LMFDB - Embedded newform 2151.2.a.d.1.3

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.80194 q^{2} +1.24698 q^{4} -0.246980 q^{5} -1.00000 q^{7} -1.35690 q^{8} +O(q^{10})$	\(q+1.80194 q^{2} +1.24698 q^{4} -0.246980 q^{5} -1.00000 q^{7} -1.35690 q^{8} -0.445042 q^{10} -1.24698 q^{11} -0.753020 q^{13} -1.80194 q^{14} -4.93900 q^{16} +2.24698 q^{17} -6.15883 q^{19} -0.307979 q^{20} -2.24698 q^{22} +2.04892 q^{23} -4.93900 q^{25} -1.35690 q^{26} -1.24698 q^{28} +6.67994 q^{29} -2.89008 q^{31} -6.18598 q^{32} +4.04892 q^{34} +0.246980 q^{35} -5.40581 q^{37} -11.0978 q^{38} +0.335126 q^{40} -9.39373 q^{41} +0.198062 q^{43} -1.55496 q^{44} +3.69202 q^{46} -8.67994 q^{47} -6.00000 q^{49} -8.89977 q^{50} -0.939001 q^{52} -5.91185 q^{53} +0.307979 q^{55} +1.35690 q^{56} +12.0368 q^{58} +0.939001 q^{59} -7.16421 q^{61} -5.20775 q^{62} -1.26875 q^{64} +0.185981 q^{65} +12.0368 q^{67} +2.80194 q^{68} +0.445042 q^{70} +2.02715 q^{71} +2.00969 q^{73} -9.74094 q^{74} -7.67994 q^{76} +1.24698 q^{77} +5.87800 q^{79} +1.21983 q^{80} -16.9269 q^{82} +0.670251 q^{83} -0.554958 q^{85} +0.356896 q^{86} +1.69202 q^{88} +15.3177 q^{89} +0.753020 q^{91} +2.55496 q^{92} -15.6407 q^{94} +1.52111 q^{95} -2.47889 q^{97} -10.8116 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} - q^{10} + q^{11} - 7 q^{13} - q^{14} - 5 q^{16} + 2 q^{17} - 10 q^{19} - 6 q^{20} - 2 q^{22} - 3 q^{23} - 5 q^{25} + q^{28} - 4 q^{29} - 8 q^{31} - 4 q^{32} + 3 q^{34} - 4 q^{35} - 3 q^{37} - 15 q^{38} + 4 q^{41} + 5 q^{43} - 5 q^{44} + 6 q^{46} - 2 q^{47} - 18 q^{49} - 4 q^{50} + 7 q^{52} - 14 q^{53} + 6 q^{55} + 8 q^{58} - 7 q^{59} - 10 q^{61} + 2 q^{62} + 4 q^{64} - 14 q^{65} + 8 q^{67} + 4 q^{68} + q^{70} - 16 q^{73} - 15 q^{74} + q^{76} - q^{77} - 2 q^{79} + 5 q^{80} - 22 q^{82} - 2 q^{85} - 3 q^{86} + 29 q^{89} + 7 q^{91} + 8 q^{92} - 10 q^{94} - 11 q^{95} - 23 q^{97} - 6 q^{98}+O(q^{100}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 - q^10 + q^11 - 7 * q^13 - q^14 - 5 * q^16 + 2 * q^17 - 10 * q^19 - 6 * q^20 - 2 * q^22 - 3 * q^23 - 5 * q^25 + q^28 - 4 * q^29 - 8 * q^31 - 4 * q^32 + 3 * q^34 - 4 * q^35 - 3 * q^37 - 15 * q^38 + 4 * q^41 + 5 * q^43 - 5 * q^44 + 6 * q^46 - 2 * q^47 - 18 * q^49 - 4 * q^50 + 7 * q^52 - 14 * q^53 + 6 * q^55 + 8 * q^58 - 7 * q^59 - 10 * q^61 + 2 * q^62 + 4 * q^64 - 14 * q^65 + 8 * q^67 + 4 * q^68 + q^70 - 16 * q^73 - 15 * q^74 + q^76 - q^77 - 2 * q^79 + 5 * q^80 - 22 * q^82 - 2 * q^85 - 3 * q^86 + 29 * q^89 + 7 * q^91 + 8 * q^92 - 10 * q^94 - 11 * q^95 - 23 * q^97 - 6 * 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 $

$2$	1.80194	1.27416	0.637081	−	0.770797i	$-0.280142\pi$
$2$	1.80194	1.27416	0.637081	+	0.770797i	$0.280142\pi$
$3$	0	0
$3$	0	0
$4$	1.24698	0.623490
$5$	−0.246980	−0.110453	−0.0552263	−	0.998474i	$-0.517588\pi$
$5$	−0.246980	−0.110453	−0.0552263	+	0.998474i	$0.517588\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.35690	−0.479735
$9$	0	0
$10$	−0.445042	−0.140735
$11$	−1.24698	−0.375978	−0.187989	−	0.982171i	$-0.560197\pi$
$11$	−1.24698	−0.375978	−0.187989	+	0.982171i	$0.560197\pi$
$12$	0	0
$13$	−0.753020	−0.208850	−0.104425	−	0.994533i	$-0.533300\pi$
$13$	−0.753020	−0.208850	−0.104425	+	0.994533i	$0.533300\pi$
$14$	−1.80194	−0.481588
$15$	0	0
$16$	−4.93900	−1.23475
$17$	2.24698	0.544973	0.272486	−	0.962160i	$-0.412154\pi$
$17$	2.24698	0.544973	0.272486	+	0.962160i	$0.412154\pi$
$18$	0	0
$19$	−6.15883	−1.41293	−0.706467	−	0.707746i	$-0.749712\pi$
$19$	−6.15883	−1.41293	−0.706467	+	0.707746i	$0.749712\pi$
$20$	−0.307979	−0.0688661
$21$	0	0
$22$	−2.24698	−0.479058
$23$	2.04892	0.427229	0.213614	−	0.976918i	$-0.431476\pi$
$23$	2.04892	0.427229	0.213614	+	0.976918i	$0.431476\pi$
$24$	0	0
$25$	−4.93900	−0.987800
$26$	−1.35690	−0.266109
$27$	0	0
$28$	−1.24698	−0.235657
$29$	6.67994	1.24043	0.620217	−	0.784430i	$-0.287045\pi$
$29$	6.67994	1.24043	0.620217	+	0.784430i	$0.287045\pi$
$30$	0	0
$31$	−2.89008	−0.519074	−0.259537	−	0.965733i	$-0.583570\pi$
$31$	−2.89008	−0.519074	−0.259537	+	0.965733i	$0.583570\pi$
$32$	−6.18598	−1.09354
$33$	0	0
$34$	4.04892	0.694384
$35$	0.246980	0.0417472
$36$	0	0
$37$	−5.40581	−0.888710	−0.444355	−	0.895851i	$-0.646567\pi$
$37$	−5.40581	−0.888710	−0.444355	+	0.895851i	$0.646567\pi$
$38$	−11.0978	−1.80031
$39$	0	0
$40$	0.335126	0.0529880
$41$	−9.39373	−1.46705	−0.733527	−	0.679660i	$-0.762127\pi$
$41$	−9.39373	−1.46705	−0.733527	+	0.679660i	$0.762127\pi$
$42$	0	0
$43$	0.198062	0.0302042	0.0151021	−	0.999886i	$-0.495193\pi$
$43$	0.198062	0.0302042	0.0151021	+	0.999886i	$0.495193\pi$
$44$	−1.55496	−0.234419
$45$	0	0
$46$	3.69202	0.544359
$47$	−8.67994	−1.26610	−0.633050	−	0.774111i	$-0.718197\pi$
$47$	−8.67994	−1.26610	−0.633050	+	0.774111i	$0.718197\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−8.89977	−1.25862
$51$	0	0
$52$	−0.939001	−0.130216
$53$	−5.91185	−0.812056	−0.406028	−	0.913861i	$-0.633086\pi$
$53$	−5.91185	−0.812056	−0.406028	+	0.913861i	$0.633086\pi$
$54$	0	0
$55$	0.307979	0.0415278
$56$	1.35690	0.181323
$57$	0	0
$58$	12.0368	1.58051
$59$	0.939001	0.122248	0.0611238	−	0.998130i	$-0.480532\pi$
