LMFDB - Embedded newform 8001.2.a.ba.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.09235 q^{2} +2.37795 q^{4} -3.96692 q^{5} +1.00000 q^{7} -0.790800 q^{8} +O(q^{10})$	\(q-2.09235 q^{2} +2.37795 q^{4} -3.96692 q^{5} +1.00000 q^{7} -0.790800 q^{8} +8.30020 q^{10} -3.36492 q^{11} -1.92494 q^{13} -2.09235 q^{14} -3.10126 q^{16} +1.09693 q^{17} +1.50219 q^{19} -9.43312 q^{20} +7.04060 q^{22} +6.96998 q^{23} +10.7364 q^{25} +4.02765 q^{26} +2.37795 q^{28} -8.74282 q^{29} +7.96336 q^{31} +8.07054 q^{32} -2.29517 q^{34} -3.96692 q^{35} +0.395595 q^{37} -3.14311 q^{38} +3.13704 q^{40} -5.12703 q^{41} +4.08571 q^{43} -8.00159 q^{44} -14.5837 q^{46} +8.64796 q^{47} +1.00000 q^{49} -22.4644 q^{50} -4.57740 q^{52} +1.42610 q^{53} +13.3483 q^{55} -0.790800 q^{56} +18.2931 q^{58} -7.32907 q^{59} -8.24548 q^{61} -16.6622 q^{62} -10.6839 q^{64} +7.63607 q^{65} -0.895891 q^{67} +2.60845 q^{68} +8.30020 q^{70} -12.9035 q^{71} -14.8018 q^{73} -0.827725 q^{74} +3.57213 q^{76} -3.36492 q^{77} -10.4969 q^{79} +12.3024 q^{80} +10.7276 q^{82} -13.0258 q^{83} -4.35144 q^{85} -8.54876 q^{86} +2.66097 q^{88} +1.78663 q^{89} -1.92494 q^{91} +16.5742 q^{92} -18.0946 q^{94} -5.95906 q^{95} +15.4051 q^{97} -2.09235 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.09235	−1.47952	−0.739759	−	0.672872i	$-0.765061\pi$
$2$	−2.09235	−1.47952	−0.739759	+	0.672872i	$0.765061\pi$
$3$	0	0
$3$	0	0
$4$	2.37795	1.18897
$5$	−3.96692	−1.77406	−0.887030	−	0.461712i	$-0.847235\pi$
$5$	−3.96692	−1.77406	−0.887030	+	0.461712i	$0.847235\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.790800	−0.279590
$9$	0	0
$10$	8.30020	2.62475
$11$	−3.36492	−1.01456	−0.507280	−	0.861781i	$-0.669349\pi$
$11$	−3.36492	−1.01456	−0.507280	+	0.861781i	$0.669349\pi$
$12$	0	0
$13$	−1.92494	−0.533881	−0.266941	−	0.963713i	$-0.586013\pi$
$13$	−1.92494	−0.533881	−0.266941	+	0.963713i	$0.586013\pi$
$14$	−2.09235	−0.559205
$15$	0	0
$16$	−3.10126	−0.775315
$17$	1.09693	0.266045	0.133023	−	0.991113i	$-0.457532\pi$
$17$	1.09693	0.266045	0.133023	+	0.991113i	$0.457532\pi$
$18$	0	0
$19$	1.50219	0.344626	0.172313	−	0.985042i	$-0.444876\pi$
$19$	1.50219	0.344626	0.172313	+	0.985042i	$0.444876\pi$
$20$	−9.43312	−2.10931
$21$	0	0
$22$	7.04060	1.50106
$23$	6.96998	1.45334	0.726670	−	0.686987i	$-0.241067\pi$
$23$	6.96998	1.45334	0.726670	+	0.686987i	$0.241067\pi$
$24$	0	0
$25$	10.7364	2.14729
$26$	4.02765	0.789887
$27$	0	0
$28$	2.37795	0.449390
$29$	−8.74282	−1.62350	−0.811751	−	0.584004i	$-0.801485\pi$
$29$	−8.74282	−1.62350	−0.811751	+	0.584004i	$0.801485\pi$
$30$	0	0
$31$	7.96336	1.43026	0.715131	−	0.698991i	$-0.246367\pi$
$31$	7.96336	1.43026	0.715131	+	0.698991i	$0.246367\pi$
$32$	8.07054	1.42668
$33$	0	0
$34$	−2.29517	−0.393619
$35$	−3.96692	−0.670532
$36$	0	0
$37$	0.395595	0.0650354	0.0325177	−	0.999471i	$-0.489647\pi$
$37$	0.395595	0.0650354	0.0325177	+	0.999471i	$0.489647\pi$
$38$	−3.14311	−0.509880
$39$	0	0
$40$	3.13704	0.496009
$41$	−5.12703	−0.800707	−0.400354	−	0.916361i	$-0.631113\pi$
$41$	−5.12703	−0.800707	−0.400354	+	0.916361i	$0.631113\pi$
$42$	0	0
$43$	4.08571	0.623065	0.311533	−	0.950235i	$-0.399158\pi$
$43$	4.08571	0.623065	0.311533	+	0.950235i	$0.399158\pi$
$44$	−8.00159	−1.20629
$45$	0	0
$46$	−14.5837	−2.15024
$47$	8.64796	1.26143	0.630717	−	0.776013i	$-0.282761\pi$
$47$	8.64796	1.26143	0.630717	+	0.776013i	$0.282761\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−22.4644	−3.17695
$51$	0	0
$52$	−4.57740	−0.634771
$53$	1.42610	0.195889	0.0979446	−	0.995192i	$-0.468773\pi$
$53$	1.42610	0.195889	0.0979446	+	0.995192i	$0.468773\pi$
$54$	0	0
$55$	13.3483	1.79989
$56$	−0.790800	−0.105675
$57$	0	0
$58$	18.2931	2.40200
$59$	−7.32907	−0.954164	−0.477082	−	0.878859i	$-0.658305\pi$
$59$	−7.32907	−0.954164	−0.477082	+	0.878859i	$0.658305\pi$
$60$	0	0
$61$	−8.24548	−1.05573	−0.527863	−	0.849330i	$-0.677006\pi$
$61$	−8.24548	−1.05573	−0.527863	+	0.849330i	$0.677006\pi$
$62$	−16.6622	−2.11610
$63$	0	0
$64$	−10.6839	−1.33549
$65$	7.63607	0.947137
$66$	0	0
$67$	−0.895891	−0.109451	−0.0547253	−	0.998501i	$-0.517428\pi$
$67$	−0.895891	−0.109451	−0.0547253	+	0.998501i	$0.517428\pi$
$68$	2.60845	0.316321
$69$	0	0
$70$	8.30020	0.992064
$71$	−12.9035	−1.53136	−0.765682	−	0.643220i	$-0.777598\pi$
$71$	−12.9035	−1.53136	−0.765682	+	0.643220i	$0.777598\pi$
$72$	0	0
$73$	−14.8018	−1.73242	−0.866209	−	0.499682i	$-0.833450\pi$
$73$	−14.8018	−1.73242	−0.866209	+	0.499682i	$0.833450\pi$
$74$	−0.827725	−0.0962210
$75$	0	0
$76$	3.57213	0.409751
$77$	−3.36492	−0.383468
$78$	0	0
$79$	−10.4969	−1.18099	−0.590495	−	0.807041i	$-0.701067\pi$
$79$	−10.4969	−1.18099	−0.590495	+	0.807041i	$0.701067\pi$
$80$	12.3024	1.37546
$81$	0	0
$82$	10.7276	1.18466
$83$	−13.0258	−1.42977	−0.714883	−	0.699244i	$-0.753520\pi$
