LMFDB - Embedded newform 1737.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1737,2,Mod(1,1737)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1737, 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("1737.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1737 = 3^{2} \cdot 193 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1737.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:	$13.8700148311$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 20 x^{11} + 39 x^{10} + 148 x^{9} - 275 x^{8} - 508 x^{7} + 865 x^{6} + 823 x^{5} + \cdots - 48 $ x^13 - 2x^12 - 20x^11 + 39x^10 + 148x^9 - 275x^8 - 508x^7 + 865x^6 + 823x^5 - 1187x^4 - 556x^3 + 576x^2 + 64x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 579)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.68916$ of defining polynomial
Character	$\chi$	$=$	1737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68916 q^{2} +0.853276 q^{4} +1.44492 q^{5} +2.45560 q^{7} +1.93701 q^{8} +O(q^{10})$	\(q-1.68916 q^{2} +0.853276 q^{4} +1.44492 q^{5} +2.45560 q^{7} +1.93701 q^{8} -2.44072 q^{10} -5.26332 q^{11} +3.78153 q^{13} -4.14792 q^{14} -4.97847 q^{16} +0.858046 q^{17} +4.03245 q^{19} +1.23292 q^{20} +8.89061 q^{22} +2.21760 q^{23} -2.91219 q^{25} -6.38762 q^{26} +2.09531 q^{28} +7.87396 q^{29} +7.07051 q^{31} +4.53544 q^{32} -1.44938 q^{34} +3.54816 q^{35} -7.45792 q^{37} -6.81147 q^{38} +2.79883 q^{40} +5.27074 q^{41} -3.38817 q^{43} -4.49106 q^{44} -3.74590 q^{46} +5.84151 q^{47} -0.970011 q^{49} +4.91917 q^{50} +3.22668 q^{52} -5.20251 q^{53} -7.60510 q^{55} +4.75652 q^{56} -13.3004 q^{58} +3.43605 q^{59} -2.93414 q^{61} -11.9433 q^{62} +2.29583 q^{64} +5.46402 q^{65} -6.89075 q^{67} +0.732150 q^{68} -5.99343 q^{70} -12.5393 q^{71} -1.19175 q^{73} +12.5976 q^{74} +3.44079 q^{76} -12.9246 q^{77} +10.0506 q^{79} -7.19352 q^{80} -8.90315 q^{82} +6.10832 q^{83} +1.23981 q^{85} +5.72318 q^{86} -10.1951 q^{88} +11.8076 q^{89} +9.28593 q^{91} +1.89223 q^{92} -9.86726 q^{94} +5.82659 q^{95} -9.60171 q^{97} +1.63851 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 18 q^{4} - 6 q^{5} + 15 q^{7} - 3 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 18 * q^4 - 6 * q^5 + 15 * q^7 - 3 * q^8	$ 13 q - 2 q^{2} + 18 q^{4} - 6 q^{5} + 15 q^{7} - 3 q^{8} - q^{10} - 3 q^{11} + 11 q^{13} + 4 q^{14} + 20 q^{16} + 2 q^{17} + 9 q^{19} + 9 q^{20} - 8 q^{22} + 8 q^{23} + 21 q^{25} + 15 q^{26} + 16 q^{28} - 5 q^{29} + 25 q^{31} + 17 q^{32} - 10 q^{34} + 10 q^{35} + 29 q^{37} + 40 q^{38} - 21 q^{40} + 11 q^{41} + 8 q^{43} + 18 q^{44} - 6 q^{46} + 12 q^{47} + 20 q^{49} + 4 q^{50} + 2 q^{52} - 14 q^{53} + 12 q^{55} + 7 q^{56} + 9 q^{58} - 10 q^{59} + 6 q^{61} + 14 q^{62} + 23 q^{64} + 15 q^{65} + 25 q^{67} + 33 q^{68} - 21 q^{70} + 8 q^{73} + 2 q^{74} + 20 q^{76} + 25 q^{77} + 7 q^{79} + 40 q^{80} - 19 q^{82} + 28 q^{83} - 3 q^{85} - 2 q^{86} - 21 q^{88} - 7 q^{89} + 7 q^{91} - 9 q^{92} - 35 q^{94} + 26 q^{95} + 26 q^{97} + 14 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 18 * q^4 - 6 * q^5 + 15 * q^7 - 3 * q^8 - q^10 - 3 * q^11 + 11 * q^13 + 4 * q^14 + 20 * q^16 + 2 * q^17 + 9 * q^19 + 9 * q^20 - 8 * q^22 + 8 * q^23 + 21 * q^25 + 15 * q^26 + 16 * q^28 - 5 * q^29 + 25 * q^31 + 17 * q^32 - 10 * q^34 + 10 * q^35 + 29 * q^37 + 40 * q^38 - 21 * q^40 + 11 * q^41 + 8 * q^43 + 18 * q^44 - 6 * q^46 + 12 * q^47 + 20 * q^49 + 4 * q^50 + 2 * q^52 - 14 * q^53 + 12 * q^55 + 7 * q^56 + 9 * q^58 - 10 * q^59 + 6 * q^61 + 14 * q^62 + 23 * q^64 + 15 * q^65 + 25 * q^67 + 33 * q^68 - 21 * q^70 + 8 * q^73 + 2 * q^74 + 20 * q^76 + 25 * q^77 + 7 * q^79 + 40 * q^80 - 19 * q^82 + 28 * q^83 - 3 * q^85 - 2 * q^86 - 21 * q^88 - 7 * q^89 + 7 * q^91 - 9 * q^92 - 35 * q^94 + 26 * q^95 + 26 * q^97 + 14 * 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 $

$2$	−1.68916	−1.19442	−0.597210	−	0.802085i	$-0.703724\pi$
$2$	−1.68916	−1.19442	−0.597210	+	0.802085i	$0.703724\pi$
$3$	0	0
$3$	0	0
$4$	0.853276	0.426638
$5$	1.44492	0.646190	0.323095	−	0.946367i	$-0.395277\pi$
$5$	1.44492	0.646190	0.323095	+	0.946367i	$0.395277\pi$
$6$	0	0
$7$	2.45560	0.928131	0.464065	−	0.885801i	$-0.346390\pi$
$7$	2.45560	0.928131	0.464065	+	0.885801i	$0.346390\pi$
$8$	1.93701	0.684835
$9$	0	0
$10$	−2.44072	−0.771822
$11$	−5.26332	−1.58695	−0.793475	−	0.608602i	$-0.791730\pi$
$11$	−5.26332	−1.58695	−0.793475	+	0.608602i	$0.791730\pi$
$12$	0	0
$13$	3.78153	1.04881	0.524403	−	0.851470i	$-0.324288\pi$
$13$	3.78153	1.04881	0.524403	+	0.851470i	$0.324288\pi$
$14$	−4.14792	−1.10858
$15$	0	0
$16$	−4.97847	−1.24462
$17$	0.858046	0.208107	0.104053	−	0.994572i	$-0.466819\pi$
$17$	0.858046	0.208107	0.104053	+	0.994572i	$0.466819\pi$
$18$	0	0
$19$	4.03245	0.925108	0.462554	−	0.886591i	$-0.346933\pi$
$19$	4.03245	0.925108	0.462554	+	0.886591i	$0.346933\pi$
$20$	1.23292	0.275689
$21$	0	0
$22$	8.89061	1.89548
$23$	2.21760	0.462403	0.231201	−	0.972906i	$-0.425734\pi$
$23$	2.21760	0.462403	0.231201	+	0.972906i	$0.425734\pi$
$24$	0	0
$25$	−2.91219	−0.582438
$26$	−6.38762	−1.25272
$27$	0	0
$28$	2.09531	0.395976
$29$	7.87396	1.46216	0.731079	−	0.682293i	$-0.239017\pi$
$29$	7.87396	1.46216	0.731079	+	0.682293i	$0.239017\pi$
$30$	0	0
$31$	7.07051	1.26990	0.634951	−	0.772553i	$-0.281020\pi$
$31$	7.07051	1.26990	0.634951	+	0.772553i	$0.281020\pi$
$32$	4.53544	0.801761
