LMFDB - Embedded newform 8004.2.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
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} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.425546$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +2.74669 q^{11} +2.75061 q^{13} +0.425546 q^{15} -0.750311 q^{17} -4.08585 q^{19} +4.01837 q^{21} +1.00000 q^{23} -4.81891 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.50717 q^{31} -2.74669 q^{33} +1.71000 q^{35} -5.25315 q^{37} -2.75061 q^{39} +11.5128 q^{41} +7.71894 q^{43} -0.425546 q^{45} +3.88894 q^{47} +9.14727 q^{49} +0.750311 q^{51} -0.561794 q^{53} -1.16884 q^{55} +4.08585 q^{57} -11.0832 q^{59} -14.0362 q^{61} -4.01837 q^{63} -1.17051 q^{65} +11.7333 q^{67} -1.00000 q^{69} +11.6376 q^{71} -8.17001 q^{73} +4.81891 q^{75} -11.0372 q^{77} -1.31646 q^{79} +1.00000 q^{81} +8.92356 q^{83} +0.319292 q^{85} +1.00000 q^{87} +2.90607 q^{89} -11.0530 q^{91} -3.50717 q^{93} +1.73872 q^{95} -17.7139 q^{97} +2.74669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * 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
$5$	−0.425546	−0.190310	−0.0951549	−	0.995462i	$-0.530335\pi$
$5$	−0.425546	−0.190310	−0.0951549	+	0.995462i	$0.530335\pi$
$6$	0	0
$7$	−4.01837	−1.51880	−0.759400	−	0.650624i	$-0.774507\pi$
$7$	−4.01837	−1.51880	−0.759400	+	0.650624i	$0.774507\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.74669	0.828158	0.414079	−	0.910241i	$-0.364104\pi$
$11$	2.74669	0.828158	0.414079	+	0.910241i	$0.364104\pi$
$12$	0	0
$13$	2.75061	0.762883	0.381441	−	0.924393i	$-0.375428\pi$
$13$	2.75061	0.762883	0.381441	+	0.924393i	$0.375428\pi$
$14$	0	0
$15$	0.425546	0.109875
$16$	0	0
$17$	−0.750311	−0.181977	−0.0909886	−	0.995852i	$-0.529003\pi$
$17$	−0.750311	−0.181977	−0.0909886	+	0.995852i	$0.529003\pi$
$18$	0	0
$19$	−4.08585	−0.937359	−0.468679	−	0.883368i	$-0.655270\pi$
$19$	−4.08585	−0.937359	−0.468679	+	0.883368i	$0.655270\pi$
$20$	0	0
$21$	4.01837	0.876880
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.81891	−0.963782
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.50717	0.629906	0.314953	−	0.949107i	$-0.398011\pi$
$31$	3.50717	0.629906	0.314953	+	0.949107i	$0.398011\pi$
$32$	0	0
$33$	−2.74669	−0.478137
$34$	0	0
$35$	1.71000	0.289043
$36$	0	0
$37$	−5.25315	−0.863612	−0.431806	−	0.901966i	$-0.642124\pi$
$37$	−5.25315	−0.863612	−0.431806	+	0.901966i	$0.642124\pi$
$38$	0	0
$39$	−2.75061	−0.440451
$40$	0	0
$41$	11.5128	1.79799	0.898997	−	0.437955i	$-0.144297\pi$
$41$	11.5128	1.79799	0.898997	+	0.437955i	$0.144297\pi$
$42$	0	0
$43$	7.71894	1.17713	0.588564	−	0.808451i	$-0.299694\pi$
$43$	7.71894	1.17713	0.588564	+	0.808451i	$0.299694\pi$
$44$	0	0
$45$	−0.425546	−0.0634366
$46$	0	0
$47$	3.88894	0.567261	0.283630	−	0.958934i	$-0.408461\pi$
$47$	3.88894	0.567261	0.283630	+	0.958934i	$0.408461\pi$
$48$	0	0
$49$	9.14727	1.30675
$50$	0	0
$51$	0.750311	0.105065
$52$	0	0
$53$	−0.561794	−0.0771683	−0.0385841	−	0.999255i	$-0.512285\pi$
$53$	−0.561794	−0.0771683	−0.0385841	+	0.999255i	$0.512285\pi$
$54$	0	0
$55$	−1.16884	−0.157607
$56$	0	0
$57$	4.08585	0.541184
$58$	0	0
$59$	−11.0832	−1.44291	−0.721456	−	0.692460i	$-0.756527\pi$
$59$	−11.0832	−1.44291	−0.721456	+	0.692460i	$0.756527\pi$
$60$	0	0
$61$	−14.0362	−1.79715	−0.898574	−	0.438821i	$-0.855396\pi$
$61$	−14.0362	−1.79715	−0.898574	+	0.438821i	$0.855396\pi$
$62$	0	0
$63$	−4.01837	−0.506267
$64$	0	0
$65$	−1.17051	−0.145184
$66$	0	0
$67$	11.7333	1.43345	0.716724	−	0.697357i	$-0.245641\pi$
$67$	11.7333	1.43345	0.716724	+	0.697357i	$0.245641\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	11.6376	1.38113	0.690563	−	0.723272i	$-0.257363\pi$
$71$	11.6376	1.38113	0.690563	+	0.723272i	$0.257363\pi$
$72$	0	0
$73$	−8.17001	−0.956228	−0.478114	−	0.878298i	$-0.658679\pi$
$73$	−8.17001	−0.956228	−0.478114	+	0.878298i	$0.658679\pi$
$74$	0	0
$75$	4.81891	0.556440
$76$	0	0
$77$	−11.0372	−1.25781
$78$	0	0
$79$	−1.31646	−0.148113	−0.0740565	−	0.997254i	$-0.523595\pi$
$79$	−1.31646	−0.148113	−0.0740565	+	0.997254i	$0.523595\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.92356	0.979488	0.489744	−	0.871866i	$-0.337090\pi$
$83$	8.92356	0.979488	0.489744	+	0.871866i	$0.337090\pi$
$84$	0	0
$85$	0.319292	0.0346321
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	2.90607	0.308043	0.154021	−	0.988068i	$-0.450778\pi$
$89$	2.90607	0.308043	0.154021	+	0.988068i	$0.450778\pi$
