LMFDB - Embedded newform 2151.2.a.e.1.5

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			1.5
Root			$-0.360520$ 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)

$f(q)$	$=$	$q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} +O(q^{10})$	\(q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} -7.35809 q^{10} +5.35035 q^{11} -5.86183 q^{13} +0.434095 q^{14} -2.28669 q^{16} -4.05336 q^{17} -2.42759 q^{19} -9.03561 q^{20} +11.5340 q^{22} +4.58506 q^{23} +6.65030 q^{25} -12.6366 q^{26} +0.533062 q^{28} -3.44243 q^{29} -7.68267 q^{31} -7.71995 q^{32} -8.73799 q^{34} -0.687317 q^{35} -0.781965 q^{37} -5.23326 q^{38} -4.76224 q^{40} -5.87100 q^{41} -8.05607 q^{43} +14.1635 q^{44} +9.88419 q^{46} +3.12717 q^{47} -6.95945 q^{49} +14.3363 q^{50} -15.5175 q^{52} +11.6536 q^{53} -18.2621 q^{55} +0.280952 q^{56} -7.42099 q^{58} -9.93358 q^{59} +5.67948 q^{61} -16.5618 q^{62} -12.0688 q^{64} +20.0079 q^{65} -6.84647 q^{67} -10.7301 q^{68} -1.48168 q^{70} -12.8677 q^{71} -7.37793 q^{73} -1.68571 q^{74} -6.42636 q^{76} +1.07739 q^{77} +14.6391 q^{79} +7.80506 q^{80} -12.6564 q^{82} +15.8846 q^{83} +13.8351 q^{85} -17.3668 q^{86} +7.46492 q^{88} +5.12607 q^{89} -1.18038 q^{91} +12.1376 q^{92} +6.74135 q^{94} +8.28599 q^{95} -11.8563 q^{97} -15.0028 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.15574	1.52434	0.762169	−	0.647378i	$-0.224135\pi$
$2$	2.15574	1.52434	0.762169	+	0.647378i	$0.224135\pi$
$3$	0	0
$3$	0	0
$4$	2.64721	1.32361
$5$	−3.41325	−1.52645	−0.763227	−	0.646131i	$-0.776386\pi$
$5$	−3.41325	−1.52645	−0.763227	+	0.646131i	$0.776386\pi$
$6$	0	0
$7$	0.201367	0.0761096	0.0380548	−	0.999276i	$-0.487884\pi$
$7$	0.201367	0.0761096	0.0380548	+	0.999276i	$0.487884\pi$
$8$	1.39522	0.493285
$9$	0	0
$10$	−7.35809	−2.32683
$11$	5.35035	1.61319	0.806596	−	0.591103i	$-0.201307\pi$
$11$	5.35035	1.61319	0.806596	+	0.591103i	$0.201307\pi$
$12$	0	0
$13$	−5.86183	−1.62578	−0.812890	−	0.582417i	$-0.802107\pi$
$13$	−5.86183	−1.62578	−0.812890	+	0.582417i	$0.802107\pi$
$14$	0.434095	0.116017
$15$	0	0
$16$	−2.28669	−0.571673
$17$	−4.05336	−0.983084	−0.491542	−	0.870854i	$-0.663567\pi$
$17$	−4.05336	−0.983084	−0.491542	+	0.870854i	$0.663567\pi$
$18$	0	0
$19$	−2.42759	−0.556928	−0.278464	−	0.960447i	$-0.589825\pi$
$19$	−2.42759	−0.556928	−0.278464	+	0.960447i	$0.589825\pi$
$20$	−9.03561	−2.02042
$21$	0	0
$22$	11.5340	2.45905
$23$	4.58506	0.956050	0.478025	−	0.878346i	$-0.341353\pi$
$23$	4.58506	0.956050	0.478025	+	0.878346i	$0.341353\pi$
$24$	0	0
$25$	6.65030	1.33006
$26$	−12.6366	−2.47824
$27$	0	0
$28$	0.533062	0.100739
$29$	−3.44243	−0.639244	−0.319622	−	0.947545i	$-0.603556\pi$
$29$	−3.44243	−0.639244	−0.319622	+	0.947545i	$0.603556\pi$
$30$	0	0
$31$	−7.68267	−1.37985	−0.689924	−	0.723881i	$-0.742356\pi$
$31$	−7.68267	−1.37985	−0.689924	+	0.723881i	$0.742356\pi$
$32$	−7.71995	−1.36471
$33$	0	0
$34$	−8.73799	−1.49855
$35$	−0.687317	−0.116178
$36$	0	0
$37$	−0.781965	−0.128554	−0.0642771	−	0.997932i	$-0.520474\pi$
$37$	−0.781965	−0.128554	−0.0642771	+	0.997932i	$0.520474\pi$
$38$	−5.23326	−0.848947
$39$	0	0
$40$	−4.76224	−0.752977
$41$	−5.87100	−0.916897	−0.458448	−	0.888721i	$-0.651595\pi$
$41$	−5.87100	−0.916897	−0.458448	+	0.888721i	$0.651595\pi$
$42$	0	0
$43$	−8.05607	−1.22854	−0.614270	−	0.789096i	$-0.710549\pi$
$43$	−8.05607	−1.22854	−0.614270	+	0.789096i	$0.710549\pi$
$44$	14.1635	2.13523
$45$	0	0
$46$	9.88419	1.45734
$47$	3.12717	0.456144	0.228072	−	0.973644i	$-0.426758\pi$
$47$	3.12717	0.456144	0.228072	+	0.973644i	$0.426758\pi$
$48$	0	0
$49$	−6.95945	−0.994207
$50$	14.3363	2.02746
$51$	0	0
$52$	−15.5175	−2.15189
$53$	11.6536	1.60075	0.800375	−	0.599499i	$-0.204634\pi$
$53$	11.6536	1.60075	0.800375	+	0.599499i	$0.204634\pi$
$54$	0	0
$55$	−18.2621	−2.46246
$56$	0.280952	0.0375437
$57$	0	0
$58$	−7.42099	−0.974423
$59$	−9.93358	−1.29324	−0.646621	−	0.762811i	$-0.723818\pi$
$59$	−9.93358	−1.29324	−0.646621	+	0.762811i	$0.723818\pi$
$60$	0	0
$61$	5.67948	0.727183	0.363591	−	0.931559i	$-0.381550\pi$
$61$	5.67948	0.727183	0.363591	+	0.931559i	$0.381550\pi$
$62$	−16.5618	−2.10336
$63$	0	0
$64$	−12.0688	−1.50860
$65$	20.0079	2.48168
$66$	0	0
$67$	−6.84647	−0.836429	−0.418215	−	0.908348i	$-0.637344\pi$
$67$	−6.84647	−0.836429	−0.418215	+	0.908348i	$0.637344\pi$
$68$	−10.7301	−1.30122
$69$	0	0
$70$	−1.48168	−0.177094
$71$	−12.8677	−1.52711	−0.763555	−	0.645743i	$-0.776548\pi$
$71$	−12.8677	−1.52711	−0.763555	+	0.645743i	$0.776548\pi$
$72$	0	0
$73$	−7.37793	−0.863521	−0.431761	−	0.901988i	$-0.642107\pi$
$73$	−7.37793	−0.863521	−0.431761	+	0.901988i	$0.642107\pi$
$74$	−1.68571	−0.195960
$75$	0	0
$76$	−6.42636	−0.737154
$77$	1.07739	0.122779
$78$	0	0
$79$	14.6391	1.64702	0.823512	−	0.567299i	$-0.192012\pi$
