LMFDB - Embedded newform 48.9.g.b.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,9,Mod(31,48)]

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([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.5541732829$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.9.g.b.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654i q^{3} +726.000 q^{5} +3055.34i q^{7} -2187.00 q^{9} +O(q^{10})$	$q+46.7654i q^{3} +726.000 q^{5} +3055.34i q^{7} -2187.00 q^{9} -13281.4i q^{11} +39034.0 q^{13} +33951.7i q^{15} -65814.0 q^{17} +130257. i q^{19} -142884. q^{21} +502073. i q^{23} +136451. q^{25} -102276. i q^{27} +202062. q^{29} +1.19563e6i q^{31} +621108. q^{33} +2.21818e6i q^{35} -1.87603e6 q^{37} +1.82544e6i q^{39} +3.09105e6 q^{41} +2.26388e6i q^{43} -1.58776e6 q^{45} -6.35672e6i q^{47} -3.57029e6 q^{49} -3.07782e6i q^{51} -1.06648e6 q^{53} -9.64227e6i q^{55} -6.09152e6 q^{57} +5.76355e6i q^{59} +1.71542e7 q^{61} -6.68202e6i q^{63} +2.83387e7 q^{65} -2.74275e7i q^{67} -2.34796e7 q^{69} -3.98336e7i q^{71} -5.32860e7 q^{73} +6.38118e6i q^{75} +4.05791e7 q^{77} +1.82696e7i q^{79} +4.78297e6 q^{81} +7.78905e6i q^{83} -4.77810e7 q^{85} +9.44950e6i q^{87} +8.66672e7 q^{89} +1.19262e8i q^{91} -5.59143e7 q^{93} +9.45667e7i q^{95} -7.39018e7 q^{97} +2.90463e7i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1452 q^{5} - 4374 q^{9}+O(q^{10}) $ 2 * q + 1452 * q^5 - 4374 * q^9	$ 2 q + 1452 q^{5} - 4374 q^{9} + 78068 q^{13} - 131628 q^{17} - 285768 q^{21} + 272902 q^{25} + 404124 q^{29} + 1242216 q^{33} - 3752060 q^{37} + 6182100 q^{41} - 3175524 q^{45} - 7140574 q^{49} - 2132964 q^{53} - 12183048 q^{57} + 34308388 q^{61} + 56677368 q^{65} - 46959264 q^{69} - 106572028 q^{73} + 81158112 q^{77} + 9565938 q^{81} - 95561928 q^{85} + 173334468 q^{89} - 111828600 q^{93} - 147803644 q^{97}+O(q^{100}) $ 2 * q + 1452 * q^5 - 4374 * q^9 + 78068 * q^13 - 131628 * q^17 - 285768 * q^21 + 272902 * q^25 + 404124 * q^29 + 1242216 * q^33 - 3752060 * q^37 + 6182100 * q^41 - 3175524 * q^45 - 7140574 * q^49 - 2132964 * q^53 - 12183048 * q^57 + 34308388 * q^61 + 56677368 * q^65 - 46959264 * q^69 - 106572028 * q^73 + 81158112 * q^77 + 9565938 * q^81 - 95561928 * q^85 + 173334468 * q^89 - 111828600 * q^93 - 147803644 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	46.7654i	0.577350i
$3$	46.7654i	0.577350i
$4$	0	0
$5$	726.000	1.16160	0.580800	−	0.814046i	$-0.302740\pi$
$5$	726.000	1.16160	0.580800	+	0.814046i	$0.302740\pi$
$6$	0	0
$7$	3055.34i	1.27253i	0.771472	+	0.636264i	$0.219521\pi$
$7$	3055.34i	1.27253i	−0.771472	+	0.636264i	$0.780479\pi$
$8$	0	0
$9$	−2187.00	−0.333333
$10$	0	0
$11$	− 13281.4i	− 0.907135i	−0.891222	−	0.453568i	$-0.850151\pi$
$11$	− 13281.4i	− 0.907135i	0.891222	−	0.453568i	$-0.149849\pi$
$12$	0	0
$13$	39034.0	1.36669	0.683344	−	0.730096i	$-0.260525\pi$
$13$	39034.0	1.36669	0.683344	+	0.730096i	$0.260525\pi$
$14$	0	0
$15$	33951.7i	0.670650i
$16$	0	0
$17$	−65814.0	−0.787993	−0.393997	−	0.919112i	$-0.628908\pi$
$17$	−65814.0	−0.787993	−0.393997	+	0.919112i	$0.628908\pi$
$18$	0	0
$19$	130257.i	0.999510i	0.866167	+	0.499755i	$0.166577\pi$
$19$	130257.i	0.999510i	−0.866167	+	0.499755i	$0.833423\pi$
$20$	0	0
$21$	−142884.	−0.734694
$22$	0	0
$23$	502073.i	1.79414i	0.441892	+	0.897068i	$0.354308\pi$
$23$	502073.i	1.79414i	−0.441892	+	0.897068i	$0.645692\pi$
$24$	0	0
$25$	136451.	0.349315
$26$	0	0
$27$	− 102276.i	− 0.192450i
$28$	0	0
$29$	202062.	0.285688	0.142844	−	0.989745i	$-0.454375\pi$
$29$	202062.	0.285688	0.142844	+	0.989745i	$0.454375\pi$
$30$	0	0
$31$	1.19563e6i	1.29465i	0.762215	+	0.647324i	$0.224112\pi$
$31$	1.19563e6i	1.29465i	−0.762215	+	0.647324i	$0.775888\pi$
$32$	0	0
$33$	621108.	0.523735
$34$	0	0
$35$	2.21818e6i	1.47817i
$36$	0	0
$37$	−1.87603e6	−1.00100	−0.500499	−	0.865737i	$-0.666850\pi$
$37$	−1.87603e6	−1.00100	−0.500499	+	0.865737i	$0.666850\pi$
$38$	0	0
$39$	1.82544e6i	0.789058i
$40$	0	0
$41$	3.09105e6	1.09388	0.546941	−	0.837171i	$-0.315792\pi$
$41$	3.09105e6	1.09388	0.546941	+	0.837171i	$0.315792\pi$
$42$	0	0
$43$	2.26388e6i	0.662186i	0.943598	+	0.331093i	$0.107417\pi$
$43$	2.26388e6i	0.662186i	−0.943598	+	0.331093i	$0.892583\pi$
$44$	0	0
$45$	−1.58776e6	−0.387200
$46$	0	0
$47$	− 6.35672e6i	− 1.30269i	−0.758781	−	0.651346i	$-0.774205\pi$
$47$	− 6.35672e6i	− 1.30269i	0.758781	−	0.651346i	$-0.225795\pi$
$48$	0	0
$49$	−3.57029e6	−0.619325
$50$	0	0
$51$	− 3.07782e6i	− 0.454948i
$52$	0	0
$53$	−1.06648e6	−0.135161	−0.0675803	−	0.997714i	$-0.521528\pi$
$53$	−1.06648e6	−0.135161	−0.0675803	+	0.997714i	$0.521528\pi$
$54$	0	0
$55$	− 9.64227e6i	− 1.05373i
$56$	0	0
$57$	−6.09152e6	−0.577067
$58$	0	0
$59$	5.76355e6i	0.475644i	0.971309	+	0.237822i	$0.0764335\pi$
$59$	5.76355e6i	0.475644i	−0.971309	+	0.237822i	$0.923566\pi$
$60$	0	0
$61$	1.71542e7	1.23894	0.619471	−	0.785020i	$-0.287347\pi$
$61$	1.71542e7	1.23894	0.619471	+	0.785020i	$0.287347\pi$
$62$	0	0
$63$	− 6.68202e6i	− 0.424176i
$64$	0	0
$65$	2.83387e7	1.58755
$66$	0	0
$67$	− 2.74275e7i	− 1.36109i	−0.732706	−	0.680546i	$-0.761743\pi$
