LMFDB - Embedded newform 8004.2.a.h.1.6

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.672529$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.672529 q^{5} -0.215981 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.672529 q^{5} -0.215981 q^{7} +1.00000 q^{9} -0.00111213 q^{11} -0.970007 q^{13} +0.672529 q^{15} -4.29799 q^{17} -6.88904 q^{19} +0.215981 q^{21} -1.00000 q^{23} -4.54771 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.68639 q^{31} +0.00111213 q^{33} +0.145253 q^{35} +10.4513 q^{37} +0.970007 q^{39} -3.03139 q^{41} -5.90932 q^{43} -0.672529 q^{45} -3.76327 q^{47} -6.95335 q^{49} +4.29799 q^{51} +10.9321 q^{53} +0.000747939 q^{55} +6.88904 q^{57} -6.67243 q^{59} -11.7474 q^{61} -0.215981 q^{63} +0.652358 q^{65} +2.91865 q^{67} +1.00000 q^{69} +14.8584 q^{71} +10.7310 q^{73} +4.54771 q^{75} +0.000240198 q^{77} -2.65534 q^{79} +1.00000 q^{81} +7.79540 q^{83} +2.89052 q^{85} +1.00000 q^{87} -6.34633 q^{89} +0.209503 q^{91} -5.68639 q^{93} +4.63307 q^{95} +16.4905 q^{97} -0.00111213 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.672529	−0.300764	−0.150382	−	0.988628i	$-0.548050\pi$
$5$	−0.672529	−0.300764	−0.150382	+	0.988628i	$0.548050\pi$
$6$	0	0
$7$	−0.215981	−0.0816330	−0.0408165	−	0.999167i	$-0.512996\pi$
$7$	−0.215981	−0.0816330	−0.0408165	+	0.999167i	$0.512996\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.00111213	−0.000335320 0	−0.000167660	−	1.00000i	$-0.500053\pi$
$11$	−0.00111213	−0.000335320 0	−0.000167660		1.00000i	$0.500053\pi$
$12$	0	0
$13$	−0.970007	−0.269032	−0.134516	−	0.990911i	$-0.542948\pi$
$13$	−0.970007	−0.269032	−0.134516	+	0.990911i	$0.542948\pi$
$14$	0	0
$15$	0.672529	0.173646
$16$	0	0
$17$	−4.29799	−1.04241	−0.521207	−	0.853430i	$-0.674518\pi$
$17$	−4.29799	−1.04241	−0.521207	+	0.853430i	$0.674518\pi$
$18$	0	0
$19$	−6.88904	−1.58045	−0.790227	−	0.612815i	$-0.790037\pi$
$19$	−6.88904	−1.58045	−0.790227	+	0.612815i	$0.790037\pi$
$20$	0	0
$21$	0.215981	0.0471309
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.54771	−0.909541
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	5.68639	1.02130	0.510652	−	0.859787i	$-0.329404\pi$
$31$	5.68639	1.02130	0.510652	+	0.859787i	$0.329404\pi$
$32$	0	0
$33$	0.00111213	0.000193597 0
$34$	0	0
$35$	0.145253	0.0245523
$36$	0	0
$37$	10.4513	1.71818	0.859089	−	0.511826i	$-0.171031\pi$
$37$	10.4513	1.71818	0.859089	+	0.511826i	$0.171031\pi$
$38$	0	0
$39$	0.970007	0.155326
$40$	0	0
$41$	−3.03139	−0.473424	−0.236712	−	0.971580i	$-0.576070\pi$
$41$	−3.03139	−0.473424	−0.236712	+	0.971580i	$0.576070\pi$
$42$	0	0
$43$	−5.90932	−0.901163	−0.450581	−	0.892735i	$-0.648783\pi$
$43$	−5.90932	−0.901163	−0.450581	+	0.892735i	$0.648783\pi$
$44$	0	0
$45$	−0.672529	−0.100255
$46$	0	0
$47$	−3.76327	−0.548930	−0.274465	−	0.961597i	$-0.588501\pi$
$47$	−3.76327	−0.548930	−0.274465	+	0.961597i	$0.588501\pi$
$48$	0	0
$49$	−6.95335	−0.993336
$50$	0	0
$51$	4.29799	0.601838
$52$	0	0
$53$	10.9321	1.50164	0.750821	−	0.660506i	$-0.229658\pi$
$53$	10.9321	1.50164	0.750821	+	0.660506i	$0.229658\pi$
$54$	0	0
$55$	0.000747939 0	0.000100852 0
$56$	0	0
$57$	6.88904	0.912475
$58$	0	0
$59$	−6.67243	−0.868677	−0.434338	−	0.900750i	$-0.643018\pi$
$59$	−6.67243	−0.868677	−0.434338	+	0.900750i	$0.643018\pi$
$60$	0	0
$61$	−11.7474	−1.50411	−0.752053	−	0.659103i	$-0.770936\pi$
$61$	−11.7474	−1.50411	−0.752053	+	0.659103i	$0.770936\pi$
$62$	0	0
$63$	−0.215981	−0.0272110
$64$	0	0
$65$	0.652358	0.0809150
$66$	0	0
$67$	2.91865	0.356570	0.178285	−	0.983979i	$-0.442945\pi$
$67$	2.91865	0.356570	0.178285	+	0.983979i	$0.442945\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	14.8584	1.76336	0.881682	−	0.471845i	$-0.156412\pi$
$71$	14.8584	1.76336	0.881682	+	0.471845i	$0.156412\pi$
$72$	0	0
$73$	10.7310	1.25597	0.627985	−	0.778225i	$-0.283880\pi$
$73$	10.7310	1.25597	0.627985	+	0.778225i	$0.283880\pi$
$74$	0	0
$75$	4.54771	0.525124
$76$	0	0
$77$	0.000240198 0	2.73731e−5 0
$78$	0	0
$79$	−2.65534	−0.298749	−0.149374	−	0.988781i	$-0.547726\pi$
$79$	−2.65534	−0.298749	−0.149374	+	0.988781i	$0.547726\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.79540	0.855656	0.427828	−	0.903860i	$-0.359279\pi$
$83$	7.79540	0.855656	0.427828	+	0.903860i	$0.359279\pi$
$84$	0	0
$85$	2.89052	0.313521
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−6.34633	−0.672709	−0.336355	−	0.941735i	$-0.609194\pi$
$89$	−6.34633	−0.672709	−0.336355	+	0.941735i	$0.609194\pi$
