LMFDB - Embedded newform 4008.2.a.l.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.93317$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.93317 q^{5} +1.35675 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.93317 q^{5} +1.35675 q^{7} +1.00000 q^{9} -0.0643015 q^{11} -2.07209 q^{13} +2.93317 q^{15} -1.73736 q^{17} +7.35449 q^{19} +1.35675 q^{21} -0.641526 q^{23} +3.60351 q^{25} +1.00000 q^{27} +3.65476 q^{29} -3.92456 q^{31} -0.0643015 q^{33} +3.97960 q^{35} +8.28523 q^{37} -2.07209 q^{39} +1.55401 q^{41} -2.77619 q^{43} +2.93317 q^{45} -4.51709 q^{47} -5.15922 q^{49} -1.73736 q^{51} -1.55065 q^{53} -0.188608 q^{55} +7.35449 q^{57} +11.0736 q^{59} +13.4007 q^{61} +1.35675 q^{63} -6.07781 q^{65} +4.50770 q^{67} -0.641526 q^{69} +10.2267 q^{71} -6.16638 q^{73} +3.60351 q^{75} -0.0872414 q^{77} +2.51122 q^{79} +1.00000 q^{81} +1.74485 q^{83} -5.09598 q^{85} +3.65476 q^{87} -17.4735 q^{89} -2.81132 q^{91} -3.92456 q^{93} +21.5720 q^{95} +17.6851 q^{97} -0.0643015 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.93317	1.31175	0.655877	−	0.754867i	$-0.272299\pi$
$5$	2.93317	1.31175	0.655877	+	0.754867i	$0.272299\pi$
$6$	0	0
$7$	1.35675	0.512805	0.256402	−	0.966570i	$-0.417463\pi$
$7$	1.35675	0.512805	0.256402	+	0.966570i	$0.417463\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.0643015	−0.0193876	−0.00969382	−	0.999953i	$-0.503086\pi$
$11$	−0.0643015	−0.0193876	−0.00969382	+	0.999953i	$0.503086\pi$
$12$	0	0
$13$	−2.07209	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$13$	−2.07209	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$14$	0	0
$15$	2.93317	0.757342
$16$	0	0
$17$	−1.73736	−0.421372	−0.210686	−	0.977554i	$-0.567570\pi$
$17$	−1.73736	−0.421372	−0.210686	+	0.977554i	$0.567570\pi$
$18$	0	0
$19$	7.35449	1.68724	0.843618	−	0.536943i	$-0.180421\pi$
$19$	7.35449	1.68724	0.843618	+	0.536943i	$0.180421\pi$
$20$	0	0
$21$	1.35675	0.296068
$22$	0	0
$23$	−0.641526	−0.133767	−0.0668837	−	0.997761i	$-0.521306\pi$
$23$	−0.641526	−0.133767	−0.0668837	+	0.997761i	$0.521306\pi$
$24$	0	0
$25$	3.60351	0.720701
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.65476	0.678672	0.339336	−	0.940665i	$-0.389798\pi$
$29$	3.65476	0.678672	0.339336	+	0.940665i	$0.389798\pi$
$30$	0	0
$31$	−3.92456	−0.704871	−0.352436	−	0.935836i	$-0.614646\pi$
$31$	−3.92456	−0.704871	−0.352436	+	0.935836i	$0.614646\pi$
$32$	0	0
$33$	−0.0643015	−0.0111935
$34$	0	0
$35$	3.97960	0.672674
$36$	0	0
$37$	8.28523	1.36208	0.681041	−	0.732245i	$-0.261527\pi$
$37$	8.28523	1.36208	0.681041	+	0.732245i	$0.261527\pi$
$38$	0	0
$39$	−2.07209	−0.331801
$40$	0	0
$41$	1.55401	0.242696	0.121348	−	0.992610i	$-0.461278\pi$
$41$	1.55401	0.242696	0.121348	+	0.992610i	$0.461278\pi$
$42$	0	0
$43$	−2.77619	−0.423364	−0.211682	−	0.977339i	$-0.567894\pi$
$43$	−2.77619	−0.423364	−0.211682	+	0.977339i	$0.567894\pi$
$44$	0	0
$45$	2.93317	0.437252
$46$	0	0
$47$	−4.51709	−0.658886	−0.329443	−	0.944176i	$-0.606861\pi$
$47$	−4.51709	−0.658886	−0.329443	+	0.944176i	$0.606861\pi$
$48$	0	0
$49$	−5.15922	−0.737031
$50$	0	0
$51$	−1.73736	−0.243279
$52$	0	0
$53$	−1.55065	−0.212998	−0.106499	−	0.994313i	$-0.533964\pi$
$53$	−1.55065	−0.212998	−0.106499	+	0.994313i	$0.533964\pi$
$54$	0	0
$55$	−0.188608	−0.0254318
$56$	0	0
$57$	7.35449	0.974127
$58$	0	0
$59$	11.0736	1.44165	0.720827	−	0.693115i	$-0.243762\pi$
$59$	11.0736	1.44165	0.720827	+	0.693115i	$0.243762\pi$
$60$	0	0
$61$	13.4007	1.71578	0.857892	−	0.513830i	$-0.171774\pi$
$61$	13.4007	1.71578	0.857892	+	0.513830i	$0.171774\pi$
$62$	0	0
$63$	1.35675	0.170935
$64$	0	0
$65$	−6.07781	−0.753860
$66$	0	0
$67$	4.50770	0.550703	0.275352	−	0.961344i	$-0.411206\pi$
$67$	4.50770	0.550703	0.275352	+	0.961344i	$0.411206\pi$
$68$	0	0
$69$	−0.641526	−0.0772307
$70$	0	0
$71$	10.2267	1.21369	0.606843	−	0.794822i	$-0.292436\pi$
$71$	10.2267	1.21369	0.606843	+	0.794822i	$0.292436\pi$
$72$	0	0
$73$	−6.16638	−0.721720	−0.360860	−	0.932620i	$-0.617517\pi$
$73$	−6.16638	−0.721720	−0.360860	+	0.932620i	$0.617517\pi$
$74$	0	0
$75$	3.60351	0.416097
$76$	0	0
$77$	−0.0872414	−0.00994208
$78$	0	0
$79$	2.51122	0.282534	0.141267	−	0.989972i	$-0.454882\pi$
$79$	2.51122	0.282534	0.141267	+	0.989972i	$0.454882\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.74485	0.191523	0.0957613	−	0.995404i	$-0.469471\pi$
$83$	1.74485	0.191523	0.0957613	+	0.995404i	$0.469471\pi$
$84$	0	0
$85$	−5.09598	−0.552736
$86$	0	0
$87$	3.65476	0.391831
$88$	0	0