$59$	0.939001	0.122248	0.0611238	+	0.998130i	$0.480532\pi$
$60$	0	0
$61$	−7.16421	−0.917283	−0.458642	−	0.888621i	$-0.651664\pi$
$61$	−7.16421	−0.917283	−0.458642	+	0.888621i	$0.651664\pi$
$62$	−5.20775	−0.661385
$63$	0	0
$64$	−1.26875	−0.158594
$65$	0.185981	0.0230681
$66$	0	0
$67$	12.0368	1.47053	0.735267	−	0.677778i	$-0.237057\pi$
$67$	12.0368	1.47053	0.735267	+	0.677778i	$0.237057\pi$
$68$	2.80194	0.339785
$69$	0	0
$70$	0.445042	0.0531927
$71$	2.02715	0.240578	0.120289	−	0.992739i	$-0.461618\pi$
$71$	2.02715	0.240578	0.120289	+	0.992739i	$0.461618\pi$
$72$	0	0
$73$	2.00969	0.235216	0.117608	−	0.993060i	$-0.462477\pi$
$73$	2.00969	0.235216	0.117608	+	0.993060i	$0.462477\pi$
$74$	−9.74094	−1.13236
$75$	0	0
$76$	−7.67994	−0.880950
$77$	1.24698	0.142107
$78$	0	0
$79$	5.87800	0.661327	0.330663	−	0.943749i	$-0.392728\pi$
$79$	5.87800	0.661327	0.330663	+	0.943749i	$0.392728\pi$
$80$	1.21983	0.136381
$81$	0	0
$82$	−16.9269	−1.86927
$83$	0.670251	0.0735696	0.0367848	−	0.999323i	$-0.488288\pi$
$83$	0.670251	0.0735696	0.0367848	+	0.999323i	$0.488288\pi$
$84$	0	0
$85$	−0.554958	−0.0601937
$86$	0.356896	0.0384851
$87$	0	0
$88$	1.69202	0.180370
$89$	15.3177	1.62367	0.811835	−	0.583887i	$-0.198469\pi$
$89$	15.3177	1.62367	0.811835	+	0.583887i	$0.198469\pi$
$90$	0	0
$91$	0.753020	0.0789380
$92$	2.55496	0.266373
$93$	0	0
$94$	−15.6407	−1.61322
$95$	1.52111	0.156062
$96$	0	0
$97$	−2.47889	−0.251694	−0.125847	−	0.992050i	$-0.540165\pi$
$97$	−2.47889	−0.251694	−0.125847	+	0.992050i	$0.540165\pi$
$98$	−10.8116	−1.09214
$99$	0	0
$100$	−6.15883	−0.615883
$101$	10.0978	1.00477	0.502386	−	0.864643i	$-0.332456\pi$
$101$	10.0978	1.00477	0.502386	+	0.864643i	$0.332456\pi$
$102$	0	0
$103$	1.49396	0.147204	0.0736021	−	0.997288i	$-0.476551\pi$
$103$	1.49396	0.147204	0.0736021	+	0.997288i	$0.476551\pi$
$104$	1.02177	0.100193
$105$	0	0
$106$	−10.6528	−1.03469
$107$	−3.25236	−0.314417	−0.157209	−	0.987565i	$-0.550249\pi$
$107$	−3.25236	−0.314417	−0.157209	+	0.987565i	$0.550249\pi$
$108$	0	0
$109$	−11.4276	−1.09456	−0.547282	−	0.836948i	$-0.684337\pi$
$109$	−11.4276	−1.09456	−0.547282	+	0.836948i	$0.684337\pi$
$110$	0.554958	0.0529132
$111$	0	0
$112$	4.93900	0.466692
$113$	−1.80194	−0.169512	−0.0847560	−	0.996402i	$-0.527011\pi$
$113$	−1.80194	−0.169512	−0.0847560	+	0.996402i	$0.527011\pi$
$114$	0	0
$115$	−0.506041	−0.0471885
$116$	8.32975	0.773398
$117$	0	0
$118$	1.69202	0.155763
$119$	−2.24698	−0.205980
$120$	0	0
$121$	−9.44504	−0.858640
$122$	−12.9095	−1.16877
$123$	0	0
$124$	−3.60388	−0.323638
$125$	2.45473	0.219558
$126$	0	0
$127$	−0.823708	−0.0730923	−0.0365461	−	0.999332i	$-0.511636\pi$
$127$	−0.823708	−0.0730923	−0.0365461	+	0.999332i	$0.511636\pi$
$128$	10.0858	0.891463
$129$	0	0
$130$	0.335126	0.0293925
$131$	−3.31336	−0.289489	−0.144745	−	0.989469i	$-0.546236\pi$
$131$	−3.31336	−0.289489	−0.144745	+	0.989469i	$0.546236\pi$
$132$	0	0
$133$	6.15883	0.534039
$134$	21.6896	1.87370
$135$	0	0
$136$	−3.04892	−0.261443
$137$	1.07606	0.0919344	0.0459672	−	0.998943i	$-0.485363\pi$
$137$	1.07606	0.0919344	0.0459672	+	0.998943i	$0.485363\pi$
$138$	0	0
$139$	10.3274	0.875955	0.437977	−	0.898986i	$-0.355695\pi$
$139$	10.3274	0.875955	0.437977	+	0.898986i	$0.355695\pi$
$140$	0.307979	0.0260289
$141$	0	0
$142$	3.65279	0.306536
$143$	0.939001	0.0785232
$144$	0	0
$145$	−1.64981	−0.137009
$146$	3.62133	0.299704
$147$	0	0
$148$	−6.74094	−0.554102
$149$	−12.5483	−1.02799	−0.513996	−	0.857792i	$-0.671836\pi$
$149$	−12.5483	−1.02799	−0.513996	+	0.857792i	$0.671836\pi$
$150$	0	0
$151$	21.3274	1.73560	0.867798	−	0.496917i	$-0.165535\pi$
$151$	21.3274	1.73560	0.867798	+	0.496917i	$0.165535\pi$
$152$	8.35690	0.677834
$153$	0	0
$154$	2.24698	0.181067
$155$	0.713792	0.0573331
$156$	0	0
$157$	−7.30127	−0.582705	−0.291353	−	0.956616i	$-0.594105\pi$
$157$	−7.30127	−0.582705	−0.291353	+	0.956616i	$0.594105\pi$
$158$	10.5918	0.842638
$159$	0	0
$160$	1.52781	0.120784
$161$	−2.04892	−0.161477
$162$	0	0
$163$	−7.17390	−0.561903	−0.280952	−	0.959722i	$-0.590650\pi$
$163$	−7.17390	−0.561903	−0.280952	+	0.959722i	$0.590650\pi$
$164$	−11.7138	−0.914693
$165$	0	0
$166$	1.20775	0.0937397
$167$	10.4940	0.812047	0.406023	−	0.913863i	$-0.366915\pi$
$167$	10.4940	0.812047	0.406023	+	0.913863i	$0.366915\pi$
$168$	0	0
$169$	−12.4330	−0.956382
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	0.246980	0.0188320
$173$	−25.1618	−1.91302	−0.956509	−	0.291704i	$-0.905778\pi$
$173$	−25.1618	−1.91302	−0.956509	+	0.291704i	$0.905778\pi$
$174$	0	0
$175$	4.93900	0.373353
$176$	6.15883	0.464240
$177$	0	0
$178$	27.6015	2.06882
$179$	7.82908	0.585173	0.292587	−	0.956239i	$-0.405484\pi$
$179$	7.82908	0.585173	0.292587	+	0.956239i	$0.405484\pi$
$180$	0	0
$181$	15.4795	1.15058	0.575291	−	0.817949i	$-0.304889\pi$
$181$	15.4795	1.15058	0.575291	+	0.817949i	$0.304889\pi$
$182$	1.35690	0.100580
$183$	0	0
$184$	−2.78017	−0.204957
$185$	1.33513	0.0981604
$186$	0	0
$187$	−2.80194	−0.204898
$188$	−10.8237	−0.789400
$189$	0	0
$190$	2.74094	0.198849
$191$	9.22282	0.667340	0.333670	−	0.942690i	$-0.391713\pi$
$191$	9.22282	0.667340	0.333670	+	0.942690i	$0.391713\pi$
$192$	0	0
