LMFDB - Embedded newform 1000.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1000,2,Mod(1,1000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1000, 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("1000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1000 = 2^{3} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1000.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:	$7.98504020213$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14896$ of defining polynomial
Character	$\chi$	$=$	1000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.14896 q^{3} -1.28990 q^{7} +6.91596 q^{9} +O(q^{10})$	$q-3.14896 q^{3} -1.28990 q^{7} +6.91596 q^{9} -2.65626 q^{11} +5.53399 q^{13} -5.89233 q^{17} +6.95418 q^{19} +4.06185 q^{21} -1.47403 q^{23} -12.3312 q^{27} +4.21244 q^{29} -1.34374 q^{31} +8.36447 q^{33} +2.48205 q^{37} -17.4263 q^{39} -12.0444 q^{41} -5.14896 q^{43} +3.04129 q^{47} -5.33615 q^{49} +18.5547 q^{51} -10.4721 q^{53} -21.8985 q^{57} -5.12840 q^{59} -8.21388 q^{61} -8.92091 q^{63} +2.35834 q^{67} +4.64166 q^{69} -8.48205 q^{71} +5.88242 q^{73} +3.42632 q^{77} -16.2979 q^{79} +18.0826 q^{81} +0.931563 q^{83} -13.2648 q^{87} -0.505675 q^{89} -7.13831 q^{91} +4.23138 q^{93} -8.37438 q^{97} -18.3706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9	$ 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{21} - 14 q^{23} - 22 q^{27} + 10 q^{29} - 16 q^{31} - 4 q^{33} - 24 q^{39} + 2 q^{41} - 12 q^{43} - 16 q^{47} + 2 q^{49} - 24 q^{53} - 24 q^{57} + 8 q^{59} + 6 q^{61} - 18 q^{63} + 16 q^{67} + 12 q^{69} - 24 q^{71} - 4 q^{73} - 32 q^{77} - 48 q^{79} + 16 q^{81} - 2 q^{83} + 22 q^{87} + 10 q^{89} - 8 q^{91} + 20 q^{93} - 4 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^21 - 14 * q^23 - 22 * q^27 + 10 * q^29 - 16 * q^31 - 4 * q^33 - 24 * q^39 + 2 * q^41 - 12 * q^43 - 16 * q^47 + 2 * q^49 - 24 * q^53 - 24 * q^57 + 8 * q^59 + 6 * q^61 - 18 * q^63 + 16 * q^67 + 12 * q^69 - 24 * q^71 - 4 * q^73 - 32 * q^77 - 48 * q^79 + 16 * q^81 - 2 * q^83 + 22 * q^87 + 10 * q^89 - 8 * q^91 + 20 * q^93 - 4 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.14896	−1.81805	−0.909027	−	0.416738i	$-0.863173\pi$
$3$	−3.14896	−1.81805	−0.909027	+	0.416738i	$0.863173\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.28990	−0.487537	−0.243769	−	0.969833i	$-0.578384\pi$
$7$	−1.28990	−0.487537	−0.243769	+	0.969833i	$0.578384\pi$
$8$	0	0
$9$	6.91596	2.30532
$10$	0	0
$11$	−2.65626	−0.800893	−0.400447	−	0.916320i	$-0.631145\pi$
$11$	−2.65626	−0.800893	−0.400447	+	0.916320i	$0.631145\pi$
$12$	0	0
$13$	5.53399	1.53485	0.767426	−	0.641137i	$-0.221537\pi$
$13$	5.53399	1.53485	0.767426	+	0.641137i	$0.221537\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.89233	−1.42910	−0.714550	−	0.699584i	$-0.753368\pi$
$17$	−5.89233	−1.42910	−0.714550	+	0.699584i	$0.753368\pi$
$18$	0	0
$19$	6.95418	1.59540	0.797700	−	0.603055i	$-0.206050\pi$
$19$	6.95418	1.59540	0.797700	+	0.603055i	$0.206050\pi$
$20$	0	0
$21$	4.06185	0.886369
$22$	0	0
$23$	−1.47403	−0.307356	−0.153678	−	0.988121i	$-0.549112\pi$
$23$	−1.47403	−0.307356	−0.153678	+	0.988121i	$0.549112\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−12.3312	−2.37314
$28$	0	0
$29$	4.21244	0.782231	0.391115	−	0.920342i	$-0.372089\pi$
$29$	4.21244	0.782231	0.391115	+	0.920342i	$0.372089\pi$
$30$	0	0
$31$	−1.34374	−0.241342	−0.120671	−	0.992693i	$-0.538505\pi$
$31$	−1.34374	−0.241342	−0.120671	+	0.992693i	$0.538505\pi$
$32$	0	0
$33$	8.36447	1.45607
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.48205	0.408046	0.204023	−	0.978966i	$-0.434598\pi$
$37$	2.48205	0.408046	0.204023	+	0.978966i	$0.434598\pi$
$38$	0	0
$39$	−17.4263	−2.79044
$40$	0	0
$41$	−12.0444	−1.88101	−0.940506	−	0.339777i	$-0.889648\pi$
$41$	−12.0444	−1.88101	−0.940506	+	0.339777i	$0.889648\pi$
$42$	0	0
$43$	−5.14896	−0.785209	−0.392605	−	0.919707i	$-0.628426\pi$
$43$	−5.14896	−0.785209	−0.392605	+	0.919707i	$0.628426\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.04129	0.443618	0.221809	−	0.975090i	$-0.428804\pi$
$47$	3.04129	0.443618	0.221809	+	0.975090i	$0.428804\pi$
$48$	0	0
$49$	−5.33615	−0.762307
$50$	0	0
$51$	18.5547	2.59818
$52$	0	0
$53$	−10.4721	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$53$	−10.4721	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−21.8985	−2.90052
$58$	0	0
$59$	−5.12840	−0.667661	−0.333830	−	0.942633i	$-0.608341\pi$
$59$	−5.12840	−0.667661	−0.333830	+	0.942633i	$0.608341\pi$
$60$	0	0
$61$	−8.21388	−1.05168	−0.525840	−	0.850584i	$-0.676249\pi$
$61$	−8.21388	−1.05168	−0.525840	+	0.850584i	$0.676249\pi$
$62$	0	0
$63$	−8.92091	−1.12393
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.35834	0.288117	0.144059	−	0.989569i	$-0.453985\pi$