$89$	−17.4735	−1.85219	−0.926095	−	0.377291i	$-0.876856\pi$
$89$	−17.4735	−1.85219	−0.926095	+	0.377291i	$0.876856\pi$
$90$	0	0
$91$	−2.81132	−0.294707
$92$	0	0
$93$	−3.92456	−0.406958
$94$	0	0
$95$	21.5720	2.21324
$96$	0	0
$97$	17.6851	1.79565	0.897824	−	0.440354i	$-0.145147\pi$
$97$	17.6851	1.79565	0.897824	+	0.440354i	$0.145147\pi$
$98$	0	0
$99$	−0.0643015	−0.00646255
$100$	0	0
$101$	10.0244	0.997464	0.498732	−	0.866756i	$-0.333799\pi$
$101$	10.0244	0.997464	0.498732	+	0.866756i	$0.333799\pi$
$102$	0	0
$103$	6.08730	0.599800	0.299900	−	0.953971i	$-0.403047\pi$
$103$	6.08730	0.599800	0.299900	+	0.953971i	$0.403047\pi$
$104$	0	0
$105$	3.97960	0.388369
$106$	0	0
$107$	12.5950	1.21760	0.608801	−	0.793323i	$-0.291651\pi$
$107$	12.5950	1.21760	0.608801	+	0.793323i	$0.291651\pi$
$108$	0	0
$109$	−17.0536	−1.63343	−0.816717	−	0.577038i	$-0.804208\pi$
$109$	−17.0536	−1.63343	−0.816717	+	0.577038i	$0.804208\pi$
$110$	0	0
$111$	8.28523	0.786399
$112$	0	0
$113$	−11.3676	−1.06937	−0.534686	−	0.845051i	$-0.679570\pi$
$113$	−11.3676	−1.06937	−0.534686	+	0.845051i	$0.679570\pi$
$114$	0	0
$115$	−1.88171	−0.175470
$116$	0	0
$117$	−2.07209	−0.191565
$118$	0	0
$119$	−2.35717	−0.216081
$120$	0	0
$121$	−10.9959	−0.999624
$122$	0	0
$123$	1.55401	0.140120
$124$	0	0
$125$	−4.09616	−0.366372
$126$	0	0
$127$	4.85124	0.430478	0.215239	−	0.976561i	$-0.430947\pi$
$127$	4.85124	0.430478	0.215239	+	0.976561i	$0.430947\pi$
$128$	0	0
$129$	−2.77619	−0.244430
$130$	0	0
$131$	−13.1490	−1.14884	−0.574419	−	0.818562i	$-0.694772\pi$
$131$	−13.1490	−1.14884	−0.574419	+	0.818562i	$0.694772\pi$
$132$	0	0
$133$	9.97824	0.865223
$134$	0	0
$135$	2.93317	0.252447
$136$	0	0
$137$	9.39248	0.802454	0.401227	−	0.915979i	$-0.368584\pi$
$137$	9.39248	0.802454	0.401227	+	0.915979i	$0.368584\pi$
$138$	0	0
$139$	−6.71091	−0.569212	−0.284606	−	0.958645i	$-0.591863\pi$
$139$	−6.71091	−0.569212	−0.284606	+	0.958645i	$0.591863\pi$
$140$	0	0
$141$	−4.51709	−0.380408
$142$	0	0
$143$	0.133239	0.0111420
$144$	0	0
$145$	10.7200	0.890251
$146$	0	0
$147$	−5.15922	−0.425525
$148$	0	0
$149$	−6.09023	−0.498931	−0.249466	−	0.968384i	$-0.580255\pi$
$149$	−6.09023	−0.498931	−0.249466	+	0.968384i	$0.580255\pi$
$150$	0	0
$151$	−13.3009	−1.08241	−0.541206	−	0.840890i	$-0.682032\pi$
$151$	−13.3009	−1.08241	−0.541206	+	0.840890i	$0.682032\pi$
$152$	0	0
$153$	−1.73736	−0.140457
$154$	0	0
$155$	−11.5114	−0.924618
$156$	0	0
$157$	−21.4309	−1.71037	−0.855186	−	0.518322i	$-0.826557\pi$
$157$	−21.4309	−1.71037	−0.855186	+	0.518322i	$0.826557\pi$
$158$	0	0
$159$	−1.55065	−0.122974
$160$	0	0
$161$	−0.870393	−0.0685966
$162$	0	0
$163$	16.4013	1.28465	0.642324	−	0.766433i	$-0.277970\pi$
$163$	16.4013	1.28465	0.642324	+	0.766433i	$0.277970\pi$
$164$	0	0
$165$	−0.188608	−0.0146831
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−8.70642	−0.669725
$170$	0	0
$171$	7.35449	0.562412
$172$	0	0
$173$	−7.77119	−0.590833	−0.295416	−	0.955369i	$-0.595458\pi$
$173$	−7.77119	−0.590833	−0.295416	+	0.955369i	$0.595458\pi$
$174$	0	0
$175$	4.88907	0.369579
$176$	0	0
$177$	11.0736	0.832339
$178$	0	0
$179$	20.5631	1.53696	0.768479	−	0.639875i	$-0.221014\pi$
$179$	20.5631	1.53696	0.768479	+	0.639875i	$0.221014\pi$
$180$	0	0
$181$	−6.27861	−0.466686	−0.233343	−	0.972395i	$-0.574966\pi$
$181$	−6.27861	−0.466686	−0.233343	+	0.972395i	$0.574966\pi$
$182$	0	0
$183$	13.4007	0.990608
$184$	0	0
$185$	24.3020	1.78672
$186$	0	0
$187$	0.111715	0.00816940
$188$	0	0
$189$	1.35675	0.0986894
$190$	0	0
$191$	−10.4618	−0.756991	−0.378495	−	0.925603i	$-0.623558\pi$
$191$	−10.4618	−0.756991	−0.378495	+	0.925603i	$0.623558\pi$
$192$	0	0
$193$	3.93031	0.282910	0.141455	−	0.989945i	$-0.454822\pi$
$193$	3.93031	0.282910	0.141455	+	0.989945i	$0.454822\pi$
$194$	0	0
$195$	−6.07781	−0.435241
$196$	0	0
$197$	−9.96631	−0.710070	−0.355035	−	0.934853i	$-0.615531\pi$
$197$	−9.96631	−0.710070	−0.355035	+	0.934853i	$0.615531\pi$
$198$	0	0
$199$	−23.2537	−1.64841	−0.824205	−	0.566291i	$-0.808378\pi$
$199$	−23.2537	−1.64841	−0.824205	+	0.566291i	$0.808378\pi$
$200$	0	0
$201$	4.50770	0.317949
$202$	0	0
$203$	4.95861	0.348026
$204$	0	0
$205$	4.55818	0.318357
$206$	0	0
$207$	−0.641526	−0.0445892
$208$	0	0
$209$	−0.472905	−0.0327115
$210$	0	0
$211$	−1.84912	−0.127298	−0.0636492	−	0.997972i	$-0.520274\pi$
$211$	−1.84912	−0.127298	−0.0636492	+	0.997972i	$0.520274\pi$
$212$	0	0
$213$	10.2267	0.700722
$214$	0	0
$215$	−8.14303	−0.555350
$216$	0	0
$217$	−5.32466	−0.361462
