LMFDB - Embedded newform 8001.2.a.t.1.15

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.37299$ of defining polynomial
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)

$f(q)$	$=$	$q+2.37299 q^{2} +3.63108 q^{4} +3.84175 q^{5} -1.00000 q^{7} +3.87053 q^{8} +O(q^{10})$	\(q+2.37299 q^{2} +3.63108 q^{4} +3.84175 q^{5} -1.00000 q^{7} +3.87053 q^{8} +9.11644 q^{10} +3.34213 q^{11} +4.24344 q^{13} -2.37299 q^{14} +1.92256 q^{16} -3.68573 q^{17} -1.16163 q^{19} +13.9497 q^{20} +7.93083 q^{22} +1.32516 q^{23} +9.75907 q^{25} +10.0696 q^{26} -3.63108 q^{28} -6.59844 q^{29} -3.12317 q^{31} -3.17883 q^{32} -8.74620 q^{34} -3.84175 q^{35} +5.94714 q^{37} -2.75654 q^{38} +14.8696 q^{40} +10.3758 q^{41} +4.75366 q^{43} +12.1355 q^{44} +3.14459 q^{46} -1.96484 q^{47} +1.00000 q^{49} +23.1582 q^{50} +15.4082 q^{52} -7.36495 q^{53} +12.8396 q^{55} -3.87053 q^{56} -15.6580 q^{58} +8.49156 q^{59} +7.06437 q^{61} -7.41126 q^{62} -11.3885 q^{64} +16.3022 q^{65} +3.62274 q^{67} -13.3832 q^{68} -9.11644 q^{70} +3.79052 q^{71} -14.4419 q^{73} +14.1125 q^{74} -4.21798 q^{76} -3.34213 q^{77} +5.35475 q^{79} +7.38601 q^{80} +24.6216 q^{82} -1.82593 q^{83} -14.1597 q^{85} +11.2804 q^{86} +12.9358 q^{88} +13.8250 q^{89} -4.24344 q^{91} +4.81176 q^{92} -4.66254 q^{94} -4.46271 q^{95} -1.00708 q^{97} +2.37299 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * 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 $

$2$	2.37299	1.67796	0.838978	−	0.544165i	$-0.183153\pi$
$2$	2.37299	1.67796	0.838978	+	0.544165i	$0.183153\pi$
$3$	0	0
$3$	0	0
$4$	3.63108	1.81554
$5$	3.84175	1.71808	0.859042	−	0.511905i	$-0.171060\pi$
$5$	3.84175	1.71808	0.859042	+	0.511905i	$0.171060\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.87053	1.36844
$9$	0	0
$10$	9.11644	2.88287
$11$	3.34213	1.00769	0.503845	−	0.863794i	$-0.331918\pi$
$11$	3.34213	1.00769	0.503845	+	0.863794i	$0.331918\pi$
$12$	0	0
$13$	4.24344	1.17692	0.588459	−	0.808527i	$-0.299735\pi$
$13$	4.24344	1.17692	0.588459	+	0.808527i	$0.299735\pi$
$14$	−2.37299	−0.634208
$15$	0	0
$16$	1.92256	0.480641
$17$	−3.68573	−0.893921	−0.446961	−	0.894554i	$-0.647494\pi$
$17$	−3.68573	−0.893921	−0.446961	+	0.894554i	$0.647494\pi$
$18$	0	0
$19$	−1.16163	−0.266497	−0.133249	−	0.991083i	$-0.542541\pi$
$19$	−1.16163	−0.266497	−0.133249	+	0.991083i	$0.542541\pi$
$20$	13.9497	3.11925
$21$	0	0
$22$	7.93083	1.69086
$23$	1.32516	0.276315	0.138157	−	0.990410i	$-0.455882\pi$
$23$	1.32516	0.276315	0.138157	+	0.990410i	$0.455882\pi$
$24$	0	0
$25$	9.75907	1.95181
$26$	10.0696	1.97482
$27$	0	0
$28$	−3.63108	−0.686209
$29$	−6.59844	−1.22530	−0.612650	−	0.790354i	$-0.709896\pi$
$29$	−6.59844	−1.22530	−0.612650	+	0.790354i	$0.709896\pi$
$30$	0	0
$31$	−3.12317	−0.560939	−0.280469	−	0.959863i	$-0.590490\pi$
$31$	−3.12317	−0.560939	−0.280469	+	0.959863i	$0.590490\pi$
$32$	−3.17883	−0.561944
$33$	0	0
$34$	−8.74620	−1.49996
$35$	−3.84175	−0.649375
$36$	0	0
$37$	5.94714	0.977704	0.488852	−	0.872367i	$-0.337416\pi$
$37$	5.94714	0.977704	0.488852	+	0.872367i	$0.337416\pi$
$38$	−2.75654	−0.447171
$39$	0	0
$40$	14.8696	2.35109
$41$	10.3758	1.62043	0.810213	−	0.586136i	$-0.199351\pi$
$41$	10.3758	1.62043	0.810213	+	0.586136i	$0.199351\pi$
$42$	0	0
$43$	4.75366	0.724926	0.362463	−	0.931998i	$-0.381936\pi$
$43$	4.75366	0.724926	0.362463	+	0.931998i	$0.381936\pi$
$44$	12.1355	1.82950
$45$	0	0
$46$	3.14459	0.463644
$47$	−1.96484	−0.286601	−0.143301	−	0.989679i	$-0.545772\pi$
$47$	−1.96484	−0.286601	−0.143301	+	0.989679i	$0.545772\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	23.1582	3.27506
$51$	0	0
$52$	15.4082	2.13674
$53$	−7.36495	−1.01165	−0.505827	−	0.862635i	$-0.668812\pi$
$53$	−7.36495	−1.01165	−0.505827	+	0.862635i	$0.668812\pi$
$54$	0	0
$55$	12.8396	1.73130
$56$	−3.87053	−0.517221
$57$	0	0
$58$	−15.6580	−2.05600
$59$	8.49156	1.10551	0.552753	−	0.833345i	$-0.313577\pi$
$59$	8.49156	1.10551	0.552753	+	0.833345i	$0.313577\pi$
$60$	0	0
$61$	7.06437	0.904500	0.452250	−	0.891891i	$-0.350621\pi$
$61$	7.06437	0.904500	0.452250	+	0.891891i	$0.350621\pi$
$62$	−7.41126	−0.941231
$63$	0	0
$64$	−11.3885	−1.42356
$65$	16.3022	2.02204
$66$	0	0
$67$	3.62274	0.442588	0.221294	−	0.975207i	$-0.428972\pi$
$67$	3.62274	0.442588	0.221294	+	0.975207i	$0.428972\pi$
$68$	−13.3832	−1.62295
$69$	0	0
$70$	−9.11644	−1.08962
$71$	3.79052	0.449851	0.224926	−	0.974376i	$-0.427786\pi$
$71$	3.79052	0.449851	0.224926	+	0.974376i	$0.427786\pi$
$72$	0	0
$73$	−14.4419	−1.69029	−0.845146	−	0.534536i	$-0.820487\pi$
$73$	−14.4419	−1.69029	−0.845146	+	0.534536i	$0.820487\pi$
$74$	14.1125	1.64055
$75$	0	0
$76$	−4.21798	−0.483836
$77$	−3.34213	−0.380871
$78$	0	0
$79$	5.35475	0.602456	0.301228	−	0.953552i	$-0.402603\pi$
$79$	5.35475	0.602456	0.301228	+	0.953552i	$0.402603\pi$
