LMFDB - Embedded newform 2151.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 28 x^{15} - x^{14} + 319 x^{13} + 17 x^{12} - 1903 x^{11} - 91 x^{10} + 6377 x^{9} + 125 x^{8} + \cdots - 49 $ x^17 - 28x^15 - x^14 + 319x^13 + 17x^12 - 1903x^11 - 91x^10 + 6377x^9 + 125x^8 - 11967x^7 + 233x^6 + 11733x^5 - 503x^4 - 5015x^3 + 94x^2 + 609x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.25955$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25955 q^{2} -0.413530 q^{4} +3.10986 q^{5} -4.47710 q^{7} -3.03997 q^{8} +O(q^{10})$	\(q+1.25955 q^{2} -0.413530 q^{4} +3.10986 q^{5} -4.47710 q^{7} -3.03997 q^{8} +3.91703 q^{10} +4.55947 q^{11} -0.199501 q^{13} -5.63914 q^{14} -3.00193 q^{16} -1.46352 q^{17} +3.73563 q^{19} -1.28602 q^{20} +5.74288 q^{22} +8.29633 q^{23} +4.67123 q^{25} -0.251281 q^{26} +1.85141 q^{28} +3.78272 q^{29} +3.54191 q^{31} +2.29884 q^{32} -1.84338 q^{34} -13.9232 q^{35} -2.34273 q^{37} +4.70522 q^{38} -9.45387 q^{40} +1.04130 q^{41} -0.609594 q^{43} -1.88548 q^{44} +10.4497 q^{46} -3.21630 q^{47} +13.0444 q^{49} +5.88365 q^{50} +0.0824995 q^{52} +13.2152 q^{53} +14.1793 q^{55} +13.6102 q^{56} +4.76454 q^{58} +11.2542 q^{59} +7.51691 q^{61} +4.46122 q^{62} +8.89938 q^{64} -0.620419 q^{65} -2.71800 q^{67} +0.605209 q^{68} -17.5369 q^{70} -12.2637 q^{71} +0.377965 q^{73} -2.95079 q^{74} -1.54480 q^{76} -20.4132 q^{77} -0.915159 q^{79} -9.33559 q^{80} +1.31157 q^{82} -7.42384 q^{83} -4.55134 q^{85} -0.767815 q^{86} -13.8606 q^{88} -4.63741 q^{89} +0.893184 q^{91} -3.43078 q^{92} -4.05110 q^{94} +11.6173 q^{95} +15.2724 q^{97} +16.4301 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	$ 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) $ 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.25955	0.890637	0.445319	−	0.895372i	$-0.353090\pi$
$2$	1.25955	0.890637	0.445319	+	0.895372i	$0.353090\pi$
$3$	0	0
$3$	0	0
$4$	−0.413530	−0.206765
$5$	3.10986	1.39077	0.695386	−	0.718637i	$-0.255233\pi$
$5$	3.10986	1.39077	0.695386	+	0.718637i	$0.255233\pi$
$6$	0	0
$7$	−4.47710	−1.69218	−0.846092	−	0.533037i	$-0.821051\pi$
$7$	−4.47710	−1.69218	−0.846092	+	0.533037i	$0.821051\pi$
$8$	−3.03997	−1.07479
$9$	0	0
$10$	3.91703	1.23867
$11$	4.55947	1.37473	0.687365	−	0.726312i	$-0.258767\pi$
$11$	4.55947	1.37473	0.687365	+	0.726312i	$0.258767\pi$
$12$	0	0
$13$	−0.199501	−0.0553315	−0.0276658	−	0.999617i	$-0.508807\pi$
$13$	−0.199501	−0.0553315	−0.0276658	+	0.999617i	$0.508807\pi$
$14$	−5.63914	−1.50712
$15$	0	0
$16$	−3.00193	−0.750483
$17$	−1.46352	−0.354955	−0.177478	−	0.984125i	$-0.556794\pi$
$17$	−1.46352	−0.354955	−0.177478	+	0.984125i	$0.556794\pi$
$18$	0	0
$19$	3.73563	0.857013	0.428507	−	0.903539i	$-0.359040\pi$
$19$	3.73563	0.857013	0.428507	+	0.903539i	$0.359040\pi$
$20$	−1.28602	−0.287563
$21$	0	0
$22$	5.74288	1.22439
$23$	8.29633	1.72991	0.864953	−	0.501853i	$-0.167348\pi$
$23$	8.29633	1.72991	0.864953	+	0.501853i	$0.167348\pi$
$24$	0	0
$25$	4.67123	0.934246
$26$	−0.251281	−0.0492803
$27$	0	0
$28$	1.85141	0.349884
$29$	3.78272	0.702434	0.351217	−	0.936294i	$-0.385768\pi$
$29$	3.78272	0.702434	0.351217	+	0.936294i	$0.385768\pi$
$30$	0	0
$31$	3.54191	0.636147	0.318073	−	0.948066i	$-0.396964\pi$
$31$	3.54191	0.636147	0.318073	+	0.948066i	$0.396964\pi$
$32$	2.29884	0.406382
$33$	0	0
$34$	−1.84338	−0.316136
$35$	−13.9232	−2.35344
$36$	0	0
$37$	−2.34273	−0.385143	−0.192572	−	0.981283i	$-0.561683\pi$
$37$	−2.34273	−0.385143	−0.192572	+	0.981283i	$0.561683\pi$
$38$	4.70522	0.763288
$39$	0	0
$40$	−9.45387	−1.49479
$41$	1.04130	0.162624	0.0813120	−	0.996689i	$-0.474089\pi$
$41$	1.04130	0.162624	0.0813120	+	0.996689i	$0.474089\pi$
$42$	0	0
$43$	−0.609594	−0.0929622	−0.0464811	−	0.998919i	$-0.514801\pi$
$43$	−0.609594	−0.0929622	−0.0464811	+	0.998919i	$0.514801\pi$
$44$	−1.88548	−0.284246
$45$	0	0
$46$	10.4497	1.54072
$47$	−3.21630	−0.469146	−0.234573	−	0.972099i	$-0.575369\pi$
$47$	−3.21630	−0.469146	−0.234573	+	0.972099i	$0.575369\pi$
$48$	0	0
$49$	13.0444	1.86349
$50$	5.88365	0.832074
$51$	0	0
$52$	0.0824995	0.0114406
$53$	13.2152	1.81525	0.907623	−	0.419786i	$-0.137895\pi$
$53$	13.2152	1.81525	0.907623	+	0.419786i	$0.137895\pi$
$54$	0	0
$55$	14.1793	1.91194
$56$	13.6102	1.81874
$57$	0	0
$58$	4.76454	0.625614
$59$	11.2542	1.46517	0.732587	−	0.680674i	$-0.238313\pi$
$59$	11.2542	1.46517	0.732587	+	0.680674i	$0.238313\pi$
$60$	0	0
$61$	7.51691	0.962442	0.481221	−	0.876599i	$-0.340194\pi$
$61$	7.51691	0.962442	0.481221	+	0.876599i	$0.340194\pi$
$62$	4.46122	0.566576
$63$	0	0
$64$	8.89938	1.11242
$65$	−0.620419	−0.0769535
$66$	0	0
$67$	−2.71800	−0.332056	−0.166028	−	0.986121i	$-0.553094\pi$
$67$	−2.71800	−0.332056	−0.166028	+	0.986121i	$0.553094\pi$
$68$	0.605209	0.0733923
$69$	0	0
$70$	−17.5369	−2.09606
$71$	−12.2637	−1.45543	−0.727717	−	0.685877i	$-0.759419\pi$
$71$	−12.2637	−1.45543	−0.727717	+	0.685877i	$0.759419\pi$
$72$	0	0
$73$	0.377965	0.0442375	0.0221188	−	0.999755i	$-0.492959\pi$
$73$	0.377965	0.0442375	0.0221188	+	0.999755i	$0.492959\pi$
$74$	−2.95079	−0.343023
$75$	0	0
$76$	−1.54480	−0.177200
