LMFDB - Embedded newform 4.38.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,38,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 38, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 38);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 38 $
Character orbit:	$[\chi]$	$=$	4.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:	$34.6856152498$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 134608389910x + 8010664803252592 $ x^3 - x^2 - 134608389910*x + 8010664803252592
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{5}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$61215.0$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31760e8 q^{3} -1.08025e13 q^{5} -4.54986e15 q^{7} -3.96571e17 q^{9} +O(q^{10})$	$q-2.31760e8 q^{3} -1.08025e13 q^{5} -4.54986e15 q^{7} -3.96571e17 q^{9} +1.61802e19 q^{11} -5.66947e20 q^{13} +2.50358e21 q^{15} -2.03513e21 q^{17} +2.44505e22 q^{19} +1.05447e24 q^{21} -1.30489e25 q^{23} +4.39339e25 q^{25} +1.96267e26 q^{27} -1.16791e27 q^{29} +6.59209e27 q^{31} -3.74992e27 q^{33} +4.91498e28 q^{35} +1.44956e29 q^{37} +1.31395e29 q^{39} +1.04384e30 q^{41} +2.02175e30 q^{43} +4.28395e30 q^{45} -7.03340e30 q^{47} +2.13911e30 q^{49} +4.71661e29 q^{51} -1.24164e32 q^{53} -1.74786e32 q^{55} -5.66664e30 q^{57} -3.71997e32 q^{59} -2.11074e32 q^{61} +1.80434e33 q^{63} +6.12443e33 q^{65} -3.63890e33 q^{67} +3.02422e33 q^{69} +3.36753e34 q^{71} -3.78227e34 q^{73} -1.01821e34 q^{75} -7.36177e34 q^{77} -1.82842e35 q^{79} +1.33083e35 q^{81} -1.67857e35 q^{83} +2.19844e34 q^{85} +2.70676e35 q^{87} +9.60208e35 q^{89} +2.57953e36 q^{91} -1.52778e36 q^{93} -2.64126e35 q^{95} +6.57484e36 q^{97} -6.41661e36 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots + 10\!\cdots\!63 q^{9}+O(q^{10}) $ 3 * q - 272163492 * q^3 + 3641045116194 * q^5 + 1514184507161592 * q^7 + 102953932588056663 * q^9	$ 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots - 14\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q - 272163492 * q^3 + 3641045116194 * q^5 + 1514184507161592 * q^7 + 102953932588056663 * q^9 - 11609057084515238700 * q^11 - 70576684907789880678 * q^13 - 2933969461740641207448 * q^15 + 21449865005452482229494 * q^17 + 158301069447345491378508 * q^19 - 1151851369077659765142432 * q^21 + 866923505554430216912616 * q^23 + 21645722139716923137352461 * q^25 + 147832683778944657999655704 * q^27 + 1010653042580699333097500298 * q^29 + 2180097844230541775332759008 * q^31 + 31048498538042201781617598480 * q^33 + 100605979662426959198661319248 * q^35 + 311298722814240927017686969602 * q^37 + 979152159930324950801131779528 * q^39 + 2635940958803787572844371014878 * q^41 + 5419069191259593947069584358580 * q^43 + 8099313551230663632839319909594 * q^45 + 4156587745521390586896893605584 * q^47 - 13495549806538130861979895910709 * q^49 - 89987135087681316961287710640264 * q^51 - 159108065600661790773554978344206 * q^53 - 501378639325706351372809856249160 * q^55 - 698546758941918182526174576638352 * q^57 - 540908935574431553277616956343644 * q^59 + 1178713661089330885859771001735786 * q^61 + 3403110131746397253746662721587992 * q^63 + 6554640894858129925078606186828668 * q^65 + 16889218571987157419703623848158492 * q^67 + 17143397087023799436715277896360224 * q^69 + 10309319797367225088925816018968504 * q^71 - 8520196523581770817793340839438658 * q^73 - 84054788877364659786396368345168412 * q^75 - 208861611906418376677492203082464480 * q^77 - 264775084715781163780628803947993936 * q^79 - 370309441007212745944873457987660853 * q^81 - 68956651634275475176080339084470004 * q^83 + 522499848715075310236425753489988836 * q^85 + 911140443393561025920716736778215048 * q^87 + 2493391276449702933740991498959962542 * q^89 + 2807109998185271068204514151779860752 * q^91 + 5409735857664632612107028249216969088 * q^93 + 3240594516611672218958902717719033864 * q^95 - 4697277125747917937176391731430481498 * q^97 - 14740636216794395787136495035497749500 * 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
$3$	−2.31760e8	−0.345378	−0.172689	−	0.984976i	$-0.555246\pi$
$3$	−2.31760e8	−0.345378	−0.172689	+	0.984976i	$0.555246\pi$
$4$	0	0
$5$	−1.08025e13	−1.26642	−0.633211	−	0.773979i	$-0.718263\pi$
$5$	−1.08025e13	−1.26642	−0.633211	+	0.773979i	$0.718263\pi$
$6$	0	0
$7$	−4.54986e15	−1.05605	−0.528025	−	0.849229i	$-0.677067\pi$
$7$	−4.54986e15	−1.05605	−0.528025	+	0.849229i	$0.677067\pi$
$8$	0	0
$9$	−3.96571e17	−0.880714
$10$	0	0
$11$	1.61802e19	0.877444	0.438722	−	0.898623i	$-0.355431\pi$
$11$	1.61802e19	0.877444	0.438722	+	0.898623i	$0.355431\pi$
$12$	0	0
$13$	−5.66947e20	−1.39827	−0.699134	−	0.714991i	$-0.746431\pi$
$13$	−5.66947e20	−1.39827	−0.699134	+	0.714991i	$0.746431\pi$
$14$	0	0
$15$	2.50358e21	0.437394
$16$	0	0
$17$	−2.03513e21	−0.0350983	−0.0175492	−	0.999846i	$-0.505586\pi$
$17$	−2.03513e21	−0.0350983	−0.0175492	+	0.999846i	$0.505586\pi$
$18$	0	0
$19$	2.44505e22	0.0538699	0.0269350	−	0.999637i	$-0.491425\pi$
$19$	2.44505e22	0.0538699	0.0269350	+	0.999637i	$0.491425\pi$
$20$	0	0
$21$	1.05447e24	0.364736
$22$	0	0
$23$	−1.30489e25	−0.838707	−0.419354	−	0.907823i	$-0.637743\pi$
$23$	−1.30489e25	−0.838707	−0.419354	+	0.907823i	$0.637743\pi$
$24$	0	0
$25$	4.39339e25	0.603824
$26$	0	0
$27$	1.96267e26	0.649557
$28$	0	0
$29$	−1.16791e27	−1.03050	−0.515250	−	0.857040i	$-0.672301\pi$