$90$	0	0
$91$	0.209503	0.0219619
$92$	0	0
$93$	−5.68639	−0.589651
$94$	0	0
$95$	4.63307	0.475343
$96$	0	0
$97$	16.4905	1.67436	0.837180	−	0.546928i	$-0.184203\pi$
$97$	16.4905	1.67436	0.837180	+	0.546928i	$0.184203\pi$
$98$	0	0
$99$	−0.00111213	−0.000111773 0
$100$	0	0
$101$	−1.38971	−0.138281	−0.0691404	−	0.997607i	$-0.522026\pi$
$101$	−1.38971	−0.138281	−0.0691404	+	0.997607i	$0.522026\pi$
$102$	0	0
$103$	−10.8097	−1.06511	−0.532557	−	0.846394i	$-0.678769\pi$
$103$	−10.8097	−1.06511	−0.532557	+	0.846394i	$0.678769\pi$
$104$	0	0
$105$	−0.145253	−0.0141753
$106$	0	0
$107$	−10.3150	−0.997192	−0.498596	−	0.866835i	$-0.666151\pi$
$107$	−10.3150	−0.997192	−0.498596	+	0.866835i	$0.666151\pi$
$108$	0	0
$109$	−2.37137	−0.227136	−0.113568	−	0.993530i	$-0.536228\pi$
$109$	−2.37137	−0.227136	−0.113568	+	0.993530i	$0.536228\pi$
$110$	0	0
$111$	−10.4513	−0.991991
$112$	0	0
$113$	5.38562	0.506637	0.253318	−	0.967383i	$-0.418478\pi$
$113$	5.38562	0.506637	0.253318	+	0.967383i	$0.418478\pi$
$114$	0	0
$115$	0.672529	0.0627136
$116$	0	0
$117$	−0.970007	−0.0896772
$118$	0	0
$119$	0.928282	0.0850955
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	3.03139	0.273331
$124$	0	0
$125$	6.42110	0.574321
$126$	0	0
$127$	6.62676	0.588030	0.294015	−	0.955801i	$-0.405008\pi$
$127$	6.62676	0.588030	0.294015	+	0.955801i	$0.405008\pi$
$128$	0	0
$129$	5.90932	0.520287
$130$	0	0
$131$	11.0773	0.967832	0.483916	−	0.875114i	$-0.339214\pi$
$131$	11.0773	0.967832	0.483916	+	0.875114i	$0.339214\pi$
$132$	0	0
$133$	1.48790	0.129017
$134$	0	0
$135$	0.672529	0.0578820
$136$	0	0
$137$	−2.23488	−0.190938	−0.0954692	−	0.995432i	$-0.530435\pi$
$137$	−2.23488	−0.190938	−0.0954692	+	0.995432i	$0.530435\pi$
$138$	0	0
$139$	1.32651	0.112513	0.0562566	−	0.998416i	$-0.482084\pi$
$139$	1.32651	0.112513	0.0562566	+	0.998416i	$0.482084\pi$
$140$	0	0
$141$	3.76327	0.316925
$142$	0	0
$143$	0.00107877	9.02116e−5 0
$144$	0	0
$145$	0.672529	0.0558505
$146$	0	0
$147$	6.95335	0.573503
$148$	0	0
$149$	16.9107	1.38538	0.692690	−	0.721235i	$-0.256425\pi$
$149$	16.9107	1.38538	0.692690	+	0.721235i	$0.256425\pi$
$150$	0	0
$151$	11.6491	0.947987	0.473994	−	0.880528i	$-0.342812\pi$
$151$	11.6491	0.947987	0.473994	+	0.880528i	$0.342812\pi$
$152$	0	0
$153$	−4.29799	−0.347471
$154$	0	0
$155$	−3.82426	−0.307172
$156$	0	0
$157$	15.5798	1.24341	0.621703	−	0.783253i	$-0.286441\pi$
$157$	15.5798	1.24341	0.621703	+	0.783253i	$0.286441\pi$
$158$	0	0
$159$	−10.9321	−0.866973
$160$	0	0
$161$	0.215981	0.0170217
$162$	0	0
$163$	−0.951286	−0.0745104	−0.0372552	−	0.999306i	$-0.511861\pi$
$163$	−0.951286	−0.0745104	−0.0372552	+	0.999306i	$0.511861\pi$
$164$	0	0
$165$	−0.000747939 0	−5.82269e−5 0
$166$	0	0
$167$	−10.5702	−0.817946	−0.408973	−	0.912547i	$-0.634113\pi$
$167$	−10.5702	−0.817946	−0.408973	+	0.912547i	$0.634113\pi$
$168$	0	0
$169$	−12.0591	−0.927622
$170$	0	0
$171$	−6.88904	−0.526818
$172$	0	0
$173$	−17.8215	−1.35494	−0.677472	−	0.735548i	$-0.736925\pi$
$173$	−17.8215	−1.35494	−0.677472	+	0.735548i	$0.736925\pi$
$174$	0	0
$175$	0.982217	0.0742486
$176$	0	0
$177$	6.67243	0.501531
$178$	0	0
$179$	16.7683	1.25332	0.626659	−	0.779293i	$-0.284422\pi$
$179$	16.7683	1.25332	0.626659	+	0.779293i	$0.284422\pi$
$180$	0	0
$181$	23.3694	1.73703	0.868517	−	0.495659i	$-0.165073\pi$
$181$	23.3694	1.73703	0.868517	+	0.495659i	$0.165073\pi$
$182$	0	0
$183$	11.7474	0.868396
$184$	0	0
$185$	−7.02878	−0.516766
$186$	0	0
$187$	0.00477991	0.000349542 0
$188$	0	0
$189$	0.215981	0.0157103
$190$	0	0
$191$	−0.839357	−0.0607337	−0.0303669	−	0.999539i	$-0.509668\pi$
$191$	−0.839357	−0.0607337	−0.0303669	+	0.999539i	$0.509668\pi$
$192$	0	0
$193$	−20.7130	−1.49096	−0.745479	−	0.666529i	$-0.767779\pi$
$193$	−20.7130	−1.49096	−0.745479	+	0.666529i	$0.767779\pi$
$194$	0	0
$195$	−0.652358	−0.0467163
$196$	0	0
$197$	9.28009	0.661179	0.330589	−	0.943775i	$-0.392752\pi$
$197$	9.28009	0.661179	0.330589	+	0.943775i	$0.392752\pi$
$198$	0	0
$199$	−21.5634	−1.52859	−0.764293	−	0.644869i	$-0.776912\pi$
$199$	−21.5634	−1.52859	−0.764293	+	0.644869i	$0.776912\pi$
$200$	0	0
$201$	−2.91865	−0.205866
$202$	0	0
$203$	0.215981	0.0151589
$204$	0	0
$205$	2.03870	0.142389
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	0.00766150	0.000529957 0
$210$	0	0
$211$	−4.73933	−0.326269	−0.163134	−	0.986604i	$-0.552160\pi$
$211$	−4.73933	−0.326269	−0.163134	+	0.986604i	$0.552160\pi$
$212$	0	0
$213$	−14.8584	−1.01808
$214$	0	0
$215$	3.97419	0.271037