$67$	2.35834	0.288117	0.144059	+	0.989569i	$0.453985\pi$
$68$	0	0
$69$	4.64166	0.558790
$70$	0	0
$71$	−8.48205	−1.00663	−0.503317	−	0.864102i	$-0.667887\pi$
$71$	−8.48205	−1.00663	−0.503317	+	0.864102i	$0.667887\pi$
$72$	0	0
$73$	5.88242	0.688485	0.344242	−	0.938881i	$-0.388136\pi$
$73$	5.88242	0.688485	0.344242	+	0.938881i	$0.388136\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.42632	0.390465
$78$	0	0
$79$	−16.2979	−1.83366	−0.916830	−	0.399278i	$-0.869261\pi$
$79$	−16.2979	−1.83366	−0.916830	+	0.399278i	$0.869261\pi$
$80$	0	0
$81$	18.0826	2.00918
$82$	0	0
$83$	0.931563	0.102252	0.0511262	−	0.998692i	$-0.483719\pi$
$83$	0.931563	0.102252	0.0511262	+	0.998692i	$0.483719\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−13.2648	−1.42214
$88$	0	0
$89$	−0.505675	−0.0536014	−0.0268007	−	0.999641i	$-0.508532\pi$
$89$	−0.505675	−0.0536014	−0.0268007	+	0.999641i	$0.508532\pi$
$90$	0	0
$91$	−7.13831	−0.748298
$92$	0	0
$93$	4.23138	0.438774
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.37438	−0.850289	−0.425145	−	0.905125i	$-0.639777\pi$
$97$	−8.37438	−0.850289	−0.425145	+	0.905125i	$0.639777\pi$
$98$	0	0
$99$	−18.3706	−1.84631
$100$	0	0
$101$	11.9924	1.19329	0.596645	−	0.802505i	$-0.296500\pi$
$101$	11.9924	1.19329	0.596645	+	0.802505i	$0.296500\pi$
$102$	0	0
$103$	17.0779	1.68273	0.841367	−	0.540464i	$-0.181751\pi$
$103$	17.0779	1.68273	0.841367	+	0.540464i	$0.181751\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.74481	0.555371	0.277686	−	0.960672i	$-0.410433\pi$
$107$	5.74481	0.555371	0.277686	+	0.960672i	$0.410433\pi$
$108$	0	0
$109$	2.61803	0.250762	0.125381	−	0.992109i	$-0.459985\pi$
$109$	2.61803	0.250762	0.125381	+	0.992109i	$0.459985\pi$
$110$	0	0
$111$	−7.81587	−0.741850
$112$	0	0
$113$	−8.22147	−0.773410	−0.386705	−	0.922203i	$-0.626387\pi$
$113$	−8.22147	−0.773410	−0.386705	+	0.922203i	$0.626387\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	38.2728	3.53832
$118$	0	0
$119$	7.60053	0.696740
$120$	0	0
$121$	−3.94427	−0.358570
$122$	0	0
$123$	37.9272	3.41978
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.74337	−0.775849	−0.387924	−	0.921691i	$-0.626808\pi$
$127$	−8.74337	−0.775849	−0.387924	+	0.921691i	$0.626808\pi$
$128$	0	0
$129$	16.2139	1.42755
$130$	0	0
$131$	−4.29792	−0.375511	−0.187756	−	0.982216i	$-0.560121\pi$
$131$	−4.29792	−0.375511	−0.187756	+	0.982216i	$0.560121\pi$
$132$	0	0
$133$	−8.97022	−0.777817
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2994	−0.965369	−0.482685	−	0.875794i	$-0.660338\pi$
$137$	−11.2994	−0.965369	−0.482685	+	0.875794i	$0.660338\pi$
$138$	0	0
$139$	−5.45753	−0.462902	−0.231451	−	0.972847i	$-0.574347\pi$
$139$	−5.45753	−0.462902	−0.231451	+	0.972847i	$0.574347\pi$
$140$	0	0
$141$	−9.57691	−0.806521
$142$	0	0
$143$	−14.6997	−1.22925
$144$	0	0
$145$	0	0
$146$	0	0
$147$	16.8033	1.38592
$148$	0	0
$149$	10.6860	0.875432	0.437716	−	0.899113i	$-0.355787\pi$
$149$	10.6860	0.875432	0.437716	+	0.899113i	$0.355787\pi$
$150$	0	0
$151$	−21.9589	−1.78699	−0.893494	−	0.449075i	$-0.851753\pi$
$151$	−21.9589	−1.78699	−0.893494	+	0.449075i	$0.851753\pi$
$152$	0	0
$153$	−40.7511	−3.29453
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.89233	−0.151024	−0.0755122	−	0.997145i	$-0.524059\pi$
$157$	−1.89233	−0.151024	−0.0755122	+	0.997145i	$0.524059\pi$
$158$	0	0
$159$	32.9763	2.61519
$160$	0	0
$161$	1.90135	0.149848
$162$	0	0
$163$	−15.3725	−1.20407	−0.602033	−	0.798471i	$-0.705642\pi$
$163$	−15.3725	−1.20407	−0.602033	+	0.798471i	$0.705642\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.6730	1.59973	0.799864	−	0.600181i	$-0.204905\pi$
$167$	20.6730	1.59973	0.799864	+	0.600181i	$0.204905\pi$
$168$	0	0
$169$	17.6250	1.35577
$170$	0	0
$171$	48.0948	3.67790
$172$	0	0
$173$	−11.2049	−0.851889	−0.425945	−	0.904749i	$-0.640058\pi$
$173$	−11.2049	−0.851889	−0.425945	+	0.904749i	$0.640058\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	16.1491	1.21384
$178$	0	0
$179$	4.77997	0.357272	0.178636	−	0.983915i	$-0.442832\pi$
$179$	4.77997	0.357272	0.178636	+	0.983915i	$0.442832\pi$
$180$	0	0
$181$	1.29560	0.0963010	0.0481505	−	0.998840i	$-0.484667\pi$
$181$	1.29560	0.0963010	0.0481505	+	0.998840i	$0.484667\pi$
$182$	0	0
$183$	25.8652	1.91201
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.6516	1.14456
$188$	0	0
$189$	15.9060	1.15699
$190$	0	0
$191$	10.9542	0.792617	0.396308	−	0.918117i	$-0.370291\pi$
$191$	10.9542	0.792617	0.396308	+	0.918117i	$0.370291\pi$
$192$	0	0
$193$	8.00991	0.576566	0.288283	−	0.957545i	$-0.406916\pi$
$193$	8.00991	0.576566	0.288283	+	0.957545i	$0.406916\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.599631	−0.0427219	−0.0213610	−	0.999772i	$-0.506800\pi$
