LMFDB - Embedded newform 16.15.c.b.15.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,15,Mod(15,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("16.15");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	16.c (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:	$19.8926349043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3395}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 849 $ x^2 - x + 849
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.2
Root			$0.500000 - 29.1333i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.15.c.b.15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2796.80i q^{3} +107850. q^{5} -1.45993e6i q^{7} -3.03911e6 q^{9} +O(q^{10})$	$q+2796.80i q^{3} +107850. q^{5} -1.45993e6i q^{7} -3.03911e6 q^{9} -2.07662e7i q^{11} +5.92754e7 q^{13} +3.01635e8i q^{15} -4.48964e7 q^{17} -8.57708e8i q^{19} +4.08313e9 q^{21} -1.37087e9i q^{23} +5.52811e9 q^{25} +4.87722e9i q^{27} +3.05202e10 q^{29} +3.46708e10i q^{31} +5.80789e10 q^{33} -1.57453e11i q^{35} -1.25972e11 q^{37} +1.65781e11i q^{39} -4.14447e10 q^{41} +1.00323e11i q^{43} -3.27768e11 q^{45} +4.42902e9i q^{47} -1.45317e12 q^{49} -1.25566e11i q^{51} +4.73374e11 q^{53} -2.23964e12i q^{55} +2.39884e12 q^{57} +3.16160e12i q^{59} -5.85267e10 q^{61} +4.43689e12i q^{63} +6.39286e12 q^{65} -4.17601e12i q^{67} +3.83406e12 q^{69} -1.56931e13i q^{71} +1.46449e13 q^{73} +1.54610e13i q^{75} -3.03172e13 q^{77} +8.37780e12i q^{79} -2.81766e13 q^{81} -2.46290e13i q^{83} -4.84207e12 q^{85} +8.53588e13i q^{87} +3.58596e12 q^{89} -8.65379e13i q^{91} -9.69672e13 q^{93} -9.25038e13i q^{95} +6.51975e13 q^{97} +6.31109e13i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 215700 q^{5} - 6078222 q^{9}+O(q^{10}) $ 2 * q + 215700 * q^5 - 6078222 * q^9	$ 2 q + 215700 q^{5} - 6078222 q^{9} + 118550900 q^{13} - 89792700 q^{17} + 8166251520 q^{21} + 11056213750 q^{25} + 61040337204 q^{29} + 116157888000 q^{33} - 251943320300 q^{37} - 82889351964 q^{41} - 655536242700 q^{45} - 2906337147742 q^{49} + 946748870100 q^{53} + 4797672768000 q^{57} - 117053344844 q^{61} + 12785714565000 q^{65} + 7668110177280 q^{69} + 29289808336100 q^{73} - 60634417536000 q^{77} - 56353140970398 q^{81} - 9684142695000 q^{85} + 7171921108644 q^{89} - 193934393856000 q^{93} + 130394938141700 q^{97}+O(q^{100}) $ 2 * q + 215700 * q^5 - 6078222 * q^9 + 118550900 * q^13 - 89792700 * q^17 + 8166251520 * q^21 + 11056213750 * q^25 + 61040337204 * q^29 + 116157888000 * q^33 - 251943320300 * q^37 - 82889351964 * q^41 - 655536242700 * q^45 - 2906337147742 * q^49 + 946748870100 * q^53 + 4797672768000 * q^57 - 117053344844 * q^61 + 12785714565000 * q^65 + 7668110177280 * q^69 + 29289808336100 * q^73 - 60634417536000 * q^77 - 56353140970398 * q^81 - 9684142695000 * q^85 + 7171921108644 * q^89 - 193934393856000 * q^93 + 130394938141700 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$.

$n$	$5$	$15$
$\chi(n)$	$1$	$-1$

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
$3$	2796.80i	1.27883i	0.768862	+	0.639414i	$0.220823\pi$
$3$	2796.80i	1.27883i	−0.768862	+	0.639414i	$0.779177\pi$
$4$	0	0
$5$	107850.	1.38048	0.690240	−	0.723581i	$-0.257505\pi$
$5$	107850.	1.38048	0.690240	+	0.723581i	$0.257505\pi$
$6$	0	0
$7$	− 1.45993e6i	− 1.77274i	−0.462976	−	0.886371i	$-0.653219\pi$
$7$	− 1.45993e6i	− 1.77274i	0.462976	−	0.886371i	$-0.346781\pi$
$8$	0	0
$9$	−3.03911e6	−0.635403
$10$	0	0
$11$	− 2.07662e7i	− 1.06564i	−0.846230	−	0.532818i	$-0.821133\pi$
$11$	− 2.07662e7i	− 1.06564i	0.846230	−	0.532818i	$-0.178867\pi$
$12$	0	0
$13$	5.92754e7	0.944651	0.472326	−	0.881424i	$-0.343415\pi$
$13$	5.92754e7	0.944651	0.472326	+	0.881424i	$0.343415\pi$
$14$	0	0
$15$	3.01635e8i	1.76540i
$16$	0	0
$17$	−4.48964e7	−0.109413	−0.0547065	−	0.998502i	$-0.517422\pi$
$17$	−4.48964e7	−0.109413	−0.0547065	+	0.998502i	$0.517422\pi$
$18$	0	0
$19$	− 8.57708e8i	− 0.959543i	−0.877394	−	0.479771i	$-0.840720\pi$
$19$	− 8.57708e8i	− 0.959543i	0.877394	−	0.479771i	$-0.159280\pi$
$20$	0	0
$21$	4.08313e9	2.26703
$22$	0	0
$23$	− 1.37087e9i	− 0.402627i	−0.979527	−	0.201313i	$-0.935479\pi$
$23$	− 1.37087e9i	− 0.402627i	0.979527	−	0.201313i	$-0.0645209\pi$
$24$	0	0
$25$	5.52811e9	0.905725
$26$	0	0
$27$	4.87722e9i	0.466258i
$28$	0	0
$29$	3.05202e10	1.76930	0.884649	−	0.466258i	$-0.154398\pi$
$29$	3.05202e10	1.76930	0.884649	+	0.466258i	$0.154398\pi$
$30$	0	0
$31$	3.46708e10i	1.26018i	0.776523	+	0.630089i	$0.216982\pi$
$31$	3.46708e10i	1.26018i	−0.776523	+	0.630089i	$0.783018\pi$
$32$	0	0
$33$	5.80789e10	1.36277
$34$	0	0
$35$	− 1.57453e11i	− 2.44723i
$36$	0	0
$37$	−1.25972e11	−1.32697	−0.663485	−	0.748190i	$-0.730923\pi$
$37$	−1.25972e11	−1.32697	−0.663485	+	0.748190i	$0.730923\pi$
$38$	0	0
$39$	1.65781e11i	1.20805i
$40$	0	0
$41$	−4.14447e10	−0.212805	−0.106402	−	0.994323i	$-0.533933\pi$
$41$	−4.14447e10	−0.212805	−0.106402	+	0.994323i	$0.533933\pi$
$42$	0	0
$43$	1.00323e11i	0.369081i	0.982825	+	0.184540i	$0.0590797\pi$
$43$	1.00323e11i	0.369081i	−0.982825	+	0.184540i	$0.940920\pi$
$44$	0	0
$45$	−3.27768e11	−0.877161
$46$	0	0
$47$	4.42902e9i	0.00874224i	0.999990	+	0.00437112i	$0.00139138\pi$
$47$	4.42902e9i	0.00874224i	−0.999990	+	0.00437112i	$0.998609\pi$
$48$	0	0
$49$	−1.45317e12	−2.14261
$50$	0	0
