LMFDB - Embedded newform 7623.2.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7623,2,Mod(1,7623)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,-1,0,-1,3,0,2,0,0,-4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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 231)
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)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +0.381966 q^{5} +1.00000 q^{7} -2.23607 q^{8} +0.236068 q^{10} -1.00000 q^{13} +0.618034 q^{14} +1.85410 q^{16} +4.23607 q^{17} -0.618034 q^{20} +3.23607 q^{23} -4.85410 q^{25} -0.618034 q^{26} -1.61803 q^{28} -6.70820 q^{29} -10.2361 q^{31} +5.61803 q^{32} +2.61803 q^{34} +0.381966 q^{35} +6.94427 q^{37} -0.854102 q^{40} -5.09017 q^{41} -1.00000 q^{43} +2.00000 q^{46} +7.32624 q^{47} +1.00000 q^{49} -3.00000 q^{50} +1.61803 q^{52} -7.61803 q^{53} -2.23607 q^{56} -4.14590 q^{58} +4.14590 q^{59} -5.76393 q^{61} -6.32624 q^{62} -0.236068 q^{64} -0.381966 q^{65} -9.23607 q^{67} -6.85410 q^{68} +0.236068 q^{70} +7.47214 q^{71} +11.5623 q^{73} +4.29180 q^{74} +10.8541 q^{79} +0.708204 q^{80} -3.14590 q^{82} +6.00000 q^{83} +1.61803 q^{85} -0.618034 q^{86} +6.38197 q^{89} -1.00000 q^{91} -5.23607 q^{92} +4.52786 q^{94} -17.0000 q^{97} +0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{7} - 4 q^{10} - 2 q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + q^{20} + 2 q^{23} - 3 q^{25} + q^{26} - q^{28} - 16 q^{31} + 9 q^{32} + 3 q^{34} + 3 q^{35} - 4 q^{37}+ \cdots - q^{98}+O(q^{100})$ 2 * q - q^2 - q^4 + 3 * q^5 + 2 * q^7 - 4 * q^10 - 2 * q^13 - q^14 - 3 * q^16 + 4 * q^17 + q^20 + 2 * q^23 - 3 * q^25 + q^26 - q^28 - 16 * q^31 + 9 * q^32 + 3 * q^34 + 3 * q^35 - 4 * q^37 + 5 * q^40 + q^41 - 2 * q^43 + 4 * q^46 - q^47 + 2 * q^49 - 6 * q^50 + q^52 - 13 * q^53 - 15 * q^58 + 15 * q^59 - 16 * q^61 + 3 * q^62 + 4 * q^64 - 3 * q^65 - 14 * q^67 - 7 * q^68 - 4 * q^70 + 6 * q^71 + 3 * q^73 + 22 * q^74 + 15 * q^79 - 12 * q^80 - 13 * q^82 + 12 * q^83 + q^85 + q^86 + 15 * q^89 - 2 * q^91 - 6 * q^92 + 18 * q^94 - 34 * q^97 - q^98

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
$5$	0.381966	0.170820	0.0854102	−	0.996346i	$-0.472780\pi$
$5$	0.381966	0.170820	0.0854102	+	0.996346i	$0.472780\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.236068	0.0746512
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	4.23607	1.02740	0.513699	−	0.857971i	$-0.328275\pi$
$17$	4.23607	1.02740	0.513699	+	0.857971i	$0.328275\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	0	0
$23$	3.23607	0.674767	0.337383	−	0.941367i	$-0.390458\pi$
$23$	3.23607	0.674767	0.337383	+	0.941367i	$0.390458\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	−0.618034	−0.121206
$27$	0	0
$28$	−1.61803	−0.305780
$29$	−6.70820	−1.24568	−0.622841	−	0.782348i	$-0.714022\pi$
$29$	−6.70820	−1.24568	−0.622841	+	0.782348i	$0.714022\pi$
$30$	0	0
$31$	−10.2361	−1.83845	−0.919226	−	0.393730i	$-0.871184\pi$
$31$	−10.2361	−1.83845	−0.919226	+	0.393730i	$0.871184\pi$
$32$	5.61803	0.993137
$33$	0	0
$34$	2.61803	0.448989
$35$	0.381966	0.0645640
$36$	0	0
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	0	0
$39$	0	0
$40$	−0.854102	−0.135045
$41$	−5.09017	−0.794951	−0.397475	−	0.917613i	$-0.630114\pi$
$41$	−5.09017	−0.794951	−0.397475	+	0.917613i	$0.630114\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	0	0
$46$	2.00000	0.294884
$47$	7.32624	1.06864	0.534321	−	0.845282i	$-0.320567\pi$
$47$	7.32624	1.06864	0.534321	+	0.845282i	$0.320567\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.00000	−0.424264
$51$	0	0
$52$	1.61803	0.224381
$53$	−7.61803	−1.04642	−0.523209	−	0.852205i	$-0.675265\pi$
$53$	−7.61803	−1.04642	−0.523209	+	0.852205i	$0.675265\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−4.14590	−0.544383
$59$	4.14590	0.539750	0.269875	−	0.962895i	$-0.413018\pi$
$59$	4.14590	0.539750	0.269875	+	0.962895i	$0.413018\pi$
$60$	0	0
$61$	−5.76393	−0.737996	−0.368998	−	0.929430i	$-0.620299\pi$
$61$	−5.76393	−0.737996	−0.368998	+	0.929430i	$0.620299\pi$
$62$	−6.32624	−0.803433
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	−0.381966	−0.0473771
$66$	0	0
$67$	−9.23607	−1.12837	−0.564183	−	0.825650i	$-0.690809\pi$
$67$	−9.23607	−1.12837	−0.564183	+	0.825650i	$0.690809\pi$
$68$	−6.85410	−0.831182
$69$	0	0
$70$	0.236068	0.0282155
$71$	7.47214	0.886779	0.443390	−	0.896329i	$-0.353776\pi$
$71$	7.47214	0.886779	0.443390	+	0.896329i	$0.353776\pi$
$72$	0	0
$73$	11.5623	1.35327	0.676633	−	0.736321i	$-0.263438\pi$
$73$	11.5623	1.35327	0.676633	+	0.736321i	$0.263438\pi$
