LMFDB - Embedded newform 8001.2.a.q.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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
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			$2.71493$ 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)

$f(q)$	$=$	$q+2.71493 q^{2} +5.37083 q^{4} -1.33828 q^{5} -1.00000 q^{7} +9.15156 q^{8} +O(q^{10})$	\(q+2.71493 q^{2} +5.37083 q^{4} -1.33828 q^{5} -1.00000 q^{7} +9.15156 q^{8} -3.63333 q^{10} -5.10834 q^{11} -0.127798 q^{13} -2.71493 q^{14} +14.1042 q^{16} -3.48298 q^{17} +0.837970 q^{19} -7.18767 q^{20} -13.8688 q^{22} -5.34786 q^{23} -3.20901 q^{25} -0.346962 q^{26} -5.37083 q^{28} +2.80116 q^{29} -4.97492 q^{31} +19.9886 q^{32} -9.45604 q^{34} +1.33828 q^{35} -5.80141 q^{37} +2.27503 q^{38} -12.2473 q^{40} -0.0808917 q^{41} -4.43627 q^{43} -27.4360 q^{44} -14.5190 q^{46} -8.49856 q^{47} +1.00000 q^{49} -8.71223 q^{50} -0.686381 q^{52} +11.0692 q^{53} +6.83639 q^{55} -9.15156 q^{56} +7.60496 q^{58} -7.65504 q^{59} +11.0659 q^{61} -13.5065 q^{62} +26.0594 q^{64} +0.171029 q^{65} -1.49689 q^{67} -18.7065 q^{68} +3.63333 q^{70} +5.89494 q^{71} -5.98304 q^{73} -15.7504 q^{74} +4.50059 q^{76} +5.10834 q^{77} +11.5082 q^{79} -18.8753 q^{80} -0.219615 q^{82} -13.7714 q^{83} +4.66120 q^{85} -12.0441 q^{86} -46.7493 q^{88} -10.9046 q^{89} +0.127798 q^{91} -28.7224 q^{92} -23.0730 q^{94} -1.12144 q^{95} +0.340451 q^{97} +2.71493 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * 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 $

$2$	2.71493	1.91974	0.959872	−	0.280439i	$-0.0904802\pi$
$2$	2.71493	1.91974	0.959872	+	0.280439i	$0.0904802\pi$
$3$	0	0
$3$	0	0
$4$	5.37083	2.68542
$5$	−1.33828	−0.598497	−0.299248	−	0.954175i	$-0.596736\pi$
$5$	−1.33828	−0.598497	−0.299248	+	0.954175i	$0.596736\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	9.15156	3.23556
$9$	0	0
$10$	−3.63333	−1.14896
$11$	−5.10834	−1.54022	−0.770111	−	0.637909i	$-0.779799\pi$
$11$	−5.10834	−1.54022	−0.770111	+	0.637909i	$0.779799\pi$
$12$	0	0
$13$	−0.127798	−0.0354448	−0.0177224	−	0.999843i	$-0.505642\pi$
$13$	−0.127798	−0.0354448	−0.0177224	+	0.999843i	$0.505642\pi$
$14$	−2.71493	−0.725595
$15$	0	0
$16$	14.1042	3.52604
$17$	−3.48298	−0.844747	−0.422373	−	0.906422i	$-0.638803\pi$
$17$	−3.48298	−0.844747	−0.422373	+	0.906422i	$0.638803\pi$
$18$	0	0
$19$	0.837970	0.192243	0.0961217	−	0.995370i	$-0.469356\pi$
$19$	0.837970	0.192243	0.0961217	+	0.995370i	$0.469356\pi$
$20$	−7.18767	−1.60721
$21$	0	0
$22$	−13.8688	−2.95683
$23$	−5.34786	−1.11511	−0.557553	−	0.830142i	$-0.688260\pi$
$23$	−5.34786	−1.11511	−0.557553	+	0.830142i	$0.688260\pi$
$24$	0	0
$25$	−3.20901	−0.641802
$26$	−0.346962	−0.0680449
$27$	0	0
$28$	−5.37083	−1.01499
$29$	2.80116	0.520163	0.260082	−	0.965587i	$-0.416251\pi$
$29$	2.80116	0.520163	0.260082	+	0.965587i	$0.416251\pi$
$30$	0	0
$31$	−4.97492	−0.893522	−0.446761	−	0.894653i	$-0.647423\pi$
$31$	−4.97492	−0.893522	−0.446761	+	0.894653i	$0.647423\pi$
$32$	19.9886	3.53353
$33$	0	0
$34$	−9.45604	−1.62170
$35$	1.33828	0.226210
$36$	0	0
$37$	−5.80141	−0.953746	−0.476873	−	0.878972i	$-0.658230\pi$
$37$	−5.80141	−0.953746	−0.476873	+	0.878972i	$0.658230\pi$
$38$	2.27503	0.369058
$39$	0	0
$40$	−12.2473	−1.93647
$41$	−0.0808917	−0.0126332	−0.00631658	−	0.999980i	$-0.502011\pi$
$41$	−0.0808917	−0.0126332	−0.00631658	+	0.999980i	$0.502011\pi$
$42$	0	0
$43$	−4.43627	−0.676524	−0.338262	−	0.941052i	$-0.609839\pi$
$43$	−4.43627	−0.676524	−0.338262	+	0.941052i	$0.609839\pi$
$44$	−27.4360	−4.13614
$45$	0	0
$46$	−14.5190	−2.14072
$47$	−8.49856	−1.23964	−0.619821	−	0.784743i	$-0.712795\pi$
$47$	−8.49856	−1.23964	−0.619821	+	0.784743i	$0.712795\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−8.71223	−1.23209
$51$	0	0
$52$	−0.686381	−0.0951839
$53$	11.0692	1.52047	0.760233	−	0.649651i	$-0.225085\pi$
$53$	11.0692	1.52047	0.760233	+	0.649651i	$0.225085\pi$
$54$	0	0
$55$	6.83639	0.921818
$56$	−9.15156	−1.22293
$57$	0	0
$58$	7.60496	0.998580
$59$	−7.65504	−0.996602	−0.498301	−	0.867004i	$-0.666043\pi$
$59$	−7.65504	−0.996602	−0.498301	+	0.867004i	$0.666043\pi$
$60$	0	0
$61$	11.0659	1.41684	0.708420	−	0.705791i	$-0.249408\pi$
$61$	11.0659	1.41684	0.708420	+	0.705791i	$0.249408\pi$
$62$	−13.5065	−1.71533
$63$	0	0
$64$	26.0594	3.25742
$65$	0.171029	0.0212136
$66$	0	0
$67$	−1.49689	−0.182874	−0.0914372	−	0.995811i	$-0.529146\pi$
$67$	−1.49689	−0.182874	−0.0914372	+	0.995811i	$0.529146\pi$
$68$	−18.7065	−2.26850
$69$	0	0
$70$	3.63333	0.434266
$71$	5.89494	0.699601	0.349800	−	0.936824i	$-0.386249\pi$
$71$	5.89494	0.699601	0.349800	+	0.936824i	$0.386249\pi$
$72$	0	0
$73$	−5.98304	−0.700261	−0.350131	−	0.936701i	$-0.613863\pi$
$73$	−5.98304	−0.700261	−0.350131	+	0.936701i	$0.613863\pi$
$74$	−15.7504	−1.83095
$75$	0	0
$76$	4.50059	0.516254