$77$	−20.4132	−2.32630
$78$	0	0
$79$	−0.915159	−0.102963	−0.0514817	−	0.998674i	$-0.516394\pi$
$79$	−0.915159	−0.102963	−0.0514817	+	0.998674i	$0.516394\pi$
$80$	−9.33559	−1.04375
$81$	0	0
$82$	1.31157	0.144839
$83$	−7.42384	−0.814872	−0.407436	−	0.913234i	$-0.633577\pi$
$83$	−7.42384	−0.814872	−0.407436	+	0.913234i	$0.633577\pi$
$84$	0	0
$85$	−4.55134	−0.493662
$86$	−0.767815	−0.0827956
$87$	0	0
$88$	−13.8606	−1.47755
$89$	−4.63741	−0.491564	−0.245782	−	0.969325i	$-0.579045\pi$
$89$	−4.63741	−0.491564	−0.245782	+	0.969325i	$0.579045\pi$
$90$	0	0
$91$	0.893184	0.0936312
$92$	−3.43078	−0.357684
$93$	0	0
$94$	−4.05110	−0.417839
$95$	11.6173	1.19191
$96$	0	0
$97$	15.2724	1.55068	0.775339	−	0.631546i	$-0.217579\pi$
$97$	15.2724	1.55068	0.775339	+	0.631546i	$0.217579\pi$
$98$	16.4301	1.65969
$99$	0	0
$100$	−1.93169	−0.193169
$101$	6.31049	0.627917	0.313959	−	0.949437i	$-0.398345\pi$
$101$	6.31049	0.627917	0.313959	+	0.949437i	$0.398345\pi$
$102$	0	0
$103$	−8.17402	−0.805410	−0.402705	−	0.915330i	$-0.631930\pi$
$103$	−8.17402	−0.805410	−0.402705	+	0.915330i	$0.631930\pi$
$104$	0.606475	0.0594698
$105$	0	0
$106$	16.6452	1.61673
$107$	−13.1056	−1.26697	−0.633483	−	0.773757i	$-0.718375\pi$
$107$	−13.1056	−1.26697	−0.633483	+	0.773757i	$0.718375\pi$
$108$	0	0
$109$	−5.84174	−0.559538	−0.279769	−	0.960067i	$-0.590258\pi$
$109$	−5.84174	−0.559538	−0.279769	+	0.960067i	$0.590258\pi$
$110$	17.8596	1.70284
$111$	0	0
$112$	13.4400	1.26996
$113$	0.893430	0.0840468	0.0420234	−	0.999117i	$-0.486620\pi$
$113$	0.893430	0.0840468	0.0420234	+	0.999117i	$0.486620\pi$
$114$	0	0
$115$	25.8004	2.40590
$116$	−1.56427	−0.145239
$117$	0	0
$118$	14.1753	1.30494
$119$	6.55232	0.600650
$120$	0	0
$121$	9.78873	0.889885
$122$	9.46794	0.857187
$123$	0	0
$124$	−1.46469	−0.131533
$125$	−1.02243	−0.0914488
$126$	0	0
$127$	2.44769	0.217198	0.108599	−	0.994086i	$-0.465364\pi$
$127$	2.44769	0.217198	0.108599	+	0.994086i	$0.465364\pi$
$128$	6.61154	0.584383
$129$	0	0
$130$	−0.781450	−0.0685377
$131$	−11.9754	−1.04629	−0.523147	−	0.852243i	$-0.675242\pi$
$131$	−11.9754	−1.04629	−0.523147	+	0.852243i	$0.675242\pi$
$132$	0	0
$133$	−16.7248	−1.45022
$134$	−3.42346	−0.295742
$135$	0	0
$136$	4.44904	0.381502
$137$	−1.31560	−0.112400	−0.0561998	−	0.998420i	$-0.517898\pi$
$137$	−1.31560	−0.112400	−0.0561998	+	0.998420i	$0.517898\pi$
$138$	0	0
$139$	−7.79609	−0.661255	−0.330628	−	0.943761i	$-0.607260\pi$
$139$	−7.79609	−0.661255	−0.330628	+	0.943761i	$0.607260\pi$
$140$	5.75764	0.486609
$141$	0	0
$142$	−15.4468	−1.29626
$143$	−0.909617	−0.0760660
$144$	0	0
$145$	11.7637	0.976926
$146$	0.476067	0.0393996
$147$	0	0
$148$	0.968790	0.0796341
$149$	9.65365	0.790858	0.395429	−	0.918497i	$-0.370596\pi$
$149$	9.65365	0.790858	0.395429	+	0.918497i	$0.370596\pi$
$150$	0	0
$151$	−3.94768	−0.321258	−0.160629	−	0.987015i	$-0.551352\pi$
$151$	−3.94768	−0.321258	−0.160629	+	0.987015i	$0.551352\pi$
$152$	−11.3562	−0.921109
$153$	0	0
$154$	−25.7115	−2.07189
$155$	11.0149	0.884735
$156$	0	0
$157$	8.37149	0.668117	0.334059	−	0.942552i	$-0.391582\pi$
$157$	8.37149	0.668117	0.334059	+	0.942552i	$0.391582\pi$
$158$	−1.15269	−0.0917031
$159$	0	0
$160$	7.14907	0.565184
$161$	−37.1435	−2.92732
$162$	0	0
$163$	−6.87702	−0.538650	−0.269325	−	0.963049i	$-0.586801\pi$
$163$	−6.87702	−0.538650	−0.269325	+	0.963049i	$0.586801\pi$
$164$	−0.430609	−0.0336249
$165$	0	0
$166$	−9.35071	−0.725756
$167$	10.5343	0.815166	0.407583	−	0.913168i	$-0.366372\pi$
$167$	10.5343	0.815166	0.407583	+	0.913168i	$0.366372\pi$
$168$	0	0
$169$	−12.9602	−0.996938
$170$	−5.73264	−0.439674
$171$	0	0
$172$	0.252085	0.0192213
$173$	19.0305	1.44686	0.723430	−	0.690398i	$-0.242564\pi$
$173$	19.0305	1.44686	0.723430	+	0.690398i	$0.242564\pi$
$174$	0	0
$175$	−20.9136	−1.58092
$176$	−13.6872	−1.03171
$177$	0	0
$178$	−5.84105	−0.437805
$179$	18.0878	1.35195	0.675974	−	0.736926i	$-0.263723\pi$
$179$	18.0878	1.35195	0.675974	+	0.736926i	$0.263723\pi$
$180$	0	0
$181$	−18.5346	−1.37767	−0.688835	−	0.724918i	$-0.741877\pi$
$181$	−18.5346	−1.37767	−0.688835	+	0.724918i	$0.741877\pi$
$182$	1.12501	0.0833914
$183$	0	0
$184$	−25.2206	−1.85929
$185$	−7.28557	−0.535646
$186$	0	0
$187$	−6.67286	−0.487968
$188$	1.33004	0.0970029
$189$	0	0
$190$	14.6326	1.06156
$191$	6.95136	0.502983	0.251491	−	0.967860i	$-0.419079\pi$
$191$	6.95136	0.502983	0.251491	+	0.967860i	$0.419079\pi$
$192$	0	0
$193$	20.0240	1.44136	0.720678	−	0.693269i	$-0.243830\pi$
$193$	20.0240	1.44136	0.720678	+	0.693269i	$0.243830\pi$
$194$	19.2364	1.38109
$195$	0	0
$196$	−5.39426	−0.385304
$197$	−10.8362	−0.772046	−0.386023	−	0.922489i	$-0.626151\pi$
$197$	−10.8362	−0.772046	−0.386023	+	0.922489i	$0.626151\pi$
$198$	0	0
$199$	17.1301	1.21432	0.607160	−	0.794580i	$-0.292309\pi$
$199$	17.1301	1.21432	0.607160	+	0.794580i	$0.292309\pi$
$200$	−14.2004	−1.00412
$201$	0	0
$202$	7.94839	0.559246
$203$	−16.9356	−1.18865
$204$	0	0
$205$	3.23830	0.226173
$206$	−10.2956	−0.717328
$207$	0	0
$208$	0.598888	0.0415254
$209$	17.0325	1.17816
$210$	0	0
