LMFDB - Embedded newform 16.15.c.c.15.3

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:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 9547x^{2} + 9546x + 91126116 $ x^4 - x^3 + 9547x^2 + 9546x + 91126116
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.3
Root			$49.1025 - 85.0479i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.15.c.c.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1141.90i q^{3} +15768.7 q^{5} +288003. i q^{7} +3.47902e6 q^{9} +O(q^{10})$	$q+1141.90i q^{3} +15768.7 q^{5} +288003. i q^{7} +3.47902e6 q^{9} +5.71462e6i q^{11} -2.57175e7 q^{13} +1.80063e7i q^{15} -2.60477e8 q^{17} +8.37875e8i q^{19} -3.28872e8 q^{21} +5.88776e9i q^{23} -5.85486e9 q^{25} +9.43440e9i q^{27} -1.96743e10 q^{29} +3.89627e10i q^{31} -6.52555e9 q^{33} +4.54143e9i q^{35} +3.78659e10 q^{37} -2.93669e10i q^{39} +5.95929e10 q^{41} -4.00278e11i q^{43} +5.48596e10 q^{45} -4.72330e11i q^{47} +5.95277e11 q^{49} -2.97440e11i q^{51} +5.00346e11 q^{53} +9.01121e10i q^{55} -9.56772e11 q^{57} +2.44119e12i q^{59} -2.89419e12 q^{61} +1.00197e12i q^{63} -4.05531e11 q^{65} +7.00135e12i q^{67} -6.72326e12 q^{69} +5.99930e12i q^{71} +1.03306e13 q^{73} -6.68569e12i q^{75} -1.64583e12 q^{77} -1.03559e13i q^{79} +5.86688e12 q^{81} -3.46617e13i q^{83} -4.10738e12 q^{85} -2.24662e13i q^{87} +6.54141e13 q^{89} -7.40671e12i q^{91} -4.44917e13 q^{93} +1.32122e13i q^{95} +7.10418e13 q^{97} +1.98813e13i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 87000 q^{5} - 20000796 q^{9}+O(q^{10}) $ 4 * q - 87000 * q^5 - 20000796 * q^9	$ 4 q - 87000 q^{5} - 20000796 q^{9} + 66864616 q^{13} + 13116552 q^{17} - 9582203904 q^{21} - 16891205140 q^{25} - 17501785752 q^{29} - 118197379584 q^{33} - 139985734360 q^{37} + 662088410568 q^{41} + 1707534576360 q^{45} + 366399422788 q^{49} - 2975173624152 q^{53} + 3730435570176 q^{57} - 6624970851736 q^{61} - 7822522322160 q^{65} - 14338843057152 q^{69} - 12033132589112 q^{73} - 28983083609088 q^{77} + 200426504614596 q^{81} - 39868454746800 q^{85} + 106307014497096 q^{89} - 166229577732096 q^{93} + 95793165936136 q^{97}+O(q^{100}) $ 4 * q - 87000 * q^5 - 20000796 * q^9 + 66864616 * q^13 + 13116552 * q^17 - 9582203904 * q^21 - 16891205140 * q^25 - 17501785752 * q^29 - 118197379584 * q^33 - 139985734360 * q^37 + 662088410568 * q^41 + 1707534576360 * q^45 + 366399422788 * q^49 - 2975173624152 * q^53 + 3730435570176 * q^57 - 6624970851736 * q^61 - 7822522322160 * q^65 - 14338843057152 * q^69 - 12033132589112 * q^73 - 28983083609088 * q^77 + 200426504614596 * q^81 - 39868454746800 * q^85 + 106307014497096 * q^89 - 166229577732096 * q^93 + 95793165936136 * 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
$3$	1141.90i	0.522133i	0.965321	+	0.261066i	$0.0840741\pi$
$3$	1141.90i	0.522133i	−0.965321	+	0.261066i	$0.915926\pi$
$4$	0	0
$5$	15768.7	0.201839	0.100920	−	0.994895i	$-0.467822\pi$
$5$	15768.7	0.201839	0.100920	+	0.994895i	$0.467822\pi$
$6$	0	0
$7$	288003.i	0.349712i	0.984594	+	0.174856i	$0.0559460\pi$
$7$	288003.i	0.349712i	−0.984594	+	0.174856i	$0.944054\pi$
$8$	0	0
$9$	3.47902e6	0.727378
$10$	0	0
$11$	5.71462e6i	0.293251i	0.989192	+	0.146625i	$0.0468412\pi$
$11$	5.71462e6i	0.293251i	−0.989192	+	0.146625i	$0.953159\pi$
$12$	0	0
$13$	−2.57175e7	−0.409850	−0.204925	−	0.978778i	$-0.565695\pi$
$13$	−2.57175e7	−0.409850	−0.204925	+	0.978778i	$0.565695\pi$
$14$	0	0
$15$	1.80063e7i	0.105387i
$16$	0	0
$17$	−2.60477e8	−0.634786	−0.317393	−	0.948294i	$-0.602807\pi$
$17$	−2.60477e8	−0.634786	−0.317393	+	0.948294i	$0.602807\pi$
$18$	0	0
$19$	8.37875e8i	0.937354i	0.883370	+	0.468677i	$0.155269\pi$
$19$	8.37875e8i	0.937354i	−0.883370	+	0.468677i	$0.844731\pi$
$20$	0	0
$21$	−3.28872e8	−0.182596
$22$	0	0
$23$	5.88776e9i	1.72924i	0.502425	+	0.864621i	$0.332441\pi$
$23$	5.88776e9i	1.72924i	−0.502425	+	0.864621i	$0.667559\pi$
$24$	0	0
$25$	−5.85486e9	−0.959261
$26$	0	0
$27$	9.43440e9i	0.901920i
$28$	0	0
$29$	−1.96743e10	−1.14055	−0.570274	−	0.821454i	$-0.693163\pi$
$29$	−1.96743e10	−1.14055	−0.570274	+	0.821454i	$0.693163\pi$
$30$	0	0
$31$	3.89627e10i	1.41618i	0.706124	+	0.708088i	$0.250442\pi$
$31$	3.89627e10i	1.41618i	−0.706124	+	0.708088i	$0.749558\pi$
$32$	0	0
$33$	−6.52555e9	−0.153116
$34$	0	0
$35$	4.54143e9i	0.0705856i
$36$	0	0
$37$	3.78659e10	0.398875	0.199437	−	0.979911i	$-0.436089\pi$
$37$	3.78659e10	0.398875	0.199437	+	0.979911i	$0.436089\pi$
$38$	0	0
$39$	− 2.93669e10i	− 0.213996i
$40$	0	0
$41$	5.95929e10	0.305990	0.152995	−	0.988227i	$-0.451108\pi$
$41$	5.95929e10	0.305990	0.152995	+	0.988227i	$0.451108\pi$
$42$	0	0
$43$	− 4.00278e11i	− 1.47259i	−0.676659	−	0.736297i	$-0.736573\pi$
$43$	− 4.00278e11i	− 1.47259i	0.676659	−	0.736297i	$-0.263427\pi$
$44$	0	0
$45$	5.48596e10	0.146813
$46$	0	0
$47$	− 4.72330e11i	− 0.932310i	−0.884703	−	0.466155i	$-0.845639\pi$
$47$	− 4.72330e11i	− 0.932310i	0.884703	−	0.466155i	$-0.154361\pi$
$48$	0	0
$49$	5.95277e11	0.877702
$50$	0	0
$51$	− 2.97440e11i	− 0.331442i
$52$	0	0
$53$	5.00346e11	0.425931	0.212966	−	0.977060i	$-0.431688\pi$
$53$	5.00346e11	0.425931	0.212966	+	0.977060i	$0.431688\pi$
$54$	0	0
$55$	9.01121e10i	0.0591895i
$56$	0	0
$57$	−9.56772e11	−0.489423
$58$	0	0
$59$	2.44119e12i	0.980930i	0.871461	+	0.490465i	$0.163173\pi$
