LMFDB - Embedded newform 72.3.m.a.41.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,3,Mod(41,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("72.41");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	72.m (of order $6$, degree $2$, 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:	$1.96185790339$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			41.1
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	72.41
Dual form			72.3.m.a.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +(-3.39898 - 1.96240i) q^{5} +(-6.39898 - 11.0834i) q^{7} +9.00000 q^{9} +O(q^{10})$	\(q-3.00000 q^{3} +(-3.39898 - 1.96240i) q^{5} +(-6.39898 - 11.0834i) q^{7} +9.00000 q^{9} +(-14.2980 + 8.25493i) q^{11} +(1.39898 - 2.42310i) q^{13} +(10.1969 + 5.88721i) q^{15} -2.54270i q^{17} +21.5959 q^{19} +(19.1969 + 33.2501i) q^{21} +(2.60102 + 1.50170i) q^{23} +(-4.79796 - 8.31031i) q^{25} -27.0000 q^{27} +(-13.1969 + 7.61926i) q^{29} +(13.6010 - 23.5577i) q^{31} +(42.8939 - 24.7648i) q^{33} +50.2295i q^{35} -10.4041 q^{37} +(-4.19694 + 7.26931i) q^{39} +(-34.5000 - 19.9186i) q^{41} +(-17.0959 - 29.6110i) q^{43} +(-30.5908 - 17.6616i) q^{45} +(58.1969 - 33.6000i) q^{47} +(-57.3939 + 99.4091i) q^{49} +7.62809i q^{51} -100.680i q^{53} +64.7980 q^{55} -64.7878 q^{57} +(5.29796 + 3.05878i) q^{59} +(-20.3990 - 35.3321i) q^{61} +(-57.5908 - 99.7502i) q^{63} +(-9.51021 + 5.49072i) q^{65} +(-54.4898 + 94.3791i) q^{67} +(-7.80306 - 4.50510i) q^{69} +52.8829i q^{71} +68.7878 q^{73} +(14.3939 + 24.9309i) q^{75} +(182.985 + 105.646i) q^{77} +(12.7929 + 22.1579i) q^{79} +81.0000 q^{81} +(52.0857 - 30.0717i) q^{83} +(-4.98979 + 8.64258i) q^{85} +(39.5908 - 22.8578i) q^{87} -7.62809i q^{89} -35.8082 q^{91} +(-40.8031 + 70.6730i) q^{93} +(-73.4041 - 42.3799i) q^{95} +(50.7020 + 87.8185i) q^{97} +(-128.682 + 74.2944i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 6 q^{5} - 6 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 6 * q^5 - 6 * q^7 + 36 * q^9	$ 4 q - 12 q^{3} + 6 q^{5} - 6 q^{7} + 36 q^{9} - 18 q^{11} - 14 q^{13} - 18 q^{15} + 8 q^{19} + 18 q^{21} + 30 q^{23} + 20 q^{25} - 108 q^{27} + 6 q^{29} + 74 q^{31} + 54 q^{33} - 120 q^{37} + 42 q^{39} - 138 q^{41} + 10 q^{43} + 54 q^{45} + 174 q^{47} - 112 q^{49} + 220 q^{55} - 24 q^{57} - 18 q^{59} - 62 q^{61} - 54 q^{63} - 234 q^{65} - 22 q^{67} - 90 q^{69} + 40 q^{73} - 60 q^{75} + 438 q^{77} - 86 q^{79} + 324 q^{81} - 66 q^{83} + 176 q^{85} - 18 q^{87} - 300 q^{91} - 222 q^{93} - 372 q^{95} + 242 q^{97} - 162 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 6 * q^5 - 6 * q^7 + 36 * q^9 - 18 * q^11 - 14 * q^13 - 18 * q^15 + 8 * q^19 + 18 * q^21 + 30 * q^23 + 20 * q^25 - 108 * q^27 + 6 * q^29 + 74 * q^31 + 54 * q^33 - 120 * q^37 + 42 * q^39 - 138 * q^41 + 10 * q^43 + 54 * q^45 + 174 * q^47 - 112 * q^49 + 220 * q^55 - 24 * q^57 - 18 * q^59 - 62 * q^61 - 54 * q^63 - 234 * q^65 - 22 * q^67 - 90 * q^69 + 40 * q^73 - 60 * q^75 + 438 * q^77 - 86 * q^79 + 324 * q^81 - 66 * q^83 + 176 * q^85 - 18 * q^87 - 300 * q^91 - 222 * q^93 - 372 * q^95 + 242 * q^97 - 162 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$	−3.00000			−1.00000
$3$	−3.00000			−1.00000
$4$		0			0
$5$	−3.39898	−	1.96240i	−0.679796	−	0.392480i	0.119982	−	0.992776i	$-0.461716\pi$
$5$	−3.39898	−	1.96240i	−0.679796	−	0.392480i	−0.799778	+	0.600296i	$0.795050\pi$
$6$		0			0
$7$	−6.39898	−	11.0834i	−0.914140	−	1.58334i	−0.808156	−	0.588969i	$-0.799534\pi$
$7$	−6.39898	−	11.0834i	−0.914140	−	1.58334i	−0.105984	−	0.994368i	$-0.533799\pi$
$8$		0			0
$9$	9.00000			1.00000
$10$		0			0
$11$	−14.2980	+	8.25493i	−1.29981	+	0.750448i	−0.980372	−	0.197157i	$-0.936829\pi$
$11$	−14.2980	+	8.25493i	−1.29981	+	0.750448i	−0.319443	+	0.947606i	$0.603496\pi$
$12$		0			0
$13$	1.39898	−	2.42310i	0.107614	−	0.186393i	−0.807189	−	0.590293i	$-0.799012\pi$
$13$	1.39898	−	2.42310i	0.107614	−	0.186393i	0.914803	+	0.403900i	$0.132346\pi$
$14$		0			0
$15$	10.1969	+	5.88721i	0.679796	+	0.392480i
$16$		0			0
$17$		−	2.54270i		−	0.149570i	−0.997200	−	0.0747852i	$-0.976173\pi$
$17$		−	2.54270i		−	0.149570i	0.997200	−	0.0747852i	$-0.0238271\pi$
$18$		0			0
$19$	21.5959			1.13663			0.568314	−	0.822812i	$-0.307596\pi$
$19$	21.5959			1.13663			0.568314	+	0.822812i	$0.307596\pi$
$20$		0			0
$21$	19.1969	+	33.2501i	0.914140	+	1.58334i
$22$		0			0
$23$	2.60102	+	1.50170i	0.113088	+	0.0652913i	0.555477	−	0.831532i	$-0.312536\pi$
$23$	2.60102	+	1.50170i	0.113088	+	0.0652913i	−0.442389	+	0.896823i	$0.645869\pi$
$24$		0			0
$25$	−4.79796	−	8.31031i	−0.191918	−	0.332412i
$26$		0			0
$27$	−27.0000			−1.00000
$28$		0			0
$29$	−13.1969	+	7.61926i	−0.455067	+	0.262733i	−0.709968	−	0.704234i	$-0.751290\pi$
$29$	−13.1969	+	7.61926i	−0.455067	+	0.262733i	0.254901	+	0.966967i	$0.417957\pi$
$30$		0			0
$31$	13.6010	−	23.5577i	0.438743	−	0.759924i	−0.558850	−	0.829269i	$-0.688757\pi$
$31$	13.6010	−	23.5577i	0.438743	−	0.759924i	0.997593	+	0.0693442i	$0.0220907\pi$
$32$		0			0
$33$	42.8939	−	24.7648i	1.29981	−	0.750448i
$34$		0			0
$35$			50.2295i			1.43513i
$36$		0			0
$37$	−10.4041			−0.281191			−0.140596	−	0.990067i	$-0.544902\pi$
$37$	−10.4041			−0.281191			−0.140596	+	0.990067i	$0.544902\pi$
$38$		0			0
$39$	−4.19694	+	7.26931i	−0.107614	+	0.186393i
$40$		0			0
$41$	−34.5000	−	19.9186i	−0.841463	−	0.485819i	0.0162980	−	0.999867i	$-0.494812\pi$
