LMFDB - Embedded newform 8004.2.a.k.1.1

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.70317$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.70317 q^{5} -1.62014 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.70317 q^{5} -1.62014 q^{7} +1.00000 q^{9} -4.14603 q^{11} -5.66360 q^{13} -3.70317 q^{15} +0.274804 q^{17} -7.65773 q^{19} -1.62014 q^{21} -1.00000 q^{23} +8.71348 q^{25} +1.00000 q^{27} +1.00000 q^{29} -9.73653 q^{31} -4.14603 q^{33} +5.99966 q^{35} -1.32885 q^{37} -5.66360 q^{39} -8.38372 q^{41} +11.4237 q^{43} -3.70317 q^{45} -7.35084 q^{47} -4.37515 q^{49} +0.274804 q^{51} -0.0615882 q^{53} +15.3534 q^{55} -7.65773 q^{57} -1.61842 q^{59} +8.30895 q^{61} -1.62014 q^{63} +20.9733 q^{65} +1.88651 q^{67} -1.00000 q^{69} -12.7519 q^{71} -1.88003 q^{73} +8.71348 q^{75} +6.71714 q^{77} -13.8611 q^{79} +1.00000 q^{81} -10.8949 q^{83} -1.01765 q^{85} +1.00000 q^{87} +13.3395 q^{89} +9.17583 q^{91} -9.73653 q^{93} +28.3579 q^{95} -5.34555 q^{97} -4.14603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * 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
$5$	−3.70317	−1.65611	−0.828054	−	0.560648i	$-0.810552\pi$
$5$	−3.70317	−1.65611	−0.828054	+	0.560648i	$0.810552\pi$
$6$	0	0
$7$	−1.62014	−0.612356	−0.306178	−	0.951974i	$-0.599050\pi$
$7$	−1.62014	−0.612356	−0.306178	+	0.951974i	$0.599050\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.14603	−1.25007	−0.625037	−	0.780595i	$-0.714916\pi$
$11$	−4.14603	−1.25007	−0.625037	+	0.780595i	$0.714916\pi$
$12$	0	0
$13$	−5.66360	−1.57080	−0.785400	−	0.618988i	$-0.787543\pi$
$13$	−5.66360	−1.57080	−0.785400	+	0.618988i	$0.787543\pi$
$14$	0	0
$15$	−3.70317	−0.956155
$16$	0	0
$17$	0.274804	0.0666499	0.0333249	−	0.999445i	$-0.489390\pi$
$17$	0.274804	0.0666499	0.0333249	+	0.999445i	$0.489390\pi$
$18$	0	0
$19$	−7.65773	−1.75680	−0.878402	−	0.477923i	$-0.841390\pi$
$19$	−7.65773	−1.75680	−0.878402	+	0.477923i	$0.841390\pi$
$20$	0	0
$21$	−1.62014	−0.353544
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.71348	1.74270
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−9.73653	−1.74873	−0.874366	−	0.485266i	$-0.838723\pi$
$31$	−9.73653	−1.74873	−0.874366	+	0.485266i	$0.838723\pi$
$32$	0	0
$33$	−4.14603	−0.721731
$34$	0	0
$35$	5.99966	1.01413
$36$	0	0
$37$	−1.32885	−0.218462	−0.109231	−	0.994016i	$-0.534839\pi$
$37$	−1.32885	−0.218462	−0.109231	+	0.994016i	$0.534839\pi$
$38$	0	0
$39$	−5.66360	−0.906902
$40$	0	0
$41$	−8.38372	−1.30932	−0.654659	−	0.755924i	$-0.727188\pi$
$41$	−8.38372	−1.30932	−0.654659	+	0.755924i	$0.727188\pi$
$42$	0	0
$43$	11.4237	1.74209	0.871045	−	0.491202i	$-0.163442\pi$
$43$	11.4237	1.74209	0.871045	+	0.491202i	$0.163442\pi$
$44$	0	0
$45$	−3.70317	−0.552036
$46$	0	0
$47$	−7.35084	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$47$	−7.35084	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$48$	0	0
$49$	−4.37515	−0.625021
$50$	0	0
$51$	0.274804	0.0384803
$52$	0	0
$53$	−0.0615882	−0.00845979	−0.00422989	−	0.999991i	$-0.501346\pi$
$53$	−0.0615882	−0.00845979	−0.00422989	+	0.999991i	$0.501346\pi$
$54$	0	0
$55$	15.3534	2.07026
$56$	0	0
$57$	−7.65773	−1.01429
$58$	0	0
$59$	−1.61842	−0.210700	−0.105350	−	0.994435i	$-0.533596\pi$
$59$	−1.61842	−0.210700	−0.105350	+	0.994435i	$0.533596\pi$
$60$	0	0
$61$	8.30895	1.06385	0.531926	−	0.846791i	$-0.321468\pi$
$61$	8.30895	1.06385	0.531926	+	0.846791i	$0.321468\pi$
$62$	0	0
$63$	−1.62014	−0.204119
$64$	0	0
$65$	20.9733	2.60142
$66$	0	0
$67$	1.88651	0.230473	0.115237	−	0.993338i	$-0.463237\pi$
$67$	1.88651	0.230473	0.115237	+	0.993338i	$0.463237\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.7519	−1.51337	−0.756686	−	0.653779i	$-0.773183\pi$
$71$	−12.7519	−1.51337	−0.756686	+	0.653779i	$0.773183\pi$
$72$	0	0
$73$	−1.88003	−0.220041	−0.110020	−	0.993929i	$-0.535092\pi$
$73$	−1.88003	−0.220041	−0.110020	+	0.993929i	$0.535092\pi$
$74$	0	0
$75$	8.71348	1.00615
$76$	0	0
$77$	6.71714	0.765490
$78$	0	0
$79$	−13.8611	−1.55949	−0.779746	−	0.626096i	$-0.784652\pi$
$79$	−13.8611	−1.55949	−0.779746	+	0.626096i	$0.784652\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.8949	−1.19587	−0.597933	−	0.801546i	$-0.704011\pi$
$83$	−10.8949	−1.19587	−0.597933	+	0.801546i	$0.704011\pi$
$84$	0	0
$85$	−1.01765	−0.110379
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	13.3395	1.41398	0.706990	−	0.707224i	$-0.250053\pi$
