LMFDB - Embedded newform 9200.2.a.ct.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.521397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 9x^{3} - 3x^{2} + 18x + 12 $ x^5 - 9x^3 - 3x^2 + 18*x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.29544$ 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)

$f(q)$	$=$	$q+2.29544 q^{3} +1.61758 q^{7} +2.26905 q^{9} +O(q^{10})$	$q+2.29544 q^{3} +1.61758 q^{7} +2.26905 q^{9} +2.79255 q^{11} +4.88663 q^{13} +3.54388 q^{17} +2.79255 q^{19} +3.71306 q^{21} +1.00000 q^{23} -1.67786 q^{27} -3.36282 q^{29} +4.84564 q^{31} +6.41013 q^{33} +3.87204 q^{37} +11.2170 q^{39} -0.327923 q^{41} +5.38343 q^{43} -0.0121087 q^{47} -4.38343 q^{49} +8.13476 q^{51} -0.866254 q^{53} +6.41013 q^{57} -2.96752 q^{59} -7.29816 q^{61} +3.67037 q^{63} -6.94082 q^{67} +2.29544 q^{69} +2.16864 q^{71} -6.02888 q^{73} +4.51718 q^{77} -6.50561 q^{79} -10.6586 q^{81} +7.88030 q^{83} -7.71915 q^{87} -9.71986 q^{89} +7.90452 q^{91} +11.1229 q^{93} +11.9952 q^{97} +6.33643 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 5 * q - 4 * q^7 + 3 * q^9	$ 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * 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
$3$	2.29544	1.32527	0.662637	−	0.748941i	$-0.269437\pi$
$3$	2.29544	1.32527	0.662637	+	0.748941i	$0.269437\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.61758	0.611388	0.305694	−	0.952130i	$-0.401111\pi$
$7$	1.61758	0.611388	0.305694	+	0.952130i	$0.401111\pi$
$8$	0	0
$9$	2.26905	0.756349
$10$	0	0
$11$	2.79255	0.841985	0.420993	−	0.907064i	$-0.361682\pi$
$11$	2.79255	0.841985	0.420993	+	0.907064i	$0.361682\pi$
$12$	0	0
$13$	4.88663	1.35531	0.677653	−	0.735381i	$-0.262997\pi$
$13$	4.88663	1.35531	0.677653	+	0.735381i	$0.262997\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.54388	0.859516	0.429758	−	0.902944i	$-0.358599\pi$
$17$	3.54388	0.859516	0.429758	+	0.902944i	$0.358599\pi$
$18$	0	0
$19$	2.79255	0.640655	0.320327	−	0.947307i	$-0.396207\pi$
$19$	2.79255	0.640655	0.320327	+	0.947307i	$0.396207\pi$
$20$	0	0
$21$	3.71306	0.810257
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.67786	−0.322904
$28$	0	0
$29$	−3.36282	−0.624460	−0.312230	−	0.950007i	$-0.601076\pi$
$29$	−3.36282	−0.624460	−0.312230	+	0.950007i	$0.601076\pi$
$30$	0	0
$31$	4.84564	0.870303	0.435152	−	0.900357i	$-0.356695\pi$
$31$	4.84564	0.870303	0.435152	+	0.900357i	$0.356695\pi$
$32$	0	0
$33$	6.41013	1.11586
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.87204	0.636559	0.318279	−	0.947997i	$-0.396895\pi$
$37$	3.87204	0.636559	0.318279	+	0.947997i	$0.396895\pi$
$38$	0	0
$39$	11.2170	1.79615
$40$	0	0
$41$	−0.327923	−0.0512130	−0.0256065	−	0.999672i	$-0.508152\pi$
$41$	−0.327923	−0.0512130	−0.0256065	+	0.999672i	$0.508152\pi$
$42$	0	0
$43$	5.38343	0.820965	0.410483	−	0.911868i	$-0.365360\pi$
$43$	5.38343	0.820965	0.410483	+	0.911868i	$0.365360\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.0121087	−0.00176623	−0.000883117	−	1.00000i	$-0.500281\pi$
$47$	−0.0121087	−0.00176623	−0.000883117		1.00000i	$0.500281\pi$
$48$	0	0
$49$	−4.38343	−0.626204
$50$	0	0
$51$	8.13476	1.13909
$52$	0	0
$53$	−0.866254	−0.118989	−0.0594946	−	0.998229i	$-0.518949\pi$
$53$	−0.866254	−0.118989	−0.0594946	+	0.998229i	$0.518949\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.41013	0.849043
$58$	0	0
$59$	−2.96752	−0.386338	−0.193169	−	0.981166i	$-0.561877\pi$
$59$	−2.96752	−0.386338	−0.193169	+	0.981166i	$0.561877\pi$
$60$	0	0
$61$	−7.29816	−0.934434	−0.467217	−	0.884143i	$-0.654743\pi$
$61$	−7.29816	−0.934434	−0.467217	+	0.884143i	$0.654743\pi$
$62$	0	0
$63$	3.67037	0.462423
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.94082	−0.847956	−0.423978	−	0.905673i	$-0.639367\pi$
$67$	−6.94082	−0.847956	−0.423978	+	0.905673i	$0.639367\pi$
$68$	0	0
$69$	2.29544	0.276339
$70$	0	0
$71$	2.16864	0.257370	0.128685	−	0.991685i	$-0.458924\pi$
$71$	2.16864	0.257370	0.128685	+	0.991685i	$0.458924\pi$
$72$	0	0
$73$	−6.02888	−0.705627	−0.352813	−	0.935694i	$-0.614775\pi$
$73$	−6.02888	−0.705627	−0.352813	+	0.935694i	$0.614775\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.51718	0.514780
$78$	0	0
$79$	−6.50561	−0.731938	−0.365969	−	0.930627i	$-0.619262\pi$
$79$	−6.50561	−0.731938	−0.365969	+	0.930627i	$0.619262\pi$
$80$	0	0
$81$	−10.6586	−1.18429
$82$	0	0
$83$	7.88030	0.864976	0.432488	−	0.901640i	$-0.357636\pi$
$83$	7.88030	0.864976	0.432488	+	0.901640i	$0.357636\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.71915	−0.827580
$88$	0	0
