LMFDB - Embedded newform 7600.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3800)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.87939 q^{3} +1.18479 q^{7} +0.532089 q^{9} +O(q^{10})$	$q-1.87939 q^{3} +1.18479 q^{7} +0.532089 q^{9} +2.18479 q^{11} +1.71688 q^{13} +0.120615 q^{17} -1.00000 q^{19} -2.22668 q^{21} +7.98545 q^{23} +4.63816 q^{27} +3.24897 q^{29} +8.41147 q^{31} -4.10607 q^{33} -3.33275 q^{37} -3.22668 q^{39} -8.98545 q^{41} +4.06418 q^{43} -1.71688 q^{47} -5.59627 q^{49} -0.226682 q^{51} +6.51754 q^{53} +1.87939 q^{57} -10.2121 q^{59} +6.53983 q^{61} +0.630415 q^{63} -2.18479 q^{67} -15.0077 q^{69} +9.12836 q^{71} -0.773318 q^{73} +2.58853 q^{77} -1.63816 q^{79} -10.3131 q^{81} +2.44831 q^{83} -6.10607 q^{87} +2.83750 q^{89} +2.03415 q^{91} -15.8084 q^{93} +2.19934 q^{97} +1.16250 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^9	$ 3 q - 3 q^{9} + 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 6 q^{23} - 3 q^{27} - 3 q^{29} + 15 q^{31} + 9 q^{37} - 3 q^{39} - 9 q^{41} + 3 q^{43} + 3 q^{47} - 3 q^{49} + 6 q^{51} - 3 q^{53} - 6 q^{59} - 9 q^{61} + 9 q^{63} - 3 q^{67} - 21 q^{69} + 9 q^{71} - 9 q^{73} + 18 q^{77} + 12 q^{79} - 9 q^{81} + 9 q^{83} - 6 q^{87} + 6 q^{89} + 27 q^{91} - 9 q^{93} + 21 q^{97} + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^9 + 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 6 * q^23 - 3 * q^27 - 3 * q^29 + 15 * q^31 + 9 * q^37 - 3 * q^39 - 9 * q^41 + 3 * q^43 + 3 * q^47 - 3 * q^49 + 6 * q^51 - 3 * q^53 - 6 * q^59 - 9 * q^61 + 9 * q^63 - 3 * q^67 - 21 * q^69 + 9 * q^71 - 9 * q^73 + 18 * q^77 + 12 * q^79 - 9 * q^81 + 9 * q^83 - 6 * q^87 + 6 * q^89 + 27 * q^91 - 9 * q^93 + 21 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.87939	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$3$	−1.87939	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.18479	0.447809	0.223905	−	0.974611i	$-0.428120\pi$
$7$	1.18479	0.447809	0.223905	+	0.974611i	$0.428120\pi$
$8$	0	0
$9$	0.532089	0.177363
$10$	0	0
$11$	2.18479	0.658740	0.329370	−	0.944201i	$-0.393164\pi$
$11$	2.18479	0.658740	0.329370	+	0.944201i	$0.393164\pi$
$12$	0	0
$13$	1.71688	0.476177	0.238089	−	0.971243i	$-0.423479\pi$
$13$	1.71688	0.476177	0.238089	+	0.971243i	$0.423479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.120615	0.0292534	0.0146267	−	0.999893i	$-0.495344\pi$
$17$	0.120615	0.0292534	0.0146267	+	0.999893i	$0.495344\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.22668	−0.485902
$22$	0	0
$23$	7.98545	1.66508	0.832541	−	0.553964i	$-0.186885\pi$
$23$	7.98545	1.66508	0.832541	+	0.553964i	$0.186885\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.63816	0.892613
$28$	0	0
$29$	3.24897	0.603319	0.301659	−	0.953416i	$-0.402459\pi$
$29$	3.24897	0.603319	0.301659	+	0.953416i	$0.402459\pi$
$30$	0	0
$31$	8.41147	1.51075	0.755373	−	0.655295i	$-0.227456\pi$
$31$	8.41147	1.51075	0.755373	+	0.655295i	$0.227456\pi$
$32$	0	0
$33$	−4.10607	−0.714774
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.33275	−0.547900	−0.273950	−	0.961744i	$-0.588330\pi$
$37$	−3.33275	−0.547900	−0.273950	+	0.961744i	$0.588330\pi$
$38$	0	0
$39$	−3.22668	−0.516683
$40$	0	0
$41$	−8.98545	−1.40329	−0.701646	−	0.712526i	$-0.747551\pi$
$41$	−8.98545	−1.40329	−0.701646	+	0.712526i	$0.747551\pi$
$42$	0	0
$43$	4.06418	0.619781	0.309891	−	0.950772i	$-0.399708\pi$
$43$	4.06418	0.619781	0.309891	+	0.950772i	$0.399708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.71688	−0.250433	−0.125216	−	0.992129i	$-0.539963\pi$
$47$	−1.71688	−0.250433	−0.125216	+	0.992129i	$0.539963\pi$
$48$	0	0
$49$	−5.59627	−0.799467
$50$	0	0
$51$	−0.226682	−0.0317418
$52$	0	0
$53$	6.51754	0.895253	0.447627	−	0.894221i	$-0.352269\pi$
$53$	6.51754	0.895253	0.447627	+	0.894221i	$0.352269\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.87939	0.248931
$58$	0	0
$59$	−10.2121	−1.32951	−0.664753	−	0.747063i	$-0.731463\pi$
$59$	−10.2121	−1.32951	−0.664753	+	0.747063i	$0.731463\pi$
$60$	0	0
$61$	6.53983	0.837339	0.418670	−	0.908139i	$-0.362497\pi$
$61$	6.53983	0.837339	0.418670	+	0.908139i	$0.362497\pi$
$62$	0	0
$63$	0.630415	0.0794248
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.18479	−0.266915	−0.133457	−	0.991055i	$-0.542608\pi$
$67$	−2.18479	−0.266915	−0.133457	+	0.991055i	$0.542608\pi$
$68$	0	0
$69$	−15.0077	−1.80672
$70$	0	0
$71$	9.12836	1.08334	0.541668	−	0.840592i	$-0.317793\pi$
$71$	9.12836	1.08334	0.541668	+	0.840592i	$0.317793\pi$
$72$	0	0
$73$	−0.773318	−0.0905101	−0.0452550	−	0.998975i	$-0.514410\pi$
$73$	−0.773318	−0.0905101	−0.0452550	+	0.998975i	$0.514410\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.58853	0.294990
