	$-0.186738\pi$
$271$	27.4191	1.66559	0.832797	+	0.553578i	$0.186738\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.41231	0.144942	0.0724709	−	0.997371i	$-0.476912\pi$
$277$	2.41231	0.144942	0.0724709	+	0.997371i	$0.476912\pi$
$278$	0	0
$279$	−0.405972	−0.0243049
$280$	0	0
$281$	−18.3879	−1.09693	−0.548464	−	0.836174i	$-0.684787\pi$
$281$	−18.3879	−1.09693	−0.548464	+	0.836174i	$0.684787\pi$
$282$	0	0
$283$	30.5623	1.81674	0.908370	−	0.418167i	$-0.137327\pi$
$283$	30.5623	1.81674	0.908370	+	0.418167i	$0.137327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.8496	−0.699457
$288$	0	0
$289$	31.4749	1.85146
$290$	0	0
$291$	−24.2882	−1.42380
$292$	0	0
$293$	−16.9829	−0.992149	−0.496075	−	0.868280i	$-0.665226\pi$
$293$	−16.9829	−0.992149	−0.496075	+	0.868280i	$0.665226\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	29.1246	1.68998
$298$	0	0
$299$	0.324869	0.0187877
$300$	0	0
$301$	−9.38646	−0.541026
$302$	0	0
$303$	−7.66291	−0.440223
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.61942	0.149498	0.0747491	−	0.997202i	$-0.476184\pi$
$307$	2.61942	0.149498	0.0747491	+	0.997202i	$0.476184\pi$
$308$	0	0
$309$	−18.3430	−1.04349
$310$	0	0
$311$	5.40105	0.306265	0.153133	−	0.988206i	$-0.451064\pi$
$311$	5.40105	0.306265	0.153133	+	0.988206i	$0.451064\pi$
$312$	0	0
$313$	−7.27504	−0.411210	−0.205605	−	0.978635i	$-0.565916\pi$
$313$	−7.27504	−0.411210	−0.205605	+	0.978635i	$0.565916\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.73084	−0.490373	−0.245186	−	0.969476i	$-0.578849\pi$
$317$	−8.73084	−0.490373	−0.245186	+	0.969476i	$0.578849\pi$
$318$	0	0
$319$	22.6253	1.26677
$320$	0	0
$321$	−34.5296	−1.92725
$322$	0	0
$323$	6.05079	0.336675
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.2374	1.00853
$328$	0	0
$329$	8.12601	0.448001
$330$	0	0
$331$	28.9199	1.58958	0.794789	−	0.606885i	$-0.207581\pi$
$331$	28.9199	1.58958	0.794789	+	0.606885i	$0.207581\pi$
$332$	0	0
$333$	−0.292182	−0.0160115
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.7988	0.642720	0.321360	−	0.946957i	$-0.395860\pi$
$337$	11.7988	0.642720	0.321360	+	0.946957i	$0.395860\pi$
$338$	0	0
$339$	8.18664	0.444637
$340$	0	0
$341$	−11.3952	−0.617083
$342$	0	0
$343$	15.0132	0.810635
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.7113	1.16553	0.582763	−	0.812642i	$-0.301972\pi$
$347$	21.7113	1.16553	0.582763	+	0.812642i	$0.301972\pi$
$348$	0	0
$349$	15.3357	0.820900	0.410450	−	0.911883i	$-0.365372\pi$
$349$	15.3357	0.820900	0.410450	+	0.911883i	$0.365372\pi$
$350$	0	0
$351$	5.35026	0.285576
$352$	0	0
$353$	20.0567	1.06751	0.533754	−	0.845640i	$-0.320781\pi$
$353$	20.0567	1.06751	0.533754	+	0.845640i	$0.320781\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.9248	0.736977
$358$	0	0
$359$	13.1573	0.694415	0.347207	−	0.937788i	$-0.387130\pi$
$359$	13.1573	0.694415	0.347207	+	0.937788i	$0.387130\pi$
$360$	0	0
$361$	−18.2447	−0.960249
$362$	0	0
$363$	31.2120	1.63821
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.50166	−0.443783	−0.221892	−	0.975071i	$-0.571223\pi$
$367$	−8.50166	−0.443783	−0.221892	+	0.975071i	$0.571223\pi$
$368$	0	0
$369$	−1.92478	−0.100200
$370$	0	0
$371$	13.5515	0.703558
$372$	0	0
$373$	11.4010	0.590324	0.295162	−	0.955447i	$-0.404626\pi$