$89$	13.3395	1.41398	0.706990	+	0.707224i	$0.250053\pi$
$90$	0	0
$91$	9.17583	0.961888
$92$	0	0
$93$	−9.73653	−1.00963
$94$	0	0
$95$	28.3579	2.90946
$96$	0	0
$97$	−5.34555	−0.542758	−0.271379	−	0.962473i	$-0.587480\pi$
$97$	−5.34555	−0.542758	−0.271379	+	0.962473i	$0.587480\pi$
$98$	0	0
$99$	−4.14603	−0.416691
$100$	0	0
$101$	−9.87708	−0.982806	−0.491403	−	0.870932i	$-0.663516\pi$
$101$	−9.87708	−0.982806	−0.491403	+	0.870932i	$0.663516\pi$
$102$	0	0
$103$	−8.52386	−0.839881	−0.419940	−	0.907552i	$-0.637949\pi$
$103$	−8.52386	−0.839881	−0.419940	+	0.907552i	$0.637949\pi$
$104$	0	0
$105$	5.99966	0.585507
$106$	0	0
$107$	0.447058	0.0432187	0.0216094	−	0.999766i	$-0.493121\pi$
$107$	0.447058	0.0432187	0.0216094	+	0.999766i	$0.493121\pi$
$108$	0	0
$109$	18.4051	1.76289	0.881443	−	0.472291i	$-0.156573\pi$
$109$	18.4051	1.76289	0.881443	+	0.472291i	$0.156573\pi$
$110$	0	0
$111$	−1.32885	−0.126129
$112$	0	0
$113$	20.7107	1.94830	0.974149	−	0.225906i	$-0.0725342\pi$
$113$	20.7107	1.94830	0.974149	+	0.225906i	$0.0725342\pi$
$114$	0	0
$115$	3.70317	0.345323
$116$	0	0
$117$	−5.66360	−0.523600
$118$	0	0
$119$	−0.445222	−0.0408134
$120$	0	0
$121$	6.18953	0.562685
$122$	0	0
$123$	−8.38372	−0.755935
$124$	0	0
$125$	−13.7517	−1.22999
$126$	0	0
$127$	−4.24437	−0.376627	−0.188313	−	0.982109i	$-0.560302\pi$
$127$	−4.24437	−0.376627	−0.188313	+	0.982109i	$0.560302\pi$
$128$	0	0
$129$	11.4237	1.00580
$130$	0	0
$131$	−9.02976	−0.788934	−0.394467	−	0.918910i	$-0.629071\pi$
$131$	−9.02976	−0.788934	−0.394467	+	0.918910i	$0.629071\pi$
$132$	0	0
$133$	12.4066	1.07579
$134$	0	0
$135$	−3.70317	−0.318718
$136$	0	0
$137$	−4.01671	−0.343171	−0.171585	−	0.985169i	$-0.554889\pi$
$137$	−4.01671	−0.343171	−0.171585	+	0.985169i	$0.554889\pi$
$138$	0	0
$139$	16.9461	1.43735	0.718676	−	0.695345i	$-0.244748\pi$
$139$	16.9461	1.43735	0.718676	+	0.695345i	$0.244748\pi$
$140$	0	0
$141$	−7.35084	−0.619053
$142$	0	0
$143$	23.4814	1.96362
$144$	0	0
$145$	−3.70317	−0.307532
$146$	0	0
$147$	−4.37515	−0.360856
$148$	0	0
$149$	−2.63519	−0.215883	−0.107942	−	0.994157i	$-0.534426\pi$
$149$	−2.63519	−0.215883	−0.107942	+	0.994157i	$0.534426\pi$
$150$	0	0
$151$	6.48742	0.527939	0.263970	−	0.964531i	$-0.414968\pi$
$151$	6.48742	0.527939	0.263970	+	0.964531i	$0.414968\pi$
$152$	0	0
$153$	0.274804	0.0222166
$154$	0	0
$155$	36.0561	2.89609
$156$	0	0
$157$	2.35301	0.187791	0.0938954	−	0.995582i	$-0.470068\pi$
$157$	2.35301	0.187791	0.0938954	+	0.995582i	$0.470068\pi$
$158$	0	0
$159$	−0.0615882	−0.00488426
$160$	0	0
$161$	1.62014	0.127685
$162$	0	0
$163$	21.5540	1.68824	0.844120	−	0.536155i	$-0.180124\pi$
$163$	21.5540	1.68824	0.844120	+	0.536155i	$0.180124\pi$
$164$	0	0
$165$	15.3534	1.19526
$166$	0	0
$167$	−6.81013	−0.526984	−0.263492	−	0.964662i	$-0.584874\pi$
$167$	−6.81013	−0.526984	−0.263492	+	0.964662i	$0.584874\pi$
$168$	0	0
$169$	19.0764	1.46741
$170$	0	0
$171$	−7.65773	−0.585601
$172$	0	0
$173$	−18.8314	−1.43173	−0.715863	−	0.698241i	$-0.753967\pi$
$173$	−18.8314	−1.43173	−0.715863	+	0.698241i	$0.753967\pi$
$174$	0	0
$175$	−14.1171	−1.06715
$176$	0	0
$177$	−1.61842	−0.121648
$178$	0	0
$179$	−11.9798	−0.895410	−0.447705	−	0.894181i	$-0.647759\pi$
$179$	−11.9798	−0.895410	−0.447705	+	0.894181i	$0.647759\pi$
$180$	0	0
$181$	18.2795	1.35870	0.679351	−	0.733814i	$-0.262262\pi$
$181$	18.2795	1.35870	0.679351	+	0.733814i	$0.262262\pi$
$182$	0	0
$183$	8.30895	0.614215
$184$	0	0
$185$	4.92097	0.361797
$186$	0	0
$187$	−1.13935	−0.0833173
$188$	0	0
$189$	−1.62014	−0.117848
$190$	0	0
$191$	8.52442	0.616805	0.308403	−	0.951256i	$-0.400206\pi$
$191$	8.52442	0.616805	0.308403	+	0.951256i	$0.400206\pi$
$192$	0	0
$193$	−8.68527	−0.625179	−0.312590	−	0.949888i	$-0.601196\pi$
$193$	−8.68527	−0.625179	−0.312590	+	0.949888i	$0.601196\pi$
$194$	0	0
$195$	20.9733	1.50193
$196$	0	0
$197$	17.0253	1.21300	0.606501	−	0.795082i	$-0.292572\pi$
$197$	17.0253	1.21300	0.606501	+	0.795082i	$0.292572\pi$
$198$	0	0
$199$	−19.4607	−1.37954	−0.689768	−	0.724031i	$-0.742287\pi$
$199$	−19.4607	−1.37954	−0.689768	+	0.724031i	$0.742287\pi$
$200$	0	0
$201$	1.88651	0.133064
$202$	0	0
$203$	−1.62014	−0.113712
$204$	0	0
$205$	31.0464	2.16837
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	31.7491	2.19613
$210$	0	0
$211$	−4.52898	−0.311788	−0.155894	−	0.987774i	$-0.549826\pi$
$211$	−4.52898	−0.311788	−0.155894	+	0.987774i	$0.549826\pi$
$212$	0	0
$213$	−12.7519	−0.873745