$41$	−34.5000	−	19.9186i	−0.841463	−	0.485819i	−0.857761	+	0.514048i	$0.828145\pi$
$42$		0			0
$43$	−17.0959	−	29.6110i	−0.397579	−	0.688628i	0.595847	−	0.803098i	$-0.296816\pi$
$43$	−17.0959	−	29.6110i	−0.397579	−	0.688628i	−0.993427	+	0.114470i	$0.963483\pi$
$44$		0			0
$45$	−30.5908	−	17.6616i	−0.679796	−	0.392480i
$46$		0			0
$47$	58.1969	−	33.6000i	1.23823	−	0.714894i	0.269500	−	0.963000i	$-0.413142\pi$
$47$	58.1969	−	33.6000i	1.23823	−	0.714894i	0.968733	+	0.248106i	$0.0798083\pi$
$48$		0			0
$49$	−57.3939	+	99.4091i	−1.17130	+	2.02876i
$50$		0			0
$51$			7.62809i			0.149570i
$52$		0			0
$53$		−	100.680i		−	1.89963i	−0.312810	−	0.949816i	$-0.601270\pi$
$53$		−	100.680i		−	1.89963i	0.312810	−	0.949816i	$-0.398730\pi$
$54$		0			0
$55$	64.7980			1.17814
$56$		0			0
$57$	−64.7878			−1.13663
$58$		0			0
$59$	5.29796	+	3.05878i	0.0897959	+	0.0518437i	0.544226	−	0.838939i	$-0.316824\pi$
$59$	5.29796	+	3.05878i	0.0897959	+	0.0518437i	−0.454430	+	0.890783i	$0.650157\pi$
$60$		0			0
$61$	−20.3990	−	35.3321i	−0.334409	−	0.579214i	0.648962	−	0.760821i	$-0.275204\pi$
$61$	−20.3990	−	35.3321i	−0.334409	−	0.579214i	−0.983371	+	0.181607i	$0.941870\pi$
$62$		0			0
$63$	−57.5908	−	99.7502i	−0.914140	−	1.58334i
$64$		0			0
$65$	−9.51021	+	5.49072i	−0.146311	+	0.0844726i
$66$		0			0
$67$	−54.4898	+	94.3791i	−0.813281	+	1.40864i	0.0972755	+	0.995257i	$0.468987\pi$
$67$	−54.4898	+	94.3791i	−0.813281	+	1.40864i	−0.910556	+	0.413386i	$0.864346\pi$
$68$		0			0
$69$	−7.80306	−	4.50510i	−0.113088	−	0.0652913i
$70$		0			0
$71$			52.8829i			0.744830i	0.928066	+	0.372415i	$0.121470\pi$
$71$			52.8829i			0.744830i	−0.928066	+	0.372415i	$0.878530\pi$
$72$		0			0
$73$	68.7878			0.942298			0.471149	−	0.882054i	$-0.343839\pi$
$73$	68.7878			0.942298			0.471149	+	0.882054i	$0.343839\pi$
$74$		0			0
$75$	14.3939	+	24.9309i	0.191918	+	0.332412i
$76$		0			0
$77$	182.985	+	105.646i	2.37642	+	1.37203i
$78$		0			0
$79$	12.7929	+	22.1579i	0.161935	+	0.280479i	0.935563	−	0.353161i	$-0.114893\pi$
$79$	12.7929	+	22.1579i	0.161935	+	0.280479i	−0.773628	+	0.633640i	$0.781560\pi$
$80$		0			0
$81$	81.0000			1.00000
$82$		0			0
$83$	52.0857	−	30.0717i	0.627539	−	0.362310i	−0.152260	−	0.988341i	$-0.548655\pi$
$83$	52.0857	−	30.0717i	0.627539	−	0.362310i	0.779798	+	0.626031i	$0.215322\pi$
$84$		0			0
$85$	−4.98979	+	8.64258i	−0.0587035	+	0.101677i
$86$		0			0
$87$	39.5908	−	22.8578i	0.455067	−	0.262733i
$88$		0			0
$89$		−	7.62809i		−	0.0857089i	−0.999081	−	0.0428545i	$-0.986355\pi$
$89$		−	7.62809i		−	0.0857089i	0.999081	−	0.0428545i	$-0.0136452\pi$
$90$		0			0
$91$	−35.8082			−0.393496
$92$		0			0
$93$	−40.8031	+	70.6730i	−0.438743	+	0.759924i
$94$		0			0
$95$	−73.4041	−	42.3799i	−0.772675	−	0.446104i
$96$		0			0
$97$	50.7020	+	87.8185i	0.522701	+	0.905345i	0.999651	+	0.0264148i	$0.00840908\pi$
$97$	50.7020	+	87.8185i	0.522701	+	0.905345i	−0.476950	+	0.878931i	$0.658258\pi$
$98$		0			0
$99$	−128.682	+	74.2944i	−1.29981	+	0.750448i
$100$		0			0
$101$	−1.19694	+	0.691053i	−0.0118509	+	0.00684211i	−0.505914	−	0.862584i	$-0.668845\pi$
$101$	−1.19694	+	0.691053i	−0.0118509	+	0.00684211i	0.494063	+	0.869426i	$0.335511\pi$
$102$		0			0
$103$	0.792856	−	1.37327i	0.00769763	−	0.0133327i	−0.862151	−	0.506652i	$-0.830883\pi$
$103$	0.792856	−	1.37327i	0.00769763	−	0.0133327i	0.869849	+	0.493319i	$0.164216\pi$
$104$		0			0
$105$		−	150.688i		−	1.43513i
$106$		0			0
$107$		−	103.223i		−	0.964702i	−0.875978	−	0.482351i	$-0.839783\pi$
$107$		−	103.223i		−	0.964702i	0.875978	−	0.482351i	$-0.160217\pi$
$108$		0			0
$109$	14.4041			0.132148			0.0660738	−	0.997815i	$-0.478953\pi$
$109$	14.4041			0.132148			0.0660738	+	0.997815i	$0.478953\pi$
$110$		0			0
$111$	31.2122			0.281191
$112$		0			0
$113$	−92.0755	−	53.1598i	−0.814828	−	0.470441i	0.0338020	−	0.999429i	$-0.489238\pi$
$113$	−92.0755	−	53.1598i	−0.814828	−	0.470441i	−0.848630	+	0.528988i	$0.822572\pi$
$114$		0			0
$115$	−5.89388	−	10.2085i	−0.0512511	−	0.0887695i
$116$		0			0
$117$	12.5908	−	21.8079i	0.107614	−	0.186393i
$118$		0			0
$119$	−28.1816	+	16.2707i	−0.236820	+	0.136728i
$120$		0			0
$121$	75.7878	−	131.268i	0.626345	−	1.08486i
$122$		0			0
$123$	103.500	+	59.7558i	0.841463	+	0.485819i
$124$		0			0
$125$			135.782i			1.08626i
$126$		0			0
$127$	−183.980			−1.44866			−0.724329	−	0.689454i	$-0.757850\pi$
$127$	−183.980			−1.44866			−0.724329	+	0.689454i	$0.757850\pi$
$128$		0			0
$129$	51.2878	+	88.8330i	0.397579	+	0.688628i
$130$		0			0
$131$	−91.6918	−	52.9383i	−0.699938	−	0.404109i	0.107387	−	0.994217i	$-0.465752\pi$
$131$	−91.6918	−	52.9383i	−0.699938	−	0.404109i	−0.807324	+	0.590108i	$0.799085\pi$
$132$		0			0
$133$	−138.192	−	239.355i	−1.03904	−	1.79966i
$134$		0			0
$135$	91.7724	+	52.9848i	0.679796	+	0.392480i
$136$		0			0
$137$	−165.288	+	95.4289i	−1.20648	+	0.696562i	−0.961989	−	0.273090i	$-0.911954\pi$
$137$	−165.288	+	95.4289i	−1.20648	+	0.696562i	−0.244491	+	0.969651i	$0.578621\pi$
$138$		0			0
$139$	47.4898	−	82.2547i	0.341653	−	0.591761i	−0.643087	−	0.765793i	$-0.722347\pi$
$139$	47.4898	−	82.2547i	0.341653	−	0.591761i	0.984740	+	0.174033i	$0.0556798\pi$
$140$		0			0
$141$	−174.591	+	100.800i	−1.23823	+	0.714894i
$142$		0			0
$143$			46.1939i			0.323034i
$144$		0			0
$145$	59.8082			0.412470
$146$		0			0
$147$	172.182	−	298.227i	1.17130	−	2.02876i