$373$	11.4010	0.590324	0.295162	+	0.955447i	$0.404626\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.15633	0.214062
$378$	0	0
$379$	12.2701	0.630273	0.315137	−	0.949046i	$-0.397950\pi$
$379$	12.2701	0.630273	0.315137	+	0.949046i	$0.397950\pi$
$380$	0	0
$381$	19.6424	1.00631
$382$	0	0
$383$	−26.7816	−1.36848	−0.684239	−	0.729258i	$-0.739865\pi$
$383$	−26.7816	−1.36848	−0.684239	+	0.729258i	$0.739865\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.52469	−0.0775041
$388$	0	0
$389$	5.84955	0.296584	0.148292	−	0.988944i	$-0.452622\pi$
$389$	5.84955	0.296584	0.148292	+	0.988944i	$0.452622\pi$
$390$	0	0
$391$	2.26187	0.114387
$392$	0	0
$393$	−2.70052	−0.136223
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−38.4445	−1.92948	−0.964738	−	0.263212i	$-0.915218\pi$
$397$	−38.4445	−1.92948	−0.964738	+	0.263212i	$0.915218\pi$
$398$	0	0
$399$	1.73813	0.0870156
$400$	0	0
$401$	12.7612	0.637262	0.318631	−	0.947879i	$-0.396777\pi$
$401$	12.7612	0.637262	0.318631	+	0.947879i	$0.396777\pi$
$402$	0	0
$403$	−2.09332	−0.104276
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.20123	−0.406520
$408$	0	0
$409$	18.5501	0.917242	0.458621	−	0.888632i	$-0.348344\pi$
$409$	18.5501	0.917242	0.458621	+	0.888632i	$0.348344\pi$
$410$	0	0
$411$	−19.8740	−0.980312
$412$	0	0
$413$	−10.9624	−0.539424
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.0132	−0.539317
$418$	0	0
$419$	−34.2374	−1.67261	−0.836304	−	0.548266i	$-0.815288\pi$
$419$	−34.2374	−1.67261	−0.836304	+	0.548266i	$0.815288\pi$
$420$	0	0
$421$	12.4387	0.606223	0.303112	−	0.952955i	$-0.401975\pi$
$421$	12.4387	0.606223	0.303112	+	0.952955i	$0.401975\pi$
$422$	0	0
$423$	1.31994	0.0641779
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.2605	−0.544931
$428$	0	0
$429$	9.11871	0.440256
$430$	0	0
$431$	2.24377	0.108078	0.0540392	−	0.998539i	$-0.482790\pi$
$431$	2.24377	0.108078	0.0540392	+	0.998539i	$0.482790\pi$
$432$	0	0
$433$	39.3258	1.88988	0.944939	−	0.327246i	$-0.106121\pi$
$433$	39.3258	1.88988	0.944939	+	0.327246i	$0.106121\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.282333	0.0135058
$438$	0	0
$439$	26.6761	1.27318	0.636590	−	0.771202i	$-0.280344\pi$
$439$	26.6761	1.27318	0.636590	+	0.771202i	$0.280344\pi$
$440$	0	0
$441$	1.08110	0.0514811
$442$	0	0
$443$	4.31502	0.205013	0.102506	−	0.994732i	$-0.467314\pi$
$443$	4.31502	0.205013	0.102506	+	0.994732i	$0.467314\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.8364	−0.796333
$448$	0	0
$449$	−22.5501	−1.06420	−0.532102	−	0.846680i	$-0.678598\pi$
$449$	−22.5501	−1.06420	−0.532102	+	0.846680i	$0.678598\pi$
$450$	0	0
$451$	−54.0263	−2.54400
$452$	0	0
$453$	−22.6458	−1.06399
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.2750	0.527424	0.263712	−	0.964601i	$-0.415053\pi$
$457$	11.2750	0.527424	0.263712	+	0.964601i	$0.415053\pi$
$458$	0	0
$459$	37.2506	1.73871
$460$	0	0
$461$	−11.9003	−0.554254	−0.277127	−	0.960833i	$-0.589382\pi$
$461$	−11.9003	−0.554254	−0.277127	+	0.960833i	$0.589382\pi$
$462$	0	0
$463$	20.5442	0.954770	0.477385	−	0.878694i	$-0.341585\pi$
$463$	20.5442	0.954770	0.477385	+	0.878694i	$0.341585\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.09920	−0.189688	−0.0948442	−	0.995492i	$-0.530235\pi$
