LMFDB - Embedded newform 9200.2.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,0,0,-3,0,-3,0,9] 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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} -4.85410 q^{7} -2.61803 q^{9} +3.38197 q^{11} +0.381966 q^{13} -5.85410 q^{17} +6.85410 q^{19} -3.00000 q^{21} -1.00000 q^{23} -3.47214 q^{27} +3.70820 q^{29} +8.85410 q^{31} +2.09017 q^{33} +3.70820 q^{37} +0.236068 q^{39} -3.38197 q^{41} -6.76393 q^{43} +11.7082 q^{47} +16.5623 q^{49} -3.61803 q^{51} -2.00000 q^{53} +4.23607 q^{57} +6.00000 q^{59} -3.85410 q^{61} +12.7082 q^{63} -0.763932 q^{67} -0.618034 q^{69} -2.61803 q^{71} -7.52786 q^{73} -16.4164 q^{77} -5.70820 q^{79} +5.70820 q^{81} +5.70820 q^{83} +2.29180 q^{87} -9.70820 q^{89} -1.85410 q^{91} +5.47214 q^{93} -16.0344 q^{97} -8.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 3 q^{7} - 3 q^{9} + 9 q^{11} + 3 q^{13} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 7 q^{33} - 6 q^{37} - 4 q^{39} - 9 q^{41} - 18 q^{43} + 10 q^{47} + 13 q^{49}+ \cdots - 11 q^{99}+O(q^{100}) $ 2 * q - q^3 - 3 * q^7 - 3 * q^9 + 9 * q^11 + 3 * q^13 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 2 * q^23 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 7 * q^33 - 6 * q^37 - 4 * q^39 - 9 * q^41 - 18 * q^43 + 10 * q^47 + 13 * q^49 - 5 * q^51 - 4 * q^53 + 4 * q^57 + 12 * q^59 - q^61 + 12 * q^63 - 6 * q^67 + q^69 - 3 * q^71 - 24 * q^73 - 6 * q^77 + 2 * q^79 - 2 * q^81 - 2 * q^83 + 18 * q^87 - 6 * q^89 + 3 * q^91 + 2 * q^93 - 3 * q^97 - 11 * 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.85410	−1.83468	−0.917339	−	0.398107i	$-0.869667\pi$
$7$	−4.85410	−1.83468	−0.917339	+	0.398107i	$0.869667\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	3.38197	1.01970	0.509851	−	0.860263i	$-0.329701\pi$
$11$	3.38197	1.01970	0.509851	+	0.860263i	$0.329701\pi$
$12$	0	0
$13$	0.381966	0.105938	0.0529692	−	0.998596i	$-0.483131\pi$
$13$	0.381966	0.105938	0.0529692	+	0.998596i	$0.483131\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.85410	−1.41983	−0.709914	−	0.704288i	$-0.751266\pi$
$17$	−5.85410	−1.41983	−0.709914	+	0.704288i	$0.751266\pi$
$18$	0	0
$19$	6.85410	1.57244	0.786219	−	0.617947i	$-0.212036\pi$
$19$	6.85410	1.57244	0.786219	+	0.617947i	$0.212036\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	3.70820	0.688596	0.344298	−	0.938860i	$-0.388117\pi$
$29$	3.70820	0.688596	0.344298	+	0.938860i	$0.388117\pi$
$30$	0	0
$31$	8.85410	1.59024	0.795122	−	0.606450i	$-0.207407\pi$
$31$	8.85410	1.59024	0.795122	+	0.606450i	$0.207407\pi$
$32$	0	0
$33$	2.09017	0.363852
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.70820	0.609625	0.304812	−	0.952412i	$-0.401406\pi$
$37$	3.70820	0.609625	0.304812	+	0.952412i	$0.401406\pi$
$38$	0	0
$39$	0.236068	0.0378011
$40$	0	0
$41$	−3.38197	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$41$	−3.38197	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$42$	0	0
$43$	−6.76393	−1.03149	−0.515745	−	0.856742i	$-0.672485\pi$
$43$	−6.76393	−1.03149	−0.515745	+	0.856742i	$0.672485\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7082	1.70782	0.853909	−	0.520423i	$-0.174226\pi$
$47$	11.7082	1.70782	0.853909	+	0.520423i	$0.174226\pi$
$48$	0	0
$49$	16.5623	2.36604
$50$	0	0
$51$	−3.61803	−0.506626
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.23607	0.561081
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−3.85410	−0.493467	−0.246734	−	0.969083i	$-0.579357\pi$
$61$	−3.85410	−0.493467	−0.246734	+	0.969083i	$0.579357\pi$
$62$	0	0
$63$	12.7082	1.60108
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.763932	−0.0933292	−0.0466646	−	0.998911i	$-0.514859\pi$
$67$	−0.763932	−0.0933292	−0.0466646	+	0.998911i	$0.514859\pi$
$68$	0	0
$69$	−0.618034	−0.0744025
$70$	0	0
$71$	−2.61803	−0.310703	−0.155352	−	0.987859i	$-0.549651\pi$
$71$	−2.61803	−0.310703	−0.155352	+	0.987859i	$0.549651\pi$
$72$	0	0
$73$	−7.52786	−0.881070	−0.440535	−	0.897735i	$-0.645211\pi$
$73$	−7.52786	−0.881070	−0.440535	+	0.897735i	$0.645211\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16.4164	−1.87082
$78$	0	0
$79$	−5.70820	−0.642223	−0.321112	−	0.947041i	$-0.604056\pi$
$79$	−5.70820	−0.642223	−0.321112	+	0.947041i	$0.604056\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	5.70820	0.626557	0.313278	−	0.949661i	$-0.398573\pi$
