LMFDB - Embedded newform 7623.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} -1.38197 q^{10} -4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} -2.00000 q^{17} +7.47214 q^{19} +3.61803 q^{20} +2.00000 q^{23} -2.61803 q^{26} -1.61803 q^{28} -5.00000 q^{29} +1.52786 q^{31} +5.61803 q^{32} -1.23607 q^{34} -2.23607 q^{35} +7.47214 q^{37} +4.61803 q^{38} +5.00000 q^{40} +6.47214 q^{41} +0.472136 q^{43} +1.23607 q^{46} -0.527864 q^{47} +1.00000 q^{49} +6.85410 q^{52} +2.47214 q^{53} -2.23607 q^{56} -3.09017 q^{58} +9.94427 q^{59} +3.52786 q^{61} +0.944272 q^{62} -0.236068 q^{64} +9.47214 q^{65} -13.1803 q^{67} +3.23607 q^{68} -1.38197 q^{70} -6.47214 q^{71} +2.70820 q^{73} +4.61803 q^{74} -12.0902 q^{76} +8.47214 q^{79} -4.14590 q^{80} +4.00000 q^{82} -16.4721 q^{83} +4.47214 q^{85} +0.291796 q^{86} -8.47214 q^{89} -4.23607 q^{91} -3.23607 q^{92} -0.326238 q^{94} -16.7082 q^{95} -17.4164 q^{97} +0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{7} - 5 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} + 6 q^{19} + 5 q^{20} + 4 q^{23} - 3 q^{26} - q^{28} - 10 q^{29} + 12 q^{31} + 9 q^{32} + 2 q^{34} + 6 q^{37} + 7 q^{38} + 10 q^{40} + 4 q^{41} - 8 q^{43} - 2 q^{46} - 10 q^{47} + 2 q^{49} + 7 q^{52} - 4 q^{53} + 5 q^{58} + 2 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} + 10 q^{65} - 4 q^{67} + 2 q^{68} - 5 q^{70} - 4 q^{71} - 8 q^{73} + 7 q^{74} - 13 q^{76} + 8 q^{79} - 15 q^{80} + 8 q^{82} - 24 q^{83} + 14 q^{86} - 8 q^{89} - 4 q^{91} - 2 q^{92} + 15 q^{94} - 20 q^{95} - 8 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^7 - 5 * q^10 - 4 * q^13 - q^14 - 3 * q^16 - 4 * q^17 + 6 * q^19 + 5 * q^20 + 4 * q^23 - 3 * q^26 - q^28 - 10 * q^29 + 12 * q^31 + 9 * q^32 + 2 * q^34 + 6 * q^37 + 7 * q^38 + 10 * q^40 + 4 * q^41 - 8 * q^43 - 2 * q^46 - 10 * q^47 + 2 * q^49 + 7 * q^52 - 4 * q^53 + 5 * q^58 + 2 * q^59 + 16 * q^61 - 16 * q^62 + 4 * q^64 + 10 * q^65 - 4 * q^67 + 2 * q^68 - 5 * q^70 - 4 * q^71 - 8 * q^73 + 7 * q^74 - 13 * q^76 + 8 * q^79 - 15 * q^80 + 8 * q^82 - 24 * q^83 + 14 * q^86 - 8 * q^89 - 4 * q^91 - 2 * q^92 + 15 * q^94 - 20 * q^95 - 8 * q^97 - q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−1.38197	−0.437016
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.23607	−1.17487	−0.587437	−	0.809270i	$-0.699863\pi$
$13$	−4.23607	−1.17487	−0.587437	+	0.809270i	$0.699863\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	7.47214	1.71423	0.857113	−	0.515129i	$-0.172256\pi$
$19$	7.47214	1.71423	0.857113	+	0.515129i	$0.172256\pi$
$20$	3.61803	0.809017
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	0	0
$26$	−2.61803	−0.513439
$27$	0	0
$28$	−1.61803	−0.305780
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	−1.23607	−0.211984
$35$	−2.23607	−0.377964
$36$	0	0
$37$	7.47214	1.22841	0.614206	−	0.789146i	$-0.289476\pi$
$37$	7.47214	1.22841	0.614206	+	0.789146i	$0.289476\pi$
$38$	4.61803	0.749144
$39$	0	0
$40$	5.00000	0.790569
$41$	6.47214	1.01078	0.505389	−	0.862892i	$-0.331349\pi$
$41$	6.47214	1.01078	0.505389	+	0.862892i	$0.331349\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	0	0
$45$	0	0
$46$	1.23607	0.182248
$47$	−0.527864	−0.0769969	−0.0384984	−	0.999259i	$-0.512257\pi$
$47$	−0.527864	−0.0769969	−0.0384984	+	0.999259i	$0.512257\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	6.85410	0.950493
$53$	2.47214	0.339574	0.169787	−	0.985481i	$-0.445692\pi$
$53$	2.47214	0.339574	0.169787	+	0.985481i	$0.445692\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−3.09017	−0.405759
$59$	9.94427	1.29463	0.647317	−	0.762221i	$-0.275891\pi$
$59$	9.94427	1.29463	0.647317	+	0.762221i	$0.275891\pi$
$60$	0	0
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	0.944272	0.119923
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	9.47214	1.17487
$66$	0	0
$67$	−13.1803	−1.61023	−0.805117	−	0.593115i	$-0.797898\pi$
$67$	−13.1803	−1.61023	−0.805117	+	0.593115i	$0.797898\pi$
$68$	3.23607	0.392431
$69$	0	0
$70$	−1.38197	−0.165177
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	2.70820	0.316971	0.158486	−	0.987361i	$-0.449339\pi$
$73$	2.70820	0.316971	0.158486	+	0.987361i	$0.449339\pi$
$74$	4.61803	0.536836
$75$	0	0
$76$	−12.0902	−1.38684
$77$	0	0
$78$	0	0
$79$	8.47214	0.953190	0.476595	−	0.879123i	$-0.341871\pi$
$79$	8.47214	0.953190	0.476595	+	0.879123i	$0.341871\pi$
$80$	−4.14590	−0.463525
$81$	0	0
$82$	4.00000	0.441726
$83$	−16.4721	−1.80805	−0.904026	−	0.427478i	$-0.859402\pi$
