LMFDB - Embedded newform 4.34.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,34,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, 34, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 34);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 34 $
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:	$27.5931315524$
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} - 65185566x - 173679864984 $ x^3 - x^2 - 65185566*x - 173679864984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9172.25$ 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)

$f(q)$	$=$	$q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +O(q^{10})$	$q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +2.12022e17 q^{11} -2.20110e18 q^{13} +1.31252e19 q^{15} -9.09205e19 q^{17} +5.28541e20 q^{19} +2.43500e21 q^{21} -2.52550e22 q^{23} +2.49568e23 q^{25} -2.31004e23 q^{27} +2.22959e24 q^{29} +5.96021e24 q^{31} +4.59998e24 q^{33} +6.78977e25 q^{35} -1.07477e25 q^{37} -4.77546e25 q^{39} +5.45950e25 q^{41} -7.34669e26 q^{43} -3.07828e27 q^{45} -2.88377e27 q^{47} +4.86547e27 q^{49} -1.97259e27 q^{51} -6.58827e27 q^{53} +1.28266e29 q^{55} +1.14671e28 q^{57} -1.55330e29 q^{59} +5.35869e29 q^{61} -5.71086e29 q^{63} -1.33159e30 q^{65} -5.20034e29 q^{67} -5.47926e29 q^{69} +2.97143e30 q^{71} -5.09019e30 q^{73} +5.41457e30 q^{75} +2.37961e31 q^{77} -1.81467e30 q^{79} +2.32747e31 q^{81} -3.91242e31 q^{83} -5.50038e31 q^{85} +4.83726e31 q^{87} -1.76955e32 q^{89} -2.47039e32 q^{91} +1.29311e32 q^{93} +3.19749e32 q^{95} +8.13793e32 q^{97} -1.07884e33 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) $ 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9	$ 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots + 22\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9 + 227617657302449700 * q^11 + 272970442217358762 * q^13 - 32142473481241105848 * q^15 + 93037188311816716854 * q^17 + 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 + 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 + 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 + 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 + 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 + 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 - 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 - 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 + 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 + 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 + 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 - 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 - 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 + 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 + 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 + 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 - 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 - 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 - 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 - 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 + 2255131944887948953089434155026900 * 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
$3$	2.16957e7	0.290987	0.145494	−	0.989359i	$-0.453523\pi$
$3$	2.16957e7	0.290987	0.145494	+	0.989359i	$0.453523\pi$
$4$	0	0
$5$	6.04966e11	1.77307	0.886535	−	0.462662i	$-0.153106\pi$
$5$	6.04966e11	1.77307	0.886535	+	0.462662i	$0.153106\pi$
$6$	0	0
$7$	1.12234e14	1.27646	0.638229	−	0.769846i	$-0.279667\pi$
$7$	1.12234e14	1.27646	0.638229	+	0.769846i	$0.279667\pi$
$8$	0	0
$9$	−5.08836e15	−0.915326
$10$	0	0
$11$	2.12022e17	1.39124	0.695620	−	0.718410i	$-0.255130\pi$
$11$	2.12022e17	1.39124	0.695620	+	0.718410i	$0.255130\pi$
$12$	0	0
$13$	−2.20110e18	−0.917435	−0.458718	−	0.888582i	$-0.651691\pi$
$13$	−2.20110e18	−0.917435	−0.458718	+	0.888582i	$0.651691\pi$
$14$	0	0
$15$	1.31252e19	0.515940
$16$	0	0
$17$	−9.09205e19	−0.453163	−0.226582	−	0.973992i	$-0.572755\pi$
$17$	−9.09205e19	−0.453163	−0.226582	+	0.973992i	$0.572755\pi$
$18$	0	0
$19$	5.28541e20	0.420382	0.210191	−	0.977660i	$-0.432591\pi$
$19$	5.28541e20	0.420382	0.210191	+	0.977660i	$0.432591\pi$
$20$	0	0
$21$	2.43500e21	0.371433
$22$	0	0
$23$	−2.52550e22	−0.858693	−0.429346	−	0.903140i	$-0.641256\pi$
$23$	−2.52550e22	−0.858693	−0.429346	+	0.903140i	$0.641256\pi$
$24$	0	0
$25$	2.49568e23	2.14377
$26$	0	0
$27$	−2.31004e23	−0.557336
$28$	0	0
$29$	2.22959e24	1.65447	0.827233	−	0.561859i	$-0.189914\pi$
$29$	2.22959e24	1.65447	0.827233	+	0.561859i	$0.189914\pi$
$30$	0	0
$31$	5.96021e24	1.47161	0.735806	−	0.677192i	$-0.236803\pi$
$31$	5.96021e24	1.47161	0.735806	+	0.677192i	$0.236803\pi$
$32$	0	0
$33$	4.59998e24	0.404833
$34$	0	0
$35$	6.78977e25	2.26325
$36$	0	0
$37$	−1.07477e25	−0.143215	−0.0716074	−	0.997433i	$-0.522813\pi$
$37$	−1.07477e25	−0.143215	−0.0716074	+	0.997433i	$0.522813\pi$
$38$	0	0
$39$	−4.77546e25	−0.266962
$40$	0	0
$41$	5.45950e25	0.133727	0.0668635	−	0.997762i	$-0.478701\pi$
$41$	5.45950e25	0.133727	0.0668635	+	0.997762i	$0.478701\pi$
$42$	0	0
$43$	−7.34669e26	−0.820092	−0.410046	−	0.912065i	$-0.634487\pi$
$43$	−7.34669e26	−0.820092	−0.410046	+	0.912065i	$0.634487\pi$
$44$	0	0