$89$	−9.71986	−1.03030	−0.515151	−	0.857099i	$-0.672264\pi$
$89$	−9.71986	−1.03030	−0.515151	+	0.857099i	$0.672264\pi$
$90$	0	0
$91$	7.90452	0.828619
$92$	0	0
$93$	11.1229	1.15339
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.9952	1.21793	0.608966	−	0.793197i	$-0.291585\pi$
$97$	11.9952	1.21793	0.608966	+	0.793197i	$0.291585\pi$
$98$	0	0
$99$	6.33643	0.636835
$100$	0	0
$101$	11.4712	1.14143	0.570713	−	0.821150i	$-0.306667\pi$
$101$	11.4712	1.14143	0.570713	+	0.821150i	$0.306667\pi$
$102$	0	0
$103$	4.02030	0.396132	0.198066	−	0.980189i	$-0.436534\pi$
$103$	4.02030	0.396132	0.198066	+	0.980189i	$0.436534\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.8318	−1.43385	−0.716923	−	0.697152i	$-0.754450\pi$
$107$	−14.8318	−1.43385	−0.716923	+	0.697152i	$0.754450\pi$
$108$	0	0
$109$	12.3573	1.18362	0.591809	−	0.806078i	$-0.298414\pi$
$109$	12.3573	1.18362	0.591809	+	0.806078i	$0.298414\pi$
$110$	0	0
$111$	8.88803	0.843614
$112$	0	0
$113$	−5.48632	−0.516109	−0.258055	−	0.966130i	$-0.583081\pi$
$113$	−5.48632	−0.516109	−0.258055	+	0.966130i	$0.583081\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	11.0880	1.02509
$118$	0	0
$119$	5.73251	0.525498
$120$	0	0
$121$	−3.20167	−0.291061
$122$	0	0
$123$	−0.752728	−0.0678712
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.14959	0.279481	0.139740	−	0.990188i	$-0.455373\pi$
$127$	3.14959	0.279481	0.139740	+	0.990188i	$0.455373\pi$
$128$	0	0
$129$	12.3573	1.08800
$130$	0	0
$131$	2.93231	0.256197	0.128099	−	0.991761i	$-0.459113\pi$
$131$	2.93231	0.256197	0.128099	+	0.991761i	$0.459113\pi$
$132$	0	0
$133$	4.51718	0.391689
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.4054	1.23073	0.615366	−	0.788241i	$-0.289008\pi$
$137$	14.4054	1.23073	0.615366	+	0.788241i	$0.289008\pi$
$138$	0	0
$139$	−18.2241	−1.54575	−0.772873	−	0.634561i	$-0.781181\pi$
$139$	−18.2241	−1.54575	−0.772873	+	0.634561i	$0.781181\pi$
$140$	0	0
$141$	−0.0277948	−0.00234074
$142$	0	0
$143$	13.6462	1.14115
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−10.0619	−0.829892
$148$	0	0
$149$	−5.71986	−0.468589	−0.234294	−	0.972166i	$-0.575278\pi$
$149$	−5.71986	−0.468589	−0.234294	+	0.972166i	$0.575278\pi$
$150$	0	0
$151$	5.43082	0.441954	0.220977	−	0.975279i	$-0.429075\pi$
$151$	5.43082	0.441954	0.220977	+	0.975279i	$0.429075\pi$
$152$	0	0
$153$	8.04122	0.650094
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−24.2004	−1.93140	−0.965701	−	0.259657i	$-0.916390\pi$
$157$	−24.2004	−1.93140	−0.965701	+	0.259657i	$0.916390\pi$
$158$	0	0
$159$	−1.98844	−0.157693
$160$	0	0
$161$	1.61758	0.127483
$162$	0	0
$163$	11.2970	0.884849	0.442425	−	0.896806i	$-0.354118\pi$
$163$	11.2970	0.884849	0.442425	+	0.896806i	$0.354118\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.6478	−1.67516	−0.837578	−	0.546318i	$-0.816029\pi$
$167$	−21.6478	−1.67516	−0.837578	+	0.546318i	$0.816029\pi$
$168$	0	0
$169$	10.8791	0.836857
$170$	0	0
$171$	6.33643	0.484559
$172$	0	0
$173$	1.39593	0.106130	0.0530652	−	0.998591i	$-0.483101\pi$
$173$	1.39593	0.106130	0.0530652	+	0.998591i	$0.483101\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.81176	−0.512003
$178$	0	0
$179$	−23.5665	−1.76144	−0.880722	−	0.473634i	$-0.842942\pi$
$179$	−23.5665	−1.76144	−0.880722	+	0.473634i	$0.842942\pi$
$180$	0	0
$181$	8.33495	0.619532	0.309766	−	0.950813i	$-0.399749\pi$
$181$	8.33495	0.619532	0.309766	+	0.950813i	$0.399749\pi$
$182$	0	0
$183$	−16.7525	−1.23838
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.89645	0.723700
$188$	0	0
$189$	−2.71407	−0.197420
$190$	0	0
$191$	−18.3074	−1.32468	−0.662340	−	0.749204i	$-0.730436\pi$
$191$	−18.3074	−1.32468	−0.662340	+	0.749204i	$0.730436\pi$
$192$	0	0
$193$	15.4434	1.11164	0.555820	−	0.831303i	$-0.312404\pi$
$193$	15.4434	1.11164	0.555820	+	0.831303i	$0.312404\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.876228	−0.0624287	−0.0312143	−	0.999513i	$-0.509937\pi$
$197$	−0.876228	−0.0624287	−0.0312143	+	0.999513i	$0.509937\pi$
$198$	0	0
$199$	13.5818	0.962788	0.481394	−	0.876504i	$-0.340131\pi$
$199$	13.5818	0.962788	0.481394	+	0.876504i	$0.340131\pi$
$200$	0	0
$201$	−15.9322	−1.12377
$202$	0	0
$203$	−5.43963	−0.381787
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.26905	0.157710
$208$	0	0
$209$	7.79833	0.539422
$210$	0	0
$211$	−12.9100	−0.888758	−0.444379	−	0.895839i	$-0.646576\pi$
$211$	−12.9100	−0.888758	−0.444379	+	0.895839i	$0.646576\pi$
$212$	0	0
$213$	4.97799	0.341086
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.83822	0.532093