$80$	7.38601	0.825781
$81$	0	0
$82$	24.6216	2.71900
$83$	−1.82593	−0.200422	−0.100211	−	0.994966i	$-0.531952\pi$
$83$	−1.82593	−0.200422	−0.100211	+	0.994966i	$0.531952\pi$
$84$	0	0
$85$	−14.1597	−1.53583
$86$	11.2804	1.21639
$87$	0	0
$88$	12.9358	1.37896
$89$	13.8250	1.46544	0.732722	−	0.680528i	$-0.238250\pi$
$89$	13.8250	1.46544	0.732722	+	0.680528i	$0.238250\pi$
$90$	0	0
$91$	−4.24344	−0.444833
$92$	4.81176	0.501660
$93$	0	0
$94$	−4.66254	−0.480905
$95$	−4.46271	−0.457865
$96$	0	0
$97$	−1.00708	−0.102253	−0.0511265	−	0.998692i	$-0.516281\pi$
$97$	−1.00708	−0.102253	−0.0511265	+	0.998692i	$0.516281\pi$
$98$	2.37299	0.239708
$99$	0	0
$100$	35.4359	3.54359
$101$	−7.06452	−0.702946	−0.351473	−	0.936198i	$-0.614319\pi$
$101$	−7.06452	−0.702946	−0.351473	+	0.936198i	$0.614319\pi$
$102$	0	0
$103$	−18.9614	−1.86832	−0.934159	−	0.356856i	$-0.883849\pi$
$103$	−18.9614	−1.86832	−0.934159	+	0.356856i	$0.883849\pi$
$104$	16.4243	1.61054
$105$	0	0
$106$	−17.4769	−1.69751
$107$	11.3317	1.09548	0.547739	−	0.836649i	$-0.315489\pi$
$107$	11.3317	1.09548	0.547739	+	0.836649i	$0.315489\pi$
$108$	0	0
$109$	−9.79562	−0.938250	−0.469125	−	0.883132i	$-0.655431\pi$
$109$	−9.79562	−0.938250	−0.469125	+	0.883132i	$0.655431\pi$
$110$	30.4683	2.90504
$111$	0	0
$112$	−1.92256	−0.181665
$113$	1.37782	0.129614	0.0648071	−	0.997898i	$-0.479357\pi$
$113$	1.37782	0.129614	0.0648071	+	0.997898i	$0.479357\pi$
$114$	0	0
$115$	5.09094	0.474732
$116$	−23.9594	−2.22458
$117$	0	0
$118$	20.1504	1.85499
$119$	3.68573	0.337870
$120$	0	0
$121$	0.169822	0.0154384
$122$	16.7637	1.51771
$123$	0	0
$124$	−11.3405	−1.01841
$125$	18.2832	1.63530
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−20.6670	−1.82672
$129$	0	0
$130$	38.6850	3.39290
$131$	−20.9839	−1.83337	−0.916684	−	0.399612i	$-0.869145\pi$
$131$	−20.9839	−1.83337	−0.916684	+	0.399612i	$0.869145\pi$
$132$	0	0
$133$	1.16163	0.100726
$134$	8.59672	0.742643
$135$	0	0
$136$	−14.2657	−1.22328
$137$	−21.5715	−1.84298	−0.921488	−	0.388408i	$-0.873025\pi$
$137$	−21.5715	−1.84298	−0.921488	+	0.388408i	$0.873025\pi$
$138$	0	0
$139$	15.6214	1.32499	0.662495	−	0.749067i	$-0.269498\pi$
$139$	15.6214	1.32499	0.662495	+	0.749067i	$0.269498\pi$
$140$	−13.9497	−1.17896
$141$	0	0
$142$	8.99485	0.754831
$143$	14.1821	1.18597
$144$	0	0
$145$	−25.3496	−2.10517
$146$	−34.2704	−2.83624
$147$	0	0
$148$	21.5945	1.77506
$149$	−10.9874	−0.900122	−0.450061	−	0.892998i	$-0.648598\pi$
$149$	−10.9874	−0.900122	−0.450061	+	0.892998i	$0.648598\pi$
$150$	0	0
$151$	16.3574	1.33114	0.665572	−	0.746334i	$-0.268188\pi$
$151$	16.3574	1.33114	0.665572	+	0.746334i	$0.268188\pi$
$152$	−4.49613	−0.364685
$153$	0	0
$154$	−7.93083	−0.639085
$155$	−11.9985	−0.963740
$156$	0	0
$157$	−16.3116	−1.30181	−0.650904	−	0.759160i	$-0.725610\pi$
$157$	−16.3116	−1.30181	−0.650904	+	0.759160i	$0.725610\pi$
$158$	12.7068	1.01090
$159$	0	0
$160$	−12.2123	−0.965467
$161$	−1.32516	−0.104437
$162$	0	0
$163$	−13.2041	−1.03423	−0.517113	−	0.855917i	$-0.672993\pi$
$163$	−13.2041	−1.03423	−0.517113	+	0.855917i	$0.672993\pi$
$164$	37.6753	2.94194
$165$	0	0
$166$	−4.33291	−0.336299
$167$	23.9498	1.85329	0.926647	−	0.375934i	$-0.122678\pi$
$167$	23.9498	1.85329	0.926647	+	0.375934i	$0.122678\pi$
$168$	0	0
$169$	5.00676	0.385136
$170$	−33.6008	−2.57706
$171$	0	0
$172$	17.2609	1.31613
$173$	−18.9427	−1.44019	−0.720093	−	0.693878i	$-0.755901\pi$
$173$	−18.9427	−1.44019	−0.720093	+	0.693878i	$0.755901\pi$
$174$	0	0
$175$	−9.75907	−0.737717
$176$	6.42545	0.484337
$177$	0	0
$178$	32.8065	2.45895
$179$	−7.04317	−0.526431	−0.263216	−	0.964737i	$-0.584783\pi$
$179$	−7.04317	−0.526431	−0.263216	+	0.964737i	$0.584783\pi$
$180$	0	0
$181$	−23.0227	−1.71126	−0.855632	−	0.517585i	$-0.826831\pi$
$181$	−23.0227	−1.71126	−0.855632	+	0.517585i	$0.826831\pi$
$182$	−10.0696	−0.746411
$183$	0	0
$184$	5.12906	0.378120
$185$	22.8475	1.67978
$186$	0	0
$187$	−12.3182	−0.900795
$188$	−7.13448	−0.520336
$189$	0	0
$190$	−10.5900	−0.768277
$191$	−1.35061	−0.0977269	−0.0488634	−	0.998805i	$-0.515560\pi$
$191$	−1.35061	−0.0977269	−0.0488634	+	0.998805i	$0.515560\pi$
$192$	0	0
$193$	15.3745	1.10668	0.553341	−	0.832955i	$-0.313353\pi$
$193$	15.3745	1.10668	0.553341	+	0.832955i	$0.313353\pi$
$194$	−2.38978	−0.171576
$195$	0	0
$196$	3.63108	0.259363
$197$	−0.925481	−0.0659378	−0.0329689	−	0.999456i	$-0.510496\pi$
$197$	−0.925481	−0.0659378	−0.0329689	+	0.999456i	$0.510496\pi$
$198$	0	0
$199$	−0.516943	−0.0366451	−0.0183226	−	0.999832i	$-0.505833\pi$
$199$	−0.516943	−0.0366451	−0.0183226	+	0.999832i	$0.505833\pi$
$200$	37.7727	2.67094
$201$	0	0
$202$	−16.7640	−1.17951
$203$	6.59844	0.463120
$204$	0	0
$205$	39.8612	2.78403
$206$	−44.9951	−3.13496
$207$	0	0
$208$	8.15827	0.565675
$209$	−3.88233	−0.268546
$210$	0	0
