LMFDB - Embedded newform 1287.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.704624$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79888 q^{2} +1.23597 q^{4} -3.97216 q^{5} +3.17328 q^{7} -1.37440 q^{8} +O(q^{10})$	\(q+1.79888 q^{2} +1.23597 q^{4} -3.97216 q^{5} +3.17328 q^{7} -1.37440 q^{8} -7.14544 q^{10} +1.00000 q^{11} +1.00000 q^{13} +5.70835 q^{14} -4.94432 q^{16} +7.80589 q^{17} +7.17328 q^{19} -4.90947 q^{20} +1.79888 q^{22} -3.36179 q^{23} +10.7780 q^{25} +1.79888 q^{26} +3.92208 q^{28} +7.61738 q^{29} +3.15366 q^{31} -6.14544 q^{32} +14.0419 q^{34} -12.6048 q^{35} +2.93731 q^{37} +12.9039 q^{38} +5.45933 q^{40} -1.64522 q^{41} -4.15245 q^{43} +1.23597 q^{44} -6.04746 q^{46} -0.660451 q^{47} +3.06970 q^{49} +19.3884 q^{50} +1.23597 q^{52} -0.0696997 q^{53} -3.97216 q^{55} -4.36135 q^{56} +13.7027 q^{58} -8.88820 q^{59} -5.47895 q^{61} +5.67306 q^{62} -1.16627 q^{64} -3.97216 q^{65} +5.50022 q^{67} +9.64784 q^{68} -22.6745 q^{70} -2.11881 q^{71} -14.7154 q^{73} +5.28387 q^{74} +8.86596 q^{76} +3.17328 q^{77} +14.4990 q^{79} +19.6396 q^{80} -2.95955 q^{82} +11.4724 q^{83} -31.0062 q^{85} -7.46975 q^{86} -1.37440 q^{88} -0.965150 q^{89} +3.17328 q^{91} -4.15507 q^{92} -1.18807 q^{94} -28.4934 q^{95} +7.31433 q^{97} +5.52202 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	$ 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * 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 $

$2$	1.79888	1.27200	0.636000	−	0.771689i	$-0.280588\pi$
$2$	1.79888	1.27200	0.636000	+	0.771689i	$0.280588\pi$
$3$	0	0
$3$	0	0
$4$	1.23597	0.617985
$5$	−3.97216	−1.77640	−0.888202	−	0.459454i	$-0.848045\pi$
$5$	−3.97216	−1.77640	−0.888202	+	0.459454i	$0.848045\pi$
$6$	0	0
$7$	3.17328	1.19939	0.599693	−	0.800230i	$-0.295289\pi$
$7$	3.17328	1.19939	0.599693	+	0.800230i	$0.295289\pi$
$8$	−1.37440	−0.485923
$9$	0	0
$10$	−7.14544	−2.25959
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.70835	1.52562
$15$	0	0
$16$	−4.94432	−1.23608
$17$	7.80589	1.89321	0.946603	−	0.322401i	$-0.104490\pi$
$17$	7.80589	1.89321	0.946603	+	0.322401i	$0.104490\pi$
$18$	0	0
$19$	7.17328	1.64566	0.822832	−	0.568285i	$-0.192393\pi$
$19$	7.17328	1.64566	0.822832	+	0.568285i	$0.192393\pi$
$20$	−4.90947	−1.09779
$21$	0	0
$22$	1.79888	0.383523
$23$	−3.36179	−0.700982	−0.350491	−	0.936566i	$-0.613985\pi$
$23$	−3.36179	−0.700982	−0.350491	+	0.936566i	$0.613985\pi$
$24$	0	0
$25$	10.7780	2.15561
$26$	1.79888	0.352789
$27$	0	0
$28$	3.92208	0.741203
$29$	7.61738	1.41451	0.707256	−	0.706958i	$-0.249933\pi$
$29$	7.61738	1.41451	0.707256	+	0.706958i	$0.249933\pi$
$30$	0	0
$31$	3.15366	0.566414	0.283207	−	0.959059i	$-0.408602\pi$
$31$	3.15366	0.566414	0.283207	+	0.959059i	$0.408602\pi$
$32$	−6.14544	−1.08637
$33$	0	0
$34$	14.0419	2.40816
$35$	−12.6048	−2.13059
$36$	0	0
$37$	2.93731	0.482891	0.241445	−	0.970414i	$-0.422379\pi$
$37$	2.93731	0.482891	0.241445	+	0.970414i	$0.422379\pi$
$38$	12.9039	2.09328
$39$	0	0
$40$	5.45933	0.863196
$41$	−1.64522	−0.256940	−0.128470	−	0.991713i	$-0.541007\pi$
$41$	−1.64522	−0.256940	−0.128470	+	0.991713i	$0.541007\pi$
$42$	0	0
$43$	−4.15245	−0.633242	−0.316621	−	0.948552i	$-0.602548\pi$
$43$	−4.15245	−0.633242	−0.316621	+	0.948552i	$0.602548\pi$
$44$	1.23597	0.186329
$45$	0	0
$46$	−6.04746	−0.891649
$47$	−0.660451	−0.0963367	−0.0481683	−	0.998839i	$-0.515338\pi$
$47$	−0.660451	−0.0963367	−0.0481683	+	0.998839i	$0.515338\pi$
$48$	0	0
$49$	3.06970	0.438529
$50$	19.3884	2.74194
$51$	0	0
$52$	1.23597	0.171398
$53$	−0.0696997	−0.00957399	−0.00478700	−	0.999989i	$-0.501524\pi$
$53$	−0.0696997	−0.00957399	−0.00478700	+	0.999989i	$0.501524\pi$
$54$	0	0
$55$	−3.97216	−0.535606
$56$	−4.36135	−0.582810
$57$	0	0
$58$	13.7027	1.79926
$59$	−8.88820	−1.15714	−0.578572	−	0.815631i	$-0.696390\pi$
$59$	−8.88820	−1.15714	−0.578572	+	0.815631i	$0.696390\pi$
$60$	0	0
$61$	−5.47895	−0.701507	−0.350754	−	0.936468i	$-0.614075\pi$
$61$	−5.47895	−0.701507	−0.350754	+	0.936468i	$0.614075\pi$
$62$	5.67306	0.720479
$63$	0	0
$64$	−1.16627	−0.145784
$65$	−3.97216	−0.492686
$66$	0	0
$67$	5.50022	0.671959	0.335979	−	0.941869i	$-0.390933\pi$
$67$	5.50022	0.671959	0.335979	+	0.941869i	$0.390933\pi$
$68$	9.64784	1.16997
$69$	0	0
$70$	−22.6745	−2.71012
$71$	−2.11881	−0.251457	−0.125728	−	0.992065i	$-0.540127\pi$
$71$	−2.11881	−0.251457	−0.125728	+	0.992065i	$0.540127\pi$
$72$	0	0
$73$	−14.7154	−1.72230	−0.861151	−	0.508349i	$-0.830256\pi$
$73$	−14.7154	−1.72230	−0.861151	+	0.508349i	$0.830256\pi$
$74$	5.28387	0.614237
$75$	0	0
$76$	8.86596	1.01699
$77$	3.17328	0.361629
$78$	0	0
$79$	14.4990	1.63127	0.815633	−	0.578570i	$-0.196389\pi$
$79$	14.4990	1.63127	0.815633	+	0.578570i	$0.196389\pi$
$80$	19.6396	2.19578
$81$	0	0
$82$	−2.95955	−0.326828
$83$	11.4724	1.25926	0.629629	−	0.776896i	$-0.283207\pi$
$83$	11.4724	1.25926	0.629629	+	0.776896i	$0.283207\pi$
$84$	0	0
$85$	−31.0062	−3.36310
$86$	−7.46975	−0.805484
$87$	0	0
$88$	−1.37440	−0.146511