$148$		0			0
$149$	79.6112	+	45.9636i	0.534304	+	0.308480i	0.742767	−	0.669550i	$-0.233513\pi$
$149$	79.6112	+	45.9636i	0.534304	+	0.308480i	−0.208464	+	0.978030i	$0.566846\pi$
$150$		0			0
$151$	8.20714	+	14.2152i	0.0543519	+	0.0941403i	0.891921	−	0.452191i	$-0.149357\pi$
$151$	8.20714	+	14.2152i	0.0543519	+	0.0941403i	−0.837569	+	0.546331i	$0.816024\pi$
$152$		0			0
$153$		−	22.8843i		−	0.149570i
$154$		0			0
$155$	−92.4592	+	53.3813i	−0.596511	+	0.344396i
$156$		0			0
$157$	105.985	−	183.571i	0.675062	−	1.16924i	−0.301389	−	0.953501i	$-0.597450\pi$
$157$	105.985	−	183.571i	0.675062	−	1.16924i	0.976451	−	0.215740i	$-0.0692164\pi$
$158$		0			0
$159$			302.041i			1.89963i
$160$		0			0
$161$		−	38.4374i		−	0.238742i
$162$		0			0
$163$	172.788			1.06005			0.530024	−	0.847983i	$-0.322183\pi$
$163$	172.788			1.06005			0.530024	+	0.847983i	$0.322183\pi$
$164$		0			0
$165$	−194.394			−1.17814
$166$		0			0
$167$	−163.197	−	94.2218i	−0.977227	−	0.564202i	−0.0757953	−	0.997123i	$-0.524150\pi$
$167$	−163.197	−	94.2218i	−0.977227	−	0.564202i	−0.901432	+	0.432921i	$0.857483\pi$
$168$		0			0
$169$	80.5857	+	139.579i	0.476839	+	0.825909i
$170$		0			0
$171$	194.363			1.13663
$172$		0			0
$173$	52.1969	−	30.1359i	0.301716	−	0.174196i	−0.341497	−	0.939883i	$-0.610934\pi$
$173$	52.1969	−	30.1359i	0.301716	−	0.174196i	0.643214	+	0.765687i	$0.277601\pi$
$174$		0			0
$175$	−61.4041	+	106.355i	−0.350880	+	0.607743i
$176$		0			0
$177$	−15.8939	−	9.17633i	−0.0897959	−	0.0518437i
$178$		0			0
$179$			72.7108i			0.406205i	0.979157	+	0.203103i	$0.0651025\pi$
$179$			72.7108i			0.406205i	−0.979157	+	0.203103i	$0.934897\pi$
$180$		0			0
$181$	−97.5959			−0.539204			−0.269602	−	0.962972i	$-0.586892\pi$
$181$	−97.5959			−0.539204			−0.269602	+	0.962972i	$0.586892\pi$
$182$		0			0
$183$	61.1969	+	105.996i	0.334409	+	0.579214i
$184$		0			0
$185$	35.3633	+	20.4170i	0.191153	+	0.110362i
$186$		0			0
$187$	20.9898	+	36.3554i	0.112245	+	0.194414i
$188$		0			0
$189$	172.772	+	299.251i	0.914140	+	1.58334i
$190$		0			0
$191$	282.560	−	163.136i	1.47937	−	0.854116i	0.479645	−	0.877462i	$-0.340765\pi$
$191$	282.560	−	163.136i	1.47937	−	0.854116i	0.999727	+	0.0233462i	$0.00743200\pi$
$192$		0			0
$193$	−10.5102	+	18.2042i	−0.0544570	+	0.0943223i	−0.891969	−	0.452097i	$-0.850676\pi$
$193$	−10.5102	+	18.2042i	−0.0544570	+	0.0943223i	0.837512	+	0.546419i	$0.184009\pi$
$194$		0			0
$195$	28.5306	−	16.4722i	0.146311	−	0.0844726i
$196$		0			0
$197$			32.5413i			0.165184i	0.996583	+	0.0825922i	$0.0263199\pi$
$197$			32.5413i			0.165184i	−0.996583	+	0.0825922i	$0.973680\pi$
$198$		0			0
$199$	−62.0000			−0.311558			−0.155779	−	0.987792i	$-0.549789\pi$
$199$	−62.0000			−0.311558			−0.155779	+	0.987792i	$0.549789\pi$
$200$		0			0
$201$	163.469	−	283.137i	0.813281	−	1.40864i
$202$		0			0
$203$	168.894	+	97.5109i	0.831990	+	0.480349i
$204$		0			0
$205$	78.1765	+	135.406i	0.381349	+	0.660516i
$206$		0			0
$207$	23.4092	+	13.5153i	0.113088	+	0.0652913i
$208$		0			0
$209$	−308.778	+	178.273i	−1.47740	+	0.852980i
$210$		0			0
$211$	84.2980	−	146.008i	0.399516	−	0.691983i	−0.594150	−	0.804354i	$-0.702511\pi$
$211$	84.2980	−	146.008i	0.399516	−	0.691983i	0.993666	+	0.112372i	$0.0358447\pi$
$212$		0			0
$213$		−	158.649i		−	0.744830i
$214$		0			0
$215$			134.196i			0.624169i
$216$		0			0
$217$	−348.131			−1.60429
$218$		0			0
$219$	−206.363			−0.942298
$220$		0			0
$221$	−6.16122	−	3.55718i	−0.0278788	−	0.0160958i
$222$		0			0
$223$	78.1867	+	135.423i	0.350613	+	0.607280i	0.986357	−	0.164620i	$-0.0526399\pi$
$223$	78.1867	+	135.423i	0.350613	+	0.607280i	−0.635744	+	0.771900i	$0.719307\pi$
$224$		0			0
$225$	−43.1816	−	74.7928i	−0.191918	−	0.332412i
$226$		0			0
$227$	86.6510	−	50.0280i	0.381723	−	0.220388i	−0.296845	−	0.954926i	$-0.595934\pi$
$227$	86.6510	−	50.0280i	0.381723	−	0.220388i	0.678567	+	0.734538i	$0.262601\pi$
$228$		0			0
$229$	104.995	−	181.856i	0.458493	−	0.794133i	−0.540389	−	0.841416i	$-0.681723\pi$
$229$	104.995	−	181.856i	0.458493	−	0.794133i	0.998882	+	0.0472824i	$0.0150561\pi$
$230$		0			0
$231$	−548.954	−	316.939i	−2.37642	−	1.37203i
$232$		0			0
$233$		−	307.127i		−	1.31814i	−0.752081	−	0.659070i	$-0.770950\pi$
$233$		−	307.127i		−	1.31814i	0.752081	−	0.659070i	$-0.229050\pi$
$234$		0			0
$235$	−263.747			−1.12233
$236$		0			0
$237$	−38.3786	−	66.4736i	−0.161935	−	0.280479i
$238$		0			0
$239$	−70.3888	−	40.6390i	−0.294514	−	0.170038i	0.345462	−	0.938433i	$-0.387722\pi$
$239$	−70.3888	−	40.6390i	−0.294514	−	0.170038i	−0.639976	+	0.768395i	$0.721056\pi$
$240$		0			0
$241$	180.096	+	311.935i	0.747286	+	1.29434i	0.949119	+	0.314917i	$0.101977\pi$
$241$	180.096	+	311.935i	0.747286	+	1.29434i	−0.201833	+	0.979420i	$0.564690\pi$
$242$		0			0
$243$	−243.000			−1.00000
$244$		0			0
$245$	390.161	−	225.260i	1.59249	−	0.919427i
$246$		0			0
$247$	30.2122	−	52.3291i	0.122317	−	0.211859i
$248$		0			0
$249$	−156.257	+	90.2151i	−0.627539	+	0.362310i
$250$		0			0
$251$		−	400.179i		−	1.59434i	−0.603755	−	0.797170i	$-0.706330\pi$
$251$		−	400.179i		−	1.59434i	0.603755	−	0.797170i	$-0.293670\pi$
$252$		0			0
$253$	−49.5857			−0.195991
$254$		0			0
$255$	14.9694	−	25.9277i	0.0587035	−	0.101677i
$256$		0			0
$257$	315.035	+	181.885i	1.22582	+	0.707725i	0.966152	−	0.257974i	$-0.0830549\pi$
