LMFDB - Embedded newform 1386.2.e.b.307.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(307,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1386.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1386.e (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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 23x^{4} + 1$ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			307.5
Root			$-1.54779 + 1.54779i$ of defining polynomial
Character	$\chi$	$=$	1386.307
Dual form			1386.2.e.b.307.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -3.09557i q^{5} +(2.44949 + 1.00000i) q^{7} -1.00000i q^{8} +3.09557 q^{10} +(1.79129 + 2.79129i) q^{11} +0.646084 q^{13} +(-1.00000 + 2.44949i) q^{14} +1.00000 q^{16} +3.74166 q^{17} -1.80341 q^{19} +3.09557i q^{20} +(-2.79129 + 1.79129i) q^{22} -4.00000 q^{23} -4.58258 q^{25} +0.646084i q^{26} +(-2.44949 - 1.00000i) q^{28} -1.58258i q^{29} -8.64064i q^{31} +1.00000i q^{32} +3.74166i q^{34} +(3.09557 - 7.58258i) q^{35} +3.58258 q^{37} -1.80341i q^{38} -3.09557 q^{40} +9.93280 q^{41} -7.16515i q^{43} +(-1.79129 - 2.79129i) q^{44} -4.00000i q^{46} +9.93280i q^{47} +(5.00000 + 4.89898i) q^{49} -4.58258i q^{50} -0.646084 q^{52} +11.5826 q^{53} +(8.64064 - 5.54506i) q^{55} +(1.00000 - 2.44949i) q^{56} +1.58258 q^{58} +0.646084i q^{59} +1.93825 q^{61} +8.64064 q^{62} -1.00000 q^{64} -2.00000i q^{65} -7.58258 q^{67} -3.74166 q^{68} +(7.58258 + 3.09557i) q^{70} -2.00000 q^{71} +16.1240 q^{73} +3.58258i q^{74} +1.80341 q^{76} +(1.59645 + 8.62852i) q^{77} -4.00000i q^{79} -3.09557i q^{80} +9.93280i q^{82} -12.8935 q^{83} -11.5826i q^{85} +7.16515 q^{86} +(2.79129 - 1.79129i) q^{88} -9.79796i q^{89} +(1.58258 + 0.646084i) q^{91} +4.00000 q^{92} -9.93280 q^{94} +5.58258i q^{95} -8.50579i q^{97} +(-4.89898 + 5.00000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 8 q^{4} - 4 q^{11} - 8 q^{14} + 8 q^{16} - 4 q^{22} - 32 q^{23} - 8 q^{37} + 4 q^{44} + 40 q^{49} + 56 q^{53} + 8 q^{56} - 24 q^{58} - 8 q^{64} - 24 q^{67} + 24 q^{70} - 16 q^{71} - 4 q^{77} - 16 q^{86}+ \cdots + 32 q^{92}+O(q^{100})$ 8 * q - 8 * q^4 - 4 * q^11 - 8 * q^14 + 8 * q^16 - 4 * q^22 - 32 * q^23 - 8 * q^37 + 4 * q^44 + 40 * q^49 + 56 * q^53 + 8 * q^56 - 24 * q^58 - 8 * q^64 - 24 * q^67 + 24 * q^70 - 16 * q^71 - 4 * q^77 - 16 * q^86 + 4 * q^88 - 24 * q^91 + 32 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$ .

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	3.09557i		−	1.38438i	−0.721714	−	0.692191i	$-0.756645\pi$
$5$		−	3.09557i		−	1.38438i	0.721714	−	0.692191i	$-0.243355\pi$
$6$		0			0
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	3.09557			0.978906
$11$	1.79129	+	2.79129i	0.540094	+	0.841605i
$11$	1.79129	+	2.79129i	0.540094	+	0.841605i
$12$		0			0
$13$	0.646084			0.179191			0.0895957	−	0.995978i	$-0.471443\pi$
$13$	0.646084			0.179191			0.0895957	+	0.995978i	$0.471443\pi$
$14$	−1.00000	+	2.44949i	−0.267261	+	0.654654i
$15$		0			0
$16$	1.00000			0.250000
$17$	3.74166			0.907485			0.453743	−	0.891133i	$-0.350089\pi$
$17$	3.74166			0.907485			0.453743	+	0.891133i	$0.350089\pi$
$18$		0			0
$19$	−1.80341			−0.413730			−0.206865	−	0.978370i	$-0.566326\pi$
$19$	−1.80341			−0.413730			−0.206865	+	0.978370i	$0.566326\pi$
$20$			3.09557i			0.692191i
$21$		0			0
$22$	−2.79129	+	1.79129i	−0.595105	+	0.381904i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−4.58258			−0.916515
$26$			0.646084i			0.126707i
$27$		0			0
$28$	−2.44949	−	1.00000i	−0.462910	−	0.188982i
$29$		−	1.58258i		−	0.293877i	−0.989146	−	0.146938i	$-0.953058\pi$
$29$		−	1.58258i		−	0.293877i	0.989146	−	0.146938i	$-0.0469419\pi$
$30$		0			0
$31$		−	8.64064i		−	1.55190i	−0.630792	−	0.775952i	$-0.717270\pi$
$31$		−	8.64064i		−	1.55190i	0.630792	−	0.775952i	$-0.282730\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			3.74166i			0.641689i
$35$	3.09557	−	7.58258i	0.523247	−	1.28169i
$36$		0			0
$37$	3.58258			0.588972			0.294486	−	0.955656i	$-0.404852\pi$
$37$	3.58258			0.588972			0.294486	+	0.955656i	$0.404852\pi$
$38$		−	1.80341i		−	0.292551i
$39$		0			0
$40$	−3.09557			−0.489453
$41$	9.93280			1.55124			0.775622	−	0.631198i	$-0.217436\pi$
$41$	9.93280			1.55124			0.775622	+	0.631198i	$0.217436\pi$
$42$		0			0
$43$		−	7.16515i		−	1.09268i	−0.837565	−	0.546338i	$-0.816022\pi$
$43$		−	7.16515i		−	1.09268i	0.837565	−	0.546338i	$-0.183978\pi$
$44$	−1.79129	−	2.79129i	−0.270047	−	0.420802i
$45$		0			0
$46$		−	4.00000i		−	0.589768i
$47$			9.93280i			1.44885i	0.689354	+	0.724424i	$0.257894\pi$
$47$			9.93280i			1.44885i	−0.689354	+	0.724424i	$0.742106\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		−	4.58258i		−	0.648074i
$51$		0			0
$52$	−0.646084			−0.0895957
$53$	11.5826			1.59099			0.795495	−	0.605961i	$-0.207211\pi$
$53$	11.5826			1.59099			0.795495	+	0.605961i	$0.207211\pi$
$54$		0			0
$55$	8.64064	−	5.54506i	1.16510	−	0.747696i
$56$	1.00000	−	2.44949i	0.133631	−	0.327327i
$57$		0			0
$58$	1.58258			0.207802
$59$			0.646084i			0.0841129i	0.999115	+	0.0420565i	$0.0133909\pi$
$59$			0.646084i			0.0841129i	−0.999115	+	0.0420565i	$0.986609\pi$
$60$		0			0
$61$	1.93825			0.248168			0.124084	−	0.992272i	$-0.460401\pi$
$61$	1.93825			0.248168			0.124084	+	0.992272i	$0.460401\pi$
$62$	8.64064			1.09736
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	2.00000i		−	0.248069i
$66$		0			0
$67$	−7.58258			−0.926359			−0.463180	−	0.886264i	$-0.653291\pi$
$67$	−7.58258			−0.926359			−0.463180	+	0.886264i	$0.653291\pi$
$68$	−3.74166			−0.453743
$69$		0			0
$70$	7.58258	+	3.09557i	0.906291	+	0.369992i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	16.1240			1.88717			0.943583	−	0.331136i	$-0.107432\pi$
$73$	16.1240			1.88717			0.943583	+	0.331136i	$0.107432\pi$
$74$			3.58258i			0.416466i
$75$		0			0
$76$	1.80341			0.206865
$77$	1.59645	+	8.62852i	0.181933	+	0.983311i
