LMFDB - Embedded newform 3249.2.a.bg.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,6,-9,0,9,-3,0,-6,-9,0,-3,-6,0,6,-15,0,0,9,0,21,-6,0,15, 0,0,-3,-15,0,0,-21,0,-33,3,0,6,0,0,-39,6,0,15,6,0,27,-9,0,9,-9,0,-18,-6, 0,-6,-39,0,-12,-15,0,3,-27,0,21,-21,0,18,-21,0,-9,-15,0,-6,-9,0,0,-48, 0,-21,21,0,-15,3,0,42,-30,0,42,-24,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.6357609.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 $ x^6 - 9x^4 - x^3 + 18x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.842316$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.842316 q^{2} -1.29050 q^{4} -1.70747 q^{5} +3.93033 q^{7} +2.77164 q^{8} +1.43823 q^{10} -4.42535 q^{11} -1.82462 q^{13} -3.31058 q^{14} +0.246406 q^{16} +5.02644 q^{17} +2.20349 q^{20} +3.72755 q^{22} -3.45831 q^{23} -2.08456 q^{25} +1.53691 q^{26} -5.07211 q^{28} -1.45022 q^{29} +2.86943 q^{31} -5.75084 q^{32} -4.23385 q^{34} -6.71091 q^{35} +4.83123 q^{37} -4.73249 q^{40} -8.80804 q^{41} +10.1017 q^{43} +5.71093 q^{44} +2.91299 q^{46} -3.43531 q^{47} +8.44752 q^{49} +1.75586 q^{50} +2.35468 q^{52} -2.11234 q^{53} +7.55614 q^{55} +10.8935 q^{56} +1.22154 q^{58} +6.40598 q^{59} +0.0766957 q^{61} -2.41696 q^{62} +4.35121 q^{64} +3.11548 q^{65} +1.04715 q^{67} -6.48664 q^{68} +5.65271 q^{70} -13.5488 q^{71} +4.01120 q^{73} -4.06942 q^{74} -17.3931 q^{77} -16.9649 q^{79} -0.420729 q^{80} +7.41916 q^{82} +13.0252 q^{83} -8.58248 q^{85} -8.50883 q^{86} -12.2655 q^{88} +12.3638 q^{89} -7.17137 q^{91} +4.46296 q^{92} +2.89362 q^{94} -0.175254 q^{97} -7.11548 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} - 9 q^{5} + 9 q^{7} - 3 q^{8} - 6 q^{10} - 9 q^{11} - 3 q^{13} - 6 q^{14} + 6 q^{16} - 15 q^{17} + 9 q^{20} + 21 q^{22} - 6 q^{23} + 15 q^{25} - 3 q^{28} - 15 q^{29} - 21 q^{32} - 33 q^{34}+ \cdots - 3 q^{98}+O(q^{100}) $ 6 * q + 6 * q^4 - 9 * q^5 + 9 * q^7 - 3 * q^8 - 6 * q^10 - 9 * q^11 - 3 * q^13 - 6 * q^14 + 6 * q^16 - 15 * q^17 + 9 * q^20 + 21 * q^22 - 6 * q^23 + 15 * q^25 - 3 * q^28 - 15 * q^29 - 21 * q^32 - 33 * q^34 + 3 * q^35 + 6 * q^37 - 39 * q^40 + 6 * q^41 + 15 * q^43 + 6 * q^44 + 27 * q^46 - 9 * q^47 + 9 * q^49 - 9 * q^50 - 18 * q^52 - 6 * q^53 - 6 * q^55 - 39 * q^56 - 12 * q^58 - 15 * q^59 + 3 * q^61 - 27 * q^62 + 21 * q^64 - 21 * q^65 + 18 * q^67 - 21 * q^68 - 9 * q^70 - 15 * q^71 - 6 * q^73 - 9 * q^74 - 48 * q^77 - 21 * q^79 + 21 * q^80 - 15 * q^82 + 3 * q^83 + 42 * q^85 - 30 * q^86 + 42 * q^88 - 24 * q^89 - 12 * q^91 - 42 * q^92 - 9 * q^94 + 3 * q^97 - 3 * q^98

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 $

$2$	−0.842316	−0.595608	−0.297804	−	0.954627i	$-0.596254\pi$
$2$	−0.842316	−0.595608	−0.297804	+	0.954627i	$0.596254\pi$
$3$	0	0
$3$	0	0
$4$	−1.29050	−0.645252
$5$	−1.70747	−0.763602	−0.381801	−	0.924244i	$-0.624696\pi$
$5$	−1.70747	−0.763602	−0.381801	+	0.924244i	$0.624696\pi$
$6$	0	0
$7$	3.93033	1.48553	0.742763	−	0.669554i	$-0.233515\pi$
$7$	3.93033	1.48553	0.742763	+	0.669554i	$0.233515\pi$
$8$	2.77164	0.979924
$9$	0	0
$10$	1.43823	0.454807
$11$	−4.42535	−1.33429	−0.667147	−	0.744926i	$-0.732485\pi$
$11$	−4.42535	−1.33429	−0.667147	+	0.744926i	$0.732485\pi$
$12$	0	0
$13$	−1.82462	−0.506059	−0.253030	−	0.967459i	$-0.581427\pi$
$13$	−1.82462	−0.506059	−0.253030	+	0.967459i	$0.581427\pi$
$14$	−3.31058	−0.884791
$15$	0	0
$16$	0.246406	0.0616014
$17$	5.02644	1.21909	0.609545	−	0.792751i	$-0.291352\pi$
$17$	5.02644	1.21909	0.609545	+	0.792751i	$0.291352\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	2.20349	0.492716
$21$	0	0
$22$	3.72755	0.794716
$23$	−3.45831	−0.721107	−0.360553	−	0.932739i	$-0.617412\pi$
$23$	−3.45831	−0.721107	−0.360553	+	0.932739i	$0.617412\pi$
$24$	0	0
$25$	−2.08456	−0.416912
$26$	1.53691	0.301413
$27$	0	0
$28$	−5.07211	−0.958538
$29$	−1.45022	−0.269299	−0.134649	−	0.990893i	$-0.542991\pi$
$29$	−1.45022	−0.269299	−0.134649	+	0.990893i	$0.542991\pi$
$30$	0	0
$31$	2.86943	0.515364	0.257682	−	0.966230i	$-0.417041\pi$
$31$	2.86943	0.515364	0.257682	+	0.966230i	$0.417041\pi$
$32$	−5.75084	−1.01661
$33$	0	0
$34$	−4.23385	−0.726100
$35$	−6.71091	−1.13435
$36$	0	0
$37$	4.83123	0.794249	0.397124	−	0.917765i	$-0.370008\pi$
$37$	4.83123	0.794249	0.397124	+	0.917765i	$0.370008\pi$
$38$	0	0
$39$	0	0
$40$	−4.73249	−0.748272
$41$	−8.80804	−1.37559	−0.687793	−	0.725907i	$-0.741420\pi$
$41$	−8.80804	−1.37559	−0.687793	+	0.725907i	$0.741420\pi$
$42$	0	0
$43$	10.1017	1.54050	0.770248	−	0.637745i	$-0.220132\pi$
$43$	10.1017	1.54050	0.770248	+	0.637745i	$0.220132\pi$
$44$	5.71093	0.860956
$45$	0	0
$46$	2.91299	0.429497
$47$	−3.43531	−0.501092	−0.250546	−	0.968105i	$-0.580610\pi$
$47$	−3.43531	−0.501092	−0.250546	+	0.968105i	$0.580610\pi$
$48$	0	0
$49$	8.44752	1.20679
$50$	1.75586	0.248316
$51$	0	0
$52$	2.35468	0.326536
$53$	−2.11234	−0.290152	−0.145076	−	0.989421i	$-0.546343\pi$
$53$	−2.11234	−0.290152	−0.145076	+	0.989421i	$0.546343\pi$
$54$	0	0
$55$	7.55614	1.01887
$56$	10.8935	1.45570
$57$	0	0
$58$	1.22154	0.160396
$59$	6.40598	0.833988	0.416994	−	0.908909i	$-0.363084\pi$
