LMFDB - Embedded newform 392.6.i.f.361.1

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	392.361
Dual form			392.6.i.f.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.0000 - 25.9808i) q^{3} +(16.0000 + 27.7128i) q^{5} +(-328.500 - 568.979i) q^{9} +O(q^{10})$	\(q+(15.0000 - 25.9808i) q^{3} +(16.0000 + 27.7128i) q^{5} +(-328.500 - 568.979i) q^{9} +(312.000 - 540.400i) q^{11} +708.000 q^{13} +960.000 q^{15} +(467.000 - 808.868i) q^{17} +(929.000 + 1609.08i) q^{19} +(560.000 + 969.948i) q^{23} +(1050.50 - 1819.52i) q^{25} -12420.0 q^{27} -1174.00 q^{29} +(1454.00 - 2518.40i) q^{31} +(-9360.00 - 16212.0i) q^{33} +(6231.00 + 10792.4i) q^{37} +(10620.0 - 18394.4i) q^{39} -2662.00 q^{41} -7144.00 q^{43} +(10512.0 - 18207.3i) q^{45} +(-3734.00 - 6467.48i) q^{47} +(-14010.0 - 24266.0i) q^{51} +(13637.0 - 23620.0i) q^{53} +19968.0 q^{55} +55740.0 q^{57} +(1245.00 - 2156.40i) q^{59} +(-5548.00 - 9609.42i) q^{61} +(11328.0 + 19620.7i) q^{65} +(-19878.0 + 34429.7i) q^{67} +33600.0 q^{69} -69888.0 q^{71} +(8225.00 - 14246.1i) q^{73} +(-31515.0 - 54585.6i) q^{75} +(-39188.0 - 67875.6i) q^{79} +(-106474. + 184419. i) q^{81} -109818. q^{83} +29888.0 q^{85} +(-17610.0 + 30501.4i) q^{87} +(-28483.0 - 49334.0i) q^{89} +(-43620.0 - 75552.1i) q^{93} +(-29728.0 + 51490.4i) q^{95} +115946. q^{97} -409968. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 30 q^{3} + 32 q^{5} - 657 q^{9}+O(q^{10}) $ 2 * q + 30 * q^3 + 32 * q^5 - 657 * q^9	$ 2 q + 30 q^{3} + 32 q^{5} - 657 q^{9} + 624 q^{11} + 1416 q^{13} + 1920 q^{15} + 934 q^{17} + 1858 q^{19} + 1120 q^{23} + 2101 q^{25} - 24840 q^{27} - 2348 q^{29} + 2908 q^{31} - 18720 q^{33} + 12462 q^{37} + 21240 q^{39} - 5324 q^{41} - 14288 q^{43} + 21024 q^{45} - 7468 q^{47} - 28020 q^{51} + 27274 q^{53} + 39936 q^{55} + 111480 q^{57} + 2490 q^{59} - 11096 q^{61} + 22656 q^{65} - 39756 q^{67} + 67200 q^{69} - 139776 q^{71} + 16450 q^{73} - 63030 q^{75} - 78376 q^{79} - 212949 q^{81} - 219636 q^{83} + 59776 q^{85} - 35220 q^{87} - 56966 q^{89} - 87240 q^{93} - 59456 q^{95} + 231892 q^{97} - 819936 q^{99}+O(q^{100}) $ 2 * q + 30 * q^3 + 32 * q^5 - 657 * q^9 + 624 * q^11 + 1416 * q^13 + 1920 * q^15 + 934 * q^17 + 1858 * q^19 + 1120 * q^23 + 2101 * q^25 - 24840 * q^27 - 2348 * q^29 + 2908 * q^31 - 18720 * q^33 + 12462 * q^37 + 21240 * q^39 - 5324 * q^41 - 14288 * q^43 + 21024 * q^45 - 7468 * q^47 - 28020 * q^51 + 27274 * q^53 + 39936 * q^55 + 111480 * q^57 + 2490 * q^59 - 11096 * q^61 + 22656 * q^65 - 39756 * q^67 + 67200 * q^69 - 139776 * q^71 + 16450 * q^73 - 63030 * q^75 - 78376 * q^79 - 212949 * q^81 - 219636 * q^83 + 59776 * q^85 - 35220 * q^87 - 56966 * q^89 - 87240 * q^93 - 59456 * q^95 + 231892 * q^97 - 819936 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	15.0000	−	25.9808i	0.962250	−	1.66667i	0.245423	−	0.969416i	$-0.421073\pi$
$3$	15.0000	−	25.9808i	0.962250	−	1.66667i	0.716827	−	0.697251i	$-0.245594\pi$
$4$		0			0
$5$	16.0000	+	27.7128i	0.286217	+	0.495742i	0.972904	−	0.231212i	$-0.0742689\pi$
$5$	16.0000	+	27.7128i	0.286217	+	0.495742i	−0.686687	+	0.726953i	$0.740936\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−328.500	−	568.979i	−1.35185	−	2.34148i
$10$		0			0
$11$	312.000	−	540.400i	0.777451	−	1.34658i	−0.155956	−	0.987764i	$-0.549846\pi$
$11$	312.000	−	540.400i	0.777451	−	1.34658i	0.933407	−	0.358820i	$-0.116821\pi$
$12$		0			0
$13$	708.000			1.16192			0.580958	−	0.813933i	$-0.302678\pi$
$13$	708.000			1.16192			0.580958	+	0.813933i	$0.302678\pi$
$14$		0			0
$15$	960.000			1.10165
$16$		0			0
$17$	467.000	−	808.868i	0.391917	−	0.678821i	−0.600785	−	0.799411i	$-0.705145\pi$
$17$	467.000	−	808.868i	0.391917	−	0.678821i	0.992702	+	0.120590i	$0.0384786\pi$
$18$		0			0
$19$	929.000	+	1609.08i	0.590380	+	1.02257i	0.994181	+	0.107721i	$0.0343555\pi$
$19$	929.000	+	1609.08i	0.590380	+	1.02257i	−0.403801	+	0.914847i	$0.632311\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	560.000	+	969.948i	0.220734	+	0.382322i	0.955031	−	0.296506i	$-0.0958215\pi$
$23$	560.000	+	969.948i	0.220734	+	0.382322i	−0.734297	+	0.678828i	$0.762488\pi$
$24$		0			0
$25$	1050.50	−	1819.52i	0.336160	−	0.582246i
$26$		0			0
$27$	−12420.0			−3.27878
$28$		0			0
$29$	−1174.00			−0.259223			−0.129611	−	0.991565i	$-0.541373\pi$
$29$	−1174.00			−0.259223			−0.129611	+	0.991565i	$0.541373\pi$
$30$		0			0
$31$	1454.00	−	2518.40i	0.271744	−	0.470675i	−0.697564	−	0.716522i	$-0.745733\pi$
$31$	1454.00	−	2518.40i	0.271744	−	0.470675i	0.969309	+	0.245847i	$0.0790662\pi$
$32$		0			0
$33$	−9360.00	−	16212.0i	−1.49620	−	2.59150i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6231.00	+	10792.4i	0.748262	+	1.29603i	0.948655	+	0.316312i	$0.102445\pi$
$37$	6231.00	+	10792.4i	0.748262	+	1.29603i	−0.200394	+	0.979715i	$0.564222\pi$
$38$		0			0
$39$	10620.0	−	18394.4i	1.11805	−	1.93653i
$40$		0			0
$41$	−2662.00			−0.247314			−0.123657	−	0.992325i	$-0.539462\pi$
$41$	−2662.00			−0.247314			−0.123657	+	0.992325i	$0.539462\pi$
$42$		0			0
$43$	−7144.00			−0.589210			−0.294605	−	0.955619i	$-0.595188\pi$
$43$	−7144.00			−0.589210			−0.294605	+	0.955619i	$0.595188\pi$
$44$		0			0
$45$	10512.0	−	18207.3i	0.773845	−	1.34034i
