LMFDB - Embedded newform 2151.2.a.e.1.1

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			1.1
Root			$0.267500$ 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)

$f(q)$	$=$	$q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} +O(q^{10})$	\(q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} -5.23885 q^{10} +4.01073 q^{11} +1.62598 q^{13} +5.20045 q^{14} -2.76312 q^{16} -4.89570 q^{17} -6.28561 q^{19} +6.16623 q^{20} -8.50391 q^{22} +3.83124 q^{23} +1.10496 q^{25} -3.44755 q^{26} -6.12103 q^{28} +0.357168 q^{29} -8.44143 q^{31} +7.96032 q^{32} +10.3803 q^{34} -6.06021 q^{35} -1.81610 q^{37} +13.3273 q^{38} -2.59648 q^{40} -10.7403 q^{41} +9.09255 q^{43} +10.0093 q^{44} -8.12332 q^{46} -11.0197 q^{47} -0.984213 q^{49} -2.34284 q^{50} +4.05783 q^{52} -5.78427 q^{53} +9.90981 q^{55} +2.57745 q^{56} -0.757299 q^{58} +1.43742 q^{59} +2.21862 q^{61} +17.8983 q^{62} -11.3519 q^{64} +4.01751 q^{65} -14.2454 q^{67} -12.2178 q^{68} +12.8494 q^{70} +4.12013 q^{71} +2.78648 q^{73} +3.85066 q^{74} -15.6865 q^{76} -9.83717 q^{77} +7.84263 q^{79} -6.82718 q^{80} +22.7725 q^{82} +9.91715 q^{83} -12.0964 q^{85} -19.2788 q^{86} -4.21471 q^{88} -0.978993 q^{89} -3.98806 q^{91} +9.56131 q^{92} +23.3649 q^{94} -15.5306 q^{95} -16.8523 q^{97} +2.08682 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.12029	−1.49927	−0.749635	−	0.661852i	$-0.769771\pi$
$2$	−2.12029	−1.49927	−0.749635	+	0.661852i	$0.769771\pi$
$3$	0	0
$3$	0	0
$4$	2.49562	1.24781
$5$	2.47082	1.10499	0.552493	−	0.833518i	$-0.313677\pi$
$5$	2.47082	1.10499	0.552493	+	0.833518i	$0.313677\pi$
$6$	0	0
$7$	−2.45271	−0.927037	−0.463519	−	0.886087i	$-0.653413\pi$
$7$	−2.45271	−0.927037	−0.463519	+	0.886087i	$0.653413\pi$
$8$	−1.05086	−0.371534
$9$	0	0
$10$	−5.23885	−1.65667
$11$	4.01073	1.20928	0.604641	−	0.796498i	$-0.293317\pi$
$11$	4.01073	1.20928	0.604641	+	0.796498i	$0.293317\pi$
$12$	0	0
$13$	1.62598	0.450966	0.225483	−	0.974247i	$-0.427604\pi$
$13$	1.62598	0.450966	0.225483	+	0.974247i	$0.427604\pi$
$14$	5.20045	1.38988
$15$	0	0
$16$	−2.76312	−0.690780
$17$	−4.89570	−1.18738	−0.593691	−	0.804693i	$-0.702330\pi$
$17$	−4.89570	−1.18738	−0.593691	+	0.804693i	$0.702330\pi$
$18$	0	0
$19$	−6.28561	−1.44202	−0.721009	−	0.692926i	$-0.756321\pi$
$19$	−6.28561	−1.44202	−0.721009	+	0.692926i	$0.756321\pi$
$20$	6.16623	1.37881
$21$	0	0
$22$	−8.50391	−1.81304
$23$	3.83124	0.798868	0.399434	−	0.916762i	$-0.369207\pi$
$23$	3.83124	0.798868	0.399434	+	0.916762i	$0.369207\pi$
$24$	0	0
$25$	1.10496	0.220992
$26$	−3.44755	−0.676120
$27$	0	0
$28$	−6.12103	−1.15677
$29$	0.357168	0.0663244	0.0331622	−	0.999450i	$-0.489442\pi$
$29$	0.357168	0.0663244	0.0331622	+	0.999450i	$0.489442\pi$
$30$	0	0
$31$	−8.44143	−1.51613	−0.758063	−	0.652181i	$-0.773854\pi$
$31$	−8.44143	−1.51613	−0.758063	+	0.652181i	$0.773854\pi$
$32$	7.96032	1.40720
$33$	0	0
$34$	10.3803	1.78020
$35$	−6.06021	−1.02436
$36$	0	0
$37$	−1.81610	−0.298566	−0.149283	−	0.988795i	$-0.547696\pi$
$37$	−1.81610	−0.298566	−0.149283	+	0.988795i	$0.547696\pi$
$38$	13.3273	2.16197
$39$	0	0
$40$	−2.59648	−0.410540
$41$	−10.7403	−1.67735	−0.838676	−	0.544630i	$-0.816670\pi$
$41$	−10.7403	−1.67735	−0.838676	+	0.544630i	$0.816670\pi$
$42$	0	0
$43$	9.09255	1.38660	0.693301	−	0.720649i	$-0.256156\pi$
$43$	9.09255	1.38660	0.693301	+	0.720649i	$0.256156\pi$
$44$	10.0093	1.50895
$45$	0	0
$46$	−8.12332	−1.19772
$47$	−11.0197	−1.60739	−0.803693	−	0.595044i	$-0.797134\pi$
$47$	−11.0197	−1.60739	−0.803693	+	0.595044i	$0.797134\pi$
$48$	0	0
$49$	−0.984213	−0.140602
$50$	−2.34284	−0.331327
$51$	0	0
$52$	4.05783	0.562720
$53$	−5.78427	−0.794530	−0.397265	−	0.917704i	$-0.630041\pi$
$53$	−5.78427	−0.794530	−0.397265	+	0.917704i	$0.630041\pi$
$54$	0	0
$55$	9.90981	1.33624
$56$	2.57745	0.344426
$57$	0	0
$58$	−0.757299	−0.0994382
$59$	1.43742	0.187136	0.0935678	−	0.995613i	$-0.470173\pi$
$59$	1.43742	0.187136	0.0935678	+	0.995613i	$0.470173\pi$
$60$	0	0
$61$	2.21862	0.284065	0.142032	−	0.989862i	$-0.454636\pi$
$61$	2.21862	0.284065	0.142032	+	0.989862i	$0.454636\pi$
$62$	17.8983	2.27308
$63$	0	0
$64$	−11.3519	−1.41899
$65$	4.01751	0.498311
$66$	0	0
$67$	−14.2454	−1.74036	−0.870178	−	0.492738i	$-0.835996\pi$
$67$	−14.2454	−1.74036	−0.870178	+	0.492738i	$0.835996\pi$
$68$	−12.2178	−1.48163
$69$	0	0
$70$	12.8494	1.53580
$71$	4.12013	0.488969	0.244485	−	0.969653i	$-0.421381\pi$
$71$	4.12013	0.488969	0.244485	+	0.969653i	$0.421381\pi$
$72$	0	0
$73$	2.78648	0.326133	0.163067	−	0.986615i	$-0.447861\pi$
$73$	2.78648	0.326133	0.163067	+	0.986615i	$0.447861\pi$
$74$	3.85066	0.447630
$75$	0	0
$76$	−15.6865	−1.79936
$77$	−9.83717	−1.12105
$78$	0	0
$79$	7.84263	0.882365	0.441182	−	0.897417i	$-0.354559\pi$
$79$	7.84263	0.882365	0.441182	+	0.897417i	$0.354559\pi$
