LMFDB - Embedded newform 8649.2.a.bj.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8649 = 3^{2} \cdot 31^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.1697203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 3x^{7} - 5x^{6} + 12x^{5} + 9x^{4} - 12x^{3} - 5x^{2} + 3x + 1$ x^8 - 3x^7 - 5x^6 + 12x^5 + 9x^4 - 12x^3 - 5x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 93)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.49568$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49568 q^{2} +4.22843 q^{4} +2.30796 q^{5} -2.75147 q^{7} +5.56145 q^{8} +5.75994 q^{10} +2.94472 q^{11} +5.94342 q^{13} -6.86680 q^{14} +5.42276 q^{16} +7.07424 q^{17} +1.97555 q^{19} +9.75905 q^{20} +7.34909 q^{22} -0.381727 q^{23} +0.326688 q^{25} +14.8329 q^{26} -11.6344 q^{28} -4.79100 q^{29} +2.41057 q^{32} +17.6550 q^{34} -6.35030 q^{35} -5.32646 q^{37} +4.93034 q^{38} +12.8356 q^{40} -9.41463 q^{41} +4.56431 q^{43} +12.4515 q^{44} -0.952670 q^{46} +3.63425 q^{47} +0.570605 q^{49} +0.815309 q^{50} +25.1313 q^{52} +1.69500 q^{53} +6.79631 q^{55} -15.3022 q^{56} -11.9568 q^{58} -9.36388 q^{59} -0.922702 q^{61} -4.82949 q^{64} +13.7172 q^{65} +13.8887 q^{67} +29.9129 q^{68} -15.8483 q^{70} +0.982560 q^{71} +6.76740 q^{73} -13.2932 q^{74} +8.35347 q^{76} -8.10232 q^{77} +2.86350 q^{79} +12.5155 q^{80} -23.4959 q^{82} -7.39597 q^{83} +16.3271 q^{85} +11.3911 q^{86} +16.3769 q^{88} -2.13672 q^{89} -16.3532 q^{91} -1.61411 q^{92} +9.06992 q^{94} +4.55949 q^{95} -18.6324 q^{97} +1.42405 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 5 q^{2} + 5 q^{4} + 6 q^{5} - 6 q^{7} + q^{10} + 12 q^{13} + 3 q^{14} + 3 q^{16} - 2 q^{17} - 8 q^{19} + 5 q^{20} + 4 q^{22} - 4 q^{23} - 12 q^{25} + 18 q^{26} - 15 q^{28} + 6 q^{29} + 17 q^{34}+ \cdots - 32 q^{98}+O(q^{100})$ 8 * q + 5 * q^2 + 5 * q^4 + 6 * q^5 - 6 * q^7 + q^10 + 12 * q^13 + 3 * q^14 + 3 * q^16 - 2 * q^17 - 8 * q^19 + 5 * q^20 + 4 * q^22 - 4 * q^23 - 12 * q^25 + 18 * q^26 - 15 * q^28 + 6 * q^29 + 17 * q^34 - 9 * q^35 + 8 * q^37 + 7 * q^38 + 13 * q^40 + 20 * q^41 + 14 * q^43 - 25 * q^44 + 9 * q^46 - 6 * q^47 - 20 * q^49 - 6 * q^50 + 30 * q^52 + 30 * q^53 + 19 * q^55 - 6 * q^56 + 19 * q^58 + 20 * q^59 + 13 * q^61 - 26 * q^64 + 27 * q^65 + 32 * q^67 + 15 * q^68 - 6 * q^70 + 17 * q^71 - 12 * q^73 + 10 * q^74 - 20 * q^76 - 18 * q^77 + 30 * q^79 + 21 * q^80 - 23 * q^82 - 12 * q^83 + 28 * q^85 + 46 * q^86 + 22 * q^88 - 6 * q^89 - 15 * q^91 + 70 * q^92 + 44 * q^94 + 36 * q^95 - 39 * q^97 - 32 * 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

)

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

$2$	2.49568	1.76471	0.882357	−	0.470581i	$-0.155956\pi$
$2$	2.49568	1.76471	0.882357	+	0.470581i	$0.155956\pi$
$3$	0	0
$3$	0	0
$4$	4.22843	2.11421
$5$	2.30796	1.03215	0.516076	−	0.856543i	$-0.327392\pi$
$5$	2.30796	1.03215	0.516076	+	0.856543i	$0.327392\pi$
$6$	0	0
$7$	−2.75147	−1.03996	−0.519980	−	0.854179i	$-0.674060\pi$
$7$	−2.75147	−1.03996	−0.519980	+	0.854179i	$0.674060\pi$
$8$	5.56145	1.96627
$9$	0	0
$10$	5.75994	1.82145
$11$	2.94472	0.887867	0.443934	−	0.896060i	$-0.353583\pi$
$11$	2.94472	0.887867	0.443934	+	0.896060i	$0.353583\pi$
$12$	0	0
$13$	5.94342	1.64841	0.824204	−	0.566293i	$-0.191623\pi$
$13$	5.94342	1.64841	0.824204	+	0.566293i	$0.191623\pi$
$14$	−6.86680	−1.83523
$15$	0	0
$16$	5.42276	1.35569
$17$	7.07424	1.71575	0.857877	−	0.513855i	$-0.171783\pi$
$17$	7.07424	1.71575	0.857877	+	0.513855i	$0.171783\pi$
$18$	0	0
$19$	1.97555	0.453222	0.226611	−	0.973985i	$-0.427235\pi$
$19$	1.97555	0.453222	0.226611	+	0.973985i	$0.427235\pi$
$20$	9.75905	2.18219
$21$	0	0
$22$	7.34909	1.56683
$23$	−0.381727	−0.0795957	−0.0397978	−	0.999208i	$-0.512671\pi$
$23$	−0.381727	−0.0795957	−0.0397978	+	0.999208i	$0.512671\pi$
$24$	0	0
$25$	0.326688	0.0653376
$26$	14.8329	2.90897
$27$	0	0
$28$	−11.6344	−2.19870
$29$	−4.79100	−0.889667	−0.444834	−	0.895613i	$-0.646737\pi$
$29$	−4.79100	−0.889667	−0.444834	+	0.895613i	$0.646737\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	2.41057	0.426133
$33$	0	0
$34$	17.6550	3.02781
$35$	−6.35030	−1.07340
$36$	0	0
$37$	−5.32646	−0.875665	−0.437832	−	0.899057i	$-0.644254\pi$
$37$	−5.32646	−0.875665	−0.437832	+	0.899057i	$0.644254\pi$
$38$	4.93034	0.799807
$39$	0	0
$40$	12.8356	2.02949
$41$	−9.41463	−1.47032	−0.735159	−	0.677895i	$-0.762892\pi$
$41$	−9.41463	−1.47032	−0.735159	+	0.677895i	$0.762892\pi$
$42$	0	0
$43$	4.56431	0.696050	0.348025	−	0.937485i	$-0.386852\pi$
$43$	4.56431	0.696050	0.348025	+	0.937485i	$0.386852\pi$
$44$	12.4515	1.87714
$45$	0	0
$46$	−0.952670	−0.140464
$47$	3.63425	0.530109	0.265055	−	0.964233i	$-0.414610\pi$
$47$	3.63425	0.530109	0.265055	+	0.964233i	$0.414610\pi$
$48$	0	0
$49$	0.570605	0.0815151
$50$	0.815309	0.115302
$51$	0	0
$52$	25.1313	3.48509