$59$	2.44119e12i	0.980930i	−0.871461	+	0.490465i	$0.836827\pi$
$60$	0	0
$61$	−2.89419e12	−0.920911	−0.460456	−	0.887683i	$-0.652314\pi$
$61$	−2.89419e12	−0.920911	−0.460456	+	0.887683i	$0.652314\pi$
$62$	0	0
$63$	1.00197e12i	0.254373i
$64$	0	0
$65$	−4.05531e11	−0.0827238
$66$	0	0
$67$	7.00135e12i	1.15520i	0.816319	+	0.577602i	$0.196011\pi$
$67$	7.00135e12i	1.15520i	−0.816319	+	0.577602i	$0.803989\pi$
$68$	0	0
$69$	−6.72326e12	−0.902893
$70$	0	0
$71$	5.99930e12i	0.659617i	0.944048	+	0.329809i	$0.106984\pi$
$71$	5.99930e12i	0.659617i	−0.944048	+	0.329809i	$0.893016\pi$
$72$	0	0
$73$	1.03306e13	0.935113	0.467556	−	0.883963i	$-0.345135\pi$
$73$	1.03306e13	0.935113	0.467556	+	0.883963i	$0.345135\pi$
$74$	0	0
$75$	− 6.68569e12i	− 0.500861i
$76$	0	0
$77$	−1.64583e12	−0.102553
$78$	0	0
$79$	− 1.03559e13i	− 0.539258i	−0.962964	−	0.269629i	$-0.913099\pi$
$79$	− 1.03559e13i	− 0.539258i	0.962964	−	0.269629i	$-0.0869010\pi$
$80$	0	0
$81$	5.86688e12	0.256456
$82$	0	0
$83$	− 3.46617e13i	− 1.27733i	−0.769484	−	0.638666i	$-0.779487\pi$
$83$	− 3.46617e13i	− 1.27733i	0.769484	−	0.638666i	$-0.220513\pi$
$84$	0	0
$85$	−4.10738e12	−0.128125
$86$	0	0
$87$	− 2.24662e13i	− 0.595518i
$88$	0	0
$89$	6.54141e13	1.47891	0.739454	−	0.673207i	$-0.235084\pi$
$89$	6.54141e13	1.47891	0.739454	+	0.673207i	$0.235084\pi$
$90$	0	0
$91$	− 7.40671e12i	− 0.143329i
$92$	0	0
$93$	−4.44917e13	−0.739432
$94$	0	0
$95$	1.32122e13i	0.189195i
$96$	0	0
$97$	7.10418e13	0.879249	0.439625	−	0.898182i	$-0.355112\pi$
$97$	7.10418e13	0.879249	0.439625	+	0.898182i	$0.355112\pi$
$98$	0	0
$99$	1.98813e13i	0.213304i
$100$	0	0
$101$	−1.37906e14	−1.28627	−0.643137	−	0.765751i	$-0.722368\pi$
$101$	−1.37906e14	−1.28627	−0.643137	+	0.765751i	$0.722368\pi$
$102$	0	0
$103$	9.27304e13i	0.753983i	0.926217	+	0.376992i	$0.123041\pi$
$103$	9.27304e13i	0.753983i	−0.926217	+	0.376992i	$0.876959\pi$
$104$	0	0
$105$	−5.18587e12	−0.0368550
$106$	0	0
$107$	1.62369e14i	1.01115i	0.862781	+	0.505577i	$0.168721\pi$
$107$	1.62369e14i	1.01115i	−0.862781	+	0.505577i	$0.831279\pi$
$108$	0	0
$109$	−2.66319e14	−1.45686	−0.728428	−	0.685122i	$-0.759749\pi$
$109$	−2.66319e14	−1.45686	−0.728428	+	0.685122i	$0.759749\pi$
$110$	0	0
$111$	4.32393e13i	0.208266i
$112$	0	0
$113$	1.93762e14	0.823607	0.411803	−	0.911273i	$-0.364899\pi$
$113$	1.93762e14	0.823607	0.411803	+	0.911273i	$0.364899\pi$
$114$	0	0
$115$	9.28423e13i	0.349029i
$116$	0	0
$117$	−8.94717e13	−0.298116
$118$	0	0
$119$	− 7.50182e13i	− 0.221992i
$120$	0	0
$121$	3.47093e14	0.914004
$122$	0	0
$123$	6.80494e13i	0.159767i
$124$	0	0
$125$	−1.88568e14	−0.395456
$126$	0	0
$127$	− 5.88082e14i	− 1.10360i	−0.833976	−	0.551800i	$-0.813941\pi$
$127$	− 5.88082e14i	− 1.10360i	0.833976	−	0.551800i	$-0.186059\pi$
$128$	0	0
$129$	4.57080e14	0.768889
$130$	0	0
$131$	− 1.08609e15i	− 1.64046i	−0.572036	−	0.820229i	$-0.693846\pi$
$131$	− 1.08609e15i	− 1.64046i	0.572036	−	0.820229i	$-0.306154\pi$
$132$	0	0
$133$	−2.41310e14	−0.327804
$134$	0	0
$135$	1.48768e14i	0.182043i
$136$	0	0
$137$	4.56130e14	0.503552	0.251776	−	0.967786i	$-0.418985\pi$
$137$	4.56130e14	0.503552	0.251776	+	0.967786i	$0.418985\pi$
$138$	0	0
$139$	− 1.75112e14i	− 0.174668i	−0.996179	−	0.0873339i	$-0.972165\pi$
$139$	− 1.75112e14i	− 0.174668i	0.996179	−	0.0873339i	$-0.0278347\pi$
$140$	0	0
$141$	5.39355e14	0.486789
$142$	0	0
$143$	− 1.46966e14i	− 0.120189i
$144$	0	0
$145$	−3.10238e14	−0.230207
$146$	0	0
$147$	6.79750e14i	0.458277i
$148$	0	0
$149$	−3.93773e14	−0.241514	−0.120757	−	0.992682i	$-0.538532\pi$
$149$	−3.93773e14	−0.241514	−0.120757	+	0.992682i	$0.538532\pi$
$150$	0	0
$151$	2.26432e15i	1.26502i	0.774550	+	0.632512i	$0.217976\pi$
$151$	2.26432e15i	1.26502i	−0.774550	+	0.632512i	$0.782024\pi$
$152$	0	0
$153$	−9.06207e14	−0.461729
$154$	0	0
$155$	6.14391e14i	0.285840i
$156$	0	0
$157$	−1.05710e15	−0.449590	−0.224795	−	0.974406i	$-0.572171\pi$
$157$	−1.05710e15	−0.449590	−0.224795	+	0.974406i	$0.572171\pi$
$158$	0	0
$159$	5.71347e14i	0.222393i
$160$	0	0
$161$	−1.69569e15	−0.604736
$162$	0	0
$163$	1.92010e15i	0.628073i	0.949411	+	0.314037i	$0.101681\pi$
$163$	1.92010e15i	0.628073i	−0.949411	+	0.314037i	$0.898319\pi$
$164$	0	0
$165$	−1.02899e14	−0.0309047
$166$	0	0
$167$	− 1.15661e15i	− 0.319279i	−0.987175	−	0.159640i	$-0.948967\pi$
$167$	− 1.15661e15i	− 0.319279i	0.987175	−	0.159640i	$-0.0510332\pi$
$168$	0	0
$169$	−3.27599e15	−0.832023
$170$	0	0
$171$	2.91499e15i	0.681810i
$172$	0	0
$173$	8.69517e15	1.87480	0.937401	−	0.348253i	$-0.113225\pi$
$173$	8.69517e15	1.87480	0.937401	+	0.348253i	$0.113225\pi$
$174$	0	0
$175$	− 1.68622e15i	− 0.335465i
$176$	0	0
$177$	−2.78761e15	−0.512176
$178$	0	0
$179$	− 8.11829e15i	− 1.37877i	−0.724393	−	0.689387i	$-0.757880\pi$
$179$	− 8.11829e15i	− 1.37877i	0.724393	−	0.689387i	$-0.242120\pi$
$180$	0	0
$181$	7.04106e15	1.10634	0.553169	−	0.833069i	$-0.313418\pi$
$181$	7.04106e15	1.10634	0.553169	+	0.833069i	$0.313418\pi$
$182$	0	0
$183$	− 3.30488e15i	− 0.480838i
$184$	0	0
$185$	5.97096e14	0.0805086
$186$	0	0
$187$	− 1.48853e15i	− 0.186151i
$188$	0	0
$189$	−2.71714e15	−0.315412
$190$	0	0
$191$	− 9.89731e15i	− 1.06729i	−0.845708	−	0.533647i	$-0.820821\pi$
$191$	− 9.89731e15i	− 1.06729i	0.845708	−	0.533647i	$-0.179179\pi$