$211$	−5.22993	−0.360043	−0.180022	−	0.983663i	$-0.557617\pi$
$211$	−5.22993	−0.360043	−0.180022	+	0.983663i	$0.557617\pi$
$212$	−26.7427	−1.83670
$213$	0	0
$214$	26.8900	1.83816
$215$	18.2624	1.24548
$216$	0	0
$217$	3.12317	0.212015
$218$	−23.2449	−1.57434
$219$	0	0
$220$	46.6217	3.14323
$221$	−15.6402	−1.05207
$222$	0	0
$223$	11.8307	0.792245	0.396123	−	0.918198i	$-0.370356\pi$
$223$	11.8307	0.792245	0.396123	+	0.918198i	$0.370356\pi$
$224$	3.17883	0.212395
$225$	0	0
$226$	3.26955	0.217487
$227$	2.88221	0.191299	0.0956496	−	0.995415i	$-0.469507\pi$
$227$	2.88221	0.191299	0.0956496	+	0.995415i	$0.469507\pi$
$228$	0	0
$229$	23.8595	1.57668	0.788340	−	0.615240i	$-0.210941\pi$
$229$	23.8595	1.57668	0.788340	+	0.615240i	$0.210941\pi$
$230$	12.0807	0.796580
$231$	0	0
$232$	−25.5394	−1.67675
$233$	−25.7400	−1.68628	−0.843141	−	0.537693i	$-0.819296\pi$
$233$	−25.7400	−1.68628	−0.843141	+	0.537693i	$0.819296\pi$
$234$	0	0
$235$	−7.54843	−0.492405
$236$	30.8335	2.00709
$237$	0	0
$238$	8.74620	0.566932
$239$	7.15909	0.463083	0.231542	−	0.972825i	$-0.425623\pi$
$239$	7.15909	0.463083	0.231542	+	0.972825i	$0.425623\pi$
$240$	0	0
$241$	20.3942	1.31370	0.656852	−	0.754020i	$-0.271888\pi$
$241$	20.3942	1.31370	0.656852	+	0.754020i	$0.271888\pi$
$242$	0.402986	0.0259049
$243$	0	0
$244$	25.6513	1.64215
$245$	3.84175	0.245441
$246$	0	0
$247$	−4.92932	−0.313645
$248$	−12.0883	−0.767609
$249$	0	0
$250$	43.3858	2.74396
$251$	19.0371	1.20161	0.600807	−	0.799394i	$-0.294846\pi$
$251$	19.0371	1.20161	0.600807	+	0.799394i	$0.294846\pi$
$252$	0	0
$253$	4.42885	0.278440
$254$	−2.37299	−0.148895
$255$	0	0
$256$	−26.2657	−1.64161
$257$	8.26607	0.515623	0.257812	−	0.966195i	$-0.416999\pi$
$257$	8.26607	0.515623	0.257812	+	0.966195i	$0.416999\pi$
$258$	0	0
$259$	−5.94714	−0.369538
$260$	59.1947	3.67110
$261$	0	0
$262$	−49.7945	−3.07631
$263$	23.2859	1.43587	0.717936	−	0.696109i	$-0.245087\pi$
$263$	23.2859	1.43587	0.717936	+	0.696109i	$0.245087\pi$
$264$	0	0
$265$	−28.2943	−1.73811
$266$	2.75654	0.169015
$267$	0	0
$268$	13.1544	0.803535
$269$	13.1257	0.800290	0.400145	−	0.916452i	$-0.368960\pi$
$269$	13.1257	0.800290	0.400145	+	0.916452i	$0.368960\pi$
$270$	0	0
$271$	5.44455	0.330733	0.165366	−	0.986232i	$-0.447119\pi$
$271$	5.44455	0.330733	0.165366	+	0.986232i	$0.447119\pi$
$272$	−7.08605	−0.429655
$273$	0	0
$274$	−51.1889	−3.09243
$275$	32.6161	1.96682
$276$	0	0
$277$	−1.14315	−0.0686851	−0.0343425	−	0.999410i	$-0.510934\pi$
$277$	−1.14315	−0.0686851	−0.0343425	+	0.999410i	$0.510934\pi$
$278$	37.0694	2.22327
$279$	0	0
$280$	−14.8696	−0.888629
$281$	−9.39254	−0.560312	−0.280156	−	0.959955i	$-0.590386\pi$
$281$	−9.39254	−0.560312	−0.280156	+	0.959955i	$0.590386\pi$
$282$	0	0
$283$	−6.65829	−0.395794	−0.197897	−	0.980223i	$-0.563411\pi$
$283$	−6.65829	−0.395794	−0.197897	+	0.980223i	$0.563411\pi$
$284$	13.7637	0.816722
$285$	0	0
$286$	33.6540	1.99000
$287$	−10.3758	−0.612463
$288$	0	0
$289$	−3.41538	−0.200905
$290$	−60.1543	−3.53238
$291$	0	0
$292$	−52.4395	−3.06879
$293$	−17.4649	−1.02031	−0.510155	−	0.860082i	$-0.670412\pi$
$293$	−17.4649	−1.02031	−0.510155	+	0.860082i	$0.670412\pi$
$294$	0	0
$295$	32.6225	1.89935
$296$	23.0186	1.33793
$297$	0	0
$298$	−26.0729	−1.51037
$299$	5.62323	0.325200
$300$	0	0
$301$	−4.75366	−0.273996
$302$	38.8158	2.23360
$303$	0	0
$304$	−2.23331	−0.128089
$305$	27.1396	1.55401
$306$	0	0
$307$	−6.50990	−0.371540	−0.185770	−	0.982593i	$-0.559478\pi$
$307$	−6.50990	−0.371540	−0.185770	+	0.982593i	$0.559478\pi$
$308$	−12.1355	−0.691486
$309$	0	0
$310$	−28.4722	−1.61711
$311$	24.4759	1.38790	0.693951	−	0.720022i	$-0.255868\pi$
$311$	24.4759	1.38790	0.693951	+	0.720022i	$0.255868\pi$
$312$	0	0
$313$	−17.5981	−0.994702	−0.497351	−	0.867549i	$-0.665694\pi$
$313$	−17.5981	−0.994702	−0.497351	+	0.867549i	$0.665694\pi$
$314$	−38.7072	−2.18438
$315$	0	0
$316$	19.4435	1.09378
$317$	13.7295	0.771124	0.385562	−	0.922682i	$-0.374008\pi$
$317$	13.7295	0.771124	0.385562	+	0.922682i	$0.374008\pi$
$318$	0	0
$319$	−22.0528	−1.23472
$320$	−43.7517	−2.44579
$321$	0	0
$322$	−3.14459	−0.175241
$323$	4.28147	0.238227
$324$	0	0
$325$	41.4120	2.29713
$326$	−31.3332	−1.73539
$327$	0	0
$328$	40.1598	2.21745
$329$	1.96484	0.108325
$330$	0	0
$331$	−21.1263	−1.16121	−0.580603	−	0.814187i	$-0.697183\pi$
$331$	−21.1263	−1.16121	−0.580603	+	0.814187i	$0.697183\pi$
$332$	−6.63010	−0.363874
$333$	0	0
$334$	56.8327	3.10975
$335$	13.9177	0.760403
$336$	0	0
$337$	−27.3273	−1.48861	−0.744307	−	0.667838i	$-0.767220\pi$
$337$	−27.3273	−1.48861	−0.744307	+	0.667838i	$0.767220\pi$
$338$	11.8810	0.646241
$339$	0	0
$340$	−51.4149	−2.78836
$341$	−10.4380	−0.565252
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	18.3992	0.992016
$345$	0	0
$346$	−44.9508	−2.41657
