LMFDB - Embedded newform 8004.2.a.g.1.2

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			1.2
Root			$3.12492$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} -3.88477 q^{11} -5.76924 q^{13} +3.12492 q^{15} +1.77786 q^{17} +1.14116 q^{19} -4.33538 q^{21} +1.00000 q^{23} +4.76514 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.76454 q^{31} +3.88477 q^{33} -13.5477 q^{35} +9.00519 q^{37} +5.76924 q^{39} -1.21895 q^{41} +6.15921 q^{43} -3.12492 q^{45} -1.33639 q^{47} +11.7955 q^{49} -1.77786 q^{51} -5.98281 q^{53} +12.1396 q^{55} -1.14116 q^{57} -7.53101 q^{59} +2.35532 q^{61} +4.33538 q^{63} +18.0284 q^{65} -3.03611 q^{67} -1.00000 q^{69} +6.64492 q^{71} +2.08303 q^{73} -4.76514 q^{75} -16.8420 q^{77} -13.0123 q^{79} +1.00000 q^{81} -14.3584 q^{83} -5.55566 q^{85} +1.00000 q^{87} -4.89965 q^{89} -25.0119 q^{91} -5.76454 q^{93} -3.56603 q^{95} +7.62077 q^{97} -3.88477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.12492	−1.39751	−0.698754	−	0.715362i	$-0.746262\pi$
$5$	−3.12492	−1.39751	−0.698754	+	0.715362i	$0.746262\pi$
$6$	0	0
$7$	4.33538	1.63862	0.819310	−	0.573351i	$-0.194357\pi$
$7$	4.33538	1.63862	0.819310	+	0.573351i	$0.194357\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.88477	−1.17130	−0.585652	−	0.810563i	$-0.699161\pi$
$11$	−3.88477	−1.17130	−0.585652	+	0.810563i	$0.699161\pi$
$12$	0	0
$13$	−5.76924	−1.60010	−0.800050	−	0.599933i	$-0.795194\pi$
$13$	−5.76924	−1.60010	−0.800050	+	0.599933i	$0.795194\pi$
$14$	0	0
$15$	3.12492	0.806851
$16$	0	0
$17$	1.77786	0.431194	0.215597	−	0.976482i	$-0.430830\pi$
$17$	1.77786	0.431194	0.215597	+	0.976482i	$0.430830\pi$
$18$	0	0
$19$	1.14116	0.261799	0.130900	−	0.991396i	$-0.458213\pi$
$19$	1.14116	0.261799	0.130900	+	0.991396i	$0.458213\pi$
$20$	0	0
$21$	−4.33538	−0.946058
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.76514	0.953028
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.76454	1.03534	0.517671	−	0.855580i	$-0.326799\pi$
$31$	5.76454	1.03534	0.517671	+	0.855580i	$0.326799\pi$
$32$	0	0
$33$	3.88477	0.676252
$34$	0	0
$35$	−13.5477	−2.28998
$36$	0	0
$37$	9.00519	1.48044	0.740222	−	0.672362i	$-0.234720\pi$
$37$	9.00519	1.48044	0.740222	+	0.672362i	$0.234720\pi$
$38$	0	0
$39$	5.76924	0.923818
$40$	0	0
$41$	−1.21895	−0.190368	−0.0951839	−	0.995460i	$-0.530344\pi$
$41$	−1.21895	−0.190368	−0.0951839	+	0.995460i	$0.530344\pi$
$42$	0	0
$43$	6.15921	0.939271	0.469636	−	0.882860i	$-0.344385\pi$
$43$	6.15921	0.939271	0.469636	+	0.882860i	$0.344385\pi$
$44$	0	0
$45$	−3.12492	−0.465836
$46$	0	0
$47$	−1.33639	−0.194932	−0.0974661	−	0.995239i	$-0.531074\pi$
$47$	−1.33639	−0.194932	−0.0974661	+	0.995239i	$0.531074\pi$
$48$	0	0
$49$	11.7955	1.68508
$50$	0	0
$51$	−1.77786	−0.248950
$52$	0	0
$53$	−5.98281	−0.821802	−0.410901	−	0.911680i	$-0.634786\pi$
$53$	−5.98281	−0.821802	−0.410901	+	0.911680i	$0.634786\pi$
$54$	0	0
$55$	12.1396	1.63691
$56$	0	0
$57$	−1.14116	−0.151150
$58$	0	0
$59$	−7.53101	−0.980454	−0.490227	−	0.871595i	$-0.663086\pi$
$59$	−7.53101	−0.980454	−0.490227	+	0.871595i	$0.663086\pi$
$60$	0	0
$61$	2.35532	0.301568	0.150784	−	0.988567i	$-0.451820\pi$
$61$	2.35532	0.301568	0.150784	+	0.988567i	$0.451820\pi$
$62$	0	0
$63$	4.33538	0.546207
$64$	0	0
$65$	18.0284	2.23615
$66$	0	0
$67$	−3.03611	−0.370920	−0.185460	−	0.982652i	$-0.559377\pi$
$67$	−3.03611	−0.370920	−0.185460	+	0.982652i	$0.559377\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	6.64492	0.788607	0.394304	−	0.918980i	$-0.370986\pi$
$71$	6.64492	0.788607	0.394304	+	0.918980i	$0.370986\pi$
$72$	0	0
$73$	2.08303	0.243800	0.121900	−	0.992542i	$-0.461101\pi$
$73$	2.08303	0.243800	0.121900	+	0.992542i	$0.461101\pi$
$74$	0	0
$75$	−4.76514	−0.550231
$76$	0	0
$77$	−16.8420	−1.91932
$78$	0	0
$79$	−13.0123	−1.46400	−0.732001	−	0.681304i	$-0.761413\pi$
$79$	−13.0123	−1.46400	−0.732001	+	0.681304i	$0.761413\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.3584	−1.57604	−0.788019	−	0.615651i	$-0.788893\pi$
$83$	−14.3584	−1.57604	−0.788019	+	0.615651i	$0.788893\pi$
$84$	0	0
$85$	−5.55566	−0.602596
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−4.89965	−0.519362	−0.259681	−	0.965694i	$-0.583617\pi$
$89$	−4.89965	−0.519362	−0.259681	+	0.965694i	$0.583617\pi$
$90$	0	0
$91$	−25.0119	−2.62196
$92$	0	0
$93$	−5.76454	−0.597755
$94$	0	0
$95$	−3.56603	−0.365866
$96$	0	0