$90$	0	0
$91$	−11.0530	−1.15867
$92$	0	0
$93$	−3.50717	−0.363677
$94$	0	0
$95$	1.73872	0.178389
$96$	0	0
$97$	−17.7139	−1.79857	−0.899286	−	0.437360i	$-0.855913\pi$
$97$	−17.7139	−1.79857	−0.899286	+	0.437360i	$0.855913\pi$
$98$	0	0
$99$	2.74669	0.276053
$100$	0	0
$101$	−10.1187	−1.00685	−0.503426	−	0.864038i	$-0.667927\pi$
$101$	−10.1187	−1.00685	−0.503426	+	0.864038i	$0.667927\pi$
$102$	0	0
$103$	14.6364	1.44217	0.721083	−	0.692849i	$-0.243645\pi$
$103$	14.6364	1.44217	0.721083	+	0.692849i	$0.243645\pi$
$104$	0	0
$105$	−1.71000	−0.166879
$106$	0	0
$107$	4.56932	0.441733	0.220866	−	0.975304i	$-0.429111\pi$
$107$	4.56932	0.441733	0.220866	+	0.975304i	$0.429111\pi$
$108$	0	0
$109$	6.65094	0.637045	0.318522	−	0.947915i	$-0.396813\pi$
$109$	6.65094	0.637045	0.318522	+	0.947915i	$0.396813\pi$
$110$	0	0
$111$	5.25315	0.498607
$112$	0	0
$113$	−7.23728	−0.680826	−0.340413	−	0.940276i	$-0.610567\pi$
$113$	−7.23728	−0.680826	−0.340413	+	0.940276i	$0.610567\pi$
$114$	0	0
$115$	−0.425546	−0.0396823
$116$	0	0
$117$	2.75061	0.254294
$118$	0	0
$119$	3.01503	0.276387
$120$	0	0
$121$	−3.45569	−0.314154
$122$	0	0
$123$	−11.5128	−1.03807
$124$	0	0
$125$	4.17840	0.373727
$126$	0	0
$127$	6.27278	0.556619	0.278310	−	0.960491i	$-0.410226\pi$
$127$	6.27278	0.556619	0.278310	+	0.960491i	$0.410226\pi$
$128$	0	0
$129$	−7.71894	−0.679615
$130$	0	0
$131$	4.38075	0.382748	0.191374	−	0.981517i	$-0.438706\pi$
$131$	4.38075	0.382748	0.191374	+	0.981517i	$0.438706\pi$
$132$	0	0
$133$	16.4185	1.42366
$134$	0	0
$135$	0.425546	0.0366251
$136$	0	0
$137$	1.73154	0.147935	0.0739676	−	0.997261i	$-0.476434\pi$
$137$	1.73154	0.147935	0.0739676	+	0.997261i	$0.476434\pi$
$138$	0	0
$139$	7.32975	0.621701	0.310851	−	0.950459i	$-0.399386\pi$
$139$	7.32975	0.621701	0.310851	+	0.950459i	$0.399386\pi$
$140$	0	0
$141$	−3.88894	−0.327508
$142$	0	0
$143$	7.55508	0.631788
$144$	0	0
$145$	0.425546	0.0353396
$146$	0	0
$147$	−9.14727	−0.754454
$148$	0	0
$149$	−2.50672	−0.205358	−0.102679	−	0.994715i	$-0.532742\pi$
$149$	−2.50672	−0.205358	−0.102679	+	0.994715i	$0.532742\pi$
$150$	0	0
$151$	−21.6115	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$151$	−21.6115	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$152$	0	0
$153$	−0.750311	−0.0606591
$154$	0	0
$155$	−1.49246	−0.119877
$156$	0	0
$157$	14.6172	1.16658	0.583289	−	0.812265i	$-0.301766\pi$
$157$	14.6172	1.16658	0.583289	+	0.812265i	$0.301766\pi$
$158$	0	0
$159$	0.561794	0.0445531
$160$	0	0
$161$	−4.01837	−0.316692
$162$	0	0
$163$	−0.0491942	−0.00385319	−0.00192659	−	0.999998i	$-0.500613\pi$
$163$	−0.0491942	−0.00385319	−0.00192659	+	0.999998i	$0.500613\pi$
$164$	0	0
$165$	1.16884	0.0909942
$166$	0	0
$167$	0.101312	0.00783979	0.00391989	−	0.999992i	$-0.498752\pi$
$167$	0.101312	0.00783979	0.00391989	+	0.999992i	$0.498752\pi$
$168$	0	0
$169$	−5.43413	−0.418010
$170$	0	0
$171$	−4.08585	−0.312453
$172$	0	0
$173$	25.4548	1.93529	0.967646	−	0.252312i	$-0.0811909\pi$
$173$	25.4548	1.93529	0.967646	+	0.252312i	$0.0811909\pi$
$174$	0	0
$175$	19.3642	1.46379
$176$	0	0
$177$	11.0832	0.833066
$178$	0	0
$179$	−21.2235	−1.58632	−0.793161	−	0.609012i	$-0.791566\pi$
$179$	−21.2235	−1.58632	−0.793161	+	0.609012i	$0.791566\pi$
$180$	0	0
$181$	−5.95457	−0.442599	−0.221300	−	0.975206i	$-0.571030\pi$
$181$	−5.95457	−0.442599	−0.221300	+	0.975206i	$0.571030\pi$
$182$	0	0
$183$	14.0362	1.03758
$184$	0	0
$185$	2.23546	0.164354
$186$	0	0
$187$	−2.06087	−0.150706
$188$	0	0
$189$	4.01837	0.292293
$190$	0	0
$191$	22.3732	1.61887	0.809434	−	0.587210i	$-0.199774\pi$
$191$	22.3732	1.61887	0.809434	+	0.587210i	$0.199774\pi$
$192$	0	0
$193$	−14.1935	−1.02167	−0.510834	−	0.859679i	$-0.670663\pi$
$193$	−14.1935	−1.02167	−0.510834	+	0.859679i	$0.670663\pi$
$194$	0	0
$195$	1.17051	0.0838221
$196$	0	0
$197$	−16.4768	−1.17392	−0.586962	−	0.809614i	$-0.699676\pi$
$197$	−16.4768	−1.17392	−0.586962	+	0.809614i	$0.699676\pi$
$198$	0	0
$199$	11.4091	0.808769	0.404384	−	0.914589i	$-0.367486\pi$
$199$	11.4091	0.808769	0.404384	+	0.914589i	$0.367486\pi$
$200$	0	0
$201$	−11.7333	−0.827602
$202$	0	0
$203$	4.01837	0.282034
$204$	0	0
$205$	−4.89921	−0.342176
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−11.2226	−0.776282
$210$	0	0
$211$	−13.0094	−0.895606	−0.447803	−	0.894132i	$-0.647793\pi$
$211$	−13.0094	−0.895606	−0.447803	+	0.894132i	$0.647793\pi$
$212$	0	0
$213$	−11.6376	−0.797393