$79$	14.6391	1.64702	0.823512	+	0.567299i	$0.192012\pi$
$80$	7.80506	0.872632
$81$	0	0
$82$	−12.6564	−1.39766
$83$	15.8846	1.74357	0.871783	−	0.489893i	$-0.162964\pi$
$83$	15.8846	1.74357	0.871783	+	0.489893i	$0.162964\pi$
$84$	0	0
$85$	13.8351	1.50063
$86$	−17.3668	−1.87271
$87$	0	0
$88$	7.46492	0.795763
$89$	5.12607	0.543362	0.271681	−	0.962387i	$-0.412420\pi$
$89$	5.12607	0.543362	0.271681	+	0.962387i	$0.412420\pi$
$90$	0	0
$91$	−1.18038	−0.123738
$92$	12.1376	1.26543
$93$	0	0
$94$	6.74135	0.695318
$95$	8.28599	0.850125
$96$	0	0
$97$	−11.8563	−1.20382	−0.601911	−	0.798563i	$-0.705594\pi$
$97$	−11.8563	−1.20382	−0.601911	+	0.798563i	$0.705594\pi$
$98$	−15.0028	−1.51551
$99$	0	0
$100$	17.6048	1.76048
$101$	15.8321	1.57535	0.787677	−	0.616089i	$-0.211284\pi$
$101$	15.8321	1.57535	0.787677	+	0.616089i	$0.211284\pi$
$102$	0	0
$103$	16.7153	1.64701	0.823505	−	0.567310i	$-0.192016\pi$
$103$	16.7153	1.64701	0.823505	+	0.567310i	$0.192016\pi$
$104$	−8.17855	−0.801973
$105$	0	0
$106$	25.1222	2.44008
$107$	3.25084	0.314271	0.157135	−	0.987577i	$-0.449774\pi$
$107$	3.25084	0.314271	0.157135	+	0.987577i	$0.449774\pi$
$108$	0	0
$109$	−2.91908	−0.279597	−0.139799	−	0.990180i	$-0.544646\pi$
$109$	−2.91908	−0.279597	−0.139799	+	0.990180i	$0.544646\pi$
$110$	−39.3684	−3.75363
$111$	0	0
$112$	−0.460465	−0.0435098
$113$	−2.24561	−0.211249	−0.105625	−	0.994406i	$-0.533684\pi$
$113$	−2.24561	−0.211249	−0.105625	+	0.994406i	$0.533684\pi$
$114$	0	0
$115$	−15.6500	−1.45937
$116$	−9.11285	−0.846107
$117$	0	0
$118$	−21.4142	−1.97134
$119$	−0.816214	−0.0748222
$120$	0	0
$121$	17.6263	1.60239
$122$	12.2435	1.10847
$123$	0	0
$124$	−20.3377	−1.82638
$125$	−5.63289	−0.503821
$126$	0	0
$127$	−8.75323	−0.776724	−0.388362	−	0.921507i	$-0.626959\pi$
$127$	−8.75323	−0.776724	−0.388362	+	0.921507i	$0.626959\pi$
$128$	−10.5773	−0.934913
$129$	0	0
$130$	43.1319	3.78292
$131$	−8.69454	−0.759646	−0.379823	−	0.925059i	$-0.624015\pi$
$131$	−8.69454	−0.759646	−0.379823	+	0.925059i	$0.624015\pi$
$132$	0	0
$133$	−0.488838	−0.0423876
$134$	−14.7592	−1.27500
$135$	0	0
$136$	−5.65533	−0.484941
$137$	8.49089	0.725426	0.362713	−	0.931901i	$-0.381851\pi$
$137$	8.49089	0.725426	0.362713	+	0.931901i	$0.381851\pi$
$138$	0	0
$139$	−1.97577	−0.167582	−0.0837912	−	0.996483i	$-0.526703\pi$
$139$	−1.97577	−0.167582	−0.0837912	+	0.996483i	$0.526703\pi$
$140$	−1.81947	−0.153774
$141$	0	0
$142$	−27.7393	−2.32783
$143$	−31.3629	−2.62270
$144$	0	0
$145$	11.7499	0.975775
$146$	−15.9049	−1.31630
$147$	0	0
$148$	−2.07003	−0.170155
$149$	−3.31898	−0.271901	−0.135951	−	0.990716i	$-0.543409\pi$
$149$	−3.31898	−0.271901	−0.135951	+	0.990716i	$0.543409\pi$
$150$	0	0
$151$	−13.3688	−1.08794	−0.543968	−	0.839106i	$-0.683079\pi$
$151$	−13.3688	−1.08794	−0.543968	+	0.839106i	$0.683079\pi$
$152$	−3.38703	−0.274724
$153$	0	0
$154$	2.32256	0.187157
$155$	26.2229	2.10628
$156$	0	0
$157$	3.64757	0.291108	0.145554	−	0.989350i	$-0.453504\pi$
$157$	3.64757	0.291108	0.145554	+	0.989350i	$0.453504\pi$
$158$	31.5580	2.51062
$159$	0	0
$160$	26.3502	2.08316
$161$	0.923280	0.0727646
$162$	0	0
$163$	−14.8589	−1.16384	−0.581920	−	0.813246i	$-0.697698\pi$
$163$	−14.8589	−1.16384	−0.581920	+	0.813246i	$0.697698\pi$
$164$	−15.5418	−1.21361
$165$	0	0
$166$	34.2431	2.65778
$167$	−12.1352	−0.939049	−0.469524	−	0.882919i	$-0.655575\pi$
$167$	−12.1352	−0.939049	−0.469524	+	0.882919i	$0.655575\pi$
$168$	0	0
$169$	21.3611	1.64316
$170$	29.8250	2.28747
$171$	0	0
$172$	−21.3261	−1.62610
$173$	13.0426	0.991610	0.495805	−	0.868434i	$-0.334873\pi$
$173$	13.0426	0.991610	0.495805	+	0.868434i	$0.334873\pi$
$174$	0	0
$175$	1.33915	0.101230
$176$	−12.2346	−0.922219
$177$	0	0
$178$	11.0505	0.828267
$179$	−5.36036	−0.400652	−0.200326	−	0.979729i	$-0.564200\pi$
$179$	−5.36036	−0.400652	−0.200326	+	0.979729i	$0.564200\pi$
$180$	0	0
$181$	5.12883	0.381223	0.190611	−	0.981666i	$-0.438953\pi$
$181$	5.12883	0.381223	0.190611	+	0.981666i	$0.438953\pi$
$182$	−2.54459	−0.188618
$183$	0	0
$184$	6.39716	0.471605
$185$	2.66905	0.196232
$186$	0	0
$187$	−21.6869	−1.58590
$188$	8.27827	0.603755
$189$	0	0
$190$	17.8624	1.29588
$191$	12.0807	0.874130	0.437065	−	0.899430i	$-0.356018\pi$
$191$	12.0807	0.874130	0.437065	+	0.899430i	$0.356018\pi$
$192$	0	0
$193$	25.6877	1.84904	0.924519	−	0.381135i	$-0.124467\pi$
$193$	25.6877	1.84904	0.924519	+	0.381135i	$0.124467\pi$
$194$	−25.5590	−1.83503
$195$	0	0
$196$	−18.4231	−1.31594
$197$	8.44159	0.601439	0.300719	−	0.953713i	$-0.402773\pi$
$197$	8.44159	0.601439	0.300719	+	0.953713i	$0.402773\pi$
$198$	0	0
$199$	4.76581	0.337839	0.168920	−	0.985630i	$-0.445972\pi$
$199$	4.76581	0.337839	0.168920	+	0.985630i	$0.445972\pi$
$200$	9.27864	0.656099
$201$	0	0
$202$	34.1299	2.40137