$53$	1.69500	0.232827	0.116413	−	0.993201i	$-0.462860\pi$
$53$	1.69500	0.232827	0.116413	+	0.993201i	$0.462860\pi$
$54$	0	0
$55$	6.79631	0.916414
$56$	−15.3022	−2.04484
$57$	0	0
$58$	−11.9568	−1.57001
$59$	−9.36388	−1.21907	−0.609537	−	0.792758i	$-0.708645\pi$
$59$	−9.36388	−1.21907	−0.609537	+	0.792758i	$0.708645\pi$
$60$	0	0
$61$	−0.922702	−0.118140	−0.0590699	−	0.998254i	$-0.518813\pi$
$61$	−0.922702	−0.118140	−0.0590699	+	0.998254i	$0.518813\pi$
$62$	0	0
$63$	0	0
$64$	−4.82949	−0.603686
$65$	13.7172	1.70141
$66$	0	0
$67$	13.8887	1.69677	0.848385	−	0.529379i	$-0.177575\pi$
$67$	13.8887	1.69677	0.848385	+	0.529379i	$0.177575\pi$
$68$	29.9129	3.62747
$69$	0	0
$70$	−15.8483	−1.89424
$71$	0.982560	0.116608	0.0583042	−	0.998299i	$-0.481431\pi$
$71$	0.982560	0.116608	0.0583042	+	0.998299i	$0.481431\pi$
$72$	0	0
$73$	6.76740	0.792065	0.396032	−	0.918237i	$-0.370387\pi$
$73$	6.76740	0.792065	0.396032	+	0.918237i	$0.370387\pi$
$74$	−13.2932	−1.54530
$75$	0	0
$76$	8.35347	0.958209
$77$	−8.10232	−0.923345
$78$	0	0
$79$	2.86350	0.322169	0.161085	−	0.986941i	$-0.448501\pi$
$79$	2.86350	0.322169	0.161085	+	0.986941i	$0.448501\pi$
$80$	12.5155	1.39928
$81$	0	0
$82$	−23.4959	−2.59469
$83$	−7.39597	−0.811813	−0.405906	−	0.913915i	$-0.633044\pi$
$83$	−7.39597	−0.811813	−0.405906	+	0.913915i	$0.633044\pi$
$84$	0	0
$85$	16.3271	1.77092
$86$	11.3911	1.22833
$87$	0	0
$88$	16.3769	1.74579
$89$	−2.13672	−0.226491	−0.113246	−	0.993567i	$-0.536125\pi$
$89$	−2.13672	−0.226491	−0.113246	+	0.993567i	$0.536125\pi$
$90$	0	0
$91$	−16.3532	−1.71428
$92$	−1.61411	−0.168282
$93$	0	0
$94$	9.06992	0.935491
$95$	4.55949	0.467794
$96$	0	0
$97$	−18.6324	−1.89184	−0.945918	−	0.324406i	$-0.894836\pi$
$97$	−18.6324	−1.89184	−0.945918	+	0.324406i	$0.894836\pi$
$98$	1.42405	0.143851
$99$	0	0
$100$	1.38138	0.138138
$101$	5.27117	0.524501	0.262250	−	0.965000i	$-0.415535\pi$
$101$	5.27117	0.524501	0.262250	+	0.965000i	$0.415535\pi$
$102$	0	0
$103$	−1.31278	−0.129352	−0.0646761	−	0.997906i	$-0.520601\pi$
$103$	−1.31278	−0.129352	−0.0646761	+	0.997906i	$0.520601\pi$
$104$	33.0540	3.24121
$105$	0	0
$106$	4.23019	0.410872
$107$	11.1946	1.08222	0.541111	−	0.840951i	$-0.318004\pi$
$107$	11.1946	1.08222	0.541111	+	0.840951i	$0.318004\pi$
$108$	0	0
$109$	3.41557	0.327152	0.163576	−	0.986531i	$-0.447697\pi$
$109$	3.41557	0.327152	0.163576	+	0.986531i	$0.447697\pi$
$110$	16.9614	1.61721
$111$	0	0
$112$	−14.9206	−1.40986
$113$	6.33897	0.596320	0.298160	−	0.954516i	$-0.403627\pi$
$113$	6.33897	0.596320	0.298160	+	0.954516i	$0.403627\pi$
$114$	0	0
$115$	−0.881012	−0.0821548
$116$	−20.2584	−1.88095
$117$	0	0
$118$	−23.3693	−2.15132
$119$	−19.4646	−1.78431
$120$	0	0
$121$	−2.32861	−0.211692
$122$	−2.30277	−0.208483
$123$	0	0
$124$	0	0
$125$	−10.7858	−0.964714
$126$	0	0
$127$	5.92881	0.526097	0.263049	−	0.964783i	$-0.415272\pi$
$127$	5.92881	0.526097	0.263049	+	0.964783i	$0.415272\pi$
$128$	−16.8740	−1.49147
$129$	0	0
$130$	34.2337	3.00250
$131$	−7.78468	−0.680151	−0.340075	−	0.940398i	$-0.610453\pi$
$131$	−7.78468	−0.680151	−0.340075	+	0.940398i	$0.610453\pi$
$132$	0	0
$133$	−5.43567	−0.471332
$134$	34.6617	2.99431
$135$	0	0
$136$	39.3430	3.37364
$137$	2.39865	0.204930	0.102465	−	0.994737i	$-0.467327\pi$
$137$	2.39865	0.204930	0.102465	+	0.994737i	$0.467327\pi$
$138$	0	0
$139$	21.8103	1.84993	0.924963	−	0.380057i	$-0.124096\pi$
$139$	21.8103	1.84993	0.924963	+	0.380057i	$0.124096\pi$
$140$	−26.8518	−2.26939
$141$	0	0
$142$	2.45216	0.205781
$143$	17.5017	1.46357
$144$	0	0
$145$	−11.0575	−0.918272
$146$	16.8893	1.39777
$147$	0	0
$148$	−22.5226	−1.85134
$149$	−12.1446	−0.994921	−0.497461	−	0.867487i	$-0.665734\pi$
$149$	−12.1446	−0.994921	−0.497461	+	0.867487i	$0.665734\pi$
$150$	0	0
$151$	3.47475	0.282771	0.141386	−	0.989955i	$-0.454844\pi$
$151$	3.47475	0.282771	0.141386	+	0.989955i	$0.454844\pi$
$152$	10.9869	0.891157
$153$	0	0
$154$	−20.2208	−1.62944
$155$	0	0
$156$	0	0
$157$	−3.57916	−0.285648	−0.142824	−	0.989748i	$-0.545618\pi$
$157$	−3.57916	−0.285648	−0.142824	+	0.989748i	$0.545618\pi$
$158$	7.14640	0.568537
$159$	0	0
$160$	5.56351	0.439834
$161$	1.05031	0.0827762
$162$	0	0
$163$	3.05486	0.239275	0.119637	−	0.992818i	$-0.461827\pi$
$163$	3.05486	0.239275	0.119637	+	0.992818i	$0.461827\pi$
$164$	−39.8091	−3.10857
$165$	0	0
$166$	−18.4580	−1.43262
$167$	2.66114	0.205926	0.102963	−	0.994685i	$-0.467168\pi$
$167$	2.66114	0.205926	0.102963	+	0.994685i	$0.467168\pi$
$168$	0	0
$169$	22.3242	1.71725
$170$	40.7472	3.12516
$171$	0	0
$172$	19.2998	1.47160
$173$	0.677941	0.0515429	0.0257714	−	0.999668i	$-0.491796\pi$
$173$	0.677941	0.0515429	0.0257714	+	0.999668i	$0.491796\pi$
$174$	0	0
$175$	−0.898873	−0.0679484
$176$	15.9685	1.20367
$177$	0	0
$178$	−5.33256	−0.399693
$179$	−7.29501	−0.545254	−0.272627	−	0.962120i	$-0.587893\pi$
