LMFDB - Embedded newform 2151.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1767625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 11x^{2} + x - 1 $ x^6 - 7x^4 - x^3 + 11x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30642$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +O(q^{10})$	\(q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +3.02110 q^{10} +1.17944 q^{11} +0.346441 q^{13} +4.32613 q^{14} -1.41230 q^{16} +3.26277 q^{17} -1.31974 q^{19} +2.56868 q^{20} -1.24031 q^{22} +9.10667 q^{23} +3.25325 q^{25} -0.364319 q^{26} +3.67827 q^{28} +0.639311 q^{29} -1.52951 q^{31} -4.60178 q^{32} -3.43115 q^{34} +11.8184 q^{35} +1.40670 q^{37} +1.38785 q^{38} -8.74345 q^{40} +2.17524 q^{41} -1.97537 q^{43} -1.05456 q^{44} -9.57663 q^{46} +1.69565 q^{47} +9.92360 q^{49} -3.42114 q^{50} -0.309761 q^{52} +0.0493538 q^{53} -3.38835 q^{55} -12.5204 q^{56} -0.672304 q^{58} -0.806525 q^{59} -1.25891 q^{61} +1.60844 q^{62} +7.66385 q^{64} -0.995271 q^{65} +4.01099 q^{67} -2.91732 q^{68} -12.4283 q^{70} -0.361936 q^{71} -11.9650 q^{73} -1.47930 q^{74} +1.18001 q^{76} -4.85201 q^{77} +4.27659 q^{79} +4.05731 q^{80} -2.28749 q^{82} -14.0559 q^{83} -9.37343 q^{85} +2.07731 q^{86} +3.58960 q^{88} +10.6550 q^{89} -1.42520 q^{91} -8.14249 q^{92} -1.78316 q^{94} +3.79142 q^{95} -10.2134 q^{97} -10.4357 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * 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 $

$2$	−1.05161	−0.743598	−0.371799	−	0.928313i	$-0.621259\pi$
$2$	−1.05161	−0.743598	−0.371799	+	0.928313i	$0.621259\pi$
$3$	0	0
$3$	0	0
$4$	−0.894124	−0.447062
$5$	−2.87285	−1.28478	−0.642388	−	0.766380i	$-0.722056\pi$
$5$	−2.87285	−1.28478	−0.642388	+	0.766380i	$0.722056\pi$
$6$	0	0
$7$	−4.11383	−1.55488	−0.777441	−	0.628956i	$-0.783483\pi$
$7$	−4.11383	−1.55488	−0.777441	+	0.628956i	$0.783483\pi$
$8$	3.04348	1.07603
$9$	0	0
$10$	3.02110	0.955357
$11$	1.17944	0.355614	0.177807	−	0.984065i	$-0.443100\pi$
$11$	1.17944	0.355614	0.177807	+	0.984065i	$0.443100\pi$
$12$	0	0
$13$	0.346441	0.0960854	0.0480427	−	0.998845i	$-0.484702\pi$
$13$	0.346441	0.0960854	0.0480427	+	0.998845i	$0.484702\pi$
$14$	4.32613	1.15621
$15$	0	0
$16$	−1.41230	−0.353074
$17$	3.26277	0.791337	0.395669	−	0.918393i	$-0.370513\pi$
$17$	3.26277	0.791337	0.395669	+	0.918393i	$0.370513\pi$
$18$	0	0
$19$	−1.31974	−0.302770	−0.151385	−	0.988475i	$-0.548373\pi$
$19$	−1.31974	−0.302770	−0.151385	+	0.988475i	$0.548373\pi$
$20$	2.56868	0.574374
$21$	0	0
$22$	−1.24031	−0.264434
$23$	9.10667	1.89887	0.949436	−	0.313960i	$-0.101656\pi$
$23$	9.10667	1.89887	0.949436	+	0.313960i	$0.101656\pi$
$24$	0	0
$25$	3.25325	0.650649
$26$	−0.364319	−0.0714489
$27$	0	0
$28$	3.67827	0.695128
$29$	0.639311	0.118717	0.0593585	−	0.998237i	$-0.481094\pi$
$29$	0.639311	0.118717	0.0593585	+	0.998237i	$0.481094\pi$
$30$	0	0
$31$	−1.52951	−0.274708	−0.137354	−	0.990522i	$-0.543860\pi$
$31$	−1.52951	−0.274708	−0.137354	+	0.990522i	$0.543860\pi$
$32$	−4.60178	−0.813487
$33$	0	0
$34$	−3.43115	−0.588437
$35$	11.8184	1.99767
$36$	0	0
$37$	1.40670	0.231260	0.115630	−	0.993292i	$-0.463111\pi$
$37$	1.40670	0.231260	0.115630	+	0.993292i	$0.463111\pi$
$38$	1.38785	0.225139
$39$	0	0
$40$	−8.74345	−1.38246
$41$	2.17524	0.339715	0.169857	−	0.985469i	$-0.445669\pi$
$41$	2.17524	0.339715	0.169857	+	0.985469i	$0.445669\pi$
$42$	0	0
$43$	−1.97537	−0.301241	−0.150621	−	0.988592i	$-0.548127\pi$
$43$	−1.97537	−0.301241	−0.150621	+	0.988592i	$0.548127\pi$
$44$	−1.05456	−0.158982
$45$	0	0
$46$	−9.57663	−1.41200
$47$	1.69565	0.247336	0.123668	−	0.992324i	$-0.460534\pi$
$47$	1.69565	0.247336	0.123668	+	0.992324i	$0.460534\pi$
$48$	0	0
$49$	9.92360	1.41766
$50$	−3.42114	−0.483822
$51$	0	0
$52$	−0.309761	−0.0429561
$53$	0.0493538	0.00677926	0.00338963	−	0.999994i	$-0.498921\pi$
$53$	0.0493538	0.00677926	0.00338963	+	0.999994i	$0.498921\pi$
$54$	0	0
$55$	−3.38835	−0.456884
$56$	−12.5204	−1.67310
$57$	0	0
$58$	−0.672304	−0.0882778
$59$	−0.806525	−0.105001	−0.0525003	−	0.998621i	$-0.516719\pi$
$59$	−0.806525	−0.105001	−0.0525003	+	0.998621i	$0.516719\pi$
$60$	0	0
$61$	−1.25891	−0.161187	−0.0805935	−	0.996747i	$-0.525682\pi$
$61$	−1.25891	−0.161187	−0.0805935	+	0.996747i	$0.525682\pi$
$62$	1.60844	0.204272
$63$	0	0
$64$	7.66385	0.957982
$65$	−0.995271	−0.123448
$66$	0	0
$67$	4.01099	0.490021	0.245010	−	0.969520i	$-0.421209\pi$
$67$	4.01099	0.490021	0.245010	+	0.969520i	$0.421209\pi$
$68$	−2.91732	−0.353777
$69$	0	0
$70$	−12.4283	−1.48547
$71$	−0.361936	−0.0429539	−0.0214770	−	0.999769i	$-0.506837\pi$
$71$	−0.361936	−0.0429539	−0.0214770	+	0.999769i	$0.506837\pi$
$72$	0	0
$73$	−11.9650	−1.40040	−0.700198	−	0.713949i	$-0.746905\pi$
$73$	−11.9650	−1.40040	−0.700198	+	0.713949i	$0.746905\pi$
$74$	−1.47930	−0.171965
$75$	0	0
$76$	1.18001	0.135357