$193$	9.54288	0.686911	0.343456	−	0.939169i	$-0.388402\pi$
$193$	9.54288	0.686911	0.343456	+	0.939169i	$0.388402\pi$
$194$	−4.46681	−0.320698
$195$	0	0
$196$	−7.48188	−0.534420
$197$	−11.1250	−0.792622	−0.396311	−	0.918116i	$-0.629710\pi$
$197$	−11.1250	−0.792622	−0.396311	+	0.918116i	$0.629710\pi$
$198$	0	0
$199$	−8.06829	−0.571946	−0.285973	−	0.958238i	$-0.592317\pi$
$199$	−8.06829	−0.571946	−0.285973	+	0.958238i	$0.592317\pi$
$200$	6.70171	0.473882
$201$	0	0
$202$	18.1957	1.28024
$203$	−6.67994	−0.468840
$204$	0	0
$205$	2.32006	0.162040
$206$	2.69202	0.187562
$207$	0	0
$208$	3.71917	0.257878
$209$	7.67994	0.531233
$210$	0	0
$211$	11.6679	0.803248	0.401624	−	0.915805i	$-0.368446\pi$
$211$	11.6679	0.803248	0.401624	+	0.915805i	$0.368446\pi$
$212$	−7.37196	−0.506308
$213$	0	0
$214$	−5.86054	−0.400619
$215$	−0.0489173	−0.00333613
$216$	0	0
$217$	2.89008	0.196192
$218$	−20.5918	−1.39465
$219$	0	0
$220$	0.384043	0.0258922
$221$	−1.69202	−0.113818
$222$	0	0
$223$	2.57673	0.172550	0.0862752	−	0.996271i	$-0.472504\pi$
$223$	2.57673	0.172550	0.0862752	+	0.996271i	$0.472504\pi$
$224$	6.18598	0.413318
$225$	0	0
$226$	−3.24698	−0.215986
$227$	8.76032	0.581443	0.290721	−	0.956808i	$-0.406105\pi$
$227$	8.76032	0.581443	0.290721	+	0.956808i	$0.406105\pi$
$228$	0	0
$229$	7.87800	0.520593	0.260297	−	0.965529i	$-0.416180\pi$
$229$	7.87800	0.520593	0.260297	+	0.965529i	$0.416180\pi$
$230$	−0.911854	−0.0601259
$231$	0	0
$232$	−9.06398	−0.595080
$233$	27.6058	1.80852	0.904258	−	0.426987i	$-0.140425\pi$
$233$	27.6058	1.80852	0.904258	+	0.426987i	$0.140425\pi$
$234$	0	0
$235$	2.14377	0.139844
$236$	1.17092	0.0762201
$237$	0	0
$238$	−4.04892	−0.262452
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−17.5593	−1.13109	−0.565546	−	0.824717i	$-0.691335\pi$
$241$	−17.5593	−1.13109	−0.565546	+	0.824717i	$0.691335\pi$
$242$	−17.0194	−1.09405
$243$	0	0
$244$	−8.93362	−0.571917
$245$	1.48188	0.0946737
$246$	0	0
$247$	4.63773	0.295092
$248$	3.92154	0.249018
$249$	0	0
$250$	4.42327	0.279752
$251$	8.41550	0.531182	0.265591	−	0.964086i	$-0.414433\pi$
$251$	8.41550	0.531182	0.265591	+	0.964086i	$0.414433\pi$
$252$	0	0
$253$	−2.55496	−0.160629
$254$	−1.48427	−0.0931314
$255$	0	0
$256$	20.7114	1.29446
$257$	−0.143768	−0.00896801	−0.00448400	−	0.999990i	$-0.501427\pi$
$257$	−0.143768	−0.00896801	−0.00448400	+	0.999990i	$0.501427\pi$
$258$	0	0
$259$	5.40581	0.335901
$260$	0.231914	0.0143827
$261$	0	0
$262$	−5.97046	−0.368856
$263$	−27.7778	−1.71285	−0.856425	−	0.516271i	$-0.827320\pi$
$263$	−27.7778	−1.71285	−0.856425	+	0.516271i	$0.827320\pi$
$264$	0	0
$265$	1.46011	0.0896937
$266$	11.0978	0.680452
$267$	0	0
$268$	15.0097	0.916863
$269$	−3.45473	−0.210639	−0.105319	−	0.994438i	$-0.533586\pi$
$269$	−3.45473	−0.210639	−0.105319	+	0.994438i	$0.533586\pi$
$270$	0	0
$271$	−22.6950	−1.37862	−0.689312	−	0.724465i	$-0.742087\pi$
$271$	−22.6950	−1.37862	−0.689312	+	0.724465i	$0.742087\pi$
$272$	−11.0978	−0.672905
$273$	0	0
$274$	1.93900	0.117139
$275$	6.15883	0.371392
$276$	0	0
$277$	−18.8853	−1.13471	−0.567354	−	0.823474i	$-0.692033\pi$
$277$	−18.8853	−1.13471	−0.567354	+	0.823474i	$0.692033\pi$
$278$	18.6093	1.11611
$279$	0	0
$280$	−0.335126	−0.0200276
$281$	22.1793	1.32310	0.661552	−	0.749899i	$-0.269898\pi$
$281$	22.1793	1.32310	0.661552	+	0.749899i	$0.269898\pi$
$282$	0	0
$283$	18.8769	1.12212	0.561059	−	0.827776i	$-0.310394\pi$
$283$	18.8769	1.12212	0.561059	+	0.827776i	$0.310394\pi$
$284$	2.52781	0.149998
$285$	0	0
$286$	1.69202	0.100051
$287$	9.39373	0.554494
$288$	0	0
$289$	−11.9511	−0.703005
$290$	−2.97285	−0.174572
$291$	0	0
$292$	2.50604	0.146655
$293$	14.2010	0.829634	0.414817	−	0.909905i	$-0.363846\pi$
$293$	14.2010	0.829634	0.414817	+	0.909905i	$0.363846\pi$
$294$	0	0
$295$	−0.231914	−0.0135026
$296$	7.33513	0.426346
$297$	0	0
$298$	−22.6112	−1.30983
$299$	−1.54288	−0.0892269
$300$	0	0
$301$	−0.198062	−0.0114161
$302$	38.4306	2.21143
$303$	0	0
$304$	30.4185	1.74462
$305$	1.76941	0.101316
$306$	0	0
$307$	−24.5338	−1.40022	−0.700108	−	0.714037i	$-0.746865\pi$
$307$	−24.5338	−1.40022	−0.700108	+	0.714037i	$0.746865\pi$
$308$	1.55496	0.0886020
$309$	0	0
$310$	1.28621	0.0730517
$311$	4.48858	0.254524	0.127262	−	0.991869i	$-0.459381\pi$
$311$	4.48858	0.254524	0.127262	+	0.991869i	$0.459381\pi$
$312$	0	0
$313$	−23.5405	−1.33059	−0.665293	−	0.746582i	$-0.731694\pi$
$313$	−23.5405	−1.33059	−0.665293	+	0.746582i	$0.731694\pi$
$314$	−13.1564	−0.742461
$315$	0	0
$316$	7.32975	0.412331
$317$	22.5036	1.26393	0.631965	−	0.774997i	$-0.282248\pi$
$317$	22.5036	1.26393	0.631965	+	0.774997i	$0.282248\pi$
$318$	0	0
$319$	−8.32975	−0.466376
$320$	0.313355	0.0175171
$321$	0	0
$322$	−3.69202	−0.205748
$323$	−13.8388	−0.770010
$324$	0	0
$325$	3.71917	0.206302
$326$	−12.9269	−0.715956
$327$	0	0
$328$	12.7463	0.703798
$329$	8.67994	0.478541
$330$	0	0
$331$	5.20237	0.285948	0.142974	−	0.989726i	$-0.454333\pi$
$331$	5.20237	0.285948	0.142974	+	0.989726i	$0.454333\pi$
$332$	0.835790	0.0458699
$333$	0	0
$334$	18.9095	1.03468
$335$	−2.97285	−0.162424
$336$	0	0
$337$	−3.20882	−0.174795	−0.0873977	−	0.996174i	$-0.527855\pi$