$83$	−13.0258	−1.42977	−0.714883	+	0.699244i	$0.753520\pi$
$84$	0	0
$85$	−4.35144	−0.471980
$86$	−8.54876	−0.921836
$87$	0	0
$88$	2.66097	0.283661
$89$	1.78663	0.189383	0.0946913	−	0.995507i	$-0.469814\pi$
$89$	1.78663	0.189383	0.0946913	+	0.995507i	$0.469814\pi$
$90$	0	0
$91$	−1.92494	−0.201788
$92$	16.5742	1.72798
$93$	0	0
$94$	−18.0946	−1.86631
$95$	−5.95906	−0.611387
$96$	0	0
$97$	15.4051	1.56415	0.782075	−	0.623185i	$-0.214162\pi$
$97$	15.4051	1.56415	0.782075	+	0.623185i	$0.214162\pi$
$98$	−2.09235	−0.211360
$99$	0	0
$100$	25.5307	2.55307
$101$	−13.4223	−1.33557	−0.667786	−	0.744353i	$-0.732758\pi$
$101$	−13.4223	−1.33557	−0.667786	+	0.744353i	$0.732758\pi$
$102$	0	0
$103$	19.8444	1.95533	0.977664	−	0.210174i	$-0.0674032\pi$
$103$	19.8444	1.95533	0.977664	+	0.210174i	$0.0674032\pi$
$104$	1.52224	0.149268
$105$	0	0
$106$	−2.98390	−0.289822
$107$	−6.88756	−0.665845	−0.332923	−	0.942954i	$-0.608035\pi$
$107$	−6.88756	−0.665845	−0.332923	+	0.942954i	$0.608035\pi$
$108$	0	0
$109$	18.2912	1.75198	0.875990	−	0.482329i	$-0.160209\pi$
$109$	18.2912	1.75198	0.875990	+	0.482329i	$0.160209\pi$
$110$	−27.9295	−2.66297
$111$	0	0
$112$	−3.10126	−0.293042
$113$	17.1246	1.61095	0.805476	−	0.592629i	$-0.201910\pi$
$113$	17.1246	1.61095	0.805476	+	0.592629i	$0.201910\pi$
$114$	0	0
$115$	−27.6493	−2.57831
$116$	−20.7900	−1.93030
$117$	0	0
$118$	15.3350	1.41170
$119$	1.09693	0.100556
$120$	0	0
$121$	0.322654	0.0293321
$122$	17.2525	1.56196
$123$	0	0
$124$	18.9364	1.70054
$125$	−22.7560	−2.03536
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	6.21344	0.549195
$129$	0	0
$130$	−15.9774	−1.40131
$131$	14.5754	1.27346	0.636729	−	0.771088i	$-0.280287\pi$
$131$	14.5754	1.27346	0.636729	+	0.771088i	$0.280287\pi$
$132$	0	0
$133$	1.50219	0.130256
$134$	1.87452	0.161934
$135$	0	0
$136$	−0.867454	−0.0743836
$137$	−5.22293	−0.446225	−0.223113	−	0.974793i	$-0.571622\pi$
$137$	−5.22293	−0.446225	−0.223113	+	0.974793i	$0.571622\pi$
$138$	0	0
$139$	−7.32979	−0.621705	−0.310852	−	0.950458i	$-0.600614\pi$
$139$	−7.32979	−0.621705	−0.310852	+	0.950458i	$0.600614\pi$
$140$	−9.43312	−0.797244
$141$	0	0
$142$	26.9987	2.26568
$143$	6.47725	0.541655
$144$	0	0
$145$	34.6821	2.88019
$146$	30.9706	2.56314
$147$	0	0
$148$	0.940704	0.0773254
$149$	21.4541	1.75759	0.878796	−	0.477198i	$-0.158347\pi$
$149$	21.4541	1.75759	0.878796	+	0.477198i	$0.158347\pi$
$150$	0	0
$151$	−8.93525	−0.727140	−0.363570	−	0.931567i	$-0.618442\pi$
$151$	−8.93525	−0.727140	−0.363570	+	0.931567i	$0.618442\pi$
$152$	−1.18793	−0.0963539
$153$	0	0
$154$	7.04060	0.567347
$155$	−31.5900	−2.53737
$156$	0	0
$157$	−20.9666	−1.67332	−0.836660	−	0.547723i	$-0.815495\pi$
$157$	−20.9666	−1.67332	−0.836660	+	0.547723i	$0.815495\pi$
$158$	21.9632	1.74730
$159$	0	0
$160$	−32.0152	−2.53102
$161$	6.96998	0.549311
$162$	0	0
$163$	−18.1296	−1.42002	−0.710008	−	0.704194i	$-0.751309\pi$
$163$	−18.1296	−1.42002	−0.710008	+	0.704194i	$0.751309\pi$
$164$	−12.1918	−0.952020
$165$	0	0
$166$	27.2546	2.11536
$167$	0.641839	0.0496670	0.0248335	−	0.999692i	$-0.492094\pi$
$167$	0.641839	0.0496670	0.0248335	+	0.999692i	$0.492094\pi$
$168$	0	0
$169$	−9.29462	−0.714971
$170$	9.10476	0.698303
$171$	0	0
$172$	9.71561	0.740808
$173$	6.95360	0.528673	0.264336	−	0.964431i	$-0.414847\pi$
$173$	6.95360	0.528673	0.264336	+	0.964431i	$0.414847\pi$
$174$	0	0
$175$	10.7364	0.811599
$176$	10.4355	0.786604
$177$	0	0
$178$	−3.73827	−0.280195
$179$	−7.79665	−0.582749	−0.291375	−	0.956609i	$-0.594113\pi$
$179$	−7.79665	−0.582749	−0.291375	+	0.956609i	$0.594113\pi$
$180$	0	0
$181$	−1.99767	−0.148486	−0.0742429	−	0.997240i	$-0.523654\pi$
$181$	−1.99767	−0.148486	−0.0742429	+	0.997240i	$0.523654\pi$
$182$	4.02765	0.298549
$183$	0	0
$184$	−5.51186	−0.406339
$185$	−1.56929	−0.115377
$186$	0	0
$187$	−3.69108	−0.269919
$188$	20.5644	1.49981
$189$	0	0
$190$	12.4685	0.904558
$191$	−3.62932	−0.262609	−0.131304	−	0.991342i	$-0.541916\pi$
$191$	−3.62932	−0.262609	−0.131304	+	0.991342i	$0.541916\pi$
$192$	0	0
$193$	26.3305	1.89531	0.947656	−	0.319293i	$-0.103445\pi$
$193$	26.3305	1.89531	0.947656	+	0.319293i	$0.103445\pi$
$194$	−32.2329	−2.31419
$195$	0	0
$196$	2.37795	0.169853
$197$	−6.30796	−0.449423	−0.224712	−	0.974425i	$-0.572144\pi$
$197$	−6.30796	−0.449423	−0.224712	+	0.974425i	$0.572144\pi$
$198$	0	0
$199$	−23.8239	−1.68883	−0.844416	−	0.535687i	$-0.820053\pi$
$199$	−23.8239	−1.68883	−0.844416	+	0.535687i	$0.820053\pi$
$200$	−8.49037	−0.600360
$201$	0	0
$202$	28.0843	1.97600
$203$	−8.74282	−0.613626
$204$	0	0
$205$	20.3385	1.42050
$206$	−41.5215	−2.89294
$207$	0	0
$208$	5.96973	0.413926
$209$	−5.05474	−0.349644
$210$	0	0
$211$	−9.03762	−0.622176	−0.311088	−	0.950381i	$-0.600693\pi$
$211$	−9.03762	−0.622176	−0.311088	+	0.950381i	$0.600693\pi$