$33$	0	0
$34$	−1.44938	−0.248567
$35$	3.54816	0.599749
$36$	0	0
$37$	−7.45792	−1.22607	−0.613037	−	0.790054i	$-0.710052\pi$
$37$	−7.45792	−1.22607	−0.613037	+	0.790054i	$0.710052\pi$
$38$	−6.81147	−1.10497
$39$	0	0
$40$	2.79883	0.442534
$41$	5.27074	0.823152	0.411576	−	0.911375i	$-0.364978\pi$
$41$	5.27074	0.823152	0.411576	+	0.911375i	$0.364978\pi$
$42$	0	0
$43$	−3.38817	−0.516691	−0.258346	−	0.966053i	$-0.583177\pi$
$43$	−3.38817	−0.516691	−0.258346	+	0.966053i	$0.583177\pi$
$44$	−4.49106	−0.677053
$45$	0	0
$46$	−3.74590	−0.552303
$47$	5.84151	0.852071	0.426036	−	0.904706i	$-0.359910\pi$
$47$	5.84151	0.852071	0.426036	+	0.904706i	$0.359910\pi$
$48$	0	0
$49$	−0.970011	−0.138573
$50$	4.91917	0.695676
$51$	0	0
$52$	3.22668	0.447461
$53$	−5.20251	−0.714620	−0.357310	−	0.933986i	$-0.616306\pi$
$53$	−5.20251	−0.714620	−0.357310	+	0.933986i	$0.616306\pi$
$54$	0	0
$55$	−7.60510	−1.02547
$56$	4.75652	0.635617
$57$	0	0
$58$	−13.3004	−1.74643
$59$	3.43605	0.447336	0.223668	−	0.974665i	$-0.428197\pi$
$59$	3.43605	0.447336	0.223668	+	0.974665i	$0.428197\pi$
$60$	0	0
$61$	−2.93414	−0.375678	−0.187839	−	0.982200i	$-0.560148\pi$
$61$	−2.93414	−0.375678	−0.187839	+	0.982200i	$0.560148\pi$
$62$	−11.9433	−1.51679
$63$	0	0
$64$	2.29583	0.286979
$65$	5.46402	0.677728
$66$	0	0
$67$	−6.89075	−0.841840	−0.420920	−	0.907098i	$-0.638293\pi$
$67$	−6.89075	−0.841840	−0.420920	+	0.907098i	$0.638293\pi$
$68$	0.732150	0.0887862
$69$	0	0
$70$	−5.99343	−0.716352
$71$	−12.5393	−1.48814	−0.744068	−	0.668104i	$-0.767106\pi$
$71$	−12.5393	−1.48814	−0.744068	+	0.668104i	$0.767106\pi$
$72$	0	0
$73$	−1.19175	−0.139484	−0.0697421	−	0.997565i	$-0.522218\pi$
$73$	−1.19175	−0.139484	−0.0697421	+	0.997565i	$0.522218\pi$
$74$	12.5976	1.46445
$75$	0	0
$76$	3.44079	0.394686
$77$	−12.9246	−1.47290
$78$	0	0
$79$	10.0506	1.13078	0.565392	−	0.824823i	$-0.308725\pi$
$79$	10.0506	1.13078	0.565392	+	0.824823i	$0.308725\pi$
$80$	−7.19352	−0.804260
$81$	0	0
$82$	−8.90315	−0.983189
$83$	6.10832	0.670475	0.335237	−	0.942134i	$-0.391183\pi$
$83$	6.10832	0.670475	0.335237	+	0.942134i	$0.391183\pi$
$84$	0	0
$85$	1.23981	0.134477
$86$	5.72318	0.617146
$87$	0	0
$88$	−10.1951	−1.08680
$89$	11.8076	1.25160	0.625801	−	0.779983i	$-0.284772\pi$
$89$	11.8076	1.25160	0.625801	+	0.779983i	$0.284772\pi$
$90$	0	0
$91$	9.28593	0.973430
$92$	1.89223	0.197278
$93$	0	0
$94$	−9.86726	−1.01773
$95$	5.82659	0.597795
$96$	0	0
$97$	−9.60171	−0.974906	−0.487453	−	0.873149i	$-0.662074\pi$
$97$	−9.60171	−0.974906	−0.487453	+	0.873149i	$0.662074\pi$
$98$	1.63851	0.165514
$99$	0	0
$100$	−2.48490	−0.248490
$101$	−7.94106	−0.790165	−0.395082	−	0.918646i	$-0.629284\pi$
$101$	−7.94106	−0.790165	−0.395082	+	0.918646i	$0.629284\pi$
$102$	0	0
$103$	15.0437	1.48230	0.741151	−	0.671339i	$-0.234280\pi$
$103$	15.0437	1.48230	0.741151	+	0.671339i	$0.234280\pi$
$104$	7.32484	0.718259
$105$	0	0
$106$	8.78790	0.853556
$107$	7.44849	0.720073	0.360036	−	0.932938i	$-0.382764\pi$
$107$	7.44849	0.720073	0.360036	+	0.932938i	$0.382764\pi$
$108$	0	0
$109$	7.38126	0.706996	0.353498	−	0.935435i	$-0.384992\pi$
$109$	7.38126	0.706996	0.353498	+	0.935435i	$0.384992\pi$
$110$	12.8463	1.22484
$111$	0	0
$112$	−12.2252	−1.15517
$113$	9.44816	0.888808	0.444404	−	0.895826i	$-0.353415\pi$
$113$	9.44816	0.888808	0.444404	+	0.895826i	$0.353415\pi$
$114$	0	0
$115$	3.20427	0.298800
$116$	6.71865	0.623811
$117$	0	0
$118$	−5.80406	−0.534307
$119$	2.10702	0.193150
$120$	0	0
$121$	16.7025	1.51841
$122$	4.95625	0.448718
$123$	0	0
$124$	6.03309	0.541788
$125$	−11.4325	−1.02256
$126$	0	0
$127$	19.9462	1.76994	0.884971	−	0.465646i	$-0.154178\pi$
$127$	19.9462	1.76994	0.884971	+	0.465646i	$0.154178\pi$
$128$	−12.9489	−1.14453
$129$	0	0
$130$	−9.22963	−0.809492
$131$	−2.21285	−0.193338	−0.0966688	−	0.995317i	$-0.530819\pi$
$131$	−2.21285	−0.193338	−0.0966688	+	0.995317i	$0.530819\pi$
$132$	0	0
$133$	9.90210	0.858621
$134$	11.6396	1.00551
$135$	0	0
$136$	1.66204	0.142519
$137$	−8.48164	−0.724636	−0.362318	−	0.932055i	$-0.618014\pi$
$137$	−8.48164	−0.724636	−0.362318	+	0.932055i	$0.618014\pi$
$138$	0	0
$139$	−7.71002	−0.653956	−0.326978	−	0.945032i	$-0.606030\pi$
$139$	−7.71002	−0.653956	−0.326978	+	0.945032i	$0.606030\pi$
$140$	3.02756	0.255876
$141$	0	0
$142$	21.1809	1.77746
$143$	−19.9034	−1.66440
$144$	0	0
$145$	11.3773	0.944831
$146$	2.01307	0.166603
$147$	0	0
$148$	−6.36366	−0.523090
$149$	15.2068	1.24579	0.622894	−	0.782306i	$-0.285957\pi$
$149$	15.2068	1.24579	0.622894	+	0.782306i	$0.285957\pi$
$150$	0	0
$151$	13.2142	1.07536	0.537679	−	0.843150i	$-0.319301\pi$
$151$	13.2142	1.07536	0.537679	+	0.843150i	$0.319301\pi$
$152$	7.81088	0.633546
$153$	0	0
$154$	21.8318	1.75926
$155$	10.2164	0.820597
$156$	0	0
$157$	−17.7105	−1.41345	−0.706727	−	0.707487i	$-0.749829\pi$
$157$	−17.7105	−1.41345	−0.706727	+	0.707487i	$0.749829\pi$
$158$	−16.9771	−1.35063
$159$	0	0
$160$	6.55338	0.518090
$161$	5.44556	0.429170
$162$	0	0
$163$	16.2387	1.27191	0.635955	−	0.771726i	$-0.280606\pi$
$163$	16.2387	1.27191	0.635955	+	0.771726i	$0.280606\pi$
$164$	4.49740	0.351188
$165$	0	0
$166$	−10.3179	−0.800828
$167$	20.6283	1.59626	0.798132	−	0.602482i	$-0.205822\pi$
$167$	20.6283	1.59626	0.798132	+	0.602482i	$0.205822\pi$
$168$	0	0
$169$	1.29994	0.0999954
$170$	−2.09425	−0.160621
$171$	0	0
$172$	−2.89104	−0.220440