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		−	3.09557i		−	0.346096i
$81$		0			0
$82$			9.93280i			1.09689i
$83$	−12.8935			−1.41525			−0.707625	−	0.706589i	$-0.750233\pi$
$83$	−12.8935			−1.41525			−0.707625	+	0.706589i	$0.750233\pi$
$84$		0			0
$85$		−	11.5826i		−	1.25631i
$86$	7.16515			0.772638
$87$		0			0
$88$	2.79129	−	1.79129i	0.297552	−	0.190952i
$89$		−	9.79796i		−	1.03858i	−0.854598	−	0.519291i	$-0.826196\pi$
$89$		−	9.79796i		−	1.03858i	0.854598	−	0.519291i	$-0.173804\pi$
$90$		0			0
$91$	1.58258	+	0.646084i	0.165899	+	0.0677280i
$92$	4.00000			0.417029
$93$		0			0
$94$	−9.93280			−1.02449
$95$			5.58258i			0.572760i
$96$		0			0
$97$		−	8.50579i		−	0.863632i	−0.901962	−	0.431816i	$-0.857873\pi$
$97$		−	8.50579i		−	0.863632i	0.901962	−	0.431816i	$-0.142127\pi$
$98$	−4.89898	+	5.00000i	−0.494872	+	0.505076i
$99$		0			0
$100$	4.58258			0.458258
$101$	11.7362			1.16780			0.583898	−	0.811827i	$-0.301527\pi$
$101$	11.7362			1.16780			0.583898	+	0.811827i	$0.301527\pi$
$102$		0			0
$103$			6.05630i			0.596745i	0.954449	+	0.298373i	$0.0964438\pi$
$103$			6.05630i			0.596745i	−0.954449	+	0.298373i	$0.903556\pi$
$104$		−	0.646084i		−	0.0633537i
$105$		0			0
$106$			11.5826i			1.12500i
$107$		−	17.5826i		−	1.69977i	−0.526967	−	0.849886i	$-0.676671\pi$
$107$		−	17.5826i		−	1.69977i	0.526967	−	0.849886i	$-0.323329\pi$
$108$		0			0
$109$			17.1652i			1.64412i	0.569398	+	0.822062i	$0.307176\pi$
$109$			17.1652i			1.64412i	−0.569398	+	0.822062i	$0.692824\pi$
$110$	5.54506	+	8.64064i	0.528701	+	0.823852i
$111$		0			0
$112$	2.44949	+	1.00000i	0.231455	+	0.0944911i
$113$	−10.7477			−1.01106			−0.505531	−	0.862809i	$-0.668703\pi$
$113$	−10.7477			−1.01106			−0.505531	+	0.862809i	$0.668703\pi$
$114$		0			0
$115$			12.3823i			1.15465i
$116$			1.58258i			0.146938i
$117$		0			0
$118$	−0.646084			−0.0594768
$119$	9.16515	+	3.74166i	0.840168	+	0.342997i
$120$		0			0
$121$	−4.58258	+	10.0000i	−0.416598	+	0.909091i
$122$			1.93825i			0.175481i
$123$		0			0
$124$			8.64064i			0.775952i
$125$		−	1.29217i		−	0.115575i
$126$		0			0
$127$		−	3.58258i		−	0.317902i	−0.987286	−	0.158951i	$-0.949189\pi$
$127$		−	3.58258i		−	0.317902i	0.987286	−	0.158951i	$-0.0508112\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	2.00000			0.175412
$131$	16.7700			1.46520			0.732602	−	0.680657i	$-0.238306\pi$
$131$	16.7700			1.46520			0.732602	+	0.680657i	$0.238306\pi$
$132$		0			0
$133$	−4.41742	−	1.80341i	−0.383039	−	0.156375i
$134$		−	7.58258i		−	0.655035i
$135$		0			0
$136$		−	3.74166i		−	0.320844i
$137$	3.16515			0.270417			0.135209	−	0.990817i	$-0.456830\pi$
$137$	3.16515			0.270417			0.135209	+	0.990817i	$0.456830\pi$
$138$		0			0
$139$	−15.4779			−1.31282			−0.656408	−	0.754406i	$-0.727925\pi$
$139$	−15.4779			−1.31282			−0.656408	+	0.754406i	$0.727925\pi$
$140$	−3.09557	+	7.58258i	−0.261624	+	0.640845i
$141$		0			0
$142$		−	2.00000i		−	0.167836i
$143$	1.15732	+	1.80341i	0.0967801	+	0.150808i
$144$		0			0
$145$	−4.89898			−0.406838
$146$			16.1240i			1.33443i
$147$		0			0
$148$	−3.58258			−0.294486
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$			5.58258i			0.454304i	0.973859	+	0.227152i	$0.0729414\pi$
$151$			5.58258i			0.454304i	−0.973859	+	0.227152i	$0.927059\pi$
$152$			1.80341i			0.146276i
$153$		0			0
$154$	−8.62852	+	1.59645i	−0.695306	+	0.128646i
$155$	−26.7477			−2.14843
$156$		0			0
$157$		−	13.1632i		−	1.05054i	−0.850936	−	0.525270i	$-0.823964\pi$
$157$		−	13.1632i		−	1.05054i	0.850936	−	0.525270i	$-0.176036\pi$
$158$	4.00000			0.318223
$159$		0			0
$160$	3.09557			0.244727
$161$	−9.79796	−	4.00000i	−0.772187	−	0.315244i
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	−9.93280			−0.775622
$165$		0			0
$166$		−	12.8935i		−	1.00073i
$167$	−1.29217			−0.0999909			−0.0499955	−	0.998749i	$-0.515921\pi$
$167$	−1.29217			−0.0999909			−0.0499955	+	0.998749i	$0.515921\pi$
$168$		0			0
$169$	−12.5826			−0.967890
$170$	11.5826			0.888343
$171$		0			0
$172$			7.16515i			0.546338i
$173$	−10.7137			−0.814550			−0.407275	−	0.913306i	$-0.633521\pi$
$173$	−10.7137			−0.814550			−0.407275	+	0.913306i	$0.633521\pi$
$174$		0			0
$175$	−11.2250	−	4.58258i	−0.848528	−	0.346410i
$176$	1.79129	+	2.79129i	0.135023	+	0.210401i
$177$		0			0
$178$	9.79796			0.734388
$179$	−11.1652			−0.834523			−0.417261	−	0.908787i	$-0.637010\pi$
$179$	−11.1652			−0.834523			−0.417261	+	0.908787i	$0.637010\pi$
$180$		0			0
$181$			5.41022i			0.402138i	0.979577	+	0.201069i	$0.0644416\pi$
$181$			5.41022i			0.402138i	−0.979577	+	0.201069i	$0.935558\pi$
$182$	−0.646084	+	1.58258i	−0.0478909	+	0.117308i
$183$		0			0
$184$			4.00000i			0.294884i
$185$		−	11.0901i		−	0.815362i
$186$		0			0
$187$	6.70239	+	10.4440i	0.490127	+	0.763744i
$188$		−	9.93280i		−	0.724424i
$189$		0			0
$190$	−5.58258			−0.405003
$191$	18.0000			1.30243			0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000			1.30243			0.651217	+	0.758891i	$0.274259\pi$
$192$		0			0
$193$			16.3303i			1.17548i	0.809050	+	0.587740i	$0.199982\pi$
$193$			16.3303i			1.17548i	−0.809050	+	0.587740i	$0.800018\pi$
$194$	8.50579			0.610680
$195$		0			0
$196$	−5.00000	−	4.89898i	−0.357143	−	0.349927i
$197$		−	8.74773i		−	0.623250i	−0.950205	−	0.311625i	$-0.899127\pi$
$197$		−	8.74773i		−	0.623250i	0.950205	−	0.311625i	$-0.100873\pi$
$198$		0			0
$199$		−	9.66311i		−	0.685000i	−0.939518	−	0.342500i	$-0.888726\pi$
$199$		−	9.66311i		−	0.685000i	0.939518	−	0.342500i	$-0.111274\pi$
$200$			4.58258i			0.324037i
$201$		0			0
$202$			11.7362i			0.825757i
$203$	1.58258	−	3.87650i	0.111075	−	0.272077i
$204$		0			0
$205$		−	30.7477i		−	2.14751i
$206$	−6.05630			−0.421963
$207$		0			0
$208$	0.646084			0.0447979
$209$	−3.23042	−	5.03383i	−0.223453	−	0.348197i
$210$		0			0
$211$			16.7477i			1.15296i	0.817111	+	0.576481i	$0.195574\pi$