$51$	− 1.25566e11i	− 0.139920i
$52$	0	0
$53$	4.73374e11	0.402971	0.201485	−	0.979492i	$-0.435423\pi$
$53$	4.73374e11	0.402971	0.201485	+	0.979492i	$0.435423\pi$
$54$	0	0
$55$	− 2.23964e12i	− 1.47109i
$56$	0	0
$57$	2.39884e12	1.22709
$58$	0	0
$59$	3.16160e12i	1.27041i	0.772344	+	0.635204i	$0.219084\pi$
$59$	3.16160e12i	1.27041i	−0.772344	+	0.635204i	$0.780916\pi$
$60$	0	0
$61$	−5.85267e10	−0.0186228	−0.00931140	−	0.999957i	$-0.502964\pi$
$61$	−5.85267e10	−0.0186228	−0.00931140	+	0.999957i	$0.502964\pi$
$62$	0	0
$63$	4.43689e12i	1.12640i
$64$	0	0
$65$	6.39286e12	1.30407
$66$	0	0
$67$	− 4.17601e12i	− 0.689029i	−0.938781	−	0.344514i	$-0.888044\pi$
$67$	− 4.17601e12i	− 0.689029i	0.938781	−	0.344514i	$-0.111956\pi$
$68$	0	0
$69$	3.83406e12	0.514890
$70$	0	0
$71$	− 1.56931e13i	− 1.72544i	−0.505680	−	0.862721i	$-0.668758\pi$
$71$	− 1.56931e13i	− 1.72544i	0.505680	−	0.862721i	$-0.331242\pi$
$72$	0	0
$73$	1.46449e13	1.32564	0.662821	−	0.748778i	$-0.269359\pi$
$73$	1.46449e13	1.32564	0.662821	+	0.748778i	$0.269359\pi$
$74$	0	0
$75$	1.54610e13i	1.15827i
$76$	0	0
$77$	−3.03172e13	−1.88910
$78$	0	0
$79$	8.37780e12i	0.436255i	0.975920	+	0.218128i	$0.0699949\pi$
$79$	8.37780e12i	0.436255i	−0.975920	+	0.218128i	$0.930005\pi$
$80$	0	0
$81$	−2.81766e13	−1.23167
$82$	0	0
$83$	− 2.46290e13i	− 0.907612i	−0.891101	−	0.453806i	$-0.850066\pi$
$83$	− 2.46290e13i	− 0.907612i	0.891101	−	0.453806i	$-0.149934\pi$
$84$	0	0
$85$	−4.84207e12	−0.151042
$86$	0	0
$87$	8.53588e13i	2.26263i
$88$	0	0
$89$	3.58596e12	0.0810729	0.0405364	−	0.999178i	$-0.487093\pi$
$89$	3.58596e12	0.0810729	0.0405364	+	0.999178i	$0.487093\pi$
$90$	0	0
$91$	− 8.65379e13i	− 1.67462i
$92$	0	0
$93$	−9.69672e13	−1.61155
$94$	0	0
$95$	− 9.25038e13i	− 1.32463i
$96$	0	0
$97$	6.51975e13	0.806917	0.403458	−	0.914998i	$-0.367808\pi$
$97$	6.51975e13	0.806917	0.403458	+	0.914998i	$0.367808\pi$
$98$	0	0
$99$	6.31109e13i	0.677108i
$100$	0	0
$101$	−4.73055e12	−0.0441227	−0.0220613	−	0.999757i	$-0.507023\pi$
$101$	−4.73055e12	−0.0441227	−0.0220613	+	0.999757i	$0.507023\pi$
$102$	0	0
$103$	7.92520e13i	0.644391i	0.946673	+	0.322196i	$0.104421\pi$
$103$	7.92520e13i	0.644391i	−0.946673	+	0.322196i	$0.895579\pi$
$104$	0	0
$105$	4.40365e14	3.12959
$106$	0	0
$107$	2.81631e14i	1.75386i	0.480622	+	0.876928i	$0.340411\pi$
$107$	2.81631e14i	1.75386i	−0.480622	+	0.876928i	$0.659589\pi$
$108$	0	0
$109$	−1.39744e14	−0.764445	−0.382223	−	0.924070i	$-0.624841\pi$
$109$	−1.39744e14	−0.764445	−0.382223	+	0.924070i	$0.624841\pi$
$110$	0	0
$111$	− 3.52317e14i	− 1.69697i
$112$	0	0
$113$	1.85436e14	0.788215	0.394107	−	0.919064i	$-0.371054\pi$
$113$	1.85436e14	0.788215	0.394107	+	0.919064i	$0.371054\pi$
$114$	0	0
$115$	− 1.47849e14i	− 0.555818i
$116$	0	0
$117$	−1.80145e14	−0.600234
$118$	0	0
$119$	6.55455e13i	0.193961i
$120$	0	0
$121$	−5.14863e13	−0.135580
$122$	0	0
$123$	− 1.15912e14i	− 0.272141i
$124$	0	0
$125$	−6.20578e13	−0.130145
$126$	0	0
$127$	2.54672e14i	0.477921i	0.971029	+	0.238960i	$0.0768066\pi$
$127$	2.54672e14i	0.477921i	−0.971029	+	0.238960i	$0.923193\pi$
$128$	0	0
$129$	−2.80583e14	−0.471991
$130$	0	0
$131$	2.40804e13i	0.0363719i	0.999835	+	0.0181859i	$0.00578908\pi$
$131$	2.40804e13i	0.0363719i	−0.999835	+	0.0181859i	$0.994211\pi$
$132$	0	0
$133$	−1.25219e15	−1.70102
$134$	0	0
$135$	5.26008e14i	0.643659i
$136$	0	0
$137$	−1.06347e15	−1.17404	−0.587020	−	0.809572i	$-0.699699\pi$
$137$	−1.06347e15	−1.17404	−0.587020	+	0.809572i	$0.699699\pi$
$138$	0	0
$139$	8.74022e14i	0.871804i	0.899994	+	0.435902i	$0.143571\pi$
$139$	8.74022e14i	0.871804i	−0.899994	+	0.435902i	$0.856429\pi$
$140$	0	0
$141$	−1.23871e13	−0.0111798
$142$	0	0
$143$	− 1.23093e15i	− 1.00665i
$144$	0	0
$145$	3.29160e15	2.44248
$146$	0	0
$147$	− 4.06422e15i	− 2.74003i
$148$	0	0
$149$	−5.99280e14	−0.367558	−0.183779	−	0.982968i	$-0.558833\pi$
$149$	−5.99280e14	−0.367558	−0.183779	+	0.982968i	$0.558833\pi$
$150$	0	0
$151$	2.04759e15i	1.14394i	0.820273	+	0.571972i	$0.193822\pi$
$151$	2.04759e15i	1.14394i	−0.820273	+	0.571972i	$0.806178\pi$
$152$	0	0
$153$	1.36445e14	0.0695213
$154$	0	0
$155$	3.73924e15i	1.73965i
$156$	0	0
$157$	3.21764e15	1.36848	0.684242	−	0.729255i	$-0.260133\pi$
$157$	3.21764e15	1.36848	0.684242	+	0.729255i	$0.260133\pi$
$158$	0	0
$159$	1.32393e15i	0.515331i
$160$	0	0
$161$	−2.00138e15	−0.713753
$162$	0	0
$163$	− 2.06319e15i	− 0.674878i	−0.941347	−	0.337439i	$-0.890439\pi$
$163$	− 2.06319e15i	− 0.674878i	0.941347	−	0.337439i	$-0.109561\pi$
$164$	0	0
$165$	6.26381e15	1.88127
$166$	0	0
$167$	− 1.41823e15i	− 0.391500i	−0.980654	−	0.195750i	$-0.937286\pi$
$167$	− 1.41823e15i	− 0.391500i	0.980654	−	0.195750i	$-0.0627141\pi$
$168$	0	0
$169$	−4.23797e14	−0.107634
$170$	0	0
$171$	2.60667e15i	0.609696i
$172$	0	0
$173$	−1.32564e14	−0.0285828	−0.0142914	−	0.999898i	$-0.504549\pi$
$173$	−1.32564e14	−0.0285828	−0.0142914	+	0.999898i	$0.504549\pi$
$174$	0	0
$175$	− 8.07064e15i	− 1.60562i
$176$	0	0
$177$	−8.84237e15	−1.62463
$178$	0	0
$179$	3.10776e15i	0.527809i	0.964549	+	0.263904i	$0.0850103\pi$
$179$	3.10776e15i	0.527809i	−0.964549	+	0.263904i	$0.914990\pi$
$180$	0	0
$181$	−3.83608e15	−0.602751	−0.301376	−	0.953506i	$-0.597446\pi$
$181$	−3.83608e15	−0.602751	−0.301376	+	0.953506i	$0.597446\pi$
$182$	0	0