$83$	−16.4721	−1.80805	−0.904026	+	0.427478i	$0.859402\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0.291796	0.0314652
$87$	0	0
$88$	0	0
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	−3.23607	−0.337383
$93$	0	0
$94$	−0.326238	−0.0336489
$95$	−16.7082	−1.71423
$96$	0	0
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	0	0
$101$	−18.4721	−1.83805	−0.919023	−	0.394204i	$-0.871020\pi$
$101$	−18.4721	−1.83805	−0.919023	+	0.394204i	$0.871020\pi$
$102$	0	0
$103$	−1.52786	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$103$	−1.52786	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$104$	9.47214	0.928819
$105$	0	0
$106$	1.52786	0.148399
$107$	11.6525	1.12649	0.563244	−	0.826291i	$-0.309553\pi$
$107$	11.6525	1.12649	0.563244	+	0.826291i	$0.309553\pi$
$108$	0	0
$109$	−1.52786	−0.146343	−0.0731714	−	0.997319i	$-0.523312\pi$
$109$	−1.52786	−0.146343	−0.0731714	+	0.997319i	$0.523312\pi$
$110$	0	0
$111$	0	0
$112$	1.85410	0.175196
$113$	14.4721	1.36142	0.680712	−	0.732551i	$-0.261671\pi$
$113$	14.4721	1.36142	0.680712	+	0.732551i	$0.261671\pi$
$114$	0	0
$115$	−4.47214	−0.417029
$116$	8.09017	0.751153
$117$	0	0
$118$	6.14590	0.565776
$119$	−2.00000	−0.183340
$120$	0	0
$121$	0	0
$122$	2.18034	0.197399
$123$	0	0
$124$	−2.47214	−0.222004
$125$	11.1803	1.00000
$126$	0	0
$127$	17.4164	1.54546	0.772728	−	0.634737i	$-0.218892\pi$
$127$	17.4164	1.54546	0.772728	+	0.634737i	$0.218892\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	5.85410	0.513439
$131$	7.41641	0.647975	0.323987	−	0.946061i	$-0.394976\pi$
$131$	7.41641	0.647975	0.323987	+	0.946061i	$0.394976\pi$
$132$	0	0
$133$	7.47214	0.647916
$134$	−8.14590	−0.703698
$135$	0	0
$136$	4.47214	0.383482
$137$	−17.4164	−1.48798	−0.743992	−	0.668188i	$-0.767070\pi$
$137$	−17.4164	−1.48798	−0.743992	+	0.668188i	$0.767070\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	3.61803	0.305780
$141$	0	0
$142$	−4.00000	−0.335673
$143$	0	0
$144$	0	0
$145$	11.1803	0.928477
$146$	1.67376	0.138522
$147$	0	0
$148$	−12.0902	−0.993806
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−16.7082	−1.35521
$153$	0	0
$154$	0	0
$155$	−3.41641	−0.274412
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	5.23607	0.416559
$159$	0	0
$160$	−12.5623	−0.993137
$161$	2.00000	0.157622
$162$	0	0
$163$	−8.23607	−0.645099	−0.322549	−	0.946553i	$-0.604540\pi$
$163$	−8.23607	−0.645099	−0.322549	+	0.946553i	$0.604540\pi$
$164$	−10.4721	−0.817736
$165$	0	0
$166$	−10.1803	−0.790148
$167$	−21.4164	−1.65725	−0.828626	−	0.559803i	$-0.810877\pi$
$167$	−21.4164	−1.65725	−0.828626	+	0.559803i	$0.810877\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	2.76393	0.211984
$171$	0	0
$172$	−0.763932	−0.0582493
$173$	16.4721	1.25235	0.626177	−	0.779681i	$-0.284619\pi$
$173$	16.4721	1.25235	0.626177	+	0.779681i	$0.284619\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−5.23607	−0.392460
$179$	−18.3607	−1.37234	−0.686171	−	0.727440i	$-0.740710\pi$
$179$	−18.3607	−1.37234	−0.686171	+	0.727440i	$0.740710\pi$
$180$	0	0
$181$	−4.94427	−0.367505	−0.183752	−	0.982973i	$-0.558824\pi$
$181$	−4.94427	−0.367505	−0.183752	+	0.982973i	$0.558824\pi$
$182$	−2.61803	−0.194062
$183$	0	0
$184$	−4.47214	−0.329690
$185$	−16.7082	−1.22841
$186$	0	0
$187$	0	0
$188$	0.854102	0.0622918
$189$	0	0
$190$	−10.3262	−0.749144
$191$	−24.4721	−1.77074	−0.885371	−	0.464886i	$-0.846095\pi$
$191$	−24.4721	−1.77074	−0.885371	+	0.464886i	$0.846095\pi$
$192$	0	0
$193$	10.4721	0.753801	0.376900	−	0.926254i	$-0.376990\pi$
$193$	10.4721	0.753801	0.376900	+	0.926254i	$0.376990\pi$
$194$	−10.7639	−0.772805
$195$	0	0
$196$	−1.61803	−0.115574
$197$	−11.8885	−0.847024	−0.423512	−	0.905891i	$-0.639203\pi$
$197$	−11.8885	−0.847024	−0.423512	+	0.905891i	$0.639203\pi$
$198$	0	0
$199$	23.8885	1.69341	0.846707	−	0.532059i	$-0.178582\pi$
$199$	23.8885	1.69341	0.846707	+	0.532059i	$0.178582\pi$
$200$	0	0
$201$	0	0
$202$	−11.4164	−0.803256
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−14.4721	−1.01078
$206$	−0.944272	−0.0657905
$207$	0	0
$208$	−7.85410	−0.544584
$209$	0	0
$210$	0	0
$211$	−18.3607	−1.26400	−0.632001	−	0.774968i	$-0.717766\pi$
$211$	−18.3607	−1.26400	−0.632001	+	0.774968i	$0.717766\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	7.20163	0.492293
$215$	−1.05573	−0.0720001
$216$	0	0
$217$	1.52786	0.103718
$218$	−0.944272	−0.0639542
$219$	0	0
$220$	0	0
$221$	8.47214	0.569898
$222$	0	0
$223$	11.8885	0.796116	0.398058	−	0.917360i	$-0.369684\pi$
$223$	11.8885	0.796116	0.398058	+	0.917360i	$0.369684\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	8.94427	0.594964