$216$	0	0
$217$	−1.22815	−0.0833722
$218$	0	0
$219$	−10.7310	−0.725135
$220$	0	0
$221$	4.16908	0.280443
$222$	0	0
$223$	−12.2177	−0.818161	−0.409081	−	0.912498i	$-0.634150\pi$
$223$	−12.2177	−0.818161	−0.409081	+	0.912498i	$0.634150\pi$
$224$	0	0
$225$	−4.54771	−0.303180
$226$	0	0
$227$	−22.4270	−1.48853	−0.744265	−	0.667885i	$-0.767200\pi$
$227$	−22.4270	−1.48853	−0.744265	+	0.667885i	$0.767200\pi$
$228$	0	0
$229$	18.8792	1.24757	0.623785	−	0.781596i	$-0.285594\pi$
$229$	18.8792	1.24757	0.623785	+	0.781596i	$0.285594\pi$
$230$	0	0
$231$	−0.000240198 0	−1.58039e−5 0
$232$	0	0
$233$	−19.4418	−1.27367	−0.636836	−	0.770999i	$-0.719757\pi$
$233$	−19.4418	−1.27367	−0.636836	+	0.770999i	$0.719757\pi$
$234$	0	0
$235$	2.53091	0.165098
$236$	0	0
$237$	2.65534	0.172483
$238$	0	0
$239$	12.1394	0.785233	0.392616	−	0.919702i	$-0.371570\pi$
$239$	12.1394	0.785233	0.392616	+	0.919702i	$0.371570\pi$
$240$	0	0
$241$	−25.3169	−1.63081	−0.815404	−	0.578892i	$-0.803485\pi$
$241$	−25.3169	−1.63081	−0.815404	+	0.578892i	$0.803485\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.67633	0.298760
$246$	0	0
$247$	6.68242	0.425192
$248$	0	0
$249$	−7.79540	−0.494013
$250$	0	0
$251$	−16.4191	−1.03636	−0.518181	−	0.855271i	$-0.673391\pi$
$251$	−16.4191	−1.03636	−0.518181	+	0.855271i	$0.673391\pi$
$252$	0	0
$253$	0.00111213	6.99190e−5 0
$254$	0	0
$255$	−2.89052	−0.181011
$256$	0	0
$257$	26.5897	1.65862	0.829311	−	0.558787i	$-0.188733\pi$
$257$	26.5897	1.65862	0.829311	+	0.558787i	$0.188733\pi$
$258$	0	0
$259$	−2.25727	−0.140260
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	25.9312	1.59899	0.799493	−	0.600676i	$-0.205102\pi$
$263$	25.9312	1.59899	0.799493	+	0.600676i	$0.205102\pi$
$264$	0	0
$265$	−7.35216	−0.451640
$266$	0	0
$267$	6.34633	0.388389
$268$	0	0
$269$	−2.26409	−0.138044	−0.0690219	−	0.997615i	$-0.521988\pi$
$269$	−2.26409	−0.138044	−0.0690219	+	0.997615i	$0.521988\pi$
$270$	0	0
$271$	17.6472	1.07199	0.535996	−	0.844221i	$-0.319936\pi$
$271$	17.6472	1.07199	0.535996	+	0.844221i	$0.319936\pi$
$272$	0	0
$273$	−0.209503	−0.0126797
$274$	0	0
$275$	0.00505764	0.000304987 0
$276$	0	0
$277$	23.2745	1.39843	0.699214	−	0.714912i	$-0.253533\pi$
$277$	23.2745	1.39843	0.699214	+	0.714912i	$0.253533\pi$
$278$	0	0
$279$	5.68639	0.340435
$280$	0	0
$281$	10.1633	0.606289	0.303144	−	0.952945i	$-0.401964\pi$
$281$	10.1633	0.606289	0.303144	+	0.952945i	$0.401964\pi$
$282$	0	0
$283$	−15.6539	−0.930528	−0.465264	−	0.885172i	$-0.654041\pi$
$283$	−15.6539	−0.930528	−0.465264	+	0.885172i	$0.654041\pi$
$284$	0	0
$285$	−4.63307	−0.274440
$286$	0	0
$287$	0.654722	0.0386470
$288$	0	0
$289$	1.47268	0.0866280
$290$	0	0
$291$	−16.4905	−0.966692
$292$	0	0
$293$	8.30021	0.484904	0.242452	−	0.970163i	$-0.422048\pi$
$293$	8.30021	0.484904	0.242452	+	0.970163i	$0.422048\pi$
$294$	0	0
$295$	4.48740	0.261267
$296$	0	0
$297$	0.00111213	6.45323e−5 0
$298$	0	0
$299$	0.970007	0.0560970
$300$	0	0
$301$	1.27630	0.0735647
$302$	0	0
$303$	1.38971	0.0798365
$304$	0	0
$305$	7.90049	0.452381
$306$	0	0
$307$	11.2228	0.640519	0.320260	−	0.947330i	$-0.396230\pi$
$307$	11.2228	0.640519	0.320260	+	0.947330i	$0.396230\pi$
$308$	0	0
$309$	10.8097	0.614943
$310$	0	0
$311$	12.0015	0.680545	0.340272	−	0.940327i	$-0.389481\pi$
$311$	12.0015	0.680545	0.340272	+	0.940327i	$0.389481\pi$
$312$	0	0
$313$	20.5901	1.16382	0.581910	−	0.813253i	$-0.302306\pi$
$313$	20.5901	1.16382	0.581910	+	0.813253i	$0.302306\pi$
$314$	0	0
$315$	0.145253	0.00818409
$316$	0	0
$317$	28.1834	1.58294	0.791468	−	0.611210i	$-0.209317\pi$
$317$	28.1834	1.58294	0.791468	+	0.611210i	$0.209317\pi$
$318$	0	0
$319$	0.00111213	6.22673e−5 0
$320$	0	0
$321$	10.3150	0.575729
$322$	0	0
$323$	29.6090	1.64749
$324$	0	0
$325$	4.41131	0.244695
$326$	0	0
$327$	2.37137	0.131137
$328$	0	0
$329$	0.812795	0.0448108
$330$	0	0
$331$	−16.4795	−0.905794	−0.452897	−	0.891563i	$-0.649609\pi$
$331$	−16.4795	−0.905794	−0.452897	+	0.891563i	$0.649609\pi$
$332$	0	0
$333$	10.4513	0.572726
$334$	0	0
$335$	−1.96288	−0.107243
$336$	0	0
$337$	9.22363	0.502443	0.251222	−	0.967930i	$-0.419168\pi$
$337$	9.22363	0.502443	0.251222	+	0.967930i	$0.419168\pi$
$338$	0	0
$339$	−5.38562	−0.292507
$340$	0	0
$341$	−0.00632399	−0.000342463 0
$342$	0	0
$343$	3.01365	0.162722
$344$	0	0
$345$	−0.672529	−0.0362077
$346$	0	0
$347$	−1.09992	−0.0590468	−0.0295234	−	0.999564i	$-0.509399\pi$
$347$	−1.09992	−0.0590468	−0.0295234	+	0.999564i	$0.509399\pi$
$348$	0	0
