LMFDB - Embedded newform 48.22.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.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:	$134.149125258$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 797544 $ x^2 - x - 797544
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 12)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-892.553$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+59049.0 q^{3} +4.01339e7 q^{5} -1.31964e9 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q+59049.0 q^{3} +4.01339e7 q^{5} -1.31964e9 q^{7} +3.48678e9 q^{9} +2.02992e10 q^{11} -5.75865e11 q^{13} +2.36987e12 q^{15} +1.03429e13 q^{17} -1.56487e13 q^{19} -7.79234e13 q^{21} +8.22113e13 q^{23} +1.13389e15 q^{25} +2.05891e14 q^{27} -3.87340e15 q^{29} -4.77913e15 q^{31} +1.19865e15 q^{33} -5.29623e16 q^{35} -1.88484e16 q^{37} -3.40043e16 q^{39} +4.71195e16 q^{41} -1.30737e17 q^{43} +1.39938e17 q^{45} -1.61273e17 q^{47} +1.18290e18 q^{49} +6.10740e17 q^{51} -2.18858e17 q^{53} +8.14688e17 q^{55} -9.24039e17 q^{57} -1.87625e18 q^{59} -9.32449e18 q^{61} -4.60130e18 q^{63} -2.31117e19 q^{65} +1.59897e19 q^{67} +4.85449e18 q^{69} -3.28177e19 q^{71} +3.60069e19 q^{73} +6.69552e19 q^{75} -2.67877e19 q^{77} +2.95603e19 q^{79} +1.21577e19 q^{81} -1.86959e20 q^{83} +4.15102e20 q^{85} -2.28720e20 q^{87} +1.93153e20 q^{89} +7.59935e20 q^{91} -2.82203e20 q^{93} -6.28043e20 q^{95} -5.20362e20 q^{97} +7.07791e19 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9}+O(q^{10}) $ 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9	$ 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9} + 76050855288 q^{11} - 809326043300 q^{13} + 1702258667100 q^{15} + 831385897668 q^{17} - 29817568652920 q^{19} - 30095487768672 q^{21} - 62680641406128 q^{23} + 784879798238750 q^{25} + 411782264189298 q^{27} - 21\!\cdots\!52 q^{29}+ \cdots + 26\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9 + 76050855288 * q^11 - 809326043300 * q^13 + 1702258667100 * q^15 + 831385897668 * q^17 - 29817568652920 * q^19 - 30095487768672 * q^21 - 62680641406128 * q^23 + 784879798238750 * q^25 + 411782264189298 * q^27 - 2113791894718452 * q^29 - 6575738314225648 * q^31 + 4490726953901112 * q^33 - 62119804481121600 * q^35 + 34811548897106284 * q^37 - 47789893530821700 * q^39 + 111628536749375220 * q^41 - 250859109258832648 * q^43 + 100516672033587900 * q^45 - 773718896888534592 * q^47 + 1280410738569171378 * q^49 + 49092505871397732 * q^51 - 589751680675248324 * q^53 + 184360807989219600 * q^55 - 1760697611386273080 * q^57 + 2865708124062989112 * q^59 - 17394704031104736356 * q^61 - 1777108457252312928 * q^63 - 20472203381314851000 * q^65 + 15757929817647896648 * q^67 - 3701229194390452272 * q^69 - 9134476240415397840 * q^71 + 14362399003114082644 * q^73 + 46346367206199948750 * q^75 + 18369452723793706368 * q^77 + 347027209795696304 * q^79 + 24315330918113857602 * q^81 - 320685874118443742328 * q^83 + 522639873698354580600 * q^85 - 124817297591229872148 * q^87 + 691379668751362059060 * q^89 + 570838488118933393600 * q^91 - 388290771716710288752 * q^93 - 467849347422771978000 * q^95 - 1174056891416588162684 * q^97 + 265172935900906762488 * 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
$3$	59049.0	0.577350
$3$	59049.0	0.577350
$4$	0	0
$5$	4.01339e7	1.83792	0.918959	−	0.394353i	$-0.129031\pi$
$5$	4.01339e7	1.83792	0.918959	+	0.394353i	$0.129031\pi$
$6$	0	0
$7$	−1.31964e9	−1.76574	−0.882869	−	0.469620i	$-0.844391\pi$
$7$	−1.31964e9	−1.76574	−0.882869	+	0.469620i	$0.844391\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	2.02992e10	0.235970	0.117985	−	0.993015i	$-0.462357\pi$
$11$	2.02992e10	0.235970	0.117985	+	0.993015i	$0.462357\pi$
$12$	0	0
$13$	−5.75865e11	−1.15855	−0.579276	−	0.815131i	$-0.696665\pi$
$13$	−5.75865e11	−1.15855	−0.579276	+	0.815131i	$0.696665\pi$
$14$	0	0
$15$	2.36987e12	1.06112
$16$	0	0
$17$	1.03429e13	1.24431	0.622157	−	0.782892i	$-0.286256\pi$
$17$	1.03429e13	1.24431	0.622157	+	0.782892i	$0.286256\pi$
$18$	0	0
$19$	−1.56487e13	−0.585551	−0.292776	−	0.956181i	$-0.594579\pi$
$19$	−1.56487e13	−0.585551	−0.292776	+	0.956181i	$0.594579\pi$
$20$	0	0
$21$	−7.79234e13	−1.01945
$22$	0	0
$23$	8.22113e13	0.413799	0.206899	−	0.978362i	$-0.433663\pi$
$23$	8.22113e13	0.413799	0.206899	+	0.978362i	$0.433663\pi$
$24$	0	0
$25$	1.13389e15	2.37794
$26$	0	0
$27$	2.05891e14	0.192450
$28$	0	0
$29$	−3.87340e15	−1.70967	−0.854836	−	0.518898i	$-0.826343\pi$
$29$	−3.87340e15	−1.70967	−0.854836	+	0.518898i	$0.826343\pi$
$30$	0	0
$31$	−4.77913e15	−1.04725	−0.523626	−	0.851948i	$-0.675421\pi$
$31$	−4.77913e15	−1.04725	−0.523626	+	0.851948i	$0.675421\pi$
$32$	0	0
$33$	1.19865e15	0.136237
$34$	0	0
$35$	−5.29623e16	−3.24528
$36$	0	0
$37$	−1.88484e16	−0.644403	−0.322201	−	0.946671i	$-0.604423\pi$
$37$	−1.88484e16	−0.644403	−0.322201	+	0.946671i	$0.604423\pi$
$38$	0	0
$39$	−3.40043e16	−0.668890
$40$	0	0
$41$	4.71195e16	0.548240	0.274120	−	0.961696i	$-0.411614\pi$
$41$	4.71195e16	0.548240	0.274120	+	0.961696i	$0.411614\pi$
$42$	0	0
$43$	−1.30737e17	−0.922531	−0.461265	−	0.887262i	$-0.652604\pi$
$43$	−1.30737e17	−0.922531	−0.461265	+	0.887262i	$0.652604\pi$
$44$	0	0
$45$	1.39938e17	0.612639