$29$	−1.16791e27	−1.03050	−0.515250	+	0.857040i	$0.672301\pi$
$30$	0	0
$31$	6.59209e27	1.69368	0.846841	−	0.531846i	$-0.178502\pi$
$31$	6.59209e27	1.69368	0.846841	+	0.531846i	$0.178502\pi$
$32$	0	0
$33$	−3.74992e27	−0.303050
$34$	0	0
$35$	4.91498e28	1.33740
$36$	0	0
$37$	1.44956e29	1.41093	0.705463	−	0.708747i	$-0.250739\pi$
$37$	1.44956e29	1.41093	0.705463	+	0.708747i	$0.250739\pi$
$38$	0	0
$39$	1.31395e29	0.482931
$40$	0	0
$41$	1.04384e30	1.52101	0.760503	−	0.649334i	$-0.224952\pi$
$41$	1.04384e30	1.52101	0.760503	+	0.649334i	$0.224952\pi$
$42$	0	0
$43$	2.02175e30	1.22056	0.610282	−	0.792185i	$-0.291056\pi$
$43$	2.02175e30	1.22056	0.610282	+	0.792185i	$0.291056\pi$
$44$	0	0
$45$	4.28395e30	1.11536
$46$	0	0
$47$	−7.03340e30	−0.819134	−0.409567	−	0.912280i	$-0.634320\pi$
$47$	−7.03340e30	−0.819134	−0.409567	+	0.912280i	$0.634320\pi$
$48$	0	0
$49$	2.13911e30	0.115241
$50$	0	0
$51$	4.71661e29	0.0121222
$52$	0	0
$53$	−1.24164e32	−1.56636	−0.783180	−	0.621795i	$-0.786404\pi$
$53$	−1.24164e32	−1.56636	−0.783180	+	0.621795i	$0.786404\pi$
$54$	0	0
$55$	−1.74786e32	−1.11121
$56$	0	0
$57$	−5.66664e30	−0.0186055
$58$	0	0
$59$	−3.71997e32	−0.645324	−0.322662	−	0.946514i	$-0.604578\pi$
$59$	−3.71997e32	−0.645324	−0.322662	+	0.946514i	$0.604578\pi$
$60$	0	0
$61$	−2.11074e32	−0.197621	−0.0988104	−	0.995106i	$-0.531504\pi$
$61$	−2.11074e32	−0.197621	−0.0988104	+	0.995106i	$0.531504\pi$
$62$	0	0
$63$	1.80434e33	0.930078
$64$	0	0
$65$	6.12443e33	1.77080
$66$	0	0
$67$	−3.63890e33	−0.600601	−0.300300	−	0.953845i	$-0.597087\pi$
$67$	−3.63890e33	−0.600601	−0.300300	+	0.953845i	$0.597087\pi$
$68$	0	0
$69$	3.02422e33	0.289671
$70$	0	0
$71$	3.36753e34	1.90122	0.950610	−	0.310388i	$-0.100459\pi$
$71$	3.36753e34	1.90122	0.950610	+	0.310388i	$0.100459\pi$
$72$	0	0
$73$	−3.78227e34	−1.27726	−0.638631	−	0.769513i	$-0.720499\pi$
$73$	−3.78227e34	−1.27726	−0.638631	+	0.769513i	$0.720499\pi$
$74$	0	0
$75$	−1.01821e34	−0.208547
$76$	0	0
$77$	−7.36177e34	−0.926625
$78$	0	0
$79$	−1.82842e35	−1.43210	−0.716052	−	0.698047i	$-0.754053\pi$
$79$	−1.82842e35	−1.43210	−0.716052	+	0.698047i	$0.754053\pi$
$80$	0	0
$81$	1.33083e35	0.656371
$82$	0	0
$83$	−1.67857e35	−0.527223	−0.263611	−	0.964629i	$-0.584914\pi$
$83$	−1.67857e35	−0.527223	−0.263611	+	0.964629i	$0.584914\pi$
$84$	0	0
$85$	2.19844e34	0.0444493
$86$	0	0
$87$	2.70676e35	0.355912
$88$	0	0
$89$	9.60208e35	0.829183	0.414591	−	0.910008i	$-0.363925\pi$
$89$	9.60208e35	0.829183	0.414591	+	0.910008i	$0.363925\pi$
$90$	0	0
$91$	2.57953e36	1.47664
$92$	0	0
$93$	−1.52778e36	−0.584960
$94$	0	0
$95$	−2.64126e35	−0.0682220
$96$	0	0
$97$	6.57484e36	1.15507	0.577535	−	0.816366i	$-0.304015\pi$
$97$	6.57484e36	1.15507	0.577535	+	0.816366i	$0.304015\pi$
$98$	0	0
$99$	−6.41661e36	−0.772778
$100$	0	0
$101$	−1.39074e37	−1.15691	−0.578455	−	0.815714i	$-0.696344\pi$
$101$	−1.39074e37	−1.15691	−0.578455	+	0.815714i	$0.696344\pi$
$102$	0	0
$103$	−2.01160e37	−1.16427	−0.582134	−	0.813093i	$-0.697782\pi$
$103$	−2.01160e37	−1.16427	−0.582134	+	0.813093i	$0.697782\pi$
$104$	0	0
$105$	−1.13909e37	−0.461910
$106$	0	0
$107$	3.07555e37	0.879676	0.439838	−	0.898077i	$-0.355036\pi$
$107$	3.07555e37	0.879676	0.439838	+	0.898077i	$0.355036\pi$
$108$	0	0
$109$	9.42599e36	0.191398	0.0956989	−	0.995410i	$-0.469491\pi$
$109$	9.42599e36	0.191398	0.0956989	+	0.995410i	$0.469491\pi$
$110$	0	0
$111$	−3.35950e37	−0.487303
$112$	0	0
$113$	2.03827e37	0.212476	0.106238	−	0.994341i	$-0.466119\pi$
$113$	2.03827e37	0.212476	0.106238	+	0.994341i	$0.466119\pi$
$114$	0	0
$115$	1.40961e38	1.06216
$116$	0	0
$117$	2.24835e38	1.23147
$118$	0	0
$119$	9.25954e36	0.0370656
$120$	0	0
$121$	−7.82401e37	−0.230091
$122$	0	0
$123$	−2.41919e38	−0.525322
$124$	0	0
$125$	3.11388e38	0.501726
$126$	0	0
$127$	−1.39428e39	−1.67487	−0.837436	−	0.546535i	$-0.815946\pi$
$127$	−1.39428e39	−1.67487	−0.837436	+	0.546535i	$0.815946\pi$
$128$	0	0
$129$	−4.68559e38	−0.421556
$130$	0	0
$131$	−1.32651e38	−0.0897827	−0.0448913	−	0.998992i	$-0.514294\pi$
$131$	−1.32651e38	−0.0897827	−0.0448913	+	0.998992i	$0.514294\pi$
$132$	0	0
$133$	−1.11246e38	−0.0568893
$134$	0	0
$135$	−2.12017e39	−0.822613
$136$	0	0
$137$	1.90101e39	0.561893	0.280947	−	0.959723i	$-0.409352\pi$
$137$	1.90101e39	0.561893	0.280947	+	0.959723i	$0.409352\pi$
$138$	0	0
$139$	8.31029e39	1.87863	0.939317	−	0.343050i	$-0.111460\pi$
$139$	8.31029e39	1.87863	0.939317	+	0.343050i	$0.111460\pi$
$140$	0	0
$141$	1.63006e39	0.282911
$142$	0	0
$143$	−9.17332e39	−1.22690
$144$	0	0
$145$	1.26164e40	1.30505
$146$	0	0
$147$	−4.95760e38	−0.0398016
$148$	0	0
$149$	2.20198e40	1.37679	0.688395	−	0.725336i	$-0.258316\pi$
$149$	2.20198e40	1.37679	0.688395	+	0.725336i	$0.258316\pi$
$150$	0	0
$151$	−2.65159e40	−1.29549	−0.647744	−	0.761858i	$-0.724287\pi$
$151$	−2.65159e40	−1.29549	−0.647744	+	0.761858i	$0.724287\pi$
$152$	0	0
$153$	8.07073e38	0.0309116
$154$	0	0
$155$	−7.12109e40	−2.14491
$156$	0	0
$157$	−2.39401e40	−0.568830	−0.284415	−	0.958701i	$-0.591799\pi$
$157$	−2.39401e40	−0.568830	−0.284415	+	0.958701i	$0.591799\pi$
$158$	0	0
$159$	2.87762e40	0.540986
$160$	0	0
$161$	5.93709e40	0.885717
$162$	0	0
$163$	−9.69475e40	−1.15098	−0.575488	−	0.817810i	$-0.695188\pi$
$163$	−9.69475e40	−1.15098	−0.575488	+	0.817810i	$0.695188\pi$
$164$	0	0
$165$	4.05084e40	0.383789
$166$	0	0
$167$	4.54501e40	0.344572	0.172286	−	0.985047i	$-0.444885\pi$