$349$	−5.63351	−0.301555	−0.150778	−	0.988568i	$-0.548178\pi$
$349$	−5.63351	−0.301555	−0.150778	+	0.988568i	$0.548178\pi$
$350$	0	0
$351$	0.970007	0.0517752
$352$	0	0
$353$	17.8632	0.950760	0.475380	−	0.879781i	$-0.342311\pi$
$353$	17.8632	0.950760	0.475380	+	0.879781i	$0.342311\pi$
$354$	0	0
$355$	−9.99268	−0.530356
$356$	0	0
$357$	−0.928282	−0.0491299
$358$	0	0
$359$	16.3583	0.863360	0.431680	−	0.902027i	$-0.357921\pi$
$359$	16.3583	0.863360	0.431680	+	0.902027i	$0.357921\pi$
$360$	0	0
$361$	28.4588	1.49783
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−7.21692	−0.377751
$366$	0	0
$367$	−3.28379	−0.171413	−0.0857063	−	0.996320i	$-0.527315\pi$
$367$	−3.28379	−0.171413	−0.0857063	+	0.996320i	$0.527315\pi$
$368$	0	0
$369$	−3.03139	−0.157808
$370$	0	0
$371$	−2.36113	−0.122584
$372$	0	0
$373$	−16.4652	−0.852535	−0.426267	−	0.904597i	$-0.640172\pi$
$373$	−16.4652	−0.852535	−0.426267	+	0.904597i	$0.640172\pi$
$374$	0	0
$375$	−6.42110	−0.331584
$376$	0	0
$377$	0.970007	0.0499579
$378$	0	0
$379$	−14.9986	−0.770427	−0.385213	−	0.922827i	$-0.625872\pi$
$379$	−14.9986	−0.770427	−0.385213	+	0.922827i	$0.625872\pi$
$380$	0	0
$381$	−6.62676	−0.339499
$382$	0	0
$383$	27.7693	1.41895	0.709473	−	0.704733i	$-0.248933\pi$
$383$	27.7693	1.41895	0.709473	+	0.704733i	$0.248933\pi$
$384$	0	0
$385$	−0.000161540 0	−8.23286e−6 0
$386$	0	0
$387$	−5.90932	−0.300388
$388$	0	0
$389$	19.9507	1.01154	0.505771	−	0.862668i	$-0.331208\pi$
$389$	19.9507	1.01154	0.505771	+	0.862668i	$0.331208\pi$
$390$	0	0
$391$	4.29799	0.217358
$392$	0	0
$393$	−11.0773	−0.558778
$394$	0	0
$395$	1.78579	0.0898528
$396$	0	0
$397$	12.9342	0.649151	0.324576	−	0.945860i	$-0.394779\pi$
$397$	12.9342	0.649151	0.324576	+	0.945860i	$0.394779\pi$
$398$	0	0
$399$	−1.48790	−0.0744881
$400$	0	0
$401$	18.2666	0.912189	0.456095	−	0.889931i	$-0.349248\pi$
$401$	18.2666	0.912189	0.456095	+	0.889931i	$0.349248\pi$
$402$	0	0
$403$	−5.51584	−0.274763
$404$	0	0
$405$	−0.672529	−0.0334182
$406$	0	0
$407$	−0.0116232	−0.000576139 0
$408$	0	0
$409$	−10.9826	−0.543052	−0.271526	−	0.962431i	$-0.587528\pi$
$409$	−10.9826	−0.543052	−0.271526	+	0.962431i	$0.587528\pi$
$410$	0	0
$411$	2.23488	0.110238
$412$	0	0
$413$	1.44112	0.0709127
$414$	0	0
$415$	−5.24263	−0.257350
$416$	0	0
$417$	−1.32651	−0.0649595
$418$	0	0
$419$	28.1122	1.37337	0.686685	−	0.726955i	$-0.259065\pi$
$419$	28.1122	1.37337	0.686685	+	0.726955i	$0.259065\pi$
$420$	0	0
$421$	25.0721	1.22194	0.610970	−	0.791653i	$-0.290779\pi$
$421$	25.0721	1.22194	0.610970	+	0.791653i	$0.290779\pi$
$422$	0	0
$423$	−3.76327	−0.182977
$424$	0	0
$425$	19.5460	0.948119
$426$	0	0
$427$	2.53722	0.122785
$428$	0	0
$429$	−0.00107877	−5.20837e−5 0
$430$	0	0
$431$	41.0304	1.97637	0.988183	−	0.153278i	$-0.0489830\pi$
$431$	41.0304	1.97637	0.988183	+	0.153278i	$0.0489830\pi$
$432$	0	0
$433$	28.8275	1.38536	0.692680	−	0.721245i	$-0.256430\pi$
$433$	28.8275	1.38536	0.692680	+	0.721245i	$0.256430\pi$
$434$	0	0
$435$	−0.672529	−0.0322453
$436$	0	0
$437$	6.88904	0.329547
$438$	0	0
$439$	0.770685	0.0367828	0.0183914	−	0.999831i	$-0.494146\pi$
$439$	0.770685	0.0367828	0.0183914	+	0.999831i	$0.494146\pi$
$440$	0	0
$441$	−6.95335	−0.331112
$442$	0	0
$443$	−10.2684	−0.487866	−0.243933	−	0.969792i	$-0.578438\pi$
$443$	−10.2684	−0.487866	−0.243933	+	0.969792i	$0.578438\pi$
$444$	0	0
$445$	4.26809	0.202327
$446$	0	0
$447$	−16.9107	−0.799849
$448$	0	0
$449$	27.1186	1.27981	0.639904	−	0.768455i	$-0.278974\pi$
$449$	27.1186	1.27981	0.639904	+	0.768455i	$0.278974\pi$
$450$	0	0
$451$	0.00337130	0.000158748 0
$452$	0	0
$453$	−11.6491	−0.547321
$454$	0	0
$455$	−0.140897	−0.00660534
$456$	0	0
$457$	5.23294	0.244787	0.122393	−	0.992482i	$-0.460943\pi$
$457$	5.23294	0.244787	0.122393	+	0.992482i	$0.460943\pi$
$458$	0	0
$459$	4.29799	0.200613
$460$	0	0
$461$	7.44599	0.346794	0.173397	−	0.984852i	$-0.444526\pi$
$461$	7.44599	0.346794	0.173397	+	0.984852i	$0.444526\pi$
$462$	0	0
$463$	−13.7500	−0.639016	−0.319508	−	0.947584i	$-0.603518\pi$
$463$	−13.7500	−0.639016	−0.319508	+	0.947584i	$0.603518\pi$
$464$	0	0
$465$	3.82426	0.177346
$466$	0	0
$467$	−25.6661	−1.18769	−0.593844	−	0.804581i	$-0.702390\pi$
$467$	−25.6661	−1.18769	−0.593844	+	0.804581i	$0.702390\pi$
$468$	0	0
$469$	−0.630372	−0.0291079
$470$	0	0
$471$	−15.5798	−0.717880
$472$	0	0
$473$	0.00657193	0.000302178 0
$474$	0	0
$475$	31.3293	1.43749
$476$	0	0
$477$	10.9321	0.500547
$478$	0	0
$479$	5.67190	0.259156	0.129578	−	0.991569i	$-0.458638\pi$