$74$	4.29180	0.498911
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.8541	1.22118	0.610591	−	0.791946i	$-0.290932\pi$
$79$	10.8541	1.22118	0.610591	+	0.791946i	$0.290932\pi$
$80$	0.708204	0.0791796
$81$	0	0
$82$	−3.14590	−0.347406
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	1.61803	0.175500
$86$	−0.618034	−0.0666443
$87$	0	0
$88$	0	0
$89$	6.38197	0.676487	0.338244	−	0.941059i	$-0.390167\pi$
$89$	6.38197	0.676487	0.338244	+	0.941059i	$0.390167\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−5.23607	−0.545898
$93$	0	0
$94$	4.52786	0.467014
$95$	0	0
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0.618034	0.0624309
$99$	0	0
$100$	7.85410	0.785410
$101$	4.18034	0.415959	0.207980	−	0.978133i	$-0.433311\pi$
$101$	4.18034	0.415959	0.207980	+	0.978133i	$0.433311\pi$
$102$	0	0
$103$	−12.7082	−1.25218	−0.626088	−	0.779752i	$-0.715345\pi$
$103$	−12.7082	−1.25218	−0.626088	+	0.779752i	$0.715345\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	−4.70820	−0.457301
$107$	−15.7639	−1.52396	−0.761978	−	0.647602i	$-0.775772\pi$
$107$	−15.7639	−1.52396	−0.761978	+	0.647602i	$0.775772\pi$
$108$	0	0
$109$	−7.56231	−0.724338	−0.362169	−	0.932113i	$-0.617964\pi$
$109$	−7.56231	−0.724338	−0.362169	+	0.932113i	$0.617964\pi$
$110$	0	0
$111$	0	0
$112$	1.85410	0.175196
$113$	11.6525	1.09617	0.548086	−	0.836422i	$-0.315357\pi$
$113$	11.6525	1.09617	0.548086	+	0.836422i	$0.315357\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	10.8541	1.00778
$117$	0	0
$118$	2.56231	0.235879
$119$	4.23607	0.388320
$120$	0	0
$121$	0	0
$122$	−3.56231	−0.322516
$123$	0	0
$124$	16.5623	1.48734
$125$	−3.76393	−0.336656
$126$	0	0
$127$	−7.32624	−0.650098	−0.325049	−	0.945697i	$-0.605381\pi$
$127$	−7.32624	−0.650098	−0.325049	+	0.945697i	$0.605381\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	−0.236068	−0.0207045
$131$	3.32624	0.290615	0.145307	−	0.989387i	$-0.453583\pi$
$131$	3.32624	0.290615	0.145307	+	0.989387i	$0.453583\pi$
$132$	0	0
$133$	0	0
$134$	−5.70820	−0.493114
$135$	0	0
$136$	−9.47214	−0.812229
$137$	3.05573	0.261068	0.130534	−	0.991444i	$-0.458331\pi$
$137$	3.05573	0.261068	0.130534	+	0.991444i	$0.458331\pi$
$138$	0	0
$139$	−17.5623	−1.48962	−0.744808	−	0.667279i	$-0.767459\pi$
$139$	−17.5623	−1.48962	−0.744808	+	0.667279i	$0.767459\pi$
$140$	−0.618034	−0.0522334
$141$	0	0
$142$	4.61803	0.387537
$143$	0	0
$144$	0	0
$145$	−2.56231	−0.212788
$146$	7.14590	0.591399
$147$	0	0
$148$	−11.2361	−0.923599
$149$	13.0902	1.07239	0.536194	−	0.844095i	$-0.319861\pi$
$149$	13.0902	1.07239	0.536194	+	0.844095i	$0.319861\pi$
$150$	0	0
$151$	−21.0902	−1.71629	−0.858147	−	0.513404i	$-0.828384\pi$
$151$	−21.0902	−1.71629	−0.858147	+	0.513404i	$0.828384\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.90983	−0.314045
$156$	0	0
$157$	−1.67376	−0.133581	−0.0667904	−	0.997767i	$-0.521276\pi$
$157$	−1.67376	−0.133581	−0.0667904	+	0.997767i	$0.521276\pi$
$158$	6.70820	0.533676
$159$	0	0
$160$	2.14590	0.169648
$161$	3.23607	0.255038
$162$	0	0
$163$	6.03444	0.472654	0.236327	−	0.971674i	$-0.424056\pi$
$163$	6.03444	0.472654	0.236327	+	0.971674i	$0.424056\pi$
$164$	8.23607	0.643129
$165$	0	0
$166$	3.70820	0.287812
$167$	−9.38197	−0.725998	−0.362999	−	0.931789i	$-0.618247\pi$
$167$	−9.38197	−0.725998	−0.362999	+	0.931789i	$0.618247\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	1.00000	0.0766965
$171$	0	0
$172$	1.61803	0.123374
$173$	14.6180	1.11139	0.555694	−	0.831387i	$-0.312453\pi$
$173$	14.6180	1.11139	0.555694	+	0.831387i	$0.312453\pi$
$174$	0	0
$175$	−4.85410	−0.366936
$176$	0	0
$177$	0	0
$178$	3.94427	0.295636
$179$	2.23607	0.167132	0.0835658	−	0.996502i	$-0.473369\pi$
$179$	2.23607	0.167132	0.0835658	+	0.996502i	$0.473369\pi$
$180$	0	0
$181$	−0.236068	−0.0175468	−0.00877340	−	0.999962i	$-0.502793\pi$
$181$	−0.236068	−0.0175468	−0.00877340	+	0.999962i	$0.502793\pi$
$182$	−0.618034	−0.0458117
$183$	0	0
$184$	−7.23607	−0.533450
$185$	2.65248	0.195014
$186$	0	0
$187$	0	0
$188$	−11.8541	−0.864549
$189$	0	0
$190$	0	0
$191$	−9.76393	−0.706493	−0.353247	−	0.935530i	$-0.614922\pi$
$191$	−9.76393	−0.706493	−0.353247	+	0.935530i	$0.614922\pi$
$192$	0	0
$193$	5.05573	0.363919	0.181960	−	0.983306i	$-0.441756\pi$
$193$	5.05573	0.363919	0.181960	+	0.983306i	$0.441756\pi$
$194$	−10.5066	−0.754328
$195$	0	0
$196$	−1.61803	−0.115574
$197$	9.23607	0.658043	0.329021	−	0.944323i	$-0.393281\pi$
$197$	9.23607	0.658043	0.329021	+	0.944323i	$0.393281\pi$
$198$	0	0