$89$	−0.965150	−0.102306	−0.0511529	−	0.998691i	$-0.516290\pi$
$89$	−0.965150	−0.102306	−0.0511529	+	0.998691i	$0.516290\pi$
$90$	0	0
$91$	3.17328	0.332650
$92$	−4.15507	−0.433196
$93$	0	0
$94$	−1.18807	−0.122540
$95$	−28.4934	−2.92336
$96$	0	0
$97$	7.31433	0.742658	0.371329	−	0.928501i	$-0.378902\pi$
$97$	7.31433	0.742658	0.371329	+	0.928501i	$0.378902\pi$
$98$	5.52202	0.557808
$99$	0	0
$100$	13.3213	1.33213
$101$	−17.3480	−1.72619	−0.863094	−	0.505044i	$-0.831476\pi$
$101$	−17.3480	−1.72619	−0.863094	+	0.505044i	$0.831476\pi$
$102$	0	0
$103$	4.47194	0.440633	0.220317	−	0.975428i	$-0.429291\pi$
$103$	4.47194	0.440633	0.220317	+	0.975428i	$0.429291\pi$
$104$	−1.37440	−0.134771
$105$	0	0
$106$	−0.125381	−0.0121781
$107$	12.0837	1.16818	0.584089	−	0.811690i	$-0.301452\pi$
$107$	12.0837	1.16818	0.584089	+	0.811690i	$0.301452\pi$
$108$	0	0
$109$	−4.36880	−0.418455	−0.209228	−	0.977867i	$-0.567095\pi$
$109$	−4.36880	−0.418455	−0.209228	+	0.977867i	$0.567095\pi$
$110$	−7.14544	−0.681291
$111$	0	0
$112$	−15.6897	−1.48254
$113$	8.77926	0.825884	0.412942	−	0.910757i	$-0.364501\pi$
$113$	8.77926	0.825884	0.412942	+	0.910757i	$0.364501\pi$
$114$	0	0
$115$	13.3536	1.24523
$116$	9.41485	0.874147
$117$	0	0
$118$	−15.9888	−1.47189
$119$	24.7703	2.27069
$120$	0	0
$121$	1.00000	0.0909091
$122$	−9.85597	−0.892318
$123$	0	0
$124$	3.89783	0.350035
$125$	−22.9513	−2.05283
$126$	0	0
$127$	2.42186	0.214905	0.107453	−	0.994210i	$-0.465731\pi$
$127$	2.42186	0.214905	0.107453	+	0.994210i	$0.465731\pi$
$128$	10.1929	0.900933
$129$	0	0
$130$	−7.14544	−0.626696
$131$	8.36058	0.730467	0.365233	−	0.930916i	$-0.380989\pi$
$131$	8.36058	0.730467	0.365233	+	0.930916i	$0.380989\pi$
$132$	0	0
$133$	22.7628	1.97379
$134$	9.89424	0.854732
$135$	0	0
$136$	−10.7284	−0.919953
$137$	9.85335	0.841828	0.420914	−	0.907100i	$-0.361709\pi$
$137$	9.85335	0.841828	0.420914	+	0.907100i	$0.361709\pi$
$138$	0	0
$139$	−2.74320	−0.232675	−0.116338	−	0.993210i	$-0.537115\pi$
$139$	−2.74320	−0.232675	−0.116338	+	0.993210i	$0.537115\pi$
$140$	−15.5791	−1.31668
$141$	0	0
$142$	−3.81149	−0.319853
$143$	1.00000	0.0836242
$144$	0	0
$145$	−30.2574	−2.51274
$146$	−26.4712	−2.19077
$147$	0	0
$148$	3.63043	0.298419
$149$	−3.99178	−0.327019	−0.163510	−	0.986542i	$-0.552281\pi$
$149$	−3.99178	−0.327019	−0.163510	+	0.986542i	$0.552281\pi$
$150$	0	0
$151$	9.18730	0.747652	0.373826	−	0.927499i	$-0.378046\pi$
$151$	9.18730	0.747652	0.373826	+	0.927499i	$0.378046\pi$
$152$	−9.85895	−0.799666
$153$	0	0
$154$	5.70835	0.459992
$155$	−12.5268	−1.00618
$156$	0	0
$157$	−18.8482	−1.50425	−0.752125	−	0.659021i	$-0.770971\pi$
$157$	−18.8482	−1.50425	−0.752125	+	0.659021i	$0.770971\pi$
$158$	26.0820	2.07497
$159$	0	0
$160$	24.4107	1.92983
$161$	−10.6679	−0.840748
$162$	0	0
$163$	−15.7027	−1.22993	−0.614967	−	0.788553i	$-0.710831\pi$
$163$	−15.7027	−1.22993	−0.614967	+	0.788553i	$0.710831\pi$
$164$	−2.03344	−0.158785
$165$	0	0
$166$	20.6374	1.60178
$167$	0.709563	0.0549076	0.0274538	−	0.999623i	$-0.491260\pi$
$167$	0.709563	0.0549076	0.0274538	+	0.999623i	$0.491260\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−55.7765	−4.27786
$171$	0	0
$172$	−5.13230	−0.391334
$173$	−6.86201	−0.521709	−0.260854	−	0.965378i	$-0.584004\pi$
$173$	−6.86201	−0.521709	−0.260854	+	0.965378i	$0.584004\pi$
$174$	0	0
$175$	34.2018	2.58541
$176$	−4.94432	−0.372692
$177$	0	0
$178$	−1.73619	−0.130133
$179$	−10.1411	−0.757978	−0.378989	−	0.925401i	$-0.623728\pi$
$179$	−10.1411	−0.757978	−0.378989	+	0.925401i	$0.623728\pi$
$180$	0	0
$181$	3.76403	0.279778	0.139889	−	0.990167i	$-0.455325\pi$
$181$	3.76403	0.279778	0.139889	+	0.990167i	$0.455325\pi$
$182$	5.70835	0.423131
$183$	0	0
$184$	4.62044	0.340623
$185$	−11.6675	−0.857809
$186$	0	0
$187$	7.80589	0.570823
$188$	−0.816297	−0.0595346
$189$	0	0
$190$	−51.2562	−3.71852
$191$	1.73716	0.125696	0.0628482	−	0.998023i	$-0.479982\pi$
$191$	1.73716	0.125696	0.0628482	+	0.998023i	$0.479982\pi$
$192$	0	0
$193$	−17.2995	−1.24525	−0.622624	−	0.782521i	$-0.713933\pi$
$193$	−17.2995	−1.24525	−0.622624	+	0.782521i	$0.713933\pi$
$194$	13.1576	0.944661
$195$	0	0
$196$	3.79406	0.271004
$197$	−21.0062	−1.49663	−0.748316	−	0.663342i	$-0.769137\pi$
$197$	−21.0062	−1.49663	−0.748316	+	0.663342i	$0.769137\pi$
$198$	0	0
$199$	−23.1791	−1.64312	−0.821560	−	0.570121i	$-0.806896\pi$
$199$	−23.1791	−1.64312	−0.821560	+	0.570121i	$0.806896\pi$
$200$	−14.8133	−1.04746
$201$	0	0
$202$	−31.2069	−2.19571
$203$	24.1721	1.69655
$204$	0	0
$205$	6.53507	0.456429
$206$	8.04448	0.560486
$207$	0	0
$208$	−4.94432	−0.342827
$209$	7.17328	0.496186
$210$	0	0
$211$	−11.6248	−0.800286	−0.400143	−	0.916453i	$-0.631040\pi$
$211$	−11.6248	−0.800286	−0.400143	+	0.916453i	$0.631040\pi$
$212$	−0.0861467	−0.00591658
$213$	0	0
$214$	21.7372	1.48592
$215$	16.4942	1.12489
$216$	0	0
$217$	10.0074	0.679350
$218$	−7.85895	−0.532275
$219$	0	0
$220$	−4.90947	−0.330996
$221$	7.80589	0.525081
$222$	0	0
$223$	5.11200	0.342325	0.171162	−	0.985243i	$-0.445248\pi$
