LMFDB - Embedded newform 2151.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 15 x^{10} + 47 x^{9} + 75 x^{8} - 256 x^{7} - 134 x^{6} + 571 x^{5} + 23 x^{4} + \cdots - 31 $ x^12 - 3x^11 - 15x^10 + 47x^9 + 75x^8 - 256x^7 - 134x^6 + 571x^5 + 23x^4 - 479x^3 + 129x^2 + 88*x - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.650058$ 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)

$f(q)$	$=$	$q-0.650058 q^{2} -1.57742 q^{4} -3.22288 q^{5} +3.97632 q^{7} +2.32553 q^{8} +O(q^{10})$	\(q-0.650058 q^{2} -1.57742 q^{4} -3.22288 q^{5} +3.97632 q^{7} +2.32553 q^{8} +2.09506 q^{10} +0.733708 q^{11} -0.452781 q^{13} -2.58484 q^{14} +1.64312 q^{16} +4.35866 q^{17} +2.01789 q^{19} +5.08385 q^{20} -0.476953 q^{22} -6.39492 q^{23} +5.38697 q^{25} +0.294334 q^{26} -6.27235 q^{28} -8.70236 q^{29} -5.41467 q^{31} -5.71919 q^{32} -2.83338 q^{34} -12.8152 q^{35} -0.547324 q^{37} -1.31175 q^{38} -7.49492 q^{40} -2.33364 q^{41} +12.8408 q^{43} -1.15737 q^{44} +4.15707 q^{46} -2.53886 q^{47} +8.81113 q^{49} -3.50184 q^{50} +0.714228 q^{52} +8.66216 q^{53} -2.36465 q^{55} +9.24707 q^{56} +5.65704 q^{58} +7.36282 q^{59} -1.63952 q^{61} +3.51985 q^{62} +0.431566 q^{64} +1.45926 q^{65} -5.86493 q^{67} -6.87546 q^{68} +8.33063 q^{70} +2.81291 q^{71} +12.3179 q^{73} +0.355792 q^{74} -3.18308 q^{76} +2.91746 q^{77} +13.7347 q^{79} -5.29558 q^{80} +1.51700 q^{82} +6.09089 q^{83} -14.0475 q^{85} -8.34724 q^{86} +1.70626 q^{88} +13.6794 q^{89} -1.80040 q^{91} +10.0875 q^{92} +1.65041 q^{94} -6.50344 q^{95} -10.5657 q^{97} -5.72775 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * 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 $

$2$	−0.650058	−0.459660	−0.229830	−	0.973231i	$-0.573817\pi$
$2$	−0.650058	−0.459660	−0.229830	+	0.973231i	$0.573817\pi$
$3$	0	0
$3$	0	0
$4$	−1.57742	−0.788712
$5$	−3.22288	−1.44132	−0.720658	−	0.693290i	$-0.756160\pi$
$5$	−3.22288	−1.44132	−0.720658	+	0.693290i	$0.756160\pi$
$6$	0	0
$7$	3.97632	1.50291	0.751454	−	0.659785i	$-0.229353\pi$
$7$	3.97632	1.50291	0.751454	+	0.659785i	$0.229353\pi$
$8$	2.32553	0.822200
$9$	0	0
$10$	2.09506	0.662516
$11$	0.733708	0.221221	0.110611	−	0.993864i	$-0.464719\pi$
$11$	0.733708	0.221221	0.110611	+	0.993864i	$0.464719\pi$
$12$	0	0
$13$	−0.452781	−0.125579	−0.0627895	−	0.998027i	$-0.520000\pi$
$13$	−0.452781	−0.125579	−0.0627895	+	0.998027i	$0.520000\pi$
$14$	−2.58484	−0.690827
$15$	0	0
$16$	1.64312	0.410780
$17$	4.35866	1.05713	0.528565	−	0.848893i	$-0.322730\pi$
$17$	4.35866	1.05713	0.528565	+	0.848893i	$0.322730\pi$
$18$	0	0
$19$	2.01789	0.462937	0.231468	−	0.972842i	$-0.425647\pi$
$19$	2.01789	0.462937	0.231468	+	0.972842i	$0.425647\pi$
$20$	5.08385	1.13678
$21$	0	0
$22$	−0.476953	−0.101687
$23$	−6.39492	−1.33343	−0.666716	−	0.745312i	$-0.732301\pi$
$23$	−6.39492	−1.33343	−0.666716	+	0.745312i	$0.732301\pi$
$24$	0	0
$25$	5.38697	1.07739
$26$	0.294334	0.0577237
$27$	0	0
$28$	−6.27235	−1.18536
$29$	−8.70236	−1.61599	−0.807994	−	0.589191i	$-0.799447\pi$
$29$	−8.70236	−1.61599	−0.807994	+	0.589191i	$0.799447\pi$
$30$	0	0
$31$	−5.41467	−0.972503	−0.486252	−	0.873819i	$-0.661636\pi$
$31$	−5.41467	−0.972503	−0.486252	+	0.873819i	$0.661636\pi$
$32$	−5.71919	−1.01102
$33$	0	0
$34$	−2.83338	−0.485921
$35$	−12.8152	−2.16617
$36$	0	0
$37$	−0.547324	−0.0899795	−0.0449897	−	0.998987i	$-0.514326\pi$
$37$	−0.547324	−0.0899795	−0.0449897	+	0.998987i	$0.514326\pi$
$38$	−1.31175	−0.212794
$39$	0	0
$40$	−7.49492	−1.18505
$41$	−2.33364	−0.364453	−0.182227	−	0.983257i	$-0.558330\pi$
$41$	−2.33364	−0.364453	−0.182227	+	0.983257i	$0.558330\pi$
$42$	0	0
$43$	12.8408	1.95820	0.979100	−	0.203382i	$-0.0651932\pi$
$43$	12.8408	1.95820	0.979100	+	0.203382i	$0.0651932\pi$
$44$	−1.15737	−0.174480
$45$	0	0
$46$	4.15707	0.612926
$47$	−2.53886	−0.370331	−0.185165	−	0.982707i	$-0.559282\pi$
$47$	−2.53886	−0.370331	−0.185165	+	0.982707i	$0.559282\pi$
$48$	0	0
$49$	8.81113	1.25873
$50$	−3.50184	−0.495235
$51$	0	0
$52$	0.714228	0.0990457
$53$	8.66216	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$53$	8.66216	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$54$	0	0
$55$	−2.36465	−0.318850
$56$	9.24707	1.23569
$57$	0	0
$58$	5.65704	0.742805
$59$	7.36282	0.958558	0.479279	−	0.877663i	$-0.340898\pi$
$59$	7.36282	0.958558	0.479279	+	0.877663i	$0.340898\pi$
$60$	0	0
$61$	−1.63952	−0.209920	−0.104960	−	0.994476i	$-0.533471\pi$
$61$	−1.63952	−0.209920	−0.104960	+	0.994476i	$0.533471\pi$
$62$	3.51985	0.447021
$63$	0	0
$64$	0.431566	0.0539457
$65$	1.45926	0.180999
$66$	0	0
$67$	−5.86493	−0.716516	−0.358258	−	0.933623i	$-0.616629\pi$
$67$	−5.86493	−0.716516	−0.358258	+	0.933623i	$0.616629\pi$
$68$	−6.87546	−0.833772
$69$	0	0
$70$	8.33063	0.995701
$71$	2.81291	0.333831	0.166916	−	0.985971i	$-0.446619\pi$
$71$	2.81291	0.333831	0.166916	+	0.985971i	$0.446619\pi$
$72$	0	0
$73$	12.3179	1.44170	0.720849	−	0.693092i	$-0.243752\pi$
$73$	12.3179	1.44170	0.720849	+	0.693092i	$0.243752\pi$
$74$	0.355792	0.0413600
$75$	0	0
$76$	−3.18308	−0.365124
$77$	2.91746	0.332475
