LMFDB - Embedded newform 240.8.h.b.191.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,8,Mod(191,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("240.191");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$240 = 2^{4} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	240.h (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:	$74.9724061162$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.6
Character	$\chi$	$=$	240.191
Dual form			240.8.h.b.191.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-42.2282 + 20.0943i) q^{3} -125.000i q^{5} +1615.14i q^{7} +(1379.44 - 1697.09i) q^{9} -504.807 q^{11} -9494.97 q^{13} +(2511.79 + 5278.52i) q^{15} -4167.95i q^{17} +43866.8i q^{19} +(-32455.1 - 68204.2i) q^{21} +22361.0 q^{23} -15625.0 q^{25} +(-24149.1 + 99384.0i) q^{27} +244935. i q^{29} -56250.1i q^{31} +(21317.1 - 10143.8i) q^{33} +201892. q^{35} +255240. q^{37} +(400955. - 190795. i) q^{39} +70497.9i q^{41} +240483. i q^{43} +(-212137. - 172430. i) q^{45} +563322. q^{47} -1.78512e6 q^{49} +(83752.1 + 176005. i) q^{51} +1.03606e6i q^{53} +63100.9i q^{55} +(-881473. - 1.85241e6i) q^{57} -2.42345e6 q^{59} +1.04708e6 q^{61} +(2.74104e6 + 2.22798e6i) q^{63} +1.18687e6i q^{65} -93046.8i q^{67} +(-944264. + 449329. i) q^{69} -5.15867e6 q^{71} -2.53785e6 q^{73} +(659815. - 313974. i) q^{75} -815332. i q^{77} -6.21039e6i q^{79} +(-977281. - 4.68206e6i) q^{81} +9.59459e6 q^{83} -520993. q^{85} +(-4.92180e6 - 1.03432e7i) q^{87} -9.31922e6i q^{89} -1.53357e7i q^{91} +(1.13031e6 + 2.37534e6i) q^{93} +5.48334e6 q^{95} +965673. q^{97} +(-696349. + 856704. i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$36 q + 3324 q^{9} - 432 q^{13} - 22032 q^{21} - 562500 q^{25} - 386232 q^{33} + 1200048 q^{37} - 1710000 q^{45} - 3237972 q^{49} - 9331200 q^{57} + 11262360 q^{61} - 11352552 q^{69} + 37210296 q^{73} - 12564252 q^{81}+ \cdots + 94537320 q^{97}+O(q^{100})$ 36 * q + 3324 * q^9 - 432 * q^13 - 22032 * q^21 - 562500 * q^25 - 386232 * q^33 + 1200048 * q^37 - 1710000 * q^45 - 3237972 * q^49 - 9331200 * q^57 + 11262360 * q^61 - 11352552 * q^69 + 37210296 * q^73 - 12564252 * q^81 + 5271000 * q^85 + 5594400 * q^93 + 94537320 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$ .

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$	−42.2282	+	20.0943i	−0.902979	+	0.429684i
$3$	−42.2282	+	20.0943i	−0.902979	+	0.429684i
$4$		0			0
$5$		−	125.000i		−	0.447214i
$5$		−	125.000i		−	0.447214i
$6$		0			0
$7$			1615.14i			1.77978i	0.456177	+	0.889889i	$0.349218\pi$
$7$			1615.14i			1.77978i	−0.456177	+	0.889889i	$0.650782\pi$
$8$		0			0
$9$	1379.44	−	1697.09i	0.630744	−	0.775991i
$10$		0			0
$11$	−504.807			−0.114354			−0.0571769	−	0.998364i	$-0.518210\pi$
$11$	−504.807			−0.114354			−0.0571769	+	0.998364i	$0.518210\pi$
$12$		0			0
$13$	−9494.97			−1.19865			−0.599324	−	0.800506i	$-0.704564\pi$
$13$	−9494.97			−1.19865			−0.599324	+	0.800506i	$0.704564\pi$
$14$		0			0
$15$	2511.79	+	5278.52i	0.192160	+	0.403825i
$16$		0			0
$17$		−	4167.95i		−	0.205755i	−0.994694	−	0.102878i	$-0.967195\pi$
$17$		−	4167.95i		−	0.205755i	0.994694	−	0.102878i	$-0.0328050\pi$
$18$		0			0
$19$			43866.8i			1.46723i	0.679566	+	0.733614i	$0.262168\pi$
$19$			43866.8i			1.46723i	−0.679566	+	0.733614i	$0.737832\pi$
$20$		0			0
$21$	−32455.1	−	68204.2i	−0.764742	−	1.60710i
$22$		0			0
$23$	22361.0			0.383216			0.191608	−	0.981472i	$-0.438630\pi$
$23$	22361.0			0.383216			0.191608	+	0.981472i	$0.438630\pi$
$24$		0			0
$25$	−15625.0			−0.200000
$26$		0			0
$27$	−24149.1	+	99384.0i	−0.236117	+	0.971725i
$28$		0			0
$29$			244935.i			1.86491i	0.361286	+	0.932455i	$0.382338\pi$
$29$			244935.i			1.86491i	−0.361286	+	0.932455i	$0.617662\pi$
$30$		0			0
$31$		−	56250.1i		−	0.339123i	−0.985520	−	0.169561i	$-0.945765\pi$
$31$		−	56250.1i		−	0.339123i	0.985520	−	0.169561i	$-0.0542351\pi$
$32$		0			0
$33$	21317.1	−	10143.8i	0.103259	−	0.0491360i
$34$		0			0
$35$	201892.			0.795941
$36$		0			0
$37$	255240.			0.828405			0.414202	−	0.910185i	$-0.364061\pi$
$37$	255240.			0.828405			0.414202	+	0.910185i	$0.364061\pi$
$38$		0			0
$39$	400955.	−	190795.i	1.08235	−	0.515040i
$40$		0			0
$41$			70497.9i			0.159747i	0.996805	+	0.0798735i	$0.0254516\pi$
$41$			70497.9i			0.159747i	−0.996805	+	0.0798735i	$0.974548\pi$
$42$		0			0
$43$			240483.i			0.461259i	0.973042	+	0.230630i	$0.0740786\pi$
$43$			240483.i			0.461259i	−0.973042	+	0.230630i	$0.925921\pi$
$44$		0			0
$45$	−212137.	−	172430.i	−0.347034	−	0.282077i
$46$		0			0
$47$	563322.			0.791433			0.395716	−	0.918373i	$-0.370496\pi$
$47$	563322.			0.791433			0.395716	+	0.918373i	$0.370496\pi$
$48$		0			0
$49$	−1.78512e6			−2.16761
$50$		0			0
$51$	83752.1	+	176005.i	0.0884097	+	0.185793i
$52$		0			0
$53$			1.03606e6i			0.955911i	0.878384	+	0.477956i	$0.158622\pi$
$53$			1.03606e6i			0.955911i	−0.878384	+	0.477956i	$0.841378\pi$
$54$		0			0
$55$			63100.9i			0.0511406i
$56$		0			0
$57$	−881473.	−	1.85241e6i	−0.630445	−	1.32488i
$58$		0			0
$59$	−2.42345e6			−1.53622			−0.768109	−	0.640319i	$-0.778802\pi$
$59$	−2.42345e6			−1.53622			−0.768109	+	0.640319i	$0.778802\pi$
$60$		0			0
$61$	1.04708e6			0.590643			0.295321	−	0.955398i	$-0.404573\pi$
$61$	1.04708e6			0.590643			0.295321	+	0.955398i	$0.404573\pi$
$62$		0			0
$63$	2.74104e6	+	2.22798e6i	1.38109	+	1.12258i
$64$		0			0
$65$			1.18687e6i			0.536052i
$66$		0			0
$67$		−	93046.8i		−	0.0377955i	−0.999821	−	0.0188977i	$-0.993984\pi$
$67$		−	93046.8i		−	0.0377955i	0.999821	−	0.0188977i	$-0.00601569\pi$
$68$		0			0
$69$	−944264.	+	449329.i	−0.346036	+	0.164662i
$70$		0			0
$71$	−5.15867e6			−1.71054			−0.855270	−	0.518183i	$-0.826609\pi$
$71$	−5.15867e6			−1.71054			−0.855270	+	0.518183i	$0.826609\pi$
