LMFDB - Embedded newform 336.6.q.c.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (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:	$53.8889634572$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	336.193
Dual form			336.6.q.c.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 7.79423i) q^{3} +(3.00000 + 5.19615i) q^{5} +(-59.5000 + 115.181i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 - 7.79423i) q^{3} +(3.00000 + 5.19615i) q^{5} +(-59.5000 + 115.181i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(-333.000 + 576.773i) q^{11} -559.000 q^{13} +54.0000 q^{15} +(870.000 - 1506.88i) q^{17} +(578.500 + 1001.99i) q^{19} +(630.000 + 982.073i) q^{21} +(-1734.00 - 3003.38i) q^{23} +(1544.50 - 2675.15i) q^{25} -729.000 q^{27} +3372.00 q^{29} +(3146.50 - 5449.90i) q^{31} +(2997.00 + 5190.96i) q^{33} +(-777.000 + 36.3731i) q^{35} +(-1565.50 - 2711.53i) q^{37} +(-2515.50 + 4356.97i) q^{39} -4866.00 q^{41} +11407.0 q^{43} +(243.000 - 420.888i) q^{45} +(1155.00 + 2000.52i) q^{47} +(-9726.50 - 13706.6i) q^{49} +(-7830.00 - 13562.0i) q^{51} +(14148.0 - 24505.1i) q^{53} -3996.00 q^{55} +10413.0 q^{57} +(10272.0 - 17791.6i) q^{59} +(2315.00 + 4009.70i) q^{61} +(10489.5 - 491.036i) q^{63} +(-1677.00 - 2904.65i) q^{65} +(-9372.50 + 16233.6i) q^{67} -31212.0 q^{69} +38226.0 q^{71} +(-35294.5 + 61131.9i) q^{73} +(-13900.5 - 24076.4i) q^{75} +(-46620.0 - 72673.4i) q^{77} +(-31146.5 - 53947.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -79818.0 q^{83} +10440.0 q^{85} +(15174.0 - 26282.1i) q^{87} +(9060.00 + 15692.4i) q^{89} +(33260.5 - 64386.4i) q^{91} +(-28318.5 - 49049.1i) q^{93} +(-3471.00 + 6011.95i) q^{95} +124754. q^{97} +53946.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9}+O(q^{10}) $ 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9	$ 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9} - 666 q^{11} - 1118 q^{13} + 108 q^{15} + 1740 q^{17} + 1157 q^{19} + 1260 q^{21} - 3468 q^{23} + 3089 q^{25} - 1458 q^{27} + 6744 q^{29} + 6293 q^{31} + 5994 q^{33} - 1554 q^{35} - 3131 q^{37} - 5031 q^{39} - 9732 q^{41} + 22814 q^{43} + 486 q^{45} + 2310 q^{47} - 19453 q^{49} - 15660 q^{51} + 28296 q^{53} - 7992 q^{55} + 20826 q^{57} + 20544 q^{59} + 4630 q^{61} + 20979 q^{63} - 3354 q^{65} - 18745 q^{67} - 62424 q^{69} + 76452 q^{71} - 70589 q^{73} - 27801 q^{75} - 93240 q^{77} - 62293 q^{79} - 6561 q^{81} - 159636 q^{83} + 20880 q^{85} + 30348 q^{87} + 18120 q^{89} + 66521 q^{91} - 56637 q^{93} - 6942 q^{95} + 249508 q^{97} + 107892 q^{99}+O(q^{100}) $ 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9 - 666 * q^11 - 1118 * q^13 + 108 * q^15 + 1740 * q^17 + 1157 * q^19 + 1260 * q^21 - 3468 * q^23 + 3089 * q^25 - 1458 * q^27 + 6744 * q^29 + 6293 * q^31 + 5994 * q^33 - 1554 * q^35 - 3131 * q^37 - 5031 * q^39 - 9732 * q^41 + 22814 * q^43 + 486 * q^45 + 2310 * q^47 - 19453 * q^49 - 15660 * q^51 + 28296 * q^53 - 7992 * q^55 + 20826 * q^57 + 20544 * q^59 + 4630 * q^61 + 20979 * q^63 - 3354 * q^65 - 18745 * q^67 - 62424 * q^69 + 76452 * q^71 - 70589 * q^73 - 27801 * q^75 - 93240 * q^77 - 62293 * q^79 - 6561 * q^81 - 159636 * q^83 + 20880 * q^85 + 30348 * q^87 + 18120 * q^89 + 66521 * q^91 - 56637 * q^93 - 6942 * q^95 + 249508 * q^97 + 107892 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$4$		0			0
$5$	3.00000	+	5.19615i	0.0536656	+	0.0929516i	0.891610	−	0.452804i	$-0.149576\pi$
$5$	3.00000	+	5.19615i	0.0536656	+	0.0929516i	−0.837945	+	0.545755i	$0.816243\pi$
$6$		0			0
$7$	−59.5000	+	115.181i	−0.458957	+	0.888459i
$7$	−59.5000	+	115.181i	−0.458957	+	0.888459i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	−333.000	+	576.773i	−0.829779	+	1.43722i	0.0684322	+	0.997656i	$0.478200\pi$
$11$	−333.000	+	576.773i	−0.829779	+	1.43722i	−0.898211	+	0.439564i	$0.855133\pi$
$12$		0			0
$13$	−559.000			−0.917389			−0.458694	−	0.888594i	$-0.651683\pi$
$13$	−559.000			−0.917389			−0.458694	+	0.888594i	$0.651683\pi$
$14$		0			0
$15$	54.0000			0.0619677
$16$		0			0
$17$	870.000	−	1506.88i	0.730125	−	1.26461i	−0.226705	−	0.973963i	$-0.572795\pi$
$17$	870.000	−	1506.88i	0.730125	−	1.26461i	0.956830	−	0.290649i	$-0.0938713\pi$
$18$		0			0
$19$	578.500	+	1001.99i	0.367637	+	0.636766i	0.989196	−	0.146602i	$-0.0468335\pi$
$19$	578.500	+	1001.99i	0.367637	+	0.636766i	−0.621558	+	0.783368i	$0.713500\pi$
$20$		0			0
$21$	630.000	+	982.073i	0.311740	+	0.485954i
$22$		0			0
$23$	−1734.00	−	3003.38i	−0.683486	−	1.18383i	−0.973910	−	0.226934i	$-0.927130\pi$
$23$	−1734.00	−	3003.38i	−0.683486	−	1.18383i	0.290424	−	0.956898i	$-0.406204\pi$
$24$		0			0
$25$	1544.50	−	2675.15i	0.494240	−	0.856049i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	3372.00			0.744548			0.372274	−	0.928123i	$-0.378578\pi$
$29$	3372.00			0.744548			0.372274	+	0.928123i	$0.378578\pi$
$30$		0			0
$31$	3146.50	−	5449.90i	0.588063	−	1.01855i	−0.406423	−	0.913685i	$-0.633224\pi$
$31$	3146.50	−	5449.90i	0.588063	−	1.01855i	0.994486	−	0.104869i	$-0.0334424\pi$
$32$		0			0
$33$	2997.00	+	5190.96i	0.479073	+	0.829779i
$34$		0			0
$35$	−777.000	+	36.3731i	−0.107214	+	0.00501891i
$36$		0			0
$37$	−1565.50	−	2711.53i	−0.187996	−	0.325619i	0.756586	−	0.653894i	$-0.226866\pi$
$37$	−1565.50	−	2711.53i	−0.187996	−	0.325619i	−0.944582	+	0.328276i	$0.893533\pi$
$38$		0			0
$39$	−2515.50	+	4356.97i	−0.264827	+	0.458694i
$40$		0			0
$41$	−4866.00			−0.452077			−0.226039	−	0.974118i	$-0.572578\pi$
$41$	−4866.00			−0.452077			−0.226039	+	0.974118i	$0.572578\pi$