$46$	0	0
$47$	−1.61273e17	−0.447233	−0.223617	−	0.974677i	$-0.571786\pi$
$47$	−1.61273e17	−0.447233	−0.223617	+	0.974677i	$0.571786\pi$
$48$	0	0
$49$	1.18290e18	2.11783
$50$	0	0
$51$	6.10740e17	0.718406
$52$	0	0
$53$	−2.18858e17	−0.171896	−0.0859479	−	0.996300i	$-0.527392\pi$
$53$	−2.18858e17	−0.171896	−0.0859479	+	0.996300i	$0.527392\pi$
$54$	0	0
$55$	8.14688e17	0.433694
$56$	0	0
$57$	−9.24039e17	−0.338068
$58$	0	0
$59$	−1.87625e18	−0.477908	−0.238954	−	0.971031i	$-0.576804\pi$
$59$	−1.87625e18	−0.477908	−0.238954	+	0.971031i	$0.576804\pi$
$60$	0	0
$61$	−9.32449e18	−1.67364	−0.836819	−	0.547479i	$-0.815588\pi$
$61$	−9.32449e18	−1.67364	−0.836819	+	0.547479i	$0.815588\pi$
$62$	0	0
$63$	−4.60130e18	−0.588579
$64$	0	0
$65$	−2.31117e19	−2.12932
$66$	0	0
$67$	1.59897e19	1.07166	0.535828	−	0.844327i	$-0.320000\pi$
$67$	1.59897e19	1.07166	0.535828	+	0.844327i	$0.320000\pi$
$68$	0	0
$69$	4.85449e18	0.238907
$70$	0	0
$71$	−3.28177e19	−1.19645	−0.598225	−	0.801328i	$-0.704127\pi$
$71$	−3.28177e19	−1.19645	−0.598225	+	0.801328i	$0.704127\pi$
$72$	0	0
$73$	3.60069e19	0.980610	0.490305	−	0.871551i	$-0.336885\pi$
$73$	3.60069e19	0.980610	0.490305	+	0.871551i	$0.336885\pi$
$74$	0	0
$75$	6.69552e19	1.37291
$76$	0	0
$77$	−2.67877e19	−0.416661
$78$	0	0
$79$	2.95603e19	0.351256	0.175628	−	0.984457i	$-0.443804\pi$
$79$	2.95603e19	0.351256	0.175628	+	0.984457i	$0.443804\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−1.86959e20	−1.32259	−0.661297	−	0.750125i	$-0.729994\pi$
$83$	−1.86959e20	−1.32259	−0.661297	+	0.750125i	$0.729994\pi$
$84$	0	0
$85$	4.15102e20	2.28695
$86$	0	0
$87$	−2.28720e20	−0.987080
$88$	0	0
$89$	1.93153e20	0.656607	0.328304	−	0.944572i	$-0.393523\pi$
$89$	1.93153e20	0.656607	0.328304	+	0.944572i	$0.393523\pi$
$90$	0	0
$91$	7.59935e20	2.04570
$92$	0	0
$93$	−2.82203e20	−0.604632
$94$	0	0
$95$	−6.28043e20	−1.07620
$96$	0	0
$97$	−5.20362e20	−0.716477	−0.358238	−	0.933630i	$-0.616623\pi$
$97$	−5.20362e20	−0.716477	−0.358238	+	0.933630i	$0.616623\pi$
$98$	0	0
$99$	7.07791e19	0.0786567
$100$	0	0
$101$	2.34380e20	0.211128	0.105564	−	0.994412i	$-0.466335\pi$
$101$	2.34380e20	0.211128	0.105564	+	0.994412i	$0.466335\pi$
$102$	0	0
$103$	8.19842e19	0.0601090	0.0300545	−	0.999548i	$-0.490432\pi$
$103$	8.19842e19	0.0601090	0.0300545	+	0.999548i	$0.490432\pi$
$104$	0	0
$105$	−3.12737e21	−1.87366
$106$	0	0
$107$	5.86850e20	0.288401	0.144201	−	0.989548i	$-0.453939\pi$
$107$	5.86850e20	0.288401	0.144201	+	0.989548i	$0.453939\pi$
$108$	0	0
$109$	−5.75556e20	−0.232868	−0.116434	−	0.993198i	$-0.537146\pi$
$109$	−5.75556e20	−0.232868	−0.116434	+	0.993198i	$0.537146\pi$
$110$	0	0
$111$	−1.11298e21	−0.372046
$112$	0	0
$113$	−1.07419e21	−0.297684	−0.148842	−	0.988861i	$-0.547555\pi$
$113$	−1.07419e21	−0.297684	−0.148842	+	0.988861i	$0.547555\pi$
$114$	0	0
$115$	3.29946e21	0.760528
$116$	0	0
$117$	−2.00792e21	−0.386184
$118$	0	0
$119$	−1.36490e22	−2.19713
$120$	0	0
$121$	−6.98819e21	−0.944318
$122$	0	0
$123$	2.78236e21	0.316526
$124$	0	0
$125$	2.63701e22	2.53255
$126$	0	0
$127$	1.18917e22	0.966731	0.483366	−	0.875419i	$-0.339414\pi$
$127$	1.18917e22	0.966731	0.483366	+	0.875419i	$0.339414\pi$
$128$	0	0
$129$	−7.71991e21	−0.532623
$130$	0	0
$131$	−2.69744e22	−1.58344	−0.791722	−	0.610881i	$-0.790815\pi$
$131$	−2.69744e22	−1.58344	−0.791722	+	0.610881i	$0.790815\pi$
$132$	0	0
$133$	2.06506e22	1.03393
$134$	0	0
$135$	8.26321e21	0.353708
$136$	0	0
$137$	−1.79757e22	−0.659355	−0.329677	−	0.944094i	$-0.606940\pi$
$137$	−1.79757e22	−0.659355	−0.329677	+	0.944094i	$0.606940\pi$
$138$	0	0
$139$	−5.87261e22	−1.85002	−0.925009	−	0.379945i	$-0.875943\pi$
$139$	−5.87261e22	−1.85002	−0.925009	+	0.379945i	$0.875943\pi$
$140$	0	0
$141$	−9.52301e21	−0.258210
$142$	0	0
$143$	−1.16896e22	−0.273384
$144$	0	0
$145$	−1.55454e23	−3.14224
$146$	0	0
$147$	6.98493e22	1.22273
$148$	0	0
$149$	−1.23375e22	−0.187400	−0.0937001	−	0.995600i	$-0.529869\pi$
$149$	−1.23375e22	−0.187400	−0.0937001	+	0.995600i	$0.529869\pi$
$150$	0	0
$151$	1.82918e22	0.241545	0.120773	−	0.992680i	$-0.461463\pi$
$151$	1.82918e22	0.241545	0.120773	+	0.992680i	$0.461463\pi$
$152$	0	0
$153$	3.60636e22	0.414772
$154$	0	0
$155$	−1.91805e23	−1.92476
$156$	0	0
$157$	−1.04789e23	−0.919114	−0.459557	−	0.888148i	$-0.651992\pi$
$157$	−1.04789e23	−0.919114	−0.459557	+	0.888148i	$0.651992\pi$
$158$	0	0
$159$	−1.29233e22	−0.0992441
$160$	0	0
$161$	−1.08489e23	−0.730660
$162$	0	0
$163$	6.83915e22	0.404606	0.202303	−	0.979323i	$-0.435157\pi$
$163$	6.83915e22	0.404606	0.202303	+	0.979323i	$0.435157\pi$
$164$	0	0
$165$	4.81065e22	0.250393
$166$	0	0
$167$	3.30867e23	1.51750	0.758752	−	0.651379i	$-0.225809\pi$
$167$	3.30867e23	1.51750	0.758752	+	0.651379i	$0.225809\pi$
$168$	0	0
$169$	8.45562e22	0.342243
$170$	0	0
$171$	−5.45636e22	−0.195184
$172$	0	0
$173$	5.65748e22	0.179118	0.0895590	−	0.995982i	$-0.471454\pi$
$173$	5.65748e22	0.179118	0.0895590	+	0.995982i	$0.471454\pi$
$174$	0	0
$175$	−1.49633e24	−4.19882
$176$	0	0
$177$	−1.10791e23	−0.275920
$178$	0	0
$179$	−2.67798e23	−0.592720	−0.296360	−	0.955076i	$-0.595773\pi$
$179$	−2.67798e23	−0.592720	−0.296360	+	0.955076i	$0.595773\pi$
$180$	0	0
$181$	−5.83024e23	−1.14832	−0.574158	−	0.818745i	$-0.694670\pi$