$479$	5.67190	0.259156	0.129578	+	0.991569i	$0.458638\pi$
$480$	0	0
$481$	−10.1378	−0.462244
$482$	0	0
$483$	−0.215981	−0.00982746
$484$	0	0
$485$	−11.0904	−0.503587
$486$	0	0
$487$	−39.3004	−1.78087	−0.890436	−	0.455108i	$-0.849601\pi$
$487$	−39.3004	−1.78087	−0.890436	+	0.455108i	$0.849601\pi$
$488$	0	0
$489$	0.951286	0.0430186
$490$	0	0
$491$	39.0127	1.76062	0.880310	−	0.474399i	$-0.157335\pi$
$491$	39.0127	1.76062	0.880310	+	0.474399i	$0.157335\pi$
$492$	0	0
$493$	4.29799	0.193572
$494$	0	0
$495$	0.000747939 0	3.36173e−5 0
$496$	0	0
$497$	−3.20912	−0.143949
$498$	0	0
$499$	−19.7792	−0.885438	−0.442719	−	0.896660i	$-0.645986\pi$
$499$	−19.7792	−0.885438	−0.442719	+	0.896660i	$0.645986\pi$
$500$	0	0
$501$	10.5702	0.472241
$502$	0	0
$503$	−33.6880	−1.50207	−0.751036	−	0.660261i	$-0.770446\pi$
$503$	−33.6880	−1.50207	−0.751036	+	0.660261i	$0.770446\pi$
$504$	0	0
$505$	0.934616	0.0415899
$506$	0	0
$507$	12.0591	0.535563
$508$	0	0
$509$	−13.3470	−0.591596	−0.295798	−	0.955251i	$-0.595586\pi$
$509$	−13.3470	−0.591596	−0.295798	+	0.955251i	$0.595586\pi$
$510$	0	0
$511$	−2.31769	−0.102529
$512$	0	0
$513$	6.88904	0.304158
$514$	0	0
$515$	7.26984	0.320348
$516$	0	0
$517$	0.00418525	0.000184067 0
$518$	0	0
$519$	17.8215	0.782278
$520$	0	0
$521$	1.84056	0.0806364	0.0403182	−	0.999187i	$-0.487163\pi$
$521$	1.84056	0.0806364	0.0403182	+	0.999187i	$0.487163\pi$
$522$	0	0
$523$	−11.0847	−0.484699	−0.242350	−	0.970189i	$-0.577918\pi$
$523$	−11.0847	−0.484699	−0.242350	+	0.970189i	$0.577918\pi$
$524$	0	0
$525$	−0.982217	−0.0428674
$526$	0	0
$527$	−24.4400	−1.06462
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.67243	−0.289559
$532$	0	0
$533$	2.94047	0.127366
$534$	0	0
$535$	6.93715	0.299919
$536$	0	0
$537$	−16.7683	−0.723604
$538$	0	0
$539$	0.00773303	0.000333085 0
$540$	0	0
$541$	40.8360	1.75568	0.877839	−	0.478955i	$-0.158984\pi$
$541$	40.8360	1.75568	0.877839	+	0.478955i	$0.158984\pi$
$542$	0	0
$543$	−23.3694	−1.00288
$544$	0	0
$545$	1.59482	0.0683145
$546$	0	0
$547$	16.3848	0.700563	0.350281	−	0.936645i	$-0.386086\pi$
$547$	16.3848	0.700563	0.350281	+	0.936645i	$0.386086\pi$
$548$	0	0
$549$	−11.7474	−0.501368
$550$	0	0
$551$	6.88904	0.293483
$552$	0	0
$553$	0.573501	0.0243877
$554$	0	0
$555$	7.02878	0.298355
$556$	0	0
$557$	1.13507	0.0480944	0.0240472	−	0.999711i	$-0.492345\pi$
$557$	1.13507	0.0480944	0.0240472	+	0.999711i	$0.492345\pi$
$558$	0	0
$559$	5.73208	0.242441
$560$	0	0
$561$	−0.00477991	−0.000201808 0
$562$	0	0
$563$	−28.9990	−1.22216	−0.611081	−	0.791568i	$-0.709265\pi$
$563$	−28.9990	−1.22216	−0.611081	+	0.791568i	$0.709265\pi$
$564$	0	0
$565$	−3.62199	−0.152378
$566$	0	0
$567$	−0.215981	−0.00907034
$568$	0	0
$569$	−21.3320	−0.894283	−0.447141	−	0.894463i	$-0.647558\pi$
$569$	−21.3320	−0.894283	−0.447141	+	0.894463i	$0.647558\pi$
$570$	0	0
$571$	38.3296	1.60404	0.802022	−	0.597294i	$-0.203758\pi$
$571$	38.3296	1.60404	0.802022	+	0.597294i	$0.203758\pi$
$572$	0	0
$573$	0.839357	0.0350646
$574$	0	0
$575$	4.54771	0.189652
$576$	0	0
$577$	4.49114	0.186968	0.0934842	−	0.995621i	$-0.470200\pi$
$577$	4.49114	0.186968	0.0934842	+	0.995621i	$0.470200\pi$
$578$	0	0
$579$	20.7130	0.860805
$580$	0	0
$581$	−1.68366	−0.0698498
$582$	0	0
$583$	−0.0121579	−0.000503530 0
$584$	0	0
$585$	0.652358	0.0269717
$586$	0	0
$587$	−8.76700	−0.361853	−0.180926	−	0.983497i	$-0.557910\pi$
$587$	−8.76700	−0.361853	−0.180926	+	0.983497i	$0.557910\pi$
$588$	0	0
$589$	−39.1737	−1.61413
$590$	0	0
$591$	−9.28009	−0.381732
$592$	0	0
$593$	24.4503	1.00405	0.502027	−	0.864852i	$-0.332588\pi$
$593$	24.4503	1.00405	0.502027	+	0.864852i	$0.332588\pi$
$594$	0	0
$595$	−0.624296	−0.0255936
$596$	0	0
$597$	21.5634	0.882529
$598$	0	0
$599$	9.86521	0.403081	0.201541	−	0.979480i	$-0.435405\pi$
$599$	9.86521	0.403081	0.201541	+	0.979480i	$0.435405\pi$
$600$	0	0
$601$	0.219265	0.00894401	0.00447200	−	0.999990i	$-0.498577\pi$
$601$	0.219265	0.00894401	0.00447200	+	0.999990i	$0.498577\pi$
$602$	0	0
$603$	2.91865	0.118857
$604$	0	0
$605$	7.39781	0.300764
$606$	0	0
$607$	−12.5454	−0.509204	−0.254602	−	0.967046i	$-0.581944\pi$
$607$	−12.5454	−0.509204	−0.254602	+	0.967046i	$0.581944\pi$
$608$	0	0
$609$	−0.215981	−0.00875198
$610$	0	0
$611$	3.65040	0.147680
$612$	0	0
$613$	1.95001	0.0787601	0.0393801	−	0.999224i	$-0.487462\pi$
$613$	1.95001	0.0787601	0.0393801	+	0.999224i	$0.487462\pi$
$614$	0	0
$615$	−2.03870	−0.0822082
$616$	0	0