$77$	5.10834	0.582150
$78$	0	0
$79$	11.5082	1.29477	0.647384	−	0.762164i	$-0.275863\pi$
$79$	11.5082	1.29477	0.647384	+	0.762164i	$0.275863\pi$
$80$	−18.8753	−2.11032
$81$	0	0
$82$	−0.219615	−0.0242524
$83$	−13.7714	−1.51161	−0.755805	−	0.654796i	$-0.772754\pi$
$83$	−13.7714	−1.51161	−0.755805	+	0.654796i	$0.772754\pi$
$84$	0	0
$85$	4.66120	0.505578
$86$	−12.0441	−1.29875
$87$	0	0
$88$	−46.7493	−4.98349
$89$	−10.9046	−1.15589	−0.577943	−	0.816077i	$-0.696145\pi$
$89$	−10.9046	−1.15589	−0.577943	+	0.816077i	$0.696145\pi$
$90$	0	0
$91$	0.127798	0.0133969
$92$	−28.7224	−2.99452
$93$	0	0
$94$	−23.0730	−2.37979
$95$	−1.12144	−0.115057
$96$	0	0
$97$	0.340451	0.0345675	0.0172838	−	0.999851i	$-0.494498\pi$
$97$	0.340451	0.0345675	0.0172838	+	0.999851i	$0.494498\pi$
$98$	2.71493	0.274249
$99$	0	0
$100$	−17.2350	−1.72350
$101$	−1.91200	−0.190251	−0.0951253	−	0.995465i	$-0.530325\pi$
$101$	−1.91200	−0.190251	−0.0951253	+	0.995465i	$0.530325\pi$
$102$	0	0
$103$	−4.46758	−0.440203	−0.220102	−	0.975477i	$-0.570639\pi$
$103$	−4.46758	−0.440203	−0.220102	+	0.975477i	$0.570639\pi$
$104$	−1.16955	−0.114684
$105$	0	0
$106$	30.0520	2.91890
$107$	−20.0087	−1.93432	−0.967158	−	0.254176i	$-0.918196\pi$
$107$	−20.0087	−1.93432	−0.967158	+	0.254176i	$0.918196\pi$
$108$	0	0
$109$	19.2099	1.83997	0.919985	−	0.391953i	$-0.128200\pi$
$109$	19.2099	1.83997	0.919985	+	0.391953i	$0.128200\pi$
$110$	18.5603	1.76965
$111$	0	0
$112$	−14.1042	−1.33272
$113$	5.19798	0.488985	0.244493	−	0.969651i	$-0.421379\pi$
$113$	5.19798	0.488985	0.244493	+	0.969651i	$0.421379\pi$
$114$	0	0
$115$	7.15693	0.667387
$116$	15.0446	1.39685
$117$	0	0
$118$	−20.7829	−1.91322
$119$	3.48298	0.319284
$120$	0	0
$121$	15.0952	1.37229
$122$	30.0430	2.71997
$123$	0	0
$124$	−26.7194	−2.39948
$125$	10.9859	0.982613
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	30.7721	2.71989
$129$	0	0
$130$	0.464332	0.0407246
$131$	−6.66474	−0.582301	−0.291151	−	0.956677i	$-0.594038\pi$
$131$	−6.66474	−0.582301	−0.291151	+	0.956677i	$0.594038\pi$
$132$	0	0
$133$	−0.837970	−0.0726612
$134$	−4.06395	−0.351072
$135$	0	0
$136$	−31.8747	−2.73323
$137$	10.6186	0.907212	0.453606	−	0.891202i	$-0.350137\pi$
$137$	10.6186	0.907212	0.453606	+	0.891202i	$0.350137\pi$
$138$	0	0
$139$	−16.6291	−1.41046	−0.705230	−	0.708979i	$-0.749156\pi$
$139$	−16.6291	−1.41046	−0.705230	+	0.708979i	$0.749156\pi$
$140$	7.18767	0.607469
$141$	0	0
$142$	16.0043	1.34305
$143$	0.652836	0.0545929
$144$	0	0
$145$	−3.74874	−0.311316
$146$	−16.2435	−1.34432
$147$	0	0
$148$	−31.1584	−2.56120
$149$	9.73938	0.797881	0.398941	−	0.916977i	$-0.369378\pi$
$149$	9.73938	0.797881	0.398941	+	0.916977i	$0.369378\pi$
$150$	0	0
$151$	20.0476	1.63145	0.815724	−	0.578441i	$-0.196339\pi$
$151$	20.0476	1.63145	0.815724	+	0.578441i	$0.196339\pi$
$152$	7.66873	0.622016
$153$	0	0
$154$	13.8688	1.11758
$155$	6.65783	0.534770
$156$	0	0
$157$	22.6761	1.80975	0.904874	−	0.425679i	$-0.139965\pi$
$157$	22.6761	1.80975	0.904874	+	0.425679i	$0.139965\pi$
$158$	31.2438	2.48562
$159$	0	0
$160$	−26.7504	−2.11480
$161$	5.34786	0.421470
$162$	0	0
$163$	2.14957	0.168368	0.0841838	−	0.996450i	$-0.473172\pi$
$163$	2.14957	0.168368	0.0841838	+	0.996450i	$0.473172\pi$
$164$	−0.434456	−0.0339253
$165$	0	0
$166$	−37.3884	−2.90190
$167$	−14.1268	−1.09316	−0.546582	−	0.837405i	$-0.684071\pi$
$167$	−14.1268	−1.09316	−0.546582	+	0.837405i	$0.684071\pi$
$168$	0	0
$169$	−12.9837	−0.998744
$170$	12.6548	0.970580
$171$	0	0
$172$	−23.8264	−1.81675
$173$	−16.7356	−1.27238	−0.636190	−	0.771532i	$-0.719491\pi$
$173$	−16.7356	−1.27238	−0.636190	+	0.771532i	$0.719491\pi$
$174$	0	0
$175$	3.20901	0.242578
$176$	−72.0488	−5.43089
$177$	0	0
$178$	−29.6052	−2.21901
$179$	−2.62893	−0.196496	−0.0982478	−	0.995162i	$-0.531324\pi$
$179$	−2.62893	−0.196496	−0.0982478	+	0.995162i	$0.531324\pi$
$180$	0	0
$181$	−16.5373	−1.22921	−0.614605	−	0.788835i	$-0.710684\pi$
$181$	−16.5373	−1.22921	−0.614605	+	0.788835i	$0.710684\pi$
$182$	0.346962	0.0257185
$183$	0	0
$184$	−48.9412	−3.60800
$185$	7.76390	0.570814
$186$	0	0
$187$	17.7923	1.30110
$188$	−45.6443	−3.32895
$189$	0	0
$190$	−3.04462	−0.220880
$191$	5.17605	0.374526	0.187263	−	0.982310i	$-0.440038\pi$
$191$	5.17605	0.374526	0.187263	+	0.982310i	$0.440038\pi$
$192$	0	0
$193$	22.3098	1.60590	0.802949	−	0.596048i	$-0.203263\pi$
$193$	22.3098	1.60590	0.802949	+	0.596048i	$0.203263\pi$
$194$	0.924299	0.0663608
$195$	0	0
$196$	5.37083	0.383631
$197$	−10.0532	−0.716262	−0.358131	−	0.933671i	$-0.616586\pi$
$197$	−10.0532	−0.716262	−0.358131	+	0.933671i	$0.616586\pi$
$198$	0	0
$199$	9.64785	0.683918	0.341959	−	0.939715i	$-0.388910\pi$
$199$	9.64785	0.683918	0.341959	+	0.939715i	$0.388910\pi$
$200$	−29.3674	−2.07659
$201$	0	0
$202$	−5.19093	−0.365232
$203$	−2.80116	−0.196603
$204$	0	0
$205$	0.108256	0.00756090
