LMFDB - Embedded newform 4815.2.a.u.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4815,2,Mod(1,4815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4815, 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("4815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4815 = 3^{2} \cdot 5 \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4815.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:	$38.4479685732$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 49 x^{9} + 71 x^{8} - 278 x^{7} - 92 x^{6} + 649 x^{5} - 127 x^{4} + \cdots - 6 $ x^12 - 3x^11 - 15x^10 + 49x^9 + 71x^8 - 278x^7 - 92x^6 + 649x^5 - 127x^4 - 529x^3 + 267x^2 + 15*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.11086$ of defining polynomial
Character	$\chi$	$=$	4815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.11086 q^{2} +2.45572 q^{4} -1.00000 q^{5} +2.18333 q^{7} +0.961964 q^{8} +O(q^{10})$	\(q+2.11086 q^{2} +2.45572 q^{4} -1.00000 q^{5} +2.18333 q^{7} +0.961964 q^{8} -2.11086 q^{10} -2.72393 q^{11} +6.21332 q^{13} +4.60870 q^{14} -2.88087 q^{16} +0.494671 q^{17} +8.10330 q^{19} -2.45572 q^{20} -5.74983 q^{22} +1.62619 q^{23} +1.00000 q^{25} +13.1154 q^{26} +5.36165 q^{28} +1.55853 q^{29} +0.301876 q^{31} -8.00504 q^{32} +1.04418 q^{34} -2.18333 q^{35} -10.8707 q^{37} +17.1049 q^{38} -0.961964 q^{40} -4.60130 q^{41} +4.77087 q^{43} -6.68922 q^{44} +3.43266 q^{46} +6.82654 q^{47} -2.23307 q^{49} +2.11086 q^{50} +15.2582 q^{52} -8.36725 q^{53} +2.72393 q^{55} +2.10029 q^{56} +3.28983 q^{58} +13.9902 q^{59} -4.89646 q^{61} +0.637218 q^{62} -11.1358 q^{64} -6.21332 q^{65} +5.35756 q^{67} +1.21478 q^{68} -4.60870 q^{70} +16.7467 q^{71} +10.7751 q^{73} -22.9465 q^{74} +19.8994 q^{76} -5.94725 q^{77} -7.47583 q^{79} +2.88087 q^{80} -9.71268 q^{82} +13.5073 q^{83} -0.494671 q^{85} +10.0706 q^{86} -2.62032 q^{88} +1.42443 q^{89} +13.5657 q^{91} +3.99348 q^{92} +14.4099 q^{94} -8.10330 q^{95} +13.8331 q^{97} -4.71368 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * 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 $

$2$	2.11086	1.49260	0.746301	−	0.665609i	$-0.231828\pi$
$2$	2.11086	1.49260	0.746301	+	0.665609i	$0.231828\pi$
$3$	0	0
$3$	0	0
$4$	2.45572	1.22786
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.18333	0.825222	0.412611	−	0.910907i	$-0.364617\pi$
$7$	2.18333	0.825222	0.412611	+	0.910907i	$0.364617\pi$
$8$	0.961964	0.340106
$9$	0	0
$10$	−2.11086	−0.667512
$11$	−2.72393	−0.821296	−0.410648	−	0.911794i	$-0.634698\pi$
$11$	−2.72393	−0.821296	−0.410648	+	0.911794i	$0.634698\pi$
$12$	0	0
$13$	6.21332	1.72326	0.861632	−	0.507534i	$-0.169443\pi$
$13$	6.21332	1.72326	0.861632	+	0.507534i	$0.169443\pi$
$14$	4.60870	1.23173
$15$	0	0
$16$	−2.88087	−0.720219
$17$	0.494671	0.119975	0.0599877	−	0.998199i	$-0.480894\pi$
$17$	0.494671	0.119975	0.0599877	+	0.998199i	$0.480894\pi$
$18$	0	0
$19$	8.10330	1.85902	0.929512	−	0.368792i	$-0.120229\pi$
$19$	8.10330	1.85902	0.929512	+	0.368792i	$0.120229\pi$
$20$	−2.45572	−0.549116
$21$	0	0
$22$	−5.74983	−1.22587
$23$	1.62619	0.339084	0.169542	−	0.985523i	$-0.445771\pi$
$23$	1.62619	0.339084	0.169542	+	0.985523i	$0.445771\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	13.1154	2.57215
$27$	0	0
$28$	5.36165	1.01326
$29$	1.55853	0.289412	0.144706	−	0.989475i	$-0.453776\pi$
$29$	1.55853	0.289412	0.144706	+	0.989475i	$0.453776\pi$
$30$	0	0
$31$	0.301876	0.0542186	0.0271093	−	0.999632i	$-0.491370\pi$
$31$	0.301876	0.0542186	0.0271093	+	0.999632i	$0.491370\pi$
$32$	−8.00504	−1.41511
$33$	0	0
$34$	1.04418	0.179076
$35$	−2.18333	−0.369050
$36$	0	0
$37$	−10.8707	−1.78713	−0.893566	−	0.448932i	$-0.851804\pi$
$37$	−10.8707	−1.78713	−0.893566	+	0.448932i	$0.851804\pi$
$38$	17.1049	2.77478
$39$	0	0
$40$	−0.961964	−0.152100
$41$	−4.60130	−0.718602	−0.359301	−	0.933222i	$-0.616985\pi$
$41$	−4.60130	−0.718602	−0.359301	+	0.933222i	$0.616985\pi$
$42$	0	0
$43$	4.77087	0.727551	0.363776	−	0.931487i	$-0.381488\pi$
$43$	4.77087	0.727551	0.363776	+	0.931487i	$0.381488\pi$
$44$	−6.68922	−1.00844
$45$	0	0
$46$	3.43266	0.506118
$47$	6.82654	0.995753	0.497876	−	0.867248i	$-0.334113\pi$
$47$	6.82654	0.995753	0.497876	+	0.867248i	$0.334113\pi$
$48$	0	0
$49$	−2.23307	−0.319009
$50$	2.11086	0.298520
$51$	0	0
$52$	15.2582	2.11593
$53$	−8.36725	−1.14933	−0.574665	−	0.818389i	$-0.694868\pi$
$53$	−8.36725	−1.14933	−0.574665	+	0.818389i	$0.694868\pi$
$54$	0	0
$55$	2.72393	0.367295
$56$	2.10029	0.280662
$57$	0	0
$58$	3.28983	0.431976
$59$	13.9902	1.82137	0.910687	−	0.413096i	$-0.135553\pi$
$59$	13.9902	1.82137	0.910687	+	0.413096i	$0.135553\pi$
$60$	0	0
$61$	−4.89646	−0.626928	−0.313464	−	0.949600i	$-0.601489\pi$
$61$	−4.89646	−0.626928	−0.313464	+	0.949600i	$0.601489\pi$
$62$	0.637218	0.0809268
$63$	0	0
$64$	−11.1358	−1.39197
$65$	−6.21332	−0.770667
$66$	0	0
$67$	5.35756	0.654530	0.327265	−	0.944933i	$-0.393873\pi$
$67$	5.35756	0.654530	0.327265	+	0.944933i	$0.393873\pi$
$68$	1.21478	0.147313
$69$	0	0
$70$	−4.60870	−0.550845
$71$	16.7467	1.98747	0.993735	−	0.111766i	$-0.0356508\pi$
