LMFDB - Embedded newform 1300.2.a.j.1.1

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			1.1
Root			$0.311108$ 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)

$f(q)$	$=$	$q-2.21432 q^{3} +0.903212 q^{7} +1.90321 q^{9} +O(q^{10})$	$q-2.21432 q^{3} +0.903212 q^{7} +1.90321 q^{9} -0.0666765 q^{11} -1.00000 q^{13} -3.37778 q^{17} +5.11753 q^{19} -2.00000 q^{21} -4.21432 q^{23} +2.42864 q^{27} +1.52543 q^{29} +4.49532 q^{31} +0.147643 q^{33} -11.9541 q^{37} +2.21432 q^{39} +2.75557 q^{41} +8.77631 q^{43} -8.90321 q^{47} -6.18421 q^{49} +7.47949 q^{51} -3.57136 q^{53} -11.3319 q^{57} -8.16839 q^{59} -11.1985 q^{61} +1.71900 q^{63} -2.14764 q^{67} +9.33185 q^{69} -5.54617 q^{71} +2.70964 q^{73} -0.0602231 q^{77} -3.18421 q^{79} -11.0874 q^{81} -9.89384 q^{83} -3.37778 q^{87} -14.4701 q^{89} -0.903212 q^{91} -9.95407 q^{93} -7.93978 q^{97} -0.126900 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.21432	−1.27844	−0.639219	−	0.769025i	$-0.720742\pi$
$3$	−2.21432	−1.27844	−0.639219	+	0.769025i	$0.720742\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.903212	0.341382	0.170691	−	0.985325i	$-0.445400\pi$
$7$	0.903212	0.341382	0.170691	+	0.985325i	$0.445400\pi$
$8$	0	0
$9$	1.90321	0.634404
$10$	0	0
$11$	−0.0666765	−0.0201037	−0.0100519	−	0.999949i	$-0.503200\pi$
$11$	−0.0666765	−0.0201037	−0.0100519	+	0.999949i	$0.503200\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.37778	−0.819233	−0.409617	−	0.912258i	$-0.634337\pi$
$17$	−3.37778	−0.819233	−0.409617	+	0.912258i	$0.634337\pi$
$18$	0	0
$19$	5.11753	1.17404	0.587021	−	0.809572i	$-0.300301\pi$
$19$	5.11753	1.17404	0.587021	+	0.809572i	$0.300301\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.21432	−0.878746	−0.439373	−	0.898305i	$-0.644799\pi$
$23$	−4.21432	−0.878746	−0.439373	+	0.898305i	$0.644799\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.42864	0.467392
$28$	0	0
$29$	1.52543	0.283265	0.141632	−	0.989919i	$-0.454765\pi$
$29$	1.52543	0.283265	0.141632	+	0.989919i	$0.454765\pi$
$30$	0	0
$31$	4.49532	0.807383	0.403691	−	0.914895i	$-0.367727\pi$
$31$	4.49532	0.807383	0.403691	+	0.914895i	$0.367727\pi$
$32$	0	0
$33$	0.147643	0.0257014
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.9541	−1.96524	−0.982618	−	0.185638i	$-0.940565\pi$
$37$	−11.9541	−1.96524	−0.982618	+	0.185638i	$0.940565\pi$
$38$	0	0
$39$	2.21432	0.354575
$40$	0	0
$41$	2.75557	0.430348	0.215174	−	0.976576i	$-0.430968\pi$
$41$	2.75557	0.430348	0.215174	+	0.976576i	$0.430968\pi$
$42$	0	0
$43$	8.77631	1.33838	0.669188	−	0.743094i	$-0.266642\pi$
$43$	8.77631	1.33838	0.669188	+	0.743094i	$0.266642\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.90321	−1.29867	−0.649333	−	0.760504i	$-0.724952\pi$
$47$	−8.90321	−1.29867	−0.649333	+	0.760504i	$0.724952\pi$
$48$	0	0
$49$	−6.18421	−0.883458
$50$	0	0
$51$	7.47949	1.04734
$52$	0	0
$53$	−3.57136	−0.490564	−0.245282	−	0.969452i	$-0.578881\pi$
$53$	−3.57136	−0.490564	−0.245282	+	0.969452i	$0.578881\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−11.3319	−1.50094
$58$	0	0
$59$	−8.16839	−1.06343	−0.531717	−	0.846922i	$-0.678453\pi$
$59$	−8.16839	−1.06343	−0.531717	+	0.846922i	$0.678453\pi$
$60$	0	0
$61$	−11.1985	−1.43382	−0.716910	−	0.697165i	$-0.754444\pi$
$61$	−11.1985	−1.43382	−0.716910	+	0.697165i	$0.754444\pi$
$62$	0	0
$63$	1.71900	0.216574
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.14764	−0.262376	−0.131188	−	0.991357i	$-0.541879\pi$
$67$	−2.14764	−0.262376	−0.131188	+	0.991357i	$0.541879\pi$
$68$	0	0
$69$	9.33185	1.12342
$70$	0	0
$71$	−5.54617	−0.658209	−0.329105	−	0.944293i	$-0.606747\pi$
$71$	−5.54617	−0.658209	−0.329105	+	0.944293i	$0.606747\pi$
$72$	0	0
$73$	2.70964	0.317139	0.158569	−	0.987348i	$-0.449312\pi$
$73$	2.70964	0.317139	0.158569	+	0.987348i	$0.449312\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.0602231	−0.00686305
$78$	0	0
$79$	−3.18421	−0.358251	−0.179126	−	0.983826i	$-0.557327\pi$
$79$	−3.18421	−0.358251	−0.179126	+	0.983826i	$0.557327\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	−9.89384	−1.08599	−0.542995	−	0.839736i	$-0.682710\pi$
$83$	−9.89384	−1.08599	−0.542995	+	0.839736i	$0.682710\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.37778	−0.362136
$88$	0	0
$89$	−14.4701	−1.53383	−0.766915	−	0.641748i	$-0.778209\pi$
$89$	−14.4701	−1.53383	−0.766915	+	0.641748i	$0.778209\pi$
$90$	0	0
$91$	−0.903212	−0.0946823
$92$	0	0
$93$	−9.95407	−1.03219
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.93978	−0.806162	−0.403081	−	0.915164i	$-0.632061\pi$