$78$	0	0
$79$	13.7347	1.54527	0.772636	−	0.634849i	$-0.218938\pi$
$79$	13.7347	1.54527	0.772636	+	0.634849i	$0.218938\pi$
$80$	−5.29558	−0.592064
$81$	0	0
$82$	1.51700	0.167525
$83$	6.09089	0.668562	0.334281	−	0.942474i	$-0.391507\pi$
$83$	6.09089	0.668562	0.334281	+	0.942474i	$0.391507\pi$
$84$	0	0
$85$	−14.0475	−1.52366
$86$	−8.34724	−0.900106
$87$	0	0
$88$	1.70626	0.181888
$89$	13.6794	1.45001	0.725006	−	0.688742i	$-0.241837\pi$
$89$	13.6794	1.45001	0.725006	+	0.688742i	$0.241837\pi$
$90$	0	0
$91$	−1.80040	−0.188734
$92$	10.0875	1.05169
$93$	0	0
$94$	1.65041	0.170226
$95$	−6.50344	−0.667238
$96$	0	0
$97$	−10.5657	−1.07279	−0.536393	−	0.843968i	$-0.680213\pi$
$97$	−10.5657	−1.07279	−0.536393	+	0.843968i	$0.680213\pi$
$98$	−5.72775	−0.578590
$99$	0	0
$100$	−8.49754	−0.849754
$101$	−3.70763	−0.368923	−0.184461	−	0.982840i	$-0.559054\pi$
$101$	−3.70763	−0.368923	−0.184461	+	0.982840i	$0.559054\pi$
$102$	0	0
$103$	1.76758	0.174165	0.0870823	−	0.996201i	$-0.472246\pi$
$103$	1.76758	0.174165	0.0870823	+	0.996201i	$0.472246\pi$
$104$	−1.05296	−0.103251
$105$	0	0
$106$	−5.63091	−0.546922
$107$	1.72991	0.167237	0.0836184	−	0.996498i	$-0.473352\pi$
$107$	1.72991	0.167237	0.0836184	+	0.996498i	$0.473352\pi$
$108$	0	0
$109$	−10.0024	−0.958061	−0.479030	−	0.877798i	$-0.659012\pi$
$109$	−10.0024	−0.958061	−0.479030	+	0.877798i	$0.659012\pi$
$110$	1.53716	0.146563
$111$	0	0
$112$	6.53357	0.617364
$113$	18.7259	1.76159	0.880794	−	0.473499i	$-0.157009\pi$
$113$	18.7259	1.76159	0.880794	+	0.473499i	$0.157009\pi$
$114$	0	0
$115$	20.6101	1.92190
$116$	13.7273	1.27455
$117$	0	0
$118$	−4.78626	−0.440611
$119$	17.3314	1.58877
$120$	0	0
$121$	−10.4617	−0.951061
$122$	1.06579	0.0964917
$123$	0	0
$124$	8.54123	0.767025
$125$	−1.24715	−0.111549
$126$	0	0
$127$	−6.22841	−0.552682	−0.276341	−	0.961060i	$-0.589122\pi$
$127$	−6.22841	−0.552682	−0.276341	+	0.961060i	$0.589122\pi$
$128$	11.1578	0.986222
$129$	0	0
$130$	−0.948604	−0.0831981
$131$	−0.724572	−0.0633062	−0.0316531	−	0.999499i	$-0.510077\pi$
$131$	−0.724572	−0.0633062	−0.0316531	+	0.999499i	$0.510077\pi$
$132$	0	0
$133$	8.02380	0.695751
$134$	3.81255	0.329354
$135$	0	0
$136$	10.1362	0.869173
$137$	18.5285	1.58300	0.791499	−	0.611171i	$-0.209301\pi$
$137$	18.5285	1.58300	0.791499	+	0.611171i	$0.209301\pi$
$138$	0	0
$139$	5.75858	0.488436	0.244218	−	0.969720i	$-0.421469\pi$
$139$	5.75858	0.488436	0.244218	+	0.969720i	$0.421469\pi$
$140$	20.2150	1.70848
$141$	0	0
$142$	−1.82855	−0.153449
$143$	−0.332209	−0.0277807
$144$	0	0
$145$	28.0467	2.32915
$146$	−8.00733	−0.662691
$147$	0	0
$148$	0.863362	0.0709679
$149$	−10.9588	−0.897779	−0.448890	−	0.893587i	$-0.648180\pi$
$149$	−10.9588	−0.897779	−0.448890	+	0.893587i	$0.648180\pi$
$150$	0	0
$151$	13.2404	1.07749	0.538745	−	0.842469i	$-0.318899\pi$
$151$	13.2404	1.07749	0.538745	+	0.842469i	$0.318899\pi$
$152$	4.69268	0.380627
$153$	0	0
$154$	−1.89652	−0.152826
$155$	17.4508	1.40169
$156$	0	0
$157$	−4.50344	−0.359414	−0.179707	−	0.983720i	$-0.557515\pi$
$157$	−4.50344	−0.359414	−0.179707	+	0.983720i	$0.557515\pi$
$158$	−8.92834	−0.710300
$159$	0	0
$160$	18.4323	1.45720
$161$	−25.4282	−2.00403
$162$	0	0
$163$	9.95349	0.779617	0.389809	−	0.920896i	$-0.372541\pi$
$163$	9.95349	0.779617	0.389809	+	0.920896i	$0.372541\pi$
$164$	3.68114	0.287449
$165$	0	0
$166$	−3.95943	−0.307311
$167$	8.92001	0.690251	0.345126	−	0.938556i	$-0.387836\pi$
$167$	8.92001	0.690251	0.345126	+	0.938556i	$0.387836\pi$
$168$	0	0
$169$	−12.7950	−0.984230
$170$	9.13165	0.700366
$171$	0	0
$172$	−20.2553	−1.54446
$173$	−9.40930	−0.715376	−0.357688	−	0.933841i	$-0.616435\pi$
$173$	−9.40930	−0.715376	−0.357688	+	0.933841i	$0.616435\pi$
$174$	0	0
$175$	21.4203	1.61922
$176$	1.20557	0.0908732
$177$	0	0
$178$	−8.89239	−0.666513
$179$	11.7239	0.876283	0.438141	−	0.898906i	$-0.355637\pi$
$179$	11.7239	0.876283	0.438141	+	0.898906i	$0.355637\pi$
$180$	0	0
$181$	−20.8680	−1.55111	−0.775554	−	0.631281i	$-0.782529\pi$
$181$	−20.8680	−1.55111	−0.775554	+	0.631281i	$0.782529\pi$
$182$	1.17037	0.0867534
$183$	0	0
$184$	−14.8716	−1.09635
$185$	1.76396	0.129689
$186$	0	0
$187$	3.19798	0.233860
$188$	4.00486	0.292085
$189$	0	0
$190$	4.22761	0.306703
$191$	−16.1989	−1.17211	−0.586057	−	0.810270i	$-0.699321\pi$
$191$	−16.1989	−1.17211	−0.586057	+	0.810270i	$0.699321\pi$
$192$	0	0
$193$	21.4989	1.54753	0.773764	−	0.633474i	$-0.218372\pi$
$193$	21.4989	1.54753	0.773764	+	0.633474i	$0.218372\pi$
$194$	6.86832	0.493117
$195$	0	0
$196$	−13.8989	−0.992779
$197$	6.18812	0.440885	0.220443	−	0.975400i	$-0.429250\pi$
$197$	6.18812	0.440885	0.220443	+	0.975400i	$0.429250\pi$
$198$	0	0
$199$	21.1906	1.50216	0.751080	−	0.660211i	$-0.229533\pi$
$199$	21.1906	1.50216	0.751080	+	0.660211i	$0.229533\pi$
$200$	12.5276	0.885833
$201$	0	0
$202$	2.41017	0.169579
$203$	−34.6034	−2.42868
$204$	0	0
$205$	7.52104	0.525292
$206$	−1.14903	−0.0800565
$207$	0	0
$208$	−0.743973	−0.0515853
$209$	1.48054	0.102411
$210$	0	0
$211$	−4.67930	−0.322136	−0.161068	−	0.986943i	$-0.551494\pi$