$227$	−12.4721	−0.827805	−0.413902	−	0.910321i	$-0.635835\pi$
$227$	−12.4721	−0.827805	−0.413902	+	0.910321i	$0.635835\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−2.76393	−0.182248
$231$	0	0
$232$	11.1803	0.734025
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	0	0
$235$	1.18034	0.0769969
$236$	−16.0902	−1.04738
$237$	0	0
$238$	−1.23607	−0.0801224
$239$	−14.1246	−0.913645	−0.456823	−	0.889558i	$-0.651013\pi$
$239$	−14.1246	−0.913645	−0.456823	+	0.889558i	$0.651013\pi$
$240$	0	0
$241$	3.18034	0.204864	0.102432	−	0.994740i	$-0.467338\pi$
$241$	3.18034	0.204864	0.102432	+	0.994740i	$0.467338\pi$
$242$	0	0
$243$	0	0
$244$	−5.70820	−0.365430
$245$	−2.23607	−0.142857
$246$	0	0
$247$	−31.6525	−2.01400
$248$	−3.41641	−0.216942
$249$	0	0
$250$	6.90983	0.437016
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	0	0
$253$	0	0
$254$	10.7639	0.675389
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−17.6525	−1.10113	−0.550566	−	0.834792i	$-0.685588\pi$
$257$	−17.6525	−1.10113	−0.550566	+	0.834792i	$0.685588\pi$
$258$	0	0
$259$	7.47214	0.464296
$260$	−15.3262	−0.950493
$261$	0	0
$262$	4.58359	0.283175
$263$	−24.7082	−1.52357	−0.761787	−	0.647828i	$-0.775678\pi$
$263$	−24.7082	−1.52357	−0.761787	+	0.647828i	$0.775678\pi$
$264$	0	0
$265$	−5.52786	−0.339574
$266$	4.61803	0.283150
$267$	0	0
$268$	21.3262	1.30271
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	0	0
$271$	−28.8885	−1.75485	−0.877427	−	0.479710i	$-0.840742\pi$
$271$	−28.8885	−1.75485	−0.877427	+	0.479710i	$0.840742\pi$
$272$	−3.70820	−0.224843
$273$	0	0
$274$	−10.7639	−0.650273
$275$	0	0
$276$	0	0
$277$	−14.3607	−0.862850	−0.431425	−	0.902149i	$-0.641989\pi$
$277$	−14.3607	−0.862850	−0.431425	+	0.902149i	$0.641989\pi$
$278$	5.52786	0.331539
$279$	0	0
$280$	5.00000	0.298807
$281$	−2.52786	−0.150800	−0.0753999	−	0.997153i	$-0.524023\pi$
$281$	−2.52786	−0.150800	−0.0753999	+	0.997153i	$0.524023\pi$
$282$	0	0
$283$	8.41641	0.500304	0.250152	−	0.968207i	$-0.419519\pi$
$283$	8.41641	0.500304	0.250152	+	0.968207i	$0.419519\pi$
$284$	10.4721	0.621407
$285$	0	0
$286$	0	0
$287$	6.47214	0.382038
$288$	0	0
$289$	−13.0000	−0.764706
$290$	6.90983	0.405759
$291$	0	0
$292$	−4.38197	−0.256435
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−22.2361	−1.29463
$296$	−16.7082	−0.971145
$297$	0	0
$298$	−12.9787	−0.751837
$299$	−8.47214	−0.489956
$300$	0	0
$301$	0.472136	0.0272135
$302$	6.18034	0.355639
$303$	0	0
$304$	13.8541	0.794587
$305$	−7.88854	−0.451697
$306$	0	0
$307$	−5.88854	−0.336077	−0.168038	−	0.985780i	$-0.553743\pi$
$307$	−5.88854	−0.336077	−0.168038	+	0.985780i	$0.553743\pi$
$308$	0	0
$309$	0	0
$310$	−2.11146	−0.119923
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−2.94427	−0.166420	−0.0832100	−	0.996532i	$-0.526517\pi$
$313$	−2.94427	−0.166420	−0.0832100	+	0.996532i	$0.526517\pi$
$314$	7.70820	0.434999
$315$	0	0
$316$	−13.7082	−0.771147
$317$	20.9443	1.17635	0.588174	−	0.808735i	$-0.299847\pi$
$317$	20.9443	1.17635	0.588174	+	0.808735i	$0.299847\pi$
$318$	0	0
$319$	0	0
$320$	0.527864	0.0295085
$321$	0	0
$322$	1.23607	0.0688834
$323$	−14.9443	−0.831522
$324$	0	0
$325$	0	0
$326$	−5.09017	−0.281918
$327$	0	0
$328$	−14.4721	−0.799090
$329$	−0.527864	−0.0291021
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	26.6525	1.46274
$333$	0	0
$334$	−13.2361	−0.724245
$335$	29.4721	1.61023
$336$	0	0
$337$	−36.4721	−1.98676	−0.993382	−	0.114858i	$-0.963359\pi$
$337$	−36.4721	−1.98676	−0.993382	+	0.114858i	$0.963359\pi$
$338$	3.05573	0.166210
$339$	0	0
$340$	−7.23607	−0.392431
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.05573	−0.0569210
$345$	0	0
$346$	10.1803	0.547298
$347$	−12.9443	−0.694885	−0.347442	−	0.937701i	$-0.612950\pi$
$347$	−12.9443	−0.694885	−0.347442	+	0.937701i	$0.612950\pi$
$348$	0	0
$349$	−22.1246	−1.18430	−0.592152	−	0.805827i	$-0.701721\pi$
$349$	−22.1246	−1.18430	−0.592152	+	0.805827i	$0.701721\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.1803	−0.914417	−0.457209	−	0.889359i	$-0.651151\pi$
$353$	−17.1803	−0.914417	−0.457209	+	0.889359i	$0.651151\pi$
$354$	0	0
$355$	14.4721	0.768101
$356$	13.7082	0.726533
$357$	0	0
$358$	−11.3475	−0.599735
$359$	−11.0557	−0.583499	−0.291750	−	0.956495i	$-0.594237\pi$
$359$	−11.0557	−0.583499	−0.291750	+	0.956495i	$0.594237\pi$
$360$	0	0
$361$	36.8328	1.93857
$362$	−3.05573	−0.160606
$363$	0	0
$364$	6.85410	0.359253
$365$	−6.05573	−0.316971
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	3.70820	0.193303