$211$			16.7477i			1.15296i	−0.817111	+	0.576481i	$0.804426\pi$
$212$	−11.5826			−0.795495
$213$		0			0
$214$	17.5826			1.20192
$215$	−22.1803			−1.51268
$216$		0			0
$217$	8.64064	−	21.1652i	0.586565	−	1.43678i
$218$	−17.1652			−1.16257
$219$		0			0
$220$	−8.64064	+	5.54506i	−0.582552	+	0.373848i
$221$	2.41742			0.162614
$222$		0			0
$223$			16.1240i			1.07974i	0.841749	+	0.539870i	$0.181527\pi$
$223$			16.1240i			1.07974i	−0.841749	+	0.539870i	$0.818473\pi$
$224$	−1.00000	+	2.44949i	−0.0668153	+	0.163663i
$225$		0			0
$226$		−	10.7477i		−	0.714928i
$227$	0.511238			0.0339321			0.0169660	−	0.999856i	$-0.494599\pi$
$227$	0.511238			0.0339321			0.0169660	+	0.999856i	$0.494599\pi$
$228$		0			0
$229$		−	5.67991i		−	0.375339i	−0.982232	−	0.187669i	$-0.939907\pi$
$229$		−	5.67991i		−	0.375339i	0.982232	−	0.187669i	$-0.0600934\pi$
$230$	−12.3823			−0.816464
$231$		0			0
$232$	−1.58258			−0.103901
$233$			27.1652i			1.77965i	0.456304	+	0.889824i	$0.349173\pi$
$233$			27.1652i			1.77965i	−0.456304	+	0.889824i	$0.650827\pi$
$234$		0			0
$235$	30.7477			2.00576
$236$		−	0.646084i		−	0.0420565i
$237$		0			0
$238$	−3.74166	+	9.16515i	−0.242536	+	0.594089i
$239$			8.41742i			0.544478i	0.962230	+	0.272239i	$0.0877641\pi$
$239$			8.41742i			0.544478i	−0.962230	+	0.272239i	$0.912236\pi$
$240$		0			0
$241$	−3.47197			−0.223649			−0.111825	−	0.993728i	$-0.535669\pi$
$241$	−3.47197			−0.223649			−0.111825	+	0.993728i	$0.535669\pi$
$242$	−10.0000	−	4.58258i	−0.642824	−	0.294579i
$243$		0			0
$244$	−1.93825			−0.124084
$245$	15.1652	−	15.4779i	0.968866	−	0.988845i
$246$		0			0
$247$	−1.16515			−0.0741368
$248$	−8.64064			−0.548681
$249$		0			0
$250$	1.29217			0.0817239
$251$		−	24.1185i		−	1.52235i	−0.648549	−	0.761173i	$-0.724624\pi$
$251$		−	24.1185i		−	1.52235i	0.648549	−	0.761173i	$-0.275376\pi$
$252$		0			0
$253$	−7.16515	−	11.1652i	−0.450469	−	0.701947i
$254$	3.58258			0.224791
$255$		0			0
$256$	1.00000			0.0625000
$257$			4.89898i			0.305590i	0.988258	+	0.152795i	$0.0488274\pi$
$257$			4.89898i			0.305590i	−0.988258	+	0.152795i	$0.951173\pi$
$258$		0			0
$259$	8.77548	+	3.58258i	0.545282	+	0.222610i
$260$			2.00000i			0.124035i
$261$		0			0
$262$			16.7700i			1.03606i
$263$			8.33030i			0.513668i	0.966455	+	0.256834i	$0.0826794\pi$
$263$			8.33030i			0.513668i	−0.966455	+	0.256834i	$0.917321\pi$
$264$		0			0
$265$		−	35.8547i		−	2.20254i
$266$	1.80341	−	4.41742i	0.110574	−	0.270850i
$267$		0			0
$268$	7.58258			0.463180
$269$			16.5003i			1.00604i	0.864274	+	0.503022i	$0.167778\pi$
$269$			16.5003i			1.00604i	−0.864274	+	0.503022i	$0.832222\pi$
$270$		0			0
$271$	−4.89898			−0.297592			−0.148796	−	0.988868i	$-0.547540\pi$
$271$	−4.89898			−0.297592			−0.148796	+	0.988868i	$0.547540\pi$
$272$	3.74166			0.226871
$273$		0			0
$274$			3.16515i			0.191214i
$275$	−8.20871	−	12.7913i	−0.495004	−	0.771344i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		−	15.4779i		−	0.928301i
$279$		0			0
$280$	−7.58258	−	3.09557i	−0.453146	−	0.184996i
$281$		−	11.1652i		−	0.666057i	−0.942917	−	0.333029i	$-0.891929\pi$
$281$		−	11.1652i		−	0.666057i	0.942917	−	0.333029i	$-0.108071\pi$
$282$		0			0
$283$	−18.0622			−1.07369			−0.536843	−	0.843682i	$-0.680383\pi$
$283$	−18.0622			−1.07369			−0.536843	+	0.843682i	$0.680383\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	−1.80341	+	1.15732i	−0.106638	+	0.0684339i
$287$	24.3303	+	9.93280i	1.43617	+	0.586315i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		−	4.89898i		−	0.287678i
$291$		0			0
$292$	−16.1240			−0.943583
$293$	−25.1410			−1.46875			−0.734376	−	0.678743i	$-0.762525\pi$
$293$	−25.1410			−1.46875			−0.734376	+	0.678743i	$0.762525\pi$
$294$		0			0
$295$	2.00000			0.116445
$296$		−	3.58258i		−	0.208233i
$297$		0			0
$298$	−14.0000			−0.810998
$299$	−2.58434			−0.149456
$300$		0			0
$301$	7.16515	−	17.5510i	0.412992	−	1.01162i
$302$	−5.58258			−0.321241
$303$		0			0
$304$	−1.80341			−0.103432
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$	−10.5789			−0.603769			−0.301885	−	0.953344i	$-0.597616\pi$
$307$	−10.5789			−0.603769			−0.301885	+	0.953344i	$0.597616\pi$
$308$	−1.59645	−	8.62852i	−0.0909664	−	0.491655i
$309$		0			0
$310$		−	26.7477i		−	1.51917i
$311$		−	5.03383i		−	0.285442i	−0.989763	−	0.142721i	$-0.954415\pi$
$311$		−	5.03383i		−	0.285442i	0.989763	−	0.142721i	$-0.0455852\pi$
$312$		0			0
$313$		−	23.4724i		−	1.32674i	−0.748292	−	0.663370i	$-0.769126\pi$
$313$		−	23.4724i		−	1.32674i	0.748292	−	0.663370i	$-0.230874\pi$
$314$	13.1632			0.742844
$315$		0			0
$316$			4.00000i			0.225018i
$317$	−9.16515			−0.514766			−0.257383	−	0.966309i	$-0.582860\pi$
$317$	−9.16515			−0.514766			−0.257383	+	0.966309i	$0.582860\pi$
$318$		0			0
$319$	4.41742	−	2.83485i	0.247328	−	0.158721i
$320$			3.09557i			0.173048i
$321$		0			0
$322$	4.00000	−	9.79796i	0.222911	−	0.546019i
$323$	−6.74773			−0.375454
$324$		0			0
$325$	−2.96073			−0.164232
$326$			4.00000i			0.221540i
$327$		0			0
$328$		−	9.93280i		−	0.548447i
$329$	−9.93280	+	24.3303i	−0.547613	+	1.34137i
$330$		0			0
$331$	−23.5826			−1.29622			−0.648108	−	0.761549i	$-0.724439\pi$
$331$	−23.5826			−1.29622			−0.648108	+	0.761549i	$0.724439\pi$
$332$	12.8935			0.707625
$333$		0			0
$334$		−	1.29217i		−	0.0707043i
$335$			23.4724i			1.28244i
$336$		0			0
$337$			10.8348i			0.590212i	0.955465	+	0.295106i	$0.0953549\pi$
$337$			10.8348i			0.590212i	−0.955465	+	0.295106i	$0.904645\pi$
$338$		−	12.5826i		−	0.684402i
$339$		0			0
$340$			11.5826i			0.628153i
$341$	24.1185	−	15.4779i	1.30609	−	0.838174i
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$	−7.16515			−0.386319
$345$		0			0
$346$		−	10.7137i		−	0.575974i
$347$			19.1652i			1.02884i	0.857539	+	0.514420i	$0.171993\pi$
$347$			19.1652i			1.02884i	−0.857539	+	0.514420i	$0.828007\pi$
