LMFDB - Embedded newform 8001.2.a.ba.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.301452 q^{2} -1.90913 q^{4} -3.64431 q^{5} +1.00000 q^{7} +1.17841 q^{8} +O(q^{10})$	\(q-0.301452 q^{2} -1.90913 q^{4} -3.64431 q^{5} +1.00000 q^{7} +1.17841 q^{8} +1.09858 q^{10} -2.79621 q^{11} +5.84660 q^{13} -0.301452 q^{14} +3.46302 q^{16} +5.05184 q^{17} +5.17562 q^{19} +6.95745 q^{20} +0.842924 q^{22} +7.69561 q^{23} +8.28099 q^{25} -1.76247 q^{26} -1.90913 q^{28} -5.38739 q^{29} +8.19880 q^{31} -3.40076 q^{32} -1.52289 q^{34} -3.64431 q^{35} -10.6757 q^{37} -1.56020 q^{38} -4.29451 q^{40} +7.27955 q^{41} +1.97821 q^{43} +5.33832 q^{44} -2.31986 q^{46} -7.60450 q^{47} +1.00000 q^{49} -2.49632 q^{50} -11.1619 q^{52} -12.8340 q^{53} +10.1903 q^{55} +1.17841 q^{56} +1.62404 q^{58} -0.730147 q^{59} +12.6793 q^{61} -2.47154 q^{62} -5.90087 q^{64} -21.3068 q^{65} -13.4905 q^{67} -9.64459 q^{68} +1.09858 q^{70} +0.0983847 q^{71} +16.7418 q^{73} +3.21822 q^{74} -9.88092 q^{76} -2.79621 q^{77} +10.9907 q^{79} -12.6203 q^{80} -2.19444 q^{82} +15.2450 q^{83} -18.4105 q^{85} -0.596336 q^{86} -3.29510 q^{88} -4.72607 q^{89} +5.84660 q^{91} -14.6919 q^{92} +2.29239 q^{94} -18.8616 q^{95} -6.92959 q^{97} -0.301452 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.301452	−0.213159	−0.106579	−	0.994304i	$-0.533990\pi$
$2$	−0.301452	−0.213159	−0.106579	+	0.994304i	$0.533990\pi$
$3$	0	0
$3$	0	0
$4$	−1.90913	−0.954563
$5$	−3.64431	−1.62978	−0.814892	−	0.579612i	$-0.803204\pi$
$5$	−3.64431	−1.62978	−0.814892	+	0.579612i	$0.803204\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.17841	0.416632
$9$	0	0
$10$	1.09858	0.347403
$11$	−2.79621	−0.843090	−0.421545	−	0.906808i	$-0.638512\pi$
$11$	−2.79621	−0.843090	−0.421545	+	0.906808i	$0.638512\pi$
$12$	0	0
$13$	5.84660	1.62155	0.810777	−	0.585355i	$-0.199045\pi$
$13$	5.84660	1.62155	0.810777	+	0.585355i	$0.199045\pi$
$14$	−0.301452	−0.0805664
$15$	0	0
$16$	3.46302	0.865754
$17$	5.05184	1.22525	0.612625	−	0.790374i	$-0.290114\pi$
$17$	5.05184	1.22525	0.612625	+	0.790374i	$0.290114\pi$
$18$	0	0
$19$	5.17562	1.18737	0.593685	−	0.804698i	$-0.297673\pi$
$19$	5.17562	1.18737	0.593685	+	0.804698i	$0.297673\pi$
$20$	6.95745	1.55573
$21$	0	0
$22$	0.842924	0.179712
$23$	7.69561	1.60465	0.802323	−	0.596890i	$-0.203597\pi$
$23$	7.69561	1.60465	0.802323	+	0.596890i	$0.203597\pi$
$24$	0	0
$25$	8.28099	1.65620
$26$	−1.76247	−0.345649
$27$	0	0
$28$	−1.90913	−0.360791
$29$	−5.38739	−1.00041	−0.500206	−	0.865906i	$-0.666742\pi$
$29$	−5.38739	−1.00041	−0.500206	+	0.865906i	$0.666742\pi$
$30$	0	0
$31$	8.19880	1.47255	0.736274	−	0.676684i	$-0.236584\pi$
$31$	8.19880	1.47255	0.736274	+	0.676684i	$0.236584\pi$
$32$	−3.40076	−0.601176
$33$	0	0
$34$	−1.52289	−0.261173
$35$	−3.64431	−0.616001
$36$	0	0
$37$	−10.6757	−1.75508	−0.877539	−	0.479505i	$-0.840816\pi$
$37$	−10.6757	−1.75508	−0.877539	+	0.479505i	$0.840816\pi$
$38$	−1.56020	−0.253098
$39$	0	0
$40$	−4.29451	−0.679021
$41$	7.27955	1.13688	0.568438	−	0.822726i	$-0.307548\pi$
$41$	7.27955	1.13688	0.568438	+	0.822726i	$0.307548\pi$
$42$	0	0
$43$	1.97821	0.301674	0.150837	−	0.988559i	$-0.451803\pi$
$43$	1.97821	0.301674	0.150837	+	0.988559i	$0.451803\pi$
$44$	5.33832	0.804783
$45$	0	0
$46$	−2.31986	−0.342044
$47$	−7.60450	−1.10923	−0.554615	−	0.832107i	$-0.687134\pi$
$47$	−7.60450	−1.10923	−0.554615	+	0.832107i	$0.687134\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.49632	−0.353033
$51$	0	0
$52$	−11.1619	−1.54788
$53$	−12.8340	−1.76288	−0.881442	−	0.472293i	$-0.843427\pi$
$53$	−12.8340	−1.76288	−0.881442	+	0.472293i	$0.843427\pi$
$54$	0	0
$55$	10.1903	1.37405
$56$	1.17841	0.157472
$57$	0	0
$58$	1.62404	0.213247
$59$	−0.730147	−0.0950570	−0.0475285	−	0.998870i	$-0.515135\pi$
$59$	−0.730147	−0.0950570	−0.0475285	+	0.998870i	$0.515135\pi$
$60$	0	0
$61$	12.6793	1.62342	0.811711	−	0.584060i	$-0.198537\pi$
$61$	12.6793	1.62342	0.811711	+	0.584060i	$0.198537\pi$
$62$	−2.47154	−0.313886
$63$	0	0
$64$	−5.90087	−0.737609
$65$	−21.3068	−2.64278
$66$	0	0
$67$	−13.4905	−1.64812	−0.824062	−	0.566499i	$-0.808297\pi$
$67$	−13.4905	−1.64812	−0.824062	+	0.566499i	$0.808297\pi$
$68$	−9.64459	−1.16958
$69$	0	0
$70$	1.09858	0.131306
$71$	0.0983847	0.0116761	0.00583806	−	0.999983i	$-0.498142\pi$
$71$	0.0983847	0.0116761	0.00583806	+	0.999983i	$0.498142\pi$
$72$	0	0
$73$	16.7418	1.95948	0.979740	−	0.200276i	$-0.0641837\pi$
$73$	16.7418	1.95948	0.979740	+	0.200276i	$0.0641837\pi$
$74$	3.21822	0.374110
$75$	0	0
$76$	−9.88092	−1.13342
$77$	−2.79621	−0.318658
$78$	0	0
$79$	10.9907	1.23656	0.618278	−	0.785960i	$-0.287831\pi$
$79$	10.9907	1.23656	0.618278	+	0.785960i	$0.287831\pi$
$80$	−12.6203	−1.41099
$81$	0	0
$82$	−2.19444	−0.242335
$83$	15.2450	1.67336	0.836679	−	0.547694i	$-0.184494\pi$