$369$	0	0
$370$	−10.3262	−0.536836
$371$	2.47214	0.128347
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	1.18034	0.0608714
$377$	21.1803	1.09084
$378$	0	0
$379$	16.2361	0.833991	0.416995	−	0.908909i	$-0.363083\pi$
$379$	16.2361	0.833991	0.416995	+	0.908909i	$0.363083\pi$
$380$	27.0344	1.38684
$381$	0	0
$382$	−15.1246	−0.773842
$383$	−16.9443	−0.865812	−0.432906	−	0.901439i	$-0.642512\pi$
$383$	−16.9443	−0.865812	−0.432906	+	0.901439i	$0.642512\pi$
$384$	0	0
$385$	0	0
$386$	6.47214	0.329423
$387$	0	0
$388$	28.1803	1.43064
$389$	34.4721	1.74781	0.873903	−	0.486100i	$-0.161581\pi$
$389$	34.4721	1.74781	0.873903	+	0.486100i	$0.161581\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	−2.23607	−0.112938
$393$	0	0
$394$	−7.34752	−0.370163
$395$	−18.9443	−0.953190
$396$	0	0
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	14.7639	0.740049
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	−6.47214	−0.322400
$404$	29.8885	1.48701
$405$	0	0
$406$	−3.09017	−0.153363
$407$	0	0
$408$	0	0
$409$	−16.4721	−0.814495	−0.407247	−	0.913318i	$-0.633511\pi$
$409$	−16.4721	−0.814495	−0.407247	+	0.913318i	$0.633511\pi$
$410$	−8.94427	−0.441726
$411$	0	0
$412$	2.47214	0.121793
$413$	9.94427	0.489326
$414$	0	0
$415$	36.8328	1.80805
$416$	−23.7984	−1.16681
$417$	0	0
$418$	0	0
$419$	−17.0000	−0.830504	−0.415252	−	0.909706i	$-0.636307\pi$
$419$	−17.0000	−0.830504	−0.415252	+	0.909706i	$0.636307\pi$
$420$	0	0
$421$	7.36068	0.358738	0.179369	−	0.983782i	$-0.442594\pi$
$421$	7.36068	0.358738	0.179369	+	0.983782i	$0.442594\pi$
$422$	−11.3475	−0.552389
$423$	0	0
$424$	−5.52786	−0.268457
$425$	0	0
$426$	0	0
$427$	3.52786	0.170725
$428$	−18.8541	−0.911347
$429$	0	0
$430$	−0.652476	−0.0314652
$431$	−15.7639	−0.759322	−0.379661	−	0.925126i	$-0.623959\pi$
$431$	−15.7639	−0.759322	−0.379661	+	0.925126i	$0.623959\pi$
$432$	0	0
$433$	33.4164	1.60589	0.802945	−	0.596053i	$-0.203265\pi$
$433$	33.4164	1.60589	0.802945	+	0.596053i	$0.203265\pi$
$434$	0.944272	0.0453265
$435$	0	0
$436$	2.47214	0.118394
$437$	14.9443	0.714881
$438$	0	0
$439$	−8.88854	−0.424227	−0.212114	−	0.977245i	$-0.568035\pi$
$439$	−8.88854	−0.424227	−0.212114	+	0.977245i	$0.568035\pi$
$440$	0	0
$441$	0	0
$442$	5.23607	0.249054
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	0	0
$445$	18.9443	0.898045
$446$	7.34752	0.347915
$447$	0	0
$448$	−0.236068	−0.0111532
$449$	8.47214	0.399825	0.199912	−	0.979814i	$-0.435934\pi$
$449$	8.47214	0.399825	0.199912	+	0.979814i	$0.435934\pi$
$450$	0	0
$451$	0	0
$452$	−23.4164	−1.10142
$453$	0	0
$454$	−7.70820	−0.361764
$455$	9.47214	0.444061
$456$	0	0
$457$	−38.3607	−1.79444	−0.897218	−	0.441587i	$-0.854416\pi$
$457$	−38.3607	−1.79444	−0.897218	+	0.441587i	$0.854416\pi$
$458$	8.65248	0.404304
$459$	0	0
$460$	7.23607	0.337383
$461$	−1.05573	−0.0491702	−0.0245851	−	0.999698i	$-0.507826\pi$
$461$	−1.05573	−0.0491702	−0.0245851	+	0.999698i	$0.507826\pi$
$462$	0	0
$463$	−6.70820	−0.311757	−0.155878	−	0.987776i	$-0.549821\pi$
$463$	−6.70820	−0.311757	−0.155878	+	0.987776i	$0.549821\pi$
$464$	−9.27051	−0.430373
$465$	0	0
$466$	1.81966	0.0842941
$467$	−27.9443	−1.29311	−0.646553	−	0.762869i	$-0.723790\pi$
$467$	−27.9443	−1.29311	−0.646553	+	0.762869i	$0.723790\pi$
$468$	0	0
$469$	−13.1803	−0.608612
$470$	0.729490	0.0336489
$471$	0	0
$472$	−22.2361	−1.02350
$473$	0	0
$474$	0	0
$475$	0	0
$476$	3.23607	0.148325
$477$	0	0
$478$	−8.72949	−0.399278
$479$	0.111456	0.00509256	0.00254628	−	0.999997i	$-0.499189\pi$
$479$	0.111456	0.00509256	0.00254628	+	0.999997i	$0.499189\pi$
$480$	0	0
$481$	−31.6525	−1.44323
$482$	1.96556	0.0895287
$483$	0	0
$484$	0	0
$485$	38.9443	1.76837
$486$	0	0
$487$	−29.8885	−1.35438	−0.677190	−	0.735809i	$-0.736802\pi$
$487$	−29.8885	−1.35438	−0.677190	+	0.735809i	$0.736802\pi$
$488$	−7.88854	−0.357098
$489$	0	0
$490$	−1.38197	−0.0624309
$491$	−28.1246	−1.26925	−0.634623	−	0.772822i	$-0.718845\pi$
$491$	−28.1246	−1.26925	−0.634623	+	0.772822i	$0.718845\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	−19.5623	−0.880150
$495$	0	0
$496$	2.83282	0.127197
$497$	−6.47214	−0.290315
$498$	0	0
$499$	24.2361	1.08496	0.542478	−	0.840070i	$-0.317486\pi$
$499$	24.2361	1.08496	0.542478	+	0.840070i	$0.317486\pi$
$500$	−18.0902	−0.809017
$501$	0	0
$502$	6.79837	0.303426
$503$	25.8885	1.15431	0.577157	−	0.816634i	$-0.304162\pi$
$503$	25.8885	1.15431	0.577157	+	0.816634i	$0.304162\pi$
$504$	0	0
$505$	41.3050	1.83805
$506$	0	0
$507$	0	0
$508$	−28.1803	−1.25030
$509$	−6.58359	−0.291813	−0.145906	−	0.989298i	$-0.546610\pi$