$257$	315.035	+	181.885i	1.22582	+	0.707725i	0.259664	+	0.965699i	$0.416388\pi$
$258$		0			0
$259$	66.5755	+	115.312i	0.257048	+	0.445221i
$260$		0			0
$261$	−118.772	+	68.5733i	−0.455067	+	0.262733i
$262$		0			0
$263$	−326.570	+	188.546i	−1.24171	+	0.716903i	−0.969443	−	0.245318i	$-0.921108\pi$
$263$	−326.570	+	188.546i	−1.24171	+	0.716903i	−0.272270	+	0.962221i	$0.587774\pi$
$264$		0			0
$265$	−197.576	+	342.211i	−0.745568	+	1.29136i
$266$		0			0
$267$			22.8843i			0.0857089i
$268$		0			0
$269$		−	170.849i		−	0.635125i	−0.948237	−	0.317562i	$-0.897136\pi$
$269$		−	170.849i		−	0.635125i	0.948237	−	0.317562i	$-0.102864\pi$
$270$		0			0
$271$	−50.4041			−0.185993			−0.0929965	−	0.995666i	$-0.529645\pi$
$271$	−50.4041			−0.185993			−0.0929965	+	0.995666i	$0.529645\pi$
$272$		0			0
$273$	107.424			0.393496
$274$		0			0
$275$	137.202	+	79.2136i	0.498917	+	0.288050i
$276$		0			0
$277$	−12.8031	−	22.1756i	−0.0462204	−	0.0800561i	0.841990	−	0.539494i	$-0.181384\pi$
$277$	−12.8031	−	22.1756i	−0.0462204	−	0.0800561i	−0.888210	+	0.459438i	$0.848051\pi$
$278$		0			0
$279$	122.409	−	212.019i	0.438743	−	0.759924i
$280$		0			0
$281$	290.267	−	167.586i	1.03298	−	0.596391i	0.115143	−	0.993349i	$-0.463267\pi$
$281$	290.267	−	167.586i	1.03298	−	0.596391i	0.917837	+	0.396958i	$0.129934\pi$
$282$		0			0
$283$	7.48979	−	12.9727i	0.0264657	−	0.0458399i	−0.852489	−	0.522745i	$-0.824908\pi$
$283$	7.48979	−	12.9727i	0.0264657	−	0.0458399i	0.878955	+	0.476905i	$0.158241\pi$
$284$		0			0
$285$	220.212	+	127.140i	0.772675	+	0.446104i
$286$		0			0
$287$			509.834i			1.77643i
$288$		0			0
$289$	282.535			0.977629
$290$		0			0
$291$	−152.106	−	263.456i	−0.522701	−	0.905345i
$292$		0			0
$293$	−261.015	−	150.697i	−0.890837	−	0.514325i	−0.0166210	−	0.999862i	$-0.505291\pi$
$293$	−261.015	−	150.697i	−0.890837	−	0.514325i	−0.874216	+	0.485537i	$0.838624\pi$
$294$		0			0
$295$	−12.0051	−	20.7934i	−0.0406953	−	0.0704863i
$296$		0			0
$297$	386.045	−	222.883i	1.29981	−	0.750448i
$298$		0			0
$299$	7.27755	−	4.20169i	0.0243396	−	0.0140525i
$300$		0			0
$301$	−218.793	+	378.960i	−0.726887	+	1.25900i
$302$		0			0
$303$	3.59082	−	2.07316i	0.0118509	−	0.00684211i
$304$		0			0
$305$			160.124i			0.524997i
$306$		0			0
$307$	−44.7469			−0.145755			−0.0728777	−	0.997341i	$-0.523218\pi$
$307$	−44.7469			−0.145755			−0.0728777	+	0.997341i	$0.523218\pi$
$308$		0			0
$309$	−2.37857	+	4.11980i	−0.00769763	+	0.0133327i
$310$		0			0
$311$	281.348	+	162.436i	0.904656	+	0.522303i	0.878708	−	0.477360i	$-0.158406\pi$
$311$	281.348	+	162.436i	0.904656	+	0.522303i	0.0259480	+	0.999663i	$0.491740\pi$
$312$		0			0
$313$	−156.894	−	271.748i	−0.501258	−	0.868205i	−0.999999	−	0.00145368i	$-0.999537\pi$
$313$	−156.894	−	271.748i	−0.501258	−	0.868205i	0.498741	−	0.866751i	$-0.333796\pi$
$314$		0			0
$315$			452.065i			1.43513i
$316$		0			0
$317$	−47.9847	+	27.7040i	−0.151371	+	0.0873942i	−0.573773	−	0.819015i	$-0.694521\pi$
$317$	−47.9847	+	27.7040i	−0.151371	+	0.0873942i	0.422401	+	0.906409i	$0.361187\pi$
$318$		0			0
$319$	125.793	−	217.880i	0.394335	−	0.683008i
$320$		0			0
$321$			309.669i			0.964702i
$322$		0			0
$323$		−	54.9119i		−	0.170006i
$324$		0			0
$325$	−26.8490			−0.0826123
$326$		0			0
$327$	−43.2122			−0.132148
$328$		0			0
$329$	−744.802	−	430.012i	−2.26384	−	1.30703i
$330$		0			0
$331$	−7.70204	−	13.3403i	−0.0232690	−	0.0403031i	0.854156	−	0.520016i	$-0.174074\pi$
$331$	−7.70204	−	13.3403i	−0.0232690	−	0.0403031i	−0.877425	+	0.479713i	$0.840741\pi$
$332$		0			0
$333$	−93.6367			−0.281191
$334$		0			0
$335$	370.419	−	213.862i	1.10573	−	0.638393i
$336$		0			0
$337$	−37.8837	+	65.6164i	−0.112414	+	0.194708i	−0.916743	−	0.399477i	$-0.869192\pi$
$337$	−37.8837	+	65.6164i	−0.112414	+	0.194708i	0.804329	+	0.594184i	$0.202525\pi$
$338$		0			0
$339$	276.227	+	159.479i	0.814828	+	0.470441i
$340$		0			0
$341$			449.102i			1.31701i
$342$		0			0
$343$	841.949			2.45466
$344$		0			0
$345$	17.6816	+	30.6255i	0.0512511	+	0.0887695i
$346$		0			0
$347$	578.651	+	334.084i	1.66758	+	0.962779i	0.968936	+	0.247312i	$0.0795472\pi$
$347$	578.651	+	334.084i	1.66758	+	0.962779i	0.698646	+	0.715467i	$0.253786\pi$
$348$		0			0
$349$	123.015	+	213.069i	0.352479	+	0.610512i	0.986683	−	0.162654i	$-0.0520053\pi$
$349$	123.015	+	213.069i	0.352479	+	0.610512i	−0.634204	+	0.773166i	$0.718672\pi$
$350$		0			0
$351$	−37.7724	+	65.4238i	−0.107614	+	0.186393i
$352$		0			0
$353$	156.510	−	90.3612i	0.443372	−	0.255981i	−0.261655	−	0.965161i	$-0.584268\pi$
$353$	156.510	−	90.3612i	0.443372	−	0.255981i	0.705027	+	0.709181i	$0.250935\pi$
$354$		0			0
$355$	103.778	−	179.748i	0.292331	−	0.506332i
$356$		0			0
$357$	84.5449	−	48.8120i	0.236820	−	0.136728i
$358$		0			0
$359$		−	256.273i		−	0.713852i	−0.934133	−	0.356926i	$-0.883825\pi$
$359$		−	256.273i		−	0.713852i	0.934133	−	0.356926i	$-0.116175\pi$
$360$		0			0
$361$	105.384			0.291922
$362$		0			0
$363$	−227.363	+	393.805i	−0.626345	+	1.08486i
$364$		0			0
$365$	−233.808	−	134.989i	−0.640570	−	0.369833i
$366$		0			0
$367$	−270.358	−	468.274i	−0.736671	−	1.27595i	−0.953986	−	0.299850i	$-0.903063\pi$
$367$	−270.358	−	468.274i	−0.736671	−	1.27595i	0.217316	−	0.976101i	$-0.430270\pi$
$368$		0			0
$369$	−310.500	−	179.267i	−0.841463	−	0.485819i
$370$		0			0
$371$	−1115.88	+	644.252i	−3.00776	+	1.73653i
$372$		0			0
