LMFDB - Embedded newform 72.4.d.d.37.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(37,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("72.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	72.d (of order $2$, degree $1$, 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:	$4.24813752041$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8248384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + x^{4} - 12x^{3} + 4x^{2} + 64 $ x^6 + x^4 - 12x^3 + 4x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.6
Root			$-1.24181 - 1.56777i$ of defining polynomial
Character	$\chi$	$=$	72.37
Dual form			72.4.d.d.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.80958 + 0.325969i) q^{2} +(7.78749 + 1.83167i) q^{4} -18.5422i q^{5} +9.32669 q^{7} +(21.2825 + 7.68472i) q^{8} +O(q^{10})$	\(q+(2.80958 + 0.325969i) q^{2} +(7.78749 + 1.83167i) q^{4} -18.5422i q^{5} +9.32669 q^{7} +(21.2825 + 7.68472i) q^{8} +(6.04419 - 52.0958i) q^{10} +39.7378i q^{11} +32.9533i q^{13} +(26.2041 + 3.04021i) q^{14} +(57.2899 + 28.5283i) q^{16} -90.5998 q^{17} -72.5998i q^{19} +(33.9633 - 144.397i) q^{20} +(-12.9533 + 111.647i) q^{22} -45.3466 q^{23} -218.813 q^{25} +(-10.7418 + 92.5849i) q^{26} +(72.6315 + 17.0835i) q^{28} +143.364i q^{29} +90.4865 q^{31} +(151.661 + 98.8272i) q^{32} +(-254.547 - 29.5327i) q^{34} -172.937i q^{35} +1.77977i q^{37} +(23.6653 - 203.975i) q^{38} +(142.492 - 394.625i) q^{40} -195.827 q^{41} +407.027i q^{43} +(-72.7866 + 309.458i) q^{44} +(-127.405 - 14.7816i) q^{46} +278.467 q^{47} -256.013 q^{49} +(-614.773 - 71.3263i) q^{50} +(-60.3597 + 256.623i) q^{52} -241.303i q^{53} +736.826 q^{55} +(198.495 + 71.6730i) q^{56} +(-46.7324 + 402.794i) q^{58} +149.724i q^{59} -508.314i q^{61} +(254.229 + 29.4958i) q^{62} +(393.890 + 327.100i) q^{64} +611.027 q^{65} -950.026i q^{67} +(-705.545 - 165.949i) q^{68} +(56.3723 - 485.882i) q^{70} +803.559 q^{71} +449.786 q^{73} +(-0.580152 + 5.00042i) q^{74} +(132.979 - 565.370i) q^{76} +370.622i q^{77} -157.220 q^{79} +(528.977 - 1062.28i) q^{80} +(-550.192 - 63.8335i) q^{82} -175.063i q^{83} +1679.92i q^{85} +(-132.678 + 1143.57i) q^{86} +(-305.374 + 845.720i) q^{88} -127.200 q^{89} +307.345i q^{91} +(-353.136 - 83.0602i) q^{92} +(782.374 + 90.7715i) q^{94} -1346.16 q^{95} +158.826 q^{97} +(-719.289 - 83.4523i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8	$ 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8} + 60 q^{10} + 100 q^{14} + 56 q^{16} - 52 q^{17} - 56 q^{20} + 224 q^{22} - 328 q^{23} - 106 q^{25} - 56 q^{26} - 352 q^{28} - 636 q^{31} + 248 q^{32} - 548 q^{34} + 776 q^{38} + 232 q^{40} - 236 q^{41} - 1152 q^{44} + 328 q^{46} + 408 q^{47} + 654 q^{49} - 1970 q^{50} - 368 q^{52} + 1024 q^{55} + 1864 q^{56} + 140 q^{58} + 2108 q^{62} + 832 q^{64} + 1744 q^{65} - 2976 q^{68} + 1352 q^{70} + 1704 q^{71} + 956 q^{73} - 1568 q^{74} - 1744 q^{76} - 44 q^{79} + 2112 q^{80} - 2236 q^{82} + 760 q^{86} + 1856 q^{88} + 220 q^{89} - 1728 q^{92} + 2088 q^{94} - 5104 q^{95} - 2444 q^{97} - 3354 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8 + 60 * q^10 + 100 * q^14 + 56 * q^16 - 52 * q^17 - 56 * q^20 + 224 * q^22 - 328 * q^23 - 106 * q^25 - 56 * q^26 - 352 * q^28 - 636 * q^31 + 248 * q^32 - 548 * q^34 + 776 * q^38 + 232 * q^40 - 236 * q^41 - 1152 * q^44 + 328 * q^46 + 408 * q^47 + 654 * q^49 - 1970 * q^50 - 368 * q^52 + 1024 * q^55 + 1864 * q^56 + 140 * q^58 + 2108 * q^62 + 832 * q^64 + 1744 * q^65 - 2976 * q^68 + 1352 * q^70 + 1704 * q^71 + 956 * q^73 - 1568 * q^74 - 1744 * q^76 - 44 * q^79 + 2112 * q^80 - 2236 * q^82 + 760 * q^86 + 1856 * q^88 + 220 * q^89 - 1728 * q^92 + 2088 * q^94 - 5104 * q^95 - 2444 * q^97 - 3354 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	2.80958	+	0.325969i	0.993337	+	0.115247i
$2$	2.80958	+	0.325969i	0.993337	+	0.115247i
$3$		0			0
$3$		0			0