$212$	3.39118	0.232907
$213$	0	0
$214$	14.4112	0.985130
$215$	−16.2077	−1.10536
$216$	0	0
$217$	7.96336	0.540588
$218$	−38.2717	−2.59209
$219$	0	0
$220$	31.7417	2.14002
$221$	−2.11153	−0.142037
$222$	0	0
$223$	−24.6428	−1.65021	−0.825103	−	0.564982i	$-0.808883\pi$
$223$	−24.6428	−1.65021	−0.825103	+	0.564982i	$0.808883\pi$
$224$	8.07054	0.539235
$225$	0	0
$226$	−35.8308	−2.38343
$227$	3.90263	0.259027	0.129513	−	0.991578i	$-0.458659\pi$
$227$	3.90263	0.259027	0.129513	+	0.991578i	$0.458659\pi$
$228$	0	0
$229$	−26.0183	−1.71934	−0.859670	−	0.510850i	$-0.829331\pi$
$229$	−26.0183	−1.71934	−0.859670	+	0.510850i	$0.829331\pi$
$230$	57.8522	3.81466
$231$	0	0
$232$	6.91382	0.453915
$233$	−2.53200	−0.165877	−0.0829386	−	0.996555i	$-0.526431\pi$
$233$	−2.53200	−0.165877	−0.0829386	+	0.996555i	$0.526431\pi$
$234$	0	0
$235$	−34.3057	−2.23786
$236$	−17.4281	−1.13448
$237$	0	0
$238$	−2.29517	−0.148774
$239$	28.8611	1.86687	0.933436	−	0.358744i	$-0.116795\pi$
$239$	28.8611	1.86687	0.933436	+	0.358744i	$0.116795\pi$
$240$	0	0
$241$	17.5365	1.12963	0.564813	−	0.825219i	$-0.308948\pi$
$241$	17.5365	1.12963	0.564813	+	0.825219i	$0.308948\pi$
$242$	−0.675106	−0.0433974
$243$	0	0
$244$	−19.6073	−1.25523
$245$	−3.96692	−0.253437
$246$	0	0
$247$	−2.89162	−0.183989
$248$	−6.29742	−0.399887
$249$	0	0
$250$	47.6136	3.01135
$251$	0.675957	0.0426660	0.0213330	−	0.999772i	$-0.493209\pi$
$251$	0.675957	0.0426660	0.0213330	+	0.999772i	$0.493209\pi$
$252$	0	0
$253$	−23.4534	−1.47450
$254$	2.09235	0.131286
$255$	0	0
$256$	8.36709	0.522943
$257$	17.2714	1.07736	0.538679	−	0.842511i	$-0.318924\pi$
$257$	17.2714	1.07736	0.538679	+	0.842511i	$0.318924\pi$
$258$	0	0
$259$	0.395595	0.0245811
$260$	18.1582	1.12612
$261$	0	0
$262$	−30.4969	−1.88410
$263$	19.5148	1.20333	0.601667	−	0.798747i	$-0.294503\pi$
$263$	19.5148	1.20333	0.601667	+	0.798747i	$0.294503\pi$
$264$	0	0
$265$	−5.65720	−0.347519
$266$	−3.14311	−0.192717
$267$	0	0
$268$	−2.13038	−0.130134
$269$	−7.20512	−0.439304	−0.219652	−	0.975578i	$-0.570492\pi$
$269$	−7.20512	−0.439304	−0.219652	+	0.975578i	$0.570492\pi$
$270$	0	0
$271$	1.50972	0.0917087	0.0458543	−	0.998948i	$-0.485399\pi$
$271$	1.50972	0.0917087	0.0458543	+	0.998948i	$0.485399\pi$
$272$	−3.40187	−0.206269
$273$	0	0
$274$	10.9282	0.660198
$275$	−36.1272	−2.17855
$276$	0	0
$277$	−7.49701	−0.450452	−0.225226	−	0.974307i	$-0.572312\pi$
$277$	−7.49701	−0.450452	−0.225226	+	0.974307i	$0.572312\pi$
$278$	15.3365	0.919823
$279$	0	0
$280$	3.13704	0.187474
$281$	10.2223	0.609813	0.304907	−	0.952382i	$-0.401375\pi$
$281$	10.2223	0.609813	0.304907	+	0.952382i	$0.401375\pi$
$282$	0	0
$283$	10.7912	0.641470	0.320735	−	0.947169i	$-0.396070\pi$
$283$	10.7912	0.641470	0.320735	+	0.947169i	$0.396070\pi$
$284$	−30.6838	−1.82075
$285$	0	0
$286$	−13.5527	−0.801388
$287$	−5.12703	−0.302639
$288$	0	0
$289$	−15.7967	−0.929220
$290$	−72.5672	−4.26129
$291$	0	0
$292$	−35.1979	−2.05980
$293$	0.963362	0.0562802	0.0281401	−	0.999604i	$-0.491042\pi$
$293$	0.963362	0.0562802	0.0281401	+	0.999604i	$0.491042\pi$
$294$	0	0
$295$	29.0738	1.69274
$296$	−0.312836	−0.0181832
$297$	0	0
$298$	−44.8897	−2.60039
$299$	−13.4168	−0.775911
$300$	0	0
$301$	4.08571	0.235497
$302$	18.6957	1.07582
$303$	0	0
$304$	−4.65868	−0.267194
$305$	32.7091	1.87292
$306$	0	0
$307$	−1.41653	−0.0808455	−0.0404227	−	0.999183i	$-0.512870\pi$
$307$	−1.41653	−0.0808455	−0.0404227	+	0.999183i	$0.512870\pi$
$308$	−8.00159	−0.455933
$309$	0	0
$310$	66.0975	3.75408
$311$	−20.8547	−1.18256	−0.591281	−	0.806466i	$-0.701377\pi$
$311$	−20.8547	−1.18256	−0.591281	+	0.806466i	$0.701377\pi$
$312$	0	0
$313$	−34.2479	−1.93581	−0.967903	−	0.251325i	$-0.919134\pi$
$313$	−34.2479	−1.93581	−0.967903	+	0.251325i	$0.919134\pi$
$314$	43.8696	2.47571
$315$	0	0
$316$	−24.9610	−1.40417
$317$	−4.40061	−0.247163	−0.123581	−	0.992334i	$-0.539438\pi$
$317$	−4.40061	−0.247163	−0.123581	+	0.992334i	$0.539438\pi$
$318$	0	0
$319$	29.4189	1.64714
$320$	42.3822	2.36924
$321$	0	0
$322$	−14.5837	−0.812716
$323$	1.64780	0.0916860
$324$	0	0
$325$	−20.6670	−1.14640
$326$	37.9334	2.10094
$327$	0	0
$328$	4.05445	0.223870
$329$	8.64796	0.476777
$330$	0	0
$331$	9.58462	0.526818	0.263409	−	0.964684i	$-0.415153\pi$
$331$	9.58462	0.526818	0.263409	+	0.964684i	$0.415153\pi$
$332$	−30.9746	−1.69995
$333$	0	0
$334$	−1.34295	−0.0734832
$335$	3.55393	0.194172
$336$	0	0
$337$	−1.04491	−0.0569201	−0.0284600	−	0.999595i	$-0.509060\pi$
$337$	−1.04491	−0.0569201	−0.0284600	+	0.999595i	$0.509060\pi$
$338$	19.4476	1.05781
$339$	0	0
$340$	−10.3475	−0.561172
$341$	−26.7960	−1.45109
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.23098	−0.174203
$345$	0	0
$346$	−14.5494	−0.782181
$347$	−24.4688	−1.31355	−0.656777	−	0.754085i	$-0.728081\pi$