$211$	−4.67930	−0.322136	−0.161068	+	0.986943i	$0.551494\pi$
$212$	−13.6639	−0.938441
$213$	0	0
$214$	−1.12454	−0.0768721
$215$	−41.3843	−2.82238
$216$	0	0
$217$	−21.5305	−1.46158
$218$	6.50217	0.440382
$219$	0	0
$220$	3.73006	0.251481
$221$	−1.97352	−0.132753
$222$	0	0
$223$	6.92175	0.463515	0.231757	−	0.972774i	$-0.425552\pi$
$223$	6.92175	0.463515	0.231757	+	0.972774i	$0.425552\pi$
$224$	−22.7413	−1.51947
$225$	0	0
$226$	−12.1729	−0.809732
$227$	9.96192	0.661196	0.330598	−	0.943772i	$-0.392750\pi$
$227$	9.96192	0.661196	0.330598	+	0.943772i	$0.392750\pi$
$228$	0	0
$229$	16.4245	1.08536	0.542679	−	0.839940i	$-0.317410\pi$
$229$	16.4245	1.08536	0.542679	+	0.839940i	$0.317410\pi$
$230$	−13.3977	−0.883420
$231$	0	0
$232$	−20.2376	−1.32867
$233$	16.1223	1.05621	0.528105	−	0.849179i	$-0.322903\pi$
$233$	16.1223	1.05621	0.528105	+	0.849179i	$0.322903\pi$
$234$	0	0
$235$	8.18245	0.533764
$236$	−11.6143	−0.756026
$237$	0	0
$238$	−11.2664	−0.730295
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−29.4412	−1.89647	−0.948237	−	0.317563i	$-0.897135\pi$
$241$	−29.4412	−1.89647	−0.948237	+	0.317563i	$0.897135\pi$
$242$	6.80069	0.437165
$243$	0	0
$244$	2.58623	0.165566
$245$	−28.3972	−1.81423
$246$	0	0
$247$	−0.913665	−0.0581351
$248$	−12.5920	−0.799592
$249$	0	0
$250$	0.810721	0.0512745
$251$	14.7377	0.930234	0.465117	−	0.885249i	$-0.346012\pi$
$251$	14.7377	0.930234	0.465117	+	0.885249i	$0.346012\pi$
$252$	0	0
$253$	−4.69200	−0.294984
$254$	4.04883	0.254046
$255$	0	0
$256$	−8.11637	−0.507273
$257$	24.7190	1.54193	0.770963	−	0.636879i	$-0.219775\pi$
$257$	24.7190	1.54193	0.770963	+	0.636879i	$0.219775\pi$
$258$	0	0
$259$	−2.17634	−0.135231
$260$	−2.30187	−0.142756
$261$	0	0
$262$	0.471014	0.0290993
$263$	11.5676	0.713291	0.356645	−	0.934240i	$-0.383920\pi$
$263$	11.5676	0.713291	0.356645	+	0.934240i	$0.383920\pi$
$264$	0	0
$265$	−27.9171	−1.71494
$266$	−5.21593	−0.319809
$267$	0	0
$268$	9.25149	0.565125
$269$	−0.847433	−0.0516689	−0.0258344	−	0.999666i	$-0.508224\pi$
$269$	−0.847433	−0.0516689	−0.0258344	+	0.999666i	$0.508224\pi$
$270$	0	0
$271$	15.0464	0.914006	0.457003	−	0.889465i	$-0.348923\pi$
$271$	15.0464	0.914006	0.457003	+	0.889465i	$0.348923\pi$
$272$	7.16180	0.434248
$273$	0	0
$274$	−12.0446	−0.727641
$275$	3.95246	0.238342
$276$	0	0
$277$	−0.244243	−0.0146752	−0.00733758	−	0.999973i	$-0.502336\pi$
$277$	−0.244243	−0.0146752	−0.00733758	+	0.999973i	$0.502336\pi$
$278$	−3.74341	−0.224515
$279$	0	0
$280$	−29.8022	−1.78102
$281$	−7.60445	−0.453643	−0.226822	−	0.973936i	$-0.572833\pi$
$281$	−7.60445	−0.453643	−0.226822	+	0.973936i	$0.572833\pi$
$282$	0	0
$283$	12.4160	0.738055	0.369028	−	0.929418i	$-0.379691\pi$
$283$	12.4160	0.738055	0.369028	+	0.929418i	$0.379691\pi$
$284$	−4.43715	−0.263297
$285$	0	0
$286$	0.215955	0.0127697
$287$	−9.27929	−0.547739
$288$	0	0
$289$	1.99793	0.117525
$290$	−18.2320	−1.07062
$291$	0	0
$292$	−19.4305	−1.13708
$293$	−8.88829	−0.519260	−0.259630	−	0.965708i	$-0.583601\pi$
$293$	−8.88829	−0.519260	−0.259630	+	0.965708i	$0.583601\pi$
$294$	0	0
$295$	−23.7295	−1.38159
$296$	−1.27282	−0.0739811
$297$	0	0
$298$	7.12385	0.412674
$299$	2.89550	0.167451
$300$	0	0
$301$	51.0590	2.94299
$302$	−8.60704	−0.495279
$303$	0	0
$304$	3.31564	0.190165
$305$	5.28399	0.302561
$306$	0	0
$307$	−5.35650	−0.305711	−0.152856	−	0.988249i	$-0.548847\pi$
$307$	−5.35650	−0.305711	−0.152856	+	0.988249i	$0.548847\pi$
$308$	−4.60207	−0.262227
$309$	0	0
$310$	−11.3441	−0.644299
$311$	24.0115	1.36157	0.680785	−	0.732484i	$-0.261639\pi$
$311$	24.0115	1.36157	0.680785	+	0.732484i	$0.261639\pi$
$312$	0	0
$313$	−17.7274	−1.00201	−0.501006	−	0.865444i	$-0.667036\pi$
$313$	−17.7274	−1.00201	−0.501006	+	0.865444i	$0.667036\pi$
$314$	2.92750	0.165208
$315$	0	0
$316$	−21.6654	−1.21878
$317$	−14.7894	−0.830655	−0.415327	−	0.909672i	$-0.636333\pi$
$317$	−14.7894	−0.830655	−0.415327	+	0.909672i	$0.636333\pi$
$318$	0	0
$319$	−6.38499	−0.357491
$320$	−1.39088	−0.0777528
$321$	0	0
$322$	16.5298	0.921172
$323$	8.79532	0.489385
$324$	0	0
$325$	−2.43912	−0.135298
$326$	−6.47034	−0.358359
$327$	0	0
$328$	−5.42695	−0.299653
$329$	−10.0953	−0.556573
$330$	0	0
$331$	3.29097	0.180888	0.0904441	−	0.995902i	$-0.471171\pi$
$331$	3.29097	0.180888	0.0904441	+	0.995902i	$0.471171\pi$
$332$	−9.60791	−0.527303
$333$	0	0
$334$	−5.79853	−0.317281
$335$	18.9020	1.03273
$336$	0	0
$337$	19.5372	1.06426	0.532130	−	0.846663i	$-0.321392\pi$
$337$	19.5372	1.06426	0.532130	+	0.846663i	$0.321392\pi$
$338$	8.31748	0.452411
$339$	0	0
$340$	22.1588	1.20173
$341$	−3.97279	−0.215138
$342$	0	0
$343$	7.20165	0.388852
$344$	29.8616	1.61003
$345$	0	0
$346$	6.11659	0.328830
$347$	−0.418259	−0.0224533	−0.0112267	−	0.999937i	$-0.503574\pi$
$347$	−0.418259	−0.0224533	−0.0112267	+	0.999937i	$0.503574\pi$
$348$	0	0
$349$	−31.7013	−1.69693	−0.848466	−	0.529250i	$-0.822473\pi$
$349$	−31.7013	−1.69693	−0.848466	+	0.529250i	$0.822473\pi$
$350$	−13.9244	−0.744293
$351$	0	0
$352$	−4.19621	−0.223659