$214$	0	0
$215$	−3.28476	−0.224019
$216$	0	0
$217$	−14.0931	−0.956701
$218$	0	0
$219$	8.17001	0.552078
$220$	0	0
$221$	−2.06382	−0.138827
$222$	0	0
$223$	−25.1539	−1.68443	−0.842214	−	0.539143i	$-0.818748\pi$
$223$	−25.1539	−1.68443	−0.842214	+	0.539143i	$0.818748\pi$
$224$	0	0
$225$	−4.81891	−0.321261
$226$	0	0
$227$	−21.9530	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$227$	−21.9530	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$228$	0	0
$229$	25.3691	1.67644	0.838220	−	0.545333i	$-0.183597\pi$
$229$	25.3691	1.67644	0.838220	+	0.545333i	$0.183597\pi$
$230$	0	0
$231$	11.0372	0.726195
$232$	0	0
$233$	12.1740	0.797546	0.398773	−	0.917050i	$-0.369436\pi$
$233$	12.1740	0.797546	0.398773	+	0.917050i	$0.369436\pi$
$234$	0	0
$235$	−1.65492	−0.107955
$236$	0	0
$237$	1.31646	0.0855131
$238$	0	0
$239$	−22.5942	−1.46150	−0.730749	−	0.682647i	$-0.760829\pi$
$239$	−22.5942	−1.46150	−0.730749	+	0.682647i	$0.760829\pi$
$240$	0	0
$241$	−9.98289	−0.643054	−0.321527	−	0.946900i	$-0.604196\pi$
$241$	−9.98289	−0.643054	−0.321527	+	0.946900i	$0.604196\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.89258	−0.248688
$246$	0	0
$247$	−11.2386	−0.715095
$248$	0	0
$249$	−8.92356	−0.565508
$250$	0	0
$251$	−9.85136	−0.621812	−0.310906	−	0.950441i	$-0.600632\pi$
$251$	−9.85136	−0.621812	−0.310906	+	0.950441i	$0.600632\pi$
$252$	0	0
$253$	2.74669	0.172683
$254$	0	0
$255$	−0.319292	−0.0199948
$256$	0	0
$257$	1.69890	0.105975	0.0529873	−	0.998595i	$-0.483126\pi$
$257$	1.69890	0.105975	0.0529873	+	0.998595i	$0.483126\pi$
$258$	0	0
$259$	21.1091	1.31165
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−7.43928	−0.458726	−0.229363	−	0.973341i	$-0.573664\pi$
$263$	−7.43928	−0.458726	−0.229363	+	0.973341i	$0.573664\pi$
$264$	0	0
$265$	0.239069	0.0146859
$266$	0	0
$267$	−2.90607	−0.177849
$268$	0	0
$269$	−29.6573	−1.80824	−0.904120	−	0.427279i	$-0.859472\pi$
$269$	−29.6573	−1.80824	−0.904120	+	0.427279i	$0.859472\pi$
$270$	0	0
$271$	17.2669	1.04889	0.524443	−	0.851445i	$-0.324273\pi$
$271$	17.2669	1.04889	0.524443	+	0.851445i	$0.324273\pi$
$272$	0	0
$273$	11.0530	0.668956
$274$	0	0
$275$	−13.2361	−0.798164
$276$	0	0
$277$	−22.6426	−1.36046	−0.680231	−	0.732998i	$-0.738121\pi$
$277$	−22.6426	−1.36046	−0.680231	+	0.732998i	$0.738121\pi$
$278$	0	0
$279$	3.50717	0.209969
$280$	0	0
$281$	−26.8833	−1.60372	−0.801861	−	0.597510i	$-0.796157\pi$
$281$	−26.8833	−1.60372	−0.801861	+	0.597510i	$0.796157\pi$
$282$	0	0
$283$	11.4876	0.682864	0.341432	−	0.939906i	$-0.389088\pi$
$283$	11.4876	0.682864	0.341432	+	0.939906i	$0.389088\pi$
$284$	0	0
$285$	−1.73872	−0.102993
$286$	0	0
$287$	−46.2626	−2.73079
$288$	0	0
$289$	−16.4370	−0.966884
$290$	0	0
$291$	17.7139	1.03841
$292$	0	0
$293$	19.4370	1.13552	0.567761	−	0.823193i	$-0.307810\pi$
$293$	19.4370	1.13552	0.567761	+	0.823193i	$0.307810\pi$
$294$	0	0
$295$	4.71642	0.274600
$296$	0	0
$297$	−2.74669	−0.159379
$298$	0	0
$299$	2.75061	0.159072
$300$	0	0
$301$	−31.0175	−1.78782
$302$	0	0
$303$	10.1187	0.581307
$304$	0	0
$305$	5.97304	0.342015
$306$	0	0
$307$	5.58977	0.319025	0.159513	−	0.987196i	$-0.449008\pi$
$307$	5.58977	0.319025	0.159513	+	0.987196i	$0.449008\pi$
$308$	0	0
$309$	−14.6364	−0.832635
$310$	0	0
$311$	−28.5683	−1.61996	−0.809981	−	0.586456i	$-0.800523\pi$
$311$	−28.5683	−1.61996	−0.809981	+	0.586456i	$0.800523\pi$
$312$	0	0
$313$	−12.0076	−0.678708	−0.339354	−	0.940659i	$-0.610208\pi$
$313$	−12.0076	−0.678708	−0.339354	+	0.940659i	$0.610208\pi$
$314$	0	0
$315$	1.71000	0.0963475
$316$	0	0
$317$	−21.6627	−1.21670	−0.608348	−	0.793670i	$-0.708168\pi$
$317$	−21.6627	−1.21670	−0.608348	+	0.793670i	$0.708168\pi$
$318$	0	0
$319$	−2.74669	−0.153785
$320$	0	0
$321$	−4.56932	−0.255035
$322$	0	0
$323$	3.06566	0.170578
$324$	0	0
$325$	−13.2550	−0.735253
$326$	0	0
$327$	−6.65094	−0.367798
$328$	0	0
$329$	−15.6272	−0.861556
$330$	0	0
$331$	−8.50054	−0.467232	−0.233616	−	0.972329i	$-0.575056\pi$
$331$	−8.50054	−0.467232	−0.233616	+	0.972329i	$0.575056\pi$
$332$	0	0
$333$	−5.25315	−0.287871
$334$	0	0
$335$	−4.99305	−0.272799
$336$	0	0
$337$	−20.8670	−1.13670	−0.568350	−	0.822787i	$-0.692418\pi$
$337$	−20.8670	−1.13670	−0.568350	+	0.822787i	$0.692418\pi$
$338$	0	0
$339$	7.23728	0.393075
$340$	0	0
$341$	9.63311	0.521662
$342$	0	0
$343$	−8.62852	−0.465896
$344$	0	0
$345$	0.425546	0.0229106
$346$	0	0
$347$	15.6962	0.842616	0.421308	−	0.906918i	$-0.361571\pi$
