LMFDB - Embedded newform 7396.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7396,2,Mod(1,7396)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7396, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7396.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7396 = 2^{2} \cdot 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7396.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:	$59.0573573349$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.26624689.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 18x^{4} + 7x^{3} + 96x^{2} - 2x - 139 $ x^6 - x^5 - 18x^4 + 7x^3 + 96x^2 - 2x - 139
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 172)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.56657$ of defining polynomial
Character	$\chi$	$=$	7396.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80194 q^{3} -1.56657 q^{5} -1.39852 q^{7} +0.246980 q^{9} +O(q^{10})$	$q+1.80194 q^{3} -1.56657 q^{5} -1.39852 q^{7} +0.246980 q^{9} -5.76703 q^{11} -0.944168 q^{13} -2.82286 q^{15} +0.676484 q^{17} +2.96999 q^{19} -2.52005 q^{21} +2.99607 q^{23} -2.54586 q^{25} -4.96077 q^{27} -5.82516 q^{29} +6.80433 q^{31} -10.3918 q^{33} +2.19088 q^{35} +0.0209206 q^{37} -1.70133 q^{39} -2.68271 q^{41} -0.386910 q^{45} +11.0771 q^{47} -5.04414 q^{49} +1.21898 q^{51} +1.29578 q^{53} +9.03444 q^{55} +5.35173 q^{57} +11.8138 q^{59} -0.785064 q^{61} -0.345406 q^{63} +1.47910 q^{65} +2.50710 q^{67} +5.39874 q^{69} +2.36621 q^{71} +12.9069 q^{73} -4.58749 q^{75} +8.06530 q^{77} +9.74642 q^{79} -9.67994 q^{81} -2.42848 q^{83} -1.05976 q^{85} -10.4966 q^{87} +13.8494 q^{89} +1.32044 q^{91} +12.2610 q^{93} -4.65268 q^{95} -14.4599 q^{97} -1.42434 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9	$ 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9} - 3 q^{11} + 5 q^{13} + 4 q^{15} - 9 q^{17} + 3 q^{19} + 7 q^{21} + 11 q^{25} - 4 q^{27} + 4 q^{29} + 6 q^{31} - q^{33} - 8 q^{35} - 12 q^{37} - 3 q^{39} - 3 q^{41} - 9 q^{45} + 18 q^{47} + 12 q^{49} + 4 q^{51} - 4 q^{53} + 24 q^{55} + 15 q^{57} + 23 q^{59} - 10 q^{61} + 7 q^{63} + 25 q^{65} + 25 q^{67} + 9 q^{71} + 37 q^{73} - q^{75} + 39 q^{77} + 7 q^{79} - 10 q^{81} + 4 q^{83} - 35 q^{85} + 6 q^{87} + 13 q^{89} + 9 q^{91} + 23 q^{93} - 40 q^{95} - 51 q^{97} + 4 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9 - 3 * q^11 + 5 * q^13 + 4 * q^15 - 9 * q^17 + 3 * q^19 + 7 * q^21 + 11 * q^25 - 4 * q^27 + 4 * q^29 + 6 * q^31 - q^33 - 8 * q^35 - 12 * q^37 - 3 * q^39 - 3 * q^41 - 9 * q^45 + 18 * q^47 + 12 * q^49 + 4 * q^51 - 4 * q^53 + 24 * q^55 + 15 * q^57 + 23 * q^59 - 10 * q^61 + 7 * q^63 + 25 * q^65 + 25 * q^67 + 9 * q^71 + 37 * q^73 - q^75 + 39 * q^77 + 7 * q^79 - 10 * q^81 + 4 * q^83 - 35 * q^85 + 6 * q^87 + 13 * q^89 + 9 * q^91 + 23 * q^93 - 40 * q^95 - 51 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.80194	1.04035	0.520175	−	0.854060i	$-0.325867\pi$
$3$	1.80194	1.04035	0.520175	+	0.854060i	$0.325867\pi$
$4$	0	0
$5$	−1.56657	−0.700591	−0.350295	−	0.936639i	$-0.613919\pi$
$5$	−1.56657	−0.700591	−0.350295	+	0.936639i	$0.613919\pi$
$6$	0	0
$7$	−1.39852	−0.528591	−0.264296	−	0.964442i	$-0.585139\pi$
$7$	−1.39852	−0.528591	−0.264296	+	0.964442i	$0.585139\pi$
$8$	0	0
$9$	0.246980	0.0823265
$10$	0	0
$11$	−5.76703	−1.73882	−0.869412	−	0.494088i	$-0.835502\pi$
$11$	−5.76703	−1.73882	−0.869412	+	0.494088i	$0.835502\pi$
$12$	0	0
$13$	−0.944168	−0.261865	−0.130933	−	0.991391i	$-0.541797\pi$
$13$	−0.944168	−0.261865	−0.130933	+	0.991391i	$0.541797\pi$
$14$	0	0
$15$	−2.82286	−0.728859
$16$	0	0
$17$	0.676484	0.164072	0.0820358	−	0.996629i	$-0.473858\pi$
$17$	0.676484	0.164072	0.0820358	+	0.996629i	$0.473858\pi$
$18$	0	0
$19$	2.96999	0.681361	0.340681	−	0.940179i	$-0.389342\pi$
$19$	2.96999	0.681361	0.340681	+	0.940179i	$0.389342\pi$
$20$	0	0
$21$	−2.52005	−0.549919
$22$	0	0
$23$	2.99607	0.624724	0.312362	−	0.949963i	$-0.398880\pi$
$23$	2.99607	0.624724	0.312362	+	0.949963i	$0.398880\pi$
$24$	0	0
$25$	−2.54586	−0.509173
$26$	0	0
$27$	−4.96077	−0.954701
$28$	0	0
$29$	−5.82516	−1.08170	−0.540852	−	0.841118i	$-0.681898\pi$
$29$	−5.82516	−1.08170	−0.540852	+	0.841118i	$0.681898\pi$
$30$	0	0
$31$	6.80433	1.22209	0.611047	−	0.791594i	$-0.290749\pi$
$31$	6.80433	1.22209	0.611047	+	0.791594i	$0.290749\pi$
$32$	0	0
$33$	−10.3918	−1.80898
$34$	0	0
$35$	2.19088	0.370326
$36$	0	0
$37$	0.0209206	0.00343932	0.00171966	−	0.999999i	$-0.499453\pi$
$37$	0.0209206	0.00343932	0.00171966	+	0.999999i	$0.499453\pi$
$38$	0	0
$39$	−1.70133	−0.272431
$40$	0	0
$41$	−2.68271	−0.418969	−0.209485	−	0.977812i	$-0.567179\pi$
$41$	−2.68271	−0.418969	−0.209485	+	0.977812i	$0.567179\pi$
$42$	0	0
$43$	0	0
$43$	0	0
$44$	0	0
$45$	−0.386910	−0.0576772
$46$	0	0
$47$	11.0771	1.61577	0.807883	−	0.589343i	$-0.200613\pi$
$47$	11.0771	1.61577	0.807883	+	0.589343i	$0.200613\pi$