$167$	4.54501e40	0.344572	0.172286	+	0.985047i	$0.444885\pi$
$168$	0	0
$169$	1.57028e41	0.955152
$170$	0	0
$171$	−9.69637e39	−0.0474440
$172$	0	0
$173$	4.12570e41	1.62796	0.813981	−	0.580891i	$-0.197296\pi$
$173$	4.12570e41	1.62796	0.813981	+	0.580891i	$0.197296\pi$
$174$	0	0
$175$	−1.99893e41	−0.637668
$176$	0	0
$177$	8.62140e40	0.222881
$178$	0	0
$179$	−6.16454e41	−1.29455	−0.647276	−	0.762256i	$-0.724092\pi$
$179$	−6.16454e41	−1.29455	−0.647276	+	0.762256i	$0.724092\pi$
$180$	0	0
$181$	4.59766e41	0.786110	0.393055	−	0.919515i	$-0.371418\pi$
$181$	4.59766e41	0.786110	0.393055	+	0.919515i	$0.371418\pi$
$182$	0	0
$183$	4.89184e40	0.0682539
$184$	0	0
$185$	−1.56588e42	−1.78683
$186$	0	0
$187$	−3.29288e40	−0.0307968
$188$	0	0
$189$	−8.92987e41	−0.685965
$190$	0	0
$191$	−3.15394e41	−0.199405	−0.0997026	−	0.995017i	$-0.531789\pi$
$191$	−3.15394e41	−0.199405	−0.0997026	+	0.995017i	$0.531789\pi$
$192$	0	0
$193$	1.24569e42	0.649532	0.324766	−	0.945794i	$-0.394714\pi$
$193$	1.24569e42	0.649532	0.324766	+	0.945794i	$0.394714\pi$
$194$	0	0
$195$	−1.41940e42	−0.611594
$196$	0	0
$197$	4.99627e42	1.78246	0.891232	−	0.453548i	$-0.149842\pi$
$197$	4.99627e42	1.78246	0.891232	+	0.453548i	$0.149842\pi$
$198$	0	0
$199$	1.57820e42	0.467069	0.233535	−	0.972348i	$-0.424971\pi$
$199$	1.57820e42	0.467069	0.233535	+	0.972348i	$0.424971\pi$
$200$	0	0
$201$	8.43350e41	0.207434
$202$	0	0
$203$	5.31385e42	1.08826
$204$	0	0
$205$	−1.12760e43	−1.92623
$206$	0	0
$207$	5.17484e42	0.738661
$208$	0	0
$209$	3.95615e41	0.0472679
$210$	0	0
$211$	1.36838e43	1.37082	0.685410	−	0.728158i	$-0.259623\pi$
$211$	1.36838e43	1.37082	0.685410	+	0.728158i	$0.259623\pi$
$212$	0	0
$213$	−7.80458e42	−0.656640
$214$	0	0
$215$	−2.18399e43	−1.54575
$216$	0	0
$217$	−2.99931e43	−1.78861
$218$	0	0
$219$	8.76579e42	0.441138
$220$	0	0
$221$	1.15381e42	0.0490769
$222$	0	0
$223$	2.02201e43	0.728019	0.364010	−	0.931395i	$-0.381408\pi$
$223$	2.02201e43	0.728019	0.364010	+	0.931395i	$0.381408\pi$
$224$	0	0
$225$	−1.74229e43	−0.531796
$226$	0	0
$227$	4.21148e42	0.109133	0.0545665	−	0.998510i	$-0.482622\pi$
$227$	4.21148e42	0.109133	0.0545665	+	0.998510i	$0.482622\pi$
$228$	0	0
$229$	3.22246e43	0.709955	0.354977	−	0.934875i	$-0.384489\pi$
$229$	3.22246e43	0.709955	0.354977	+	0.934875i	$0.384489\pi$
$230$	0	0
$231$	1.70616e43	0.320036
$232$	0	0
$233$	1.32663e43	0.212160	0.106080	−	0.994358i	$-0.466170\pi$
$233$	1.32663e43	0.212160	0.106080	+	0.994358i	$0.466170\pi$
$234$	0	0
$235$	7.59782e43	1.03737
$236$	0	0
$237$	4.23754e43	0.494617
$238$	0	0
$239$	5.49078e43	0.548622	0.274311	−	0.961641i	$-0.411550\pi$
$239$	5.49078e43	0.548622	0.274311	+	0.961641i	$0.411550\pi$
$240$	0	0
$241$	−1.31742e44	−1.12826	−0.564128	−	0.825688i	$-0.690787\pi$
$241$	−1.31742e44	−1.12826	−0.564128	+	0.825688i	$0.690787\pi$
$242$	0	0
$243$	−1.19219e44	−0.876254
$244$	0	0
$245$	−2.31077e43	−0.145943
$246$	0	0
$247$	−1.38621e43	−0.0753246
$248$	0	0
$249$	3.89024e43	0.182091
$250$	0	0
$251$	−1.67421e44	−0.675842	−0.337921	−	0.941174i	$-0.609724\pi$
$251$	−1.67421e44	−0.675842	−0.337921	+	0.941174i	$0.609724\pi$
$252$	0	0
$253$	−2.11135e44	−0.735919
$254$	0	0
$255$	−5.09510e42	−0.0153518
$256$	0	0
$257$	1.88763e44	0.492215	0.246108	−	0.969243i	$-0.420848\pi$
$257$	1.88763e44	0.492215	0.246108	+	0.969243i	$0.420848\pi$
$258$	0	0
$259$	−6.59529e44	−1.49001
$260$	0	0
$261$	4.63161e44	0.907576
$262$	0	0
$263$	1.47596e44	0.251127	0.125564	−	0.992086i	$-0.459926\pi$
$263$	1.47596e44	0.251127	0.125564	+	0.992086i	$0.459926\pi$
$264$	0	0
$265$	1.34128e45	1.98367
$266$	0	0
$267$	−2.22538e44	−0.286381
$268$	0	0
$269$	7.36652e44	0.825744	0.412872	−	0.910789i	$-0.364526\pi$
$269$	7.36652e44	0.825744	0.412872	+	0.910789i	$0.364526\pi$
$270$	0	0
$271$	2.39411e44	0.233998	0.116999	−	0.993132i	$-0.462673\pi$
$271$	2.39411e44	0.233998	0.116999	+	0.993132i	$0.462673\pi$
$272$	0	0
$273$	−5.97831e44	−0.509999
$274$	0	0
$275$	7.10861e44	0.529822
$276$	0	0
$277$	−1.20053e45	−0.782522	−0.391261	−	0.920280i	$-0.627961\pi$
$277$	−1.20053e45	−0.782522	−0.391261	+	0.920280i	$0.627961\pi$
$278$	0	0
$279$	−2.61423e45	−1.49165
$280$	0	0
$281$	1.31149e45	0.655688	0.327844	−	0.944732i	$-0.393678\pi$
$281$	1.31149e45	0.655688	0.327844	+	0.944732i	$0.393678\pi$
$282$	0	0
$283$	1.28691e45	0.564287	0.282143	−	0.959372i	$-0.408955\pi$
$283$	1.28691e45	0.564287	0.282143	+	0.959372i	$0.408955\pi$
$284$	0	0
$285$	6.12138e43	0.0235624
$286$	0	0
$287$	−4.74931e45	−1.60626
$288$	0	0
$289$	−3.35795e45	−0.998768
$290$	0	0
$291$	−1.52378e45	−0.398936
$292$	0	0
$293$	2.72231e45	0.627894	0.313947	−	0.949440i	$-0.398349\pi$
$293$	2.72231e45	0.627894	0.313947	+	0.949440i	$0.398349\pi$
$294$	0	0
$295$	4.01849e45	0.817252
$296$	0	0
$297$	3.17564e45	0.569950
$298$	0	0
$299$	7.39806e45	1.17274
$300$	0	0
$301$	−9.19866e45	−1.28898
$302$	0	0
$303$	3.22317e45	0.399572
$304$	0	0
$305$	2.28012e45	0.250271
$306$	0	0
$307$	6.62882e45	0.644727	0.322364	−	0.946616i	$-0.395523\pi$
$307$	6.62882e45	0.644727	0.322364	+	0.946616i	$0.395523\pi$
$308$	0	0
$309$	4.66207e45	0.402112
$310$	0	0
$311$	2.44200e46	1.86930	0.934649	−	0.355571i	$-0.115714\pi$
$311$	2.44200e46	1.86930	0.934649	+	0.355571i	$0.115714\pi$
$312$	0	0
$313$	−1.81885e46	−1.23659	−0.618297	−	0.785944i	$-0.712177\pi$
$313$	−1.81885e46	−1.23659	−0.618297	+	0.785944i	$0.712177\pi$
$314$	0	0
$315$	−1.94914e46	−1.17787
$316$	0	0