$347$	15.6962	0.842616	0.421308	+	0.906918i	$0.361571\pi$
$348$	0	0
$349$	−36.1278	−1.93388	−0.966938	−	0.255011i	$-0.917921\pi$
$349$	−36.1278	−1.93388	−0.966938	+	0.255011i	$0.917921\pi$
$350$	0	0
$351$	−2.75061	−0.146817
$352$	0	0
$353$	16.4759	0.876924	0.438462	−	0.898750i	$-0.355523\pi$
$353$	16.4759	0.876924	0.438462	+	0.898750i	$0.355523\pi$
$354$	0	0
$355$	−4.95232	−0.262842
$356$	0	0
$357$	−3.01503	−0.159572
$358$	0	0
$359$	9.04694	0.477479	0.238740	−	0.971084i	$-0.423266\pi$
$359$	9.04694	0.477479	0.238740	+	0.971084i	$0.423266\pi$
$360$	0	0
$361$	−2.30581	−0.121358
$362$	0	0
$363$	3.45569	0.181377
$364$	0	0
$365$	3.47671	0.181980
$366$	0	0
$367$	28.9960	1.51358	0.756791	−	0.653657i	$-0.226766\pi$
$367$	28.9960	1.51358	0.756791	+	0.653657i	$0.226766\pi$
$368$	0	0
$369$	11.5128	0.599331
$370$	0	0
$371$	2.25749	0.117203
$372$	0	0
$373$	24.6416	1.27589	0.637946	−	0.770081i	$-0.279784\pi$
$373$	24.6416	1.27589	0.637946	+	0.770081i	$0.279784\pi$
$374$	0	0
$375$	−4.17840	−0.215771
$376$	0	0
$377$	−2.75061	−0.141664
$378$	0	0
$379$	8.85063	0.454627	0.227313	−	0.973822i	$-0.427006\pi$
$379$	8.85063	0.454627	0.227313	+	0.973822i	$0.427006\pi$
$380$	0	0
$381$	−6.27278	−0.321364
$382$	0	0
$383$	21.9176	1.11994	0.559969	−	0.828513i	$-0.310813\pi$
$383$	21.9176	1.11994	0.559969	+	0.828513i	$0.310813\pi$
$384$	0	0
$385$	4.69684	0.239373
$386$	0	0
$387$	7.71894	0.392376
$388$	0	0
$389$	−23.2463	−1.17863	−0.589316	−	0.807902i	$-0.700603\pi$
$389$	−23.2463	−1.17863	−0.589316	+	0.807902i	$0.700603\pi$
$390$	0	0
$391$	−0.750311	−0.0379449
$392$	0	0
$393$	−4.38075	−0.220980
$394$	0	0
$395$	0.560213	0.0281874
$396$	0	0
$397$	−19.2387	−0.965563	−0.482781	−	0.875741i	$-0.660373\pi$
$397$	−19.2387	−0.965563	−0.482781	+	0.875741i	$0.660373\pi$
$398$	0	0
$399$	−16.4185	−0.821951
$400$	0	0
$401$	−8.72345	−0.435628	−0.217814	−	0.975990i	$-0.569893\pi$
$401$	−8.72345	−0.435628	−0.217814	+	0.975990i	$0.569893\pi$
$402$	0	0
$403$	9.64687	0.480545
$404$	0	0
$405$	−0.425546	−0.0211455
$406$	0	0
$407$	−14.4288	−0.715208
$408$	0	0
$409$	16.3605	0.808977	0.404489	−	0.914543i	$-0.367450\pi$
$409$	16.3605	0.808977	0.404489	+	0.914543i	$0.367450\pi$
$410$	0	0
$411$	−1.73154	−0.0854104
$412$	0	0
$413$	44.5364	2.19149
$414$	0	0
$415$	−3.79738	−0.186406
$416$	0	0
$417$	−7.32975	−0.358940
$418$	0	0
$419$	−19.4575	−0.950563	−0.475282	−	0.879834i	$-0.657654\pi$
$419$	−19.4575	−0.950563	−0.475282	+	0.879834i	$0.657654\pi$
$420$	0	0
$421$	−6.28820	−0.306468	−0.153234	−	0.988190i	$-0.548969\pi$
$421$	−6.28820	−0.306468	−0.153234	+	0.988190i	$0.548969\pi$
$422$	0	0
$423$	3.88894	0.189087
$424$	0	0
$425$	3.61568	0.175386
$426$	0	0
$427$	56.4025	2.72951
$428$	0	0
$429$	−7.55508	−0.364763
$430$	0	0
$431$	−7.41978	−0.357398	−0.178699	−	0.983904i	$-0.557189\pi$
$431$	−7.41978	−0.357398	−0.178699	+	0.983904i	$0.557189\pi$
$432$	0	0
$433$	−18.8000	−0.903471	−0.451736	−	0.892152i	$-0.649195\pi$
$433$	−18.8000	−0.903471	−0.451736	+	0.892152i	$0.649195\pi$
$434$	0	0
$435$	−0.425546	−0.0204034
$436$	0	0
$437$	−4.08585	−0.195453
$438$	0	0
$439$	−19.3463	−0.923350	−0.461675	−	0.887049i	$-0.652751\pi$
$439$	−19.3463	−0.923350	−0.461675	+	0.887049i	$0.652751\pi$
$440$	0	0
$441$	9.14727	0.435584
$442$	0	0
$443$	−21.8430	−1.03779	−0.518896	−	0.854837i	$-0.673657\pi$
$443$	−21.8430	−1.03779	−0.518896	+	0.854837i	$0.673657\pi$
$444$	0	0
$445$	−1.23667	−0.0586236
$446$	0	0
$447$	2.50672	0.118564
$448$	0	0
$449$	−1.37864	−0.0650618	−0.0325309	−	0.999471i	$-0.510357\pi$
$449$	−1.37864	−0.0650618	−0.0325309	+	0.999471i	$0.510357\pi$
$450$	0	0
$451$	31.6220	1.48902
$452$	0	0
$453$	21.6115	1.01539
$454$	0	0
$455$	4.70355	0.220506
$456$	0	0
$457$	8.11178	0.379453	0.189727	−	0.981837i	$-0.439240\pi$
$457$	8.11178	0.379453	0.189727	+	0.981837i	$0.439240\pi$
$458$	0	0
$459$	0.750311	0.0350215
$460$	0	0
$461$	−12.4751	−0.581022	−0.290511	−	0.956872i	$-0.593825\pi$
$461$	−12.4751	−0.581022	−0.290511	+	0.956872i	$0.593825\pi$
$462$	0	0
$463$	−33.1911	−1.54252	−0.771261	−	0.636519i	$-0.780374\pi$
$463$	−33.1911	−1.54252	−0.771261	+	0.636519i	$0.780374\pi$
$464$	0	0
$465$	1.49246	0.0692112
$466$	0	0
$467$	−5.38720	−0.249290	−0.124645	−	0.992201i	$-0.539779\pi$
$467$	−5.38720	−0.249290	−0.124645	+	0.992201i	$0.539779\pi$
$468$	0	0
$469$	−47.1486	−2.17712
$470$	0	0
$471$	−14.6172	−0.673524
$472$	0	0
$473$	21.2015	0.974848