$179$	−7.29501	−0.545254	−0.272627	+	0.962120i	$0.587893\pi$
$180$	0	0
$181$	6.21931	0.462278	0.231139	−	0.972921i	$-0.425755\pi$
$181$	6.21931	0.462278	0.231139	+	0.972921i	$0.425755\pi$
$182$	−40.8123	−3.02521
$183$	0	0
$184$	−2.12296	−0.156507
$185$	−12.2933	−0.903819
$186$	0	0
$187$	20.8317	1.52336
$188$	15.3671	1.12076
$189$	0	0
$190$	11.3790	0.825523
$191$	3.48813	0.252392	0.126196	−	0.992005i	$-0.459723\pi$
$191$	3.48813	0.252392	0.126196	+	0.992005i	$0.459723\pi$
$192$	0	0
$193$	−5.40685	−0.389193	−0.194597	−	0.980883i	$-0.562340\pi$
$193$	−5.40685	−0.389193	−0.194597	+	0.980883i	$0.562340\pi$
$194$	−46.5006	−3.33855
$195$	0	0
$196$	2.41276	0.172340
$197$	−5.08826	−0.362523	−0.181262	−	0.983435i	$-0.558018\pi$
$197$	−5.08826	−0.362523	−0.181262	+	0.983435i	$0.558018\pi$
$198$	0	0
$199$	7.25137	0.514036	0.257018	−	0.966407i	$-0.417260\pi$
$199$	7.25137	0.514036	0.257018	+	0.966407i	$0.417260\pi$
$200$	1.81686	0.128471
$201$	0	0
$202$	13.1552	0.925594
$203$	13.1823	0.925218
$204$	0	0
$205$	−21.7286	−1.51759
$206$	−3.27628	−0.228269
$207$	0	0
$208$	32.2297	2.23473
$209$	5.81744	0.402401
$210$	0	0
$211$	−8.79653	−0.605578	−0.302789	−	0.953058i	$-0.597918\pi$
$211$	−8.79653	−0.605578	−0.302789	+	0.953058i	$0.597918\pi$
$212$	7.16720	0.492245
$213$	0	0
$214$	27.9382	1.90981
$215$	10.5342	0.718429
$216$	0	0
$217$	0	0
$218$	8.52418	0.577330
$219$	0	0
$220$	28.7377	1.93750
$221$	42.0451	2.82826
$222$	0	0
$223$	6.40331	0.428797	0.214399	−	0.976746i	$-0.431221\pi$
$223$	6.40331	0.428797	0.214399	+	0.976746i	$0.431221\pi$
$224$	−6.63263	−0.443161
$225$	0	0
$226$	15.8201	1.05233
$227$	13.2247	0.877755	0.438878	−	0.898547i	$-0.355376\pi$
$227$	13.2247	0.877755	0.438878	+	0.898547i	$0.355376\pi$
$228$	0	0
$229$	−26.6674	−1.76223	−0.881117	−	0.472899i	$-0.843207\pi$
$229$	−26.6674	−1.76223	−0.881117	+	0.472899i	$0.843207\pi$
$230$	−2.19873	−0.144980
$231$	0	0
$232$	−26.6449	−1.74933
$233$	25.3765	1.66247	0.831235	−	0.555922i	$-0.187635\pi$
$233$	25.3765	1.66247	0.831235	+	0.555922i	$0.187635\pi$
$234$	0	0
$235$	8.38770	0.547153
$236$	−39.5945	−2.57738
$237$	0	0
$238$	−48.5774	−3.14880
$239$	9.74780	0.630533	0.315266	−	0.949003i	$-0.397906\pi$
$239$	9.74780	0.630533	0.315266	+	0.949003i	$0.397906\pi$
$240$	0	0
$241$	−28.0310	−1.80564	−0.902818	−	0.430024i	$-0.858505\pi$
$241$	−28.0310	−1.80564	−0.902818	+	0.430024i	$0.858505\pi$
$242$	−5.81148	−0.373576
$243$	0	0
$244$	−3.90158	−0.249773
$245$	1.31694	0.0841359
$246$	0	0
$247$	11.7415	0.747095
$248$	0	0
$249$	0	0
$250$	−26.9180	−1.70244
$251$	10.8984	0.687901	0.343950	−	0.938988i	$-0.388235\pi$
$251$	10.8984	0.687901	0.343950	+	0.938988i	$0.388235\pi$
$252$	0	0
$253$	−1.12408	−0.0706704
$254$	14.7964	0.928411
$255$	0	0
$256$	−32.4532	−2.02832
$257$	14.2707	0.890181	0.445090	−	0.895486i	$-0.353172\pi$
$257$	14.2707	0.890181	0.445090	+	0.895486i	$0.353172\pi$
$258$	0	0
$259$	14.6556	0.910655
$260$	58.0021	3.59714
$261$	0	0
$262$	−19.4281	−1.20027
$263$	−25.0617	−1.54537	−0.772686	−	0.634788i	$-0.781087\pi$
$263$	−25.0617	−1.54537	−0.772686	+	0.634788i	$0.781087\pi$
$264$	0	0
$265$	3.91200	0.240312
$266$	−13.5657	−0.831767
$267$	0	0
$268$	58.7272	3.58734
$269$	3.37486	0.205769	0.102884	−	0.994693i	$-0.467193\pi$
$269$	3.37486	0.205769	0.102884	+	0.994693i	$0.467193\pi$
$270$	0	0
$271$	−0.106962	−0.00649746	−0.00324873	−	0.999995i	$-0.501034\pi$
$271$	−0.106962	−0.00649746	−0.00324873	+	0.999995i	$0.501034\pi$
$272$	38.3619	2.32603
$273$	0	0
$274$	5.98626	0.361643
$275$	0.962005	0.0580111
$276$	0	0
$277$	3.27048	0.196504	0.0982521	−	0.995162i	$-0.468675\pi$
$277$	3.27048	0.196504	0.0982521	+	0.995162i	$0.468675\pi$
$278$	54.4316	3.26459
$279$	0	0
$280$	−35.3169	−2.11059
$281$	1.11598	0.0665737	0.0332868	−	0.999446i	$-0.489403\pi$
$281$	1.11598	0.0665737	0.0332868	+	0.999446i	$0.489403\pi$
$282$	0	0
$283$	−18.1331	−1.07790	−0.538951	−	0.842337i	$-0.681179\pi$
$283$	−18.1331	−1.07790	−0.538951	+	0.842337i	$0.681179\pi$
$284$	4.15469	0.246535
$285$	0	0
$286$	43.6787	2.58278
$287$	25.9041	1.52907
$288$	0	0
$289$	33.0448	1.94381
$290$	−27.5959	−1.62049
$291$	0	0
$292$	28.6155	1.67459
$293$	31.0121	1.81175	0.905874	−	0.423548i	$-0.139215\pi$
$293$	31.0121	1.81175	0.905874	+	0.423548i	$0.139215\pi$
$294$	0	0
$295$	−21.6115	−1.25827
$296$	−29.6228	−1.72179
$297$	0	0
$298$	−30.3090	−1.75575
$299$	−2.26876	−0.131206
$300$	0	0
$301$	−12.5586	−0.723864
$302$	8.67188	0.499011
$303$	0	0
$304$	10.7129	0.614428
$305$	−2.12956	−0.121938
$306$	0	0
$307$	−29.2654	−1.67026	−0.835131	−	0.550051i	$-0.814608\pi$
$307$	−29.2654	−1.67026	−0.835131	+	0.550051i	$0.814608\pi$
$308$	−34.2601	−1.95215
$309$	0	0
$310$	0	0
$311$	31.6759	1.79618	0.898088	−	0.439815i	$-0.144956\pi$
$311$	31.6759	1.79618	0.898088	+	0.439815i	$0.144956\pi$
$312$	0	0