$199$	−7.56231	−0.536078	−0.268039	−	0.963408i	$-0.586376\pi$
$199$	−7.56231	−0.536078	−0.268039	+	0.963408i	$0.586376\pi$
$200$	10.8541	0.767501
$201$	0	0
$202$	2.58359	0.181781
$203$	−6.70820	−0.470824
$204$	0	0
$205$	−1.94427	−0.135794
$206$	−7.85410	−0.547221
$207$	0	0
$208$	−1.85410	−0.128559
$209$	0	0
$210$	0	0
$211$	5.29180	0.364302	0.182151	−	0.983271i	$-0.441694\pi$
$211$	5.29180	0.364302	0.182151	+	0.983271i	$0.441694\pi$
$212$	12.3262	0.846569
$213$	0	0
$214$	−9.74265	−0.665994
$215$	−0.381966	−0.0260499
$216$	0	0
$217$	−10.2361	−0.694870
$218$	−4.67376	−0.316547
$219$	0	0
$220$	0	0
$221$	−4.23607	−0.284949
$222$	0	0
$223$	−12.7082	−0.851004	−0.425502	−	0.904957i	$-0.639903\pi$
$223$	−12.7082	−0.851004	−0.425502	+	0.904957i	$0.639903\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	7.20163	0.479045
$227$	−18.9787	−1.25966	−0.629831	−	0.776732i	$-0.716876\pi$
$227$	−18.9787	−1.25966	−0.629831	+	0.776732i	$0.716876\pi$
$228$	0	0
$229$	−11.7082	−0.773700	−0.386850	−	0.922143i	$-0.626437\pi$
$229$	−11.7082	−0.773700	−0.386850	+	0.922143i	$0.626437\pi$
$230$	0.763932	0.0503722
$231$	0	0
$232$	15.0000	0.984798
$233$	−6.23607	−0.408538	−0.204269	−	0.978915i	$-0.565482\pi$
$233$	−6.23607	−0.408538	−0.204269	+	0.978915i	$0.565482\pi$
$234$	0	0
$235$	2.79837	0.182546
$236$	−6.70820	−0.436667
$237$	0	0
$238$	2.61803	0.169702
$239$	−19.1459	−1.23845	−0.619223	−	0.785215i	$-0.712552\pi$
$239$	−19.1459	−1.23845	−0.619223	+	0.785215i	$0.712552\pi$
$240$	0	0
$241$	−24.7082	−1.59160	−0.795798	−	0.605563i	$-0.792948\pi$
$241$	−24.7082	−1.59160	−0.795798	+	0.605563i	$0.792948\pi$
$242$	0	0
$243$	0	0
$244$	9.32624	0.597051
$245$	0.381966	0.0244029
$246$	0	0
$247$	0	0
$248$	22.8885	1.45342
$249$	0	0
$250$	−2.32624	−0.147124
$251$	−26.5967	−1.67877	−0.839386	−	0.543536i	$-0.817085\pi$
$251$	−26.5967	−1.67877	−0.839386	+	0.543536i	$0.817085\pi$
$252$	0	0
$253$	0	0
$254$	−4.52786	−0.284103
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−26.9443	−1.68074	−0.840369	−	0.542015i	$-0.817662\pi$
$257$	−26.9443	−1.68074	−0.840369	+	0.542015i	$0.817662\pi$
$258$	0	0
$259$	6.94427	0.431496
$260$	0.618034	0.0383288
$261$	0	0
$262$	2.05573	0.127003
$263$	−0.708204	−0.0436697	−0.0218349	−	0.999762i	$-0.506951\pi$
$263$	−0.708204	−0.0436697	−0.0218349	+	0.999762i	$0.506951\pi$
$264$	0	0
$265$	−2.90983	−0.178749
$266$	0	0
$267$	0	0
$268$	14.9443	0.912867
$269$	−23.9443	−1.45991	−0.729954	−	0.683496i	$-0.760459\pi$
$269$	−23.9443	−1.45991	−0.729954	+	0.683496i	$0.760459\pi$
$270$	0	0
$271$	−6.41641	−0.389769	−0.194885	−	0.980826i	$-0.562433\pi$
$271$	−6.41641	−0.389769	−0.194885	+	0.980826i	$0.562433\pi$
$272$	7.85410	0.476225
$273$	0	0
$274$	1.88854	0.114091
$275$	0	0
$276$	0	0
$277$	−25.4164	−1.52712	−0.763562	−	0.645735i	$-0.776551\pi$
$277$	−25.4164	−1.52712	−0.763562	+	0.645735i	$0.776551\pi$
$278$	−10.8541	−0.650986
$279$	0	0
$280$	−0.854102	−0.0510424
$281$	−15.6180	−0.931694	−0.465847	−	0.884865i	$-0.654250\pi$
$281$	−15.6180	−0.931694	−0.465847	+	0.884865i	$0.654250\pi$
$282$	0	0
$283$	7.29180	0.433452	0.216726	−	0.976232i	$-0.430462\pi$
$283$	7.29180	0.433452	0.216726	+	0.976232i	$0.430462\pi$
$284$	−12.0902	−0.717420
$285$	0	0
$286$	0	0
$287$	−5.09017	−0.300463
$288$	0	0
$289$	0.944272	0.0555454
$290$	−1.58359	−0.0929917
$291$	0	0
$292$	−18.7082	−1.09481
$293$	13.2361	0.773259	0.386630	−	0.922235i	$-0.373639\pi$
$293$	13.2361	0.773259	0.386630	+	0.922235i	$0.373639\pi$
$294$	0	0
$295$	1.58359	0.0922003
$296$	−15.5279	−0.902539
$297$	0	0
$298$	8.09017	0.468651
$299$	−3.23607	−0.187147
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	−13.0344	−0.750048
$303$	0	0
$304$	0	0
$305$	−2.20163	−0.126065
$306$	0	0
$307$	19.1803	1.09468	0.547340	−	0.836910i	$-0.315640\pi$
$307$	19.1803	1.09468	0.547340	+	0.836910i	$0.315640\pi$
$308$	0	0
$309$	0	0
$310$	−2.41641	−0.137243
$311$	−16.7984	−0.952548	−0.476274	−	0.879297i	$-0.658013\pi$
$311$	−16.7984	−0.952548	−0.476274	+	0.879297i	$0.658013\pi$
$312$	0	0
$313$	−15.7984	−0.892977	−0.446488	−	0.894789i	$-0.647326\pi$
$313$	−15.7984	−0.892977	−0.446488	+	0.894789i	$0.647326\pi$
$314$	−1.03444	−0.0583769
$315$	0	0
$316$	−17.5623	−0.987957
$317$	−25.3607	−1.42440	−0.712199	−	0.701978i	$-0.752301\pi$
$317$	−25.3607	−1.42440	−0.712199	+	0.701978i	$0.752301\pi$
$318$	0	0
$319$	0	0
$320$	−0.0901699	−0.00504065
$321$	0	0
$322$	2.00000	0.111456
$323$	0	0
$324$	0	0
$325$	4.85410	0.269257
$326$	3.72949	0.206557
$327$	0	0
$328$	11.3820	0.628464
$329$	7.32624	0.403909
$330$	0	0