$337$	−3.20882	−0.174795	−0.0873977	+	0.996174i	$0.527855\pi$
$338$	−22.4034	−1.21859
$339$	0	0
$340$	−0.692021	−0.0375301
$341$	3.60388	0.195161
$342$	0	0
$343$	13.0000	0.701934
$344$	−0.268750	−0.0144900
$345$	0	0
$346$	−45.3400	−2.43750
$347$	−26.5786	−1.42682	−0.713408	−	0.700749i	$-0.752849\pi$
$347$	−26.5786	−1.42682	−0.713408	+	0.700749i	$0.752849\pi$
$348$	0	0
$349$	26.3317	1.40950	0.704751	−	0.709455i	$-0.251059\pi$
$349$	26.3317	1.40950	0.704751	+	0.709455i	$0.251059\pi$
$350$	8.89977	0.475713
$351$	0	0
$352$	7.71379	0.411146
$353$	−18.9909	−1.01078	−0.505392	−	0.862890i	$-0.668652\pi$
$353$	−18.9909	−1.01078	−0.505392	+	0.862890i	$0.668652\pi$
$354$	0	0
$355$	−0.500664	−0.0265725
$356$	19.1008	1.01234
$357$	0	0
$358$	14.1075	0.745606
$359$	−27.0930	−1.42992	−0.714958	−	0.699167i	$-0.753554\pi$
$359$	−27.0930	−1.42992	−0.714958	+	0.699167i	$0.753554\pi$
$360$	0	0
$361$	18.9312	0.996381
$362$	27.8931	1.46603
$363$	0	0
$364$	0.939001	0.0492170
$365$	−0.496352	−0.0259803
$366$	0	0
$367$	−4.61058	−0.240670	−0.120335	−	0.992733i	$-0.538397\pi$
$367$	−4.61058	−0.240670	−0.120335	+	0.992733i	$0.538397\pi$
$368$	−10.1196	−0.527521
$369$	0	0
$370$	2.40581	0.125072
$371$	5.91185	0.306928
$372$	0	0
$373$	10.0881	0.522344	0.261172	−	0.965292i	$-0.415891\pi$
$373$	10.0881	0.522344	0.261172	+	0.965292i	$0.415891\pi$
$374$	−5.04892	−0.261073
$375$	0	0
$376$	11.7778	0.607392
$377$	−5.03013	−0.259065
$378$	0	0
$379$	−21.5405	−1.10646	−0.553230	−	0.833028i	$-0.686605\pi$
$379$	−21.5405	−1.10646	−0.553230	+	0.833028i	$0.686605\pi$
$380$	1.89679	0.0973032
$381$	0	0
$382$	16.6189	0.850299
$383$	18.9903	0.970360	0.485180	−	0.874414i	$-0.338754\pi$
$383$	18.9903	0.970360	0.485180	+	0.874414i	$0.338754\pi$
$384$	0	0
$385$	−0.307979	−0.0156960
$386$	17.1957	0.875237
$387$	0	0
$388$	−3.09113	−0.156928
$389$	−12.7071	−0.644275	−0.322137	−	0.946693i	$-0.604401\pi$
$389$	−12.7071	−0.644275	−0.322137	+	0.946693i	$0.604401\pi$
$390$	0	0
$391$	4.60388	0.232828
$392$	8.14138	0.411202
$393$	0	0
$394$	−20.0465	−1.00993
$395$	−1.45175	−0.0730453
$396$	0	0
$397$	−16.9312	−0.849754	−0.424877	−	0.905251i	$-0.639683\pi$
$397$	−16.9312	−0.849754	−0.424877	+	0.905251i	$0.639683\pi$
$398$	−14.5386	−0.728752
$399$	0	0
$400$	24.3937	1.21969
$401$	6.60281	0.329729	0.164864	−	0.986316i	$-0.447281\pi$
$401$	6.60281	0.329729	0.164864	+	0.986316i	$0.447281\pi$
$402$	0	0
$403$	2.17629	0.108409
$404$	12.5918	0.626465
$405$	0	0
$406$	−12.0368	−0.597378
$407$	6.74094	0.334136
$408$	0	0
$409$	20.9124	1.03405	0.517027	−	0.855969i	$-0.327039\pi$
$409$	20.9124	1.03405	0.517027	+	0.855969i	$0.327039\pi$
$410$	4.18060	0.206465
$411$	0	0
$412$	1.86294	0.0917803
$413$	−0.939001	−0.0462052
$414$	0	0
$415$	−0.165538	−0.00812596
$416$	4.65817	0.228386
$417$	0	0
$418$	13.8388	0.676877
$419$	−1.67755	−0.0819535	−0.0409768	−	0.999160i	$-0.513047\pi$
$419$	−1.67755	−0.0819535	−0.0409768	+	0.999160i	$0.513047\pi$
$420$	0	0
$421$	9.33811	0.455112	0.227556	−	0.973765i	$-0.426927\pi$
$421$	9.33811	0.455112	0.227556	+	0.973765i	$0.426927\pi$
$422$	21.0248	1.02347
$423$	0	0
$424$	8.02177	0.389572
$425$	−11.0978	−0.538324
$426$	0	0
$427$	7.16421	0.346700
$428$	−4.05562	−0.196036
$429$	0	0
$430$	−0.0881460	−0.00425078
$431$	23.4058	1.12742	0.563709	−	0.825973i	$-0.309374\pi$
$431$	23.4058	1.12742	0.563709	+	0.825973i	$0.309374\pi$
$432$	0	0
$433$	−21.2543	−1.02142	−0.510708	−	0.859754i	$-0.670617\pi$
$433$	−21.2543	−1.02142	−0.510708	+	0.859754i	$0.670617\pi$
$434$	5.20775	0.249980
$435$	0	0
$436$	−14.2500	−0.682449
$437$	−12.6189	−0.603646
$438$	0	0
$439$	−30.2476	−1.44364	−0.721819	−	0.692082i	$-0.756694\pi$
$439$	−30.2476	−1.44364	−0.721819	+	0.692082i	$0.756694\pi$
$440$	−0.417895	−0.0199224
$441$	0	0
$442$	−3.04892	−0.145022
$443$	−4.08815	−0.194234	−0.0971168	−	0.995273i	$-0.530962\pi$
$443$	−4.08815	−0.194234	−0.0971168	+	0.995273i	$0.530962\pi$
$444$	0	0
$445$	−3.78315	−0.179339
$446$	4.64310	0.219857
$447$	0	0
$448$	1.26875	0.0599428
$449$	4.85192	0.228976	0.114488	−	0.993425i	$-0.463477\pi$
$449$	4.85192	0.228976	0.114488	+	0.993425i	$0.463477\pi$
$450$	0	0
$451$	11.7138	0.551581
$452$	−2.24698	−0.105689
$453$	0	0
$454$	15.7855	0.740852
$455$	−0.185981	−0.00871891
$456$	0	0
$457$	−17.1957	−0.804379	−0.402190	−	0.915556i	$-0.631751\pi$
$457$	−17.1957	−0.804379	−0.402190	+	0.915556i	$0.631751\pi$
$458$	14.1957	0.663320
$459$	0	0
$460$	−0.631023	−0.0294216
$461$	−4.19806	−0.195523	−0.0977616	−	0.995210i	$-0.531168\pi$
$461$	−4.19806	−0.195523	−0.0977616	+	0.995210i	$0.531168\pi$
$462$	0	0
$463$	9.50125	0.441561	0.220780	−	0.975324i	$-0.429140\pi$
$463$	9.50125	0.441561	0.220780	+	0.975324i	$0.429140\pi$
$464$	−32.9922	−1.53163
$465$	0	0
$466$	49.7439	2.30434
$467$	34.4873	1.59588	0.797940	−	0.602737i	$-0.205923\pi$
$467$	34.4873	1.59588	0.797940	+	0.602737i	$0.205923\pi$
$468$	0	0
$469$	−12.0368	−0.555809
$470$	3.86294	0.178184
$471$	0	0
$472$	−1.27413	−0.0586464
$473$	−0.246980	−0.0113561
$474$	0	0
$475$	30.4185	1.39570
$476$	−2.80194	−0.128427
$477$	0	0
$478$	1.80194	0.0824187
$479$	9.47219	0.432795	0.216398	−	0.976305i	$-0.430569\pi$