$97$	−7.93978	−0.806162	−0.403081	+	0.915164i	$0.632061\pi$
$98$	0	0
$99$	−0.126900	−0.0127539
$100$	0	0
$101$	−5.18421	−0.515848	−0.257924	−	0.966165i	$-0.583038\pi$
$101$	−5.18421	−0.515848	−0.257924	+	0.966165i	$0.583038\pi$
$102$	0	0
$103$	7.88739	0.777168	0.388584	−	0.921413i	$-0.372964\pi$
$103$	7.88739	0.777168	0.388584	+	0.921413i	$0.372964\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.3669	1.67892	0.839460	−	0.543421i	$-0.182871\pi$
$107$	17.3669	1.67892	0.839460	+	0.543421i	$0.182871\pi$
$108$	0	0
$109$	0.133353	0.0127729	0.00638645	−	0.999980i	$-0.497967\pi$
$109$	0.133353	0.0127729	0.00638645	+	0.999980i	$0.497967\pi$
$110$	0	0
$111$	26.4701	2.51243
$112$	0	0
$113$	−5.86665	−0.551888	−0.275944	−	0.961174i	$-0.588990\pi$
$113$	−5.86665	−0.551888	−0.275944	+	0.961174i	$0.588990\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.90321	−0.175952
$118$	0	0
$119$	−3.05086	−0.279671
$120$	0	0
$121$	−10.9956	−0.999596
$122$	0	0
$123$	−6.10171	−0.550173
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.5002	−1.37542	−0.687712	−	0.725984i	$-0.741385\pi$
$127$	−15.5002	−1.37542	−0.687712	+	0.725984i	$0.741385\pi$
$128$	0	0
$129$	−19.4336	−1.71103
$130$	0	0
$131$	−5.80642	−0.507310	−0.253655	−	0.967295i	$-0.581633\pi$
$131$	−5.80642	−0.507310	−0.253655	+	0.967295i	$0.581633\pi$
$132$	0	0
$133$	4.62222	0.400797
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.2766	1.39060	0.695300	−	0.718720i	$-0.255272\pi$
$137$	16.2766	1.39060	0.695300	+	0.718720i	$0.255272\pi$
$138$	0	0
$139$	−7.18421	−0.609357	−0.304678	−	0.952455i	$-0.598549\pi$
$139$	−7.18421	−0.609357	−0.304678	+	0.952455i	$0.598549\pi$
$140$	0	0
$141$	19.7146	1.66027
$142$	0	0
$143$	0.0666765	0.00557577
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.6938	1.12945
$148$	0	0
$149$	13.2859	1.08842	0.544212	−	0.838947i	$-0.316829\pi$
$149$	13.2859	1.08842	0.544212	+	0.838947i	$0.316829\pi$
$150$	0	0
$151$	−15.3111	−1.24600	−0.623000	−	0.782222i	$-0.714086\pi$
$151$	−15.3111	−1.24600	−0.623000	+	0.782222i	$0.714086\pi$
$152$	0	0
$153$	−6.42864	−0.519725
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.4701	1.47408	0.737038	−	0.675851i	$-0.236224\pi$
$157$	18.4701	1.47408	0.737038	+	0.675851i	$0.236224\pi$
$158$	0	0
$159$	7.90813	0.627156
$160$	0	0
$161$	−3.80642	−0.299988
$162$	0	0
$163$	12.7699	1.00021	0.500106	−	0.865964i	$-0.333294\pi$
$163$	12.7699	1.00021	0.500106	+	0.865964i	$0.333294\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.43356	−0.575226	−0.287613	−	0.957747i	$-0.592862\pi$
$167$	−7.43356	−0.575226	−0.287613	+	0.957747i	$0.592862\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	9.73975	0.744817
$172$	0	0
$173$	−20.4286	−1.55316	−0.776580	−	0.630018i	$-0.783047\pi$
$173$	−20.4286	−1.55316	−0.776580	+	0.630018i	$0.783047\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.0874	1.35953
$178$	0	0
$179$	24.6035	1.83895	0.919475	−	0.393148i	$-0.128614\pi$
$179$	24.6035	1.83895	0.919475	+	0.393148i	$0.128614\pi$
$180$	0	0
$181$	13.9956	1.04028	0.520141	−	0.854081i	$-0.325880\pi$
$181$	13.9956	1.04028	0.520141	+	0.854081i	$0.325880\pi$
$182$	0	0
$183$	24.7971	1.83305
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.225219	0.0164696
$188$	0	0
$189$	2.19358	0.159559
$190$	0	0
$191$	−10.7556	−0.778246	−0.389123	−	0.921186i	$-0.627222\pi$
$191$	−10.7556	−0.778246	−0.389123	+	0.921186i	$0.627222\pi$
$192$	0	0
$193$	−15.4193	−1.10990	−0.554952	−	0.831883i	$-0.687263\pi$
$193$	−15.4193	−1.10990	−0.554952	+	0.831883i	$0.687263\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.2257	−1.51227	−0.756134	−	0.654417i	$-0.772914\pi$
$197$	−21.2257	−1.51227	−0.756134	+	0.654417i	$0.772914\pi$
$198$	0	0
$199$	20.3368	1.44164	0.720818	−	0.693125i	$-0.243766\pi$
$199$	20.3368	1.44164	0.720818	+	0.693125i	$0.243766\pi$
$200$	0	0
$201$	4.75557	0.335432
$202$	0	0
$203$	1.37778	0.0967015
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.02074	−0.557480
$208$	0	0
$209$	−0.341219	−0.0236026
$210$	0	0
$211$	−20.7239	−1.42669	−0.713347	−	0.700811i	$-0.752822\pi$
$211$	−20.7239	−1.42669	−0.713347	+	0.700811i	$0.752822\pi$
$212$	0	0
$213$	12.2810	0.841480
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.06022	0.275626
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	3.37778	0.227214
$222$	0	0
$223$	20.2494	1.35600	0.677999	−	0.735063i	$-0.262848\pi$