$348$		0			0
$349$	24.1185			1.29103			0.645517	−	0.763746i	$-0.276642\pi$
$349$	24.1185			1.29103			0.645517	+	0.763746i	$0.276642\pi$
$350$	4.58258	−	11.2250i	0.244949	−	0.600000i
$351$		0			0
$352$	−2.79129	+	1.79129i	−0.148776	+	0.0954760i
$353$		−	7.48331i		−	0.398297i	−0.979969	−	0.199148i	$-0.936182\pi$
$353$		−	7.48331i		−	0.398297i	0.979969	−	0.199148i	$-0.0638176\pi$
$354$		0			0
$355$			6.19115i			0.328592i
$356$			9.79796i			0.519291i
$357$		0			0
$358$		−	11.1652i		−	0.590097i
$359$			9.58258i			0.505749i	0.967499	+	0.252875i	$0.0813760\pi$
$359$			9.58258i			0.505749i	−0.967499	+	0.252875i	$0.918624\pi$
$360$		0			0
$361$	−15.7477			−0.828828
$362$	−5.41022			−0.284355
$363$		0			0
$364$	−1.58258	−	0.646084i	−0.0829495	−	0.0338640i
$365$		−	49.9129i		−	2.61256i
$366$		0			0
$367$			21.0229i			1.09739i	0.836023	+	0.548694i	$0.184875\pi$
$367$			21.0229i			1.09739i	−0.836023	+	0.548694i	$0.815125\pi$
$368$	−4.00000			−0.208514
$369$		0			0
$370$	11.0901			0.576548
$371$	28.3714	+	11.5826i	1.47297	+	0.601337i
$372$		0			0
$373$			9.58258i			0.496167i	0.968739	+	0.248083i	$0.0798007\pi$
$373$			9.58258i			0.496167i	−0.968739	+	0.248083i	$0.920199\pi$
$374$	−10.4440	+	6.70239i	−0.540049	+	0.346572i
$375$		0			0
$376$	9.93280			0.512245
$377$		−	1.02248i		−	0.0526602i
$378$		0			0
$379$	−22.3303			−1.14703			−0.573515	−	0.819195i	$-0.694421\pi$
$379$	−22.3303			−1.14703			−0.573515	+	0.819195i	$0.694421\pi$
$380$		−	5.58258i		−	0.286380i
$381$		0			0
$382$			18.0000i			0.920960i
$383$		−	19.7308i		−	1.00819i	−0.863647	−	0.504097i	$-0.831825\pi$
$383$		−	19.7308i		−	1.00819i	0.863647	−	0.504097i	$-0.168175\pi$
$384$		0			0
$385$	26.7102	−	4.94194i	1.36128	−	0.251865i
$386$	−16.3303			−0.831191
$387$		0			0
$388$			8.50579i			0.431816i
$389$	−10.0000			−0.507020			−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000			−0.507020			−0.253510	+	0.967333i	$0.581585\pi$
$390$		0			0
$391$	−14.9666			−0.756895
$392$	4.89898	−	5.00000i	0.247436	−	0.252538i
$393$		0			0
$394$	8.74773			0.440704
$395$	−12.3823			−0.623021
$396$		0			0
$397$		−	27.5905i		−	1.38473i	−0.721549	−	0.692363i	$-0.756570\pi$
$397$		−	27.5905i		−	1.38473i	0.721549	−	0.692363i	$-0.243430\pi$
$398$	9.66311			0.484368
$399$		0			0
$400$	−4.58258			−0.229129
$401$	−7.58258			−0.378656			−0.189328	−	0.981914i	$-0.560631\pi$
$401$	−7.58258			−0.378656			−0.189328	+	0.981914i	$0.560631\pi$
$402$		0			0
$403$		−	5.58258i		−	0.278088i
$404$	−11.7362			−0.583898
$405$		0			0
$406$	3.87650	+	1.58258i	0.192388	+	0.0785419i
$407$	6.41742	+	10.0000i	0.318100	+	0.495682i
$408$		0			0
$409$	−5.03383			−0.248907			−0.124453	−	0.992225i	$-0.539718\pi$
$409$	−5.03383			−0.248907			−0.124453	+	0.992225i	$0.539718\pi$
$410$	30.7477			1.51852
$411$		0			0
$412$		−	6.05630i		−	0.298373i
$413$	−0.646084	+	1.58258i	−0.0317917	+	0.0778735i
$414$		0			0
$415$			39.9129i			1.95925i
$416$			0.646084i			0.0316769i
$417$		0			0
$418$	5.03383	−	3.23042i	0.246212	−	0.158005i
$419$		−	13.0284i		−	0.636478i	−0.948011	−	0.318239i	$-0.896909\pi$
$419$		−	13.0284i		−	0.636478i	0.948011	−	0.318239i	$-0.103091\pi$
$420$		0			0
$421$	7.58258			0.369552			0.184776	−	0.982781i	$-0.440844\pi$
$421$	7.58258			0.369552			0.184776	+	0.982781i	$0.440844\pi$
$422$	−16.7477			−0.815267
$423$		0			0
$424$		−	11.5826i		−	0.562500i
$425$	−17.1464			−0.831724
$426$		0			0
$427$	4.74773	+	1.93825i	0.229759	+	0.0937986i
$428$			17.5826i			0.849886i
$429$		0			0
$430$		−	22.1803i		−	1.06963i
$431$			27.9129i			1.34452i	0.740317	+	0.672258i	$0.234675\pi$
$431$			27.9129i			1.34452i	−0.740317	+	0.672258i	$0.765325\pi$
$432$		0			0
$433$			35.5850i			1.71011i	0.518540	+	0.855054i	$0.326476\pi$
$433$			35.5850i			1.71011i	−0.518540	+	0.855054i	$0.673524\pi$
$434$	21.1652	+	8.64064i	1.01596	+	0.414764i
$435$		0			0
$436$		−	17.1652i		−	0.822062i
$437$	7.21362			0.345074
$438$		0			0
$439$	17.2813			0.824790			0.412395	−	0.911005i	$-0.364692\pi$
$439$	17.2813			0.824790			0.412395	+	0.911005i	$0.364692\pi$
$440$	−5.54506	−	8.64064i	−0.264351	−	0.411926i
$441$		0			0
$442$			2.41742i			0.114985i
$443$	−15.1652			−0.720518			−0.360259	−	0.932852i	$-0.617312\pi$
$443$	−15.1652			−0.720518			−0.360259	+	0.932852i	$0.617312\pi$
$444$		0			0
$445$	−30.3303			−1.43779
$446$	−16.1240			−0.763491
$447$		0			0
$448$	−2.44949	−	1.00000i	−0.115728	−	0.0472456i
$449$	−31.1652			−1.47077			−0.735387	−	0.677647i	$-0.763000\pi$
$449$	−31.1652			−1.47077			−0.735387	+	0.677647i	$0.763000\pi$
$450$		0			0
$451$	17.7925	+	27.7253i	0.837817	+	1.30553i
$452$	10.7477			0.505531
$453$		0			0
$454$			0.511238i			0.0239936i
$455$	2.00000	−	4.89898i	0.0937614	−	0.229668i
$456$		0			0
$457$			33.1652i			1.55140i	0.631102	+	0.775700i	$0.282603\pi$
$457$			33.1652i			1.55140i	−0.631102	+	0.775700i	$0.717397\pi$
$458$	5.67991			0.265405
$459$		0			0
$460$		−	12.3823i		−	0.577327i
$461$	−30.0400			−1.39910			−0.699550	−	0.714583i	$-0.746616\pi$
$461$	−30.0400			−1.39910			−0.699550	+	0.714583i	$0.746616\pi$
$462$		0			0
$463$	5.16515			0.240045			0.120022	−	0.992771i	$-0.461703\pi$
$463$	5.16515			0.240045			0.120022	+	0.992771i	$0.461703\pi$
$464$		−	1.58258i		−	0.0734692i
$465$		0			0
$466$	−27.1652			−1.25840
$467$			11.7362i			0.543087i	0.962426	+	0.271544i	$0.0875341\pi$
$467$			11.7362i			0.543087i	−0.962426	+	0.271544i	$0.912466\pi$
$468$		0			0
$469$	−18.5734	−	7.58258i	−0.857642	−	0.350131i
$470$			30.7477i			1.41829i
$471$		0			0
$472$	0.646084			0.0297384
$473$	20.0000	−	12.8348i	0.919601	−	0.590147i
$474$		0			0
$475$	8.26424			0.379190
$476$	−9.16515	−	3.74166i	−0.420084	−	0.171499i
$477$		0			0
$478$	−8.41742			−0.385004
$479$	−30.9557			−1.41440			−0.707202	−	0.707012i	$-0.750043\pi$
$479$	−30.9557			−1.41440			−0.707202	+	0.707012i	$0.750043\pi$