$183$	− 1.63687e14i	− 0.0238154i
$184$	0	0
$185$	−1.35860e16	−1.83185
$186$	0	0
$187$	9.32328e14i	0.116594i
$188$	0	0
$189$	7.12039e15	0.826554
$190$	0	0
$191$	− 1.29125e15i	− 0.139244i	−0.997573	−	0.0696220i	$-0.977821\pi$
$191$	− 1.29125e15i	− 0.139244i	0.997573	−	0.0696220i	$-0.0221793\pi$
$192$	0	0
$193$	−1.23474e15	−0.123787	−0.0618935	−	0.998083i	$-0.519714\pi$
$193$	−1.23474e15	−0.123787	−0.0618935	+	0.998083i	$0.519714\pi$
$194$	0	0
$195$	1.78795e16i	1.66768i
$196$	0	0
$197$	−2.94966e15	−0.256158	−0.128079	−	0.991764i	$-0.540881\pi$
$197$	−2.94966e15	−0.256158	−0.128079	+	0.991764i	$0.540881\pi$
$198$	0	0
$199$	1.56820e16i	1.26891i	0.772960	+	0.634455i	$0.218775\pi$
$199$	1.56820e16i	1.26891i	−0.772960	+	0.634455i	$0.781225\pi$
$200$	0	0
$201$	1.16794e16	0.881150
$202$	0	0
$203$	− 4.45573e16i	− 3.13651i
$204$	0	0
$205$	−4.46981e15	−0.293773
$206$	0	0
$207$	4.16624e15i	0.255830i
$208$	0	0
$209$	−1.78114e16	−1.02252
$210$	0	0
$211$	− 2.49019e16i	− 1.33738i	−0.743542	−	0.668690i	$-0.766855\pi$
$211$	− 2.49019e16i	− 1.33738i	0.743542	−	0.668690i	$-0.233145\pi$
$212$	0	0
$213$	4.38904e16	2.20654
$214$	0	0
$215$	1.08198e16i	0.509509i
$216$	0	0
$217$	5.06169e16	2.23397
$218$	0	0
$219$	4.09588e16i	1.69527i
$220$	0	0
$221$	−2.66125e15	−0.103357
$222$	0	0
$223$	7.53906e15i	0.274905i	0.990508	+	0.137453i	$0.0438915\pi$
$223$	7.53906e15i	0.274905i	−0.990508	+	0.137453i	$0.956109\pi$
$224$	0	0
$225$	−1.68005e16	−0.575500
$226$	0	0
$227$	3.46735e16i	1.11639i	0.829709	+	0.558196i	$0.188506\pi$
$227$	3.46735e16i	1.11639i	−0.829709	+	0.558196i	$0.811494\pi$
$228$	0	0
$229$	1.71050e16	0.517935	0.258967	−	0.965886i	$-0.416618\pi$
$229$	1.71050e16	0.517935	0.258967	+	0.965886i	$0.416618\pi$
$230$	0	0
$231$	− 8.47911e16i	− 2.41583i
$232$	0	0
$233$	−7.03354e16	−1.88661	−0.943306	−	0.331924i	$-0.892302\pi$
$233$	−7.03354e16	−1.88661	−0.943306	+	0.331924i	$0.892302\pi$
$234$	0	0
$235$	4.77670e14i	0.0120685i
$236$	0	0
$237$	−2.34310e16	−0.557895
$238$	0	0
$239$	7.88011e16i	1.76908i	0.466467	+	0.884539i	$0.345527\pi$
$239$	7.88011e16i	1.76908i	−0.466467	+	0.884539i	$0.654473\pi$
$240$	0	0
$241$	7.23787e16	1.53282	0.766410	−	0.642351i	$-0.222041\pi$
$241$	7.23787e16	1.53282	0.766410	+	0.642351i	$0.222041\pi$
$242$	0	0
$243$	− 5.54766e16i	− 1.10883i
$244$	0	0
$245$	−1.56724e17	−2.95783
$246$	0	0
$247$	− 5.08410e16i	− 0.906433i
$248$	0	0
$249$	6.88823e16	1.16068
$250$	0	0
$251$	− 3.27715e16i	− 0.522133i	−0.965321	−	0.261066i	$-0.915926\pi$
$251$	− 3.27715e16i	− 0.522133i	0.965321	−	0.261066i	$-0.0840741\pi$
$252$	0	0
$253$	−2.84679e16	−0.429053
$254$	0	0
$255$	− 1.35423e16i	− 0.193157i
$256$	0	0
$257$	−5.73037e16	−0.773839	−0.386920	−	0.922113i	$-0.626461\pi$
$257$	−5.73037e16	−0.773839	−0.386920	+	0.922113i	$0.626461\pi$
$258$	0	0
$259$	1.83910e17i	2.35237i
$260$	0	0
$261$	−9.27542e16	−1.12422
$262$	0	0
$263$	8.19452e15i	0.0941528i	0.998891	+	0.0470764i	$0.0149904\pi$
$263$	8.19452e15i	0.0941528i	−0.998891	+	0.0470764i	$0.985010\pi$
$264$	0	0
$265$	5.10534e16	0.556293
$266$	0	0
$267$	1.00292e16i	0.103678i
$268$	0	0
$269$	4.28752e16	0.420669	0.210335	−	0.977629i	$-0.432545\pi$
$269$	4.28752e16	0.420669	0.210335	+	0.977629i	$0.432545\pi$
$270$	0	0
$271$	3.85772e16i	0.359373i	0.983724	+	0.179687i	$0.0575084\pi$
$271$	3.85772e16i	0.359373i	−0.983724	+	0.179687i	$0.942492\pi$
$272$	0	0
$273$	2.42029e17	2.14155
$274$	0	0
$275$	− 1.14798e17i	− 0.965173i
$276$	0	0
$277$	7.03533e16	0.562246	0.281123	−	0.959672i	$-0.409293\pi$
$277$	7.03533e16	0.562246	0.281123	+	0.959672i	$0.409293\pi$
$278$	0	0
$279$	− 1.05368e17i	− 0.800720i
$280$	0	0
$281$	1.19040e16	0.0860495	0.0430248	−	0.999074i	$-0.486301\pi$
$281$	1.19040e16	0.0860495	0.0430248	+	0.999074i	$0.486301\pi$
$282$	0	0
$283$	− 2.30157e17i	− 1.58314i	−0.611078	−	0.791570i	$-0.709264\pi$
$283$	− 2.30157e17i	− 1.58314i	0.611078	−	0.791570i	$-0.290736\pi$
$284$	0	0
$285$	2.58715e17	1.69397
$286$	0	0
$287$	6.05063e16i	0.377248i
$288$	0	0
$289$	−1.66362e17	−0.988029
$290$	0	0
$291$	1.82344e17i	1.03191i
$292$	0	0
$293$	−2.26382e17	−1.22115	−0.610575	−	0.791958i	$-0.709062\pi$
$293$	−2.26382e17	−1.22115	−0.610575	+	0.791958i	$0.709062\pi$
$294$	0	0
$295$	3.40979e17i	1.75377i
$296$	0	0
$297$	1.01281e17	0.496861
$298$	0	0
$299$	− 8.12591e16i	− 0.380342i
$300$	0	0
$301$	1.46464e17	0.654285
$302$	0	0
$303$	− 1.32304e16i	− 0.0564254i
$304$	0	0
$305$	−6.31210e15	−0.0257084
$306$	0	0
$307$	− 3.14208e17i	− 1.22250i	−0.791437	−	0.611250i	$-0.790667\pi$
$307$	− 3.14208e17i	− 1.22250i	0.791437	−	0.611250i	$-0.209333\pi$
$308$	0	0
$309$	−2.21652e17	−0.824066
$310$	0	0
$311$	1.47839e17i	0.525370i	0.964882	+	0.262685i	$0.0846080\pi$
$311$	1.47839e17i	0.525370i	−0.964882	+	0.262685i	$0.915392\pi$
$312$	0	0
$313$	−1.28447e16	−0.0436427	−0.0218214	−	0.999762i	$-0.506947\pi$
$313$	−1.28447e16	−0.0436427	−0.0218214	+	0.999762i	$0.506947\pi$
$314$	0	0
$315$	4.78518e17i	1.55498i
$316$	0	0
$317$	−5.17156e17	−1.60771	−0.803854	−	0.594827i	$-0.797220\pi$
$317$	−5.17156e17	−1.60771	−0.803854	+	0.594827i	$0.797220\pi$
$318$	0	0
$319$	− 6.33789e17i	− 1.88543i
$320$	0	0
$321$	−7.87665e17	−2.24288
$322$	0	0
$323$	3.85080e16i	0.104986i
$324$	0	0
$325$	3.27681e17	0.855594
$326$	0	0
$327$	− 3.90835e17i	− 0.977595i