$211$	−11.4195	−0.786149	−0.393074	−	0.919507i	$-0.628589\pi$
$211$	−11.4195	−0.786149	−0.393074	+	0.919507i	$0.628589\pi$
$212$	−5.46488	−0.375329
$213$	0	0
$214$	−16.5072	−1.12841
$215$	−1.89575	−0.129289
$216$	0	0
$217$	−15.8575	−1.07648
$218$	−7.35798	−0.498345
$219$	0	0
$220$	−5.86357	−0.395322
$221$	0.291973	0.0196402
$222$	0	0
$223$	−24.4592	−1.63791	−0.818956	−	0.573856i	$-0.805447\pi$
$223$	−24.4592	−1.63791	−0.818956	+	0.573856i	$0.805447\pi$
$224$	−10.2921	−0.687673
$225$	0	0
$226$	1.12532	0.0748552
$227$	−17.7505	−1.17814	−0.589070	−	0.808082i	$-0.700506\pi$
$227$	−17.7505	−1.17814	−0.589070	+	0.808082i	$0.700506\pi$
$228$	0	0
$229$	11.3131	0.747589	0.373794	−	0.927512i	$-0.378057\pi$
$229$	11.3131	0.747589	0.373794	+	0.927512i	$0.378057\pi$
$230$	32.4970	2.14279
$231$	0	0
$232$	−11.4993	−0.754969
$233$	16.0222	1.04965	0.524826	−	0.851209i	$-0.324130\pi$
$233$	16.0222	1.04965	0.524826	+	0.851209i	$0.324130\pi$
$234$	0	0
$235$	−10.0022	−0.652475
$236$	−4.65395	−0.302947
$237$	0	0
$238$	8.25298	0.534961
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	11.0668	0.712876	0.356438	−	0.934319i	$-0.383991\pi$
$241$	11.0668	0.712876	0.356438	+	0.934319i	$0.383991\pi$
$242$	12.3294	0.792565
$243$	0	0
$244$	−3.10847	−0.198999
$245$	40.5663	2.59169
$246$	0	0
$247$	−0.745262	−0.0474199
$248$	−10.7673	−0.683724
$249$	0	0
$250$	−1.28780	−0.0814477
$251$	24.6773	1.55762	0.778810	−	0.627260i	$-0.215824\pi$
$251$	24.6773	1.55762	0.778810	+	0.627260i	$0.215824\pi$
$252$	0	0
$253$	37.8269	2.37815
$254$	3.08300	0.193444
$255$	0	0
$256$	−9.47118	−0.591948
$257$	−18.1791	−1.13398	−0.566992	−	0.823724i	$-0.691893\pi$
$257$	−18.1791	−1.13398	−0.566992	+	0.823724i	$0.691893\pi$
$258$	0	0
$259$	10.4886	0.651733
$260$	0.256562	0.0159113
$261$	0	0
$262$	−15.0836	−0.931868
$263$	−23.5248	−1.45060	−0.725302	−	0.688431i	$-0.758300\pi$
$263$	−23.5248	−1.45060	−0.725302	+	0.688431i	$0.758300\pi$
$264$	0	0
$265$	41.0974	2.52459
$266$	−21.0658	−1.29162
$267$	0	0
$268$	1.12397	0.0686576
$269$	8.31276	0.506838	0.253419	−	0.967357i	$-0.418445\pi$
$269$	8.31276	0.506838	0.253419	+	0.967357i	$0.418445\pi$
$270$	0	0
$271$	−8.98469	−0.545781	−0.272891	−	0.962045i	$-0.587980\pi$
$271$	−8.98469	−0.545781	−0.272891	+	0.962045i	$0.587980\pi$
$272$	4.39338	0.266388
$273$	0	0
$274$	−1.65707	−0.100107
$275$	21.2983	1.28434
$276$	0	0
$277$	1.12445	0.0675617	0.0337809	−	0.999429i	$-0.489245\pi$
$277$	1.12445	0.0675617	0.0337809	+	0.999429i	$0.489245\pi$
$278$	−9.81957	−0.588939
$279$	0	0
$280$	42.3259	2.52946
$281$	−15.9640	−0.952335	−0.476167	−	0.879355i	$-0.657974\pi$
$281$	−15.9640	−0.952335	−0.476167	+	0.879355i	$0.657974\pi$
$282$	0	0
$283$	8.23057	0.489256	0.244628	−	0.969617i	$-0.421334\pi$
$283$	8.23057	0.489256	0.244628	+	0.969617i	$0.421334\pi$
$284$	5.07141	0.300933
$285$	0	0
$286$	−1.14571	−0.0677472
$287$	−4.66201	−0.275190
$288$	0	0
$289$	−14.8581	−0.874007
$290$	14.8170	0.870087
$291$	0	0
$292$	−0.156300	−0.00914677
$293$	−28.2040	−1.64770	−0.823848	−	0.566810i	$-0.808177\pi$
$293$	−28.2040	−1.64770	−0.823848	+	0.566810i	$0.808177\pi$
$294$	0	0
$295$	34.9990	2.03772
$296$	7.12183	0.413948
$297$	0	0
$298$	12.1593	0.704368
$299$	−1.65512	−0.0957183
$300$	0	0
$301$	2.72921	0.157309
$302$	−4.97231	−0.286124
$303$	0	0
$304$	−11.2141	−0.643174
$305$	23.3765	1.33854
$306$	0	0
$307$	16.4229	0.937306	0.468653	−	0.883382i	$-0.344740\pi$
$307$	16.4229	0.937306	0.468653	+	0.883382i	$0.344740\pi$
$308$	8.44146	0.480997
$309$	0	0
$310$	13.8738	0.787978
$311$	−26.8586	−1.52301	−0.761507	−	0.648157i	$-0.775540\pi$
$311$	−26.8586	−1.52301	−0.761507	+	0.648157i	$0.775540\pi$
$312$	0	0
$313$	−21.5546	−1.21834	−0.609168	−	0.793041i	$-0.708497\pi$
$313$	−21.5546	−1.21834	−0.609168	+	0.793041i	$0.708497\pi$
$314$	10.5443	0.595050
$315$	0	0
$316$	0.378446	0.0212892
$317$	−3.68396	−0.206912	−0.103456	−	0.994634i	$-0.532990\pi$
$317$	−3.68396	−0.206912	−0.103456	+	0.994634i	$0.532990\pi$
$318$	0	0
$319$	17.2472	0.965658
$320$	27.6758	1.54712
$321$	0	0
$322$	−46.7842	−2.60718
$323$	−5.46717	−0.304201
$324$	0	0
$325$	−0.931914	−0.0516933
$326$	−8.66196	−0.479742
$327$	0	0
$328$	−3.16552	−0.174787
$329$	14.3997	0.793881
$330$	0	0
$331$	12.4309	0.683266	0.341633	−	0.939833i	$-0.389020\pi$
$331$	12.4309	0.683266	0.341633	+	0.939833i	$0.389020\pi$
$332$	3.06998	0.168487
$333$	0	0
$334$	13.2684	0.726017
$335$	−8.45259	−0.461815
$336$	0	0
$337$	−3.84850	−0.209641	−0.104820	−	0.994491i	$-0.533427\pi$
$337$	−3.84850	−0.209641	−0.104820	+	0.994491i	$0.533427\pi$
$338$	−16.3240	−0.887911
$339$	0	0
$340$	1.88211	0.102072
$341$	16.1492	0.874530
$342$	0	0
$343$	−27.0615	−1.46118
$344$	1.85314	0.0999148
$345$	0	0
$346$	23.9699	1.28863
$347$	−29.5596	−1.58684	−0.793422	−	0.608672i	$-0.791702\pi$
$347$	−29.5596	−1.58684	−0.793422	+	0.608672i	$0.791702\pi$
$348$	0	0
$349$	12.2155	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$349$	12.2155	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$350$	−26.3417	−1.40802
$351$	0	0
$352$	10.4815	0.558665