$467$	−4.09920	−0.189688	−0.0948442	+	0.995492i	$0.530235\pi$
$468$	0	0
$469$	−8.49929	−0.392461
$470$	0	0
$471$	−9.17347	−0.422691
$472$	0	0
$473$	−42.7962	−1.96777
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.20123	0.100787
$478$	0	0
$479$	−9.49437	−0.433809	−0.216904	−	0.976193i	$-0.569596\pi$
$479$	−9.49437	−0.433809	−0.216904	+	0.976193i	$0.569596\pi$
$480$	0	0
$481$	−1.50659	−0.0686945
$482$	0	0
$483$	0.649738	0.0295641
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−33.6326	−1.52404	−0.762019	−	0.647554i	$-0.775792\pi$
$487$	−33.6326	−1.52404	−0.762019	+	0.647554i	$0.775792\pi$
$488$	0	0
$489$	−0.135857	−0.00614367
$490$	0	0
$491$	24.2882	1.09611	0.548056	−	0.836442i	$-0.315368\pi$
$491$	24.2882	1.09611	0.548056	+	0.836442i	$0.315368\pi$
$492$	0	0
$493$	28.9380	1.30330
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.81336	−0.440189
$498$	0	0
$499$	6.98049	0.312490	0.156245	−	0.987718i	$-0.450061\pi$
$499$	6.98049	0.312490	0.156245	+	0.987718i	$0.450061\pi$
$500$	0	0
$501$	42.1622	1.88367
$502$	0	0
$503$	−29.1368	−1.29915	−0.649573	−	0.760299i	$-0.725052\pi$
$503$	−29.1368	−1.29915	−0.649573	+	0.760299i	$0.725052\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.67513	0.0743952
$508$	0	0
$509$	−7.90034	−0.350176	−0.175088	−	0.984553i	$-0.556021\pi$
$509$	−7.90034	−0.350176	−0.175088	+	0.984553i	$0.556021\pi$
$510$	0	0
$511$	4.27645	0.189179
$512$	0	0
$513$	4.64974	0.205291
$514$	0	0
$515$	0	0
$516$	0	0
$517$	37.0494	1.62943
$518$	0	0
$519$	−21.1900	−0.930136
$520$	0	0
$521$	21.6180	0.947102	0.473551	−	0.880766i	$-0.342972\pi$
$521$	21.6180	0.947102	0.473551	+	0.880766i	$0.342972\pi$
$522$	0	0
$523$	−30.4264	−1.33046	−0.665228	−	0.746641i	$-0.731666\pi$
$523$	−30.4264	−1.33046	−0.665228	+	0.746641i	$0.731666\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.5745	−0.634876
$528$	0	0
$529$	−22.8945	−0.995411
$530$	0	0
$531$	−1.78067	−0.0772745
$532$	0	0
$533$	−9.92478	−0.429890
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.1147	0.824862
$538$	0	0
$539$	30.3453	1.30707
$540$	0	0
$541$	−41.7235	−1.79384	−0.896918	−	0.442198i	$-0.854199\pi$
$541$	−41.7235	−1.79384	−0.896918	+	0.442198i	$0.854199\pi$
$542$	0	0
$543$	−26.1866	−1.12378
$544$	0	0
$545$	0	0
$546$	0	0
$547$	30.4119	1.30032	0.650158	−	0.759799i	$-0.274703\pi$
$547$	30.4119	1.30032	0.650158	+	0.759799i	$0.274703\pi$
$548$	0	0
$549$	−1.82909	−0.0780635
$550$	0	0
$551$	3.61213	0.153882
$552$	0	0
$553$	3.07381	0.130712
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.1939	−1.23699	−0.618493	−	0.785790i	$-0.712257\pi$
$557$	−29.1939	−1.23699	−0.618493	+	0.785790i	$0.712257\pi$
$558$	0	0
$559$	−7.86177	−0.332518
$560$	0	0
$561$	63.4880	2.68047
$562$	0	0
$563$	−18.5379	−0.781278	−0.390639	−	0.920544i	$-0.627746\pi$
$563$	−18.5379	−0.781278	−0.390639	+	0.920544i	$0.627746\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.0059	0.420207
$568$	0	0
$569$	−7.50659	−0.314692	−0.157346	−	0.987543i	$-0.550294\pi$
$569$	−7.50659	−0.314692	−0.157346	+	0.987543i	$0.550294\pi$
$570$	0	0
$571$	10.0508	0.420612	0.210306	−	0.977636i	$-0.432554\pi$
$571$	10.0508	0.420612	0.210306	+	0.977636i	$0.432554\pi$
$572$	0	0
$573$	−30.0263	−1.25437
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.98286	−0.373961	−0.186981	−	0.982364i	$-0.559870\pi$