$479$	9.47219	0.432795	0.216398	+	0.976305i	$0.430569\pi$
$480$	0	0
$481$	4.07069	0.185607
$482$	−31.6407	−1.44119
$483$	0	0
$484$	−11.7778	−0.535353
$485$	0.612236	0.0278002
$486$	0	0
$487$	−36.4040	−1.64962	−0.824812	−	0.565408i	$-0.808719\pi$
$487$	−36.4040	−1.64962	−0.824812	+	0.565408i	$0.808719\pi$
$488$	9.72109	0.440053
$489$	0	0
$490$	2.67025	0.120630
$491$	−13.6437	−0.615731	−0.307866	−	0.951430i	$-0.599615\pi$
$491$	−13.6437	−0.615731	−0.307866	+	0.951430i	$0.599615\pi$
$492$	0	0
$493$	15.0097	0.676002
$494$	8.35690	0.375995
$495$	0	0
$496$	14.2741	0.640927
$497$	−2.02715	−0.0909300
$498$	0	0
$499$	32.0097	1.43295	0.716475	−	0.697613i	$-0.245754\pi$
$499$	32.0097	1.43295	0.716475	+	0.697613i	$0.245754\pi$
$500$	3.06100	0.136892
$501$	0	0
$502$	15.1642	0.676812
$503$	−29.2760	−1.30535	−0.652677	−	0.757636i	$-0.726354\pi$
$503$	−29.2760	−1.30535	−0.652677	+	0.757636i	$0.726354\pi$
$504$	0	0
$505$	−2.49396	−0.110980
$506$	−4.60388	−0.204667
$507$	0	0
$508$	−1.02715	−0.0455723
$509$	−30.1758	−1.33752	−0.668760	−	0.743479i	$-0.733175\pi$
$509$	−30.1758	−1.33752	−0.668760	+	0.743479i	$0.733175\pi$
$510$	0	0
$511$	−2.00969	−0.0889034
$512$	17.1491	0.757892
$513$	0	0
$514$	−0.259061	−0.0114267
$515$	−0.368977	−0.0162591
$516$	0	0
$517$	10.8237	0.476026
$518$	9.74094	0.427992
$519$	0	0
$520$	−0.252356	−0.0110666
$521$	−3.47458	−0.152224	−0.0761121	−	0.997099i	$-0.524251\pi$
$521$	−3.47458	−0.152224	−0.0761121	+	0.997099i	$0.524251\pi$
$522$	0	0
$523$	−11.1655	−0.488235	−0.244117	−	0.969746i	$-0.578498\pi$
$523$	−11.1655	−0.488235	−0.244117	+	0.969746i	$0.578498\pi$
$524$	−4.13169	−0.180494
$525$	0	0
$526$	−50.0538	−2.18245
$527$	−6.49396	−0.282881
$528$	0	0
$529$	−18.8019	−0.817476
$530$	2.63102	0.114284
$531$	0	0
$532$	7.67994	0.332968
$533$	7.07367	0.306395
$534$	0	0
$535$	0.803266	0.0347282
$536$	−16.3327	−0.705467
$537$	0	0
$538$	−6.22521	−0.268388
$539$	7.48188	0.322267
$540$	0	0
$541$	−4.17523	−0.179507	−0.0897535	−	0.995964i	$-0.528608\pi$
$541$	−4.17523	−0.179507	−0.0897535	+	0.995964i	$0.528608\pi$
$542$	−40.8950	−1.75659
$543$	0	0
$544$	−13.8998	−0.595948
$545$	2.82238	0.120897
$546$	0	0
$547$	39.5663	1.69173	0.845866	−	0.533395i	$-0.179084\pi$
$547$	39.5663	1.69173	0.845866	+	0.533395i	$0.179084\pi$
$548$	1.34183	0.0573202
$549$	0	0
$550$	11.0978	0.473213
$551$	−41.1406	−1.75265
$552$	0	0
$553$	−5.87800	−0.249958
$554$	−34.0301	−1.44580
$555$	0	0
$556$	12.8780	0.546149
$557$	−33.4838	−1.41875	−0.709377	−	0.704829i	$-0.751024\pi$
$557$	−33.4838	−1.41875	−0.709377	+	0.704829i	$0.751024\pi$
$558$	0	0
$559$	−0.149145	−0.00630816
$560$	−1.21983	−0.0515473
$561$	0	0
$562$	39.9657	1.68585
$563$	33.5351	1.41334	0.706668	−	0.707545i	$-0.250197\pi$
$563$	33.5351	1.41334	0.706668	+	0.707545i	$0.250197\pi$
$564$	0	0
$565$	0.445042	0.0187231
$566$	34.0151	1.42976
$567$	0	0
$568$	−2.75063	−0.115414
$569$	−21.1002	−0.884568	−0.442284	−	0.896875i	$-0.645832\pi$
$569$	−21.1002	−0.884568	−0.442284	+	0.896875i	$0.645832\pi$
$570$	0	0
$571$	28.6256	1.19795	0.598973	−	0.800769i	$-0.295576\pi$
$571$	28.6256	1.19795	0.598973	+	0.800769i	$0.295576\pi$
$572$	1.17092	0.0489584
$573$	0	0
$574$	16.9269	0.706516
$575$	−10.1196	−0.422017
$576$	0	0
$577$	2.66786	0.111064	0.0555322	−	0.998457i	$-0.482314\pi$
$577$	2.66786	0.111064	0.0555322	+	0.998457i	$0.482314\pi$
$578$	−21.5351	−0.895742
$579$	0	0
$580$	−2.05728	−0.0854238
$581$	−0.670251	−0.0278067
$582$	0	0
$583$	7.37196	0.305315
$584$	−2.72694	−0.112842
$585$	0	0
$586$	25.5894	1.05709
$587$	17.0901	0.705382	0.352691	−	0.935740i	$-0.385267\pi$
$587$	17.0901	0.705382	0.352691	+	0.935740i	$0.385267\pi$
$588$	0	0
$589$	17.7995	0.733417
$590$	−0.417895	−0.0172045
$591$	0	0
$592$	26.6993	1.09734
$593$	4.80433	0.197290	0.0986451	−	0.995123i	$-0.468549\pi$
$593$	4.80433	0.197290	0.0986451	+	0.995123i	$0.468549\pi$
$594$	0	0
$595$	0.554958	0.0227511
$596$	−15.6474	−0.640943
$597$	0	0
$598$	−2.78017	−0.113690
$599$	37.5405	1.53386	0.766931	−	0.641729i	$-0.221783\pi$
$599$	37.5405	1.53386	0.766931	+	0.641729i	$0.221783\pi$
$600$	0	0
$601$	−35.3937	−1.44374	−0.721870	−	0.692028i	$-0.756717\pi$
$601$	−35.3937	−1.44374	−0.721870	+	0.692028i	$0.756717\pi$
$602$	−0.356896	−0.0145460
$603$	0	0
$604$	26.5948	1.08213
$605$	2.33273	0.0948391
$606$	0	0
$607$	−20.7023	−0.840280	−0.420140	−	0.907459i	$-0.638019\pi$
$607$	−20.7023	−0.840280	−0.420140	+	0.907459i	$0.638019\pi$
$608$	38.0984	1.54510
$609$	0	0
$610$	3.18837	0.129093
$611$	6.53617	0.264425
$612$	0	0
$613$	25.3196	1.02265	0.511324	−	0.859388i	$-0.329155\pi$
$613$	25.3196	1.02265	0.511324	+	0.859388i	$0.329155\pi$
$614$	−44.2083	−1.78410
$615$	0	0
$616$	−1.69202	−0.0681735
$617$	11.3797	0.458131	0.229065	−	0.973411i	$-0.426433\pi$
$617$	11.3797	0.458131	0.229065	+	0.973411i	$0.426433\pi$
$618$	0	0
$619$	−38.3317	−1.54068	−0.770340	−	0.637633i	$-0.779913\pi$
$619$	−38.3317	−1.54068	−0.770340	+	0.637633i	$0.779913\pi$
$620$	0.890084	0.0357466
$621$	0	0
$622$	8.08815	0.324305
$623$	−15.3177	−0.613689
$624$	0	0
$625$	24.0887	0.963549
$626$	−42.4185	−1.69538
$627$	0	0
$628$	−9.10454	−0.363311
$629$	−12.1468	−0.484323