$72$		0			0
$73$	−2.53785e6			−0.763548			−0.381774	−	0.924256i	$-0.624687\pi$
$73$	−2.53785e6			−0.763548			−0.381774	+	0.924256i	$0.624687\pi$
$74$		0			0
$75$	659815.	−	313974.i	0.180596	−	0.0859368i
$76$		0			0
$77$		−	815332.i		−	0.203525i
$78$		0			0
$79$		−	6.21039e6i		−	1.41718i	−0.705622	−	0.708589i	$-0.749332\pi$
$79$		−	6.21039e6i		−	1.41718i	0.705622	−	0.708589i	$-0.250668\pi$
$80$		0			0
$81$	−977281.	−	4.68206e6i	−0.204325	−	0.978903i
$82$		0			0
$83$	9.59459e6			1.84184			0.920922	−	0.389747i	$-0.127437\pi$
$83$	9.59459e6			1.84184			0.920922	+	0.389747i	$0.127437\pi$
$84$		0			0
$85$	−520993.			−0.0920165
$86$		0			0
$87$	−4.92180e6	−	1.03432e7i	−0.801322	−	1.68398i
$88$		0			0
$89$		−	9.31922e6i		−	1.40125i	−0.713531	−	0.700623i	$-0.752905\pi$
$89$		−	9.31922e6i		−	1.40125i	0.713531	−	0.700623i	$-0.247095\pi$
$90$		0			0
$91$		−	1.53357e7i		−	2.13333i
$92$		0			0
$93$	1.13031e6	+	2.37534e6i	0.145716	+	0.306221i
$94$		0			0
$95$	5.48334e6			0.656165
$96$		0			0
$97$	965673.			0.107431			0.0537154	−	0.998556i	$-0.482894\pi$
$97$	965673.			0.107431			0.0537154	+	0.998556i	$0.482894\pi$
$98$		0			0
$99$	−696349.	+	856704.i	−0.0721280	+	0.0887376i
$100$		0			0
$101$		−	1.60909e7i		−	1.55402i	−0.629490	−	0.777008i	$-0.716736\pi$
$101$		−	1.60909e7i		−	1.55402i	0.629490	−	0.777008i	$-0.283264\pi$
$102$		0			0
$103$			1.76166e6i			0.158852i	0.996841	+	0.0794259i	$0.0253087\pi$
$103$			1.76166e6i			0.158852i	−0.996841	+	0.0794259i	$0.974691\pi$
$104$		0			0
$105$	−8.52553e6	+	4.05688e6i	−0.718718	+	0.342003i
$106$		0			0
$107$	−770387.			−0.0607947			−0.0303973	−	0.999538i	$-0.509677\pi$
$107$	−770387.			−0.0607947			−0.0303973	+	0.999538i	$0.509677\pi$
$108$		0			0
$109$	−1.64442e7			−1.21624			−0.608122	−	0.793843i	$-0.708077\pi$
$109$	−1.64442e7			−1.21624			−0.608122	+	0.793843i	$0.708077\pi$
$110$		0			0
$111$	−1.07783e7	+	5.12887e6i	−0.748032	+	0.355952i
$112$		0			0
$113$		−	7.88226e6i		−	0.513897i	−0.966425	−	0.256948i	$-0.917283\pi$
$113$		−	7.88226e6i		−	0.513897i	0.966425	−	0.256948i	$-0.0827171\pi$
$114$		0			0
$115$		−	2.79513e6i		−	0.171380i
$116$		0			0
$117$	−1.30977e7	+	1.61138e7i	−0.756040	+	0.930141i
$118$		0			0
$119$	6.73180e6			0.366199
$120$		0			0
$121$	−1.92323e7			−0.986923
$122$		0			0
$123$	−1.41661e6	−	2.97700e6i	−0.0686407	−	0.144248i
$124$		0			0
$125$			1.95312e6i			0.0894427i
$126$		0			0
$127$		−	1.47847e7i		−	0.640470i	−0.947338	−	0.320235i	$-0.896238\pi$
$127$		−	1.47847e7i		−	0.640470i	0.947338	−	0.320235i	$-0.103762\pi$
$128$		0			0
$129$	−4.83235e6	−	1.01552e7i	−0.198196	−	0.416508i
$130$		0			0
$131$	1.54930e6			0.0602125			0.0301063	−	0.999547i	$-0.490415\pi$
$131$	1.54930e6			0.0602125			0.0301063	+	0.999547i	$0.490415\pi$
$132$		0			0
$133$	−7.08508e7			−2.61134
$134$		0			0
$135$	1.24230e7	+	3.01864e6i	0.434568	+	0.105595i
$136$		0			0
$137$		−	2.36832e7i		−	0.786899i	−0.919346	−	0.393450i	$-0.871282\pi$
$137$		−	2.36832e7i		−	0.786899i	0.919346	−	0.393450i	$-0.128718\pi$
$138$		0			0
$139$			4.99958e7i			1.57900i	0.613751	+	0.789499i	$0.289660\pi$
$139$			4.99958e7i			1.57900i	−0.613751	+	0.789499i	$0.710340\pi$
$140$		0			0
$141$	−2.37880e7	+	1.13196e7i	−0.714647	+	0.340066i
$142$		0			0
$143$	4.79313e6			0.137070
$144$		0			0
$145$	3.06169e7			0.834013
$146$		0			0
$147$	7.53824e7	−	3.58708e7i	1.95731	−	0.931387i
$148$		0			0
$149$		−	2.27672e7i		−	0.563841i	−0.959438	−	0.281921i	$-0.909028\pi$
$149$		−	2.27672e7i		−	0.563841i	0.959438	−	0.281921i	$-0.0909715\pi$
$150$		0			0
$151$		−	2.66407e7i		−	0.629689i	−0.949143	−	0.314845i	$-0.898048\pi$
$151$		−	2.66407e7i		−	0.629689i	0.949143	−	0.314845i	$-0.101952\pi$
$152$		0			0
$153$	−7.07339e6	−	5.74942e6i	−0.159664	−	0.129779i
$154$		0			0
$155$	−7.03126e6			−0.151660
$156$		0			0
$157$	−5.26226e7			−1.08523			−0.542617	−	0.839980i	$-0.682567\pi$
$157$	−5.26226e7			−1.08523			−0.542617	+	0.839980i	$0.682567\pi$
$158$		0			0
$159$	−2.08188e7	−	4.37507e7i	−0.410740	−	0.863168i
$160$		0			0
$161$			3.61161e7i			0.682040i
$162$		0			0
$163$		−	4.98522e7i		−	0.901628i	−0.892618	−	0.450814i	$-0.851134\pi$
$163$		−	4.98522e7i		−	0.901628i	0.892618	−	0.450814i	$-0.148866\pi$
$164$		0			0
$165$	−1.26797e6	−	2.66463e6i	−0.0219743	−	0.0461789i
$166$		0			0
$167$	8.34222e7			1.38603			0.693017	−	0.720921i	$-0.256281\pi$
$167$	8.34222e7			1.38603			0.693017	+	0.720921i	$0.256281\pi$
$168$		0			0
$169$	2.74059e7			0.436758
$170$		0			0
$171$	7.44460e7	+	6.05114e7i	1.13856	+	0.925445i
$172$		0			0
$173$		−	3.83060e7i		−	0.562478i	−0.959638	−	0.281239i	$-0.909255\pi$
$173$		−	3.83060e7i		−	0.562478i	0.959638	−	0.281239i	$-0.0907453\pi$
$174$		0			0
$175$		−	2.52365e7i		−	0.355956i
$176$		0			0
$177$	1.02338e8	−	4.86977e7i	1.38717	−	0.660088i
$178$		0			0
$179$	−1.12691e8			−1.46859			−0.734297	−	0.678828i	$-0.762488\pi$
$179$	−1.12691e8			−1.46859			−0.734297	+	0.678828i	$0.762488\pi$
$180$		0			0
$181$	−6.48076e7			−0.812364			−0.406182	−	0.913792i	$-0.633140\pi$
$181$	−6.48076e7			−0.812364			−0.406182	+	0.913792i	$0.633140\pi$
$182$		0			0
$183$	−4.42162e7	+	2.10403e7i	−0.533338	+	0.253790i
$184$		0			0
$185$		−	3.19050e7i		−	0.370474i
$186$		0			0
$187$			2.10401e6i			0.0235289i
$188$		0			0
$189$	−1.60519e8	−	3.90041e7i	−1.72945	−	0.420237i
$190$		0			0
$191$	4.13575e7			0.429474			0.214737	−	0.976672i	$-0.431111\pi$
$191$	4.13575e7			0.429474			0.214737	+	0.976672i	$0.431111\pi$
$192$		0			0
$193$	2.40767e6			0.0241072			0.0120536	−	0.999927i	$-0.496163\pi$
$193$	2.40767e6			0.0241072			0.0120536	+	0.999927i	$0.496163\pi$
$194$		0			0
$195$	−2.38494e7	−	5.01194e7i	−0.230333	−	0.484044i
$196$		0			0