$80$	−6.82718	−0.763302
$81$	0	0
$82$	22.7725	2.51480
$83$	9.91715	1.08855	0.544274	−	0.838907i	$-0.316805\pi$
$83$	9.91715	1.08855	0.544274	+	0.838907i	$0.316805\pi$
$84$	0	0
$85$	−12.0964	−1.31204
$86$	−19.2788	−2.07889
$87$	0	0
$88$	−4.21471	−0.449290
$89$	−0.978993	−0.103773	−0.0518865	−	0.998653i	$-0.516523\pi$
$89$	−0.978993	−0.103773	−0.0518865	+	0.998653i	$0.516523\pi$
$90$	0	0
$91$	−3.98806	−0.418062
$92$	9.56131	0.996836
$93$	0	0
$94$	23.3649	2.40991
$95$	−15.5306	−1.59341
$96$	0	0
$97$	−16.8523	−1.71109	−0.855545	−	0.517728i	$-0.826778\pi$
$97$	−16.8523	−1.71109	−0.855545	+	0.517728i	$0.826778\pi$
$98$	2.08682	0.210800
$99$	0	0
$100$	2.75756	0.275756
$101$	−3.80431	−0.378543	−0.189271	−	0.981925i	$-0.560613\pi$
$101$	−3.80431	−0.378543	−0.189271	+	0.981925i	$0.560613\pi$
$102$	0	0
$103$	8.01017	0.789266	0.394633	−	0.918839i	$-0.370872\pi$
$103$	8.01017	0.789266	0.394633	+	0.918839i	$0.370872\pi$
$104$	−1.70867	−0.167549
$105$	0	0
$106$	12.2643	1.19122
$107$	12.6857	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$107$	12.6857	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$108$	0	0
$109$	−18.7766	−1.79847	−0.899237	−	0.437461i	$-0.855878\pi$
$109$	−18.7766	−1.79847	−0.899237	+	0.437461i	$0.855878\pi$
$110$	−21.0116	−2.00338
$111$	0	0
$112$	6.77713	0.640379
$113$	7.61332	0.716201	0.358101	−	0.933683i	$-0.383425\pi$
$113$	7.61332	0.716201	0.358101	+	0.933683i	$0.383425\pi$
$114$	0	0
$115$	9.46630	0.882737
$116$	0.891355	0.0827603
$117$	0	0
$118$	−3.04774	−0.280567
$119$	12.0077	1.10075
$120$	0	0
$121$	5.08598	0.462362
$122$	−4.70410	−0.425890
$123$	0	0
$124$	−21.0666	−1.89184
$125$	−9.62395	−0.860792
$126$	0	0
$127$	14.0067	1.24290	0.621448	−	0.783455i	$-0.286545\pi$
$127$	14.0067	1.24290	0.621448	+	0.783455i	$0.286545\pi$
$128$	8.14873	0.720253
$129$	0	0
$130$	−8.51828	−0.747102
$131$	4.05565	0.354344	0.177172	−	0.984180i	$-0.443305\pi$
$131$	4.05565	0.354344	0.177172	+	0.984180i	$0.443305\pi$
$132$	0	0
$133$	15.4168	1.33680
$134$	30.2044	2.60926
$135$	0	0
$136$	5.14468	0.441153
$137$	15.5346	1.32721	0.663604	−	0.748084i	$-0.269026\pi$
$137$	15.5346	1.32721	0.663604	+	0.748084i	$0.269026\pi$
$138$	0	0
$139$	1.09201	0.0926232	0.0463116	−	0.998927i	$-0.485253\pi$
$139$	1.09201	0.0926232	0.0463116	+	0.998927i	$0.485253\pi$
$140$	−15.1240	−1.27821
$141$	0	0
$142$	−8.73586	−0.733097
$143$	6.52138	0.545345
$144$	0	0
$145$	0.882498	0.0732875
$146$	−5.90815	−0.488962
$147$	0	0
$148$	−4.53230	−0.372553
$149$	15.9702	1.30833	0.654167	−	0.756350i	$-0.273019\pi$
$149$	15.9702	1.30833	0.654167	+	0.756350i	$0.273019\pi$
$150$	0	0
$151$	−8.44617	−0.687339	−0.343670	−	0.939091i	$-0.611670\pi$
$151$	−8.44617	−0.687339	−0.343670	+	0.939091i	$0.611670\pi$
$152$	6.60528	0.535759
$153$	0	0
$154$	20.8576	1.68076
$155$	−20.8573	−1.67530
$156$	0	0
$157$	8.21810	0.655876	0.327938	−	0.944699i	$-0.393646\pi$
$157$	8.21810	0.655876	0.327938	+	0.944699i	$0.393646\pi$
$158$	−16.6286	−1.32290
$159$	0	0
$160$	19.6685	1.55494
$161$	−9.39691	−0.740580
$162$	0	0
$163$	3.58850	0.281073	0.140537	−	0.990075i	$-0.455117\pi$
$163$	3.58850	0.281073	0.140537	+	0.990075i	$0.455117\pi$
$164$	−26.8037	−2.09302
$165$	0	0
$166$	−21.0272	−1.63203
$167$	−14.0773	−1.08934	−0.544668	−	0.838652i	$-0.683344\pi$
$167$	−14.0773	−1.08934	−0.544668	+	0.838652i	$0.683344\pi$
$168$	0	0
$169$	−10.3562	−0.796630
$170$	25.6478	1.96710
$171$	0	0
$172$	22.6916	1.73021
$173$	−5.90842	−0.449208	−0.224604	−	0.974450i	$-0.572109\pi$
$173$	−5.90842	−0.449208	−0.224604	+	0.974450i	$0.572109\pi$
$174$	0	0
$175$	−2.71015	−0.204868
$176$	−11.0821	−0.835348
$177$	0	0
$178$	2.07575	0.155584
$179$	−18.5491	−1.38642	−0.693211	−	0.720735i	$-0.743805\pi$
$179$	−18.5491	−1.38642	−0.693211	+	0.720735i	$0.743805\pi$
$180$	0	0
$181$	−0.921544	−0.0684978	−0.0342489	−	0.999413i	$-0.510904\pi$
$181$	−0.921544	−0.0684978	−0.0342489	+	0.999413i	$0.510904\pi$
$182$	8.45584	0.626788
$183$	0	0
$184$	−4.02608	−0.296807
$185$	−4.48727	−0.329911
$186$	0	0
$187$	−19.6353	−1.43588
$188$	−27.5009	−2.00571
$189$	0	0
$190$	32.9294	2.38895
$191$	−15.1285	−1.09466	−0.547329	−	0.836918i	$-0.684355\pi$
$191$	−15.1285	−1.09466	−0.547329	+	0.836918i	$0.684355\pi$
$192$	0	0
$193$	−22.7285	−1.63603	−0.818015	−	0.575197i	$-0.804925\pi$
$193$	−22.7285	−1.63603	−0.818015	+	0.575197i	$0.804925\pi$
$194$	35.7317	2.56539
$195$	0	0
$196$	−2.45622	−0.175444
$197$	2.47078	0.176036	0.0880179	−	0.996119i	$-0.471947\pi$
$197$	2.47078	0.176036	0.0880179	+	0.996119i	$0.471947\pi$
$198$	0	0
$199$	−5.34461	−0.378869	−0.189435	−	0.981893i	$-0.560666\pi$
$199$	−5.34461	−0.378869	−0.189435	+	0.981893i	$0.560666\pi$
$200$	−1.16116	−0.0821062
$201$	0	0
$202$	8.06622	0.567538
$203$	−0.876029	−0.0614852
$204$	0	0
$205$	−26.5374	−1.85345