$77$	−4.85201	−0.552938
$78$	0	0
$79$	4.27659	0.481154	0.240577	−	0.970630i	$-0.422663\pi$
$79$	4.27659	0.481154	0.240577	+	0.970630i	$0.422663\pi$
$80$	4.05731	0.453621
$81$	0	0
$82$	−2.28749	−0.252611
$83$	−14.0559	−1.54284	−0.771418	−	0.636329i	$-0.780452\pi$
$83$	−14.0559	−1.54284	−0.771418	+	0.636329i	$0.780452\pi$
$84$	0	0
$85$	−9.37343	−1.01669
$86$	2.07731	0.224003
$87$	0	0
$88$	3.58960	0.382652
$89$	10.6550	1.12943	0.564714	−	0.825287i	$-0.308987\pi$
$89$	10.6550	1.12943	0.564714	+	0.825287i	$0.308987\pi$
$90$	0	0
$91$	−1.42520	−0.149401
$92$	−8.14249	−0.848913
$93$	0	0
$94$	−1.78316	−0.183919
$95$	3.79142	0.388991
$96$	0	0
$97$	−10.2134	−1.03701	−0.518505	−	0.855075i	$-0.673511\pi$
$97$	−10.2134	−1.03701	−0.518505	+	0.855075i	$0.673511\pi$
$98$	−10.4357	−1.05417
$99$	0	0
$100$	−2.90881	−0.290881
$101$	−13.3232	−1.32571	−0.662855	−	0.748748i	$-0.730655\pi$
$101$	−13.3232	−1.32571	−0.662855	+	0.748748i	$0.730655\pi$
$102$	0	0
$103$	−16.3775	−1.61372	−0.806860	−	0.590743i	$-0.798835\pi$
$103$	−16.3775	−1.61372	−0.806860	+	0.590743i	$0.798835\pi$
$104$	1.05439	0.103391
$105$	0	0
$106$	−0.0519008	−0.00504105
$107$	10.8242	1.04641	0.523205	−	0.852207i	$-0.324736\pi$
$107$	10.8242	1.04641	0.523205	+	0.852207i	$0.324736\pi$
$108$	0	0
$109$	17.1432	1.64202	0.821012	−	0.570910i	$-0.193410\pi$
$109$	17.1432	1.64202	0.821012	+	0.570910i	$0.193410\pi$
$110$	3.56321	0.339738
$111$	0	0
$112$	5.80994	0.548988
$113$	−13.1922	−1.24102	−0.620511	−	0.784198i	$-0.713075\pi$
$113$	−13.1922	−1.24102	−0.620511	+	0.784198i	$0.713075\pi$
$114$	0	0
$115$	−26.1621	−2.43963
$116$	−0.571623	−0.0530739
$117$	0	0
$118$	0.848147	0.0780782
$119$	−13.4225	−1.23044
$120$	0	0
$121$	−9.60892	−0.873539
$122$	1.32388	0.119858
$123$	0	0
$124$	1.36757	0.122812
$125$	5.01815	0.448837
$126$	0	0
$127$	13.0405	1.15716	0.578580	−	0.815625i	$-0.303607\pi$
$127$	13.0405	1.15716	0.578580	+	0.815625i	$0.303607\pi$
$128$	1.14420	0.101134
$129$	0	0
$130$	1.04663	0.0917959
$131$	−20.5593	−1.79628	−0.898138	−	0.439714i	$-0.855080\pi$
$131$	−20.5593	−1.79628	−0.898138	+	0.439714i	$0.855080\pi$
$132$	0	0
$133$	5.42920	0.470771
$134$	−4.21799	−0.364378
$135$	0	0
$136$	9.93016	0.851504
$137$	−20.2062	−1.72633	−0.863165	−	0.504923i	$-0.831521\pi$
$137$	−20.2062	−1.72633	−0.863165	+	0.504923i	$0.831521\pi$
$138$	0	0
$139$	14.9894	1.27139	0.635694	−	0.771941i	$-0.280714\pi$
$139$	14.9894	1.27139	0.635694	+	0.771941i	$0.280714\pi$
$140$	−10.5671	−0.893084
$141$	0	0
$142$	0.380614	0.0319404
$143$	0.408606	0.0341693
$144$	0	0
$145$	−1.83664	−0.152525
$146$	12.5825	1.04133
$147$	0	0
$148$	−1.25776	−0.103388
$149$	−5.24611	−0.429778	−0.214889	−	0.976638i	$-0.568939\pi$
$149$	−5.24611	−0.429778	−0.214889	+	0.976638i	$0.568939\pi$
$150$	0	0
$151$	5.59004	0.454911	0.227456	−	0.973788i	$-0.426959\pi$
$151$	5.59004	0.454911	0.227456	+	0.973788i	$0.426959\pi$
$152$	−4.01661	−0.325790
$153$	0	0
$154$	5.10241	0.411164
$155$	4.39405	0.352938
$156$	0	0
$157$	17.6356	1.40747	0.703736	−	0.710461i	$-0.251514\pi$
$157$	17.6356	1.40747	0.703736	+	0.710461i	$0.251514\pi$
$158$	−4.49729	−0.357785
$159$	0	0
$160$	13.2202	1.04515
$161$	−37.4633	−2.95252
$162$	0	0
$163$	−12.8967	−1.01015	−0.505073	−	0.863077i	$-0.668534\pi$
$163$	−12.8967	−1.01015	−0.505073	+	0.863077i	$0.668534\pi$
$164$	−1.94493	−0.151874
$165$	0	0
$166$	14.7813	1.14725
$167$	4.91384	0.380244	0.190122	−	0.981760i	$-0.439112\pi$
$167$	4.91384	0.380244	0.190122	+	0.981760i	$0.439112\pi$
$168$	0	0
$169$	−12.8800	−0.990768
$170$	9.85716	0.756010
$171$	0	0
$172$	1.76623	0.134674
$173$	15.5398	1.18147	0.590736	−	0.806865i	$-0.298837\pi$
$173$	15.5398	1.18147	0.590736	+	0.806865i	$0.298837\pi$
$174$	0	0
$175$	−13.3833	−1.01168
$176$	−1.66572	−0.125558
$177$	0	0
$178$	−11.2049	−0.839840
$179$	16.9641	1.26796	0.633979	−	0.773350i	$-0.281420\pi$
$179$	16.9641	1.26796	0.633979	+	0.773350i	$0.281420\pi$
$180$	0	0
$181$	3.30222	0.245452	0.122726	−	0.992441i	$-0.460836\pi$
$181$	3.30222	0.245452	0.122726	+	0.992441i	$0.460836\pi$
$182$	1.49875	0.111095
$183$	0	0
$184$	27.7160	2.04325
$185$	−4.04124	−0.297118
$186$	0	0
$187$	3.84823	0.281411
$188$	−1.51612	−0.110575
$189$	0	0
$190$	−3.98708	−0.289253
$191$	12.2407	0.885704	0.442852	−	0.896595i	$-0.353967\pi$
$191$	12.2407	0.885704	0.442852	+	0.896595i	$0.353967\pi$
$192$	0	0
$193$	−15.3502	−1.10493	−0.552465	−	0.833536i	$-0.686313\pi$
$193$	−15.3502	−1.10493	−0.552465	+	0.833536i	$0.686313\pi$
$194$	10.7404	0.771119
$195$	0	0
$196$	−8.87293	−0.633781
$197$	20.6755	1.47307	0.736533	−	0.676402i	$-0.236462\pi$
$197$	20.6755	1.47307	0.736533	+	0.676402i	$0.236462\pi$
$198$	0	0
$199$	−24.9536	−1.76892	−0.884459	−	0.466619i	$-0.845472\pi$
$199$	−24.9536	−1.76892	−0.884459	+	0.466619i	$0.845472\pi$
$200$	9.90119	0.700120
$201$	0	0
