LMFDB - Embedded newform 18.15.b.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [18,15,Mod(17,18)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(18, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("18.17");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$N$	$=$	$18 = 2 \cdot 3^{2}$
Weight:	$k$	$=$	$15$
Character orbit:	$[\chi]$	$=$	18.b (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:	$22.3792142673$
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:	$x^{4} - 2x^{3} - 44875x^{2} + 44876x + 503643366$ x^4 - 2x^3 - 44875x^2 + 44876*x + 503643366
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{7}\cdot 3^{8}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$150.301 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	18.17
Dual form			18.15.b.b.17.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-90.5097i q^{2} -8192.00 q^{4} +141636. i q^{5} -305793. q^{7} +741455. i q^{8} +1.28194e7 q^{10} -3.26518e7i q^{11} -3.64244e7 q^{13} +2.76772e7i q^{14} +6.71089e7 q^{16} -4.11602e8i q^{17} +5.42773e8 q^{19} -1.16028e9i q^{20} -2.95530e9 q^{22} +1.04179e9i q^{23} -1.39573e10 q^{25} +3.29676e9i q^{26} +2.50505e9 q^{28} -3.04021e10i q^{29} -3.36638e10 q^{31} -6.07400e9i q^{32} -3.72540e10 q^{34} -4.33113e10i q^{35} -2.33075e10 q^{37} -4.91262e10i q^{38} -1.05017e11 q^{40} -1.96280e11i q^{41} +4.39368e11 q^{43} +2.67483e11i q^{44} +9.42917e10 q^{46} +6.50358e11i q^{47} -5.84714e11 q^{49} +1.26327e12i q^{50} +2.98389e11 q^{52} -9.88636e11i q^{53} +4.62467e12 q^{55} -2.26731e11i q^{56} -2.75169e12 q^{58} -3.21661e11i q^{59} +2.51765e12 q^{61} +3.04690e12i q^{62} -5.49756e11 q^{64} -5.15901e12i q^{65} -7.51002e12 q^{67} +3.37184e12i q^{68} -3.92009e12 q^{70} -1.69163e13i q^{71} -5.98095e12 q^{73} +2.10955e12i q^{74} -4.44639e12 q^{76} +9.98467e12i q^{77} -2.06435e13 q^{79} +9.50504e12i q^{80} -1.77652e13 q^{82} -1.20159e12i q^{83} +5.82977e13 q^{85} -3.97671e13i q^{86} +2.42098e13 q^{88} -7.62718e12i q^{89} +1.11383e13 q^{91} -8.53431e12i q^{92} +5.88637e13 q^{94} +7.68762e13i q^{95} -4.73358e13 q^{97} +5.29223e13i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 32768 q^{4} + 2659664 q^{7} + 1577472 q^{10} + 60092672 q^{13} + 268435456 q^{16} - 220734784 q^{19} - 6354180096 q^{22} - 51043883876 q^{25} - 21787967488 q^{28} - 106337705584 q^{31} - 139572822528 q^{34}+ \cdots + 67539358987904 q^{97}+O(q^{100})$ 4 * q - 32768 * q^4 + 2659664 * q^7 + 1577472 * q^10 + 60092672 * q^13 + 268435456 * q^16 - 220734784 * q^19 - 6354180096 * q^22 - 51043883876 * q^25 - 21787967488 * q^28 - 106337705584 * q^31 - 139572822528 * q^34 - 130303339336 * q^37 - 12922650624 * q^40 + 769936374560 * q^43 - 549743314944 * q^46 + 2824661151228 * q^49 - 492279169024 * q^52 + 8597914848864 * q^55 - 3060909556224 * q^58 + 13516759183880 * q^61 - 2199023255552 * q^64 - 17563798808608 * q^67 - 47195597395968 * q^70 - 40575625349824 * q^73 + 1808259350528 * q^76 - 22017761826064 * q^79 - 1842848730624 * q^82 + 21041704028232 * q^85 + 52053443346432 * q^88 + 239718894698752 * q^91 + 126064058603520 * q^94 + 67539358987904 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$ .

$n$	$11$
$\chi(n)$	$-1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	− 90.5097i	− 0.707107i
$2$	− 90.5097i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8192.00	−0.500000
$5$	141636.i	1.81294i	0.422268	+	0.906471i	$0.361234\pi$
$5$	141636.i	1.81294i	−0.422268	+	0.906471i	$0.638766\pi$
$6$	0	0
$7$	−305793.	−0.371313	−0.185657	−	0.982615i	$-0.559441\pi$
$7$	−305793.	−0.371313	−0.185657	+	0.982615i	$0.559441\pi$
$8$	741455.i	0.353553i
$9$	0	0
$10$	1.28194e7	1.28194
$11$	− 3.26518e7i	− 1.67555i	−0.546013	−	0.837777i	$-0.683855\pi$
$11$	− 3.26518e7i	− 1.67555i	0.546013	−	0.837777i	$-0.316145\pi$
$12$	0	0
$13$	−3.64244e7	−0.580482	−0.290241	−	0.956954i	$-0.593735\pi$
$13$	−3.64244e7	−0.580482	−0.290241	+	0.956954i	$0.593735\pi$
$14$	2.76772e7i	0.262558i
$15$	0	0
$16$	6.71089e7	0.250000
$17$	− 4.11602e8i	− 1.00308i	−0.865135	−	0.501539i	$-0.832767\pi$
$17$	− 4.11602e8i	− 1.00308i	0.865135	−	0.501539i	$-0.167233\pi$
$18$	0	0
$19$	5.42773e8	0.607215	0.303608	−	0.952797i	$-0.401809\pi$
$19$	5.42773e8	0.607215	0.303608	+	0.952797i	$0.401809\pi$
$20$	− 1.16028e9i	− 0.906471i
$21$	0	0
$22$	−2.95530e9	−1.18479
$23$	1.04179e9i	0.305973i	0.988228	+	0.152987i	$0.0488892\pi$
$23$	1.04179e9i	0.305973i	−0.988228	+	0.152987i	$0.951111\pi$
$24$	0	0
$25$	−1.39573e10	−2.28676
$26$	3.29676e9i	0.410463i
$27$	0	0
$28$	2.50505e9	0.185657
$29$	− 3.04021e10i	− 1.76245i	−0.472693	−	0.881227i	$-0.656718\pi$
$29$	− 3.04021e10i	− 1.76245i	0.472693	−	0.881227i	$-0.343282\pi$
$30$	0	0
$31$	−3.36638e10	−1.22358	−0.611789	−	0.791021i	$-0.709550\pi$
$31$	−3.36638e10	−1.22358	−0.611789	+	0.791021i	$0.709550\pi$
$32$	− 6.07400e9i	− 0.176777i
$33$	0	0
$34$	−3.72540e10	−0.709284
$35$	− 4.33113e10i	− 0.673170i
$36$	0	0
$37$	−2.33075e10	−0.245518	−0.122759	−	0.992436i	$-0.539174\pi$
$37$	−2.33075e10	−0.245518	−0.122759	+	0.992436i	$0.539174\pi$
$38$	− 4.91262e10i	− 0.429366i
$39$	0	0
$40$	−1.05017e11	−0.640972
$41$	− 1.96280e11i	− 1.00783i	−0.863752	−	0.503917i	$-0.831892\pi$
$41$	− 1.96280e11i	− 1.00783i	0.863752	−	0.503917i	$-0.168108\pi$
$42$	0	0
$43$	4.39368e11	1.61640	0.808201	−	0.588906i	$-0.200441\pi$
$43$	4.39368e11	1.61640	0.808201	+	0.588906i	$0.200441\pi$
$44$	2.67483e11i	0.837777i
