LMFDB - Embedded newform 2151.2.a.i.1.11

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 $ x^17 - 28x^15 - x^14 + 319x^13 + 17x^12 - 1903x^11 - 91x^10 + 6377x^9 + 125x^8 - 11967x^7 + 233x^6 + 11733x^5 - 503x^4 - 5015x^3 + 94x^2 + 609x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.17471$ 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)

$f(q)$	$=$	$q+1.17471 q^{2} -0.620061 q^{4} -4.24400 q^{5} -3.49274 q^{7} -3.07781 q^{8} +O(q^{10})$	\(q+1.17471 q^{2} -0.620061 q^{4} -4.24400 q^{5} -3.49274 q^{7} -3.07781 q^{8} -4.98546 q^{10} -1.65775 q^{11} -4.59282 q^{13} -4.10295 q^{14} -2.37540 q^{16} +5.91628 q^{17} +1.31642 q^{19} +2.63154 q^{20} -1.94737 q^{22} +1.05388 q^{23} +13.0116 q^{25} -5.39522 q^{26} +2.16571 q^{28} -7.26679 q^{29} -2.73764 q^{31} +3.36521 q^{32} +6.94990 q^{34} +14.8232 q^{35} -3.39423 q^{37} +1.54641 q^{38} +13.0622 q^{40} -4.00124 q^{41} -0.430509 q^{43} +1.02791 q^{44} +1.23800 q^{46} +2.62558 q^{47} +5.19923 q^{49} +15.2848 q^{50} +2.84783 q^{52} +2.43027 q^{53} +7.03549 q^{55} +10.7500 q^{56} -8.53636 q^{58} +8.73888 q^{59} -5.43530 q^{61} -3.21593 q^{62} +8.70394 q^{64} +19.4919 q^{65} -3.16898 q^{67} -3.66845 q^{68} +17.4129 q^{70} +7.09644 q^{71} +1.45109 q^{73} -3.98723 q^{74} -0.816263 q^{76} +5.79009 q^{77} -12.1010 q^{79} +10.0812 q^{80} -4.70029 q^{82} +1.76218 q^{83} -25.1087 q^{85} -0.505722 q^{86} +5.10223 q^{88} -7.16013 q^{89} +16.0415 q^{91} -0.653470 q^{92} +3.08429 q^{94} -5.58691 q^{95} +18.8569 q^{97} +6.10758 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * 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 $

$2$	1.17471	0.830644	0.415322	−	0.909674i	$-0.363669\pi$
$2$	1.17471	0.830644	0.415322	+	0.909674i	$0.363669\pi$
$3$	0	0
$3$	0	0
$4$	−0.620061	−0.310030
$5$	−4.24400	−1.89798	−0.948988	−	0.315313i	$-0.897890\pi$
$5$	−4.24400	−1.89798	−0.948988	+	0.315313i	$0.897890\pi$
$6$	0	0
$7$	−3.49274	−1.32013	−0.660066	−	0.751208i	$-0.729472\pi$
$7$	−3.49274	−1.32013	−0.660066	+	0.751208i	$0.729472\pi$
$8$	−3.07781	−1.08817
$9$	0	0
$10$	−4.98546	−1.57654
$11$	−1.65775	−0.499830	−0.249915	−	0.968268i	$-0.580403\pi$
$11$	−1.65775	−0.499830	−0.249915	+	0.968268i	$0.580403\pi$
$12$	0	0
$13$	−4.59282	−1.27382	−0.636910	−	0.770938i	$-0.719788\pi$
$13$	−4.59282	−1.27382	−0.636910	+	0.770938i	$0.719788\pi$
$14$	−4.10295	−1.09656
$15$	0	0
$16$	−2.37540	−0.593851
$17$	5.91628	1.43491	0.717454	−	0.696606i	$-0.245307\pi$
$17$	5.91628	1.43491	0.717454	+	0.696606i	$0.245307\pi$
$18$	0	0
$19$	1.31642	0.302008	0.151004	−	0.988533i	$-0.451749\pi$
$19$	1.31642	0.302008	0.151004	+	0.988533i	$0.451749\pi$
$20$	2.63154	0.588430
$21$	0	0
$22$	−1.94737	−0.415181
$23$	1.05388	0.219749	0.109875	−	0.993945i	$-0.464955\pi$
$23$	1.05388	0.219749	0.109875	+	0.993945i	$0.464955\pi$
$24$	0	0
$25$	13.0116	2.60231
$26$	−5.39522	−1.05809
$27$	0	0
$28$	2.16571	0.409281
$29$	−7.26679	−1.34941	−0.674704	−	0.738088i	$-0.735729\pi$
$29$	−7.26679	−1.34941	−0.674704	+	0.738088i	$0.735729\pi$
$30$	0	0
$31$	−2.73764	−0.491695	−0.245848	−	0.969308i	$-0.579066\pi$
$31$	−2.73764	−0.491695	−0.245848	+	0.969308i	$0.579066\pi$
$32$	3.36521	0.594890
$33$	0	0
$34$	6.94990	1.19190
$35$	14.8232	2.50558
$36$	0	0
$37$	−3.39423	−0.558009	−0.279004	−	0.960290i	$-0.590004\pi$
$37$	−3.39423	−0.558009	−0.279004	+	0.960290i	$0.590004\pi$
$38$	1.54641	0.250862
$39$	0	0
$40$	13.0622	2.06532
$41$	−4.00124	−0.624889	−0.312445	−	0.949936i	$-0.601148\pi$
$41$	−4.00124	−0.624889	−0.312445	+	0.949936i	$0.601148\pi$
$42$	0	0
$43$	−0.430509	−0.0656520	−0.0328260	−	0.999461i	$-0.510451\pi$
$43$	−0.430509	−0.0656520	−0.0328260	+	0.999461i	$0.510451\pi$
$44$	1.02791	0.154963
$45$	0	0
$46$	1.23800	0.182533
$47$	2.62558	0.382980	0.191490	−	0.981495i	$-0.438668\pi$
$47$	2.62558	0.382980	0.191490	+	0.981495i	$0.438668\pi$
$48$	0	0
$49$	5.19923	0.742748
$50$	15.2848	2.16159
$51$	0	0
$52$	2.84783	0.394923
$53$	2.43027	0.333823	0.166911	−	0.985972i	$-0.446621\pi$
$53$	2.43027	0.333823	0.166911	+	0.985972i	$0.446621\pi$
$54$	0	0
$55$	7.03549	0.948666
$56$	10.7500	1.43653
$57$	0	0
$58$	−8.53636	−1.12088
$59$	8.73888	1.13771	0.568853	−	0.822439i	$-0.307387\pi$
$59$	8.73888	1.13771	0.568853	+	0.822439i	$0.307387\pi$
$60$	0	0
$61$	−5.43530	−0.695919	−0.347959	−	0.937510i	$-0.613125\pi$
$61$	−5.43530	−0.695919	−0.347959	+	0.937510i	$0.613125\pi$
$62$	−3.21593	−0.408424
$63$	0	0
$64$	8.70394	1.08799
$65$	19.4919	2.41768
$66$	0	0
$67$	−3.16898	−0.387153	−0.193576	−	0.981085i	$-0.562009\pi$
$67$	−3.16898	−0.387153	−0.193576	+	0.981085i	$0.562009\pi$
$68$	−3.66845	−0.444865
$69$	0	0
$70$	17.4129	2.08124
$71$	7.09644	0.842192	0.421096	−	0.907016i	$-0.361646\pi$
$71$	7.09644	0.842192	0.421096	+	0.907016i	$0.361646\pi$
$72$	0	0
$73$	1.45109	0.169837	0.0849187	−	0.996388i	$-0.472937\pi$
$73$	1.45109	0.169837	0.0849187	+	0.996388i	$0.472937\pi$