$317$	−5.86315e45	−0.315163	−0.157581	−	0.987506i	$-0.550370\pi$
$317$	−5.86315e45	−0.315163	−0.157581	+	0.987506i	$0.550370\pi$
$318$	0	0
$319$	−1.88971e46	−0.904207
$320$	0	0
$321$	−7.12789e45	−0.303821
$322$	0	0
$323$	−4.97599e43	−0.00189074
$324$	0	0
$325$	−2.49082e46	−0.844307
$326$	0	0
$327$	−2.18457e45	−0.0661046
$328$	0	0
$329$	3.20010e46	0.865047
$330$	0	0
$331$	3.22607e46	0.779572	0.389786	−	0.920906i	$-0.372549\pi$
$331$	3.22607e46	0.779572	0.389786	+	0.920906i	$0.372549\pi$
$332$	0	0
$333$	−5.74854e46	−1.24262
$334$	0	0
$335$	3.93091e46	0.760614
$336$	0	0
$337$	−1.79861e46	−0.311733	−0.155867	−	0.987778i	$-0.549817\pi$
$337$	−1.79861e46	−0.311733	−0.155867	+	0.987778i	$0.549817\pi$
$338$	0	0
$339$	−4.72388e45	−0.0733845
$340$	0	0
$341$	1.06661e47	1.48611
$342$	0	0
$343$	7.47224e46	0.934350
$344$	0	0
$345$	−3.26691e46	−0.366846
$346$	0	0
$347$	2.03561e46	0.205399	0.102700	−	0.994712i	$-0.467252\pi$
$347$	2.03561e46	0.205399	0.102700	+	0.994712i	$0.467252\pi$
$348$	0	0
$349$	7.55636e46	0.685555	0.342777	−	0.939417i	$-0.388632\pi$
$349$	7.55636e46	0.685555	0.342777	+	0.939417i	$0.388632\pi$
$350$	0	0
$351$	−1.11273e47	−0.908255
$352$	0	0
$353$	1.78872e47	1.31435	0.657176	−	0.753737i	$-0.271751\pi$
$353$	1.78872e47	1.31435	0.657176	+	0.753737i	$0.271751\pi$
$354$	0	0
$355$	−3.63777e47	−2.40775
$356$	0	0
$357$	−2.14599e45	−0.0128016
$358$	0	0
$359$	−6.12637e46	−0.329576	−0.164788	−	0.986329i	$-0.552694\pi$
$359$	−6.12637e46	−0.329576	−0.164788	+	0.986329i	$0.552694\pi$
$360$	0	0
$361$	−2.05410e47	−0.997098
$362$	0	0
$363$	1.81329e46	0.0794685
$364$	0	0
$365$	4.08579e47	1.61755
$366$	0	0
$367$	−4.59083e47	−1.64274	−0.821371	−	0.570395i	$-0.806790\pi$
$367$	−4.59083e47	−1.64274	−0.821371	+	0.570395i	$0.806790\pi$
$368$	0	0
$369$	−4.13955e47	−1.33957
$370$	0	0
$371$	5.64928e47	1.65415
$372$	0	0
$373$	−1.54916e47	−0.410661	−0.205330	−	0.978693i	$-0.565827\pi$
$373$	−1.54916e47	−0.410661	−0.205330	+	0.978693i	$0.565827\pi$
$374$	0	0
$375$	−7.21672e46	−0.173285
$376$	0	0
$377$	6.62145e47	1.44091
$378$	0	0
$379$	3.30180e47	0.651516	0.325758	−	0.945453i	$-0.394381\pi$
$379$	3.30180e47	0.651516	0.325758	+	0.945453i	$0.394381\pi$
$380$	0	0
$381$	3.23138e47	0.578464
$382$	0	0
$383$	6.41516e46	0.104239	0.0521195	−	0.998641i	$-0.483402\pi$
$383$	6.41516e46	0.104239	0.0521195	+	0.998641i	$0.483402\pi$
$384$	0	0
$385$	7.95254e47	1.17350
$386$	0	0
$387$	−8.01767e47	−1.07497
$388$	0	0
$389$	−1.03221e48	−1.25806	−0.629030	−	0.777381i	$-0.716548\pi$
$389$	−1.03221e48	−1.25806	−0.629030	+	0.777381i	$0.716548\pi$
$390$	0	0
$391$	2.65563e46	0.0294372
$392$	0	0
$393$	3.07432e46	0.0310090
$394$	0	0
$395$	1.97515e48	1.81365
$396$	0	0
$397$	−4.62836e47	−0.387082	−0.193541	−	0.981092i	$-0.561997\pi$
$397$	−4.62836e47	−0.387082	−0.193541	+	0.981092i	$0.561997\pi$
$398$	0	0
$399$	2.57824e46	0.0196483
$400$	0	0
$401$	−1.52846e48	−1.06190	−0.530949	−	0.847404i	$-0.678164\pi$
$401$	−1.52846e48	−1.06190	−0.530949	+	0.847404i	$0.678164\pi$
$402$	0	0
$403$	−3.73736e48	−2.36822
$404$	0	0
$405$	−1.43763e48	−0.831243
$406$	0	0
$407$	2.34542e48	1.23801
$408$	0	0
$409$	−4.46279e47	−0.215142	−0.107571	−	0.994197i	$-0.534307\pi$
$409$	−4.46279e47	−0.215142	−0.107571	+	0.994197i	$0.534307\pi$
$410$	0	0
$411$	−4.40578e47	−0.194066
$412$	0	0
$413$	1.69254e48	0.681494
$414$	0	0
$415$	1.81327e48	0.667686
$416$	0	0
$417$	−1.92599e48	−0.648839
$418$	0	0
$419$	5.73559e48	1.76856	0.884278	−	0.466960i	$-0.154651\pi$
$419$	5.73559e48	1.76856	0.884278	+	0.466960i	$0.154651\pi$
$420$	0	0
$421$	−1.64196e48	−0.463600	−0.231800	−	0.972763i	$-0.574462\pi$
$421$	−1.64196e48	−0.463600	−0.231800	+	0.972763i	$0.574462\pi$
$422$	0	0
$423$	2.78925e48	0.721423
$424$	0	0
$425$	−8.94112e46	−0.0211932
$426$	0	0
$427$	9.60357e47	0.208697
$428$	0	0
$429$	2.12601e48	0.423745
$430$	0	0
$431$	−2.79980e48	−0.512033	−0.256017	−	0.966672i	$-0.582410\pi$
$431$	−2.79980e48	−0.512033	−0.256017	+	0.966672i	$0.582410\pi$
$432$	0	0
$433$	3.94771e48	0.662704	0.331352	−	0.943507i	$-0.392495\pi$
$433$	3.94771e48	0.662704	0.331352	+	0.943507i	$0.392495\pi$
$434$	0	0
$435$	−2.92397e48	−0.450735
$436$	0	0
$437$	−3.19053e47	−0.0451811
$438$	0	0
$439$	−2.54774e48	−0.331559	−0.165780	−	0.986163i	$-0.553014\pi$
$439$	−2.54774e48	−0.331559	−0.165780	+	0.986163i	$0.553014\pi$
$440$	0	0
$441$	−8.48310e47	−0.101494
$442$	0	0
$443$	−6.78938e48	−0.747070	−0.373535	−	0.927616i	$-0.621854\pi$
$443$	−6.78938e48	−0.747070	−0.373535	+	0.927616i	$0.621854\pi$
$444$	0	0
$445$	−1.03726e49	−1.05009
$446$	0	0
$447$	−5.10331e48	−0.475513
$448$	0	0
$449$	−5.78360e48	−0.496182	−0.248091	−	0.968737i	$-0.579803\pi$
$449$	−5.78360e48	−0.496182	−0.248091	+	0.968737i	$0.579803\pi$
$450$	0	0
$451$	1.68895e49	1.33460
$452$	0	0
$453$	6.14532e48	0.447433
$454$	0	0
$455$	−2.78653e49	−1.87005
$456$	0	0
$457$	1.70414e49	1.05452	0.527260	−	0.849704i	$-0.323219\pi$
$457$	1.70414e49	1.05452	0.527260	+	0.849704i	$0.323219\pi$
$458$	0	0
$459$	−3.99428e47	−0.0227984
$460$	0	0
$461$	8.95615e48	0.471687	0.235844	−	0.971791i	$-0.424215\pi$
$461$	8.95615e48	0.471687	0.235844	+	0.971791i	$0.424215\pi$
$462$	0	0
$463$	7.71924e46	0.00375255	0.00187627	−	0.999998i	$-0.499403\pi$
$463$	7.71924e46	0.00375255	0.00187627	+	0.999998i	$0.499403\pi$
$464$	0	0
$465$	1.65038e49	0.740807
$466$	0	0
$467$	9.47410e47	0.0392804	0.0196402	−	0.999807i	$-0.493748\pi$