$71$	16.7467	1.98747	0.993735	+	0.111766i	$0.0356508\pi$
$72$	0	0
$73$	10.7751	1.26113	0.630563	−	0.776138i	$-0.282824\pi$
$73$	10.7751	1.26113	0.630563	+	0.776138i	$0.282824\pi$
$74$	−22.9465	−2.66748
$75$	0	0
$76$	19.8994	2.28262
$77$	−5.94725	−0.677751
$78$	0	0
$79$	−7.47583	−0.841096	−0.420548	−	0.907270i	$-0.638162\pi$
$79$	−7.47583	−0.841096	−0.420548	+	0.907270i	$0.638162\pi$
$80$	2.88087	0.322092
$81$	0	0
$82$	−9.71268	−1.07259
$83$	13.5073	1.48262	0.741311	−	0.671162i	$-0.234204\pi$
$83$	13.5073	1.48262	0.741311	+	0.671162i	$0.234204\pi$
$84$	0	0
$85$	−0.494671	−0.0536546
$86$	10.0706	1.08594
$87$	0	0
$88$	−2.62032	−0.279327
$89$	1.42443	0.150990	0.0754948	−	0.997146i	$-0.475946\pi$
$89$	1.42443	0.150990	0.0754948	+	0.997146i	$0.475946\pi$
$90$	0	0
$91$	13.5657	1.42207
$92$	3.99348	0.416349
$93$	0	0
$94$	14.4099	1.48626
$95$	−8.10330	−0.831381
$96$	0	0
$97$	13.8331	1.40454	0.702271	−	0.711910i	$-0.252169\pi$
$97$	13.8331	1.40454	0.702271	+	0.711910i	$0.252169\pi$
$98$	−4.71368	−0.476154
$99$	0	0
$100$	2.45572	0.245572
$101$	13.3450	1.32788	0.663941	−	0.747785i	$-0.268883\pi$
$101$	13.3450	1.32788	0.663941	+	0.747785i	$0.268883\pi$
$102$	0	0
$103$	−1.48546	−0.146366	−0.0731832	−	0.997319i	$-0.523316\pi$
$103$	−1.48546	−0.146366	−0.0731832	+	0.997319i	$0.523316\pi$
$104$	5.97698	0.586092
$105$	0	0
$106$	−17.6621	−1.71549
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	0	0
$109$	−11.0608	−1.05943	−0.529716	−	0.848175i	$-0.677701\pi$
$109$	−11.0608	−1.05943	−0.529716	+	0.848175i	$0.677701\pi$
$110$	5.74983	0.548225
$111$	0	0
$112$	−6.28990	−0.594340
$113$	5.46105	0.513733	0.256866	−	0.966447i	$-0.417310\pi$
$113$	5.46105	0.513733	0.256866	+	0.966447i	$0.417310\pi$
$114$	0	0
$115$	−1.62619	−0.151643
$116$	3.82731	0.355357
$117$	0	0
$118$	29.5314	2.71859
$119$	1.08003	0.0990063
$120$	0	0
$121$	−3.58019	−0.325472
$122$	−10.3357	−0.935754
$123$	0	0
$124$	0.741324	0.0665729
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.3400	1.27247	0.636233	−	0.771497i	$-0.280492\pi$
$127$	14.3400	1.27247	0.636233	+	0.771497i	$0.280492\pi$
$128$	−7.49593	−0.662553
$129$	0	0
$130$	−13.1154	−1.15030
$131$	9.47983	0.828257	0.414128	−	0.910218i	$-0.364086\pi$
$131$	9.47983	0.828257	0.414128	+	0.910218i	$0.364086\pi$
$132$	0	0
$133$	17.6922	1.53411
$134$	11.3090	0.976952
$135$	0	0
$136$	0.475856	0.0408043
$137$	6.19886	0.529604	0.264802	−	0.964303i	$-0.414693\pi$
$137$	6.19886	0.529604	0.264802	+	0.964303i	$0.414693\pi$
$138$	0	0
$139$	20.1745	1.71118	0.855591	−	0.517653i	$-0.173194\pi$
$139$	20.1745	1.71118	0.855591	+	0.517653i	$0.173194\pi$
$140$	−5.36165	−0.453142
$141$	0	0
$142$	35.3499	2.96650
$143$	−16.9246	−1.41531
$144$	0	0
$145$	−1.55853	−0.129429
$146$	22.7446	1.88236
$147$	0	0
$148$	−26.6954	−2.19435
$149$	5.52613	0.452718	0.226359	−	0.974044i	$-0.427318\pi$
$149$	5.52613	0.452718	0.226359	+	0.974044i	$0.427318\pi$
$150$	0	0
$151$	−14.3017	−1.16385	−0.581926	−	0.813242i	$-0.697701\pi$
$151$	−14.3017	−1.16385	−0.581926	+	0.813242i	$0.697701\pi$
$152$	7.79508	0.632264
$153$	0	0
$154$	−12.5538	−1.01161
$155$	−0.301876	−0.0242473
$156$	0	0
$157$	−1.61398	−0.128809	−0.0644047	−	0.997924i	$-0.520515\pi$
$157$	−1.61398	−0.128809	−0.0644047	+	0.997924i	$0.520515\pi$
$158$	−15.7804	−1.25542
$159$	0	0
$160$	8.00504	0.632854
$161$	3.55052	0.279820
$162$	0	0
$163$	3.13866	0.245839	0.122920	−	0.992417i	$-0.460774\pi$
$163$	3.13866	0.245839	0.122920	+	0.992417i	$0.460774\pi$
$164$	−11.2995	−0.882343
$165$	0	0
$166$	28.5121	2.21296
$167$	−17.8137	−1.37846	−0.689231	−	0.724542i	$-0.742052\pi$
$167$	−17.8137	−1.37846	−0.689231	+	0.724542i	$0.742052\pi$
$168$	0	0
$169$	25.6053	1.96964
$170$	−1.04418	−0.0800850
$171$	0	0
$172$	11.7159	0.893332
$173$	−20.0547	−1.52473	−0.762364	−	0.647148i	$-0.775962\pi$
$173$	−20.0547	−1.52473	−0.762364	+	0.647148i	$0.775962\pi$
$174$	0	0
$175$	2.18333	0.165044
$176$	7.84731	0.591513
$177$	0	0
$178$	3.00678	0.225367
$179$	−24.9921	−1.86799	−0.933997	−	0.357280i	$-0.883704\pi$
$179$	−24.9921	−1.86799	−0.933997	+	0.357280i	$0.883704\pi$
$180$	0	0
$181$	−15.3702	−1.14246	−0.571228	−	0.820792i	$-0.693533\pi$
$181$	−15.3702	−1.14246	−0.571228	+	0.820792i	$0.693533\pi$
$182$	28.6353	2.12259
$183$	0	0
$184$	1.56434	0.115325
$185$	10.8707	0.799229
$186$	0	0
$187$	−1.34745	−0.0985354
$188$	16.7641	1.22265
$189$	0	0
$190$	−17.1049	−1.24092
$191$	−6.84344	−0.495174	−0.247587	−	0.968866i	$-0.579638\pi$
$191$	−6.84344	−0.495174	−0.247587	+	0.968866i	$0.579638\pi$
$192$	0	0
$193$	−23.0591	−1.65983	−0.829917	−	0.557887i	$-0.811612\pi$
$193$	−23.0591	−1.65983	−0.829917	+	0.557887i	$0.811612\pi$
$194$	29.1998	2.09642
$195$	0	0
$196$	−5.48379	−0.391699
$197$	−8.45699	−0.602536	−0.301268	−	0.953540i	$-0.597410\pi$
$197$	−8.45699	−0.602536	−0.301268	+	0.953540i	$0.597410\pi$
$198$	0	0
$199$	9.00696	0.638486	0.319243	−	0.947673i	$-0.396571\pi$
$199$	9.00696	0.638486	0.319243	+	0.947673i	$0.396571\pi$
$200$	0.961964	0.0680211
$201$	0	0
$202$	28.1695	1.98200
$203$	3.40279	0.238829
$204$	0	0
$205$	4.60130	0.321368
$206$	−3.13559	−0.218467
$207$	0	0