$181$	−5.83024e23	−1.14832	−0.574158	+	0.818745i	$0.694670\pi$
$182$	0	0
$183$	−5.50602e23	−0.966276
$184$	0	0
$185$	−7.56461e23	−1.18436
$186$	0	0
$187$	2.09954e23	0.293621
$188$	0	0
$189$	−2.71702e23	−0.339816
$190$	0	0
$191$	−6.43954e23	−0.721115	−0.360558	−	0.932737i	$-0.617414\pi$
$191$	−6.43954e23	−0.721115	−0.360558	+	0.932737i	$0.617414\pi$
$192$	0	0
$193$	2.03840e23	0.204615	0.102307	−	0.994753i	$-0.467377\pi$
$193$	2.03840e23	0.204615	0.102307	+	0.994753i	$0.467377\pi$
$194$	0	0
$195$	−1.36472e24	−1.22937
$196$	0	0
$197$	−2.00965e24	−1.62639	−0.813197	−	0.581988i	$-0.802275\pi$
$197$	−2.00965e24	−1.62639	−0.813197	+	0.581988i	$0.802275\pi$
$198$	0	0
$199$	1.37340e24	0.999634	0.499817	−	0.866131i	$-0.333401\pi$
$199$	1.37340e24	0.999634	0.499817	+	0.866131i	$0.333401\pi$
$200$	0	0
$201$	9.44176e23	0.618720
$202$	0	0
$203$	5.11149e24	3.01883
$204$	0	0
$205$	1.89109e24	1.00762
$206$	0	0
$207$	2.86653e23	0.137933
$208$	0	0
$209$	−3.17657e23	−0.138173
$210$	0	0
$211$	2.05044e24	0.807015	0.403507	−	0.914976i	$-0.367791\pi$
$211$	2.05044e24	0.807015	0.403507	+	0.914976i	$0.367791\pi$
$212$	0	0
$213$	−1.93785e24	−0.690771
$214$	0	0
$215$	−5.24700e24	−1.69554
$216$	0	0
$217$	6.30674e24	1.84917
$218$	0	0
$219$	2.12617e24	0.566155
$220$	0	0
$221$	−5.95614e24	−1.44160
$222$	0	0
$223$	−8.67249e23	−0.190960	−0.0954801	−	0.995431i	$-0.530439\pi$
$223$	−8.67249e23	−0.190960	−0.0954801	+	0.995431i	$0.530439\pi$
$224$	0	0
$225$	3.95364e24	0.792648
$226$	0	0
$227$	−9.09438e24	−1.66151	−0.830753	−	0.556642i	$-0.812090\pi$
$227$	−9.09438e24	−1.66151	−0.830753	+	0.556642i	$0.812090\pi$
$228$	0	0
$229$	−7.98819e24	−1.33099	−0.665497	−	0.746401i	$-0.731780\pi$
$229$	−7.98819e24	−1.33099	−0.665497	+	0.746401i	$0.731780\pi$
$230$	0	0
$231$	−1.58179e24	−0.240559
$232$	0	0
$233$	3.92892e24	0.545803	0.272901	−	0.962042i	$-0.412017\pi$
$233$	3.92892e24	0.545803	0.272901	+	0.962042i	$0.412017\pi$
$234$	0	0
$235$	−6.47251e24	−0.821978
$236$	0	0
$237$	1.74550e24	0.202798
$238$	0	0
$239$	7.33250e24	0.779963	0.389982	−	0.920823i	$-0.372481\pi$
$239$	7.33250e24	0.779963	0.389982	+	0.920823i	$0.372481\pi$
$240$	0	0
$241$	−4.87813e24	−0.475417	−0.237708	−	0.971337i	$-0.576396\pi$
$241$	−4.87813e24	−0.475417	−0.237708	+	0.971337i	$0.576396\pi$
$242$	0	0
$243$	7.17898e23	0.0641500
$244$	0	0
$245$	4.74746e25	3.89240
$246$	0	0
$247$	9.01153e24	0.678392
$248$	0	0
$249$	−1.10397e25	−0.763600
$250$	0	0
$251$	1.63408e25	1.03920	0.519601	−	0.854409i	$-0.326081\pi$
$251$	1.63408e25	1.03920	0.519601	+	0.854409i	$0.326081\pi$
$252$	0	0
$253$	1.66883e24	0.0976441
$254$	0	0
$255$	2.45114e25	1.32037
$256$	0	0
$257$	−1.73200e25	−0.859509	−0.429754	−	0.902946i	$-0.641400\pi$
$257$	−1.73200e25	−0.859509	−0.429754	+	0.902946i	$0.641400\pi$
$258$	0	0
$259$	2.48732e25	1.13785
$260$	0	0
$261$	−1.35057e25	−0.569891
$262$	0	0
$263$	1.29572e25	0.504635	0.252317	−	0.967645i	$-0.418807\pi$
$263$	1.29572e25	0.504635	0.252317	+	0.967645i	$0.418807\pi$
$264$	0	0
$265$	−8.78361e24	−0.315931
$266$	0	0
$267$	1.14055e25	0.379092
$268$	0	0
$269$	6.42213e25	1.97370	0.986848	−	0.161648i	$-0.0516810\pi$
$269$	6.42213e25	1.97370	0.986848	+	0.161648i	$0.0516810\pi$
$270$	0	0
$271$	−1.90451e25	−0.541510	−0.270755	−	0.962648i	$-0.587273\pi$
$271$	−1.90451e25	−0.541510	−0.270755	+	0.962648i	$0.587273\pi$
$272$	0	0
$273$	4.48734e25	1.18108
$274$	0	0
$275$	2.30171e25	0.561123
$276$	0	0
$277$	−2.75396e25	−0.622186	−0.311093	−	0.950379i	$-0.600695\pi$
$277$	−2.75396e25	−0.622186	−0.311093	+	0.950379i	$0.600695\pi$
$278$	0	0
$279$	−1.66638e25	−0.349084
$280$	0	0
$281$	−7.58454e24	−0.147405	−0.0737026	−	0.997280i	$-0.523482\pi$
$281$	−7.58454e24	−0.147405	−0.0737026	+	0.997280i	$0.523482\pi$
$282$	0	0
$283$	4.43414e25	0.799929	0.399965	−	0.916531i	$-0.369022\pi$
$283$	4.43414e25	0.799929	0.399965	+	0.916531i	$0.369022\pi$
$284$	0	0
$285$	−3.70853e25	−0.621342
$286$	0	0
$287$	−6.21808e25	−0.968047
$288$	0	0
$289$	3.78845e25	0.548320
$290$	0	0
$291$	−3.07268e25	−0.413658
$292$	0	0
$293$	9.05056e25	1.13388	0.566938	−	0.823761i	$-0.308128\pi$
$293$	9.05056e25	1.13388	0.566938	+	0.823761i	$0.308128\pi$
$294$	0	0
$295$	−7.53011e25	−0.878356
$296$	0	0
$297$	4.17944e24	0.0454124
$298$	0	0
$299$	−4.73426e25	−0.479407
$300$	0	0
$301$	1.72526e26	1.62895
$302$	0	0
$303$	1.38399e25	0.121895
$304$	0	0
$305$	−3.74228e26	−3.07601
$306$	0	0
$307$	8.72375e25	0.669500	0.334750	−	0.942307i	$-0.391348\pi$
$307$	8.72375e25	0.669500	0.334750	+	0.942307i	$0.391348\pi$
$308$	0	0
$309$	4.84109e24	0.0347039
$310$	0	0
$311$	1.58933e26	1.06471	0.532354	−	0.846522i	$-0.321307\pi$
$311$	1.58933e26	1.06471	0.532354	+	0.846522i	$0.321307\pi$
$312$	0	0
$313$	2.53086e26	1.58509	0.792544	−	0.609815i	$-0.208756\pi$
$313$	2.53086e26	1.58509	0.792544	+	0.609815i	$0.208756\pi$
$314$	0	0
$315$	−1.84668e26	−1.08176
$316$	0	0
$317$	−3.09254e26	−1.69509	−0.847547	−	0.530721i	$-0.821921\pi$
$317$	−3.09254e26	−1.69509	−0.847547	+	0.530721i	$0.821921\pi$
$318$	0	0
$319$	−7.86270e25	−0.403431
$320$	0	0
$321$	3.46529e25	0.166509
$322$	0	0
$323$	−1.61853e26	−0.728610
$324$	0	0
$325$	−6.52969e26	−2.75497
$326$	0	0
$327$	−3.39860e25	−0.134446
