LMFDB - Embedded newform 7440.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} -2.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} -3.12311 q^{17} +7.68466 q^{19} -2.56155 q^{21} -1.43845 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.12311 q^{29} -1.00000 q^{31} -2.56155 q^{33} -2.56155 q^{35} -3.12311 q^{37} +2.00000 q^{39} +7.12311 q^{41} -12.8078 q^{43} +1.00000 q^{45} -5.12311 q^{47} -0.438447 q^{49} -3.12311 q^{51} +7.43845 q^{53} -2.56155 q^{55} +7.68466 q^{57} +13.1231 q^{59} +6.00000 q^{61} -2.56155 q^{63} +2.00000 q^{65} +15.3693 q^{67} -1.43845 q^{69} +7.68466 q^{71} -10.8078 q^{73} +1.00000 q^{75} +6.56155 q^{77} +4.31534 q^{79} +1.00000 q^{81} -14.2462 q^{83} -3.12311 q^{85} +7.12311 q^{87} -13.6847 q^{89} -5.12311 q^{91} -1.00000 q^{93} +7.68466 q^{95} -6.00000 q^{97} -2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 + 3 * q^19 - q^21 - 7 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^31 - q^33 - q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 - 5 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 + 2 * q^51 + 19 * q^53 - q^55 + 3 * q^57 + 18 * q^59 + 12 * q^61 - q^63 + 4 * q^65 + 6 * q^67 - 7 * q^69 + 3 * q^71 - q^73 + 2 * q^75 + 9 * q^77 + 21 * q^79 + 2 * q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 15 * q^89 - 2 * q^91 - 2 * q^93 + 3 * q^95 - 12 * q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	7.68466	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$19$	7.68466	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	−1.43845	−0.299937	−0.149968	−	0.988691i	$-0.547917\pi$
$23$	−1.43845	−0.299937	−0.149968	+	0.988691i	$0.547917\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−2.56155	−0.445909
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	−12.8078	−1.95317	−0.976583	−	0.215142i	$-0.930979\pi$
$43$	−12.8078	−1.95317	−0.976583	+	0.215142i	$0.930979\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−5.12311	−0.747282	−0.373641	−	0.927573i	$-0.621891\pi$
$47$	−5.12311	−0.747282	−0.373641	+	0.927573i	$0.621891\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	7.43845	1.02175	0.510875	−	0.859655i	$-0.329322\pi$
$53$	7.43845	1.02175	0.510875	+	0.859655i	$0.329322\pi$
$54$	0	0
$55$	−2.56155	−0.345400
$56$	0	0
$57$	7.68466	1.01786
$58$	0	0
$59$	13.1231	1.70848	0.854241	−	0.519877i	$-0.174022\pi$
$59$	13.1231	1.70848	0.854241	+	0.519877i	$0.174022\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−2.56155	−0.322725
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	15.3693	1.87766	0.938830	−	0.344380i	$-0.111911\pi$
$67$	15.3693	1.87766	0.938830	+	0.344380i	$0.111911\pi$
$68$	0	0
$69$	−1.43845	−0.173169
$70$	0	0
$71$	7.68466	0.912001	0.456001	−	0.889979i	$-0.349281\pi$
$71$	7.68466	0.912001	0.456001	+	0.889979i	$0.349281\pi$
$72$	0	0
$73$	−10.8078	−1.26495	−0.632477	−	0.774580i	$-0.717962\pi$
$73$	−10.8078	−1.26495	−0.632477	+	0.774580i	$0.717962\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.56155	0.747758
$78$	0	0
$79$	4.31534	0.485514	0.242757	−	0.970087i	$-0.421948\pi$
$79$	4.31534	0.485514	0.242757	+	0.970087i	$0.421948\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.2462	−1.56372	−0.781862	−	0.623451i	$-0.785730\pi$
$83$	−14.2462	−1.56372	−0.781862	+	0.623451i	$0.785730\pi$
$84$	0	0
$85$	−3.12311	−0.338748
$86$	0	0
$87$	7.12311	0.763677
$88$	0	0
$89$	−13.6847	−1.45057	−0.725285	−	0.688448i	$-0.758292\pi$
$89$	−13.6847	−1.45057	−0.725285	+	0.688448i	$0.758292\pi$
$90$	0	0
$91$	−5.12311	−0.537047
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	7.68466	0.788429
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−2.56155	−0.257446
$100$	0	0
$101$	−0.561553	−0.0558766	−0.0279383	−	0.999610i	$-0.508894\pi$
$101$	−0.561553	−0.0558766	−0.0279383	+	0.999610i	$0.508894\pi$
$102$	0	0
$103$	1.75379	0.172806	0.0864030	−	0.996260i	$-0.472463\pi$
$103$	1.75379	0.172806	0.0864030	+	0.996260i	$0.472463\pi$
$104$	0	0
$105$	−2.56155	−0.249982
$106$	0	0
$107$	2.56155	0.247635	0.123817	−	0.992305i	$-0.460486\pi$
$107$	2.56155	0.247635	0.123817	+	0.992305i	$0.460486\pi$
$108$	0	0
