LMFDB - Embedded newform 1872.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.4833 q^{5} -7.48331 q^{7} +O(q^{10})$	$q-19.4833 q^{5} -7.48331 q^{7} +22.8999 q^{11} -13.0000 q^{13} -67.0334 q^{17} -16.5167 q^{19} -175.600 q^{23} +254.600 q^{25} -291.800 q^{29} -117.283 q^{31} +145.800 q^{35} -154.766 q^{37} +251.716 q^{41} +502.566 q^{43} -281.733 q^{47} -287.000 q^{49} -366.999 q^{53} -446.166 q^{55} -79.6663 q^{59} -194.865 q^{61} +253.283 q^{65} -400.082 q^{67} +528.299 q^{71} -734.366 q^{73} -171.367 q^{77} -113.266 q^{79} -933.466 q^{83} +1306.03 q^{85} -1190.91 q^{89} +97.2831 q^{91} +321.800 q^{95} +557.165 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 24 q^{5}+O(q^{10}) $ 2 * q - 24 * q^5	$ 2 q - 24 q^{5} - 44 q^{11} - 26 q^{13} - 164 q^{17} - 48 q^{19} + 8 q^{23} + 150 q^{25} - 404 q^{29} - 40 q^{31} + 112 q^{35} - 100 q^{37} - 200 q^{41} + 616 q^{43} - 324 q^{47} - 574 q^{49} + 164 q^{53} - 144 q^{55} + 140 q^{59} + 628 q^{61} + 312 q^{65} + 472 q^{67} + 428 q^{71} - 900 q^{73} - 672 q^{77} + 432 q^{79} - 1388 q^{83} + 1744 q^{85} - 960 q^{89} + 464 q^{95} - 532 q^{97}+O(q^{100}) $ 2 * q - 24 * q^5 - 44 * q^11 - 26 * q^13 - 164 * q^17 - 48 * q^19 + 8 * q^23 + 150 * q^25 - 404 * q^29 - 40 * q^31 + 112 * q^35 - 100 * q^37 - 200 * q^41 + 616 * q^43 - 324 * q^47 - 574 * q^49 + 164 * q^53 - 144 * q^55 + 140 * q^59 + 628 * q^61 + 312 * q^65 + 472 * q^67 + 428 * q^71 - 900 * q^73 - 672 * q^77 + 432 * q^79 - 1388 * q^83 + 1744 * q^85 - 960 * q^89 + 464 * q^95 - 532 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−19.4833	−1.74264	−0.871320	−	0.490715i	$-0.836736\pi$
$5$	−19.4833	−1.74264	−0.871320	+	0.490715i	$0.836736\pi$
$6$	0	0
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	22.8999	0.627689	0.313844	−	0.949474i	$-0.398383\pi$
$11$	22.8999	0.627689	0.313844	+	0.949474i	$0.398383\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−67.0334	−0.956352	−0.478176	−	0.878264i	$-0.658702\pi$
$17$	−67.0334	−0.956352	−0.478176	+	0.878264i	$0.658702\pi$
$18$	0	0
$19$	−16.5167	−0.199431	−0.0997155	−	0.995016i	$-0.531793\pi$
$19$	−16.5167	−0.199431	−0.0997155	+	0.995016i	$0.531793\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−175.600	−1.59196	−0.795979	−	0.605324i	$-0.793044\pi$
$23$	−175.600	−1.59196	−0.795979	+	0.605324i	$0.793044\pi$
$24$	0	0
$25$	254.600	2.03680
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−291.800	−1.86848	−0.934239	−	0.356648i	$-0.883920\pi$
$29$	−291.800	−1.86848	−0.934239	+	0.356648i	$0.883920\pi$
$30$	0	0
$31$	−117.283	−0.679505	−0.339753	−	0.940515i	$-0.610343\pi$
$31$	−117.283	−0.679505	−0.339753	+	0.940515i	$0.610343\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	145.800	0.704133
$36$	0	0
$37$	−154.766	−0.687661	−0.343830	−	0.939032i	$-0.611724\pi$
$37$	−154.766	−0.687661	−0.343830	+	0.939032i	$0.611724\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	251.716	0.958815	0.479407	−	0.877592i	$-0.340852\pi$
$41$	251.716	0.958815	0.479407	+	0.877592i	$0.340852\pi$
$42$	0	0
$43$	502.566	1.78234	0.891170	−	0.453669i	$-0.149885\pi$
$43$	502.566	1.78234	0.891170	+	0.453669i	$0.149885\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−281.733	−0.874361	−0.437181	−	0.899374i	$-0.644023\pi$
$47$	−281.733	−0.874361	−0.437181	+	0.899374i	$0.644023\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−366.999	−0.951154	−0.475577	−	0.879674i	$-0.657761\pi$
$53$	−366.999	−0.951154	−0.475577	+	0.879674i	$0.657761\pi$
$54$	0	0
$55$	−446.166	−1.09384
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−79.6663	−0.175791	−0.0878955	−	0.996130i	$-0.528014\pi$
$59$	−79.6663	−0.175791	−0.0878955	+	0.996130i	$0.528014\pi$
$60$	0	0
$61$	−194.865	−0.409016	−0.204508	−	0.978865i	$-0.565559\pi$
$61$	−194.865	−0.409016	−0.204508	+	0.978865i	$0.565559\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	253.283	0.483322
$66$	0	0
$67$	−400.082	−0.729519	−0.364759	−	0.931102i	$-0.618849\pi$
$67$	−400.082	−0.729519	−0.364759	+	0.931102i	$0.618849\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	528.299	0.883065	0.441532	−	0.897245i	$-0.354435\pi$
$71$	528.299	0.883065	0.441532	+	0.897245i	$0.354435\pi$
$72$	0	0
$73$	−734.366	−1.17741	−0.588706	−	0.808347i	$-0.700362\pi$
$73$	−734.366	−1.17741	−0.588706	+	0.808347i	$0.700362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−171.367	−0.253625
$78$	0	0
$79$	−113.266	−0.161309	−0.0806545	−	0.996742i	$-0.525701\pi$
$79$	−113.266	−0.161309	−0.0806545	+	0.996742i	$0.525701\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−933.466	−1.23447	−0.617236	−	0.786778i	$-0.711748\pi$
