LMFDB - Embedded newform 2925.2.a.bo.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2925,2,Mod(1,2925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,4,0,0,10,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.12730624.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 13x^{2} - 4 $ x^6 - 8x^4 + 13x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.43255$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43255 q^{2} +3.91729 q^{4} +4.51056 q^{7} +4.66389 q^{8} +5.03230 q^{11} -1.00000 q^{13} +10.9721 q^{14} +3.51056 q^{16} -2.06414 q^{17} -2.81346 q^{19} +12.2413 q^{22} +4.04291 q^{23} -2.43255 q^{26} +17.6691 q^{28} -7.83457 q^{31} -0.788180 q^{32} -5.02112 q^{34} -6.34513 q^{37} -6.84387 q^{38} +1.47716 q^{41} +6.81346 q^{43} +19.7129 q^{44} +9.83457 q^{46} -9.07521 q^{47} +13.3451 q^{49} -3.91729 q^{52} -9.81556 q^{53} +21.0367 q^{56} -13.2035 q^{59} +8.34513 q^{61} -19.0580 q^{62} -8.93840 q^{64} +15.8346 q^{67} -8.08582 q^{68} +0.569614 q^{71} +0.373086 q^{73} -15.4348 q^{74} -11.0211 q^{76} +22.6985 q^{77} +6.13747 q^{79} +3.59327 q^{82} +0.654981 q^{83} +16.5741 q^{86} +23.4701 q^{88} -5.93985 q^{89} -4.51056 q^{91} +15.8372 q^{92} -22.0759 q^{94} +2.67599 q^{97} +32.4627 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 10 q^{7} - 6 q^{13} + 4 q^{16} - 12 q^{19} + 32 q^{22} + 28 q^{28} - 8 q^{31} + 4 q^{34} + 18 q^{37} + 36 q^{43} + 20 q^{46} + 24 q^{49} - 4 q^{52} - 6 q^{61} + 56 q^{67} + 12 q^{73} - 32 q^{76}+ \cdots + 38 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 + 10 * q^7 - 6 * q^13 + 4 * q^16 - 12 * q^19 + 32 * q^22 + 28 * q^28 - 8 * q^31 + 4 * q^34 + 18 * q^37 + 36 * q^43 + 20 * q^46 + 24 * q^49 - 4 * q^52 - 6 * q^61 + 56 * q^67 + 12 * q^73 - 32 * q^76 + 10 * q^79 + 24 * q^82 + 36 * q^88 - 10 * q^91 - 52 * q^94 + 38 * q^97

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 $

$2$	2.43255	1.72007	0.860035	−	0.510235i	$-0.170441\pi$
$2$	2.43255	1.72007	0.860035	+	0.510235i	$0.170441\pi$
$3$	0	0
$3$	0	0
$4$	3.91729	1.95864
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.51056	1.70483	0.852415	−	0.522865i	$-0.175137\pi$
$7$	4.51056	1.70483	0.852415	+	0.522865i	$0.175137\pi$
$8$	4.66389	1.64893
$9$	0	0
$10$	0	0
$11$	5.03230	1.51729	0.758647	−	0.651502i	$-0.225861\pi$
$11$	5.03230	1.51729	0.758647	+	0.651502i	$0.225861\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	10.9721	2.93243
$15$	0	0
$16$	3.51056	0.877639
$17$	−2.06414	−0.500627	−0.250314	−	0.968165i	$-0.580534\pi$
$17$	−2.06414	−0.500627	−0.250314	+	0.968165i	$0.580534\pi$
$18$	0	0
$19$	−2.81346	−0.645451	−0.322726	−	0.946493i	$-0.604599\pi$
$19$	−2.81346	−0.645451	−0.322726	+	0.946493i	$0.604599\pi$
$20$	0	0
$21$	0	0
$22$	12.2413	2.60985
$23$	4.04291	0.843005	0.421503	−	0.906827i	$-0.361503\pi$
$23$	4.04291	0.843005	0.421503	+	0.906827i	$0.361503\pi$
$24$	0	0
$25$	0	0
$26$	−2.43255	−0.477062
$27$	0	0
$28$	17.6691	3.33915
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−7.83457	−1.40713	−0.703565	−	0.710631i	$-0.748410\pi$
$31$	−7.83457	−1.40713	−0.703565	+	0.710631i	$0.748410\pi$
$32$	−0.788180	−0.139332
$33$	0	0
$34$	−5.02112	−0.861114
$35$	0	0
$36$	0	0
$37$	−6.34513	−1.04313	−0.521566	−	0.853211i	$-0.674652\pi$
$37$	−6.34513	−1.04313	−0.521566	+	0.853211i	$0.674652\pi$
$38$	−6.84387	−1.11022
$39$	0	0
$40$	0	0
$41$	1.47716	0.230694	0.115347	−	0.993325i	$-0.463202\pi$
$41$	1.47716	0.230694	0.115347	+	0.993325i	$0.463202\pi$
$42$	0	0
$43$	6.81346	1.03904	0.519521	−	0.854458i	$-0.326110\pi$
$43$	6.81346	1.03904	0.519521	+	0.854458i	$0.326110\pi$
$44$	19.7129	2.97184
$45$	0	0
$46$	9.83457	1.45003
$47$	−9.07521	−1.32376	−0.661878	−	0.749612i	$-0.730240\pi$
$47$	−9.07521	−1.32376	−0.661878	+	0.749612i	$0.730240\pi$
$48$	0	0
$49$	13.3451	1.90645
$50$	0	0
$51$	0	0
$52$	−3.91729	−0.543230
$53$	−9.81556	−1.34827	−0.674135	−	0.738608i	$-0.735483\pi$
$53$	−9.81556	−1.34827	−0.674135	+	0.738608i	$0.735483\pi$
$54$	0	0
$55$	0	0
$56$	21.0367	2.81115
$57$	0	0
$58$	0	0
$59$	−13.2035	−1.71895	−0.859474	−	0.511180i	$-0.829209\pi$
$59$	−13.2035	−1.71895	−0.859474	+	0.511180i	$0.829209\pi$
$60$	0	0
$61$	8.34513	1.06848	0.534242	−	0.845331i	$-0.320597\pi$
$61$	8.34513	1.06848	0.534242	+	0.845331i	$0.320597\pi$
$62$	−19.0580	−2.42036
$63$	0	0
$64$	−8.93840	−1.11730
$65$	0	0
$66$	0	0
$67$	15.8346	1.93450	0.967250	−	0.253824i	$-0.0816884\pi$
$67$	15.8346	1.93450	0.967250	+	0.253824i	$0.0816884\pi$
$68$	−8.08582	−0.980550
$69$	0	0
$70$	0	0
$71$	0.569614	0.0676008	0.0338004	−	0.999429i	$-0.489239\pi$
$71$	0.569614	0.0676008	0.0338004	+	0.999429i	$0.489239\pi$
$72$	0	0
$73$	0.373086	0.0436664	0.0218332	−	0.999762i	$-0.493050\pi$