$223$	5.11200	0.342325	0.171162	+	0.985243i	$0.445248\pi$
$224$	−19.5012	−1.30298
$225$	0	0
$226$	15.7928	1.05052
$227$	−15.2290	−1.01078	−0.505391	−	0.862891i	$-0.668652\pi$
$227$	−15.2290	−1.01078	−0.505391	+	0.862891i	$0.668652\pi$
$228$	0	0
$229$	24.4237	1.61396	0.806982	−	0.590576i	$-0.201099\pi$
$229$	24.4237	1.61396	0.806982	+	0.590576i	$0.201099\pi$
$230$	24.0215	1.58393
$231$	0	0
$232$	−10.4693	−0.687344
$233$	27.4223	1.79649	0.898247	−	0.439491i	$-0.144841\pi$
$233$	27.4223	1.79649	0.898247	+	0.439491i	$0.144841\pi$
$234$	0	0
$235$	2.62342	0.171133
$236$	−10.9855	−0.715098
$237$	0	0
$238$	44.5587	2.88831
$239$	−21.6036	−1.39742	−0.698709	−	0.715406i	$-0.746242\pi$
$239$	−21.6036	−1.39742	−0.698709	+	0.715406i	$0.746242\pi$
$240$	0	0
$241$	6.32134	0.407193	0.203597	−	0.979055i	$-0.434737\pi$
$241$	6.32134	0.407193	0.203597	+	0.979055i	$0.434737\pi$
$242$	1.79888	0.115636
$243$	0	0
$244$	−6.77181	−0.433521
$245$	−12.1933	−0.779004
$246$	0	0
$247$	7.17328	0.456425
$248$	−4.33439	−0.275234
$249$	0	0
$250$	−41.2867	−2.61120
$251$	9.48640	0.598776	0.299388	−	0.954131i	$-0.403217\pi$
$251$	9.48640	0.598776	0.299388	+	0.954131i	$0.403217\pi$
$252$	0	0
$253$	−3.36179	−0.211354
$254$	4.35663	0.273359
$255$	0	0
$256$	20.6683	1.29177
$257$	11.6370	0.725896	0.362948	−	0.931809i	$-0.381770\pi$
$257$	11.6370	0.725896	0.362948	+	0.931809i	$0.381770\pi$
$258$	0	0
$259$	9.32090	0.579173
$260$	−4.90947	−0.304472
$261$	0	0
$262$	15.0397	0.929154
$263$	9.95834	0.614057	0.307029	−	0.951700i	$-0.400665\pi$
$263$	9.95834	0.614057	0.307029	+	0.951700i	$0.400665\pi$
$264$	0	0
$265$	0.276858	0.0170073
$266$	40.9476	2.51066
$267$	0	0
$268$	6.79811	0.415260
$269$	16.4303	1.00177	0.500886	−	0.865513i	$-0.333008\pi$
$269$	16.4303	1.00177	0.500886	+	0.865513i	$0.333008\pi$
$270$	0	0
$271$	−29.1761	−1.77232	−0.886161	−	0.463378i	$-0.846637\pi$
$271$	−29.1761	−1.77232	−0.886161	+	0.463378i	$0.846637\pi$
$272$	−38.5948	−2.34015
$273$	0	0
$274$	17.7250	1.07081
$275$	10.7780	0.649941
$276$	0	0
$277$	−4.20716	−0.252784	−0.126392	−	0.991980i	$-0.540340\pi$
$277$	−4.20716	−0.252784	−0.126392	+	0.991980i	$0.540340\pi$
$278$	−4.93469	−0.295963
$279$	0	0
$280$	17.3240	1.03531
$281$	32.9113	1.96332	0.981662	−	0.190628i	$-0.0610525\pi$
$281$	32.9113	1.96332	0.981662	+	0.190628i	$0.0610525\pi$
$282$	0	0
$283$	−19.8690	−1.18109	−0.590545	−	0.807004i	$-0.701087\pi$
$283$	−19.8690	−1.18109	−0.590545	+	0.807004i	$0.701087\pi$
$284$	−2.61879	−0.155396
$285$	0	0
$286$	1.79888	0.106370
$287$	−5.22074	−0.308170
$288$	0	0
$289$	43.9319	2.58423
$290$	−54.4295	−3.19621
$291$	0	0
$292$	−18.1877	−1.06436
$293$	−7.00536	−0.409257	−0.204629	−	0.978840i	$-0.565599\pi$
$293$	−7.00536	−0.409257	−0.204629	+	0.978840i	$0.565599\pi$
$294$	0	0
$295$	35.3053	2.05556
$296$	−4.03703	−0.234648
$297$	0	0
$298$	−7.18073	−0.415968
$299$	−3.36179	−0.194417
$300$	0	0
$301$	−13.1769	−0.759502
$302$	16.5268	0.951013
$303$	0	0
$304$	−35.4670	−2.03417
$305$	21.7633	1.24616
$306$	0	0
$307$	−8.67087	−0.494873	−0.247436	−	0.968904i	$-0.579588\pi$
$307$	−8.67087	−0.494873	−0.247436	+	0.968904i	$0.579588\pi$
$308$	3.92208	0.223481
$309$	0	0
$310$	−22.5343	−1.27986
$311$	30.0689	1.70505	0.852526	−	0.522685i	$-0.175069\pi$
$311$	30.0689	1.70505	0.852526	+	0.522685i	$0.175069\pi$
$312$	0	0
$313$	−30.6662	−1.73336	−0.866679	−	0.498866i	$-0.833750\pi$
$313$	−30.6662	−1.73336	−0.866679	+	0.498866i	$0.833750\pi$
$314$	−33.9056	−1.91341
$315$	0	0
$316$	17.9203	1.00810
$317$	−7.49559	−0.420994	−0.210497	−	0.977594i	$-0.567508\pi$
$317$	−7.49559	−0.420994	−0.210497	+	0.977594i	$0.567508\pi$
$318$	0	0
$319$	7.61738	0.426491
$320$	4.63261	0.258971
$321$	0	0
$322$	−19.1903	−1.06943
$323$	55.9938	3.11558
$324$	0	0
$325$	10.7780	0.597859
$326$	−28.2474	−1.56448
$327$	0	0
$328$	2.26119	0.124853
$329$	−2.09580	−0.115545
$330$	0	0
$331$	−10.8538	−0.596578	−0.298289	−	0.954476i	$-0.596416\pi$
$331$	−10.8538	−0.596578	−0.298289	+	0.954476i	$0.596416\pi$
$332$	14.1795	0.778202
$333$	0	0
$334$	1.27642	0.0698425
$335$	−21.8477	−1.19367
$336$	0	0
$337$	10.0837	0.549295	0.274648	−	0.961545i	$-0.411439\pi$
$337$	10.0837	0.549295	0.274648	+	0.961545i	$0.411439\pi$
$338$	1.79888	0.0978462
$339$	0	0
$340$	−38.3228	−2.07834
$341$	3.15366	0.170780
$342$	0	0
$343$	−12.4719	−0.673421
$344$	5.70712	0.307707
$345$	0	0
$346$	−12.3439	−0.663614
$347$	0.104794	0.00562563	0.00281281	−	0.999996i	$-0.499105\pi$
$347$	0.104794	0.00562563	0.00281281	+	0.999996i	$0.499105\pi$
$348$	0	0
$349$	−12.0566	−0.645373	−0.322686	−	0.946506i	$-0.604586\pi$
$349$	−12.0566	−0.645373	−0.322686	+	0.946506i	$0.604586\pi$
$350$	61.5249	3.28864
$351$	0	0
$352$	−6.14544	−0.327553
$353$	−6.16812	−0.328296	−0.164148	−	0.986436i	$-0.552487\pi$
$353$	−6.16812	−0.328296	−0.164148	+	0.986436i	$0.552487\pi$
$354$	0	0
$355$	8.41626	0.446689
$356$	−1.19290	−0.0632234
$357$	0	0
$358$	−18.2425	−0.964148
$359$	−8.67612	−0.457908	−0.228954	−	0.973437i	$-0.573531\pi$