$67$	− 2.74275e7i	− 1.36109i	0.732706	−	0.680546i	$-0.238257\pi$
$68$	0	0
$69$	−2.34796e7	−1.03585
$70$	0	0
$71$	− 3.98336e7i	− 1.56753i	−0.621056	−	0.783766i	$-0.713296\pi$
$71$	− 3.98336e7i	− 1.56753i	0.621056	−	0.783766i	$-0.286704\pi$
$72$	0	0
$73$	−5.32860e7	−1.87638	−0.938192	−	0.346115i	$-0.887501\pi$
$73$	−5.32860e7	−1.87638	−0.938192	+	0.346115i	$0.887501\pi$
$74$	0	0
$75$	6.38118e6i	0.201677i
$76$	0	0
$77$	4.05791e7	1.15435
$78$	0	0
$79$	1.82696e7i	0.469052i	0.972110	+	0.234526i	$0.0753538\pi$
$79$	1.82696e7i	0.469052i	−0.972110	+	0.234526i	$0.924646\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	7.78905e6i	0.164124i	0.996627	+	0.0820620i	$0.0261506\pi$
$83$	7.78905e6i	0.164124i	−0.996627	+	0.0820620i	$0.973849\pi$
$84$	0	0
$85$	−4.77810e7	−0.915333
$86$	0	0
$87$	9.44950e6i	0.164942i
$88$	0	0
$89$	8.66672e7	1.38132	0.690661	−	0.723179i	$-0.257320\pi$
$89$	8.66672e7	1.38132	0.690661	+	0.723179i	$0.257320\pi$
$90$	0	0
$91$	1.19262e8i	1.73915i
$92$	0	0
$93$	−5.59143e7	−0.747465
$94$	0	0
$95$	9.45667e7i	1.16103i
$96$	0	0
$97$	−7.39018e7	−0.834773	−0.417386	−	0.908729i	$-0.637054\pi$
$97$	−7.39018e7	−0.834773	−0.417386	+	0.908729i	$0.637054\pi$
$98$	0	0
$99$	2.90463e7i	0.302378i
$100$	0	0
$101$	1.91310e8	1.83845	0.919227	−	0.393727i	$-0.128815\pi$
$101$	1.91310e8	1.83845	0.919227	+	0.393727i	$0.128815\pi$
$102$	0	0
$103$	− 1.62781e8i	− 1.44629i	−0.690699	−	0.723143i	$-0.742697\pi$
$103$	− 1.62781e8i	− 1.44629i	0.690699	−	0.723143i	$-0.257303\pi$
$104$	0	0
$105$	−1.03734e8	−0.853420
$106$	0	0
$107$	2.00810e8i	1.53197i	0.642857	+	0.765986i	$0.277749\pi$
$107$	2.00810e8i	1.53197i	−0.642857	+	0.765986i	$0.722251\pi$
$108$	0	0
$109$	6.86083e7	0.486039	0.243019	−	0.970021i	$-0.421862\pi$
$109$	6.86083e7	0.486039	0.243019	+	0.970021i	$0.421862\pi$
$110$	0	0
$111$	− 8.77332e7i	− 0.577926i
$112$	0	0
$113$	3.30831e7	0.202905	0.101452	−	0.994840i	$-0.467651\pi$
$113$	3.30831e7	0.202905	0.101452	+	0.994840i	$0.467651\pi$
$114$	0	0
$115$	3.64505e8i	2.08407i
$116$	0	0
$117$	−8.53674e7	−0.455563
$118$	0	0
$119$	− 2.01084e8i	− 1.00274i
$120$	0	0
$121$	3.79642e7	0.177106
$122$	0	0
$123$	1.44554e8i	0.631553i
$124$	0	0
$125$	−1.84530e8	−0.755836
$126$	0	0
$127$	− 2.70471e8i	− 1.03970i	−0.854259	−	0.519848i	$-0.825989\pi$
$127$	− 2.70471e8i	− 1.03970i	0.854259	−	0.519848i	$-0.174011\pi$
$128$	0	0
$129$	−1.05871e8	−0.382313
$130$	0	0
$131$	− 3.02851e8i	− 1.02836i	−0.857683	−	0.514178i	$-0.828097\pi$
$131$	− 3.02851e8i	− 1.02836i	0.857683	−	0.514178i	$-0.171903\pi$
$132$	0	0
$133$	−3.97980e8	−1.27190
$134$	0	0
$135$	− 7.42523e7i	− 0.223550i
$136$	0	0
$137$	6.40316e8	1.81766	0.908828	−	0.417170i	$-0.136978\pi$
$137$	6.40316e8	1.81766	0.908828	+	0.417170i	$0.136978\pi$
$138$	0	0
$139$	− 4.90714e8i	− 1.31453i	−0.753661	−	0.657263i	$-0.771714\pi$
$139$	− 4.90714e8i	− 1.31453i	0.753661	−	0.657263i	$-0.228286\pi$
$140$	0	0
$141$	2.97275e8	0.752110
$142$	0	0
$143$	− 5.18425e8i	− 1.23977i
$144$	0	0
$145$	1.46697e8	0.331856
$146$	0	0
$147$	− 1.66966e8i	− 0.357568i
$148$	0	0
$149$	−8.11121e7	−0.164566	−0.0822831	−	0.996609i	$-0.526221\pi$
$149$	−8.11121e7	−0.164566	−0.0822831	+	0.996609i	$0.526221\pi$
$150$	0	0
$151$	− 1.77325e8i	− 0.341086i	−0.985350	−	0.170543i	$-0.945448\pi$
$151$	− 1.77325e8i	− 0.341086i	0.985350	−	0.170543i	$-0.0545521\pi$
$152$	0	0
$153$	1.43935e8	0.262664
$154$	0	0
$155$	8.68031e8i	1.50386i
$156$	0	0
$157$	−2.14784e7	−0.0353511	−0.0176755	−	0.999844i	$-0.505627\pi$
$157$	−2.14784e7	−0.0353511	−0.0176755	+	0.999844i	$0.505627\pi$
$158$	0	0
$159$	− 4.98744e7i	− 0.0780350i
$160$	0	0
$161$	−1.53400e9	−2.28309
$162$	0	0
$163$	2.42230e8i	0.343144i	0.985172	+	0.171572i	$0.0548847\pi$
$163$	2.42230e8i	0.343144i	−0.985172	+	0.171572i	$0.945115\pi$
$164$	0	0
$165$	4.50924e8	0.608370
$166$	0	0
$167$	− 3.89012e8i	− 0.500146i	−0.968227	−	0.250073i	$-0.919545\pi$
$167$	− 3.89012e8i	− 0.500146i	0.968227	−	0.250073i	$-0.0804547\pi$
$168$	0	0
$169$	7.07922e8	0.867838
$170$	0	0
$171$	− 2.84872e8i	− 0.333170i
$172$	0	0
$173$	−6.35072e7	−0.0708988	−0.0354494	−	0.999371i	$-0.511286\pi$
$173$	−6.35072e7	−0.0708988	−0.0354494	+	0.999371i	$0.511286\pi$
$174$	0	0
$175$	4.16904e8i	0.444512i
$176$	0	0
$177$	−2.69535e8	−0.274613
$178$	0	0
$179$	5.33629e8i	0.519789i	0.965637	+	0.259895i	$0.0836878\pi$
$179$	5.33629e8i	0.519789i	−0.965637	+	0.259895i	$0.916312\pi$
$180$	0	0
$181$	8.56360e8	0.797888	0.398944	−	0.916975i	$-0.369377\pi$
$181$	8.56360e8	0.797888	0.398944	+	0.916975i	$0.369377\pi$
$182$	0	0
$183$	8.02222e8i	0.715303i
$184$	0	0
$185$	−1.36200e9	−1.16276
$186$	0	0
$187$	8.74100e8i	0.714817i
$188$	0	0
$189$	3.12487e8	0.244898
$190$	0	0
$191$	4.75759e8i	0.357481i	0.983896	+	0.178741i	$0.0572023\pi$
$191$	4.75759e8i	0.357481i	−0.983896	+	0.178741i	$0.942798\pi$
$192$	0	0
$193$	8.76708e8	0.631867	0.315933	−	0.948781i	$-0.397682\pi$
$193$	8.76708e8	0.631867	0.315933	+	0.948781i	$0.397682\pi$
$194$	0	0
$195$	1.32527e9i	0.916570i
$196$	0	0
$197$	−2.76762e9	−1.83756	−0.918780	−	0.394771i	$-0.870824\pi$
$197$	−2.76762e9	−1.83756	−0.918780	+	0.394771i	$0.870824\pi$
$198$	0	0