$206$	−12.1291	−0.845077
$207$	0	0
$208$	−1.80248	−0.124980
$209$	−4.28064	−0.296098
$210$	0	0
$211$	−22.9603	−1.58065	−0.790324	−	0.612689i	$-0.790088\pi$
$211$	−22.9603	−1.58065	−0.790324	+	0.612689i	$0.790088\pi$
$212$	59.4506	4.08308
$213$	0	0
$214$	−54.3222	−3.71339
$215$	5.93696	0.404898
$216$	0	0
$217$	4.97492	0.337720
$218$	52.1534	3.53227
$219$	0	0
$220$	36.7171	2.47546
$221$	0.445118	0.0299419
$222$	0	0
$223$	28.7076	1.92240	0.961200	−	0.275852i	$-0.0889599\pi$
$223$	28.7076	1.92240	0.961200	+	0.275852i	$0.0889599\pi$
$224$	−19.9886	−1.33555
$225$	0	0
$226$	14.1122	0.938726
$227$	26.2241	1.74055	0.870277	−	0.492563i	$-0.163940\pi$
$227$	26.2241	1.74055	0.870277	+	0.492563i	$0.163940\pi$
$228$	0	0
$229$	−5.85240	−0.386737	−0.193369	−	0.981126i	$-0.561941\pi$
$229$	−5.85240	−0.386737	−0.193369	+	0.981126i	$0.561941\pi$
$230$	19.4305	1.28121
$231$	0	0
$232$	25.6350	1.68302
$233$	−18.2853	−1.19791	−0.598953	−	0.800784i	$-0.704417\pi$
$233$	−18.2853	−1.19791	−0.598953	+	0.800784i	$0.704417\pi$
$234$	0	0
$235$	11.3734	0.741921
$236$	−41.1139	−2.67629
$237$	0	0
$238$	9.45604	0.612944
$239$	−16.5422	−1.07003	−0.535014	−	0.844843i	$-0.679694\pi$
$239$	−16.5422	−1.07003	−0.535014	+	0.844843i	$0.679694\pi$
$240$	0	0
$241$	−16.1993	−1.04349	−0.521745	−	0.853101i	$-0.674719\pi$
$241$	−16.1993	−1.04349	−0.521745	+	0.853101i	$0.674719\pi$
$242$	40.9822	2.63444
$243$	0	0
$244$	59.4329	3.80480
$245$	−1.33828	−0.0854995
$246$	0	0
$247$	−0.107091	−0.00681403
$248$	−45.5283	−2.89105
$249$	0	0
$250$	29.8260	1.88636
$251$	−22.5140	−1.42107	−0.710534	−	0.703662i	$-0.751547\pi$
$251$	−22.5140	−1.42107	−0.710534	+	0.703662i	$0.751547\pi$
$252$	0	0
$253$	27.3187	1.71751
$254$	2.71493	0.170350
$255$	0	0
$256$	31.4252	1.96407
$257$	−15.6268	−0.974775	−0.487387	−	0.873186i	$-0.662050\pi$
$257$	−15.6268	−0.974775	−0.487387	+	0.873186i	$0.662050\pi$
$258$	0	0
$259$	5.80141	0.360482
$260$	0.918569	0.0569673
$261$	0	0
$262$	−18.0943	−1.11787
$263$	26.9718	1.66315	0.831575	−	0.555413i	$-0.187440\pi$
$263$	26.9718	1.66315	0.831575	+	0.555413i	$0.187440\pi$
$264$	0	0
$265$	−14.8136	−0.909994
$266$	−2.27503	−0.139491
$267$	0	0
$268$	−8.03955	−0.491094
$269$	−11.1413	−0.679295	−0.339648	−	0.940553i	$-0.610308\pi$
$269$	−11.1413	−0.679295	−0.339648	+	0.940553i	$0.610308\pi$
$270$	0	0
$271$	−5.08370	−0.308813	−0.154406	−	0.988007i	$-0.549347\pi$
$271$	−5.08370	−0.308813	−0.154406	+	0.988007i	$0.549347\pi$
$272$	−49.1245	−2.97861
$273$	0	0
$274$	28.8288	1.74161
$275$	16.3927	0.988518
$276$	0	0
$277$	27.5780	1.65700	0.828502	−	0.559986i	$-0.189194\pi$
$277$	27.5780	1.65700	0.828502	+	0.559986i	$0.189194\pi$
$278$	−45.1467	−2.70772
$279$	0	0
$280$	12.2473	0.731919
$281$	7.82181	0.466610	0.233305	−	0.972404i	$-0.425046\pi$
$281$	7.82181	0.466610	0.233305	+	0.972404i	$0.425046\pi$
$282$	0	0
$283$	−12.8276	−0.762524	−0.381262	−	0.924467i	$-0.624510\pi$
$283$	−12.8276	−0.762524	−0.381262	+	0.924467i	$0.624510\pi$
$284$	31.6607	1.87872
$285$	0	0
$286$	1.77240	0.104804
$287$	0.0808917	0.00477489
$288$	0	0
$289$	−4.86885	−0.286403
$290$	−10.1776	−0.597647
$291$	0	0
$292$	−32.1339	−1.88049
$293$	−10.1462	−0.592749	−0.296374	−	0.955072i	$-0.595778\pi$
$293$	−10.1462	−0.592749	−0.296374	+	0.955072i	$0.595778\pi$
$294$	0	0
$295$	10.2446	0.596463
$296$	−53.0919	−3.08591
$297$	0	0
$298$	26.4417	1.53173
$299$	0.683445	0.0395247
$300$	0	0
$301$	4.43627	0.255702
$302$	54.4277	3.13196
$303$	0	0
$304$	11.8189	0.677858
$305$	−14.8092	−0.847973
$306$	0	0
$307$	34.8397	1.98841	0.994204	−	0.107508i	$-0.0342872\pi$
$307$	34.8397	1.98841	0.994204	+	0.107508i	$0.0342872\pi$
$308$	27.4360	1.56331
$309$	0	0
$310$	18.0755	1.02662
$311$	−10.7073	−0.607158	−0.303579	−	0.952806i	$-0.598182\pi$
$311$	−10.7073	−0.607158	−0.303579	+	0.952806i	$0.598182\pi$
$312$	0	0
$313$	−7.30795	−0.413069	−0.206535	−	0.978439i	$-0.566219\pi$
$313$	−7.30795	−0.413069	−0.206535	+	0.978439i	$0.566219\pi$
$314$	61.5639	3.47425
$315$	0	0
$316$	61.8083	3.47699
$317$	10.9031	0.612382	0.306191	−	0.951970i	$-0.400945\pi$
$317$	10.9031	0.612382	0.306191	+	0.951970i	$0.400945\pi$
$318$	0	0
$319$	−14.3093	−0.801167
$320$	−34.8747	−1.94956
$321$	0	0
$322$	14.5190	0.809115
$323$	−2.91863	−0.162397
$324$	0	0
$325$	0.410105	0.0227485
$326$	5.83594	0.323223
$327$	0	0
$328$	−0.740285	−0.0408754
$329$	8.49856	0.468541
$330$	0	0
$331$	−10.2085	−0.561111	−0.280555	−	0.959838i	$-0.590519\pi$
$331$	−10.2085	−0.561111	−0.280555	+	0.959838i	$0.590519\pi$
$332$	−73.9640	−4.05930
$333$	0	0
$334$	−38.3532	−2.09860
$335$	2.00326	0.109450
$336$	0	0
$337$	−27.2038	−1.48189	−0.740943	−	0.671568i	$-0.765621\pi$
$337$	−27.2038	−1.48189	−0.740943	+	0.671568i	$0.765621\pi$
$338$	−35.2497	−1.91733
$339$	0	0
$340$	25.0345	1.35769
$341$	25.4136	1.37622