$373$	85.9847	−	148.930i	0.230522	−	0.399276i	−0.727440	−	0.686171i	$-0.759290\pi$
$373$	85.9847	−	148.930i	0.230522	−	0.399276i	0.957962	+	0.286896i	$0.0926233\pi$
$374$		0			0
$375$		−	407.347i		−	1.08626i
$376$		0			0
$377$			42.6367i			0.113095i
$378$		0			0
$379$	−170.849			−0.450789			−0.225394	−	0.974268i	$-0.572367\pi$
$379$	−170.849			−0.450789			−0.225394	+	0.974268i	$0.572367\pi$
$380$		0			0
$381$	551.939			1.44866
$382$		0			0
$383$	−505.681	−	291.955i	−1.32031	−	0.762284i	−0.336536	−	0.941671i	$-0.609255\pi$
$383$	−505.681	−	291.955i	−1.32031	−	0.762284i	−0.983779	+	0.179386i	$0.942589\pi$
$384$		0			0
$385$	−414.641	−	718.179i	−1.07699	−	1.86540i
$386$		0			0
$387$	−153.863	−	266.499i	−0.397579	−	0.688628i
$388$		0			0
$389$	−384.752	+	222.137i	−0.989080	+	0.571045i	−0.904999	−	0.425413i	$-0.860129\pi$
$389$	−384.752	+	222.137i	−0.989080	+	0.571045i	−0.0840807	+	0.996459i	$0.526795\pi$
$390$		0			0
$391$	3.81837	−	6.61361i	0.00976565	−	0.0169146i
$392$		0			0
$393$	275.076	+	158.815i	0.699938	+	0.404109i
$394$		0			0
$395$		−	100.419i		−	0.254225i
$396$		0			0
$397$	−575.090			−1.44859			−0.724294	−	0.689491i	$-0.757834\pi$
$397$	−575.090			−1.44859			−0.724294	+	0.689491i	$0.757834\pi$
$398$		0			0
$399$	414.576	+	718.066i	1.03904	+	1.79966i
$400$		0			0
$401$	141.318	+	81.5902i	0.352415	+	0.203467i	0.665748	−	0.746176i	$-0.268112\pi$
$401$	141.318	+	81.5902i	0.352415	+	0.203467i	−0.313333	+	0.949643i	$0.601446\pi$
$402$		0			0
$403$	−38.0551	−	65.9134i	−0.0944295	−	0.163557i
$404$		0			0
$405$	−275.317	−	158.955i	−0.679796	−	0.392480i
$406$		0			0
$407$	148.757	−	85.8850i	0.365497	−	0.211020i
$408$		0			0
$409$	300.641	−	520.725i	0.735063	−	1.27317i	−0.219633	−	0.975583i	$-0.570486\pi$
$409$	300.641	−	520.725i	0.735063	−	1.27317i	0.954696	−	0.297584i	$-0.0961808\pi$
$410$		0			0
$411$	495.863	−	286.287i	1.20648	−	0.696562i
$412$		0			0
$413$		−	78.2922i		−	0.189570i
$414$		0			0
$415$	−236.051			−0.568798
$416$		0			0
$417$	−142.469	+	246.764i	−0.341653	+	0.591761i
$418$		0			0
$419$	−158.620	−	91.5795i	−0.378569	−	0.218567i	0.298626	−	0.954370i	$-0.403472\pi$
$419$	−158.620	−	91.5795i	−0.378569	−	0.218567i	−0.677195	+	0.735803i	$0.736805\pi$
$420$		0			0
$421$	−83.7724	−	145.098i	−0.198984	−	0.344651i	0.749215	−	0.662327i	$-0.230431\pi$
$421$	−83.7724	−	145.098i	−0.198984	−	0.344651i	−0.948199	+	0.317676i	$0.897098\pi$
$422$		0			0
$423$	523.772	−	302.400i	1.23823	−	0.714894i
$424$		0			0
$425$	−21.1306	+	12.1998i	−0.0497191	+	0.0287053i
$426$		0			0
$427$	−261.065	+	452.178i	−0.611394	+	1.05897i
$428$		0			0
$429$		−	138.582i		−	0.323034i
$430$		0			0
$431$		−	644.766i		−	1.49598i	−0.663712	−	0.747988i	$-0.731020\pi$
$431$		−	644.766i		−	1.49598i	0.663712	−	0.747988i	$-0.268980\pi$
$432$		0			0
$433$	133.514			0.308347			0.154174	−	0.988044i	$-0.450729\pi$
$433$	133.514			0.308347			0.154174	+	0.988044i	$0.450729\pi$
$434$		0			0
$435$	−179.424			−0.412470
$436$		0			0
$437$	56.1714	+	32.4306i	0.128539	+	0.0742119i
$438$		0			0
$439$	46.2276	+	80.0685i	0.105302	+	0.182388i	0.913862	−	0.406026i	$-0.133086\pi$
$439$	46.2276	+	80.0685i	0.105302	+	0.182388i	−0.808560	+	0.588414i	$0.799752\pi$
$440$		0			0
$441$	−516.545	+	894.682i	−1.17130	+	2.02876i
$442$		0			0
$443$	−150.157	+	86.6933i	−0.338955	+	0.195696i	−0.659810	−	0.751433i	$-0.729363\pi$
$443$	−150.157	+	86.6933i	−0.338955	+	0.195696i	0.320855	+	0.947128i	$0.396030\pi$
$444$		0			0
$445$	−14.9694	+	25.9277i	−0.0336391	+	0.0582646i
$446$		0			0
$447$	−238.834	−	137.891i	−0.534304	−	0.308480i
$448$		0			0
$449$			430.692i			0.959224i	0.877481	+	0.479612i	$0.159223\pi$
$449$			430.692i			0.959224i	−0.877481	+	0.479612i	$0.840777\pi$
$450$		0			0
$451$	657.706			1.45833
$452$		0			0
$453$	−24.6214	−	42.6456i	−0.0543519	−	0.0941403i
$454$		0			0
$455$	121.711	+	70.2700i	0.267497	+	0.154440i
$456$		0			0
$457$	262.843	+	455.257i	0.575148	+	0.996186i	0.996025	+	0.0890687i	$0.0283891\pi$
$457$	262.843	+	455.257i	0.575148	+	0.996186i	−0.420877	+	0.907118i	$0.638278\pi$
$458$		0			0
$459$			68.6528i			0.149570i
$460$		0			0
$461$	1.54999	−	0.894890i	0.00336224	−	0.00194119i	−0.498318	−	0.866994i	$-0.666049\pi$
$461$	1.54999	−	0.894890i	0.00336224	−	0.00194119i	0.501680	+	0.865053i	$0.332715\pi$
$462$		0			0
$463$	−263.166	+	455.817i	−0.568394	+	0.984487i	0.428331	+	0.903622i	$0.359102\pi$
$463$	−263.166	+	455.817i	−0.568394	+	0.984487i	−0.996725	+	0.0808651i	$0.974232\pi$
$464$		0			0
$465$	277.378	−	160.144i	0.596511	−	0.344396i
$466$		0			0
$467$			409.322i			0.876494i	0.898855	+	0.438247i	$0.144400\pi$
$467$			409.322i			0.876494i	−0.898855	+	0.438247i	$0.855600\pi$
$468$		0			0
$469$	1394.72			2.97381
$470$		0			0
$471$	−317.954	+	550.713i	−0.675062	+	1.16924i
$472$		0			0
$473$	488.873	+	282.251i	1.03356	+	0.596726i
$474$		0			0
$475$	−103.616	−	179.469i	−0.218140	−	0.377829i
$476$		0			0
$477$		−	906.124i		−	1.89963i
$478$		0			0
$479$	525.207	−	303.228i	1.09647	−	0.633045i	0.161175	−	0.986926i	$-0.448472\pi$
$479$	525.207	−	303.228i	1.09647	−	0.633045i	0.935290	+	0.353881i	$0.115138\pi$
$480$		0			0
$481$	−14.5551	+	25.2102i	−0.0302601	+	0.0524120i
$482$		0			0
$483$			115.312i			0.238742i
$484$		0			0
$485$		−	397.991i		−	0.820600i
$486$		0			0
$487$	−275.131			−0.564950			−0.282475	−	0.959275i	$-0.591155\pi$