$199$	1.42932e9i	0.911420i	0.890128	+	0.455710i	$0.150615\pi$
$199$	1.42932e9i	0.911420i	−0.890128	+	0.455710i	$0.849385\pi$
$200$	0	0
$201$	1.28266e9	0.785826
$202$	0	0
$203$	6.17368e8i	0.363546i
$204$	0	0
$205$	2.24410e9	1.27065
$206$	0	0
$207$	− 1.09803e9i	− 0.598046i
$208$	0	0
$209$	1.72999e9	0.906691
$210$	0	0
$211$	4.61738e8i	0.232952i	0.993194	+	0.116476i	$0.0371598\pi$
$211$	4.61738e8i	0.232952i	−0.993194	+	0.116476i	$0.962840\pi$
$212$	0	0
$213$	1.86283e9	0.905015
$214$	0	0
$215$	1.64358e9i	0.769195i
$216$	0	0
$217$	−3.65307e9	−1.64747
$218$	0	0
$219$	− 2.49194e9i	− 1.08333i
$220$	0	0
$221$	−2.56898e9	−1.07694
$222$	0	0
$223$	− 3.40037e9i	− 1.37501i	−0.726179	−	0.687506i	$-0.758705\pi$
$223$	− 3.40037e9i	− 1.37501i	0.726179	−	0.687506i	$-0.241295\pi$
$224$	0	0
$225$	−2.98418e8	−0.116438
$226$	0	0
$227$	− 4.52697e9i	− 1.70492i	−0.522792	−	0.852460i	$-0.675109\pi$
$227$	− 4.52697e9i	− 1.70492i	0.522792	−	0.852460i	$-0.324891\pi$
$228$	0	0
$229$	−9.90176e8	−0.360056	−0.180028	−	0.983661i	$-0.557619\pi$
$229$	−9.90176e8	−0.360056	−0.180028	+	0.983661i	$0.557619\pi$
$230$	0	0
$231$	1.89769e9i	0.666467i
$232$	0	0
$233$	2.23709e9	0.759032	0.379516	−	0.925185i	$-0.376091\pi$
$233$	2.23709e9	0.759032	0.379516	+	0.925185i	$0.376091\pi$
$234$	0	0
$235$	− 4.61498e9i	− 1.51321i
$236$	0	0
$237$	−8.54385e8	−0.270807
$238$	0	0
$239$	2.63524e9i	0.807659i	0.914834	+	0.403830i	$0.132321\pi$
$239$	2.63524e9i	0.807659i	−0.914834	+	0.403830i	$0.867679\pi$
$240$	0	0
$241$	−6.19651e8	−0.183687	−0.0918436	−	0.995773i	$-0.529276\pi$
$241$	−6.19651e8	−0.183687	−0.0918436	+	0.995773i	$0.529276\pi$
$242$	0	0
$243$	2.23677e8i	0.0641500i
$244$	0	0
$245$	−2.59203e9	−0.719408
$246$	0	0
$247$	5.08446e9i	1.36602i
$248$	0	0
$249$	−3.64258e8	−0.0947571
$250$	0	0
$251$	− 6.81334e9i	− 1.71659i	−0.513161	−	0.858293i	$-0.671526\pi$
$251$	− 6.81334e9i	− 1.71659i	0.513161	−	0.858293i	$-0.328474\pi$
$252$	0	0
$253$	6.66822e9	1.62752
$254$	0	0
$255$	− 2.23449e9i	− 0.528468i
$256$	0	0
$257$	3.95756e9	0.907184	0.453592	−	0.891209i	$-0.350142\pi$
$257$	3.95756e9	0.907184	0.453592	+	0.891209i	$0.350142\pi$
$258$	0	0
$259$	− 5.73191e9i	− 1.27380i
$260$	0	0
$261$	−4.41910e8	−0.0952295
$262$	0	0
$263$	− 1.86129e9i	− 0.389037i	−0.980899	−	0.194518i	$-0.937686\pi$
$263$	− 1.86129e9i	− 0.389037i	0.980899	−	0.194518i	$-0.0623144\pi$
$264$	0	0
$265$	−7.74266e8	−0.157003
$266$	0	0
$267$	4.05303e9i	0.797507i
$268$	0	0
$269$	1.17367e9	0.224149	0.112074	−	0.993700i	$-0.464250\pi$
$269$	1.17367e9	0.224149	0.112074	+	0.993700i	$0.464250\pi$
$270$	0	0
$271$	1.90505e9i	0.353207i	0.984282	+	0.176604i	$0.0565110\pi$
$271$	1.90505e9i	0.353207i	−0.984282	+	0.176604i	$0.943489\pi$
$272$	0	0
$273$	−5.57733e9	−1.00410
$274$	0	0
$275$	− 1.81226e9i	− 0.316876i
$276$	0	0
$277$	−5.03752e9	−0.855654	−0.427827	−	0.903861i	$-0.640721\pi$
$277$	−5.03752e9	−0.855654	−0.427827	+	0.903861i	$0.640721\pi$
$278$	0	0
$279$	− 2.61485e9i	− 0.431549i
$280$	0	0
$281$	6.66317e9	1.06870	0.534350	−	0.845264i	$-0.320557\pi$
$281$	6.66317e9	1.06870	0.534350	+	0.845264i	$0.320557\pi$
$282$	0	0
$283$	5.54295e9i	0.864162i	0.901835	+	0.432081i	$0.142221\pi$
$283$	5.54295e9i	0.864162i	−0.901835	+	0.432081i	$0.857779\pi$
$284$	0	0
$285$	−4.42245e9	−0.670321
$286$	0	0
$287$	9.44420e9i	1.39199i
$288$	0	0
$289$	−2.64427e9	−0.379066
$290$	0	0
$291$	− 3.45605e9i	− 0.481956i
$292$	0	0
$293$	6.67390e9	0.905543	0.452772	−	0.891627i	$-0.350435\pi$
$293$	6.67390e9	0.905543	0.452772	+	0.891627i	$0.350435\pi$
$294$	0	0
$295$	4.18434e9i	0.552508i
$296$	0	0
$297$	−1.35836e9	−0.174578
$298$	0	0
$299$	1.95979e10i	2.45203i
$300$	0	0
$301$	−6.91692e9	−0.842649
$302$	0	0
$303$	8.94670e9i	1.06143i
$304$	0	0
$305$	1.24539e10	1.43916
$306$	0	0
$307$	− 6.49752e8i	− 0.0731466i	−0.999331	−	0.0365733i	$-0.988356\pi$
$307$	− 6.49752e8i	− 0.0731466i	0.999331	−	0.0365733i	$-0.0116442\pi$
$308$	0	0
$309$	7.61250e9	0.835013
$310$	0	0
$311$	− 3.97832e8i	− 0.0425264i	−0.999774	−	0.0212632i	$-0.993231\pi$
$311$	− 3.97832e8i	− 0.0425264i	0.999774	−	0.0212632i	$-0.00676879\pi$
$312$	0	0
$313$	−1.58217e10	−1.64845	−0.824223	−	0.566266i	$-0.808388\pi$
$313$	−1.58217e10	−1.64845	−0.824223	+	0.566266i	$0.808388\pi$
$314$	0	0
$315$	− 4.85115e9i	− 0.492723i
$316$	0	0
$317$	−9.60836e9	−0.951507	−0.475754	−	0.879579i	$-0.657825\pi$
$317$	−9.60836e9	−0.951507	−0.475754	+	0.879579i	$0.657825\pi$
$318$	0	0
$319$	− 2.68366e9i	− 0.259158i
$320$	0	0
$321$	−9.39097e9	−0.884485
$322$	0	0
$323$	− 8.57274e9i	− 0.787607i
$324$	0	0
$325$	5.32623e9	0.477404
$326$	0	0
$327$	3.20849e9i	0.280615i
$328$	0	0
$329$	1.94219e10	1.65771
$330$	0	0
$331$	1.41991e10i	1.18290i	0.806341	+	0.591451i	$0.201445\pi$
$331$	1.41991e10i	1.18290i	−0.806341	+	0.591451i	$0.798555\pi$
$332$	0	0
$333$	4.10288e9	0.333666
$334$	0	0
$335$	− 1.99124e10i	− 1.58104i
$336$	0	0
$337$	3.39383e9	0.263130	0.131565	−	0.991308i	$-0.458000\pi$
$337$	3.39383e9	0.263130	0.131565	+	0.991308i	$0.458000\pi$
$338$	0	0
$339$	1.54714e9i	0.117147i
$340$	0	0
$341$	1.58797e10	1.17442
$342$	0	0
$343$	6.70498e9i	0.484419i
