LMFDB - Embedded newform 4.38.a.a.1.1

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			1.1
Root			$332433.$ 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)

$f(q)$	$=$	$q-8.56646e8 q^{3} +1.02977e13 q^{5} +4.27767e15 q^{7} +2.83558e17 q^{9} +O(q^{10})$	$q-8.56646e8 q^{3} +1.02977e13 q^{5} +4.27767e15 q^{7} +2.83558e17 q^{9} -3.43605e19 q^{11} -2.64572e20 q^{13} -8.82151e21 q^{15} +6.55323e22 q^{17} +4.79491e23 q^{19} -3.66444e24 q^{21} -1.65012e24 q^{23} +3.32838e25 q^{25} +1.42825e26 q^{27} +6.80125e26 q^{29} -6.29975e27 q^{31} +2.94347e28 q^{33} +4.40503e28 q^{35} +2.92536e28 q^{37} +2.26644e29 q^{39} +1.17322e30 q^{41} +2.37316e30 q^{43} +2.92001e30 q^{45} -4.28443e30 q^{47} -2.63698e29 q^{49} -5.61380e31 q^{51} +1.05927e32 q^{53} -3.53835e32 q^{55} -4.10754e32 q^{57} -3.39980e32 q^{59} +1.98820e33 q^{61} +1.21297e33 q^{63} -2.72449e33 q^{65} +1.05982e34 q^{67} +1.41357e33 q^{69} +2.78180e33 q^{71} +2.22277e34 q^{73} -2.85124e34 q^{75} -1.46983e35 q^{77} -3.94375e34 q^{79} -2.50032e35 q^{81} +2.31621e35 q^{83} +6.74835e35 q^{85} -5.82626e35 q^{87} +7.56957e35 q^{89} -1.13175e36 q^{91} +5.39665e36 q^{93} +4.93767e36 q^{95} -6.01498e36 q^{97} -9.74319e36 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−8.56646e8	−1.27661	−0.638305	−	0.769784i	$-0.720364\pi$
$3$	−8.56646e8	−1.27661	−0.638305	+	0.769784i	$0.720364\pi$
$4$	0	0
$5$	1.02977e13	1.20725	0.603624	−	0.797269i	$-0.293723\pi$
$5$	1.02977e13	1.20725	0.603624	+	0.797269i	$0.293723\pi$
$6$	0	0
$7$	4.27767e15	0.992871	0.496436	−	0.868073i	$-0.334642\pi$
$7$	4.27767e15	0.992871	0.496436	+	0.868073i	$0.334642\pi$
$8$	0	0
$9$	2.83558e17	0.629732
$10$	0	0
$11$	−3.43605e19	−1.86335	−0.931675	−	0.363294i	$-0.881652\pi$
$11$	−3.43605e19	−1.86335	−0.931675	+	0.363294i	$0.881652\pi$
$12$	0	0
$13$	−2.64572e20	−0.652516	−0.326258	−	0.945281i	$-0.605788\pi$
$13$	−2.64572e20	−0.652516	−0.326258	+	0.945281i	$0.605788\pi$
$14$	0	0
$15$	−8.82151e21	−1.54119
$16$	0	0
$17$	6.55323e22	1.13019	0.565094	−	0.825027i	$-0.308840\pi$
$17$	6.55323e22	1.13019	0.565094	+	0.825027i	$0.308840\pi$
$18$	0	0
$19$	4.79491e23	1.05643	0.528213	−	0.849112i	$-0.322862\pi$
$19$	4.79491e23	1.05643	0.528213	+	0.849112i	$0.322862\pi$
$20$	0	0
$21$	−3.66444e24	−1.26751
$22$	0	0
$23$	−1.65012e24	−0.106060	−0.0530298	−	0.998593i	$-0.516888\pi$
$23$	−1.65012e24	−0.106060	−0.0530298	+	0.998593i	$0.516888\pi$
$24$	0	0
$25$	3.32838e25	0.457449
$26$	0	0
$27$	1.42825e26	0.472688
$28$	0	0
$29$	6.80125e26	0.600103	0.300052	−	0.953923i	$-0.402996\pi$
$29$	6.80125e26	0.600103	0.300052	+	0.953923i	$0.402996\pi$
$30$	0	0
$31$	−6.29975e27	−1.61857	−0.809286	−	0.587415i	$-0.800145\pi$
$31$	−6.29975e27	−1.61857	−0.809286	+	0.587415i	$0.800145\pi$
$32$	0	0
$33$	2.94347e28	2.37877
$34$	0	0
$35$	4.40503e28	1.19864
$36$	0	0
$37$	2.92536e28	0.284739	0.142370	−	0.989814i	$-0.454528\pi$
$37$	2.92536e28	0.284739	0.142370	+	0.989814i	$0.454528\pi$
$38$	0	0
$39$	2.26644e29	0.833008
$40$	0	0
$41$	1.17322e30	1.70954	0.854768	−	0.519010i	$-0.173699\pi$
$41$	1.17322e30	1.70954	0.854768	+	0.519010i	$0.173699\pi$
$42$	0	0
$43$	2.37316e30	1.43272	0.716360	−	0.697731i	$-0.245807\pi$
$43$	2.37316e30	1.43272	0.716360	+	0.697731i	$0.245807\pi$
$44$	0	0
$45$	2.92001e30	0.760243
$46$	0	0
$47$	−4.28443e30	−0.498980	−0.249490	−	0.968377i	$-0.580263\pi$
$47$	−4.28443e30	−0.498980	−0.249490	+	0.968377i	$0.580263\pi$
$48$	0	0
$49$	−2.63698e29	−0.0142062
$50$	0	0
$51$	−5.61380e31	−1.44281
$52$	0	0
$53$	1.05927e32	1.33629	0.668147	−	0.744029i	$-0.267088\pi$
$53$	1.05927e32	1.33629	0.668147	+	0.744029i	$0.267088\pi$
$54$	0	0
$55$	−3.53835e32	−2.24953
$56$	0	0
$57$	−4.10754e32	−1.34864
$58$	0	0
$59$	−3.39980e32	−0.589781	−0.294891	−	0.955531i	$-0.595283\pi$
$59$	−3.39980e32	−0.589781	−0.294891	+	0.955531i	$0.595283\pi$
$60$	0	0
$61$	1.98820e33	1.86148	0.930742	−	0.365677i	$-0.119163\pi$
$61$	1.98820e33	1.86148	0.930742	+	0.365677i	$0.119163\pi$
$62$	0	0
$63$	1.21297e33	0.625243
$64$	0	0
$65$	−2.72449e33	−0.787749
$66$	0	0
$67$	1.05982e34	1.74924	0.874618	−	0.484813i	$-0.161112\pi$
$67$	1.05982e34	1.74924	0.874618	+	0.484813i	$0.161112\pi$
$68$	0	0
$69$	1.41357e33	0.135397
$70$	0	0
$71$	2.78180e33	0.157053	0.0785264	−	0.996912i	$-0.474978\pi$
$71$	2.78180e33	0.157053	0.0785264	+	0.996912i	$0.474978\pi$
$72$	0	0
$73$	2.22277e34	0.750623	0.375312	−	0.926899i	$-0.377536\pi$
$73$	2.22277e34	0.750623	0.375312	+	0.926899i	$0.377536\pi$
$74$	0	0
$75$	−2.85124e34	−0.583984
$76$	0	0
$77$	−1.46983e35	−1.85007
$78$	0	0
$79$	−3.94375e34	−0.308893	−0.154447	−	0.988001i	$-0.549359\pi$
$79$	−3.94375e34	−0.308893	−0.154447	+	0.988001i	$0.549359\pi$
$80$	0	0
$81$	−2.50032e35	−1.23317
$82$	0	0