$73$	0.373086	0.0436664	0.0218332	+	0.999762i	$0.493050\pi$
$74$	−15.4348	−1.79426
$75$	0	0
$76$	−11.0211	−1.26421
$77$	22.6985	2.58673
$78$	0	0
$79$	6.13747	0.690519	0.345260	−	0.938507i	$-0.387791\pi$
$79$	6.13747	0.690519	0.345260	+	0.938507i	$0.387791\pi$
$80$	0	0
$81$	0	0
$82$	3.59327	0.396810
$83$	0.654981	0.0718935	0.0359467	−	0.999354i	$-0.488555\pi$
$83$	0.654981	0.0718935	0.0359467	+	0.999354i	$0.488555\pi$
$84$	0	0
$85$	0	0
$86$	16.5741	1.78723
$87$	0	0
$88$	23.4701	2.50192
$89$	−5.93985	−0.629623	−0.314811	−	0.949154i	$-0.601941\pi$
$89$	−5.93985	−0.629623	−0.314811	+	0.949154i	$0.601941\pi$
$90$	0	0
$91$	−4.51056	−0.472835
$92$	15.8372	1.65115
$93$	0	0
$94$	−22.0759	−2.27695
$95$	0	0
$96$	0	0
$97$	2.67599	0.271705	0.135853	−	0.990729i	$-0.456623\pi$
$97$	2.67599	0.271705	0.135853	+	0.990729i	$0.456623\pi$
$98$	32.4627	3.27922
$99$	0	0
$100$	0	0
$101$	11.8797	1.18207	0.591037	−	0.806645i	$-0.298719\pi$
$101$	11.8797	1.18207	0.591037	+	0.806645i	$0.298719\pi$
$102$	0	0
$103$	−2.20766	−0.217527	−0.108764	−	0.994068i	$-0.534689\pi$
$103$	−2.20766	−0.217527	−0.108764	+	0.994068i	$0.534689\pi$
$104$	−4.66389	−0.457332
$105$	0	0
$106$	−23.8768	−2.31912
$107$	−10.1500	−0.981234	−0.490617	−	0.871375i	$-0.663229\pi$
$107$	−10.1500	−0.981234	−0.490617	+	0.871375i	$0.663229\pi$
$108$	0	0
$109$	0.373086	0.0357352	0.0178676	−	0.999840i	$-0.494312\pi$
$109$	0.373086	0.0357352	0.0178676	+	0.999840i	$0.494312\pi$
$110$	0	0
$111$	0	0
$112$	15.8346	1.49623
$113$	6.91187	0.650214	0.325107	−	0.945677i	$-0.394600\pi$
$113$	6.91187	0.650214	0.325107	+	0.945677i	$0.394600\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−32.1181	−2.95671
$119$	−9.31042	−0.853485
$120$	0	0
$121$	14.3240	1.30218
$122$	20.2999	1.83787
$123$	0	0
$124$	−30.6903	−2.75607
$125$	0	0
$126$	0	0
$127$	−3.66914	−0.325584	−0.162792	−	0.986660i	$-0.552050\pi$
$127$	−3.66914	−0.325584	−0.162792	+	0.986660i	$0.552050\pi$
$128$	−20.1667	−1.78250
$129$	0	0
$130$	0	0
$131$	−18.9900	−1.65916	−0.829580	−	0.558387i	$-0.811420\pi$
$131$	−18.9900	−1.65916	−0.829580	+	0.558387i	$0.811420\pi$
$132$	0	0
$133$	−12.6903	−1.10039
$134$	38.5183	3.32748
$135$	0	0
$136$	−9.62691	−0.825501
$137$	7.02843	0.600479	0.300240	−	0.953864i	$-0.402933\pi$
$137$	7.02843	0.600479	0.300240	+	0.953864i	$0.402933\pi$
$138$	0	0
$139$	18.9932	1.61098	0.805489	−	0.592610i	$-0.201903\pi$
$139$	18.9932	1.61098	0.805489	+	0.592610i	$0.201903\pi$
$140$	0	0
$141$	0	0
$142$	1.38561	0.116278
$143$	−5.03230	−0.420822
$144$	0	0
$145$	0	0
$146$	0.907550	0.0751094
$147$	0	0
$148$	−24.8557	−2.04312
$149$	−18.9588	−1.55316	−0.776581	−	0.630017i	$-0.783048\pi$
$149$	−18.9588	−1.55316	−0.776581	+	0.630017i	$0.783048\pi$
$150$	0	0
$151$	−16.8557	−1.37170	−0.685848	−	0.727745i	$-0.740569\pi$
$151$	−16.8557	−1.37170	−0.685848	+	0.727745i	$0.740569\pi$
$152$	−13.1216	−1.06431
$153$	0	0
$154$	55.2151	4.44936
$155$	0	0
$156$	0	0
$157$	9.02112	0.719963	0.359982	−	0.932959i	$-0.382783\pi$
$157$	9.02112	0.719963	0.359982	+	0.932959i	$0.382783\pi$
$158$	14.9297	1.18774
$159$	0	0
$160$	0	0
$161$	18.2358	1.43718
$162$	0	0
$163$	0.841414	0.0659046	0.0329523	−	0.999457i	$-0.489509\pi$
$163$	0.841414	0.0659046	0.0329523	+	0.999457i	$0.489509\pi$
$164$	5.78647	0.451848
$165$	0	0
$166$	1.59327	0.123662
$167$	9.91475	0.767227	0.383613	−	0.923494i	$-0.374680\pi$
$167$	9.91475	0.767227	0.383613	+	0.923494i	$0.374680\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	26.6903	2.03511
$173$	−4.79709	−0.364716	−0.182358	−	0.983232i	$-0.558373\pi$
$173$	−4.79709	−0.364716	−0.182358	+	0.983232i	$0.558373\pi$
$174$	0	0
$175$	0	0
$176$	17.6662	1.33164
$177$	0	0
$178$	−14.4490	−1.08300
$179$	7.41701	0.554373	0.277187	−	0.960816i	$-0.410598\pi$
$179$	7.41701	0.554373	0.277187	+	0.960816i	$0.410598\pi$
$180$	0	0
$181$	3.69710	0.274803	0.137402	−	0.990515i	$-0.456125\pi$
$181$	3.69710	0.274803	0.137402	+	0.990515i	$0.456125\pi$
$182$	−10.9721	−0.813309
$183$	0	0
$184$	18.8557	1.39006
$185$	0	0
$186$	0	0
$187$	−10.3874	−0.759599
$188$	−35.5502	−2.59276
$189$	0	0
$190$	0	0
$191$	−8.55624	−0.619108	−0.309554	−	0.950882i	$-0.600180\pi$
$191$	−8.55624	−0.619108	−0.309554	+	0.950882i	$0.600180\pi$
$192$	0	0
$193$	15.3662	1.10609	0.553043	−	0.833153i	$-0.313466\pi$
$193$	15.3662	1.10609	0.553043	+	0.833153i	$0.313466\pi$
$194$	6.50946	0.467352
$195$	0	0
$196$	52.2767	3.73405
$197$	11.2927	0.804573	0.402286	−	0.915514i	$-0.368216\pi$
$197$	11.2927	0.804573	0.402286	+	0.915514i	$0.368216\pi$
$198$	0	0
$199$	−17.8768	−1.26725	−0.633626	−	0.773639i	$-0.718434\pi$
$199$	−17.8768	−1.26725	−0.633626	+	0.773639i	$0.718434\pi$