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−40.5988	−2.18894
$345$	0	0
$346$	−45.4358	−2.44264
$347$	19.1046	1.02559	0.512796	−	0.858511i	$-0.328610\pi$
$347$	19.1046	1.02559	0.512796	+	0.858511i	$0.328610\pi$
$348$	0	0
$349$	−28.2885	−1.51425	−0.757124	−	0.653272i	$-0.773396\pi$
$349$	−28.2885	−1.51425	−0.757124	+	0.653272i	$0.773396\pi$
$350$	8.71223	0.465688
$351$	0	0
$352$	−102.109	−5.44242
$353$	29.2223	1.55535	0.777673	−	0.628668i	$-0.216400\pi$
$353$	29.2223	1.55535	0.777673	+	0.628668i	$0.216400\pi$
$354$	0	0
$355$	−7.88907	−0.418708
$356$	−58.5668	−3.10404
$357$	0	0
$358$	−7.13736	−0.377221
$359$	−15.9790	−0.843337	−0.421669	−	0.906750i	$-0.638555\pi$
$359$	−15.9790	−0.843337	−0.421669	+	0.906750i	$0.638555\pi$
$360$	0	0
$361$	−18.2978	−0.963042
$362$	−44.8976	−2.35977
$363$	0	0
$364$	0.686381	0.0359761
$365$	8.00697	0.419104
$366$	0	0
$367$	25.3999	1.32587	0.662933	−	0.748679i	$-0.269311\pi$
$367$	25.3999	1.32587	0.662933	+	0.748679i	$0.269311\pi$
$368$	−75.4270	−3.93191
$369$	0	0
$370$	21.0784	1.09582
$371$	−11.0692	−0.574682
$372$	0	0
$373$	12.8845	0.667133	0.333566	−	0.942727i	$-0.391748\pi$
$373$	12.8845	0.667133	0.333566	+	0.942727i	$0.391748\pi$
$374$	48.3047	2.49778
$375$	0	0
$376$	−77.7750	−4.01094
$377$	−0.357983	−0.0184371
$378$	0	0
$379$	−10.0187	−0.514626	−0.257313	−	0.966328i	$-0.582837\pi$
$379$	−10.0187	−0.514626	−0.257313	+	0.966328i	$0.582837\pi$
$380$	−6.02305	−0.308976
$381$	0	0
$382$	14.0526	0.718994
$383$	−5.81802	−0.297287	−0.148644	−	0.988891i	$-0.547491\pi$
$383$	−5.81802	−0.297287	−0.148644	+	0.988891i	$0.547491\pi$
$384$	0	0
$385$	−6.83639	−0.348414
$386$	60.5696	3.08291
$387$	0	0
$388$	1.82850	0.0928282
$389$	−4.43491	−0.224859	−0.112429	−	0.993660i	$-0.535863\pi$
$389$	−4.43491	−0.224859	−0.112429	+	0.993660i	$0.535863\pi$
$390$	0	0
$391$	18.6265	0.941982
$392$	9.15156	0.462224
$393$	0	0
$394$	−27.2938	−1.37504
$395$	−15.4011	−0.774915
$396$	0	0
$397$	−19.9182	−0.999667	−0.499834	−	0.866121i	$-0.666606\pi$
$397$	−19.9182	−0.999667	−0.499834	+	0.866121i	$0.666606\pi$
$398$	26.1932	1.31295
$399$	0	0
$400$	−45.2604	−2.26302
$401$	26.6075	1.32871	0.664357	−	0.747415i	$-0.268705\pi$
$401$	26.6075	1.32871	0.664357	+	0.747415i	$0.268705\pi$
$402$	0	0
$403$	0.635785	0.0316707
$404$	−10.2690	−0.510902
$405$	0	0
$406$	−7.60496	−0.377428
$407$	29.6356	1.46898
$408$	0	0
$409$	29.3110	1.44933	0.724667	−	0.689099i	$-0.241994\pi$
$409$	29.3110	1.44933	0.724667	+	0.689099i	$0.241994\pi$
$410$	0.293906	0.0145150
$411$	0	0
$412$	−23.9946	−1.18213
$413$	7.65504	0.376680
$414$	0	0
$415$	18.4300	0.904694
$416$	−2.55451	−0.125245
$417$	0	0
$418$	−11.6216	−0.568432
$419$	26.7010	1.30443	0.652214	−	0.758035i	$-0.273841\pi$
$419$	26.7010	1.30443	0.652214	+	0.758035i	$0.273841\pi$
$420$	0	0
$421$	6.58038	0.320708	0.160354	−	0.987060i	$-0.448736\pi$
$421$	6.58038	0.320708	0.160354	+	0.987060i	$0.448736\pi$
$422$	−62.3354	−3.03444
$423$	0	0
$424$	101.300	4.91957
$425$	11.1769	0.542160
$426$	0	0
$427$	−11.0659	−0.535515
$428$	−107.463	−5.19444
$429$	0	0
$430$	16.1184	0.777299
$431$	−12.3844	−0.596538	−0.298269	−	0.954482i	$-0.596409\pi$
$431$	−12.3844	−0.596538	−0.298269	+	0.954482i	$0.596409\pi$
$432$	0	0
$433$	37.8545	1.81917	0.909585	−	0.415519i	$-0.136400\pi$
$433$	37.8545	1.81917	0.909585	+	0.415519i	$0.136400\pi$
$434$	13.5065	0.648335
$435$	0	0
$436$	103.173	4.94109
$437$	−4.48134	−0.214372
$438$	0	0
$439$	−19.3809	−0.925002	−0.462501	−	0.886619i	$-0.653048\pi$
$439$	−19.3809	−0.925002	−0.462501	+	0.886619i	$0.653048\pi$
$440$	62.5636	2.98260
$441$	0	0
$442$	1.20846	0.0574807
$443$	−16.4270	−0.780472	−0.390236	−	0.920715i	$-0.627606\pi$
$443$	−16.4270	−0.780472	−0.390236	+	0.920715i	$0.627606\pi$
$444$	0	0
$445$	14.5934	0.691794
$446$	77.9389	3.69052
$447$	0	0
$448$	−26.0594	−1.23119
$449$	−36.8109	−1.73721	−0.868607	−	0.495502i	$-0.834984\pi$
$449$	−36.8109	−1.73721	−0.868607	+	0.495502i	$0.834984\pi$
$450$	0	0
$451$	0.413222	0.0194579
$452$	27.9175	1.31313
$453$	0	0
$454$	71.1965	3.34142
$455$	−0.171029	−0.00801798
$456$	0	0
$457$	23.8171	1.11412	0.557059	−	0.830473i	$-0.311930\pi$
$457$	23.8171	1.11412	0.557059	+	0.830473i	$0.311930\pi$
$458$	−15.8888	−0.742437
$459$	0	0
$460$	38.4386	1.79221
$461$	7.34713	0.342190	0.171095	−	0.985255i	$-0.445269\pi$
$461$	7.34713	0.342190	0.171095	+	0.985255i	$0.445269\pi$
$462$	0	0
$463$	−1.82291	−0.0847177	−0.0423589	−	0.999102i	$-0.513487\pi$
$463$	−1.82291	−0.0847177	−0.0423589	+	0.999102i	$0.513487\pi$
$464$	39.5081	1.83412
$465$	0	0
$466$	−49.6431	−2.29967
$467$	2.22986	0.103185	0.0515927	−	0.998668i	$-0.483570\pi$
$467$	2.22986	0.103185	0.0515927	+	0.998668i	$0.483570\pi$
$468$	0	0
$469$	1.49689	0.0691200
$470$	30.8781	1.42430
$471$	0	0
$472$	−70.0556	−3.22457
$473$	22.6620	1.04200
$474$	0	0
$475$	−2.68905	−0.123382