$74$	−3.98723	−0.463507
$75$	0	0
$76$	−0.816263	−0.0936318
$77$	5.79009	0.659842
$78$	0	0
$79$	−12.1010	−1.36147	−0.680737	−	0.732528i	$-0.738340\pi$
$79$	−12.1010	−1.36147	−0.680737	+	0.732528i	$0.738340\pi$
$80$	10.0812	1.12711
$81$	0	0
$82$	−4.70029	−0.519061
$83$	1.76218	0.193425	0.0967123	−	0.995312i	$-0.469167\pi$
$83$	1.76218	0.193425	0.0967123	+	0.995312i	$0.469167\pi$
$84$	0	0
$85$	−25.1087	−2.72342
$86$	−0.505722	−0.0545334
$87$	0	0
$88$	5.10223	0.543900
$89$	−7.16013	−0.758972	−0.379486	−	0.925197i	$-0.623899\pi$
$89$	−7.16013	−0.758972	−0.379486	+	0.925197i	$0.623899\pi$
$90$	0	0
$91$	16.0415	1.68161
$92$	−0.653470	−0.0681290
$93$	0	0
$94$	3.08429	0.318120
$95$	−5.58691	−0.573205
$96$	0	0
$97$	18.8569	1.91463	0.957314	−	0.289049i	$-0.0933392\pi$
$97$	18.8569	1.91463	0.957314	+	0.289049i	$0.0933392\pi$
$98$	6.10758	0.616959
$99$	0	0
$100$	−8.06795	−0.806795
$101$	−4.51657	−0.449415	−0.224708	−	0.974426i	$-0.572143\pi$
$101$	−4.51657	−0.449415	−0.224708	+	0.974426i	$0.572143\pi$
$102$	0	0
$103$	−8.95725	−0.882584	−0.441292	−	0.897363i	$-0.645480\pi$
$103$	−8.95725	−0.882584	−0.441292	+	0.897363i	$0.645480\pi$
$104$	14.1358	1.38613
$105$	0	0
$106$	2.85485	0.277288
$107$	−14.0385	−1.35715	−0.678576	−	0.734531i	$-0.737402\pi$
$107$	−14.0385	−1.35715	−0.678576	+	0.734531i	$0.737402\pi$
$108$	0	0
$109$	16.9133	1.62000	0.809999	−	0.586431i	$-0.199467\pi$
$109$	16.9133	1.62000	0.809999	+	0.586431i	$0.199467\pi$
$110$	8.26465	0.788004
$111$	0	0
$112$	8.29667	0.783961
$113$	−5.76860	−0.542664	−0.271332	−	0.962486i	$-0.587464\pi$
$113$	−5.76860	−0.542664	−0.271332	+	0.962486i	$0.587464\pi$
$114$	0	0
$115$	−4.47267	−0.417079
$116$	4.50585	0.418358
$117$	0	0
$118$	10.2656	0.945028
$119$	−20.6640	−1.89427
$120$	0	0
$121$	−8.25187	−0.750170
$122$	−6.38489	−0.578061
$123$	0	0
$124$	1.69751	0.152440
$125$	−34.0011	−3.04115
$126$	0	0
$127$	−5.53392	−0.491056	−0.245528	−	0.969390i	$-0.578961\pi$
$127$	−5.53392	−0.491056	−0.245528	+	0.969390i	$0.578961\pi$
$128$	3.49418	0.308845
$129$	0	0
$130$	22.8973	2.00823
$131$	−14.2017	−1.24081	−0.620403	−	0.784283i	$-0.713031\pi$
$131$	−14.2017	−1.24081	−0.620403	+	0.784283i	$0.713031\pi$
$132$	0	0
$133$	−4.59793	−0.398691
$134$	−3.72263	−0.321586
$135$	0	0
$136$	−18.2092	−1.56142
$137$	22.7765	1.94593	0.972966	−	0.230949i	$-0.0741830\pi$
$137$	22.7765	1.94593	0.972966	+	0.230949i	$0.0741830\pi$
$138$	0	0
$139$	−5.55856	−0.471471	−0.235735	−	0.971817i	$-0.575750\pi$
$139$	−5.55856	−0.471471	−0.235735	+	0.971817i	$0.575750\pi$
$140$	−9.19128	−0.776805
$141$	0	0
$142$	8.33624	0.699562
$143$	7.61375	0.636694
$144$	0	0
$145$	30.8403	2.56114
$146$	1.70461	0.141074
$147$	0	0
$148$	2.10463	0.173000
$149$	−2.37340	−0.194437	−0.0972183	−	0.995263i	$-0.530995\pi$
$149$	−2.37340	−0.194437	−0.0972183	+	0.995263i	$0.530995\pi$
$150$	0	0
$151$	7.15649	0.582387	0.291193	−	0.956664i	$-0.405948\pi$
$151$	7.15649	0.582387	0.291193	+	0.956664i	$0.405948\pi$
$152$	−4.05170	−0.328636
$153$	0	0
$154$	6.80167	0.548094
$155$	11.6186	0.933226
$156$	0	0
$157$	8.12774	0.648664	0.324332	−	0.945943i	$-0.394860\pi$
$157$	8.12774	0.648664	0.324332	+	0.945943i	$0.394860\pi$
$158$	−14.2152	−1.13090
$159$	0	0
$160$	−14.2819	−1.12909
$161$	−3.68093	−0.290098
$162$	0	0
$163$	23.0502	1.80543	0.902717	−	0.430235i	$-0.141569\pi$
$163$	23.0502	1.80543	0.902717	+	0.430235i	$0.141569\pi$
$164$	2.48101	0.193735
$165$	0	0
$166$	2.07005	0.160667
$167$	25.6773	1.98697	0.993485	−	0.113962i	$-0.0363543\pi$
$167$	25.6773	1.98697	0.993485	+	0.113962i	$0.0363543\pi$
$168$	0	0
$169$	8.09401	0.622616
$170$	−29.4954	−2.26219
$171$	0	0
$172$	0.266942	0.0203541
$173$	−12.3899	−0.941990	−0.470995	−	0.882136i	$-0.656105\pi$
$173$	−12.3899	−0.941990	−0.470995	+	0.882136i	$0.656105\pi$
$174$	0	0
$175$	−45.4460	−3.43539
$176$	3.93782	0.296825
$177$	0	0
$178$	−8.41107	−0.630436
$179$	4.88679	0.365256	0.182628	−	0.983182i	$-0.441540\pi$
$179$	4.88679	0.365256	0.182628	+	0.983182i	$0.441540\pi$
$180$	0	0
$181$	8.65099	0.643023	0.321512	−	0.946906i	$-0.395809\pi$
$181$	8.65099	0.643023	0.321512	+	0.946906i	$0.395809\pi$
$182$	18.8441	1.39682
$183$	0	0
$184$	−3.24364	−0.239124
$185$	14.4051	1.05909
$186$	0	0
$187$	−9.80771	−0.717210
$188$	−1.62802	−0.118735
$189$	0	0
$190$	−6.56299	−0.476129
$191$	−7.45421	−0.539368	−0.269684	−	0.962949i	$-0.586919\pi$
$191$	−7.45421	−0.539368	−0.269684	+	0.962949i	$0.586919\pi$
$192$	0	0
$193$	−10.9167	−0.785800	−0.392900	−	0.919581i	$-0.628528\pi$
$193$	−10.9167	−0.785800	−0.392900	+	0.919581i	$0.628528\pi$
$194$	22.1514	1.59037
$195$	0	0
$196$	−3.22384	−0.230274
$197$	−11.1115	−0.791660	−0.395830	−	0.918324i	$-0.629543\pi$
$197$	−11.1115	−0.791660	−0.395830	+	0.918324i	$0.629543\pi$
$198$	0	0
$199$	3.61220	0.256062	0.128031	−	0.991770i	$-0.459134\pi$