$83$	2.31621e35	0.727501	0.363750	−	0.931497i	$-0.381496\pi$
$83$	2.31621e35	0.727501	0.363750	+	0.931497i	$0.381496\pi$
$84$	0	0
$85$	6.74835e35	1.36442
$86$	0	0
$87$	−5.82626e35	−0.766098
$88$	0	0
$89$	7.56957e35	0.653666	0.326833	−	0.945082i	$-0.394018\pi$
$89$	7.56957e35	0.653666	0.326833	+	0.945082i	$0.394018\pi$
$90$	0	0
$91$	−1.13175e36	−0.647865
$92$	0	0
$93$	5.39665e36	2.06628
$94$	0	0
$95$	4.93767e36	1.27537
$96$	0	0
$97$	−6.01498e36	−1.05671	−0.528357	−	0.849023i	$-0.677192\pi$
$97$	−6.01498e36	−1.05671	−0.528357	+	0.849023i	$0.677192\pi$
$98$	0	0
$99$	−9.74319e36	−1.17341
$100$	0	0
$101$	−1.27647e37	−1.06186	−0.530928	−	0.847417i	$-0.678157\pi$
$101$	−1.27647e37	−1.06186	−0.530928	+	0.847417i	$0.678157\pi$
$102$	0	0
$103$	8.70044e36	0.503561	0.251781	−	0.967784i	$-0.418984\pi$
$103$	8.70044e36	0.503561	0.251781	+	0.967784i	$0.418984\pi$
$104$	0	0
$105$	−3.77355e37	−1.53020
$106$	0	0
$107$	1.34226e37	0.383916	0.191958	−	0.981403i	$-0.438516\pi$
$107$	1.34226e37	0.383916	0.191958	+	0.981403i	$0.438516\pi$
$108$	0	0
$109$	3.67405e37	0.746027	0.373014	−	0.927826i	$-0.378324\pi$
$109$	3.67405e37	0.746027	0.373014	+	0.927826i	$0.378324\pi$
$110$	0	0
$111$	−2.50600e37	−0.363501
$112$	0	0
$113$	1.12367e38	1.17135	0.585677	−	0.810544i	$-0.300829\pi$
$113$	1.12367e38	1.17135	0.585677	+	0.810544i	$0.300829\pi$
$114$	0	0
$115$	−1.69925e37	−0.128040
$116$	0	0
$117$	−7.50214e37	−0.410910
$118$	0	0
$119$	2.80325e38	1.12213
$120$	0	0
$121$	8.40602e38	2.47207
$122$	0	0
$123$	−1.00503e39	−2.18241
$124$	0	0
$125$	−4.06511e38	−0.654994
$126$	0	0
$127$	−2.89074e38	−0.347248	−0.173624	−	0.984812i	$-0.555548\pi$
$127$	−2.89074e38	−0.347248	−0.173624	+	0.984812i	$0.555548\pi$
$128$	0	0
$129$	−2.03296e39	−1.82902
$130$	0	0
$131$	−3.40496e38	−0.230459	−0.115229	−	0.993339i	$-0.536760\pi$
$131$	−3.40496e38	−0.230459	−0.115229	+	0.993339i	$0.536760\pi$
$132$	0	0
$133$	2.05110e39	1.04889
$134$	0	0
$135$	1.47077e39	0.570652
$136$	0	0
$137$	1.89610e39	0.560443	0.280222	−	0.959935i	$-0.409592\pi$
$137$	1.89610e39	0.560443	0.280222	+	0.959935i	$0.409592\pi$
$138$	0	0
$139$	−6.45321e39	−1.45882	−0.729410	−	0.684076i	$-0.760205\pi$
$139$	−6.45321e39	−1.45882	−0.729410	+	0.684076i	$0.760205\pi$
$140$	0	0
$141$	3.67024e39	0.637003
$142$	0	0
$143$	9.09080e39	1.21587
$144$	0	0
$145$	7.00375e39	0.724474
$146$	0	0
$147$	2.25895e38	0.0181358
$148$	0	0
$149$	7.48615e39	0.468072	0.234036	−	0.972228i	$-0.424807\pi$
$149$	7.48615e39	0.468072	0.234036	+	0.972228i	$0.424807\pi$
$150$	0	0
$151$	−2.71291e40	−1.32545	−0.662723	−	0.748865i	$-0.730599\pi$
$151$	−2.71291e40	−1.32545	−0.662723	+	0.748865i	$0.730599\pi$
$152$	0	0
$153$	1.85822e40	0.711715
$154$	0	0
$155$	−6.48731e40	−1.95402
$156$	0	0
$157$	1.51323e40	0.359552	0.179776	−	0.983708i	$-0.442463\pi$
$157$	1.51323e40	0.359552	0.179776	+	0.983708i	$0.442463\pi$
$158$	0	0
$159$	−9.07417e40	−1.70593
$160$	0	0
$161$	−7.05865e39	−0.105303
$162$	0	0
$163$	3.19018e40	0.378742	0.189371	−	0.981906i	$-0.439355\pi$
$163$	3.19018e40	0.378742	0.189371	+	0.981906i	$0.439355\pi$
$164$	0	0
$165$	3.03111e41	2.87177
$166$	0	0
$167$	−9.14876e40	−0.693598	−0.346799	−	0.937940i	$-0.612731\pi$
$167$	−9.14876e40	−0.693598	−0.346799	+	0.937940i	$0.612731\pi$
$168$	0	0
$169$	−9.44027e40	−0.574223
$170$	0	0
$171$	1.35964e41	0.665265
$172$	0	0
$173$	−8.81872e40	−0.347979	−0.173989	−	0.984748i	$-0.555666\pi$
$173$	−8.81872e40	−0.347979	−0.173989	+	0.984748i	$0.555666\pi$
$174$	0	0
$175$	1.42377e41	0.454188
$176$	0	0
$177$	2.91242e41	0.752920
$178$	0	0
$179$	2.89793e41	0.608564	0.304282	−	0.952582i	$-0.401583\pi$
$179$	2.89793e41	0.608564	0.304282	+	0.952582i	$0.401583\pi$
$180$	0	0
$181$	6.33412e40	0.108301	0.0541506	−	0.998533i	$-0.482755\pi$
$181$	6.33412e40	0.108301	0.0541506	+	0.998533i	$0.482755\pi$
$182$	0	0
$183$	−1.70319e42	−2.37639
$184$	0	0
$185$	3.01246e41	0.343751
$186$	0	0
$187$	−2.25172e42	−2.10593
$188$	0	0
$189$	6.10957e41	0.469318
$190$	0	0
$191$	1.02416e41	0.0647513	0.0323757	−	0.999476i	$-0.489693\pi$
$191$	1.02416e41	0.0647513	0.0323757	+	0.999476i	$0.489693\pi$
$192$	0	0
$193$	5.05238e40	0.0263443	0.0131721	−	0.999913i	$-0.495807\pi$
$193$	5.05238e40	0.0263443	0.0131721	+	0.999913i	$0.495807\pi$
$194$	0	0
$195$	2.33392e42	1.00565
$196$	0	0
$197$	1.87295e42	0.668192	0.334096	−	0.942539i	$-0.391569\pi$
$197$	1.87295e42	0.668192	0.334096	+	0.942539i	$0.391569\pi$
$198$	0	0
$199$	5.14543e42	1.52279	0.761394	−	0.648289i	$-0.224515\pi$
$199$	5.14543e42	1.52279	0.761394	+	0.648289i	$0.224515\pi$
$200$	0	0
$201$	−9.07891e42	−2.23309
$202$	0	0
$203$	2.90935e42	0.595825
$204$	0	0
$205$	1.20815e43	2.06384
$206$	0	0
$207$	−4.67904e41	−0.0667891
$208$	0	0
$209$	−1.64755e43	−1.96849
$210$	0	0