$347$	−20.8386	−1.11867	−0.559336	−	0.828941i	$-0.688944\pi$
$347$	−20.8386	−1.11867	−0.559336	+	0.828941i	$0.688944\pi$
$348$	0	0
$349$	2.69975	0.144514	0.0722571	−	0.997386i	$-0.476980\pi$
$349$	2.69975	0.144514	0.0722571	+	0.997386i	$0.476980\pi$
$350$	−23.1582	−1.23786
$351$	0	0
$352$	−10.6241	−0.566265
$353$	6.29122	0.334848	0.167424	−	0.985885i	$-0.446455\pi$
$353$	6.29122	0.334848	0.167424	+	0.985885i	$0.446455\pi$
$354$	0	0
$355$	14.5622	0.772883
$356$	50.1995	2.66057
$357$	0	0
$358$	−16.7134	−0.883329
$359$	32.8406	1.73326	0.866631	−	0.498949i	$-0.166280\pi$
$359$	32.8406	1.73326	0.866631	+	0.498949i	$0.166280\pi$
$360$	0	0
$361$	−17.6506	−0.928979
$362$	−54.6326	−2.87143
$363$	0	0
$364$	−15.4082	−0.807612
$365$	−55.4821	−2.90406
$366$	0	0
$367$	−26.9378	−1.40614	−0.703070	−	0.711121i	$-0.748188\pi$
$367$	−26.9378	−1.40614	−0.703070	+	0.711121i	$0.748188\pi$
$368$	2.54770	0.132808
$369$	0	0
$370$	54.2168	2.81860
$371$	7.36495	0.382369
$372$	0	0
$373$	−26.2356	−1.35843	−0.679214	−	0.733941i	$-0.737679\pi$
$373$	−26.2356	−1.35843	−0.679214	+	0.733941i	$0.737679\pi$
$374$	−29.2309	−1.51150
$375$	0	0
$376$	−7.60496	−0.392196
$377$	−28.0001	−1.44208
$378$	0	0
$379$	3.05423	0.156885	0.0784426	−	0.996919i	$-0.475005\pi$
$379$	3.05423	0.156885	0.0784426	+	0.996919i	$0.475005\pi$
$380$	−16.2044	−0.831271
$381$	0	0
$382$	−3.20499	−0.163981
$383$	−18.0499	−0.922309	−0.461154	−	0.887320i	$-0.652565\pi$
$383$	−18.0499	−0.922309	−0.461154	+	0.887320i	$0.652565\pi$
$384$	0	0
$385$	−12.8396	−0.654368
$386$	36.4836	1.85697
$387$	0	0
$388$	−3.65677	−0.185644
$389$	25.4202	1.28885	0.644427	−	0.764666i	$-0.277096\pi$
$389$	25.4202	1.28885	0.644427	+	0.764666i	$0.277096\pi$
$390$	0	0
$391$	−4.88418	−0.247004
$392$	3.87053	0.195491
$393$	0	0
$394$	−2.19616	−0.110641
$395$	20.5716	1.03507
$396$	0	0
$397$	−17.8438	−0.895555	−0.447778	−	0.894145i	$-0.647784\pi$
$397$	−17.8438	−0.895555	−0.447778	+	0.894145i	$0.647784\pi$
$398$	−1.22670	−0.0614889
$399$	0	0
$400$	18.7624	0.938121
$401$	−39.0920	−1.95216	−0.976079	−	0.217414i	$-0.930238\pi$
$401$	−39.0920	−1.95216	−0.976079	+	0.217414i	$0.930238\pi$
$402$	0	0
$403$	−13.2530	−0.660179
$404$	−25.6518	−1.27623
$405$	0	0
$406$	15.6580	0.777095
$407$	19.8761	0.985223
$408$	0	0
$409$	−16.3722	−0.809556	−0.404778	−	0.914415i	$-0.632651\pi$
$409$	−16.3722	−0.809556	−0.404778	+	0.914415i	$0.632651\pi$
$410$	94.5902	4.67148
$411$	0	0
$412$	−68.8502	−3.39200
$413$	−8.49156	−0.417842
$414$	0	0
$415$	−7.01478	−0.344342
$416$	−13.4892	−0.661361
$417$	0	0
$418$	−9.21272	−0.450609
$419$	16.1807	0.790481	0.395240	−	0.918578i	$-0.370661\pi$
$419$	16.1807	0.790481	0.395240	+	0.918578i	$0.370661\pi$
$420$	0	0
$421$	−6.95082	−0.338762	−0.169381	−	0.985551i	$-0.554177\pi$
$421$	−6.95082	−0.338762	−0.169381	+	0.985551i	$0.554177\pi$
$422$	−12.4106	−0.604137
$423$	0	0
$424$	−28.5062	−1.38438
$425$	−35.9693	−1.74477
$426$	0	0
$427$	−7.06437	−0.341869
$428$	41.1463	1.98888
$429$	0	0
$430$	43.3364	2.08987
$431$	−17.1337	−0.825303	−0.412651	−	0.910889i	$-0.635397\pi$
$431$	−17.1337	−0.825303	−0.412651	+	0.910889i	$0.635397\pi$
$432$	0	0
$433$	32.8315	1.57778	0.788891	−	0.614533i	$-0.210656\pi$
$433$	32.8315	1.57778	0.788891	+	0.614533i	$0.210656\pi$
$434$	7.41126	0.355752
$435$	0	0
$436$	−35.5687	−1.70343
$437$	−1.53935	−0.0736371
$438$	0	0
$439$	−2.42246	−0.115618	−0.0578088	−	0.998328i	$-0.518411\pi$
$439$	−2.42246	−0.115618	−0.0578088	+	0.998328i	$0.518411\pi$
$440$	49.6961	2.36917
$441$	0	0
$442$	−37.1140	−1.76533
$443$	6.20966	0.295030	0.147515	−	0.989060i	$-0.452873\pi$
$443$	6.20966	0.295030	0.147515	+	0.989060i	$0.452873\pi$
$444$	0	0
$445$	53.1121	2.51776
$446$	28.0742	1.32935
$447$	0	0
$448$	11.3885	0.538054
$449$	16.1943	0.764256	0.382128	−	0.924109i	$-0.375191\pi$
$449$	16.1943	0.764256	0.382128	+	0.924109i	$0.375191\pi$
$450$	0	0
$451$	34.6772	1.63289
$452$	5.00296	0.235320
$453$	0	0
$454$	6.83946	0.320992
$455$	−16.3022	−0.764261
$456$	0	0
$457$	35.9764	1.68290	0.841452	−	0.540331i	$-0.181701\pi$
$457$	35.9764	1.68290	0.841452	+	0.540331i	$0.181701\pi$
$458$	56.6183	2.64560
$459$	0	0
$460$	18.4856	0.861895
$461$	14.3856	0.670002	0.335001	−	0.942218i	$-0.391263\pi$
$461$	14.3856	0.670002	0.335001	+	0.942218i	$0.391263\pi$
$462$	0	0
$463$	4.39760	0.204374	0.102187	−	0.994765i	$-0.467416\pi$
$463$	4.39760	0.204374	0.102187	+	0.994765i	$0.467416\pi$
$464$	−12.6859	−0.588929
$465$	0	0
$466$	−61.0807	−2.82951
$467$	−32.3297	−1.49604	−0.748019	−	0.663677i	$-0.768995\pi$
$467$	−32.3297	−1.49604	−0.748019	+	0.663677i	$0.768995\pi$
$468$	0	0
$469$	−3.62274	−0.167282
$470$	−17.9123	−0.826235
$471$	0	0
$472$	32.8668	1.51282
$473$	15.8873	0.730500
$474$	0	0
$475$	−11.3365	−0.520153
$476$	13.3832	0.613417
$477$	0	0
$478$	16.9884	0.777034