$577$	−8.98286	−0.373961	−0.186981	+	0.982364i	$0.559870\pi$
$578$	0	0
$579$	−4.75131	−0.197458
$580$	0	0
$581$	3.57310	0.148237
$582$	0	0
$583$	61.7861	2.55892
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−4.08110	−0.168445	−0.0842226	−	0.996447i	$-0.526841\pi$
$587$	−4.08110	−0.168445	−0.0842226	+	0.996447i	$0.526841\pi$
$588$	0	0
$589$	−1.81924	−0.0749604
$590$	0	0
$591$	−7.45183	−0.306527
$592$	0	0
$593$	−31.3766	−1.28848	−0.644241	−	0.764822i	$-0.722827\pi$
$593$	−31.3766	−1.28848	−0.644241	+	0.764822i	$0.722827\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.0606	−0.984736
$598$	0	0
$599$	−22.1768	−0.906119	−0.453060	−	0.891480i	$-0.649668\pi$
$599$	−22.1768	−0.906119	−0.453060	+	0.891480i	$0.649668\pi$
$600$	0	0
$601$	−43.5026	−1.77451	−0.887254	−	0.461280i	$-0.847390\pi$
$601$	−43.5026	−1.77451	−0.887254	+	0.461280i	$0.847390\pi$
$602$	0	0
$603$	−1.38058	−0.0562215
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.7880	−0.519047	−0.259524	−	0.965737i	$-0.583566\pi$
$607$	−12.7880	−0.519047	−0.259524	+	0.965737i	$0.583566\pi$
$608$	0	0
$609$	8.31265	0.336846
$610$	0	0
$611$	6.80606	0.275344
$612$	0	0
$613$	25.1392	1.01536	0.507681	−	0.861545i	$-0.330503\pi$
$613$	25.1392	1.01536	0.507681	+	0.861545i	$0.330503\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.4142	0.741329	0.370664	−	0.928767i	$-0.379130\pi$
$617$	18.4142	0.741329	0.370664	+	0.928767i	$0.379130\pi$
$618$	0	0
$619$	−39.5452	−1.58945	−0.794727	−	0.606967i	$-0.792386\pi$
$619$	−39.5452	−1.58945	−0.794727	+	0.606967i	$0.792386\pi$
$620$	0	0
$621$	1.73813	0.0697489
$622$	0	0
$623$	−11.3141	−0.453288
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.92478	0.316485
$628$	0	0
$629$	−10.4894	−0.418242
$630$	0	0
$631$	36.1441	1.43887	0.719437	−	0.694558i	$-0.244400\pi$
$631$	36.1441	1.43887	0.719437	+	0.694558i	$0.244400\pi$
$632$	0	0
$633$	9.36011	0.372031
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.57452	0.220870
$638$	0	0
$639$	−1.59403	−0.0630588
$640$	0	0
$641$	−33.8007	−1.33505	−0.667523	−	0.744589i	$-0.732646\pi$
$641$	−33.8007	−1.33505	−0.667523	+	0.744589i	$0.732646\pi$
$642$	0	0
$643$	3.64244	0.143644	0.0718220	−	0.997417i	$-0.477119\pi$
$643$	3.64244	0.143644	0.0718220	+	0.997417i	$0.477119\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.1514	−1.46057	−0.730286	−	0.683141i	$-0.760613\pi$
$647$	−37.1514	−1.46057	−0.730286	+	0.683141i	$0.760613\pi$
$648$	0	0
$649$	−49.9814	−1.96194
$650$	0	0
$651$	−4.18664	−0.164087
$652$	0	0
$653$	−29.2243	−1.14363	−0.571817	−	0.820381i	$-0.693761\pi$
$653$	−29.2243	−1.14363	−0.571817	+	0.820381i	$0.693761\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.694644	0.0271006
$658$	0	0
$659$	27.1147	1.05624	0.528120	−	0.849170i	$-0.322897\pi$
$659$	27.1147	1.05624	0.528120	+	0.849170i	$0.322897\pi$
$660$	0	0
$661$	−6.94780	−0.270238	−0.135119	−	0.990829i	$-0.543142\pi$
$661$	−6.94780	−0.270238	−0.135119	+	0.990829i	$0.543142\pi$
$662$	0	0
$663$	11.6629	0.452950
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.35026	0.0522823
$668$	0	0
$669$	−19.6728	−0.760593
$670$	0	0
$671$	−51.3404	−1.98197
$672$	0	0
$673$	−17.5731	−0.677393	−0.338697	−	0.940896i	$-0.609986\pi$