$45$	−3.07828e27	−1.62294
$46$	0	0
$47$	−2.88377e27	−0.741901	−0.370950	−	0.928653i	$-0.620968\pi$
$47$	−2.88377e27	−0.741901	−0.370950	+	0.928653i	$0.620968\pi$
$48$	0	0
$49$	4.86547e27	0.629346
$50$	0	0
$51$	−1.97259e27	−0.131865
$52$	0	0
$53$	−6.58827e27	−0.233464	−0.116732	−	0.993163i	$-0.537242\pi$
$53$	−6.58827e27	−0.233464	−0.116732	+	0.993163i	$0.537242\pi$
$54$	0	0
$55$	1.28266e29	2.46676
$56$	0	0
$57$	1.14671e28	0.122326
$58$	0	0
$59$	−1.55330e29	−0.937990	−0.468995	−	0.883201i	$-0.655384\pi$
$59$	−1.55330e29	−0.937990	−0.468995	+	0.883201i	$0.655384\pi$
$60$	0	0
$61$	5.35869e29	1.86688	0.933440	−	0.358733i	$-0.116791\pi$
$61$	5.35869e29	1.86688	0.933440	+	0.358733i	$0.116791\pi$
$62$	0	0
$63$	−5.71086e29	−1.16838
$64$	0	0
$65$	−1.33159e30	−1.62668
$66$	0	0
$67$	−5.20034e29	−0.385298	−0.192649	−	0.981268i	$-0.561708\pi$
$67$	−5.20034e29	−0.385298	−0.192649	+	0.981268i	$0.561708\pi$
$68$	0	0
$69$	−5.47926e29	−0.249869
$70$	0	0
$71$	2.97143e30	0.845675	0.422837	−	0.906206i	$-0.361034\pi$
$71$	2.97143e30	0.845675	0.422837	+	0.906206i	$0.361034\pi$
$72$	0	0
$73$	−5.09019e30	−0.916024	−0.458012	−	0.888946i	$-0.651438\pi$
$73$	−5.09019e30	−0.916024	−0.458012	+	0.888946i	$0.651438\pi$
$74$	0	0
$75$	5.41457e30	0.623811
$76$	0	0
$77$	2.37961e31	1.77586
$78$	0	0
$79$	−1.81467e30	−0.0887054	−0.0443527	−	0.999016i	$-0.514123\pi$
$79$	−1.81467e30	−0.0887054	−0.0443527	+	0.999016i	$0.514123\pi$
$80$	0	0
$81$	2.32747e31	0.753149
$82$	0	0
$83$	−3.91242e31	−0.846559	−0.423279	−	0.905999i	$-0.639121\pi$
$83$	−3.91242e31	−0.846559	−0.423279	+	0.905999i	$0.639121\pi$
$84$	0	0
$85$	−5.50038e31	−0.803490
$86$	0	0
$87$	4.83726e31	0.481428
$88$	0	0
$89$	−1.76955e32	−1.21040	−0.605199	−	0.796075i	$-0.706906\pi$
$89$	−1.76955e32	−1.21040	−0.605199	+	0.796075i	$0.706906\pi$
$90$	0	0
$91$	−2.47039e32	−1.17107
$92$	0	0
$93$	1.29311e32	0.428220
$94$	0	0
$95$	3.19749e32	0.745367
$96$	0	0
$97$	8.13793e32	1.34518	0.672590	−	0.740016i	$-0.265182\pi$
$97$	8.13793e32	1.34518	0.672590	+	0.740016i	$0.265182\pi$
$98$	0	0
$99$	−1.07884e33	−1.27344
$100$	0	0
$101$	6.19750e32	0.525913	0.262956	−	0.964808i	$-0.415302\pi$
$101$	6.19750e32	0.525913	0.262956	+	0.964808i	$0.415302\pi$
$102$	0	0
$103$	2.96344e33	1.81962	0.909812	−	0.415020i	$-0.136225\pi$
$103$	2.96344e33	1.81962	0.909812	+	0.415020i	$0.136225\pi$
$104$	0	0
$105$	1.47309e33	0.658577
$106$	0	0
$107$	−5.09640e33	−1.66890	−0.834451	−	0.551082i	$-0.814215\pi$
$107$	−5.09640e33	−1.66890	−0.834451	+	0.551082i	$0.814215\pi$
$108$	0	0
$109$	−5.31361e33	−1.28189	−0.640947	−	0.767585i	$-0.721458\pi$
$109$	−5.31361e33	−1.28189	−0.640947	+	0.767585i	$0.721458\pi$
$110$	0	0
$111$	−2.33180e32	−0.0416737
$112$	0	0
$113$	3.75830e33	0.500262	0.250131	−	0.968212i	$-0.419526\pi$
$113$	3.75830e33	0.500262	0.250131	+	0.968212i	$0.419526\pi$
$114$	0	0
$115$	−1.52784e34	−1.52252
$116$	0	0
$117$	1.12000e34	0.839753
$118$	0	0
$119$	−1.02044e34	−0.578444
$120$	0	0
$121$	2.17282e34	0.935546
$122$	0	0
$123$	1.18448e33	0.0389128
$124$	0	0
$125$	8.05529e34	2.02799
$126$	0	0
$127$	−2.74550e34	−0.531937	−0.265968	−	0.963982i	$-0.585692\pi$
$127$	−2.74550e34	−0.531937	−0.265968	+	0.963982i	$0.585692\pi$
$128$	0	0
$129$	−1.59392e34	−0.238636
$130$	0	0
$131$	−1.61531e34	−0.187620	−0.0938099	−	0.995590i	$-0.529905\pi$
$131$	−1.61531e34	−0.187620	−0.0938099	+	0.995590i	$0.529905\pi$
$132$	0	0
$133$	5.93203e34	0.536601
$134$	0	0
$135$	−1.39749e35	−0.988194
$136$	0	0
$137$	−1.20499e35	−0.668490	−0.334245	−	0.942486i	$-0.608481\pi$
$137$	−1.20499e35	−0.668490	−0.334245	+	0.942486i	$0.608481\pi$
$138$	0	0
$139$	−1.71862e35	−0.750648	−0.375324	−	0.926894i	$-0.622469\pi$
$139$	−1.71862e35	−0.750648	−0.375324	+	0.926894i	$0.622469\pi$
$140$	0	0
$141$	−6.25655e34	−0.215884
$142$	0	0
$143$	−4.66683e35	−1.27637
$144$	0	0
$145$	1.34882e36	2.93348
$146$	0	0
$147$	1.05560e35	0.183132
$148$	0	0
$149$	−3.16022e33	−0.00438676	−0.00219338	−	0.999998i	$-0.500698\pi$
$149$	−3.16022e33	−0.00438676	−0.00219338	+	0.999998i	$0.500698\pi$
$150$	0	0
$151$	−8.88992e35	−0.990326	−0.495163	−	0.868800i	$-0.664892\pi$
$151$	−8.88992e35	−0.990326	−0.495163	+	0.868800i	$0.664892\pi$
$152$	0	0
$153$	4.62636e35	0.414792
$154$	0	0
$155$	3.60572e36	2.60927
$156$	0	0
$157$	−1.61326e36	−0.944842	−0.472421	−	0.881373i	$-0.656620\pi$
$157$	−1.61326e36	−0.944842	−0.472421	+	0.881373i	$0.656620\pi$
$158$	0	0
$159$	−1.42937e35	−0.0679349
$160$	0	0
$161$	−2.83447e36	−1.09609
$162$	0	0
$163$	−5.86098e36	−1.84874	−0.924368	−	0.381501i	$-0.875407\pi$
$163$	−5.86098e36	−1.84874	−0.924368	+	0.381501i	$0.875407\pi$
$164$	0	0
$165$	2.78283e36	0.717797
$166$	0	0
$167$	7.82436e35	0.165435	0.0827175	−	0.996573i	$-0.473640\pi$
$167$	7.82436e35	0.165435	0.0827175	+	0.996573i	$0.473640\pi$
$168$	0	0
$169$	−9.11269e35	−0.158313
$170$	0	0
$171$	−2.68941e36	−0.384787
$172$	0	0
$173$	−1.05668e37	−1.24791	−0.623957	−	0.781459i	$-0.714476\pi$
$173$	−1.05668e37	−1.24791	−0.623957	+	0.781459i	$0.714476\pi$
$174$	0	0
$175$	2.80100e37	2.73644
$176$	0	0
$177$	−3.37001e36	−0.272943
$178$	0	0
$179$	1.19253e37	0.802405	0.401202	−	0.915989i	$-0.368592\pi$
$179$	1.19253e37	0.802405	0.401202	+	0.915989i	$0.368592\pi$