$197$			1.73789e8i			1.61953i	0.586752	+	0.809767i	$0.300406\pi$
$197$			1.73789e8i			1.61953i	−0.586752	+	0.809767i	$0.699594\pi$
$198$		0			0
$199$			1.59487e8i			1.43463i	0.696750	+	0.717314i	$0.254629\pi$
$199$			1.59487e8i			1.43463i	−0.696750	+	0.717314i	$0.745371\pi$
$200$		0			0
$201$	1.86971e6	+	3.92920e6i	0.0162401	+	0.0341285i
$202$		0			0
$203$	−3.95603e8			−3.31913
$204$		0			0
$205$	8.81224e6			0.0714410
$206$		0			0
$207$	3.08456e7	−	3.79487e7i	0.241711	−	0.297373i
$208$		0			0
$209$		−	2.21442e7i		−	0.167783i
$210$		0			0
$211$		−	2.41030e7i		−	0.176637i	−0.996092	−	0.0883187i	$-0.971851\pi$
$211$		−	2.41030e7i		−	0.176637i	0.996092	−	0.0883187i	$-0.0281494\pi$
$212$		0			0
$213$	2.17841e8	−	1.03660e8i	1.54458	−	0.734991i
$214$		0			0
$215$	3.00604e7			0.206281
$216$		0			0
$217$	9.08515e7			0.603564
$218$		0			0
$219$	1.07169e8	−	5.09964e7i	0.689468	−	0.328084i
$220$		0			0
$221$			3.95745e7i			0.246628i
$222$		0			0
$223$		−	6.15469e7i		−	0.371655i	−0.982582	−	0.185827i	$-0.940504\pi$
$223$		−	6.15469e7i		−	0.371655i	0.982582	−	0.185827i	$-0.0594965\pi$
$224$		0			0
$225$	−2.15537e7	+	2.65171e7i	−0.126149	+	0.155198i
$226$		0			0
$227$	1.46794e8			0.832950			0.416475	−	0.909147i	$-0.363265\pi$
$227$	1.46794e8			0.832950			0.416475	+	0.909147i	$0.363265\pi$
$228$		0			0
$229$	3.39135e8			1.86616			0.933079	−	0.359672i	$-0.117112\pi$
$229$	3.39135e8			1.86616			0.933079	+	0.359672i	$0.117112\pi$
$230$		0			0
$231$	1.63835e7	+	3.44300e7i	0.0874512	+	0.183778i
$232$		0			0
$233$		−	2.49003e8i		−	1.28961i	−0.764347	−	0.644805i	$-0.776939\pi$
$233$		−	2.49003e8i		−	1.28961i	0.764347	−	0.644805i	$-0.223061\pi$
$234$		0			0
$235$		−	7.04152e7i		−	0.353939i
$236$		0			0
$237$	1.24794e8	+	2.62254e8i	0.608938	+	1.27968i
$238$		0			0
$239$	−2.34396e8			−1.11060			−0.555299	−	0.831651i	$-0.687396\pi$
$239$	−2.34396e8			−1.11060			−0.555299	+	0.831651i	$0.687396\pi$
$240$		0			0
$241$	1.10391e8			0.508012			0.254006	−	0.967203i	$-0.418252\pi$
$241$	1.10391e8			0.508012			0.254006	+	0.967203i	$0.418252\pi$
$242$		0			0
$243$	1.35352e8	+	1.78077e8i	0.605120	+	0.796134i
$244$		0			0
$245$			2.23140e8i			0.969385i
$246$		0			0
$247$		−	4.16513e8i		−	1.75869i
$248$		0			0
$249$	−4.05162e8	+	1.92797e8i	−1.66315	+	0.791411i
$250$		0			0
$251$	−3.46521e8			−1.38316			−0.691580	−	0.722300i	$-0.743085\pi$
$251$	−3.46521e8			−1.38316			−0.691580	+	0.722300i	$0.743085\pi$
$252$		0			0
$253$	−1.12880e7			−0.0438223
$254$		0			0
$255$	2.20006e7	−	1.04690e7i	0.0830890	−	0.0395380i
$256$		0			0
$257$		−	1.60246e8i		−	0.588871i	−0.955671	−	0.294435i	$-0.904868\pi$
$257$		−	1.60246e8i		−	0.588871i	0.955671	−	0.294435i	$-0.0951316\pi$
$258$		0			0
$259$			4.12247e8i			1.47438i
$260$		0			0
$261$	4.15677e8	+	3.37872e8i	1.44715	+	1.17628i
$262$		0			0
$263$	3.15723e8			1.07019			0.535096	−	0.844792i	$-0.320276\pi$
$263$	3.15723e8			1.07019			0.535096	+	0.844792i	$0.320276\pi$
$264$		0			0
$265$	1.29507e8			0.427496
$266$		0			0
$267$	1.87264e8	+	3.93534e8i	0.602093	+	1.26530i
$268$		0			0
$269$		−	5.48213e8i		−	1.71718i	−0.512661	−	0.858591i	$-0.671340\pi$
$269$		−	5.48213e8i		−	1.71718i	0.512661	−	0.858591i	$-0.328660\pi$
$270$		0			0
$271$			3.67600e8i			1.12198i	0.827824	+	0.560988i	$0.189578\pi$
$271$			3.67600e8i			1.12198i	−0.827824	+	0.560988i	$0.810422\pi$
$272$		0			0
$273$	3.08160e8	+	6.47597e8i	0.916657	+	1.92635i
$274$		0			0
$275$	7.88761e6			0.0228708
$276$		0			0
$277$	3.52920e8			0.997693			0.498846	−	0.866690i	$-0.333757\pi$
$277$	3.52920e8			0.997693			0.498846	+	0.866690i	$0.333757\pi$
$278$		0			0
$279$	−9.54616e7	−	7.75934e7i	−0.263156	−	0.213900i
$280$		0			0
$281$			1.28285e7i			0.0344907i	0.999851	+	0.0172454i	$0.00548964\pi$
$281$			1.28285e7i			0.0344907i	−0.999851	+	0.0172454i	$0.994510\pi$
$282$		0			0
$283$			5.85502e8i			1.53559i	0.640694	+	0.767796i	$0.278647\pi$
$283$			5.85502e8i			1.53559i	−0.640694	+	0.767796i	$0.721353\pi$
$284$		0			0
$285$	−2.31552e8	+	1.10184e8i	−0.592503	+	0.281943i
$286$		0			0
$287$	−1.13864e8			−0.284314
$288$		0			0
$289$	3.92967e8			0.957665
$290$		0			0
$291$	−4.07786e7	+	1.94046e7i	−0.0970079	+	0.0461613i
$292$		0			0
$293$			2.73481e8i			0.635170i	0.948230	+	0.317585i	$0.102872\pi$
$293$			2.73481e8i			0.635170i	−0.948230	+	0.317585i	$0.897128\pi$
$294$		0			0
$295$			3.02932e8i			0.687017i
$296$		0			0
$297$	1.21906e7	−	5.01697e7i	0.0270009	−	0.111120i
$298$		0			0
$299$	−2.12317e8			−0.459342
$300$		0			0
$301$	−3.88413e8			−0.820939
$302$		0			0
$303$	3.23336e8	+	6.79489e8i	0.667736	+	1.40324i
$304$		0			0
$305$		−	1.30885e8i		−	0.264143i
$306$		0			0
$307$			4.53362e8i			0.894253i	0.894471	+	0.447127i	$0.147553\pi$
$307$			4.53362e8i			0.894253i	−0.894471	+	0.447127i	$0.852447\pi$
$308$		0			0
$309$	−3.53994e7	−	7.43918e7i	−0.0682561	−	0.143440i
$310$		0			0
$311$	−8.02691e8			−1.51317			−0.756584	−	0.653897i	$-0.773133\pi$
$311$	−8.02691e8			−1.51317			−0.756584	+	0.653897i	$0.773133\pi$
$312$		0			0
$313$	−6.97229e8			−1.28520			−0.642599	−	0.766202i	$-0.722144\pi$
$313$	−6.97229e8			−1.28520			−0.642599	+	0.766202i	$0.722144\pi$
$314$		0			0
$315$	2.78497e8	−	3.42629e8i	0.502035	−	0.617643i
$316$		0			0
$317$		−	5.73682e8i		−	1.01150i	−0.862681	−	0.505748i	$-0.831217\pi$
$317$		−	5.73682e8i		−	1.01150i	0.862681	−	0.505748i	$-0.168783\pi$
$318$		0			0
$319$		−	1.23645e8i		−	0.213260i
$320$		0			0
$321$	3.25320e7	−	1.54804e7i	0.0548963	−	0.0261225i
$322$		0			0
$323$	1.82834e8			0.301890
$324$		0			0
$325$	1.48359e8			0.239730
$326$		0			0
$327$	6.94410e8	−	3.30436e8i	1.09824	−	0.522601i
$328$		0			0
$329$			9.09841e8i			1.40857i
$330$		0			0
$331$		−	8.91706e8i		−	1.35152i	−0.737120	−	0.675762i	$-0.763815\pi$