$328$	0	0
$329$	2.12822e26	0.789696
$330$	0	0
$331$	2.69621e26	0.938772	0.469386	−	0.882993i	$-0.344475\pi$
$331$	2.69621e26	0.938772	0.469386	+	0.882993i	$0.344475\pi$
$332$	0	0
$333$	−6.57204e25	−0.214801
$334$	0	0
$335$	6.41729e26	1.96961
$336$	0	0
$337$	−4.36686e25	−0.125909	−0.0629543	−	0.998016i	$-0.520052\pi$
$337$	−4.36686e25	−0.125909	−0.0629543	+	0.998016i	$0.520052\pi$
$338$	0	0
$339$	−6.34296e25	−0.171868
$340$	0	0
$341$	−9.70128e25	−0.247120
$342$	0	0
$343$	−8.23929e26	−1.97379
$344$	0	0
$345$	1.94830e26	0.439091
$346$	0	0
$347$	−7.46835e26	−1.58404	−0.792019	−	0.610497i	$-0.790970\pi$
$347$	−7.46835e26	−1.58404	−0.792019	+	0.610497i	$0.790970\pi$
$348$	0	0
$349$	1.28747e26	0.257082	0.128541	−	0.991704i	$-0.458971\pi$
$349$	1.28747e26	0.257082	0.128541	+	0.991704i	$0.458971\pi$
$350$	0	0
$351$	−1.18566e26	−0.222963
$352$	0	0
$353$	−9.85985e24	−0.0174677	−0.00873386	−	0.999962i	$-0.502780\pi$
$353$	−9.85985e24	−0.0174677	−0.00873386	+	0.999962i	$0.502780\pi$
$354$	0	0
$355$	−1.31710e27	−2.19898
$356$	0	0
$357$	−8.05957e26	−1.26852
$358$	0	0
$359$	7.41498e26	1.10057	0.550286	−	0.834976i	$-0.314519\pi$
$359$	7.41498e26	1.10057	0.550286	+	0.834976i	$0.314519\pi$
$360$	0	0
$361$	−4.69328e26	−0.657130
$362$	0	0
$363$	−4.12646e26	−0.545202
$364$	0	0
$365$	1.44510e27	1.80228
$366$	0	0
$367$	−1.28742e26	−0.151610	−0.0758048	−	0.997123i	$-0.524153\pi$
$367$	−1.28742e26	−0.151610	−0.0758048	+	0.997123i	$0.524153\pi$
$368$	0	0
$369$	1.64296e26	0.182747
$370$	0	0
$371$	2.88814e26	0.303523
$372$	0	0
$373$	−1.64605e27	−1.63494	−0.817468	−	0.575974i	$-0.804623\pi$
$373$	−1.64605e27	−1.63494	−0.817468	+	0.575974i	$0.804623\pi$
$374$	0	0
$375$	1.55713e27	1.46217
$376$	0	0
$377$	2.23055e27	1.98075
$378$	0	0
$379$	1.52121e27	1.27785	0.638923	−	0.769271i	$-0.279380\pi$
$379$	1.52121e27	1.27785	0.638923	+	0.769271i	$0.279380\pi$
$380$	0	0
$381$	7.02195e26	0.558143
$382$	0	0
$383$	−2.00310e27	−1.50701	−0.753506	−	0.657442i	$-0.771639\pi$
$383$	−2.00310e27	−1.50701	−0.753506	+	0.657442i	$0.771639\pi$
$384$	0	0
$385$	−1.07509e27	−0.765789
$386$	0	0
$387$	−4.55853e26	−0.307510
$388$	0	0
$389$	−7.67505e26	−0.490468	−0.245234	−	0.969464i	$-0.578865\pi$
$389$	−7.67505e26	−0.490468	−0.245234	+	0.969464i	$0.578865\pi$
$390$	0	0
$391$	8.50306e26	0.514896
$392$	0	0
$393$	−1.59281e27	−0.914202
$394$	0	0
$395$	1.18637e27	0.645580
$396$	0	0
$397$	1.71894e27	0.887073	0.443537	−	0.896256i	$-0.353724\pi$
$397$	1.71894e27	0.887073	0.443537	+	0.896256i	$0.353724\pi$
$398$	0	0
$399$	1.21940e27	0.596940
$400$	0	0
$401$	−3.52628e27	−1.63795	−0.818975	−	0.573829i	$-0.805458\pi$
$401$	−3.52628e27	−1.63795	−0.818975	+	0.573829i	$0.805458\pi$
$402$	0	0
$403$	2.75214e27	1.21330
$404$	0	0
$405$	4.87934e26	0.204213
$406$	0	0
$407$	−3.82609e26	−0.152060
$408$	0	0
$409$	−2.63682e26	−0.0995373	−0.0497687	−	0.998761i	$-0.515848\pi$
$409$	−2.63682e26	−0.0995373	−0.0497687	+	0.998761i	$0.515848\pi$
$410$	0	0
$411$	−1.06145e27	−0.380679
$412$	0	0
$413$	2.47597e27	0.843860
$414$	0	0
$415$	−7.50339e27	−2.43082
$416$	0	0
$417$	−3.46772e27	−1.06811
$418$	0	0
$419$	−5.07702e26	−0.148717	−0.0743586	−	0.997232i	$-0.523691\pi$
$419$	−5.07702e26	−0.148717	−0.0743586	+	0.997232i	$0.523691\pi$
$420$	0	0
$421$	4.51718e27	1.25865	0.629325	−	0.777142i	$-0.283331\pi$
$421$	4.51718e27	1.25865	0.629325	+	0.777142i	$0.283331\pi$
$422$	0	0
$423$	−5.62324e26	−0.149078
$424$	0	0
$425$	1.17278e28	2.95891
$426$	0	0
$427$	1.23050e28	2.95521
$428$	0	0
$429$	−6.90261e26	−0.157838
$430$	0	0
$431$	8.21802e27	1.78960	0.894799	−	0.446468i	$-0.147319\pi$
$431$	8.21802e27	1.78960	0.894799	+	0.446468i	$0.147319\pi$
$432$	0	0
$433$	2.29776e27	0.476631	0.238315	−	0.971188i	$-0.423405\pi$
$433$	2.29776e27	0.476631	0.238315	+	0.971188i	$0.423405\pi$
$434$	0	0
$435$	−9.17943e27	−1.81417
$436$	0	0
$437$	−1.28650e27	−0.242300
$438$	0	0
$439$	−4.02595e27	−0.722755	−0.361377	−	0.932420i	$-0.617693\pi$
$439$	−4.02595e27	−0.722755	−0.361377	+	0.932420i	$0.617693\pi$
$440$	0	0
$441$	4.12453e27	0.705943
$442$	0	0
$443$	5.20834e27	0.850081	0.425040	−	0.905174i	$-0.360260\pi$
$443$	5.20834e27	0.850081	0.425040	+	0.905174i	$0.360260\pi$
$444$	0	0
$445$	7.75197e27	1.20679
$446$	0	0
$447$	−7.28515e26	−0.108196
$448$	0	0
$449$	8.47734e27	1.20136	0.600680	−	0.799489i	$-0.294897\pi$
$449$	8.47734e27	1.20136	0.600680	+	0.799489i	$0.294897\pi$
$450$	0	0
$451$	9.56491e26	0.129368
$452$	0	0
$453$	1.08011e27	0.139456
$454$	0	0
$455$	3.04991e28	3.75983
$456$	0	0
$457$	−8.54951e27	−1.00652	−0.503258	−	0.864136i	$-0.667866\pi$
$457$	−8.54951e27	−1.00652	−0.503258	+	0.864136i	$0.667866\pi$
$458$	0	0
$459$	2.12952e27	0.239469
$460$	0	0
$461$	−1.06642e28	−1.14569	−0.572847	−	0.819663i	$-0.694161\pi$
$461$	−1.06642e28	−1.14569	−0.572847	+	0.819663i	$0.694161\pi$
$462$	0	0
$463$	−1.74522e28	−1.79164	−0.895818	−	0.444421i	$-0.853409\pi$
$463$	−1.74522e28	−1.79164	−0.895818	+	0.444421i	$0.853409\pi$
$464$	0	0
$465$	−1.13259e28	−1.11126
$466$	0	0
$467$	−6.43044e27	−0.603134	−0.301567	−	0.953445i	$-0.597510\pi$
$467$	−6.43044e27	−0.603134	−0.301567	+	0.953445i	$0.597510\pi$
$468$	0	0
$469$	−2.11007e28	−1.89226
$470$	0	0
$471$	−6.18769e27	−0.530651