$328$	0	0
$329$	6.46605e15	0.0154977
$330$	0	0
$331$	1.43997e17i	0.330793i	0.986227	+	0.165397i	$0.0528905\pi$
$331$	1.43997e17i	0.330793i	−0.986227	+	0.165397i	$0.947110\pi$
$332$	0	0
$333$	3.82842e17	0.843160
$334$	0	0
$335$	− 4.50382e17i	− 0.951191i
$336$	0	0
$337$	−3.56377e17	−0.721938	−0.360969	−	0.932578i	$-0.617554\pi$
$337$	−3.56377e17	−0.721938	−0.360969	+	0.932578i	$0.617554\pi$
$338$	0	0
$339$	5.18627e17i	1.00799i
$340$	0	0
$341$	7.19981e17	1.34289
$342$	0	0
$343$	1.13137e18i	2.02555i
$344$	0	0
$345$	4.13503e17	0.710796
$346$	0	0
$347$	4.83852e17i	0.798741i	0.916790	+	0.399370i	$0.130771\pi$
$347$	4.83852e17i	0.798741i	−0.916790	+	0.399370i	$0.869229\pi$
$348$	0	0
$349$	9.72710e17	1.54243	0.771215	−	0.636575i	$-0.219649\pi$
$349$	9.72710e17	1.54243	0.771215	+	0.636575i	$0.219649\pi$
$350$	0	0
$351$	2.89099e17i	0.440451i
$352$	0	0
$353$	−6.43469e17	−0.942117	−0.471059	−	0.882102i	$-0.656128\pi$
$353$	−6.43469e17	−0.942117	−0.471059	+	0.882102i	$0.656128\pi$
$354$	0	0
$355$	− 1.69250e18i	− 2.38194i
$356$	0	0
$357$	−1.83317e17	−0.248043
$358$	0	0
$359$	− 9.80595e17i	− 1.27593i	−0.770064	−	0.637967i	$-0.779776\pi$
$359$	− 9.80595e17i	− 1.27593i	0.770064	−	0.637967i	$-0.220224\pi$
$360$	0	0
$361$	6.33435e16	0.0792779
$362$	0	0
$363$	− 1.43997e17i	− 0.173383i
$364$	0	0
$365$	1.57945e18	1.83002
$366$	0	0
$367$	1.02823e18i	1.14665i	0.819329	+	0.573323i	$0.194346\pi$
$367$	1.02823e18i	1.14665i	−0.819329	+	0.573323i	$0.805654\pi$
$368$	0	0
$369$	1.25955e17	0.135217
$370$	0	0
$371$	− 6.91093e17i	− 0.714363i
$372$	0	0
$373$	−9.02085e17	−0.898019	−0.449009	−	0.893527i	$-0.648223\pi$
$373$	−9.02085e17	−0.898019	−0.449009	+	0.893527i	$0.648223\pi$
$374$	0	0
$375$	− 1.73563e17i	− 0.166433i
$376$	0	0
$377$	1.80910e18	1.67137
$378$	0	0
$379$	7.39981e17i	0.658789i	0.944192	+	0.329394i	$0.106845\pi$
$379$	7.39981e17i	0.658789i	−0.944192	+	0.329394i	$0.893155\pi$
$380$	0	0
$381$	−7.12267e17	−0.611179
$382$	0	0
$383$	1.48824e18i	1.23107i	0.788110	+	0.615535i	$0.211060\pi$
$383$	1.48824e18i	1.23107i	−0.788110	+	0.615535i	$0.788940\pi$
$384$	0	0
$385$	−3.26971e18	−2.60786
$386$	0	0
$387$	− 3.04893e17i	− 0.234515i
$388$	0	0
$389$	−1.45205e18	−1.07729	−0.538647	−	0.842532i	$-0.681064\pi$
$389$	−1.45205e18	−1.07729	−0.538647	+	0.842532i	$0.681064\pi$
$390$	0	0
$391$	6.15472e16i	0.0440525i
$392$	0	0
$393$	−6.73481e16	−0.0465134
$394$	0	0
$395$	9.03546e17i	0.602241i
$396$	0	0
$397$	−1.91661e18	−1.23311	−0.616553	−	0.787314i	$-0.711471\pi$
$397$	−1.91661e18	−1.23311	−0.616553	+	0.787314i	$0.711471\pi$
$398$	0	0
$399$	− 3.50213e18i	− 2.17531i
$400$	0	0
$401$	2.28415e18	1.36998	0.684989	−	0.728554i	$-0.259807\pi$
$401$	2.28415e18	1.36998	0.684989	+	0.728554i	$0.259807\pi$
$402$	0	0
$403$	2.05513e18i	1.19043i
$404$	0	0
$405$	−3.03884e18	−1.70029
$406$	0	0
$407$	2.61596e18i	1.41407i
$408$	0	0
$409$	1.55202e18	0.810650	0.405325	−	0.914173i	$-0.367158\pi$
$409$	1.55202e18	0.810650	0.405325	+	0.914173i	$0.367158\pi$
$410$	0	0
$411$	− 2.97432e18i	− 1.50140i
$412$	0	0
$413$	4.61572e18	2.25210
$414$	0	0
$415$	− 2.65624e18i	− 1.25294i
$416$	0	0
$417$	−2.44446e18	−1.11489
$418$	0	0
$419$	− 1.47382e18i	− 0.650050i	−0.945705	−	0.325025i	$-0.894627\pi$
$419$	− 1.47382e18i	− 0.650050i	0.945705	−	0.325025i	$-0.105373\pi$
$420$	0	0
$421$	−2.36093e18	−1.00718	−0.503592	−	0.863941i	$-0.667989\pi$
$421$	−2.36093e18	−1.00718	−0.503592	+	0.863941i	$0.667989\pi$
$422$	0	0
$423$	− 1.34603e16i	− 0.00555484i
$424$	0	0
$425$	−2.48192e17	−0.0990980
$426$	0	0
$427$	8.54448e16i	0.0330134i
$428$	0	0
$429$	3.44266e18	1.28734
$430$	0	0
$431$	4.06637e18i	1.47186i	0.677059	+	0.735929i	$0.263254\pi$
$431$	4.06637e18i	1.47186i	−0.677059	+	0.735929i	$0.736746\pi$
$432$	0	0
$433$	4.35475e18	1.52598	0.762989	−	0.646412i	$-0.223731\pi$
$433$	4.35475e18	1.52598	0.762989	+	0.646412i	$0.223731\pi$
$434$	0	0
$435$	9.20594e18i	3.12351i
$436$	0	0
$437$	−1.17581e18	−0.386337
$438$	0	0
$439$	− 2.44446e18i	− 0.777915i	−0.921256	−	0.388957i	$-0.872835\pi$
$439$	− 2.44446e18i	− 0.777915i	0.921256	−	0.388957i	$-0.127165\pi$
$440$	0	0
$441$	4.41634e18	1.36142
$442$	0	0
$443$	− 3.86146e18i	− 1.15325i	−0.817007	−	0.576627i	$-0.804369\pi$
$443$	− 3.86146e18i	− 1.15325i	0.817007	−	0.576627i	$-0.195631\pi$
$444$	0	0
$445$	3.86746e17	0.111919
$446$	0	0
$447$	− 1.67607e18i	− 0.470044i
$448$	0	0
$449$	5.13640e18	1.39616	0.698078	−	0.716021i	$-0.254039\pi$
$449$	5.13640e18	1.39616	0.698078	+	0.716021i	$0.254039\pi$
$450$	0	0
$451$	8.60650e17i	0.226773i
$452$	0	0
$453$	−5.72671e18	−1.46291
$454$	0	0
$455$	− 9.33312e18i	− 2.31178i
$456$	0	0
$457$	3.37145e18	0.809848	0.404924	−	0.914350i	$-0.367298\pi$
$457$	3.37145e18	0.809848	0.404924	+	0.914350i	$0.367298\pi$
$458$	0	0
$459$	− 2.18969e17i	− 0.0510146i
$460$	0	0
$461$	−6.50856e17	−0.147088	−0.0735442	−	0.997292i	$-0.523431\pi$
$461$	−6.50856e17	−0.147088	−0.0735442	+	0.997292i	$0.523431\pi$
$462$	0	0
$463$	6.94572e18i	1.52283i	0.648267	+	0.761413i	$0.275494\pi$
$463$	6.94572e18i	1.52283i	−0.648267	+	0.761413i	$0.724506\pi$
$464$	0	0
$465$	−1.04579e19	−2.22471
$466$	0	0
$467$	− 6.37124e18i	− 1.31524i	−0.753349	−	0.657621i	$-0.771563\pi$
$467$	− 6.37124e18i	− 1.31524i	0.753349	−	0.657621i	$-0.228437\pi$
$468$	0	0