$48$	0	0
$49$	−5.04414	−0.720592
$50$	0	0
$51$	1.21898	0.170692
$52$	0	0
$53$	1.29578	0.177989	0.0889943	−	0.996032i	$-0.471635\pi$
$53$	1.29578	0.177989	0.0889943	+	0.996032i	$0.471635\pi$
$54$	0	0
$55$	9.03444	1.21820
$56$	0	0
$57$	5.35173	0.708854
$58$	0	0
$59$	11.8138	1.53802	0.769011	−	0.639236i	$-0.220749\pi$
$59$	11.8138	1.53802	0.769011	+	0.639236i	$0.220749\pi$
$60$	0	0
$61$	−0.785064	−0.100517	−0.0502586	−	0.998736i	$-0.516005\pi$
$61$	−0.785064	−0.100517	−0.0502586	+	0.998736i	$0.516005\pi$
$62$	0	0
$63$	−0.345406	−0.0435171
$64$	0	0
$65$	1.47910	0.183460
$66$	0	0
$67$	2.50710	0.306291	0.153146	−	0.988204i	$-0.451060\pi$
$67$	2.50710	0.306291	0.153146	+	0.988204i	$0.451060\pi$
$68$	0	0
$69$	5.39874	0.649932
$70$	0	0
$71$	2.36621	0.280817	0.140409	−	0.990094i	$-0.455158\pi$
$71$	2.36621	0.280817	0.140409	+	0.990094i	$0.455158\pi$
$72$	0	0
$73$	12.9069	1.51064	0.755319	−	0.655357i	$-0.227482\pi$
$73$	12.9069	1.51064	0.755319	+	0.655357i	$0.227482\pi$
$74$	0	0
$75$	−4.58749	−0.529718
$76$	0	0
$77$	8.06530	0.919127
$78$	0	0
$79$	9.74642	1.09656	0.548279	−	0.836296i	$-0.315283\pi$
$79$	9.74642	1.09656	0.548279	+	0.836296i	$0.315283\pi$
$80$	0	0
$81$	−9.67994	−1.07555
$82$	0	0
$83$	−2.42848	−0.266560	−0.133280	−	0.991078i	$-0.542551\pi$
$83$	−2.42848	−0.266560	−0.133280	+	0.991078i	$0.542551\pi$
$84$	0	0
$85$	−1.05976	−0.114947
$86$	0	0
$87$	−10.4966	−1.12535
$88$	0	0
$89$	13.8494	1.46803	0.734017	−	0.679131i	$-0.237643\pi$
$89$	13.8494	1.46803	0.734017	+	0.679131i	$0.237643\pi$
$90$	0	0
$91$	1.32044	0.138420
$92$	0	0
$93$	12.2610	1.27140
$94$	0	0
$95$	−4.65268	−0.477355
$96$	0	0
$97$	−14.4599	−1.46818	−0.734090	−	0.679052i	$-0.762391\pi$
$97$	−14.4599	−1.46818	−0.734090	+	0.679052i	$0.762391\pi$
$98$	0	0
$99$	−1.42434	−0.143151
$100$	0	0
$101$	12.6664	1.26036	0.630178	−	0.776451i	$-0.282982\pi$
$101$	12.6664	1.26036	0.630178	+	0.776451i	$0.282982\pi$
$102$	0	0
$103$	−11.8929	−1.17184	−0.585920	−	0.810369i	$-0.699267\pi$
$103$	−11.8929	−1.17184	−0.585920	+	0.810369i	$0.699267\pi$
$104$	0	0
$105$	3.94782	0.385268
$106$	0	0
$107$	1.06206	0.102673	0.0513365	−	0.998681i	$-0.483652\pi$
$107$	1.06206	0.102673	0.0513365	+	0.998681i	$0.483652\pi$
$108$	0	0
$109$	−20.0281	−1.91835	−0.959174	−	0.282818i	$-0.908731\pi$
$109$	−20.0281	−1.91835	−0.959174	+	0.282818i	$0.908731\pi$
$110$	0	0
$111$	0.0376975	0.00357809
$112$	0	0
$113$	−12.0821	−1.13659	−0.568295	−	0.822825i	$-0.692397\pi$
$113$	−12.0821	−1.13659	−0.568295	+	0.822825i	$0.692397\pi$
$114$	0	0
$115$	−4.69355	−0.437676
$116$	0	0
$117$	−0.233190	−0.0215584
$118$	0	0
$119$	−0.946077	−0.0867268
$120$	0	0
$121$	22.2586	2.02351
$122$	0	0
$123$	−4.83408	−0.435874
$124$	0	0
$125$	11.8211	1.05731
$126$	0	0
$127$	−5.36227	−0.475825	−0.237912	−	0.971287i	$-0.576463\pi$
$127$	−5.36227	−0.475825	−0.237912	+	0.971287i	$0.576463\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.23910	0.719853	0.359927	−	0.932981i	$-0.382802\pi$
$131$	8.23910	0.719853	0.359927	+	0.932981i	$0.382802\pi$
$132$	0	0
$133$	−4.15359	−0.360162
$134$	0	0
$135$	7.77139	0.668854
$136$	0	0
$137$	10.3752	0.886415	0.443207	−	0.896419i	$-0.353841\pi$
$137$	10.3752	0.886415	0.443207	+	0.896419i	$0.353841\pi$
$138$	0	0
$139$	7.73568	0.656132	0.328066	−	0.944655i	$-0.393603\pi$
$139$	7.73568	0.656132	0.328066	+	0.944655i	$0.393603\pi$
$140$	0	0
$141$	19.9603	1.68096
$142$	0	0
$143$	5.44504	0.455337
$144$	0	0
$145$	9.12551	0.757832
$146$	0	0
$147$	−9.08923	−0.749667
$148$	0	0
$149$	18.7558	1.53653	0.768266	−	0.640131i	$-0.221120\pi$
$149$	18.7558	1.53653	0.768266	+	0.640131i	$0.221120\pi$
$150$	0	0
$151$	−14.1424	−1.15090	−0.575448	−	0.817838i	$-0.695172\pi$
$151$	−14.1424	−1.15090	−0.575448	+	0.817838i	$0.695172\pi$
$152$	0	0
$153$	0.167078	0.0135074
$154$	0	0
$155$	−10.6595	−0.856188
$156$	0	0
$157$	14.8314	1.18367	0.591837	−	0.806058i	$-0.298403\pi$
$157$	14.8314	1.18367	0.591837	+	0.806058i	$0.298403\pi$
$158$	0	0
$159$	2.33491	0.185170
$160$	0	0
$161$	−4.19007	−0.330224
$162$	0	0
$163$	24.8646	1.94754	0.973772	−	0.227527i	$-0.0730641\pi$
$163$	24.8646	1.94754	0.973772	+	0.227527i	$0.0730641\pi$
$164$	0	0
$165$	16.2795	1.26736
$166$	0	0
$167$	−6.75170	−0.522462	−0.261231	−	0.965276i	$-0.584128\pi$
$167$	−6.75170	−0.522462	−0.261231	+	0.965276i	$0.584128\pi$
$168$	0	0
$169$	−12.1085	−0.931427
$170$	0	0
$171$	0.733526	0.0560941
$172$	0	0
$173$	−11.0273	−0.838391	−0.419196	−	0.907896i	$-0.637688\pi$
$173$	−11.0273	−0.838391	−0.419196	+	0.907896i	$0.637688\pi$
$174$	0	0
$175$	3.56044	0.269144
$176$	0	0
$177$	21.2877	1.60008
$178$	0	0
$179$	13.5509	1.01284	0.506419	−	0.862287i	$-0.330969\pi$
$179$	13.5509	1.01284	0.506419	+	0.862287i	$0.330969\pi$
$180$	0	0
$181$	12.1350	0.901990	0.450995	−	0.892526i	$-0.351069\pi$
$181$	12.1350	0.901990	0.450995	+	0.892526i	$0.351069\pi$
$182$	0	0