$206$	−16.9839	−1.18332
$207$	0	0
$208$	−4.49278	−0.311518
$209$	−25.2099	−1.74381
$210$	0	0
$211$	−10.3377	−0.711676	−0.355838	−	0.934548i	$-0.615804\pi$
$211$	−10.3377	−0.711676	−0.355838	+	0.934548i	$0.615804\pi$
$212$	−14.4353	−0.991423
$213$	0	0
$214$	−26.8974	−1.83867
$215$	22.4661	1.53217
$216$	0	0
$217$	20.7044	1.40551
$218$	39.8119	2.69640
$219$	0	0
$220$	24.7311	1.66737
$221$	−7.96031	−0.535469
$222$	0	0
$223$	−10.2702	−0.687745	−0.343873	−	0.939016i	$-0.611739\pi$
$223$	−10.2702	−0.687745	−0.343873	+	0.939016i	$0.611739\pi$
$224$	−19.5244	−1.30453
$225$	0	0
$226$	−16.1424	−1.07378
$227$	16.9467	1.12479	0.562396	−	0.826868i	$-0.309880\pi$
$227$	16.9467	1.12479	0.562396	+	0.826868i	$0.309880\pi$
$228$	0	0
$229$	−29.0495	−1.91964	−0.959822	−	0.280609i	$-0.909464\pi$
$229$	−29.0495	−1.91964	−0.959822	+	0.280609i	$0.909464\pi$
$230$	−20.0713	−1.32346
$231$	0	0
$232$	−0.375332	−0.0246418
$233$	19.7368	1.29300	0.646500	−	0.762914i	$-0.276232\pi$
$233$	19.7368	1.29300	0.646500	+	0.762914i	$0.276232\pi$
$234$	0	0
$235$	−27.2277	−1.77614
$236$	3.58725	0.233510
$237$	0	0
$238$	−25.4598	−1.65032
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	10.4837	0.675311	0.337656	−	0.941270i	$-0.390366\pi$
$241$	10.4837	0.675311	0.337656	+	0.941270i	$0.390366\pi$
$242$	−10.7837	−0.693205
$243$	0	0
$244$	5.53682	0.354459
$245$	−2.43182	−0.155363
$246$	0	0
$247$	−10.2203	−0.650301
$248$	8.87074	0.563293
$249$	0	0
$250$	20.4055	1.29056
$251$	−12.1227	−0.765177	−0.382588	−	0.923919i	$-0.624967\pi$
$251$	−12.1227	−0.765177	−0.382588	+	0.923919i	$0.624967\pi$
$252$	0	0
$253$	15.3661	0.966056
$254$	−29.6983	−1.86344
$255$	0	0
$256$	5.42623	0.339139
$257$	−28.3873	−1.77075	−0.885377	−	0.464874i	$-0.846100\pi$
$257$	−28.3873	−1.77075	−0.885377	+	0.464874i	$0.846100\pi$
$258$	0	0
$259$	4.45438	0.276781
$260$	10.0262	0.621797
$261$	0	0
$262$	−8.59914	−0.531257
$263$	14.0844	0.868482	0.434241	−	0.900797i	$-0.357017\pi$
$263$	14.0844	0.868482	0.434241	+	0.900797i	$0.357017\pi$
$264$	0	0
$265$	−14.2919	−0.877944
$266$	−32.6880	−2.00423
$267$	0	0
$268$	−35.5512	−2.17163
$269$	−22.1224	−1.34883	−0.674415	−	0.738353i	$-0.735604\pi$
$269$	−22.1224	−1.34883	−0.674415	+	0.738353i	$0.735604\pi$
$270$	0	0
$271$	−0.185181	−0.0112490	−0.00562449	−	0.999984i	$-0.501790\pi$
$271$	−0.185181	−0.0112490	−0.00562449	+	0.999984i	$0.501790\pi$
$272$	13.5274	0.820219
$273$	0	0
$274$	−32.9377	−1.98984
$275$	4.43171	0.267242
$276$	0	0
$277$	−15.8869	−0.954549	−0.477274	−	0.878754i	$-0.658375\pi$
$277$	−15.8869	−0.954549	−0.477274	+	0.878754i	$0.658375\pi$
$278$	−2.31538	−0.138867
$279$	0	0
$280$	6.36842	0.380586
$281$	−19.9963	−1.19288	−0.596440	−	0.802658i	$-0.703419\pi$
$281$	−19.9963	−1.19288	−0.596440	+	0.802658i	$0.703419\pi$
$282$	0	0
$283$	−30.1737	−1.79364	−0.896819	−	0.442398i	$-0.854128\pi$
$283$	−30.1737	−1.79364	−0.896819	+	0.442398i	$0.854128\pi$
$284$	10.2823	0.610141
$285$	0	0
$286$	−13.8272	−0.817619
$287$	26.3428	1.55497
$288$	0	0
$289$	6.96786	0.409874
$290$	−1.87115	−0.109878
$291$	0	0
$292$	6.95401	0.406953
$293$	0.459157	0.0268242	0.0134121	−	0.999910i	$-0.495731\pi$
$293$	0.459157	0.0268242	0.0134121	+	0.999910i	$0.495731\pi$
$294$	0	0
$295$	3.55160	0.206782
$296$	1.90847	0.110927
$297$	0	0
$298$	−33.8615	−1.96155
$299$	6.22952	0.360262
$300$	0	0
$301$	−22.3014	−1.28543
$302$	17.9083	1.03051
$303$	0	0
$304$	17.3679	0.996117
$305$	5.48181	0.313887
$306$	0	0
$307$	−11.2133	−0.639978	−0.319989	−	0.947421i	$-0.603679\pi$
$307$	−11.2133	−0.639978	−0.319989	+	0.947421i	$0.603679\pi$
$308$	−24.5498	−1.39886
$309$	0	0
$310$	44.2234	2.51172
$311$	−12.1228	−0.687423	−0.343712	−	0.939075i	$-0.611684\pi$
$311$	−12.1228	−0.687423	−0.343712	+	0.939075i	$0.611684\pi$
$312$	0	0
$313$	12.8050	0.723782	0.361891	−	0.932220i	$-0.382131\pi$
$313$	12.8050	0.723782	0.361891	+	0.932220i	$0.382131\pi$
$314$	−17.4247	−0.983335
$315$	0	0
$316$	19.5722	1.10102
$317$	2.96789	0.166693	0.0833467	−	0.996521i	$-0.473439\pi$
$317$	2.96789	0.166693	0.0833467	+	0.996521i	$0.473439\pi$
$318$	0	0
$319$	1.43250	0.0802049
$320$	−28.0486	−1.56797
$321$	0	0
$322$	19.9242	1.11033
$323$	30.7725	1.71223
$324$	0	0
$325$	1.79665	0.0996600
$326$	−7.60865	−0.421404
$327$	0	0
$328$	11.2865	0.623194
$329$	27.0281	1.49011
$330$	0	0
$331$	35.0721	1.92773	0.963867	−	0.266383i	$-0.0858286\pi$
$331$	35.0721	1.92773	0.963867	+	0.266383i	$0.0858286\pi$
$332$	24.7494	1.35830
$333$	0	0
$334$	29.8480	1.63321
$335$	−35.1979	−1.92307
$336$	0	0
$337$	12.4068	0.675840	0.337920	−	0.941175i	$-0.390277\pi$
$337$	12.4068	0.675840	0.337920	+	0.941175i	$0.390277\pi$
$338$	21.9581	1.19436
$339$	0	0
$340$	−30.1880	−1.63718
$341$	−33.8563	−1.83342
$342$	0	0
$343$	19.5830	1.05738
$344$	−9.55498	−0.515170
$345$	0	0
$346$	12.5275	0.673485