$83$	5.70820	0.626557	0.313278	+	0.949661i	$0.398573\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.29180	0.245706
$88$	0	0
$89$	−9.70820	−1.02907	−0.514534	−	0.857470i	$-0.672035\pi$
$89$	−9.70820	−1.02907	−0.514534	+	0.857470i	$0.672035\pi$
$90$	0	0
$91$	−1.85410	−0.194363
$92$	0	0
$93$	5.47214	0.567434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.0344	−1.62805	−0.814025	−	0.580829i	$-0.802728\pi$
$97$	−16.0344	−1.62805	−0.814025	+	0.580829i	$0.802728\pi$
$98$	0	0
$99$	−8.85410	−0.889871
$100$	0	0
$101$	−10.4721	−1.04202	−0.521008	−	0.853552i	$-0.674444\pi$
$101$	−10.4721	−1.04202	−0.521008	+	0.853552i	$0.674444\pi$
$102$	0	0
$103$	−4.14590	−0.408507	−0.204254	−	0.978918i	$-0.565477\pi$
$103$	−4.14590	−0.408507	−0.204254	+	0.978918i	$0.565477\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.70820	0.165138	0.0825692	−	0.996585i	$-0.473687\pi$
$107$	1.70820	0.165138	0.0825692	+	0.996585i	$0.473687\pi$
$108$	0	0
$109$	−12.5623	−1.20325	−0.601625	−	0.798778i	$-0.705480\pi$
$109$	−12.5623	−1.20325	−0.601625	+	0.798778i	$0.705480\pi$
$110$	0	0
$111$	2.29180	0.217528
$112$	0	0
$113$	13.4164	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	28.4164	2.60493
$120$	0	0
$121$	0.437694	0.0397904
$122$	0	0
$123$	−2.09017	−0.188464
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.70820	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$127$	−3.70820	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$128$	0	0
$129$	−4.18034	−0.368058
$130$	0	0
$131$	−8.18034	−0.714720	−0.357360	−	0.933967i	$-0.616323\pi$
$131$	−8.18034	−0.714720	−0.357360	+	0.933967i	$0.616323\pi$
$132$	0	0
$133$	−33.2705	−2.88492
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.14590	−0.610515	−0.305258	−	0.952270i	$-0.598743\pi$
$137$	−7.14590	−0.610515	−0.305258	+	0.952270i	$0.598743\pi$
$138$	0	0
$139$	−17.7082	−1.50199	−0.750995	−	0.660308i	$-0.770426\pi$
$139$	−17.7082	−1.50199	−0.750995	+	0.660308i	$0.770426\pi$
$140$	0	0
$141$	7.23607	0.609387
$142$	0	0
$143$	1.29180	0.108025
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.2361	0.844257
$148$	0	0
$149$	−0.381966	−0.0312919	−0.0156459	−	0.999878i	$-0.504980\pi$
$149$	−0.381966	−0.0312919	−0.0156459	+	0.999878i	$0.504980\pi$
$150$	0	0
$151$	19.2705	1.56821	0.784106	−	0.620627i	$-0.213122\pi$
$151$	19.2705	1.56821	0.784106	+	0.620627i	$0.213122\pi$
$152$	0	0
$153$	15.3262	1.23905
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.7082	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$157$	15.7082	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$158$	0	0
$159$	−1.23607	−0.0980266
$160$	0	0
$161$	4.85410	0.382557
$162$	0	0
$163$	−7.03444	−0.550980	−0.275490	−	0.961304i	$-0.588840\pi$
$163$	−7.03444	−0.550980	−0.275490	+	0.961304i	$0.588840\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.70820	−0.286949	−0.143475	−	0.989654i	$-0.545828\pi$
$167$	−3.70820	−0.286949	−0.143475	+	0.989654i	$0.545828\pi$
$168$	0	0
$169$	−12.8541	−0.988777
$170$	0	0
$171$	−17.9443	−1.37223
$172$	0	0
$173$	−3.56231	−0.270837	−0.135419	−	0.990788i	$-0.543238\pi$
$173$	−3.56231	−0.270837	−0.135419	+	0.990788i	$0.543238\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.70820	0.278726
$178$	0	0
$179$	−16.4721	−1.23119	−0.615593	−	0.788065i	$-0.711083\pi$
$179$	−16.4721	−1.23119	−0.615593	+	0.788065i	$0.711083\pi$
$180$	0	0
$181$	−4.56231	−0.339114	−0.169557	−	0.985520i	$-0.554234\pi$
$181$	−4.56231	−0.339114	−0.169557	+	0.985520i	$0.554234\pi$
$182$	0	0
$183$	−2.38197	−0.176080
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−19.7984	−1.44780
$188$	0	0
$189$	16.8541	1.22596
$190$	0	0
$191$	1.41641	0.102488	0.0512438	−	0.998686i	$-0.483681\pi$
$191$	1.41641	0.102488	0.0512438	+	0.998686i	$0.483681\pi$
$192$	0	0
$193$	−2.29180	−0.164967	−0.0824835	−	0.996592i	$-0.526285\pi$
$193$	−2.29180	−0.164967	−0.0824835	+	0.996592i	$0.526285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.437694	−0.0311844	−0.0155922	−	0.999878i	$-0.504963\pi$
$197$	−0.437694	−0.0311844	−0.0155922	+	0.999878i	$0.504963\pi$
$198$	0	0
$199$	15.4164	1.09284	0.546420	−	0.837511i	$-0.315990\pi$
$199$	15.4164	1.09284	0.546420	+	0.837511i	$0.315990\pi$
$200$	0	0
$201$	−0.472136	−0.0333019
$202$	0	0
$203$	−18.0000	−1.26335
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.61803	0.181966