$4$	7.78749	+	1.83167i	0.973436	+	0.228959i
$5$		−	18.5422i		−	1.65846i	−0.558904	−	0.829232i	$-0.688778\pi$
$5$		−	18.5422i		−	1.65846i	0.558904	−	0.829232i	$-0.311222\pi$
$6$		0			0
$7$	9.32669			0.503594			0.251797	−	0.967780i	$-0.418978\pi$
$7$	9.32669			0.503594			0.251797	+	0.967780i	$0.418978\pi$
$8$	21.2825	+	7.68472i	0.940563	+	0.339620i
$9$		0			0
$10$	6.04419	−	52.0958i	0.191134	−	1.64741i
$11$			39.7378i			1.08922i	0.838690	+	0.544609i	$0.183322\pi$
$11$			39.7378i			1.08922i	−0.838690	+	0.544609i	$0.816678\pi$
$12$		0			0
$13$			32.9533i			0.703046i	0.936179	+	0.351523i	$0.114336\pi$
$13$			32.9533i			0.703046i	−0.936179	+	0.351523i	$0.885664\pi$
$14$	26.2041	+	3.04021i	0.500239	+	0.0580380i
$15$		0			0
$16$	57.2899	+	28.5283i	0.895155	+	0.445754i
$17$	−90.5998			−1.29257			−0.646285	−	0.763096i	$-0.723678\pi$
$17$	−90.5998			−1.29257			−0.646285	+	0.763096i	$0.723678\pi$
$18$		0			0
$19$		−	72.5998i		−	0.876607i	−0.898827	−	0.438304i	$-0.855579\pi$
$19$		−	72.5998i		−	0.876607i	0.898827	−	0.438304i	$-0.144421\pi$
$20$	33.9633	−	144.397i	0.379721	−	1.61441i
$21$		0			0
$22$	−12.9533	+	111.647i	−0.125530	+	1.08196i
$23$	−45.3466			−0.411105			−0.205553	−	0.978646i	$-0.565899\pi$
$23$	−45.3466			−0.411105			−0.205553	+	0.978646i	$0.565899\pi$
$24$		0			0
$25$	−218.813			−1.75051
$26$	−10.7418	+	92.5849i	−0.0810243	+	0.698362i
$27$		0			0
$28$	72.6315	+	17.0835i	0.490217	+	0.115302i
$29$			143.364i			0.918003i	0.888435	+	0.459002i	$0.151793\pi$
$29$			143.364i			0.918003i	−0.888435	+	0.459002i	$0.848207\pi$
$30$		0			0
$31$	90.4865			0.524253			0.262127	−	0.965033i	$-0.415576\pi$
$31$	90.4865			0.524253			0.262127	+	0.965033i	$0.415576\pi$
$32$	151.661	+	98.8272i	0.837819	+	0.545948i
$33$		0			0
$34$	−254.547	−	29.5327i	−1.28396	−	0.148965i
$35$		−	172.937i		−	0.835193i
$36$		0			0
$37$			1.77977i			0.00790792i	0.999992	+	0.00395396i	$0.00125859\pi$
$37$			1.77977i			0.00790792i	−0.999992	+	0.00395396i	$0.998741\pi$
$38$	23.6653	−	203.975i	0.101027	−	0.870766i
$39$		0			0
$40$	142.492	−	394.625i	0.563247	−	1.55989i
$41$	−195.827			−0.745927			−0.372964	−	0.927846i	$-0.621658\pi$
$41$	−195.827			−0.745927			−0.372964	+	0.927846i	$0.621658\pi$
$42$		0			0
$43$			407.027i			1.44351i	0.692148	+	0.721755i	$0.256664\pi$
$43$			407.027i			1.44351i	−0.692148	+	0.721755i	$0.743336\pi$
$44$	−72.7866	+	309.458i	−0.249386	+	1.06028i
$45$		0			0
$46$	−127.405	−	14.7816i	−0.408366	−	0.0473789i
$47$	278.467			0.864224			0.432112	−	0.901820i	$-0.357769\pi$
$47$	278.467			0.864224			0.432112	+	0.901820i	$0.357769\pi$
$48$		0			0
$49$	−256.013			−0.746393
$50$	−614.773	−	71.3263i	−1.73884	−	0.201741i
$51$		0			0
$52$	−60.3597	+	256.623i	−0.160969	+	0.684370i
$53$		−	241.303i		−	0.625386i	−0.949854	−	0.312693i	$-0.898769\pi$
$53$		−	241.303i		−	0.625386i	0.949854	−	0.312693i	$-0.101231\pi$
$54$		0			0
$55$	736.826			1.80643
$56$	198.495	+	71.6730i	0.473662	+	0.171030i
$57$		0			0
$58$	−46.7324	+	402.794i	−0.105798	+	0.911887i
$59$			149.724i			0.330380i	0.986262	+	0.165190i	$0.0528237\pi$
$59$			149.724i			0.330380i	−0.986262	+	0.165190i	$0.947176\pi$
$60$		0			0
$61$		−	508.314i		−	1.06693i	−0.845821	−	0.533466i	$-0.820889\pi$
$61$		−	508.314i		−	1.06693i	0.845821	−	0.533466i	$-0.179111\pi$
$62$	254.229	+	29.4958i	0.520760	+	0.0604189i
$63$		0			0
$64$	393.890	+	327.100i	0.769317	+	0.638867i
$65$	611.027			1.16598
$66$		0			0
$67$		−	950.026i		−	1.73230i	−0.499784	−	0.866150i	$-0.666587\pi$
$67$		−	950.026i		−	1.73230i	0.499784	−	0.866150i	$-0.333413\pi$
$68$	−705.545	−	165.949i	−1.25823	−	0.295946i
$69$		0			0
$70$	56.3723	−	485.882i	0.0962539	−	0.829628i
$71$	803.559			1.34317			0.671584	−	0.740929i	$-0.265614\pi$
$71$	803.559			1.34317			0.671584	+	0.740929i	$0.265614\pi$
$72$		0			0
$73$	449.786			0.721143			0.360571	−	0.932732i	$-0.382582\pi$
