LMFDB - Embedded newform 304.10.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,10,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	304.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:	$156.570894194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 $ x^4 - x^3 - 34433x^2 - 2723303x - 48270488
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-124.888$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+140.229 q^{3} +1263.95 q^{5} +3487.42 q^{7} -18.7042 q^{9} +O(q^{10})$	$q+140.229 q^{3} +1263.95 q^{5} +3487.42 q^{7} -18.7042 q^{9} +49259.2 q^{11} -68065.3 q^{13} +177243. q^{15} +505716. q^{17} -130321. q^{19} +489039. q^{21} +585044. q^{23} -355562. q^{25} -2.76276e6 q^{27} +2.62021e6 q^{29} -3.53093e6 q^{31} +6.90759e6 q^{33} +4.40791e6 q^{35} +1.85431e7 q^{37} -9.54476e6 q^{39} +2.18141e7 q^{41} +1.12704e7 q^{43} -23641.1 q^{45} -1.54092e7 q^{47} -2.81915e7 q^{49} +7.09162e7 q^{51} +7.05741e6 q^{53} +6.22611e7 q^{55} -1.82748e7 q^{57} -2.90730e7 q^{59} +1.44284e8 q^{61} -65229.3 q^{63} -8.60310e7 q^{65} -2.65266e7 q^{67} +8.20404e7 q^{69} +4.27081e7 q^{71} +3.24644e8 q^{73} -4.98602e7 q^{75} +1.71787e8 q^{77} +8.88210e7 q^{79} -3.87052e8 q^{81} -6.29830e7 q^{83} +6.39198e8 q^{85} +3.67431e8 q^{87} -4.54329e7 q^{89} -2.37372e8 q^{91} -4.95140e8 q^{93} -1.64719e8 q^{95} +1.64830e9 q^{97} -921353. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) $ 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9	$ 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9} + 104249 q^{11} + 120486 q^{13} + 591090 q^{15} - 412139 q^{17} - 521284 q^{19} + 2437006 q^{21} - 3010300 q^{23} + 9760585 q^{25} - 12387978 q^{27} + 6153240 q^{29} - 12774024 q^{31} - 3258022 q^{33} - 9823425 q^{35} + 20506048 q^{37} - 69881444 q^{39} + 11620300 q^{41} - 7698327 q^{43} - 124015815 q^{45} + 31581083 q^{47} + 18970383 q^{49} + 8594812 q^{51} + 72549422 q^{53} - 21332505 q^{55} + 10946964 q^{57} + 149234120 q^{59} + 129004373 q^{61} - 102967551 q^{63} + 124691700 q^{65} - 132595266 q^{67} - 45529972 q^{69} + 47138482 q^{71} - 39332795 q^{73} - 824627010 q^{75} - 165933719 q^{77} + 307010840 q^{79} + 1305551744 q^{81} + 746568232 q^{83} - 105005985 q^{85} + 82148208 q^{87} + 286943482 q^{89} - 3155781114 q^{91} + 1151901596 q^{93} + 181797795 q^{95} + 793519958 q^{97} + 1681833809 q^{99}+O(q^{100}) $ 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9 + 104249 * q^11 + 120486 * q^13 + 591090 * q^15 - 412139 * q^17 - 521284 * q^19 + 2437006 * q^21 - 3010300 * q^23 + 9760585 * q^25 - 12387978 * q^27 + 6153240 * q^29 - 12774024 * q^31 - 3258022 * q^33 - 9823425 * q^35 + 20506048 * q^37 - 69881444 * q^39 + 11620300 * q^41 - 7698327 * q^43 - 124015815 * q^45 + 31581083 * q^47 + 18970383 * q^49 + 8594812 * q^51 + 72549422 * q^53 - 21332505 * q^55 + 10946964 * q^57 + 149234120 * q^59 + 129004373 * q^61 - 102967551 * q^63 + 124691700 * q^65 - 132595266 * q^67 - 45529972 * q^69 + 47138482 * q^71 - 39332795 * q^73 - 824627010 * q^75 - 165933719 * q^77 + 307010840 * q^79 + 1305551744 * q^81 + 746568232 * q^83 - 105005985 * q^85 + 82148208 * q^87 + 286943482 * q^89 - 3155781114 * q^91 + 1151901596 * q^93 + 181797795 * q^95 + 793519958 * q^97 + 1681833809 * 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
$3$	140.229	0.999525	0.499762	−	0.866163i	$-0.333421\pi$
$3$	140.229	0.999525	0.499762	+	0.866163i	$0.333421\pi$
$4$	0	0
$5$	1263.95	0.904407	0.452204	−	0.891915i	$-0.350638\pi$
$5$	1263.95	0.904407	0.452204	+	0.891915i	$0.350638\pi$
$6$	0	0
$7$	3487.42	0.548988	0.274494	−	0.961589i	$-0.411490\pi$
$7$	3487.42	0.548988	0.274494	+	0.961589i	$0.411490\pi$
$8$	0	0
$9$	−18.7042	−0.000950271 0
$10$	0	0
$11$	49259.2	1.01443	0.507213	−	0.861821i	$-0.330676\pi$
$11$	49259.2	1.01443	0.507213	+	0.861821i	$0.330676\pi$
$12$	0	0
$13$	−68065.3	−0.660969	−0.330484	−	0.943811i	$-0.607212\pi$
$13$	−68065.3	−0.660969	−0.330484	+	0.943811i	$0.607212\pi$
$14$	0	0
$15$	177243.	0.903977
$16$	0	0
$17$	505716.	1.46854	0.734271	−	0.678857i	$-0.237524\pi$
$17$	505716.	1.46854	0.734271	+	0.678857i	$0.237524\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	0	0
$21$	489039.	0.548727
$22$	0	0
$23$	585044.	0.435926	0.217963	−	0.975957i	$-0.430059\pi$
$23$	585044.	0.435926	0.217963	+	0.975957i	$0.430059\pi$
$24$	0	0
$25$	−355562.	−0.182048
$26$	0	0
$27$	−2.76276e6	−1.00047
$28$	0	0
$29$	2.62021e6	0.687932	0.343966	−	0.938982i	$-0.388229\pi$
$29$	2.62021e6	0.687932	0.343966	+	0.938982i	$0.388229\pi$
$30$	0	0
$31$	−3.53093e6	−0.686691	−0.343345	−	0.939209i	$-0.611560\pi$
$31$	−3.53093e6	−0.686691	−0.343345	+	0.939209i	$0.611560\pi$
$32$	0	0
$33$	6.90759e6	1.01394
$34$	0	0
$35$	4.40791e6	0.496509
$36$	0	0
$37$	1.85431e7	1.62657	0.813287	−	0.581863i	$-0.197676\pi$
$37$	1.85431e7	1.62657	0.813287	+	0.581863i	$0.197676\pi$
$38$	0	0
$39$	−9.54476e6	−0.660655
$40$	0	0
$41$	2.18141e7	1.20562	0.602811	−	0.797884i	$-0.294047\pi$
$41$	2.18141e7	1.20562	0.602811	+	0.797884i	$0.294047\pi$
$42$	0	0
$43$	1.12704e7	0.502725	0.251362	−	0.967893i	$-0.419121\pi$
$43$	1.12704e7	0.502725	0.251362	+	0.967893i	$0.419121\pi$
$44$	0	0
$45$	−23641.1	−0.000859432 0
$46$	0	0