$347$	−24.4688	−1.31355	−0.656777	+	0.754085i	$0.728081\pi$
$348$	0	0
$349$	2.10143	0.112487	0.0562436	−	0.998417i	$-0.482088\pi$
$349$	2.10143	0.112487	0.0562436	+	0.998417i	$0.482088\pi$
$350$	−22.4644	−1.20077
$351$	0	0
$352$	−27.1567	−1.44746
$353$	6.53053	0.347585	0.173793	−	0.984782i	$-0.444398\pi$
$353$	6.53053	0.347585	0.173793	+	0.984782i	$0.444398\pi$
$354$	0	0
$355$	51.1871	2.71673
$356$	4.24852	0.225171
$357$	0	0
$358$	16.3134	0.862188
$359$	4.49262	0.237111	0.118556	−	0.992947i	$-0.462174\pi$
$359$	4.49262	0.237111	0.118556	+	0.992947i	$0.462174\pi$
$360$	0	0
$361$	−16.7434	−0.881233
$362$	4.17984	0.219688
$363$	0	0
$364$	−4.57740	−0.239921
$365$	58.7175	3.07341
$366$	0	0
$367$	0.923799	0.0482219	0.0241110	−	0.999709i	$-0.492325\pi$
$367$	0.923799	0.0482219	0.0241110	+	0.999709i	$0.492325\pi$
$368$	−21.6157	−1.12680
$369$	0	0
$370$	3.28352	0.170702
$371$	1.42610	0.0740392
$372$	0	0
$373$	−10.2352	−0.529961	−0.264980	−	0.964254i	$-0.585366\pi$
$373$	−10.2352	−0.529961	−0.264980	+	0.964254i	$0.585366\pi$
$374$	7.72306	0.399350
$375$	0	0
$376$	−6.83880	−0.352684
$377$	16.8294	0.866757
$378$	0	0
$379$	24.0858	1.23720	0.618602	−	0.785704i	$-0.287699\pi$
$379$	24.0858	1.23720	0.618602	+	0.785704i	$0.287699\pi$
$380$	−14.1703	−0.726923
$381$	0	0
$382$	7.59383	0.388534
$383$	−5.47064	−0.279537	−0.139768	−	0.990184i	$-0.544636\pi$
$383$	−5.47064	−0.279537	−0.139768	+	0.990184i	$0.544636\pi$
$384$	0	0
$385$	13.3483	0.680295
$386$	−55.0928	−2.80415
$387$	0	0
$388$	36.6325	1.85973
$389$	−14.5911	−0.739798	−0.369899	−	0.929072i	$-0.620608\pi$
$389$	−14.5911	−0.739798	−0.369899	+	0.929072i	$0.620608\pi$
$390$	0	0
$391$	7.64559	0.386654
$392$	−0.790800	−0.0399414
$393$	0	0
$394$	13.1985	0.664930
$395$	41.6402	2.09515
$396$	0	0
$397$	32.9307	1.65274	0.826371	−	0.563126i	$-0.190401\pi$
$397$	32.9307	1.65274	0.826371	+	0.563126i	$0.190401\pi$
$398$	49.8481	2.49866
$399$	0	0
$400$	−33.2965	−1.66483
$401$	26.3376	1.31524	0.657619	−	0.753351i	$-0.271564\pi$
$401$	26.3376	1.31524	0.657619	+	0.753351i	$0.271564\pi$
$402$	0	0
$403$	−15.3290	−0.763590
$404$	−31.9176	−1.58796
$405$	0	0
$406$	18.2931	0.907871
$407$	−1.33114	−0.0659823
$408$	0	0
$409$	−0.402040	−0.0198796	−0.00993979	−	0.999951i	$-0.503164\pi$
$409$	−0.402040	−0.0198796	−0.00993979	+	0.999951i	$0.503164\pi$
$410$	−42.5554	−2.10166
$411$	0	0
$412$	47.1890	2.32483
$413$	−7.32907	−0.360640
$414$	0	0
$415$	51.6722	2.53649
$416$	−15.5353	−0.761679
$417$	0	0
$418$	10.5763	0.517304
$419$	13.6019	0.664495	0.332248	−	0.943192i	$-0.392193\pi$
$419$	13.6019	0.664495	0.332248	+	0.943192i	$0.392193\pi$
$420$	0	0
$421$	24.9947	1.21817	0.609084	−	0.793106i	$-0.291537\pi$
$421$	24.9947	1.21817	0.609084	+	0.793106i	$0.291537\pi$
$422$	18.9099	0.920520
$423$	0	0
$424$	−1.12776	−0.0547687
$425$	11.7771	0.571276
$426$	0	0
$427$	−8.24548	−0.399027
$428$	−16.3783	−0.791673
$429$	0	0
$430$	33.9122	1.63539
$431$	−33.8505	−1.63052	−0.815260	−	0.579095i	$-0.803406\pi$
$431$	−33.8505	−1.63052	−0.815260	+	0.579095i	$0.803406\pi$
$432$	0	0
$433$	25.8901	1.24420	0.622100	−	0.782937i	$-0.286280\pi$
$433$	25.8901	1.24420	0.622100	+	0.782937i	$0.286280\pi$
$434$	−16.6622	−0.799810
$435$	0	0
$436$	43.4955	2.08306
$437$	10.4702	0.500858
$438$	0	0
$439$	24.4846	1.16859	0.584294	−	0.811542i	$-0.301371\pi$
$439$	24.4846	1.16859	0.584294	+	0.811542i	$0.301371\pi$
$440$	−10.5559	−0.503231
$441$	0	0
$442$	4.41806	0.210146
$443$	20.9781	0.996701	0.498350	−	0.866976i	$-0.333939\pi$
$443$	20.9781	0.996701	0.498350	+	0.866976i	$0.333939\pi$
$444$	0	0
$445$	−7.08743	−0.335976
$446$	51.5615	2.44151
$447$	0	0
$448$	−10.6839	−0.504767
$449$	6.21606	0.293354	0.146677	−	0.989184i	$-0.453142\pi$
$449$	6.21606	0.293354	0.146677	+	0.989184i	$0.453142\pi$
$450$	0	0
$451$	17.2520	0.812366
$452$	40.7215	1.91538
$453$	0	0
$454$	−8.16568	−0.383234
$455$	7.63607	0.357984
$456$	0	0
$457$	4.14940	0.194101	0.0970504	−	0.995279i	$-0.469059\pi$
$457$	4.14940	0.194101	0.0970504	+	0.995279i	$0.469059\pi$
$458$	54.4396	2.54379
$459$	0	0
$460$	−65.7486	−3.06555
$461$	18.7754	0.874455	0.437228	−	0.899351i	$-0.355960\pi$
$461$	18.7754	0.874455	0.437228	+	0.899351i	$0.355960\pi$
$462$	0	0
$463$	−7.26619	−0.337688	−0.168844	−	0.985643i	$-0.554003\pi$
$463$	−7.26619	−0.337688	−0.168844	+	0.985643i	$0.554003\pi$
$464$	27.1138	1.25873
$465$	0	0
$466$	5.29785	0.245418
$467$	7.37026	0.341055	0.170527	−	0.985353i	$-0.445453\pi$
$467$	7.37026	0.341055	0.170527	+	0.985353i	$0.445453\pi$
$468$	0	0
$469$	−0.895891	−0.0413684
$470$	71.7798	3.31095
$471$	0	0
$472$	5.79583	0.266775
$473$	−13.7481	−0.632137
$474$	0	0
$475$	16.1282	0.740011
$476$	2.60845	0.119558
$477$	0	0
$478$	−60.3877	−2.76207
$479$	13.8941	0.634839	0.317419	−	0.948285i	$-0.397184\pi$