$673$	−17.5731	−0.677393	−0.338697	+	0.940896i	$0.609986\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.41090	−0.131091	−0.0655457	−	0.997850i	$-0.520879\pi$
$677$	−3.41090	−0.131091	−0.0655457	+	0.997850i	$0.520879\pi$
$678$	0	0
$679$	17.3112	0.664344
$680$	0	0
$681$	24.8021	0.950418
$682$	0	0
$683$	17.6688	0.676078	0.338039	−	0.941132i	$-0.390237\pi$
$683$	17.6688	0.676078	0.338039	+	0.941132i	$0.390237\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−49.4372	−1.88615
$688$	0	0
$689$	11.3503	0.432411
$690$	0	0
$691$	15.9527	0.606870	0.303435	−	0.952852i	$-0.401866\pi$
$691$	15.9527	0.606870	0.303435	+	0.952852i	$0.401866\pi$
$692$	0	0
$693$	−1.26045	−0.0478806
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−69.1002	−2.61736
$698$	0	0
$699$	−42.6516	−1.61323
$700$	0	0
$701$	−17.0982	−0.645792	−0.322896	−	0.946434i	$-0.604656\pi$
$701$	−17.0982	−0.645792	−0.322896	+	0.946434i	$0.604656\pi$
$702$	0	0
$703$	−1.30933	−0.0493822
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.46168	0.205408
$708$	0	0
$709$	11.1735	0.419628	0.209814	−	0.977741i	$-0.432714\pi$
$709$	11.1735	0.419628	0.209814	+	0.977741i	$0.432714\pi$
$710$	0	0
$711$	0.499293	0.0187249
$712$	0	0
$713$	−0.680055	−0.0254683
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−18.8324	−0.703309
$718$	0	0
$719$	34.8627	1.30016	0.650080	−	0.759866i	$-0.274735\pi$
$719$	34.8627	1.30016	0.650080	+	0.759866i	$0.274735\pi$
$720$	0	0
$721$	13.0738	0.486894
$722$	0	0
$723$	2.95254	0.109806
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.0748	−1.26376	−0.631882	−	0.775065i	$-0.717717\pi$
$727$	−34.0748	−1.26376	−0.631882	+	0.775065i	$0.717717\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−54.7367	−2.02451
$732$	0	0
$733$	−5.47627	−0.202271	−0.101135	−	0.994873i	$-0.532248\pi$
$733$	−5.47627	−0.202271	−0.101135	+	0.994873i	$0.532248\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.7513	−1.42742
$738$	0	0
$739$	−47.5959	−1.75084	−0.875422	−	0.483359i	$-0.839417\pi$
$739$	−47.5959	−1.75084	−0.875422	+	0.483359i	$0.839417\pi$
$740$	0	0
$741$	1.45580	0.0534802
$742$	0	0
$743$	−44.1827	−1.62091	−0.810453	−	0.585804i	$-0.800779\pi$
$743$	−44.1827	−1.62091	−0.810453	+	0.585804i	$0.800779\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.580395	0.0212355
$748$	0	0
$749$	24.6107	0.899256
$750$	0	0
$751$	−42.1886	−1.53948	−0.769741	−	0.638356i	$-0.779615\pi$
$751$	−42.1886	−1.53948	−0.769741	+	0.638356i	$0.779615\pi$
$752$	0	0
$753$	39.4274	1.43681
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.6747	1.73277	0.866383	−	0.499381i	$-0.166439\pi$
$757$	47.6747	1.73277	0.866383	+	0.499381i	$0.166439\pi$
$758$	0	0
$759$	2.96239	0.107528
$760$	0	0
$761$	43.7137	1.58462	0.792310	−	0.610119i	$-0.208878\pi$
$761$	43.7137	1.58462	0.792310	+	0.610119i	$0.208878\pi$
$762$	0	0
$763$	−12.9986	−0.470580
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.18172	−0.331533
$768$	0	0
$769$	31.4617	1.13454	0.567269	−	0.823533i	$-0.308000\pi$
$769$	31.4617	1.13454	0.567269	+	0.823533i	$0.308000\pi$
$770$	0	0
$771$	13.0475	0.469893
$772$	0	0
$773$	−27.4920	−0.988818	−0.494409	−	0.869229i	$-0.664616\pi$
$773$	−27.4920	−0.988818	−0.494409	+	0.869229i	$0.664616\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.01317	−0.108097