$344$	0	0
$345$	−1.70462e10	−1.20324
$346$	0	0
$347$	1.35188e8i	0.00932439i	0.999989	+	0.00466219i	$0.00148403\pi$
$347$	1.35188e8i	0.00932439i	−0.999989	+	0.00466219i	$0.998516\pi$
$348$	0	0
$349$	1.13213e10	0.763122	0.381561	−	0.924344i	$-0.375387\pi$
$349$	1.13213e10	0.763122	0.381561	+	0.924344i	$0.375387\pi$
$350$	0	0
$351$	− 3.99224e9i	− 0.263019i
$352$	0	0
$353$	−1.42650e10	−0.918697	−0.459348	−	0.888256i	$-0.651917\pi$
$353$	−1.42650e10	−0.918697	−0.459348	+	0.888256i	$0.651917\pi$
$354$	0	0
$355$	− 2.89192e10i	− 1.82085i
$356$	0	0
$357$	9.40377e9	0.578934
$358$	0	0
$359$	8.15636e9i	0.491042i	0.969391	+	0.245521i	$0.0789590\pi$
$359$	8.15636e9i	0.491042i	−0.969391	+	0.245521i	$0.921041\pi$
$360$	0	0
$361$	1.66382e7	0.000979664 0
$362$	0	0
$363$	1.77541e9i	0.102252i
$364$	0	0
$365$	−3.86856e10	−2.17961
$366$	0	0
$367$	− 2.06760e10i	− 1.13973i	−0.821738	−	0.569865i	$-0.806995\pi$
$367$	− 2.06760e10i	− 1.13973i	0.821738	−	0.569865i	$-0.193005\pi$
$368$	0	0
$369$	−6.76013e9	−0.364627
$370$	0	0
$371$	− 3.25846e9i	− 0.171996i
$372$	0	0
$373$	−7.71358e9	−0.398493	−0.199247	−	0.979949i	$-0.563849\pi$
$373$	−7.71358e9	−0.398493	−0.199247	+	0.979949i	$0.563849\pi$
$374$	0	0
$375$	− 8.62963e9i	− 0.436382i
$376$	0	0
$377$	7.88729e9	0.390447
$378$	0	0
$379$	1.53767e9i	0.0745256i	0.999306	+	0.0372628i	$0.0118639\pi$
$379$	1.53767e9i	0.0745256i	−0.999306	+	0.0372628i	$0.988136\pi$
$380$	0	0
$381$	1.26487e10	0.600268
$382$	0	0
$383$	− 3.20555e10i	− 1.48973i	−0.667216	−	0.744864i	$-0.732514\pi$
$383$	− 3.20555e10i	− 1.48973i	0.667216	−	0.744864i	$-0.267486\pi$
$384$	0	0
$385$	2.94604e10	1.34090
$386$	0	0
$387$	− 4.95111e9i	− 0.220729i
$388$	0	0
$389$	−2.99296e10	−1.30708	−0.653541	−	0.756891i	$-0.726717\pi$
$389$	−2.99296e10	−1.30708	−0.653541	+	0.756891i	$0.726717\pi$
$390$	0	0
$391$	− 3.30434e10i	− 1.41377i
$392$	0	0
$393$	1.41629e10	0.593722
$394$	0	0
$395$	1.32637e10i	0.544851i
$396$	0	0
$397$	1.32156e10	0.532016	0.266008	−	0.963971i	$-0.414295\pi$
$397$	1.32156e10	0.532016	0.266008	+	0.963971i	$0.414295\pi$
$398$	0	0
$399$	− 1.86117e10i	− 0.734334i
$400$	0	0
$401$	2.51637e10	0.973190	0.486595	−	0.873628i	$-0.338239\pi$
$401$	2.51637e10	0.973190	0.486595	+	0.873628i	$0.338239\pi$
$402$	0	0
$403$	4.66704e10i	1.76938i
$404$	0	0
$405$	3.47244e9	0.129067
$406$	0	0
$407$	2.49162e10i	0.908040i
$408$	0	0
$409$	−3.78473e10	−1.35251	−0.676257	−	0.736666i	$-0.736399\pi$
$409$	−3.78473e10	−1.35251	−0.676257	+	0.736666i	$0.736399\pi$
$410$	0	0
$411$	2.99446e10i	1.04942i
$412$	0	0
$413$	−1.76096e10	−0.605270
$414$	0	0
$415$	5.65485e9i	0.190647i
$416$	0	0
$417$	2.29484e10	0.758942
$418$	0	0
$419$	1.88088e10i	0.610246i	0.952313	+	0.305123i	$0.0986976\pi$
$419$	1.88088e10i	0.610246i	−0.952313	+	0.305123i	$0.901302\pi$
$420$	0	0
$421$	6.04555e9	0.192445	0.0962227	−	0.995360i	$-0.469324\pi$
$421$	6.04555e9	0.192445	0.0962227	+	0.995360i	$0.469324\pi$
$422$	0	0
$423$	1.39022e10i	0.434231i
$424$	0	0
$425$	−8.98039e9	−0.275258
$426$	0	0
$427$	5.24119e10i	1.57659i
$428$	0	0
$429$	2.42443e10	0.715782
$430$	0	0
$431$	− 4.60486e10i	− 1.33447i	−0.744849	−	0.667233i	$-0.767479\pi$
$431$	− 4.60486e10i	− 1.33447i	0.744849	−	0.667233i	$-0.232521\pi$
$432$	0	0
$433$	1.85654e9	0.0528145	0.0264072	−	0.999651i	$-0.491593\pi$
$433$	1.85654e9	0.0528145	0.0264072	+	0.999651i	$0.491593\pi$
$434$	0	0
$435$	6.86034e9i	0.191597i
$436$	0	0
$437$	−6.53986e10	−1.79326
$438$	0	0
$439$	1.24165e10i	0.334303i	0.985931	+	0.167152i	$0.0534569\pi$
$439$	1.24165e10i	0.334303i	−0.985931	+	0.167152i	$0.946543\pi$
$440$	0	0
$441$	7.80822e9	0.206442
$442$	0	0
$443$	7.57779e9i	0.196756i	0.995149	+	0.0983779i	$0.0313654\pi$
$443$	7.57779e9i	0.196756i	−0.995149	+	0.0983779i	$0.968635\pi$
$444$	0	0
$445$	6.29204e10	1.60454
$446$	0	0
$447$	− 3.79324e9i	− 0.0950123i
$448$	0	0
$449$	−3.37970e10	−0.831559	−0.415780	−	0.909465i	$-0.636491\pi$
$449$	−3.37970e10	−0.831559	−0.415780	+	0.909465i	$0.636491\pi$
$450$	0	0
$451$	− 4.10534e10i	− 0.992299i
$452$	0	0
$453$	8.29269e9	0.196926
$454$	0	0
$455$	8.65842e10i	2.02020i
$456$	0	0
$457$	−2.01366e10	−0.461659	−0.230829	−	0.972994i	$-0.574144\pi$
$457$	−2.01366e10	−0.461659	−0.230829	+	0.972994i	$0.574144\pi$
$458$	0	0
$459$	6.73118e9i	0.151649i
$460$	0	0
$461$	−2.54155e10	−0.562724	−0.281362	−	0.959602i	$-0.590786\pi$
$461$	−2.54155e10	−0.562724	−0.281362	+	0.959602i	$0.590786\pi$
$462$	0	0
$463$	− 1.19712e9i	− 0.0260504i	−0.999915	−	0.0130252i	$-0.995854\pi$
$463$	− 1.19712e9i	− 0.0260504i	0.999915	−	0.0130252i	$-0.00414617\pi$
$464$	0	0
$465$	−4.05938e10	−0.868256
$466$	0	0
$467$	2.92676e10i	0.615347i	0.951492	+	0.307673i	$0.0995504\pi$
$467$	2.92676e10i	0.615347i	−0.951492	+	0.307673i	$0.900450\pi$
$468$	0	0
$469$	8.38003e10	1.73203
$470$	0	0
$471$	− 1.00444e9i	− 0.0204100i
$472$	0	0
$473$	3.00674e10	0.600692
$474$	0	0
$475$	1.77737e10i	0.349143i
$476$	0	0
$477$	2.33240e9	0.0450535
$478$	0	0
$479$	− 2.51066e10i	− 0.476919i	−0.971152	−	0.238460i	$-0.923357\pi$
$479$	− 2.51066e10i	− 0.476919i	0.971152	−	0.238460i	$-0.0766425\pi$
$480$	0	0