$353$	14.4243	0.767727	0.383864	−	0.923390i	$-0.374593\pi$
$353$	14.4243	0.767727	0.383864	+	0.923390i	$0.374593\pi$
$354$	0	0
$355$	−38.1384	−2.02418
$356$	1.91771	0.101638
$357$	0	0
$358$	22.7825	1.20409
$359$	−14.9235	−0.787630	−0.393815	−	0.919190i	$-0.628845\pi$
$359$	−14.9235	−0.787630	−0.393815	+	0.919190i	$0.628845\pi$
$360$	0	0
$361$	−5.04503	−0.265528
$362$	−23.3453	−1.22700
$363$	0	0
$364$	−0.369358	−0.0193596
$365$	1.17542	0.0615243
$366$	0	0
$367$	1.49609	0.0780950	0.0390475	−	0.999237i	$-0.487568\pi$
$367$	1.49609	0.0780950	0.0390475	+	0.999237i	$0.487568\pi$
$368$	−24.9050	−1.29827
$369$	0	0
$370$	−9.17656	−0.477066
$371$	−59.1657	−3.07173
$372$	0	0
$373$	19.3983	1.00441	0.502203	−	0.864750i	$-0.332523\pi$
$373$	19.3983	1.00441	0.502203	+	0.864750i	$0.332523\pi$
$374$	−8.40481	−0.434603
$375$	0	0
$376$	9.77744	0.504233
$377$	−0.754656	−0.0388668
$378$	0	0
$379$	25.2651	1.29778	0.648891	−	0.760881i	$-0.275233\pi$
$379$	25.2651	1.29778	0.648891	+	0.760881i	$0.275233\pi$
$380$	−4.80410	−0.246445
$381$	0	0
$382$	8.75559	0.447975
$383$	−14.1890	−0.725026	−0.362513	−	0.931979i	$-0.618081\pi$
$383$	−14.1890	−0.725026	−0.362513	+	0.931979i	$0.618081\pi$
$384$	0	0
$385$	−63.4821	−3.23535
$386$	25.2212	1.28373
$387$	0	0
$388$	−6.31559	−0.320626
$389$	11.7091	0.593674	0.296837	−	0.954928i	$-0.404068\pi$
$389$	11.7091	0.593674	0.296837	+	0.954928i	$0.404068\pi$
$390$	0	0
$391$	−12.1418	−0.614039
$392$	−39.6546	−2.00286
$393$	0	0
$394$	−13.6487	−0.687613
$395$	−2.84602	−0.143199
$396$	0	0
$397$	2.05183	0.102978	0.0514891	−	0.998674i	$-0.483603\pi$
$397$	2.05183	0.102978	0.0514891	+	0.998674i	$0.483603\pi$
$398$	21.5762	1.08152
$399$	0	0
$400$	−14.0227	−0.701136
$401$	36.5830	1.82687	0.913434	−	0.406986i	$-0.133420\pi$
$401$	36.5830	1.82687	0.913434	+	0.406986i	$0.133420\pi$
$402$	0	0
$403$	−0.706614	−0.0351990
$404$	−2.60958	−0.129831
$405$	0	0
$406$	−21.3313	−1.05865
$407$	−10.6816	−0.529468
$408$	0	0
$409$	10.9750	0.542679	0.271340	−	0.962484i	$-0.412533\pi$
$409$	10.9750	0.542679	0.271340	+	0.962484i	$0.412533\pi$
$410$	4.07881	0.201438
$411$	0	0
$412$	3.38020	0.166531
$413$	−50.3862	−2.47934
$414$	0	0
$415$	−23.0871	−1.13330
$416$	−0.458620	−0.0224857
$417$	0	0
$418$	21.4533	1.04932
$419$	−6.81357	−0.332865	−0.166432	−	0.986053i	$-0.553225\pi$
$419$	−6.81357	−0.332865	−0.166432	+	0.986053i	$0.553225\pi$
$420$	0	0
$421$	38.0913	1.85646	0.928228	−	0.372012i	$-0.121332\pi$
$421$	38.0913	1.85646	0.928228	+	0.372012i	$0.121332\pi$
$422$	−14.3834	−0.700174
$423$	0	0
$424$	−40.1737	−1.95101
$425$	−6.83643	−0.331616
$426$	0	0
$427$	−33.6540	−1.62863
$428$	5.41955	0.261964
$429$	0	0
$430$	−2.38780	−0.115150
$431$	−29.0393	−1.39877	−0.699386	−	0.714744i	$-0.746543\pi$
$431$	−29.0393	−1.39877	−0.699386	+	0.714744i	$0.746543\pi$
$432$	0	0
$433$	−19.6825	−0.945880	−0.472940	−	0.881095i	$-0.656807\pi$
$433$	−19.6825	−0.945880	−0.472940	+	0.881095i	$0.656807\pi$
$434$	−19.9733	−0.958751
$435$	0	0
$436$	2.41574	0.115693
$437$	30.9921	1.48255
$438$	0	0
$439$	−32.0714	−1.53068	−0.765341	−	0.643625i	$-0.777430\pi$
$439$	−32.0714	−1.53068	−0.765341	+	0.643625i	$0.777430\pi$
$440$	−43.1046	−2.05493
$441$	0	0
$442$	0.367755	0.0174923
$443$	−18.2343	−0.866336	−0.433168	−	0.901313i	$-0.642604\pi$
$443$	−18.2343	−0.866336	−0.433168	+	0.901313i	$0.642604\pi$
$444$	0	0
$445$	−14.4217	−0.683653
$446$	−30.8077	−1.45879
$447$	0	0
$448$	−39.8434	−1.88242
$449$	20.2550	0.955895	0.477947	−	0.878389i	$-0.341381\pi$
$449$	20.2550	0.955895	0.477947	+	0.878389i	$0.341381\pi$
$450$	0	0
$451$	4.74778	0.223564
$452$	−0.369460	−0.0173779
$453$	0	0
$454$	−22.3576	−1.04930
$455$	2.77768	0.130220
$456$	0	0
$457$	−0.933151	−0.0436509	−0.0218255	−	0.999762i	$-0.506948\pi$
$457$	−0.933151	−0.0436509	−0.0218255	+	0.999762i	$0.506948\pi$
$458$	14.2494	0.665831
$459$	0	0
$460$	−10.6693	−0.497457
$461$	26.2046	1.22047	0.610234	−	0.792221i	$-0.291075\pi$
$461$	26.2046	1.22047	0.610234	+	0.792221i	$0.291075\pi$
$462$	0	0
$463$	−21.3324	−0.991403	−0.495701	−	0.868493i	$-0.665089\pi$
$463$	−21.3324	−0.991403	−0.495701	+	0.868493i	$0.665089\pi$
$464$	−11.3555	−0.527165
$465$	0	0
$466$	20.1808	0.934859
$467$	2.71857	0.125801	0.0629003	−	0.998020i	$-0.479965\pi$
$467$	2.71857	0.125801	0.0629003	+	0.998020i	$0.479965\pi$
$468$	0	0
$469$	12.1687	0.561901
$470$	−12.5983	−0.581118
$471$	0	0
$472$	−34.2124	−1.57475
$473$	−2.77942	−0.127798
$474$	0	0
$475$	17.4500	0.800661
$476$	−2.70958	−0.124193
$477$	0	0
$478$	−1.25955	−0.0576105
$479$	8.15691	0.372699	0.186349	−	0.982484i	$-0.440334\pi$
$479$	8.15691	0.372699	0.186349	+	0.982484i	$0.440334\pi$
$480$	0	0
$481$	0.467377	0.0213106
$482$	13.9392	0.634914
$483$	0	0
$484$	−4.04793	−0.183997
$485$	47.4950	2.15664
$486$	0	0
$487$	−26.7719	−1.21315	−0.606575	−	0.795027i	$-0.707457\pi$
$487$	−26.7719	−1.21315	−0.606575	+	0.795027i	$0.707457\pi$
$488$	−22.8511	−1.03442
$489$	0	0
$490$	51.0954	2.30825
$491$	−2.95038	−0.133149	−0.0665744	−	0.997781i	$-0.521207\pi$
$491$	−2.95038	−0.133149	−0.0665744	+	0.997781i	$0.521207\pi$
$492$	0	0