$197$	−0.599631	−0.0427219	−0.0213610	+	0.999772i	$0.506800\pi$
$198$	0	0
$199$	19.1818	1.35976	0.679880	−	0.733323i	$-0.262032\pi$
$199$	19.1818	1.35976	0.679880	+	0.733323i	$0.262032\pi$
$200$	0	0
$201$	−7.42632	−0.523812
$202$	0	0
$203$	−5.43364	−0.381367
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−10.1943	−0.708554
$208$	0	0
$209$	−18.4721	−1.27774
$210$	0	0
$211$	−7.36590	−0.507090	−0.253545	−	0.967324i	$-0.581597\pi$
$211$	−7.36590	−0.507090	−0.253545	+	0.967324i	$0.581597\pi$
$212$	0	0
$213$	26.7096	1.83011
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.73329	0.117663
$218$	0	0
$219$	−18.5235	−1.25170
$220$	0	0
$221$	−32.6081	−2.19346
$222$	0	0
$223$	10.3751	0.694769	0.347385	−	0.937723i	$-0.387070\pi$
$223$	10.3751	0.694769	0.347385	+	0.937723i	$0.387070\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.7360	−1.11081	−0.555405	−	0.831580i	$-0.687437\pi$
$227$	−16.7360	−1.11081	−0.555405	+	0.831580i	$0.687437\pi$
$228$	0	0
$229$	−15.4088	−1.01824	−0.509122	−	0.860695i	$-0.670030\pi$
$229$	−15.4088	−1.01824	−0.509122	+	0.860695i	$0.670030\pi$
$230$	0	0
$231$	−10.7893	−0.709887
$232$	0	0
$233$	14.6411	0.959169	0.479585	−	0.877496i	$-0.340787\pi$
$233$	14.6411	0.959169	0.479585	+	0.877496i	$0.340787\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	51.3215	3.33369
$238$	0	0
$239$	−3.70208	−0.239468	−0.119734	−	0.992806i	$-0.538204\pi$
$239$	−3.70208	−0.239468	−0.119734	+	0.992806i	$0.538204\pi$
$240$	0	0
$241$	20.2492	1.30437	0.652183	−	0.758061i	$-0.273853\pi$
$241$	20.2492	1.30437	0.652183	+	0.758061i	$0.273853\pi$
$242$	0	0
$243$	−19.9478	−1.27965
$244$	0	0
$245$	0	0
$246$	0	0
$247$	38.4844	2.44870
$248$	0	0
$249$	−2.93346	−0.185900
$250$	0	0
$251$	4.34843	0.274470	0.137235	−	0.990538i	$-0.456178\pi$
$251$	4.34843	0.274470	0.137235	+	0.990538i	$0.456178\pi$
$252$	0	0
$253$	3.91541	0.246160
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.4721	−0.902747	−0.451374	−	0.892335i	$-0.649066\pi$
$257$	−14.4721	−0.902747	−0.451374	+	0.892335i	$0.649066\pi$
$258$	0	0
$259$	−3.20160	−0.198938
$260$	0	0
$261$	29.1331	1.80329
$262$	0	0
$263$	−6.91758	−0.426556	−0.213278	−	0.976992i	$-0.568414\pi$
$263$	−6.91758	−0.426556	−0.213278	+	0.976992i	$0.568414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.59235	0.0974502
$268$	0	0
$269$	−14.9443	−0.911168	−0.455584	−	0.890193i	$-0.650570\pi$
$269$	−14.9443	−0.911168	−0.455584	+	0.890193i	$0.650570\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	22.4783	1.36045
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.8366	−1.37212	−0.686059	−	0.727546i	$-0.740661\pi$
$277$	−22.8366	−1.37212	−0.686059	+	0.727546i	$0.740661\pi$
$278$	0	0
$279$	−9.29323	−0.556371
$280$	0	0
$281$	−18.0397	−1.07616	−0.538078	−	0.842895i	$-0.680849\pi$
$281$	−18.0397	−1.07616	−0.538078	+	0.842895i	$0.680849\pi$
$282$	0	0
$283$	16.8305	1.00047	0.500234	−	0.865890i	$-0.333247\pi$
$283$	16.8305	1.00047	0.500234	+	0.865890i	$0.333247\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.5360	0.917064
$288$	0	0
$289$	17.7196	1.04233
$290$	0	0
$291$	26.3706	1.54587
$292$	0	0
$293$	−20.9504	−1.22394	−0.611968	−	0.790883i	$-0.709622\pi$
$293$	−20.9504	−1.22394	−0.611968	+	0.790883i	$0.709622\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	32.7549	1.90063
$298$	0	0
$299$	−8.15726	−0.471747
$300$	0	0
$301$	6.64166	0.382819
$302$	0	0
$303$	−37.7636	−2.16946
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.08916	0.461673	0.230837	−	0.972993i	$-0.425854\pi$
$307$	8.08916	0.461673	0.230837	+	0.972993i	$0.425854\pi$
$308$	0	0
$309$	−53.7776	−3.05930
$310$	0	0
$311$	1.14822	0.0651097	0.0325549	−	0.999470i	$-0.489636\pi$
$311$	1.14822	0.0651097	0.0325549	+	0.999470i	$0.489636\pi$
$312$	0	0
$313$	11.0680	0.625599	0.312800	−	0.949819i	$-0.398733\pi$
$313$	11.0680	0.625599	0.312800	+	0.949819i	$0.398733\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.60576	−0.483347	−0.241674	−	0.970358i	$-0.577696\pi$
$317$	−8.60576	−0.483347	−0.241674	+	0.970358i	$0.577696\pi$
$318$	0	0
$319$	−11.1893	−0.626483
$320$	0	0
$321$	−18.0902	−1.00969
$322$	0	0
$323$	−40.9763	−2.27999
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.24409	−0.455899
$328$	0	0
$329$	−3.92297	−0.216280
$330$	0	0
$331$	−14.7502	−0.810746	−0.405373	−	0.914151i	$-0.632858\pi$
$331$	−14.7502	−0.810746	−0.405373	+	0.914151i	$0.632858\pi$
$332$	0	0
$333$	17.1657	0.940677
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.0901	−1.25780	−0.628900	−	0.777486i	$-0.716495\pi$
$337$	−23.0901	−1.25780	−0.628900	+	0.777486i	$0.716495\pi$
$338$	0	0
$339$	25.8891	1.40610
$340$	0	0
$341$	3.56932	0.193290
$342$	0	0