$109$	5.36932	0.514287	0.257144	−	0.966373i	$-0.417219\pi$
$109$	5.36932	0.514287	0.257144	+	0.966373i	$0.417219\pi$
$110$	0	0
$111$	−3.12311	−0.296432
$112$	0	0
$113$	−1.68466	−0.158479	−0.0792397	−	0.996856i	$-0.525249\pi$
$113$	−1.68466	−0.158479	−0.0792397	+	0.996856i	$0.525249\pi$
$114$	0	0
$115$	−1.43845	−0.134136
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	7.12311	0.642269
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−12.8078	−1.12766
$130$	0	0
$131$	15.3693	1.34282	0.671412	−	0.741085i	$-0.265688\pi$
$131$	15.3693	1.34282	0.671412	+	0.741085i	$0.265688\pi$
$132$	0	0
$133$	−19.6847	−1.70688
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	20.2462	1.72975	0.864875	−	0.501987i	$-0.167397\pi$
$137$	20.2462	1.72975	0.864875	+	0.501987i	$0.167397\pi$
$138$	0	0
$139$	17.1231	1.45236	0.726181	−	0.687503i	$-0.241293\pi$
$139$	17.1231	1.45236	0.726181	+	0.687503i	$0.241293\pi$
$140$	0	0
$141$	−5.12311	−0.431443
$142$	0	0
$143$	−5.12311	−0.428416
$144$	0	0
$145$	7.12311	0.591542
$146$	0	0
$147$	−0.438447	−0.0361625
$148$	0	0
$149$	17.0540	1.39712	0.698558	−	0.715553i	$-0.253825\pi$
$149$	17.0540	1.39712	0.698558	+	0.715553i	$0.253825\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−3.12311	−0.252488
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	17.0540	1.36106	0.680528	−	0.732722i	$-0.261751\pi$
$157$	17.0540	1.36106	0.680528	+	0.732722i	$0.261751\pi$
$158$	0	0
$159$	7.43845	0.589907
$160$	0	0
$161$	3.68466	0.290392
$162$	0	0
$163$	15.3693	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$163$	15.3693	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$164$	0	0
$165$	−2.56155	−0.199417
$166$	0	0
$167$	6.56155	0.507748	0.253874	−	0.967237i	$-0.418295\pi$
$167$	6.56155	0.507748	0.253874	+	0.967237i	$0.418295\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	7.68466	0.587661
$172$	0	0
$173$	8.24621	0.626948	0.313474	−	0.949597i	$-0.398507\pi$
$173$	8.24621	0.626948	0.313474	+	0.949597i	$0.398507\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	0	0
$177$	13.1231	0.986393
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−13.0540	−0.970294	−0.485147	−	0.874433i	$-0.661234\pi$
$181$	−13.0540	−0.970294	−0.485147	+	0.874433i	$0.661234\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−3.12311	−0.229615
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−2.56155	−0.186326
$190$	0	0
$191$	14.2462	1.03082	0.515410	−	0.856944i	$-0.327640\pi$
$191$	14.2462	1.03082	0.515410	+	0.856944i	$0.327640\pi$
$192$	0	0
$193$	14.4924	1.04319	0.521594	−	0.853194i	$-0.325338\pi$
$193$	14.4924	1.04319	0.521594	+	0.853194i	$0.325338\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	16.8078	1.19147	0.595735	−	0.803181i	$-0.296861\pi$
$199$	16.8078	1.19147	0.595735	+	0.803181i	$0.296861\pi$
$200$	0	0
$201$	15.3693	1.08407
$202$	0	0
$203$	−18.2462	−1.28063
$204$	0	0
$205$	7.12311	0.497499
$206$	0	0
$207$	−1.43845	−0.0999790
$208$	0	0
$209$	−19.6847	−1.36162
$210$	0	0
$211$	23.6847	1.63052	0.815260	−	0.579096i	$-0.196594\pi$
$211$	23.6847	1.63052	0.815260	+	0.579096i	$0.196594\pi$
$212$	0	0
$213$	7.68466	0.526544
$214$	0	0
$215$	−12.8078	−0.873482
$216$	0	0
$217$	2.56155	0.173890
$218$	0	0
$219$	−10.8078	−0.730321
$220$	0	0
$221$	−6.24621	−0.420166
$222$	0	0
$223$	21.1231	1.41451	0.707254	−	0.706960i	$-0.249934\pi$
$223$	21.1231	1.41451	0.707254	+	0.706960i	$0.249934\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.68466	−0.510049	−0.255024	−	0.966935i	$-0.582083\pi$
$227$	−7.68466	−0.510049	−0.255024	+	0.966935i	$0.582083\pi$
$228$	0	0
$229$	−16.5616	−1.09442	−0.547209	−	0.836996i	$-0.684310\pi$
$229$	−16.5616	−1.09442	−0.547209	+	0.836996i	$0.684310\pi$
$230$	0	0
$231$	6.56155	0.431718
$232$	0	0
$233$	−17.0540	−1.11724	−0.558622	−	0.829423i	$-0.688670\pi$
$233$	−17.0540	−1.11724	−0.558622	+	0.829423i	$0.688670\pi$
$234$	0	0
$235$	−5.12311	−0.334195
$236$	0	0
$237$	4.31534	0.280312
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	12.2462	0.788848	0.394424	−	0.918929i	$-0.370944\pi$
$241$	12.2462	0.788848	0.394424	+	0.918929i	$0.370944\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.438447	−0.0280114