$42$		0			0
$43$	11407.0			0.940806			0.470403	−	0.882452i	$-0.344108\pi$
$43$	11407.0			0.940806			0.470403	+	0.882452i	$0.344108\pi$
$44$		0			0
$45$	243.000	−	420.888i	0.0178885	−	0.0309839i
$46$		0			0
$47$	1155.00	+	2000.52i	0.0762671	+	0.132099i	0.901637	−	0.432494i	$-0.142366\pi$
$47$	1155.00	+	2000.52i	0.0762671	+	0.132099i	−0.825369	+	0.564593i	$0.809033\pi$
$48$		0			0
$49$	−9726.50	−	13706.6i	−0.578717	−	0.815528i
$50$		0			0
$51$	−7830.00	−	13562.0i	−0.421538	−	0.730125i
$52$		0			0
$53$	14148.0	−	24505.1i	0.691840	−	1.19830i	−0.279395	−	0.960176i	$-0.590134\pi$
$53$	14148.0	−	24505.1i	0.691840	−	1.19830i	0.971235	−	0.238125i	$-0.0765328\pi$
$54$		0			0
$55$	−3996.00			−0.178122
$56$		0			0
$57$	10413.0			0.424511
$58$		0			0
$59$	10272.0	−	17791.6i	0.384171	−	0.665404i	−0.607483	−	0.794333i	$-0.707821\pi$
$59$	10272.0	−	17791.6i	0.384171	−	0.665404i	0.991654	+	0.128929i	$0.0411539\pi$
$60$		0			0
$61$	2315.00	+	4009.70i	0.0796575	+	0.137971i	0.903102	−	0.429426i	$-0.141284\pi$
$61$	2315.00	+	4009.70i	0.0796575	+	0.137971i	−0.823445	+	0.567397i	$0.807951\pi$
$62$		0			0
$63$	10489.5	−	491.036i	0.332969	−	0.0155870i
$64$		0			0
$65$	−1677.00	−	2904.65i	−0.0492322	−	0.0852728i
$66$		0			0
$67$	−9372.50	+	16233.6i	−0.255075	+	0.441803i	−0.964916	−	0.262559i	$-0.915434\pi$
$67$	−9372.50	+	16233.6i	−0.255075	+	0.441803i	0.709841	+	0.704362i	$0.248767\pi$
$68$		0			0
$69$	−31212.0			−0.789221
$70$		0			0
$71$	38226.0			0.899939			0.449969	−	0.893044i	$-0.351435\pi$
$71$	38226.0			0.899939			0.449969	+	0.893044i	$0.351435\pi$
$72$		0			0
$73$	−35294.5	+	61131.9i	−0.775175	+	1.34264i	0.159521	+	0.987195i	$0.449005\pi$
$73$	−35294.5	+	61131.9i	−0.775175	+	1.34264i	−0.934696	+	0.355448i	$0.884328\pi$
$74$		0			0
$75$	−13900.5	−	24076.4i	−0.285350	−	0.494240i
$76$		0			0
$77$	−46620.0	−	72673.4i	−0.896077	−	1.39685i
$78$		0			0
$79$	−31146.5	−	53947.3i	−0.561489	−	0.972528i	−0.997367	−	0.0725221i	$-0.976895\pi$
$79$	−31146.5	−	53947.3i	−0.561489	−	0.972528i	0.435877	−	0.900006i	$-0.356438\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−79818.0			−1.27176			−0.635881	−	0.771787i	$-0.719363\pi$
$83$	−79818.0			−1.27176			−0.635881	+	0.771787i	$0.719363\pi$
$84$		0			0
$85$	10440.0			0.156730
$86$		0			0
$87$	15174.0	−	26282.1i	0.214932	−	0.372274i
$88$		0			0
$89$	9060.00	+	15692.4i	0.121242	+	0.209997i	0.920258	−	0.391313i	$-0.127979\pi$
$89$	9060.00	+	15692.4i	0.121242	+	0.209997i	−0.799016	+	0.601310i	$0.794646\pi$
$90$		0			0
$91$	33260.5	−	64386.4i	0.421042	−	0.815062i
$92$		0			0
$93$	−28318.5	−	49049.1i	−0.339518	−	0.588063i
$94$		0			0
$95$	−3471.00	+	6011.95i	−0.0394590	+	0.0683449i
$96$		0			0
$97$	124754.			1.34625			0.673124	−	0.739530i	$-0.264952\pi$
$97$	124754.			1.34625			0.673124	+	0.739530i	$0.264952\pi$
$98$		0			0
$99$	53946.0			0.553186
$100$		0			0
$101$	46695.0	−	80878.1i	0.455478	−	0.788910i	−0.543238	−	0.839579i	$-0.682802\pi$
$101$	46695.0	−	80878.1i	0.455478	−	0.788910i	0.998716	+	0.0506685i	$0.0161352\pi$
$102$		0			0
$103$	−83865.5	−	145259.i	−0.778915	−	1.34912i	−0.932567	−	0.360996i	$-0.882437\pi$
$103$	−83865.5	−	145259.i	−0.778915	−	1.34912i	0.153652	−	0.988125i	$-0.450897\pi$
$104$		0			0
$105$	−3213.00	+	6219.79i	−0.0284405	+	0.0550558i
$106$		0			0
$107$	34590.0	+	59911.6i	0.292073	+	0.505885i	0.974300	−	0.225256i	$-0.0723217\pi$
$107$	34590.0	+	59911.6i	0.292073	+	0.505885i	−0.682227	+	0.731141i	$0.738988\pi$
$108$		0			0
$109$	109779.	−	190144.i	0.885024	−	1.53291i	0.0393377	−	0.999226i	$-0.487475\pi$
$109$	109779.	−	190144.i	0.885024	−	1.53291i	0.845686	−	0.533680i	$-0.179191\pi$
$110$		0			0
$111$	−28179.0			−0.217079
$112$		0			0
$113$	−39354.0			−0.289930			−0.144965	−	0.989437i	$-0.546307\pi$
$113$	−39354.0			−0.289930			−0.144965	+	0.989437i	$0.546307\pi$
$114$		0			0
$115$	10404.0	−	18020.3i	0.0733594	−	0.127062i
$116$		0			0
$117$	22639.5	+	39212.8i	0.152898	+	0.264827i
$118$		0			0
$119$	121800.	+	189867.i	0.788460	+	1.22909i
$120$		0			0
$121$	−141252.	−	244657.i	−0.877067	−	1.51912i
$122$		0			0
$123$	−21897.0	+	37926.7i	−0.130503	+	0.226039i
$124$		0			0
$125$	37284.0			0.213426
$126$		0			0
$127$	−317093.			−1.74453			−0.872263	−	0.489037i	$-0.837348\pi$
$127$	−317093.			−1.74453			−0.872263	+	0.489037i	$0.837348\pi$
$128$		0			0
$129$	51331.5	−	88908.8i	0.271587	−	0.470403i
$130$		0			0
$131$	−77415.0	−	134087.i	−0.394137	−	0.682665i	0.598854	−	0.800858i	$-0.295623\pi$
$131$	−77415.0	−	134087.i	−0.394137	−	0.682665i	−0.992991	+	0.118194i	$0.962290\pi$
$132$		0			0
$133$	−149832.	+	7013.94i	−0.734470	+	0.0343821i
$134$		0			0
$135$	−2187.00	−	3788.00i	−0.0103280	−	0.0178885i
$136$		0			0
$137$	−33666.0	+	58311.2i	−0.153246	+	0.265430i	−0.932419	−	0.361379i	$-0.882306\pi$
$137$	−33666.0	+	58311.2i	−0.153246	+	0.265430i	0.779173	+	0.626809i	$0.215639\pi$
$138$		0			0
$139$	365215.			1.60329			0.801644	−	0.597802i	$-0.203959\pi$
$139$	365215.			1.60329			0.801644	+	0.597802i	$0.203959\pi$
$140$		0			0
$141$	20790.0			0.0880657
$142$		0			0
$143$	186147.	−	322416.i	0.761230	−	1.31849i
$144$		0			0
$145$	10116.0	+	17521.4i	0.0399566	+	0.0692069i
$146$		0			0
$147$	−150602.	+	14130.9i	−0.574825	+	0.0539359i
$148$		0			0
$149$	84030.0	+	145544.i	0.310076	+	0.537068i	0.978379	−	0.206822i	$-0.0663120\pi$
$149$	84030.0	+	145544.i	0.310076	+	0.537068i	−0.668302	+	0.743890i	$0.732979\pi$
$150$		0			0
$151$	76768.0	−	132966.i	0.273992	−	0.474568i	−0.695888	−	0.718150i	$-0.744989\pi$
