LMFDB - Embedded newform 5200.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5200,2,Mod(1,5200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-4,0,0,0,-2,0,3,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.5222090511$
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 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.481194 q^{3} -0.806063 q^{7} -2.76845 q^{9} +3.67513 q^{11} -1.00000 q^{13} -1.35026 q^{17} +1.67513 q^{19} -0.387873 q^{21} -6.48119 q^{23} -2.77575 q^{27} +2.41819 q^{29} +5.28726 q^{31} +1.76845 q^{33} +3.76845 q^{37} -0.481194 q^{39} -8.31265 q^{41} +6.79384 q^{43} -3.19394 q^{47} -6.35026 q^{49} -0.649738 q^{51} +5.73813 q^{53} +0.806063 q^{57} -5.98778 q^{59} -1.76845 q^{61} +2.23155 q^{63} -9.89446 q^{67} -3.11871 q^{69} -8.56230 q^{71} -11.7685 q^{73} -2.96239 q^{77} +2.26187 q^{79} +6.96968 q^{81} -3.84367 q^{83} +1.16362 q^{87} +2.77575 q^{89} +0.806063 q^{91} +2.54420 q^{93} -1.87399 q^{97} -10.1744 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * q^99

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
$3$	0.481194	0.277818	0.138909	−	0.990305i	$-0.455641\pi$
$3$	0.481194	0.277818	0.138909	+	0.990305i	$0.455641\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.806063	−0.304663	−0.152332	−	0.988329i	$-0.548678\pi$
$7$	−0.806063	−0.304663	−0.152332	+	0.988329i	$0.548678\pi$
$8$	0	0
$9$	−2.76845	−0.922817
$10$	0	0
$11$	3.67513	1.10809	0.554047	−	0.832486i	$-0.313083\pi$
$11$	3.67513	1.10809	0.554047	+	0.832486i	$0.313083\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.35026	−0.327487	−0.163743	−	0.986503i	$-0.552357\pi$
$17$	−1.35026	−0.327487	−0.163743	+	0.986503i	$0.552357\pi$
$18$	0	0
$19$	1.67513	0.384301	0.192151	−	0.981365i	$-0.438454\pi$
$19$	1.67513	0.384301	0.192151	+	0.981365i	$0.438454\pi$
$20$	0	0
$21$	−0.387873	−0.0846409
$22$	0	0
$23$	−6.48119	−1.35142	−0.675711	−	0.737166i	$-0.736163\pi$
$23$	−6.48119	−1.35142	−0.675711	+	0.737166i	$0.736163\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.77575	−0.534193
$28$	0	0
$29$	2.41819	0.449047	0.224523	−	0.974469i	$-0.427917\pi$
$29$	2.41819	0.449047	0.224523	+	0.974469i	$0.427917\pi$
$30$	0	0
$31$	5.28726	0.949620	0.474810	−	0.880088i	$-0.342517\pi$
$31$	5.28726	0.949620	0.474810	+	0.880088i	$0.342517\pi$
$32$	0	0
$33$	1.76845	0.307848
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.76845	0.619530	0.309765	−	0.950813i	$-0.399750\pi$
$37$	3.76845	0.619530	0.309765	+	0.950813i	$0.399750\pi$
$38$	0	0
$39$	−0.481194	−0.0770528
$40$	0	0
$41$	−8.31265	−1.29822	−0.649109	−	0.760695i	$-0.724858\pi$
$41$	−8.31265	−1.29822	−0.649109	+	0.760695i	$0.724858\pi$
$42$	0	0
$43$	6.79384	1.03605	0.518026	−	0.855365i	$-0.326667\pi$
$43$	6.79384	1.03605	0.518026	+	0.855365i	$0.326667\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.19394	−0.465884	−0.232942	−	0.972491i	$-0.574835\pi$
$47$	−3.19394	−0.465884	−0.232942	+	0.972491i	$0.574835\pi$
$48$	0	0
$49$	−6.35026	−0.907180
$50$	0	0
$51$	−0.649738	−0.0909816
$52$	0	0
$53$	5.73813	0.788193	0.394097	−	0.919069i	$-0.371057\pi$
$53$	5.73813	0.788193	0.394097	+	0.919069i	$0.371057\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.806063	0.106766
$58$	0	0
$59$	−5.98778	−0.779543	−0.389771	−	0.920912i	$-0.627446\pi$
$59$	−5.98778	−0.779543	−0.389771	+	0.920912i	$0.627446\pi$
$60$	0	0
$61$	−1.76845	−0.226427	−0.113214	−	0.993571i	$-0.536114\pi$
$61$	−1.76845	−0.226427	−0.113214	+	0.993571i	$0.536114\pi$
$62$	0	0
$63$	2.23155	0.281149
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.89446	−1.20880	−0.604400	−	0.796681i	$-0.706587\pi$
$67$	−9.89446	−1.20880	−0.604400	+	0.796681i	$0.706587\pi$
$68$	0	0
$69$	−3.11871	−0.375449
$70$	0	0
$71$	−8.56230	−1.01616	−0.508079	−	0.861311i	$-0.669644\pi$
$71$	−8.56230	−1.01616	−0.508079	+	0.861311i	$0.669644\pi$
$72$	0	0
$73$	−11.7685	−1.37739	−0.688697	−	0.725050i	$-0.741817\pi$
$73$	−11.7685	−1.37739	−0.688697	+	0.725050i	$0.741817\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.96239	−0.337596
$78$	0	0
$79$	2.26187	0.254480	0.127240	−	0.991872i	$-0.459388\pi$
$79$	2.26187	0.254480	0.127240	+	0.991872i	$0.459388\pi$
$80$	0	0
$81$	6.96968	0.774409
$82$	0	0
$83$	−3.84367	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$83$	−3.84367	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.16362	0.124753
$88$	0	0
$89$	2.77575	0.294229	0.147114	−	0.989120i	$-0.453001\pi$
$89$	2.77575	0.294229	0.147114	+	0.989120i	$0.453001\pi$
$90$	0	0
$91$	0.806063	0.0844984
$92$	0	0
$93$	2.54420	0.263821
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.87399	−0.190275	−0.0951375	−	0.995464i	$-0.530329\pi$