$467$	9.47410e47	0.0392804	0.0196402	+	0.999807i	$0.493748\pi$
$468$	0	0
$469$	1.65565e49	0.634264
$470$	0	0
$471$	5.54835e48	0.196461
$472$	0	0
$473$	3.27123e49	1.07098
$474$	0	0
$475$	1.07421e48	0.0325279
$476$	0	0
$477$	4.92398e49	1.37951
$478$	0	0
$479$	−4.93800e49	−1.28040	−0.640198	−	0.768210i	$-0.721148\pi$
$479$	−4.93800e49	−1.28040	−0.640198	+	0.768210i	$0.721148\pi$
$480$	0	0
$481$	−8.21823e49	−1.97285
$482$	0	0
$483$	−1.37598e49	−0.305907
$484$	0	0
$485$	−7.10246e49	−1.46281
$486$	0	0
$487$	5.14970e49	0.982871	0.491435	−	0.870914i	$-0.336472\pi$
$487$	5.14970e49	0.982871	0.491435	+	0.870914i	$0.336472\pi$
$488$	0	0
$489$	2.24685e49	0.397522
$490$	0	0
$491$	−2.61978e49	−0.429792	−0.214896	−	0.976637i	$-0.568941\pi$
$491$	−2.61978e49	−0.429792	−0.214896	+	0.976637i	$0.568941\pi$
$492$	0	0
$493$	2.37685e48	0.0361688
$494$	0	0
$495$	6.93153e49	0.978662
$496$	0	0
$497$	−1.53218e50	−2.00778
$498$	0	0
$499$	3.10414e49	0.377643	0.188822	−	0.982011i	$-0.439533\pi$
$499$	3.10414e49	0.377643	0.188822	+	0.982011i	$0.439533\pi$
$500$	0	0
$501$	−1.05335e49	−0.119008
$502$	0	0
$503$	−1.24708e50	−1.30884	−0.654420	−	0.756131i	$-0.727087\pi$
$503$	−1.24708e50	−1.30884	−0.654420	+	0.756131i	$0.727087\pi$
$504$	0	0
$505$	1.50234e50	1.46514
$506$	0	0
$507$	−3.63927e49	−0.329888
$508$	0	0
$509$	1.23242e50	1.03868	0.519338	−	0.854569i	$-0.326178\pi$
$509$	1.23242e50	1.03868	0.519338	+	0.854569i	$0.326178\pi$
$510$	0	0
$511$	1.72088e50	1.34885
$512$	0	0
$513$	4.79883e48	0.0349916
$514$	0	0
$515$	2.17302e50	1.47445
$516$	0	0
$517$	−1.13802e50	−0.718745
$518$	0	0
$519$	−9.56170e49	−0.562262
$520$	0	0
$521$	−1.52770e50	−0.836641	−0.418320	−	0.908300i	$-0.637381\pi$
$521$	−1.52770e50	−0.836641	−0.418320	+	0.908300i	$0.637381\pi$
$522$	0	0
$523$	2.24295e50	1.14430	0.572148	−	0.820151i	$-0.306110\pi$
$523$	2.24295e50	1.14430	0.572148	+	0.820151i	$0.306110\pi$
$524$	0	0
$525$	4.63272e49	0.220236
$526$	0	0
$527$	−1.34157e49	−0.0594454
$528$	0	0
$529$	−7.17889e49	−0.296570
$530$	0	0
$531$	1.47523e50	0.568346
$532$	0	0
$533$	−5.91799e50	−2.12677
$534$	0	0
$535$	−3.32236e50	−1.11404
$536$	0	0
$537$	1.42869e50	0.447110
$538$	0	0
$539$	3.46113e49	0.101117
$540$	0	0
$541$	−4.16928e49	−0.113739	−0.0568697	−	0.998382i	$-0.518112\pi$
$541$	−4.16928e49	−0.113739	−0.0568697	+	0.998382i	$0.518112\pi$
$542$	0	0
$543$	−1.06555e50	−0.271505
$544$	0	0
$545$	−1.01824e50	−0.242390
$546$	0	0
$547$	−2.87730e50	−0.640059	−0.320029	−	0.947408i	$-0.603693\pi$
$547$	−2.87730e50	−0.640059	−0.320029	+	0.947408i	$0.603693\pi$
$548$	0	0
$549$	8.37059e49	0.174047
$550$	0	0
$551$	−2.85561e49	−0.0555130
$552$	0	0
$553$	8.31905e50	1.51237
$554$	0	0
$555$	3.62909e50	0.617131
$556$	0	0
$557$	1.05807e51	1.68343	0.841715	−	0.539922i	$-0.181546\pi$
$557$	1.05807e51	1.68343	0.841715	+	0.539922i	$0.181546\pi$
$558$	0	0
$559$	−1.14622e51	−1.70667
$560$	0	0
$561$	7.63157e48	0.0106366
$562$	0	0
$563$	8.79551e50	1.14777	0.573886	−	0.818935i	$-0.305435\pi$
$563$	8.79551e50	1.14777	0.573886	+	0.818935i	$0.305435\pi$
$564$	0	0
$565$	−2.20183e50	−0.269084
$566$	0	0
$567$	−6.05509e50	−0.693160
$568$	0	0
$569$	3.05710e50	0.327894	0.163947	−	0.986469i	$-0.447577\pi$
$569$	3.05710e50	0.327894	0.163947	+	0.986469i	$0.447577\pi$
$570$	0	0
$571$	1.13121e51	1.13704	0.568521	−	0.822668i	$-0.307516\pi$
$571$	1.13121e51	1.13704	0.568521	+	0.822668i	$0.307516\pi$
$572$	0	0
$573$	7.30957e49	0.0688702
$574$	0	0
$575$	−5.73292e50	−0.506431
$576$	0	0
$577$	2.65515e50	0.219957	0.109978	−	0.993934i	$-0.464922\pi$
$577$	2.65515e50	0.219957	0.109978	+	0.993934i	$0.464922\pi$
$578$	0	0
$579$	−2.88701e50	−0.224334
$580$	0	0
$581$	7.63725e50	0.556773
$582$	0	0
$583$	−2.00900e51	−1.37439
$584$	0	0
$585$	−2.42877e51	−1.55956
$586$	0	0
$587$	−1.35709e51	−0.818093	−0.409047	−	0.912513i	$-0.634139\pi$
$587$	−1.35709e51	−0.818093	−0.409047	+	0.912513i	$0.634139\pi$
$588$	0	0
$589$	1.61180e50	0.0912385
$590$	0	0
$591$	−1.15793e51	−0.615624
$592$	0	0
$593$	1.19816e51	0.598418	0.299209	−	0.954188i	$-0.403277\pi$
$593$	1.19816e51	0.598418	0.299209	+	0.954188i	$0.403277\pi$
$594$	0	0
$595$	−1.00026e50	−0.0469407
$596$	0	0
$597$	−3.65764e50	−0.161315
$598$	0	0
$599$	3.71203e51	1.53892	0.769458	−	0.638698i	$-0.220526\pi$
$599$	3.71203e51	1.53892	0.769458	+	0.638698i	$0.220526\pi$
$600$	0	0
$601$	2.98939e49	0.0116521	0.00582605	−	0.999983i	$-0.498146\pi$
$601$	2.98939e49	0.0116521	0.00582605	+	0.999983i	$0.498146\pi$
$602$	0	0
$603$	1.44308e51	0.528957
$604$	0	0
$605$	8.45187e50	0.291393
$606$	0	0
$607$	−1.82791e51	−0.592875	−0.296438	−	0.955052i	$-0.595799\pi$
$607$	−1.82791e51	−0.592875	−0.296438	+	0.955052i	$0.595799\pi$
$608$	0	0
$609$	−1.23154e51	−0.375861
$610$	0	0
$611$	3.98756e51	1.14537
$612$	0	0
$613$	−3.48635e51	−0.942654	−0.471327	−	0.881959i	$-0.656225\pi$
$613$	−3.48635e51	−0.942654	−0.471327	+	0.881959i	$0.656225\pi$
$614$	0	0
$615$	2.61333e51	0.665279
$616$	0	0
$617$	6.90579e51	1.65553	0.827766	−	0.561073i	$-0.189611\pi$
$617$	6.90579e51	1.65553	0.827766	+	0.561073i	$0.189611\pi$
$618$	0	0
$619$	6.48084e51	1.46337	0.731685	−	0.681643i	$-0.238734\pi$
$619$	6.48084e51	1.46337	0.731685	+	0.681643i	$0.238734\pi$
$620$	0	0
$621$	−2.56108e51	−0.544789
$622$	0	0
$623$	−4.36881e51	−0.875658
$624$	0	0
$625$	−6.56038e51	−1.23922
$626$	0	0
$627$	−9.16875e49	−0.0163253
$628$	0	0