$246$	0	0
$247$	15.3693	0.977926
$248$	0	0
$249$	−14.2462	−0.902817
$250$	0	0
$251$	−16.4924	−1.04099	−0.520496	−	0.853864i	$-0.674253\pi$
$251$	−16.4924	−1.04099	−0.520496	+	0.853864i	$0.674253\pi$
$252$	0	0
$253$	3.68466	0.231652
$254$	0	0
$255$	−3.12311	−0.195576
$256$	0	0
$257$	−1.68466	−0.105086	−0.0525431	−	0.998619i	$-0.516733\pi$
$257$	−1.68466	−0.105086	−0.0525431	+	0.998619i	$0.516733\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	7.12311	0.440909
$262$	0	0
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	7.43845	0.456940
$266$	0	0
$267$	−13.6847	−0.837487
$268$	0	0
$269$	−0.246211	−0.0150118	−0.00750588	−	0.999972i	$-0.502389\pi$
$269$	−0.246211	−0.0150118	−0.00750588	+	0.999972i	$0.502389\pi$
$270$	0	0
$271$	−27.0540	−1.64341	−0.821706	−	0.569912i	$-0.806977\pi$
$271$	−27.0540	−1.64341	−0.821706	+	0.569912i	$0.806977\pi$
$272$	0	0
$273$	−5.12311	−0.310064
$274$	0	0
$275$	−2.56155	−0.154467
$276$	0	0
$277$	−5.36932	−0.322611	−0.161305	−	0.986905i	$-0.551570\pi$
$277$	−5.36932	−0.322611	−0.161305	+	0.986905i	$0.551570\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	4.87689	0.290931	0.145466	−	0.989363i	$-0.453532\pi$
$281$	4.87689	0.290931	0.145466	+	0.989363i	$0.453532\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	7.68466	0.455200
$286$	0	0
$287$	−18.2462	−1.07704
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	26.4924	1.54770	0.773852	−	0.633367i	$-0.218327\pi$
$293$	26.4924	1.54770	0.773852	+	0.633367i	$0.218327\pi$
$294$	0	0
$295$	13.1231	0.764057
$296$	0	0
$297$	−2.56155	−0.148636
$298$	0	0
$299$	−2.87689	−0.166375
$300$	0	0
$301$	32.8078	1.89101
$302$	0	0
$303$	−0.561553	−0.0322604
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	1.75379	0.0997696
$310$	0	0
$311$	1.75379	0.0994482	0.0497241	−	0.998763i	$-0.484166\pi$
$311$	1.75379	0.0994482	0.0497241	+	0.998763i	$0.484166\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	−2.56155	−0.144327
$316$	0	0
$317$	16.8769	0.947901	0.473950	−	0.880552i	$-0.342828\pi$
$317$	16.8769	0.947901	0.473950	+	0.880552i	$0.342828\pi$
$318$	0	0
$319$	−18.2462	−1.02159
$320$	0	0
$321$	2.56155	0.142972
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	5.36932	0.296924
$328$	0	0
$329$	13.1231	0.723500
$330$	0	0
$331$	6.24621	0.343323	0.171661	−	0.985156i	$-0.445086\pi$
$331$	6.24621	0.343323	0.171661	+	0.985156i	$0.445086\pi$
$332$	0	0
$333$	−3.12311	−0.171145
$334$	0	0
$335$	15.3693	0.839715
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	−1.68466	−0.0914981
$340$	0	0
$341$	2.56155	0.138716
$342$	0	0
$343$	19.0540	1.02882
$344$	0	0
$345$	−1.43845	−0.0774434
$346$	0	0
$347$	−14.2462	−0.764777	−0.382388	−	0.924002i	$-0.624898\pi$
$347$	−14.2462	−0.764777	−0.382388	+	0.924002i	$0.624898\pi$
$348$	0	0
$349$	5.36932	0.287413	0.143706	−	0.989620i	$-0.454098\pi$
$349$	5.36932	0.287413	0.143706	+	0.989620i	$0.454098\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−0.246211	−0.0131045	−0.00655225	−	0.999979i	$-0.502086\pi$
$353$	−0.246211	−0.0131045	−0.00655225	+	0.999979i	$0.502086\pi$
$354$	0	0
$355$	7.68466	0.407859
$356$	0	0
$357$	8.00000	0.423405
$358$	0	0
$359$	31.6847	1.67225	0.836126	−	0.548537i	$-0.184815\pi$
$359$	31.6847	1.67225	0.836126	+	0.548537i	$0.184815\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	0	0
$363$	−4.43845	−0.232958
$364$	0	0
$365$	−10.8078	−0.565704
$366$	0	0
$367$	−15.3693	−0.802272	−0.401136	−	0.916019i	$-0.631384\pi$
$367$	−15.3693	−0.802272	−0.401136	+	0.916019i	$0.631384\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	−19.0540	−0.989233
$372$	0	0
$373$	−5.68466	−0.294340	−0.147170	−	0.989111i	$-0.547017\pi$
$373$	−5.68466	−0.294340	−0.147170	+	0.989111i	$0.547017\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	14.2462	0.733717
$378$	0	0
$379$	7.05398	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$379$	7.05398	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	0	0
$385$	6.56155	0.334408
$386$	0	0
$387$	−12.8078	−0.651055
$388$	0	0
$389$	7.75379	0.393133	0.196566	−	0.980491i	$-0.437021\pi$