$313$	−10.2334	−0.578427	−0.289214	−	0.957265i	$-0.593394\pi$
$313$	−10.2334	−0.578427	−0.289214	+	0.957265i	$0.593394\pi$
$314$	−8.93246	−0.504088
$315$	0	0
$316$	12.1081	0.681135
$317$	19.2692	1.08226	0.541132	−	0.840938i	$-0.317996\pi$
$317$	19.2692	1.08226	0.541132	+	0.840938i	$0.317996\pi$
$318$	0	0
$319$	−14.1082	−0.789906
$320$	−11.1463	−0.623096
$321$	0	0
$322$	2.62125	0.146076
$323$	13.9755	0.777618
$324$	0	0
$325$	1.94164	0.107703
$326$	7.62395	0.422251
$327$	0	0
$328$	−52.3590	−2.89104
$329$	−9.99953	−0.551292
$330$	0	0
$331$	14.8307	0.815171	0.407586	−	0.913167i	$-0.366371\pi$
$331$	14.8307	0.815171	0.407586	+	0.913167i	$0.366371\pi$
$332$	−31.2733	−1.71635
$333$	0	0
$334$	6.64137	0.363400
$335$	32.0545	1.75132
$336$	0	0
$337$	−10.6059	−0.577743	−0.288871	−	0.957368i	$-0.593280\pi$
$337$	−10.6059	−0.577743	−0.288871	+	0.957368i	$0.593280\pi$
$338$	55.7141	3.03045
$339$	0	0
$340$	69.0378	3.74410
$341$	0	0
$342$	0	0
$343$	17.6903	0.955187
$344$	25.3842	1.36862
$345$	0	0
$346$	1.69192	0.0909584
$347$	−10.1749	−0.546215	−0.273107	−	0.961984i	$-0.588051\pi$
$347$	−10.1749	−0.546215	−0.273107	+	0.961984i	$0.588051\pi$
$348$	0	0
$349$	−19.1169	−1.02330	−0.511652	−	0.859193i	$-0.670966\pi$
$349$	−19.1169	−1.02330	−0.511652	+	0.859193i	$0.670966\pi$
$350$	−2.24330	−0.119909
$351$	0	0
$352$	7.09847	0.378350
$353$	−5.88028	−0.312976	−0.156488	−	0.987680i	$-0.550017\pi$
$353$	−5.88028	−0.312976	−0.156488	+	0.987680i	$0.550017\pi$
$354$	0	0
$355$	2.26771	0.120358
$356$	−9.03495	−0.478852
$357$	0	0
$358$	−18.2060	−0.962218
$359$	−16.1855	−0.854237	−0.427118	−	0.904196i	$-0.640471\pi$
$359$	−16.1855	−0.854237	−0.427118	+	0.904196i	$0.640471\pi$
$360$	0	0
$361$	−15.0972	−0.794590
$362$	15.5214	0.815788
$363$	0	0
$364$	−69.1482	−3.62435
$365$	15.6189	0.817531
$366$	0	0
$367$	−25.2403	−1.31753	−0.658766	−	0.752348i	$-0.728921\pi$
$367$	−25.2403	−1.31753	−0.658766	+	0.752348i	$0.728921\pi$
$368$	−2.07001	−0.107907
$369$	0	0
$370$	−30.6801	−1.59498
$371$	−4.66375	−0.242130
$372$	0	0
$373$	−21.2632	−1.10096	−0.550482	−	0.834847i	$-0.685556\pi$
$373$	−21.2632	−1.10096	−0.550482	+	0.834847i	$0.685556\pi$
$374$	51.9892	2.68830
$375$	0	0
$376$	20.2117	1.04234
$377$	−28.4749	−1.46653
$378$	0	0
$379$	−26.4025	−1.35621	−0.678103	−	0.734967i	$-0.737198\pi$
$379$	−26.4025	−1.35621	−0.678103	+	0.734967i	$0.737198\pi$
$380$	19.2795	0.989017
$381$	0	0
$382$	8.70526	0.445400
$383$	1.96541	0.100428	0.0502139	−	0.998738i	$-0.484010\pi$
$383$	1.96541	0.100428	0.0502139	+	0.998738i	$0.484010\pi$
$384$	0	0
$385$	−18.6999	−0.953033
$386$	−13.4938	−0.686815
$387$	0	0
$388$	−78.7859	−3.99975
$389$	−27.7973	−1.40938	−0.704690	−	0.709515i	$-0.748914\pi$
$389$	−27.7973	−1.40938	−0.704690	+	0.709515i	$0.748914\pi$
$390$	0	0
$391$	−2.70043	−0.136567
$392$	3.17339	0.160281
$393$	0	0
$394$	−12.6987	−0.639750
$395$	6.60886	0.332528
$396$	0	0
$397$	−4.00908	−0.201210	−0.100605	−	0.994926i	$-0.532078\pi$
$397$	−4.00908	−0.201210	−0.100605	+	0.994926i	$0.532078\pi$
$398$	18.0971	0.907127
$399$	0	0
$400$	1.77155	0.0885774
$401$	−18.2338	−0.910551	−0.455275	−	0.890351i	$-0.650459\pi$
$401$	−18.2338	−0.910551	−0.455275	+	0.890351i	$0.650459\pi$
$402$	0	0
$403$	0	0
$404$	22.2888	1.10891
$405$	0	0
$406$	32.8989	1.63274
$407$	−15.6849	−0.777474
$408$	0	0
$409$	−28.4102	−1.40479	−0.702397	−	0.711785i	$-0.747887\pi$
$409$	−28.4102	−1.40479	−0.702397	+	0.711785i	$0.747887\pi$
$410$	−54.2277	−2.67811
$411$	0	0
$412$	−5.55100	−0.273478
$413$	25.7645	1.26779
$414$	0	0
$415$	−17.0696	−0.837914
$416$	14.3270	0.702441
$417$	0	0
$418$	14.5185	0.710122
$419$	−28.6598	−1.40012	−0.700060	−	0.714084i	$-0.746844\pi$
$419$	−28.6598	−1.40012	−0.700060	+	0.714084i	$0.746844\pi$
$420$	0	0
$421$	25.1273	1.22463	0.612315	−	0.790614i	$-0.290238\pi$
$421$	25.1273	1.22463	0.612315	+	0.790614i	$0.290238\pi$
$422$	−21.9534	−1.06867
$423$	0	0
$424$	9.42667	0.457800
$425$	2.31107	0.112103
$426$	0	0
$427$	2.53879	0.122861
$428$	47.3356	2.28805
$429$	0	0
$430$	26.2901	1.26782
$431$	0.373748	0.0180028	0.00900140	−	0.999959i	$-0.497135\pi$
$431$	0.373748	0.0180028	0.00900140	+	0.999959i	$0.497135\pi$
$432$	0	0
$433$	1.00490	0.0482926	0.0241463	−	0.999708i	$-0.492313\pi$
$433$	1.00490	0.0482926	0.0241463	+	0.999708i	$0.492313\pi$
$434$	0	0
$435$	0	0
$436$	14.4425	0.691671
$437$	−0.754121	−0.0360745
$438$	0	0
$439$	−5.41147	−0.258275	−0.129138	−	0.991627i	$-0.541221\pi$
$439$	−5.41147	−0.258275	−0.129138	+	0.991627i	$0.541221\pi$
$440$	37.7973	1.80192
$441$	0	0
$442$	104.931	4.99107
$443$	15.1374	0.719201	0.359600	−	0.933106i	$-0.382913\pi$
$443$	15.1374	0.719201	0.359600	+	0.933106i	$0.382913\pi$
$444$	0	0
$445$	−4.93146	−0.233774
$446$	15.9806	0.756704
$447$	0	0
$448$	13.2882	0.627809
$449$	−10.3025	−0.486206	−0.243103	−	0.970001i	$-0.578165\pi$