$474$	0	0
$475$	19.6894	0.903410
$476$	0	0
$477$	−0.561794	−0.0257228
$478$	0	0
$479$	−26.5235	−1.21189	−0.605945	−	0.795506i	$-0.707205\pi$
$479$	−26.5235	−1.21189	−0.605945	+	0.795506i	$0.707205\pi$
$480$	0	0
$481$	−14.4494	−0.658835
$482$	0	0
$483$	4.01837	0.182842
$484$	0	0
$485$	7.53807	0.342286
$486$	0	0
$487$	−34.1450	−1.54726	−0.773630	−	0.633638i	$-0.781561\pi$
$487$	−34.1450	−1.54726	−0.773630	+	0.633638i	$0.781561\pi$
$488$	0	0
$489$	0.0491942	0.00222464
$490$	0	0
$491$	−29.3107	−1.32277	−0.661387	−	0.750044i	$-0.730032\pi$
$491$	−29.3107	−1.32277	−0.661387	+	0.750044i	$0.730032\pi$
$492$	0	0
$493$	0.750311	0.0337923
$494$	0	0
$495$	−1.16884	−0.0525356
$496$	0	0
$497$	−46.7640	−2.09765
$498$	0	0
$499$	−5.79016	−0.259203	−0.129602	−	0.991566i	$-0.541370\pi$
$499$	−5.79016	−0.259203	−0.129602	+	0.991566i	$0.541370\pi$
$500$	0	0
$501$	−0.101312	−0.00452630
$502$	0	0
$503$	−35.0276	−1.56180	−0.780902	−	0.624653i	$-0.785240\pi$
$503$	−35.0276	−1.56180	−0.780902	+	0.624653i	$0.785240\pi$
$504$	0	0
$505$	4.30599	0.191614
$506$	0	0
$507$	5.43413	0.241338
$508$	0	0
$509$	28.0283	1.24233	0.621166	−	0.783679i	$-0.286659\pi$
$509$	28.0283	1.24233	0.621166	+	0.783679i	$0.286659\pi$
$510$	0	0
$511$	32.8301	1.45232
$512$	0	0
$513$	4.08585	0.180395
$514$	0	0
$515$	−6.22845	−0.274458
$516$	0	0
$517$	10.6817	0.469782
$518$	0	0
$519$	−25.4548	−1.11734
$520$	0	0
$521$	5.66552	0.248211	0.124106	−	0.992269i	$-0.460394\pi$
$521$	5.66552	0.248211	0.124106	+	0.992269i	$0.460394\pi$
$522$	0	0
$523$	0.708170	0.0309661	0.0154831	−	0.999880i	$-0.495071\pi$
$523$	0.708170	0.0309661	0.0154831	+	0.999880i	$0.495071\pi$
$524$	0	0
$525$	−19.3642	−0.845121
$526$	0	0
$527$	−2.63147	−0.114629
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.0832	−0.480971
$532$	0	0
$533$	31.6672	1.37166
$534$	0	0
$535$	−1.94445	−0.0840661
$536$	0	0
$537$	21.2235	0.915863
$538$	0	0
$539$	25.1247	1.08220
$540$	0	0
$541$	38.9948	1.67652	0.838260	−	0.545271i	$-0.183573\pi$
$541$	38.9948	1.67652	0.838260	+	0.545271i	$0.183573\pi$
$542$	0	0
$543$	5.95457	0.255535
$544$	0	0
$545$	−2.83028	−0.121236
$546$	0	0
$547$	−10.6089	−0.453605	−0.226802	−	0.973941i	$-0.572827\pi$
$547$	−10.6089	−0.453605	−0.226802	+	0.973941i	$0.572827\pi$
$548$	0	0
$549$	−14.0362	−0.599050
$550$	0	0
$551$	4.08585	0.174063
$552$	0	0
$553$	5.29001	0.224954
$554$	0	0
$555$	−2.23546	−0.0948898
$556$	0	0
$557$	14.3143	0.606517	0.303259	−	0.952908i	$-0.401925\pi$
$557$	14.3143	0.606517	0.303259	+	0.952908i	$0.401925\pi$
$558$	0	0
$559$	21.2318	0.898010
$560$	0	0
$561$	2.06087	0.0870101
$562$	0	0
$563$	−9.43658	−0.397704	−0.198852	−	0.980029i	$-0.563721\pi$
$563$	−9.43658	−0.397704	−0.198852	+	0.980029i	$0.563721\pi$
$564$	0	0
$565$	3.07979	0.129568
$566$	0	0
$567$	−4.01837	−0.168756
$568$	0	0
$569$	−8.71454	−0.365332	−0.182666	−	0.983175i	$-0.558473\pi$
$569$	−8.71454	−0.365332	−0.182666	+	0.983175i	$0.558473\pi$
$570$	0	0
$571$	−34.4748	−1.44272	−0.721362	−	0.692558i	$-0.756483\pi$
$571$	−34.4748	−1.44272	−0.721362	+	0.692558i	$0.756483\pi$
$572$	0	0
$573$	−22.3732	−0.934654
$574$	0	0
$575$	−4.81891	−0.200962
$576$	0	0
$577$	−6.60547	−0.274989	−0.137495	−	0.990502i	$-0.543905\pi$
$577$	−6.60547	−0.274989	−0.137495	+	0.990502i	$0.543905\pi$
$578$	0	0
$579$	14.1935	0.589860
$580$	0	0
$581$	−35.8581	−1.48765
$582$	0	0
$583$	−1.54307	−0.0639076
$584$	0	0
$585$	−1.17051	−0.0483947
$586$	0	0
$587$	−36.2328	−1.49549	−0.747744	−	0.663987i	$-0.768863\pi$
$587$	−36.2328	−1.49549	−0.747744	+	0.663987i	$0.768863\pi$
$588$	0	0
$589$	−14.3298	−0.590448
$590$	0	0
$591$	16.4768	0.677766
$592$	0	0
$593$	2.12562	0.0872888	0.0436444	−	0.999047i	$-0.486103\pi$
$593$	2.12562	0.0872888	0.0436444	+	0.999047i	$0.486103\pi$
$594$	0	0
$595$	−1.28303	−0.0525992
$596$	0	0
$597$	−11.4091	−0.466943
$598$	0	0
$599$	−1.29513	−0.0529177	−0.0264588	−	0.999650i	$-0.508423\pi$
$599$	−1.29513	−0.0529177	−0.0264588	+	0.999650i	$0.508423\pi$
$600$	0	0
$601$	0.407705	0.0166306	0.00831532	−	0.999965i	$-0.497353\pi$
$601$	0.407705	0.0166306	0.00831532	+	0.999965i	$0.497353\pi$
$602$	0	0
$603$	11.7333	0.477816
$604$	0	0
$605$	1.47055	0.0597866
$606$	0	0
$607$	23.5993	0.957865	0.478932	−	0.877852i	$-0.341024\pi$
$607$	23.5993	0.957865	0.478932	+	0.877852i	$0.341024\pi$
$608$	0	0
$609$	−4.01837	−0.162832
$610$	0	0
$611$	10.6970	0.432754
$612$	0	0
$613$	26.3821	1.06556	0.532782	−	0.846253i	$-0.321147\pi$