$83$	−933.466	−1.23447	−0.617236	+	0.786778i	$0.711748\pi$
$84$	0	0
$85$	1306.03	1.66658
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1190.91	−1.41839	−0.709195	−	0.705012i	$-0.750941\pi$
$89$	−1190.91	−1.41839	−0.709195	+	0.705012i	$0.750941\pi$
$90$	0	0
$91$	97.2831	0.112066
$92$	0	0
$93$	0	0
$94$	0	0
$95$	321.800	0.347536
$96$	0	0
$97$	557.165	0.583211	0.291606	−	0.956539i	$-0.405811\pi$
$97$	557.165	0.583211	0.291606	+	0.956539i	$0.405811\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	286.766	0.282518	0.141259	−	0.989973i	$-0.454885\pi$
$101$	286.766	0.282518	0.141259	+	0.989973i	$0.454885\pi$
$102$	0	0
$103$	1911.36	1.82847	0.914234	−	0.405187i	$-0.132794\pi$
$103$	1911.36	1.82847	0.914234	+	0.405187i	$0.132794\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	834.334	0.753814	0.376907	−	0.926251i	$-0.376988\pi$
$107$	834.334	0.753814	0.376907	+	0.926251i	$0.376988\pi$
$108$	0	0
$109$	−1077.66	−0.946986	−0.473493	−	0.880798i	$-0.657007\pi$
$109$	−1077.66	−0.946986	−0.473493	+	0.880798i	$0.657007\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	166.065	0.138248	0.0691241	−	0.997608i	$-0.477980\pi$
$113$	166.065	0.138248	0.0691241	+	0.997608i	$0.477980\pi$
$114$	0	0
$115$	3421.26	2.77421
$116$	0	0
$117$	0	0
$118$	0	0
$119$	501.632	0.386424
$120$	0	0
$121$	−806.595	−0.606007
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2525.03	−1.80676
$126$	0	0
$127$	−1296.16	−0.905637	−0.452819	−	0.891603i	$-0.649581\pi$
$127$	−1296.16	−0.905637	−0.452819	+	0.891603i	$0.649581\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−197.201	−0.131523	−0.0657617	−	0.997835i	$-0.520948\pi$
$131$	−197.201	−0.131523	−0.0657617	+	0.997835i	$0.520948\pi$
$132$	0	0
$133$	123.600	0.0805823
$134$	0	0
$135$	0	0
$136$	0	0
$137$	546.915	0.341066	0.170533	−	0.985352i	$-0.445451\pi$
$137$	546.915	0.341066	0.170533	+	0.985352i	$0.445451\pi$
$138$	0	0
$139$	−609.666	−0.372023	−0.186012	−	0.982548i	$-0.559556\pi$
$139$	−609.666	−0.372023	−0.186012	+	0.982548i	$0.559556\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−297.699	−0.174090
$144$	0	0
$145$	5685.23	3.25609
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2165.08	1.19040	0.595202	−	0.803576i	$-0.297072\pi$
$149$	2165.08	1.19040	0.595202	+	0.803576i	$0.297072\pi$
$150$	0	0
$151$	846.549	0.456233	0.228116	−	0.973634i	$-0.426743\pi$
$151$	846.549	0.456233	0.228116	+	0.973634i	$0.426743\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2285.06	1.18413
$156$	0	0
$157$	1653.60	0.840581	0.420291	−	0.907390i	$-0.361928\pi$
$157$	1653.60	0.840581	0.420291	+	0.907390i	$0.361928\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1314.07	0.643248
$162$	0	0
$163$	2866.51	1.37744	0.688720	−	0.725027i	$-0.258173\pi$
$163$	2866.51	1.37744	0.688720	+	0.725027i	$0.258173\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	729.066	0.337825	0.168913	−	0.985631i	$-0.445974\pi$
$167$	729.066	0.337825	0.168913	+	0.985631i	$0.445974\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3834.83	1.68530	0.842650	−	0.538462i	$-0.180995\pi$
$173$	3834.83	1.68530	0.842650	+	0.538462i	$0.180995\pi$
$174$	0	0
$175$	−1905.25	−0.822990
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−283.862	−0.118530	−0.0592649	−	0.998242i	$-0.518876\pi$
$179$	−283.862	−0.118530	−0.0592649	+	0.998242i	$0.518876\pi$
$180$	0	0
$181$	2363.60	0.970634	0.485317	−	0.874338i	$-0.338704\pi$
$181$	2363.60	0.970634	0.485317	+	0.874338i	$0.338704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3015.36	1.19835
$186$	0	0
$187$	−1535.06	−0.600291
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2514.26	0.952491	0.476246	−	0.879312i	$-0.341997\pi$
$191$	2514.26	0.952491	0.476246	+	0.879312i	$0.341997\pi$
$192$	0	0
$193$	2420.73	0.902839	0.451420	−	0.892312i	$-0.350918\pi$
$193$	2420.73	0.902839	0.451420	+	0.892312i	$0.350918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4633.65	1.67581	0.837903	−	0.545819i	$-0.183781\pi$
$197$	4633.65	1.67581	0.837903	+	0.545819i	$0.183781\pi$
$198$	0	0
$199$	−3054.17	−1.08796	−0.543980	−	0.839098i	$-0.683083\pi$
$199$	−3054.17	−1.08796	−0.543980	+	0.839098i	$0.683083\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2183.63	0.754979
$204$	0	0
$205$	−4904.26	−1.67087
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−378.230	−0.125181
$210$	0	0
$211$	4031.60	1.31539	0.657694	−	0.753285i	$-0.271532\pi$
$211$	4031.60	1.31539	0.657694	+	0.753285i	$0.271532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9791.66	−3.10598
$216$	0	0
$217$	877.666	0.274562