$78$	0	0
$79$	−1.63816	−0.184307	−0.0921535	−	0.995745i	$-0.529375\pi$
$79$	−1.63816	−0.184307	−0.0921535	+	0.995745i	$0.529375\pi$
$80$	0	0
$81$	−10.3131	−1.14591
$82$	0	0
$83$	2.44831	0.268737	0.134369	−	0.990931i	$-0.457099\pi$
$83$	2.44831	0.268737	0.134369	+	0.990931i	$0.457099\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.10607	−0.654639
$88$	0	0
$89$	2.83750	0.300774	0.150387	−	0.988627i	$-0.451948\pi$
$89$	2.83750	0.300774	0.150387	+	0.988627i	$0.451948\pi$
$90$	0	0
$91$	2.03415	0.213237
$92$	0	0
$93$	−15.8084	−1.63925
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.19934	0.223309	0.111655	−	0.993747i	$-0.464385\pi$
$97$	2.19934	0.223309	0.111655	+	0.993747i	$0.464385\pi$
$98$	0	0
$99$	1.16250	0.116836
$100$	0	0
$101$	−10.3327	−1.02815	−0.514073	−	0.857746i	$-0.671864\pi$
$101$	−10.3327	−1.02815	−0.514073	+	0.857746i	$0.671864\pi$
$102$	0	0
$103$	17.5253	1.72682	0.863409	−	0.504505i	$-0.168325\pi$
$103$	17.5253	1.72682	0.863409	+	0.504505i	$0.168325\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.16250	0.305731	0.152865	−	0.988247i	$-0.451150\pi$
$107$	3.16250	0.305731	0.152865	+	0.988247i	$0.451150\pi$
$108$	0	0
$109$	−8.77332	−0.840331	−0.420166	−	0.907447i	$-0.638028\pi$
$109$	−8.77332	−0.840331	−0.420166	+	0.907447i	$0.638028\pi$
$110$	0	0
$111$	6.26352	0.594507
$112$	0	0
$113$	15.0128	1.41228	0.706142	−	0.708070i	$-0.250434\pi$
$113$	15.0128	1.41228	0.706142	+	0.708070i	$0.250434\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.913534	0.0844562
$118$	0	0
$119$	0.142903	0.0130999
$120$	0	0
$121$	−6.22668	−0.566062
$122$	0	0
$123$	16.8871	1.52266
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−6.35504	−0.563918	−0.281959	−	0.959426i	$-0.590984\pi$
$127$	−6.35504	−0.563918	−0.281959	+	0.959426i	$0.590984\pi$
$128$	0	0
$129$	−7.63816	−0.672502
$130$	0	0
$131$	−8.41921	−0.735590	−0.367795	−	0.929907i	$-0.619887\pi$
$131$	−8.41921	−0.735590	−0.367795	+	0.929907i	$0.619887\pi$
$132$	0	0
$133$	−1.18479	−0.102735
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.644963	0.0551029	0.0275514	−	0.999620i	$-0.491229\pi$
$137$	0.644963	0.0551029	0.0275514	+	0.999620i	$0.491229\pi$
$138$	0	0
$139$	−10.0351	−0.851165	−0.425582	−	0.904920i	$-0.639931\pi$
$139$	−10.0351	−0.851165	−0.425582	+	0.904920i	$0.639931\pi$
$140$	0	0
$141$	3.22668	0.271736
$142$	0	0
$143$	3.75103	0.313677
$144$	0	0
$145$	0	0
$146$	0	0
$147$	10.5175	0.867472
$148$	0	0
$149$	−4.24628	−0.347869	−0.173934	−	0.984757i	$-0.555648\pi$
$149$	−4.24628	−0.347869	−0.173934	+	0.984757i	$0.555648\pi$
$150$	0	0
$151$	17.3378	1.41093	0.705465	−	0.708745i	$-0.250738\pi$
$151$	17.3378	1.41093	0.705465	+	0.708745i	$0.250738\pi$
$152$	0	0
$153$	0.0641778	0.00518847
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.27126	−0.181266	−0.0906331	−	0.995884i	$-0.528889\pi$
$157$	−2.27126	−0.181266	−0.0906331	+	0.995884i	$0.528889\pi$
$158$	0	0
$159$	−12.2490	−0.971407
$160$	0	0
$161$	9.46110	0.745639
$162$	0	0
$163$	−0.0418891	−0.00328100	−0.00164050	−	0.999999i	$-0.500522\pi$
$163$	−0.0418891	−0.00328100	−0.00164050	+	0.999999i	$0.500522\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.246282	−0.0190579	−0.00952893	−	0.999955i	$-0.503033\pi$
$167$	−0.246282	−0.0190579	−0.00952893	+	0.999955i	$0.503033\pi$
$168$	0	0
$169$	−10.0523	−0.773255
$170$	0	0
$171$	−0.532089	−0.0406899
$172$	0	0
$173$	−9.50980	−0.723017	−0.361508	−	0.932369i	$-0.617738\pi$
$173$	−9.50980	−0.723017	−0.361508	+	0.932369i	$0.617738\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	19.1925	1.44260
$178$	0	0
$179$	2.10338	0.157214	0.0786069	−	0.996906i	$-0.474953\pi$
$179$	2.10338	0.157214	0.0786069	+	0.996906i	$0.474953\pi$
$180$	0	0
$181$	−7.86753	−0.584789	−0.292394	−	0.956298i	$-0.594452\pi$
$181$	−7.86753	−0.584789	−0.292394	+	0.956298i	$0.594452\pi$
$182$	0	0
$183$	−12.2909	−0.908566
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.263518	0.0192704
$188$	0	0
$189$	5.49525	0.399721
$190$	0	0
$191$	20.0719	1.45235	0.726177	−	0.687508i	$-0.241296\pi$
$191$	20.0719	1.45235	0.726177	+	0.687508i	$0.241296\pi$
$192$	0	0
$193$	15.6800	1.12867	0.564337	−	0.825544i	$-0.309132\pi$
$193$	15.6800	1.12867	0.564337	+	0.825544i	$0.309132\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.65776	0.688087	0.344043	−	0.938954i	$-0.388203\pi$
$197$	9.65776	0.688087	0.344043	+	0.938954i	$0.388203\pi$
$198$	0	0
$199$	28.1712	1.99700	0.998501	−	0.0547346i	$-0.0174313\pi$
$199$	28.1712	1.99700	0.998501	+	0.0547346i	$0.0174313\pi$
$200$	0	0
$201$	4.10607	0.289620
$202$	0	0
$203$	3.84936	0.270172