$613$	26.3821	1.06556	0.532782	+	0.846253i	$0.321147\pi$
$614$	0	0
$615$	4.89921	0.197555
$616$	0	0
$617$	−11.7704	−0.473857	−0.236928	−	0.971527i	$-0.576141\pi$
$617$	−11.7704	−0.473857	−0.236928	+	0.971527i	$0.576141\pi$
$618$	0	0
$619$	21.5097	0.864550	0.432275	−	0.901742i	$-0.357711\pi$
$619$	21.5097	0.864550	0.432275	+	0.901742i	$0.357711\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−11.6777	−0.467855
$624$	0	0
$625$	22.3165	0.892658
$626$	0	0
$627$	11.2226	0.448186
$628$	0	0
$629$	3.94150	0.157158
$630$	0	0
$631$	35.0626	1.39582	0.697910	−	0.716186i	$-0.254114\pi$
$631$	35.0626	1.39582	0.697910	+	0.716186i	$0.254114\pi$
$632$	0	0
$633$	13.0094	0.517078
$634$	0	0
$635$	−2.66935	−0.105930
$636$	0	0
$637$	25.1606	0.996899
$638$	0	0
$639$	11.6376	0.460375
$640$	0	0
$641$	−34.8561	−1.37673	−0.688367	−	0.725362i	$-0.741672\pi$
$641$	−34.8561	−1.37673	−0.688367	+	0.725362i	$0.741672\pi$
$642$	0	0
$643$	33.0448	1.30316	0.651579	−	0.758581i	$-0.274107\pi$
$643$	33.0448	1.30316	0.651579	+	0.758581i	$0.274107\pi$
$644$	0	0
$645$	3.28476	0.129337
$646$	0	0
$647$	3.65962	0.143874	0.0719372	−	0.997409i	$-0.477082\pi$
$647$	3.65962	0.143874	0.0719372	+	0.997409i	$0.477082\pi$
$648$	0	0
$649$	−30.4422	−1.19496
$650$	0	0
$651$	14.0931	0.552352
$652$	0	0
$653$	15.3181	0.599444	0.299722	−	0.954027i	$-0.403106\pi$
$653$	15.3181	0.599444	0.299722	+	0.954027i	$0.403106\pi$
$654$	0	0
$655$	−1.86421	−0.0728407
$656$	0	0
$657$	−8.17001	−0.318743
$658$	0	0
$659$	19.1016	0.744093	0.372046	−	0.928214i	$-0.378656\pi$
$659$	19.1016	0.744093	0.372046	+	0.928214i	$0.378656\pi$
$660$	0	0
$661$	−15.0959	−0.587164	−0.293582	−	0.955934i	$-0.594847\pi$
$661$	−15.0959	−0.587164	−0.293582	+	0.955934i	$0.594847\pi$
$662$	0	0
$663$	2.06382	0.0801520
$664$	0	0
$665$	−6.98680	−0.270937
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	25.1539	0.972505
$670$	0	0
$671$	−38.5530	−1.48832
$672$	0	0
$673$	44.6238	1.72012	0.860061	−	0.510191i	$-0.170425\pi$
$673$	44.6238	1.72012	0.860061	+	0.510191i	$0.170425\pi$
$674$	0	0
$675$	4.81891	0.185480
$676$	0	0
$677$	35.6671	1.37080	0.685400	−	0.728167i	$-0.259628\pi$
$677$	35.6671	1.37080	0.685400	+	0.728167i	$0.259628\pi$
$678$	0	0
$679$	71.1809	2.73167
$680$	0	0
$681$	21.9530	0.841240
$682$	0	0
$683$	−18.3501	−0.702146	−0.351073	−	0.936348i	$-0.614183\pi$
$683$	−18.3501	−0.702146	−0.351073	+	0.936348i	$0.614183\pi$
$684$	0	0
$685$	−0.736848	−0.0281535
$686$	0	0
$687$	−25.3691	−0.967893
$688$	0	0
$689$	−1.54528	−0.0588704
$690$	0	0
$691$	45.4845	1.73031	0.865155	−	0.501504i	$-0.167220\pi$
$691$	45.4845	1.73031	0.865155	+	0.501504i	$0.167220\pi$
$692$	0	0
$693$	−11.0372	−0.419269
$694$	0	0
$695$	−3.11914	−0.118316
$696$	0	0
$697$	−8.63816	−0.327194
$698$	0	0
$699$	−12.1740	−0.460463
$700$	0	0
$701$	−19.9987	−0.755339	−0.377670	−	0.925940i	$-0.623275\pi$
$701$	−19.9987	−0.755339	−0.377670	+	0.925940i	$0.623275\pi$
$702$	0	0
$703$	21.4636	0.809515
$704$	0	0
$705$	1.65492	0.0623280
$706$	0	0
$707$	40.6608	1.52921
$708$	0	0
$709$	1.71473	0.0643979	0.0321989	−	0.999481i	$-0.489749\pi$
$709$	1.71473	0.0643979	0.0321989	+	0.999481i	$0.489749\pi$
$710$	0	0
$711$	−1.31646	−0.0493710
$712$	0	0
$713$	3.50717	0.131345
$714$	0	0
$715$	−3.21503	−0.120235
$716$	0	0
$717$	22.5942	0.843796
$718$	0	0
$719$	−12.0054	−0.447725	−0.223862	−	0.974621i	$-0.571867\pi$
$719$	−12.0054	−0.447725	−0.223862	+	0.974621i	$0.571867\pi$
$720$	0	0
$721$	−58.8143	−2.19036
$722$	0	0
$723$	9.98289	0.371268
$724$	0	0
$725$	4.81891	0.178970
$726$	0	0
$727$	−48.7484	−1.80798	−0.903989	−	0.427557i	$-0.859374\pi$
$727$	−48.7484	−1.80798	−0.903989	+	0.427557i	$0.859374\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.79161	−0.214210
$732$	0	0
$733$	17.4743	0.645428	0.322714	−	0.946497i	$-0.395405\pi$
$733$	17.4743	0.645428	0.322714	+	0.946497i	$0.395405\pi$
$734$	0	0
$735$	3.89258	0.143580
$736$	0	0
$737$	32.2277	1.18712
$738$	0	0
$739$	−7.07816	−0.260374	−0.130187	−	0.991489i	$-0.541558\pi$
$739$	−7.07816	−0.260374	−0.130187	+	0.991489i	$0.541558\pi$
$740$	0	0
$741$	11.2386	0.412860
$742$	0	0
$743$	31.5073	1.15589	0.577945	−	0.816076i	$-0.303855\pi$
$743$	31.5073	1.15589	0.577945	+	0.816076i	$0.303855\pi$
$744$	0	0
$745$	1.06672	0.0390817
$746$	0	0
$747$	8.92356	0.326496
$748$	0	0
$749$	−18.3612	−0.670904
$750$	0	0
$751$	17.9447	0.654812	0.327406	−	0.944884i	$-0.393825\pi$