$630$	0	0
$631$	−39.1487	−1.55848	−0.779242	−	0.626723i	$-0.784396\pi$
$631$	−39.1487	−1.55848	−0.779242	+	0.626723i	$0.784396\pi$
$632$	−7.97584	−0.317262
$633$	0	0
$634$	40.5502	1.61045
$635$	0.203439	0.00807323
$636$	0	0
$637$	4.51812	0.179015
$638$	−15.0097	−0.594239
$639$	0	0
$640$	−2.49098	−0.0984644
$641$	26.9420	1.06414	0.532072	−	0.846699i	$-0.321413\pi$
$641$	26.9420	1.06414	0.532072	+	0.846699i	$0.321413\pi$
$642$	0	0
$643$	−42.9922	−1.69545	−0.847724	−	0.530438i	$-0.822028\pi$
$643$	−42.9922	−1.69545	−0.847724	+	0.530438i	$0.822028\pi$
$644$	−2.55496	−0.100679
$645$	0	0
$646$	−24.9366	−0.981118
$647$	−27.3612	−1.07568	−0.537840	−	0.843047i	$-0.680760\pi$
$647$	−27.3612	−1.07568	−0.537840	+	0.843047i	$0.680760\pi$
$648$	0	0
$649$	−1.17092	−0.0459624
$650$	6.70171	0.262863
$651$	0	0
$652$	−8.94571	−0.350341
$653$	12.8025	0.501002	0.250501	−	0.968116i	$-0.419405\pi$
$653$	12.8025	0.501002	0.250501	+	0.968116i	$0.419405\pi$
$654$	0	0
$655$	0.818331	0.0319748
$656$	46.3957	1.81145
$657$	0	0
$658$	15.6407	0.609738
$659$	−27.1946	−1.05935	−0.529676	−	0.848200i	$-0.677686\pi$
$659$	−27.1946	−1.05935	−0.529676	+	0.848200i	$0.677686\pi$
$660$	0	0
$661$	−10.0696	−0.391663	−0.195831	−	0.980638i	$-0.562741\pi$
$661$	−10.0696	−0.391663	−0.195831	+	0.980638i	$0.562741\pi$
$662$	9.37435	0.364345
$663$	0	0
$664$	−0.909461	−0.0352939
$665$	−1.52111	−0.0589860
$666$	0	0
$667$	13.6866	0.529949
$668$	13.0858	0.506303
$669$	0	0
$670$	−5.35690	−0.206955
$671$	8.93362	0.344879
$672$	0	0
$673$	−37.8310	−1.45828	−0.729139	−	0.684366i	$-0.760079\pi$
$673$	−37.8310	−1.45828	−0.729139	+	0.684366i	$0.760079\pi$
$674$	−5.78209	−0.222718
$675$	0	0
$676$	−15.5036	−0.596294
$677$	−5.47458	−0.210405	−0.105203	−	0.994451i	$-0.533549\pi$
$677$	−5.47458	−0.210405	−0.105203	+	0.994451i	$0.533549\pi$
$678$	0	0
$679$	2.47889	0.0951312
$680$	0.753020	0.0288770
$681$	0	0
$682$	6.49396	0.248667
$683$	39.8582	1.52513	0.762565	−	0.646912i	$-0.223940\pi$
$683$	39.8582	1.52513	0.762565	+	0.646912i	$0.223940\pi$
$684$	0	0
$685$	−0.265766	−0.0101544
$686$	23.4252	0.894378
$687$	0	0
$688$	−0.978230	−0.0372947
$689$	4.45175	0.169598
$690$	0	0
$691$	−44.3086	−1.68558	−0.842789	−	0.538245i	$-0.819088\pi$
$691$	−44.3086	−1.68558	−0.842789	+	0.538245i	$0.819088\pi$
$692$	−31.3763	−1.19275
$693$	0	0
$694$	−47.8931	−1.81800
$695$	−2.55065	−0.0967515
$696$	0	0
$697$	−21.1075	−0.799504
$698$	47.4480	1.79593
$699$	0	0
$700$	6.15883	0.232782
$701$	−46.7904	−1.76725	−0.883625	−	0.468195i	$-0.844905\pi$
$701$	−46.7904	−1.76725	−0.883625	+	0.468195i	$0.844905\pi$
$702$	0	0
$703$	33.2935	1.25569
$704$	1.58211	0.0596278
$705$	0	0
$706$	−34.2204	−1.28790
$707$	−10.0978	−0.379768
$708$	0	0
$709$	−28.0140	−1.05209	−0.526044	−	0.850457i	$-0.676325\pi$
$709$	−28.0140	−1.05209	−0.526044	+	0.850457i	$0.676325\pi$
$710$	−0.902165	−0.0338577
$711$	0	0
$712$	−20.7845	−0.778931
$713$	−5.92154	−0.221764
$714$	0	0
$715$	−0.231914	−0.00867310
$716$	9.76271	0.364850
$717$	0	0
$718$	−48.8200	−1.82195
$719$	15.3166	0.571213	0.285606	−	0.958347i	$-0.407805\pi$
$719$	15.3166	0.571213	0.285606	+	0.958347i	$0.407805\pi$
$720$	0	0
$721$	−1.49396	−0.0556379
$722$	34.1129	1.26955
$723$	0	0
$724$	19.3026	0.717376
$725$	−32.9922	−1.22530
$726$	0	0
$727$	10.1260	0.375554	0.187777	−	0.982212i	$-0.439872\pi$
$727$	10.1260	0.375554	0.187777	+	0.982212i	$0.439872\pi$
$728$	−1.02177	−0.0378693
$729$	0	0
$730$	−0.894396	−0.0331031
$731$	0.445042	0.0164605
$732$	0	0
$733$	37.2669	1.37649	0.688243	−	0.725480i	$-0.258382\pi$
$733$	37.2669	1.37649	0.688243	+	0.725480i	$0.258382\pi$
$734$	−8.30798	−0.306653
$735$	0	0
$736$	−12.6746	−0.467191
$737$	−15.0097	−0.552889
$738$	0	0
$739$	−32.2215	−1.18529	−0.592643	−	0.805465i	$-0.701916\pi$
$739$	−32.2215	−1.18529	−0.592643	+	0.805465i	$0.701916\pi$
$740$	1.66487	0.0612020
$741$	0	0
$742$	10.6528	0.391076
$743$	35.3437	1.29664	0.648318	−	0.761369i	$-0.275473\pi$
$743$	35.3437	1.29664	0.648318	+	0.761369i	$0.275473\pi$
$744$	0	0
$745$	3.09916	0.113545
$746$	18.1782	0.665552
$747$	0	0
$748$	−3.49396	−0.127752
$749$	3.25236	0.118839
$750$	0	0
$751$	54.5333	1.98995	0.994974	−	0.100131i	$-0.0319261\pi$
$751$	54.5333	1.98995	0.994974	+	0.100131i	$0.0319261\pi$
$752$	42.8702	1.56332
$753$	0	0
$754$	−9.06398	−0.330091
$755$	−5.26742	−0.191701
$756$	0	0
$757$	32.6775	1.18769	0.593843	−	0.804581i	$-0.297610\pi$
$757$	32.6775	1.18769	0.593843	+	0.804581i	$0.297610\pi$
$758$	−38.8146	−1.40981
$759$	0	0
$760$	−2.06398	−0.0748685
$761$	5.36227	0.194382	0.0971911	−	0.995266i	$-0.469014\pi$
$761$	5.36227	0.194382	0.0971911	+	0.995266i	$0.469014\pi$
$762$	0	0
$763$	11.4276	0.413706
$764$	11.5007	0.416079
$765$	0	0
$766$	34.2194	1.23640
$767$	−0.707087	−0.0255314
$768$	0	0
$769$	−45.3889	−1.63677	−0.818384	−	0.574672i	$-0.805130\pi$
$769$	−45.3889	−1.63677	−0.818384	+	0.574672i	$0.805130\pi$
$770$	−0.554958	−0.0199993
$771$	0	0
$772$	11.8998	0.428282
$773$	48.4392	1.74224	0.871118	−	0.491073i	$-0.163395\pi$
$773$	48.4392	1.74224	0.871118	+	0.491073i	$0.163395\pi$
$774$	0	0
$775$	14.2741	0.512742
$776$	3.36360	0.120746
$777$	0	0
$778$	−22.8974	−0.820911