$218$	0	0
$219$	−6.16638	−0.416685
$220$	0	0
$221$	3.59997	0.242160
$222$	0	0
$223$	19.7420	1.32202	0.661010	−	0.750377i	$-0.270128\pi$
$223$	19.7420	1.32202	0.661010	+	0.750377i	$0.270128\pi$
$224$	0	0
$225$	3.60351	0.240234
$226$	0	0
$227$	−4.07103	−0.270204	−0.135102	−	0.990832i	$-0.543136\pi$
$227$	−4.07103	−0.270204	−0.135102	+	0.990832i	$0.543136\pi$
$228$	0	0
$229$	16.3832	1.08263	0.541315	−	0.840820i	$-0.317926\pi$
$229$	16.3832	1.08263	0.541315	+	0.840820i	$0.317926\pi$
$230$	0	0
$231$	−0.0872414	−0.00574006
$232$	0	0
$233$	−14.9192	−0.977392	−0.488696	−	0.872454i	$-0.662527\pi$
$233$	−14.9192	−0.977392	−0.488696	+	0.872454i	$0.662527\pi$
$234$	0	0
$235$	−13.2494	−0.864297
$236$	0	0
$237$	2.51122	0.163121
$238$	0	0
$239$	−10.0543	−0.650362	−0.325181	−	0.945652i	$-0.605425\pi$
$239$	−10.0543	−0.650362	−0.325181	+	0.945652i	$0.605425\pi$
$240$	0	0
$241$	17.4465	1.12383	0.561915	−	0.827195i	$-0.310065\pi$
$241$	17.4465	1.12383	0.561915	+	0.827195i	$0.310065\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−15.1329	−0.966804
$246$	0	0
$247$	−15.2392	−0.969648
$248$	0	0
$249$	1.74485	0.110576
$250$	0	0
$251$	−9.90197	−0.625007	−0.312504	−	0.949917i	$-0.601168\pi$
$251$	−9.90197	−0.625007	−0.312504	+	0.949917i	$0.601168\pi$
$252$	0	0
$253$	0.0412511	0.00259344
$254$	0	0
$255$	−5.09598	−0.319123
$256$	0	0
$257$	−19.8372	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$257$	−19.8372	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$258$	0	0
$259$	11.2410	0.698483
$260$	0	0
$261$	3.65476	0.226224
$262$	0	0
$263$	10.8242	0.667447	0.333723	−	0.942671i	$-0.391695\pi$
$263$	10.8242	0.667447	0.333723	+	0.942671i	$0.391695\pi$
$264$	0	0
$265$	−4.54832	−0.279401
$266$	0	0
$267$	−17.4735	−1.06936
$268$	0	0
$269$	−8.67721	−0.529059	−0.264530	−	0.964378i	$-0.585217\pi$
$269$	−8.67721	−0.529059	−0.264530	+	0.964378i	$0.585217\pi$
$270$	0	0
$271$	18.8503	1.14507	0.572536	−	0.819879i	$-0.305960\pi$
$271$	18.8503	1.14507	0.572536	+	0.819879i	$0.305960\pi$
$272$	0	0
$273$	−2.81132	−0.170149
$274$	0	0
$275$	−0.231711	−0.0139727
$276$	0	0
$277$	5.13118	0.308303	0.154151	−	0.988047i	$-0.450736\pi$
$277$	5.13118	0.308303	0.154151	+	0.988047i	$0.450736\pi$
$278$	0	0
$279$	−3.92456	−0.234957
$280$	0	0
$281$	−17.8779	−1.06650	−0.533252	−	0.845956i	$-0.679030\pi$
$281$	−17.8779	−1.06650	−0.533252	+	0.845956i	$0.679030\pi$
$282$	0	0
$283$	13.0771	0.777351	0.388676	−	0.921375i	$-0.372933\pi$
$283$	13.0771	0.777351	0.388676	+	0.921375i	$0.372933\pi$
$284$	0	0
$285$	21.5720	1.27782
$286$	0	0
$287$	2.10841	0.124456
$288$	0	0
$289$	−13.9816	−0.822446
$290$	0	0
$291$	17.6851	1.03672
$292$	0	0
$293$	16.1704	0.944682	0.472341	−	0.881416i	$-0.343409\pi$
$293$	16.1704	0.944682	0.472341	+	0.881416i	$0.343409\pi$
$294$	0	0
$295$	32.4806	1.89110
$296$	0	0
$297$	−0.0643015	−0.00373115
$298$	0	0
$299$	1.32930	0.0768756
$300$	0	0
$301$	−3.76660	−0.217103
$302$	0	0
$303$	10.0244	0.575886
$304$	0	0
$305$	39.3066	2.25069
$306$	0	0
$307$	12.5563	0.716623	0.358312	−	0.933602i	$-0.383353\pi$
$307$	12.5563	0.716623	0.358312	+	0.933602i	$0.383353\pi$
$308$	0	0
$309$	6.08730	0.346295
$310$	0	0
$311$	−2.18454	−0.123874	−0.0619368	−	0.998080i	$-0.519728\pi$
$311$	−2.18454	−0.123874	−0.0619368	+	0.998080i	$0.519728\pi$
$312$	0	0
$313$	29.7286	1.68036	0.840179	−	0.542310i	$-0.182450\pi$
$313$	29.7286	1.68036	0.840179	+	0.542310i	$0.182450\pi$
$314$	0	0
$315$	3.97960	0.224225
$316$	0	0
$317$	24.6910	1.38678	0.693392	−	0.720561i	$-0.256116\pi$
$317$	24.6910	1.38678	0.693392	+	0.720561i	$0.256116\pi$
$318$	0	0
$319$	−0.235007	−0.0131578
$320$	0	0
$321$	12.5950	0.702983
$322$	0	0
$323$	−12.7774	−0.710954
$324$	0	0
$325$	−7.46680	−0.414184
$326$	0	0
$327$	−17.0536	−0.943064
$328$	0	0
$329$	−6.12859	−0.337880
$330$	0	0
$331$	25.3306	1.39230	0.696148	−	0.717899i	$-0.254896\pi$
$331$	25.3306	1.39230	0.696148	+	0.717899i	$0.254896\pi$
$332$	0	0
$333$	8.28523	0.454028
$334$	0	0
$335$	13.2219	0.722387
$336$	0	0
$337$	1.00882	0.0549542	0.0274771	−	0.999622i	$-0.491253\pi$
$337$	1.00882	0.0549542	0.0274771	+	0.999622i	$0.491253\pi$
$338$	0	0
$339$	−11.3676	−0.617402
$340$	0	0
$341$	0.252355	0.0136658
$342$	0	0
$343$	−16.4971	−0.890758
$344$	0	0
$345$	−1.88171	−0.101308
$346$	0	0
$347$	−5.37742	−0.288675	−0.144337	−	0.989529i	$-0.546105\pi$
$347$	−5.37742	−0.288675	−0.144337	+	0.989529i	$0.546105\pi$
$348$	0	0
$349$	−34.8774	−1.86694	−0.933471	−	0.358653i	$-0.883236\pi$