$218$	0	0
$219$	0	0
$220$	0	0
$221$	871.434	0.265244
$222$	0	0
$223$	−3784.95	−1.13659	−0.568294	−	0.822826i	$-0.692396\pi$
$223$	−3784.95	−1.13659	−0.568294	+	0.822826i	$0.692396\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2013.83	0.588821	0.294411	−	0.955679i	$-0.404877\pi$
$227$	2013.83	0.588821	0.294411	+	0.955679i	$0.404877\pi$
$228$	0	0
$229$	−3050.73	−0.880340	−0.440170	−	0.897915i	$-0.645082\pi$
$229$	−3050.73	−0.880340	−0.440170	+	0.897915i	$0.645082\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5587.49	−1.57103	−0.785513	−	0.618846i	$-0.787601\pi$
$233$	−5587.49	−1.57103	−0.785513	+	0.618846i	$0.787601\pi$
$234$	0	0
$235$	5489.09	1.52370
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1335.69	−0.361501	−0.180750	−	0.983529i	$-0.557853\pi$
$239$	−1335.69	−0.361501	−0.180750	+	0.983529i	$0.557853\pi$
$240$	0	0
$241$	−571.558	−0.152769	−0.0763845	−	0.997078i	$-0.524338\pi$
$241$	−571.558	−0.152769	−0.0763845	+	0.997078i	$0.524338\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5591.71	1.45813
$246$	0	0
$247$	214.717	0.0553122
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4088.60	1.02817	0.514084	−	0.857740i	$-0.328132\pi$
$251$	4088.60	1.02817	0.514084	+	0.857740i	$0.328132\pi$
$252$	0	0
$253$	−4021.21	−0.999254
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3050.23	−0.740342	−0.370171	−	0.928964i	$-0.620701\pi$
$257$	−3050.23	−0.740342	−0.370171	+	0.928964i	$0.620701\pi$
$258$	0	0
$259$	1158.17	0.277857
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5770.99	1.35306	0.676530	−	0.736415i	$-0.263483\pi$
$263$	5770.99	1.35306	0.676530	+	0.736415i	$0.263483\pi$
$264$	0	0
$265$	7150.35	1.65752
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2079.40	0.471314	0.235657	−	0.971836i	$-0.424276\pi$
$269$	2079.40	0.471314	0.235657	+	0.971836i	$0.424276\pi$
$270$	0	0
$271$	−6012.00	−1.34761	−0.673807	−	0.738908i	$-0.735342\pi$
$271$	−6012.00	−1.34761	−0.673807	+	0.738908i	$0.735342\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5830.30	1.27847
$276$	0	0
$277$	−735.201	−0.159473	−0.0797364	−	0.996816i	$-0.525408\pi$
$277$	−735.201	−0.159473	−0.0797364	+	0.996816i	$0.525408\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1902.92	0.403981	0.201990	−	0.979387i	$-0.435259\pi$
$281$	1902.92	0.403981	0.201990	+	0.979387i	$0.435259\pi$
$282$	0	0
$283$	−2125.71	−0.446502	−0.223251	−	0.974761i	$-0.571667\pi$
$283$	−2125.71	−0.446502	−0.223251	+	0.974761i	$0.571667\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1883.67	−0.387420
$288$	0	0
$289$	−419.527	−0.0853913
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1641.03	0.327200	0.163600	−	0.986527i	$-0.447689\pi$
$293$	1641.03	0.327200	0.163600	+	0.986527i	$0.447689\pi$
$294$	0	0
$295$	1552.16	0.306341
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2282.79	0.441530
$300$	0	0
$301$	−3760.86	−0.720174
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3796.62	0.712767
$306$	0	0
$307$	3373.27	0.627111	0.313555	−	0.949570i	$-0.398480\pi$
$307$	3373.27	0.627111	0.313555	+	0.949570i	$0.398480\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−868.525	−0.158359	−0.0791793	−	0.996860i	$-0.525230\pi$
$311$	−868.525	−0.158359	−0.0791793	+	0.996860i	$0.525230\pi$
$312$	0	0
$313$	−4343.19	−0.784319	−0.392159	−	0.919897i	$-0.628272\pi$
$313$	−4343.19	−0.784319	−0.392159	+	0.919897i	$0.628272\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3277.65	0.580730	0.290365	−	0.956916i	$-0.406223\pi$
$317$	3277.65	0.580730	0.290365	+	0.956916i	$0.406223\pi$
$318$	0	0
$319$	−6682.18	−1.17282
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1107.17	0.190726
$324$	0	0
$325$	−3309.79	−0.564906
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2108.30	0.353295
$330$	0	0
$331$	−5589.62	−0.928197	−0.464099	−	0.885784i	$-0.653622\pi$
$331$	−5589.62	−0.928197	−0.464099	+	0.885784i	$0.653622\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7794.92	1.27129
$336$	0	0
$337$	901.544	0.145728	0.0728638	−	0.997342i	$-0.476786\pi$
$337$	901.544	0.145728	0.0728638	+	0.997342i	$0.476786\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2685.77	−0.426518
$342$	0	0
$343$	4714.49	0.742153
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−812.318	−0.125670	−0.0628350	−	0.998024i	$-0.520014\pi$
$347$	−812.318	−0.125670	−0.0628350	+	0.998024i	$0.520014\pi$
$348$	0	0
$349$	4437.96	0.680683	0.340342	−	0.940302i	$-0.389457\pi$
$349$	4437.96	0.680683	0.340342	+	0.940302i	$0.389457\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7115.35	−1.07284	−0.536419	−	0.843952i	$-0.680223\pi$