$480$		0			0
$481$	2.31464			0.105539
$482$		−	3.47197i		−	0.158144i
$483$		0			0
$484$	4.58258	−	10.0000i	0.208299	−	0.454545i
$485$	−26.3303			−1.19560
$486$		0			0
$487$	21.4955			0.974052			0.487026	−	0.873387i	$-0.338082\pi$
$487$	21.4955			0.974052			0.487026	+	0.873387i	$0.338082\pi$
$488$		−	1.93825i		−	0.0877405i
$489$		0			0
$490$	15.4779	+	15.1652i	0.699219	+	0.685092i
$491$		−	33.4955i		−	1.51163i	−0.654786	−	0.755814i	$-0.727241\pi$
$491$		−	33.4955i		−	1.51163i	0.654786	−	0.755814i	$-0.272759\pi$
$492$		0			0
$493$		−	5.92146i		−	0.266689i
$494$		−	1.16515i		−	0.0524226i
$495$		0			0
$496$		−	8.64064i		−	0.387976i
$497$	−4.89898	−	2.00000i	−0.219749	−	0.0897123i
$498$		0			0
$499$	−17.9129			−0.801891			−0.400945	−	0.916102i	$-0.631318\pi$
$499$	−17.9129			−0.801891			−0.400945	+	0.916102i	$0.631318\pi$
$500$			1.29217i			0.0577875i
$501$		0			0
$502$	24.1185			1.07646
$503$	17.0116			0.758509			0.379254	−	0.925292i	$-0.376180\pi$
$503$	17.0116			0.758509			0.379254	+	0.925292i	$0.376180\pi$
$504$		0			0
$505$		−	36.3303i		−	1.61668i
$506$	11.1652	−	7.16515i	0.496352	−	0.318530i
$507$		0			0
$508$			3.58258i			0.158951i
$509$		−	0.780929i		−	0.0346141i	−0.999850	−	0.0173070i	$-0.994491\pi$
$509$		−	0.780929i		−	0.0346141i	0.999850	−	0.0173070i	$-0.00550928\pi$
$510$		0			0
$511$	39.4955	+	16.1240i	1.74718	+	0.713282i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−4.89898			−0.216085
$515$	18.7477			0.826124
$516$		0			0
$517$	−27.7253	+	17.7925i	−1.21936	+	0.782514i
$518$	−3.58258	+	8.77548i	−0.157409	+	0.385573i
$519$		0			0
$520$	−2.00000			−0.0877058
$521$			28.1017i			1.23116i	0.788075	+	0.615579i	$0.211078\pi$
$521$			28.1017i			1.23116i	−0.788075	+	0.615579i	$0.788922\pi$
$522$		0			0
$523$	−14.4554			−0.632090			−0.316045	−	0.948744i	$-0.602355\pi$
$523$	−14.4554			−0.632090			−0.316045	+	0.948744i	$0.602355\pi$
$524$	−16.7700			−0.732602
$525$		0			0
$526$	−8.33030			−0.363218
$527$		−	32.3303i		−	1.40833i
$528$		0			0
$529$	−7.00000			−0.304348
$530$	35.8547			1.55743
$531$		0			0
$532$	4.41742	+	1.80341i	0.191520	+	0.0781876i
$533$	6.41742			0.277970
$534$		0			0
$535$	−54.4282			−2.35313
$536$			7.58258i			0.327517i
$537$		0			0
$538$	−16.5003			−0.711380
$539$	−4.71802	+	22.7319i	−0.203220	+	0.979133i
$540$		0			0
$541$		−	36.7477i		−	1.57991i	−0.613166	−	0.789954i	$-0.710104\pi$
$541$		−	36.7477i		−	1.57991i	0.613166	−	0.789954i	$-0.289896\pi$
$542$		−	4.89898i		−	0.210429i
$543$		0			0
$544$			3.74166i			0.160422i
$545$	53.1360			2.27610
$546$		0			0
$547$		−	24.7477i		−	1.05814i	−0.848579	−	0.529068i	$-0.822542\pi$
$547$		−	24.7477i		−	1.05814i	0.848579	−	0.529068i	$-0.177458\pi$
$548$	−3.16515			−0.135209
$549$		0			0
$550$	12.7913	−	8.20871i	0.545422	−	0.350021i
$551$			2.85403i			0.121586i
$552$		0			0
$553$	4.00000	−	9.79796i	0.170097	−	0.416652i
$554$	−2.00000			−0.0849719
$555$		0			0
$556$	15.4779			0.656408
$557$			35.9129i			1.52168i	0.648941	+	0.760839i	$0.275212\pi$
$557$			35.9129i			1.52168i	−0.648941	+	0.760839i	$0.724788\pi$
$558$		0			0
$559$		−	4.62929i		−	0.195798i
$560$	3.09557	−	7.58258i	0.130812	−	0.320422i
$561$		0			0
$562$	11.1652			0.470973
$563$	−23.7140			−0.999425			−0.499712	−	0.866191i	$-0.666561\pi$
$563$	−23.7140			−0.999425			−0.499712	+	0.866191i	$0.666561\pi$
$564$		0			0
$565$			33.2704i			1.39970i
$566$		−	18.0622i		−	0.759211i
$567$		0			0
$568$			2.00000i			0.0839181i
$569$		−	39.4955i		−	1.65574i	−0.560923	−	0.827868i	$-0.689554\pi$
$569$		−	39.4955i		−	1.65574i	0.560923	−	0.827868i	$-0.310446\pi$
$570$		0			0
$571$		−	5.58258i		−	0.233624i	−0.993154	−	0.116812i	$-0.962733\pi$
$571$		−	5.58258i		−	0.233624i	0.993154	−	0.116812i	$-0.0372674\pi$
$572$	−1.15732	−	1.80341i	−0.0483901	−	0.0754042i
$573$		0			0
$574$	−9.93280	+	24.3303i	−0.414587	+	1.01553i
$575$	18.3303			0.764426
$576$		0			0
$577$			29.6636i			1.23491i	0.786606	+	0.617455i	$0.211836\pi$
$577$			29.6636i			1.23491i	−0.786606	+	0.617455i	$0.788164\pi$
$578$		−	3.00000i		−	0.124784i
$579$		0			0
$580$	4.89898			0.203419
$581$	−31.5826	−	12.8935i	−1.31027	−	0.534914i
$582$		0			0
$583$	20.7477	+	32.3303i	0.859283	+	1.33898i
$584$		−	16.1240i		−	0.667214i
$585$		0			0
$586$		−	25.1410i		−	1.03856i
$587$			39.0851i			1.61322i	0.591087	+	0.806608i	$0.298699\pi$
$587$			39.0851i			1.61322i	−0.591087	+	0.806608i	$0.701301\pi$
$588$		0			0
$589$			15.5826i			0.642069i
$590$			2.00000i			0.0823387i
$591$		0			0
$592$	3.58258			0.147243
$593$	14.8318			0.609068			0.304534	−	0.952501i	$-0.401499\pi$
$593$	14.8318			0.609068			0.304534	+	0.952501i	$0.401499\pi$
$594$		0			0
$595$	11.5826	−	28.3714i	0.474839	−	1.16311i
$596$		−	14.0000i		−	0.573462i
$597$		0			0
$598$		−	2.58434i		−	0.105681i
$599$	21.1652			0.864785			0.432392	−	0.901686i	$-0.357670\pi$
$599$	21.1652			0.864785			0.432392	+	0.901686i	$0.357670\pi$
$600$		0			0
$601$	21.0229			0.857543			0.428772	−	0.903413i	$-0.358947\pi$
$601$	21.0229			0.857543			0.428772	+	0.903413i	$0.358947\pi$
$602$	17.5510	+	7.16515i	0.715324	+	0.292030i
$603$		0			0
$604$		−	5.58258i		−	0.227152i
$605$	30.9557	+	14.1857i	1.25853	+	0.576731i
$606$		0			0
$607$	2.04495			0.0830021			0.0415010	−	0.999138i	$-0.486786\pi$
$607$	2.04495			0.0830021			0.0415010	+	0.999138i	$0.486786\pi$
$608$		−	1.80341i		−	0.0731378i
$609$		0			0
$610$	6.00000			0.242933
$611$			6.41742i			0.259621i
$612$		0			0
$613$			31.9129i			1.28895i	0.764626	+	0.644475i	$0.222924\pi$
$613$			31.9129i			1.28895i	−0.764626	+	0.644475i	$0.777076\pi$
$614$		−	10.5789i		−	0.426929i
$615$		0			0
$616$	8.62852	−	1.59645i	0.347653	−	0.0643229i
$617$	19.9129			0.801662			0.400831	−	0.916152i	$-0.368721\pi$
$617$	19.9129			0.801662			0.400831	+	0.916152i	$0.368721\pi$
$618$		0			0
$619$		−	17.9274i		−	0.720561i	−0.932844	−	0.360281i	$-0.882681\pi$