$199$	3.61220	0.256062	0.128031	+	0.991770i	$0.459134\pi$
$200$	−40.0470	−2.83175
$201$	0	0
$202$	−5.30565	−0.373304
$203$	25.3810	1.78140
$204$	0	0
$205$	16.9813	1.18602
$206$	−10.5222	−0.733114
$207$	0	0
$208$	10.9098	0.756459
$209$	−2.18230	−0.150953
$210$	0	0
$211$	−2.95298	−0.203292	−0.101646	−	0.994821i	$-0.532411\pi$
$211$	−2.95298	−0.203292	−0.101646	+	0.994821i	$0.532411\pi$
$212$	−1.50691	−0.103495
$213$	0	0
$214$	−16.4911	−1.12731
$215$	1.82708	0.124606
$216$	0	0
$217$	9.56188	0.649103
$218$	19.8682	1.34564
$219$	0	0
$220$	−4.36243	−0.294115
$221$	−27.1724	−1.82781
$222$	0	0
$223$	−2.22067	−0.148707	−0.0743536	−	0.997232i	$-0.523689\pi$
$223$	−2.22067	−0.148707	−0.0743536	+	0.997232i	$0.523689\pi$
$224$	−11.7538	−0.785334
$225$	0	0
$226$	−6.77642	−0.450761
$227$	−29.0338	−1.92704	−0.963519	−	0.267639i	$-0.913756\pi$
$227$	−29.0338	−1.92704	−0.963519	+	0.267639i	$0.913756\pi$
$228$	0	0
$229$	20.3033	1.34168	0.670841	−	0.741602i	$-0.265933\pi$
$229$	20.3033	1.34168	0.670841	+	0.741602i	$0.265933\pi$
$230$	−5.25408	−0.346444
$231$	0	0
$232$	22.3658	1.46838
$233$	−9.78116	−0.640785	−0.320393	−	0.947285i	$-0.603815\pi$
$233$	−9.78116	−0.640785	−0.320393	+	0.947285i	$0.603815\pi$
$234$	0	0
$235$	−11.1430	−0.726887
$236$	−5.41864	−0.352723
$237$	0	0
$238$	−24.2742	−1.57346
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	12.7466	0.821079	0.410539	−	0.911843i	$-0.365340\pi$
$241$	12.7466	0.821079	0.410539	+	0.911843i	$0.365340\pi$
$242$	−9.69353	−0.623124
$243$	0	0
$244$	3.37022	0.215756
$245$	−22.0656	−1.40972
$246$	0	0
$247$	−6.04610	−0.384704
$248$	8.42594	0.535048
$249$	0	0
$250$	−39.9413	−2.52611
$251$	5.65259	0.356788	0.178394	−	0.983959i	$-0.442910\pi$
$251$	5.65259	0.356788	0.178394	+	0.983959i	$0.442910\pi$
$252$	0	0
$253$	−1.74707	−0.109837
$254$	−6.50074	−0.407893
$255$	0	0
$256$	−13.3032	−0.831453
$257$	11.5121	0.718104	0.359052	−	0.933318i	$-0.383100\pi$
$257$	11.5121	0.718104	0.359052	+	0.933318i	$0.383100\pi$
$258$	0	0
$259$	11.8552	0.736645
$260$	−12.0862	−0.749554
$261$	0	0
$262$	−16.6828	−1.03067
$263$	28.4676	1.75539	0.877694	−	0.479221i	$-0.159081\pi$
$263$	28.4676	1.75539	0.877694	+	0.479221i	$0.159081\pi$
$264$	0	0
$265$	−10.3141	−0.633587
$266$	−5.40122	−0.331170
$267$	0	0
$268$	1.96496	0.120029
$269$	6.52324	0.397729	0.198865	−	0.980027i	$-0.436275\pi$
$269$	6.52324	0.397729	0.198865	+	0.980027i	$0.436275\pi$
$270$	0	0
$271$	18.4879	1.12306	0.561531	−	0.827456i	$-0.310213\pi$
$271$	18.4879	1.12306	0.561531	+	0.827456i	$0.310213\pi$
$272$	−14.0535	−0.852121
$273$	0	0
$274$	26.7558	1.61638
$275$	−21.5699	−1.30071
$276$	0	0
$277$	−11.4890	−0.690306	−0.345153	−	0.938546i	$-0.612173\pi$
$277$	−11.4890	−0.690306	−0.345153	+	0.938546i	$0.612173\pi$
$278$	−6.52969	−0.391625
$279$	0	0
$280$	−45.6229	−2.72649
$281$	−8.29737	−0.494980	−0.247490	−	0.968890i	$-0.579606\pi$
$281$	−8.29737	−0.494980	−0.247490	+	0.968890i	$0.579606\pi$
$282$	0	0
$283$	16.7394	0.995055	0.497528	−	0.867448i	$-0.334241\pi$
$283$	16.7394	0.995055	0.497528	+	0.867448i	$0.334241\pi$
$284$	−4.40022	−0.261105
$285$	0	0
$286$	8.94393	0.528866
$287$	13.9753	0.824936
$288$	0	0
$289$	18.0023	1.05896
$290$	36.2283	2.12740
$291$	0	0
$292$	−0.899765	−0.0526547
$293$	−12.8035	−0.747988	−0.373994	−	0.927431i	$-0.622012\pi$
$293$	−12.8035	−0.747988	−0.373994	+	0.927431i	$0.622012\pi$
$294$	0	0
$295$	−37.0878	−2.15934
$296$	10.4468	0.607208
$297$	0	0
$298$	−2.78805	−0.161508
$299$	−4.84029	−0.279921
$300$	0	0
$301$	1.50366	0.0866692
$302$	8.40679	0.483756
$303$	0	0
$304$	−3.12704	−0.179348
$305$	23.0674	1.32084
$306$	0	0
$307$	−8.85775	−0.505539	−0.252769	−	0.967527i	$-0.581341\pi$
$307$	−8.85775	−0.505539	−0.252769	+	0.967527i	$0.581341\pi$
$308$	−3.59021	−0.204571
$309$	0	0
$310$	13.6484	0.775178
$311$	26.6500	1.51118	0.755591	−	0.655044i	$-0.227350\pi$
$311$	26.6500	1.51118	0.755591	+	0.655044i	$0.227350\pi$
$312$	0	0
$313$	18.0530	1.02041	0.510207	−	0.860052i	$-0.329569\pi$
$313$	18.0530	1.02041	0.510207	+	0.860052i	$0.329569\pi$
$314$	9.54772	0.538809
$315$	0	0
$316$	7.50338	0.422098
$317$	8.03830	0.451476	0.225738	−	0.974188i	$-0.427521\pi$
$317$	8.03830	0.451476	0.225738	+	0.974188i	$0.427521\pi$
$318$	0	0
$319$	12.0465	0.674475
$320$	−36.9396	−2.06498
$321$	0	0
$322$	−4.32402	−0.240968
$323$	7.78833	0.433354
$324$	0	0
$325$	−59.7597	−3.31487
$326$	27.0773	1.49967
$327$	0	0
$328$	12.3151	0.679985
$329$	−9.17047	−0.505584
$330$	0	0
$331$	20.4372	1.12333	0.561664	−	0.827365i	$-0.310161\pi$
$331$	20.4372	1.12333	0.561664	+	0.827365i	$0.310161\pi$
$332$	−1.09266	−0.0599675
$333$	0	0
$334$	30.1633	1.65047
$335$	13.4492	0.734806
$336$	0	0
$337$	−20.6458	−1.12465	−0.562325	−	0.826917i	$-0.690093\pi$
$337$	−20.6458	−1.12465	−0.562325	+	0.826917i	$0.690093\pi$
$338$	9.50810	0.517172
$339$	0	0
$340$	15.5689	0.844343