$192$	0	0
$193$	−3.30454e15	−0.331291	−0.165645	−	0.986185i	$-0.552971\pi$
$193$	−3.30454e15	−0.331291	−0.165645	+	0.986185i	$0.552971\pi$
$194$	0	0
$195$	− 4.63077e14i	− 0.0431928i
$196$	0	0
$197$	1.15838e16	1.00597	0.502987	−	0.864294i	$-0.332234\pi$
$197$	1.15838e16	1.00597	0.502987	+	0.864294i	$0.332234\pi$
$198$	0	0
$199$	7.10856e15i	0.575188i	0.957752	+	0.287594i	$0.0928554\pi$
$199$	7.10856e15i	0.575188i	−0.957752	+	0.287594i	$0.907145\pi$
$200$	0	0
$201$	−7.99487e15	−0.603169
$202$	0	0
$203$	− 5.66626e15i	− 0.398864i
$204$	0	0
$205$	9.39702e14	0.0617608
$206$	0	0
$207$	2.04837e16i	1.25781i
$208$	0	0
$209$	−4.78814e15	−0.274880
$210$	0	0
$211$	2.21036e16i	1.18710i	0.804798	+	0.593548i	$0.202273\pi$
$211$	2.21036e16i	1.18710i	−0.804798	+	0.593548i	$0.797727\pi$
$212$	0	0
$213$	−6.85062e15	−0.344408
$214$	0	0
$215$	− 6.31186e15i	− 0.297227i
$216$	0	0
$217$	−1.12214e16	−0.495254
$218$	0	0
$219$	1.17965e16i	0.488253i
$220$	0	0
$221$	6.69882e15	0.260167
$222$	0	0
$223$	3.24859e16i	1.18457i	0.805729	+	0.592285i	$0.201774\pi$
$223$	3.24859e16i	1.18457i	−0.805729	+	0.592285i	$0.798226\pi$
$224$	0	0
$225$	−2.03692e16	−0.697745
$226$	0	0
$227$	− 2.89720e16i	− 0.932818i	−0.884569	−	0.466409i	$-0.845548\pi$
$227$	− 2.89720e16i	− 0.932818i	0.884569	−	0.466409i	$-0.154452\pi$
$228$	0	0
$229$	−4.20154e16	−1.27221	−0.636107	−	0.771601i	$-0.719456\pi$
$229$	−4.20154e16	−1.27221	−0.636107	+	0.771601i	$0.719456\pi$
$230$	0	0
$231$	− 1.87938e15i	− 0.0535464i
$232$	0	0
$233$	6.06295e16	1.62627	0.813135	−	0.582075i	$-0.197759\pi$
$233$	6.06295e16	1.62627	0.813135	+	0.582075i	$0.197759\pi$
$234$	0	0
$235$	− 7.44802e15i	− 0.188177i
$236$	0	0
$237$	1.18254e16	0.281564
$238$	0	0
$239$	− 7.31918e16i	− 1.64315i	−0.570101	−	0.821575i	$-0.693096\pi$
$239$	− 7.31918e16i	− 1.64315i	0.570101	−	0.821575i	$-0.306904\pi$
$240$	0	0
$241$	2.88532e16	0.611048	0.305524	−	0.952184i	$-0.401168\pi$
$241$	2.88532e16	0.611048	0.305524	+	0.952184i	$0.401168\pi$
$242$	0	0
$243$	5.18239e16i	1.03582i
$244$	0	0
$245$	9.38674e15	0.177155
$246$	0	0
$247$	− 2.15480e16i	− 0.384175i
$248$	0	0
$249$	3.95804e16	0.666937
$250$	0	0
$251$	− 9.75150e16i	− 1.55366i	−0.629712	−	0.776829i	$-0.716827\pi$
$251$	− 9.75150e16i	− 1.55366i	0.629712	−	0.776829i	$-0.283173\pi$
$252$	0	0
$253$	−3.36464e16	−0.507101
$254$	0	0
$255$	− 4.69024e15i	− 0.0668981i
$256$	0	0
$257$	5.63983e16	0.761613	0.380807	−	0.924655i	$-0.375646\pi$
$257$	5.63983e16	0.761613	0.380807	+	0.924655i	$0.375646\pi$
$258$	0	0
$259$	1.09055e16i	0.139491i
$260$	0	0
$261$	−6.84475e16	−0.829610
$262$	0	0
$263$	− 7.14679e16i	− 0.821146i	−0.911828	−	0.410573i	$-0.865329\pi$
$263$	− 7.14679e16i	− 0.821146i	0.911828	−	0.410573i	$-0.134671\pi$
$264$	0	0
$265$	7.88980e15	0.0859696
$266$	0	0
$267$	7.46966e16i	0.772187i
$268$	0	0
$269$	−4.60431e16	−0.451750	−0.225875	−	0.974156i	$-0.572524\pi$
$269$	−4.60431e16	−0.451750	−0.225875	+	0.974156i	$0.572524\pi$
$270$	0	0
$271$	1.53117e17i	1.42639i	0.700964	+	0.713196i	$0.252753\pi$
$271$	1.53117e17i	1.42639i	−0.700964	+	0.713196i	$0.747247\pi$
$272$	0	0
$273$	8.45775e15	0.0748370
$274$	0	0
$275$	− 3.34583e16i	− 0.281304i
$276$	0	0
$277$	−1.20366e17	−0.961931	−0.480965	−	0.876740i	$-0.659714\pi$
$277$	−1.20366e17	−0.961931	−0.480965	+	0.876740i	$0.659714\pi$
$278$	0	0
$279$	1.35552e17i	1.03009i
$280$	0	0
$281$	8.82065e16	0.637612	0.318806	−	0.947820i	$-0.396718\pi$
$281$	8.82065e16	0.637612	0.318806	+	0.947820i	$0.396718\pi$
$282$	0	0
$283$	9.07764e16i	0.624407i	0.950015	+	0.312204i	$0.101067\pi$
$283$	9.07764e16i	0.624407i	−0.950015	+	0.312204i	$0.898933\pi$
$284$	0	0
$285$	−1.50870e16	−0.0987848
$286$	0	0
$287$	1.71629e16i	0.107008i
$288$	0	0
$289$	−1.00529e17	−0.597047
$290$	0	0
$291$	8.11229e16i	0.459085i
$292$	0	0
$293$	−1.73623e15	−0.00936555	−0.00468278	−	0.999989i	$-0.501491\pi$
$293$	−1.73623e15	−0.00936555	−0.00468278	+	0.999989i	$0.501491\pi$
$294$	0	0
$295$	3.84944e16i	0.197990i
$296$	0	0
$297$	−5.39141e16	−0.264489
$298$	0	0
$299$	− 1.51418e17i	− 0.708730i
$300$	0	0
$301$	1.15281e17	0.514984
$302$	0	0
$303$	− 1.57475e17i	− 0.671606i
$304$	0	0
$305$	−4.56375e16	−0.185876
$306$	0	0
$307$	− 1.45941e17i	− 0.567819i	−0.958851	−	0.283909i	$-0.908369\pi$
$307$	− 1.45941e17i	− 0.567819i	0.958851	−	0.283909i	$-0.0916315\pi$
$308$	0	0
$309$	−1.05889e17	−0.393679
$310$	0	0
$311$	− 1.44131e17i	− 0.512194i	−0.966651	−	0.256097i	$-0.917563\pi$
$311$	− 1.44131e17i	− 0.512194i	0.966651	−	0.256097i	$-0.0824366\pi$
$312$	0	0
$313$	2.03286e17	0.690711	0.345356	−	0.938472i	$-0.387758\pi$
$313$	2.03286e17	0.690711	0.345356	+	0.938472i	$0.387758\pi$
$314$	0	0
$315$	1.57997e16i	0.0513424i
$316$	0	0
$317$	5.63149e17	1.75069	0.875344	−	0.483501i	$-0.160635\pi$
$317$	5.63149e17	1.75069	0.875344	+	0.483501i	$0.160635\pi$
$318$	0	0
$319$	− 1.12431e17i	− 0.334467i
$320$	0	0
$321$	−1.85410e17	−0.527957
$322$	0	0
$323$	− 2.18247e17i	− 0.595019i
$324$	0	0
$325$	1.50572e17	0.393153
$326$	0	0
$327$	− 3.04111e17i	− 0.760672i
$328$	0	0
$329$	1.36032e17	0.326040
$330$	0	0
$331$	1.06191e17i	0.243944i	0.992534	+	0.121972i	$0.0389218\pi$
$331$	1.06191e17i	0.243944i	−0.992534	+	0.121972i	$0.961078\pi$
$332$	0	0
$333$	1.31737e17	0.290133
$334$	0	0