$180$	0	0
$181$	2.20113e36	0.123296	0.0616481	−	0.998098i	$-0.480364\pi$
$181$	2.20113e36	0.123296	0.0616481	+	0.998098i	$0.480364\pi$
$182$	0	0
$183$	1.16261e37	0.543238
$184$	0	0
$185$	−6.50200e36	−0.253930
$186$	0	0
$187$	−1.92772e37	−0.630458
$188$	0	0
$189$	−2.59265e37	−0.711416
$190$	0	0
$191$	1.41890e37	0.327265	0.163633	−	0.986521i	$-0.447679\pi$
$191$	1.41890e37	0.327265	0.163633	+	0.986521i	$0.447679\pi$
$192$	0	0
$193$	−3.71061e36	−0.0720693	−0.0360346	−	0.999351i	$-0.511473\pi$
$193$	−3.71061e36	−0.0720693	−0.0360346	+	0.999351i	$0.511473\pi$
$194$	0	0
$195$	−2.88899e37	−0.473342
$196$	0	0
$197$	−4.85371e37	−0.672018	−0.336009	−	0.941859i	$-0.609077\pi$
$197$	−4.85371e37	−0.672018	−0.336009	+	0.941859i	$0.609077\pi$
$198$	0	0
$199$	−1.20691e38	−1.41449	−0.707245	−	0.706968i	$-0.750062\pi$
$199$	−1.20691e38	−1.41449	−0.707245	+	0.706968i	$0.750062\pi$
$200$	0	0
$201$	−1.12825e37	−0.112117
$202$	0	0
$203$	2.50236e38	2.11186
$204$	0	0
$205$	3.30281e37	0.237107
$206$	0	0
$207$	1.28506e38	0.785984
$208$	0	0
$209$	1.12062e38	0.584852
$210$	0	0
$211$	−8.47609e37	−0.378038	−0.189019	−	0.981973i	$-0.560531\pi$
$211$	−8.47609e37	−0.378038	−0.189019	+	0.981973i	$0.560531\pi$
$212$	0	0
$213$	6.44675e37	0.246081
$214$	0	0
$215$	−4.44450e38	−1.45408
$216$	0	0
$217$	6.68938e38	1.87845
$218$	0	0
$219$	−1.10435e38	−0.266551
$220$	0	0
$221$	2.00126e38	0.415748
$222$	0	0
$223$	−9.55613e38	−1.71101	−0.855504	−	0.517797i	$-0.826752\pi$
$223$	−9.55613e38	−1.71101	−0.855504	+	0.517797i	$0.826752\pi$
$224$	0	0
$225$	−1.26989e39	−1.96225
$226$	0	0
$227$	−1.31422e38	−0.175486	−0.0877430	−	0.996143i	$-0.527965\pi$
$227$	−1.31422e38	−0.175486	−0.0877430	+	0.996143i	$0.527965\pi$
$228$	0	0
$229$	1.01200e39	1.16922	0.584610	−	0.811314i	$-0.301247\pi$
$229$	1.01200e39	1.16922	0.584610	+	0.811314i	$0.301247\pi$
$230$	0	0
$231$	5.16274e38	0.516752
$232$	0	0
$233$	4.43939e38	0.385435	0.192717	−	0.981254i	$-0.438270\pi$
$233$	4.43939e38	0.385435	0.192717	+	0.981254i	$0.438270\pi$
$234$	0	0
$235$	−1.74458e39	−1.31544
$236$	0	0
$237$	−3.93706e37	−0.0258121
$238$	0	0
$239$	2.91733e39	1.66502	0.832512	−	0.554008i	$-0.186902\pi$
$239$	2.91733e39	1.66502	0.832512	+	0.554008i	$0.186902\pi$
$240$	0	0
$241$	4.88758e36	0.00243115	0.00121558	−	0.999999i	$-0.499613\pi$
$241$	4.88758e36	0.00243115	0.00121558	+	0.999999i	$0.499613\pi$
$242$	0	0
$243$	1.78912e39	0.776492
$244$	0	0
$245$	2.94344e39	1.11587
$246$	0	0
$247$	−1.16337e39	−0.385674
$248$	0	0
$249$	−8.48829e38	−0.246338
$250$	0	0
$251$	−2.40071e39	−0.610553	−0.305276	−	0.952264i	$-0.598749\pi$
$251$	−2.40071e39	−0.610553	−0.305276	+	0.952264i	$0.598749\pi$
$252$	0	0
$253$	−5.35461e39	−1.19465
$254$	0	0
$255$	−1.19335e39	−0.233805
$256$	0	0
$257$	1.55258e39	0.267398	0.133699	−	0.991022i	$-0.457315\pi$
$257$	1.55258e39	0.267398	0.133699	+	0.991022i	$0.457315\pi$
$258$	0	0
$259$	−1.20626e39	−0.182808
$260$	0	0
$261$	−1.13449e40	−1.51438
$262$	0	0
$263$	1.41636e40	1.66688	0.833442	−	0.552607i	$-0.186367\pi$
$263$	1.41636e40	1.66688	0.833442	+	0.552607i	$0.186367\pi$
$264$	0	0
$265$	−3.98567e39	−0.413947
$266$	0	0
$267$	−3.83917e39	−0.352210
$268$	0	0
$269$	−4.46974e39	−0.362552	−0.181276	−	0.983432i	$-0.558023\pi$
$269$	−4.46974e39	−0.362552	−0.181276	+	0.983432i	$0.558023\pi$
$270$	0	0
$271$	4.70466e39	0.337703	0.168851	−	0.985642i	$-0.445994\pi$
$271$	4.70466e39	0.337703	0.168851	+	0.985642i	$0.445994\pi$
$272$	0	0
$273$	−5.35969e39	−0.340766
$274$	0	0
$275$	5.29140e40	2.98250
$276$	0	0
$277$	2.48781e40	1.24423	0.622116	−	0.782925i	$-0.286273\pi$
$277$	2.48781e40	1.24423	0.622116	+	0.782925i	$0.286273\pi$
$278$	0	0
$279$	−3.03276e40	−1.34701
$280$	0	0
$281$	6.82873e38	0.0269579	0.0134789	−	0.999909i	$-0.495709\pi$
$281$	6.82873e38	0.0269579	0.0134789	+	0.999909i	$0.495709\pi$
$282$	0	0
$283$	−3.13055e40	−1.09937	−0.549686	−	0.835371i	$-0.685253\pi$
$283$	−3.13055e40	−1.09937	−0.549686	+	0.835371i	$0.685253\pi$
$284$	0	0
$285$	6.93720e39	0.216892
$286$	0	0
$287$	6.12741e39	0.170697
$288$	0	0
$289$	−3.19880e40	−0.794643
$290$	0	0
$291$	1.76558e40	0.391430
$292$	0	0
$293$	4.43484e38	0.00878138	0.00439069	−	0.999990i	$-0.498602\pi$
$293$	4.43484e38	0.00878138	0.00439069	+	0.999990i	$0.498602\pi$
$294$	0	0
$295$	−9.39696e40	−1.66312
$296$	0	0
$297$	−4.89779e40	−0.775387
$298$	0	0
$299$	5.55889e40	0.787795
$300$	0	0
$301$	−8.24549e40	−1.04681
$302$	0	0
$303$	1.34459e40	0.153034
$304$	0	0
$305$	3.24182e41	3.31011
$306$	0	0
$307$	−9.15172e40	−0.838918	−0.419459	−	0.907774i	$-0.637780\pi$
$307$	−9.15172e40	−0.838918	−0.419459	+	0.907774i	$0.637780\pi$
$308$	0	0
$309$	6.42940e40	0.529487
$310$	0	0
$311$	−4.61111e40	−0.341397	−0.170698	−	0.985323i	$-0.554602\pi$
$311$	−4.61111e40	−0.341397	−0.170698	+	0.985323i	$0.554602\pi$
$312$	0	0
$313$	8.08893e40	0.538778	0.269389	−	0.963031i	$-0.413178\pi$
$313$	8.08893e40	0.538778	0.269389	+	0.963031i	$0.413178\pi$
$314$	0	0
$315$	−3.45488e41	−2.07161
$316$	0	0
$317$	2.72792e41	1.47351	0.736756	−	0.676159i	$-0.236357\pi$
$317$	2.72792e41	1.47351	0.736756	+	0.676159i	$0.236357\pi$
$318$	0	0
$319$	4.72722e41	2.30176
$320$	0	0
$321$	−1.10570e41	−0.485629