$173$	19.8341	1.50796	0.753980	−	0.656898i	$-0.228132\pi$
$173$	19.8341	1.50796	0.753980	+	0.656898i	$0.228132\pi$
$174$	0	0
$175$	−7.15119	−0.540579
$176$	26.2033	1.97515
$177$	0	0
$178$	−19.9450	−1.49494
$179$	−5.15575	−0.385359	−0.192680	−	0.981262i	$-0.561718\pi$
$179$	−5.15575	−0.385359	−0.192680	+	0.981262i	$0.561718\pi$
$180$	0	0
$181$	−7.70169	−0.572462	−0.286231	−	0.958161i	$-0.592402\pi$
$181$	−7.70169	−0.572462	−0.286231	+	0.958161i	$0.592402\pi$
$182$	−15.6855	−1.16268
$183$	0	0
$184$	4.29551	0.316669
$185$	−10.7761	−0.792277
$186$	0	0
$187$	−4.51617	−0.330255
$188$	4.98441	0.363526
$189$	0	0
$190$	−9.84206	−0.714018
$191$	2.46329	0.178237	0.0891185	−	0.996021i	$-0.471595\pi$
$191$	2.46329	0.178237	0.0891185	+	0.996021i	$0.471595\pi$
$192$	0	0
$193$	−1.00000	−0.0719816
$193$	−1.00000	−0.0719816
$194$	16.2189	1.16445
$195$	0	0
$196$	−0.827686	−0.0591205
$197$	−9.40645	−0.670181	−0.335091	−	0.942186i	$-0.608767\pi$
$197$	−9.40645	−0.670181	−0.335091	+	0.942186i	$0.608767\pi$
$198$	0	0
$199$	5.84728	0.414503	0.207251	−	0.978288i	$-0.433548\pi$
$199$	5.84728	0.414503	0.207251	+	0.978288i	$0.433548\pi$
$200$	−5.64093	−0.398874
$201$	0	0
$202$	13.4137	0.943788
$203$	19.3353	1.35707
$204$	0	0
$205$	7.61583	0.531913
$206$	−25.4113	−1.77049
$207$	0	0
$208$	−18.8262	−1.30536
$209$	−21.2241	−1.46810
$210$	0	0
$211$	13.3109	0.916358	0.458179	−	0.888860i	$-0.348502\pi$
$211$	13.3109	0.916358	0.458179	+	0.888860i	$0.348502\pi$
$212$	−4.43918	−0.304884
$213$	0	0
$214$	−12.5817	−0.860069
$215$	−4.89565	−0.333881
$216$	0	0
$217$	17.3624	1.17863
$218$	−12.4682	−0.844450
$219$	0	0
$220$	−6.48925	−0.437505
$221$	3.24472	0.218264
$222$	0	0
$223$	−28.8013	−1.92868	−0.964340	−	0.264666i	$-0.914738\pi$
$223$	−28.8013	−1.92868	−0.964340	+	0.264666i	$0.914738\pi$
$224$	11.1373	0.744139
$225$	0	0
$226$	−15.9595	−1.06161
$227$	19.1850	1.27335	0.636675	−	0.771132i	$-0.280309\pi$
$227$	19.1850	1.27335	0.636675	+	0.771132i	$0.280309\pi$
$228$	0	0
$229$	5.06674	0.334820	0.167410	−	0.985887i	$-0.446460\pi$
$229$	5.06674	0.334820	0.167410	+	0.985887i	$0.446460\pi$
$230$	−5.41254	−0.356892
$231$	0	0
$232$	15.2519	1.00134
$233$	−22.8573	−1.49743	−0.748716	−	0.662891i	$-0.769329\pi$
$233$	−22.8573	−1.49743	−0.748716	+	0.662891i	$0.769329\pi$
$234$	0	0
$235$	8.44054	0.550600
$236$	2.93190	0.190850
$237$	0	0
$238$	−3.55911	−0.230703
$239$	1.26308	0.0817022	0.0408511	−	0.999165i	$-0.486993\pi$
$239$	1.26308	0.0817022	0.0408511	+	0.999165i	$0.486993\pi$
$240$	0	0
$241$	16.4700	1.06093	0.530463	−	0.847708i	$-0.322018\pi$
$241$	16.4700	1.06093	0.530463	+	0.847708i	$0.322018\pi$
$242$	−28.2133	−1.81362
$243$	0	0
$244$	−2.50363	−0.160279
$245$	−1.40159	−0.0895445
$246$	0	0
$247$	15.2488	0.970259
$248$	13.6956	0.869673
$249$	0	0
$250$	19.3114	1.22136
$251$	11.9847	0.756466	0.378233	−	0.925710i	$-0.376532\pi$
$251$	11.9847	0.756466	0.378233	+	0.925710i	$0.376532\pi$
$252$	0	0
$253$	−11.6720	−0.733810
$254$	−33.6925	−2.11405
$255$	0	0
$256$	17.2812	1.08007
$257$	1.41069	0.0879967	0.0439984	−	0.999032i	$-0.485990\pi$
$257$	1.41069	0.0879967	0.0439984	+	0.999032i	$0.485990\pi$
$258$	0	0
$259$	−18.3137	−1.13796
$260$	4.66232	0.289145
$261$	0	0
$262$	3.73787	0.230926
$263$	16.8917	1.04159	0.520795	−	0.853682i	$-0.325636\pi$
$263$	16.8917	1.04159	0.520795	+	0.853682i	$0.325636\pi$
$264$	0	0
$265$	−7.51724	−0.461780
$266$	−16.7263	−1.02555
$267$	0	0
$268$	−5.87971	−0.359161
$269$	2.25119	0.137258	0.0686288	−	0.997642i	$-0.478138\pi$
$269$	2.25119	0.137258	0.0686288	+	0.997642i	$0.478138\pi$
$270$	0	0
$271$	−22.3890	−1.36003	−0.680017	−	0.733197i	$-0.738027\pi$
$271$	−22.3890	−1.36003	−0.680017	+	0.733197i	$0.738027\pi$
$272$	−4.27176	−0.259013
$273$	0	0
$274$	14.3269	0.865519
$275$	15.3278	0.924301
$276$	0	0
$277$	21.7579	1.30730	0.653651	−	0.756796i	$-0.273236\pi$
$277$	21.7579	1.30730	0.653651	+	0.756796i	$0.273236\pi$
$278$	13.0235	0.781097
$279$	0	0
$280$	6.87281	0.410729
$281$	18.4006	1.09769	0.548843	−	0.835925i	$-0.315068\pi$
$281$	18.4006	1.09769	0.548843	+	0.835925i	$0.315068\pi$
$282$	0	0
$283$	11.9137	0.708199	0.354099	−	0.935208i	$-0.384787\pi$
$283$	11.9137	0.708199	0.354099	+	0.935208i	$0.384787\pi$
$284$	−10.6994	−0.634895
$285$	0	0
$286$	33.6201	1.98800
$287$	12.9429	0.763993
$288$	0	0
$289$	−16.2638	−0.956692
$290$	−19.2181	−1.12852
$291$	0	0
$292$	−1.01689	−0.0595092
$293$	−32.1770	−1.87980	−0.939899	−	0.341451i	$-0.889082\pi$
$293$	−32.1770	−1.87980	−0.939899	+	0.341451i	$0.889082\pi$
$294$	0	0
$295$	4.96484	0.289064
$296$	−14.4460	−0.839658
$297$	0	0
$298$	−25.6868	−1.48799
$299$	8.38593	0.484971
$300$	0	0
$301$	−8.32001	−0.479557
$302$	−22.3210	−1.28443
$303$	0	0
$304$	−20.0754	−1.15141
$305$	−4.23962	−0.242760
$306$	0	0
$307$	27.3311	1.55987	0.779935	−	0.625860i	$-0.215252\pi$
$307$	27.3311	1.55987	0.779935	+	0.625860i	$0.215252\pi$
$308$	−11.0283	−0.628394
$309$	0	0
$310$	−17.2571	−0.980138
$311$	−20.2842	−1.15021	−0.575104	−	0.818080i	$-0.695039\pi$
$311$	−20.2842	−1.15021	−0.575104	+	0.818080i	$0.695039\pi$
$312$	0	0
$313$	3.96033	0.223851	0.111926	−	0.993717i	$-0.464298\pi$
$313$	3.96033	0.223851	0.111926	+	0.993717i	$0.464298\pi$
$314$	29.9160	1.68826
$315$	0	0
$316$	8.57595	0.482435
$317$	−26.2360	−1.47356	−0.736781	−	0.676132i	$-0.763655\pi$