$151$	76768.0	−	132966.i	0.273992	−	0.474568i	0.969880	+	0.243582i	$0.0783225\pi$
$152$		0			0
$153$	−140940.			−0.486750
$154$		0			0
$155$	37758.0			0.126235
$156$		0			0
$157$	−101209.	+	175299.i	−0.327695	+	0.567585i	−0.982054	−	0.188599i	$-0.939605\pi$
$157$	−101209.	+	175299.i	−0.327695	+	0.567585i	0.654359	+	0.756184i	$0.272939\pi$
$158$		0			0
$159$	−127332.	−	220545.i	−0.399434	−	0.691840i
$160$		0			0
$161$	449106.	−	21023.6i	1.36548	−	0.0639209i
$162$		0			0
$163$	−89882.0	−	155680.i	−0.264974	−	0.458949i	0.702583	−	0.711602i	$-0.252030\pi$
$163$	−89882.0	−	155680.i	−0.264974	−	0.458949i	−0.967557	+	0.252653i	$0.918697\pi$
$164$		0			0
$165$	−17982.0	+	31145.7i	−0.0514195	+	0.0890612i
$166$		0			0
$167$	−217302.			−0.602938			−0.301469	−	0.953476i	$-0.597477\pi$
$167$	−217302.			−0.602938			−0.301469	+	0.953476i	$0.597477\pi$
$168$		0			0
$169$	−58812.0			−0.158398
$170$		0			0
$171$	46858.5	−	81161.3i	0.122546	−	0.212255i
$172$		0			0
$173$	36990.0	+	64068.6i	0.0939656	+	0.162753i	0.909176	−	0.416411i	$-0.136712\pi$
$173$	36990.0	+	64068.6i	0.0939656	+	0.162753i	−0.815211	+	0.579164i	$0.803379\pi$
$174$		0			0
$175$	216230.	+	337069.i	0.533729	+	0.832001i
$176$		0			0
$177$	−92448.0	−	160125.i	−0.221801	−	0.384171i
$178$		0			0
$179$	394683.	−	683611.i	0.920695	−	1.59469i	0.122353	−	0.992487i	$-0.460956\pi$
$179$	394683.	−	683611.i	0.920695	−	1.59469i	0.798342	−	0.602204i	$-0.205711\pi$
$180$		0			0
$181$	−477739.			−1.08391			−0.541956	−	0.840407i	$-0.682316\pi$
$181$	−477739.			−1.08391			−0.541956	+	0.840407i	$0.682316\pi$
$182$		0			0
$183$	41670.0			0.0919805
$184$		0			0
$185$	9393.00	−	16269.2i	0.0201779	−	0.0349491i
$186$		0			0
$187$	579420.	+	1.00358e6i	1.21168	+	2.09870i
$188$		0			0
$189$	43375.5	−	83967.2i	0.0883263	−	0.170984i
$190$		0			0
$191$	179487.	+	310881.i	0.356000	+	0.616609i	0.987289	−	0.158938i	$-0.0508071\pi$
$191$	179487.	+	310881.i	0.356000	+	0.616609i	−0.631289	+	0.775548i	$0.717474\pi$
$192$		0			0
$193$	90966.5	−	157559.i	0.175788	−	0.304473i	−0.764646	−	0.644451i	$-0.777086\pi$
$193$	90966.5	−	157559.i	0.175788	−	0.304473i	0.940434	+	0.339978i	$0.110419\pi$
$194$		0			0
$195$	−30186.0			−0.0568485
$196$		0			0
$197$	717924.			1.31799			0.658996	−	0.752146i	$-0.270981\pi$
$197$	717924.			1.31799			0.658996	+	0.752146i	$0.270981\pi$
$198$		0			0
$199$	101548.	−	175886.i	0.181777	−	0.314847i	−0.760709	−	0.649093i	$-0.775148\pi$
$199$	101548.	−	175886.i	0.181777	−	0.314847i	0.942486	+	0.334246i	$0.108482\pi$
$200$		0			0
$201$	84352.5	+	146103.i	0.147268	+	0.255075i
$202$		0			0
$203$	−200634.	+	388392.i	−0.341715	+	0.661500i
$204$		0			0
$205$	−14598.0	−	25284.5i	−0.0242610	−	0.0420213i
$206$		0			0
$207$	−140454.	+	243273.i	−0.227829	+	0.394611i
$208$		0			0
$209$	−770562.			−1.22023
$210$		0			0
$211$	−1.17098e6			−1.81069			−0.905343	−	0.424680i	$-0.860387\pi$
$211$	−1.17098e6			−1.81069			−0.905343	+	0.424680i	$0.860387\pi$
$212$		0			0
$213$	172017.	−	297942.i	0.259790	−	0.449969i
$214$		0			0
$215$	34221.0	+	59272.5i	0.0504890	+	0.0874495i
$216$		0			0
$217$	440510.	+	686687.i	0.635048	+	0.989942i
$218$		0			0
$219$	317651.	+	550187.i	0.447548	+	0.775175i
$220$		0			0
$221$	−486330.	+	842348.i	−0.669808	+	1.16014i
$222$		0			0
$223$	−1.24635e6			−1.67833			−0.839167	−	0.543873i	$-0.816957\pi$
$223$	−1.24635e6			−1.67833			−0.839167	+	0.543873i	$0.816957\pi$
$224$		0			0
$225$	−250209.			−0.329493
$226$		0			0
$227$	−459471.	+	795827.i	−0.591825	+	1.02507i	0.402161	+	0.915569i	$0.368259\pi$
$227$	−459471.	+	795827.i	−0.591825	+	1.02507i	−0.993987	+	0.109503i	$0.965074\pi$
$228$		0			0
$229$	−601874.	−	1.04248e6i	−0.758433	−	1.31364i	−0.943649	−	0.330947i	$-0.892632\pi$
$229$	−601874.	−	1.04248e6i	−0.758433	−	1.31364i	0.185216	−	0.982698i	$-0.440701\pi$
$230$		0			0
$231$	−776223.	+	36336.7i	−0.957098	+	0.0448039i
$232$		0			0
$233$	459531.	+	795931.i	0.554530	+	0.960474i	0.997940	+	0.0641551i	$0.0204353\pi$
$233$	459531.	+	795931.i	0.554530	+	0.960474i	−0.443410	+	0.896319i	$0.646231\pi$
$234$		0			0
$235$	−6930.00	+	12003.1i	−0.00818585	+	0.0141783i
$236$		0			0
$237$	−560637.			−0.648352
$238$		0			0
$239$	625338.			0.708142			0.354071	−	0.935219i	$-0.384797\pi$
$239$	625338.			0.708142			0.354071	+	0.935219i	$0.384797\pi$
$240$		0			0
$241$	−626911.	+	1.08584e6i	−0.695286	+	1.20427i	0.274799	+	0.961502i	$0.411389\pi$
$241$	−626911.	+	1.08584e6i	−0.695286	+	1.20427i	−0.970084	+	0.242768i	$0.921945\pi$
$242$		0			0
$243$	29524.5	+	51137.9i	0.0320750	+	0.0555556i
$244$		0			0
$245$	42042.0	−	91660.1i	0.0447474	−	0.0975585i
$246$		0			0
$247$	−323382.	−	560113.i	−0.337266	−	0.584162i
$248$		0			0
$249$	−359181.	+	622120.i	−0.367126	+	0.635881i
$250$		0			0
$251$	1.51333e6			1.51618			0.758089	−	0.652152i	$-0.226133\pi$
$251$	1.51333e6			1.51618			0.758089	+	0.652152i	$0.226133\pi$
$252$		0			0
$253$	2.30969e6			2.26857
$254$		0			0
$255$	46980.0	−	81371.7i	0.0452442	−	0.0783652i
$256$		0			0
$257$	777465.	+	1.34661e6i	0.734257	+	1.27177i	0.955049	+	0.296449i	$0.0958026\pi$
$257$	777465.	+	1.34661e6i	0.734257	+	1.27177i	−0.220792	+	0.975321i	$0.570864\pi$
$258$		0			0
$259$	405464.	−	18980.7i	0.375581	−	0.0175818i
$260$		0			0
$261$	−136566.	−	236539.i	−0.124091	−	0.214932i
$262$		0			0
$263$	−557658.	+	965892.i	−0.497140	+	0.861071i	−0.999995	−	0.00329949i	$-0.998950\pi$
$263$	−557658.	+	965892.i	−0.497140	+	0.861071i	0.502855	+	0.864371i	$0.332283\pi$