$469$	−6.09667e18	−1.22147
$470$	0	0
$471$	8.99909e18i	1.75006i
$472$	0	0
$473$	2.08333e18	0.393306
$474$	0	0
$475$	− 4.74150e18i	− 0.869082i
$476$	0	0
$477$	−1.43864e18	−0.256049
$478$	0	0
$479$	8.59353e18i	1.48533i	0.669663	+	0.742665i	$0.266439\pi$
$479$	8.59353e18i	1.48533i	−0.669663	+	0.742665i	$0.733561\pi$
$480$	0	0
$481$	−7.46703e18	−1.25352
$482$	0	0
$483$	− 5.59745e18i	− 0.912767i
$484$	0	0
$485$	7.03155e18	1.11393
$486$	0	0
$487$	− 4.80934e18i	− 0.740257i	−0.928981	−	0.370128i	$-0.879314\pi$
$487$	− 4.80934e18i	− 0.740257i	0.928981	−	0.370128i	$-0.120686\pi$
$488$	0	0
$489$	5.77032e18	0.863054
$490$	0	0
$491$	8.36318e18i	1.21563i	0.794079	+	0.607815i	$0.207954\pi$
$491$	8.36318e18i	1.21563i	−0.794079	+	0.607815i	$0.792046\pi$
$492$	0	0
$493$	−1.37024e18	−0.193584
$494$	0	0
$495$	6.80651e18i	0.934734i
$496$	0	0
$497$	−2.29108e19	−3.05876
$498$	0	0
$499$	3.25910e18i	0.423052i	0.977372	+	0.211526i	$0.0678433\pi$
$499$	3.25910e18i	0.423052i	−0.977372	+	0.211526i	$0.932157\pi$
$500$	0	0
$501$	3.96651e18	0.500662
$502$	0	0
$503$	− 9.95082e18i	− 1.22147i	−0.791835	−	0.610735i	$-0.790874\pi$
$503$	− 9.95082e18i	− 1.22147i	0.791835	−	0.610735i	$-0.209126\pi$
$504$	0	0
$505$	−5.10190e17	−0.0609105
$506$	0	0
$507$	− 1.18528e18i	− 0.137646i
$508$	0	0
$509$	1.28715e19	1.45413	0.727066	−	0.686567i	$-0.240883\pi$
$509$	1.28715e19	1.45413	0.727066	+	0.686567i	$0.240883\pi$
$510$	0	0
$511$	− 2.13805e19i	− 2.35002i
$512$	0	0
$513$	4.18323e18	0.447394
$514$	0	0
$515$	8.54733e18i	0.889569i
$516$	0	0
$517$	9.19740e16	0.00931604
$518$	0	0
$519$	− 3.70756e17i	− 0.0365524i
$520$	0	0
$521$	9.55060e18	0.916572	0.458286	−	0.888805i	$-0.348463\pi$
$521$	9.55060e18	0.916572	0.458286	+	0.888805i	$0.348463\pi$
$522$	0	0
$523$	2.83524e17i	0.0264898i	0.999912	+	0.0132449i	$0.00421610\pi$
$523$	2.83524e17i	0.0264898i	−0.999912	+	0.0132449i	$0.995784\pi$
$524$	0	0
$525$	2.25720e19	2.05331
$526$	0	0
$527$	− 1.55659e18i	− 0.137880i
$528$	0	0
$529$	9.71354e18	0.837892
$530$	0	0
$531$	− 9.60846e18i	− 0.807221i
$532$	0	0
$533$	−2.45665e18	−0.201026
$534$	0	0
$535$	3.03739e19i	2.42116i
$536$	0	0
$537$	−8.69178e18	−0.674977
$538$	0	0
$539$	3.01768e19i	2.28324i
$540$	0	0
$541$	2.91409e18	0.214843	0.107422	−	0.994214i	$-0.465741\pi$
$541$	2.91409e18	0.214843	0.107422	+	0.994214i	$0.465741\pi$
$542$	0	0
$543$	− 1.07288e19i	− 0.770815i
$544$	0	0
$545$	−1.50713e19	−1.05530
$546$	0	0
$547$	8.20773e18i	0.560160i	0.959977	+	0.280080i	$0.0903610\pi$
$547$	8.20773e18i	0.560160i	−0.959977	+	0.280080i	$0.909639\pi$
$548$	0	0
$549$	1.77869e17	0.0118330
$550$	0	0
$551$	− 2.61774e19i	− 1.69772i
$552$	0	0
$553$	1.22310e19	0.773367
$554$	0	0
$555$	− 3.79974e19i	− 2.34263i
$556$	0	0
$557$	−2.68669e19	−1.61522	−0.807609	−	0.589718i	$-0.799239\pi$
$557$	−2.68669e19	−1.61522	−0.807609	+	0.589718i	$0.799239\pi$
$558$	0	0
$559$	5.94669e18i	0.348653i
$560$	0	0
$561$	−2.60753e18	−0.149104
$562$	0	0
$563$	− 1.99464e19i	− 1.11252i	−0.831009	−	0.556259i	$-0.812236\pi$
$563$	− 1.99464e19i	− 1.11252i	0.831009	−	0.556259i	$-0.187764\pi$
$564$	0	0
$565$	1.99993e19	1.08811
$566$	0	0
$567$	4.11358e19i	2.18343i
$568$	0	0
$569$	−2.74316e19	−1.42058	−0.710288	−	0.703911i	$-0.751435\pi$
$569$	−2.74316e19	−1.42058	−0.710288	+	0.703911i	$0.751435\pi$
$570$	0	0
$571$	4.93324e18i	0.249275i	0.992202	+	0.124638i	$0.0397768\pi$
$571$	4.93324e18i	0.249275i	−0.992202	+	0.124638i	$0.960223\pi$
$572$	0	0
$573$	3.61136e18	0.178069
$574$	0	0
$575$	− 7.57833e18i	− 0.364669i
$576$	0	0
$577$	−1.26460e19	−0.593914	−0.296957	−	0.954891i	$-0.595972\pi$
$577$	−1.26460e19	−0.593914	−0.296957	+	0.954891i	$0.595972\pi$
$578$	0	0
$579$	− 3.45332e18i	− 0.158302i
$580$	0	0
$581$	−3.59566e19	−1.60896
$582$	0	0
$583$	− 9.83020e18i	− 0.429420i
$584$	0	0
$585$	−1.94286e19	−0.828611
$586$	0	0
$587$	1.44862e19i	0.603236i	0.953429	+	0.301618i	$0.0975268\pi$
$587$	1.44862e19i	0.603236i	−0.953429	+	0.301618i	$0.902473\pi$
$588$	0	0
$589$	2.97374e19	1.20919
$590$	0	0
$591$	− 8.24959e18i	− 0.327582i
$592$	0	0
$593$	−4.62271e18	−0.179273	−0.0896364	−	0.995975i	$-0.528570\pi$
$593$	−4.62271e18	−0.179273	−0.0896364	+	0.995975i	$0.528570\pi$
$594$	0	0
$595$	7.06908e18i	0.267759i
$596$	0	0
$597$	−4.38595e19	−1.62272
$598$	0	0
$599$	4.34190e19i	1.56925i	0.619971	+	0.784625i	$0.287144\pi$
$599$	4.34190e19i	1.56925i	−0.619971	+	0.784625i	$0.712856\pi$
$600$	0	0
$601$	1.62162e19	0.572570	0.286285	−	0.958144i	$-0.407580\pi$
$601$	1.62162e19	0.572570	0.286285	+	0.958144i	$0.407580\pi$
$602$	0	0
$603$	1.26913e19i	0.437811i
$604$	0	0
$605$	−5.55280e18	−0.187165
$606$	0	0
$607$	− 2.11384e19i	− 0.696226i	−0.937453	−	0.348113i	$-0.886823\pi$
$607$	− 2.11384e19i	− 0.696226i	0.937453	−	0.348113i	$-0.113177\pi$
$608$	0	0
$609$	1.24618e20	4.01106
$610$	0	0
$611$	2.62532e17i	0.00825837i
$612$	0	0
$613$	3.27599e19	1.00721	0.503604	−	0.863935i	$-0.332007\pi$
$613$	3.27599e19	1.00721	0.503604	+	0.863935i	$0.332007\pi$
$614$	0	0
$615$	− 1.25012e19i	− 0.375685i
$616$	0	0
$617$	−1.09864e19	−0.322744	−0.161372	−	0.986894i	$-0.551592\pi$
$617$	−1.09864e19	−0.322744	−0.161372	+	0.986894i	$0.551592\pi$
$618$	0	0
$619$	− 4.04696e19i	− 1.16224i	−0.813819	−	0.581118i	$-0.802616\pi$
$619$	− 4.04696e19i	− 1.16224i	0.813819	−	0.581118i	$-0.197384\pi$