$202$	14.0108	0.985795
$203$	−2.63002	−0.184591
$204$	0	0
$205$	−6.24912	−0.436457
$206$	17.2226	1.19996
$207$	0	0
$208$	−0.489277	−0.0339252
$209$	−1.55655	−0.107669
$210$	0	0
$211$	−3.86876	−0.266336	−0.133168	−	0.991093i	$-0.542515\pi$
$211$	−3.86876	−0.266336	−0.133168	+	0.991093i	$0.542515\pi$
$212$	−0.0441284	−0.00303075
$213$	0	0
$214$	−11.3828	−0.778109
$215$	5.67494	0.387028
$216$	0	0
$217$	6.29214	0.427139
$218$	−18.0279	−1.22101
$219$	0	0
$220$	3.02960	0.204256
$221$	1.13036	0.0760360
$222$	0	0
$223$	9.50369	0.636414	0.318207	−	0.948021i	$-0.396919\pi$
$223$	9.50369	0.636414	0.318207	+	0.948021i	$0.396919\pi$
$224$	18.9309	1.26488
$225$	0	0
$226$	13.8730	0.922821
$227$	−7.95147	−0.527758	−0.263879	−	0.964556i	$-0.585002\pi$
$227$	−7.95147	−0.527758	−0.263879	+	0.964556i	$0.585002\pi$
$228$	0	0
$229$	−11.1993	−0.740070	−0.370035	−	0.929018i	$-0.620654\pi$
$229$	−11.1993	−0.740070	−0.370035	+	0.929018i	$0.620654\pi$
$230$	27.5122	1.81410
$231$	0	0
$232$	1.94573	0.127743
$233$	15.6891	1.02783	0.513914	−	0.857842i	$-0.328195\pi$
$233$	15.6891	1.02783	0.513914	+	0.857842i	$0.328195\pi$
$234$	0	0
$235$	−4.87135	−0.317772
$236$	0.721133	0.0469418
$237$	0	0
$238$	14.1152	0.914950
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−25.5643	−1.64674	−0.823371	−	0.567503i	$-0.807910\pi$
$241$	−25.5643	−1.64674	−0.823371	+	0.567503i	$0.807910\pi$
$242$	10.1048	0.649562
$243$	0	0
$244$	1.12562	0.0720605
$245$	−28.5090	−1.82137
$246$	0	0
$247$	−0.457213	−0.0290917
$248$	−4.65503	−0.295595
$249$	0	0
$250$	−5.27712	−0.333754
$251$	−16.7833	−1.05935	−0.529676	−	0.848200i	$-0.677686\pi$
$251$	−16.7833	−1.05935	−0.529676	+	0.848200i	$0.677686\pi$
$252$	0	0
$253$	10.7408	0.675266
$254$	−13.7135	−0.860462
$255$	0	0
$256$	−16.5310	−1.03318
$257$	6.98956	0.435996	0.217998	−	0.975949i	$-0.430047\pi$
$257$	6.98956	0.435996	0.217998	+	0.975949i	$0.430047\pi$
$258$	0	0
$259$	−5.78693	−0.359582
$260$	0.889896	0.0551890
$261$	0	0
$262$	21.6203	1.33571
$263$	13.7752	0.849413	0.424707	−	0.905331i	$-0.360377\pi$
$263$	13.7752	0.849413	0.424707	+	0.905331i	$0.360377\pi$
$264$	0	0
$265$	−0.141786	−0.00870984
$266$	−5.70938	−0.350064
$267$	0	0
$268$	−3.58632	−0.219070
$269$	−11.2058	−0.683228	−0.341614	−	0.939840i	$-0.610974\pi$
$269$	−11.2058	−0.683228	−0.341614	+	0.939840i	$0.610974\pi$
$270$	0	0
$271$	−20.5528	−1.24850	−0.624248	−	0.781226i	$-0.714595\pi$
$271$	−20.5528	−1.24850	−0.624248	+	0.781226i	$0.714595\pi$
$272$	−4.60799	−0.279400
$273$	0	0
$274$	21.2489	1.28370
$275$	3.83701	0.231380
$276$	0	0
$277$	−17.2928	−1.03902	−0.519511	−	0.854463i	$-0.673886\pi$
$277$	−17.2928	−1.03902	−0.519511	+	0.854463i	$0.673886\pi$
$278$	−15.7630	−0.945401
$279$	0	0
$280$	35.9691	2.14956
$281$	−24.1530	−1.44085	−0.720425	−	0.693533i	$-0.756053\pi$
$281$	−24.1530	−1.44085	−0.720425	+	0.693533i	$0.756053\pi$
$282$	0	0
$283$	−12.4777	−0.741722	−0.370861	−	0.928688i	$-0.620937\pi$
$283$	−12.4777	−0.741722	−0.370861	+	0.928688i	$0.620937\pi$
$284$	0.323616	0.0192031
$285$	0	0
$286$	−0.429692	−0.0254082
$287$	−8.94855	−0.528216
$288$	0	0
$289$	−6.35435	−0.373785
$290$	1.93143	0.113417
$291$	0	0
$292$	10.6982	0.626063
$293$	24.0095	1.40265	0.701325	−	0.712842i	$-0.252592\pi$
$293$	24.0095	1.40265	0.701325	+	0.712842i	$0.252592\pi$
$294$	0	0
$295$	2.31702	0.134902
$296$	4.28126	0.248843
$297$	0	0
$298$	5.51685	0.319582
$299$	3.15492	0.182454
$300$	0	0
$301$	8.12635	0.468395
$302$	−5.87853	−0.338271
$303$	0	0
$304$	1.86387	0.106900
$305$	3.61666	0.207089
$306$	0	0
$307$	7.54396	0.430557	0.215278	−	0.976553i	$-0.430934\pi$
$307$	7.54396	0.430557	0.215278	+	0.976553i	$0.430934\pi$
$308$	4.33830	0.247197
$309$	0	0
$310$	−4.62081	−0.262444
$311$	7.22920	0.409930	0.204965	−	0.978769i	$-0.434292\pi$
$311$	7.22920	0.409930	0.204965	+	0.978769i	$0.434292\pi$
$312$	0	0
$313$	−21.2712	−1.20232	−0.601161	−	0.799128i	$-0.705295\pi$
$313$	−21.2712	−1.20232	−0.601161	+	0.799128i	$0.705295\pi$
$314$	−18.5457	−1.04659
$315$	0	0
$316$	−3.82380	−0.215106
$317$	−31.0998	−1.74674	−0.873369	−	0.487059i	$-0.838070\pi$
$317$	−31.0998	−1.74674	−0.873369	+	0.487059i	$0.838070\pi$
$318$	0	0
$319$	0.754028	0.0422175
$320$	−22.0171	−1.23079
$321$	0	0
$322$	39.3967	2.19549
$323$	−4.30601	−0.239593
$324$	0	0
$325$	1.12706	0.0625179
$326$	13.5622	0.751142
$327$	0	0
$328$	6.62029	0.365544
$329$	−6.97562	−0.384579
$330$	0	0
$331$	−4.80463	−0.264086	−0.132043	−	0.991244i	$-0.542154\pi$
$331$	−4.80463	−0.264086	−0.132043	+	0.991244i	$0.542154\pi$
$332$	12.5677	0.689743
$333$	0	0
$334$	−5.16743	−0.282749
$335$	−11.5230	−0.629567
$336$	0	0
$337$	−1.20117	−0.0654317	−0.0327159	−	0.999465i	$-0.510416\pi$
$337$	−1.20117	−0.0654317	−0.0327159	+	0.999465i	$0.510416\pi$
$338$	13.5447	0.736733
$339$	0	0
$340$	8.38100	0.454524