$493$	−5.53609	−0.249333
$494$	−0.938695	−0.0422339
$495$	0	0
$496$	−10.6326	−0.477417
$497$	54.9058	2.46286
$498$	0	0
$499$	21.3316	0.954935	0.477468	−	0.878649i	$-0.341555\pi$
$499$	21.3316	0.954935	0.477468	+	0.878649i	$0.341555\pi$
$500$	0.422805	0.0189084
$501$	0	0
$502$	31.0824	1.38727
$503$	−4.23137	−0.188667	−0.0943337	−	0.995541i	$-0.530072\pi$
$503$	−4.23137	−0.188667	−0.0943337	+	0.995541i	$0.530072\pi$
$504$	0	0
$505$	19.6247	0.873289
$506$	47.6449	2.11807
$507$	0	0
$508$	−1.01220	−0.0449089
$509$	−16.7456	−0.742236	−0.371118	−	0.928586i	$-0.621025\pi$
$509$	−16.7456	−0.742236	−0.371118	+	0.928586i	$0.621025\pi$
$510$	0	0
$511$	−1.69219	−0.0748580
$512$	−25.1525	−1.11159
$513$	0	0
$514$	−22.8976	−1.00997
$515$	−25.4201	−1.12014
$516$	0	0
$517$	−14.6646	−0.644949
$518$	13.2110	0.580458
$519$	0	0
$520$	1.88605	0.0827089
$521$	−18.7797	−0.822751	−0.411376	−	0.911466i	$-0.634952\pi$
$521$	−18.7797	−0.822751	−0.411376	+	0.911466i	$0.634952\pi$
$522$	0	0
$523$	−0.595252	−0.0260286	−0.0130143	−	0.999915i	$-0.504143\pi$
$523$	−0.595252	−0.0260286	−0.0130143	+	0.999915i	$0.504143\pi$
$524$	4.95218	0.216337
$525$	0	0
$526$	−29.6307	−1.29196
$527$	−5.18366	−0.225804
$528$	0	0
$529$	45.8292	1.99257
$530$	51.7643	2.24850
$531$	0	0
$532$	6.91621	0.299856
$533$	−0.207740	−0.00899823
$534$	0	0
$535$	−40.7565	−1.76206
$536$	8.26262	0.356891
$537$	0	0
$538$	10.4704	0.451409
$539$	59.4756	2.56179
$540$	0	0
$541$	−36.4122	−1.56548	−0.782742	−	0.622346i	$-0.786180\pi$
$541$	−36.4122	−1.56548	−0.782742	+	0.622346i	$0.786180\pi$
$542$	−11.3167	−0.486093
$543$	0	0
$544$	−3.36440	−0.144247
$545$	−18.1670	−0.778189
$546$	0	0
$547$	−13.2489	−0.566482	−0.283241	−	0.959049i	$-0.591410\pi$
$547$	−13.2489	−0.566482	−0.283241	+	0.959049i	$0.591410\pi$
$548$	0.544041	0.0232403
$549$	0	0
$550$	26.8263	1.14388
$551$	14.1309	0.601995
$552$	0	0
$553$	4.09726	0.174233
$554$	1.41630	0.0601730
$555$	0	0
$556$	3.22391	0.136724
$557$	−7.58479	−0.321378	−0.160689	−	0.987005i	$-0.551372\pi$
$557$	−7.58479	−0.321378	−0.160689	+	0.987005i	$0.551372\pi$
$558$	0	0
$559$	0.121614	0.00514374
$560$	41.7964	1.76622
$561$	0	0
$562$	−20.1075	−0.848185
$563$	−39.4482	−1.66254	−0.831271	−	0.555867i	$-0.812386\pi$
$563$	−39.4482	−1.66254	−0.831271	+	0.555867i	$0.812386\pi$
$564$	0	0
$565$	2.77844	0.116890
$566$	10.3668	0.435750
$567$	0	0
$568$	37.2812	1.56429
$569$	−22.1854	−0.930060	−0.465030	−	0.885295i	$-0.653956\pi$
$569$	−22.1854	−0.930060	−0.465030	+	0.885295i	$0.653956\pi$
$570$	0	0
$571$	9.49231	0.397241	0.198620	−	0.980076i	$-0.436354\pi$
$571$	9.49231	0.397241	0.198620	+	0.980076i	$0.436354\pi$
$572$	0.376154	0.0157278
$573$	0	0
$574$	−5.87204	−0.245094
$575$	38.7541	1.61616
$576$	0	0
$577$	4.36130	0.181563	0.0907816	−	0.995871i	$-0.471063\pi$
$577$	4.36130	0.181563	0.0907816	+	0.995871i	$0.471063\pi$
$578$	−18.7146	−0.778423
$579$	0	0
$580$	−4.86466	−0.201994
$581$	33.2373	1.37891
$582$	0	0
$583$	60.2542	2.49547
$584$	−1.14900	−0.0475460
$585$	0	0
$586$	−35.5244	−1.46750
$587$	34.3501	1.41778	0.708891	−	0.705318i	$-0.249196\pi$
$587$	34.3501	1.41778	0.708891	+	0.705318i	$0.249196\pi$
$588$	0	0
$589$	13.2313	0.545186
$590$	44.0831	1.81487
$591$	0	0
$592$	7.03273	0.289043
$593$	31.9700	1.31285	0.656425	−	0.754392i	$-0.272068\pi$
$593$	31.9700	1.31285	0.656425	+	0.754392i	$0.272068\pi$
$594$	0	0
$595$	20.3768	0.835367
$596$	−3.99207	−0.163522
$597$	0	0
$598$	−2.08471	−0.0852503
$599$	11.1506	0.455600	0.227800	−	0.973708i	$-0.426847\pi$
$599$	11.1506	0.455600	0.227800	+	0.973708i	$0.426847\pi$
$600$	0	0
$601$	−20.8773	−0.851604	−0.425802	−	0.904816i	$-0.640008\pi$
$601$	−20.8773	−0.851604	−0.425802	+	0.904816i	$0.640008\pi$
$602$	3.43758	0.140105
$603$	0	0
$604$	1.63249	0.0664249
$605$	30.4416	1.23763
$606$	0	0
$607$	−18.3802	−0.746031	−0.373015	−	0.927825i	$-0.621676\pi$
$607$	−18.3802	−0.746031	−0.373015	+	0.927825i	$0.621676\pi$
$608$	8.58763	0.348274
$609$	0	0
$610$	29.4440	1.19215
$611$	0.641654	0.0259586
$612$	0	0
$613$	−32.3361	−1.30604	−0.653021	−	0.757340i	$-0.726499\pi$
$613$	−32.3361	−1.30604	−0.653021	+	0.757340i	$0.726499\pi$
$614$	20.6855	0.834800
$615$	0	0
$616$	62.0554	2.50028
$617$	−26.4782	−1.06597	−0.532986	−	0.846124i	$-0.678930\pi$
$617$	−26.4782	−1.06597	−0.532986	+	0.846124i	$0.678930\pi$
$618$	0	0
$619$	14.5984	0.586761	0.293380	−	0.955996i	$-0.405220\pi$
$619$	14.5984	0.586761	0.293380	+	0.955996i	$0.405220\pi$
$620$	−4.55497	−0.182932
$621$	0	0
$622$	−33.8298	−1.35645
$623$	20.7621	0.831817
$624$	0	0
$625$	−26.5358	−1.06143
$626$	−27.1491	−1.08510
$627$	0	0
$628$	−3.46186	−0.138143
$629$	3.42863	0.136709
$630$	0	0
$631$	−7.83781	−0.312018	−0.156009	−	0.987756i	$-0.549863\pi$
$631$	−7.83781	−0.312018	−0.156009	+	0.987756i	$0.549863\pi$
$632$	2.78205	0.110664
$633$	0	0
$634$	−4.64014	−0.184283
$635$	7.61199	0.302073
$636$	0	0
$637$	−2.60237	−0.103110
$638$	21.7237	0.860051
$639$	0	0
$640$	20.5610	0.812743
$641$	18.0022	0.711045	0.355523	−	0.934668i	$-0.384303\pi$
$641$	18.0022	0.711045	0.355523	+	0.934668i	$0.384303\pi$