$203$	−0.693193	−0.0486526
$204$	0	0
$205$	20.0392	1.39960
$206$	36.0339	2.51060
$207$	0	0
$208$	13.4042	0.929415
$209$	−12.9885	−0.898432
$210$	0	0
$211$	7.98569	0.549758	0.274879	−	0.961479i	$-0.411362\pi$
$211$	7.98569	0.549758	0.274879	+	0.961479i	$0.411362\pi$
$212$	30.8497	2.11876
$213$	0	0
$214$	7.00797	0.479055
$215$	27.4974	1.87531
$216$	0	0
$217$	−1.54704	−0.105020
$218$	−6.29278	−0.426201
$219$	0	0
$220$	−48.3437	−3.25933
$221$	23.7601	1.59828
$222$	0	0
$223$	−24.9471	−1.67058	−0.835291	−	0.549808i	$-0.814701\pi$
$223$	−24.9471	−1.67058	−0.835291	+	0.549808i	$0.814701\pi$
$224$	−1.55455	−0.103867
$225$	0	0
$226$	−4.84095	−0.322015
$227$	−25.4484	−1.68907	−0.844536	−	0.535499i	$-0.820123\pi$
$227$	−25.4484	−1.68907	−0.844536	+	0.535499i	$0.820123\pi$
$228$	0	0
$229$	−8.84963	−0.584800	−0.292400	−	0.956296i	$-0.594454\pi$
$229$	−8.84963	−0.584800	−0.292400	+	0.956296i	$0.594454\pi$
$230$	−33.7372	−2.22457
$231$	0	0
$232$	−4.80295	−0.315329
$233$	−23.9242	−1.56733	−0.783665	−	0.621184i	$-0.786652\pi$
$233$	−23.9242	−1.56733	−0.783665	+	0.621184i	$0.786652\pi$
$234$	0	0
$235$	−10.6738	−0.696283
$236$	−26.2963	−1.71174
$237$	0	0
$238$	−1.75954	−0.114054
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−5.57115	−0.358870	−0.179435	−	0.983770i	$-0.557427\pi$
$241$	−5.57115	−0.358870	−0.179435	+	0.983770i	$0.557427\pi$
$242$	37.9977	2.44258
$243$	0	0
$244$	15.0348	0.962503
$245$	23.7544	1.51761
$246$	0	0
$247$	14.2302	0.905443
$248$	−10.7190	−0.680659
$249$	0	0
$250$	−12.1430	−0.767994
$251$	24.0752	1.51961	0.759806	−	0.650150i	$-0.225294\pi$
$251$	24.0752	1.51961	0.759806	+	0.650150i	$0.225294\pi$
$252$	0	0
$253$	24.5317	1.54229
$254$	−18.8697	−1.18399
$255$	0	0
$256$	1.33568	0.0834801
$257$	−1.65551	−0.103268	−0.0516339	−	0.998666i	$-0.516443\pi$
$257$	−1.65551	−0.103268	−0.0516339	+	0.998666i	$0.516443\pi$
$258$	0	0
$259$	−0.157462	−0.00978422
$260$	52.9652	3.28476
$261$	0	0
$262$	−18.7432	−1.15796
$263$	13.7969	0.850751	0.425375	−	0.905017i	$-0.360142\pi$
$263$	13.7969	0.850751	0.425375	+	0.905017i	$0.360142\pi$
$264$	0	0
$265$	−39.7768	−2.44347
$266$	−1.05381	−0.0646130
$267$	0	0
$268$	−18.1241	−1.10710
$269$	2.37746	0.144956	0.0724782	−	0.997370i	$-0.476909\pi$
$269$	2.37746	0.144956	0.0724782	+	0.997370i	$0.476909\pi$
$270$	0	0
$271$	3.47333	0.210990	0.105495	−	0.994420i	$-0.466357\pi$
$271$	3.47333	0.210990	0.105495	+	0.994420i	$0.466357\pi$
$272$	9.26879	0.562003
$273$	0	0
$274$	18.3042	1.10579
$275$	35.5815	2.14564
$276$	0	0
$277$	−2.25803	−0.135672	−0.0678359	−	0.997696i	$-0.521609\pi$
$277$	−2.25803	−0.135672	−0.0678359	+	0.997696i	$0.521609\pi$
$278$	−4.25924	−0.255452
$279$	0	0
$280$	−0.958959	−0.0573088
$281$	0.502713	0.0299894	0.0149947	−	0.999888i	$-0.495227\pi$
$281$	0.502713	0.0299894	0.0149947	+	0.999888i	$0.495227\pi$
$282$	0	0
$283$	21.6234	1.28538	0.642690	−	0.766127i	$-0.277818\pi$
$283$	21.6234	1.28538	0.642690	+	0.766127i	$0.277818\pi$
$284$	−34.0634	−2.02129
$285$	0	0
$286$	−67.6102	−3.99787
$287$	−1.18223	−0.0697847
$288$	0	0
$289$	−0.570276	−0.0335456
$290$	25.3297	1.48741
$291$	0	0
$292$	−19.5309	−1.14296
$293$	−18.7307	−1.09426	−0.547129	−	0.837048i	$-0.684279\pi$
$293$	−18.7307	−1.09426	−0.547129	+	0.837048i	$0.684279\pi$
$294$	0	0
$295$	33.9058	1.97407
$296$	−1.09101	−0.0634139
$297$	0	0
$298$	−7.15485	−0.414469
$299$	−26.8768	−1.55433
$300$	0	0
$301$	−1.62223	−0.0935037
$302$	−28.8196	−1.65838
$303$	0	0
$304$	5.55116	0.318381
$305$	−19.3855	−1.11001
$306$	0	0
$307$	4.73274	0.270112	0.135056	−	0.990838i	$-0.456879\pi$
$307$	4.73274	0.270112	0.135056	+	0.990838i	$0.456879\pi$
$308$	2.85207	0.162512
$309$	0	0
$310$	56.5298	3.21067
$311$	21.6404	1.22711	0.613556	−	0.789651i	$-0.289738\pi$
$311$	21.6404	1.22711	0.613556	+	0.789651i	$0.289738\pi$
$312$	0	0
$313$	24.4364	1.38123	0.690615	−	0.723223i	$-0.257340\pi$
$313$	24.4364	1.38123	0.690615	+	0.723223i	$0.257340\pi$
$314$	7.86321	0.443747
$315$	0	0
$316$	38.7527	2.18001
$317$	−2.93779	−0.165003	−0.0825013	−	0.996591i	$-0.526291\pi$
$317$	−2.93779	−0.165003	−0.0825013	+	0.996591i	$0.526291\pi$
$318$	0	0
$319$	−18.4182	−1.03122
$320$	41.1940	2.30281
$321$	0	0
$322$	1.99035	0.110918
$323$	9.83991	0.547507
$324$	0	0
$325$	−38.9830	−2.16239
$326$	−32.0319	−1.77408
$327$	0	0
$328$	−8.19135	−0.452291
$329$	0.629709	0.0347170
$330$	0	0
$331$	−13.2517	−0.728379	−0.364190	−	0.931325i	$-0.618654\pi$
$331$	−13.2517	−0.728379	−0.364190	+	0.931325i	$0.618654\pi$
$332$	42.0500	2.30779
$333$	0	0
$334$	−26.1603	−1.43143
$335$	23.3687	1.27677
$336$	0	0
$337$	−6.76419	−0.368469	−0.184234	−	0.982882i	$-0.558981\pi$
$337$	−6.76419	−0.368469	−0.184234	+	0.982882i	$0.558981\pi$
$338$	46.0490	2.50473
$339$	0	0
$340$	36.6246	1.98625