$341$	4.53833	0.245764
$342$	0	0
$343$	6.28961	0.339607
$344$	1.32502	0.0714404
$345$	0	0
$346$	−14.5546	−0.782458
$347$	−17.8392	−0.957660	−0.478830	−	0.877908i	$-0.658939\pi$
$347$	−17.8392	−0.957660	−0.478830	+	0.877908i	$0.658939\pi$
$348$	0	0
$349$	18.6415	0.997859	0.498930	−	0.866643i	$-0.333727\pi$
$349$	18.6415	0.997859	0.498930	+	0.866643i	$0.333727\pi$
$350$	−53.3858	−2.85359
$351$	0	0
$352$	−5.57867	−0.297344
$353$	4.27639	0.227609	0.113805	−	0.993503i	$-0.463696\pi$
$353$	4.27639	0.227609	0.113805	+	0.993503i	$0.463696\pi$
$354$	0	0
$355$	−30.1173	−1.59846
$356$	4.43972	0.235304
$357$	0	0
$358$	5.74056	0.303398
$359$	−6.02148	−0.317802	−0.158901	−	0.987295i	$-0.550795\pi$
$359$	−6.02148	−0.317802	−0.158901	+	0.987295i	$0.550795\pi$
$360$	0	0
$361$	−17.2670	−0.908791
$362$	10.1624	0.534124
$363$	0	0
$364$	−9.94672	−0.521350
$365$	−6.15843	−0.322347
$366$	0	0
$367$	−21.6838	−1.13189	−0.565943	−	0.824444i	$-0.691488\pi$
$367$	−21.6838	−1.13189	−0.565943	+	0.824444i	$0.691488\pi$
$368$	−2.50339	−0.130498
$369$	0	0
$370$	16.9218	0.879724
$371$	−8.48829	−0.440690
$372$	0	0
$373$	−18.5549	−0.960735	−0.480368	−	0.877067i	$-0.659497\pi$
$373$	−18.5549	−0.960735	−0.480368	+	0.877067i	$0.659497\pi$
$374$	−11.5212	−0.595747
$375$	0	0
$376$	−8.08103	−0.416747
$377$	33.3751	1.71890
$378$	0	0
$379$	18.8514	0.968334	0.484167	−	0.874976i	$-0.339123\pi$
$379$	18.8514	0.968334	0.484167	+	0.874976i	$0.339123\pi$
$380$	3.46422	0.177711
$381$	0	0
$382$	−8.75653	−0.448023
$383$	−20.3517	−1.03992	−0.519962	−	0.854190i	$-0.674054\pi$
$383$	−20.3517	−1.03992	−0.519962	+	0.854190i	$0.674054\pi$
$384$	0	0
$385$	−24.5732	−1.25236
$386$	−12.8239	−0.652720
$387$	0	0
$388$	−11.6924	−0.593593
$389$	31.2215	1.58299	0.791496	−	0.611175i	$-0.209303\pi$
$389$	31.2215	1.58299	0.791496	+	0.611175i	$0.209303\pi$
$390$	0	0
$391$	6.23505	0.315320
$392$	−16.0022	−0.808235
$393$	0	0
$394$	−13.0527	−0.657588
$395$	51.3569	2.58404
$396$	0	0
$397$	−6.03972	−0.303125	−0.151562	−	0.988448i	$-0.548430\pi$
$397$	−6.03972	−0.303125	−0.151562	+	0.988448i	$0.548430\pi$
$398$	4.24328	0.212697
$399$	0	0
$400$	−30.9077	−1.54538
$401$	−20.5769	−1.02756	−0.513781	−	0.857922i	$-0.671755\pi$
$401$	−20.5769	−1.02756	−0.513781	+	0.857922i	$0.671755\pi$
$402$	0	0
$403$	12.5735	0.626331
$404$	2.80055	0.139332
$405$	0	0
$406$	29.8153	1.47971
$407$	5.62679	0.278910
$408$	0	0
$409$	−1.97031	−0.0974258	−0.0487129	−	0.998813i	$-0.515512\pi$
$409$	−1.97031	−0.0974258	−0.0487129	+	0.998813i	$0.515512\pi$
$410$	19.9481	0.985164
$411$	0	0
$412$	5.55404	0.273628
$413$	−30.5226	−1.50192
$414$	0	0
$415$	−7.47871	−0.367115
$416$	−15.4558	−0.757783
$417$	0	0
$418$	−2.56357	−0.125388
$419$	29.9620	1.46374	0.731871	−	0.681443i	$-0.238647\pi$
$419$	29.9620	1.46374	0.731871	+	0.681443i	$0.238647\pi$
$420$	0	0
$421$	−11.0964	−0.540807	−0.270404	−	0.962747i	$-0.587157\pi$
$421$	−11.0964	−0.540807	−0.270404	+	0.962747i	$0.587157\pi$
$422$	−3.46889	−0.168863
$423$	0	0
$424$	−7.47989	−0.363256
$425$	76.9799	3.73408
$426$	0	0
$427$	18.9841	0.918704
$428$	8.70471	0.420758
$429$	0	0
$430$	2.14629	0.103503
$431$	25.3432	1.22074	0.610370	−	0.792116i	$-0.291021\pi$
$431$	25.3432	1.22074	0.610370	+	0.792116i	$0.291021\pi$
$432$	0	0
$433$	37.3323	1.79407	0.897037	−	0.441956i	$-0.145715\pi$
$433$	37.3323	1.79407	0.897037	+	0.441956i	$0.145715\pi$
$434$	11.2324	0.539173
$435$	0	0
$436$	−10.4873	−0.502249
$437$	1.38735	0.0663661
$438$	0	0
$439$	25.3643	1.21057	0.605287	−	0.796007i	$-0.293058\pi$
$439$	25.3643	1.21057	0.605287	+	0.796007i	$0.293058\pi$
$440$	−21.6539	−1.03231
$441$	0	0
$442$	−31.9196	−1.51826
$443$	−15.5254	−0.737634	−0.368817	−	0.929502i	$-0.620237\pi$
$443$	−15.5254	−0.737634	−0.368817	+	0.929502i	$0.620237\pi$
$444$	0	0
$445$	30.3876	1.44051
$446$	−2.60864	−0.123523
$447$	0	0
$448$	−30.4006	−1.43629
$449$	25.6036	1.20831	0.604154	−	0.796868i	$-0.293511\pi$
$449$	25.6036	1.20831	0.604154	+	0.796868i	$0.293511\pi$
$450$	0	0
$451$	6.63306	0.312339
$452$	3.57688	0.168242
$453$	0	0
$454$	−34.1062	−1.60068
$455$	−68.0803	−3.19165
$456$	0	0
$457$	−0.509124	−0.0238158	−0.0119079	−	0.999929i	$-0.503790\pi$
$457$	−0.509124	−0.0238158	−0.0119079	+	0.999929i	$0.503790\pi$
$458$	23.8505	1.11446
$459$	0	0
$460$	2.77333	0.129307
$461$	12.0585	0.561620	0.280810	−	0.959763i	$-0.409397\pi$
$461$	12.0585	0.561620	0.280810	+	0.959763i	$0.409397\pi$
$462$	0	0
$463$	−8.83985	−0.410823	−0.205411	−	0.978676i	$-0.565853\pi$
$463$	−8.83985	−0.410823	−0.205411	+	0.978676i	$0.565853\pi$
$464$	17.2616	0.801348
$465$	0	0
$466$	−11.4900	−0.532265
$467$	13.0612	0.604401	0.302201	−	0.953244i	$-0.402279\pi$
$467$	13.0612	0.604401	0.302201	+	0.953244i	$0.402279\pi$
$468$	0	0
$469$	11.0684	0.511093
$470$	−13.0897	−0.603784
$471$	0	0
$472$	−26.8966	−1.23802
$473$	0.713676	0.0328148
$474$	0	0