$331$	−14.7082	−0.808436	−0.404218	−	0.914663i	$-0.632456\pi$
$331$	−14.7082	−0.808436	−0.404218	+	0.914663i	$0.632456\pi$
$332$	−9.70820	−0.532807
$333$	0	0
$334$	−5.79837	−0.317273
$335$	−3.52786	−0.192748
$336$	0	0
$337$	1.29180	0.0703686	0.0351843	−	0.999381i	$-0.488798\pi$
$337$	1.29180	0.0703686	0.0351843	+	0.999381i	$0.488798\pi$
$338$	−7.41641	−0.403399
$339$	0	0
$340$	−2.61803	−0.141983
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.23607	0.120561
$345$	0	0
$346$	9.03444	0.485695
$347$	25.4164	1.36442	0.682212	−	0.731154i	$-0.261018\pi$
$347$	25.4164	1.36442	0.682212	+	0.731154i	$0.261018\pi$
$348$	0	0
$349$	34.9230	1.86938	0.934692	−	0.355458i	$-0.115675\pi$
$349$	34.9230	1.86938	0.934692	+	0.355458i	$0.115675\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	0	0
$353$	25.4721	1.35574	0.677872	−	0.735179i	$-0.262902\pi$
$353$	25.4721	1.35574	0.677872	+	0.735179i	$0.262902\pi$
$354$	0	0
$355$	2.85410	0.151480
$356$	−10.3262	−0.547290
$357$	0	0
$358$	1.38197	0.0730392
$359$	33.2148	1.75301	0.876505	−	0.481394i	$-0.159869\pi$
$359$	33.2148	1.75301	0.876505	+	0.481394i	$0.159869\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−0.145898	−0.00766823
$363$	0	0
$364$	1.61803	0.0848080
$365$	4.41641	0.231165
$366$	0	0
$367$	11.2918	0.589427	0.294713	−	0.955586i	$-0.404776\pi$
$367$	11.2918	0.589427	0.294713	+	0.955586i	$0.404776\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	1.63932	0.0852242
$371$	−7.61803	−0.395509
$372$	0	0
$373$	30.1803	1.56268	0.781339	−	0.624106i	$-0.214537\pi$
$373$	30.1803	1.56268	0.781339	+	0.624106i	$0.214537\pi$
$374$	0	0
$375$	0	0
$376$	−16.3820	−0.844835
$377$	6.70820	0.345490
$378$	0	0
$379$	−10.8541	−0.557538	−0.278769	−	0.960358i	$-0.589926\pi$
$379$	−10.8541	−0.557538	−0.278769	+	0.960358i	$0.589926\pi$
$380$	0	0
$381$	0	0
$382$	−6.03444	−0.308749
$383$	7.90983	0.404173	0.202087	−	0.979368i	$-0.435228\pi$
$383$	7.90983	0.404173	0.202087	+	0.979368i	$0.435228\pi$
$384$	0	0
$385$	0	0
$386$	3.12461	0.159039
$387$	0	0
$388$	27.5066	1.39643
$389$	−20.6525	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$389$	−20.6525	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$390$	0	0
$391$	13.7082	0.693254
$392$	−2.23607	−0.112938
$393$	0	0
$394$	5.70820	0.287575
$395$	4.14590	0.208603
$396$	0	0
$397$	25.6869	1.28919	0.644595	−	0.764524i	$-0.277026\pi$
$397$	25.6869	1.28919	0.644595	+	0.764524i	$0.277026\pi$
$398$	−4.67376	−0.234275
$399$	0	0
$400$	−9.00000	−0.450000
$401$	−11.6738	−0.582960	−0.291480	−	0.956577i	$-0.594148\pi$
$401$	−11.6738	−0.582960	−0.291480	+	0.956577i	$0.594148\pi$
$402$	0	0
$403$	10.2361	0.509895
$404$	−6.76393	−0.336518
$405$	0	0
$406$	−4.14590	−0.205757
$407$	0	0
$408$	0	0
$409$	−26.3820	−1.30450	−0.652252	−	0.758002i	$-0.726176\pi$
$409$	−26.3820	−1.30450	−0.652252	+	0.758002i	$0.726176\pi$
$410$	−1.20163	−0.0593441
$411$	0	0
$412$	20.5623	1.01303
$413$	4.14590	0.204006
$414$	0	0
$415$	2.29180	0.112500
$416$	−5.61803	−0.275447
$417$	0	0
$418$	0	0
$419$	15.3262	0.748736	0.374368	−	0.927280i	$-0.377860\pi$
$419$	15.3262	0.748736	0.374368	+	0.927280i	$0.377860\pi$
$420$	0	0
$421$	2.72949	0.133027	0.0665136	−	0.997786i	$-0.478812\pi$
$421$	2.72949	0.133027	0.0665136	+	0.997786i	$0.478812\pi$
$422$	3.27051	0.159206
$423$	0	0
$424$	17.0344	0.827266
$425$	−20.5623	−0.997418
$426$	0	0
$427$	−5.76393	−0.278936
$428$	25.5066	1.23291
$429$	0	0
$430$	−0.236068	−0.0113842
$431$	37.7984	1.82068	0.910342	−	0.413857i	$-0.135819\pi$
$431$	37.7984	1.82068	0.910342	+	0.413857i	$0.135819\pi$
$432$	0	0
$433$	−2.58359	−0.124160	−0.0620798	−	0.998071i	$-0.519773\pi$
$433$	−2.58359	−0.124160	−0.0620798	+	0.998071i	$0.519773\pi$
$434$	−6.32624	−0.303669
$435$	0	0
$436$	12.2361	0.586001
$437$	0	0
$438$	0	0
$439$	0.527864	0.0251936	0.0125968	−	0.999921i	$-0.495990\pi$
$439$	0.527864	0.0251936	0.0125968	+	0.999921i	$0.495990\pi$
$440$	0	0
$441$	0	0
$442$	−2.61803	−0.124527
$443$	36.6525	1.74141	0.870706	−	0.491804i	$-0.163662\pi$
$443$	36.6525	1.74141	0.870706	+	0.491804i	$0.163662\pi$
$444$	0	0
$445$	2.43769	0.115558
$446$	−7.85410	−0.371903
$447$	0	0
$448$	−0.236068	−0.0111532
$449$	−15.6525	−0.738686	−0.369343	−	0.929293i	$-0.620417\pi$
$449$	−15.6525	−0.738686	−0.369343	+	0.929293i	$0.620417\pi$
$450$	0	0
$451$	0	0
$452$	−18.8541	−0.886822
$453$	0	0
$454$	−11.7295	−0.550492
$455$	−0.381966	−0.0179068
$456$	0	0
$457$	12.2705	0.573990	0.286995	−	0.957932i	$-0.407344\pi$
$457$	12.2705	0.573990	0.286995	+	0.957932i	$0.407344\pi$
$458$	−7.23607	−0.338119