$331$		−	8.91706e8i		−	1.35152i	0.737120	−	0.675762i	$-0.236185\pi$
$332$		0			0
$333$	3.52087e8	−	4.33166e8i	0.522511	−	0.642835i
$334$		0			0
$335$	−1.16309e7			−0.0169026
$336$		0			0
$337$	1.07474e9			1.52968			0.764839	−	0.644221i	$-0.222818\pi$
$337$	1.07474e9			1.52968			0.764839	+	0.644221i	$0.222818\pi$
$338$		0			0
$339$	1.58389e8	+	3.32853e8i	0.220813	+	0.464038i
$340$		0			0
$341$			2.83954e7i			0.0387800i
$342$		0			0
$343$		−	1.55308e9i		−	2.07809i
$344$		0			0
$345$	5.61662e7	+	1.18033e8i	0.0736390	+	0.154752i
$346$		0			0
$347$	−1.19579e9			−1.53639			−0.768193	−	0.640218i	$-0.778844\pi$
$347$	−1.19579e9			−1.53639			−0.768193	+	0.640218i	$0.778844\pi$
$348$		0			0
$349$	1.26415e9			1.59188			0.795939	−	0.605377i	$-0.206978\pi$
$349$	1.26415e9			1.59188			0.795939	+	0.605377i	$0.206978\pi$
$350$		0			0
$351$	2.29295e8	−	9.43648e8i	0.283022	−	1.16476i
$352$		0			0
$353$			9.52367e8i			1.15237i	0.817318	+	0.576186i	$0.195460\pi$
$353$			9.52367e8i			1.15237i	−0.817318	+	0.576186i	$0.804540\pi$
$354$		0			0
$355$			6.44833e8i			0.764977i
$356$		0			0
$357$	−2.84272e8	+	1.35271e8i	−0.330670	+	0.157350i
$358$		0			0
$359$	1.90430e7			0.0217223			0.0108611	−	0.999941i	$-0.496543\pi$
$359$	1.90430e7			0.0217223			0.0108611	+	0.999941i	$0.496543\pi$
$360$		0			0
$361$	−1.03042e9			−1.15276
$362$		0			0
$363$	8.12147e8	−	3.86461e8i	0.891171	−	0.424065i
$364$		0			0
$365$			3.17232e8i			0.341469i
$366$		0			0
$367$		−	1.91773e8i		−	0.202514i	−0.994860	−	0.101257i	$-0.967714\pi$
$367$		−	1.91773e8i		−	0.202514i	0.994860	−	0.101257i	$-0.0322865\pi$
$368$		0			0
$369$	1.19642e8	+	9.72474e7i	0.123962	+	0.100759i
$370$		0			0
$371$	−1.67337e9			−1.70131
$372$		0			0
$373$	−9.06692e8			−0.904647			−0.452323	−	0.891854i	$-0.649405\pi$
$373$	−9.06692e8			−0.904647			−0.452323	+	0.891854i	$0.649405\pi$
$374$		0			0
$375$	−3.92467e7	−	8.24769e7i	−0.0384321	−	0.0807649i
$376$		0			0
$377$		−	2.32565e9i		−	2.23537i
$378$		0			0
$379$			4.04161e8i			0.381344i	0.981654	+	0.190672i	$0.0610667\pi$
$379$			4.04161e8i			0.381344i	−0.981654	+	0.190672i	$0.938933\pi$
$380$		0			0
$381$	2.97088e8	+	6.24330e8i	0.275200	+	0.578331i
$382$		0			0
$383$	−1.70087e9			−1.54695			−0.773473	−	0.633829i	$-0.781482\pi$
$383$	−1.70087e9			−1.54695			−0.773473	+	0.633829i	$0.781482\pi$
$384$		0			0
$385$	−1.01916e8			−0.0910189
$386$		0			0
$387$	4.08122e8	+	3.31731e8i	0.357933	+	0.290936i
$388$		0			0
$389$		−	2.31704e9i		−	1.99577i	−0.0649941	−	0.997886i	$-0.520703\pi$
$389$		−	2.31704e9i		−	1.99577i	0.0649941	−	0.997886i	$-0.479297\pi$
$390$		0			0
$391$		−	9.31995e7i		−	0.0788488i
$392$		0			0
$393$	−6.54242e7	+	3.11322e7i	−0.0543707	+	0.0258723i
$394$		0			0
$395$	−7.76299e8			−0.633781
$396$		0			0
$397$	−1.68423e9			−1.35093			−0.675466	−	0.737391i	$-0.736057\pi$
$397$	−1.68423e9			−1.35093			−0.675466	+	0.737391i	$0.736057\pi$
$398$		0			0
$399$	2.99190e9	−	1.42370e9i	2.35799	−	1.12205i
$400$		0			0
$401$			1.56306e9i			1.21052i	0.796029	+	0.605259i	$0.206931\pi$
$401$			1.56306e9i			1.21052i	−0.796029	+	0.605259i	$0.793069\pi$
$402$		0			0
$403$			5.34093e8i			0.406489i
$404$		0			0
$405$	−5.85258e8	+	1.22160e8i	−0.437779	+	0.0913770i
$406$		0			0
$407$	−1.28847e8			−0.0947313
$408$		0			0
$409$	−1.70121e9			−1.22949			−0.614746	−	0.788725i	$-0.710741\pi$
$409$	−1.70121e9			−1.22949			−0.614746	+	0.788725i	$0.710741\pi$
$410$		0			0
$411$	4.75899e8	+	1.00010e9i	0.338118	+	0.710554i
$412$		0			0
$413$		−	3.91421e9i		−	2.73413i
$414$		0			0
$415$		−	1.19932e9i		−	0.823698i
$416$		0			0
$417$	−1.00463e9	−	2.11123e9i	−0.678470	−	1.42580i
$418$		0			0
$419$	1.94507e8			0.129177			0.0645887	−	0.997912i	$-0.479426\pi$
$419$	1.94507e8			0.129177			0.0645887	+	0.997912i	$0.479426\pi$
$420$		0			0
$421$	−6.02035e8			−0.393219			−0.196610	−	0.980482i	$-0.562993\pi$
$421$	−6.02035e8			−0.393219			−0.196610	+	0.980482i	$0.562993\pi$
$422$		0			0
$423$	7.77066e8	−	9.56009e8i	0.499191	−	0.614145i
$424$		0			0
$425$			6.51242e7i			0.0411510i
$426$		0			0
$427$			1.69117e9i			1.05121i
$428$		0			0
$429$	−2.02405e8	+	9.63146e7i	−0.123771	+	0.0588968i
$430$		0			0
$431$	2.04141e9			1.22817			0.614086	−	0.789239i	$-0.289525\pi$
$431$	2.04141e9			1.22817			0.614086	+	0.789239i	$0.289525\pi$
$432$		0			0
$433$	1.25079e9			0.740416			0.370208	−	0.928949i	$-0.379286\pi$
$433$	1.25079e9			0.740416			0.370208	+	0.928949i	$0.379286\pi$
$434$		0			0
$435$	−1.29289e9	+	6.15225e8i	−0.753097	+	0.358362i
$436$		0			0
$437$			9.80905e8i			0.562266i
$438$		0			0
$439$		−	2.45450e9i		−	1.38464i	−0.721590	−	0.692321i	$-0.756588\pi$
$439$		−	2.45450e9i		−	1.38464i	0.721590	−	0.692321i	$-0.243412\pi$
$440$		0			0
$441$	−2.46246e9	+	3.02952e9i	−1.36721	+	1.68205i
$442$		0			0
$443$	4.11036e8			0.224629			0.112315	−	0.993673i	$-0.464174\pi$
$443$	4.11036e8			0.224629			0.112315	+	0.993673i	$0.464174\pi$
$444$		0			0
$445$	−1.16490e9			−0.626657
$446$		0			0
$447$	4.57491e8	+	9.61415e8i	0.242274	+	0.509137i
$448$		0			0
$449$		−	1.03090e9i		−	0.537470i	−0.963214	−	0.268735i	$-0.913394\pi$
$449$		−	1.03090e9i		−	0.537470i	0.963214	−	0.268735i	$-0.0866055\pi$
$450$		0			0
$451$		−	3.55878e7i		−	0.0182677i
$452$		0			0
$453$	5.35327e8	+	1.12499e9i	0.270567	+	0.568597i
$454$		0			0
$455$	−1.91696e9			−0.954053
$456$		0			0
$457$	−1.37832e9			−0.675526			−0.337763	−	0.941231i	$-0.609670\pi$
$457$	−1.37832e9			−0.675526			−0.337763	+	0.941231i	$0.609670\pi$
$458$		0			0
$459$	4.14227e8	+	1.00652e8i	0.199937	+	0.0485824i
$460$		0			0
$461$			1.44371e9i			0.686321i	0.939277	+	0.343161i	$0.111497\pi$
$461$			1.44371e9i			0.686321i	−0.939277	+	0.343161i	$0.888503\pi$
$462$		0			0