$322$	0	0
$323$	−4.80552e40	−0.190502
$324$	0	0
$325$	−5.49326e41	−1.96677
$326$	0	0
$327$	−1.15283e41	−0.373015
$328$	0	0
$329$	−3.23657e41	−0.947006
$330$	0	0
$331$	3.98182e40	0.105419	0.0527096	−	0.998610i	$-0.483214\pi$
$331$	3.98182e40	0.105419	0.0527096	+	0.998610i	$0.483214\pi$
$332$	0	0
$333$	5.46882e40	0.131088
$334$	0	0
$335$	−3.14603e41	−0.683161
$336$	0	0
$337$	−1.02227e41	−0.201220	−0.100610	−	0.994926i	$-0.532079\pi$
$337$	−1.02227e41	−0.201220	−0.100610	+	0.994926i	$0.532079\pi$
$338$	0	0
$339$	8.15391e40	0.145570
$340$	0	0
$341$	1.26370e42	2.04736
$342$	0	0
$343$	−3.21609e41	−0.473124
$344$	0	0
$345$	−3.31476e41	−0.443034
$346$	0	0
$347$	1.19238e42	1.44870	0.724348	−	0.689434i	$-0.242141\pi$
$347$	1.19238e42	1.44870	0.724348	+	0.689434i	$0.242141\pi$
$348$	0	0
$349$	−9.44292e41	−1.04349	−0.521743	−	0.853103i	$-0.674718\pi$
$349$	−9.44292e41	−1.04349	−0.521743	+	0.853103i	$0.674718\pi$
$350$	0	0
$351$	5.08463e41	0.511319
$352$	0	0
$353$	3.78690e41	0.346738	0.173369	−	0.984857i	$-0.444535\pi$
$353$	3.78690e41	0.346738	0.173369	+	0.984857i	$0.444535\pi$
$354$	0	0
$355$	1.79762e42	1.49944
$356$	0	0
$357$	−2.21391e41	−0.168320
$358$	0	0
$359$	−3.89278e41	−0.269899	−0.134950	−	0.990852i	$-0.543087\pi$
$359$	−3.89278e41	−0.269899	−0.134950	+	0.990852i	$0.543087\pi$
$360$	0	0
$361$	−1.30141e42	−0.823279
$362$	0	0
$363$	4.71410e41	0.272232
$364$	0	0
$365$	−3.07939e42	−1.62417
$366$	0	0
$367$	−1.36076e42	−0.655829	−0.327914	−	0.944707i	$-0.606346\pi$
$367$	−1.36076e42	−0.655829	−0.327914	+	0.944707i	$0.606346\pi$
$368$	0	0
$369$	−2.77799e41	−0.122404
$370$	0	0
$371$	−7.39427e41	−0.298007
$372$	0	0
$373$	3.42419e42	1.26288	0.631440	−	0.775425i	$-0.282464\pi$
$373$	3.42419e42	1.26288	0.631440	+	0.775425i	$0.282464\pi$
$374$	0	0
$375$	1.74765e42	0.590119
$376$	0	0
$377$	−4.90756e42	−1.51786
$378$	0	0
$379$	−1.97551e42	−0.559928	−0.279964	−	0.960011i	$-0.590322\pi$
$379$	−1.97551e42	−0.559928	−0.279964	+	0.960011i	$0.590322\pi$
$380$	0	0
$381$	−5.95656e41	−0.154787
$382$	0	0
$383$	−4.75888e42	−1.13429	−0.567146	−	0.823617i	$-0.691952\pi$
$383$	−4.75888e42	−1.13429	−0.567146	+	0.823617i	$0.691952\pi$
$384$	0	0
$385$	1.43958e43	3.14872
$386$	0	0
$387$	3.73826e42	0.750652
$388$	0	0
$389$	3.85205e42	0.710434	0.355217	−	0.934784i	$-0.384407\pi$
$389$	3.85205e42	0.710434	0.355217	+	0.934784i	$0.384407\pi$
$390$	0	0
$391$	2.29620e42	0.389128
$392$	0	0
$393$	−3.50453e41	−0.0545949
$394$	0	0
$395$	−1.09781e42	−0.157281
$396$	0	0
$397$	−7.07957e42	−0.933176	−0.466588	−	0.884475i	$-0.654517\pi$
$397$	−7.07957e42	−0.933176	−0.466588	+	0.884475i	$0.654517\pi$
$398$	0	0
$399$	1.28700e42	0.156144
$400$	0	0
$401$	−3.99621e42	−0.446444	−0.223222	−	0.974768i	$-0.571657\pi$
$401$	−3.99621e42	−0.446444	−0.223222	+	0.974768i	$0.571657\pi$
$402$	0	0
$403$	−1.31190e43	−1.35011
$404$	0	0
$405$	1.40804e43	1.33539
$406$	0	0
$407$	−2.27875e42	−0.199246
$408$	0	0
$409$	4.22070e42	0.340369	0.170184	−	0.985412i	$-0.445564\pi$
$409$	4.22070e42	0.340369	0.170184	+	0.985412i	$0.445564\pi$
$410$	0	0
$411$	−2.61432e42	−0.194522
$412$	0	0
$413$	−1.74334e43	−1.19731
$414$	0	0
$415$	−2.36688e43	−1.50101
$416$	0	0
$417$	−3.72868e42	−0.218429
$418$	0	0
$419$	−5.25752e42	−0.284610	−0.142305	−	0.989823i	$-0.545451\pi$
$419$	−5.25752e42	−0.284610	−0.142305	+	0.989823i	$0.545451\pi$
$420$	0	0
$421$	3.56053e43	1.78180	0.890902	−	0.454195i	$-0.150073\pi$
$421$	3.56053e43	1.78180	0.890902	+	0.454195i	$0.150073\pi$
$422$	0	0
$423$	1.46736e43	0.679081
$424$	0	0
$425$	−2.26909e43	−0.971480
$426$	0	0
$427$	6.01427e43	2.38300
$428$	0	0
$429$	−1.01250e43	−0.371408
$430$	0	0
$431$	−3.50406e43	−1.19041	−0.595205	−	0.803574i	$-0.702929\pi$
$431$	−3.50406e43	−1.19041	−0.595205	+	0.803574i	$0.702929\pi$
$432$	0	0
$433$	4.65228e43	1.46425	0.732125	−	0.681170i	$-0.238529\pi$
$433$	4.65228e43	1.46425	0.732125	+	0.681170i	$0.238529\pi$
$434$	0	0
$435$	2.92638e43	0.853606
$436$	0	0
$437$	−1.33483e43	−0.360979
$438$	0	0
$439$	−4.97854e42	−0.124864	−0.0624320	−	0.998049i	$-0.519886\pi$
$439$	−4.97854e42	−0.124864	−0.0624320	+	0.998049i	$0.519886\pi$
$440$	0	0
$441$	−2.47573e43	−0.576057
$442$	0	0
$443$	6.46629e43	1.39635	0.698174	−	0.715928i	$-0.253996\pi$
$443$	6.46629e43	1.39635	0.698174	+	0.715928i	$0.253996\pi$
$444$	0	0
$445$	−1.07052e44	−2.14612
$446$	0	0
$447$	−6.85633e40	−0.00127649
$448$	0	0
$449$	2.27700e43	0.393820	0.196910	−	0.980422i	$-0.436909\pi$
$449$	2.27700e43	0.393820	0.196910	+	0.980422i	$0.436909\pi$
$450$	0	0
$451$	1.15753e43	0.186046
$452$	0	0
$453$	−1.92874e43	−0.288172
$454$	0	0
$455$	−1.49450e44	−2.07638
$456$	0	0
$457$	−9.89272e43	−1.27849	−0.639247	−	0.769002i	$-0.720754\pi$
$457$	−9.89272e43	−1.27849	−0.639247	+	0.769002i	$0.720754\pi$
$458$	0	0
$459$	2.10030e43	0.252564
$460$	0	0
$461$	1.21100e44	1.35544	0.677720	−	0.735320i	$-0.262968\pi$
$461$	1.21100e44	1.35544	0.677720	+	0.735320i	$0.262968\pi$
$462$	0	0
$463$	1.96429e43	0.204701	0.102350	−	0.994748i	$-0.467364\pi$
$463$	1.96429e43	0.204701	0.102350	+	0.994748i	$0.467364\pi$
$464$	0	0
$465$	7.82288e43	0.759264
$466$	0	0
$467$	1.33961e44	1.21130	0.605650	−	0.795731i	$-0.292913\pi$