$617$	−29.7838	−1.19905	−0.599525	−	0.800356i	$-0.704644\pi$
$617$	−29.7838	−1.19905	−0.599525	+	0.800356i	$0.704644\pi$
$618$	0	0
$619$	38.2142	1.53596	0.767979	−	0.640475i	$-0.221263\pi$
$619$	38.2142	1.53596	0.767979	+	0.640475i	$0.221263\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	1.37068	0.0549153
$624$	0	0
$625$	18.4202	0.736806
$626$	0	0
$627$	−0.00766150	−0.000305971 0
$628$	0	0
$629$	−44.9194	−1.79105
$630$	0	0
$631$	−17.9415	−0.714238	−0.357119	−	0.934059i	$-0.616241\pi$
$631$	−17.9415	−0.714238	−0.357119	+	0.934059i	$0.616241\pi$
$632$	0	0
$633$	4.73933	0.188371
$634$	0	0
$635$	−4.45668	−0.176858
$636$	0	0
$637$	6.74480	0.267239
$638$	0	0
$639$	14.8584	0.587788
$640$	0	0
$641$	−6.40740	−0.253077	−0.126539	−	0.991962i	$-0.540387\pi$
$641$	−6.40740	−0.253077	−0.126539	+	0.991962i	$0.540387\pi$
$642$	0	0
$643$	−8.90466	−0.351166	−0.175583	−	0.984465i	$-0.556181\pi$
$643$	−8.90466	−0.351166	−0.175583	+	0.984465i	$0.556181\pi$
$644$	0	0
$645$	−3.97419	−0.156483
$646$	0	0
$647$	−1.57095	−0.0617604	−0.0308802	−	0.999523i	$-0.509831\pi$
$647$	−1.57095	−0.0617604	−0.0308802	+	0.999523i	$0.509831\pi$
$648$	0	0
$649$	0.00742061	0.000291284 0
$650$	0	0
$651$	1.22815	0.0481350
$652$	0	0
$653$	−32.9258	−1.28849	−0.644243	−	0.764821i	$-0.722827\pi$
$653$	−32.9258	−1.28849	−0.644243	+	0.764821i	$0.722827\pi$
$654$	0	0
$655$	−7.44983	−0.291089
$656$	0	0
$657$	10.7310	0.418657
$658$	0	0
$659$	42.3482	1.64965	0.824826	−	0.565387i	$-0.191273\pi$
$659$	42.3482	1.64965	0.824826	+	0.565387i	$0.191273\pi$
$660$	0	0
$661$	20.6983	0.805070	0.402535	−	0.915405i	$-0.368129\pi$
$661$	20.6983	0.805070	0.402535	+	0.915405i	$0.368129\pi$
$662$	0	0
$663$	−4.16908	−0.161914
$664$	0	0
$665$	−1.00065	−0.0388037
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	12.2177	0.472365
$670$	0	0
$671$	0.0130647	0.000504356 0
$672$	0	0
$673$	−27.7099	−1.06814	−0.534069	−	0.845441i	$-0.679338\pi$
$673$	−27.7099	−1.06814	−0.534069	+	0.845441i	$0.679338\pi$
$674$	0	0
$675$	4.54771	0.175041
$676$	0	0
$677$	18.7860	0.722003	0.361002	−	0.932565i	$-0.382435\pi$
$677$	18.7860	0.722003	0.361002	+	0.932565i	$0.382435\pi$
$678$	0	0
$679$	−3.56164	−0.136683
$680$	0	0
$681$	22.4270	0.859403
$682$	0	0
$683$	−13.5708	−0.519271	−0.259636	−	0.965707i	$-0.583602\pi$
$683$	−13.5708	−0.519271	−0.259636	+	0.965707i	$0.583602\pi$
$684$	0	0
$685$	1.50302	0.0574274
$686$	0	0
$687$	−18.8792	−0.720285
$688$	0	0
$689$	−10.6042	−0.403989
$690$	0	0
$691$	−5.93307	−0.225705	−0.112852	−	0.993612i	$-0.535999\pi$
$691$	−5.93307	−0.225705	−0.112852	+	0.993612i	$0.535999\pi$
$692$	0	0
$693$	0.000240198 0	9.12438e−6 0
$694$	0	0
$695$	−0.892117	−0.0338399
$696$	0	0
$697$	13.0289	0.493504
$698$	0	0
$699$	19.4418	0.735355
$700$	0	0
$701$	15.4021	0.581730	0.290865	−	0.956764i	$-0.406057\pi$
$701$	15.4021	0.581730	0.290865	+	0.956764i	$0.406057\pi$
$702$	0	0
$703$	−71.9992	−2.71550
$704$	0	0
$705$	−2.53091	−0.0953196
$706$	0	0
$707$	0.300149	0.0112883
$708$	0	0
$709$	39.0910	1.46809	0.734046	−	0.679100i	$-0.237630\pi$
$709$	39.0910	1.46809	0.734046	+	0.679100i	$0.237630\pi$
$710$	0	0
$711$	−2.65534	−0.0995828
$712$	0	0
$713$	−5.68639	−0.212957
$714$	0	0
$715$	−0.000725506 0	−2.71324e−5 0
$716$	0	0
$717$	−12.1394	−0.453354
$718$	0	0
$719$	19.8393	0.739879	0.369940	−	0.929056i	$-0.379378\pi$
$719$	19.8393	0.739879	0.369940	+	0.929056i	$0.379378\pi$
$720$	0	0
$721$	2.33469	0.0869484
$722$	0	0
$723$	25.3169	0.941547
$724$	0	0
$725$	4.54771	0.168898
$726$	0	0
$727$	−44.5491	−1.65224	−0.826118	−	0.563497i	$-0.809456\pi$
$727$	−44.5491	−1.65224	−0.826118	+	0.563497i	$0.809456\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.3982	0.939385
$732$	0	0
$733$	25.7263	0.950223	0.475112	−	0.879926i	$-0.342408\pi$
$733$	25.7263	0.950223	0.475112	+	0.879926i	$0.342408\pi$
$734$	0	0
$735$	−4.67633	−0.172489
$736$	0	0
$737$	−0.00324592	−0.000119565 0
$738$	0	0
$739$	37.1724	1.36741	0.683705	−	0.729758i	$-0.260367\pi$
$739$	37.1724	1.36741	0.683705	+	0.729758i	$0.260367\pi$
$740$	0	0
$741$	−6.68242	−0.245485
$742$	0	0
$743$	−7.20719	−0.264406	−0.132203	−	0.991223i	$-0.542205\pi$
$743$	−7.20719	−0.264406	−0.132203	+	0.991223i	$0.542205\pi$
$744$	0	0
$745$	−11.3729	−0.416672
$746$	0	0
$747$	7.79540	0.285219
$748$	0	0
$749$	2.22785	0.0814038
$750$	0	0
$751$	33.6763	1.22886	0.614432	−	0.788970i	$-0.289385\pi$
$751$	33.6763	1.22886	0.614432	+	0.788970i	$0.289385\pi$
$752$	0	0
$753$	16.4191	0.598343
$754$	0	0