$642$	0	0
$643$	26.8938	1.06059	0.530293	−	0.847814i	$-0.322082\pi$
$643$	26.8938	1.06059	0.530293	+	0.847814i	$0.322082\pi$
$644$	15.3600	0.605267
$645$	0	0
$646$	−6.88618	−0.270933
$647$	−47.5514	−1.86944	−0.934719	−	0.355389i	$-0.884349\pi$
$647$	−47.5514	−1.86944	−0.934719	+	0.355389i	$0.884349\pi$
$648$	0	0
$649$	51.3132	2.01422
$650$	−1.17379	−0.0460400
$651$	0	0
$652$	2.84385	0.111374
$653$	8.23592	0.322296	0.161148	−	0.986930i	$-0.448480\pi$
$653$	8.23592	0.322296	0.161148	+	0.986930i	$0.448480\pi$
$654$	0	0
$655$	−37.2417	−1.45516
$656$	−3.12592	−0.122047
$657$	0	0
$658$	18.1372	0.707060
$659$	21.0323	0.819301	0.409651	−	0.912243i	$-0.365651\pi$
$659$	21.0323	0.819301	0.409651	+	0.912243i	$0.365651\pi$
$660$	0	0
$661$	−26.4466	−1.02865	−0.514327	−	0.857594i	$-0.671958\pi$
$661$	−26.4466	−1.02865	−0.514327	+	0.857594i	$0.671958\pi$
$662$	15.6574	0.608543
$663$	0	0
$664$	22.5682	0.875817
$665$	−52.0118	−2.01693
$666$	0	0
$667$	31.3827	1.21514
$668$	−4.35623	−0.168548
$669$	0	0
$670$	−10.6465	−0.411309
$671$	34.2731	1.32310
$672$	0	0
$673$	10.6948	0.412255	0.206127	−	0.978525i	$-0.433914\pi$
$673$	10.6948	0.412255	0.206127	+	0.978525i	$0.433914\pi$
$674$	−4.84738	−0.186714
$675$	0	0
$676$	5.35943	0.206132
$677$	31.0009	1.19146	0.595731	−	0.803184i	$-0.296863\pi$
$677$	31.0009	1.19146	0.595731	+	0.803184i	$0.296863\pi$
$678$	0	0
$679$	−68.3760	−2.62403
$680$	13.8359	0.530583
$681$	0	0
$682$	20.3408	0.778889
$683$	9.79903	0.374950	0.187475	−	0.982269i	$-0.439970\pi$
$683$	9.79903	0.374950	0.187475	+	0.982269i	$0.439970\pi$
$684$	0	0
$685$	−4.09134	−0.156322
$686$	−34.0853	−1.30138
$687$	0	0
$688$	1.82996	0.0697665
$689$	−2.63644	−0.100440
$690$	0	0
$691$	14.6078	0.555706	0.277853	−	0.960624i	$-0.410377\pi$
$691$	14.6078	0.555706	0.277853	+	0.960624i	$0.410377\pi$
$692$	−7.86967	−0.299160
$693$	0	0
$694$	−37.2319	−1.41330
$695$	−24.2447	−0.919655
$696$	0	0
$697$	−1.52396	−0.0577242
$698$	15.3861	0.582372
$699$	0	0
$700$	8.64838	0.326878
$701$	29.4910	1.11386	0.556930	−	0.830559i	$-0.311979\pi$
$701$	29.4910	1.11386	0.556930	+	0.830559i	$0.311979\pi$
$702$	0	0
$703$	−8.75160	−0.330073
$704$	40.5764	1.52928
$705$	0	0
$706$	18.1681	0.683767
$707$	−28.2527	−1.06255
$708$	0	0
$709$	48.6303	1.82635	0.913174	−	0.407570i	$-0.133624\pi$
$709$	48.6303	1.82635	0.913174	+	0.407570i	$0.133624\pi$
$710$	−48.0373	−1.80281
$711$	0	0
$712$	14.0976	0.528328
$713$	29.3849	1.10047
$714$	0	0
$715$	−2.82878	−0.105790
$716$	−7.47986	−0.279535
$717$	0	0
$718$	−18.7969	−0.701493
$719$	−33.5651	−1.25177	−0.625884	−	0.779916i	$-0.715262\pi$
$719$	−33.5651	−1.25177	−0.625884	+	0.779916i	$0.715262\pi$
$720$	0	0
$721$	36.5959	1.36290
$722$	−6.35448	−0.236489
$723$	0	0
$724$	7.66463	0.284854
$725$	17.6700	0.656246
$726$	0	0
$727$	3.74648	0.138949	0.0694747	−	0.997584i	$-0.477868\pi$
$727$	3.74648	0.138949	0.0694747	+	0.997584i	$0.477868\pi$
$728$	−2.71525	−0.100634
$729$	0	0
$730$	1.48050	0.0547958
$731$	0.892151	0.0329974
$732$	0	0
$733$	−37.7977	−1.39609	−0.698045	−	0.716054i	$-0.745946\pi$
$733$	−37.7977	−1.39609	−0.698045	+	0.716054i	$0.745946\pi$
$734$	1.88440	0.0695543
$735$	0	0
$736$	19.0720	0.703002
$737$	−12.3926	−0.456488
$738$	0	0
$739$	31.0636	1.14269	0.571346	−	0.820709i	$-0.306421\pi$
$739$	31.0636	1.14269	0.571346	+	0.820709i	$0.306421\pi$
$740$	3.01280	0.110753
$741$	0	0
$742$	−74.5223	−2.73580
$743$	−37.3113	−1.36882	−0.684409	−	0.729098i	$-0.739940\pi$
$743$	−37.3113	−1.36882	−0.684409	+	0.729098i	$0.739940\pi$
$744$	0	0
$745$	30.0215	1.09990
$746$	24.4332	0.894561
$747$	0	0
$748$	2.75943	0.100895
$749$	58.6750	2.14394
$750$	0	0
$751$	44.6016	1.62754	0.813768	−	0.581190i	$-0.197413\pi$
$751$	44.6016	1.62754	0.813768	+	0.581190i	$0.197413\pi$
$752$	9.65512	0.352086
$753$	0	0
$754$	−0.950528	−0.0346162
$755$	−12.2767	−0.446796
$756$	0	0
$757$	25.4150	0.923723	0.461862	−	0.886952i	$-0.347182\pi$
$757$	25.4150	0.923723	0.461862	+	0.886952i	$0.347182\pi$
$758$	31.8227	1.15585
$759$	0	0
$760$	−35.3162	−1.28105
$761$	−9.76708	−0.354056	−0.177028	−	0.984206i	$-0.556648\pi$
$761$	−9.76708	−0.354056	−0.177028	+	0.984206i	$0.556648\pi$
$762$	0	0
$763$	26.1541	0.946841
$764$	−2.87459	−0.103999
$765$	0	0
$766$	−17.8718	−0.645735
$767$	−2.24522	−0.0810703
$768$	0	0
$769$	−31.0668	−1.12030	−0.560148	−	0.828393i	$-0.689256\pi$
$769$	−31.0668	−1.12030	−0.560148	+	0.828393i	$0.689256\pi$
$770$	−79.9590	−2.88152
$771$	0	0
$772$	−8.28051	−0.298022
$773$	−34.3852	−1.23675	−0.618374	−	0.785884i	$-0.712208\pi$
$773$	−34.3852	−1.23675	−0.618374	+	0.785884i	$0.712208\pi$
$774$	0	0
$775$	16.5451	0.594317
$776$	−46.4276	−1.66665
$777$	0	0
$778$	14.7482	0.528748
$779$	3.88992	0.139371
$780$	0	0
$781$	−55.9160	−2.00083
$782$	−15.2933	−0.546886
$783$	0	0
$784$	−39.1585	−1.39852
$785$	26.0341	0.929199
$786$	0	0
$787$	3.92388	0.139871	0.0699356	−	0.997552i	$-0.477721\pi$
$787$	3.92388	0.139871	0.0699356	+	0.997552i	$0.477721\pi$
$788$	4.48108	0.159632
$789$	0	0
$790$	−3.58470	−0.127538
$791$	−3.99997	−0.142223
$792$	0	0