$59$	6.40598	0.833988	0.416994	+	0.908909i	$0.363084\pi$
$60$	0	0
$61$	0.0766957	0.00981988	0.00490994	−	0.999988i	$-0.498437\pi$
$61$	0.0766957	0.00981988	0.00490994	+	0.999988i	$0.498437\pi$
$62$	−2.41696	−0.306955
$63$	0	0
$64$	4.35121	0.543902
$65$	3.11548	0.386428
$66$	0	0
$67$	1.04715	0.127930	0.0639650	−	0.997952i	$-0.479625\pi$
$67$	1.04715	0.127930	0.0639650	+	0.997952i	$0.479625\pi$
$68$	−6.48664	−0.786620
$69$	0	0
$70$	5.65271	0.675628
$71$	−13.5488	−1.60794	−0.803971	−	0.594668i	$-0.797283\pi$
$71$	−13.5488	−1.60794	−0.803971	+	0.594668i	$0.797283\pi$
$72$	0	0
$73$	4.01120	0.469475	0.234738	−	0.972059i	$-0.424577\pi$
$73$	4.01120	0.469475	0.234738	+	0.972059i	$0.424577\pi$
$74$	−4.06942	−0.473061
$75$	0	0
$76$	0	0
$77$	−17.3931	−1.98213
$78$	0	0
$79$	−16.9649	−1.90870	−0.954348	−	0.298697i	$-0.903448\pi$
$79$	−16.9649	−1.90870	−0.954348	+	0.298697i	$0.903448\pi$
$80$	−0.420729	−0.0470390
$81$	0	0
$82$	7.41916	0.819309
$83$	13.0252	1.42970	0.714850	−	0.699278i	$-0.246495\pi$
$83$	13.0252	1.42970	0.714850	+	0.699278i	$0.246495\pi$
$84$	0	0
$85$	−8.58248	−0.930900
$86$	−8.50883	−0.917531
$87$	0	0
$88$	−12.2655	−1.30751
$89$	12.3638	1.31056	0.655279	−	0.755387i	$-0.272551\pi$
$89$	12.3638	1.31056	0.655279	+	0.755387i	$0.272551\pi$
$90$	0	0
$91$	−7.17137	−0.751764
$92$	4.46296	0.465295
$93$	0	0
$94$	2.89362	0.298454
$95$	0	0
$96$	0	0
$97$	−0.175254	−0.0177944	−0.00889718	−	0.999960i	$-0.502832\pi$
$97$	−0.175254	−0.0177944	−0.00889718	+	0.999960i	$0.502832\pi$
$98$	−7.11548	−0.718772
$99$	0	0
$100$	2.69013	0.269013
$101$	11.2135	1.11578	0.557891	−	0.829914i	$-0.311611\pi$
$101$	11.2135	1.11578	0.557891	+	0.829914i	$0.311611\pi$
$102$	0	0
$103$	−9.52010	−0.938043	−0.469022	−	0.883187i	$-0.655393\pi$
$103$	−9.52010	−0.938043	−0.469022	+	0.883187i	$0.655393\pi$
$104$	−5.05720	−0.495900
$105$	0	0
$106$	1.77925	0.172816
$107$	−10.2690	−0.992742	−0.496371	−	0.868111i	$-0.665334\pi$
$107$	−10.2690	−0.992742	−0.496371	+	0.868111i	$0.665334\pi$
$108$	0	0
$109$	2.04164	0.195554	0.0977770	−	0.995208i	$-0.468827\pi$
$109$	2.04164	0.195554	0.0977770	+	0.995208i	$0.468827\pi$
$110$	−6.36466	−0.606847
$111$	0	0
$112$	0.968456	0.0915105
$113$	−13.9973	−1.31675	−0.658375	−	0.752690i	$-0.728756\pi$
$113$	−13.9973	−1.31675	−0.658375	+	0.752690i	$0.728756\pi$
$114$	0	0
$115$	5.90494	0.550639
$116$	1.87151	0.173765
$117$	0	0
$118$	−5.39587	−0.496730
$119$	19.7556	1.81099
$120$	0	0
$121$	8.58375	0.780341
$122$	−0.0646020	−0.00584879
$123$	0	0
$124$	−3.70300	−0.332539
$125$	12.0966	1.08196
$126$	0	0
$127$	−1.95928	−0.173858	−0.0869288	−	0.996215i	$-0.527705\pi$
$127$	−1.95928	−0.173858	−0.0869288	+	0.996215i	$0.527705\pi$
$128$	7.83658	0.692662
$129$	0	0
$130$	−2.62422	−0.230159
$131$	1.56004	0.136302	0.0681508	−	0.997675i	$-0.478290\pi$
$131$	1.56004	0.136302	0.0681508	+	0.997675i	$0.478290\pi$
$132$	0	0
$133$	0	0
$134$	−0.882034	−0.0761961
$135$	0	0
$136$	13.9315	1.19462
$137$	−15.3618	−1.31244	−0.656222	−	0.754568i	$-0.727846\pi$
$137$	−15.3618	−1.31244	−0.656222	+	0.754568i	$0.727846\pi$
$138$	0	0
$139$	9.30803	0.789496	0.394748	−	0.918789i	$-0.370832\pi$
$139$	9.30803	0.789496	0.394748	+	0.918789i	$0.370832\pi$
$140$	8.66046	0.731942
$141$	0	0
$142$	11.4123	0.957703
$143$	8.07460	0.675232
$144$	0	0
$145$	2.47620	0.205637
$146$	−3.37870	−0.279623
$147$	0	0
$148$	−6.23472	−0.512490
$149$	−5.18414	−0.424702	−0.212351	−	0.977194i	$-0.568112\pi$
$149$	−5.18414	−0.424702	−0.212351	+	0.977194i	$0.568112\pi$
$150$	0	0
$151$	−13.0922	−1.06543	−0.532713	−	0.846296i	$-0.678827\pi$
$151$	−13.0922	−1.06543	−0.532713	+	0.846296i	$0.678827\pi$
$152$	0	0
$153$	0	0
$154$	14.6505	1.18057
$155$	−4.89945	−0.393533
$156$	0	0
$157$	−7.36968	−0.588164	−0.294082	−	0.955780i	$-0.595014\pi$
$157$	−7.36968	−0.588164	−0.294082	+	0.955780i	$0.595014\pi$
$158$	14.2898	1.13683
$159$	0	0
$160$	9.81937	0.776289
$161$	−13.5923	−1.07122
$162$	0	0
$163$	−16.0281	−1.25542	−0.627708	−	0.778449i	$-0.716007\pi$
$163$	−16.0281	−1.25542	−0.627708	+	0.778449i	$0.716007\pi$
$164$	11.3668	0.887599
$165$	0	0
$166$	−10.9713	−0.851540
$167$	−13.1273	−1.01582	−0.507911	−	0.861409i	$-0.669582\pi$
$167$	−13.1273	−1.01582	−0.507911	+	0.861409i	$0.669582\pi$
$168$	0	0
$169$	−9.67075	−0.743904
$170$	7.22916	0.554451
$171$	0	0
$172$	−13.0363	−0.994007
$173$	−18.4720	−1.40440	−0.702201	−	0.711978i	$-0.747799\pi$
$173$	−18.4720	−1.40440	−0.702201	+	0.711978i	$0.747799\pi$
$174$	0	0
$175$	−8.19301	−0.619333
$176$	−1.09043	−0.0821944
$177$	0	0
$178$	−10.4142	−0.780578
$179$	11.8588	0.886370	0.443185	−	0.896430i	$-0.353849\pi$
$179$	11.8588	0.886370	0.443185	+	0.896430i	$0.353849\pi$
$180$	0	0
$181$	−0.273724	−0.0203457	−0.0101729	−	0.999948i	$-0.503238\pi$
$181$	−0.273724	−0.0203457	−0.0101729	+	0.999948i	$0.503238\pi$
$182$	6.04056	0.447756
$183$	0	0
$184$	−9.58520	−0.706630
$185$	−8.24916	−0.606490
$186$	0	0
$187$	−22.2438	−1.62663
$188$	4.43328	0.323330
$189$	0	0
$190$	0	0
$191$	12.1649	0.880218	0.440109	−	0.897944i	$-0.354940\pi$
$191$	12.1649	0.880218	0.440109	+	0.897944i	$0.354940\pi$
$192$	0	0