$472$	0	0
$473$	−2.65387e27	−0.217690
$474$	0	0
$475$	−1.77439e28	−1.39241
$476$	0	0
$477$	−7.63110e26	−0.0572986
$478$	0	0
$479$	−1.20115e28	−0.863127	−0.431564	−	0.902082i	$-0.642038\pi$
$479$	−1.20115e28	−0.863127	−0.431564	+	0.902082i	$0.642038\pi$
$480$	0	0
$481$	1.08542e28	0.746574
$482$	0	0
$483$	−6.40618e27	−0.421847
$484$	0	0
$485$	−2.08841e28	−1.31683
$486$	0	0
$487$	1.05192e28	0.635228	0.317614	−	0.948220i	$-0.397118\pi$
$487$	1.05192e28	0.635228	0.317614	+	0.948220i	$0.397118\pi$
$488$	0	0
$489$	4.03845e27	0.233600
$490$	0	0
$491$	1.79589e28	0.995234	0.497617	−	0.867397i	$-0.334209\pi$
$491$	1.79589e28	0.995234	0.497617	+	0.867397i	$0.334209\pi$
$492$	0	0
$493$	−4.00623e28	−2.12737
$494$	0	0
$495$	2.84064e27	0.144565
$496$	0	0
$497$	4.33075e28	2.11262
$498$	0	0
$499$	−9.17159e27	−0.428932	−0.214466	−	0.976731i	$-0.568801\pi$
$499$	−9.17159e27	−0.428932	−0.214466	+	0.976731i	$0.568801\pi$
$500$	0	0
$501$	1.95373e28	0.876132
$502$	0	0
$503$	−3.64266e27	−0.156659	−0.0783293	−	0.996928i	$-0.524959\pi$
$503$	−3.64266e27	−0.156659	−0.0783293	+	0.996928i	$0.524959\pi$
$504$	0	0
$505$	9.40659e27	0.388037
$506$	0	0
$507$	4.99296e27	0.197594
$508$	0	0
$509$	4.44508e28	1.68789	0.843943	−	0.536433i	$-0.180229\pi$
$509$	4.44508e28	1.68789	0.843943	+	0.536433i	$0.180229\pi$
$510$	0	0
$511$	−4.75162e28	−1.73150
$512$	0	0
$513$	−3.22193e27	−0.112689
$514$	0	0
$515$	3.29035e27	0.110475
$516$	0	0
$517$	−3.27372e27	−0.105534
$518$	0	0
$519$	3.34069e27	0.103414
$520$	0	0
$521$	3.30994e28	0.984067	0.492033	−	0.870576i	$-0.336254\pi$
$521$	3.30994e28	0.984067	0.492033	+	0.870576i	$0.336254\pi$
$522$	0	0
$523$	6.29283e27	0.179713	0.0898563	−	0.995955i	$-0.471359\pi$
$523$	6.29283e27	0.179713	0.0898563	+	0.995955i	$0.471359\pi$
$524$	0	0
$525$	−8.83567e28	−2.42419
$526$	0	0
$527$	−4.94303e28	−1.30311
$528$	0	0
$529$	−3.27129e28	−0.828771
$530$	0	0
$531$	−6.54207e27	−0.159303
$532$	0	0
$533$	−2.71345e28	−0.635164
$534$	0	0
$535$	2.35526e28	0.530058
$536$	0	0
$537$	−1.58132e28	−0.342207
$538$	0	0
$539$	2.40121e28	0.499744
$540$	0	0
$541$	−3.41666e28	−0.683959	−0.341980	−	0.939707i	$-0.611097\pi$
$541$	−3.41666e28	−0.683959	−0.341980	+	0.939707i	$0.611097\pi$
$542$	0	0
$543$	−3.44270e28	−0.662980
$544$	0	0
$545$	−2.30993e28	−0.427992
$546$	0	0
$547$	−2.44339e28	−0.435637	−0.217819	−	0.975989i	$-0.569894\pi$
$547$	−2.44339e28	−0.435637	−0.217819	+	0.975989i	$0.569894\pi$
$548$	0	0
$549$	−3.25125e28	−0.557880
$550$	0	0
$551$	6.06136e28	1.00110
$552$	0	0
$553$	−3.90089e28	−0.620226
$554$	0	0
$555$	−4.46683e28	−0.683790
$556$	0	0
$557$	7.45899e28	1.09951	0.549757	−	0.835325i	$-0.314720\pi$
$557$	7.45899e28	1.09951	0.549757	+	0.835325i	$0.314720\pi$
$558$	0	0
$559$	7.52871e28	1.06880
$560$	0	0
$561$	1.23976e28	0.169522
$562$	0	0
$563$	−6.74101e28	−0.887947	−0.443973	−	0.896040i	$-0.646432\pi$
$563$	−6.74101e28	−0.887947	−0.443973	+	0.896040i	$0.646432\pi$
$564$	0	0
$565$	−4.31112e28	−0.547119
$566$	0	0
$567$	−1.60437e28	−0.196193
$568$	0	0
$569$	7.85930e27	0.0926200	0.0463100	−	0.998927i	$-0.485254\pi$
$569$	7.85930e27	0.0926200	0.0463100	+	0.998927i	$0.485254\pi$
$570$	0	0
$571$	7.84020e28	0.890529	0.445264	−	0.895399i	$-0.353110\pi$
$571$	7.84020e28	0.890529	0.445264	+	0.895399i	$0.353110\pi$
$572$	0	0
$573$	−3.80249e28	−0.416336
$574$	0	0
$575$	9.32187e28	0.983990
$576$	0	0
$577$	1.80598e29	1.83809	0.919043	−	0.394158i	$-0.128964\pi$
$577$	1.80598e29	1.83809	0.919043	+	0.394158i	$0.128964\pi$
$578$	0	0
$579$	1.20365e28	0.118134
$580$	0	0
$581$	2.46719e29	2.33535
$582$	0	0
$583$	−4.44265e27	−0.0405623
$584$	0	0
$585$	−8.05855e28	−0.709775
$586$	0	0
$587$	2.19952e29	1.86908	0.934539	−	0.355860i	$-0.115812\pi$
$587$	2.19952e29	1.86908	0.934539	+	0.355860i	$0.115812\pi$
$588$	0	0
$589$	7.47872e28	0.613220
$590$	0	0
$591$	−1.18668e29	−0.938999
$592$	0	0
$593$	−8.21606e27	−0.0627465	−0.0313732	−	0.999508i	$-0.509988\pi$
$593$	−8.21606e27	−0.0627465	−0.0313732	+	0.999508i	$0.509988\pi$
$594$	0	0
$595$	−5.47786e29	−4.03815
$596$	0	0
$597$	8.10981e28	0.577139
$598$	0	0
$599$	1.00575e29	0.691049	0.345525	−	0.938410i	$-0.387701\pi$
$599$	1.00575e29	0.691049	0.345525	+	0.938410i	$0.387701\pi$
$600$	0	0
$601$	9.44955e27	0.0626944	0.0313472	−	0.999509i	$-0.490020\pi$
$601$	9.44955e27	0.0626944	0.0313472	+	0.999509i	$0.490020\pi$
$602$	0	0
$603$	5.57527e28	0.357218
$604$	0	0
$605$	−2.80463e29	−1.73558
$606$	0	0
$607$	2.63217e29	1.57338	0.786688	−	0.617351i	$-0.211794\pi$
$607$	2.63217e29	1.57338	0.786688	+	0.617351i	$0.211794\pi$
$608$	0	0
$609$	3.01828e29	1.74292
$610$	0	0
$611$	9.28715e28	0.518143
$612$	0	0
$613$	−1.82634e29	−0.984572	−0.492286	−	0.870434i	$-0.663839\pi$
$613$	−1.82634e29	−0.984572	−0.492286	+	0.870434i	$0.663839\pi$
$614$	0	0
$615$	1.11667e29	0.581749
$616$	0	0
$617$	8.99712e28	0.453011	0.226506	−	0.974010i	$-0.427270\pi$
$617$	8.99712e28	0.453011	0.226506	+	0.974010i	$0.427270\pi$
$618$	0	0
$619$	1.95648e28	0.0952188	0.0476094	−	0.998866i	$-0.484840\pi$
$619$	1.95648e28	0.0952188	0.0476094	+	0.998866i	$0.484840\pi$
$620$	0	0
$621$	1.69266e28	0.0796356
$622$	0	0
$623$	−2.54892e29	−1.15940
$624$	0	0
$625$	5.17655e29	2.27667
$626$	0	0