$629$	−2.95004e50	−0.0495212
$630$	0	0
$631$	−9.04373e51	−1.43154	−0.715772	−	0.698335i	$-0.753925\pi$
$631$	−9.04373e51	−1.43154	−0.715772	+	0.698335i	$0.753925\pi$
$632$	0	0
$633$	−3.17135e51	−0.473451
$634$	0	0
$635$	1.50617e52	2.12109
$636$	0	0
$637$	−1.21276e51	−0.161137
$638$	0	0
$639$	−1.33547e52	−1.67443
$640$	0	0
$641$	−1.02640e52	−1.21463	−0.607314	−	0.794462i	$-0.707753\pi$
$641$	−1.02640e52	−1.21463	−0.607314	+	0.794462i	$0.707753\pi$
$642$	0	0
$643$	1.12581e52	1.25766	0.628830	−	0.777543i	$-0.283534\pi$
$643$	1.12581e52	1.25766	0.628830	+	0.777543i	$0.283534\pi$
$644$	0	0
$645$	5.06160e51	0.533867
$646$	0	0
$647$	−1.32899e52	−1.32370	−0.661852	−	0.749634i	$-0.730229\pi$
$647$	−1.32899e52	−1.32370	−0.661852	+	0.749634i	$0.730229\pi$
$648$	0	0
$649$	−6.01900e51	−0.566236
$650$	0	0
$651$	6.95119e51	0.617747
$652$	0	0
$653$	−1.11156e51	−0.0933335	−0.0466668	−	0.998911i	$-0.514860\pi$
$653$	−1.11156e51	−0.0933335	−0.0466668	+	0.998911i	$0.514860\pi$
$654$	0	0
$655$	1.43296e51	0.113703
$656$	0	0
$657$	1.49994e52	1.12490
$658$	0	0
$659$	2.43589e51	0.172694	0.0863469	−	0.996265i	$-0.472481\pi$
$659$	2.43589e51	0.172694	0.0863469	+	0.996265i	$0.472481\pi$
$660$	0	0
$661$	2.08839e52	1.39986	0.699929	−	0.714212i	$-0.253215\pi$
$661$	2.08839e52	1.39986	0.699929	+	0.714212i	$0.253215\pi$
$662$	0	0
$663$	−2.67406e50	−0.0169501
$664$	0	0
$665$	1.20174e51	0.0720459
$666$	0	0
$667$	1.52401e52	0.864288
$668$	0	0
$669$	−4.68620e51	−0.251442
$670$	0	0
$671$	−3.41522e51	−0.173401
$672$	0	0
$673$	8.00239e51	0.384540	0.192270	−	0.981342i	$-0.438415\pi$
$673$	8.00239e51	0.384540	0.192270	+	0.981342i	$0.438415\pi$
$674$	0	0
$675$	8.62278e51	0.392218
$676$	0	0
$677$	−1.17793e52	−0.507258	−0.253629	−	0.967302i	$-0.581624\pi$
$677$	−1.17793e52	−0.507258	−0.253629	+	0.967302i	$0.581624\pi$
$678$	0	0
$679$	−2.99146e52	−1.21981
$680$	0	0
$681$	−9.76051e50	−0.0376922
$682$	0	0
$683$	−3.05105e52	−1.11600	−0.558002	−	0.829840i	$-0.688432\pi$
$683$	−3.05105e52	−1.11600	−0.558002	+	0.829840i	$0.688432\pi$
$684$	0	0
$685$	−2.05356e52	−0.711594
$686$	0	0
$687$	−7.46836e51	−0.245203
$688$	0	0
$689$	7.03943e52	2.19019
$690$	0	0
$691$	−1.38514e52	−0.408459	−0.204230	−	0.978923i	$-0.565469\pi$
$691$	−1.38514e52	−0.408459	−0.204230	+	0.978923i	$0.565469\pi$
$692$	0	0
$693$	2.91947e52	0.816091
$694$	0	0
$695$	−8.97717e52	−2.37914
$696$	0	0
$697$	−2.12434e51	−0.0533848
$698$	0	0
$699$	−3.07459e51	−0.0732755
$700$	0	0
$701$	−4.94343e52	−1.11749	−0.558747	−	0.829338i	$-0.688718\pi$
$701$	−4.94343e52	−1.11749	−0.558747	+	0.829338i	$0.688718\pi$
$702$	0	0
$703$	3.54425e51	0.0760065
$704$	0	0
$705$	−1.76087e52	−0.358285
$706$	0	0
$707$	6.32766e52	1.22175
$708$	0	0
$709$	6.70722e52	1.22910	0.614550	−	0.788878i	$-0.289337\pi$
$709$	6.70722e52	1.22910	0.614550	+	0.788878i	$0.289337\pi$
$710$	0	0
$711$	7.25099e52	1.26127
$712$	0	0
$713$	−8.60198e52	−1.42050
$714$	0	0
$715$	9.90946e52	1.55377
$716$	0	0
$717$	−1.27254e52	−0.189482
$718$	0	0
$719$	−1.50366e52	−0.212649	−0.106325	−	0.994331i	$-0.533908\pi$
$719$	−1.50366e52	−0.212649	−0.106325	+	0.994331i	$0.533908\pi$
$720$	0	0
$721$	9.15249e52	1.22952
$722$	0	0
$723$	3.05324e52	0.389675
$724$	0	0
$725$	−5.13111e52	−0.622240
$726$	0	0
$727$	−4.05250e52	−0.467021	−0.233510	−	0.972354i	$-0.575021\pi$
$727$	−4.05250e52	−0.467021	−0.233510	+	0.972354i	$0.575021\pi$
$728$	0	0
$729$	−3.22949e52	−0.353732
$730$	0	0
$731$	−4.11451e51	−0.0428397
$732$	0	0
$733$	7.49902e52	0.742303	0.371151	−	0.928572i	$-0.378963\pi$
$733$	7.49902e52	0.742303	0.371151	+	0.928572i	$0.378963\pi$
$734$	0	0
$735$	5.35544e51	0.0504056
$736$	0	0
$737$	−5.88781e52	−0.526994
$738$	0	0
$739$	3.81728e52	0.324962	0.162481	−	0.986712i	$-0.448050\pi$
$739$	3.81728e52	0.324962	0.162481	+	0.986712i	$0.448050\pi$
$740$	0	0
$741$	3.21269e51	0.0260154
$742$	0	0
$743$	−1.98672e53	−1.53054	−0.765269	−	0.643710i	$-0.777394\pi$
$743$	−1.98672e53	−1.53054	−0.765269	+	0.643710i	$0.777394\pi$
$744$	0	0
$745$	−2.37869e53	−1.74360
$746$	0	0
$747$	6.65672e52	0.464332
$748$	0	0
$749$	−1.39933e53	−0.928982
$750$	0	0
$751$	−8.32961e52	−0.526364	−0.263182	−	0.964746i	$-0.584772\pi$
$751$	−8.32961e52	−0.526364	−0.263182	+	0.964746i	$0.584772\pi$
$752$	0	0
$753$	3.88014e52	0.233421
$754$	0	0
$755$	2.86438e53	1.64063
$756$	0	0
$757$	−4.52303e52	−0.246692	−0.123346	−	0.992364i	$-0.539363\pi$
$757$	−4.52303e52	−0.246692	−0.123346	+	0.992364i	$0.539363\pi$
$758$	0	0
$759$	4.89325e52	0.254170
$760$	0	0
$761$	1.94307e53	0.961329	0.480664	−	0.876905i	$-0.340396\pi$
$761$	1.94307e53	0.961329	0.480664	+	0.876905i	$0.340396\pi$
$762$	0	0
$763$	−4.28869e52	−0.202126
$764$	0	0
$765$	−8.71839e51	−0.0391471
$766$	0	0
$767$	2.10903e53	0.902335
$768$	0	0
$769$	−2.45326e53	−1.00025	−0.500123	−	0.865954i	$-0.666712\pi$
$769$	−2.45326e53	−1.00025	−0.500123	+	0.865954i	$0.666712\pi$
$770$	0	0
$771$	−4.37478e52	−0.170000
$772$	0	0
$773$	−1.97446e52	−0.0731356	−0.0365678	−	0.999331i	$-0.511642\pi$
$773$	−1.97446e52	−0.0731356	−0.0365678	+	0.999331i	$0.511642\pi$
$774$	0	0
$775$	2.89617e53	1.02268
$776$	0	0
$777$	1.52852e53	0.514616
$778$	0	0
$779$	2.55223e52	0.0819365
$780$	0	0
$781$	5.44874e53	1.66821
$782$	0	0
$783$	−2.29223e53	−0.669369
$784$	0	0
$785$	2.58612e53	0.720379
$786$	0	0
$787$	−4.81731e53	−1.28018	−0.640091	−	0.768299i	$-0.721103\pi$
$787$	−4.81731e53	−1.28018	−0.640091	+	0.768299i	$0.721103\pi$