$45$	0	0
$46$	9.42917e10	0.216356
$47$	6.50358e11i	1.28371i	0.766825	+	0.641856i	$0.221835\pi$
$47$	6.50358e11i	1.28371i	−0.766825	+	0.641856i	$0.778165\pi$
$48$	0	0
$49$	−5.84714e11	−0.862126
$50$	1.26327e12i	1.61698i
$51$	0	0
$52$	2.98389e11	0.290241
$53$	− 9.88636e11i	− 0.841599i	−0.907154	−	0.420800i	$-0.861750\pi$
$53$	− 9.88636e11i	− 0.841599i	0.907154	−	0.420800i	$-0.138250\pi$
$54$	0	0
$55$	4.62467e12	3.03768
$56$	− 2.26731e11i	− 0.131279i
$57$	0	0
$58$	−2.75169e12	−1.24624
$59$	− 3.21661e11i	− 0.129251i	−0.997910	−	0.0646255i	$-0.979415\pi$
$59$	− 3.21661e11i	− 0.129251i	0.997910	−	0.0646255i	$-0.0205853\pi$
$60$	0	0
$61$	2.51765e12	0.801101	0.400551	−	0.916275i	$-0.368819\pi$
$61$	2.51765e12	0.801101	0.400551	+	0.916275i	$0.368819\pi$
$62$	3.04690e12i	0.865200i
$63$	0	0
$64$	−5.49756e11	−0.125000
$65$	− 5.15901e12i	− 1.05238i
$66$	0	0
$67$	−7.51002e12	−1.23913	−0.619566	−	0.784945i	$-0.712691\pi$
$67$	−7.51002e12	−1.23913	−0.619566	+	0.784945i	$0.712691\pi$
$68$	3.37184e12i	0.501539i
$69$	0	0
$70$	−3.92009e12	−0.476003
$71$	− 1.69163e13i	− 1.85993i	−0.367646	−	0.929966i	$-0.619836\pi$
$71$	− 1.69163e13i	− 1.85993i	0.367646	−	0.929966i	$-0.380164\pi$
$72$	0	0
$73$	−5.98095e12	−0.541390	−0.270695	−	0.962665i	$-0.587254\pi$
$73$	−5.98095e12	−0.541390	−0.270695	+	0.962665i	$0.587254\pi$
$74$	2.10955e12i	0.173608i
$75$	0	0
$76$	−4.44639e12	−0.303608
$77$	9.98467e12i	0.622155i
$78$	0	0
$79$	−2.06435e13	−1.07496	−0.537482	−	0.843275i	$-0.680624\pi$
$79$	−2.06435e13	−1.07496	−0.537482	+	0.843275i	$0.680624\pi$
$80$	9.50504e12i	0.453236i
$81$	0	0
$82$	−1.77652e13	−0.712646
$83$	− 1.20159e12i	− 0.0442800i	−0.999755	−	0.0221400i	$-0.992952\pi$
$83$	− 1.20159e12i	− 0.0442800i	0.999755	−	0.0221400i	$-0.00704796\pi$
$84$	0	0
$85$	5.82977e13	1.81852
$86$	− 3.97671e13i	− 1.14297i
$87$	0	0
$88$	2.42098e13	0.592397
$89$	− 7.62718e12i	− 0.172438i	−0.996276	−	0.0862192i	$-0.972521\pi$
$89$	− 7.62718e12i	− 0.172438i	0.996276	−	0.0862192i	$-0.0274786\pi$
$90$	0	0
$91$	1.11383e13	0.215541
$92$	− 8.53431e12i	− 0.152987i
$93$	0	0
$94$	5.88637e13	0.907722
$95$	7.68762e13i	1.10085i
$96$	0	0
$97$	−4.73358e13	−0.585852	−0.292926	−	0.956135i	$-0.594629\pi$
$97$	−4.73358e13	−0.585852	−0.292926	+	0.956135i	$0.594629\pi$
$98$	5.29223e13i	0.609615i
$99$	0	0
$100$	1.14338e14	1.14338
$101$	1.72827e14i	1.61198i	0.591926	+	0.805992i	$0.298368\pi$
$101$	1.72827e14i	1.61198i	−0.591926	+	0.805992i	$0.701632\pi$
$102$	0	0
$103$	1.89084e13	0.153743	0.0768715	−	0.997041i	$-0.475507\pi$
$103$	1.89084e13	0.153743	0.0768715	+	0.997041i	$0.475507\pi$
$104$	− 2.70070e13i	− 0.205231i
$105$	0	0
$106$	−8.94811e13	−0.595101
$107$	5.00344e13i	0.311589i	0.987790	+	0.155794i	$0.0497937\pi$
$107$	5.00344e13i	0.311589i	−0.987790	+	0.155794i	$0.950206\pi$
$108$	0	0
$109$	−2.57177e13	−0.140684	−0.0703422	−	0.997523i	$-0.522409\pi$
$109$	−2.57177e13	−0.140684	−0.0703422	+	0.997523i	$0.522409\pi$
$110$	− 4.18578e14i	− 2.14796i
$111$	0	0
$112$	−2.05214e13	−0.0928283
$113$	− 6.88029e13i	− 0.292454i	−0.989251	−	0.146227i	$-0.953287\pi$
$113$	− 6.88029e13i	− 0.292454i	0.989251	−	0.146227i	$-0.0467130\pi$
$114$	0	0
$115$	−1.47555e14	−0.554712
$116$	2.49054e14i	0.881227i
$117$	0	0
$118$	−2.91134e13	−0.0913942
$119$	1.25865e14i	0.372457i
$120$	0	0
$121$	−6.86390e14	−1.80748
$122$	− 2.27872e14i	− 0.566464i
$123$	0	0
$124$	2.75774e14	0.611789
$125$	− 1.11238e15i	− 2.33282i
$126$	0	0
$127$	5.48895e14	1.03006	0.515031	−	0.857171i	$-0.327780\pi$
$127$	5.48895e14	1.03006	0.515031	+	0.857171i	$0.327780\pi$
$128$	4.97582e13i	0.0883883i
$129$	0	0
$130$	−4.66940e14	−0.744145
$131$	− 1.05117e14i	− 0.158773i	−0.996844	−	0.0793863i	$-0.974704\pi$
$131$	− 1.05117e14i	− 0.158773i	0.996844	−	0.0793863i	$-0.0252961\pi$
$132$	0	0
$133$	−1.65976e14	−0.225467
$134$	6.79729e14i	0.876198i
$135$	0	0
$136$	3.05185e14	0.354642
$137$	6.72926e14i	0.742888i	0.928455	+	0.371444i	$0.121137\pi$
$137$	6.72926e14i	0.742888i	−0.928455	+	0.371444i	$0.878863\pi$
$138$	0	0
$139$	−7.24414e14	−0.722575	−0.361288	−	0.932454i	$-0.617663\pi$
$139$	−7.24414e14	−0.722575	−0.361288	+	0.932454i	$0.617663\pi$
$140$	3.54806e14i	0.336585i
$141$	0	0
$142$	−1.53109e15	−1.31517
$143$	1.18932e15i	0.972628i
$144$	0	0
$145$	4.30604e15	3.19523
$146$	5.41334e14i	0.382821i
$147$	0	0
$148$	1.90935e14	0.122759
$149$	3.96437e14i	0.243148i	0.992582	+	0.121574i	$0.0387941\pi$
$149$	3.96437e14i	0.243148i	−0.992582	+	0.121574i	$0.961206\pi$
$150$	0	0
$151$	7.45907e14	0.416722	0.208361	−	0.978052i	$-0.433187\pi$
$151$	7.45907e14	0.416722	0.208361	+	0.978052i	$0.433187\pi$
$152$	4.02442e14i	0.214683i
$153$	0	0
$154$	9.03710e14	0.439930
$155$	− 4.76801e15i	− 2.21827i
$156$	0	0
$157$	1.86875e15	0.794791	0.397395	−	0.917648i	$-0.369914\pi$
$157$	1.86875e15	0.794791	0.397395	+	0.917648i	$0.369914\pi$
$158$	1.86844e15i	0.760114i
$159$	0	0
$160$	8.60298e14	0.320486
$161$	− 3.18570e14i	− 0.113612i
$162$	0	0
$163$	−5.26008e15	−1.72060	−0.860299	−	0.509790i	$-0.829723\pi$
$163$	−5.26008e15	−1.72060	−0.860299	+	0.509790i	$0.829723\pi$
$164$	1.60793e15i	0.503917i
$165$	0	0
$166$	−1.08755e14	−0.0313107
$167$	3.79444e15i	1.04745i	0.851888	+	0.523723i	$0.175457\pi$
$167$	3.79444e15i	1.04745i	−0.851888	+	0.523723i	$0.824543\pi$
$168$	0	0
$169$	−2.61064e15	−0.663041
$170$	− 5.27651e15i	− 1.28589i
$171$	0	0
$172$	−3.59930e15	−0.808201