$487$	−275.131			−0.564950			−0.282475	+	0.959275i	$0.591155\pi$
$488$		0			0
$489$	−518.363			−1.06005
$490$		0			0
$491$	−127.469	−	73.5945i	−0.259612	−	0.149887i	0.364546	−	0.931186i	$-0.381224\pi$
$491$	−127.469	−	73.5945i	−0.259612	−	0.149887i	−0.624157	+	0.781299i	$0.714558\pi$
$492$		0			0
$493$	19.3735	+	33.5558i	0.0392971	+	0.0680646i
$494$		0			0
$495$	583.182			1.17814
$496$		0			0
$497$	586.120	−	338.397i	1.17932	−	0.680879i
$498$		0			0
$499$	−226.308	+	391.977i	−0.453523	+	0.785526i	−0.998602	−	0.0528594i	$-0.983166\pi$
$499$	−226.308	+	391.977i	−0.453523	+	0.785526i	0.545079	+	0.838385i	$0.316500\pi$
$500$		0			0
$501$	489.591	+	282.665i	0.977227	+	0.564202i
$502$		0			0
$503$			571.541i			1.13627i	0.822937	+	0.568133i	$0.192334\pi$
$503$			571.541i			1.13627i	−0.822937	+	0.568133i	$0.807666\pi$
$504$		0			0
$505$	5.42449			0.0107416
$506$		0			0
$507$	−241.757	−	418.736i	−0.476839	−	0.825909i
$508$		0			0
$509$	394.136	+	227.554i	0.774333	+	0.447062i	0.834418	−	0.551132i	$-0.185804\pi$
$509$	394.136	+	227.554i	0.774333	+	0.447062i	−0.0600849	+	0.998193i	$0.519137\pi$
$510$		0			0
$511$	−440.171	−	762.399i	−0.861392	−	1.49198i
$512$		0			0
$513$	−583.090			−1.13663
$514$		0			0
$515$	−5.38981	+	3.11181i	−0.0104656	+	0.00604234i
$516$		0			0
$517$	−554.732	+	960.823i	−1.07298	+	1.85846i
$518$		0			0
$519$	−156.591	+	90.4077i	−0.301716	+	0.174196i
$520$		0			0
$521$			846.127i			1.62404i	0.583627	+	0.812022i	$0.301633\pi$
$521$			846.127i			1.62404i	−0.583627	+	0.812022i	$0.698367\pi$
$522$		0			0
$523$	−487.616			−0.932345			−0.466172	−	0.884694i	$-0.654367\pi$
$523$	−487.616			−0.932345			−0.466172	+	0.884694i	$0.654367\pi$
$524$		0			0
$525$	184.212	−	319.065i	0.350880	−	0.607743i
$526$		0			0
$527$	−59.9000	−	34.5833i	−0.113662	−	0.0656229i
$528$		0			0
$529$	−259.990	−	450.316i	−0.491474	−	0.851258i
$530$		0			0
$531$	47.6816	+	27.5290i	0.0897959	+	0.0518437i
$532$		0			0
$533$	−96.5296	+	55.7314i	−0.181106	+	0.104562i
$534$		0			0
$535$	−202.565	+	350.853i	−0.378627	+	0.655801i
$536$		0			0
$537$		−	218.132i		−	0.406205i
$538$		0			0
$539$		−	1895.13i		−	3.51601i
$540$		0			0
$541$	608.302			1.12440			0.562202	−	0.827000i	$-0.309955\pi$
$541$	608.302			1.12440			0.562202	+	0.827000i	$0.309955\pi$
$542$		0			0
$543$	292.788			0.539204
$544$		0			0
$545$	−48.9592	−	28.2666i	−0.0898334	−	0.0518653i
$546$		0			0
$547$	431.671	+	747.677i	0.789162	+	1.36687i	0.926481	+	0.376341i	$0.122818\pi$
$547$	431.671	+	747.677i	0.789162	+	1.36687i	−0.137319	+	0.990527i	$0.543849\pi$
$548$		0			0
$549$	−183.591	−	317.989i	−0.334409	−	0.579214i
$550$		0			0
$551$	−285.000	+	164.545i	−0.517241	+	0.298629i
$552$		0			0
$553$	163.722	−	283.576i	0.296062	−	0.512795i
$554$		0			0
$555$	−106.090	−	61.2510i	−0.191153	−	0.110362i
$556$		0			0
$557$			331.526i			0.595200i	0.954691	+	0.297600i	$0.0961861\pi$
$557$			331.526i			0.595200i	−0.954691	+	0.297600i	$0.903814\pi$
$558$		0			0
$559$	−95.6674			−0.171140
$560$		0			0
$561$	−62.9694	−	109.066i	−0.112245	−	0.194414i
$562$		0			0
$563$	514.024	+	296.772i	0.913010	+	0.527126i	0.881398	−	0.472374i	$-0.156603\pi$
$563$	514.024	+	296.772i	0.913010	+	0.527126i	0.0316114	+	0.999500i	$0.489936\pi$
$564$		0			0
$565$	208.642	+	361.378i	0.369278	+	0.639608i
$566$		0			0
$567$	−518.317	−	897.752i	−0.914140	−	1.58334i
$568$		0			0
$569$	889.155	−	513.354i	1.56266	−	0.902204i	0.565676	−	0.824627i	$-0.308615\pi$
$569$	889.155	−	513.354i	1.56266	−	0.902204i	0.996986	−	0.0775764i	$-0.0247182\pi$
$570$		0			0
$571$	−191.843	+	332.282i	−0.335977	+	0.581929i	−0.983672	−	0.179971i	$-0.942400\pi$
$571$	−191.843	+	332.282i	−0.335977	+	0.581929i	0.647695	+	0.761900i	$0.275733\pi$
$572$		0			0
$573$	−847.681	+	489.409i	−1.47937	+	0.854116i
$574$		0			0
$575$		−	28.8204i		−	0.0501224i
$576$		0			0
$577$	−777.433			−1.34737			−0.673685	−	0.739019i	$-0.735290\pi$
$577$	−777.433			−1.34737			−0.673685	+	0.739019i	$0.735290\pi$
$578$		0			0
$579$	31.5306	−	54.6126i	0.0544570	−	0.0943223i
$580$		0			0
$581$	−666.591	−	384.856i	−1.14732	−	0.662403i
$582$		0			0
$583$	831.110	+	1439.53i	1.42557	+	2.46917i
$584$		0			0
$585$	−85.5918	+	49.4165i	−0.146311	+	0.0844726i
$586$		0			0
$587$	581.520	−	335.741i	0.990665	−	0.571961i	0.0851921	−	0.996365i	$-0.472850\pi$
$587$	581.520	−	335.741i	0.990665	−	0.571961i	0.905473	+	0.424404i	$0.139516\pi$
$588$		0			0
$589$	293.727	−	508.749i	0.498687	−	0.863751i
$590$		0			0
$591$		−	97.6240i		−	0.165184i
$592$		0			0
$593$		−	536.457i		−	0.904650i	−0.891853	−	0.452325i	$-0.850595\pi$
$593$		−	536.457i		−	0.904650i	0.891853	−	0.452325i	$-0.149405\pi$
$594$		0			0
$595$	127.718			0.214653
$596$		0			0
$597$	186.000			0.311558
$598$		0			0
$599$	−735.438	−	424.605i	−1.22778	−	0.708857i	−0.261212	−	0.965282i	$-0.584122\pi$
$599$	−735.438	−	424.605i	−1.22778	−	0.708857i	−0.966564	+	0.256425i	$0.917455\pi$
$600$		0			0
$601$	−330.692	−	572.775i	−0.550236	−	0.953037i	−0.998257	−	0.0590138i	$-0.981204\pi$
$601$	−330.692	−	572.775i	−0.550236	−	0.953037i	0.448021	−	0.894023i	$-0.352129\pi$
$602$		0			0
$603$	−490.408	+	849.412i	−0.813281	+	1.40864i
$604$		0			0
$605$	−515.202	+	297.452i	−0.851574	+	0.491656i
$606$		0			0
$607$	10.9541	−	18.9730i	0.0180463	−	0.0312570i	−0.856861	−	0.515547i	$-0.827589\pi$
$607$	10.9541	−	18.9730i	0.0180463	−	0.0312570i	0.874908	+	0.484290i	$0.160922\pi$