$214$	0	0
$215$	−42.3038	−2.88509
$216$	0	0
$217$	15.7745	1.07085
$218$	0	0
$219$	−1.88003	−0.127040
$220$	0	0
$221$	−1.55638	−0.104694
$222$	0	0
$223$	29.4109	1.96950	0.984748	−	0.173984i	$-0.0556642\pi$
$223$	29.4109	1.96950	0.984748	+	0.173984i	$0.0556642\pi$
$224$	0	0
$225$	8.71348	0.580899
$226$	0	0
$227$	−2.89069	−0.191862	−0.0959309	−	0.995388i	$-0.530583\pi$
$227$	−2.89069	−0.191862	−0.0959309	+	0.995388i	$0.530583\pi$
$228$	0	0
$229$	−18.1411	−1.19880	−0.599399	−	0.800450i	$-0.704594\pi$
$229$	−18.1411	−1.19880	−0.599399	+	0.800450i	$0.704594\pi$
$230$	0	0
$231$	6.71714	0.441956
$232$	0	0
$233$	−8.24266	−0.539995	−0.269997	−	0.962861i	$-0.587023\pi$
$233$	−8.24266	−0.539995	−0.269997	+	0.962861i	$0.587023\pi$
$234$	0	0
$235$	27.2214	1.77573
$236$	0	0
$237$	−13.8611	−0.900373
$238$	0	0
$239$	0.715010	0.0462502	0.0231251	−	0.999733i	$-0.492638\pi$
$239$	0.715010	0.0462502	0.0231251	+	0.999733i	$0.492638\pi$
$240$	0	0
$241$	−14.8598	−0.957201	−0.478601	−	0.878033i	$-0.658856\pi$
$241$	−14.8598	−0.957201	−0.478601	+	0.878033i	$0.658856\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.2019	1.03510
$246$	0	0
$247$	43.3703	2.75959
$248$	0	0
$249$	−10.8949	−0.690434
$250$	0	0
$251$	−17.5738	−1.10925	−0.554625	−	0.832101i	$-0.687138\pi$
$251$	−17.5738	−1.10925	−0.554625	+	0.832101i	$0.687138\pi$
$252$	0	0
$253$	4.14603	0.260658
$254$	0	0
$255$	−1.01765	−0.0637276
$256$	0	0
$257$	−14.8820	−0.928314	−0.464157	−	0.885753i	$-0.653643\pi$
$257$	−14.8820	−0.928314	−0.464157	+	0.885753i	$0.653643\pi$
$258$	0	0
$259$	2.15293	0.133777
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−16.5865	−1.02277	−0.511384	−	0.859352i	$-0.670867\pi$
$263$	−16.5865	−1.02277	−0.511384	+	0.859352i	$0.670867\pi$
$264$	0	0
$265$	0.228072	0.0140103
$266$	0	0
$267$	13.3395	0.816362
$268$	0	0
$269$	−30.8549	−1.88126	−0.940630	−	0.339435i	$-0.889764\pi$
$269$	−30.8549	−1.88126	−0.940630	+	0.339435i	$0.889764\pi$
$270$	0	0
$271$	21.1166	1.28274	0.641372	−	0.767230i	$-0.278366\pi$
$271$	21.1166	1.28274	0.641372	+	0.767230i	$0.278366\pi$
$272$	0	0
$273$	9.17583	0.555346
$274$	0	0
$275$	−36.1263	−2.17850
$276$	0	0
$277$	13.6044	0.817409	0.408705	−	0.912667i	$-0.365981\pi$
$277$	13.6044	0.817409	0.408705	+	0.912667i	$0.365981\pi$
$278$	0	0
$279$	−9.73653	−0.582911
$280$	0	0
$281$	−2.38601	−0.142338	−0.0711689	−	0.997464i	$-0.522673\pi$
$281$	−2.38601	−0.142338	−0.0711689	+	0.997464i	$0.522673\pi$
$282$	0	0
$283$	17.9545	1.06728	0.533641	−	0.845711i	$-0.320823\pi$
$283$	17.9545	1.06728	0.533641	+	0.845711i	$0.320823\pi$
$284$	0	0
$285$	28.3579	1.67978
$286$	0	0
$287$	13.5828	0.801768
$288$	0	0
$289$	−16.9245	−0.995558
$290$	0	0
$291$	−5.34555	−0.313362
$292$	0	0
$293$	−19.7308	−1.15268	−0.576342	−	0.817209i	$-0.695520\pi$
$293$	−19.7308	−1.15268	−0.576342	+	0.817209i	$0.695520\pi$
$294$	0	0
$295$	5.99328	0.348942
$296$	0	0
$297$	−4.14603	−0.240577
$298$	0	0
$299$	5.66360	0.327534
$300$	0	0
$301$	−18.5079	−1.06678
$302$	0	0
$303$	−9.87708	−0.567424
$304$	0	0
$305$	−30.7695	−1.76186
$306$	0	0
$307$	4.30725	0.245828	0.122914	−	0.992417i	$-0.460776\pi$
$307$	4.30725	0.245828	0.122914	+	0.992417i	$0.460776\pi$
$308$	0	0
$309$	−8.52386	−0.484905
$310$	0	0
$311$	−21.4124	−1.21419	−0.607094	−	0.794630i	$-0.707665\pi$
$311$	−21.4124	−1.21419	−0.607094	+	0.794630i	$0.707665\pi$
$312$	0	0
$313$	14.1128	0.797705	0.398853	−	0.917015i	$-0.369409\pi$
$313$	14.1128	0.797705	0.398853	+	0.917015i	$0.369409\pi$
$314$	0	0
$315$	5.99966	0.338042
$316$	0	0
$317$	8.95458	0.502939	0.251470	−	0.967865i	$-0.419086\pi$
$317$	8.95458	0.502939	0.251470	+	0.967865i	$0.419086\pi$
$318$	0	0
$319$	−4.14603	−0.232133
$320$	0	0
$321$	0.447058	0.0249523
$322$	0	0
$323$	−2.10438	−0.117091
$324$	0	0
$325$	−49.3497	−2.73743
$326$	0	0
$327$	18.4051	1.01780
$328$	0	0
$329$	11.9094	0.656586
$330$	0	0
$331$	2.38872	0.131296	0.0656481	−	0.997843i	$-0.479089\pi$
$331$	2.38872	0.131296	0.0656481	+	0.997843i	$0.479089\pi$
$332$	0	0
$333$	−1.32885	−0.0728207
$334$	0	0
$335$	−6.98606	−0.381689
$336$	0	0
$337$	−31.2494	−1.70226	−0.851132	−	0.524952i	$-0.824083\pi$
$337$	−31.2494	−1.70226	−0.851132	+	0.524952i	$0.824083\pi$
$338$	0	0
$339$	20.7107	1.12485
$340$	0	0
$341$	40.3679	2.18605
$342$	0	0
$343$	18.4293	0.995090
$344$	0	0
$345$	3.70317	0.199372
$346$	0	0
$347$	11.0720	0.594376	0.297188	−	0.954819i	$-0.403951\pi$