$97$	−1.87399	−0.190275	−0.0951375	+	0.995464i	$0.530329\pi$
$98$	0	0
$99$	−10.1744	−1.02257
$100$	0	0
$101$	10.4993	1.04472	0.522359	−	0.852725i	$-0.325052\pi$
$101$	10.4993	1.04472	0.522359	+	0.852725i	$0.325052\pi$
$102$	0	0
$103$	−15.3684	−1.51429	−0.757145	−	0.653247i	$-0.773406\pi$
$103$	−15.3684	−1.51429	−0.757145	+	0.653247i	$0.773406\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1309	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$107$	−11.1309	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$108$	0	0
$109$	9.58769	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$109$	9.58769	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$110$	0	0
$111$	1.81336	0.172116
$112$	0	0
$113$	0.574515	0.0540459	0.0270229	−	0.999635i	$-0.491397\pi$
$113$	0.574515	0.0540459	0.0270229	+	0.999635i	$0.491397\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.76845	0.255943
$118$	0	0
$119$	1.08840	0.0997732
$120$	0	0
$121$	2.50659	0.227872
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.29455	0.381080	0.190540	−	0.981679i	$-0.438976\pi$
$127$	4.29455	0.381080	0.190540	+	0.981679i	$0.438976\pi$
$128$	0	0
$129$	3.26916	0.287833
$130$	0	0
$131$	0.836381	0.0730749	0.0365375	−	0.999332i	$-0.488367\pi$
$131$	0.836381	0.0730749	0.0365375	+	0.999332i	$0.488367\pi$
$132$	0	0
$133$	−1.35026	−0.117083
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.9380	−1.27624	−0.638118	−	0.769939i	$-0.720287\pi$
$137$	−14.9380	−1.27624	−0.638118	+	0.769939i	$0.720287\pi$
$138$	0	0
$139$	8.43866	0.715758	0.357879	−	0.933768i	$-0.383500\pi$
$139$	8.43866	0.715758	0.357879	+	0.933768i	$0.383500\pi$
$140$	0	0
$141$	−1.53690	−0.129431
$142$	0	0
$143$	−3.67513	−0.307330
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.05571	−0.252031
$148$	0	0
$149$	−11.3503	−0.929850	−0.464925	−	0.885350i	$-0.653919\pi$
$149$	−11.3503	−0.929850	−0.464925	+	0.885350i	$0.653919\pi$
$150$	0	0
$151$	−13.9878	−1.13831	−0.569155	−	0.822230i	$-0.692729\pi$
$151$	−13.9878	−1.13831	−0.569155	+	0.822230i	$0.692729\pi$
$152$	0	0
$153$	3.73813	0.302210
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.77575	0.221529	0.110764	−	0.993847i	$-0.464670\pi$
$157$	2.77575	0.221529	0.110764	+	0.993847i	$0.464670\pi$
$158$	0	0
$159$	2.76116	0.218974
$160$	0	0
$161$	5.22425	0.411729
$162$	0	0
$163$	−2.23155	−0.174788	−0.0873942	−	0.996174i	$-0.527854\pi$
$163$	−2.23155	−0.174788	−0.0873942	+	0.996174i	$0.527854\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.6932	1.21438	0.607189	−	0.794557i	$-0.292297\pi$
$167$	15.6932	1.21438	0.607189	+	0.794557i	$0.292297\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.63752	−0.354640
$172$	0	0
$173$	−25.5877	−1.94540	−0.972698	−	0.232075i	$-0.925449\pi$
$173$	−25.5877	−1.94540	−0.972698	+	0.232075i	$0.925449\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.88129	−0.216571
$178$	0	0
$179$	−12.1260	−0.906340	−0.453170	−	0.891424i	$-0.649707\pi$
$179$	−12.1260	−0.906340	−0.453170	+	0.891424i	$0.649707\pi$
$180$	0	0
$181$	−2.73084	−0.202982	−0.101491	−	0.994836i	$-0.532361\pi$
$181$	−2.73084	−0.202982	−0.101491	+	0.994836i	$0.532361\pi$
$182$	0	0
$183$	−0.850969	−0.0629054
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.96239	−0.362886
$188$	0	0
$189$	2.23743	0.162749
$190$	0	0
$191$	−20.6253	−1.49239	−0.746197	−	0.665725i	$-0.768122\pi$
$191$	−20.6253	−1.49239	−0.746197	+	0.665725i	$0.768122\pi$
$192$	0	0
$193$	−21.7889	−1.56840	−0.784200	−	0.620508i	$-0.786927\pi$
$193$	−21.7889	−1.56840	−0.784200	+	0.620508i	$0.786927\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	16.7513	1.18747	0.593734	−	0.804661i	$-0.297653\pi$
$199$	16.7513	1.18747	0.593734	+	0.804661i	$0.297653\pi$
$200$	0	0
$201$	−4.76116	−0.335826
$202$	0	0
$203$	−1.94921	−0.136808
$204$	0	0
$205$	0	0
$206$	0	0
$207$	17.9429	1.24712
$208$	0	0
$209$	6.15633	0.425842
$210$	0	0
$211$	4.90175	0.337451	0.168725	−	0.985663i	$-0.446035\pi$
$211$	4.90175	0.337451	0.168725	+	0.985663i	$0.446035\pi$
$212$	0	0
$213$	−4.12013	−0.282307
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.26187	−0.289314
$218$	0	0
$219$	−5.66291	−0.382664
$220$	0	0
$221$	1.35026	0.0908284
$222$	0	0
$223$	−24.9076	−1.66794	−0.833969	−	0.551811i	$-0.813937\pi$
$223$	−24.9076	−1.66794	−0.833969	+	0.551811i	$0.813937\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.95509	−0.660743	−0.330371	−	0.943851i	$-0.607174\pi$
$227$	−9.95509	−0.660743	−0.330371	+	0.943851i	$0.607174\pi$
$228$	0	0
$229$	5.35026	0.353555	0.176778	−	0.984251i	$-0.443433\pi$
$229$	5.35026	0.353555	0.176778	+	0.984251i	$0.443433\pi$
$230$	0	0
$231$	−1.42548	−0.0937900