$341$	−41.1050	−2.22596
$342$	0	0
$343$	−2.81098	−0.151778
$344$	−11.2400	−0.606020
$345$	0	0
$346$	28.1164	1.51155
$347$	−28.8872	−1.55075	−0.775374	−	0.631503i	$-0.782439\pi$
$347$	−28.8872	−1.55075	−0.775374	+	0.631503i	$0.782439\pi$
$348$	0	0
$349$	8.86947	0.474772	0.237386	−	0.971415i	$-0.423709\pi$
$349$	8.86947	0.474772	0.237386	+	0.971415i	$0.423709\pi$
$350$	2.88686	0.154309
$351$	0	0
$352$	−41.3045	−2.20154
$353$	−1.97699	−0.105224	−0.0526122	−	0.998615i	$-0.516755\pi$
$353$	−1.97699	−0.105224	−0.0526122	+	0.998615i	$0.516755\pi$
$354$	0	0
$355$	43.9206	2.33106
$356$	13.5698	0.719197
$357$	0	0
$358$	−11.5555	−0.610730
$359$	−16.3404	−0.862413	−0.431206	−	0.902253i	$-0.641912\pi$
$359$	−16.3404	−0.862413	−0.431206	+	0.902253i	$0.641912\pi$
$360$	0	0
$361$	−13.1068	−0.689831
$362$	11.0564	0.581112
$363$	0	0
$364$	−3.12472	−0.163780
$365$	25.1827	1.31812
$366$	0	0
$367$	−32.2353	−1.68267	−0.841333	−	0.540517i	$-0.818229\pi$
$367$	−32.2353	−1.68267	−0.841333	+	0.540517i	$0.818229\pi$
$368$	−10.4846	−0.546548
$369$	0	0
$370$	5.75377	0.299124
$371$	2.34666	0.121833
$372$	0	0
$373$	−0.382479	−0.0198040	−0.00990201	−	0.999951i	$-0.503152\pi$
$373$	−0.382479	−0.0198040	−0.00990201	+	0.999951i	$0.503152\pi$
$374$	−46.7513	−2.41745
$375$	0	0
$376$	4.36309	0.225009
$377$	20.1790	1.03927
$378$	0	0
$379$	−3.06644	−0.157512	−0.0787562	−	0.996894i	$-0.525095\pi$
$379$	−3.06644	−0.157512	−0.0787562	+	0.996894i	$0.525095\pi$
$380$	21.9348	1.12523
$381$	0	0
$382$	26.0429	1.33247
$383$	7.87059	0.402168	0.201084	−	0.979574i	$-0.435554\pi$
$383$	7.87059	0.402168	0.201084	+	0.979574i	$0.435554\pi$
$384$	0	0
$385$	−3.67739	−0.187417
$386$	55.3759	2.81856
$387$	0	0
$388$	−31.3861	−1.59339
$389$	−9.22062	−0.467504	−0.233752	−	0.972296i	$-0.575100\pi$
$389$	−9.22062	−0.467504	−0.233752	+	0.972296i	$0.575100\pi$
$390$	0	0
$391$	−18.5849	−0.939878
$392$	−9.70997	−0.490428
$393$	0	0
$394$	18.1979	0.916796
$395$	−49.9669	−2.51411
$396$	0	0
$397$	29.5873	1.48495	0.742473	−	0.669876i	$-0.233653\pi$
$397$	29.5873	1.48495	0.742473	+	0.669876i	$0.233653\pi$
$398$	10.2738	0.514981
$399$	0	0
$400$	−15.2072	−0.760360
$401$	−33.1480	−1.65533	−0.827666	−	0.561221i	$-0.810332\pi$
$401$	−33.1480	−1.65533	−0.827666	+	0.561221i	$0.810332\pi$
$402$	0	0
$403$	45.0346	2.24333
$404$	41.9109	2.08515
$405$	0	0
$406$	−1.49434	−0.0741630
$407$	−4.18379	−0.207383
$408$	0	0
$409$	−27.5969	−1.36458	−0.682290	−	0.731081i	$-0.739016\pi$
$409$	−27.5969	−1.36458	−0.682290	+	0.731081i	$0.739016\pi$
$410$	43.1993	2.13346
$411$	0	0
$412$	44.2490	2.17999
$413$	−2.00030	−0.0984282
$414$	0	0
$415$	−54.2183	−2.66147
$416$	45.2531	2.21872
$417$	0	0
$418$	−27.9998	−1.36951
$419$	23.6207	1.15394	0.576972	−	0.816764i	$-0.304234\pi$
$419$	23.6207	1.15394	0.576972	+	0.816764i	$0.304234\pi$
$420$	0	0
$421$	−19.6014	−0.955315	−0.477657	−	0.878546i	$-0.658514\pi$
$421$	−19.6014	−0.955315	−0.477657	+	0.878546i	$0.658514\pi$
$422$	17.2151	0.838017
$423$	0	0
$424$	16.2594	0.789626
$425$	−26.9561	−1.30756
$426$	0	0
$427$	1.14366	0.0553456
$428$	8.60567	0.415971
$429$	0	0
$430$	59.2773	2.85860
$431$	−34.8589	−1.67910	−0.839548	−	0.543286i	$-0.817180\pi$
$431$	−34.8589	−1.67910	−0.839548	+	0.543286i	$0.817180\pi$
$432$	0	0
$433$	−11.7316	−0.563787	−0.281893	−	0.959446i	$-0.590962\pi$
$433$	−11.7316	−0.563787	−0.281893	+	0.959446i	$0.590962\pi$
$434$	−3.33501	−0.160086
$435$	0	0
$436$	−7.72743	−0.370077
$437$	−11.1307	−0.532451
$438$	0	0
$439$	37.1023	1.77079	0.885397	−	0.464835i	$-0.153886\pi$
$439$	37.1023	1.77079	0.885397	+	0.464835i	$0.153886\pi$
$440$	−25.4797	−1.21470
$441$	0	0
$442$	51.2206	2.43632
$443$	−5.25818	−0.249824	−0.124912	−	0.992168i	$-0.539865\pi$
$443$	−5.25818	−0.249824	−0.124912	+	0.992168i	$0.539865\pi$
$444$	0	0
$445$	−17.4966	−0.829417
$446$	−53.7794	−2.54653
$447$	0	0
$448$	−2.43026	−0.114819
$449$	−21.0799	−0.994822	−0.497411	−	0.867515i	$-0.665716\pi$
$449$	−21.0799	−0.994822	−0.497411	+	0.867515i	$0.665716\pi$
$450$	0	0
$451$	−31.4119	−1.47913
$452$	−5.94460	−0.279611
$453$	0	0
$454$	−54.8602	−2.57472
$455$	4.02894	0.188880
$456$	0	0
$457$	−31.7180	−1.48371	−0.741853	−	0.670563i	$-0.766053\pi$
$457$	−31.7180	−1.48371	−0.741853	+	0.670563i	$0.766053\pi$
$458$	−19.0775	−0.891433
$459$	0	0
$460$	−41.4288	−1.93163
$461$	−14.1119	−0.657258	−0.328629	−	0.944459i	$-0.606587\pi$
$461$	−14.1119	−0.657258	−0.328629	+	0.944459i	$0.606587\pi$
$462$	0	0
$463$	5.51783	0.256435	0.128218	−	0.991746i	$-0.459074\pi$
$463$	5.51783	0.256435	0.128218	+	0.991746i	$0.459074\pi$
$464$	7.87178	0.365438
$465$	0	0
$466$	−51.5744	−2.38914
$467$	11.0132	0.509630	0.254815	−	0.966990i	$-0.417985\pi$
$467$	11.0132	0.509630	0.254815	+	0.966990i	$0.417985\pi$