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.24897	0.295324
$208$	0	0
$209$	−2.18479	−0.151125
$210$	0	0
$211$	2.17705	0.149874	0.0749372	−	0.997188i	$-0.476124\pi$
$211$	2.17705	0.149874	0.0749372	+	0.997188i	$0.476124\pi$
$212$	0	0
$213$	−17.1557	−1.17549
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.96585	0.676526
$218$	0	0
$219$	1.45336	0.0982092
$220$	0	0
$221$	0.207081	0.0139298
$222$	0	0
$223$	−12.2371	−0.819458	−0.409729	−	0.912207i	$-0.634377\pi$
$223$	−12.2371	−0.819458	−0.409729	+	0.912207i	$0.634377\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.5476	−1.62928	−0.814640	−	0.579967i	$-0.803065\pi$
$227$	−24.5476	−1.62928	−0.814640	+	0.579967i	$0.803065\pi$
$228$	0	0
$229$	−16.1061	−1.06432	−0.532159	−	0.846644i	$-0.678619\pi$
$229$	−16.1061	−1.06432	−0.532159	+	0.846644i	$0.678619\pi$
$230$	0	0
$231$	−4.86484	−0.320083
$232$	0	0
$233$	−7.46110	−0.488793	−0.244397	−	0.969675i	$-0.578590\pi$
$233$	−7.46110	−0.488793	−0.244397	+	0.969675i	$0.578590\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.07873	0.199985
$238$	0	0
$239$	−14.1702	−0.916597	−0.458298	−	0.888798i	$-0.651541\pi$
$239$	−14.1702	−0.916597	−0.458298	+	0.888798i	$0.651541\pi$
$240$	0	0
$241$	−4.40373	−0.283669	−0.141835	−	0.989890i	$-0.545300\pi$
$241$	−4.40373	−0.283669	−0.141835	+	0.989890i	$0.545300\pi$
$242$	0	0
$243$	5.46791	0.350767
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.71688	−0.109243
$248$	0	0
$249$	−4.60132	−0.291597
$250$	0	0
$251$	−13.7520	−0.868016	−0.434008	−	0.900909i	$-0.642901\pi$
$251$	−13.7520	−0.868016	−0.434008	+	0.900909i	$0.642901\pi$
$252$	0	0
$253$	17.4466	1.09686
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.0719	1.31443	0.657215	−	0.753703i	$-0.271734\pi$
$257$	21.0719	1.31443	0.657215	+	0.753703i	$0.271734\pi$
$258$	0	0
$259$	−3.94862	−0.245355
$260$	0	0
$261$	1.72874	0.107006
$262$	0	0
$263$	−4.66819	−0.287853	−0.143926	−	0.989588i	$-0.545973\pi$
$263$	−4.66819	−0.287853	−0.143926	+	0.989588i	$0.545973\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.33275	−0.326359
$268$	0	0
$269$	7.08647	0.432069	0.216035	−	0.976386i	$-0.430688\pi$
$269$	7.08647	0.432069	0.216035	+	0.976386i	$0.430688\pi$
$270$	0	0
$271$	−6.28405	−0.381729	−0.190864	−	0.981616i	$-0.561129\pi$
$271$	−6.28405	−0.381729	−0.190864	+	0.981616i	$0.561129\pi$
$272$	0	0
$273$	−3.82295	−0.231375
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.8675	1.07356	0.536778	−	0.843724i	$-0.319641\pi$
$277$	17.8675	1.07356	0.536778	+	0.843724i	$0.319641\pi$
$278$	0	0
$279$	4.47565	0.267950
$280$	0	0
$281$	−8.36959	−0.499288	−0.249644	−	0.968338i	$-0.580314\pi$
$281$	−8.36959	−0.499288	−0.249644	+	0.968338i	$0.580314\pi$
$282$	0	0
$283$	−7.19759	−0.427852	−0.213926	−	0.976850i	$-0.568625\pi$
$283$	−7.19759	−0.427852	−0.213926	+	0.976850i	$0.568625\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.6459	−0.628407
$288$	0	0
$289$	−16.9855	−0.999144
$290$	0	0
$291$	−4.13341	−0.242305
$292$	0	0
$293$	4.39424	0.256714	0.128357	−	0.991728i	$-0.459030\pi$
$293$	4.39424	0.256714	0.128357	+	0.991728i	$0.459030\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.1334	0.588000
$298$	0	0
$299$	13.7101	0.792874
$300$	0	0
$301$	4.81521	0.277544
$302$	0	0
$303$	19.4192	1.11560
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.19160	0.0680082	0.0340041	−	0.999422i	$-0.489174\pi$
$307$	1.19160	0.0680082	0.0340041	+	0.999422i	$0.489174\pi$
$308$	0	0
$309$	−32.9368	−1.87371
$310$	0	0
$311$	12.5844	0.713596	0.356798	−	0.934182i	$-0.383868\pi$
$311$	12.5844	0.713596	0.356798	+	0.934182i	$0.383868\pi$
$312$	0	0
$313$	4.34224	0.245438	0.122719	−	0.992441i	$-0.460839\pi$
$313$	4.34224	0.245438	0.122719	+	0.992441i	$0.460839\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.7050	0.825916	0.412958	−	0.910750i	$-0.364496\pi$
$317$	14.7050	0.825916	0.412958	+	0.910750i	$0.364496\pi$
$318$	0	0
$319$	7.09833	0.397430
$320$	0	0
$321$	−5.94356	−0.331737
$322$	0	0
$323$	−0.120615	−0.00671118
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.4884	0.911813
$328$	0	0
$329$	−2.03415	−0.112146
$330$	0	0
$331$	32.1215	1.76556	0.882780	−	0.469787i	$-0.155669\pi$
$331$	32.1215	1.76556	0.882780	+	0.469787i	$0.155669\pi$
$332$	0	0
$333$	−1.77332	−0.0971772
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.27900	0.505459	0.252730	−	0.967537i	$-0.418672\pi$
$337$	9.27900	0.505459	0.252730	+	0.967537i	$0.418672\pi$
$338$	0	0
$339$	−28.2148	−1.53242
$340$	0	0
$341$	18.3773	0.995188
$342$	0	0
$343$	−14.9240	−0.805818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	33.3928	1.79262	0.896310	−	0.443428i	$-0.146238\pi$