$47$	−1.54092e7	−0.460616	−0.230308	−	0.973118i	$-0.573973\pi$
$47$	−1.54092e7	−0.460616	−0.230308	+	0.973118i	$0.573973\pi$
$48$	0	0
$49$	−2.81915e7	−0.698612
$50$	0	0
$51$	7.09162e7	1.46784
$52$	0	0
$53$	7.05741e6	0.122858	0.0614291	−	0.998111i	$-0.480434\pi$
$53$	7.05741e6	0.122858	0.0614291	+	0.998111i	$0.480434\pi$
$54$	0	0
$55$	6.22611e7	0.917454
$56$	0	0
$57$	−1.82748e7	−0.229307
$58$	0	0
$59$	−2.90730e7	−0.312360	−0.156180	−	0.987729i	$-0.549918\pi$
$59$	−2.90730e7	−0.312360	−0.156180	+	0.987729i	$0.549918\pi$
$60$	0	0
$61$	1.44284e8	1.33424	0.667121	−	0.744949i	$-0.267526\pi$
$61$	1.44284e8	1.33424	0.667121	+	0.744949i	$0.267526\pi$
$62$	0	0
$63$	−65229.3	−0.000521687 0
$64$	0	0
$65$	−8.60310e7	−0.597785
$66$	0	0
$67$	−2.65266e7	−0.160822	−0.0804110	−	0.996762i	$-0.525623\pi$
$67$	−2.65266e7	−0.160822	−0.0804110	+	0.996762i	$0.525623\pi$
$68$	0	0
$69$	8.20404e7	0.435719
$70$	0	0
$71$	4.27081e7	0.199456	0.0997281	−	0.995015i	$-0.468203\pi$
$71$	4.27081e7	0.199456	0.0997281	+	0.995015i	$0.468203\pi$
$72$	0	0
$73$	3.24644e8	1.33800	0.668998	−	0.743264i	$-0.266723\pi$
$73$	3.24644e8	1.33800	0.668998	+	0.743264i	$0.266723\pi$
$74$	0	0
$75$	−4.98602e7	−0.181961
$76$	0	0
$77$	1.71787e8	0.556908
$78$	0	0
$79$	8.88210e7	0.256563	0.128281	−	0.991738i	$-0.459054\pi$
$79$	8.88210e7	0.256563	0.128281	+	0.991738i	$0.459054\pi$
$80$	0	0
$81$	−3.87052e8	−0.999049
$82$	0	0
$83$	−6.29830e7	−0.145671	−0.0728353	−	0.997344i	$-0.523205\pi$
$83$	−6.29830e7	−0.145671	−0.0728353	+	0.997344i	$0.523205\pi$
$84$	0	0
$85$	6.39198e8	1.32816
$86$	0	0
$87$	3.67431e8	0.687605
$88$	0	0
$89$	−4.54329e7	−0.0767565	−0.0383783	−	0.999263i	$-0.512219\pi$
$89$	−4.54329e7	−0.0767565	−0.0383783	+	0.999263i	$0.512219\pi$
$90$	0	0
$91$	−2.37372e8	−0.362864
$92$	0	0
$93$	−4.95140e8	−0.686365
$94$	0	0
$95$	−1.64719e8	−0.207485
$96$	0	0
$97$	1.64830e9	1.89044	0.945219	−	0.326438i	$-0.105848\pi$
$97$	1.64830e9	1.89044	0.945219	+	0.326438i	$0.105848\pi$
$98$	0	0
$99$	−921353.	−0.000963980 0
$100$	0	0
$101$	−8.97226e8	−0.857938	−0.428969	−	0.903319i	$-0.641123\pi$
$101$	−8.97226e8	−0.857938	−0.428969	+	0.903319i	$0.641123\pi$
$102$	0	0
$103$	−1.67622e9	−1.46745	−0.733726	−	0.679445i	$-0.762221\pi$
$103$	−1.67622e9	−1.46745	−0.733726	+	0.679445i	$0.762221\pi$
$104$	0	0
$105$	6.18119e8	0.496273
$106$	0	0
$107$	−3.08053e8	−0.227195	−0.113597	−	0.993527i	$-0.536237\pi$
$107$	−3.08053e8	−0.227195	−0.113597	+	0.993527i	$0.536237\pi$
$108$	0	0
$109$	−1.79569e8	−0.121846	−0.0609230	−	0.998142i	$-0.519404\pi$
$109$	−1.79569e8	−0.121846	−0.0609230	+	0.998142i	$0.519404\pi$
$110$	0	0
$111$	2.60028e9	1.62580
$112$	0	0
$113$	−2.30787e9	−1.33155	−0.665776	−	0.746152i	$-0.731899\pi$
$113$	−2.30787e9	−1.33155	−0.665776	+	0.746152i	$0.731899\pi$
$114$	0	0
$115$	7.39465e8	0.394255
$116$	0	0
$117$	1.27311e6	0.000628099 0
$118$	0	0
$119$	1.76364e9	0.806212
$120$	0	0
$121$	6.85222e7	0.0290601
$122$	0	0
$123$	3.05899e9	1.20505
$124$	0	0
$125$	−2.91806e9	−1.06905
$126$	0	0
$127$	4.51313e9	1.53943	0.769717	−	0.638385i	$-0.220397\pi$
$127$	4.51313e9	1.53943	0.769717	+	0.638385i	$0.220397\pi$
$128$	0	0
$129$	1.58044e9	0.502486
$130$	0	0
$131$	2.53716e9	0.752708	0.376354	−	0.926476i	$-0.377178\pi$
$131$	2.53716e9	0.752708	0.376354	+	0.926476i	$0.377178\pi$
$132$	0	0
$133$	−4.54484e8	−0.125946
$134$	0	0
$135$	−3.49198e9	−0.904836
$136$	0	0
$137$	4.04090e9	0.980021	0.490010	−	0.871717i	$-0.336993\pi$
$137$	4.04090e9	0.980021	0.490010	+	0.871717i	$0.336993\pi$
$138$	0	0
$139$	1.86522e9	0.423803	0.211901	−	0.977291i	$-0.432034\pi$
$139$	1.86522e9	0.423803	0.211901	+	0.977291i	$0.432034\pi$
$140$	0	0
$141$	−2.16082e9	−0.460397
$142$	0	0
$143$	−3.35284e9	−0.670504
$144$	0	0
$145$	3.31181e9	0.622171
$146$	0	0
$147$	−3.95328e9	−0.698280
$148$	0	0
$149$	−7.60481e9	−1.26401	−0.632004	−	0.774965i	$-0.717768\pi$
$149$	−7.60481e9	−1.26401	−0.632004	+	0.774965i	$0.717768\pi$
$150$	0	0
$151$	−1.74653e9	−0.273388	−0.136694	−	0.990613i	$-0.543648\pi$
$151$	−1.74653e9	−0.273388	−0.136694	+	0.990613i	$0.543648\pi$
$152$	0	0
$153$	−9.45900e6	−0.00139551
$154$	0	0
$155$	−4.46291e9	−0.621048
$156$	0	0
$157$	1.06564e10	1.39979	0.699893	−	0.714248i	$-0.253231\pi$
$157$	1.06564e10	1.39979	0.699893	+	0.714248i	$0.253231\pi$
$158$	0	0
$159$	9.89657e8	0.122800
$160$	0	0
$161$	2.04029e9	0.239318
$162$	0	0
$163$	−9.11348e9	−1.01121	−0.505603	−	0.862766i	$-0.668730\pi$
$163$	−9.11348e9	−1.01121	−0.505603	+	0.862766i	$0.668730\pi$
$164$	0	0
$165$	8.73083e9	0.917018
$166$	0	0
$167$	−1.15546e10	−1.14956	−0.574778	−	0.818309i	$-0.694912\pi$
$167$	−1.15546e10	−1.14956	−0.574778	+	0.818309i	$0.694912\pi$
$168$	0	0
$169$	−5.97161e9	−0.563120
$170$	0	0
$171$	2.43755e6	0.000218007 0
$172$	0	0
$173$	−2.52786e9	−0.214558	−0.107279	−	0.994229i	$-0.534214\pi$
$173$	−2.52786e9	−0.214558	−0.107279	+	0.994229i	$0.534214\pi$
$174$	0	0
$175$	−1.23999e9	−0.0999419
$176$	0	0
$177$	−4.07689e9	−0.312212
$178$	0	0
$179$	1.73068e10	1.26002	0.630009	−	0.776588i	$-0.283051\pi$
$179$	1.73068e10	1.26002	0.630009	+	0.776588i	$0.283051\pi$
$180$	0	0
$181$	1.08688e10	0.752711	0.376355	−	0.926475i	$-0.377177\pi$
$181$	1.08688e10	0.752711	0.376355	+	0.926475i	$0.377177\pi$
$182$	0	0
$183$	2.02329e10	1.33361