$479$	24.9525	1.14011	0.570054	−	0.821607i	$-0.306922\pi$
$479$	24.9525	1.14011	0.570054	+	0.821607i	$0.306922\pi$
$480$	0	0
$481$	25.2363	1.15068
$482$	48.3951	2.20434
$483$	0	0
$484$	0.616637	0.0280289
$485$	−3.86894	−0.175679
$486$	0	0
$487$	1.35849	0.0615591	0.0307795	−	0.999526i	$-0.490201\pi$
$487$	1.35849	0.0615591	0.0307795	+	0.999526i	$0.490201\pi$
$488$	27.3428	1.23775
$489$	0	0
$490$	9.11644	0.411839
$491$	11.1338	0.502463	0.251232	−	0.967927i	$-0.419164\pi$
$491$	11.1338	0.502463	0.251232	+	0.967927i	$0.419164\pi$
$492$	0	0
$493$	24.3201	1.09532
$494$	−11.6972	−0.526283
$495$	0	0
$496$	−6.00450	−0.269610
$497$	−3.79052	−0.170028
$498$	0	0
$499$	−32.9447	−1.47481	−0.737403	−	0.675453i	$-0.763948\pi$
$499$	−32.9447	−1.47481	−0.737403	+	0.675453i	$0.763948\pi$
$500$	66.3877	2.96895
$501$	0	0
$502$	45.1749	2.01626
$503$	12.7999	0.570721	0.285361	−	0.958420i	$-0.407887\pi$
$503$	12.7999	0.570721	0.285361	+	0.958420i	$0.407887\pi$
$504$	0	0
$505$	−27.1402	−1.20772
$506$	10.5096	0.467210
$507$	0	0
$508$	−3.63108	−0.161103
$509$	−10.3847	−0.460293	−0.230146	−	0.973156i	$-0.573920\pi$
$509$	−10.3847	−0.460293	−0.230146	+	0.973156i	$0.573920\pi$
$510$	0	0
$511$	14.4419	0.638870
$512$	−20.9942	−0.927820
$513$	0	0
$514$	19.6153	0.865193
$515$	−72.8449	−3.20993
$516$	0	0
$517$	−6.56675	−0.288805
$518$	−14.1125	−0.620068
$519$	0	0
$520$	63.0983	2.76704
$521$	45.2510	1.98248	0.991241	−	0.132067i	$-0.0421615\pi$
$521$	45.2510	1.98248	0.991241	+	0.132067i	$0.0421615\pi$
$522$	0	0
$523$	25.7000	1.12378	0.561891	−	0.827212i	$-0.310074\pi$
$523$	25.7000	1.12378	0.561891	+	0.827212i	$0.310074\pi$
$524$	−76.1940	−3.32855
$525$	0	0
$526$	55.2573	2.40933
$527$	11.5112	0.501435
$528$	0	0
$529$	−21.2440	−0.923650
$530$	−67.1421	−2.91647
$531$	0	0
$532$	4.21798	0.182873
$533$	44.0290	1.90711
$534$	0	0
$535$	43.5337	1.88212
$536$	14.0219	0.605654
$537$	0	0
$538$	31.1472	1.34285
$539$	3.34213	0.143956
$540$	0	0
$541$	−26.8765	−1.15551	−0.577755	−	0.816210i	$-0.696071\pi$
$541$	−26.8765	−1.15551	−0.577755	+	0.816210i	$0.696071\pi$
$542$	12.9198	0.554955
$543$	0	0
$544$	11.7163	0.502333
$545$	−37.6324	−1.61199
$546$	0	0
$547$	25.1919	1.07713	0.538565	−	0.842584i	$-0.318967\pi$
$547$	25.1919	1.07713	0.538565	+	0.842584i	$0.318967\pi$
$548$	−78.3277	−3.34599
$549$	0	0
$550$	77.3976	3.30024
$551$	7.66497	0.326539
$552$	0	0
$553$	−5.35475	−0.227707
$554$	−2.71268	−0.115251
$555$	0	0
$556$	56.7225	2.40557
$557$	−0.688944	−0.0291915	−0.0145957	−	0.999893i	$-0.504646\pi$
$557$	−0.688944	−0.0291915	−0.0145957	+	0.999893i	$0.504646\pi$
$558$	0	0
$559$	20.1718	0.853178
$560$	−7.38601	−0.312116
$561$	0	0
$562$	−22.2884	−0.940179
$563$	−9.21162	−0.388223	−0.194112	−	0.980979i	$-0.562182\pi$
$563$	−9.21162	−0.388223	−0.194112	+	0.980979i	$0.562182\pi$
$564$	0	0
$565$	5.29324	0.222688
$566$	−15.8000	−0.664125
$567$	0	0
$568$	14.6713	0.615594
$569$	−5.10961	−0.214206	−0.107103	−	0.994248i	$-0.534157\pi$
$569$	−5.10961	−0.214206	−0.107103	+	0.994248i	$0.534157\pi$
$570$	0	0
$571$	−26.3025	−1.10073	−0.550363	−	0.834925i	$-0.685511\pi$
$571$	−26.3025	−1.10073	−0.550363	+	0.834925i	$0.685511\pi$
$572$	51.4963	2.15317
$573$	0	0
$574$	−24.6216	−1.02769
$575$	12.9323	0.539315
$576$	0	0
$577$	33.5999	1.39878	0.699391	−	0.714740i	$-0.253455\pi$
$577$	33.5999	1.39878	0.699391	+	0.714740i	$0.253455\pi$
$578$	−8.10466	−0.337109
$579$	0	0
$580$	−92.0463	−3.82201
$581$	1.82593	0.0757524
$582$	0	0
$583$	−24.6146	−1.01943
$584$	−55.8976	−2.31306
$585$	0	0
$586$	−41.4440	−1.71204
$587$	3.98468	0.164466	0.0822328	−	0.996613i	$-0.473795\pi$
$587$	3.98468	0.164466	0.0822328	+	0.996613i	$0.473795\pi$
$588$	0	0
$589$	3.62798	0.149489
$590$	77.4128	3.18703
$591$	0	0
$592$	11.4338	0.469924
$593$	−26.5497	−1.09027	−0.545133	−	0.838349i	$-0.683521\pi$
$593$	−26.5497	−1.09027	−0.545133	+	0.838349i	$0.683521\pi$
$594$	0	0
$595$	14.1597	0.580490
$596$	−39.8960	−1.63421
$597$	0	0
$598$	13.3439	0.545671
$599$	−20.7801	−0.849051	−0.424525	−	0.905416i	$-0.639559\pi$
$599$	−20.7801	−0.849051	−0.424525	+	0.905416i	$0.639559\pi$
$600$	0	0
$601$	−16.1839	−0.660155	−0.330078	−	0.943954i	$-0.607075\pi$
$601$	−16.1839	−0.660155	−0.330078	+	0.943954i	$0.607075\pi$
$602$	−11.2804	−0.459754
$603$	0	0
$604$	59.3948	2.41674
$605$	0.652415	0.0265244
$606$	0	0
$607$	−23.8171	−0.966705	−0.483353	−	0.875426i	$-0.660581\pi$
$607$	−23.8171	−0.966705	−0.483353	+	0.875426i	$0.660581\pi$
$608$	3.69264	0.149756
$609$	0	0
$610$	64.4019	2.60756
$611$	−8.33768	−0.337306
$612$	0	0
$613$	5.08138	0.205235	0.102618	−	0.994721i	$-0.467278\pi$
$613$	5.08138	0.205235	0.102618	+	0.994721i	$0.467278\pi$
$614$	−15.4479	−0.623428
$615$	0	0
$616$	−12.9358	−0.521198
$617$	−24.9861	−1.00590	−0.502950	−	0.864315i	$-0.667752\pi$
$617$	−24.9861	−1.00590	−0.502950	+	0.864315i	$0.667752\pi$