$793$	−1.49963	−0.0532534
$794$	2.58438	0.0917162
$795$	0	0
$796$	−7.08380	−0.251079
$797$	−41.3822	−1.46583	−0.732916	−	0.680320i	$-0.761841\pi$
$797$	−41.3822	−1.46583	−0.732916	+	0.680320i	$0.761841\pi$
$798$	0	0
$799$	4.70712	0.166526
$800$	10.7384	0.379660
$801$	0	0
$802$	46.0782	1.62708
$803$	1.72332	0.0608147
$804$	0	0
$805$	−115.511	−4.07123
$806$	−0.890017	−0.0313495
$807$	0	0
$808$	−19.1837	−0.674879
$809$	11.2741	0.396377	0.198188	−	0.980164i	$-0.436494\pi$
$809$	11.2741	0.396377	0.198188	+	0.980164i	$0.436494\pi$
$810$	0	0
$811$	16.6758	0.585568	0.292784	−	0.956179i	$-0.405418\pi$
$811$	16.6758	0.585568	0.292784	+	0.956179i	$0.405418\pi$
$812$	7.00339	0.245771
$813$	0	0
$814$	−13.4540	−0.471564
$815$	−21.3866	−0.749139
$816$	0	0
$817$	−2.27722	−0.0796698
$818$	13.8236	0.483330
$819$	0	0
$820$	−1.33913	−0.0467646
$821$	−35.8361	−1.25069	−0.625344	−	0.780349i	$-0.715041\pi$
$821$	−35.8361	−1.25069	−0.625344	+	0.780349i	$0.715041\pi$
$822$	0	0
$823$	−26.5404	−0.925142	−0.462571	−	0.886582i	$-0.653073\pi$
$823$	−26.5404	−0.925142	−0.462571	+	0.886582i	$0.653073\pi$
$824$	24.8487	0.865647
$825$	0	0
$826$	−63.4640	−2.20820
$827$	34.7410	1.20806	0.604031	−	0.796961i	$-0.293560\pi$
$827$	34.7410	1.20806	0.604031	+	0.796961i	$0.293560\pi$
$828$	0	0
$829$	30.4494	1.05755	0.528776	−	0.848761i	$-0.322651\pi$
$829$	30.4494	1.05755	0.528776	+	0.848761i	$0.322651\pi$
$830$	−29.0794	−1.00936
$831$	0	0
$832$	−1.77543	−0.0615520
$833$	−19.0907	−0.661455
$834$	0	0
$835$	32.7601	1.13371
$836$	−7.04345	−0.243603
$837$	0	0
$838$	−8.58205	−0.296462
$839$	10.4551	0.360949	0.180475	−	0.983580i	$-0.442237\pi$
$839$	10.4551	0.360949	0.180475	+	0.983580i	$0.442237\pi$
$840$	0	0
$841$	−14.6910	−0.506586
$842$	47.9779	1.65343
$843$	0	0
$844$	4.72229	0.162548
$845$	−40.3044	−1.38651
$846$	0	0
$847$	−43.8251	−1.50585
$848$	−39.6711	−1.36231
$849$	0	0
$850$	−8.61084	−0.295349
$851$	−19.4361	−0.666261
$852$	0	0
$853$	34.3924	1.17757	0.588787	−	0.808288i	$-0.299606\pi$
$853$	34.3924	1.17757	0.588787	+	0.808288i	$0.299606\pi$
$854$	−42.3889	−1.45052
$855$	0	0
$856$	39.8405	1.36172
$857$	20.3003	0.693446	0.346723	−	0.937968i	$-0.387294\pi$
$857$	20.3003	0.693446	0.346723	+	0.937968i	$0.387294\pi$
$858$	0	0
$859$	−37.0169	−1.26300	−0.631500	−	0.775376i	$-0.717560\pi$
$859$	−37.0169	−1.26300	−0.631500	+	0.775376i	$0.717560\pi$
$860$	0.783950	0.0267325
$861$	0	0
$862$	−36.5764	−1.24580
$863$	13.6389	0.464274	0.232137	−	0.972683i	$-0.425428\pi$
$863$	13.6389	0.464274	0.232137	+	0.972683i	$0.425428\pi$
$864$	0	0
$865$	59.1821	2.01225
$866$	−24.7911	−0.842436
$867$	0	0
$868$	6.55755	0.222578
$869$	−4.17264	−0.141547
$870$	0	0
$871$	0.542242	0.0183732
$872$	17.7587	0.601385
$873$	0	0
$874$	39.0361	1.32042
$875$	4.57751	0.154748
$876$	0	0
$877$	−24.1630	−0.815927	−0.407964	−	0.912998i	$-0.633761\pi$
$877$	−24.1630	−0.815927	−0.407964	+	0.912998i	$0.633761\pi$
$878$	−40.3955	−1.36328
$879$	0	0
$880$	−42.5653	−1.43488
$881$	−12.1619	−0.409746	−0.204873	−	0.978789i	$-0.565678\pi$
$881$	−12.1619	−0.409746	−0.204873	+	0.978789i	$0.565678\pi$
$882$	0	0
$883$	47.1676	1.58732	0.793659	−	0.608363i	$-0.208174\pi$
$883$	47.1676	1.58732	0.793659	+	0.608363i	$0.208174\pi$
$884$	−0.120740	−0.00406091
$885$	0	0
$886$	−22.9670	−0.771591
$887$	7.02348	0.235825	0.117913	−	0.993024i	$-0.462380\pi$
$887$	7.02348	0.235825	0.117913	+	0.993024i	$0.462380\pi$
$888$	0	0
$889$	−10.9586	−0.367539
$890$	−18.1649	−0.608887
$891$	0	0
$892$	10.1146	0.338663
$893$	−12.0149	−0.402064
$894$	0	0
$895$	56.2506	1.88025
$896$	−29.6005	−0.988884
$897$	0	0
$898$	25.5123	0.851356
$899$	13.3981	0.446851
$900$	0	0
$901$	−19.3407	−0.644331
$902$	5.98007	0.199115
$903$	0	0
$904$	−2.71600	−0.0903327
$905$	−57.6402	−1.91602
$906$	0	0
$907$	14.1521	0.469913	0.234957	−	0.972006i	$-0.424505\pi$
$907$	14.1521	0.469913	0.234957	+	0.972006i	$0.424505\pi$
$908$	7.34035	0.243598
$909$	0	0
$910$	3.49863	0.115978
$911$	−3.04227	−0.100795	−0.0503975	−	0.998729i	$-0.516049\pi$
$911$	−3.04227	−0.100795	−0.0503975	+	0.998729i	$0.516049\pi$
$912$	0	0
$913$	−33.8488	−1.12023
$914$	−1.17535	−0.0388772
$915$	0	0
$916$	−4.67829	−0.154575
$917$	53.6149	1.77052
$918$	0	0
$919$	−56.1637	−1.85267	−0.926334	−	0.376703i	$-0.877058\pi$
$919$	−56.1637	−1.85267	−0.926334	+	0.376703i	$0.877058\pi$
$920$	−78.4324	−2.58584
$921$	0	0
$922$	33.0060	1.08699
$923$	2.44662	0.0805314
$924$	0	0
$925$	−10.9434	−0.359818
$926$	−26.8693	−0.882980
$927$	0	0
$928$	8.69588	0.285456
$929$	5.30442	0.174032	0.0870162	−	0.996207i	$-0.472267\pi$
$929$	5.30442	0.174032	0.0870162	+	0.996207i	$0.472267\pi$
$930$	0	0
$931$	48.7292	1.59703
$932$	−6.62568	−0.217031
$933$	0	0
$934$	3.42418	0.112043
$935$	−20.7517	−0.678652
$936$	0	0
$937$	17.6569	0.576824	0.288412	−	0.957506i	$-0.406873\pi$
$937$	17.6569	0.576824	0.288412	+	0.957506i	$0.406873\pi$
$938$	15.3272	0.500450
$939$	0	0
$940$	4.13623	0.134909
$941$	22.1265	0.721305	0.360652	−	0.932700i	$-0.382554\pi$
$941$	22.1265	0.721305	0.360652	+	0.932700i	$0.382554\pi$
$942$	0	0
$943$	8.63899	0.281324