$467$	1.33961e44	1.21130	0.605650	+	0.795731i	$0.292913\pi$
$468$	0	0
$469$	−5.83655e43	−0.491817
$470$	0	0
$471$	−3.50008e43	−0.274937
$472$	0	0
$473$	−1.55766e44	−1.14094
$474$	0	0
$475$	1.31907e44	0.901205
$476$	0	0
$477$	3.35234e43	0.213695
$478$	0	0
$479$	2.06153e44	1.22646	0.613232	−	0.789903i	$-0.289869\pi$
$479$	2.06153e44	1.22646	0.613232	+	0.789903i	$0.289869\pi$
$480$	0	0
$481$	2.36568e43	0.131390
$482$	0	0
$483$	−6.14959e43	−0.318947
$484$	0	0
$485$	4.92317e44	2.38510
$486$	0	0
$487$	−1.57095e44	−0.711108	−0.355554	−	0.934656i	$-0.615708\pi$
$487$	−1.57095e44	−0.711108	−0.355554	+	0.934656i	$0.615708\pi$
$488$	0	0
$489$	−1.27158e44	−0.537959
$490$	0	0
$491$	−4.22476e44	−1.67093	−0.835464	−	0.549545i	$-0.814801\pi$
$491$	−4.22476e44	−1.67093	−0.835464	+	0.549545i	$0.814801\pi$
$492$	0	0
$493$	−2.02715e44	−0.749743
$494$	0	0
$495$	−6.52663e44	−2.25789
$496$	0	0
$497$	3.33496e44	1.07947
$498$	0	0
$499$	1.23722e44	0.374791	0.187395	−	0.982285i	$-0.439995\pi$
$499$	1.23722e44	0.374791	0.187395	+	0.982285i	$0.439995\pi$
$500$	0	0
$501$	1.69755e43	0.0481395
$502$	0	0
$503$	−1.11832e44	−0.296957	−0.148478	−	0.988916i	$-0.547438\pi$
$503$	−1.11832e44	−0.296957	−0.148478	+	0.988916i	$0.547438\pi$
$504$	0	0
$505$	3.74927e44	0.932479
$506$	0	0
$507$	−1.97707e43	−0.0460670
$508$	0	0
$509$	−2.64785e44	−0.578162	−0.289081	−	0.957305i	$-0.593350\pi$
$509$	−2.64785e44	−0.578162	−0.289081	+	0.957305i	$0.593350\pi$
$510$	0	0
$511$	−5.71292e44	−1.16927
$512$	0	0
$513$	−1.22095e44	−0.234294
$514$	0	0
$515$	1.79278e45	3.22632
$516$	0	0
$517$	−6.11423e44	−1.03216
$518$	0	0
$519$	−2.29256e44	−0.363127
$520$	0	0
$521$	6.23370e44	0.926666	0.463333	−	0.886184i	$-0.346653\pi$
$521$	6.23370e44	0.926666	0.463333	+	0.886184i	$0.346653\pi$
$522$	0	0
$523$	8.27079e44	1.15417	0.577084	−	0.816685i	$-0.304191\pi$
$523$	8.27079e44	1.15417	0.577084	+	0.816685i	$0.304191\pi$
$524$	0	0
$525$	6.07698e44	0.796269
$526$	0	0
$527$	−5.41905e44	−0.666881
$528$	0	0
$529$	−2.27191e44	−0.262647
$530$	0	0
$531$	7.90376e44	0.858567
$532$	0	0
$533$	−1.20169e44	−0.122686
$534$	0	0
$535$	−3.08314e45	−2.95908
$536$	0	0
$537$	2.58728e44	0.233489
$538$	0	0
$539$	1.03159e45	0.875571
$540$	0	0
$541$	1.03419e45	0.825745	0.412873	−	0.910789i	$-0.364525\pi$
$541$	1.03419e45	0.825745	0.412873	+	0.910789i	$0.364525\pi$
$542$	0	0
$543$	4.77552e43	0.0358776
$544$	0	0
$545$	−3.21455e45	−2.27289
$546$	0	0
$547$	1.01840e45	0.677844	0.338922	−	0.940815i	$-0.389938\pi$
$547$	1.01840e45	0.677844	0.338922	+	0.940815i	$0.389938\pi$
$548$	0	0
$549$	−2.72669e45	−1.70880
$550$	0	0
$551$	1.17843e45	0.695508
$552$	0	0
$553$	−2.03667e44	−0.113229
$554$	0	0
$555$	−1.41066e44	−0.0738903
$556$	0	0
$557$	−2.74214e44	−0.135357	−0.0676783	−	0.997707i	$-0.521559\pi$
$557$	−2.74214e44	−0.135357	−0.0676783	+	0.997707i	$0.521559\pi$
$558$	0	0
$559$	1.61708e45	0.752381
$560$	0	0
$561$	−4.18232e44	−0.183455
$562$	0	0
$563$	−1.73021e44	−0.0715666	−0.0357833	−	0.999360i	$-0.511393\pi$
$563$	−1.73021e44	−0.0715666	−0.0357833	+	0.999360i	$0.511393\pi$
$564$	0	0
$565$	2.27364e45	0.886998
$566$	0	0
$567$	2.61221e45	0.961363
$568$	0	0
$569$	3.46685e45	1.20388	0.601939	−	0.798542i	$-0.294395\pi$
$569$	3.46685e45	1.20388	0.601939	+	0.798542i	$0.294395\pi$
$570$	0	0
$571$	3.55607e45	1.16540	0.582699	−	0.812688i	$-0.301997\pi$
$571$	3.55607e45	1.16540	0.582699	+	0.812688i	$0.301997\pi$
$572$	0	0
$573$	3.07840e44	0.0952300
$574$	0	0
$575$	−6.30284e45	−1.84084
$576$	0	0
$577$	−1.33881e45	−0.369249	−0.184625	−	0.982809i	$-0.559107\pi$
$577$	−1.33881e45	−0.369249	−0.184625	+	0.982809i	$0.559107\pi$
$578$	0	0
$579$	−8.05046e43	−0.0209712
$580$	0	0
$581$	−4.39107e45	−1.08060
$582$	0	0
$583$	−1.39686e45	−0.324804
$584$	0	0
$585$	6.77562e45	1.48894
$586$	0	0
$587$	1.27700e45	0.265255	0.132628	−	0.991166i	$-0.457659\pi$
$587$	1.27700e45	0.265255	0.132628	+	0.991166i	$0.457659\pi$
$588$	0	0
$589$	3.15021e45	0.618640
$590$	0	0
$591$	−1.05305e45	−0.195549
$592$	0	0
$593$	−7.49368e45	−1.31611	−0.658056	−	0.752969i	$-0.728621\pi$
$593$	−7.49368e45	−1.31611	−0.658056	+	0.752969i	$0.728621\pi$
$594$	0	0
$595$	−6.17329e45	−1.02562
$596$	0	0
$597$	−2.61849e45	−0.411599
$598$	0	0
$599$	−1.19178e46	−1.77278	−0.886388	−	0.462943i	$-0.846793\pi$
$599$	−1.19178e46	−1.77278	−0.886388	+	0.462943i	$0.846793\pi$
$600$	0	0
$601$	−1.25892e45	−0.177243	−0.0886214	−	0.996065i	$-0.528246\pi$
$601$	−1.25892e45	−0.177243	−0.0886214	+	0.996065i	$0.528246\pi$
$602$	0	0
$603$	2.64612e45	0.352674
$604$	0	0
$605$	1.31448e46	1.65879
$606$	0	0
$607$	5.28539e44	0.0631632	0.0315816	−	0.999501i	$-0.489946\pi$
$607$	5.28539e44	0.0631632	0.0315816	+	0.999501i	$0.489946\pi$
$608$	0	0
$609$	5.42905e45	0.614523
$610$	0	0
$611$	6.34748e45	0.680646
$612$	0	0
$613$	−9.85915e44	−0.100171	−0.0500854	−	0.998745i	$-0.515949\pi$
$613$	−9.85915e44	−0.100171	−0.0500854	+	0.998745i	$0.515949\pi$
$614$	0	0
$615$	7.16569e44	0.0689952
$616$	0	0
$617$	−2.14287e45	−0.195565	−0.0977826	−	0.995208i	$-0.531175\pi$
$617$	−2.14287e45	−0.195565	−0.0977826	+	0.995208i	$0.531175\pi$
$618$	0	0
$619$	9.99540e45	0.864778	0.432389	−	0.901687i	$-0.357671\pi$