$463$		−	1.41936e9i		−	0.664596i	−0.943174	−	0.332298i	$-0.892176\pi$
$463$		−	1.41936e9i		−	0.664596i	0.943174	−	0.332298i	$-0.107824\pi$
$464$		0			0
$465$	2.96917e8	−	1.41288e8i	0.136946	−	0.0651660i
$466$		0			0
$467$	1.96275e9			0.891777			0.445889	−	0.895088i	$-0.352888\pi$
$467$	1.96275e9			0.891777			0.445889	+	0.895088i	$0.352888\pi$
$468$		0			0
$469$	1.50283e8			0.0672676
$470$		0			0
$471$	2.22216e9	−	1.05742e9i	0.979945	−	0.466308i
$472$		0			0
$473$		−	1.21398e8i		−	0.0527468i
$474$		0			0
$475$		−	6.85418e8i		−	0.293446i
$476$		0			0
$477$	1.75828e9	+	1.42917e9i	0.741779	+	0.602935i
$478$		0			0
$479$	2.17684e9			0.905010			0.452505	−	0.891762i	$-0.350531\pi$
$479$	2.17684e9			0.905010			0.452505	+	0.891762i	$0.350531\pi$
$480$		0			0
$481$	−2.42350e9			−0.992966
$482$		0			0
$483$	−7.25728e8	−	1.52512e9i	−0.293062	−	0.615868i
$484$		0			0
$485$		−	1.20709e8i		−	0.0480446i
$486$		0			0
$487$			2.14419e9i			0.841226i	0.907240	+	0.420613i	$0.138185\pi$
$487$			2.14419e9i			0.841226i	−0.907240	+	0.420613i	$0.861815\pi$
$488$		0			0
$489$	1.00175e9	+	2.10517e9i	0.387415	+	0.814151i
$490$		0			0
$491$	−1.00071e9			−0.381524			−0.190762	−	0.981636i	$-0.561096\pi$
$491$	−1.00071e9			−0.381524			−0.190762	+	0.981636i	$0.561096\pi$
$492$		0			0
$493$	1.02088e9			0.383715
$494$		0			0
$495$	1.07088e8	+	8.70436e7i	0.0396847	+	0.0322566i
$496$		0			0
$497$		−	8.33195e9i		−	3.04438i
$498$		0			0
$499$		−	5.29357e8i		−	0.190720i	−0.995443	−	0.0953601i	$-0.969600\pi$
$499$		−	5.29357e8i		−	0.190720i	0.995443	−	0.0953601i	$-0.0304003\pi$
$500$		0			0
$501$	−3.52277e9	+	1.67631e9i	−1.25156	+	0.595557i
$502$		0			0
$503$	−8.79475e8			−0.308131			−0.154066	−	0.988061i	$-0.549237\pi$
$503$	−8.79475e8			−0.308131			−0.154066	+	0.988061i	$0.549237\pi$
$504$		0			0
$505$	−2.01136e9			−0.694977
$506$		0			0
$507$	−1.15730e9	+	5.50704e8i	−0.394384	+	0.187668i
$508$		0			0
$509$			1.10437e9i			0.371196i	0.982626	+	0.185598i	$0.0594221\pi$
$509$			1.10437e9i			0.371196i	−0.982626	+	0.185598i	$0.940578\pi$
$510$		0			0
$511$		−	4.09898e9i		−	1.35895i
$512$		0			0
$513$	−4.35965e9	−	1.05934e9i	−1.42574	−	0.346438i
$514$		0			0
$515$	2.20208e8			0.0710407
$516$		0			0
$517$	−2.84369e8			−0.0905034
$518$		0			0
$519$	7.69733e8	+	1.61759e9i	0.241688	+	0.507906i
$520$		0			0
$521$			1.09267e9i			0.338497i	0.985573	+	0.169249i	$0.0541341\pi$
$521$			1.09267e9i			0.338497i	−0.985573	+	0.169249i	$0.945866\pi$
$522$		0			0
$523$			4.10862e9i			1.25586i	0.778271	+	0.627929i	$0.216097\pi$
$523$			4.10862e9i			1.25586i	−0.778271	+	0.627929i	$0.783903\pi$
$524$		0			0
$525$	5.07110e8	+	1.06569e9i	0.152948	+	0.321421i
$526$		0			0
$527$	−2.34447e8			−0.0697763
$528$		0			0
$529$	−2.90481e9			−0.853145
$530$		0			0
$531$	−3.34300e9	+	4.11283e9i	−0.968959	+	1.19209i
$532$		0			0
$533$		−	6.69375e8i		−	0.191481i
$534$		0			0
$535$			9.62983e7i			0.0271882i
$536$		0			0
$537$	4.75872e9	−	2.26444e9i	1.32611	−	0.631031i
$538$		0			0
$539$	9.01141e8			0.247875
$540$		0			0
$541$	3.93556e9			1.06860			0.534301	−	0.845294i	$-0.320575\pi$
$541$	3.93556e9			1.06860			0.534301	+	0.845294i	$0.320575\pi$
$542$		0			0
$543$	2.73670e9	−	1.30226e9i	0.733548	−	0.349060i
$544$		0			0
$545$			2.05553e9i			0.543921i
$546$		0			0
$547$			2.60747e9i			0.681183i	0.940211	+	0.340591i	$0.110627\pi$
$547$			2.60747e9i			0.681183i	−0.940211	+	0.340591i	$0.889373\pi$
$548$		0			0
$549$	1.44438e9	−	1.77699e9i	0.372544	−	0.458334i
$550$		0			0
$551$	−1.07445e10			−2.73625
$552$		0			0
$553$	1.00306e10			2.52226
$554$		0			0
$555$	6.41109e8	+	1.34729e9i	0.159187	+	0.334530i
$556$		0			0
$557$		−	7.17366e9i		−	1.75893i	−0.475967	−	0.879463i	$-0.657902\pi$
$557$		−	7.17366e9i		−	1.75893i	0.475967	−	0.879463i	$-0.342098\pi$
$558$		0			0
$559$		−	2.28338e9i		−	0.552888i
$560$		0			0
$561$	−4.22786e7	−	8.88484e7i	−0.0101100	−	0.0212461i
$562$		0			0
$563$	−1.65421e9			−0.390670			−0.195335	−	0.980737i	$-0.562579\pi$
$563$	−1.65421e9			−0.390670			−0.195335	+	0.980737i	$0.562579\pi$
$564$		0			0
$565$	−9.85282e8			−0.229822
$566$		0			0
$567$	7.56217e9	−	1.57844e9i	1.74223	−	0.363653i
$568$		0			0
$569$		−	2.33458e9i		−	0.531270i	−0.964074	−	0.265635i	$-0.914418\pi$
$569$		−	2.33458e9i		−	0.531270i	0.964074	−	0.265635i	$-0.0855815\pi$
$570$		0			0
$571$		−	1.86071e9i		−	0.418267i	−0.977887	−	0.209133i	$-0.932936\pi$
$571$		−	1.86071e9i		−	0.418267i	0.977887	−	0.209133i	$-0.0670643\pi$
$572$		0			0
$573$	−1.74645e9	+	8.31050e8i	−0.387806	+	0.184538i
$574$		0			0
$575$	−3.49391e8			−0.0766433
$576$		0			0
$577$	5.80765e9			1.25859			0.629296	−	0.777166i	$-0.283343\pi$
$577$	5.80765e9			1.25859			0.629296	+	0.777166i	$0.283343\pi$
$578$		0			0
$579$	−1.01672e8	+	4.83806e7i	−0.0217683	+	0.0103585i
$580$		0			0
$581$			1.54966e10i			3.27807i
$582$		0			0
$583$		−	5.23008e8i		−	0.109312i
$584$		0			0
$585$	2.01423e9	+	1.63721e9i	0.415972	+	0.338111i
$586$		0			0
$587$	−5.36539e9			−1.09488			−0.547441	−	0.836844i	$-0.684398\pi$
$587$	−5.36539e9			−1.09488			−0.547441	+	0.836844i	$0.684398\pi$
$588$		0			0
$589$	2.46751e9			0.497571
$590$		0			0
$591$	−3.49217e9	−	7.33878e9i	−0.695887	−	1.46241i
$592$		0			0
$593$			7.09204e9i			1.39662i	0.715794	+	0.698312i	$0.246065\pi$
$593$			7.09204e9i			1.39662i	−0.715794	+	0.698312i	$0.753935\pi$
$594$		0			0
$595$		−	8.41475e8i		−	0.163769i
$596$		0			0
$597$	−3.20478e9	−	6.73484e9i	−0.616437	−	1.29544i
$598$		0			0
$599$	9.80130e8			0.186333			0.0931665	−	0.995651i	$-0.470301\pi$
$599$	9.80130e8			0.186333			0.0931665	+	0.995651i	$0.470301\pi$
$600$		0			0
$601$	2.89051e9			0.543142			0.271571	−	0.962418i	$-0.412457\pi$
$601$	2.89051e9			0.543142			0.271571	+	0.962418i	$0.412457\pi$