$475$	17.1287	0.785920
$476$	12.8129	0.587280
$477$	0	0
$478$	−1.17471	−0.0537299
$479$	−1.44631	−0.0660835	−0.0330418	−	0.999454i	$-0.510519\pi$
$479$	−1.44631	−0.0660835	−0.0330418	+	0.999454i	$0.510519\pi$
$480$	0	0
$481$	15.5891	0.710802
$482$	14.9735	0.682024
$483$	0	0
$484$	5.11666	0.232575
$485$	−80.0287	−3.63392
$486$	0	0
$487$	−8.61040	−0.390174	−0.195087	−	0.980786i	$-0.562499\pi$
$487$	−8.61040	−0.390174	−0.195087	+	0.980786i	$0.562499\pi$
$488$	16.7288	0.757277
$489$	0	0
$490$	−25.9206	−1.17097
$491$	5.85303	0.264144	0.132072	−	0.991240i	$-0.457837\pi$
$491$	5.85303	0.264144	0.132072	+	0.991240i	$0.457837\pi$
$492$	0	0
$493$	−42.9923	−1.93628
$494$	−7.10240	−0.319552
$495$	0	0
$496$	6.50301	0.291994
$497$	−24.7860	−1.11180
$498$	0	0
$499$	−8.27596	−0.370483	−0.185241	−	0.982693i	$-0.559307\pi$
$499$	−8.27596	−0.370483	−0.185241	+	0.982693i	$0.559307\pi$
$500$	21.0827	0.942848
$501$	0	0
$502$	6.64014	0.296364
$503$	−4.32356	−0.192778	−0.0963890	−	0.995344i	$-0.530729\pi$
$503$	−4.32356	−0.192778	−0.0963890	+	0.995344i	$0.530729\pi$
$504$	0	0
$505$	19.1683	0.852980
$506$	−2.05230	−0.0912358
$507$	0	0
$508$	3.43136	0.152242
$509$	−2.42438	−0.107459	−0.0537293	−	0.998556i	$-0.517111\pi$
$509$	−2.42438	−0.107459	−0.0537293	+	0.998556i	$0.517111\pi$
$510$	0	0
$511$	−5.06828	−0.224208
$512$	−22.6158	−0.999486
$513$	0	0
$514$	13.5233	0.596489
$515$	38.0146	1.67512
$516$	0	0
$517$	−4.35255	−0.191425
$518$	13.9264	0.611890
$519$	0	0
$520$	−59.9924	−2.63084
$521$	−30.9049	−1.35397	−0.676985	−	0.735997i	$-0.736714\pi$
$521$	−30.9049	−1.35397	−0.676985	+	0.735997i	$0.736714\pi$
$522$	0	0
$523$	−16.6684	−0.728858	−0.364429	−	0.931231i	$-0.618736\pi$
$523$	−16.6684	−0.728858	−0.364429	+	0.931231i	$0.618736\pi$
$524$	8.80590	0.384688
$525$	0	0
$526$	33.4411	1.45810
$527$	−16.1967	−0.705537
$528$	0	0
$529$	−21.8893	−0.951710
$530$	−12.1160	−0.526286
$531$	0	0
$532$	2.85099	0.123606
$533$	18.3770	0.795996
$534$	0	0
$535$	59.5793	2.57584
$536$	9.75351	0.421288
$537$	0	0
$538$	7.66291	0.330371
$539$	−8.61903	−0.371248
$540$	0	0
$541$	17.2504	0.741654	0.370827	−	0.928702i	$-0.379074\pi$
$541$	17.2504	0.741654	0.370827	+	0.928702i	$0.379074\pi$
$542$	21.7179	0.932864
$543$	0	0
$544$	19.9095	0.853613
$545$	−71.7800	−3.07472
$546$	0	0
$547$	−3.29649	−0.140948	−0.0704738	−	0.997514i	$-0.522451\pi$
$547$	−3.29649	−0.140948	−0.0704738	+	0.997514i	$0.522451\pi$
$548$	−14.1228	−0.603298
$549$	0	0
$550$	−25.3383	−1.08043
$551$	−9.56618	−0.407533
$552$	0	0
$553$	42.2658	1.79733
$554$	−13.4962	−0.573399
$555$	0	0
$556$	3.44664	0.146170
$557$	−33.0021	−1.39834	−0.699171	−	0.714954i	$-0.746448\pi$
$557$	−33.0021	−1.39834	−0.699171	+	0.714954i	$0.746448\pi$
$558$	0	0
$559$	1.97725	0.0836287
$560$	−35.2111	−1.48794
$561$	0	0
$562$	−9.74699	−0.411152
$563$	−40.1518	−1.69220	−0.846098	−	0.533028i	$-0.821054\pi$
$563$	−40.1518	−1.69220	−0.846098	+	0.533028i	$0.821054\pi$
$564$	0	0
$565$	24.4819	1.02996
$566$	19.6639	0.826537
$567$	0	0
$568$	−21.8415	−0.916447
$569$	36.6551	1.53666	0.768332	−	0.640052i	$-0.221087\pi$
$569$	36.6551	1.53666	0.768332	+	0.640052i	$0.221087\pi$
$570$	0	0
$571$	−27.7251	−1.16026	−0.580129	−	0.814525i	$-0.696998\pi$
$571$	−27.7251	−1.16026	−0.580129	+	0.814525i	$0.696998\pi$
$572$	−4.72099	−0.197394
$573$	0	0
$574$	16.4169	0.685228
$575$	13.7126	0.571856
$576$	0	0
$577$	32.6152	1.35779	0.678893	−	0.734237i	$-0.262460\pi$
$577$	32.6152	1.35779	0.678893	+	0.734237i	$0.262460\pi$
$578$	21.1475	0.879619
$579$	0	0
$580$	−19.1228	−0.794033
$581$	−6.15485	−0.255346
$582$	0	0
$583$	−4.02877	−0.166855
$584$	−4.46618	−0.184812
$585$	0	0
$586$	−15.0404	−0.621312
$587$	−17.8116	−0.735164	−0.367582	−	0.929991i	$-0.619814\pi$
$587$	−17.8116	−0.735164	−0.367582	+	0.929991i	$0.619814\pi$
$588$	0	0
$589$	−3.60390	−0.148496
$590$	−43.5674	−1.79364
$591$	0	0
$592$	8.06267	0.331374
$593$	14.3955	0.591154	0.295577	−	0.955319i	$-0.404488\pi$
$593$	14.3955	0.591154	0.295577	+	0.955319i	$0.404488\pi$
$594$	0	0
$595$	87.6981	3.59527
$596$	1.47165	0.0602813
$597$	0	0
$598$	−5.68592	−0.232515
$599$	38.7266	1.58233	0.791163	−	0.611605i	$-0.209476\pi$
$599$	38.7266	1.58233	0.791163	+	0.611605i	$0.209476\pi$
$600$	0	0
$601$	11.8964	0.485263	0.242631	−	0.970119i	$-0.421989\pi$
$601$	11.8964	0.485263	0.242631	+	0.970119i	$0.421989\pi$
$602$	1.76636	0.0719913
$603$	0	0
$604$	−4.43746	−0.180558
$605$	35.0209	1.42380
$606$	0	0
$607$	−34.9424	−1.41827	−0.709133	−	0.705074i	$-0.750914\pi$
$607$	−34.9424	−1.41827	−0.709133	+	0.705074i	$0.750914\pi$
$608$	4.43004	0.179662
$609$	0	0
$610$	27.0975	1.09715
$611$	−12.0588	−0.487847
$612$	0	0
$613$	−15.7293	−0.635302	−0.317651	−	0.948208i	$-0.602894\pi$
$613$	−15.7293	−0.635302	−0.317651	+	0.948208i	$0.602894\pi$
$614$	−10.4053	−0.419923
$615$	0	0
$616$	−17.8208	−0.718020