$218$	0	0
$219$	−13.8389	−0.935148
$220$	0	0
$221$	17.3176	1.16491
$222$	0	0
$223$	8.05527	0.539421	0.269710	−	0.962941i	$-0.413072\pi$
$223$	8.05527	0.539421	0.269710	+	0.962941i	$0.413072\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.41490	−0.293027	−0.146514	−	0.989209i	$-0.546805\pi$
$227$	−4.41490	−0.293027	−0.146514	+	0.989209i	$0.546805\pi$
$228$	0	0
$229$	17.5197	1.15773	0.578866	−	0.815423i	$-0.303495\pi$
$229$	17.5197	1.15773	0.578866	+	0.815423i	$0.303495\pi$
$230$	0	0
$231$	10.3689	0.682224
$232$	0	0
$233$	15.8080	1.03562	0.517808	−	0.855497i	$-0.326748\pi$
$233$	15.8080	1.03562	0.517808	+	0.855497i	$0.326748\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−14.9332	−0.970018
$238$	0	0
$239$	8.93972	0.578263	0.289131	−	0.957289i	$-0.406634\pi$
$239$	8.93972	0.578263	0.289131	+	0.957289i	$0.406634\pi$
$240$	0	0
$241$	3.50992	0.226094	0.113047	−	0.993590i	$-0.463939\pi$
$241$	3.50992	0.226094	0.113047	+	0.993590i	$0.463939\pi$
$242$	0	0
$243$	−19.4325	−1.24660
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.6462	0.868284
$248$	0	0
$249$	18.0888	1.14633
$250$	0	0
$251$	−4.33837	−0.273836	−0.136918	−	0.990582i	$-0.543720\pi$
$251$	−4.33837	−0.273836	−0.136918	+	0.990582i	$0.543720\pi$
$252$	0	0
$253$	2.79255	0.175566
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.6252	−1.03705	−0.518524	−	0.855063i	$-0.673518\pi$
$257$	−16.6252	−1.03705	−0.518524	+	0.855063i	$0.673518\pi$
$258$	0	0
$259$	6.26333	0.389185
$260$	0	0
$261$	−7.63039	−0.472310
$262$	0	0
$263$	−13.0084	−0.802134	−0.401067	−	0.916049i	$-0.631360\pi$
$263$	−13.0084	−0.802134	−0.401067	+	0.916049i	$0.631360\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−22.3114	−1.36543
$268$	0	0
$269$	−8.30995	−0.506667	−0.253333	−	0.967379i	$-0.581527\pi$
$269$	−8.30995	−0.506667	−0.253333	+	0.967379i	$0.581527\pi$
$270$	0	0
$271$	8.35009	0.507232	0.253616	−	0.967305i	$-0.418380\pi$
$271$	8.35009	0.507232	0.253616	+	0.967305i	$0.418380\pi$
$272$	0	0
$273$	18.1444	1.09815
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.23195	0.374441	0.187221	−	0.982318i	$-0.440052\pi$
$277$	6.23195	0.374441	0.187221	+	0.982318i	$0.440052\pi$
$278$	0	0
$279$	10.9950	0.658253
$280$	0	0
$281$	8.01836	0.478335	0.239168	−	0.970978i	$-0.423125\pi$
$281$	8.01836	0.478335	0.239168	+	0.970978i	$0.423125\pi$
$282$	0	0
$283$	9.11741	0.541974	0.270987	−	0.962583i	$-0.412650\pi$
$283$	9.11741	0.541974	0.270987	+	0.962583i	$0.412650\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.530443	−0.0313110
$288$	0	0
$289$	−4.44094	−0.261232
$290$	0	0
$291$	27.5343	1.61409
$292$	0	0
$293$	17.5323	1.02425	0.512124	−	0.858911i	$-0.328859\pi$
$293$	17.5323	1.02425	0.512124	+	0.858911i	$0.328859\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.68550	−0.271881
$298$	0	0
$299$	4.88663	0.282601
$300$	0	0
$301$	8.70814	0.501929
$302$	0	0
$303$	26.3314	1.51270
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−30.3404	−1.73162	−0.865809	−	0.500375i	$-0.833196\pi$
$307$	−30.3404	−1.73162	−0.865809	+	0.500375i	$0.833196\pi$
$308$	0	0
$309$	9.22837	0.524984
$310$	0	0
$311$	13.7703	0.780840	0.390420	−	0.920637i	$-0.372330\pi$
$311$	13.7703	0.780840	0.390420	+	0.920637i	$0.372330\pi$
$312$	0	0
$313$	14.3715	0.812328	0.406164	−	0.913800i	$-0.366866\pi$
$313$	14.3715	0.812328	0.406164	+	0.913800i	$0.366866\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.5726	1.49247	0.746233	−	0.665685i	$-0.231860\pi$
$317$	26.5726	1.49247	0.746233	+	0.665685i	$0.231860\pi$
$318$	0	0
$319$	−9.39084	−0.525786
$320$	0	0
$321$	−34.0456	−1.90024
$322$	0	0
$323$	9.89645	0.550653
$324$	0	0
$325$	0	0
$326$	0	0
$327$	28.3655	1.56862
$328$	0	0
$329$	−0.0195868	−0.00107986
$330$	0	0
$331$	12.2379	0.672655	0.336327	−	0.941745i	$-0.390815\pi$
$331$	12.2379	0.672655	0.336327	+	0.941745i	$0.390815\pi$
$332$	0	0
$333$	8.78583	0.481461
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.7474	0.912290	0.456145	−	0.889906i	$-0.349230\pi$
$337$	16.7474	0.912290	0.456145	+	0.889906i	$0.349230\pi$
$338$	0	0
$339$	−12.5935	−0.683986
$340$	0	0
$341$	13.5317	0.732783
$342$	0	0
$343$	−18.4136	−0.994242
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.6122	−1.10652	−0.553260	−	0.833008i	$-0.686617\pi$
$347$	−20.6122	−1.10652	−0.553260	+	0.833008i	$0.686617\pi$
$348$	0	0
$349$	35.5885	1.90501	0.952505	−	0.304523i	$-0.0984970\pi$
$349$	35.5885	1.90501	0.952505	+	0.304523i	$0.0984970\pi$
$350$	0	0
$351$	−8.19907	−0.437634
$352$	0	0
$353$	−0.574309	−0.0305674	−0.0152837	−	0.999883i	$-0.504865\pi$