$602$		0			0
$603$	−1.57909e8	−	1.28352e8i	−0.0293290	−	0.0238392i
$604$		0			0
$605$			2.40404e9i			0.441365i
$606$		0			0
$607$		−	3.14712e9i		−	0.571153i	−0.958356	−	0.285576i	$-0.907815\pi$
$607$		−	3.14712e9i		−	0.571153i	0.958356	−	0.285576i	$-0.0921851\pi$
$608$		0			0
$609$	1.67056e10	−	7.94938e9i	2.99710	−	1.42617i
$610$		0			0
$611$	−5.34872e9			−0.948649
$612$		0			0
$613$	−9.58526e9			−1.68071			−0.840354	−	0.542038i	$-0.817653\pi$
$613$	−9.58526e9			−1.68071			−0.840354	+	0.542038i	$0.817653\pi$
$614$		0			0
$615$	−3.72125e8	+	1.77076e8i	−0.0645098	+	0.0306971i
$616$		0			0
$617$			6.12833e9i			1.05037i	0.850987	+	0.525187i	$0.176005\pi$
$617$			6.12833e9i			1.05037i	−0.850987	+	0.525187i	$0.823995\pi$
$618$		0			0
$619$			6.74204e9i			1.14255i	0.820760	+	0.571274i	$0.193550\pi$
$619$			6.74204e9i			1.14255i	−0.820760	+	0.571274i	$0.806450\pi$
$620$		0			0
$621$	−5.39999e8	+	2.22233e9i	−0.0904841	+	0.372381i
$622$		0			0
$623$	1.50518e10			2.49391
$624$		0			0
$625$	2.44141e8			0.0400000
$626$		0			0
$627$	4.44974e8	+	9.35111e8i	0.0720938	+	0.151505i
$628$		0			0
$629$		−	1.06383e9i		−	0.170449i
$630$		0			0
$631$		−	1.01052e9i		−	0.160119i	−0.996790	−	0.0800596i	$-0.974489\pi$
$631$		−	1.01052e9i		−	0.160119i	0.996790	−	0.0800596i	$-0.0255111\pi$
$632$		0			0
$633$	4.84333e8	+	1.01783e9i	0.0758982	+	0.159500i
$634$		0			0
$635$	−1.84809e9			−0.286427
$636$		0			0
$637$	1.69497e10			2.59820
$638$		0			0
$639$	−7.11605e9	+	8.75474e9i	−1.07891	+	1.32736i
$640$		0			0
$641$		−	9.57550e8i		−	0.143601i	−0.997419	−	0.0718007i	$-0.977125\pi$
$641$		−	9.57550e8i		−	0.143601i	0.997419	−	0.0718007i	$-0.0228745\pi$
$642$		0			0
$643$			1.30492e10i			1.93573i	0.251466	+	0.967866i	$0.419087\pi$
$643$			1.30492e10i			1.93573i	−0.251466	+	0.967866i	$0.580913\pi$
$644$		0			0
$645$	−1.26940e9	+	6.04043e8i	−0.186268	+	0.0886358i
$646$		0			0
$647$	1.01818e10			1.47796			0.738978	−	0.673729i	$-0.235309\pi$
$647$	1.01818e10			1.47796			0.738978	+	0.673729i	$0.235309\pi$
$648$		0			0
$649$	1.22338e9			0.175672
$650$		0			0
$651$	−3.83649e9	+	1.82560e9i	−0.545005	+	0.259342i
$652$		0			0
$653$		−	7.76251e8i		−	0.109095i	−0.998511	−	0.0545476i	$-0.982628\pi$
$653$		−	7.76251e8i		−	0.109095i	0.998511	−	0.0545476i	$-0.0173717\pi$
$654$		0			0
$655$		−	1.93663e8i		−	0.0269279i
$656$		0			0
$657$	−3.50081e9	+	4.30697e9i	−0.481603	+	0.592507i
$658$		0			0
$659$	−4.11575e9			−0.560210			−0.280105	−	0.959969i	$-0.590369\pi$
$659$	−4.11575e9			−0.560210			−0.280105	+	0.959969i	$0.590369\pi$
$660$		0			0
$661$	−1.03507e10			−1.39400			−0.697000	−	0.717071i	$-0.745482\pi$
$661$	−1.03507e10			−1.39400			−0.697000	+	0.717071i	$0.745482\pi$
$662$		0			0
$663$	−7.95223e8	−	1.67116e9i	−0.105972	−	0.222700i
$664$		0			0
$665$			8.85634e9i			1.16783i
$666$		0			0
$667$			5.47699e9i			0.714664i
$668$		0			0
$669$	1.23674e9	+	2.59901e9i	0.159694	+	0.335596i
$670$		0			0
$671$	−5.28572e8			−0.0675423
$672$		0			0
$673$	5.33940e9			0.675211			0.337605	−	0.941288i	$-0.390383\pi$
$673$	5.33940e9			0.675211			0.337605	+	0.941288i	$0.390383\pi$
$674$		0			0
$675$	3.77330e8	−	1.55287e9i	0.0472235	−	0.194345i
$676$		0			0
$677$		−	2.48269e9i		−	0.307512i	−0.988109	−	0.153756i	$-0.950863\pi$
$677$		−	2.48269e9i		−	0.307512i	0.988109	−	0.153756i	$-0.0491369\pi$
$678$		0			0
$679$			1.55969e9i			0.191203i
$680$		0			0
$681$	−6.19886e9	+	2.94973e9i	−0.752136	+	0.357905i
$682$		0			0
$683$	−4.31053e9			−0.517676			−0.258838	−	0.965921i	$-0.583340\pi$
$683$	−4.31053e9			−0.517676			−0.258838	+	0.965921i	$0.583340\pi$
$684$		0			0
$685$	−2.96041e9			−0.351912
$686$		0			0
$687$	−1.43210e10	+	6.81469e9i	−1.68510	+	0.801858i
$688$		0			0
$689$		−	9.83732e9i		−	1.14580i
$690$		0			0
$691$			8.48943e9i			0.978826i	0.872052	+	0.489413i	$0.162789\pi$
$691$			8.48943e9i			0.978826i	−0.872052	+	0.489413i	$0.837211\pi$
$692$		0			0
$693$	−1.38369e9	−	1.12470e9i	−0.157933	−	0.128372i
$694$		0			0
$695$	6.24948e9			0.706150
$696$		0			0
$697$	2.93832e8			0.0328688
$698$		0			0
$699$	5.00354e9	+	1.05149e10i	0.554124	+	1.16449i
$700$		0			0
$701$			1.03504e10i			1.13487i	0.823418	+	0.567435i	$0.192064\pi$
$701$			1.03504e10i			1.13487i	−0.823418	+	0.567435i	$0.807936\pi$
$702$		0			0
$703$			1.11965e10i			1.21546i
$704$		0			0
$705$	1.41495e9	+	2.97350e9i	0.152082	+	0.319600i
$706$		0			0
$707$	2.59890e10			2.76580
$708$		0			0
$709$	−6.78725e9			−0.715208			−0.357604	−	0.933873i	$-0.616406\pi$
$709$	−6.78725e9			−0.715208			−0.357604	+	0.933873i	$0.616406\pi$
$710$		0			0
$711$	−1.05396e10	−	8.56684e9i	−1.09972	−	0.893876i
$712$		0			0
$713$		−	1.25781e9i		−	0.129957i
$714$		0			0
$715$		−	5.99141e8i		−	0.0612996i
$716$		0			0
$717$	9.89810e9	−	4.71002e9i	1.00285	−	0.477206i
$718$		0			0
$719$	5.72358e9			0.574271			0.287135	−	0.957890i	$-0.407297\pi$
$719$	5.72358e9			0.574271			0.287135	+	0.957890i	$0.407297\pi$
$720$		0			0
$721$	−2.84532e9			−0.282721
$722$		0			0
$723$	−4.66161e9	+	2.21823e9i	−0.458724	+	0.218285i
$724$		0			0
$725$		−	3.82711e9i		−	0.372982i
$726$		0			0
$727$		−	9.50328e9i		−	0.917282i	−0.888622	−	0.458641i	$-0.848336\pi$
$727$		−	9.50328e9i		−	0.917282i	0.888622	−	0.458641i	$-0.151664\pi$
$728$		0			0
$729$	−9.29399e9	−	4.80007e9i	−0.888497	−	0.458882i
$730$		0			0
$731$	1.00232e9			0.0949065
$732$		0			0
$733$	7.87003e9			0.738096			0.369048	−	0.929410i	$-0.379684\pi$
$733$	7.87003e9			0.738096			0.369048	+	0.929410i	$0.379684\pi$
$734$		0			0
$735$	−4.48385e9	−	9.42280e9i	−0.416529	−	0.875335i
$736$		0			0
$737$			4.69707e7i			0.00432206i
$738$		0			0
$739$		−	5.91622e9i		−	0.539248i	−0.962966	−	0.269624i	$-0.913101\pi$