$476$	18.7065	0.857411
$477$	0	0
$478$	−44.9110	−2.05418
$479$	−31.0876	−1.42043	−0.710215	−	0.703985i	$-0.751402\pi$
$479$	−31.0876	−1.42043	−0.710215	+	0.703985i	$0.751402\pi$
$480$	0	0
$481$	0.741408	0.0338053
$482$	−43.9800	−2.00323
$483$	0	0
$484$	81.0735	3.68516
$485$	−0.455618	−0.0206886
$486$	0	0
$487$	18.7015	0.847447	0.423723	−	0.905792i	$-0.360723\pi$
$487$	18.7015	0.847447	0.423723	+	0.905792i	$0.360723\pi$
$488$	101.270	4.58428
$489$	0	0
$490$	−3.63333	−0.164137
$491$	4.29515	0.193838	0.0969188	−	0.995292i	$-0.469101\pi$
$491$	4.29515	0.193838	0.0969188	+	0.995292i	$0.469101\pi$
$492$	0	0
$493$	−9.75640	−0.439406
$494$	−0.290744	−0.0130812
$495$	0	0
$496$	−70.1670	−3.15059
$497$	−5.89494	−0.264424
$498$	0	0
$499$	−15.7256	−0.703975	−0.351988	−	0.936005i	$-0.614494\pi$
$499$	−15.7256	−0.703975	−0.351988	+	0.936005i	$0.614494\pi$
$500$	59.0036	2.63872
$501$	0	0
$502$	−61.1238	−2.72809
$503$	10.6736	0.475914	0.237957	−	0.971276i	$-0.423522\pi$
$503$	10.6736	0.475914	0.237957	+	0.971276i	$0.423522\pi$
$504$	0	0
$505$	2.55878	0.113864
$506$	74.1682	3.29718
$507$	0	0
$508$	5.37083	0.238292
$509$	17.2020	0.762466	0.381233	−	0.924479i	$-0.375499\pi$
$509$	17.2020	0.762466	0.381233	+	0.924479i	$0.375499\pi$
$510$	0	0
$511$	5.98304	0.264674
$512$	23.7729	1.05062
$513$	0	0
$514$	−42.4257	−1.87132
$515$	5.97886	0.263460
$516$	0	0
$517$	43.4135	1.90932
$518$	15.7504	0.692033
$519$	0	0
$520$	1.56518	0.0686379
$521$	20.1056	0.880840	0.440420	−	0.897792i	$-0.354829\pi$
$521$	20.1056	0.880840	0.440420	+	0.897792i	$0.354829\pi$
$522$	0	0
$523$	2.74019	0.119820	0.0599102	−	0.998204i	$-0.480919\pi$
$523$	2.74019	0.119820	0.0599102	+	0.998204i	$0.480919\pi$
$524$	−35.7952	−1.56372
$525$	0	0
$526$	73.2263	3.19282
$527$	17.3275	0.754800
$528$	0	0
$529$	5.59958	0.243460
$530$	−40.2179	−1.74695
$531$	0	0
$532$	−4.50059	−0.195126
$533$	0.0103378	0.000447780 0
$534$	0	0
$535$	26.7773	1.15768
$536$	−13.6989	−0.591702
$537$	0	0
$538$	−30.2477	−1.30407
$539$	−5.10834	−0.220032
$540$	0	0
$541$	−14.9334	−0.642036	−0.321018	−	0.947073i	$-0.604025\pi$
$541$	−14.9334	−0.642036	−0.321018	+	0.947073i	$0.604025\pi$
$542$	−13.8019	−0.592842
$543$	0	0
$544$	−69.6200	−2.98493
$545$	−25.7082	−1.10122
$546$	0	0
$547$	−15.3207	−0.655064	−0.327532	−	0.944840i	$-0.606217\pi$
$547$	−15.3207	−0.655064	−0.327532	+	0.944840i	$0.606217\pi$
$548$	57.0309	2.43624
$549$	0	0
$550$	44.5050	1.89770
$551$	2.34729	0.0999980
$552$	0	0
$553$	−11.5082	−0.489377
$554$	74.8724	3.18102
$555$	0	0
$556$	−89.3119	−3.78767
$557$	13.4859	0.571414	0.285707	−	0.958317i	$-0.407772\pi$
$557$	13.4859	0.571414	0.285707	+	0.958317i	$0.407772\pi$
$558$	0	0
$559$	0.566946	0.0239793
$560$	18.8753	0.797627
$561$	0	0
$562$	21.2356	0.895772
$563$	2.00440	0.0844752	0.0422376	−	0.999108i	$-0.486551\pi$
$563$	2.00440	0.0844752	0.0422376	+	0.999108i	$0.486551\pi$
$564$	0	0
$565$	−6.95635	−0.292656
$566$	−34.8261	−1.46385
$567$	0	0
$568$	53.9479	2.26360
$569$	20.3335	0.852424	0.426212	−	0.904623i	$-0.359848\pi$
$569$	20.3335	0.852424	0.426212	+	0.904623i	$0.359848\pi$
$570$	0	0
$571$	−9.05000	−0.378731	−0.189365	−	0.981907i	$-0.560643\pi$
$571$	−9.05000	−0.378731	−0.189365	+	0.981907i	$0.560643\pi$
$572$	3.50627	0.146604
$573$	0	0
$574$	0.219615	0.00916656
$575$	17.1613	0.715677
$576$	0	0
$577$	16.4836	0.686221	0.343111	−	0.939295i	$-0.388519\pi$
$577$	16.4836	0.686221	0.343111	+	0.939295i	$0.388519\pi$
$578$	−13.2186	−0.549820
$579$	0	0
$580$	−20.1338	−0.836012
$581$	13.7714	0.571335
$582$	0	0
$583$	−56.5450	−2.34186
$584$	−54.7541	−2.26574
$585$	0	0
$586$	−27.5463	−1.13793
$587$	12.6956	0.524005	0.262002	−	0.965067i	$-0.415617\pi$
$587$	12.6956	0.524005	0.262002	+	0.965067i	$0.415617\pi$
$588$	0	0
$589$	−4.16883	−0.171774
$590$	27.8133	1.14506
$591$	0	0
$592$	−81.8240	−3.36294
$593$	−33.5627	−1.37825	−0.689127	−	0.724640i	$-0.742006\pi$
$593$	−33.5627	−1.37825	−0.689127	+	0.724640i	$0.742006\pi$
$594$	0	0
$595$	−4.66120	−0.191091
$596$	52.3086	2.14264
$597$	0	0
$598$	1.85550	0.0758772
$599$	−37.4865	−1.53166	−0.765829	−	0.643044i	$-0.777671\pi$
$599$	−37.4865	−1.53166	−0.765829	+	0.643044i	$0.777671\pi$
$600$	0	0
$601$	36.0447	1.47029	0.735147	−	0.677908i	$-0.237113\pi$
$601$	36.0447	1.47029	0.735147	+	0.677908i	$0.237113\pi$
$602$	12.0441	0.490883
$603$	0	0
$604$	107.672	4.38112
$605$	−20.2015	−0.821309
$606$	0	0
$607$	25.9670	1.05397	0.526985	−	0.849875i	$-0.323323\pi$
$607$	25.9670	1.05397	0.526985	+	0.849875i	$0.323323\pi$
$608$	16.7499	0.679297
$609$	0	0
$610$	−40.2060	−1.62789
$611$	1.08610	0.0439388
$612$	0	0
$613$	−18.7943	−0.759096	−0.379548	−	0.925172i	$-0.623920\pi$
$613$	−18.7943	−0.759096	−0.379548	+	0.925172i	$0.623920\pi$
$614$	94.5873	3.81723
$615$	0	0
$616$	46.7493	1.88358
$617$	−20.2800	−0.816442	−0.408221	−	0.912883i	$-0.633851\pi$