$73$	449.786			0.721143			0.360571	+	0.932732i	$0.382582\pi$
$74$	−0.580152	+	5.00042i	−0.000911368	+	0.00785523i
$75$		0			0
$76$	132.979	−	565.370i	0.200707	−	0.853321i
$77$			370.622i			0.548524i
$78$		0			0
$79$	−157.220			−0.223906			−0.111953	−	0.993713i	$-0.535711\pi$
$79$	−157.220			−0.223906			−0.111953	+	0.993713i	$0.535711\pi$
$80$	528.977	−	1062.28i	0.739268	−	1.48458i
$81$		0			0
$82$	−550.192	−	63.8335i	−0.740957	−	0.0859663i
$83$		−	175.063i		−	0.231513i	−0.993278	−	0.115757i	$-0.963071\pi$
$83$		−	175.063i		−	0.231513i	0.993278	−	0.115757i	$-0.0369293\pi$
$84$		0			0
$85$			1679.92i			2.14368i
$86$	−132.678	+	1143.57i	−0.166361	+	1.43389i
$87$		0			0
$88$	−305.374	+	845.720i	−0.369920	+	1.02448i
$89$	−127.200			−0.151496			−0.0757479	−	0.997127i	$-0.524134\pi$
$89$	−127.200			−0.151496			−0.0757479	+	0.997127i	$0.524134\pi$
$90$		0			0
$91$			307.345i			0.354050i
$92$	−353.136	−	83.0602i	−0.400185	−	0.0941263i
$93$		0			0
$94$	782.374	+	90.7715i	0.858465	+	0.0995996i
$95$	−1346.16			−1.45382
$96$		0			0
$97$	158.826			0.166251			0.0831254	−	0.996539i	$-0.473510\pi$
$97$	158.826			0.166251			0.0831254	+	0.996539i	$0.473510\pi$
$98$	−719.289	−	83.4523i	−0.741420	−	0.0860199i
$99$		0			0
$100$	−1704.00	−	400.794i	−1.70400	−	0.400794i
$101$		−	1366.26i		−	1.34602i	−0.739633	−	0.673010i	$-0.765001\pi$
$101$		−	1366.26i		−	1.34602i	0.739633	−	0.673010i	$-0.234999\pi$
$102$		0			0
$103$	−1741.30			−1.66578			−0.832889	−	0.553440i	$-0.813315\pi$
$103$	−1741.30			−1.66578			−0.832889	+	0.553440i	$0.813315\pi$
$104$	−253.237	+	701.329i	−0.238768	+	0.661259i
$105$		0			0
$106$	78.6572	−	677.960i	0.0720742	−	0.621219i
$107$			649.378i			0.586708i	0.956004	+	0.293354i	$0.0947714\pi$
$107$			649.378i			0.586708i	−0.956004	+	0.293354i	$0.905229\pi$
$108$		0			0
$109$		−	1413.18i		−	1.24182i	−0.783883	−	0.620908i	$-0.786764\pi$
$109$		−	1413.18i		−	1.24182i	0.783883	−	0.620908i	$-0.213236\pi$
$110$	2070.17	+	240.183i	1.79439	+	0.208186i
$111$		0			0
$112$	534.326	+	266.074i	0.450795	+	0.224479i
$113$	−1096.43			−0.912771			−0.456386	−	0.889782i	$-0.650856\pi$
$113$	−1096.43			−0.912771			−0.456386	+	0.889782i	$0.650856\pi$
$114$		0			0
$115$			840.826i			0.681804i
$116$	−262.597	+	1116.45i	−0.210185	+	0.893618i
$117$		0			0
$118$	−48.8054	+	420.662i	−0.0380755	+	0.328178i
$119$	−844.997			−0.650930
$120$		0			0
$121$	−248.092			−0.186395
$122$	165.695	−	1428.15i	0.122961	−	1.05982i
$123$		0			0
$124$	704.662	+	165.742i	0.510327	+	0.120033i
$125$			1739.50i			1.24469i
$126$		0			0
$127$	−737.794			−0.515501			−0.257751	−	0.966211i	$-0.582981\pi$
$127$	−737.794			−0.515501			−0.257751	+	0.966211i	$0.582981\pi$
$128$	1000.04	+	1047.41i	0.690563	+	0.723272i
$129$		0			0
$130$	1716.73	+	199.176i	1.15821	+	0.134376i
$131$			147.698i			0.0985074i	0.998786	+	0.0492537i	$0.0156843\pi$
$131$			147.698i			0.0985074i	−0.998786	+	0.0492537i	$0.984316\pi$
$132$		0			0
$133$		−	677.116i		−	0.441454i
$134$	309.679	−	2669.17i	0.199643	−	1.72076i
$135$		0			0
$136$	−1928.19	−	696.234i	−1.21574	−	0.438982i
$137$	1880.79			1.17289			0.586447	−	0.809988i	$-0.300526\pi$
$137$	1880.79			1.17289			0.586447	+	0.809988i	$0.300526\pi$
$138$		0			0
$139$			629.333i			0.384024i	0.981393	+	0.192012i	$0.0615013\pi$
$139$			629.333i			0.384024i	−0.981393	+	0.192012i	$0.938499\pi$
$140$	316.765	−	1346.75i	0.191225	−	0.813007i
$141$		0			0
$142$	2257.66	+	261.935i	1.33422	+	0.154797i
$143$	−1309.49			−0.765770
$144$		0			0
$145$	2658.29			1.52248
$146$	1263.71	+	146.616i	0.716338	+	0.0831099i
$147$		0			0
$148$	−3.25997	+	13.8600i	−0.00181059	+	0.00769786i
$149$			429.457i			0.236124i	0.993006	+	0.118062i	$0.0376681\pi$
$149$			429.457i			0.236124i	−0.993006	+	0.118062i	$0.962332\pi$
$150$		0			0
$151$	−27.3124			−0.0147196			−0.00735978	−	0.999973i	$-0.502343\pi$