$389$	7.75379	0.393133	0.196566	+	0.980491i	$0.437021\pi$
$390$	0	0
$391$	4.49242	0.227192
$392$	0	0
$393$	15.3693	0.775279
$394$	0	0
$395$	4.31534	0.217128
$396$	0	0
$397$	9.05398	0.454406	0.227203	−	0.973847i	$-0.427042\pi$
$397$	9.05398	0.454406	0.227203	+	0.973847i	$0.427042\pi$
$398$	0	0
$399$	−19.6847	−0.985466
$400$	0	0
$401$	−13.6847	−0.683379	−0.341690	−	0.939813i	$-0.610999\pi$
$401$	−13.6847	−0.683379	−0.341690	+	0.939813i	$0.610999\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−37.3693	−1.84779	−0.923897	−	0.382642i	$-0.875014\pi$
$409$	−37.3693	−1.84779	−0.923897	+	0.382642i	$0.875014\pi$
$410$	0	0
$411$	20.2462	0.998672
$412$	0	0
$413$	−33.6155	−1.65411
$414$	0	0
$415$	−14.2462	−0.699319
$416$	0	0
$417$	17.1231	0.838522
$418$	0	0
$419$	−5.75379	−0.281091	−0.140545	−	0.990074i	$-0.544886\pi$
$419$	−5.75379	−0.281091	−0.140545	+	0.990074i	$0.544886\pi$
$420$	0	0
$421$	−14.4924	−0.706317	−0.353159	−	0.935563i	$-0.614892\pi$
$421$	−14.4924	−0.706317	−0.353159	+	0.935563i	$0.614892\pi$
$422$	0	0
$423$	−5.12311	−0.249094
$424$	0	0
$425$	−3.12311	−0.151493
$426$	0	0
$427$	−15.3693	−0.743773
$428$	0	0
$429$	−5.12311	−0.247346
$430$	0	0
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	35.3002	1.69642	0.848209	−	0.529661i	$-0.177681\pi$
$433$	35.3002	1.69642	0.848209	+	0.529661i	$0.177681\pi$
$434$	0	0
$435$	7.12311	0.341527
$436$	0	0
$437$	−11.0540	−0.528783
$438$	0	0
$439$	−9.61553	−0.458924	−0.229462	−	0.973318i	$-0.573697\pi$
$439$	−9.61553	−0.458924	−0.229462	+	0.973318i	$0.573697\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	−23.0540	−1.09533	−0.547664	−	0.836699i	$-0.684483\pi$
$443$	−23.0540	−1.09533	−0.547664	+	0.836699i	$0.684483\pi$
$444$	0	0
$445$	−13.6847	−0.648715
$446$	0	0
$447$	17.0540	0.806625
$448$	0	0
$449$	10.4924	0.495168	0.247584	−	0.968866i	$-0.420363\pi$
$449$	10.4924	0.495168	0.247584	+	0.968866i	$0.420363\pi$
$450$	0	0
$451$	−18.2462	−0.859181
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	0	0
$459$	−3.12311	−0.145774
$460$	0	0
$461$	9.36932	0.436373	0.218186	−	0.975907i	$-0.429986\pi$
$461$	9.36932	0.436373	0.218186	+	0.975907i	$0.429986\pi$
$462$	0	0
$463$	−30.7386	−1.42855	−0.714273	−	0.699867i	$-0.753242\pi$
$463$	−30.7386	−1.42855	−0.714273	+	0.699867i	$0.753242\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	9.75379	0.451352	0.225676	−	0.974202i	$-0.427541\pi$
$467$	9.75379	0.451352	0.225676	+	0.974202i	$0.427541\pi$
$468$	0	0
$469$	−39.3693	−1.81791
$470$	0	0
$471$	17.0540	0.785806
$472$	0	0
$473$	32.8078	1.50850
$474$	0	0
$475$	7.68466	0.352596
$476$	0	0
$477$	7.43845	0.340583
$478$	0	0
$479$	−10.5616	−0.482570	−0.241285	−	0.970454i	$-0.577569\pi$
$479$	−10.5616	−0.482570	−0.241285	+	0.970454i	$0.577569\pi$
$480$	0	0
$481$	−6.24621	−0.284803
$482$	0	0
$483$	3.68466	0.167658
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	15.3693	0.695025
$490$	0	0
$491$	25.9309	1.17024	0.585122	−	0.810945i	$-0.301047\pi$
$491$	25.9309	1.17024	0.585122	+	0.810945i	$0.301047\pi$
$492$	0	0
$493$	−22.2462	−1.00192
$494$	0	0
$495$	−2.56155	−0.115133
$496$	0	0
$497$	−19.6847	−0.882978
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	6.56155	0.293149
$502$	0	0
$503$	−26.2462	−1.17026	−0.585130	−	0.810939i	$-0.698957\pi$
$503$	−26.2462	−1.17026	−0.585130	+	0.810939i	$0.698957\pi$
$504$	0	0
$505$	−0.561553	−0.0249888
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	19.6155	0.869443	0.434721	−	0.900565i	$-0.356847\pi$
$509$	19.6155	0.869443	0.434721	+	0.900565i	$0.356847\pi$
$510$	0	0
$511$	27.6847	1.22470
$512$	0	0
$513$	7.68466	0.339286
$514$	0	0
$515$	1.75379	0.0772812
$516$	0	0
$517$	13.1231	0.577154
$518$	0	0
$519$	8.24621	0.361968
$520$	0	0
$521$	−32.2462	−1.41273	−0.706366	−	0.707847i	$-0.749667\pi$
$521$	−32.2462	−1.41273	−0.706366	+	0.707847i	$0.749667\pi$
$522$	0	0
$523$	22.4233	0.980502	0.490251	−	0.871581i	$-0.336905\pi$
$523$	22.4233	0.980502	0.490251	+	0.871581i	$0.336905\pi$
$524$	0	0
$525$	−2.56155	−0.111795
$526$	0	0
$527$	3.12311	0.136045
$528$	0	0
$529$	−20.9309	−0.910038
$530$	0	0
$531$	13.1231	0.569494
$532$	0	0
$533$	14.2462	0.617072
$534$	0	0