$347$	11.0720	0.594376	0.297188	+	0.954819i	$0.403951\pi$
$348$	0	0
$349$	−15.7963	−0.845558	−0.422779	−	0.906233i	$-0.638945\pi$
$349$	−15.7963	−0.845558	−0.422779	+	0.906233i	$0.638945\pi$
$350$	0	0
$351$	−5.66360	−0.302301
$352$	0	0
$353$	8.91243	0.474361	0.237180	−	0.971466i	$-0.423777\pi$
$353$	8.91243	0.474361	0.237180	+	0.971466i	$0.423777\pi$
$354$	0	0
$355$	47.2225	2.50631
$356$	0	0
$357$	−0.445222	−0.0235636
$358$	0	0
$359$	−6.30371	−0.332697	−0.166349	−	0.986067i	$-0.553198\pi$
$359$	−6.30371	−0.332697	−0.166349	+	0.986067i	$0.553198\pi$
$360$	0	0
$361$	39.6408	2.08636
$362$	0	0
$363$	6.18953	0.324866
$364$	0	0
$365$	6.96207	0.364411
$366$	0	0
$367$	−10.7036	−0.558723	−0.279362	−	0.960186i	$-0.590123\pi$
$367$	−10.7036	−0.558723	−0.279362	+	0.960186i	$0.590123\pi$
$368$	0	0
$369$	−8.38372	−0.436439
$370$	0	0
$371$	0.0997815	0.00518040
$372$	0	0
$373$	−16.7454	−0.867042	−0.433521	−	0.901143i	$-0.642729\pi$
$373$	−16.7454	−0.867042	−0.433521	+	0.901143i	$0.642729\pi$
$374$	0	0
$375$	−13.7517	−0.710133
$376$	0	0
$377$	−5.66360	−0.291690
$378$	0	0
$379$	8.17817	0.420084	0.210042	−	0.977692i	$-0.432640\pi$
$379$	8.17817	0.420084	0.210042	+	0.977692i	$0.432640\pi$
$380$	0	0
$381$	−4.24437	−0.217446
$382$	0	0
$383$	14.4935	0.740582	0.370291	−	0.928916i	$-0.379258\pi$
$383$	14.4935	0.740582	0.370291	+	0.928916i	$0.379258\pi$
$384$	0	0
$385$	−24.8747	−1.26773
$386$	0	0
$387$	11.4237	0.580697
$388$	0	0
$389$	−11.0941	−0.562492	−0.281246	−	0.959636i	$-0.590748\pi$
$389$	−11.0941	−0.562492	−0.281246	+	0.959636i	$0.590748\pi$
$390$	0	0
$391$	−0.274804	−0.0138975
$392$	0	0
$393$	−9.02976	−0.455491
$394$	0	0
$395$	51.3299	2.58269
$396$	0	0
$397$	23.7957	1.19427	0.597135	−	0.802141i	$-0.296306\pi$
$397$	23.7957	1.19427	0.597135	+	0.802141i	$0.296306\pi$
$398$	0	0
$399$	12.4066	0.621107
$400$	0	0
$401$	0.400682	0.0200091	0.0100046	−	0.999950i	$-0.496815\pi$
$401$	0.400682	0.0200091	0.0100046	+	0.999950i	$0.496815\pi$
$402$	0	0
$403$	55.1438	2.74691
$404$	0	0
$405$	−3.70317	−0.184012
$406$	0	0
$407$	5.50946	0.273094
$408$	0	0
$409$	−30.9882	−1.53227	−0.766135	−	0.642680i	$-0.777822\pi$
$409$	−30.9882	−1.53227	−0.766135	+	0.642680i	$0.777822\pi$
$410$	0	0
$411$	−4.01671	−0.198130
$412$	0	0
$413$	2.62206	0.129023
$414$	0	0
$415$	40.3456	1.98049
$416$	0	0
$417$	16.9461	0.829856
$418$	0	0
$419$	32.6516	1.59514	0.797568	−	0.603229i	$-0.206120\pi$
$419$	32.6516	1.59514	0.797568	+	0.603229i	$0.206120\pi$
$420$	0	0
$421$	16.3602	0.797347	0.398674	−	0.917093i	$-0.369471\pi$
$421$	16.3602	0.797347	0.398674	+	0.917093i	$0.369471\pi$
$422$	0	0
$423$	−7.35084	−0.357410
$424$	0	0
$425$	2.39450	0.116151
$426$	0	0
$427$	−13.4617	−0.651456
$428$	0	0
$429$	23.4814	1.13369
$430$	0	0
$431$	10.7337	0.517025	0.258513	−	0.966008i	$-0.416768\pi$
$431$	10.7337	0.517025	0.258513	+	0.966008i	$0.416768\pi$
$432$	0	0
$433$	21.6523	1.04054	0.520272	−	0.854001i	$-0.325831\pi$
$433$	21.6523	1.04054	0.520272	+	0.854001i	$0.325831\pi$
$434$	0	0
$435$	−3.70317	−0.177554
$436$	0	0
$437$	7.65773	0.366319
$438$	0	0
$439$	−18.7896	−0.896781	−0.448390	−	0.893838i	$-0.648003\pi$
$439$	−18.7896	−0.896781	−0.448390	+	0.893838i	$0.648003\pi$
$440$	0	0
$441$	−4.37515	−0.208340
$442$	0	0
$443$	12.3997	0.589125	0.294563	−	0.955632i	$-0.404826\pi$
$443$	12.3997	0.589125	0.294563	+	0.955632i	$0.404826\pi$
$444$	0	0
$445$	−49.3983	−2.34170
$446$	0	0
$447$	−2.63519	−0.124640
$448$	0	0
$449$	2.60536	0.122955	0.0614773	−	0.998108i	$-0.480419\pi$
$449$	2.60536	0.122955	0.0614773	+	0.998108i	$0.480419\pi$
$450$	0	0
$451$	34.7591	1.63674
$452$	0	0
$453$	6.48742	0.304806
$454$	0	0
$455$	−33.9797	−1.59299
$456$	0	0
$457$	10.5896	0.495359	0.247679	−	0.968842i	$-0.420332\pi$
$457$	10.5896	0.495359	0.247679	+	0.968842i	$0.420332\pi$
$458$	0	0
$459$	0.274804	0.0128268
$460$	0	0
$461$	23.5697	1.09775	0.548874	−	0.835905i	$-0.315057\pi$
$461$	23.5697	1.09775	0.548874	+	0.835905i	$0.315057\pi$
$462$	0	0
$463$	−34.8933	−1.62163	−0.810815	−	0.585303i	$-0.800976\pi$
$463$	−34.8933	−1.62163	−0.810815	+	0.585303i	$0.800976\pi$
$464$	0	0
$465$	36.0561	1.67206
$466$	0	0
$467$	−41.0521	−1.89966	−0.949832	−	0.312761i	$-0.898746\pi$
$467$	−41.0521	−1.89966	−0.949832	+	0.312761i	$0.898746\pi$
$468$	0	0
$469$	−3.05640	−0.141132
$470$	0	0
$471$	2.35301	0.108421
$472$	0	0
$473$	−47.3628	−2.17774
$474$	0	0
$475$	−66.7255	−3.06158
$476$	0	0