$264$		0			0
$265$	169776.			0.148512
$266$		0			0
$267$	163080.			0.139998
$268$		0			0
$269$	−17835.0	+	30891.1i	−0.0150277	+	0.0260287i	−0.873441	−	0.486929i	$-0.838117\pi$
$269$	−17835.0	+	30891.1i	−0.0150277	+	0.0260287i	0.858414	+	0.512958i	$0.171450\pi$
$270$		0			0
$271$	−146384.	−	253545.i	−0.121079	−	0.209716i	0.799114	−	0.601179i	$-0.205302\pi$
$271$	−146384.	−	253545.i	−0.121079	−	0.209716i	−0.920194	+	0.391464i	$0.871969\pi$
$272$		0			0
$273$	−352170.	−	548979.i	−0.285987	−	0.445809i
$274$		0			0
$275$	1.02864e6	+	1.78165e6i	0.820220	+	1.42066i
$276$		0			0
$277$	−431607.	+	747564.i	−0.337978	+	0.585395i	−0.984052	−	0.177879i	$-0.943076\pi$
$277$	−431607.	+	747564.i	−0.337978	+	0.585395i	0.646074	+	0.763275i	$0.276410\pi$
$278$		0			0
$279$	−509733.			−0.392042
$280$		0			0
$281$	1.47110e6			1.11142			0.555709	−	0.831377i	$-0.312447\pi$
$281$	1.47110e6			1.11142			0.555709	+	0.831377i	$0.312447\pi$
$282$		0			0
$283$	344420.	−	596554.i	0.255637	−	0.442775i	−0.709432	−	0.704774i	$-0.751048\pi$
$283$	344420.	−	596554.i	0.255637	−	0.442775i	0.965068	+	0.261999i	$0.0843816\pi$
$284$		0			0
$285$	31239.0	+	54107.5i	0.0227816	+	0.0394590i
$286$		0			0
$287$	289527.	−	560473.i	0.207484	−	0.401652i
$288$		0			0
$289$	−803872.	−	1.39235e6i	−0.566164	−	0.980624i
$290$		0			0
$291$	561393.	−	972361.i	0.388628	−	0.673124i
$292$		0			0
$293$	722832.			0.491890			0.245945	−	0.969284i	$-0.420902\pi$
$293$	722832.			0.491890			0.245945	+	0.969284i	$0.420902\pi$
$294$		0			0
$295$	123264.			0.0824672
$296$		0			0
$297$	242757.	−	420467.i	0.159691	−	0.276593i
$298$		0			0
$299$	969306.	+	1.67889e6i	0.627022	+	1.08603i
$300$		0			0
$301$	−678716.	+	1.31387e6i	−0.431790	+	0.835868i
$302$		0			0
$303$	−420255.	−	727903.i	−0.262970	−	0.455478i
$304$		0			0
$305$	−13890.0	+	24058.2i	−0.00854973	+	0.0148086i
$306$		0			0
$307$	20125.0			0.0121868			0.00609340	−	0.999981i	$-0.498060\pi$
$307$	20125.0			0.0121868			0.00609340	+	0.999981i	$0.498060\pi$
$308$		0			0
$309$	−1.50958e6			−0.899414
$310$		0			0
$311$	−871779.	+	1.50997e6i	−0.511099	+	0.885250i	0.488818	+	0.872386i	$0.337428\pi$
$311$	−871779.	+	1.50997e6i	−0.511099	+	0.885250i	−0.999917	+	0.0128643i	$0.995905\pi$
$312$		0			0
$313$	−904272.	−	1.56624e6i	−0.521721	−	0.903647i	−0.999681	−	0.0252651i	$-0.991957\pi$
$313$	−904272.	−	1.56624e6i	−0.521721	−	0.903647i	0.477960	−	0.878382i	$-0.341376\pi$
$314$		0			0
$315$	34020.0	+	53031.9i	0.0193178	+	0.0301135i
$316$		0			0
$317$	−511776.	−	886422.i	−0.286043	−	0.495442i	0.686818	−	0.726829i	$-0.259007\pi$
$317$	−511776.	−	886422.i	−0.286043	−	0.495442i	−0.972862	+	0.231388i	$0.925673\pi$
$318$		0			0
$319$	−1.12288e6	+	1.94488e6i	−0.617810	+	1.07008i
$320$		0			0
$321$	622620.			0.337257
$322$		0			0
$323$	2.01318e6			1.07368
$324$		0			0
$325$	−863376.	+	1.49541e6i	−0.453410	+	0.785330i
$326$		0			0
$327$	−988016.	−	1.71129e6i	−0.510969	−	0.885024i
$328$		0			0
$329$	−299145.	+	14003.6i	−0.152367	+	0.00713265i
$330$		0			0
$331$	−503766.	−	872549.i	−0.252731	−	0.437744i	0.711545	−	0.702640i	$-0.247996\pi$
$331$	−503766.	−	872549.i	−0.252731	−	0.437744i	−0.964277	+	0.264896i	$0.914662\pi$
$332$		0			0
$333$	−126806.	+	219634.i	−0.0626654	+	0.108540i
$334$		0			0
$335$	−112470.			−0.0547551
$336$		0			0
$337$	−1.56571e6			−0.750993			−0.375496	−	0.926824i	$-0.622528\pi$
$337$	−1.56571e6			−0.750993			−0.375496	+	0.926824i	$0.622528\pi$
$338$		0			0
$339$	−177093.	+	306734.i	−0.0836955	+	0.144965i
$340$		0			0
$341$	2.09557e6	+	3.62963e6i	0.975924	+	1.69035i
$342$		0			0
$343$	2.15747e6	−	304770.i	0.990169	−	0.139874i
$344$		0			0
$345$	−93636.0	−	162182.i	−0.0423541	−	0.0733594i
$346$		0			0
$347$	378642.	−	655827.i	0.168813	−	0.292392i	−0.769190	−	0.639020i	$-0.779340\pi$
$347$	378642.	−	655827.i	0.168813	−	0.292392i	0.938003	+	0.346628i	$0.112673\pi$
$348$		0			0
$349$	−455638.			−0.200243			−0.100121	−	0.994975i	$-0.531923\pi$
$349$	−455638.			−0.200243			−0.100121	+	0.994975i	$0.531923\pi$
$350$		0			0
$351$	407511.			0.176552
$352$		0			0
$353$	1.81569e6	−	3.14487e6i	0.775543	−	1.34328i	−0.158946	−	0.987287i	$-0.550810\pi$
$353$	1.81569e6	−	3.14487e6i	0.775543	−	1.34328i	0.934489	−	0.355992i	$-0.115857\pi$
$354$		0			0
$355$	114678.	+	198628.i	0.0482958	+	0.0836508i
$356$		0			0
$357$	2.02797e6	−	94933.7i	0.842153	−	0.0394230i
$358$		0			0
$359$	−2.01242e6	−	3.48561e6i	−0.824104	−	1.42739i	−0.902602	−	0.430476i	$-0.858346\pi$
$359$	−2.01242e6	−	3.48561e6i	−0.824104	−	1.42739i	0.0784980	−	0.996914i	$-0.474988\pi$
$360$		0			0
$361$	568725.	−	985061.i	0.229686	−	0.397828i
$362$		0			0
$363$	−2.54254e6			−1.01275
$364$		0			0
$365$	−423534.			−0.166401
$366$		0			0
$367$	1.28894e6	−	2.23251e6i	0.499536	−	0.865222i	−0.500464	−	0.865757i	$-0.666837\pi$
$367$	1.28894e6	−	2.23251e6i	0.499536	−	0.865222i	1.00000		0.000535822i	$0.000170558\pi$
$368$		0			0
$369$	197073.	+	341340.i	0.0753462	+	0.130503i
$370$		0			0
$371$	1.98072e6	+	3.08764e6i	0.747116	+	1.16464i
$372$		0			0
$373$	1.26566e6	+	2.19220e6i	0.471028	+	0.815844i	0.999451	−	0.0331372i	$-0.0105498\pi$
$373$	1.26566e6	+	2.19220e6i	0.471028	+	0.815844i	−0.528423	+	0.848981i	$0.677216\pi$
$374$		0			0
$375$	167778.	−	290600.i	0.0616108	−	0.106713i
$376$		0			0
$377$	−1.88495e6			−0.683040
$378$		0			0
$379$	3.06677e6			1.09669			0.548344	−	0.836253i	$-0.315258\pi$
$379$	3.06677e6			1.09669			0.548344	+	0.836253i	$0.315258\pi$
$380$		0			0