$211$	1.39241e42	0.139490	0.0697449	−	0.997565i	$-0.477781\pi$
$211$	1.39241e42	0.139490	0.0697449	+	0.997565i	$0.477781\pi$
$212$	0	0
$213$	−2.38301e42	−0.200495
$214$	0	0
$215$	2.44382e43	1.72965
$216$	0	0
$217$	−2.69482e43	−1.60703
$218$	0	0
$219$	−1.90413e43	−0.958253
$220$	0	0
$221$	−1.73380e43	−0.737466
$222$	0	0
$223$	−2.79345e43	−1.00577	−0.502887	−	0.864352i	$-0.667729\pi$
$223$	−2.79345e43	−1.00577	−0.502887	+	0.864352i	$0.667729\pi$
$224$	0	0
$225$	9.43790e42	0.288070
$226$	0	0
$227$	3.47813e43	0.901297	0.450648	−	0.892702i	$-0.351193\pi$
$227$	3.47813e43	0.901297	0.450648	+	0.892702i	$0.351193\pi$
$228$	0	0
$229$	4.60002e43	1.01345	0.506727	−	0.862107i	$-0.330855\pi$
$229$	4.60002e43	1.01345	0.506727	+	0.862107i	$0.330855\pi$
$230$	0	0
$231$	1.25912e44	2.36181
$232$	0	0
$233$	−2.53760e43	−0.405825	−0.202913	−	0.979197i	$-0.565041\pi$
$233$	−2.53760e43	−0.405825	−0.202913	+	0.979197i	$0.565041\pi$
$234$	0	0
$235$	−4.41200e43	−0.602393
$236$	0	0
$237$	3.37840e43	0.394336
$238$	0	0
$239$	−9.07981e43	−0.907226	−0.453613	−	0.891199i	$-0.649865\pi$
$239$	−9.07981e43	−0.907226	−0.453613	+	0.891199i	$0.649865\pi$
$240$	0	0
$241$	−5.44644e43	−0.466441	−0.233221	−	0.972424i	$-0.574926\pi$
$241$	−5.44644e43	−0.466441	−0.233221	+	0.972424i	$0.574926\pi$
$242$	0	0
$243$	1.49877e44	1.10159
$244$	0	0
$245$	−2.71549e42	−0.0171504
$246$	0	0
$247$	−1.26860e44	−0.689335
$248$	0	0
$249$	−1.98417e44	−0.928734
$250$	0	0
$251$	6.84014e43	0.276122	0.138061	−	0.990424i	$-0.455913\pi$
$251$	6.84014e43	0.276122	0.138061	+	0.990424i	$0.455913\pi$
$252$	0	0
$253$	5.66988e43	0.197626
$254$	0	0
$255$	−5.78094e44	−1.74183
$256$	0	0
$257$	−3.50915e43	−0.0915037	−0.0457519	−	0.998953i	$-0.514568\pi$
$257$	−3.50915e43	−0.0915037	−0.0457519	+	0.998953i	$0.514568\pi$
$258$	0	0
$259$	1.25137e44	0.282709
$260$	0	0
$261$	1.92855e44	0.377904
$262$	0	0
$263$	2.11632e44	0.360081	0.180040	−	0.983659i	$-0.442377\pi$
$263$	2.11632e44	0.360081	0.180040	+	0.983659i	$0.442377\pi$
$264$	0	0
$265$	1.09081e45	1.61324
$266$	0	0
$267$	−6.48444e44	−0.834477
$268$	0	0
$269$	1.52864e45	1.71351	0.856756	−	0.515723i	$-0.172477\pi$
$269$	1.52864e45	1.71351	0.856756	+	0.515723i	$0.172477\pi$
$270$	0	0
$271$	7.15929e43	0.0699742	0.0349871	−	0.999388i	$-0.488861\pi$
$271$	7.15929e43	0.0699742	0.0349871	+	0.999388i	$0.488861\pi$
$272$	0	0
$273$	9.69508e44	0.827070
$274$	0	0
$275$	−1.14365e45	−0.852388
$276$	0	0
$277$	−1.39354e45	−0.908331	−0.454165	−	0.890917i	$-0.650063\pi$
$277$	−1.39354e45	−0.908331	−0.454165	+	0.890917i	$0.650063\pi$
$278$	0	0
$279$	−1.78634e45	−1.01927
$280$	0	0
$281$	1.71378e45	0.856815	0.428407	−	0.903586i	$-0.359075\pi$
$281$	1.71378e45	0.856815	0.428407	+	0.903586i	$0.359075\pi$
$282$	0	0
$283$	−2.90166e45	−1.27232	−0.636161	−	0.771556i	$-0.719479\pi$
$283$	−2.90166e45	−1.27232	−0.636161	+	0.771556i	$0.719479\pi$
$284$	0	0
$285$	−4.22984e45	−1.62815
$286$	0	0
$287$	5.01865e45	1.69735
$288$	0	0
$289$	9.32390e44	0.277324
$290$	0	0
$291$	5.15271e45	1.34901
$292$	0	0
$293$	−7.32879e45	−1.69037	−0.845184	−	0.534475i	$-0.820509\pi$
$293$	−7.32879e45	−1.69037	−0.845184	+	0.534475i	$0.820509\pi$
$294$	0	0
$295$	−3.50102e45	−0.712012
$296$	0	0
$297$	−4.90753e45	−0.880783
$298$	0	0
$299$	4.36574e44	0.0692056
$300$	0	0
$301$	1.01516e46	1.42251
$302$	0	0
$303$	1.09348e46	1.35558
$304$	0	0
$305$	2.04740e46	2.24727
$306$	0	0
$307$	−9.38849e45	−0.913137	−0.456569	−	0.889688i	$-0.650922\pi$
$307$	−9.38849e45	−0.913137	−0.456569	+	0.889688i	$0.650922\pi$
$308$	0	0
$309$	−7.45319e45	−0.642851
$310$	0	0
$311$	6.86772e45	0.525709	0.262855	−	0.964835i	$-0.415336\pi$
$311$	6.86772e45	0.525709	0.262855	+	0.964835i	$0.415336\pi$
$312$	0	0
$313$	−8.87418e45	−0.603334	−0.301667	−	0.953413i	$-0.597543\pi$
$313$	−8.87418e45	−0.603334	−0.301667	+	0.953413i	$0.597543\pi$
$314$	0	0
$315$	1.24908e46	0.754824
$316$	0	0
$317$	−3.42964e46	−1.84354	−0.921770	−	0.387738i	$-0.873257\pi$
$317$	−3.42964e46	−1.84354	−0.921770	+	0.387738i	$0.873257\pi$
$318$	0	0
$319$	−2.33694e46	−1.11820
$320$	0	0
$321$	−1.14984e46	−0.490111
$322$	0	0
$323$	3.14222e46	1.19396
$324$	0	0
$325$	−8.80595e45	−0.298493
$326$	0	0
$327$	−3.14736e46	−0.952385
$328$	0	0
$329$	−1.83274e46	−0.495423
$330$	0	0
$331$	4.52646e46	1.09381	0.546904	−	0.837195i	$-0.315806\pi$
$331$	4.52646e46	1.09381	0.546904	+	0.837195i	$0.315806\pi$
$332$	0	0
$333$	8.29509e45	0.179309
$334$	0	0
$335$	1.09138e47	2.11176
$336$	0	0
$337$	−9.58299e46	−1.66092	−0.830459	−	0.557080i	$-0.811921\pi$
$337$	−9.58299e46	−1.66092	−0.830459	+	0.557080i	$0.811921\pi$
$338$	0	0
$339$	−9.62589e46	−1.49536
$340$	0	0
$341$	2.16462e47	3.01596
$342$	0	0
$343$	−8.05305e46	−1.00698
$344$	0	0
$345$	1.45565e46	0.163457
$346$	0	0