$627$	−1.87573e28	−0.0797740
$628$	0	0
$629$	−1.94948e29	−0.801840
$630$	0	0
$631$	4.75564e29	1.89191	0.945955	−	0.324298i	$-0.105128\pi$
$631$	4.75564e29	1.89191	0.945955	+	0.324298i	$0.105128\pi$
$632$	0	0
$633$	1.21076e29	0.465930
$634$	0	0
$635$	4.77261e29	1.77677
$636$	0	0
$637$	−6.81193e29	−2.45362
$638$	0	0
$639$	−1.14428e29	−0.398817
$640$	0	0
$641$	3.15320e29	1.06351	0.531756	−	0.846898i	$-0.321532\pi$
$641$	3.15320e29	1.06351	0.531756	+	0.846898i	$0.321532\pi$
$642$	0	0
$643$	1.20958e29	0.394837	0.197418	−	0.980319i	$-0.436744\pi$
$643$	1.20958e29	0.394837	0.197418	+	0.980319i	$0.436744\pi$
$644$	0	0
$645$	−3.09830e29	−0.978918
$646$	0	0
$647$	3.71198e29	1.13530	0.567650	−	0.823270i	$-0.307853\pi$
$647$	3.71198e29	1.13530	0.567650	+	0.823270i	$0.307853\pi$
$648$	0	0
$649$	−3.80864e28	−0.112772
$650$	0	0
$651$	3.72407e29	1.06762
$652$	0	0
$653$	−2.54435e29	−0.706298	−0.353149	−	0.935567i	$-0.614889\pi$
$653$	−2.54435e29	−0.706298	−0.353149	+	0.935567i	$0.614889\pi$
$654$	0	0
$655$	−1.08259e30	−2.91024
$656$	0	0
$657$	1.25548e29	0.326870
$658$	0	0
$659$	−7.78343e28	−0.196279	−0.0981396	−	0.995173i	$-0.531289\pi$
$659$	−7.78343e28	−0.196279	−0.0981396	+	0.995173i	$0.531289\pi$
$660$	0	0
$661$	2.62927e29	0.642274	0.321137	−	0.947033i	$-0.395935\pi$
$661$	2.62927e29	0.642274	0.321137	+	0.947033i	$0.395935\pi$
$662$	0	0
$663$	−3.51704e29	−0.832310
$664$	0	0
$665$	8.28790e29	1.90028
$666$	0	0
$667$	−3.18437e29	−0.707460
$668$	0	0
$669$	−5.12102e28	−0.110251
$670$	0	0
$671$	−1.89280e29	−0.394929
$672$	0	0
$673$	2.87103e29	0.580604	0.290302	−	0.956935i	$-0.406244\pi$
$673$	2.87103e29	0.580604	0.290302	+	0.956935i	$0.406244\pi$
$674$	0	0
$675$	2.33458e29	0.457635
$676$	0	0
$677$	−4.09174e29	−0.777547	−0.388773	−	0.921333i	$-0.627101\pi$
$677$	−4.09174e29	−0.777547	−0.388773	+	0.921333i	$0.627101\pi$
$678$	0	0
$679$	6.86690e29	1.26511
$680$	0	0
$681$	−5.37014e29	−0.959271
$682$	0	0
$683$	−8.59043e29	−1.48798	−0.743991	−	0.668190i	$-0.767069\pi$
$683$	−8.59043e29	−1.48798	−0.743991	+	0.668190i	$0.767069\pi$
$684$	0	0
$685$	−7.21433e29	−1.21184
$686$	0	0
$687$	−4.71695e29	−0.768449
$688$	0	0
$689$	1.26033e29	0.199150
$690$	0	0
$691$	8.76762e29	1.34389	0.671943	−	0.740603i	$-0.265460\pi$
$691$	8.76762e29	1.34389	0.671943	+	0.740603i	$0.265460\pi$
$692$	0	0
$693$	−9.34029e28	−0.138887
$694$	0	0
$695$	−2.35691e30	−3.40018
$696$	0	0
$697$	4.87354e29	0.682183
$698$	0	0
$699$	2.31999e29	0.315119
$700$	0	0
$701$	3.08423e29	0.406544	0.203272	−	0.979122i	$-0.434842\pi$
$701$	3.08423e29	0.406544	0.203272	+	0.979122i	$0.434842\pi$
$702$	0	0
$703$	2.94953e29	0.377331
$704$	0	0
$705$	−3.82195e29	−0.474569
$706$	0	0
$707$	−3.09298e29	−0.372797
$708$	0	0
$709$	−7.25205e29	−0.848546	−0.424273	−	0.905534i	$-0.639470\pi$
$709$	−7.25205e29	−0.848546	−0.424273	+	0.905534i	$0.639470\pi$
$710$	0	0
$711$	1.03070e29	0.117085
$712$	0	0
$713$	−3.92899e29	−0.433352
$714$	0	0
$715$	−4.69150e29	−0.502457
$716$	0	0
$717$	4.32977e29	0.450312
$718$	0	0
$719$	8.52565e29	0.861141	0.430570	−	0.902557i	$-0.358312\pi$
$719$	8.52565e29	0.861141	0.430570	+	0.902557i	$0.358312\pi$
$720$	0	0
$721$	−1.08190e29	−0.106137
$722$	0	0
$723$	−2.88049e29	−0.274482
$724$	0	0
$725$	−4.39201e30	−4.06550
$726$	0	0
$727$	3.00541e29	0.270266	0.135133	−	0.990827i	$-0.456854\pi$
$727$	3.00541e29	0.270266	0.135133	+	0.990827i	$0.456854\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−1.35221e30	−1.14792
$732$	0	0
$733$	1.63142e30	1.34578	0.672889	−	0.739744i	$-0.265053\pi$
$733$	1.63142e30	1.34578	0.672889	+	0.739744i	$0.265053\pi$
$734$	0	0
$735$	2.80332e30	2.24728
$736$	0	0
$737$	3.24579e29	0.252878
$738$	0	0
$739$	3.85984e29	0.292283	0.146141	−	0.989264i	$-0.453315\pi$
$739$	3.85984e29	0.292283	0.146141	+	0.989264i	$0.453315\pi$
$740$	0	0
$741$	5.32122e29	0.391670
$742$	0	0
$743$	−8.44890e29	−0.604530	−0.302265	−	0.953224i	$-0.597743\pi$
$743$	−8.44890e29	−0.604530	−0.302265	+	0.953224i	$0.597743\pi$
$744$	0	0
$745$	−4.95150e29	−0.344426
$746$	0	0
$747$	−6.51886e29	−0.440864
$748$	0	0
$749$	−7.74431e29	−0.509241
$750$	0	0
$751$	1.21111e30	0.774395	0.387198	−	0.921997i	$-0.373443\pi$
$751$	1.21111e30	0.774395	0.387198	+	0.921997i	$0.373443\pi$
$752$	0	0
$753$	9.64909e29	0.599983
$754$	0	0
$755$	7.34120e29	0.443940
$756$	0	0
$757$	1.35627e30	0.797699	0.398849	−	0.917016i	$-0.369410\pi$
$757$	1.35627e30	0.797699	0.398849	+	0.917016i	$0.369410\pi$
$758$	0	0
$759$	9.85426e28	0.0563748
$760$	0	0
$761$	3.06597e30	1.70620	0.853098	−	0.521750i	$-0.174721\pi$
$761$	3.06597e30	1.70620	0.853098	+	0.521750i	$0.174721\pi$
$762$	0	0
$763$	7.59527e29	0.411183
$764$	0	0
$765$	1.44737e30	0.762316
$766$	0	0
$767$	1.08047e30	0.553681
$768$	0	0
$769$	−1.14606e30	−0.571455	−0.285728	−	0.958311i	$-0.592235\pi$
$769$	−1.14606e30	−0.571455	−0.285728	+	0.958311i	$0.592235\pi$
$770$	0	0
$771$	−1.02273e30	−0.496238
$772$	0	0
$773$	−7.86838e29	−0.371536	−0.185768	−	0.982594i	$-0.559477\pi$
$773$	−7.86838e29	−0.371536	−0.185768	+	0.982594i	$0.559477\pi$
$774$	0	0
$775$	−5.41902e30	−2.49031
$776$	0	0
$777$	1.46873e30	0.656936
$778$	0	0
$779$	−7.37359e29	−0.321022
$780$	0	0
$781$	−6.66174e29	−0.282326
$782$	0	0
$783$	−7.97498e29	−0.329027
