LMFDB - Embedded newform 216.3.r.a.115.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(43,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("216.43");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	216.r (of order $18$, degree $6$, 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:	$5.88557371018$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			115.1
Root			$-1.39273 + 0.245576i$ of defining polynomial
Character	$\chi$	$=$	216.115
Dual form			216.3.r.a.139.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87939 - 0.684040i) q^{2} +(-2.95911 + 0.493657i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-5.22362 + 2.95192i) q^{6} +(4.00000 - 6.92820i) q^{8} +(8.51261 - 2.92156i) q^{9} +O(q^{10})$	\(q+(1.87939 - 0.684040i) q^{2} +(-2.95911 + 0.493657i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-5.22362 + 2.95192i) q^{6} +(4.00000 - 6.92820i) q^{8} +(8.51261 - 2.92156i) q^{9} +(3.75655 - 21.3044i) q^{11} +(-7.79796 + 9.12096i) q^{12} +(2.77837 - 15.7569i) q^{16} +(6.74397 + 11.6809i) q^{17} +(14.0000 - 11.3137i) q^{18} +(11.3084 - 19.5867i) q^{19} +(-7.51309 - 42.6089i) q^{22} +(-8.41627 + 22.4759i) q^{24} +(-23.4923 + 8.55050i) q^{25} +(-23.7474 + 12.8475i) q^{27} +(-5.55674 - 31.5138i) q^{32} +(-0.598943 + 64.8965i) q^{33} +(20.6647 + 17.3398i) q^{34} +(18.5724 - 30.8394i) q^{36} +(7.85473 - 44.5464i) q^{38} +(70.3255 + 25.5964i) q^{41} +(-7.32397 + 41.5363i) q^{43} +(-43.2662 - 74.9392i) q^{44} +(-0.442983 + 47.9980i) q^{48} +(8.50876 + 48.2556i) q^{49} +(-38.3022 + 32.1394i) q^{50} +(-25.7225 - 31.2358i) q^{51} +(-35.8424 + 40.3896i) q^{54} +(-23.7936 + 63.5416i) q^{57} +(12.0381 + 68.2714i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(43.2662 + 122.375i) q^{66} +(-116.385 - 42.3606i) q^{67} +(50.6980 + 18.4526i) q^{68} +(13.8092 - 70.6633i) q^{72} +(-60.9139 + 105.506i) q^{73} +(65.2952 - 36.8990i) q^{75} +(-15.7095 - 89.0927i) q^{76} +(63.9289 - 49.7402i) q^{81} +149.678 q^{82} +(32.8039 - 11.9396i) q^{83} +(14.6479 + 83.0726i) q^{86} +(-132.575 - 111.244i) q^{88} +(80.5908 - 139.587i) q^{89} +(32.0000 + 90.5097i) q^{96} +(6.43827 - 36.5132i) q^{97} +(49.0000 + 84.8705i) q^{98} +(-30.2642 - 192.331i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 48 q^{8}+O(q^{10}) $ 12 * q + 48 * q^8	$ 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67} + 48 q^{68} - 816 q^{76} + 84 q^{86} - 672 q^{88} + 438 q^{89} + 384 q^{96} + 282 q^{97} + 588 q^{98}+O(q^{100}) $ 12 * q + 48 * q^8 - 42 * q^11 + 24 * q^12 + 168 * q^18 + 84 * q^22 - 138 * q^27 + 186 * q^33 + 12 * q^34 + 408 * q^38 + 138 * q^41 - 42 * q^43 - 588 * q^51 - 246 * q^57 - 492 * q^59 - 384 * q^64 - 186 * q^67 + 48 * q^68 - 816 * q^76 + 84 * q^86 - 672 * q^88 + 438 * q^89 + 384 * q^96 + 282 * q^97 + 588 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{8}{9}\right)$

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 $

$2$	1.87939	−	0.684040i	0.939693	−	0.342020i
$2$	1.87939	−	0.684040i	0.939693	−	0.342020i
$3$	−2.95911	+	0.493657i	−0.986368	+	0.164552i
$3$	−2.95911	+	0.493657i	−0.986368	+	0.164552i
$4$	3.06418	−	2.57115i	0.766044	−	0.642788i
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$	−5.22362	+	2.95192i	−0.870603	+	0.491986i
$7$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$7$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$8$	4.00000	−	6.92820i	0.500000	−	0.866025i
$9$	8.51261	−	2.92156i	0.945845	−	0.324618i
$10$		0			0
$11$	3.75655	−	21.3044i	0.341504	−	1.93677i	−0.00835660	−	0.999965i	$-0.502660\pi$
$11$	3.75655	−	21.3044i	0.341504	−	1.93677i	0.349861	−	0.936802i	$-0.386229\pi$
$12$	−7.79796	+	9.12096i	−0.649830	+	0.760080i
$13$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$13$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$14$		0			0
$15$		0			0
$16$	2.77837	−	15.7569i	0.173648	−	0.984808i
$17$	6.74397	+	11.6809i	0.396704	+	0.687111i	0.993317	−	0.115418i	$-0.0368206\pi$
$17$	6.74397	+	11.6809i	0.396704	+	0.687111i	−0.596613	+	0.802529i	$0.703487\pi$
$18$	14.0000	−	11.3137i	0.777778	−	0.628539i
$19$	11.3084	−	19.5867i	0.595178	−	1.03088i	−0.398343	−	0.917236i	$-0.630415\pi$
$19$	11.3084	−	19.5867i	0.595178	−	1.03088i	0.993522	−	0.113643i	$-0.0362520\pi$
$20$		0			0
$21$		0			0
$22$	−7.51309	−	42.6089i	−0.341504	−	1.93677i
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$	−8.41627	+	22.4759i	−0.350678	+	0.936496i
$25$	−23.4923	+	8.55050i	−0.939693	+	0.342020i
$26$		0			0
$27$	−23.7474	+	12.8475i	−0.879535	+	0.475834i
$28$		0			0
$29$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$29$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$30$		0			0
$31$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$31$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$32$	−5.55674	−	31.5138i	−0.173648	−	0.984808i
$33$	−0.598943	+	64.8965i	−0.0181498	+	1.96656i
$34$	20.6647	+	17.3398i	0.607786	+	0.509993i
$35$		0			0
$36$	18.5724	−	30.8394i	0.515899	−	0.856649i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	7.85473	−	44.5464i	0.206703	−	1.17227i
$39$		0			0
$40$		0			0
$41$	70.3255	+	25.5964i	1.71525	+	0.624302i	0.997412	−	0.0719030i	$-0.0229072\pi$
$41$	70.3255	+	25.5964i	1.71525	+	0.624302i	0.717843	+	0.696205i	$0.245129\pi$
$42$		0			0
$43$	−7.32397	+	41.5363i	−0.170325	+	0.965961i	0.773078	+	0.634311i	$0.218716\pi$
$43$	−7.32397	+	41.5363i	−0.170325	+	0.965961i	−0.943403	+	0.331649i	$0.892395\pi$
$44$	−43.2662	−	74.9392i	−0.983322	−	1.70316i
$45$		0			0
$46$		0			0
$47$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$47$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$48$	−0.442983	+	47.9980i	−0.00922881	+	0.999957i