$468$	0	0
$469$	−1.37865	−0.0636603
$470$	−23.0100	−1.06137
$471$	0	0
$472$	−13.8595	−0.637937
$473$	−43.1028	−1.98187
$474$	0	0
$475$	−16.1442	−0.740748
$476$	−2.16069	−0.0990351
$477$	0	0
$478$	2.15574	0.0986012
$479$	−18.8603	−0.861747	−0.430874	−	0.902412i	$-0.641795\pi$
$479$	−18.8603	−0.861747	−0.430874	+	0.902412i	$0.641795\pi$
$480$	0	0
$481$	4.58375	0.209001
$482$	−12.0100	−0.547039
$483$	0	0
$484$	46.6605	2.12093
$485$	40.4685	1.83758
$486$	0	0
$487$	30.7402	1.39297	0.696485	−	0.717571i	$-0.254746\pi$
$487$	30.7402	1.39297	0.696485	+	0.717571i	$0.254746\pi$
$488$	7.92412	0.358708
$489$	0	0
$490$	51.2082	2.31335
$491$	−31.9405	−1.44146	−0.720728	−	0.693218i	$-0.756192\pi$
$491$	−31.9405	−1.44146	−0.720728	+	0.693218i	$0.756192\pi$
$492$	0	0
$493$	13.9534	0.628430
$494$	30.6765	1.38020
$495$	0	0
$496$	17.5679	0.788822
$497$	−2.59112	−0.116228
$498$	0	0
$499$	−21.7380	−0.973127	−0.486564	−	0.873645i	$-0.661750\pi$
$499$	−21.7380	−0.973127	−0.486564	+	0.873645i	$0.661750\pi$
$500$	−14.9115	−0.666861
$501$	0	0
$502$	51.8998	2.31640
$503$	13.3905	0.597052	0.298526	−	0.954401i	$-0.403505\pi$
$503$	13.3905	0.597052	0.298526	+	0.954401i	$0.403505\pi$
$504$	0	0
$505$	−54.0390	−2.40470
$506$	52.8839	2.35098
$507$	0	0
$508$	−23.1717	−1.02808
$509$	15.7825	0.699547	0.349774	−	0.936834i	$-0.386259\pi$
$509$	15.7825	0.699547	0.349774	+	0.936834i	$0.386259\pi$
$510$	0	0
$511$	−1.48567	−0.0657223
$512$	24.0340	1.06216
$513$	0	0
$514$	−3.56884	−0.157415
$515$	−57.0536	−2.51408
$516$	0	0
$517$	16.7314	0.735848
$518$	−0.339447	−0.0149145
$519$	0	0
$520$	27.9155	1.22417
$521$	3.01206	0.131961	0.0659803	−	0.997821i	$-0.478983\pi$
$521$	3.01206	0.131961	0.0659803	+	0.997821i	$0.478983\pi$
$522$	0	0
$523$	21.0704	0.921342	0.460671	−	0.887571i	$-0.347609\pi$
$523$	21.0704	0.921342	0.460671	+	0.887571i	$0.347609\pi$
$524$	−23.0163	−1.00547
$525$	0	0
$526$	29.7424	1.29683
$527$	31.1406	1.35651
$528$	0	0
$529$	−1.97726	−0.0859679
$530$	−85.7485	−3.72468
$531$	0	0
$532$	−1.29406	−0.0561045
$533$	34.4148	1.49067
$534$	0	0
$535$	−11.0960	−0.479720
$536$	−9.55234	−0.412598
$537$	0	0
$538$	5.12519	0.220963
$539$	−37.2355	−1.60385
$540$	0	0
$541$	34.6808	1.49104	0.745522	−	0.666481i	$-0.232200\pi$
$541$	34.6808	1.49104	0.745522	+	0.666481i	$0.232200\pi$
$542$	7.48760	0.321620
$543$	0	0
$544$	31.2917	1.34162
$545$	9.96357	0.426792
$546$	0	0
$547$	−26.6304	−1.13863	−0.569317	−	0.822118i	$-0.692792\pi$
$547$	−26.6304	−1.13863	−0.569317	+	0.822118i	$0.692792\pi$
$548$	22.4772	0.960178
$549$	0	0
$550$	76.7043	3.27068
$551$	8.35683	0.356013
$552$	0	0
$553$	2.94783	0.125354
$554$	−4.86772	−0.206810
$555$	0	0
$556$	−5.23027	−0.221813
$557$	29.6425	1.25599	0.627996	−	0.778217i	$-0.283875\pi$
$557$	29.6425	1.25599	0.627996	+	0.778217i	$0.283875\pi$
$558$	0	0
$559$	47.2234	1.99734
$560$	1.57168	0.0664157
$561$	0	0
$562$	1.08372	0.0457139
$563$	11.2332	0.473422	0.236711	−	0.971580i	$-0.423931\pi$
$563$	11.2332	0.473422	0.236711	+	0.971580i	$0.423931\pi$
$564$	0	0
$565$	7.66483	0.322462
$566$	46.6145	1.95935
$567$	0	0
$568$	−17.9532	−0.753300
$569$	−41.8311	−1.75365	−0.876825	−	0.480810i	$-0.840343\pi$
$569$	−41.8311	−1.75365	−0.876825	+	0.480810i	$0.840343\pi$
$570$	0	0
$571$	10.8247	0.453001	0.226501	−	0.974011i	$-0.427271\pi$
$571$	10.8247	0.453001	0.226501	+	0.974011i	$0.427271\pi$
$572$	−83.0242	−3.47142
$573$	0	0
$574$	−2.54857	−0.106375
$575$	30.4920	1.27160
$576$	0	0
$577$	10.1277	0.421621	0.210810	−	0.977527i	$-0.432390\pi$
$577$	10.1277	0.421621	0.210810	+	0.977527i	$0.432390\pi$
$578$	−1.22937	−0.0511349
$579$	0	0
$580$	31.1045	1.29154
$581$	3.19864	0.132702
$582$	0	0
$583$	62.3511	2.58232
$584$	−10.2938	−0.425962
$585$	0	0
$586$	−40.3785	−1.66802
$587$	−10.6488	−0.439521	−0.219761	−	0.975554i	$-0.570528\pi$
$587$	−10.6488	−0.439521	−0.219761	+	0.975554i	$0.570528\pi$
$588$	0	0
$589$	18.6504	0.768477
$590$	73.0921	3.00915
$591$	0	0
$592$	1.78811	0.0734910
$593$	16.3794	0.672620	0.336310	−	0.941751i	$-0.390821\pi$
$593$	16.3794	0.672620	0.336310	+	0.941751i	$0.390821\pi$
$594$	0	0
$595$	2.78594	0.114213
$596$	−8.78603	−0.359890
$597$	0	0
$598$	−57.9395	−2.36932
$599$	−36.6709	−1.49833	−0.749167	−	0.662381i	$-0.769546\pi$
$599$	−36.6709	−1.49833	−0.749167	+	0.662381i	$0.769546\pi$
$600$	0	0
$601$	−31.4184	−1.28158	−0.640792	−	0.767714i	$-0.721394\pi$
$601$	−31.4184	−1.28158	−0.640792	+	0.767714i	$0.721394\pi$
$602$	−3.49710	−0.142531
$603$	0	0
$604$	−35.3900	−1.44000
$605$	−60.1630	−2.44597
$606$	0	0
$607$	−25.9196	−1.05204	−0.526021	−	0.850471i	$-0.676317\pi$
$607$	−25.9196	−1.05204	−0.526021	+	0.850471i	$0.676317\pi$
$608$	18.7409	0.760044
$609$	0	0
$610$	−41.7901	−1.69203
$611$	−18.3309	−0.741590
$612$	0	0