$343$	15.9124	0.859191
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.5359	−0.565596	−0.282798	−	0.959180i	$-0.591263\pi$
$347$	−10.5359	−0.565596	−0.282798	+	0.959180i	$0.591263\pi$
$348$	0	0
$349$	5.10477	0.273252	0.136626	−	0.990623i	$-0.456374\pi$
$349$	5.10477	0.273252	0.136626	+	0.990623i	$0.456374\pi$
$350$	0	0
$351$	−68.2407	−3.64242
$352$	0	0
$353$	−28.5075	−1.51730	−0.758650	−	0.651499i	$-0.774141\pi$
$353$	−28.5075	−1.51730	−0.758650	+	0.651499i	$0.774141\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−23.9338	−1.26671
$358$	0	0
$359$	−15.8480	−0.836423	−0.418211	−	0.908350i	$-0.637343\pi$
$359$	−15.8480	−0.836423	−0.418211	+	0.908350i	$0.637343\pi$
$360$	0	0
$361$	29.3607	1.54530
$362$	0	0
$363$	12.4204	0.651900
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.68558	−0.192386	−0.0961929	−	0.995363i	$-0.530667\pi$
$367$	−3.68558	−0.192386	−0.0961929	+	0.995363i	$0.530667\pi$
$368$	0	0
$369$	−83.2982	−4.33633
$370$	0	0
$371$	13.5080	0.701302
$372$	0	0
$373$	9.89846	0.512523	0.256261	−	0.966608i	$-0.417509\pi$
$373$	9.89846	0.512523	0.256261	+	0.966608i	$0.417509\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.3116	1.20061
$378$	0	0
$379$	20.8234	1.06963	0.534814	−	0.844970i	$-0.320382\pi$
$379$	20.8234	1.06963	0.534814	+	0.844970i	$0.320382\pi$
$380$	0	0
$381$	27.5325	1.41053
$382$	0	0
$383$	−35.0734	−1.79217	−0.896083	−	0.443887i	$-0.853599\pi$
$383$	−35.0734	−1.79217	−0.896083	+	0.443887i	$0.853599\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−35.6100	−1.81016
$388$	0	0
$389$	6.05427	0.306964	0.153482	−	0.988151i	$-0.450951\pi$
$389$	6.05427	0.306964	0.153482	+	0.988151i	$0.450951\pi$
$390$	0	0
$391$	8.68547	0.439243
$392$	0	0
$393$	13.5340	0.682699
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.2894	−0.767355	−0.383678	−	0.923467i	$-0.625343\pi$
$397$	−15.2894	−0.767355	−0.383678	+	0.923467i	$0.625343\pi$
$398$	0	0
$399$	28.2469	1.41411
$400$	0	0
$401$	−3.29881	−0.164735	−0.0823674	−	0.996602i	$-0.526248\pi$
$401$	−3.29881	−0.164735	−0.0823674	+	0.996602i	$0.526248\pi$
$402$	0	0
$403$	−7.43623	−0.370425
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.59297	−0.326801
$408$	0	0
$409$	−28.2492	−1.39683	−0.698417	−	0.715691i	$-0.746112\pi$
$409$	−28.2492	−1.39683	−0.698417	+	0.715691i	$0.746112\pi$
$410$	0	0
$411$	35.5812	1.75509
$412$	0	0
$413$	6.61514	0.325510
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.1856	0.841581
$418$	0	0
$419$	14.8404	0.725000	0.362500	−	0.931984i	$-0.381923\pi$
$419$	14.8404	0.725000	0.362500	+	0.931984i	$0.381923\pi$
$420$	0	0
$421$	−3.33471	−0.162524	−0.0812620	−	0.996693i	$-0.525895\pi$
$421$	−3.33471	−0.162524	−0.0812620	+	0.996693i	$0.525895\pi$
$422$	0	0
$423$	21.0334	1.02268
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.5951	0.512733
$428$	0	0
$429$	46.2889	2.23485
$430$	0	0
$431$	−14.1865	−0.683338	−0.341669	−	0.939820i	$-0.610992\pi$
$431$	−14.1865	−0.683338	−0.341669	+	0.939820i	$0.610992\pi$
$432$	0	0
$433$	27.3055	1.31222	0.656109	−	0.754666i	$-0.272201\pi$
$433$	27.3055	1.31222	0.656109	+	0.754666i	$0.272201\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.2507	−0.490356
$438$	0	0
$439$	−5.76250	−0.275029	−0.137514	−	0.990500i	$-0.543911\pi$
$439$	−5.76250	−0.275029	−0.137514	+	0.990500i	$0.543911\pi$
$440$	0	0
$441$	−36.9046	−1.75736
$442$	0	0
$443$	18.2762	0.868327	0.434163	−	0.900834i	$-0.357044\pi$
$443$	18.2762	0.868327	0.434163	+	0.900834i	$0.357044\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−33.6498	−1.59158
$448$	0	0
$449$	18.7610	0.885387	0.442693	−	0.896673i	$-0.354023\pi$
$449$	18.7610	0.885387	0.442693	+	0.896673i	$0.354023\pi$
$450$	0	0
$451$	31.9930	1.50649
$452$	0	0
$453$	69.1476	3.24884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	14.8526	0.694777	0.347389	−	0.937721i	$-0.387068\pi$
$457$	14.8526	0.694777	0.347389	+	0.937721i	$0.387068\pi$
$458$	0	0
$459$	72.6595	3.39145
$460$	0	0
$461$	−11.2285	−0.522962	−0.261481	−	0.965209i	$-0.584211\pi$
$461$	−11.2285	−0.522962	−0.261481	+	0.965209i	$0.584211\pi$
$462$	0	0
$463$	−16.5420	−0.768772	−0.384386	−	0.923172i	$-0.625587\pi$
$463$	−16.5420	−0.768772	−0.384386	+	0.923172i	$0.625587\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.5934	1.41569	0.707846	−	0.706366i	$-0.249667\pi$
$467$	30.5934	1.41569	0.707846	+	0.706366i	$0.249667\pi$
$468$	0	0
$469$	−3.04203	−0.140468
$470$	0	0
$471$	5.95887	0.274570
$472$	0	0
$473$	13.6770	0.628869
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−72.4248	−3.31611
$478$	0	0
$479$	−7.63965	−0.349065	−0.174532	−	0.984651i	$-0.555841\pi$
$479$	−7.63965	−0.349065	−0.174532	+	0.984651i	$0.555841\pi$
$480$	0	0
$481$	13.7356	0.626291