$335$	1.10402e17i	0.233165i
$336$	0	0
$337$	−7.37356e17	−1.49372	−0.746858	−	0.664983i	$-0.768439\pi$
$337$	−7.37356e17	−1.49372	−0.746858	+	0.664983i	$0.768439\pi$
$338$	0	0
$339$	2.21258e17i	0.430032i
$340$	0	0
$341$	−2.22657e17	−0.415295
$342$	0	0
$343$	3.66772e17i	0.656655i
$344$	0	0
$345$	−1.06017e17	−0.182239
$346$	0	0
$347$	3.68924e17i	0.609019i	0.952509	+	0.304510i	$0.0984925\pi$
$347$	3.68924e17i	0.609019i	−0.952509	+	0.304510i	$0.901507\pi$
$348$	0	0
$349$	6.77040e17	1.07358	0.536792	−	0.843715i	$-0.319636\pi$
$349$	6.77040e17	1.07358	0.536792	+	0.843715i	$0.319636\pi$
$350$	0	0
$351$	− 2.42629e17i	− 0.369652i
$352$	0	0
$353$	−1.30089e18	−1.90466	−0.952332	−	0.305063i	$-0.901322\pi$
$353$	−1.30089e18	−1.90466	−0.952332	+	0.305063i	$0.901322\pi$
$354$	0	0
$355$	9.46011e16i	0.133137i
$356$	0	0
$357$	8.56636e16	0.115909
$358$	0	0
$359$	1.30712e18i	1.70080i	0.526135	+	0.850401i	$0.323641\pi$
$359$	1.30712e18i	1.70080i	−0.526135	+	0.850401i	$0.676359\pi$
$360$	0	0
$361$	9.69729e16	0.121367
$362$	0	0
$363$	3.96347e17i	0.477231i
$364$	0	0
$365$	1.62899e17	0.188742
$366$	0	0
$367$	− 2.10275e17i	− 0.234491i	−0.993103	−	0.117245i	$-0.962594\pi$
$367$	− 2.10275e17i	− 0.234491i	0.993103	−	0.117245i	$-0.0374064\pi$
$368$	0	0
$369$	2.07325e17	0.222570
$370$	0	0
$371$	1.44101e17i	0.148953i
$372$	0	0
$373$	−5.35975e17	−0.533560	−0.266780	−	0.963758i	$-0.585960\pi$
$373$	−5.35975e17	−0.533560	−0.266780	+	0.963758i	$0.585960\pi$
$374$	0	0
$375$	− 2.15326e17i	− 0.206480i
$376$	0	0
$377$	5.05974e17	0.467454
$378$	0	0
$379$	− 9.79940e17i	− 0.872418i	−0.899845	−	0.436209i	$-0.856321\pi$
$379$	− 9.79940e17i	− 0.872418i	0.899845	−	0.436209i	$-0.143679\pi$
$380$	0	0
$381$	6.71533e17	0.576226
$382$	0	0
$383$	2.01068e17i	0.166323i	0.996536	+	0.0831615i	$0.0265017\pi$
$383$	2.01068e17i	0.166323i	−0.996536	+	0.0831615i	$0.973498\pi$
$384$	0	0
$385$	−2.59525e16	−0.0206993
$386$	0	0
$387$	− 1.39258e18i	− 1.07113i
$388$	0	0
$389$	−1.43160e18	−1.06212	−0.531062	−	0.847333i	$-0.678207\pi$
$389$	−1.43160e18	−1.06212	−0.531062	+	0.847333i	$0.678207\pi$
$390$	0	0
$391$	− 1.53363e18i	− 1.09770i
$392$	0	0
$393$	1.24021e18	0.856536
$394$	0	0
$395$	− 1.63298e17i	− 0.108843i
$396$	0	0
$397$	1.71391e18	1.10269	0.551347	−	0.834276i	$-0.314114\pi$
$397$	1.71391e18	1.10269	0.551347	+	0.834276i	$0.314114\pi$
$398$	0	0
$399$	− 2.75553e17i	− 0.171157i
$400$	0	0
$401$	−7.21821e17	−0.432931	−0.216466	−	0.976290i	$-0.569453\pi$
$401$	−7.21821e17	−0.432931	−0.216466	+	0.976290i	$0.569453\pi$
$402$	0	0
$403$	− 1.00202e18i	− 0.580420i
$404$	0	0
$405$	9.25130e16	0.0517628
$406$	0	0
$407$	2.16390e17i	0.116970i
$408$	0	0
$409$	2.49040e18	1.30079	0.650394	−	0.759597i	$-0.274604\pi$
$409$	2.49040e18	1.30079	0.650394	+	0.759597i	$0.274604\pi$
$410$	0	0
$411$	5.20857e17i	0.262921i
$412$	0	0
$413$	−7.03071e17	−0.343043
$414$	0	0
$415$	− 5.46570e17i	− 0.257816i
$416$	0	0
$417$	1.99961e17	0.0911997
$418$	0	0
$419$	4.23796e18i	1.86921i	0.355686	+	0.934605i	$0.384247\pi$
$419$	4.23796e18i	1.86921i	−0.355686	+	0.934605i	$0.615753\pi$
$420$	0	0
$421$	−4.15253e18	−1.77149	−0.885744	−	0.464174i	$-0.846351\pi$
$421$	−4.15253e18	−1.77149	−0.885744	+	0.464174i	$0.846351\pi$
$422$	0	0
$423$	− 1.64325e18i	− 0.678141i
$424$	0	0
$425$	1.52506e18	0.608925
$426$	0	0
$427$	− 8.33534e17i	− 0.322054i
$428$	0	0
$429$	1.67821e17	0.0627545
$430$	0	0
$431$	3.21020e18i	1.16196i	0.813918	+	0.580980i	$0.197331\pi$
$431$	3.21020e18i	1.16196i	−0.813918	+	0.580980i	$0.802669\pi$
$432$	0	0
$433$	−1.13912e18	−0.399168	−0.199584	−	0.979881i	$-0.563959\pi$
$433$	−1.13912e18	−0.399168	−0.199584	+	0.979881i	$0.563959\pi$
$434$	0	0
$435$	− 3.54262e17i	− 0.120199i
$436$	0	0
$437$	−4.93321e18	−1.62091
$438$	0	0
$439$	2.63163e18i	0.837479i	0.908106	+	0.418739i	$0.137528\pi$
$439$	2.63163e18i	0.837479i	−0.908106	+	0.418739i	$0.862472\pi$
$440$	0	0
$441$	2.07098e18	0.638420
$442$	0	0
$443$	3.31528e18i	0.990133i	0.868855	+	0.495067i	$0.164856\pi$
$443$	3.31528e18i	0.990133i	−0.868855	+	0.495067i	$0.835144\pi$
$444$	0	0
$445$	1.03149e18	0.298502
$446$	0	0
$447$	− 4.49651e17i	− 0.126102i
$448$	0	0
$449$	−4.31048e18	−1.17166	−0.585829	−	0.810434i	$-0.699231\pi$
$449$	−4.31048e18	−1.17166	−0.585829	+	0.810434i	$0.699231\pi$
$450$	0	0
$451$	3.40551e17i	0.0897318i
$452$	0	0
$453$	−2.58563e18	−0.660510
$454$	0	0
$455$	− 1.16794e17i	− 0.0289295i
$456$	0	0
$457$	7.30926e18	1.75574	0.877870	−	0.478899i	$-0.158964\pi$
$457$	7.30926e18	1.75574	0.877870	+	0.478899i	$0.158964\pi$
$458$	0	0
$459$	− 2.45745e18i	− 0.572526i
$460$	0	0
$461$	4.51567e18	1.02051	0.510254	−	0.860024i	$-0.329552\pi$
$461$	4.51567e18	1.02051	0.510254	+	0.860024i	$0.329552\pi$
$462$	0	0
$463$	2.94637e17i	0.0645982i	0.999478	+	0.0322991i	$0.0102829\pi$
$463$	2.94637e17i	0.0645982i	−0.999478	+	0.0322991i	$0.989717\pi$
$464$	0	0
$465$	−7.01575e17	−0.149246
$466$	0	0
$467$	− 2.49459e18i	− 0.514969i	−0.966282	−	0.257484i	$-0.917106\pi$
$467$	− 2.49459e18i	− 0.514969i	0.966282	−	0.257484i	$-0.0828936\pi$
$468$	0	0
$469$	−2.01641e18	−0.403988
$470$	0	0
$471$	− 1.20710e18i	− 0.234746i
$472$	0	0
$473$	2.28744e18	0.431839
$474$	0	0
$475$	− 4.90564e18i	− 0.899167i
$476$	0	0
$477$	1.74072e18	0.309813
$478$	0	0