$778$	0	0
$779$	−8.62530	−0.309033
$780$	0	0
$781$	−44.7426	−1.60102
$782$	0	0
$783$	22.2374	0.794701
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−0.352817	−0.0125766	−0.00628828	−	0.999980i	$-0.502002\pi$
$787$	−0.352817	−0.0125766	−0.00628828	+	0.999980i	$0.502002\pi$
$788$	0	0
$789$	5.81924	0.207170
$790$	0	0
$791$	−5.83497	−0.207468
$792$	0	0
$793$	−9.43136	−0.334918
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.61071	0.305007	0.152504	−	0.988303i	$-0.451266\pi$
$797$	8.61071	0.305007	0.152504	+	0.988303i	$0.451266\pi$
$798$	0	0
$799$	47.3865	1.67641
$800$	0	0
$801$	−1.83780	−0.0649353
$802$	0	0
$803$	19.4979	0.688065
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−34.8872	−1.22809
$808$	0	0
$809$	−24.8423	−0.873407	−0.436704	−	0.899605i	$-0.643854\pi$
$809$	−24.8423	−0.873407	−0.436704	+	0.899605i	$0.643854\pi$
$810$	0	0
$811$	14.1295	0.496154	0.248077	−	0.968740i	$-0.420201\pi$
$811$	14.1295	0.496154	0.248077	+	0.968740i	$0.420201\pi$
$812$	0	0
$813$	45.9307	1.61086
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.83241	−0.239036
$818$	0	0
$819$	−0.231548	−0.00809094
$820$	0	0
$821$	40.2276	1.40395	0.701976	−	0.712201i	$-0.252301\pi$
$821$	40.2276	1.40395	0.701976	+	0.712201i	$0.252301\pi$
$822$	0	0
$823$	−35.4133	−1.23443	−0.617214	−	0.786795i	$-0.711739\pi$
$823$	−35.4133	−1.23443	−0.617214	+	0.786795i	$0.711739\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.1563	0.909545	0.454772	−	0.890608i	$-0.349721\pi$
$827$	26.1563	0.909545	0.454772	+	0.890608i	$0.349721\pi$
$828$	0	0
$829$	26.6820	0.926703	0.463351	−	0.886175i	$-0.346647\pi$
$829$	26.6820	0.926703	0.463351	+	0.886175i	$0.346647\pi$
$830$	0	0
$831$	4.04094	0.140179
$832$	0	0
$833$	38.8119	1.34475
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−11.1998	−0.387122
$838$	0	0
$839$	−8.29455	−0.286360	−0.143180	−	0.989697i	$-0.545733\pi$
$839$	−8.29455	−0.286360	−0.143180	+	0.989697i	$0.545733\pi$
$840$	0	0
$841$	−11.7250	−0.404309
$842$	0	0
$843$	−30.8021	−1.06088
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−22.2461	−0.764386
$848$	0	0
$849$	51.1958	1.75704
$850$	0	0
$851$	−0.489444	−0.0167779
$852$	0	0
$853$	26.7163	0.914747	0.457374	−	0.889275i	$-0.348790\pi$
$853$	26.7163	0.914747	0.457374	+	0.889275i	$0.348790\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.7005	−0.365523	−0.182761	−	0.983157i	$-0.558504\pi$
$857$	−10.7005	−0.365523	−0.182761	+	0.983157i	$0.558504\pi$
$858$	0	0
$859$	−1.08366	−0.0369739	−0.0184870	−	0.999829i	$-0.505885\pi$
$859$	−1.08366	−0.0369739	−0.0184870	+	0.999829i	$0.505885\pi$
$860$	0	0
$861$	−19.8496	−0.676471
$862$	0	0
$863$	31.2809	1.06481	0.532407	−	0.846488i	$-0.321287\pi$
$863$	31.2809	1.06481	0.532407	+	0.846488i	$0.321287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	52.7245	1.79062
$868$	0	0
$869$	14.0146	0.475412
$870$	0	0
$871$	−7.11871	−0.241208
$872$	0	0
$873$	2.81194	0.0951699
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.7104	0.631804	0.315902	−	0.948792i	$-0.397693\pi$
$877$	18.7104	0.631804	0.315902	+	0.948792i	$0.397693\pi$
$878$	0	0
$879$	−28.4485	−0.959544
$880$	0	0
$881$	−31.6688	−1.06695	−0.533474	−	0.845816i	$-0.679114\pi$
$881$	−31.6688	−1.06695	−0.533474	+	0.845816i	$0.679114\pi$