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−23.1803	−1.07962	−0.539808	−	0.841788i	$-0.681503\pi$
$461$	−23.1803	−1.07962	−0.539808	+	0.841788i	$0.681503\pi$
$462$	0	0
$463$	−35.9230	−1.66948	−0.834741	−	0.550642i	$-0.814383\pi$
$463$	−35.9230	−1.66948	−0.834741	+	0.550642i	$0.814383\pi$
$464$	−12.4377	−0.577405
$465$	0	0
$466$	−3.85410	−0.178538
$467$	−31.9443	−1.47820	−0.739102	−	0.673593i	$-0.764750\pi$
$467$	−31.9443	−1.47820	−0.739102	+	0.673593i	$0.764750\pi$
$468$	0	0
$469$	−9.23607	−0.426482
$470$	1.72949	0.0797754
$471$	0	0
$472$	−9.27051	−0.426710
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−6.85410	−0.314157
$477$	0	0
$478$	−11.8328	−0.541220
$479$	−21.5066	−0.982661	−0.491330	−	0.870973i	$-0.663489\pi$
$479$	−21.5066	−0.982661	−0.491330	+	0.870973i	$0.663489\pi$
$480$	0	0
$481$	−6.94427	−0.316632
$482$	−15.2705	−0.695553
$483$	0	0
$484$	0	0
$485$	−6.49342	−0.294851
$486$	0	0
$487$	16.6180	0.753035	0.376518	−	0.926410i	$-0.377121\pi$
$487$	16.6180	0.753035	0.376518	+	0.926410i	$0.377121\pi$
$488$	12.8885	0.583437
$489$	0	0
$490$	0.236068	0.0106645
$491$	−3.70820	−0.167349	−0.0836745	−	0.996493i	$-0.526666\pi$
$491$	−3.70820	−0.167349	−0.0836745	+	0.996493i	$0.526666\pi$
$492$	0	0
$493$	−28.4164	−1.27981
$494$	0	0
$495$	0	0
$496$	−18.9787	−0.852169
$497$	7.47214	0.335171
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	6.09017	0.272361
$501$	0	0
$502$	−16.4377	−0.733650
$503$	32.3050	1.44041	0.720203	−	0.693763i	$-0.244049\pi$
$503$	32.3050	1.44041	0.720203	+	0.693763i	$0.244049\pi$
$504$	0	0
$505$	1.59675	0.0710543
$506$	0	0
$507$	0	0
$508$	11.8541	0.525941
$509$	−41.8328	−1.85421	−0.927103	−	0.374805i	$-0.877709\pi$
$509$	−41.8328	−1.85421	−0.927103	+	0.374805i	$0.877709\pi$
$510$	0	0
$511$	11.5623	0.511486
$512$	18.7082	0.826794
$513$	0	0
$514$	−16.6525	−0.734509
$515$	−4.85410	−0.213897
$516$	0	0
$517$	0	0
$518$	4.29180	0.188571
$519$	0	0
$520$	0.854102	0.0374548
$521$	19.1803	0.840306	0.420153	−	0.907453i	$-0.361976\pi$
$521$	19.1803	0.840306	0.420153	+	0.907453i	$0.361976\pi$
$522$	0	0
$523$	20.3050	0.887874	0.443937	−	0.896058i	$-0.353581\pi$
$523$	20.3050	0.887874	0.443937	+	0.896058i	$0.353581\pi$
$524$	−5.38197	−0.235112
$525$	0	0
$526$	−0.437694	−0.0190844
$527$	−43.3607	−1.88882
$528$	0	0
$529$	−12.5279	−0.544690
$530$	−1.79837	−0.0781164
$531$	0	0
$532$	0	0
$533$	5.09017	0.220480
$534$	0	0
$535$	−6.02129	−0.260323
$536$	20.6525	0.892051
$537$	0	0
$538$	−14.7984	−0.638003
$539$	0	0
$540$	0	0
$541$	20.0902	0.863744	0.431872	−	0.901935i	$-0.357853\pi$
$541$	20.0902	0.863744	0.431872	+	0.901935i	$0.357853\pi$
$542$	−3.96556	−0.170335
$543$	0	0
$544$	23.7984	1.02035
$545$	−2.88854	−0.123732
$546$	0	0
$547$	22.5967	0.966167	0.483084	−	0.875574i	$-0.339517\pi$
$547$	22.5967	0.966167	0.483084	+	0.875574i	$0.339517\pi$
$548$	−4.94427	−0.211209
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	10.8541	0.461563
$554$	−15.7082	−0.667378
$555$	0	0
$556$	28.4164	1.20512
$557$	−0.763932	−0.0323688	−0.0161844	−	0.999869i	$-0.505152\pi$
$557$	−0.763932	−0.0323688	−0.0161844	+	0.999869i	$0.505152\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0.708204	0.0299271
$561$	0	0
$562$	−9.65248	−0.407165
$563$	45.0689	1.89943	0.949713	−	0.313120i	$-0.101374\pi$
$563$	45.0689	1.89943	0.949713	+	0.313120i	$0.101374\pi$
$564$	0	0
$565$	4.45085	0.187249
$566$	4.50658	0.189426
$567$	0	0
$568$	−16.7082	−0.701061
$569$	−39.0689	−1.63785	−0.818926	−	0.573899i	$-0.805430\pi$
$569$	−39.0689	−1.63785	−0.818926	+	0.573899i	$0.805430\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	0	0
$573$	0	0
$574$	−3.14590	−0.131307
$575$	−15.7082	−0.655077
$576$	0	0
$577$	−9.43769	−0.392896	−0.196448	−	0.980514i	$-0.562941\pi$
$577$	−9.43769	−0.392896	−0.196448	+	0.980514i	$0.562941\pi$
$578$	0.583592	0.0242742
$579$	0	0
$580$	4.14590	0.172149
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	−25.8541	−1.06985
$585$	0	0
$586$	8.18034	0.337927
$587$	−35.8885	−1.48128	−0.740639	−	0.671903i	$-0.765477\pi$
$587$	−35.8885	−1.48128	−0.740639	+	0.671903i	$0.765477\pi$
$588$	0	0
$589$	0	0
$590$	0.978714	0.0402930
$591$	0	0
$592$	12.8754	0.529175
$593$	23.8885	0.980985	0.490492	−	0.871445i	$-0.336817\pi$
$593$	23.8885	0.980985	0.490492	+	0.871445i	$0.336817\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	−21.1803	−0.867581
$597$	0	0
$598$	−2.00000	−0.0817861
$599$	−31.5066	−1.28732	−0.643662	−	0.765310i	$-0.722586\pi$
$599$	−31.5066	−1.28732	−0.643662	+	0.765310i	$0.722586\pi$