$353$	−7115.35	−1.07284	−0.536419	+	0.843952i	$0.680223\pi$
$354$	0	0
$355$	−10293.0	−1.53886
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4693.98	0.690081	0.345040	−	0.938588i	$-0.387865\pi$
$359$	4693.98	0.690081	0.345040	+	0.938588i	$0.387865\pi$
$360$	0	0
$361$	−6586.20	−0.960227
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14307.9	2.05181
$366$	0	0
$367$	−9243.98	−1.31480	−0.657400	−	0.753542i	$-0.728344\pi$
$367$	−9243.98	−1.31480	−0.657400	+	0.753542i	$0.728344\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2746.37	0.384324
$372$	0	0
$373$	−4311.99	−0.598569	−0.299285	−	0.954164i	$-0.596748\pi$
$373$	−4311.99	−0.598569	−0.299285	+	0.954164i	$0.596748\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3793.40	0.518223
$378$	0	0
$379$	2382.73	0.322936	0.161468	−	0.986878i	$-0.448377\pi$
$379$	2382.73	0.322936	0.161468	+	0.986878i	$0.448377\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4845.81	0.646499	0.323250	−	0.946314i	$-0.395225\pi$
$383$	4845.81	0.646499	0.323250	+	0.946314i	$0.395225\pi$
$384$	0	0
$385$	3338.80	0.441976
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9561.50	−1.24624	−0.623120	−	0.782127i	$-0.714135\pi$
$389$	−9561.50	−1.24624	−0.623120	+	0.782127i	$0.714135\pi$
$390$	0	0
$391$	11771.0	1.52247
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2206.79	0.281103
$396$	0	0
$397$	−7440.11	−0.940575	−0.470287	−	0.882513i	$-0.655850\pi$
$397$	−7440.11	−0.940575	−0.470287	+	0.882513i	$0.655850\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8687.80	1.08192	0.540958	−	0.841050i	$-0.318062\pi$
$401$	8687.80	1.08192	0.540958	+	0.841050i	$0.318062\pi$
$402$	0	0
$403$	1524.68	0.188461
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3544.13	−0.431637
$408$	0	0
$409$	2556.10	0.309024	0.154512	−	0.987991i	$-0.450619\pi$
$409$	2556.10	0.309024	0.154512	+	0.987991i	$0.450619\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	596.168	0.0710303
$414$	0	0
$415$	18187.0	2.15124
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3347.46	−0.390296	−0.195148	−	0.980774i	$-0.562519\pi$
$419$	−3347.46	−0.390296	−0.195148	+	0.980774i	$0.562519\pi$
$420$	0	0
$421$	−1854.48	−0.214684	−0.107342	−	0.994222i	$-0.534234\pi$
$421$	−1854.48	−0.214684	−0.107342	+	0.994222i	$0.534234\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−17066.7	−1.94789
$426$	0	0
$427$	1458.24	0.165267
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14043.1	−1.56945	−0.784725	−	0.619844i	$-0.787196\pi$
$431$	−14043.1	−1.56945	−0.784725	+	0.619844i	$0.787196\pi$
$432$	0	0
$433$	3086.47	0.342555	0.171278	−	0.985223i	$-0.445210\pi$
$433$	3086.47	0.342555	0.171278	+	0.985223i	$0.445210\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2900.32	0.317486
$438$	0	0
$439$	−2837.68	−0.308508	−0.154254	−	0.988031i	$-0.549297\pi$
$439$	−2837.68	−0.308508	−0.154254	+	0.988031i	$0.549297\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18309.4	1.96367	0.981834	−	0.189744i	$-0.0607658\pi$
$443$	18309.4	1.96367	0.981834	+	0.189744i	$0.0607658\pi$
$444$	0	0
$445$	23203.0	2.47174
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13861.2	−1.45690	−0.728451	−	0.685098i	$-0.759759\pi$
$449$	−13861.2	−1.45690	−0.728451	+	0.685098i	$0.759759\pi$
$450$	0	0
$451$	5764.26	0.601837
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1895.40	−0.195291
$456$	0	0
$457$	−8990.36	−0.920243	−0.460122	−	0.887856i	$-0.652194\pi$
$457$	−8990.36	−0.920243	−0.460122	+	0.887856i	$0.652194\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3406.90	0.344198	0.172099	−	0.985080i	$-0.444945\pi$
$461$	3406.90	0.344198	0.172099	+	0.985080i	$0.444945\pi$
$462$	0	0
$463$	7498.45	0.752662	0.376331	−	0.926485i	$-0.377186\pi$
$463$	7498.45	0.752662	0.376331	+	0.926485i	$0.377186\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7711.38	0.764112	0.382056	−	0.924139i	$-0.375216\pi$
$467$	7711.38	0.764112	0.382056	+	0.924139i	$0.375216\pi$
$468$	0	0
$469$	2993.94	0.294770
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11508.7	1.11875
$474$	0	0
$475$	−4205.14	−0.406200
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9439.82	−0.900451	−0.450226	−	0.892915i	$-0.648656\pi$
$479$	−9439.82	−0.900451	−0.450226	+	0.892915i	$0.648656\pi$
$480$	0	0
$481$	2011.96	0.190723
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10855.4	−1.01633
$486$	0	0
$487$	6156.20	0.572821	0.286411	−	0.958107i	$-0.407538\pi$
$487$	6156.20	0.572821	0.286411	+	0.958107i	$0.407538\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3842.74	0.353198	0.176599	−	0.984283i	$-0.443490\pi$
