LMFDB - Embedded newform 392.6.i.i.177.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,6,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not 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:	$62.8704573667$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-115})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 28x^{2} - 29x + 841 $ x^4 - x^3 - 28x^2 - 29x + 841
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			177.2
Root			$4.89354 - 2.24794i$ of defining polynomial
Character	$\chi$	$=$	392.177
Dual form			392.6.i.i.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.78709 + 13.4876i) q^{3} +(48.3613 - 83.7642i) q^{5} +(0.222527 - 0.385428i) q^{9} +O(q^{10})$	\(q+(7.78709 + 13.4876i) q^{3} +(48.3613 - 83.7642i) q^{5} +(0.222527 - 0.385428i) q^{9} +(-140.723 - 243.739i) q^{11} +269.393 q^{13} +1506.37 q^{15} +(859.723 + 1489.08i) q^{17} +(-586.419 + 1015.71i) q^{19} +(-392.670 + 680.125i) q^{23} +(-3115.12 - 5395.55i) q^{25} +3791.46 q^{27} +6149.46 q^{29} +(-3503.29 - 6067.87i) q^{31} +(2191.64 - 3796.03i) q^{33} +(5749.96 - 9959.23i) q^{37} +(2097.79 + 3633.47i) q^{39} +13245.3 q^{41} -19824.2 q^{43} +(-21.5234 - 37.2796i) q^{45} +(3443.98 - 5965.15i) q^{47} +(-13389.5 + 23191.2i) q^{51} +(-4327.48 - 7495.42i) q^{53} -27222.1 q^{55} -18266.0 q^{57} +(-23928.3 - 41445.0i) q^{59} +(26381.0 - 45693.3i) q^{61} +(13028.2 - 22565.5i) q^{65} +(12020.1 + 20819.4i) q^{67} -12231.0 q^{69} +12540.9 q^{71} +(2039.64 + 3532.76i) q^{73} +(48515.5 - 84031.3i) q^{75} +(5959.54 - 10322.2i) q^{79} +(29470.3 + 51044.1i) q^{81} -81916.2 q^{83} +166309. q^{85} +(47886.4 + 82941.6i) q^{87} +(-48484.4 + 83977.4i) q^{89} +(54560.8 - 94502.1i) q^{93} +(56719.9 + 98241.8i) q^{95} +26410.7 q^{97} -125.258 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{3} + 82 q^{5} - 222 q^{9}+O(q^{10}) $ 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9	$ 4 q - 6 q^{3} + 82 q^{5} - 222 q^{9} - 340 q^{11} - 1820 q^{13} + 3648 q^{15} + 3216 q^{17} - 674 q^{19} + 1104 q^{23} - 3322 q^{25} + 6696 q^{27} + 16128 q^{29} - 6212 q^{31} + 3120 q^{33} + 8512 q^{37} + 29640 q^{39} + 2608 q^{41} - 20008 q^{43} - 3318 q^{45} - 12748 q^{47} + 5508 q^{51} + 11220 q^{53} - 52720 q^{55} - 58056 q^{57} - 12018 q^{59} + 102738 q^{61} + 43420 q^{65} - 24136 q^{67} - 105984 q^{69} + 179440 q^{71} - 55588 q^{73} + 159774 q^{75} - 48824 q^{79} + 122562 q^{81} - 71564 q^{83} + 288552 q^{85} + 54468 q^{87} - 18300 q^{89} + 126264 q^{93} + 120784 q^{95} + 139968 q^{97} + 25800 q^{99}+O(q^{100}) $ 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9 - 340 * q^11 - 1820 * q^13 + 3648 * q^15 + 3216 * q^17 - 674 * q^19 + 1104 * q^23 - 3322 * q^25 + 6696 * q^27 + 16128 * q^29 - 6212 * q^31 + 3120 * q^33 + 8512 * q^37 + 29640 * q^39 + 2608 * q^41 - 20008 * q^43 - 3318 * q^45 - 12748 * q^47 + 5508 * q^51 + 11220 * q^53 - 52720 * q^55 - 58056 * q^57 - 12018 * q^59 + 102738 * q^61 + 43420 * q^65 - 24136 * q^67 - 105984 * q^69 + 179440 * q^71 - 55588 * q^73 + 159774 * q^75 - 48824 * q^79 + 122562 * q^81 - 71564 * q^83 + 288552 * q^85 + 54468 * q^87 - 18300 * q^89 + 126264 * q^93 + 120784 * q^95 + 139968 * q^97 + 25800 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	7.78709	+	13.4876i	0.499542	+	0.865232i	1.00000		0.000528869i	$-0.000168344\pi$
$3$	7.78709	+	13.4876i	0.499542	+	0.865232i	−0.500458	+	0.865761i	$0.666835\pi$
$4$		0			0
$5$	48.3613	−	83.7642i	0.865113	−	1.49842i	−0.00182277	−	0.999998i	$-0.500580\pi$
$5$	48.3613	−	83.7642i	0.865113	−	1.49842i	0.866935	−	0.498421i	$-0.166086\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.222527	−	0.385428i	0.000915748	−	0.00158612i
$10$		0			0
$11$	−140.723	−	243.739i	−0.350657	−	0.607355i	0.635708	−	0.771930i	$-0.280708\pi$
$11$	−140.723	−	243.739i	−0.350657	−	0.607355i	−0.986365	+	0.164575i	$0.947375\pi$
$12$		0			0
$13$	269.393			0.442107			0.221054	−	0.975262i	$-0.429050\pi$
$13$	269.393			0.442107			0.221054	+	0.975262i	$0.429050\pi$
$14$		0			0
$15$	1506.37			1.72864
$16$		0			0
$17$	859.723	+	1489.08i	0.721499	+	1.24967i	0.960399	+	0.278629i	$0.0898801\pi$
$17$	859.723	+	1489.08i	0.721499	+	1.24967i	−0.238899	+	0.971044i	$0.576787\pi$
$18$		0			0
$19$	−586.419	+	1015.71i	−0.372670	+	0.645483i	−0.989975	−	0.141241i	$-0.954891\pi$
$19$	−586.419	+	1015.71i	−0.372670	+	0.645483i	0.617306	+	0.786723i	$0.288224\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−392.670	+	680.125i	−0.154778	+	0.268083i	−0.932978	−	0.359933i	$-0.882799\pi$
$23$	−392.670	+	680.125i	−0.154778	+	0.268083i	0.778200	+	0.628016i	$0.216133\pi$
$24$		0			0
$25$	−3115.12	−	5395.55i	−0.996840	−	1.72658i
$26$		0			0
$27$	3791.46			1.00091
$28$		0			0
$29$	6149.46			1.35782			0.678909	−	0.734222i	$-0.262453\pi$
$29$	6149.46			1.35782			0.678909	+	0.734222i	$0.262453\pi$
$30$		0			0
$31$	−3503.29	−	6067.87i	−0.654744	−	1.13405i	−0.981958	−	0.189100i	$-0.939443\pi$
$31$	−3503.29	−	6067.87i	−0.654744	−	1.13405i	0.327214	−	0.944950i	$-0.393890\pi$
$32$		0			0
$33$	2191.64	−	3796.03i	0.350335	−	0.606798i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5749.96	−	9959.23i	0.690495	−	1.19597i	−0.281180	−	0.959655i	$-0.590726\pi$
$37$	5749.96	−	9959.23i	0.690495	−	1.19597i	0.971676	−	0.236318i	$-0.0759408\pi$