$151$	−27.3124			−0.0147196			−0.00735978	+	0.999973i	$0.502343\pi$
$152$	557.909	−	1545.11i	0.297713	−	0.824504i
$153$		0			0
$154$	−120.811	+	1041.29i	−0.0632160	+	0.544869i
$155$		−	1677.82i		−	0.869456i
$156$		0			0
$157$		−	1251.08i		−	0.635970i	−0.948096	−	0.317985i	$-0.896994\pi$
$157$		−	1251.08i		−	0.635970i	0.948096	−	0.317985i	$-0.103006\pi$
$158$	−441.721	−	51.2487i	−0.222414	−	0.0258046i
$159$		0			0
$160$	1832.47	−	2812.14i	0.905436	−	1.38949i
$161$	−422.934			−0.207030
$162$		0			0
$163$		−	127.884i		−	0.0614517i	−0.999528	−	0.0307258i	$-0.990218\pi$
$163$		−	127.884i		−	0.0614517i	0.999528	−	0.0307258i	$-0.00978188\pi$
$164$	−1525.00	−	358.691i	−0.726113	−	0.170787i
$165$		0			0
$166$	57.0650	−	491.852i	0.0266813	−	0.229971i
$167$	2079.65			0.963642			0.481821	−	0.876270i	$-0.339976\pi$
$167$	2079.65			0.963642			0.481821	+	0.876270i	$0.339976\pi$
$168$		0			0
$169$	1111.08			0.505726
$170$	−547.602	+	4719.87i	−0.247054	+	2.12940i
$171$		0			0
$172$	−745.540	+	3169.71i	−0.330505	+	1.40517i
$173$			685.140i			0.301099i	0.988602	+	0.150550i	$0.0481044\pi$
$173$			685.140i			0.301099i	−0.988602	+	0.150550i	$0.951896\pi$
$174$		0			0
$175$	−2040.80			−0.881544
$176$	−1133.65	+	2276.58i	−0.485523	+	0.975019i
$177$		0			0
$178$	−357.378	−	41.4632i	−0.150486	−	0.0174595i
$179$		−	429.423i		−	0.179310i	−0.995973	−	0.0896552i	$-0.971424\pi$
$179$		−	429.423i		−	0.179310i	0.995973	−	0.0896552i	$-0.0285765\pi$
$180$		0			0
$181$			2842.85i			1.16745i	0.811953	+	0.583723i	$0.198404\pi$
$181$			2842.85i			1.16745i	−0.811953	+	0.583723i	$0.801596\pi$
$182$	−100.185	+	863.511i	−0.0408034	+	0.351691i
$183$		0			0
$184$	−965.090	−	348.476i	−0.386670	−	0.139619i
$185$	33.0009			0.0131150
$186$		0			0
$187$		−	3600.24i		−	1.40789i
$188$	2168.56	+	510.060i	0.841267	+	0.197872i
$189$		0			0
$190$	−3782.15	−	438.807i	−1.44414	−	0.167549i
$191$	−2546.78			−0.964808			−0.482404	−	0.875949i	$-0.660236\pi$
$191$	−2546.78			−0.964808			−0.482404	+	0.875949i	$0.660236\pi$
$192$		0			0
$193$	−3579.97			−1.33519			−0.667596	−	0.744524i	$-0.732677\pi$
$193$	−3579.97			−1.33519			−0.667596	+	0.744524i	$0.732677\pi$
$194$	446.234	+	51.7724i	0.165143	+	0.0191600i
$195$		0			0
$196$	−1993.70	−	468.932i	−0.726566	−	0.170894i
$197$		−	3872.58i		−	1.40056i	−0.713869	−	0.700280i	$-0.753059\pi$
$197$		−	3872.58i		−	1.40056i	0.713869	−	0.700280i	$-0.246941\pi$
$198$		0			0
$199$	5558.64			1.98011			0.990054	−	0.140686i	$-0.0449307\pi$
$199$	5558.64			1.98011			0.990054	+	0.140686i	$0.0449307\pi$
$200$	−4656.89	−	1681.52i	−1.64646	−	0.594506i
$201$		0			0
$202$	445.359	−	3838.62i	0.155125	−	1.33705i
$203$			1337.12i			0.462301i
$204$		0			0
$205$			3631.06i			1.23709i
$206$	−4892.32	−	567.609i	−1.65468	−	0.191977i
$207$		0			0
$208$	−940.100	+	1887.89i	−0.313386	+	0.629335i
$209$	2884.96			0.954816
$210$		0			0
$211$			4658.93i			1.52007i	0.649884	+	0.760034i	$0.274818\pi$
$211$			4658.93i			1.52007i	−0.649884	+	0.760034i	$0.725182\pi$
$212$	441.988	−	1879.14i	0.143188	−	0.608774i
$213$		0			0
$214$	−211.677	+	1824.48i	−0.0676166	+	0.582798i
$215$	7547.17			2.39401
$216$		0			0
$217$	843.940			0.264011
$218$	460.653	−	3970.44i	0.143116	−	1.23354i
$219$		0			0
$220$	5738.02	+	1349.62i	1.75844	+	0.413598i
$221$		−	2985.56i		−	0.908736i
$222$		0			0
$223$	−1545.42			−0.464077			−0.232038	−	0.972707i	$-0.574540\pi$
$223$	−1545.42			−0.464077			−0.232038	+	0.972707i	$0.574540\pi$
$224$	1414.50	+	921.731i	0.421921	+	0.274936i
$225$		0			0
$226$	−3080.50	−	357.401i	−0.906689	−	0.105195i
$227$			6545.76i			1.91391i	0.290240	+	0.956954i	$0.406265\pi$
$227$			6545.76i			1.91391i	−0.290240	+	0.956954i	$0.593735\pi$
$228$		0			0
$229$		−	5463.48i		−	1.57658i	−0.615303	−	0.788291i	$-0.710966\pi$
$229$		−	5463.48i		−	1.57658i	0.615303	−	0.788291i	$-0.289034\pi$
$230$	−274.083	+	2362.37i	−0.0785762	+	0.677261i