$208$	0	0
$209$	23.1803	1.60342
$210$	0	0
$211$	−5.70820	−0.392969	−0.196484	−	0.980507i	$-0.562953\pi$
$211$	−5.70820	−0.392969	−0.196484	+	0.980507i	$0.562953\pi$
$212$	0	0
$213$	−1.61803	−0.110866
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−42.9787	−2.91759
$218$	0	0
$219$	−4.65248	−0.314385
$220$	0	0
$221$	−2.23607	−0.150414
$222$	0	0
$223$	22.3607	1.49738	0.748691	−	0.662919i	$-0.230683\pi$
$223$	22.3607	1.49738	0.748691	+	0.662919i	$0.230683\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.7082	−1.57357	−0.786784	−	0.617228i	$-0.788256\pi$
$227$	−23.7082	−1.57357	−0.786784	+	0.617228i	$0.788256\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−10.1459	−0.667551
$232$	0	0
$233$	−19.1246	−1.25289	−0.626447	−	0.779464i	$-0.715492\pi$
$233$	−19.1246	−1.25289	−0.626447	+	0.779464i	$0.715492\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.52786	−0.229159
$238$	0	0
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.61803	0.166582
$248$	0	0
$249$	3.52786	0.223569
$250$	0	0
$251$	−7.79837	−0.492229	−0.246114	−	0.969241i	$-0.579154\pi$
$251$	−7.79837	−0.492229	−0.246114	+	0.969241i	$0.579154\pi$
$252$	0	0
$253$	−3.38197	−0.212622
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.4164	0.961649	0.480825	−	0.876817i	$-0.340337\pi$
$257$	15.4164	0.961649	0.480825	+	0.876817i	$0.340337\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−9.70820	−0.600923
$262$	0	0
$263$	−15.2705	−0.941620	−0.470810	−	0.882235i	$-0.656038\pi$
$263$	−15.2705	−0.941620	−0.470810	+	0.882235i	$0.656038\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−10.4721	−0.638497	−0.319249	−	0.947671i	$-0.603431\pi$
$269$	−10.4721	−0.638497	−0.319249	+	0.947671i	$0.603431\pi$
$270$	0	0
$271$	−19.1459	−1.16303	−0.581515	−	0.813536i	$-0.697540\pi$
$271$	−19.1459	−1.16303	−0.581515	+	0.813536i	$0.697540\pi$
$272$	0	0
$273$	−1.14590	−0.0693529
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.4721	0.989715	0.494857	−	0.868974i	$-0.335220\pi$
$277$	16.4721	0.989715	0.494857	+	0.868974i	$0.335220\pi$
$278$	0	0
$279$	−23.1803	−1.38777
$280$	0	0
$281$	−0.652476	−0.0389234	−0.0194617	−	0.999811i	$-0.506195\pi$
$281$	−0.652476	−0.0389234	−0.0194617	+	0.999811i	$0.506195\pi$
$282$	0	0
$283$	−3.05573	−0.181644	−0.0908221	−	0.995867i	$-0.528949\pi$
$283$	−3.05573	−0.181644	−0.0908221	+	0.995867i	$0.528949\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.4164	0.969030
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	−9.90983	−0.580925
$292$	0	0
$293$	−27.7082	−1.61873	−0.809365	−	0.587306i	$-0.800189\pi$
$293$	−27.7082	−1.61873	−0.809365	+	0.587306i	$0.800189\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−11.7426	−0.681377
$298$	0	0
$299$	−0.381966	−0.0220897
$300$	0	0
$301$	32.8328	1.89245
$302$	0	0
$303$	−6.47214	−0.371814
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.1459	−0.579057	−0.289528	−	0.957169i	$-0.593498\pi$
$307$	−10.1459	−0.579057	−0.289528	+	0.957169i	$0.593498\pi$
$308$	0	0
$309$	−2.56231	−0.145764
$310$	0	0
$311$	13.5279	0.767095	0.383547	−	0.923521i	$-0.374702\pi$
$311$	13.5279	0.767095	0.383547	+	0.923521i	$0.374702\pi$
$312$	0	0
$313$	19.1459	1.08219	0.541095	−	0.840961i	$-0.318010\pi$
$313$	19.1459	1.08219	0.541095	+	0.840961i	$0.318010\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.2705	1.47550	0.737749	−	0.675075i	$-0.235889\pi$
$317$	26.2705	1.47550	0.737749	+	0.675075i	$0.235889\pi$
$318$	0	0
$319$	12.5410	0.702162
$320$	0	0
$321$	1.05573	0.0589250
$322$	0	0
$323$	−40.1246	−2.23259
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.76393	−0.429346
$328$	0	0
$329$	−56.8328	−3.13329
$330$	0	0
$331$	−15.1246	−0.831324	−0.415662	−	0.909519i	$-0.636450\pi$
$331$	−15.1246	−0.831324	−0.415662	+	0.909519i	$0.636450\pi$
$332$	0	0
$333$	−9.70820	−0.532006
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−5.61803	−0.306034	−0.153017	−	0.988224i	$-0.548899\pi$
$337$	−5.61803	−0.306034	−0.153017	+	0.988224i	$0.548899\pi$
$338$	0	0
$339$	8.29180	0.450349
$340$	0	0
$341$	29.9443	1.62157
$342$	0	0
$343$	−46.4164	−2.50625
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.56231	0.298600	0.149300	−	0.988792i	$-0.452298\pi$
$347$	5.56231	0.298600	0.149300	+	0.988792i	$0.452298\pi$
$348$	0	0
$349$	14.2918	0.765022	0.382511	−	0.923951i	$-0.375059\pi$