$341$	−1.80396	−0.0976901
$342$	0	0
$343$	−12.0272	−0.649408
$344$	−6.01200	−0.324146
$345$	0	0
$346$	−16.3418	−0.878541
$347$	1.61746	0.0868296	0.0434148	−	0.999057i	$-0.486176\pi$
$347$	1.61746	0.0868296	0.0434148	+	0.999057i	$0.486176\pi$
$348$	0	0
$349$	31.2253	1.67145	0.835727	−	0.549145i	$-0.185047\pi$
$349$	31.2253	1.67145	0.835727	+	0.549145i	$0.185047\pi$
$350$	14.0740	0.752286
$351$	0	0
$352$	−5.42752	−0.289288
$353$	14.6883	0.781779	0.390889	−	0.920438i	$-0.372168\pi$
$353$	14.6883	0.781779	0.390889	+	0.920438i	$0.372168\pi$
$354$	0	0
$355$	1.03979	0.0551861
$356$	−9.52689	−0.504924
$357$	0	0
$358$	−17.8396	−0.942851
$359$	−6.81390	−0.359624	−0.179812	−	0.983701i	$-0.557549\pi$
$359$	−6.81390	−0.359624	−0.179812	+	0.983701i	$0.557549\pi$
$360$	0	0
$361$	−17.2583	−0.908331
$362$	−3.47264	−0.182518
$363$	0	0
$364$	1.27430	0.0667917
$365$	34.3736	1.79919
$366$	0	0
$367$	9.51165	0.496504	0.248252	−	0.968695i	$-0.420144\pi$
$367$	9.51165	0.496504	0.248252	+	0.968695i	$0.420144\pi$
$368$	−12.8613	−0.670442
$369$	0	0
$370$	4.24979	0.220936
$371$	−0.203033	−0.0105410
$372$	0	0
$373$	−12.7503	−0.660186	−0.330093	−	0.943949i	$-0.607080\pi$
$373$	−12.7503	−0.660186	−0.330093	+	0.943949i	$0.607080\pi$
$374$	−4.04683	−0.209256
$375$	0	0
$376$	5.16068	0.266142
$377$	0.221483	0.0114070
$378$	0	0
$379$	−10.2372	−0.525852	−0.262926	−	0.964816i	$-0.584688\pi$
$379$	−10.2372	−0.525852	−0.262926	+	0.964816i	$0.584688\pi$
$380$	−3.39000	−0.173903
$381$	0	0
$382$	−12.8724	−0.658608
$383$	24.8455	1.26954	0.634772	−	0.772699i	$-0.281094\pi$
$383$	24.8455	1.26954	0.634772	+	0.772699i	$0.281094\pi$
$384$	0	0
$385$	13.9391	0.710401
$386$	16.1423	0.821624
$387$	0	0
$388$	9.13201	0.463608
$389$	−3.01669	−0.152952	−0.0764760	−	0.997071i	$-0.524367\pi$
$389$	−3.01669	−0.152952	−0.0764760	+	0.997071i	$0.524367\pi$
$390$	0	0
$391$	29.7129	1.50265
$392$	30.2023	1.52545
$393$	0	0
$394$	−21.7424	−1.09537
$395$	−12.2860	−0.618175
$396$	0	0
$397$	0.428312	0.0214964	0.0107482	−	0.999942i	$-0.496579\pi$
$397$	0.428312	0.0214964	0.0107482	+	0.999942i	$0.496579\pi$
$398$	26.2414	1.31536
$399$	0	0
$400$	−4.59455	−0.229727
$401$	14.0036	0.699305	0.349653	−	0.936879i	$-0.386300\pi$
$401$	14.0036	0.699305	0.349653	+	0.936879i	$0.386300\pi$
$402$	0	0
$403$	−0.529885	−0.0263954
$404$	11.9126	0.592674
$405$	0	0
$406$	2.76574	0.137262
$407$	1.65912	0.0822394
$408$	0	0
$409$	−39.9092	−1.97338	−0.986692	−	0.162600i	$-0.948012\pi$
$409$	−39.9092	−1.97338	−0.986692	+	0.162600i	$0.948012\pi$
$410$	6.57162	0.324549
$411$	0	0
$412$	14.6435	0.721432
$413$	3.31791	0.163264
$414$	0	0
$415$	40.3805	1.98220
$416$	−1.59424	−0.0781643
$417$	0	0
$418$	1.63688	0.0800626
$419$	−23.6201	−1.15392	−0.576959	−	0.816773i	$-0.695761\pi$
$419$	−23.6201	−1.15392	−0.576959	+	0.816773i	$0.695761\pi$
$420$	0	0
$421$	−16.4261	−0.800559	−0.400279	−	0.916393i	$-0.631087\pi$
$421$	−16.4261	−0.800559	−0.400279	+	0.916393i	$0.631087\pi$
$422$	4.06841	0.198047
$423$	0	0
$424$	0.150207	0.00729471
$425$	10.6146	0.514883
$426$	0	0
$427$	5.17894	0.250627
$428$	−9.67814	−0.467810
$429$	0	0
$430$	−5.96780	−0.287793
$431$	−24.1771	−1.16457	−0.582285	−	0.812985i	$-0.697841\pi$
$431$	−24.1771	−1.16457	−0.582285	+	0.812985i	$0.697841\pi$
$432$	0	0
$433$	−7.20673	−0.346333	−0.173167	−	0.984893i	$-0.555400\pi$
$433$	−7.20673	−0.346333	−0.173167	+	0.984893i	$0.555400\pi$
$434$	−6.61686	−0.317619
$435$	0	0
$436$	−15.3282	−0.734087
$437$	−12.0185	−0.574921
$438$	0	0
$439$	21.1400	1.00896	0.504479	−	0.863424i	$-0.331684\pi$
$439$	21.1400	1.00896	0.504479	+	0.863424i	$0.331684\pi$
$440$	−10.3124	−0.491623
$441$	0	0
$442$	−1.18869	−0.0565402
$443$	11.1509	0.529797	0.264899	−	0.964276i	$-0.414662\pi$
$443$	11.1509	0.529797	0.264899	+	0.964276i	$0.414662\pi$
$444$	0	0
$445$	−30.6102	−1.45106
$446$	−9.99415	−0.473237
$447$	0	0
$448$	−31.5278	−1.48955
$449$	2.12495	0.100283	0.0501414	−	0.998742i	$-0.484033\pi$
$449$	2.12495	0.100283	0.0501414	+	0.998742i	$0.484033\pi$
$450$	0	0
$451$	2.56556	0.120807
$452$	11.7955	0.554814
$453$	0	0
$454$	8.36182	0.392440
$455$	4.09438	0.191947
$456$	0	0
$457$	37.1322	1.73697	0.868485	−	0.495715i	$-0.165094\pi$
$457$	37.1322	1.73697	0.868485	+	0.495715i	$0.165094\pi$
$458$	11.7772	0.550315
$459$	0	0
$460$	23.3921	1.09066
$461$	−29.8834	−1.39181	−0.695905	−	0.718134i	$-0.744996\pi$
$461$	−29.8834	−1.39181	−0.695905	+	0.718134i	$0.744996\pi$
$462$	0	0
$463$	20.4052	0.948311	0.474156	−	0.880441i	$-0.342753\pi$
$463$	20.4052	0.948311	0.474156	+	0.880441i	$0.342753\pi$
$464$	−0.902896	−0.0419159
$465$	0	0
$466$	−16.4988	−0.764291
$467$	−4.72528	−0.218660	−0.109330	−	0.994006i	$-0.534871\pi$
$467$	−4.72528	−0.218660	−0.109330	+	0.994006i	$0.534871\pi$
$468$	0	0
$469$	−16.5005	−0.761924
$470$	5.12274	0.236294
$471$	0	0
$472$	−2.45464	−0.112984