$83$	15.2450	1.67336	0.836679	+	0.547694i	$0.184494\pi$
$84$	0	0
$85$	−18.4105	−1.99689
$86$	−0.596336	−0.0643045
$87$	0	0
$88$	−3.29510	−0.351258
$89$	−4.72607	−0.500962	−0.250481	−	0.968122i	$-0.580589\pi$
$89$	−4.72607	−0.500962	−0.250481	+	0.968122i	$0.580589\pi$
$90$	0	0
$91$	5.84660	0.612890
$92$	−14.6919	−1.53174
$93$	0	0
$94$	2.29239	0.236442
$95$	−18.8616	−1.93516
$96$	0	0
$97$	−6.92959	−0.703593	−0.351797	−	0.936076i	$-0.614429\pi$
$97$	−6.92959	−0.703593	−0.351797	+	0.936076i	$0.614429\pi$
$98$	−0.301452	−0.0304513
$99$	0	0
$100$	−15.8095	−1.58095
$101$	10.9160	1.08618	0.543090	−	0.839674i	$-0.317254\pi$
$101$	10.9160	1.08618	0.543090	+	0.839674i	$0.317254\pi$
$102$	0	0
$103$	−9.98495	−0.983846	−0.491923	−	0.870639i	$-0.663706\pi$
$103$	−9.98495	−0.983846	−0.491923	+	0.870639i	$0.663706\pi$
$104$	6.88971	0.675592
$105$	0	0
$106$	3.86883	0.375774
$107$	−2.74531	−0.265399	−0.132700	−	0.991156i	$-0.542365\pi$
$107$	−2.74531	−0.265399	−0.132700	+	0.991156i	$0.542365\pi$
$108$	0	0
$109$	−17.6466	−1.69024	−0.845121	−	0.534576i	$-0.820471\pi$
$109$	−17.6466	−1.69024	−0.845121	+	0.534576i	$0.820471\pi$
$110$	−3.07188	−0.292892
$111$	0	0
$112$	3.46302	0.327224
$113$	4.84737	0.456002	0.228001	−	0.973661i	$-0.426781\pi$
$113$	4.84737	0.456002	0.228001	+	0.973661i	$0.426781\pi$
$114$	0	0
$115$	−28.0452	−2.61523
$116$	10.2852	0.954957
$117$	0	0
$118$	0.220104	0.0202622
$119$	5.05184	0.463101
$120$	0	0
$121$	−3.18119	−0.289200
$122$	−3.82221	−0.346047
$123$	0	0
$124$	−15.6525	−1.40564
$125$	−11.9569	−1.06946
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	8.58035	0.758403
$129$	0	0
$130$	6.42298	0.563333
$131$	−17.4969	−1.52871	−0.764354	−	0.644797i	$-0.776942\pi$
$131$	−17.4969	−1.52871	−0.764354	+	0.644797i	$0.776942\pi$
$132$	0	0
$133$	5.17562	0.448784
$134$	4.06673	0.351312
$135$	0	0
$136$	5.95316	0.510479
$137$	12.4960	1.06760	0.533801	−	0.845610i	$-0.320763\pi$
$137$	12.4960	1.06760	0.533801	+	0.845610i	$0.320763\pi$
$138$	0	0
$139$	−2.97736	−0.252536	−0.126268	−	0.991996i	$-0.540300\pi$
$139$	−2.97736	−0.252536	−0.126268	+	0.991996i	$0.540300\pi$
$140$	6.95745	0.588012
$141$	0	0
$142$	−0.0296583	−0.00248887
$143$	−16.3483	−1.36712
$144$	0	0
$145$	19.6333	1.63046
$146$	−5.04685	−0.417680
$147$	0	0
$148$	20.3813	1.67533
$149$	21.3175	1.74640	0.873198	−	0.487365i	$-0.162042\pi$
$149$	21.3175	1.74640	0.873198	+	0.487365i	$0.162042\pi$
$150$	0	0
$151$	14.9011	1.21264	0.606318	−	0.795222i	$-0.292646\pi$
$151$	14.9011	1.21264	0.606318	+	0.795222i	$0.292646\pi$
$152$	6.09903	0.494697
$153$	0	0
$154$	0.842924	0.0679248
$155$	−29.8790	−2.39994
$156$	0	0
$157$	2.07880	0.165906	0.0829531	−	0.996553i	$-0.473565\pi$
$157$	2.07880	0.165906	0.0829531	+	0.996553i	$0.473565\pi$
$158$	−3.31318	−0.263583
$159$	0	0
$160$	12.3934	0.979787
$161$	7.69561	0.606499
$162$	0	0
$163$	−5.68064	−0.444942	−0.222471	−	0.974939i	$-0.571412\pi$
$163$	−5.68064	−0.444942	−0.222471	+	0.974939i	$0.571412\pi$
$164$	−13.8976	−1.08522
$165$	0	0
$166$	−4.59564	−0.356691
$167$	9.28314	0.718351	0.359176	−	0.933270i	$-0.383058\pi$
$167$	9.28314	0.718351	0.359176	+	0.933270i	$0.383058\pi$
$168$	0	0
$169$	21.1827	1.62944
$170$	5.54987	0.425655
$171$	0	0
$172$	−3.77666	−0.287967
$173$	−5.61444	−0.426858	−0.213429	−	0.976959i	$-0.568463\pi$
$173$	−5.61444	−0.426858	−0.213429	+	0.976959i	$0.568463\pi$
$174$	0	0
$175$	8.28099	0.625984
$176$	−9.68333	−0.729909
$177$	0	0
$178$	1.42468	0.106784
$179$	−16.7860	−1.25465	−0.627323	−	0.778759i	$-0.715849\pi$
$179$	−16.7860	−1.25465	−0.627323	+	0.778759i	$0.715849\pi$
$180$	0	0
$181$	23.0245	1.71140	0.855700	−	0.517473i	$-0.173127\pi$
$181$	23.0245	1.71140	0.855700	+	0.517473i	$0.173127\pi$
$182$	−1.76247	−0.130643
$183$	0	0
$184$	9.06862	0.668547
$185$	38.9056	2.86040
$186$	0	0
$187$	−14.1260	−1.03300
$188$	14.5179	1.05883
$189$	0	0
$190$	5.68586	0.412496
$191$	18.6884	1.35224	0.676122	−	0.736789i	$-0.263659\pi$
$191$	18.6884	1.35224	0.676122	+	0.736789i	$0.263659\pi$
$192$	0	0
$193$	−9.94568	−0.715906	−0.357953	−	0.933740i	$-0.616525\pi$
$193$	−9.94568	−0.715906	−0.357953	+	0.933740i	$0.616525\pi$
$194$	2.08894	0.149977
$195$	0	0
$196$	−1.90913	−0.136366
$197$	4.30924	0.307021	0.153510	−	0.988147i	$-0.450942\pi$
$197$	4.30924	0.307021	0.153510	+	0.988147i	$0.450942\pi$
$198$	0	0
$199$	3.15272	0.223491	0.111745	−	0.993737i	$-0.464356\pi$
$199$	3.15272	0.223491	0.111745	+	0.993737i	$0.464356\pi$
$200$	9.75844	0.690026
$201$	0	0
$202$	−3.29064	−0.231529
$203$	−5.38739	−0.378120
$204$	0	0
$205$	−26.5289	−1.85286
$206$	3.00998	0.209715
$207$	0	0
$208$	20.2469	1.40387
$209$	−14.4721	−1.00106
$210$	0	0
$211$	19.8813	1.36869	0.684343	−	0.729160i	$-0.260089\pi$
$211$	19.8813	1.36869	0.684343	+	0.729160i	$0.260089\pi$
$212$	24.5017	1.68278