$46$		0			0
$47$	−3734.00	−	6467.48i	−0.246564	−	0.427061i	0.716006	−	0.698094i	$-0.245968\pi$
$47$	−3734.00	−	6467.48i	−0.246564	−	0.427061i	−0.962570	+	0.271033i	$0.912635\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−14010.0	−	24266.0i	−0.754245	−	1.30639i
$52$		0			0
$53$	13637.0	−	23620.0i	0.666852	−	1.15502i	−0.311928	−	0.950106i	$-0.600975\pi$
$53$	13637.0	−	23620.0i	0.666852	−	1.15502i	0.978780	−	0.204915i	$-0.0656918\pi$
$54$		0			0
$55$	19968.0			0.890078
$56$		0			0
$57$	55740.0			2.27237
$58$		0			0
$59$	1245.00	−	2156.40i	0.0465628	−	0.0806492i	−0.841805	−	0.539782i	$-0.818507\pi$
$59$	1245.00	−	2156.40i	0.0465628	−	0.0806492i	0.888367	+	0.459133i	$0.151840\pi$
$60$		0			0
$61$	−5548.00	−	9609.42i	−0.190903	−	0.330653i	0.754647	−	0.656131i	$-0.227808\pi$
$61$	−5548.00	−	9609.42i	−0.190903	−	0.330653i	−0.945550	+	0.325478i	$0.894475\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	11328.0	+	19620.7i	0.332560	+	0.576011i
$66$		0			0
$67$	−19878.0	+	34429.7i	−0.540986	+	0.937014i	0.457862	+	0.889023i	$0.348615\pi$
$67$	−19878.0	+	34429.7i	−0.540986	+	0.937014i	−0.998848	+	0.0479913i	$0.984718\pi$
$68$		0			0
$69$	33600.0			0.849604
$70$		0			0
$71$	−69888.0			−1.64534			−0.822672	−	0.568516i	$-0.807518\pi$
$71$	−69888.0			−1.64534			−0.822672	+	0.568516i	$0.807518\pi$
$72$		0			0
$73$	8225.00	−	14246.1i	0.180646	−	0.312888i	−0.761455	−	0.648218i	$-0.775514\pi$
$73$	8225.00	−	14246.1i	0.180646	−	0.312888i	0.942101	+	0.335330i	$0.108848\pi$
$74$		0			0
$75$	−31515.0	−	54585.6i	−0.646940	−	1.12053i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−39188.0	−	67875.6i	−0.706456	−	1.22362i	−0.966163	−	0.257931i	$-0.916959\pi$
$79$	−39188.0	−	67875.6i	−0.706456	−	1.22362i	0.259707	−	0.965687i	$-0.416374\pi$
$80$		0			0
$81$	−106474.	+	184419.i	−1.80316	+	3.12316i
$82$		0			0
$83$	−109818.			−1.74976			−0.874880	−	0.484340i	$-0.839060\pi$
$83$	−109818.			−1.74976			−0.874880	+	0.484340i	$0.839060\pi$
$84$		0			0
$85$	29888.0			0.448693
$86$		0			0
$87$	−17610.0	+	30501.4i	−0.249437	+	0.432038i
$88$		0			0
$89$	−28483.0	−	49334.0i	−0.381163	−	0.660194i	0.610066	−	0.792351i	$-0.291143\pi$
$89$	−28483.0	−	49334.0i	−0.381163	−	0.660194i	−0.991229	+	0.132157i	$0.957810\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−43620.0	−	75552.1i	−0.522972	−	0.905814i
$94$		0			0
$95$	−29728.0	+	51490.4i	−0.337953	+	0.585352i
$96$		0			0
$97$	115946.			1.25120			0.625600	−	0.780144i	$-0.284854\pi$
$97$	115946.			1.25120			0.625600	+	0.780144i	$0.284854\pi$
$98$		0			0
$99$	−409968.			−4.20399
$100$		0			0
$101$	4176.00	−	7233.04i	0.0407340	−	0.0705534i	−0.844940	−	0.534862i	$-0.820364\pi$
$101$	4176.00	−	7233.04i	0.0407340	−	0.0705534i	0.885674	+	0.464308i	$0.153697\pi$
$102$		0			0
$103$	89742.0	+	155438.i	0.833494	+	1.44365i	0.895250	+	0.445563i	$0.146997\pi$
$103$	89742.0	+	155438.i	0.833494	+	1.44365i	−0.0617560	+	0.998091i	$0.519670\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	26946.0	+	46671.8i	0.227528	+	0.394090i	0.957075	−	0.289841i	$-0.0936023\pi$
$107$	26946.0	+	46671.8i	0.227528	+	0.394090i	−0.729547	+	0.683931i	$0.760269\pi$
$108$		0			0
$109$	−52985.0	+	91772.7i	−0.427156	+	0.739856i	−0.996619	−	0.0821608i	$-0.973818\pi$
$109$	−52985.0	+	91772.7i	−0.427156	+	0.739856i	0.569463	+	0.822017i	$0.307151\pi$
$110$		0			0
$111$	373860.			2.88006
$112$		0			0
$113$	2502.00			0.0184328			0.00921640	−	0.999958i	$-0.497066\pi$
$113$	2502.00			0.0184328			0.00921640	+	0.999958i	$0.497066\pi$
$114$		0			0
$115$	−17920.0	+	31038.4i	−0.126355	+	0.218854i
$116$		0			0
$117$	−232578.	−	402837.i	−1.57074	−	2.72060i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−114162.	−	197735.i	−0.708859	−	1.22778i
$122$		0			0
$123$	−39930.0	+	69160.8i	−0.237978	+	0.412190i
$124$		0			0
$125$	167232.			0.957292
$126$		0			0
$127$	287792.			1.58332			0.791661	−	0.610960i	$-0.209216\pi$
$127$	287792.			1.58332			0.791661	+	0.610960i	$0.209216\pi$
$128$		0			0
$129$	−107160.	+	185607.i	−0.566968	+	0.982017i
$130$		0			0
$131$	−23831.0	−	41276.5i	−0.121329	−	0.210148i	0.798963	−	0.601380i	$-0.205382\pi$
$131$	−23831.0	−	41276.5i	−0.121329	−	0.210148i	−0.920292	+	0.391232i	$0.872049\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−198720.	−	344193.i	−0.938441	−	1.62543i
$136$		0			0
$137$	−111577.	+	193257.i	−0.507894	+	0.879699i	0.492064	+	0.870559i	$0.336243\pi$
$137$	−111577.	+	193257.i	−0.507894	+	0.879699i	−0.999958	+	0.00913956i	$0.997091\pi$
$138$		0			0
$139$	−250542.			−1.09988			−0.549938	−	0.835206i	$-0.685349\pi$
$139$	−250542.			−1.09988			−0.549938	+	0.835206i	$0.685349\pi$
$140$		0			0
$141$	−224040.			−0.949025
$142$		0			0
$143$	220896.	−	382603.i	0.903333	−	1.56462i
$144$		0			0
$145$	−18784.0	−	32534.8i	−0.0741939	−	0.128508i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	243697.	+	422096.i	0.899258	+	1.55756i	0.828444	+	0.560072i	$0.189226\pi$
$149$	243697.	+	422096.i	0.899258	+	1.55756i	0.0708143	+	0.997490i	$0.477440\pi$
$150$		0			0
$151$	27340.0	−	47354.3i	0.0975790	−	0.169012i	−0.813103	−	0.582120i	$-0.802223\pi$
$151$	27340.0	−	47354.3i	0.0975790	−	0.169012i	0.910682	+	0.413108i	$0.135557\pi$