$184$	0	0
$185$	2.34375e10	1.47108
$186$	0	0
$187$	2.49112e10	1.48973
$188$	0	0
$189$	−9.63489e9	−0.549248
$190$	0	0
$191$	5.33871e9	0.290259	0.145130	−	0.989413i	$-0.453640\pi$
$191$	5.33871e9	0.290259	0.145130	+	0.989413i	$0.453640\pi$
$192$	0	0
$193$	2.75948e10	1.43159	0.715797	−	0.698309i	$-0.246064\pi$
$193$	2.75948e10	1.43159	0.715797	+	0.698309i	$0.246064\pi$
$194$	0	0
$195$	−1.20641e10	−0.597501
$196$	0	0
$197$	−9.61053e9	−0.454621	−0.227310	−	0.973822i	$-0.572993\pi$
$197$	−9.61053e9	−0.454621	−0.227310	+	0.973822i	$0.572993\pi$
$198$	0	0
$199$	−2.88815e10	−1.30551	−0.652756	−	0.757568i	$-0.726388\pi$
$199$	−2.88815e10	−1.30551	−0.652756	+	0.757568i	$0.726388\pi$
$200$	0	0
$201$	−3.71982e9	−0.160746
$202$	0	0
$203$	9.13778e9	0.377666
$204$	0	0
$205$	2.75719e10	1.09037
$206$	0	0
$207$	−1.09428e7	−0.000414248 0
$208$	0	0
$209$	−6.41951e9	−0.232725
$210$	0	0
$211$	−2.26589e10	−0.786986	−0.393493	−	0.919328i	$-0.628733\pi$
$211$	−2.26589e10	−0.786986	−0.393493	+	0.919328i	$0.628733\pi$
$212$	0	0
$213$	5.98893e9	0.199361
$214$	0	0
$215$	1.42452e10	0.454668
$216$	0	0
$217$	−1.23138e10	−0.376985
$218$	0	0
$219$	4.55247e10	1.33736
$220$	0	0
$221$	−3.44217e10	−0.970660
$222$	0	0
$223$	5.84566e10	1.58293	0.791465	−	0.611214i	$-0.209319\pi$
$223$	5.84566e10	1.58293	0.791465	+	0.611214i	$0.209319\pi$
$224$	0	0
$225$	6.65049e6	0.000172995 0
$226$	0	0
$227$	4.82187e9	0.120531	0.0602656	−	0.998182i	$-0.480805\pi$
$227$	4.82187e9	0.120531	0.0602656	+	0.998182i	$0.480805\pi$
$228$	0	0
$229$	−7.15833e9	−0.172009	−0.0860046	−	0.996295i	$-0.527410\pi$
$229$	−7.15833e9	−0.172009	−0.0860046	+	0.996295i	$0.527410\pi$
$230$	0	0
$231$	2.40897e10	0.556643
$232$	0	0
$233$	4.52737e10	1.00634	0.503169	−	0.864188i	$-0.332167\pi$
$233$	4.52737e10	1.00634	0.503169	+	0.864188i	$0.332167\pi$
$234$	0	0
$235$	−1.94764e10	−0.416585
$236$	0	0
$237$	1.24553e10	0.256441
$238$	0	0
$239$	7.05422e10	1.39849	0.699244	−	0.714883i	$-0.253520\pi$
$239$	7.05422e10	1.39849	0.699244	+	0.714883i	$0.253520\pi$
$240$	0	0
$241$	7.83138e10	1.49542	0.747708	−	0.664028i	$-0.231154\pi$
$241$	7.83138e10	1.49542	0.747708	+	0.664028i	$0.231154\pi$
$242$	0	0
$243$	1.03301e8	0.00190054
$244$	0	0
$245$	−3.56326e10	−0.631830
$246$	0	0
$247$	8.87034e9	0.151637
$248$	0	0
$249$	−8.83207e9	−0.145601
$250$	0	0
$251$	−1.03661e11	−1.64848	−0.824240	−	0.566241i	$-0.808397\pi$
$251$	−1.03661e11	−1.64848	−0.824240	+	0.566241i	$0.808397\pi$
$252$	0	0
$253$	2.88188e10	0.442215
$254$	0	0
$255$	8.96344e10	1.32753
$256$	0	0
$257$	1.07294e11	1.53419	0.767093	−	0.641536i	$-0.221703\pi$
$257$	1.07294e11	1.53419	0.767093	+	0.641536i	$0.221703\pi$
$258$	0	0
$259$	6.46674e10	0.892969
$260$	0	0
$261$	−4.90090e7	−0.000653722 0
$262$	0	0
$263$	1.18240e11	1.52393	0.761963	−	0.647620i	$-0.224236\pi$
$263$	1.18240e11	1.52393	0.761963	+	0.647620i	$0.224236\pi$
$264$	0	0
$265$	8.92019e9	0.111114
$266$	0	0
$267$	−6.37103e9	−0.0767201
$268$	0	0
$269$	1.08142e11	1.25924	0.629620	−	0.776903i	$-0.283211\pi$
$269$	1.08142e11	1.25924	0.629620	+	0.776903i	$0.283211\pi$
$270$	0	0
$271$	1.22183e11	1.37609	0.688047	−	0.725667i	$-0.258468\pi$
$271$	1.22183e11	1.37609	0.688047	+	0.725667i	$0.258468\pi$
$272$	0	0
$273$	−3.32866e10	−0.362691
$274$	0	0
$275$	−1.75147e10	−0.184674
$276$	0	0
$277$	1.18944e11	1.21390	0.606951	−	0.794740i	$-0.292393\pi$
$277$	1.18944e11	1.21390	0.606951	+	0.794740i	$0.292393\pi$
$278$	0	0
$279$	6.60431e7	0.000652542 0
$280$	0	0
$281$	−4.73680e10	−0.453218	−0.226609	−	0.973986i	$-0.572764\pi$
$281$	−4.73680e10	−0.453218	−0.226609	+	0.973986i	$0.572764\pi$
$282$	0	0
$283$	−1.59829e11	−1.48121	−0.740603	−	0.671943i	$-0.765460\pi$
$283$	−1.59829e11	−1.48121	−0.740603	+	0.671943i	$0.765460\pi$
$284$	0	0
$285$	−2.30984e10	−0.207387
$286$	0	0
$287$	7.60750e10	0.661871
$288$	0	0
$289$	1.37161e11	1.15662
$290$	0	0
$291$	2.31140e11	1.88954
$292$	0	0
$293$	1.32731e11	1.05212	0.526062	−	0.850446i	$-0.323668\pi$
$293$	1.32731e11	1.05212	0.526062	+	0.850446i	$0.323668\pi$
$294$	0	0
$295$	−3.67467e10	−0.282501
$296$	0	0
$297$	−1.36091e11	−1.01491
$298$	0	0
$299$	−3.98212e10	−0.288134
$300$	0	0
$301$	3.93045e10	0.275990
$302$	0	0
$303$	−1.25818e11	−0.857530
$304$	0	0
$305$	1.82368e11	1.20670
$306$	0	0
$307$	−2.41951e11	−1.55455	−0.777274	−	0.629162i	$-0.783398\pi$
$307$	−2.41951e11	−1.55455	−0.777274	+	0.629162i	$0.783398\pi$
$308$	0	0
$309$	−2.35056e11	−1.46676
$310$	0	0
$311$	1.75276e11	1.06243	0.531217	−	0.847236i	$-0.321735\pi$
$311$	1.75276e11	1.06243	0.531217	+	0.847236i	$0.321735\pi$
$312$	0	0
$313$	−2.56460e11	−1.51032	−0.755161	−	0.655539i	$-0.772442\pi$
$313$	−2.56460e11	−1.51032	−0.755161	+	0.655539i	$0.772442\pi$
$314$	0	0
$315$	−8.24464e7	−0.000471818 0
$316$	0	0
$317$	−2.91020e10	−0.161866	−0.0809331	−	0.996720i	$-0.525790\pi$
$317$	−2.91020e10	−0.161866	−0.0809331	+	0.996720i	$0.525790\pi$
$318$	0	0
$319$	1.29070e11	0.697856
$320$	0	0
$321$	−4.31980e10	−0.227087
$322$	0	0
$323$	−6.59054e10	−0.336907
$324$	0	0
$325$	2.42014e10	0.120328
$326$	0	0
$327$	−2.51808e10	−0.121788
$328$	0	0
$329$	−5.37382e10	−0.252873
$330$	0	0
$331$	1.45683e11	0.667088	0.333544	−	0.942734i	$-0.391755\pi$