$349$	14.2918	0.765022	0.382511	+	0.923951i	$0.375059\pi$
$350$	0	0
$351$	−1.32624	−0.0707893
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	17.5623	0.929496
$358$	0	0
$359$	−13.4164	−0.708091	−0.354045	−	0.935228i	$-0.615194\pi$
$359$	−13.4164	−0.708091	−0.354045	+	0.935228i	$0.615194\pi$
$360$	0	0
$361$	27.9787	1.47256
$362$	0	0
$363$	0.270510	0.0141981
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	8.85410	0.460926
$370$	0	0
$371$	9.70820	0.504025
$372$	0	0
$373$	−3.05573	−0.158220	−0.0791098	−	0.996866i	$-0.525208\pi$
$373$	−3.05573	−0.158220	−0.0791098	+	0.996866i	$0.525208\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.41641	0.0729487
$378$	0	0
$379$	−9.27051	−0.476194	−0.238097	−	0.971241i	$-0.576524\pi$
$379$	−9.27051	−0.476194	−0.238097	+	0.971241i	$0.576524\pi$
$380$	0	0
$381$	−2.29180	−0.117412
$382$	0	0
$383$	−27.4164	−1.40091	−0.700456	−	0.713695i	$-0.747020\pi$
$383$	−27.4164	−1.40091	−0.700456	+	0.713695i	$0.747020\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.7082	0.900159
$388$	0	0
$389$	−5.67376	−0.287671	−0.143836	−	0.989602i	$-0.545944\pi$
$389$	−5.67376	−0.287671	−0.143836	+	0.989602i	$0.545944\pi$
$390$	0	0
$391$	5.85410	0.296055
$392$	0	0
$393$	−5.05573	−0.255028
$394$	0	0
$395$	0	0
$396$	0	0
$397$	35.5623	1.78482	0.892410	−	0.451224i	$-0.149013\pi$
$397$	35.5623	1.78482	0.892410	+	0.451224i	$0.149013\pi$
$398$	0	0
$399$	−20.5623	−1.02940
$400$	0	0
$401$	−34.4721	−1.72146	−0.860728	−	0.509065i	$-0.829991\pi$
$401$	−34.4721	−1.72146	−0.860728	+	0.509065i	$0.829991\pi$
$402$	0	0
$403$	3.38197	0.168468
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.5410	0.621635
$408$	0	0
$409$	−6.56231	−0.324485	−0.162243	−	0.986751i	$-0.551873\pi$
$409$	−6.56231	−0.324485	−0.162243	+	0.986751i	$0.551873\pi$
$410$	0	0
$411$	−4.41641	−0.217845
$412$	0	0
$413$	−29.1246	−1.43313
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.9443	−0.535943
$418$	0	0
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	6.85410	0.334048	0.167024	−	0.985953i	$-0.446584\pi$
$421$	6.85410	0.334048	0.167024	+	0.985953i	$0.446584\pi$
$422$	0	0
$423$	−30.6525	−1.49037
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.7082	0.905353
$428$	0	0
$429$	0.798374	0.0385459
$430$	0	0
$431$	6.76393	0.325807	0.162904	−	0.986642i	$-0.447914\pi$
$431$	6.76393	0.325807	0.162904	+	0.986642i	$0.447914\pi$
$432$	0	0
$433$	−29.6180	−1.42335	−0.711676	−	0.702508i	$-0.752064\pi$
$433$	−29.6180	−1.42335	−0.711676	+	0.702508i	$0.752064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.85410	−0.327876
$438$	0	0
$439$	0.270510	0.0129107	0.00645536	−	0.999979i	$-0.497945\pi$
$439$	0.270510	0.0129107	0.00645536	+	0.999979i	$0.497945\pi$
$440$	0	0
$441$	−43.3607	−2.06479
$442$	0	0
$443$	−6.85410	−0.325648	−0.162824	−	0.986655i	$-0.552060\pi$
$443$	−6.85410	−0.325648	−0.162824	+	0.986655i	$0.552060\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.236068	−0.0111656
$448$	0	0
$449$	2.56231	0.120923	0.0604613	−	0.998171i	$-0.480743\pi$
$449$	2.56231	0.120923	0.0604613	+	0.998171i	$0.480743\pi$
$450$	0	0
$451$	−11.4377	−0.538580
$452$	0	0
$453$	11.9098	0.559573
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.4164	−1.18893	−0.594465	−	0.804122i	$-0.702636\pi$
$457$	−25.4164	−1.18893	−0.594465	+	0.804122i	$0.702636\pi$
$458$	0	0
$459$	20.3262	0.948748
$460$	0	0
$461$	−25.3050	−1.17857	−0.589285	−	0.807926i	$-0.700590\pi$
$461$	−25.3050	−1.17857	−0.589285	+	0.807926i	$0.700590\pi$
$462$	0	0
$463$	2.29180	0.106509	0.0532544	−	0.998581i	$-0.483041\pi$
$463$	2.29180	0.106509	0.0532544	+	0.998581i	$0.483041\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.1246	0.792433	0.396216	−	0.918157i	$-0.370323\pi$
$467$	17.1246	0.792433	0.396216	+	0.918157i	$0.370323\pi$
$468$	0	0
$469$	3.70820	0.171229
$470$	0	0
$471$	9.70820	0.447330
$472$	0	0
$473$	−22.8754	−1.05181
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.23607	0.239743
$478$	0	0
$479$	19.5279	0.892251	0.446125	−	0.894970i	$-0.352804\pi$
$479$	19.5279	0.892251	0.446125	+	0.894970i	$0.352804\pi$
$480$	0	0
$481$	1.41641	0.0645826
$482$	0	0
$483$	3.00000	0.136505
$484$	0	0
$485$	0	0
$486$	0	0
$487$	21.7082	0.983693	0.491846	−	0.870682i	$-0.336322\pi$
$487$	21.7082	0.983693	0.491846	+	0.870682i	$0.336322\pi$
$488$	0	0
$489$	−4.34752	−0.196602