$617$	−20.2800	−0.816442	−0.408221	+	0.912883i	$0.633851\pi$
$618$	0	0
$619$	20.8542	0.838200	0.419100	−	0.907940i	$-0.362346\pi$
$619$	20.8542	0.838200	0.419100	+	0.907940i	$0.362346\pi$
$620$	35.7581	1.43608
$621$	0	0
$622$	−29.0697	−1.16559
$623$	10.9046	0.436884
$624$	0	0
$625$	1.34279	0.0537115
$626$	−19.8405	−0.792987
$627$	0	0
$628$	121.789	4.85993
$629$	20.2062	0.805674
$630$	0	0
$631$	6.62011	0.263542	0.131771	−	0.991280i	$-0.457934\pi$
$631$	6.62011	0.263542	0.131771	+	0.991280i	$0.457934\pi$
$632$	105.318	4.18931
$633$	0	0
$634$	29.6012	1.17562
$635$	−1.33828	−0.0531080
$636$	0	0
$637$	−0.127798	−0.00506354
$638$	−38.8487	−1.53804
$639$	0	0
$640$	−41.1816	−1.62785
$641$	−19.0995	−0.754386	−0.377193	−	0.926135i	$-0.623111\pi$
$641$	−19.0995	−0.754386	−0.377193	+	0.926135i	$0.623111\pi$
$642$	0	0
$643$	−41.2276	−1.62586	−0.812928	−	0.582364i	$-0.802128\pi$
$643$	−41.2276	−1.62586	−0.812928	+	0.582364i	$0.802128\pi$
$644$	28.7224	1.13182
$645$	0	0
$646$	−7.92388	−0.311761
$647$	15.0210	0.590534	0.295267	−	0.955415i	$-0.404591\pi$
$647$	15.0210	0.590534	0.295267	+	0.955415i	$0.404591\pi$
$648$	0	0
$649$	39.1046	1.53499
$650$	1.11340	0.0436713
$651$	0	0
$652$	11.5450	0.452137
$653$	−7.59068	−0.297046	−0.148523	−	0.988909i	$-0.547452\pi$
$653$	−7.59068	−0.297046	−0.148523	+	0.988909i	$0.547452\pi$
$654$	0	0
$655$	8.91928	0.348505
$656$	−1.14091	−0.0445450
$657$	0	0
$658$	23.0730	0.899478
$659$	13.8974	0.541364	0.270682	−	0.962669i	$-0.412751\pi$
$659$	13.8974	0.541364	0.270682	+	0.962669i	$0.412751\pi$
$660$	0	0
$661$	−0.257929	−0.0100323	−0.00501614	−	0.999987i	$-0.501597\pi$
$661$	−0.257929	−0.0100323	−0.00501614	+	0.999987i	$0.501597\pi$
$662$	−27.7154	−1.07719
$663$	0	0
$664$	−126.030	−4.89091
$665$	1.12144	0.0434875
$666$	0	0
$667$	−14.9802	−0.580037
$668$	−75.8726	−2.93560
$669$	0	0
$670$	5.43870	0.210115
$671$	−56.5282	−2.18225
$672$	0	0
$673$	−24.3744	−0.939564	−0.469782	−	0.882782i	$-0.655668\pi$
$673$	−24.3744	−0.939564	−0.469782	+	0.882782i	$0.655668\pi$
$674$	−73.8563	−2.84484
$675$	0	0
$676$	−69.7331	−2.68204
$677$	−3.15632	−0.121307	−0.0606536	−	0.998159i	$-0.519319\pi$
$677$	−3.15632	−0.121307	−0.0606536	+	0.998159i	$0.519319\pi$
$678$	0	0
$679$	−0.340451	−0.0130653
$680$	42.6572	1.63583
$681$	0	0
$682$	68.9960	2.64200
$683$	−7.57329	−0.289784	−0.144892	−	0.989447i	$-0.546283\pi$
$683$	−7.57329	−0.289784	−0.144892	+	0.989447i	$0.546283\pi$
$684$	0	0
$685$	−14.2107	−0.542963
$686$	−2.71493	−0.103656
$687$	0	0
$688$	−62.5698	−2.38545
$689$	−1.41462	−0.0538926
$690$	0	0
$691$	−5.29158	−0.201301	−0.100651	−	0.994922i	$-0.532092\pi$
$691$	−5.29158	−0.201301	−0.100651	+	0.994922i	$0.532092\pi$
$692$	−89.8838	−3.41687
$693$	0	0
$694$	51.8677	1.96887
$695$	22.2543	0.844155
$696$	0	0
$697$	0.281744	0.0106718
$698$	−76.8011	−2.90697
$699$	0	0
$700$	17.2350	0.651423
$701$	18.8465	0.711821	0.355910	−	0.934520i	$-0.384171\pi$
$701$	18.8465	0.711821	0.355910	+	0.934520i	$0.384171\pi$
$702$	0	0
$703$	−4.86141	−0.183351
$704$	−133.120	−5.01716
$705$	0	0
$706$	79.3365	2.98587
$707$	1.91200	0.0719080
$708$	0	0
$709$	1.96929	0.0739581	0.0369791	−	0.999316i	$-0.488227\pi$
$709$	1.96929	0.0739581	0.0369791	+	0.999316i	$0.488227\pi$
$710$	−21.4183	−0.803813
$711$	0	0
$712$	−99.7942	−3.73995
$713$	26.6052	0.996371
$714$	0	0
$715$	−0.873676	−0.0326736
$716$	−14.1195	−0.527672
$717$	0	0
$718$	−43.3817	−1.61899
$719$	−6.82275	−0.254446	−0.127223	−	0.991874i	$-0.540606\pi$
$719$	−6.82275	−0.254446	−0.127223	+	0.991874i	$0.540606\pi$
$720$	0	0
$721$	4.46758	0.166381
$722$	−49.6772	−1.84879
$723$	0	0
$724$	−88.8192	−3.30094
$725$	−8.98896	−0.333842
$726$	0	0
$727$	−8.47424	−0.314292	−0.157146	−	0.987575i	$-0.550229\pi$
$727$	−8.47424	−0.314292	−0.157146	+	0.987575i	$0.550229\pi$
$728$	1.16955	0.0433464
$729$	0	0
$730$	21.7383	0.804572
$731$	15.4514	0.571492
$732$	0	0
$733$	−19.6518	−0.725855	−0.362928	−	0.931817i	$-0.618223\pi$
$733$	−19.6518	−0.725855	−0.362928	+	0.931817i	$0.618223\pi$
$734$	68.9590	2.54532
$735$	0	0
$736$	−106.896	−3.94025
$737$	7.64663	0.281667
$738$	0	0
$739$	−11.4296	−0.420444	−0.210222	−	0.977654i	$-0.567419\pi$
$739$	−11.4296	−0.420444	−0.210222	+	0.977654i	$0.567419\pi$
$740$	41.6986	1.53287
$741$	0	0
$742$	−30.0520	−1.10324
$743$	23.4883	0.861702	0.430851	−	0.902423i	$-0.358213\pi$
$743$	23.4883	0.861702	0.430851	+	0.902423i	$0.358213\pi$
$744$	0	0
$745$	−13.0340	−0.477529
$746$	34.9804	1.28072
$747$	0	0
$748$	95.5592	3.49399
$749$	20.0087	0.731103
$750$	0	0
$751$	−15.6742	−0.571958	−0.285979	−	0.958236i	$-0.592319\pi$
$751$	−15.6742	−0.571958	−0.285979	+	0.958236i	$0.592319\pi$
$752$	−119.865	−4.37103
$753$	0	0
$754$	−0.971898	−0.0353944
$755$	−26.8293	−0.976416
$756$	0	0
$757$	7.94903	0.288912	0.144456	−	0.989511i	$-0.453857\pi$