$97$	7.62077	0.773772	0.386886	−	0.922128i	$-0.373551\pi$
$97$	7.62077	0.773772	0.386886	+	0.922128i	$0.373551\pi$
$98$	0	0
$99$	−3.88477	−0.390435
$100$	0	0
$101$	16.8588	1.67752	0.838758	−	0.544504i	$-0.183282\pi$
$101$	16.8588	1.67752	0.838758	+	0.544504i	$0.183282\pi$
$102$	0	0
$103$	−1.87256	−0.184509	−0.0922543	−	0.995735i	$-0.529407\pi$
$103$	−1.87256	−0.184509	−0.0922543	+	0.995735i	$0.529407\pi$
$104$	0	0
$105$	13.5477	1.32212
$106$	0	0
$107$	12.8890	1.24603	0.623015	−	0.782210i	$-0.285907\pi$
$107$	12.8890	1.24603	0.623015	+	0.782210i	$0.285907\pi$
$108$	0	0
$109$	0.163721	0.0156817	0.00784083	−	0.999969i	$-0.497504\pi$
$109$	0.163721	0.0156817	0.00784083	+	0.999969i	$0.497504\pi$
$110$	0	0
$111$	−9.00519	−0.854735
$112$	0	0
$113$	−1.58635	−0.149231	−0.0746155	−	0.997212i	$-0.523773\pi$
$113$	−1.58635	−0.149231	−0.0746155	+	0.997212i	$0.523773\pi$
$114$	0	0
$115$	−3.12492	−0.291401
$116$	0	0
$117$	−5.76924	−0.533367
$118$	0	0
$119$	7.70769	0.706563
$120$	0	0
$121$	4.09147	0.371952
$122$	0	0
$123$	1.21895	0.109909
$124$	0	0
$125$	0.733922	0.0656440
$126$	0	0
$127$	−19.0449	−1.68996	−0.844981	−	0.534796i	$-0.820388\pi$
$127$	−19.0449	−1.68996	−0.844981	+	0.534796i	$0.820388\pi$
$128$	0	0
$129$	−6.15921	−0.542289
$130$	0	0
$131$	13.9702	1.22058	0.610292	−	0.792176i	$-0.291052\pi$
$131$	13.9702	1.22058	0.610292	+	0.792176i	$0.291052\pi$
$132$	0	0
$133$	4.94735	0.428990
$134$	0	0
$135$	3.12492	0.268950
$136$	0	0
$137$	6.51252	0.556402	0.278201	−	0.960523i	$-0.410262\pi$
$137$	6.51252	0.556402	0.278201	+	0.960523i	$0.410262\pi$
$138$	0	0
$139$	7.91579	0.671409	0.335704	−	0.941967i	$-0.391026\pi$
$139$	7.91579	0.671409	0.335704	+	0.941967i	$0.391026\pi$
$140$	0	0
$141$	1.33639	0.112544
$142$	0	0
$143$	22.4122	1.87420
$144$	0	0
$145$	3.12492	0.259511
$146$	0	0
$147$	−11.7955	−0.972879
$148$	0	0
$149$	−17.3544	−1.42173	−0.710865	−	0.703328i	$-0.751696\pi$
$149$	−17.3544	−1.42173	−0.710865	+	0.703328i	$0.751696\pi$
$150$	0	0
$151$	−20.6742	−1.68244	−0.841220	−	0.540693i	$-0.818162\pi$
$151$	−20.6742	−1.68244	−0.841220	+	0.540693i	$0.818162\pi$
$152$	0	0
$153$	1.77786	0.143731
$154$	0	0
$155$	−18.0137	−1.44690
$156$	0	0
$157$	1.53395	0.122422	0.0612112	−	0.998125i	$-0.480504\pi$
$157$	1.53395	0.122422	0.0612112	+	0.998125i	$0.480504\pi$
$158$	0	0
$159$	5.98281	0.474468
$160$	0	0
$161$	4.33538	0.341676
$162$	0	0
$163$	3.92716	0.307599	0.153800	−	0.988102i	$-0.450849\pi$
$163$	3.92716	0.307599	0.153800	+	0.988102i	$0.450849\pi$
$164$	0	0
$165$	−12.1396	−0.945068
$166$	0	0
$167$	19.2364	1.48855	0.744277	−	0.667871i	$-0.232794\pi$
$167$	19.2364	1.48855	0.744277	+	0.667871i	$0.232794\pi$
$168$	0	0
$169$	20.2842	1.56032
$170$	0	0
$171$	1.14116	0.0872664
$172$	0	0
$173$	−16.6806	−1.26820	−0.634102	−	0.773249i	$-0.718630\pi$
$173$	−16.6806	−1.26820	−0.634102	+	0.773249i	$0.718630\pi$
$174$	0	0
$175$	20.6587	1.56165
$176$	0	0
$177$	7.53101	0.566066
$178$	0	0
$179$	−9.82298	−0.734204	−0.367102	−	0.930181i	$-0.619650\pi$
$179$	−9.82298	−0.734204	−0.367102	+	0.930181i	$0.619650\pi$
$180$	0	0
$181$	15.3057	1.13766	0.568831	−	0.822454i	$-0.307396\pi$
$181$	15.3057	1.13766	0.568831	+	0.822454i	$0.307396\pi$
$182$	0	0
$183$	−2.35532	−0.174110
$184$	0	0
$185$	−28.1405	−2.06893
$186$	0	0
$187$	−6.90657	−0.505059
$188$	0	0
$189$	−4.33538	−0.315353
$190$	0	0
$191$	−15.6137	−1.12977	−0.564884	−	0.825171i	$-0.691079\pi$
$191$	−15.6137	−1.12977	−0.564884	+	0.825171i	$0.691079\pi$
$192$	0	0
$193$	−4.76207	−0.342781	−0.171391	−	0.985203i	$-0.554826\pi$
$193$	−4.76207	−0.342781	−0.171391	+	0.985203i	$0.554826\pi$
$194$	0	0
$195$	−18.0284	−1.29104
$196$	0	0
$197$	−7.66400	−0.546037	−0.273019	−	0.962009i	$-0.588022\pi$
$197$	−7.66400	−0.546037	−0.273019	+	0.962009i	$0.588022\pi$
$198$	0	0
$199$	26.8524	1.90351	0.951757	−	0.306853i	$-0.0992761\pi$
$199$	26.8524	1.90351	0.951757	+	0.306853i	$0.0992761\pi$
$200$	0	0
$201$	3.03611	0.214151
$202$	0	0
$203$	−4.33538	−0.304284
$204$	0	0
$205$	3.80912	0.266040
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−4.43314	−0.306646
$210$	0	0
$211$	−14.9556	−1.02959	−0.514793	−	0.857314i	$-0.672131\pi$
$211$	−14.9556	−1.02959	−0.514793	+	0.857314i	$0.672131\pi$
$212$	0	0
$213$	−6.64492	−0.455303
$214$	0	0
$215$	−19.2471	−1.31264
$216$	0	0
$217$	24.9915	1.69653
$218$	0	0
$219$	−2.08303	−0.140758
$220$	0	0