$491$	3842.74	0.353198	0.176599	+	0.984283i	$0.443490\pi$
$492$	0	0
$493$	19560.3	1.78692
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3953.43	−0.356812
$498$	0	0
$499$	12842.4	1.15211	0.576056	−	0.817410i	$-0.304591\pi$
$499$	12842.4	1.15211	0.576056	+	0.817410i	$0.304591\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8580.11	0.760573	0.380287	−	0.924869i	$-0.375825\pi$
$503$	8580.11	0.760573	0.380287	+	0.924869i	$0.375825\pi$
$504$	0	0
$505$	−5587.16	−0.492327
$506$	0	0
$507$	0	0
$508$	0	0
$509$	43.5957	0.00379635	0.00189818	−	0.999998i	$-0.499396\pi$
$509$	43.5957	0.00379635	0.00189818	+	0.999998i	$0.499396\pi$
$510$	0	0
$511$	5495.49	0.475746
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−37239.7	−3.18636
$516$	0	0
$517$	−6451.66	−0.548827
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11368.1	−0.955939	−0.477969	−	0.878377i	$-0.658627\pi$
$521$	−11368.1	−0.955939	−0.477969	+	0.878377i	$0.658627\pi$
$522$	0	0
$523$	5229.53	0.437230	0.218615	−	0.975811i	$-0.429846\pi$
$523$	5229.53	0.437230	0.218615	+	0.975811i	$0.429846\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7861.88	0.649846
$528$	0	0
$529$	18668.2	1.53433
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3272.31	−0.265927
$534$	0	0
$535$	−16255.6	−1.31363
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6572.27	−0.525209
$540$	0	0
$541$	−6567.99	−0.521959	−0.260980	−	0.965344i	$-0.584046\pi$
$541$	−6567.99	−0.521959	−0.260980	+	0.965344i	$0.584046\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20996.5	1.65026
$546$	0	0
$547$	13675.7	1.06897	0.534487	−	0.845177i	$-0.320505\pi$
$547$	13675.7	1.06897	0.534487	+	0.845177i	$0.320505\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4819.57	0.372632
$552$	0	0
$553$	847.604	0.0651786
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4527.96	−0.344445	−0.172222	−	0.985058i	$-0.555095\pi$
$557$	−4527.96	−0.344445	−0.172222	+	0.985058i	$0.555095\pi$
$558$	0	0
$559$	−6533.36	−0.494332
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18441.8	1.38051	0.690256	−	0.723566i	$-0.257498\pi$
$563$	18441.8	1.38051	0.690256	+	0.723566i	$0.257498\pi$
$564$	0	0
$565$	−3235.49	−0.240917
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13553.5	0.998578	0.499289	−	0.866436i	$-0.333595\pi$
$569$	13553.5	0.998578	0.499289	+	0.866436i	$0.333595\pi$
$570$	0	0
$571$	−14815.5	−1.08583	−0.542915	−	0.839788i	$-0.682679\pi$
$571$	−14815.5	−1.08583	−0.542915	+	0.839788i	$0.682679\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−44707.6	−3.24249
$576$	0	0
$577$	21596.2	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$577$	21596.2	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6985.42	0.498802
$582$	0	0
$583$	−8404.23	−0.597029
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−918.801	−0.0646047	−0.0323024	−	0.999478i	$-0.510284\pi$
$587$	−918.801	−0.0646047	−0.0323024	+	0.999478i	$0.510284\pi$
$588$	0	0
$589$	1937.13	0.135514
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19816.0	−1.37226	−0.686128	−	0.727481i	$-0.740691\pi$
$593$	−19816.0	−1.37226	−0.686128	+	0.727481i	$0.740691\pi$
$594$	0	0
$595$	−9773.45	−0.673399
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5141.86	−0.350736	−0.175368	−	0.984503i	$-0.556111\pi$
$599$	−5141.86	−0.350736	−0.175368	+	0.984503i	$0.556111\pi$
$600$	0	0
$601$	12380.9	0.840312	0.420156	−	0.907452i	$-0.361975\pi$
$601$	12380.9	0.840312	0.420156	+	0.907452i	$0.361975\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15715.1	1.05605
$606$	0	0
$607$	23717.0	1.58590	0.792951	−	0.609286i	$-0.208544\pi$
$607$	23717.0	1.58590	0.792951	+	0.609286i	$0.208544\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3662.53	0.242504
$612$	0	0
$613$	−26157.1	−1.72345	−0.861726	−	0.507373i	$-0.830617\pi$
$613$	−26157.1	−1.72345	−0.861726	+	0.507373i	$0.830617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23613.9	−1.54077	−0.770387	−	0.637576i	$-0.779937\pi$
$617$	−23613.9	−1.54077	−0.770387	+	0.637576i	$0.779937\pi$
$618$	0	0
$619$	−23345.4	−1.51588	−0.757940	−	0.652324i	$-0.773794\pi$
$619$	−23345.4	−1.51588	−0.757940	+	0.652324i	$0.773794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8911.99	0.573116
$624$	0	0
$625$	17371.0	1.11174
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10374.5	0.657645
$630$	0	0
$631$	−15245.7	−0.961841	−0.480921	−	0.876764i	$-0.659698\pi$
$631$	−15245.7	−0.961841	−0.480921	+	0.876764i	$0.659698\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	25253.6	1.57820