$619$		−	17.9274i		−	0.720561i	0.932844	−	0.360281i	$-0.117319\pi$
$620$	26.7477			1.07421
$621$		0			0
$622$	5.03383			0.201838
$623$	9.79796	−	24.0000i	0.392547	−	0.961540i
$624$		0			0
$625$	−26.9129			−1.07652
$626$	23.4724			0.938147
$627$		0			0
$628$			13.1632i			0.525270i
$629$	13.4048			0.534483
$630$		0			0
$631$	13.1652			0.524096			0.262048	−	0.965055i	$-0.415602\pi$
$631$	13.1652			0.524096			0.262048	+	0.965055i	$0.415602\pi$
$632$	−4.00000			−0.159111
$633$		0			0
$634$		−	9.16515i		−	0.363995i
$635$	−11.0901			−0.440098
$636$		0			0
$637$	3.23042	+	3.16515i	0.127994	+	0.125408i
$638$	2.83485	+	4.41742i	0.112233	+	0.174888i
$639$		0			0
$640$	−3.09557			−0.122363
$641$	15.9129			0.628521			0.314260	−	0.949337i	$-0.398243\pi$
$641$	15.9129			0.628521			0.314260	+	0.949337i	$0.398243\pi$
$642$		0			0
$643$		−	9.42157i		−	0.371550i	−0.982592	−	0.185775i	$-0.940520\pi$
$643$		−	9.42157i		−	0.371550i	0.982592	−	0.185775i	$-0.0594796\pi$
$644$	9.79796	+	4.00000i	0.386094	+	0.157622i
$645$		0			0
$646$		−	6.74773i		−	0.265486i
$647$			30.0681i			1.18210i	0.806635	+	0.591050i	$0.201286\pi$
$647$			30.0681i			1.18210i	−0.806635	+	0.591050i	$0.798714\pi$
$648$		0			0
$649$	−1.80341	+	1.15732i	−0.0707899	+	0.0454289i
$650$		−	2.96073i		−	0.116129i
$651$		0			0
$652$	−4.00000			−0.156652
$653$	6.00000			0.234798			0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000			0.234798			0.117399	+	0.993085i	$0.462544\pi$
$654$		0			0
$655$		−	51.9129i		−	2.02840i
$656$	9.93280			0.387811
$657$		0			0
$658$	−24.3303	−	9.93280i	−0.948494	−	0.387221i
$659$			6.33030i			0.246594i	0.992370	+	0.123297i	$0.0393467\pi$
$659$			6.33030i			0.246594i	−0.992370	+	0.123297i	$0.960653\pi$
$660$		0			0
$661$			8.26424i			0.321442i	0.987000	+	0.160721i	$0.0513819\pi$
$661$			8.26424i			0.321442i	−0.987000	+	0.160721i	$0.948618\pi$
$662$		−	23.5826i		−	0.916563i
$663$		0			0
$664$			12.8935i			0.500366i
$665$	−5.58258	+	13.6745i	−0.216483	+	0.530273i
$666$		0			0
$667$			6.33030i			0.245110i
$668$	1.29217			0.0499955
$669$		0			0
$670$	−23.4724			−0.906819
$671$	3.47197	+	5.41022i	0.134034	+	0.208859i
$672$		0			0
$673$		−	18.3303i		−	0.706581i	−0.935514	−	0.353291i	$-0.885063\pi$
$673$		−	18.3303i		−	0.706581i	0.935514	−	0.353291i	$-0.114937\pi$
$674$	−10.8348			−0.417343
$675$		0			0
$676$	12.5826			0.483945
$677$	−5.27537			−0.202749			−0.101375	−	0.994848i	$-0.532324\pi$
$677$	−5.27537			−0.202749			−0.101375	+	0.994848i	$0.532324\pi$
$678$		0			0
$679$	8.50579	−	20.8348i	0.326422	−	0.799568i
$680$	−11.5826			−0.444172
$681$		0			0
$682$	15.4779	+	24.1185i	0.592678	+	0.923545i
$683$	9.49545			0.363333			0.181667	−	0.983360i	$-0.441851\pi$
$683$	9.49545			0.363333			0.181667	+	0.983360i	$0.441851\pi$
$684$		0			0
$685$		−	9.79796i		−	0.374361i
$686$	−17.0000	+	7.34847i	−0.649063	+	0.280566i
$687$		0			0
$688$		−	7.16515i		−	0.273169i
$689$	7.48331			0.285092
$690$		0			0
$691$		−	3.98320i		−	0.151528i	−0.997126	−	0.0757641i	$-0.975860\pi$
$691$		−	3.98320i		−	0.151528i	0.997126	−	0.0757641i	$-0.0241396\pi$
$692$	10.7137			0.407275
$693$		0			0
$694$	−19.1652			−0.727499
$695$			47.9129i			1.81744i
$696$		0			0
$697$	37.1652			1.40773
$698$			24.1185i			0.912899i
$699$		0			0
$700$	11.2250	+	4.58258i	0.424264	+	0.173205i
$701$		−	12.3303i		−	0.465709i	−0.972512	−	0.232855i	$-0.925193\pi$
$701$		−	12.3303i		−	0.465709i	0.972512	−	0.232855i	$-0.0748066\pi$
$702$		0			0
$703$	−6.46084			−0.243675
$704$	−1.79129	−	2.79129i	−0.0675117	−	0.105201i
$705$		0			0
$706$	7.48331			0.281638
$707$	28.7477	+	11.7362i	1.08117	+	0.441386i
$708$		0			0
$709$	52.3303			1.96531			0.982653	−	0.185454i	$-0.0593757\pi$
$709$	52.3303			1.96531			0.982653	+	0.185454i	$0.0593757\pi$
$710$	−6.19115			−0.232350
$711$		0			0
$712$	−9.79796			−0.367194
$713$			34.5625i			1.29438i
$714$		0			0
$715$	5.58258	−	3.58258i	0.208776	−	0.133981i
$716$	11.1652			0.417261
$717$		0			0
$718$	−9.58258			−0.357619
$719$			23.3376i			0.870345i	0.900347	+	0.435172i	$0.143313\pi$
$719$			23.3376i			0.870345i	−0.900347	+	0.435172i	$0.856687\pi$
$720$		0			0
$721$	−6.05630	+	14.8348i	−0.225548	+	0.552479i
$722$		−	15.7477i		−	0.586070i
$723$		0			0
$724$		−	5.41022i		−	0.201069i
$725$			7.25227i			0.269343i
$726$		0			0
$727$			37.5514i			1.39271i	0.717700	+	0.696353i	$0.245195\pi$
$727$			37.5514i			1.39271i	−0.717700	+	0.696353i	$0.754805\pi$
$728$	0.646084	−	1.58258i	0.0239455	−	0.0586542i
$729$		0			0
$730$	49.9129			1.84736
$731$		−	26.8095i		−	0.991587i
$732$		0			0
$733$	10.7137			0.395721			0.197860	−	0.980230i	$-0.436601\pi$
$733$	10.7137			0.395721			0.197860	+	0.980230i	$0.436601\pi$
$734$	−21.0229			−0.775971
$735$		0			0
$736$		−	4.00000i		−	0.147442i
$737$	−13.5826	−	21.1652i	−0.500321	−	0.779628i
$738$		0			0
$739$			1.58258i			0.0582160i	0.999576	+	0.0291080i	$0.00926667\pi$
$739$			1.58258i			0.0582160i	−0.999576	+	0.0291080i	$0.990733\pi$
$740$			11.0901i			0.407681i
$741$		0			0
$742$	−11.5826	+	28.3714i	−0.425210	+	1.04155i
$743$		−	1.91288i		−	0.0701767i	−0.999384	−	0.0350884i	$-0.988829\pi$
$743$		−	1.91288i		−	0.0701767i	0.999384	−	0.0350884i	$-0.0111713\pi$
$744$		0			0
$745$	43.3380			1.58778
$746$	−9.58258			−0.350843
$747$		0			0
$748$	−6.70239	−	10.4440i	−0.245063	−	0.381872i
$749$	17.5826	−	43.0683i	0.642453	−	1.57368i
$750$		0			0
$751$	−41.4955			−1.51419			−0.757095	−	0.653304i	$-0.773382\pi$
$751$	−41.4955			−1.51419			−0.757095	+	0.653304i	$0.773382\pi$
$752$			9.93280i			0.362212i
$753$		0			0
$754$	1.02248			0.0372364
$755$	17.2813			0.628930
$756$		0			0
$757$	−30.8348			−1.12071			−0.560356	−	0.828252i	$-0.689336\pi$
$757$	−30.8348			−1.12071			−0.560356	+	0.828252i	$0.689336\pi$
$758$		−	22.3303i		−	0.811073i
$759$		0			0
$760$	5.58258			0.202501