$173$	− 5.74435e15i	− 1.23856i	−0.785169	−	0.619281i	$-0.787424\pi$
$173$	− 5.74435e15i	− 1.23856i	0.785169	−	0.619281i	$-0.212576\pi$
$174$	0	0
$175$	4.26803e15	0.849104
$176$	− 2.19122e15i	− 0.418888i
$177$	0	0
$178$	−6.90334e14	−0.121932
$179$	7.92944e15i	1.34670i	0.739323	+	0.673351i	$0.235146\pi$
$179$	7.92944e15i	1.34670i	−0.739323	+	0.673351i	$0.764854\pi$
$180$	0	0
$181$	−6.66038e15	−1.04652	−0.523262	−	0.852172i	$-0.675285\pi$
$181$	−6.66038e15	−1.04652	−0.523262	+	0.852172i	$0.675285\pi$
$182$	− 1.00812e15i	− 0.152410i
$183$	0	0
$184$	−7.72438e14	−0.108178
$185$	− 3.30119e15i	− 0.445110i
$186$	0	0
$187$	−1.34395e16	−1.68071
$188$	− 5.32774e15i	− 0.641856i
$189$	0	0
$190$	6.95804e15	0.778416
$191$	6.78546e15i	0.731721i	0.930670	+	0.365861i	$0.119225\pi$
$191$	6.78546e15i	0.731721i	−0.930670	+	0.365861i	$0.880775\pi$
$192$	0	0
$193$	−4.51288e15	−0.452431	−0.226216	−	0.974077i	$-0.572635\pi$
$193$	−4.51288e15	−0.452431	−0.226216	+	0.974077i	$0.572635\pi$
$194$	4.28435e15i	0.414260i
$195$	0	0
$196$	4.78998e15	0.431063
$197$	− 2.05504e15i	− 0.178466i	−0.996011	−	0.0892332i	$-0.971558\pi$
$197$	− 2.05504e15i	− 0.178466i	0.996011	−	0.0892332i	$-0.0284416\pi$
$198$	0	0
$199$	1.42689e16	1.15457	0.577285	−	0.816543i	$-0.304112\pi$
$199$	1.42689e16	1.15457	0.577285	+	0.816543i	$0.304112\pi$
$200$	− 1.03487e16i	− 0.808492i
$201$	0	0
$202$	1.56425e16	1.13985
$203$	9.29674e15i	0.654423i
$204$	0	0
$205$	2.78003e16	1.82714
$206$	− 1.71140e15i	− 0.108713i
$207$	0	0
$208$	−2.44440e15	−0.145121
$209$	− 1.77225e16i	− 1.01742i
$210$	0	0
$211$	−1.56308e16	−0.839465	−0.419733	−	0.907648i	$-0.637876\pi$
$211$	−1.56308e16	−0.839465	−0.419733	+	0.907648i	$0.637876\pi$
$212$	8.09891e15i	0.420800i
$213$	0	0
$214$	4.52859e15	0.220327
$215$	6.22304e16i	2.93044i
$216$	0	0
$217$	1.02941e16	0.454331
$218$	2.32770e15i	0.0994789i
$219$	0	0
$220$	−3.78853e16	−1.51884
$221$	1.49924e16i	0.582269i
$222$	0	0
$223$	7.21422e15	0.263060	0.131530	−	0.991312i	$-0.458011\pi$
$223$	7.21422e15	0.263060	0.131530	+	0.991312i	$0.458011\pi$
$224$	1.85738e15i	0.0656396i
$225$	0	0
$226$	−6.22732e15	−0.206796
$227$	− 5.84982e16i	− 1.88348i	−0.336339	−	0.941741i	$-0.609189\pi$
$227$	− 5.84982e16i	− 1.88348i	0.336339	−	0.941741i	$-0.390811\pi$
$228$	0	0
$229$	2.09468e16	0.634264	0.317132	−	0.948381i	$-0.397280\pi$
$229$	2.09468e16	0.634264	0.317132	+	0.948381i	$0.397280\pi$
$230$	1.33551e16i	0.392241i
$231$	0	0
$232$	2.25418e16	0.623122
$233$	− 6.38524e16i	− 1.71272i	−0.516382	−	0.856359i	$-0.672721\pi$
$233$	− 6.38524e16i	− 1.71272i	0.516382	−	0.856359i	$-0.327279\pi$
$234$	0	0
$235$	−9.21142e16	−2.32730
$236$	2.63504e15i	0.0646255i
$237$	0	0
$238$	1.13920e16	0.263367
$239$	1.11462e16i	0.250231i	0.992142	+	0.125115i	$0.0399301\pi$
$239$	1.11462e16i	0.250231i	−0.992142	+	0.125115i	$0.960070\pi$
$240$	0	0
$241$	3.07220e16	0.650623	0.325312	−	0.945607i	$-0.394531\pi$
$241$	3.07220e16	0.650623	0.325312	+	0.945607i	$0.394531\pi$
$242$	6.21249e16i	1.27808i
$243$	0	0
$244$	−2.06246e16	−0.400551
$245$	− 8.28166e16i	− 1.56299i
$246$	0	0
$247$	−1.97702e16	−0.352478
$248$	− 2.49602e16i	− 0.432600i
$249$	0	0
$250$	−1.00681e17	−1.64955
$251$	− 2.22039e16i	− 0.353764i	−0.984232	−	0.176882i	$-0.943399\pi$
$251$	− 2.22039e16i	− 0.353764i	0.984232	−	0.176882i	$-0.0566011\pi$
$252$	0	0
$253$	3.40162e16	0.512675
$254$	− 4.96803e16i	− 0.728364i
$255$	0	0
$256$	4.50360e15	0.0625000
$257$	2.15775e16i	0.291386i	0.989330	+	0.145693i	$0.0465412\pi$
$257$	2.15775e16i	0.291386i	−0.989330	+	0.145693i	$0.953459\pi$
$258$	0	0
$259$	7.12726e15	0.0911642
$260$	4.22626e16i	0.526190i
$261$	0	0
$262$	−9.51414e15	−0.112269
$263$	6.35073e15i	0.0729682i	0.999334	+	0.0364841i	$0.0116158\pi$
$263$	6.35073e15i	0.0729682i	−0.999334	+	0.0364841i	$0.988384\pi$
$264$	0	0
$265$	1.40027e17	1.52577
$266$	1.50224e16i	0.159429i
$267$	0	0
$268$	6.15221e16	0.619566
$269$	− 2.46504e16i	− 0.241856i	−0.992661	−	0.120928i	$-0.961413\pi$
$269$	− 2.46504e16i	− 0.241856i	0.992661	−	0.120928i	$-0.0385871\pi$
$270$	0	0
$271$	−7.97252e16	−0.742695	−0.371348	−	0.928494i	$-0.621104\pi$
$271$	−7.97252e16	−0.742695	−0.371348	+	0.928494i	$0.621104\pi$
$272$	− 2.76221e16i	− 0.250770i
$273$	0	0
$274$	6.09063e16	0.525301
$275$	4.55730e17i	3.83159i
$276$	0	0
$277$	−1.03675e17	−0.828544	−0.414272	−	0.910153i	$-0.635964\pi$
$277$	−1.03675e17	−0.828544	−0.414272	+	0.910153i	$0.635964\pi$
$278$	6.55664e16i	0.510938i
$279$	0	0
$280$	3.21134e16	0.238001
$281$	− 1.68452e17i	− 1.21768i	−0.793294	−	0.608838i	$-0.791636\pi$
$281$	− 1.68452e17i	− 1.21768i	0.793294	−	0.608838i	$-0.208364\pi$
$282$	0	0
$283$	1.39346e17	0.958495	0.479247	−	0.877680i	$-0.340910\pi$
$283$	1.39346e17	0.958495	0.479247	+	0.877680i	$0.340910\pi$
$284$	1.38578e17i	0.929966i
$285$	0	0
$286$	1.07645e17	0.687752
$287$	6.00209e16i	0.374222i
$288$	0	0
$289$	−1.03846e15	−0.00616742
$290$	− 3.89738e17i	− 2.25937i
$291$	0	0
$292$	4.89960e16	0.270695
$293$	− 1.86786e17i	− 1.00756i	−0.863832	−	0.503780i	$-0.831942\pi$
$293$	− 1.86786e17i	− 1.00756i	0.863832	−	0.503780i	$-0.168058\pi$
$294$	0	0
$295$	4.55587e16	0.234324
$296$	− 1.72815e16i	− 0.0868038i
$297$	0	0
$298$	3.58814e16	0.171931
$299$	− 3.79464e16i	− 0.177612i
$300$	0	0
$301$	−1.34356e17	−0.600192
$302$	− 6.75118e16i	− 0.294667i
$303$	0	0
$304$	3.64249e16	0.151804
$305$	3.56591e17i	1.45235i
$306$	0	0
$307$	6.73138e16	0.261900	0.130950	−	0.991389i	$-0.458197\pi$