$481$	−7.32290e10	−1.36805
$482$	0	0
$483$	− 7.17382e10i	− 1.31814i
$484$	0	0
$485$	−5.36527e10	−0.969672
$486$	0	0
$487$	5.82581e10i	1.03571i	0.855467	+	0.517857i	$0.173270\pi$
$487$	5.82581e10i	1.03571i	−0.855467	+	0.517857i	$0.826730\pi$
$488$	0	0
$489$	−1.13280e10	−0.198115
$490$	0	0
$491$	− 3.61816e10i	− 0.622532i	−0.950323	−	0.311266i	$-0.899247\pi$
$491$	− 3.61816e10i	− 0.622532i	0.950323	−	0.311266i	$-0.100753\pi$
$492$	0	0
$493$	−1.32985e10	−0.225121
$494$	0	0
$495$	2.10876e10i	0.351243i
$496$	0	0
$497$	1.21705e11	1.99473
$498$	0	0
$499$	− 5.58440e10i	− 0.900687i	−0.892855	−	0.450344i	$-0.851301\pi$
$499$	− 5.58440e10i	− 0.900687i	0.892855	−	0.450344i	$-0.148699\pi$
$500$	0	0
$501$	1.81923e10	0.288760
$502$	0	0
$503$	− 1.84340e10i	− 0.287971i	−0.989580	−	0.143985i	$-0.954008\pi$
$503$	− 1.84340e10i	− 0.287971i	0.989580	−	0.143985i	$-0.0459918\pi$
$504$	0	0
$505$	1.38891e11	2.13555
$506$	0	0
$507$	3.31063e10i	0.501047i
$508$	0	0
$509$	1.42165e10	0.211798	0.105899	−	0.994377i	$-0.466228\pi$
$509$	1.42165e10	0.211798	0.105899	+	0.994377i	$0.466228\pi$
$510$	0	0
$511$	− 1.62807e11i	− 2.38775i
$512$	0	0
$513$	1.33222e10	0.192356
$514$	0	0
$515$	− 1.18179e11i	− 1.68001i
$516$	0	0
$517$	−8.44260e10	−1.18172
$518$	0	0
$519$	− 2.96994e9i	− 0.0409334i
$520$	0	0
$521$	−6.81614e10	−0.925098	−0.462549	−	0.886594i	$-0.653065\pi$
$521$	−6.81614e10	−0.925098	−0.462549	+	0.886594i	$0.653065\pi$
$522$	0	0
$523$	− 5.63922e10i	− 0.753724i	−0.926269	−	0.376862i	$-0.877003\pi$
$523$	− 5.63922e10i	− 0.753724i	0.926269	−	0.376862i	$-0.122997\pi$
$524$	0	0
$525$	−1.94967e10	−0.256639
$526$	0	0
$527$	− 7.86895e10i	− 1.02017i
$528$	0	0
$529$	−1.73766e11	−2.21893
$530$	0	0
$531$	− 1.26049e10i	− 0.158548i
$532$	0	0
$533$	1.20656e11	1.49500
$534$	0	0
$535$	1.45788e11i	1.77954i
$536$	0	0
$537$	−2.49554e10	−0.300100
$538$	0	0
$539$	4.74183e10i	0.561812i
$540$	0	0
$541$	−7.61478e10	−0.888932	−0.444466	−	0.895796i	$-0.646607\pi$
$541$	−7.61478e10	−0.888932	−0.444466	+	0.895796i	$0.646607\pi$
$542$	0	0
$543$	4.00480e10i	0.460661i
$544$	0	0
$545$	4.98096e10	0.564583
$546$	0	0
$547$	− 5.84939e10i	− 0.653373i	−0.945133	−	0.326686i	$-0.894068\pi$
$547$	− 5.84939e10i	− 0.653373i	0.945133	−	0.326686i	$-0.105932\pi$
$548$	0	0
$549$	−3.75162e10	−0.412981
$550$	0	0
$551$	2.63200e10i	0.285548i
$552$	0	0
$553$	−5.58198e10	−0.596881
$554$	0	0
$555$	− 6.36943e10i	− 0.671319i
$556$	0	0
$557$	1.61301e11	1.67577	0.837887	−	0.545844i	$-0.183791\pi$
$557$	1.61301e11	1.67577	0.837887	+	0.545844i	$0.183791\pi$
$558$	0	0
$559$	8.83683e10i	0.905002i
$560$	0	0
$561$	−4.08776e10	−0.412700
$562$	0	0
$563$	− 6.88172e9i	− 0.0684957i	−0.999413	−	0.0342479i	$-0.989096\pi$
$563$	− 6.88172e9i	− 0.0684957i	0.999413	−	0.0342479i	$-0.0109036\pi$
$564$	0	0
$565$	2.40183e10	0.235694
$566$	0	0
$567$	1.46136e10i	0.141392i
$568$	0	0
$569$	9.38382e10	0.895221	0.447611	−	0.894229i	$-0.352275\pi$
$569$	9.38382e10	0.895221	0.447611	+	0.894229i	$0.352275\pi$
$570$	0	0
$571$	1.92744e10i	0.181316i	0.995882	+	0.0906582i	$0.0288971\pi$
$571$	1.92744e10i	0.181316i	−0.995882	+	0.0906582i	$0.971103\pi$
$572$	0	0
$573$	−2.22490e10	−0.206392
$574$	0	0
$575$	6.85084e10i	0.626718i
$576$	0	0
$577$	1.65488e11	1.49301	0.746507	−	0.665378i	$-0.231730\pi$
$577$	1.65488e11	1.49301	0.746507	+	0.665378i	$0.231730\pi$
$578$	0	0
$579$	4.09996e10i	0.364809i
$580$	0	0
$581$	−2.37982e10	−0.208852
$582$	0	0
$583$	1.41643e10i	0.122609i
$584$	0	0
$585$	−6.19767e10	−0.529182
$586$	0	0
$587$	2.09633e11i	1.76566i	0.469695	+	0.882829i	$0.344364\pi$
$587$	2.09633e11i	1.76566i	−0.469695	+	0.882829i	$0.655636\pi$
$588$	0	0
$589$	−1.55740e11	−1.29401
$590$	0	0
$591$	− 1.29429e11i	− 1.06092i
$592$	0	0
$593$	−7.33746e10	−0.593372	−0.296686	−	0.954975i	$-0.595881\pi$
$593$	−7.33746e10	−0.593372	−0.296686	+	0.954975i	$0.595881\pi$
$594$	0	0
$595$	− 1.45987e11i	− 1.16479i
$596$	0	0
$597$	−6.68429e10	−0.526209
$598$	0	0
$599$	1.33326e11i	1.03563i	0.855491	+	0.517817i	$0.173255\pi$
$599$	1.33326e11i	1.03563i	−0.855491	+	0.517817i	$0.826745\pi$
$600$	0	0
$601$	2.01691e11	1.54593	0.772965	−	0.634449i	$-0.218773\pi$
$601$	2.01691e11	1.54593	0.772965	+	0.634449i	$0.218773\pi$
$602$	0	0
$603$	5.99840e10i	0.453697i
$604$	0	0
$605$	2.75620e10	0.205726
$606$	0	0
$607$	1.55515e11i	1.14556i	0.819710	+	0.572779i	$0.194134\pi$
$607$	1.55515e11i	1.14556i	−0.819710	+	0.572779i	$0.805866\pi$
$608$	0	0
$609$	−2.88714e10	−0.209894
$610$	0	0
$611$	− 2.48128e11i	− 1.78038i
$612$	0	0
$613$	3.19775e10	0.226466	0.113233	−	0.993568i	$-0.463879\pi$
$613$	3.19775e10	0.226466	0.113233	+	0.993568i	$0.463879\pi$
$614$	0	0
$615$	1.04946e11i	0.733612i
$616$	0	0
$617$	5.63108e10	0.388553	0.194277	−	0.980947i	$-0.437764\pi$
$617$	5.63108e10	0.388553	0.194277	+	0.980947i	$0.437764\pi$
$618$	0	0
$619$	2.66432e11i	1.81478i	0.420287	+	0.907391i	$0.361929\pi$
$619$	2.66432e11i	1.81478i	−0.420287	+	0.907391i	$0.638071\pi$
$620$	0	0
$621$	5.13500e10	0.345282
$622$	0	0
$623$	2.64798e11i	1.75777i
$624$	0	0
$625$	−1.87270e11	−1.22729
$626$	0	0
$627$	8.09038e10i	0.523478i