$347$	33.3928	1.79262	0.896310	+	0.443428i	$0.146238\pi$
$348$	0	0
$349$	4.31820	0.231148	0.115574	−	0.993299i	$-0.463129\pi$
$349$	4.31820	0.231148	0.115574	+	0.993299i	$0.463129\pi$
$350$	0	0
$351$	7.96316	0.425042
$352$	0	0
$353$	−9.82026	−0.522680	−0.261340	−	0.965247i	$-0.584164\pi$
$353$	−9.82026	−0.522680	−0.261340	+	0.965247i	$0.584164\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.268571	−0.0142143
$358$	0	0
$359$	−14.7365	−0.777762	−0.388881	−	0.921288i	$-0.627138\pi$
$359$	−14.7365	−0.777762	−0.388881	+	0.921288i	$0.627138\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.7023	0.614213
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.9017	0.986659	0.493330	−	0.869842i	$-0.335780\pi$
$367$	18.9017	0.986659	0.493330	+	0.869842i	$0.335780\pi$
$368$	0	0
$369$	−4.78106	−0.248892
$370$	0	0
$371$	7.72193	0.400903
$372$	0	0
$373$	23.7033	1.22731	0.613654	−	0.789575i	$-0.289699\pi$
$373$	23.7033	1.22731	0.613654	+	0.789575i	$0.289699\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.57810	0.287287
$378$	0	0
$379$	−15.1061	−0.775947	−0.387973	−	0.921671i	$-0.626825\pi$
$379$	−15.1061	−0.775947	−0.387973	+	0.921671i	$0.626825\pi$
$380$	0	0
$381$	11.9436	0.611887
$382$	0	0
$383$	35.5357	1.81579	0.907895	−	0.419198i	$-0.137689\pi$
$383$	35.5357	1.81579	0.907895	+	0.419198i	$0.137689\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.16250	0.109926
$388$	0	0
$389$	0.0641778	0.00325394	0.00162697	−	0.999999i	$-0.499482\pi$
$389$	0.0641778	0.00325394	0.00162697	+	0.999999i	$0.499482\pi$
$390$	0	0
$391$	0.963163	0.0487093
$392$	0	0
$393$	15.8229	0.798162
$394$	0	0
$395$	0	0
$396$	0	0
$397$	31.9195	1.60199	0.800997	−	0.598668i	$-0.204303\pi$
$397$	31.9195	1.60199	0.800997	+	0.598668i	$0.204303\pi$
$398$	0	0
$399$	2.22668	0.111474
$400$	0	0
$401$	25.7374	1.28527	0.642633	−	0.766174i	$-0.277842\pi$
$401$	25.7374	1.28527	0.642633	+	0.766174i	$0.277842\pi$
$402$	0	0
$403$	14.4415	0.719383
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.28136	−0.360924
$408$	0	0
$409$	29.5631	1.46180	0.730899	−	0.682485i	$-0.239101\pi$
$409$	29.5631	1.46180	0.730899	+	0.682485i	$0.239101\pi$
$410$	0	0
$411$	−1.21213	−0.0597901
$412$	0	0
$413$	−12.0993	−0.595366
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.8598	0.923568
$418$	0	0
$419$	16.8452	0.822944	0.411472	−	0.911422i	$-0.365015\pi$
$419$	16.8452	0.822944	0.411472	+	0.911422i	$0.365015\pi$
$420$	0	0
$421$	−1.76146	−0.0858483	−0.0429241	−	0.999078i	$-0.513667\pi$
$421$	−1.76146	−0.0858483	−0.0429241	+	0.999078i	$0.513667\pi$
$422$	0	0
$423$	−0.913534	−0.0444175
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.74834	0.374969
$428$	0	0
$429$	−7.04963	−0.340359
$430$	0	0
$431$	13.4766	0.649144	0.324572	−	0.945861i	$-0.394780\pi$
$431$	13.4766	0.649144	0.324572	+	0.945861i	$0.394780\pi$
$432$	0	0
$433$	21.4037	1.02860	0.514299	−	0.857611i	$-0.328052\pi$
$433$	21.4037	1.02860	0.514299	+	0.857611i	$0.328052\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.98545	−0.381996
$438$	0	0
$439$	16.9786	0.810347	0.405173	−	0.914240i	$-0.367211\pi$
$439$	16.9786	0.810347	0.405173	+	0.914240i	$0.367211\pi$
$440$	0	0
$441$	−2.97771	−0.141796
$442$	0	0
$443$	−26.5921	−1.26343	−0.631716	−	0.775200i	$-0.717649\pi$
$443$	−26.5921	−1.26343	−0.631716	+	0.775200i	$0.717649\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.98040	0.377460
$448$	0	0
$449$	25.7425	1.21486	0.607431	−	0.794372i	$-0.292200\pi$
$449$	25.7425	1.21486	0.607431	+	0.794372i	$0.292200\pi$
$450$	0	0
$451$	−19.6313	−0.924404
$452$	0	0
$453$	−32.5844	−1.53095
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−32.4715	−1.51895	−0.759477	−	0.650534i	$-0.774545\pi$
$457$	−32.4715	−1.51895	−0.759477	+	0.650534i	$0.774545\pi$
$458$	0	0
$459$	0.559430	0.0261120
$460$	0	0
$461$	11.4115	0.531485	0.265743	−	0.964044i	$-0.414383\pi$
$461$	11.4115	0.531485	0.265743	+	0.964044i	$0.414383\pi$
$462$	0	0
$463$	−8.59863	−0.399612	−0.199806	−	0.979835i	$-0.564031\pi$
$463$	−8.59863	−0.399612	−0.199806	+	0.979835i	$0.564031\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.4056	1.22191	0.610953	−	0.791667i	$-0.290787\pi$
$467$	26.4056	1.22191	0.610953	+	0.791667i	$0.290787\pi$
$468$	0	0
$469$	−2.58853	−0.119527
$470$	0	0
$471$	4.26857	0.196685
$472$	0	0
$473$	8.87939	0.408275
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.46791	0.158785
$478$	0	0
$479$	19.2472	0.879428	0.439714	−	0.898138i	$-0.355080\pi$
$479$	19.2472	0.879428	0.439714	+	0.898138i	$0.355080\pi$
$480$	0	0
$481$	−5.72193	−0.260898
$482$	0	0
$483$	−17.7811	−0.809066