$317$	−26.2360	−1.47356	−0.736781	+	0.676132i	$0.763655\pi$
$318$	0	0
$319$	−41.4432	−2.32037
$320$	3.31731	0.185443
$321$	0	0
$322$	−9.19844	−0.512609
$323$	3.46003	0.192521
$324$	0	0
$325$	−11.0125	−0.610865
$326$	−27.4298	−1.51919
$327$	0	0
$328$	10.2095	0.563723
$329$	14.3444	0.790834
$330$	0	0
$331$	22.3835	1.23031	0.615154	−	0.788407i	$-0.289094\pi$
$331$	22.3835	1.23031	0.615154	+	0.788407i	$0.289094\pi$
$332$	5.21208	0.286050
$333$	0	0
$334$	−34.8446	−1.90661
$335$	−9.95662	−0.543988
$336$	0	0
$337$	32.4389	1.76706	0.883529	−	0.468376i	$-0.155161\pi$
$337$	32.4389	1.76706	0.883529	+	0.468376i	$0.155161\pi$
$338$	−2.19581	−0.119436
$339$	0	0
$340$	1.05790	0.0573728
$341$	−37.2144	−2.01527
$342$	0	0
$343$	−19.5712	−1.05674
$344$	−6.56291	−0.353848
$345$	0	0
$346$	−33.5031	−1.80114
$347$	1.22305	0.0656569	0.0328285	−	0.999461i	$-0.489549\pi$
$347$	1.22305	0.0656569	0.0328285	+	0.999461i	$0.489549\pi$
$348$	0	0
$349$	20.6268	1.10412	0.552062	−	0.833803i	$-0.313841\pi$
$349$	20.6268	1.10412	0.552062	+	0.833803i	$0.313841\pi$
$350$	12.0795	0.645678
$351$	0	0
$352$	−23.8715	−1.27235
$353$	−3.21291	−0.171006	−0.0855029	−	0.996338i	$-0.527250\pi$
$353$	−3.21291	−0.171006	−0.0855029	+	0.996338i	$0.527250\pi$
$354$	0	0
$355$	−18.1183	−0.961619
$356$	10.0751	0.533981
$357$	0	0
$358$	8.70891	0.460280
$359$	7.58991	0.400580	0.200290	−	0.979737i	$-0.435812\pi$
$359$	7.58991	0.400580	0.200290	+	0.979737i	$0.435812\pi$
$360$	0	0
$361$	−2.73934	−0.144176
$362$	13.0094	0.683760
$363$	0	0
$364$	7.92346	0.415302
$365$	−1.72199	−0.0901332
$366$	0	0
$367$	−11.2532	−0.587414	−0.293707	−	0.955896i	$-0.594889\pi$
$367$	−11.2532	−0.587414	−0.293707	+	0.955896i	$0.594889\pi$
$368$	−11.0403	−0.575515
$369$	0	0
$370$	18.2027	0.946311
$371$	−12.7753	−0.663261
$372$	0	0
$373$	17.1534	0.888169	0.444084	−	0.895985i	$-0.353529\pi$
$373$	17.1534	0.888169	0.444084	+	0.895985i	$0.353529\pi$
$374$	7.62856	0.394463
$375$	0	0
$376$	11.3150	0.583528
$377$	29.7756	1.53352
$378$	0	0
$379$	−17.9398	−0.921507	−0.460754	−	0.887528i	$-0.652421\pi$
$379$	−17.9398	−0.921507	−0.460754	+	0.887528i	$0.652421\pi$
$380$	4.97169	0.255042
$381$	0	0
$382$	−4.16089	−0.212890
$383$	−11.4267	−0.583878	−0.291939	−	0.956437i	$-0.594300\pi$
$383$	−11.4267	−0.583878	−0.291939	+	0.956437i	$0.594300\pi$
$384$	0	0
$385$	−18.6751	−0.951772
$386$	1.68916	0.0859762
$387$	0	0
$388$	−8.19291	−0.415932
$389$	−28.3638	−1.43810	−0.719050	−	0.694959i	$-0.755423\pi$
$389$	−28.3638	−1.43810	−0.719050	+	0.694959i	$0.755423\pi$
$390$	0	0
$391$	1.90281	0.0962291
$392$	−1.87892	−0.0948996
$393$	0	0
$394$	15.8890	0.800478
$395$	14.5224	0.730701
$396$	0	0
$397$	−3.94999	−0.198244	−0.0991221	−	0.995075i	$-0.531603\pi$
$397$	−3.94999	−0.198244	−0.0991221	+	0.995075i	$0.531603\pi$
$398$	−9.87702	−0.495090
$399$	0	0
$400$	14.4983	0.724913
$401$	9.73822	0.486304	0.243152	−	0.969988i	$-0.421819\pi$
$401$	9.73822	0.486304	0.243152	+	0.969988i	$0.421819\pi$
$402$	0	0
$403$	26.7373	1.33188
$404$	−6.77591	−0.337114
$405$	0	0
$406$	−32.6605	−1.62091
$407$	39.2534	1.94572
$408$	0	0
$409$	−0.410667	−0.0203062	−0.0101531	−	0.999948i	$-0.503232\pi$
$409$	−0.410667	−0.0203062	−0.0101531	+	0.999948i	$0.503232\pi$
$410$	−12.8644	−0.635327
$411$	0	0
$412$	12.8364	0.632406
$413$	8.43758	0.415186
$414$	0	0
$415$	8.82606	0.433254
$416$	17.1509	0.840892
$417$	0	0
$418$	35.8510	1.75353
$419$	−40.4009	−1.97371	−0.986857	−	0.161598i	$-0.948335\pi$
$419$	−40.4009	−1.97371	−0.986857	+	0.161598i	$0.948335\pi$
$420$	0	0
$421$	−18.8200	−0.917231	−0.458616	−	0.888635i	$-0.651655\pi$
$421$	−18.8200	−0.917231	−0.458616	+	0.888635i	$0.651655\pi$
$422$	−22.4842	−1.09452
$423$	0	0
$424$	−10.0773	−0.489397
$425$	−2.49880	−0.121209
$426$	0	0
$427$	−7.20509	−0.348679
$428$	6.35561	0.307210
$429$	0	0
$430$	8.26956	0.398794
$431$	−31.8176	−1.53260	−0.766299	−	0.642484i	$-0.777904\pi$
$431$	−31.8176	−1.53260	−0.766299	+	0.642484i	$0.777904\pi$
$432$	0	0
$433$	16.6983	0.802468	0.401234	−	0.915976i	$-0.368581\pi$
$433$	16.6983	0.802468	0.401234	+	0.915976i	$0.368581\pi$
$434$	−29.3279	−1.40778
$435$	0	0
$436$	6.29825	0.301631
$437$	8.94238	0.427772
$438$	0	0
$439$	−13.2585	−0.632792	−0.316396	−	0.948627i	$-0.602473\pi$
$439$	−13.2585	−0.632792	−0.316396	+	0.948627i	$0.602473\pi$
$440$	−14.7311	−0.702279
$441$	0	0
$442$	−5.48087	−0.260699
$443$	7.62687	0.362363	0.181182	−	0.983450i	$-0.442008\pi$
$443$	7.62687	0.362363	0.181182	+	0.983450i	$0.442008\pi$
$444$	0	0
$445$	17.0611	0.808773
$446$	48.6502	2.30365
$447$	0	0
$448$	5.63766	0.266354
$449$	24.2798	1.14583	0.572917	−	0.819613i	$-0.305812\pi$
$449$	24.2798	1.14583	0.572917	+	0.819613i	$0.305812\pi$
$450$	0	0
$451$	−27.7416	−1.30630
$452$	8.06189	0.379199
$453$	0	0
$454$	−32.4065	−1.52091
$455$	13.4175	0.629021
$456$	0	0
$457$	−5.15367	−0.241079	−0.120539	−	0.992709i	$-0.538462\pi$
$457$	−5.15367	−0.241079	−0.120539	+	0.992709i	$0.538462\pi$
$458$	−8.55856	−0.399915
$459$	0	0
$460$	2.73413	0.127479
$461$	−13.8519	−0.645148	−0.322574	−	0.946544i	$-0.604548\pi$
$461$	−13.8519	−0.645148	−0.322574	+	0.946544i	$0.604548\pi$
$462$	0	0
$463$	−15.7013	−0.729700	−0.364850	−	0.931066i	$-0.618880\pi$
$463$	−15.7013	−0.729700	−0.364850	+	0.931066i	$0.618880\pi$
$464$	−39.2003	−1.81983
$465$	0	0
$466$	38.6098	1.78856
$467$	−32.4824	−1.50311	−0.751553	−	0.659673i	$-0.770695\pi$