$307$	6.73138e16	0.261900	0.130950	+	0.991389i	$0.458197\pi$
$308$	− 8.17944e16i	− 0.311078i
$309$	0	0
$310$	−4.31551e17	−1.56856
$311$	1.58977e17i	0.564953i	0.959274	+	0.282476i	$0.0911559\pi$
$311$	1.58977e17i	0.564953i	−0.959274	+	0.282476i	$0.908844\pi$
$312$	0	0
$313$	−1.88505e17	−0.640492	−0.320246	−	0.947334i	$-0.603766\pi$
$313$	−1.88505e17	−0.640492	−0.320246	+	0.947334i	$0.603766\pi$
$314$	− 1.69140e17i	− 0.562002i
$315$	0	0
$316$	1.69112e17	0.537482
$317$	− 2.12672e17i	− 0.661142i	−0.943781	−	0.330571i	$-0.892759\pi$
$317$	− 2.12672e17i	− 0.661142i	0.943781	−	0.330571i	$-0.107241\pi$
$318$	0	0
$319$	−9.92683e17	−2.95309
$320$	− 7.78653e16i	− 0.226618i
$321$	0	0
$322$	−2.88337e16	−0.0803358
$323$	− 2.23406e17i	− 0.609085i
$324$	0	0
$325$	5.08385e17	1.32742
$326$	4.76088e17i	1.21665i
$327$	0	0
$328$	1.45533e17	0.356323
$329$	− 1.98875e17i	− 0.476660i
$330$	0	0
$331$	−3.85352e17	−0.885241	−0.442621	−	0.896709i	$-0.645951\pi$
$331$	−3.85352e17	−0.885241	−0.442621	+	0.896709i	$0.645951\pi$
$332$	9.84339e15i	0.0221400i
$333$	0	0
$334$	3.43433e17	0.740656
$335$	− 1.06369e18i	− 2.24647i
$336$	0	0
$337$	2.35431e17	0.476929	0.238465	−	0.971151i	$-0.423356\pi$
$337$	2.35431e17	0.476929	0.238465	+	0.971151i	$0.423356\pi$
$338$	2.36288e17i	0.468841i
$339$	0	0
$340$	−4.77575e17	−0.909262
$341$	1.09918e18i	2.05017i
$342$	0	0
$343$	3.86197e17	0.691432
$344$	3.25772e17i	0.571485i
$345$	0	0
$346$	−5.19919e17	−0.875796
$347$	7.65501e17i	1.26369i	0.775096	+	0.631844i	$0.217701\pi$
$347$	7.65501e17i	1.26369i	−0.775096	+	0.631844i	$0.782299\pi$
$348$	0	0
$349$	6.36950e17	1.01001	0.505007	−	0.863115i	$-0.331490\pi$
$349$	6.36950e17	1.01001	0.505007	+	0.863115i	$0.331490\pi$
$350$	− 3.86298e17i	− 0.600407i
$351$	0	0
$352$	−1.98327e17	−0.296199
$353$	− 5.61287e17i	− 0.821792i	−0.911682	−	0.410896i	$-0.865216\pi$
$353$	− 5.61287e17i	− 0.821792i	0.911682	−	0.410896i	$-0.134784\pi$
$354$	0	0
$355$	2.39596e18	3.37195
$356$	6.24819e16i	0.0862192i
$357$	0	0
$358$	7.17691e17	0.952262
$359$	− 4.77260e17i	− 0.621003i	−0.950573	−	0.310502i	$-0.899503\pi$
$359$	− 4.77260e17i	− 0.621003i	0.950573	−	0.310502i	$-0.100497\pi$
$360$	0	0
$361$	−5.04404e17	−0.631289
$362$	6.02829e17i	0.740004i
$363$	0	0
$364$	−9.12450e16	−0.107770
$365$	− 8.47119e17i	− 0.981509i
$366$	0	0
$367$	−1.43733e17	−0.160285	−0.0801425	−	0.996783i	$-0.525538\pi$
$367$	−1.43733e17	−0.160285	−0.0801425	+	0.996783i	$0.525538\pi$
$368$	6.99131e16i	0.0764934i
$369$	0	0
$370$	−2.98789e17	−0.314741
$371$	3.02318e17i	0.312497i
$372$	0	0
$373$	−3.64226e17	−0.362584	−0.181292	−	0.983429i	$-0.558028\pi$
$373$	−3.64226e17	−0.362584	−0.181292	+	0.983429i	$0.558028\pi$
$374$	1.21641e18i	1.18844i
$375$	0	0
$376$	−4.82212e17	−0.453861
$377$	1.10738e18i	1.02307i
$378$	0	0
$379$	−7.31187e17	−0.650959	−0.325480	−	0.945549i	$-0.605526\pi$
$379$	−7.31187e17	−0.650959	−0.325480	+	0.945549i	$0.605526\pi$
$380$	− 6.29770e17i	− 0.550423i
$381$	0	0
$382$	6.14150e17	0.517405
$383$	− 1.78744e17i	− 0.147856i	−0.997264	−	0.0739281i	$-0.976446\pi$
$383$	− 1.78744e17i	− 0.147856i	0.997264	−	0.0739281i	$-0.0235535\pi$
$384$	0	0
$385$	−1.41419e18	−1.12793
$386$	4.08459e17i	0.319917i
$387$	0	0
$388$	3.87775e17	0.292926
$389$	1.31562e18i	0.976078i	0.872822	+	0.488039i	$0.162288\pi$
$389$	1.31562e18i	0.976078i	−0.872822	+	0.488039i	$0.837712\pi$
$390$	0	0
$391$	4.28801e17	0.306916
$392$	− 4.33539e17i	− 0.304808i
$393$	0	0
$394$	−1.86001e17	−0.126195
$395$	− 2.92386e18i	− 1.94885i
$396$	0	0
$397$	−2.49426e18	−1.60475	−0.802375	−	0.596820i	$-0.796431\pi$
$397$	−2.49426e18	−1.60475	−0.802375	+	0.596820i	$0.796431\pi$
$398$	− 1.29148e18i	− 0.816404i
$399$	0	0
$400$	−9.36657e17	−0.571690
$401$	− 7.73789e17i	− 0.464100i	−0.972704	−	0.232050i	$-0.925457\pi$
$401$	− 7.73789e17i	− 0.464100i	0.972704	−	0.232050i	$-0.0745433\pi$
$402$	0	0
$403$	1.22618e18	0.710264
$404$	− 1.41580e18i	− 0.805992i
$405$	0	0
$406$	8.41445e17	0.462747
$407$	7.61032e17i	0.411379i
$408$	0	0
$409$	−4.10157e16	−0.0214233	−0.0107117	−	0.999943i	$-0.503410\pi$
$409$	−4.10157e16	−0.0214233	−0.0107117	+	0.999943i	$0.503410\pi$
$410$	− 2.51620e18i	− 1.29199i
$411$	0	0
$412$	−1.54898e17	−0.0768715
$413$	9.83614e16i	0.0479926i
$414$	0	0
$415$	1.70188e17	0.0802772
$416$	2.21242e17i	0.102616i
$417$	0	0
$418$	−1.60406e18	−0.719426
$419$	7.76233e17i	0.342369i	0.985239	+	0.171184i	$0.0547594\pi$
$419$	7.76233e17i	0.342369i	−0.985239	+	0.171184i	$0.945241\pi$
$420$	0	0
$421$	−2.20820e18	−0.942027	−0.471014	−	0.882126i	$-0.656112\pi$
$421$	−2.20820e18	−0.942027	−0.471014	+	0.882126i	$0.656112\pi$
$422$	1.41473e18i	0.593592i
$423$	0	0
$424$	7.33030e17	0.297550
$425$	5.74484e18i	2.29380i
$426$	0	0
$427$	−7.69880e17	−0.297460
$428$	− 4.09881e17i	− 0.155794i
$429$	0	0
$430$	5.63245e18	2.07214
$431$	− 3.01395e18i	− 1.09093i	−0.838135	−	0.545463i	$-0.816354\pi$
$431$	− 3.01395e18i	− 1.09093i	0.838135	−	0.545463i	$-0.183646\pi$
$432$	0	0
$433$	6.88938e17	0.241415	0.120708	−	0.992688i	$-0.461484\pi$
$433$	6.88938e17	0.241415	0.120708	+	0.992688i	$0.461484\pi$
$434$	− 9.31719e17i	− 0.321260i
$435$	0	0
$436$	2.10679e17	0.0703422
$437$	5.65453e17i	0.185792i
$438$	0	0
$439$	4.08732e18	1.30073	0.650365	−	0.759622i	$-0.274616\pi$
$439$	4.08732e18	1.30073	0.650365	+	0.759622i	$0.274616\pi$
$440$	3.42899e18i	1.07398i
$441$	0	0
$442$	1.35695e18	0.411727