$779$	57.8544	2.07285
$780$	0	0
$781$	−2.52781	−0.0904522
$782$	8.29590	0.296661
$783$	0	0
$784$	29.6340	1.05836
$785$	1.80327	0.0643613
$786$	0	0
$787$	−31.6926	−1.12972	−0.564860	−	0.825187i	$-0.691070\pi$
$787$	−31.6926	−1.12972	−0.564860	+	0.825187i	$0.691070\pi$
$788$	−13.8726	−0.494192
$789$	0	0
$790$	−2.61596	−0.0930716
$791$	1.80194	0.0640695
$792$	0	0
$793$	5.39480	0.191575
$794$	−30.5090	−1.08272
$795$	0	0
$796$	−10.0610	−0.356603
$797$	−23.9920	−0.849839	−0.424920	−	0.905231i	$-0.639698\pi$
$797$	−23.9920	−0.849839	−0.424920	+	0.905231i	$0.639698\pi$
$798$	0	0
$799$	−19.5036	−0.689989
$800$	30.5526	1.08020
$801$	0	0
$802$	11.8979	0.420128
$803$	−2.50604	−0.0884363
$804$	0	0
$805$	0.506041	0.0178356
$806$	3.92154	0.138130
$807$	0	0
$808$	−13.7017	−0.482024
$809$	−50.0581	−1.75995	−0.879975	−	0.475020i	$-0.842441\pi$
$809$	−50.0581	−1.75995	−0.879975	+	0.475020i	$0.842441\pi$
$810$	0	0
$811$	−51.8031	−1.81905	−0.909527	−	0.415645i	$-0.863556\pi$
$811$	−51.8031	−1.81905	−0.909527	+	0.415645i	$0.863556\pi$
$812$	−8.32975	−0.292317
$813$	0	0
$814$	12.1468	0.425743
$815$	1.77181	0.0620637
$816$	0	0
$817$	−1.21983	−0.0426765
$818$	37.6829	1.31755
$819$	0	0
$820$	2.89307	0.101030
$821$	−11.6334	−0.406009	−0.203004	−	0.979178i	$-0.565071\pi$
$821$	−11.6334	−0.406009	−0.203004	+	0.979178i	$0.565071\pi$
$822$	0	0
$823$	4.69335	0.163600	0.0817999	−	0.996649i	$-0.473933\pi$
$823$	4.69335	0.163600	0.0817999	+	0.996649i	$0.473933\pi$
$824$	−2.02715	−0.0706190
$825$	0	0
$826$	−1.69202	−0.0588730
$827$	−45.6819	−1.58851	−0.794257	−	0.607582i	$-0.792140\pi$
$827$	−45.6819	−1.58851	−0.794257	+	0.607582i	$0.792140\pi$
$828$	0	0
$829$	−24.6665	−0.856704	−0.428352	−	0.903612i	$-0.640906\pi$
$829$	−24.6665	−0.856704	−0.428352	+	0.903612i	$0.640906\pi$
$830$	−0.298290	−0.0103538
$831$	0	0
$832$	0.955395	0.0331223
$833$	−13.4819	−0.467119
$834$	0	0
$835$	−2.59179	−0.0896927
$836$	9.57673	0.331218
$837$	0	0
$838$	−3.02284	−0.104422
$839$	−40.4892	−1.39784	−0.698921	−	0.715199i	$-0.746336\pi$
$839$	−40.4892	−1.39784	−0.698921	+	0.715199i	$0.746336\pi$
$840$	0	0
$841$	15.6216	0.538676
$842$	16.8267	0.579886
$843$	0	0
$844$	14.5496	0.500817
$845$	3.07069	0.105635
$846$	0	0
$847$	9.44504	0.324535
$848$	29.1987	1.00269
$849$	0	0
$850$	−19.9976	−0.685912
$851$	−11.0761	−0.379683
$852$	0	0
$853$	15.5996	0.534119	0.267059	−	0.963680i	$-0.413948\pi$
$853$	15.5996	0.534119	0.267059	+	0.963680i	$0.413948\pi$
$854$	12.9095	0.441753
$855$	0	0
$856$	4.41311	0.150837
$857$	−13.4862	−0.460679	−0.230340	−	0.973110i	$-0.573984\pi$
$857$	−13.4862	−0.460679	−0.230340	+	0.973110i	$0.573984\pi$
$858$	0	0
$859$	−2.93841	−0.100257	−0.0501286	−	0.998743i	$-0.515963\pi$
$859$	−2.93841	−0.100257	−0.0501286	+	0.998743i	$0.515963\pi$
$860$	−0.0609989	−0.00208005
$861$	0	0
$862$	42.1758	1.43651
$863$	26.5080	0.902341	0.451171	−	0.892438i	$-0.351007\pi$
$863$	26.5080	0.902341	0.451171	+	0.892438i	$0.351007\pi$
$864$	0	0
$865$	6.21446	0.211298
$866$	−38.2989	−1.30145
$867$	0	0
$868$	3.60388	0.122324
$869$	−7.32975	−0.248645
$870$	0	0
$871$	−9.06398	−0.307121
$872$	15.5060	0.525101
$873$	0	0
$874$	−22.7385	−0.769143
$875$	−2.45473	−0.0829850
$876$	0	0
$877$	−6.28813	−0.212335	−0.106167	−	0.994348i	$-0.533858\pi$
$877$	−6.28813	−0.212335	−0.106167	+	0.994348i	$0.533858\pi$
$878$	−54.5042	−1.83943
$879$	0	0
$880$	−1.52111	−0.0512765
$881$	−24.5469	−0.827007	−0.413503	−	0.910503i	$-0.635695\pi$
$881$	−24.5469	−0.827007	−0.413503	+	0.910503i	$0.635695\pi$
$882$	0	0
$883$	14.6431	0.492780	0.246390	−	0.969171i	$-0.420756\pi$
$883$	14.6431	0.492780	0.246390	+	0.969171i	$0.420756\pi$
$884$	−2.10992	−0.0709642
$885$	0	0
$886$	−7.36658	−0.247485
$887$	−41.0998	−1.37999	−0.689997	−	0.723812i	$-0.742388\pi$
$887$	−41.0998	−1.37999	−0.689997	+	0.723812i	$0.742388\pi$
$888$	0	0
$889$	0.823708	0.0276263
$890$	−6.81700	−0.228506
$891$	0	0
$892$	3.21313	0.107583
$893$	53.4583	1.78891
$894$	0	0
$895$	−1.93362	−0.0646339
$896$	−10.0858	−0.336941
$897$	0	0
$898$	8.74286	0.291753
$899$	−19.3056	−0.643877
$900$	0	0
$901$	−13.2838	−0.442548
$902$	21.1075	0.702804
$903$	0	0
$904$	2.44504	0.0813209
$905$	−3.82312	−0.127085
$906$	0	0
$907$	45.1885	1.50046	0.750230	−	0.661177i	$-0.229943\pi$
$907$	45.1885	1.50046	0.750230	+	0.661177i	$0.229943\pi$
$908$	10.9239	0.362524
$909$	0	0
$910$	−0.335126	−0.0111093
$911$	20.5314	0.680235	0.340118	−	0.940383i	$-0.389533\pi$
$911$	20.5314	0.680235	0.340118	+	0.940383i	$0.389533\pi$
$912$	0	0
$913$	−0.835790	−0.0276606
$914$	−30.9855	−1.02491
$915$	0	0
$916$	9.82371	0.324584
$917$	3.31336	0.109417
$918$	0	0
$919$	53.6883	1.77101	0.885507	−	0.464626i	$-0.153811\pi$
$919$	53.6883	1.77101	0.885507	+	0.464626i	$0.153811\pi$
$920$	0.686645	0.0226380
$921$	0	0
$922$	−7.56465	−0.249128
$923$	−1.52648	−0.0502448
$924$	0	0
$925$	26.6993	0.877868
$926$	17.1207	0.562620
$927$	0	0
$928$	−41.3220	−1.35646
$929$	−58.6708	−1.92493	−0.962464	−	0.271410i	$-0.912510\pi$
$929$	−58.6708	−1.92493	−0.962464	+	0.271410i	$0.912510\pi$
$930$	0	0
$931$	36.9530	1.21109
$932$	34.4239	1.12759
$933$	0	0
$934$	62.1439	2.03341