$223$	20.2494	1.35600	0.677999	+	0.735063i	$0.262848\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.9032	1.12191	0.560953	−	0.827848i	$-0.310435\pi$
$227$	16.9032	1.12191	0.560953	+	0.827848i	$0.310435\pi$
$228$	0	0
$229$	3.96836	0.262236	0.131118	−	0.991367i	$-0.458143\pi$
$229$	3.96836	0.262236	0.131118	+	0.991367i	$0.458143\pi$
$230$	0	0
$231$	0.133353	0.00877399
$232$	0	0
$233$	−15.3176	−1.00349	−0.501743	−	0.865017i	$-0.667308\pi$
$233$	−15.3176	−1.00349	−0.501743	+	0.865017i	$0.667308\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.05086	0.458002
$238$	0	0
$239$	−14.8637	−0.961455	−0.480727	−	0.876870i	$-0.659627\pi$
$239$	−14.8637	−0.961455	−0.480727	+	0.876870i	$0.659627\pi$
$240$	0	0
$241$	20.2953	1.30733	0.653667	−	0.756782i	$-0.273230\pi$
$241$	20.2953	1.30733	0.653667	+	0.756782i	$0.273230\pi$
$242$	0	0
$243$	17.2652	1.10756
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.11753	−0.325621
$248$	0	0
$249$	21.9081	1.38837
$250$	0	0
$251$	20.5620	1.29786	0.648931	−	0.760847i	$-0.275216\pi$
$251$	20.5620	1.29786	0.648931	+	0.760847i	$0.275216\pi$
$252$	0	0
$253$	0.280996	0.0176661
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.5210	−1.71671	−0.858356	−	0.513055i	$-0.828514\pi$
$257$	−27.5210	−1.71671	−0.858356	+	0.513055i	$0.828514\pi$
$258$	0	0
$259$	−10.7971	−0.670896
$260$	0	0
$261$	2.90321	0.179704
$262$	0	0
$263$	8.58274	0.529234	0.264617	−	0.964354i	$-0.414754\pi$
$263$	8.58274	0.529234	0.264617	+	0.964354i	$0.414754\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	32.0415	1.96091
$268$	0	0
$269$	10.8988	0.664510	0.332255	−	0.943190i	$-0.392191\pi$
$269$	10.8988	0.664510	0.332255	+	0.943190i	$0.392191\pi$
$270$	0	0
$271$	−8.46367	−0.514132	−0.257066	−	0.966394i	$-0.582756\pi$
$271$	−8.46367	−0.514132	−0.257066	+	0.966394i	$0.582756\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.7239	1.72585	0.862927	−	0.505329i	$-0.168629\pi$
$277$	28.7239	1.72585	0.862927	+	0.505329i	$0.168629\pi$
$278$	0	0
$279$	8.55554	0.512207
$280$	0	0
$281$	−14.1936	−0.846718	−0.423359	−	0.905962i	$-0.639149\pi$
$281$	−14.1936	−0.846718	−0.423359	+	0.905962i	$0.639149\pi$
$282$	0	0
$283$	15.9190	0.946288	0.473144	−	0.880985i	$-0.343119\pi$
$283$	15.9190	0.946288	0.473144	+	0.880985i	$0.343119\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.48886	0.146913
$288$	0	0
$289$	−5.59057	−0.328857
$290$	0	0
$291$	17.5812	1.03063
$292$	0	0
$293$	20.4242	1.19319	0.596597	−	0.802541i	$-0.296519\pi$
$293$	20.4242	1.19319	0.596597	+	0.802541i	$0.296519\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.161933	−0.00939632
$298$	0	0
$299$	4.21432	0.243720
$300$	0	0
$301$	7.92687	0.456897
$302$	0	0
$303$	11.4795	0.659480
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−0.0874201	−0.00498933	−0.00249467	−	0.999997i	$-0.500794\pi$
$307$	−0.0874201	−0.00498933	−0.00249467	+	0.999997i	$0.500794\pi$
$308$	0	0
$309$	−17.4652	−0.993561
$310$	0	0
$311$	−25.7146	−1.45814	−0.729069	−	0.684440i	$-0.760047\pi$
$311$	−25.7146	−1.45814	−0.729069	+	0.684440i	$0.760047\pi$
$312$	0	0
$313$	7.67307	0.433708	0.216854	−	0.976204i	$-0.430421\pi$
$313$	7.67307	0.433708	0.216854	+	0.976204i	$0.430421\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.65878	−0.205498	−0.102749	−	0.994707i	$-0.532764\pi$
$317$	−3.65878	−0.205498	−0.102749	+	0.994707i	$0.532764\pi$
$318$	0	0
$319$	−0.101710	−0.00569468
$320$	0	0
$321$	−38.4558	−2.14640
$322$	0	0
$323$	−17.2859	−0.961814
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.295286	−0.0163294
$328$	0	0
$329$	−8.04149	−0.443342
$330$	0	0
$331$	−0.403450	−0.0221756	−0.0110878	−	0.999939i	$-0.503529\pi$
$331$	−0.403450	−0.0221756	−0.0110878	+	0.999939i	$0.503529\pi$
$332$	0	0
$333$	−22.7511	−1.24675
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.7971	1.13289	0.566444	−	0.824100i	$-0.308319\pi$
$337$	20.7971	1.13289	0.566444	+	0.824100i	$0.308319\pi$
$338$	0	0
$339$	12.9906	0.705554
$340$	0	0
$341$	−0.299732	−0.0162314
$342$	0	0
$343$	−11.9081	−0.642979
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.28745	0.444894	0.222447	−	0.974945i	$-0.428596\pi$
$347$	8.28745	0.444894	0.222447	+	0.974945i	$0.428596\pi$
$348$	0	0
$349$	21.3590	1.14332	0.571662	−	0.820489i	$-0.306299\pi$
$349$	21.3590	1.14332	0.571662	+	0.820489i	$0.306299\pi$
$350$	0	0
$351$	−2.42864	−0.129631
$352$	0	0
$353$	−23.3002	−1.24014	−0.620072	−	0.784545i	$-0.712897\pi$
$353$	−23.3002	−1.24014	−0.620072	+	0.784545i	$0.712897\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.75557	0.357543
$358$	0	0