$381$	−1.42692e6	+	2.47150e6i	−0.503601	+	0.872263i
$382$		0			0
$383$	−1.96260e6	−	3.39932e6i	−0.683652	−	1.18412i	−0.973859	−	0.227155i	$-0.927057\pi$
$383$	−1.96260e6	−	3.39932e6i	−0.683652	−	1.18412i	0.290207	−	0.956964i	$-0.406276\pi$
$384$		0			0
$385$	237762.	−	460265.i	0.0817505	−	0.158254i
$386$		0			0
$387$	−461984.	−	800179.i	−0.156801	−	0.271587i
$388$		0			0
$389$	2.01334e6	−	3.48722e6i	0.674597	−	1.16844i	−0.301990	−	0.953311i	$-0.597651\pi$
$389$	2.01334e6	−	3.48722e6i	0.674597	−	1.16844i	0.976587	−	0.215125i	$-0.0690158\pi$
$390$		0			0
$391$	−6.03432e6			−1.99612
$392$		0			0
$393$	−1.39347e6			−0.455110
$394$		0			0
$395$	186879.	−	323684.i	0.0602654	−	0.104383i
$396$		0			0
$397$	−2.28720e6	−	3.96155e6i	−0.728329	−	1.26150i	−0.957589	−	0.288138i	$-0.906964\pi$
$397$	−2.28720e6	−	3.96155e6i	−0.728329	−	1.26150i	0.229260	−	0.973365i	$-0.426370\pi$
$398$		0			0
$399$	−619574.	+	1.19938e6i	−0.194832	+	0.377160i
$400$		0			0
$401$	1.13472e6	+	1.96539e6i	0.352393	+	0.610363i	0.986668	−	0.162744i	$-0.0520346\pi$
$401$	1.13472e6	+	1.96539e6i	0.352393	+	0.610363i	−0.634275	+	0.773108i	$0.718701\pi$
$402$		0			0
$403$	−1.75889e6	+	3.04649e6i	−0.539482	+	0.934410i
$404$		0			0
$405$	−39366.0			−0.0119257
$406$		0			0
$407$	2.08525e6			0.623981
$408$		0			0
$409$	2.02298e6	−	3.50391e6i	0.597976	−	1.03572i	−0.395144	−	0.918619i	$-0.629305\pi$
$409$	2.02298e6	−	3.50391e6i	0.597976	−	1.03572i	0.993120	−	0.117105i	$-0.0373615\pi$
$410$		0			0
$411$	302994.	+	524801.i	0.0884768	+	0.153246i
$412$		0			0
$413$	1.43808e6	+	2.24174e6i	0.414866	+	0.646712i
$414$		0			0
$415$	−239454.	−	414746.i	−0.0682499	−	0.118212i
$416$		0			0
$417$	1.64347e6	−	2.84657e6i	0.462829	−	0.801644i
$418$		0			0
$419$	3.91281e6			1.08881			0.544407	−	0.838821i	$-0.316755\pi$
$419$	3.91281e6			1.08881			0.544407	+	0.838821i	$0.316755\pi$
$420$		0			0
$421$	−2.78086e6			−0.764671			−0.382335	−	0.924024i	$-0.624880\pi$
$421$	−2.78086e6			−0.764671			−0.382335	+	0.924024i	$0.624880\pi$
$422$		0			0
$423$	93555.0	−	162042.i	0.0254224	−	0.0440328i
$424$		0			0
$425$	−2.68743e6	−	4.65477e6i	−0.721714	−	1.25004i
$426$		0			0
$427$	−599585.	+	28067.9i	−0.159141	+	0.00744972i
$428$		0			0
$429$	−1.67532e6	−	2.90174e6i	−0.439496	−	0.761230i
$430$		0			0
$431$	−2.19104e6	+	3.79498e6i	−0.568141	+	0.984049i	0.428609	+	0.903490i	$0.359004\pi$
$431$	−2.19104e6	+	3.79498e6i	−0.568141	+	0.984049i	−0.996750	+	0.0805589i	$0.974329\pi$
$432$		0			0
$433$	1.24946e6			0.320261			0.160130	−	0.987096i	$-0.448809\pi$
$433$	1.24946e6			0.320261			0.160130	+	0.987096i	$0.448809\pi$
$434$		0			0
$435$	182088.			0.0461379
$436$		0			0
$437$	2.00624e6	−	3.47491e6i	0.502550	−	0.870441i
$438$		0			0
$439$	−3.37210e6	−	5.84066e6i	−0.835102	−	1.44644i	−0.893947	−	0.448172i	$-0.852075\pi$
$439$	−3.37210e6	−	5.84066e6i	−0.835102	−	1.44644i	0.0588449	−	0.998267i	$-0.481258\pi$
$440$		0			0
$441$	−567567.	+	1.23741e6i	−0.138970	+	0.302983i
$442$		0			0
$443$	239448.	+	414736.i	0.0579698	+	0.100407i	0.893554	−	0.448956i	$-0.148204\pi$
$443$	239448.	+	414736.i	0.0579698	+	0.100407i	−0.835584	+	0.549362i	$0.814871\pi$
$444$		0			0
$445$	−54360.0	+	94154.3i	−0.0130131	+	0.0225393i
$446$		0			0
$447$	1.51254e6			0.358045
$448$		0			0
$449$	724506.			0.169600			0.0848001	−	0.996398i	$-0.472975\pi$
$449$	724506.			0.169600			0.0848001	+	0.996398i	$0.472975\pi$
$450$		0			0
$451$	1.62038e6	−	2.80658e6i	0.375124	−	0.649734i
$452$		0			0
$453$	−690912.	−	1.19669e6i	−0.158189	−	0.273992i
$454$		0			0
$455$	434343.	−	20332.5i	0.0983568	−	0.00460430i
$456$		0			0
$457$	1.16978e6	+	2.02612e6i	0.262008	+	0.453811i	0.966775	−	0.255627i	$-0.0822820\pi$
$457$	1.16978e6	+	2.02612e6i	0.262008	+	0.453811i	−0.704768	+	0.709438i	$0.748949\pi$
$458$		0			0
$459$	−634230.	+	1.09852e6i	−0.140513	+	0.243375i
$460$		0			0
$461$	2.98247e6			0.653617			0.326809	−	0.945091i	$-0.394027\pi$
$461$	2.98247e6			0.653617			0.326809	+	0.945091i	$0.394027\pi$
$462$		0			0
$463$	−4.28423e6			−0.928795			−0.464398	−	0.885627i	$-0.653729\pi$
$463$	−4.28423e6			−0.928795			−0.464398	+	0.885627i	$0.653729\pi$
$464$		0			0
$465$	169911.	−	294294.i	0.0364409	−	0.0631175i
$466$		0			0
$467$	−2.87018e6	−	4.97129e6i	−0.608998	−	1.05482i	−0.991406	−	0.130822i	$-0.958238\pi$
$467$	−2.87018e6	−	4.97129e6i	−0.608998	−	1.05482i	0.382407	−	0.923994i	$-0.375095\pi$
$468$		0			0
$469$	−1.31215e6	−	2.04544e6i	−0.275455	−	0.429393i
$470$		0			0
$471$	910881.	+	1.57769e6i	0.189195	+	0.327695i
$472$		0			0
$473$	−3.79853e6	+	6.57925e6i	−0.780662	+	1.35215i
$474$		0			0
$475$	3.57397e6			0.726804
$476$		0			0
$477$	−2.29198e6			−0.461226
$478$		0			0
$479$	1.32526e6	−	2.29541e6i	0.263913	−	0.457111i	−0.703365	−	0.710829i	$-0.748320\pi$
$479$	1.32526e6	−	2.29541e6i	0.263913	−	0.457111i	0.967278	+	0.253718i	$0.0816534\pi$
$480$		0			0
$481$	875114.	+	1.51574e6i	0.172465	+	0.298719i
$482$		0			0
$483$	1.85711e6	−	3.59504e6i	0.362219	−	0.701191i
$484$		0			0
$485$	374262.	+	648241.i	0.0722473	+	0.125136i
$486$		0			0
$487$	1.40277e6	−	2.42967e6i	0.268018	−	0.464221i	−0.700332	−	0.713817i	$-0.746965\pi$
$487$	1.40277e6	−	2.42967e6i	0.268018	−	0.464221i	0.968350	+	0.249597i	$0.0802980\pi$
$488$		0			0
$489$	−1.61788e6			−0.305966
$490$		0			0
$491$	4.68450e6			0.876919			0.438460	−	0.898751i	$-0.355524\pi$
$491$	4.68450e6			0.876919			0.438460	+	0.898751i	$0.355524\pi$
$492$		0			0
$493$	2.93364e6	−	5.08121e6i	0.543613	−	0.941565i
$494$		0			0