$200$	0	0
$201$	0	0
$202$	28.8979	2.03325
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−5.37023	−0.374162
$207$	0	0
$208$	−3.51056	−0.243413
$209$	−14.1582	−0.979340
$210$	0	0
$211$	15.8346	1.09010	0.545048	−	0.838405i	$-0.316511\pi$
$211$	15.8346	1.09010	0.545048	+	0.838405i	$0.316511\pi$
$212$	−38.4503	−2.64078
$213$	0	0
$214$	−24.6903	−1.68779
$215$	0	0
$216$	0	0
$217$	−35.3383	−2.39892
$218$	0.907550	0.0614670
$219$	0	0
$220$	0	0
$221$	2.06414	0.138849
$222$	0	0
$223$	−1.46149	−0.0978683	−0.0489342	−	0.998802i	$-0.515582\pi$
$223$	−1.46149	−0.0978683	−0.0489342	+	0.998802i	$0.515582\pi$
$224$	−3.55513	−0.237537
$225$	0	0
$226$	16.8135	1.11841
$227$	−12.2279	−0.811596	−0.405798	−	0.913963i	$-0.633006\pi$
$227$	−12.2279	−0.811596	−0.405798	+	0.913963i	$0.633006\pi$
$228$	0	0
$229$	−24.3172	−1.60692	−0.803462	−	0.595356i	$-0.797011\pi$
$229$	−24.3172	−1.60692	−0.803462	+	0.595356i	$0.797011\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.9470	0.979214	0.489607	−	0.871943i	$-0.337140\pi$
$233$	14.9470	0.979214	0.489607	+	0.871943i	$0.337140\pi$
$234$	0	0
$235$	0	0
$236$	−51.7218	−3.36680
$237$	0	0
$238$	−22.6480	−1.46805
$239$	9.49498	0.614179	0.307090	−	0.951681i	$-0.400645\pi$
$239$	9.49498	0.614179	0.307090	+	0.951681i	$0.400645\pi$
$240$	0	0
$241$	−2.60580	−0.167854	−0.0839271	−	0.996472i	$-0.526746\pi$
$241$	−2.60580	−0.167854	−0.0839271	+	0.996472i	$0.526746\pi$
$242$	34.8438	2.23985
$243$	0	0
$244$	32.6903	2.09278
$245$	0	0
$246$	0	0
$247$	2.81346	0.179016
$248$	−36.5396	−2.32027
$249$	0	0
$250$	0	0
$251$	1.50835	0.0952065	0.0476033	−	0.998866i	$-0.484842\pi$
$251$	1.50835	0.0952065	0.0476033	+	0.998866i	$0.484842\pi$
$252$	0	0
$253$	20.3451	1.27909
$254$	−8.92537	−0.560027
$255$	0	0
$256$	−31.1797	−1.94873
$257$	−7.41701	−0.462660	−0.231330	−	0.972875i	$-0.574308\pi$
$257$	−7.41701	−0.462660	−0.231330	+	0.972875i	$0.574308\pi$
$258$	0	0
$259$	−28.6201	−1.77836
$260$	0	0
$261$	0	0
$262$	−46.1940	−2.85387
$263$	−22.7491	−1.40277	−0.701385	−	0.712782i	$-0.747435\pi$
$263$	−22.7491	−1.40277	−0.701385	+	0.712782i	$0.747435\pi$
$264$	0	0
$265$	0	0
$266$	−30.8697	−1.89274
$267$	0	0
$268$	62.0285	3.78900
$269$	−8.92537	−0.544189	−0.272095	−	0.962271i	$-0.587716\pi$
$269$	−8.92537	−0.544189	−0.272095	+	0.962271i	$0.587716\pi$
$270$	0	0
$271$	−8.16543	−0.496014	−0.248007	−	0.968758i	$-0.579776\pi$
$271$	−8.16543	−0.496014	−0.248007	+	0.968758i	$0.579776\pi$
$272$	−7.24628	−0.439370
$273$	0	0
$274$	17.0970	1.03287
$275$	0	0
$276$	0	0
$277$	26.3172	1.58125	0.790623	−	0.612303i	$-0.209757\pi$
$277$	26.3172	1.58125	0.790623	+	0.612303i	$0.209757\pi$
$278$	46.2018	2.77100
$279$	0	0
$280$	0	0
$281$	−19.7129	−1.17598	−0.587988	−	0.808870i	$-0.700080\pi$
$281$	−19.7129	−1.17598	−0.587988	+	0.808870i	$0.700080\pi$
$282$	0	0
$283$	27.0633	1.60875	0.804374	−	0.594123i	$-0.202501\pi$
$283$	27.0633	1.60875	0.804374	+	0.594123i	$0.202501\pi$
$284$	2.23134	0.132406
$285$	0	0
$286$	−12.2413	−0.723843
$287$	6.66283	0.393295
$288$	0	0
$289$	−12.7393	−0.749372
$290$	0	0
$291$	0	0
$292$	1.46149	0.0855270
$293$	−4.87892	−0.285030	−0.142515	−	0.989793i	$-0.545519\pi$
$293$	−4.87892	−0.285030	−0.142515	+	0.989793i	$0.545519\pi$
$294$	0	0
$295$	0	0
$296$	−29.5930	−1.72006
$297$	0	0
$298$	−46.1181	−2.67155
$299$	−4.04291	−0.233808
$300$	0	0
$301$	30.7325	1.77139
$302$	−41.0023	−2.35941
$303$	0	0
$304$	−9.87680	−0.566473
$305$	0	0
$306$	0	0
$307$	−12.7855	−0.729707	−0.364854	−	0.931065i	$-0.618881\pi$
$307$	−12.7855	−0.729707	−0.364854	+	0.931065i	$0.618881\pi$
$308$	88.9164	5.06648
$309$	0	0
$310$	0	0
$311$	−27.5462	−1.56200	−0.781001	−	0.624530i	$-0.785291\pi$
$311$	−27.5462	−1.56200	−0.781001	+	0.624530i	$0.785291\pi$
$312$	0	0
$313$	31.2961	1.76896	0.884479	−	0.466580i	$-0.154514\pi$
$313$	31.2961	1.76896	0.884479	+	0.466580i	$0.154514\pi$
$314$	21.9443	1.23839
$315$	0	0
$316$	24.0422	1.35248
$317$	3.70498	0.208092	0.104046	−	0.994572i	$-0.466821\pi$
$317$	3.70498	0.208092	0.104046	+	0.994572i	$0.466821\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	44.3594	2.47205
$323$	5.80737	0.323130
$324$	0	0
$325$	0	0
$326$	2.04678	0.113361
$327$	0	0
$328$	6.88933	0.380400
$329$	−40.9342	−2.25678
$330$	0	0
$331$	−16.5807	−0.911360	−0.455680	−	0.890144i	$-0.650604\pi$
$331$	−16.5807	−0.911360	−0.455680	+	0.890144i	$0.650604\pi$
$332$	2.56575	0.140814
$333$	0	0
$334$	24.1181	1.31968
$335$	0	0
$336$	0	0
$337$	4.04223	0.220194	0.110097	−	0.993921i	$-0.464884\pi$
$337$	4.04223	0.220194	0.110097	+	0.993921i	$0.464884\pi$
$338$	2.43255	0.132313
$339$	0	0
$340$	0	0
$341$	−39.4259	−2.13503
$342$	0	0