$473$	−2.32983	−0.107126
$474$	0	0
$475$	−4.29345	−0.196997
$476$	12.0013	0.550081
$477$	0	0
$478$	−1.05161	−0.0480994
$479$	−39.4927	−1.80447	−0.902234	−	0.431247i	$-0.858074\pi$
$479$	−39.4927	−1.80447	−0.902234	+	0.431247i	$0.858074\pi$
$480$	0	0
$481$	0.487339	0.0222207
$482$	26.8836	1.22451
$483$	0	0
$484$	8.59157	0.390526
$485$	29.3414	1.33233
$486$	0	0
$487$	−26.1172	−1.18349	−0.591743	−	0.806127i	$-0.701560\pi$
$487$	−26.1172	−1.18349	−0.591743	+	0.806127i	$0.701560\pi$
$488$	−3.83147	−0.173442
$489$	0	0
$490$	29.9802	1.35437
$491$	20.1522	0.909454	0.454727	−	0.890631i	$-0.349737\pi$
$491$	20.1522	0.909454	0.454727	+	0.890631i	$0.349737\pi$
$492$	0	0
$493$	2.08592	0.0939453
$494$	0.480808	0.0216326
$495$	0	0
$496$	2.16012	0.0969922
$497$	1.48894	0.0667882
$498$	0	0
$499$	16.6298	0.744452	0.372226	−	0.928142i	$-0.378595\pi$
$499$	16.6298	0.744452	0.372226	+	0.928142i	$0.378595\pi$
$500$	−4.48685	−0.200658
$501$	0	0
$502$	17.6494	0.787732
$503$	13.0732	0.582906	0.291453	−	0.956585i	$-0.405861\pi$
$503$	13.0732	0.582906	0.291453	+	0.956585i	$0.405861\pi$
$504$	0	0
$505$	38.2755	1.70324
$506$	−11.2951	−0.502126
$507$	0	0
$508$	−11.6599	−0.517322
$509$	12.4973	0.553934	0.276967	−	0.960879i	$-0.410671\pi$
$509$	12.4973	0.553934	0.276967	+	0.960879i	$0.410671\pi$
$510$	0	0
$511$	49.2219	2.17745
$512$	15.0957	0.667140
$513$	0	0
$514$	−7.35026	−0.324206
$515$	47.0499	2.07327
$516$	0	0
$517$	1.99992	0.0879563
$518$	6.08557	0.267385
$519$	0	0
$520$	−3.02909	−0.132834
$521$	−3.31677	−0.145310	−0.0726552	−	0.997357i	$-0.523147\pi$
$521$	−3.31677	−0.145310	−0.0726552	+	0.997357i	$0.523147\pi$
$522$	0	0
$523$	32.7979	1.43415	0.717075	−	0.696996i	$-0.245480\pi$
$523$	32.7979	1.43415	0.717075	+	0.696996i	$0.245480\pi$
$524$	18.3826	0.803047
$525$	0	0
$526$	−14.4861	−0.631622
$527$	−4.99043	−0.217387
$528$	0	0
$529$	59.9315	2.60572
$530$	0.149103	0.00647662
$531$	0	0
$532$	−4.85437	−0.210464
$533$	0.753591	0.0326416
$534$	0	0
$535$	−31.0961	−1.34440
$536$	12.2074	0.527278
$537$	0	0
$538$	11.7841	0.508047
$539$	11.7043	0.504139
$540$	0	0
$541$	−5.52107	−0.237369	−0.118685	−	0.992932i	$-0.537868\pi$
$541$	−5.52107	−0.237369	−0.118685	+	0.992932i	$0.537868\pi$
$542$	21.6135	0.928380
$543$	0	0
$544$	−15.0145	−0.643743
$545$	−49.2499	−2.10963
$546$	0	0
$547$	12.7057	0.543258	0.271629	−	0.962402i	$-0.412438\pi$
$547$	12.7057	0.543258	0.271629	+	0.962402i	$0.412438\pi$
$548$	18.0668	0.771776
$549$	0	0
$550$	−4.03502	−0.172054
$551$	−0.843726	−0.0359439
$552$	0	0
$553$	−17.5932	−0.748138
$554$	18.1852	0.772615
$555$	0	0
$556$	−13.4024	−0.568389
$557$	−42.9858	−1.82137	−0.910683	−	0.413105i	$-0.864444\pi$
$557$	−42.9858	−1.82137	−0.910683	+	0.413105i	$0.864444\pi$
$558$	0	0
$559$	−0.684350	−0.0289449
$560$	−16.6911	−0.705327
$561$	0	0
$562$	25.3995	1.07141
$563$	−20.3703	−0.858504	−0.429252	−	0.903185i	$-0.641223\pi$
$563$	−20.3703	−0.858504	−0.429252	+	0.903185i	$0.641223\pi$
$564$	0	0
$565$	37.8993	1.59444
$566$	13.1216	0.551543
$567$	0	0
$568$	−1.10154	−0.0462198
$569$	31.4900	1.32013	0.660064	−	0.751210i	$-0.270529\pi$
$569$	31.4900	1.32013	0.660064	+	0.751210i	$0.270529\pi$
$570$	0	0
$571$	−31.3410	−1.31158	−0.655789	−	0.754944i	$-0.727664\pi$
$571$	−31.3410	−1.31158	−0.655789	+	0.754944i	$0.727664\pi$
$572$	−0.365344	−0.0152758
$573$	0	0
$574$	9.41036	0.392781
$575$	29.6263	1.23550
$576$	0	0
$577$	23.8525	0.992992	0.496496	−	0.868039i	$-0.334620\pi$
$577$	23.8525	0.992992	0.496496	+	0.868039i	$0.334620\pi$
$578$	6.68228	0.277946
$579$	0	0
$580$	1.64219	0.0681881
$581$	57.8236	2.39893
$582$	0	0
$583$	0.0582098	0.00241080
$584$	−36.4152	−1.50687
$585$	0	0
$586$	−25.2485	−1.04301
$587$	−8.14382	−0.336131	−0.168066	−	0.985776i	$-0.553752\pi$
$587$	−8.14382	−0.336131	−0.168066	+	0.985776i	$0.553752\pi$
$588$	0	0
$589$	2.01856	0.0831733
$590$	−2.43660	−0.100313
$591$	0	0
$592$	−1.98668	−0.0816519
$593$	−1.77048	−0.0727049	−0.0363524	−	0.999339i	$-0.511574\pi$
$593$	−1.77048	−0.0727049	−0.0363524	+	0.999339i	$0.511574\pi$
$594$	0	0
$595$	38.5607	1.58083
$596$	4.69067	0.192138
$597$	0	0
$598$	−3.31774	−0.135672
$599$	24.1665	0.987415	0.493707	−	0.869628i	$-0.335641\pi$
$599$	24.1665	0.987415	0.493707	+	0.869628i	$0.335641\pi$
$600$	0	0
$601$	10.2259	0.417123	0.208561	−	0.978009i	$-0.433122\pi$
$601$	10.2259	0.417123	0.208561	+	0.978009i	$0.433122\pi$
$602$	−8.54572	−0.348297
$603$	0	0
$604$	−4.99819	−0.203373
$605$	27.6050	1.12230
$606$	0	0
$607$	−21.0640	−0.854960	−0.427480	−	0.904025i	$-0.640599\pi$
$607$	−21.0640	−0.854960	−0.427480	+	0.904025i	$0.640599\pi$
$608$	6.07316	0.246299
$609$	0	0
$610$	−3.80330	−0.153991
$611$	0.587443	0.0237654
$612$	0	0
$613$	−6.72545	−0.271638	−0.135819	−	0.990734i	$-0.543367\pi$
$613$	−6.72545	−0.271638	−0.135819	+	0.990734i	$0.543367\pi$