$221$	−10.2569	−0.689953
$222$	0	0
$223$	−25.6301	−1.71632	−0.858160	−	0.513383i	$-0.828392\pi$
$223$	−25.6301	−1.71632	−0.858160	+	0.513383i	$0.828392\pi$
$224$	0	0
$225$	4.76514	0.317676
$226$	0	0
$227$	−21.3625	−1.41788	−0.708939	−	0.705270i	$-0.750826\pi$
$227$	−21.3625	−1.41788	−0.708939	+	0.705270i	$0.750826\pi$
$228$	0	0
$229$	−25.1208	−1.66003	−0.830015	−	0.557741i	$-0.811668\pi$
$229$	−25.1208	−1.66003	−0.830015	+	0.557741i	$0.811668\pi$
$230$	0	0
$231$	16.8420	1.10812
$232$	0	0
$233$	−5.74498	−0.376366	−0.188183	−	0.982134i	$-0.560260\pi$
$233$	−5.74498	−0.376366	−0.188183	+	0.982134i	$0.560260\pi$
$234$	0	0
$235$	4.17611	0.272419
$236$	0	0
$237$	13.0123	0.845241
$238$	0	0
$239$	9.74972	0.630657	0.315328	−	0.948983i	$-0.397885\pi$
$239$	9.74972	0.630657	0.315328	+	0.948983i	$0.397885\pi$
$240$	0	0
$241$	−0.660892	−0.0425718	−0.0212859	−	0.999773i	$-0.506776\pi$
$241$	−0.660892	−0.0425718	−0.0212859	+	0.999773i	$0.506776\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−36.8601	−2.35491
$246$	0	0
$247$	−6.58361	−0.418905
$248$	0	0
$249$	14.3584	0.909926
$250$	0	0
$251$	3.07690	0.194212	0.0971062	−	0.995274i	$-0.469041\pi$
$251$	3.07690	0.194212	0.0971062	+	0.995274i	$0.469041\pi$
$252$	0	0
$253$	−3.88477	−0.244234
$254$	0	0
$255$	5.55566	0.347909
$256$	0	0
$257$	1.30458	0.0813772	0.0406886	−	0.999172i	$-0.487045\pi$
$257$	1.30458	0.0813772	0.0406886	+	0.999172i	$0.487045\pi$
$258$	0	0
$259$	39.0409	2.42589
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	14.7719	0.910874	0.455437	−	0.890268i	$-0.349483\pi$
$263$	14.7719	0.910874	0.455437	+	0.890268i	$0.349483\pi$
$264$	0	0
$265$	18.6958	1.14847
$266$	0	0
$267$	4.89965	0.299854
$268$	0	0
$269$	13.6664	0.833253	0.416627	−	0.909078i	$-0.363212\pi$
$269$	13.6664	0.833253	0.416627	+	0.909078i	$0.363212\pi$
$270$	0	0
$271$	3.00165	0.182337	0.0911687	−	0.995835i	$-0.470940\pi$
$271$	3.00165	0.182337	0.0911687	+	0.995835i	$0.470940\pi$
$272$	0	0
$273$	25.0119	1.51379
$274$	0	0
$275$	−18.5115	−1.11628
$276$	0	0
$277$	18.1613	1.09121	0.545603	−	0.838044i	$-0.316301\pi$
$277$	18.1613	1.09121	0.545603	+	0.838044i	$0.316301\pi$
$278$	0	0
$279$	5.76454	0.345114
$280$	0	0
$281$	3.73451	0.222782	0.111391	−	0.993777i	$-0.464469\pi$
$281$	3.73451	0.222782	0.111391	+	0.993777i	$0.464469\pi$
$282$	0	0
$283$	−23.6268	−1.40447	−0.702235	−	0.711945i	$-0.747814\pi$
$283$	−23.6268	−1.40447	−0.702235	+	0.711945i	$0.747814\pi$
$284$	0	0
$285$	3.56603	0.211233
$286$	0	0
$287$	−5.28461	−0.311941
$288$	0	0
$289$	−13.8392	−0.814072
$290$	0	0
$291$	−7.62077	−0.446737
$292$	0	0
$293$	−7.17862	−0.419379	−0.209690	−	0.977768i	$-0.567245\pi$
$293$	−7.17862	−0.419379	−0.209690	+	0.977768i	$0.567245\pi$
$294$	0	0
$295$	23.5338	1.37019
$296$	0	0
$297$	3.88477	0.225417
$298$	0	0
$299$	−5.76924	−0.333644
$300$	0	0
$301$	26.7025	1.53911
$302$	0	0
$303$	−16.8588	−0.968514
$304$	0	0
$305$	−7.36019	−0.421443
$306$	0	0
$307$	10.3673	0.591690	0.295845	−	0.955236i	$-0.404399\pi$
$307$	10.3673	0.591690	0.295845	+	0.955236i	$0.404399\pi$
$308$	0	0
$309$	1.87256	0.106526
$310$	0	0
$311$	−29.7060	−1.68447	−0.842236	−	0.539109i	$-0.818761\pi$
$311$	−29.7060	−1.68447	−0.842236	+	0.539109i	$0.818761\pi$
$312$	0	0
$313$	10.6960	0.604571	0.302286	−	0.953217i	$-0.402250\pi$
$313$	10.6960	0.604571	0.302286	+	0.953217i	$0.402250\pi$
$314$	0	0
$315$	−13.5477	−0.763328
$316$	0	0
$317$	11.5388	0.648083	0.324041	−	0.946043i	$-0.394958\pi$
$317$	11.5388	0.648083	0.324041	+	0.946043i	$0.394958\pi$
$318$	0	0
$319$	3.88477	0.217506
$320$	0	0
$321$	−12.8890	−0.719396
$322$	0	0
$323$	2.02881	0.112886
$324$	0	0
$325$	−27.4913	−1.52494
$326$	0	0
$327$	−0.163721	−0.00905381
$328$	0	0
$329$	−5.79375	−0.319420
$330$	0	0
$331$	12.4593	0.684827	0.342414	−	0.939549i	$-0.388756\pi$
$331$	12.4593	0.684827	0.342414	+	0.939549i	$0.388756\pi$
$332$	0	0
$333$	9.00519	0.493481
$334$	0	0
$335$	9.48761	0.518363
$336$	0	0
$337$	−17.0473	−0.928623	−0.464312	−	0.885672i	$-0.653698\pi$
$337$	−17.0473	−0.928623	−0.464312	+	0.885672i	$0.653698\pi$
$338$	0	0
$339$	1.58635	0.0861585
$340$	0	0
$341$	−22.3939	−1.21270
$342$	0	0
$343$	20.7905	1.12258
$344$	0	0
$345$	3.12492	0.168240
$346$	0	0
$347$	−22.8400	−1.22612	−0.613059	−	0.790037i	$-0.710061\pi$
$347$	−22.8400	−1.22612	−0.613059	+	0.790037i	$0.710061\pi$
$348$	0	0
$349$	−1.40175	−0.0750337	−0.0375169	−	0.999296i	$-0.511945\pi$