$232$	0	0
$233$	−10.7612	−0.704987	−0.352493	−	0.935814i	$-0.614666\pi$
$233$	−10.7612	−0.704987	−0.352493	+	0.935814i	$0.614666\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.08840	0.0706990
$238$	0	0
$239$	11.8618	0.767274	0.383637	−	0.923484i	$-0.374671\pi$
$239$	11.8618	0.767274	0.383637	+	0.923484i	$0.374671\pi$
$240$	0	0
$241$	−28.6253	−1.84392	−0.921959	−	0.387288i	$-0.873412\pi$
$241$	−28.6253	−1.84392	−0.921959	+	0.387288i	$0.873412\pi$
$242$	0	0
$243$	11.6810	0.749337
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.67513	−0.106586
$248$	0	0
$249$	−1.84955	−0.117211
$250$	0	0
$251$	19.3865	1.22366	0.611831	−	0.790988i	$-0.290433\pi$
$251$	19.3865	1.22366	0.611831	+	0.790988i	$0.290433\pi$
$252$	0	0
$253$	−23.8192	−1.49750
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.8627	1.42614	0.713069	−	0.701094i	$-0.247305\pi$
$257$	22.8627	1.42614	0.713069	+	0.701094i	$0.247305\pi$
$258$	0	0
$259$	−3.03761	−0.188748
$260$	0	0
$261$	−6.69464	−0.414388
$262$	0	0
$263$	−21.8822	−1.34932	−0.674658	−	0.738130i	$-0.735709\pi$
$263$	−21.8822	−1.34932	−0.674658	+	0.738130i	$0.735709\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.33567	0.0817419
$268$	0	0
$269$	22.7513	1.38717	0.693586	−	0.720374i	$-0.256030\pi$
$269$	22.7513	1.38717	0.693586	+	0.720374i	$0.256030\pi$
$270$	0	0
$271$	−0.123638	−0.00751049	−0.00375525	−	0.999993i	$-0.501195\pi$
$271$	−0.123638	−0.00751049	−0.00375525	+	0.999993i	$0.501195\pi$
$272$	0	0
$273$	0.387873	0.0234751
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.3503	−0.922308	−0.461154	−	0.887320i	$-0.652564\pi$
$277$	−15.3503	−0.922308	−0.461154	+	0.887320i	$0.652564\pi$
$278$	0	0
$279$	−14.6375	−0.876325
$280$	0	0
$281$	13.9248	0.830683	0.415341	−	0.909666i	$-0.363662\pi$
$281$	13.9248	0.830683	0.415341	+	0.909666i	$0.363662\pi$
$282$	0	0
$283$	−20.3815	−1.21156	−0.605778	−	0.795634i	$-0.707138\pi$
$283$	−20.3815	−1.21156	−0.605778	+	0.795634i	$0.707138\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.70052	0.395519
$288$	0	0
$289$	−15.1768	−0.892753
$290$	0	0
$291$	−0.901754	−0.0528618
$292$	0	0
$293$	5.38058	0.314337	0.157168	−	0.987572i	$-0.449763\pi$
$293$	5.38058	0.314337	0.157168	+	0.987572i	$0.449763\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−10.2012	−0.591935
$298$	0	0
$299$	6.48119	0.374817
$300$	0	0
$301$	−5.47627	−0.315647
$302$	0	0
$303$	5.05220	0.290241
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.1695	1.09406	0.547031	−	0.837113i	$-0.315758\pi$
$307$	19.1695	1.09406	0.547031	+	0.837113i	$0.315758\pi$
$308$	0	0
$309$	−7.39517	−0.420696
$310$	0	0
$311$	25.2506	1.43183	0.715915	−	0.698187i	$-0.246010\pi$
$311$	25.2506	1.43183	0.715915	+	0.698187i	$0.246010\pi$
$312$	0	0
$313$	2.81194	0.158940	0.0794702	−	0.996837i	$-0.474677\pi$
$313$	2.81194	0.158940	0.0794702	+	0.996837i	$0.474677\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.7685	−1.33497	−0.667485	−	0.744624i	$-0.732629\pi$
$317$	−23.7685	−1.33497	−0.667485	+	0.744624i	$0.732629\pi$
$318$	0	0
$319$	8.88717	0.497586
$320$	0	0
$321$	−5.35614	−0.298951
$322$	0	0
$323$	−2.26187	−0.125854
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.61354	0.255129
$328$	0	0
$329$	2.57452	0.141938
$330$	0	0
$331$	11.8011	0.648649	0.324325	−	0.945946i	$-0.394863\pi$
$331$	11.8011	0.648649	0.324325	+	0.945946i	$0.394863\pi$
$332$	0	0
$333$	−10.4328	−0.571713
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.1114	0.877645	0.438822	−	0.898574i	$-0.355396\pi$
$337$	16.1114	0.877645	0.438822	+	0.898574i	$0.355396\pi$
$338$	0	0
$339$	0.276454	0.0150149
$340$	0	0
$341$	19.4314	1.05227
$342$	0	0
$343$	10.7612	0.581048
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.4944	−1.47598	−0.737988	−	0.674814i	$-0.764224\pi$
$347$	−27.4944	−1.47598	−0.737988	+	0.674814i	$0.764224\pi$
$348$	0	0
$349$	−17.6023	−0.942228	−0.471114	−	0.882072i	$-0.656148\pi$
$349$	−17.6023	−0.942228	−0.471114	+	0.882072i	$0.656148\pi$
$350$	0	0
$351$	2.77575	0.148158
$352$	0	0
$353$	15.7685	0.839270	0.419635	−	0.907693i	$-0.362158\pi$
$353$	15.7685	0.839270	0.419635	+	0.907693i	$0.362158\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.523730	0.0277187
$358$	0	0
$359$	14.8242	0.782389	0.391195	−	0.920308i	$-0.372062\pi$
$359$	14.8242	0.782389	0.391195	+	0.920308i	$0.372062\pi$
$360$	0	0
$361$	−16.1939	−0.852312
$362$	0	0
$363$	1.20616	0.0633067
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.0313	1.41102	0.705510	−	0.708700i	$-0.250718\pi$
$367$	27.0313	1.41102	0.705510	+	0.708700i	$0.250718\pi$
$368$	0	0
$369$	23.0132	1.19802
$370$	0	0
$371$	−4.62530	−0.240134
$372$	0	0