$49$	8.50876	+	48.2556i	0.173648	+	0.984808i
$50$	−38.3022	+	32.1394i	−0.766044	+	0.642788i
$51$	−25.7225	−	31.2358i	−0.504362	−	0.612466i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−35.8424	+	40.3896i	−0.663748	+	0.747956i
$55$		0			0
$56$		0			0
$57$	−23.7936	+	63.5416i	−0.417432	+	1.11476i
$58$		0			0
$59$	12.0381	+	68.2714i	0.204035	+	1.15714i	0.898951	+	0.438050i	$0.144331\pi$
$59$	12.0381	+	68.2714i	0.204035	+	1.15714i	−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$61$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$62$		0			0
$63$		0			0
$64$	−32.0000	−	55.4256i	−0.500000	−	0.866025i
$65$		0			0
$66$	43.2662	+	122.375i	0.655548	+	1.85417i
$67$	−116.385	−	42.3606i	−1.73709	−	0.632247i	−0.737992	−	0.674810i	$-0.764226\pi$
$67$	−116.385	−	42.3606i	−1.73709	−	0.632247i	−0.999094	+	0.0425626i	$0.986448\pi$
$68$	50.6980	+	18.4526i	0.745560	+	0.271361i
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$	13.8092	−	70.6633i	0.191795	−	0.981435i
$73$	−60.9139	+	105.506i	−0.834437	+	1.44529i	0.0600510	+	0.998195i	$0.480874\pi$
$73$	−60.9139	+	105.506i	−0.834437	+	1.44529i	−0.894488	+	0.447092i	$0.852460\pi$
$74$		0			0
$75$	65.2952	−	36.8990i	0.870603	−	0.491986i
$76$	−15.7095	−	89.0927i	−0.206703	−	1.17227i
$77$		0			0
$78$		0			0
$79$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$79$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$80$		0			0
$81$	63.9289	−	49.7402i	0.789246	−	0.614077i
$82$	149.678			1.82534
$83$	32.8039	−	11.9396i	0.395228	−	0.143851i	−0.136758	−	0.990604i	$-0.543668\pi$
$83$	32.8039	−	11.9396i	0.395228	−	0.143851i	0.531986	+	0.846753i	$0.321446\pi$
$84$		0			0
$85$		0			0
$86$	14.6479	+	83.0726i	0.170325	+	0.965961i
$87$		0			0
$88$	−132.575	−	111.244i	−1.50654	−	1.26413i
$89$	80.5908	−	139.587i	0.905515	−	1.56840i	0.0852901	−	0.996356i	$-0.472818\pi$
$89$	80.5908	−	139.587i	0.905515	−	1.56840i	0.820225	−	0.572041i	$-0.193848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	32.0000	+	90.5097i	0.333333	+	0.942809i
$97$	6.43827	−	36.5132i	0.0663739	−	0.376425i	−0.933468	−	0.358660i	$-0.883234\pi$
$97$	6.43827	−	36.5132i	0.0663739	−	0.376425i	0.999842	−	0.0177651i	$-0.00565510\pi$
$98$	49.0000	+	84.8705i	0.500000	+	0.866025i
$99$	−30.2642	−	192.331i	−0.305699	−	1.94274i
$100$	−50.0000	+	86.6025i	−0.500000	+	0.866025i
$101$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$101$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$102$	−69.7089	−	41.1089i	−0.683421	−	0.403028i
$103$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$103$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	207.895			1.94295			0.971473	−	0.237150i	$-0.0762134\pi$
$107$	207.895			1.94295			0.971473	+	0.237150i	$0.0762134\pi$
$108$	−39.7335	+	100.425i	−0.367903	+	0.929864i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	22.1164	+	125.428i	0.195720	+	1.10999i	0.911389	+	0.411545i	$0.135011\pi$
$113$	22.1164	+	125.428i	0.195720	+	1.10999i	−0.715669	+	0.698440i	$0.753878\pi$
$114$	−1.25236	+	135.695i	−0.0109856	+	1.19031i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	69.3246	+	120.074i	0.587496	+	1.01757i
$119$		0			0
$120$		0			0
$121$	−326.064	−	118.678i	−2.69475	−	0.980808i
$122$		0			0
$123$	−220.736	−	41.0257i	−1.79460	−	0.333543i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	−98.0537	−	82.2768i	−0.766044	−	0.642788i
$129$	1.16773	−	126.526i	0.00905219	−	0.980820i
$130$		0			0
$131$	−192.625	+	161.632i	−1.47042	+	1.23383i	−0.554676	+	0.832067i	$0.687158\pi$
$131$	−192.625	+	161.632i	−1.47042	+	1.23383i	−0.915744	+	0.401762i	$0.868398\pi$
$132$	165.023	+	200.394i	1.25018	+	1.51814i
$133$		0			0
$134$	−247.708			−1.84857
$135$		0			0
$136$	107.903			0.793408
$137$	253.328	−	92.2040i	1.84911	−	0.673022i	0.863414	−	0.504496i	$-0.168322\pi$
$137$	253.328	−	92.2040i	1.84911	−	0.673022i	0.985697	−	0.168526i	$-0.0539007\pi$
$138$		0			0
$139$	28.9657	−	24.3051i	0.208386	−	0.174857i	−0.532621	−	0.846354i	$-0.678793\pi$
$139$	28.9657	−	24.3051i	0.208386	−	0.174857i	0.741007	+	0.671497i	$0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−22.3837	−	142.250i	−0.155442	−	0.987845i
$145$		0			0
$146$	−42.3104	+	239.954i	−0.289797	+	1.64352i
$147$	−49.0000	−	138.593i	−0.333333	−	0.942809i
$148$		0			0
$149$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$149$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$150$	97.4745	−	114.012i	0.649830	−	0.760080i
$151$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$151$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$152$	−90.4671	−	156.694i	−0.595178	−	1.03088i
$153$	91.5352	+	79.7319i	0.598269	+	0.521124i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$157$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	86.1228	−	137.211i	0.531622	−	0.846982i
$163$	205.091			1.25823			0.629113	−	0.777314i	$-0.283418\pi$
$163$	205.091			1.25823			0.629113	+	0.777314i	$0.283418\pi$
$164$	281.302	−	102.385i	1.71525	−	0.624302i
$165$		0			0
$166$	53.4840	−	44.8784i	0.322193	−	0.270352i
$167$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$167$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$168$		0			0
$169$	129.462	+	108.631i	0.766044	+	0.642788i
$170$		0			0
$171$	39.0401	−	199.772i	0.228305	−	1.16826i
$172$	84.3541	+	146.106i	0.490431	+	0.849451i
$173$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$173$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$174$		0			0
$175$		0			0
$176$	−325.255	−	118.383i	−1.84804	−	0.672632i