$349$	−34.8774	−1.86694	−0.933471	+	0.358653i	$0.883236\pi$
$350$	0	0
$351$	−2.07209	−0.110600
$352$	0	0
$353$	−36.1276	−1.92288	−0.961438	−	0.275020i	$-0.911315\pi$
$353$	−36.1276	−1.92288	−0.961438	+	0.275020i	$0.911315\pi$
$354$	0	0
$355$	29.9967	1.59206
$356$	0	0
$357$	−2.35717	−0.124755
$358$	0	0
$359$	−23.1328	−1.22090	−0.610451	−	0.792054i	$-0.709012\pi$
$359$	−23.1328	−1.22090	−0.610451	+	0.792054i	$0.709012\pi$
$360$	0	0
$361$	35.0886	1.84677
$362$	0	0
$363$	−10.9959	−0.577133
$364$	0	0
$365$	−18.0871	−0.946720
$366$	0	0
$367$	8.93252	0.466274	0.233137	−	0.972444i	$-0.425101\pi$
$367$	8.93252	0.466274	0.233137	+	0.972444i	$0.425101\pi$
$368$	0	0
$369$	1.55401	0.0808986
$370$	0	0
$371$	−2.10385	−0.109226
$372$	0	0
$373$	16.2017	0.838891	0.419445	−	0.907781i	$-0.362225\pi$
$373$	16.2017	0.838891	0.419445	+	0.907781i	$0.362225\pi$
$374$	0	0
$375$	−4.09616	−0.211525
$376$	0	0
$377$	−7.57301	−0.390030
$378$	0	0
$379$	−2.95663	−0.151872	−0.0759361	−	0.997113i	$-0.524194\pi$
$379$	−2.95663	−0.151872	−0.0759361	+	0.997113i	$0.524194\pi$
$380$	0	0
$381$	4.85124	0.248536
$382$	0	0
$383$	10.0666	0.514381	0.257191	−	0.966361i	$-0.417203\pi$
$383$	10.0666	0.514381	0.257191	+	0.966361i	$0.417203\pi$
$384$	0	0
$385$	−0.255894	−0.0130416
$386$	0	0
$387$	−2.77619	−0.141121
$388$	0	0
$389$	13.7618	0.697749	0.348874	−	0.937170i	$-0.386564\pi$
$389$	13.7618	0.697749	0.348874	+	0.937170i	$0.386564\pi$
$390$	0	0
$391$	1.11456	0.0563658
$392$	0	0
$393$	−13.1490	−0.663281
$394$	0	0
$395$	7.36584	0.370615
$396$	0	0
$397$	−29.8647	−1.49887	−0.749433	−	0.662081i	$-0.769674\pi$
$397$	−29.8647	−1.49887	−0.749433	+	0.662081i	$0.769674\pi$
$398$	0	0
$399$	9.97824	0.499537
$400$	0	0
$401$	−25.3485	−1.26584	−0.632922	−	0.774215i	$-0.718145\pi$
$401$	−25.3485	−1.26584	−0.632922	+	0.774215i	$0.718145\pi$
$402$	0	0
$403$	8.13205	0.405086
$404$	0	0
$405$	2.93317	0.145751
$406$	0	0
$407$	−0.532753	−0.0264076
$408$	0	0
$409$	11.5335	0.570293	0.285147	−	0.958484i	$-0.407958\pi$
$409$	11.5335	0.570293	0.285147	+	0.958484i	$0.407958\pi$
$410$	0	0
$411$	9.39248	0.463297
$412$	0	0
$413$	15.0241	0.739287
$414$	0	0
$415$	5.11796	0.251231
$416$	0	0
$417$	−6.71091	−0.328635
$418$	0	0
$419$	30.1036	1.47066	0.735328	−	0.677711i	$-0.237028\pi$
$419$	30.1036	1.47066	0.735328	+	0.677711i	$0.237028\pi$
$420$	0	0
$421$	2.97021	0.144759	0.0723795	−	0.997377i	$-0.476941\pi$
$421$	2.97021	0.144759	0.0723795	+	0.997377i	$0.476941\pi$
$422$	0	0
$423$	−4.51709	−0.219629
$424$	0	0
$425$	−6.26059	−0.303683
$426$	0	0
$427$	18.1815	0.879862
$428$	0	0
$429$	0.133239	0.00643283
$430$	0	0
$431$	−0.290617	−0.0139985	−0.00699927	−	0.999976i	$-0.502228\pi$
$431$	−0.290617	−0.0139985	−0.00699927	+	0.999976i	$0.502228\pi$
$432$	0	0
$433$	−8.66027	−0.416186	−0.208093	−	0.978109i	$-0.566726\pi$
$433$	−8.66027	−0.416186	−0.208093	+	0.978109i	$0.566726\pi$
$434$	0	0
$435$	10.7200	0.513987
$436$	0	0
$437$	−4.71810	−0.225697
$438$	0	0
$439$	36.4829	1.74123	0.870616	−	0.491963i	$-0.163721\pi$
$439$	36.4829	1.74123	0.870616	+	0.491963i	$0.163721\pi$
$440$	0	0
$441$	−5.15922	−0.245677
$442$	0	0
$443$	−38.6330	−1.83551	−0.917754	−	0.397149i	$-0.870000\pi$
$443$	−38.6330	−1.83551	−0.917754	+	0.397149i	$0.870000\pi$
$444$	0	0
$445$	−51.2529	−2.42962
$446$	0	0
$447$	−6.09023	−0.288058
$448$	0	0
$449$	−26.1961	−1.23627	−0.618136	−	0.786071i	$-0.712112\pi$
$449$	−26.1961	−1.23627	−0.618136	+	0.786071i	$0.712112\pi$
$450$	0	0
$451$	−0.0999253	−0.00470530
$452$	0	0
$453$	−13.3009	−0.624931
$454$	0	0
$455$	−8.24610	−0.386583
$456$	0	0
$457$	5.98759	0.280088	0.140044	−	0.990145i	$-0.455276\pi$
$457$	5.98759	0.280088	0.140044	+	0.990145i	$0.455276\pi$
$458$	0	0
$459$	−1.73736	−0.0810930
$460$	0	0
$461$	−33.6863	−1.56893	−0.784464	−	0.620174i	$-0.787062\pi$
$461$	−33.6863	−1.56893	−0.784464	+	0.620174i	$0.787062\pi$
$462$	0	0
$463$	−15.4825	−0.719532	−0.359766	−	0.933043i	$-0.617143\pi$
$463$	−15.4825	−0.719532	−0.359766	+	0.933043i	$0.617143\pi$
$464$	0	0
$465$	−11.5114	−0.533829
$466$	0	0
$467$	−4.98854	−0.230842	−0.115421	−	0.993317i	$-0.536822\pi$
$467$	−4.98854	−0.230842	−0.115421	+	0.993317i	$0.536822\pi$
$468$	0	0
$469$	6.11584	0.282403
$470$	0	0
$471$	−21.4309	−0.987483
$472$	0	0
$473$	0.178513	0.00820804
$474$	0	0
$475$	26.5020	1.21599
$476$	0	0
$477$	−1.55065	−0.0709993
$478$	0	0
$479$	15.2655	0.697499	0.348749	−	0.937216i	$-0.386606\pi$
$479$	15.2655	0.697499	0.348749	+	0.937216i	$0.386606\pi$