$751$	17.9447	0.654812	0.327406	+	0.944884i	$0.393825\pi$
$752$	0	0
$753$	9.85136	0.359003
$754$	0	0
$755$	9.19666	0.334701
$756$	0	0
$757$	4.40530	0.160113	0.0800566	−	0.996790i	$-0.474490\pi$
$757$	4.40530	0.160113	0.0800566	+	0.996790i	$0.474490\pi$
$758$	0	0
$759$	−2.74669	−0.0996985
$760$	0	0
$761$	−4.46186	−0.161742	−0.0808712	−	0.996725i	$-0.525770\pi$
$761$	−4.46186	−0.161742	−0.0808712	+	0.996725i	$0.525770\pi$
$762$	0	0
$763$	−26.7259	−0.967544
$764$	0	0
$765$	0.319292	0.0115440
$766$	0	0
$767$	−30.4856	−1.10077
$768$	0	0
$769$	24.2390	0.874080	0.437040	−	0.899442i	$-0.356027\pi$
$769$	24.2390	0.874080	0.437040	+	0.899442i	$0.356027\pi$
$770$	0	0
$771$	−1.69890	−0.0611844
$772$	0	0
$773$	−5.22313	−0.187863	−0.0939315	−	0.995579i	$-0.529943\pi$
$773$	−5.22313	−0.187863	−0.0939315	+	0.995579i	$0.529943\pi$
$774$	0	0
$775$	−16.9007	−0.607092
$776$	0	0
$777$	−21.1091	−0.757284
$778$	0	0
$779$	−47.0395	−1.68536
$780$	0	0
$781$	31.9648	1.14379
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−6.22028	−0.222011
$786$	0	0
$787$	−26.6085	−0.948492	−0.474246	−	0.880392i	$-0.657279\pi$
$787$	−26.6085	−0.948492	−0.474246	+	0.880392i	$0.657279\pi$
$788$	0	0
$789$	7.43928	0.264845
$790$	0	0
$791$	29.0821	1.03404
$792$	0	0
$793$	−38.6081	−1.37101
$794$	0	0
$795$	−0.239069	−0.00847890
$796$	0	0
$797$	−24.7448	−0.876507	−0.438253	−	0.898852i	$-0.644403\pi$
$797$	−24.7448	−0.876507	−0.438253	+	0.898852i	$0.644403\pi$
$798$	0	0
$799$	−2.91792	−0.103229
$800$	0	0
$801$	2.90607	0.102681
$802$	0	0
$803$	−22.4405	−0.791908
$804$	0	0
$805$	1.71000	0.0602695
$806$	0	0
$807$	29.6573	1.04399
$808$	0	0
$809$	−6.39850	−0.224959	−0.112480	−	0.993654i	$-0.535879\pi$
$809$	−6.39850	−0.224959	−0.112480	+	0.993654i	$0.535879\pi$
$810$	0	0
$811$	12.3958	0.435276	0.217638	−	0.976030i	$-0.430165\pi$
$811$	12.3958	0.435276	0.217638	+	0.976030i	$0.430165\pi$
$812$	0	0
$813$	−17.2669	−0.605575
$814$	0	0
$815$	0.0209344	0.000733300 0
$816$	0	0
$817$	−31.5385	−1.10339
$818$	0	0
$819$	−11.0530	−0.386222
$820$	0	0
$821$	−14.8206	−0.517242	−0.258621	−	0.965979i	$-0.583268\pi$
$821$	−14.8206	−0.517242	−0.258621	+	0.965979i	$0.583268\pi$
$822$	0	0
$823$	−10.5541	−0.367894	−0.183947	−	0.982936i	$-0.558888\pi$
$823$	−10.5541	−0.367894	−0.183947	+	0.982936i	$0.558888\pi$
$824$	0	0
$825$	13.2361	0.460820
$826$	0	0
$827$	−45.1063	−1.56850	−0.784250	−	0.620445i	$-0.786952\pi$
$827$	−45.1063	−1.56850	−0.784250	+	0.620445i	$0.786952\pi$
$828$	0	0
$829$	20.4287	0.709518	0.354759	−	0.934958i	$-0.384563\pi$
$829$	20.4287	0.709518	0.354759	+	0.934958i	$0.384563\pi$
$830$	0	0
$831$	22.6426	0.785463
$832$	0	0
$833$	−6.86330	−0.237799
$834$	0	0
$835$	−0.0431130	−0.00149199
$836$	0	0
$837$	−3.50717	−0.121226
$838$	0	0
$839$	4.66753	0.161141	0.0805705	−	0.996749i	$-0.474326\pi$
$839$	4.66753	0.161141	0.0805705	+	0.996749i	$0.474326\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	26.8833	0.925910
$844$	0	0
$845$	2.31247	0.0795514
$846$	0	0
$847$	13.8862	0.477137
$848$	0	0
$849$	−11.4876	−0.394252
$850$	0	0
$851$	−5.25315	−0.180076
$852$	0	0
$853$	−44.5932	−1.52684	−0.763421	−	0.645901i	$-0.776482\pi$
$853$	−44.5932	−1.52684	−0.763421	+	0.645901i	$0.776482\pi$
$854$	0	0
$855$	1.73872	0.0594629
$856$	0	0
$857$	3.26798	0.111632	0.0558160	−	0.998441i	$-0.482224\pi$
$857$	3.26798	0.111632	0.0558160	+	0.998441i	$0.482224\pi$
$858$	0	0
$859$	−21.7938	−0.743596	−0.371798	−	0.928314i	$-0.621259\pi$
$859$	−21.7938	−0.743596	−0.371798	+	0.928314i	$0.621259\pi$
$860$	0	0
$861$	46.2626	1.57662
$862$	0	0
$863$	−31.5587	−1.07427	−0.537135	−	0.843496i	$-0.680494\pi$
$863$	−31.5587	−1.07427	−0.537135	+	0.843496i	$0.680494\pi$
$864$	0	0
$865$	−10.8322	−0.368305
$866$	0	0
$867$	16.4370	0.558231
$868$	0	0
$869$	−3.61590	−0.122661
$870$	0	0
$871$	32.2737	1.09355
$872$	0	0
$873$	−17.7139	−0.599524
$874$	0	0
$875$	−16.7903	−0.567617
$876$	0	0
$877$	−3.24755	−0.109662	−0.0548310	−	0.998496i	$-0.517462\pi$
$877$	−3.24755	−0.109662	−0.0548310	+	0.998496i	$0.517462\pi$
$878$	0	0
$879$	−19.4370	−0.655594
$880$	0	0
$881$	7.16127	0.241269	0.120635	−	0.992697i	$-0.461507\pi$
$881$	7.16127	0.241269	0.120635	+	0.992697i	$0.461507\pi$
$882$	0	0
$883$	−24.0619	−0.809747	−0.404874	−	0.914373i	$-0.632684\pi$
$883$	−24.0619	−0.809747	−0.404874	+	0.914373i	$0.632684\pi$
$884$	0	0
$885$	−4.71642	−0.158541
$886$	0	0