$535$	2.56155	0.110746
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	1.12311	0.0483756
$540$	0	0
$541$	19.7538	0.849282	0.424641	−	0.905362i	$-0.360400\pi$
$541$	19.7538	0.849282	0.424641	+	0.905362i	$0.360400\pi$
$542$	0	0
$543$	−13.0540	−0.560200
$544$	0	0
$545$	5.36932	0.229996
$546$	0	0
$547$	−26.8769	−1.14917	−0.574587	−	0.818444i	$-0.694837\pi$
$547$	−26.8769	−1.14917	−0.574587	+	0.818444i	$0.694837\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	54.7386	2.33194
$552$	0	0
$553$	−11.0540	−0.470063
$554$	0	0
$555$	−3.12311	−0.132568
$556$	0	0
$557$	−26.8078	−1.13588	−0.567941	−	0.823069i	$-0.692260\pi$
$557$	−26.8078	−1.13588	−0.567941	+	0.823069i	$0.692260\pi$
$558$	0	0
$559$	−25.6155	−1.08342
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−16.4924	−0.695073	−0.347536	−	0.937667i	$-0.612982\pi$
$563$	−16.4924	−0.695073	−0.347536	+	0.937667i	$0.612982\pi$
$564$	0	0
$565$	−1.68466	−0.0708741
$566$	0	0
$567$	−2.56155	−0.107575
$568$	0	0
$569$	30.8078	1.29153	0.645764	−	0.763537i	$-0.276539\pi$
$569$	30.8078	1.29153	0.645764	+	0.763537i	$0.276539\pi$
$570$	0	0
$571$	−41.1231	−1.72095	−0.860474	−	0.509494i	$-0.829833\pi$
$571$	−41.1231	−1.72095	−0.860474	+	0.509494i	$0.829833\pi$
$572$	0	0
$573$	14.2462	0.595144
$574$	0	0
$575$	−1.43845	−0.0599874
$576$	0	0
$577$	15.1231	0.629583	0.314792	−	0.949161i	$-0.398065\pi$
$577$	15.1231	0.629583	0.314792	+	0.949161i	$0.398065\pi$
$578$	0	0
$579$	14.4924	0.602285
$580$	0	0
$581$	36.4924	1.51396
$582$	0	0
$583$	−19.0540	−0.789135
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	0.492423	0.0203245	0.0101622	−	0.999948i	$-0.496765\pi$
$587$	0.492423	0.0203245	0.0101622	+	0.999948i	$0.496765\pi$
$588$	0	0
$589$	−7.68466	−0.316641
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	16.8078	0.687896
$598$	0	0
$599$	−38.4233	−1.56993	−0.784967	−	0.619538i	$-0.787320\pi$
$599$	−38.4233	−1.56993	−0.784967	+	0.619538i	$0.787320\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	15.3693	0.625887
$604$	0	0
$605$	−4.43845	−0.180449
$606$	0	0
$607$	−7.05398	−0.286312	−0.143156	−	0.989700i	$-0.545725\pi$
$607$	−7.05398	−0.286312	−0.143156	+	0.989700i	$0.545725\pi$
$608$	0	0
$609$	−18.2462	−0.739374
$610$	0	0
$611$	−10.2462	−0.414517
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	7.12311	0.287231
$616$	0	0
$617$	−15.4384	−0.621528	−0.310764	−	0.950487i	$-0.600585\pi$
$617$	−15.4384	−0.621528	−0.310764	+	0.950487i	$0.600585\pi$
$618$	0	0
$619$	−11.3693	−0.456971	−0.228486	−	0.973547i	$-0.573377\pi$
$619$	−11.3693	−0.456971	−0.228486	+	0.973547i	$0.573377\pi$
$620$	0	0
$621$	−1.43845	−0.0577229
$622$	0	0
$623$	35.0540	1.40441
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−19.6847	−0.786130
$628$	0	0
$629$	9.75379	0.388909
$630$	0	0
$631$	−29.3002	−1.16642	−0.583211	−	0.812321i	$-0.698204\pi$
$631$	−29.3002	−1.16642	−0.583211	+	0.812321i	$0.698204\pi$
$632$	0	0
$633$	23.6847	0.941381
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.876894	−0.0347438
$638$	0	0
$639$	7.68466	0.304000
$640$	0	0
$641$	−5.50758	−0.217536	−0.108768	−	0.994067i	$-0.534691\pi$
$641$	−5.50758	−0.217536	−0.108768	+	0.994067i	$0.534691\pi$
$642$	0	0
$643$	7.05398	0.278182	0.139091	−	0.990280i	$-0.455582\pi$
$643$	7.05398	0.278182	0.139091	+	0.990280i	$0.455582\pi$
$644$	0	0
$645$	−12.8078	−0.504305
$646$	0	0
$647$	45.9309	1.80573	0.902864	−	0.429925i	$-0.141460\pi$
$647$	45.9309	1.80573	0.902864	+	0.429925i	$0.141460\pi$
$648$	0	0
$649$	−33.6155	−1.31952
$650$	0	0
$651$	2.56155	0.100395
$652$	0	0
$653$	40.2462	1.57496	0.787478	−	0.616343i	$-0.211386\pi$
$653$	40.2462	1.57496	0.787478	+	0.616343i	$0.211386\pi$
$654$	0	0
$655$	15.3693	0.600529
$656$	0	0
$657$	−10.8078	−0.421651
$658$	0	0
$659$	30.7386	1.19741	0.598704	−	0.800971i	$-0.295683\pi$
$659$	30.7386	1.19741	0.598704	+	0.800971i	$0.295683\pi$
$660$	0	0
$661$	26.4924	1.03044	0.515218	−	0.857059i	$-0.327711\pi$
$661$	26.4924	1.03044	0.515218	+	0.857059i	$0.327711\pi$
$662$	0	0
$663$	−6.24621	−0.242583
$664$	0	0
$665$	−19.6847	−0.763338
$666$	0	0
$667$	−10.2462	−0.396735
$668$	0	0
$669$	21.1231	0.816666