$359$	−8.67612	−0.457908	−0.228954	+	0.973437i	$0.573531\pi$
$360$	0	0
$361$	32.4559	1.70821
$362$	6.77104	0.355878
$363$	0	0
$364$	3.92208	0.205573
$365$	58.4517	3.05950
$366$	0	0
$367$	19.9443	1.04108	0.520542	−	0.853836i	$-0.325730\pi$
$367$	19.9443	1.04108	0.520542	+	0.853836i	$0.325730\pi$
$368$	16.6218	0.866469
$369$	0	0
$370$	−20.9884	−1.09113
$371$	−0.221177	−0.0114829
$372$	0	0
$373$	−5.56730	−0.288264	−0.144132	−	0.989558i	$-0.546039\pi$
$373$	−5.56730	−0.288264	−0.144132	+	0.989558i	$0.546039\pi$
$374$	14.0419	0.726087
$375$	0	0
$376$	0.907723	0.0468122
$377$	7.61738	0.392315
$378$	0	0
$379$	8.64663	0.444147	0.222074	−	0.975030i	$-0.428717\pi$
$379$	8.64663	0.444147	0.222074	+	0.975030i	$0.428717\pi$
$380$	−35.2170	−1.80659
$381$	0	0
$382$	3.12494	0.159886
$383$	−8.41626	−0.430051	−0.215025	−	0.976608i	$-0.568983\pi$
$383$	−8.41626	−0.430051	−0.215025	+	0.976608i	$0.568983\pi$
$384$	0	0
$385$	−12.6048	−0.642399
$386$	−31.1198	−1.58396
$387$	0	0
$388$	9.04029	0.458951
$389$	−33.8330	−1.71540	−0.857699	−	0.514151i	$-0.828107\pi$
$389$	−33.8330	−1.71540	−0.857699	+	0.514151i	$0.828107\pi$
$390$	0	0
$391$	−26.2418	−1.32710
$392$	−4.21899	−0.213091
$393$	0	0
$394$	−37.7877	−1.90372
$395$	−57.5924	−2.89779
$396$	0	0
$397$	−25.5660	−1.28312	−0.641560	−	0.767073i	$-0.721712\pi$
$397$	−25.5660	−1.28312	−0.641560	+	0.767073i	$0.721712\pi$
$398$	−41.6964	−2.09005
$399$	0	0
$400$	−53.2901	−2.66451
$401$	−0.122757	−0.00613020	−0.00306510	−	0.999995i	$-0.500976\pi$
$401$	−0.122757	−0.00613020	−0.00306510	+	0.999995i	$0.500976\pi$
$402$	0	0
$403$	3.15366	0.157095
$404$	−21.4416	−1.06676
$405$	0	0
$406$	43.4826	2.15801
$407$	2.93731	0.145597
$408$	0	0
$409$	−12.4637	−0.616291	−0.308146	−	0.951339i	$-0.599708\pi$
$409$	−12.4637	−0.616291	−0.308146	+	0.951339i	$0.599708\pi$
$410$	11.7558	0.580578
$411$	0	0
$412$	5.52718	0.272305
$413$	−28.2047	−1.38786
$414$	0	0
$415$	−45.5701	−2.23695
$416$	−6.14544	−0.301305
$417$	0	0
$418$	12.9039	0.631149
$419$	15.4324	0.753921	0.376960	−	0.926229i	$-0.376969\pi$
$419$	15.4324	0.753921	0.376960	+	0.926229i	$0.376969\pi$
$420$	0	0
$421$	−2.61134	−0.127269	−0.0636344	−	0.997973i	$-0.520269\pi$
$421$	−2.61134	−0.127269	−0.0636344	+	0.997973i	$0.520269\pi$
$422$	−20.9117	−1.01796
$423$	0	0
$424$	0.0957952	0.00465223
$425$	84.1322	4.08101
$426$	0	0
$427$	−17.3862	−0.841379
$428$	14.9351	0.721916
$429$	0	0
$430$	29.6711	1.43087
$431$	2.31852	0.111679	0.0558396	−	0.998440i	$-0.482216\pi$
$431$	2.31852	0.111679	0.0558396	+	0.998440i	$0.482216\pi$
$432$	0	0
$433$	−17.0842	−0.821012	−0.410506	−	0.911858i	$-0.634648\pi$
$433$	−17.0842	−0.821012	−0.410506	+	0.911858i	$0.634648\pi$
$434$	18.0022	0.864133
$435$	0	0
$436$	−5.39970	−0.258599
$437$	−24.1151	−1.15358
$438$	0	0
$439$	1.62395	0.0775068	0.0387534	−	0.999249i	$-0.487661\pi$
$439$	1.62395	0.0775068	0.0387534	+	0.999249i	$0.487661\pi$
$440$	5.45933	0.260263
$441$	0	0
$442$	14.0419	0.667903
$443$	−2.30208	−0.109375	−0.0546874	−	0.998504i	$-0.517416\pi$
$443$	−2.30208	−0.109375	−0.0546874	+	0.998504i	$0.517416\pi$
$444$	0	0
$445$	3.83373	0.181736
$446$	9.19587	0.435437
$447$	0	0
$448$	−3.70090	−0.174851
$449$	39.4726	1.86283	0.931413	−	0.363964i	$-0.118577\pi$
$449$	39.4726	1.86283	0.931413	+	0.363964i	$0.118577\pi$
$450$	0	0
$451$	−1.64522	−0.0774703
$452$	10.8509	0.510384
$453$	0	0
$454$	−27.3951	−1.28571
$455$	−12.6048	−0.590921
$456$	0	0
$457$	19.9191	0.931776	0.465888	−	0.884844i	$-0.345735\pi$
$457$	19.9191	0.931776	0.465888	+	0.884844i	$0.345735\pi$
$458$	43.9353	2.05296
$459$	0	0
$460$	16.5046	0.769531
$461$	−31.7855	−1.48040	−0.740199	−	0.672388i	$-0.765269\pi$
$461$	−31.7855	−1.48040	−0.740199	+	0.672388i	$0.765269\pi$
$462$	0	0
$463$	−5.75561	−0.267486	−0.133743	−	0.991016i	$-0.542700\pi$
$463$	−5.75561	−0.267486	−0.133743	+	0.991016i	$0.542700\pi$
$464$	−37.6627	−1.74845
$465$	0	0
$466$	49.3294	2.28514
$467$	4.38745	0.203027	0.101513	−	0.994834i	$-0.467632\pi$
$467$	4.38745	0.203027	0.101513	+	0.994834i	$0.467632\pi$
$468$	0	0
$469$	17.4537	0.805938
$470$	4.71921	0.217681
$471$	0	0
$472$	12.2159	0.562284
$473$	−4.15245	−0.190930
$474$	0	0
$475$	77.3139	3.54741
$476$	30.6153	1.40325
$477$	0	0
$478$	−38.8622	−1.77752
$479$	−27.0062	−1.23395	−0.616973	−	0.786984i	$-0.711641\pi$
$479$	−27.0062	−1.23395	−0.616973	+	0.786984i	$0.711641\pi$
$480$	0	0
$481$	2.93731	0.133930
$482$	11.3713	0.517950
$483$	0	0
$484$	1.23597	0.0561804
$485$	−29.0537	−1.31926
$486$	0	0
$487$	−0.800534	−0.0362757	−0.0181378	−	0.999835i	$-0.505774\pi$
$487$	−0.800534	−0.0362757	−0.0181378	+	0.999835i	$0.505774\pi$
$488$	7.53026	0.340879
$489$	0	0
$490$	−21.9343	−0.990893
$491$	−9.26985	−0.418342	−0.209171	−	0.977879i	$-0.567077\pi$
$491$	−9.26985	−0.418342	−0.209171	+	0.977879i	$0.567077\pi$
$492$	0	0
$493$	59.4604	2.67796
$494$	12.9039	0.580573
$495$	0	0
$496$	−15.5927	−0.700133
$497$	−6.72358	−0.301594
$498$	0	0
$499$	14.1116	0.631720	0.315860	−	0.948806i	$-0.397707\pi$