$882$	0	0
$883$	−44.4020	−1.49425	−0.747123	−	0.664686i	$-0.768565\pi$
$883$	−44.4020	−1.49425	−0.747123	+	0.664686i	$0.768565\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.8651	0.364814	0.182407	−	0.983223i	$-0.441611\pi$
$887$	10.8651	0.364814	0.182407	+	0.983223i	$0.441611\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	45.6204	1.52834
$892$	0	0
$893$	5.91493	0.197936
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.544198	0.0181703
$898$	0	0
$899$	−8.70052	−0.290179
$900$	0	0
$901$	79.0249	2.63270
$902$	0	0
$903$	−15.7235	−0.523247
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−3.47390	−0.115349	−0.0576745	−	0.998335i	$-0.518369\pi$
$907$	−3.47390	−0.115349	−0.0576745	+	0.998335i	$0.518369\pi$
$908$	0	0
$909$	0.887166	0.0294254
$910$	0	0
$911$	23.5125	0.779003	0.389501	−	0.921026i	$-0.372647\pi$
$911$	23.5125	0.779003	0.389501	+	0.921026i	$0.372647\pi$
$912$	0	0
$913$	16.2910	0.539155
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.92478	0.0635617
$918$	0	0
$919$	15.3258	0.505552	0.252776	−	0.967525i	$-0.418656\pi$
$919$	15.3258	0.505552	0.252776	+	0.967525i	$0.418656\pi$
$920$	0	0
$921$	4.38787	0.144585
$922$	0	0
$923$	−8.21933	−0.270543
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.12364	0.0697494
$928$	0	0
$929$	−11.6873	−0.383450	−0.191725	−	0.981449i	$-0.561408\pi$
$929$	−11.6873	−0.383450	−0.191725	+	0.981449i	$0.561408\pi$
$930$	0	0
$931$	4.84463	0.158776
$932$	0	0
$933$	9.04746	0.296201
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21.5633	0.704441	0.352220	−	0.935917i	$-0.385427\pi$
$937$	21.5633	0.704441	0.352220	+	0.935917i	$0.385427\pi$
$938$	0	0
$939$	−12.1866	−0.397696
$940$	0	0
$941$	20.8872	0.680902	0.340451	−	0.940262i	$-0.389420\pi$
$941$	20.8872	0.680902	0.340451	+	0.940262i	$0.389420\pi$
$942$	0	0
$943$	−3.22425	−0.104996
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.6801	0.477038	0.238519	−	0.971138i	$-0.423338\pi$
$947$	14.6801	0.477038	0.238519	+	0.971138i	$0.423338\pi$
$948$	0	0
$949$	3.58181	0.116270
$950$	0	0
$951$	−14.6253	−0.474258
$952$	0	0
$953$	−21.1490	−0.685084	−0.342542	−	0.939502i	$-0.611288\pi$
$953$	−21.1490	−0.685084	−0.342542	+	0.939502i	$0.611288\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37.9003	1.22514
$958$	0	0
$959$	14.1650	0.457413
$960$	0	0
$961$	−26.6180	−0.858645
$962$	0	0
$963$	3.99763	0.128822
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−30.8178	−0.991034	−0.495517	−	0.868598i	$-0.665021\pi$
$967$	−30.8178	−0.991034	−0.495517	+	0.868598i	$0.665021\pi$
$968$	0	0
$969$	10.1359	0.325611
$970$	0	0
$971$	−30.5139	−0.979237	−0.489619	−	0.871937i	$-0.662864\pi$
$971$	−30.5139	−0.979237	−0.489619	+	0.871937i	$0.662864\pi$
$972$	0	0
$973$	7.84955	0.251645
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.40360	−0.0768980	−0.0384490	−	0.999261i	$-0.512242\pi$
$977$	−2.40360	−0.0768980	−0.0384490	+	0.999261i	$0.512242\pi$
$978$	0	0
$979$	−51.5849	−1.64866
$980$	0	0
$981$	−2.11142	−0.0674124
$982$	0	0
$983$	−16.6048	−0.529612	−0.264806	−	0.964302i	$-0.585308\pi$
$983$	−16.6048	−0.529612	−0.264806	+	0.964302i	$0.585308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.6121	0.433279
$988$	0	0
$989$	−2.55405	−0.0812140