$373$	12.9525	0.670657	0.335329	−	0.942101i	$-0.391153\pi$
$373$	12.9525	0.670657	0.335329	+	0.942101i	$0.391153\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.41819	−0.124543
$378$	0	0
$379$	30.2858	1.55568	0.777840	−	0.628463i	$-0.216316\pi$
$379$	30.2858	1.55568	0.777840	+	0.628463i	$0.216316\pi$
$380$	0	0
$381$	2.06651	0.105871
$382$	0	0
$383$	21.0943	1.07787	0.538934	−	0.842348i	$-0.318827\pi$
$383$	21.0943	1.07787	0.538934	+	0.842348i	$0.318827\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−18.8084	−0.956086
$388$	0	0
$389$	−6.77575	−0.343544	−0.171772	−	0.985137i	$-0.554949\pi$
$389$	−6.77575	−0.343544	−0.171772	+	0.985137i	$0.554949\pi$
$390$	0	0
$391$	8.75131	0.442573
$392$	0	0
$393$	0.402462	0.0203015
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4690	0.525423	0.262711	−	0.964874i	$-0.415383\pi$
$397$	10.4690	0.525423	0.262711	+	0.964874i	$0.415383\pi$
$398$	0	0
$399$	−0.649738	−0.0325276
$400$	0	0
$401$	5.01317	0.250346	0.125173	−	0.992135i	$-0.460051\pi$
$401$	5.01317	0.250346	0.125173	+	0.992135i	$0.460051\pi$
$402$	0	0
$403$	−5.28726	−0.263377
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13.8496	0.686497
$408$	0	0
$409$	−14.3879	−0.711435	−0.355717	−	0.934594i	$-0.615763\pi$
$409$	−14.3879	−0.711435	−0.355717	+	0.934594i	$0.615763\pi$
$410$	0	0
$411$	−7.18806	−0.354561
$412$	0	0
$413$	4.82653	0.237498
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.06063	0.198850
$418$	0	0
$419$	−17.4617	−0.853059	−0.426529	−	0.904474i	$-0.640264\pi$
$419$	−17.4617	−0.853059	−0.426529	+	0.904474i	$0.640264\pi$
$420$	0	0
$421$	2.88717	0.140712	0.0703559	−	0.997522i	$-0.477586\pi$
$421$	2.88717	0.140712	0.0703559	+	0.997522i	$0.477586\pi$
$422$	0	0
$423$	8.84226	0.429925
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.42548	0.0689840
$428$	0	0
$429$	−1.76845	−0.0853817
$430$	0	0
$431$	−0.889535	−0.0428474	−0.0214237	−	0.999770i	$-0.506820\pi$
$431$	−0.889535	−0.0428474	−0.0214237	+	0.999770i	$0.506820\pi$
$432$	0	0
$433$	−25.2506	−1.21347	−0.606733	−	0.794906i	$-0.707520\pi$
$433$	−25.2506	−1.21347	−0.606733	+	0.794906i	$0.707520\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.8568	−0.519354
$438$	0	0
$439$	−28.8119	−1.37512	−0.687560	−	0.726128i	$-0.741318\pi$
$439$	−28.8119	−1.37512	−0.687560	+	0.726128i	$0.741318\pi$
$440$	0	0
$441$	17.5804	0.837162
$442$	0	0
$443$	−36.9805	−1.75700	−0.878498	−	0.477746i	$-0.841454\pi$
$443$	−36.9805	−1.75700	−0.878498	+	0.477746i	$0.841454\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.46168	−0.258329
$448$	0	0
$449$	−12.6859	−0.598686	−0.299343	−	0.954146i	$-0.596768\pi$
$449$	−12.6859	−0.598686	−0.299343	+	0.954146i	$0.596768\pi$
$450$	0	0
$451$	−30.5501	−1.43855
$452$	0	0
$453$	−6.73084	−0.316242
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.0494	−1.17176	−0.585880	−	0.810398i	$-0.699251\pi$
$457$	−25.0494	−1.17176	−0.585880	+	0.810398i	$0.699251\pi$
$458$	0	0
$459$	3.74798	0.174941
$460$	0	0
$461$	−36.8872	−1.71801	−0.859003	−	0.511970i	$-0.828916\pi$
$461$	−36.8872	−1.71801	−0.859003	+	0.511970i	$0.828916\pi$
$462$	0	0
$463$	39.0191	1.81337	0.906685	−	0.421809i	$-0.138605\pi$
$463$	39.0191	1.81337	0.906685	+	0.421809i	$0.138605\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.7694	−1.51639	−0.758194	−	0.652029i	$-0.773918\pi$
$467$	−32.7694	−1.51639	−0.758194	+	0.652029i	$0.773918\pi$
$468$	0	0
$469$	7.97556	0.368277
$470$	0	0
$471$	1.33567	0.0615446
$472$	0	0
$473$	24.9683	1.14804
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.8858	−0.727359
$478$	0	0
$479$	−16.8749	−0.771036	−0.385518	−	0.922700i	$-0.625977\pi$
$479$	−16.8749	−0.771036	−0.385518	+	0.922700i	$0.625977\pi$
$480$	0	0
$481$	−3.76845	−0.171827
$482$	0	0
$483$	2.51388	0.114386
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.24472	0.418918	0.209459	−	0.977817i	$-0.432830\pi$
$487$	9.24472	0.418918	0.209459	+	0.977817i	$0.432830\pi$
$488$	0	0
$489$	−1.07381	−0.0485593
$490$	0	0
$491$	25.7499	1.16208	0.581038	−	0.813876i	$-0.302647\pi$
$491$	25.7499	1.16208	0.581038	+	0.813876i	$0.302647\pi$
$492$	0	0
$493$	−3.26519	−0.147057
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.90175	0.309586
$498$	0	0
$499$	27.7015	1.24009	0.620044	−	0.784567i	$-0.287115\pi$
$499$	27.7015	1.24009	0.620044	+	0.784567i	$0.287115\pi$
$500$	0	0
$501$	7.55149	0.337376
$502$	0	0
$503$	−2.35519	−0.105013	−0.0525063	−	0.998621i	$-0.516721\pi$
$503$	−2.35519	−0.105013	−0.0525063	+	0.998621i	$0.516721\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.481194	0.0213706
$508$	0	0
$509$	21.5125	0.953523	0.476762	−	0.879033i	$-0.341810\pi$
$509$	21.5125	0.953523	0.476762	+	0.879033i	$0.341810\pi$