$177$	−69.3246	−	196.079i	−0.391664	−	1.10779i
$178$	55.9778	−	317.466i	0.314482	−	1.78352i
$179$	−162.818	−	282.009i	−0.909597	−	1.57547i	−0.814625	−	0.579988i	$-0.803057\pi$
$179$	−162.818	−	282.009i	−0.909597	−	1.57547i	−0.0949721	−	0.995480i	$-0.530276\pi$
$180$		0			0
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	274.189	−	99.7966i	1.46625	−	0.533672i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$191$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$192$	122.053	+	148.213i	0.635691	+	0.771944i
$193$	−268.623	+	225.402i	−1.39183	+	1.16788i	−0.427238	+	0.904139i	$0.640513\pi$
$193$	−268.623	+	225.402i	−1.39183	+	1.16788i	−0.964592	+	0.263745i	$0.915042\pi$
$194$	−12.8765	−	73.0264i	−0.0663739	−	0.376425i
$195$		0			0
$196$	150.145	+	125.986i	0.766044	+	0.642788i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$	−188.441	−	340.763i	−0.951720	−	1.72102i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−34.7296	+	196.962i	−0.173648	+	0.984808i
$201$	365.306	+	67.8953i	1.81744	+	0.337788i
$202$		0			0
$203$		0			0
$204$	−159.130	−	29.5757i	−0.780049	−	0.144979i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−374.803	−	314.497i	−1.79332	−	1.50477i
$210$		0			0
$211$	73.2163	+	415.230i	0.346997	+	1.96792i	0.214409	+	0.976744i	$0.431217\pi$
$211$	73.2163	+	415.230i	0.346997	+	1.96792i	0.132587	+	0.991171i	$0.457672\pi$
$212$		0			0
$213$		0			0
$214$	390.715	−	142.209i	1.82577	−	0.664527i
$215$		0			0
$216$	−5.97959	+	215.917i	−0.0276833	+	0.999617i
$217$		0			0
$218$		0			0
$219$	128.167	−	342.274i	0.585237	−	1.56289i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$223$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$224$		0			0
$225$	−175.000	+	141.421i	−0.777778	+	0.628539i
$226$	127.363	+	220.600i	0.563554	+	0.976105i
$227$	−77.8160	+	441.316i	−0.342802	+	1.94412i	−0.0134727	+	0.999909i	$0.504289\pi$
$227$	−77.8160	+	441.316i	−0.342802	+	1.94412i	−0.329329	+	0.944215i	$0.606822\pi$
$228$	90.4671	+	255.880i	0.396786	+	1.12228i
$229$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$229$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	45.8821	+	79.4700i	0.196919	+	0.341073i	0.947528	−	0.319673i	$-0.103573\pi$
$233$	45.8821	+	79.4700i	0.196919	+	0.341073i	−0.750609	+	0.660746i	$0.770240\pi$
$234$		0			0
$235$		0			0
$236$	212.423	+	178.244i	0.900096	+	0.755271i
$237$		0			0
$238$		0			0
$239$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$239$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$240$		0			0
$241$	126.866	−	46.1753i	0.526413	−	0.191599i	−0.0651229	−	0.997877i	$-0.520744\pi$
$241$	126.866	−	46.1753i	0.526413	−	0.191599i	0.591536	+	0.806279i	$0.298522\pi$
$242$	−693.981			−2.86769
$243$	−164.618	+	178.746i	−0.677440	+	0.735578i
$244$		0			0
$245$		0			0
$246$	−442.912	+	73.8893i	−1.80045	+	0.300363i
$247$		0			0
$248$		0			0
$249$	−91.1761	+	51.5245i	−0.366169	+	0.206926i
$250$		0			0
$251$	−187.025	+	323.937i	−0.745119	+	1.29058i	0.205020	+	0.978758i	$0.434274\pi$
$251$	−187.025	+	323.937i	−0.745119	+	1.29058i	−0.950139	+	0.311826i	$0.899059\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−240.561	−	87.5572i	−0.939693	−	0.342020i
$257$	−72.8485	−	26.5147i	−0.283457	−	0.103170i	0.196379	−	0.980528i	$-0.437082\pi$
$257$	−72.8485	−	26.5147i	−0.283457	−	0.103170i	−0.479836	+	0.877358i	$0.659304\pi$
$258$	−84.3541	−	238.590i	−0.326954	−	0.924766i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−251.454	+	435.531i	−0.959748	+	1.66233i
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$	447.220	+	263.736i	1.69402	+	0.998999i
$265$		0			0
$266$		0			0
$267$	−169.568	+	452.838i	−0.635088	+	1.69602i
$268$	−465.539	+	169.442i	−1.73709	+	0.632247i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	202.792	−	73.8103i	0.745560	−	0.271361i
$273$		0			0
$274$	413.030	−	346.573i	1.50741	−	1.26487i
$275$	93.9137	+	532.611i	0.341504	+	1.93677i
$276$		0			0
$277$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$277$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$278$	37.8120	−	65.4923i	0.136014	−	0.235584i
$279$		0			0
$280$		0			0
$281$	97.2270	−	551.402i	0.346004	−	1.96228i	0.0881114	−	0.996111i	$-0.471917\pi$
$281$	97.2270	−	551.402i	0.346004	−	1.96228i	0.257892	−	0.966174i	$-0.416972\pi$
$282$		0			0
$283$	417.223	+	151.857i	1.47428	+	0.536596i	0.949260	−	0.314491i	$-0.101834\pi$
$283$	417.223	+	151.857i	1.47428	+	0.536596i	0.525024	+	0.851087i	$0.324056\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−139.372	−	252.031i	−0.483931	−	0.875106i
$289$	53.5378	−	92.7302i	0.185252	−	0.320866i
$290$		0			0
$291$	−1.02652	+	111.225i	−0.00352755	+	0.382216i
$292$	84.6207	+	479.908i	0.289797	+	1.64352i
$293$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$293$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$294$	−186.893	−	226.952i	−0.635691	−	0.771944i
$295$		0			0
$296$		0			0
$297$	184.501	+	554.188i	0.621214	+	1.86595i
$298$		0			0
$299$		0			0
$300$	105.203	−	280.949i	0.350678	−	0.936496i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−277.207	−	232.605i	−0.911866	−	0.765147i
$305$		0			0
$306$	226.570	+	87.2332i	0.740424	+	0.285076i
$307$	−189.117	−	327.560i	−0.616016	−	1.06697i	−0.990205	−	0.139619i	$-0.955412\pi$
$307$	−189.117	−	327.560i	−0.616016	−	1.06697i	0.374189	−	0.927352i	$-0.377921\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$311$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$312$		0			0
$313$	42.1819	−	239.225i	0.134766	−	0.764299i	−0.840256	−	0.542191i	$-0.817595\pi$