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9.48339	−0.429734	−0.214867	−	0.976643i	$-0.568932\pi$
$487$	−9.48339	−0.429734	−0.214867	+	0.976643i	$0.568932\pi$
$488$	0	0
$489$	0.0787257	0.00356010
$490$	0	0
$491$	18.2618	0.824142	0.412071	−	0.911152i	$-0.364806\pi$
$491$	18.2618	0.824142	0.412071	+	0.911152i	$0.364806\pi$
$492$	0	0
$493$	0.391874	0.0176491
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.8152	0.485128
$498$	0	0
$499$	10.9828	0.491656	0.245828	−	0.969313i	$-0.420940\pi$
$499$	10.9828	0.491656	0.245828	+	0.969313i	$0.420940\pi$
$500$	0	0
$501$	0.462859	0.0206790
$502$	0	0
$503$	18.9590	0.845342	0.422671	−	0.906283i	$-0.361093\pi$
$503$	18.9590	0.845342	0.422671	+	0.906283i	$0.361093\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	18.8922	0.839031
$508$	0	0
$509$	12.6705	0.561612	0.280806	−	0.959765i	$-0.409398\pi$
$509$	12.6705	0.561612	0.280806	+	0.959765i	$0.409398\pi$
$510$	0	0
$511$	−0.916222	−0.0405313
$512$	0	0
$513$	−4.63816	−0.204780
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−3.75103	−0.164970
$518$	0	0
$519$	17.8726	0.784519
$520$	0	0
$521$	41.3756	1.81270	0.906348	−	0.422531i	$-0.138858\pi$
$521$	41.3756	1.81270	0.906348	+	0.422531i	$0.138858\pi$
$522$	0	0
$523$	−35.2104	−1.53964	−0.769821	−	0.638260i	$-0.779655\pi$
$523$	−35.2104	−1.53964	−0.769821	+	0.638260i	$0.779655\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.01455	0.0441944
$528$	0	0
$529$	40.7674	1.77250
$530$	0	0
$531$	−5.43376	−0.235805
$532$	0	0
$533$	−15.4270	−0.668216
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.95306	−0.170587
$538$	0	0
$539$	−12.2267	−0.526640
$540$	0	0
$541$	25.8307	1.11055	0.555274	−	0.831667i	$-0.312613\pi$
$541$	25.8307	1.11055	0.555274	+	0.831667i	$0.312613\pi$
$542$	0	0
$543$	14.7861	0.634533
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.6878	1.22660	0.613301	−	0.789850i	$-0.289841\pi$
$547$	28.6878	1.22660	0.613301	+	0.789850i	$0.289841\pi$
$548$	0	0
$549$	3.47977	0.148513
$550$	0	0
$551$	−3.24897	−0.138411
$552$	0	0
$553$	−1.94087	−0.0825344
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.50980	−0.318200	−0.159100	−	0.987262i	$-0.550859\pi$
$557$	−7.50980	−0.318200	−0.159100	+	0.987262i	$0.550859\pi$
$558$	0	0
$559$	6.97771	0.295126
$560$	0	0
$561$	−0.495252	−0.0209096
$562$	0	0
$563$	−29.6287	−1.24870	−0.624350	−	0.781145i	$-0.714636\pi$
$563$	−29.6287	−1.24870	−0.624350	+	0.781145i	$0.714636\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−12.2189	−0.513147
$568$	0	0
$569$	9.34461	0.391746	0.195873	−	0.980629i	$-0.437246\pi$
$569$	9.34461	0.391746	0.195873	+	0.980629i	$0.437246\pi$
$570$	0	0
$571$	−27.1239	−1.13510	−0.567550	−	0.823339i	$-0.692109\pi$
$571$	−27.1239	−1.13510	−0.567550	+	0.823339i	$0.692109\pi$
$572$	0	0
$573$	−37.7229	−1.57590
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.7273	−0.488214	−0.244107	−	0.969748i	$-0.578495\pi$
$577$	−11.7273	−0.488214	−0.244107	+	0.969748i	$0.578495\pi$
$578$	0	0
$579$	−29.4688	−1.22468
$580$	0	0
$581$	2.90074	0.120343
$582$	0	0
$583$	14.2395	0.589739
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.37195	0.304273	0.152136	−	0.988359i	$-0.451385\pi$
$587$	7.37195	0.304273	0.152136	+	0.988359i	$0.451385\pi$
$588$	0	0
$589$	−8.41147	−0.346589
$590$	0	0
$591$	−18.1506	−0.746618
$592$	0	0
$593$	22.2831	0.915058	0.457529	−	0.889195i	$-0.348735\pi$
$593$	22.2831	0.915058	0.457529	+	0.889195i	$0.348735\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−52.9445	−2.16687
$598$	0	0
$599$	−28.7766	−1.17578	−0.587890	−	0.808941i	$-0.700041\pi$
$599$	−28.7766	−1.17578	−0.587890	+	0.808941i	$0.700041\pi$
$600$	0	0
$601$	−36.2719	−1.47956	−0.739780	−	0.672849i	$-0.765071\pi$
$601$	−36.2719	−1.47956	−0.739780	+	0.672849i	$0.765071\pi$
$602$	0	0
$603$	−1.16250	−0.0473408
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−11.2867	−0.458115	−0.229057	−	0.973413i	$-0.573564\pi$
$607$	−11.2867	−0.458115	−0.229057	+	0.973413i	$0.573564\pi$
$608$	0	0
$609$	−7.23442	−0.293154
$610$	0	0
$611$	−2.94768	−0.119250
$612$	0	0
$613$	−1.45067	−0.0585922	−0.0292961	−	0.999571i	$-0.509327\pi$
$613$	−1.45067	−0.0585922	−0.0292961	+	0.999571i	$0.509327\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.65951	0.0668094	0.0334047	−	0.999442i	$-0.489365\pi$
$617$	1.65951	0.0668094	0.0334047	+	0.999442i	$0.489365\pi$
$618$	0	0
$619$	−9.66456	−0.388452	−0.194226	−	0.980957i	$-0.562219\pi$
$619$	−9.66456	−0.388452	−0.194226	+	0.980957i	$0.562219\pi$
$620$	0	0
$621$	37.0378	1.48627
$622$	0	0
$623$	3.36184	0.134689
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.10607	0.163981
$628$	0	0