$479$	− 1.00530e18i	− 0.173758i	−0.996219	−	0.0868790i	$-0.972311\pi$
$479$	− 1.00530e18i	− 0.173758i	0.996219	−	0.0868790i	$-0.0276894\pi$
$480$	0	0
$481$	−9.73817e17	−0.163479
$482$	0	0
$483$	− 1.93632e18i	− 0.315753i
$484$	0	0
$485$	1.12024e18	0.177467
$486$	0	0
$487$	− 9.28333e18i	− 1.42890i	−0.699688	−	0.714449i	$-0.746677\pi$
$487$	− 9.28333e18i	− 1.42890i	0.699688	−	0.714449i	$-0.253323\pi$
$488$	0	0
$489$	−2.19257e18	−0.327938
$490$	0	0
$491$	− 2.08135e18i	− 0.302534i	−0.988493	−	0.151267i	$-0.951665\pi$
$491$	− 2.08135e18i	− 0.302534i	0.988493	−	0.151267i	$-0.0483354\pi$
$492$	0	0
$493$	5.12471e18	0.724004
$494$	0	0
$495$	3.13502e17i	0.0430531i
$496$	0	0
$497$	−1.72782e18	−0.230676
$498$	0	0
$499$	6.01687e18i	0.781029i	0.920597	+	0.390514i	$0.127703\pi$
$499$	6.01687e18i	0.781029i	−0.920597	+	0.390514i	$0.872297\pi$
$500$	0	0
$501$	1.32073e18	0.166706
$502$	0	0
$503$	6.61523e18i	0.812025i	0.913868	+	0.406012i	$0.133081\pi$
$503$	6.61523e18i	0.812025i	−0.913868	+	0.406012i	$0.866919\pi$
$504$	0	0
$505$	−2.17460e18	−0.259621
$506$	0	0
$507$	− 3.74086e18i	− 0.434426i
$508$	0	0
$509$	5.98516e18	0.676163	0.338081	−	0.941117i	$-0.390222\pi$
$509$	5.98516e18	0.676163	0.338081	+	0.941117i	$0.390222\pi$
$510$	0	0
$511$	2.97523e18i	0.327020i
$512$	0	0
$513$	−7.90485e18	−0.845419
$514$	0	0
$515$	1.46224e18i	0.152183i
$516$	0	0
$517$	2.69919e18	0.273400
$518$	0	0
$519$	9.92905e18i	0.978895i
$520$	0	0
$521$	−4.89021e18	−0.469314	−0.234657	−	0.972078i	$-0.575397\pi$
$521$	−4.89021e18	−0.469314	−0.234657	+	0.972078i	$0.575397\pi$
$522$	0	0
$523$	1.58741e19i	1.48312i	0.670884	+	0.741562i	$0.265915\pi$
$523$	1.58741e19i	1.48312i	−0.670884	+	0.741562i	$0.734085\pi$
$524$	0	0
$525$	1.92550e18	0.175157
$526$	0	0
$527$	− 1.01489e19i	− 0.898969i
$528$	0	0
$529$	−2.30729e19	−1.99028
$530$	0	0
$531$	8.49297e18i	0.713507i
$532$	0	0
$533$	−1.53258e18	−0.125410
$534$	0	0
$535$	2.56035e18i	0.204091i
$536$	0	0
$537$	9.27031e18	0.719903
$538$	0	0
$539$	3.40179e18i	0.257386i
$540$	0	0
$541$	1.51715e18	0.111853	0.0559266	−	0.998435i	$-0.482189\pi$
$541$	1.51715e18	0.111853	0.0559266	+	0.998435i	$0.482189\pi$
$542$	0	0
$543$	8.04021e18i	0.577655i
$544$	0	0
$545$	−4.19950e18	−0.294051
$546$	0	0
$547$	− 1.28798e19i	− 0.879020i	−0.898238	−	0.439510i	$-0.855152\pi$
$547$	− 1.28798e19i	− 0.879020i	0.898238	−	0.439510i	$-0.144848\pi$
$548$	0	0
$549$	−1.00689e19	−0.669850
$550$	0	0
$551$	− 1.64846e19i	− 1.06910i
$552$	0	0
$553$	2.98252e18	0.188585
$554$	0	0
$555$	6.81827e17i	0.0420362i
$556$	0	0
$557$	2.45410e19	1.47539	0.737693	−	0.675136i	$-0.235915\pi$
$557$	2.45410e19	1.47539	0.737693	+	0.675136i	$0.235915\pi$
$558$	0	0
$559$	1.02942e19i	0.603543i
$560$	0	0
$561$	1.69976e18	0.0971957
$562$	0	0
$563$	− 1.70247e19i	− 0.949555i	−0.880106	−	0.474778i	$-0.842529\pi$
$563$	− 1.70247e19i	− 0.949555i	0.880106	−	0.474778i	$-0.157471\pi$
$564$	0	0
$565$	3.05537e18	0.166236
$566$	0	0
$567$	1.68968e18i	0.0896856i
$568$	0	0
$569$	−5.30932e17	−0.0274949	−0.0137475	−	0.999905i	$-0.504376\pi$
$569$	−5.30932e17	−0.0274949	−0.0137475	+	0.999905i	$0.504376\pi$
$570$	0	0
$571$	5.04407e18i	0.254876i	0.991847	+	0.127438i	$0.0406753\pi$
$571$	5.04407e18i	0.254876i	−0.991847	+	0.127438i	$0.959325\pi$
$572$	0	0
$573$	1.13018e19	0.557269
$574$	0	0
$575$	− 3.44721e19i	− 1.65879i
$576$	0	0
$577$	1.49383e19	0.701567	0.350784	−	0.936457i	$-0.385915\pi$
$577$	1.49383e19	0.701567	0.350784	+	0.936457i	$0.385915\pi$
$578$	0	0
$579$	− 3.77347e18i	− 0.172978i
$580$	0	0
$581$	9.98268e18	0.446698
$582$	0	0
$583$	2.85929e18i	0.124905i
$584$	0	0
$585$	−1.41085e18	−0.0601714
$586$	0	0
$587$	− 4.37167e19i	− 1.82046i	−0.414107	−	0.910228i	$-0.635906\pi$
$587$	− 4.37167e19i	− 1.82046i	0.414107	−	0.910228i	$-0.364094\pi$
$588$	0	0
$589$	−3.26459e19	−1.32746
$590$	0	0
$591$	1.32276e19i	0.525252i
$592$	0	0
$593$	6.97741e18	0.270590	0.135295	−	0.990805i	$-0.456802\pi$
$593$	6.97741e18	0.270590	0.135295	+	0.990805i	$0.456802\pi$
$594$	0	0
$595$	− 1.18294e18i	− 0.0448067i
$596$	0	0
$597$	−8.11729e18	−0.300325
$598$	0	0
$599$	2.44252e19i	0.882776i	0.897316	+	0.441388i	$0.145514\pi$
$599$	2.44252e19i	0.882776i	−0.897316	+	0.441388i	$0.854486\pi$
$600$	0	0
$601$	−3.31228e18	−0.116952	−0.0584758	−	0.998289i	$-0.518624\pi$
$601$	−3.31228e18	−0.116952	−0.0584758	+	0.998289i	$0.518624\pi$
$602$	0	0
$603$	2.43579e19i	0.840269i
$604$	0	0
$605$	5.47320e18	0.184482
$606$	0	0
$607$	− 1.34355e19i	− 0.442521i	−0.975215	−	0.221260i	$-0.928983\pi$
$607$	− 1.34355e19i	− 0.442521i	0.975215	−	0.221260i	$-0.0710170\pi$
$608$	0	0
$609$	6.47033e18	0.208260
$610$	0	0
$611$	1.21471e19i	0.382107i
$612$	0	0
$613$	−2.65225e19	−0.815438	−0.407719	−	0.913107i	$-0.633676\pi$
$613$	−2.65225e19	−0.815438	−0.407719	+	0.913107i	$0.633676\pi$
$614$	0	0
$615$	1.07305e18i	0.0322473i
$616$	0	0
$617$	2.80647e19	0.824449	0.412225	−	0.911082i	$-0.364752\pi$
$617$	2.80647e19	0.824449	0.412225	+	0.911082i	$0.364752\pi$
$618$	0	0
$619$	5.98325e19i	1.71832i	0.511711	+	0.859158i	$0.329012\pi$
$619$	5.98325e19i	1.71832i	−0.511711	+	0.859158i	$0.670988\pi$
$620$	0	0
$621$	−5.55475e19	−1.55964
$622$	0	0
$623$	1.88395e19i	0.517192i
$624$	0	0
$625$	3.27618e19	0.879442
$626$	0	0
$627$	− 5.46759e18i	− 0.143524i
$628$	0	0