$617$	14.0914	0.567298	0.283649	−	0.958928i	$-0.408455\pi$
$617$	14.0914	0.567298	0.283649	+	0.958928i	$0.408455\pi$
$618$	0	0
$619$	15.9313	0.640333	0.320167	−	0.947361i	$-0.396261\pi$
$619$	15.9313	0.640333	0.320167	+	0.947361i	$0.396261\pi$
$620$	−7.20422	−0.289328
$621$	0	0
$622$	31.3060	1.25525
$623$	25.0085	1.00194
$624$	0	0
$625$	79.2428	3.16971
$626$	21.2070	0.847601
$627$	0	0
$628$	−5.03969	−0.201106
$629$	−20.0812	−0.800691
$630$	0	0
$631$	−20.2867	−0.807602	−0.403801	−	0.914847i	$-0.632311\pi$
$631$	−20.2867	−0.807602	−0.403801	+	0.914847i	$0.632311\pi$
$632$	37.2447	1.48151
$633$	0	0
$634$	9.44266	0.375016
$635$	23.4860	0.932012
$636$	0	0
$637$	−23.8792	−0.946127
$638$	14.1511	0.560249
$639$	0	0
$640$	−14.8293	−0.586180
$641$	−34.3083	−1.35510	−0.677548	−	0.735478i	$-0.736957\pi$
$641$	−34.3083	−1.35510	−0.677548	+	0.735478i	$0.736957\pi$
$642$	0	0
$643$	−45.5146	−1.79492	−0.897460	−	0.441097i	$-0.854590\pi$
$643$	−45.5146	−1.79492	−0.897460	+	0.441097i	$0.854590\pi$
$644$	2.28240	0.0899392
$645$	0	0
$646$	9.14901	0.359963
$647$	5.51888	0.216969	0.108485	−	0.994098i	$-0.465400\pi$
$647$	5.51888	0.216969	0.108485	+	0.994098i	$0.465400\pi$
$648$	0	0
$649$	−14.4869	−0.568660
$650$	−70.2003	−2.75348
$651$	0	0
$652$	−14.2925	−0.559739
$653$	−2.09786	−0.0820955	−0.0410478	−	0.999157i	$-0.513070\pi$
$653$	−2.09786	−0.0820955	−0.0410478	+	0.999157i	$0.513070\pi$
$654$	0	0
$655$	60.2720	2.35502
$656$	9.50457	0.371091
$657$	0	0
$658$	−10.7726	−0.419961
$659$	−13.2935	−0.517839	−0.258920	−	0.965899i	$-0.583366\pi$
$659$	−13.2935	−0.517839	−0.258920	+	0.965899i	$0.583366\pi$
$660$	0	0
$661$	21.4867	0.835734	0.417867	−	0.908508i	$-0.362778\pi$
$661$	21.4867	0.835734	0.417867	+	0.908508i	$0.362778\pi$
$662$	24.0077	0.933086
$663$	0	0
$664$	−5.42366	−0.210479
$665$	19.5136	0.756705
$666$	0	0
$667$	−7.65833	−0.296532
$668$	−15.9215	−0.616021
$669$	0	0
$670$	15.7988	0.610363
$671$	9.01037	0.347841
$672$	0	0
$673$	−10.7493	−0.414356	−0.207178	−	0.978303i	$-0.566428\pi$
$673$	−10.7493	−0.414356	−0.207178	+	0.978303i	$0.566428\pi$
$674$	−24.2528	−0.934183
$675$	0	0
$676$	−5.01878	−0.193030
$677$	17.4102	0.669127	0.334563	−	0.942373i	$-0.391411\pi$
$677$	17.4102	0.669127	0.334563	+	0.942373i	$0.391411\pi$
$678$	0	0
$679$	−65.8623	−2.52756
$680$	77.2797	2.96354
$681$	0	0
$682$	5.33121	0.204143
$683$	−8.18146	−0.313055	−0.156527	−	0.987674i	$-0.550030\pi$
$683$	−8.18146	−0.313055	−0.156527	+	0.987674i	$0.550030\pi$
$684$	0	0
$685$	−96.6637	−3.69333
$686$	7.38845	0.282092
$687$	0	0
$688$	1.02263	0.0389875
$689$	−11.1618	−0.425230
$690$	0	0
$691$	15.3945	0.585633	0.292816	−	0.956169i	$-0.405408\pi$
$691$	15.3945	0.585633	0.292816	+	0.956169i	$0.405408\pi$
$692$	7.68251	0.292045
$693$	0	0
$694$	−20.9559	−0.795474
$695$	23.5905	0.894840
$696$	0	0
$697$	−23.6725	−0.896658
$698$	21.8984	0.828866
$699$	0	0
$700$	28.1793	1.06508
$701$	33.7538	1.27486	0.637432	−	0.770506i	$-0.279996\pi$
$701$	33.7538	1.27486	0.637432	+	0.770506i	$0.279996\pi$
$702$	0	0
$703$	−4.46825	−0.168523
$704$	−14.4290	−0.543812
$705$	0	0
$706$	5.02351	0.189062
$707$	15.7752	0.593288
$708$	0	0
$709$	0.400804	0.0150525	0.00752625	−	0.999972i	$-0.497604\pi$
$709$	0.400804	0.0150525	0.00752625	+	0.999972i	$0.497604\pi$
$710$	−35.3790	−1.32775
$711$	0	0
$712$	22.0375	0.825890
$713$	−2.88515	−0.108050
$714$	0	0
$715$	−32.3128	−1.20843
$716$	−3.03011	−0.113241
$717$	0	0
$718$	−7.07349	−0.263980
$719$	−25.8696	−0.964774	−0.482387	−	0.875958i	$-0.660230\pi$
$719$	−25.8696	−0.964774	−0.482387	+	0.875958i	$0.660230\pi$
$720$	0	0
$721$	31.2854	1.16513
$722$	−20.2837	−0.754882
$723$	0	0
$724$	−5.36414	−0.199357
$725$	−94.5522	−3.51158
$726$	0	0
$727$	−9.85196	−0.365389	−0.182694	−	0.983170i	$-0.558482\pi$
$727$	−9.85196	−0.365389	−0.182694	+	0.983170i	$0.558482\pi$
$728$	−49.3727	−1.82988
$729$	0	0
$730$	−7.23436	−0.267756
$731$	−2.54701	−0.0942045
$732$	0	0
$733$	51.5739	1.90493	0.952463	−	0.304653i	$-0.0985406\pi$
$733$	51.5739	1.90493	0.952463	+	0.304653i	$0.0985406\pi$
$734$	−25.4722	−0.940194
$735$	0	0
$736$	3.54653	0.130727
$737$	5.25338	0.193511
$738$	0	0
$739$	15.1696	0.558021	0.279011	−	0.960288i	$-0.409994\pi$
$739$	15.1696	0.558021	0.279011	+	0.960288i	$0.409994\pi$
$740$	−8.93206	−0.328349
$741$	0	0
$742$	−9.97126	−0.366057
$743$	1.50092	0.0550632	0.0275316	−	0.999621i	$-0.491235\pi$
$743$	1.50092	0.0550632	0.0275316	+	0.999621i	$0.491235\pi$
$744$	0	0
$745$	10.0727	0.369036
$746$	−21.7966	−0.798029
$747$	0	0
$748$	6.08137	0.222357
$749$	49.0328	1.79162
$750$	0	0
$751$	22.8702	0.834545	0.417273	−	0.908781i	$-0.362986\pi$
$751$	22.8702	0.834545	0.417273	+	0.908781i	$0.362986\pi$
$752$	−6.23681	−0.227433
$753$	0	0
$754$	39.2060	1.42780
$755$	−30.3722	−1.10536
$756$	0	0
$757$	2.61218	0.0949412	0.0474706	−	0.998873i	$-0.484884\pi$