$208$	−17.8998	−1.24113
$209$	−22.0728	−1.52681
$210$	0	0
$211$	5.45535	0.375562	0.187781	−	0.982211i	$-0.439870\pi$
$211$	5.45535	0.375562	0.187781	+	0.982211i	$0.439870\pi$
$212$	−20.5476	−1.41122
$213$	0	0
$214$	−2.11086	−0.144295
$215$	−4.77087	−0.325371
$216$	0	0
$217$	0.659096	0.0447423
$218$	−23.3478	−1.58131
$219$	0	0
$220$	6.68922	0.450987
$221$	3.07355	0.206749
$222$	0	0
$223$	−22.1012	−1.48000	−0.740001	−	0.672605i	$-0.765175\pi$
$223$	−22.1012	−1.48000	−0.740001	+	0.672605i	$0.765175\pi$
$224$	−17.4777	−1.16778
$225$	0	0
$226$	11.5275	0.766799
$227$	−8.72467	−0.579077	−0.289538	−	0.957166i	$-0.593502\pi$
$227$	−8.72467	−0.579077	−0.289538	+	0.957166i	$0.593502\pi$
$228$	0	0
$229$	20.9636	1.38531	0.692655	−	0.721269i	$-0.256441\pi$
$229$	20.9636	1.38531	0.692655	+	0.721269i	$0.256441\pi$
$230$	−3.43266	−0.226343
$231$	0	0
$232$	1.49925	0.0984305
$233$	−18.4278	−1.20724	−0.603622	−	0.797271i	$-0.706276\pi$
$233$	−18.4278	−1.20724	−0.603622	+	0.797271i	$0.706276\pi$
$234$	0	0
$235$	−6.82654	−0.445314
$236$	34.3561	2.23639
$237$	0	0
$238$	2.27979	0.147777
$239$	−27.3147	−1.76684	−0.883420	−	0.468582i	$-0.844765\pi$
$239$	−27.3147	−1.76684	−0.883420	+	0.468582i	$0.844765\pi$
$240$	0	0
$241$	18.7160	1.20560	0.602801	−	0.797891i	$-0.294051\pi$
$241$	18.7160	1.20560	0.602801	+	0.797891i	$0.294051\pi$
$242$	−7.55728	−0.485801
$243$	0	0
$244$	−12.0244	−0.769780
$245$	2.23307	0.142665
$246$	0	0
$247$	50.3483	3.20359
$248$	0.290394	0.0184400
$249$	0	0
$250$	−2.11086	−0.133502
$251$	10.4264	0.658110	0.329055	−	0.944311i	$-0.393270\pi$
$251$	10.4264	0.658110	0.329055	+	0.944311i	$0.393270\pi$
$252$	0	0
$253$	−4.42964	−0.278489
$254$	30.2696	1.89928
$255$	0	0
$256$	6.44869	0.403043
$257$	−0.283910	−0.0177098	−0.00885491	−	0.999961i	$-0.502819\pi$
$257$	−0.283910	−0.0177098	−0.00885491	+	0.999961i	$0.502819\pi$
$258$	0	0
$259$	−23.7343	−1.47478
$260$	−15.2582	−0.946272
$261$	0	0
$262$	20.0106	1.23626
$263$	−18.4782	−1.13942	−0.569708	−	0.821847i	$-0.692944\pi$
$263$	−18.4782	−1.13942	−0.569708	+	0.821847i	$0.692944\pi$
$264$	0	0
$265$	8.36725	0.513996
$266$	37.3457	2.28981
$267$	0	0
$268$	13.1567	0.803671
$269$	20.7054	1.26243	0.631216	−	0.775607i	$-0.282556\pi$
$269$	20.7054	1.26243	0.631216	+	0.775607i	$0.282556\pi$
$270$	0	0
$271$	−12.1623	−0.738810	−0.369405	−	0.929269i	$-0.620438\pi$
$271$	−12.1623	−0.738810	−0.369405	+	0.929269i	$0.620438\pi$
$272$	−1.42509	−0.0864085
$273$	0	0
$274$	13.0849	0.790489
$275$	−2.72393	−0.164259
$276$	0	0
$277$	−11.4743	−0.689422	−0.344711	−	0.938709i	$-0.612023\pi$
$277$	−11.4743	−0.689422	−0.344711	+	0.938709i	$0.612023\pi$
$278$	42.5856	2.55411
$279$	0	0
$280$	−2.10029	−0.125516
$281$	9.03945	0.539248	0.269624	−	0.962966i	$-0.413101\pi$
$281$	9.03945	0.539248	0.269624	+	0.962966i	$0.413101\pi$
$282$	0	0
$283$	−27.1966	−1.61667	−0.808335	−	0.588723i	$-0.799631\pi$
$283$	−27.1966	−1.61667	−0.808335	+	0.588723i	$0.799631\pi$
$284$	41.1253	2.44034
$285$	0	0
$286$	−35.7255	−2.11249
$287$	−10.0462	−0.593006
$288$	0	0
$289$	−16.7553	−0.985606
$290$	−3.28983	−0.193186
$291$	0	0
$292$	26.4605	1.54849
$293$	−8.78843	−0.513425	−0.256713	−	0.966488i	$-0.582639\pi$
$293$	−8.78843	−0.513425	−0.256713	+	0.966488i	$0.582639\pi$
$294$	0	0
$295$	−13.9902	−0.814543
$296$	−10.4572	−0.607813
$297$	0	0
$298$	11.6649	0.675728
$299$	10.1040	0.584332
$300$	0	0
$301$	10.4164	0.600391
$302$	−30.1888	−1.73717
$303$	0	0
$304$	−23.3446	−1.33890
$305$	4.89646	0.280371
$306$	0	0
$307$	−24.7915	−1.41493	−0.707464	−	0.706750i	$-0.750161\pi$
$307$	−24.7915	−1.41493	−0.707464	+	0.706750i	$0.750161\pi$
$308$	−14.6048	−0.832185
$309$	0	0
$310$	−0.637218	−0.0361915
$311$	6.51398	0.369374	0.184687	−	0.982797i	$-0.440873\pi$
$311$	6.51398	0.369374	0.184687	+	0.982797i	$0.440873\pi$
$312$	0	0
$313$	−12.3449	−0.697773	−0.348887	−	0.937165i	$-0.613440\pi$
$313$	−12.3449	−0.697773	−0.348887	+	0.937165i	$0.613440\pi$
$314$	−3.40687	−0.192261
$315$	0	0
$316$	−18.3586	−1.03275
$317$	−7.08442	−0.397901	−0.198950	−	0.980010i	$-0.563753\pi$
$317$	−7.08442	−0.397901	−0.198950	+	0.980010i	$0.563753\pi$
$318$	0	0
$319$	−4.24533	−0.237693
$320$	11.1358	0.622508
$321$	0	0
$322$	7.49463	0.417660
$323$	4.00847	0.223037
$324$	0	0
$325$	6.21332	0.344653
$326$	6.62527	0.366940
$327$	0	0
$328$	−4.42628	−0.244400
$329$	14.9046	0.821717
$330$	0	0
$331$	−26.9719	−1.48251	−0.741254	−	0.671224i	$-0.765769\pi$
$331$	−26.9719	−1.48251	−0.741254	+	0.671224i	$0.765769\pi$
$332$	33.1702	1.82045
$333$	0	0
$334$	−37.6021	−2.05750
$335$	−5.35756	−0.292715
$336$	0	0
$337$	−7.24132	−0.394460	−0.197230	−	0.980357i	$-0.563195\pi$
$337$	−7.24132	−0.394460	−0.197230	+	0.980357i	$0.563195\pi$
$338$	54.0491	2.93989
$339$	0	0
$340$	−1.21478	−0.0658804
$341$	−0.822290	−0.0445295
$342$	0	0
$343$	−20.1588	−1.08847
$344$	4.58941	0.247444
$345$	0	0
$346$	−42.3326	−2.27581
$347$	−9.81239	−0.526757	−0.263378	−	0.964693i	$-0.584837\pi$
$347$	−9.81239	−0.526757	−0.263378	+	0.964693i	$0.584837\pi$
$348$	0	0
$349$	−2.17814	−0.116593	−0.0582966	−	0.998299i	$-0.518567\pi$