$499$	14.1116	0.631720	0.315860	+	0.948806i	$0.397707\pi$
$500$	−28.3671	−1.26862
$501$	0	0
$502$	17.0649	0.761643
$503$	15.2512	0.680017	0.340009	−	0.940422i	$-0.389570\pi$
$503$	15.2512	0.680017	0.340009	+	0.940422i	$0.389570\pi$
$504$	0	0
$505$	68.9089	3.06641
$506$	−6.04746	−0.268842
$507$	0	0
$508$	2.99334	0.132808
$509$	−7.69993	−0.341293	−0.170647	−	0.985332i	$-0.554586\pi$
$509$	−7.69993	−0.341293	−0.170647	+	0.985332i	$0.554586\pi$
$510$	0	0
$511$	−46.6959	−2.06571
$512$	16.7941	0.742200
$513$	0	0
$514$	20.9336	0.923340
$515$	−17.7633	−0.782743
$516$	0	0
$517$	−0.660451	−0.0290466
$518$	16.7672	0.736708
$519$	0	0
$520$	5.45933	0.239408
$521$	7.30776	0.320159	0.160079	−	0.987104i	$-0.448825\pi$
$521$	7.30776	0.320159	0.160079	+	0.987104i	$0.448825\pi$
$522$	0	0
$523$	21.5716	0.943259	0.471630	−	0.881797i	$-0.343666\pi$
$523$	21.5716	0.943259	0.471630	+	0.881797i	$0.343666\pi$
$524$	10.3334	0.451417
$525$	0	0
$526$	17.9139	0.781081
$527$	24.6171	1.07234
$528$	0	0
$529$	−11.6984	−0.508625
$530$	0.498035	0.0216333
$531$	0	0
$532$	28.1342	1.21977
$533$	−1.64522	−0.0712623
$534$	0	0
$535$	−47.9984	−2.07515
$536$	−7.55950	−0.326520
$537$	0	0
$538$	29.5561	1.27425
$539$	3.06970	0.132221
$540$	0	0
$541$	18.1394	0.779874	0.389937	−	0.920842i	$-0.372497\pi$
$541$	18.1394	0.779874	0.389937	+	0.920842i	$0.372497\pi$
$542$	−52.4843	−2.25439
$543$	0	0
$544$	−47.9706	−2.05672
$545$	17.3536	0.743345
$546$	0	0
$547$	35.5757	1.52111	0.760554	−	0.649275i	$-0.224928\pi$
$547$	35.5757	1.52111	0.760554	+	0.649275i	$0.224928\pi$
$548$	12.1784	0.520237
$549$	0	0
$550$	19.3884	0.826725
$551$	54.6416	2.32781
$552$	0	0
$553$	46.0094	1.95652
$554$	−7.56817	−0.321541
$555$	0	0
$556$	−3.39051	−0.143790
$557$	−37.4670	−1.58753	−0.793763	−	0.608227i	$-0.791881\pi$
$557$	−37.4670	−1.58753	−0.793763	+	0.608227i	$0.791881\pi$
$558$	0	0
$559$	−4.15245	−0.175630
$560$	62.3220	2.63358
$561$	0	0
$562$	59.2035	2.49735
$563$	−19.4447	−0.819498	−0.409749	−	0.912198i	$-0.634384\pi$
$563$	−19.4447	−0.819498	−0.409749	+	0.912198i	$0.634384\pi$
$564$	0	0
$565$	−34.8726	−1.46710
$566$	−35.7420	−1.50235
$567$	0	0
$568$	2.91209	0.122189
$569$	−2.59635	−0.108845	−0.0544223	−	0.998518i	$-0.517332\pi$
$569$	−2.59635	−0.108845	−0.0544223	+	0.998518i	$0.517332\pi$
$570$	0	0
$571$	−15.3427	−0.642073	−0.321036	−	0.947067i	$-0.604031\pi$
$571$	−15.3427	−0.642073	−0.321036	+	0.947067i	$0.604031\pi$
$572$	1.23597	0.0516785
$573$	0	0
$574$	−9.39148	−0.391993
$575$	−36.2335	−1.51104
$576$	0	0
$577$	9.29044	0.386766	0.193383	−	0.981123i	$-0.438054\pi$
$577$	9.29044	0.386766	0.193383	+	0.981123i	$0.438054\pi$
$578$	79.0282	3.28714
$579$	0	0
$580$	−37.3973	−1.55284
$581$	36.4051	1.51034
$582$	0	0
$583$	−0.0696997	−0.00288667
$584$	20.2248	0.836907
$585$	0	0
$586$	−12.6018	−0.520575
$587$	−20.1819	−0.832998	−0.416499	−	0.909136i	$-0.636743\pi$
$587$	−20.1819	−0.832998	−0.416499	+	0.909136i	$0.636743\pi$
$588$	0	0
$589$	22.6221	0.932127
$590$	63.5101	2.61467
$591$	0	0
$592$	−14.5230	−0.596891
$593$	−28.8800	−1.18596	−0.592979	−	0.805218i	$-0.702048\pi$
$593$	−28.8800	−1.18596	−0.592979	+	0.805218i	$0.702048\pi$
$594$	0	0
$595$	−98.3914	−4.03366
$596$	−4.93371	−0.202093
$597$	0	0
$598$	−6.04746	−0.247299
$599$	−22.1515	−0.905085	−0.452542	−	0.891743i	$-0.649483\pi$
$599$	−22.1515	−0.905085	−0.452542	+	0.891743i	$0.649483\pi$
$600$	0	0
$601$	−39.9307	−1.62881	−0.814403	−	0.580299i	$-0.802936\pi$
$601$	−39.9307	−1.62881	−0.814403	+	0.580299i	$0.802936\pi$
$602$	−23.7036	−0.966087
$603$	0	0
$604$	11.3552	0.462037
$605$	−3.97216	−0.161491
$606$	0	0
$607$	10.2320	0.415305	0.207653	−	0.978203i	$-0.433418\pi$
$607$	10.2320	0.415305	0.207653	+	0.978203i	$0.433418\pi$
$608$	−44.0829	−1.78780
$609$	0	0
$610$	39.1495	1.58512
$611$	−0.660451	−0.0267190
$612$	0	0
$613$	17.2013	0.694755	0.347377	−	0.937725i	$-0.387072\pi$
$613$	17.2013	0.694755	0.347377	+	0.937725i	$0.387072\pi$
$614$	−15.5979	−0.629479
$615$	0	0
$616$	−4.36135	−0.175724
$617$	1.30426	0.0525075	0.0262538	−	0.999655i	$-0.491642\pi$
$617$	1.30426	0.0525075	0.0262538	+	0.999655i	$0.491642\pi$
$618$	0	0
$619$	−20.3698	−0.818730	−0.409365	−	0.912371i	$-0.634250\pi$
$619$	−20.3698	−0.818730	−0.409365	+	0.912371i	$0.634250\pi$
$620$	−15.4828	−0.621804
$621$	0	0
$622$	54.0904	2.16883
$623$	−3.06269	−0.122704
$624$	0	0
$625$	37.2761	1.49104
$626$	−55.1649	−2.20483
$627$	0	0
$628$	−23.2958	−0.929603
$629$	22.9283	0.914212
$630$	0	0
$631$	6.67184	0.265602	0.132801	−	0.991143i	$-0.457603\pi$
$631$	6.67184	0.265602	0.132801	+	0.991143i	$0.457603\pi$
$632$	−19.9274	−0.792670
$633$	0	0
$634$	−13.4837	−0.535505
$635$	−9.62000	−0.381758
$636$	0	0
$637$	3.06970	0.121626
$638$	13.7027	0.542497
$639$	0	0
$640$	−40.4878	−1.60042
$641$	17.9836	0.710308	0.355154	−	0.934808i	$-0.384428\pi$
$641$	17.9836	0.710308	0.355154	+	0.934808i	$0.384428\pi$
$642$	0	0
$643$	−13.9557	−0.550360	−0.275180	−	0.961393i	$-0.588737\pi$
$643$	−13.9557	−0.550360	−0.275180	+	0.961393i	$0.588737\pi$