$784$	0	0
$785$	−4.20559e30	−1.68926
$786$	0	0
$787$	3.18569e29	0.124586	0.0622930	−	0.998058i	$-0.480159\pi$
$787$	3.18569e29	0.124586	0.0622930	+	0.998058i	$0.480159\pi$
$788$	0	0
$789$	7.65112e29	0.291351
$790$	0	0
$791$	1.41754e30	0.525632
$792$	0	0
$793$	5.36965e30	1.93900
$794$	0	0
$795$	−5.18664e29	−0.182403
$796$	0	0
$797$	9.85629e29	0.337599	0.168799	−	0.985650i	$-0.446011\pi$
$797$	9.85629e29	0.337599	0.168799	+	0.985650i	$0.446011\pi$
$798$	0	0
$799$	−1.66804e30	−0.556499
$800$	0	0
$801$	6.73482e29	0.218869
$802$	0	0
$803$	7.30914e29	0.231394
$804$	0	0
$805$	−4.35410e30	−1.34289
$806$	0	0
$807$	3.79221e30	1.13951
$808$	0	0
$809$	−1.90393e30	−0.557432	−0.278716	−	0.960374i	$-0.589909\pi$
$809$	−1.90393e30	−0.557432	−0.278716	+	0.960374i	$0.589909\pi$
$810$	0	0
$811$	−4.93041e28	−0.0140658	−0.00703289	−	0.999975i	$-0.502239\pi$
$811$	−4.93041e28	−0.0140658	−0.00703289	+	0.999975i	$0.502239\pi$
$812$	0	0
$813$	−1.12460e30	−0.312641
$814$	0	0
$815$	2.74482e30	0.743633
$816$	0	0
$817$	2.04587e30	0.540189
$818$	0	0
$819$	2.64973e30	0.681900
$820$	0	0
$821$	−5.83238e30	−1.46300	−0.731498	−	0.681844i	$-0.761178\pi$
$821$	−5.83238e30	−1.46300	−0.731498	+	0.681844i	$0.761178\pi$
$822$	0	0
$823$	2.49211e30	0.609353	0.304676	−	0.952456i	$-0.401452\pi$
$823$	2.49211e30	0.609353	0.304676	+	0.952456i	$0.401452\pi$
$824$	0	0
$825$	1.35914e30	0.323965
$826$	0	0
$827$	−1.15577e30	−0.268573	−0.134286	−	0.990943i	$-0.542874\pi$
$827$	−1.15577e30	−0.268573	−0.134286	+	0.990943i	$0.542874\pi$
$828$	0	0
$829$	9.17045e29	0.207763	0.103882	−	0.994590i	$-0.466874\pi$
$829$	9.17045e29	0.207763	0.103882	+	0.994590i	$0.466874\pi$
$830$	0	0
$831$	−1.62619e30	−0.359219
$832$	0	0
$833$	1.22347e31	2.63525
$834$	0	0
$835$	1.32790e31	2.78905
$836$	0	0
$837$	−9.83981e29	−0.201544
$838$	0	0
$839$	1.07861e30	0.215458	0.107729	−	0.994180i	$-0.465642\pi$
$839$	1.07861e30	0.215458	0.107729	+	0.994180i	$0.465642\pi$
$840$	0	0
$841$	9.87036e30	1.92298
$842$	0	0
$843$	−4.47859e29	−0.0851044
$844$	0	0
$845$	3.39357e30	0.629015
$846$	0	0
$847$	9.22190e30	1.66742
$848$	0	0
$849$	2.61831e30	0.461839
$850$	0	0
$851$	−1.54955e30	−0.266653
$852$	0	0
$853$	−3.30433e30	−0.554776	−0.277388	−	0.960758i	$-0.589469\pi$
$853$	−3.30433e30	−0.554776	−0.277388	+	0.960758i	$0.589469\pi$
$854$	0	0
$855$	−2.18985e30	−0.358732
$856$	0	0
$857$	−1.19979e31	−1.91782	−0.958909	−	0.283715i	$-0.908433\pi$
$857$	−1.19979e31	−1.91782	−0.958909	+	0.283715i	$0.908433\pi$
$858$	0	0
$859$	5.05979e30	0.789231	0.394615	−	0.918846i	$-0.370878\pi$
$859$	5.05979e30	0.789231	0.394615	+	0.918846i	$0.370878\pi$
$860$	0	0
$861$	−3.67172e30	−0.558902
$862$	0	0
$863$	2.32114e29	0.0344816	0.0172408	−	0.999851i	$-0.494512\pi$
$863$	2.32114e29	0.0344816	0.0172408	+	0.999851i	$0.494512\pi$
$864$	0	0
$865$	2.27057e30	0.329204
$866$	0	0
$867$	2.23704e30	0.316573
$868$	0	0
$869$	6.00051e29	0.0828859
$870$	0	0
$871$	−9.20792e30	−1.24157
$872$	0	0
$873$	−1.81439e30	−0.238826
$874$	0	0
$875$	−3.47991e31	−4.47181
$876$	0	0
$877$	1.02892e31	1.29088	0.645439	−	0.763812i	$-0.276675\pi$
$877$	1.02892e31	1.29088	0.645439	+	0.763812i	$0.276675\pi$
$878$	0	0
$879$	5.34427e30	0.654643
$880$	0	0
$881$	4.57535e30	0.547239	0.273620	−	0.961838i	$-0.411779\pi$
$881$	4.57535e30	0.547239	0.273620	+	0.961838i	$0.411779\pi$
$882$	0	0
$883$	−8.32468e30	−0.972255	−0.486127	−	0.873888i	$-0.661591\pi$
$883$	−8.32468e30	−0.972255	−0.486127	+	0.873888i	$0.661591\pi$
$884$	0	0
$885$	−4.44646e30	−0.507119
$886$	0	0
$887$	−1.68222e31	−1.87364	−0.936818	−	0.349817i	$-0.886244\pi$
$887$	−1.68222e31	−1.87364	−0.936818	+	0.349817i	$0.886244\pi$
$888$	0	0
$889$	−1.56928e31	−1.70699
$890$	0	0
$891$	2.46791e29	0.0262189
$892$	0	0
$893$	2.52371e30	0.261878
$894$	0	0
$895$	−1.07478e31	−1.08937
$896$	0	0
$897$	−2.79553e30	−0.276786
$898$	0	0
$899$	1.85115e31	1.79046
$900$	0	0
$901$	−2.26363e30	−0.213893
$902$	0	0
$903$	1.01875e31	0.940473
$904$	0	0
$905$	−2.33990e31	−2.11051
$906$	0	0
$907$	7.32473e30	0.645528	0.322764	−	0.946479i	$-0.395388\pi$
$907$	7.32473e30	0.645528	0.322764	+	0.946479i	$0.395388\pi$
$908$	0	0
$909$	8.17234e29	0.0703761
$910$	0	0
$911$	7.32814e30	0.616667	0.308333	−	0.951278i	$-0.400229\pi$
$911$	7.32814e30	0.616667	0.308333	+	0.951278i	$0.400229\pi$
$912$	0	0
$913$	−3.79513e30	−0.312092
$914$	0	0
$915$	−2.20978e31	−1.77594
$916$	0	0
$917$	3.55965e31	2.79595
$918$	0	0
$919$	7.09983e30	0.545048	0.272524	−	0.962149i	$-0.412142\pi$
$919$	7.09983e30	0.545048	0.272524	+	0.962149i	$0.412142\pi$
$920$	0	0
$921$	5.15129e30	0.386536
$922$	0	0
$923$	1.88985e31	1.38615
$924$	0	0
$925$	−2.13721e31	−1.53235
$926$	0	0
$927$	2.85861e29	0.0200363
$928$	0	0
$929$	−1.29312e31	−0.886080	−0.443040	−	0.896502i	$-0.646100\pi$
$929$	−1.29312e31	−0.886080	−0.443040	+	0.896502i	$0.646100\pi$
$930$	0	0
$931$	−1.85109e31	−1.24010
$932$	0	0
$933$	9.38485e30	0.614710
$934$	0	0
$935$	8.42627e30	0.539651
$936$	0	0
$937$	−1.25772e31	−0.787621	−0.393811	−	0.919192i	$-0.628843\pi$
$937$	−1.25772e31	−0.787621	−0.393811	+	0.919192i	$0.628843\pi$
$938$	0	0
$939$	1.49445e31	0.915150
$940$	0	0
$941$	−3.86885e30	−0.231681	−0.115841	−	0.993268i	$-0.536956\pi$