$349$	−2.17814	−0.116593	−0.0582966	+	0.998299i	$0.518567\pi$
$350$	4.60870	0.246345
$351$	0	0
$352$	21.8052	1.16222
$353$	6.35833	0.338420	0.169210	−	0.985580i	$-0.445878\pi$
$353$	6.35833	0.338420	0.169210	+	0.985580i	$0.445878\pi$
$354$	0	0
$355$	−16.7467	−0.888823
$356$	3.49801	0.185394
$357$	0	0
$358$	−52.7547	−2.78817
$359$	−16.0105	−0.845003	−0.422501	−	0.906362i	$-0.638848\pi$
$359$	−16.0105	−0.845003	−0.422501	+	0.906362i	$0.638848\pi$
$360$	0	0
$361$	46.6634	2.45597
$362$	−32.4442	−1.70523
$363$	0	0
$364$	33.3136	1.74611
$365$	−10.7751	−0.563992
$366$	0	0
$367$	9.71531	0.507135	0.253568	−	0.967318i	$-0.418396\pi$
$367$	9.71531	0.507135	0.253568	+	0.967318i	$0.418396\pi$
$368$	−4.68486	−0.244215
$369$	0	0
$370$	22.9465	1.19293
$371$	−18.2685	−0.948452
$372$	0	0
$373$	0.183187	0.00948509	0.00474254	−	0.999989i	$-0.498490\pi$
$373$	0.183187	0.00948509	0.00474254	+	0.999989i	$0.498490\pi$
$374$	−2.84428	−0.147074
$375$	0	0
$376$	6.56688	0.338661
$377$	9.68363	0.498733
$378$	0	0
$379$	−5.09169	−0.261543	−0.130771	−	0.991413i	$-0.541745\pi$
$379$	−5.09169	−0.261543	−0.130771	+	0.991413i	$0.541745\pi$
$380$	−19.8994	−1.02082
$381$	0	0
$382$	−14.4455	−0.739098
$383$	0.924733	0.0472516	0.0236258	−	0.999721i	$-0.492479\pi$
$383$	0.924733	0.0472516	0.0236258	+	0.999721i	$0.492479\pi$
$384$	0	0
$385$	5.94725	0.303100
$386$	−48.6746	−2.47747
$387$	0	0
$388$	33.9703	1.72458
$389$	24.9413	1.26457	0.632287	−	0.774734i	$-0.282116\pi$
$389$	24.9413	1.26457	0.632287	+	0.774734i	$0.282116\pi$
$390$	0	0
$391$	0.804431	0.0406818
$392$	−2.14813	−0.108497
$393$	0	0
$394$	−17.8515	−0.899346
$395$	7.47583	0.376150
$396$	0	0
$397$	−21.9904	−1.10367	−0.551834	−	0.833954i	$-0.686072\pi$
$397$	−21.9904	−1.10367	−0.551834	+	0.833954i	$0.686072\pi$
$398$	19.0124	0.953006
$399$	0	0
$400$	−2.88087	−0.144044
$401$	−22.1398	−1.10561	−0.552804	−	0.833311i	$-0.686442\pi$
$401$	−22.1398	−1.10561	−0.552804	+	0.833311i	$0.686442\pi$
$402$	0	0
$403$	1.87565	0.0934329
$404$	32.7717	1.63045
$405$	0	0
$406$	7.18280	0.356476
$407$	29.6110	1.46776
$408$	0	0
$409$	6.16231	0.304707	0.152353	−	0.988326i	$-0.451315\pi$
$409$	6.16231	0.304707	0.152353	+	0.988326i	$0.451315\pi$
$410$	9.71268	0.479675
$411$	0	0
$412$	−3.64787	−0.179718
$413$	30.5453	1.50304
$414$	0	0
$415$	−13.5073	−0.663049
$416$	−49.7379	−2.43860
$417$	0	0
$418$	−46.5926	−2.27892
$419$	−1.87957	−0.0918230	−0.0459115	−	0.998946i	$-0.514619\pi$
$419$	−1.87957	−0.0918230	−0.0459115	+	0.998946i	$0.514619\pi$
$420$	0	0
$421$	−25.2279	−1.22953	−0.614766	−	0.788710i	$-0.710749\pi$
$421$	−25.2279	−1.22953	−0.614766	+	0.788710i	$0.710749\pi$
$422$	11.5155	0.560565
$423$	0	0
$424$	−8.04899	−0.390894
$425$	0.494671	0.0239951
$426$	0	0
$427$	−10.6906	−0.517354
$428$	−2.45572	−0.118702
$429$	0	0
$430$	−10.0706	−0.485649
$431$	18.9656	0.913539	0.456770	−	0.889585i	$-0.349006\pi$
$431$	18.9656	0.913539	0.456770	+	0.889585i	$0.349006\pi$
$432$	0	0
$433$	21.9148	1.05316	0.526579	−	0.850126i	$-0.323474\pi$
$433$	21.9148	1.05316	0.526579	+	0.850126i	$0.323474\pi$
$434$	1.39126	0.0667825
$435$	0	0
$436$	−27.1622	−1.30083
$437$	13.1775	0.630366
$438$	0	0
$439$	−16.8390	−0.803679	−0.401840	−	0.915710i	$-0.631629\pi$
$439$	−16.8390	−0.803679	−0.401840	+	0.915710i	$0.631629\pi$
$440$	2.62032	0.124919
$441$	0	0
$442$	6.48783	0.308594
$443$	7.42954	0.352988	0.176494	−	0.984302i	$-0.443524\pi$
$443$	7.42954	0.352988	0.176494	+	0.984302i	$0.443524\pi$
$444$	0	0
$445$	−1.42443	−0.0675246
$446$	−46.6524	−2.20906
$447$	0	0
$448$	−24.3131	−1.14868
$449$	10.4408	0.492734	0.246367	−	0.969177i	$-0.420763\pi$
$449$	10.4408	0.492734	0.246367	+	0.969177i	$0.420763\pi$
$450$	0	0
$451$	12.5336	0.590185
$452$	13.4108	0.630792
$453$	0	0
$454$	−18.4165	−0.864331
$455$	−13.5657	−0.635971
$456$	0	0
$457$	31.7309	1.48431	0.742155	−	0.670229i	$-0.233804\pi$
$457$	31.7309	1.48431	0.742155	+	0.670229i	$0.233804\pi$
$458$	44.2511	2.06772
$459$	0	0
$460$	−3.99348	−0.186197
$461$	33.4552	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$461$	33.4552	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$462$	0	0
$463$	34.8913	1.62154	0.810769	−	0.585367i	$-0.199050\pi$
$463$	34.8913	1.62154	0.810769	+	0.585367i	$0.199050\pi$
$464$	−4.48993	−0.208440
$465$	0	0
$466$	−38.8984	−1.80193
$467$	28.8028	1.33283	0.666417	−	0.745579i	$-0.267827\pi$
$467$	28.8028	1.33283	0.666417	+	0.745579i	$0.267827\pi$
$468$	0	0
$469$	11.6973	0.540132
$470$	−14.4099	−0.664677
$471$	0	0
$472$	13.4581	0.619460
$473$	−12.9955	−0.597535
$474$	0	0
$475$	8.10330	0.371805
$476$	2.65226	0.121566
$477$	0	0
$478$	−57.6574	−2.63719
$479$	2.12521	0.0971035	0.0485517	−	0.998821i	$-0.484539\pi$
$479$	2.12521	0.0971035	0.0485517	+	0.998821i	$0.484539\pi$
$480$	0	0
$481$	−67.5431	−3.07970
$482$	39.5068	1.79949
$483$	0	0
$484$	−8.79196	−0.399635
$485$	−13.8331	−0.628130
$486$	0	0
$487$	22.7686	1.03174	0.515871	−	0.856666i	$-0.327468\pi$
$487$	22.7686	1.03174	0.515871	+	0.856666i	$0.327468\pi$
$488$	−4.71022	−0.213222
$489$	0	0
$490$	4.71368	0.212943
$491$	−2.19054	−0.0988578	−0.0494289	−	0.998778i	$-0.515740\pi$