$359$	−34.6987	−1.83133	−0.915665	−	0.401943i	$-0.868335\pi$
$359$	−34.6987	−1.83133	−0.915665	+	0.401943i	$0.868335\pi$
$360$	0	0
$361$	7.18913	0.378375
$362$	0	0
$363$	24.3477	1.27792
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.1131	1.41529	0.707646	−	0.706567i	$-0.249757\pi$
$367$	27.1131	1.41529	0.707646	+	0.706567i	$0.249757\pi$
$368$	0	0
$369$	5.24443	0.273014
$370$	0	0
$371$	−3.22570	−0.167470
$372$	0	0
$373$	−19.7146	−1.02078	−0.510391	−	0.859943i	$-0.670499\pi$
$373$	−19.7146	−1.02078	−0.510391	+	0.859943i	$0.670499\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.52543	−0.0785635
$378$	0	0
$379$	−24.8321	−1.27554	−0.637769	−	0.770227i	$-0.720143\pi$
$379$	−24.8321	−1.27554	−0.637769	+	0.770227i	$0.720143\pi$
$380$	0	0
$381$	34.3225	1.75839
$382$	0	0
$383$	1.62714	0.0831429	0.0415714	−	0.999136i	$-0.486764\pi$
$383$	1.62714	0.0831429	0.0415714	+	0.999136i	$0.486764\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.7032	0.849070
$388$	0	0
$389$	−8.48886	−0.430402	−0.215201	−	0.976570i	$-0.569041\pi$
$389$	−8.48886	−0.430402	−0.215201	+	0.976570i	$0.569041\pi$
$390$	0	0
$391$	14.2351	0.719898
$392$	0	0
$393$	12.8573	0.648564
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.10663	0.457049	0.228524	−	0.973538i	$-0.426610\pi$
$397$	9.10663	0.457049	0.228524	+	0.973538i	$0.426610\pi$
$398$	0	0
$399$	−10.2351	−0.512394
$400$	0	0
$401$	18.1748	0.907608	0.453804	−	0.891101i	$-0.350067\pi$
$401$	18.1748	0.907608	0.453804	+	0.891101i	$0.350067\pi$
$402$	0	0
$403$	−4.49532	−0.223928
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.797056	0.0395086
$408$	0	0
$409$	−11.3461	−0.561031	−0.280515	−	0.959850i	$-0.590505\pi$
$409$	−11.3461	−0.561031	−0.280515	+	0.959850i	$0.590505\pi$
$410$	0	0
$411$	−36.0415	−1.77780
$412$	0	0
$413$	−7.37778	−0.363037
$414$	0	0
$415$	0	0
$416$	0	0
$417$	15.9081	0.779025
$418$	0	0
$419$	−15.7047	−0.767225	−0.383613	−	0.923494i	$-0.625320\pi$
$419$	−15.7047	−0.767225	−0.383613	+	0.923494i	$0.625320\pi$
$420$	0	0
$421$	−15.0923	−0.735556	−0.367778	−	0.929914i	$-0.619881\pi$
$421$	−15.0923	−0.735556	−0.367778	+	0.929914i	$0.619881\pi$
$422$	0	0
$423$	−16.9447	−0.823879
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.1146	−0.489481
$428$	0	0
$429$	−0.147643	−0.00712828
$430$	0	0
$431$	18.9842	0.914436	0.457218	−	0.889355i	$-0.348846\pi$
$431$	18.9842	0.914436	0.457218	+	0.889355i	$0.348846\pi$
$432$	0	0
$433$	1.04101	0.0500278	0.0250139	−	0.999687i	$-0.492037\pi$
$433$	1.04101	0.0500278	0.0250139	+	0.999687i	$0.492037\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−21.5669	−1.03169
$438$	0	0
$439$	−19.3876	−0.925321	−0.462661	−	0.886536i	$-0.653105\pi$
$439$	−19.3876	−0.925321	−0.462661	+	0.886536i	$0.653105\pi$
$440$	0	0
$441$	−11.7699	−0.560469
$442$	0	0
$443$	−36.1037	−1.71534	−0.857670	−	0.514201i	$-0.828089\pi$
$443$	−36.1037	−1.71534	−0.857670	+	0.514201i	$0.828089\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−29.4193	−1.39148
$448$	0	0
$449$	7.34614	0.346686	0.173343	−	0.984862i	$-0.444543\pi$
$449$	7.34614	0.346686	0.173343	+	0.984862i	$0.444543\pi$
$450$	0	0
$451$	−0.183732	−0.00865159
$452$	0	0
$453$	33.9037	1.59293
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.67307	−0.171819	−0.0859095	−	0.996303i	$-0.527380\pi$
$457$	−3.67307	−0.171819	−0.0859095	+	0.996303i	$0.527380\pi$
$458$	0	0
$459$	−8.20342	−0.382903
$460$	0	0
$461$	25.7748	1.20045	0.600226	−	0.799831i	$-0.295077\pi$
$461$	25.7748	1.20045	0.600226	+	0.799831i	$0.295077\pi$
$462$	0	0
$463$	10.6681	0.495791	0.247895	−	0.968787i	$-0.420261\pi$
$463$	10.6681	0.495791	0.247895	+	0.968787i	$0.420261\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.5190	0.625584	0.312792	−	0.949822i	$-0.398736\pi$
$467$	13.5190	0.625584	0.312792	+	0.949822i	$0.398736\pi$
$468$	0	0
$469$	−1.93978	−0.0895706
$470$	0	0
$471$	−40.8988	−1.88452
$472$	0	0
$473$	−0.585174	−0.0269063
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.79706	−0.311216
$478$	0	0
$479$	19.2192	0.878150	0.439075	−	0.898451i	$-0.355306\pi$
$479$	19.2192	0.878150	0.439075	+	0.898451i	$0.355306\pi$
$480$	0	0
$481$	11.9541	0.545059
$482$	0	0
$483$	8.42864	0.383516
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.00445	−0.181459	−0.0907294	−	0.995876i	$-0.528920\pi$
$487$	−4.00445	−0.181459	−0.0907294	+	0.995876i	$0.528920\pi$
$488$	0	0
$489$	−28.2766	−1.27871
$490$	0	0
$491$	−17.5812	−0.793429	−0.396714	−	0.917942i	$-0.629850\pi$
$491$	−17.5812	−0.793429	−0.396714	+	0.917942i	$0.629850\pi$