$331$	1.45683e11	0.667088	0.333544	+	0.942734i	$0.391755\pi$
$332$	0	0
$333$	−3.46833e8	−0.00154569
$334$	0	0
$335$	−3.35283e10	−0.145449
$336$	0	0
$337$	−4.16729e10	−0.176002	−0.0880012	−	0.996120i	$-0.528048\pi$
$337$	−4.16729e10	−0.176002	−0.0880012	+	0.996120i	$0.528048\pi$
$338$	0	0
$339$	−3.23631e11	−1.33092
$340$	0	0
$341$	−1.73931e11	−0.696597
$342$	0	0
$343$	−2.39045e11	−0.932518
$344$	0	0
$345$	1.03695e11	0.394067
$346$	0	0
$347$	5.79909e10	0.214722	0.107361	−	0.994220i	$-0.465760\pi$
$347$	5.79909e10	0.214722	0.107361	+	0.994220i	$0.465760\pi$
$348$	0	0
$349$	−1.06762e11	−0.385214	−0.192607	−	0.981276i	$-0.561694\pi$
$349$	−1.06762e11	−0.385214	−0.192607	+	0.981276i	$0.561694\pi$
$350$	0	0
$351$	1.88048e11	0.661282
$352$	0	0
$353$	−4.27202e11	−1.46436	−0.732180	−	0.681112i	$-0.761497\pi$
$353$	−4.27202e11	−1.46436	−0.732180	+	0.681112i	$0.761497\pi$
$354$	0	0
$355$	5.39808e10	0.180390
$356$	0	0
$357$	2.47314e11	0.805828
$358$	0	0
$359$	−4.29892e11	−1.36595	−0.682975	−	0.730442i	$-0.739314\pi$
$359$	−4.29892e11	−1.36595	−0.682975	+	0.730442i	$0.739314\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	0	0
$363$	9.60883e9	0.0290463
$364$	0	0
$365$	4.10333e11	1.21009
$366$	0	0
$367$	−1.40582e11	−0.404512	−0.202256	−	0.979333i	$-0.564827\pi$
$367$	−1.40582e11	−0.404512	−0.202256	+	0.979333i	$0.564827\pi$
$368$	0	0
$369$	−4.08016e8	−0.00114567
$370$	0	0
$371$	2.46121e10	0.0674476
$372$	0	0
$373$	4.31647e11	1.15462	0.577310	−	0.816525i	$-0.304102\pi$
$373$	4.31647e11	1.15462	0.577310	+	0.816525i	$0.304102\pi$
$374$	0	0
$375$	−4.09198e11	−1.06854
$376$	0	0
$377$	−1.78346e11	−0.454702
$378$	0	0
$379$	−7.16332e11	−1.78336	−0.891678	−	0.452670i	$-0.850472\pi$
$379$	−7.16332e11	−1.78336	−0.891678	+	0.452670i	$0.850472\pi$
$380$	0	0
$381$	6.32874e11	1.53870
$382$	0	0
$383$	1.03271e11	0.245236	0.122618	−	0.992454i	$-0.460871\pi$
$383$	1.03271e11	0.245236	0.122618	+	0.992454i	$0.460871\pi$
$384$	0	0
$385$	2.17130e11	0.503671
$386$	0	0
$387$	−2.10803e8	−0.000477725 0
$388$	0	0
$389$	−3.49935e11	−0.774845	−0.387422	−	0.921902i	$-0.626634\pi$
$389$	−3.49935e11	−0.774845	−0.387422	+	0.921902i	$0.626634\pi$
$390$	0	0
$391$	2.95866e11	0.640176
$392$	0	0
$393$	3.55784e11	0.752351
$394$	0	0
$395$	1.12265e11	0.232037
$396$	0	0
$397$	−2.83640e11	−0.573073	−0.286536	−	0.958069i	$-0.592504\pi$
$397$	−2.83640e11	−0.573073	−0.286536	+	0.958069i	$0.592504\pi$
$398$	0	0
$399$	−6.37320e10	−0.125887
$400$	0	0
$401$	−9.94065e11	−1.91984	−0.959920	−	0.280275i	$-0.909574\pi$
$401$	−9.94065e11	−1.91984	−0.959920	+	0.280275i	$0.909574\pi$
$402$	0	0
$403$	2.40334e11	0.453881
$404$	0	0
$405$	−4.89213e11	−0.903547
$406$	0	0
$407$	9.13417e11	1.65004
$408$	0	0
$409$	2.41111e11	0.426051	0.213026	−	0.977047i	$-0.431668\pi$
$409$	2.41111e11	0.426051	0.213026	+	0.977047i	$0.431668\pi$
$410$	0	0
$411$	5.66653e11	0.979555
$412$	0	0
$413$	−1.01390e11	−0.171482
$414$	0	0
$415$	−7.96072e10	−0.131745
$416$	0	0
$417$	2.61559e11	0.423602
$418$	0	0
$419$	−7.03102e11	−1.11444	−0.557218	−	0.830366i	$-0.688131\pi$
$419$	−7.03102e11	−1.11444	−0.557218	+	0.830366i	$0.688131\pi$
$420$	0	0
$421$	−2.68270e11	−0.416200	−0.208100	−	0.978108i	$-0.566728\pi$
$421$	−2.68270e11	−0.416200	−0.208100	+	0.978108i	$0.566728\pi$
$422$	0	0
$423$	2.88216e8	0.000437710 0
$424$	0	0
$425$	−1.79813e11	−0.267344
$426$	0	0
$427$	5.03180e11	0.732483
$428$	0	0
$429$	−4.70168e11	−0.670185
$430$	0	0
$431$	−5.95908e10	−0.0831825	−0.0415912	−	0.999135i	$-0.513243\pi$
$431$	−5.95908e10	−0.0831825	−0.0415912	+	0.999135i	$0.513243\pi$
$432$	0	0
$433$	−1.16290e12	−1.58982	−0.794909	−	0.606729i	$-0.792481\pi$
$433$	−1.16290e12	−1.58982	−0.794909	+	0.606729i	$0.792481\pi$
$434$	0	0
$435$	4.64414e11	0.621875
$436$	0	0
$437$	−7.62435e10	−0.100008
$438$	0	0
$439$	1.32465e12	1.70220	0.851100	−	0.525004i	$-0.175936\pi$
$439$	1.32465e12	1.70220	0.851100	+	0.525004i	$0.175936\pi$
$440$	0	0
$441$	5.27300e8	0.000663871 0
$442$	0	0
$443$	−7.48682e11	−0.923593	−0.461797	−	0.886986i	$-0.652795\pi$
$443$	−7.48682e11	−0.923593	−0.461797	+	0.886986i	$0.652795\pi$
$444$	0	0
$445$	−5.74248e10	−0.0694192
$446$	0	0
$447$	−1.06642e12	−1.26341
$448$	0	0
$449$	8.07450e11	0.937577	0.468788	−	0.883310i	$-0.344691\pi$
$449$	8.07450e11	0.937577	0.468788	+	0.883310i	$0.344691\pi$
$450$	0	0
$451$	1.07455e12	1.22301
$452$	0	0
$453$	−2.44914e11	−0.273258
$454$	0	0
$455$	−3.00026e11	−0.328177
$456$	0	0
$457$	−2.09593e11	−0.224778	−0.112389	−	0.993664i	$-0.535850\pi$
$457$	−2.09593e11	−0.224778	−0.112389	+	0.993664i	$0.535850\pi$
$458$	0	0
$459$	−1.39717e12	−1.46924
$460$	0	0
$461$	1.12823e12	1.16343	0.581717	−	0.813391i	$-0.302381\pi$
$461$	1.12823e12	1.16343	0.581717	+	0.813391i	$0.302381\pi$
$462$	0	0
$463$	9.75512e11	0.986548	0.493274	−	0.869874i	$-0.335800\pi$
$463$	9.75512e11	0.986548	0.493274	+	0.869874i	$0.335800\pi$
$464$	0	0
$465$	−6.25831e11	−0.620753
$466$	0	0
$467$	5.15878e11	0.501904	0.250952	−	0.968000i	$-0.419256\pi$
$467$	5.15878e11	0.501904	0.250952	+	0.968000i	$0.419256\pi$
$468$	0	0
$469$	−9.25095e10	−0.0882894
$470$	0	0
$471$	1.49434e12	1.39912
$472$	0	0
$473$	5.55170e11	0.509977
$474$	0	0
$475$	4.63372e10	0.0417646
$476$	0	0
$477$	−1.32003e8	−0.000116749 0