$152$		0			0
$153$	−613638.			−2.11926
$154$		0			0
$155$	93056.0			0.311111
$156$		0			0
$157$	105534.	−	182790.i	0.341699	−	0.591839i	−0.643050	−	0.765825i	$-0.722331\pi$
$157$	105534.	−	182790.i	0.341699	−	0.591839i	0.984748	+	0.173985i	$0.0556644\pi$
$158$		0			0
$159$	−409110.	−	708599.i	−1.28336	−	2.22284i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10096.0	+	17486.8i	0.0297632	+	0.0515515i	0.880523	−	0.474003i	$-0.157191\pi$
$163$	10096.0	+	17486.8i	0.0297632	+	0.0515515i	−0.850760	+	0.525554i	$0.823858\pi$
$164$		0			0
$165$	299520.	−	518784.i	0.856478	−	1.48346i
$166$		0			0
$167$	4524.00			0.0125525			0.00627627	−	0.999980i	$-0.498002\pi$
$167$	4524.00			0.0125525			0.00627627	+	0.999980i	$0.498002\pi$
$168$		0			0
$169$	129971.			0.350050
$170$		0			0
$171$	610353.	−	1.05716e6i	1.59621	−	2.76472i
$172$		0			0
$173$	−52166.0	−	90354.2i	−0.132517	−	0.229527i	0.792129	−	0.610354i	$-0.208973\pi$
$173$	−52166.0	−	90354.2i	−0.132517	−	0.229527i	−0.924646	+	0.380827i	$0.875639\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−37350.0	−	64692.1i	−0.0896102	−	0.155209i
$178$		0			0
$179$	100862.	−	174698.i	0.235285	−	0.407526i	−0.724070	−	0.689726i	$-0.757731\pi$
$179$	100862.	−	174698.i	0.235285	−	0.407526i	0.959356	+	0.282200i	$0.0910642\pi$
$180$		0			0
$181$	−655700.			−1.48768			−0.743839	−	0.668359i	$-0.766997\pi$
$181$	−655700.			−1.48768			−0.743839	+	0.668359i	$0.766997\pi$
$182$		0			0
$183$	−332880.			−0.734784
$184$		0			0
$185$	−199392.	+	345357.i	−0.428330	+	0.741889i
$186$		0			0
$187$	−291408.	−	504733.i	−0.609393	−	1.05550i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	75748.0	+	131199.i	0.150241	+	0.260225i	0.931316	−	0.364212i	$-0.118662\pi$
$191$	75748.0	+	131199.i	0.150241	+	0.260225i	−0.781075	+	0.624437i	$0.785328\pi$
$192$		0			0
$193$	−114663.	+	198602.i	−0.221580	+	0.383787i	−0.955288	−	0.295677i	$-0.904455\pi$
$193$	−114663.	+	198602.i	−0.221580	+	0.383787i	0.733708	+	0.679465i	$0.237788\pi$
$194$		0			0
$195$	679680.			1.28002
$196$		0			0
$197$	421086.			0.773046			0.386523	−	0.922280i	$-0.373676\pi$
$197$	421086.			0.773046			0.386523	+	0.922280i	$0.373676\pi$
$198$		0			0
$199$	98650.0	−	170867.i	0.176589	−	0.305862i	−0.764121	−	0.645073i	$-0.776827\pi$
$199$	98650.0	−	170867.i	0.176589	−	0.305862i	0.940710	+	0.339211i	$0.110160\pi$
$200$		0			0
$201$	596340.	+	1.03289e6i	1.04113	+	1.80329i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−42592.0	−	73771.5i	−0.0707853	−	0.122604i
$206$		0			0
$207$	367920.	−	637256.i	0.596798	−	1.03368i
$208$		0			0
$209$	1.15939e6			1.83597
$210$		0			0
$211$	679052.			1.05002			0.525009	−	0.851097i	$-0.324062\pi$
$211$	679052.			1.05002			0.525009	+	0.851097i	$0.324062\pi$
$212$		0			0
$213$	−1.04832e6	+	1.81574e6i	−1.58323	+	2.74224i
$214$		0			0
$215$	−114304.	−	197980.i	−0.168642	−	0.292096i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−246750.	−	427384.i	−0.347654	−	0.602154i
$220$		0			0
$221$	330636.	−	572678.i	0.455375	−	0.788733i
$222$		0			0
$223$	184440.			0.248366			0.124183	−	0.992259i	$-0.460369\pi$
$223$	184440.			0.248366			0.124183	+	0.992259i	$0.460369\pi$
$224$		0			0
$225$	−1.38036e6			−1.81775
$226$		0			0
$227$	−434039.	+	751778.i	−0.559067	+	0.968333i	0.438507	+	0.898728i	$0.355507\pi$
$227$	−434039.	+	751778.i	−0.559067	+	0.968333i	−0.997575	+	0.0696054i	$0.977826\pi$
$228$		0			0
$229$	−296930.	−	514298.i	−0.374167	−	0.648076i	0.616035	−	0.787719i	$-0.288738\pi$
$229$	−296930.	−	514298.i	−0.374167	−	0.648076i	−0.990202	+	0.139643i	$0.955405\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24109.0	−	41758.0i	−0.0290931	−	0.0503907i	0.851112	−	0.524984i	$-0.175929\pi$
$233$	−24109.0	−	41758.0i	−0.0290931	−	0.0503907i	−0.880205	+	0.474593i	$0.842595\pi$
$234$		0			0
$235$	119488.	−	206959.i	0.141141	−	0.244464i
$236$		0			0
$237$	−2.35128e6			−2.71915
$238$		0			0
$239$	241688.			0.273691			0.136845	−	0.990592i	$-0.456304\pi$
$239$	241688.			0.273691			0.136845	+	0.990592i	$0.456304\pi$
$240$		0			0
$241$	282635.	−	489538.i	0.313461	−	0.542930i	−0.665648	−	0.746266i	$-0.731845\pi$
$241$	282635.	−	489538.i	0.313461	−	0.542930i	0.979109	+	0.203335i	$0.0651782\pi$
$242$		0			0
$243$	1.68520e6	+	2.91886e6i	1.83078	+	3.17101i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	657732.	+	1.13923e6i	0.685972	+	1.18814i
$248$		0			0
$249$	−1.64727e6	+	2.85316e6i	−1.68371	+	2.91627i
$250$		0			0
$251$	1.43775e6			1.44045			0.720224	−	0.693741i	$-0.244039\pi$
$251$	1.43775e6			1.44045			0.720224	+	0.693741i	$0.244039\pi$
$252$		0			0
$253$	698880.			0.686438
$254$		0			0
$255$	448320.	−	776513.i	0.431755	−	0.747822i
$256$		0			0
$257$	247401.	+	428511.i	0.233652	+	0.404696i	0.958880	−	0.283812i	$-0.0915992\pi$
$257$	247401.	+	428511.i	0.233652	+	0.404696i	−0.725228	+	0.688508i	$0.758266\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	385659.	+	667981.i	0.350431	+	0.606964i
$262$		0			0
$263$	−778272.	+	1.34801e6i	−0.693812	+	1.20172i	0.276767	+	0.960937i	$0.410737\pi$
$263$	−778272.	+	1.34801e6i	−0.693812	+	1.20172i	−0.970579	+	0.240781i	$0.922596\pi$
$264$		0			0