$343$	28.6201	1.54534
$344$	31.7772	1.71331
$345$	0	0
$346$	−11.6691	−0.627337
$347$	−6.86123	−0.368330	−0.184165	−	0.982895i	$-0.558958\pi$
$347$	−6.86123	−0.368330	−0.184165	+	0.982895i	$0.558958\pi$
$348$	0	0
$349$	26.9652	1.44341	0.721707	−	0.692199i	$-0.243358\pi$
$349$	26.9652	1.44341	0.721707	+	0.692199i	$0.243358\pi$
$350$	0	0
$351$	0	0
$352$	−3.96636	−0.211408
$353$	13.4352	0.715082	0.357541	−	0.933898i	$-0.383615\pi$
$353$	13.4352	0.715082	0.357541	+	0.933898i	$0.383615\pi$
$354$	0	0
$355$	0	0
$356$	−23.2681	−1.23321
$357$	0	0
$358$	18.0422	0.953561
$359$	−11.3884	−0.601056	−0.300528	−	0.953773i	$-0.597163\pi$
$359$	−11.3884	−0.601056	−0.300528	+	0.953773i	$0.597163\pi$
$360$	0	0
$361$	−11.0845	−0.583393
$362$	8.99337	0.472681
$363$	0	0
$364$	−17.6691	−0.926115
$365$	0	0
$366$	0	0
$367$	−6.81346	−0.355660	−0.177830	−	0.984061i	$-0.556908\pi$
$367$	−6.81346	−0.355660	−0.177830	+	0.984061i	$0.556908\pi$
$368$	14.1929	0.739855
$369$	0	0
$370$	0	0
$371$	−44.2736	−2.29857
$372$	0	0
$373$	−9.39420	−0.486413	−0.243207	−	0.969974i	$-0.578199\pi$
$373$	−9.39420	−0.486413	−0.243207	+	0.969974i	$0.578199\pi$
$374$	−25.2677	−1.30656
$375$	0	0
$376$	−42.3258	−2.18278
$377$	0	0
$378$	0	0
$379$	−29.8768	−1.53467	−0.767334	−	0.641247i	$-0.778417\pi$
$379$	−29.8768	−1.53467	−0.767334	+	0.641247i	$0.778417\pi$
$380$	0	0
$381$	0	0
$382$	−20.8135	−1.06491
$383$	1.65820	0.0847299	0.0423650	−	0.999102i	$-0.486511\pi$
$383$	1.65820	0.0847299	0.0423650	+	0.999102i	$0.486511\pi$
$384$	0	0
$385$	0	0
$386$	37.3791	1.90255
$387$	0	0
$388$	10.4826	0.532173
$389$	−23.0835	−1.17038	−0.585190	−	0.810896i	$-0.698980\pi$
$389$	−23.0835	−1.17038	−0.585190	+	0.810896i	$0.698980\pi$
$390$	0	0
$391$	−8.34513	−0.422031
$392$	62.2402	3.14360
$393$	0	0
$394$	27.4701	1.38392
$395$	0	0
$396$	0	0
$397$	−7.09130	−0.355902	−0.177951	−	0.984039i	$-0.556947\pi$
$397$	−7.09130	−0.355902	−0.177951	+	0.984039i	$0.556947\pi$
$398$	−43.4862	−2.17976
$399$	0	0
$400$	0	0
$401$	33.1010	1.65298	0.826492	−	0.562948i	$-0.190333\pi$
$401$	33.1010	1.65298	0.826492	+	0.562948i	$0.190333\pi$
$402$	0	0
$403$	7.83457	0.390268
$404$	46.5362	2.31526
$405$	0	0
$406$	0	0
$407$	−31.9306	−1.58274
$408$	0	0
$409$	−16.0422	−0.793237	−0.396619	−	0.917983i	$-0.629816\pi$
$409$	−16.0422	−0.793237	−0.396619	+	0.917983i	$0.629816\pi$
$410$	0	0
$411$	0	0
$412$	−8.64803	−0.426058
$413$	−59.5551	−2.93051
$414$	0	0
$415$	0	0
$416$	0.788180	0.0386437
$417$	0	0
$418$	−34.4404	−1.68453
$419$	2.64758	0.129343	0.0646714	−	0.997907i	$-0.479400\pi$
$419$	2.64758	0.129343	0.0646714	+	0.997907i	$0.479400\pi$
$420$	0	0
$421$	−29.0074	−1.41374	−0.706868	−	0.707346i	$-0.749892\pi$
$421$	−29.0074	−1.41374	−0.706868	+	0.707346i	$0.749892\pi$
$422$	38.5183	1.87504
$423$	0	0
$424$	−45.7787	−2.22321
$425$	0	0
$426$	0	0
$427$	37.6412	1.82158
$428$	−39.7603	−1.92189
$429$	0	0
$430$	0	0
$431$	17.2970	0.833169	0.416585	−	0.909097i	$-0.363227\pi$
$431$	17.2970	0.833169	0.416585	+	0.909097i	$0.363227\pi$
$432$	0	0
$433$	−21.3383	−1.02545	−0.512726	−	0.858552i	$-0.671364\pi$
$433$	−21.3383	−1.02545	−0.512726	+	0.858552i	$0.671364\pi$
$434$	−85.9621	−4.12631
$435$	0	0
$436$	1.46149	0.0699925
$437$	−11.3746	−0.544119
$438$	0	0
$439$	−0.675986	−0.0322630	−0.0161315	−	0.999870i	$-0.505135\pi$
$439$	−0.675986	−0.0322630	−0.0161315	+	0.999870i	$0.505135\pi$
$440$	0	0
$441$	0	0
$442$	5.02112	0.238830
$443$	11.6583	0.553903	0.276952	−	0.960884i	$-0.410676\pi$
$443$	11.6583	0.553903	0.276952	+	0.960884i	$0.410676\pi$
$444$	0	0
$445$	0	0
$446$	−3.55513	−0.168340
$447$	0	0
$448$	−40.3172	−1.90481
$449$	−2.98552	−0.140895	−0.0704477	−	0.997515i	$-0.522443\pi$
$449$	−2.98552	−0.140895	−0.0704477	+	0.997515i	$0.522443\pi$
$450$	0	0
$451$	7.43353	0.350031
$452$	27.0758	1.27354
$453$	0	0
$454$	−29.7450	−1.39600
$455$	0	0
$456$	0	0
$457$	28.0565	1.31243	0.656214	−	0.754575i	$-0.272157\pi$
$457$	28.0565	1.31243	0.656214	+	0.754575i	$0.272157\pi$
$458$	−59.1527	−2.76402
$459$	0	0
$460$	0	0
$461$	23.4215	1.09085	0.545423	−	0.838161i	$-0.316369\pi$
$461$	23.4215	1.09085	0.545423	+	0.838161i	$0.316369\pi$
$462$	0	0
$463$	4.51056	0.209623	0.104812	−	0.994492i	$-0.466576\pi$
$463$	4.51056	0.209623	0.104812	+	0.994492i	$0.466576\pi$
$464$	0	0
$465$	0	0
$466$	36.3594	1.68432
$467$	−24.3428	−1.12645	−0.563226	−	0.826303i	$-0.690440\pi$
$467$	−24.3428	−1.12645	−0.563226	+	0.826303i	$0.690440\pi$
$468$	0	0
$469$	71.4227	3.29800
$470$	0	0
$471$	0	0
$472$	−61.5796	−2.83443
$473$	34.2873	1.57653
$474$	0	0
$475$	0	0
$476$	−36.4716	−1.67167
$477$	0	0
$478$	23.0970	1.05643
$479$	16.9120	0.772729	0.386364	−	0.922346i	$-0.373731\pi$