$492$	0	0
$493$	−5.15257	−0.232060
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.00937	−0.224701
$498$	0	0
$499$	−1.37133	−0.0613892	−0.0306946	−	0.999529i	$-0.509772\pi$
$499$	−1.37133	−0.0613892	−0.0306946	+	0.999529i	$0.509772\pi$
$500$	0	0
$501$	16.4603	0.735391
$502$	0	0
$503$	−15.1032	−0.673420	−0.336710	−	0.941608i	$-0.609314\pi$
$503$	−15.1032	−0.673420	−0.336710	+	0.941608i	$0.609314\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.21432	−0.0983414
$508$	0	0
$509$	29.7748	1.31974	0.659872	−	0.751378i	$-0.270610\pi$
$509$	29.7748	1.31974	0.659872	+	0.751378i	$0.270610\pi$
$510$	0	0
$511$	2.44738	0.108266
$512$	0	0
$513$	12.4286	0.548738
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.593635	0.0261081
$518$	0	0
$519$	45.2355	1.98562
$520$	0	0
$521$	5.79213	0.253758	0.126879	−	0.991918i	$-0.459504\pi$
$521$	5.79213	0.253758	0.126879	+	0.991918i	$0.459504\pi$
$522$	0	0
$523$	12.3575	0.540356	0.270178	−	0.962810i	$-0.412917\pi$
$523$	12.3575	0.540356	0.270178	+	0.962810i	$0.412917\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.1842	−0.661434
$528$	0	0
$529$	−5.23951	−0.227805
$530$	0	0
$531$	−15.5462	−0.674646
$532$	0	0
$533$	−2.75557	−0.119357
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−54.4800	−2.35098
$538$	0	0
$539$	0.412342	0.0177608
$540$	0	0
$541$	−43.5526	−1.87247	−0.936237	−	0.351370i	$-0.885716\pi$
$541$	−43.5526	−1.87247	−0.936237	+	0.351370i	$0.885716\pi$
$542$	0	0
$543$	−30.9906	−1.32994
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.43017	0.0611497	0.0305748	−	0.999532i	$-0.490266\pi$
$547$	1.43017	0.0611497	0.0305748	+	0.999532i	$0.490266\pi$
$548$	0	0
$549$	−21.3131	−0.909622
$550$	0	0
$551$	7.80642	0.332565
$552$	0	0
$553$	−2.87601	−0.122301
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.0968	−1.14813	−0.574064	−	0.818811i	$-0.694634\pi$
$557$	−27.0968	−1.14813	−0.574064	+	0.818811i	$0.694634\pi$
$558$	0	0
$559$	−8.77631	−0.371198
$560$	0	0
$561$	−0.498707	−0.0210554
$562$	0	0
$563$	26.6113	1.12153	0.560767	−	0.827974i	$-0.310506\pi$
$563$	26.6113	1.12153	0.560767	+	0.827974i	$0.310506\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.0143	−0.420561
$568$	0	0
$569$	5.95407	0.249607	0.124804	−	0.992181i	$-0.460170\pi$
$569$	5.95407	0.249607	0.124804	+	0.992181i	$0.460170\pi$
$570$	0	0
$571$	−13.2859	−0.555998	−0.277999	−	0.960581i	$-0.589671\pi$
$571$	−13.2859	−0.555998	−0.277999	+	0.960581i	$0.589671\pi$
$572$	0	0
$573$	23.8163	0.994939
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.4242	1.18331	0.591657	−	0.806190i	$-0.298474\pi$
$577$	28.4242	1.18331	0.591657	+	0.806190i	$0.298474\pi$
$578$	0	0
$579$	34.1432	1.41894
$580$	0	0
$581$	−8.93624	−0.370738
$582$	0	0
$583$	0.238126	0.00986217
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.76986	0.361971	0.180985	−	0.983486i	$-0.442071\pi$
$587$	8.76986	0.361971	0.180985	+	0.983486i	$0.442071\pi$
$588$	0	0
$589$	23.0049	0.947901
$590$	0	0
$591$	47.0005	1.93334
$592$	0	0
$593$	30.2449	1.24201	0.621005	−	0.783807i	$-0.286725\pi$
$593$	30.2449	1.24201	0.621005	+	0.783807i	$0.286725\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−45.0321	−1.84304
$598$	0	0
$599$	17.3274	0.707979	0.353989	−	0.935249i	$-0.384825\pi$
$599$	17.3274	0.707979	0.353989	+	0.935249i	$0.384825\pi$
$600$	0	0
$601$	34.2864	1.39857	0.699286	−	0.714842i	$-0.253502\pi$
$601$	34.2864	1.39857	0.699286	+	0.714842i	$0.253502\pi$
$602$	0	0
$603$	−4.08742	−0.166453
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19.6523	−0.797663	−0.398832	−	0.917024i	$-0.630584\pi$
$607$	−19.6523	−0.797663	−0.398832	+	0.917024i	$0.630584\pi$
$608$	0	0
$609$	−3.05086	−0.123627
$610$	0	0
$611$	8.90321	0.360185
$612$	0	0
$613$	−17.9496	−0.724978	−0.362489	−	0.931988i	$-0.618073\pi$
$613$	−17.9496	−0.724978	−0.362489	+	0.931988i	$0.618073\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.6227	−1.59515	−0.797575	−	0.603220i	$-0.793884\pi$
$617$	−39.6227	−1.59515	−0.797575	+	0.603220i	$0.793884\pi$
$618$	0	0
$619$	12.5052	0.502625	0.251312	−	0.967906i	$-0.419138\pi$
$619$	12.5052	0.502625	0.251312	+	0.967906i	$0.419138\pi$
$620$	0	0
$621$	−10.2351	−0.410719
$622$	0	0
$623$	−13.0696	−0.523622
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.755569	0.0301745
$628$	0	0
$629$	40.3783	1.60999
$630$	0	0
$631$	15.2094	0.605477	0.302738	−	0.953074i	$-0.402099\pi$
$631$	15.2094	0.605477	0.302738	+	0.953074i	$0.402099\pi$
$632$	0	0
$633$	45.8894	1.82394