$449$	−10.3025	−0.486206	−0.243103	+	0.970001i	$0.578165\pi$
$450$	0	0
$451$	−27.7235	−1.30545
$452$	26.8039	1.26075
$453$	0	0
$454$	33.0047	1.54899
$455$	−37.7425	−1.76939
$456$	0	0
$457$	22.9697	1.07448	0.537238	−	0.843431i	$-0.319468\pi$
$457$	22.9697	1.07448	0.537238	+	0.843431i	$0.319468\pi$
$458$	−66.5534	−3.10984
$459$	0	0
$460$	−3.72530	−0.173693
$461$	−18.6097	−0.866740	−0.433370	−	0.901216i	$-0.642676\pi$
$461$	−18.6097	−0.866740	−0.433370	+	0.901216i	$0.642676\pi$
$462$	0	0
$463$	−11.5651	−0.537478	−0.268739	−	0.963213i	$-0.586607\pi$
$463$	−11.5651	−0.537478	−0.268739	+	0.963213i	$0.586607\pi$
$464$	−25.9805	−1.20611
$465$	0	0
$466$	63.3317	2.93378
$467$	8.74826	0.404821	0.202411	−	0.979301i	$-0.435122\pi$
$467$	8.74826	0.404821	0.202411	+	0.979301i	$0.435122\pi$
$468$	0	0
$469$	−38.2143	−1.76457
$470$	20.9330	0.965569
$471$	0	0
$472$	−52.0768	−2.39703
$473$	13.4406	0.618000
$474$	0	0
$475$	0.645388	0.0296124
$476$	−82.3046	−3.77242
$477$	0	0
$478$	24.3274	1.11271
$479$	−10.4647	−0.478145	−0.239072	−	0.971002i	$-0.576843\pi$
$479$	−10.4647	−0.478145	−0.239072	+	0.971002i	$0.576843\pi$
$480$	0	0
$481$	−31.6574	−1.44345
$482$	−69.9565	−3.18643
$483$	0	0
$484$	−9.84638	−0.447563
$485$	−43.0029	−1.95266
$486$	0	0
$487$	37.3362	1.69186	0.845932	−	0.533291i	$-0.179045\pi$
$487$	37.3362	1.69186	0.845932	+	0.533291i	$0.179045\pi$
$488$	−5.13156	−0.232295
$489$	0	0
$490$	3.28665	0.148476
$491$	26.7746	1.20832	0.604160	−	0.796863i	$-0.293509\pi$
$491$	26.7746	1.20832	0.604160	+	0.796863i	$0.293509\pi$
$492$	0	0
$493$	−33.8927	−1.52645
$494$	29.3031	1.31841
$495$	0	0
$496$	0	0
$497$	−2.70349	−0.121268
$498$	0	0
$499$	−27.9698	−1.25210	−0.626049	−	0.779783i	$-0.715329\pi$
$499$	−27.9698	−1.25210	−0.626049	+	0.779783i	$0.715329\pi$
$500$	−45.6071	−2.03961
$501$	0	0
$502$	27.1989	1.21395
$503$	−9.76347	−0.435332	−0.217666	−	0.976023i	$-0.569844\pi$
$503$	−9.76347	−0.435332	−0.217666	+	0.976023i	$0.569844\pi$
$504$	0	0
$505$	12.1657	0.541365
$506$	−2.80535	−0.124713
$507$	0	0
$508$	25.0696	1.11228
$509$	−35.3335	−1.56613	−0.783064	−	0.621941i	$-0.786344\pi$
$509$	−35.3335	−1.56613	−0.783064	+	0.621941i	$0.786344\pi$
$510$	0	0
$511$	−18.6203	−0.823715
$512$	−47.2448	−2.08795
$513$	0	0
$514$	35.6151	1.57091
$515$	−3.02985	−0.133511
$516$	0	0
$517$	10.7018	0.470666
$518$	36.5758	1.60705
$519$	0	0
$520$	76.2874	3.34543
$521$	2.53654	0.111128	0.0555638	−	0.998455i	$-0.482304\pi$
$521$	2.53654	0.111128	0.0555638	+	0.998455i	$0.482304\pi$
$522$	0	0
$523$	16.3163	0.713463	0.356732	−	0.934207i	$-0.383891\pi$
$523$	16.3163	0.713463	0.356732	+	0.934207i	$0.383891\pi$
$524$	−32.9170	−1.43798
$525$	0	0
$526$	−62.5461	−2.72714
$527$	0	0
$528$	0	0
$529$	−22.8543	−0.993665
$530$	9.76311	0.424082
$531$	0	0
$532$	−22.9843	−0.996498
$533$	−55.9551	−2.42368
$534$	0	0
$535$	25.8367	1.11702
$536$	77.2411	3.33631
$537$	0	0
$538$	8.42258	0.363123
$539$	1.68027	0.0723745
$540$	0	0
$541$	−31.2262	−1.34252	−0.671260	−	0.741222i	$-0.734247\pi$
$541$	−31.2262	−1.34252	−0.671260	+	0.741222i	$0.734247\pi$
$542$	−0.266942	−0.0114662
$543$	0	0
$544$	17.0530	0.731140
$545$	7.88301	0.337671
$546$	0	0
$547$	−26.7801	−1.14503	−0.572517	−	0.819893i	$-0.694033\pi$
$547$	−26.7801	−1.14503	−0.572517	+	0.819893i	$0.694033\pi$
$548$	10.1425	0.433266
$549$	0	0
$550$	2.40086	0.102373
$551$	−9.46486	−0.403217
$552$	0	0
$553$	−7.87886	−0.335043
$554$	8.16208	0.346774
$555$	0	0
$556$	92.2234	3.91114
$557$	43.0564	1.82436	0.912178	−	0.409794i	$-0.134399\pi$
$557$	43.0564	1.82436	0.912178	+	0.409794i	$0.134399\pi$
$558$	0	0
$559$	27.1276	1.14737
$560$	−34.4361	−1.45519
$561$	0	0
$562$	2.78513	0.117483
$563$	−37.6900	−1.58844	−0.794222	−	0.607628i	$-0.792121\pi$
$563$	−37.6900	−1.58844	−0.794222	+	0.607628i	$0.792121\pi$
$564$	0	0
$565$	14.6301	0.615493
$566$	−45.2545	−1.90219
$567$	0	0
$568$	5.46446	0.229284
$569$	20.2946	0.850794	0.425397	−	0.905007i	$-0.360134\pi$
$569$	20.2946	0.850794	0.425397	+	0.905007i	$0.360134\pi$
$570$	0	0
$571$	−28.8149	−1.20586	−0.602932	−	0.797792i	$-0.706001\pi$
$571$	−28.8149	−1.20586	−0.602932	+	0.797792i	$0.706001\pi$
$572$	74.0048	3.09429
$573$	0	0
$574$	64.6484	2.69837
$575$	−0.124706	−0.00520059
$576$	0	0
$577$	31.3490	1.30507	0.652537	−	0.757757i	$-0.273705\pi$
$577$	31.3490	1.30507	0.652537	+	0.757757i	$0.273705\pi$
$578$	82.4693	3.43027
$579$	0	0
$580$	−46.7557	−1.94142
$581$	20.3498	0.844252
$582$	0	0
$583$	4.99131	0.206719
$584$	37.6366	1.55741
$585$	0	0
$586$	77.3964	3.19722
$587$	16.5500	0.683092	0.341546	−	0.939865i	$-0.389050\pi$
$587$	16.5500	0.683092	0.341546	+	0.939865i	$0.389050\pi$
$588$	0	0
$589$	0	0
$590$	−53.9354	−2.22048
$591$	0	0
$592$	−28.8841	−1.18713
$593$	−22.0312	−0.904712	−0.452356	−	0.891837i	$-0.649416\pi$
$593$	−22.0312	−0.904712	−0.452356	+	0.891837i	$0.649416\pi$