$347$	25.6920	1.37922	0.689610	−	0.724181i	$-0.257782\pi$
$347$	25.6920	1.37922	0.689610	+	0.724181i	$0.257782\pi$
$348$	0	0
$349$	−16.4397	−0.879996	−0.439998	−	0.897999i	$-0.645021\pi$
$349$	−16.4397	−0.879996	−0.439998	+	0.897999i	$0.645021\pi$
$350$	5.74630	0.307153
$351$	0	0
$352$	31.9267	1.70170
$353$	−9.96346	−0.530302	−0.265151	−	0.964207i	$-0.585422\pi$
$353$	−9.96346	−0.530302	−0.265151	+	0.964207i	$0.585422\pi$
$354$	0	0
$355$	10.1801	0.540304
$356$	−2.44319	−0.129489
$357$	0	0
$358$	39.3293	2.07862
$359$	−5.08801	−0.268535	−0.134267	−	0.990945i	$-0.542868\pi$
$359$	−5.08801	−0.268535	−0.134267	+	0.990945i	$0.542868\pi$
$360$	0	0
$361$	20.5089	1.07942
$362$	1.95394	0.102697
$363$	0	0
$364$	−9.95269	−0.521662
$365$	6.88491	0.360373
$366$	0	0
$367$	15.5615	0.812302	0.406151	−	0.913806i	$-0.366871\pi$
$367$	15.5615	0.812302	0.406151	+	0.913806i	$0.366871\pi$
$368$	−10.5862	−0.551842
$369$	0	0
$370$	9.51430	0.494625
$371$	14.1871	0.736559
$372$	0	0
$373$	−17.8554	−0.924519	−0.462259	−	0.886745i	$-0.652961\pi$
$373$	−17.8554	−0.924519	−0.462259	+	0.886745i	$0.652961\pi$
$374$	41.6326	2.15277
$375$	0	0
$376$	11.5801	0.597199
$377$	0.580748	0.0299101
$378$	0	0
$379$	25.1729	1.29304	0.646522	−	0.762896i	$-0.276223\pi$
$379$	25.1729	1.29304	0.646522	+	0.762896i	$0.276223\pi$
$380$	−38.7586	−1.98827
$381$	0	0
$382$	32.0767	1.64119
$383$	21.4662	1.09687	0.548435	−	0.836194i	$-0.315224\pi$
$383$	21.4662	1.09687	0.548435	+	0.836194i	$0.315224\pi$
$384$	0	0
$385$	−24.3059	−1.23874
$386$	48.1909	2.45285
$387$	0	0
$388$	−42.0569	−2.13512
$389$	−14.1828	−0.719096	−0.359548	−	0.933127i	$-0.617069\pi$
$389$	−14.1828	−0.719096	−0.359548	+	0.933127i	$0.617069\pi$
$390$	0	0
$391$	−18.7566	−0.948561
$392$	1.03427	0.0522384
$393$	0	0
$394$	−5.23877	−0.263925
$395$	19.3777	0.975000
$396$	0	0
$397$	−8.60119	−0.431681	−0.215841	−	0.976429i	$-0.569249\pi$
$397$	−8.60119	−0.431681	−0.215841	+	0.976429i	$0.569249\pi$
$398$	11.3321	0.568027
$399$	0	0
$400$	−3.05314	−0.152657
$401$	13.5162	0.674969	0.337485	−	0.941331i	$-0.390424\pi$
$401$	13.5162	0.674969	0.337485	+	0.941331i	$0.390424\pi$
$402$	0	0
$403$	−13.7256	−0.683721
$404$	−9.49410	−0.472349
$405$	0	0
$406$	1.85743	0.0921829
$407$	−7.28391	−0.361050
$408$	0	0
$409$	26.6663	1.31856	0.659281	−	0.751896i	$-0.270861\pi$
$409$	26.6663	1.31856	0.659281	+	0.751896i	$0.270861\pi$
$410$	56.2668	2.77882
$411$	0	0
$412$	19.9904	0.984854
$413$	−3.52557	−0.173482
$414$	0	0
$415$	24.5035	1.20283
$416$	12.9433	0.634599
$417$	0	0
$418$	53.4523	2.61444
$419$	−7.37267	−0.360179	−0.180089	−	0.983650i	$-0.557639\pi$
$419$	−7.37267	−0.360179	−0.180089	+	0.983650i	$0.557639\pi$
$420$	0	0
$421$	36.1603	1.76235	0.881174	−	0.472793i	$-0.156754\pi$
$421$	36.1603	1.76235	0.881174	+	0.472793i	$0.156754\pi$
$422$	21.9189	1.06699
$423$	0	0
$424$	6.07844	0.295195
$425$	−5.40956	−0.262402
$426$	0	0
$427$	−5.44162	−0.263339
$428$	31.6588	1.53028
$429$	0	0
$430$	−47.6345	−2.29714
$431$	13.4750	0.649066	0.324533	−	0.945874i	$-0.394793\pi$
$431$	13.4750	0.649066	0.324533	+	0.945874i	$0.394793\pi$
$432$	0	0
$433$	−8.22943	−0.395481	−0.197741	−	0.980254i	$-0.563360\pi$
$433$	−8.22943	−0.395481	−0.197741	+	0.980254i	$0.563360\pi$
$434$	−43.8992	−2.10723
$435$	0	0
$436$	−46.8593	−2.24416
$437$	−24.0817	−1.15198
$438$	0	0
$439$	−1.65529	−0.0790028	−0.0395014	−	0.999220i	$-0.512577\pi$
$439$	−1.65529	−0.0790028	−0.0395014	+	0.999220i	$0.512577\pi$
$440$	−10.4138	−0.496458
$441$	0	0
$442$	16.8782	0.802812
$443$	15.9770	0.759092	0.379546	−	0.925173i	$-0.376080\pi$
$443$	15.9770	0.759092	0.379546	+	0.925173i	$0.376080\pi$
$444$	0	0
$445$	−2.41892	−0.114668
$446$	21.7758	1.03112
$447$	0	0
$448$	27.8430	1.31546
$449$	−13.7005	−0.646566	−0.323283	−	0.946302i	$-0.604787\pi$
$449$	−13.7005	−0.646566	−0.323283	+	0.946302i	$0.604787\pi$
$450$	0	0
$451$	−43.0765	−2.02839
$452$	19.0000	0.893683
$453$	0	0
$454$	−35.9319	−1.68637
$455$	−9.85379	−0.461953
$456$	0	0
$457$	34.4962	1.61367	0.806833	−	0.590779i	$-0.201180\pi$
$457$	34.4962	1.61367	0.806833	+	0.590779i	$0.201180\pi$
$458$	61.5933	2.87806
$459$	0	0
$460$	23.6243	1.10149
$461$	30.3840	1.41513	0.707563	−	0.706651i	$-0.249795\pi$
$461$	30.3840	1.41513	0.707563	+	0.706651i	$0.249795\pi$
$462$	0	0
$463$	−19.9606	−0.927648	−0.463824	−	0.885927i	$-0.653523\pi$
$463$	−19.9606	−0.927648	−0.463824	+	0.885927i	$0.653523\pi$
$464$	−0.986897	−0.0458156
$465$	0	0
$466$	−41.8477	−1.93856
$467$	−8.39058	−0.388270	−0.194135	−	0.980975i	$-0.562190\pi$
$467$	−8.39058	−0.388270	−0.194135	+	0.980975i	$0.562190\pi$
$468$	0	0
$469$	34.9399	1.61337
$470$	57.7305	2.66291
$471$	0	0
$472$	−1.51052	−0.0695273
$473$	36.4678	1.67679
$474$	0	0
$475$	−6.94536	−0.318675
$476$	29.9667	1.37352
$477$	0	0