$231$		0			0
$232$	−1101.71	+	3051.15i	−0.311772	+	0.863440i
$233$	−4722.40			−1.32779			−0.663894	−	0.747827i	$-0.731097\pi$
$233$	−4722.40			−1.32779			−0.663894	+	0.747827i	$0.731097\pi$
$234$		0			0
$235$		−	5163.38i		−	1.43328i
$236$	−274.246	+	1165.97i	−0.0756435	+	0.321604i
$237$		0			0
$238$	−2374.09	−	275.443i	−0.646593	−	0.0750181i
$239$	1054.38			0.285363			0.142682	−	0.989769i	$-0.454427\pi$
$239$	1054.38			0.285363			0.142682	+	0.989769i	$0.454427\pi$
$240$		0			0
$241$	−3134.40			−0.837777			−0.418888	−	0.908038i	$-0.637580\pi$
$241$	−3134.40			−0.837777			−0.418888	+	0.908038i	$0.637580\pi$
$242$	−697.033	−	80.8702i	−0.185153	−	0.0214815i
$243$		0			0
$244$	931.065	−	3958.49i	0.244284	−	1.03859i
$245$			4747.04i			1.23787i
$246$		0			0
$247$	2392.40			0.616295
$248$	1925.78	+	695.363i	0.493093	+	0.178047i
$249$		0			0
$250$	−567.024	+	4887.27i	−0.143447	+	1.23639i
$251$		−	4881.91i		−	1.22766i	−0.789437	−	0.613831i	$-0.789628\pi$
$251$		−	4881.91i		−	1.22766i	0.789437	−	0.613831i	$-0.210372\pi$
$252$		0			0
$253$		−	1801.97i		−	0.447783i
$254$	−2072.89	−	240.498i	−0.512066	−	0.0594102i
$255$		0			0
$256$	2468.28	+	3268.77i	0.602606	+	0.798039i
$257$	540.458			0.131178			0.0655892	−	0.997847i	$-0.479107\pi$
$257$	540.458			0.131178			0.0655892	+	0.997847i	$0.479107\pi$
$258$		0			0
$259$			16.5994i			0.00398238i
$260$	4758.36	+	1119.20i	1.13500	+	0.266961i
$261$		0			0
$262$	−48.1451	+	414.971i	−0.0113527	+	0.0978511i
$263$	−4800.12			−1.12543			−0.562715	−	0.826651i	$-0.690243\pi$
$263$	−4800.12			−1.12543			−0.562715	+	0.826651i	$0.690243\pi$
$264$		0			0
$265$	−4474.28			−1.03718
$266$	220.719	−	1902.41i	0.0508765	−	0.438513i
$267$		0			0
$268$	1740.14	−	7398.31i	0.396626	−	1.68628i
$269$			3321.28i			0.752795i	0.926458	+	0.376397i	$0.122837\pi$
$269$			3321.28i			0.752795i	−0.926458	+	0.376397i	$0.877163\pi$
$270$		0			0
$271$	5274.04			1.18220			0.591098	−	0.806600i	$-0.298695\pi$
$271$	5274.04			1.18220			0.591098	+	0.806600i	$0.298695\pi$
$272$	−5190.46	−	2584.66i	−1.15705	−	0.576168i
$273$		0			0
$274$	5284.22	+	613.078i	1.16508	+	0.135173i
$275$		−	8695.15i		−	1.90668i
$276$		0			0
$277$			3190.24i			0.691996i	0.938235	+	0.345998i	$0.112460\pi$
$277$			3190.24i			0.691996i	−0.938235	+	0.345998i	$0.887540\pi$
$278$	−205.143	+	1768.16i	−0.0442578	+	0.381465i
$279$		0			0
$280$	1328.97	−	3680.54i	0.283648	−	0.785552i
$281$	545.619			0.115832			0.0579162	−	0.998321i	$-0.481554\pi$
$281$	545.619			0.115832			0.0579162	+	0.998321i	$0.481554\pi$
$282$		0			0
$283$			5927.74i			1.24511i	0.782574	+	0.622557i	$0.213906\pi$
$283$			5927.74i			1.24511i	−0.782574	+	0.622557i	$0.786094\pi$
$284$	6257.71	+	1471.86i	1.30749	+	0.307531i
$285$		0			0
$286$	−3679.12	−	426.854i	−0.760668	−	0.0882531i
$287$	−1826.42			−0.375645
$288$		0			0
$289$	3295.33			0.670736
$290$	7468.68	+	866.521i	1.51233	+	0.175462i
$291$		0			0
$292$	3502.70	+	823.860i	0.701987	+	0.165112i
$293$			5406.01i			1.07789i	0.842340	+	0.538946i	$0.181177\pi$
$293$			5406.01i			1.07789i	−0.842340	+	0.538946i	$0.818823\pi$
$294$		0			0
$295$	2776.21			0.547923
$296$	−13.6771	+	37.8781i	−0.00268569	+	0.00743790i
$297$		0			0
$298$	−139.990	+	1206.59i	−0.0272127	+	0.234551i
$299$		−	1494.32i		−	0.289026i
$300$		0			0
$301$			3796.21i			0.726944i
$302$	−76.7365	−	8.90301i	−0.0146215	−	0.00169639i
$303$		0			0
$304$	2071.15	−	4159.24i	0.390751	−	0.784700i
$305$	−9425.25			−1.76947
$306$		0			0
$307$			1558.56i			0.289745i	0.989450	+	0.144873i	$0.0462772\pi$
$307$			1558.56i			0.289745i	−0.989450	+	0.144873i	$0.953723\pi$
$308$	−678.859	+	2886.22i	−0.125589	+	0.533953i
$309$		0			0
$310$	546.917	−	4713.97i	0.100203	−	0.863662i
$311$	8348.21			1.52213			0.761067	−	0.648673i	$-0.224676\pi$
$311$	8348.21			1.52213			0.761067	+	0.648673i	$0.224676\pi$
$312$		0			0