$38$		0			0
$39$	2097.79	+	3633.47i	0.220851	+	0.382525i
$40$		0			0
$41$	13245.3			1.23056			0.615279	−	0.788310i	$-0.289043\pi$
$41$	13245.3			1.23056			0.615279	+	0.788310i	$0.289043\pi$
$42$		0			0
$43$	−19824.2			−1.63502			−0.817512	−	0.575911i	$-0.804647\pi$
$43$	−19824.2			−1.63502			−0.817512	+	0.575911i	$0.804647\pi$
$44$		0			0
$45$	−21.5234	−	37.2796i	−0.00158445	−	0.00274435i
$46$		0			0
$47$	3443.98	−	5965.15i	0.227413	−	0.393892i	−0.729627	−	0.683845i	$-0.760306\pi$
$47$	3443.98	−	5965.15i	0.227413	−	0.393892i	0.957041	+	0.289953i	$0.0936398\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−13389.5	+	23191.2i	−0.720838	+	1.24853i
$52$		0			0
$53$	−4327.48	−	7495.42i	−0.211615	−	0.366527i	0.740605	−	0.671940i	$-0.234539\pi$
$53$	−4327.48	−	7495.42i	−0.211615	−	0.366527i	−0.952220	+	0.305413i	$0.901206\pi$
$54$		0			0
$55$	−27222.1			−1.21343
$56$		0			0
$57$	−18266.0			−0.744656
$58$		0			0
$59$	−23928.3	−	41445.0i	−0.894915	−	1.55004i	−0.833909	−	0.551902i	$-0.813902\pi$
$59$	−23928.3	−	41445.0i	−0.894915	−	1.55004i	−0.0610061	−	0.998137i	$-0.519431\pi$
$60$		0			0
$61$	26381.0	−	45693.3i	0.907752	−	1.57227i	0.0905720	−	0.995890i	$-0.471130\pi$
$61$	26381.0	−	45693.3i	0.907752	−	1.57227i	0.817180	−	0.576383i	$-0.195536\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	13028.2	−	22565.5i	0.382473	−	0.662462i
$66$		0			0
$67$	12020.1	+	20819.4i	0.327130	+	0.566607i	0.981941	−	0.189186i	$-0.0605850\pi$
$67$	12020.1	+	20819.4i	0.327130	+	0.566607i	−0.654811	+	0.755793i	$0.727252\pi$
$68$		0			0
$69$	−12231.0			−0.309272
$70$		0			0
$71$	12540.9			0.295246			0.147623	−	0.989044i	$-0.452838\pi$
$71$	12540.9			0.295246			0.147623	+	0.989044i	$0.452838\pi$
$72$		0			0
$73$	2039.64	+	3532.76i	0.0447968	+	0.0775903i	0.887554	−	0.460703i	$-0.152403\pi$
$73$	2039.64	+	3532.76i	0.0447968	+	0.0775903i	−0.842758	+	0.538293i	$0.819069\pi$
$74$		0			0
$75$	48515.5	−	84031.3i	0.995926	−	1.72499i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5959.54	−	10322.2i	0.107435	−	0.186083i	−0.807296	−	0.590147i	$-0.799070\pi$
$79$	5959.54	−	10322.2i	0.107435	−	0.186083i	0.914730	+	0.404065i	$0.132403\pi$
$80$		0			0
$81$	29470.3	+	51044.1i	0.499083	+	0.864436i
$82$		0			0
$83$	−81916.2			−1.30519			−0.652596	−	0.757706i	$-0.726320\pi$
$83$	−81916.2			−1.30519			−0.652596	+	0.757706i	$0.726320\pi$
$84$		0			0
$85$	166309.			2.49671
$86$		0			0
$87$	47886.4	+	82941.6i	0.678287	+	1.17483i
$88$		0			0
$89$	−48484.4	+	83977.4i	−0.648824	+	1.12380i	0.334581	+	0.942367i	$0.391405\pi$
$89$	−48484.4	+	83977.4i	−0.648824	+	1.12380i	−0.983404	+	0.181428i	$0.941928\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	54560.8	−	94502.1i	0.654144	−	1.13301i
$94$		0			0
$95$	56719.9	+	98241.8i	0.644802	+	1.11683i
$96$		0			0
$97$	26410.7			0.285004			0.142502	−	0.989795i	$-0.454485\pi$
$97$	26410.7			0.285004			0.142502	+	0.989795i	$0.454485\pi$
$98$		0			0
$99$	−125.258			−0.00128445
$100$		0			0
$101$	−36568.8	−	63339.1i	−0.356704	−	0.617829i	0.630704	−	0.776023i	$-0.282766\pi$
$101$	−36568.8	−	63339.1i	−0.356704	−	0.617829i	−0.987408	+	0.158194i	$0.949433\pi$
$102$		0			0
$103$	43824.7	−	75906.7i	0.407030	−	0.704996i	−0.587526	−	0.809206i	$-0.699898\pi$
$103$	43824.7	−	75906.7i	0.407030	−	0.704996i	0.994555	+	0.104209i	$0.0332312\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	57719.1	−	99972.5i	0.487372	−	0.844153i	−0.512523	−	0.858674i	$-0.671289\pi$
$107$	57719.1	−	99972.5i	0.487372	−	0.844153i	0.999895	+	0.0145210i	$0.00462235\pi$
$108$		0			0
$109$	58651.4	+	101587.i	0.472837	+	0.818979i	0.999517	−	0.0310856i	$-0.00989644\pi$
$109$	58651.4	+	101587.i	0.472837	+	0.818979i	−0.526679	+	0.850064i	$0.676563\pi$
$110$		0			0
$111$	179102.			1.37973
$112$		0			0
$113$	181535.			1.33741			0.668705	−	0.743528i	$-0.266849\pi$
$113$	181535.			1.33741			0.668705	+	0.743528i	$0.266849\pi$
$114$		0			0
$115$	37980.1	+	65783.4i	0.267800	+	0.463844i
$116$		0			0
$117$	59.9471	−	103.832i	0.000404859	−	0.000701236i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	40919.8	−	70875.2i	0.254080	−	0.440079i
$122$		0			0
$123$	103142.	+	178648.i	0.614715	+	1.06472i
$124$		0			0
$125$	−300347.			−1.71929
$126$		0			0
$127$	106111.			0.583780			0.291890	−	0.956452i	$-0.405716\pi$
$127$	106111.			0.583780			0.291890	+	0.956452i	$0.405716\pi$
$128$		0			0
$129$	−154373.	−	267381.i	−0.816763	−	1.41468i
$130$		0			0
$131$	150167.	−	260097.i	0.764534	−	1.32421i	−0.175959	−	0.984397i	$-0.556303\pi$
$131$	150167.	−	260097.i	0.764534	−	1.32421i	0.940493	−	0.339814i	$-0.110364\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	183360.	−	317588.i	0.865903	−	1.49979i
$136$		0			0
$137$	45946.2	+	79581.2i	0.209146	+	0.362251i	0.951446	−	0.307817i	$-0.0995984\pi$
$137$	45946.2	+	79581.2i	0.209146	+	0.362251i	−0.742300	+	0.670068i	$0.766265\pi$
$138$		0			0
$139$	−153133.			−0.672252			−0.336126	−	0.941817i	$-0.609117\pi$
$139$	−153133.			−0.672252			−0.336126	+	0.941817i	$0.609117\pi$
$140$		0			0
$141$	107274.			0.454410
$142$		0			0
$143$	−37909.6	−	65661.4i	−0.155028	−	0.268516i
$144$		0			0
$145$	297395.	−	515104.i	1.17467	−	2.03458i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−249301.	+	431802.i	−0.919937	+	1.59338i	−0.120429	+	0.992722i	$0.538427\pi$