$213$	0	0
$214$	0.827580	0.0565722
$215$	−7.20921	−0.491664
$216$	0	0
$217$	8.19880	0.556571
$218$	5.31961	0.360290
$219$	0	0
$220$	−19.4545	−1.31162
$221$	29.5361	1.98681
$222$	0	0
$223$	7.71751	0.516802	0.258401	−	0.966038i	$-0.416804\pi$
$223$	7.71751	0.516802	0.258401	+	0.966038i	$0.416804\pi$
$224$	−3.40076	−0.227223
$225$	0	0
$226$	−1.46125	−0.0972009
$227$	−10.4906	−0.696282	−0.348141	−	0.937442i	$-0.613187\pi$
$227$	−10.4906	−0.696282	−0.348141	+	0.937442i	$0.613187\pi$
$228$	0	0
$229$	−1.30953	−0.0865359	−0.0432679	−	0.999064i	$-0.513777\pi$
$229$	−1.30953	−0.0865359	−0.0432679	+	0.999064i	$0.513777\pi$
$230$	8.45428	0.557459
$231$	0	0
$232$	−6.34857	−0.416804
$233$	−5.21251	−0.341483	−0.170742	−	0.985316i	$-0.554616\pi$
$233$	−5.21251	−0.341483	−0.170742	+	0.985316i	$0.554616\pi$
$234$	0	0
$235$	27.7131	1.80781
$236$	1.39394	0.0907380
$237$	0	0
$238$	−1.52289	−0.0987141
$239$	−10.3271	−0.668004	−0.334002	−	0.942572i	$-0.608399\pi$
$239$	−10.3271	−0.668004	−0.334002	+	0.942572i	$0.608399\pi$
$240$	0	0
$241$	18.0399	1.16206	0.581028	−	0.813884i	$-0.302651\pi$
$241$	18.0399	1.16206	0.581028	+	0.813884i	$0.302651\pi$
$242$	0.958978	0.0616454
$243$	0	0
$244$	−24.2064	−1.54966
$245$	−3.64431	−0.232826
$246$	0	0
$247$	30.2598	1.92538
$248$	9.66158	0.613511
$249$	0	0
$250$	3.60444	0.227965
$251$	1.48618	0.0938071	0.0469036	−	0.998899i	$-0.485065\pi$
$251$	1.48618	0.0938071	0.0469036	+	0.998899i	$0.485065\pi$
$252$	0	0
$253$	−21.5186	−1.35286
$254$	0.301452	0.0189148
$255$	0	0
$256$	9.21517	0.575948
$257$	−7.01325	−0.437474	−0.218737	−	0.975784i	$-0.570194\pi$
$257$	−7.01325	−0.437474	−0.218737	+	0.975784i	$0.570194\pi$
$258$	0	0
$259$	−10.6757	−0.663357
$260$	40.6774	2.52271
$261$	0	0
$262$	5.27446	0.325857
$263$	8.27601	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$263$	8.27601	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$264$	0	0
$265$	46.7710	2.87312
$266$	−1.56020	−0.0956622
$267$	0	0
$268$	25.7550	1.57324
$269$	−15.8680	−0.967491	−0.483745	−	0.875209i	$-0.660724\pi$
$269$	−15.8680	−0.967491	−0.483745	+	0.875209i	$0.660724\pi$
$270$	0	0
$271$	3.13351	0.190347	0.0951736	−	0.995461i	$-0.469659\pi$
$271$	3.13351	0.190347	0.0951736	+	0.995461i	$0.469659\pi$
$272$	17.4946	1.06077
$273$	0	0
$274$	−3.76694	−0.227569
$275$	−23.1554	−1.39632
$276$	0	0
$277$	−7.94076	−0.477114	−0.238557	−	0.971128i	$-0.576674\pi$
$277$	−7.94076	−0.477114	−0.238557	+	0.971128i	$0.576674\pi$
$278$	0.897530	0.0538303
$279$	0	0
$280$	−4.29451	−0.256646
$281$	31.3317	1.86909	0.934547	−	0.355839i	$-0.115805\pi$
$281$	31.3317	1.86909	0.934547	+	0.355839i	$0.115805\pi$
$282$	0	0
$283$	−9.12230	−0.542264	−0.271132	−	0.962542i	$-0.587398\pi$
$283$	−9.12230	−0.542264	−0.271132	+	0.962542i	$0.587398\pi$
$284$	−0.187829	−0.0111456
$285$	0	0
$286$	4.92824	0.291413
$287$	7.27955	0.429698
$288$	0	0
$289$	8.52105	0.501238
$290$	−5.91850	−0.347546
$291$	0	0
$292$	−31.9622	−1.87045
$293$	3.11400	0.181922	0.0909608	−	0.995854i	$-0.471006\pi$
$293$	3.11400	0.181922	0.0909608	+	0.995854i	$0.471006\pi$
$294$	0	0
$295$	2.66088	0.154923
$296$	−12.5804	−0.731222
$297$	0	0
$298$	−6.42620	−0.372260
$299$	44.9932	2.60202
$300$	0	0
$301$	1.97821	0.114022
$302$	−4.49197	−0.258484
$303$	0	0
$304$	17.9233	1.02797
$305$	−46.2074	−2.64583
$306$	0	0
$307$	1.49715	0.0854468	0.0427234	−	0.999087i	$-0.486397\pi$
$307$	1.49715	0.0854468	0.0427234	+	0.999087i	$0.486397\pi$
$308$	5.33832	0.304179
$309$	0	0
$310$	9.00707	0.511567
$311$	−14.9225	−0.846180	−0.423090	−	0.906088i	$-0.639055\pi$
$311$	−14.9225	−0.846180	−0.423090	+	0.906088i	$0.639055\pi$
$312$	0	0
$313$	−12.9789	−0.733613	−0.366806	−	0.930297i	$-0.619549\pi$
$313$	−12.9789	−0.733613	−0.366806	+	0.930297i	$0.619549\pi$
$314$	−0.626658	−0.0353644
$315$	0	0
$316$	−20.9827	−1.18037
$317$	9.87730	0.554765	0.277382	−	0.960760i	$-0.410533\pi$
$317$	9.87730	0.554765	0.277382	+	0.960760i	$0.410533\pi$
$318$	0	0
$319$	15.0643	0.843438
$320$	21.5046	1.20214
$321$	0	0
$322$	−2.31986	−0.129281
$323$	26.1464	1.45482
$324$	0	0
$325$	48.4156	2.68562
$326$	1.71244	0.0948433
$327$	0	0
$328$	8.57833	0.473659
$329$	−7.60450	−0.419249
$330$	0	0
$331$	−25.9685	−1.42736	−0.713679	−	0.700473i	$-0.752972\pi$
$331$	−25.9685	−1.42736	−0.713679	+	0.700473i	$0.752972\pi$
$332$	−29.1047	−1.59733
$333$	0	0
$334$	−2.79842	−0.153123
$335$	49.1635	2.68609
$336$	0	0
$337$	−22.2512	−1.21210	−0.606049	−	0.795427i	$-0.707247\pi$
$337$	−22.2512	−1.21210	−0.606049	+	0.795427i	$0.707247\pi$
$338$	−6.38557	−0.347329
$339$	0	0
$340$	35.1479	1.90616
$341$	−22.9256	−1.24149
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.33115	0.125687
$345$	0	0
$346$	1.69248	0.0909885
$347$	−14.5176	−0.779345	−0.389673	−	0.920953i	$-0.627412\pi$