$490$	0	0
$491$	26.8328	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$491$	26.8328	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$492$	0	0
$493$	−21.7082	−0.977688
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.7082	0.570041
$498$	0	0
$499$	−11.4164	−0.511069	−0.255534	−	0.966800i	$-0.582251\pi$
$499$	−11.4164	−0.511069	−0.255534	+	0.966800i	$0.582251\pi$
$500$	0	0
$501$	−2.29180	−0.102390
$502$	0	0
$503$	−16.8541	−0.751487	−0.375744	−	0.926724i	$-0.622613\pi$
$503$	−16.8541	−0.751487	−0.375744	+	0.926724i	$0.622613\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.94427	−0.352818
$508$	0	0
$509$	−24.6525	−1.09270	−0.546351	−	0.837556i	$-0.683983\pi$
$509$	−24.6525	−1.09270	−0.546351	+	0.837556i	$0.683983\pi$
$510$	0	0
$511$	36.5410	1.61648
$512$	0	0
$513$	−23.7984	−1.05072
$514$	0	0
$515$	0	0
$516$	0	0
$517$	39.5967	1.74146
$518$	0	0
$519$	−2.20163	−0.0966407
$520$	0	0
$521$	20.0689	0.879234	0.439617	−	0.898185i	$-0.355114\pi$
$521$	20.0689	0.879234	0.439617	+	0.898185i	$0.355114\pi$
$522$	0	0
$523$	40.3607	1.76485	0.882425	−	0.470454i	$-0.155910\pi$
$523$	40.3607	1.76485	0.882425	+	0.470454i	$0.155910\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−51.8328	−2.25787
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−15.7082	−0.681678
$532$	0	0
$533$	−1.29180	−0.0559539
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.1803	−0.439314
$538$	0	0
$539$	56.0132	2.41266
$540$	0	0
$541$	−27.4164	−1.17872	−0.589362	−	0.807869i	$-0.700621\pi$
$541$	−27.4164	−1.17872	−0.589362	+	0.807869i	$0.700621\pi$
$542$	0	0
$543$	−2.81966	−0.121003
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.14590	0.0489951	0.0244975	−	0.999700i	$-0.492201\pi$
$547$	1.14590	0.0489951	0.0244975	+	0.999700i	$0.492201\pi$
$548$	0	0
$549$	10.0902	0.430638
$550$	0	0
$551$	25.4164	1.08278
$552$	0	0
$553$	27.7082	1.17827
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−2.58359	−0.109274
$560$	0	0
$561$	−12.2361	−0.516607
$562$	0	0
$563$	2.29180	0.0965877	0.0482938	−	0.998833i	$-0.484622\pi$
$563$	2.29180	0.0965877	0.0482938	+	0.998833i	$0.484622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−27.7082	−1.16364
$568$	0	0
$569$	28.3607	1.18894	0.594471	−	0.804117i	$-0.297362\pi$
$569$	28.3607	1.18894	0.594471	+	0.804117i	$0.297362\pi$
$570$	0	0
$571$	25.2705	1.05754	0.528769	−	0.848766i	$-0.322654\pi$
$571$	25.2705	1.05754	0.528769	+	0.848766i	$0.322654\pi$
$572$	0	0
$573$	0.875388	0.0365699
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−26.2918	−1.09454	−0.547271	−	0.836956i	$-0.684333\pi$
$577$	−26.2918	−1.09454	−0.547271	+	0.836956i	$0.684333\pi$
$578$	0	0
$579$	−1.41641	−0.0588639
$580$	0	0
$581$	−27.7082	−1.14953
$582$	0	0
$583$	−6.76393	−0.280133
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.14590	0.0885707	0.0442853	−	0.999019i	$-0.485899\pi$
$587$	2.14590	0.0885707	0.0442853	+	0.999019i	$0.485899\pi$
$588$	0	0
$589$	60.6869	2.50056
$590$	0	0
$591$	−0.270510	−0.0111273
$592$	0	0
$593$	33.4164	1.37225	0.686124	−	0.727485i	$-0.259311\pi$
$593$	33.4164	1.37225	0.686124	+	0.727485i	$0.259311\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.52786	0.389950
$598$	0	0
$599$	−21.3820	−0.873643	−0.436822	−	0.899548i	$-0.643896\pi$
$599$	−21.3820	−0.873643	−0.436822	+	0.899548i	$0.643896\pi$
$600$	0	0
$601$	−3.14590	−0.128324	−0.0641619	−	0.997940i	$-0.520437\pi$
$601$	−3.14590	−0.128324	−0.0641619	+	0.997940i	$0.520437\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−38.9443	−1.58070	−0.790350	−	0.612656i	$-0.790101\pi$
$607$	−38.9443	−1.58070	−0.790350	+	0.612656i	$0.790101\pi$
$608$	0	0
$609$	−11.1246	−0.450792
$610$	0	0
$611$	4.47214	0.180923
$612$	0	0
$613$	−5.23607	−0.211483	−0.105741	−	0.994394i	$-0.533722\pi$
$613$	−5.23607	−0.211483	−0.105741	+	0.994394i	$0.533722\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.5623	0.707032	0.353516	−	0.935429i	$-0.384986\pi$
$617$	17.5623	0.707032	0.353516	+	0.935429i	$0.384986\pi$
$618$	0	0
$619$	21.1459	0.849925	0.424963	−	0.905211i	$-0.360287\pi$
$619$	21.1459	0.849925	0.424963	+	0.905211i	$0.360287\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	0	0
$623$	47.1246	1.88801
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.3262	0.572135
$628$	0	0
$629$	−21.7082	−0.865563
$630$	0	0