$761$	−28.2366			−1.02357			−0.511787	−	0.859112i	$-0.671016\pi$
$761$	−28.2366			−1.02357			−0.511787	+	0.859112i	$0.671016\pi$
$762$		0			0
$763$	−17.1652	+	42.0459i	−0.621420	+	1.52216i
$764$	−18.0000			−0.651217
$765$		0			0
$766$	19.7308			0.712901
$767$			0.417424i			0.0150723i
$768$		0			0
$769$	−15.1015			−0.544573			−0.272287	−	0.962216i	$-0.587780\pi$
$769$	−15.1015			−0.544573			−0.272287	+	0.962216i	$0.587780\pi$
$770$	4.94194	+	26.7102i	0.178095	+	0.962569i
$771$		0			0
$772$		−	16.3303i		−	0.587740i
$773$		−	10.0395i		−	0.361096i	−0.983566	−	0.180548i	$-0.942213\pi$
$773$		−	10.0395i		−	0.361096i	0.983566	−	0.180548i	$-0.0577871\pi$
$774$		0			0
$775$			39.5964i			1.42234i
$776$	−8.50579			−0.305340
$777$		0			0
$778$		−	10.0000i		−	0.358517i
$779$	−17.9129			−0.641795
$780$		0			0
$781$	−3.58258	−	5.58258i	−0.128195	−	0.199760i
$782$		−	14.9666i		−	0.535206i
$783$		0			0
$784$	5.00000	+	4.89898i	0.178571	+	0.174964i
$785$	−40.7477			−1.45435
$786$		0			0
$787$	23.2309			0.828091			0.414046	−	0.910256i	$-0.364115\pi$
$787$	23.2309			0.828091			0.414046	+	0.910256i	$0.364115\pi$
$788$			8.74773i			0.311625i
$789$		0			0
$790$		−	12.3823i		−	0.440542i
$791$	−26.3264	−	10.7477i	−0.936061	−	0.382145i
$792$		0			0
$793$	1.25227			0.0444695
$794$	27.5905			0.979149
$795$		0			0
$796$			9.66311i			0.342500i
$797$		−	25.0061i		−	0.885763i	−0.896580	−	0.442881i	$-0.853956\pi$
$797$		−	25.0061i		−	0.885763i	0.896580	−	0.442881i	$-0.146044\pi$
$798$		0			0
$799$			37.1652i			1.31481i
$800$		−	4.58258i		−	0.162019i
$801$		0			0
$802$		−	7.58258i		−	0.267750i
$803$	28.8826	+	45.0066i	1.01925	+	1.58825i
$804$		0			0
$805$	−12.3823	+	30.3303i	−0.436419	+	1.06900i
$806$	5.58258			0.196638
$807$		0			0
$808$		−	11.7362i		−	0.412878i
$809$		−	14.8348i		−	0.521566i	−0.965398	−	0.260783i	$-0.916019\pi$
$809$		−	14.8348i		−	0.521566i	0.965398	−	0.260783i	$-0.0839806\pi$
$810$		0			0
$811$	3.36526			0.118170			0.0590852	−	0.998253i	$-0.481182\pi$
$811$	3.36526			0.118170			0.0590852	+	0.998253i	$0.481182\pi$
$812$	−1.58258	+	3.87650i	−0.0555375	+	0.136039i
$813$		0			0
$814$	−10.0000	+	6.41742i	−0.350500	+	0.224931i
$815$		−	12.3823i		−	0.433733i
$816$		0			0
$817$			12.9217i			0.452072i
$818$		−	5.03383i		−	0.176004i
$819$		0			0
$820$			30.7477i			1.07376i
$821$			43.4955i			1.51800i	0.651090	+	0.759001i	$0.274312\pi$
$821$			43.4955i			1.51800i	−0.651090	+	0.759001i	$0.725688\pi$
$822$		0			0
$823$	−3.66970			−0.127918			−0.0639588	−	0.997953i	$-0.520373\pi$
$823$	−3.66970			−0.127918			−0.0639588	+	0.997953i	$0.520373\pi$
$824$	6.05630			0.210981
$825$		0			0
$826$	−1.58258	−	0.646084i	−0.0550649	−	0.0224801i
$827$		−	3.16515i		−	0.110063i	−0.998485	−	0.0550315i	$-0.982474\pi$
$827$		−	3.16515i		−	0.110063i	0.998485	−	0.0550315i	$-0.0175259\pi$
$828$		0			0
$829$			7.99455i			0.277662i	0.990316	+	0.138831i	$0.0443345\pi$
$829$			7.99455i			0.277662i	−0.990316	+	0.138831i	$0.955665\pi$
$830$	−39.9129			−1.38540
$831$		0			0
$832$	−0.646084			−0.0223989
$833$	18.7083	+	18.3303i	0.648204	+	0.635107i
$834$		0			0
$835$			4.00000i			0.138426i
$836$	3.23042	+	5.03383i	0.111726	+	0.174098i
$837$		0			0
$838$	13.0284			0.450058
$839$			9.66311i			0.333608i	0.985990	+	0.166804i	$0.0533447\pi$
$839$			9.66311i			0.333608i	−0.985990	+	0.166804i	$0.946655\pi$
$840$		0			0
$841$	26.4955			0.913636
$842$			7.58258i			0.261313i
$843$		0			0
$844$		−	16.7477i		−	0.576481i
$845$			38.9503i			1.33993i
$846$		0			0
$847$	−21.2250	+	19.9123i	−0.729299	+	0.684195i
$848$	11.5826			0.397747
$849$		0			0
$850$		−	17.1464i		−	0.588118i
$851$	−14.3303			−0.491236
$852$		0			0
$853$	11.4665			0.392606			0.196303	−	0.980543i	$-0.437106\pi$
$853$	11.4665			0.392606			0.196303	+	0.980543i	$0.437106\pi$
$854$	−1.93825	+	4.74773i	−0.0663256	+	0.162464i
$855$		0			0
$856$	−17.5826			−0.600960
$857$	31.0906			1.06203			0.531017	−	0.847361i	$-0.321810\pi$
$857$	31.0906			1.06203			0.531017	+	0.847361i	$0.321810\pi$
$858$		0			0
$859$		−	24.3882i		−	0.832115i	−0.909338	−	0.416057i	$-0.863412\pi$
$859$		−	24.3882i		−	0.832115i	0.909338	−	0.416057i	$-0.136588\pi$
$860$	22.1803			0.756340
$861$		0			0
$862$	−27.9129			−0.950716
$863$	14.8348			0.504984			0.252492	−	0.967599i	$-0.418750\pi$
$863$	14.8348			0.504984			0.252492	+	0.967599i	$0.418750\pi$
$864$		0			0
$865$			33.1652i			1.12765i
$866$	−35.5850			−1.20923
$867$		0			0
$868$	−8.64064	+	21.1652i	−0.293282	+	0.718392i
$869$	11.1652	−	7.16515i	0.378752	−	0.243061i
$870$		0			0
$871$	−4.89898			−0.165996
$872$	17.1652			0.581285
$873$		0			0
$874$			7.21362i			0.244004i
$875$	1.29217	−	3.16515i	0.0436832	−	0.107002i
$876$		0			0
$877$		−	6.83485i		−	0.230796i	−0.993319	−	0.115398i	$-0.963186\pi$
$877$		−	6.83485i		−	0.230796i	0.993319	−	0.115398i	$-0.0368144\pi$
$878$			17.2813i			0.583215i
$879$		0			0
$880$	8.64064	−	5.54506i	0.291276	−	0.186924i
$881$		−	7.21362i		−	0.243033i	−0.992589	−	0.121517i	$-0.961224\pi$
$881$		−	7.21362i		−	0.243033i	0.992589	−	0.121517i	$-0.0387758\pi$
$882$		0			0
$883$	12.8348			0.431927			0.215964	−	0.976401i	$-0.430711\pi$
$883$	12.8348			0.431927			0.215964	+	0.976401i	$0.430711\pi$
$884$	−2.41742			−0.0813068
$885$		0			0
$886$		−	15.1652i		−	0.509483i
$887$	53.4057			1.79319			0.896594	−	0.442854i	$-0.146034\pi$
$887$	53.4057			1.79319			0.896594	+	0.442854i	$0.146034\pi$
$888$		0			0
$889$	3.58258	−	8.77548i	0.120156	−	0.294320i
$890$		−	30.3303i		−	1.01667i
$891$		0			0
$892$		−	16.1240i		−	0.539870i
$893$		−	17.9129i		−	0.599432i
$894$		0			0
$895$			34.5625i			1.15530i
$896$	1.00000	−	2.44949i	0.0334077	−	0.0818317i
$897$		0			0
$898$		−	31.1652i		−	1.03999i
$899$	−13.6745			−0.456069
$900$		0			0
$901$	43.3380			1.44380
$902$	−27.7253	+	17.7925i	−0.923152	+	0.592426i
$903$		0			0