$478$	−2.12029	−0.0969797
$479$	−21.2295	−0.969999	−0.485000	−	0.874514i	$-0.661180\pi$
$479$	−21.2295	−0.969999	−0.485000	+	0.874514i	$0.661180\pi$
$480$	0	0
$481$	−2.95295	−0.134643
$482$	−22.2284	−1.01247
$483$	0	0
$484$	12.6927	0.576940
$485$	−41.6390	−1.89073
$486$	0	0
$487$	5.32666	0.241374	0.120687	−	0.992691i	$-0.461490\pi$
$487$	5.32666	0.241374	0.120687	+	0.992691i	$0.461490\pi$
$488$	−2.33145	−0.105540
$489$	0	0
$490$	5.15615	0.232931
$491$	−24.0997	−1.08760	−0.543802	−	0.839213i	$-0.683016\pi$
$491$	−24.0997	−1.08760	−0.543802	+	0.839213i	$0.683016\pi$
$492$	0	0
$493$	−1.74859	−0.0787523
$494$	21.6700	0.974977
$495$	0	0
$496$	23.3247	1.04731
$497$	−10.1055	−0.453293
$498$	0	0
$499$	−34.1034	−1.52668	−0.763339	−	0.645998i	$-0.776441\pi$
$499$	−34.1034	−1.52668	−0.763339	+	0.645998i	$0.776441\pi$
$500$	−24.0177	−1.07410
$501$	0	0
$502$	25.7036	1.14721
$503$	10.6016	0.472704	0.236352	−	0.971667i	$-0.424048\pi$
$503$	10.6016	0.472704	0.236352	+	0.971667i	$0.424048\pi$
$504$	0	0
$505$	−9.39976	−0.418284
$506$	−32.5805	−1.44838
$507$	0	0
$508$	34.9555	1.55090
$509$	−20.2990	−0.899736	−0.449868	−	0.893095i	$-0.648529\pi$
$509$	−20.2990	−0.899736	−0.449868	+	0.893095i	$0.648529\pi$
$510$	0	0
$511$	−6.83444	−0.302338
$512$	−27.8026	−1.22871
$513$	0	0
$514$	60.1893	2.65484
$515$	19.7917	0.872127
$516$	0	0
$517$	−44.1970	−1.94378
$518$	−9.44456	−0.414970
$519$	0	0
$520$	−4.22183	−0.185140
$521$	1.16040	0.0508378	0.0254189	−	0.999677i	$-0.491908\pi$
$521$	1.16040	0.0508378	0.0254189	+	0.999677i	$0.491908\pi$
$522$	0	0
$523$	−19.3796	−0.847412	−0.423706	−	0.905800i	$-0.639271\pi$
$523$	−19.3796	−0.847412	−0.423706	+	0.905800i	$0.639271\pi$
$524$	10.1214	0.442154
$525$	0	0
$526$	−29.8630	−1.30209
$527$	41.3267	1.80022
$528$	0	0
$529$	−8.32163	−0.361810
$530$	30.3029	1.31628
$531$	0	0
$532$	38.4744	1.66808
$533$	−17.4635	−0.756429
$534$	0	0
$535$	31.3442	1.35513
$536$	14.9699	0.646602
$537$	0	0
$538$	46.9059	2.02226
$539$	−3.94742	−0.170027
$540$	0	0
$541$	−4.77487	−0.205288	−0.102644	−	0.994718i	$-0.532730\pi$
$541$	−4.77487	−0.205288	−0.102644	+	0.994718i	$0.532730\pi$
$542$	0.392638	0.0168652
$543$	0	0
$544$	−38.9713	−1.67088
$545$	−46.3937	−1.98729
$546$	0	0
$547$	−12.9501	−0.553704	−0.276852	−	0.960913i	$-0.589291\pi$
$547$	−12.9501	−0.553704	−0.276852	+	0.960913i	$0.589291\pi$
$548$	38.7684	1.65610
$549$	0	0
$550$	−9.39649	−0.400668
$551$	−2.24502	−0.0956410
$552$	0	0
$553$	−19.2357	−0.817985
$554$	33.6847	1.43113
$555$	0	0
$556$	2.72525	0.115576
$557$	−15.2371	−0.645618	−0.322809	−	0.946464i	$-0.604627\pi$
$557$	−15.2371	−0.645618	−0.322809	+	0.946464i	$0.604627\pi$
$558$	0	0
$559$	14.7843	0.625310
$560$	16.7451	0.707609
$561$	0	0
$562$	42.3979	1.78845
$563$	44.4294	1.87247	0.936237	−	0.351368i	$-0.114284\pi$
$563$	44.4294	1.87247	0.936237	+	0.351368i	$0.114284\pi$
$564$	0	0
$565$	18.8112	0.791392
$566$	63.9768	2.68915
$567$	0	0
$568$	−4.32967	−0.181669
$569$	32.3049	1.35429	0.677146	−	0.735848i	$-0.263216\pi$
$569$	32.3049	1.35429	0.677146	+	0.735848i	$0.263216\pi$
$570$	0	0
$571$	−17.6597	−0.739036	−0.369518	−	0.929224i	$-0.620477\pi$
$571$	−17.6597	−0.739036	−0.369518	+	0.929224i	$0.620477\pi$
$572$	16.2749	0.680487
$573$	0	0
$574$	−55.8544	−2.33132
$575$	4.23337	0.176544
$576$	0	0
$577$	−5.92655	−0.246726	−0.123363	−	0.992362i	$-0.539368\pi$
$577$	−5.92655	−0.246726	−0.123363	+	0.992362i	$0.539368\pi$
$578$	−14.7739	−0.614512
$579$	0	0
$580$	2.20238	0.0914489
$581$	−24.3239	−1.00913
$582$	0	0
$583$	−23.1992	−0.960811
$584$	−2.92820	−0.121170
$585$	0	0
$586$	−0.973544	−0.0402167
$587$	−25.8268	−1.06599	−0.532993	−	0.846120i	$-0.678933\pi$
$587$	−25.8268	−1.06599	−0.532993	+	0.846120i	$0.678933\pi$
$588$	0	0
$589$	53.0596	2.18628
$590$	−7.53041	−0.310022
$591$	0	0
$592$	5.01811	0.206243
$593$	9.21060	0.378234	0.189117	−	0.981955i	$-0.439437\pi$
$593$	9.21060	0.378234	0.189117	+	0.981955i	$0.439437\pi$
$594$	0	0
$595$	29.6690	1.21631
$596$	39.8557	1.63255
$597$	0	0
$598$	−13.2084	−0.540131
$599$	43.4301	1.77450	0.887252	−	0.461285i	$-0.152611\pi$
$599$	43.4301	1.77450	0.887252	+	0.461285i	$0.152611\pi$
$600$	0	0
$601$	−10.2416	−0.417765	−0.208883	−	0.977941i	$-0.566983\pi$
$601$	−10.2416	−0.417765	−0.208883	+	0.977941i	$0.566983\pi$
$602$	47.2854	1.92721
$603$	0	0
$604$	−21.0784	−0.857669
$605$	12.5666	0.510903
$606$	0	0
$607$	45.2285	1.83577	0.917884	−	0.396849i	$-0.129896\pi$
$607$	45.2285	1.83577	0.917884	+	0.396849i	$0.129896\pi$
$608$	−50.0355	−2.02921
$609$	0	0
$610$	−11.6230	−0.470602
$611$	−17.9178	−0.724876
$612$	0	0
$613$	20.4029	0.824064	0.412032	−	0.911169i	$-0.364819\pi$
$613$	20.4029	0.824064	0.412032	+	0.911169i	$0.364819\pi$
$614$	23.7755	0.959500
$615$	0	0
$616$	10.3375	0.416508
$617$	−4.40978	−0.177531	−0.0887655	−	0.996053i	$-0.528292\pi$