$482$	0	0
$483$	−5.98729	−0.272431
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−21.9444	−0.994397	−0.497199	−	0.867637i	$-0.665638\pi$
$487$	−21.9444	−0.994397	−0.497199	+	0.867637i	$0.665638\pi$
$488$	0	0
$489$	48.4074	2.18906
$490$	0	0
$491$	34.9664	1.57801	0.789007	−	0.614385i	$-0.210596\pi$
$491$	34.9664	1.57801	0.789007	+	0.614385i	$0.210596\pi$
$492$	0	0
$493$	−24.8211	−1.11789
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.9410	0.490772
$498$	0	0
$499$	19.7654	0.884819	0.442410	−	0.896813i	$-0.354124\pi$
$499$	19.7654	0.884819	0.442410	+	0.896813i	$0.354124\pi$
$500$	0	0
$501$	−65.0986	−2.90839
$502$	0	0
$503$	−9.39757	−0.419017	−0.209509	−	0.977807i	$-0.567186\pi$
$503$	−9.39757	−0.419017	−0.209509	+	0.977807i	$0.567186\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−55.5006	−2.46487
$508$	0	0
$509$	17.6638	0.782935	0.391468	−	0.920192i	$-0.371967\pi$
$509$	17.6638	0.782935	0.391468	+	0.920192i	$0.371967\pi$
$510$	0	0
$511$	−7.58775	−0.335662
$512$	0	0
$513$	−85.7534	−3.78610
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8.07847	−0.355291
$518$	0	0
$519$	35.2837	1.54878
$520$	0	0
$521$	21.8295	0.956370	0.478185	−	0.878259i	$-0.341295\pi$
$521$	21.8295	0.956370	0.478185	+	0.878259i	$0.341295\pi$
$522$	0	0
$523$	31.3062	1.36893	0.684463	−	0.729048i	$-0.260037\pi$
$523$	31.3062	1.36893	0.684463	+	0.729048i	$0.260037\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.91775	0.344902
$528$	0	0
$529$	−20.8272	−0.905532
$530$	0	0
$531$	−35.4678	−1.53917
$532$	0	0
$533$	−66.6533	−2.88708
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−15.0519	−0.649539
$538$	0	0
$539$	14.1742	0.610527
$540$	0	0
$541$	32.1681	1.38301	0.691507	−	0.722370i	$-0.256947\pi$
$541$	32.1681	1.38301	0.691507	+	0.722370i	$0.256947\pi$
$542$	0	0
$543$	−4.07979	−0.175080
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−17.9523	−0.767585	−0.383792	−	0.923419i	$-0.625382\pi$
$547$	−17.9523	−0.767585	−0.383792	+	0.923419i	$0.625382\pi$
$548$	0	0
$549$	−56.8068	−2.42446
$550$	0	0
$551$	29.2941	1.24797
$552$	0	0
$553$	21.0227	0.893978
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.92153	0.166161	0.0830804	−	0.996543i	$-0.473524\pi$
$557$	3.92153	0.166161	0.0830804	+	0.996543i	$0.473524\pi$
$558$	0	0
$559$	−28.4943	−1.20518
$560$	0	0
$561$	−49.2862	−2.08086
$562$	0	0
$563$	7.39712	0.311751	0.155876	−	0.987777i	$-0.450180\pi$
$563$	7.39712	0.311751	0.155876	+	0.987777i	$0.450180\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−23.3248	−0.979549
$568$	0	0
$569$	23.3277	0.977947	0.488973	−	0.872299i	$-0.337371\pi$
$569$	23.3277	0.977947	0.488973	+	0.872299i	$0.337371\pi$
$570$	0	0
$571$	−37.2039	−1.55694	−0.778468	−	0.627684i	$-0.784003\pi$
$571$	−37.2039	−1.55694	−0.778468	+	0.627684i	$0.784003\pi$
$572$	0	0
$573$	−34.4943	−1.44102
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.8045	0.990994	0.495497	−	0.868610i	$-0.334986\pi$
$577$	23.8045	0.990994	0.495497	+	0.868610i	$0.334986\pi$
$578$	0	0
$579$	−25.2229	−1.04823
$580$	0	0
$581$	−1.20163	−0.0498519
$582$	0	0
$583$	27.8167	1.15205
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.6539	0.563557	0.281779	−	0.959479i	$-0.409076\pi$
$587$	13.6539	0.563557	0.281779	+	0.959479i	$0.409076\pi$
$588$	0	0
$589$	−9.34460	−0.385038
$590$	0	0
$591$	1.88821	0.0776707
$592$	0	0
$593$	−12.3904	−0.508813	−0.254407	−	0.967097i	$-0.581880\pi$
$593$	−12.3904	−0.508813	−0.254407	+	0.967097i	$0.581880\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−60.4027	−2.47212
$598$	0	0
$599$	21.2422	0.867933	0.433966	−	0.900929i	$-0.357114\pi$
$599$	21.2422	0.867933	0.433966	+	0.900929i	$0.357114\pi$
$600$	0	0
$601$	27.9253	1.13910	0.569548	−	0.821958i	$-0.307118\pi$
$601$	27.9253	1.13910	0.569548	+	0.821958i	$0.307118\pi$
$602$	0	0
$603$	16.3102	0.664202
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.6484	0.432204	0.216102	−	0.976371i	$-0.430666\pi$
$607$	10.6484	0.432204	0.216102	+	0.976371i	$0.430666\pi$
$608$	0	0
$609$	17.1103	0.693345
$610$	0	0
$611$	16.8305	0.680888
$612$	0	0
$613$	−21.7555	−0.878695	−0.439347	−	0.898317i	$-0.644790\pi$
$613$	−21.7555	−0.878695	−0.439347	+	0.898317i	$0.644790\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.3645	−1.30294	−0.651472	−	0.758673i	$-0.725848\pi$
$617$	−32.3645	−1.30294	−0.651472	+	0.758673i	$0.725848\pi$
$618$	0	0
$619$	26.7388	1.07472	0.537362	−	0.843351i	$-0.319421\pi$
$619$	26.7388	1.07472	0.537362	+	0.843351i	$0.319421\pi$
$620$	0	0
$621$	18.1765	0.729399
$622$	0	0
$623$	0.652271	0.0261327
$624$	0	0
$625$	0	0
$626$	0	0
$627$	58.1680	2.32301
$628$	0	0
$629$	−14.6250	−0.583139
$630$	0	0
$631$	−35.1313	−1.39855	−0.699277	−	0.714851i	$-0.746495\pi$