$265$	872768.			0.763456
$266$		0			0
$267$	−1.70898e6			−1.46710
$268$		0			0
$269$	681018.	−	1.17956e6i	0.573823	−	0.993890i	−0.422346	−	0.906435i	$-0.638793\pi$
$269$	681018.	−	1.17956e6i	0.573823	−	0.993890i	0.996168	−	0.0874555i	$-0.0278736\pi$
$270$		0			0
$271$	279160.	+	483519.i	0.230903	+	0.399936i	0.958074	−	0.286520i	$-0.0924986\pi$
$271$	279160.	+	483519.i	0.230903	+	0.399936i	−0.727171	+	0.686456i	$0.759165\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−655512.	−	1.13538e6i	−0.522696	−	0.905335i
$276$		0			0
$277$	−293171.	+	507787.i	−0.229573	+	0.397633i	−0.957682	−	0.287829i	$-0.907066\pi$
$277$	−293171.	+	507787.i	−0.229573	+	0.397633i	0.728108	+	0.685462i	$0.240400\pi$
$278$		0			0
$279$	−1.91056e6			−1.46943
$280$		0			0
$281$	606234.			0.458010			0.229005	−	0.973425i	$-0.426453\pi$
$281$	606234.			0.458010			0.229005	+	0.973425i	$0.426453\pi$
$282$		0			0
$283$	−432587.	+	749263.i	−0.321076	+	0.556119i	−0.980710	−	0.195467i	$-0.937378\pi$
$283$	−432587.	+	749263.i	−0.321076	+	0.556119i	0.659635	+	0.751586i	$0.270711\pi$
$284$		0			0
$285$	891840.	+	1.54471e6i	0.650391	+	1.12651i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	273750.	+	474150.i	0.192801	+	0.333942i
$290$		0			0
$291$	1.73919e6	−	3.01237e6i	1.20397	−	2.08533i
$292$		0			0
$293$	353352.			0.240458			0.120229	−	0.992746i	$-0.461637\pi$
$293$	353352.			0.240458			0.120229	+	0.992746i	$0.461637\pi$
$294$		0			0
$295$	79680.0			0.0533082
$296$		0			0
$297$	−3.87504e6	+	6.71177e6i	−2.54909	+	4.41515i
$298$		0			0
$299$	396480.	+	686724.i	0.256474	+	0.444226i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−125280.	−	216991.i	−0.0783926	−	0.135780i
$304$		0			0
$305$	177536.	−	307501.i	0.109279	−	0.189277i
$306$		0			0
$307$	1.95904e6			1.18631			0.593153	−	0.805090i	$-0.297883\pi$
$307$	1.95904e6			1.18631			0.593153	+	0.805090i	$0.297883\pi$
$308$		0			0
$309$	5.38452e6			3.20812
$310$		0			0
$311$	1.53128e6	−	2.65226e6i	0.897749	−	1.55495i	0.0673832	−	0.997727i	$-0.478535\pi$
$311$	1.53128e6	−	2.65226e6i	0.897749	−	1.55495i	0.830365	−	0.557219i	$-0.188132\pi$
$312$		0			0
$313$	−291317.	−	504576.i	−0.168076	−	0.291116i	0.769668	−	0.638445i	$-0.220422\pi$
$313$	−291317.	−	504576.i	−0.168076	−	0.291116i	−0.937743	+	0.347329i	$0.887089\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.54793e6	+	2.68108e6i	0.865171	+	1.49852i	0.866878	+	0.498521i	$0.166123\pi$
$317$	1.54793e6	+	2.68108e6i	0.865171	+	1.49852i	−0.00170706	+	0.999999i	$0.500543\pi$
$318$		0			0
$319$	−366288.	+	634429.i	−0.201533	+	0.349065i
$320$		0			0
$321$	1.61676e6			0.875756
$322$		0			0
$323$	1.73537e6			0.925521
$324$		0			0
$325$	743754.	−	1.28822e6i	0.390590	−	0.676521i
$326$		0			0
$327$	1.58955e6	+	2.75318e6i	0.822062	+	1.42385i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−312748.	−	541695.i	−0.156901	−	0.271760i	0.776849	−	0.629687i	$-0.216817\pi$
$331$	−312748.	−	541695.i	−0.156901	−	0.271760i	−0.933749	+	0.357927i	$0.883484\pi$
$332$		0			0
$333$	4.09377e6	−	7.09061e6i	2.02308	−	3.50407i
$334$		0			0
$335$	−1.27219e6			−0.619356
$336$		0			0
$337$	2.32494e6			1.11516			0.557580	−	0.830123i	$-0.311730\pi$
$337$	2.32494e6			1.11516			0.557580	+	0.830123i	$0.311730\pi$
$338$		0			0
$339$	37530.0	−	65003.9i	0.0177370	−	0.0307213i
$340$		0			0
$341$	−907296.	−	1.57148e6i	−0.422535	−	0.731853i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	537600.	+	931151.i	0.243171	+	0.421184i
$346$		0			0
$347$	390564.	−	676477.i	0.174128	−	0.301598i	−0.765731	−	0.643161i	$-0.777623\pi$
$347$	390564.	−	676477.i	0.174128	−	0.301598i	0.939859	+	0.341562i	$0.110956\pi$
$348$		0			0
$349$	−1.48586e6			−0.653002			−0.326501	−	0.945197i	$-0.605870\pi$
$349$	−1.48586e6			−0.653002			−0.326501	+	0.945197i	$0.605870\pi$
$350$		0			0
$351$	−8.79336e6			−3.80967
$352$		0			0
$353$	722313.	−	1.25108e6i	0.308524	−	0.534379i	−0.669516	−	0.742798i	$-0.733498\pi$
$353$	722313.	−	1.25108e6i	0.308524	−	0.534379i	0.978040	+	0.208419i	$0.0668317\pi$
$354$		0			0
$355$	−1.11821e6	−	1.93679e6i	−0.470925	−	0.815666i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−202020.	−	349909.i	−0.0827291	−	0.143291i	0.821692	−	0.569932i	$-0.193030\pi$
$359$	−202020.	−	349909.i	−0.0827291	−	0.143291i	−0.904421	+	0.426641i	$0.859697\pi$
$360$		0			0
$361$	−488032.	+	845297.i	−0.197097	+	0.341383i
$362$		0			0
$363$	−6.84975e6			−2.72840
$364$		0			0
$365$	526400.			0.206816
$366$		0			0
$367$	−1.35849e6	+	2.35298e6i	−0.526492	+	0.911911i	0.473031	+	0.881046i	$0.343160\pi$
$367$	−1.35849e6	+	2.35298e6i	−0.526492	+	0.911911i	−0.999524	+	0.0308656i	$0.990174\pi$
$368$		0			0
$369$	874467.	+	1.51462e6i	0.334332	+	0.579079i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	896993.	+	1.55364e6i	0.333824	+	0.578199i	0.983258	−	0.182218i	$-0.0583277\pi$
$373$	896993.	+	1.55364e6i	0.333824	+	0.578199i	−0.649435	+	0.760417i	$0.724994\pi$
$374$		0			0
$375$	2.50848e6	−	4.34481e6i	0.921154	−	1.59549i
$376$		0			0
$377$	−831192.			−0.301195
$378$		0			0
$379$	−18624.0			−0.00666001			−0.00333001	−	0.999994i	$-0.501060\pi$
$379$	−18624.0			−0.00666001			−0.00333001	+	0.999994i	$0.501060\pi$
$380$		0			0