$617$	−4.40978	−0.177531	−0.0887655	+	0.996053i	$0.528292\pi$
$618$	0	0
$619$	20.8034	0.836160	0.418080	−	0.908410i	$-0.362703\pi$
$619$	20.8034	0.836160	0.418080	+	0.908410i	$0.362703\pi$
$620$	−52.0518	−2.09045
$621$	0	0
$622$	25.7039	1.03063
$623$	2.40119	0.0962015
$624$	0	0
$625$	−29.3039	−1.17215
$626$	−27.1503	−1.08514
$627$	0	0
$628$	20.5093	0.818409
$629$	8.89109	0.354511
$630$	0	0
$631$	33.5580	1.33592	0.667962	−	0.744196i	$-0.267167\pi$
$631$	33.5580	1.33592	0.667962	+	0.744196i	$0.267167\pi$
$632$	−8.24149	−0.327829
$633$	0	0
$634$	−6.29278	−0.249918
$635$	34.6081	1.37338
$636$	0	0
$637$	−1.60031	−0.0634067
$638$	−3.03732	−0.120249
$639$	0	0
$640$	20.1341	0.795869
$641$	−24.5383	−0.969206	−0.484603	−	0.874734i	$-0.661036\pi$
$641$	−24.5383	−0.969206	−0.484603	+	0.874734i	$0.661036\pi$
$642$	0	0
$643$	9.18123	0.362073	0.181036	−	0.983476i	$-0.442055\pi$
$643$	9.18123	0.362073	0.181036	+	0.983476i	$0.442055\pi$
$644$	−23.4511	−0.924104
$645$	0	0
$646$	−65.2465	−2.56709
$647$	−33.3174	−1.30984	−0.654922	−	0.755697i	$-0.727298\pi$
$647$	−33.3174	−1.30984	−0.654922	+	0.755697i	$0.727298\pi$
$648$	0	0
$649$	5.76509	0.226300
$650$	−3.80941	−0.149417
$651$	0	0
$652$	8.95553	0.350726
$653$	8.78496	0.343782	0.171891	−	0.985116i	$-0.445012\pi$
$653$	8.78496	0.343782	0.171891	+	0.985116i	$0.445012\pi$
$654$	0	0
$655$	10.0208	0.391545
$656$	29.6767	1.15868
$657$	0	0
$658$	−57.3073	−2.23407
$659$	27.9953	1.09054	0.545270	−	0.838260i	$-0.316427\pi$
$659$	27.9953	1.09054	0.545270	+	0.838260i	$0.316427\pi$
$660$	0	0
$661$	−31.7412	−1.23459	−0.617294	−	0.786732i	$-0.711771\pi$
$661$	−31.7412	−1.23459	−0.617294	+	0.786732i	$0.711771\pi$
$662$	−74.3628	−2.89019
$663$	0	0
$664$	−10.4215	−0.404433
$665$	38.0921	1.47715
$666$	0	0
$667$	1.36839	0.0529844
$668$	−35.1316	−1.35928
$669$	0	0
$670$	74.6297	2.88320
$671$	8.89828	0.343514
$672$	0	0
$673$	−0.521763	−0.0201125	−0.0100562	−	0.999949i	$-0.503201\pi$
$673$	−0.521763	−0.0201125	−0.0100562	+	0.999949i	$0.503201\pi$
$674$	−26.3059	−1.01327
$675$	0	0
$676$	−25.8451	−0.994042
$677$	2.22663	0.0855765	0.0427882	−	0.999084i	$-0.486376\pi$
$677$	2.22663	0.0855765	0.0427882	+	0.999084i	$0.486376\pi$
$678$	0	0
$679$	41.3338	1.58624
$680$	12.7116	0.487467
$681$	0	0
$682$	71.7852	2.74880
$683$	18.9101	0.723577	0.361788	−	0.932260i	$-0.382166\pi$
$683$	18.9101	0.723577	0.361788	+	0.932260i	$0.382166\pi$
$684$	0	0
$685$	38.3831	1.46654
$686$	−41.5215	−1.58530
$687$	0	0
$688$	−25.1238	−0.957836
$689$	−9.40511	−0.358306
$690$	0	0
$691$	−18.7431	−0.713022	−0.356511	−	0.934291i	$-0.616034\pi$
$691$	−18.7431	−0.713022	−0.356511	+	0.934291i	$0.616034\pi$
$692$	−14.7452	−0.560527
$693$	0	0
$694$	−54.4745	−2.06782
$695$	2.69817	0.102347
$696$	0	0
$697$	52.5812	1.99166
$698$	34.8568	1.31935
$699$	0	0
$700$	−6.76351	−0.255637
$701$	−16.1798	−0.611102	−0.305551	−	0.952176i	$-0.598841\pi$
$701$	−16.1798	−0.611102	−0.305551	+	0.952176i	$0.598841\pi$
$702$	0	0
$703$	11.4153	0.430537
$704$	−45.5296	−1.71596
$705$	0	0
$706$	21.1254	0.795065
$707$	9.33086	0.350923
$708$	0	0
$709$	−8.57907	−0.322194	−0.161097	−	0.986939i	$-0.551503\pi$
$709$	−8.57907	−0.322194	−0.161097	+	0.986939i	$0.551503\pi$
$710$	−21.5848	−0.810062
$711$	0	0
$712$	1.02878	0.0385552
$713$	−32.3411	−1.21118
$714$	0	0
$715$	16.1132	0.602598
$716$	−46.2914	−1.72999
$717$	0	0
$718$	10.7880	0.402606
$719$	32.5343	1.21333	0.606663	−	0.794959i	$-0.292508\pi$
$719$	32.5343	1.21333	0.606663	+	0.794959i	$0.292508\pi$
$720$	0	0
$721$	−19.6466	−0.731679
$722$	−43.4848	−1.61834
$723$	0	0
$724$	−2.29982	−0.0854723
$725$	0.394657	0.0146572
$726$	0	0
$727$	−4.65810	−0.172759	−0.0863797	−	0.996262i	$-0.527530\pi$
$727$	−4.65810	−0.172759	−0.0863797	+	0.996262i	$0.527530\pi$
$728$	4.19088	0.155325
$729$	0	0
$730$	−14.5980	−0.540296
$731$	−44.5144	−1.64642
$732$	0	0
$733$	8.64374	0.319264	0.159632	−	0.987177i	$-0.448969\pi$
$733$	8.64374	0.319264	0.159632	+	0.987177i	$0.448969\pi$
$734$	−32.9948	−1.21786
$735$	0	0
$736$	30.4979	1.12417
$737$	−57.1346	−2.10458
$738$	0	0
$739$	42.9217	1.57890	0.789450	−	0.613815i	$-0.210366\pi$
$739$	42.9217	1.57890	0.789450	+	0.613815i	$0.210366\pi$
$740$	−11.1985	−0.411666
$741$	0	0
$742$	−30.0808	−1.10430
$743$	−39.0878	−1.43399	−0.716997	−	0.697076i	$-0.754484\pi$
$743$	−39.0878	−1.43399	−0.716997	+	0.697076i	$0.754484\pi$
$744$	0	0
$745$	39.4596	1.44569
$746$	37.8586	1.38610
$747$	0	0
$748$	−49.0024	−1.79170
$749$	−31.1144	−1.13690
$750$	0	0
$751$	33.4998	1.22243	0.611213	−	0.791466i	$-0.290682\pi$
$751$	33.4998	1.22243	0.611213	+	0.791466i	$0.290682\pi$
$752$	30.4487	1.11035
$753$	0	0
$754$	−1.23135	−0.0448432
$755$	−20.8690	−0.759500
$756$	0	0
$757$	42.7926	1.55533	0.777663	−	0.628682i	$-0.216405\pi$