$600$	0	0
$601$	−14.8328	−0.605043	−0.302522	−	0.953143i	$-0.597828\pi$
$601$	−14.8328	−0.605043	−0.302522	+	0.953143i	$0.597828\pi$
$602$	−0.618034	−0.0251892
$603$	0	0
$604$	34.1246	1.38851
$605$	0	0
$606$	0	0
$607$	−16.1459	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$607$	−16.1459	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$608$	0	0
$609$	0	0
$610$	−1.36068	−0.0550923
$611$	−7.32624	−0.296388
$612$	0	0
$613$	34.4508	1.39146	0.695728	−	0.718305i	$-0.255082\pi$
$613$	34.4508	1.39146	0.695728	+	0.718305i	$0.255082\pi$
$614$	11.8541	0.478393
$615$	0	0
$616$	0	0
$617$	10.0902	0.406215	0.203107	−	0.979156i	$-0.434896\pi$
$617$	10.0902	0.406215	0.203107	+	0.979156i	$0.434896\pi$
$618$	0	0
$619$	35.1246	1.41178	0.705889	−	0.708323i	$-0.250548\pi$
$619$	35.1246	1.41178	0.705889	+	0.708323i	$0.250548\pi$
$620$	6.32624	0.254068
$621$	0	0
$622$	−10.3820	−0.416279
$623$	6.38197	0.255688
$624$	0	0
$625$	22.8328	0.913313
$626$	−9.76393	−0.390245
$627$	0	0
$628$	2.70820	0.108069
$629$	29.4164	1.17291
$630$	0	0
$631$	−9.18034	−0.365464	−0.182732	−	0.983163i	$-0.558494\pi$
$631$	−9.18034	−0.365464	−0.182732	+	0.983163i	$0.558494\pi$
$632$	−24.2705	−0.965429
$633$	0	0
$634$	−15.6738	−0.622485
$635$	−2.79837	−0.111050
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0	0
$640$	−4.34752	−0.171851
$641$	−27.9787	−1.10509	−0.552546	−	0.833482i	$-0.686344\pi$
$641$	−27.9787	−1.10509	−0.552546	+	0.833482i	$0.686344\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	−5.23607	−0.206330
$645$	0	0
$646$	0	0
$647$	−2.34752	−0.0922907	−0.0461453	−	0.998935i	$-0.514694\pi$
$647$	−2.34752	−0.0922907	−0.0461453	+	0.998935i	$0.514694\pi$
$648$	0	0
$649$	0	0
$650$	3.00000	0.117670
$651$	0	0
$652$	−9.76393	−0.382385
$653$	−0.708204	−0.0277142	−0.0138571	−	0.999904i	$-0.504411\pi$
$653$	−0.708204	−0.0277142	−0.0138571	+	0.999904i	$0.504411\pi$
$654$	0	0
$655$	1.27051	0.0496429
$656$	−9.43769	−0.368480
$657$	0	0
$658$	4.52786	0.176515
$659$	−41.8328	−1.62958	−0.814788	−	0.579760i	$-0.803146\pi$
$659$	−41.8328	−1.62958	−0.814788	+	0.579760i	$0.803146\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	−9.09017	−0.353299
$663$	0	0
$664$	−13.4164	−0.520658
$665$	0	0
$666$	0	0
$667$	−21.7082	−0.840545
$668$	15.1803	0.587345
$669$	0	0
$670$	−2.18034	−0.0842339
$671$	0	0
$672$	0	0
$673$	−7.58359	−0.292326	−0.146163	−	0.989261i	$-0.546692\pi$
$673$	−7.58359	−0.292326	−0.146163	+	0.989261i	$0.546692\pi$
$674$	0.798374	0.0307522
$675$	0	0
$676$	19.4164	0.746785
$677$	−23.4508	−0.901289	−0.450645	−	0.892703i	$-0.648806\pi$
$677$	−23.4508	−0.901289	−0.450645	+	0.892703i	$0.648806\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	−3.61803	−0.138745
$681$	0	0
$682$	0	0
$683$	32.3050	1.23611	0.618057	−	0.786133i	$-0.287920\pi$
$683$	32.3050	1.23611	0.618057	+	0.786133i	$0.287920\pi$
$684$	0	0
$685$	1.16718	0.0445958
$686$	0.618034	0.0235966
$687$	0	0
$688$	−1.85410	−0.0706870
$689$	7.61803	0.290224
$690$	0	0
$691$	14.5623	0.553976	0.276988	−	0.960873i	$-0.410664\pi$
$691$	14.5623	0.553976	0.276988	+	0.960873i	$0.410664\pi$
$692$	−23.6525	−0.899132
$693$	0	0
$694$	15.7082	0.596275
$695$	−6.70820	−0.254457
$696$	0	0
$697$	−21.5623	−0.816731
$698$	21.5836	0.816951
$699$	0	0
$700$	7.85410	0.296857
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	15.7426	0.592482
$707$	4.18034	0.157218
$708$	0	0
$709$	−50.6525	−1.90229	−0.951147	−	0.308739i	$-0.900093\pi$
$709$	−50.6525	−1.90229	−0.951147	+	0.308739i	$0.900093\pi$
$710$	1.76393	0.0661992
$711$	0	0
$712$	−14.2705	−0.534810
$713$	−33.1246	−1.24053
$714$	0	0
$715$	0	0
$716$	−3.61803	−0.135212
$717$	0	0
$718$	20.5279	0.766093
$719$	8.61803	0.321398	0.160699	−	0.987003i	$-0.448625\pi$
$719$	8.61803	0.321398	0.160699	+	0.987003i	$0.448625\pi$
$720$	0	0
$721$	−12.7082	−0.473278
$722$	−11.7426	−0.437016
$723$	0	0
$724$	0.381966	0.0141957
$725$	32.5623	1.20933
$726$	0	0
$727$	22.1459	0.821346	0.410673	−	0.911783i	$-0.365294\pi$
$727$	22.1459	0.821346	0.410673	+	0.911783i	$0.365294\pi$
$728$	2.23607	0.0828742
$729$	0	0
$730$	2.72949	0.101023
$731$	−4.23607	−0.156677
$732$	0	0
$733$	4.12461	0.152346	0.0761730	−	0.997095i	$-0.475730\pi$
$733$	4.12461	0.152346	0.0761730	+	0.997095i	$0.475730\pi$
$734$	6.97871	0.257589
$735$	0	0
$736$	18.1803	0.670136
$737$	0	0
$738$	0	0
$739$	42.6869	1.57026	0.785132	−	0.619329i	$-0.212595\pi$
$739$	42.6869	1.57026	0.785132	+	0.619329i	$0.212595\pi$
$740$	−4.29180	−0.157770
$741$	0	0
$742$	−4.70820	−0.172844