$443$	− 5.29962e17i	− 0.158277i	−0.996864	−	0.0791387i	$-0.974783\pi$
$443$	− 5.29962e17i	− 0.158277i	0.996864	−	0.0791387i	$-0.0252170\pi$
$444$	0	0
$445$	1.08028e18	0.312621
$446$	− 6.52957e17i	− 0.186012i
$447$	0	0
$448$	1.68111e17	0.0464142
$449$	4.27608e18i	1.16231i	0.813794	+	0.581154i	$0.197399\pi$
$449$	4.27608e18i	1.16231i	−0.813794	+	0.581154i	$0.802601\pi$
$450$	0	0
$451$	−6.40889e18	−1.68868
$452$	5.63633e17i	0.146227i
$453$	0	0
$454$	−5.29465e18	−1.33182
$455$	1.57759e18i	0.390763i
$456$	0	0
$457$	5.18206e18	1.24477	0.622385	−	0.782712i	$-0.286164\pi$
$457$	5.18206e18	1.24477	0.622385	+	0.782712i	$0.286164\pi$
$458$	− 1.89589e18i	− 0.448493i
$459$	0	0
$460$	1.20877e18	0.277356
$461$	8.17907e18i	1.84841i	0.381901	+	0.924203i	$0.375270\pi$
$461$	8.17907e18i	1.84841i	−0.381901	+	0.924203i	$0.624730\pi$
$462$	0	0
$463$	8.52422e18	1.86891	0.934454	−	0.356085i	$-0.115889\pi$
$463$	8.52422e18	1.86891	0.934454	+	0.356085i	$0.115889\pi$
$464$	− 2.04025e18i	− 0.440614i
$465$	0	0
$466$	−5.77926e18	−1.21107
$467$	− 7.76114e18i	− 1.60217i	−0.598554	−	0.801083i	$-0.704258\pi$
$467$	− 7.76114e18i	− 1.60217i	0.598554	−	0.801083i	$-0.295742\pi$
$468$	0	0
$469$	2.29651e18	0.460106
$470$	8.33723e18i	1.64565i
$471$	0	0
$472$	2.38497e17	0.0456971
$473$	− 1.43462e19i	− 2.70837i
$474$	0	0
$475$	−7.57563e18	−1.38856
$476$	− 1.03108e18i	− 0.186228i
$477$	0	0
$478$	1.00884e18	0.176940
$479$	− 1.65919e18i	− 0.286780i	−0.989666	−	0.143390i	$-0.954200\pi$
$479$	− 1.65919e18i	− 0.286780i	0.989666	−	0.143390i	$-0.0458003\pi$
$480$	0	0
$481$	8.48962e17	0.142519
$482$	− 2.78063e18i	− 0.460060i
$483$	0	0
$484$	5.62290e18	0.903739
$485$	− 6.70447e18i	− 1.06212i
$486$	0	0
$487$	3.18312e18	0.489948	0.244974	−	0.969530i	$-0.421221\pi$
$487$	3.18312e18	0.489948	0.244974	+	0.969530i	$0.421221\pi$
$488$	1.86673e18i	0.283232i
$489$	0	0
$490$	−7.49570e18	−1.10520
$491$	− 5.12632e18i	− 0.745136i	−0.928005	−	0.372568i	$-0.878477\pi$
$491$	− 5.12632e18i	− 0.745136i	0.928005	−	0.372568i	$-0.121523\pi$
$492$	0	0
$493$	−1.25136e19	−1.76788
$494$	1.78939e18i	0.249239i
$495$	0	0
$496$	−2.25914e18	−0.305894
$497$	5.17288e18i	0.690618i
$498$	0	0
$499$	4.15318e18	0.539110	0.269555	−	0.962985i	$-0.413123\pi$
$499$	4.15318e18	0.539110	0.269555	+	0.962985i	$0.413123\pi$
$500$	9.11258e18i	1.16641i
$501$	0	0
$502$	−2.00967e18	−0.250149
$503$	7.46116e18i	0.915863i	0.888987	+	0.457932i	$0.151410\pi$
$503$	7.46116e18i	0.915863i	−0.888987	+	0.457932i	$0.848590\pi$
$504$	0	0
$505$	−2.44785e19	−2.92244
$506$	− 3.07879e18i	− 0.362516i
$507$	0	0
$508$	−4.49655e18	−0.515031
$509$	− 9.48090e18i	− 1.07109i	−0.844507	−	0.535544i	$-0.820107\pi$
$509$	− 9.48090e18i	− 1.07109i	0.844507	−	0.535544i	$-0.179893\pi$
$510$	0	0
$511$	1.82893e18	0.201025
$512$	− 4.07619e17i	− 0.0441942i
$513$	0	0
$514$	1.95297e18	0.206041
$515$	2.67812e18i	0.278727i
$516$	0	0
$517$	2.12354e19	2.15093
$518$	− 6.45086e17i	− 0.0644628i
$519$	0	0
$520$	3.82517e18	0.372073
$521$	− 2.62455e18i	− 0.251878i	−0.992038	−	0.125939i	$-0.959806\pi$
$521$	− 2.62455e18i	− 0.251878i	0.992038	−	0.125939i	$-0.0401944\pi$
$522$	0	0
$523$	1.43596e19	1.34162	0.670812	−	0.741627i	$-0.265946\pi$
$523$	1.43596e19	1.34162	0.670812	+	0.741627i	$0.265946\pi$
$524$	8.61122e17i	0.0793863i
$525$	0	0
$526$	5.74803e17	0.0515963
$527$	1.38561e19i	1.22734i
$528$	0	0
$529$	1.05075e19	0.906380
$530$	− 1.26738e19i	− 1.07888i
$531$	0	0
$532$	1.35967e18	0.112734
$533$	7.14938e18i	0.585029i
$534$	0	0
$535$	−7.08667e18	−0.564892
$536$	− 5.56834e18i	− 0.438099i
$537$	0	0
$538$	−2.23110e18	−0.171018
$539$	1.90920e19i	1.44454i
$540$	0	0
$541$	7.66465e18	0.565082	0.282541	−	0.959255i	$-0.408823\pi$
$541$	7.66465e18	0.565082	0.282541	+	0.959255i	$0.408823\pi$
$542$	7.21590e18i	0.525165i
$543$	0	0
$544$	−2.50007e18	−0.177321
$545$	− 3.64255e18i	− 0.255053i
$546$	0	0
$547$	−1.82201e19	−1.24348	−0.621741	−	0.783223i	$-0.713574\pi$
$547$	−1.82201e19	−1.24348	−0.621741	+	0.783223i	$0.713574\pi$
$548$	− 5.51261e18i	− 0.371444i
$549$	0	0
$550$	4.12480e19	2.70934
$551$	− 1.65014e19i	− 1.07019i
$552$	0	0
$553$	6.31263e18	0.399148
$554$	9.38359e18i	0.585869i
$555$	0	0
$556$	5.93440e18	0.361288
$557$	6.47551e18i	0.389303i	0.980872	+	0.194651i	$0.0623576\pi$
$557$	6.47551e18i	0.389303i	−0.980872	+	0.194651i	$0.937642\pi$
$558$	0	0
$559$	−1.60037e19	−0.938293
$560$	− 2.90657e18i	− 0.168292i
$561$	0	0
$562$	−1.52465e19	−0.861027
$563$	2.07968e19i	1.15994i	0.814636	+	0.579972i	$0.196937\pi$
$563$	2.07968e19i	1.15994i	−0.814636	+	0.579972i	$0.803063\pi$
$564$	0	0
$565$	9.74497e18	0.530202
$566$	− 1.26122e19i	− 0.677758i
$567$	0	0
$568$	1.25427e19	0.657585
$569$	− 4.21373e18i	− 0.218213i	−0.994030	−	0.109106i	$-0.965201\pi$
$569$	− 4.21373e18i	− 0.218213i	0.994030	−	0.109106i	$-0.0347989\pi$
$570$	0	0
$571$	−2.22631e19	−1.12495	−0.562476	−	0.826814i	$-0.690151\pi$
$571$	−2.22631e19	−1.12495	−0.562476	+	0.826814i	$0.690151\pi$
$572$	− 9.74292e18i	− 0.486314i
$573$	0	0
$574$	5.43247e18	0.264615
$575$	− 1.45405e19i	− 0.699688i
$576$	0	0
$577$	3.10071e19	1.45623	0.728116	−	0.685454i	$-0.240396\pi$
$577$	3.10071e19	1.45623	0.728116	+	0.685454i	$0.240396\pi$
$578$	9.39904e16i	0.00436102i
$579$	0	0
$580$	−3.52751e19	−1.59761
$581$	3.67436e17i	0.0164418i
$582$	0	0
$583$	−3.22807e19	−1.41014
$584$	− 4.43461e18i	− 0.191410i
$585$	0	0
$586$	−1.69059e19	−0.712453
$587$	3.87372e19i	1.61310i	0.591164	+	0.806551i	$0.298669\pi$