$347$	−14.5176	−0.779345	−0.389673	+	0.920953i	$0.627412\pi$
$348$	0	0
$349$	8.81898	0.472069	0.236035	−	0.971745i	$-0.424152\pi$
$349$	8.81898	0.472069	0.236035	+	0.971745i	$0.424152\pi$
$350$	−2.49632	−0.133434
$351$	0	0
$352$	9.50925	0.506845
$353$	−31.4148	−1.67204	−0.836021	−	0.548697i	$-0.815124\pi$
$353$	−31.4148	−1.67204	−0.836021	+	0.548697i	$0.815124\pi$
$354$	0	0
$355$	−0.358544	−0.0190296
$356$	9.02266	0.478200
$357$	0	0
$358$	5.06018	0.267439
$359$	−1.34910	−0.0712029	−0.0356014	−	0.999366i	$-0.511335\pi$
$359$	−1.34910	−0.0712029	−0.0356014	+	0.999366i	$0.511335\pi$
$360$	0	0
$361$	7.78709	0.409847
$362$	−6.94079	−0.364800
$363$	0	0
$364$	−11.1619	−0.585042
$365$	−61.0123	−3.19353
$366$	0	0
$367$	7.97102	0.416084	0.208042	−	0.978120i	$-0.433291\pi$
$367$	7.97102	0.416084	0.208042	+	0.978120i	$0.433291\pi$
$368$	26.6500	1.38923
$369$	0	0
$370$	−11.7282	−0.609719
$371$	−12.8340	−0.666307
$372$	0	0
$373$	−13.8930	−0.719354	−0.359677	−	0.933077i	$-0.617113\pi$
$373$	−13.8930	−0.719354	−0.359677	+	0.933077i	$0.617113\pi$
$374$	4.25831	0.220192
$375$	0	0
$376$	−8.96125	−0.462141
$377$	−31.4979	−1.62222
$378$	0	0
$379$	−12.1351	−0.623340	−0.311670	−	0.950190i	$-0.600888\pi$
$379$	−12.1351	−0.623340	−0.311670	+	0.950190i	$0.600888\pi$
$380$	36.0091	1.84723
$381$	0	0
$382$	−5.63365	−0.288243
$383$	−3.43085	−0.175308	−0.0876541	−	0.996151i	$-0.527937\pi$
$383$	−3.43085	−0.175308	−0.0876541	+	0.996151i	$0.527937\pi$
$384$	0	0
$385$	10.1903	0.519344
$386$	2.99815	0.152602
$387$	0	0
$388$	13.2295	0.671624
$389$	−25.7052	−1.30331	−0.651654	−	0.758517i	$-0.725924\pi$
$389$	−25.7052	−1.30331	−0.651654	+	0.758517i	$0.725924\pi$
$390$	0	0
$391$	38.8770	1.96609
$392$	1.17841	0.0595189
$393$	0	0
$394$	−1.29903	−0.0654442
$395$	−40.0537	−2.01532
$396$	0	0
$397$	2.94458	0.147784	0.0738922	−	0.997266i	$-0.476458\pi$
$397$	2.94458	0.147784	0.0738922	+	0.997266i	$0.476458\pi$
$398$	−0.950395	−0.0476390
$399$	0	0
$400$	28.6772	1.43386
$401$	−23.7461	−1.18582	−0.592912	−	0.805267i	$-0.702022\pi$
$401$	−23.7461	−1.18582	−0.592912	+	0.805267i	$0.702022\pi$
$402$	0	0
$403$	47.9351	2.38782
$404$	−20.8400	−1.03683
$405$	0	0
$406$	1.62404	0.0805997
$407$	29.8516	1.47969
$408$	0	0
$409$	19.8512	0.981577	0.490788	−	0.871279i	$-0.336709\pi$
$409$	19.8512	0.981577	0.490788	+	0.871279i	$0.336709\pi$
$410$	7.99720	0.394954
$411$	0	0
$412$	19.0625	0.939143
$413$	−0.730147	−0.0359282
$414$	0	0
$415$	−55.5575	−2.72721
$416$	−19.8829	−0.974839
$417$	0	0
$418$	4.36266	0.213385
$419$	−5.56462	−0.271849	−0.135925	−	0.990719i	$-0.543401\pi$
$419$	−5.56462	−0.271849	−0.135925	+	0.990719i	$0.543401\pi$
$420$	0	0
$421$	9.36926	0.456630	0.228315	−	0.973587i	$-0.426678\pi$
$421$	9.36926	0.456630	0.228315	+	0.973587i	$0.426678\pi$
$422$	−5.99326	−0.291748
$423$	0	0
$424$	−15.1238	−0.734474
$425$	41.8342	2.02926
$426$	0	0
$427$	12.6793	0.613596
$428$	5.24115	0.253340
$429$	0	0
$430$	2.17323	0.104803
$431$	−8.20250	−0.395101	−0.197550	−	0.980293i	$-0.563299\pi$
$431$	−8.20250	−0.395101	−0.197550	+	0.980293i	$0.563299\pi$
$432$	0	0
$433$	7.29896	0.350766	0.175383	−	0.984500i	$-0.443884\pi$
$433$	7.29896	0.350766	0.175383	+	0.984500i	$0.443884\pi$
$434$	−2.47154	−0.118638
$435$	0	0
$436$	33.6897	1.61344
$437$	39.8296	1.90531
$438$	0	0
$439$	28.3571	1.35341	0.676705	−	0.736255i	$-0.263407\pi$
$439$	28.3571	1.35341	0.676705	+	0.736255i	$0.263407\pi$
$440$	12.0084	0.572476
$441$	0	0
$442$	−8.90370	−0.423506
$443$	12.9155	0.613635	0.306818	−	0.951768i	$-0.400736\pi$
$443$	12.9155	0.613635	0.306818	+	0.951768i	$0.400736\pi$
$444$	0	0
$445$	17.2232	0.816460
$446$	−2.32646	−0.110161
$447$	0	0
$448$	−5.90087	−0.278790
$449$	0.345574	0.0163087	0.00815433	−	0.999967i	$-0.497404\pi$
$449$	0.345574	0.0163087	0.00815433	+	0.999967i	$0.497404\pi$
$450$	0	0
$451$	−20.3552	−0.958488
$452$	−9.25425	−0.435283
$453$	0	0
$454$	3.16240	0.148419
$455$	−21.3068	−0.998879
$456$	0	0
$457$	−12.1051	−0.566254	−0.283127	−	0.959082i	$-0.591372\pi$
$457$	−12.1051	−0.566254	−0.283127	+	0.959082i	$0.591372\pi$
$458$	0.394759	0.0184459
$459$	0	0
$460$	53.5418	2.49640
$461$	−27.1151	−1.26288	−0.631439	−	0.775426i	$-0.717535\pi$
$461$	−27.1151	−1.26288	−0.631439	+	0.775426i	$0.717535\pi$
$462$	0	0
$463$	9.32540	0.433388	0.216694	−	0.976240i	$-0.430473\pi$
$463$	9.32540	0.433388	0.216694	+	0.976240i	$0.430473\pi$
$464$	−18.6566	−0.866112
$465$	0	0
$466$	1.57132	0.0727901
$467$	−0.862773	−0.0399244	−0.0199622	−	0.999801i	$-0.506355\pi$
$467$	−0.862773	−0.0399244	−0.0199622	+	0.999801i	$0.506355\pi$
$468$	0	0
$469$	−13.4905	−0.622933
$470$	−8.35418	−0.385350
$471$	0	0
$472$	−0.860416	−0.0396038
$473$	−5.53150	−0.254339
$474$	0	0
$475$	42.8593	1.96652
$476$	−9.64459	−0.442059
$477$	0	0
$478$	3.11312	0.142391