$313$	42.1819	−	239.225i	0.134766	−	0.764299i	0.975022	−	0.222108i	$-0.0712938\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$317$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−615.184	+	102.629i	−1.91646	+	0.319716i
$322$		0			0
$323$	305.054			0.944438
$324$	68.0000	−	316.784i	0.209877	−	0.977728i
$325$		0			0
$326$	385.445	−	140.290i	1.18235	−	0.430339i
$327$		0			0
$328$	458.639	−	384.844i	1.39829	−	1.17330i
$329$		0			0
$330$		0			0
$331$	433.720	+	363.934i	1.31033	+	1.09950i	0.988260	+	0.152783i	$0.0488234\pi$
$331$	433.720	+	363.934i	1.31033	+	1.09950i	0.322071	+	0.946716i	$0.395621\pi$
$332$	69.8184	−	120.929i	0.210296	−	0.364244i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−574.803	−	209.211i	−1.70565	−	0.620805i	−0.709199	−	0.705009i	$-0.750943\pi$
$337$	−574.803	−	209.211i	−1.70565	−	0.620805i	−0.996449	+	0.0842037i	$0.973165\pi$
$338$	317.616	+	115.603i	0.939693	+	0.342020i
$339$	−127.363	−	360.238i	−0.375703	−	1.06265i
$340$		0			0
$341$		0			0
$342$	−63.2808	−	402.154i	−0.185032	−	1.17589i
$343$		0			0
$344$	258.476	+	216.887i	0.751384	+	0.630486i
$345$		0			0
$346$		0			0
$347$	−253.964	+	213.101i	−0.731884	+	0.614124i	−0.930645	−	0.365924i	$-0.880753\pi$
$347$	−253.964	+	213.101i	−0.731884	+	0.614124i	0.198761	+	0.980048i	$0.436308\pi$
$348$		0			0
$349$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$349$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$350$		0			0
$351$		0			0
$352$	−692.259			−1.96664
$353$	−46.8630	+	17.0568i	−0.132756	+	0.0483194i	−0.407544	−	0.913186i	$-0.633615\pi$
$353$	−46.8630	+	17.0568i	−0.132756	+	0.0483194i	0.274788	+	0.961505i	$0.411392\pi$
$354$	−264.414	−	321.088i	−0.746932	−	0.907028i
$355$		0			0
$356$	−111.956	−	634.932i	−0.314482	−	1.78352i
$357$		0			0
$358$	−498.903	−	418.629i	−1.39358	−	1.16936i
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−75.2594	−	130.353i	−0.208475	−	0.361089i
$362$		0			0
$363$	1023.45	+	190.216i	2.81941	+	0.524011i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$367$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$368$		0			0
$369$	673.434	+	12.4316i	1.82503	+	0.0336899i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$373$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$374$	447.042	−	375.112i	1.19530	−	1.00297i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−198.419			−0.523534			−0.261767	−	0.965131i	$-0.584305\pi$
$379$	−198.419			−0.523534			−0.261767	+	0.965131i	$0.584305\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$383$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$384$	330.768	+	195.061i	0.861374	+	0.507971i
$385$		0			0
$386$	−350.663	+	607.366i	−0.908453	+	1.57349i
$387$	59.0049	+	374.980i	0.152467	+	0.968940i
$388$	−74.1530	−	128.437i	−0.191116	−	0.331023i
$389$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$389$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$390$		0			0
$391$		0			0
$392$	368.360	+	134.072i	0.939693	+	0.342020i
$393$	490.207	−	573.375i	1.24735	−	1.45897i
$394$		0			0
$395$		0			0
$396$	−587.248	−	511.523i	−1.48295	−	1.29173i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	69.4593	+	393.923i	0.173648	+	0.984808i
$401$	77.3428	−	64.8983i	0.192875	−	0.161841i	−0.541236	−	0.840871i	$-0.682044\pi$
$401$	77.3428	−	64.8983i	0.192875	−	0.161841i	0.734111	+	0.679029i	$0.237599\pi$
$402$	732.994	−	122.283i	1.82337	−	0.304186i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−319.298	+	53.2673i	−0.782592	+	0.130557i
$409$	44.7901	−	37.5833i	0.109511	−	0.0918908i	−0.586388	−	0.810030i	$-0.699451\pi$
$409$	44.7901	−	37.5833i	0.109511	−	0.0918908i	0.695899	+	0.718140i	$0.255006\pi$
$410$		0			0
$411$	−704.108	+	397.898i	−1.71316	+	0.968122i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−73.7141	+	86.2204i	−0.176772	+	0.206764i
$418$	−919.528	−	334.681i	−2.19983	−	0.800672i
$419$	−780.115	−	283.938i	−1.86185	−	0.677657i	−0.977538	−	0.210758i	$-0.932407\pi$
$419$	−780.115	−	283.938i	−1.86185	−	0.677657i	−0.884310	−	0.466900i	$-0.845371\pi$
$420$		0			0
$421$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$421$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$422$	421.636	+	730.294i	0.999137	+	1.73056i
$423$		0			0
$424$		0			0
$425$	−258.309	−	216.747i	−0.607786	−	0.509993i
$426$		0			0
$427$		0			0
$428$	637.028	−	534.530i	1.48838	−	1.24890i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	136.458	+	409.882i	0.315875	+	0.948801i
$433$	−763.705			−1.76375			−0.881876	−	0.471481i	$-0.843720\pi$
$433$	−763.705			−1.76375			−0.881876	+	0.471481i	$0.843720\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	6.74595	−	730.936i	0.0154017	−	1.66880i
$439$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$439$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$440$		0			0
$441$	213.413	+	385.922i	0.483931	+	0.875106i
$442$		0			0
$443$	6.15998	−	34.9350i	0.0139051	−	0.0788600i	−0.977066	−	0.212939i	$-0.931696\pi$
$443$	6.15998	−	34.9350i	0.0139051	−	0.0788600i	0.990971	+	0.134079i	$0.0428076\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−255.338	−	442.258i	−0.568681	−	0.984985i	−0.996697	−	0.0812137i	$-0.974120\pi$
$449$	−255.338	−	442.258i	−0.568681	−	0.984985i	0.428015	−	0.903772i	$-0.359213\pi$
$450$	−232.155	+	385.492i	−0.515899	+	0.856649i
$451$	809.497	−	1402.09i	1.79489	−	3.10885i
$452$	390.264	+	327.470i	0.863415	+	0.724491i
$453$		0			0
$454$	155.632	+	882.633i	0.342802	+	1.94412i
$455$		0			0
$456$	345.055	+	419.013i	0.756699	+	0.918889i
$457$	855.488	−	311.372i	1.87196	−	0.681339i	0.905617	−	0.424096i	$-0.139408\pi$