$347$	9.21856e46	0.930182	0.465091	−	0.885263i	$-0.346022\pi$
$347$	9.21856e46	0.930182	0.465091	+	0.885263i	$0.346022\pi$
$348$	0	0
$349$	5.45881e46	0.495254	0.247627	−	0.968855i	$-0.420349\pi$
$349$	5.45881e46	0.495254	0.247627	+	0.968855i	$0.420349\pi$
$350$	0	0
$351$	−3.77874e46	−0.308436
$352$	0	0
$353$	6.45915e46	0.474617	0.237309	−	0.971434i	$-0.423735\pi$
$353$	6.45915e46	0.474617	0.237309	+	0.971434i	$0.423735\pi$
$354$	0	0
$355$	2.86462e46	0.189602
$356$	0	0
$357$	−2.40140e47	−1.43252
$358$	0	0
$359$	−1.50173e47	−0.807876	−0.403938	−	0.914786i	$-0.632359\pi$
$359$	−1.50173e47	−0.807876	−0.403938	+	0.914786i	$0.632359\pi$
$360$	0	0
$361$	2.39040e46	0.116034
$362$	0	0
$363$	−7.20098e47	−3.15587
$364$	0	0
$365$	2.28895e47	0.906189
$366$	0	0
$367$	−1.68884e47	−0.604320	−0.302160	−	0.953257i	$-0.597708\pi$
$367$	−1.68884e47	−0.604320	−0.302160	+	0.953257i	$0.597708\pi$
$368$	0	0
$369$	3.32676e47	1.07655
$370$	0	0
$371$	4.53119e47	1.32677
$372$	0	0
$373$	2.16154e47	0.572994	0.286497	−	0.958081i	$-0.407509\pi$
$373$	2.16154e47	0.572994	0.286497	+	0.958081i	$0.407509\pi$
$374$	0	0
$375$	3.48236e47	0.836171
$376$	0	0
$377$	−1.79942e47	−0.391577
$378$	0	0
$379$	−5.27976e47	−1.04181	−0.520905	−	0.853615i	$-0.674405\pi$
$379$	−5.27976e47	−1.04181	−0.520905	+	0.853615i	$0.674405\pi$
$380$	0	0
$381$	2.47634e47	0.443300
$382$	0	0
$383$	9.60431e47	1.56059	0.780295	−	0.625412i	$-0.215069\pi$
$383$	9.60431e47	1.56059	0.780295	+	0.625412i	$0.215069\pi$
$384$	0	0
$385$	−1.51359e48	−2.23349
$386$	0	0
$387$	6.72930e47	0.902229
$388$	0	0
$389$	1.45162e48	1.76923	0.884614	−	0.466323i	$-0.154422\pi$
$389$	1.45162e48	1.76923	0.884614	+	0.466323i	$0.154422\pi$
$390$	0	0
$391$	−1.08136e47	−0.119867
$392$	0	0
$393$	2.91685e47	0.294206
$394$	0	0
$395$	−4.06117e47	−0.372911
$396$	0	0
$397$	−1.91740e48	−1.60357	−0.801786	−	0.597611i	$-0.796117\pi$
$397$	−1.91740e48	−1.60357	−0.801786	+	0.597611i	$0.796117\pi$
$398$	0	0
$399$	−1.75707e48	−1.33903
$400$	0	0
$401$	1.24268e48	0.863356	0.431678	−	0.902028i	$-0.357922\pi$
$401$	1.24268e48	0.863356	0.431678	+	0.902028i	$0.357922\pi$
$402$	0	0
$403$	1.66673e48	1.05614
$404$	0	0
$405$	−2.57476e48	−1.48874
$406$	0	0
$407$	−1.00517e48	−0.530569
$408$	0	0
$409$	−9.53274e47	−0.459554	−0.229777	−	0.973243i	$-0.573800\pi$
$409$	−9.53274e47	−0.459554	−0.229777	+	0.973243i	$0.573800\pi$
$410$	0	0
$411$	−1.62429e48	−0.715467
$412$	0	0
$413$	−1.45432e48	−0.585577
$414$	0	0
$415$	2.38517e48	0.878274
$416$	0	0
$417$	5.52811e48	1.86234
$418$	0	0
$419$	5.34024e48	1.64665	0.823326	−	0.567569i	$-0.192116\pi$
$419$	5.34024e48	1.64665	0.823326	+	0.567569i	$0.192116\pi$
$420$	0	0
$421$	−1.64063e48	−0.463225	−0.231613	−	0.972808i	$-0.574400\pi$
$421$	−1.64063e48	−0.463225	−0.231613	+	0.972808i	$0.574400\pi$
$422$	0	0
$423$	−1.21489e48	−0.314224
$424$	0	0
$425$	2.18117e48	0.517003
$426$	0	0
$427$	8.50487e48	1.84821
$428$	0	0
$429$	−7.78760e48	−1.55219
$430$	0	0
$431$	1.22740e48	0.224468	0.112234	−	0.993682i	$-0.464199\pi$
$431$	1.22740e48	0.224468	0.112234	+	0.993682i	$0.464199\pi$
$432$	0	0
$433$	−5.52514e48	−0.927507	−0.463753	−	0.885964i	$-0.653498\pi$
$433$	−5.52514e48	−0.927507	−0.463753	+	0.885964i	$0.653498\pi$
$434$	0	0
$435$	−5.99973e48	−0.924870
$436$	0	0
$437$	−7.91216e47	−0.112044
$438$	0	0
$439$	4.95015e48	0.644206	0.322103	−	0.946705i	$-0.395610\pi$
$439$	4.95015e48	0.644206	0.322103	+	0.946705i	$0.395610\pi$
$440$	0	0
$441$	−7.47736e46	−0.00894611
$442$	0	0
$443$	−1.08464e48	−0.119348	−0.0596741	−	0.998218i	$-0.519006\pi$
$443$	−1.08464e48	−0.119348	−0.0596741	+	0.998218i	$0.519006\pi$
$444$	0	0
$445$	7.79495e48	0.789138
$446$	0	0
$447$	−6.41298e48	−0.597545
$448$	0	0
$449$	−1.22533e48	−0.105123	−0.0525614	−	0.998618i	$-0.516739\pi$
$449$	−1.22533e48	−0.105123	−0.0525614	+	0.998618i	$0.516739\pi$
$450$	0	0
$451$	−4.03124e49	−3.18546
$452$	0	0
$453$	2.32400e49	1.69208
$454$	0	0
$455$	−1.16545e49	−0.782134
$456$	0	0
$457$	1.72466e48	0.106722	0.0533610	−	0.998575i	$-0.483007\pi$
$457$	1.72466e48	0.106722	0.0533610	+	0.998575i	$0.483007\pi$
$458$	0	0
$459$	9.35965e48	0.534226
$460$	0	0
$461$	−1.87709e49	−0.988596	−0.494298	−	0.869293i	$-0.664575\pi$
$461$	−1.87709e49	−0.988596	−0.494298	+	0.869293i	$0.664575\pi$
$462$	0	0
$463$	1.48932e49	0.724004	0.362002	−	0.932177i	$-0.382093\pi$
$463$	1.48932e49	0.724004	0.362002	+	0.932177i	$0.382093\pi$
$464$	0	0
$465$	5.55733e49	2.49452
$466$	0	0
$467$	1.94902e49	0.808078	0.404039	−	0.914742i	$-0.367606\pi$
$467$	1.94902e49	0.808078	0.404039	+	0.914742i	$0.367606\pi$
$468$	0	0
$469$	4.53356e49	1.73677
$470$	0	0
$471$	−1.29630e49	−0.459008
$472$	0	0
$473$	−8.15430e49	−2.66966
$474$	0	0
$475$	1.59593e49	0.483261
$476$	0	0
$477$	3.00364e49	0.841507