$644$	−13.1852	−0.519570
$645$	0	0
$646$	100.726	3.96302
$647$	−23.7380	−0.933239	−0.466619	−	0.884458i	$-0.654528\pi$
$647$	−23.7380	−0.933239	−0.466619	+	0.884458i	$0.654528\pi$
$648$	0	0
$649$	−8.88820	−0.348892
$650$	19.3884	0.760476
$651$	0	0
$652$	−19.4081	−0.760081
$653$	10.8325	0.423909	0.211955	−	0.977280i	$-0.432017\pi$
$653$	10.8325	0.423909	0.211955	+	0.977280i	$0.432017\pi$
$654$	0	0
$655$	−33.2095	−1.29760
$656$	8.13448	0.317598
$657$	0	0
$658$	−3.77008	−0.146973
$659$	31.7315	1.23608	0.618041	−	0.786146i	$-0.287926\pi$
$659$	31.7315	1.23608	0.618041	+	0.786146i	$0.287926\pi$
$660$	0	0
$661$	−15.1225	−0.588198	−0.294099	−	0.955775i	$-0.595019\pi$
$661$	−15.1225	−0.588198	−0.294099	+	0.955775i	$0.595019\pi$
$662$	−19.5247	−0.758848
$663$	0	0
$664$	−15.7676	−0.611903
$665$	−90.4175	−3.50624
$666$	0	0
$667$	−25.6080	−0.991547
$668$	0.876998	0.0339321
$669$	0	0
$670$	−39.3015	−1.51835
$671$	−5.47895	−0.211512
$672$	0	0
$673$	10.4209	0.401696	0.200848	−	0.979622i	$-0.435630\pi$
$673$	10.4209	0.401696	0.200848	+	0.979622i	$0.435630\pi$
$674$	18.1394	0.698704
$675$	0	0
$676$	1.23597	0.0475373
$677$	22.5229	0.865625	0.432813	−	0.901484i	$-0.357521\pi$
$677$	22.5229	0.865625	0.432813	+	0.901484i	$0.357521\pi$
$678$	0	0
$679$	23.2104	0.890734
$680$	42.6149	1.63421
$681$	0	0
$682$	5.67306	0.217233
$683$	−40.6164	−1.55414	−0.777071	−	0.629413i	$-0.783295\pi$
$683$	−40.6164	−1.55414	−0.777071	+	0.629413i	$0.783295\pi$
$684$	0	0
$685$	−39.1391	−1.49543
$686$	−22.4355	−0.856592
$687$	0	0
$688$	20.5310	0.782738
$689$	−0.0696997	−0.00265535
$690$	0	0
$691$	−4.00725	−0.152443	−0.0762215	−	0.997091i	$-0.524286\pi$
$691$	−4.00725	−0.152443	−0.0762215	+	0.997091i	$0.524286\pi$
$692$	−8.48124	−0.322408
$693$	0	0
$694$	0.188512	0.00715580
$695$	10.8964	0.413325
$696$	0	0
$697$	−12.8424	−0.486440
$698$	−21.6883	−0.820915
$699$	0	0
$700$	42.2723	1.59774
$701$	18.0018	0.679920	0.339960	−	0.940440i	$-0.389586\pi$
$701$	18.0018	0.679920	0.339960	+	0.940440i	$0.389586\pi$
$702$	0	0
$703$	21.0701	0.794675
$704$	−1.16627	−0.0439555
$705$	0	0
$706$	−11.0957	−0.417592
$707$	−55.0499	−2.07037
$708$	0	0
$709$	−44.1696	−1.65883	−0.829413	−	0.558636i	$-0.811325\pi$
$709$	−44.1696	−1.65883	−0.829413	+	0.558636i	$0.811325\pi$
$710$	15.1398	0.568188
$711$	0	0
$712$	1.32650	0.0497127
$713$	−10.6019	−0.397046
$714$	0	0
$715$	−3.97216	−0.148550
$716$	−12.5340	−0.468419
$717$	0	0
$718$	−15.6073	−0.582459
$719$	−36.6931	−1.36842	−0.684211	−	0.729284i	$-0.739853\pi$
$719$	−36.6931	−1.36842	−0.684211	+	0.729284i	$0.739853\pi$
$720$	0	0
$721$	14.1907	0.528490
$722$	58.3843	2.17284
$723$	0	0
$724$	4.65223	0.172899
$725$	82.1005	3.04913
$726$	0	0
$727$	3.89984	0.144637	0.0723184	−	0.997382i	$-0.476960\pi$
$727$	3.89984	0.144637	0.0723184	+	0.997382i	$0.476960\pi$
$728$	−4.36135	−0.161642
$729$	0	0
$730$	105.148	3.89169
$731$	−32.4135	−1.19886
$732$	0	0
$733$	3.01986	0.111541	0.0557706	−	0.998444i	$-0.482238\pi$
$733$	3.01986	0.111541	0.0557706	+	0.998444i	$0.482238\pi$
$734$	35.8774	1.32426
$735$	0	0
$736$	20.6597	0.761526
$737$	5.50022	0.202603
$738$	0	0
$739$	4.95298	0.182198	0.0910992	−	0.995842i	$-0.470962\pi$
$739$	4.95298	0.182198	0.0910992	+	0.995842i	$0.470962\pi$
$740$	−14.4206	−0.530113
$741$	0	0
$742$	−0.397870	−0.0146063
$743$	34.8891	1.27996	0.639978	−	0.768393i	$-0.278943\pi$
$743$	34.8891	1.27996	0.639978	+	0.768393i	$0.278943\pi$
$744$	0	0
$745$	15.8560	0.580918
$746$	−10.0149	−0.366671
$747$	0	0
$748$	9.64784	0.352760
$749$	38.3450	1.40110
$750$	0	0
$751$	21.1539	0.771915	0.385958	−	0.922517i	$-0.373871\pi$
$751$	21.1539	0.771915	0.385958	+	0.922517i	$0.373871\pi$
$752$	3.26548	0.119080
$753$	0	0
$754$	13.7027	0.499025
$755$	−36.4934	−1.32813
$756$	0	0
$757$	−19.5264	−0.709699	−0.354850	−	0.934923i	$-0.615468\pi$
$757$	−19.5264	−0.709699	−0.354850	+	0.934923i	$0.615468\pi$
$758$	15.5542	0.564956
$759$	0	0
$760$	39.1613	1.42053
$761$	−22.2546	−0.806729	−0.403365	−	0.915039i	$-0.632159\pi$
$761$	−22.2546	−0.806729	−0.403365	+	0.915039i	$0.632159\pi$
$762$	0	0
$763$	−13.8634	−0.501889
$764$	2.14708	0.0776785
$765$	0	0
$766$	−15.1398	−0.547025
$767$	−8.88820	−0.320934
$768$	0	0
$769$	−46.4424	−1.67475	−0.837377	−	0.546626i	$-0.815912\pi$
$769$	−46.4424	−1.67475	−0.837377	+	0.546626i	$0.815912\pi$
$770$	−22.6745	−0.817131
$771$	0	0
$772$	−21.3817	−0.769544
$773$	49.6587	1.78610	0.893049	−	0.449960i	$-0.148562\pi$
$773$	49.6587	1.78610	0.893049	+	0.449960i	$0.148562\pi$
$774$	0	0
$775$	33.9903	1.22097
$776$	−10.0528	−0.360875
$777$	0	0
$778$	−60.8614	−2.18199
$779$	−11.8016	−0.422837
$780$	0	0
$781$	−2.11881	−0.0758170
$782$	−47.2058	−1.68808
$783$	0	0
$784$	−15.1776	−0.542056
$785$	74.8680	2.67215
$786$	0	0
$787$	4.95298	0.176555	0.0882774	−	0.996096i	$-0.471864\pi$
$787$	4.95298	0.176555	0.0882774	+	0.996096i	$0.471864\pi$
$788$	−25.9631	−0.924896
$789$	0	0
$790$	−103.602	−3.68598
$791$	27.8591	0.990554
$792$	0	0
$793$	−5.47895	−0.194563
$794$	−45.9901	−1.63213