$477$	−0.0615882	−0.00281993
$478$	0	0
$479$	−28.0218	−1.28035	−0.640175	−	0.768229i	$-0.721138\pi$
$479$	−28.0218	−1.28035	−0.640175	+	0.768229i	$0.721138\pi$
$480$	0	0
$481$	7.52610	0.343160
$482$	0	0
$483$	1.62014	0.0737189
$484$	0	0
$485$	19.7955	0.898867
$486$	0	0
$487$	6.83799	0.309859	0.154930	−	0.987926i	$-0.450485\pi$
$487$	6.83799	0.309859	0.154930	+	0.987926i	$0.450485\pi$
$488$	0	0
$489$	21.5540	0.974706
$490$	0	0
$491$	−40.6010	−1.83230	−0.916148	−	0.400839i	$-0.868719\pi$
$491$	−40.6010	−1.83230	−0.916148	+	0.400839i	$0.868719\pi$
$492$	0	0
$493$	0.274804	0.0123766
$494$	0	0
$495$	15.3534	0.690086
$496$	0	0
$497$	20.6599	0.926721
$498$	0	0
$499$	25.6044	1.14621	0.573105	−	0.819482i	$-0.305739\pi$
$499$	25.6044	1.14621	0.573105	+	0.819482i	$0.305739\pi$
$500$	0	0
$501$	−6.81013	−0.304254
$502$	0	0
$503$	32.5231	1.45013	0.725067	−	0.688679i	$-0.241809\pi$
$503$	32.5231	1.45013	0.725067	+	0.688679i	$0.241809\pi$
$504$	0	0
$505$	36.5765	1.62763
$506$	0	0
$507$	19.0764	0.847211
$508$	0	0
$509$	−1.18085	−0.0523401	−0.0261700	−	0.999658i	$-0.508331\pi$
$509$	−1.18085	−0.0523401	−0.0261700	+	0.999658i	$0.508331\pi$
$510$	0	0
$511$	3.04591	0.134743
$512$	0	0
$513$	−7.65773	−0.338097
$514$	0	0
$515$	31.5653	1.39093
$516$	0	0
$517$	30.4768	1.34037
$518$	0	0
$519$	−18.8314	−0.826608
$520$	0	0
$521$	−29.2622	−1.28200	−0.641000	−	0.767541i	$-0.721480\pi$
$521$	−29.2622	−1.28200	−0.641000	+	0.767541i	$0.721480\pi$
$522$	0	0
$523$	36.6658	1.60328	0.801642	−	0.597805i	$-0.203960\pi$
$523$	36.6658	1.60328	0.801642	+	0.597805i	$0.203960\pi$
$524$	0	0
$525$	−14.1171	−0.616119
$526$	0	0
$527$	−2.67564	−0.116553
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.61842	−0.0702333
$532$	0	0
$533$	47.4821	2.05668
$534$	0	0
$535$	−1.65553	−0.0715749
$536$	0	0
$537$	−11.9798	−0.516965
$538$	0	0
$539$	18.1395	0.781322
$540$	0	0
$541$	−6.77548	−0.291301	−0.145650	−	0.989336i	$-0.546527\pi$
$541$	−6.77548	−0.291301	−0.145650	+	0.989336i	$0.546527\pi$
$542$	0	0
$543$	18.2795	0.784447
$544$	0	0
$545$	−68.1571	−2.91953
$546$	0	0
$547$	12.9117	0.552063	0.276032	−	0.961149i	$-0.410981\pi$
$547$	12.9117	0.552063	0.276032	+	0.961149i	$0.410981\pi$
$548$	0	0
$549$	8.30895	0.354617
$550$	0	0
$551$	−7.65773	−0.326230
$552$	0	0
$553$	22.4569	0.954964
$554$	0	0
$555$	4.92097	0.208884
$556$	0	0
$557$	12.9192	0.547405	0.273703	−	0.961814i	$-0.411752\pi$
$557$	12.9192	0.547405	0.273703	+	0.961814i	$0.411752\pi$
$558$	0	0
$559$	−64.6990	−2.73648
$560$	0	0
$561$	−1.13935	−0.0481032
$562$	0	0
$563$	37.1517	1.56576	0.782878	−	0.622176i	$-0.213751\pi$
$563$	37.1517	1.56576	0.782878	+	0.622176i	$0.213751\pi$
$564$	0	0
$565$	−76.6953	−3.22659
$566$	0	0
$567$	−1.62014	−0.0680395
$568$	0	0
$569$	23.9045	1.00213	0.501065	−	0.865410i	$-0.332942\pi$
$569$	23.9045	1.00213	0.501065	+	0.865410i	$0.332942\pi$
$570$	0	0
$571$	−42.2800	−1.76936	−0.884682	−	0.466195i	$-0.845624\pi$
$571$	−42.2800	−1.76936	−0.884682	+	0.466195i	$0.845624\pi$
$572$	0	0
$573$	8.52442	0.356113
$574$	0	0
$575$	−8.71348	−0.363377
$576$	0	0
$577$	4.46519	0.185888	0.0929442	−	0.995671i	$-0.470372\pi$
$577$	4.46519	0.185888	0.0929442	+	0.995671i	$0.470372\pi$
$578$	0	0
$579$	−8.68527	−0.360947
$580$	0	0
$581$	17.6512	0.732296
$582$	0	0
$583$	0.255346	0.0105754
$584$	0	0
$585$	20.9733	0.867139
$586$	0	0
$587$	−6.75742	−0.278908	−0.139454	−	0.990229i	$-0.544535\pi$
$587$	−6.75742	−0.278908	−0.139454	+	0.990229i	$0.544535\pi$
$588$	0	0
$589$	74.5597	3.07218
$590$	0	0
$591$	17.0253	0.700328
$592$	0	0
$593$	−26.3588	−1.08243	−0.541213	−	0.840886i	$-0.682035\pi$
$593$	−26.3588	−1.08243	−0.541213	+	0.840886i	$0.682035\pi$
$594$	0	0
$595$	1.64873	0.0675915
$596$	0	0
$597$	−19.4607	−0.796475
$598$	0	0
$599$	−39.5395	−1.61554	−0.807771	−	0.589497i	$-0.799326\pi$
$599$	−39.5395	−1.61554	−0.807771	+	0.589497i	$0.799326\pi$
$600$	0	0
$601$	20.0367	0.817314	0.408657	−	0.912688i	$-0.365997\pi$
$601$	20.0367	0.817314	0.408657	+	0.912688i	$0.365997\pi$
$602$	0	0
$603$	1.88651	0.0768245
$604$	0	0
$605$	−22.9209	−0.931867
$606$	0	0
$607$	−37.4468	−1.51992	−0.759959	−	0.649971i	$-0.774781\pi$
$607$	−37.4468	−1.51992	−0.759959	+	0.649971i	$0.774781\pi$
$608$	0	0
$609$	−1.62014	−0.0656514
$610$	0	0
$611$	41.6322	1.68426
$612$	0	0
$613$	16.2695	0.657117	0.328559	−	0.944484i	$-0.393437\pi$
$613$	16.2695	0.657117	0.328559	+	0.944484i	$0.393437\pi$
$614$	0	0