$457$	855.488	−	311.372i	1.87196	−	0.681339i	0.966347	−	0.257244i	$-0.0828143\pi$
$458$		0			0
$459$	−310.222	−	190.748i	−0.675866	−	0.415573i
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$463$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$464$		0			0
$465$		0			0
$466$	140.591	+	117.970i	0.301697	+	0.253154i
$467$	462.552	−	801.164i	0.990476	−	1.71556i	0.376001	−	0.926619i	$-0.377299\pi$
$467$	462.552	−	801.164i	0.990476	−	1.71556i	0.614476	−	0.788936i	$-0.289368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	521.150	+	189.683i	1.10413	+	0.401871i
$473$	857.395	+	312.066i	1.81267	+	0.659759i
$474$		0			0
$475$	−98.1841	+	556.830i	−0.206703	+	1.17227i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$479$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$480$		0			0
$481$		0			0
$482$	206.844	−	173.562i	0.429136	−	0.360088i
$483$		0			0
$484$	−1304.26	+	474.711i	−2.69475	+	0.980808i
$485$		0			0
$486$	−187.111	+	448.537i	−0.385003	+	0.922916i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−606.885	+	101.244i	−1.24107	+	0.207044i
$490$		0			0
$491$	−162.880	−	923.739i	−0.331731	−	1.88134i	−0.457386	−	0.889268i	$-0.651214\pi$
$491$	−162.880	−	923.739i	−0.331731	−	1.88134i	0.125655	−	0.992074i	$-0.459897\pi$
$492$	−781.858	+	441.836i	−1.58914	+	0.898040i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−136.110	+	159.203i	−0.273314	+	0.319684i
$499$	−517.286	−	188.277i	−1.03664	−	0.377308i	−0.233037	−	0.972468i	$-0.574866\pi$
$499$	−517.286	−	188.277i	−1.03664	−	0.377308i	−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$	−129.906	+	736.734i	−0.258777	+	1.46760i
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−436.717	−	257.541i	−0.861374	−	0.507971i
$508$		0			0
$509$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$509$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$	−16.9049	+	610.419i	−0.0329530	+	1.18990i
$514$	−155.048			−0.301649
$515$		0			0
$516$	−321.739	−	390.700i	−0.623525	−	0.757171i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−519.100	+	899.107i	−0.996352	+	1.72573i	−0.424274	+	0.905534i	$0.639471\pi$
$521$	−519.100	+	899.107i	−0.996352	+	1.72573i	−0.572078	+	0.820199i	$0.693863\pi$
$522$		0			0
$523$	−319.363	−	553.153i	−0.610636	−	1.05765i	−0.991133	−	0.132871i	$-0.957580\pi$
$523$	−319.363	−	553.153i	−0.610636	−	1.05765i	0.380497	−	0.924782i	$-0.375753\pi$
$524$	−174.658	+	990.536i	−0.333317	+	1.89034i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1020.91	+	189.744i	1.93353	+	0.359364i
$529$	91.8599	−	520.963i	0.173648	−	0.984808i
$530$		0			0
$531$	301.935	+	545.997i	0.568615	+	1.02824i
$532$		0			0
$533$		0			0
$534$	−8.92509	+	967.049i	−0.0167136	+	1.81095i
$535$		0			0
$536$	−759.022	+	636.895i	−1.41608	+	1.18824i
$537$	621.011	+	754.118i	1.15644	+	1.40432i
$538$		0			0
$539$	1060.02			1.96664
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	330.635	−	277.436i	0.607786	−	0.509993i
$545$		0			0
$546$		0			0
$547$	−837.949	−	703.122i	−1.53190	−	1.28542i	−0.776078	−	0.630637i	$-0.782794\pi$
$547$	−837.949	−	703.122i	−1.53190	−	1.28542i	−0.755820	−	0.654779i	$-0.772762\pi$
$548$	539.173	−	933.874i	0.983892	−	1.70415i
$549$		0			0
$550$	540.827	+	936.740i	0.983322	+	1.70316i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	26.2639	−	148.950i	0.0472373	−	0.267896i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−762.088	+	430.664i	−1.35845	+	0.767671i
$562$	−194.454	−	1102.80i	−0.346004	−	1.96228i
$563$	−675.786	+	567.052i	−1.20033	+	1.00720i	−0.200710	+	0.979651i	$0.564325\pi$
$563$	−675.786	+	567.052i	−1.20033	+	1.00720i	−0.999621	+	0.0275467i	$0.991231\pi$
$564$		0			0
$565$		0			0
$566$	887.998			1.56890
$567$		0			0
$568$		0			0
$569$	−981.619	+	357.280i	−1.72517	+	0.627909i	−0.998268	−	0.0588367i	$-0.981261\pi$
$569$	−981.619	+	357.280i	−1.72517	+	0.627909i	−0.726898	+	0.686746i	$0.759039\pi$
$570$		0			0
$571$	718.207	−	602.647i	1.25781	−	1.05542i	0.261894	−	0.965097i	$-0.415653\pi$
$571$	718.207	−	602.647i	1.25781	−	1.05542i	0.995912	−	0.0903277i	$-0.0287914\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−434.333	−	378.327i	−0.754050	−	0.656817i
$577$	370.122	+	641.070i	0.641459	+	1.11104i	0.985107	+	0.171941i	$0.0550039\pi$
$577$	370.122	+	641.070i	0.641459	+	1.11104i	−0.343648	+	0.939098i	$0.611663\pi$
$578$	37.1870	−	210.898i	0.0643373	−	0.364875i
$579$	683.614	−	799.595i	1.18068	−	1.38099i
$580$		0			0
$581$		0			0
$582$	74.1530	+	209.736i	0.127411	+	0.360372i
$583$		0			0
$584$	487.311	+	844.048i	0.834437	+	1.44529i
$585$		0			0
$586$		0			0
$587$	−809.864	−	679.557i	−1.37967	−	1.15768i	−0.969336	−	0.245741i	$-0.920969\pi$
$587$	−809.864	−	679.557i	−1.37967	−	1.15768i	−0.410331	−	0.911937i	$-0.634587\pi$
$588$	−506.488	−	298.687i	−0.861374	−	0.507971i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1136.45			1.91645			0.958223	−	0.286021i	$-0.0923326\pi$
$593$	1136.45			1.91645			0.958223	+	0.286021i	$0.0923326\pi$
$594$	725.835	+	915.327i	1.22194	+	1.54096i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$599$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$600$	5.53729	−	599.974i	0.00922881	−	0.999957i
$601$	920.750	+	772.601i	1.53203	+	1.28553i	0.771631	+	0.636070i	$0.219441\pi$
$601$	920.750	+	772.601i	1.53203	+	1.28553i	0.760399	+	0.649456i	$0.225003\pi$
$602$		0			0
$603$	−1114.50	−	20.5736i	−1.84825	−	0.0341187i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$607$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$608$	−680.090	−	247.533i	−1.11857	−	0.407126i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	485.483	+	8.96200i	0.793273	+	0.0146438i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−579.488	−	486.248i	−0.943792	−	0.791935i