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.18421	0.245027
$638$	0	0
$639$	−10.5555	−0.417571
$640$	0	0
$641$	41.5496	1.64111	0.820555	−	0.571568i	$-0.193665\pi$
$641$	41.5496	1.64111	0.820555	+	0.571568i	$0.193665\pi$
$642$	0	0
$643$	18.3225	0.722568	0.361284	−	0.932456i	$-0.382338\pi$
$643$	18.3225	0.722568	0.361284	+	0.932456i	$0.382338\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.31555	−0.366232	−0.183116	−	0.983091i	$-0.558618\pi$
$647$	−9.31555	−0.366232	−0.183116	+	0.983091i	$0.558618\pi$
$648$	0	0
$649$	0.544640	0.0213790
$650$	0	0
$651$	−8.99063	−0.352371
$652$	0	0
$653$	−37.6128	−1.47190	−0.735952	−	0.677033i	$-0.763265\pi$
$653$	−37.6128	−1.47190	−0.735952	+	0.677033i	$0.763265\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.15701	0.201194
$658$	0	0
$659$	−46.4800	−1.81060	−0.905301	−	0.424770i	$-0.860355\pi$
$659$	−46.4800	−1.81060	−0.905301	+	0.424770i	$0.860355\pi$
$660$	0	0
$661$	−17.1655	−0.667659	−0.333830	−	0.942633i	$-0.608341\pi$
$661$	−17.1655	−0.667659	−0.333830	+	0.942633i	$0.608341\pi$
$662$	0	0
$663$	−7.47949	−0.290480
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.42864	−0.248918
$668$	0	0
$669$	−44.8385	−1.73356
$670$	0	0
$671$	0.746677	0.0288252
$672$	0	0
$673$	−5.06376	−0.195194	−0.0975968	−	0.995226i	$-0.531116\pi$
$673$	−5.06376	−0.195194	−0.0975968	+	0.995226i	$0.531116\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.6035	−0.638124	−0.319062	−	0.947734i	$-0.603368\pi$
$677$	−16.6035	−0.638124	−0.319062	+	0.947734i	$0.603368\pi$
$678$	0	0
$679$	−7.17130	−0.275209
$680$	0	0
$681$	−37.4291	−1.43429
$682$	0	0
$683$	−21.4938	−0.822437	−0.411218	−	0.911537i	$-0.634897\pi$
$683$	−21.4938	−0.822437	−0.411218	+	0.911537i	$0.634897\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.78721	−0.335253
$688$	0	0
$689$	3.57136	0.136058
$690$	0	0
$691$	48.3245	1.83835	0.919175	−	0.393849i	$-0.128857\pi$
$691$	48.3245	1.83835	0.919175	+	0.393849i	$0.128857\pi$
$692$	0	0
$693$	−0.114617	−0.00435395
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.30772	−0.352555
$698$	0	0
$699$	33.9180	1.28290
$700$	0	0
$701$	−41.6543	−1.57326	−0.786631	−	0.617423i	$-0.788177\pi$
$701$	−41.6543	−1.57326	−0.786631	+	0.617423i	$0.788177\pi$
$702$	0	0
$703$	−61.1753	−2.30727
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.68244	−0.176101
$708$	0	0
$709$	42.8988	1.61110	0.805548	−	0.592530i	$-0.201871\pi$
$709$	42.8988	1.61110	0.805548	+	0.592530i	$0.201871\pi$
$710$	0	0
$711$	−6.06022	−0.227276
$712$	0	0
$713$	−18.9447	−0.709485
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32.9131	1.22916
$718$	0	0
$719$	−6.39700	−0.238568	−0.119284	−	0.992860i	$-0.538060\pi$
$719$	−6.39700	−0.238568	−0.119284	+	0.992860i	$0.538060\pi$
$720$	0	0
$721$	7.12399	0.265311
$722$	0	0
$723$	−44.9403	−1.67135
$724$	0	0
$725$	0	0
$726$	0	0
$727$	14.0493	0.521061	0.260530	−	0.965466i	$-0.416103\pi$
$727$	14.0493	0.521061	0.260530	+	0.965466i	$0.416103\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	−29.6445	−1.09644
$732$	0	0
$733$	18.4701	0.682210	0.341105	−	0.940025i	$-0.389199\pi$
$733$	18.4701	0.682210	0.341105	+	0.940025i	$0.389199\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.143197	0.00527475
$738$	0	0
$739$	27.7911	1.02231	0.511156	−	0.859488i	$-0.329218\pi$
$739$	27.7911	1.02231	0.511156	+	0.859488i	$0.329218\pi$
$740$	0	0
$741$	11.3319	0.416286
$742$	0	0
$743$	15.3417	0.562832	0.281416	−	0.959586i	$-0.409196\pi$
$743$	15.3417	0.562832	0.281416	+	0.959586i	$0.409196\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18.8301	−0.688957
$748$	0	0
$749$	15.6860	0.573153
$750$	0	0
$751$	37.3560	1.36314	0.681570	−	0.731753i	$-0.261298\pi$
$751$	37.3560	1.36314	0.681570	+	0.731753i	$0.261298\pi$
$752$	0	0
$753$	−45.5308	−1.65924
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.5081	−0.418268	−0.209134	−	0.977887i	$-0.567065\pi$
$757$	−11.5081	−0.418268	−0.209134	+	0.977887i	$0.567065\pi$
$758$	0	0
$759$	−0.622216	−0.0225850
$760$	0	0
$761$	1.23459	0.0447537	0.0223769	−	0.999750i	$-0.492877\pi$
$761$	1.23459	0.0447537	0.0223769	+	0.999750i	$0.492877\pi$
$762$	0	0
$763$	0.120446	0.00436044
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.16839	0.294943
$768$	0	0
$769$	21.3176	0.768731	0.384365	−	0.923181i	$-0.374420\pi$
$769$	21.3176	0.768731	0.384365	+	0.923181i	$0.374420\pi$
$770$	0	0
$771$	60.9403	2.19471
$772$	0	0
$773$	−27.8336	−1.00111	−0.500553	−	0.865706i	$-0.666870\pi$