$613$	42.5971	1.72048	0.860240	−	0.509890i	$-0.170314\pi$
$613$	42.5971	1.72048	0.860240	+	0.509890i	$0.170314\pi$
$614$	10.2026	0.411741
$615$	0	0
$616$	1.50319	0.0605653
$617$	31.0455	1.24984	0.624922	−	0.780687i	$-0.285131\pi$
$617$	31.0455	1.24984	0.624922	+	0.780687i	$0.285131\pi$
$618$	0	0
$619$	2.71096	0.108963	0.0544813	−	0.998515i	$-0.482649\pi$
$619$	2.71096	0.108963	0.0544813	+	0.998515i	$0.482649\pi$
$620$	69.4176	2.78788
$621$	0	0
$622$	46.6510	1.87053
$623$	1.03222	0.0413551
$624$	0	0
$625$	−14.0250	−0.561000
$626$	52.6786	2.10546
$627$	0	0
$628$	9.65589	0.385312
$629$	3.16959	0.126380
$630$	0	0
$631$	−22.5598	−0.898093	−0.449047	−	0.893508i	$-0.648236\pi$
$631$	−22.5598	−0.898093	−0.449047	+	0.893508i	$0.648236\pi$
$632$	20.4247	0.812452
$633$	0	0
$634$	−6.33311	−0.251520
$635$	29.8770	1.18563
$636$	0	0
$637$	40.7951	1.61636
$638$	−39.7049	−1.57193
$639$	0	0
$640$	36.1031	1.42710
$641$	−45.7883	−1.80853	−0.904265	−	0.426971i	$-0.859580\pi$
$641$	−45.7883	−1.80853	−0.904265	+	0.426971i	$0.859580\pi$
$642$	0	0
$643$	−16.8011	−0.662571	−0.331285	−	0.943531i	$-0.607482\pi$
$643$	−16.8011	−0.662571	−0.331285	+	0.943531i	$0.607482\pi$
$644$	2.44412	0.0963117
$645$	0	0
$646$	21.2123	0.834586
$647$	−30.7434	−1.20865	−0.604323	−	0.796739i	$-0.706557\pi$
$647$	−30.7434	−1.20865	−0.604323	+	0.796739i	$0.706557\pi$
$648$	0	0
$649$	−53.1482	−2.08625
$650$	−84.0371	−3.29621
$651$	0	0
$652$	−39.3347	−1.54046
$653$	−18.9143	−0.740172	−0.370086	−	0.928997i	$-0.620672\pi$
$653$	−18.9143	−0.740172	−0.370086	+	0.928997i	$0.620672\pi$
$654$	0	0
$655$	29.6767	1.15956
$656$	13.4252	0.524165
$657$	0	0
$658$	1.35749	0.0529204
$659$	29.5539	1.15126	0.575629	−	0.817711i	$-0.304757\pi$
$659$	29.5539	1.15126	0.575629	+	0.817711i	$0.304757\pi$
$660$	0	0
$661$	16.6977	0.649465	0.324733	−	0.945806i	$-0.394726\pi$
$661$	16.6977	0.649465	0.324733	+	0.945806i	$0.394726\pi$
$662$	−28.5672	−1.11030
$663$	0	0
$664$	22.1626	0.860075
$665$	1.66853	0.0647027
$666$	0	0
$667$	−15.7837	−0.611149
$668$	−32.1244	−1.24293
$669$	0	0
$670$	50.3769	1.94623
$671$	30.3872	1.17309
$672$	0	0
$673$	45.9061	1.76955	0.884775	−	0.466018i	$-0.154312\pi$
$673$	45.9061	1.76955	0.884775	+	0.466018i	$0.154312\pi$
$674$	−14.5818	−0.561671
$675$	0	0
$676$	56.5473	2.17490
$677$	−12.4084	−0.476892	−0.238446	−	0.971156i	$-0.576638\pi$
$677$	−12.4084	−0.476892	−0.238446	+	0.971156i	$0.576638\pi$
$678$	0	0
$679$	−2.38746	−0.0916225
$680$	19.3031	0.740239
$681$	0	0
$682$	−88.6117	−3.39312
$683$	12.9318	0.494820	0.247410	−	0.968911i	$-0.420421\pi$
$683$	12.9318	0.494820	0.247410	+	0.968911i	$0.420421\pi$
$684$	0	0
$685$	−28.9816	−1.10733
$686$	−6.05973	−0.231362
$687$	0	0
$688$	18.4218	0.702323
$689$	−68.3117	−2.60247
$690$	0	0
$691$	7.42986	0.282645	0.141323	−	0.989964i	$-0.454865\pi$
$691$	7.42986	0.282645	0.141323	+	0.989964i	$0.454865\pi$
$692$	34.5265	1.31250
$693$	0	0
$694$	−62.2733	−2.36386
$695$	6.74379	0.255807
$696$	0	0
$697$	23.7973	0.901387
$698$	19.1203	0.723713
$699$	0	0
$700$	3.54502	0.133989
$701$	13.2033	0.498682	0.249341	−	0.968416i	$-0.419786\pi$
$701$	13.2033	0.498682	0.249341	+	0.968416i	$0.419786\pi$
$702$	0	0
$703$	1.89829	0.0715955
$704$	−64.5725	−2.43367
$705$	0	0
$706$	−4.26187	−0.160398
$707$	3.18807	0.119900
$708$	0	0
$709$	−38.7599	−1.45566	−0.727830	−	0.685758i	$-0.759471\pi$
$709$	−38.7599	−1.45566	−0.727830	+	0.685758i	$0.759471\pi$
$710$	94.6813	3.55333
$711$	0	0
$712$	7.15199	0.268032
$713$	−35.2255	−1.31920
$714$	0	0
$715$	107.049	4.00342
$716$	−14.1900	−0.530306
$717$	0	0
$718$	−35.2256	−1.31461
$719$	−37.8102	−1.41008	−0.705041	−	0.709167i	$-0.749071\pi$
$719$	−37.8102	−1.41008	−0.705041	+	0.709167i	$0.749071\pi$
$720$	0	0
$721$	3.36592	0.125353
$722$	−28.2548	−1.05154
$723$	0	0
$724$	13.5771	0.504589
$725$	−22.8932	−0.850232
$726$	0	0
$727$	−7.34986	−0.272591	−0.136296	−	0.990668i	$-0.543520\pi$
$727$	−7.34986	−0.272591	−0.136296	+	0.990668i	$0.543520\pi$
$728$	−1.64689	−0.0610379
$729$	0	0
$730$	54.2874	2.00927
$731$	32.6542	1.20776
$732$	0	0
$733$	−30.6192	−1.13095	−0.565473	−	0.824767i	$-0.691306\pi$
$733$	−30.6192	−1.13095	−0.565473	+	0.824767i	$0.691306\pi$
$734$	−69.4908	−2.56495
$735$	0	0
$736$	−35.3964	−1.30473
$737$	−36.6310	−1.34932
$738$	0	0
$739$	17.9779	0.661328	0.330664	−	0.943749i	$-0.392727\pi$
$739$	17.9779	0.661328	0.330664	+	0.943749i	$0.392727\pi$
$740$	7.06553	0.259734
$741$	0	0
$742$	5.05879	0.185714
$743$	15.7997	0.579634	0.289817	−	0.957082i	$-0.406405\pi$
$743$	15.7997	0.579634	0.289817	+	0.957082i	$0.406405\pi$
$744$	0	0
$745$	11.3285	0.415044
$746$	−0.824525	−0.0301880
$747$	0	0
$748$	−57.4098	−2.09911
$749$	0.654613	0.0239190
$750$	0	0
$751$	−18.1005	−0.660495	−0.330248	−	0.943894i	$-0.607132\pi$