$509$	−6.58359	−0.291813	−0.145906	+	0.989298i	$0.546610\pi$
$510$	0	0
$511$	2.70820	0.119804
$512$	18.7082	0.826794
$513$	0	0
$514$	−10.9098	−0.481212
$515$	3.41641	0.150545
$516$	0	0
$517$	0	0
$518$	4.61803	0.202905
$519$	0	0
$520$	−21.1803	−0.928819
$521$	30.1246	1.31978	0.659892	−	0.751361i	$-0.270602\pi$
$521$	30.1246	1.31978	0.659892	+	0.751361i	$0.270602\pi$
$522$	0	0
$523$	−28.4164	−1.24256	−0.621281	−	0.783588i	$-0.713388\pi$
$523$	−28.4164	−1.24256	−0.621281	+	0.783588i	$0.713388\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−15.2705	−0.665826
$527$	−3.05573	−0.133110
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−3.41641	−0.148399
$531$	0	0
$532$	−12.0902	−0.524175
$533$	−27.4164	−1.18754
$534$	0	0
$535$	−26.0557	−1.12649
$536$	29.4721	1.27300
$537$	0	0
$538$	8.29180	0.357485
$539$	0	0
$540$	0	0
$541$	20.3607	0.875374	0.437687	−	0.899127i	$-0.355798\pi$
$541$	20.3607	0.875374	0.437687	+	0.899127i	$0.355798\pi$
$542$	−17.8541	−0.766899
$543$	0	0
$544$	−11.2361	−0.481742
$545$	3.41641	0.146343
$546$	0	0
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	28.1803	1.20380
$549$	0	0
$550$	0	0
$551$	−37.3607	−1.59162
$552$	0	0
$553$	8.47214	0.360272
$554$	−8.87539	−0.377079
$555$	0	0
$556$	−14.4721	−0.613755
$557$	37.8328	1.60303	0.801514	−	0.597976i	$-0.204028\pi$
$557$	37.8328	1.60303	0.801514	+	0.597976i	$0.204028\pi$
$558$	0	0
$559$	−2.00000	−0.0845910
$560$	−4.14590	−0.175196
$561$	0	0
$562$	−1.56231	−0.0659019
$563$	13.8885	0.585332	0.292666	−	0.956215i	$-0.405458\pi$
$563$	13.8885	0.585332	0.292666	+	0.956215i	$0.405458\pi$
$564$	0	0
$565$	−32.3607	−1.36142
$566$	5.20163	0.218641
$567$	0	0
$568$	14.4721	0.607237
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	−8.83282	−0.369642	−0.184821	−	0.982772i	$-0.559171\pi$
$571$	−8.83282	−0.369642	−0.184821	+	0.982772i	$0.559171\pi$
$572$	0	0
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	0	0
$577$	17.5279	0.729695	0.364847	−	0.931067i	$-0.381121\pi$
$577$	17.5279	0.729695	0.364847	+	0.931067i	$0.381121\pi$
$578$	−8.03444	−0.334189
$579$	0	0
$580$	−18.0902	−0.751153
$581$	−16.4721	−0.683379
$582$	0	0
$583$	0	0
$584$	−6.05573	−0.250588
$585$	0	0
$586$	0	0
$587$	−36.7771	−1.51795	−0.758976	−	0.651118i	$-0.774300\pi$
$587$	−36.7771	−1.51795	−0.758976	+	0.651118i	$0.774300\pi$
$588$	0	0
$589$	11.4164	0.470405
$590$	−13.7426	−0.565776
$591$	0	0
$592$	13.8541	0.569400
$593$	−39.8885	−1.63803	−0.819013	−	0.573775i	$-0.805478\pi$
$593$	−39.8885	−1.63803	−0.819013	+	0.573775i	$0.805478\pi$
$594$	0	0
$595$	4.47214	0.183340
$596$	33.9787	1.39182
$597$	0	0
$598$	−5.23607	−0.214119
$599$	4.47214	0.182727	0.0913633	−	0.995818i	$-0.470878\pi$
$599$	4.47214	0.182727	0.0913633	+	0.995818i	$0.470878\pi$
$600$	0	0
$601$	−5.29180	−0.215857	−0.107928	−	0.994159i	$-0.534422\pi$
$601$	−5.29180	−0.215857	−0.107928	+	0.994159i	$0.534422\pi$
$602$	0.291796	0.0118927
$603$	0	0
$604$	−16.1803	−0.658369
$605$	0	0
$606$	0	0
$607$	−10.0557	−0.408149	−0.204075	−	0.978955i	$-0.565419\pi$
$607$	−10.0557	−0.408149	−0.204075	+	0.978955i	$0.565419\pi$
$608$	41.9787	1.70246
$609$	0	0
$610$	−4.87539	−0.197399
$611$	2.23607	0.0904616
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	−3.63932	−0.146871
$615$	0	0
$616$	0	0
$617$	26.8328	1.08025	0.540124	−	0.841585i	$-0.318377\pi$
$617$	26.8328	1.08025	0.540124	+	0.841585i	$0.318377\pi$
$618$	0	0
$619$	−29.8885	−1.20132	−0.600661	−	0.799504i	$-0.705096\pi$
$619$	−29.8885	−1.20132	−0.600661	+	0.799504i	$0.705096\pi$
$620$	5.52786	0.222004
$621$	0	0
$622$	0	0
$623$	−8.47214	−0.339429
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−1.81966	−0.0727282
$627$	0	0
$628$	−20.1803	−0.805283
$629$	−14.9443	−0.595867
$630$	0	0
$631$	−26.8328	−1.06820	−0.534099	−	0.845422i	$-0.679349\pi$
$631$	−26.8328	−1.06820	−0.534099	+	0.845422i	$0.679349\pi$
$632$	−18.9443	−0.753563
$633$	0	0
$634$	12.9443	0.514083
$635$	−38.9443	−1.54546
$636$	0	0
$637$	−4.23607	−0.167839
$638$	0	0
$639$	0	0
$640$	25.4508	1.00603
$641$	11.4164	0.450921	0.225460	−	0.974252i	$-0.427611\pi$
$641$	11.4164	0.450921	0.225460	+	0.974252i	$0.427611\pi$
$642$	0	0
$643$	33.7771	1.33204	0.666019	−	0.745935i	$-0.267997\pi$
$643$	33.7771	1.33204	0.666019	+	0.745935i	$0.267997\pi$
$644$	−3.23607	−0.127519
$645$	0	0
$646$	−9.23607	−0.363388
$647$	12.3050	0.483758	0.241879	−	0.970306i	$-0.422236\pi$
$647$	12.3050	0.483758	0.241879	+	0.970306i	$0.422236\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	13.3262	0.521896