$495$	161838.	+	280312.i	0.0296871	+	0.0514195i
$496$		0			0
$497$	−2.27445e6	+	4.40292e6i	−0.413033	+	0.799558i
$498$		0			0
$499$	737876.	+	1.27804e6i	0.132658	+	0.229770i	0.924700	−	0.380696i	$-0.124316\pi$
$499$	737876.	+	1.27804e6i	0.132658	+	0.229770i	−0.792043	+	0.610466i	$0.790982\pi$
$500$		0			0
$501$	−977859.	+	1.69370e6i	−0.174053	+	0.301469i
$502$		0			0
$503$	−63606.0			−0.0112093			−0.00560465	−	0.999984i	$-0.501784\pi$
$503$	−63606.0			−0.0112093			−0.00560465	+	0.999984i	$0.501784\pi$
$504$		0			0
$505$	560340.			0.0977740
$506$		0			0
$507$	−264654.	+	458394.i	−0.0457255	+	0.0791989i
$508$		0			0
$509$	−3.10578e6	−	5.37937e6i	−0.531345	−	0.920317i	−0.999331	−	0.0365806i	$-0.988353\pi$
$509$	−3.10578e6	−	5.37937e6i	−0.531345	−	0.920317i	0.467986	−	0.883736i	$-0.344980\pi$
$510$		0			0
$511$	−4.94123e6	−	7.70262e6i	−0.837111	−	1.30493i
$512$		0			0
$513$	−421726.	−	730452.i	−0.0707518	−	0.122546i
$514$		0			0
$515$	503193.	−	871556.i	0.0836020	−	0.144803i
$516$		0			0
$517$	−1.53846e6			−0.253139
$518$		0			0
$519$	665820.			0.108502
$520$		0			0
$521$	−706026.	+	1.22287e6i	−0.113953	+	0.197373i	−0.917361	−	0.398057i	$-0.869685\pi$
$521$	−706026.	+	1.22287e6i	−0.113953	+	0.197373i	0.803408	+	0.595429i	$0.203018\pi$
$522$		0			0
$523$	2.61467e6	+	4.52875e6i	0.417987	+	0.723976i	0.995737	−	0.0922386i	$-0.0294023\pi$
$523$	2.61467e6	+	4.52875e6i	0.417987	+	0.723976i	−0.577749	+	0.816214i	$0.696069\pi$
$524$		0			0
$525$	3.60023e6	−	168535.i	0.570075	−	0.0266865i
$526$		0			0
$527$	−5.47491e6	−	9.48282e6i	−0.858718	−	1.48734i
$528$		0			0
$529$	−2.79534e6	+	4.84167e6i	−0.434306	+	0.752240i
$530$		0			0
$531$	−1.66406e6			−0.256114
$532$		0			0
$533$	2.72009e6			0.414730
$534$		0			0
$535$	−207540.	+	359470.i	−0.0313485	+	0.0542973i
$536$		0			0
$537$	−3.55215e6	−	6.15250e6i	−0.531564	−	0.920695i
$538$		0			0
$539$	1.11445e7	−	1.04569e6i	1.65230	−	0.155035i
$540$		0			0
$541$	−2.20686e6	−	3.82240e6i	−0.324177	−	0.561491i	0.657169	−	0.753744i	$-0.271754\pi$
$541$	−2.20686e6	−	3.82240e6i	−0.324177	−	0.561491i	−0.981345	+	0.192253i	$0.938421\pi$
$542$		0			0
$543$	−2.14983e6	+	3.72361e6i	−0.312899	+	0.541956i
$544$		0			0
$545$	1.31735e6			0.189981
$546$		0			0
$547$	1.19038e7			1.70105			0.850523	−	0.525938i	$-0.176286\pi$
$547$	1.19038e7			1.70105			0.850523	+	0.525938i	$0.176286\pi$
$548$		0			0
$549$	187515.	−	324786.i	0.0265525	−	0.0459903i
$550$		0			0
$551$	1.95070e6	+	3.37871e6i	0.273723	+	0.474103i
$552$		0			0
$553$	8.06694e6	−	377631.i	1.12175	−	0.0525116i
$554$		0			0
$555$	−84537.0	−	146422.i	−0.0116497	−	0.0201779i
$556$		0			0
$557$	−6.45665e6	+	1.11832e7i	−0.881798	+	1.52732i	−0.0324587	+	0.999473i	$0.510334\pi$
$557$	−6.45665e6	+	1.11832e7i	−0.881798	+	1.52732i	−0.849340	+	0.527847i	$0.823000\pi$
$558$		0			0
$559$	−6.37651e6			−0.863085
$560$		0			0
$561$	1.04296e7			1.39913
$562$		0			0
$563$	−5.68492e6	+	9.84657e6i	−0.755881	+	1.30922i	0.189055	+	0.981967i	$0.439458\pi$
$563$	−5.68492e6	+	9.84657e6i	−0.755881	+	1.30922i	−0.944935	+	0.327257i	$0.893876\pi$
$564$		0			0
$565$	−118062.	−	204489.i	−0.0155593	−	0.0269494i
$566$		0			0
$567$	−459270.	−	715931.i	−0.0599944	−	0.0935220i
$568$		0			0
$569$	2.84898e6	+	4.93457e6i	0.368900	+	0.638953i	0.989394	−	0.145258i	$-0.0464013\pi$
$569$	2.84898e6	+	4.93457e6i	0.368900	+	0.638953i	−0.620494	+	0.784211i	$0.713068\pi$
$570$		0			0
$571$	−3.52110e6	+	6.09873e6i	−0.451948	+	0.782797i	−0.998507	−	0.0546236i	$-0.982604\pi$
$571$	−3.52110e6	+	6.09873e6i	−0.451948	+	0.782797i	0.546559	+	0.837421i	$0.315937\pi$
$572$		0			0
$573$	3.23077e6			0.411073
$574$		0			0
$575$	−1.07127e7			−1.35122
$576$		0			0
$577$	1.29098e6	−	2.23605e6i	0.161429	−	0.279603i	−0.773952	−	0.633244i	$-0.781723\pi$
$577$	1.29098e6	−	2.23605e6i	0.161429	−	0.279603i	0.935381	+	0.353641i	$0.115056\pi$
$578$		0			0
$579$	−818698.	−	1.41803e6i	−0.101491	−	0.175788i
$580$		0			0
$581$	4.74917e6	−	9.19355e6i	0.583684	−	1.12991i
$582$		0			0
$583$	9.42257e6	+	1.63204e7i	1.14815	+	1.98865i
$584$		0			0
$585$	−135837.	+	235277.i	−0.0164107	+	0.0284243i
$586$		0			0
$587$	−4.69459e6			−0.562345			−0.281172	−	0.959657i	$-0.590723\pi$
$587$	−4.69459e6			−0.562345			−0.281172	+	0.959657i	$0.590723\pi$
$588$		0			0
$589$	7.28100e6			0.864774
$590$		0			0
$591$	3.23066e6	−	5.59566e6i	0.380472	−	0.658996i
$592$		0			0
$593$	−6.71175e6	−	1.16251e7i	−0.783789	−	1.35756i	−0.929720	−	0.368268i	$-0.879951\pi$
$593$	−6.71175e6	−	1.16251e7i	−0.783789	−	1.35756i	0.145931	−	0.989295i	$-0.453382\pi$
$594$		0			0
$595$	−621180.	+	1.20249e6i	−0.0719325	+	0.139248i
$596$		0			0
$597$	−913932.	−	1.58298e6i	−0.104949	−	0.181777i
$598$		0			0
$599$	2.52301e6	−	4.36997e6i	0.287310	−	0.497636i	−0.685856	−	0.727737i	$-0.740572\pi$
$599$	2.52301e6	−	4.36997e6i	0.287310	−	0.497636i	0.973167	+	0.230101i	$0.0739056\pi$
$600$		0			0
$601$	−1.06391e7			−1.20148			−0.600742	−	0.799443i	$-0.705128\pi$
$601$	−1.06391e7			−1.20148			−0.600742	+	0.799443i	$0.705128\pi$
$602$		0			0
$603$	1.51834e6			0.170050
$604$		0			0
$605$	847515.	−	1.46794e6i	0.0941367	−	0.163050i
$606$		0			0
$607$	708041.	+	1.22636e6i	0.0779986	+	0.135098i	0.902386	−	0.430928i	$-0.141814\pi$
$607$	708041.	+	1.22636e6i	0.0779986	+	0.135098i	−0.824388	+	0.566026i	$0.808480\pi$
$608$		0			0
$609$	2.12436e6	+	3.31155e6i	0.232105	+	0.361816i
$610$		0			0
$611$	−645645.	−	1.11829e6i	−0.0699666	−	0.121186i
$612$		0			0
$613$	−4.73152e6	+	8.19523e6i	−0.508568	+	0.880866i	0.491383	+	0.870944i	$0.336492\pi$