$480$	0	0
$481$	−17.1678	−0.782783
$482$	0	0
$483$	−0.870393	−0.0396043
$484$	0	0
$485$	51.8734	2.35545
$486$	0	0
$487$	29.7295	1.34717	0.673587	−	0.739108i	$-0.264753\pi$
$487$	29.7295	1.34717	0.673587	+	0.739108i	$0.264753\pi$
$488$	0	0
$489$	16.4013	0.741692
$490$	0	0
$491$	−37.9213	−1.71136	−0.855682	−	0.517502i	$-0.826862\pi$
$491$	−37.9213	−1.71136	−0.855682	+	0.517502i	$0.826862\pi$
$492$	0	0
$493$	−6.34963	−0.285973
$494$	0	0
$495$	−0.188608	−0.00847728
$496$	0	0
$497$	13.8751	0.622384
$498$	0	0
$499$	1.26368	0.0565699	0.0282850	−	0.999600i	$-0.490995\pi$
$499$	1.26368	0.0565699	0.0282850	+	0.999600i	$0.490995\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	11.1166	0.495665	0.247833	−	0.968803i	$-0.420282\pi$
$503$	11.1166	0.495665	0.247833	+	0.968803i	$0.420282\pi$
$504$	0	0
$505$	29.4033	1.30843
$506$	0	0
$507$	−8.70642	−0.386666
$508$	0	0
$509$	36.4271	1.61460	0.807301	−	0.590140i	$-0.200927\pi$
$509$	36.4271	1.61460	0.807301	+	0.590140i	$0.200927\pi$
$510$	0	0
$511$	−8.36626	−0.370102
$512$	0	0
$513$	7.35449	0.324709
$514$	0	0
$515$	17.8551	0.786790
$516$	0	0
$517$	0.290456	0.0127742
$518$	0	0
$519$	−7.77119	−0.341118
$520$	0	0
$521$	−28.0491	−1.22885	−0.614427	−	0.788974i	$-0.710613\pi$
$521$	−28.0491	−1.22885	−0.614427	+	0.788974i	$0.710613\pi$
$522$	0	0
$523$	−35.4652	−1.55078	−0.775392	−	0.631481i	$-0.782447\pi$
$523$	−35.4652	−1.55078	−0.775392	+	0.631481i	$0.782447\pi$
$524$	0	0
$525$	4.88907	0.213377
$526$	0	0
$527$	6.81837	0.297013
$528$	0	0
$529$	−22.5884	−0.982106
$530$	0	0
$531$	11.0736	0.480551
$532$	0	0
$533$	−3.22006	−0.139476
$534$	0	0
$535$	36.9432	1.59720
$536$	0	0
$537$	20.5631	0.887363
$538$	0	0
$539$	0.331746	0.0142893
$540$	0	0
$541$	7.84741	0.337387	0.168693	−	0.985669i	$-0.446045\pi$
$541$	7.84741	0.337387	0.168693	+	0.985669i	$0.446045\pi$
$542$	0	0
$543$	−6.27861	−0.269441
$544$	0	0
$545$	−50.0210	−2.14267
$546$	0	0
$547$	−26.0303	−1.11297	−0.556487	−	0.830856i	$-0.687851\pi$
$547$	−26.0303	−1.11297	−0.556487	+	0.830856i	$0.687851\pi$
$548$	0	0
$549$	13.4007	0.571928
$550$	0	0
$551$	26.8789	1.14508
$552$	0	0
$553$	3.40711	0.144885
$554$	0	0
$555$	24.3020	1.03156
$556$	0	0
$557$	23.5772	0.998999	0.499499	−	0.866314i	$-0.333517\pi$
$557$	23.5772	0.998999	0.499499	+	0.866314i	$0.333517\pi$
$558$	0	0
$559$	5.75252	0.243306
$560$	0	0
$561$	0.111715	0.00471661
$562$	0	0
$563$	1.45924	0.0614997	0.0307498	−	0.999527i	$-0.490210\pi$
$563$	1.45924	0.0614997	0.0307498	+	0.999527i	$0.490210\pi$
$564$	0	0
$565$	−33.3431	−1.40275
$566$	0	0
$567$	1.35675	0.0569783
$568$	0	0
$569$	−18.4501	−0.773470	−0.386735	−	0.922191i	$-0.626397\pi$
$569$	−18.4501	−0.773470	−0.386735	+	0.922191i	$0.626397\pi$
$570$	0	0
$571$	27.0806	1.13329	0.566644	−	0.823963i	$-0.308242\pi$
$571$	27.0806	1.13329	0.566644	+	0.823963i	$0.308242\pi$
$572$	0	0
$573$	−10.4618	−0.437049
$574$	0	0
$575$	−2.31174	−0.0964063
$576$	0	0
$577$	11.7436	0.488894	0.244447	−	0.969663i	$-0.421394\pi$
$577$	11.7436	0.488894	0.244447	+	0.969663i	$0.421394\pi$
$578$	0	0
$579$	3.93031	0.163338
$580$	0	0
$581$	2.36734	0.0982138
$582$	0	0
$583$	0.0997091	0.00412953
$584$	0	0
$585$	−6.07781	−0.251287
$586$	0	0
$587$	−17.5381	−0.723874	−0.361937	−	0.932203i	$-0.617884\pi$
$587$	−17.5381	−0.723874	−0.361937	+	0.932203i	$0.617884\pi$
$588$	0	0
$589$	−28.8631	−1.18928
$590$	0	0
$591$	−9.96631	−0.409959
$592$	0	0
$593$	−22.8036	−0.936433	−0.468216	−	0.883614i	$-0.655103\pi$
$593$	−22.8036	−0.936433	−0.468216	+	0.883614i	$0.655103\pi$
$594$	0	0
$595$	−6.91399	−0.283446
$596$	0	0
$597$	−23.2537	−0.951710
$598$	0	0
$599$	−21.3655	−0.872971	−0.436485	−	0.899711i	$-0.643777\pi$
$599$	−21.3655	−0.872971	−0.436485	+	0.899711i	$0.643777\pi$
$600$	0	0
$601$	−2.09186	−0.0853288	−0.0426644	−	0.999089i	$-0.513585\pi$
$601$	−2.09186	−0.0853288	−0.0426644	+	0.999089i	$0.513585\pi$
$602$	0	0
$603$	4.50770	0.183568
$604$	0	0
$605$	−32.2528	−1.31126
$606$	0	0
$607$	0.887942	0.0360405	0.0180202	−	0.999838i	$-0.494264\pi$
$607$	0.887942	0.0360405	0.0180202	+	0.999838i	$0.494264\pi$
$608$	0	0
$609$	4.95861	0.200933
$610$	0	0
$611$	9.35984	0.378659
$612$	0	0
$613$	31.2654	1.26280	0.631399	−	0.775458i	$-0.282481\pi$
$613$	31.2654	1.26280	0.631399	+	0.775458i	$0.282481\pi$
$614$	0	0
$615$	4.55818	0.183804
$616$	0	0
$617$	19.7468	0.794975	0.397488	−	0.917608i	$-0.369882\pi$
$617$	19.7468	0.794975	0.397488	+	0.917608i	$0.369882\pi$
$618$	0	0