$353$	30.7909	1.63883	0.819416	−	0.573199i	$-0.194298\pi$
$353$	30.7909	1.63883	0.819416	+	0.573199i	$0.194298\pi$
$354$	0	0
$355$	−9.06568	−0.481156
$356$	−21.5782	−1.14364
$357$	0	0
$358$	−7.62119	−0.402792
$359$	28.5950	1.50919	0.754594	−	0.656192i	$-0.227834\pi$
$359$	28.5950	1.50919	0.754594	+	0.656192i	$0.227834\pi$
$360$	0	0
$361$	−14.9281	−0.785690
$362$	13.5654	0.712983
$363$	0	0
$364$	2.84000	0.148857
$365$	−39.6990	−2.07794
$366$	0	0
$367$	36.0122	1.87982	0.939912	−	0.341418i	$-0.110907\pi$
$367$	36.0122	1.87982	0.939912	+	0.341418i	$0.110907\pi$
$368$	−10.5076	−0.547747
$369$	0	0
$370$	−1.14668	−0.0596129
$371$	34.4436	1.78822
$372$	0	0
$373$	7.61476	0.394277	0.197139	−	0.980376i	$-0.436835\pi$
$373$	7.61476	0.394277	0.197139	+	0.980376i	$0.436835\pi$
$374$	−2.07887	−0.107496
$375$	0	0
$376$	−5.90420	−0.304486
$377$	3.94027	0.202934
$378$	0	0
$379$	19.9471	1.02462	0.512308	−	0.858802i	$-0.328791\pi$
$379$	19.9471	1.02462	0.512308	+	0.858802i	$0.328791\pi$
$380$	10.2587	0.526259
$381$	0	0
$382$	10.5302	0.538774
$383$	−7.77685	−0.397378	−0.198689	−	0.980063i	$-0.563668\pi$
$383$	−7.77685	−0.397378	−0.198689	+	0.980063i	$0.563668\pi$
$384$	0	0
$385$	−9.40262	−0.479202
$386$	−13.9756	−0.711337
$387$	0	0
$388$	16.6666	0.846119
$389$	−8.72581	−0.442416	−0.221208	−	0.975227i	$-0.571000\pi$
$389$	−8.72581	−0.442416	−0.221208	+	0.975227i	$0.571000\pi$
$390$	0	0
$391$	−27.8733	−1.40961
$392$	20.4906	1.03493
$393$	0	0
$394$	−4.02263	−0.202657
$395$	−44.2652	−2.22723
$396$	0	0
$397$	14.1567	0.710503	0.355252	−	0.934771i	$-0.384395\pi$
$397$	14.1567	0.710503	0.355252	+	0.934771i	$0.384395\pi$
$398$	−13.7751	−0.690483
$399$	0	0
$400$	8.85143	0.442571
$401$	29.2198	1.45917	0.729584	−	0.683891i	$-0.239714\pi$
$401$	29.2198	1.45917	0.729584	+	0.683891i	$0.239714\pi$
$402$	0	0
$403$	2.45166	0.122126
$404$	5.84850	0.290974
$405$	0	0
$406$	22.4942	1.11637
$407$	−0.401576	−0.0199054
$408$	0	0
$409$	9.80672	0.484911	0.242456	−	0.970163i	$-0.422047\pi$
$409$	9.80672	0.484911	0.242456	+	0.970163i	$0.422047\pi$
$410$	−4.88911	−0.241456
$411$	0	0
$412$	−2.78822	−0.137366
$413$	29.2770	1.44062
$414$	0	0
$415$	−19.6302	−0.963609
$416$	2.58954	0.126963
$417$	0	0
$418$	−0.962440	−0.0470745
$419$	−10.3025	−0.503311	−0.251656	−	0.967817i	$-0.580975\pi$
$419$	−10.3025	−0.503311	−0.251656	+	0.967817i	$0.580975\pi$
$420$	0	0
$421$	38.4868	1.87573	0.937866	−	0.346999i	$-0.112799\pi$
$421$	38.4868	1.87573	0.937866	+	0.346999i	$0.112799\pi$
$422$	3.04182	0.148073
$423$	0	0
$424$	20.1441	0.978286
$425$	23.4800	1.13895
$426$	0	0
$427$	−6.51928	−0.315490
$428$	−2.72881	−0.131902
$429$	0	0
$430$	26.9022	1.29734
$431$	6.44756	0.310568	0.155284	−	0.987870i	$-0.450371\pi$
$431$	6.44756	0.310568	0.155284	+	0.987870i	$0.450371\pi$
$432$	0	0
$433$	22.8393	1.09759	0.548793	−	0.835959i	$-0.315088\pi$
$433$	22.8393	1.09759	0.548793	+	0.835959i	$0.315088\pi$
$434$	13.9960	0.671832
$435$	0	0
$436$	15.7781	0.755634
$437$	−12.9043	−0.617295
$438$	0	0
$439$	−37.1115	−1.77123	−0.885617	−	0.464417i	$-0.846264\pi$
$439$	−37.1115	−1.77123	−0.885617	+	0.464417i	$0.846264\pi$
$440$	−5.49908	−0.262158
$441$	0	0
$442$	1.28290	0.0610214
$443$	−20.3905	−0.968781	−0.484391	−	0.874852i	$-0.660959\pi$
$443$	−20.3905	−0.968781	−0.484391	+	0.874852i	$0.660959\pi$
$444$	0	0
$445$	−44.0871	−2.08993
$446$	−4.49954	−0.213059
$447$	0	0
$448$	1.71604	0.0810754
$449$	−34.2489	−1.61631	−0.808153	−	0.588973i	$-0.799532\pi$
$449$	−34.2489	−1.61631	−0.808153	+	0.588973i	$0.799532\pi$
$450$	0	0
$451$	−1.71221	−0.0806247
$452$	−29.5388	−1.38939
$453$	0	0
$454$	−6.47582	−0.303925
$455$	5.80249	0.272025
$456$	0	0
$457$	−12.8055	−0.599014	−0.299507	−	0.954094i	$-0.596822\pi$
$457$	−12.8055	−0.599014	−0.299507	+	0.954094i	$0.596822\pi$
$458$	−10.6768	−0.498896
$459$	0	0
$460$	−32.5108	−1.51583
$461$	−27.4846	−1.28008	−0.640042	−	0.768340i	$-0.721083\pi$
$461$	−27.4846	−1.28008	−0.640042	+	0.768340i	$0.721083\pi$
$462$	0	0
$463$	24.3067	1.12963	0.564815	−	0.825217i	$-0.308947\pi$
$463$	24.3067	1.12963	0.564815	+	0.825217i	$0.308947\pi$
$464$	−14.2990	−0.663815
$465$	0	0
$466$	−10.4805	−0.485498
$467$	−25.2164	−1.16687	−0.583437	−	0.812158i	$-0.698293\pi$
$467$	−25.2164	−1.16687	−0.583437	+	0.812158i	$0.698293\pi$
$468$	0	0
$469$	−23.3209	−1.07686
$470$	−5.31906	−0.245350
$471$	0	0
$472$	17.1225	0.788126
$473$	9.42137	0.433195
$474$	0	0
$475$	10.8703	0.498765
$476$	−27.3390	−1.25308
$477$	0	0
$478$	0.650058	0.0297330
$479$	1.18058	0.0539419	0.0269709	−	0.999636i	$-0.491414\pi$
$479$	1.18058	0.0539419	0.0269709	+	0.999636i	$0.491414\pi$
$480$	0	0
$481$	0.247818	0.0112995
$482$	19.1385	0.871734
$483$	0	0
$484$	16.5025	0.750114
$485$	34.0520	1.54622
$486$	0	0
$487$	−11.5977	−0.525540	−0.262770	−	0.964859i	$-0.584636\pi$
$487$	−11.5977	−0.525540	−0.262770	+	0.964859i	$0.584636\pi$
$488$	−3.81277	−0.172596
$489$	0	0
$490$	18.4598	0.833931
$491$	37.3597	1.68602	0.843011	−	0.537896i	$-0.180781\pi$
$491$	37.3597	1.68602	0.843011	+	0.537896i	$0.180781\pi$
$492$	0	0