$193$	−23.0762	−1.66106	−0.830532	−	0.556970i	$-0.811964\pi$
$193$	−23.0762	−1.66106	−0.830532	+	0.556970i	$0.811964\pi$
$194$	0.147619	0.0105985
$195$	0	0
$196$	−10.9016	−0.778682
$197$	−6.46873	−0.460878	−0.230439	−	0.973087i	$-0.574016\pi$
$197$	−6.46873	−0.460878	−0.230439	+	0.973087i	$0.574016\pi$
$198$	0	0
$199$	0.307088	0.0217689	0.0108845	−	0.999941i	$-0.496535\pi$
$199$	0.307088	0.0217689	0.0108845	+	0.999941i	$0.496535\pi$
$200$	−5.77765	−0.408542
$201$	0	0
$202$	−9.44529	−0.664569
$203$	−5.69984	−0.400050
$204$	0	0
$205$	15.0394	1.05040
$206$	8.01893	0.558706
$207$	0	0
$208$	−0.449597	−0.0311740
$209$	0	0
$210$	0	0
$211$	−10.5856	−0.728741	−0.364371	−	0.931254i	$-0.618716\pi$
$211$	−10.5856	−0.728741	−0.364371	+	0.931254i	$0.618716\pi$
$212$	2.72598	0.187221
$213$	0	0
$214$	8.64975	0.591284
$215$	−17.2483	−1.17633
$216$	0	0
$217$	11.2778	0.765587
$218$	−1.71971	−0.116473
$219$	0	0
$220$	−9.75123	−0.657428
$221$	−9.17135	−0.616932
$222$	0	0
$223$	−7.43507	−0.497889	−0.248944	−	0.968518i	$-0.580084\pi$
$223$	−7.43507	−0.497889	−0.248944	+	0.968518i	$0.580084\pi$
$224$	−22.6027	−1.51021
$225$	0	0
$226$	11.7901	0.784267
$227$	3.54389	0.235216	0.117608	−	0.993060i	$-0.462477\pi$
$227$	3.54389	0.235216	0.117608	+	0.993060i	$0.462477\pi$
$228$	0	0
$229$	−11.3492	−0.749973	−0.374987	−	0.927030i	$-0.622353\pi$
$229$	−11.3492	−0.749973	−0.374987	+	0.927030i	$0.622353\pi$
$230$	−4.97383	−0.327965
$231$	0	0
$232$	−4.01949	−0.263892
$233$	5.92767	0.388335	0.194167	−	0.980968i	$-0.437800\pi$
$233$	5.92767	0.388335	0.194167	+	0.980968i	$0.437800\pi$
$234$	0	0
$235$	5.86568	0.382635
$236$	−8.26695	−0.538132
$237$	0	0
$238$	−16.6404	−1.07864
$239$	−15.1684	−0.981165	−0.490583	−	0.871395i	$-0.663216\pi$
$239$	−15.1684	−0.981165	−0.490583	+	0.871395i	$0.663216\pi$
$240$	0	0
$241$	−8.86254	−0.570887	−0.285443	−	0.958396i	$-0.592141\pi$
$241$	−8.86254	−0.570887	−0.285443	+	0.958396i	$0.592141\pi$
$242$	−7.23023	−0.464777
$243$	0	0
$244$	−0.0989760	−0.00633629
$245$	−14.4239	−0.921506
$246$	0	0
$247$	0	0
$248$	7.95303	0.505018
$249$	0	0
$250$	−10.1892	−0.644422
$251$	−11.7071	−0.738948	−0.369474	−	0.929241i	$-0.620462\pi$
$251$	−11.7071	−0.738948	−0.369474	+	0.929241i	$0.620462\pi$
$252$	0	0
$253$	15.3042	0.962169
$254$	1.65033	0.103551
$255$	0	0
$256$	−15.3033	−0.956457
$257$	−9.91597	−0.618541	−0.309271	−	0.950974i	$-0.600085\pi$
$257$	−9.91597	−0.618541	−0.309271	+	0.950974i	$0.600085\pi$
$258$	0	0
$259$	18.9883	1.17988
$260$	−4.02054	−0.249343
$261$	0	0
$262$	−1.31405	−0.0811823
$263$	6.73892	0.415540	0.207770	−	0.978178i	$-0.433379\pi$
$263$	6.73892	0.415540	0.207770	+	0.978178i	$0.433379\pi$
$264$	0	0
$265$	3.60674	0.221560
$266$	0	0
$267$	0	0
$268$	−1.35135	−0.0825471
$269$	0.718290	0.0437949	0.0218975	−	0.999760i	$-0.493029\pi$
$269$	0.718290	0.0437949	0.0218975	+	0.999760i	$0.493029\pi$
$270$	0	0
$271$	−28.8411	−1.75197	−0.875985	−	0.482339i	$-0.839787\pi$
$271$	−28.8411	−1.75197	−0.875985	+	0.482339i	$0.839787\pi$
$272$	1.23854	0.0750977
$273$	0	0
$274$	12.9395	0.781701
$275$	9.22490	0.556283
$276$	0	0
$277$	−3.83134	−0.230203	−0.115101	−	0.993354i	$-0.536719\pi$
$277$	−3.83134	−0.230203	−0.115101	+	0.993354i	$0.536719\pi$
$278$	−7.84030	−0.470230
$279$	0	0
$280$	−18.6003	−1.11158
$281$	−22.0921	−1.31791	−0.658953	−	0.752184i	$-0.729000\pi$
$281$	−22.0921	−1.31791	−0.658953	+	0.752184i	$0.729000\pi$
$282$	0	0
$283$	−25.1465	−1.49481	−0.747403	−	0.664371i	$-0.768699\pi$
$283$	−25.1465	−1.49481	−0.747403	+	0.664371i	$0.768699\pi$
$284$	17.4847	1.03753
$285$	0	0
$286$	−6.80136	−0.402173
$287$	−34.6185	−2.04347
$288$	0	0
$289$	8.26510	0.486182
$290$	−2.08574	−0.122479
$291$	0	0
$292$	−5.17647	−0.302930
$293$	−5.92712	−0.346266	−0.173133	−	0.984898i	$-0.555389\pi$
$293$	−5.92712	−0.346266	−0.173133	+	0.984898i	$0.555389\pi$
$294$	0	0
$295$	−10.9380	−0.636835
$296$	13.3904	0.778304
$297$	0	0
$298$	4.36669	0.252955
$299$	6.31010	0.364923
$300$	0	0
$301$	39.7031	2.28845
$302$	11.0277	0.634575
$303$	0	0
$304$	0	0
$305$	−0.130955	−0.00749848
$306$	0	0
$307$	17.5842	1.00359	0.501793	−	0.864988i	$-0.332674\pi$
$307$	17.5842	1.00359	0.501793	+	0.864988i	$0.332674\pi$
$308$	22.4459	1.27897
$309$	0	0
$310$	4.12688	0.234391
$311$	6.02866	0.341854	0.170927	−	0.985284i	$-0.445324\pi$
$311$	6.02866	0.341854	0.170927	+	0.985284i	$0.445324\pi$
$312$	0	0
$313$	16.6980	0.943829	0.471914	−	0.881644i	$-0.343563\pi$
$313$	16.6980	0.943829	0.471914	+	0.881644i	$0.343563\pi$
$314$	6.20760	0.350315
$315$	0	0
$316$	21.8932	1.23159
$317$	−2.19305	−0.123174	−0.0615870	−	0.998102i	$-0.519616\pi$
$317$	−2.19305	−0.123174	−0.0615870	+	0.998102i	$0.519616\pi$
$318$	0	0
$319$	6.41772	0.359323
$320$	−7.42955	−0.415325
$321$	0	0
$322$	11.4490	0.638029
$323$	0	0
$324$	0	0
$325$	3.80353	0.210982
$326$	13.5007	0.747735
$327$	0	0
$328$	−24.4128	−1.34797
$329$	−13.5019	−0.744385
$330$	0	0
$331$	−15.5408	−0.854199	−0.427099	−	0.904205i	$-0.640465\pi$
$331$	−15.5408	−0.854199	−0.427099	+	0.904205i	$0.640465\pi$
$332$	−16.8091	−0.922516
$333$	0	0