$944$	−33.7844	−1.09959
$945$	0	0
$946$	−3.50082	−0.113822
$947$	57.2551	1.86054	0.930271	−	0.366873i	$-0.119572\pi$
$947$	57.2551	1.86054	0.930271	+	0.366873i	$0.119572\pi$
$948$	0	0
$949$	−0.0754044	−0.00244773
$950$	21.9792	0.713099
$951$	0	0
$952$	−19.9188	−0.645572
$953$	−21.3996	−0.693202	−0.346601	−	0.938013i	$-0.612664\pi$
$953$	−21.3996	−0.693202	−0.346601	+	0.938013i	$0.612664\pi$
$954$	0	0
$955$	21.6177	0.699534
$956$	0.413530	0.0133745
$957$	0	0
$958$	10.2741	0.331940
$959$	5.89009	0.190201
$960$	0	0
$961$	−18.4548	−0.595318
$962$	0.588685	0.0189800
$963$	0	0
$964$	−4.57646	−0.147398
$965$	62.2717	2.00460
$966$	0	0
$967$	20.9679	0.674281	0.337140	−	0.941454i	$-0.390540\pi$
$967$	20.9679	0.674281	0.337140	+	0.941454i	$0.390540\pi$
$968$	−29.7574	−0.956439
$969$	0	0
$970$	59.8224	1.92078
$971$	43.8466	1.40710	0.703552	−	0.710644i	$-0.251596\pi$
$971$	43.8466	1.40710	0.703552	+	0.710644i	$0.251596\pi$
$972$	0	0
$973$	34.9038	1.11897
$974$	−33.7206	−1.08048
$975$	0	0
$976$	−22.5653	−0.722296
$977$	−9.95507	−0.318491	−0.159245	−	0.987239i	$-0.550906\pi$
$977$	−9.95507	−0.318491	−0.159245	+	0.987239i	$0.550906\pi$
$978$	0	0
$979$	−21.1441	−0.675768
$980$	−16.7754	−0.535870
$981$	0	0
$982$	−3.71616	−0.118587
$983$	57.7272	1.84121	0.920605	−	0.390495i	$-0.127696\pi$
$983$	57.7272	1.84121	0.920605	+	0.390495i	$0.127696\pi$
$984$	0	0
$985$	−33.6990	−1.07374
$986$	−6.97298	−0.222065
$987$	0	0
$988$	0.308188	0.00980477
$989$	−5.05739	−0.160816
$990$	0	0
$991$	−5.86776	−0.186396	−0.0931978	−	0.995648i	$-0.529709\pi$
$991$	−5.86776	−0.186396	−0.0931978	+	0.995648i	$0.529709\pi$
$992$	8.14230	0.258518
$993$	0	0
$994$	69.1567	2.19352
$995$	53.2722	1.68884
$996$	0	0
$997$	44.4483	1.40769	0.703846	−	0.710353i	$-0.251465\pi$
$997$	44.4483	1.40769	0.703846	+	0.710353i	$0.251465\pi$
$998$	26.8683	0.850501
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.i.1.12		17
3.2	odd	2		239.2.a.b.1.6	✓	17
12.11	even	2		3824.2.a.p.1.16		17
15.14	odd	2		5975.2.a.g.1.12		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.b.1.6	✓	17	3.2	odd	2
2151.2.a.i.1.12		17	1.1	even	1	trivial
3824.2.a.p.1.16		17	12.11	even	2
5975.2.a.g.1.12		17	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.25955 q^{2} -0.413530 q^{4} +3.10986 q^{5} -4.47710 q^{7} -3.03997 q^{8} +O(q^{10})\)	\(q+1.25955 q^{2} -0.413530 q^{4} +3.10986 q^{5} -4.47710 q^{7} -3.03997 q^{8} +3.91703 q^{10} +4.55947 q^{11} -0.199501 q^{13} -5.63914 q^{14} -3.00193 q^{16} -1.46352 q^{17} +3.73563 q^{19} -1.28602 q^{20} +5.74288 q^{22} +8.29633 q^{23} +4.67123 q^{25} -0.251281 q^{26} +1.85141 q^{28} +3.78272 q^{29} +3.54191 q^{31} +2.29884 q^{32} -1.84338 q^{34} -13.9232 q^{35} -2.34273 q^{37} +4.70522 q^{38} -9.45387 q^{40} +1.04130 q^{41} -0.609594 q^{43} -1.88548 q^{44} +10.4497 q^{46} -3.21630 q^{47} +13.0444 q^{49} +5.88365 q^{50} +0.0824995 q^{52} +13.2152 q^{53} +14.1793 q^{55} +13.6102 q^{56} +4.76454 q^{58} +11.2542 q^{59} +7.51691 q^{61} +4.46122 q^{62} +8.89938 q^{64} -0.620419 q^{65} -2.71800 q^{67} +0.605209 q^{68} -17.5369 q^{70} -12.2637 q^{71} +0.377965 q^{73} -2.95079 q^{74} -1.54480 q^{76} -20.4132 q^{77} -0.915159 q^{79} -9.33559 q^{80} +1.31157 q^{82} -7.42384 q^{83} -4.55134 q^{85} -0.767815 q^{86} -13.8606 q^{88} -4.63741 q^{89} +0.893184 q^{91} -3.43078 q^{92} -4.05110 q^{94} +11.6173 q^{95} +15.2724 q^{97} +16.4301 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8	\( 17 q + 22 q^{4} - 6 q^{5} + 5 q^{7} + 3 q^{8} + 5 q^{10} + q^{11} + 15 q^{13} + 3 q^{14} + 24 q^{16} - 4 q^{17} + 24 q^{19} - 4 q^{20} - 10 q^{22} + 9 q^{23} + 39 q^{25} + 12 q^{26} - 7 q^{28} + 2 q^{29} + 28 q^{31} + 31 q^{32} + 29 q^{34} + 24 q^{35} + 11 q^{37} + 19 q^{38} - 18 q^{40} - 20 q^{41} - 9 q^{43} + 43 q^{44} - 18 q^{46} + 18 q^{47} + 60 q^{49} + 61 q^{50} - q^{52} + 12 q^{53} - 10 q^{55} + 60 q^{56} - 38 q^{58} - q^{59} + 24 q^{61} + 33 q^{62} + 21 q^{64} - 2 q^{65} + 16 q^{67} + 10 q^{68} + 7 q^{70} - 12 q^{71} + 30 q^{73} + 21 q^{74} + 75 q^{76} + 15 q^{77} - 10 q^{79} - 32 q^{80} + 50 q^{82} + 16 q^{83} - 18 q^{85} + 3 q^{86} - 28 q^{88} - 65 q^{89} + 47 q^{91} - 24 q^{92} + 32 q^{94} + 37 q^{95} + 87 q^{97} + 9 q^{98}+O(q^{100}) \) 17 * q + 22 * q^4 - 6 * q^5 + 5 * q^7 + 3 * q^8 + 5 * q^10 + q^11 + 15 * q^13 + 3 * q^14 + 24 * q^16 - 4 * q^17 + 24 * q^19 - 4 * q^20 - 10 * q^22 + 9 * q^23 + 39 * q^25 + 12 * q^26 - 7 * q^28 + 2 * q^29 + 28 * q^31 + 31 * q^32 + 29 * q^34 + 24 * q^35 + 11 * q^37 + 19 * q^38 - 18 * q^40 - 20 * q^41 - 9 * q^43 + 43 * q^44 - 18 * q^46 + 18 * q^47 + 60 * q^49 + 61 * q^50 - q^52 + 12 * q^53 - 10 * q^55 + 60 * q^56 - 38 * q^58 - q^59 + 24 * q^61 + 33 * q^62 + 21 * q^64 - 2 * q^65 + 16 * q^67 + 10 * q^68 + 7 * q^70 - 12 * q^71 + 30 * q^73 + 21 * q^74 + 75 * q^76 + 15 * q^77 - 10 * q^79 - 32 * q^80 + 50 * q^82 + 16 * q^83 - 18 * q^85 + 3 * q^86 - 28 * q^88 - 65 * q^89 + 47 * q^91 - 24 * q^92 + 32 * q^94 + 37 * q^95 + 87 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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