$353$	−0.574309	−0.0305674	−0.0152837	+	0.999883i	$0.504865\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	13.1586	0.696429
$358$	0	0
$359$	−4.12882	−0.217911	−0.108955	−	0.994047i	$-0.534751\pi$
$359$	−4.12882	−0.217911	−0.108955	+	0.994047i	$0.534751\pi$
$360$	0	0
$361$	−11.2017	−0.589561
$362$	0	0
$363$	−7.34924	−0.385735
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.16724	−0.217528	−0.108764	−	0.994068i	$-0.534689\pi$
$367$	−4.16724	−0.217528	−0.108764	+	0.994068i	$0.534689\pi$
$368$	0	0
$369$	−0.744073	−0.0387349
$370$	0	0
$371$	−1.40124	−0.0727486
$372$	0	0
$373$	−25.3579	−1.31298	−0.656491	−	0.754334i	$-0.727960\pi$
$373$	−25.3579	−1.31298	−0.656491	+	0.754334i	$0.727960\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.4328	−0.846335
$378$	0	0
$379$	−14.2580	−0.732382	−0.366191	−	0.930540i	$-0.619338\pi$
$379$	−14.2580	−0.732382	−0.366191	+	0.930540i	$0.619338\pi$
$380$	0	0
$381$	7.22969	0.370388
$382$	0	0
$383$	−16.6805	−0.852334	−0.426167	−	0.904645i	$-0.640136\pi$
$383$	−16.6805	−0.852334	−0.426167	+	0.904645i	$0.640136\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.2153	0.620936
$388$	0	0
$389$	26.3853	1.33779	0.668894	−	0.743358i	$-0.266768\pi$
$389$	26.3853	1.33779	0.668894	+	0.743358i	$0.266768\pi$
$390$	0	0
$391$	3.54388	0.179222
$392$	0	0
$393$	6.73095	0.339532
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.44231	−0.323330	−0.161665	−	0.986846i	$-0.551686\pi$
$397$	−6.44231	−0.323330	−0.161665	+	0.986846i	$0.551686\pi$
$398$	0	0
$399$	10.3689	0.519095
$400$	0	0
$401$	21.6326	1.08028	0.540141	−	0.841574i	$-0.318371\pi$
$401$	21.6326	1.08028	0.540141	+	0.841574i	$0.318371\pi$
$402$	0	0
$403$	23.6789	1.17953
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.8129	0.535973
$408$	0	0
$409$	8.07200	0.399135	0.199567	−	0.979884i	$-0.436046\pi$
$409$	8.07200	0.399135	0.199567	+	0.979884i	$0.436046\pi$
$410$	0	0
$411$	33.0666	1.63106
$412$	0	0
$413$	−4.80020	−0.236202
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−41.8323	−2.04853
$418$	0	0
$419$	25.4538	1.24350	0.621751	−	0.783215i	$-0.286422\pi$
$419$	25.4538	1.24350	0.621751	+	0.783215i	$0.286422\pi$
$420$	0	0
$421$	31.4111	1.53089	0.765443	−	0.643504i	$-0.222520\pi$
$421$	31.4111	1.53089	0.765443	+	0.643504i	$0.222520\pi$
$422$	0	0
$423$	−0.0274752	−0.00133589
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−11.8054	−0.571302
$428$	0	0
$429$	31.3239	1.51233
$430$	0	0
$431$	−3.35572	−0.161639	−0.0808196	−	0.996729i	$-0.525754\pi$
$431$	−3.35572	−0.161639	−0.0808196	+	0.996729i	$0.525754\pi$
$432$	0	0
$433$	18.0365	0.866777	0.433388	−	0.901207i	$-0.357318\pi$
$433$	18.0365	0.866777	0.433388	+	0.901207i	$0.357318\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.79255	0.133586
$438$	0	0
$439$	−19.8080	−0.945384	−0.472692	−	0.881228i	$-0.656718\pi$
$439$	−19.8080	−0.945384	−0.472692	+	0.881228i	$0.656718\pi$
$440$	0	0
$441$	−9.94621	−0.473629
$442$	0	0
$443$	−11.5628	−0.549364	−0.274682	−	0.961535i	$-0.588573\pi$
$443$	−11.5628	−0.549364	−0.274682	+	0.961535i	$0.588573\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.1296	−0.621008
$448$	0	0
$449$	−19.6506	−0.927368	−0.463684	−	0.886001i	$-0.653473\pi$
$449$	−19.6506	−0.927368	−0.463684	+	0.886001i	$0.653473\pi$
$450$	0	0
$451$	−0.915742	−0.0431206
$452$	0	0
$453$	12.4661	0.585710
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.0682	−1.07908	−0.539542	−	0.841959i	$-0.681402\pi$
$457$	−23.0682	−1.07908	−0.539542	+	0.841959i	$0.681402\pi$
$458$	0	0
$459$	−5.94613	−0.277541
$460$	0	0
$461$	1.86556	0.0868876	0.0434438	−	0.999056i	$-0.486167\pi$
$461$	1.86556	0.0868876	0.0434438	+	0.999056i	$0.486167\pi$
$462$	0	0
$463$	12.9981	0.604071	0.302035	−	0.953297i	$-0.402334\pi$
$463$	12.9981	0.604071	0.302035	+	0.953297i	$0.402334\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.7712	−1.28510	−0.642548	−	0.766245i	$-0.722123\pi$
$467$	−27.7712	−1.28510	−0.642548	+	0.766245i	$0.722123\pi$
$468$	0	0
$469$	−11.2273	−0.518430
$470$	0	0
$471$	−55.5506	−2.55963
$472$	0	0
$473$	15.0335	0.691241
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.96557	−0.0899974
$478$	0	0
$479$	5.25003	0.239880	0.119940	−	0.992781i	$-0.461730\pi$
$479$	5.25003	0.239880	0.119940	+	0.992781i	$0.461730\pi$
$480$	0	0
$481$	18.9212	0.862733
$482$	0	0
$483$	3.71306	0.168950
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.90856	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$487$	9.90856	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$488$	0	0