$493$	−37.9306	−1.70831
$494$	0.593935	0.0267224
$495$	0	0
$496$	−8.89694	−0.399485
$497$	11.1850	0.501717
$498$	0	0
$499$	−8.71800	−0.390271	−0.195136	−	0.980776i	$-0.562515\pi$
$499$	−8.71800	−0.390271	−0.195136	+	0.980776i	$0.562515\pi$
$500$	1.96729	0.0879798
$501$	0	0
$502$	−9.58034	−0.427592
$503$	−11.5856	−0.516577	−0.258288	−	0.966068i	$-0.583158\pi$
$503$	−11.5856	−0.516577	−0.258288	+	0.966068i	$0.583158\pi$
$504$	0	0
$505$	11.9492	0.531734
$506$	3.05007	0.135592
$507$	0	0
$508$	9.82484	0.435907
$509$	3.05672	0.135487	0.0677434	−	0.997703i	$-0.478420\pi$
$509$	3.05672	0.135487	0.0677434	+	0.997703i	$0.478420\pi$
$510$	0	0
$511$	48.9798	2.16674
$512$	−17.0396	−0.753049
$513$	0	0
$514$	−16.0688	−0.708763
$515$	−5.69669	−0.251026
$516$	0	0
$517$	−1.86278	−0.0819250
$518$	1.41474	0.0621603
$519$	0	0
$520$	3.39356	0.148817
$521$	−11.8689	−0.519986	−0.259993	−	0.965610i	$-0.583720\pi$
$521$	−11.8689	−0.519986	−0.259993	+	0.965610i	$0.583720\pi$
$522$	0	0
$523$	18.8494	0.824228	0.412114	−	0.911132i	$-0.364791\pi$
$523$	18.8494	0.824228	0.412114	+	0.911132i	$0.364791\pi$
$524$	1.14296	0.0499304
$525$	0	0
$526$	−7.51963	−0.327871
$527$	−23.6007	−1.02806
$528$	0	0
$529$	17.8950	0.778042
$530$	18.1478	0.788288
$531$	0	0
$532$	−12.6569	−0.548748
$533$	1.05663	0.0457676
$534$	0	0
$535$	−5.57530	−0.241041
$536$	−13.6391	−0.589119
$537$	0	0
$538$	0.550880	0.0237501
$539$	6.46480	0.278459
$540$	0	0
$541$	−1.75688	−0.0755342	−0.0377671	−	0.999287i	$-0.512024\pi$
$541$	−1.75688	−0.0755342	−0.0377671	+	0.999287i	$0.512024\pi$
$542$	−9.78105	−0.420132
$543$	0	0
$544$	−24.9280	−1.06878
$545$	32.2367	1.38087
$546$	0	0
$547$	11.6504	0.498133	0.249067	−	0.968486i	$-0.419876\pi$
$547$	11.6504	0.498133	0.249067	+	0.968486i	$0.419876\pi$
$548$	−29.2273	−1.24853
$549$	0	0
$550$	−2.56933	−0.109557
$551$	−17.5604	−0.748100
$552$	0	0
$553$	54.6135	2.32240
$554$	0.158772	0.00674559
$555$	0	0
$556$	−9.08372	−0.385236
$557$	−33.7694	−1.43086	−0.715428	−	0.698686i	$-0.753768\pi$
$557$	−33.7694	−1.43086	−0.715428	+	0.698686i	$0.753768\pi$
$558$	0	0
$559$	−5.81406	−0.245909
$560$	−21.0569	−0.889817
$561$	0	0
$562$	4.94333	0.208522
$563$	33.5568	1.41425	0.707126	−	0.707087i	$-0.249991\pi$
$563$	33.5568	1.41425	0.707126	+	0.707087i	$0.249991\pi$
$564$	0	0
$565$	−60.3515	−2.53901
$566$	−8.07113	−0.339255
$567$	0	0
$568$	6.54152	0.274476
$569$	4.05967	0.170190	0.0850950	−	0.996373i	$-0.472881\pi$
$569$	4.05967	0.170190	0.0850950	+	0.996373i	$0.472881\pi$
$570$	0	0
$571$	−42.8024	−1.79122	−0.895611	−	0.444837i	$-0.853262\pi$
$571$	−42.8024	−1.79122	−0.895611	+	0.444837i	$0.853262\pi$
$572$	0.524035	0.0219110
$573$	0	0
$574$	6.03208	0.251774
$575$	−34.4492	−1.43663
$576$	0	0
$577$	−24.4340	−1.01720	−0.508600	−	0.861003i	$-0.669837\pi$
$577$	−24.4340	−1.01720	−0.508600	+	0.861003i	$0.669837\pi$
$578$	−1.29877	−0.0540216
$579$	0	0
$580$	−44.2415	−1.83703
$581$	24.2193	1.00479
$582$	0	0
$583$	6.35550	0.263218
$584$	28.6456	1.18536
$585$	0	0
$586$	5.77790	0.238683
$587$	−30.4606	−1.25724	−0.628621	−	0.777712i	$-0.716380\pi$
$587$	−30.4606	−1.25724	−0.628621	+	0.777712i	$0.716380\pi$
$588$	0	0
$589$	−10.9262	−0.450208
$590$	15.4256	0.635060
$591$	0	0
$592$	−0.899318	−0.0369617
$593$	39.5984	1.62611	0.813056	−	0.582186i	$-0.197802\pi$
$593$	39.5984	1.62611	0.813056	+	0.582186i	$0.197802\pi$
$594$	0	0
$595$	−55.8572	−2.28992
$596$	17.2867	0.708090
$597$	0	0
$598$	−1.88224	−0.0769706
$599$	27.4391	1.12113	0.560565	−	0.828110i	$-0.310584\pi$
$599$	27.4391	1.12113	0.560565	+	0.828110i	$0.310584\pi$
$600$	0	0
$601$	−12.7275	−0.519164	−0.259582	−	0.965721i	$-0.583585\pi$
$601$	−12.7275	−0.519164	−0.259582	+	0.965721i	$0.583585\pi$
$602$	−33.1913	−1.35278
$603$	0	0
$604$	−20.8858	−0.849830
$605$	33.7167	1.37078
$606$	0	0
$607$	34.4002	1.39626	0.698131	−	0.715970i	$-0.254015\pi$
$607$	34.4002	1.39626	0.698131	+	0.715970i	$0.254015\pi$
$608$	−11.5407	−0.468038
$609$	0	0
$610$	−3.43490	−0.139075
$611$	1.14955	0.0465058
$612$	0	0
$613$	42.1812	1.70368	0.851842	−	0.523799i	$-0.175486\pi$
$613$	42.1812	1.70368	0.851842	+	0.523799i	$0.175486\pi$
$614$	3.48203	0.140523
$615$	0	0
$616$	6.78465	0.273361
$617$	−45.5257	−1.83280	−0.916398	−	0.400268i	$-0.868917\pi$
$617$	−45.5257	−1.83280	−0.916398	+	0.400268i	$0.868917\pi$
$618$	0	0
$619$	−34.3740	−1.38161	−0.690805	−	0.723041i	$-0.742744\pi$
$619$	−34.3740	−1.38161	−0.690805	+	0.723041i	$0.742744\pi$
$620$	−27.5274	−1.10553
$621$	0	0
$622$	−15.6089	−0.625859
$623$	54.3937	2.17924
$624$	0	0
$625$	−22.9154	−0.916617
$626$	11.5238	0.460585
$627$	0	0
$628$	7.10384	0.283474
$629$	−2.38560	−0.0951201
$630$	0	0
$631$	23.9834	0.954762	0.477381	−	0.878696i	$-0.341586\pi$
$631$	23.9834	0.954762	0.477381	+	0.878696i	$0.341586\pi$
$632$	31.9404	1.27052
$633$	0	0
$634$	9.61396	0.381819
$635$	20.0734	0.796589
$636$	0	0
$637$	−3.98952	−0.158070
$638$	4.15061	0.164324
$639$	0	0
$640$	−35.9604	−1.42146
$641$	19.2396	0.759918	0.379959	−	0.925003i	$-0.375938\pi$
$641$	19.2396	0.759918	0.379959	+	0.925003i	$0.375938\pi$