$619$	−1.73668	−0.0698032	−0.0349016	−	0.999391i	$-0.511112\pi$
$619$	−1.73668	−0.0698032	−0.0349016	+	0.999391i	$0.511112\pi$
$620$	0	0
$621$	−0.641526	−0.0257436
$622$	0	0
$623$	−23.7073	−0.949812
$624$	0	0
$625$	−30.0323	−1.20129
$626$	0	0
$627$	−0.472905	−0.0188860
$628$	0	0
$629$	−14.3944	−0.573943
$630$	0	0
$631$	−3.51023	−0.139740	−0.0698700	−	0.997556i	$-0.522258\pi$
$631$	−3.51023	−0.139740	−0.0698700	+	0.997556i	$0.522258\pi$
$632$	0	0
$633$	−1.84912	−0.0734958
$634$	0	0
$635$	14.2295	0.564681
$636$	0	0
$637$	10.6904	0.423569
$638$	0	0
$639$	10.2267	0.404562
$640$	0	0
$641$	−27.8251	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$641$	−27.8251	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$642$	0	0
$643$	−13.8299	−0.545399	−0.272700	−	0.962099i	$-0.587916\pi$
$643$	−13.8299	−0.545399	−0.272700	+	0.962099i	$0.587916\pi$
$644$	0	0
$645$	−8.14303	−0.320632
$646$	0	0
$647$	−16.6414	−0.654241	−0.327120	−	0.944983i	$-0.606078\pi$
$647$	−16.6414	−0.654241	−0.327120	+	0.944983i	$0.606078\pi$
$648$	0	0
$649$	−0.712046	−0.0279503
$650$	0	0
$651$	−5.32466	−0.208690
$652$	0	0
$653$	26.1625	1.02382	0.511909	−	0.859040i	$-0.328938\pi$
$653$	26.1625	1.02382	0.511909	+	0.859040i	$0.328938\pi$
$654$	0	0
$655$	−38.5684	−1.50699
$656$	0	0
$657$	−6.16638	−0.240573
$658$	0	0
$659$	−41.9458	−1.63398	−0.816989	−	0.576653i	$-0.804358\pi$
$659$	−41.9458	−1.63398	−0.816989	+	0.576653i	$0.804358\pi$
$660$	0	0
$661$	−25.6903	−0.999236	−0.499618	−	0.866246i	$-0.666526\pi$
$661$	−25.6903	−0.999236	−0.499618	+	0.866246i	$0.666526\pi$
$662$	0	0
$663$	3.59997	0.139811
$664$	0	0
$665$	29.2679	1.13496
$666$	0	0
$667$	−2.34462	−0.0907842
$668$	0	0
$669$	19.7420	0.763269
$670$	0	0
$671$	−0.861686	−0.0332650
$672$	0	0
$673$	−0.468458	−0.0180577	−0.00902887	−	0.999959i	$-0.502874\pi$
$673$	−0.468458	−0.0180577	−0.00902887	+	0.999959i	$0.502874\pi$
$674$	0	0
$675$	3.60351	0.138699
$676$	0	0
$677$	5.12748	0.197065	0.0985325	−	0.995134i	$-0.468585\pi$
$677$	5.12748	0.197065	0.0985325	+	0.995134i	$0.468585\pi$
$678$	0	0
$679$	23.9943	0.920817
$680$	0	0
$681$	−4.07103	−0.156002
$682$	0	0
$683$	−1.24490	−0.0476349	−0.0238174	−	0.999716i	$-0.507582\pi$
$683$	−1.24490	−0.0476349	−0.0238174	+	0.999716i	$0.507582\pi$
$684$	0	0
$685$	27.5498	1.05262
$686$	0	0
$687$	16.3832	0.625057
$688$	0	0
$689$	3.21309	0.122409
$690$	0	0
$691$	−12.8788	−0.489934	−0.244967	−	0.969531i	$-0.578777\pi$
$691$	−12.8788	−0.489934	−0.244967	+	0.969531i	$0.578777\pi$
$692$	0	0
$693$	−0.0872414	−0.00331403
$694$	0	0
$695$	−19.6843	−0.746666
$696$	0	0
$697$	−2.69988	−0.102265
$698$	0	0
$699$	−14.9192	−0.564298
$700$	0	0
$701$	−12.0111	−0.453654	−0.226827	−	0.973935i	$-0.572835\pi$
$701$	−12.0111	−0.453654	−0.226827	+	0.973935i	$0.572835\pi$
$702$	0	0
$703$	60.9337	2.29816
$704$	0	0
$705$	−13.2494	−0.499002
$706$	0	0
$707$	13.6006	0.511505
$708$	0	0
$709$	0.333173	0.0125126	0.00625628	−	0.999980i	$-0.498009\pi$
$709$	0.333173	0.0125126	0.00625628	+	0.999980i	$0.498009\pi$
$710$	0	0
$711$	2.51122	0.0941780
$712$	0	0
$713$	2.51771	0.0942889
$714$	0	0
$715$	0.390813	0.0146156
$716$	0	0
$717$	−10.0543	−0.375486
$718$	0	0
$719$	−4.69923	−0.175252	−0.0876258	−	0.996153i	$-0.527928\pi$
$719$	−4.69923	−0.175252	−0.0876258	+	0.996153i	$0.527928\pi$
$720$	0	0
$721$	8.25898	0.307580
$722$	0	0
$723$	17.4465	0.648843
$724$	0	0
$725$	13.1699	0.489119
$726$	0	0
$727$	16.9255	0.627733	0.313866	−	0.949467i	$-0.398376\pi$
$727$	16.9255	0.627733	0.313866	+	0.949467i	$0.398376\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.82323	0.178394
$732$	0	0
$733$	20.7928	0.767998	0.383999	−	0.923333i	$-0.374547\pi$
$733$	20.7928	0.767998	0.383999	+	0.923333i	$0.374547\pi$
$734$	0	0
$735$	−15.1329	−0.558185
$736$	0	0
$737$	−0.289852	−0.0106768
$738$	0	0
$739$	38.1583	1.40368	0.701838	−	0.712336i	$-0.252363\pi$
$739$	38.1583	1.40368	0.701838	+	0.712336i	$0.252363\pi$
$740$	0	0
$741$	−15.2392	−0.559826
$742$	0	0
$743$	1.00712	0.0369476	0.0184738	−	0.999829i	$-0.494119\pi$
$743$	1.00712	0.0369476	0.0184738	+	0.999829i	$0.494119\pi$
$744$	0	0
$745$	−17.8637	−0.654475
$746$	0	0
$747$	1.74485	0.0638409
$748$	0	0
$749$	17.0883	0.624392
$750$	0	0
$751$	−10.4846	−0.382589	−0.191295	−	0.981533i	$-0.561269\pi$
$751$	−10.4846	−0.382589	−0.191295	+	0.981533i	$0.561269\pi$
$752$	0	0
$753$	−9.90197	−0.360848
$754$	0	0
$755$	−39.0139	−1.41986
$756$	0	0
$757$	22.4109	0.814537	0.407269	−	0.913308i	$-0.366481\pi$