$614$	−7.93328	−0.320161
$615$	0	0
$616$	−14.7670	−0.594979
$617$	0.401847	0.0161778	0.00808888	−	0.999967i	$-0.497425\pi$
$617$	0.401847	0.0161778	0.00808888	+	0.999967i	$0.497425\pi$
$618$	0	0
$619$	−20.6279	−0.829104	−0.414552	−	0.910026i	$-0.636062\pi$
$619$	−20.6279	−0.829104	−0.414552	+	0.910026i	$0.636062\pi$
$620$	−3.92882	−0.157785
$621$	0	0
$622$	−7.60227	−0.304823
$623$	−43.8329	−1.75613
$624$	0	0
$625$	−30.6826	−1.22730
$626$	22.3690	0.894044
$627$	0	0
$628$	−15.7684	−0.629227
$629$	4.58974	0.183005
$630$	0	0
$631$	−29.6371	−1.17984	−0.589918	−	0.807463i	$-0.700840\pi$
$631$	−29.6371	−1.17984	−0.589918	+	0.807463i	$0.700840\pi$
$632$	13.0157	0.517737
$633$	0	0
$634$	32.7047	1.29887
$635$	−37.4635	−1.48669
$636$	0	0
$637$	3.43794	0.136216
$638$	−0.792941	−0.0313928
$639$	0	0
$640$	−3.28712	−0.129935
$641$	−11.3866	−0.449745	−0.224872	−	0.974388i	$-0.572197\pi$
$641$	−11.3866	−0.449745	−0.224872	+	0.974388i	$0.572197\pi$
$642$	0	0
$643$	−33.4917	−1.32079	−0.660393	−	0.750921i	$-0.729610\pi$
$643$	−33.4917	−1.32079	−0.660393	+	0.750921i	$0.729610\pi$
$644$	33.4968	1.31996
$645$	0	0
$646$	4.52823	0.178161
$647$	−48.9050	−1.92265	−0.961326	−	0.275412i	$-0.911186\pi$
$647$	−48.9050	−1.92265	−0.961326	+	0.275412i	$0.911186\pi$
$648$	0	0
$649$	−0.951247	−0.0373397
$650$	−1.18522	−0.0464882
$651$	0	0
$652$	11.5312	0.451598
$653$	−5.23254	−0.204765	−0.102383	−	0.994745i	$-0.532647\pi$
$653$	−5.23254	−0.204765	−0.102383	+	0.994745i	$0.532647\pi$
$654$	0	0
$655$	59.0638	2.30781
$656$	−3.07208	−0.119944
$657$	0	0
$658$	7.33561	0.285972
$659$	3.59290	0.139960	0.0699798	−	0.997548i	$-0.477707\pi$
$659$	3.59290	0.139960	0.0699798	+	0.997548i	$0.477707\pi$
$660$	0	0
$661$	−15.9161	−0.619066	−0.309533	−	0.950889i	$-0.600173\pi$
$661$	−15.9161	−0.619066	−0.309533	+	0.950889i	$0.600173\pi$
$662$	5.05258	0.196374
$663$	0	0
$664$	−42.7789	−1.66014
$665$	−15.5972	−0.604835
$666$	0	0
$667$	5.82200	0.225429
$668$	−4.39358	−0.169993
$669$	0	0
$670$	12.1176	0.468145
$671$	−1.48481	−0.0573204
$672$	0	0
$673$	34.3829	1.32536	0.662682	−	0.748901i	$-0.269418\pi$
$673$	34.3829	1.32536	0.662682	+	0.748901i	$0.269418\pi$
$674$	1.26315	0.0486549
$675$	0	0
$676$	11.5163	0.442934
$677$	27.8827	1.07162	0.535809	−	0.844339i	$-0.320007\pi$
$677$	27.8827	1.07162	0.535809	+	0.844339i	$0.320007\pi$
$678$	0	0
$679$	42.0161	1.61243
$680$	−28.5278	−1.09399
$681$	0	0
$682$	1.89706	0.0726422
$683$	−1.84211	−0.0704864	−0.0352432	−	0.999379i	$-0.511221\pi$
$683$	−1.84211	−0.0704864	−0.0352432	+	0.999379i	$0.511221\pi$
$684$	0	0
$685$	58.0492	2.21795
$686$	12.6479	0.482898
$687$	0	0
$688$	2.78981	0.106360
$689$	0.0170982	0.000651388 0
$690$	0	0
$691$	18.0372	0.686166	0.343083	−	0.939305i	$-0.388529\pi$
$691$	18.0372	0.686166	0.343083	+	0.939305i	$0.388529\pi$
$692$	−13.8945	−0.528191
$693$	0	0
$694$	−1.70093	−0.0645663
$695$	−43.0623	−1.63345
$696$	0	0
$697$	7.09729	0.268829
$698$	−32.8368	−1.24289
$699$	0	0
$700$	11.9663	0.452285
$701$	6.10724	0.230667	0.115334	−	0.993327i	$-0.463206\pi$
$701$	6.10724	0.230667	0.115334	+	0.993327i	$0.463206\pi$
$702$	0	0
$703$	−1.85648	−0.0700186
$704$	9.03904	0.340672
$705$	0	0
$706$	−15.4463	−0.581329
$707$	54.8094	2.06132
$708$	0	0
$709$	−29.3897	−1.10375	−0.551877	−	0.833926i	$-0.686088\pi$
$709$	−29.3897	−1.10375	−0.551877	+	0.833926i	$0.686088\pi$
$710$	−1.09345	−0.0410363
$711$	0	0
$712$	32.4283	1.21530
$713$	−13.9287	−0.521636
$714$	0	0
$715$	−1.17386	−0.0438999
$716$	−15.1680	−0.566856
$717$	0	0
$718$	7.16554	0.267416
$719$	38.2319	1.42581	0.712904	−	0.701262i	$-0.247379\pi$
$719$	38.2319	1.42581	0.712904	+	0.701262i	$0.247379\pi$
$720$	0	0
$721$	67.3741	2.50914
$722$	18.1489	0.675433
$723$	0	0
$724$	−2.95260	−0.109732
$725$	2.07984	0.0772432
$726$	0	0
$727$	16.8660	0.625526	0.312763	−	0.949831i	$-0.398745\pi$
$727$	16.8660	0.625526	0.312763	+	0.949831i	$0.398745\pi$
$728$	−4.33756	−0.160761
$729$	0	0
$730$	−36.1475	−1.33788
$731$	−6.44518	−0.238384
$732$	0	0
$733$	−9.13669	−0.337471	−0.168736	−	0.985661i	$-0.553968\pi$
$733$	−9.13669	−0.337471	−0.168736	+	0.985661i	$0.553968\pi$
$734$	−10.0025	−0.369200
$735$	0	0
$736$	−41.9069	−1.54471
$737$	4.73072	0.174258
$738$	0	0
$739$	−28.4826	−1.04775	−0.523875	−	0.851795i	$-0.675514\pi$
$739$	−28.4826	−1.04775	−0.523875	+	0.851795i	$0.675514\pi$
$740$	3.61336	0.132830
$741$	0	0
$742$	0.213511	0.00783823
$743$	37.7287	1.38413	0.692066	−	0.721834i	$-0.256701\pi$
$743$	37.7287	1.38413	0.692066	+	0.721834i	$0.256701\pi$
$744$	0	0
$745$	15.0713	0.552169
$746$	13.4083	0.490913
$747$	0	0
$748$	−3.44080	−0.125808
$749$	−44.5288	−1.62705
$750$	0	0
$751$	−32.0210	−1.16846	−0.584231	−	0.811588i	$-0.698604\pi$
$751$	−32.0210	−1.16846	−0.584231	+	0.811588i	$0.698604\pi$
$752$	−2.39476	−0.0873280
$753$	0	0