$149$	−249301.	+	431802.i	−0.919937	+	1.59338i	−0.799508	+	0.600656i	$0.794906\pi$
$150$		0			0
$151$	−104805.	−	181527.i	−0.374057	−	0.647886i	0.616128	−	0.787646i	$-0.288700\pi$
$151$	−104805.	−	181527.i	−0.374057	−	0.647886i	−0.990185	+	0.139760i	$0.955367\pi$
$152$		0			0
$153$	765.245			0.00264285
$154$		0			0
$155$	−677694.			−2.26571
$156$		0			0
$157$	−3690.86	−	6392.76i	−0.0119503	−	0.0206985i	0.859988	−	0.510314i	$-0.170471\pi$
$157$	−3690.86	−	6392.76i	−0.0119503	−	0.0206985i	−0.871939	+	0.489615i	$0.837137\pi$
$158$		0			0
$159$	67397.0	−	116735.i	0.211421	−	0.366192i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−194077.	+	336152.i	−0.572144	+	0.990983i	0.424201	+	0.905568i	$0.360555\pi$
$163$	−194077.	+	336152.i	−0.572144	+	0.990983i	−0.996345	+	0.0854149i	$0.972778\pi$
$164$		0			0
$165$	−211981.	−	367161.i	−0.606159	−	1.04990i
$166$		0			0
$167$	168240.			0.466808			0.233404	−	0.972380i	$-0.425013\pi$
$167$	168240.			0.466808			0.233404	+	0.972380i	$0.425013\pi$
$168$		0			0
$169$	−298720.			−0.804541
$170$		0			0
$171$	260.988	+	452.044i	0.000682543	+	0.00118220i
$172$		0			0
$173$	160757.	−	278439.i	0.408370	−	0.707317i	−0.586337	−	0.810067i	$-0.699431\pi$
$173$	160757.	−	278439.i	0.408370	−	0.707317i	0.994707	+	0.102750i	$0.0327641\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	372664.	−	645472.i	0.894095	−	1.54862i
$178$		0			0
$179$	305953.	+	529926.i	0.713710	+	1.23618i	0.963455	+	0.267870i	$0.0863199\pi$
$179$	305953.	+	529926.i	0.713710	+	1.23618i	−0.249745	+	0.968312i	$0.580347\pi$
$180$		0			0
$181$	−261474.			−0.593242			−0.296621	−	0.954995i	$-0.595860\pi$
$181$	−261474.			−0.593242			−0.296621	+	0.954995i	$0.595860\pi$
$182$		0			0
$183$	821726.			1.81384
$184$		0			0
$185$	−556151.	−	963282.i	−1.19471	−	2.06930i
$186$		0			0
$187$	241965.	−	419095.i	0.505997	−	0.876412i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−351651.	+	609077.i	−0.697474	+	1.20806i	0.271866	+	0.962335i	$0.412359\pi$
$191$	−351651.	+	609077.i	−0.697474	+	1.20806i	−0.969340	+	0.245725i	$0.920974\pi$
$192$		0			0
$193$	−258205.	−	447225.i	−0.498967	−	0.864236i	0.501032	−	0.865429i	$-0.332954\pi$
$193$	−258205.	−	447225.i	−0.498967	−	0.864236i	−0.999999	+	0.00119237i	$0.999620\pi$
$194$		0			0
$195$	405806.			0.764244
$196$		0			0
$197$	142579.			0.261752			0.130876	−	0.991399i	$-0.458221\pi$
$197$	142579.			0.261752			0.130876	+	0.991399i	$0.458221\pi$
$198$		0			0
$199$	492419.	+	852894.i	0.881458	+	1.52673i	0.849720	+	0.527234i	$0.176771\pi$
$199$	492419.	+	852894.i	0.881458	+	1.52673i	0.0317383	+	0.999496i	$0.489896\pi$
$200$		0			0
$201$	−187203.	+	324245.i	−0.326831	+	0.566088i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	640559.	−	1.10948e6i	1.06457	−	1.84389i
$206$		0			0
$207$	174.759	+	302.692i	0.000283475	+	0.000490993i
$208$		0			0
$209$	330089.			0.522716
$210$		0			0
$211$	−242836.			−0.375497			−0.187748	−	0.982217i	$-0.560119\pi$
$211$	−242836.			−0.375497			−0.187748	+	0.982217i	$0.560119\pi$
$212$		0			0
$213$	97657.4	+	169148.i	0.147488	+	0.255456i
$214$		0			0
$215$	−958723.	+	1.66056e6i	−1.41448	+	2.44995i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−31765.8	+	55019.9i	−0.0447558	+	0.0775192i
$220$		0			0
$221$	231603.	+	401148.i	0.318980	+	0.552490i
$222$		0			0
$223$	797063.			1.07332			0.536661	−	0.843798i	$-0.319685\pi$
$223$	797063.			1.07332			0.536661	+	0.843798i	$0.319685\pi$
$224$		0			0
$225$	−2772.79			−0.00365142
$226$		0			0
$227$	−13060.7	−	22621.8i	−0.0168229	−	0.0291382i	0.857491	−	0.514498i	$-0.172022\pi$
$227$	−13060.7	−	22621.8i	−0.0168229	−	0.0291382i	−0.874314	+	0.485360i	$0.838688\pi$
$228$		0			0
$229$	−380796.	+	659558.i	−0.479848	+	0.831121i	−0.999733	−	0.0231156i	$-0.992641\pi$
$229$	−380796.	+	659558.i	−0.479848	+	0.831121i	0.519885	+	0.854236i	$0.325975\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−328926.	+	569716.i	−0.396925	+	0.687494i	−0.993345	−	0.115179i	$-0.963256\pi$
$233$	−328926.	+	569716.i	−0.396925	+	0.687494i	0.596420	+	0.802673i	$0.296589\pi$
$234$		0			0
$235$	−333111.	−	576964.i	−0.393476	−	0.681521i
$236$		0			0
$237$	185630.			0.214673
$238$		0			0
$239$	965388.			1.09322			0.546609	−	0.837388i	$-0.315918\pi$
$239$	965388.			1.09322			0.546609	+	0.837388i	$0.315918\pi$
$240$		0			0
$241$	17532.1	+	30366.5i	0.0194443	+	0.0336784i	0.875584	−	0.483066i	$-0.160477\pi$
$241$	17532.1	+	30366.5i	0.0194443	+	0.0336784i	−0.856140	+	0.516745i	$0.827144\pi$
$242$		0			0
$243$	1685.86	−	2919.99i	0.00183149	−	0.00317224i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−157977.	+	273624.i	−0.164760	+	0.285373i
$248$		0			0
$249$	−637889.	−	1.10486e6i	−0.651998	−	1.12929i
$250$		0			0
$251$	−1.31610e6			−1.31857			−0.659286	−	0.751892i	$-0.729141\pi$
$251$	−1.31610e6			−1.31857			−0.659286	+	0.751892i	$0.729141\pi$
$252$		0			0
$253$	221030.			0.217095
$254$		0			0
$255$	1.29506e6	+	2.24312e6i	1.24721	+	2.16024i
$256$		0			0
$257$	−264901.	+	458822.i	−0.250179	+	0.433323i	−0.963575	−	0.267439i	$-0.913823\pi$
$257$	−264901.	+	458822.i	−0.250179	+	0.433323i	0.713396	+	0.700761i	$0.247156\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1368.42	−	2370.17i	0.00124342	−	0.00215367i