$904$			10.7477i			0.357464i
$905$	16.7477			0.556713
$906$		0			0
$907$	5.91288			0.196334			0.0981670	−	0.995170i	$-0.468702\pi$
$907$	5.91288			0.196334			0.0981670	+	0.995170i	$0.468702\pi$
$908$	−0.511238			−0.0169660
$909$		0			0
$910$	4.89898	+	2.00000i	0.162400	+	0.0662994i
$911$	−3.16515			−0.104866			−0.0524331	−	0.998624i	$-0.516698\pi$
$911$	−3.16515			−0.104866			−0.0524331	+	0.998624i	$0.516698\pi$
$912$		0			0
$913$	−23.0960	−	35.9896i	−0.764367	−	1.19108i
$914$	−33.1652			−1.09701
$915$		0			0
$916$			5.67991i			0.187669i
$917$	41.0780	+	16.7700i	1.35652	+	0.553795i
$918$		0			0
$919$			19.4955i			0.643096i	0.946893	+	0.321548i	$0.104203\pi$
$919$			19.4955i			0.643096i	−0.946893	+	0.321548i	$0.895797\pi$
$920$	12.3823			0.408232
$921$		0			0
$922$		−	30.0400i		−	0.989313i
$923$	−1.29217			−0.0425322
$924$		0			0
$925$	−16.4174			−0.539802
$926$			5.16515i			0.169737i
$927$		0			0
$928$	1.58258			0.0519506
$929$		−	43.6077i		−	1.43072i	−0.698755	−	0.715361i	$-0.746262\pi$
$929$		−	43.6077i		−	1.43072i	0.698755	−	0.715361i	$-0.253738\pi$
$930$		0			0
$931$	−9.01703	−	8.83485i	−0.295521	−	0.289550i
$932$		−	27.1652i		−	0.889824i
$933$		0			0
$934$	−11.7362			−0.384021
$935$	32.3303	−	20.7477i	1.05731	−	0.678523i
$936$		0			0
$937$	30.0681			0.982282			0.491141	−	0.871080i	$-0.336580\pi$
$937$	30.0681			0.982282			0.491141	+	0.871080i	$0.336580\pi$
$938$	7.58258	−	18.5734i	0.247580	−	0.606444i
$939$		0			0
$940$	−30.7477			−1.00288
$941$	5.27537			0.171972			0.0859861	−	0.996296i	$-0.472596\pi$
$941$	5.27537			0.171972			0.0859861	+	0.996296i	$0.472596\pi$
$942$		0			0
$943$	−39.7312			−1.29383
$944$			0.646084i			0.0210282i
$945$		0			0
$946$	12.8348	+	20.0000i	0.417297	+	0.650256i
$947$	−3.58258			−0.116418			−0.0582090	−	0.998304i	$-0.518539\pi$
$947$	−3.58258			−0.116418			−0.0582090	+	0.998304i	$0.518539\pi$
$948$		0			0
$949$	10.4174			0.338164
$950$			8.26424i			0.268127i
$951$		0			0
$952$	3.74166	−	9.16515i	0.121268	−	0.297044i
$953$			18.3303i			0.593777i	0.954912	+	0.296888i	$0.0959489\pi$
$953$			18.3303i			0.593777i	−0.954912	+	0.296888i	$0.904051\pi$
$954$		0			0
$955$		−	55.7203i		−	1.80307i
$956$		−	8.41742i		−	0.272239i
$957$		0			0
$958$		−	30.9557i		−	1.00013i
$959$	7.75301	+	3.16515i	0.250358	+	0.102208i
$960$		0			0
$961$	−43.6606			−1.40841
$962$			2.31464i			0.0746271i
$963$		0			0
$964$	3.47197			0.111825
$965$	50.5517			1.62732
$966$		0			0
$967$		−	22.4174i		−	0.720896i	−0.932779	−	0.360448i	$-0.882624\pi$
$967$		−	22.4174i		−	0.720896i	0.932779	−	0.360448i	$-0.117376\pi$
$968$	10.0000	+	4.58258i	0.321412	+	0.147290i
$969$		0			0
$970$		−	26.3303i		−	0.845415i
$971$		−	14.0509i		−	0.450913i	−0.974253	−	0.225457i	$-0.927613\pi$
$971$		−	14.0509i		−	0.450913i	0.974253	−	0.225457i	$-0.0723874\pi$
$972$		0			0
$973$	−37.9129	−	15.4779i	−1.21543	−	0.496198i
$974$			21.4955i			0.688759i
$975$		0			0
$976$	1.93825			0.0620419
$977$	−9.16515			−0.293219			−0.146610	−	0.989194i	$-0.546836\pi$
$977$	−9.16515			−0.293219			−0.146610	+	0.989194i	$0.546836\pi$
$978$		0			0
$979$	27.3489	−	17.5510i	0.874075	−	0.560931i
$980$	−15.1652	+	15.4779i	−0.484433	+	0.494422i
$981$		0			0
$982$	33.4955			1.06888
$983$		−	50.6865i		−	1.61665i	−0.588738	−	0.808324i	$-0.700375\pi$
$983$		−	50.6865i		−	1.61665i	0.588738	−	0.808324i	$-0.299625\pi$
$984$		0			0
$985$	−27.0792			−0.862816
$986$	5.92146			0.188578
$987$		0			0
$988$	1.16515			0.0370684
$989$			28.6606i			0.911354i
$990$		0			0
$991$	46.6606			1.48222			0.741111	−	0.671382i	$-0.234299\pi$
$991$	46.6606			1.48222			0.741111	+	0.671382i	$0.234299\pi$
$992$	8.64064			0.274340
$993$		0			0
$994$	2.00000	−	4.89898i	0.0634361	−	0.155386i
$995$	−29.9129			−0.948302
$996$		0			0
$997$	22.5566			0.714376			0.357188	−	0.934032i	$-0.383736\pi$
$997$	22.5566			0.714376			0.357188	+	0.934032i	$0.383736\pi$
$998$		−	17.9129i		−	0.567022i
$999$		0			0

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	1386.2.e.b.307.5		8
3.2	odd	2		154.2.c.a.153.3	yes	8
7.6	odd	2	inner	1386.2.e.b.307.8		8
11.10	odd	2	inner	1386.2.e.b.307.1		8
12.11	even	2		1232.2.e.e.769.3		8
21.2	odd	6		1078.2.i.b.1011.2		16
21.5	even	6		1078.2.i.b.1011.3		16
21.11	odd	6		1078.2.i.b.901.7		16
21.17	even	6		1078.2.i.b.901.6		16
21.20	even	2		154.2.c.a.153.2	✓	8
33.32	even	2		154.2.c.a.153.7	yes	8
77.76	even	2	inner	1386.2.e.b.307.4		8
84.83	odd	2		1232.2.e.e.769.6		8
132.131	odd	2		1232.2.e.e.769.4		8
231.32	even	6		1078.2.i.b.901.3		16
231.65	even	6		1078.2.i.b.1011.6		16
231.131	odd	6		1078.2.i.b.1011.7		16
231.164	odd	6		1078.2.i.b.901.2		16
231.230	odd	2		154.2.c.a.153.6	yes	8
924.923	even	2		1232.2.e.e.769.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.c.a.153.2	✓	8	21.20	even	2
154.2.c.a.153.3	yes	8	3.2	odd	2
154.2.c.a.153.6	yes	8	231.230	odd	2
154.2.c.a.153.7	yes	8	33.32	even	2
1078.2.i.b.901.2		16	231.164	odd	6
1078.2.i.b.901.3		16	231.32	even	6
1078.2.i.b.901.6		16	21.17	even	6
1078.2.i.b.901.7		16	21.11	odd	6
1078.2.i.b.1011.2		16	21.2	odd	6
1078.2.i.b.1011.3		16	21.5	even	6
1078.2.i.b.1011.6		16	231.65	even	6
1078.2.i.b.1011.7		16	231.131	odd	6
1232.2.e.e.769.3		8	12.11	even	2
1232.2.e.e.769.4		8	132.131	odd	2
1232.2.e.e.769.5		8	924.923	even	2
1232.2.e.e.769.6		8	84.83	odd	2
1386.2.e.b.307.1		8	11.10	odd	2	inner
1386.2.e.b.307.4		8	77.76	even	2	inner
1386.2.e.b.307.5		8	1.1	even	1	trivial
1386.2.e.b.307.8		8	7.6	odd	2	inner

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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

Newform invariants

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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	1386.2.e.b.307.5

Level	$1386$
Weight	$2$
Character	1386.307
Analytic conductor	$11.067$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$