$628$	0	0
$629$	1.23469e11	0.788779
$630$	0	0
$631$	8.55727e9i	0.0539781i	0.999636	+	0.0269891i	$0.00859193\pi$
$631$	8.55727e9i	0.0539781i	−0.999636	+	0.0269891i	$0.991408\pi$
$632$	0	0
$633$	−2.15934e10	−0.134495
$634$	0	0
$635$	− 1.96362e11i	− 1.20771i
$636$	0	0
$637$	−1.39363e11	−0.846425
$638$	0	0
$639$	8.71161e10i	0.522511i
$640$	0	0
$641$	2.90248e10	0.171924	0.0859620	−	0.996298i	$-0.472604\pi$
$641$	2.90248e10	0.171924	0.0859620	+	0.996298i	$0.472604\pi$
$642$	0	0
$643$	− 5.13563e10i	− 0.300435i	−0.988653	−	0.150217i	$-0.952003\pi$
$643$	− 5.13563e10i	− 0.300435i	0.988653	−	0.150217i	$-0.0479973\pi$
$644$	0	0
$645$	−7.68625e10	−0.444095
$646$	0	0
$647$	5.58175e10i	0.318532i	0.987236	+	0.159266i	$0.0509128\pi$
$647$	5.58175e10i	0.318532i	−0.987236	+	0.159266i	$0.949087\pi$
$648$	0	0
$649$	7.65478e10	0.431473
$650$	0	0
$651$	− 1.70837e11i	− 0.951170i
$652$	0	0
$653$	−6.40717e10	−0.352382	−0.176191	−	0.984356i	$-0.556378\pi$
$653$	−6.40717e10	−0.352382	−0.176191	+	0.984356i	$0.556378\pi$
$654$	0	0
$655$	− 2.19870e11i	− 1.19454i
$656$	0	0
$657$	1.16537e11	0.625461
$658$	0	0
$659$	− 3.17581e11i	− 1.68389i	−0.539567	−	0.841943i	$-0.681412\pi$
$659$	− 3.17581e11i	− 1.68389i	0.539567	−	0.841943i	$-0.318588\pi$
$660$	0	0
$661$	−1.33716e11	−0.700449	−0.350224	−	0.936666i	$-0.613895\pi$
$661$	−1.33716e11	−0.700449	−0.350224	+	0.936666i	$0.613895\pi$
$662$	0	0
$663$	− 1.20139e11i	− 0.621773i
$664$	0	0
$665$	−2.88933e11	−1.47744
$666$	0	0
$667$	1.01450e11i	0.512564i
$668$	0	0
$669$	1.59019e11	0.793864
$670$	0	0
$671$	− 2.27831e11i	− 1.12389i
$672$	0	0
$673$	−4.12429e10	−0.201043	−0.100521	−	0.994935i	$-0.532051\pi$
$673$	−4.12429e10	−0.201043	−0.100521	+	0.994935i	$0.532051\pi$
$674$	0	0
$675$	− 1.39556e10i	− 0.0672256i
$676$	0	0
$677$	−5.06159e10	−0.240953	−0.120476	−	0.992716i	$-0.538442\pi$
$677$	−5.06159e10	−0.240953	−0.120476	+	0.992716i	$0.538442\pi$
$678$	0	0
$679$	− 2.25795e11i	− 1.06227i
$680$	0	0
$681$	2.11705e11	0.984337
$682$	0	0
$683$	3.59716e11i	1.65302i	0.562924	+	0.826508i	$0.309676\pi$
$683$	3.59716e11i	1.65302i	−0.562924	+	0.826508i	$0.690324\pi$
$684$	0	0
$685$	4.64869e11	2.11139
$686$	0	0
$687$	− 4.63059e10i	− 0.207879i
$688$	0	0
$689$	−4.16291e10	−0.184722
$690$	0	0
$691$	− 1.38563e11i	− 0.607763i	−0.952710	−	0.303882i	$-0.901717\pi$
$691$	− 1.38563e11i	− 0.607763i	0.952710	−	0.303882i	$-0.0982827\pi$
$692$	0	0
$693$	−8.87464e10	−0.384785
$694$	0	0
$695$	− 3.56258e11i	− 1.52695i
$696$	0	0
$697$	−2.03434e11	−0.861972
$698$	0	0
$699$	1.04618e11i	0.438227i
$700$	0	0
$701$	2.39409e11	0.991445	0.495722	−	0.868481i	$-0.334903\pi$
$701$	2.39409e11	0.991445	0.495722	+	0.868481i	$0.334903\pi$
$702$	0	0
$703$	− 2.44366e11i	− 1.00051i
$704$	0	0
$705$	2.15821e11	0.873651
$706$	0	0
$707$	5.84518e11i	2.33948i
$708$	0	0
$709$	1.08904e11	0.430981	0.215490	−	0.976506i	$-0.430865\pi$
$709$	1.08904e11	0.430981	0.215490	+	0.976506i	$0.430865\pi$
$710$	0	0
$711$	− 3.99556e10i	− 0.156351i
$712$	0	0
$713$	−6.00296e11	−2.32278
$714$	0	0
$715$	− 3.76376e11i	− 1.44012i
$716$	0	0
$717$	−1.23238e11	−0.466302
$718$	0	0
$719$	− 2.60327e11i	− 0.974101i	−0.873374	−	0.487051i	$-0.838073\pi$
$719$	− 2.60327e11i	− 0.974101i	0.873374	−	0.487051i	$-0.161927\pi$
$720$	0	0
$721$	4.97350e11	1.84044
$722$	0	0
$723$	− 2.89782e10i	− 0.106052i
$724$	0	0
$725$	2.75716e10	0.0997951
$726$	0	0
$727$	4.81879e11i	1.72504i	0.506020	+	0.862522i	$0.331116\pi$
$727$	4.81879e11i	1.72504i	−0.506020	+	0.862522i	$0.668884\pi$
$728$	0	0
$729$	−1.04604e10	−0.0370370
$730$	0	0
$731$	− 1.48995e11i	− 0.521798i
$732$	0	0
$733$	−1.82274e11	−0.631405	−0.315702	−	0.948858i	$-0.602240\pi$
$733$	−1.82274e11	−0.631405	−0.315702	+	0.948858i	$0.602240\pi$
$734$	0	0
$735$	− 1.21217e11i	− 0.415351i
$736$	0	0
$737$	−3.64275e11	−1.23469
$738$	0	0
$739$	5.49813e11i	1.84347i	0.387815	+	0.921737i	$0.373230\pi$
$739$	5.49813e11i	1.84347i	−0.387815	+	0.921737i	$0.626770\pi$
$740$	0	0
$741$	−2.37777e11	−0.788672
$742$	0	0
$743$	− 2.24817e11i	− 0.737690i	−0.929491	−	0.368845i	$-0.879753\pi$
$743$	− 2.24817e11i	− 0.737690i	0.929491	−	0.368845i	$-0.120247\pi$
$744$	0	0
$745$	−5.88874e10	−0.191160
$746$	0	0
$747$	− 1.70347e10i	− 0.0547080i
$748$	0	0
$749$	−6.13543e11	−1.94948
$750$	0	0
$751$	− 4.17556e11i	− 1.31267i	−0.754470	−	0.656334i	$-0.772106\pi$
$751$	− 4.17556e11i	− 1.31267i	0.754470	−	0.656334i	$-0.227894\pi$
$752$	0	0
$753$	3.18629e11	0.991071
$754$	0	0
$755$	− 1.28738e11i	− 0.396205i
$756$	0	0
$757$	6.29371e11	1.91656	0.958282	−	0.285826i	$-0.0922679\pi$
$757$	6.29371e11	1.91656	0.958282	+	0.285826i	$0.0922679\pi$
$758$	0	0
$759$	3.11842e11i	0.939652i
$760$	0	0
$761$	−1.72289e11	−0.513710	−0.256855	−	0.966450i	$-0.582686\pi$
$761$	−1.72289e11	−0.513710	−0.256855	+	0.966450i	$0.582686\pi$
$762$	0	0
$763$	2.09622e11i	0.618497i
$764$	0	0
$765$	1.04497e11	0.305111
$766$	0	0
$767$	2.24974e11i	0.650058i
$768$	0	0
$769$	9.21192e10	0.263418	0.131709	−	0.991288i	$-0.457954\pi$
$769$	9.21192e10	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$770$	0	0
$771$	1.85077e11i	0.523763i
$772$	0	0