$613$	−4.73152e6	+	8.19523e6i	−0.508568	+	0.880866i	−0.999951	+	0.00992215i	$0.996842\pi$
$614$		0			0
$615$	−262764.			−0.0280142
$616$		0			0
$617$	1.29388e7			1.36830			0.684148	−	0.729343i	$-0.260174\pi$
$617$	1.29388e7			1.36830			0.684148	+	0.729343i	$0.260174\pi$
$618$		0			0
$619$	−1.90188e6	+	3.29415e6i	−0.199506	+	0.345555i	−0.948368	−	0.317171i	$-0.897267\pi$
$619$	−1.90188e6	+	3.29415e6i	−0.199506	+	0.345555i	0.748862	+	0.662726i	$0.230600\pi$
$620$		0			0
$621$	1.26409e6	+	2.18946e6i	0.131537	+	0.227829i
$622$		0			0
$623$	−2.34654e6	+	109847.i	−0.242219	+	0.0113388i
$624$		0			0
$625$	−4.71471e6	−	8.16612e6i	−0.482786	−	0.836210i
$626$		0			0
$627$	−3.46753e6	+	6.00594e6i	−0.352250	+	0.610115i
$628$		0			0
$629$	−5.44794e6			−0.549042
$630$		0			0
$631$	9.17498e6			0.917343			0.458671	−	0.888606i	$-0.348326\pi$
$631$	9.17498e6			0.917343			0.458671	+	0.888606i	$0.348326\pi$
$632$		0			0
$633$	−5.26941e6	+	9.12689e6i	−0.522700	+	0.905343i
$634$		0			0
$635$	−951279.	−	1.64766e6i	−0.0936211	−	0.162156i
$636$		0			0
$637$	5.43711e6	+	7.66198e6i	0.530909	+	0.748157i
$638$		0			0
$639$	−1.54815e6	−	2.68148e6i	−0.149990	−	0.259790i
$640$		0			0
$641$	−5.12269e6	+	8.87275e6i	−0.492439	+	0.852930i	−0.999962	−	0.00870851i	$-0.997228\pi$
$641$	−5.12269e6	+	8.87275e6i	−0.492439	+	0.852930i	0.507523	+	0.861638i	$0.330561\pi$
$642$		0			0
$643$	5.72346e6			0.545922			0.272961	−	0.962025i	$-0.411997\pi$
$643$	5.72346e6			0.545922			0.272961	+	0.962025i	$0.411997\pi$
$644$		0			0
$645$	615978.			0.0582996
$646$		0			0
$647$	−4.99397e6	+	8.64981e6i	−0.469013	+	0.812355i	−0.999373	−	0.0354179i	$-0.988724\pi$
$647$	−4.99397e6	+	8.64981e6i	−0.469013	+	0.812355i	0.530359	+	0.847773i	$0.322057\pi$
$648$		0			0
$649$	6.84115e6	+	1.18492e7i	0.637555	+	1.10428i
$650$		0			0
$651$	7.33449e6	−	343344.i	0.678293	−	0.0317524i
$652$		0			0
$653$	599439.	+	1.03826e6i	0.0550126	+	0.0952846i	0.892220	−	0.451601i	$-0.149147\pi$
$653$	599439.	+	1.03826e6i	0.0550126	+	0.0952846i	−0.837208	+	0.546885i	$0.815813\pi$
$654$		0			0
$655$	464490.	−	804520.i	0.0423032	−	0.0732713i
$656$		0			0
$657$	5.71771e6			0.516783
$658$		0			0
$659$	−1.18065e7			−1.05903			−0.529516	−	0.848300i	$-0.677626\pi$
$659$	−1.18065e7			−1.05903			−0.529516	+	0.848300i	$0.677626\pi$
$660$		0			0
$661$	−2.36020e6	+	4.08798e6i	−0.210109	+	0.363919i	−0.951748	−	0.306879i	$-0.900715\pi$
$661$	−2.36020e6	+	4.08798e6i	−0.210109	+	0.363919i	0.741640	+	0.670799i	$0.234049\pi$
$662$		0			0
$663$	4.37697e6	+	7.58113e6i	0.386714	+	0.669808i
$664$		0			0
$665$	−485940.	−	757505.i	−0.0426117	−	0.0664250i
$666$		0			0
$667$	−5.84705e6	−	1.01274e7i	−0.508888	−	0.881420i
$668$		0			0
$669$	−5.60858e6	+	9.71435e6i	−0.484493	+	0.839167i
$670$		0			0
$671$	−3.08358e6			−0.264392
$672$		0			0
$673$	−8.70826e6			−0.741129			−0.370564	−	0.928807i	$-0.620836\pi$
$673$	−8.70826e6			−0.741129			−0.370564	+	0.928807i	$0.620836\pi$
$674$		0			0
$675$	−1.12594e6	+	1.95019e6i	−0.0951165	+	0.164747i
$676$		0			0
$677$	−2.55553e6	−	4.42630e6i	−0.214293	−	0.371167i	0.738760	−	0.673968i	$-0.235412\pi$
$677$	−2.55553e6	−	4.42630e6i	−0.214293	−	0.371167i	−0.953054	+	0.302801i	$0.902078\pi$
$678$		0			0
$679$	−7.42286e6	+	1.43693e7i	−0.617870	+	1.19609i
$680$		0			0
$681$	4.13524e6	+	7.16244e6i	0.341690	+	0.591825i
$682$		0			0
$683$	−8.85989e6	+	1.53458e7i	−0.726736	+	1.25874i	0.231520	+	0.972830i	$0.425630\pi$
$683$	−8.85989e6	+	1.53458e7i	−0.726736	+	1.25874i	−0.958256	+	0.285913i	$0.907703\pi$
$684$		0			0
$685$	−403992.			−0.0328962
$686$		0			0
$687$	−1.08337e7			−0.875763
$688$		0			0
$689$	−7.90873e6	+	1.36983e7i	−0.634686	+	1.09931i
$690$		0			0
$691$	−1.12997e7	−	1.95716e7i	−0.900265	−	1.55931i	−0.827149	−	0.561982i	$-0.810039\pi$
$691$	−1.12997e7	−	1.95716e7i	−0.900265	−	1.55931i	−0.0731160	−	0.997323i	$-0.523294\pi$
$692$		0			0
$693$	−3.20979e6	+	6.21357e6i	−0.253889	+	0.491483i
$694$		0			0
$695$	1.09564e6	+	1.89771e6i	0.0860415	+	0.149028i
$696$		0			0
$697$	−4.23342e6	+	7.33250e6i	−0.330073	+	0.571702i
$698$		0			0
$699$	8.27156e6			0.640316
$700$		0			0
$701$	−818148.			−0.0628835			−0.0314418	−	0.999506i	$-0.510010\pi$
$701$	−818148.			−0.0628835			−0.0314418	+	0.999506i	$0.510010\pi$
$702$		0			0
$703$	1.81128e6	−	3.13724e6i	0.138229	−	0.239419i
$704$		0			0
$705$	62370.0	+	108028.i	0.00472610	+	0.00818585i
$706$		0			0
$707$	6.53730e6	+	1.01906e7i	0.491869	+	0.766749i
$708$		0			0
$709$	2.54591e6	+	4.40965e6i	0.190208	+	0.329449i	0.945319	−	0.326147i	$-0.105751\pi$
$709$	2.54591e6	+	4.40965e6i	0.190208	+	0.329449i	−0.755111	+	0.655597i	$0.772417\pi$
$710$		0			0
$711$	−2.52287e6	+	4.36973e6i	−0.187163	+	0.324176i
$712$		0			0
$713$	−2.18241e7			−1.60773
$714$		0			0
$715$	2.23376e6			0.163408
$716$		0			0
$717$	2.81402e6	−	4.87403e6i	0.204423	−	0.354071i
$718$		0			0
$719$	−240429.	−	416435.i	−0.0173446	−	0.0300418i	0.857223	−	0.514946i	$-0.172188\pi$
$719$	−240429.	−	416435.i	−0.0173446	−	0.0300418i	−0.874567	+	0.484904i	$0.838855\pi$
$720$		0			0
$721$	2.17212e7	−	1.01682e6i	1.55613	−	0.0728457i
$722$		0			0
$723$	5.64220e6	+	9.77258e6i	0.401423	+	0.695286i
$724$		0			0
$725$	5.20805e6	−	9.02061e6i	0.367985	−	0.637369i
$726$		0			0
$727$	1.40783e7			0.987905			0.493952	−	0.869489i	$-0.335552\pi$
$727$	1.40783e7			0.987905			0.493952	+	0.869489i	$0.335552\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	9.92409e6	−	1.71890e7i	0.686906	−	1.18976i