$489$	25.9316	1.17267
$490$	0	0
$491$	24.2168	1.09289	0.546444	−	0.837495i	$-0.315981\pi$
$491$	24.2168	1.09289	0.546444	+	0.837495i	$0.315981\pi$
$492$	0	0
$493$	−11.9174	−0.536733
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.50795	0.157353
$498$	0	0
$499$	26.6722	1.19401	0.597007	−	0.802236i	$-0.296357\pi$
$499$	26.6722	1.19401	0.597007	+	0.802236i	$0.296357\pi$
$500$	0	0
$501$	−49.6912	−2.22004
$502$	0	0
$503$	−2.38303	−0.106254	−0.0531271	−	0.998588i	$-0.516919\pi$
$503$	−2.38303	−0.106254	−0.0531271	+	0.998588i	$0.516919\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	24.9724	1.10906
$508$	0	0
$509$	36.2807	1.60811	0.804056	−	0.594554i	$-0.202671\pi$
$509$	36.2807	1.60811	0.804056	+	0.594554i	$0.202671\pi$
$510$	0	0
$511$	−9.75220	−0.431412
$512$	0	0
$513$	−4.68550	−0.206870
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.0338141	−0.00148714
$518$	0	0
$519$	3.20427	0.140652
$520$	0	0
$521$	−13.8498	−0.606773	−0.303386	−	0.952868i	$-0.598117\pi$
$521$	−13.8498	−0.606773	−0.303386	+	0.952868i	$0.598117\pi$
$522$	0	0
$523$	11.3982	0.498411	0.249205	−	0.968451i	$-0.419831\pi$
$523$	11.3982	0.498411	0.249205	+	0.968451i	$0.419831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.1724	0.748040
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.73344	−0.292206
$532$	0	0
$533$	−1.60244	−0.0694094
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−54.0955	−2.33439
$538$	0	0
$539$	−12.2409	−0.527255
$540$	0	0
$541$	−15.6397	−0.672402	−0.336201	−	0.941790i	$-0.609142\pi$
$541$	−15.6397	−0.672402	−0.336201	+	0.941790i	$0.609142\pi$
$542$	0	0
$543$	19.1324	0.821050
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−45.7911	−1.95789	−0.978944	−	0.204131i	$-0.934563\pi$
$547$	−45.7911	−1.95789	−0.978944	+	0.204131i	$0.934563\pi$
$548$	0	0
$549$	−16.5599	−0.706758
$550$	0	0
$551$	−9.39084	−0.400063
$552$	0	0
$553$	−10.5234	−0.447499
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.8143	−1.30564	−0.652821	−	0.757512i	$-0.726415\pi$
$557$	−30.8143	−1.30564	−0.652821	+	0.757512i	$0.726415\pi$
$558$	0	0
$559$	26.3068	1.11266
$560$	0	0
$561$	22.7167	0.959100
$562$	0	0
$563$	−27.8036	−1.17178	−0.585891	−	0.810390i	$-0.699255\pi$
$563$	−27.8036	−1.17178	−0.585891	+	0.810390i	$0.699255\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.2411	−0.724058
$568$	0	0
$569$	7.89706	0.331062	0.165531	−	0.986205i	$-0.447066\pi$
$569$	7.89706	0.331062	0.165531	+	0.986205i	$0.447066\pi$
$570$	0	0
$571$	−6.12319	−0.256248	−0.128124	−	0.991758i	$-0.540895\pi$
$571$	−6.12319	−0.256248	−0.128124	+	0.991758i	$0.540895\pi$
$572$	0	0
$573$	−42.0236	−1.75556
$574$	0	0
$575$	0	0
$576$	0	0
$577$	27.2174	1.13307	0.566537	−	0.824036i	$-0.308283\pi$
$577$	27.2174	1.13307	0.566537	+	0.824036i	$0.308283\pi$
$578$	0	0
$579$	35.4494	1.47323
$580$	0	0
$581$	12.7470	0.528836
$582$	0	0
$583$	−2.41906	−0.100187
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.7305	−1.10329	−0.551644	−	0.834080i	$-0.685999\pi$
$587$	−26.7305	−1.10329	−0.551644	+	0.834080i	$0.685999\pi$
$588$	0	0
$589$	13.5317	0.557564
$590$	0	0
$591$	−2.01133	−0.0827350
$592$	0	0
$593$	−0.0562817	−0.00231121	−0.00115561	−	0.999999i	$-0.500368\pi$
$593$	−0.0562817	−0.00231121	−0.00115561	+	0.999999i	$0.500368\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.1762	1.27596
$598$	0	0
$599$	39.4760	1.61295	0.806473	−	0.591271i	$-0.201374\pi$
$599$	39.4760	1.61295	0.806473	+	0.591271i	$0.201374\pi$
$600$	0	0
$601$	23.2118	0.946829	0.473415	−	0.880840i	$-0.343021\pi$
$601$	23.2118	0.946829	0.473415	+	0.880840i	$0.343021\pi$
$602$	0	0
$603$	−15.7490	−0.641350
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.7212	0.800459	0.400230	−	0.916415i	$-0.368930\pi$
$607$	19.7212	0.800459	0.400230	+	0.916415i	$0.368930\pi$
$608$	0	0
$609$	−12.4864	−0.505973
$610$	0	0
$611$	−0.0591707	−0.00239379
$612$	0	0
$613$	−10.4609	−0.422512	−0.211256	−	0.977431i	$-0.567755\pi$
$613$	−10.4609	−0.422512	−0.211256	+	0.977431i	$0.567755\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.4969	−0.583622	−0.291811	−	0.956476i	$-0.594258\pi$
$617$	−14.4969	−0.583622	−0.291811	+	0.956476i	$0.594258\pi$
$618$	0	0
$619$	1.30293	0.0523692	0.0261846	−	0.999657i	$-0.491664\pi$
$619$	1.30293	0.0523692	0.0261846	+	0.999657i	$0.491664\pi$
$620$	0	0
$621$	−1.67786	−0.0673302
$622$	0	0
$623$	−15.7227	−0.629915
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.9006	0.714881
$628$	0	0
$629$	13.7220	0.547133