$615$		0			0
$616$		0			0
$617$	388.947	−	326.365i	0.630384	−	0.528955i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$	388.947	−	326.365i	0.630384	−	0.528955i	0.901048	+	0.433719i	$0.142799\pi$
$618$		0			0
$619$	−929.110	+	338.168i	−1.50099	+	0.546314i	−0.956315	−	0.292337i	$-0.905567\pi$
$619$	−929.110	+	338.168i	−1.50099	+	0.546314i	−0.544670	+	0.838651i	$0.683345\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	478.778	−	401.742i	0.766044	−	0.642788i
$626$	−84.3638	−	478.451i	−0.134766	−	0.764299i
$627$	1264.34	+	745.606i	2.01648	+	1.18916i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−421.636	−	1192.57i	−0.666091	−	1.88399i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−832.072	−	698.192i	−1.29809	−	1.08922i	−0.990472	−	0.137711i	$-0.956025\pi$
$641$	−832.072	−	698.192i	−1.29809	−	1.08922i	−0.307613	−	0.951512i	$-0.599530\pi$
$642$	−1085.97	+	613.690i	−1.69153	+	0.955903i
$643$	208.845	+	1184.42i	0.324798	+	1.84202i	0.511094	+	0.859525i	$0.329241\pi$
$643$	208.845	+	1184.42i	0.324798	+	1.84202i	−0.186296	+	0.982494i	$0.559648\pi$
$644$		0			0
$645$		0			0
$646$	573.313	−	208.669i	0.887482	−	0.323017i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−88.8947	−	641.874i	−0.137183	−	0.990546i
$649$	1499.70			2.31079
$650$		0			0
$651$		0			0
$652$	628.435	−	527.319i	0.963857	−	0.808772i
$653$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$653$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$654$		0			0
$655$		0			0
$656$	598.710	−	1037.00i	0.912668	−	1.58079i
$657$	−210.294	+	1076.09i	−0.320082	+	1.63789i
$658$		0			0
$659$	−43.8538	+	248.707i	−0.0665460	+	0.377401i	0.933287	+	0.359131i	$0.116927\pi$
$659$	−43.8538	+	248.707i	−0.0665460	+	0.377401i	−0.999833	+	0.0182698i	$0.994184\pi$
$660$		0			0
$661$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$661$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$662$	1064.07	+	387.290i	1.60736	+	0.585031i
$663$		0			0
$664$	48.4953	−	275.031i	0.0730351	−	0.414203i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−999.747	+	363.878i	−1.48551	+	0.540681i	−0.952263	−	0.305280i	$-0.901250\pi$
$673$	−999.747	+	363.878i	−1.48551	+	0.540681i	−0.533245	+	0.845961i	$0.679028\pi$
$674$	−1223.39			−1.81511
$675$	448.030	−	504.871i	0.663748	−	0.747956i
$676$	676.000			1.00000
$677$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$677$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$678$	−485.782	−	589.904i	−0.716492	−	0.870065i
$679$		0			0
$680$		0			0
$681$	12.4070	−	1344.32i	0.0182187	−	1.97403i
$682$		0			0
$683$	267.533	−	463.381i	0.391703	−	0.678450i	−0.600971	−	0.799271i	$-0.705219\pi$
$683$	267.533	−	463.381i	0.391703	−	0.678450i	0.992674	+	0.120821i	$0.0385527\pi$
$684$	−394.018	−	712.515i	−0.576050	−	1.04169i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	634.136	+	230.807i	0.921709	+	0.335475i
$689$		0			0
$690$		0			0
$691$	−239.824	+	1360.11i	−0.347067	+	1.96832i	−0.137777	+	0.990463i	$0.543996\pi$
$691$	−239.824	+	1360.11i	−0.347067	+	1.96832i	−0.209290	+	0.977854i	$0.567115\pi$
$692$		0			0
$693$		0			0
$694$	−331.526	+	574.220i	−0.477703	+	0.827407i
$695$		0			0
$696$		0			0
$697$	175.284	+	994.085i	0.251484	+	1.42623i
$698$		0			0
$699$	−175.001	−	212.510i	−0.250359	−	0.304020i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−1301.02	+	473.533i	−1.84804	+	0.672632i
$705$		0			0
$706$	−76.4062	+	64.1124i	−0.108224	+	0.0908108i
$707$		0			0
$708$	−716.573	−	422.578i	−1.01211	−	0.596862i
$709$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$709$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$710$		0			0
$711$		0			0
$712$	−644.727	−	1116.70i	−0.905515	−	1.56840i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1223.99	−	445.496i	−1.70948	−	0.622201i
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−230.608	−	193.503i	−0.319402	−	0.268010i
$723$	−352.614	+	199.266i	−0.487709	+	0.275609i
$724$		0			0
$725$		0			0
$726$	2053.56	−	342.588i	2.82860	−	0.471885i
$727$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$727$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$728$		0			0
$729$	398.883	−	610.191i	0.547164	−	0.837025i
$730$		0			0
$731$	−534.574	+	194.569i	−0.731291	+	0.266168i
$732$		0			0
$733$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$733$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1339.67	+	2320.38i	−1.81774	+	3.14841i
$738$	1274.15	−	437.292i	1.72649	−	0.592537i
$739$	−340.277	−	589.377i	−0.460456	−	0.797533i	0.538528	−	0.842608i	$-0.318981\pi$
$739$	−340.277	−	589.377i	−0.460456	−	0.797533i	−0.998984	+	0.0450751i	$0.985647\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$743$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	244.364	−	197.476i	0.327128	−	0.264359i
$748$	583.571	−	1010.78i	0.780176	−	1.35130i
$749$		0			0
$750$		0			0
$751$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$751$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$752$		0			0
$753$	393.513	−	1050.89i	0.522593	−	1.39560i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−372.907	+	135.727i	−0.491961	+	0.179059i
$759$		0			0
$760$		0			0
$761$	242.066	+	1372.82i	0.318089	+	1.80397i	0.554352	+	0.832282i	$0.312966\pi$
$761$	242.066	+	1372.82i	0.318089	+	1.80397i	−0.236264	+	0.971689i	$0.575923\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	755.069	+	140.336i	0.983163	+	0.182729i
$769$	−1406.72	−	512.003i	−1.82928	−	0.665804i	−0.993088	−	0.117370i	$-0.962554\pi$