$467$	−32.4824	−1.50311	−0.751553	+	0.659673i	$0.770695\pi$
$468$	0	0
$469$	−16.9210	−0.781337
$470$	−14.2575	−0.657647
$471$	0	0
$472$	6.65565	0.306351
$473$	17.8330	0.819964
$474$	0	0
$475$	−11.7433	−0.538818
$476$	1.79787	0.0824052
$477$	0	0
$478$	−2.13356	−0.0975866
$479$	−1.32294	−0.0604466	−0.0302233	−	0.999543i	$-0.509622\pi$
$479$	−1.32294	−0.0604466	−0.0302233	+	0.999543i	$0.509622\pi$
$480$	0	0
$481$	−28.2023	−1.28591
$482$	−27.8205	−1.26719
$483$	0	0
$484$	14.2519	0.647812
$485$	−13.8738	−0.629975
$486$	0	0
$487$	−37.1267	−1.68237	−0.841185	−	0.540747i	$-0.818141\pi$
$487$	−37.1267	−1.68237	−0.841185	+	0.540747i	$0.818141\pi$
$488$	−5.68345	−0.257278
$489$	0	0
$490$	2.36752	0.106954
$491$	−18.7450	−0.845952	−0.422976	−	0.906141i	$-0.639015\pi$
$491$	−18.7450	−0.845952	−0.422976	+	0.906141i	$0.639015\pi$
$492$	0	0
$493$	6.75622	0.304285
$494$	−25.7578	−1.15890
$495$	0	0
$496$	−35.2003	−1.58054
$497$	−30.7915	−1.38119
$498$	0	0
$499$	−21.9520	−0.982704	−0.491352	−	0.870961i	$-0.663497\pi$
$499$	−21.9520	−0.982704	−0.491352	+	0.870961i	$0.663497\pi$
$500$	−9.75509	−0.436261
$501$	0	0
$502$	−20.2441	−0.903538
$503$	−31.8471	−1.41999	−0.709996	−	0.704206i	$-0.751303\pi$
$503$	−31.8471	−1.41999	−0.709996	+	0.704206i	$0.751303\pi$
$504$	0	0
$505$	−11.4742	−0.510596
$506$	19.7159	0.876477
$507$	0	0
$508$	17.0196	0.755124
$509$	−21.1302	−0.936579	−0.468290	−	0.883575i	$-0.655130\pi$
$509$	−21.1302	−0.936579	−0.468290	+	0.883575i	$0.655130\pi$
$510$	0	0
$511$	−2.92647	−0.129460
$512$	−3.29293	−0.145528
$513$	0	0
$514$	−2.38289	−0.105105
$515$	21.7370	0.957848
$516$	0	0
$517$	−30.7457	−1.35219
$518$	30.9348	1.35920
$519$	0	0
$520$	10.5838	0.464132
$521$	11.8767	0.520330	0.260165	−	0.965564i	$-0.416223\pi$
$521$	11.8767	0.520330	0.260165	+	0.965564i	$0.416223\pi$
$522$	0	0
$523$	−21.4741	−0.938996	−0.469498	−	0.882933i	$-0.655565\pi$
$523$	−21.4741	−0.938996	−0.469498	+	0.882933i	$0.655565\pi$
$524$	−1.88817	−0.0824851
$525$	0	0
$526$	−28.5329	−1.24409
$527$	6.06682	0.264275
$528$	0	0
$529$	−18.0822	−0.786184
$530$	12.6979	0.551560
$531$	0	0
$532$	8.44922	0.366320
$533$	19.9315	0.863327
$534$	0	0
$535$	10.7625	0.465304
$536$	−13.3474	−0.576521
$537$	0	0
$538$	−3.80263	−0.163943
$539$	5.10548	0.219908
$540$	0	0
$541$	6.99342	0.300671	0.150335	−	0.988635i	$-0.451965\pi$
$541$	6.99342	0.300671	0.150335	+	0.988635i	$0.451965\pi$
$542$	37.8186	1.62445
$543$	0	0
$544$	3.89162	0.166852
$545$	10.6654	0.456854
$546$	0	0
$547$	−15.0870	−0.645074	−0.322537	−	0.946557i	$-0.604536\pi$
$547$	−15.0870	−0.645074	−0.322537	+	0.946557i	$0.604536\pi$
$548$	−7.23718	−0.309157
$549$	0	0
$550$	−25.8912	−1.10400
$551$	31.7513	1.35265
$552$	0	0
$553$	24.6803	1.04951
$554$	−36.7526	−1.56147
$555$	0	0
$556$	−6.57877	−0.279002
$557$	−38.5179	−1.63206	−0.816029	−	0.578011i	$-0.803829\pi$
$557$	−38.5179	−1.63206	−0.816029	+	0.578011i	$0.803829\pi$
$558$	0	0
$559$	−12.8125	−0.541909
$560$	−17.6644	−0.746458
$561$	0	0
$562$	−31.0816	−1.31110
$563$	34.0670	1.43575	0.717876	−	0.696171i	$-0.245114\pi$
$563$	34.0670	1.43575	0.717876	+	0.696171i	$0.245114\pi$
$564$	0	0
$565$	13.6519	0.574339
$566$	−20.1243	−0.845886
$567$	0	0
$568$	−24.2886	−1.01913
$569$	−3.22359	−0.135140	−0.0675699	−	0.997715i	$-0.521525\pi$
$569$	−3.22359	−0.135140	−0.0675699	+	0.997715i	$0.521525\pi$
$570$	0	0
$571$	34.0417	1.42460	0.712301	−	0.701874i	$-0.247653\pi$
$571$	34.0417	1.42460	0.712301	+	0.701874i	$0.247653\pi$
$572$	−16.9831	−0.710098
$573$	0	0
$574$	−21.8626	−0.912528
$575$	−6.45809	−0.269321
$576$	0	0
$577$	5.45841	0.227237	0.113618	−	0.993524i	$-0.463756\pi$
$577$	5.45841	0.227237	0.113618	+	0.993524i	$0.463756\pi$
$578$	27.4722	1.14269
$579$	0	0
$580$	9.70795	0.403101
$581$	14.9996	0.622288
$582$	0	0
$583$	27.3825	1.13407
$584$	−2.30843	−0.0955236
$585$	0	0
$586$	54.3522	2.24527
$587$	36.8175	1.51962	0.759811	−	0.650144i	$-0.225291\pi$
$587$	36.8175	1.51962	0.759811	+	0.650144i	$0.225291\pi$
$588$	0	0
$589$	28.5115	1.17480
$590$	−8.38642	−0.345264
$591$	0	0
$592$	37.1290	1.52599
$593$	−4.42551	−0.181734	−0.0908670	−	0.995863i	$-0.528964\pi$
$593$	−4.42551	−0.181734	−0.0908670	+	0.995863i	$0.528964\pi$
$594$	0	0
$595$	3.04449	0.124812
$596$	12.9756	0.531500
$597$	0	0
$598$	−14.1652	−0.579259
$599$	35.7470	1.46058	0.730291	−	0.683136i	$-0.239385\pi$
$599$	35.7470	1.46058	0.730291	+	0.683136i	$0.239385\pi$
$600$	0	0
$601$	−25.8654	−1.05507	−0.527535	−	0.849533i	$-0.676884\pi$
$601$	−25.8654	−1.05507	−0.527535	+	0.849533i	$0.676884\pi$
$602$	14.0539	0.572792
$603$	0	0
$604$	11.2754	0.458788
$605$	24.1339	0.981183
$606$	0	0
$607$	19.4234	0.788371	0.394186	−	0.919031i	$-0.371027\pi$
$607$	19.4234	0.788371	0.394186	+	0.919031i	$0.371027\pi$
$608$	18.2890	0.741715
$609$	0	0
$610$	7.16141	0.289957
$611$	22.0898	0.893658
$612$	0	0
$613$	0.530563	0.0214292	0.0107146	−	0.999943i	$-0.496589\pi$
$613$	0.530563	0.0214292	0.0107146	+	0.999943i	$0.496589\pi$
$614$	−46.1668	−1.86314
$615$	0	0
$616$	−25.0351	−1.00869
$617$	43.9644	1.76994	0.884970	−	0.465649i	$-0.154179\pi$
$617$	43.9644	1.76994	0.884970	+	0.465649i	$0.154179\pi$
$618$	0	0
$619$	−36.6443	−1.47286	−0.736430	−	0.676513i	$-0.763490\pi$
$619$	−36.6443	−1.47286	−0.736430	+	0.676513i	$0.763490\pi$
$620$	8.71737	0.350098
$621$	0	0
$622$	34.2633	1.37383