$183$	−1.41464	−0.104573
$184$	0	0
$185$	−0.0327735	−0.00240955
$186$	0	0
$187$	−3.90130	−0.285292
$188$	0	0
$189$	6.93774	0.504646
$190$	0	0
$191$	16.0070	1.15822	0.579112	−	0.815248i	$-0.303399\pi$
$191$	16.0070	1.15822	0.579112	+	0.815248i	$0.303399\pi$
$192$	0	0
$193$	23.8225	1.71478	0.857389	−	0.514669i	$-0.172085\pi$
$193$	23.8225	1.71478	0.857389	+	0.514669i	$0.172085\pi$
$194$	0	0
$195$	2.66525	0.190863
$196$	0	0
$197$	3.61282	0.257403	0.128701	−	0.991683i	$-0.458919\pi$
$197$	3.61282	0.257403	0.128701	+	0.991683i	$0.458919\pi$
$198$	0	0
$199$	−10.8239	−0.767284	−0.383642	−	0.923482i	$-0.625330\pi$
$199$	−10.8239	−0.767284	−0.383642	+	0.923482i	$0.625330\pi$
$200$	0	0
$201$	4.51764	0.318650
$202$	0	0
$203$	8.14660	0.571779
$204$	0	0
$205$	4.20265	0.293526
$206$	0	0
$207$	0.739969	0.0514314
$208$	0	0
$209$	−17.1280	−1.18477
$210$	0	0
$211$	−16.1842	−1.11416	−0.557081	−	0.830458i	$-0.688079\pi$
$211$	−16.1842	−1.11416	−0.557081	+	0.830458i	$0.688079\pi$
$212$	0	0
$213$	4.26376	0.292148
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.51600	−0.645988
$218$	0	0
$219$	23.2574	1.57159
$220$	0	0
$221$	−0.638715	−0.0429646
$222$	0	0
$223$	−26.1912	−1.75389	−0.876945	−	0.480591i	$-0.840422\pi$
$223$	−26.1912	−1.75389	−0.876945	+	0.480591i	$0.840422\pi$
$224$	0	0
$225$	−0.628777	−0.0419184
$226$	0	0
$227$	24.5625	1.63027	0.815136	−	0.579269i	$-0.196662\pi$
$227$	24.5625	1.63027	0.815136	+	0.579269i	$0.196662\pi$
$228$	0	0
$229$	9.49734	0.627602	0.313801	−	0.949489i	$-0.398398\pi$
$229$	9.49734	0.627602	0.313801	+	0.949489i	$0.398398\pi$
$230$	0	0
$231$	14.5332	0.956213
$232$	0	0
$233$	−0.511946	−0.0335387	−0.0167693	−	0.999859i	$-0.505338\pi$
$233$	−0.511946	−0.0335387	−0.0167693	+	0.999859i	$0.505338\pi$
$234$	0	0
$235$	−17.3531	−1.13199
$236$	0	0
$237$	17.5624	1.14080
$238$	0	0
$239$	17.9382	1.16033	0.580164	−	0.814500i	$-0.302989\pi$
$239$	17.9382	1.16033	0.580164	+	0.814500i	$0.302989\pi$
$240$	0	0
$241$	−7.03757	−0.453330	−0.226665	−	0.973973i	$-0.572782\pi$
$241$	−7.03757	−0.453330	−0.226665	+	0.973973i	$0.572782\pi$
$242$	0	0
$243$	−2.56033	−0.164246
$244$	0	0
$245$	7.90199	0.504840
$246$	0	0
$247$	−2.80417	−0.178425
$248$	0	0
$249$	−4.37597	−0.277316
$250$	0	0
$251$	−7.71181	−0.486765	−0.243383	−	0.969930i	$-0.578257\pi$
$251$	−7.71181	−0.486765	−0.243383	+	0.969930i	$0.578257\pi$
$252$	0	0
$253$	−17.2784	−1.08629
$254$	0	0
$255$	−1.90962	−0.119585
$256$	0	0
$257$	4.74348	0.295890	0.147945	−	0.988996i	$-0.452734\pi$
$257$	4.74348	0.295890	0.147945	+	0.988996i	$0.452734\pi$
$258$	0	0
$259$	−0.0292578	−0.00181799
$260$	0	0
$261$	−1.43870	−0.0890530
$262$	0	0
$263$	8.74444	0.539205	0.269603	−	0.962972i	$-0.413108\pi$
$263$	8.74444	0.539205	0.269603	+	0.962972i	$0.413108\pi$
$264$	0	0
$265$	−2.02992	−0.124697
$266$	0	0
$267$	24.9558	1.52727
$268$	0	0
$269$	−14.0899	−0.859076	−0.429538	−	0.903049i	$-0.641324\pi$
$269$	−14.0899	−0.859076	−0.429538	+	0.903049i	$0.641324\pi$
$270$	0	0
$271$	−10.4208	−0.633021	−0.316511	−	0.948589i	$-0.602511\pi$
$271$	−10.4208	−0.633021	−0.316511	+	0.948589i	$0.602511\pi$
$272$	0	0
$273$	2.37935	0.144005
$274$	0	0
$275$	14.6821	0.885362
$276$	0	0
$277$	25.7322	1.54610	0.773050	−	0.634345i	$-0.218730\pi$
$277$	25.7322	1.54610	0.773050	+	0.634345i	$0.218730\pi$
$278$	0	0
$279$	1.68053	0.100611
$280$	0	0
$281$	22.0286	1.31412	0.657058	−	0.753840i	$-0.271800\pi$
$281$	22.0286	1.31412	0.657058	+	0.753840i	$0.271800\pi$
$282$	0	0
$283$	3.85437	0.229118	0.114559	−	0.993416i	$-0.463454\pi$
$283$	3.85437	0.229118	0.114559	+	0.993416i	$0.463454\pi$
$284$	0	0
$285$	−8.38385	−0.496616
$286$	0	0
$287$	3.75183	0.221463
$288$	0	0
$289$	−16.5424	−0.973081
$290$	0	0
$291$	−26.0558	−1.52742
$292$	0	0
$293$	17.4970	1.02219	0.511094	−	0.859525i	$-0.329240\pi$
$293$	17.4970	1.02219	0.511094	+	0.859525i	$0.329240\pi$
$294$	0	0
$295$	−18.5071	−1.07752
$296$	0	0
$297$	28.6089	1.66006
$298$	0	0
$299$	−2.82880	−0.163594
$300$	0	0
$301$	0	0
$302$	0	0
$303$	22.8241	1.31121
$304$	0	0
$305$	1.22986	0.0704214
$306$	0	0
$307$	−21.5606	−1.23053	−0.615265	−	0.788320i	$-0.710951\pi$
$307$	−21.5606	−1.23053	−0.615265	+	0.788320i	$0.710951\pi$
$308$	0	0
$309$	−21.4302	−1.21912
$310$	0	0
$311$	−1.89392	−0.107394	−0.0536972	−	0.998557i	$-0.517101\pi$
$311$	−1.89392	−0.107394	−0.0536972	+	0.998557i	$0.517101\pi$
$312$	0	0
$313$	−10.6438	−0.601622	−0.300811	−	0.953684i	$-0.597257\pi$
$313$	−10.6438	−0.601622	−0.300811	+	0.953684i	$0.597257\pi$
$314$	0	0
$315$	0.541102	0.0304876
$316$	0	0
$317$	−30.5712	−1.71705	−0.858524	−	0.512774i	$-0.828618\pi$
$317$	−30.5712	−1.71705	−0.858524	+	0.512774i	$0.828618\pi$
$318$	0	0
$319$	33.5938	1.88089
$320$	0	0
$321$	1.91376	0.106816
$322$	0	0
$323$	2.00915	0.111792
$324$	0	0
$325$	2.40372	0.133335
$326$	0	0
$327$	−36.0894	−1.99575
$328$	0	0