$751$	−18.1005	−0.660495	−0.330248	+	0.943894i	$0.607132\pi$
$752$	−7.15087	−0.260765
$753$	0	0
$754$	43.5006	1.58420
$755$	45.6310	1.66068
$756$	0	0
$757$	−29.8230	−1.08394	−0.541968	−	0.840399i	$-0.682321\pi$
$757$	−29.8230	−1.08394	−0.541968	+	0.840399i	$0.682321\pi$
$758$	−6.61044	−0.240102
$759$	0	0
$760$	11.5608	0.419354
$761$	13.1033	0.474996	0.237498	−	0.971388i	$-0.423673\pi$
$761$	13.1033	0.474996	0.237498	+	0.971388i	$0.423673\pi$
$762$	0	0
$763$	−0.587807	−0.0212801
$764$	31.9802	1.15700
$765$	0	0
$766$	16.9669	0.613040
$767$	58.2290	2.10253
$768$	0	0
$769$	−23.6108	−0.851428	−0.425714	−	0.904858i	$-0.639977\pi$
$769$	−23.6108	−0.851428	−0.425714	+	0.904858i	$0.639977\pi$
$770$	−7.92749	−0.285687
$771$	0	0
$772$	68.0007	2.44740
$773$	35.3954	1.27308	0.636542	−	0.771242i	$-0.280364\pi$
$773$	35.3954	1.27308	0.636542	+	0.771242i	$0.280364\pi$
$774$	0	0
$775$	−51.0921	−1.83528
$776$	−16.5421	−0.593828
$777$	0	0
$778$	−19.8773	−0.712634
$779$	14.2524	0.510646
$780$	0	0
$781$	−68.8465	−2.46352
$782$	−40.0642	−1.43269
$783$	0	0
$784$	15.9141	0.568362
$785$	−12.4501	−0.444362
$786$	0	0
$787$	8.17427	0.291381	0.145691	−	0.989330i	$-0.453460\pi$
$787$	8.17427	0.291381	0.145691	+	0.989330i	$0.453460\pi$
$788$	22.3467	0.796068
$789$	0	0
$790$	−107.716	−3.83235
$791$	−0.452192	−0.0160781
$792$	0	0
$793$	−33.2922	−1.18224
$794$	63.7826	2.26356
$795$	0	0
$796$	12.6161	0.447166
$797$	1.26503	0.0448098	0.0224049	−	0.999749i	$-0.492868\pi$
$797$	1.26503	0.0448098	0.0224049	+	0.999749i	$0.492868\pi$
$798$	0	0
$799$	−12.6755	−0.448428
$800$	−51.3400	−1.81514
$801$	0	0
$802$	−71.4584	−2.52328
$803$	−39.4745	−1.39303
$804$	0	0
$805$	−3.15139	−0.111072
$806$	97.0828	3.41959
$807$	0	0
$808$	22.0893	0.777098
$809$	43.0690	1.51423	0.757113	−	0.653284i	$-0.226609\pi$
$809$	43.0690	1.51423	0.757113	+	0.653284i	$0.226609\pi$
$810$	0	0
$811$	−21.0119	−0.737829	−0.368914	−	0.929463i	$-0.620270\pi$
$811$	−21.0119	−0.737829	−0.368914	+	0.929463i	$0.620270\pi$
$812$	−1.83503	−0.0643969
$813$	0	0
$814$	−9.01916	−0.316121
$815$	50.7172	1.77655
$816$	0	0
$817$	19.5569	0.684208
$818$	−59.4918	−2.08008
$819$	0	0
$820$	53.0481	1.85252
$821$	39.9137	1.39300	0.696498	−	0.717559i	$-0.254740\pi$
$821$	39.9137	1.39300	0.696498	+	0.717559i	$0.254740\pi$
$822$	0	0
$823$	−40.1020	−1.39787	−0.698934	−	0.715186i	$-0.746342\pi$
$823$	−40.1020	−1.39787	−0.698934	+	0.715186i	$0.746342\pi$
$824$	23.3216	0.812445
$825$	0	0
$826$	−4.31212	−0.150038
$827$	12.4405	0.432599	0.216299	−	0.976327i	$-0.430601\pi$
$827$	12.4405	0.432599	0.216299	+	0.976327i	$0.430601\pi$
$828$	0	0
$829$	−37.5224	−1.30321	−0.651603	−	0.758560i	$-0.725903\pi$
$829$	−37.5224	−1.30321	−0.651603	+	0.758560i	$0.725903\pi$
$830$	−116.881	−4.05698
$831$	0	0
$832$	70.7454	2.45266
$833$	28.2092	0.977389
$834$	0	0
$835$	41.4205	1.43341
$836$	−34.3833	−1.18917
$837$	0	0
$838$	50.9200	1.75900
$839$	16.9559	0.585381	0.292691	−	0.956207i	$-0.405449\pi$
$839$	16.9559	0.585381	0.292691	+	0.956207i	$0.405449\pi$
$840$	0	0
$841$	−17.1497	−0.591368
$842$	−42.2556	−1.45622
$843$	0	0
$844$	21.1398	0.727663
$845$	−72.9108	−2.50821
$846$	0	0
$847$	3.54935	0.121957
$848$	−26.6483	−0.915106
$849$	0	0
$850$	−58.1102	−1.99316
$851$	−3.58535	−0.122904
$852$	0	0
$853$	7.61916	0.260875	0.130438	−	0.991457i	$-0.458362\pi$
$853$	7.61916	0.260875	0.130438	+	0.991457i	$0.458362\pi$
$854$	2.46543	0.0843654
$855$	0	0
$856$	4.53564	0.155025
$857$	39.0722	1.33468	0.667341	−	0.744753i	$-0.267433\pi$
$857$	39.0722	1.33468	0.667341	+	0.744753i	$0.267433\pi$
$858$	0	0
$859$	19.9870	0.681949	0.340974	−	0.940073i	$-0.389243\pi$
$859$	19.9870	0.681949	0.340974	+	0.940073i	$0.389243\pi$
$860$	72.7915	2.48217
$861$	0	0
$862$	−75.1467	−2.55951
$863$	−24.0312	−0.818032	−0.409016	−	0.912527i	$-0.634128\pi$
$863$	−24.0312	−0.818032	−0.409016	+	0.912527i	$0.634128\pi$
$864$	0	0
$865$	−44.5177	−1.51365
$866$	−25.2904	−0.859402
$867$	0	0
$868$	−4.09534	−0.139005
$869$	78.3242	2.65697
$870$	0	0
$871$	40.1329	1.35985
$872$	−4.07276	−0.137921
$873$	0	0
$874$	−23.9948	−0.811636
$875$	−1.13428	−0.0383457
$876$	0	0
$877$	−53.1210	−1.79377	−0.896884	−	0.442266i	$-0.854175\pi$
$877$	−53.1210	−1.79377	−0.896884	+	0.442266i	$0.854175\pi$
$878$	79.9828	2.69929
$879$	0	0
$880$	41.7598	1.40772
$881$	22.0782	0.743834	0.371917	−	0.928266i	$-0.378701\pi$
$881$	22.0782	0.743834	0.371917	+	0.928266i	$0.378701\pi$
$882$	0	0
$883$	34.6218	1.16512	0.582559	−	0.812789i	$-0.302052\pi$
$883$	34.6218	1.16512	0.582559	+	0.812789i	$0.302052\pi$
$884$	62.8981	2.11549
$885$	0	0
$886$	−11.3353	−0.380816
$887$	19.0942	0.641121	0.320561	−	0.947228i	$-0.396129\pi$
$887$	19.0942	0.641121	0.320561	+	0.947228i	$0.396129\pi$
$888$	0	0
$889$	−1.76261	−0.0591162