$619$	9.99540e45	0.864778	0.432389	+	0.901687i	$0.357671\pi$
$620$	0	0
$621$	5.83399e45	0.478580
$622$	0	0
$623$	−1.98604e46	−1.54502
$624$	0	0
$625$	1.96782e46	1.45199
$626$	0	0
$627$	2.43128e45	0.170185
$628$	0	0
$629$	9.77188e44	0.0648996
$630$	0	0
$631$	1.87168e46	1.17964	0.589818	−	0.807536i	$-0.299199\pi$
$631$	1.87168e46	1.17964	0.589818	+	0.807536i	$0.299199\pi$
$632$	0	0
$633$	−1.83895e45	−0.110004
$634$	0	0
$635$	−1.66093e46	−0.943160
$636$	0	0
$637$	−1.07094e46	−0.577385
$638$	0	0
$639$	−1.51197e46	−0.774068
$640$	0	0
$641$	−3.73888e46	−1.81796	−0.908980	−	0.416841i	$-0.863137\pi$
$641$	−3.73888e46	−1.81796	−0.908980	+	0.416841i	$0.863137\pi$
$642$	0	0
$643$	3.70674e45	0.171203	0.0856015	−	0.996329i	$-0.472719\pi$
$643$	3.70674e45	0.171203	0.0856015	+	0.996329i	$0.472719\pi$
$644$	0	0
$645$	−9.64267e45	−0.423118
$646$	0	0
$647$	3.44855e46	1.43786	0.718928	−	0.695084i	$-0.244633\pi$
$647$	3.44855e46	1.43786	0.718928	+	0.695084i	$0.244633\pi$
$648$	0	0
$649$	−3.29335e46	−1.30497
$650$	0	0
$651$	1.45131e46	0.546606
$652$	0	0
$653$	−8.14942e45	−0.291782	−0.145891	−	0.989301i	$-0.546605\pi$
$653$	−8.14942e45	−0.291782	−0.145891	+	0.989301i	$0.546605\pi$
$654$	0	0
$655$	−9.77205e45	−0.332663
$656$	0	0
$657$	2.59007e46	0.838461
$658$	0	0
$659$	2.21169e46	0.680948	0.340474	−	0.940254i	$-0.389412\pi$
$659$	2.21169e46	0.680948	0.340474	+	0.940254i	$0.389412\pi$
$660$	0	0
$661$	−1.77870e46	−0.520928	−0.260464	−	0.965484i	$-0.583876\pi$
$661$	−1.77870e46	−0.520928	−0.260464	+	0.965484i	$0.583876\pi$
$662$	0	0
$663$	4.34187e45	0.120977
$664$	0	0
$665$	3.58867e46	0.951430
$666$	0	0
$667$	−5.63082e46	−1.42068
$668$	0	0
$669$	−2.07327e46	−0.497881
$670$	0	0
$671$	1.13616e47	2.59728
$672$	0	0
$673$	1.15371e46	0.251100	0.125550	−	0.992087i	$-0.459930\pi$
$673$	1.15371e46	0.251100	0.125550	+	0.992087i	$0.459930\pi$
$674$	0	0
$675$	−5.76511e46	−1.19480
$676$	0	0
$677$	8.04683e46	1.58822	0.794112	−	0.607771i	$-0.207936\pi$
$677$	8.04683e46	1.58822	0.794112	+	0.607771i	$0.207936\pi$
$678$	0	0
$679$	9.13352e46	1.71707
$680$	0	0
$681$	−2.85130e45	−0.0510642
$682$	0	0
$683$	−8.01067e46	−1.36687	−0.683435	−	0.730011i	$-0.739515\pi$
$683$	−8.01067e46	−1.36687	−0.683435	+	0.730011i	$0.739515\pi$
$684$	0	0
$685$	−7.28979e46	−1.18528
$686$	0	0
$687$	2.19562e46	0.340228
$688$	0	0
$689$	1.45015e46	0.214188
$690$	0	0
$691$	−1.09406e46	−0.154046	−0.0770232	−	0.997029i	$-0.524542\pi$
$691$	−1.09406e46	−0.154046	−0.0770232	+	0.997029i	$0.524542\pi$
$692$	0	0
$693$	−1.21083e47	−1.62549
$694$	0	0
$695$	−1.03971e47	−1.33095
$696$	0	0
$697$	−4.96380e45	−0.0606001
$698$	0	0
$699$	9.63159e45	0.112157
$700$	0	0
$701$	6.08980e46	0.676481	0.338240	−	0.941060i	$-0.390168\pi$
$701$	6.08980e46	0.676481	0.338240	+	0.941060i	$0.390168\pi$
$702$	0	0
$703$	−5.68061e45	−0.0602049
$704$	0	0
$705$	−3.78500e46	−0.382777
$706$	0	0
$707$	6.95570e46	0.671306
$708$	0	0
$709$	1.22329e47	1.12685	0.563425	−	0.826167i	$-0.309484\pi$
$709$	1.22329e47	1.12685	0.563425	+	0.826167i	$0.309484\pi$
$710$	0	0
$711$	9.23368e45	0.0811944
$712$	0	0
$713$	−1.50525e47	−1.26366
$714$	0	0
$715$	−2.82327e47	−2.26310
$716$	0	0
$717$	6.32937e46	0.484501
$718$	0	0
$719$	4.92469e46	0.360042	0.180021	−	0.983663i	$-0.442384\pi$
$719$	4.92469e46	0.360042	0.180021	+	0.983663i	$0.442384\pi$
$720$	0	0
$721$	3.32599e47	2.32268
$722$	0	0
$723$	1.06040e44	0.000707435 0
$724$	0	0
$725$	5.56434e47	3.54680
$726$	0	0
$727$	−1.32921e47	−0.809611	−0.404806	−	0.914403i	$-0.632661\pi$
$727$	−1.32921e47	−0.809611	−0.404806	+	0.914403i	$0.632661\pi$
$728$	0	0
$729$	−9.05689e46	−0.527200
$730$	0	0
$731$	6.67965e46	0.371635
$732$	0	0
$733$	−3.32699e47	−1.76944	−0.884721	−	0.466121i	$-0.845651\pi$
$733$	−3.32699e47	−1.76944	−0.884721	+	0.466121i	$0.845651\pi$
$734$	0	0
$735$	6.38602e46	0.324705
$736$	0	0
$737$	−1.10259e47	−0.536042
$738$	0	0
$739$	−1.02617e47	−0.477074	−0.238537	−	0.971133i	$-0.576668\pi$
$739$	−1.02617e47	−0.477074	−0.238537	+	0.971133i	$0.576668\pi$
$740$	0	0
$741$	−2.52403e46	−0.112226
$742$	0	0
$743$	2.30680e47	0.981059	0.490530	−	0.871424i	$-0.336803\pi$
$743$	2.30680e47	0.981059	0.490530	+	0.871424i	$0.336803\pi$
$744$	0	0
$745$	−1.91182e45	−0.00777803
$746$	0	0
$747$	1.99078e47	0.774878
$748$	0	0
$749$	−5.71989e47	−2.13028
$750$	0	0
$751$	2.87651e47	1.02519	0.512597	−	0.858629i	$-0.328683\pi$
$751$	2.87651e47	1.02519	0.512597	+	0.858629i	$0.328683\pi$
$752$	0	0
$753$	−5.20852e46	−0.177663
$754$	0	0
$755$	−5.37810e47	−1.75592
$756$	0	0
$757$	−1.30633e47	−0.408292	−0.204146	−	0.978940i	$-0.565442\pi$
$757$	−1.30633e47	−0.408292	−0.204146	+	0.978940i	$0.565442\pi$
$758$	0	0
$759$	−1.16172e47	−0.347627
$760$	0	0
$761$	1.00969e47	0.289295	0.144648	−	0.989483i	$-0.453795\pi$
$761$	1.00969e47	0.289295	0.144648	+	0.989483i	$0.453795\pi$
$762$	0	0
$763$	−5.96367e47	−1.63628
$764$	0	0
$765$	2.79879e47	0.735455
$766$	0	0
$767$	3.41898e47	0.860545
$768$	0	0
$769$	4.46600e47	1.07680	0.538399	−	0.842690i	$-0.319029\pi$
$769$	4.46600e47	1.07680	0.538399	+	0.842690i	$0.319029\pi$
$770$	0	0
$771$	3.36844e46	0.0778093
$772$	0	0
$773$	−5.83555e47	−1.29158	−0.645788	−	0.763516i	$-0.723471\pi$