$990$	0	0
$991$	−30.8265	−0.979237	−0.489619	−	0.871937i	$-0.662864\pi$
$991$	−30.8265	−0.979237	−0.489619	+	0.871937i	$0.662864\pi$
$992$	0	0
$993$	48.4445	1.53734
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.9149	1.01076	0.505378	−	0.862898i	$-0.331353\pi$
$997$	31.9149	1.01076	0.505378	+	0.862898i	$0.331353\pi$
$998$	0	0
$999$	−8.06063	−0.255027

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.j.1.3		3
4.3	odd	2		5200.2.a.cg.1.1		3
5.2	odd	4		260.2.c.a.209.2	✓	6
5.3	odd	4		260.2.c.a.209.5	yes	6
5.4	even	2		1300.2.a.k.1.1		3
15.2	even	4		2340.2.h.e.469.6		6
15.8	even	4		2340.2.h.e.469.5		6
20.3	even	4		1040.2.d.d.209.2		6
20.7	even	4		1040.2.d.d.209.5		6
20.19	odd	2		5200.2.a.cd.1.3		3
65.8	even	4		3380.2.d.b.1689.5		6
65.12	odd	4		3380.2.c.c.2029.2		6
65.18	even	4		3380.2.d.a.1689.5		6
65.38	odd	4		3380.2.c.c.2029.5		6
65.47	even	4		3380.2.d.a.1689.2		6
65.57	even	4		3380.2.d.b.1689.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.c.a.209.2	✓	6	5.2	odd	4
260.2.c.a.209.5	yes	6	5.3	odd	4
1040.2.d.d.209.2		6	20.3	even	4
1040.2.d.d.209.5		6	20.7	even	4
1300.2.a.j.1.3		3	1.1	even	1	trivial
1300.2.a.k.1.1		3	5.4	even	2
2340.2.h.e.469.5		6	15.8	even	4
2340.2.h.e.469.6		6	15.2	even	4
3380.2.c.c.2029.2		6	65.12	odd	4
3380.2.c.c.2029.5		6	65.38	odd	4
3380.2.d.a.1689.2		6	65.47	even	4
3380.2.d.a.1689.5		6	65.18	even	4
3380.2.d.b.1689.2		6	65.57	even	4
3380.2.d.b.1689.5		6	65.8	even	4
5200.2.a.cd.1.3		3	20.19	odd	2
5200.2.a.cg.1.1		3	4.3	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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67513 q^{3} -1.19394 q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q+1.67513 q^{3} -1.19394 q^{7} -0.193937 q^{9} -5.44358 q^{11} -1.00000 q^{13} -6.96239 q^{17} -0.869067 q^{19} -2.00000 q^{21} -0.324869 q^{23} -5.35026 q^{27} -4.15633 q^{29} +2.09332 q^{31} -9.11871 q^{33} +1.50659 q^{37} -1.67513 q^{39} +9.92478 q^{41} +7.86177 q^{43} -6.80606 q^{47} -5.57452 q^{49} -11.6629 q^{51} -11.3503 q^{53} -1.45580 q^{57} +9.18172 q^{59} +9.43136 q^{61} +0.231548 q^{63} +7.11871 q^{67} -0.544198 q^{69} +8.21933 q^{71} -3.58181 q^{73} +6.49929 q^{77} -2.57452 q^{79} -8.38058 q^{81} -2.99271 q^{83} -6.96239 q^{87} +9.47627 q^{89} +1.19394 q^{91} +3.50659 q^{93} -14.4993 q^{97} +1.05571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 4 * q^7 - q^9	\( 3 q - 4 q^{7} - q^{9} - 3 q^{13} - 10 q^{17} + 2 q^{19} - 6 q^{21} - 6 q^{23} - 6 q^{27} - 2 q^{29} - 6 q^{33} - 16 q^{37} + 8 q^{41} + 6 q^{43} - 20 q^{47} - 5 q^{49} - 4 q^{51} - 24 q^{53} - 14 q^{57} + 2 q^{59} - 14 q^{61} + 12 q^{63} + 8 q^{69} + 10 q^{71} - 12 q^{73} - 14 q^{77} + 4 q^{79} - 13 q^{81} + 4 q^{83} - 10 q^{87} + 10 q^{89} + 4 q^{91} - 10 q^{93} - 10 q^{97} - 14 q^{99}+O(q^{100}) \) 3 * q - 4 * q^7 - q^9 - 3 * q^13 - 10 * q^17 + 2 * q^19 - 6 * q^21 - 6 * q^23 - 6 * q^27 - 2 * q^29 - 6 * q^33 - 16 * q^37 + 8 * q^41 + 6 * q^43 - 20 * q^47 - 5 * q^49 - 4 * q^51 - 24 * q^53 - 14 * q^57 + 2 * q^59 - 14 * q^61 + 12 * q^63 + 8 * q^69 + 10 * q^71 - 12 * q^73 - 14 * q^77 + 4 * q^79 - 13 * q^81 + 4 * q^83 - 10 * q^87 + 10 * q^89 + 4 * q^91 - 10 * q^93 - 10 * q^97 - 14 * q^99

Introduction

L-functions

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