$629$	−9.86322e18	−0.253200
$630$	0	0
$631$	2.82990e19i	0.710503i	0.934771	+	0.355252i	$0.115605\pi$
$631$	2.82990e19i	0.710503i	−0.934771	+	0.355252i	$0.884395\pi$
$632$	0	0
$633$	−2.52402e19	−0.619822
$634$	0	0
$635$	− 9.27328e18i	− 0.222750i
$636$	0	0
$637$	−1.53090e19	−0.359726
$638$	0	0
$639$	2.08717e19i	0.479791i
$640$	0	0
$641$	1.73385e19	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$641$	1.73385e19	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$642$	0	0
$643$	2.40048e19i	0.528228i	0.964491	+	0.264114i	$0.0850795\pi$
$643$	2.40048e19i	0.528228i	−0.964491	+	0.264114i	$0.914921\pi$
$644$	0	0
$645$	7.20754e18	0.155192
$646$	0	0
$647$	− 8.12447e19i	− 1.71185i	−0.517103	−	0.855923i	$-0.672989\pi$
$647$	− 8.12447e19i	− 1.71185i	0.517103	−	0.855923i	$-0.327011\pi$
$648$	0	0
$649$	−1.39505e19	−0.287658
$650$	0	0
$651$	− 1.28137e19i	− 0.258588i
$652$	0	0
$653$	−3.14655e19	−0.621503	−0.310751	−	0.950491i	$-0.600581\pi$
$653$	−3.14655e19	−0.621503	−0.310751	+	0.950491i	$0.600581\pi$
$654$	0	0
$655$	− 1.71261e19i	− 0.331109i
$656$	0	0
$657$	3.59403e19	0.680180
$658$	0	0
$659$	4.94164e19i	0.915532i	0.889073	+	0.457766i	$0.151350\pi$
$659$	4.94164e19i	0.915532i	−0.889073	+	0.457766i	$0.848650\pi$
$660$	0	0
$661$	2.92188e19	0.529971	0.264986	−	0.964252i	$-0.414633\pi$
$661$	2.92188e19	0.529971	0.264986	+	0.964252i	$0.414633\pi$
$662$	0	0
$663$	7.64941e18i	0.135842i
$664$	0	0
$665$	−3.80515e18	−0.0661637
$666$	0	0
$667$	− 1.15838e20i	− 1.97228i
$668$	0	0
$669$	−3.70957e19	−0.618502
$670$	0	0
$671$	− 1.65392e19i	− 0.270058i
$672$	0	0
$673$	5.22348e19	0.835323	0.417662	−	0.908603i	$-0.362850\pi$
$673$	5.22348e19	0.835323	0.417662	+	0.908603i	$0.362850\pi$
$674$	0	0
$675$	− 5.52372e19i	− 0.865177i
$676$	0	0
$677$	9.70740e19	1.48930	0.744650	−	0.667455i	$-0.232616\pi$
$677$	9.70740e19	1.48930	0.744650	+	0.667455i	$0.232616\pi$
$678$	0	0
$679$	2.04602e19i	0.307484i
$680$	0	0
$681$	3.30832e19	0.487055
$682$	0	0
$683$	6.09514e16i	0 0.000879100i	1.00000		0.000439550i	$0.000139913\pi$
$683$	6.09514e16i	0 0.000879100i	−1.00000		0.000439550i	$0.999860\pi$
$684$	0	0
$685$	7.19257e18	0.101637
$686$	0	0
$687$	− 4.79775e19i	− 0.664264i
$688$	0	0
$689$	−1.28676e19	−0.174568
$690$	0	0
$691$	− 1.24977e19i	− 0.166143i	−0.996544	−	0.0830717i	$-0.973527\pi$
$691$	− 1.24977e19i	− 0.166143i	0.996544	−	0.0830717i	$-0.0264731\pi$
$692$	0	0
$693$	−5.72588e18	−0.0745949
$694$	0	0
$695$	− 2.76129e18i	− 0.0352548i
$696$	0	0
$697$	−1.55226e19	−0.194238
$698$	0	0
$699$	6.92331e19i	0.849128i
$700$	0	0
$701$	−2.81385e19	−0.338278	−0.169139	−	0.985592i	$-0.554099\pi$
$701$	−2.81385e19	−0.338278	−0.169139	+	0.985592i	$0.554099\pi$
$702$	0	0
$703$	3.17269e19i	0.373887i
$704$	0	0
$705$	8.50492e18	0.0982532
$706$	0	0
$707$	− 3.97173e19i	− 0.449826i
$708$	0	0
$709$	−2.81394e18	−0.0312457	−0.0156229	−	0.999878i	$-0.504973\pi$
$709$	−2.81394e18	−0.0312457	−0.0156229	+	0.999878i	$0.504973\pi$
$710$	0	0
$711$	− 3.60283e19i	− 0.392244i
$712$	0	0
$713$	−2.29403e20	−2.44891
$714$	0	0
$715$	− 2.31746e18i	− 0.0242588i
$716$	0	0
$717$	8.35780e19	0.857942
$718$	0	0
$719$	1.89059e20i	1.90324i	0.307272	+	0.951622i	$0.400584\pi$
$719$	1.89059e20i	1.90324i	−0.307272	+	0.951622i	$0.599416\pi$
$720$	0	0
$721$	−2.67066e19	−0.263677
$722$	0	0
$723$	3.29476e19i	0.319048i
$724$	0	0
$725$	1.15191e20	1.09408
$726$	0	0
$727$	− 4.59690e19i	− 0.428277i	−0.976803	−	0.214138i	$-0.931306\pi$
$727$	− 4.59690e19i	− 0.428277i	0.976803	−	0.214138i	$-0.0686943\pi$
$728$	0	0
$729$	−3.11168e19	−0.284382
$730$	0	0
$731$	1.04263e20i	0.934782i
$732$	0	0
$733$	−1.11814e20	−0.983488	−0.491744	−	0.870740i	$-0.663640\pi$
$733$	−1.11814e20	−0.983488	−0.491744	+	0.870740i	$0.663640\pi$
$734$	0	0
$735$	1.07188e19i	0.0924982i
$736$	0	0
$737$	−4.00101e19	−0.338764
$738$	0	0
$739$	1.16782e20i	0.970206i	0.874457	+	0.485103i	$0.161218\pi$
$739$	1.16782e20i	0.970206i	−0.874457	+	0.485103i	$0.838782\pi$
$740$	0	0
$741$	2.46058e19	0.200590
$742$	0	0
$743$	1.53371e20i	1.22694i	0.789720	+	0.613468i	$0.210226\pi$
$743$	1.53371e20i	1.22694i	−0.789720	+	0.613468i	$0.789774\pi$
$744$	0	0
$745$	−6.20929e18	−0.0487470
$746$	0	0
$747$	− 1.20589e20i	− 0.929103i
$748$	0	0
$749$	−4.67629e19	−0.353613
$750$	0	0
$751$	− 7.49724e19i	− 0.556444i	−0.960517	−	0.278222i	$-0.910255\pi$
$751$	− 7.49724e19i	− 0.556444i	0.960517	−	0.278222i	$-0.0897451\pi$
$752$	0	0
$753$	1.11353e20	0.811216
$754$	0	0
$755$	3.57053e19i	0.255331i
$756$	0	0
$757$	2.06086e20	1.44669	0.723347	−	0.690484i	$-0.242603\pi$
$757$	2.06086e20	1.44669	0.723347	+	0.690484i	$0.242603\pi$
$758$	0	0
$759$	− 3.84209e19i	− 0.264774i
$760$	0	0
$761$	−2.72514e20	−1.84373	−0.921863	−	0.387516i	$-0.873333\pi$
$761$	−2.72514e20	−1.84373	−0.921863	+	0.387516i	$0.873333\pi$
$762$	0	0
$763$	− 7.67006e19i	− 0.509480i
$764$	0	0
$765$	−1.42897e19	−0.0931950
$766$	0	0
$767$	− 6.27814e19i	− 0.402034i
$768$	0	0
$769$	4.53426e19	0.285116	0.142558	−	0.989786i	$-0.454467\pi$
$769$	4.53426e19	0.285116	0.142558	+	0.989786i	$0.454467\pi$
$770$	0	0
$771$	6.44015e19i	0.397663i
$772$	0	0
$773$	1.63435e20	0.991033	0.495517	−	0.868598i	$-0.334979\pi$
$773$	1.63435e20	0.991033	0.495517	+	0.868598i	$0.334979\pi$
$774$	0	0