$620$	0	0
$621$	6.68605e18	0.187728
$622$	0	0
$623$	− 5.23525e18i	− 0.143721i
$624$	0	0
$625$	−4.04338e19	−1.08539
$626$	0	0
$627$	− 4.98148e19i	− 1.30763i
$628$	0	0
$629$	5.65567e18	0.145188
$630$	0	0
$631$	− 2.48779e19i	− 0.624610i	−0.949982	−	0.312305i	$-0.898899\pi$
$631$	− 2.48779e19i	− 0.624610i	0.949982	−	0.312305i	$-0.101101\pi$
$632$	0	0
$633$	6.96455e19	1.71028
$634$	0	0
$635$	2.74664e19i	0.659760i
$636$	0	0
$637$	−8.61372e19	−2.02402
$638$	0	0
$639$	4.76931e19i	1.09635i
$640$	0	0
$641$	6.84682e19	1.53987	0.769933	−	0.638125i	$-0.220290\pi$
$641$	6.84682e19	1.53987	0.769933	+	0.638125i	$0.220290\pi$
$642$	0	0
$643$	7.12580e19i	1.56804i	0.620736	+	0.784020i	$0.286834\pi$
$643$	7.12580e19i	1.56804i	−0.620736	+	0.784020i	$0.713166\pi$
$644$	0	0
$645$	−3.02609e19	−0.651574
$646$	0	0
$647$	− 5.66219e19i	− 1.19304i	−0.802599	−	0.596519i	$-0.796550\pi$
$647$	− 5.66219e19i	− 1.19304i	0.802599	−	0.596519i	$-0.203450\pi$
$648$	0	0
$649$	6.56546e19	1.35379
$650$	0	0
$651$	1.41565e20i	2.85686i
$652$	0	0
$653$	−3.54681e18	−0.0700561	−0.0350280	−	0.999386i	$-0.511152\pi$
$653$	−3.54681e18	−0.0700561	−0.0350280	+	0.999386i	$0.511152\pi$
$654$	0	0
$655$	2.59708e18i	0.0502106i
$656$	0	0
$657$	−4.45075e19	−0.842317
$658$	0	0
$659$	− 9.55319e19i	− 1.76991i	−0.465679	−	0.884954i	$-0.654190\pi$
$659$	− 9.55319e19i	− 1.76991i	0.465679	−	0.884954i	$-0.345810\pi$
$660$	0	0
$661$	3.45571e19	0.626798	0.313399	−	0.949622i	$-0.398532\pi$
$661$	3.45571e19	0.626798	0.313399	+	0.949622i	$0.398532\pi$
$662$	0	0
$663$	− 7.44298e18i	− 0.132176i
$664$	0	0
$665$	−1.35049e20	−2.34823
$666$	0	0
$667$	− 4.18393e19i	− 0.712366i
$668$	0	0
$669$	−2.10852e19	−0.351557
$670$	0	0
$671$	1.21538e18i	0.0198451i
$672$	0	0
$673$	−2.72208e19	−0.435307	−0.217653	−	0.976026i	$-0.569840\pi$
$673$	−2.72208e19	−0.435307	−0.217653	+	0.976026i	$0.569840\pi$
$674$	0	0
$675$	2.69618e19i	0.422301i
$676$	0	0
$677$	2.69677e17	0.00413736	0.00206868	−	0.999998i	$-0.499342\pi$
$677$	2.69677e17	0.00413736	0.00206868	+	0.999998i	$0.499342\pi$
$678$	0	0
$679$	− 9.51837e19i	− 1.43045i
$680$	0	0
$681$	−9.69747e19	−1.42767
$682$	0	0
$683$	− 6.59143e19i	− 0.950681i	−0.879802	−	0.475340i	$-0.842325\pi$
$683$	− 6.59143e19i	− 0.950681i	0.879802	−	0.475340i	$-0.157675\pi$
$684$	0	0
$685$	−1.14696e20	−1.62074
$686$	0	0
$687$	4.78393e19i	0.662350i
$688$	0	0
$689$	2.80595e19	0.380667
$690$	0	0
$691$	1.00152e20i	1.33142i	0.746211	+	0.665710i	$0.231871\pi$
$691$	1.00152e20i	1.33142i	−0.746211	+	0.665710i	$0.768129\pi$
$692$	0	0
$693$	9.21374e19	1.20034
$694$	0	0
$695$	9.42633e19i	1.20351i
$696$	0	0
$697$	1.86071e18	0.0232836
$698$	0	0
$699$	− 1.96714e20i	− 2.41265i
$700$	0	0
$701$	5.37939e19	0.646706	0.323353	−	0.946278i	$-0.395190\pi$
$701$	5.37939e19	0.646706	0.323353	+	0.946278i	$0.395190\pi$
$702$	0	0
$703$	1.08047e20i	1.27328i
$704$	0	0
$705$	−1.33595e18	−0.0154335
$706$	0	0
$707$	6.90627e18i	0.0782181i
$708$	0	0
$709$	1.50460e20	1.67070	0.835348	−	0.549722i	$-0.185266\pi$
$709$	1.50460e20	1.67070	0.835348	+	0.549722i	$0.185266\pi$
$710$	0	0
$711$	− 2.54611e19i	− 0.277198i
$712$	0	0
$713$	4.75292e19	0.507381
$714$	0	0
$715$	− 1.32756e20i	− 1.38967i
$716$	0	0
$717$	−2.20391e20	−2.26235
$718$	0	0
$719$	1.67575e20i	1.68697i	0.537150	+	0.843487i	$0.319501\pi$
$719$	1.67575e20i	1.68697i	−0.537150	+	0.843487i	$0.680499\pi$
$720$	0	0
$721$	1.15702e20	1.14234
$722$	0	0
$723$	2.02429e20i	1.96021i
$724$	0	0
$725$	1.68719e20	1.60250
$726$	0	0
$727$	1.54859e20i	1.44277i	0.692536	+	0.721383i	$0.256493\pi$
$727$	1.54859e20i	1.44277i	−0.692536	+	0.721383i	$0.743507\pi$
$728$	0	0
$729$	2.03892e19	0.186340
$730$	0	0
$731$	− 4.50414e18i	− 0.0403822i
$732$	0	0
$733$	−3.75815e19	−0.330557	−0.165278	−	0.986247i	$-0.552852\pi$
$733$	−3.75815e19	−0.330557	−0.165278	+	0.986247i	$0.552852\pi$
$734$	0	0
$735$	− 4.38326e20i	− 3.78256i
$736$	0	0
$737$	−8.67199e19	−0.734254
$738$	0	0
$739$	− 1.10818e20i	− 0.920663i	−0.887747	−	0.460332i	$-0.847730\pi$
$739$	− 1.10818e20i	− 0.920663i	0.887747	−	0.460332i	$-0.152270\pi$
$740$	0	0
$741$	1.42192e20	1.15917
$742$	0	0
$743$	− 1.46597e20i	− 1.17274i	−0.810042	−	0.586372i	$-0.800556\pi$
$743$	− 1.46597e20i	− 1.17274i	0.810042	−	0.586372i	$-0.199444\pi$
$744$	0	0
$745$	−6.46324e19	−0.507407
$746$	0	0
$747$	7.48503e19i	0.576699i
$748$	0	0
$749$	4.11161e20	3.10913
$750$	0	0
$751$	− 8.39520e19i	− 0.623091i	−0.950231	−	0.311546i	$-0.899153\pi$
$751$	− 8.39520e19i	− 0.623091i	0.950231	−	0.311546i	$-0.100847\pi$
$752$	0	0
$753$	9.16554e19	0.667718
$754$	0	0
$755$	2.20833e20i	1.57919i
$756$	0	0
$757$	−1.77335e20	−1.24487	−0.622435	−	0.782672i	$-0.713856\pi$
$757$	−1.77335e20	−1.24487	−0.622435	+	0.782672i	$0.713856\pi$
$758$	0	0
$759$	− 7.96189e19i	− 0.548686i
$760$	0	0
$761$	−2.59568e20	−1.75613	−0.878067	−	0.478537i	$-0.841167\pi$
$761$	−2.59568e20	−1.75613	−0.878067	+	0.478537i	$0.841167\pi$
$762$	0	0
$763$	2.04016e20i	1.35516i
$764$	0	0
$765$	1.47156e19	0.0959727
$766$	0	0
$767$	1.87405e20i	1.20009i
$768$	0	0
$769$	−1.07657e20	−0.676954	−0.338477	−	0.940975i	$-0.609912\pi$
$769$	−1.07657e20	−0.676954	−0.338477	+	0.940975i	$0.609912\pi$
$770$	0	0
$771$	− 1.60267e20i	− 0.989608i
$772$	0	0
$773$	4.19968e19	0.254660	0.127330	−	0.991860i	$-0.459359\pi$