$670$	0	0
$671$	−15.3693	−0.593326
$672$	0	0
$673$	11.7538	0.453075	0.226538	−	0.974002i	$-0.427259\pi$
$673$	11.7538	0.453075	0.226538	+	0.974002i	$0.427259\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	32.4233	1.24613	0.623064	−	0.782171i	$-0.285888\pi$
$677$	32.4233	1.24613	0.623064	+	0.782171i	$0.285888\pi$
$678$	0	0
$679$	15.3693	0.589820
$680$	0	0
$681$	−7.68466	−0.294477
$682$	0	0
$683$	−31.6847	−1.21238	−0.606190	−	0.795320i	$-0.707303\pi$
$683$	−31.6847	−1.21238	−0.606190	+	0.795320i	$0.707303\pi$
$684$	0	0
$685$	20.2462	0.773568
$686$	0	0
$687$	−16.5616	−0.631863
$688$	0	0
$689$	14.8769	0.566765
$690$	0	0
$691$	−15.0540	−0.572680	−0.286340	−	0.958128i	$-0.592439\pi$
$691$	−15.0540	−0.572680	−0.286340	+	0.958128i	$0.592439\pi$
$692$	0	0
$693$	6.56155	0.249253
$694$	0	0
$695$	17.1231	0.649516
$696$	0	0
$697$	−22.2462	−0.842635
$698$	0	0
$699$	−17.0540	−0.645041
$700$	0	0
$701$	5.19224	0.196108	0.0980540	−	0.995181i	$-0.468738\pi$
$701$	5.19224	0.196108	0.0980540	+	0.995181i	$0.468738\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	−5.12311	−0.192947
$706$	0	0
$707$	1.43845	0.0540984
$708$	0	0
$709$	17.0540	0.640475	0.320238	−	0.947337i	$-0.396237\pi$
$709$	17.0540	0.640475	0.320238	+	0.947337i	$0.396237\pi$
$710$	0	0
$711$	4.31534	0.161838
$712$	0	0
$713$	1.43845	0.0538703
$714$	0	0
$715$	−5.12311	−0.191593
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	34.8769	1.30069	0.650344	−	0.759640i	$-0.274625\pi$
$719$	34.8769	1.30069	0.650344	+	0.759640i	$0.274625\pi$
$720$	0	0
$721$	−4.49242	−0.167307
$722$	0	0
$723$	12.2462	0.455441
$724$	0	0
$725$	7.12311	0.264546
$726$	0	0
$727$	−23.0540	−0.855025	−0.427512	−	0.904010i	$-0.640610\pi$
$727$	−23.0540	−0.855025	−0.427512	+	0.904010i	$0.640610\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	−6.49242	−0.239803	−0.119902	−	0.992786i	$-0.538258\pi$
$733$	−6.49242	−0.239803	−0.119902	+	0.992786i	$0.538258\pi$
$734$	0	0
$735$	−0.438447	−0.0161724
$736$	0	0
$737$	−39.3693	−1.45019
$738$	0	0
$739$	−52.9848	−1.94908	−0.974540	−	0.224216i	$-0.928018\pi$
$739$	−52.9848	−1.94908	−0.974540	+	0.224216i	$0.928018\pi$
$740$	0	0
$741$	15.3693	0.564606
$742$	0	0
$743$	50.4233	1.84985	0.924926	−	0.380148i	$-0.124127\pi$
$743$	50.4233	1.84985	0.924926	+	0.380148i	$0.124127\pi$
$744$	0	0
$745$	17.0540	0.624809
$746$	0	0
$747$	−14.2462	−0.521242
$748$	0	0
$749$	−6.56155	−0.239754
$750$	0	0
$751$	−45.1231	−1.64657	−0.823283	−	0.567631i	$-0.807860\pi$
$751$	−45.1231	−1.64657	−0.823283	+	0.567631i	$0.807860\pi$
$752$	0	0
$753$	−16.4924	−0.601017
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	3.68466	0.133745
$760$	0	0
$761$	−20.4233	−0.740344	−0.370172	−	0.928963i	$-0.620701\pi$
$761$	−20.4233	−0.740344	−0.370172	+	0.928963i	$0.620701\pi$
$762$	0	0
$763$	−13.7538	−0.497921
$764$	0	0
$765$	−3.12311	−0.112916
$766$	0	0
$767$	26.2462	0.947696
$768$	0	0
$769$	−20.5616	−0.741469	−0.370734	−	0.928739i	$-0.620894\pi$
$769$	−20.5616	−0.741469	−0.370734	+	0.928739i	$0.620894\pi$
$770$	0	0
$771$	−1.68466	−0.0606715
$772$	0	0
$773$	−11.4384	−0.411412	−0.205706	−	0.978614i	$-0.565949\pi$
$773$	−11.4384	−0.411412	−0.205706	+	0.978614i	$0.565949\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	54.7386	1.96122
$780$	0	0
$781$	−19.6847	−0.704372
$782$	0	0
$783$	7.12311	0.254559
$784$	0	0
$785$	17.0540	0.608682
$786$	0	0
$787$	17.9309	0.639166	0.319583	−	0.947558i	$-0.396457\pi$
$787$	17.9309	0.639166	0.319583	+	0.947558i	$0.396457\pi$
$788$	0	0
$789$	20.4924	0.729550
$790$	0	0
$791$	4.31534	0.153436
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	7.43845	0.263815
$796$	0	0
$797$	−26.9848	−0.955852	−0.477926	−	0.878400i	$-0.658611\pi$
$797$	−26.9848	−0.955852	−0.477926	+	0.878400i	$0.658611\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−13.6847	−0.483524
$802$	0	0
$803$	27.6847	0.976970
$804$	0	0
$805$	3.68466	0.129867
$806$	0	0
$807$	−0.246211	−0.00866705
$808$	0	0
$809$	−37.6847	−1.32492	−0.662461	−	0.749096i	$-0.730488\pi$
$809$	−37.6847	−1.32492	−0.662461	+	0.749096i	$0.730488\pi$
$810$	0	0
$811$	24.6695	0.866263	0.433132	−	0.901331i	$-0.357408\pi$