$757$	42.7926	1.55533	0.777663	+	0.628682i	$0.216405\pi$
$758$	−53.3737	−1.93862
$759$	0	0
$760$	16.3205	0.592006
$761$	−3.50389	−0.127016	−0.0635079	−	0.997981i	$-0.520229\pi$
$761$	−3.50389	−0.127016	−0.0635079	+	0.997981i	$0.520229\pi$
$762$	0	0
$763$	46.0536	1.66725
$764$	−37.7549	−1.36592
$765$	0	0
$766$	−45.5144	−1.64450
$767$	2.33721	0.0843919
$768$	0	0
$769$	−33.9827	−1.22545	−0.612724	−	0.790297i	$-0.709926\pi$
$769$	−33.9827	−1.22545	−0.612724	+	0.790297i	$0.709926\pi$
$770$	51.5355	1.85721
$771$	0	0
$772$	−56.7216	−2.04146
$773$	−16.2523	−0.584553	−0.292277	−	0.956334i	$-0.594413\pi$
$773$	−16.2523	−0.584553	−0.292277	+	0.956334i	$0.594413\pi$
$774$	0	0
$775$	−9.32746	−0.335052
$776$	17.7094	0.635729
$777$	0	0
$778$	30.0716	1.07812
$779$	67.5093	2.41877
$780$	0	0
$781$	16.5247	0.591302
$782$	39.7693	1.42215
$783$	0	0
$784$	2.71950	0.0971250
$785$	20.3055	0.724733
$786$	0	0
$787$	−10.9298	−0.389606	−0.194803	−	0.980842i	$-0.562407\pi$
$787$	−10.9298	−0.389606	−0.194803	+	0.980842i	$0.562407\pi$
$788$	6.16613	0.219659
$789$	0	0
$790$	−41.0864	−1.46179
$791$	−18.6733	−0.663945
$792$	0	0
$793$	3.60743	0.128104
$794$	18.2370	0.647207
$795$	0	0
$796$	−13.3381	−0.472757
$797$	17.4122	0.616772	0.308386	−	0.951261i	$-0.400211\pi$
$797$	17.4122	0.616772	0.308386	+	0.951261i	$0.400211\pi$
$798$	0	0
$799$	53.9490	1.90858
$800$	8.79585	0.310980
$801$	0	0
$802$	−28.6583	−1.01196
$803$	11.1758	0.394387
$804$	0	0
$805$	−23.2181	−0.818330
$806$	29.1022	1.02508
$807$	0	0
$808$	3.99778	0.140642
$809$	21.8841	0.769403	0.384702	−	0.923041i	$-0.374304\pi$
$809$	21.8841	0.769403	0.384702	+	0.923041i	$0.374304\pi$
$810$	0	0
$811$	41.8415	1.46925	0.734627	−	0.678471i	$-0.237357\pi$
$811$	41.8415	1.46925	0.734627	+	0.678471i	$0.237357\pi$
$812$	−2.18624	−0.0767218
$813$	0	0
$814$	15.4440	0.541311
$815$	8.86655	0.310582
$816$	0	0
$817$	−57.1522	−1.99950
$818$	−56.5402	−1.97688
$819$	0	0
$820$	−66.2272	−2.31275
$821$	−36.7094	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$821$	−36.7094	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$822$	0	0
$823$	−31.0617	−1.08274	−0.541371	−	0.840784i	$-0.682095\pi$
$823$	−31.0617	−1.08274	−0.541371	+	0.840784i	$0.682095\pi$
$824$	−8.41755	−0.293239
$825$	0	0
$826$	7.47521	0.260096
$827$	−55.1422	−1.91748	−0.958741	−	0.284281i	$-0.908245\pi$
$827$	−55.1422	−1.91748	−0.958741	+	0.284281i	$0.908245\pi$
$828$	0	0
$829$	11.6426	0.404366	0.202183	−	0.979348i	$-0.435196\pi$
$829$	11.6426	0.404366	0.202183	+	0.979348i	$0.435196\pi$
$830$	−51.9545	−1.80337
$831$	0	0
$832$	−18.4580	−0.639917
$833$	4.81841	0.166948
$834$	0	0
$835$	−34.7825	−1.20370
$836$	−62.9144	−2.17594
$837$	0	0
$838$	15.6322	0.540005
$839$	−9.99713	−0.345139	−0.172570	−	0.984997i	$-0.555207\pi$
$839$	−9.99713	−0.345139	−0.172570	+	0.984997i	$0.555207\pi$
$840$	0	0
$841$	−28.8724	−0.995601
$842$	−76.6703	−2.64223
$843$	0	0
$844$	−25.7989	−0.888036
$845$	−25.5883	−0.880264
$846$	0	0
$847$	−12.4744	−0.428627
$848$	15.9826	0.548846
$849$	0	0
$850$	11.4698	0.393412
$851$	−6.95792	−0.238514
$852$	0	0
$853$	41.0682	1.40615	0.703073	−	0.711117i	$-0.251811\pi$
$853$	41.0682	1.40615	0.703073	+	0.711117i	$0.251811\pi$
$854$	11.5378	0.394816
$855$	0	0
$856$	−13.3309	−0.455641
$857$	22.0503	0.753224	0.376612	−	0.926371i	$-0.377089\pi$
$857$	22.0503	0.753224	0.376612	+	0.926371i	$0.377089\pi$
$858$	0	0
$859$	5.68153	0.193851	0.0969256	−	0.995292i	$-0.469099\pi$
$859$	5.68153	0.193851	0.0969256	+	0.995292i	$0.469099\pi$
$860$	56.0668	1.91186
$861$	0	0
$862$	−28.5708	−0.973125
$863$	−27.0640	−0.921269	−0.460635	−	0.887590i	$-0.652378\pi$
$863$	−27.0640	−0.921269	−0.460635	+	0.887590i	$0.652378\pi$
$864$	0	0
$865$	−14.5986	−0.496369
$866$	17.4488	0.592933
$867$	0	0
$868$	51.6703	1.75380
$869$	31.4547	1.06703
$870$	0	0
$871$	−23.1628	−0.784841
$872$	19.7316	0.668195
$873$	0	0
$874$	51.0601	1.72713
$875$	23.6048	0.797986
$876$	0	0
$877$	−31.5393	−1.06501	−0.532503	−	0.846428i	$-0.678749\pi$
$877$	−31.5393	−1.06501	−0.532503	+	0.846428i	$0.678749\pi$
$878$	3.50970	0.118447
$879$	0	0
$880$	−27.3820	−0.923047
$881$	9.23845	0.311251	0.155626	−	0.987816i	$-0.450261\pi$
$881$	9.23845	0.311251	0.155626	+	0.987816i	$0.450261\pi$
$882$	0	0
$883$	−22.5049	−0.757350	−0.378675	−	0.925530i	$-0.623620\pi$
$883$	−22.5049	−0.757350	−0.378675	+	0.925530i	$0.623620\pi$
$884$	−19.8659	−0.668163
$885$	0	0
$886$	−33.8759	−1.13808
$887$	30.2960	1.01724	0.508621	−	0.860991i	$-0.330156\pi$
$887$	30.2960	1.01724	0.508621	+	0.860991i	$0.330156\pi$
$888$	0	0
$889$	−34.3545	−1.15221
$890$	5.12880	0.171918
$891$	0	0
$892$	−25.6306	−0.858176
$893$	69.2654	2.31788
$894$	0	0
$895$	−45.8314	−1.53198
$896$	−19.9865	−0.667701
$897$	0	0
$898$	29.0490	0.969377
$899$	−3.01501	−0.100556
$900$	0	0