$754$	−0.232913	−0.00848221
$755$	−16.0593	−0.584459
$756$	0	0
$757$	−13.7562	−0.499976	−0.249988	−	0.968249i	$-0.580427\pi$
$757$	−13.7562	−0.499976	−0.249988	+	0.968249i	$0.580427\pi$
$758$	10.7656	0.391023
$759$	0	0
$760$	11.5391	0.418567
$761$	−45.3489	−1.64390	−0.821948	−	0.569563i	$-0.807112\pi$
$761$	−45.3489	−1.64390	−0.821948	+	0.569563i	$0.807112\pi$
$762$	0	0
$763$	−70.5244	−2.55315
$764$	−10.9447	−0.395965
$765$	0	0
$766$	−26.1277	−0.944031
$767$	−0.279413	−0.0100890
$768$	0	0
$769$	−15.6445	−0.564153	−0.282077	−	0.959392i	$-0.591023\pi$
$769$	−15.6445	−0.564153	−0.282077	+	0.959392i	$0.591023\pi$
$770$	−14.6584	−0.528253
$771$	0	0
$772$	13.7250	0.493972
$773$	22.9648	0.825988	0.412994	−	0.910734i	$-0.364483\pi$
$773$	22.9648	0.825988	0.412994	+	0.910734i	$0.364483\pi$
$774$	0	0
$775$	−4.97587	−0.178739
$776$	−31.0842	−1.11586
$777$	0	0
$778$	3.17237	0.113735
$779$	−2.87075	−0.102855
$780$	0	0
$781$	−0.426881	−0.0152750
$782$	−31.2463	−1.11737
$783$	0	0
$784$	−14.0151	−0.500538
$785$	−50.6643	−1.80829
$786$	0	0
$787$	11.3151	0.403341	0.201671	−	0.979453i	$-0.435363\pi$
$787$	11.3151	0.403341	0.201671	+	0.979453i	$0.435363\pi$
$788$	−18.4864	−0.658551
$789$	0	0
$790$	12.9200	0.459674
$791$	54.2707	1.92964
$792$	0	0
$793$	−0.436138	−0.0154877
$794$	−0.450416	−0.0159847
$795$	0	0
$796$	22.3116	0.790815
$797$	−13.4591	−0.476745	−0.238372	−	0.971174i	$-0.576614\pi$
$797$	−13.4591	−0.476745	−0.238372	+	0.971174i	$0.576614\pi$
$798$	0	0
$799$	5.53252	0.195726
$800$	−14.9707	−0.529295
$801$	0	0
$802$	−14.7263	−0.520002
$803$	−14.1120	−0.498000
$804$	0	0
$805$	107.626	3.79333
$806$	0.557230	0.0196276
$807$	0	0
$808$	−40.5489	−1.42651
$809$	−29.8582	−1.04976	−0.524878	−	0.851177i	$-0.675889\pi$
$809$	−29.8582	−1.04976	−0.524878	+	0.851177i	$0.675889\pi$
$810$	0	0
$811$	23.1835	0.814083	0.407042	−	0.913410i	$-0.366560\pi$
$811$	23.1835	0.814083	0.407042	+	0.913410i	$0.366560\pi$
$812$	2.35156	0.0825236
$813$	0	0
$814$	−1.74474	−0.0611531
$815$	37.0502	1.29781
$816$	0	0
$817$	2.60698	0.0912067
$818$	41.9688	1.46740
$819$	0	0
$820$	5.58749	0.195123
$821$	−16.7615	−0.584980	−0.292490	−	0.956269i	$-0.594484\pi$
$821$	−16.7615	−0.584980	−0.292490	+	0.956269i	$0.594484\pi$
$822$	0	0
$823$	−9.77979	−0.340902	−0.170451	−	0.985366i	$-0.554522\pi$
$823$	−9.77979	−0.340902	−0.170451	+	0.985366i	$0.554522\pi$
$824$	−49.8445	−1.73641
$825$	0	0
$826$	−3.48913	−0.121402
$827$	38.6934	1.34550	0.672751	−	0.739869i	$-0.265113\pi$
$827$	38.6934	1.34550	0.672751	+	0.739869i	$0.265113\pi$
$828$	0	0
$829$	7.90150	0.274430	0.137215	−	0.990541i	$-0.456185\pi$
$829$	7.90150	0.274430	0.137215	+	0.990541i	$0.456185\pi$
$830$	−42.4644	−1.47396
$831$	0	0
$832$	2.65507	0.0920480
$833$	32.3784	1.12184
$834$	0	0
$835$	−14.1167	−0.488529
$836$	1.39175	0.0481348
$837$	0	0
$838$	24.8391	0.858051
$839$	1.29253	0.0446230	0.0223115	−	0.999751i	$-0.492897\pi$
$839$	1.29253	0.0446230	0.0223115	+	0.999751i	$0.492897\pi$
$840$	0	0
$841$	−28.5913	−0.985906
$842$	17.2738	0.595294
$843$	0	0
$844$	3.45915	0.119069
$845$	37.0022	1.27291
$846$	0	0
$847$	39.5295	1.35825
$848$	−0.0697021	−0.00239358
$849$	0	0
$850$	−11.1624	−0.382866
$851$	12.8104	0.439134
$852$	0	0
$853$	47.2923	1.61926	0.809629	−	0.586942i	$-0.199668\pi$
$853$	47.2923	1.61926	0.809629	+	0.586942i	$0.199668\pi$
$854$	−5.44621	−0.186366
$855$	0	0
$856$	32.9431	1.12597
$857$	−41.5916	−1.42074	−0.710371	−	0.703827i	$-0.751473\pi$
$857$	−41.5916	−1.42074	−0.710371	+	0.703827i	$0.751473\pi$
$858$	0	0
$859$	39.2094	1.33781	0.668904	−	0.743349i	$-0.266764\pi$
$859$	39.2094	1.33781	0.668904	+	0.743349i	$0.266764\pi$
$860$	−5.07410	−0.173025
$861$	0	0
$862$	25.4248	0.865972
$863$	−37.3913	−1.27281	−0.636407	−	0.771354i	$-0.719580\pi$
$863$	−37.3913	−1.27281	−0.636407	+	0.771354i	$0.719580\pi$
$864$	0	0
$865$	−44.6436	−1.51793
$866$	7.57864	0.257533
$867$	0	0
$868$	−5.62596	−0.190957
$869$	5.04398	0.171105
$870$	0	0
$871$	1.38957	0.0470838
$872$	52.1751	1.76687
$873$	0	0
$874$	12.6387	0.427510
$875$	−20.6438	−0.697889
$876$	0	0
$877$	−42.5083	−1.43540	−0.717701	−	0.696351i	$-0.754806\pi$
$877$	−42.5083	−1.43540	−0.717701	+	0.696351i	$0.754806\pi$
$878$	−22.2310	−0.750260
$879$	0	0
$880$	4.78535	0.161314
$881$	−30.1193	−1.01475	−0.507373	−	0.861726i	$-0.669383\pi$
$881$	−30.1193	−1.01475	−0.507373	+	0.861726i	$0.669383\pi$
$882$	0	0
$883$	−34.4505	−1.15935	−0.579676	−	0.814847i	$-0.696821\pi$
$883$	−34.4505	−1.15935	−0.579676	+	0.814847i	$0.696821\pi$
$884$	−1.01068	−0.0339928
$885$	0	0
$886$	−11.7264	−0.393956
$887$	−47.0599	−1.58012	−0.790059	−	0.613031i	$-0.789950\pi$
$887$	−47.0599	−1.58012	−0.790059	+	0.613031i	$0.789950\pi$
$888$	0	0
$889$	−53.6466	−1.79925
$890$	32.1899	1.07901
$891$	0	0
$892$	−8.49748	−0.284517
$893$	−2.23782	−0.0748859
$894$	0	0