$381$	4.31688e6	−	7.47706e6i	1.52355	−	2.63887i
$382$		0			0
$383$	−665022.	−	1.15185e6i	−0.231654	−	0.401236i	0.726641	−	0.687017i	$-0.241080\pi$
$383$	−665022.	−	1.15185e6i	−0.231654	−	0.401236i	−0.958295	+	0.285781i	$0.907747\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.34680e6	+	4.06478e6i	0.796525	+	1.37962i
$388$		0			0
$389$	−1.13253e6	+	1.96160e6i	−0.379468	+	0.657258i	−0.990985	−	0.133973i	$-0.957226\pi$
$389$	−1.13253e6	+	1.96160e6i	−0.379468	+	0.657258i	0.611517	+	0.791231i	$0.290560\pi$
$390$		0			0
$391$	1.04608e6			0.346037
$392$		0			0
$393$	−1.42986e6			−0.466995
$394$		0			0
$395$	1.25402e6	−	2.17202e6i	0.404399	−	0.700440i
$396$		0			0
$397$	−2.24450e6	−	3.88759e6i	−0.714733	−	1.23795i	−0.963062	−	0.269278i	$-0.913215\pi$
$397$	−2.24450e6	−	3.88759e6i	−0.714733	−	1.23795i	0.248330	−	0.968676i	$-0.420118\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	47721.0	+	82655.2i	0.0148200	+	0.0256690i	0.873340	−	0.487111i	$-0.161949\pi$
$401$	47721.0	+	82655.2i	0.0148200	+	0.0256690i	−0.858520	+	0.512780i	$0.828616\pi$
$402$		0			0
$403$	1.02943e6	−	1.78303e6i	0.315744	−	0.546885i
$404$		0			0
$405$	−6.81437e6			−2.06437
$406$		0			0
$407$	7.77629e6			2.32695
$408$		0			0
$409$	−1.49502e6	+	2.58945e6i	−0.441914	+	0.765418i	−0.997832	−	0.0658198i	$-0.979034\pi$
$409$	−1.49502e6	+	2.58945e6i	−0.441914	+	0.765418i	0.555917	+	0.831238i	$0.312367\pi$
$410$		0			0
$411$	3.34731e6	+	5.79771e6i	0.977443	+	1.69298i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.75709e6	−	3.04337e6i	−0.500810	−	0.867429i
$416$		0			0
$417$	−3.75813e6	+	6.50927e6i	−1.05836	+	1.83313i
$418$		0			0
$419$	−3.39037e6			−0.943436			−0.471718	−	0.881749i	$-0.656366\pi$
$419$	−3.39037e6			−0.943436			−0.471718	+	0.881749i	$0.656366\pi$
$420$		0			0
$421$	3.38397e6			0.930512			0.465256	−	0.885176i	$-0.345962\pi$
$421$	3.38397e6			0.930512			0.465256	+	0.885176i	$0.345962\pi$
$422$		0			0
$423$	−2.45324e6	+	4.24913e6i	−0.666636	+	1.15465i
$424$		0			0
$425$	−981167.	−	1.69943e6i	−0.263494	−	0.456385i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.62688e6	−	1.14781e7i	−1.73846	−	3.01111i
$430$		0			0
$431$	991764.	−	1.71779e6i	0.257167	−	0.445426i	−0.708315	−	0.705897i	$-0.750544\pi$
$431$	991764.	−	1.71779e6i	0.257167	−	0.445426i	0.965482	+	0.260470i	$0.0838776\pi$
$432$		0			0
$433$	7.17581e6			1.83929			0.919647	−	0.392746i	$-0.128475\pi$
$433$	7.17581e6			1.83929			0.919647	+	0.392746i	$0.128475\pi$
$434$		0			0
$435$	−1.12704e6			−0.285572
$436$		0			0
$437$	−1.04048e6	+	1.80216e6i	−0.260633	+	0.451430i
$438$		0			0
$439$	−1.22195e6	−	2.11648e6i	−0.302616	−	0.524146i	0.674112	−	0.738629i	$-0.264527\pi$
$439$	−1.22195e6	−	2.11648e6i	−0.302616	−	0.524146i	−0.976728	+	0.214483i	$0.931193\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−115858.	−	200672.i	−0.0280490	−	0.0485822i	0.851660	−	0.524094i	$-0.175596\pi$
$443$	−115858.	−	200672.i	−0.0280490	−	0.0485822i	−0.879709	+	0.475512i	$0.842263\pi$
$444$		0			0
$445$	911456.	−	1.57869e6i	0.218190	−	0.377917i
$446$		0			0
$447$	1.46218e7			3.46125
$448$		0			0
$449$	−4.73637e6			−1.10874			−0.554370	−	0.832271i	$-0.687041\pi$
$449$	−4.73637e6			−1.10874			−0.554370	+	0.832271i	$0.687041\pi$
$450$		0			0
$451$	−830544.	+	1.43854e6i	−0.192274	+	0.333029i
$452$		0			0
$453$	−820200.	−	1.42063e6i	−0.187791	−	0.325263i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.93743e6	+	6.81983e6i	0.881906	+	1.52751i	0.849219	+	0.528040i	$0.177073\pi$
$457$	3.93743e6	+	6.81983e6i	0.881906	+	1.52751i	0.0326865	+	0.999466i	$0.489594\pi$
$458$		0			0
$459$	−5.80014e6	+	1.00461e7i	−1.28501	+	2.22570i
$460$		0			0
$461$	8.23218e6			1.80411			0.902054	−	0.431623i	$-0.142059\pi$
$461$	8.23218e6			1.80411			0.902054	+	0.431623i	$0.142059\pi$
$462$		0			0
$463$	2.36038e6			0.511717			0.255859	−	0.966714i	$-0.417642\pi$
$463$	2.36038e6			0.511717			0.255859	+	0.966714i	$0.417642\pi$
$464$		0			0
$465$	1.39584e6	−	2.41767e6i	0.299367	−	0.518518i
$466$		0			0
$467$	−3.15850e6	−	5.47068e6i	−0.670175	−	1.16078i	−0.977854	−	0.209288i	$-0.932885\pi$
$467$	−3.15850e6	−	5.47068e6i	−0.670175	−	1.16078i	0.307679	−	0.951490i	$-0.400448\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−3.16602e6	−	5.48371e6i	−0.657599	−	1.13900i
$472$		0			0
$473$	−2.22893e6	+	3.86062e6i	−0.458082	+	0.793421i
$474$		0			0
$475$	3.90366e6			0.793849
$476$		0			0
$477$	−1.79190e7			−3.60594
$478$		0			0
$479$	729278.	−	1.26315e6i	0.145229	−	0.251545i	−0.784229	−	0.620471i	$-0.786941\pi$
$479$	729278.	−	1.26315e6i	0.145229	−	0.251545i	0.929458	+	0.368927i	$0.120275\pi$
$480$		0			0
$481$	4.41155e6	+	7.64103e6i	0.869417	+	1.50588i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.85514e6	+	3.21319e6i	0.358114	+	0.620272i
$486$		0			0
$487$	−4.64391e6	+	8.04349e6i	−0.887282	+	1.53682i	−0.0442060	+	0.999022i	$0.514076\pi$
$487$	−4.64391e6	+	8.04349e6i	−0.887282	+	1.53682i	−0.843076	+	0.537795i	$0.819258\pi$
$488$		0			0
$489$	605760.			0.114559
$490$		0			0
$491$	−234972.			−0.0439858			−0.0219929	−	0.999758i	$-0.507001\pi$
$491$	−234972.			−0.0439858			−0.0219929	+	0.999758i	$0.507001\pi$
$492$		0			0