$629$	−0.401979	−0.0160279
$630$	0	0
$631$	16.0205	0.637767	0.318884	−	0.947794i	$-0.396692\pi$
$631$	16.0205	0.637767	0.318884	+	0.947794i	$0.396692\pi$
$632$	0	0
$633$	−4.09152	−0.162623
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.60813	−0.380688
$638$	0	0
$639$	4.85710	0.192144
$640$	0	0
$641$	−39.8093	−1.57237	−0.786187	−	0.617989i	$-0.787948\pi$
$641$	−39.8093	−1.57237	−0.786187	+	0.617989i	$0.787948\pi$
$642$	0	0
$643$	0.295912	0.0116696	0.00583481	−	0.999983i	$-0.498143\pi$
$643$	0.295912	0.0116696	0.00583481	+	0.999983i	$0.498143\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.3651	1.74417	0.872087	−	0.489351i	$-0.162766\pi$
$647$	44.3651	1.74417	0.872087	+	0.489351i	$0.162766\pi$
$648$	0	0
$649$	−22.3114	−0.875799
$650$	0	0
$651$	−18.7297	−0.734074
$652$	0	0
$653$	38.7110	1.51488	0.757439	−	0.652905i	$-0.226450\pi$
$653$	38.7110	1.51488	0.757439	+	0.652905i	$0.226450\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.411474	−0.0160531
$658$	0	0
$659$	9.17562	0.357431	0.178716	−	0.983901i	$-0.442806\pi$
$659$	9.17562	0.357431	0.178716	+	0.983901i	$0.442806\pi$
$660$	0	0
$661$	−5.04727	−0.196316	−0.0981579	−	0.995171i	$-0.531295\pi$
$661$	−5.04727	−0.196316	−0.0981579	+	0.995171i	$0.531295\pi$
$662$	0	0
$663$	−0.389185	−0.0151147
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.9445	1.00457
$668$	0	0
$669$	22.9982	0.889164
$670$	0	0
$671$	14.2882	0.551589
$672$	0	0
$673$	−6.24216	−0.240618	−0.120309	−	0.992737i	$-0.538388\pi$
$673$	−6.24216	−0.240618	−0.120309	+	0.992737i	$0.538388\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.2044	−0.507486	−0.253743	−	0.967272i	$-0.581662\pi$
$677$	−13.2044	−0.507486	−0.253743	+	0.967272i	$0.581662\pi$
$678$	0	0
$679$	2.60576	0.100000
$680$	0	0
$681$	46.1343	1.76787
$682$	0	0
$683$	−41.3988	−1.58408	−0.792040	−	0.610469i	$-0.790981\pi$
$683$	−41.3988	−1.58408	−0.792040	+	0.610469i	$0.790981\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.2695	1.15485
$688$	0	0
$689$	11.1898	0.426299
$690$	0	0
$691$	18.1985	0.692304	0.346152	−	0.938178i	$-0.387488\pi$
$691$	18.1985	0.692304	0.346152	+	0.938178i	$0.387488\pi$
$692$	0	0
$693$	1.37733	0.0523203
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.08378	−0.0410510
$698$	0	0
$699$	14.0223	0.530372
$700$	0	0
$701$	−23.7383	−0.896585	−0.448293	−	0.893887i	$-0.647968\pi$
$701$	−23.7383	−0.896585	−0.448293	+	0.893887i	$0.647968\pi$
$702$	0	0
$703$	3.33275	0.125697
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.2422	−0.460414
$708$	0	0
$709$	−11.9426	−0.448515	−0.224257	−	0.974530i	$-0.571996\pi$
$709$	−11.9426	−0.448515	−0.224257	+	0.974530i	$0.571996\pi$
$710$	0	0
$711$	−0.871644	−0.0326892
$712$	0	0
$713$	67.1694	2.51551
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.6313	0.994566
$718$	0	0
$719$	−13.1352	−0.489859	−0.244929	−	0.969541i	$-0.578765\pi$
$719$	−13.1352	−0.489859	−0.244929	+	0.969541i	$0.578765\pi$
$720$	0	0
$721$	20.7638	0.773285
$722$	0	0
$723$	8.27631	0.307799
$724$	0	0
$725$	0	0
$726$	0	0
$727$	13.5253	0.501625	0.250812	−	0.968036i	$-0.419302\pi$
$727$	13.5253	0.501625	0.250812	+	0.968036i	$0.419302\pi$
$728$	0	0
$729$	20.6631	0.765301
$730$	0	0
$731$	0.490200	0.0181307
$732$	0	0
$733$	−49.7256	−1.83666	−0.918328	−	0.395821i	$-0.870460\pi$
$733$	−49.7256	−1.83666	−0.918328	+	0.395821i	$0.870460\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.77332	−0.175827
$738$	0	0
$739$	−39.7282	−1.46143	−0.730714	−	0.682684i	$-0.760812\pi$
$739$	−39.7282	−1.46143	−0.730714	+	0.682684i	$0.760812\pi$
$740$	0	0
$741$	3.22668	0.118535
$742$	0	0
$743$	38.6955	1.41960	0.709801	−	0.704403i	$-0.248785\pi$
$743$	38.6955	1.41960	0.709801	+	0.704403i	$0.248785\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.30272	0.0476640
$748$	0	0
$749$	3.74691	0.136909
$750$	0	0
$751$	53.1353	1.93893	0.969467	−	0.245222i	$-0.0788610\pi$
$751$	53.1353	1.93893	0.969467	+	0.245222i	$0.0788610\pi$
$752$	0	0
$753$	25.8452	0.941853
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.3259	−1.10222	−0.551108	−	0.834434i	$-0.685795\pi$
$757$	−30.3259	−1.10222	−0.551108	+	0.834434i	$0.685795\pi$
$758$	0	0
$759$	−32.7888	−1.19016
$760$	0	0
$761$	−37.4935	−1.35914	−0.679569	−	0.733611i	$-0.737833\pi$
$761$	−37.4935	−1.35914	−0.679569	+	0.733611i	$0.737833\pi$
$762$	0	0
$763$	−10.3946	−0.376308
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−17.5330	−0.633081
$768$	0	0
$769$	9.36009	0.337533	0.168767	−	0.985656i	$-0.446022\pi$
$769$	9.36009	0.337533	0.168767	+	0.985656i	$0.446022\pi$
$770$	0	0
$771$	−39.6023	−1.42624
$772$	0	0
$773$	−35.8881	−1.29080	−0.645402	−	0.763843i	$-0.723310\pi$