$653$	−19.0557	−0.745708	−0.372854	−	0.927890i	$-0.621621\pi$
$653$	−19.0557	−0.745708	−0.372854	+	0.927890i	$0.621621\pi$
$654$	0	0
$655$	−16.5836	−0.647975
$656$	12.0000	0.468521
$657$	0	0
$658$	−0.326238	−0.0127181
$659$	−16.1246	−0.628126	−0.314063	−	0.949402i	$-0.601690\pi$
$659$	−16.1246	−0.628126	−0.314063	+	0.949402i	$0.601690\pi$
$660$	0	0
$661$	37.3050	1.45099	0.725497	−	0.688225i	$-0.241610\pi$
$661$	37.3050	1.45099	0.725497	+	0.688225i	$0.241610\pi$
$662$	−9.88854	−0.384329
$663$	0	0
$664$	36.8328	1.42939
$665$	−16.7082	−0.647916
$666$	0	0
$667$	−10.0000	−0.387202
$668$	34.6525	1.34074
$669$	0	0
$670$	18.2148	0.703698
$671$	0	0
$672$	0	0
$673$	7.05573	0.271978	0.135989	−	0.990710i	$-0.456579\pi$
$673$	7.05573	0.271978	0.135989	+	0.990710i	$0.456579\pi$
$674$	−22.5410	−0.868248
$675$	0	0
$676$	−8.00000	−0.307692
$677$	−26.4721	−1.01741	−0.508703	−	0.860942i	$-0.669875\pi$
$677$	−26.4721	−1.01741	−0.508703	+	0.860942i	$0.669875\pi$
$678$	0	0
$679$	−17.4164	−0.668380
$680$	−10.0000	−0.383482
$681$	0	0
$682$	0	0
$683$	−27.0557	−1.03526	−0.517629	−	0.855605i	$-0.673185\pi$
$683$	−27.0557	−1.03526	−0.517629	+	0.855605i	$0.673185\pi$
$684$	0	0
$685$	38.9443	1.48798
$686$	0.618034	0.0235966
$687$	0	0
$688$	0.875388	0.0333739
$689$	−10.4721	−0.398957
$690$	0	0
$691$	38.8328	1.47727	0.738635	−	0.674106i	$-0.235471\pi$
$691$	38.8328	1.47727	0.738635	+	0.674106i	$0.235471\pi$
$692$	−26.6525	−1.01318
$693$	0	0
$694$	−8.00000	−0.303676
$695$	−20.0000	−0.758643
$696$	0	0
$697$	−12.9443	−0.490299
$698$	−13.6738	−0.517560
$699$	0	0
$700$	0	0
$701$	47.8885	1.80873	0.904363	−	0.426765i	$-0.140347\pi$
$701$	47.8885	1.80873	0.904363	+	0.426765i	$0.140347\pi$
$702$	0	0
$703$	55.8328	2.10577
$704$	0	0
$705$	0	0
$706$	−10.6180	−0.399615
$707$	−18.4721	−0.694716
$708$	0	0
$709$	−43.4721	−1.63263	−0.816315	−	0.577607i	$-0.803987\pi$
$709$	−43.4721	−1.63263	−0.816315	+	0.577607i	$0.803987\pi$
$710$	8.94427	0.335673
$711$	0	0
$712$	18.9443	0.709967
$713$	3.05573	0.114438
$714$	0	0
$715$	0	0
$716$	29.7082	1.11025
$717$	0	0
$718$	−6.83282	−0.254998
$719$	23.3607	0.871206	0.435603	−	0.900139i	$-0.356535\pi$
$719$	23.3607	0.871206	0.435603	+	0.900139i	$0.356535\pi$
$720$	0	0
$721$	−1.52786	−0.0569006
$722$	22.7639	0.847186
$723$	0	0
$724$	8.00000	0.297318
$725$	0	0
$726$	0	0
$727$	−12.4721	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$727$	−12.4721	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$728$	9.47214	0.351061
$729$	0	0
$730$	−3.74265	−0.138522
$731$	−0.944272	−0.0349252
$732$	0	0
$733$	18.3607	0.678167	0.339084	−	0.940756i	$-0.389883\pi$
$733$	18.3607	0.678167	0.339084	+	0.940756i	$0.389883\pi$
$734$	−9.88854	−0.364993
$735$	0	0
$736$	11.2361	0.414167
$737$	0	0
$738$	0	0
$739$	16.1115	0.592669	0.296335	−	0.955084i	$-0.404236\pi$
$739$	16.1115	0.592669	0.296335	+	0.955084i	$0.404236\pi$
$740$	27.0344	0.993806
$741$	0	0
$742$	1.52786	0.0560897
$743$	−18.5967	−0.682249	−0.341124	−	0.940018i	$-0.610808\pi$
$743$	−18.5967	−0.682249	−0.341124	+	0.940018i	$0.610808\pi$
$744$	0	0
$745$	46.9574	1.72039
$746$	3.70820	0.135767
$747$	0	0
$748$	0	0
$749$	11.6525	0.425772
$750$	0	0
$751$	15.1803	0.553938	0.276969	−	0.960879i	$-0.410670\pi$
$751$	15.1803	0.553938	0.276969	+	0.960879i	$0.410670\pi$
$752$	−0.978714	−0.0356900
$753$	0	0
$754$	13.0902	0.476716
$755$	−22.3607	−0.813788
$756$	0	0
$757$	−42.4164	−1.54165	−0.770825	−	0.637047i	$-0.780156\pi$
$757$	−42.4164	−1.54165	−0.770825	+	0.637047i	$0.780156\pi$
$758$	10.0344	0.364467
$759$	0	0
$760$	37.3607	1.35521
$761$	−15.5279	−0.562885	−0.281442	−	0.959578i	$-0.590813\pi$
$761$	−15.5279	−0.562885	−0.281442	+	0.959578i	$0.590813\pi$
$762$	0	0
$763$	−1.52786	−0.0553124
$764$	39.5967	1.43256
$765$	0	0
$766$	−10.4721	−0.378374
$767$	−42.1246	−1.52103
$768$	0	0
$769$	7.18034	0.258930	0.129465	−	0.991584i	$-0.458674\pi$
$769$	7.18034	0.258930	0.129465	+	0.991584i	$0.458674\pi$
$770$	0	0
$771$	0	0
$772$	−16.9443	−0.609838
$773$	−3.18034	−0.114389	−0.0571944	−	0.998363i	$-0.518215\pi$
$773$	−3.18034	−0.114389	−0.0571944	+	0.998363i	$0.518215\pi$
$774$	0	0
$775$	0	0
$776$	38.9443	1.39802
$777$	0	0
$778$	21.3050	0.763820
$779$	48.3607	1.73270
$780$	0	0
$781$	0	0
$782$	−2.47214	−0.0884034
$783$	0	0
$784$	1.85410	0.0662179
$785$	−27.8885	−0.995385
$786$	0	0
$787$	−35.2492	−1.25650	−0.628250	−	0.778012i	$-0.716228\pi$
$787$	−35.2492	−1.25650	−0.628250	+	0.778012i	$0.716228\pi$
$788$	19.2361	0.685257
$789$	0	0
$790$	−11.7082	−0.416559
$791$	14.4721	0.514570