$890$	−37.7180	−1.26431
$891$	0	0
$892$	−66.0403	−2.21119
$893$	−7.59149	−0.254039
$894$	0	0
$895$	18.2963	0.611577
$896$	−2.12993	−0.0711559
$897$	0	0
$898$	−45.4428	−1.51644
$899$	26.4471	0.882059
$900$	0	0
$901$	−47.2364	−1.57367
$902$	−67.7160	−2.25470
$903$	0	0
$904$	−3.13312	−0.104206
$905$	−17.5060	−0.581919
$906$	0	0
$907$	−41.6275	−1.38222	−0.691109	−	0.722751i	$-0.742877\pi$
$907$	−41.6275	−1.38222	−0.691109	+	0.722751i	$0.742877\pi$
$908$	−67.3674	−2.23567
$909$	0	0
$910$	8.68534	0.287916
$911$	21.5111	0.712693	0.356347	−	0.934354i	$-0.384022\pi$
$911$	21.5111	0.712693	0.356347	+	0.934354i	$0.384022\pi$
$912$	0	0
$913$	84.9884	2.81271
$914$	−68.3757	−2.26167
$915$	0	0
$916$	−23.4269	−0.774045
$917$	−1.75080	−0.0578163
$918$	0	0
$919$	31.8930	1.05205	0.526027	−	0.850468i	$-0.323681\pi$
$919$	31.8930	1.05205	0.526027	+	0.850468i	$0.323681\pi$
$920$	−21.8351	−0.719883
$921$	0	0
$922$	−30.4216	−1.00188
$923$	75.4281	2.48275
$924$	0	0
$925$	−5.20030	−0.170985
$926$	11.8950	0.390894
$927$	0	0
$928$	26.5754	0.872381
$929$	−27.5226	−0.902986	−0.451493	−	0.892275i	$-0.649108\pi$
$929$	−27.5226	−0.902986	−0.451493	+	0.892275i	$0.649108\pi$
$930$	0	0
$931$	16.8947	0.553702
$932$	−63.3326	−2.07453
$933$	0	0
$934$	23.7416	0.776848
$935$	74.0229	2.42081
$936$	0	0
$937$	−25.9600	−0.848076	−0.424038	−	0.905644i	$-0.639388\pi$
$937$	−25.9600	−0.848076	−0.424038	+	0.905644i	$0.639388\pi$
$938$	−2.97202	−0.0970399
$939$	0	0
$940$	−28.2558	−0.921604
$941$	−11.4091	−0.371925	−0.185963	−	0.982557i	$-0.559540\pi$
$941$	−11.4091	−0.371925	−0.185963	+	0.982557i	$0.559540\pi$
$942$	0	0
$943$	−26.9189	−0.876599
$944$	22.7150	0.739312
$945$	0	0
$946$	−92.9185	−3.02104
$947$	−41.0600	−1.33427	−0.667136	−	0.744936i	$-0.732480\pi$
$947$	−41.0600	−1.33427	−0.667136	+	0.744936i	$0.732480\pi$
$948$	0	0
$949$	43.2482	1.40390
$950$	−34.8027	−1.12915
$951$	0	0
$952$	−1.13880	−0.0369087
$953$	1.67519	0.0542647	0.0271324	−	0.999632i	$-0.491362\pi$
$953$	1.67519	0.0542647	0.0271324	+	0.999632i	$0.491362\pi$
$954$	0	0
$955$	−41.2346	−1.33432
$956$	2.64721	0.0856170
$957$	0	0
$958$	−40.6578	−1.31359
$959$	1.70979	0.0552119
$960$	0	0
$961$	28.0235	0.903983
$962$	9.88137	0.318588
$963$	0	0
$964$	−14.7480	−0.475002
$965$	−87.6785	−2.82247
$966$	0	0
$967$	−5.42710	−0.174524	−0.0872618	−	0.996185i	$-0.527812\pi$
$967$	−5.42710	−0.174524	−0.0872618	+	0.996185i	$0.527812\pi$
$968$	24.5925	0.790434
$969$	0	0
$970$	87.2395	2.80109
$971$	−20.7489	−0.665865	−0.332932	−	0.942951i	$-0.608038\pi$
$971$	−20.7489	−0.665865	−0.332932	+	0.942951i	$0.608038\pi$
$972$	0	0
$973$	−0.397854	−0.0127546
$974$	66.2678	2.12336
$975$	0	0
$976$	−12.9872	−0.415711
$977$	4.41802	0.141345	0.0706725	−	0.997500i	$-0.477485\pi$
$977$	4.41802	0.141345	0.0706725	+	0.997500i	$0.477485\pi$
$978$	0	0
$979$	27.4263	0.876547
$980$	62.8829	2.00872
$981$	0	0
$982$	−68.8554	−2.19727
$983$	10.9017	0.347710	0.173855	−	0.984771i	$-0.444378\pi$
$983$	10.9017	0.347710	0.173855	+	0.984771i	$0.444378\pi$
$984$	0	0
$985$	−28.8133	−0.918068
$986$	30.0799	0.957940
$987$	0	0
$988$	37.6702	1.19845
$989$	−36.9375	−1.17455
$990$	0	0
$991$	−2.15506	−0.0684576	−0.0342288	−	0.999414i	$-0.510897\pi$
$991$	−2.15506	−0.0684576	−0.0342288	+	0.999414i	$0.510897\pi$
$992$	59.3099	1.88309
$993$	0	0
$994$	−5.58579	−0.177170
$995$	−16.2669	−0.515696
$996$	0	0
$997$	36.3803	1.15217	0.576087	−	0.817388i	$-0.304579\pi$
$997$	36.3803	1.15217	0.576087	+	0.817388i	$0.304579\pi$
$998$	−46.8615	−1.48338
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} +O(q^{10})\)	\(q+2.15574 q^{2} +2.64721 q^{4} -3.41325 q^{5} +0.201367 q^{7} +1.39522 q^{8} -7.35809 q^{10} +5.35035 q^{11} -5.86183 q^{13} +0.434095 q^{14} -2.28669 q^{16} -4.05336 q^{17} -2.42759 q^{19} -9.03561 q^{20} +11.5340 q^{22} +4.58506 q^{23} +6.65030 q^{25} -12.6366 q^{26} +0.533062 q^{28} -3.44243 q^{29} -7.68267 q^{31} -7.71995 q^{32} -8.73799 q^{34} -0.687317 q^{35} -0.781965 q^{37} -5.23326 q^{38} -4.76224 q^{40} -5.87100 q^{41} -8.05607 q^{43} +14.1635 q^{44} +9.88419 q^{46} +3.12717 q^{47} -6.95945 q^{49} +14.3363 q^{50} -15.5175 q^{52} +11.6536 q^{53} -18.2621 q^{55} +0.280952 q^{56} -7.42099 q^{58} -9.93358 q^{59} +5.67948 q^{61} -16.5618 q^{62} -12.0688 q^{64} +20.0079 q^{65} -6.84647 q^{67} -10.7301 q^{68} -1.48168 q^{70} -12.8677 q^{71} -7.37793 q^{73} -1.68571 q^{74} -6.42636 q^{76} +1.07739 q^{77} +14.6391 q^{79} +7.80506 q^{80} -12.6564 q^{82} +15.8846 q^{83} +13.8351 q^{85} -17.3668 q^{86} +7.46492 q^{88} +5.12607 q^{89} -1.18038 q^{91} +12.1376 q^{92} +6.74135 q^{94} +8.28599 q^{95} -11.8563 q^{97} -15.0028 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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