$478$	0	0
$479$	−1.89215e12	−1.64228	−0.821139	−	0.570728i	$-0.806661\pi$
$479$	−1.89215e12	−1.64228	−0.821139	+	0.570728i	$0.806661\pi$
$480$	0	0
$481$	−1.26214e12	−1.07511
$482$	0	0
$483$	2.86109e11	0.239204
$484$	0	0
$485$	2.08336e12	1.70973
$486$	0	0
$487$	−1.35254e12	−1.08960	−0.544802	−	0.838565i	$-0.683395\pi$
$487$	−1.35254e12	−1.08960	−0.544802	+	0.838565i	$0.683395\pi$
$488$	0	0
$489$	−1.27798e12	−1.01073
$490$	0	0
$491$	−9.23267e11	−0.716903	−0.358452	−	0.933548i	$-0.616695\pi$
$491$	−9.23267e11	−0.716903	−0.358452	+	0.933548i	$0.616695\pi$
$492$	0	0
$493$	1.32508e12	1.01026
$494$	0	0
$495$	−1.16454e9	−0.000871830 0
$496$	0	0
$497$	1.48941e11	0.109499
$498$	0	0
$499$	−1.98116e12	−1.43043	−0.715214	−	0.698905i	$-0.753671\pi$
$499$	−1.98116e12	−1.43043	−0.715214	+	0.698905i	$0.753671\pi$
$500$	0	0
$501$	−1.62029e12	−1.14901
$502$	0	0
$503$	−2.56937e12	−1.78966	−0.894830	−	0.446406i	$-0.852704\pi$
$503$	−2.56937e12	−1.78966	−0.894830	+	0.446406i	$0.852704\pi$
$504$	0	0
$505$	−1.13405e12	−0.775925
$506$	0	0
$507$	−8.37395e11	−0.562853
$508$	0	0
$509$	1.56900e11	0.103608	0.0518040	−	0.998657i	$-0.483503\pi$
$509$	1.56900e11	0.103608	0.0518040	+	0.998657i	$0.483503\pi$
$510$	0	0
$511$	1.13217e12	0.734544
$512$	0	0
$513$	3.60046e11	0.229525
$514$	0	0
$515$	−2.11866e12	−1.32717
$516$	0	0
$517$	−7.59044e11	−0.467261
$518$	0	0
$519$	−3.54480e11	−0.214456
$520$	0	0
$521$	−1.30379e12	−0.775241	−0.387621	−	0.921819i	$-0.626703\pi$
$521$	−1.30379e12	−0.775241	−0.387621	+	0.921819i	$0.626703\pi$
$522$	0	0
$523$	−2.30458e12	−1.34690	−0.673449	−	0.739234i	$-0.735188\pi$
$523$	−2.30458e12	−1.34690	−0.673449	+	0.739234i	$0.735188\pi$
$524$	0	0
$525$	−1.73883e11	−0.0998944
$526$	0	0
$527$	−1.78565e12	−1.00843
$528$	0	0
$529$	−1.45888e12	−0.809968
$530$	0	0
$531$	5.43786e8	0.000296827 0
$532$	0	0
$533$	−1.48479e12	−0.796878
$534$	0	0
$535$	−3.89362e11	−0.205476
$536$	0	0
$537$	2.42692e12	1.25942
$538$	0	0
$539$	−1.38869e12	−0.708691
$540$	0	0
$541$	5.93525e11	0.297887	0.148943	−	0.988846i	$-0.452413\pi$
$541$	5.93525e11	0.297887	0.148943	+	0.988846i	$0.452413\pi$
$542$	0	0
$543$	1.52413e12	0.752353
$544$	0	0
$545$	−2.26965e11	−0.110198
$546$	0	0
$547$	2.57843e12	1.23144	0.615719	−	0.787966i	$-0.288866\pi$
$547$	2.57843e12	1.23144	0.615719	+	0.787966i	$0.288866\pi$
$548$	0	0
$549$	−2.69872e9	−0.00126789
$550$	0	0
$551$	−3.41469e11	−0.157822
$552$	0	0
$553$	3.09756e11	0.140850
$554$	0	0
$555$	3.28662e12	1.47039
$556$	0	0
$557$	−3.49163e12	−1.53702	−0.768510	−	0.639838i	$-0.779001\pi$
$557$	−3.49163e12	−1.53702	−0.768510	+	0.639838i	$0.779001\pi$
$558$	0	0
$559$	−7.67122e11	−0.332285
$560$	0	0
$561$	3.49328e12	1.48902
$562$	0	0
$563$	1.40676e12	0.590109	0.295054	−	0.955480i	$-0.404662\pi$
$563$	1.40676e12	0.590109	0.295054	+	0.955480i	$0.404662\pi$
$564$	0	0
$565$	−2.91702e12	−1.20427
$566$	0	0
$567$	−1.34981e12	−0.548466
$568$	0	0
$569$	2.81635e12	1.12637	0.563185	−	0.826331i	$-0.309576\pi$
$569$	2.81635e12	1.12637	0.563185	+	0.826331i	$0.309576\pi$
$570$	0	0
$571$	1.85246e11	0.0729266	0.0364633	−	0.999335i	$-0.488391\pi$
$571$	1.85246e11	0.0729266	0.0364633	+	0.999335i	$0.488391\pi$
$572$	0	0
$573$	7.48645e11	0.290121
$574$	0	0
$575$	−2.08019e11	−0.0793593
$576$	0	0
$577$	−2.36440e12	−0.888034	−0.444017	−	0.896018i	$-0.646447\pi$
$577$	−2.36440e12	−0.888034	−0.444017	+	0.896018i	$0.646447\pi$
$578$	0	0
$579$	3.86961e12	1.43091
$580$	0	0
$581$	−2.19648e11	−0.0799714
$582$	0	0
$583$	3.47642e11	0.124630
$584$	0	0
$585$	1.60914e9	0.000568058 0
$586$	0	0
$587$	−8.75403e11	−0.304324	−0.152162	−	0.988356i	$-0.548624\pi$
$587$	−8.75403e11	−0.304324	−0.152162	+	0.988356i	$0.548624\pi$
$588$	0	0
$589$	4.60154e11	0.157538
$590$	0	0
$591$	−1.34768e12	−0.454405
$592$	0	0
$593$	−2.10579e12	−0.699307	−0.349654	−	0.936879i	$-0.613701\pi$
$593$	−2.10579e12	−0.699307	−0.349654	+	0.936879i	$0.613701\pi$
$594$	0	0
$595$	2.22915e12	0.729144
$596$	0	0
$597$	−4.05004e12	−1.30489
$598$	0	0
$599$	−5.86647e12	−1.86190	−0.930950	−	0.365147i	$-0.881019\pi$
$599$	−5.86647e12	−1.86190	−0.930950	+	0.365147i	$0.881019\pi$
$600$	0	0
$601$	−8.97472e11	−0.280599	−0.140299	−	0.990109i	$-0.544807\pi$
$601$	−8.97472e11	−0.280599	−0.140299	+	0.990109i	$0.544807\pi$
$602$	0	0
$603$	4.96159e8	0.000152825 0
$604$	0	0
$605$	8.66084e10	0.0262822
$606$	0	0
$607$	4.20534e12	1.25734	0.628669	−	0.777673i	$-0.283600\pi$
$607$	4.20534e12	1.25734	0.628669	+	0.777673i	$0.283600\pi$
$608$	0	0
$609$	1.28139e12	0.377487
$610$	0	0
$611$	1.04883e12	0.304453
$612$	0	0
$613$	4.66333e12	1.33390	0.666951	−	0.745102i	$-0.267599\pi$
$613$	4.66333e12	1.33390	0.666951	+	0.745102i	$0.267599\pi$
$614$	0	0
$615$	3.86640e12	1.08985
$616$	0	0
$617$	3.80910e12	1.05813	0.529065	−	0.848581i	$-0.322543\pi$
$617$	3.80910e12	1.05813	0.529065	+	0.848581i	$0.322543\pi$
$618$	0	0
$619$	3.62001e12	0.991063	0.495532	−	0.868590i	$-0.334973\pi$
$619$	3.62001e12	0.991063	0.495532	+	0.868590i	$0.334973\pi$
$620$	0	0
$621$	−1.61633e12	−0.436133
$622$	0	0
$623$	−1.58443e11	−0.0421384
$624$	0	0
$625$	−2.99382e12	−0.784811
$626$	0	0
$627$	−9.00204e11	−0.232615
$628$	0	0
$629$	9.37752e12	2.38869
$630$	0	0
$631$	7.16358e11	0.179886	0.0899431	−	0.995947i	$-0.471331\pi$