$887$	−41.5681	−1.39572	−0.697860	−	0.716234i	$-0.745864\pi$
$887$	−41.5681	−1.39572	−0.697860	+	0.716234i	$0.745864\pi$
$888$	0	0
$889$	−25.2063	−0.845393
$890$	0	0
$891$	2.74669	0.0920176
$892$	0	0
$893$	−15.8897	−0.531727
$894$	0	0
$895$	9.03159	0.301893
$896$	0	0
$897$	−2.75061	−0.0918403
$898$	0	0
$899$	−3.50717	−0.116971
$900$	0	0
$901$	0.421520	0.0140429
$902$	0	0
$903$	31.0175	1.03220
$904$	0	0
$905$	2.53394	0.0842310
$906$	0	0
$907$	40.6957	1.35128	0.675640	−	0.737232i	$-0.263868\pi$
$907$	40.6957	1.35128	0.675640	+	0.737232i	$0.263868\pi$
$908$	0	0
$909$	−10.1187	−0.335617
$910$	0	0
$911$	56.4805	1.87128	0.935642	−	0.352951i	$-0.114822\pi$
$911$	56.4805	1.87128	0.935642	+	0.352951i	$0.114822\pi$
$912$	0	0
$913$	24.5103	0.811171
$914$	0	0
$915$	−5.97304	−0.197462
$916$	0	0
$917$	−17.6035	−0.581318
$918$	0	0
$919$	−27.7336	−0.914847	−0.457424	−	0.889249i	$-0.651228\pi$
$919$	−27.7336	−0.914847	−0.457424	+	0.889249i	$0.651228\pi$
$920$	0	0
$921$	−5.58977	−0.184189
$922$	0	0
$923$	32.0105	1.05364
$924$	0	0
$925$	25.3145	0.832334
$926$	0	0
$927$	14.6364	0.480722
$928$	0	0
$929$	−25.2779	−0.829342	−0.414671	−	0.909971i	$-0.636103\pi$
$929$	−25.2779	−0.829342	−0.414671	+	0.909971i	$0.636103\pi$
$930$	0	0
$931$	−37.3744	−1.22490
$932$	0	0
$933$	28.5683	0.935286
$934$	0	0
$935$	0.876996	0.0286808
$936$	0	0
$937$	35.3248	1.15401	0.577006	−	0.816740i	$-0.304221\pi$
$937$	35.3248	1.15401	0.577006	+	0.816740i	$0.304221\pi$
$938$	0	0
$939$	12.0076	0.391852
$940$	0	0
$941$	−23.6560	−0.771164	−0.385582	−	0.922674i	$-0.625999\pi$
$941$	−23.6560	−0.771164	−0.385582	+	0.922674i	$0.625999\pi$
$942$	0	0
$943$	11.5128	0.374908
$944$	0	0
$945$	−1.71000	−0.0556263
$946$	0	0
$947$	43.9569	1.42841	0.714203	−	0.699938i	$-0.246789\pi$
$947$	43.9569	1.42841	0.714203	+	0.699938i	$0.246789\pi$
$948$	0	0
$949$	−22.4726	−0.729490
$950$	0	0
$951$	21.6627	0.702460
$952$	0	0
$953$	−11.9240	−0.386256	−0.193128	−	0.981174i	$-0.561863\pi$
$953$	−11.9240	−0.386256	−0.193128	+	0.981174i	$0.561863\pi$
$954$	0	0
$955$	−9.52082	−0.308087
$956$	0	0
$957$	2.74669	0.0887879
$958$	0	0
$959$	−6.95795	−0.224684
$960$	0	0
$961$	−18.6998	−0.603218
$962$	0	0
$963$	4.56932	0.147244
$964$	0	0
$965$	6.03997	0.194433
$966$	0	0
$967$	35.6155	1.14532	0.572658	−	0.819794i	$-0.305912\pi$
$967$	35.6155	1.14532	0.572658	+	0.819794i	$0.305912\pi$
$968$	0	0
$969$	−3.06566	−0.0984832
$970$	0	0
$971$	−53.8649	−1.72861	−0.864304	−	0.502970i	$-0.832241\pi$
$971$	−53.8649	−1.72861	−0.864304	+	0.502970i	$0.832241\pi$
$972$	0	0
$973$	−29.4536	−0.944240
$974$	0	0
$975$	13.2550	0.424498
$976$	0	0
$977$	−33.3196	−1.06599	−0.532994	−	0.846119i	$-0.678933\pi$
$977$	−33.3196	−1.06599	−0.532994	+	0.846119i	$0.678933\pi$
$978$	0	0
$979$	7.98207	0.255108
$980$	0	0
$981$	6.65094	0.212348
$982$	0	0
$983$	9.83804	0.313785	0.156892	−	0.987616i	$-0.449852\pi$
$983$	9.83804	0.313785	0.156892	+	0.987616i	$0.449852\pi$
$984$	0	0
$985$	7.01164	0.223409
$986$	0	0
$987$	15.6272	0.497419
$988$	0	0
$989$	7.71894	0.245448
$990$	0	0
$991$	−2.64379	−0.0839827	−0.0419914	−	0.999118i	$-0.513370\pi$
$991$	−2.64379	−0.0839827	−0.0419914	+	0.999118i	$0.513370\pi$
$992$	0	0
$993$	8.50054	0.269756
$994$	0	0
$995$	−4.85509	−0.153917
$996$	0	0
$997$	−44.8077	−1.41908	−0.709538	−	0.704668i	$-0.751096\pi$
$997$	−44.8077	−1.41908	−0.709538	+	0.704668i	$0.751096\pi$
$998$	0	0
$999$	5.25315	0.166202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.5	✓	12	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.425546 q^{5} -4.01837 q^{7} +1.00000 q^{9} +2.74669 q^{11} +2.75061 q^{13} +0.425546 q^{15} -0.750311 q^{17} -4.08585 q^{19} +4.01837 q^{21} +1.00000 q^{23} -4.81891 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.50717 q^{31} -2.74669 q^{33} +1.71000 q^{35} -5.25315 q^{37} -2.75061 q^{39} +11.5128 q^{41} +7.71894 q^{43} -0.425546 q^{45} +3.88894 q^{47} +9.14727 q^{49} +0.750311 q^{51} -0.561794 q^{53} -1.16884 q^{55} +4.08585 q^{57} -11.0832 q^{59} -14.0362 q^{61} -4.01837 q^{63} -1.17051 q^{65} +11.7333 q^{67} -1.00000 q^{69} +11.6376 q^{71} -8.17001 q^{73} +4.81891 q^{75} -11.0372 q^{77} -1.31646 q^{79} +1.00000 q^{81} +8.92356 q^{83} +0.319292 q^{85} +1.00000 q^{87} +2.90607 q^{89} -11.0530 q^{91} -3.50717 q^{93} +1.73872 q^{95} -17.7139 q^{97} +2.74669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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