$792$	0	0
$793$	−14.9443	−0.530687
$794$	5.59675	0.198621
$795$	0	0
$796$	−38.6525	−1.37000
$797$	42.2361	1.49608	0.748039	−	0.663655i	$-0.230996\pi$
$797$	42.2361	1.49608	0.748039	+	0.663655i	$0.230996\pi$
$798$	0	0
$799$	1.05573	0.0373490
$800$	0	0
$801$	0	0
$802$	16.0689	0.567412
$803$	0	0
$804$	0	0
$805$	−4.47214	−0.157622
$806$	−4.00000	−0.140894
$807$	0	0
$808$	41.3050	1.45310
$809$	−4.63932	−0.163110	−0.0815549	−	0.996669i	$-0.525989\pi$
$809$	−4.63932	−0.163110	−0.0815549	+	0.996669i	$0.525989\pi$
$810$	0	0
$811$	49.4721	1.73720	0.868601	−	0.495512i	$-0.165020\pi$
$811$	49.4721	1.73720	0.868601	+	0.495512i	$0.165020\pi$
$812$	8.09017	0.283909
$813$	0	0
$814$	0	0
$815$	18.4164	0.645099
$816$	0	0
$817$	3.52786	0.123424
$818$	−10.1803	−0.355947
$819$	0	0
$820$	23.4164	0.817736
$821$	51.8328	1.80898	0.904489	−	0.426497i	$-0.140253\pi$
$821$	51.8328	1.80898	0.904489	+	0.426497i	$0.140253\pi$
$822$	0	0
$823$	25.0689	0.873846	0.436923	−	0.899499i	$-0.356068\pi$
$823$	25.0689	0.873846	0.436923	+	0.899499i	$0.356068\pi$
$824$	3.41641	0.119016
$825$	0	0
$826$	6.14590	0.213843
$827$	21.7639	0.756806	0.378403	−	0.925641i	$-0.376473\pi$
$827$	21.7639	0.756806	0.378403	+	0.925641i	$0.376473\pi$
$828$	0	0
$829$	−3.05573	−0.106130	−0.0530649	−	0.998591i	$-0.516899\pi$
$829$	−3.05573	−0.106130	−0.0530649	+	0.998591i	$0.516899\pi$
$830$	22.7639	0.790148
$831$	0	0
$832$	1.00000	0.0346688
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	47.8885	1.65725
$836$	0	0
$837$	0	0
$838$	−10.5066	−0.362944
$839$	51.2492	1.76932	0.884660	−	0.466237i	$-0.154391\pi$
$839$	51.2492	1.76932	0.884660	+	0.466237i	$0.154391\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	4.54915	0.156774
$843$	0	0
$844$	29.7082	1.02260
$845$	−11.0557	−0.380329
$846$	0	0
$847$	0	0
$848$	4.58359	0.157401
$849$	0	0
$850$	0	0
$851$	14.9443	0.512283
$852$	0	0
$853$	−2.58359	−0.0884605	−0.0442303	−	0.999021i	$-0.514084\pi$
$853$	−2.58359	−0.0884605	−0.0442303	+	0.999021i	$0.514084\pi$
$854$	2.18034	0.0746097
$855$	0	0
$856$	−26.0557	−0.890566
$857$	41.4164	1.41476	0.707379	−	0.706835i	$-0.249878\pi$
$857$	41.4164	1.41476	0.707379	+	0.706835i	$0.249878\pi$
$858$	0	0
$859$	26.9443	0.919327	0.459663	−	0.888093i	$-0.347970\pi$
$859$	26.9443	0.919327	0.459663	+	0.888093i	$0.347970\pi$
$860$	1.70820	0.0582493
$861$	0	0
$862$	−9.74265	−0.331836
$863$	9.88854	0.336610	0.168305	−	0.985735i	$-0.446171\pi$
$863$	9.88854	0.336610	0.168305	+	0.985735i	$0.446171\pi$
$864$	0	0
$865$	−36.8328	−1.25235
$866$	20.6525	0.701800
$867$	0	0
$868$	−2.47214	−0.0839098
$869$	0	0
$870$	0	0
$871$	55.8328	1.89182
$872$	3.41641	0.115694
$873$	0	0
$874$	9.23607	0.312415
$875$	11.1803	0.377964
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−5.49342	−0.185394
$879$	0	0
$880$	0	0
$881$	−35.7639	−1.20492	−0.602459	−	0.798150i	$-0.705812\pi$
$881$	−35.7639	−1.20492	−0.602459	+	0.798150i	$0.705812\pi$
$882$	0	0
$883$	−2.81966	−0.0948891	−0.0474446	−	0.998874i	$-0.515108\pi$
$883$	−2.81966	−0.0948891	−0.0474446	+	0.998874i	$0.515108\pi$
$884$	−13.7082	−0.461057
$885$	0	0
$886$	−8.65248	−0.290686
$887$	0.944272	0.0317055	0.0158528	−	0.999874i	$-0.494954\pi$
$887$	0.944272	0.0317055	0.0158528	+	0.999874i	$0.494954\pi$
$888$	0	0
$889$	17.4164	0.584128
$890$	11.7082	0.392460
$891$	0	0
$892$	−19.2361	−0.644071
$893$	−3.94427	−0.131990
$894$	0	0
$895$	41.0557	1.37234
$896$	−11.3820	−0.380245
$897$	0	0
$898$	5.23607	0.174730
$899$	−7.63932	−0.254786
$900$	0	0
$901$	−4.94427	−0.164718
$902$	0	0
$903$	0	0
$904$	−32.3607	−1.07630
$905$	11.0557	0.367505
$906$	0	0
$907$	−15.7771	−0.523870	−0.261935	−	0.965086i	$-0.584361\pi$
$907$	−15.7771	−0.523870	−0.261935	+	0.965086i	$0.584361\pi$
$908$	20.1803	0.669708
$909$	0	0
$910$	5.85410	0.194062
$911$	−26.2492	−0.869676	−0.434838	−	0.900509i	$-0.643194\pi$
$911$	−26.2492	−0.869676	−0.434838	+	0.900509i	$0.643194\pi$
$912$	0	0
$913$	0	0
$914$	−23.7082	−0.784198
$915$	0	0
$916$	−22.6525	−0.748459
$917$	7.41641	0.244911
$918$	0	0
$919$	13.5279	0.446243	0.223122	−	0.974791i	$-0.428375\pi$
$919$	13.5279	0.446243	0.223122	+	0.974791i	$0.428375\pi$
$920$	10.0000	0.329690
$921$	0	0
$922$	−0.652476	−0.0214881
$923$	27.4164	0.902422
$924$	0	0
$925$	0	0
$926$	−4.14590	−0.136243
$927$	0	0
$928$	−28.0902	−0.922105
$929$	0.708204	0.0232354	0.0116177	−	0.999933i	$-0.496302\pi$
$929$	0.708204	0.0232354	0.0116177	+	0.999933i	$0.496302\pi$
$930$	0	0
$931$	7.47214	0.244889
$932$	−4.76393	−0.156048
$933$	0	0
$934$	−17.2705	−0.565108