$478$	0	0
$479$	−1.52674e49	−0.395876	−0.197938	−	0.980215i	$-0.563424\pi$
$479$	−1.52674e49	−0.395876	−0.197938	+	0.980215i	$0.563424\pi$
$480$	0	0
$481$	−7.73967e48	−0.185797
$482$	0	0
$483$	6.04676e48	0.134431
$484$	0	0
$485$	−6.19407e49	−1.27572
$486$	0	0
$487$	−1.37059e49	−0.261591	−0.130795	−	0.991409i	$-0.541753\pi$
$487$	−1.37059e49	−0.261591	−0.130795	+	0.991409i	$0.541753\pi$
$488$	0	0
$489$	−2.73285e49	−0.483506
$490$	0	0
$491$	−2.20103e49	−0.361093	−0.180546	−	0.983566i	$-0.557787\pi$
$491$	−2.20103e49	−0.361093	−0.180546	+	0.983566i	$0.557787\pi$
$492$	0	0
$493$	4.45702e49	0.678229
$494$	0	0
$495$	−1.00333e50	−1.41660
$496$	0	0
$497$	1.18996e49	0.155933
$498$	0	0
$499$	7.98127e49	0.970984	0.485492	−	0.874241i	$-0.338640\pi$
$499$	7.98127e49	0.970984	0.485492	+	0.874241i	$0.338640\pi$
$500$	0	0
$501$	7.83725e49	0.885453
$502$	0	0
$503$	2.34298e49	0.245901	0.122950	−	0.992413i	$-0.460764\pi$
$503$	2.34298e49	0.245901	0.122950	+	0.992413i	$0.460764\pi$
$504$	0	0
$505$	−1.31448e50	−1.28192
$506$	0	0
$507$	8.08697e49	0.733058
$508$	0	0
$509$	−1.79227e50	−1.51052	−0.755258	−	0.655428i	$-0.772488\pi$
$509$	−1.79227e50	−1.51052	−0.755258	+	0.655428i	$0.772488\pi$
$510$	0	0
$511$	9.50827e49	0.745272
$512$	0	0
$513$	6.84833e49	0.499359
$514$	0	0
$515$	8.95948e49	0.607924
$516$	0	0
$517$	1.47215e50	0.929774
$518$	0	0
$519$	7.55452e49	0.444233
$520$	0	0
$521$	2.11640e50	1.15905	0.579523	−	0.814956i	$-0.303239\pi$
$521$	2.11640e50	1.15905	0.579523	+	0.814956i	$0.303239\pi$
$522$	0	0
$523$	2.50874e50	1.27989	0.639947	−	0.768419i	$-0.278956\pi$
$523$	2.50874e50	1.27989	0.639947	+	0.768419i	$0.278956\pi$
$524$	0	0
$525$	−1.21967e50	−0.579821
$526$	0	0
$527$	−4.12837e50	−1.82929
$528$	0	0
$529$	−2.39341e50	−0.988751
$530$	0	0
$531$	−9.64040e49	−0.371404
$532$	0	0
$533$	−3.10401e50	−1.11550
$534$	0	0
$535$	1.38222e50	0.463483
$536$	0	0
$537$	−2.48250e50	−0.776899
$538$	0	0
$539$	9.06077e48	0.0264712
$540$	0	0
$541$	−3.79895e50	−1.03637	−0.518184	−	0.855269i	$-0.673392\pi$
$541$	−3.79895e50	−1.03637	−0.518184	+	0.855269i	$0.673392\pi$
$542$	0	0
$543$	−5.42610e49	−0.138258
$544$	0	0
$545$	3.78344e50	0.900640
$546$	0	0
$547$	2.04942e50	0.455896	0.227948	−	0.973673i	$-0.426798\pi$
$547$	2.04942e50	0.455896	0.227948	+	0.973673i	$0.426798\pi$
$548$	0	0
$549$	5.63772e50	1.17224
$550$	0	0
$551$	3.26114e50	0.633964
$552$	0	0
$553$	−1.68701e50	−0.306691
$554$	0	0
$555$	−2.58061e50	−0.438836
$556$	0	0
$557$	−7.33052e50	−1.16631	−0.583155	−	0.812361i	$-0.698182\pi$
$557$	−7.33052e50	−1.16631	−0.583155	+	0.812361i	$0.698182\pi$
$558$	0	0
$559$	−6.27872e50	−0.934872
$560$	0	0
$561$	1.92893e51	2.68846
$562$	0	0
$563$	2.37166e49	0.0309491	0.0154745	−	0.999880i	$-0.495074\pi$
$563$	2.37166e49	0.0309491	0.0154745	+	0.999880i	$0.495074\pi$
$564$	0	0
$565$	1.15713e51	1.41412
$566$	0	0
$567$	−1.06955e51	−1.22438
$568$	0	0
$569$	7.05699e50	0.756907	0.378453	−	0.925620i	$-0.376456\pi$
$569$	7.05699e50	0.756907	0.378453	+	0.925620i	$0.376456\pi$
$570$	0	0
$571$	1.62902e50	0.163741	0.0818705	−	0.996643i	$-0.473911\pi$
$571$	1.62902e50	0.163741	0.0818705	+	0.996643i	$0.473911\pi$
$572$	0	0
$573$	−8.77339e49	−0.0826622
$574$	0	0
$575$	−5.49222e49	−0.0485168
$576$	0	0
$577$	−4.13399e50	−0.342466	−0.171233	−	0.985231i	$-0.554775\pi$
$577$	−4.13399e50	−0.342466	−0.171233	+	0.985231i	$0.554775\pi$
$578$	0	0
$579$	−4.32810e49	−0.0336313
$580$	0	0
$581$	9.90797e50	0.722315
$582$	0	0
$583$	−3.63969e51	−2.48998
$584$	0	0
$585$	−7.72551e50	−0.496071
$586$	0	0
$587$	2.19997e51	1.32621	0.663105	−	0.748526i	$-0.269238\pi$
$587$	2.19997e51	1.32621	0.663105	+	0.748526i	$0.269238\pi$
$588$	0	0
$589$	−3.02067e51	−1.70990
$590$	0	0
$591$	−1.60446e51	−0.853020
$592$	0	0
$593$	5.79566e50	0.289462	0.144731	−	0.989471i	$-0.453768\pi$
$593$	5.79566e50	0.289462	0.144731	+	0.989471i	$0.453768\pi$
$594$	0	0
$595$	2.88672e51	1.35469
$596$	0	0
$597$	−4.40781e51	−1.94401
$598$	0	0
$599$	3.34743e50	0.138776	0.0693882	−	0.997590i	$-0.477895\pi$
$599$	3.34743e50	0.138776	0.0693882	+	0.997590i	$0.477895\pi$
$600$	0	0
$601$	−1.39433e51	−0.543484	−0.271742	−	0.962370i	$-0.587600\pi$
$601$	−1.39433e51	−0.543484	−0.271742	+	0.962370i	$0.587600\pi$
$602$	0	0
$603$	3.00521e51	1.10155
$604$	0	0
$605$	8.65629e51	2.98440
$606$	0	0
$607$	3.63186e51	1.17798	0.588991	−	0.808140i	$-0.299525\pi$
$607$	3.63186e51	1.17798	0.588991	+	0.808140i	$0.299525\pi$
$608$	0	0
$609$	−2.49228e51	−0.760636
$610$	0	0
$611$	1.13354e51	0.325593
$612$	0	0
$613$	6.30097e51	1.70368	0.851841	−	0.523800i	$-0.175486\pi$
$613$	6.30097e51	1.70368	0.851841	+	0.523800i	$0.175486\pi$
$614$	0	0
$615$	−1.03496e52	−2.63471
$616$	0	0
$617$	−4.87751e51	−1.16929	−0.584646	−	0.811288i	$-0.698767\pi$