$608$		0			0
$609$	−506.682	−	292.533i	−0.831990	−	0.480349i
$610$		0			0
$611$		−	188.023i		−	0.307730i
$612$		0			0
$613$	−1164.75			−1.90008			−0.950038	−	0.312133i	$-0.898956\pi$
$613$	−1164.75			−1.90008			−0.950038	+	0.312133i	$0.898956\pi$
$614$		0			0
$615$	−234.530	−	406.217i	−0.381349	−	0.660516i
$616$		0			0
$617$	558.227	+	322.292i	0.904743	+	0.522354i	0.878736	−	0.477308i	$-0.158387\pi$
$617$	558.227	+	322.292i	0.904743	+	0.522354i	0.0260071	+	0.999662i	$0.491721\pi$
$618$		0			0
$619$	−564.469	−	977.690i	−0.911905	−	1.57947i	−0.811370	−	0.584533i	$-0.801278\pi$
$619$	−564.469	−	977.690i	−0.911905	−	1.57947i	−0.100535	−	0.994933i	$-0.532056\pi$
$620$		0			0
$621$	−70.2276	−	40.5459i	−0.113088	−	0.0652913i
$622$		0			0
$623$	−84.5449	+	48.8120i	−0.135706	+	0.0783499i
$624$		0			0
$625$	146.510	−	253.763i	0.234416	−	0.406021i
$626$		0			0
$627$	926.333	−	534.818i	1.47740	−	0.852980i
$628$		0			0
$629$			26.4544i			0.0420579i
$630$		0			0
$631$	143.212			0.226961			0.113480	−	0.993540i	$-0.463800\pi$
$631$	143.212			0.226961			0.113480	+	0.993540i	$0.463800\pi$
$632$		0			0
$633$	−252.894	+	438.025i	−0.399516	+	0.691983i
$634$		0			0
$635$	625.343	+	361.042i	0.984792	+	0.568570i
$636$		0			0
$637$	160.586	+	278.143i	0.252097	+	0.436645i
$638$		0			0
$639$			475.946i			0.744830i
$640$		0			0
$641$	−198.418	+	114.557i	−0.309545	+	0.178716i	−0.646723	−	0.762725i	$-0.723861\pi$
$641$	−198.418	+	114.557i	−0.309545	+	0.178716i	0.337178	+	0.941441i	$0.390528\pi$
$642$		0			0
$643$	554.682	−	960.737i	0.862646	−	1.49415i	−0.00671893	−	0.999977i	$-0.502139\pi$
$643$	554.682	−	960.737i	0.862646	−	1.49415i	0.869365	−	0.494170i	$-0.164528\pi$
$644$		0			0
$645$		−	402.589i		−	0.624169i
$646$		0			0
$647$			1052.57i			1.62685i	0.581669	+	0.813426i	$0.302400\pi$
$647$			1052.57i			1.62685i	−0.581669	+	0.813426i	$0.697600\pi$
$648$		0			0
$649$	−101.000			−0.155624
$650$		0			0
$651$	1044.39			1.60429
$652$		0			0
$653$	−795.499	−	459.282i	−1.21822	−	0.703341i	−0.253685	−	0.967287i	$-0.581643\pi$
$653$	−795.499	−	459.282i	−1.21822	−	0.703341i	−0.964537	+	0.263946i	$0.914976\pi$
$654$		0			0
$655$	207.772	+	359.872i	0.317210	+	0.549424i
$656$		0			0
$657$	619.090			0.942298
$658$		0			0
$659$	−207.288	+	119.678i	−0.314549	+	0.181605i	−0.648960	−	0.760822i	$-0.724796\pi$
$659$	−207.288	+	119.678i	−0.314549	+	0.181605i	0.334411	+	0.942427i	$0.391463\pi$
$660$		0			0
$661$	−177.793	+	307.946i	−0.268976	+	0.465879i	−0.968598	−	0.248634i	$-0.920018\pi$
$661$	−177.793	+	307.946i	−0.268976	+	0.465879i	0.699622	+	0.714513i	$0.253352\pi$
$662$		0			0
$663$	18.4837	+	10.6715i	0.0278788	+	0.0160958i
$664$		0			0
$665$			1084.75i			1.63121i
$666$		0			0
$667$	−45.7673			−0.0686167
$668$		0			0
$669$	−234.560	−	406.270i	−0.350613	−	0.607280i
$670$		0			0
$671$	583.328	+	336.784i	0.869341	+	0.501914i
$672$		0			0
$673$	291.429	+	504.769i	0.433029	+	0.750028i	0.997132	−	0.0756758i	$-0.0241114\pi$
$673$	291.429	+	504.769i	0.433029	+	0.750028i	−0.564103	+	0.825704i	$0.690778\pi$
$674$		0			0
$675$	129.545	+	224.378i	0.191918	+	0.332412i
$676$		0			0
$677$	−154.450	+	89.1718i	−0.228139	+	0.131716i	−0.609713	−	0.792622i	$-0.708715\pi$
$677$	−154.450	+	89.1718i	−0.228139	+	0.131716i	0.381574	+	0.924338i	$0.375382\pi$
$678$		0			0
$679$	648.883	−	1123.90i	0.955645	−	1.65522i
$680$		0			0
$681$	−259.953	+	150.084i	−0.381723	+	0.220388i
$682$		0			0
$683$			660.510i			0.967072i	0.875325	+	0.483536i	$0.160648\pi$
$683$			660.510i			0.967072i	−0.875325	+	0.483536i	$0.839352\pi$
$684$		0			0
$685$	749.080			1.09355
$686$		0			0
$687$	−314.985	+	545.569i	−0.458493	+	0.794133i
$688$		0			0
$689$	−243.959	−	140.850i	−0.354077	−	0.204427i
$690$		0			0
$691$	−43.2571	−	74.9236i	−0.0626008	−	0.108428i	0.833026	−	0.553233i	$-0.186606\pi$
$691$	−43.2571	−	74.9236i	−0.0626008	−	0.108428i	−0.895627	+	0.444805i	$0.853273\pi$
$692$		0			0
$693$	1646.86	+	950.816i	2.37642	+	1.37203i
$694$		0			0
$695$	−322.834	+	186.388i	−0.464509	+	0.268184i
$696$		0			0
$697$	−50.6469	+	87.7231i	−0.0726642	+	0.125858i
$698$		0			0
$699$			921.380i			1.31814i
$700$		0			0
$701$		−	1204.11i		−	1.71770i	−0.512227	−	0.858850i	$-0.671179\pi$
$701$		−	1204.11i		−	1.71770i	0.512227	−	0.858850i	$-0.328821\pi$
$702$		0			0
$703$	−224.686			−0.319610
$704$		0			0
$705$	791.241			1.12233
$706$		0			0
$707$	15.3184	+	8.84406i	0.0216667	+	0.0125093i
$708$		0			0
$709$	401.944	+	696.187i	0.566917	+	0.981928i	0.996869	+	0.0790766i	$0.0251972\pi$
$709$	401.944	+	696.187i	0.566917	+	0.981928i	−0.429952	+	0.902852i	$0.641470\pi$
$710$		0			0
$711$	115.136	+	199.421i	0.161935	+	0.280479i
$712$		0			0
$713$	70.7531	−	40.8493i	0.0992329	−	0.0572921i
$714$		0			0
$715$	90.6510	−	157.012i	0.126785	−	0.219597i
$716$		0			0
$717$	211.166	+	121.917i	0.294514	+	0.170038i
$718$		0			0
$719$		−	66.1101i		−	0.0919474i	−0.998943	−	0.0459737i	$-0.985361\pi$
$719$		−	66.1101i		−	0.0919474i	0.998943	−	0.0459737i	$-0.0146390\pi$
$720$		0			0
$721$	−20.2939			−0.0281469
$722$		0			0
$723$	−540.288	−	935.806i	−0.747286	−	1.29434i
$724$		0			0
$725$	126.637	+	73.1138i	0.174671	+	0.100847i
$726$		0			0
$727$	−259.834	−	450.045i	−0.357405	−	0.619044i	0.630121	−	0.776497i	$-0.283005\pi$