$757$	2.61218	0.0949412	0.0474706	+	0.998873i	$0.484884\pi$
$758$	22.1449	0.804341
$759$	0	0
$760$	17.1954	0.623743
$761$	36.2628	1.31452	0.657262	−	0.753662i	$-0.271714\pi$
$761$	36.2628	1.31452	0.657262	+	0.753662i	$0.271714\pi$
$762$	0	0
$763$	−59.0737	−2.13861
$764$	4.62206	0.167220
$765$	0	0
$766$	−23.9073	−0.863806
$767$	−40.1361	−1.44923
$768$	0	0
$769$	−35.4277	−1.27755	−0.638777	−	0.769392i	$-0.720559\pi$
$769$	−35.4277	−1.27755	−0.638777	+	0.769392i	$0.720559\pi$
$770$	−28.8663	−1.04027
$771$	0	0
$772$	6.76901	0.243622
$773$	44.6800	1.60703	0.803514	−	0.595285i	$-0.202961\pi$
$773$	44.6800	1.60703	0.803514	+	0.595285i	$0.202961\pi$
$774$	0	0
$775$	−35.6210	−1.27954
$776$	−58.0379	−2.08344
$777$	0	0
$778$	36.6761	1.31490
$779$	−5.26733	−0.188722
$780$	0	0
$781$	−11.7641	−0.420953
$782$	7.32436	0.261919
$783$	0	0
$784$	−12.3503	−0.441081
$785$	−34.4941	−1.23115
$786$	0	0
$787$	−2.24342	−0.0799694	−0.0399847	−	0.999200i	$-0.512731\pi$
$787$	−2.24342	−0.0799694	−0.0399847	+	0.999200i	$0.512731\pi$
$788$	6.88979	0.245439
$789$	0	0
$790$	60.3293	2.14642
$791$	20.1482	0.716388
$792$	0	0
$793$	24.9634	0.886475
$794$	−7.09490	−0.251789
$795$	0	0
$796$	−2.23979	−0.0793871
$797$	7.53191	0.266794	0.133397	−	0.991063i	$-0.457412\pi$
$797$	7.53191	0.266794	0.133397	+	0.991063i	$0.457412\pi$
$798$	0	0
$799$	15.5337	0.549541
$800$	43.7866	1.54809
$801$	0	0
$802$	−24.1719	−0.853538
$803$	−2.40555	−0.0848899
$804$	0	0
$805$	15.6219	0.550599
$806$	14.7702	0.520258
$807$	0	0
$808$	13.9011	0.489040
$809$	−10.7272	−0.377149	−0.188574	−	0.982059i	$-0.560387\pi$
$809$	−10.7272	−0.377149	−0.188574	+	0.982059i	$0.560387\pi$
$810$	0	0
$811$	−43.6078	−1.53128	−0.765638	−	0.643271i	$-0.777577\pi$
$811$	−43.6078	−1.53128	−0.765638	+	0.643271i	$0.777577\pi$
$812$	−15.7378	−0.552287
$813$	0	0
$814$	6.60984	0.231675
$815$	−97.8252	−3.42667
$816$	0	0
$817$	−0.566732	−0.0198274
$818$	−2.31454	−0.0809261
$819$	0	0
$820$	−10.5294	−0.367704
$821$	−6.08763	−0.212460	−0.106230	−	0.994342i	$-0.533878\pi$
$821$	−6.08763	−0.212460	−0.106230	+	0.994342i	$0.533878\pi$
$822$	0	0
$823$	10.7806	0.375788	0.187894	−	0.982189i	$-0.439834\pi$
$823$	10.7806	0.375788	0.187894	+	0.982189i	$0.439834\pi$
$824$	27.5687	0.960401
$825$	0	0
$826$	−35.8552	−1.24756
$827$	−39.5266	−1.37448	−0.687238	−	0.726432i	$-0.741177\pi$
$827$	−39.5266	−1.37448	−0.687238	+	0.726432i	$0.741177\pi$
$828$	0	0
$829$	−12.4152	−0.431197	−0.215599	−	0.976482i	$-0.569170\pi$
$829$	−12.4152	−0.431197	−0.215599	+	0.976482i	$0.569170\pi$
$830$	−8.78530	−0.304942
$831$	0	0
$832$	−39.9757	−1.38591
$833$	30.7601	1.06577
$834$	0	0
$835$	−108.975	−3.77122
$836$	1.35316	0.0468000
$837$	0	0
$838$	35.1967	1.21585
$839$	−20.6530	−0.713019	−0.356510	−	0.934292i	$-0.616033\pi$
$839$	−20.6530	−0.713019	−0.356510	+	0.934292i	$0.616033\pi$
$840$	0	0
$841$	23.8062	0.820904
$842$	−13.0351	−0.449218
$843$	0	0
$844$	1.83103	0.0630265
$845$	−34.3510	−1.18171
$846$	0	0
$847$	28.8216	0.990323
$848$	−5.77286	−0.198241
$849$	0	0
$850$	90.4290	3.10169
$851$	−3.57712	−0.122622
$852$	0	0
$853$	16.0152	0.548350	0.274175	−	0.961680i	$-0.411595\pi$
$853$	16.0152	0.548350	0.274175	+	0.961680i	$0.411595\pi$
$854$	22.3008	0.763116
$855$	0	0
$856$	43.2077	1.47681
$857$	48.8529	1.66878	0.834392	−	0.551171i	$-0.185819\pi$
$857$	48.8529	1.66878	0.834392	+	0.551171i	$0.185819\pi$
$858$	0	0
$859$	40.4419	1.37986	0.689931	−	0.723876i	$-0.257641\pi$
$859$	40.4419	1.37986	0.689931	+	0.723876i	$0.257641\pi$
$860$	−1.13290	−0.0386316
$861$	0	0
$862$	29.7709	1.01400
$863$	0.485673	0.0165325	0.00826625	−	0.999966i	$-0.497369\pi$
$863$	0.485673	0.0165325	0.00826625	+	0.999966i	$0.497369\pi$
$864$	0	0
$865$	52.5829	1.78787
$866$	43.8545	1.49024
$867$	0	0
$868$	−5.92894	−0.201241
$869$	20.0605	0.680506
$870$	0	0
$871$	14.5546	0.493163
$872$	−52.0558	−1.76283
$873$	0	0
$874$	1.62974	0.0551266
$875$	118.757	4.01471
$876$	0	0
$877$	−27.8626	−0.940854	−0.470427	−	0.882439i	$-0.655900\pi$
$877$	−27.8626	−0.940854	−0.470427	+	0.882439i	$0.655900\pi$
$878$	29.7957	1.00556
$879$	0	0
$880$	−16.7121	−0.563366
$881$	12.3587	0.416375	0.208187	−	0.978089i	$-0.433244\pi$
$881$	12.3587	0.416375	0.208187	+	0.978089i	$0.433244\pi$
$882$	0	0
$883$	−12.6974	−0.427301	−0.213650	−	0.976910i	$-0.568535\pi$
$883$	−12.6974	−0.427301	−0.213650	+	0.976910i	$0.568535\pi$
$884$	16.8485	0.566678
$885$	0	0
$886$	−18.2378	−0.612711
$887$	50.1954	1.68540	0.842699	−	0.538385i	$-0.180966\pi$
$887$	50.1954	1.68540	0.842699	+	0.538385i	$0.180966\pi$
$888$	0	0
$889$	19.3285	0.648258
$890$	35.6966	1.19655
$891$	0	0
$892$	1.37695	0.0461038
$893$	3.45638	0.115663
$894$	0	0
$895$	−20.7396	−0.693247
$896$	−12.2043	−0.407716
$897$	0	0
$898$	30.0767	1.00367
$899$	19.8939	0.663498
$900$	0	0
$901$	14.3781	0.479005
$902$	7.79191	0.259442
$903$	0	0
$904$	17.7546	0.590510