$493$	−548258.	+	949611.i	−0.101594	+	0.175966i
$494$		0			0
$495$	−6.55949e6	−	1.13614e7i	−1.20325	−	2.08410i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.50396e6	+	6.06904e6i	0.629953	+	1.09111i	0.987561	+	0.157238i	$0.0502591\pi$
$499$	3.50396e6	+	6.06904e6i	0.629953	+	1.09111i	−0.357608	+	0.933872i	$0.616408\pi$
$500$		0			0
$501$	67860.0	−	117537.i	0.0120787	−	0.0209209i
$502$		0			0
$503$	−4.94752e6			−0.871902			−0.435951	−	0.899970i	$-0.643588\pi$
$503$	−4.94752e6			−0.871902			−0.435951	+	0.899970i	$0.643588\pi$
$504$		0			0
$505$	267264.			0.0466350
$506$		0			0
$507$	1.94956e6	−	3.37675e6i	0.336835	−	0.583416i
$508$		0			0
$509$	2.75320e6	+	4.76869e6i	0.471025	+	0.815839i	0.999451	−	0.0331407i	$-0.0105510\pi$
$509$	2.75320e6	+	4.76869e6i	0.471025	+	0.815839i	−0.528426	+	0.848979i	$0.677218\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.15382e7	−	1.99847e7i	−1.93573	−	3.35278i
$514$		0			0
$515$	−2.87174e6	+	4.97401e6i	−0.477120	+	0.826396i
$516$		0			0
$517$	−4.66003e6			−0.766765
$518$		0			0
$519$	−3.12996e6			−0.510059
$520$		0			0
$521$	815379.	−	1.41228e6i	0.131603	−	0.227943i	−0.792692	−	0.609623i	$-0.791321\pi$
$521$	815379.	−	1.41228e6i	0.131603	−	0.227943i	0.924295	+	0.381680i	$0.124654\pi$
$522$		0			0
$523$	−5.03827e6	−	8.72654e6i	−0.805429	−	1.39504i	−0.916001	−	0.401176i	$-0.868602\pi$
$523$	−5.03827e6	−	8.72654e6i	−0.805429	−	1.39504i	0.110572	−	0.993868i	$-0.464732\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.35804e6	−	2.35219e6i	−0.213003	−	0.368931i
$528$		0			0
$529$	2.59097e6	−	4.48769e6i	0.402553	−	0.697243i
$530$		0			0
$531$	−1.63593e6			−0.251784
$532$		0			0
$533$	−1.88470e6			−0.287358
$534$		0			0
$535$	−862272.	+	1.49350e6i	−0.130245	+	0.225590i
$536$		0			0
$537$	−3.02586e6	−	5.24094e6i	−0.452807	−	0.784285i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	6.26125e6	+	1.08448e7i	0.919746	+	1.59305i	0.799800	+	0.600267i	$0.204939\pi$
$541$	6.26125e6	+	1.08448e7i	0.919746	+	1.59305i	0.119947	+	0.992780i	$0.461728\pi$
$542$		0			0
$543$	−9.83550e6	+	1.70356e7i	−1.43152	+	2.47946i
$544$		0			0
$545$	−3.39104e6			−0.489037
$546$		0			0
$547$	6.67430e6			0.953756			0.476878	−	0.878970i	$-0.341768\pi$
$547$	6.67430e6			0.953756			0.476878	+	0.878970i	$0.341768\pi$
$548$		0			0
$549$	−3.64504e6	+	6.31339e6i	−0.516144	+	0.893988i
$550$		0			0
$551$	−1.09065e6	−	1.88905e6i	−0.153040	−	0.265073i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	5.98176e6	+	1.03607e7i	0.824321	+	1.42777i
$556$		0			0
$557$	2.30821e6	−	3.99794e6i	0.315238	−	0.546007i	−0.664250	−	0.747510i	$-0.731249\pi$
$557$	2.30821e6	−	3.99794e6i	0.315238	−	0.546007i	0.979488	+	0.201503i	$0.0645825\pi$
$558$		0			0
$559$	−5.05795e6			−0.684613
$560$		0			0
$561$	−1.74845e7			−2.34555
$562$		0			0
$563$	−4.42109e6	+	7.65755e6i	−0.587839	+	1.01817i	0.406676	+	0.913572i	$0.366688\pi$
$563$	−4.42109e6	+	7.65755e6i	−0.587839	+	1.01817i	−0.994515	+	0.104594i	$0.966646\pi$
$564$		0			0
$565$	40032.0	+	69337.5i	0.00527577	+	0.00913791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.78856e6	−	1.00261e7i	−0.749532	−	1.29823i	−0.948047	−	0.318129i	$-0.896945\pi$
$569$	−5.78856e6	−	1.00261e7i	−0.749532	−	1.29823i	0.198515	−	0.980098i	$-0.436388\pi$
$570$		0			0
$571$	−2.24034e6	+	3.88039e6i	−0.287557	+	0.498064i	−0.973226	−	0.229850i	$-0.926177\pi$
$571$	−2.24034e6	+	3.88039e6i	−0.287557	+	0.498064i	0.685669	+	0.727914i	$0.259510\pi$
$572$		0			0
$573$	4.54488e6			0.578277
$574$		0			0
$575$	2.35312e6			0.296807
$576$		0			0
$577$	6.63939e6	−	1.14998e7i	0.830212	−	1.43797i	−0.0676585	−	0.997709i	$-0.521553\pi$
$577$	6.63939e6	−	1.14998e7i	0.830212	−	1.43797i	0.897870	−	0.440260i	$-0.145114\pi$
$578$		0			0
$579$	3.43989e6	+	5.95806e6i	0.426430	+	0.738599i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−8.50949e6	−	1.47389e7i	−1.03689	−	1.79594i
$584$		0			0
$585$	7.44250e6	−	1.28908e7i	0.899143	−	1.55736i
$586$		0			0
$587$	1.11188e7			1.33187			0.665936	−	0.746009i	$-0.268032\pi$
$587$	1.11188e7			1.33187			0.665936	+	0.746009i	$0.268032\pi$
$588$		0			0
$589$	5.40306e6			0.641729
$590$		0			0
$591$	6.31629e6	−	1.09401e7i	0.743864	−	1.28841i
$592$		0			0
$593$	−1.96368e6	−	3.40120e6i	−0.229316	−	0.397187i	0.728289	−	0.685270i	$-0.240316\pi$
$593$	−1.96368e6	−	3.40120e6i	−0.229316	−	0.397187i	−0.957606	+	0.288082i	$0.906982\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.95950e6	−	5.12600e6i	−0.339846	−	0.588631i
$598$		0			0
$599$	8.70497e6	−	1.50775e7i	0.991289	−	1.71696i	0.381585	−	0.924334i	$-0.375378\pi$
$599$	8.70497e6	−	1.50775e7i	0.991289	−	1.71696i	0.609704	−	0.792630i	$-0.291288\pi$
$600$		0			0
$601$	−7.46243e6			−0.842740			−0.421370	−	0.906889i	$-0.638451\pi$
$601$	−7.46243e6			−0.842740			−0.421370	+	0.906889i	$0.638451\pi$
$602$		0			0
$603$	2.61197e7			2.92533
$604$		0			0
$605$	3.65320e6	−	6.32753e6i	0.405775	−	0.702822i
$606$		0			0
$607$	350576.	+	607215.i	0.0386198	+	0.0668915i	0.884689	−	0.466181i	$-0.154371\pi$
$607$	350576.	+	607215.i	0.0386198	+	0.0668915i	−0.846069	+	0.533073i	$0.821037\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.64367e6	−	4.57897e6i	−0.286487	−	0.496210i