$769$	−1406.72	−	512.003i	−1.82928	−	0.665804i	−0.836194	−	0.548434i	$-0.815224\pi$
$770$		0			0
$771$	228.656	+	42.4976i	0.296570	+	0.0551201i
$772$	−243.568	+	1381.34i	−0.315502	+	1.78930i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$	367.394	+	664.370i	0.474669	+	0.858359i
$775$		0			0
$776$	−227.218	−	190.659i	−0.292807	−	0.245694i
$777$		0			0
$778$		0			0
$779$	1296.62	−	1087.99i	1.66446	−	1.39665i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$		0			0
$786$	529.076	−	1412.91i	0.673125	−	1.79760i
$787$	489.710	−	410.915i	0.622248	−	0.522128i	−0.276261	−	0.961083i	$-0.589095\pi$
$787$	489.710	−	410.915i	0.622248	−	0.522128i	0.898509	+	0.438954i	$0.144651\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1453.57	−	559.648i	−1.83531	−	0.706626i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$797$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$798$		0			0
$799$		0			0
$800$	400.000	+	692.820i	0.500000	+	0.866025i
$801$	278.225	−	1423.70i	0.347346	−	1.77741i
$802$	100.964	−	174.875i	0.125890	−	0.218048i
$803$	2018.92	+	1694.07i	2.51422	+	2.10968i
$804$	1293.93	−	731.214i	1.60937	−	0.909470i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1370.51			1.69408			0.847038	−	0.531532i	$-0.178383\pi$
$809$	1370.51			1.69408			0.847038	+	0.531532i	$0.178383\pi$
$810$		0			0
$811$	−1172.66			−1.44594			−0.722972	−	0.690878i	$-0.757224\pi$
$811$	−1172.66			−1.44594			−0.722972	+	0.690878i	$0.757224\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−563.646	+	318.522i	−0.690743	+	0.390346i
$817$	730.737	+	613.161i	0.894415	+	0.750503i
$818$	58.4693	−	101.272i	0.0714783	−	0.123804i
$819$		0			0
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$	−1051.11	+	1229.44i	−1.27872	+	1.49567i
$823$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$823$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$824$		0			0
$825$	−540.827	−	1529.69i	−0.655548	−	1.85417i
$826$		0			0
$827$	−631.000	−	1092.92i	−0.762999	−	1.32155i	−0.941298	−	0.337576i	$-0.890393\pi$
$827$	−631.000	−	1092.92i	−0.762999	−	1.32155i	0.178299	−	0.983976i	$-0.442940\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−506.285	+	424.824i	−0.607786	+	0.509993i
$834$	−79.5590	+	212.465i	−0.0953945	+	0.254754i
$835$		0			0
$836$	−1957.08			−2.34101
$837$		0			0
$838$	−1660.36			−1.98134
$839$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$839$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$840$		0			0
$841$	644.243	−	540.584i	0.766044	−	0.642788i
$842$		0			0
$843$	−15.5019	+	1679.65i	−0.0183889	+	1.99247i
$844$	1291.97	+	1084.09i	1.53077	+	1.28447i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1309.57	−	243.395i	−1.54249	−	0.286684i
$850$	−633.726	−	230.657i	−0.745560	−	0.271361i
$851$		0			0
$852$		0			0
$853$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$853$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$854$		0			0
$855$		0			0
$856$	831.581	−	1440.34i	0.971473	−	1.68264i
$857$	920.785	+	772.631i	1.07443	+	0.901553i	0.995446	−	0.0953233i	$-0.0303885\pi$
$857$	920.785	+	772.631i	1.07443	+	0.901553i	0.0789824	+	0.996876i	$0.474833\pi$
$858$		0			0
$859$	164.022	+	930.214i	0.190945	+	1.08290i	0.918076	+	0.396404i	$0.129742\pi$
$859$	164.022	+	930.214i	0.190945	+	1.08290i	−0.727131	+	0.686499i	$0.759147\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	536.833	+	676.983i	0.621335	+	0.783545i
$865$		0			0
$866$	−1435.30	+	522.405i	−1.65739	+	0.603239i
$867$	−112.647	+	300.828i	−0.129928	+	0.346976i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−51.8693	−	329.632i	−0.0594150	−	0.377586i
$874$		0			0
$875$		0			0
$876$	−487.311	−	1378.32i	−0.556291	−	1.57343i
$877$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$877$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	81.4082	+	141.003i	0.0924043	+	0.160049i	0.908522	−	0.417836i	$-0.137211\pi$
$881$	81.4082	+	141.003i	0.0924043	+	0.160049i	−0.816118	+	0.577885i	$0.803878\pi$
$882$	665.072	+	579.312i	0.754050	+	0.656817i
$883$	−848.184	+	1469.10i	−0.960571	+	1.66376i	−0.239500	+	0.970896i	$0.576984\pi$
$883$	−848.184	+	1469.10i	−0.960571	+	1.66376i	−0.721071	+	0.692861i	$0.756350\pi$
$884$		0			0
$885$		0			0
$886$	−12.3200	−	69.8700i	−0.0139051	−	0.0788600i
$887$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$887$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−819.536	−	1548.82i	−0.919793	−	1.73830i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−782.401	−	656.512i	−0.871271	−	0.731083i
$899$		0			0
$900$	−172.616	+	883.292i	−0.191795	+	0.981435i
$901$		0			0
$902$	562.271	−	3188.80i	0.623360	−	3.53525i
$903$		0			0
$904$	957.459	+	348.486i	1.05914	+	0.385494i
$905$		0			0
$906$		0			0
$907$	−294.298	+	1669.05i	−0.324474	+	1.84019i	0.188868	+	0.982002i	$0.439518\pi$
$907$	−294.298	+	1669.05i	−0.324474	+	1.84019i	−0.513343	+	0.858184i	$0.671593\pi$
$908$	896.249	+	1552.35i	0.987058	+	1.70963i
$909$		0			0
$910$		0			0
$911$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$911$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$912$	935.112	+	551.456i	1.02534	+	0.604667i
$913$	−131.138	−	743.720i	−0.143634	−	0.814590i
$914$	1394.80	−	1170.38i	1.52604	−	1.28050i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−713.507	−	146.285i	−0.777241	−	0.159351i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	721.319	+	875.926i	0.783191	+	0.951060i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	183.720	−	1041.93i	0.197761	−	1.12156i	−0.710671	−	0.703524i	$-0.751609\pi$