$479$	13.8941	0.634839	0.317419	+	0.948285i	$0.397184\pi$
$480$	0	0
$481$	−0.761495	−0.0347212
$482$	−36.6926	−1.67130
$483$	0	0
$484$	0.767253	0.0348751
$485$	−61.1107	−2.77489
$486$	0	0
$487$	9.72227	0.440558	0.220279	−	0.975437i	$-0.429303\pi$
$487$	9.72227	0.440558	0.220279	+	0.975437i	$0.429303\pi$
$488$	6.52052	0.295170
$489$	0	0
$490$	8.30020	0.374965
$491$	−15.8783	−0.716580	−0.358290	−	0.933610i	$-0.616640\pi$
$491$	−15.8783	−0.716580	−0.358290	+	0.933610i	$0.616640\pi$
$492$	0	0
$493$	−9.59029	−0.431925
$494$	6.05029	0.272215
$495$	0	0
$496$	−24.6965	−1.10890
$497$	−12.9035	−0.578801
$498$	0	0
$499$	36.7007	1.64295	0.821474	−	0.570246i	$-0.193152\pi$
$499$	36.7007	1.64295	0.821474	+	0.570246i	$0.193152\pi$
$500$	−54.1125	−2.41999
$501$	0	0
$502$	−1.41434	−0.0631252
$503$	−10.2539	−0.457198	−0.228599	−	0.973521i	$-0.573414\pi$
$503$	−10.2539	−0.457198	−0.228599	+	0.973521i	$0.573414\pi$
$504$	0	0
$505$	53.2453	2.36938
$506$	49.0728	2.18155
$507$	0	0
$508$	−2.37795	−0.105504
$509$	−5.05177	−0.223916	−0.111958	−	0.993713i	$-0.535712\pi$
$509$	−5.05177	−0.223916	−0.111958	+	0.993713i	$0.535712\pi$
$510$	0	0
$511$	−14.8018	−0.654792
$512$	−29.9338	−1.32290
$513$	0	0
$514$	−36.1378	−1.59397
$515$	−78.7212	−3.46887
$516$	0	0
$517$	−29.0996	−1.27980
$518$	−0.827725	−0.0363681
$519$	0	0
$520$	−6.03860	−0.264810
$521$	19.0791	0.835869	0.417935	−	0.908477i	$-0.362754\pi$
$521$	19.0791	0.835869	0.417935	+	0.908477i	$0.362754\pi$
$522$	0	0
$523$	1.59855	0.0698997	0.0349498	−	0.999389i	$-0.488873\pi$
$523$	1.59855	0.0698997	0.0349498	+	0.999389i	$0.488873\pi$
$524$	34.6595	1.51411
$525$	0	0
$526$	−40.8319	−1.78036
$527$	8.73527	0.380514
$528$	0	0
$529$	25.5806	1.11220
$530$	11.8369	0.514161
$531$	0	0
$532$	3.57213	0.154871
$533$	9.86920	0.427483
$534$	0	0
$535$	27.3224	1.18125
$536$	0.708471	0.0306013
$537$	0	0
$538$	15.0757	0.649958
$539$	−3.36492	−0.144937
$540$	0	0
$541$	30.5583	1.31380	0.656902	−	0.753976i	$-0.271867\pi$
$541$	30.5583	1.31380	0.656902	+	0.753976i	$0.271867\pi$
$542$	−3.15886	−0.135685
$543$	0	0
$544$	8.85283	0.379562
$545$	−72.5597	−3.10812
$546$	0	0
$547$	14.6932	0.628235	0.314118	−	0.949384i	$-0.398291\pi$
$547$	14.6932	0.628235	0.314118	+	0.949384i	$0.398291\pi$
$548$	−12.4199	−0.530550
$549$	0	0
$550$	75.5909	3.22321
$551$	−13.1334	−0.559500
$552$	0	0
$553$	−10.4969	−0.446372
$554$	15.6864	0.666452
$555$	0	0
$556$	−17.4299	−0.739190
$557$	27.1940	1.15225	0.576124	−	0.817362i	$-0.304565\pi$
$557$	27.1940	1.15225	0.576124	+	0.817362i	$0.304565\pi$
$558$	0	0
$559$	−7.86474	−0.332643
$560$	12.3024	0.519873
$561$	0	0
$562$	−21.3887	−0.902230
$563$	24.2738	1.02302	0.511510	−	0.859277i	$-0.329086\pi$
$563$	24.2738	1.02302	0.511510	+	0.859277i	$0.329086\pi$
$564$	0	0
$565$	−67.9321	−2.85792
$566$	−22.5790	−0.949066
$567$	0	0
$568$	10.2041	0.428154
$569$	10.7187	0.449350	0.224675	−	0.974434i	$-0.427868\pi$
$569$	10.7187	0.449350	0.224675	+	0.974434i	$0.427868\pi$
$570$	0	0
$571$	6.74727	0.282365	0.141182	−	0.989984i	$-0.454910\pi$
$571$	6.74727	0.282365	0.141182	+	0.989984i	$0.454910\pi$
$572$	15.4026	0.644013
$573$	0	0
$574$	10.7276	0.447760
$575$	74.8327	3.12074
$576$	0	0
$577$	−14.0711	−0.585788	−0.292894	−	0.956145i	$-0.594618\pi$
$577$	−14.0711	−0.585788	−0.292894	+	0.956145i	$0.594618\pi$
$578$	33.0524	1.37480
$579$	0	0
$580$	82.4721	3.42447
$581$	−13.0258	−0.540400
$582$	0	0
$583$	−4.79869	−0.198741
$584$	11.7053	0.484367
$585$	0	0
$586$	−2.01569	−0.0832676
$587$	2.55038	0.105265	0.0526327	−	0.998614i	$-0.483239\pi$
$587$	2.55038	0.105265	0.0526327	+	0.998614i	$0.483239\pi$
$588$	0	0
$589$	11.9625	0.492905
$590$	−60.8327	−2.50444
$591$	0	0
$592$	−1.22684	−0.0504229
$593$	44.5795	1.83066	0.915331	−	0.402702i	$-0.131929\pi$
$593$	44.5795	1.83066	0.915331	+	0.402702i	$0.131929\pi$
$594$	0	0
$595$	−4.35144	−0.178392
$596$	51.0168	2.08973
$597$	0	0
$598$	28.0726	1.14797
$599$	−27.4284	−1.12069	−0.560347	−	0.828258i	$-0.689332\pi$
$599$	−27.4284	−1.12069	−0.560347	+	0.828258i	$0.689332\pi$
$600$	0	0
$601$	−39.6149	−1.61593	−0.807963	−	0.589233i	$-0.799430\pi$
$601$	−39.6149	−1.61593	−0.807963	+	0.589233i	$0.799430\pi$
$602$	−8.54876	−0.348421
$603$	0	0
$604$	−21.2476	−0.864551
$605$	−1.27994	−0.0520370
$606$	0	0
$607$	30.6434	1.24378	0.621889	−	0.783106i	$-0.286366\pi$
$607$	30.6434	1.24378	0.621889	+	0.783106i	$0.286366\pi$
$608$	12.1235	0.491672
$609$	0	0
$610$	−68.4391	−2.77102
$611$	−16.6468	−0.673456
$612$	0	0
$613$	32.6688	1.31948	0.659741	−	0.751493i	$-0.270666\pi$
$613$	32.6688	1.31948	0.659741	+	0.751493i	$0.270666\pi$
$614$	2.96388	0.119612
$615$	0	0
$616$	2.66097	0.107214
$617$	−30.1631	−1.21432	−0.607160	−	0.794580i	$-0.707691\pi$
$617$	−30.1631	−1.21432	−0.607160	+	0.794580i	$0.707691\pi$
$618$	0	0