$262$		0			0
$263$	647286.	+	1.12113e6i	0.577041	+	0.999465i	0.995817	+	0.0913747i	$0.0291261\pi$
$263$	647286.	+	1.12113e6i	0.577041	+	0.999465i	−0.418775	+	0.908090i	$0.637541\pi$
$264$		0			0
$265$	−837130.			−0.732282
$266$		0			0
$267$	−1.51021e6			−1.29646
$268$		0			0
$269$	−222168.	−	384807.i	−0.187198	−	0.324236i	0.757117	−	0.653279i	$-0.226607\pi$
$269$	−222168.	−	384807.i	−0.187198	−	0.324236i	−0.944315	+	0.329043i	$0.893274\pi$
$270$		0			0
$271$	312379.	−	541056.i	0.258379	−	0.447526i	−0.707428	−	0.706785i	$-0.750145\pi$
$271$	312379.	−	541056.i	0.258379	−	0.447526i	0.965808	+	0.259259i	$0.0834782\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−876736.	+	1.51855e6i	−0.699097	+	1.21087i
$276$		0			0
$277$	1.00101e6	+	1.73380e6i	0.783860	+	1.35768i	0.929678	+	0.368373i	$0.120085\pi$
$277$	1.00101e6	+	1.73380e6i	0.783860	+	1.35768i	−0.145818	+	0.989311i	$0.546582\pi$
$278$		0			0
$279$	−3118.30			−0.00239832
$280$		0			0
$281$	249316.			0.188358			0.0941792	−	0.995555i	$-0.469977\pi$
$281$	249316.			0.188358			0.0941792	+	0.995555i	$0.469977\pi$
$282$		0			0
$283$	322568.	+	558705.i	0.239417	+	0.414683i	0.960547	−	0.278117i	$-0.0897103\pi$
$283$	322568.	+	558705.i	0.239417	+	0.414683i	−0.721130	+	0.692800i	$0.756377\pi$
$284$		0			0
$285$	−883366.	+	1.53003e6i	−0.644212	+	1.11581i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−768317.	+	1.33076e6i	−0.541123	+	0.937252i
$290$		0			0
$291$	205663.	+	356218.i	0.142371	+	0.246595i
$292$		0			0
$293$	−2.00830e6			−1.36666			−0.683328	−	0.730112i	$-0.739468\pi$
$293$	−2.00830e6			−1.36666			−0.683328	+	0.730112i	$0.739468\pi$
$294$		0			0
$295$	−4.62881e6			−3.09681
$296$		0			0
$297$	−533543.	−	924124.i	−0.350977	−	0.607910i
$298$		0			0
$299$	−105783.	+	183221.i	−0.0684284	+	0.118521i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	569530.	−	986454.i	0.356377	−	0.617263i
$304$		0			0
$305$	−2.55164e6	−	4.41957e6i	−1.57062	−	2.72039i
$306$		0			0
$307$	1.71113e6			1.03618			0.518092	−	0.855325i	$-0.326642\pi$
$307$	1.71113e6			1.03618			0.518092	+	0.855325i	$0.326642\pi$
$308$		0			0
$309$	1.36507e6			0.813314
$310$		0			0
$311$	−301025.	−	521391.i	−0.176483	−	0.305677i	0.764191	−	0.644990i	$-0.223139\pi$
$311$	−301025.	−	521391.i	−0.176483	−	0.305677i	−0.940673	+	0.339313i	$0.889805\pi$
$312$		0			0
$313$	−671042.	+	1.16228e6i	−0.387159	+	0.670579i	−0.992066	−	0.125717i	$-0.959877\pi$
$313$	−671042.	+	1.16228e6i	−0.387159	+	0.670579i	0.604907	+	0.796296i	$0.293210\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−509016.	+	881641.i	−0.284500	+	0.492769i	−0.972488	−	0.232953i	$-0.925161\pi$
$317$	−509016.	+	881641.i	−0.284500	+	0.492769i	0.687987	+	0.725723i	$0.258494\pi$
$318$		0			0
$319$	−865367.	−	1.49886e6i	−0.476128	−	0.824678i
$320$		0			0
$321$	1.79786e6			0.973850
$322$		0			0
$323$	−2.01663e6			−1.07552
$324$		0			0
$325$	−839192.	−	1.45352e6i	−0.440710	−	0.763332i
$326$		0			0
$327$	−913447.	+	1.58214e6i	−0.472404	+	0.818228i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	266185.	−	461047.i	0.133541	−	0.231300i	−0.791498	−	0.611171i	$-0.790699\pi$
$331$	266185.	−	461047.i	0.133541	−	0.231300i	0.925039	+	0.379872i	$0.124032\pi$
$332$		0			0
$333$	−2559.04	−	4432.39i	−0.00126464	−	0.00219042i
$334$		0			0
$335$	2.32523e6			1.13202
$336$		0			0
$337$	−234947.			−0.112693			−0.0563463	−	0.998411i	$-0.517945\pi$
$337$	−234947.			−0.112693			−0.0563463	+	0.998411i	$0.517945\pi$
$338$		0			0
$339$	1.41363e6	+	2.44848e6i	0.668092	+	1.15717i
$340$		0			0
$341$	−985983.	+	1.70777e6i	−0.459181	+	0.795324i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−591508.	+	1.02452e6i	−0.267555	+	0.463419i
$346$		0			0
$347$	950268.	+	1.64591e6i	0.423665	+	0.733809i	0.996295	−	0.0860050i	$-0.0274101\pi$
$347$	950268.	+	1.64591e6i	0.423665	+	0.733809i	−0.572630	+	0.819814i	$0.694077\pi$
$348$		0			0
$349$	−341162.			−0.149933			−0.0749665	−	0.997186i	$-0.523885\pi$
$349$	−341162.			−0.149933			−0.0749665	+	0.997186i	$0.523885\pi$
$350$		0			0
$351$	1.02139e6			0.442511
$352$		0			0
$353$	336561.	+	582941.i	0.143756	+	0.248993i	0.928908	−	0.370310i	$-0.120749\pi$
$353$	336561.	+	582941.i	0.143756	+	0.248993i	−0.785152	+	0.619303i	$0.787415\pi$
$354$		0			0
$355$	606495.	−	1.05048e6i	0.255421	−	0.442402i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−716507.	+	1.24103e6i	−0.293417	+	0.508212i	−0.974615	−	0.223886i	$-0.928126\pi$
$359$	−716507.	+	1.24103e6i	−0.293417	+	0.508212i	0.681199	+	0.732099i	$0.261459\pi$
$360$		0			0
$361$	550275.	+	953104.i	0.222235	+	0.384922i
$362$		0			0
$363$	1.27459e6			0.507694
$364$		0			0
$365$	394559.			0.155017
$366$		0			0
$367$	−1.02391e6	−	1.77346e6i	−0.396822	−	0.687315i	0.596510	−	0.802606i	$-0.296554\pi$
$367$	−1.02391e6	−	1.77346e6i	−0.396822	−	0.687315i	−0.993332	+	0.115290i	$0.963220\pi$
$368$		0			0
$369$	2947.43	−	5105.10i	0.00112688	−	0.00195182i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.66789e6	−	2.88887e6i	0.620719	−	1.07512i	−0.368633	−	0.929575i	$-0.620174\pi$
$373$	1.66789e6	−	2.88887e6i	0.620719	−	1.07512i	0.989352	−	0.145542i	$-0.0464925\pi$
$374$		0			0