$510$	0	0
$511$	9.48612	0.419641
$512$	0	0
$513$	−4.64974	−0.205291
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.7381	−0.516243
$518$	0	0
$519$	−12.3127	−0.540465
$520$	0	0
$521$	37.7440	1.65360	0.826798	−	0.562499i	$-0.190160\pi$
$521$	37.7440	1.65360	0.826798	+	0.562499i	$0.190160\pi$
$522$	0	0
$523$	−23.7416	−1.03815	−0.519075	−	0.854729i	$-0.673723\pi$
$523$	−23.7416	−1.03815	−0.519075	+	0.854729i	$0.673723\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.13918	−0.310988
$528$	0	0
$529$	19.0059	0.826343
$530$	0	0
$531$	16.5769	0.719376
$532$	0	0
$533$	8.31265	0.360061
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.83497	−0.251797
$538$	0	0
$539$	−23.3380	−1.00524
$540$	0	0
$541$	13.0376	0.560531	0.280265	−	0.959923i	$-0.409578\pi$
$541$	13.0376	0.560531	0.280265	+	0.959923i	$0.409578\pi$
$542$	0	0
$543$	−1.31406	−0.0563919
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.43041	−0.360458	−0.180229	−	0.983625i	$-0.557684\pi$
$547$	−8.43041	−0.360458	−0.180229	+	0.983625i	$0.557684\pi$
$548$	0	0
$549$	4.89587	0.208951
$550$	0	0
$551$	4.05079	0.172569
$552$	0	0
$553$	−1.82321	−0.0775306
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.6932	0.580201	0.290100	−	0.956996i	$-0.406311\pi$
$557$	13.6932	0.580201	0.290100	+	0.956996i	$0.406311\pi$
$558$	0	0
$559$	−6.79384	−0.287349
$560$	0	0
$561$	−2.38787	−0.100816
$562$	0	0
$563$	8.86907	0.373787	0.186893	−	0.982380i	$-0.440158\pi$
$563$	8.86907	0.373787	0.186893	+	0.982380i	$0.440158\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5.61801	−0.235934
$568$	0	0
$569$	−32.7816	−1.37428	−0.687139	−	0.726526i	$-0.741134\pi$
$569$	−32.7816	−1.37428	−0.687139	+	0.726526i	$0.741134\pi$
$570$	0	0
$571$	−40.2882	−1.68601	−0.843005	−	0.537906i	$-0.819215\pi$
$571$	−40.2882	−1.68601	−0.843005	+	0.537906i	$0.819215\pi$
$572$	0	0
$573$	−9.92478	−0.414614
$574$	0	0
$575$	0	0
$576$	0	0
$577$	28.8568	1.20133	0.600663	−	0.799502i	$-0.294903\pi$
$577$	28.8568	1.20133	0.600663	+	0.799502i	$0.294903\pi$
$578$	0	0
$579$	−10.4847	−0.435729
$580$	0	0
$581$	3.09825	0.128537
$582$	0	0
$583$	21.0884	0.873392
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.6786	−1.72026	−0.860131	−	0.510074i	$-0.829618\pi$
$587$	−41.6786	−1.72026	−0.860131	+	0.510074i	$0.829618\pi$
$588$	0	0
$589$	8.85685	0.364940
$590$	0	0
$591$	0.962389	0.0395874
$592$	0	0
$593$	22.4993	0.923935	0.461968	−	0.886897i	$-0.347144\pi$
$593$	22.4993	0.923935	0.461968	+	0.886897i	$0.347144\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.06063	0.329900
$598$	0	0
$599$	−4.15045	−0.169583	−0.0847913	−	0.996399i	$-0.527022\pi$
$599$	−4.15045	−0.169583	−0.0847913	+	0.996399i	$0.527022\pi$
$600$	0	0
$601$	27.9248	1.13908	0.569538	−	0.821965i	$-0.307122\pi$
$601$	27.9248	1.13908	0.569538	+	0.821965i	$0.307122\pi$
$602$	0	0
$603$	27.3923	1.11550
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.19489	−0.332620	−0.166310	−	0.986073i	$-0.553185\pi$
$607$	−8.19489	−0.332620	−0.166310	+	0.986073i	$0.553185\pi$
$608$	0	0
$609$	−0.937951	−0.0380077
$610$	0	0
$611$	3.19394	0.129213
$612$	0	0
$613$	33.1392	1.33848	0.669239	−	0.743047i	$-0.266620\pi$
$613$	33.1392	1.33848	0.669239	+	0.743047i	$0.266620\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.0132	−1.16803	−0.584013	−	0.811744i	$-0.698518\pi$
$617$	−29.0132	−1.16803	−0.584013	+	0.811744i	$0.698518\pi$
$618$	0	0
$619$	−12.2134	−0.490900	−0.245450	−	0.969409i	$-0.578936\pi$
$619$	−12.2134	−0.490900	−0.245450	+	0.969409i	$0.578936\pi$
$620$	0	0
$621$	17.9902	0.721920
$622$	0	0
$623$	−2.23743	−0.0896406
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.96239	0.118306
$628$	0	0
$629$	−5.08840	−0.202888
$630$	0	0
$631$	1.22188	0.0486424	0.0243212	−	0.999704i	$-0.492258\pi$
$631$	1.22188	0.0486424	0.0243212	+	0.999704i	$0.492258\pi$
$632$	0	0
$633$	2.35870	0.0937498
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.35026	0.251607
$638$	0	0
$639$	23.7043	0.937728
$640$	0	0
$641$	−22.1016	−0.872960	−0.436480	−	0.899714i	$-0.643775\pi$
$641$	−22.1016	−0.872960	−0.436480	+	0.899714i	$0.643775\pi$
$642$	0	0
$643$	−11.6688	−0.460172	−0.230086	−	0.973170i	$-0.573901\pi$
$643$	−11.6688	−0.460172	−0.230086	+	0.973170i	$0.573901\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.9575	−0.470096	−0.235048	−	0.971984i	$-0.575525\pi$
$647$	−11.9575	−0.470096	−0.235048	+	0.971984i	$0.575525\pi$
$648$	0	0
$649$	−22.0059	−0.863806
$650$	0	0
$651$	−2.05079	−0.0803766
$652$	0	0
$653$	−10.9986	−0.430408	−0.215204	−	0.976569i	$-0.569042\pi$
$653$	−10.9986	−0.430408	−0.215204	+	0.976569i	$0.569042\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	32.5804	1.27108