$757$	7.94903	0.288912	0.144456	+	0.989511i	$0.453857\pi$
$758$	−27.2001	−0.987951
$759$	0	0
$760$	−10.2629	−0.372275
$761$	−25.7371	−0.932968	−0.466484	−	0.884530i	$-0.654480\pi$
$761$	−25.7371	−0.932968	−0.466484	+	0.884530i	$0.654480\pi$
$762$	0	0
$763$	−19.2099	−0.695444
$764$	27.7997	1.00576
$765$	0	0
$766$	−15.7955	−0.570715
$767$	0.978299	0.0353243
$768$	0	0
$769$	18.0824	0.652067	0.326034	−	0.945358i	$-0.394288\pi$
$769$	18.0824	0.652067	0.326034	+	0.945358i	$0.394288\pi$
$770$	−18.5603	−0.668866
$771$	0	0
$772$	119.822	4.31250
$773$	−10.1587	−0.365384	−0.182692	−	0.983170i	$-0.558481\pi$
$773$	−10.1587	−0.365384	−0.182692	+	0.983170i	$0.558481\pi$
$774$	0	0
$775$	15.9646	0.573464
$776$	3.11566	0.111846
$777$	0	0
$778$	−12.0404	−0.431671
$779$	−0.0677848	−0.00242864
$780$	0	0
$781$	−30.1134	−1.07754
$782$	50.5695	1.80836
$783$	0	0
$784$	14.1042	0.503720
$785$	−30.3469	−1.08313
$786$	0	0
$787$	37.3251	1.33050	0.665248	−	0.746623i	$-0.268326\pi$
$787$	37.3251	1.33050	0.665248	+	0.746623i	$0.268326\pi$
$788$	−53.9941	−1.92346
$789$	0	0
$790$	−41.8129	−1.48764
$791$	−5.19798	−0.184819
$792$	0	0
$793$	−1.41420	−0.0502196
$794$	−54.0765	−1.91910
$795$	0	0
$796$	51.8169	1.83660
$797$	15.5289	0.550062	0.275031	−	0.961435i	$-0.411312\pi$
$797$	15.5289	0.550062	0.275031	+	0.961435i	$0.411312\pi$
$798$	0	0
$799$	29.6003	1.04718
$800$	−64.1437	−2.26782
$801$	0	0
$802$	72.2374	2.55079
$803$	30.5634	1.07856
$804$	0	0
$805$	−7.15693	−0.252248
$806$	1.72611	0.0607996
$807$	0	0
$808$	−17.4977	−0.615568
$809$	−36.2909	−1.27592	−0.637960	−	0.770069i	$-0.720222\pi$
$809$	−36.2909	−1.27592	−0.637960	+	0.770069i	$0.720222\pi$
$810$	0	0
$811$	−32.4055	−1.13791	−0.568956	−	0.822368i	$-0.692652\pi$
$811$	−32.4055	−1.13791	−0.568956	+	0.822368i	$0.692652\pi$
$812$	−15.0446	−0.527961
$813$	0	0
$814$	80.4584	2.82007
$815$	−2.87673	−0.100767
$816$	0	0
$817$	−3.71746	−0.130057
$818$	79.5772	2.78235
$819$	0	0
$820$	0.581423	0.0203042
$821$	−33.9858	−1.18611	−0.593056	−	0.805161i	$-0.702079\pi$
$821$	−33.9858	−1.18611	−0.593056	+	0.805161i	$0.702079\pi$
$822$	0	0
$823$	−26.8813	−0.937025	−0.468512	−	0.883457i	$-0.655210\pi$
$823$	−26.8813	−0.937025	−0.468512	+	0.883457i	$0.655210\pi$
$824$	−40.8853	−1.42431
$825$	0	0
$826$	20.7829	0.723129
$827$	−18.7645	−0.652506	−0.326253	−	0.945283i	$-0.605786\pi$
$827$	−18.7645	−0.652506	−0.326253	+	0.945283i	$0.605786\pi$
$828$	0	0
$829$	10.1239	0.351618	0.175809	−	0.984424i	$-0.443746\pi$
$829$	10.1239	0.351618	0.175809	+	0.984424i	$0.443746\pi$
$830$	50.0361	1.73678
$831$	0	0
$832$	−3.33034	−0.115459
$833$	−3.48298	−0.120678
$834$	0	0
$835$	18.9056	0.654255
$836$	−22.9906	−0.795146
$837$	0	0
$838$	72.4912	2.50417
$839$	9.42481	0.325381	0.162690	−	0.986677i	$-0.447983\pi$
$839$	9.42481	0.325381	0.162690	+	0.986677i	$0.447983\pi$
$840$	0	0
$841$	−21.1535	−0.729430
$842$	17.8653	0.615677
$843$	0	0
$844$	−123.316	−4.24470
$845$	17.3758	0.597745
$846$	0	0
$847$	−15.0952	−0.518676
$848$	156.121	5.36122
$849$	0	0
$850$	30.3445	1.04081
$851$	31.0251	1.06353
$852$	0	0
$853$	−49.0195	−1.67840	−0.839199	−	0.543825i	$-0.816976\pi$
$853$	−49.0195	−1.67840	−0.839199	+	0.543825i	$0.816976\pi$
$854$	−30.0430	−1.02805
$855$	0	0
$856$	−183.111	−6.25860
$857$	34.2759	1.17084	0.585420	−	0.810730i	$-0.300929\pi$
$857$	34.2759	1.17084	0.585420	+	0.810730i	$0.300929\pi$
$858$	0	0
$859$	23.2301	0.792600	0.396300	−	0.918121i	$-0.370294\pi$
$859$	23.2301	0.792600	0.396300	+	0.918121i	$0.370294\pi$
$860$	31.8864	1.08732
$861$	0	0
$862$	−33.6229	−1.14520
$863$	23.3333	0.794274	0.397137	−	0.917759i	$-0.370004\pi$
$863$	23.3333	0.794274	0.397137	+	0.917759i	$0.370004\pi$
$864$	0	0
$865$	22.3968	0.761515
$866$	102.772	3.49234
$867$	0	0
$868$	26.7194	0.906917
$869$	−58.7876	−1.99423
$870$	0	0
$871$	0.191300	0.00648194
$872$	175.800	5.95334
$873$	0	0
$874$	−12.1665	−0.411539
$875$	−10.9859	−0.371393
$876$	0	0
$877$	30.0833	1.01584	0.507921	−	0.861404i	$-0.330414\pi$
$877$	30.0833	1.01584	0.507921	+	0.861404i	$0.330414\pi$
$878$	−52.6178	−1.77577
$879$	0	0
$880$	96.4215	3.25037
$881$	0.974375	0.0328275	0.0164138	−	0.999865i	$-0.494775\pi$
$881$	0.974375	0.0328275	0.0164138	+	0.999865i	$0.494775\pi$
$882$	0	0
$883$	4.60468	0.154960	0.0774799	−	0.996994i	$-0.475313\pi$
$883$	4.60468	0.154960	0.0774799	+	0.996994i	$0.475313\pi$
$884$	2.39065	0.0804063
$885$	0	0
$886$	−44.5982	−1.49831
$887$	−48.7742	−1.63768	−0.818839	−	0.574024i	$-0.805382\pi$
$887$	−48.7742	−1.63768	−0.818839	+	0.574024i	$0.805382\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	39.6201	1.32807
$891$	0	0
$892$	154.183	5.16244
$893$	−7.12153	−0.238313
$894$	0	0
$895$	3.51824	0.117602
$896$	−30.7721	−1.02802
$897$	0	0
$898$	−99.9389	−3.33500
$899$	−13.9356	−0.464777
$900$	0	0
$901$	−38.5537	−1.28441