$334$	11.0574	0.605032
$335$	−1.78798	−0.0976877
$336$	0	0
$337$	−12.3767	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$337$	−12.3767	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$338$	8.14583	0.443075
$339$	0	0
$340$	11.0757	0.600665
$341$	−12.6982	−0.687647
$342$	0	0
$343$	5.68923	0.307189
$344$	27.9983	1.50957
$345$	0	0
$346$	15.5593	0.836473
$347$	22.2056	1.19206	0.596030	−	0.802962i	$-0.296744\pi$
$347$	22.2056	1.19206	0.596030	+	0.802962i	$0.296744\pi$
$348$	0	0
$349$	−19.9292	−1.06679	−0.533393	−	0.845867i	$-0.679083\pi$
$349$	−19.9292	−1.06679	−0.533393	+	0.845867i	$0.679083\pi$
$350$	6.90110	0.368879
$351$	0	0
$352$	25.4495	1.35646
$353$	5.20604	0.277089	0.138545	−	0.990356i	$-0.455758\pi$
$353$	5.20604	0.277089	0.138545	+	0.990356i	$0.455758\pi$
$354$	0	0
$355$	23.1341	1.22783
$356$	−15.9555	−0.845640
$357$	0	0
$358$	−9.98887	−0.527928
$359$	5.16826	0.272771	0.136385	−	0.990656i	$-0.456451\pi$
$359$	5.16826	0.272771	0.136385	+	0.990656i	$0.456451\pi$
$360$	0	0
$361$	0	0
$362$	0.230562	0.0121181
$363$	0	0
$364$	9.25468	0.485077
$365$	−6.84899	−0.358493
$366$	0	0
$367$	−6.06224	−0.316447	−0.158223	−	0.987403i	$-0.550577\pi$
$367$	−6.06224	−0.316447	−0.158223	+	0.987403i	$0.550577\pi$
$368$	−0.852146	−0.0444212
$369$	0	0
$370$	6.94840	0.361230
$371$	−8.30218	−0.431028
$372$	0	0
$373$	−9.98545	−0.517027	−0.258514	−	0.966008i	$-0.583233\pi$
$373$	−9.98545	−0.517027	−0.258514	+	0.966008i	$0.583233\pi$
$374$	18.7363	0.968830
$375$	0	0
$376$	−9.52147	−0.491032
$377$	2.64610	0.136281
$378$	0	0
$379$	13.7799	0.707826	0.353913	−	0.935278i	$-0.384851\pi$
$379$	13.7799	0.707826	0.353913	+	0.935278i	$0.384851\pi$
$380$	0	0
$381$	0	0
$382$	−10.2467	−0.524265
$383$	16.3124	0.833524	0.416762	−	0.909016i	$-0.363165\pi$
$383$	16.3124	0.833524	0.416762	+	0.909016i	$0.363165\pi$
$384$	0	0
$385$	29.6982	1.51356
$386$	19.4375	0.989343
$387$	0	0
$388$	0.226166	0.0114818
$389$	11.4713	0.581617	0.290808	−	0.956781i	$-0.406076\pi$
$389$	11.4713	0.581617	0.290808	+	0.956781i	$0.406076\pi$
$390$	0	0
$391$	−17.3830	−0.879095
$392$	23.4135	1.18256
$393$	0	0
$394$	5.44871	0.274502
$395$	28.9669	1.45748
$396$	0	0
$397$	−0.448109	−0.0224899	−0.0112450	−	0.999937i	$-0.503579\pi$
$397$	−0.448109	−0.0224899	−0.0112450	+	0.999937i	$0.503579\pi$
$398$	−0.258665	−0.0129657
$399$	0	0
$400$	−0.513647	−0.0256823
$401$	14.5064	0.724416	0.362208	−	0.932097i	$-0.382023\pi$
$401$	14.5064	0.724416	0.362208	+	0.932097i	$0.382023\pi$
$402$	0	0
$403$	−5.23562	−0.260805
$404$	−14.4710	−0.719961
$405$	0	0
$406$	4.80106	0.238273
$407$	−21.3799	−1.05976
$408$	0	0
$409$	13.1990	0.652647	0.326323	−	0.945258i	$-0.394190\pi$
$409$	13.1990	0.652647	0.326323	+	0.945258i	$0.394190\pi$
$410$	−12.6680	−0.625626
$411$	0	0
$412$	12.2857	0.605274
$413$	25.1777	1.23891
$414$	0	0
$415$	−22.2401	−1.09172
$416$	10.4931	0.514467
$417$	0	0
$418$	0	0
$419$	−11.4229	−0.558047	−0.279024	−	0.960284i	$-0.590011\pi$
$419$	−11.4229	−0.558047	−0.279024	+	0.960284i	$0.590011\pi$
$420$	0	0
$421$	16.5837	0.808240	0.404120	−	0.914706i	$-0.367578\pi$
$421$	16.5837	0.808240	0.404120	+	0.914706i	$0.367578\pi$
$422$	8.91641	0.434044
$423$	0	0
$424$	−5.85464	−0.284327
$425$	−10.4779	−0.508253
$426$	0	0
$427$	0.301440	0.0145877
$428$	13.2522	0.640568
$429$	0	0
$430$	14.5285	0.700629
$431$	28.4987	1.37274	0.686368	−	0.727255i	$-0.259204\pi$
$431$	28.4987	1.37274	0.686368	+	0.727255i	$0.259204\pi$
$432$	0	0
$433$	−13.9975	−0.672677	−0.336339	−	0.941741i	$-0.609189\pi$
$433$	−13.9975	−0.672677	−0.336339	+	0.941741i	$0.609189\pi$
$434$	−9.49947	−0.455989
$435$	0	0
$436$	−2.63475	−0.126182
$437$	0	0
$438$	0	0
$439$	26.8291	1.28048	0.640242	−	0.768174i	$-0.278834\pi$
$439$	26.8291	1.28048	0.640242	+	0.768174i	$0.278834\pi$
$440$	20.9429	0.998416
$441$	0	0
$442$	7.72518	0.367449
$443$	13.9183	0.661277	0.330638	−	0.943758i	$-0.392736\pi$
$443$	13.9183	0.661277	0.330638	+	0.943758i	$0.392736\pi$
$444$	0	0
$445$	−21.1107	−1.00075
$446$	6.26268	0.296546
$447$	0	0
$448$	17.1017	0.807981
$449$	21.2841	1.00446	0.502229	−	0.864735i	$-0.332513\pi$
$449$	21.2841	1.00446	0.502229	+	0.864735i	$0.332513\pi$
$450$	0	0
$451$	38.9787	1.83544
$452$	18.0635	0.849636
$453$	0	0
$454$	−2.98507	−0.140097
$455$	12.2449	0.574049
$456$	0	0
$457$	−0.745738	−0.0348842	−0.0174421	−	0.999848i	$-0.505552\pi$
$457$	−0.745738	−0.0348842	−0.0174421	+	0.999848i	$0.505552\pi$
$458$	9.55958	0.446690
$459$	0	0
$460$	−7.62035	−0.355301
$461$	−28.1942	−1.31314	−0.656568	−	0.754267i	$-0.727993\pi$
$461$	−28.1942	−1.31314	−0.656568	+	0.754267i	$0.727993\pi$
$462$	0	0
$463$	8.85705	0.411622	0.205811	−	0.978592i	$-0.434017\pi$
$463$	8.85705	0.411622	0.205811	+	0.978592i	$0.434017\pi$
$464$	−0.357342	−0.0165892
$465$	0	0
$466$	−4.99298	−0.231295
$467$	−37.2776	−1.72500	−0.862500	−	0.506057i	$-0.831102\pi$
$467$	−37.2776	−1.72500	−0.862500	+	0.506057i	$0.831102\pi$
$468$	0	0
$469$	4.11566	0.190044
$470$	−4.94076	−0.227900
$471$	0	0
$472$	17.7551	0.817245
$473$	−44.7036	−2.05547
$474$	0	0
$475$	0	0
$476$	−25.4946	−1.16855
$477$	0	0