$619$	−4.57715	−0.183971	−0.0919856	−	0.995760i	$-0.529321\pi$
$619$	−4.57715	−0.183971	−0.0919856	+	0.995760i	$0.529321\pi$
$620$	−75.1193	−3.01687
$621$	0	0
$622$	43.6354	1.74962
$623$	1.78663	0.0715799
$624$	0	0
$625$	36.5889	1.46356
$626$	71.6587	2.86406
$627$	0	0
$628$	−49.8575	−1.98953
$629$	0.433941	0.0173024
$630$	0	0
$631$	−15.8483	−0.630911	−0.315455	−	0.948940i	$-0.602157\pi$
$631$	−15.8483	−0.630911	−0.315455	+	0.948940i	$0.602157\pi$
$632$	8.30092	0.330193
$633$	0	0
$634$	9.20763	0.365682
$635$	3.96692	0.157422
$636$	0	0
$637$	−1.92494	−0.0762688
$638$	−61.5547	−2.43697
$639$	0	0
$640$	−24.6482	−0.974306
$641$	−22.9792	−0.907624	−0.453812	−	0.891097i	$-0.649936\pi$
$641$	−22.9792	−0.907624	−0.453812	+	0.891097i	$0.649936\pi$
$642$	0	0
$643$	−8.56313	−0.337697	−0.168848	−	0.985642i	$-0.554005\pi$
$643$	−8.56313	−0.337697	−0.168848	+	0.985642i	$0.554005\pi$
$644$	16.5742	0.653116
$645$	0	0
$646$	−3.44778	−0.135651
$647$	−37.9948	−1.49373	−0.746865	−	0.664976i	$-0.768442\pi$
$647$	−37.9948	−1.49373	−0.746865	+	0.664976i	$0.768442\pi$
$648$	0	0
$649$	24.6617	0.968056
$650$	43.2426	1.69611
$651$	0	0
$652$	−43.1111	−1.68836
$653$	26.3329	1.03048	0.515242	−	0.857045i	$-0.327702\pi$
$653$	26.3329	1.03048	0.515242	+	0.857045i	$0.327702\pi$
$654$	0	0
$655$	−57.8194	−2.25919
$656$	15.9003	0.620801
$657$	0	0
$658$	−18.0946	−0.705401
$659$	−9.50560	−0.370286	−0.185143	−	0.982712i	$-0.559275\pi$
$659$	−9.50560	−0.370286	−0.185143	+	0.982712i	$0.559275\pi$
$660$	0	0
$661$	17.6702	0.687292	0.343646	−	0.939099i	$-0.388338\pi$
$661$	17.6702	0.687292	0.343646	+	0.939099i	$0.388338\pi$
$662$	−20.0544	−0.779437
$663$	0	0
$664$	10.3008	0.399748
$665$	−5.95906	−0.231082
$666$	0	0
$667$	−60.9373	−2.35950
$668$	1.52626	0.0590528
$669$	0	0
$670$	−7.43608	−0.287281
$671$	27.7453	1.07110
$672$	0	0
$673$	−40.0668	−1.54446	−0.772231	−	0.635342i	$-0.780859\pi$
$673$	−40.0668	−1.54446	−0.772231	+	0.635342i	$0.780859\pi$
$674$	2.18633	0.0842142
$675$	0	0
$676$	−22.1021	−0.850081
$677$	45.5736	1.75153	0.875767	−	0.482734i	$-0.160356\pi$
$677$	45.5736	1.75153	0.875767	+	0.482734i	$0.160356\pi$
$678$	0	0
$679$	15.4051	0.591193
$680$	3.44112	0.131961
$681$	0	0
$682$	56.0668	2.14691
$683$	−8.24883	−0.315633	−0.157816	−	0.987468i	$-0.550445\pi$
$683$	−8.24883	−0.315633	−0.157816	+	0.987468i	$0.550445\pi$
$684$	0	0
$685$	20.7189	0.791630
$686$	−2.09235	−0.0798865
$687$	0	0
$688$	−12.6709	−0.483072
$689$	−2.74514	−0.104582
$690$	0	0
$691$	−21.3610	−0.812610	−0.406305	−	0.913738i	$-0.633183\pi$
$691$	−21.3610	−0.812610	−0.406305	+	0.913738i	$0.633183\pi$
$692$	16.5353	0.628578
$693$	0	0
$694$	51.1974	1.94343
$695$	29.0767	1.10294
$696$	0	0
$697$	−5.62400	−0.213024
$698$	−4.39695	−0.166427
$699$	0	0
$700$	25.5307	0.964969
$701$	−19.5387	−0.737968	−0.368984	−	0.929436i	$-0.620294\pi$
$701$	−19.5387	−0.737968	−0.368984	+	0.929436i	$0.620294\pi$
$702$	0	0
$703$	0.594258	0.0224129
$704$	35.9504	1.35493
$705$	0	0
$706$	−13.6642	−0.514258
$707$	−13.4223	−0.504799
$708$	0	0
$709$	37.3661	1.40331	0.701657	−	0.712515i	$-0.252444\pi$
$709$	37.3661	1.40331	0.701657	+	0.712515i	$0.252444\pi$
$710$	−107.102	−4.01945
$711$	0	0
$712$	−1.41287	−0.0529495
$713$	55.5044	2.07866
$714$	0	0
$715$	−25.6947	−0.960928
$716$	−18.5400	−0.692874
$717$	0	0
$718$	−9.40015	−0.350811
$719$	−6.79800	−0.253523	−0.126761	−	0.991933i	$-0.540458\pi$
$719$	−6.79800	−0.253523	−0.126761	+	0.991933i	$0.540458\pi$
$720$	0	0
$721$	19.8444	0.739044
$722$	35.0332	1.30380
$723$	0	0
$724$	−4.75036	−0.176546
$725$	−93.8668	−3.48613
$726$	0	0
$727$	−11.9071	−0.441608	−0.220804	−	0.975318i	$-0.570868\pi$
$727$	−11.9071	−0.441608	−0.220804	+	0.975318i	$0.570868\pi$
$728$	1.52224	0.0564179
$729$	0	0
$730$	−122.858	−4.54717
$731$	4.48175	0.165764
$732$	0	0
$733$	−15.9489	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$733$	−15.9489	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$734$	−1.93291	−0.0713452
$735$	0	0
$736$	56.2514	2.07346
$737$	3.01460	0.111044
$738$	0	0
$739$	32.3318	1.18934	0.594671	−	0.803969i	$-0.297282\pi$
$739$	32.3318	1.18934	0.594671	+	0.803969i	$0.297282\pi$
$740$	−3.73169	−0.137180
$741$	0	0
$742$	−2.98390	−0.109542
$743$	47.8459	1.75530	0.877648	−	0.479306i	$-0.159112\pi$
$743$	47.8459	1.75530	0.877648	+	0.479306i	$0.159112\pi$
$744$	0	0
$745$	−85.1068	−3.11807
$746$	21.4158	0.784087
$747$	0	0
$748$	−8.77720	−0.320926
$749$	−6.88756	−0.251666
$750$	0	0
$751$	4.64951	0.169663	0.0848315	−	0.996395i	$-0.472965\pi$
$751$	4.64951	0.169663	0.0848315	+	0.996395i	$0.472965\pi$
$752$	−26.8196	−0.978009
$753$	0	0
$754$	−35.2130	−1.28238
$755$	35.4454	1.28999
$756$	0	0
$757$	3.97451	0.144456	0.0722279	−	0.997388i	$-0.476989\pi$
$757$	3.97451	0.144456	0.0722279	+	0.997388i	$0.476989\pi$
$758$	−50.3960	−1.83047