$479$	16.9120	0.772729	0.386364	+	0.922346i	$0.373731\pi$
$480$	0	0
$481$	6.34513	0.289313
$482$	−6.33873	−0.288721
$483$	0	0
$484$	56.1113	2.55051
$485$	0	0
$486$	0	0
$487$	31.5739	1.43075	0.715375	−	0.698741i	$-0.246256\pi$
$487$	31.5739	1.43075	0.715375	+	0.698741i	$0.246256\pi$
$488$	38.9208	1.76186
$489$	0	0
$490$	0	0
$491$	−14.8340	−0.669450	−0.334725	−	0.942316i	$-0.608643\pi$
$491$	−14.8340	−0.669450	−0.334725	+	0.942316i	$0.608643\pi$
$492$	0	0
$493$	0	0
$494$	6.84387	0.307920
$495$	0	0
$496$	−27.5037	−1.23495
$497$	2.56928	0.115248
$498$	0	0
$499$	0.771227	0.0345248	0.0172624	−	0.999851i	$-0.494505\pi$
$499$	0.771227	0.0345248	0.0172624	+	0.999851i	$0.494505\pi$
$500$	0	0
$501$	0	0
$502$	3.66914	0.163762
$503$	41.5407	1.85221	0.926104	−	0.377269i	$-0.123137\pi$
$503$	41.5407	1.85221	0.926104	+	0.377269i	$0.123137\pi$
$504$	0	0
$505$	0	0
$506$	49.4905	2.20012
$507$	0	0
$508$	−14.3731	−0.637703
$509$	33.8552	1.50060	0.750302	−	0.661095i	$-0.229908\pi$
$509$	33.8552	1.50060	0.750302	+	0.661095i	$0.229908\pi$
$510$	0	0
$511$	1.68283	0.0744439
$512$	−35.5127	−1.56945
$513$	0	0
$514$	−18.0422	−0.795809
$515$	0	0
$516$	0	0
$517$	−45.6691	−2.00853
$518$	−69.6197	−3.05891
$519$	0	0
$520$	0	0
$521$	−25.2677	−1.10700	−0.553500	−	0.832849i	$-0.686708\pi$
$521$	−25.2677	−1.10700	−0.553500	+	0.832849i	$0.686708\pi$
$522$	0	0
$523$	11.2288	0.491000	0.245500	−	0.969397i	$-0.421048\pi$
$523$	11.2288	0.491000	0.245500	+	0.969397i	$0.421048\pi$
$524$	−74.3891	−3.24970
$525$	0	0
$526$	−55.3383	−2.41286
$527$	16.1716	0.704448
$528$	0	0
$529$	−6.65487	−0.289342
$530$	0	0
$531$	0	0
$532$	−49.7114	−2.15526
$533$	−1.47716	−0.0639831
$534$	0	0
$535$	0	0
$536$	73.8507	3.18986
$537$	0	0
$538$	−21.7114	−0.936044
$539$	67.1567	2.89264
$540$	0	0
$541$	6.41532	0.275816	0.137908	−	0.990445i	$-0.455962\pi$
$541$	6.41532	0.275816	0.137908	+	0.990445i	$0.455962\pi$
$542$	−19.8628	−0.853180
$543$	0	0
$544$	1.62691	0.0697533
$545$	0	0
$546$	0	0
$547$	24.8557	1.06275	0.531376	−	0.847136i	$-0.321675\pi$
$547$	24.8557	1.06275	0.531376	+	0.847136i	$0.321675\pi$
$548$	27.5324	1.17612
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	27.6834	1.17722
$554$	64.0178	2.71985
$555$	0	0
$556$	74.4016	3.15533
$557$	−35.0450	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$557$	−35.0450	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$558$	0	0
$559$	−6.81346	−0.288179
$560$	0	0
$561$	0	0
$562$	−47.9527	−2.02276
$563$	31.7598	1.33852	0.669259	−	0.743029i	$-0.266612\pi$
$563$	31.7598	1.33852	0.669259	+	0.743029i	$0.266612\pi$
$564$	0	0
$565$	0	0
$566$	65.8329	2.76716
$567$	0	0
$568$	2.65662	0.111469
$569$	−2.95433	−0.123852	−0.0619259	−	0.998081i	$-0.519724\pi$
$569$	−2.95433	−0.123852	−0.0619259	+	0.998081i	$0.519724\pi$
$570$	0	0
$571$	−12.7855	−0.535057	−0.267528	−	0.963550i	$-0.586207\pi$
$571$	−12.7855	−0.535057	−0.267528	+	0.963550i	$0.586207\pi$
$572$	−19.7129	−0.824240
$573$	0	0
$574$	16.2077	0.676495
$575$	0	0
$576$	0	0
$577$	10.0143	0.416900	0.208450	−	0.978033i	$-0.433158\pi$
$577$	10.0143	0.416900	0.208450	+	0.978033i	$0.433158\pi$
$578$	−30.9890	−1.28897
$579$	0	0
$580$	0	0
$581$	2.95433	0.122566
$582$	0	0
$583$	−49.3948	−2.04572
$584$	1.74003	0.0720031
$585$	0	0
$586$	−11.8682	−0.490271
$587$	17.9659	0.741530	0.370765	−	0.928727i	$-0.379096\pi$
$587$	17.9659	0.741530	0.370765	+	0.928727i	$0.379096\pi$
$588$	0	0
$589$	22.0422	0.908234
$590$	0	0
$591$	0	0
$592$	−22.2749	−0.915494
$593$	6.18889	0.254147	0.127074	−	0.991893i	$-0.459442\pi$
$593$	6.18889	0.254147	0.127074	+	0.991893i	$0.459442\pi$
$594$	0	0
$595$	0	0
$596$	−74.2669	−3.04209
$597$	0	0
$598$	−9.83457	−0.402166
$599$	−37.6108	−1.53674	−0.768368	−	0.640009i	$-0.778931\pi$
$599$	−37.6108	−1.53674	−0.768368	+	0.640009i	$0.778931\pi$
$600$	0	0
$601$	7.32401	0.298753	0.149376	−	0.988780i	$-0.452273\pi$
$601$	7.32401	0.298753	0.149376	+	0.988780i	$0.452273\pi$
$602$	74.7582	3.04692
$603$	0	0
$604$	−66.0285	−2.68666
$605$	0	0
$606$	0	0
$607$	18.7575	0.761345	0.380673	−	0.924710i	$-0.375692\pi$
$607$	18.7575	0.761345	0.380673	+	0.924710i	$0.375692\pi$
$608$	2.21751	0.0899320
$609$	0	0
$610$	0	0
$611$	9.07521	0.367144
$612$	0	0
$613$	−1.73933	−0.0702509	−0.0351255	−	0.999383i	$-0.511183\pi$
$613$	−1.73933	−0.0702509	−0.0351255	+	0.999383i	$0.511183\pi$
$614$	−31.1013	−1.25515
$615$	0	0
$616$	105.863	4.26535
$617$	−1.39180	−0.0560317	−0.0280158	−	0.999607i	$-0.508919\pi$
$617$	−1.39180	−0.0560317	−0.0280158	+	0.999607i	$0.508919\pi$
$618$	0	0
$619$	14.2077	0.571054	0.285527	−	0.958371i	$-0.407831\pi$
$619$	14.2077	0.571054	0.285527	+	0.958371i	$0.407831\pi$
$620$	0	0
$621$	0	0