$941$	−3.86885e30	−0.231681	−0.115841	+	0.993268i	$0.536956\pi$
$942$	0	0
$943$	3.87376e30	0.226861
$944$	0	0
$945$	−1.09045e31	−0.624555
$946$	0	0
$947$	−2.33673e30	−0.130898	−0.0654491	−	0.997856i	$-0.520848\pi$
$947$	−2.33673e30	−0.130898	−0.0654491	+	0.997856i	$0.520848\pi$
$948$	0	0
$949$	−2.07351e31	−1.13609
$950$	0	0
$951$	−1.82612e31	−0.978663
$952$	0	0
$953$	6.04194e30	0.316738	0.158369	−	0.987380i	$-0.449376\pi$
$953$	6.04194e30	0.316738	0.158369	+	0.987380i	$0.449376\pi$
$954$	0	0
$955$	−2.58444e31	−1.32535
$956$	0	0
$957$	−4.64285e30	−0.232921
$958$	0	0
$959$	2.37214e31	1.16425
$960$	0	0
$961$	2.01461e30	0.0967378
$962$	0	0
$963$	2.04622e30	0.0961338
$964$	0	0
$965$	8.18088e30	0.376065
$966$	0	0
$967$	−3.71796e31	−1.67235	−0.836174	−	0.548464i	$-0.815213\pi$
$967$	−3.71796e31	−1.67235	−0.836174	+	0.548464i	$0.815213\pi$
$968$	0	0
$969$	−9.55728e30	−0.420663
$970$	0	0
$971$	4.01669e31	1.73008	0.865040	−	0.501704i	$-0.167293\pi$
$971$	4.01669e31	1.73008	0.865040	+	0.501704i	$0.167293\pi$
$972$	0	0
$973$	7.74974e31	3.26665
$974$	0	0
$975$	−3.85572e31	−1.59058
$976$	0	0
$977$	−1.28635e31	−0.519356	−0.259678	−	0.965695i	$-0.583616\pi$
$977$	−1.28635e31	−0.519356	−0.259678	+	0.965695i	$0.583616\pi$
$978$	0	0
$979$	3.92085e30	0.154940
$980$	0	0
$981$	−2.00684e30	−0.0776226
$982$	0	0
$983$	−4.51422e31	−1.70911	−0.854556	−	0.519359i	$-0.826171\pi$
$983$	−4.51422e31	−1.70911	−0.854556	+	0.519359i	$0.826171\pi$
$984$	0	0
$985$	−8.06552e31	−2.98918
$986$	0	0
$987$	1.25669e31	0.455931
$988$	0	0
$989$	−1.07481e31	−0.381742
$990$	0	0
$991$	−5.33643e31	−1.85557	−0.927785	−	0.373115i	$-0.878290\pi$
$991$	−5.33643e31	−1.85557	−0.927785	+	0.373115i	$0.878290\pi$
$992$	0	0
$993$	1.59209e31	0.542000
$994$	0	0
$995$	5.51200e31	1.83725
$996$	0	0
$997$	2.92841e31	0.955725	0.477862	−	0.878435i	$-0.341412\pi$
$997$	2.92841e31	0.955725	0.477862	+	0.878435i	$0.341412\pi$
$998$	0	0
$999$	−3.88073e30	−0.124015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.i.1.2		2
4.3	odd	2		12.22.a.b.1.2	✓	2
12.11	even	2		36.22.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.22.a.b.1.2	✓	2	4.3	odd	2
36.22.a.b.1.1		2	12.11	even	2
48.22.a.i.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+59049.0 q^{3} +4.01339e7 q^{5} -1.31964e9 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q+59049.0 q^{3} +4.01339e7 q^{5} -1.31964e9 q^{7} +3.48678e9 q^{9} +2.02992e10 q^{11} -5.75865e11 q^{13} +2.36987e12 q^{15} +1.03429e13 q^{17} -1.56487e13 q^{19} -7.79234e13 q^{21} +8.22113e13 q^{23} +1.13389e15 q^{25} +2.05891e14 q^{27} -3.87340e15 q^{29} -4.77913e15 q^{31} +1.19865e15 q^{33} -5.29623e16 q^{35} -1.88484e16 q^{37} -3.40043e16 q^{39} +4.71195e16 q^{41} -1.30737e17 q^{43} +1.39938e17 q^{45} -1.61273e17 q^{47} +1.18290e18 q^{49} +6.10740e17 q^{51} -2.18858e17 q^{53} +8.14688e17 q^{55} -9.24039e17 q^{57} -1.87625e18 q^{59} -9.32449e18 q^{61} -4.60130e18 q^{63} -2.31117e19 q^{65} +1.59897e19 q^{67} +4.85449e18 q^{69} -3.28177e19 q^{71} +3.60069e19 q^{73} +6.69552e19 q^{75} -2.67877e19 q^{77} +2.95603e19 q^{79} +1.21577e19 q^{81} -1.86959e20 q^{83} +4.15102e20 q^{85} -2.28720e20 q^{87} +1.93153e20 q^{89} +7.59935e20 q^{91} -2.82203e20 q^{93} -6.28043e20 q^{95} -5.20362e20 q^{97} +7.07791e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9}+O(q^{10}) \) 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9	\( 2 q + 118098 q^{3} + 28827900 q^{5} - 509669728 q^{7} + 6973568802 q^{9} + 76050855288 q^{11} - 809326043300 q^{13} + 1702258667100 q^{15} + 831385897668 q^{17} - 29817568652920 q^{19} - 30095487768672 q^{21} - 62680641406128 q^{23} + 784879798238750 q^{25} + 411782264189298 q^{27} - 21\!\cdots\!52 q^{29}+ \cdots + 26\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q + 118098 * q^3 + 28827900 * q^5 - 509669728 * q^7 + 6973568802 * q^9 + 76050855288 * q^11 - 809326043300 * q^13 + 1702258667100 * q^15 + 831385897668 * q^17 - 29817568652920 * q^19 - 30095487768672 * q^21 - 62680641406128 * q^23 + 784879798238750 * q^25 + 411782264189298 * q^27 - 2113791894718452 * q^29 - 6575738314225648 * q^31 + 4490726953901112 * q^33 - 62119804481121600 * q^35 + 34811548897106284 * q^37 - 47789893530821700 * q^39 + 111628536749375220 * q^41 - 250859109258832648 * q^43 + 100516672033587900 * q^45 - 773718896888534592 * q^47 + 1280410738569171378 * q^49 + 49092505871397732 * q^51 - 589751680675248324 * q^53 + 184360807989219600 * q^55 - 1760697611386273080 * q^57 + 2865708124062989112 * q^59 - 17394704031104736356 * q^61 - 1777108457252312928 * q^63 - 20472203381314851000 * q^65 + 15757929817647896648 * q^67 - 3701229194390452272 * q^69 - 9134476240415397840 * q^71 + 14362399003114082644 * q^73 + 46346367206199948750 * q^75 + 18369452723793706368 * q^77 + 347027209795696304 * q^79 + 24315330918113857602 * q^81 - 320685874118443742328 * q^83 + 522639873698354580600 * q^85 - 124817297591229872148 * q^87 + 691379668751362059060 * q^89 + 570838488118933393600 * q^91 - 388290771716710288752 * q^93 - 467849347422771978000 * q^95 - 1174056891416588162684 * q^97 + 265172935900906762488 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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