$773$	−5.83555e47	−1.29158	−0.645788	+	0.763516i	$0.723471\pi$
$774$	0	0
$775$	1.48748e48	3.15480
$776$	0	0
$777$	−2.61707e46	−0.0531947
$778$	0	0
$779$	2.88557e46	0.0562165
$780$	0	0
$781$	6.30010e47	1.17654
$782$	0	0
$783$	−5.15043e47	−0.922092
$784$	0	0
$785$	−9.75965e47	−1.67527
$786$	0	0
$787$	7.26390e47	1.19560	0.597801	−	0.801644i	$-0.296041\pi$
$787$	7.26390e47	1.19560	0.597801	+	0.801644i	$0.296041\pi$
$788$	0	0
$789$	3.07291e47	0.485042
$790$	0	0
$791$	4.21809e47	0.638563
$792$	0	0
$793$	−1.17950e48	−1.71274
$794$	0	0
$795$	−8.64722e46	−0.120453
$796$	0	0
$797$	5.44974e47	0.728306	0.364153	−	0.931339i	$-0.381359\pi$
$797$	5.44974e47	0.728306	0.364153	+	0.931339i	$0.381359\pi$
$798$	0	0
$799$	2.62194e47	0.336202
$800$	0	0
$801$	9.00410e47	1.10791
$802$	0	0
$803$	−1.07923e48	−1.27441
$804$	0	0
$805$	−1.71476e48	−1.94344
$806$	0	0
$807$	−9.69744e46	−0.105498
$808$	0	0
$809$	7.45844e47	0.778928	0.389464	−	0.921042i	$-0.372660\pi$
$809$	7.45844e47	0.778928	0.389464	+	0.921042i	$0.372660\pi$
$810$	0	0
$811$	2.28203e47	0.228811	0.114406	−	0.993434i	$-0.463504\pi$
$811$	2.28203e47	0.228811	0.114406	+	0.993434i	$0.463504\pi$
$812$	0	0
$813$	1.02071e47	0.0982672
$814$	0	0
$815$	−3.54569e48	−3.27794
$816$	0	0
$817$	−3.88303e47	−0.344752
$818$	0	0
$819$	1.25702e48	1.07191
$820$	0	0
$821$	7.49028e47	0.613530	0.306765	−	0.951785i	$-0.400753\pi$
$821$	7.49028e47	0.613530	0.306765	+	0.951785i	$0.400753\pi$
$822$	0	0
$823$	−1.29285e48	−1.01730	−0.508652	−	0.860972i	$-0.669856\pi$
$823$	−1.29285e48	−1.01730	−0.508652	+	0.860972i	$0.669856\pi$
$824$	0	0
$825$	1.14801e48	0.867870
$826$	0	0
$827$	1.67722e48	1.21828	0.609141	−	0.793062i	$-0.291514\pi$
$827$	1.67722e48	1.21828	0.609141	+	0.793062i	$0.291514\pi$
$828$	0	0
$829$	1.42330e48	0.993451	0.496725	−	0.867908i	$-0.334536\pi$
$829$	1.42330e48	0.993451	0.496725	+	0.867908i	$0.334536\pi$
$830$	0	0
$831$	5.39750e47	0.362056
$832$	0	0
$833$	−4.42371e47	−0.285197
$834$	0	0
$835$	4.73347e47	0.293328
$836$	0	0
$837$	−1.37683e48	−0.820182
$838$	0	0
$839$	−1.05174e48	−0.602329	−0.301165	−	0.953572i	$-0.597375\pi$
$839$	−1.05174e48	−0.602329	−0.301165	+	0.953572i	$0.597375\pi$
$840$	0	0
$841$	3.15499e48	1.73726
$842$	0	0
$843$	1.48154e46	0.00784440
$844$	0	0
$845$	−5.51286e47	−0.280700
$846$	0	0
$847$	2.43864e48	1.19419
$848$	0	0
$849$	−6.79196e47	−0.319903
$850$	0	0
$851$	2.71433e47	0.122977
$852$	0	0
$853$	1.01265e48	0.441368	0.220684	−	0.975345i	$-0.429171\pi$
$853$	1.01265e48	0.441368	0.220684	+	0.975345i	$0.429171\pi$
$854$	0	0
$855$	−1.62700e48	−0.682254
$856$	0	0
$857$	3.70633e48	1.49541	0.747705	−	0.664031i	$-0.231156\pi$
$857$	3.70633e48	1.49541	0.747705	+	0.664031i	$0.231156\pi$
$858$	0	0
$859$	−3.78239e48	−1.46852	−0.734259	−	0.678869i	$-0.762470\pi$
$859$	−3.78239e48	−1.46852	−0.734259	+	0.678869i	$0.762470\pi$
$860$	0	0
$861$	1.32939e47	0.0496706
$862$	0	0
$863$	−4.35247e48	−1.56516	−0.782578	−	0.622552i	$-0.786096\pi$
$863$	−4.35247e48	−1.56516	−0.782578	+	0.622552i	$0.786096\pi$
$864$	0	0
$865$	−6.39258e48	−2.21264
$866$	0	0
$867$	−6.94003e47	−0.231231
$868$	0	0
$869$	−3.84750e47	−0.123410
$870$	0	0
$871$	1.14465e48	0.353486
$872$	0	0
$873$	−4.14087e48	−1.23128
$874$	0	0
$875$	9.04077e48	2.58865
$876$	0	0
$877$	4.09991e48	1.13053	0.565264	−	0.824910i	$-0.308774\pi$
$877$	4.09991e48	1.13053	0.565264	+	0.824910i	$0.308774\pi$
$878$	0	0
$879$	9.62172e45	0.00255527
$880$	0	0
$881$	2.48110e48	0.634660	0.317330	−	0.948315i	$-0.397214\pi$
$881$	2.48110e48	0.634660	0.317330	+	0.948315i	$0.397214\pi$
$882$	0	0
$883$	−6.54154e48	−1.61186	−0.805932	−	0.592008i	$-0.798336\pi$
$883$	−6.54154e48	−1.61186	−0.805932	+	0.592008i	$0.798336\pi$
$884$	0	0
$885$	−2.03874e48	−0.483947
$886$	0	0
$887$	−3.77603e48	−0.863566	−0.431783	−	0.901977i	$-0.642115\pi$
$887$	−3.77603e48	−0.863566	−0.431783	+	0.901977i	$0.642115\pi$
$888$	0	0
$889$	−3.08138e48	−0.678995
$890$	0	0
$891$	4.93475e48	1.04781
$892$	0	0
$893$	−1.52419e48	−0.311882
$894$	0	0
$895$	7.21438e48	1.42272
$896$	0	0
$897$	1.20604e48	0.229238
$898$	0	0
$899$	1.32888e49	2.43473
$900$	0	0
$901$	5.99008e47	0.105797
$902$	0	0
$903$	−1.78892e48	−0.304609
$904$	0	0
$905$	1.33161e48	0.218613
$906$	0	0
$907$	−1.08989e48	−0.172530	−0.0862649	−	0.996272i	$-0.527493\pi$
$907$	−1.08989e48	−0.172530	−0.0862649	+	0.996272i	$0.527493\pi$
$908$	0	0
$909$	−3.15351e48	−0.481382
$910$	0	0
$911$	5.98086e48	0.880462	0.440231	−	0.897885i	$-0.354897\pi$
$911$	5.98086e48	0.880462	0.440231	+	0.897885i	$0.354897\pi$
$912$	0	0
$913$	−8.29520e48	−1.17777
$914$	0	0
$915$	7.03338e48	0.963199
$916$	0	0
$917$	−1.81292e48	−0.239489
$918$	0	0
$919$	−1.03793e49	−1.32271	−0.661353	−	0.750075i	$-0.730018\pi$
$919$	−1.03793e49	−1.32271	−0.661353	+	0.750075i	$0.730018\pi$
$920$	0	0
$921$	−1.98553e48	−0.244114
$922$	0	0
$923$	−6.54044e48	−0.775852
$924$	0	0
$925$	−2.68229e48	−0.307020
$926$	0	0
$927$	−1.50790e49	−1.66555
$928$	0	0
$929$	1.49398e49	1.59252	0.796261	−	0.604953i	$-0.206808\pi$
$929$	1.49398e49	1.59252	0.796261	+	0.604953i	$0.206808\pi$
$930$	0	0
$931$	2.57160e48	0.264566
$932$	0	0
$933$	−1.00041e48	−0.0993420
$934$	0	0
$935$	−1.16620e49	−1.11785
$936$	0	0