$773$	4.19968e19	0.254660	0.127330	+	0.991860i	$0.459359\pi$
$774$	0	0
$775$	1.91664e20i	1.14137i
$776$	0	0
$777$	−5.14358e20	−3.00828
$778$	0	0
$779$	3.55474e19i	0.204195i
$780$	0	0
$781$	−3.25886e20	−1.83869
$782$	0	0
$783$	1.48854e20i	0.824949i
$784$	0	0
$785$	3.47023e20	1.88917
$786$	0	0
$787$	− 2.86283e20i	− 1.53099i	−0.643441	−	0.765496i	$-0.722494\pi$
$787$	− 2.86283e20i	− 1.53099i	0.643441	−	0.765496i	$-0.277506\pi$
$788$	0	0
$789$	−2.29184e19	−0.120405
$790$	0	0
$791$	− 2.70723e20i	− 1.39730i
$792$	0	0
$793$	−3.46919e18	−0.0175920
$794$	0	0
$795$	1.42786e20i	0.711404i
$796$	0	0
$797$	1.69043e20	0.827542	0.413771	−	0.910381i	$-0.364211\pi$
$797$	1.69043e20	0.827542	0.413771	+	0.910381i	$0.364211\pi$
$798$	0	0
$799$	− 1.98847e17i	0 0.000956514i
$800$	0	0
$801$	−1.08981e19	−0.0515139
$802$	0	0
$803$	− 3.04119e20i	− 1.41265i
$804$	0	0
$805$	−2.15848e20	−0.985321
$806$	0	0
$807$	1.19913e20i	0.537964i
$808$	0	0
$809$	−1.12555e19	−0.0496280	−0.0248140	−	0.999692i	$-0.507899\pi$
$809$	−1.12555e19	−0.0496280	−0.0248140	+	0.999692i	$0.507899\pi$
$810$	0	0
$811$	1.56739e20i	0.679251i	0.940561	+	0.339626i	$0.110300\pi$
$811$	1.56739e20i	0.679251i	−0.940561	+	0.339626i	$0.889700\pi$
$812$	0	0
$813$	−1.07893e20	−0.459577
$814$	0	0
$815$	− 2.22515e20i	− 0.931656i
$816$	0	0
$817$	8.60479e19	0.354149
$818$	0	0
$819$	2.62998e20i	1.06406i
$820$	0	0
$821$	2.14514e20	0.853204	0.426602	−	0.904440i	$-0.359711\pi$
$821$	2.14514e20	0.853204	0.426602	+	0.904440i	$0.359711\pi$
$822$	0	0
$823$	− 1.39731e20i	− 0.546377i	−0.961960	−	0.273189i	$-0.911922\pi$
$823$	− 1.39731e20i	− 0.546377i	0.961960	−	0.273189i	$-0.0880783\pi$
$824$	0	0
$825$	3.21067e20	1.23429
$826$	0	0
$827$	− 2.26109e20i	− 0.854634i	−0.904102	−	0.427317i	$-0.859459\pi$
$827$	− 2.26109e20i	− 0.854634i	0.904102	−	0.427317i	$-0.140541\pi$
$828$	0	0
$829$	−2.94368e20	−1.09398	−0.546990	−	0.837139i	$-0.684226\pi$
$829$	−2.94368e20	−1.09398	−0.546990	+	0.837139i	$0.684226\pi$
$830$	0	0
$831$	1.96764e20i	0.719016i
$832$	0	0
$833$	6.52420e19	0.234429
$834$	0	0
$835$	− 1.52956e20i	− 0.540458i
$836$	0	0
$837$	−1.69097e20	−0.587567
$838$	0	0
$839$	4.76155e20i	1.62710i	0.581494	+	0.813551i	$0.302469\pi$
$839$	4.76155e20i	1.62710i	−0.581494	+	0.813551i	$0.697531\pi$
$840$	0	0
$841$	6.33922e20	2.13041
$842$	0	0
$843$	3.32931e19i	0.110043i
$844$	0	0
$845$	−4.57066e19	−0.148587
$846$	0	0
$847$	7.51664e19i	0.240348i
$848$	0	0
$849$	6.43703e20	2.02457
$850$	0	0
$851$	1.72691e20i	0.534273i
$852$	0	0
$853$	5.72548e20	1.74248	0.871241	−	0.490855i	$-0.163315\pi$
$853$	5.72548e20	1.74248	0.871241	+	0.490855i	$0.163315\pi$
$854$	0	0
$855$	2.81129e20i	0.841673i
$856$	0	0
$857$	−3.62001e20	−1.06621	−0.533107	−	0.846048i	$-0.678976\pi$
$857$	−3.62001e20	−1.06621	−0.533107	+	0.846048i	$0.678976\pi$
$858$	0	0
$859$	− 2.82382e19i	− 0.0818248i	−0.999163	−	0.0409124i	$-0.986974\pi$
$859$	− 2.82382e19i	− 0.0818248i	0.999163	−	0.0409124i	$-0.0130265\pi$
$860$	0	0
$861$	−1.69224e20	−0.482436
$862$	0	0
$863$	− 2.81609e20i	− 0.789899i	−0.918703	−	0.394949i	$-0.870762\pi$
$863$	− 2.81609e20i	− 0.789899i	0.918703	−	0.394949i	$-0.129238\pi$
$864$	0	0
$865$	−1.42971e19	−0.0394579
$866$	0	0
$867$	− 4.65281e20i	− 1.26352i
$868$	0	0
$869$	1.73975e20	0.464889
$870$	0	0
$871$	− 2.47535e20i	− 0.650892i
$872$	0	0
$873$	−1.98142e20	−0.512717
$874$	0	0
$875$	9.06000e19i	0.230713i
$876$	0	0
$877$	−2.22663e20	−0.558021	−0.279011	−	0.960288i	$-0.590006\pi$
$877$	−2.22663e20	−0.558021	−0.279011	+	0.960288i	$0.590006\pi$
$878$	0	0
$879$	− 6.33146e20i	− 1.56164i
$880$	0	0
$881$	−5.26202e20	−1.27738	−0.638691	−	0.769463i	$-0.720524\pi$
$881$	−5.26202e20	−1.27738	−0.638691	+	0.769463i	$0.720524\pi$
$882$	0	0
$883$	− 5.34787e20i	− 1.27778i	−0.769298	−	0.638890i	$-0.779394\pi$
$883$	− 5.34787e20i	− 1.27778i	0.769298	−	0.638890i	$-0.220606\pi$
$884$	0	0
$885$	−9.53649e20	−2.24278
$886$	0	0
$887$	1.06525e20i	0.246595i	0.992370	+	0.123298i	$0.0393470\pi$
$887$	1.06525e20i	0.246595i	−0.992370	+	0.123298i	$0.960653\pi$
$888$	0	0
$889$	3.71804e20	0.847230
$890$	0	0
$891$	5.85121e20i	1.31251i
$892$	0	0
$893$	3.79881e18	0.00838855
$894$	0	0
$895$	3.35172e20i	0.728629i
$896$	0	0
$897$	2.27265e20	0.486392
$898$	0	0
$899$	1.05816e21i	2.22963i
$900$	0	0
$901$	−2.12528e19	−0.0440902
$902$	0	0
$903$	4.09632e20i	0.836718i
$904$	0	0
$905$	−4.13722e20	−0.832086
$906$	0	0
$907$	1.32642e20i	0.262681i	0.991337	+	0.131340i	$0.0419281\pi$
$907$	1.32642e20i	0.262681i	−0.991337	+	0.131340i	$0.958072\pi$
$908$	0	0
$909$	1.43767e19	0.0280357
$910$	0	0
$911$	− 5.87712e20i	− 1.12859i	−0.825573	−	0.564295i	$-0.809148\pi$
$911$	− 5.87712e20i	− 1.12859i	0.825573	−	0.564295i	$-0.190852\pi$
$912$	0	0
$913$	−5.11451e20	−0.967184
$914$	0	0
$915$	− 1.76537e19i	− 0.0328766i
$916$	0	0
$917$	3.51557e19	0.0644779
$918$	0	0
$919$	− 3.16028e20i	− 0.570843i	−0.958402	−	0.285421i	$-0.907866\pi$
$919$	− 3.16028e20i	− 0.570843i	0.958402	−	0.285421i	$-0.0921336\pi$
$920$	0	0
$921$	8.78777e20	1.56337
$922$	0	0
$923$	− 9.30216e20i	− 1.62994i
$924$	0	0
$925$	−6.96385e20	−1.20187
$926$	0	0
$927$	− 2.40856e20i	− 0.409448i
$928$	0	0
$929$	−1.10387e21	−1.84845	−0.924224	−	0.381852i	$-0.875286\pi$
$929$	−1.10387e21	−1.84845	−0.924224	+	0.381852i	$0.875286\pi$