$329$	−15.4916	−0.854079
$330$	0	0
$331$	7.03282	0.386559	0.193279	−	0.981144i	$-0.438088\pi$
$331$	7.03282	0.386559	0.193279	+	0.981144i	$0.438088\pi$
$332$	0	0
$333$	0.00516695	0.000283147 0
$334$	0	0
$335$	−3.92754	−0.214585
$336$	0	0
$337$	16.1271	0.878497	0.439249	−	0.898366i	$-0.355245\pi$
$337$	16.1271	0.878497	0.439249	+	0.898366i	$0.355245\pi$
$338$	0	0
$339$	−21.7712	−1.18245
$340$	0	0
$341$	−39.2408	−2.12501
$342$	0	0
$343$	16.8440	0.909489
$344$	0	0
$345$	−8.45749	−0.455336
$346$	0	0
$347$	28.3967	1.52442	0.762208	−	0.647332i	$-0.224115\pi$
$347$	28.3967	1.52442	0.762208	+	0.647332i	$0.224115\pi$
$348$	0	0
$349$	−22.9571	−1.22887	−0.614433	−	0.788969i	$-0.710615\pi$
$349$	−22.9571	−1.22887	−0.614433	+	0.788969i	$0.710615\pi$
$350$	0	0
$351$	4.68380	0.250003
$352$	0	0
$353$	4.37909	0.233076	0.116538	−	0.993186i	$-0.462820\pi$
$353$	4.37909	0.233076	0.116538	+	0.993186i	$0.462820\pi$
$354$	0	0
$355$	−3.70682	−0.196738
$356$	0	0
$357$	−1.70477	−0.0902261
$358$	0	0
$359$	−28.9945	−1.53027	−0.765136	−	0.643869i	$-0.777328\pi$
$359$	−28.9945	−1.53027	−0.765136	+	0.643869i	$0.777328\pi$
$360$	0	0
$361$	−10.1792	−0.535747
$362$	0	0
$363$	40.1086	2.10516
$364$	0	0
$365$	−20.2195	−1.05834
$366$	0	0
$367$	27.3724	1.42883	0.714413	−	0.699724i	$-0.246694\pi$
$367$	27.3724	1.42883	0.714413	+	0.699724i	$0.246694\pi$
$368$	0	0
$369$	−0.662575	−0.0344923
$370$	0	0
$371$	−1.81217	−0.0940832
$372$	0	0
$373$	19.4558	1.00738	0.503691	−	0.863884i	$-0.331975\pi$
$373$	19.4558	1.00738	0.503691	+	0.863884i	$0.331975\pi$
$374$	0	0
$375$	21.3009	1.09997
$376$	0	0
$377$	5.49993	0.283261
$378$	0	0
$379$	−0.411696	−0.0211474	−0.0105737	−	0.999944i	$-0.503366\pi$
$379$	−0.411696	−0.0211474	−0.0105737	+	0.999944i	$0.503366\pi$
$380$	0	0
$381$	−9.66248	−0.495024
$382$	0	0
$383$	−25.8203	−1.31936	−0.659678	−	0.751548i	$-0.729307\pi$
$383$	−25.8203	−1.31936	−0.659678	+	0.751548i	$0.729307\pi$
$384$	0	0
$385$	−12.6348	−0.643932
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.5084	1.44543	0.722716	−	0.691146i	$-0.242894\pi$
$389$	28.5084	1.44543	0.722716	+	0.691146i	$0.242894\pi$
$390$	0	0
$391$	2.02680	0.102500
$392$	0	0
$393$	14.8463	0.748899
$394$	0	0
$395$	−15.2684	−0.768238
$396$	0	0
$397$	−16.0168	−0.803861	−0.401931	−	0.915670i	$-0.631661\pi$
$397$	−16.0168	−0.803861	−0.401931	+	0.915670i	$0.631661\pi$
$398$	0	0
$399$	−7.48450	−0.374694
$400$	0	0
$401$	−11.5121	−0.574885	−0.287443	−	0.957798i	$-0.592805\pi$
$401$	−11.5121	−0.574885	−0.287443	+	0.957798i	$0.592805\pi$
$402$	0	0
$403$	−6.42443	−0.320024
$404$	0	0
$405$	15.1643	0.753519
$406$	0	0
$407$	−0.120649	−0.00598037
$408$	0	0
$409$	−19.3481	−0.956700	−0.478350	−	0.878169i	$-0.658765\pi$
$409$	−19.3481	−0.956700	−0.478350	+	0.878169i	$0.658765\pi$
$410$	0	0
$411$	18.6955	0.922181
$412$	0	0
$413$	−16.5218	−0.812984
$414$	0	0
$415$	3.80438	0.186750
$416$	0	0
$417$	13.9392	0.682606
$418$	0	0
$419$	10.8126	0.528229	0.264115	−	0.964491i	$-0.414920\pi$
$419$	10.8126	0.528229	0.264115	+	0.964491i	$0.414920\pi$
$420$	0	0
$421$	17.2411	0.840282	0.420141	−	0.907459i	$-0.361981\pi$
$421$	17.2411	0.840282	0.420141	+	0.907459i	$0.361981\pi$
$422$	0	0
$423$	2.73583	0.133020
$424$	0	0
$425$	−1.72224	−0.0835408
$426$	0	0
$427$	1.09793	0.0531325
$428$	0	0
$429$	9.81163	0.473710
$430$	0	0
$431$	−2.81155	−0.135428	−0.0677139	−	0.997705i	$-0.521571\pi$
$431$	−2.81155	−0.135428	−0.0677139	+	0.997705i	$0.521571\pi$
$432$	0	0
$433$	−8.41362	−0.404333	−0.202166	−	0.979351i	$-0.564798\pi$
$433$	−8.41362	−0.404333	−0.202166	+	0.979351i	$0.564798\pi$
$434$	0	0
$435$	16.4436	0.788410
$436$	0	0
$437$	8.89829	0.425663
$438$	0	0
$439$	26.1135	1.24633	0.623166	−	0.782090i	$-0.285846\pi$
$439$	26.1135	1.24633	0.623166	+	0.782090i	$0.285846\pi$
$440$	0	0
$441$	−1.24580	−0.0593238
$442$	0	0
$443$	−9.34528	−0.444008	−0.222004	−	0.975046i	$-0.571260\pi$
$443$	−9.34528	−0.444008	−0.222004	+	0.975046i	$0.571260\pi$
$444$	0	0
$445$	−21.6961	−1.02849
$446$	0	0
$447$	33.7967	1.59853
$448$	0	0
$449$	−18.5289	−0.874435	−0.437217	−	0.899356i	$-0.644036\pi$
$449$	−18.5289	−0.874435	−0.437217	+	0.899356i	$0.644036\pi$
$450$	0	0
$451$	15.4713	0.728513
$452$	0	0
$453$	−25.4838	−1.19733
$454$	0	0
$455$	−2.06856	−0.0969754
$456$	0	0
$457$	16.2986	0.762415	0.381208	−	0.924489i	$-0.375508\pi$
$457$	16.2986	0.762415	0.381208	+	0.924489i	$0.375508\pi$
$458$	0	0
$459$	−3.35588	−0.156639
$460$	0	0
$461$	−26.8563	−1.25082	−0.625411	−	0.780296i	$-0.715068\pi$
$461$	−26.8563	−1.25082	−0.625411	+	0.780296i	$0.715068\pi$
$462$	0	0
$463$	−6.75981	−0.314155	−0.157078	−	0.987586i	$-0.550207\pi$
$463$	−6.75981	−0.314155	−0.157078	+	0.987586i	$0.550207\pi$
$464$	0	0
$465$	−19.2077	−0.890734
$466$	0	0
$467$	6.82480	0.315814	0.157907	−	0.987454i	$-0.449525\pi$
$467$	6.82480	0.315814	0.157907	+	0.987454i	$0.449525\pi$