$623$	28.9948	1.16165
$624$	0	0
$625$	−1.95818	−0.0783270
$626$	−6.68965	−0.267372
$627$	0	0
$628$	−15.1119	−0.603032
$629$	−6.39924	−0.255154
$630$	0	0
$631$	−0.938392	−0.0373568	−0.0186784	−	0.999826i	$-0.505946\pi$
$631$	−0.938392	−0.0373568	−0.0186784	+	0.999826i	$0.505946\pi$
$632$	19.4681	0.774400
$633$	0	0
$634$	44.3169	1.76005
$635$	28.8208	1.14372
$636$	0	0
$637$	−3.66812	−0.145336
$638$	70.0043	2.77150
$639$	0	0
$640$	−18.7102	−0.739587
$641$	2.10376	0.0830936	0.0415468	−	0.999137i	$-0.486771\pi$
$641$	2.10376	0.0830936	0.0415468	+	0.999137i	$0.486771\pi$
$642$	0	0
$643$	0.157043	0.00619317	0.00309658	−	0.999995i	$-0.499014\pi$
$643$	0.157043	0.00619317	0.00309658	+	0.999995i	$0.499014\pi$
$644$	4.64656	0.183100
$645$	0	0
$646$	−5.84456	−0.229951
$647$	23.1168	0.908815	0.454407	−	0.890794i	$-0.349851\pi$
$647$	23.1168	0.908815	0.454407	+	0.890794i	$0.349851\pi$
$648$	0	0
$649$	−18.0850	−0.709900
$650$	18.6020	0.729629
$651$	0	0
$652$	13.8561	0.542645
$653$	35.5847	1.39254	0.696269	−	0.717781i	$-0.254842\pi$
$653$	35.5847	1.39254	0.696269	+	0.717781i	$0.254842\pi$
$654$	0	0
$655$	−3.19740	−0.124933
$656$	−26.2403	−1.02451
$657$	0	0
$658$	−24.2301	−0.944587
$659$	4.02769	0.156896	0.0784482	−	0.996918i	$-0.475003\pi$
$659$	4.02769	0.156896	0.0784482	+	0.996918i	$0.475003\pi$
$660$	0	0
$661$	−21.8047	−0.848103	−0.424052	−	0.905638i	$-0.639393\pi$
$661$	−21.8047	−0.848103	−0.424052	+	0.905638i	$0.639393\pi$
$662$	−37.8094	−1.46950
$663$	0	0
$664$	11.8318	0.459165
$665$	14.3078	0.554832
$666$	0	0
$667$	17.4613	0.676105
$668$	17.6016	0.681027
$669$	0	0
$670$	16.8184	0.649750
$671$	15.4433	0.596183
$672$	0	0
$673$	−32.8310	−1.26554	−0.632772	−	0.774338i	$-0.718083\pi$
$673$	−32.8310	−1.26554	−0.632772	+	0.774338i	$0.718083\pi$
$674$	−54.7946	−2.11061
$675$	0	0
$676$	1.10921	0.0426618
$677$	−26.0138	−0.999790	−0.499895	−	0.866086i	$-0.666628\pi$
$677$	−26.0138	−0.999790	−0.499895	+	0.866086i	$0.666628\pi$
$678$	0	0
$679$	−23.5780	−0.904841
$680$	2.40152	0.0920942
$681$	0	0
$682$	62.8612	2.40708
$683$	−23.3530	−0.893578	−0.446789	−	0.894639i	$-0.647433\pi$
$683$	−23.3530	−0.893578	−0.446789	+	0.894639i	$0.647433\pi$
$684$	0	0
$685$	−12.2553	−0.468252
$686$	33.0589	1.26220
$687$	0	0
$688$	16.8679	0.643083
$689$	−19.6734	−0.749498
$690$	0	0
$691$	37.7486	1.43602	0.718012	−	0.696031i	$-0.245052\pi$
$691$	37.7486	1.43602	0.718012	+	0.696031i	$0.245052\pi$
$692$	16.9240	0.643352
$693$	0	0
$694$	−2.06594	−0.0784219
$695$	−11.1404	−0.422580
$696$	0	0
$697$	4.52254	0.171303
$698$	−34.8420	−1.31879
$699$	0	0
$700$	−6.10194	−0.230631
$701$	−29.3166	−1.10727	−0.553635	−	0.832759i	$-0.686760\pi$
$701$	−29.3166	−1.10727	−0.553635	+	0.832759i	$0.686760\pi$
$702$	0	0
$703$	−30.0737	−1.13425
$704$	−12.0837	−0.455422
$705$	0	0
$706$	5.42713	0.204253
$707$	−19.5001	−0.733376
$708$	0	0
$709$	47.3213	1.77719	0.888594	−	0.458694i	$-0.151683\pi$
$709$	47.3213	1.77719	0.888594	+	0.458694i	$0.151683\pi$
$710$	30.6048	1.14858
$711$	0	0
$712$	22.8714	0.857141
$713$	15.6796	0.587206
$714$	0	0
$715$	−28.7589	−1.07552
$716$	−4.39928	−0.164409
$717$	0	0
$718$	−12.8206	−0.478461
$719$	−14.3255	−0.534252	−0.267126	−	0.963662i	$-0.586074\pi$
$719$	−14.3255	−0.534252	−0.267126	+	0.963662i	$0.586074\pi$
$720$	0	0
$721$	36.9414	1.37577
$722$	4.62720	0.172206
$723$	0	0
$724$	−6.57166	−0.244234
$725$	−22.9305	−0.851616
$726$	0	0
$727$	45.2485	1.67818	0.839088	−	0.543996i	$-0.183089\pi$
$727$	45.2485	1.67818	0.839088	+	0.543996i	$0.183089\pi$
$728$	17.9869	0.666639
$729$	0	0
$730$	2.90873	0.107657
$731$	−2.90721	−0.107527
$732$	0	0
$733$	−21.5786	−0.797022	−0.398511	−	0.917164i	$-0.630473\pi$
$733$	−21.5786	−0.797022	−0.398511	+	0.917164i	$0.630473\pi$
$734$	19.0086	0.701618
$735$	0	0
$736$	10.0578	0.370736
$737$	36.2682	1.33596
$738$	0	0
$739$	3.15424	0.116031	0.0580154	−	0.998316i	$-0.481523\pi$
$739$	3.15424	0.116031	0.0580154	+	0.998316i	$0.481523\pi$
$740$	−9.19501	−0.338015
$741$	0	0
$742$	21.5796	0.792212
$743$	28.4790	1.04479	0.522397	−	0.852703i	$-0.325038\pi$
$743$	28.4790	1.04479	0.522397	+	0.852703i	$0.325038\pi$
$744$	0	0
$745$	21.9727	0.805016
$746$	−28.9749	−1.06085
$747$	0	0
$748$	−3.85354	−0.140899
$749$	18.2905	0.668322
$750$	0	0
$751$	−11.3748	−0.415071	−0.207536	−	0.978227i	$-0.566544\pi$
$751$	−11.3748	−0.415071	−0.207536	+	0.978227i	$0.566544\pi$
$752$	−29.0818	−1.06050
$753$	0	0
$754$	−50.2958	−1.83167
$755$	19.0935	0.694885
$756$	0	0
$757$	39.3892	1.43163	0.715813	−	0.698293i	$-0.246057\pi$
$757$	39.3892	1.43163	0.715813	+	0.698293i	$0.246057\pi$
$758$	30.3033	1.10067
$759$	0	0
$760$	11.2861	0.409391
$761$	28.6305	1.03785	0.518927	−	0.854819i	$-0.326332\pi$
$761$	28.6305	1.03785	0.518927	+	0.854819i	$0.326332\pi$
$762$	0	0
$763$	18.1254	0.656185
$764$	2.10186	0.0760427
$765$	0	0
$766$	19.3016	0.697395
$767$	12.9935	0.469169
$768$	0	0
$769$	−44.0196	−1.58739	−0.793694	−	0.608317i	$-0.791845\pi$
$769$	−44.0196	−1.58739	−0.793694	+	0.608317i	$0.791845\pi$
$770$	31.5453	1.13681
$771$	0	0
$772$	−0.853276	−0.0307101
$773$	−10.3466	−0.372143	−0.186071	−	0.982536i	$-0.559576\pi$
$773$	−10.3466	−0.372143	−0.186071	+	0.982536i	$0.559576\pi$
$774$	0	0
$775$	−20.5907	−0.739639
$776$	−18.5986	−0.667650
$777$	0	0
$778$	47.9110	1.71769
$779$	21.2540	0.761504
$780$	0	0
$781$	65.9981	2.36160