$759$	0	0
$760$	4.71242	0.170938
$761$	−24.8801	−0.901903	−0.450952	−	0.892548i	$-0.648915\pi$
$761$	−24.8801	−0.901903	−0.450952	+	0.892548i	$0.648915\pi$
$762$	0	0
$763$	18.2912	0.662186
$764$	−8.63034	−0.312235
$765$	0	0
$766$	11.4465	0.413580
$767$	14.1080	0.509410
$768$	0	0
$769$	−50.9477	−1.83722	−0.918611	−	0.395164i	$-0.870688\pi$
$769$	−50.9477	−1.83722	−0.918611	+	0.395164i	$0.870688\pi$
$770$	−27.9295	−1.00651
$771$	0	0
$772$	62.6126	2.25348
$773$	3.65460	0.131447	0.0657233	−	0.997838i	$-0.479065\pi$
$773$	3.65460	0.131447	0.0657233	+	0.997838i	$0.479065\pi$
$774$	0	0
$775$	85.4981	3.07118
$776$	−12.1823	−0.437320
$777$	0	0
$778$	30.5298	1.09455
$779$	−7.70176	−0.275944
$780$	0	0
$781$	43.4192	1.55366
$782$	−15.9973	−0.572062
$783$	0	0
$784$	−3.10126	−0.110759
$785$	83.1729	2.96857
$786$	0	0
$787$	−40.4237	−1.44095	−0.720474	−	0.693482i	$-0.756076\pi$
$787$	−40.4237	−1.44095	−0.720474	+	0.693482i	$0.756076\pi$
$788$	−15.0000	−0.534353
$789$	0	0
$790$	−87.1261	−3.09981
$791$	17.1246	0.608882
$792$	0	0
$793$	15.8720	0.563632
$794$	−68.9026	−2.44526
$795$	0	0
$796$	−56.6520	−2.00798
$797$	8.44675	0.299199	0.149600	−	0.988747i	$-0.452202\pi$
$797$	8.44675	0.299199	0.149600	+	0.988747i	$0.452202\pi$
$798$	0	0
$799$	9.48623	0.335599
$800$	86.6488	3.06350
$801$	0	0
$802$	−55.1076	−1.94592
$803$	49.8068	1.75764
$804$	0	0
$805$	−27.6493	−0.974511
$806$	32.0736	1.12975
$807$	0	0
$808$	10.6144	0.373413
$809$	23.3390	0.820556	0.410278	−	0.911960i	$-0.365432\pi$
$809$	23.3390	0.820556	0.410278	+	0.911960i	$0.365432\pi$
$810$	0	0
$811$	15.5167	0.544864	0.272432	−	0.962175i	$-0.412172\pi$
$811$	15.5167	0.544864	0.272432	+	0.962175i	$0.412172\pi$
$812$	−20.7900	−0.729585
$813$	0	0
$814$	2.78522	0.0976220
$815$	71.9184	2.51919
$816$	0	0
$817$	6.13751	0.214724
$818$	0.841209	0.0294122
$819$	0	0
$820$	48.3639	1.68894
$821$	41.1428	1.43590	0.717948	−	0.696097i	$-0.245082\pi$
$821$	41.1428	1.43590	0.717948	+	0.696097i	$0.245082\pi$
$822$	0	0
$823$	−36.6572	−1.27779	−0.638895	−	0.769294i	$-0.720608\pi$
$823$	−36.6572	−1.27779	−0.638895	+	0.769294i	$0.720608\pi$
$824$	−15.6930	−0.546690
$825$	0	0
$826$	15.3350	0.533573
$827$	37.3911	1.30022	0.650109	−	0.759841i	$-0.274723\pi$
$827$	37.3911	1.30022	0.650109	+	0.759841i	$0.274723\pi$
$828$	0	0
$829$	−20.0071	−0.694877	−0.347438	−	0.937703i	$-0.612948\pi$
$829$	−20.0071	−0.694877	−0.347438	+	0.937703i	$0.612948\pi$
$830$	−108.117	−3.75278
$831$	0	0
$832$	20.5658	0.712992
$833$	1.09693	0.0380065
$834$	0	0
$835$	−2.54612	−0.0881122
$836$	−12.0199	−0.415717
$837$	0	0
$838$	−28.4600	−0.983133
$839$	−18.1768	−0.627534	−0.313767	−	0.949500i	$-0.601591\pi$
$839$	−18.1768	−0.627534	−0.313767	+	0.949500i	$0.601591\pi$
$840$	0	0
$841$	47.4370	1.63576
$842$	−52.2978	−1.80230
$843$	0	0
$844$	−21.4910	−0.739751
$845$	36.8710	1.26840
$846$	0	0
$847$	0.322654	0.0110865
$848$	−4.42269	−0.151876
$849$	0	0
$850$	−24.6420	−0.845213
$851$	2.75729	0.0945186
$852$	0	0
$853$	−14.0356	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$853$	−14.0356	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$854$	17.2525	0.590367
$855$	0	0
$856$	5.44668	0.186164
$857$	47.3584	1.61773	0.808866	−	0.587993i	$-0.200082\pi$
$857$	47.3584	1.61773	0.808866	+	0.587993i	$0.200082\pi$
$858$	0	0
$859$	42.4759	1.44926	0.724629	−	0.689139i	$-0.242011\pi$
$859$	42.4759	1.44926	0.724629	+	0.689139i	$0.242011\pi$
$860$	−38.5410	−1.31424
$861$	0	0
$862$	70.8272	2.41238
$863$	−17.4580	−0.594276	−0.297138	−	0.954834i	$-0.596032\pi$
$863$	−17.4580	−0.594276	−0.297138	+	0.954834i	$0.596032\pi$
$864$	0	0
$865$	−27.5844	−0.937897
$866$	−54.1714	−1.84082
$867$	0	0
$868$	18.9364	0.642745
$869$	35.3211	1.19819
$870$	0	0
$871$	1.72453	0.0584336
$872$	−14.4647	−0.489836
$873$	0	0
$874$	−21.9074	−0.741029
$875$	−22.7560	−0.769293
$876$	0	0
$877$	24.2333	0.818299	0.409150	−	0.912467i	$-0.365825\pi$
$877$	24.2333	0.818299	0.409150	+	0.912467i	$0.365825\pi$
$878$	−51.2306	−1.72895
$879$	0	0
$880$	−41.3967	−1.39548
$881$	31.1584	1.04975	0.524877	−	0.851178i	$-0.324111\pi$
$881$	31.1584	1.04975	0.524877	+	0.851178i	$0.324111\pi$
$882$	0	0
$883$	−47.0978	−1.58497	−0.792483	−	0.609894i	$-0.791212\pi$
$883$	−47.0978	−1.58497	−0.792483	+	0.609894i	$0.791212\pi$
$884$	−5.02110	−0.168878
$885$	0	0
$886$	−43.8937	−1.47464
$887$	51.6181	1.73317	0.866583	−	0.499034i	$-0.166312\pi$
$887$	51.6181	1.73317	0.866583	+	0.499034i	$0.166312\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	14.8294	0.497083
$891$	0	0
$892$	−58.5994	−1.96205
$893$	12.9909	0.434723
$894$	0	0
$895$	30.9287	1.03383
$896$	6.21344	0.207576
$897$	0	0
$898$	−13.0062	−0.434023
$899$	−69.6223	−2.32203
$900$	0	0
$901$	1.56433	0.0521154
$902$	−36.0973	−1.20191
$903$	0	0
$904$	−13.5422	−0.450406