$631$	7.16358e11	0.179886	0.0899431	+	0.995947i	$0.471331\pi$
$632$	0	0
$633$	−3.17744e12	−0.786612
$634$	0	0
$635$	5.70436e12	1.39228
$636$	0	0
$637$	1.91887e12	0.461761
$638$	0	0
$639$	−7.98820e8	−0.000189537 0
$640$	0	0
$641$	−4.92458e12	−1.15215	−0.576074	−	0.817397i	$-0.695416\pi$
$641$	−4.92458e12	−1.15215	−0.576074	+	0.817397i	$0.695416\pi$
$642$	0	0
$643$	−3.16432e12	−0.730013	−0.365007	−	0.931005i	$-0.618933\pi$
$643$	−3.16432e12	−0.730013	−0.365007	+	0.931005i	$0.618933\pi$
$644$	0	0
$645$	1.99759e12	0.454452
$646$	0	0
$647$	−8.63261e12	−1.93675	−0.968374	−	0.249504i	$-0.919732\pi$
$647$	−8.63261e12	−1.93675	−0.968374	+	0.249504i	$0.919732\pi$
$648$	0	0
$649$	−1.43211e12	−0.316866
$650$	0	0
$651$	−1.72676e12	−0.376806
$652$	0	0
$653$	−4.53346e12	−0.975710	−0.487855	−	0.872925i	$-0.662220\pi$
$653$	−4.53346e12	−0.975710	−0.487855	+	0.872925i	$0.662220\pi$
$654$	0	0
$655$	3.20684e12	0.680755
$656$	0	0
$657$	−6.07221e9	−0.00127146
$658$	0	0
$659$	−2.67609e12	−0.552734	−0.276367	−	0.961052i	$-0.589131\pi$
$659$	−2.67609e12	−0.552734	−0.276367	+	0.961052i	$0.589131\pi$
$660$	0	0
$661$	1.23915e12	0.252475	0.126238	−	0.992000i	$-0.459710\pi$
$661$	1.23915e12	0.252475	0.126238	+	0.992000i	$0.459710\pi$
$662$	0	0
$663$	−4.82694e12	−0.970199
$664$	0	0
$665$	−5.74444e11	−0.113907
$666$	0	0
$667$	1.53294e12	0.299888
$668$	0	0
$669$	8.19734e12	1.58218
$670$	0	0
$671$	7.10733e12	1.35349
$672$	0	0
$673$	−5.79692e11	−0.108926	−0.0544628	−	0.998516i	$-0.517345\pi$
$673$	−5.79692e11	−0.108926	−0.0544628	+	0.998516i	$0.517345\pi$
$674$	0	0
$675$	9.82331e11	0.182134
$676$	0	0
$677$	6.09981e12	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$677$	6.09981e12	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$678$	0	0
$679$	5.74829e12	1.03783
$680$	0	0
$681$	6.76168e11	0.120474
$682$	0	0
$683$	1.92035e12	0.337665	0.168833	−	0.985645i	$-0.446000\pi$
$683$	1.92035e12	0.337665	0.168833	+	0.985645i	$0.446000\pi$
$684$	0	0
$685$	5.10748e12	0.886338
$686$	0	0
$687$	−1.00381e12	−0.171928
$688$	0	0
$689$	−4.80365e11	−0.0812054
$690$	0	0
$691$	1.22588e12	0.204549	0.102275	−	0.994756i	$-0.467388\pi$
$691$	1.22588e12	0.204549	0.102275	+	0.994756i	$0.467388\pi$
$692$	0	0
$693$	−3.21314e9	−0.000529213 0
$694$	0	0
$695$	2.35754e12	0.383290
$696$	0	0
$697$	1.10318e13	1.77051
$698$	0	0
$699$	6.34870e12	1.00586
$700$	0	0
$701$	−7.54657e10	−0.0118037	−0.00590186	−	0.999983i	$-0.501879\pi$
$701$	−7.54657e10	−0.0118037	−0.00590186	+	0.999983i	$0.501879\pi$
$702$	0	0
$703$	−2.41655e12	−0.373162
$704$	0	0
$705$	−2.73116e12	−0.416387
$706$	0	0
$707$	−3.12900e12	−0.470997
$708$	0	0
$709$	−2.48251e12	−0.368964	−0.184482	−	0.982836i	$-0.559061\pi$
$709$	−2.48251e12	−0.368964	−0.184482	+	0.982836i	$0.559061\pi$
$710$	0	0
$711$	−1.66132e9	−0.000243804 0
$712$	0	0
$713$	−2.06575e12	−0.299347
$714$	0	0
$715$	−4.23782e12	−0.606409
$716$	0	0
$717$	9.89210e12	1.39782
$718$	0	0
$719$	5.21696e12	0.728010	0.364005	−	0.931397i	$-0.381409\pi$
$719$	5.21696e12	0.728010	0.364005	+	0.931397i	$0.381409\pi$
$720$	0	0
$721$	−5.84569e12	−0.805614
$722$	0	0
$723$	1.09819e13	1.49470
$724$	0	0
$725$	−9.31648e11	−0.125236
$726$	0	0
$727$	−1.23462e13	−1.63919	−0.819595	−	0.572944i	$-0.805801\pi$
$727$	−1.23462e13	−1.63919	−0.819595	+	0.572944i	$0.805801\pi$
$728$	0	0
$729$	7.63283e12	1.00095
$730$	0	0
$731$	5.69961e12	0.738273
$732$	0	0
$733$	−7.23773e12	−0.926050	−0.463025	−	0.886345i	$-0.653236\pi$
$733$	−7.23773e12	−0.926050	−0.463025	+	0.886345i	$0.653236\pi$
$734$	0	0
$735$	−4.99674e12	−0.631530
$736$	0	0
$737$	−1.30668e12	−0.163142
$738$	0	0
$739$	1.58616e12	0.195636	0.0978179	−	0.995204i	$-0.468814\pi$
$739$	1.58616e12	0.195636	0.0978179	+	0.995204i	$0.468814\pi$
$740$	0	0
$741$	1.24388e12	0.151565
$742$	0	0
$743$	1.32982e13	1.60082	0.800408	−	0.599455i	$-0.204616\pi$
$743$	1.32982e13	1.60082	0.800408	+	0.599455i	$0.204616\pi$
$744$	0	0
$745$	−9.61208e12	−1.14318
$746$	0	0
$747$	1.17805e9	0.000138426 0
$748$	0	0
$749$	−1.07431e12	−0.124727
$750$	0	0
$751$	7.69495e12	0.882726	0.441363	−	0.897329i	$-0.354495\pi$
$751$	7.69495e12	0.882726	0.441363	+	0.897329i	$0.354495\pi$
$752$	0	0
$753$	−1.45363e13	−1.64770
$754$	0	0
$755$	−2.20752e12	−0.247254
$756$	0	0
$757$	1.46042e13	1.61640	0.808198	−	0.588911i	$-0.200443\pi$
$757$	1.46042e13	1.61640	0.808198	+	0.588911i	$0.200443\pi$
$758$	0	0
$759$	4.04124e12	0.442005
$760$	0	0
$761$	−2.14254e12	−0.231578	−0.115789	−	0.993274i	$-0.536940\pi$
$761$	−2.14254e12	−0.231578	−0.115789	+	0.993274i	$0.536940\pi$
$762$	0	0
$763$	−6.26231e11	−0.0668920
$764$	0	0
$765$	−1.19557e10	−0.00126211
$766$	0	0
$767$	1.97886e12	0.206460
$768$	0	0
$769$	1.37576e13	1.41865	0.709326	−	0.704881i	$-0.249000\pi$
$769$	1.37576e13	1.41865	0.709326	+	0.704881i	$0.249000\pi$
$770$	0	0
$771$	1.50458e13	1.53346
$772$	0	0
$773$	1.78934e13	1.80254	0.901270	−	0.433257i	$-0.142636\pi$
$773$	1.78934e13	1.80254	0.901270	+	0.433257i	$0.142636\pi$
$774$	0	0
$775$	1.25546e12	0.125010
$776$	0	0
$777$	9.06827e12	0.892545
$778$	0	0
$779$	−2.84284e12	−0.276588
$780$	0	0
$781$	2.10377e12	0.202334
$782$	0	0
$783$	−7.23902e12	−0.688259
$784$	0	0
$785$	1.34691e13	1.26598