$732$		0			0
$733$	−1.01966e6	−	1.76610e6i	−0.0700964	−	0.121411i	0.828847	−	0.559475i	$-0.188997\pi$
$733$	−1.01966e6	−	1.76610e6i	−0.0700964	−	0.121411i	−0.898943	+	0.438065i	$0.855664\pi$
$734$		0			0
$735$	−525231.	−	740156.i	−0.0358618	−	0.0505364i
$736$		0			0
$737$	−6.24209e6	−	1.08116e7i	−0.423312	−	0.733199i
$738$		0			0
$739$	−8.24785e6	+	1.42857e7i	−0.555558	+	0.962255i	0.442302	+	0.896866i	$0.354162\pi$
$739$	−8.24785e6	+	1.42857e7i	−0.555558	+	0.962255i	−0.997860	+	0.0653888i	$0.979171\pi$
$740$		0			0
$741$	−5.82087e6			−0.389441
$742$		0			0
$743$	2.38121e7			1.58243			0.791217	−	0.611536i	$-0.209448\pi$
$743$	2.38121e7			1.58243			0.791217	+	0.611536i	$0.209448\pi$
$744$		0			0
$745$	−504180.	+	873265.i	−0.0332809	+	0.0576442i
$746$		0			0
$747$	3.23263e6	+	5.59908e6i	0.211960	+	0.367126i
$748$		0			0
$749$	−8.95881e6	+	419381.i	−0.583507	+	0.0273152i
$750$		0			0
$751$	962480.	+	1.66707e6i	0.0622719	+	0.107858i	0.895481	−	0.445101i	$-0.146832\pi$
$751$	962480.	+	1.66707e6i	0.0622719	+	0.107858i	−0.833209	+	0.552959i	$0.813499\pi$
$752$		0			0
$753$	6.80999e6	−	1.17953e7i	0.437683	−	0.758089i
$754$		0			0
$755$	921216.			0.0588158
$756$		0			0
$757$	8.98092e6			0.569615			0.284807	−	0.958585i	$-0.408070\pi$
$757$	8.98092e6			0.569615			0.284807	+	0.958585i	$0.408070\pi$
$758$		0			0
$759$	1.03936e7	−	1.80022e7i	0.654879	−	1.13428i
$760$		0			0
$761$	−7.29955e6	−	1.26432e7i	−0.456914	−	0.791398i	0.541882	−	0.840454i	$-0.317712\pi$
$761$	−7.29955e6	−	1.26432e7i	−0.456914	−	0.791398i	−0.998796	+	0.0490566i	$0.984379\pi$
$762$		0			0
$763$	1.53691e7	+	2.39581e7i	0.955736	+	1.48984i
$764$		0			0
$765$	−422820.	−	732346.i	−0.0261217	−	0.0452442i
$766$		0			0
$767$	−5.74205e6	+	9.94552e6i	−0.352434	+	0.610434i
$768$		0			0
$769$	2.78381e7			1.69755			0.848776	−	0.528753i	$-0.177340\pi$
$769$	2.78381e7			1.69755			0.848776	+	0.528753i	$0.177340\pi$
$770$		0			0
$771$	1.39944e7			0.847847
$772$		0			0
$773$	1.41429e7	−	2.44962e7i	0.851312	−	1.47451i	−0.0287135	−	0.999588i	$-0.509141\pi$
$773$	1.41429e7	−	2.44962e7i	0.851312	−	1.47451i	0.880025	−	0.474927i	$-0.157526\pi$
$774$		0			0
$775$	−9.71954e6	−	1.68347e7i	−0.581288	−	1.00682i
$776$		0			0
$777$	1.67665e6	−	3.24570e6i	0.0996300	−	0.192866i
$778$		0			0
$779$	−2.81498e6	−	4.87569e6i	−0.166200	−	0.287867i
$780$		0			0
$781$	−1.27293e7	+	2.20477e7i	−0.746750	+	1.29341i
$782$		0			0

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 7.79423i) q^{3} +(3.00000 + 5.19615i) q^{5} +(-59.5000 + 115.181i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 7.79423i) q^{3} +(3.00000 + 5.19615i) q^{5} +(-59.5000 + 115.181i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(-333.000 + 576.773i) q^{11} -559.000 q^{13} +54.0000 q^{15} +(870.000 - 1506.88i) q^{17} +(578.500 + 1001.99i) q^{19} +(630.000 + 982.073i) q^{21} +(-1734.00 - 3003.38i) q^{23} +(1544.50 - 2675.15i) q^{25} -729.000 q^{27} +3372.00 q^{29} +(3146.50 - 5449.90i) q^{31} +(2997.00 + 5190.96i) q^{33} +(-777.000 + 36.3731i) q^{35} +(-1565.50 - 2711.53i) q^{37} +(-2515.50 + 4356.97i) q^{39} -4866.00 q^{41} +11407.0 q^{43} +(243.000 - 420.888i) q^{45} +(1155.00 + 2000.52i) q^{47} +(-9726.50 - 13706.6i) q^{49} +(-7830.00 - 13562.0i) q^{51} +(14148.0 - 24505.1i) q^{53} -3996.00 q^{55} +10413.0 q^{57} +(10272.0 - 17791.6i) q^{59} +(2315.00 + 4009.70i) q^{61} +(10489.5 - 491.036i) q^{63} +(-1677.00 - 2904.65i) q^{65} +(-9372.50 + 16233.6i) q^{67} -31212.0 q^{69} +38226.0 q^{71} +(-35294.5 + 61131.9i) q^{73} +(-13900.5 - 24076.4i) q^{75} +(-46620.0 - 72673.4i) q^{77} +(-31146.5 - 53947.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -79818.0 q^{83} +10440.0 q^{85} +(15174.0 - 26282.1i) q^{87} +(9060.00 + 15692.4i) q^{89} +(33260.5 - 64386.4i) q^{91} +(-28318.5 - 49049.1i) q^{93} +(-3471.00 + 6011.95i) q^{95} +124754. q^{97} +53946.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9}+O(q^{10}) \) 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9	\( 2 q + 9 q^{3} + 6 q^{5} - 119 q^{7} - 81 q^{9} - 666 q^{11} - 1118 q^{13} + 108 q^{15} + 1740 q^{17} + 1157 q^{19} + 1260 q^{21} - 3468 q^{23} + 3089 q^{25} - 1458 q^{27} + 6744 q^{29} + 6293 q^{31} + 5994 q^{33} - 1554 q^{35} - 3131 q^{37} - 5031 q^{39} - 9732 q^{41} + 22814 q^{43} + 486 q^{45} + 2310 q^{47} - 19453 q^{49} - 15660 q^{51} + 28296 q^{53} - 7992 q^{55} + 20826 q^{57} + 20544 q^{59} + 4630 q^{61} + 20979 q^{63} - 3354 q^{65} - 18745 q^{67} - 62424 q^{69} + 76452 q^{71} - 70589 q^{73} - 27801 q^{75} - 93240 q^{77} - 62293 q^{79} - 6561 q^{81} - 159636 q^{83} + 20880 q^{85} + 30348 q^{87} + 18120 q^{89} + 66521 q^{91} - 56637 q^{93} - 6942 q^{95} + 249508 q^{97} + 107892 q^{99}+O(q^{100}) \) 2 * q + 9 * q^3 + 6 * q^5 - 119 * q^7 - 81 * q^9 - 666 * q^11 - 1118 * q^13 + 108 * q^15 + 1740 * q^17 + 1157 * q^19 + 1260 * q^21 - 3468 * q^23 + 3089 * q^25 - 1458 * q^27 + 6744 * q^29 + 6293 * q^31 + 5994 * q^33 - 1554 * q^35 - 3131 * q^37 - 5031 * q^39 - 9732 * q^41 + 22814 * q^43 + 486 * q^45 + 2310 * q^47 - 19453 * q^49 - 15660 * q^51 + 28296 * q^53 - 7992 * q^55 + 20826 * q^57 + 20544 * q^59 + 4630 * q^61 + 20979 * q^63 - 3354 * q^65 - 18745 * q^67 - 62424 * q^69 + 76452 * q^71 - 70589 * q^73 - 27801 * q^75 - 93240 * q^77 - 62293 * q^79 - 6561 * q^81 - 159636 * q^83 + 20880 * q^85 + 30348 * q^87 + 18120 * q^89 + 66521 * q^91 - 56637 * q^93 - 6942 * q^95 + 249508 * q^97 + 107892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more