$935$	0.692021	0.0226315
$936$	0	0
$937$	−40.1420	−1.31138	−0.655690	−	0.755030i	$-0.727622\pi$
$937$	−40.1420	−1.31138	−0.655690	+	0.755030i	$0.727622\pi$
$938$	−21.6896	−0.708191
$939$	0	0
$940$	2.67324	0.0871913
$941$	43.4292	1.41575	0.707876	−	0.706336i	$-0.249653\pi$
$941$	43.4292	1.41575	0.707876	+	0.706336i	$0.249653\pi$
$942$	0	0
$943$	−19.2470	−0.626768
$944$	−4.63773	−0.150945
$945$	0	0
$946$	−0.445042	−0.0144696
$947$	23.3803	0.759758	0.379879	−	0.925036i	$-0.375966\pi$
$947$	23.3803	0.759758	0.379879	+	0.925036i	$0.375966\pi$
$948$	0	0
$949$	−1.51334	−0.0491250
$950$	54.8122	1.77834
$951$	0	0
$952$	3.04892	0.0988160
$953$	−12.3897	−0.401341	−0.200671	−	0.979659i	$-0.564312\pi$
$953$	−12.3897	−0.401341	−0.200671	+	0.979659i	$0.564312\pi$
$954$	0	0
$955$	−2.27785	−0.0737094
$956$	1.24698	0.0403302
$957$	0	0
$958$	17.0683	0.551452
$959$	−1.07606	−0.0347479
$960$	0	0
$961$	−22.6474	−0.730562
$962$	7.33513	0.236494
$963$	0	0
$964$	−21.8961	−0.705224
$965$	−2.35690	−0.0758712
$966$	0	0
$967$	30.0325	0.965781	0.482890	−	0.875681i	$-0.339587\pi$
$967$	30.0325	0.965781	0.482890	+	0.875681i	$0.339587\pi$
$968$	12.8159	0.411920
$969$	0	0
$970$	1.10321	0.0354220
$971$	−27.7127	−0.889344	−0.444672	−	0.895694i	$-0.646680\pi$
$971$	−27.7127	−0.889344	−0.444672	+	0.895694i	$0.646680\pi$
$972$	0	0
$973$	−10.3274	−0.331080
$974$	−65.5978	−2.10189
$975$	0	0
$976$	35.3840	1.13262
$977$	−47.8786	−1.53177	−0.765886	−	0.642976i	$-0.777699\pi$
$977$	−47.8786	−1.53177	−0.765886	+	0.642976i	$0.777699\pi$
$978$	0	0
$979$	−19.1008	−0.610465
$980$	1.84787	0.0590281
$981$	0	0
$982$	−24.5851	−0.784542
$983$	−39.3123	−1.25387	−0.626934	−	0.779073i	$-0.715690\pi$
$983$	−39.3123	−1.25387	−0.626934	+	0.779073i	$0.715690\pi$
$984$	0	0
$985$	2.74764	0.0875472
$986$	27.0465	0.861337
$987$	0	0
$988$	5.78315	0.183987
$989$	0.405813	0.0129041
$990$	0	0
$991$	25.2006	0.800523	0.400261	−	0.916401i	$-0.368919\pi$
$991$	25.2006	0.800523	0.400261	+	0.916401i	$0.368919\pi$
$992$	17.8780	0.567627
$993$	0	0
$994$	−3.65279	−0.115860
$995$	1.99270	0.0631730
$996$	0	0
$997$	36.3726	1.15193	0.575965	−	0.817474i	$-0.304626\pi$
$997$	36.3726	1.15193	0.575965	+	0.817474i	$0.304626\pi$
$998$	57.6795	1.82581
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.d.1.3		3
3.2	odd	2		239.2.a.a.1.1	✓	3
12.11	even	2		3824.2.a.e.1.2		3
15.14	odd	2		5975.2.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.2.a.a.1.1	✓	3	3.2	odd	2
2151.2.a.d.1.3		3	1.1	even	1	trivial
3824.2.a.e.1.2		3	12.11	even	2
5975.2.a.d.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.80194 q^{2} +1.24698 q^{4} -0.246980 q^{5} -1.00000 q^{7} -1.35690 q^{8} +O(q^{10})\)	\(q+1.80194 q^{2} +1.24698 q^{4} -0.246980 q^{5} -1.00000 q^{7} -1.35690 q^{8} -0.445042 q^{10} -1.24698 q^{11} -0.753020 q^{13} -1.80194 q^{14} -4.93900 q^{16} +2.24698 q^{17} -6.15883 q^{19} -0.307979 q^{20} -2.24698 q^{22} +2.04892 q^{23} -4.93900 q^{25} -1.35690 q^{26} -1.24698 q^{28} +6.67994 q^{29} -2.89008 q^{31} -6.18598 q^{32} +4.04892 q^{34} +0.246980 q^{35} -5.40581 q^{37} -11.0978 q^{38} +0.335126 q^{40} -9.39373 q^{41} +0.198062 q^{43} -1.55496 q^{44} +3.69202 q^{46} -8.67994 q^{47} -6.00000 q^{49} -8.89977 q^{50} -0.939001 q^{52} -5.91185 q^{53} +0.307979 q^{55} +1.35690 q^{56} +12.0368 q^{58} +0.939001 q^{59} -7.16421 q^{61} -5.20775 q^{62} -1.26875 q^{64} +0.185981 q^{65} +12.0368 q^{67} +2.80194 q^{68} +0.445042 q^{70} +2.02715 q^{71} +2.00969 q^{73} -9.74094 q^{74} -7.67994 q^{76} +1.24698 q^{77} +5.87800 q^{79} +1.21983 q^{80} -16.9269 q^{82} +0.670251 q^{83} -0.554958 q^{85} +0.356896 q^{86} +1.69202 q^{88} +15.3177 q^{89} +0.753020 q^{91} +2.55496 q^{92} -15.6407 q^{94} +1.52111 q^{95} -2.47889 q^{97} -10.8116 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} - q^{10} + q^{11} - 7 q^{13} - q^{14} - 5 q^{16} + 2 q^{17} - 10 q^{19} - 6 q^{20} - 2 q^{22} - 3 q^{23} - 5 q^{25} + q^{28} - 4 q^{29} - 8 q^{31} - 4 q^{32} + 3 q^{34} - 4 q^{35} - 3 q^{37} - 15 q^{38} + 4 q^{41} + 5 q^{43} - 5 q^{44} + 6 q^{46} - 2 q^{47} - 18 q^{49} - 4 q^{50} + 7 q^{52} - 14 q^{53} + 6 q^{55} + 8 q^{58} - 7 q^{59} - 10 q^{61} + 2 q^{62} + 4 q^{64} - 14 q^{65} + 8 q^{67} + 4 q^{68} + q^{70} - 16 q^{73} - 15 q^{74} + q^{76} - q^{77} - 2 q^{79} + 5 q^{80} - 22 q^{82} - 2 q^{85} - 3 q^{86} + 29 q^{89} + 7 q^{91} + 8 q^{92} - 10 q^{94} - 11 q^{95} - 23 q^{97} - 6 q^{98}+O(q^{100}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 3 * q^7 - q^10 + q^11 - 7 * q^13 - q^14 - 5 * q^16 + 2 * q^17 - 10 * q^19 - 6 * q^20 - 2 * q^22 - 3 * q^23 - 5 * q^25 + q^28 - 4 * q^29 - 8 * q^31 - 4 * q^32 + 3 * q^34 - 4 * q^35 - 3 * q^37 - 15 * q^38 + 4 * q^41 + 5 * q^43 - 5 * q^44 + 6 * q^46 - 2 * q^47 - 18 * q^49 - 4 * q^50 + 7 * q^52 - 14 * q^53 + 6 * q^55 + 8 * q^58 - 7 * q^59 - 10 * q^61 + 2 * q^62 + 4 * q^64 - 14 * q^65 + 8 * q^67 + 4 * q^68 + q^70 - 16 * q^73 - 15 * q^74 + q^76 - q^77 - 2 * q^79 + 5 * q^80 - 22 * q^82 - 2 * q^85 - 3 * q^86 + 29 * q^89 + 7 * q^91 + 8 * q^92 - 10 * q^94 - 11 * q^95 - 23 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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