$631$	−3.41641	−0.136005	−0.0680025	−	0.997685i	$-0.521663\pi$
$631$	−3.41641	−0.136005	−0.0680025	+	0.997685i	$0.521663\pi$
$632$	0	0
$633$	−3.52786	−0.140220
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.32624	0.250655
$638$	0	0
$639$	6.85410	0.271144
$640$	0	0
$641$	30.6525	1.21070	0.605350	−	0.795959i	$-0.293033\pi$
$641$	30.6525	1.21070	0.605350	+	0.795959i	$0.293033\pi$
$642$	0	0
$643$	−35.0132	−1.38078	−0.690392	−	0.723435i	$-0.742562\pi$
$643$	−35.0132	−1.38078	−0.690392	+	0.723435i	$0.742562\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.291796	0.0114717	0.00573584	−	0.999984i	$-0.498174\pi$
$647$	0.291796	0.0114717	0.00573584	+	0.999984i	$0.498174\pi$
$648$	0	0
$649$	20.2918	0.796523
$650$	0	0
$651$	−26.5623	−1.04106
$652$	0	0
$653$	−11.9787	−0.468763	−0.234382	−	0.972145i	$-0.575307\pi$
$653$	−11.9787	−0.468763	−0.234382	+	0.972145i	$0.575307\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	19.7082	0.768890
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	34.3951	1.33782	0.668908	−	0.743346i	$-0.266762\pi$
$661$	34.3951	1.33782	0.668908	+	0.743346i	$0.266762\pi$
$662$	0	0
$663$	−1.38197	−0.0536711
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.70820	−0.143582
$668$	0	0
$669$	13.8197	0.534299
$670$	0	0
$671$	−13.0344	−0.503189
$672$	0	0
$673$	−14.2918	−0.550908	−0.275454	−	0.961314i	$-0.588828\pi$
$673$	−14.2918	−0.550908	−0.275454	+	0.961314i	$0.588828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	77.8328	2.98695
$680$	0	0
$681$	−14.6525	−0.561484
$682$	0	0
$683$	−35.9787	−1.37669	−0.688344	−	0.725385i	$-0.741662\pi$
$683$	−35.9787	−1.37669	−0.688344	+	0.725385i	$0.741662\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.18034	−0.235795
$688$	0	0
$689$	−0.763932	−0.0291035
$690$	0	0
$691$	−19.1246	−0.727535	−0.363767	−	0.931490i	$-0.618510\pi$
$691$	−19.1246	−0.727535	−0.363767	+	0.931490i	$0.618510\pi$
$692$	0	0
$693$	42.9787	1.63263
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.7984	0.749917
$698$	0	0
$699$	−11.8197	−0.447061
$700$	0	0
$701$	36.9230	1.39456	0.697281	−	0.716798i	$-0.254393\pi$
$701$	36.9230	1.39456	0.697281	+	0.716798i	$0.254393\pi$
$702$	0	0
$703$	25.4164	0.958598
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.8328	1.91176
$708$	0	0
$709$	−11.5623	−0.434232	−0.217116	−	0.976146i	$-0.569665\pi$
$709$	−11.5623	−0.434232	−0.217116	+	0.976146i	$0.569665\pi$
$710$	0	0
$711$	14.9443	0.560454
$712$	0	0
$713$	−8.85410	−0.331589
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.944272	0.0352645
$718$	0	0
$719$	32.4508	1.21021	0.605106	−	0.796145i	$-0.293131\pi$
$719$	32.4508	1.21021	0.605106	+	0.796145i	$0.293131\pi$
$720$	0	0
$721$	20.1246	0.749480
$722$	0	0
$723$	−1.23607	−0.0459699
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.74265	0.0646312	0.0323156	−	0.999478i	$-0.489712\pi$
$727$	1.74265	0.0646312	0.0323156	+	0.999478i	$0.489712\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	39.5967	1.46454
$732$	0	0
$733$	35.8885	1.32557	0.662787	−	0.748808i	$-0.269374\pi$
$733$	35.8885	1.32557	0.662787	+	0.748808i	$0.269374\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.58359	−0.0951678
$738$	0	0
$739$	32.5410	1.19704	0.598520	−	0.801108i	$-0.295756\pi$
$739$	32.5410	1.19704	0.598520	+	0.801108i	$0.295756\pi$
$740$	0	0
$741$	1.61803	0.0594400
$742$	0	0
$743$	9.97871	0.366084	0.183042	−	0.983105i	$-0.441406\pi$
$743$	9.97871	0.366084	0.183042	+	0.983105i	$0.441406\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−14.9443	−0.546782
$748$	0	0
$749$	−8.29180	−0.302976
$750$	0	0
$751$	41.1246	1.50066	0.750329	−	0.661064i	$-0.229895\pi$
$751$	41.1246	1.50066	0.750329	+	0.661064i	$0.229895\pi$
$752$	0	0
$753$	−4.81966	−0.175638
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−47.0132	−1.70872	−0.854361	−	0.519680i	$-0.826051\pi$
$757$	−47.0132	−1.70872	−0.854361	+	0.519680i	$0.826051\pi$
$758$	0	0
$759$	−2.09017	−0.0758684
$760$	0	0
$761$	−11.5066	−0.417113	−0.208557	−	0.978010i	$-0.566877\pi$
$761$	−11.5066	−0.417113	−0.208557	+	0.978010i	$0.566877\pi$
$762$	0	0
$763$	60.9787	2.20758
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.29180	0.0827520
$768$	0	0
$769$	−34.2918	−1.23659	−0.618297	−	0.785945i	$-0.712177\pi$
$769$	−34.2918	−1.23659	−0.618297	+	0.785945i	$0.712177\pi$
$770$	0	0
$771$	9.52786	0.343138