$587$	3.87372e19i	1.61310i	−0.591164	+	0.806551i	$0.701331\pi$
$588$	0	0
$589$	−1.82718e19	−0.742975
$590$	− 4.12351e18i	− 0.165692i
$591$	0	0
$592$	−1.56414e18	−0.0613796
$593$	1.35053e19i	0.523747i	0.965102	+	0.261874i	$0.0843403\pi$
$593$	1.35053e19i	0.523747i	−0.965102	+	0.261874i	$0.915660\pi$
$594$	0	0
$595$	−1.78270e19	−0.675242
$596$	− 3.24761e18i	− 0.121574i
$597$	0	0
$598$	−3.43452e18	−0.125591
$599$	− 8.77507e18i	− 0.317149i	−0.987347	−	0.158574i	$-0.949310\pi$
$599$	− 8.77507e18i	− 0.317149i	0.987347	−	0.158574i	$-0.0506897\pi$
$600$	0	0
$601$	−2.76952e19	−0.977874	−0.488937	−	0.872319i	$-0.662615\pi$
$601$	−2.76952e19	−0.977874	−0.488937	+	0.872319i	$0.662615\pi$
$602$	1.21605e19i	0.424400i
$603$	0	0
$604$	−6.11047e18	−0.208361
$605$	− 9.72175e19i	− 3.27685i
$606$	0	0
$607$	4.45270e19	1.46657	0.733285	−	0.679922i	$-0.237986\pi$
$607$	4.45270e19	1.46657	0.733285	+	0.679922i	$0.237986\pi$
$608$	− 3.29680e18i	− 0.107342i
$609$	0	0
$610$	3.22749e19	1.02697
$611$	− 2.36889e19i	− 0.745172i
$612$	0	0
$613$	−2.30024e19	−0.707214	−0.353607	−	0.935394i	$-0.615045\pi$
$613$	−2.30024e19	−0.707214	−0.353607	+	0.935394i	$0.615045\pi$
$614$	− 6.09255e18i	− 0.185191i
$615$	0	0
$616$	−7.40319e18	−0.219965
$617$	− 2.09667e19i	− 0.615934i	−0.951397	−	0.307967i	$-0.900351\pi$
$617$	− 2.09667e19i	− 0.615934i	0.951397	−	0.307967i	$-0.0996486\pi$
$618$	0	0
$619$	4.78696e19	1.37475	0.687377	−	0.726301i	$-0.258762\pi$
$619$	4.78696e19	1.37475	0.687377	+	0.726301i	$0.258762\pi$
$620$	3.90595e19i	1.10914i
$621$	0	0
$622$	1.43890e19	0.399482
$623$	2.33234e18i	0.0640287i
$624$	0	0
$625$	7.23641e19	1.94251
$626$	1.70616e19i	0.452896i
$627$	0	0
$628$	−1.53088e19	−0.397395
$629$	9.59342e18i	0.246274i
$630$	0	0
$631$	−6.94907e19	−1.74470	−0.872351	−	0.488881i	$-0.837405\pi$
$631$	−6.94907e19	−1.74470	−0.872351	+	0.488881i	$0.837405\pi$
$632$	− 1.53062e19i	− 0.380057i
$633$	0	0
$634$	−1.92488e19	−0.467498
$635$	7.77434e19i	1.86744i
$636$	0	0
$637$	2.12978e19	0.500449
$638$	8.98475e19i	2.08815i
$639$	0	0
$640$	−7.04756e18	−0.160243
$641$	− 3.16065e19i	− 0.710837i	−0.934707	−	0.355419i	$-0.884338\pi$
$641$	− 3.16065e19i	− 0.710837i	0.934707	−	0.355419i	$-0.115662\pi$
$642$	0	0
$643$	7.05010e19	1.55138	0.775691	−	0.631113i	$-0.217402\pi$
$643$	7.05010e19	1.55138	0.775691	+	0.631113i	$0.217402\pi$
$644$	2.60973e18i	0.0568060i
$645$	0	0
$646$	−2.02204e19	−0.430688
$647$	− 6.78089e18i	− 0.142875i	−0.997445	−	0.0714376i	$-0.977241\pi$
$647$	− 6.78089e18i	− 0.142875i	0.997445	−	0.0714376i	$-0.0227587\pi$
$648$	0	0
$649$	−1.05028e19	−0.216567
$650$	− 4.60138e19i	− 0.938630i
$651$	0	0
$652$	4.30906e19	0.860299
$653$	6.63074e19i	1.30969i	0.755762	+	0.654847i	$0.227267\pi$
$653$	6.63074e19i	1.30969i	−0.755762	+	0.654847i	$0.772733\pi$
$654$	0	0
$655$	1.48884e19	0.287846
$656$	− 1.31721e19i	− 0.251958i
$657$	0	0
$658$	−1.80001e19	−0.337049
$659$	3.49308e19i	0.647159i	0.946201	+	0.323579i	$0.104886\pi$
$659$	3.49308e19i	0.647159i	−0.946201	+	0.323579i	$0.895114\pi$
$660$	0	0
$661$	4.67299e19	0.847587	0.423793	−	0.905759i	$-0.360698\pi$
$661$	4.67299e19	0.847587	0.423793	+	0.905759i	$0.360698\pi$
$662$	3.48781e19i	0.625960i
$663$	0	0
$664$	8.90922e17	0.0156554
$665$	− 2.35082e19i	− 0.408759i
$666$	0	0
$667$	3.16725e19	0.539264
$668$	− 3.10840e19i	− 0.523723i
$669$	0	0
$670$	−9.62742e19	−1.58850
$671$	− 8.22059e19i	− 1.34229i
$672$	0	0
$673$	−3.45172e19	−0.551989	−0.275994	−	0.961159i	$-0.589007\pi$
$673$	−3.45172e19	−0.551989	−0.275994	+	0.961159i	$0.589007\pi$
$674$	− 2.13088e19i	− 0.337240i
$675$	0	0
$676$	2.13864e19	0.331520
$677$	2.12460e18i	0.0325954i	0.999867	+	0.0162977i	$0.00518795\pi$
$677$	2.12460e18i	0.0325954i	−0.999867	+	0.0162977i	$0.994812\pi$
$678$	0	0
$679$	1.44749e19	0.217535
$680$	4.32251e19i	0.642945i
$681$	0	0
$682$	9.94867e19	1.44969
$683$	7.50880e19i	1.08299i	0.840703	+	0.541497i	$0.182142\pi$
$683$	7.50880e19i	1.08299i	−0.840703	+	0.541497i	$0.817858\pi$
$684$	0	0
$685$	−9.53106e19	−1.34681
$686$	− 3.49545e19i	− 0.488917i
$687$	0	0
$688$	2.94855e19	0.404101
$689$	3.60105e19i	0.488533i
$690$	0	0
$691$	−1.37137e20	−1.82310	−0.911548	−	0.411194i	$-0.865112\pi$
$691$	−1.37137e20	−1.82310	−0.911548	+	0.411194i	$0.865112\pi$
$692$	4.70577e19i	0.619281i
$693$	0	0
$694$	6.92853e19	0.893562
$695$	− 1.02603e20i	− 1.30999i
$696$	0	0
$697$	−8.07892e19	−1.01094
$698$	− 5.76501e19i	− 0.714187i
$699$	0	0
$700$	−3.49637e19	−0.424552
$701$	− 1.11819e20i	− 1.34428i	−0.740423	−	0.672141i	$-0.765375\pi$
$701$	− 1.11819e20i	− 1.34428i	0.740423	−	0.672141i	$-0.234625\pi$
$702$	0	0
$703$	−1.26507e19	−0.149082
$704$	1.79505e19i	0.209444i
$705$	0	0
$706$	−5.08019e19	−0.581094
$707$	− 5.28491e19i	− 0.598552i
$708$	0	0
$709$	1.37741e20	1.52947	0.764735	−	0.644345i	$-0.222870\pi$
$709$	1.37741e20	1.52947	0.764735	+	0.644345i	$0.222870\pi$
$710$	− 2.16858e20i	− 2.38433i
$711$	0	0
$712$	5.65522e18	0.0609662
$713$	− 3.50705e19i	− 0.374382i
$714$	0	0
$715$	−1.68451e20	−1.76332
$716$	− 6.49580e19i	− 0.673351i
$717$	0	0
$718$	−4.31966e19	−0.439116
$719$	8.44075e19i	0.849726i	0.905258	+	0.424863i	$0.139678\pi$
$719$	8.44075e19i	0.849726i	−0.905258	+	0.424863i	$0.860322\pi$
$720$	0	0
$721$	−5.78206e18	−0.0570868
$722$	4.56535e19i	0.446389i
$723$	0	0
$724$	5.45618e19	0.523262
$725$	4.24331e20i	4.03031i
$726$	0	0
$727$	1.69431e20	1.57853	0.789265	−	0.614053i	$-0.210462\pi$