$631$	−35.1313	−1.39855	−0.699277	+	0.714851i	$0.746495\pi$
$632$	0	0
$633$	23.1949	0.921916
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−29.5302	−1.17003
$638$	0	0
$639$	−58.6615	−2.32061
$640$	0	0
$641$	5.37493	0.212297	0.106148	−	0.994350i	$-0.466148\pi$
$641$	5.37493	0.212297	0.106148	+	0.994350i	$0.466148\pi$
$642$	0	0
$643$	2.14958	0.0847713	0.0423856	−	0.999101i	$-0.486504\pi$
$643$	2.14958	0.0847713	0.0423856	+	0.999101i	$0.486504\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.9107	−1.56905	−0.784526	−	0.620096i	$-0.787094\pi$
$647$	−39.9107	−1.56905	−0.784526	+	0.620096i	$0.787094\pi$
$648$	0	0
$649$	13.6224	0.534725
$650$	0	0
$651$	−5.45807	−0.213919
$652$	0	0
$653$	−22.3291	−0.873807	−0.436903	−	0.899508i	$-0.643925\pi$
$653$	−22.3291	−0.873807	−0.436903	+	0.899508i	$0.643925\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	40.6825	1.58718
$658$	0	0
$659$	−7.10710	−0.276853	−0.138427	−	0.990373i	$-0.544204\pi$
$659$	−7.10710	−0.276853	−0.138427	+	0.990373i	$0.544204\pi$
$660$	0	0
$661$	41.4065	1.61053	0.805263	−	0.592918i	$-0.202024\pi$
$661$	41.4065	1.61053	0.805263	+	0.592918i	$0.202024\pi$
$662$	0	0
$663$	102.682	3.98782
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.20926	−0.240424
$668$	0	0
$669$	−32.6708	−1.26313
$670$	0	0
$671$	21.8182	0.842283
$672$	0	0
$673$	33.6992	1.29901	0.649503	−	0.760359i	$-0.274977\pi$
$673$	33.6992	1.29901	0.649503	+	0.760359i	$0.274977\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.8847	−1.18700	−0.593498	−	0.804835i	$-0.702254\pi$
$677$	−30.8847	−1.18700	−0.593498	+	0.804835i	$0.702254\pi$
$678$	0	0
$679$	10.8021	0.414548
$680$	0	0
$681$	52.7012	2.01951
$682$	0	0
$683$	13.2315	0.506291	0.253145	−	0.967428i	$-0.418535\pi$
$683$	13.2315	0.506291	0.253145	+	0.967428i	$0.418535\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	48.5218	1.85122
$688$	0	0
$689$	−57.9527	−2.20782
$690$	0	0
$691$	−18.9854	−0.722239	−0.361119	−	0.932520i	$-0.617605\pi$
$691$	−18.9854	−0.722239	−0.361119	+	0.932520i	$0.617605\pi$
$692$	0	0
$693$	23.6963	0.900147
$694$	0	0
$695$	0	0
$696$	0	0
$697$	70.9693	2.68815
$698$	0	0
$699$	−46.1042	−1.74382
$700$	0	0
$701$	31.1917	1.17809	0.589047	−	0.808099i	$-0.299503\pi$
$701$	31.1917	1.17809	0.589047	+	0.808099i	$0.299503\pi$
$702$	0	0
$703$	17.2606	0.650997
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.4690	−0.581773
$708$	0	0
$709$	−2.56556	−0.0963516	−0.0481758	−	0.998839i	$-0.515341\pi$
$709$	−2.56556	−0.0963516	−0.0481758	+	0.998839i	$0.515341\pi$
$710$	0	0
$711$	−112.716	−4.22717
$712$	0	0
$713$	1.98071	0.0741781
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.6577	0.435365
$718$	0	0
$719$	49.6974	1.85340	0.926700	−	0.375803i	$-0.122633\pi$
$719$	49.6974	1.85340	0.926700	+	0.375803i	$0.122633\pi$
$720$	0	0
$721$	−22.0288	−0.820396
$722$	0	0
$723$	−63.7640	−2.37141
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.6999	−1.47239	−0.736194	−	0.676771i	$-0.763379\pi$
$727$	−39.6999	−1.47239	−0.736194	+	0.676771i	$0.763379\pi$
$728$	0	0
$729$	8.56700	0.317296
$730$	0	0
$731$	30.3394	1.12214
$732$	0	0
$733$	−38.8710	−1.43573	−0.717867	−	0.696180i	$-0.754881\pi$
$733$	−38.8710	−1.43573	−0.717867	+	0.696180i	$0.754881\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.26437	−0.230751
$738$	0	0
$739$	−42.4537	−1.56168	−0.780842	−	0.624728i	$-0.785210\pi$
$739$	−42.4537	−1.56168	−0.780842	+	0.624728i	$0.785210\pi$
$740$	0	0
$741$	−121.186	−4.45187
$742$	0	0
$743$	11.0487	0.405337	0.202669	−	0.979247i	$-0.435039\pi$
$743$	11.0487	0.405337	0.202669	+	0.979247i	$0.435039\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.44265	0.235724
$748$	0	0
$749$	−7.41024	−0.270764
$750$	0	0
$751$	31.8993	1.16402	0.582011	−	0.813181i	$-0.302266\pi$
$751$	31.8993	1.16402	0.582011	+	0.813181i	$0.302266\pi$
$752$	0	0
$753$	−13.6930	−0.499002
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.2752	1.68190	0.840950	−	0.541113i	$-0.181997\pi$
$757$	46.2752	1.68190	0.840950	+	0.541113i	$0.181997\pi$
$758$	0	0
$759$	−12.3295	−0.447531
$760$	0	0
$761$	33.2544	1.20547	0.602736	−	0.797941i	$-0.294077\pi$
$761$	33.2544	1.20547	0.602736	+	0.797941i	$0.294077\pi$
$762$	0	0
$763$	−3.37701	−0.122256
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28.3805	−1.02476
$768$	0	0
$769$	6.81820	0.245871	0.122935	−	0.992415i	$-0.460769\pi$
$769$	6.81820	0.245871	0.122935	+	0.992415i	$0.460769\pi$
$770$	0	0
$771$	45.5722	1.64124
$772$	0	0
$773$	6.70730	0.241245	0.120622	−	0.992698i	$-0.461511\pi$
$773$	6.70730	0.241245	0.120622	+	0.992698i	$0.461511\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	10.0817	0.361680
$778$	0	0
$779$	−83.7587	−3.00097
$780$	0	0