$478$	12.7766	0.584389
$479$	29.8552	1.36412	0.682059	−	0.731297i	$-0.261085\pi$
$479$	29.8552	1.36412	0.682059	+	0.731297i	$0.261085\pi$
$480$	0	0
$481$	−8.81517	−0.401937
$482$	7.46506	0.340024
$483$	0	0
$484$	−11.0774	−0.503516
$485$	0.299241	0.0135878
$486$	0	0
$487$	−30.8808	−1.39934	−0.699671	−	0.714465i	$-0.746670\pi$
$487$	−30.8808	−1.39934	−0.699671	+	0.714465i	$0.746670\pi$
$488$	0.212573	0.00962273
$489$	0	0
$490$	12.1494	0.548856
$491$	−10.1276	−0.457051	−0.228525	−	0.973538i	$-0.573390\pi$
$491$	−10.1276	−0.457051	−0.228525	+	0.973538i	$0.573390\pi$
$492$	0	0
$493$	−7.28943	−0.328299
$494$	0	0
$495$	0	0
$496$	0.707042	0.0317471
$497$	−53.2512	−2.38864
$498$	0	0
$499$	8.59460	0.384747	0.192374	−	0.981322i	$-0.438381\pi$
$499$	8.59460	0.384747	0.192374	+	0.981322i	$0.438381\pi$
$500$	−15.6108	−0.698135
$501$	0	0
$502$	9.86111	0.440123
$503$	18.0671	0.805570	0.402785	−	0.915295i	$-0.368042\pi$
$503$	18.0671	0.805570	0.402785	+	0.915295i	$0.368042\pi$
$504$	0	0
$505$	−19.1466	−0.852014
$506$	−12.8910	−0.573075
$507$	0	0
$508$	2.52845	0.112182
$509$	31.4464	1.39384	0.696919	−	0.717150i	$-0.254554\pi$
$509$	31.4464	1.39384	0.696919	+	0.717150i	$0.254554\pi$
$510$	0	0
$511$	15.7654	0.697418
$512$	−2.78294	−0.122990
$513$	0	0
$514$	8.35239	0.368408
$515$	16.2553	0.716292
$516$	0	0
$517$	15.2025	0.668604
$518$	−15.9942	−0.702744
$519$	0	0
$520$	8.63501	0.378670
$521$	19.9501	0.874030	0.437015	−	0.899454i	$-0.356036\pi$
$521$	19.9501	0.874030	0.437015	+	0.899454i	$0.356036\pi$
$522$	0	0
$523$	−3.16211	−0.138269	−0.0691346	−	0.997607i	$-0.522024\pi$
$523$	−3.16211	−0.138269	−0.0691346	+	0.997607i	$0.522024\pi$
$524$	−2.01324	−0.0879488
$525$	0	0
$526$	−5.67630	−0.247499
$527$	14.4230	0.628275
$528$	0	0
$529$	−11.0401	−0.480005
$530$	−3.03802	−0.131963
$531$	0	0
$532$	0	0
$533$	16.0714	0.696128
$534$	0	0
$535$	17.5340	0.758060
$536$	2.90234	0.125362
$537$	0	0
$538$	−0.605028	−0.0260846
$539$	−37.3832	−1.61021
$540$	0	0
$541$	0.594406	0.0255555	0.0127778	−	0.999918i	$-0.495933\pi$
$541$	0.594406	0.0255555	0.0127778	+	0.999918i	$0.495933\pi$
$542$	24.2933	1.04349
$543$	0	0
$544$	−28.9063	−1.23935
$545$	−3.48604	−0.149325
$546$	0	0
$547$	−26.5738	−1.13622	−0.568108	−	0.822954i	$-0.692324\pi$
$547$	−26.5738	−1.13622	−0.568108	+	0.822954i	$0.692324\pi$
$548$	19.8244	0.846856
$549$	0	0
$550$	−7.77029	−0.331326
$551$	0	0
$552$	0	0
$553$	−66.6776	−2.83542
$554$	3.22720	0.137111
$555$	0	0
$556$	−12.0120	−0.509424
$557$	−34.1718	−1.44791	−0.723953	−	0.689849i	$-0.757677\pi$
$557$	−34.1718	−1.44791	−0.723953	+	0.689849i	$0.757677\pi$
$558$	0	0
$559$	−18.4318	−0.779582
$560$	−1.65361	−0.0698776
$561$	0	0
$562$	18.6086	0.784955
$563$	−30.0716	−1.26737	−0.633683	−	0.773593i	$-0.718457\pi$
$563$	−30.0716	−1.26737	−0.633683	+	0.773593i	$0.718457\pi$
$564$	0	0
$565$	23.8998	1.00547
$566$	21.1813	0.890317
$567$	0	0
$568$	−37.5524	−1.57566
$569$	−24.7511	−1.03762	−0.518809	−	0.854890i	$-0.673624\pi$
$569$	−24.7511	−1.03762	−0.518809	+	0.854890i	$0.673624\pi$
$570$	0	0
$571$	31.5869	1.32187	0.660934	−	0.750444i	$-0.270160\pi$
$571$	31.5869	1.32187	0.660934	+	0.750444i	$0.270160\pi$
$572$	−10.4203	−0.435694
$573$	0	0
$574$	29.1598	1.21711
$575$	7.20904	0.300638
$576$	0	0
$577$	−30.8109	−1.28267	−0.641337	−	0.767260i	$-0.721620\pi$
$577$	−30.8109	−1.28267	−0.641337	+	0.767260i	$0.721620\pi$
$578$	−6.96182	−0.289574
$579$	0	0
$580$	−3.19554	−0.132688
$581$	51.1933	2.12386
$582$	0	0
$583$	9.34783	0.387148
$584$	11.1176	0.460050
$585$	0	0
$586$	4.99251	0.206239
$587$	14.1246	0.582986	0.291493	−	0.956573i	$-0.405848\pi$
$587$	14.1246	0.582986	0.291493	+	0.956573i	$0.405848\pi$
$588$	0	0
$589$	0	0
$590$	9.21326	0.379304
$591$	0	0
$592$	1.19044	0.0489268
$593$	−37.7151	−1.54877	−0.774387	−	0.632712i	$-0.781942\pi$
$593$	−37.7151	−1.54877	−0.774387	+	0.632712i	$0.781942\pi$
$594$	0	0
$595$	−33.7320	−1.38288
$596$	6.69015	0.274039
$597$	0	0
$598$	−5.31510	−0.217351
$599$	0.0927993	0.00379167	0.00189584	−	0.999998i	$-0.499397\pi$
$599$	0.0927993	0.00379167	0.00189584	+	0.999998i	$0.499397\pi$
$600$	0	0
$601$	23.0380	0.939740	0.469870	−	0.882736i	$-0.344301\pi$
$601$	23.0380	0.939740	0.469870	+	0.882736i	$0.344301\pi$
$602$	−33.4425	−1.36302
$603$	0	0
$604$	16.8955	0.687468
$605$	−14.6565	−0.595870
$606$	0	0
$607$	−31.9363	−1.29625	−0.648127	−	0.761532i	$-0.724447\pi$
$607$	−31.9363	−1.29625	−0.648127	+	0.761532i	$0.724447\pi$
$608$	0	0
$609$	0	0
$610$	0.110306	0.00446615
$611$	6.26815	0.253582
$612$	0	0
$613$	43.7113	1.76548	0.882741	−	0.469859i	$-0.155695\pi$
$613$	43.7113	1.76548	0.882741	+	0.469859i	$0.155695\pi$
$614$	−14.8115	−0.597743
$615$	0	0
$616$	−48.2075	−1.94234
$617$	13.4123	0.539957	0.269978	−	0.962866i	$-0.412983\pi$
$617$	13.4123	0.539957	0.269978	+	0.962866i	$0.412983\pi$
$618$	0	0
$619$	22.9340	0.921797	0.460899	−	0.887453i	$-0.347527\pi$
$619$	22.9340	0.921797	0.460899	+	0.887453i	$0.347527\pi$
$620$	6.32275	0.253928
$621$	0	0
$622$	−5.07804	−0.203611
$623$	48.5938	1.94687
$624$	0	0
$625$	−10.2318	−0.409273
$626$	−14.0650	−0.562152