$775$	− 2.28121e20i	− 1.35848i
$776$	0	0
$777$	−1.24530e19	−0.0728330
$778$	0	0
$779$	4.99314e19i	0.286821i
$780$	0	0
$781$	−3.42837e19	−0.193433
$782$	0	0
$783$	− 1.85616e20i	− 1.02868i
$784$	0	0
$785$	−1.66690e19	−0.0907449
$786$	0	0
$787$	− 3.43175e20i	− 1.83524i	−0.397459	−	0.917620i	$-0.630108\pi$
$787$	− 3.43175e20i	− 1.83524i	0.397459	−	0.917620i	$-0.369892\pi$
$788$	0	0
$789$	8.16094e19	0.428747
$790$	0	0
$791$	5.58041e19i	0.288025i
$792$	0	0
$793$	7.44312e19	0.377436
$794$	0	0
$795$	9.00940e18i	0.0448875i
$796$	0	0
$797$	7.32861e19	0.358768	0.179384	−	0.983779i	$-0.442590\pi$
$797$	7.32861e19	0.358768	0.179384	+	0.983779i	$0.442590\pi$
$798$	0	0
$799$	1.23031e20i	0.591817i
$800$	0	0
$801$	2.27577e20	1.07573
$802$	0	0
$803$	5.90353e19i	0.274222i
$804$	0	0
$805$	−2.67389e19	−0.122060
$806$	0	0
$807$	− 5.25768e19i	− 0.235874i
$808$	0	0
$809$	−2.14949e20	−0.947754	−0.473877	−	0.880591i	$-0.657146\pi$
$809$	−2.14949e20	−0.947754	−0.473877	+	0.880591i	$0.657146\pi$
$810$	0	0
$811$	5.26836e19i	0.228312i	0.993463	+	0.114156i	$0.0364164\pi$
$811$	5.26836e19i	0.228312i	−0.993463	+	0.114156i	$0.963584\pi$
$812$	0	0
$813$	−1.74845e20	−0.744766
$814$	0	0
$815$	3.02774e19i	0.126770i
$816$	0	0
$817$	3.35383e20	1.38034
$818$	0	0
$819$	− 2.57681e19i	− 0.104255i
$820$	0	0
$821$	1.38293e19	0.0550046	0.0275023	−	0.999622i	$-0.491245\pi$
$821$	1.38293e19	0.0550046	0.0275023	+	0.999622i	$0.491245\pi$
$822$	0	0
$823$	2.64322e20i	1.03356i	0.856119	+	0.516779i	$0.172869\pi$
$823$	2.64322e20i	1.03356i	−0.856119	+	0.516779i	$0.827131\pi$
$824$	0	0
$825$	3.82062e19	0.146878
$826$	0	0
$827$	1.39882e19i	0.0528718i	0.999651	+	0.0264359i	$0.00841580\pi$
$827$	1.39882e19i	0.0528718i	−0.999651	+	0.0264359i	$0.991584\pi$
$828$	0	0
$829$	−3.02439e20	−1.12397	−0.561987	−	0.827146i	$-0.689963\pi$
$829$	−3.02439e20	−1.12397	−0.561987	+	0.827146i	$0.689963\pi$
$830$	0	0
$831$	− 1.37446e20i	− 0.502255i
$832$	0	0
$833$	−1.55056e20	−0.557153
$834$	0	0
$835$	− 1.82382e19i	− 0.0644430i
$836$	0	0
$837$	−3.67590e20	−1.27728
$838$	0	0
$839$	4.37264e20i	1.49420i	0.664710	+	0.747102i	$0.268555\pi$
$839$	4.37264e20i	1.49420i	−0.664710	+	0.747102i	$0.731445\pi$
$840$	0	0
$841$	8.95210e19	0.300852
$842$	0	0
$843$	1.00723e20i	0.332918i
$844$	0	0
$845$	−5.16580e19	−0.167935
$846$	0	0
$847$	9.99638e19i	0.319638i
$848$	0	0
$849$	−1.03658e20	−0.326024
$850$	0	0
$851$	2.22946e20i	0.689751i
$852$	0	0
$853$	−1.99355e20	−0.606714	−0.303357	−	0.952877i	$-0.598107\pi$
$853$	−1.99355e20	−0.606714	−0.303357	+	0.952877i	$0.598107\pi$
$854$	0	0
$855$	4.59655e19i	0.137616i
$856$	0	0
$857$	1.85598e20	0.546648	0.273324	−	0.961922i	$-0.411877\pi$
$857$	1.85598e20	0.546648	0.273324	+	0.961922i	$0.411877\pi$
$858$	0	0
$859$	9.94715e19i	0.288235i	0.989561	+	0.144117i	$0.0460343\pi$
$859$	9.94715e19i	0.288235i	−0.989561	+	0.144117i	$0.953966\pi$
$860$	0	0
$861$	−1.95984e19	−0.0558726
$862$	0	0
$863$	1.02123e19i	0.0286450i	0.999897	+	0.0143225i	$0.00455915\pi$
$863$	1.02123e19i	0.0286450i	−0.999897	+	0.0143225i	$0.995441\pi$
$864$	0	0
$865$	1.37111e20	0.378408
$866$	0	0
$867$	− 1.14795e20i	− 0.311738i
$868$	0	0
$869$	5.91798e19	0.158138
$870$	0	0
$871$	− 1.80057e20i	− 0.473460i
$872$	0	0
$873$	2.47156e20	0.639546
$874$	0	0
$875$	− 5.43081e19i	− 0.138296i
$876$	0	0
$877$	−1.42181e20	−0.356322	−0.178161	−	0.984001i	$-0.557015\pi$
$877$	−1.42181e20	−0.356322	−0.178161	+	0.984001i	$0.557015\pi$
$878$	0	0
$879$	− 1.98261e18i	− 0.00489006i
$880$	0	0
$881$	5.76942e19	0.140056	0.0700278	−	0.997545i	$-0.477691\pi$
$881$	5.76942e19	0.140056	0.0700278	+	0.997545i	$0.477691\pi$
$882$	0	0
$883$	1.96085e20i	0.468511i	0.972175	+	0.234256i	$0.0752652\pi$
$883$	1.96085e20i	0.468511i	−0.972175	+	0.234256i	$0.924735\pi$
$884$	0	0
$885$	−4.39569e19	−0.103377
$886$	0	0
$887$	− 5.69967e20i	− 1.31942i	−0.751518	−	0.659712i	$-0.770678\pi$
$887$	− 5.69967e20i	− 1.31942i	0.751518	−	0.659712i	$-0.229322\pi$
$888$	0	0
$889$	1.69369e20	0.385942
$890$	0	0
$891$	3.35270e19i	0.0752058i
$892$	0	0
$893$	3.95753e20	0.873905
$894$	0	0
$895$	− 1.28015e20i	− 0.278291i
$896$	0	0
$897$	1.72905e20	0.370051
$898$	0	0
$899$	− 7.66565e20i	− 1.61522i
$900$	0	0
$901$	−1.30329e20	−0.270375
$902$	0	0
$903$	1.31640e20i	0.268890i
$904$	0	0
$905$	1.11028e20	0.223302
$906$	0	0
$907$	− 4.52441e19i	− 0.0896007i	−0.998996	−	0.0448004i	$-0.985735\pi$
$907$	− 4.52441e19i	− 0.0896007i	0.998996	−	0.0448004i	$-0.0142652\pi$
$908$	0	0
$909$	−4.79779e20	−0.935607
$910$	0	0
$911$	− 4.23762e20i	− 0.813755i	−0.913483	−	0.406877i	$-0.866618\pi$
$911$	− 4.23762e20i	− 0.813755i	0.913483	−	0.406877i	$-0.133382\pi$
$912$	0	0
$913$	1.98079e20	0.374578
$914$	0	0
$915$	− 5.21137e19i	− 0.0970519i
$916$	0	0
$917$	3.12796e20	0.573688
$918$	0	0
$919$	− 4.46742e20i	− 0.806952i	−0.914990	−	0.403476i	$-0.867802\pi$
$919$	− 4.46742e20i	− 0.806952i	0.914990	−	0.403476i	$-0.132198\pi$
$920$	0	0
$921$	1.66651e20	0.296477
$922$	0	0
$923$	− 1.54287e20i	− 0.270344i
$924$	0	0
$925$	−2.21700e20	−0.382625
$926$	0	0
$927$	3.22611e20i	0.548430i
$928$	0	0
$929$	−4.65421e20	−0.779356	−0.389678	−	0.920951i	$-0.627414\pi$
$929$	−4.65421e20	−0.779356	−0.389678	+	0.920951i	$0.627414\pi$
$930$	0	0
$931$	4.98768e20i	0.822717i
$932$	0	0
$933$	1.64584e20	0.267433