$491$	−2.19054	−0.0988578	−0.0494289	+	0.998778i	$0.515740\pi$
$492$	0	0
$493$	0.770960	0.0347223
$494$	106.278	4.78168
$495$	0	0
$496$	−0.869668	−0.0390492
$497$	36.5636	1.64010
$498$	0	0
$499$	−25.2165	−1.12884	−0.564422	−	0.825486i	$-0.690901\pi$
$499$	−25.2165	−1.12884	−0.564422	+	0.825486i	$0.690901\pi$
$500$	−2.45572	−0.109823
$501$	0	0
$502$	22.0087	0.982296
$503$	8.71501	0.388583	0.194292	−	0.980944i	$-0.437759\pi$
$503$	8.71501	0.388583	0.194292	+	0.980944i	$0.437759\pi$
$504$	0	0
$505$	−13.3450	−0.593847
$506$	−9.35033	−0.415673
$507$	0	0
$508$	35.2149	1.56241
$509$	−36.9240	−1.63663	−0.818315	−	0.574770i	$-0.805091\pi$
$509$	−36.9240	−1.63663	−0.818315	+	0.574770i	$0.805091\pi$
$510$	0	0
$511$	23.5255	1.04071
$512$	28.6041	1.26414
$513$	0	0
$514$	−0.599294	−0.0264337
$515$	1.48546	0.0654570
$516$	0	0
$517$	−18.5950	−0.817808
$518$	−50.0998	−2.20126
$519$	0	0
$520$	−5.97698	−0.262108
$521$	−4.97591	−0.217998	−0.108999	−	0.994042i	$-0.534765\pi$
$521$	−4.97591	−0.217998	−0.108999	+	0.994042i	$0.534765\pi$
$522$	0	0
$523$	−10.7511	−0.470114	−0.235057	−	0.971982i	$-0.575528\pi$
$523$	−10.7511	−0.470114	−0.235057	+	0.971982i	$0.575528\pi$
$524$	23.2798	1.01698
$525$	0	0
$526$	−39.0049	−1.70069
$527$	0.149330	0.00650490
$528$	0	0
$529$	−20.3555	−0.885022
$530$	17.6621	0.767192
$531$	0	0
$532$	43.4471	1.88367
$533$	−28.5893	−1.23834
$534$	0	0
$535$	1.00000	0.0432338
$536$	5.15377	0.222609
$537$	0	0
$538$	43.7062	1.88431
$539$	6.08272	0.262001
$540$	0	0
$541$	4.89035	0.210253	0.105126	−	0.994459i	$-0.466475\pi$
$541$	4.89035	0.210253	0.105126	+	0.994459i	$0.466475\pi$
$542$	−25.6730	−1.10275
$543$	0	0
$544$	−3.95987	−0.169778
$545$	11.0608	0.473792
$546$	0	0
$547$	−20.7357	−0.886593	−0.443297	−	0.896375i	$-0.646191\pi$
$547$	−20.7357	−0.886593	−0.443297	+	0.896375i	$0.646191\pi$
$548$	15.2227	0.650281
$549$	0	0
$550$	−5.74983	−0.245174
$551$	12.6292	0.538023
$552$	0	0
$553$	−16.3222	−0.694091
$554$	−24.2206	−1.02903
$555$	0	0
$556$	49.5430	2.10109
$557$	38.7336	1.64120	0.820599	−	0.571505i	$-0.193640\pi$
$557$	38.7336	1.64120	0.820599	+	0.571505i	$0.193640\pi$
$558$	0	0
$559$	29.6429	1.25376
$560$	6.28990	0.265797
$561$	0	0
$562$	19.0810	0.804883
$563$	−31.3041	−1.31931	−0.659655	−	0.751569i	$-0.729297\pi$
$563$	−31.3041	−1.31931	−0.659655	+	0.751569i	$0.729297\pi$
$564$	0	0
$565$	−5.46105	−0.229748
$566$	−57.4081	−2.41304
$567$	0	0
$568$	16.1097	0.675949
$569$	18.5578	0.777983	0.388992	−	0.921241i	$-0.372823\pi$
$569$	18.5578	0.777983	0.388992	+	0.921241i	$0.372823\pi$
$570$	0	0
$571$	37.0056	1.54864	0.774319	−	0.632796i	$-0.218093\pi$
$571$	37.0056	1.54864	0.774319	+	0.632796i	$0.218093\pi$
$572$	−41.5622	−1.73780
$573$	0	0
$574$	−21.2060	−0.885121
$575$	1.62619	0.0678169
$576$	0	0
$577$	−38.0378	−1.58354	−0.791768	−	0.610823i	$-0.790839\pi$
$577$	−38.0378	−1.58354	−0.791768	+	0.610823i	$0.790839\pi$
$578$	−35.3681	−1.47112
$579$	0	0
$580$	−3.82731	−0.158921
$581$	29.4910	1.22349
$582$	0	0
$583$	22.7918	0.943941
$584$	10.3652	0.428916
$585$	0	0
$586$	−18.5511	−0.766340
$587$	−43.3243	−1.78819	−0.894093	−	0.447882i	$-0.852178\pi$
$587$	−43.3243	−1.78819	−0.894093	+	0.447882i	$0.852178\pi$
$588$	0	0
$589$	2.44619	0.100794
$590$	−29.5314	−1.21579
$591$	0	0
$592$	31.3171	1.28713
$593$	10.4702	0.429958	0.214979	−	0.976619i	$-0.431032\pi$
$593$	10.4702	0.429958	0.214979	+	0.976619i	$0.431032\pi$
$594$	0	0
$595$	−1.08003	−0.0442770
$596$	13.5706	0.555875
$597$	0	0
$598$	21.3282	0.872175
$599$	−45.7036	−1.86740	−0.933700	−	0.358056i	$-0.883440\pi$
$599$	−45.7036	−1.86740	−0.933700	+	0.358056i	$0.883440\pi$
$600$	0	0
$601$	−2.78019	−0.113406	−0.0567032	−	0.998391i	$-0.518059\pi$
$601$	−2.78019	−0.113406	−0.0567032	+	0.998391i	$0.518059\pi$
$602$	21.9875	0.896145
$603$	0	0
$604$	−35.1209	−1.42905
$605$	3.58019	0.145556
$606$	0	0
$607$	−26.9100	−1.09224	−0.546122	−	0.837705i	$-0.683897\pi$
$607$	−26.9100	−1.09224	−0.546122	+	0.837705i	$0.683897\pi$
$608$	−64.8672	−2.63071
$609$	0	0
$610$	10.3357	0.418482
$611$	42.4154	1.71595
$612$	0	0
$613$	33.8684	1.36793	0.683966	−	0.729514i	$-0.260254\pi$
$613$	33.8684	1.36793	0.683966	+	0.729514i	$0.260254\pi$
$614$	−52.3314	−2.11192
$615$	0	0
$616$	−5.72103	−0.230507
$617$	7.61794	0.306687	0.153343	−	0.988173i	$-0.450996\pi$
$617$	7.61794	0.306687	0.153343	+	0.988173i	$0.450996\pi$
$618$	0	0
$619$	4.24435	0.170595	0.0852974	−	0.996356i	$-0.472816\pi$
$619$	4.24435	0.170595	0.0852974	+	0.996356i	$0.472816\pi$
$620$	−0.741324	−0.0297723
$621$	0	0
$622$	13.7501	0.551329
$623$	3.11001	0.124600
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.0583	−1.04150
$627$	0	0
$628$	−3.96348	−0.158160
$629$	−5.37742	−0.214412
$630$	0	0
$631$	5.79749	0.230794	0.115397	−	0.993319i	$-0.463186\pi$
$631$	5.79749	0.230794	0.115397	+	0.993319i	$0.463186\pi$
$632$	−7.19148	−0.286062
$633$	0	0
$634$	−14.9542	−0.593907
$635$	−14.3400	−0.569064
$636$	0	0
$637$	−13.8747	−0.549737
$638$	−8.96128	−0.354781
$639$	0	0
$640$	7.49593	0.296303
$641$	−4.38116	−0.173045	−0.0865227	−	0.996250i	$-0.527575\pi$