$786$	0	0
$787$	−1.90983e13	−1.77464	−0.887318	−	0.461158i	$-0.847434\pi$
$787$	−1.90983e13	−1.77464	−0.887318	+	0.461158i	$0.847434\pi$
$788$	0	0
$789$	1.65807e13	1.52320
$790$	0	0
$791$	−8.04850e12	−0.731006
$792$	0	0
$793$	−9.82076e12	−0.881893
$794$	0	0
$795$	1.25087e12	0.111061
$796$	0	0
$797$	−4.39907e10	−0.00386188	−0.00193094	−	0.999998i	$-0.500615\pi$
$797$	−4.39907e10	−0.00386188	−0.00193094	+	0.999998i	$0.500615\pi$
$798$	0	0
$799$	−7.79267e12	−0.676434
$800$	0	0
$801$	8.49785e8	7.29395e−5 0
$802$	0	0
$803$	1.59917e13	1.35730
$804$	0	0
$805$	2.57882e12	0.216441
$806$	0	0
$807$	1.51647e13	1.25864
$808$	0	0
$809$	−1.85839e13	−1.52535	−0.762675	−	0.646782i	$-0.776114\pi$
$809$	−1.85839e13	−1.52535	−0.762675	+	0.646782i	$0.776114\pi$
$810$	0	0
$811$	−1.29810e13	−1.05369	−0.526847	−	0.849960i	$-0.676626\pi$
$811$	−1.29810e13	−1.05369	−0.526847	+	0.849960i	$0.676626\pi$
$812$	0	0
$813$	1.71336e13	1.37544
$814$	0	0
$815$	−1.15190e13	−0.914543
$816$	0	0
$817$	−1.46877e12	−0.115333
$818$	0	0
$819$	4.43985e9	0.000344819 0
$820$	0	0
$821$	−4.47787e12	−0.343975	−0.171987	−	0.985099i	$-0.555019\pi$
$821$	−4.47787e12	−0.343975	−0.171987	+	0.985099i	$0.555019\pi$
$822$	0	0
$823$	−1.13315e13	−0.860973	−0.430487	−	0.902597i	$-0.641658\pi$
$823$	−1.13315e13	−0.860973	−0.430487	+	0.902597i	$0.641658\pi$
$824$	0	0
$825$	−2.45607e12	−0.184586
$826$	0	0
$827$	1.46669e13	1.09034	0.545171	−	0.838325i	$-0.316465\pi$
$827$	1.46669e13	1.09034	0.545171	+	0.838325i	$0.316465\pi$
$828$	0	0
$829$	−2.45530e13	−1.80555	−0.902774	−	0.430115i	$-0.858473\pi$
$829$	−2.45530e13	−1.80555	−0.902774	+	0.430115i	$0.858473\pi$
$830$	0	0
$831$	1.66794e13	1.21332
$832$	0	0
$833$	−1.42569e13	−1.02594
$834$	0	0
$835$	−1.46044e13	−1.03967
$836$	0	0
$837$	9.75510e12	0.687017
$838$	0	0
$839$	−8.71743e12	−0.607379	−0.303689	−	0.952771i	$-0.598219\pi$
$839$	−8.71743e12	−0.607379	−0.303689	+	0.952771i	$0.598219\pi$
$840$	0	0
$841$	−7.64163e12	−0.526749
$842$	0	0
$843$	−6.64239e12	−0.453002
$844$	0	0
$845$	−7.54780e12	−0.509290
$846$	0	0
$847$	2.38965e11	0.0159536
$848$	0	0
$849$	−2.24127e13	−1.48050
$850$	0	0
$851$	1.08485e13	0.709066
$852$	0	0
$853$	1.30680e12	0.0845159	0.0422580	−	0.999107i	$-0.486545\pi$
$853$	1.30680e12	0.0845159	0.0422580	+	0.999107i	$0.486545\pi$
$854$	0	0
$855$	3.08093e9	0.000197167 0
$856$	0	0
$857$	−1.57408e12	−0.0996811	−0.0498405	−	0.998757i	$-0.515871\pi$
$857$	−1.57408e12	−0.0996811	−0.0498405	+	0.998757i	$0.515871\pi$
$858$	0	0
$859$	−1.39034e13	−0.871265	−0.435633	−	0.900125i	$-0.643475\pi$
$859$	−1.39034e13	−0.871265	−0.435633	+	0.900125i	$0.643475\pi$
$860$	0	0
$861$	1.06680e13	0.661557
$862$	0	0
$863$	−2.53344e11	−0.0155476	−0.00777378	−	0.999970i	$-0.502474\pi$
$863$	−2.53344e11	−0.0155476	−0.00777378	+	0.999970i	$0.502474\pi$
$864$	0	0
$865$	−3.19508e12	−0.194048
$866$	0	0
$867$	1.92339e13	1.15607
$868$	0	0
$869$	4.37525e12	0.260264
$870$	0	0
$871$	1.80554e12	0.106298
$872$	0	0
$873$	−3.08300e10	−0.00179643
$874$	0	0
$875$	−1.01765e13	−0.586897
$876$	0	0
$877$	−4.95849e12	−0.283042	−0.141521	−	0.989935i	$-0.545199\pi$
$877$	−4.95849e12	−0.283042	−0.141521	+	0.989935i	$0.545199\pi$
$878$	0	0
$879$	1.86127e13	1.05162
$880$	0	0
$881$	−1.32981e13	−0.743702	−0.371851	−	0.928292i	$-0.621277\pi$
$881$	−1.32981e13	−0.743702	−0.371851	+	0.928292i	$0.621277\pi$
$882$	0	0
$883$	−1.27657e13	−0.706677	−0.353338	−	0.935496i	$-0.614954\pi$
$883$	−1.27657e13	−0.706677	−0.353338	+	0.935496i	$0.614954\pi$
$884$	0	0
$885$	−5.15297e12	−0.282366
$886$	0	0
$887$	−1.14862e13	−0.623044	−0.311522	−	0.950239i	$-0.600839\pi$
$887$	−1.14862e13	−0.623044	−0.311522	+	0.950239i	$0.600839\pi$
$888$	0	0
$889$	1.57392e13	0.845131
$890$	0	0
$891$	−1.90659e13	−1.01346
$892$	0	0
$893$	2.00814e12	0.105673
$894$	0	0
$895$	2.18748e13	1.13957
$896$	0	0
$897$	−5.58410e12	−0.287997
$898$	0	0
$899$	−9.25179e12	−0.472397
$900$	0	0
$901$	3.56904e12	0.180422
$902$	0	0
$903$	5.51165e12	0.275859
$904$	0	0
$905$	1.37376e13	0.680757
$906$	0	0
$907$	−1.87998e13	−0.922400	−0.461200	−	0.887296i	$-0.652581\pi$
$907$	−1.87998e13	−0.922400	−0.461200	+	0.887296i	$0.652581\pi$
$908$	0	0
$909$	1.67819e10	0.000815273 0
$910$	0	0
$911$	1.56523e12	0.0752914	0.0376457	−	0.999291i	$-0.488014\pi$
$911$	1.56523e12	0.0752914	0.0376457	+	0.999291i	$0.488014\pi$
$912$	0	0
$913$	−3.10249e12	−0.147772
$914$	0	0
$915$	2.55733e13	1.20613
$916$	0	0
$917$	8.84813e12	0.413228
$918$	0	0
$919$	−2.72891e13	−1.26203	−0.631014	−	0.775771i	$-0.717361\pi$
$919$	−2.72891e13	−1.26203	−0.631014	+	0.775771i	$0.717361\pi$
$920$	0	0
$921$	−3.39286e13	−1.55381
$922$	0	0
$923$	−2.90694e12	−0.131834
$924$	0	0
$925$	−6.59320e12	−0.296114
$926$	0	0
$927$	3.13524e10	0.00139448
$928$	0	0
$929$	−3.27413e13	−1.44220	−0.721100	−	0.692831i	$-0.756363\pi$
$929$	−3.27413e13	−1.44220	−0.721100	+	0.692831i	$0.756363\pi$
$930$	0	0
$931$	3.67395e12	0.160273
$932$	0	0
$933$	2.45789e13	1.06193
$934$	0	0
$935$	3.14864e13	1.34732
$936$	0	0
$937$	−2.41823e13	−1.02487	−0.512435	−	0.858726i	$-0.671256\pi$
$937$	−2.41823e13	−1.02487	−0.512435	+	0.858726i	$0.671256\pi$
$938$	0	0
$939$	−3.59632e13	−1.50961
$940$	0	0
$941$	7.48541e12	0.311216	0.155608	−	0.987819i	$-0.450266\pi$