$781$	22.5305	0.806206
$782$	0	0
$783$	−51.9444	−1.85634
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.3378	0.368502	0.184251	−	0.982879i	$-0.441014\pi$
$787$	10.3378	0.368502	0.184251	+	0.982879i	$0.441014\pi$
$788$	0	0
$789$	21.7832	0.775502
$790$	0	0
$791$	10.6049	0.377067
$792$	0	0
$793$	−45.4555	−1.61417
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.9116	0.421931	0.210966	−	0.977493i	$-0.432339\pi$
$797$	11.9116	0.421931	0.210966	+	0.977493i	$0.432339\pi$
$798$	0	0
$799$	−17.9203	−0.633974
$800$	0	0
$801$	−3.49722	−0.123568
$802$	0	0
$803$	−15.6252	−0.551403
$804$	0	0
$805$	0	0
$806$	0	0
$807$	47.0589	1.65655
$808$	0	0
$809$	−21.1138	−0.742321	−0.371160	−	0.928569i	$-0.621040\pi$
$809$	−21.1138	−0.742321	−0.371160	+	0.928569i	$0.621040\pi$
$810$	0	0
$811$	30.0925	1.05669	0.528345	−	0.849030i	$-0.322813\pi$
$811$	30.0925	1.05669	0.528345	+	0.849030i	$0.322813\pi$
$812$	0	0
$813$	50.3834	1.76702
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−35.8068	−1.25272
$818$	0	0
$819$	−49.3682	−1.72507
$820$	0	0
$821$	−29.5066	−1.02979	−0.514893	−	0.857254i	$-0.672168\pi$
$821$	−29.5066	−1.02979	−0.514893	+	0.857254i	$0.672168\pi$
$822$	0	0
$823$	−55.2261	−1.92506	−0.962531	−	0.271173i	$-0.912588\pi$
$823$	−55.2261	−1.92506	−0.962531	+	0.271173i	$0.912588\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.0873	1.35920	0.679599	−	0.733584i	$-0.262154\pi$
$827$	39.0873	1.35920	0.679599	+	0.733584i	$0.262154\pi$
$828$	0	0
$829$	25.9014	0.899591	0.449796	−	0.893132i	$-0.351497\pi$
$829$	25.9014	0.899591	0.449796	+	0.893132i	$0.351497\pi$
$830$	0	0
$831$	71.9116	2.49458
$832$	0	0
$833$	31.4424	1.08941
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.5699	0.572739
$838$	0	0
$839$	21.0467	0.726612	0.363306	−	0.931670i	$-0.381648\pi$
$839$	21.0467	0.726612	0.363306	+	0.931670i	$0.381648\pi$
$840$	0	0
$841$	−11.2553	−0.388115
$842$	0	0
$843$	56.8062	1.95651
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.08773	0.174816
$848$	0	0
$849$	−52.9985	−1.81890
$850$	0	0
$851$	−3.65861	−0.125416
$852$	0	0
$853$	41.7677	1.43010	0.715050	−	0.699074i	$-0.246404\pi$
$853$	41.7677	1.43010	0.715050	+	0.699074i	$0.246404\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.3158	0.933089	0.466545	−	0.884498i	$-0.345499\pi$
$857$	27.3158	0.933089	0.466545	+	0.884498i	$0.345499\pi$
$858$	0	0
$859$	34.7120	1.18436	0.592179	−	0.805807i	$-0.298268\pi$
$859$	34.7120	1.18436	0.592179	+	0.805807i	$0.298268\pi$
$860$	0	0
$861$	−48.9224	−1.66727
$862$	0	0
$863$	−25.8668	−0.880516	−0.440258	−	0.897871i	$-0.645113\pi$
$863$	−25.8668	−0.880516	−0.440258	+	0.897871i	$0.645113\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−55.7982	−1.89501
$868$	0	0
$869$	43.2916	1.46857
$870$	0	0
$871$	13.0510	0.442217
$872$	0	0
$873$	−57.9168	−1.96019
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21.4876	−0.725585	−0.362792	−	0.931870i	$-0.618177\pi$
$877$	−21.4876	−0.725585	−0.362792	+	0.931870i	$0.618177\pi$
$878$	0	0
$879$	65.9720	2.22518
$880$	0	0
$881$	−40.3589	−1.35973	−0.679863	−	0.733339i	$-0.737961\pi$
$881$	−40.3589	−1.35973	−0.679863	+	0.733339i	$0.737961\pi$
$882$	0	0
$883$	−13.7483	−0.462668	−0.231334	−	0.972874i	$-0.574309\pi$
$883$	−13.7483	−0.462668	−0.231334	+	0.972874i	$0.574309\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.7819	1.63793	0.818967	−	0.573840i	$-0.194547\pi$
$887$	48.7819	1.63793	0.818967	+	0.573840i	$0.194547\pi$
$888$	0	0
$889$	11.2781	0.378255
$890$	0	0
$891$	−48.0321	−1.60914
$892$	0	0
$893$	21.1497	0.707748
$894$	0	0
$895$	0	0
$896$	0	0
$897$	25.6869	0.857661
$898$	0	0
$899$	−5.66042	−0.188786
$900$	0	0
$901$	61.7053	2.05570
$902$	0	0
$903$	−20.9143	−0.695985
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−6.41682	−0.213067	−0.106534	−	0.994309i	$-0.533975\pi$
$907$	−6.41682	−0.213067	−0.106534	+	0.994309i	$0.533975\pi$
$908$	0	0
$909$	82.9390	2.75091
$910$	0	0
$911$	−51.7855	−1.71573	−0.857865	−	0.513874i	$-0.828210\pi$
$911$	−51.7855	−1.71573	−0.857865	+	0.513874i	$0.828210\pi$
$912$	0	0
$913$	−2.47448	−0.0818932
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.54390	0.183076
$918$	0	0
$919$	−8.60288	−0.283783	−0.141891	−	0.989882i	$-0.545318\pi$
$919$	−8.60288	−0.283783	−0.141891	+	0.989882i	$0.545318\pi$
$920$	0	0
$921$	−25.4725	−0.839346
$922$	0	0
$923$	−46.9396	−1.54503
$924$	0	0
$925$	0	0
$926$	0	0
$927$	118.110	3.87924
$928$	0	0
$929$	51.1491	1.67815	0.839074	−	0.544018i	$-0.183098\pi$
$929$	51.1491	1.67815	0.839074	+	0.544018i	$0.183098\pi$
$930$	0	0
$931$	−37.1086	−1.21618
$932$	0	0
$933$	−3.61571	−0.118373
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7.18882	−0.234848	−0.117424	−	0.993082i	$-0.537464\pi$