$349$	−1.40175	−0.0750337	−0.0375169	+	0.999296i	$0.511945\pi$
$350$	0	0
$351$	5.76924	0.307939
$352$	0	0
$353$	1.34698	0.0716923	0.0358462	−	0.999357i	$-0.488587\pi$
$353$	1.34698	0.0716923	0.0358462	+	0.999357i	$0.488587\pi$
$354$	0	0
$355$	−20.7649	−1.10208
$356$	0	0
$357$	−7.70769	−0.407934
$358$	0	0
$359$	−10.1026	−0.533195	−0.266597	−	0.963808i	$-0.585899\pi$
$359$	−10.1026	−0.533195	−0.266597	+	0.963808i	$0.585899\pi$
$360$	0	0
$361$	−17.6978	−0.931461
$362$	0	0
$363$	−4.09147	−0.214747
$364$	0	0
$365$	−6.50930	−0.340712
$366$	0	0
$367$	2.38990	0.124752	0.0623758	−	0.998053i	$-0.480132\pi$
$367$	2.38990	0.124752	0.0623758	+	0.998053i	$0.480132\pi$
$368$	0	0
$369$	−1.21895	−0.0634559
$370$	0	0
$371$	−25.9378	−1.34662
$372$	0	0
$373$	−35.2159	−1.82341	−0.911705	−	0.410845i	$-0.865234\pi$
$373$	−35.2159	−1.82341	−0.911705	+	0.410845i	$0.865234\pi$
$374$	0	0
$375$	−0.733922	−0.0378996
$376$	0	0
$377$	5.76924	0.297131
$378$	0	0
$379$	9.92302	0.509711	0.254856	−	0.966979i	$-0.417972\pi$
$379$	9.92302	0.509711	0.254856	+	0.966979i	$0.417972\pi$
$380$	0	0
$381$	19.0449	0.975700
$382$	0	0
$383$	−23.2260	−1.18680	−0.593398	−	0.804909i	$-0.702214\pi$
$383$	−23.2260	−1.18680	−0.593398	+	0.804909i	$0.702214\pi$
$384$	0	0
$385$	52.6299	2.68227
$386$	0	0
$387$	6.15921	0.313090
$388$	0	0
$389$	−37.3840	−1.89545	−0.947723	−	0.319094i	$-0.896622\pi$
$389$	−37.3840	−1.89545	−0.947723	+	0.319094i	$0.896622\pi$
$390$	0	0
$391$	1.77786	0.0899101
$392$	0	0
$393$	−13.9702	−0.704705
$394$	0	0
$395$	40.6625	2.04595
$396$	0	0
$397$	18.5902	0.933016	0.466508	−	0.884517i	$-0.345512\pi$
$397$	18.5902	0.933016	0.466508	+	0.884517i	$0.345512\pi$
$398$	0	0
$399$	−4.94735	−0.247677
$400$	0	0
$401$	−12.2176	−0.610119	−0.305059	−	0.952333i	$-0.598676\pi$
$401$	−12.2176	−0.610119	−0.305059	+	0.952333i	$0.598676\pi$
$402$	0	0
$403$	−33.2570	−1.65665
$404$	0	0
$405$	−3.12492	−0.155279
$406$	0	0
$407$	−34.9831	−1.73405
$408$	0	0
$409$	−1.97213	−0.0975158	−0.0487579	−	0.998811i	$-0.515526\pi$
$409$	−1.97213	−0.0975158	−0.0487579	+	0.998811i	$0.515526\pi$
$410$	0	0
$411$	−6.51252	−0.321239
$412$	0	0
$413$	−32.6498	−1.60659
$414$	0	0
$415$	44.8688	2.20252
$416$	0	0
$417$	−7.91579	−0.387638
$418$	0	0
$419$	13.3906	0.654174	0.327087	−	0.944994i	$-0.393933\pi$
$419$	13.3906	0.654174	0.327087	+	0.944994i	$0.393933\pi$
$420$	0	0
$421$	−25.8993	−1.26226	−0.631128	−	0.775678i	$-0.717408\pi$
$421$	−25.8993	−1.26226	−0.631128	+	0.775678i	$0.717408\pi$
$422$	0	0
$423$	−1.33639	−0.0649774
$424$	0	0
$425$	8.47173	0.410939
$426$	0	0
$427$	10.2112	0.494155
$428$	0	0
$429$	−22.4122	−1.08207
$430$	0	0
$431$	−17.1828	−0.827667	−0.413834	−	0.910353i	$-0.635810\pi$
$431$	−17.1828	−0.827667	−0.413834	+	0.910353i	$0.635810\pi$
$432$	0	0
$433$	−0.495336	−0.0238043	−0.0119022	−	0.999929i	$-0.503789\pi$
$433$	−0.495336	−0.0238043	−0.0119022	+	0.999929i	$0.503789\pi$
$434$	0	0
$435$	−3.12492	−0.149829
$436$	0	0
$437$	1.14116	0.0545889
$438$	0	0
$439$	−22.2342	−1.06118	−0.530589	−	0.847629i	$-0.678029\pi$
$439$	−22.2342	−1.06118	−0.530589	+	0.847629i	$0.678029\pi$
$440$	0	0
$441$	11.7955	0.561692
$442$	0	0
$443$	−17.4641	−0.829745	−0.414873	−	0.909880i	$-0.636174\pi$
$443$	−17.4641	−0.829745	−0.414873	+	0.909880i	$0.636174\pi$
$444$	0	0
$445$	15.3110	0.725813
$446$	0	0
$447$	17.3544	0.820836
$448$	0	0
$449$	−32.9362	−1.55436	−0.777178	−	0.629280i	$-0.783350\pi$
$449$	−32.9362	−1.55436	−0.777178	+	0.629280i	$0.783350\pi$
$450$	0	0
$451$	4.73534	0.222978
$452$	0	0
$453$	20.6742	0.971357
$454$	0	0
$455$	78.1602	3.66421
$456$	0	0
$457$	12.8131	0.599370	0.299685	−	0.954038i	$-0.403118\pi$
$457$	12.8131	0.599370	0.299685	+	0.954038i	$0.403118\pi$
$458$	0	0
$459$	−1.77786	−0.0829832
$460$	0	0
$461$	10.7073	0.498688	0.249344	−	0.968415i	$-0.419785\pi$
$461$	10.7073	0.498688	0.249344	+	0.968415i	$0.419785\pi$
$462$	0	0
$463$	−33.9367	−1.57717	−0.788587	−	0.614923i	$-0.789187\pi$
$463$	−33.9367	−1.57717	−0.788587	+	0.614923i	$0.789187\pi$
$464$	0	0
$465$	18.0137	0.835367
$466$	0	0
$467$	−21.4185	−0.991129	−0.495565	−	0.868571i	$-0.665039\pi$
$467$	−21.4185	−0.991129	−0.495565	+	0.868571i	$0.665039\pi$
$468$	0	0
$469$	−13.1627	−0.607797
$470$	0	0
$471$	−1.53395	−0.0706807
$472$	0	0
$473$	−23.9272	−1.10017
$474$	0	0
$475$	5.43777	0.249502
$476$	0	0
$477$	−5.98281	−0.273934