$594$	0	0
$595$	−44.9235	−1.84168
$596$	−51.3524	−2.10348
$597$	0	0
$598$	−5.66212	−0.231541
$599$	−7.11684	−0.290786	−0.145393	−	0.989374i	$-0.546445\pi$
$599$	−7.11684	−0.290786	−0.145393	+	0.989374i	$0.546445\pi$
$600$	0	0
$601$	29.1376	1.18855	0.594274	−	0.804262i	$-0.297439\pi$
$601$	29.1376	1.18855	0.594274	+	0.804262i	$0.297439\pi$
$602$	−31.3422	−1.27741
$603$	0	0
$604$	14.6927	0.597839
$605$	−5.37435	−0.218498
$606$	0	0
$607$	−26.9747	−1.09487	−0.547435	−	0.836848i	$-0.684396\pi$
$607$	−26.9747	−1.09487	−0.547435	+	0.836848i	$0.684396\pi$
$608$	4.76221	0.193133
$609$	0	0
$610$	−5.31471	−0.215186
$611$	21.5998	0.873836
$612$	0	0
$613$	8.62862	0.348507	0.174253	−	0.984701i	$-0.444249\pi$
$613$	8.62862	0.348507	0.174253	+	0.984701i	$0.444249\pi$
$614$	−73.0371	−2.94754
$615$	0	0
$616$	−45.0607	−1.81555
$617$	−42.8673	−1.72577	−0.862886	−	0.505399i	$-0.831345\pi$
$617$	−42.8673	−1.72577	−0.862886	+	0.505399i	$0.831345\pi$
$618$	0	0
$619$	38.0726	1.53027	0.765134	−	0.643871i	$-0.222673\pi$
$619$	38.0726	1.53027	0.765134	+	0.643871i	$0.222673\pi$
$620$	0	0
$621$	0	0
$622$	79.0530	3.16974
$623$	5.87912	0.235542
$624$	0	0
$625$	−26.5267	−1.06107
$626$	−25.5394	−1.02076
$627$	0	0
$628$	−15.1342	−0.603922
$629$	−37.6806	−1.50243
$630$	0	0
$631$	12.5530	0.499729	0.249864	−	0.968281i	$-0.419614\pi$
$631$	12.5530	0.499729	0.249864	+	0.968281i	$0.419614\pi$
$632$	15.9252	0.633472
$633$	0	0
$634$	48.0897	1.90989
$635$	13.6835	0.543012
$636$	0	0
$637$	3.39135	0.134370
$638$	−35.2095	−1.39396
$639$	0	0
$640$	−38.9446	−1.53942
$641$	41.5549	1.64132	0.820659	−	0.571418i	$-0.193606\pi$
$641$	41.5549	1.64132	0.820659	+	0.571418i	$0.193606\pi$
$642$	0	0
$643$	36.2977	1.43144	0.715721	−	0.698386i	$-0.246098\pi$
$643$	36.2977	1.43144	0.715721	+	0.698386i	$0.246098\pi$
$644$	4.44117	0.175007
$645$	0	0
$646$	34.8784	1.37227
$647$	50.0421	1.96736	0.983678	−	0.179936i	$-0.0575889\pi$
$647$	50.0421	1.96736	0.983678	+	0.179936i	$0.0575889\pi$
$648$	0	0
$649$	−27.5740	−1.08238
$650$	4.84572	0.190065
$651$	0	0
$652$	12.9172	0.505878
$653$	−29.1570	−1.14100	−0.570501	−	0.821297i	$-0.693251\pi$
$653$	−29.1570	−1.14100	−0.570501	+	0.821297i	$0.693251\pi$
$654$	0	0
$655$	−17.9667	−0.702019
$656$	−51.0532	−1.99329
$657$	0	0
$658$	−24.9556	−0.972872
$659$	16.3913	0.638515	0.319258	−	0.947668i	$-0.396566\pi$
$659$	16.3913	0.638515	0.319258	+	0.947668i	$0.396566\pi$
$660$	0	0
$661$	0.750582	0.0291943	0.0145971	−	0.999893i	$-0.495353\pi$
$661$	0.750582	0.0291943	0.0145971	+	0.999893i	$0.495353\pi$
$662$	37.0128	1.43854
$663$	0	0
$664$	−41.1323	−1.59624
$665$	−12.5453	−0.486487
$666$	0	0
$667$	1.82886	0.0708136
$668$	11.2525	0.435371
$669$	0	0
$670$	79.9979	3.09059
$671$	−2.71710	−0.104893
$672$	0	0
$673$	−16.4315	−0.633388	−0.316694	−	0.948528i	$-0.602573\pi$
$673$	−16.4315	−0.633388	−0.316694	+	0.948528i	$0.602573\pi$
$674$	−26.4691	−1.01955
$675$	0	0
$676$	94.3964	3.63063
$677$	3.47627	0.133604	0.0668020	−	0.997766i	$-0.478720\pi$
$677$	3.47627	0.133604	0.0668020	+	0.997766i	$0.478720\pi$
$678$	0	0
$679$	51.2666	1.96743
$680$	90.8022	3.48210
$681$	0	0
$682$	0	0
$683$	−2.33721	−0.0894308	−0.0447154	−	0.999000i	$-0.514238\pi$
$683$	−2.33721	−0.0894308	−0.0447154	+	0.999000i	$0.514238\pi$
$684$	0	0
$685$	5.53598	0.211519
$686$	44.1494	1.68563
$687$	0	0
$688$	24.7511	0.943628
$689$	10.0741	0.383793
$690$	0	0
$691$	−45.8502	−1.74422	−0.872111	−	0.489308i	$-0.837249\pi$
$691$	−45.8502	−1.74422	−0.872111	+	0.489308i	$0.837249\pi$
$692$	2.86662	0.108973
$693$	0	0
$694$	−25.3932	−0.963913
$695$	50.3374	1.90941
$696$	0	0
$697$	−66.6013	−2.52270
$698$	−47.7097	−1.80584
$699$	0	0
$700$	−3.80082	−0.143658
$701$	23.0395	0.870191	0.435095	−	0.900384i	$-0.356715\pi$
$701$	23.0395	0.870191	0.435095	+	0.900384i	$0.356715\pi$
$702$	0	0
$703$	−10.5227	−0.396871
$704$	−14.2215	−0.535993
$705$	0	0
$706$	−14.6753	−0.552313
$707$	−14.5035	−0.545460
$708$	0	0
$709$	−23.5750	−0.885379	−0.442689	−	0.896675i	$-0.645976\pi$
$709$	−23.5750	−0.885379	−0.442689	+	0.896675i	$0.645976\pi$
$710$	5.65949	0.212397
$711$	0	0
$712$	−11.8832	−0.445343
$713$	0	0
$714$	0	0
$715$	40.3933	1.51062
$716$	−30.8464	−1.15278
$717$	0	0
$718$	−40.3938	−1.50748
$719$	21.1285	0.787961	0.393980	−	0.919119i	$-0.371098\pi$
$719$	21.1285	0.787961	0.393980	+	0.919119i	$0.371098\pi$
$720$	0	0
$721$	3.61208	0.134521
$722$	−37.6778	−1.40222
$723$	0	0
$724$	26.2979	0.977354
$725$	−1.56516	−0.0581287
$726$	0	0
$727$	46.4512	1.72278	0.861391	−	0.507943i	$-0.169594\pi$
$727$	46.4512	1.72278	0.861391	+	0.507943i	$0.169594\pi$
$728$	−90.9473	−3.37073
$729$	0	0
$730$	38.9798	1.44271
$731$	32.2890	1.19425
$732$	0	0
$733$	−16.9886	−0.627488	−0.313744	−	0.949508i	$-0.601583\pi$
$733$	−16.9886	−0.627488	−0.313744	+	0.949508i	$0.601583\pi$
$734$	−62.9917	−2.32507