$618$	0	0
$619$	−39.9378	−1.60523	−0.802617	−	0.596494i	$-0.796560\pi$
$619$	−39.9378	−1.60523	−0.802617	+	0.596494i	$0.796560\pi$
$620$	−43.5673	−1.74971
$621$	0	0
$622$	58.0811	2.32884
$623$	−13.8250	−0.553886
$624$	0	0
$625$	21.4442	0.857766
$626$	−41.7601	−1.66907
$627$	0	0
$628$	−59.2287	−2.36348
$629$	−21.9196	−0.873991
$630$	0	0
$631$	−35.7932	−1.42490	−0.712452	−	0.701721i	$-0.752415\pi$
$631$	−35.7932	−1.42490	−0.712452	+	0.701721i	$0.752415\pi$
$632$	20.7257	0.824424
$633$	0	0
$634$	32.5799	1.29391
$635$	−3.84175	−0.152455
$636$	0	0
$637$	4.24344	0.168131
$638$	−52.3311	−2.07181
$639$	0	0
$640$	−79.3976	−3.13847
$641$	−10.4769	−0.413812	−0.206906	−	0.978361i	$-0.566339\pi$
$641$	−10.4769	−0.413812	−0.206906	+	0.978361i	$0.566339\pi$
$642$	0	0
$643$	18.2686	0.720445	0.360222	−	0.932866i	$-0.382701\pi$
$643$	18.2686	0.720445	0.360222	+	0.932866i	$0.382701\pi$
$644$	−4.81176	−0.189610
$645$	0	0
$646$	10.1599	0.399735
$647$	5.94994	0.233916	0.116958	−	0.993137i	$-0.462686\pi$
$647$	5.94994	0.233916	0.116958	+	0.993137i	$0.462686\pi$
$648$	0	0
$649$	28.3799	1.11401
$650$	98.2703	3.85448
$651$	0	0
$652$	−47.9452	−1.87768
$653$	11.0374	0.431926	0.215963	−	0.976402i	$-0.430711\pi$
$653$	11.0374	0.431926	0.215963	+	0.976402i	$0.430711\pi$
$654$	0	0
$655$	−80.6149	−3.14988
$656$	19.9481	0.778842
$657$	0	0
$658$	4.66254	0.181765
$659$	2.45283	0.0955486	0.0477743	−	0.998858i	$-0.484787\pi$
$659$	2.45283	0.0955486	0.0477743	+	0.998858i	$0.484787\pi$
$660$	0	0
$661$	4.63826	0.180407	0.0902036	−	0.995923i	$-0.471248\pi$
$661$	4.63826	0.180407	0.0902036	+	0.995923i	$0.471248\pi$
$662$	−50.1324	−1.94845
$663$	0	0
$664$	−7.06731	−0.274265
$665$	4.46271	0.173057
$666$	0	0
$667$	−8.74399	−0.338569
$668$	86.9636	3.36472
$669$	0	0
$670$	33.0265	1.27592
$671$	23.6100	0.911455
$672$	0	0
$673$	10.6571	0.410800	0.205400	−	0.978678i	$-0.434151\pi$
$673$	10.6571	0.410800	0.205400	+	0.978678i	$0.434151\pi$
$674$	−64.8474	−2.49783
$675$	0	0
$676$	18.1799	0.699229
$677$	9.18895	0.353160	0.176580	−	0.984286i	$-0.443497\pi$
$677$	9.18895	0.353160	0.176580	+	0.984286i	$0.443497\pi$
$678$	0	0
$679$	1.00708	0.0386480
$680$	−54.8054	−2.10169
$681$	0	0
$682$	−24.7694	−0.948468
$683$	20.1647	0.771581	0.385790	−	0.922586i	$-0.373929\pi$
$683$	20.1647	0.771581	0.385790	+	0.922586i	$0.373929\pi$
$684$	0	0
$685$	−82.8723	−3.16639
$686$	−2.37299	−0.0906011
$687$	0	0
$688$	9.13920	0.348429
$689$	−31.2527	−1.19063
$690$	0	0
$691$	22.0786	0.839908	0.419954	−	0.907545i	$-0.362046\pi$
$691$	22.0786	0.839908	0.419954	+	0.907545i	$0.362046\pi$
$692$	−68.7823	−2.61471
$693$	0	0
$694$	−49.4497	−1.87708
$695$	60.0135	2.27644
$696$	0	0
$697$	−38.2424	−1.44853
$698$	6.40647	0.242488
$699$	0	0
$700$	−35.4359	−1.33935
$701$	48.8689	1.84575	0.922876	−	0.385096i	$-0.125832\pi$
$701$	48.8689	1.84575	0.922876	+	0.385096i	$0.125832\pi$
$702$	0	0
$703$	−6.90840	−0.260555
$704$	−38.0617	−1.43450
$705$	0	0
$706$	14.9290	0.561860
$707$	7.06452	0.265689
$708$	0	0
$709$	3.64500	0.136891	0.0684454	−	0.997655i	$-0.478196\pi$
$709$	3.64500	0.136891	0.0684454	+	0.997655i	$0.478196\pi$
$710$	34.5560	1.29686
$711$	0	0
$712$	53.5099	2.00537
$713$	−4.13870	−0.154996
$714$	0	0
$715$	54.4842	2.03759
$716$	−25.5743	−0.955756
$717$	0	0
$718$	77.9305	2.90834
$719$	22.7474	0.848337	0.424168	−	0.905583i	$-0.360567\pi$
$719$	22.7474	0.848337	0.424168	+	0.905583i	$0.360567\pi$
$720$	0	0
$721$	18.9614	0.706158
$722$	−41.8847	−1.55879
$723$	0	0
$724$	−83.5972	−3.10687
$725$	−64.3947	−2.39156
$726$	0	0
$727$	25.4578	0.944177	0.472088	−	0.881551i	$-0.343500\pi$
$727$	25.4578	0.944177	0.472088	+	0.881551i	$0.343500\pi$
$728$	−16.4243	−0.608726
$729$	0	0
$730$	−131.658	−4.87289
$731$	−17.5207	−0.648027
$732$	0	0
$733$	−34.6298	−1.27908	−0.639540	−	0.768758i	$-0.720875\pi$
$733$	−34.6298	−1.27908	−0.639540	+	0.768758i	$0.720875\pi$
$734$	−63.9230	−2.35944
$735$	0	0
$736$	−4.21246	−0.155273
$737$	12.1077	0.445991
$738$	0	0
$739$	−14.0400	−0.516470	−0.258235	−	0.966082i	$-0.583141\pi$
$739$	−14.0400	−0.516470	−0.258235	+	0.966082i	$0.583141\pi$
$740$	82.9609	3.04970
$741$	0	0
$742$	17.4769	0.641599
$743$	−50.5342	−1.85392	−0.926959	−	0.375162i	$-0.877587\pi$
$743$	−50.5342	−1.85392	−0.926959	+	0.375162i	$0.877587\pi$
$744$	0	0
$745$	−42.2108	−1.54649
$746$	−62.2568	−2.27938
$747$	0	0
$748$	−44.7283	−1.63543
$749$	−11.3317	−0.414052
$750$	0	0
$751$	19.8309	0.723642	0.361821	−	0.932248i	$-0.382155\pi$
$751$	19.8309	0.723642	0.361821	+	0.932248i	$0.382155\pi$
$752$	−3.77753	−0.137752
$753$	0	0
$754$	−66.4439	−2.41974
$755$	62.8410	2.28702
$756$	0	0
$757$	20.0864	0.730054	0.365027	−	0.930997i	$-0.381060\pi$
$757$	20.0864	0.730054	0.365027	+	0.930997i	$0.381060\pi$
$758$	7.24765	0.263246
$759$	0	0