$757$	22.4109	0.814537	0.407269	+	0.913308i	$0.366481\pi$
$758$	0	0
$759$	0.0412511	0.00149732
$760$	0	0
$761$	2.38127	0.0863211	0.0431606	−	0.999068i	$-0.486257\pi$
$761$	2.38127	0.0863211	0.0431606	+	0.999068i	$0.486257\pi$
$762$	0	0
$763$	−23.1375	−0.837633
$764$	0	0
$765$	−5.09598	−0.184245
$766$	0	0
$767$	−22.9454	−0.828512
$768$	0	0
$769$	6.71515	0.242154	0.121077	−	0.992643i	$-0.461365\pi$
$769$	6.71515	0.242154	0.121077	+	0.992643i	$0.461365\pi$
$770$	0	0
$771$	−19.8372	−0.714418
$772$	0	0
$773$	−18.7417	−0.674091	−0.337046	−	0.941488i	$-0.609428\pi$
$773$	−18.7417	−0.674091	−0.337046	+	0.941488i	$0.609428\pi$
$774$	0	0
$775$	−14.1422	−0.508002
$776$	0	0
$777$	11.2410	0.403269
$778$	0	0
$779$	11.4290	0.409485
$780$	0	0
$781$	−0.657593	−0.0235305
$782$	0	0
$783$	3.65476	0.130610
$784$	0	0
$785$	−62.8605	−2.24359
$786$	0	0
$787$	−7.60078	−0.270939	−0.135469	−	0.990782i	$-0.543254\pi$
$787$	−7.60078	−0.270939	−0.135469	+	0.990782i	$0.543254\pi$
$788$	0	0
$789$	10.8242	0.385351
$790$	0	0
$791$	−15.4230	−0.548379
$792$	0	0
$793$	−27.7675	−0.986054
$794$	0	0
$795$	−4.54832	−0.161312
$796$	0	0
$797$	20.3082	0.719352	0.359676	−	0.933077i	$-0.382887\pi$
$797$	20.3082	0.719352	0.359676	+	0.933077i	$0.382887\pi$
$798$	0	0
$799$	7.84782	0.277636
$800$	0	0
$801$	−17.4735	−0.617397
$802$	0	0
$803$	0.396508	0.0139925
$804$	0	0
$805$	−2.55301	−0.0899819
$806$	0	0
$807$	−8.67721	−0.305452
$808$	0	0
$809$	−4.84896	−0.170480	−0.0852402	−	0.996360i	$-0.527166\pi$
$809$	−4.84896	−0.170480	−0.0852402	+	0.996360i	$0.527166\pi$
$810$	0	0
$811$	−0.215265	−0.00755897	−0.00377948	−	0.999993i	$-0.501203\pi$
$811$	−0.215265	−0.00755897	−0.00377948	+	0.999993i	$0.501203\pi$
$812$	0	0
$813$	18.8503	0.661108
$814$	0	0
$815$	48.1078	1.68514
$816$	0	0
$817$	−20.4174	−0.714316
$818$	0	0
$819$	−2.81132	−0.0982356
$820$	0	0
$821$	−22.6835	−0.791659	−0.395829	−	0.918324i	$-0.629543\pi$
$821$	−22.6835	−0.791659	−0.395829	+	0.918324i	$0.629543\pi$
$822$	0	0
$823$	9.75812	0.340147	0.170073	−	0.985431i	$-0.445600\pi$
$823$	9.75812	0.340147	0.170073	+	0.985431i	$0.445600\pi$
$824$	0	0
$825$	−0.231711	−0.00806714
$826$	0	0
$827$	−29.6125	−1.02973	−0.514863	−	0.857272i	$-0.672157\pi$
$827$	−29.6125	−1.02973	−0.514863	+	0.857272i	$0.672157\pi$
$828$	0	0
$829$	−43.3806	−1.50667	−0.753336	−	0.657636i	$-0.771557\pi$
$829$	−43.3806	−1.50667	−0.753336	+	0.657636i	$0.771557\pi$
$830$	0	0
$831$	5.13118	0.177999
$832$	0	0
$833$	8.96342	0.310564
$834$	0	0
$835$	2.93317	0.101507
$836$	0	0
$837$	−3.92456	−0.135653
$838$	0	0
$839$	−39.6211	−1.36787	−0.683936	−	0.729542i	$-0.739733\pi$
$839$	−39.6211	−1.36787	−0.683936	+	0.729542i	$0.739733\pi$
$840$	0	0
$841$	−15.6427	−0.539405
$842$	0	0
$843$	−17.8779	−0.615746
$844$	0	0
$845$	−25.5375	−0.878515
$846$	0	0
$847$	−14.9187	−0.512612
$848$	0	0
$849$	13.0771	0.448804
$850$	0	0
$851$	−5.31519	−0.182202
$852$	0	0
$853$	−34.6643	−1.18688	−0.593442	−	0.804877i	$-0.702231\pi$
$853$	−34.6643	−1.18688	−0.593442	+	0.804877i	$0.702231\pi$
$854$	0	0
$855$	21.5720	0.737747
$856$	0	0
$857$	19.6878	0.672522	0.336261	−	0.941769i	$-0.390838\pi$
$857$	19.6878	0.672522	0.336261	+	0.941769i	$0.390838\pi$
$858$	0	0
$859$	11.1405	0.380108	0.190054	−	0.981774i	$-0.439134\pi$
$859$	11.1405	0.380108	0.190054	+	0.981774i	$0.439134\pi$
$860$	0	0
$861$	2.10841	0.0718545
$862$	0	0
$863$	−28.5360	−0.971377	−0.485688	−	0.874132i	$-0.661431\pi$
$863$	−28.5360	−0.971377	−0.485688	+	0.874132i	$0.661431\pi$
$864$	0	0
$865$	−22.7943	−0.775028
$866$	0	0
$867$	−13.9816	−0.474839
$868$	0	0
$869$	−0.161475	−0.00547767
$870$	0	0
$871$	−9.34038	−0.316487
$872$	0	0
$873$	17.6851	0.598549
$874$	0	0
$875$	−5.55748	−0.187877
$876$	0	0
$877$	33.5568	1.13313	0.566566	−	0.824016i	$-0.308272\pi$
$877$	33.5568	1.13313	0.566566	+	0.824016i	$0.308272\pi$
$878$	0	0
$879$	16.1704	0.545413
$880$	0	0
$881$	−34.9200	−1.17648	−0.588242	−	0.808685i	$-0.700180\pi$
$881$	−34.9200	−1.17648	−0.588242	+	0.808685i	$0.700180\pi$
$882$	0	0
$883$	−3.89129	−0.130952	−0.0654761	−	0.997854i	$-0.520857\pi$
$883$	−3.89129	−0.130952	−0.0654761	+	0.997854i	$0.520857\pi$
$884$	0	0
$885$	32.4806	1.09182
$886$	0	0
$887$	18.7256	0.628745	0.314372	−	0.949300i	$-0.398206\pi$
$887$	18.7256	0.628745	0.314372	+	0.949300i	$0.398206\pi$
$888$	0	0
$889$	6.58194	0.220751
$890$	0	0
$891$	−0.0643015	−0.00215418
$892$	0	0
$893$	−33.2209	−1.11170
$894$	0	0
$895$	60.3151	2.01611
$896$	0	0
$897$	1.32930	0.0443841
$898$	0	0