$617$	−4.87751e51	−1.16929	−0.584646	+	0.811288i	$0.698767\pi$
$618$	0	0
$619$	1.82138e51	0.411267	0.205634	−	0.978629i	$-0.434074\pi$
$619$	1.82138e51	0.411267	0.205634	+	0.978629i	$0.434074\pi$
$620$	0	0
$621$	−2.35678e50	−0.0501330
$622$	0	0
$623$	3.23801e51	0.649007
$624$	0	0
$625$	−6.60786e51	−1.24819
$626$	0	0
$627$	1.41137e52	2.51299
$628$	0	0
$629$	1.91706e51	0.321809
$630$	0	0
$631$	−9.55000e51	−1.51168	−0.755841	−	0.654755i	$-0.772772\pi$
$631$	−9.55000e51	−1.51168	−0.755841	+	0.654755i	$0.772772\pi$
$632$	0	0
$633$	−1.19280e51	−0.178074
$634$	0	0
$635$	−2.97681e51	−0.419215
$636$	0	0
$637$	6.97669e49	0.00926979
$638$	0	0
$639$	7.88801e50	0.0989012
$640$	0	0
$641$	−2.92473e51	−0.346109	−0.173054	−	0.984912i	$-0.555364\pi$
$641$	−2.92473e51	−0.346109	−0.173054	+	0.984912i	$0.555364\pi$
$642$	0	0
$643$	−1.58763e52	−1.77356	−0.886782	−	0.462189i	$-0.847064\pi$
$643$	−1.58763e52	−1.77356	−0.886782	+	0.462189i	$0.847064\pi$
$644$	0	0
$645$	−2.09349e52	−2.20809
$646$	0	0
$647$	7.40224e51	0.737283	0.368641	−	0.929572i	$-0.379823\pi$
$647$	7.40224e51	0.737283	0.368641	+	0.929572i	$0.379823\pi$
$648$	0	0
$649$	1.16819e52	1.09897
$650$	0	0
$651$	2.30851e52	2.05155
$652$	0	0
$653$	−1.39335e52	−1.16995	−0.584975	−	0.811051i	$-0.698896\pi$
$653$	−1.39335e52	−1.16995	−0.584975	+	0.811051i	$0.698896\pi$
$654$	0	0
$655$	−3.50634e51	−0.278221
$656$	0	0
$657$	6.30285e51	0.472691
$658$	0	0
$659$	1.66820e52	1.18268	0.591340	−	0.806422i	$-0.298599\pi$
$659$	1.66820e52	1.18268	0.591340	+	0.806422i	$0.298599\pi$
$660$	0	0
$661$	2.75700e51	0.184803	0.0924015	−	0.995722i	$-0.470546\pi$
$661$	2.75700e51	0.184803	0.0924015	+	0.995722i	$0.470546\pi$
$662$	0	0
$663$	1.48525e52	0.941456
$664$	0	0
$665$	2.11217e52	1.26628
$666$	0	0
$667$	−1.12229e51	−0.0636467
$668$	0	0
$669$	2.39300e52	1.28398
$670$	0	0
$671$	−6.83156e52	−3.46859
$672$	0	0
$673$	1.49652e52	0.719123	0.359562	−	0.933121i	$-0.382926\pi$
$673$	1.49652e52	0.719123	0.359562	+	0.933121i	$0.382926\pi$
$674$	0	0
$675$	4.75376e51	0.216231
$676$	0	0
$677$	1.87714e52	0.808362	0.404181	−	0.914679i	$-0.367557\pi$
$677$	1.87714e52	0.808362	0.404181	+	0.914679i	$0.367557\pi$
$678$	0	0
$679$	−2.57301e52	−1.04918
$680$	0	0
$681$	−2.97953e52	−1.15060
$682$	0	0
$683$	9.46464e51	0.346195	0.173098	−	0.984905i	$-0.444622\pi$
$683$	9.46464e51	0.346195	0.173098	+	0.984905i	$0.444622\pi$
$684$	0	0
$685$	1.95256e52	0.676594
$686$	0	0
$687$	−3.94059e52	−1.29378
$688$	0	0
$689$	−2.80252e52	−0.871953
$690$	0	0
$691$	4.78608e52	1.41135	0.705677	−	0.708534i	$-0.250643\pi$
$691$	4.78608e52	1.41135	0.705677	+	0.708534i	$0.250643\pi$
$692$	0	0
$693$	−4.16781e52	−1.16505
$694$	0	0
$695$	−6.64535e52	−1.76116
$696$	0	0
$697$	7.68839e52	1.93210
$698$	0	0
$699$	2.17383e52	0.518080
$700$	0	0
$701$	−6.12605e51	−0.138483	−0.0692416	−	0.997600i	$-0.522058\pi$
$701$	−6.12605e51	−0.138483	−0.0692416	+	0.997600i	$0.522058\pi$
$702$	0	0
$703$	1.40268e52	0.300806
$704$	0	0
$705$	3.77952e52	0.769021
$706$	0	0
$707$	−5.46032e52	−1.05429
$708$	0	0
$709$	−9.65451e52	−1.76919	−0.884596	−	0.466359i	$-0.845566\pi$
$709$	−9.65451e52	−1.76919	−0.884596	+	0.466359i	$0.845566\pi$
$710$	0	0
$711$	−1.11828e52	−0.194520
$712$	0	0
$713$	1.03953e52	0.171665
$714$	0	0
$715$	9.36147e52	1.46785
$716$	0	0
$717$	7.77818e52	1.15817
$718$	0	0
$719$	7.65140e52	1.08207	0.541037	−	0.840999i	$-0.318032\pi$
$719$	7.65140e52	1.08207	0.541037	+	0.840999i	$0.318032\pi$
$720$	0	0
$721$	3.72176e52	0.499972
$722$	0	0
$723$	4.66567e52	0.595464
$724$	0	0
$725$	2.26372e52	0.274517
$726$	0	0
$727$	−2.45484e52	−0.282903	−0.141451	−	0.989945i	$-0.545177\pi$
$727$	−2.45484e52	−0.282903	−0.141451	+	0.989945i	$0.545177\pi$
$728$	0	0
$729$	−1.58062e52	−0.173128
$730$	0	0
$731$	1.55519e53	1.61924
$732$	0	0
$733$	−1.57488e53	−1.55892	−0.779462	−	0.626450i	$-0.784507\pi$
$733$	−1.57488e53	−1.55892	−0.779462	+	0.626450i	$0.784507\pi$
$734$	0	0
$735$	2.32621e51	0.0218944
$736$	0	0
$737$	−3.64159e53	−3.25944
$738$	0	0
$739$	−1.37347e53	−1.16922	−0.584611	−	0.811314i	$-0.698753\pi$
$739$	−1.37347e53	−1.16922	−0.584611	+	0.811314i	$0.698753\pi$
$740$	0	0
$741$	1.08674e53	0.880011
$742$	0	0
$743$	−1.57364e53	−1.21231	−0.606154	−	0.795348i	$-0.707288\pi$
$743$	−1.57364e53	−1.21231	−0.606154	+	0.795348i	$0.707288\pi$
$744$	0	0
$745$	7.70905e52	0.565080
$746$	0	0
$747$	6.56780e52	0.458130
$748$	0	0
$749$	5.74174e52	0.381180
$750$	0	0
$751$	2.38806e53	1.50906	0.754531	−	0.656265i	$-0.227864\pi$
$751$	2.38806e53	1.50906	0.754531	+	0.656265i	$0.227864\pi$
$752$	0	0
$753$	−5.85958e52	−0.352500
$754$	0	0
$755$	−2.79368e53	−1.60014
$756$	0	0
$757$	5.97971e51	0.0326142	0.0163071	−	0.999867i	$-0.494809\pi$