$641$	−4.38116	−0.173045	−0.0865227	+	0.996250i	$0.527575\pi$
$642$	0	0
$643$	12.7375	0.502318	0.251159	−	0.967946i	$-0.419188\pi$
$643$	12.7375	0.502318	0.251159	+	0.967946i	$0.419188\pi$
$644$	8.71908	0.343580
$645$	0	0
$646$	8.46131	0.332906
$647$	17.0725	0.671191	0.335595	−	0.942006i	$-0.391063\pi$
$647$	17.0725	0.671191	0.335595	+	0.942006i	$0.391063\pi$
$648$	0	0
$649$	−38.1085	−1.49589
$650$	13.1154	0.514429
$651$	0	0
$652$	7.70768	0.301856
$653$	0.739442	0.0289366	0.0144683	−	0.999895i	$-0.495394\pi$
$653$	0.739442	0.0289366	0.0144683	+	0.999895i	$0.495394\pi$
$654$	0	0
$655$	−9.47983	−0.370408
$656$	13.2558	0.517550
$657$	0	0
$658$	31.4615	1.22650
$659$	−21.8685	−0.851877	−0.425939	−	0.904752i	$-0.640056\pi$
$659$	−21.8685	−0.851877	−0.425939	+	0.904752i	$0.640056\pi$
$660$	0	0
$661$	15.9869	0.621818	0.310909	−	0.950440i	$-0.399366\pi$
$661$	15.9869	0.621818	0.310909	+	0.950440i	$0.399366\pi$
$662$	−56.9338	−2.21280
$663$	0	0
$664$	12.9936	0.504248
$665$	−17.6922	−0.686073
$666$	0	0
$667$	2.53447	0.0981350
$668$	−43.7454	−1.69256
$669$	0	0
$670$	−11.3090	−0.436906
$671$	13.3376	0.514894
$672$	0	0
$673$	19.4731	0.750633	0.375316	−	0.926897i	$-0.377534\pi$
$673$	19.4731	0.750633	0.375316	+	0.926897i	$0.377534\pi$
$674$	−15.2854	−0.588772
$675$	0	0
$676$	62.8795	2.41844
$677$	17.3115	0.665334	0.332667	−	0.943044i	$-0.392051\pi$
$677$	17.3115	0.665334	0.332667	+	0.943044i	$0.392051\pi$
$678$	0	0
$679$	30.2023	1.15906
$680$	−0.475856	−0.0182482
$681$	0	0
$682$	−1.73574	−0.0664648
$683$	−34.0244	−1.30191	−0.650954	−	0.759117i	$-0.725631\pi$
$683$	−34.0244	−1.30191	−0.650954	+	0.759117i	$0.725631\pi$
$684$	0	0
$685$	−6.19886	−0.236846
$686$	−42.5524	−1.62466
$687$	0	0
$688$	−13.7443	−0.523996
$689$	−51.9884	−1.98060
$690$	0	0
$691$	40.5782	1.54367	0.771834	−	0.635824i	$-0.219340\pi$
$691$	40.5782	1.54367	0.771834	+	0.635824i	$0.219340\pi$
$692$	−49.2487	−1.87215
$693$	0	0
$694$	−20.7126	−0.786238
$695$	−20.1745	−0.765264
$696$	0	0
$697$	−2.27613	−0.0862145
$698$	−4.59774	−0.174027
$699$	0	0
$700$	5.36165	0.202651
$701$	0.341079	0.0128824	0.00644118	−	0.999979i	$-0.497950\pi$
$701$	0.341079	0.0128824	0.00644118	+	0.999979i	$0.497950\pi$
$702$	0	0
$703$	−88.0885	−3.32232
$704$	30.3331	1.14322
$705$	0	0
$706$	13.4215	0.505126
$707$	29.1367	1.09580
$708$	0	0
$709$	−40.8883	−1.53559	−0.767796	−	0.640694i	$-0.778647\pi$
$709$	−40.8883	−1.53559	−0.767796	+	0.640694i	$0.778647\pi$
$710$	−35.3499	−1.32666
$711$	0	0
$712$	1.37025	0.0513524
$713$	0.490909	0.0183847
$714$	0	0
$715$	16.9246	0.632946
$716$	−61.3735	−2.29364
$717$	0	0
$718$	−33.7959	−1.26125
$719$	23.3315	0.870118	0.435059	−	0.900402i	$-0.356728\pi$
$719$	23.3315	0.870118	0.435059	+	0.900402i	$0.356728\pi$
$720$	0	0
$721$	−3.24324	−0.120785
$722$	98.4998	3.66578
$723$	0	0
$724$	−37.7448	−1.40278
$725$	1.55853	0.0578823
$726$	0	0
$727$	31.9707	1.18573	0.592864	−	0.805302i	$-0.297997\pi$
$727$	31.9707	1.18573	0.592864	+	0.805302i	$0.297997\pi$
$728$	13.0497	0.483655
$729$	0	0
$730$	−22.7446	−0.841816
$731$	2.36001	0.0872883
$732$	0	0
$733$	−30.8033	−1.13775	−0.568873	−	0.822425i	$-0.692620\pi$
$733$	−30.8033	−1.13775	−0.568873	+	0.822425i	$0.692620\pi$
$734$	20.5076	0.756951
$735$	0	0
$736$	−13.0177	−0.479840
$737$	−14.5936	−0.537563
$738$	0	0
$739$	−27.1312	−0.998036	−0.499018	−	0.866592i	$-0.666306\pi$
$739$	−27.1312	−0.998036	−0.499018	+	0.866592i	$0.666306\pi$
$740$	26.6954	0.981343
$741$	0	0
$742$	−38.5622	−1.41566
$743$	−43.3260	−1.58948	−0.794739	−	0.606951i	$-0.792392\pi$
$743$	−43.3260	−1.58948	−0.794739	+	0.606951i	$0.792392\pi$
$744$	0	0
$745$	−5.52613	−0.202462
$746$	0.386683	0.0141575
$747$	0	0
$748$	−3.30896	−0.120988
$749$	−2.18333	−0.0797772
$750$	0	0
$751$	5.88157	0.214622	0.107311	−	0.994226i	$-0.465776\pi$
$751$	5.88157	0.214622	0.107311	+	0.994226i	$0.465776\pi$
$752$	−19.6664	−0.717160
$753$	0	0
$754$	20.4408	0.744409
$755$	14.3017	0.520490
$756$	0	0
$757$	25.1745	0.914981	0.457491	−	0.889214i	$-0.348748\pi$
$757$	25.1745	0.914981	0.457491	+	0.889214i	$0.348748\pi$
$758$	−10.7478	−0.390379
$759$	0	0
$760$	−7.79508	−0.282757
$761$	−26.2482	−0.951495	−0.475747	−	0.879582i	$-0.657822\pi$
$761$	−26.2482	−0.951495	−0.475747	+	0.879582i	$0.657822\pi$
$762$	0	0
$763$	−24.1494	−0.874266
$764$	−16.8056	−0.608005
$765$	0	0
$766$	1.95198	0.0705279
$767$	86.9258	3.13871
$768$	0	0
$769$	13.9109	0.501641	0.250820	−	0.968034i	$-0.419300\pi$
$769$	13.9109	0.501641	0.250820	+	0.968034i	$0.419300\pi$
$770$	12.5538	0.452407
$771$	0	0
$772$	−56.6268	−2.03804
$773$	24.0720	0.865810	0.432905	−	0.901439i	$-0.357488\pi$
$773$	24.0720	0.865810	0.432905	+	0.901439i	$0.357488\pi$
$774$	0	0
$775$	0.301876	0.0108437
$776$	13.3070	0.477693
$777$	0	0
$778$	52.6476	1.88751
$779$	−37.2857	−1.33590
$780$	0	0
$781$	−45.6169	−1.63230
$782$	1.69804	0.0607217
$783$	0	0
$784$	6.43318	0.229756
$785$	1.61398	0.0576053
$786$	0	0
$787$	23.4427	0.835641	0.417820	−	0.908530i	$-0.362794\pi$
$787$	23.4427	0.835641	0.417820	+	0.908530i	$0.362794\pi$
$788$	−20.7680	−0.739830
$789$	0	0
$790$	15.7804	0.561442