$627$	0	0
$628$	9.51059	0.379514
$629$	24.2839	0.968261
$630$	0	0
$631$	−21.3573	−0.850219	−0.425109	−	0.905142i	$-0.639764\pi$
$631$	−21.3573	−0.850219	−0.425109	+	0.905142i	$0.639764\pi$
$632$	−47.0206	−1.87038
$633$	0	0
$634$	1.84724	0.0733634
$635$	3.34540	0.132758
$636$	0	0
$637$	−15.4135	−0.610706
$638$	−5.40575	−0.214016
$639$	0	0
$640$	−13.3807	−0.528919
$641$	−29.5018	−1.16525	−0.582625	−	0.812741i	$-0.697974\pi$
$641$	−29.5018	−1.16525	−0.582625	+	0.812741i	$0.697974\pi$
$642$	0	0
$643$	6.83309	0.269471	0.134735	−	0.990882i	$-0.456982\pi$
$643$	6.83309	0.269471	0.134735	+	0.990882i	$0.456982\pi$
$644$	17.5409	0.691209
$645$	0	0
$646$	0	0
$647$	18.9452	0.744812	0.372406	−	0.928070i	$-0.378533\pi$
$647$	18.9452	0.744812	0.372406	+	0.928070i	$0.378533\pi$
$648$	0	0
$649$	−28.3487	−1.11279
$650$	−3.20378	−0.125662
$651$	0	0
$652$	20.6843	0.810060
$653$	−27.1399	−1.06207	−0.531033	−	0.847351i	$-0.678196\pi$
$653$	−27.1399	−1.06207	−0.531033	+	0.847351i	$0.678196\pi$
$654$	0	0
$655$	−2.66372	−0.104080
$656$	−2.17035	−0.0847380
$657$	0	0
$658$	11.3729	0.443361
$659$	−39.9075	−1.55457	−0.777287	−	0.629146i	$-0.783405\pi$
$659$	−39.9075	−1.55457	−0.777287	+	0.629146i	$0.783405\pi$
$660$	0	0
$661$	50.3466	1.95826	0.979129	−	0.203241i	$-0.0651476\pi$
$661$	50.3466	1.95826	0.979129	+	0.203241i	$0.0651476\pi$
$662$	13.0903	0.508767
$663$	0	0
$664$	36.1012	1.40100
$665$	0	0
$666$	0	0
$667$	5.01530	0.194193
$668$	16.9409	0.655461
$669$	0	0
$670$	1.50604	0.0581835
$671$	−0.339405	−0.0131026
$672$	0	0
$673$	47.8104	1.84295	0.921477	−	0.388432i	$-0.126983\pi$
$673$	47.8104	1.84295	0.921477	+	0.388432i	$0.126983\pi$
$674$	10.4251	0.401559
$675$	0	0
$676$	12.4801	0.480005
$677$	41.2610	1.58579	0.792894	−	0.609359i	$-0.208573\pi$
$677$	41.2610	1.58579	0.792894	+	0.609359i	$0.208573\pi$
$678$	0	0
$679$	−0.688807	−0.0264340
$680$	−23.7876	−0.912212
$681$	0	0
$682$	10.6959	0.409568
$683$	−17.8805	−0.684178	−0.342089	−	0.939668i	$-0.611134\pi$
$683$	−17.8805	−0.684178	−0.342089	+	0.939668i	$0.611134\pi$
$684$	0	0
$685$	26.2297	1.00218
$686$	−4.79213	−0.182964
$687$	0	0
$688$	2.48912	0.0948967
$689$	3.85421	0.146834
$690$	0	0
$691$	−11.3331	−0.431132	−0.215566	−	0.976489i	$-0.569160\pi$
$691$	−11.3331	−0.431132	−0.215566	+	0.976489i	$0.569160\pi$
$692$	23.8382	0.906193
$693$	0	0
$694$	−18.7042	−0.710000
$695$	−15.8931	−0.602861
$696$	0	0
$697$	−44.2731	−1.67696
$698$	16.7867	0.635386
$699$	0	0
$700$	10.5731	0.399626
$701$	−6.89362	−0.260368	−0.130184	−	0.991490i	$-0.541557\pi$
$701$	−6.89362	−0.260368	−0.130184	+	0.991490i	$0.541557\pi$
$702$	0	0
$703$	0	0
$704$	−19.2557	−0.725725
$705$	0	0
$706$	−4.38513	−0.165037
$707$	44.0727	1.65752
$708$	0	0
$709$	−50.6485	−1.90214	−0.951072	−	0.308971i	$-0.900015\pi$
$709$	−50.6485	−1.90214	−0.951072	+	0.308971i	$0.900015\pi$
$710$	−19.4862	−0.731304
$711$	0	0
$712$	34.2680	1.28425
$713$	−9.92335	−0.371632
$714$	0	0
$715$	−13.7871	−0.515609
$716$	−15.3038	−0.571931
$717$	0	0
$718$	−4.35331	−0.162464
$719$	−19.6429	−0.732556	−0.366278	−	0.930505i	$-0.619368\pi$
$719$	−19.6429	−0.732556	−0.366278	+	0.930505i	$0.619368\pi$
$720$	0	0
$721$	−37.4172	−1.39349
$722$	0	0
$723$	0	0
$724$	0.353242	0.0131281
$725$	3.02306	0.112274
$726$	0	0
$727$	−17.8693	−0.662734	−0.331367	−	0.943502i	$-0.607510\pi$
$727$	−17.8693	−0.662734	−0.331367	+	0.943502i	$0.607510\pi$
$728$	−19.8765	−0.736672
$729$	0	0
$730$	5.76902	0.213521
$731$	50.7756	1.87800
$732$	0	0
$733$	39.1472	1.44594	0.722968	−	0.690882i	$-0.242777\pi$
$733$	39.1472	1.44594	0.722968	+	0.690882i	$0.242777\pi$
$734$	5.10633	0.188478
$735$	0	0
$736$	19.8882	0.733088
$737$	−4.63402	−0.170696
$738$	0	0
$739$	30.9676	1.13916	0.569581	−	0.821935i	$-0.307105\pi$
$739$	30.9676	1.13916	0.569581	+	0.821935i	$0.307105\pi$
$740$	10.6456	0.391339
$741$	0	0
$742$	6.99306	0.256723
$743$	−12.2426	−0.449136	−0.224568	−	0.974458i	$-0.572097\pi$
$743$	−12.2426	−0.449136	−0.224568	+	0.974458i	$0.572097\pi$
$744$	0	0
$745$	8.85175	0.324303
$746$	8.41091	0.307945
$747$	0	0
$748$	28.7057	1.04958
$749$	−40.3606	−1.47474
$750$	0	0
$751$	44.9133	1.63891	0.819455	−	0.573143i	$-0.194276\pi$
$751$	44.9133	1.63891	0.819455	+	0.573143i	$0.194276\pi$
$752$	−0.846480	−0.0308680
$753$	0	0
$754$	−2.22885	−0.0811700
$755$	22.3544	0.813561
$756$	0	0
$757$	−15.2301	−0.553549	−0.276775	−	0.960935i	$-0.589266\pi$
$757$	−15.2301	−0.553549	−0.276775	+	0.960935i	$0.589266\pi$
$758$	−11.6070	−0.421587
$759$	0	0
$760$	0	0
$761$	32.2300	1.16834	0.584169	−	0.811632i	$-0.301421\pi$
$761$	32.2300	1.16834	0.584169	+	0.811632i	$0.301421\pi$
$762$	0	0
$763$	8.02434	0.290501
$764$	−15.6988	−0.567962
$765$	0	0
$766$	−13.7402	−0.496453
$767$	−11.6885	−0.422047
$768$	0	0
$769$	10.7275	0.386843	0.193422	−	0.981116i	$-0.438041\pi$
$769$	10.7275	0.386843	0.193422	+	0.981116i	$0.438041\pi$
$770$	−25.0152	−0.901487
$771$	0	0
$772$	29.7800	1.07180
$773$	0.826686	0.0297338	0.0148669	−	0.999889i	$-0.495268\pi$
$773$	0.826686	0.0297338	0.0148669	+	0.999889i	$0.495268\pi$
$774$	0	0
$775$	−5.98148	−0.214861