$773$	−35.8881	−1.29080	−0.645402	+	0.763843i	$0.723310\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.42097	0.266226
$778$	0	0
$779$	8.98545	0.321937
$780$	0	0
$781$	19.9436	0.713637
$782$	0	0
$783$	15.0692	0.538530
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0.781059	0.0278418	0.0139209	−	0.999903i	$-0.495569\pi$
$787$	0.781059	0.0278418	0.0139209	+	0.999903i	$0.495569\pi$
$788$	0	0
$789$	8.77332	0.312338
$790$	0	0
$791$	17.7870	0.632435
$792$	0	0
$793$	11.2281	0.398722
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.0240434	−0.000851661 0	−0.000425830	−	1.00000i	$-0.500136\pi$
$797$	−0.0240434	−0.000851661 0	−0.000425830		1.00000i	$0.500136\pi$
$798$	0	0
$799$	−0.207081	−0.00732601
$800$	0	0
$801$	1.50980	0.0533462
$802$	0	0
$803$	−1.68954	−0.0596226
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13.3182	−0.468823
$808$	0	0
$809$	39.3919	1.38494	0.692472	−	0.721444i	$-0.256521\pi$
$809$	39.3919	1.38494	0.692472	+	0.721444i	$0.256521\pi$
$810$	0	0
$811$	−11.4216	−0.401066	−0.200533	−	0.979687i	$-0.564267\pi$
$811$	−11.4216	−0.401066	−0.200533	+	0.979687i	$0.564267\pi$
$812$	0	0
$813$	11.8102	0.414200
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4.06418	−0.142188
$818$	0	0
$819$	1.08235	0.0378203
$820$	0	0
$821$	53.9778	1.88384	0.941920	−	0.335839i	$-0.109020\pi$
$821$	53.9778	1.88384	0.941920	+	0.335839i	$0.109020\pi$
$822$	0	0
$823$	−33.4216	−1.16500	−0.582502	−	0.812830i	$-0.697926\pi$
$823$	−33.4216	−1.16500	−0.582502	+	0.812830i	$0.697926\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.0716	−0.628411	−0.314205	−	0.949355i	$-0.601738\pi$
$827$	−18.0716	−0.628411	−0.314205	+	0.949355i	$0.601738\pi$
$828$	0	0
$829$	−47.0642	−1.63461	−0.817303	−	0.576208i	$-0.804532\pi$
$829$	−47.0642	−1.63461	−0.817303	+	0.576208i	$0.804532\pi$
$830$	0	0
$831$	−33.5800	−1.16488
$832$	0	0
$833$	−0.674992	−0.0233871
$834$	0	0
$835$	0	0
$836$	0	0
$837$	39.0137	1.34851
$838$	0	0
$839$	10.1102	0.349042	0.174521	−	0.984653i	$-0.444162\pi$
$839$	10.1102	0.349042	0.174521	+	0.984653i	$0.444162\pi$
$840$	0	0
$841$	−18.4442	−0.636007
$842$	0	0
$843$	15.7297	0.541759
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.37733	−0.253488
$848$	0	0
$849$	13.5270	0.464247
$850$	0	0
$851$	−26.6135	−0.912299
$852$	0	0
$853$	13.1908	0.451644	0.225822	−	0.974169i	$-0.427493\pi$
$853$	13.1908	0.451644	0.225822	+	0.974169i	$0.427493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.7419	0.947644	0.473822	−	0.880621i	$-0.342874\pi$
$857$	27.7419	0.947644	0.473822	+	0.880621i	$0.342874\pi$
$858$	0	0
$859$	14.3301	0.488935	0.244468	−	0.969657i	$-0.421387\pi$
$859$	14.3301	0.488935	0.244468	+	0.969657i	$0.421387\pi$
$860$	0	0
$861$	20.0077	0.681862
$862$	0	0
$863$	−5.05819	−0.172183	−0.0860914	−	0.996287i	$-0.527438\pi$
$863$	−5.05819	−0.172183	−0.0860914	+	0.996287i	$0.527438\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	31.9222	1.08414
$868$	0	0
$869$	−3.57903	−0.121410
$870$	0	0
$871$	−3.75103	−0.127099
$872$	0	0
$873$	1.17024	0.0396068
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−35.0351	−1.18305	−0.591525	−	0.806286i	$-0.701474\pi$
$877$	−35.0351	−1.18305	−0.591525	+	0.806286i	$0.701474\pi$
$878$	0	0
$879$	−8.25847	−0.278551
$880$	0	0
$881$	3.36926	0.113513	0.0567566	−	0.998388i	$-0.481924\pi$
$881$	3.36926	0.113513	0.0567566	+	0.998388i	$0.481924\pi$
$882$	0	0
$883$	−36.9436	−1.24325	−0.621625	−	0.783315i	$-0.713527\pi$
$883$	−36.9436	−1.24325	−0.621625	+	0.783315i	$0.713527\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.05232	0.136064	0.0680318	−	0.997683i	$-0.478328\pi$
$887$	4.05232	0.136064	0.0680318	+	0.997683i	$0.478328\pi$
$888$	0	0
$889$	−7.52940	−0.252528
$890$	0	0
$891$	−22.5321	−0.754853
$892$	0	0
$893$	1.71688	0.0574532
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−25.7665	−0.860319
$898$	0	0
$899$	27.3286	0.911461
$900$	0	0
$901$	0.786112	0.0261892
$902$	0	0
$903$	−9.04963	−0.301153
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.3550	1.27356	0.636779	−	0.771046i	$-0.280266\pi$
$907$	38.3550	1.27356	0.636779	+	0.771046i	$0.280266\pi$
$908$	0	0
$909$	−5.49794	−0.182355
$910$	0	0
$911$	−3.98721	−0.132102	−0.0660510	−	0.997816i	$-0.521040\pi$
$911$	−3.98721	−0.132102	−0.0660510	+	0.997816i	$0.521040\pi$
$912$	0	0
$913$	5.34905	0.177028
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.97502	−0.329404
$918$	0	0
$919$	10.1453	0.334661	0.167331	−	0.985901i	$-0.446485\pi$
$919$	10.1453	0.334661	0.167331	+	0.985901i	$0.446485\pi$
$920$	0	0
$921$	−2.23947	−0.0737932
$922$	0	0
$923$	15.6723	0.515860