$658$	0	0
$659$	2.63989	0.102835	0.0514177	−	0.998677i	$-0.483626\pi$
$659$	2.63989	0.102835	0.0514177	+	0.998677i	$0.483626\pi$
$660$	0	0
$661$	18.3028	0.711896	0.355948	−	0.934506i	$-0.384158\pi$
$661$	18.3028	0.711896	0.355948	+	0.934506i	$0.384158\pi$
$662$	0	0
$663$	0.649738	0.0252337
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15.6728	−0.606852
$668$	0	0
$669$	−11.9854	−0.463383
$670$	0	0
$671$	−6.49929	−0.250902
$672$	0	0
$673$	6.71037	0.258666	0.129333	−	0.991601i	$-0.458716\pi$
$673$	6.71037	0.258666	0.129333	+	0.991601i	$0.458716\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.57593	−0.0605679	−0.0302840	−	0.999541i	$-0.509641\pi$
$677$	−1.57593	−0.0605679	−0.0302840	+	0.999541i	$0.509641\pi$
$678$	0	0
$679$	1.51056	0.0579698
$680$	0	0
$681$	−4.79033	−0.183566
$682$	0	0
$683$	−15.1939	−0.581380	−0.290690	−	0.956817i	$-0.593885\pi$
$683$	−15.1939	−0.581380	−0.290690	+	0.956817i	$0.593885\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.57452	0.0982239
$688$	0	0
$689$	−5.73813	−0.218606
$690$	0	0
$691$	18.7127	0.711866	0.355933	−	0.934511i	$-0.384163\pi$
$691$	18.7127	0.711866	0.355933	+	0.934511i	$0.384163\pi$
$692$	0	0
$693$	8.20123	0.311539
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.2243	0.425149
$698$	0	0
$699$	−5.17821	−0.195858
$700$	0	0
$701$	24.3028	0.917904	0.458952	−	0.888461i	$-0.348225\pi$
$701$	24.3028	0.917904	0.458952	+	0.888461i	$0.348225\pi$
$702$	0	0
$703$	6.31265	0.238086
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.46310	−0.318287
$708$	0	0
$709$	−9.66291	−0.362898	−0.181449	−	0.983400i	$-0.558079\pi$
$709$	−9.66291	−0.362898	−0.181449	+	0.983400i	$0.558079\pi$
$710$	0	0
$711$	−6.26187	−0.234838
$712$	0	0
$713$	−34.2677	−1.28334
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.70782	0.213162
$718$	0	0
$719$	−28.4142	−1.05967	−0.529836	−	0.848100i	$-0.677746\pi$
$719$	−28.4142	−1.05967	−0.529836	+	0.848100i	$0.677746\pi$
$720$	0	0
$721$	12.3879	0.461349
$722$	0	0
$723$	−13.7743	−0.512273
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.8545	1.29268	0.646341	−	0.763049i	$-0.276299\pi$
$727$	34.8545	1.29268	0.646341	+	0.763049i	$0.276299\pi$
$728$	0	0
$729$	−15.2882	−0.566230
$730$	0	0
$731$	−9.17347	−0.339293
$732$	0	0
$733$	6.25202	0.230923	0.115462	−	0.993312i	$-0.463165\pi$
$733$	6.25202	0.230923	0.115462	+	0.993312i	$0.463165\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.3634	−1.33946
$738$	0	0
$739$	32.0846	1.18025	0.590126	−	0.807311i	$-0.299078\pi$
$739$	32.0846	1.18025	0.590126	+	0.807311i	$0.299078\pi$
$740$	0	0
$741$	−0.806063	−0.0296115
$742$	0	0
$743$	30.5442	1.12056	0.560279	−	0.828304i	$-0.310694\pi$
$743$	30.5442	1.12056	0.560279	+	0.828304i	$0.310694\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10.6410	0.389335
$748$	0	0
$749$	8.97224	0.327838
$750$	0	0
$751$	−28.1622	−1.02765	−0.513827	−	0.857894i	$-0.671773\pi$
$751$	−28.1622	−1.02765	−0.513827	+	0.857894i	$0.671773\pi$
$752$	0	0
$753$	9.32865	0.339955
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.4109	1.28703	0.643515	−	0.765433i	$-0.277475\pi$
$757$	35.4109	1.28703	0.643515	+	0.765433i	$0.277475\pi$
$758$	0	0
$759$	−11.4617	−0.416033
$760$	0	0
$761$	−19.2388	−0.697407	−0.348704	−	0.937233i	$-0.613378\pi$
$761$	−19.2388	−0.697407	−0.348704	+	0.937233i	$0.613378\pi$
$762$	0	0
$763$	−7.72829	−0.279783
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.98778	0.216206
$768$	0	0
$769$	48.9643	1.76570	0.882849	−	0.469657i	$-0.155622\pi$
$769$	48.9643	1.76570	0.882849	+	0.469657i	$0.155622\pi$
$770$	0	0
$771$	11.0014	0.396206
$772$	0	0
$773$	46.1681	1.66055	0.830275	−	0.557354i	$-0.188183\pi$
$773$	46.1681	1.66055	0.830275	+	0.557354i	$0.188183\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.46168	−0.0524375
$778$	0	0
$779$	−13.9248	−0.498907
$780$	0	0
$781$	−31.4676	−1.12600
$782$	0	0
$783$	−6.71228	−0.239877
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.6458	0.807234	0.403617	−	0.914928i	$-0.367753\pi$
$787$	22.6458	0.807234	0.403617	+	0.914928i	$0.367753\pi$
$788$	0	0
$789$	−10.5296	−0.374864
$790$	0	0
$791$	−0.463096	−0.0164658
$792$	0	0
$793$	1.76845	0.0627996
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.23743	0.291785	0.145892	−	0.989300i	$-0.453395\pi$
$797$	8.23743	0.291785	0.145892	+	0.989300i	$0.453395\pi$
$798$	0	0
$799$	4.31265	0.152571
$800$	0	0
$801$	−7.68452	−0.271519
$802$	0	0
$803$	−43.2506	−1.52628
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.9478	0.385381
$808$	0	0
$809$	44.1319	1.55159	0.775797	−	0.630982i	$-0.217348\pi$