$615$	31.0464	1.25191
$616$	0	0
$617$	−16.2920	−0.655890	−0.327945	−	0.944697i	$-0.606356\pi$
$617$	−16.2920	−0.655890	−0.327945	+	0.944697i	$0.606356\pi$
$618$	0	0
$619$	34.9575	1.40506	0.702531	−	0.711653i	$-0.252053\pi$
$619$	34.9575	1.40506	0.702531	+	0.711653i	$0.252053\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−21.6118	−0.865858
$624$	0	0
$625$	7.35736	0.294295
$626$	0	0
$627$	31.7491	1.26794
$628$	0	0
$629$	−0.365175	−0.0145605
$630$	0	0
$631$	−23.0492	−0.917573	−0.458787	−	0.888546i	$-0.651716\pi$
$631$	−23.0492	−0.917573	−0.458787	+	0.888546i	$0.651716\pi$
$632$	0	0
$633$	−4.52898	−0.180011
$634$	0	0
$635$	15.7176	0.623735
$636$	0	0
$637$	24.7791	0.981783
$638$	0	0
$639$	−12.7519	−0.504457
$640$	0	0
$641$	−24.1548	−0.954058	−0.477029	−	0.878888i	$-0.658286\pi$
$641$	−24.1548	−0.954058	−0.477029	+	0.878888i	$0.658286\pi$
$642$	0	0
$643$	−18.8473	−0.743266	−0.371633	−	0.928380i	$-0.621202\pi$
$643$	−18.8473	−0.743266	−0.371633	+	0.928380i	$0.621202\pi$
$644$	0	0
$645$	−42.3038	−1.66571
$646$	0	0
$647$	−30.4301	−1.19633	−0.598166	−	0.801372i	$-0.704104\pi$
$647$	−30.4301	−1.19633	−0.598166	+	0.801372i	$0.704104\pi$
$648$	0	0
$649$	6.71000	0.263390
$650$	0	0
$651$	15.7745	0.618253
$652$	0	0
$653$	−30.2060	−1.18205	−0.591025	−	0.806653i	$-0.701277\pi$
$653$	−30.2060	−1.18205	−0.591025	+	0.806653i	$0.701277\pi$
$654$	0	0
$655$	33.4388	1.30656
$656$	0	0
$657$	−1.88003	−0.0733469
$658$	0	0
$659$	14.9830	0.583653	0.291827	−	0.956471i	$-0.405737\pi$
$659$	14.9830	0.583653	0.291827	+	0.956471i	$0.405737\pi$
$660$	0	0
$661$	−40.6718	−1.58195	−0.790974	−	0.611850i	$-0.790426\pi$
$661$	−40.6718	−1.58195	−0.790974	+	0.611850i	$0.790426\pi$
$662$	0	0
$663$	−1.55638	−0.0604449
$664$	0	0
$665$	−45.9438	−1.78162
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	29.4109	1.13709
$670$	0	0
$671$	−34.4491	−1.32989
$672$	0	0
$673$	−28.4682	−1.09737	−0.548685	−	0.836029i	$-0.684871\pi$
$673$	−28.4682	−1.09737	−0.548685	+	0.836029i	$0.684871\pi$
$674$	0	0
$675$	8.71348	0.335382
$676$	0	0
$677$	40.3759	1.55177	0.775885	−	0.630874i	$-0.217303\pi$
$677$	40.3759	1.55177	0.775885	+	0.630874i	$0.217303\pi$
$678$	0	0
$679$	8.66054	0.332361
$680$	0	0
$681$	−2.89069	−0.110772
$682$	0	0
$683$	−48.5861	−1.85909	−0.929547	−	0.368704i	$-0.879802\pi$
$683$	−48.5861	−1.85909	−0.929547	+	0.368704i	$0.879802\pi$
$684$	0	0
$685$	14.8746	0.568328
$686$	0	0
$687$	−18.1411	−0.692127
$688$	0	0
$689$	0.348811	0.0132886
$690$	0	0
$691$	29.2514	1.11278	0.556388	−	0.830922i	$-0.312187\pi$
$691$	29.2514	1.11278	0.556388	+	0.830922i	$0.312187\pi$
$692$	0	0
$693$	6.71714	0.255163
$694$	0	0
$695$	−62.7545	−2.38041
$696$	0	0
$697$	−2.30388	−0.0872659
$698$	0	0
$699$	−8.24266	−0.311766
$700$	0	0
$701$	19.9639	0.754027	0.377013	−	0.926208i	$-0.376951\pi$
$701$	19.9639	0.754027	0.377013	+	0.926208i	$0.376951\pi$
$702$	0	0
$703$	10.1760	0.383795
$704$	0	0
$705$	27.2214	1.02522
$706$	0	0
$707$	16.0023	0.601827
$708$	0	0
$709$	25.9453	0.974395	0.487198	−	0.873292i	$-0.338019\pi$
$709$	25.9453	0.974395	0.487198	+	0.873292i	$0.338019\pi$
$710$	0	0
$711$	−13.8611	−0.519831
$712$	0	0
$713$	9.73653	0.364636
$714$	0	0
$715$	−86.9558	−3.25196
$716$	0	0
$717$	0.715010	0.0267026
$718$	0	0
$719$	25.3540	0.945544	0.472772	−	0.881185i	$-0.343253\pi$
$719$	25.3540	0.945544	0.472772	+	0.881185i	$0.343253\pi$
$720$	0	0
$721$	13.8098	0.514306
$722$	0	0
$723$	−14.8598	−0.552641
$724$	0	0
$725$	8.71348	0.323611
$726$	0	0
$727$	−33.8817	−1.25660	−0.628302	−	0.777969i	$-0.716250\pi$
$727$	−33.8817	−1.25660	−0.628302	+	0.777969i	$0.716250\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.13927	0.116110
$732$	0	0
$733$	46.1803	1.70571	0.852855	−	0.522148i	$-0.174869\pi$
$733$	46.1803	1.70571	0.852855	+	0.522148i	$0.174869\pi$
$734$	0	0
$735$	16.2019	0.597617
$736$	0	0
$737$	−7.82150	−0.288109
$738$	0	0
$739$	−45.8663	−1.68722	−0.843610	−	0.536957i	$-0.819574\pi$
$739$	−45.8663	−1.68722	−0.843610	+	0.536957i	$0.819574\pi$
$740$	0	0
$741$	43.3703	1.59325
$742$	0	0
$743$	−13.4807	−0.494560	−0.247280	−	0.968944i	$-0.579537\pi$
$743$	−13.4807	−0.494560	−0.247280	+	0.968944i	$0.579537\pi$
$744$	0	0
$745$	9.75857	0.357526
$746$	0	0
$747$	−10.8949	−0.398622
$748$	0	0
$749$	−0.724297	−0.0264652
$750$	0	0
$751$	−21.2766	−0.776396	−0.388198	−	0.921576i	$-0.626902\pi$
$751$	−21.2766	−0.776396	−0.388198	+	0.921576i	$0.626902\pi$
$752$	0	0