$782$	−3.21415	−0.114938
$783$	0	0
$784$	4.82917	0.172470
$785$	−25.5904	−0.913359
$786$	0	0
$787$	16.1088	0.574218	0.287109	−	0.957898i	$-0.407306\pi$
$787$	16.1088	0.574218	0.287109	+	0.957898i	$0.407306\pi$
$788$	−8.02629	−0.285925
$789$	0	0
$790$	−24.5307	−0.872763
$791$	23.2009	0.824931
$792$	0	0
$793$	−11.0955	−0.394014
$794$	6.67217	0.236787
$795$	0	0
$796$	4.98934	0.176843
$797$	10.9570	0.388118	0.194059	−	0.980990i	$-0.437835\pi$
$797$	10.9570	0.388118	0.194059	+	0.980990i	$0.437835\pi$
$798$	0	0
$799$	5.01228	0.177322
$800$	−13.2081	−0.466976
$801$	0	0
$802$	−16.4495	−0.580851
$803$	6.27258	0.221354
$804$	0	0
$805$	7.86842	0.277325
$806$	−45.1637	−1.59082
$807$	0	0
$808$	−15.3819	−0.541132
$809$	−16.9641	−0.596427	−0.298213	−	0.954499i	$-0.596391\pi$
$809$	−16.9641	−0.596427	−0.298213	+	0.954499i	$0.596391\pi$
$810$	0	0
$811$	−5.88415	−0.206621	−0.103310	−	0.994649i	$-0.532943\pi$
$811$	−5.88415	−0.206621	−0.103310	+	0.994649i	$0.532943\pi$
$812$	16.4984	0.578979
$813$	0	0
$814$	−66.3054	−2.32400
$815$	23.4637	0.821896
$816$	0	0
$817$	−13.6626	−0.477995
$818$	0.693683	0.0242541
$819$	0	0
$820$	6.49840	0.226934
$821$	−54.4044	−1.89873	−0.949363	−	0.314182i	$-0.898270\pi$
$821$	−54.4044	−1.89873	−0.949363	+	0.314182i	$0.898270\pi$
$822$	0	0
$823$	−20.4999	−0.714580	−0.357290	−	0.933993i	$-0.616299\pi$
$823$	−20.4999	−0.714580	−0.357290	+	0.933993i	$0.616299\pi$
$824$	29.1398	1.01513
$825$	0	0
$826$	−14.2525	−0.495906
$827$	−17.6988	−0.615449	−0.307724	−	0.951476i	$-0.599568\pi$
$827$	−17.6988	−0.615449	−0.307724	+	0.951476i	$0.599568\pi$
$828$	0	0
$829$	−55.0939	−1.91349	−0.956744	−	0.290930i	$-0.906035\pi$
$829$	−55.0939	−1.91349	−0.956744	+	0.290930i	$0.906035\pi$
$830$	−14.9087	−0.517487
$831$	0	0
$832$	8.68175	0.300986
$833$	−0.832314	−0.0288380
$834$	0	0
$835$	29.8063	1.03149
$836$	−18.1100	−0.626347
$837$	0	0
$838$	68.2438	2.35744
$839$	−2.52000	−0.0869999	−0.0435000	−	0.999053i	$-0.513851\pi$
$839$	−2.52000	−0.0869999	−0.0435000	+	0.999053i	$0.513851\pi$
$840$	0	0
$841$	32.9992	1.13790
$842$	31.7901	1.09556
$843$	0	0
$844$	11.3578	0.390953
$845$	1.87832	0.0646160
$846$	0	0
$847$	41.0148	1.40929
$848$	25.9006	0.889429
$849$	0	0
$850$	4.22088	0.144775
$851$	−16.5387	−0.566940
$852$	0	0
$853$	−26.7427	−0.915653	−0.457826	−	0.889042i	$-0.651372\pi$
$853$	−26.7427	−0.915653	−0.457826	+	0.889042i	$0.651372\pi$
$854$	12.1706	0.416469
$855$	0	0
$856$	14.4278	0.493131
$857$	−16.1901	−0.553043	−0.276521	−	0.961008i	$-0.589182\pi$
$857$	−16.1901	−0.553043	−0.276521	+	0.961008i	$0.589182\pi$
$858$	0	0
$859$	−28.1540	−0.960603	−0.480302	−	0.877103i	$-0.659473\pi$
$859$	−28.1540	−0.960603	−0.480302	+	0.877103i	$0.659473\pi$
$860$	−4.17734	−0.142446
$861$	0	0
$862$	53.7451	1.83057
$863$	−32.6647	−1.11192	−0.555959	−	0.831210i	$-0.687649\pi$
$863$	−32.6647	−1.11192	−0.555959	+	0.831210i	$0.687649\pi$
$864$	0	0
$865$	28.6588	0.974428
$866$	−28.2061	−0.958483
$867$	0	0
$868$	14.8149	0.502850
$869$	−52.8996	−1.79450
$870$	0	0
$871$	−26.0576	−0.882927
$872$	14.2975	0.484176
$873$	0	0
$874$	−15.1052	−0.510939
$875$	−28.0737	−0.949066
$876$	0	0
$877$	25.7920	0.870934	0.435467	−	0.900205i	$-0.356583\pi$
$877$	25.7920	0.870934	0.435467	+	0.900205i	$0.356583\pi$
$878$	22.3957	0.755819
$879$	0	0
$880$	37.8618	1.27632
$881$	−52.5016	−1.76882	−0.884412	−	0.466708i	$-0.845440\pi$
$881$	−52.5016	−1.76882	−0.884412	+	0.466708i	$0.845440\pi$
$882$	0	0
$883$	44.5711	1.49994	0.749969	−	0.661473i	$-0.230068\pi$
$883$	44.5711	1.49994	0.749969	+	0.661473i	$0.230068\pi$
$884$	2.76864	0.0931196
$885$	0	0
$886$	−12.8830	−0.432814
$887$	22.8894	0.768550	0.384275	−	0.923219i	$-0.374452\pi$
$887$	22.8894	0.768550	0.384275	+	0.923219i	$0.374452\pi$
$888$	0	0
$889$	48.9800	1.64274
$890$	−28.8190	−0.966014
$891$	0	0
$892$	−24.5755	−0.822848
$893$	23.5556	0.788258
$894$	0	0
$895$	−7.44968	−0.249015
$896$	−31.7974	−1.06228
$897$	0	0
$898$	−41.0126	−1.36861
$899$	55.6729	1.85679
$900$	0	0
$901$	−4.46400	−0.148717
$902$	46.8601	1.56027
$903$	0	0
$904$	18.3011	0.608687
$905$	−11.1284	−0.369919
$906$	0	0
$907$	−53.7367	−1.78430	−0.892149	−	0.451742i	$-0.850803\pi$
$907$	−53.7367	−1.78430	−0.892149	+	0.451742i	$0.850803\pi$
$908$	16.3701	0.543259
$909$	0	0
$910$	−22.6643	−0.751315
$911$	−39.0107	−1.29248	−0.646241	−	0.763134i	$-0.723660\pi$
$911$	−39.0107	−1.29248	−0.646241	+	0.763134i	$0.723660\pi$
$912$	0	0
$913$	−32.1500	−1.06401
$914$	8.70540	0.287949
$915$	0	0
$916$	4.32333	0.142847
$917$	−5.43388	−0.179443
$918$	0	0
$919$	41.6686	1.37452	0.687260	−	0.726411i	$-0.258813\pi$
$919$	41.6686	1.37452	0.687260	+	0.726411i	$0.258813\pi$
$920$	6.20669	0.204629
$921$	0	0
$922$	23.3982	0.770578
$923$	−47.4175	−1.56077
$924$	0	0
$925$	21.7189	0.714113
$926$	26.5220	0.871568
$927$	0	0
$928$	35.7119	1.17230
$929$	−40.9634	−1.34396	−0.671982	−	0.740567i	$-0.734557\pi$
$929$	−40.9634	−1.34396	−0.671982	+	0.740567i	$0.734557\pi$
$930$	0	0
$931$	−3.91152	−0.128195
$932$	−19.5036	−0.638861
$933$	0	0
$934$	54.8681	1.79534
$935$	−6.52553	−0.213408
$936$	0	0
$937$	−21.9291	−0.716392	−0.358196	−	0.933647i	$-0.616608\pi$
$937$	−21.9291	−0.716392	−0.358196	+	0.933647i	$0.616608\pi$
$938$	28.5823	0.933245
$939$	0	0
$940$	7.20210	0.234907
$941$	−7.65724	−0.249619	−0.124809	−	0.992181i	$-0.539832\pi$