$902$	1.12187	0.0373541
$903$	0	0
$904$	47.5697	1.58214
$905$	22.1316	0.735678
$906$	0	0
$907$	−30.8391	−1.02399	−0.511997	−	0.858987i	$-0.671094\pi$
$907$	−30.8391	−1.02399	−0.511997	+	0.858987i	$0.671094\pi$
$908$	140.845	4.67411
$909$	0	0
$910$	−0.464332	−0.0153925
$911$	−16.3651	−0.542201	−0.271101	−	0.962551i	$-0.587388\pi$
$911$	−16.3651	−0.542201	−0.271101	+	0.962551i	$0.587388\pi$
$912$	0	0
$913$	70.3492	2.32822
$914$	64.6617	2.13882
$915$	0	0
$916$	−31.4322	−1.03855
$917$	6.66474	0.220089
$918$	0	0
$919$	50.4600	1.66452	0.832262	−	0.554383i	$-0.187046\pi$
$919$	50.4600	1.66452	0.832262	+	0.554383i	$0.187046\pi$
$920$	65.4970	2.15937
$921$	0	0
$922$	19.9469	0.656917
$923$	−0.753361	−0.0247972
$924$	0	0
$925$	18.6168	0.612116
$926$	−4.94906	−0.162636
$927$	0	0
$928$	55.9915	1.83801
$929$	−15.2041	−0.498832	−0.249416	−	0.968396i	$-0.580239\pi$
$929$	−15.2041	−0.498832	−0.249416	+	0.968396i	$0.580239\pi$
$930$	0	0
$931$	0.837970	0.0274634
$932$	−98.2070	−3.21688
$933$	0	0
$934$	6.05390	0.198090
$935$	−23.8110	−0.778703
$936$	0	0
$937$	−8.67775	−0.283490	−0.141745	−	0.989903i	$-0.545271\pi$
$937$	−8.67775	−0.283490	−0.141745	+	0.989903i	$0.545271\pi$
$938$	4.06395	0.132693
$939$	0	0
$940$	61.0848	1.99237
$941$	2.13017	0.0694417	0.0347208	−	0.999397i	$-0.488946\pi$
$941$	2.13017	0.0694417	0.0347208	+	0.999397i	$0.488946\pi$
$942$	0	0
$943$	0.432597	0.0140873
$944$	−107.968	−3.51406
$945$	0	0
$946$	61.5256	2.00037
$947$	19.9973	0.649826	0.324913	−	0.945744i	$-0.394665\pi$
$947$	19.9973	0.649826	0.324913	+	0.945744i	$0.394665\pi$
$948$	0	0
$949$	0.764620	0.0248206
$950$	−7.30058	−0.236862
$951$	0	0
$952$	31.8747	1.03306
$953$	51.3207	1.66244	0.831220	−	0.555944i	$-0.187643\pi$
$953$	51.3207	1.66244	0.831220	+	0.555944i	$0.187643\pi$
$954$	0	0
$955$	−6.92700	−0.224152
$956$	−88.8455	−2.87347
$957$	0	0
$958$	−84.4007	−2.72686
$959$	−10.6186	−0.342894
$960$	0	0
$961$	−6.25017	−0.201619
$962$	2.01287	0.0648975
$963$	0	0
$964$	−87.0039	−2.80221
$965$	−29.8568	−0.961124
$966$	0	0
$967$	−20.2228	−0.650320	−0.325160	−	0.945659i	$-0.605418\pi$
$967$	−20.2228	−0.650320	−0.325160	+	0.945659i	$0.605418\pi$
$968$	138.144	4.44012
$969$	0	0
$970$	−1.23697	−0.0397167
$971$	18.1890	0.583714	0.291857	−	0.956462i	$-0.405727\pi$
$971$	18.1890	0.583714	0.291857	+	0.956462i	$0.405727\pi$
$972$	0	0
$973$	16.6291	0.533104
$974$	50.7733	1.62688
$975$	0	0
$976$	156.075	4.99583
$977$	−4.12930	−0.132108	−0.0660540	−	0.997816i	$-0.521041\pi$
$977$	−4.12930	−0.132108	−0.0660540	+	0.997816i	$0.521041\pi$
$978$	0	0
$979$	55.7045	1.78032
$980$	−7.18767	−0.229602
$981$	0	0
$982$	11.6610	0.372118
$983$	16.3835	0.522551	0.261276	−	0.965264i	$-0.415857\pi$
$983$	16.3835	0.522551	0.261276	+	0.965264i	$0.415857\pi$
$984$	0	0
$985$	13.4540	0.428680
$986$	−26.4879	−0.843547
$987$	0	0
$988$	−0.575167	−0.0182985
$989$	23.7245	0.754396
$990$	0	0
$991$	−40.8613	−1.29800	−0.649001	−	0.760787i	$-0.724813\pi$
$991$	−40.8613	−1.29800	−0.649001	+	0.760787i	$0.724813\pi$
$992$	−99.4419	−3.15728
$993$	0	0
$994$	−16.0043	−0.507627
$995$	−12.9115	−0.409322
$996$	0	0
$997$	10.3119	0.326581	0.163290	−	0.986578i	$-0.447789\pi$
$997$	10.3119	0.326581	0.163290	+	0.986578i	$0.447789\pi$
$998$	−42.6939	−1.35145
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.15		15
3.2	odd	2		889.2.a.b.1.1	✓	15
21.20	even	2		6223.2.a.j.1.1		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.1	✓	15	3.2	odd	2
6223.2.a.j.1.1		15	21.20	even	2
8001.2.a.q.1.15		15	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)

\(f(q)\)	\(=\)	\(q+2.71493 q^{2} +5.37083 q^{4} -1.33828 q^{5} -1.00000 q^{7} +9.15156 q^{8} +O(q^{10})\)	\(q+2.71493 q^{2} +5.37083 q^{4} -1.33828 q^{5} -1.00000 q^{7} +9.15156 q^{8} -3.63333 q^{10} -5.10834 q^{11} -0.127798 q^{13} -2.71493 q^{14} +14.1042 q^{16} -3.48298 q^{17} +0.837970 q^{19} -7.18767 q^{20} -13.8688 q^{22} -5.34786 q^{23} -3.20901 q^{25} -0.346962 q^{26} -5.37083 q^{28} +2.80116 q^{29} -4.97492 q^{31} +19.9886 q^{32} -9.45604 q^{34} +1.33828 q^{35} -5.80141 q^{37} +2.27503 q^{38} -12.2473 q^{40} -0.0808917 q^{41} -4.43627 q^{43} -27.4360 q^{44} -14.5190 q^{46} -8.49856 q^{47} +1.00000 q^{49} -8.71223 q^{50} -0.686381 q^{52} +11.0692 q^{53} +6.83639 q^{55} -9.15156 q^{56} +7.60496 q^{58} -7.65504 q^{59} +11.0659 q^{61} -13.5065 q^{62} +26.0594 q^{64} +0.171029 q^{65} -1.49689 q^{67} -18.7065 q^{68} +3.63333 q^{70} +5.89494 q^{71} -5.98304 q^{73} -15.7504 q^{74} +4.50059 q^{76} +5.10834 q^{77} +11.5082 q^{79} -18.8753 q^{80} -0.219615 q^{82} -13.7714 q^{83} +4.66120 q^{85} -12.0441 q^{86} -46.7493 q^{88} -10.9046 q^{89} +0.127798 q^{91} -28.7224 q^{92} -23.0730 q^{94} -1.12144 q^{95} +0.340451 q^{97} +2.71493 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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