$642$	0	0
$643$	−26.5425	−1.04673	−0.523367	−	0.852108i	$-0.675324\pi$
$643$	−26.5425	−1.04673	−0.523367	+	0.852108i	$0.675324\pi$
$644$	40.1112	1.58060
$645$	0	0
$646$	−5.71747	−0.224951
$647$	−21.6788	−0.852282	−0.426141	−	0.904657i	$-0.640127\pi$
$647$	−21.6788	−0.852282	−0.426141	+	0.904657i	$0.640127\pi$
$648$	0	0
$649$	5.40216	0.212053
$650$	1.58557	0.0621911
$651$	0	0
$652$	−15.7009	−0.614894
$653$	−49.0451	−1.91928	−0.959642	−	0.281223i	$-0.909260\pi$
$653$	−49.0451	−1.91928	−0.959642	+	0.281223i	$0.909260\pi$
$654$	0	0
$655$	2.33521	0.0912443
$656$	−3.83444	−0.149710
$657$	0	0
$658$	6.56255	0.255835
$659$	−31.3254	−1.22027	−0.610133	−	0.792299i	$-0.708884\pi$
$659$	−31.3254	−1.22027	−0.610133	+	0.792299i	$0.708884\pi$
$660$	0	0
$661$	47.6059	1.85165	0.925827	−	0.377948i	$-0.123370\pi$
$661$	47.6059	1.85165	0.925827	+	0.377948i	$0.123370\pi$
$662$	−2.13932	−0.0831471
$663$	0	0
$664$	14.1646	0.549691
$665$	−25.8598	−1.00280
$666$	0	0
$667$	55.6509	2.15481
$668$	−14.0707	−0.544410
$669$	0	0
$670$	−12.2874	−0.474703
$671$	−1.20293	−0.0464387
$672$	0	0
$673$	−4.31055	−0.166159	−0.0830797	−	0.996543i	$-0.526476\pi$
$673$	−4.31055	−0.166159	−0.0830797	+	0.996543i	$0.526476\pi$
$674$	−12.7003	−0.489198
$675$	0	0
$676$	20.1831	0.776274
$677$	39.5857	1.52140	0.760701	−	0.649102i	$-0.224855\pi$
$677$	39.5857	1.52140	0.760701	+	0.649102i	$0.224855\pi$
$678$	0	0
$679$	−42.0127	−1.61230
$680$	−32.6678	−1.25275
$681$	0	0
$682$	2.58254	0.0988906
$683$	−22.0957	−0.845467	−0.422733	−	0.906254i	$-0.638929\pi$
$683$	−22.0957	−0.845467	−0.422733	+	0.906254i	$0.638929\pi$
$684$	0	0
$685$	−59.7152	−2.28160
$686$	−4.68149	−0.178740
$687$	0	0
$688$	21.0989	0.804388
$689$	−3.92207	−0.149419
$690$	0	0
$691$	−44.0889	−1.67722	−0.838611	−	0.544731i	$-0.816632\pi$
$691$	−44.0889	−1.67722	−0.838611	+	0.544731i	$0.816632\pi$
$692$	14.8425	0.564226
$693$	0	0
$694$	0.271893	0.0103209
$695$	−18.5592	−0.703991
$696$	0	0
$697$	−10.1715	−0.385274
$698$	20.6077	0.780012
$699$	0	0
$700$	−33.7889	−1.27710
$701$	−13.4256	−0.507079	−0.253539	−	0.967325i	$-0.581595\pi$
$701$	−13.4256	−0.507079	−0.253539	+	0.967325i	$0.581595\pi$
$702$	0	0
$703$	−1.10444	−0.0416548
$704$	0.316643	0.0119339
$705$	0	0
$706$	−20.0158	−0.753306
$707$	−14.7427	−0.554457
$708$	0	0
$709$	−17.4495	−0.655329	−0.327664	−	0.944794i	$-0.606262\pi$
$709$	−17.4495	−0.655329	−0.327664	+	0.944794i	$0.606262\pi$
$710$	5.89322	0.221168
$711$	0	0
$712$	31.8119	1.19220
$713$	34.6264	1.29677
$714$	0	0
$715$	1.07067	0.0400408
$716$	−18.4935	−0.691135
$717$	0	0
$718$	−18.5884	−0.693714
$719$	−26.6745	−0.994791	−0.497396	−	0.867524i	$-0.665710\pi$
$719$	−26.6745	−0.994791	−0.497396	+	0.867524i	$0.665710\pi$
$720$	0	0
$721$	7.02846	0.261753
$722$	9.70413	0.361150
$723$	0	0
$724$	32.9177	1.22338
$725$	−46.8793	−1.74106
$726$	0	0
$727$	−35.9045	−1.33162	−0.665812	−	0.746119i	$-0.731915\pi$
$727$	−35.9045	−1.33162	−0.665812	+	0.746119i	$0.731915\pi$
$728$	−4.18690	−0.155177
$729$	0	0
$730$	25.8067	0.955148
$731$	55.9686	2.07007
$732$	0	0
$733$	24.0226	0.887296	0.443648	−	0.896201i	$-0.353684\pi$
$733$	24.0226	0.887296	0.443648	+	0.896201i	$0.353684\pi$
$734$	−23.4100	−0.864080
$735$	0	0
$736$	36.5737	1.34813
$737$	−4.30315	−0.158508
$738$	0	0
$739$	−40.4153	−1.48670	−0.743351	−	0.668901i	$-0.766765\pi$
$739$	−40.4153	−1.48670	−0.743351	+	0.668901i	$0.766765\pi$
$740$	−2.78251	−0.102287
$741$	0	0
$742$	−22.3903	−0.821974
$743$	18.5103	0.679076	0.339538	−	0.940592i	$-0.389729\pi$
$743$	18.5103	0.679076	0.339538	+	0.940592i	$0.389729\pi$
$744$	0	0
$745$	35.3189	1.29398
$746$	−4.95003	−0.181234
$747$	0	0
$748$	−5.04458	−0.184448
$749$	6.87868	0.251342
$750$	0	0
$751$	−25.9244	−0.945996	−0.472998	−	0.881064i	$-0.656828\pi$
$751$	−25.9244	−0.945996	−0.472998	+	0.881064i	$0.656828\pi$
$752$	−4.17165	−0.152124
$753$	0	0
$754$	−2.56140	−0.0932807
$755$	−42.6723	−1.55300
$756$	0	0
$757$	−15.0239	−0.546054	−0.273027	−	0.962006i	$-0.588025\pi$
$757$	−15.0239	−0.546054	−0.273027	+	0.962006i	$0.588025\pi$
$758$	−12.9668	−0.470975
$759$	0	0
$760$	−15.1240	−0.548603
$761$	−13.4524	−0.487650	−0.243825	−	0.969819i	$-0.578402\pi$
$761$	−13.4524	−0.487650	−0.243825	+	0.969819i	$0.578402\pi$
$762$	0	0
$763$	−39.7729	−1.43988
$764$	25.5526	0.924461
$765$	0	0
$766$	5.05540	0.182659
$767$	−3.33375	−0.120375
$768$	0	0
$769$	16.1463	0.582252	0.291126	−	0.956685i	$-0.405970\pi$
$769$	16.1463	0.582252	0.291126	+	0.956685i	$0.405970\pi$
$770$	6.11225	0.220270
$771$	0	0
$772$	−33.9130	−1.22055
$773$	−42.3626	−1.52368	−0.761838	−	0.647768i	$-0.775703\pi$
$773$	−42.3626	−1.52368	−0.761838	+	0.647768i	$0.775703\pi$
$774$	0	0
$775$	−29.1687	−1.04777
$776$	−24.5709	−0.882044
$777$	0	0
$778$	5.67228	0.203361
$779$	−4.70903	−0.168719
$780$	0	0
$781$	2.06385	0.0738505
$782$	18.1192	0.647943
$783$	0	0
$784$	14.4777	0.517062
$785$	14.5141	0.518029
$786$	0	0
$787$	46.6838	1.66410	0.832049	−	0.554702i	$-0.187168\pi$
$787$	46.6838	1.66410	0.832049	+	0.554702i	$0.187168\pi$