$375$	−2.33883e6	−	4.05097e6i	−0.858857	−	1.48758i
$376$		0			0
$377$	1.65662e6			0.600301
$378$		0			0
$379$	2.29135e6			0.819394			0.409697	−	0.912222i	$-0.365634\pi$
$379$	2.29135e6			0.819394			0.409697	+	0.912222i	$0.365634\pi$
$380$		0			0
$381$	826292.	+	1.43118e6i	0.291623	+	0.505105i
$382$		0			0
$383$	922109.	−	1.59714e6i	0.321207	−	0.556347i	−0.659530	−	0.751678i	$-0.729245\pi$
$383$	922109.	−	1.59714e6i	0.321207	−	0.556347i	0.980737	+	0.195331i	$0.0625781\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4411.42	+	7640.80i	−0.00149727	+	0.00259335i
$388$		0			0
$389$	−1.83301e6	−	3.17486e6i	−0.614172	−	1.06378i	−0.990529	−	0.137302i	$-0.956157\pi$
$389$	−1.83301e6	−	3.17486e6i	−0.614172	−	1.06378i	0.376357	−	0.926475i	$-0.377177\pi$
$390$		0			0
$391$	−1.35035e6			−0.446688
$392$		0			0
$393$	4.67746e6			1.52767
$394$		0			0
$395$	−576422.	−	998392.i	−0.185887	−	0.321965i
$396$		0			0
$397$	−2.29691e6	+	3.97836e6i	−0.731421	+	1.26686i	0.224855	+	0.974392i	$0.427809\pi$
$397$	−2.29691e6	+	3.97836e6i	−0.731421	+	1.26686i	−0.956276	+	0.292466i	$0.905524\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	554918.	−	961146.i	0.172333	−	0.298489i	−0.766902	−	0.641764i	$-0.778203\pi$
$401$	554918.	−	961146.i	0.172333	−	0.298489i	0.939235	+	0.343275i	$0.111536\pi$
$402$		0			0
$403$	−943761.	−	1.63464e6i	−0.289467	−	0.501372i
$404$		0			0
$405$	5.70089e6			1.72705
$406$		0			0
$407$	−3.23660e6			−0.968507
$408$		0			0
$409$	1.80345e6	+	3.12366e6i	0.533083	+	0.923327i	0.999254	+	0.0386321i	$0.0123000\pi$
$409$	1.80345e6	+	3.12366e6i	0.533083	+	0.923327i	−0.466170	+	0.884695i	$0.654367\pi$
$410$		0			0
$411$	−715575.	+	1.23941e6i	−0.208954	+	0.361919i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−3.96157e6	+	6.86164e6i	−1.12914	+	1.95572i
$416$		0			0
$417$	−1.19246e6	−	2.06540e6i	−0.335818	−	0.581654i
$418$		0			0
$419$	−5.51790e6			−1.53546			−0.767731	−	0.640773i	$-0.778614\pi$
$419$	−5.51790e6			−1.53546			−0.767731	+	0.640773i	$0.778614\pi$
$420$		0			0
$421$	−2.59092e6			−0.712440			−0.356220	−	0.934402i	$-0.615935\pi$
$421$	−2.59092e6			−0.712440			−0.356220	+	0.934402i	$0.615935\pi$
$422$		0			0
$423$	−1532.76	−	2654.81i	−0.000416507	−	0.000721411i
$424$		0			0
$425$	5.35628e6	−	9.27736e6i	1.43844	−	2.49145i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	590411.	−	1.02262e6i	0.154886	−	0.268270i
$430$		0			0
$431$	1.72682e6	+	2.99094e6i	0.447769	+	0.775558i	0.998240	−	0.0592958i	$-0.0188855\pi$
$431$	1.72682e6	+	2.99094e6i	0.447769	+	0.775558i	−0.550472	+	0.834854i	$0.685552\pi$
$432$		0			0
$433$	−397522.			−0.101892			−0.0509461	−	0.998701i	$-0.516224\pi$
$433$	−397522.			−0.101892			−0.0509461	+	0.998701i	$0.516224\pi$
$434$		0			0
$435$	9.26338e6			2.34718
$436$		0			0
$437$	−460539.	−	797676.i	−0.115362	−	0.199813i
$438$		0			0
$439$	804514.	−	1.39346e6i	0.199238	−	0.345090i	−0.749044	−	0.662521i	$-0.769487\pi$
$439$	804514.	−	1.39346e6i	0.199238	−	0.345090i	0.948282	+	0.317430i	$0.102820\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.54091e6	−	6.13304e6i	0.857246	−	1.48479i	−0.0172989	−	0.999850i	$-0.505507\pi$
$443$	3.54091e6	−	6.13304e6i	0.857246	−	1.48479i	0.874545	−	0.484944i	$-0.161160\pi$
$444$		0			0
$445$	4.68953e6	+	8.12250e6i	1.12261	+	1.94442i
$446$		0			0
$447$	−7.76531e6			−1.83819
$448$		0			0
$449$	−6.46156e6			−1.51259			−0.756295	−	0.654230i	$-0.772993\pi$
$449$	−6.46156e6			−1.51259			−0.756295	+	0.654230i	$0.772993\pi$
$450$		0			0
$451$	−1.86391e6	−	3.22839e6i	−0.431503	−	0.747385i
$452$		0			0
$453$	1.63224e6	−	2.82713e6i	0.373714	−	0.647292i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.68536e6	−	2.91914e6i	0.377488	−	0.653828i	−0.613208	−	0.789921i	$-0.710121\pi$
$457$	1.68536e6	−	2.91914e6i	0.377488	−	0.653828i	0.990696	+	0.136093i	$0.0434546\pi$
$458$		0			0
$459$	3.25960e6	+	5.64579e6i	0.722159	+	1.25082i
$460$		0			0
$461$	−6.20821e6			−1.36055			−0.680274	−	0.732958i	$-0.738139\pi$
$461$	−6.20821e6			−1.36055			−0.680274	+	0.732958i	$0.738139\pi$
$462$		0			0
$463$	−5.82940e6			−1.26378			−0.631890	−	0.775058i	$-0.717721\pi$
$463$	−5.82940e6			−1.26378			−0.631890	+	0.775058i	$0.717721\pi$
$464$		0			0
$465$	−5.27726e6	−	9.14048e6i	−1.13182	−	1.96036i
$466$		0			0
$467$	−87370.8	+	151331.i	−0.0185385	+	0.0321096i	−0.875146	−	0.483859i	$-0.839235\pi$
$467$	−87370.8	+	151331.i	−0.0185385	+	0.0321096i	0.856607	+	0.515969i	$0.172568\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	57482.1	−	99561.9i	0.0119393	−	0.0206795i
$472$		0			0
$473$	2.78971e6	+	4.83192e6i	0.573332	+	0.993040i
$474$		0			0
$475$	7.30707e6			1.48597
$476$		0			0
$477$	−3851.93			−0.000775143
$478$		0			0
$479$	185873.	+	321941.i	0.0370150	+	0.0641118i	0.883939	−	0.467601i	$-0.154882\pi$
$479$	185873.	+	321941.i	0.0370150	+	0.0641118i	−0.846924	+	0.531713i	$0.821548\pi$
$480$		0			0
$481$	1.54900e6	−	2.68295e6i	0.305273	−	0.528748i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.27726e6	−	2.21227e6i	0.246561	−	0.427056i
$486$		0			0
$487$	4.35654e6	+	7.54574e6i	0.832375	+	1.44172i	0.896150	+	0.443751i	$0.146353\pi$
$487$	4.35654e6	+	7.54574e6i	0.832375	+	1.44172i	−0.0637754	+	0.997964i	$0.520314\pi$
$488$		0			0
$489$	−6.04519e6			−1.14324