$478$	0	0
$479$	18.9177	0.864373	0.432186	−	0.901784i	$-0.357742\pi$
$479$	18.9177	0.864373	0.432186	+	0.901784i	$0.357742\pi$
$480$	0	0
$481$	−51.9531	−2.36886
$482$	0	0
$483$	−4.33538	−0.197267
$484$	0	0
$485$	−23.8143	−1.08135
$486$	0	0
$487$	29.4226	1.33327	0.666634	−	0.745386i	$-0.267735\pi$
$487$	29.4226	1.33327	0.666634	+	0.745386i	$0.267735\pi$
$488$	0	0
$489$	−3.92716	−0.177592
$490$	0	0
$491$	36.0670	1.62768	0.813841	−	0.581088i	$-0.197373\pi$
$491$	36.0670	1.62768	0.813841	+	0.581088i	$0.197373\pi$
$492$	0	0
$493$	−1.77786	−0.0800706
$494$	0	0
$495$	12.1396	0.545635
$496$	0	0
$497$	28.8083	1.29223
$498$	0	0
$499$	16.9469	0.758649	0.379325	−	0.925264i	$-0.376156\pi$
$499$	16.9469	0.758649	0.379325	+	0.925264i	$0.376156\pi$
$500$	0	0
$501$	−19.2364	−0.859417
$502$	0	0
$503$	8.80429	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$503$	8.80429	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$504$	0	0
$505$	−52.6825	−2.34434
$506$	0	0
$507$	−20.2842	−0.900852
$508$	0	0
$509$	26.2927	1.16540	0.582701	−	0.812687i	$-0.301996\pi$
$509$	26.2927	1.16540	0.582701	+	0.812687i	$0.301996\pi$
$510$	0	0
$511$	9.03072	0.399495
$512$	0	0
$513$	−1.14116	−0.0503833
$514$	0	0
$515$	5.85160	0.257852
$516$	0	0
$517$	5.19157	0.228325
$518$	0	0
$519$	16.6806	0.732198
$520$	0	0
$521$	−8.01412	−0.351105	−0.175552	−	0.984470i	$-0.556171\pi$
$521$	−8.01412	−0.351105	−0.175552	+	0.984470i	$0.556171\pi$
$522$	0	0
$523$	−37.7616	−1.65120	−0.825599	−	0.564258i	$-0.809162\pi$
$523$	−37.7616	−1.65120	−0.825599	+	0.564258i	$0.809162\pi$
$524$	0	0
$525$	−20.6587	−0.901619
$526$	0	0
$527$	10.2485	0.446433
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.53101	−0.326818
$532$	0	0
$533$	7.03241	0.304608
$534$	0	0
$535$	−40.2772	−1.74134
$536$	0	0
$537$	9.82298	0.423893
$538$	0	0
$539$	−45.8230	−1.97374
$540$	0	0
$541$	−32.1345	−1.38157	−0.690785	−	0.723060i	$-0.742735\pi$
$541$	−32.1345	−1.38157	−0.690785	+	0.723060i	$0.742735\pi$
$542$	0	0
$543$	−15.3057	−0.656830
$544$	0	0
$545$	−0.511616	−0.0219152
$546$	0	0
$547$	16.9345	0.724069	0.362034	−	0.932165i	$-0.382082\pi$
$547$	16.9345	0.724069	0.362034	+	0.932165i	$0.382082\pi$
$548$	0	0
$549$	2.35532	0.100523
$550$	0	0
$551$	−1.14116	−0.0486149
$552$	0	0
$553$	−56.4134	−2.39894
$554$	0	0
$555$	28.1405	1.19450
$556$	0	0
$557$	3.86058	0.163578	0.0817890	−	0.996650i	$-0.473937\pi$
$557$	3.86058	0.163578	0.0817890	+	0.996650i	$0.473937\pi$
$558$	0	0
$559$	−35.5340	−1.50293
$560$	0	0
$561$	6.90657	0.291596
$562$	0	0
$563$	−32.7287	−1.37935	−0.689676	−	0.724118i	$-0.742247\pi$
$563$	−32.7287	−1.37935	−0.689676	+	0.724118i	$0.742247\pi$
$564$	0	0
$565$	4.95721	0.208551
$566$	0	0
$567$	4.33538	0.182069
$568$	0	0
$569$	−27.1935	−1.14001	−0.570005	−	0.821641i	$-0.693059\pi$
$569$	−27.1935	−1.14001	−0.570005	+	0.821641i	$0.693059\pi$
$570$	0	0
$571$	−9.01198	−0.377140	−0.188570	−	0.982060i	$-0.560385\pi$
$571$	−9.01198	−0.377140	−0.188570	+	0.982060i	$0.560385\pi$
$572$	0	0
$573$	15.6137	0.652271
$574$	0	0
$575$	4.76514	0.198720
$576$	0	0
$577$	−16.5121	−0.687408	−0.343704	−	0.939078i	$-0.611682\pi$
$577$	−16.5121	−0.687408	−0.343704	+	0.939078i	$0.611682\pi$
$578$	0	0
$579$	4.76207	0.197905
$580$	0	0
$581$	−62.2491	−2.58253
$582$	0	0
$583$	23.2419	0.962579
$584$	0	0
$585$	18.0284	0.745384
$586$	0	0
$587$	1.61443	0.0666346	0.0333173	−	0.999445i	$-0.489393\pi$
$587$	1.61443	0.0666346	0.0333173	+	0.999445i	$0.489393\pi$
$588$	0	0
$589$	6.57824	0.271052
$590$	0	0
$591$	7.66400	0.315255
$592$	0	0
$593$	−11.2398	−0.461562	−0.230781	−	0.973006i	$-0.574128\pi$
$593$	−11.2398	−0.461562	−0.230781	+	0.973006i	$0.574128\pi$
$594$	0	0
$595$	−24.0859	−0.987427
$596$	0	0
$597$	−26.8524	−1.09899
$598$	0	0
$599$	13.3436	0.545204	0.272602	−	0.962127i	$-0.412116\pi$
$599$	13.3436	0.545204	0.272602	+	0.962127i	$0.412116\pi$
$600$	0	0
$601$	−31.6297	−1.29020	−0.645101	−	0.764097i	$-0.723185\pi$
$601$	−31.6297	−1.29020	−0.645101	+	0.764097i	$0.723185\pi$
$602$	0	0
$603$	−3.03611	−0.123640
$604$	0	0
$605$	−12.7855	−0.519806
$606$	0	0
$607$	−28.1421	−1.14225	−0.571127	−	0.820862i	$-0.693493\pi$
$607$	−28.1421	−1.14225	−0.571127	+	0.820862i	$0.693493\pi$
$608$	0	0
$609$	4.33538	0.175679
$610$	0	0
$611$	7.70995	0.311911
$612$	0	0
$613$	27.5963	1.11460	0.557301	−	0.830310i	$-0.311837\pi$
$613$	27.5963	1.11460	0.557301	+	0.830310i	$0.311837\pi$