$479$	−0.908068	−0.0414907	−0.0207453	−	0.999785i	$-0.506604\pi$
$479$	−0.908068	−0.0414907	−0.0207453	+	0.999785i	$0.506604\pi$
$480$	0	0
$481$	−62.4167	−2.84596
$482$	−5.43818	−0.247702
$483$	0	0
$484$	6.07330	0.276059
$485$	25.2536	1.14671
$486$	0	0
$487$	−19.4160	−0.879821	−0.439911	−	0.898042i	$-0.644990\pi$
$487$	−19.4160	−0.879821	−0.439911	+	0.898042i	$0.644990\pi$
$488$	14.9415	0.676370
$489$	0	0
$490$	1.09858	0.0496290
$491$	35.1889	1.58805	0.794026	−	0.607884i	$-0.207982\pi$
$491$	35.1889	1.58805	0.794026	+	0.607884i	$0.207982\pi$
$492$	0	0
$493$	−27.2162	−1.22576
$494$	−9.12188	−0.410413
$495$	0	0
$496$	28.3926	1.27486
$497$	0.0983847	0.00441316
$498$	0	0
$499$	21.3752	0.956886	0.478443	−	0.878119i	$-0.341201\pi$
$499$	21.3752	0.956886	0.478443	+	0.878119i	$0.341201\pi$
$500$	22.8273	1.02087
$501$	0	0
$502$	−0.448013	−0.0199958
$503$	31.8366	1.41952	0.709762	−	0.704442i	$-0.248802\pi$
$503$	31.8366	1.41952	0.709762	+	0.704442i	$0.248802\pi$
$504$	0	0
$505$	−39.7812	−1.77024
$506$	6.48682	0.288374
$507$	0	0
$508$	1.90913	0.0847038
$509$	−6.63882	−0.294261	−0.147130	−	0.989117i	$-0.547004\pi$
$509$	−6.63882	−0.294261	−0.147130	+	0.989117i	$0.547004\pi$
$510$	0	0
$511$	16.7418	0.740614
$512$	−19.9386	−0.881172
$513$	0	0
$514$	2.11416	0.0932515
$515$	36.3882	1.60346
$516$	0	0
$517$	21.2638	0.935180
$518$	3.21822	0.141400
$519$	0	0
$520$	−25.1083	−1.10107
$521$	−11.7541	−0.514956	−0.257478	−	0.966284i	$-0.582891\pi$
$521$	−11.7541	−0.514956	−0.257478	+	0.966284i	$0.582891\pi$
$522$	0	0
$523$	−18.0075	−0.787415	−0.393707	−	0.919236i	$-0.628808\pi$
$523$	−18.0075	−0.787415	−0.393707	+	0.919236i	$0.628808\pi$
$524$	33.4037	1.45925
$525$	0	0
$526$	−2.49482	−0.108779
$527$	41.4190	1.80424
$528$	0	0
$529$	36.2224	1.57489
$530$	−14.0992	−0.612431
$531$	0	0
$532$	−9.88092	−0.428392
$533$	42.5606	1.84351
$534$	0	0
$535$	10.0048	0.432544
$536$	−15.8974	−0.686662
$537$	0	0
$538$	4.78345	0.206229
$539$	−2.79621	−0.120441
$540$	0	0
$541$	−31.3534	−1.34799	−0.673995	−	0.738736i	$-0.735423\pi$
$541$	−31.3534	−1.34799	−0.673995	+	0.738736i	$0.735423\pi$
$542$	−0.944603	−0.0405742
$543$	0	0
$544$	−17.1801	−0.736590
$545$	64.3098	2.75473
$546$	0	0
$547$	46.3437	1.98151	0.990756	−	0.135653i	$-0.0433132\pi$
$547$	46.3437	1.98151	0.990756	+	0.135653i	$0.0433132\pi$
$548$	−23.8564	−1.01909
$549$	0	0
$550$	6.98024	0.297639
$551$	−27.8831	−1.18786
$552$	0	0
$553$	10.9907	0.467374
$554$	2.39376	0.101701
$555$	0	0
$556$	5.68415	0.241062
$557$	38.9915	1.65212	0.826060	−	0.563582i	$-0.190577\pi$
$557$	38.9915	1.65212	0.826060	+	0.563582i	$0.190577\pi$
$558$	0	0
$559$	11.5658	0.489181
$560$	−12.6203	−0.533305
$561$	0	0
$562$	−9.44502	−0.398414
$563$	−31.7181	−1.33676	−0.668379	−	0.743821i	$-0.733011\pi$
$563$	−31.7181	−1.33676	−0.668379	+	0.743821i	$0.733011\pi$
$564$	0	0
$565$	−17.6653	−0.743186
$566$	2.74993	0.115588
$567$	0	0
$568$	0.115938	0.00486465
$569$	9.36060	0.392417	0.196208	−	0.980562i	$-0.437137\pi$
$569$	9.36060	0.392417	0.196208	+	0.980562i	$0.437137\pi$
$570$	0	0
$571$	0.295299	0.0123579	0.00617894	−	0.999981i	$-0.498033\pi$
$571$	0.295299	0.0123579	0.00617894	+	0.999981i	$0.498033\pi$
$572$	31.2110	1.30500
$573$	0	0
$574$	−2.19444	−0.0915940
$575$	63.7273	2.65761
$576$	0	0
$577$	1.38603	0.0577012	0.0288506	−	0.999584i	$-0.490815\pi$
$577$	1.38603	0.0577012	0.0288506	+	0.999584i	$0.490815\pi$
$578$	−2.56869	−0.106843
$579$	0	0
$580$	−37.4825	−1.55637
$581$	15.2450	0.632470
$582$	0	0
$583$	35.8866	1.48627
$584$	19.7288	0.816382
$585$	0	0
$586$	−0.938721	−0.0387782
$587$	−22.5141	−0.929257	−0.464629	−	0.885506i	$-0.653812\pi$
$587$	−22.5141	−0.929257	−0.464629	+	0.885506i	$0.653812\pi$
$588$	0	0
$589$	42.4339	1.74846
$590$	−0.802128	−0.0330231
$591$	0	0
$592$	−36.9702	−1.51947
$593$	1.94152	0.0797287	0.0398643	−	0.999205i	$-0.487307\pi$
$593$	1.94152	0.0797287	0.0398643	+	0.999205i	$0.487307\pi$
$594$	0	0
$595$	−18.4105	−0.754755
$596$	−40.6978	−1.66705
$597$	0	0
$598$	−13.5633	−0.554644
$599$	26.2602	1.07296	0.536482	−	0.843912i	$-0.319753\pi$
$599$	26.2602	1.07296	0.536482	+	0.843912i	$0.319753\pi$
$600$	0	0
$601$	2.58479	0.105436	0.0527179	−	0.998609i	$-0.483212\pi$
$601$	2.58479	0.105436	0.0527179	+	0.998609i	$0.483212\pi$
$602$	−0.596336	−0.0243048
$603$	0	0
$604$	−28.4481	−1.15754
$605$	11.5933	0.471333
$606$	0	0
$607$	−6.49564	−0.263650	−0.131825	−	0.991273i	$-0.542084\pi$
$607$	−6.49564	−0.263650	−0.131825	+	0.991273i	$0.542084\pi$
$608$	−17.6011	−0.713818
$609$	0	0
$610$	13.9293	0.563981
$611$	−44.4604	−1.79868
$612$	0	0
$613$	2.73620	0.110514	0.0552570	−	0.998472i	$-0.482402\pi$
$613$	2.73620	0.110514	0.0552570	+	0.998472i	$0.482402\pi$
$614$	−0.451318	−0.0182137
$615$	0	0
$616$	−3.29510	−0.132763
$617$	43.7474	1.76120	0.880601	−	0.473858i	$-0.157139\pi$