$490$		0			0
$491$	6.88530e6			1.28890			0.644450	−	0.764646i	$-0.277086\pi$
$491$	6.88530e6			1.28890			0.644450	+	0.764646i	$0.277086\pi$
$492$		0			0
$493$	5.28683e6	+	9.15705e6i	0.979665	+	1.69683i
$494$		0			0
$495$	−6057.64	+	10492.1i	−0.00111120	+	0.00192465i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	22721.3	−	39354.5i	0.00408491	−	0.00707527i	−0.863976	−	0.503533i	$-0.832033\pi$
$499$	22721.3	−	39354.5i	0.00408491	−	0.00707527i	0.868061	+	0.496458i	$0.165366\pi$
$500$		0			0
$501$	1.31010e6	+	2.26916e6i	0.233190	+	0.403898i
$502$		0			0
$503$	623417.			0.109865			0.0549324	−	0.998490i	$-0.482506\pi$
$503$	623417.			0.109865			0.0549324	+	0.998490i	$0.482506\pi$
$504$		0			0
$505$	−7.07406e6			−1.23436
$506$		0			0
$507$	−2.32616e6	−	4.02903e6i	−0.401902	−	0.696115i
$508$		0			0
$509$	3.84546e6	−	6.66053e6i	0.657891	−	1.13950i	−0.323270	−	0.946307i	$-0.604782\pi$
$509$	3.84546e6	−	6.66053e6i	0.657891	−	1.13950i	0.981161	−	0.193193i	$-0.0618845\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−2.22338e6	+	3.85101e6i	−0.373010	+	0.646072i
$514$		0			0
$515$	−4.23884e6	−	7.34188e6i	−0.704253	−	1.21980i
$516$		0			0
$517$	−1.93858e6			−0.318976
$518$		0			0
$519$	5.00730e6			0.815991
$520$		0			0
$521$	3.66134e6	+	6.34163e6i	0.590943	+	1.02354i	0.994106	+	0.108415i	$0.0345776\pi$
$521$	3.66134e6	+	6.34163e6i	0.590943	+	1.02354i	−0.403162	+	0.915128i	$0.632089\pi$
$522$		0			0
$523$	2.58988e6	−	4.48580e6i	0.414024	−	0.717110i	−0.581302	−	0.813688i	$-0.697456\pi$
$523$	2.58988e6	−	4.48580e6i	0.414024	−	0.717110i	0.995325	+	0.0965781i	$0.0307897\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.02371e6	−	1.04334e7i	0.944795	−	1.63643i
$528$		0			0
$529$	2.90979e6	+	5.03991e6i	0.452088	+	0.783039i
$530$		0			0
$531$	−21298.8			−0.00327807
$532$		0			0
$533$	3.56819e6			0.544038
$534$		0			0
$535$	−5.58274e6	−	9.66959e6i	−0.843263	−	1.46057i
$536$		0			0
$537$	−4.76496e6	+	8.25316e6i	−0.713056	+	1.23505i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−4.65902e6	+	8.06966e6i	−0.684387	+	1.18539i	0.289243	+	0.957256i	$0.406597\pi$
$541$	−4.65902e6	+	8.06966e6i	−0.684387	+	1.18539i	−0.973629	+	0.228136i	$0.926737\pi$
$542$		0			0
$543$	−2.03612e6	−	3.52666e6i	−0.296349	−	0.513292i
$544$		0			0
$545$	1.13458e7			1.63623
$546$		0			0
$547$	1.26417e7			1.80650			0.903250	−	0.429114i	$-0.141174\pi$
$547$	1.26417e7			1.80650			0.903250	+	0.429114i	$0.141174\pi$
$548$		0			0
$549$	−11741.0	−	20336.0i	−0.00166254	−	0.00287961i
$550$		0			0
$551$	−3.60616e6	+	6.24605e6i	−0.506018	+	0.876448i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8.66159e6	−	1.50023e7i	1.19362	−	2.06741i
$556$		0			0
$557$	1.92347e6	+	3.33154e6i	0.262692	+	0.454996i	0.966956	−	0.254942i	$-0.0820564\pi$
$557$	1.92347e6	+	3.33154e6i	0.262692	+	0.454996i	−0.704264	+	0.709938i	$0.748723\pi$
$558$		0			0
$559$	−5.34050e6			−0.722856
$560$		0			0
$561$	7.53680e6			1.01107
$562$		0			0
$563$	1.84597e6	+	3.19731e6i	0.245444	+	0.425122i	0.962256	−	0.272144i	$-0.0877329\pi$
$563$	1.84597e6	+	3.19731e6i	0.245444	+	0.425122i	−0.716812	+	0.697266i	$0.754400\pi$
$564$		0			0
$565$	8.77927e6	−	1.52061e7i	1.15701	−	2.00400i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.32087e6	−	5.75191e6i	0.430002	−	0.744786i	−0.566871	−	0.823807i	$-0.691846\pi$
$569$	3.32087e6	−	5.75191e6i	0.430002	−	0.744786i	0.996873	+	0.0790209i	$0.0251794\pi$
$570$		0			0
$571$	−146299.	−	253398.i	−0.0187781	−	0.0325246i	0.856484	−	0.516174i	$-0.172644\pi$
$571$	−146299.	−	253398.i	−0.0187781	−	0.0325246i	−0.875262	+	0.483650i	$0.839311\pi$
$572$		0			0
$573$	−1.09533e7			−1.39367
$574$		0			0
$575$	4.89287e6			0.617154
$576$		0			0
$577$	1.77784e6	+	3.07931e6i	0.222307	+	0.385048i	0.955508	−	0.294964i	$-0.0953078\pi$
$577$	1.77784e6	+	3.07931e6i	0.222307	+	0.385048i	−0.733201	+	0.680012i	$0.761974\pi$
$578$		0			0
$579$	4.02133e6	−	6.96516e6i	0.498510	−	0.863444i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.21795e6	+	2.10955e6i	−0.148408	+	0.257050i
$584$		0			0
$585$	−5798.24	−	10042.8i	−0.000700497	−	0.00121330i
$586$		0			0
$587$	2.64810e6			0.317204			0.158602	−	0.987343i	$-0.449301\pi$
$587$	2.64810e6			0.317204			0.158602	+	0.987343i	$0.449301\pi$
$588$		0			0
$589$	8.21758e6			0.976013
$590$		0			0
$591$	1.11027e6	+	1.92305e6i	0.130756	+	0.226476i
$592$		0			0
$593$	526782.	−	912413.i	0.0615168	−	0.106550i	−0.833627	−	0.552328i	$-0.813740\pi$
$593$	526782.	−	912413.i	0.0615168	−	0.106550i	0.895144	+	0.445778i	$0.147073\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.66901e6	+	1.32831e7i	−0.880651	+	1.52533i
$598$		0			0
$599$	−4.59506e6	−	7.95888e6i	−0.523268	−	0.906326i	−0.999633	−	0.0270790i	$-0.991379\pi$
$599$	−4.59506e6	−	7.95888e6i	−0.523268	−	0.906326i	0.476366	−	0.879247i	$-0.341954\pi$
$600$		0			0
$601$	−3.46924e6			−0.391785			−0.195893	−	0.980625i	$-0.562760\pi$
$601$	−3.46924e6			−0.391785			−0.195893	+	0.980625i	$0.562760\pi$
$602$		0			0
$603$	10699.2			0.00119828
$604$		0			0
$605$	−3.95787e6	−	6.85523e6i	−0.439616	−	0.761437i
$606$		0			0
$607$	−372075.	+	644453.i	−0.0409882	+	0.0709936i	−0.885792	−	0.464083i	$-0.846384\pi$