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−11.1246	−0.399093
$778$	0	0
$779$	−23.1803	−0.830522
$780$	0	0
$781$	−8.85410	−0.316825
$782$	0	0
$783$	−12.8754	−0.460129
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−31.4164	−1.11987	−0.559937	−	0.828535i	$-0.689175\pi$
$787$	−31.4164	−1.11987	−0.559937	+	0.828535i	$0.689175\pi$
$788$	0	0
$789$	−9.43769	−0.335991
$790$	0	0
$791$	−65.1246	−2.31556
$792$	0	0
$793$	−1.47214	−0.0522771
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.5836	0.658265	0.329132	−	0.944284i	$-0.393244\pi$
$797$	18.5836	0.658265	0.329132	+	0.944284i	$0.393244\pi$
$798$	0	0
$799$	−68.5410	−2.42481
$800$	0	0
$801$	25.4164	0.898045
$802$	0	0
$803$	−25.4590	−0.898428
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.47214	−0.227830
$808$	0	0
$809$	27.1591	0.954861	0.477431	−	0.878669i	$-0.341568\pi$
$809$	27.1591	0.954861	0.477431	+	0.878669i	$0.341568\pi$
$810$	0	0
$811$	7.70820	0.270672	0.135336	−	0.990800i	$-0.456789\pi$
$811$	7.70820	0.270672	0.135336	+	0.990800i	$0.456789\pi$
$812$	0	0
$813$	−11.8328	−0.414995
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−46.3607	−1.62195
$818$	0	0
$819$	4.85410	0.169616
$820$	0	0
$821$	−2.94427	−0.102756	−0.0513779	−	0.998679i	$-0.516361\pi$
$821$	−2.94427	−0.102756	−0.0513779	+	0.998679i	$0.516361\pi$
$822$	0	0
$823$	−20.8328	−0.726186	−0.363093	−	0.931753i	$-0.618279\pi$
$823$	−20.8328	−0.726186	−0.363093	+	0.931753i	$0.618279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−48.2492	−1.67779	−0.838895	−	0.544293i	$-0.816798\pi$
$827$	−48.2492	−1.67779	−0.838895	+	0.544293i	$0.816798\pi$
$828$	0	0
$829$	32.5410	1.13020	0.565098	−	0.825024i	$-0.308838\pi$
$829$	32.5410	1.13020	0.565098	+	0.825024i	$0.308838\pi$
$830$	0	0
$831$	10.1803	0.353152
$832$	0	0
$833$	−96.9574	−3.35938
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−30.7426	−1.06262
$838$	0	0
$839$	6.76393	0.233517	0.116758	−	0.993160i	$-0.462750\pi$
$839$	6.76393	0.233517	0.116758	+	0.993160i	$0.462750\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	0	0
$843$	−0.403252	−0.0138887
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.12461	−0.0730025
$848$	0	0
$849$	−1.88854	−0.0648147
$850$	0	0
$851$	−3.70820	−0.127116
$852$	0	0
$853$	−51.2148	−1.75356	−0.876780	−	0.480891i	$-0.840313\pi$
$853$	−51.2148	−1.75356	−0.876780	+	0.480891i	$0.840313\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	10.1459	0.345771
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	10.6738	0.362500
$868$	0	0
$869$	−19.3050	−0.654876
$870$	0	0
$871$	−0.291796	−0.00988713
$872$	0	0
$873$	41.9787	1.42076
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.6869	−1.17129	−0.585647	−	0.810566i	$-0.699160\pi$
$877$	−34.6869	−1.17129	−0.585647	+	0.810566i	$0.699160\pi$
$878$	0	0
$879$	−17.1246	−0.577599
$880$	0	0
$881$	−38.2918	−1.29008	−0.645042	−	0.764147i	$-0.723160\pi$
$881$	−38.2918	−1.29008	−0.645042	+	0.764147i	$0.723160\pi$
$882$	0	0
$883$	17.7295	0.596645	0.298322	−	0.954465i	$-0.403573\pi$
$883$	17.7295	0.596645	0.298322	+	0.954465i	$0.403573\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.5410	1.22693	0.613464	−	0.789723i	$-0.289776\pi$
$887$	36.5410	1.22693	0.613464	+	0.789723i	$0.289776\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	19.3050	0.646740
$892$	0	0
$893$	80.2492	2.68544
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.236068	−0.00788208
$898$	0	0
$899$	32.8328	1.09504
$900$	0	0
$901$	11.7082	0.390057
$902$	0	0
$903$	20.2918	0.675269
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.5410	−1.01410	−0.507049	−	0.861917i	$-0.669264\pi$
$907$	−30.5410	−1.01410	−0.507049	+	0.861917i	$0.669264\pi$
$908$	0	0
$909$	27.4164	0.909345
$910$	0	0
$911$	−22.3607	−0.740842	−0.370421	−	0.928864i	$-0.620787\pi$
$911$	−22.3607	−0.740842	−0.370421	+	0.928864i	$0.620787\pi$
$912$	0	0
$913$	19.3050	0.638901
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39.7082	1.31128
$918$	0	0
$919$	14.5836	0.481068	0.240534	−	0.970641i	$-0.422677\pi$
$919$	14.5836	0.481068	0.240534	+	0.970641i	$0.422677\pi$
$920$	0	0
$921$	−6.27051	−0.206620
$922$	0	0
$923$	−1.00000	−0.0329154
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.8541	0.356495
$928$	0	0
$929$	19.3050	0.633375	0.316687	−	0.948530i	$-0.397429\pi$
$929$	19.3050	0.633375	0.316687	+	0.948530i	$0.397429\pi$