$727$	−259.834	−	450.045i	−0.357405	−	0.619044i	−0.987527	+	0.157453i	$0.949672\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$	−75.2918	+	43.4698i	−0.102998	+	0.0594662i
$732$		0			0
$733$	80.1459	−	138.817i	0.109340	−	0.189382i	−0.806163	−	0.591693i	$-0.798460\pi$
$733$	80.1459	−	138.817i	0.109340	−	0.189382i	0.915503	+	0.402311i	$0.131793\pi$
$734$		0			0
$735$	−1170.48	+	675.779i	−1.59249	+	0.919427i
$736$		0			0
$737$		−	1799.24i		−	2.44130i
$738$		0			0
$739$	−122.849			−0.166237			−0.0831184	−	0.996540i	$-0.526488\pi$
$739$	−122.849			−0.166237			−0.0831184	+	0.996540i	$0.526488\pi$
$740$		0			0
$741$	−90.6367	+	156.987i	−0.122317	+	0.211859i
$742$		0			0
$743$	−585.944	−	338.295i	−0.788619	−	0.455309i	0.0508572	−	0.998706i	$-0.483805\pi$
$743$	−585.944	−	338.295i	−0.788619	−	0.455309i	−0.839476	+	0.543397i	$0.817138\pi$
$744$		0			0
$745$	−180.398	−	312.458i	−0.242145	−	0.419407i
$746$		0			0
$747$	468.771	−	270.645i	0.627539	−	0.362310i
$748$		0			0
$749$	−1144.06	+	660.523i	−1.52745	+	0.881873i
$750$		0			0
$751$	−171.430	+	296.925i	−0.228268	+	0.395373i	−0.957295	−	0.289113i	$-0.906640\pi$
$751$	−171.430	+	296.925i	−0.228268	+	0.395373i	0.729027	+	0.684485i	$0.239973\pi$
$752$		0			0
$753$			1200.54i			1.59434i
$754$		0			0
$755$		−	64.4229i		−	0.0853283i
$756$		0			0
$757$	−452.220			−0.597385			−0.298692	−	0.954349i	$-0.596550\pi$
$757$	−452.220			−0.597385			−0.298692	+	0.954349i	$0.596550\pi$
$758$		0			0
$759$	148.757			0.195991
$760$		0			0
$761$	203.520	+	117.503i	0.267438	+	0.154405i	0.627723	−	0.778437i	$-0.283987\pi$
$761$	203.520	+	117.503i	0.267438	+	0.154405i	−0.360285	+	0.932842i	$0.617320\pi$
$762$		0			0
$763$	−92.1714	−	159.646i	−0.120801	−	0.209234i
$764$		0			0
$765$	−44.9082	+	77.7832i	−0.0587035	+	0.101677i
$766$		0			0
$767$	14.8235	−	8.55834i	0.0193266	−	0.0111582i
$768$		0			0
$769$	318.439	−	551.552i	0.414095	−	0.717233i	−0.581238	−	0.813733i	$-0.697432\pi$
$769$	318.439	−	551.552i	0.414095	−	0.717233i	0.995333	+	0.0965005i	$0.0307649\pi$
$770$		0			0
$771$	−945.104	−	545.656i	−1.22582	−	0.707725i
$772$		0			0
$773$			592.885i			0.766992i	0.923543	+	0.383496i	$0.125280\pi$
$773$			592.885i			0.766992i	−0.923543	+	0.383496i	$0.874720\pi$
$774$		0			0
$775$	−261.029			−0.336811
$776$		0			0
$777$	−199.727	−	345.936i	−0.257048	−	0.445221i
$778$		0			0
$779$	−745.059	−	430.160i	−0.956430	−	0.552195i
$780$		0			0
$781$	−436.545	−	756.118i	−0.558956	−	0.968141i
$782$		0			0
$783$	356.317	−	205.720i	0.455067	−	0.262733i
$784$		0			0
$785$	−720.480	+	415.969i	−0.917808	+	0.529897i
$786$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	72.m (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +(-3.39898 - 1.96240i) q^{5} +(-6.39898 - 11.0834i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +(-3.39898 - 1.96240i) q^{5} +(-6.39898 - 11.0834i) q^{7} +9.00000 q^{9} +(-14.2980 + 8.25493i) q^{11} +(1.39898 - 2.42310i) q^{13} +(10.1969 + 5.88721i) q^{15} -2.54270i q^{17} +21.5959 q^{19} +(19.1969 + 33.2501i) q^{21} +(2.60102 + 1.50170i) q^{23} +(-4.79796 - 8.31031i) q^{25} -27.0000 q^{27} +(-13.1969 + 7.61926i) q^{29} +(13.6010 - 23.5577i) q^{31} +(42.8939 - 24.7648i) q^{33} +50.2295i q^{35} -10.4041 q^{37} +(-4.19694 + 7.26931i) q^{39} +(-34.5000 - 19.9186i) q^{41} +(-17.0959 - 29.6110i) q^{43} +(-30.5908 - 17.6616i) q^{45} +(58.1969 - 33.6000i) q^{47} +(-57.3939 + 99.4091i) q^{49} +7.62809i q^{51} -100.680i q^{53} +64.7980 q^{55} -64.7878 q^{57} +(5.29796 + 3.05878i) q^{59} +(-20.3990 - 35.3321i) q^{61} +(-57.5908 - 99.7502i) q^{63} +(-9.51021 + 5.49072i) q^{65} +(-54.4898 + 94.3791i) q^{67} +(-7.80306 - 4.50510i) q^{69} +52.8829i q^{71} +68.7878 q^{73} +(14.3939 + 24.9309i) q^{75} +(182.985 + 105.646i) q^{77} +(12.7929 + 22.1579i) q^{79} +81.0000 q^{81} +(52.0857 - 30.0717i) q^{83} +(-4.98979 + 8.64258i) q^{85} +(39.5908 - 22.8578i) q^{87} -7.62809i q^{89} -35.8082 q^{91} +(-40.8031 + 70.6730i) q^{93} +(-73.4041 - 42.3799i) q^{95} +(50.7020 + 87.8185i) q^{97} +(-128.682 + 74.2944i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 6 q^{5} - 6 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 6 * q^5 - 6 * q^7 + 36 * q^9	\( 4 q - 12 q^{3} + 6 q^{5} - 6 q^{7} + 36 q^{9} - 18 q^{11} - 14 q^{13} - 18 q^{15} + 8 q^{19} + 18 q^{21} + 30 q^{23} + 20 q^{25} - 108 q^{27} + 6 q^{29} + 74 q^{31} + 54 q^{33} - 120 q^{37} + 42 q^{39} - 138 q^{41} + 10 q^{43} + 54 q^{45} + 174 q^{47} - 112 q^{49} + 220 q^{55} - 24 q^{57} - 18 q^{59} - 62 q^{61} - 54 q^{63} - 234 q^{65} - 22 q^{67} - 90 q^{69} + 40 q^{73} - 60 q^{75} + 438 q^{77} - 86 q^{79} + 324 q^{81} - 66 q^{83} + 176 q^{85} - 18 q^{87} - 300 q^{91} - 222 q^{93} - 372 q^{95} + 242 q^{97} - 162 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 6 * q^5 - 6 * q^7 + 36 * q^9 - 18 * q^11 - 14 * q^13 - 18 * q^15 + 8 * q^19 + 18 * q^21 + 30 * q^23 + 20 * q^25 - 108 * q^27 + 6 * q^29 + 74 * q^31 + 54 * q^33 - 120 * q^37 + 42 * q^39 - 138 * q^41 + 10 * q^43 + 54 * q^45 + 174 * q^47 - 112 * q^49 + 220 * q^55 - 24 * q^57 - 18 * q^59 - 62 * q^61 - 54 * q^63 - 234 * q^65 - 22 * q^67 - 90 * q^69 + 40 * q^73 - 60 * q^75 + 438 * q^77 - 86 * q^79 + 324 * q^81 - 66 * q^83 + 176 * q^85 - 18 * q^87 - 300 * q^91 - 222 * q^93 - 372 * q^95 + 242 * q^97 - 162 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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