$607$	−372075.	+	644453.i	−0.0409882	+	0.0709936i	0.844804	+	0.535077i	$0.179717\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	927784.	−	1.60697e6i	0.100541	−	0.174142i
$612$		0			0
$613$	−5.72889e6	−	9.92274e6i	−0.615772	−	1.06655i	−0.990249	−	0.139312i	$-0.955511\pi$
$613$	−5.72889e6	−	9.92274e6i	−0.615772	−	1.06655i	0.374477	−	0.927236i	$-0.377822\pi$
$614$		0			0
$615$	1.99524e7			2.12719
$616$		0			0
$617$	−3.90052e6			−0.412486			−0.206243	−	0.978501i	$-0.566124\pi$
$617$	−3.90052e6			−0.412486			−0.206243	+	0.978501i	$0.566124\pi$
$618$		0			0
$619$	1.32515e6	+	2.29522e6i	0.139007	+	0.240767i	0.927121	−	0.374762i	$-0.122276\pi$
$619$	1.32515e6	+	2.29522e6i	0.139007	+	0.240767i	−0.788114	+	0.615529i	$0.788942\pi$
$620$		0			0
$621$	−1.48879e6	+	2.57866e6i	−0.154919	+	0.268328i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.79042e6	+	8.29724e6i	−0.490539	+	0.849638i
$626$		0			0
$627$	2.57044e6	+	4.45212e6i	0.261119	+	0.452271i
$628$		0			0
$629$	1.97735e7			1.99277
$630$		0			0
$631$	−8.97599e6			−0.897447			−0.448723	−	0.893671i	$-0.648121\pi$
$631$	−8.97599e6			−0.897447			−0.448723	+	0.893671i	$0.648121\pi$
$632$		0			0
$633$	−1.89098e6	−	3.27528e6i	−0.187576	−	0.324892i
$634$		0			0
$635$	5.13164e6	−	8.88826e6i	0.505035	−	0.874747i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2790.69	−	4833.63i	0.000270371	−	0.000468296i
$640$		0			0
$641$	1.43576e6	+	2.48681e6i	0.138018	+	0.239054i	0.926746	−	0.375688i	$-0.122593\pi$
$641$	1.43576e6	+	2.48681e6i	0.138018	+	0.239054i	−0.788728	+	0.614742i	$0.789260\pi$
$642$		0			0
$643$	−1.38761e7			−1.32354			−0.661772	−	0.749705i	$-0.730195\pi$
$643$	−1.38761e7			−1.32354			−0.661772	+	0.749705i	$0.730195\pi$
$644$		0			0
$645$	−2.98626e7			−2.82637
$646$		0			0
$647$	−3.62426e6	−	6.27741e6i	−0.340376	−	0.589549i	0.644126	−	0.764919i	$-0.277221\pi$
$647$	−3.62426e6	−	6.27741e6i	−0.340376	−	0.589549i	−0.984503	+	0.175370i	$0.943888\pi$
$648$		0			0
$649$	−6.73450e6	+	1.16645e7i	−0.627616	+	1.08706i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.18595e6	−	1.07144e7i	0.567706	−	0.983296i	−0.429086	−	0.903264i	$-0.641164\pi$
$653$	6.18595e6	−	1.07144e7i	0.567706	−	0.983296i	0.996792	−	0.0800326i	$-0.0255024\pi$
$654$		0			0
$655$	−1.45245e7	−	2.51572e7i	−1.32282	−	2.29118i
$656$		0			0
$657$	1815.50			0.000164090
$658$		0			0
$659$	−572294.			−0.0513341			−0.0256670	−	0.999671i	$-0.508171\pi$
$659$	−572294.			−0.0513341			−0.0256670	+	0.999671i	$0.508171\pi$
$660$		0			0
$661$	3.36840e6	+	5.83423e6i	0.299861	+	0.519374i	0.976104	−	0.217305i	$-0.0697265\pi$
$661$	3.36840e6	+	5.83423e6i	0.299861	+	0.519374i	−0.676243	+	0.736678i	$0.736393\pi$
$662$		0			0
$663$	−3.60703e6	+	6.24755e6i	−0.318688	+	0.551984i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.41471e6	+	4.18240e6i	−0.210160	+	0.364008i
$668$		0			0
$669$	6.20680e6	+	1.07505e7i	0.536170	+	0.928673i
$670$		0			0
$671$	−1.48496e7			−1.27324
$672$		0			0
$673$	−1.22608e7			−1.04348			−0.521738	−	0.853106i	$-0.674716\pi$
$673$	−1.22608e7			−1.04348			−0.521738	+	0.853106i	$0.674716\pi$
$674$		0			0
$675$	−1.18109e7	−	2.04570e7i	−0.997750	−	1.72815i
$676$		0			0
$677$	−231043.	+	400178.i	−0.0193741	+	0.0335568i	−0.875550	−	0.483128i	$-0.839501\pi$
$677$	−231043.	+	400178.i	−0.0193741	+	0.0335568i	0.856176	+	0.516685i	$0.172834\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	203410.	−	352316.i	0.0168075	−	0.0291115i
$682$		0			0
$683$	5.76890e6	+	9.99203e6i	0.473196	+	0.819600i	0.999529	−	0.0306786i	$-0.00976682\pi$
$683$	5.76890e6	+	9.99203e6i	0.473196	+	0.819600i	−0.526333	+	0.850278i	$0.676433\pi$
$684$		0			0
$685$	8.88807e6			0.723738
$686$		0			0
$687$	−1.18612e7			−0.958816
$688$		0			0
$689$	−1.16579e6	−	2.01921e6i	−0.0935564	−	0.162044i
$690$		0			0
$691$	1.54961e6	−	2.68400e6i	0.123460	−	0.213839i	−0.797670	−	0.603094i	$-0.793934\pi$
$691$	1.54961e6	−	2.68400e6i	0.123460	−	0.213839i	0.921130	+	0.389255i	$0.127268\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.40571e6	+	1.28271e7i	−0.581574	+	1.00732i
$696$		0			0
$697$	1.13873e7	+	1.97233e7i	0.887846	+	1.53780i
$698$		0			0
$699$	−1.02455e7			−0.793122
$700$		0			0
$701$	5.83889e6			0.448782			0.224391	−	0.974499i	$-0.427961\pi$
$701$	5.83889e6			0.448782			0.224391	+	0.974499i	$0.427961\pi$
$702$		0			0
$703$	6.74378e6	+	1.16806e7i	0.514653	+	0.891406i
$704$		0			0
$705$	5.18792e6	−	8.98574e6i	0.393116	−	0.680897i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37717.8	+	65329.2i	−0.00281794	+	0.00488081i	−0.867431	−	0.497558i	$-0.834230\pi$
$709$	−37717.8	+	65329.2i	−0.00281794	+	0.00488081i	0.864613	+	0.502438i	$0.167564\pi$
$710$		0			0
$711$	−2652.32	−	4593.95i	−0.000196767	−	0.000340810i
$712$		0			0
$713$	5.50255e6			0.405359
$714$		0			0
$715$	−7.33343e6			−0.536466
$716$		0			0
$717$	7.51756e6	+	1.30208e7i	0.546109	+	0.945888i
$718$		0			0
$719$	−6.54344e6	+	1.13336e7i	−0.472045	+	0.817607i	−0.999488	−	0.0319839i	$-0.989817\pi$
$719$	−6.54344e6	+	1.13336e7i	−0.472045	+	0.817607i	0.527443	+	0.849590i	$0.323151\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−273048.	+	472933.i	−0.0194264	+	0.0336476i
$724$		0			0
$725$	−1.91563e7	−	3.31797e7i	−1.35353	−	2.34438i
$726$		0			0
$727$	3.58160e6			0.251328			0.125664	−	0.992073i	$-0.459894\pi$