$622$	−67.0074	−2.68675
$623$	−26.7920	−1.07340
$624$	0	0
$625$	0	0
$626$	76.1291	3.04273
$627$	0	0
$628$	35.3383	1.41015
$629$	13.0972	0.522221
$630$	0	0
$631$	−35.2288	−1.40244	−0.701218	−	0.712947i	$-0.747360\pi$
$631$	−35.2288	−1.40244	−0.701218	+	0.712947i	$0.747360\pi$
$632$	28.6245	1.13862
$633$	0	0
$634$	9.01253	0.357933
$635$	0	0
$636$	0	0
$637$	−13.3451	−0.528753
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−35.7957	−1.41384	−0.706922	−	0.707291i	$-0.749917\pi$
$641$	−35.7957	−1.41384	−0.706922	+	0.707291i	$0.749917\pi$
$642$	0	0
$643$	21.8066	0.859969	0.429984	−	0.902836i	$-0.358519\pi$
$643$	21.8066	0.859969	0.429984	+	0.902836i	$0.358519\pi$
$644$	71.4348	2.81492
$645$	0	0
$646$	14.1267	0.555807
$647$	−27.4608	−1.07960	−0.539798	−	0.841794i	$-0.681500\pi$
$647$	−27.4608	−1.07960	−0.539798	+	0.841794i	$0.681500\pi$
$648$	0	0
$649$	−66.4439	−2.60815
$650$	0	0
$651$	0	0
$652$	3.29606	0.129084
$653$	1.34468	0.0526215	0.0263107	−	0.999654i	$-0.491624\pi$
$653$	1.34468	0.0526215	0.0263107	+	0.999654i	$0.491624\pi$
$654$	0	0
$655$	0	0
$656$	5.18567	0.202466
$657$	0	0
$658$	−99.5745	−3.88182
$659$	24.8986	0.969912	0.484956	−	0.874538i	$-0.338836\pi$
$659$	24.8986	0.969912	0.484956	+	0.874538i	$0.338836\pi$
$660$	0	0
$661$	22.3594	0.869680	0.434840	−	0.900508i	$-0.356805\pi$
$661$	22.3594	0.869680	0.434840	+	0.900508i	$0.356805\pi$
$662$	−40.3334	−1.56760
$663$	0	0
$664$	3.05476	0.118548
$665$	0	0
$666$	0	0
$667$	0	0
$668$	38.8389	1.50272
$669$	0	0
$670$	0	0
$671$	41.9952	1.62121
$672$	0	0
$673$	−48.7747	−1.88013	−0.940064	−	0.340999i	$-0.889235\pi$
$673$	−48.7747	−1.88013	−0.940064	+	0.340999i	$0.889235\pi$
$674$	9.83292	0.378750
$675$	0	0
$676$	3.91729	0.150665
$677$	21.1901	0.814402	0.407201	−	0.913339i	$-0.366505\pi$
$677$	21.1901	0.814402	0.407201	+	0.913339i	$0.366505\pi$
$678$	0	0
$679$	12.0702	0.463211
$680$	0	0
$681$	0	0
$682$	−95.9053	−3.67241
$683$	26.0517	0.996840	0.498420	−	0.866936i	$-0.333914\pi$
$683$	26.0517	0.996840	0.498420	+	0.866936i	$0.333914\pi$
$684$	0	0
$685$	0	0
$686$	69.6197	2.65809
$687$	0	0
$688$	23.9190	0.911905
$689$	9.81556	0.373943
$690$	0	0
$691$	19.8346	0.754543	0.377271	−	0.926103i	$-0.376862\pi$
$691$	19.8346	0.754543	0.377271	+	0.926103i	$0.376862\pi$
$692$	−18.7916	−0.714348
$693$	0	0
$694$	−16.6903	−0.633554
$695$	0	0
$696$	0	0
$697$	−3.04907	−0.115492
$698$	65.5941	2.48277
$699$	0	0
$700$	0	0
$701$	50.6297	1.91226	0.956129	−	0.292946i	$-0.0946356\pi$
$701$	50.6297	1.91226	0.956129	+	0.292946i	$0.0946356\pi$
$702$	0	0
$703$	17.8517	0.673291
$704$	−44.9807	−1.69527
$705$	0	0
$706$	32.6817	1.22999
$707$	53.5840	2.01524
$708$	0	0
$709$	47.1056	1.76909	0.884544	−	0.466458i	$-0.154470\pi$
$709$	47.1056	1.76909	0.884544	+	0.466458i	$0.154470\pi$
$710$	0	0
$711$	0	0
$712$	−27.7028	−1.03821
$713$	−31.6745	−1.18622
$714$	0	0
$715$	0	0
$716$	29.0546	1.08582
$717$	0	0
$718$	−27.7028	−1.03386
$719$	−10.4337	−0.389112	−0.194556	−	0.980891i	$-0.562327\pi$
$719$	−10.4337	−0.389112	−0.194556	+	0.980891i	$0.562327\pi$
$720$	0	0
$721$	−9.95777	−0.370847
$722$	−26.9635	−1.00348
$723$	0	0
$724$	14.4826	0.538242
$725$	0	0
$726$	0	0
$727$	−3.66914	−0.136081	−0.0680405	−	0.997683i	$-0.521675\pi$
$727$	−3.66914	−0.136081	−0.0680405	+	0.997683i	$0.521675\pi$
$728$	−21.0367	−0.779673
$729$	0	0
$730$	0	0
$731$	−14.0639	−0.520173
$732$	0	0
$733$	−15.5567	−0.574601	−0.287300	−	0.957841i	$-0.592758\pi$
$733$	−15.5567	−0.574601	−0.287300	+	0.957841i	$0.592758\pi$
$734$	−16.5741	−0.611760
$735$	0	0
$736$	−3.18654	−0.117458
$737$	79.6843	2.93521
$738$	0	0
$739$	13.5459	0.498296	0.249148	−	0.968465i	$-0.419849\pi$
$739$	13.5459	0.498296	0.249148	+	0.968465i	$0.419849\pi$
$740$	0	0
$741$	0	0
$742$	−107.698	−3.95371
$743$	6.08616	0.223280	0.111640	−	0.993749i	$-0.464390\pi$
$743$	6.08616	0.223280	0.111640	+	0.993749i	$0.464390\pi$
$744$	0	0
$745$	0	0
$746$	−22.8518	−0.836665
$747$	0	0
$748$	−40.6903	−1.48778
$749$	−45.7820	−1.67284
$750$	0	0
$751$	20.0702	0.732372	0.366186	−	0.930542i	$-0.380663\pi$
$751$	20.0702	0.732372	0.366186	+	0.930542i	$0.380663\pi$
$752$	−31.8590	−1.16178
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−9.39420	−0.341438	−0.170719	−	0.985320i	$-0.554609\pi$
$757$	−9.39420	−0.341438	−0.170719	+	0.985320i	$0.554609\pi$
$758$	−72.6767	−2.63974
$759$	0	0
$760$	0	0
$761$	−8.50912	−0.308456	−0.154228	−	0.988035i	$-0.549289\pi$
$761$	−8.50912	−0.308456	−0.154228	+	0.988035i	$0.549289\pi$
$762$	0	0
$763$	1.68283	0.0609224
$764$	−33.5172	−1.21261
$765$	0	0
$766$	4.03364	0.145741
$767$	13.2035	0.476750
$768$	0	0
$769$	−32.6480	−1.17732	−0.588659	−	0.808381i	$-0.700344\pi$