$929$	183.720	−	1041.93i	0.197761	−	1.12156i	0.908432	−	0.418033i	$-0.137280\pi$
$930$		0			0
$931$	1041.39	+	379.034i	1.11857	+	0.407126i
$932$	344.920	+	125.541i	0.370086	+	0.134700i
$933$		0			0
$934$	321.286	−	1822.10i	0.343989	−	1.95086i
$935$		0			0
$936$		0			0
$937$	−929.044	+	1609.15i	−0.991509	+	1.71734i	−0.383138	+	0.923691i	$0.625157\pi$
$937$	−929.044	+	1609.15i	−0.991509	+	1.71734i	−0.608371	+	0.793653i	$0.708177\pi$
$938$		0			0
$939$	−6.72547	+	728.717i	−0.00716238	+	0.776056i
$940$		0			0
$941$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$941$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$942$		0			0
$943$		0			0
$944$	1109.19			1.17499
$945$		0			0
$946$	1824.84			1.92901
$947$	−359.572	+	130.873i	−0.379696	+	0.138198i	−0.524815	−	0.851216i	$-0.675866\pi$
$947$	−359.572	+	130.873i	−0.379696	+	0.138198i	0.145119	+	0.989414i	$0.453643\pi$
$948$		0			0
$949$		0			0
$950$	196.368	+	1113.66i	0.206703	+	1.17227i
$951$		0			0
$952$		0			0
$953$	−556.485	+	963.860i	−0.583930	+	1.01140i	0.411078	+	0.911600i	$0.365152\pi$
$953$	−556.485	+	963.860i	−0.583930	+	1.01140i	−0.995008	+	0.0997959i	$0.968181\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	166.876	−	946.400i	0.173648	−	0.984808i
$962$		0			0
$963$	1769.73	−	607.379i	1.83773	−	0.630715i
$964$	270.015	−	467.680i	0.280099	−	0.485145i
$965$		0			0
$966$		0			0
$967$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$967$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$968$	−2126.48	+	1784.33i	−2.19678	+	1.84332i
$969$	−902.686	+	150.592i	−0.931564	+	0.155409i
$970$		0			0
$971$	974.000			1.00309			0.501545	−	0.865132i	$-0.332765\pi$
$971$	974.000			1.00309			0.501545	+	0.865132i	$0.332765\pi$
$972$	−44.8367	+	970.965i	−0.0461283	+	0.998936i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−121.682	−	690.092i	−0.124546	−	0.706338i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−121.682	−	690.092i	−0.124546	−	0.706338i	0.857030	−	0.515267i	$-0.172307\pi$
$978$	−1071.32	+	605.411i	−1.09542	+	0.619030i
$979$	−2671.09	−	2241.31i	−2.72838	−	2.28939i
$980$		0			0
$981$		0			0
$982$	−937.989	−	1624.65i	−0.955183	−	1.65442i
$983$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$983$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$984$	−1167.18	+	1365.20i	−1.18616	+	1.38740i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−1463.08	−	862.810i	−1.47339	−	0.868892i
$994$		0			0
$995$		0			0
$996$	−146.903	+	392.308i	−0.147492	+	0.393883i
$997$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$997$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$998$	−1100.97			−1.10317
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.3.r.a.115.1	✓	12
8.3	odd	2	CM	216.3.r.a.115.1	✓	12
27.4	even	9	inner	216.3.r.a.139.1	yes	12
216.139	odd	18	inner	216.3.r.a.139.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.r.a.115.1	✓	12	1.1	even	1	trivial
216.3.r.a.115.1	✓	12	8.3	odd	2	CM
216.3.r.a.139.1	yes	12	27.4	even	9	inner
216.3.r.a.139.1	yes	12	216.139	odd	18	inner

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	216.r (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.87939 - 0.684040i) q^{2} +(-2.95911 + 0.493657i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-5.22362 + 2.95192i) q^{6} +(4.00000 - 6.92820i) q^{8} +(8.51261 - 2.92156i) q^{9} +O(q^{10})\)	\(q+(1.87939 - 0.684040i) q^{2} +(-2.95911 + 0.493657i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-5.22362 + 2.95192i) q^{6} +(4.00000 - 6.92820i) q^{8} +(8.51261 - 2.92156i) q^{9} +(3.75655 - 21.3044i) q^{11} +(-7.79796 + 9.12096i) q^{12} +(2.77837 - 15.7569i) q^{16} +(6.74397 + 11.6809i) q^{17} +(14.0000 - 11.3137i) q^{18} +(11.3084 - 19.5867i) q^{19} +(-7.51309 - 42.6089i) q^{22} +(-8.41627 + 22.4759i) q^{24} +(-23.4923 + 8.55050i) q^{25} +(-23.7474 + 12.8475i) q^{27} +(-5.55674 - 31.5138i) q^{32} +(-0.598943 + 64.8965i) q^{33} +(20.6647 + 17.3398i) q^{34} +(18.5724 - 30.8394i) q^{36} +(7.85473 - 44.5464i) q^{38} +(70.3255 + 25.5964i) q^{41} +(-7.32397 + 41.5363i) q^{43} +(-43.2662 - 74.9392i) q^{44} +(-0.442983 + 47.9980i) q^{48} +(8.50876 + 48.2556i) q^{49} +(-38.3022 + 32.1394i) q^{50} +(-25.7225 - 31.2358i) q^{51} +(-35.8424 + 40.3896i) q^{54} +(-23.7936 + 63.5416i) q^{57} +(12.0381 + 68.2714i) q^{59} +(-32.0000 - 55.4256i) q^{64} +(43.2662 + 122.375i) q^{66} +(-116.385 - 42.3606i) q^{67} +(50.6980 + 18.4526i) q^{68} +(13.8092 - 70.6633i) q^{72} +(-60.9139 + 105.506i) q^{73} +(65.2952 - 36.8990i) q^{75} +(-15.7095 - 89.0927i) q^{76} +(63.9289 - 49.7402i) q^{81} +149.678 q^{82} +(32.8039 - 11.9396i) q^{83} +(14.6479 + 83.0726i) q^{86} +(-132.575 - 111.244i) q^{88} +(80.5908 - 139.587i) q^{89} +(32.0000 + 90.5097i) q^{96} +(6.43827 - 36.5132i) q^{97} +(49.0000 + 84.8705i) q^{98} +(-30.2642 - 192.331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 48 q^{8}+O(q^{10}) \) 12 * q + 48 * q^8	\( 12 q + 48 q^{8} - 42 q^{11} + 24 q^{12} + 168 q^{18} + 84 q^{22} - 138 q^{27} + 186 q^{33} + 12 q^{34} + 408 q^{38} + 138 q^{41} - 42 q^{43} - 588 q^{51} - 246 q^{57} - 492 q^{59} - 384 q^{64} - 186 q^{67} + 48 q^{68} - 816 q^{76} + 84 q^{86} - 672 q^{88} + 438 q^{89} + 384 q^{96} + 282 q^{97} + 588 q^{98}+O(q^{100}) \) 12 * q + 48 * q^8 - 42 * q^11 + 24 * q^12 + 168 * q^18 + 84 * q^22 - 138 * q^27 + 186 * q^33 + 12 * q^34 + 408 * q^38 + 138 * q^41 - 42 * q^43 - 588 * q^51 - 246 * q^57 - 492 * q^59 - 384 * q^64 - 186 * q^67 + 48 * q^68 - 816 * q^76 + 84 * q^86 - 672 * q^88 + 438 * q^89 + 384 * q^96 + 282 * q^97 + 588 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more