$811$	24.6695	0.866263	0.433132	+	0.901331i	$0.357408\pi$
$812$	0	0
$813$	−27.0540	−0.948824
$814$	0	0
$815$	15.3693	0.538364
$816$	0	0
$817$	−98.4233	−3.44340
$818$	0	0
$819$	−5.12311	−0.179016
$820$	0	0
$821$	19.6155	0.684587	0.342293	−	0.939593i	$-0.388796\pi$
$821$	19.6155	0.684587	0.342293	+	0.939593i	$0.388796\pi$
$822$	0	0
$823$	25.6155	0.892901	0.446451	−	0.894808i	$-0.352688\pi$
$823$	25.6155	0.892901	0.446451	+	0.894808i	$0.352688\pi$
$824$	0	0
$825$	−2.56155	−0.0891818
$826$	0	0
$827$	−1.75379	−0.0609852	−0.0304926	−	0.999535i	$-0.509708\pi$
$827$	−1.75379	−0.0609852	−0.0304926	+	0.999535i	$0.509708\pi$
$828$	0	0
$829$	24.4233	0.848256	0.424128	−	0.905602i	$-0.360581\pi$
$829$	24.4233	0.848256	0.424128	+	0.905602i	$0.360581\pi$
$830$	0	0
$831$	−5.36932	−0.186260
$832$	0	0
$833$	1.36932	0.0474440
$834$	0	0
$835$	6.56155	0.227072
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−6.06913	−0.209530	−0.104765	−	0.994497i	$-0.533409\pi$
$839$	−6.06913	−0.209530	−0.104765	+	0.994497i	$0.533409\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	4.87689	0.167969
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	11.3693	0.390654
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.49242	0.153998
$852$	0	0
$853$	47.7926	1.63639	0.818194	−	0.574942i	$-0.194976\pi$
$853$	47.7926	1.63639	0.818194	+	0.574942i	$0.194976\pi$
$854$	0	0
$855$	7.68466	0.262810
$856$	0	0
$857$	29.2311	0.998514	0.499257	−	0.866454i	$-0.333606\pi$
$857$	29.2311	0.998514	0.499257	+	0.866454i	$0.333606\pi$
$858$	0	0
$859$	−26.7386	−0.912310	−0.456155	−	0.889900i	$-0.650774\pi$
$859$	−26.7386	−0.912310	−0.456155	+	0.889900i	$0.650774\pi$
$860$	0	0
$861$	−18.2462	−0.621829
$862$	0	0
$863$	−39.5464	−1.34618	−0.673088	−	0.739563i	$-0.735032\pi$
$863$	−39.5464	−1.34618	−0.673088	+	0.739563i	$0.735032\pi$
$864$	0	0
$865$	8.24621	0.280380
$866$	0	0
$867$	−7.24621	−0.246094
$868$	0	0
$869$	−11.0540	−0.374980
$870$	0	0
$871$	30.7386	1.04154
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−2.56155	−0.0865963
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	26.4924	0.893567
$880$	0	0
$881$	9.50758	0.320318	0.160159	−	0.987091i	$-0.448799\pi$
$881$	9.50758	0.320318	0.160159	+	0.987091i	$0.448799\pi$
$882$	0	0
$883$	−30.4233	−1.02383	−0.511913	−	0.859038i	$-0.671063\pi$
$883$	−30.4233	−1.02383	−0.511913	+	0.859038i	$0.671063\pi$
$884$	0	0
$885$	13.1231	0.441128
$886$	0	0
$887$	32.6307	1.09563	0.547816	−	0.836599i	$-0.315460\pi$
$887$	32.6307	1.09563	0.547816	+	0.836599i	$0.315460\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.56155	−0.0858152
$892$	0	0
$893$	−39.3693	−1.31744
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−2.87689	−0.0960567
$898$	0	0
$899$	−7.12311	−0.237569
$900$	0	0
$901$	−23.2311	−0.773939
$902$	0	0
$903$	32.8078	1.09177
$904$	0	0
$905$	−13.0540	−0.433929
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	0	0
$909$	−0.561553	−0.0186255
$910$	0	0
$911$	11.5076	0.381263	0.190632	−	0.981662i	$-0.438946\pi$
$911$	11.5076	0.381263	0.190632	+	0.981662i	$0.438946\pi$
$912$	0	0
$913$	36.4924	1.20772
$914$	0	0
$915$	6.00000	0.198354
$916$	0	0
$917$	−39.3693	−1.30009
$918$	0	0
$919$	1.61553	0.0532914	0.0266457	−	0.999645i	$-0.491517\pi$
$919$	1.61553	0.0532914	0.0266457	+	0.999645i	$0.491517\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.3693	0.505887
$924$	0	0
$925$	−3.12311	−0.102687
$926$	0	0
$927$	1.75379	0.0576020
$928$	0	0
$929$	7.43845	0.244048	0.122024	−	0.992527i	$-0.461062\pi$
$929$	7.43845	0.244048	0.122024	+	0.992527i	$0.461062\pi$
$930$	0	0
$931$	−3.36932	−0.110425
$932$	0	0
$933$	1.75379	0.0574165
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	41.3693	1.35148	0.675738	−	0.737142i	$-0.263825\pi$
$937$	41.3693	1.35148	0.675738	+	0.737142i	$0.263825\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	2.63068	0.0857578	0.0428789	−	0.999080i	$-0.486347\pi$
$941$	2.63068	0.0857578	0.0428789	+	0.999080i	$0.486347\pi$
$942$	0	0
$943$	−10.2462	−0.333663
$944$	0	0
$945$	−2.56155	−0.0833273
$946$	0	0
$947$	9.12311	0.296461	0.148231	−	0.988953i	$-0.452642\pi$