$727$	1.69431e20	1.57853	0.789265	+	0.614053i	$0.210462\pi$
$728$	8.25855e18i	0.0762052i
$729$	0	0
$730$	−7.66725e19	−0.694032
$731$	− 1.80845e20i	− 1.62138i
$732$	0	0
$733$	−1.02023e20	−0.897366	−0.448683	−	0.893691i	$-0.648107\pi$
$733$	−1.02023e20	−0.897366	−0.448683	+	0.893691i	$0.648107\pi$
$734$	1.30092e19i	0.113339i
$735$	0	0
$736$	6.32781e18	0.0540890
$737$	2.45215e20i	2.07623i
$738$	0	0
$739$	−7.22807e19	−0.600498	−0.300249	−	0.953861i	$-0.597070\pi$
$739$	−7.22807e19	−0.600498	−0.300249	+	0.953861i	$0.597070\pi$
$740$	2.70433e19i	0.222555i
$741$	0	0
$742$	2.73627e19	0.220969
$743$	− 1.06884e20i	− 0.855047i	−0.904004	−	0.427524i	$-0.859386\pi$
$743$	− 1.06884e20i	− 0.855047i	0.904004	−	0.427524i	$-0.140614\pi$
$744$	0	0
$745$	−5.61498e19	−0.440813
$746$	3.29660e19i	0.256386i
$747$	0	0
$748$	1.10097e20	0.840356
$749$	− 1.53001e19i	− 0.115697i
$750$	0	0
$751$	1.37977e20	1.02406	0.512031	−	0.858967i	$-0.328893\pi$
$751$	1.37977e20	1.02406	0.512031	+	0.858967i	$0.328893\pi$
$752$	4.36448e19i	0.320928i
$753$	0	0
$754$	1.00228e20	0.723422
$755$	1.05647e20i	0.755493i
$756$	0	0
$757$	1.88357e20	1.32224	0.661122	−	0.750279i	$-0.270081\pi$
$757$	1.88357e20	1.32224	0.661122	+	0.750279i	$0.270081\pi$
$758$	6.61795e19i	0.460298i
$759$	0	0
$760$	−5.70003e19	−0.389208
$761$	− 2.00035e20i	− 1.35336i	−0.736278	−	0.676680i	$-0.763418\pi$
$761$	− 2.00035e20i	− 1.35336i	0.736278	−	0.676680i	$-0.236582\pi$
$762$	0	0
$763$	7.86427e18	0.0522380
$764$	− 5.55865e19i	− 0.365861i
$765$	0	0
$766$	−1.61780e19	−0.104550
$767$	1.17163e19i	0.0750278i
$768$	0	0
$769$	6.35757e19	0.399767	0.199883	−	0.979820i	$-0.435944\pi$
$769$	6.35757e19	0.399767	0.199883	+	0.979820i	$0.435944\pi$
$770$	1.27998e20i	0.797568i
$771$	0	0
$772$	3.69695e19	0.226216
$773$	− 2.19200e20i	− 1.32918i	−0.747208	−	0.664591i	$-0.768606\pi$
$773$	− 2.19200e20i	− 1.32918i	0.747208	−	0.664591i	$-0.231394\pi$
$774$	0	0
$775$	4.69855e20	2.79803
$776$	− 3.50974e19i	− 0.207130i
$777$	0	0
$778$	1.19077e20	0.690192
$779$	− 1.06535e20i	− 0.611972i
$780$	0	0
$781$	−5.52348e20	−3.11641
$782$	− 3.88107e19i	− 0.217022i
$783$	0	0
$784$	−3.92395e19	−0.215532
$785$	2.64682e20i	1.44091i
$786$	0	0
$787$	2.03403e20	1.08776	0.543880	−	0.839163i	$-0.316955\pi$
$787$	2.03403e20	1.08776	0.543880	+	0.839163i	$0.316955\pi$
$788$	1.68349e19i	0.0892332i
$789$	0	0
$790$	−2.64638e20	−1.37804
$791$	2.10394e19i	0.108592i
$792$	0	0
$793$	−9.17040e19	−0.465025
$794$	2.25754e20i	1.13473i
$795$	0	0
$796$	−1.16891e20	−0.577285
$797$	− 8.46452e19i	− 0.414375i	−0.978301	−	0.207188i	$-0.933569\pi$
$797$	− 8.46452e19i	− 0.414375i	0.978301	−	0.207188i	$-0.0664311\pi$
$798$	0	0
$799$	2.67689e20	1.28766
$800$	8.47765e19i	0.404246i
$801$	0	0
$802$	−7.00353e19	−0.328168
$803$	1.95289e20i	0.907128i
$804$	0	0
$805$	4.51211e19	0.205972
$806$	− 1.10981e20i	− 0.502233i
$807$	0	0
$808$	−1.28143e20	−0.569923
$809$	− 1.68171e20i	− 0.741502i	−0.928732	−	0.370751i	$-0.879100\pi$
$809$	− 1.68171e20i	− 0.741502i	0.928732	−	0.370751i	$-0.120900\pi$
$810$	0	0
$811$	−4.27825e20	−1.85405	−0.927023	−	0.375004i	$-0.877641\pi$
$811$	−4.27825e20	−1.85405	−0.927023	+	0.375004i	$0.877641\pi$
$812$	− 7.61589e19i	− 0.327211i
$813$	0	0
$814$	6.88807e19	0.290889
$815$	− 7.45018e20i	− 3.11935i
$816$	0	0
$817$	2.38477e20	0.981505
$818$	3.71232e18i	0.0151486i
$819$	0	0
$820$	−2.27740e20	−0.913572
$821$	2.05817e19i	0.0818613i	0.999162	+	0.0409307i	$0.0130323\pi$
$821$	2.05817e19i	0.0818613i	−0.999162	+	0.0409307i	$0.986968\pi$
$822$	0	0
$823$	−8.31992e19	−0.325327	−0.162664	−	0.986682i	$-0.552009\pi$
$823$	−8.31992e19	−0.325327	−0.162664	+	0.986682i	$0.552009\pi$
$824$	1.40198e19i	0.0543563i
$825$	0	0
$826$	8.90266e18	0.0339359
$827$	7.97132e19i	0.301295i	0.988588	+	0.150647i	$0.0481358\pi$
$827$	7.97132e19i	0.301295i	−0.988588	+	0.150647i	$0.951864\pi$
$828$	0	0
$829$	−1.10972e20	−0.412412	−0.206206	−	0.978509i	$-0.566112\pi$
$829$	−1.10972e20	−0.412412	−0.206206	+	0.978509i	$0.566112\pi$
$830$	− 1.54036e19i	− 0.0567645i
$831$	0	0
$832$	2.00245e19	0.0725603
$833$	2.40670e20i	0.864781i
$834$	0	0
$835$	−5.37429e20	−1.89896
$836$	1.45183e20i	0.508711i
$837$	0	0
$838$	7.02566e19	0.242091
$839$	1.12401e20i	0.384094i	0.981386	+	0.192047i	$0.0615125\pi$
$839$	1.12401e20i	0.384094i	−0.981386	+	0.192047i	$0.938487\pi$
$840$	0	0
$841$	−6.26730e20	−2.10624
$842$	1.99863e20i	0.666114i
$843$	0	0
$844$	1.28047e20	0.419733
$845$	− 3.69761e20i	− 1.20205i
$846$	0	0
$847$	2.09893e20	0.671141
$848$	− 6.63463e19i	− 0.210400i
$849$	0	0
$850$	5.19964e20	1.62196
$851$	− 2.42814e19i	− 0.0751221i
$852$	0	0
$853$	4.76544e20	1.45031	0.725153	−	0.688588i	$-0.241769\pi$
$853$	4.76544e20	1.45031	0.725153	+	0.688588i	$0.241769\pi$
$854$	6.96816e19i	0.210336i
$855$	0	0
$856$	−3.70982e19	−0.110163
$857$	4.54483e20i	1.33860i	0.742990	+	0.669302i	$0.233407\pi$
$857$	4.54483e20i	1.33860i	−0.742990	+	0.669302i	$0.766593\pi$
$858$	0	0
$859$	−1.14099e20	−0.330621	−0.165311	−	0.986242i	$-0.552863\pi$
$859$	−1.14099e20	−0.330621	−0.165311	+	0.986242i	$0.552863\pi$
$860$	− 5.09792e20i	− 1.46522i
$861$	0	0
$862$	−2.72791e20	−0.771401
$863$	− 6.19780e20i	− 1.73845i	−0.494417	−	0.869225i	$-0.664618\pi$
$863$	− 6.19780e20i	− 1.73845i	0.494417	−	0.869225i	$-0.335382\pi$
$864$	0	0
$865$	8.13607e20	2.24544
$866$	− 6.23555e19i	− 0.170706i
$867$	0	0
$868$	−8.43296e19	−0.227165
$869$	6.74047e20i	1.80116i