$617$	43.7474	1.76120	0.880601	+	0.473858i	$0.157139\pi$
$618$	0	0
$619$	3.44195	0.138344	0.0691718	−	0.997605i	$-0.477964\pi$
$619$	3.44195	0.138344	0.0691718	+	0.997605i	$0.477964\pi$
$620$	57.0427	2.29089
$621$	0	0
$622$	4.49843	0.180371
$623$	−4.72607	−0.189346
$624$	0	0
$625$	2.16984	0.0867937
$626$	3.91253	0.156376
$627$	0	0
$628$	−3.96869	−0.158368
$629$	−53.9320	−2.15041
$630$	0	0
$631$	−18.2285	−0.725665	−0.362833	−	0.931854i	$-0.618190\pi$
$631$	−18.2285	−0.725665	−0.362833	+	0.931854i	$0.618190\pi$
$632$	12.9516	0.515189
$633$	0	0
$634$	−2.97753	−0.118253
$635$	3.64431	0.144620
$636$	0	0
$637$	5.84660	0.231651
$638$	−4.54116	−0.179786
$639$	0	0
$640$	−31.2695	−1.23603
$641$	−26.2479	−1.03673	−0.518365	−	0.855160i	$-0.673459\pi$
$641$	−26.2479	−1.03673	−0.518365	+	0.855160i	$0.673459\pi$
$642$	0	0
$643$	25.7160	1.01414	0.507069	−	0.861905i	$-0.330729\pi$
$643$	25.7160	1.01414	0.507069	+	0.861905i	$0.330729\pi$
$644$	−14.6919	−0.578942
$645$	0	0
$646$	−7.88189	−0.310109
$647$	−40.1288	−1.57763	−0.788813	−	0.614633i	$-0.789304\pi$
$647$	−40.1288	−1.57763	−0.788813	+	0.614633i	$0.789304\pi$
$648$	0	0
$649$	2.04165	0.0801416
$650$	−14.5950	−0.572463
$651$	0	0
$652$	10.8451	0.424726
$653$	31.1598	1.21938	0.609689	−	0.792641i	$-0.291294\pi$
$653$	31.1598	1.21938	0.609689	+	0.792641i	$0.291294\pi$
$654$	0	0
$655$	63.7640	2.49146
$656$	25.2092	0.984255
$657$	0	0
$658$	2.29239	0.0893667
$659$	11.9231	0.464458	0.232229	−	0.972661i	$-0.425398\pi$
$659$	11.9231	0.464458	0.232229	+	0.972661i	$0.425398\pi$
$660$	0	0
$661$	24.1640	0.939870	0.469935	−	0.882701i	$-0.344277\pi$
$661$	24.1640	0.939870	0.469935	+	0.882701i	$0.344277\pi$
$662$	7.82826	0.304254
$663$	0	0
$664$	17.9649	0.697175
$665$	−18.8616	−0.731421
$666$	0	0
$667$	−41.4592	−1.60531
$668$	−17.7227	−0.685712
$669$	0	0
$670$	−14.8204	−0.572563
$671$	−35.4541	−1.36869
$672$	0	0
$673$	19.6668	0.758099	0.379050	−	0.925376i	$-0.376251\pi$
$673$	19.6668	0.758099	0.379050	+	0.925376i	$0.376251\pi$
$674$	6.70766	0.258369
$675$	0	0
$676$	−40.4405	−1.55540
$677$	33.7236	1.29610	0.648051	−	0.761597i	$-0.275584\pi$
$677$	33.7236	1.29610	0.648051	+	0.761597i	$0.275584\pi$
$678$	0	0
$679$	−6.92959	−0.265933
$680$	−21.6951	−0.831971
$681$	0	0
$682$	6.91096	0.264634
$683$	−17.1935	−0.657890	−0.328945	−	0.944349i	$-0.606693\pi$
$683$	−17.1935	−0.657890	−0.328945	+	0.944349i	$0.606693\pi$
$684$	0	0
$685$	−45.5392	−1.73996
$686$	−0.301452	−0.0115095
$687$	0	0
$688$	6.85058	0.261176
$689$	−75.0352	−2.85861
$690$	0	0
$691$	34.6430	1.31788	0.658940	−	0.752196i	$-0.271005\pi$
$691$	34.6430	1.31788	0.658940	+	0.752196i	$0.271005\pi$
$692$	10.7187	0.407463
$693$	0	0
$694$	4.37636	0.166124
$695$	10.8504	0.411580
$696$	0	0
$697$	36.7751	1.39296
$698$	−2.65850	−0.100626
$699$	0	0
$700$	−15.8095	−0.597541
$701$	7.34877	0.277559	0.138780	−	0.990323i	$-0.455682\pi$
$701$	7.34877	0.277559	0.138780	+	0.990323i	$0.455682\pi$
$702$	0	0
$703$	−55.2535	−2.08393
$704$	16.5001	0.621870
$705$	0	0
$706$	9.47007	0.356411
$707$	10.9160	0.410538
$708$	0	0
$709$	−16.7194	−0.627910	−0.313955	−	0.949438i	$-0.601654\pi$
$709$	−16.7194	−0.627910	−0.313955	+	0.949438i	$0.601654\pi$
$710$	0.108084	0.00405632
$711$	0	0
$712$	−5.56926	−0.208717
$713$	63.0948	2.36292
$714$	0	0
$715$	59.5784	2.22811
$716$	32.0466	1.19764
$717$	0	0
$718$	0.406689	0.0151775
$719$	−34.1452	−1.27340	−0.636700	−	0.771112i	$-0.719701\pi$
$719$	−34.1452	−1.27340	−0.636700	+	0.771112i	$0.719701\pi$
$720$	0	0
$721$	−9.98495	−0.371859
$722$	−2.34743	−0.0873624
$723$	0	0
$724$	−43.9567	−1.63364
$725$	−44.6129	−1.65688
$726$	0	0
$727$	−0.121370	−0.00450138	−0.00225069	−	0.999997i	$-0.500716\pi$
$727$	−0.121370	−0.00450138	−0.00225069	+	0.999997i	$0.500716\pi$
$728$	6.88971	0.255350
$729$	0	0
$730$	18.3923	0.680729
$731$	9.99360	0.369627
$732$	0	0
$733$	7.30459	0.269801	0.134901	−	0.990859i	$-0.456928\pi$
$733$	7.30459	0.269801	0.134901	+	0.990859i	$0.456928\pi$
$734$	−2.40288	−0.0886920
$735$	0	0
$736$	−26.1709	−0.964674
$737$	37.7223	1.38952
$738$	0	0
$739$	36.1905	1.33129	0.665644	−	0.746269i	$-0.268157\pi$
$739$	36.1905	1.33129	0.665644	+	0.746269i	$0.268157\pi$
$740$	−74.2758	−2.73043
$741$	0	0
$742$	3.86883	0.142029
$743$	40.9744	1.50320	0.751602	−	0.659617i	$-0.229282\pi$
$743$	40.9744	1.50320	0.751602	+	0.659617i	$0.229282\pi$
$744$	0	0
$745$	−77.6875	−2.84625
$746$	4.18809	0.153337
$747$	0	0
$748$	26.9683	0.986060
$749$	−2.74531	−0.100311
$750$	0	0
$751$	−24.5773	−0.896839	−0.448420	−	0.893823i	$-0.648013\pi$
$751$	−24.5773	−0.896839	−0.448420	+	0.893823i	$0.648013\pi$
$752$	−26.3345	−0.960321
$753$	0	0
$754$	9.49510	0.345791
$755$	−54.3043	−1.97634
$756$	0	0
$757$	40.6291	1.47669	0.738346	−	0.674423i	$-0.235607\pi$
$757$	40.6291	1.47669	0.738346	+	0.674423i	$0.235607\pi$