$937$	8.00299e48	0.740541	0.370270	−	0.928924i	$-0.379265\pi$
$937$	8.00299e48	0.740541	0.370270	+	0.928924i	$0.379265\pi$
$938$	0	0
$939$	1.75495e48	0.156777
$940$	0	0
$941$	−5.01488e48	−0.432545	−0.216273	−	0.976333i	$-0.569390\pi$
$941$	−5.01488e48	−0.432545	−0.216273	+	0.976333i	$0.569390\pi$
$942$	0	0
$943$	−1.37880e48	−0.114830
$944$	0	0
$945$	−1.56846e49	−1.26139
$946$	0	0
$947$	7.00367e48	0.543940	0.271970	−	0.962306i	$-0.412325\pi$
$947$	7.00367e48	0.543940	0.271970	+	0.962306i	$0.412325\pi$
$948$	0	0
$949$	1.12040e49	0.840392
$950$	0	0
$951$	5.91842e48	0.428773
$952$	0	0
$953$	1.82156e48	0.127471	0.0637353	−	0.997967i	$-0.479699\pi$
$953$	1.82156e48	0.127471	0.0637353	+	0.997967i	$0.479699\pi$
$954$	0	0
$955$	8.58383e48	0.580264
$956$	0	0
$957$	1.02561e49	0.669782
$958$	0	0
$959$	−1.35241e49	−0.853300
$960$	0	0
$961$	1.91206e49	1.16564
$962$	0	0
$963$	2.59323e49	1.52759
$964$	0	0
$965$	−2.24479e48	−0.127784
$966$	0	0
$967$	4.21827e48	0.232059	0.116029	−	0.993246i	$-0.462983\pi$
$967$	4.21827e48	0.232059	0.116029	+	0.993246i	$0.462983\pi$
$968$	0	0
$969$	−1.04259e48	−0.0554336
$970$	0	0
$971$	−1.43768e49	−0.738831	−0.369415	−	0.929264i	$-0.620442\pi$
$971$	−1.43768e49	−0.738831	−0.369415	+	0.929264i	$0.620442\pi$
$972$	0	0
$973$	−1.92888e49	−0.958171
$974$	0	0
$975$	−1.19180e49	−0.572306
$976$	0	0
$977$	3.59531e49	1.66908	0.834539	−	0.550949i	$-0.185734\pi$
$977$	3.59531e49	1.66908	0.834539	+	0.550949i	$0.185734\pi$
$978$	0	0
$979$	−3.75184e49	−1.68395
$980$	0	0
$981$	2.70375e49	1.17335
$982$	0	0
$983$	−1.32449e49	−0.555797	−0.277898	−	0.960610i	$-0.589638\pi$
$983$	−1.32449e49	−0.555797	−0.277898	+	0.960610i	$0.589638\pi$
$984$	0	0
$985$	−2.93633e49	−1.19153
$986$	0	0
$987$	−7.02198e48	−0.275567
$988$	0	0
$989$	1.85541e49	0.704207
$990$	0	0
$991$	4.86373e49	1.78548	0.892738	−	0.450576i	$-0.148781\pi$
$991$	4.86373e49	1.78548	0.892738	+	0.450576i	$0.148781\pi$
$992$	0	0
$993$	8.63886e47	0.0306756
$994$	0	0
$995$	−7.30141e49	−2.50799
$996$	0	0
$997$	−3.00393e49	−0.998206	−0.499103	−	0.866543i	$-0.666337\pi$
$997$	−3.00393e49	−0.998206	−0.499103	+	0.866543i	$0.666337\pi$
$998$	0	0
$999$	2.48276e48	0.0798187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.34.a.a.1.2	✓	3
4.3	odd	2		16.34.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.34.a.a.1.2	✓	3	1.1	even	1	trivial
16.34.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:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +O(q^{10})\)	\(q+2.16957e7 q^{3} +6.04966e11 q^{5} +1.12234e14 q^{7} -5.08836e15 q^{9} +2.12022e17 q^{11} -2.20110e18 q^{13} +1.31252e19 q^{15} -9.09205e19 q^{17} +5.28541e20 q^{19} +2.43500e21 q^{21} -2.52550e22 q^{23} +2.49568e23 q^{25} -2.31004e23 q^{27} +2.22959e24 q^{29} +5.96021e24 q^{31} +4.59998e24 q^{33} +6.78977e25 q^{35} -1.07477e25 q^{37} -4.77546e25 q^{39} +5.45950e25 q^{41} -7.34669e26 q^{43} -3.07828e27 q^{45} -2.88377e27 q^{47} +4.86547e27 q^{49} -1.97259e27 q^{51} -6.58827e27 q^{53} +1.28266e29 q^{55} +1.14671e28 q^{57} -1.55330e29 q^{59} +5.35869e29 q^{61} -5.71086e29 q^{63} -1.33159e30 q^{65} -5.20034e29 q^{67} -5.47926e29 q^{69} +2.97143e30 q^{71} -5.09019e30 q^{73} +5.41457e30 q^{75} +2.37961e31 q^{77} -1.81467e30 q^{79} +2.32747e31 q^{81} -3.91242e31 q^{83} -5.50038e31 q^{85} +4.83726e31 q^{87} -1.76955e32 q^{89} -2.47039e32 q^{91} +1.29311e32 q^{93} +3.19749e32 q^{95} +8.13793e32 q^{97} -1.07884e33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) \) 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9	\( 3 q + 92491788 q^{3} - 53880683886 q^{5} + 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots + 22\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q + 92491788 * q^3 - 53880683886 * q^5 + 4541009914392 * q^7 + 6032364433690023 * q^9 + 227617657302449700 * q^11 + 272970442217358762 * q^13 - 32142473481241105848 * q^15 + 93037188311816716854 * q^17 + 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 + 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 + 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 + 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 + 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 + 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 - 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 - 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 + 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 + 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 + 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 - 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 - 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 + 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 + 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 + 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 - 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 - 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 - 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 - 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 + 2255131944887948953089434155026900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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