$760$	−17.2730	−0.626559
$761$	−21.7523	−0.788522	−0.394261	−	0.918999i	$-0.628999\pi$
$761$	−21.7523	−0.788522	−0.394261	+	0.918999i	$0.628999\pi$
$762$	0	0
$763$	9.79562	0.354625
$764$	−4.90418	−0.177427
$765$	0	0
$766$	−42.8323	−1.54759
$767$	36.0334	1.30109
$768$	0	0
$769$	38.5525	1.39024	0.695119	−	0.718894i	$-0.255351\pi$
$769$	38.5525	1.39024	0.695119	+	0.718894i	$0.255351\pi$
$770$	−30.4683	−1.09800
$771$	0	0
$772$	55.8261	2.00922
$773$	−0.568111	−0.0204335	−0.0102168	−	0.999948i	$-0.503252\pi$
$773$	−0.568111	−0.0204335	−0.0102168	+	0.999948i	$0.503252\pi$
$774$	0	0
$775$	−30.4793	−1.09485
$776$	−3.89791	−0.139927
$777$	0	0
$778$	60.3217	2.16264
$779$	−12.0529	−0.431839
$780$	0	0
$781$	12.6684	0.453311
$782$	−11.5901	−0.414462
$783$	0	0
$784$	1.92256	0.0686629
$785$	−62.6651	−2.23662
$786$	0	0
$787$	−40.1767	−1.43214	−0.716072	−	0.698026i	$-0.754062\pi$
$787$	−40.1767	−1.43214	−0.716072	+	0.698026i	$0.754062\pi$
$788$	−3.36049	−0.119713
$789$	0	0
$790$	48.8163	1.73680
$791$	−1.37782	−0.0489896
$792$	0	0
$793$	29.9772	1.06452
$794$	−42.3432	−1.50270
$795$	0	0
$796$	−1.87706	−0.0665306
$797$	41.3562	1.46491	0.732457	−	0.680814i	$-0.238374\pi$
$797$	41.3562	1.46491	0.732457	+	0.680814i	$0.238374\pi$
$798$	0	0
$799$	7.24187	0.256199
$800$	−31.0225	−1.09681
$801$	0	0
$802$	−92.7648	−3.27564
$803$	−48.2666	−1.70329
$804$	0	0
$805$	−5.09094	−0.179432
$806$	−31.4492	−1.10775
$807$	0	0
$808$	−27.3434	−0.961938
$809$	−27.0123	−0.949702	−0.474851	−	0.880066i	$-0.657498\pi$
$809$	−27.0123	−0.949702	−0.474851	+	0.880066i	$0.657498\pi$
$810$	0	0
$811$	7.36115	0.258485	0.129242	−	0.991613i	$-0.458745\pi$
$811$	7.36115	0.258485	0.129242	+	0.991613i	$0.458745\pi$
$812$	23.9594	0.840812
$813$	0	0
$814$	47.1658	1.65316
$815$	−50.7270	−1.77689
$816$	0	0
$817$	−5.52201	−0.193191
$818$	−38.8512	−1.35840
$819$	0	0
$820$	144.739	5.05451
$821$	−15.5190	−0.541617	−0.270809	−	0.962633i	$-0.587291\pi$
$821$	−15.5190	−0.541617	−0.270809	+	0.962633i	$0.587291\pi$
$822$	0	0
$823$	−11.0622	−0.385603	−0.192801	−	0.981238i	$-0.561757\pi$
$823$	−11.0622	−0.385603	−0.192801	+	0.981238i	$0.561757\pi$
$824$	−73.3904	−2.55668
$825$	0	0
$826$	−20.1504	−0.701121
$827$	44.3859	1.54345	0.771725	−	0.635957i	$-0.219394\pi$
$827$	44.3859	1.54345	0.771725	+	0.635957i	$0.219394\pi$
$828$	0	0
$829$	−23.1103	−0.802652	−0.401326	−	0.915935i	$-0.631451\pi$
$829$	−23.1103	−0.802652	−0.401326	+	0.915935i	$0.631451\pi$
$830$	−16.6460	−0.577791
$831$	0	0
$832$	−48.3262	−1.67541
$833$	−3.68573	−0.127703
$834$	0	0
$835$	92.0093	3.18411
$836$	−14.0970	−0.487556
$837$	0	0
$838$	38.3967	1.32639
$839$	29.9520	1.03406	0.517029	−	0.855968i	$-0.327038\pi$
$839$	29.9520	1.03406	0.517029	+	0.855968i	$0.327038\pi$
$840$	0	0
$841$	14.5394	0.501360
$842$	−16.4942	−0.568429
$843$	0	0
$844$	−18.9903	−0.653673
$845$	19.2348	0.661696
$846$	0	0
$847$	−0.169822	−0.00583516
$848$	−14.1596	−0.486242
$849$	0	0
$850$	−85.3548	−2.92765
$851$	7.88091	0.270154
$852$	0	0
$853$	38.5942	1.32144	0.660720	−	0.750633i	$-0.270251\pi$
$853$	38.5942	1.32144	0.660720	+	0.750633i	$0.270251\pi$
$854$	−16.7637	−0.573641
$855$	0	0
$856$	43.8597	1.49909
$857$	51.6243	1.76345	0.881727	−	0.471760i	$-0.156381\pi$
$857$	51.6243	1.76345	0.881727	+	0.471760i	$0.156381\pi$
$858$	0	0
$859$	−29.5073	−1.00678	−0.503388	−	0.864060i	$-0.667913\pi$
$859$	−29.5073	−1.00678	−0.503388	+	0.864060i	$0.667913\pi$
$860$	66.3121	2.26122
$861$	0	0
$862$	−40.6582	−1.38482
$863$	−46.8109	−1.59346	−0.796730	−	0.604335i	$-0.793439\pi$
$863$	−46.8109	−1.59346	−0.796730	+	0.604335i	$0.793439\pi$
$864$	0	0
$865$	−72.7731	−2.47436
$866$	77.9088	2.64745
$867$	0	0
$868$	11.3405	0.384921
$869$	17.8963	0.607089
$870$	0	0
$871$	15.3729	0.520890
$872$	−37.9142	−1.28394
$873$	0	0
$874$	−3.65286	−0.123560
$875$	−18.2832	−0.618085
$876$	0	0
$877$	−31.3460	−1.05848	−0.529240	−	0.848472i	$-0.677523\pi$
$877$	−31.3460	−1.05848	−0.529240	+	0.848472i	$0.677523\pi$
$878$	−5.74846	−0.194001
$879$	0	0
$880$	24.6850	0.832131
$881$	−56.2123	−1.89384	−0.946920	−	0.321470i	$-0.895823\pi$
$881$	−56.2123	−1.89384	−0.946920	+	0.321470i	$0.895823\pi$
$882$	0	0
$883$	−33.4652	−1.12619	−0.563097	−	0.826391i	$-0.690390\pi$
$883$	−33.4652	−1.12619	−0.563097	+	0.826391i	$0.690390\pi$
$884$	−56.7907	−1.91008
$885$	0	0
$886$	14.7355	0.495047
$887$	48.4496	1.62678	0.813388	−	0.581721i	$-0.197620\pi$
$887$	48.4496	1.62678	0.813388	+	0.581721i	$0.197620\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	126.034	4.22469
$891$	0	0
$892$	42.9583	1.43835
$893$	2.28242	0.0763784
$894$	0	0
$895$	−27.0581	−0.904453
$896$	20.6670	0.690437
$897$	0	0
$898$	38.4289	1.28239
$899$	20.6081	0.687318
$900$	0	0
$901$	27.1452	0.904338
$902$	82.2886	2.73991
$903$	0	0
$904$	5.33288	0.177369
$905$	−88.4476	−2.94010