$753$	−17.5738	−0.640425
$754$	0	0
$755$	−24.0240	−0.874325
$756$	0	0
$757$	14.3000	0.519744	0.259872	−	0.965643i	$-0.416320\pi$
$757$	14.3000	0.519744	0.259872	+	0.965643i	$0.416320\pi$
$758$	0	0
$759$	4.14603	0.150491
$760$	0	0
$761$	−12.3065	−0.446109	−0.223054	−	0.974806i	$-0.571603\pi$
$761$	−12.3065	−0.446109	−0.223054	+	0.974806i	$0.571603\pi$
$762$	0	0
$763$	−29.8188	−1.07951
$764$	0	0
$765$	−1.01765	−0.0367931
$766$	0	0
$767$	9.16607	0.330968
$768$	0	0
$769$	10.4114	0.375446	0.187723	−	0.982222i	$-0.439889\pi$
$769$	10.4114	0.375446	0.187723	+	0.982222i	$0.439889\pi$
$770$	0	0
$771$	−14.8820	−0.535963
$772$	0	0
$773$	−1.18411	−0.0425896	−0.0212948	−	0.999773i	$-0.506779\pi$
$773$	−1.18411	−0.0425896	−0.0212948	+	0.999773i	$0.506779\pi$
$774$	0	0
$775$	−84.8391	−3.04751
$776$	0	0
$777$	2.15293	0.0772359
$778$	0	0
$779$	64.2003	2.30021
$780$	0	0
$781$	52.8697	1.89183
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−8.71361	−0.311002
$786$	0	0
$787$	−34.4139	−1.22672	−0.613362	−	0.789802i	$-0.710183\pi$
$787$	−34.4139	−1.22672	−0.613362	+	0.789802i	$0.710183\pi$
$788$	0	0
$789$	−16.5865	−0.590496
$790$	0	0
$791$	−33.5542	−1.19305
$792$	0	0
$793$	−47.0586	−1.67110
$794$	0	0
$795$	0.228072	0.00808887
$796$	0	0
$797$	16.8428	0.596604	0.298302	−	0.954472i	$-0.403580\pi$
$797$	16.8428	0.596604	0.298302	+	0.954472i	$0.403580\pi$
$798$	0	0
$799$	−2.02004	−0.0714640
$800$	0	0
$801$	13.3395	0.471327
$802$	0	0
$803$	7.79464	0.275067
$804$	0	0
$805$	−5.99966	−0.211460
$806$	0	0
$807$	−30.8549	−1.08615
$808$	0	0
$809$	21.5272	0.756857	0.378429	−	0.925630i	$-0.376465\pi$
$809$	21.5272	0.756857	0.378429	+	0.925630i	$0.376465\pi$
$810$	0	0
$811$	14.4033	0.505767	0.252883	−	0.967497i	$-0.418621\pi$
$811$	14.4033	0.505767	0.252883	+	0.967497i	$0.418621\pi$
$812$	0	0
$813$	21.1166	0.740592
$814$	0	0
$815$	−79.8182	−2.79591
$816$	0	0
$817$	−87.4793	−3.06051
$818$	0	0
$819$	9.17583	0.320629
$820$	0	0
$821$	−16.0760	−0.561056	−0.280528	−	0.959846i	$-0.590510\pi$
$821$	−16.0760	−0.561056	−0.280528	+	0.959846i	$0.590510\pi$
$822$	0	0
$823$	29.7607	1.03739	0.518697	−	0.854958i	$-0.326417\pi$
$823$	29.7607	1.03739	0.518697	+	0.854958i	$0.326417\pi$
$824$	0	0
$825$	−36.1263	−1.25776
$826$	0	0
$827$	−56.2046	−1.95443	−0.977213	−	0.212261i	$-0.931917\pi$
$827$	−56.2046	−1.95443	−0.977213	+	0.212261i	$0.931917\pi$
$828$	0	0
$829$	−1.57759	−0.0547921	−0.0273960	−	0.999625i	$-0.508722\pi$
$829$	−1.57759	−0.0547921	−0.0273960	+	0.999625i	$0.508722\pi$
$830$	0	0
$831$	13.6044	0.471931
$832$	0	0
$833$	−1.20231	−0.0416576
$834$	0	0
$835$	25.2191	0.872742
$836$	0	0
$837$	−9.73653	−0.336544
$838$	0	0
$839$	−16.7234	−0.577357	−0.288679	−	0.957426i	$-0.593216\pi$
$839$	−16.7234	−0.577357	−0.288679	+	0.957426i	$0.593216\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−2.38601	−0.0821787
$844$	0	0
$845$	−70.6431	−2.43020
$846$	0	0
$847$	−10.0279	−0.344563
$848$	0	0
$849$	17.9545	0.616195
$850$	0	0
$851$	1.32885	0.0455525
$852$	0	0
$853$	−45.0565	−1.54271	−0.771353	−	0.636407i	$-0.780420\pi$
$853$	−45.0565	−1.54271	−0.771353	+	0.636407i	$0.780420\pi$
$854$	0	0
$855$	28.3579	0.969819
$856$	0	0
$857$	31.1371	1.06362	0.531812	−	0.846862i	$-0.321511\pi$
$857$	31.1371	1.06362	0.531812	+	0.846862i	$0.321511\pi$
$858$	0	0
$859$	5.31326	0.181286	0.0906431	−	0.995883i	$-0.471108\pi$
$859$	5.31326	0.181286	0.0906431	+	0.995883i	$0.471108\pi$
$860$	0	0
$861$	13.5828	0.462901
$862$	0	0
$863$	−17.5466	−0.597294	−0.298647	−	0.954364i	$-0.596535\pi$
$863$	−17.5466	−0.597294	−0.298647	+	0.954364i	$0.596535\pi$
$864$	0	0
$865$	69.7360	2.37109
$866$	0	0
$867$	−16.9245	−0.574786
$868$	0	0
$869$	57.4684	1.94948
$870$	0	0
$871$	−10.6844	−0.362028
$872$	0	0
$873$	−5.34555	−0.180919
$874$	0	0
$875$	22.2796	0.753189
$876$	0	0
$877$	−48.2010	−1.62763	−0.813817	−	0.581122i	$-0.802614\pi$
$877$	−48.2010	−1.62763	−0.813817	+	0.581122i	$0.802614\pi$
$878$	0	0
$879$	−19.7308	−0.665502
$880$	0	0
$881$	−10.6743	−0.359626	−0.179813	−	0.983701i	$-0.557549\pi$
$881$	−10.6743	−0.359626	−0.179813	+	0.983701i	$0.557549\pi$
$882$	0	0
$883$	−15.4526	−0.520021	−0.260011	−	0.965606i	$-0.583726\pi$
$883$	−15.4526	−0.520021	−0.260011	+	0.965606i	$0.583726\pi$
$884$	0	0
$885$	5.99328	0.201462
$886$	0	0
$887$	45.6758	1.53364	0.766821	−	0.641861i	$-0.221837\pi$
$887$	45.6758	1.53364	0.766821	+	0.641861i	$0.221837\pi$
$888$	0	0