$776$	−0.485742	−0.0174371
$777$	0	0
$778$	−9.66244	−0.346415
$779$	0	0
$780$	0	0
$781$	59.9581	2.14547
$782$	14.6420	0.523595
$783$	0	0
$784$	2.08152	0.0743398
$785$	12.5835	0.449124
$786$	0	0
$787$	13.0266	0.464348	0.232174	−	0.972674i	$-0.425416\pi$
$787$	13.0266	0.464348	0.232174	+	0.972674i	$0.425416\pi$
$788$	8.34791	0.297382
$789$	0	0
$790$	−24.3993	−0.868089
$791$	−55.0139	−1.95607
$792$	0	0
$793$	−0.139941	−0.00496944
$794$	0.377449	0.0133952
$795$	0	0
$796$	−0.396298	−0.0140464
$797$	−47.0808	−1.66769	−0.833844	−	0.552000i	$-0.813865\pi$
$797$	−47.0808	−1.66769	−0.833844	+	0.552000i	$0.813865\pi$
$798$	0	0
$799$	−17.2674	−0.610877
$800$	11.9880	0.423838
$801$	0	0
$802$	−12.2190	−0.431468
$803$	−17.7510	−0.626418
$804$	0	0
$805$	23.2084	0.817988
$806$	4.41004	0.155337
$807$	0	0
$808$	31.0798	1.09338
$809$	10.0150	0.352107	0.176053	−	0.984381i	$-0.443667\pi$
$809$	10.0150	0.352107	0.176053	+	0.984381i	$0.443667\pi$
$810$	0	0
$811$	31.5550	1.10804	0.554022	−	0.832502i	$-0.313092\pi$
$811$	31.5550	1.10804	0.554022	+	0.832502i	$0.313092\pi$
$812$	7.35566	0.258133
$813$	0	0
$814$	18.0086	0.631202
$815$	27.3674	0.958639
$816$	0	0
$817$	0	0
$818$	−11.1177	−0.388721
$819$	0	0
$820$	−19.4085	−0.677772
$821$	38.3465	1.33830	0.669151	−	0.743126i	$-0.266658\pi$
$821$	38.3465	1.33830	0.669151	+	0.743126i	$0.266658\pi$
$822$	0	0
$823$	49.1374	1.71282	0.856410	−	0.516296i	$-0.172689\pi$
$823$	49.1374	1.71282	0.856410	+	0.516296i	$0.172689\pi$
$824$	−26.3863	−0.919211
$825$	0	0
$826$	−21.2075	−0.737905
$827$	16.0238	0.557201	0.278600	−	0.960407i	$-0.410130\pi$
$827$	16.0238	0.557201	0.278600	+	0.960407i	$0.410130\pi$
$828$	0	0
$829$	−34.4952	−1.19807	−0.599033	−	0.800724i	$-0.704448\pi$
$829$	−34.4952	−1.19807	−0.599033	+	0.800724i	$0.704448\pi$
$830$	18.7332	0.650238
$831$	0	0
$832$	−7.93932	−0.275247
$833$	42.4609	1.47118
$834$	0	0
$835$	22.4145	0.775685
$836$	0	0
$837$	0	0
$838$	9.62173	0.332377
$839$	−17.0321	−0.588013	−0.294007	−	0.955803i	$-0.594989\pi$
$839$	−17.0321	−0.588013	−0.294007	+	0.955803i	$0.594989\pi$
$840$	0	0
$841$	−26.8969	−0.927478
$842$	−13.9687	−0.481394
$843$	0	0
$844$	13.6607	0.470222
$845$	16.5125	0.568047
$846$	0	0
$847$	33.7370	1.15922
$848$	−0.520491	−0.0178737
$849$	0	0
$850$	8.82571	0.302719
$851$	−16.7079	−0.572738
$852$	0	0
$853$	34.2140	1.17146	0.585732	−	0.810505i	$-0.300807\pi$
$853$	34.2140	1.17146	0.585732	+	0.810505i	$0.300807\pi$
$854$	−0.253907	−0.00868853
$855$	0	0
$856$	−28.4620	−0.972812
$857$	8.26690	0.282392	0.141196	−	0.989982i	$-0.454905\pi$
$857$	8.26690	0.282392	0.141196	+	0.989982i	$0.454905\pi$
$858$	0	0
$859$	−50.4943	−1.72284	−0.861421	−	0.507891i	$-0.830425\pi$
$859$	−50.4943	−1.72284	−0.861421	+	0.507891i	$0.830425\pi$
$860$	22.2590	0.759026
$861$	0	0
$862$	−24.0049	−0.817611
$863$	−33.0070	−1.12357	−0.561785	−	0.827283i	$-0.689885\pi$
$863$	−33.0070	−1.12357	−0.561785	+	0.827283i	$0.689885\pi$
$864$	0	0
$865$	31.5404	1.07241
$866$	11.7903	0.400652
$867$	0	0
$868$	−14.5540	−0.493996
$869$	75.0755	2.54676
$870$	0	0
$871$	−1.91066	−0.0647402
$872$	5.65871	0.191628
$873$	0	0
$874$	0	0
$875$	47.5438	1.60728
$876$	0	0
$877$	−2.11604	−0.0714535	−0.0357267	−	0.999362i	$-0.511375\pi$
$877$	−2.11604	−0.0714535	−0.0357267	+	0.999362i	$0.511375\pi$
$878$	−22.5986	−0.762666
$879$	0	0
$880$	1.86188	0.0627638
$881$	53.0406	1.78698	0.893492	−	0.449080i	$-0.148248\pi$
$881$	53.0406	1.78698	0.893492	+	0.449080i	$0.148248\pi$
$882$	0	0
$883$	17.1836	0.578275	0.289137	−	0.957288i	$-0.406632\pi$
$883$	17.1836	0.578275	0.289137	+	0.957288i	$0.406632\pi$
$884$	11.8357	0.398076
$885$	0	0
$886$	−11.7236	−0.393861
$887$	−3.89561	−0.130802	−0.0654008	−	0.997859i	$-0.520833\pi$
$887$	−3.89561	−0.130802	−0.0654008	+	0.997859i	$0.520833\pi$
$888$	0	0
$889$	−7.70060	−0.258270
$890$	17.7819	0.596051
$891$	0	0
$892$	9.59498	0.321264
$893$	0	0
$894$	0	0
$895$	−20.2485	−0.676834
$896$	30.8004	1.02897
$897$	0	0
$898$	−17.9279	−0.598263
$899$	−4.16129	−0.138787
$900$	0	0
$901$	−10.6175	−0.353721
$902$	−32.8324	−1.09320
$903$	0	0
$904$	−38.7954	−1.29032
$905$	0.467375	0.0155361
$906$	0	0
$907$	8.84164	0.293582	0.146791	−	0.989168i	$-0.453106\pi$
$907$	8.84164	0.293582	0.146791	+	0.989168i	$0.453106\pi$
$908$	−4.57340	−0.151774
$909$	0	0
$910$	−10.3141	−0.341908
$911$	18.4416	0.610997	0.305498	−	0.952193i	$-0.401177\pi$
$911$	18.4416	0.610997	0.305498	+	0.952193i	$0.401177\pi$
$912$	0	0
$913$	−57.6411	−1.90764
$914$	0.628147	0.0207773
$915$	0	0
$916$	14.6461	0.483922
$917$	6.13149	0.202480
$918$	0	0
$919$	−45.5362	−1.50210	−0.751051	−	0.660245i	$-0.770453\pi$
$919$	−45.5362	−1.50210	−0.751051	+	0.660245i	$0.770453\pi$
$920$	16.3664	0.539584
$921$	0	0
$922$	23.7485	0.782114
$923$	24.7214	0.813714
$924$	0	0
$925$	−10.0710	−0.331132
$926$	−7.46044	−0.245165
$927$	0	0
$928$	8.33997	0.273773
$929$	1.51634	0.0497494	0.0248747	−	0.999691i	$-0.492081\pi$
$929$	1.51634	0.0497494	0.0248747	+	0.999691i	$0.492081\pi$
$930$	0	0
$931$	0	0
$932$	−7.64968	−0.250574
$933$	0	0
$934$	31.3995	1.02742