$937$	−7.18882	−0.234848	−0.117424	+	0.993082i	$0.537464\pi$
$938$	0	0
$939$	−34.8526	−1.13737
$940$	0	0
$941$	34.4003	1.12142	0.560709	−	0.828013i	$-0.310529\pi$
$941$	34.4003	1.12142	0.560709	+	0.828013i	$0.310529\pi$
$942$	0	0
$943$	17.7537	0.578141
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.77688	0.252715	0.126357	−	0.991985i	$-0.459671\pi$
$947$	7.77688	0.252715	0.126357	+	0.991985i	$0.459671\pi$
$948$	0	0
$949$	32.5532	1.05672
$950$	0	0
$951$	27.0992	0.878752
$952$	0	0
$953$	33.3446	1.08014	0.540069	−	0.841621i	$-0.318398\pi$
$953$	33.3446	1.08014	0.540069	+	0.841621i	$0.318398\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	35.2348	1.13898
$958$	0	0
$959$	14.5751	0.470654
$960$	0	0
$961$	−29.1944	−0.941754
$962$	0	0
$963$	39.7308	1.28031
$964$	0	0
$965$	0	0
$966$	0	0
$967$	56.0591	1.80274	0.901370	−	0.433051i	$-0.142563\pi$
$967$	56.0591	1.80274	0.901370	+	0.433051i	$0.142563\pi$
$968$	0	0
$969$	129.033	4.14513
$970$	0	0
$971$	−26.3394	−0.845271	−0.422635	−	0.906300i	$-0.638895\pi$
$971$	−26.3394	−0.845271	−0.422635	+	0.906300i	$0.638895\pi$
$972$	0	0
$973$	7.03969	0.225682
$974$	0	0
$975$	0	0
$976$	0	0
$977$	57.8036	1.84930	0.924650	−	0.380818i	$-0.124358\pi$
$977$	57.8036	1.84930	0.924650	+	0.380818i	$0.124358\pi$
$978$	0	0
$979$	1.34320	0.0429290
$980$	0	0
$981$	18.1062	0.578087
$982$	0	0
$983$	25.5594	0.815218	0.407609	−	0.913156i	$-0.366363\pi$
$983$	25.5594	0.815218	0.407609	+	0.913156i	$0.366363\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	12.3533	0.393209
$988$	0	0
$989$	7.58972	0.241339
$990$	0	0
$991$	−21.0972	−0.670174	−0.335087	−	0.942187i	$-0.608766\pi$
$991$	−21.0972	−0.670174	−0.335087	+	0.942187i	$0.608766\pi$
$992$	0	0
$993$	46.4479	1.47398
$994$	0	0
$995$	0	0
$996$	0	0
$997$	35.4523	1.12278	0.561392	−	0.827550i	$-0.310266\pi$
$997$	35.4523	1.12278	0.561392	+	0.827550i	$0.310266\pi$
$998$	0	0
$999$	−30.6066	−0.968351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1000.2.a.e.1.1	✓	4
3.2	odd	2		9000.2.a.r.1.3		4
4.3	odd	2		2000.2.a.r.1.4		4
5.2	odd	4		1000.2.c.d.249.8		8
5.3	odd	4		1000.2.c.d.249.1		8
5.4	even	2		1000.2.a.h.1.4	yes	4
8.3	odd	2		8000.2.a.bb.1.1		4
8.5	even	2		8000.2.a.br.1.4		4
15.14	odd	2		9000.2.a.ba.1.2		4
20.3	even	4		2000.2.c.j.1249.8		8
20.7	even	4		2000.2.c.j.1249.1		8
20.19	odd	2		2000.2.a.m.1.1		4
40.19	odd	2		8000.2.a.bq.1.4		4
40.29	even	2		8000.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.e.1.1	✓	4	1.1	even	1	trivial
1000.2.a.h.1.4	yes	4	5.4	even	2
1000.2.c.d.249.1		8	5.3	odd	4
1000.2.c.d.249.8		8	5.2	odd	4
2000.2.a.m.1.1		4	20.19	odd	2
2000.2.a.r.1.4		4	4.3	odd	2
2000.2.c.j.1249.1		8	20.7	even	4
2000.2.c.j.1249.8		8	20.3	even	4
8000.2.a.ba.1.1		4	40.29	even	2
8000.2.a.bb.1.1		4	8.3	odd	2
8000.2.a.bq.1.4		4	40.19	odd	2
8000.2.a.br.1.4		4	8.5	even	2
9000.2.a.r.1.3		4	3.2	odd	2
9000.2.a.ba.1.2		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1000 = 2^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1000.a (trivial)

\(f(q)\)	\(=\)	\(q-3.14896 q^{3} -1.28990 q^{7} +6.91596 q^{9} +O(q^{10})\)	\(q-3.14896 q^{3} -1.28990 q^{7} +6.91596 q^{9} -2.65626 q^{11} +5.53399 q^{13} -5.89233 q^{17} +6.95418 q^{19} +4.06185 q^{21} -1.47403 q^{23} -12.3312 q^{27} +4.21244 q^{29} -1.34374 q^{31} +8.36447 q^{33} +2.48205 q^{37} -17.4263 q^{39} -12.0444 q^{41} -5.14896 q^{43} +3.04129 q^{47} -5.33615 q^{49} +18.5547 q^{51} -10.4721 q^{53} -21.8985 q^{57} -5.12840 q^{59} -8.21388 q^{61} -8.92091 q^{63} +2.35834 q^{67} +4.64166 q^{69} -8.48205 q^{71} +5.88242 q^{73} +3.42632 q^{77} -16.2979 q^{79} +18.0826 q^{81} +0.931563 q^{83} -13.2648 q^{87} -0.505675 q^{89} -7.13831 q^{91} +4.23138 q^{93} -8.37438 q^{97} -18.3706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9	\( 4 q - 4 q^{3} - 6 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{21} - 14 q^{23} - 22 q^{27} + 10 q^{29} - 16 q^{31} - 4 q^{33} - 24 q^{39} + 2 q^{41} - 12 q^{43} - 16 q^{47} + 2 q^{49} - 24 q^{53} - 24 q^{57} + 8 q^{59} + 6 q^{61} - 18 q^{63} + 16 q^{67} + 12 q^{69} - 24 q^{71} - 4 q^{73} - 32 q^{77} - 48 q^{79} + 16 q^{81} - 2 q^{83} + 22 q^{87} + 10 q^{89} - 8 q^{91} + 20 q^{93} - 4 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 6 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^21 - 14 * q^23 - 22 * q^27 + 10 * q^29 - 16 * q^31 - 4 * q^33 - 24 * q^39 + 2 * q^41 - 12 * q^43 - 16 * q^47 + 2 * q^49 - 24 * q^53 - 24 * q^57 + 8 * q^59 + 6 * q^61 - 18 * q^63 + 16 * q^67 + 12 * q^69 - 24 * q^71 - 4 * q^73 - 32 * q^77 - 48 * q^79 + 16 * q^81 - 2 * q^83 + 22 * q^87 + 10 * q^89 - 8 * q^91 + 20 * q^93 - 4 * q^97 + 8 * q^99

Introduction

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