$788$	0	0
$789$	−3.42069e52	−0.0867339
$790$	0	0
$791$	−9.27383e52	−0.224385
$792$	0	0
$793$	1.19668e53	0.276327
$794$	0	0
$795$	−3.10854e53	−0.685117
$796$	0	0
$797$	−1.66157e53	−0.349575	−0.174788	−	0.984606i	$-0.555924\pi$
$797$	−1.66157e53	−0.349575	−0.174788	+	0.984606i	$0.555924\pi$
$798$	0	0
$799$	1.43139e52	0.0287503
$800$	0	0
$801$	−3.80791e53	−0.730273
$802$	0	0
$803$	−6.11980e53	−1.12073
$804$	0	0
$805$	−6.41353e53	−1.12169
$806$	0	0
$807$	−1.70726e53	−0.285194
$808$	0	0
$809$	7.10187e53	1.13325	0.566624	−	0.823976i	$-0.308249\pi$
$809$	7.10187e53	1.13325	0.566624	+	0.823976i	$0.308249\pi$
$810$	0	0
$811$	−5.23793e52	−0.0798498	−0.0399249	−	0.999203i	$-0.512712\pi$
$811$	−5.23793e52	−0.0798498	−0.0399249	+	0.999203i	$0.512712\pi$
$812$	0	0
$813$	−5.54858e52	−0.0808178
$814$	0	0
$815$	1.04727e54	1.45762
$816$	0	0
$817$	4.94327e52	0.0657516
$818$	0	0
$819$	−1.02297e54	−1.30050
$820$	0	0
$821$	1.36447e54	1.65811	0.829057	−	0.559164i	$-0.188878\pi$
$821$	1.36447e54	1.65811	0.829057	+	0.559164i	$0.188878\pi$
$822$	0	0
$823$	−4.74606e53	−0.551361	−0.275680	−	0.961249i	$-0.588903\pi$
$823$	−4.74606e53	−0.551361	−0.275680	+	0.961249i	$0.588903\pi$
$824$	0	0
$825$	−1.64749e53	−0.182989
$826$	0	0
$827$	6.60680e53	0.701680	0.350840	−	0.936435i	$-0.385896\pi$
$827$	6.60680e53	0.701680	0.350840	+	0.936435i	$0.385896\pi$
$828$	0	0
$829$	3.19800e53	0.324803	0.162402	−	0.986725i	$-0.448076\pi$
$829$	3.19800e53	0.324803	0.162402	+	0.986725i	$0.448076\pi$
$830$	0	0
$831$	2.78234e53	0.270266
$832$	0	0
$833$	−4.35336e51	−0.00404476
$834$	0	0
$835$	−4.90974e53	−0.436374
$836$	0	0
$837$	1.29381e54	1.10014
$838$	0	0
$839$	2.08226e54	1.69410	0.847048	−	0.531516i	$-0.178378\pi$
$839$	2.08226e54	1.69410	0.847048	+	0.531516i	$0.178378\pi$
$840$	0	0
$841$	7.95482e52	0.0619305
$842$	0	0
$843$	−3.03950e53	−0.226460
$844$	0	0
$845$	−1.69629e54	−1.20962
$846$	0	0
$847$	3.55982e53	0.242988
$848$	0	0
$849$	−2.98254e53	−0.194892
$850$	0	0
$851$	−1.89152e54	−1.18335
$852$	0	0
$853$	−3.18441e54	−1.90753	−0.953766	−	0.300550i	$-0.902830\pi$
$853$	−3.18441e54	−1.90753	−0.953766	+	0.300550i	$0.902830\pi$
$854$	0	0
$855$	1.04745e53	0.0600841
$856$	0	0
$857$	1.50624e54	0.827463	0.413732	−	0.910399i	$-0.364225\pi$
$857$	1.50624e54	0.827463	0.413732	+	0.910399i	$0.364225\pi$
$858$	0	0
$859$	1.69349e54	0.891067	0.445533	−	0.895265i	$-0.353014\pi$
$859$	1.69349e54	0.891067	0.445533	+	0.895265i	$0.353014\pi$
$860$	0	0
$861$	1.10070e54	0.554766
$862$	0	0
$863$	6.71066e52	0.0324015	0.0162008	−	0.999869i	$-0.494843\pi$
$863$	6.71066e52	0.0324015	0.0162008	+	0.999869i	$0.494843\pi$
$864$	0	0
$865$	−4.45677e54	−2.06169
$866$	0	0
$867$	7.78238e53	0.344953
$868$	0	0
$869$	−2.95842e54	−1.25659
$870$	0	0
$871$	2.06306e54	0.839800
$872$	0	0
$873$	−2.60739e54	−1.01729
$874$	0	0
$875$	−1.41677e54	−0.529848
$876$	0	0
$877$	2.24921e54	0.806375	0.403187	−	0.915117i	$-0.367902\pi$
$877$	2.24921e54	0.806375	0.403187	+	0.915117i	$0.367902\pi$
$878$	0	0
$879$	−6.30921e53	−0.216861
$880$	0	0
$881$	4.31934e54	1.42352	0.711759	−	0.702423i	$-0.247899\pi$
$881$	4.31934e54	1.42352	0.711759	+	0.702423i	$0.247899\pi$
$882$	0	0
$883$	5.08099e53	0.160574	0.0802870	−	0.996772i	$-0.474416\pi$
$883$	5.08099e53	0.160574	0.0802870	+	0.996772i	$0.474416\pi$
$884$	0	0
$885$	−9.31325e53	−0.282261
$886$	0	0
$887$	1.25491e54	0.364775	0.182388	−	0.983227i	$-0.441617\pi$
$887$	1.25491e54	0.364775	0.182388	+	0.983227i	$0.441617\pi$
$888$	0	0
$889$	6.34379e54	1.76875
$890$	0	0
$891$	2.15331e54	0.575929
$892$	0	0
$893$	−1.71970e53	−0.0441267
$894$	0	0
$895$	6.65923e54	1.63945
$896$	0	0
$897$	−1.71457e54	−0.405038
$898$	0	0
$899$	−7.69900e54	−1.74534
$900$	0	0
$901$	2.52689e53	0.0549766
$902$	0	0
$903$	2.13188e54	0.445184
$904$	0	0
$905$	−4.96661e54	−0.995547
$906$	0	0
$907$	−8.42855e54	−1.62188	−0.810939	−	0.585131i	$-0.801043\pi$
$907$	−8.42855e54	−1.62188	−0.810939	+	0.585131i	$0.801043\pi$
$908$	0	0
$909$	5.51527e54	1.01891
$910$	0	0
$911$	8.47562e54	1.50342	0.751712	−	0.659491i	$-0.229228\pi$
$911$	8.47562e54	1.50342	0.751712	+	0.659491i	$0.229228\pi$
$912$	0	0
$913$	−2.71596e54	−0.462609
$914$	0	0
$915$	−5.28440e53	−0.0864382
$916$	0	0
$917$	6.03545e53	0.0948149
$918$	0	0
$919$	−5.38417e54	−0.812423	−0.406211	−	0.913779i	$-0.633150\pi$
$919$	−5.38417e54	−0.812423	−0.406211	+	0.913779i	$0.633150\pi$
$920$	0	0
$921$	−1.53629e54	−0.222675
$922$	0	0
$923$	−1.90921e55	−2.65841
$924$	0	0
$925$	6.36849e54	0.851950
$926$	0	0
$927$	7.97742e54	1.02539
$928$	0	0
$929$	−7.45967e54	−0.921360	−0.460680	−	0.887566i	$-0.652394\pi$
$929$	−7.45967e54	−0.921360	−0.460680	+	0.887566i	$0.652394\pi$
$930$	0	0
$931$	5.23024e52	0.00620801
$932$	0	0
$933$	−5.65957e54	−0.645615
$934$	0	0
$935$	3.55713e53	0.0390018
$936$	0	0
$937$	8.19621e54	0.863835	0.431917	−	0.901913i	$-0.357837\pi$
$937$	8.19621e54	0.863835	0.431917	+	0.901913i	$0.357837\pi$
$938$	0	0
$939$	4.21537e54	0.427093
$940$	0	0
$941$	−6.20640e53	−0.0604550	−0.0302275	−	0.999543i	$-0.509623\pi$
$941$	−6.20640e53	−0.0604550	−0.0302275	+	0.999543i	$0.509623\pi$
$942$	0	0
$943$	−1.36210e55	−1.27568
$944$	0	0
$945$	9.64647e54	0.868721
$946$	0	0
$947$	7.79896e54	0.675401	0.337701	−	0.941254i	$-0.390351\pi$
$947$	7.79896e54	0.675401	0.337701	+	0.941254i	$0.390351\pi$
$948$	0	0
$949$	2.14435e55	1.78595
$950$	0	0
$951$	1.35884e54	0.108850