$941$	7.48541e12	0.311216	0.155608	+	0.987819i	$0.450266\pi$
$942$	0	0
$943$	1.27622e13	0.525562
$944$	0	0
$945$	−1.21780e13	−0.496744
$946$	0	0
$947$	4.31138e13	1.74197	0.870986	−	0.491307i	$-0.163481\pi$
$947$	4.31138e13	1.74197	0.870986	+	0.491307i	$0.163481\pi$
$948$	0	0
$949$	−2.20970e13	−0.884374
$950$	0	0
$951$	−4.08096e12	−0.161789
$952$	0	0
$953$	−9.69686e12	−0.380814	−0.190407	−	0.981705i	$-0.560981\pi$
$953$	−9.69686e12	−0.380814	−0.190407	+	0.981705i	$0.560981\pi$
$954$	0	0
$955$	6.74785e12	0.262513
$956$	0	0
$957$	1.80994e13	0.697525
$958$	0	0
$959$	1.40923e13	0.538019
$960$	0	0
$961$	−1.39722e13	−0.528456
$962$	0	0
$963$	5.76187e9	0.000215896 0
$964$	0	0
$965$	3.48784e13	1.29474
$966$	0	0
$967$	−2.84710e13	−1.04709	−0.523545	−	0.851998i	$-0.675391\pi$
$967$	−2.84710e13	−1.04709	−0.523545	+	0.851998i	$0.675391\pi$
$968$	0	0
$969$	−9.24187e12	−0.336746
$970$	0	0
$971$	−3.67091e13	−1.32522	−0.662609	−	0.748965i	$-0.730551\pi$
$971$	−3.67091e13	−1.32522	−0.662609	+	0.748965i	$0.730551\pi$
$972$	0	0
$973$	6.50481e12	0.232663
$974$	0	0
$975$	3.39375e12	0.120271
$976$	0	0
$977$	−4.89209e13	−1.71778	−0.858892	−	0.512157i	$-0.828847\pi$
$977$	−4.89209e13	−1.71778	−0.858892	+	0.512157i	$0.828847\pi$
$978$	0	0
$979$	−2.23799e12	−0.0778638
$980$	0	0
$981$	3.35868e9	0.000115787 0
$982$	0	0
$983$	1.06278e13	0.363038	0.181519	−	0.983387i	$-0.441899\pi$
$983$	1.06278e13	0.363038	0.181519	+	0.983387i	$0.441899\pi$
$984$	0	0
$985$	−1.21472e13	−0.411162
$986$	0	0
$987$	−7.53568e12	−0.252753
$988$	0	0
$989$	6.59366e12	0.219151
$990$	0	0
$991$	9.73679e12	0.320689	0.160345	−	0.987061i	$-0.448739\pi$
$991$	9.73679e12	0.320689	0.160345	+	0.987061i	$0.448739\pi$
$992$	0	0
$993$	2.04291e13	0.666771
$994$	0	0
$995$	−3.65047e13	−1.18072
$996$	0	0
$997$	1.04996e13	0.336547	0.168273	−	0.985740i	$-0.446181\pi$
$997$	1.04996e13	0.336547	0.168273	+	0.985740i	$0.446181\pi$
$998$	0	0
$999$	−5.12300e13	−1.62735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.10.a.e.1.4		4
4.3	odd	2		38.10.a.d.1.1	✓	4
12.11	even	2		342.10.a.l.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.a.d.1.1	✓	4	4.3	odd	2
304.10.a.e.1.4		4	1.1	even	1	trivial
342.10.a.l.1.2		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q+140.229 q^{3} +1263.95 q^{5} +3487.42 q^{7} -18.7042 q^{9} +O(q^{10})\)	\(q+140.229 q^{3} +1263.95 q^{5} +3487.42 q^{7} -18.7042 q^{9} +49259.2 q^{11} -68065.3 q^{13} +177243. q^{15} +505716. q^{17} -130321. q^{19} +489039. q^{21} +585044. q^{23} -355562. q^{25} -2.76276e6 q^{27} +2.62021e6 q^{29} -3.53093e6 q^{31} +6.90759e6 q^{33} +4.40791e6 q^{35} +1.85431e7 q^{37} -9.54476e6 q^{39} +2.18141e7 q^{41} +1.12704e7 q^{43} -23641.1 q^{45} -1.54092e7 q^{47} -2.81915e7 q^{49} +7.09162e7 q^{51} +7.05741e6 q^{53} +6.22611e7 q^{55} -1.82748e7 q^{57} -2.90730e7 q^{59} +1.44284e8 q^{61} -65229.3 q^{63} -8.60310e7 q^{65} -2.65266e7 q^{67} +8.20404e7 q^{69} +4.27081e7 q^{71} +3.24644e8 q^{73} -4.98602e7 q^{75} +1.71787e8 q^{77} +8.88210e7 q^{79} -3.87052e8 q^{81} -6.29830e7 q^{83} +6.39198e8 q^{85} +3.67431e8 q^{87} -4.54329e7 q^{89} -2.37372e8 q^{91} -4.95140e8 q^{93} -1.64719e8 q^{95} +1.64830e9 q^{97} -921353. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9}+O(q^{10}) \) 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9	\( 4 q - 84 q^{3} - 1395 q^{5} - 12307 q^{7} + 16538 q^{9} + 104249 q^{11} + 120486 q^{13} + 591090 q^{15} - 412139 q^{17} - 521284 q^{19} + 2437006 q^{21} - 3010300 q^{23} + 9760585 q^{25} - 12387978 q^{27} + 6153240 q^{29} - 12774024 q^{31} - 3258022 q^{33} - 9823425 q^{35} + 20506048 q^{37} - 69881444 q^{39} + 11620300 q^{41} - 7698327 q^{43} - 124015815 q^{45} + 31581083 q^{47} + 18970383 q^{49} + 8594812 q^{51} + 72549422 q^{53} - 21332505 q^{55} + 10946964 q^{57} + 149234120 q^{59} + 129004373 q^{61} - 102967551 q^{63} + 124691700 q^{65} - 132595266 q^{67} - 45529972 q^{69} + 47138482 q^{71} - 39332795 q^{73} - 824627010 q^{75} - 165933719 q^{77} + 307010840 q^{79} + 1305551744 q^{81} + 746568232 q^{83} - 105005985 q^{85} + 82148208 q^{87} + 286943482 q^{89} - 3155781114 q^{91} + 1151901596 q^{93} + 181797795 q^{95} + 793519958 q^{97} + 1681833809 q^{99}+O(q^{100}) \) 4 * q - 84 * q^3 - 1395 * q^5 - 12307 * q^7 + 16538 * q^9 + 104249 * q^11 + 120486 * q^13 + 591090 * q^15 - 412139 * q^17 - 521284 * q^19 + 2437006 * q^21 - 3010300 * q^23 + 9760585 * q^25 - 12387978 * q^27 + 6153240 * q^29 - 12774024 * q^31 - 3258022 * q^33 - 9823425 * q^35 + 20506048 * q^37 - 69881444 * q^39 + 11620300 * q^41 - 7698327 * q^43 - 124015815 * q^45 + 31581083 * q^47 + 18970383 * q^49 + 8594812 * q^51 + 72549422 * q^53 - 21332505 * q^55 + 10946964 * q^57 + 149234120 * q^59 + 129004373 * q^61 - 102967551 * q^63 + 124691700 * q^65 - 132595266 * q^67 - 45529972 * q^69 + 47138482 * q^71 - 39332795 * q^73 - 824627010 * q^75 - 165933719 * q^77 + 307010840 * q^79 + 1305551744 * q^81 + 746568232 * q^83 - 105005985 * q^85 + 82148208 * q^87 + 286943482 * q^89 - 3155781114 * q^91 + 1151901596 * q^93 + 181797795 * q^95 + 793519958 * q^97 + 1681833809 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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