$895$	−48.7353	−1.62904
$896$	−4.70705	−0.157252
$897$	0	0
$898$	−2.23462	−0.0745701
$899$	−0.977833	−0.0326125
$900$	0	0
$901$	0.161030	0.00536468
$902$	−2.69796	−0.0898321
$903$	0	0
$904$	−40.1503	−1.33538
$905$	−9.48678	−0.315351
$906$	0	0
$907$	53.8831	1.78916	0.894579	−	0.446910i	$-0.147476\pi$
$907$	53.8831	1.78916	0.894579	+	0.446910i	$0.147476\pi$
$908$	7.10960	0.235940
$909$	0	0
$910$	−4.30567	−0.142732
$911$	−2.77566	−0.0919616	−0.0459808	−	0.998942i	$-0.514641\pi$
$911$	−2.77566	−0.0919616	−0.0459808	+	0.998942i	$0.514641\pi$
$912$	0	0
$913$	−16.5781	−0.548654
$914$	−39.0484	−1.29161
$915$	0	0
$916$	10.0136	0.330857
$917$	84.5776	2.79300
$918$	0	0
$919$	19.6148	0.647031	0.323516	−	0.946223i	$-0.395135\pi$
$919$	19.6148	0.647031	0.323516	+	0.946223i	$0.395135\pi$
$920$	−79.6237	−2.62512
$921$	0	0
$922$	31.4256	1.03495
$923$	−0.125389	−0.00412724
$924$	0	0
$925$	4.57635	0.150469
$926$	−21.4583	−0.705163
$927$	0	0
$928$	−2.94197	−0.0965749
$929$	−53.7834	−1.76458	−0.882289	−	0.470709i	$-0.843998\pi$
$929$	−53.7834	−1.76458	−0.882289	+	0.470709i	$0.843998\pi$
$930$	0	0
$931$	−13.0966	−0.429224
$932$	−14.0280	−0.459503
$933$	0	0
$934$	4.96914	0.162595
$935$	−11.0554	−0.361550
$936$	0	0
$937$	31.1352	1.01714	0.508572	−	0.861020i	$-0.330174\pi$
$937$	31.1352	1.01714	0.508572	+	0.861020i	$0.330174\pi$
$938$	17.3521	0.566565
$939$	0	0
$940$	4.35559	0.142064
$941$	−29.4881	−0.961286	−0.480643	−	0.876916i	$-0.659597\pi$
$941$	−29.4881	−0.961286	−0.480643	+	0.876916i	$0.659597\pi$
$942$	0	0
$943$	19.8092	0.645075
$944$	1.13905	0.0370730
$945$	0	0
$946$	2.45006	0.0796585
$947$	43.4174	1.41087	0.705437	−	0.708772i	$-0.250751\pi$
$947$	43.4174	1.41087	0.705437	+	0.708772i	$0.250751\pi$
$948$	0	0
$949$	−4.14516	−0.134558
$950$	4.51502	0.146487
$951$	0	0
$952$	−40.8510	−1.32399
$953$	−14.4497	−0.468071	−0.234035	−	0.972228i	$-0.575193\pi$
$953$	−14.4497	−0.468071	−0.234035	+	0.972228i	$0.575193\pi$
$954$	0	0
$955$	−35.1656	−1.13793
$956$	−0.894124	−0.0289180
$957$	0	0
$958$	41.5308	1.34180
$959$	83.1247	2.68424
$960$	0	0
$961$	−28.6606	−0.924535
$962$	−0.512488	−0.0165233
$963$	0	0
$964$	22.8577	0.736196
$965$	44.0987	1.41959
$966$	0	0
$967$	6.88995	0.221566	0.110783	−	0.993845i	$-0.464664\pi$
$967$	6.88995	0.221566	0.110783	+	0.993845i	$0.464664\pi$
$968$	−29.2446	−0.939956
$969$	0	0
$970$	−30.8556	−0.990715
$971$	1.06821	0.0342806	0.0171403	−	0.999853i	$-0.494544\pi$
$971$	1.06821	0.0342806	0.0171403	+	0.999853i	$0.494544\pi$
$972$	0	0
$973$	−61.6640	−1.97686
$974$	27.4651	0.880037
$975$	0	0
$976$	1.77795	0.0569109
$977$	36.3679	1.16351	0.581755	−	0.813364i	$-0.302366\pi$
$977$	36.3679	1.16351	0.581755	+	0.813364i	$0.302366\pi$
$978$	0	0
$979$	12.5669	0.401640
$980$	25.4906	0.814266
$981$	0	0
$982$	−21.1921	−0.676268
$983$	−5.70415	−0.181934	−0.0909671	−	0.995854i	$-0.528996\pi$
$983$	−5.70415	−0.181934	−0.0909671	+	0.995854i	$0.528996\pi$
$984$	0	0
$985$	−59.3974	−1.89256
$986$	−2.19357	−0.0698575
$987$	0	0
$988$	0.408805	0.0130058
$989$	−17.9891	−0.572019
$990$	0	0
$991$	−6.98693	−0.221947	−0.110974	−	0.993823i	$-0.535397\pi$
$991$	−6.98693	−0.221947	−0.110974	+	0.993823i	$0.535397\pi$
$992$	7.03847	0.223472
$993$	0	0
$994$	−1.56578	−0.0496636
$995$	71.6880	2.27266
$996$	0	0
$997$	−28.4466	−0.900914	−0.450457	−	0.892798i	$-0.648739\pi$
$997$	−28.4466	−0.900914	−0.450457	+	0.892798i	$0.648739\pi$
$998$	−17.4880	−0.553573
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.e.1.2		6
3.2	odd	2		717.2.a.d.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.d.1.5	✓	6	3.2	odd	2
2151.2.a.e.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +O(q^{10})\)	\(q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +3.02110 q^{10} +1.17944 q^{11} +0.346441 q^{13} +4.32613 q^{14} -1.41230 q^{16} +3.26277 q^{17} -1.31974 q^{19} +2.56868 q^{20} -1.24031 q^{22} +9.10667 q^{23} +3.25325 q^{25} -0.364319 q^{26} +3.67827 q^{28} +0.639311 q^{29} -1.52951 q^{31} -4.60178 q^{32} -3.43115 q^{34} +11.8184 q^{35} +1.40670 q^{37} +1.38785 q^{38} -8.74345 q^{40} +2.17524 q^{41} -1.97537 q^{43} -1.05456 q^{44} -9.57663 q^{46} +1.69565 q^{47} +9.92360 q^{49} -3.42114 q^{50} -0.309761 q^{52} +0.0493538 q^{53} -3.38835 q^{55} -12.5204 q^{56} -0.672304 q^{58} -0.806525 q^{59} -1.25891 q^{61} +1.60844 q^{62} +7.66385 q^{64} -0.995271 q^{65} +4.01099 q^{67} -2.91732 q^{68} -12.4283 q^{70} -0.361936 q^{71} -11.9650 q^{73} -1.47930 q^{74} +1.18001 q^{76} -4.85201 q^{77} +4.27659 q^{79} +4.05731 q^{80} -2.28749 q^{82} -14.0559 q^{83} -9.37343 q^{85} +2.07731 q^{86} +3.58960 q^{88} +10.6550 q^{89} -1.42520 q^{91} -8.14249 q^{92} -1.78316 q^{94} +3.79142 q^{95} -10.2134 q^{97} -10.4357 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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