$612$		0			0
$613$	−5.47876e6	+	9.48949e6i	−0.588886	+	1.01998i	0.405493	+	0.914098i	$0.367100\pi$
$613$	−5.47876e6	+	9.48949e6i	−0.588886	+	1.01998i	−0.994379	+	0.105882i	$0.966233\pi$
$614$		0			0
$615$	−2.55552e6			−0.272453
$616$		0			0
$617$	1.90666e6			0.201633			0.100816	−	0.994905i	$-0.467855\pi$
$617$	1.90666e6			0.201633			0.100816	+	0.994905i	$0.467855\pi$
$618$		0			0
$619$	−1.11173e6	+	1.92557e6i	−0.116620	+	0.201992i	−0.918426	−	0.395592i	$-0.870539\pi$
$619$	−1.11173e6	+	1.92557e6i	−0.116620	+	0.201992i	0.801806	+	0.597584i	$0.203873\pi$
$620$		0			0
$621$	−6.95520e6	−	1.20468e7i	−0.723737	−	1.25355i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−607100.	−	1.05153e6i	−0.0621671	−	0.107677i
$626$		0			0
$627$	1.73909e7	−	3.01219e7i	1.76666	−	3.05994i
$628$		0			0
$629$	1.16395e7			1.17303
$630$		0			0
$631$	752624.			0.0752497			0.0376248	−	0.999292i	$-0.488021\pi$
$631$	752624.			0.0752497			0.0376248	+	0.999292i	$0.488021\pi$
$632$		0			0
$633$	1.01858e7	−	1.76423e7i	1.01038	−	1.75003i
$634$		0			0
$635$	4.60467e6	+	7.97553e6i	0.453173	+	0.784919i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	2.29582e7	+	3.97648e7i	2.22426	+	3.85253i
$640$		0			0
$641$	1.22714e6	−	2.12548e6i	0.117964	−	0.204320i	−0.800996	−	0.598669i	$-0.795696\pi$
$641$	1.22714e6	−	2.12548e6i	0.117964	−	0.204320i	0.918961	+	0.394349i	$0.129030\pi$
$642$		0			0
$643$	−1.58237e7			−1.50932			−0.754660	−	0.656116i	$-0.772198\pi$
$643$	−1.58237e7			−1.50932			−0.754660	+	0.656116i	$0.772198\pi$
$644$		0			0
$645$	−6.85824e6			−0.649103
$646$		0			0
$647$	−1.32745e6	+	2.29920e6i	−0.124668	+	0.215932i	−0.921603	−	0.388133i	$-0.873120\pi$
$647$	−1.32745e6	+	2.29920e6i	−0.124668	+	0.215932i	0.796935	+	0.604065i	$0.206453\pi$
$648$		0			0
$649$	−776880.	−	1.34560e6i	−0.0724006	−	0.125402i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.63450e6	−	8.02718e6i	−0.425324	−	0.736682i	0.571127	−	0.820862i	$-0.306506\pi$
$653$	−4.63450e6	−	8.02718e6i	−0.425324	−	0.736682i	−0.996451	+	0.0841794i	$0.973173\pi$
$654$		0			0
$655$	762592.	−	1.32085e6i	0.0694527	−	0.120296i
$656$		0			0
$657$	−1.08076e7			−0.976827
$658$		0			0
$659$	−1.68242e7			−1.50911			−0.754556	−	0.656235i	$-0.772148\pi$
$659$	−1.68242e7			−1.50911			−0.754556	+	0.656235i	$0.772148\pi$
$660$		0			0
$661$	−3.38608e6	+	5.86487e6i	−0.301435	+	0.522101i	−0.976461	−	0.215693i	$-0.930799\pi$
$661$	−3.38608e6	+	5.86487e6i	−0.301435	+	0.522101i	0.675026	+	0.737794i	$0.264132\pi$
$662$		0			0
$663$	−9.91908e6	−	1.71804e7i	−0.876370	−	1.51792i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−657440.	−	1.13872e6i	−0.0572192	−	0.0991065i
$668$		0			0
$669$	2.76660e6	−	4.79189e6i	0.238991	−	0.413944i
$670$		0			0
$671$	−6.92390e6			−0.593669
$672$		0			0
$673$	7.61315e6			0.647928			0.323964	−	0.946069i	$-0.394984\pi$
$673$	7.61315e6			0.647928			0.323964	+	0.946069i	$0.394984\pi$
$674$		0			0
$675$	−1.30472e7	+	2.25984e7i	−1.10219	+	1.90906i
$676$		0			0
$677$	3.05836e6	+	5.29724e6i	0.256459	+	0.444199i	0.965291	−	0.261178i	$-0.0841109\pi$
$677$	3.05836e6	+	5.29724e6i	0.256459	+	0.444199i	−0.708832	+	0.705377i	$0.750778\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.30212e7	+	2.25533e7i	1.07593	+	1.86356i
$682$		0			0
$683$	5.62942e6	−	9.75044e6i	0.461755	−	0.799784i	−0.537293	−	0.843396i	$-0.680553\pi$
$683$	5.62942e6	−	9.75044e6i	0.461755	−	0.799784i	0.999049	+	0.0436117i	$0.0138864\pi$
$684$		0			0
$685$	−7.14093e6			−0.581471
$686$		0			0
$687$	−1.78158e7			−1.44017
$688$		0			0
$689$	9.65500e6	−	1.67229e7i	0.774826	−	1.34204i
$690$		0			0
$691$	4.75476e6	+	8.23549e6i	0.378821	+	0.656137i	0.990891	−	0.134667i	$-0.0429964\pi$
$691$	4.75476e6	+	8.23549e6i	0.378821	+	0.656137i	−0.612070	+	0.790803i	$0.709663\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.00867e6	−	6.94322e6i	−0.314803	−	0.545254i
$696$		0			0
$697$	−1.24315e6	+	2.15321e6i	−0.0969266	+	0.167882i
$698$		0			0
$699$	−1.44654e6			−0.111979
$700$		0			0
$701$	1.53868e7			1.18264			0.591322	−	0.806436i	$-0.298606\pi$
$701$	1.53868e7			1.18264			0.591322	+	0.806436i	$0.298606\pi$
$702$		0			0
$703$	−1.15772e7	+	2.00523e7i	−0.883517	+	1.53030i
$704$		0			0
$705$	−3.58464e6	−	6.20878e6i	−0.271627	−	0.470472i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	606895.	+	1.05117e6i	0.0453417	+	0.0785342i	0.887806	−	0.460219i	$-0.152229\pi$
$709$	606895.	+	1.05117e6i	0.0453417	+	0.0785342i	−0.842464	+	0.538753i	$0.818896\pi$
$710$		0			0
$711$	−2.57465e7	+	4.45943e7i	−1.91005	+	3.30830i
$712$		0			0
$713$	3.25696e6			0.239932
$714$		0			0
$715$	1.41373e7			1.03420
$716$		0			0
$717$	3.62532e6	−	6.27924e6i	0.263359	−	0.456151i
$718$		0			0
$719$	−5.00011e6	−	8.66044e6i	−0.360709	−	0.624766i	0.627369	−	0.778722i	$-0.284132\pi$
$719$	−5.00011e6	−	8.66044e6i	−0.360709	−	0.624766i	−0.988078	+	0.153956i	$0.950799\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−8.47905e6	−	1.46861e7i	−0.603256	−	1.04487i
$724$		0			0
$725$	−1.23329e6	+	2.13612e6i	−0.0871403	+	0.150931i
$726$		0			0
$727$	−1.33745e7			−0.938514			−0.469257	−	0.883062i	$-0.655478\pi$
$727$	−1.33745e7			−0.938514			−0.469257	+	0.883062i	$0.655478\pi$