$313$	−5213.09			−0.941410			−0.470705	−	0.882291i	$-0.656000\pi$
$313$	−5213.09			−0.941410			−0.470705	+	0.882291i	$0.656000\pi$
$314$	407.815	−	3515.02i	0.0732940	−	0.631733i
$315$		0			0
$316$	−1224.35	−	287.975i	−0.217958	−	0.0512653i
$317$		−	9070.57i		−	1.60711i	−0.595230	−	0.803555i	$-0.702939\pi$
$317$		−	9070.57i		−	1.60711i	0.595230	−	0.803555i	$-0.297061\pi$
$318$		0			0
$319$	−5696.98			−0.999905
$320$	6065.15	−	7303.59i	1.05954	−	1.27589i
$321$		0			0
$322$	−1188.27	−	137.863i	−0.205651	−	0.0238597i
$323$			6577.53i			1.13308i
$324$		0			0
$325$		−	7210.61i		−	1.23069i
$326$	41.6861	−	359.300i	0.00708215	−	0.0610422i
$327$		0			0
$328$	−4167.69	−	1504.87i	−0.701592	−	0.253332i
$329$	2597.17			0.435218
$330$		0			0
$331$		−	186.537i		−	0.0309758i	−0.999880	−	0.0154879i	$-0.995070\pi$
$331$		−	186.537i		−	0.0309758i	0.999880	−	0.0154879i	$-0.00493014\pi$
$332$	320.657	−	1363.30i	0.0530071	−	0.225364i
$333$		0			0
$334$	5842.95	+	677.902i	0.957221	+	0.111057i
$335$	−17615.6			−2.87296
$336$		0			0
$337$	829.350			0.134058			0.0670290	−	0.997751i	$-0.478648\pi$
$337$	829.350			0.134058			0.0670290	+	0.997751i	$0.478648\pi$
$338$	3121.67	+	362.178i	0.502356	+	0.0582837i
$339$		0			0
$340$	−3077.06	+	13082.4i	−0.490815	+	2.08674i
$341$			3595.73i			0.571026i
$342$		0			0
$343$	−5586.81			−0.879473
$344$	−3127.88	+	8662.55i	−0.490245	+	1.35771i
$345$		0			0
$346$	−223.334	+	1924.96i	−0.0347010	+	0.299093i
$347$			12005.6i			1.85734i	0.370908	+	0.928669i	$0.379046\pi$
$347$			12005.6i			1.85734i	−0.370908	+	0.928669i	$0.620954\pi$
$348$		0			0
$349$			77.8551i			0.0119412i	0.999982	+	0.00597062i	$0.00190052\pi$
$349$			77.8551i			0.0119412i	−0.999982	+	0.00597062i	$0.998099\pi$
$350$	−5733.80	−	665.239i	−0.875670	−	0.101596i
$351$		0			0
$352$	−3927.18	+	6026.69i	−0.594657	+	0.912567i
$353$	4925.06			0.742591			0.371296	−	0.928515i	$-0.378914\pi$
$353$	4925.06			0.742591			0.371296	+	0.928515i	$0.378914\pi$
$354$		0			0
$355$		−	14899.8i		−	2.22760i
$356$	−990.566	−	232.988i	−0.147472	−	0.0346864i
$357$		0			0
$358$	139.978	−	1206.50i	0.0206651	−	0.178116i
$359$	12260.2			1.80242			0.901209	−	0.433384i	$-0.142681\pi$
$359$	12260.2			1.80242			0.901209	+	0.433384i	$0.142681\pi$
$360$		0			0
$361$	1588.27			0.231560
$362$	−926.683	+	7987.23i	−0.134545	+	1.15967i
$363$		0			0
$364$	−562.956	+	2393.45i	−0.0810630	+	0.344645i
$365$		−	8340.02i		−	1.19599i
$366$		0			0
$367$	−8600.86			−1.22333			−0.611664	−	0.791118i	$-0.709500\pi$
$367$	−8600.86			−1.22333			−0.611664	+	0.791118i	$0.709500\pi$
$368$	−2597.91	−	1293.66i	−0.368003	−	0.183252i
$369$		0			0
$370$	92.7188	+	10.7573i	0.0130276	+	0.00151147i
$371$		−	2250.56i		−	0.314941i
$372$		0			0
$373$			3996.47i			0.554770i	0.960759	+	0.277385i	$0.0894678\pi$
$373$			3996.47i			0.554770i	−0.960759	+	0.277385i	$0.910532\pi$
$374$	1173.57	−	10115.2i	0.162256	−	1.39851i
$375$		0			0
$376$	5926.47	+	2139.94i	0.812857	+	0.293507i
$377$	−4724.33			−0.645399
$378$		0			0
$379$			10404.7i			1.41017i	0.709121	+	0.705087i	$0.249092\pi$
$379$			10404.7i			1.41017i	−0.709121	+	0.705087i	$0.750908\pi$
$380$	−10483.2	−	2465.73i	−1.41520	−	0.332866i
$381$		0			0
$382$	−7155.38	−	830.171i	−0.958379	−	0.111192i
$383$	−6814.19			−0.909109			−0.454554	−	0.890719i	$-0.650202\pi$
$383$	−6814.19			−0.909109			−0.454554	+	0.890719i	$0.650202\pi$
$384$		0			0
$385$	6872.15			0.909707
$386$	−10058.2	−	1166.96i	−1.32630	−	0.153878i
$387$		0			0
$388$	1236.86	+	290.917i	0.161835	+	0.0380647i
$389$		−	779.329i		−	0.101577i	−0.998709	−	0.0507886i	$-0.983827\pi$
$389$		−	779.329i		−	0.101577i	0.998709	−	0.0507886i	$-0.0161735\pi$
$390$		0			0
$391$	4108.39			0.531382
$392$	−5448.59	−	1967.39i	−0.702030	−	0.253490i
$393$		0			0
$394$	1262.34	−	10880.3i	0.161411	−	1.39123i
$395$			2915.20i			0.371340i
$396$		0			0