$755$	−7.83433	−0.285120
$756$	0	0
$757$	−41.1042	−1.49396	−0.746979	−	0.664847i	$-0.768497\pi$
$757$	−41.1042	−1.49396	−0.746979	+	0.664847i	$0.768497\pi$
$758$	0	0
$759$	−0.00111213	−4.03677e−5 0
$760$	0	0
$761$	−5.85886	−0.212383	−0.106192	−	0.994346i	$-0.533866\pi$
$761$	−5.85886	−0.212383	−0.106192	+	0.994346i	$0.533866\pi$
$762$	0	0
$763$	0.512171	0.0185418
$764$	0	0
$765$	2.89052	0.104507
$766$	0	0
$767$	6.47231	0.233702
$768$	0	0
$769$	−37.3377	−1.34643	−0.673216	−	0.739446i	$-0.735088\pi$
$769$	−37.3377	−1.34643	−0.673216	+	0.739446i	$0.735088\pi$
$770$	0	0
$771$	−26.5897	−0.957606
$772$	0	0
$773$	41.6643	1.49856	0.749280	−	0.662253i	$-0.230400\pi$
$773$	41.6643	1.49856	0.749280	+	0.662253i	$0.230400\pi$
$774$	0	0
$775$	−25.8600	−0.928919
$776$	0	0
$777$	2.25727	0.0809792
$778$	0	0
$779$	20.8834	0.748225
$780$	0	0
$781$	−0.0165244	−0.000591290 0
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−10.4779	−0.373971
$786$	0	0
$787$	4.91623	0.175245	0.0876223	−	0.996154i	$-0.472073\pi$
$787$	4.91623	0.175245	0.0876223	+	0.996154i	$0.472073\pi$
$788$	0	0
$789$	−25.9312	−0.923175
$790$	0	0
$791$	−1.16319	−0.0413583
$792$	0	0
$793$	11.3951	0.404652
$794$	0	0
$795$	7.35216	0.260754
$796$	0	0
$797$	18.6806	0.661701	0.330851	−	0.943683i	$-0.392664\pi$
$797$	18.6806	0.661701	0.330851	+	0.943683i	$0.392664\pi$
$798$	0	0
$799$	16.1745	0.572213
$800$	0	0
$801$	−6.34633	−0.224236
$802$	0	0
$803$	−0.0119343	−0.000421152 0
$804$	0	0
$805$	−0.145253	−0.00511950
$806$	0	0
$807$	2.26409	0.0796997
$808$	0	0
$809$	25.8819	0.909961	0.454980	−	0.890501i	$-0.349646\pi$
$809$	25.8819	0.909961	0.454980	+	0.890501i	$0.349646\pi$
$810$	0	0
$811$	−29.2774	−1.02807	−0.514034	−	0.857770i	$-0.671849\pi$
$811$	−29.2774	−1.02807	−0.514034	+	0.857770i	$0.671849\pi$
$812$	0	0
$813$	−17.6472	−0.618914
$814$	0	0
$815$	0.639767	0.0224101
$816$	0	0
$817$	40.7095	1.42425
$818$	0	0
$819$	0.209503	0.00732062
$820$	0	0
$821$	46.8324	1.63446	0.817231	−	0.576311i	$-0.195508\pi$
$821$	46.8324	1.63446	0.817231	+	0.576311i	$0.195508\pi$
$822$	0	0
$823$	6.57336	0.229133	0.114566	−	0.993416i	$-0.463452\pi$
$823$	6.57336	0.229133	0.114566	+	0.993416i	$0.463452\pi$
$824$	0	0
$825$	−0.00505764	−0.000176084 0
$826$	0	0
$827$	6.15070	0.213881	0.106940	−	0.994265i	$-0.465895\pi$
$827$	6.15070	0.213881	0.106940	+	0.994265i	$0.465895\pi$
$828$	0	0
$829$	12.3770	0.429871	0.214936	−	0.976628i	$-0.431046\pi$
$829$	12.3770	0.429871	0.214936	+	0.976628i	$0.431046\pi$
$830$	0	0
$831$	−23.2745	−0.807383
$832$	0	0
$833$	29.8854	1.03547
$834$	0	0
$835$	7.10876	0.246009
$836$	0	0
$837$	−5.68639	−0.196550
$838$	0	0
$839$	−23.5253	−0.812183	−0.406091	−	0.913832i	$-0.633109\pi$
$839$	−23.5253	−0.812183	−0.406091	+	0.913832i	$0.633109\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−10.1633	−0.350041
$844$	0	0
$845$	8.11008	0.278995
$846$	0	0
$847$	2.37579	0.0816330
$848$	0	0
$849$	15.6539	0.537241
$850$	0	0
$851$	−10.4513	−0.358265
$852$	0	0
$853$	−14.8583	−0.508739	−0.254370	−	0.967107i	$-0.581868\pi$
$853$	−14.8583	−0.508739	−0.254370	+	0.967107i	$0.581868\pi$
$854$	0	0
$855$	4.63307	0.158448
$856$	0	0
$857$	−29.0764	−0.993231	−0.496616	−	0.867971i	$-0.665424\pi$
$857$	−29.0764	−0.993231	−0.496616	+	0.867971i	$0.665424\pi$
$858$	0	0
$859$	−5.44504	−0.185782	−0.0928912	−	0.995676i	$-0.529611\pi$
$859$	−5.44504	−0.185782	−0.0928912	+	0.995676i	$0.529611\pi$
$860$	0	0
$861$	−0.654722	−0.0223129
$862$	0	0
$863$	−21.4326	−0.729574	−0.364787	−	0.931091i	$-0.618858\pi$
$863$	−21.4326	−0.729574	−0.364787	+	0.931091i	$0.618858\pi$
$864$	0	0
$865$	11.9855	0.407518
$866$	0	0
$867$	−1.47268	−0.0500147
$868$	0	0
$869$	0.00295308	0.000100176 0
$870$	0	0
$871$	−2.83111	−0.0959286
$872$	0	0
$873$	16.4905	0.558120
$874$	0	0
$875$	−1.38683	−0.0468836
$876$	0	0
$877$	−11.1551	−0.376683	−0.188341	−	0.982104i	$-0.560311\pi$
$877$	−11.1551	−0.376683	−0.188341	+	0.982104i	$0.560311\pi$
$878$	0	0
$879$	−8.30021	−0.279959
$880$	0	0
$881$	−32.2540	−1.08666	−0.543332	−	0.839518i	$-0.682838\pi$
$881$	−32.2540	−1.08666	−0.543332	+	0.839518i	$0.682838\pi$
$882$	0	0
$883$	−50.0797	−1.68532	−0.842658	−	0.538448i	$-0.819011\pi$
$883$	−50.0797	−1.68532	−0.842658	+	0.538448i	$0.819011\pi$
$884$	0	0
$885$	−4.48740	−0.150842
$886$	0	0
$887$	−33.7337	−1.13267	−0.566334	−	0.824176i	$-0.691639\pi$
$887$	−33.7337	−1.13267	−0.566334	+	0.824176i	$0.691639\pi$
$888$	0	0
$889$	−1.43125	−0.0480026
$890$	0	0