$773$	4.70219e11	1.31699	0.658495	−	0.752585i	$-0.271194\pi$
$773$	4.70219e11	1.31699	0.658495	+	0.752585i	$0.271194\pi$
$774$	0	0
$775$	1.63146e11i	0.452239i
$776$	0	0
$777$	2.68055e11	0.735427
$778$	0	0
$779$	4.02631e11i	1.09335i
$780$	0	0
$781$	−5.29045e11	−1.42196
$782$	0	0
$783$	− 2.06661e10i	− 0.0549808i
$784$	0	0
$785$	−1.55933e10	−0.0410638
$786$	0	0
$787$	5.86015e11i	1.52760i	0.645452	+	0.763801i	$0.276669\pi$
$787$	5.86015e11i	1.52760i	−0.645452	+	0.763801i	$0.723331\pi$
$788$	0	0
$789$	8.70439e10	0.224611
$790$	0	0
$791$	1.01080e11i	0.258202i
$792$	0	0
$793$	6.69597e11	1.69325
$794$	0	0
$795$	− 3.62088e10i	− 0.0906455i
$796$	0	0
$797$	−2.08165e11	−0.515912	−0.257956	−	0.966157i	$-0.583049\pi$
$797$	−2.08165e11	−0.515912	−0.257956	+	0.966157i	$0.583049\pi$
$798$	0	0
$799$	4.18361e11i	1.02651i
$800$	0	0
$801$	−1.89541e11	−0.460441
$802$	0	0
$803$	7.07711e11i	1.70213i
$804$	0	0
$805$	−1.11369e12	−2.65203
$806$	0	0
$807$	5.48871e10i	0.129412i
$808$	0	0
$809$	3.24105e11	0.756645	0.378322	−	0.925674i	$-0.376501\pi$
$809$	3.24105e11	0.756645	0.378322	+	0.925674i	$0.376501\pi$
$810$	0	0
$811$	− 6.94202e11i	− 1.60473i	−0.596833	−	0.802366i	$-0.703574\pi$
$811$	− 6.94202e11i	− 1.60473i	0.596833	−	0.802366i	$-0.296426\pi$
$812$	0	0
$813$	−8.90904e10	−0.203924
$814$	0	0
$815$	1.75859e11i	0.398597i
$816$	0	0
$817$	−2.94887e11	−0.661861
$818$	0	0
$819$	− 2.60826e11i	− 0.579716i
$820$	0	0
$821$	4.79835e11	1.05614	0.528068	−	0.849202i	$-0.322917\pi$
$821$	4.79835e11	1.05614	0.528068	+	0.849202i	$0.322917\pi$
$822$	0	0
$823$	− 3.34155e11i	− 0.728365i	−0.931328	−	0.364183i	$-0.881348\pi$
$823$	− 3.34155e11i	− 0.728365i	0.931328	−	0.364183i	$-0.118652\pi$
$824$	0	0
$825$	8.47508e10	0.182948
$826$	0	0
$827$	6.22860e9i	0.0133158i	0.999978	+	0.00665792i	$0.00211930\pi$
$827$	6.22860e9i	0.0133158i	−0.999978	+	0.00665792i	$0.997881\pi$
$828$	0	0
$829$	6.97808e11	1.47747	0.738733	−	0.673998i	$-0.235424\pi$
$829$	6.97808e11	1.47747	0.738733	+	0.673998i	$0.235424\pi$
$830$	0	0
$831$	− 2.35582e11i	− 0.494012i
$832$	0	0
$833$	2.34975e11	0.488024
$834$	0	0
$835$	− 2.82423e11i	− 0.580970i
$836$	0	0
$837$	1.22285e11	0.249155
$838$	0	0
$839$	− 1.01762e11i	− 0.205370i	−0.994714	−	0.102685i	$-0.967257\pi$
$839$	− 1.01762e11i	− 0.205370i	0.994714	−	0.102685i	$-0.0327433\pi$
$840$	0	0
$841$	−4.59417e11	−0.918382
$842$	0	0
$843$	3.11606e11i	0.617014i
$844$	0	0
$845$	5.13952e11	1.00808
$846$	0	0
$847$	1.15993e11i	0.225372i
$848$	0	0
$849$	−2.59218e11	−0.498924
$850$	0	0
$851$	− 9.41904e11i	− 1.79593i
$852$	0	0
$853$	−8.81799e10	−0.166561	−0.0832805	−	0.996526i	$-0.526540\pi$
$853$	−8.81799e10	−0.166561	−0.0832805	+	0.996526i	$0.526540\pi$
$854$	0	0
$855$	− 2.06817e11i	− 0.387010i
$856$	0	0
$857$	−7.95365e11	−1.47449	−0.737247	−	0.675623i	$-0.763875\pi$
$857$	−7.95365e11	−1.47449	−0.737247	+	0.675623i	$0.763875\pi$
$858$	0	0
$859$	7.01767e11i	1.28890i	0.764645	+	0.644452i	$0.222914\pi$
$859$	7.01767e11i	1.28890i	−0.764645	+	0.644452i	$0.777086\pi$
$860$	0	0
$861$	−4.41662e11	−0.803669
$862$	0	0
$863$	− 2.80009e11i	− 0.504811i	−0.967622	−	0.252405i	$-0.918778\pi$
$863$	− 2.80009e11i	− 0.504811i	0.967622	−	0.252405i	$-0.0812217\pi$
$864$	0	0
$865$	−4.61062e10	−0.0823560
$866$	0	0
$867$	− 1.23660e11i	− 0.218854i
$868$	0	0
$869$	2.42645e11	0.425493
$870$	0	0
$871$	− 1.07061e12i	− 1.86019i
$872$	0	0
$873$	1.61623e11	0.278258
$874$	0	0
$875$	− 5.63802e11i	− 0.961822i
$876$	0	0
$877$	−6.13288e11	−1.03673	−0.518365	−	0.855159i	$-0.673459\pi$
$877$	−6.13288e11	−1.03673	−0.518365	+	0.855159i	$0.673459\pi$
$878$	0	0
$879$	3.12107e11i	0.522816i
$880$	0	0
$881$	−2.48326e11	−0.412210	−0.206105	−	0.978530i	$-0.566079\pi$
$881$	−2.48326e11	−0.412210	−0.206105	+	0.978530i	$0.566079\pi$
$882$	0	0
$883$	3.43124e11i	0.564428i	0.959351	+	0.282214i	$0.0910689\pi$
$883$	3.43124e11i	0.564428i	−0.959351	+	0.282214i	$0.908931\pi$
$884$	0	0
$885$	−1.95682e11	−0.318991
$886$	0	0
$887$	1.46020e11i	0.235895i	0.993020	+	0.117947i	$0.0376314\pi$
$887$	1.46020e11i	0.235895i	−0.993020	+	0.117947i	$0.962369\pi$
$888$	0	0
$889$	8.26381e11	1.32304
$890$	0	0
$891$	− 6.35244e10i	− 0.100793i
$892$	0	0
$893$	8.28009e11	1.30205
$894$	0	0
$895$	3.87415e11i	0.603787i
$896$	0	0
$897$	−9.16504e11	−1.41568
$898$	0	0
$899$	2.41592e11i	0.369866i
$900$	0	0
$901$	7.01894e10	0.106506
$902$	0	0
$903$	− 3.23472e11i	− 0.486504i
$904$	0	0
$905$	6.21717e11	0.926827
$906$	0	0
$907$	− 9.39725e11i	− 1.38858i	−0.719695	−	0.694291i	$-0.755718\pi$
$907$	− 9.39725e11i	− 1.38858i	0.719695	−	0.694291i	$-0.244282\pi$
$908$	0	0
$909$	−4.18396e11	−0.612818
$910$	0	0
$911$	− 4.25662e11i	− 0.618004i	−0.951061	−	0.309002i	$-0.900005\pi$
$911$	− 4.25662e11i	− 0.618004i	0.951061	−	0.309002i	$-0.0999950\pi$
$912$	0	0
$913$	1.03449e11	0.148883
$914$	0	0
$915$	5.82413e11i	0.830897i
$916$	0	0
$917$	9.25312e11	1.30861
$918$	0	0
$919$	− 1.21782e12i	− 1.70734i	−0.520815	−	0.853670i	$-0.674372\pi$
$919$	− 1.21782e12i	− 1.70734i	0.520815	−	0.853670i	$-0.325628\pi$
$920$	0	0
$921$	3.03859e10	0.0422312
$922$	0	0
$923$	− 1.55487e12i	− 2.14233i