$614$	0	0
$615$	−3.80912	−0.153599
$616$	0	0
$617$	−29.9733	−1.20668	−0.603341	−	0.797484i	$-0.706164\pi$
$617$	−29.9733	−1.20668	−0.603341	+	0.797484i	$0.706164\pi$
$618$	0	0
$619$	23.8123	0.957097	0.478548	−	0.878061i	$-0.341163\pi$
$619$	23.8123	0.957097	0.478548	+	0.878061i	$0.341163\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−21.2419	−0.851038
$624$	0	0
$625$	−26.1191	−1.04477
$626$	0	0
$627$	4.43314	0.177042
$628$	0	0
$629$	16.0099	0.638358
$630$	0	0
$631$	10.7203	0.426769	0.213384	−	0.976968i	$-0.431551\pi$
$631$	10.7203	0.426769	0.213384	+	0.976968i	$0.431551\pi$
$632$	0	0
$633$	14.9556	0.594432
$634$	0	0
$635$	59.5138	2.36173
$636$	0	0
$637$	−68.0513	−2.69629
$638$	0	0
$639$	6.64492	0.262869
$640$	0	0
$641$	−27.7824	−1.09734	−0.548670	−	0.836039i	$-0.684866\pi$
$641$	−27.7824	−1.09734	−0.548670	+	0.836039i	$0.684866\pi$
$642$	0	0
$643$	−14.7862	−0.583109	−0.291555	−	0.956554i	$-0.594173\pi$
$643$	−14.7862	−0.583109	−0.291555	+	0.956554i	$0.594173\pi$
$644$	0	0
$645$	19.2471	0.757853
$646$	0	0
$647$	−21.4419	−0.842969	−0.421485	−	0.906836i	$-0.638491\pi$
$647$	−21.4419	−0.842969	−0.421485	+	0.906836i	$0.638491\pi$
$648$	0	0
$649$	29.2563	1.14841
$650$	0	0
$651$	−24.9915	−0.979493
$652$	0	0
$653$	1.45978	0.0571255	0.0285627	−	0.999592i	$-0.490907\pi$
$653$	1.45978	0.0571255	0.0285627	+	0.999592i	$0.490907\pi$
$654$	0	0
$655$	−43.6559	−1.70578
$656$	0	0
$657$	2.08303	0.0812666
$658$	0	0
$659$	−46.9684	−1.82963	−0.914815	−	0.403872i	$-0.867664\pi$
$659$	−46.9684	−1.82963	−0.914815	+	0.403872i	$0.867664\pi$
$660$	0	0
$661$	−0.829538	−0.0322653	−0.0161326	−	0.999870i	$-0.505135\pi$
$661$	−0.829538	−0.0322653	−0.0161326	+	0.999870i	$0.505135\pi$
$662$	0	0
$663$	10.2569	0.398345
$664$	0	0
$665$	−15.4601	−0.599516
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	25.6301	0.990917
$670$	0	0
$671$	−9.14988	−0.353227
$672$	0	0
$673$	−3.40741	−0.131346	−0.0656729	−	0.997841i	$-0.520919\pi$
$673$	−3.40741	−0.131346	−0.0656729	+	0.997841i	$0.520919\pi$
$674$	0	0
$675$	−4.76514	−0.183410
$676$	0	0
$677$	48.8238	1.87645	0.938225	−	0.346027i	$-0.112469\pi$
$677$	48.8238	1.87645	0.938225	+	0.346027i	$0.112469\pi$
$678$	0	0
$679$	33.0389	1.26792
$680$	0	0
$681$	21.3625	0.818612
$682$	0	0
$683$	−41.0929	−1.57238	−0.786188	−	0.617987i	$-0.787948\pi$
$683$	−41.0929	−1.57238	−0.786188	+	0.617987i	$0.787948\pi$
$684$	0	0
$685$	−20.3511	−0.777577
$686$	0	0
$687$	25.1208	0.958419
$688$	0	0
$689$	34.5163	1.31497
$690$	0	0
$691$	34.2118	1.30148	0.650739	−	0.759301i	$-0.274459\pi$
$691$	34.2118	1.30148	0.650739	+	0.759301i	$0.274459\pi$
$692$	0	0
$693$	−16.8420	−0.639774
$694$	0	0
$695$	−24.7362	−0.938299
$696$	0	0
$697$	−2.16712	−0.0820854
$698$	0	0
$699$	5.74498	0.217295
$700$	0	0
$701$	−3.58511	−0.135408	−0.0677038	−	0.997705i	$-0.521567\pi$
$701$	−3.58511	−0.135408	−0.0677038	+	0.997705i	$0.521567\pi$
$702$	0	0
$703$	10.2763	0.387579
$704$	0	0
$705$	−4.17611	−0.157281
$706$	0	0
$707$	73.0894	2.74881
$708$	0	0
$709$	8.14577	0.305921	0.152960	−	0.988232i	$-0.451119\pi$
$709$	8.14577	0.305921	0.152960	+	0.988232i	$0.451119\pi$
$710$	0	0
$711$	−13.0123	−0.488000
$712$	0	0
$713$	5.76454	0.215884
$714$	0	0
$715$	−70.0364	−2.61921
$716$	0	0
$717$	−9.74972	−0.364110
$718$	0	0
$719$	1.60257	0.0597658	0.0298829	−	0.999553i	$-0.490487\pi$
$719$	1.60257	0.0597658	0.0298829	+	0.999553i	$0.490487\pi$
$720$	0	0
$721$	−8.11825	−0.302339
$722$	0	0
$723$	0.660892	0.0245788
$724$	0	0
$725$	−4.76514	−0.176973
$726$	0	0
$727$	8.45074	0.313421	0.156710	−	0.987645i	$-0.449911\pi$
$727$	8.45074	0.313421	0.156710	+	0.987645i	$0.449911\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.9502	0.405008
$732$	0	0
$733$	20.1821	0.745442	0.372721	−	0.927943i	$-0.378425\pi$
$733$	20.1821	0.745442	0.372721	+	0.927943i	$0.378425\pi$
$734$	0	0
$735$	36.8601	1.35961
$736$	0	0
$737$	11.7946	0.434460
$738$	0	0
$739$	12.0703	0.444015	0.222007	−	0.975045i	$-0.428739\pi$
$739$	12.0703	0.444015	0.222007	+	0.975045i	$0.428739\pi$
$740$	0	0
$741$	6.58361	0.241855
$742$	0	0
$743$	−27.2302	−0.998980	−0.499490	−	0.866320i	$-0.666479\pi$
$743$	−27.2302	−0.998980	−0.499490	+	0.866320i	$0.666479\pi$
$744$	0	0
$745$	54.2312	1.98688
$746$	0	0
$747$	−14.3584	−0.525346
$748$	0	0
$749$	55.8789	2.04177
$750$	0	0
$751$	35.2040	1.28461	0.642306	−	0.766448i	$-0.277978\pi$