$468$	0	0
$469$	−3.50623	−0.161903
$470$	0	0
$471$	26.7252	1.23143
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.56118	−0.346931
$476$	0	0
$477$	0.320030	0.0146532
$478$	0	0
$479$	−37.5954	−1.71778	−0.858888	−	0.512164i	$-0.828844\pi$
$479$	−37.5954	−1.71778	−0.858888	+	0.512164i	$0.828844\pi$
$480$	0	0
$481$	−0.0197525	−0.000900638 0
$482$	0	0
$483$	−7.55024	−0.343548
$484$	0	0
$485$	22.6524	1.02859
$486$	0	0
$487$	−41.9676	−1.90173	−0.950867	−	0.309601i	$-0.899805\pi$
$487$	−41.9676	−1.90173	−0.950867	+	0.309601i	$0.899805\pi$
$488$	0	0
$489$	44.8044	2.02613
$490$	0	0
$491$	40.3911	1.82283	0.911413	−	0.411492i	$-0.134992\pi$
$491$	40.3911	1.82283	0.911413	+	0.411492i	$0.134992\pi$
$492$	0	0
$493$	−3.94063	−0.177477
$494$	0	0
$495$	2.23132	0.100290
$496$	0	0
$497$	−3.30919	−0.148437
$498$	0	0
$499$	−2.51532	−0.112601	−0.0563006	−	0.998414i	$-0.517931\pi$
$499$	−2.51532	−0.112601	−0.0563006	+	0.998414i	$0.517931\pi$
$500$	0	0
$501$	−12.1661	−0.543543
$502$	0	0
$503$	−15.1930	−0.677421	−0.338710	−	0.940891i	$-0.609991\pi$
$503$	−15.1930	−0.677421	−0.338710	+	0.940891i	$0.609991\pi$
$504$	0	0
$505$	−19.8428	−0.882994
$506$	0	0
$507$	−21.8188	−0.969009
$508$	0	0
$509$	29.0855	1.28919	0.644597	−	0.764523i	$-0.277025\pi$
$509$	29.0855	1.28919	0.644597	+	0.764523i	$0.277025\pi$
$510$	0	0
$511$	−18.0506	−0.798510
$512$	0	0
$513$	−14.7334	−0.650496
$514$	0	0
$515$	18.6310	0.820980
$516$	0	0
$517$	−63.8821	−2.80953
$518$	0	0
$519$	−19.8705	−0.872220
$520$	0	0
$521$	26.0823	1.14269	0.571343	−	0.820711i	$-0.306423\pi$
$521$	26.0823	1.14269	0.571343	+	0.820711i	$0.306423\pi$
$522$	0	0
$523$	9.81372	0.429124	0.214562	−	0.976710i	$-0.431168\pi$
$523$	9.81372	0.429124	0.214562	+	0.976710i	$0.431168\pi$
$524$	0	0
$525$	6.41570	0.280004
$526$	0	0
$527$	4.60303	0.200511
$528$	0	0
$529$	−14.0235	−0.609719
$530$	0	0
$531$	2.91776	0.126620
$532$	0	0
$533$	2.53293	0.109713
$534$	0	0
$535$	−1.66379	−0.0719318
$536$	0	0
$537$	24.4178	1.05371
$538$	0	0
$539$	29.0897	1.25298
$540$	0	0
$541$	30.2174	1.29915	0.649574	−	0.760298i	$-0.274947\pi$
$541$	30.2174	1.29915	0.649574	+	0.760298i	$0.274947\pi$
$542$	0	0
$543$	21.8666	0.938385
$544$	0	0
$545$	31.3754	1.34398
$546$	0	0
$547$	9.51445	0.406808	0.203404	−	0.979095i	$-0.434799\pi$
$547$	9.51445	0.406808	0.203404	+	0.979095i	$0.434799\pi$
$548$	0	0
$549$	−0.193895	−0.00827523
$550$	0	0
$551$	−17.3006	−0.737032
$552$	0	0
$553$	−13.6306	−0.579631
$554$	0	0
$555$	−0.0590558	−0.00250678
$556$	0	0
$557$	0.0346217	0.00146697	0.000733485	−	1.00000i	$-0.499767\pi$
$557$	0.0346217	0.00146697	0.000733485		1.00000i	$0.499767\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.02991	−0.296803
$562$	0	0
$563$	−11.6936	−0.492825	−0.246412	−	0.969165i	$-0.579252\pi$
$563$	−11.6936	−0.492825	−0.246412	+	0.969165i	$0.579252\pi$
$564$	0	0
$565$	18.9275	0.796285
$566$	0	0
$567$	13.5376	0.568525
$568$	0	0
$569$	17.9847	0.753959	0.376980	−	0.926222i	$-0.376963\pi$
$569$	17.9847	0.753959	0.376980	+	0.926222i	$0.376963\pi$
$570$	0	0
$571$	−18.0668	−0.756071	−0.378036	−	0.925791i	$-0.623400\pi$
$571$	−18.0668	−0.756071	−0.378036	+	0.925791i	$0.623400\pi$
$572$	0	0
$573$	28.8436	1.20496
$574$	0	0
$575$	−7.62759	−0.318093
$576$	0	0
$577$	−28.6260	−1.19172	−0.595859	−	0.803089i	$-0.703188\pi$
$577$	−28.6260	−1.19172	−0.595859	+	0.803089i	$0.703188\pi$
$578$	0	0
$579$	42.9266	1.78397
$580$	0	0
$581$	3.39628	0.140901
$582$	0	0
$583$	−7.47278	−0.309491
$584$	0	0
$585$	0.365308	0.0151036
$586$	0	0
$587$	32.8506	1.35589	0.677945	−	0.735113i	$-0.262871\pi$
$587$	32.8506	1.35589	0.677945	+	0.735113i	$0.262871\pi$
$588$	0	0
$589$	20.2088	0.832688
$590$	0	0
$591$	6.51008	0.267789
$592$	0	0
$593$	5.22438	0.214539	0.107270	−	0.994230i	$-0.465789\pi$
$593$	5.22438	0.214539	0.107270	+	0.994230i	$0.465789\pi$
$594$	0	0
$595$	1.48209	0.0607600
$596$	0	0
$597$	−19.5039	−0.798243
$598$	0	0
$599$	−18.0175	−0.736177	−0.368089	−	0.929791i	$-0.619988\pi$
$599$	−18.0175	−0.736177	−0.368089	+	0.929791i	$0.619988\pi$
$600$	0	0
$601$	3.57570	0.145856	0.0729280	−	0.997337i	$-0.476766\pi$
$601$	3.57570	0.145856	0.0729280	+	0.997337i	$0.476766\pi$
$602$	0	0
$603$	0.619203	0.0252159
$604$	0	0
$605$	−34.8696	−1.41765
$606$	0	0
$607$	6.69714	0.271828	0.135914	−	0.990721i	$-0.456603\pi$
$607$	6.69714	0.271828	0.135914	+	0.990721i	$0.456603\pi$
$608$	0	0
$609$	14.6797	0.594850
$610$	0	0
$611$	−10.4587	−0.423113
$612$	0	0
$613$	−2.71797	−0.109778	−0.0548888	−	0.998492i	$-0.517480\pi$
$613$	−2.71797	−0.109778	−0.0548888	+	0.998492i	$0.517480\pi$
$614$	0	0
$615$	7.57291	0.305369
$616$	0	0
$617$	16.2951	0.656016	0.328008	−	0.944675i	$-0.393623\pi$
$617$	16.2951	0.656016	0.328008	+	0.944675i	$0.393623\pi$
$618$	0	0
$619$	−0.691963	−0.0278123	−0.0139062	−	0.999903i	$-0.504427\pi$
$619$	−0.691963	−0.0278123	−0.0139062	+	0.999903i	$0.504427\pi$