$630$	0	0
$631$	0.653161	0.0260019	0.0130010	−	0.999915i	$-0.495862\pi$
$631$	0.653161	0.0260019	0.0130010	+	0.999915i	$0.495862\pi$
$632$	0	0
$633$	−29.6340	−1.17785
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−21.4202	−0.848699
$638$	0	0
$639$	4.92075	0.194662
$640$	0	0
$641$	−13.2594	−0.523714	−0.261857	−	0.965107i	$-0.584335\pi$
$641$	−13.2594	−0.523714	−0.261857	+	0.965107i	$0.584335\pi$
$642$	0	0
$643$	−45.9135	−1.81065	−0.905326	−	0.424716i	$-0.860374\pi$
$643$	−45.9135	−1.81065	−0.905326	+	0.424716i	$0.860374\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.36368	−0.132240	−0.0661199	−	0.997812i	$-0.521062\pi$
$647$	−3.36368	−0.132240	−0.0661199	+	0.997812i	$0.521062\pi$
$648$	0	0
$649$	−8.28694	−0.325291
$650$	0	0
$651$	17.9922	0.705169
$652$	0	0
$653$	−25.4704	−0.996734	−0.498367	−	0.866966i	$-0.666067\pi$
$653$	−25.4704	−0.996734	−0.498367	+	0.866966i	$0.666067\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.6798	−0.533700
$658$	0	0
$659$	−9.03335	−0.351890	−0.175945	−	0.984400i	$-0.556298\pi$
$659$	−9.03335	−0.351890	−0.175945	+	0.984400i	$0.556298\pi$
$660$	0	0
$661$	−3.20679	−0.124730	−0.0623648	−	0.998053i	$-0.519864\pi$
$661$	−3.20679	−0.124730	−0.0623648	+	0.998053i	$0.519864\pi$
$662$	0	0
$663$	39.7515	1.54382
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.36282	−0.130209
$668$	0	0
$669$	18.4904	0.714880
$670$	0	0
$671$	−20.3805	−0.786779
$672$	0	0
$673$	−16.4923	−0.635733	−0.317866	−	0.948135i	$-0.602966\pi$
$673$	−16.4923	−0.635733	−0.317866	+	0.948135i	$0.602966\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.9260	1.34232	0.671158	−	0.741314i	$-0.265797\pi$
$677$	34.9260	1.34232	0.671158	+	0.741314i	$0.265797\pi$
$678$	0	0
$679$	19.4033	0.744629
$680$	0	0
$681$	−10.1341	−0.388341
$682$	0	0
$683$	−23.1448	−0.885612	−0.442806	−	0.896618i	$-0.646017\pi$
$683$	−23.1448	−0.885612	−0.442806	+	0.896618i	$0.646017\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	40.2153	1.53431
$688$	0	0
$689$	−4.23306	−0.161267
$690$	0	0
$691$	−3.24273	−0.123359	−0.0616795	−	0.998096i	$-0.519646\pi$
$691$	−3.24273	−0.123359	−0.0616795	+	0.998096i	$0.519646\pi$
$692$	0	0
$693$	10.2497	0.389353
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.16212	−0.0440184
$698$	0	0
$699$	36.2863	1.37247
$700$	0	0
$701$	19.9466	0.753373	0.376686	−	0.926341i	$-0.377063\pi$
$701$	19.9466	0.753373	0.376686	+	0.926341i	$0.377063\pi$
$702$	0	0
$703$	10.8129	0.407814
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.5556	0.697854
$708$	0	0
$709$	14.3774	0.539955	0.269977	−	0.962867i	$-0.412984\pi$
$709$	14.3774	0.539955	0.269977	+	0.962867i	$0.412984\pi$
$710$	0	0
$711$	−14.7615	−0.553601
$712$	0	0
$713$	4.84564	0.181471
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.5206	0.766356
$718$	0	0
$719$	51.4356	1.91822	0.959111	−	0.283029i	$-0.0913394\pi$
$719$	51.4356	1.91822	0.959111	+	0.283029i	$0.0913394\pi$
$720$	0	0
$721$	6.50317	0.242191
$722$	0	0
$723$	8.05682	0.299636
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−17.4611	−0.647597	−0.323799	−	0.946126i	$-0.604960\pi$
$727$	−17.4611	−0.647597	−0.323799	+	0.946126i	$0.604960\pi$
$728$	0	0
$729$	−12.6305	−0.467797
$730$	0	0
$731$	19.0782	0.705633
$732$	0	0
$733$	−39.6443	−1.46429	−0.732147	−	0.681147i	$-0.761482\pi$
$733$	−39.6443	−1.46429	−0.732147	+	0.681147i	$0.761482\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.3826	−0.713966
$738$	0	0
$739$	43.1149	1.58601	0.793003	−	0.609218i	$-0.208517\pi$
$739$	43.1149	1.58601	0.793003	+	0.609218i	$0.208517\pi$
$740$	0	0
$741$	31.3239	1.15071
$742$	0	0
$743$	18.8929	0.693112	0.346556	−	0.938029i	$-0.387351\pi$
$743$	18.8929	0.693112	0.346556	+	0.938029i	$0.387351\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17.8808	0.654223
$748$	0	0
$749$	−23.9917	−0.876637
$750$	0	0
$751$	−27.5146	−1.00402	−0.502012	−	0.864861i	$-0.667407\pi$
$751$	−27.5146	−1.00402	−0.502012	+	0.864861i	$0.667407\pi$
$752$	0	0
$753$	−9.95847	−0.362907
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.6895	0.533899	0.266949	−	0.963711i	$-0.413984\pi$
$757$	14.6895	0.533899	0.266949	+	0.963711i	$0.413984\pi$
$758$	0	0
$759$	6.41013	0.232673
$760$	0	0
$761$	−37.5884	−1.36258	−0.681289	−	0.732015i	$-0.738580\pi$
$761$	−37.5884	−1.36258	−0.681289	+	0.732015i	$0.738580\pi$
$762$	0	0
$763$	19.9890	0.723651
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.5012	−0.523606
$768$	0	0
$769$	−43.6879	−1.57543	−0.787713	−	0.616042i	$-0.788735\pi$
$769$	−43.6879	−1.57543	−0.787713	+	0.616042i	$0.788735\pi$