$795$	0	0
$796$	−28.6486	−1.01542
$797$	−31.3325	−1.10985	−0.554927	−	0.831899i	$-0.687254\pi$
$797$	−31.3325	−1.10985	−0.554927	+	0.831899i	$0.687254\pi$
$798$	0	0
$799$	−5.15541	−0.182385
$800$	−66.2358	−2.34179
$801$	0	0
$802$	−0.220826	−0.00779762
$803$	−14.7154	−0.519294
$804$	0	0
$805$	42.3746	1.49351
$806$	5.67306	0.199825
$807$	0	0
$808$	23.8430	0.838795
$809$	41.9985	1.47659	0.738295	−	0.674478i	$-0.235631\pi$
$809$	41.9985	1.47659	0.738295	+	0.674478i	$0.235631\pi$
$810$	0	0
$811$	45.2186	1.58784	0.793921	−	0.608021i	$-0.208037\pi$
$811$	45.2186	1.58784	0.793921	+	0.608021i	$0.208037\pi$
$812$	29.8759	1.04844
$813$	0	0
$814$	5.28387	0.185199
$815$	62.3738	2.18486
$816$	0	0
$817$	−29.7867	−1.04210
$818$	−22.4207	−0.783922
$819$	0	0
$820$	8.07715	0.282066
$821$	−21.2850	−0.742853	−0.371426	−	0.928462i	$-0.621131\pi$
$821$	−21.2850	−0.742853	−0.371426	+	0.928462i	$0.621131\pi$
$822$	0	0
$823$	18.9467	0.660440	0.330220	−	0.943904i	$-0.392877\pi$
$823$	18.9467	0.660440	0.330220	+	0.943904i	$0.392877\pi$
$824$	−6.14623	−0.214114
$825$	0	0
$826$	−50.7369	−1.76536
$827$	46.8411	1.62883	0.814413	−	0.580286i	$-0.197059\pi$
$827$	46.8411	1.62883	0.814413	+	0.580286i	$0.197059\pi$
$828$	0	0
$829$	4.69224	0.162968	0.0814841	−	0.996675i	$-0.474034\pi$
$829$	4.69224	0.162968	0.0814841	+	0.996675i	$0.474034\pi$
$830$	−81.9752	−2.84540
$831$	0	0
$832$	−1.16627	−0.0404331
$833$	23.9617	0.830225
$834$	0	0
$835$	−2.81850	−0.0975381
$836$	8.86596	0.306635
$837$	0	0
$838$	27.7610	0.958987
$839$	0.234316	0.00808948	0.00404474	−	0.999992i	$-0.498713\pi$
$839$	0.234316	0.00808948	0.00404474	+	0.999992i	$0.498713\pi$
$840$	0	0
$841$	29.0244	1.00084
$842$	−4.69749	−0.161886
$843$	0	0
$844$	−14.3679	−0.494565
$845$	−3.97216	−0.136646
$846$	0	0
$847$	3.17328	0.109035
$848$	0.344618	0.0118342
$849$	0	0
$850$	151.344	5.19105
$851$	−9.87462	−0.338498
$852$	0	0
$853$	22.8770	0.783294	0.391647	−	0.920116i	$-0.371906\pi$
$853$	22.8770	0.783294	0.391647	+	0.920116i	$0.371906\pi$
$854$	−31.2757	−1.07023
$855$	0	0
$856$	−16.6078	−0.567645
$857$	−25.3545	−0.866094	−0.433047	−	0.901371i	$-0.642562\pi$
$857$	−25.3545	−0.866094	−0.433047	+	0.901371i	$0.642562\pi$
$858$	0	0
$859$	46.2580	1.57830	0.789151	−	0.614199i	$-0.210521\pi$
$859$	46.2580	1.57830	0.789151	+	0.614199i	$0.210521\pi$
$860$	20.3863	0.695167
$861$	0	0
$862$	4.17074	0.142056
$863$	48.4921	1.65069	0.825345	−	0.564629i	$-0.190981\pi$
$863$	48.4921	1.65069	0.825345	+	0.564629i	$0.190981\pi$
$864$	0	0
$865$	27.2570	0.926766
$866$	−30.7324	−1.04433
$867$	0	0
$868$	12.3689	0.419828
$869$	14.4990	0.491845
$870$	0	0
$871$	5.50022	0.186368
$872$	6.00447	0.203337
$873$	0	0
$874$	−43.3801	−1.46735
$875$	−72.8310	−2.46214
$876$	0	0
$877$	−32.7628	−1.10632	−0.553161	−	0.833074i	$-0.686579\pi$
$877$	−32.7628	−1.10632	−0.553161	+	0.833074i	$0.686579\pi$
$878$	2.92129	0.0985886
$879$	0	0
$880$	19.6396	0.662051
$881$	21.7773	0.733695	0.366847	−	0.930281i	$-0.380437\pi$
$881$	21.7773	0.733695	0.366847	+	0.930281i	$0.380437\pi$
$882$	0	0
$883$	−34.3137	−1.15475	−0.577373	−	0.816480i	$-0.695922\pi$
$883$	−34.3137	−1.15475	−0.577373	+	0.816480i	$0.695922\pi$
$884$	9.64784	0.324492
$885$	0	0
$886$	−4.14116	−0.139125
$887$	−48.9064	−1.64211	−0.821057	−	0.570846i	$-0.806615\pi$
$887$	−48.9064	−1.64211	−0.821057	+	0.570846i	$0.806615\pi$
$888$	0	0
$889$	7.68523	0.257754
$890$	6.89642	0.231169
$891$	0	0
$892$	6.31827	0.211551
$893$	−4.73760	−0.158538
$894$	0	0
$895$	40.2819	1.34647
$896$	32.3449	1.08057
$897$	0	0
$898$	71.0064	2.36952
$899$	24.0226	0.801200
$900$	0	0
$901$	−0.544068	−0.0181255
$902$	−2.95955	−0.0985423
$903$	0	0
$904$	−12.0662	−0.401316
$905$	−14.9513	−0.496999
$906$	0	0
$907$	−49.9956	−1.66008	−0.830039	−	0.557706i	$-0.811682\pi$
$907$	−49.9956	−1.66008	−0.830039	+	0.557706i	$0.811682\pi$
$908$	−18.8225	−0.624648
$909$	0	0
$910$	−22.6745	−0.751651
$911$	8.52762	0.282533	0.141266	−	0.989972i	$-0.454883\pi$
$911$	8.52762	0.282533	0.141266	+	0.989972i	$0.454883\pi$
$912$	0	0
$913$	11.4724	0.379680
$914$	35.8321	1.18522
$915$	0	0
$916$	30.1870	0.997405
$917$	26.5304	0.876112
$918$	0	0
$919$	1.49112	0.0491874	0.0245937	−	0.999698i	$-0.492171\pi$
$919$	1.49112	0.0491874	0.0245937	+	0.999698i	$0.492171\pi$
$920$	−18.3531	−0.605085
$921$	0	0
$922$	−57.1783	−1.88307
$923$	−2.11881	−0.0697415
$924$	0	0
$925$	31.6585	1.04092
$926$	−10.3537	−0.340242
$927$	0	0
$928$	−46.8121	−1.53668
$929$	31.1447	1.02182	0.510912	−	0.859633i	$-0.329308\pi$
$929$	31.1447	1.02182	0.510912	+	0.859633i	$0.329308\pi$
$930$	0	0
$931$	22.0198	0.721670
$932$	33.8931	1.11021
$933$	0	0
$934$	7.89249	0.258250
$935$	−31.0062	−1.01401
$936$	0	0
$937$	−2.88357	−0.0942021	−0.0471010	−	0.998890i	$-0.514998\pi$
$937$	−2.88357	−0.0942021	−0.0471010	+	0.998890i	$0.514998\pi$
$938$	31.3972	1.02515
$939$	0	0
$940$	3.24246	0.105757
$941$	−1.78462	−0.0581769	−0.0290884	−	0.999577i	$-0.509260\pi$
$941$	−1.78462	−0.0581769	−0.0290884	+	0.999577i	$0.509260\pi$