$924$	0	0
$925$	−2.55986e11	−0.349663
$926$	0	0
$927$	3.56001e11i	0.482095i
$928$	0	0
$929$	9.97179e11	1.33878	0.669392	−	0.742910i	$-0.266555\pi$
$929$	9.97179e11	1.33878	0.669392	+	0.742910i	$0.266555\pi$
$930$	0	0
$931$	− 4.65055e11i	− 0.619022i
$932$	0	0
$933$	1.86048e10	0.0245526
$934$	0	0
$935$	6.34596e11i	0.830331i
$936$	0	0
$937$	3.43206e11	0.445243	0.222621	−	0.974905i	$-0.428539\pi$
$937$	3.43206e11	0.445243	0.222621	+	0.974905i	$0.428539\pi$
$938$	0	0
$939$	− 7.39906e11i	− 0.951730i
$940$	0	0
$941$	1.72310e11	0.219762	0.109881	−	0.993945i	$-0.464953\pi$
$941$	1.72310e11	0.219762	0.109881	+	0.993945i	$0.464953\pi$
$942$	0	0
$943$	1.55193e12i	1.96257i
$944$	0	0
$945$	2.26866e11	0.284473
$946$	0	0
$947$	8.75107e11i	1.08808i	0.839059	+	0.544041i	$0.183106\pi$
$947$	8.75107e11i	1.08808i	−0.839059	+	0.544041i	$0.816894\pi$
$948$	0	0
$949$	−2.07997e12	−2.56443
$950$	0	0
$951$	− 4.49339e11i	− 0.549353i
$952$	0	0
$953$	−1.43349e12	−1.73789	−0.868946	−	0.494907i	$-0.835202\pi$
$953$	−1.43349e12	−1.73789	−0.868946	+	0.494907i	$0.835202\pi$
$954$	0	0
$955$	3.45401e11i	0.415250i
$956$	0	0
$957$	1.25502e11	0.149625
$958$	0	0
$959$	1.95638e12i	2.31302i
$960$	0	0
$961$	−5.76651e11	−0.676114
$962$	0	0
$963$	− 4.39172e11i	− 0.510657i
$964$	0	0
$965$	6.36490e11	0.733977
$966$	0	0
$967$	1.53627e12i	1.75696i	0.477777	+	0.878481i	$0.341443\pi$
$967$	1.53627e12i	1.75696i	−0.477777	+	0.878481i	$0.658557\pi$
$968$	0	0
$969$	4.00908e11	0.454725
$970$	0	0
$971$	− 2.88911e11i	− 0.325003i	−0.986708	−	0.162501i	$-0.948044\pi$
$971$	− 2.88911e11i	− 0.325003i	0.986708	−	0.162501i	$-0.0519562\pi$
$972$	0	0
$973$	1.49930e12	1.67277
$974$	0	0
$975$	2.49083e11i	0.275630i
$976$	0	0
$977$	4.59815e11	0.504667	0.252333	−	0.967640i	$-0.418802\pi$
$977$	4.59815e11	0.504667	0.252333	+	0.967640i	$0.418802\pi$
$978$	0	0
$979$	− 1.15106e12i	− 1.25305i
$980$	0	0
$981$	−1.50046e11	−0.162013
$982$	0	0
$983$	3.07463e11i	0.329291i	0.986353	+	0.164645i	$0.0526479\pi$
$983$	3.07463e11i	0.329291i	−0.986353	+	0.164645i	$0.947352\pi$
$984$	0	0
$985$	−2.00929e12	−2.13451
$986$	0	0
$987$	9.08274e11i	0.957080i
$988$	0	0
$989$	−1.13663e12	−1.18805
$990$	0	0
$991$	1.10256e12i	1.14316i	0.820547	+	0.571579i	$0.193669\pi$
$991$	1.10256e12i	1.14316i	−0.820547	+	0.571579i	$0.806331\pi$
$992$	0	0
$993$	−6.64026e11	−0.682949
$994$	0	0
$995$	1.03769e12i	1.05871i
$996$	0	0
$997$	5.02913e10	0.0508993	0.0254497	−	0.999676i	$-0.491898\pi$
$997$	5.02913e10	0.0508993	0.0254497	+	0.999676i	$0.491898\pi$
$998$	0	0
$999$	1.91873e11i	0.192642i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.9.g.b.31.2	yes	2
3.2	odd	2		144.9.g.c.127.2		2
4.3	odd	2	inner	48.9.g.b.31.1	✓	2
8.3	odd	2		192.9.g.a.127.2		2
8.5	even	2		192.9.g.a.127.1		2
12.11	even	2		144.9.g.c.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.9.g.b.31.1	✓	2	4.3	odd	2	inner
48.9.g.b.31.2	yes	2	1.1	even	1	trivial
144.9.g.c.127.1		2	12.11	even	2
144.9.g.c.127.2		2	3.2	odd	2
192.9.g.a.127.1		2	8.5	even	2
192.9.g.a.127.2		2	8.3	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654i q^{3} +726.000 q^{5} +3055.34i q^{7} -2187.00 q^{9} +O(q^{10})\)	\(q+46.7654i q^{3} +726.000 q^{5} +3055.34i q^{7} -2187.00 q^{9} -13281.4i q^{11} +39034.0 q^{13} +33951.7i q^{15} -65814.0 q^{17} +130257. i q^{19} -142884. q^{21} +502073. i q^{23} +136451. q^{25} -102276. i q^{27} +202062. q^{29} +1.19563e6i q^{31} +621108. q^{33} +2.21818e6i q^{35} -1.87603e6 q^{37} +1.82544e6i q^{39} +3.09105e6 q^{41} +2.26388e6i q^{43} -1.58776e6 q^{45} -6.35672e6i q^{47} -3.57029e6 q^{49} -3.07782e6i q^{51} -1.06648e6 q^{53} -9.64227e6i q^{55} -6.09152e6 q^{57} +5.76355e6i q^{59} +1.71542e7 q^{61} -6.68202e6i q^{63} +2.83387e7 q^{65} -2.74275e7i q^{67} -2.34796e7 q^{69} -3.98336e7i q^{71} -5.32860e7 q^{73} +6.38118e6i q^{75} +4.05791e7 q^{77} +1.82696e7i q^{79} +4.78297e6 q^{81} +7.78905e6i q^{83} -4.77810e7 q^{85} +9.44950e6i q^{87} +8.66672e7 q^{89} +1.19262e8i q^{91} -5.59143e7 q^{93} +9.45667e7i q^{95} -7.39018e7 q^{97} +2.90463e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1452 q^{5} - 4374 q^{9}+O(q^{10}) \) 2 * q + 1452 * q^5 - 4374 * q^9	\( 2 q + 1452 q^{5} - 4374 q^{9} + 78068 q^{13} - 131628 q^{17} - 285768 q^{21} + 272902 q^{25} + 404124 q^{29} + 1242216 q^{33} - 3752060 q^{37} + 6182100 q^{41} - 3175524 q^{45} - 7140574 q^{49} - 2132964 q^{53} - 12183048 q^{57} + 34308388 q^{61} + 56677368 q^{65} - 46959264 q^{69} - 106572028 q^{73} + 81158112 q^{77} + 9565938 q^{81} - 95561928 q^{85} + 173334468 q^{89} - 111828600 q^{93} - 147803644 q^{97}+O(q^{100}) \) 2 * q + 1452 * q^5 - 4374 * q^9 + 78068 * q^13 - 131628 * q^17 - 285768 * q^21 + 272902 * q^25 + 404124 * q^29 + 1242216 * q^33 - 3752060 * q^37 + 6182100 * q^41 - 3175524 * q^45 - 7140574 * q^49 - 2132964 * q^53 - 12183048 * q^57 + 34308388 * q^61 + 56677368 * q^65 - 46959264 * q^69 - 106572028 * q^73 + 81158112 * q^77 + 9565938 * q^81 - 95561928 * q^85 + 173334468 * q^89 - 111828600 * q^93 - 147803644 * q^97

Introduction

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