$905$	−36.7148	−1.22044
$906$	0	0
$907$	12.0685	0.400726	0.200363	−	0.979722i	$-0.435788\pi$
$907$	12.0685	0.400726	0.200363	+	0.979722i	$0.435788\pi$
$908$	18.0027	0.597440
$909$	0	0
$910$	−79.9745	−2.65113
$911$	10.3974	0.344481	0.172241	−	0.985055i	$-0.444899\pi$
$911$	10.3974	0.344481	0.172241	+	0.985055i	$0.444899\pi$
$912$	0	0
$913$	−2.92126	−0.0966795
$914$	−0.598072	−0.0197825
$915$	0	0
$916$	−12.5893	−0.415962
$917$	49.6028	1.63803
$918$	0	0
$919$	−0.896641	−0.0295775	−0.0147887	−	0.999891i	$-0.504708\pi$
$919$	−0.896641	−0.0295775	−0.0147887	+	0.999891i	$0.504708\pi$
$920$	13.7660	0.453852
$921$	0	0
$922$	14.1652	0.466506
$923$	−32.5927	−1.07280
$924$	0	0
$925$	−44.1643	−1.45211
$926$	−10.3842	−0.341247
$927$	0	0
$928$	−24.4543	−0.802750
$929$	−0.418202	−0.0137208	−0.00686038	−	0.999976i	$-0.502184\pi$
$929$	−0.418202	−0.0137208	−0.00686038	+	0.999976i	$0.502184\pi$
$930$	0	0
$931$	6.84440	0.224316
$932$	6.06491	0.198663
$933$	0	0
$934$	15.3431	0.502042
$935$	41.6239	1.36125
$936$	0	0
$937$	−51.4401	−1.68047	−0.840237	−	0.542219i	$-0.817584\pi$
$937$	−51.4401	−1.68047	−0.840237	+	0.542219i	$0.817584\pi$
$938$	13.0022	0.424536
$939$	0	0
$940$	6.90931	0.225357
$941$	45.3711	1.47906	0.739528	−	0.673126i	$-0.235049\pi$
$941$	45.3711	1.47906	0.739528	+	0.673126i	$0.235049\pi$
$942$	0	0
$943$	−4.21683	−0.137319
$944$	−20.7584	−0.675627
$945$	0	0
$946$	0.838361	0.0272575
$947$	−38.2907	−1.24428	−0.622141	−	0.782905i	$-0.713737\pi$
$947$	−38.2907	−1.24428	−0.622141	+	0.782905i	$0.713737\pi$
$948$	0	0
$949$	−6.66460	−0.216342
$950$	20.1213	0.652820
$951$	0	0
$952$	63.5998	2.06128
$953$	26.2534	0.850431	0.425215	−	0.905092i	$-0.360198\pi$
$953$	26.2534	0.850431	0.425215	+	0.905092i	$0.360198\pi$
$954$	0	0
$955$	31.6357	1.02371
$956$	0.620061	0.0200542
$957$	0	0
$958$	−1.69899	−0.0548919
$959$	−79.5526	−2.56889
$960$	0	0
$961$	−23.5053	−0.758236
$962$	18.3127	0.590424
$963$	0	0
$964$	−7.90365	−0.254559
$965$	46.3304	1.49143
$966$	0	0
$967$	−41.3131	−1.32854	−0.664270	−	0.747492i	$-0.731258\pi$
$967$	−41.3131	−1.32854	−0.664270	+	0.747492i	$0.731258\pi$
$968$	25.3976	0.816311
$969$	0	0
$970$	−94.0104	−3.01849
$971$	−15.8126	−0.507450	−0.253725	−	0.967276i	$-0.581656\pi$
$971$	−15.8126	−0.507450	−0.253725	+	0.967276i	$0.581656\pi$
$972$	0	0
$973$	19.4146	0.622404
$974$	−10.1147	−0.324096
$975$	0	0
$976$	12.9110	0.413272
$977$	42.7527	1.36778	0.683890	−	0.729585i	$-0.260287\pi$
$977$	42.7527	1.36778	0.683890	+	0.729585i	$0.260287\pi$
$978$	0	0
$979$	11.8697	0.379357
$980$	13.6820	0.437055
$981$	0	0
$982$	6.87560	0.219409
$983$	−22.2445	−0.709490	−0.354745	−	0.934963i	$-0.615432\pi$
$983$	−22.2445	−0.709490	−0.354745	+	0.934963i	$0.615432\pi$
$984$	0	0
$985$	47.1572	1.50255
$986$	−50.5034	−1.60836
$987$	0	0
$988$	3.74895	0.119270
$989$	−0.453705	−0.0144270
$990$	0	0
$991$	−22.6452	−0.719350	−0.359675	−	0.933078i	$-0.617112\pi$
$991$	−22.6452	−0.719350	−0.359675	+	0.933078i	$0.617112\pi$
$992$	−9.21274	−0.292505
$993$	0	0
$994$	−29.1163	−0.923514
$995$	−15.3302	−0.486000
$996$	0	0
$997$	−9.82730	−0.311234	−0.155617	−	0.987817i	$-0.549737\pi$
$997$	−9.82730	−0.311234	−0.155617	+	0.987817i	$0.549737\pi$
$998$	−9.72183	−0.307739
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.11		17
3.2	odd	2		239.2.a.b.1.7	✓	17
12.11	even	2		3824.2.a.p.1.2		17
15.14	odd	2		5975.2.a.g.1.11		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.7	✓	17	3.2	odd	2
2151.2.a.i.1.11		17	1.1	even	1	trivial
3824.2.a.p.1.2		17	12.11	even	2
5975.2.a.g.1.11		17	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17471 q^{2} -0.620061 q^{4} -4.24400 q^{5} -3.49274 q^{7} -3.07781 q^{8} +O(q^{10})\)	\(q+1.17471 q^{2} -0.620061 q^{4} -4.24400 q^{5} -3.49274 q^{7} -3.07781 q^{8} -4.98546 q^{10} -1.65775 q^{11} -4.59282 q^{13} -4.10295 q^{14} -2.37540 q^{16} +5.91628 q^{17} +1.31642 q^{19} +2.63154 q^{20} -1.94737 q^{22} +1.05388 q^{23} +13.0116 q^{25} -5.39522 q^{26} +2.16571 q^{28} -7.26679 q^{29} -2.73764 q^{31} +3.36521 q^{32} +6.94990 q^{34} +14.8232 q^{35} -3.39423 q^{37} +1.54641 q^{38} +13.0622 q^{40} -4.00124 q^{41} -0.430509 q^{43} +1.02791 q^{44} +1.23800 q^{46} +2.62558 q^{47} +5.19923 q^{49} +15.2848 q^{50} +2.84783 q^{52} +2.43027 q^{53} +7.03549 q^{55} +10.7500 q^{56} -8.53636 q^{58} +8.73888 q^{59} -5.43530 q^{61} -3.21593 q^{62} +8.70394 q^{64} +19.4919 q^{65} -3.16898 q^{67} -3.66845 q^{68} +17.4129 q^{70} +7.09644 q^{71} +1.45109 q^{73} -3.98723 q^{74} -0.816263 q^{76} +5.79009 q^{77} -12.1010 q^{79} +10.0812 q^{80} -4.70029 q^{82} +1.76218 q^{83} -25.1087 q^{85} -0.505722 q^{86} +5.10223 q^{88} -7.16013 q^{89} +16.0415 q^{91} -0.653470 q^{92} +3.08429 q^{94} -5.58691 q^{95} +18.8569 q^{97} +6.10758 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

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