$891$	−0.00111213	−3.72577e−5 0
$892$	0	0
$893$	25.9253	0.867558
$894$	0	0
$895$	−11.2771	−0.376953
$896$	0	0
$897$	−0.970007	−0.0323876
$898$	0	0
$899$	−5.68639	−0.189652
$900$	0	0
$901$	−46.9861	−1.56533
$902$	0	0
$903$	−1.27630	−0.0424726
$904$	0	0
$905$	−15.7166	−0.522437
$906$	0	0
$907$	22.5121	0.747503	0.373751	−	0.927529i	$-0.378071\pi$
$907$	22.5121	0.747503	0.373751	+	0.927529i	$0.378071\pi$
$908$	0	0
$909$	−1.38971	−0.0460936
$910$	0	0
$911$	21.4013	0.709055	0.354528	−	0.935045i	$-0.384642\pi$
$911$	21.4013	0.709055	0.354528	+	0.935045i	$0.384642\pi$
$912$	0	0
$913$	−0.00866949	−0.000286918 0
$914$	0	0
$915$	−7.90049	−0.261182
$916$	0	0
$917$	−2.39249	−0.0790071
$918$	0	0
$919$	−8.95322	−0.295339	−0.147670	−	0.989037i	$-0.547177\pi$
$919$	−8.95322	−0.295339	−0.147670	+	0.989037i	$0.547177\pi$
$920$	0	0
$921$	−11.2228	−0.369804
$922$	0	0
$923$	−14.4127	−0.474401
$924$	0	0
$925$	−47.5293	−1.56275
$926$	0	0
$927$	−10.8097	−0.355038
$928$	0	0
$929$	19.5805	0.642416	0.321208	−	0.947009i	$-0.395911\pi$
$929$	19.5805	0.642416	0.321208	+	0.947009i	$0.395911\pi$
$930$	0	0
$931$	47.9019	1.56992
$932$	0	0
$933$	−12.0015	−0.392913
$934$	0	0
$935$	−0.00321463	−0.000105130 0
$936$	0	0
$937$	15.6646	0.511740	0.255870	−	0.966711i	$-0.417638\pi$
$937$	15.6646	0.511740	0.255870	+	0.966711i	$0.417638\pi$
$938$	0	0
$939$	−20.5901	−0.671932
$940$	0	0
$941$	18.0915	0.589766	0.294883	−	0.955533i	$-0.404719\pi$
$941$	18.0915	0.589766	0.294883	+	0.955533i	$0.404719\pi$
$942$	0	0
$943$	3.03139	0.0987157
$944$	0	0
$945$	−0.145253	−0.00472509
$946$	0	0
$947$	−11.6603	−0.378910	−0.189455	−	0.981889i	$-0.560672\pi$
$947$	−11.6603	−0.378910	−0.189455	+	0.981889i	$0.560672\pi$
$948$	0	0
$949$	−10.4092	−0.337896
$950$	0	0
$951$	−28.1834	−0.913909
$952$	0	0
$953$	−21.0426	−0.681637	−0.340818	−	0.940129i	$-0.610704\pi$
$953$	−21.0426	−0.681637	−0.340818	+	0.940129i	$0.610704\pi$
$954$	0	0
$955$	0.564491	0.0182665
$956$	0	0
$957$	−0.00111213	−3.59500e−5 0
$958$	0	0
$959$	0.482690	0.0155869
$960$	0	0
$961$	1.33498	0.0430638
$962$	0	0
$963$	−10.3150	−0.332397
$964$	0	0
$965$	13.9301	0.448426
$966$	0	0
$967$	−7.52731	−0.242062	−0.121031	−	0.992649i	$-0.538620\pi$
$967$	−7.52731	−0.242062	−0.121031	+	0.992649i	$0.538620\pi$
$968$	0	0
$969$	−29.6090	−0.951177
$970$	0	0
$971$	14.0093	0.449578	0.224789	−	0.974407i	$-0.427831\pi$
$971$	14.0093	0.449578	0.224789	+	0.974407i	$0.427831\pi$
$972$	0	0
$973$	−0.286501	−0.00918479
$974$	0	0
$975$	−4.41131	−0.141275
$976$	0	0
$977$	46.4064	1.48467	0.742336	−	0.670028i	$-0.233718\pi$
$977$	46.4064	1.48467	0.742336	+	0.670028i	$0.233718\pi$
$978$	0	0
$979$	0.00705793	0.000225573 0
$980$	0	0
$981$	−2.37137	−0.0757122
$982$	0	0
$983$	41.9047	1.33655	0.668277	−	0.743913i	$-0.267032\pi$
$983$	41.9047	1.33655	0.668277	+	0.743913i	$0.267032\pi$
$984$	0	0
$985$	−6.24112	−0.198859
$986$	0	0
$987$	−0.812795	−0.0258715
$988$	0	0
$989$	5.90932	0.187905
$990$	0	0
$991$	−53.2313	−1.69095	−0.845474	−	0.534016i	$-0.820682\pi$
$991$	−53.2313	−1.69095	−0.845474	+	0.534016i	$0.820682\pi$
$992$	0	0
$993$	16.4795	0.522960
$994$	0	0
$995$	14.5020	0.459743
$996$	0	0
$997$	−2.17080	−0.0687500	−0.0343750	−	0.999409i	$-0.510944\pi$
$997$	−2.17080	−0.0687500	−0.0343750	+	0.999409i	$0.510944\pi$
$998$	0	0
$999$	−10.4513	−0.330664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.6	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.6	✓	13	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.672529 q^{5} -0.215981 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.672529 q^{5} -0.215981 q^{7} +1.00000 q^{9} -0.00111213 q^{11} -0.970007 q^{13} +0.672529 q^{15} -4.29799 q^{17} -6.88904 q^{19} +0.215981 q^{21} -1.00000 q^{23} -4.54771 q^{25} -1.00000 q^{27} -1.00000 q^{29} +5.68639 q^{31} +0.00111213 q^{33} +0.145253 q^{35} +10.4513 q^{37} +0.970007 q^{39} -3.03139 q^{41} -5.90932 q^{43} -0.672529 q^{45} -3.76327 q^{47} -6.95335 q^{49} +4.29799 q^{51} +10.9321 q^{53} +0.000747939 q^{55} +6.88904 q^{57} -6.67243 q^{59} -11.7474 q^{61} -0.215981 q^{63} +0.652358 q^{65} +2.91865 q^{67} +1.00000 q^{69} +14.8584 q^{71} +10.7310 q^{73} +4.54771 q^{75} +0.000240198 q^{77} -2.65534 q^{79} +1.00000 q^{81} +7.79540 q^{83} +2.89052 q^{85} +1.00000 q^{87} -6.34633 q^{89} +0.209503 q^{91} -5.68639 q^{93} +4.63307 q^{95} +16.4905 q^{97} -0.00111213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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