$751$	35.2040	1.28461	0.642306	+	0.766448i	$0.277978\pi$
$752$	0	0
$753$	−3.07690	−0.112129
$754$	0	0
$755$	64.6052	2.35122
$756$	0	0
$757$	−28.0464	−1.01936	−0.509682	−	0.860363i	$-0.670237\pi$
$757$	−28.0464	−1.01936	−0.509682	+	0.860363i	$0.670237\pi$
$758$	0	0
$759$	3.88477	0.141008
$760$	0	0
$761$	21.7067	0.786866	0.393433	−	0.919353i	$-0.371287\pi$
$761$	21.7067	0.786866	0.393433	+	0.919353i	$0.371287\pi$
$762$	0	0
$763$	0.709794	0.0256963
$764$	0	0
$765$	−5.55566	−0.200865
$766$	0	0
$767$	43.4483	1.56883
$768$	0	0
$769$	23.4309	0.844941	0.422471	−	0.906377i	$-0.361163\pi$
$769$	23.4309	0.844941	0.422471	+	0.906377i	$0.361163\pi$
$770$	0	0
$771$	−1.30458	−0.0469831
$772$	0	0
$773$	−25.0981	−0.902717	−0.451358	−	0.892343i	$-0.649060\pi$
$773$	−25.0981	−0.902717	−0.451358	+	0.892343i	$0.649060\pi$
$774$	0	0
$775$	27.4688	0.986710
$776$	0	0
$777$	−39.0409	−1.40059
$778$	0	0
$779$	−1.39101	−0.0498381
$780$	0	0
$781$	−25.8140	−0.923698
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−4.79347	−0.171086
$786$	0	0
$787$	−44.9599	−1.60265	−0.801325	−	0.598230i	$-0.795871\pi$
$787$	−44.9599	−1.60265	−0.801325	+	0.598230i	$0.795871\pi$
$788$	0	0
$789$	−14.7719	−0.525894
$790$	0	0
$791$	−6.87742	−0.244533
$792$	0	0
$793$	−13.5884	−0.482538
$794$	0	0
$795$	−18.6958	−0.663072
$796$	0	0
$797$	−24.0899	−0.853310	−0.426655	−	0.904415i	$-0.640308\pi$
$797$	−24.0899	−0.853310	−0.426655	+	0.904415i	$0.640308\pi$
$798$	0	0
$799$	−2.37591	−0.0840535
$800$	0	0
$801$	−4.89965	−0.173121
$802$	0	0
$803$	−8.09209	−0.285564
$804$	0	0
$805$	−13.5477	−0.477495
$806$	0	0
$807$	−13.6664	−0.481079
$808$	0	0
$809$	−49.4644	−1.73908	−0.869538	−	0.493866i	$-0.835583\pi$
$809$	−49.4644	−1.73908	−0.869538	+	0.493866i	$0.835583\pi$
$810$	0	0
$811$	9.55785	0.335622	0.167811	−	0.985819i	$-0.446330\pi$
$811$	9.55785	0.335622	0.167811	+	0.985819i	$0.446330\pi$
$812$	0	0
$813$	−3.00165	−0.105273
$814$	0	0
$815$	−12.2721	−0.429872
$816$	0	0
$817$	7.02863	0.245901
$818$	0	0
$819$	−25.0119	−0.873986
$820$	0	0
$821$	−18.1610	−0.633824	−0.316912	−	0.948455i	$-0.602646\pi$
$821$	−18.1610	−0.633824	−0.316912	+	0.948455i	$0.602646\pi$
$822$	0	0
$823$	43.5039	1.51645	0.758226	−	0.651992i	$-0.226067\pi$
$823$	43.5039	1.51645	0.758226	+	0.651992i	$0.226067\pi$
$824$	0	0
$825$	18.5115	0.644487
$826$	0	0
$827$	−3.09541	−0.107638	−0.0538190	−	0.998551i	$-0.517139\pi$
$827$	−3.09541	−0.107638	−0.0538190	+	0.998551i	$0.517139\pi$
$828$	0	0
$829$	−39.8485	−1.38399	−0.691997	−	0.721900i	$-0.743269\pi$
$829$	−39.8485	−1.38399	−0.691997	+	0.721900i	$0.743269\pi$
$830$	0	0
$831$	−18.1613	−0.630009
$832$	0	0
$833$	20.9708	0.726594
$834$	0	0
$835$	−60.1121	−2.08027
$836$	0	0
$837$	−5.76454	−0.199252
$838$	0	0
$839$	27.0042	0.932289	0.466144	−	0.884709i	$-0.345643\pi$
$839$	27.0042	0.932289	0.466144	+	0.884709i	$0.345643\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.73451	−0.128623
$844$	0	0
$845$	−63.3865	−2.18056
$846$	0	0
$847$	17.7381	0.609488
$848$	0	0
$849$	23.6268	0.810871
$850$	0	0
$851$	9.00519	0.308694
$852$	0	0
$853$	19.2754	0.659978	0.329989	−	0.943985i	$-0.392955\pi$
$853$	19.2754	0.659978	0.329989	+	0.943985i	$0.392955\pi$
$854$	0	0
$855$	−3.56603	−0.121955
$856$	0	0
$857$	21.5762	0.737030	0.368515	−	0.929622i	$-0.379866\pi$
$857$	21.5762	0.737030	0.368515	+	0.929622i	$0.379866\pi$
$858$	0	0
$859$	−18.2042	−0.621121	−0.310560	−	0.950554i	$-0.600517\pi$
$859$	−18.2042	−0.621121	−0.310560	+	0.950554i	$0.600517\pi$
$860$	0	0
$861$	5.28461	0.180099
$862$	0	0
$863$	26.0426	0.886500	0.443250	−	0.896398i	$-0.353825\pi$
$863$	26.0426	0.886500	0.443250	+	0.896398i	$0.353825\pi$
$864$	0	0
$865$	52.1256	1.77233
$866$	0	0
$867$	13.8392	0.470005
$868$	0	0
$869$	50.5500	1.71479
$870$	0	0
$871$	17.5161	0.593509
$872$	0	0
$873$	7.62077	0.257924
$874$	0	0
$875$	3.18183	0.107566
$876$	0	0
$877$	−32.1393	−1.08527	−0.542633	−	0.839970i	$-0.682573\pi$
$877$	−32.1393	−1.08527	−0.542633	+	0.839970i	$0.682573\pi$
$878$	0	0
$879$	7.17862	0.242129
$880$	0	0
$881$	1.44207	0.0485846	0.0242923	−	0.999705i	$-0.492267\pi$
$881$	1.44207	0.0485846	0.0242923	+	0.999705i	$0.492267\pi$
$882$	0	0
$883$	−6.05643	−0.203815	−0.101908	−	0.994794i	$-0.532495\pi$
$883$	−6.05643	−0.203815	−0.101908	+	0.994794i	$0.532495\pi$
$884$	0	0
$885$	−23.5338	−0.791081
$886$	0	0