$397$			12514.1i			1.58202i	0.611802	+	0.791011i	$0.290445\pi$
$397$			12514.1i			1.58202i	−0.611802	+	0.791011i	$0.709555\pi$
$398$	15617.5	+	1811.95i	1.96691	+	0.228203i
$399$		0			0
$400$	−12535.8	−	6242.36i	−1.56697	−	0.780295i
$401$	7949.68			0.989995			0.494998	−	0.868894i	$-0.335169\pi$
$401$	7949.68			0.989995			0.494998	+	0.868894i	$0.335169\pi$
$402$		0			0
$403$			2981.83i			0.368574i
$404$	2502.54	−	10639.7i	0.308184	−	1.31026i
$405$		0			0
$406$	−435.858	+	3756.73i	−0.0532790	+	0.459221i
$407$	−70.7243			−0.00861345
$408$		0			0
$409$	15183.9			1.83569			0.917846	−	0.396937i	$-0.129927\pi$
$409$	15183.9			1.83569			0.917846	+	0.396937i	$0.129927\pi$
$410$	−1183.61	+	10201.8i	−0.142572	+	1.22885i
$411$		0			0
$412$	−13560.3	−	3189.49i	−1.62153	−	0.381395i
$413$			1396.43i			0.166377i
$414$		0			0
$415$	−3246.05			−0.383957
$416$	−3256.68	+	4997.74i	−0.383827	+	0.589025i
$417$		0			0
$418$	8105.52	+	940.407i	0.948454	+	0.110040i
$419$		−	11767.2i		−	1.37199i	−0.727606	−	0.685996i	$-0.759367\pi$
$419$		−	11767.2i		−	1.37199i	0.727606	−	0.685996i	$-0.240633\pi$
$420$		0			0
$421$		−	14459.8i		−	1.67394i	−0.547248	−	0.836970i	$-0.684325\pi$
$421$		−	14459.8i		−	1.67394i	0.547248	−	0.836970i	$-0.315675\pi$
$422$	−1518.67	+	13089.7i	−0.175184	+	1.50994i
$423$		0			0
$424$	1854.34	−	5135.53i	0.212394	−	0.588215i
$425$	19824.4			2.26265
$426$		0			0
$427$		−	4740.89i		−	0.537301i
$428$	−1189.45	+	5057.02i	−0.134332	+	0.571122i
$429$		0			0
$430$	21204.4	+	2460.14i	2.37806	+	0.275904i
$431$	9248.50			1.03361			0.516804	−	0.856104i	$-0.327122\pi$
$431$	9248.50			1.03361			0.516804	+	0.856104i	$0.327122\pi$
$432$		0			0
$433$	−3456.12			−0.383581			−0.191790	−	0.981436i	$-0.561429\pi$
$433$	−3456.12			−0.383581			−0.191790	+	0.981436i	$0.561429\pi$
$434$	2371.12	+	275.098i	0.262252	+	0.0304266i
$435$		0			0
$436$	2588.48	−	11005.1i	0.284325	−	1.20883i
$437$			3292.16i			0.360378i
$438$		0			0
$439$	8075.68			0.877975			0.438988	−	0.898493i	$-0.355337\pi$
$439$	8075.68			0.877975			0.438988	+	0.898493i	$0.355337\pi$
$440$	15681.5	+	5662.30i	1.69906	+	0.613499i
$441$		0			0
$442$	973.201	−	8388.18i	0.104730	−	0.902681i
$443$		−	11447.7i		−	1.22776i	−0.789401	−	0.613878i	$-0.789609\pi$
$443$		−	11447.7i		−	1.22776i	0.789401	−	0.613878i	$-0.210391\pi$
$444$		0			0
$445$			2358.56i			0.251251i
$446$	−4341.99	−	503.760i	−0.460985	−	0.0534837i
$447$		0			0
$448$	3673.69	+	3050.76i	0.387424	+	0.321730i
$449$	1010.18			0.106177			0.0530886	−	0.998590i	$-0.483093\pi$
$449$	1010.18			0.106177			0.0530886	+	0.998590i	$0.483093\pi$
$450$		0			0
$451$		−	7781.73i		−	0.812477i
$452$	−8538.41	−	2008.30i	−0.888524	−	0.208987i
$453$		0			0
$454$	−2133.71	+	18390.8i	−0.220573	+	1.90116i
$455$	5698.86			0.587179
$456$		0			0
$457$	15949.2			1.63254			0.816272	−	0.577667i	$-0.196037\pi$
$457$	15949.2			1.63254			0.816272	+	0.577667i	$0.196037\pi$
$458$	1780.93	−	15350.1i	0.181697	−	1.56608i
$459$		0			0
$460$	−1540.12	+	6547.92i	−0.156105	+	0.663692i
$461$			6737.61i			0.680698i	0.940299	+	0.340349i	$0.110545\pi$
$461$			6737.61i			0.680698i	−0.940299	+	0.340349i	$0.889455\pi$
$462$		0			0
$463$	−2602.69			−0.261246			−0.130623	−	0.991432i	$-0.541698\pi$
$463$	−2602.69			−0.261246			−0.130623	+	0.991432i	$0.541698\pi$
$464$	−4089.94	+	8213.34i	−0.409204	+	0.821756i
$465$		0			0
$466$	−13268.0	−	1539.36i	−1.31894	−	0.153024i
$467$		−	9326.18i		−	0.924120i	−0.886849	−	0.462060i	$-0.847110\pi$
$467$		−	9326.18i		−	0.924120i	0.886849	−	0.462060i	$-0.152890\pi$
$468$		0			0
$469$		−	8860.60i		−	0.872376i
$470$	1683.10	−	14506.9i	0.165182	−	1.42373i
$471$		0			0
$472$	−1150.59	+	3186.50i	−0.112203	+	0.310743i
$473$	−16174.3			−1.57230
$474$		0			0
$475$			15885.8i			1.53451i
$476$	−6580.40	−	1547.76i	−0.633639	−	0.149036i
$477$		0			0
$478$	2962.35	+	343.694i	0.283462	+	0.0328874i