$757$	5.97971e51	0.0326142	0.0163071	+	0.999867i	$0.494809\pi$
$758$	0	0
$759$	−4.85708e52	−0.252291
$760$	0	0
$761$	−1.88945e52	−0.0934801	−0.0467400	−	0.998907i	$-0.514883\pi$
$761$	−1.88945e52	−0.0934801	−0.0467400	+	0.998907i	$0.514883\pi$
$762$	0	0
$763$	1.57163e53	0.740709
$764$	0	0
$765$	1.91355e53	0.859217
$766$	0	0
$767$	8.99490e52	0.384842
$768$	0	0
$769$	−1.33712e52	−0.0545171	−0.0272585	−	0.999628i	$-0.508678\pi$
$769$	−1.33712e52	−0.0545171	−0.0272585	+	0.999628i	$0.508678\pi$
$770$	0	0
$771$	3.00610e52	0.116815
$772$	0	0
$773$	1.38025e53	0.511256	0.255628	−	0.966775i	$-0.417718\pi$
$773$	1.38025e53	0.511256	0.255628	+	0.966775i	$0.417718\pi$
$774$	0	0
$775$	−2.09680e53	−0.740414
$776$	0	0
$777$	−1.07198e53	−0.360910
$778$	0	0
$779$	5.62549e53	1.80600
$780$	0	0
$781$	−9.55838e52	−0.292644
$782$	0	0
$783$	9.71389e52	0.283661
$784$	0	0
$785$	1.55829e53	0.434069
$786$	0	0
$787$	1.00310e53	0.266572	0.133286	−	0.991078i	$-0.457447\pi$
$787$	1.00310e53	0.266572	0.133286	+	0.991078i	$0.457447\pi$
$788$	0	0
$789$	−1.81294e53	−0.459683
$790$	0	0
$791$	4.80669e53	1.16300
$792$	0	0
$793$	−5.26023e53	−1.21465
$794$	0	0
$795$	−9.34434e53	−2.05948
$796$	0	0
$797$	4.83153e53	1.01650	0.508248	−	0.861211i	$-0.330293\pi$
$797$	4.83153e53	1.01650	0.508248	+	0.861211i	$0.330293\pi$
$798$	0	0
$799$	−2.80769e53	−0.563941
$800$	0	0
$801$	2.14641e53	0.411635
$802$	0	0
$803$	−7.63754e53	−1.39867
$804$	0	0
$805$	−7.26881e52	−0.127127
$806$	0	0
$807$	−1.30950e54	−2.18748
$808$	0	0
$809$	4.25989e53	0.679753	0.339877	−	0.940470i	$-0.389615\pi$
$809$	4.25989e53	0.679753	0.339877	+	0.940470i	$0.389615\pi$
$810$	0	0
$811$	−1.34482e53	−0.205012	−0.102506	−	0.994732i	$-0.532686\pi$
$811$	−1.34482e53	−0.205012	−0.102506	+	0.994732i	$0.532686\pi$
$812$	0	0
$813$	−6.13297e52	−0.0893297
$814$	0	0
$815$	3.28516e53	0.457236
$816$	0	0
$817$	1.13791e54	1.51356
$818$	0	0
$819$	−3.20917e53	−0.407981
$820$	0	0
$821$	6.82298e53	0.829136	0.414568	−	0.910018i	$-0.363933\pi$
$821$	6.82298e53	0.829136	0.414568	+	0.910018i	$0.363933\pi$
$822$	0	0
$823$	−8.54283e52	−0.0992441	−0.0496220	−	0.998768i	$-0.515802\pi$
$823$	−8.54283e52	−0.0992441	−0.0496220	+	0.998768i	$0.515802\pi$
$824$	0	0
$825$	9.79700e53	1.08817
$826$	0	0
$827$	1.49932e54	1.59236	0.796182	−	0.605057i	$-0.206850\pi$
$827$	1.49932e54	1.59236	0.796182	+	0.605057i	$0.206850\pi$
$828$	0	0
$829$	−1.04682e54	−1.06320	−0.531600	−	0.846995i	$-0.678409\pi$
$829$	−1.04682e54	−1.06320	−0.531600	+	0.846995i	$0.678409\pi$
$830$	0	0
$831$	1.19377e54	1.15958
$832$	0	0
$833$	−1.72807e52	−0.0160557
$834$	0	0
$835$	−9.42115e53	−0.837345
$836$	0	0
$837$	−8.99761e53	−0.765079
$838$	0	0
$839$	3.09433e53	0.251751	0.125875	−	0.992046i	$-0.459826\pi$
$839$	3.09433e53	0.251751	0.125875	+	0.992046i	$0.459826\pi$
$840$	0	0
$841$	−8.21905e53	−0.639876
$842$	0	0
$843$	−1.46810e54	−1.09382
$844$	0	0
$845$	−9.72134e53	−0.693229
$846$	0	0
$847$	3.59581e54	2.45445
$848$	0	0
$849$	2.48569e54	1.62426
$850$	0	0
$851$	−4.82718e52	−0.0301993
$852$	0	0
$853$	−5.28067e53	−0.316324	−0.158162	−	0.987413i	$-0.550557\pi$
$853$	−5.28067e53	−0.316324	−0.158162	+	0.987413i	$0.550557\pi$
$854$	0	0
$855$	1.40012e54	0.803140
$856$	0	0
$857$	−2.72634e54	−1.49773	−0.748866	−	0.662722i	$-0.769401\pi$
$857$	−2.72634e54	−1.49773	−0.748866	+	0.662722i	$0.769401\pi$
$858$	0	0
$859$	−2.66016e54	−1.39970	−0.699850	−	0.714290i	$-0.746750\pi$
$859$	−2.66016e54	−1.39970	−0.699850	+	0.714290i	$0.746750\pi$
$860$	0	0
$861$	−4.29920e54	−2.16685
$862$	0	0
$863$	2.08753e54	1.00793	0.503967	−	0.863723i	$-0.331873\pi$
$863$	2.08753e54	1.00793	0.503967	+	0.863723i	$0.331873\pi$
$864$	0	0
$865$	−9.08129e53	−0.420097
$866$	0	0
$867$	−7.98728e53	−0.354035
$868$	0	0
$869$	1.35509e54	0.575576
$870$	0	0
$871$	−2.80398e54	−1.14140
$872$	0	0
$873$	−1.70560e54	−0.665446
$874$	0	0
$875$	−1.73892e54	−0.650325
$876$	0	0
$877$	4.92176e54	1.76452	0.882262	−	0.470759i	$-0.156020\pi$
$877$	4.92176e54	1.76452	0.882262	+	0.470759i	$0.156020\pi$
$878$	0	0
$879$	6.27818e54	2.15794
$880$	0	0
$881$	−2.15926e53	−0.0711624	−0.0355812	−	0.999367i	$-0.511328\pi$
$881$	−2.15926e53	−0.0711624	−0.0355812	+	0.999367i	$0.511328\pi$
$882$	0	0
$883$	−4.36807e54	−1.38044	−0.690219	−	0.723600i	$-0.742486\pi$
$883$	−4.36807e54	−1.38044	−0.690219	+	0.723600i	$0.742486\pi$
$884$	0	0
$885$	2.99914e54	0.908962
$886$	0	0
$887$	1.83972e54	0.534768	0.267384	−	0.963590i	$-0.413841\pi$
$887$	1.83972e54	0.534768	0.267384	+	0.963590i	$0.413841\pi$
$888$	0	0
$889$	−1.23656e54	−0.344773
$890$	0	0
$891$	8.59122e54	2.29783
$892$	0	0
$893$	−2.05435e54	−0.527135
$894$	0	0
$895$	2.98421e54	0.734688
$896$	0	0
$897$	−3.73989e53	−0.0883485
$898$	0	0
$899$	−4.28462e54	−0.971310
$900$	0	0