$735$	0	0
$736$	−0.920182	−0.0339184
$737$	40.8983	1.50651
$738$	0	0
$739$	39.3088	1.44600	0.723000	−	0.690849i	$-0.242763\pi$
$739$	39.3088	1.44600	0.723000	+	0.690849i	$0.242763\pi$
$740$	−51.9812	−1.91087
$741$	0	0
$742$	−11.6392	−0.427290
$743$	−45.3657	−1.66430	−0.832152	−	0.554547i	$-0.812891\pi$
$743$	−45.3657	−1.66430	−0.832152	+	0.554547i	$0.812891\pi$
$744$	0	0
$745$	−28.0292	−1.02691
$746$	−53.0661	−1.94289
$747$	0	0
$748$	88.0852	3.22071
$749$	−30.8016	−1.12547
$750$	0	0
$751$	25.3877	0.926410	0.463205	−	0.886251i	$-0.346699\pi$
$751$	25.3877	0.926410	0.463205	+	0.886251i	$0.346699\pi$
$752$	19.7076	0.718663
$753$	0	0
$754$	−71.0644	−2.58801
$755$	8.01960	0.291863
$756$	0	0
$757$	−35.6806	−1.29683	−0.648417	−	0.761285i	$-0.724569\pi$
$757$	−35.6806	−1.29683	−0.648417	+	0.761285i	$0.724569\pi$
$758$	−65.8923	−2.39332
$759$	0	0
$760$	25.3574	0.919809
$761$	−34.1139	−1.23663	−0.618314	−	0.785931i	$-0.712184\pi$
$761$	−34.1139	−1.23663	−0.618314	+	0.785931i	$0.712184\pi$
$762$	0	0
$763$	−9.39786	−0.340225
$764$	14.7493	0.533611
$765$	0	0
$766$	4.90504	0.177226
$767$	−55.6535	−2.00953
$768$	0	0
$769$	−37.6212	−1.35665	−0.678327	−	0.734760i	$-0.737295\pi$
$769$	−37.6212	−1.35665	−0.678327	+	0.734760i	$0.737295\pi$
$770$	−46.6689	−1.68183
$771$	0	0
$772$	−22.8625	−0.822838
$773$	32.0230	1.15179	0.575893	−	0.817525i	$-0.304655\pi$
$773$	32.0230	1.15179	0.575893	+	0.817525i	$0.304655\pi$
$774$	0	0
$775$	0	0
$776$	−103.623	−3.71986
$777$	0	0
$778$	−69.3733	−2.48715
$779$	−18.5991	−0.666380
$780$	0	0
$781$	2.89337	0.103533
$782$	−6.73941	−0.241001
$783$	0	0
$784$	3.09425	0.110509
$785$	−8.26057	−0.294833
$786$	0	0
$787$	3.51289	0.125221	0.0626106	−	0.998038i	$-0.480057\pi$
$787$	3.51289	0.125221	0.0626106	+	0.998038i	$0.480057\pi$
$788$	−21.5153	−0.766452
$789$	0	0
$790$	16.4936	0.586816
$791$	−17.4415	−0.620149
$792$	0	0
$793$	−5.48400	−0.194743
$794$	−10.0054	−0.355078
$795$	0	0
$796$	30.6619	1.08678
$797$	−16.0363	−0.568035	−0.284017	−	0.958819i	$-0.591667\pi$
$797$	−16.0363	−0.568035	−0.284017	+	0.958819i	$0.591667\pi$
$798$	0	0
$799$	25.7095	0.909537
$800$	0.787505	0.0278425
$801$	0	0
$802$	−45.5057	−1.60686
$803$	19.9281	0.703248
$804$	0	0
$805$	2.42408	0.0854376
$806$	0	0
$807$	0	0
$808$	29.3154	1.03131
$809$	3.19038	0.112168	0.0560839	−	0.998426i	$-0.482139\pi$
$809$	3.19038	0.112168	0.0560839	+	0.998426i	$0.482139\pi$
$810$	0	0
$811$	17.1263	0.601384	0.300692	−	0.953721i	$-0.402782\pi$
$811$	17.1263	0.601384	0.300692	+	0.953721i	$0.402782\pi$
$812$	55.7405	1.95611
$813$	0	0
$814$	−39.1446	−1.37202
$815$	7.05049	0.246968
$816$	0	0
$817$	9.01701	0.315465
$818$	−70.9028	−2.47906
$819$	0	0
$820$	−91.8779	−3.20851
$821$	−28.8055	−1.00532	−0.502659	−	0.864485i	$-0.667645\pi$
$821$	−28.8055	−1.00532	−0.502659	+	0.864485i	$0.667645\pi$
$822$	0	0
$823$	−43.8981	−1.53019	−0.765096	−	0.643916i	$-0.777308\pi$
$823$	−43.8981	−1.53019	−0.765096	+	0.643916i	$0.777308\pi$
$824$	−7.30097	−0.254341
$825$	0	0
$826$	64.2999	2.23728
$827$	21.7448	0.756142	0.378071	−	0.925777i	$-0.376587\pi$
$827$	21.7448	0.756142	0.378071	+	0.925777i	$0.376587\pi$
$828$	0	0
$829$	−11.1449	−0.387079	−0.193539	−	0.981093i	$-0.561997\pi$
$829$	−11.1449	−0.387079	−0.193539	+	0.981093i	$0.561997\pi$
$830$	−42.6003	−1.47868
$831$	0	0
$832$	−28.7037	−0.995120
$833$	4.03660	0.139860
$834$	0	0
$835$	6.14182	0.212546
$836$	24.5986	0.850762
$837$	0	0
$838$	−71.5256	−2.47081
$839$	−22.7431	−0.785180	−0.392590	−	0.919714i	$-0.628421\pi$
$839$	−22.7431	−0.785180	−0.392590	+	0.919714i	$0.628421\pi$
$840$	0	0
$841$	−6.04628	−0.208492
$842$	62.7098	2.16112
$843$	0	0
$844$	−37.1955	−1.28032
$845$	51.5234	1.77246
$846$	0	0
$847$	6.40712	0.220151
$848$	9.19159	0.315640
$849$	0	0
$850$	5.76769	0.197830
$851$	2.03326	0.0696991
$852$	0	0
$853$	33.9052	1.16089	0.580446	−	0.814299i	$-0.302878\pi$
$853$	33.9052	1.16089	0.580446	+	0.814299i	$0.302878\pi$
$854$	6.33601	0.216814
$855$	0	0
$856$	62.2582	2.12794
$857$	−24.5096	−0.837233	−0.418617	−	0.908163i	$-0.637485\pi$
$857$	−24.5096	−0.837233	−0.418617	+	0.908163i	$0.637485\pi$
$858$	0	0
$859$	15.1535	0.517031	0.258516	−	0.966007i	$-0.416767\pi$
$859$	15.1535	0.517031	0.258516	+	0.966007i	$0.416767\pi$
$860$	44.5433	1.51891
$861$	0	0
$862$	0.932756	0.0317698
$863$	−4.68932	−0.159626	−0.0798131	−	0.996810i	$-0.525432\pi$
$863$	−4.68932	−0.159626	−0.0798131	+	0.996810i	$0.525432\pi$
$864$	0	0
$865$	1.56466	0.0532001
$866$	2.50792	0.0852226
$867$	0	0
$868$	0	0
$869$	8.43222	0.286044
$870$	0	0
$871$	82.5461	2.79697
$872$	18.9955	0.643270
$873$	0	0
$874$	−1.88205	−0.0636612
$875$	29.6769	1.00326
$876$	0	0
$877$	−18.9112	−0.638585	−0.319292	−	0.947656i	$-0.603445\pi$
$877$	−18.9112	−0.638585	−0.319292	+	0.947656i	$0.603445\pi$
$878$	−13.5053	−0.455782
$879$	0	0