$930$	0	0
$931$	113.520	3.72046
$932$	0	0
$933$	8.36068	0.273716
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.67376	−0.283359	−0.141680	−	0.989913i	$-0.545250\pi$
$937$	−8.67376	−0.283359	−0.141680	+	0.989913i	$0.545250\pi$
$938$	0	0
$939$	11.8328	0.386149
$940$	0	0
$941$	−24.2148	−0.789379	−0.394690	−	0.918814i	$-0.629148\pi$
$941$	−24.2148	−0.789379	−0.394690	+	0.918814i	$0.629148\pi$
$942$	0	0
$943$	3.38197	0.110132
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.3951	1.54013	0.770067	−	0.637963i	$-0.220223\pi$
$947$	47.3951	1.54013	0.770067	+	0.637963i	$0.220223\pi$
$948$	0	0
$949$	−2.87539	−0.0933391
$950$	0	0
$951$	16.2361	0.526491
$952$	0	0
$953$	−36.3951	−1.17895	−0.589477	−	0.807785i	$-0.700666\pi$
$953$	−36.3951	−1.17895	−0.589477	+	0.807785i	$0.700666\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.75078	0.250547
$958$	0	0
$959$	34.6869	1.12010
$960$	0	0
$961$	47.3951	1.52887
$962$	0	0
$963$	−4.47214	−0.144113
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−24.7639	−0.796354	−0.398177	−	0.917309i	$-0.630357\pi$
$967$	−24.7639	−0.796354	−0.398177	+	0.917309i	$0.630357\pi$
$968$	0	0
$969$	−24.7984	−0.796639
$970$	0	0
$971$	−25.8541	−0.829698	−0.414849	−	0.909890i	$-0.636166\pi$
$971$	−25.8541	−0.829698	−0.414849	+	0.909890i	$0.636166\pi$
$972$	0	0
$973$	85.9574	2.75567
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32.2705	−1.03243	−0.516213	−	0.856461i	$-0.672659\pi$
$977$	−32.2705	−1.03243	−0.516213	+	0.856461i	$0.672659\pi$
$978$	0	0
$979$	−32.8328	−1.04934
$980$	0	0
$981$	32.8885	1.05005
$982$	0	0
$983$	−26.3951	−0.841874	−0.420937	−	0.907090i	$-0.638299\pi$
$983$	−26.3951	−0.841874	−0.420937	+	0.907090i	$0.638299\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−35.1246	−1.11803
$988$	0	0
$989$	6.76393	0.215081
$990$	0	0
$991$	−51.2705	−1.62866	−0.814331	−	0.580401i	$-0.802896\pi$
$991$	−51.2705	−1.62866	−0.814331	+	0.580401i	$0.802896\pi$
$992$	0	0
$993$	−9.34752	−0.296635
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.7214	0.846274	0.423137	−	0.906066i	$-0.360929\pi$
$997$	26.7214	0.846274	0.423137	+	0.906066i	$0.360929\pi$
$998$	0	0
$999$	−12.8754	−0.407359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.bo.1.2		2
4.3	odd	2		1150.2.a.l.1.1		2
5.2	odd	4		1840.2.e.c.369.2		4
5.3	odd	4		1840.2.e.c.369.3		4
5.4	even	2		9200.2.a.by.1.1		2
20.3	even	4		230.2.b.a.139.3	yes	4
20.7	even	4		230.2.b.a.139.2	✓	4
20.19	odd	2		1150.2.a.n.1.2		2
60.23	odd	4		2070.2.d.c.829.1		4
60.47	odd	4		2070.2.d.c.829.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.a.139.2	✓	4	20.7	even	4
230.2.b.a.139.3	yes	4	20.3	even	4
1150.2.a.l.1.1		2	4.3	odd	2
1150.2.a.n.1.2		2	20.19	odd	2
1840.2.e.c.369.2		4	5.2	odd	4
1840.2.e.c.369.3		4	5.3	odd	4
2070.2.d.c.829.1		4	60.23	odd	4
2070.2.d.c.829.3		4	60.47	odd	4
9200.2.a.bo.1.2		2	1.1	even	1	trivial
9200.2.a.by.1.1		2	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} -4.85410 q^{7} -2.61803 q^{9} +3.38197 q^{11} +0.381966 q^{13} -5.85410 q^{17} +6.85410 q^{19} -3.00000 q^{21} -1.00000 q^{23} -3.47214 q^{27} +3.70820 q^{29} +8.85410 q^{31} +2.09017 q^{33} +3.70820 q^{37} +0.236068 q^{39} -3.38197 q^{41} -6.76393 q^{43} +11.7082 q^{47} +16.5623 q^{49} -3.61803 q^{51} -2.00000 q^{53} +4.23607 q^{57} +6.00000 q^{59} -3.85410 q^{61} +12.7082 q^{63} -0.763932 q^{67} -0.618034 q^{69} -2.61803 q^{71} -7.52786 q^{73} -16.4164 q^{77} -5.70820 q^{79} +5.70820 q^{81} +5.70820 q^{83} +2.29180 q^{87} -9.70820 q^{89} -1.85410 q^{91} +5.47214 q^{93} -16.0344 q^{97} -8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{7} - 3 q^{9} + 9 q^{11} + 3 q^{13} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 7 q^{33} - 6 q^{37} - 4 q^{39} - 9 q^{41} - 18 q^{43} + 10 q^{47} + 13 q^{49}+ \cdots - 11 q^{99}+O(q^{100}) \) 2 * q - q^3 - 3 * q^7 - 3 * q^9 + 9 * q^11 + 3 * q^13 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 2 * q^23 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 7 * q^33 - 6 * q^37 - 4 * q^39 - 9 * q^41 - 18 * q^43 + 10 * q^47 + 13 * q^49 - 5 * q^51 - 4 * q^53 + 4 * q^57 + 12 * q^59 - q^61 + 12 * q^63 - 6 * q^67 + q^69 - 3 * q^71 - 24 * q^73 - 6 * q^77 + 2 * q^79 - 2 * q^81 - 2 * q^83 + 18 * q^87 - 6 * q^89 + 3 * q^91 + 2 * q^93 - 3 * q^97 - 11 * q^99

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