$770$	0	0
$771$	−38.1620	−1.37437
$772$	0	0
$773$	7.01583	0.252342	0.126171	−	0.992009i	$-0.459731\pi$
$773$	7.01583	0.252342	0.126171	+	0.992009i	$0.459731\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.3771	0.515776
$778$	0	0
$779$	−0.915742	−0.0328099
$780$	0	0
$781$	6.05604	0.216702
$782$	0	0
$783$	5.64234	0.201641
$784$	0	0
$785$	0	0
$786$	0	0
$787$	23.6386	0.842624	0.421312	−	0.906916i	$-0.361570\pi$
$787$	23.6386	0.842624	0.421312	+	0.906916i	$0.361570\pi$
$788$	0	0
$789$	−29.8601	−1.06305
$790$	0	0
$791$	−8.87457	−0.315543
$792$	0	0
$793$	−35.6634	−1.26644
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.9000	1.12996	0.564978	−	0.825106i	$-0.308885\pi$
$797$	31.9000	1.12996	0.564978	+	0.825106i	$0.308885\pi$
$798$	0	0
$799$	−0.0429117	−0.00151811
$800$	0	0
$801$	−22.0548	−0.779268
$802$	0	0
$803$	−16.8359	−0.594127
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.0750	−0.671472
$808$	0	0
$809$	−5.68831	−0.199990	−0.0999951	−	0.994988i	$-0.531883\pi$
$809$	−5.68831	−0.199990	−0.0999951	+	0.994988i	$0.531883\pi$
$810$	0	0
$811$	16.1989	0.568821	0.284410	−	0.958703i	$-0.408202\pi$
$811$	16.1989	0.568821	0.284410	+	0.958703i	$0.408202\pi$
$812$	0	0
$813$	19.1671	0.672221
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.0335	0.525955
$818$	0	0
$819$	17.9357	0.626725
$820$	0	0
$821$	−26.9329	−0.939965	−0.469982	−	0.882676i	$-0.655740\pi$
$821$	−26.9329	−0.939965	−0.469982	+	0.882676i	$0.655740\pi$
$822$	0	0
$823$	−42.0280	−1.46500	−0.732502	−	0.680765i	$-0.761648\pi$
$823$	−42.0280	−1.46500	−0.732502	+	0.680765i	$0.761648\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.1311	0.387065	0.193532	−	0.981094i	$-0.438006\pi$
$827$	11.1311	0.387065	0.193532	+	0.981094i	$0.438006\pi$
$828$	0	0
$829$	−20.1506	−0.699859	−0.349929	−	0.936776i	$-0.613794\pi$
$829$	−20.1506	−0.699859	−0.349929	+	0.936776i	$0.613794\pi$
$830$	0	0
$831$	14.3051	0.496237
$832$	0	0
$833$	−15.5343	−0.538233
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.13031	−0.281024
$838$	0	0
$839$	12.5573	0.433526	0.216763	−	0.976224i	$-0.430450\pi$
$839$	12.5573	0.433526	0.216763	+	0.976224i	$0.430450\pi$
$840$	0	0
$841$	−17.6914	−0.610050
$842$	0	0
$843$	18.4057	0.633925
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.17896	−0.177951
$848$	0	0
$849$	20.9285	0.718263
$850$	0	0
$851$	3.87204	0.132732
$852$	0	0
$853$	−50.2016	−1.71887	−0.859434	−	0.511246i	$-0.829184\pi$
$853$	−50.2016	−1.71887	−0.859434	+	0.511246i	$0.829184\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.9126	−1.26091	−0.630456	−	0.776225i	$-0.717132\pi$
$857$	−36.9126	−1.26091	−0.630456	+	0.776225i	$0.717132\pi$
$858$	0	0
$859$	−7.29638	−0.248949	−0.124475	−	0.992223i	$-0.539725\pi$
$859$	−7.29638	−0.248949	−0.124475	+	0.992223i	$0.539725\pi$
$860$	0	0
$861$	−1.21760	−0.0414957
$862$	0	0
$863$	52.3903	1.78339	0.891694	−	0.452640i	$-0.149518\pi$
$863$	52.3903	1.78339	0.891694	+	0.452640i	$0.149518\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.1939	−0.346203
$868$	0	0
$869$	−18.1672	−0.616281
$870$	0	0
$871$	−33.9172	−1.14924
$872$	0	0
$873$	27.2177	0.921181
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.1120	−0.949276	−0.474638	−	0.880181i	$-0.657421\pi$
$877$	−28.1120	−0.949276	−0.474638	+	0.880181i	$0.657421\pi$
$878$	0	0
$879$	40.2444	1.35741
$880$	0	0
$881$	41.3936	1.39459	0.697293	−	0.716787i	$-0.254388\pi$
$881$	41.3936	1.39459	0.697293	+	0.716787i	$0.254388\pi$
$882$	0	0
$883$	36.7906	1.23810	0.619052	−	0.785350i	$-0.287517\pi$
$883$	36.7906	1.23810	0.619052	+	0.785350i	$0.287517\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.7836	−1.47011	−0.735055	−	0.678007i	$-0.762844\pi$
$887$	−43.7836	−1.47011	−0.735055	+	0.678007i	$0.762844\pi$
$888$	0	0
$889$	5.09471	0.170871
$890$	0	0
$891$	−29.7646	−0.997151
$892$	0	0
$893$	−0.0338141	−0.00113155
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.2170	0.374524
$898$	0	0
$899$	−16.2950	−0.543469
$900$	0	0
$901$	−3.06990	−0.102273
$902$	0	0
$903$	19.9890	0.665193
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.8370	0.691882	0.345941	−	0.938256i	$-0.387560\pi$
$907$	20.8370	0.691882	0.345941	+	0.938256i	$0.387560\pi$
$908$	0	0
$909$	26.0287	0.863316
$910$	0	0
$911$	−45.8193	−1.51806	−0.759031	−	0.651054i	$-0.774327\pi$
$911$	−45.8193	−1.51806	−0.759031	+	0.651054i	$0.774327\pi$
$912$	0	0
$913$	22.0061	0.728297
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.74326	0.156636
$918$	0	0