$769$	−32.6480	−1.17732	−0.588659	+	0.808381i	$0.700344\pi$
$770$	0	0
$771$	0	0
$772$	60.1940	2.16643
$773$	−7.66252	−0.275602	−0.137801	−	0.990460i	$-0.544003\pi$
$773$	−7.66252	−0.275602	−0.137801	+	0.990460i	$0.544003\pi$
$774$	0	0
$775$	0	0
$776$	12.4805	0.448024
$777$	0	0
$778$	−56.1517	−2.01314
$779$	−4.15594	−0.148902
$780$	0	0
$781$	2.86647	0.102570
$782$	−20.2999	−0.725924
$783$	0	0
$784$	46.8488	1.67317
$785$	0	0
$786$	0	0
$787$	−41.2151	−1.46916	−0.734580	−	0.678522i	$-0.762621\pi$
$787$	−41.2151	−1.46916	−0.734580	+	0.678522i	$0.762621\pi$
$788$	44.2368	1.57587
$789$	0	0
$790$	0	0
$791$	31.1764	1.10851
$792$	0	0
$793$	−8.34513	−0.296344
$794$	−17.2499	−0.612177
$795$	0	0
$796$	−70.0285	−2.48210
$797$	−8.47087	−0.300054	−0.150027	−	0.988682i	$-0.547936\pi$
$797$	−8.47087	−0.300054	−0.150027	+	0.988682i	$0.547936\pi$
$798$	0	0
$799$	18.7325	0.662708
$800$	0	0
$801$	0	0
$802$	80.5197	2.84325
$803$	1.87748	0.0662549
$804$	0	0
$805$	0	0
$806$	19.0580	0.671288
$807$	0	0
$808$	55.4056	1.94916
$809$	47.6754	1.67618	0.838089	−	0.545534i	$-0.183673\pi$
$809$	47.6754	1.67618	0.838089	+	0.545534i	$0.183673\pi$
$810$	0	0
$811$	21.5174	0.755578	0.377789	−	0.925892i	$-0.376684\pi$
$811$	21.5174	0.755578	0.377789	+	0.925892i	$0.376684\pi$
$812$	0	0
$813$	0	0
$814$	−77.6726	−2.72242
$815$	0	0
$816$	0	0
$817$	−19.1694	−0.670651
$818$	−39.0235	−1.36442
$819$	0	0
$820$	0	0
$821$	−10.7093	−0.373756	−0.186878	−	0.982383i	$-0.559837\pi$
$821$	−10.7093	−0.373756	−0.186878	+	0.982383i	$0.559837\pi$
$822$	0	0
$823$	−3.89049	−0.135614	−0.0678069	−	0.997698i	$-0.521600\pi$
$823$	−3.89049	−0.135614	−0.0678069	+	0.997698i	$0.521600\pi$
$824$	−10.2963	−0.358688
$825$	0	0
$826$	−144.871	−5.04069
$827$	−38.6072	−1.34251	−0.671253	−	0.741229i	$-0.734243\pi$
$827$	−38.6072	−1.34251	−0.671253	+	0.741229i	$0.734243\pi$
$828$	0	0
$829$	−20.6903	−0.718602	−0.359301	−	0.933222i	$-0.616985\pi$
$829$	−20.6903	−0.718602	−0.359301	+	0.933222i	$0.616985\pi$
$830$	0	0
$831$	0	0
$832$	8.93840	0.309883
$833$	−27.5462	−0.954419
$834$	0	0
$835$	0	0
$836$	−55.4615	−1.91818
$837$	0	0
$838$	6.44037	0.222479
$839$	9.49498	0.327803	0.163902	−	0.986477i	$-0.447592\pi$
$839$	9.49498	0.327803	0.163902	+	0.986477i	$0.447592\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	−70.5619	−2.43173
$843$	0	0
$844$	62.0285	2.13511
$845$	0	0
$846$	0	0
$847$	64.6093	2.22000
$848$	−34.4581	−1.18330
$849$	0	0
$850$	0	0
$851$	−25.6528	−0.879366
$852$	0	0
$853$	24.3874	0.835007	0.417504	−	0.908675i	$-0.362905\pi$
$853$	24.3874	0.835007	0.417504	+	0.908675i	$0.362905\pi$
$854$	91.5640	3.13325
$855$	0	0
$856$	−47.3383	−1.61799
$857$	28.1020	0.959945	0.479973	−	0.877283i	$-0.340647\pi$
$857$	28.1020	0.959945	0.479973	+	0.877283i	$0.340647\pi$
$858$	0	0
$859$	39.1586	1.33607	0.668037	−	0.744128i	$-0.267135\pi$
$859$	39.1586	1.33607	0.668037	+	0.744128i	$0.267135\pi$
$860$	0	0
$861$	0	0
$862$	42.0759	1.43311
$863$	−6.62602	−0.225552	−0.112776	−	0.993620i	$-0.535974\pi$
$863$	−6.62602	−0.225552	−0.112776	+	0.993620i	$0.535974\pi$
$864$	0	0
$865$	0	0
$866$	−51.9064	−1.76385
$867$	0	0
$868$	−138.430	−4.69863
$869$	30.8856	1.04772
$870$	0	0
$871$	−15.8346	−0.536534
$872$	1.74003	0.0589250
$873$	0	0
$874$	−27.6691	−0.935923
$875$	0	0
$876$	0	0
$877$	21.3383	0.720543	0.360271	−	0.932848i	$-0.382684\pi$
$877$	21.3383	0.720543	0.360271	+	0.932848i	$0.382684\pi$
$878$	−1.64437	−0.0554947
$879$	0	0
$880$	0	0
$881$	49.1214	1.65494	0.827470	−	0.561509i	$-0.189779\pi$
$881$	49.1214	1.65494	0.827470	+	0.561509i	$0.189779\pi$
$882$	0	0
$883$	6.28863	0.211629	0.105815	−	0.994386i	$-0.466255\pi$
$883$	6.28863	0.211629	0.105815	+	0.994386i	$0.466255\pi$
$884$	8.08582	0.271956
$885$	0	0
$886$	28.3594	0.952753
$887$	42.4932	1.42678	0.713392	−	0.700765i	$-0.247158\pi$
$887$	42.4932	1.42678	0.713392	+	0.700765i	$0.247158\pi$
$888$	0	0
$889$	−16.5499	−0.555065
$890$	0	0
$891$	0	0
$892$	−5.72506	−0.191689
$893$	25.5327	0.854419
$894$	0	0
$895$	0	0
$896$	−90.9632	−3.03887
$897$	0	0
$898$	−7.26242	−0.242350
$899$	0	0
$900$	0	0
$901$	20.2607	0.674981
$902$	18.0824	0.602078
$903$	0	0
$904$	32.2362	1.07216
$905$	0	0
$906$	0	0
$907$	43.6132	1.44815	0.724077	−	0.689719i	$-0.242266\pi$
$907$	43.6132	1.44815	0.724077	+	0.689719i	$0.242266\pi$
$908$	−47.9003	−1.58963
$909$	0	0
$910$	0	0
$911$	10.0646	0.333455	0.166727	−	0.986003i	$-0.446680\pi$
$911$	10.0646	0.333455	0.166727	+	0.986003i	$0.446680\pi$
$912$	0	0
$913$	3.29606	0.109084
$914$	68.2488	2.25747
$915$	0	0
$916$	−95.2573	−3.14739
$917$	−85.6553	−2.82859
$918$	0	0
$919$	17.3913	0.573686	0.286843	−	0.957978i	$-0.407394\pi$