$934$	0	0
$935$	− 2.34722e19i	− 0.0375726i
$936$	0	0
$937$	5.17813e18	0.00816575	0.00408287	−	0.999992i	$-0.498700\pi$
$937$	5.17813e18	0.00816575	0.00408287	+	0.999992i	$0.498700\pi$
$938$	0	0
$939$	2.32133e20i	0.360643i
$940$	0	0
$941$	9.54113e20	1.46040	0.730201	−	0.683232i	$-0.239426\pi$
$941$	9.54113e20	1.46040	0.730201	+	0.683232i	$0.239426\pi$
$942$	0	0
$943$	3.50869e20i	0.529131i
$944$	0	0
$945$	−4.28457e19	−0.0636626
$946$	0	0
$947$	− 2.26045e19i	− 0.0330937i	−0.999863	−	0.0165468i	$-0.994733\pi$
$947$	− 2.26045e19i	− 0.0330937i	0.999863	−	0.0165468i	$-0.00526726\pi$
$948$	0	0
$949$	−2.65676e20	−0.383256
$950$	0	0
$951$	6.43062e20i	0.914091i
$952$	0	0
$953$	−8.21994e20	−1.15138	−0.575689	−	0.817668i	$-0.695266\pi$
$953$	−8.21994e20	−1.15138	−0.575689	+	0.817668i	$0.695266\pi$
$954$	0	0
$955$	− 1.56068e20i	− 0.215422i
$956$	0	0
$957$	1.28386e20	0.174636
$958$	0	0
$959$	1.31367e20i	0.176098i
$960$	0	0
$961$	−7.61149e20	−1.00556
$962$	0	0
$963$	5.64887e20i	0.735491i
$964$	0	0
$965$	−5.21082e19	−0.0668675
$966$	0	0
$967$	1.31338e21i	1.66114i	0.556916	+	0.830569i	$0.311985\pi$
$967$	1.31338e21i	1.66114i	−0.556916	+	0.830569i	$0.688015\pi$
$968$	0	0
$969$	2.49217e20	0.310679
$970$	0	0
$971$	− 1.21674e20i	− 0.149507i	−0.997202	−	0.0747537i	$-0.976183\pi$
$971$	− 1.21674e20i	− 0.149507i	0.997202	−	0.0747537i	$-0.0238170\pi$
$972$	0	0
$973$	5.04328e19	0.0610834
$974$	0	0
$975$	1.71939e20i	0.205278i
$976$	0	0
$977$	−1.41280e21	−1.66272	−0.831358	−	0.555737i	$-0.812436\pi$
$977$	−1.41280e21	−1.66272	−0.831358	+	0.555737i	$0.812436\pi$
$978$	0	0
$979$	3.73817e20i	0.433691i
$980$	0	0
$981$	−9.26530e20	−1.05968
$982$	0	0
$983$	− 1.73927e21i	− 1.96107i	−0.196352	−	0.980534i	$-0.562909\pi$
$983$	− 1.73927e21i	− 1.96107i	0.196352	−	0.980534i	$-0.437091\pi$
$984$	0	0
$985$	1.82661e20	0.203045
$986$	0	0
$987$	1.55336e20i	0.170236i
$988$	0	0
$989$	2.35675e21	2.54647
$990$	0	0
$991$	2.60257e20i	0.277260i	0.990344	+	0.138630i	$0.0442699\pi$
$991$	2.60257e20i	0.277260i	−0.990344	+	0.138630i	$0.955730\pi$
$992$	0	0
$993$	−1.21259e20	−0.127371
$994$	0	0
$995$	1.12093e20i	0.116096i
$996$	0	0
$997$	−7.30975e20	−0.746511	−0.373256	−	0.927728i	$-0.621759\pi$
$997$	−7.30975e20	−0.746511	−0.373256	+	0.927728i	$0.621759\pi$
$998$	0	0
$999$	3.57243e20i	0.359753i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.15.c.c.15.3	yes	4
3.2	odd	2		144.15.g.g.127.2		4
4.3	odd	2	inner	16.15.c.c.15.2	✓	4
8.3	odd	2		64.15.c.c.63.3		4
8.5	even	2		64.15.c.c.63.2		4
12.11	even	2		144.15.g.g.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.15.c.c.15.2	✓	4	4.3	odd	2	inner
16.15.c.c.15.3	yes	4	1.1	even	1	trivial
64.15.c.c.63.2		4	8.5	even	2
64.15.c.c.63.3		4	8.3	odd	2
144.15.g.g.127.1		4	12.11	even	2
144.15.g.g.127.2		4	3.2	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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1141.90i q^{3} +15768.7 q^{5} +288003. i q^{7} +3.47902e6 q^{9} +O(q^{10})\)	\(q+1141.90i q^{3} +15768.7 q^{5} +288003. i q^{7} +3.47902e6 q^{9} +5.71462e6i q^{11} -2.57175e7 q^{13} +1.80063e7i q^{15} -2.60477e8 q^{17} +8.37875e8i q^{19} -3.28872e8 q^{21} +5.88776e9i q^{23} -5.85486e9 q^{25} +9.43440e9i q^{27} -1.96743e10 q^{29} +3.89627e10i q^{31} -6.52555e9 q^{33} +4.54143e9i q^{35} +3.78659e10 q^{37} -2.93669e10i q^{39} +5.95929e10 q^{41} -4.00278e11i q^{43} +5.48596e10 q^{45} -4.72330e11i q^{47} +5.95277e11 q^{49} -2.97440e11i q^{51} +5.00346e11 q^{53} +9.01121e10i q^{55} -9.56772e11 q^{57} +2.44119e12i q^{59} -2.89419e12 q^{61} +1.00197e12i q^{63} -4.05531e11 q^{65} +7.00135e12i q^{67} -6.72326e12 q^{69} +5.99930e12i q^{71} +1.03306e13 q^{73} -6.68569e12i q^{75} -1.64583e12 q^{77} -1.03559e13i q^{79} +5.86688e12 q^{81} -3.46617e13i q^{83} -4.10738e12 q^{85} -2.24662e13i q^{87} +6.54141e13 q^{89} -7.40671e12i q^{91} -4.44917e13 q^{93} +1.32122e13i q^{95} +7.10418e13 q^{97} +1.98813e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 87000 q^{5} - 20000796 q^{9}+O(q^{10}) \) 4 * q - 87000 * q^5 - 20000796 * q^9	\( 4 q - 87000 q^{5} - 20000796 q^{9} + 66864616 q^{13} + 13116552 q^{17} - 9582203904 q^{21} - 16891205140 q^{25} - 17501785752 q^{29} - 118197379584 q^{33} - 139985734360 q^{37} + 662088410568 q^{41} + 1707534576360 q^{45} + 366399422788 q^{49} - 2975173624152 q^{53} + 3730435570176 q^{57} - 6624970851736 q^{61} - 7822522322160 q^{65} - 14338843057152 q^{69} - 12033132589112 q^{73} - 28983083609088 q^{77} + 200426504614596 q^{81} - 39868454746800 q^{85} + 106307014497096 q^{89} - 166229577732096 q^{93} + 95793165936136 q^{97}+O(q^{100}) \) 4 * q - 87000 * q^5 - 20000796 * q^9 + 66864616 * q^13 + 13116552 * q^17 - 9582203904 * q^21 - 16891205140 * q^25 - 17501785752 * q^29 - 118197379584 * q^33 - 139985734360 * q^37 + 662088410568 * q^41 + 1707534576360 * q^45 + 366399422788 * q^49 - 2975173624152 * q^53 + 3730435570176 * q^57 - 6624970851736 * q^61 - 7822522322160 * q^65 - 14338843057152 * q^69 - 12033132589112 * q^73 - 28983083609088 * q^77 + 200426504614596 * q^81 - 39868454746800 * q^85 + 106307014497096 * q^89 - 166229577732096 * q^93 + 95793165936136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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