$880$	36.8547	1.24237
$881$	−7.21014	−0.242916	−0.121458	−	0.992597i	$-0.538757\pi$
$881$	−7.21014	−0.242916	−0.121458	+	0.992597i	$0.538757\pi$
$882$	0	0
$883$	53.2347	1.79149	0.895746	−	0.444566i	$-0.146642\pi$
$883$	53.2347	1.79149	0.895746	+	0.444566i	$0.146642\pi$
$884$	177.785	5.97955
$885$	0	0
$886$	37.7782	1.26918
$887$	26.1035	0.876470	0.438235	−	0.898860i	$-0.355604\pi$
$887$	26.1035	0.876470	0.438235	+	0.898860i	$0.355604\pi$
$888$	0	0
$889$	−16.3130	−0.547120
$890$	−12.3074	−0.412543
$891$	0	0
$892$	27.0759	0.906569
$893$	7.17963	0.240257
$894$	0	0
$895$	−16.8366	−0.562785
$896$	46.4284	1.55106
$897$	0	0
$898$	−25.7118	−0.858014
$899$	0	0
$900$	0	0
$901$	11.9908	0.399473
$902$	−69.1889	−2.30374
$903$	0	0
$904$	35.2539	1.17253
$905$	14.3539	0.477141
$906$	0	0
$907$	−7.74580	−0.257195	−0.128598	−	0.991697i	$-0.541048\pi$
$907$	−7.74580	−0.257195	−0.128598	+	0.991697i	$0.541048\pi$
$908$	55.9198	1.85576
$909$	0	0
$910$	−94.1932	−3.12247
$911$	0.765193	0.0253520	0.0126760	−	0.999920i	$-0.495965\pi$
$911$	0.765193	0.0253520	0.0126760	+	0.999920i	$0.495965\pi$
$912$	0	0
$913$	−21.7791	−0.720782
$914$	57.3250	1.89614
$915$	0	0
$916$	−112.761	−3.72574
$917$	21.4193	0.707329
$918$	0	0
$919$	−40.3869	−1.33224	−0.666121	−	0.745844i	$-0.732046\pi$
$919$	−40.3869	−1.33224	−0.666121	+	0.745844i	$0.732046\pi$
$920$	−4.89971	−0.161539
$921$	0	0
$922$	−46.4439	−1.52955
$923$	5.83977	0.192218
$924$	0	0
$925$	−1.74009	−0.0572138
$926$	−28.8629	−0.948495
$927$	0	0
$928$	−11.5491	−0.379117
$929$	−26.6328	−0.873795	−0.436898	−	0.899511i	$-0.643923\pi$
$929$	−26.6328	−0.873795	−0.436898	+	0.899511i	$0.643923\pi$
$930$	0	0
$931$	1.12726	0.0369444
$932$	107.303	3.51482
$933$	0	0
$934$	21.8329	0.714394
$935$	48.0787	1.57234
$936$	0	0
$937$	27.8372	0.909403	0.454701	−	0.890644i	$-0.349746\pi$
$937$	27.8372	0.909403	0.454701	+	0.890644i	$0.349746\pi$
$938$	−95.3707	−3.11396
$939$	0	0
$940$	35.4668	1.15680
$941$	−48.4866	−1.58062	−0.790310	−	0.612708i	$-0.790080\pi$
$941$	−48.4866	−1.58062	−0.790310	+	0.612708i	$0.790080\pi$
$942$	0	0
$943$	3.59382	0.117031
$944$	−50.7780	−1.65268
$945$	0	0
$946$	33.5435	1.09059
$947$	17.5569	0.570523	0.285262	−	0.958450i	$-0.407920\pi$
$947$	17.5569	0.570523	0.285262	+	0.958450i	$0.407920\pi$
$948$	0	0
$949$	40.2215	1.30565
$950$	1.61068	0.0522575
$951$	0	0
$952$	−108.251	−3.50844
$953$	19.0395	0.616748	0.308374	−	0.951265i	$-0.400215\pi$
$953$	19.0395	0.616748	0.308374	+	0.951265i	$0.400215\pi$
$954$	0	0
$955$	8.05047	0.260507
$956$	41.2179	1.33308
$957$	0	0
$958$	−26.1166	−0.843789
$959$	−6.59981	−0.213119
$960$	0	0
$961$	0	0
$962$	−79.0068	−2.54728
$963$	0	0
$964$	−118.527	−3.81750
$965$	−12.4788	−0.401707
$966$	0	0
$967$	−28.6941	−0.922738	−0.461369	−	0.887208i	$-0.652642\pi$
$967$	−28.6941	−0.922738	−0.461369	+	0.887208i	$0.652642\pi$
$968$	−12.9505	−0.416244
$969$	0	0
$970$	−107.322	−3.44589
$971$	−13.4537	−0.431750	−0.215875	−	0.976421i	$-0.569260\pi$
$971$	−13.4537	−0.431750	−0.215875	+	0.976421i	$0.569260\pi$
$972$	0	0
$973$	−60.0105	−1.92385
$974$	93.1792	2.98566
$975$	0	0
$976$	−5.00359	−0.160161
$977$	8.40056	0.268758	0.134379	−	0.990930i	$-0.457096\pi$
$977$	8.40056	0.268758	0.134379	+	0.990930i	$0.457096\pi$
$978$	0	0
$979$	−6.29203	−0.201094
$980$	5.56857	0.177881
$981$	0	0
$982$	66.8209	2.13234
$983$	−24.0579	−0.767327	−0.383663	−	0.923473i	$-0.625338\pi$
$983$	−24.0579	−0.767327	−0.383663	+	0.923473i	$0.625338\pi$
$984$	0	0
$985$	−11.7435	−0.374179
$986$	−84.5854	−2.69375
$987$	0	0
$988$	49.6482	1.57952
$989$	−1.74232	−0.0554026
$990$	0	0
$991$	−36.7862	−1.16855	−0.584276	−	0.811555i	$-0.698621\pi$
$991$	−36.7862	−1.16855	−0.584276	+	0.811555i	$0.698621\pi$
$992$	0	0
$993$	0	0
$994$	−6.74705	−0.214003
$995$	16.7359	0.530563
$996$	0	0
$997$	9.68136	0.306612	0.153306	−	0.988179i	$-0.451008\pi$
$997$	9.68136	0.306612	0.153306	+	0.988179i	$0.451008\pi$
$998$	−69.8037	−2.20960
$999$	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	8649.2.a.bj.1.8		8
3.2	odd	2		2883.2.a.m.1.1		8
31.10	even	15		279.2.y.b.100.2		16
31.28	even	15		279.2.y.b.226.2		16
31.30	odd	2		8649.2.a.bi.1.8		8
93.41	odd	30		93.2.m.a.7.1	✓	16
93.59	odd	30		93.2.m.a.40.1	yes	16
93.92	even	2		2883.2.a.n.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
93.2.m.a.7.1	✓	16	93.41	odd	30
93.2.m.a.40.1	yes	16	93.59	odd	30
279.2.y.b.100.2		16	31.10	even	15
279.2.y.b.226.2		16	31.28	even	15
2883.2.a.m.1.1		8	3.2	odd	2
2883.2.a.n.1.1		8	93.92	even	2
8649.2.a.bi.1.8		8	31.30	odd	2
8649.2.a.bj.1.8		8	1.1	even	1	trivial

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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	8649.2.a.bj.1.8

Level	$8649$
Weight	$2$
Character	8649.1
Self dual	yes
Analytic conductor	$69.063$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$1$