$743$	−25.9098	−0.950539	−0.475270	−	0.879840i	$-0.657650\pi$
$743$	−25.9098	−0.950539	−0.475270	+	0.879840i	$0.657650\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	18.6525	0.682916
$747$	0	0
$748$	0	0
$749$	−15.7639	−0.576002
$750$	0	0
$751$	−28.8541	−1.05290	−0.526451	−	0.850206i	$-0.676477\pi$
$751$	−28.8541	−1.05290	−0.526451	+	0.850206i	$0.676477\pi$
$752$	13.5836	0.495343
$753$	0	0
$754$	4.14590	0.150985
$755$	−8.05573	−0.293178
$756$	0	0
$757$	45.2361	1.64413	0.822066	−	0.569392i	$-0.192821\pi$
$757$	45.2361	1.64413	0.822066	+	0.569392i	$0.192821\pi$
$758$	−6.70820	−0.243653
$759$	0	0
$760$	0	0
$761$	−0.0901699	−0.00326866	−0.00163433	−	0.999999i	$-0.500520\pi$
$761$	−0.0901699	−0.00326866	−0.00163433	+	0.999999i	$0.500520\pi$
$762$	0	0
$763$	−7.56231	−0.273774
$764$	15.7984	0.571565
$765$	0	0
$766$	4.88854	0.176630
$767$	−4.14590	−0.149700
$768$	0	0
$769$	44.4721	1.60371	0.801853	−	0.597521i	$-0.203848\pi$
$769$	44.4721	1.60371	0.801853	+	0.597521i	$0.203848\pi$
$770$	0	0
$771$	0	0
$772$	−8.18034	−0.294417
$773$	−10.5836	−0.380665	−0.190333	−	0.981720i	$-0.560957\pi$
$773$	−10.5836	−0.380665	−0.190333	+	0.981720i	$0.560957\pi$
$774$	0	0
$775$	49.6869	1.78481
$776$	38.0132	1.36459
$777$	0	0
$778$	−12.7639	−0.457609
$779$	0	0
$780$	0	0
$781$	0	0
$782$	8.47214	0.302963
$783$	0	0
$784$	1.85410	0.0662179
$785$	−0.639320	−0.0228183
$786$	0	0
$787$	−32.8541	−1.17112	−0.585561	−	0.810628i	$-0.699126\pi$
$787$	−32.8541	−1.17112	−0.585561	+	0.810628i	$0.699126\pi$
$788$	−14.9443	−0.532368
$789$	0	0
$790$	2.56231	0.0911628
$791$	11.6525	0.414314
$792$	0	0
$793$	5.76393	0.204683
$794$	15.8754	0.563396
$795$	0	0
$796$	12.2361	0.433696
$797$	32.4508	1.14947	0.574734	−	0.818340i	$-0.305106\pi$
$797$	32.4508	1.14947	0.574734	+	0.818340i	$0.305106\pi$
$798$	0	0
$799$	31.0344	1.09792
$800$	−27.2705	−0.964158
$801$	0	0
$802$	−7.21478	−0.254763
$803$	0	0
$804$	0	0
$805$	1.23607	0.0435657
$806$	6.32624	0.222832
$807$	0	0
$808$	−9.34752	−0.328845
$809$	−30.9787	−1.08915	−0.544577	−	0.838711i	$-0.683310\pi$
$809$	−30.9787	−1.08915	−0.544577	+	0.838711i	$0.683310\pi$
$810$	0	0
$811$	−23.1246	−0.812015	−0.406007	−	0.913870i	$-0.633079\pi$
$811$	−23.1246	−0.812015	−0.406007	+	0.913870i	$0.633079\pi$
$812$	10.8541	0.380904
$813$	0	0
$814$	0	0
$815$	2.30495	0.0807389
$816$	0	0
$817$	0	0
$818$	−16.3050	−0.570089
$819$	0	0
$820$	3.14590	0.109860
$821$	−5.81966	−0.203108	−0.101554	−	0.994830i	$-0.532381\pi$
$821$	−5.81966	−0.203108	−0.101554	+	0.994830i	$0.532381\pi$
$822$	0	0
$823$	−26.2492	−0.914990	−0.457495	−	0.889212i	$-0.651253\pi$
$823$	−26.2492	−0.914990	−0.457495	+	0.889212i	$0.651253\pi$
$824$	28.4164	0.989932
$825$	0	0
$826$	2.56231	0.0891540
$827$	−5.11146	−0.177743	−0.0888714	−	0.996043i	$-0.528326\pi$
$827$	−5.11146	−0.177743	−0.0888714	+	0.996043i	$0.528326\pi$
$828$	0	0
$829$	37.6869	1.30892	0.654460	−	0.756096i	$-0.272896\pi$
$829$	37.6869	1.30892	0.654460	+	0.756096i	$0.272896\pi$
$830$	1.41641	0.0491642
$831$	0	0
$832$	0.236068	0.00818418
$833$	4.23607	0.146771
$834$	0	0
$835$	−3.58359	−0.124015
$836$	0	0
$837$	0	0
$838$	9.47214	0.327210
$839$	−29.0689	−1.00357	−0.501785	−	0.864993i	$-0.667323\pi$
$839$	−29.0689	−1.00357	−0.501785	+	0.864993i	$0.667323\pi$
$840$	0	0
$841$	16.0000	0.551724
$842$	1.68692	0.0581350
$843$	0	0
$844$	−8.56231	−0.294727
$845$	−4.58359	−0.157680
$846$	0	0
$847$	0	0
$848$	−14.1246	−0.485041
$849$	0	0
$850$	−12.7082	−0.435888
$851$	22.4721	0.770335
$852$	0	0
$853$	9.20163	0.315058	0.157529	−	0.987514i	$-0.449647\pi$
$853$	9.20163	0.315058	0.157529	+	0.987514i	$0.449647\pi$
$854$	−3.56231	−0.121900
$855$	0	0
$856$	35.2492	1.20479
$857$	54.2361	1.85267	0.926334	−	0.376702i	$-0.122942\pi$
$857$	54.2361	1.85267	0.926334	+	0.376702i	$0.122942\pi$
$858$	0	0
$859$	−41.8328	−1.42732	−0.713659	−	0.700494i	$-0.752963\pi$
$859$	−41.8328	−1.42732	−0.713659	+	0.700494i	$0.752963\pi$
$860$	0.618034	0.0210748
$861$	0	0
$862$	23.3607	0.795668
$863$	−31.3607	−1.06753	−0.533765	−	0.845633i	$-0.679223\pi$
$863$	−31.3607	−1.06753	−0.533765	+	0.845633i	$0.679223\pi$
$864$	0	0
$865$	5.58359	0.189848
$866$	−1.59675	−0.0542597
$867$	0	0
$868$	16.5623	0.562161
$869$	0	0
$870$	0	0
$871$	9.23607	0.312952
$872$	16.9098	0.572639
$873$	0	0
$874$	0	0
$875$	−3.76393	−0.127244
$876$	0	0
$877$	41.7426	1.40955	0.704774	−	0.709431i	$-0.251048\pi$
$877$	41.7426	1.40955	0.704774	+	0.709431i	$0.251048\pi$
$878$	0.326238	0.0110100
$879$	0	0
$880$	0	0
$881$	11.9443	0.402413	0.201206	−	0.979549i	$-0.435514\pi$