$870$	0	0
$871$	2.73548e20	0.719293
$872$	− 1.90685e19i	− 0.0497395i
$873$	0	0
$874$	5.11790e19	0.131375
$875$	3.40156e20i	0.866208i
$876$	0	0
$877$	1.97307e20	0.494476	0.247238	−	0.968955i	$-0.420477\pi$
$877$	1.97307e20	0.494476	0.247238	+	0.968955i	$0.420477\pi$
$878$	− 3.69942e20i	− 0.919755i
$879$	0	0
$880$	3.10357e20	0.759420
$881$	2.14957e18i	0.00521820i	0.999997	+	0.00260910i	$0.000830503\pi$
$881$	2.14957e18i	0.00521820i	−0.999997	+	0.00260910i	$0.999169\pi$
$882$	0	0
$883$	4.69149e20	1.12095	0.560474	−	0.828172i	$-0.310619\pi$
$883$	4.69149e20	1.12095	0.560474	+	0.828172i	$0.310619\pi$
$884$	− 1.22817e20i	− 0.291135i
$885$	0	0
$886$	−4.79667e19	−0.111919
$887$	2.81772e20i	0.652278i	0.945322	+	0.326139i	$0.105748\pi$
$887$	2.81772e20i	0.652278i	−0.945322	+	0.326139i	$0.894252\pi$
$888$	0	0
$889$	−1.67848e20	−0.382476
$890$	− 9.77762e19i	− 0.221056i
$891$	0	0
$892$	−5.90989e19	−0.131530
$893$	3.52997e20i	0.779490i
$894$	0	0
$895$	−1.12310e21	−2.44149
$896$	− 1.52157e19i	− 0.0328198i
$897$	0	0
$898$	3.87026e20	0.821876
$899$	1.02345e21i	2.15650i
$900$	0	0
$901$	−4.06925e20	−0.844191
$902$	5.80067e20i	1.19408i
$903$	0	0
$904$	5.10142e19	0.103398
$905$	− 9.43350e20i	− 1.89729i
$906$	0	0
$907$	1.74753e20	0.346077	0.173039	−	0.984915i	$-0.444641\pi$
$907$	1.74753e20	0.346077	0.173039	+	0.984915i	$0.444641\pi$
$908$	4.79217e20i	0.941741i
$909$	0	0
$910$	1.42787e20	0.276311
$911$	6.03814e20i	1.15951i	0.814791	+	0.579755i	$0.196852\pi$
$911$	6.03814e20i	1.15951i	−0.814791	+	0.579755i	$0.803148\pi$
$912$	0	0
$913$	−3.92339e19	−0.0741936
$914$	− 4.69026e20i	− 0.880185i
$915$	0	0
$916$	−1.71597e20	−0.317132
$917$	3.21441e19i	0.0589544i
$918$	0	0
$919$	9.04375e19	0.163358	0.0816790	−	0.996659i	$-0.473972\pi$
$919$	9.04375e19	0.163358	0.0816790	+	0.996659i	$0.473972\pi$
$920$	− 1.09405e20i	− 0.196120i
$921$	0	0
$922$	7.40285e20	1.30702
$923$	6.16166e20i	1.07966i
$924$	0	0
$925$	3.25309e20	0.561441
$926$	− 7.71524e20i	− 1.32152i
$927$	0	0
$928$	−1.84662e20	−0.311561
$929$	− 7.02021e20i	− 1.17555i	−0.809025	−	0.587774i	$-0.800005\pi$
$929$	− 7.02021e20i	− 1.17555i	0.809025	−	0.587774i	$-0.199995\pi$
$930$	0	0
$931$	−3.17367e20	−0.523496
$932$	5.23079e20i	0.856359i
$933$	0	0
$934$	−7.02458e20	−1.13290
$935$	− 1.90352e21i	− 3.04703i
$936$	0	0
$937$	−8.05744e20	−1.27063	−0.635316	−	0.772252i	$-0.719130\pi$
$937$	−8.05744e20	−1.27063	−0.635316	+	0.772252i	$0.719130\pi$
$938$	− 2.07856e20i	− 0.325344i
$939$	0	0
$940$	7.54600e20	1.16365
$941$	− 1.02365e21i	− 1.56684i	−0.621494	−	0.783419i	$-0.713474\pi$
$941$	− 1.02365e21i	− 1.56684i	0.621494	−	0.783419i	$-0.286526\pi$
$942$	0	0
$943$	2.04482e20	0.308370
$944$	− 2.15863e19i	− 0.0323127i
$945$	0	0
$946$	−1.29847e21	−1.91511
$947$	9.74103e20i	1.42612i	0.701104	+	0.713059i	$0.252691\pi$
$947$	9.74103e20i	1.42612i	−0.701104	+	0.713059i	$0.747309\pi$
$948$	0	0
$949$	2.17853e20	0.314267
$950$	6.85668e20i	0.981857i
$951$	0	0
$952$	−9.33231e19	−0.131683
$953$	1.36890e20i	0.191743i	0.995394	+	0.0958716i	$0.0305638\pi$
$953$	1.36890e20i	0.191743i	−0.995394	+	0.0958716i	$0.969436\pi$
$954$	0	0
$955$	−9.61066e20	−1.32657
$956$	− 9.13095e19i	− 0.125115i
$957$	0	0
$958$	−1.50173e20	−0.202784
$959$	− 2.05776e20i	− 0.275844i
$960$	0	0
$961$	3.76308e20	0.497141
$962$	− 7.68392e19i	− 0.100776i
$963$	0	0
$964$	−2.51674e20	−0.325312
$965$	− 6.39186e20i	− 0.820231i
$966$	0	0
$967$	9.51200e20	1.20306	0.601529	−	0.798851i	$-0.294558\pi$
$967$	9.51200e20	1.20306	0.601529	+	0.798851i	$0.294558\pi$
$968$	− 5.08927e20i	− 0.639040i
$969$	0	0
$970$	−6.06819e20	−0.751030
$971$	1.16400e21i	1.43027i	0.698987	+	0.715134i	$0.253634\pi$
$971$	1.16400e21i	1.43027i	−0.698987	+	0.715134i	$0.746366\pi$
$972$	0	0
$973$	2.21520e20	0.268302
$974$	− 2.88103e20i	− 0.346445i
$975$	0	0
$976$	1.68957e20	0.200275
$977$	− 1.27929e21i	− 1.50559i	−0.658253	−	0.752796i	$-0.728704\pi$
$977$	− 1.27929e21i	− 1.50559i	0.658253	−	0.752796i	$-0.271296\pi$
$978$	0	0
$979$	−2.49041e20	−0.288930
$980$	6.78434e20i	0.781493i
$981$	0	0
$982$	−4.63982e20	−0.526890
$983$	− 1.39777e21i	− 1.57602i	−0.615660	−	0.788012i	$-0.711111\pi$
$983$	− 1.39777e21i	− 1.57602i	0.615660	−	0.788012i	$-0.288889\pi$
$984$	0	0
$985$	2.91068e20	0.323549
$986$	1.13260e21i	1.25008i
$987$	0	0
$988$	1.61957e20	0.176239
$989$	4.57728e20i	0.494576i
$990$	0	0
$991$	2.30863e20	0.245945	0.122972	−	0.992410i	$-0.460757\pi$
$991$	2.30863e20	0.245945	0.122972	+	0.992410i	$0.460757\pi$
$992$	2.04474e20i	0.216300i
$993$	0	0
$994$	4.68196e20	0.488340
$995$	2.02100e21i	2.09317i
$996$	0	0
$997$	6.42123e20	0.655771	0.327885	−	0.944718i	$-0.393664\pi$
$997$	6.42123e20	0.655771	0.327885	+	0.944718i	$0.393664\pi$
$998$	− 3.75903e20i	− 0.381208i
$999$	0	0

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

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a_n

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	18.15.b.b.17.2	✓	4
3.2	odd	2	inner	18.15.b.b.17.3	yes	4
4.3	odd	2		144.15.e.b.17.4		4
12.11	even	2		144.15.e.b.17.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.15.b.b.17.2	✓	4	1.1	even	1	trivial
18.15.b.b.17.3	yes	4	3.2	odd	2	inner
144.15.e.b.17.1		4	12.11	even	2
144.15.e.b.17.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	18.15.b.b.17.2

Level	$18$
Weight	$15$
Character	18.17
Analytic conductor	$22.379$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$