$941$	−7.65724	−0.249619	−0.124809	+	0.992181i	$0.539832\pi$
$942$	0	0
$943$	11.6884	0.380628
$944$	−17.1063	−0.556762
$945$	0	0
$946$	−30.1229	−0.979380
$947$	−55.9980	−1.81969	−0.909846	−	0.414946i	$-0.863800\pi$
$947$	−55.9980	−1.81969	−0.909846	+	0.414946i	$0.863800\pi$
$948$	0	0
$949$	−4.50664	−0.146292
$950$	19.8363	0.643575
$951$	0	0
$952$	4.08131	0.132276
$953$	27.3337	0.885424	0.442712	−	0.896664i	$-0.354016\pi$
$953$	27.3337	0.885424	0.442712	+	0.896664i	$0.354016\pi$
$954$	0	0
$955$	3.55926	0.115175
$956$	1.07776	0.0348572
$957$	0	0
$958$	2.23466	0.0721985
$959$	−20.8276	−0.672557
$960$	0	0
$961$	18.9921	0.612649
$962$	47.6383	1.53592
$963$	0	0
$964$	14.0534	0.452631
$965$	−1.44492	−0.0465138
$966$	0	0
$967$	−1.24090	−0.0399046	−0.0199523	−	0.999801i	$-0.506351\pi$
$967$	−1.24090	−0.0399046	−0.0199523	+	0.999801i	$0.506351\pi$
$968$	32.3529	1.03986
$969$	0	0
$970$	23.4350	0.752454
$971$	25.9199	0.831810	0.415905	−	0.909408i	$-0.363465\pi$
$971$	25.9199	0.831810	0.415905	+	0.909408i	$0.363465\pi$
$972$	0	0
$973$	−18.9328	−0.606956
$974$	62.7130	2.00946
$975$	0	0
$976$	14.6075	0.467576
$977$	−17.6037	−0.563191	−0.281596	−	0.959533i	$-0.590864\pi$
$977$	−17.6037	−0.563191	−0.281596	+	0.959533i	$0.590864\pi$
$978$	0	0
$979$	−62.1471	−1.98623
$980$	−1.19594	−0.0382031
$981$	0	0
$982$	31.6635	1.01042
$983$	−4.01123	−0.127938	−0.0639692	−	0.997952i	$-0.520376\pi$
$983$	−4.01123	−0.127938	−0.0639692	+	0.997952i	$0.520376\pi$
$984$	0	0
$985$	−13.5916	−0.433065
$986$	−11.4124	−0.363444
$987$	0	0
$988$	13.0114	0.413949
$989$	−7.51363	−0.238919
$990$	0	0
$991$	58.0671	1.84456	0.922281	−	0.386519i	$-0.126323\pi$
$991$	58.0671	1.84456	0.922281	+	0.386519i	$0.126323\pi$
$992$	32.0679	1.01816
$993$	0	0
$994$	52.0118	1.64971
$995$	8.44889	0.267848
$996$	0	0
$997$	21.8914	0.693307	0.346653	−	0.937993i	$-0.387318\pi$
$997$	21.8914	0.693307	0.346653	+	0.937993i	$0.387318\pi$
$998$	37.0804	1.17376
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1737.2.a.j.1.5		13
3.2	odd	2		579.2.a.g.1.9	✓	13
12.11	even	2		9264.2.a.bp.1.4		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
579.2.a.g.1.9	✓	13	3.2	odd	2
1737.2.a.j.1.5		13	1.1	even	1	trivial
9264.2.a.bp.1.4		13	12.11	even	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:	\( N \)	\(=\)	\( 1737 = 3^{2} \cdot 193 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1737.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68916 q^{2} +0.853276 q^{4} +1.44492 q^{5} +2.45560 q^{7} +1.93701 q^{8} +O(q^{10})\)	\(q-1.68916 q^{2} +0.853276 q^{4} +1.44492 q^{5} +2.45560 q^{7} +1.93701 q^{8} -2.44072 q^{10} -5.26332 q^{11} +3.78153 q^{13} -4.14792 q^{14} -4.97847 q^{16} +0.858046 q^{17} +4.03245 q^{19} +1.23292 q^{20} +8.89061 q^{22} +2.21760 q^{23} -2.91219 q^{25} -6.38762 q^{26} +2.09531 q^{28} +7.87396 q^{29} +7.07051 q^{31} +4.53544 q^{32} -1.44938 q^{34} +3.54816 q^{35} -7.45792 q^{37} -6.81147 q^{38} +2.79883 q^{40} +5.27074 q^{41} -3.38817 q^{43} -4.49106 q^{44} -3.74590 q^{46} +5.84151 q^{47} -0.970011 q^{49} +4.91917 q^{50} +3.22668 q^{52} -5.20251 q^{53} -7.60510 q^{55} +4.75652 q^{56} -13.3004 q^{58} +3.43605 q^{59} -2.93414 q^{61} -11.9433 q^{62} +2.29583 q^{64} +5.46402 q^{65} -6.89075 q^{67} +0.732150 q^{68} -5.99343 q^{70} -12.5393 q^{71} -1.19175 q^{73} +12.5976 q^{74} +3.44079 q^{76} -12.9246 q^{77} +10.0506 q^{79} -7.19352 q^{80} -8.90315 q^{82} +6.10832 q^{83} +1.23981 q^{85} +5.72318 q^{86} -10.1951 q^{88} +11.8076 q^{89} +9.28593 q^{91} +1.89223 q^{92} -9.86726 q^{94} +5.82659 q^{95} -9.60171 q^{97} +1.63851 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 18 q^{4} - 6 q^{5} + 15 q^{7} - 3 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 18 * q^4 - 6 * q^5 + 15 * q^7 - 3 * q^8	\( 13 q - 2 q^{2} + 18 q^{4} - 6 q^{5} + 15 q^{7} - 3 q^{8} - q^{10} - 3 q^{11} + 11 q^{13} + 4 q^{14} + 20 q^{16} + 2 q^{17} + 9 q^{19} + 9 q^{20} - 8 q^{22} + 8 q^{23} + 21 q^{25} + 15 q^{26} + 16 q^{28} - 5 q^{29} + 25 q^{31} + 17 q^{32} - 10 q^{34} + 10 q^{35} + 29 q^{37} + 40 q^{38} - 21 q^{40} + 11 q^{41} + 8 q^{43} + 18 q^{44} - 6 q^{46} + 12 q^{47} + 20 q^{49} + 4 q^{50} + 2 q^{52} - 14 q^{53} + 12 q^{55} + 7 q^{56} + 9 q^{58} - 10 q^{59} + 6 q^{61} + 14 q^{62} + 23 q^{64} + 15 q^{65} + 25 q^{67} + 33 q^{68} - 21 q^{70} + 8 q^{73} + 2 q^{74} + 20 q^{76} + 25 q^{77} + 7 q^{79} + 40 q^{80} - 19 q^{82} + 28 q^{83} - 3 q^{85} - 2 q^{86} - 21 q^{88} - 7 q^{89} + 7 q^{91} - 9 q^{92} - 35 q^{94} + 26 q^{95} + 26 q^{97} + 14 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 18 * q^4 - 6 * q^5 + 15 * q^7 - 3 * q^8 - q^10 - 3 * q^11 + 11 * q^13 + 4 * q^14 + 20 * q^16 + 2 * q^17 + 9 * q^19 + 9 * q^20 - 8 * q^22 + 8 * q^23 + 21 * q^25 + 15 * q^26 + 16 * q^28 - 5 * q^29 + 25 * q^31 + 17 * q^32 - 10 * q^34 + 10 * q^35 + 29 * q^37 + 40 * q^38 - 21 * q^40 + 11 * q^41 + 8 * q^43 + 18 * q^44 - 6 * q^46 + 12 * q^47 + 20 * q^49 + 4 * q^50 + 2 * q^52 - 14 * q^53 + 12 * q^55 + 7 * q^56 + 9 * q^58 - 10 * q^59 + 6 * q^61 + 14 * q^62 + 23 * q^64 + 15 * q^65 + 25 * q^67 + 33 * q^68 - 21 * q^70 + 8 * q^73 + 2 * q^74 + 20 * q^76 + 25 * q^77 + 7 * q^79 + 40 * q^80 - 19 * q^82 + 28 * q^83 - 3 * q^85 - 2 * q^86 - 21 * q^88 - 7 * q^89 + 7 * q^91 - 9 * q^92 - 35 * q^94 + 26 * q^95 + 26 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more