$773$	−27.8336	−1.00111	−0.500553	+	0.865706i	$0.666870\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	23.9081	0.857700
$778$	0	0
$779$	14.1017	0.505246
$780$	0	0
$781$	0.369800	0.0132325
$782$	0	0
$783$	3.70471	0.132396
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−43.7832	−1.56070	−0.780352	−	0.625340i	$-0.784960\pi$
$787$	−43.7832	−1.56070	−0.780352	+	0.625340i	$0.784960\pi$
$788$	0	0
$789$	−19.0049	−0.676593
$790$	0	0
$791$	−5.29883	−0.188405
$792$	0	0
$793$	11.1985	0.397670
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.314022	−0.0111232	−0.00556162	−	0.999985i	$-0.501770\pi$
$797$	−0.314022	−0.0111232	−0.00556162	+	0.999985i	$0.501770\pi$
$798$	0	0
$799$	30.0731	1.06391
$800$	0	0
$801$	−27.5397	−0.973068
$802$	0	0
$803$	−0.180669	−0.00637568
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.1334	−0.849534
$808$	0	0
$809$	−17.4050	−0.611927	−0.305963	−	0.952043i	$-0.598978\pi$
$809$	−17.4050	−0.611927	−0.305963	+	0.952043i	$0.598978\pi$
$810$	0	0
$811$	6.99709	0.245701	0.122850	−	0.992425i	$-0.460796\pi$
$811$	6.99709	0.245701	0.122850	+	0.992425i	$0.460796\pi$
$812$	0	0
$813$	18.7413	0.657285
$814$	0	0
$815$	0	0
$816$	0	0
$817$	44.9131	1.57131
$818$	0	0
$819$	−1.71900	−0.0600669
$820$	0	0
$821$	−22.6133	−0.789210	−0.394605	−	0.918851i	$-0.629119\pi$
$821$	−22.6133	−0.789210	−0.394605	+	0.918851i	$0.629119\pi$
$822$	0	0
$823$	−19.5506	−0.681492	−0.340746	−	0.940155i	$-0.610680\pi$
$823$	−19.5506	−0.681492	−0.340746	+	0.940155i	$0.610680\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.4746	0.711971	0.355985	−	0.934492i	$-0.384145\pi$
$827$	20.4746	0.711971	0.355985	+	0.934492i	$0.384145\pi$
$828$	0	0
$829$	−39.4019	−1.36849	−0.684243	−	0.729254i	$-0.739867\pi$
$829$	−39.4019	−1.36849	−0.684243	+	0.729254i	$0.739867\pi$
$830$	0	0
$831$	−63.6040	−2.20640
$832$	0	0
$833$	20.8889	0.723758
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.9175	0.377364
$838$	0	0
$839$	−1.69826	−0.0586305	−0.0293152	−	0.999570i	$-0.509333\pi$
$839$	−1.69826	−0.0586305	−0.0293152	+	0.999570i	$0.509333\pi$
$840$	0	0
$841$	−26.6731	−0.919761
$842$	0	0
$843$	31.4291	1.08248
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9.93132	−0.341244
$848$	0	0
$849$	−35.2498	−1.20977
$850$	0	0
$851$	50.3783	1.72694
$852$	0	0
$853$	35.4465	1.21366	0.606832	−	0.794830i	$-0.292440\pi$
$853$	35.4465	1.21366	0.606832	+	0.794830i	$0.292440\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.85728	0.165921	0.0829607	−	0.996553i	$-0.473562\pi$
$857$	4.85728	0.165921	0.0829607	+	0.996553i	$0.473562\pi$
$858$	0	0
$859$	−39.4420	−1.34574	−0.672872	−	0.739759i	$-0.734940\pi$
$859$	−39.4420	−1.34574	−0.672872	+	0.739759i	$0.734940\pi$
$860$	0	0
$861$	−5.51114	−0.187819
$862$	0	0
$863$	−3.68736	−0.125519	−0.0627596	−	0.998029i	$-0.519990\pi$
$863$	−3.68736	−0.125519	−0.0627596	+	0.998029i	$0.519990\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.3793	0.420424
$868$	0	0
$869$	0.212312	0.00720219
$870$	0	0
$871$	2.14764	0.0727701
$872$	0	0
$873$	−15.1111	−0.511433
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.4608	1.60264	0.801318	−	0.598239i	$-0.204133\pi$
$877$	47.4608	1.60264	0.801318	+	0.598239i	$0.204133\pi$
$878$	0	0
$879$	−45.2257	−1.52542
$880$	0	0
$881$	7.49378	0.252472	0.126236	−	0.992000i	$-0.459710\pi$
$881$	7.49378	0.252472	0.126236	+	0.992000i	$0.459710\pi$
$882$	0	0
$883$	28.8879	0.972154	0.486077	−	0.873916i	$-0.338427\pi$
$883$	28.8879	0.972154	0.486077	+	0.873916i	$0.338427\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−59.4499	−1.99613	−0.998065	−	0.0621718i	$-0.980197\pi$
$887$	−59.4499	−1.99613	−0.998065	+	0.0621718i	$0.980197\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0.739271	0.0247665
$892$	0	0
$893$	−45.5625	−1.52469
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−9.33185	−0.311581
$898$	0	0
$899$	6.85728	0.228703
$900$	0	0
$901$	12.0633	0.401886
$902$	0	0
$903$	−17.5526	−0.584115
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−8.58274	−0.284985	−0.142493	−	0.989796i	$-0.545512\pi$
$907$	−8.58274	−0.284985	−0.142493	+	0.989796i	$0.545512\pi$
$908$	0	0
$909$	−9.86665	−0.327256
$910$	0	0
$911$	−9.96836	−0.330266	−0.165133	−	0.986271i	$-0.552805\pi$
$911$	−9.96836	−0.330266	−0.165133	+	0.986271i	$0.552805\pi$
$912$	0	0
$913$	0.659687	0.0218325
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.24443	−0.173186
$918$	0	0
$919$	−22.9590	−0.757347	−0.378674	−	0.925530i	$-0.623620\pi$
$919$	−22.9590	−0.757347	−0.378674	+	0.925530i	$0.623620\pi$
$920$	0	0