$901$	28.3180	0.943410
$902$	91.3345	3.04111
$903$	0	0
$904$	−8.00052	−0.266093
$905$	−2.27697	−0.0756891
$906$	0	0
$907$	−34.6786	−1.15148	−0.575742	−	0.817632i	$-0.695287\pi$
$907$	−34.6786	−1.15148	−0.575742	+	0.817632i	$0.695287\pi$
$908$	42.2925	1.40353
$909$	0	0
$910$	20.8929	0.692592
$911$	29.6688	0.982970	0.491485	−	0.870886i	$-0.336454\pi$
$911$	29.6688	0.982970	0.491485	+	0.870886i	$0.336454\pi$
$912$	0	0
$913$	39.7750	1.31636
$914$	−73.1420	−2.41932
$915$	0	0
$916$	−72.4965	−2.39535
$917$	−9.94733	−0.328490
$918$	0	0
$919$	3.41218	0.112558	0.0562788	−	0.998415i	$-0.482076\pi$
$919$	3.41218	0.112558	0.0562788	+	0.998415i	$0.482076\pi$
$920$	−9.94774	−0.327967
$921$	0	0
$922$	−64.4229	−2.12165
$923$	6.69925	0.220509
$924$	0	0
$925$	−2.00672	−0.0659807
$926$	42.3222	1.39079
$927$	0	0
$928$	2.84317	0.0933317
$929$	5.27469	0.173057	0.0865285	−	0.996249i	$-0.472423\pi$
$929$	5.27469	0.173057	0.0865285	+	0.996249i	$0.472423\pi$
$930$	0	0
$931$	6.18638	0.202750
$932$	49.2555	1.61342
$933$	0	0
$934$	17.7905	0.582121
$935$	−48.5154	−1.58662
$936$	0	0
$937$	−11.6027	−0.379043	−0.189521	−	0.981877i	$-0.560694\pi$
$937$	−11.6027	−0.379043	−0.189521	+	0.981877i	$0.560694\pi$
$938$	−74.0826	−2.41888
$939$	0	0
$940$	−67.9499	−2.21628
$941$	28.2271	0.920178	0.460089	−	0.887873i	$-0.347818\pi$
$941$	28.2271	0.920178	0.460089	+	0.887873i	$0.347818\pi$
$942$	0	0
$943$	−41.1486	−1.33998
$944$	−3.97175	−0.129270
$945$	0	0
$946$	−77.3222	−2.51396
$947$	41.9537	1.36331	0.681657	−	0.731672i	$-0.261260\pi$
$947$	41.9537	1.36331	0.681657	+	0.731672i	$0.261260\pi$
$948$	0	0
$949$	4.53077	0.147075
$950$	14.7262	0.477780
$951$	0	0
$952$	−12.6184	−0.408965
$953$	51.3228	1.66251	0.831255	−	0.555892i	$-0.187623\pi$
$953$	51.3228	1.66251	0.831255	+	0.555892i	$0.187623\pi$
$954$	0	0
$955$	−37.3798	−1.20958
$956$	2.49562	0.0807141
$957$	0	0
$958$	45.0126	1.45429
$959$	−38.1018	−1.23037
$960$	0	0
$961$	40.2578	1.29864
$962$	6.26111	0.201866
$963$	0	0
$964$	26.1632	0.842660
$965$	−56.1580	−1.80779
$966$	0	0
$967$	−15.0557	−0.484158	−0.242079	−	0.970257i	$-0.577829\pi$
$967$	−15.0557	−0.484158	−0.242079	+	0.970257i	$0.577829\pi$
$968$	−5.34464	−0.171783
$969$	0	0
$970$	88.2867	2.83471
$971$	−18.2675	−0.586232	−0.293116	−	0.956077i	$-0.594692\pi$
$971$	−18.2675	−0.586232	−0.293116	+	0.956077i	$0.594692\pi$
$972$	0	0
$973$	−2.67839	−0.0858651
$974$	−11.2940	−0.361885
$975$	0	0
$976$	−6.13030	−0.196226
$977$	−11.8112	−0.377873	−0.188936	−	0.981989i	$-0.560504\pi$
$977$	−11.8112	−0.377873	−0.188936	+	0.981989i	$0.560504\pi$
$978$	0	0
$979$	−3.92648	−0.125491
$980$	−6.06889	−0.193864
$981$	0	0
$982$	51.0983	1.63061
$983$	30.4350	0.970725	0.485363	−	0.874313i	$-0.338688\pi$
$983$	30.4350	0.970725	0.485363	+	0.874313i	$0.338688\pi$
$984$	0	0
$985$	6.10486	0.194517
$986$	3.70750	0.118071
$987$	0	0
$988$	−25.5060	−0.811453
$989$	34.8357	1.10771
$990$	0	0
$991$	54.3952	1.72792	0.863960	−	0.503561i	$-0.167977\pi$
$991$	54.3952	1.72792	0.863960	+	0.503561i	$0.167977\pi$
$992$	−67.1965	−2.13349
$993$	0	0
$994$	21.4265	0.679608
$995$	−13.2056	−0.418645
$996$	0	0
$997$	14.6453	0.463821	0.231910	−	0.972737i	$-0.425502\pi$
$997$	14.6453	0.463821	0.231910	+	0.972737i	$0.425502\pi$
$998$	72.3090	2.28890
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.e.1.1		6
3.2	odd	2		717.2.a.d.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.d.1.6	✓	6	3.2	odd	2
2151.2.a.e.1.1		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)

\(f(q)\)	\(=\)	\(q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} +O(q^{10})\)	\(q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} -5.23885 q^{10} +4.01073 q^{11} +1.62598 q^{13} +5.20045 q^{14} -2.76312 q^{16} -4.89570 q^{17} -6.28561 q^{19} +6.16623 q^{20} -8.50391 q^{22} +3.83124 q^{23} +1.10496 q^{25} -3.44755 q^{26} -6.12103 q^{28} +0.357168 q^{29} -8.44143 q^{31} +7.96032 q^{32} +10.3803 q^{34} -6.06021 q^{35} -1.81610 q^{37} +13.3273 q^{38} -2.59648 q^{40} -10.7403 q^{41} +9.09255 q^{43} +10.0093 q^{44} -8.12332 q^{46} -11.0197 q^{47} -0.984213 q^{49} -2.34284 q^{50} +4.05783 q^{52} -5.78427 q^{53} +9.90981 q^{55} +2.57745 q^{56} -0.757299 q^{58} +1.43742 q^{59} +2.21862 q^{61} +17.8983 q^{62} -11.3519 q^{64} +4.01751 q^{65} -14.2454 q^{67} -12.2178 q^{68} +12.8494 q^{70} +4.12013 q^{71} +2.78648 q^{73} +3.85066 q^{74} -15.6865 q^{76} -9.83717 q^{77} +7.84263 q^{79} -6.82718 q^{80} +22.7725 q^{82} +9.91715 q^{83} -12.0964 q^{85} -19.2788 q^{86} -4.21471 q^{88} -0.978993 q^{89} -3.98806 q^{91} +9.56131 q^{92} +23.3649 q^{94} -15.5306 q^{95} -16.8523 q^{97} +2.08682 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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Groups

Database

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