$728$		0			0
$729$	4.93657e7			3.44038
$730$		0			0
$731$	−3.33625e6	+	5.77855e6i	−0.230922	+	0.399968i
$732$		0			0
$733$	8.06899e6	+	1.39759e7i	0.554701	+	0.960771i	0.997927	+	0.0643606i	$0.0205008\pi$
$733$	8.06899e6	+	1.39759e7i	0.554701	+	0.960771i	−0.443225	+	0.896410i	$0.646166\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.24039e7	+	2.14841e7i	0.841179	+	1.45697i
$738$		0			0
$739$	−4.30530e6	+	7.45699e6i	−0.289996	+	0.502288i	−0.973808	−	0.227370i	$-0.926987\pi$
$739$	−4.30530e6	+	7.45699e6i	−0.289996	+	0.502288i	0.683812	+	0.729658i	$0.260321\pi$
$740$		0			0
$741$	3.94639e7			2.64031
$742$		0			0
$743$	−2.85027e7			−1.89415			−0.947075	−	0.321012i	$-0.895977\pi$
$743$	−2.85027e7			−1.89415			−0.947075	+	0.321012i	$0.895977\pi$
$744$		0			0
$745$	−7.79830e6	+	1.35071e7i	−0.514766	+	0.891600i
$746$		0			0
$747$	3.60752e7	+	6.24841e7i	2.36542	+	4.09702i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.64360e6	+	6.31091e6i	0.235739	+	0.408312i	0.959487	−	0.281752i	$-0.0909156\pi$
$751$	3.64360e6	+	6.31091e6i	0.235739	+	0.408312i	−0.723748	+	0.690064i	$0.757582\pi$
$752$		0			0
$753$	2.15662e7	−	3.73537e7i	1.38607	−	2.40075i
$754$		0			0
$755$	1.74976e6			0.111715
$756$		0			0
$757$	−2.77165e7			−1.75792			−0.878958	−	0.476899i	$-0.841761\pi$
$757$	−2.77165e7			−1.75792			−0.878958	+	0.476899i	$0.841761\pi$
$758$		0			0
$759$	1.04832e7	−	1.81574e7i	0.660525	−	1.14406i
$760$		0			0
$761$	1.53813e7	+	2.66411e7i	0.962787	+	1.66760i	0.715447	+	0.698667i	$0.246223\pi$
$761$	1.53813e7	+	2.66411e7i	0.962787	+	1.66760i	0.247340	+	0.968929i	$0.420444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−9.81821e6	−	1.70056e7i	−0.606567	−	1.05060i
$766$		0			0
$767$	881460.	−	1.52673e6i	0.0541021	−	0.0937076i
$768$		0			0
$769$	1.83665e7			1.11998			0.559990	−	0.828499i	$-0.310805\pi$
$769$	1.83665e7			1.11998			0.559990	+	0.828499i	$0.310805\pi$
$770$		0			0
$771$	1.48441e7			0.899325
$772$		0			0
$773$	−5.06986e6	+	8.78126e6i	−0.305174	+	0.528577i	−0.977300	−	0.211860i	$-0.932048\pi$
$773$	−5.06986e6	+	8.78126e6i	−0.305174	+	0.528577i	0.672126	+	0.740437i	$0.265381\pi$
$774$		0			0
$775$	−3.05485e6	−	5.29116e6i	−0.182699	−	0.316444i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.47300e6	−	4.28336e6i	−0.146009	−	0.252895i
$780$		0			0
$781$	−2.18051e7	+	3.77675e7i	−1.27917	+	2.21559i
$782$		0			0
$783$	1.45811e7			0.849934
$784$		0			0
$785$

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)

\(f(q)\)	\(=\)	\(q+(15.0000 - 25.9808i) q^{3} +(16.0000 + 27.7128i) q^{5} +(-328.500 - 568.979i) q^{9} +O(q^{10})\)	\(q+(15.0000 - 25.9808i) q^{3} +(16.0000 + 27.7128i) q^{5} +(-328.500 - 568.979i) q^{9} +(312.000 - 540.400i) q^{11} +708.000 q^{13} +960.000 q^{15} +(467.000 - 808.868i) q^{17} +(929.000 + 1609.08i) q^{19} +(560.000 + 969.948i) q^{23} +(1050.50 - 1819.52i) q^{25} -12420.0 q^{27} -1174.00 q^{29} +(1454.00 - 2518.40i) q^{31} +(-9360.00 - 16212.0i) q^{33} +(6231.00 + 10792.4i) q^{37} +(10620.0 - 18394.4i) q^{39} -2662.00 q^{41} -7144.00 q^{43} +(10512.0 - 18207.3i) q^{45} +(-3734.00 - 6467.48i) q^{47} +(-14010.0 - 24266.0i) q^{51} +(13637.0 - 23620.0i) q^{53} +19968.0 q^{55} +55740.0 q^{57} +(1245.00 - 2156.40i) q^{59} +(-5548.00 - 9609.42i) q^{61} +(11328.0 + 19620.7i) q^{65} +(-19878.0 + 34429.7i) q^{67} +33600.0 q^{69} -69888.0 q^{71} +(8225.00 - 14246.1i) q^{73} +(-31515.0 - 54585.6i) q^{75} +(-39188.0 - 67875.6i) q^{79} +(-106474. + 184419. i) q^{81} -109818. q^{83} +29888.0 q^{85} +(-17610.0 + 30501.4i) q^{87} +(-28483.0 - 49334.0i) q^{89} +(-43620.0 - 75552.1i) q^{93} +(-29728.0 + 51490.4i) q^{95} +115946. q^{97} -409968. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 30 q^{3} + 32 q^{5} - 657 q^{9}+O(q^{10}) \) 2 * q + 30 * q^3 + 32 * q^5 - 657 * q^9	\( 2 q + 30 q^{3} + 32 q^{5} - 657 q^{9} + 624 q^{11} + 1416 q^{13} + 1920 q^{15} + 934 q^{17} + 1858 q^{19} + 1120 q^{23} + 2101 q^{25} - 24840 q^{27} - 2348 q^{29} + 2908 q^{31} - 18720 q^{33} + 12462 q^{37} + 21240 q^{39} - 5324 q^{41} - 14288 q^{43} + 21024 q^{45} - 7468 q^{47} - 28020 q^{51} + 27274 q^{53} + 39936 q^{55} + 111480 q^{57} + 2490 q^{59} - 11096 q^{61} + 22656 q^{65} - 39756 q^{67} + 67200 q^{69} - 139776 q^{71} + 16450 q^{73} - 63030 q^{75} - 78376 q^{79} - 212949 q^{81} - 219636 q^{83} + 59776 q^{85} - 35220 q^{87} - 56966 q^{89} - 87240 q^{93} - 59456 q^{95} + 231892 q^{97} - 819936 q^{99}+O(q^{100}) \) 2 * q + 30 * q^3 + 32 * q^5 - 657 * q^9 + 624 * q^11 + 1416 * q^13 + 1920 * q^15 + 934 * q^17 + 1858 * q^19 + 1120 * q^23 + 2101 * q^25 - 24840 * q^27 - 2348 * q^29 + 2908 * q^31 - 18720 * q^33 + 12462 * q^37 + 21240 * q^39 - 5324 * q^41 - 14288 * q^43 + 21024 * q^45 - 7468 * q^47 - 28020 * q^51 + 27274 * q^53 + 39936 * q^55 + 111480 * q^57 + 2490 * q^59 - 11096 * q^61 + 22656 * q^65 - 39756 * q^67 + 67200 * q^69 - 139776 * q^71 + 16450 * q^73 - 63030 * q^75 - 78376 * q^79 - 212949 * q^81 - 219636 * q^83 + 59776 * q^85 - 35220 * q^87 - 56966 * q^89 - 87240 * q^93 - 59456 * q^95 + 231892 * q^97 - 819936 * q^99

Introduction

L-functions

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