$809$	44.1319	1.55159	0.775797	+	0.630982i	$0.217348\pi$
$810$	0	0
$811$	−22.6883	−0.796694	−0.398347	−	0.917235i	$-0.630416\pi$
$811$	−22.6883	−0.796694	−0.398347	+	0.917235i	$0.630416\pi$
$812$	0	0
$813$	−0.0594941	−0.00208655
$814$	0	0
$815$	0	0
$816$	0	0
$817$	11.3806	0.398156
$818$	0	0
$819$	−2.23155	−0.0779766
$820$	0	0
$821$	−50.2736	−1.75456	−0.877281	−	0.479978i	$-0.840645\pi$
$821$	−50.2736	−1.75456	−0.877281	+	0.479978i	$0.840645\pi$
$822$	0	0
$823$	−5.13093	−0.178853	−0.0894265	−	0.995993i	$-0.528503\pi$
$823$	−5.13093	−0.178853	−0.0894265	+	0.995993i	$0.528503\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.6946	−0.650076	−0.325038	−	0.945701i	$-0.605377\pi$
$827$	−18.6946	−0.650076	−0.325038	+	0.945701i	$0.605377\pi$
$828$	0	0
$829$	−3.44121	−0.119518	−0.0597591	−	0.998213i	$-0.519033\pi$
$829$	−3.44121	−0.119518	−0.0597591	+	0.998213i	$0.519033\pi$
$830$	0	0
$831$	−7.38646	−0.256233
$832$	0	0
$833$	8.57452	0.297089
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−14.6761	−0.507280
$838$	0	0
$839$	52.6248	1.81681	0.908406	−	0.418090i	$-0.137300\pi$
$839$	52.6248	1.81681	0.908406	+	0.418090i	$0.137300\pi$
$840$	0	0
$841$	−23.1524	−0.798357
$842$	0	0
$843$	6.70052	0.230778
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.02047	−0.0694241
$848$	0	0
$849$	−9.80748	−0.336592
$850$	0	0
$851$	−24.4241	−0.837246
$852$	0	0
$853$	−6.31853	−0.216342	−0.108171	−	0.994132i	$-0.534499\pi$
$853$	−6.31853	−0.216342	−0.108171	+	0.994132i	$0.534499\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.775746	−0.0264990	−0.0132495	−	0.999912i	$-0.504218\pi$
$857$	−0.775746	−0.0264990	−0.0132495	+	0.999912i	$0.504218\pi$
$858$	0	0
$859$	−3.24869	−0.110844	−0.0554220	−	0.998463i	$-0.517650\pi$
$859$	−3.24869	−0.110844	−0.0554220	+	0.998463i	$0.517650\pi$
$860$	0	0
$861$	3.22425	0.109882
$862$	0	0
$863$	19.9208	0.678112	0.339056	−	0.940766i	$-0.389892\pi$
$863$	19.9208	0.678112	0.339056	+	0.940766i	$0.389892\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.30299	−0.248022
$868$	0	0
$869$	8.31265	0.281987
$870$	0	0
$871$	9.89446	0.335261
$872$	0	0
$873$	5.18806	0.175589
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.1378	0.747539	0.373770	−	0.927522i	$-0.378065\pi$
$877$	22.1378	0.747539	0.373770	+	0.927522i	$0.378065\pi$
$878$	0	0
$879$	2.58910	0.0873283
$880$	0	0
$881$	2.23155	0.0751828	0.0375914	−	0.999293i	$-0.488031\pi$
$881$	2.23155	0.0751828	0.0375914	+	0.999293i	$0.488031\pi$
$882$	0	0
$883$	4.30440	0.144855	0.0724273	−	0.997374i	$-0.476925\pi$
$883$	4.30440	0.144855	0.0724273	+	0.997374i	$0.476925\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.9330	−0.534979	−0.267489	−	0.963561i	$-0.586194\pi$
$887$	−15.9330	−0.534979	−0.267489	+	0.963561i	$0.586194\pi$
$888$	0	0
$889$	−3.46168	−0.116101
$890$	0	0
$891$	25.6145	0.858118
$892$	0	0
$893$	−5.35026	−0.179040
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.11871	0.104131
$898$	0	0
$899$	12.7856	0.426423
$900$	0	0
$901$	−7.74798	−0.258123
$902$	0	0
$903$	−2.63515	−0.0876923
$904$	0	0
$905$	0	0
$906$	0	0
$907$	51.9086	1.72360	0.861798	−	0.507251i	$-0.169338\pi$
$907$	51.9086	1.72360	0.861798	+	0.507251i	$0.169338\pi$
$908$	0	0
$909$	−29.0668	−0.964085
$910$	0	0
$911$	9.67750	0.320630	0.160315	−	0.987066i	$-0.448749\pi$
$911$	9.67750	0.320630	0.160315	+	0.987066i	$0.448749\pi$
$912$	0	0
$913$	−14.1260	−0.467503
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.674176	−0.0222632
$918$	0	0
$919$	−13.5515	−0.447022	−0.223511	−	0.974701i	$-0.571752\pi$
$919$	−13.5515	−0.447022	−0.223511	+	0.974701i	$0.571752\pi$
$920$	0	0
$921$	9.22425	0.303949
$922$	0	0
$923$	8.56230	0.281831
$924$	0	0
$925$	0	0
$926$	0	0
$927$	42.5466	1.39741
$928$	0	0
$929$	−9.44992	−0.310042	−0.155021	−	0.987911i	$-0.549545\pi$
$929$	−9.44992	−0.310042	−0.155021	+	0.987911i	$0.549545\pi$
$930$	0	0
$931$	−10.6375	−0.348631
$932$	0	0
$933$	12.1504	0.397788
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.0409	0.524035	0.262017	−	0.965063i	$-0.415612\pi$
$937$	16.0409	0.524035	0.262017	+	0.965063i	$0.415612\pi$
$938$	0	0
$939$	1.35309	0.0441565
$940$	0	0
$941$	−21.6747	−0.706574	−0.353287	−	0.935515i	$-0.614936\pi$
$941$	−21.6747	−0.706574	−0.353287	+	0.935515i	$0.614936\pi$
$942$	0	0
$943$	53.8759	1.75444
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.63118	0.150493	0.0752466	−	0.997165i	$-0.476026\pi$
$947$	4.63118	0.150493	0.0752466	+	0.997165i	$0.476026\pi$
$948$	0	0
$949$	11.7685	0.382020
$950$	0	0
$951$	−11.4372	−0.370878
$952$	0	0
$953$	26.2981	0.851878	0.425939	−	0.904752i	$-0.359944\pi$
$953$	26.2981	0.851878	0.425939	+	0.904752i	$0.359944\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.27645	0.138238