$620$	0	0
$621$	−14.8628	−0.596425
$622$	0	0
$623$	−19.3687	−0.775990
$624$	0	0
$625$	−5.78925	−0.231570
$626$	0	0
$627$	−30.8636	−1.23257
$628$	0	0
$629$	0.0141524	0.000564294 0
$630$	0	0
$631$	3.07617	0.122461	0.0612303	−	0.998124i	$-0.480498\pi$
$631$	3.07617	0.122461	0.0612303	+	0.998124i	$0.480498\pi$
$632$	0	0
$633$	−29.1628	−1.15912
$634$	0	0
$635$	8.40037	0.333358
$636$	0	0
$637$	4.76252	0.188698
$638$	0	0
$639$	0.584405	0.0231187
$640$	0	0
$641$	36.3504	1.43575	0.717877	−	0.696170i	$-0.245114\pi$
$641$	36.3504	1.43575	0.717877	+	0.696170i	$0.245114\pi$
$642$	0	0
$643$	32.4952	1.28149	0.640743	−	0.767755i	$-0.278626\pi$
$643$	32.4952	1.28149	0.640743	+	0.767755i	$0.278626\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.03618	−0.0800507	−0.0400253	−	0.999199i	$-0.512744\pi$
$647$	−2.03618	−0.0800507	−0.0400253	+	0.999199i	$0.512744\pi$
$648$	0	0
$649$	−68.1303	−2.67435
$650$	0	0
$651$	−17.1472	−0.672053
$652$	0	0
$653$	9.28836	0.363481	0.181741	−	0.983346i	$-0.441827\pi$
$653$	9.28836	0.363481	0.181741	+	0.983346i	$0.441827\pi$
$654$	0	0
$655$	−12.9071	−0.504322
$656$	0	0
$657$	3.18774	0.124366
$658$	0	0
$659$	13.1192	0.511053	0.255527	−	0.966802i	$-0.417751\pi$
$659$	13.1192	0.511053	0.255527	+	0.966802i	$0.417751\pi$
$660$	0	0
$661$	0.431688	0.0167907	0.00839536	−	0.999965i	$-0.497328\pi$
$661$	0.431688	0.0167907	0.00839536	+	0.999965i	$0.497328\pi$
$662$	0	0
$663$	−1.15092	−0.0446982
$664$	0	0
$665$	6.50687	0.252326
$666$	0	0
$667$	−17.4526	−0.675767
$668$	0	0
$669$	−47.1948	−1.82466
$670$	0	0
$671$	4.52749	0.174782
$672$	0	0
$673$	34.4407	1.32759	0.663796	−	0.747914i	$-0.268944\pi$
$673$	34.4407	1.32759	0.663796	+	0.747914i	$0.268944\pi$
$674$	0	0
$675$	12.6295	0.486108
$676$	0	0
$677$	−49.8706	−1.91668	−0.958342	−	0.285624i	$-0.907799\pi$
$677$	−49.8706	−1.91668	−0.958342	+	0.285624i	$0.907799\pi$
$678$	0	0
$679$	20.2225	0.776067
$680$	0	0
$681$	44.2601	1.69605
$682$	0	0
$683$	7.44154	0.284742	0.142371	−	0.989813i	$-0.454527\pi$
$683$	7.44154	0.284742	0.142371	+	0.989813i	$0.454527\pi$
$684$	0	0
$685$	−16.2535	−0.621014
$686$	0	0
$687$	17.1136	0.652925
$688$	0	0
$689$	−1.22343	−0.0466090
$690$	0	0
$691$	26.1983	0.996632	0.498316	−	0.866995i	$-0.333952\pi$
$691$	26.1983	0.996632	0.498316	+	0.866995i	$0.333952\pi$
$692$	0	0
$693$	1.99197	0.0756685
$694$	0	0
$695$	−12.1185	−0.459680
$696$	0	0
$697$	−1.81481	−0.0687409
$698$	0	0
$699$	−0.922494	−0.0348919
$700$	0	0
$701$	−5.36090	−0.202478	−0.101239	−	0.994862i	$-0.532281\pi$
$701$	−5.36090	−0.202478	−0.101239	+	0.994862i	$0.532281\pi$
$702$	0	0
$703$	0.0621338	0.00234342
$704$	0	0
$705$	−31.2692	−1.17767
$706$	0	0
$707$	−17.7142	−0.666213
$708$	0	0
$709$	−27.5305	−1.03393	−0.516964	−	0.856007i	$-0.672938\pi$
$709$	−27.5305	−1.03393	−0.516964	+	0.856007i	$0.672938\pi$
$710$	0	0
$711$	2.40717	0.0902758
$712$	0	0
$713$	20.3863	0.763472
$714$	0	0
$715$	−8.53003	−0.319005
$716$	0	0
$717$	32.3236	1.20715
$718$	0	0
$719$	−25.3794	−0.946491	−0.473245	−	0.880931i	$-0.656918\pi$
$719$	−25.3794	−0.946491	−0.473245	+	0.880931i	$0.656918\pi$
$720$	0	0
$721$	16.6324	0.619424
$722$	0	0
$723$	−12.6813	−0.471621
$724$	0	0
$725$	14.8301	0.550775
$726$	0	0
$727$	8.29879	0.307785	0.153893	−	0.988088i	$-0.450819\pi$
$727$	8.29879	0.307785	0.153893	+	0.988088i	$0.450819\pi$
$728$	0	0
$729$	24.4263	0.904676
$730$	0	0
$731$	0	0
$732$	0	0
$733$	21.0288	0.776715	0.388358	−	0.921509i	$-0.373043\pi$
$733$	21.0288	0.776715	0.388358	+	0.921509i	$0.373043\pi$
$734$	0	0
$735$	14.2389	0.525210
$736$	0	0
$737$	−14.4585	−0.532586
$738$	0	0
$739$	−20.4899	−0.753735	−0.376867	−	0.926267i	$-0.622999\pi$
$739$	−20.4899	−0.753735	−0.376867	+	0.926267i	$0.622999\pi$
$740$	0	0
$741$	−5.05293	−0.185624
$742$	0	0
$743$	0.0918602	0.00337002	0.00168501	−	0.999999i	$-0.499464\pi$
$743$	0.0918602	0.00337002	0.00168501	+	0.999999i	$0.499464\pi$
$744$	0	0
$745$	−29.3822	−1.07648
$746$	0	0
$747$	−0.599785	−0.0219450
$748$	0	0
$749$	−1.48531	−0.0542721
$750$	0	0
$751$	−38.3932	−1.40099	−0.700494	−	0.713658i	$-0.747037\pi$
$751$	−38.3932	−1.40099	−0.700494	+	0.713658i	$0.747037\pi$
$752$	0	0
$753$	−13.8962	−0.506406
$754$	0	0
$755$	22.1551	0.806307
$756$	0	0
$757$	−7.72570	−0.280795	−0.140398	−	0.990095i	$-0.544838\pi$
$757$	−7.72570	−0.280795	−0.140398	+	0.990095i	$0.544838\pi$
$758$	0	0
$759$	−31.1347	−1.13012
$760$	0	0
$761$	−29.7723	−1.07925	−0.539623	−	0.841907i	$-0.681433\pi$
$761$	−29.7723	−1.07925	−0.539623	+	0.841907i	$0.681433\pi$
$762$	0	0
$763$	28.0098	1.01402
$764$	0	0
$765$	−0.261739	−0.00946319
$766$	0	0
$767$	−11.1542	−0.402754
$768$	0	0
$769$	17.7220	0.639070	0.319535	−	0.947574i	$-0.396473\pi$
$769$	17.7220	0.639070	0.319535	+	0.947574i	$0.396473\pi$
$770$	0	0
$771$	8.54746	0.307829
$772$	0	0
$773$	27.8535	1.00182	0.500910	−	0.865500i	$-0.332999\pi$