$636$	0	0
$637$	3731.00	0.232068
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10192.7	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$641$	−10192.7	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$642$	0	0
$643$	5506.31	0.337710	0.168855	−	0.985641i	$-0.445993\pi$
$643$	5506.31	0.337710	0.168855	+	0.985641i	$0.445993\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13297.5	−0.808005	−0.404003	−	0.914758i	$-0.632381\pi$
$647$	−13297.5	−0.808005	−0.404003	+	0.914758i	$0.632381\pi$
$648$	0	0
$649$	−1824.35	−0.110342
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12440.2	0.745519	0.372760	−	0.927928i	$-0.378412\pi$
$653$	12440.2	0.745519	0.372760	+	0.927928i	$0.378412\pi$
$654$	0	0
$655$	3842.14	0.229198
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9562.87	−0.565276	−0.282638	−	0.959227i	$-0.591209\pi$
$659$	−9562.87	−0.565276	−0.282638	+	0.959227i	$0.591209\pi$
$660$	0	0
$661$	2409.69	0.141795	0.0708973	−	0.997484i	$-0.477414\pi$
$661$	2409.69	0.141795	0.0708973	+	0.997484i	$0.477414\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2408.13	−0.140426
$666$	0	0
$667$	51239.9	2.97454
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4462.40	−0.256735
$672$	0	0
$673$	7929.02	0.454147	0.227074	−	0.973878i	$-0.427084\pi$
$673$	7929.02	0.454147	0.227074	+	0.973878i	$0.427084\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2628.26	0.149206	0.0746030	−	0.997213i	$-0.476231\pi$
$677$	2628.26	0.149206	0.0746030	+	0.997213i	$0.476231\pi$
$678$	0	0
$679$	−4169.44	−0.235653
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10021.5	0.561437	0.280719	−	0.959790i	$-0.409427\pi$
$683$	10021.5	0.561437	0.280719	+	0.959790i	$0.409427\pi$
$684$	0	0
$685$	−10655.7	−0.594356
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4770.99	0.263803
$690$	0	0
$691$	−23987.2	−1.32057	−0.660286	−	0.751014i	$-0.729565\pi$
$691$	−23987.2	−1.32057	−0.660286	+	0.751014i	$0.729565\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11878.3	0.648303
$696$	0	0
$697$	−16873.4	−0.916964
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3763.71	0.202787	0.101393	−	0.994846i	$-0.467670\pi$
$701$	3763.71	0.202787	0.101393	+	0.994846i	$0.467670\pi$
$702$	0	0
$703$	2556.23	0.137141
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2145.96	−0.114155
$708$	0	0
$709$	−36047.8	−1.90946	−0.954728	−	0.297479i	$-0.903854\pi$
$709$	−36047.8	−1.90946	−0.954728	+	0.297479i	$0.903854\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20594.9	1.08174
$714$	0	0
$715$	5800.15	0.303376
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3944.18	−0.204580	−0.102290	−	0.994755i	$-0.532617\pi$
$719$	−3944.18	−0.204580	−0.102290	+	0.994755i	$0.532617\pi$
$720$	0	0
$721$	−14303.3	−0.738812
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−74292.1	−3.80571
$726$	0	0
$727$	20447.8	1.04315	0.521573	−	0.853206i	$-0.325345\pi$
$727$	20447.8	1.04315	0.521573	+	0.853206i	$0.325345\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−33688.7	−1.70454
$732$	0	0
$733$	−13536.2	−0.682089	−0.341045	−	0.940047i	$-0.610781\pi$
$733$	−13536.2	−0.682089	−0.341045	+	0.940047i	$0.610781\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9161.83	−0.457911
$738$	0	0
$739$	−15839.1	−0.788433	−0.394217	−	0.919018i	$-0.628984\pi$
$739$	−15839.1	−0.788433	−0.394217	+	0.919018i	$0.628984\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1664.92	−0.0822075	−0.0411037	−	0.999155i	$-0.513087\pi$
$743$	−1664.92	−0.0822075	−0.0411037	+	0.999155i	$0.513087\pi$
$744$	0	0
$745$	−42182.9	−2.07445
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−6243.58	−0.304587
$750$	0	0
$751$	−22399.1	−1.08835	−0.544177	−	0.838970i	$-0.683158\pi$
$751$	−22399.1	−1.08835	−0.544177	+	0.838970i	$0.683158\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16493.6	−0.795050
$756$	0	0
$757$	23798.9	1.14265	0.571326	−	0.820723i	$-0.306429\pi$
$757$	23798.9	1.14265	0.571326	+	0.820723i	$0.306429\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13693.5	−0.652285	−0.326142	−	0.945321i	$-0.605749\pi$
$761$	−13693.5	−0.652285	−0.326142	+	0.945321i	$0.605749\pi$
$762$	0	0
$763$	8064.50	0.382640
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1035.66	0.0487556
$768$	0	0
$769$	16299.9	0.764358	0.382179	−	0.924088i	$-0.375174\pi$
$769$	16299.9	0.764358	0.382179	+	0.924088i	$0.375174\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33532.2	−1.56024	−0.780122	−	0.625628i	$-0.784843\pi$
$773$	−33532.2	−1.56024	−0.780122	+	0.625628i	$0.784843\pi$
$774$	0	0
$775$	−29860.2	−1.38401