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.32501	0.306273
$928$	0	0
$929$	51.2267	1.68069	0.840346	−	0.542050i	$-0.182352\pi$
$929$	51.2267	1.68069	0.840346	+	0.542050i	$0.182352\pi$
$930$	0	0
$931$	5.59627	0.183410
$932$	0	0
$933$	−23.6509	−0.774297
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.7469	−1.26581	−0.632903	−	0.774231i	$-0.718137\pi$
$937$	−38.7469	−1.26581	−0.632903	+	0.774231i	$0.718137\pi$
$938$	0	0
$939$	−8.16075	−0.266316
$940$	0	0
$941$	4.90167	0.159790	0.0798950	−	0.996803i	$-0.474541\pi$
$941$	4.90167	0.159790	0.0798950	+	0.996803i	$0.474541\pi$
$942$	0	0
$943$	−71.7529	−2.33660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.00505	−0.162642	−0.0813212	−	0.996688i	$-0.525914\pi$
$947$	−5.00505	−0.162642	−0.0813212	+	0.996688i	$0.525914\pi$
$948$	0	0
$949$	−1.32770	−0.0430988
$950$	0	0
$951$	−27.6364	−0.896172
$952$	0	0
$953$	60.1489	1.94841	0.974207	−	0.225657i	$-0.0724528\pi$
$953$	60.1489	1.94841	0.974207	+	0.225657i	$0.0724528\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.3405	−0.431237
$958$	0	0
$959$	0.764147	0.0246756
$960$	0	0
$961$	39.7529	1.28235
$962$	0	0
$963$	1.68273	0.0542253
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.8648	−0.960388	−0.480194	−	0.877162i	$-0.659434\pi$
$967$	−29.8648	−0.960388	−0.480194	+	0.877162i	$0.659434\pi$
$968$	0	0
$969$	0.226682	0.00728206
$970$	0	0
$971$	−37.9709	−1.21854	−0.609272	−	0.792961i	$-0.708538\pi$
$971$	−37.9709	−1.21854	−0.609272	+	0.792961i	$0.708538\pi$
$972$	0	0
$973$	−11.8895	−0.381160
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.0223	0.416620	0.208310	−	0.978063i	$-0.433204\pi$
$977$	13.0223	0.416620	0.208310	+	0.978063i	$0.433204\pi$
$978$	0	0
$979$	6.19934	0.198132
$980$	0	0
$981$	−4.66819	−0.149044
$982$	0	0
$983$	5.86215	0.186974	0.0934868	−	0.995621i	$-0.470199\pi$
$983$	5.86215	0.186974	0.0934868	+	0.995621i	$0.470199\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.82295	0.121686
$988$	0	0
$989$	32.4543	1.03199
$990$	0	0
$991$	1.71925	0.0546136	0.0273068	−	0.999627i	$-0.491307\pi$
$991$	1.71925	0.0546136	0.0273068	+	0.999627i	$0.491307\pi$
$992$	0	0
$993$	−60.3688	−1.91574
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−55.4347	−1.75563	−0.877817	−	0.478996i	$-0.841001\pi$
$997$	−55.4347	−1.75563	−0.877817	+	0.478996i	$0.841001\pi$
$998$	0	0
$999$	−15.4578	−0.489063

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.br.1.1		3
4.3	odd	2		3800.2.a.s.1.3	✓	3
5.4	even	2		7600.2.a.bs.1.3		3
20.3	even	4		3800.2.d.o.3649.6		6
20.7	even	4		3800.2.d.o.3649.1		6
20.19	odd	2		3800.2.a.t.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3800.2.a.s.1.3	✓	3	4.3	odd	2
3800.2.a.t.1.1	yes	3	20.19	odd	2
3800.2.d.o.3649.1		6	20.7	even	4
3800.2.d.o.3649.6		6	20.3	even	4
7600.2.a.br.1.1		3	1.1	even	1	trivial
7600.2.a.bs.1.3		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-1.87939 q^{3} +1.18479 q^{7} +0.532089 q^{9} +O(q^{10})\)	\(q-1.87939 q^{3} +1.18479 q^{7} +0.532089 q^{9} +2.18479 q^{11} +1.71688 q^{13} +0.120615 q^{17} -1.00000 q^{19} -2.22668 q^{21} +7.98545 q^{23} +4.63816 q^{27} +3.24897 q^{29} +8.41147 q^{31} -4.10607 q^{33} -3.33275 q^{37} -3.22668 q^{39} -8.98545 q^{41} +4.06418 q^{43} -1.71688 q^{47} -5.59627 q^{49} -0.226682 q^{51} +6.51754 q^{53} +1.87939 q^{57} -10.2121 q^{59} +6.53983 q^{61} +0.630415 q^{63} -2.18479 q^{67} -15.0077 q^{69} +9.12836 q^{71} -0.773318 q^{73} +2.58853 q^{77} -1.63816 q^{79} -10.3131 q^{81} +2.44831 q^{83} -6.10607 q^{87} +2.83750 q^{89} +2.03415 q^{91} -15.8084 q^{93} +2.19934 q^{97} +1.16250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^9	\( 3 q - 3 q^{9} + 3 q^{11} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 6 q^{23} - 3 q^{27} - 3 q^{29} + 15 q^{31} + 9 q^{37} - 3 q^{39} - 9 q^{41} + 3 q^{43} + 3 q^{47} - 3 q^{49} + 6 q^{51} - 3 q^{53} - 6 q^{59} - 9 q^{61} + 9 q^{63} - 3 q^{67} - 21 q^{69} + 9 q^{71} - 9 q^{73} + 18 q^{77} + 12 q^{79} - 9 q^{81} + 9 q^{83} - 6 q^{87} + 6 q^{89} + 27 q^{91} - 9 q^{93} + 21 q^{97} + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^9 + 3 * q^11 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 6 * q^23 - 3 * q^27 - 3 * q^29 + 15 * q^31 + 9 * q^37 - 3 * q^39 - 9 * q^41 + 3 * q^43 + 3 * q^47 - 3 * q^49 + 6 * q^51 - 3 * q^53 - 6 * q^59 - 9 * q^61 + 9 * q^63 - 3 * q^67 - 21 * q^69 + 9 * q^71 - 9 * q^73 + 18 * q^77 + 12 * q^79 - 9 * q^81 + 9 * q^83 - 6 * q^87 + 6 * q^89 + 27 * q^91 - 9 * q^93 + 21 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

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Database

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