$930$	0	0
$931$	1.24639e21i	2.05593i
$932$	0	0
$933$	−4.13475e20	−0.671858
$934$	0	0
$935$	1.00552e20i	0.160956i
$936$	0	0
$937$	4.84376e20	0.763845	0.381923	−	0.924194i	$-0.375262\pi$
$937$	4.84376e20	0.763845	0.381923	+	0.924194i	$0.375262\pi$
$938$	0	0
$939$	− 3.59239e19i	− 0.0558116i
$940$	0	0
$941$	1.05841e20	0.162005	0.0810024	−	0.996714i	$-0.474188\pi$
$941$	1.05841e20	0.162005	0.0810024	+	0.996714i	$0.474188\pi$
$942$	0	0
$943$	5.68154e19i	0.0856809i
$944$	0	0
$945$	7.67934e20	1.14104
$946$	0	0
$947$	− 7.15352e20i	− 1.04730i	−0.851934	−	0.523649i	$-0.824570\pi$
$947$	− 7.15352e20i	− 1.04730i	0.851934	−	0.523649i	$-0.175430\pi$
$948$	0	0
$949$	8.68083e20	1.25227
$950$	0	0
$951$	− 1.44638e21i	− 2.05598i
$952$	0	0
$953$	5.12146e20	0.717370	0.358685	−	0.933459i	$-0.383225\pi$
$953$	5.12146e20	0.717370	0.358685	+	0.933459i	$0.383225\pi$
$954$	0	0
$955$	− 1.39261e20i	− 0.192224i
$956$	0	0
$957$	1.77258e21	2.41114
$958$	0	0
$959$	1.55260e21i	2.08127i
$960$	0	0
$961$	−4.45120e20	−0.588048
$962$	0	0
$963$	− 8.55908e20i	− 1.11440i
$964$	0	0
$965$	−1.33167e20	−0.170886
$966$	0	0
$967$	1.26510e21i	1.60007i	0.599952	+	0.800036i	$0.295186\pi$
$967$	1.26510e21i	1.60007i	−0.599952	+	0.800036i	$0.704814\pi$
$968$	0	0
$969$	−1.07699e20	−0.134260
$970$	0	0
$971$	− 9.95209e20i	− 1.22287i	−0.791295	−	0.611434i	$-0.790593\pi$
$971$	− 9.95209e20i	− 1.22287i	0.791295	−	0.611434i	$-0.209407\pi$
$972$	0	0
$973$	1.27601e21	1.54548
$974$	0	0
$975$	9.16458e20i	1.09416i
$976$	0	0
$977$	−8.03795e20	−0.945983	−0.472992	−	0.881067i	$-0.656826\pi$
$977$	−8.03795e20	−0.945983	−0.472992	+	0.881067i	$0.656826\pi$
$978$	0	0
$979$	− 7.44669e19i	− 0.0863941i
$980$	0	0
$981$	4.24696e20	0.485731
$982$	0	0
$983$	6.82160e20i	0.769151i	0.923094	+	0.384575i	$0.125652\pi$
$983$	6.82160e20i	0.769151i	−0.923094	+	0.384575i	$0.874348\pi$
$984$	0	0
$985$	−3.18120e20	−0.353621
$986$	0	0
$987$	1.80842e19i	0.0198189i
$988$	0	0
$989$	1.37530e20	0.148602
$990$	0	0
$991$	2.25284e20i	0.240002i	0.992774	+	0.120001i	$0.0382898\pi$
$991$	2.25284e20i	0.240002i	−0.992774	+	0.120001i	$0.961710\pi$
$992$	0	0
$993$	−4.02730e20	−0.423028
$994$	0	0
$995$	1.69131e21i	1.75170i
$996$	0	0
$997$	−1.55466e20	−0.158770	−0.0793852	−	0.996844i	$-0.525296\pi$
$997$	−1.55466e20	−0.158770	−0.0793852	+	0.996844i	$0.525296\pi$
$998$	0	0
$999$	− 6.14391e20i	− 0.618709i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.c.b.15.2	yes	2
3.2	odd	2		144.15.g.b.127.1		2
4.3	odd	2	inner	16.15.c.b.15.1	✓	2
8.3	odd	2		64.15.c.b.63.2		2
8.5	even	2		64.15.c.b.63.1		2
12.11	even	2		144.15.g.b.127.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.c.b.15.1	✓	2	4.3	odd	2	inner
16.15.c.b.15.2	yes	2	1.1	even	1	trivial
64.15.c.b.63.1		2	8.5	even	2
64.15.c.b.63.2		2	8.3	odd	2
144.15.g.b.127.1		2	3.2	odd	2
144.15.g.b.127.2		2	12.11	even	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2796.80i q^{3} +107850. q^{5} -1.45993e6i q^{7} -3.03911e6 q^{9} +O(q^{10})\)	\(q+2796.80i q^{3} +107850. q^{5} -1.45993e6i q^{7} -3.03911e6 q^{9} -2.07662e7i q^{11} +5.92754e7 q^{13} +3.01635e8i q^{15} -4.48964e7 q^{17} -8.57708e8i q^{19} +4.08313e9 q^{21} -1.37087e9i q^{23} +5.52811e9 q^{25} +4.87722e9i q^{27} +3.05202e10 q^{29} +3.46708e10i q^{31} +5.80789e10 q^{33} -1.57453e11i q^{35} -1.25972e11 q^{37} +1.65781e11i q^{39} -4.14447e10 q^{41} +1.00323e11i q^{43} -3.27768e11 q^{45} +4.42902e9i q^{47} -1.45317e12 q^{49} -1.25566e11i q^{51} +4.73374e11 q^{53} -2.23964e12i q^{55} +2.39884e12 q^{57} +3.16160e12i q^{59} -5.85267e10 q^{61} +4.43689e12i q^{63} +6.39286e12 q^{65} -4.17601e12i q^{67} +3.83406e12 q^{69} -1.56931e13i q^{71} +1.46449e13 q^{73} +1.54610e13i q^{75} -3.03172e13 q^{77} +8.37780e12i q^{79} -2.81766e13 q^{81} -2.46290e13i q^{83} -4.84207e12 q^{85} +8.53588e13i q^{87} +3.58596e12 q^{89} -8.65379e13i q^{91} -9.69672e13 q^{93} -9.25038e13i q^{95} +6.51975e13 q^{97} +6.31109e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 215700 q^{5} - 6078222 q^{9}+O(q^{10}) \) 2 * q + 215700 * q^5 - 6078222 * q^9	\( 2 q + 215700 q^{5} - 6078222 q^{9} + 118550900 q^{13} - 89792700 q^{17} + 8166251520 q^{21} + 11056213750 q^{25} + 61040337204 q^{29} + 116157888000 q^{33} - 251943320300 q^{37} - 82889351964 q^{41} - 655536242700 q^{45} - 2906337147742 q^{49} + 946748870100 q^{53} + 4797672768000 q^{57} - 117053344844 q^{61} + 12785714565000 q^{65} + 7668110177280 q^{69} + 29289808336100 q^{73} - 60634417536000 q^{77} - 56353140970398 q^{81} - 9684142695000 q^{85} + 7171921108644 q^{89} - 193934393856000 q^{93} + 130394938141700 q^{97}+O(q^{100}) \) 2 * q + 215700 * q^5 - 6078222 * q^9 + 118550900 * q^13 - 89792700 * q^17 + 8166251520 * q^21 + 11056213750 * q^25 + 61040337204 * q^29 + 116157888000 * q^33 - 251943320300 * q^37 - 82889351964 * q^41 - 655536242700 * q^45 - 2906337147742 * q^49 + 946748870100 * q^53 + 4797672768000 * q^57 - 117053344844 * q^61 + 12785714565000 * q^65 + 7668110177280 * q^69 + 29289808336100 * q^73 - 60634417536000 * q^77 - 56353140970398 * q^81 - 9684142695000 * q^85 + 7171921108644 * q^89 - 193934393856000 * q^93 + 130394938141700 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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