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4157.51	−0.191217
$780$	0	0
$781$	12098.0	0.554290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−32217.5	−1.46483
$786$	0	0
$787$	−16163.3	−0.732097	−0.366049	−	0.930596i	$-0.619290\pi$
$787$	−16163.3	−0.732097	−0.366049	+	0.930596i	$0.619290\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1242.71	−0.0558607
$792$	0	0
$793$	2533.25	0.113441
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39636.4	1.76160	0.880798	−	0.473492i	$-0.157007\pi$
$797$	39636.4	1.76160	0.880798	+	0.473492i	$0.157007\pi$
$798$	0	0
$799$	18885.5	0.836197
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16816.9	−0.739048
$804$	0	0
$805$	−25602.4	−1.12095
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23811.2	1.03481	0.517403	−	0.855742i	$-0.326899\pi$
$809$	23811.2	1.03481	0.517403	+	0.855742i	$0.326899\pi$
$810$	0	0
$811$	−27218.6	−1.17851	−0.589256	−	0.807946i	$-0.700579\pi$
$811$	−27218.6	−1.17851	−0.589256	+	0.807946i	$0.700579\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−55849.2	−2.40038
$816$	0	0
$817$	−8300.73	−0.355454
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43094.8	1.83193	0.915967	−	0.401253i	$-0.131425\pi$
$821$	43094.8	1.83193	0.915967	+	0.401253i	$0.131425\pi$
$822$	0	0
$823$	−26541.1	−1.12414	−0.562068	−	0.827091i	$-0.689994\pi$
$823$	−26541.1	−1.12414	−0.562068	+	0.827091i	$0.689994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44898.7	−1.88788	−0.943942	−	0.330112i	$-0.892913\pi$
$827$	−44898.7	−1.88788	−0.943942	+	0.330112i	$0.892913\pi$
$828$	0	0
$829$	−7137.48	−0.299029	−0.149514	−	0.988760i	$-0.547771\pi$
$829$	−7137.48	−0.299029	−0.149514	+	0.988760i	$0.547771\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	19238.6	0.800213
$834$	0	0
$835$	−14204.6	−0.588708
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4387.17	−0.180527	−0.0902634	−	0.995918i	$-0.528771\pi$
$839$	−4387.17	−0.180527	−0.0902634	+	0.995918i	$0.528771\pi$
$840$	0	0
$841$	60758.1	2.49121
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3292.68	−0.134049
$846$	0	0
$847$	6036.01	0.244864
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27176.9	1.09473
$852$	0	0
$853$	−9328.85	−0.374459	−0.187230	−	0.982316i	$-0.559951\pi$
$853$	−9328.85	−0.374459	−0.187230	+	0.982316i	$0.559951\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5010.39	0.199710	0.0998552	−	0.995002i	$-0.468162\pi$
$857$	5010.39	0.199710	0.0998552	+	0.995002i	$0.468162\pi$
$858$	0	0
$859$	−30233.4	−1.20088	−0.600438	−	0.799672i	$-0.705007\pi$
$859$	−30233.4	−1.20088	−0.600438	+	0.799672i	$0.705007\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4334.93	0.170988	0.0854940	−	0.996339i	$-0.472753\pi$
$863$	4334.93	0.170988	0.0854940	+	0.996339i	$0.472753\pi$
$864$	0	0
$865$	−74715.2	−2.93687
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2593.78	−0.101252
$870$	0	0
$871$	5201.06	0.202332
$872$	0	0
$873$	0	0
$874$	0	0
$875$	18895.6	0.730043
$876$	0	0
$877$	34683.3	1.33543	0.667716	−	0.744416i	$-0.267272\pi$
$877$	34683.3	1.33543	0.667716	+	0.744416i	$0.267272\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18269.2	0.698642	0.349321	−	0.937003i	$-0.386412\pi$
$881$	18269.2	0.698642	0.349321	+	0.937003i	$0.386412\pi$
$882$	0	0
$883$	14592.0	0.556128	0.278064	−	0.960563i	$-0.410307\pi$
$883$	14592.0	0.556128	0.278064	+	0.960563i	$0.410307\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30459.3	1.15301	0.576507	−	0.817092i	$-0.304415\pi$
$887$	30459.3	1.15301	0.576507	+	0.817092i	$0.304415\pi$
$888$	0	0
$889$	9699.60	0.365933
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4653.30	0.174375
$894$	0	0
$895$	5530.57	0.206555
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34223.2	1.26964
$900$	0	0
$901$	24601.2	0.909638
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46050.7	−1.69147
$906$	0	0
$907$	9364.89	0.342840	0.171420	−	0.985198i	$-0.445164\pi$
$907$	9364.89	0.342840	0.171420	+	0.985198i	$0.445164\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32479.8	1.18123	0.590616	−	0.806952i	$-0.298885\pi$
$911$	32479.8	1.18123	0.590616	+	0.806952i	$0.298885\pi$
$912$	0	0
$913$	−21376.3	−0.774864
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1475.72	0.0531435
$918$	0	0
$919$	−295.958	−0.0106232	−0.00531161	−	0.999986i	$-0.501691\pi$
$919$	−295.958	−0.0106232	−0.00531161	+	0.999986i	$0.501691\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6867.89	−0.244918
$924$	0	0
$925$	−39403.5	−1.40062
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5620.38	0.198492	0.0992458	−	0.995063i	$-0.468357\pi$