$788$	−9.76129	−0.347731
$789$	0	0
$790$	28.7750	1.02377
$791$	74.4604	2.64751
$792$	0	0
$793$	0.742346	0.0263615
$794$	−9.20266	−0.326590
$795$	0	0
$796$	−33.4265	−1.18477
$797$	26.7659	0.948096	0.474048	−	0.880499i	$-0.342792\pi$
$797$	26.7659	0.948096	0.474048	+	0.880499i	$0.342792\pi$
$798$	0	0
$799$	−11.0660	−0.391488
$800$	−30.8091	−1.08927
$801$	0	0
$802$	−18.9946	−0.670721
$803$	9.03772	0.318934
$804$	0	0
$805$	81.9522	2.88844
$806$	−1.59372	−0.0561364
$807$	0	0
$808$	−8.62221	−0.303328
$809$	37.0469	1.30250	0.651250	−	0.758864i	$-0.274245\pi$
$809$	37.0469	1.30250	0.651250	+	0.758864i	$0.274245\pi$
$810$	0	0
$811$	−38.3708	−1.34738	−0.673690	−	0.739014i	$-0.735292\pi$
$811$	−38.3708	−1.34738	−0.673690	+	0.739014i	$0.735292\pi$
$812$	54.5842	1.91553
$813$	0	0
$814$	0.261048	0.00914971
$815$	−32.0789	−1.12368
$816$	0	0
$817$	25.9113	0.906522
$818$	−6.37494	−0.222894
$819$	0	0
$820$	−11.8639	−0.414304
$821$	28.1412	0.982134	0.491067	−	0.871122i	$-0.336607\pi$
$821$	28.1412	0.982134	0.491067	+	0.871122i	$0.336607\pi$
$822$	0	0
$823$	31.1233	1.08489	0.542445	−	0.840091i	$-0.317499\pi$
$823$	31.1233	1.08489	0.542445	+	0.840091i	$0.317499\pi$
$824$	4.11056	0.143198
$825$	0	0
$826$	−19.0317	−0.662198
$827$	−26.9531	−0.937252	−0.468626	−	0.883397i	$-0.655251\pi$
$827$	−26.9531	−0.937252	−0.468626	+	0.883397i	$0.655251\pi$
$828$	0	0
$829$	44.0294	1.52920	0.764602	−	0.644503i	$-0.222936\pi$
$829$	44.0294	1.52920	0.764602	+	0.644503i	$0.222936\pi$
$830$	12.7608	0.442933
$831$	0	0
$832$	−0.195405	−0.00677444
$833$	38.4047	1.33065
$834$	0	0
$835$	−28.7482	−0.994871
$836$	−2.33545	−0.0807732
$837$	0	0
$838$	6.69724	0.231352
$839$	−12.7460	−0.440040	−0.220020	−	0.975495i	$-0.570612\pi$
$839$	−12.7460	−0.440040	−0.220020	+	0.975495i	$0.570612\pi$
$840$	0	0
$841$	46.7311	1.61142
$842$	−25.0186	−0.862199
$843$	0	0
$844$	7.38125	0.254073
$845$	41.2367	1.41859
$846$	0	0
$847$	−41.5990	−1.42936
$848$	14.2330	0.488762
$849$	0	0
$850$	−15.2633	−0.523528
$851$	3.50009	0.119982
$852$	0	0
$853$	11.5188	0.394395	0.197197	−	0.980364i	$-0.436816\pi$
$853$	11.5188	0.394395	0.197197	+	0.980364i	$0.436816\pi$
$854$	4.23791	0.145018
$855$	0	0
$856$	4.02297	0.137502
$857$	6.34623	0.216783	0.108392	−	0.994108i	$-0.465430\pi$
$857$	6.34623	0.216783	0.108392	+	0.994108i	$0.465430\pi$
$858$	0	0
$859$	10.2640	0.350204	0.175102	−	0.984550i	$-0.443974\pi$
$859$	10.2640	0.350204	0.175102	+	0.984550i	$0.443974\pi$
$860$	65.2806	2.22605
$861$	0	0
$862$	−4.19129	−0.142756
$863$	−26.1665	−0.890719	−0.445360	−	0.895352i	$-0.646924\pi$
$863$	−26.1665	−0.890719	−0.445360	+	0.895352i	$0.646924\pi$
$864$	0	0
$865$	30.3251	1.03108
$866$	−14.8468	−0.504516
$867$	0	0
$868$	33.9627	1.15277
$869$	10.0772	0.341847
$870$	0	0
$871$	2.65553	0.0899793
$872$	−23.2610	−0.787717
$873$	0	0
$874$	8.38852	0.283746
$875$	−4.95908	−0.167647
$876$	0	0
$877$	21.0880	0.712093	0.356046	−	0.934468i	$-0.384125\pi$
$877$	21.0880	0.712093	0.356046	+	0.934468i	$0.384125\pi$
$878$	24.1246	0.814166
$879$	0	0
$880$	−3.88541	−0.130977
$881$	23.0660	0.777113	0.388557	−	0.921425i	$-0.372974\pi$
$881$	23.0660	0.777113	0.388557	+	0.921425i	$0.372974\pi$
$882$	0	0
$883$	−27.3641	−0.920875	−0.460437	−	0.887692i	$-0.652307\pi$
$883$	−27.3641	−0.920875	−0.460437	+	0.887692i	$0.652307\pi$
$884$	3.11308	0.104704
$885$	0	0
$886$	13.2550	0.445310
$887$	26.8062	0.900064	0.450032	−	0.893012i	$-0.351413\pi$
$887$	26.8062	0.900064	0.450032	+	0.893012i	$0.351413\pi$
$888$	0	0
$889$	−24.7662	−0.830630
$890$	28.6591	0.960656
$891$	0	0
$892$	−10.9185	−0.365580
$893$	−5.12315	−0.171440
$894$	0	0
$895$	−37.7846	−1.26300
$896$	44.3671	1.48220
$897$	0	0
$898$	22.2638	0.742951
$899$	47.1204	1.57155
$900$	0	0
$901$	37.7554	1.25782
$902$	1.11303	0.0370600
$903$	0	0
$904$	43.5478	1.44838
$905$	67.2552	2.23564
$906$	0	0
$907$	−48.9444	−1.62517	−0.812586	−	0.582841i	$-0.801941\pi$
$907$	−48.9444	−1.62517	−0.812586	+	0.582841i	$0.801941\pi$
$908$	−15.7142	−0.521493
$909$	0	0
$910$	−3.77195	−0.125039
$911$	−15.0460	−0.498497	−0.249248	−	0.968440i	$-0.580184\pi$
$911$	−15.0460	−0.498497	−0.249248	+	0.968440i	$0.580184\pi$
$912$	0	0
$913$	4.46893	0.147900
$914$	8.32429	0.275343
$915$	0	0
$916$	−25.9084	−0.856036
$917$	−2.88113	−0.0951434
$918$	0	0
$919$	−20.3422	−0.671028	−0.335514	−	0.942035i	$-0.608910\pi$
$919$	−20.3422	−0.671028	−0.335514	+	0.942035i	$0.608910\pi$
$920$	47.9294	1.58019
$921$	0	0
$922$	17.8666	0.588404
$923$	−1.27363	−0.0419222
$924$	0	0
$925$	−2.94842	−0.0969433
$926$	−15.8008	−0.519246
$927$	0	0
$928$	49.7704	1.63379
$929$	−1.77379	−0.0581963	−0.0290982	−	0.999577i	$-0.509264\pi$
$929$	−1.77379	−0.0581963	−0.0290982	+	0.999577i	$0.509264\pi$
$930$	0	0
$931$	17.7799	0.582714
$932$	−25.4318	−0.833046
$933$	0	0
$934$	16.3921	0.536366
$935$	−10.3067	−0.337066
$936$	0	0
$937$	−25.6666	−0.838490	−0.419245	−	0.907873i	$-0.637705\pi$
$937$	−25.6666	−0.838490	−0.419245	+	0.907873i	$0.637705\pi$
$938$	15.1599	0.494988
$939$	0	0
$940$	−12.9072	−0.420986