$773$	27.8535	1.00182	0.500910	+	0.865500i	$0.332999\pi$
$774$	0	0
$775$	−17.3229	−0.622257
$776$	0	0
$777$	−0.0527208	−0.00189135
$778$	0	0
$779$	−7.96761	−0.285469
$780$	0	0
$781$	−13.6460	−0.488291
$782$	0	0
$783$	28.8973	1.03270
$784$	0	0
$785$	−23.2344	−0.829271
$786$	0	0
$787$	24.9754	0.890275	0.445138	−	0.895462i	$-0.353155\pi$
$787$	24.9754	0.890275	0.445138	+	0.895462i	$0.353155\pi$
$788$	0	0
$789$	15.7569	0.560962
$790$	0	0
$791$	16.8971	0.600792
$792$	0	0
$793$	0.741233	0.0263219
$794$	0	0
$795$	−3.65779	−0.129729
$796$	0	0
$797$	−27.5198	−0.974801	−0.487400	−	0.873179i	$-0.662055\pi$
$797$	−27.5198	−0.974801	−0.487400	+	0.873179i	$0.662055\pi$
$798$	0	0
$799$	7.49351	0.265101
$800$	0	0
$801$	3.42052	0.120858
$802$	0	0
$803$	−74.4345	−2.62673
$804$	0	0
$805$	6.56403	0.231352
$806$	0	0
$807$	−25.3891	−0.893739
$808$	0	0
$809$	−3.60262	−0.126661	−0.0633306	−	0.997993i	$-0.520172\pi$
$809$	−3.60262	−0.126661	−0.0633306	+	0.997993i	$0.520172\pi$
$810$	0	0
$811$	12.7468	0.447600	0.223800	−	0.974635i	$-0.428154\pi$
$811$	12.7468	0.447600	0.223800	+	0.974635i	$0.428154\pi$
$812$	0	0
$813$	−18.7777	−0.658563
$814$	0	0
$815$	−38.9520	−1.36443
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.326121	0.0113956
$820$	0	0
$821$	2.16064	0.0754070	0.0377035	−	0.999289i	$-0.487996\pi$
$821$	2.16064	0.0754070	0.0377035	+	0.999289i	$0.487996\pi$
$822$	0	0
$823$	−3.83997	−0.133853	−0.0669264	−	0.997758i	$-0.521319\pi$
$823$	−3.83997	−0.133853	−0.0669264	+	0.997758i	$0.521319\pi$
$824$	0	0
$825$	26.4562	0.921086
$826$	0	0
$827$	−33.7840	−1.17479	−0.587393	−	0.809302i	$-0.699846\pi$
$827$	−33.7840	−1.17479	−0.587393	+	0.809302i	$0.699846\pi$
$828$	0	0
$829$	−11.9129	−0.413752	−0.206876	−	0.978367i	$-0.566330\pi$
$829$	−11.9129	−0.413752	−0.206876	+	0.978367i	$0.566330\pi$
$830$	0	0
$831$	46.3679	1.60848
$832$	0	0
$833$	−3.41228	−0.118229
$834$	0	0
$835$	10.5770	0.366032
$836$	0	0
$837$	−33.7547	−1.16673
$838$	0	0
$839$	0.855523	0.0295359	0.0147680	−	0.999891i	$-0.495299\pi$
$839$	0.855523	0.0295359	0.0147680	+	0.999891i	$0.495299\pi$
$840$	0	0
$841$	4.93246	0.170085
$842$	0	0
$843$	39.6942	1.36714
$844$	0	0
$845$	18.9689	0.652549
$846$	0	0
$847$	−31.1291	−1.06961
$848$	0	0
$849$	6.94533	0.238363
$850$	0	0
$851$	0.0626795	0.00214863
$852$	0	0
$853$	−40.2350	−1.37762	−0.688810	−	0.724942i	$-0.741866\pi$
$853$	−40.2350	−1.37762	−0.688810	+	0.724942i	$0.741866\pi$
$854$	0	0
$855$	−1.14912	−0.0392990
$856$	0	0
$857$	22.0874	0.754490	0.377245	−	0.926113i	$-0.376871\pi$
$857$	22.0874	0.754490	0.377245	+	0.926113i	$0.376871\pi$
$858$	0	0
$859$	−50.9454	−1.73824	−0.869118	−	0.494605i	$-0.835313\pi$
$859$	−50.9454	−1.73824	−0.869118	+	0.494605i	$0.835313\pi$
$860$	0	0
$861$	6.76056	0.230399
$862$	0	0
$863$	14.4605	0.492241	0.246120	−	0.969239i	$-0.420844\pi$
$863$	14.4605	0.492241	0.246120	+	0.969239i	$0.420844\pi$
$864$	0	0
$865$	17.2750	0.587369
$866$	0	0
$867$	−29.8083	−1.01234
$868$	0	0
$869$	−56.2079	−1.90672
$870$	0	0
$871$	−2.36712	−0.0802069
$872$	0	0
$873$	−3.57130	−0.120870
$874$	0	0
$875$	−16.5321	−0.558886
$876$	0	0
$877$	12.0527	0.406991	0.203495	−	0.979076i	$-0.434770\pi$
$877$	12.0527	0.406991	0.203495	+	0.979076i	$0.434770\pi$
$878$	0	0
$879$	31.5286	1.06343
$880$	0	0
$881$	12.7787	0.430525	0.215262	−	0.976556i	$-0.430939\pi$
$881$	12.7787	0.430525	0.215262	+	0.976556i	$0.430939\pi$
$882$	0	0
$883$	11.3132	0.380721	0.190360	−	0.981714i	$-0.439034\pi$
$883$	11.3132	0.380721	0.190360	+	0.981714i	$0.439034\pi$
$884$	0	0
$885$	−33.3486	−1.12100
$886$	0	0
$887$	18.5120	0.621574	0.310787	−	0.950480i	$-0.399407\pi$
$887$	18.5120	0.621574	0.310787	+	0.950480i	$0.399407\pi$
$888$	0	0
$889$	7.49925	0.251517
$890$	0	0
$891$	55.8245	1.87019
$892$	0	0
$893$	32.8989	1.10092
$894$	0	0
$895$	−21.2283	−0.709585
$896$	0	0
$897$	−5.09731	−0.170194
$898$	0	0
$899$	−39.6363	−1.32195
$900$	0	0
$901$	0.876573	0.0292029
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.0104	−0.631926
$906$	0	0
$907$	51.5593	1.71200	0.856000	−	0.516976i	$-0.172943\pi$
$907$	51.5593	1.71200	0.856000	+	0.516976i	$0.172943\pi$
$908$	0	0
$909$	3.12835	0.103761
$910$	0	0
$911$	−10.9305	−0.362144	−0.181072	−	0.983470i	$-0.557957\pi$
$911$	−10.9305	−0.362144	−0.181072	+	0.983470i	$0.557957\pi$
$912$	0	0
$913$	14.0051	0.463502
$914$	0	0
$915$	2.21613	0.0732628
$916$	0	0
$917$	−11.5225	−0.380508
$918$	0	0
$919$	4.00657	0.132165	0.0660823	−	0.997814i	$-0.478950\pi$
$919$	4.00657	0.132165	0.0660823	+	0.997814i	$0.478950\pi$
$920$	0	0
$921$	−38.8509	−1.28018
$922$	0	0
$923$	−2.23410	−0.0735362
$924$	0	0
$925$	−0.0532609	−0.00175121
$926$	0	0
$927$	−2.93730	−0.0964735
$928$	0	0
$929$	−4.63529	−0.152079	−0.0760394	−	0.997105i	$-0.524227\pi$
$929$	−4.63529	−0.152079	−0.0760394	+	0.997105i	$0.524227\pi$
$930$	0	0