$791$	11.9233	0.423943
$792$	0	0
$793$	−30.4233	−1.08036
$794$	−46.4187	−1.64734
$795$	0	0
$796$	22.1186	0.783972
$797$	3.09099	0.109489	0.0547443	−	0.998500i	$-0.482566\pi$
$797$	3.09099	0.109489	0.0547443	+	0.998500i	$0.482566\pi$
$798$	0	0
$799$	3.37689	0.119466
$800$	−8.00504	−0.283021
$801$	0	0
$802$	−46.7339	−1.65023
$803$	−29.3505	−1.03576
$804$	0	0
$805$	−3.55052	−0.125139
$806$	3.95924	0.139458
$807$	0	0
$808$	12.8375	0.451620
$809$	−9.38057	−0.329803	−0.164902	−	0.986310i	$-0.552731\pi$
$809$	−9.38057	−0.329803	−0.164902	+	0.986310i	$0.552731\pi$
$810$	0	0
$811$	−36.0866	−1.26717	−0.633585	−	0.773673i	$-0.718417\pi$
$811$	−36.0866	−1.26717	−0.633585	+	0.773673i	$0.718417\pi$
$812$	8.35629	0.293248
$813$	0	0
$814$	62.5047	2.19079
$815$	−3.13866	−0.109943
$816$	0	0
$817$	38.6598	1.35253
$818$	13.0078	0.454806
$819$	0	0
$820$	11.2995	0.394596
$821$	−5.91128	−0.206305	−0.103153	−	0.994666i	$-0.532893\pi$
$821$	−5.91128	−0.206305	−0.103153	+	0.994666i	$0.532893\pi$
$822$	0	0
$823$	38.6739	1.34809	0.674044	−	0.738691i	$-0.264556\pi$
$823$	38.6739	1.34809	0.674044	+	0.738691i	$0.264556\pi$
$824$	−1.42896	−0.0497800
$825$	0	0
$826$	64.4769	2.24344
$827$	1.46011	0.0507731	0.0253866	−	0.999678i	$-0.491918\pi$
$827$	1.46011	0.0507731	0.0253866	+	0.999678i	$0.491918\pi$
$828$	0	0
$829$	39.1733	1.36055	0.680273	−	0.732959i	$-0.261861\pi$
$829$	39.1733	1.36055	0.680273	+	0.732959i	$0.261861\pi$
$830$	−28.5121	−0.989668
$831$	0	0
$832$	−69.1900	−2.39873
$833$	−1.10463	−0.0382733
$834$	0	0
$835$	17.8137	0.616467
$836$	−54.2047	−1.87471
$837$	0	0
$838$	−3.96750	−0.137055
$839$	−46.1007	−1.59157	−0.795787	−	0.605576i	$-0.792943\pi$
$839$	−46.1007	−1.59157	−0.795787	+	0.605576i	$0.792943\pi$
$840$	0	0
$841$	−26.5710	−0.916241
$842$	−53.2525	−1.83520
$843$	0	0
$844$	13.3968	0.461138
$845$	−25.6053	−0.880849
$846$	0	0
$847$	−7.81675	−0.268587
$848$	24.1050	0.827769
$849$	0	0
$850$	1.04418	0.0358151
$851$	−17.6778	−0.605989
$852$	0	0
$853$	−19.0735	−0.653063	−0.326531	−	0.945186i	$-0.605880\pi$
$853$	−19.0735	−0.653063	−0.326531	+	0.945186i	$0.605880\pi$
$854$	−22.5663	−0.772204
$855$	0	0
$856$	−0.961964	−0.0328792
$857$	−4.48921	−0.153348	−0.0766742	−	0.997056i	$-0.524430\pi$
$857$	−4.48921	−0.153348	−0.0766742	+	0.997056i	$0.524430\pi$
$858$	0	0
$859$	−27.0246	−0.922067	−0.461034	−	0.887383i	$-0.652521\pi$
$859$	−27.0246	−0.922067	−0.461034	+	0.887383i	$0.652521\pi$
$860$	−11.7159	−0.399510
$861$	0	0
$862$	40.0336	1.36355
$863$	6.85364	0.233301	0.116650	−	0.993173i	$-0.462784\pi$
$863$	6.85364	0.233301	0.116650	+	0.993173i	$0.462784\pi$
$864$	0	0
$865$	20.0547	0.681879
$866$	46.2590	1.57195
$867$	0	0
$868$	1.61856	0.0549374
$869$	20.3636	0.690789
$870$	0	0
$871$	33.2882	1.12793
$872$	−10.6401	−0.360319
$873$	0	0
$874$	27.8159	0.940886
$875$	−2.18333	−0.0738101
$876$	0	0
$877$	−24.0995	−0.813783	−0.406891	−	0.913477i	$-0.633387\pi$
$877$	−24.0995	−0.813783	−0.406891	+	0.913477i	$0.633387\pi$
$878$	−35.5446	−1.19957
$879$	0	0
$880$	−7.84731	−0.264533
$881$	19.8077	0.667337	0.333669	−	0.942690i	$-0.391713\pi$
$881$	19.8077	0.667337	0.333669	+	0.942690i	$0.391713\pi$
$882$	0	0
$883$	32.9581	1.10913	0.554565	−	0.832141i	$-0.312885\pi$
$883$	32.9581	1.10913	0.554565	+	0.832141i	$0.312885\pi$
$884$	7.54778	0.253859
$885$	0	0
$886$	15.6827	0.526871
$887$	8.13648	0.273196	0.136598	−	0.990627i	$-0.456383\pi$
$887$	8.13648	0.273196	0.136598	+	0.990627i	$0.456383\pi$
$888$	0	0
$889$	31.3089	1.05007
$890$	−3.00678	−0.100787
$891$	0	0
$892$	−54.2743	−1.81724
$893$	55.3175	1.85113
$894$	0	0
$895$	24.9921	0.835392
$896$	−16.3661	−0.546753
$897$	0	0
$898$	22.0391	0.735455
$899$	0.470483	0.0156915
$900$	0	0
$901$	−4.13904	−0.137891
$902$	26.4567	0.880911
$903$	0	0
$904$	5.25334	0.174723
$905$	15.3702	0.510922
$906$	0	0
$907$	10.6235	0.352746	0.176373	−	0.984323i	$-0.443563\pi$
$907$	10.6235	0.352746	0.176373	+	0.984323i	$0.443563\pi$
$908$	−21.4254	−0.711026
$909$	0	0
$910$	−28.6353	−0.949252
$911$	36.6249	1.21344	0.606719	−	0.794916i	$-0.292485\pi$
$911$	36.6249	1.21344	0.606719	+	0.794916i	$0.292485\pi$
$912$	0	0
$913$	−36.7930	−1.21767
$914$	66.9794	2.21548
$915$	0	0
$916$	51.4807	1.70097
$917$	20.6976	0.683496
$918$	0	0
$919$	20.8132	0.686563	0.343281	−	0.939233i	$-0.388462\pi$
$919$	20.8132	0.686563	0.343281	+	0.939233i	$0.388462\pi$
$920$	−1.56434	−0.0515747
$921$	0	0
$922$	70.6191	2.32572
$923$	104.053	3.42493
$924$	0	0
$925$	−10.8707	−0.357426
$926$	73.6506	2.42031
$927$	0	0
$928$	−12.4761	−0.409548
$929$	13.3716	0.438708	0.219354	−	0.975645i	$-0.429605\pi$
$929$	13.3716	0.438708	0.219354	+	0.975645i	$0.429605\pi$
$930$	0	0
$931$	−18.0952	−0.593046
$932$	−45.2535	−1.48233
$933$	0	0
$934$	60.7986	1.98939
$935$	1.34745	0.0440664
$936$	0	0
$937$	−1.78815	−0.0584164	−0.0292082	−	0.999573i	$-0.509299\pi$
$937$	−1.78815	−0.0584164	−0.0292082	+	0.999573i	$0.509299\pi$
$938$	24.6914	0.806202
$939$	0	0
$940$	−16.7641	−0.546784
$941$	−41.1918	−1.34281	−0.671407	−	0.741089i	$-0.734310\pi$
$941$	−41.1918	−1.34281	−0.671407	+	0.741089i	$0.734310\pi$