$935$	0	0
$936$	0	0
$937$	6.36068	0.207794	0.103897	−	0.994588i	$-0.466869\pi$
$937$	6.36068	0.207794	0.103897	+	0.994588i	$0.466869\pi$
$938$	−8.14590	−0.265973
$939$	0	0
$940$	−1.90983	−0.0622918
$941$	−53.4164	−1.74133	−0.870663	−	0.491881i	$-0.836310\pi$
$941$	−53.4164	−1.74133	−0.870663	+	0.491881i	$0.836310\pi$
$942$	0	0
$943$	12.9443	0.421523
$944$	18.4377	0.600096
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	−11.4721	−0.372401
$950$	0	0
$951$	0	0
$952$	4.47214	0.144943
$953$	18.4164	0.596566	0.298283	−	0.954477i	$-0.403586\pi$
$953$	18.4164	0.596566	0.298283	+	0.954477i	$0.403586\pi$
$954$	0	0
$955$	54.7214	1.77074
$956$	22.8541	0.739154
$957$	0	0
$958$	0.0688837	0.00222553
$959$	−17.4164	−0.562405
$960$	0	0
$961$	−28.6656	−0.924698
$962$	−19.5623	−0.630714
$963$	0	0
$964$	−5.14590	−0.165738
$965$	−23.4164	−0.753801
$966$	0	0
$967$	−49.3050	−1.58554	−0.792770	−	0.609521i	$-0.791362\pi$
$967$	−49.3050	−1.58554	−0.792770	+	0.609521i	$0.791362\pi$
$968$	0	0
$969$	0	0
$970$	24.0689	0.772805
$971$	−29.9443	−0.960957	−0.480479	−	0.877006i	$-0.659537\pi$
$971$	−29.9443	−0.960957	−0.480479	+	0.877006i	$0.659537\pi$
$972$	0	0
$973$	8.94427	0.286740
$974$	−18.4721	−0.591885
$975$	0	0
$976$	6.54102	0.209373
$977$	−7.52786	−0.240838	−0.120419	−	0.992723i	$-0.538424\pi$
$977$	−7.52786	−0.240838	−0.120419	+	0.992723i	$0.538424\pi$
$978$	0	0
$979$	0	0
$980$	3.61803	0.115574
$981$	0	0
$982$	−17.3820	−0.554681
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	26.5836	0.847024
$986$	6.18034	0.196822
$987$	0	0
$988$	51.2148	1.62936
$989$	0.944272	0.0300261
$990$	0	0
$991$	11.1803	0.355155	0.177578	−	0.984107i	$-0.443174\pi$
$991$	11.1803	0.355155	0.177578	+	0.984107i	$0.443174\pi$
$992$	8.58359	0.272529
$993$	0	0
$994$	−4.00000	−0.126872
$995$	−53.4164	−1.69341
$996$	0	0
$997$	41.1935	1.30461	0.652306	−	0.757956i	$-0.273802\pi$
$997$	41.1935	1.30461	0.652306	+	0.757956i	$0.273802\pi$
$998$	14.9787	0.474143
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.y.1.2		2
3.2	odd	2		2541.2.a.y.1.1	yes	2
11.10	odd	2		7623.2.a.bn.1.1		2
33.32	even	2		2541.2.a.q.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.q.1.2	✓	2	33.32	even	2
2541.2.a.y.1.1	yes	2	3.2	odd	2
7623.2.a.y.1.2		2	1.1	even	1	trivial
7623.2.a.bn.1.1		2	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} +1.00000 q^{7} -2.23607 q^{8} -1.38197 q^{10} -4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} -2.00000 q^{17} +7.47214 q^{19} +3.61803 q^{20} +2.00000 q^{23} -2.61803 q^{26} -1.61803 q^{28} -5.00000 q^{29} +1.52786 q^{31} +5.61803 q^{32} -1.23607 q^{34} -2.23607 q^{35} +7.47214 q^{37} +4.61803 q^{38} +5.00000 q^{40} +6.47214 q^{41} +0.472136 q^{43} +1.23607 q^{46} -0.527864 q^{47} +1.00000 q^{49} +6.85410 q^{52} +2.47214 q^{53} -2.23607 q^{56} -3.09017 q^{58} +9.94427 q^{59} +3.52786 q^{61} +0.944272 q^{62} -0.236068 q^{64} +9.47214 q^{65} -13.1803 q^{67} +3.23607 q^{68} -1.38197 q^{70} -6.47214 q^{71} +2.70820 q^{73} +4.61803 q^{74} -12.0902 q^{76} +8.47214 q^{79} -4.14590 q^{80} +4.00000 q^{82} -16.4721 q^{83} +4.47214 q^{85} +0.291796 q^{86} -8.47214 q^{89} -4.23607 q^{91} -3.23607 q^{92} -0.326238 q^{94} -16.7082 q^{95} -17.4164 q^{97} +0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{7} - 5 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} - 4 q^{17} + 6 q^{19} + 5 q^{20} + 4 q^{23} - 3 q^{26} - q^{28} - 10 q^{29} + 12 q^{31} + 9 q^{32} + 2 q^{34} + 6 q^{37} + 7 q^{38} + 10 q^{40} + 4 q^{41} - 8 q^{43} - 2 q^{46} - 10 q^{47} + 2 q^{49} + 7 q^{52} - 4 q^{53} + 5 q^{58} + 2 q^{59} + 16 q^{61} - 16 q^{62} + 4 q^{64} + 10 q^{65} - 4 q^{67} + 2 q^{68} - 5 q^{70} - 4 q^{71} - 8 q^{73} + 7 q^{74} - 13 q^{76} + 8 q^{79} - 15 q^{80} + 8 q^{82} - 24 q^{83} + 14 q^{86} - 8 q^{89} - 4 q^{91} - 2 q^{92} + 15 q^{94} - 20 q^{95} - 8 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^7 - 5 * q^10 - 4 * q^13 - q^14 - 3 * q^16 - 4 * q^17 + 6 * q^19 + 5 * q^20 + 4 * q^23 - 3 * q^26 - q^28 - 10 * q^29 + 12 * q^31 + 9 * q^32 + 2 * q^34 + 6 * q^37 + 7 * q^38 + 10 * q^40 + 4 * q^41 - 8 * q^43 - 2 * q^46 - 10 * q^47 + 2 * q^49 + 7 * q^52 - 4 * q^53 + 5 * q^58 + 2 * q^59 + 16 * q^61 - 16 * q^62 + 4 * q^64 + 10 * q^65 - 4 * q^67 + 2 * q^68 - 5 * q^70 - 4 * q^71 - 8 * q^73 + 7 * q^74 - 13 * q^76 + 8 * q^79 - 15 * q^80 + 8 * q^82 - 24 * q^83 + 14 * q^86 - 8 * q^89 - 4 * q^91 - 2 * q^92 + 15 * q^94 - 20 * q^95 - 8 * q^97 - q^98

Introduction

L-functions

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