$958$	0	0
$959$	12.0409	0.388822
$960$	0	0
$961$	−3.04491	−0.0982228
$962$	0	0
$963$	30.8155	0.993014
$964$	0	0
$965$	0	0
$966$	0	0
$967$	11.9405	0.383981	0.191990	−	0.981397i	$-0.438506\pi$
$967$	11.9405	0.383981	0.191990	+	0.981397i	$0.438506\pi$
$968$	0	0
$969$	−1.08840	−0.0349643
$970$	0	0
$971$	−30.1524	−0.967635	−0.483818	−	0.875169i	$-0.660750\pi$
$971$	−30.1524	−0.967635	−0.483818	+	0.875169i	$0.660750\pi$
$972$	0	0
$973$	−6.80209	−0.218065
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.9321	0.861633	0.430817	−	0.902439i	$-0.358226\pi$
$977$	26.9321	0.861633	0.430817	+	0.902439i	$0.358226\pi$
$978$	0	0
$979$	10.2012	0.326033
$980$	0	0
$981$	−26.5431	−0.847455
$982$	0	0
$983$	20.5902	0.656727	0.328363	−	0.944551i	$-0.393503\pi$
$983$	20.5902	0.656727	0.328363	+	0.944551i	$0.393503\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.23884	0.0394328
$988$	0	0
$989$	−44.0322	−1.40014
$990$	0	0
$991$	48.1378	1.52915	0.764573	−	0.644537i	$-0.222950\pi$
$991$	48.1378	1.52915	0.764573	+	0.644537i	$0.222950\pi$
$992$	0	0
$993$	5.67864	0.180206
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.4255	1.05860	0.529298	−	0.848436i	$-0.322455\pi$
$997$	33.4255	1.05860	0.529298	+	0.848436i	$0.322455\pi$
$998$	0	0
$999$	−10.4603	−0.330948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.cb.1.3		3
4.3	odd	2		325.2.a.k.1.3		3
5.2	odd	4		1040.2.d.c.209.3		6
5.3	odd	4		1040.2.d.c.209.4		6
5.4	even	2		5200.2.a.cj.1.1		3
12.11	even	2		2925.2.a.bf.1.1		3
20.3	even	4		65.2.b.a.14.1	✓	6
20.7	even	4		65.2.b.a.14.6	yes	6
20.19	odd	2		325.2.a.j.1.1		3
52.51	odd	2		4225.2.a.ba.1.1		3
60.23	odd	4		585.2.c.b.469.6		6
60.47	odd	4		585.2.c.b.469.1		6
60.59	even	2		2925.2.a.bj.1.3		3
260.3	even	12		845.2.n.f.529.6		12
260.7	odd	12		845.2.l.e.699.6		12
260.23	even	12		845.2.n.g.529.1		12
260.43	even	12		845.2.n.g.484.6		12
260.47	odd	4		845.2.d.a.844.2		6
260.63	odd	12		845.2.l.d.654.2		12
260.67	odd	12		845.2.l.d.654.1		12
260.83	odd	4		845.2.d.a.844.1		6
260.87	even	12		845.2.n.f.484.6		12
260.103	even	4		845.2.b.c.339.6		6
260.107	even	12		845.2.n.f.529.1		12
260.123	odd	12		845.2.l.e.699.5		12
260.127	even	12		845.2.n.g.529.6		12
260.147	even	12		845.2.n.g.484.1		12
260.163	odd	12		845.2.l.d.699.1		12
260.167	odd	12		845.2.l.e.654.5		12
260.187	odd	4		845.2.d.b.844.6		6
260.203	odd	4		845.2.d.b.844.5		6
260.207	even	4		845.2.b.c.339.1		6
260.223	odd	12		845.2.l.e.654.6		12
260.227	odd	12		845.2.l.d.699.2		12
260.243	even	12		845.2.n.f.484.1		12
260.259	odd	2		4225.2.a.bh.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.1	✓	6	20.3	even	4
65.2.b.a.14.6	yes	6	20.7	even	4
325.2.a.j.1.1		3	20.19	odd	2
325.2.a.k.1.3		3	4.3	odd	2
585.2.c.b.469.1		6	60.47	odd	4
585.2.c.b.469.6		6	60.23	odd	4
845.2.b.c.339.1		6	260.207	even	4
845.2.b.c.339.6		6	260.103	even	4
845.2.d.a.844.1		6	260.83	odd	4
845.2.d.a.844.2		6	260.47	odd	4
845.2.d.b.844.5		6	260.203	odd	4
845.2.d.b.844.6		6	260.187	odd	4
845.2.l.d.654.1		12	260.67	odd	12
845.2.l.d.654.2		12	260.63	odd	12
845.2.l.d.699.1		12	260.163	odd	12
845.2.l.d.699.2		12	260.227	odd	12
845.2.l.e.654.5		12	260.167	odd	12
845.2.l.e.654.6		12	260.223	odd	12
845.2.l.e.699.5		12	260.123	odd	12
845.2.l.e.699.6		12	260.7	odd	12
845.2.n.f.484.1		12	260.243	even	12
845.2.n.f.484.6		12	260.87	even	12
845.2.n.f.529.1		12	260.107	even	12
845.2.n.f.529.6		12	260.3	even	12
845.2.n.g.484.1		12	260.147	even	12
845.2.n.g.484.6		12	260.43	even	12
845.2.n.g.529.1		12	260.23	even	12
845.2.n.g.529.6		12	260.127	even	12
1040.2.d.c.209.3		6	5.2	odd	4
1040.2.d.c.209.4		6	5.3	odd	4
2925.2.a.bf.1.1		3	12.11	even	2
2925.2.a.bj.1.3		3	60.59	even	2
4225.2.a.ba.1.1		3	52.51	odd	2
4225.2.a.bh.1.3		3	260.259	odd	2
5200.2.a.cb.1.3		3	1.1	even	1	trivial
5200.2.a.cj.1.1		3	5.4	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q+0.481194 q^{3} -0.806063 q^{7} -2.76845 q^{9} +3.67513 q^{11} -1.00000 q^{13} -1.35026 q^{17} +1.67513 q^{19} -0.387873 q^{21} -6.48119 q^{23} -2.77575 q^{27} +2.41819 q^{29} +5.28726 q^{31} +1.76845 q^{33} +3.76845 q^{37} -0.481194 q^{39} -8.31265 q^{41} +6.79384 q^{43} -3.19394 q^{47} -6.35026 q^{49} -0.649738 q^{51} +5.73813 q^{53} +0.806063 q^{57} -5.98778 q^{59} -1.76845 q^{61} +2.23155 q^{63} -9.89446 q^{67} -3.11871 q^{69} -8.56230 q^{71} -11.7685 q^{73} -2.96239 q^{77} +2.26187 q^{79} +6.96968 q^{81} -3.84367 q^{83} +1.16362 q^{87} +2.77575 q^{89} +0.806063 q^{91} +2.54420 q^{93} -1.87399 q^{97} -10.1744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * q^99

Introduction

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