$881$	11.9443	0.402413	0.201206	+	0.979549i	$0.435514\pi$
$882$	0	0
$883$	−33.0344	−1.11170	−0.555849	−	0.831283i	$-0.687607\pi$
$883$	−33.0344	−1.11170	−0.555849	+	0.831283i	$0.687607\pi$
$884$	6.85410	0.230528
$885$	0	0
$886$	22.6525	0.761025
$887$	21.4721	0.720964	0.360482	−	0.932766i	$-0.382612\pi$
$887$	21.4721	0.720964	0.360482	+	0.932766i	$0.382612\pi$
$888$	0	0
$889$	−7.32624	−0.245714
$890$	1.50658	0.0505006
$891$	0	0
$892$	20.5623	0.688477
$893$	0	0
$894$	0	0
$895$	0.854102	0.0285495
$896$	−11.3820	−0.380245
$897$	0	0
$898$	−9.67376	−0.322818
$899$	68.6656	2.29013
$900$	0	0
$901$	−32.2705	−1.07509
$902$	0	0
$903$	0	0
$904$	−26.0557	−0.866601
$905$	−0.0901699	−0.00299735
$906$	0	0
$907$	18.9787	0.630178	0.315089	−	0.949062i	$-0.397966\pi$
$907$	18.9787	0.630178	0.315089	+	0.949062i	$0.397966\pi$
$908$	30.7082	1.01909
$909$	0	0
$910$	−0.236068	−0.00782558
$911$	32.6738	1.08253	0.541265	−	0.840852i	$-0.317946\pi$
$911$	32.6738	1.08253	0.541265	+	0.840852i	$0.317946\pi$
$912$	0	0
$913$	0	0
$914$	7.58359	0.250843
$915$	0	0
$916$	18.9443	0.625936
$917$	3.32624	0.109842
$918$	0	0
$919$	−16.5836	−0.547042	−0.273521	−	0.961866i	$-0.588188\pi$
$919$	−16.5836	−0.547042	−0.273521	+	0.961866i	$0.588188\pi$
$920$	−2.76393	−0.0911241
$921$	0	0
$922$	−14.3262	−0.471810
$923$	−7.47214	−0.245948
$924$	0	0
$925$	−33.7082	−1.10832
$926$	−22.2016	−0.729591
$927$	0	0
$928$	−37.6869	−1.23713
$929$	15.7771	0.517629	0.258815	−	0.965927i	$-0.416668\pi$
$929$	15.7771	0.517629	0.258815	+	0.965927i	$0.416668\pi$
$930$	0	0
$931$	0	0
$932$	10.0902	0.330515
$933$	0	0
$934$	−19.7426	−0.645999
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	−5.70820	−0.186379
$939$	0	0
$940$	−4.52786	−0.147683
$941$	54.1803	1.76623	0.883114	−	0.469158i	$-0.155442\pi$
$941$	54.1803	1.76623	0.883114	+	0.469158i	$0.155442\pi$
$942$	0	0
$943$	−16.4721	−0.536407
$944$	7.68692	0.250188
$945$	0	0
$946$	0	0
$947$	0.618034	0.0200834	0.0100417	−	0.999950i	$-0.496804\pi$
$947$	0.618034	0.0200834	0.0100417	+	0.999950i	$0.496804\pi$
$948$	0	0
$949$	−11.5623	−0.375328
$950$	0	0
$951$	0	0
$952$	−9.47214	−0.306994
$953$	−19.2016	−0.622002	−0.311001	−	0.950410i	$-0.600664\pi$
$953$	−19.2016	−0.622002	−0.311001	+	0.950410i	$0.600664\pi$
$954$	0	0
$955$	−3.72949	−0.120683
$956$	30.9787	1.00192
$957$	0	0
$958$	−13.2918	−0.429438
$959$	3.05573	0.0986746
$960$	0	0
$961$	73.7771	2.37991
$962$	−4.29180	−0.138373
$963$	0	0
$964$	39.9787	1.28763
$965$	1.93112	0.0621648
$966$	0	0
$967$	36.2918	1.16707	0.583533	−	0.812090i	$-0.301670\pi$
$967$	36.2918	1.16707	0.583533	+	0.812090i	$0.301670\pi$
$968$	0	0
$969$	0	0
$970$	−4.01316	−0.128855
$971$	24.7082	0.792924	0.396462	−	0.918051i	$-0.370238\pi$
$971$	24.7082	0.792924	0.396462	+	0.918051i	$0.370238\pi$
$972$	0	0
$973$	−17.5623	−0.563022
$974$	10.2705	0.329088
$975$	0	0
$976$	−10.6869	−0.342080
$977$	−33.0000	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$977$	−33.0000	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$978$	0	0
$979$	0	0
$980$	−0.618034	−0.0197424
$981$	0	0
$982$	−2.29180	−0.0731342
$983$	29.9443	0.955074	0.477537	−	0.878612i	$-0.341530\pi$
$983$	29.9443	0.955074	0.477537	+	0.878612i	$0.341530\pi$
$984$	0	0
$985$	3.52786	0.112407
$986$	−17.5623	−0.559298
$987$	0	0
$988$	0	0
$989$	−3.23607	−0.102901
$990$	0	0
$991$	33.6312	1.06833	0.534165	−	0.845380i	$-0.320626\pi$
$991$	33.6312	1.06833	0.534165	+	0.845380i	$0.320626\pi$
$992$	−57.5066	−1.82584
$993$	0	0
$994$	4.61803	0.146475
$995$	−2.88854	−0.0915730
$996$	0	0
$997$	1.41641	0.0448581	0.0224290	−	0.999748i	$-0.492860\pi$
$997$	1.41641	0.0448581	0.0224290	+	0.999748i	$0.492860\pi$
$998$	−3.09017	−0.0978176
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ba.1.2		2
3.2	odd	2		2541.2.a.bb.1.1		2
11.5	even	5		693.2.m.b.190.1		4
11.9	even	5		693.2.m.b.631.1		4
11.10	odd	2		7623.2.a.bp.1.1		2
33.5	odd	10		231.2.j.d.190.1	yes	4
33.20	odd	10		231.2.j.d.169.1	✓	4
33.32	even	2		2541.2.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.d.169.1	✓	4	33.20	odd	10
231.2.j.d.190.1	yes	4	33.5	odd	10
693.2.m.b.190.1		4	11.5	even	5
693.2.m.b.631.1		4	11.9	even	5
2541.2.a.s.1.2		2	33.32	even	2
2541.2.a.bb.1.1		2	3.2	odd	2
7623.2.a.ba.1.2		2	1.1	even	1	trivial
7623.2.a.bp.1.1		2	11.10	odd	2

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

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7623.2.a.ba.1.2

Level	$7623$
Weight	$2$
Character	7623.1
Self dual	yes
Analytic conductor	$60.870$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$