$758$	3.65816	0.132870
$759$	0	0
$760$	−22.2267	−0.806249
$761$	19.8021	0.717826	0.358913	−	0.933371i	$-0.383147\pi$
$761$	19.8021	0.717826	0.358913	+	0.933371i	$0.383147\pi$
$762$	0	0
$763$	−17.6466	−0.638851
$764$	−35.6785	−1.29080
$765$	0	0
$766$	1.03424	0.0373685
$767$	−4.26888	−0.154140
$768$	0	0
$769$	6.52612	0.235338	0.117669	−	0.993053i	$-0.462458\pi$
$769$	6.52612	0.235338	0.117669	+	0.993053i	$0.462458\pi$
$770$	−3.07188	−0.110703
$771$	0	0
$772$	18.9876	0.683378
$773$	33.9687	1.22177	0.610884	−	0.791720i	$-0.290814\pi$
$773$	33.9687	1.22177	0.610884	+	0.791720i	$0.290814\pi$
$774$	0	0
$775$	67.8942	2.43883
$776$	−8.16593	−0.293140
$777$	0	0
$778$	7.74890	0.277811
$779$	37.6762	1.34989
$780$	0	0
$781$	−0.275105	−0.00984401
$782$	−11.7195	−0.419090
$783$	0	0
$784$	3.46302	0.123679
$785$	−7.57579	−0.270391
$786$	0	0
$787$	8.30012	0.295867	0.147934	−	0.988997i	$-0.452738\pi$
$787$	8.30012	0.295867	0.147934	+	0.988997i	$0.452738\pi$
$788$	−8.22689	−0.293071
$789$	0	0
$790$	12.0743	0.429583
$791$	4.84737	0.172353
$792$	0	0
$793$	74.1309	2.63247
$794$	−0.887651	−0.0315016
$795$	0	0
$796$	−6.01895	−0.213336
$797$	−22.4433	−0.794984	−0.397492	−	0.917606i	$-0.630119\pi$
$797$	−22.4433	−0.794984	−0.397492	+	0.917606i	$0.630119\pi$
$798$	0	0
$799$	−38.4167	−1.35908
$800$	−28.1617	−0.995666
$801$	0	0
$802$	7.15832	0.252769
$803$	−46.8136	−1.65202
$804$	0	0
$805$	−28.0452	−0.988463
$806$	−14.4501	−0.508984
$807$	0	0
$808$	12.8635	0.452538
$809$	−13.1049	−0.460743	−0.230372	−	0.973103i	$-0.573994\pi$
$809$	−13.1049	−0.460743	−0.230372	+	0.973103i	$0.573994\pi$
$810$	0	0
$811$	−7.04487	−0.247379	−0.123689	−	0.992321i	$-0.539473\pi$
$811$	−7.04487	−0.247379	−0.123689	+	0.992321i	$0.539473\pi$
$812$	10.2852	0.360940
$813$	0	0
$814$	−8.99883	−0.315409
$815$	20.7020	0.725160
$816$	0	0
$817$	10.2385	0.358199
$818$	−5.98417	−0.209232
$819$	0	0
$820$	50.6471	1.76867
$821$	29.7330	1.03769	0.518845	−	0.854868i	$-0.326362\pi$
$821$	29.7330	1.03769	0.518845	+	0.854868i	$0.326362\pi$
$822$	0	0
$823$	−5.23230	−0.182387	−0.0911933	−	0.995833i	$-0.529068\pi$
$823$	−5.23230	−0.182387	−0.0911933	+	0.995833i	$0.529068\pi$
$824$	−11.7664	−0.409902
$825$	0	0
$826$	0.220104	0.00765841
$827$	−7.62254	−0.265062	−0.132531	−	0.991179i	$-0.542310\pi$
$827$	−7.62254	−0.265062	−0.132531	+	0.991179i	$0.542310\pi$
$828$	0	0
$829$	8.71294	0.302613	0.151306	−	0.988487i	$-0.451652\pi$
$829$	8.71294	0.302613	0.151306	+	0.988487i	$0.451652\pi$
$830$	16.7479	0.581329
$831$	0	0
$832$	−34.5000	−1.19607
$833$	5.05184	0.175036
$834$	0	0
$835$	−33.8306	−1.17076
$836$	27.6292	0.955574
$837$	0	0
$838$	1.67746	0.0579471
$839$	−6.48274	−0.223809	−0.111904	−	0.993719i	$-0.535695\pi$
$839$	−6.48274	−0.223809	−0.111904	+	0.993719i	$0.535695\pi$
$840$	0	0
$841$	0.0239358	0.000825371 0
$842$	−2.82438	−0.0973347
$843$	0	0
$844$	−37.9560	−1.30650
$845$	−77.1964	−2.65564
$846$	0	0
$847$	−3.18119	−0.109307
$848$	−44.4443	−1.52622
$849$	0	0
$850$	−12.6110	−0.432554
$851$	−82.1562	−2.81628
$852$	0	0
$853$	47.7385	1.63454	0.817268	−	0.576257i	$-0.195487\pi$
$853$	47.7385	1.63454	0.817268	+	0.576257i	$0.195487\pi$
$854$	−3.82221	−0.130793
$855$	0	0
$856$	−3.23511	−0.110574
$857$	53.0612	1.81254	0.906269	−	0.422702i	$-0.138918\pi$
$857$	53.0612	1.81254	0.906269	+	0.422702i	$0.138918\pi$
$858$	0	0
$859$	9.97525	0.340351	0.170176	−	0.985414i	$-0.445567\pi$
$859$	9.97525	0.340351	0.170176	+	0.985414i	$0.445567\pi$
$860$	13.7633	0.469325
$861$	0	0
$862$	2.47266	0.0842192
$863$	−37.3759	−1.27229	−0.636145	−	0.771569i	$-0.719472\pi$
$863$	−37.3759	−1.27229	−0.636145	+	0.771569i	$0.719472\pi$
$864$	0	0
$865$	20.4608	0.695686
$866$	−2.20029	−0.0747687
$867$	0	0
$868$	−15.6525	−0.531282
$869$	−30.7325	−1.04253
$870$	0	0
$871$	−78.8734	−2.67252
$872$	−20.7950	−0.704209
$873$	0	0
$874$	−12.0067	−0.406133
$875$	−11.9569	−0.404218
$876$	0	0
$877$	22.6211	0.763859	0.381929	−	0.924192i	$-0.375260\pi$
$877$	22.6211	0.763859	0.381929	+	0.924192i	$0.375260\pi$
$878$	−8.54830	−0.288491
$879$	0	0
$880$	35.2891	1.18959
$881$	30.1262	1.01498	0.507489	−	0.861658i	$-0.330574\pi$
$881$	30.1262	1.01498	0.507489	+	0.861658i	$0.330574\pi$
$882$	0	0
$883$	−2.00270	−0.0673962	−0.0336981	−	0.999432i	$-0.510728\pi$
$883$	−2.00270	−0.0673962	−0.0336981	+	0.999432i	$0.510728\pi$
$884$	−56.3881	−1.89654
$885$	0	0
$886$	−3.89341	−0.130802
$887$	43.8591	1.47265	0.736323	−	0.676630i	$-0.236560\pi$
$887$	43.8591	1.47265	0.736323	+	0.676630i	$0.236560\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−5.19198	−0.174036
$891$	0	0
$892$	−14.7337	−0.493321
$893$	−39.3580	−1.31707
$894$	0	0
$895$	61.1735	2.04480
$896$	8.58035	0.286649
$897$	0	0
$898$	−0.104174	−0.00347634
$899$	−44.1701	−1.47316
$900$	0	0