$887$	53.2715	1.78868	0.894340	−	0.447388i	$-0.147646\pi$
$887$	53.2715	1.78868	0.894340	+	0.447388i	$0.147646\pi$
$888$	0	0
$889$	−82.5669	−2.76921
$890$	0	0
$891$	−3.88477	−0.130145
$892$	0	0
$893$	−1.52503	−0.0510331
$894$	0	0
$895$	30.6961	1.02606
$896$	0	0
$897$	5.76924	0.192629
$898$	0	0
$899$	−5.76454	−0.192258
$900$	0	0
$901$	−10.6366	−0.354356
$902$	0	0
$903$	−26.7025	−0.888605
$904$	0	0
$905$	−47.8291	−1.58989
$906$	0	0
$907$	26.7289	0.887517	0.443759	−	0.896146i	$-0.353645\pi$
$907$	26.7289	0.887517	0.443759	+	0.896146i	$0.353645\pi$
$908$	0	0
$909$	16.8588	0.559172
$910$	0	0
$911$	9.93260	0.329082	0.164541	−	0.986370i	$-0.447386\pi$
$911$	9.93260	0.329082	0.164541	+	0.986370i	$0.447386\pi$
$912$	0	0
$913$	55.7791	1.84602
$914$	0	0
$915$	7.36019	0.243320
$916$	0	0
$917$	60.5663	2.00007
$918$	0	0
$919$	42.8801	1.41448	0.707242	−	0.706971i	$-0.249939\pi$
$919$	42.8801	1.41448	0.707242	+	0.706971i	$0.249939\pi$
$920$	0	0
$921$	−10.3673	−0.341613
$922$	0	0
$923$	−38.3362	−1.26185
$924$	0	0
$925$	42.9110	1.41090
$926$	0	0
$927$	−1.87256	−0.0615028
$928$	0	0
$929$	55.6843	1.82694	0.913472	−	0.406903i	$-0.133391\pi$
$929$	55.6843	1.82694	0.913472	+	0.406903i	$0.133391\pi$
$930$	0	0
$931$	13.4606	0.441152
$932$	0	0
$933$	29.7060	0.972530
$934$	0	0
$935$	21.5825	0.705823
$936$	0	0
$937$	55.2344	1.80443	0.902215	−	0.431286i	$-0.141940\pi$
$937$	55.2344	1.80443	0.902215	+	0.431286i	$0.141940\pi$
$938$	0	0
$939$	−10.6960	−0.349049
$940$	0	0
$941$	13.1855	0.429835	0.214918	−	0.976632i	$-0.431052\pi$
$941$	13.1855	0.429835	0.214918	+	0.976632i	$0.431052\pi$
$942$	0	0
$943$	−1.21895	−0.0396944
$944$	0	0
$945$	13.5477	0.440708
$946$	0	0
$947$	12.4167	0.403488	0.201744	−	0.979438i	$-0.435339\pi$
$947$	12.4167	0.403488	0.201744	+	0.979438i	$0.435339\pi$
$948$	0	0
$949$	−12.0175	−0.390104
$950$	0	0
$951$	−11.5388	−0.374171
$952$	0	0
$953$	−9.96101	−0.322669	−0.161334	−	0.986900i	$-0.551580\pi$
$953$	−9.96101	−0.322669	−0.161334	+	0.986900i	$0.551580\pi$
$954$	0	0
$955$	48.7916	1.57886
$956$	0	0
$957$	−3.88477	−0.125577
$958$	0	0
$959$	28.2343	0.911732
$960$	0	0
$961$	2.22992	0.0719328
$962$	0	0
$963$	12.8890	0.415343
$964$	0	0
$965$	14.8811	0.479040
$966$	0	0
$967$	−42.7790	−1.37568	−0.687839	−	0.725863i	$-0.741441\pi$
$967$	−42.7790	−1.37568	−0.687839	+	0.725863i	$0.741441\pi$
$968$	0	0
$969$	−2.02881	−0.0651748
$970$	0	0
$971$	−8.09189	−0.259681	−0.129841	−	0.991535i	$-0.541447\pi$
$971$	−8.09189	−0.259681	−0.129841	+	0.991535i	$0.541447\pi$
$972$	0	0
$973$	34.3180	1.10018
$974$	0	0
$975$	27.4913	0.880425
$976$	0	0
$977$	14.6916	0.470025	0.235013	−	0.971992i	$-0.424487\pi$
$977$	14.6916	0.470025	0.235013	+	0.971992i	$0.424487\pi$
$978$	0	0
$979$	19.0341	0.608331
$980$	0	0
$981$	0.163721	0.00522722
$982$	0	0
$983$	45.8602	1.46271	0.731357	−	0.681995i	$-0.238887\pi$
$983$	45.8602	1.46271	0.731357	+	0.681995i	$0.238887\pi$
$984$	0	0
$985$	23.9494	0.763091
$986$	0	0
$987$	5.79375	0.184417
$988$	0	0
$989$	6.15921	0.195852
$990$	0	0
$991$	−13.0647	−0.415012	−0.207506	−	0.978234i	$-0.566535\pi$
$991$	−13.0647	−0.415012	−0.207506	+	0.978234i	$0.566535\pi$
$992$	0	0
$993$	−12.4593	−0.395385
$994$	0	0
$995$	−83.9116	−2.66018
$996$	0	0
$997$	−8.55526	−0.270948	−0.135474	−	0.990781i	$-0.543256\pi$
$997$	−8.55526	−0.270948	−0.135474	+	0.990781i	$0.543256\pi$
$998$	0	0
$999$	−9.00519	−0.284912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.12492 q^{5} +4.33538 q^{7} +1.00000 q^{9} -3.88477 q^{11} -5.76924 q^{13} +3.12492 q^{15} +1.77786 q^{17} +1.14116 q^{19} -4.33538 q^{21} +1.00000 q^{23} +4.76514 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.76454 q^{31} +3.88477 q^{33} -13.5477 q^{35} +9.00519 q^{37} +5.76924 q^{39} -1.21895 q^{41} +6.15921 q^{43} -3.12492 q^{45} -1.33639 q^{47} +11.7955 q^{49} -1.77786 q^{51} -5.98281 q^{53} +12.1396 q^{55} -1.14116 q^{57} -7.53101 q^{59} +2.35532 q^{61} +4.33538 q^{63} +18.0284 q^{65} -3.03611 q^{67} -1.00000 q^{69} +6.64492 q^{71} +2.08303 q^{73} -4.76514 q^{75} -16.8420 q^{77} -13.0123 q^{79} +1.00000 q^{81} -14.3584 q^{83} -5.55566 q^{85} +1.00000 q^{87} -4.89965 q^{89} -25.0119 q^{91} -5.76454 q^{93} -3.56603 q^{95} +7.62077 q^{97} -3.88477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 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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