$952$	0	0
$953$	−3.57034e54	−0.275100	−0.137550	−	0.990495i	$-0.543923\pi$
$953$	−3.57034e54	−0.275100	−0.137550	+	0.990495i	$0.543923\pi$
$954$	0	0
$955$	3.40704e54	0.252531
$956$	0	0
$957$	4.37959e54	0.312293
$958$	0	0
$959$	−8.64933e54	−0.593387
$960$	0	0
$961$	2.83067e55	1.86856
$962$	0	0
$963$	−1.21968e55	−0.774743
$964$	0	0
$965$	−1.34566e55	−0.822581
$966$	0	0
$967$	−5.24792e54	−0.308743	−0.154372	−	0.988013i	$-0.549335\pi$
$967$	−5.24792e54	−0.308743	−0.154372	+	0.988013i	$0.549335\pi$
$968$	0	0
$969$	1.15323e52	0.000653022 0
$970$	0	0
$971$	3.07962e55	1.67858	0.839288	−	0.543688i	$-0.182972\pi$
$971$	3.07962e55	1.67858	0.839288	+	0.543688i	$0.182972\pi$
$972$	0	0
$973$	−3.78106e55	−1.98393
$974$	0	0
$975$	5.77272e54	0.291605
$976$	0	0
$977$	2.24607e54	0.109238	0.0546191	−	0.998507i	$-0.482606\pi$
$977$	2.24607e54	0.109238	0.0546191	+	0.998507i	$0.482606\pi$
$978$	0	0
$979$	1.55364e55	0.727562
$980$	0	0
$981$	−3.73808e54	−0.168567
$982$	0	0
$983$	−3.49118e55	−1.51611	−0.758057	−	0.652188i	$-0.773851\pi$
$983$	−3.49118e55	−1.51611	−0.758057	+	0.652188i	$0.773851\pi$
$984$	0	0
$985$	−5.39721e55	−2.25735
$986$	0	0
$987$	−7.41654e54	−0.298768
$988$	0	0
$989$	−2.63817e55	−1.02370
$990$	0	0
$991$	4.70743e55	1.75963	0.879815	−	0.475317i	$-0.157667\pi$
$991$	4.70743e55	1.75963	0.879815	+	0.475317i	$0.157667\pi$
$992$	0	0
$993$	−7.47673e54	−0.269247
$994$	0	0
$995$	−1.70485e55	−0.591507
$996$	0	0
$997$	2.07899e55	0.695012	0.347506	−	0.937678i	$-0.387029\pi$
$997$	2.07899e55	0.695012	0.347506	+	0.937678i	$0.387029\pi$
$998$	0	0
$999$	2.84501e55	0.916477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.38.a.a.1.2	✓	3
4.3	odd	2		16.38.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.38.a.a.1.2	✓	3	1.1	even	1	trivial
16.38.a.d.1.2		3	4.3	odd	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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 38 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31760e8 q^{3} -1.08025e13 q^{5} -4.54986e15 q^{7} -3.96571e17 q^{9} +O(q^{10})\)	\(q-2.31760e8 q^{3} -1.08025e13 q^{5} -4.54986e15 q^{7} -3.96571e17 q^{9} +1.61802e19 q^{11} -5.66947e20 q^{13} +2.50358e21 q^{15} -2.03513e21 q^{17} +2.44505e22 q^{19} +1.05447e24 q^{21} -1.30489e25 q^{23} +4.39339e25 q^{25} +1.96267e26 q^{27} -1.16791e27 q^{29} +6.59209e27 q^{31} -3.74992e27 q^{33} +4.91498e28 q^{35} +1.44956e29 q^{37} +1.31395e29 q^{39} +1.04384e30 q^{41} +2.02175e30 q^{43} +4.28395e30 q^{45} -7.03340e30 q^{47} +2.13911e30 q^{49} +4.71661e29 q^{51} -1.24164e32 q^{53} -1.74786e32 q^{55} -5.66664e30 q^{57} -3.71997e32 q^{59} -2.11074e32 q^{61} +1.80434e33 q^{63} +6.12443e33 q^{65} -3.63890e33 q^{67} +3.02422e33 q^{69} +3.36753e34 q^{71} -3.78227e34 q^{73} -1.01821e34 q^{75} -7.36177e34 q^{77} -1.82842e35 q^{79} +1.33083e35 q^{81} -1.67857e35 q^{83} +2.19844e34 q^{85} +2.70676e35 q^{87} +9.60208e35 q^{89} +2.57953e36 q^{91} -1.52778e36 q^{93} -2.64126e35 q^{95} +6.57484e36 q^{97} -6.41661e36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots + 10\!\cdots\!63 q^{9}+O(q^{10}) \) 3 * q - 272163492 * q^3 + 3641045116194 * q^5 + 1514184507161592 * q^7 + 102953932588056663 * q^9	\( 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots - 14\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q - 272163492 * q^3 + 3641045116194 * q^5 + 1514184507161592 * q^7 + 102953932588056663 * q^9 - 11609057084515238700 * q^11 - 70576684907789880678 * q^13 - 2933969461740641207448 * q^15 + 21449865005452482229494 * q^17 + 158301069447345491378508 * q^19 - 1151851369077659765142432 * q^21 + 866923505554430216912616 * q^23 + 21645722139716923137352461 * q^25 + 147832683778944657999655704 * q^27 + 1010653042580699333097500298 * q^29 + 2180097844230541775332759008 * q^31 + 31048498538042201781617598480 * q^33 + 100605979662426959198661319248 * q^35 + 311298722814240927017686969602 * q^37 + 979152159930324950801131779528 * q^39 + 2635940958803787572844371014878 * q^41 + 5419069191259593947069584358580 * q^43 + 8099313551230663632839319909594 * q^45 + 4156587745521390586896893605584 * q^47 - 13495549806538130861979895910709 * q^49 - 89987135087681316961287710640264 * q^51 - 159108065600661790773554978344206 * q^53 - 501378639325706351372809856249160 * q^55 - 698546758941918182526174576638352 * q^57 - 540908935574431553277616956343644 * q^59 + 1178713661089330885859771001735786 * q^61 + 3403110131746397253746662721587992 * q^63 + 6554640894858129925078606186828668 * q^65 + 16889218571987157419703623848158492 * q^67 + 17143397087023799436715277896360224 * q^69 + 10309319797367225088925816018968504 * q^71 - 8520196523581770817793340839438658 * q^73 - 84054788877364659786396368345168412 * q^75 - 208861611906418376677492203082464480 * q^77 - 264775084715781163780628803947993936 * q^79 - 370309441007212745944873457987660853 * q^81 - 68956651634275475176080339084470004 * q^83 + 522499848715075310236425753489988836 * q^85 + 911140443393561025920716736778215048 * q^87 + 2493391276449702933740991498959962542 * q^89 + 2807109998185271068204514151779860752 * q^91 + 5409735857664632612107028249216969088 * q^93 + 3240594516611672218958902717719033864 * q^95 - 4697277125747917937176391731430481498 * q^97 - 14740636216794395787136495035497749500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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