$919$	−30.3282	−1.00044	−0.500218	−	0.865900i	$-0.666747\pi$
$919$	−30.3282	−1.00044	−0.500218	+	0.865900i	$0.666747\pi$
$920$	0	0
$921$	−69.6446	−2.29487
$922$	0	0
$923$	10.5973	0.348816
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.12226	0.299614
$928$	0	0
$929$	−35.7602	−1.17325	−0.586626	−	0.809858i	$-0.699544\pi$
$929$	−35.7602	−1.17325	−0.586626	+	0.809858i	$0.699544\pi$
$930$	0	0
$931$	−12.2409	−0.401181
$932$	0	0
$933$	31.6088	1.03483
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.6768	−0.871494	−0.435747	−	0.900069i	$-0.643516\pi$
$937$	−26.6768	−0.871494	−0.435747	+	0.900069i	$0.643516\pi$
$938$	0	0
$939$	32.9890	1.07656
$940$	0	0
$941$	−9.52987	−0.310665	−0.155332	−	0.987862i	$-0.549645\pi$
$941$	−9.52987	−0.310665	−0.155332	+	0.987862i	$0.549645\pi$
$942$	0	0
$943$	−0.327923	−0.0106787
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.5209	−1.05679	−0.528393	−	0.849000i	$-0.677205\pi$
$947$	−32.5209	−1.05679	−0.528393	+	0.849000i	$0.677205\pi$
$948$	0	0
$949$	−29.4609	−0.956341
$950$	0	0
$951$	60.9958	1.97793
$952$	0	0
$953$	8.80252	0.285141	0.142571	−	0.989785i	$-0.454463\pi$
$953$	8.80252	0.285141	0.142571	+	0.989785i	$0.454463\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−21.5561	−0.696810
$958$	0	0
$959$	23.3018	0.752456
$960$	0	0
$961$	−7.51974	−0.242572
$962$	0	0
$963$	−33.6541	−1.08449
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.0307	1.51241	0.756203	−	0.654337i	$-0.227052\pi$
$967$	47.0307	1.51241	0.756203	+	0.654337i	$0.227052\pi$
$968$	0	0
$969$	22.7167	0.729766
$970$	0	0
$971$	−33.3450	−1.07009	−0.535046	−	0.844823i	$-0.679706\pi$
$971$	−33.3450	−1.07009	−0.535046	+	0.844823i	$0.679706\pi$
$972$	0	0
$973$	−29.4789	−0.945050
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.8915	−1.78813	−0.894064	−	0.447939i	$-0.852158\pi$
$977$	−55.8915	−1.78813	−0.894064	+	0.447939i	$0.852158\pi$
$978$	0	0
$979$	−27.1432	−0.867500
$980$	0	0
$981$	28.0394	0.895229
$982$	0	0
$983$	−44.2540	−1.41148	−0.705742	−	0.708469i	$-0.749386\pi$
$983$	−44.2540	−1.41148	−0.705742	+	0.708469i	$0.749386\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.0449603	−0.00143110
$988$	0	0
$989$	5.38343	0.171183
$990$	0	0
$991$	28.2426	0.897157	0.448579	−	0.893743i	$-0.351930\pi$
$991$	28.2426	0.897157	0.448579	+	0.893743i	$0.351930\pi$
$992$	0	0
$993$	28.0913	0.891451
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.7730	0.626218	0.313109	−	0.949717i	$-0.398629\pi$
$997$	19.7730	0.626218	0.313109	+	0.949717i	$0.398629\pi$
$998$	0	0
$999$	−6.49673	−0.205547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ct.1.5		5
4.3	odd	2		4600.2.a.bf.1.1	yes	5
5.4	even	2		9200.2.a.cv.1.1		5
20.3	even	4		4600.2.e.w.4049.2		10
20.7	even	4		4600.2.e.w.4049.9		10
20.19	odd	2		4600.2.a.bd.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bd.1.5	✓	5	20.19	odd	2
4600.2.a.bf.1.1	yes	5	4.3	odd	2
4600.2.e.w.4049.2		10	20.3	even	4
4600.2.e.w.4049.9		10	20.7	even	4
9200.2.a.ct.1.5		5	1.1	even	1	trivial
9200.2.a.cv.1.1		5	5.4	even	2

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

\(f(q)\)	\(=\)	\(q+2.29544 q^{3} +1.61758 q^{7} +2.26905 q^{9} +O(q^{10})\)	\(q+2.29544 q^{3} +1.61758 q^{7} +2.26905 q^{9} +2.79255 q^{11} +4.88663 q^{13} +3.54388 q^{17} +2.79255 q^{19} +3.71306 q^{21} +1.00000 q^{23} -1.67786 q^{27} -3.36282 q^{29} +4.84564 q^{31} +6.41013 q^{33} +3.87204 q^{37} +11.2170 q^{39} -0.327923 q^{41} +5.38343 q^{43} -0.0121087 q^{47} -4.38343 q^{49} +8.13476 q^{51} -0.866254 q^{53} +6.41013 q^{57} -2.96752 q^{59} -7.29816 q^{61} +3.67037 q^{63} -6.94082 q^{67} +2.29544 q^{69} +2.16864 q^{71} -6.02888 q^{73} +4.51718 q^{77} -6.50561 q^{79} -10.6586 q^{81} +7.88030 q^{83} -7.71915 q^{87} -9.71986 q^{89} +7.90452 q^{91} +11.1229 q^{93} +11.9952 q^{97} +6.33643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 5 * q - 4 * q^7 + 3 * q^9	\( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57} + q^{59} + 10 q^{61} - 8 q^{67} - 8 q^{71} + 6 q^{73} - 27 q^{81} + 2 q^{83} - 39 q^{87} + 14 q^{89} + 46 q^{91} - 3 q^{93} + 6 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q - 4 * q^7 + 3 * q^9 + 4 * q^13 + 6 * q^17 + 5 * q^23 - 9 * q^27 + 12 * q^29 + 18 * q^31 + 6 * q^33 + 10 * q^37 - 9 * q^39 - 6 * q^41 - 10 * q^43 - 22 * q^47 + 15 * q^49 + 6 * q^51 + 10 * q^53 + 6 * q^57 + q^59 + 10 * q^61 - 8 * q^67 - 8 * q^71 + 6 * q^73 - 27 * q^81 + 2 * q^83 - 39 * q^87 + 14 * q^89 + 46 * q^91 - 3 * q^93 + 6 * q^97 + 6 * q^99

Introduction

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