$935$	37.9805	1.24209
$936$	0	0
$937$	7.13697	0.233155	0.116577	−	0.993182i	$-0.462808\pi$
$937$	7.13697	0.233155	0.116577	+	0.993182i	$0.462808\pi$
$938$	−3.46669	−0.113191
$939$	0	0
$940$	−7.56968	−0.246896
$941$	49.7958	1.62330	0.811648	−	0.584147i	$-0.198571\pi$
$941$	49.7958	1.62330	0.811648	+	0.584147i	$0.198571\pi$
$942$	0	0
$943$	30.4609	0.991944
$944$	1.57847	0.0513748
$945$	0	0
$946$	37.6546	1.22426
$947$	0.349552	0.0113589	0.00567945	−	0.999984i	$-0.498192\pi$
$947$	0.349552	0.0113589	0.00567945	+	0.999984i	$0.498192\pi$
$948$	0	0
$949$	−7.31892	−0.237582
$950$	0	0
$951$	0	0
$952$	54.7554	1.77463
$953$	21.0086	0.680534	0.340267	−	0.940329i	$-0.389483\pi$
$953$	21.0086	0.680534	0.340267	+	0.940329i	$0.389483\pi$
$954$	0	0
$955$	−20.7711	−0.672137
$956$	19.5749	0.633099
$957$	0	0
$958$	−25.1475	−0.812479
$959$	−60.3768	−1.94967
$960$	0	0
$961$	−22.7664	−0.734400
$962$	7.42516	0.239397
$963$	0	0
$964$	11.4371	0.368366
$965$	39.4019	1.26839
$966$	0	0
$967$	−36.8410	−1.18473	−0.592364	−	0.805670i	$-0.701805\pi$
$967$	−36.8410	−1.18473	−0.592364	+	0.805670i	$0.701805\pi$
$968$	23.7911	0.764675
$969$	0	0
$970$	−0.252055	−0.00809300
$971$	5.39119	0.173012	0.0865058	−	0.996251i	$-0.472430\pi$
$971$	5.39119	0.173012	0.0865058	+	0.996251i	$0.472430\pi$
$972$	0	0
$973$	36.5836	1.17282
$974$	26.0114	0.833459
$975$	0	0
$976$	0.0188982	0.000604918 0
$977$	−40.7182	−1.30269	−0.651345	−	0.758782i	$-0.725795\pi$
$977$	−40.7182	−1.30269	−0.651345	+	0.758782i	$0.725795\pi$
$978$	0	0
$979$	−54.7141	−1.74867
$980$	18.6140	0.594603
$981$	0	0
$982$	8.53061	0.272223
$983$	43.9907	1.40309	0.701543	−	0.712627i	$-0.252495\pi$
$983$	43.9907	1.40309	0.701543	+	0.712627i	$0.252495\pi$
$984$	0	0
$985$	11.0451	0.351927
$986$	6.14000	0.195538
$987$	0	0
$988$	0	0
$989$	−34.9348	−1.11086
$990$	0	0
$991$	−33.2649	−1.05669	−0.528347	−	0.849029i	$-0.677188\pi$
$991$	−33.2649	−1.05669	−0.528347	+	0.849029i	$0.677188\pi$
$992$	−16.5016	−0.523926
$993$	0	0
$994$	44.8543	1.42269
$995$	−0.524343	−0.0166228
$996$	0	0
$997$	27.0800	0.857633	0.428817	−	0.903392i	$-0.358931\pi$
$997$	27.0800	0.857633	0.428817	+	0.903392i	$0.358931\pi$
$998$	−7.23937	−0.229158
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bg.1.3		6
3.2	odd	2		1083.2.a.p.1.4		6
19.2	odd	18		171.2.u.e.118.1		12
19.10	odd	18		171.2.u.e.100.1		12
19.18	odd	2		3249.2.a.bh.1.4		6
57.2	even	18		57.2.i.b.4.2	✓	12
57.29	even	18		57.2.i.b.43.2	yes	12
57.56	even	2		1083.2.a.q.1.3		6
228.59	odd	18		912.2.bo.j.289.1		12
228.143	odd	18		912.2.bo.j.385.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.i.b.4.2	✓	12	57.2	even	18
57.2.i.b.43.2	yes	12	57.29	even	18
171.2.u.e.100.1		12	19.10	odd	18
171.2.u.e.118.1		12	19.2	odd	18
912.2.bo.j.289.1		12	228.59	odd	18
912.2.bo.j.385.1		12	228.143	odd	18
1083.2.a.p.1.4		6	3.2	odd	2
1083.2.a.q.1.3		6	57.56	even	2
3249.2.a.bg.1.3		6	1.1	even	1	trivial
3249.2.a.bh.1.4		6	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-0.842316 q^{2} -1.29050 q^{4} -1.70747 q^{5} +3.93033 q^{7} +2.77164 q^{8} +1.43823 q^{10} -4.42535 q^{11} -1.82462 q^{13} -3.31058 q^{14} +0.246406 q^{16} +5.02644 q^{17} +2.20349 q^{20} +3.72755 q^{22} -3.45831 q^{23} -2.08456 q^{25} +1.53691 q^{26} -5.07211 q^{28} -1.45022 q^{29} +2.86943 q^{31} -5.75084 q^{32} -4.23385 q^{34} -6.71091 q^{35} +4.83123 q^{37} -4.73249 q^{40} -8.80804 q^{41} +10.1017 q^{43} +5.71093 q^{44} +2.91299 q^{46} -3.43531 q^{47} +8.44752 q^{49} +1.75586 q^{50} +2.35468 q^{52} -2.11234 q^{53} +7.55614 q^{55} +10.8935 q^{56} +1.22154 q^{58} +6.40598 q^{59} +0.0766957 q^{61} -2.41696 q^{62} +4.35121 q^{64} +3.11548 q^{65} +1.04715 q^{67} -6.48664 q^{68} +5.65271 q^{70} -13.5488 q^{71} +4.01120 q^{73} -4.06942 q^{74} -17.3931 q^{77} -16.9649 q^{79} -0.420729 q^{80} +7.41916 q^{82} +13.0252 q^{83} -8.58248 q^{85} -8.50883 q^{86} -12.2655 q^{88} +12.3638 q^{89} -7.17137 q^{91} +4.46296 q^{92} +2.89362 q^{94} -0.175254 q^{97} -7.11548 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} - 9 q^{5} + 9 q^{7} - 3 q^{8} - 6 q^{10} - 9 q^{11} - 3 q^{13} - 6 q^{14} + 6 q^{16} - 15 q^{17} + 9 q^{20} + 21 q^{22} - 6 q^{23} + 15 q^{25} - 3 q^{28} - 15 q^{29} - 21 q^{32} - 33 q^{34}+ \cdots - 3 q^{98}+O(q^{100}) \) 6 * q + 6 * q^4 - 9 * q^5 + 9 * q^7 - 3 * q^8 - 6 * q^10 - 9 * q^11 - 3 * q^13 - 6 * q^14 + 6 * q^16 - 15 * q^17 + 9 * q^20 + 21 * q^22 - 6 * q^23 + 15 * q^25 - 3 * q^28 - 15 * q^29 - 21 * q^32 - 33 * q^34 + 3 * q^35 + 6 * q^37 - 39 * q^40 + 6 * q^41 + 15 * q^43 + 6 * q^44 + 27 * q^46 - 9 * q^47 + 9 * q^49 - 9 * q^50 - 18 * q^52 - 6 * q^53 - 6 * q^55 - 39 * q^56 - 12 * q^58 - 15 * q^59 + 3 * q^61 - 27 * q^62 + 21 * q^64 - 21 * q^65 + 18 * q^67 - 21 * q^68 - 9 * q^70 - 15 * q^71 - 6 * q^73 - 9 * q^74 - 48 * q^77 - 21 * q^79 + 21 * q^80 - 15 * q^82 + 3 * q^83 + 42 * q^85 - 30 * q^86 + 42 * q^88 - 24 * q^89 - 12 * q^91 - 42 * q^92 - 9 * q^94 + 3 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

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Database

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