$899$	−14.3433	−0.478376
$900$	0	0
$901$	2.69403	0.0897513
$902$	0	0
$903$	−3.76660	−0.125345
$904$	0	0
$905$	−18.4163	−0.612177
$906$	0	0
$907$	4.23006	0.140457	0.0702285	−	0.997531i	$-0.477627\pi$
$907$	4.23006	0.140457	0.0702285	+	0.997531i	$0.477627\pi$
$908$	0	0
$909$	10.0244	0.332488
$910$	0	0
$911$	−32.4195	−1.07411	−0.537053	−	0.843549i	$-0.680462\pi$
$911$	−32.4195	−1.07411	−0.537053	+	0.843549i	$0.680462\pi$
$912$	0	0
$913$	−0.112197	−0.00371317
$914$	0	0
$915$	39.3066	1.29944
$916$	0	0
$917$	−17.8400	−0.589129
$918$	0	0
$919$	7.50842	0.247680	0.123840	−	0.992302i	$-0.460479\pi$
$919$	7.50842	0.247680	0.123840	+	0.992302i	$0.460479\pi$
$920$	0	0
$921$	12.5563	0.413743
$922$	0	0
$923$	−21.1907	−0.697500
$924$	0	0
$925$	29.8559	0.981655
$926$	0	0
$927$	6.08730	0.199933
$928$	0	0
$929$	14.6459	0.480516	0.240258	−	0.970709i	$-0.422768\pi$
$929$	14.6459	0.480516	0.240258	+	0.970709i	$0.422768\pi$
$930$	0	0
$931$	−37.9434	−1.24355
$932$	0	0
$933$	−2.18454	−0.0715185
$934$	0	0
$935$	0.327679	0.0107163
$936$	0	0
$937$	−33.5660	−1.09655	−0.548277	−	0.836297i	$-0.684716\pi$
$937$	−33.5660	−1.09655	−0.548277	+	0.836297i	$0.684716\pi$
$938$	0	0
$939$	29.7286	0.970155
$940$	0	0
$941$	−9.09313	−0.296428	−0.148214	−	0.988955i	$-0.547352\pi$
$941$	−9.09313	−0.296428	−0.148214	+	0.988955i	$0.547352\pi$
$942$	0	0
$943$	−0.996939	−0.0324648
$944$	0	0
$945$	3.97960	0.129456
$946$	0	0
$947$	−33.3348	−1.08323	−0.541617	−	0.840625i	$-0.682188\pi$
$947$	−33.3348	−1.08323	−0.541617	+	0.840625i	$0.682188\pi$
$948$	0	0
$949$	12.7773	0.414769
$950$	0	0
$951$	24.6910	0.800660
$952$	0	0
$953$	−26.3126	−0.852349	−0.426174	−	0.904641i	$-0.640139\pi$
$953$	−26.3126	−0.852349	−0.426174	+	0.904641i	$0.640139\pi$
$954$	0	0
$955$	−30.6863	−0.992987
$956$	0	0
$957$	−0.235007	−0.00759669
$958$	0	0
$959$	12.7433	0.411502
$960$	0	0
$961$	−15.5978	−0.503156
$962$	0	0
$963$	12.5950	0.405867
$964$	0	0
$965$	11.5283	0.371109
$966$	0	0
$967$	55.3925	1.78130	0.890651	−	0.454687i	$-0.150249\pi$
$967$	55.3925	1.78130	0.890651	+	0.454687i	$0.150249\pi$
$968$	0	0
$969$	−12.7774	−0.410469
$970$	0	0
$971$	−16.9200	−0.542990	−0.271495	−	0.962440i	$-0.587518\pi$
$971$	−16.9200	−0.542990	−0.271495	+	0.962440i	$0.587518\pi$
$972$	0	0
$973$	−9.10505	−0.291895
$974$	0	0
$975$	−7.46680	−0.239129
$976$	0	0
$977$	−5.54245	−0.177319	−0.0886594	−	0.996062i	$-0.528258\pi$
$977$	−5.54245	−0.177319	−0.0886594	+	0.996062i	$0.528258\pi$
$978$	0	0
$979$	1.12357	0.0359096
$980$	0	0
$981$	−17.0536	−0.544478
$982$	0	0
$983$	20.9074	0.666842	0.333421	−	0.942778i	$-0.391797\pi$
$983$	20.9074	0.666842	0.333421	+	0.942778i	$0.391797\pi$
$984$	0	0
$985$	−29.2329	−0.931438
$986$	0	0
$987$	−6.12859	−0.195075
$988$	0	0
$989$	1.78100	0.0566324
$990$	0	0
$991$	45.9402	1.45934	0.729669	−	0.683800i	$-0.239674\pi$
$991$	45.9402	1.45934	0.729669	+	0.683800i	$0.239674\pi$
$992$	0	0
$993$	25.3306	0.803842
$994$	0	0
$995$	−68.2071	−2.16231
$996$	0	0
$997$	−29.5386	−0.935498	−0.467749	−	0.883861i	$-0.654935\pi$
$997$	−29.5386	−0.935498	−0.467749	+	0.883861i	$0.654935\pi$
$998$	0	0
$999$	8.28523	0.262133

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.10	✓	12
4.3	odd	2		8016.2.a.bf.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.10	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.10		12	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.93317 q^{5} +1.35675 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.93317 q^{5} +1.35675 q^{7} +1.00000 q^{9} -0.0643015 q^{11} -2.07209 q^{13} +2.93317 q^{15} -1.73736 q^{17} +7.35449 q^{19} +1.35675 q^{21} -0.641526 q^{23} +3.60351 q^{25} +1.00000 q^{27} +3.65476 q^{29} -3.92456 q^{31} -0.0643015 q^{33} +3.97960 q^{35} +8.28523 q^{37} -2.07209 q^{39} +1.55401 q^{41} -2.77619 q^{43} +2.93317 q^{45} -4.51709 q^{47} -5.15922 q^{49} -1.73736 q^{51} -1.55065 q^{53} -0.188608 q^{55} +7.35449 q^{57} +11.0736 q^{59} +13.4007 q^{61} +1.35675 q^{63} -6.07781 q^{65} +4.50770 q^{67} -0.641526 q^{69} +10.2267 q^{71} -6.16638 q^{73} +3.60351 q^{75} -0.0872414 q^{77} +2.51122 q^{79} +1.00000 q^{81} +1.74485 q^{83} -5.09598 q^{85} +3.65476 q^{87} -17.4735 q^{89} -2.81132 q^{91} -3.92456 q^{93} +21.5720 q^{95} +17.6851 q^{97} -0.0643015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

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Varieties

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Database

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