$901$	6.94162e54	1.51026
$902$	0	0
$903$	−8.69632e54	−1.81598
$904$	0	0
$905$	6.52271e53	0.130746
$906$	0	0
$907$	7.26656e54	1.39828	0.699140	−	0.714985i	$-0.253566\pi$
$907$	7.26656e54	1.39828	0.699140	+	0.714985i	$0.253566\pi$
$908$	0	0
$909$	−3.61954e54	−0.668685
$910$	0	0
$911$	−1.33361e54	−0.236559	−0.118280	−	0.992980i	$-0.537738\pi$
$911$	−1.33361e54	−0.236559	−0.118280	+	0.992980i	$0.537738\pi$
$912$	0	0
$913$	−7.95861e54	−1.35559
$914$	0	0
$915$	−1.75390e55	−2.86889
$916$	0	0
$917$	−1.45653e54	−0.228816
$918$	0	0
$919$	1.13652e55	1.71490	0.857449	−	0.514568i	$-0.172048\pi$
$919$	1.13652e55	1.71490	0.857449	+	0.514568i	$0.172048\pi$
$920$	0	0
$921$	8.04261e54	1.16572
$922$	0	0
$923$	−7.35985e53	−0.102480
$924$	0	0
$925$	9.73671e53	0.130254
$926$	0	0
$927$	2.46708e54	0.317109
$928$	0	0
$929$	−2.74976e54	−0.339628	−0.169814	−	0.985476i	$-0.554317\pi$
$929$	−2.74976e54	−0.339628	−0.169814	+	0.985476i	$0.554317\pi$
$930$	0	0
$931$	−1.26441e53	−0.0150078
$932$	0	0
$933$	−5.88321e54	−0.671125
$934$	0	0
$935$	−2.31876e55	−2.54239
$936$	0	0
$937$	2.69240e54	0.283764	0.141882	−	0.989884i	$-0.454685\pi$
$937$	2.69240e54	0.283764	0.141882	+	0.989884i	$0.454685\pi$
$938$	0	0
$939$	7.60203e54	0.770222
$940$	0	0
$941$	1.72393e55	1.67924	0.839620	−	0.543174i	$-0.182778\pi$
$941$	1.72393e55	1.67924	0.839620	+	0.543174i	$0.182778\pi$
$942$	0	0
$943$	−1.93595e54	−0.181313
$944$	0	0
$945$	6.29148e54	0.566584
$946$	0	0
$947$	−2.08729e55	−1.80763	−0.903814	−	0.427926i	$-0.859244\pi$
$947$	−2.08729e55	−1.80763	−0.903814	+	0.427926i	$0.859244\pi$
$948$	0	0
$949$	−5.88082e54	−0.489794
$950$	0	0
$951$	2.93799e55	2.35348
$952$	0	0
$953$	1.02561e54	0.0790251	0.0395126	−	0.999219i	$-0.487419\pi$
$953$	1.02561e54	0.0790251	0.0395126	+	0.999219i	$0.487419\pi$
$954$	0	0
$955$	1.05465e54	0.0781709
$956$	0	0
$957$	2.00193e55	1.42751
$958$	0	0
$959$	8.11089e54	0.556448
$960$	0	0
$961$	2.45379e55	1.61977
$962$	0	0
$963$	3.80609e54	0.241764
$964$	0	0
$965$	5.20281e53	0.0318041
$966$	0	0
$967$	−8.82729e54	−0.519323	−0.259662	−	0.965700i	$-0.583611\pi$
$967$	−8.82729e54	−0.519323	−0.259662	+	0.965700i	$0.583611\pi$
$968$	0	0
$969$	−2.69177e55	−1.52422
$970$	0	0
$971$	−8.94384e54	−0.487493	−0.243746	−	0.969839i	$-0.578376\pi$
$971$	−8.94384e54	−0.487493	−0.243746	+	0.969839i	$0.578376\pi$
$972$	0	0
$973$	−2.76047e55	−1.44842
$974$	0	0
$975$	7.54358e54	0.381059
$976$	0	0
$977$	−1.98871e55	−0.967211	−0.483606	−	0.875286i	$-0.660673\pi$
$977$	−1.98871e55	−0.967211	−0.483606	+	0.875286i	$0.660673\pi$
$978$	0	0
$979$	−2.60094e55	−1.21801
$980$	0	0
$981$	1.04181e55	0.469797
$982$	0	0
$983$	−2.11570e55	−0.918785	−0.459393	−	0.888233i	$-0.651933\pi$
$983$	−2.11570e55	−0.918785	−0.459393	+	0.888233i	$0.651933\pi$
$984$	0	0
$985$	1.92872e55	0.806674
$986$	0	0
$987$	1.57001e55	0.632462
$988$	0	0
$989$	−3.91600e54	−0.151954
$990$	0	0
$991$	1.15442e55	0.431520	0.215760	−	0.976446i	$-0.430777\pi$
$991$	1.15442e55	0.431520	0.215760	+	0.976446i	$0.430777\pi$
$992$	0	0
$993$	−3.87758e55	−1.39637
$994$	0	0
$995$	5.29863e55	1.83838
$996$	0	0
$997$	−3.36086e55	−1.12355	−0.561773	−	0.827292i	$-0.689880\pi$
$997$	−3.36086e55	−1.12355	−0.561773	+	0.827292i	$0.689880\pi$
$998$	0	0
$999$	4.17814e54	0.134593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.38.a.a.1.1	✓	3
4.3	odd	2		16.38.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.38.a.a.1.1	✓	3	1.1	even	1	trivial
16.38.a.d.1.3		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)

\(f(q)\)	\(=\)	\(q-8.56646e8 q^{3} +1.02977e13 q^{5} +4.27767e15 q^{7} +2.83558e17 q^{9} +O(q^{10})\)	\(q-8.56646e8 q^{3} +1.02977e13 q^{5} +4.27767e15 q^{7} +2.83558e17 q^{9} -3.43605e19 q^{11} -2.64572e20 q^{13} -8.82151e21 q^{15} +6.55323e22 q^{17} +4.79491e23 q^{19} -3.66444e24 q^{21} -1.65012e24 q^{23} +3.32838e25 q^{25} +1.42825e26 q^{27} +6.80125e26 q^{29} -6.29975e27 q^{31} +2.94347e28 q^{33} +4.40503e28 q^{35} +2.92536e28 q^{37} +2.26644e29 q^{39} +1.17322e30 q^{41} +2.37316e30 q^{43} +2.92001e30 q^{45} -4.28443e30 q^{47} -2.63698e29 q^{49} -5.61380e31 q^{51} +1.05927e32 q^{53} -3.53835e32 q^{55} -4.10754e32 q^{57} -3.39980e32 q^{59} +1.98820e33 q^{61} +1.21297e33 q^{63} -2.72449e33 q^{65} +1.05982e34 q^{67} +1.41357e33 q^{69} +2.78180e33 q^{71} +2.22277e34 q^{73} -2.85124e34 q^{75} -1.46983e35 q^{77} -3.94375e34 q^{79} -2.50032e35 q^{81} +2.31621e35 q^{83} +6.74835e35 q^{85} -5.82626e35 q^{87} +7.56957e35 q^{89} -1.13175e36 q^{91} +5.39665e36 q^{93} +4.93767e36 q^{95} -6.01498e36 q^{97} -9.74319e36 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 272163492 q^{3} + 3641045116194 q^{5} + 15\!\cdots\!92 q^{7}+ \cdots + 10\!\cdots\!63 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more