$931$	−14.9810	−0.490983
$932$	0	0
$933$	−3.41273	−0.111728
$934$	0	0
$935$	6.11166	0.199873
$936$	0	0
$937$	−18.0994	−0.591282	−0.295641	−	0.955299i	$-0.595533\pi$
$937$	−18.0994	−0.591282	−0.295641	+	0.955299i	$0.595533\pi$
$938$	0	0
$939$	−19.1794	−0.625897
$940$	0	0
$941$	4.57870	0.149261	0.0746306	−	0.997211i	$-0.476222\pi$
$941$	4.57870	0.149261	0.0746306	+	0.997211i	$0.476222\pi$
$942$	0	0
$943$	−8.03760	−0.261740
$944$	0	0
$945$	−10.8684	−0.353550
$946$	0	0
$947$	41.6556	1.35363	0.676813	−	0.736155i	$-0.263361\pi$
$947$	41.6556	1.35363	0.676813	+	0.736155i	$0.263361\pi$
$948$	0	0
$949$	−12.1863	−0.395584
$950$	0	0
$951$	−55.0873	−1.78633
$952$	0	0
$953$	−5.34710	−0.173209	−0.0866047	−	0.996243i	$-0.527602\pi$
$953$	−5.34710	−0.173209	−0.0866047	+	0.996243i	$0.527602\pi$
$954$	0	0
$955$	−25.0760	−0.811441
$956$	0	0
$957$	60.5340	1.95679
$958$	0	0
$959$	−14.5100	−0.468551
$960$	0	0
$961$	15.2990	0.493515
$962$	0	0
$963$	0.262307	0.00845272
$964$	0	0
$965$	−37.3195	−1.20136
$966$	0	0
$967$	−29.1334	−0.936868	−0.468434	−	0.883499i	$-0.655182\pi$
$967$	−29.1334	−0.936868	−0.468434	+	0.883499i	$0.655182\pi$
$968$	0	0
$969$	3.62036	0.116303
$970$	0	0
$971$	0.689254	0.0221192	0.0110596	−	0.999939i	$-0.496480\pi$
$971$	0.689254	0.0221192	0.0110596	+	0.999939i	$0.496480\pi$
$972$	0	0
$973$	−10.8185	−0.346825
$974$	0	0
$975$	4.33136	0.138715
$976$	0	0
$977$	35.4845	1.13525	0.567625	−	0.823288i	$-0.307863\pi$
$977$	35.4845	1.13525	0.567625	+	0.823288i	$0.307863\pi$
$978$	0	0
$979$	−79.8699	−2.55265
$980$	0	0
$981$	−4.94654	−0.157931
$982$	0	0
$983$	−29.0806	−0.927528	−0.463764	−	0.885959i	$-0.653501\pi$
$983$	−29.0806	−0.927528	−0.463764	+	0.885959i	$0.653501\pi$
$984$	0	0
$985$	−5.65973	−0.180334
$986$	0	0
$987$	−27.9149	−0.888541
$988$	0	0
$989$	0	0
$990$	0	0
$991$	19.1830	0.609369	0.304684	−	0.952453i	$-0.401449\pi$
$991$	19.1830	0.609369	0.304684	+	0.952453i	$0.401449\pi$
$992$	0	0
$993$	12.6727	0.402156
$994$	0	0
$995$	16.9563	0.537552
$996$	0	0
$997$	33.9769	1.07606	0.538030	−	0.842926i	$-0.319169\pi$
$997$	33.9769	1.07606	0.538030	+	0.842926i	$0.319169\pi$
$998$	0	0
$999$	−0.103782	−0.00328352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7396.2.a.j.1.5		6
43.16	even	7		172.2.i.b.41.2	yes	12
43.35	even	7		172.2.i.b.21.2	✓	12
43.42	odd	2		7396.2.a.i.1.2		6
172.35	odd	14		688.2.u.e.193.2		12
172.59	odd	14		688.2.u.e.385.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
172.2.i.b.21.2	✓	12	43.35	even	7
172.2.i.b.41.2	yes	12	43.16	even	7
688.2.u.e.193.2		12	172.35	odd	14
688.2.u.e.385.2		12	172.59	odd	14
7396.2.a.i.1.2		6	43.42	odd	2
7396.2.a.j.1.5		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7396 = 2^{2} \cdot 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7396.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80194 q^{3} -1.56657 q^{5} -1.39852 q^{7} +0.246980 q^{9} +O(q^{10})\)	\(q+1.80194 q^{3} -1.56657 q^{5} -1.39852 q^{7} +0.246980 q^{9} -5.76703 q^{11} -0.944168 q^{13} -2.82286 q^{15} +0.676484 q^{17} +2.96999 q^{19} -2.52005 q^{21} +2.99607 q^{23} -2.54586 q^{25} -4.96077 q^{27} -5.82516 q^{29} +6.80433 q^{31} -10.3918 q^{33} +2.19088 q^{35} +0.0209206 q^{37} -1.70133 q^{39} -2.68271 q^{41} -0.386910 q^{45} +11.0771 q^{47} -5.04414 q^{49} +1.21898 q^{51} +1.29578 q^{53} +9.03444 q^{55} +5.35173 q^{57} +11.8138 q^{59} -0.785064 q^{61} -0.345406 q^{63} +1.47910 q^{65} +2.50710 q^{67} +5.39874 q^{69} +2.36621 q^{71} +12.9069 q^{73} -4.58749 q^{75} +8.06530 q^{77} +9.74642 q^{79} -9.67994 q^{81} -2.42848 q^{83} -1.05976 q^{85} -10.4966 q^{87} +13.8494 q^{89} +1.32044 q^{91} +12.2610 q^{93} -4.65268 q^{95} -14.4599 q^{97} -1.42434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9	\( 6 q + 2 q^{3} + 5 q^{5} - 8 q^{9} - 3 q^{11} + 5 q^{13} + 4 q^{15} - 9 q^{17} + 3 q^{19} + 7 q^{21} + 11 q^{25} - 4 q^{27} + 4 q^{29} + 6 q^{31} - q^{33} - 8 q^{35} - 12 q^{37} - 3 q^{39} - 3 q^{41} - 9 q^{45} + 18 q^{47} + 12 q^{49} + 4 q^{51} - 4 q^{53} + 24 q^{55} + 15 q^{57} + 23 q^{59} - 10 q^{61} + 7 q^{63} + 25 q^{65} + 25 q^{67} + 9 q^{71} + 37 q^{73} - q^{75} + 39 q^{77} + 7 q^{79} - 10 q^{81} + 4 q^{83} - 35 q^{85} + 6 q^{87} + 13 q^{89} + 9 q^{91} + 23 q^{93} - 40 q^{95} - 51 q^{97} + 4 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 5 * q^5 - 8 * q^9 - 3 * q^11 + 5 * q^13 + 4 * q^15 - 9 * q^17 + 3 * q^19 + 7 * q^21 + 11 * q^25 - 4 * q^27 + 4 * q^29 + 6 * q^31 - q^33 - 8 * q^35 - 12 * q^37 - 3 * q^39 - 3 * q^41 - 9 * q^45 + 18 * q^47 + 12 * q^49 + 4 * q^51 - 4 * q^53 + 24 * q^55 + 15 * q^57 + 23 * q^59 - 10 * q^61 + 7 * q^63 + 25 * q^65 + 25 * q^67 + 9 * q^71 + 37 * q^73 - q^75 + 39 * q^77 + 7 * q^79 - 10 * q^81 + 4 * q^83 - 35 * q^85 + 6 * q^87 + 13 * q^89 + 9 * q^91 + 23 * q^93 - 40 * q^95 - 51 * q^97 + 4 * q^99

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