$739$		−	5.91622e9i		−	0.539248i	0.962966	−	0.269624i	$-0.0868995\pi$
$740$		0			0
$741$	8.36956e9	+	1.75886e10i	0.755681	+	1.58806i
$742$		0			0
$743$	−1.35359e10			−1.21067			−0.605336	−	0.795970i	$-0.706961\pi$
$743$	−1.35359e10			−1.21067			−0.605336	+	0.795970i	$0.706961\pi$
$744$		0			0
$745$	−2.84590e9			−0.252158
$746$		0			0
$747$	1.32351e10	−	1.62829e10i	1.16173	−	1.42926i
$748$		0			0
$749$		−	1.24428e9i		−	0.108201i
$750$		0			0
$751$		−	6.39015e7i		−	0.00550517i	−0.999996	−	0.00275259i	$-0.999124\pi$
$751$		−	6.39015e7i		−	0.00550517i	0.999996	−	0.00275259i	$-0.000876177\pi$
$752$		0			0
$753$	1.46330e10	−	6.96312e9i	1.24896	−	0.594321i
$754$		0			0
$755$	−3.33009e9			−0.281606
$756$		0			0
$757$	1.43927e10			1.20589			0.602945	−	0.797783i	$-0.293994\pi$
$757$	1.43927e10			1.20589			0.602945	+	0.797783i	$0.293994\pi$
$758$		0			0
$759$	4.76671e8	−	2.26825e8i	0.0395706	−	0.0188297i
$760$		0			0
$761$			1.66521e10i			1.36969i	0.728688	+	0.684846i	$0.240131\pi$
$761$			1.66521e10i			1.36969i	−0.728688	+	0.684846i	$0.759869\pi$
$762$		0			0
$763$		−	2.65597e10i		−	2.16465i
$764$		0			0
$765$	−7.18677e8	+	8.84174e8i	−0.0580388	+	0.0714040i
$766$		0			0
$767$	2.30106e10			1.84138
$768$		0			0
$769$	−1.01628e10			−0.805879			−0.402939	−	0.915227i	$-0.632011\pi$
$769$	−1.01628e10			−0.805879			−0.402939	+	0.915227i	$0.632011\pi$
$770$		0			0
$771$	3.22003e9	+	6.76688e9i	0.253028	+	0.531738i
$772$		0			0
$773$		−	2.25356e10i		−	1.75485i	−0.479711	−	0.877426i	$-0.659259\pi$
$773$		−	2.25356e10i		−	1.75485i	0.479711	−	0.877426i	$-0.340741\pi$
$774$		0			0
$775$			8.78907e8i			0.0678246i
$776$		0			0
$777$	−8.28383e9	−	1.74084e10i	−0.633516	−	1.33133i
$778$		0			0
$779$	−3.09251e9			−0.234385
$780$		0			0
$781$	2.60413e9			0.195607
$782$		0			0
$783$	−2.43426e10	−	5.91496e9i	−1.81218	−	0.440338i
$784$		0			0
$785$			6.57783e9i			0.485332i
$786$		0			0
$787$		−	2.81862e6i		−	0.000206122i	−1.00000		0.000103061i	$-0.999967\pi$
$787$		−	2.81862e6i		−	0.000206122i	1.00000		0.000103061i	$-3.28054e-5\pi$
$788$		0			0
$789$	−1.33324e10	+	6.34424e9i	−0.966361	+	0.459844i
$790$		0			0
$791$	1.27309e10			0.914623
$792$		0			0
$793$	−9.94198e9			−0.707973
$794$		0			0
$795$	−5.46884e9	+	2.60235e9i	−0.386020	+	0.183688i
$796$		0			0
$797$		−	2.60269e10i		−	1.82103i	−0.413472	−	0.910517i	$-0.635684\pi$
$797$		−	2.60269e10i		−	1.82103i	0.413472	−	0.910517i	$-0.364316\pi$
$798$		0			0
$799$		−	2.34789e9i		−	0.162841i
$800$		0			0
$801$	−1.58156e10	−	1.28553e10i	−1.08736	−	0.883828i
$802$		0			0
$803$	1.28113e9			0.0873147
$804$		0			0
$805$	4.51451e9			0.305018
$806$		0			0
$807$	1.10160e10	+	2.31500e10i	0.737845	+	1.55058i
$808$		0			0
$809$		−	2.65307e9i		−	0.176168i	−0.996113	−	0.0880842i	$-0.971926\pi$
$809$		−	2.65307e9i		−	0.176168i	0.996113	−	0.0880842i	$-0.0280745\pi$
$810$		0			0
$811$		−	1.63597e10i		−	1.07697i	−0.842637	−	0.538483i	$-0.818998\pi$
$811$		−	1.63597e10i		−	1.07697i	0.842637	−	0.538483i	$-0.181002\pi$
$812$		0			0
$813$	−7.38668e9	−	1.55231e10i	−0.482095	−	1.01312i
$814$		0			0
$815$	−6.23152e9			−0.403220
$816$		0			0
$817$	−1.05492e10			−0.676773
$818$		0			0
$819$	−2.60260e10	−	2.11546e10i	−1.65544	−	1.34558i
$820$		0			0
$821$			1.58613e10i			1.00032i	0.865933	+	0.500160i	$0.166725\pi$
$821$			1.58613e10i			1.00032i	−0.865933	+	0.500160i	$0.833275\pi$
$822$		0			0
$823$			1.96724e9i			0.123015i	0.998107	+	0.0615075i	$0.0195908\pi$
$823$			1.96724e9i			0.123015i	−0.998107	+	0.0615075i	$0.980409\pi$
$824$		0			0
$825$	−3.33079e8	+	1.58496e8i	−0.0206518	+	0.00982720i
$826$		0			0
$827$	2.14499e9			0.131873			0.0659365	−	0.997824i	$-0.478997\pi$
$827$	2.14499e9			0.131873			0.0659365	+	0.997824i	$0.478997\pi$
$828$		0			0
$829$	1.64960e10			1.00563			0.502815	−	0.864394i	$-0.332298\pi$
$829$	1.64960e10			1.00563			0.502815	+	0.864394i	$0.332298\pi$
$830$		0			0
$831$	−1.49032e10	+	7.09168e9i	−0.900896	+	0.428692i
$832$		0			0
$833$			7.44029e9i			0.445997i
$834$		0			0
$835$		−	1.04278e10i		−	0.619853i
$836$		0			0
$837$	5.59036e9	+	1.35839e9i	0.329534	+	0.0800729i
$838$		0			0
$839$	1.19250e10			0.697093			0.348547	−	0.937291i	$-0.386675\pi$
$839$	1.19250e10			0.697093			0.348547	+	0.937291i	$0.386675\pi$
$840$		0			0
$841$	−4.27433e10			−2.47789
$842$		0			0
$843$	−2.57779e8	−	5.41722e8i	−0.0148201	−	0.0311444i
$844$		0			0
$845$		−	3.42574e9i		−	0.195324i
$846$		0			0
$847$		−	3.10628e10i		−	1.75650i
$848$		0			0
$849$	−1.17653e10	−	2.47247e10i	−0.659819	−	1.38661i
$850$		0			0
$851$	5.70742e9			0.317458
$852$		0			0
$853$	−2.38426e10			−1.31532			−0.657660	−	0.753315i	$-0.728454\pi$
$853$	−2.38426e10			−1.31532			−0.657660	+	0.753315i	$0.728454\pi$
$854$		0			0
$855$	7.56392e9	−	9.30575e9i	0.413872	−	0.509178i
$856$		0			0
$857$			2.02537e10i			1.09919i	0.835432	+	0.549594i	$0.185217\pi$
$857$			2.02537e10i			1.09919i	−0.835432	+	0.549594i	$0.814783\pi$
$858$		0			0
$859$			2.37832e10i			1.28025i	0.768271	+	0.640125i	$0.221117\pi$
$859$			2.37832e10i			1.28025i	−0.768271	+	0.640125i	$0.778883\pi$
$860$		0			0
$861$	4.80826e9	−	2.28801e9i	0.256730	−	0.122165i
$862$		0			0
$863$	5.90368e9			0.312669			0.156335	−	0.987704i	$-0.450032\pi$
$863$	5.90368e9			0.312669			0.156335	+	0.987704i	$0.450032\pi$
$864$		0			0
$865$	−4.78825e9			−0.251548
$866$		0			0
$867$	−1.65943e10	+	7.89640e9i	−0.864752	+	0.411493i
$868$		0			0
$869$			3.13505e9i			0.162060i
$870$		0			0
$871$			8.83477e8i			0.0453035i
$872$		0			0
$873$	1.33208e9	−	1.63884e9i	0.0677613	−	0.0833654i
$874$		0			0
$875$	−3.15456e9			−0.159188
$876$		0			0
$877$	−1.26583e10			−0.633689			−0.316844	−	0.948478i	$-0.602623\pi$
$877$	−1.26583e10			−0.633689			−0.316844	+	0.948478i	$0.602623\pi$