$942$	0	0
$943$	−7.48259	−0.243667
$944$	−40.3041	−1.31179
$945$	0	0
$946$	−27.4317	−0.891882
$947$	0.918602	0.0298505	0.0149253	−	0.999889i	$-0.495249\pi$
$947$	0.918602	0.0298505	0.0149253	+	0.999889i	$0.495249\pi$
$948$	0	0
$949$	66.9488	2.17325
$950$	17.1049	0.554956
$951$	0	0
$952$	1.03895	0.0336726
$953$	14.1048	0.456901	0.228450	−	0.973556i	$-0.426634\pi$
$953$	14.1048	0.456901	0.228450	+	0.973556i	$0.426634\pi$
$954$	0	0
$955$	6.84344	0.221449
$956$	−67.0773	−2.16943
$957$	0	0
$958$	4.48602	0.144937
$959$	13.5342	0.437041
$960$	0	0
$961$	−30.9089	−0.997060
$962$	−142.574	−4.59676
$963$	0	0
$964$	45.9613	1.48031
$965$	23.0591	0.742300
$966$	0	0
$967$	−39.7962	−1.27976	−0.639880	−	0.768475i	$-0.721016\pi$
$967$	−39.7962	−1.27976	−0.639880	+	0.768475i	$0.721016\pi$
$968$	−3.44402	−0.110695
$969$	0	0
$970$	−29.1998	−0.937549
$971$	36.7806	1.18034	0.590172	−	0.807277i	$-0.299060\pi$
$971$	36.7806	1.18034	0.590172	+	0.807277i	$0.299060\pi$
$972$	0	0
$973$	44.0477	1.41210
$974$	48.0612	1.53998
$975$	0	0
$976$	14.1061	0.451525
$977$	10.1568	0.324946	0.162473	−	0.986713i	$-0.448053\pi$
$977$	10.1568	0.324946	0.162473	+	0.986713i	$0.448053\pi$
$978$	0	0
$979$	−3.88006	−0.124007
$980$	5.48379	0.175173
$981$	0	0
$982$	−4.62393	−0.147555
$983$	−54.4941	−1.73809	−0.869046	−	0.494731i	$-0.835267\pi$
$983$	−54.4941	−1.73809	−0.869046	+	0.494731i	$0.835267\pi$
$984$	0	0
$985$	8.45699	0.269462
$986$	1.62739	0.0518265
$987$	0	0
$988$	123.642	3.93356
$989$	7.75835	0.246701
$990$	0	0
$991$	−23.6607	−0.751606	−0.375803	−	0.926699i	$-0.622633\pi$
$991$	−23.6607	−0.751606	−0.375803	+	0.926699i	$0.622633\pi$
$992$	−2.41653	−0.0767250
$993$	0	0
$994$	77.1806	2.44802
$995$	−9.00696	−0.285540
$996$	0	0
$997$	8.36113	0.264800	0.132400	−	0.991196i	$-0.457732\pi$
$997$	8.36113	0.264800	0.132400	+	0.991196i	$0.457732\pi$
$998$	−53.2284	−1.68491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4815.2.a.u.1.11		12
3.2	odd	2		1605.2.a.n.1.2	✓	12
15.14	odd	2		8025.2.a.bf.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.n.1.2	✓	12	3.2	odd	2
4815.2.a.u.1.11		12	1.1	even	1	trivial
8025.2.a.bf.1.11		12	15.14	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:	\( N \)	\(=\)	\( 4815 = 3^{2} \cdot 5 \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4815.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11086 q^{2} +2.45572 q^{4} -1.00000 q^{5} +2.18333 q^{7} +0.961964 q^{8} +O(q^{10})\)	\(q+2.11086 q^{2} +2.45572 q^{4} -1.00000 q^{5} +2.18333 q^{7} +0.961964 q^{8} -2.11086 q^{10} -2.72393 q^{11} +6.21332 q^{13} +4.60870 q^{14} -2.88087 q^{16} +0.494671 q^{17} +8.10330 q^{19} -2.45572 q^{20} -5.74983 q^{22} +1.62619 q^{23} +1.00000 q^{25} +13.1154 q^{26} +5.36165 q^{28} +1.55853 q^{29} +0.301876 q^{31} -8.00504 q^{32} +1.04418 q^{34} -2.18333 q^{35} -10.8707 q^{37} +17.1049 q^{38} -0.961964 q^{40} -4.60130 q^{41} +4.77087 q^{43} -6.68922 q^{44} +3.43266 q^{46} +6.82654 q^{47} -2.23307 q^{49} +2.11086 q^{50} +15.2582 q^{52} -8.36725 q^{53} +2.72393 q^{55} +2.10029 q^{56} +3.28983 q^{58} +13.9902 q^{59} -4.89646 q^{61} +0.637218 q^{62} -11.1358 q^{64} -6.21332 q^{65} +5.35756 q^{67} +1.21478 q^{68} -4.60870 q^{70} +16.7467 q^{71} +10.7751 q^{73} -22.9465 q^{74} +19.8994 q^{76} -5.94725 q^{77} -7.47583 q^{79} +2.88087 q^{80} -9.71268 q^{82} +13.5073 q^{83} -0.494671 q^{85} +10.0706 q^{86} -2.62032 q^{88} +1.42443 q^{89} +13.5657 q^{91} +3.99348 q^{92} +14.4099 q^{94} -8.10330 q^{95} +13.8331 q^{97} -4.71368 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} - 12 q^{5} + 7 q^{7} - 3 q^{8} + 3 q^{10} - 4 q^{11} + 13 q^{13} - 4 q^{14} + 13 q^{16} + 4 q^{17} + 14 q^{19} - 15 q^{20} + 15 q^{22} - 11 q^{23} + 12 q^{25} + 8 q^{26} + 16 q^{28} + 7 q^{29} + 4 q^{31} - 4 q^{32} + q^{34} - 7 q^{35} + 24 q^{37} + 11 q^{38} + 3 q^{40} - 13 q^{41} + 25 q^{43} - 10 q^{44} - 22 q^{46} - 19 q^{47} + 9 q^{49} - 3 q^{50} + 20 q^{52} - 11 q^{53} + 4 q^{55} + 37 q^{56} - 2 q^{58} - 8 q^{59} + 7 q^{61} + 11 q^{62} - 19 q^{64} - 13 q^{65} + 33 q^{67} + 24 q^{68} + 4 q^{70} + 34 q^{73} + 27 q^{74} - 9 q^{76} + 29 q^{77} - 13 q^{80} + q^{82} + 24 q^{83} - 4 q^{85} + 36 q^{86} - 6 q^{88} + 10 q^{89} + 30 q^{91} + 28 q^{92} - 8 q^{94} - 14 q^{95} + 16 q^{97} + 36 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 - 12 * q^5 + 7 * q^7 - 3 * q^8 + 3 * q^10 - 4 * q^11 + 13 * q^13 - 4 * q^14 + 13 * q^16 + 4 * q^17 + 14 * q^19 - 15 * q^20 + 15 * q^22 - 11 * q^23 + 12 * q^25 + 8 * q^26 + 16 * q^28 + 7 * q^29 + 4 * q^31 - 4 * q^32 + q^34 - 7 * q^35 + 24 * q^37 + 11 * q^38 + 3 * q^40 - 13 * q^41 + 25 * q^43 - 10 * q^44 - 22 * q^46 - 19 * q^47 + 9 * q^49 - 3 * q^50 + 20 * q^52 - 11 * q^53 + 4 * q^55 + 37 * q^56 - 2 * q^58 - 8 * q^59 + 7 * q^61 + 11 * q^62 - 19 * q^64 - 13 * q^65 + 33 * q^67 + 24 * q^68 + 4 * q^70 + 34 * q^73 + 27 * q^74 - 9 * q^76 + 29 * q^77 - 13 * q^80 + q^82 + 24 * q^83 - 4 * q^85 + 36 * q^86 - 6 * q^88 + 10 * q^89 + 30 * q^91 + 28 * q^92 - 8 * q^94 - 14 * q^95 + 16 * q^97 + 36 * q^98

Introduction

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