$921$	0.193576	0.00637855
$922$	0	0
$923$	5.54617	0.182554
$924$	0	0
$925$	0	0
$926$	0	0
$927$	15.0114	0.493038
$928$	0	0
$929$	−23.0509	−0.756274	−0.378137	−	0.925750i	$-0.623435\pi$
$929$	−23.0509	−0.756274	−0.378137	+	0.925750i	$0.623435\pi$
$930$	0	0
$931$	−31.6479	−1.03722
$932$	0	0
$933$	56.9403	1.86414
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.2543	−1.15171	−0.575853	−	0.817553i	$-0.695330\pi$
$937$	−35.2543	−1.15171	−0.575853	+	0.817553i	$0.695330\pi$
$938$	0	0
$939$	−16.9906	−0.554468
$940$	0	0
$941$	10.1334	0.330338	0.165169	−	0.986265i	$-0.447183\pi$
$941$	10.1334	0.330338	0.165169	+	0.986265i	$0.447183\pi$
$942$	0	0
$943$	−11.6128	−0.378166
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.9447	1.07056	0.535279	−	0.844675i	$-0.320206\pi$
$947$	32.9447	1.07056	0.535279	+	0.844675i	$0.320206\pi$
$948$	0	0
$949$	−2.70964	−0.0879585
$950$	0	0
$951$	8.10171	0.262716
$952$	0	0
$953$	−22.3684	−0.724584	−0.362292	−	0.932065i	$-0.618006\pi$
$953$	−22.3684	−0.724584	−0.362292	+	0.932065i	$0.618006\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.225219	0.00728030
$958$	0	0
$959$	14.7012	0.474726
$960$	0	0
$961$	−10.7921	−0.348133
$962$	0	0
$963$	33.0529	1.06511
$964$	0	0
$965$	0	0
$966$	0	0
$967$	7.12537	0.229136	0.114568	−	0.993415i	$-0.463452\pi$
$967$	7.12537	0.229136	0.114568	+	0.993415i	$0.463452\pi$
$968$	0	0
$969$	38.2766	1.22962
$970$	0	0
$971$	−10.1521	−0.325796	−0.162898	−	0.986643i	$-0.552084\pi$
$971$	−10.1521	−0.325796	−0.162898	+	0.986643i	$0.552084\pi$
$972$	0	0
$973$	−6.48886	−0.208023
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.4973	−0.719753	−0.359877	−	0.933000i	$-0.617181\pi$
$977$	−22.4973	−0.719753	−0.359877	+	0.933000i	$0.617181\pi$
$978$	0	0
$979$	0.964818	0.0308357
$980$	0	0
$981$	0.253799	0.00810318
$982$	0	0
$983$	−27.7003	−0.883501	−0.441751	−	0.897138i	$-0.645642\pi$
$983$	−27.7003	−0.883501	−0.441751	+	0.897138i	$0.645642\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.8064	0.566785
$988$	0	0
$989$	−36.9862	−1.17609
$990$	0	0
$991$	0.898766	0.0285502	0.0142751	−	0.999898i	$-0.495456\pi$
$991$	0.898766	0.0285502	0.0142751	+	0.999898i	$0.495456\pi$
$992$	0	0
$993$	0.893368	0.0283502
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−19.5625	−0.619550	−0.309775	−	0.950810i	$-0.600254\pi$
$997$	−19.5625	−0.619550	−0.309775	+	0.950810i	$0.600254\pi$
$998$	0	0
$999$	−29.0321	−0.918536

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1300.2.a.j.1.1		3
4.3	odd	2		5200.2.a.cg.1.3		3
5.2	odd	4		260.2.c.a.209.6	yes	6
5.3	odd	4		260.2.c.a.209.1	✓	6
5.4	even	2		1300.2.a.k.1.3		3
15.2	even	4		2340.2.h.e.469.1		6
15.8	even	4		2340.2.h.e.469.2		6
20.3	even	4		1040.2.d.d.209.6		6
20.7	even	4		1040.2.d.d.209.1		6
20.19	odd	2		5200.2.a.cd.1.1		3
65.8	even	4		3380.2.d.b.1689.1		6
65.12	odd	4		3380.2.c.c.2029.6		6
65.18	even	4		3380.2.d.a.1689.1		6
65.38	odd	4		3380.2.c.c.2029.1		6
65.47	even	4		3380.2.d.a.1689.6		6
65.57	even	4		3380.2.d.b.1689.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
260.2.c.a.209.1	✓	6	5.3	odd	4
260.2.c.a.209.6	yes	6	5.2	odd	4
1040.2.d.d.209.1		6	20.7	even	4
1040.2.d.d.209.6		6	20.3	even	4
1300.2.a.j.1.1		3	1.1	even	1	trivial
1300.2.a.k.1.3		3	5.4	even	2
2340.2.h.e.469.1		6	15.2	even	4
2340.2.h.e.469.2		6	15.8	even	4
3380.2.c.c.2029.1		6	65.38	odd	4
3380.2.c.c.2029.6		6	65.12	odd	4
3380.2.d.a.1689.1		6	65.18	even	4
3380.2.d.a.1689.6		6	65.47	even	4
3380.2.d.b.1689.1		6	65.8	even	4
3380.2.d.b.1689.6		6	65.57	even	4
5200.2.a.cd.1.1		3	20.19	odd	2
5200.2.a.cg.1.3		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)

\(f(q)\)	\(=\)	\(q-2.21432 q^{3} +0.903212 q^{7} +1.90321 q^{9} +O(q^{10})\)	\(q-2.21432 q^{3} +0.903212 q^{7} +1.90321 q^{9} -0.0666765 q^{11} -1.00000 q^{13} -3.37778 q^{17} +5.11753 q^{19} -2.00000 q^{21} -4.21432 q^{23} +2.42864 q^{27} +1.52543 q^{29} +4.49532 q^{31} +0.147643 q^{33} -11.9541 q^{37} +2.21432 q^{39} +2.75557 q^{41} +8.77631 q^{43} -8.90321 q^{47} -6.18421 q^{49} +7.47949 q^{51} -3.57136 q^{53} -11.3319 q^{57} -8.16839 q^{59} -11.1985 q^{61} +1.71900 q^{63} -2.14764 q^{67} +9.33185 q^{69} -5.54617 q^{71} +2.70964 q^{73} -0.0602231 q^{77} -3.18421 q^{79} -11.0874 q^{81} -9.89384 q^{83} -3.37778 q^{87} -14.4701 q^{89} -0.903212 q^{91} -9.95407 q^{93} -7.93978 q^{97} -0.126900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{7} - q^{9}+O(q^{10}) \) 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

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