$889$	6.87647	0.230629
$890$	0	0
$891$	−4.14603	−0.138897
$892$	0	0
$893$	56.2908	1.88370
$894$	0	0
$895$	44.3632	1.48290
$896$	0	0
$897$	5.66360	0.189102
$898$	0	0
$899$	−9.73653	−0.324732
$900$	0	0
$901$	−0.0169247	−0.000563844 0
$902$	0	0
$903$	−18.5079	−0.615905
$904$	0	0
$905$	−67.6920	−2.25016
$906$	0	0
$907$	40.7445	1.35290	0.676450	−	0.736488i	$-0.263517\pi$
$907$	40.7445	1.35290	0.676450	+	0.736488i	$0.263517\pi$
$908$	0	0
$909$	−9.87708	−0.327602
$910$	0	0
$911$	25.1934	0.834695	0.417347	−	0.908747i	$-0.362960\pi$
$911$	25.1934	0.834695	0.417347	+	0.908747i	$0.362960\pi$
$912$	0	0
$913$	45.1704	1.49492
$914$	0	0
$915$	−30.7695	−1.01721
$916$	0	0
$917$	14.6295	0.483108
$918$	0	0
$919$	2.83961	0.0936702	0.0468351	−	0.998903i	$-0.485086\pi$
$919$	2.83961	0.0936702	0.0468351	+	0.998903i	$0.485086\pi$
$920$	0	0
$921$	4.30725	0.141929
$922$	0	0
$923$	72.2216	2.37720
$924$	0	0
$925$	−11.5789	−0.380713
$926$	0	0
$927$	−8.52386	−0.279960
$928$	0	0
$929$	−9.98095	−0.327464	−0.163732	−	0.986505i	$-0.552353\pi$
$929$	−9.98095	−0.327464	−0.163732	+	0.986505i	$0.552353\pi$
$930$	0	0
$931$	33.5037	1.09804
$932$	0	0
$933$	−21.4124	−0.701011
$934$	0	0
$935$	4.21920	0.137982
$936$	0	0
$937$	−15.1448	−0.494759	−0.247380	−	0.968919i	$-0.579569\pi$
$937$	−15.1448	−0.494759	−0.247380	+	0.968919i	$0.579569\pi$
$938$	0	0
$939$	14.1128	0.460555
$940$	0	0
$941$	−26.1721	−0.853188	−0.426594	−	0.904443i	$-0.640287\pi$
$941$	−26.1721	−0.853188	−0.426594	+	0.904443i	$0.640287\pi$
$942$	0	0
$943$	8.38372	0.273012
$944$	0	0
$945$	5.99966	0.195169
$946$	0	0
$947$	−7.64135	−0.248311	−0.124155	−	0.992263i	$-0.539622\pi$
$947$	−7.64135	−0.248311	−0.124155	+	0.992263i	$0.539622\pi$
$948$	0	0
$949$	10.6477	0.345640
$950$	0	0
$951$	8.95458	0.290372
$952$	0	0
$953$	39.9506	1.29413	0.647063	−	0.762436i	$-0.275997\pi$
$953$	39.9506	1.29413	0.647063	+	0.762436i	$0.275997\pi$
$954$	0	0
$955$	−31.5674	−1.02150
$956$	0	0
$957$	−4.14603	−0.134022
$958$	0	0
$959$	6.50763	0.210142
$960$	0	0
$961$	63.8000	2.05807
$962$	0	0
$963$	0.447058	0.0144062
$964$	0	0
$965$	32.1630	1.03536
$966$	0	0
$967$	22.2393	0.715166	0.357583	−	0.933881i	$-0.383601\pi$
$967$	22.2393	0.715166	0.357583	+	0.933881i	$0.383601\pi$
$968$	0	0
$969$	−2.10438	−0.0676024
$970$	0	0
$971$	−1.18687	−0.0380885	−0.0190442	−	0.999819i	$-0.506062\pi$
$971$	−1.18687	−0.0380885	−0.0190442	+	0.999819i	$0.506062\pi$
$972$	0	0
$973$	−27.4551	−0.880171
$974$	0	0
$975$	−49.3497	−1.58045
$976$	0	0
$977$	24.8198	0.794055	0.397027	−	0.917807i	$-0.370042\pi$
$977$	24.8198	0.794055	0.397027	+	0.917807i	$0.370042\pi$
$978$	0	0
$979$	−55.3057	−1.76758
$980$	0	0
$981$	18.4051	0.587628
$982$	0	0
$983$	−55.6777	−1.77584	−0.887921	−	0.459995i	$-0.847851\pi$
$983$	−55.6777	−1.77584	−0.887921	+	0.459995i	$0.847851\pi$
$984$	0	0
$985$	−63.0476	−2.00886
$986$	0	0
$987$	11.9094	0.379080
$988$	0	0
$989$	−11.4237	−0.363251
$990$	0	0
$991$	33.1315	1.05246	0.526229	−	0.850343i	$-0.323605\pi$
$991$	33.1315	1.05246	0.526229	+	0.850343i	$0.323605\pi$
$992$	0	0
$993$	2.38872	0.0758039
$994$	0	0
$995$	72.0665	2.28466
$996$	0	0
$997$	−30.4973	−0.965859	−0.482930	−	0.875659i	$-0.660427\pi$
$997$	−30.4973	−0.965859	−0.482930	+	0.875659i	$0.660427\pi$
$998$	0	0
$999$	−1.32885	−0.0420431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.1	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.1	✓	18	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.70317 q^{5} -1.62014 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.70317 q^{5} -1.62014 q^{7} +1.00000 q^{9} -4.14603 q^{11} -5.66360 q^{13} -3.70317 q^{15} +0.274804 q^{17} -7.65773 q^{19} -1.62014 q^{21} -1.00000 q^{23} +8.71348 q^{25} +1.00000 q^{27} +1.00000 q^{29} -9.73653 q^{31} -4.14603 q^{33} +5.99966 q^{35} -1.32885 q^{37} -5.66360 q^{39} -8.38372 q^{41} +11.4237 q^{43} -3.70317 q^{45} -7.35084 q^{47} -4.37515 q^{49} +0.274804 q^{51} -0.0615882 q^{53} +15.3534 q^{55} -7.65773 q^{57} -1.61842 q^{59} +8.30895 q^{61} -1.62014 q^{63} +20.9733 q^{65} +1.88651 q^{67} -1.00000 q^{69} -12.7519 q^{71} -1.88003 q^{73} +8.71348 q^{75} +6.71714 q^{77} -13.8611 q^{79} +1.00000 q^{81} -10.8949 q^{83} -1.01765 q^{85} +1.00000 q^{87} +13.3395 q^{89} +9.17583 q^{91} -9.73653 q^{93} +28.3579 q^{95} -5.34555 q^{97} -4.14603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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