$941$	33.0843	1.07852	0.539258	−	0.842140i	$-0.318705\pi$
$941$	33.0843	1.07852	0.539258	+	0.842140i	$0.318705\pi$
$942$	0	0
$943$	14.9234	0.485973
$944$	12.0980	0.393756
$945$	0	0
$946$	−6.12444	−0.199123
$947$	43.9115	1.42693	0.713466	−	0.700690i	$-0.247124\pi$
$947$	43.9115	1.42693	0.713466	+	0.700690i	$0.247124\pi$
$948$	0	0
$949$	−5.57730	−0.181047
$950$	−7.06634	−0.229262
$951$	0	0
$952$	40.3048	1.30629
$953$	−49.4188	−1.60083	−0.800416	−	0.599445i	$-0.795388\pi$
$953$	−49.4188	−1.60083	−0.800416	+	0.599445i	$0.795388\pi$
$954$	0	0
$955$	52.2073	1.68939
$956$	1.57742	0.0510176
$957$	0	0
$958$	−0.767442	−0.0247949
$959$	73.6753	2.37910
$960$	0	0
$961$	−1.68135	−0.0542372
$962$	−0.161096	−0.00519394
$963$	0	0
$964$	46.4413	1.49577
$965$	−69.2886	−2.23048
$966$	0	0
$967$	26.9264	0.865896	0.432948	−	0.901419i	$-0.357473\pi$
$967$	26.9264	0.865896	0.432948	+	0.901419i	$0.357473\pi$
$968$	−24.3290	−0.781963
$969$	0	0
$970$	−22.1358	−0.710738
$971$	−7.78989	−0.249990	−0.124995	−	0.992157i	$-0.539891\pi$
$971$	−7.78989	−0.249990	−0.124995	+	0.992157i	$0.539891\pi$
$972$	0	0
$973$	22.8979	0.734075
$974$	7.53914	0.241570
$975$	0	0
$976$	−2.69393	−0.0862307
$977$	−27.9584	−0.894469	−0.447234	−	0.894417i	$-0.647591\pi$
$977$	−27.9584	−0.894469	−0.447234	+	0.894417i	$0.647591\pi$
$978$	0	0
$979$	10.0367	0.320774
$980$	44.7945	1.43091
$981$	0	0
$982$	−24.2860	−0.774997
$983$	10.3120	0.328900	0.164450	−	0.986385i	$-0.447415\pi$
$983$	10.3120	0.328900	0.164450	+	0.986385i	$0.447415\pi$
$984$	0	0
$985$	−19.9436	−0.635455
$986$	24.6571	0.785242
$987$	0	0
$988$	1.44124	0.0458519
$989$	−82.1157	−2.61113
$990$	0	0
$991$	23.7424	0.754201	0.377100	−	0.926172i	$-0.376921\pi$
$991$	23.7424	0.754201	0.377100	+	0.926172i	$0.376921\pi$
$992$	30.9675	0.983220
$993$	0	0
$994$	−7.27092	−0.230620
$995$	−68.2947	−2.16509
$996$	0	0
$997$	9.98660	0.316279	0.158139	−	0.987417i	$-0.449450\pi$
$997$	9.98660	0.316279	0.158139	+	0.987417i	$0.449450\pi$
$998$	5.66721	0.179392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.h.1.6		12
3.2	odd	2		717.2.a.g.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.g.1.7	✓	12	3.2	odd	2
2151.2.a.h.1.6		12	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-0.650058 q^{2} -1.57742 q^{4} -3.22288 q^{5} +3.97632 q^{7} +2.32553 q^{8} +O(q^{10})\)	\(q-0.650058 q^{2} -1.57742 q^{4} -3.22288 q^{5} +3.97632 q^{7} +2.32553 q^{8} +2.09506 q^{10} +0.733708 q^{11} -0.452781 q^{13} -2.58484 q^{14} +1.64312 q^{16} +4.35866 q^{17} +2.01789 q^{19} +5.08385 q^{20} -0.476953 q^{22} -6.39492 q^{23} +5.38697 q^{25} +0.294334 q^{26} -6.27235 q^{28} -8.70236 q^{29} -5.41467 q^{31} -5.71919 q^{32} -2.83338 q^{34} -12.8152 q^{35} -0.547324 q^{37} -1.31175 q^{38} -7.49492 q^{40} -2.33364 q^{41} +12.8408 q^{43} -1.15737 q^{44} +4.15707 q^{46} -2.53886 q^{47} +8.81113 q^{49} -3.50184 q^{50} +0.714228 q^{52} +8.66216 q^{53} -2.36465 q^{55} +9.24707 q^{56} +5.65704 q^{58} +7.36282 q^{59} -1.63952 q^{61} +3.51985 q^{62} +0.431566 q^{64} +1.45926 q^{65} -5.86493 q^{67} -6.87546 q^{68} +8.33063 q^{70} +2.81291 q^{71} +12.3179 q^{73} +0.355792 q^{74} -3.18308 q^{76} +2.91746 q^{77} +13.7347 q^{79} -5.29558 q^{80} +1.51700 q^{82} +6.09089 q^{83} -14.0475 q^{85} -8.34724 q^{86} +1.70626 q^{88} +13.6794 q^{89} -1.80040 q^{91} +10.0875 q^{92} +1.65041 q^{94} -6.50344 q^{95} -10.5657 q^{97} -5.72775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 15 q^{4} + q^{5} + 11 q^{7} - 9 q^{8} - 15 q^{11} + 7 q^{13} + 6 q^{14} + 21 q^{16} + 3 q^{17} + 10 q^{19} + 4 q^{20} + 23 q^{22} - 20 q^{23} + 19 q^{25} + 10 q^{26} + 34 q^{28} - 2 q^{29} + 10 q^{31} - 26 q^{32} + 12 q^{34} - 7 q^{35} + 30 q^{37} + 3 q^{38} + 25 q^{40} + 28 q^{41} + 48 q^{43} - 25 q^{44} + 22 q^{46} - 13 q^{47} + 19 q^{49} - 12 q^{50} + 24 q^{52} + 2 q^{53} + 8 q^{55} + 7 q^{56} + 42 q^{58} + 14 q^{59} + 14 q^{61} - 8 q^{62} + 9 q^{64} + 35 q^{65} + 52 q^{67} - 3 q^{68} - 33 q^{70} + 7 q^{71} + 14 q^{73} + 13 q^{74} - 12 q^{76} + 6 q^{77} + 15 q^{79} + 8 q^{80} - 61 q^{82} - 29 q^{83} + 8 q^{85} + 9 q^{86} + 11 q^{88} + 71 q^{89} + 13 q^{91} - 2 q^{92} - 22 q^{94} - 2 q^{95} - 2 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 15 * q^4 + q^5 + 11 * q^7 - 9 * q^8 - 15 * q^11 + 7 * q^13 + 6 * q^14 + 21 * q^16 + 3 * q^17 + 10 * q^19 + 4 * q^20 + 23 * q^22 - 20 * q^23 + 19 * q^25 + 10 * q^26 + 34 * q^28 - 2 * q^29 + 10 * q^31 - 26 * q^32 + 12 * q^34 - 7 * q^35 + 30 * q^37 + 3 * q^38 + 25 * q^40 + 28 * q^41 + 48 * q^43 - 25 * q^44 + 22 * q^46 - 13 * q^47 + 19 * q^49 - 12 * q^50 + 24 * q^52 + 2 * q^53 + 8 * q^55 + 7 * q^56 + 42 * q^58 + 14 * q^59 + 14 * q^61 - 8 * q^62 + 9 * q^64 + 35 * q^65 + 52 * q^67 - 3 * q^68 - 33 * q^70 + 7 * q^71 + 14 * q^73 + 13 * q^74 - 12 * q^76 + 6 * q^77 + 15 * q^79 + 8 * q^80 - 61 * q^82 - 29 * q^83 + 8 * q^85 + 9 * q^86 + 11 * q^88 + 71 * q^89 + 13 * q^91 - 2 * q^92 - 22 * q^94 - 2 * q^95 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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