$947$	9.12311	0.296461	0.148231	+	0.988953i	$0.452642\pi$
$948$	0	0
$949$	−21.6155	−0.701670
$950$	0	0
$951$	16.8769	0.547271
$952$	0	0
$953$	16.1080	0.521788	0.260894	−	0.965367i	$-0.415983\pi$
$953$	16.1080	0.521788	0.260894	+	0.965367i	$0.415983\pi$
$954$	0	0
$955$	14.2462	0.460997
$956$	0	0
$957$	−18.2462	−0.589816
$958$	0	0
$959$	−51.8617	−1.67470
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	2.56155	0.0825449
$964$	0	0
$965$	14.4924	0.466528
$966$	0	0
$967$	42.2462	1.35855	0.679273	−	0.733885i	$-0.262295\pi$
$967$	42.2462	1.35855	0.679273	+	0.733885i	$0.262295\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	37.1231	1.19134	0.595669	−	0.803230i	$-0.296887\pi$
$971$	37.1231	1.19134	0.595669	+	0.803230i	$0.296887\pi$
$972$	0	0
$973$	−43.8617	−1.40614
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	42.9848	1.37521	0.687604	−	0.726086i	$-0.258663\pi$
$977$	42.9848	1.37521	0.687604	+	0.726086i	$0.258663\pi$
$978$	0	0
$979$	35.0540	1.12033
$980$	0	0
$981$	5.36932	0.171429
$982$	0	0
$983$	−40.9848	−1.30721	−0.653607	−	0.756834i	$-0.726745\pi$
$983$	−40.9848	−1.30721	−0.653607	+	0.756834i	$0.726745\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	13.1231	0.417713
$988$	0	0
$989$	18.4233	0.585827
$990$	0	0
$991$	−35.6847	−1.13356	−0.566780	−	0.823869i	$-0.691811\pi$
$991$	−35.6847	−1.13356	−0.566780	+	0.823869i	$0.691811\pi$
$992$	0	0
$993$	6.24621	0.198218
$994$	0	0
$995$	16.8078	0.532842
$996$	0	0
$997$	−29.5076	−0.934514	−0.467257	−	0.884121i	$-0.654758\pi$
$997$	−29.5076	−0.934514	−0.467257	+	0.884121i	$0.654758\pi$
$998$	0	0
$999$	−3.12311	−0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bl.1.1		2
4.3	odd	2		930.2.a.p.1.2	✓	2
12.11	even	2		2790.2.a.be.1.2		2
20.3	even	4		4650.2.d.bd.3349.1		4
20.7	even	4		4650.2.d.bd.3349.4		4
20.19	odd	2		4650.2.a.ce.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.p.1.2	✓	2	4.3	odd	2
2790.2.a.be.1.2		2	12.11	even	2
4650.2.a.ce.1.1		2	20.19	odd	2
4650.2.d.bd.3349.1		4	20.3	even	4
4650.2.d.bd.3349.4		4	20.7	even	4
7440.2.a.bl.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -2.56155 q^{7} +1.00000 q^{9} -2.56155 q^{11} +2.00000 q^{13} +1.00000 q^{15} -3.12311 q^{17} +7.68466 q^{19} -2.56155 q^{21} -1.43845 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.12311 q^{29} -1.00000 q^{31} -2.56155 q^{33} -2.56155 q^{35} -3.12311 q^{37} +2.00000 q^{39} +7.12311 q^{41} -12.8078 q^{43} +1.00000 q^{45} -5.12311 q^{47} -0.438447 q^{49} -3.12311 q^{51} +7.43845 q^{53} -2.56155 q^{55} +7.68466 q^{57} +13.1231 q^{59} +6.00000 q^{61} -2.56155 q^{63} +2.00000 q^{65} +15.3693 q^{67} -1.43845 q^{69} +7.68466 q^{71} -10.8078 q^{73} +1.00000 q^{75} +6.56155 q^{77} +4.31534 q^{79} +1.00000 q^{81} -14.2462 q^{83} -3.12311 q^{85} +7.12311 q^{87} -13.6847 q^{89} -5.12311 q^{91} -1.00000 q^{93} +7.68466 q^{95} -6.00000 q^{97} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - q^{7} + 2 q^{9} - q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} + 3 q^{19} - q^{21} - 7 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 2 q^{31} - q^{33} - q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} - 5 q^{43} + 2 q^{45} - 2 q^{47} - 5 q^{49} + 2 q^{51} + 19 q^{53} - q^{55} + 3 q^{57} + 18 q^{59} + 12 q^{61} - q^{63} + 4 q^{65} + 6 q^{67} - 7 q^{69} + 3 q^{71} - q^{73} + 2 q^{75} + 9 q^{77} + 21 q^{79} + 2 q^{81} - 12 q^{83} + 2 q^{85} + 6 q^{87} - 15 q^{89} - 2 q^{91} - 2 q^{93} + 3 q^{95} - 12 q^{97} - q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - q^7 + 2 * q^9 - q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 + 3 * q^19 - q^21 - 7 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 2 * q^31 - q^33 - q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 - 5 * q^43 + 2 * q^45 - 2 * q^47 - 5 * q^49 + 2 * q^51 + 19 * q^53 - q^55 + 3 * q^57 + 18 * q^59 + 12 * q^61 - q^63 + 4 * q^65 + 6 * q^67 - 7 * q^69 + 3 * q^71 - q^73 + 2 * q^75 + 9 * q^77 + 21 * q^79 + 2 * q^81 - 12 * q^83 + 2 * q^85 + 6 * q^87 - 15 * q^89 - 2 * q^91 - 2 * q^93 + 3 * q^95 - 12 * q^97 - q^99

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