$727$	3.58160e6			0.251328			0.125664	+	0.992073i	$0.459894\pi$
$728$		0			0
$729$	1.43751e7			1.00182
$730$		0			0
$731$	−1.70433e7	−	2.95199e7i	−1.17967	−	2.04325i
$732$		0			0
$733$	−4.61913e6	+	8.00057e6i	−0.317541	+	0.549998i	−0.979974	−	0.199123i	$-0.936191\pi$
$733$	−4.61913e6	+	8.00057e6i	−0.317541	+	0.549998i	0.662433	+	0.749121i	$0.269524\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.38300e6	−	5.85952e6i	0.229421	−	0.397369i
$738$		0			0
$739$	9.52221e6	+	1.64929e7i	0.641397	+	1.11093i	0.985121	+	0.171861i	$0.0549780\pi$
$739$	9.52221e6	+	1.64929e7i	0.641397	+	1.11093i	−0.343725	+	0.939070i	$0.611689\pi$
$740$		0			0
$741$	−4.92073e6			−0.329218
$742$		0			0
$743$	9.30669e6			0.618477			0.309238	−	0.950985i	$-0.399926\pi$
$743$	9.30669e6			0.618477			0.309238	+	0.950985i	$0.399926\pi$
$744$		0			0
$745$	2.41130e7	+	4.17650e7i	1.59170	+	2.75690i
$746$		0			0
$747$	−18228.6	+	31572.8i	−0.00119523	+	0.00207020i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.28122e7	+	2.21913e7i	−0.828939	+	1.43576i	0.0699319	+	0.997552i	$0.477722\pi$
$751$	−1.28122e7	+	2.21913e7i	−0.828939	+	1.43576i	−0.898871	+	0.438213i	$0.855612\pi$
$752$		0			0
$753$	−1.02486e7	−	1.77510e7i	−0.658682	−	1.14087i
$754$		0			0
$755$	−2.02739e7			−1.29441
$756$		0			0
$757$	5.60956e6			0.355786			0.177893	−	0.984050i	$-0.443072\pi$
$757$	5.60956e6			0.355786			0.177893	+	0.984050i	$0.443072\pi$
$758$		0			0
$759$	1.72118e6	+	2.98117e6i	0.108448	+	0.187838i
$760$		0			0
$761$	−1.41992e7	+	2.45938e7i	−0.888799	+	1.53944i	−0.0475020	+	0.998871i	$0.515126\pi$
$761$	−1.41992e7	+	2.45938e7i	−0.888799	+	1.53944i	−0.841297	+	0.540574i	$0.818207\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	37008.2	−	64100.1i	0.00228636	−	0.00396009i
$766$		0			0
$767$	−6.44612e6	−	1.11650e7i	−0.395649	−	0.685284i
$768$		0			0
$769$	2.41205e6			0.147086			0.0735429	−	0.997292i	$-0.476569\pi$
$769$	2.41205e6			0.147086			0.0735429	+	0.997292i	$0.476569\pi$
$770$		0			0
$771$	−8.25123e6			−0.499899
$772$		0			0
$773$	2.96946e6	+	5.14325e6i	0.178743	+	0.309591i	0.941450	−	0.337152i	$-0.109464\pi$
$773$	2.96946e6	+	5.14325e6i	0.178743	+	0.309591i	−0.762707	+	0.646744i	$0.776130\pi$
$774$		0			0
$775$	−2.18264e7	+	3.78044e7i	−1.30535	+	2.26093i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.76729e6	+	1.34533e7i	−0.458591	+	0.794304i
$780$		0			0
$781$	−1.76479e6	−	3.05671e6i	−0.103530	−	0.179319i
$782$		0			0
$783$	2.33154e7			1.35906
$784$		0			0
$785$	−713979.			−0.0413534
$786$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(7.78709 + 13.4876i) q^{3} +(48.3613 - 83.7642i) q^{5} +(0.222527 - 0.385428i) q^{9} +O(q^{10})\)	\(q+(7.78709 + 13.4876i) q^{3} +(48.3613 - 83.7642i) q^{5} +(0.222527 - 0.385428i) q^{9} +(-140.723 - 243.739i) q^{11} +269.393 q^{13} +1506.37 q^{15} +(859.723 + 1489.08i) q^{17} +(-586.419 + 1015.71i) q^{19} +(-392.670 + 680.125i) q^{23} +(-3115.12 - 5395.55i) q^{25} +3791.46 q^{27} +6149.46 q^{29} +(-3503.29 - 6067.87i) q^{31} +(2191.64 - 3796.03i) q^{33} +(5749.96 - 9959.23i) q^{37} +(2097.79 + 3633.47i) q^{39} +13245.3 q^{41} -19824.2 q^{43} +(-21.5234 - 37.2796i) q^{45} +(3443.98 - 5965.15i) q^{47} +(-13389.5 + 23191.2i) q^{51} +(-4327.48 - 7495.42i) q^{53} -27222.1 q^{55} -18266.0 q^{57} +(-23928.3 - 41445.0i) q^{59} +(26381.0 - 45693.3i) q^{61} +(13028.2 - 22565.5i) q^{65} +(12020.1 + 20819.4i) q^{67} -12231.0 q^{69} +12540.9 q^{71} +(2039.64 + 3532.76i) q^{73} +(48515.5 - 84031.3i) q^{75} +(5959.54 - 10322.2i) q^{79} +(29470.3 + 51044.1i) q^{81} -81916.2 q^{83} +166309. q^{85} +(47886.4 + 82941.6i) q^{87} +(-48484.4 + 83977.4i) q^{89} +(54560.8 - 94502.1i) q^{93} +(56719.9 + 98241.8i) q^{95} +26410.7 q^{97} -125.258 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{3} + 82 q^{5} - 222 q^{9}+O(q^{10}) \) 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9	\( 4 q - 6 q^{3} + 82 q^{5} - 222 q^{9} - 340 q^{11} - 1820 q^{13} + 3648 q^{15} + 3216 q^{17} - 674 q^{19} + 1104 q^{23} - 3322 q^{25} + 6696 q^{27} + 16128 q^{29} - 6212 q^{31} + 3120 q^{33} + 8512 q^{37} + 29640 q^{39} + 2608 q^{41} - 20008 q^{43} - 3318 q^{45} - 12748 q^{47} + 5508 q^{51} + 11220 q^{53} - 52720 q^{55} - 58056 q^{57} - 12018 q^{59} + 102738 q^{61} + 43420 q^{65} - 24136 q^{67} - 105984 q^{69} + 179440 q^{71} - 55588 q^{73} + 159774 q^{75} - 48824 q^{79} + 122562 q^{81} - 71564 q^{83} + 288552 q^{85} + 54468 q^{87} - 18300 q^{89} + 126264 q^{93} + 120784 q^{95} + 139968 q^{97} + 25800 q^{99}+O(q^{100}) \) 4 * q - 6 * q^3 + 82 * q^5 - 222 * q^9 - 340 * q^11 - 1820 * q^13 + 3648 * q^15 + 3216 * q^17 - 674 * q^19 + 1104 * q^23 - 3322 * q^25 + 6696 * q^27 + 16128 * q^29 - 6212 * q^31 + 3120 * q^33 + 8512 * q^37 + 29640 * q^39 + 2608 * q^41 - 20008 * q^43 - 3318 * q^45 - 12748 * q^47 + 5508 * q^51 + 11220 * q^53 - 52720 * q^55 - 58056 * q^57 - 12018 * q^59 + 102738 * q^61 + 43420 * q^65 - 24136 * q^67 - 105984 * q^69 + 179440 * q^71 - 55588 * q^73 + 159774 * q^75 - 48824 * q^79 + 122562 * q^81 - 71564 * q^83 + 288552 * q^85 + 54468 * q^87 - 18300 * q^89 + 126264 * q^93 + 120784 * q^95 + 139968 * q^97 + 25800 * q^99

Introduction

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