$942$	0	0
$943$	5.53088	0.180110
$944$	43.9461	1.43032
$945$	0	0
$946$	−7.46975	−0.242863
$947$	−17.9518	−0.583354	−0.291677	−	0.956517i	$-0.594213\pi$
$947$	−17.9518	−0.583354	−0.291677	+	0.956517i	$0.594213\pi$
$948$	0	0
$949$	−14.7154	−0.477681
$950$	139.079	4.51230
$951$	0	0
$952$	−34.0442	−1.10338
$953$	33.2857	1.07823	0.539115	−	0.842232i	$-0.318759\pi$
$953$	33.2857	1.07823	0.539115	+	0.842232i	$0.318759\pi$
$954$	0	0
$955$	−6.90027	−0.223288
$956$	−26.7013	−0.863583
$957$	0	0
$958$	−48.5810	−1.56958
$959$	31.2674	1.00968
$960$	0	0
$961$	−21.0544	−0.679175
$962$	5.28387	0.170359
$963$	0	0
$964$	7.81299	0.251639
$965$	68.7165	2.21206
$966$	0	0
$967$	22.4525	0.722024	0.361012	−	0.932561i	$-0.382431\pi$
$967$	22.4525	0.722024	0.361012	+	0.932561i	$0.382431\pi$
$968$	−1.37440	−0.0441749
$969$	0	0
$970$	−52.2641	−1.67810
$971$	−36.2108	−1.16206	−0.581029	−	0.813883i	$-0.697350\pi$
$971$	−36.2108	−1.16206	−0.581029	+	0.813883i	$0.697350\pi$
$972$	0	0
$973$	−8.70493	−0.279067
$974$	−1.44006	−0.0461427
$975$	0	0
$976$	27.0897	0.867119
$977$	32.7929	1.04914	0.524568	−	0.851368i	$-0.324227\pi$
$977$	32.7929	1.04914	0.524568	+	0.851368i	$0.324227\pi$
$978$	0	0
$979$	−0.965150	−0.0308463
$980$	−15.0706	−0.481412
$981$	0	0
$982$	−16.6753	−0.532132
$983$	8.04448	0.256579	0.128290	−	0.991737i	$-0.459051\pi$
$983$	8.04448	0.256579	0.128290	+	0.991737i	$0.459051\pi$
$984$	0	0
$985$	83.4401	2.65862
$986$	106.962	3.40637
$987$	0	0
$988$	8.86596	0.282064
$989$	13.9597	0.443891
$990$	0	0
$991$	−26.1679	−0.831251	−0.415626	−	0.909536i	$-0.636437\pi$
$991$	−26.1679	−0.831251	−0.415626	+	0.909536i	$0.636437\pi$
$992$	−19.3806	−0.615336
$993$	0	0
$994$	−12.0949	−0.383627
$995$	92.0710	2.91885
$996$	0	0
$997$	48.1911	1.52623	0.763114	−	0.646264i	$-0.223670\pi$
$997$	48.1911	1.52623	0.763114	+	0.646264i	$0.223670\pi$
$998$	25.3850	0.803548
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.m.1.3		4
3.2	odd	2		429.2.a.h.1.2	✓	4
12.11	even	2		6864.2.a.bz.1.4		4
33.32	even	2		4719.2.a.z.1.3		4
39.38	odd	2		5577.2.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.2	✓	4	3.2	odd	2
1287.2.a.m.1.3		4	1.1	even	1	trivial
4719.2.a.z.1.3		4	33.32	even	2
5577.2.a.m.1.3		4	39.38	odd	2
6864.2.a.bz.1.4		4	12.11	even	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79888 q^{2} +1.23597 q^{4} -3.97216 q^{5} +3.17328 q^{7} -1.37440 q^{8} +O(q^{10})\)	\(q+1.79888 q^{2} +1.23597 q^{4} -3.97216 q^{5} +3.17328 q^{7} -1.37440 q^{8} -7.14544 q^{10} +1.00000 q^{11} +1.00000 q^{13} +5.70835 q^{14} -4.94432 q^{16} +7.80589 q^{17} +7.17328 q^{19} -4.90947 q^{20} +1.79888 q^{22} -3.36179 q^{23} +10.7780 q^{25} +1.79888 q^{26} +3.92208 q^{28} +7.61738 q^{29} +3.15366 q^{31} -6.14544 q^{32} +14.0419 q^{34} -12.6048 q^{35} +2.93731 q^{37} +12.9039 q^{38} +5.45933 q^{40} -1.64522 q^{41} -4.15245 q^{43} +1.23597 q^{44} -6.04746 q^{46} -0.660451 q^{47} +3.06970 q^{49} +19.3884 q^{50} +1.23597 q^{52} -0.0696997 q^{53} -3.97216 q^{55} -4.36135 q^{56} +13.7027 q^{58} -8.88820 q^{59} -5.47895 q^{61} +5.67306 q^{62} -1.16627 q^{64} -3.97216 q^{65} +5.50022 q^{67} +9.64784 q^{68} -22.6745 q^{70} -2.11881 q^{71} -14.7154 q^{73} +5.28387 q^{74} +8.86596 q^{76} +3.17328 q^{77} +14.4990 q^{79} +19.6396 q^{80} -2.95955 q^{82} +11.4724 q^{83} -31.0062 q^{85} -7.46975 q^{86} -1.37440 q^{88} -0.965150 q^{89} +3.17328 q^{91} -4.15507 q^{92} -1.18807 q^{94} -28.4934 q^{95} +7.31433 q^{97} +5.52202 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7	\( 4 q + 2 q^{2} + 8 q^{4} + 2 q^{7} - 2 q^{10} + 4 q^{11} + 4 q^{13} - 12 q^{14} + 12 q^{16} + 8 q^{17} + 18 q^{19} + 10 q^{20} + 2 q^{22} + 4 q^{25} + 2 q^{26} - 6 q^{28} + 10 q^{29} + 12 q^{31} + 2 q^{32} + 36 q^{34} - 22 q^{35} - 2 q^{37} - 4 q^{38} + 20 q^{40} - 2 q^{41} + 28 q^{43} + 8 q^{44} - 30 q^{46} - 6 q^{47} + 8 q^{49} + 36 q^{50} + 8 q^{52} + 4 q^{53} - 48 q^{56} - 6 q^{58} - 16 q^{59} - 10 q^{61} + 34 q^{62} - 12 q^{64} + 34 q^{68} - 58 q^{70} - 10 q^{71} - 6 q^{73} - 14 q^{74} + 26 q^{76} + 2 q^{77} - 8 q^{79} + 48 q^{80} + 12 q^{82} + 8 q^{83} - 18 q^{85} + 8 q^{86} - 6 q^{89} + 2 q^{91} + 28 q^{92} - 46 q^{94} - 22 q^{95} + 10 q^{97} + 34 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 2 * q^7 - 2 * q^10 + 4 * q^11 + 4 * q^13 - 12 * q^14 + 12 * q^16 + 8 * q^17 + 18 * q^19 + 10 * q^20 + 2 * q^22 + 4 * q^25 + 2 * q^26 - 6 * q^28 + 10 * q^29 + 12 * q^31 + 2 * q^32 + 36 * q^34 - 22 * q^35 - 2 * q^37 - 4 * q^38 + 20 * q^40 - 2 * q^41 + 28 * q^43 + 8 * q^44 - 30 * q^46 - 6 * q^47 + 8 * q^49 + 36 * q^50 + 8 * q^52 + 4 * q^53 - 48 * q^56 - 6 * q^58 - 16 * q^59 - 10 * q^61 + 34 * q^62 - 12 * q^64 + 34 * q^68 - 58 * q^70 - 10 * q^71 - 6 * q^73 - 14 * q^74 + 26 * q^76 + 2 * q^77 - 8 * q^79 + 48 * q^80 + 12 * q^82 + 8 * q^83 - 18 * q^85 + 8 * q^86 - 6 * q^89 + 2 * q^91 + 28 * q^92 - 46 * q^94 - 22 * q^95 + 10 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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