$919$	17.3913	0.573686	0.286843	+	0.957978i	$0.407394\pi$
$920$	0	0
$921$	0	0
$922$	56.9738	1.87633
$923$	−0.569614	−0.0187491
$924$	0	0
$925$	0	0
$926$	10.9721	0.360567
$927$	0	0
$928$	0	0
$929$	−19.3279	−0.634128	−0.317064	−	0.948404i	$-0.602697\pi$
$929$	−19.3279	−0.634128	−0.317064	+	0.948404i	$0.602697\pi$
$930$	0	0
$931$	−37.5459	−1.23052
$932$	58.5519	1.91793
$933$	0	0
$934$	−59.2151	−1.93758
$935$	0	0
$936$	0	0
$937$	−15.9863	−0.522250	−0.261125	−	0.965305i	$-0.584094\pi$
$937$	−15.9863	−0.522250	−0.261125	+	0.965305i	$0.584094\pi$
$938$	173.739	5.67279
$939$	0	0
$940$	0	0
$941$	16.0044	0.521730	0.260865	−	0.965375i	$-0.415992\pi$
$941$	16.0044	0.521730	0.260865	+	0.965375i	$0.415992\pi$
$942$	0	0
$943$	5.97204	0.194476
$944$	−46.3516	−1.50862
$945$	0	0
$946$	83.4056	2.71175
$947$	−46.0519	−1.49649	−0.748243	−	0.663425i	$-0.769102\pi$
$947$	−46.0519	−1.49649	−0.748243	+	0.663425i	$0.769102\pi$
$948$	0	0
$949$	−0.373086	−0.0121109
$950$	0	0
$951$	0	0
$952$	−43.4227	−1.40734
$953$	50.7498	1.64395	0.821974	−	0.569525i	$-0.192873\pi$
$953$	50.7498	1.64395	0.821974	+	0.569525i	$0.192873\pi$
$954$	0	0
$955$	0	0
$956$	37.1946	1.20296
$957$	0	0
$958$	41.1392	1.32915
$959$	31.7021	1.02372
$960$	0	0
$961$	30.3805	0.980017
$962$	15.4348	0.497639
$963$	0	0
$964$	−10.2077	−0.328767
$965$	0	0
$966$	0	0
$967$	−15.0884	−0.485210	−0.242605	−	0.970125i	$-0.578002\pi$
$967$	−15.0884	−0.485210	−0.242605	+	0.970125i	$0.578002\pi$
$968$	66.8056	2.14721
$969$	0	0
$970$	0	0
$971$	30.1938	0.968965	0.484482	−	0.874801i	$-0.339008\pi$
$971$	30.1938	0.968965	0.484482	+	0.874801i	$0.339008\pi$
$972$	0	0
$973$	85.6697	2.74645
$974$	76.8050	2.46099
$975$	0	0
$976$	29.2961	0.937744
$977$	−23.3637	−0.747472	−0.373736	−	0.927535i	$-0.621923\pi$
$977$	−23.3637	−0.747472	−0.373736	+	0.927535i	$0.621923\pi$
$978$	0	0
$979$	−29.8911	−0.955323
$980$	0	0
$981$	0	0
$982$	−36.0845	−1.15150
$983$	25.3899	0.809813	0.404906	−	0.914358i	$-0.367304\pi$
$983$	25.3899	0.809813	0.404906	+	0.914358i	$0.367304\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	11.0211	0.350628
$989$	27.5462	0.875918
$990$	0	0
$991$	1.69710	0.0539102	0.0269551	−	0.999637i	$-0.491419\pi$
$991$	1.69710	0.0539102	0.0269551	+	0.999637i	$0.491419\pi$
$992$	6.17506	0.196058
$993$	0	0
$994$	6.24989	0.198234
$995$	0	0
$996$	0	0
$997$	−31.6691	−1.00297	−0.501486	−	0.865166i	$-0.667213\pi$
$997$	−31.6691	−1.00297	−0.501486	+	0.865166i	$0.667213\pi$
$998$	1.87605	0.0593852
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bo.1.6		6
3.2	odd	2	inner	2925.2.a.bo.1.1		6
5.2	odd	4		585.2.c.d.469.12	yes	12
5.3	odd	4		585.2.c.d.469.2	yes	12
5.4	even	2		2925.2.a.bn.1.1		6
15.2	even	4		585.2.c.d.469.1	✓	12
15.8	even	4		585.2.c.d.469.11	yes	12
15.14	odd	2		2925.2.a.bn.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.c.d.469.1	✓	12	15.2	even	4
585.2.c.d.469.2	yes	12	5.3	odd	4
585.2.c.d.469.11	yes	12	15.8	even	4
585.2.c.d.469.12	yes	12	5.2	odd	4
2925.2.a.bn.1.1		6	5.4	even	2
2925.2.a.bn.1.6		6	15.14	odd	2
2925.2.a.bo.1.1		6	3.2	odd	2	inner
2925.2.a.bo.1.6		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43255 q^{2} +3.91729 q^{4} +4.51056 q^{7} +4.66389 q^{8} +5.03230 q^{11} -1.00000 q^{13} +10.9721 q^{14} +3.51056 q^{16} -2.06414 q^{17} -2.81346 q^{19} +12.2413 q^{22} +4.04291 q^{23} -2.43255 q^{26} +17.6691 q^{28} -7.83457 q^{31} -0.788180 q^{32} -5.02112 q^{34} -6.34513 q^{37} -6.84387 q^{38} +1.47716 q^{41} +6.81346 q^{43} +19.7129 q^{44} +9.83457 q^{46} -9.07521 q^{47} +13.3451 q^{49} -3.91729 q^{52} -9.81556 q^{53} +21.0367 q^{56} -13.2035 q^{59} +8.34513 q^{61} -19.0580 q^{62} -8.93840 q^{64} +15.8346 q^{67} -8.08582 q^{68} +0.569614 q^{71} +0.373086 q^{73} -15.4348 q^{74} -11.0211 q^{76} +22.6985 q^{77} +6.13747 q^{79} +3.59327 q^{82} +0.654981 q^{83} +16.5741 q^{86} +23.4701 q^{88} -5.93985 q^{89} -4.51056 q^{91} +15.8372 q^{92} -22.0759 q^{94} +2.67599 q^{97} +32.4627 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 10 q^{7} - 6 q^{13} + 4 q^{16} - 12 q^{19} + 32 q^{22} + 28 q^{28} - 8 q^{31} + 4 q^{34} + 18 q^{37} + 36 q^{43} + 20 q^{46} + 24 q^{49} - 4 q^{52} - 6 q^{61} + 56 q^{67} + 12 q^{73} - 32 q^{76}+ \cdots + 38 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 + 10 * q^7 - 6 * q^13 + 4 * q^16 - 12 * q^19 + 32 * q^22 + 28 * q^28 - 8 * q^31 + 4 * q^34 + 18 * q^37 + 36 * q^43 + 20 * q^46 + 24 * q^49 - 4 * q^52 - 6 * q^61 + 56 * q^67 + 12 * q^73 - 32 * q^76 + 10 * q^79 + 24 * q^82 + 36 * q^88 - 10 * q^91 - 52 * q^94 + 38 * q^97

Introduction

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Database

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