$878$		0			0
$879$	−5.49541e9	−	1.15486e10i	−0.272922	−	0.573546i
$880$		0			0
$881$			1.61475e10i			0.795588i	0.917475	+	0.397794i	$0.130224\pi$
$881$			1.61475e10i			0.795588i	−0.917475	+	0.397794i	$0.869776\pi$
$882$		0			0
$883$			7.86495e9i			0.384445i	0.981351	+	0.192222i	$0.0615695\pi$
$883$			7.86495e9i			0.384445i	−0.981351	+	0.192222i	$0.938431\pi$
$884$		0			0
$885$	−6.08721e9	−	1.27923e10i	−0.295200	−	0.620363i
$886$		0			0
$887$	−7.42118e9			−0.357059			−0.178530	−	0.983935i	$-0.557134\pi$
$887$	−7.42118e9			−0.357059			−0.178530	+	0.983935i	$0.557134\pi$
$888$		0			0
$889$	2.38793e10			1.13990
$890$		0			0
$891$	4.93338e8	+	2.36354e9i	0.0233654	+	0.111941i
$892$		0			0
$893$			2.47111e10i			1.16121i
$894$		0			0
$895$			1.40863e10i			0.656775i
$896$		0			0
$897$	8.96576e9	−	4.26637e9i	0.414776	−	0.197372i
$898$		0			0
$899$	1.37776e10			0.632434
$900$		0			0
$901$	4.31822e9			0.196684
$902$		0			0
$903$	1.64020e10	−	7.80490e9i	0.741291	−	0.352744i
$904$		0			0
$905$			8.10095e9i			0.363300i
$906$		0			0
$907$			8.93722e9i			0.397720i	0.980028	+	0.198860i	$0.0637238\pi$
$907$			8.93722e9i			0.397720i	−0.980028	+	0.198860i	$0.936276\pi$
$908$		0			0
$909$	−2.73078e10	−	2.21964e10i	−1.20590	−	0.980186i
$910$		0			0
$911$	3.26257e9			0.142970			0.0714852	−	0.997442i	$-0.477226\pi$
$911$	3.26257e9			0.142970			0.0714852	+	0.997442i	$0.477226\pi$
$912$		0			0
$913$	−4.84341e9			−0.210622
$914$		0			0
$915$	2.63004e9	+	5.52703e9i	0.113498	+	0.238516i
$916$		0			0
$917$			2.50233e9i			0.107165i
$918$		0			0
$919$			1.09530e10i			0.465508i	0.972536	+	0.232754i	$0.0747738\pi$
$919$			1.09530e10i			0.465508i	−0.972536	+	0.232754i	$0.925226\pi$
$920$		0			0
$921$	−9.11000e9	−	1.91446e10i	−0.384246	−	0.807492i
$922$		0			0
$923$	4.89814e10			2.05034
$924$		0			0
$925$	−3.98812e9			−0.165681
$926$		0			0
$927$	2.98970e9	+	2.43010e9i	0.123268	+	0.100195i
$928$		0			0
$929$			8.03507e9i			0.328802i	0.986394	+	0.164401i	$0.0525692\pi$
$929$			8.03507e9i			0.328802i	−0.986394	+	0.164401i	$0.947431\pi$
$930$		0			0
$931$		−	7.83074e10i		−	3.18038i
$932$		0			0
$933$	3.38962e10	−	1.61295e10i	1.36636	−	0.650183i
$934$		0			0
$935$	2.63001e8			0.0105224
$936$		0			0
$937$	−2.30635e10			−0.915876			−0.457938	−	0.888984i	$-0.651412\pi$
$937$	−2.30635e10			−0.915876			−0.457938	+	0.888984i	$0.651412\pi$
$938$		0			0
$939$	2.94427e10	−	1.40103e10i	1.16051	−	0.552229i
$940$		0			0
$941$		−	3.74186e10i		−	1.46394i	−0.681336	−	0.731971i	$-0.738601\pi$
$941$		−	3.74186e10i		−	1.46394i	0.681336	−	0.731971i	$-0.261399\pi$
$942$		0			0
$943$			1.57640e9i			0.0612177i
$944$		0			0
$945$	−4.87551e9	+	2.00648e10i	−0.187936	+	0.773435i
$946$		0			0
$947$	−3.38155e9			−0.129387			−0.0646936	−	0.997905i	$-0.520607\pi$
$947$	−3.38155e9			−0.129387			−0.0646936	+	0.997905i	$0.520607\pi$
$948$		0			0
$949$	2.40968e10			0.915226
$950$		0			0
$951$	1.15278e10	+	2.42255e10i	0.434623	+	0.913359i
$952$		0			0
$953$			1.32817e10i			0.497081i	0.968621	+	0.248541i	$0.0799510\pi$
$953$			1.32817e10i			0.497081i	−0.968621	+	0.248541i	$0.920049\pi$
$954$		0			0
$955$		−	5.16968e9i		−	0.192067i
$956$		0			0
$957$	2.48456e9	+	5.22130e9i	0.0916342	+	0.192569i
$958$		0			0
$959$	3.82516e10			1.40051
$960$		0			0
$961$	2.43485e10			0.884996
$962$		0			0
$963$	−1.06270e9	+	1.30742e9i	−0.0383458	+	0.0471761i
$964$		0			0
$965$		−	3.00959e8i		−	0.0107811i
$966$		0			0
$967$			2.27063e10i			0.807520i	0.914865	+	0.403760i	$0.132297\pi$
$967$			2.27063e10i			0.807520i	−0.914865	+	0.403760i	$0.867703\pi$
$968$		0			0
$969$	−7.72076e9	+	3.67393e9i	−0.272600	+	0.129717i
$970$		0			0
$971$	−2.81831e10			−0.987920			−0.493960	−	0.869485i	$-0.664451\pi$
$971$	−2.81831e10			−0.987920			−0.493960	+	0.869485i	$0.664451\pi$
$972$		0			0
$973$	−8.07500e10			−2.81027
$974$		0			0
$975$	−6.26492e9	+	2.98117e9i	−0.216471	+	0.103008i
$976$		0			0
$977$		−	3.61046e10i		−	1.23860i	−0.785154	−	0.619301i	$-0.787416\pi$
$977$		−	3.61046e10i		−	1.23860i	0.785154	−	0.619301i	$-0.212584\pi$
$978$		0			0
$979$			4.70441e9i			0.160238i
$980$		0			0
$981$	−2.26838e10	+	2.79074e10i	−0.767139	+	0.943795i
$982$		0			0
$983$	−1.28740e10			−0.432292			−0.216146	−	0.976361i	$-0.569349\pi$
$983$	−1.28740e10			−0.432292			−0.216146	+	0.976361i	$0.569349\pi$
$984$		0			0
$985$	2.17236e10			0.724277
$986$		0			0
$987$	−1.82826e10	−	3.84209e10i	−0.605242	−	1.27191i
$988$		0			0
$989$			5.37745e9i			0.176762i
$990$		0			0
$991$			4.85482e10i			1.58458i	0.610142	+	0.792292i	$0.291112\pi$
$991$			4.85482e10i			1.58458i	−0.610142	+	0.792292i	$0.708888\pi$
$992$		0			0
$993$	1.79182e10	+	3.76551e10i	0.580728	+	1.22040i
$994$		0			0
$995$	1.99359e10			0.641585
$996$		0			0
$997$	−2.65563e10			−0.848661			−0.424331	−	0.905507i	$-0.639491\pi$
$997$	−2.65563e10			−0.848661			−0.424331	+	0.905507i	$0.639491\pi$
$998$		0			0
$999$	−6.16382e9	+	2.53668e10i	−0.195601	+	0.804981i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.8.h.b.191.6	yes	36
3.2	odd	2	inner	240.8.h.b.191.32	yes	36
4.3	odd	2	inner	240.8.h.b.191.31	yes	36
12.11	even	2	inner	240.8.h.b.191.5	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.8.h.b.191.5	✓	36	12.11	even	2	inner
240.8.h.b.191.6	yes	36	1.1	even	1	trivial
240.8.h.b.191.31	yes	36	4.3	odd	2	inner
240.8.h.b.191.32	yes	36	3.2	odd	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

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$q$ -expansion

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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	240.8.h.b.191.6

Level	$240$
Weight	$8$
Character	240.191
Analytic conductor	$74.972$
Analytic rank	$0$
Dimension	$36$
Inner twists	$4$