$929$	5620.38	0.198492	0.0992458	+	0.995063i	$0.468357\pi$
$930$	0	0
$931$	4740.29	0.166871
$932$	0	0
$933$	0	0
$934$	0	0
$935$	29908.0	1.04609
$936$	0	0
$937$	−32583.1	−1.13601	−0.568006	−	0.823024i	$-0.692285\pi$
$937$	−32583.1	−1.13601	−0.568006	+	0.823024i	$0.692285\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8812.99	−0.305308	−0.152654	−	0.988280i	$-0.548782\pi$
$941$	−8812.99	−0.305308	−0.152654	+	0.988280i	$0.548782\pi$
$942$	0	0
$943$	−44201.2	−1.52639
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13426.8	0.460732	0.230366	−	0.973104i	$-0.426008\pi$
$947$	13426.8	0.460732	0.230366	+	0.973104i	$0.426008\pi$
$948$	0	0
$949$	9546.76	0.326555
$950$	0	0
$951$	0	0
$952$	0	0
$953$	13394.6	0.455293	0.227647	−	0.973744i	$-0.426897\pi$
$953$	13394.6	0.455293	0.227647	+	0.973744i	$0.426897\pi$
$954$	0	0
$955$	−48986.2	−1.65985
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4092.74	−0.137812
$960$	0	0
$961$	−16035.7	−0.538273
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−47163.8	−1.57332
$966$	0	0
$967$	−45590.8	−1.51613	−0.758066	−	0.652178i	$-0.773856\pi$
$967$	−45590.8	−1.51613	−0.758066	+	0.652178i	$0.773856\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	264.763	0.00875041	0.00437521	−	0.999990i	$-0.498607\pi$
$971$	264.763	0.00875041	0.00437521	+	0.999990i	$0.498607\pi$
$972$	0	0
$973$	4562.32	0.150320
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−610.521	−0.0199921	−0.00999606	−	0.999950i	$-0.503182\pi$
$977$	−610.521	−0.0199921	−0.00999606	+	0.999950i	$0.503182\pi$
$978$	0	0
$979$	−27271.8	−0.890308
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57829.7	−1.87638	−0.938190	−	0.346121i	$-0.887499\pi$
$983$	−57829.7	−1.87638	−0.938190	+	0.346121i	$0.887499\pi$
$984$	0	0
$985$	−90278.8	−2.92033
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−88250.4	−2.83741
$990$	0	0
$991$	56780.7	1.82008	0.910039	−	0.414522i	$-0.136051\pi$
$991$	56780.7	1.82008	0.910039	+	0.414522i	$0.136051\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	59505.3	1.89592
$996$	0	0
$997$	18616.6	0.591369	0.295684	−	0.955286i	$-0.404452\pi$
$997$	18616.6	0.591369	0.295684	+	0.955286i	$0.404452\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.t.1.1		2
3.2	odd	2		624.4.a.r.1.2		2
4.3	odd	2		117.4.a.c.1.2		2
12.11	even	2		39.4.a.b.1.1	✓	2
24.5	odd	2		2496.4.a.s.1.1		2
24.11	even	2		2496.4.a.bc.1.1		2
52.51	odd	2		1521.4.a.s.1.1		2
60.59	even	2		975.4.a.j.1.2		2
84.83	odd	2		1911.4.a.h.1.1		2
156.47	odd	4		507.4.b.f.337.2		4
156.83	odd	4		507.4.b.f.337.3		4
156.155	even	2		507.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.1	✓	2	12.11	even	2
117.4.a.c.1.2		2	4.3	odd	2
507.4.a.f.1.2		2	156.155	even	2
507.4.b.f.337.2		4	156.47	odd	4
507.4.b.f.337.3		4	156.83	odd	4
624.4.a.r.1.2		2	3.2	odd	2
975.4.a.j.1.2		2	60.59	even	2
1521.4.a.s.1.1		2	52.51	odd	2
1872.4.a.t.1.1		2	1.1	even	1	trivial
1911.4.a.h.1.1		2	84.83	odd	2
2496.4.a.s.1.1		2	24.5	odd	2
2496.4.a.bc.1.1		2	24.11	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q-19.4833 q^{5} -7.48331 q^{7} +O(q^{10})\)	\(q-19.4833 q^{5} -7.48331 q^{7} +22.8999 q^{11} -13.0000 q^{13} -67.0334 q^{17} -16.5167 q^{19} -175.600 q^{23} +254.600 q^{25} -291.800 q^{29} -117.283 q^{31} +145.800 q^{35} -154.766 q^{37} +251.716 q^{41} +502.566 q^{43} -281.733 q^{47} -287.000 q^{49} -366.999 q^{53} -446.166 q^{55} -79.6663 q^{59} -194.865 q^{61} +253.283 q^{65} -400.082 q^{67} +528.299 q^{71} -734.366 q^{73} -171.367 q^{77} -113.266 q^{79} -933.466 q^{83} +1306.03 q^{85} -1190.91 q^{89} +97.2831 q^{91} +321.800 q^{95} +557.165 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{5}+O(q^{10}) \) 2 * q - 24 * q^5	\( 2 q - 24 q^{5} - 44 q^{11} - 26 q^{13} - 164 q^{17} - 48 q^{19} + 8 q^{23} + 150 q^{25} - 404 q^{29} - 40 q^{31} + 112 q^{35} - 100 q^{37} - 200 q^{41} + 616 q^{43} - 324 q^{47} - 574 q^{49} + 164 q^{53} - 144 q^{55} + 140 q^{59} + 628 q^{61} + 312 q^{65} + 472 q^{67} + 428 q^{71} - 900 q^{73} - 672 q^{77} + 432 q^{79} - 1388 q^{83} + 1744 q^{85} - 960 q^{89} + 464 q^{95} - 532 q^{97}+O(q^{100}) \) 2 * q - 24 * q^5 - 44 * q^11 - 26 * q^13 - 164 * q^17 - 48 * q^19 + 8 * q^23 + 150 * q^25 - 404 * q^29 - 40 * q^31 + 112 * q^35 - 100 * q^37 - 200 * q^41 + 616 * q^43 - 324 * q^47 - 574 * q^49 + 164 * q^53 - 144 * q^55 + 140 * q^59 + 628 * q^61 + 312 * q^65 + 472 * q^67 + 428 * q^71 - 900 * q^73 - 672 * q^77 + 432 * q^79 - 1388 * q^83 + 1744 * q^85 - 960 * q^89 + 464 * q^95 - 532 * q^97

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