$906$	0	0
$907$	−31.8491	−1.05753	−0.528767	−	0.848767i	$-0.677345\pi$
$907$	−31.8491	−1.05753	−0.528767	+	0.848767i	$0.677345\pi$
$908$	10.4655	0.347311
$909$	0	0
$910$	−38.6850	−1.28240
$911$	−47.0670	−1.55940	−0.779700	−	0.626153i	$-0.784628\pi$
$911$	−47.0670	−1.55940	−0.779700	+	0.626153i	$0.784628\pi$
$912$	0	0
$913$	−6.10250	−0.201963
$914$	85.3716	2.82384
$915$	0	0
$916$	86.6357	2.86252
$917$	20.9839	0.692948
$918$	0	0
$919$	−5.29140	−0.174547	−0.0872736	−	0.996184i	$-0.527815\pi$
$919$	−5.29140	−0.174547	−0.0872736	+	0.996184i	$0.527815\pi$
$920$	19.7046	0.649642
$921$	0	0
$922$	34.1368	1.12423
$923$	16.0848	0.529438
$924$	0	0
$925$	58.0386	1.90830
$926$	10.4355	0.342930
$927$	0	0
$928$	20.9753	0.688549
$929$	53.8424	1.76651	0.883255	−	0.468892i	$-0.155347\pi$
$929$	53.8424	1.76651	0.883255	+	0.468892i	$0.155347\pi$
$930$	0	0
$931$	−1.16163	−0.0380710
$932$	−93.4638	−3.06151
$933$	0	0
$934$	−76.7179	−2.51029
$935$	−47.3235	−1.54764
$936$	0	0
$937$	5.01442	0.163814	0.0819069	−	0.996640i	$-0.473899\pi$
$937$	5.01442	0.163814	0.0819069	+	0.996640i	$0.473899\pi$
$938$	−8.59672	−0.280693
$939$	0	0
$940$	−27.4089	−0.893981
$941$	−22.6920	−0.739737	−0.369869	−	0.929084i	$-0.620597\pi$
$941$	−22.6920	−0.739737	−0.369869	+	0.929084i	$0.620597\pi$
$942$	0	0
$943$	13.7496	0.447748
$944$	16.3256	0.531351
$945$	0	0
$946$	37.7005	1.22575
$947$	33.0389	1.07362	0.536810	−	0.843703i	$-0.319629\pi$
$947$	33.0389	1.07362	0.536810	+	0.843703i	$0.319629\pi$
$948$	0	0
$949$	−61.2831	−1.98933
$950$	−26.9013	−0.872794
$951$	0	0
$952$	14.2657	0.462355
$953$	8.77455	0.284236	0.142118	−	0.989850i	$-0.454609\pi$
$953$	8.77455	0.284236	0.142118	+	0.989850i	$0.454609\pi$
$954$	0	0
$955$	−5.18872	−0.167903
$956$	25.9952	0.840745
$957$	0	0
$958$	59.2120	1.91305
$959$	21.5715	0.696579
$960$	0	0
$961$	−21.2458	−0.685348
$962$	59.8855	1.93079
$963$	0	0
$964$	74.0528	2.38508
$965$	59.0652	1.90137
$966$	0	0
$967$	−28.6408	−0.921026	−0.460513	−	0.887653i	$-0.652335\pi$
$967$	−28.6408	−0.921026	−0.460513	+	0.887653i	$0.652335\pi$
$968$	0.657301	0.0211264
$969$	0	0
$970$	−9.18094	−0.294782
$971$	−48.9093	−1.56957	−0.784787	−	0.619765i	$-0.787228\pi$
$971$	−48.9093	−1.56957	−0.784787	+	0.619765i	$0.787228\pi$
$972$	0	0
$973$	−15.6214	−0.500799
$974$	3.22368	0.103293
$975$	0	0
$976$	13.5817	0.434739
$977$	−1.62490	−0.0519852	−0.0259926	−	0.999662i	$-0.508275\pi$
$977$	−1.62490	−0.0519852	−0.0259926	+	0.999662i	$0.508275\pi$
$978$	0	0
$979$	46.2048	1.47671
$980$	13.9497	0.445607
$981$	0	0
$982$	26.4205	0.843111
$983$	5.64658	0.180098	0.0900489	−	0.995937i	$-0.471298\pi$
$983$	5.64658	0.180098	0.0900489	+	0.995937i	$0.471298\pi$
$984$	0	0
$985$	−3.55547	−0.113287
$986$	57.7113	1.83790
$987$	0	0
$988$	−17.8987	−0.569435
$989$	6.29935	0.200308
$990$	0	0
$991$	−22.8860	−0.726997	−0.363498	−	0.931595i	$-0.618418\pi$
$991$	−22.8860	−0.726997	−0.363498	+	0.931595i	$0.618418\pi$
$992$	9.92805	0.315216
$993$	0	0
$994$	−8.99485	−0.285299
$995$	−1.98597	−0.0629594
$996$	0	0
$997$	−5.93892	−0.188087	−0.0940437	−	0.995568i	$-0.529979\pi$
$997$	−5.93892	−0.188087	−0.0940437	+	0.995568i	$0.529979\pi$
$998$	−78.1773	−2.47466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.15		16
3.2	odd	2		889.2.a.c.1.2	✓	16
21.20	even	2		6223.2.a.k.1.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.2	✓	16	3.2	odd	2
6223.2.a.k.1.2		16	21.20	even	2
8001.2.a.t.1.15		16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.37299 q^{2} +3.63108 q^{4} +3.84175 q^{5} -1.00000 q^{7} +3.87053 q^{8} +O(q^{10})\)	\(q+2.37299 q^{2} +3.63108 q^{4} +3.84175 q^{5} -1.00000 q^{7} +3.87053 q^{8} +9.11644 q^{10} +3.34213 q^{11} +4.24344 q^{13} -2.37299 q^{14} +1.92256 q^{16} -3.68573 q^{17} -1.16163 q^{19} +13.9497 q^{20} +7.93083 q^{22} +1.32516 q^{23} +9.75907 q^{25} +10.0696 q^{26} -3.63108 q^{28} -6.59844 q^{29} -3.12317 q^{31} -3.17883 q^{32} -8.74620 q^{34} -3.84175 q^{35} +5.94714 q^{37} -2.75654 q^{38} +14.8696 q^{40} +10.3758 q^{41} +4.75366 q^{43} +12.1355 q^{44} +3.14459 q^{46} -1.96484 q^{47} +1.00000 q^{49} +23.1582 q^{50} +15.4082 q^{52} -7.36495 q^{53} +12.8396 q^{55} -3.87053 q^{56} -15.6580 q^{58} +8.49156 q^{59} +7.06437 q^{61} -7.41126 q^{62} -11.3885 q^{64} +16.3022 q^{65} +3.62274 q^{67} -13.3832 q^{68} -9.11644 q^{70} +3.79052 q^{71} -14.4419 q^{73} +14.1125 q^{74} -4.21798 q^{76} -3.34213 q^{77} +5.35475 q^{79} +7.38601 q^{80} +24.6216 q^{82} -1.82593 q^{83} -14.1597 q^{85} +11.2804 q^{86} +12.9358 q^{88} +13.8250 q^{89} -4.24344 q^{91} +4.81176 q^{92} -4.66254 q^{94} -4.46271 q^{95} -1.00708 q^{97} +2.37299 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

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Database

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