$901$	−64.8352	−2.15997
$902$	6.13611	0.204310
$903$	0	0
$904$	5.71221	0.189985
$905$	−83.9085	−2.78921
$906$	0	0
$907$	−39.7509	−1.31991	−0.659954	−	0.751306i	$-0.729424\pi$
$907$	−39.7509	−1.31991	−0.659954	+	0.751306i	$0.729424\pi$
$908$	20.0278	0.664646
$909$	0	0
$910$	6.42298	0.212920
$911$	−40.8034	−1.35188	−0.675939	−	0.736957i	$-0.736262\pi$
$911$	−40.8034	−1.35188	−0.675939	+	0.736957i	$0.736262\pi$
$912$	0	0
$913$	−42.6283	−1.41079
$914$	3.64911	0.120702
$915$	0	0
$916$	2.50005	0.0826040
$917$	−17.4969	−0.577797
$918$	0	0
$919$	12.3664	0.407928	0.203964	−	0.978978i	$-0.434617\pi$
$919$	12.3664	0.407928	0.203964	+	0.978978i	$0.434617\pi$
$920$	−33.0488	−1.08959
$921$	0	0
$922$	8.17391	0.269193
$923$	0.575216	0.0189335
$924$	0	0
$925$	−88.4056	−2.90676
$926$	−2.81116	−0.0923805
$927$	0	0
$928$	18.3212	0.601424
$929$	28.9296	0.949148	0.474574	−	0.880216i	$-0.342602\pi$
$929$	28.9296	0.949148	0.474574	+	0.880216i	$0.342602\pi$
$930$	0	0
$931$	5.17562	0.169624
$932$	9.95135	0.325967
$933$	0	0
$934$	0.260085	0.00851023
$935$	51.4795	1.68356
$936$	0	0
$937$	2.73472	0.0893396	0.0446698	−	0.999002i	$-0.485776\pi$
$937$	2.73472	0.0893396	0.0446698	+	0.999002i	$0.485776\pi$
$938$	4.06673	0.132784
$939$	0	0
$940$	−52.9079	−1.72566
$941$	53.5706	1.74635	0.873176	−	0.487405i	$-0.162057\pi$
$941$	53.5706	1.74635	0.873176	+	0.487405i	$0.162057\pi$
$942$	0	0
$943$	56.0206	1.82428
$944$	−2.52851	−0.0822961
$945$	0	0
$946$	1.66748	0.0542145
$947$	31.1412	1.01195	0.505976	−	0.862548i	$-0.331132\pi$
$947$	31.1412	1.01195	0.505976	+	0.862548i	$0.331132\pi$
$948$	0	0
$949$	97.8826	3.17740
$950$	−12.9200	−0.419181
$951$	0	0
$952$	5.95316	0.192943
$953$	−6.38173	−0.206725	−0.103362	−	0.994644i	$-0.532960\pi$
$953$	−6.38173	−0.206725	−0.103362	+	0.994644i	$0.532960\pi$
$954$	0	0
$955$	−68.1063	−2.20387
$956$	19.7157	0.637652
$957$	0	0
$958$	0.273739	0.00884411
$959$	12.4960	0.403516
$960$	0	0
$961$	36.2203	1.16840
$962$	18.8156	0.606640
$963$	0	0
$964$	−34.4405	−1.10926
$965$	36.2451	1.16677
$966$	0	0
$967$	−0.616173	−0.0198148	−0.00990739	−	0.999951i	$-0.503154\pi$
$967$	−0.616173	−0.0198148	−0.00990739	+	0.999951i	$0.503154\pi$
$968$	−3.74877	−0.120490
$969$	0	0
$970$	−7.61274	−0.244430
$971$	41.0816	1.31837	0.659185	−	0.751981i	$-0.270901\pi$
$971$	41.0816	1.31837	0.659185	+	0.751981i	$0.270901\pi$
$972$	0	0
$973$	−2.97736	−0.0954497
$974$	5.85298	0.187542
$975$	0	0
$976$	43.9087	1.40548
$977$	35.6388	1.14019	0.570093	−	0.821580i	$-0.306907\pi$
$977$	35.6388	1.14019	0.570093	+	0.821580i	$0.306907\pi$
$978$	0	0
$979$	13.2151	0.422356
$980$	6.95745	0.222248
$981$	0	0
$982$	−10.6078	−0.338507
$983$	−11.1874	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$983$	−11.1874	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$984$	0	0
$985$	−15.7042	−0.500378
$986$	8.20438	0.261281
$987$	0	0
$988$	−57.7698	−1.83790
$989$	15.2235	0.484081
$990$	0	0
$991$	−25.9132	−0.823161	−0.411580	−	0.911374i	$-0.635023\pi$
$991$	−25.9132	−0.823161	−0.411580	+	0.911374i	$0.635023\pi$
$992$	−27.8822	−0.885260
$993$	0	0
$994$	−0.0296583	−0.000940703 0
$995$	−11.4895	−0.364242
$996$	0	0
$997$	−4.61761	−0.146241	−0.0731206	−	0.997323i	$-0.523296\pi$
$997$	−4.61761	−0.146241	−0.0731206	+	0.997323i	$0.523296\pi$
$998$	−6.44360	−0.203969
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.19	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.22	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.19	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.22	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.301452 q^{2} -1.90913 q^{4} -3.64431 q^{5} +1.00000 q^{7} +1.17841 q^{8} +O(q^{10})\)	\(q-0.301452 q^{2} -1.90913 q^{4} -3.64431 q^{5} +1.00000 q^{7} +1.17841 q^{8} +1.09858 q^{10} -2.79621 q^{11} +5.84660 q^{13} -0.301452 q^{14} +3.46302 q^{16} +5.05184 q^{17} +5.17562 q^{19} +6.95745 q^{20} +0.842924 q^{22} +7.69561 q^{23} +8.28099 q^{25} -1.76247 q^{26} -1.90913 q^{28} -5.38739 q^{29} +8.19880 q^{31} -3.40076 q^{32} -1.52289 q^{34} -3.64431 q^{35} -10.6757 q^{37} -1.56020 q^{38} -4.29451 q^{40} +7.27955 q^{41} +1.97821 q^{43} +5.33832 q^{44} -2.31986 q^{46} -7.60450 q^{47} +1.00000 q^{49} -2.49632 q^{50} -11.1619 q^{52} -12.8340 q^{53} +10.1903 q^{55} +1.17841 q^{56} +1.62404 q^{58} -0.730147 q^{59} +12.6793 q^{61} -2.47154 q^{62} -5.90087 q^{64} -21.3068 q^{65} -13.4905 q^{67} -9.64459 q^{68} +1.09858 q^{70} +0.0983847 q^{71} +16.7418 q^{73} +3.21822 q^{74} -9.88092 q^{76} -2.79621 q^{77} +10.9907 q^{79} -12.6203 q^{80} -2.19444 q^{82} +15.2450 q^{83} -18.4105 q^{85} -0.596336 q^{86} -3.29510 q^{88} -4.72607 q^{89} +5.84660 q^{91} -14.6919 q^{92} +2.29239 q^{94} -18.8616 q^{95} -6.92959 q^{97} -0.301452 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more