$905$	7.92461	0.263423
$906$	0	0
$907$	56.8898	1.88899	0.944497	−	0.328519i	$-0.106550\pi$
$907$	56.8898	1.88899	0.944497	+	0.328519i	$0.106550\pi$
$908$	9.28024	0.307976
$909$	0	0
$910$	−15.9774	−0.529644
$911$	38.1640	1.26443	0.632214	−	0.774794i	$-0.282146\pi$
$911$	38.1640	1.26443	0.632214	+	0.774794i	$0.282146\pi$
$912$	0	0
$913$	43.8307	1.45058
$914$	−8.68202	−0.287176
$915$	0	0
$916$	−61.8702	−2.04425
$917$	14.5754	0.481322
$918$	0	0
$919$	19.8403	0.654473	0.327236	−	0.944943i	$-0.393883\pi$
$919$	19.8403	0.654473	0.327236	+	0.944943i	$0.393883\pi$
$920$	21.8651	0.720870
$921$	0	0
$922$	−39.2847	−1.29377
$923$	24.8384	0.817566
$924$	0	0
$925$	4.24728	0.139650
$926$	15.2034	0.499616
$927$	0	0
$928$	−70.5593	−2.31622
$929$	25.0484	0.821813	0.410906	−	0.911678i	$-0.365212\pi$
$929$	25.0484	0.821813	0.410906	+	0.911678i	$0.365212\pi$
$930$	0	0
$931$	1.50219	0.0492322
$932$	−6.02097	−0.197224
$933$	0	0
$934$	−15.4212	−0.504597
$935$	14.6422	0.478852
$936$	0	0
$937$	−10.5239	−0.343800	−0.171900	−	0.985114i	$-0.554991\pi$
$937$	−10.5239	−0.343800	−0.171900	+	0.985114i	$0.554991\pi$
$938$	1.87452	0.0612053
$939$	0	0
$940$	−81.5772	−2.66076
$941$	44.8694	1.46270	0.731351	−	0.682001i	$-0.238890\pi$
$941$	44.8694	1.46270	0.731351	+	0.682001i	$0.238890\pi$
$942$	0	0
$943$	−35.7353	−1.16370
$944$	22.7294	0.739778
$945$	0	0
$946$	28.7659	0.935259
$947$	−22.9147	−0.744629	−0.372315	−	0.928107i	$-0.621436\pi$
$947$	−22.9147	−0.744629	−0.372315	+	0.928107i	$0.621436\pi$
$948$	0	0
$949$	28.4925	0.924906
$950$	−33.7458	−1.09486
$951$	0	0
$952$	−0.867454	−0.0281143
$953$	−22.6679	−0.734285	−0.367143	−	0.930165i	$-0.619664\pi$
$953$	−22.6679	−0.734285	−0.367143	+	0.930165i	$0.619664\pi$
$954$	0	0
$955$	14.3972	0.465883
$956$	68.6303	2.21966
$957$	0	0
$958$	−29.0714	−0.939255
$959$	−5.22293	−0.168657
$960$	0	0
$961$	32.4151	1.04565
$962$	1.59332	0.0513706
$963$	0	0
$964$	41.7009	1.34309
$965$	−104.451	−3.36240
$966$	0	0
$967$	37.7740	1.21473	0.607365	−	0.794423i	$-0.292226\pi$
$967$	37.7740	1.21473	0.607365	+	0.794423i	$0.292226\pi$
$968$	−0.255154	−0.00820097
$969$	0	0
$970$	127.865	4.10551
$971$	56.9817	1.82863	0.914315	−	0.405005i	$-0.132730\pi$
$971$	56.9817	1.82863	0.914315	+	0.405005i	$0.132730\pi$
$972$	0	0
$973$	−7.32979	−0.234982
$974$	−20.3424	−0.651814
$975$	0	0
$976$	25.5714	0.818520
$977$	36.8275	1.17822	0.589109	−	0.808054i	$-0.299479\pi$
$977$	36.8275	1.17822	0.589109	+	0.808054i	$0.299479\pi$
$978$	0	0
$979$	−6.01187	−0.192140
$980$	−9.43312	−0.301330
$981$	0	0
$982$	33.2231	1.06019
$983$	−8.77102	−0.279752	−0.139876	−	0.990169i	$-0.544670\pi$
$983$	−8.77102	−0.279752	−0.139876	+	0.990169i	$0.544670\pi$
$984$	0	0
$985$	25.0232	0.797304
$986$	20.0663	0.639041
$987$	0	0
$988$	−6.87612	−0.218758
$989$	28.4773	0.905526
$990$	0	0
$991$	53.2754	1.69235	0.846175	−	0.532905i	$-0.178900\pi$
$991$	53.2754	1.69235	0.846175	+	0.532905i	$0.178900\pi$
$992$	64.2686	2.04053
$993$	0	0
$994$	26.9987	0.856347
$995$	94.5075	2.99609
$996$	0	0
$997$	−43.2333	−1.36921	−0.684606	−	0.728914i	$-0.740026\pi$
$997$	−43.2333	−1.36921	−0.684606	+	0.728914i	$0.740026\pi$
$998$	−76.7909	−2.43077
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.9	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.32	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.9	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.32	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)

\(f(q)\)	\(=\)	\(q-2.09235 q^{2} +2.37795 q^{4} -3.96692 q^{5} +1.00000 q^{7} -0.790800 q^{8} +O(q^{10})\)	\(q-2.09235 q^{2} +2.37795 q^{4} -3.96692 q^{5} +1.00000 q^{7} -0.790800 q^{8} +8.30020 q^{10} -3.36492 q^{11} -1.92494 q^{13} -2.09235 q^{14} -3.10126 q^{16} +1.09693 q^{17} +1.50219 q^{19} -9.43312 q^{20} +7.04060 q^{22} +6.96998 q^{23} +10.7364 q^{25} +4.02765 q^{26} +2.37795 q^{28} -8.74282 q^{29} +7.96336 q^{31} +8.07054 q^{32} -2.29517 q^{34} -3.96692 q^{35} +0.395595 q^{37} -3.14311 q^{38} +3.13704 q^{40} -5.12703 q^{41} +4.08571 q^{43} -8.00159 q^{44} -14.5837 q^{46} +8.64796 q^{47} +1.00000 q^{49} -22.4644 q^{50} -4.57740 q^{52} +1.42610 q^{53} +13.3483 q^{55} -0.790800 q^{56} +18.2931 q^{58} -7.32907 q^{59} -8.24548 q^{61} -16.6622 q^{62} -10.6839 q^{64} +7.63607 q^{65} -0.895891 q^{67} +2.60845 q^{68} +8.30020 q^{70} -12.9035 q^{71} -14.8018 q^{73} -0.827725 q^{74} +3.57213 q^{76} -3.36492 q^{77} -10.4969 q^{79} +12.3024 q^{80} +10.7276 q^{82} -13.0258 q^{83} -4.35144 q^{85} -8.54876 q^{86} +2.66097 q^{88} +1.78663 q^{89} -1.92494 q^{91} +16.5742 q^{92} -18.0946 q^{94} -5.95906 q^{95} +15.4051 q^{97} -2.09235 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 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

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