$479$	−11472.0			−1.09430			−0.547149	−	0.837035i	$-0.684287\pi$
$479$	−11472.0			−1.09430			−0.547149	+	0.837035i	$0.684287\pi$
$480$		0			0
$481$	−58.6494			−0.00555963
$482$	−8806.34	−	1021.72i	−0.832194	−	0.0965517i
$483$		0			0
$484$	−1932.01	−	454.423i	−0.181444	−	0.0426768i
$485$		−	2944.98i		−	0.275721i
$486$		0			0
$487$	−15048.0			−1.40018			−0.700090	−	0.714054i	$-0.746857\pi$
$487$	−15048.0			−1.40018			−0.700090	+	0.714054i	$0.746857\pi$
$488$	3906.25	−	10818.2i	0.362351	−	1.00352i
$489$		0			0
$490$	−1547.39	+	13337.2i	−0.142661	+	1.22962i
$491$			14373.5i			1.32112i	0.750775	+	0.660558i	$0.229680\pi$
$491$			14373.5i			1.32112i	−0.750775	+	0.660558i	$0.770320\pi$
$492$		0			0
$493$		−	12988.8i		−	1.18658i
$494$	6721.65	+	779.850i	0.612189	+	0.0710265i
$495$		0			0
$496$	5183.97	+	2581.42i	0.469288	+	0.233688i
$497$	7494.55			0.676411
$498$		0			0
$499$			9167.08i			0.822395i	0.911546	+	0.411197i	$0.134889\pi$
$499$			9167.08i			0.822395i	−0.911546	+	0.411197i	$0.865111\pi$
$500$	−3186.20	+	13546.4i	−0.284982	+	1.21162i
$501$		0			0
$502$	1591.35	−	13716.1i	0.141485	−	1.21948i
$503$	−7690.02			−0.681672			−0.340836	−	0.940123i	$-0.610710\pi$
$503$	−7690.02			−0.681672			−0.340836	+	0.940123i	$0.610710\pi$
$504$		0			0
$505$	−25333.5			−2.23233
$506$	587.388	−	5062.79i	0.0516059	−	0.444799i
$507$		0			0
$508$	−5745.56	−	1351.40i	−0.501807	−	0.118029i
$509$			6854.21i			0.596872i	0.954430	+	0.298436i	$0.0964649\pi$
$509$			6854.21i			0.596872i	−0.954430	+	0.298436i	$0.903535\pi$
$510$		0			0
$511$	4195.01			0.363163
$512$	5869.30	+	9988.44i	0.506619	+	0.862170i
$513$		0			0
$514$	1518.46	+	176.173i	0.130304	+	0.0151180i
$515$			32287.5i			2.76264i
$516$		0			0
$517$			11065.6i			0.941328i
$518$	−5.41090	+	46.6374i	−0.000458960	+	0.00395585i
$519$		0			0
$520$	13004.2	+	4695.56i	1.09667	+	0.395989i
$521$	−1641.63			−0.138044			−0.0690221	−	0.997615i	$-0.521988\pi$
$521$	−1641.63			−0.138044			−0.0690221	+	0.997615i	$0.521988\pi$
$522$		0			0
$523$		−	1976.34i		−	0.165238i	−0.996581	−	0.0826188i	$-0.973672\pi$
$523$		−	1976.34i		−	0.165238i	0.996581	−	0.0826188i	$-0.0263284\pi$
$524$	−270.535	+	1150.20i	−0.0225542	+	0.0958907i
$525$		0			0
$526$	−13486.3	−	1564.69i	−1.11793	−	0.129703i
$527$	−8198.06			−0.677634
$528$		0			0
$529$	−10110.7			−0.830992
$530$	−12570.9	−	1458.48i	−1.03027	−	0.119533i
$531$		0			0
$532$	1240.26	−	5273.03i	0.101075	−	0.429727i
$533$		−	6453.14i		−	0.524421i
$534$		0			0
$535$	12040.9			0.973034
$536$	7300.68	−	20218.9i	0.588323	−	1.62934i
$537$		0			0
$538$	−1082.63	+	9331.39i	−0.0867577	+	0.747779i
$539$		−	10173.4i		−	0.812984i
$540$		0			0
$541$			2892.17i			0.229841i	0.993375	+	0.114921i	$0.0366613\pi$
$541$			2892.17i			0.229841i	−0.993375	+	0.114921i	$0.963339\pi$
$542$	14817.8	+	1719.17i	1.17432	+	0.136245i
$543$		0			0
$544$	−13740.5	−	8953.73i	−1.08294	−	0.705676i
$545$	−26203.5			−2.05951
$546$		0			0
$547$		−	7033.62i		−	0.549791i	−0.961474	−	0.274896i	$-0.911357\pi$
$547$		−	7033.62i		−	0.549791i	0.961474	−	0.274896i	$-0.0886433\pi$
$548$	14646.6	+	3444.99i	1.14174	+	0.268545i
$549$		0			0
$550$	2834.35	−	24429.7i	0.219740	−	1.89398i
$551$	10408.2			0.804728
$552$		0			0
$553$	−1466.34			−0.112758
$554$	−1039.92	+	8963.24i	−0.0797509	+	0.687386i
$555$		0			0
$556$	−1152.73	+	4900.92i	−0.0879258	+	0.373823i
$557$		−	8702.92i		−	0.662037i	−0.943624	−	0.331019i	$-0.892608\pi$
$557$		−	8702.92i		−	0.662037i	0.943624	−	0.331019i	$-0.107392\pi$
$558$		0			0
$559$	−13412.9			−1.01485
$560$	4933.60	−	9907.57i	0.372291	−	0.747628i
$561$		0			0
$562$	1532.96	+	177.855i	0.115061	+	0.0133494i
$563$		−	14119.7i		−	1.05697i	−0.848943	−	0.528484i	$-0.822761\pi$
$563$		−	14119.7i		−	1.05697i	0.848943	−	0.528484i	$-0.177239\pi$
$564$		0			0
$565$			20330.2i			1.51380i
$566$	−1932.26	+	16654.5i	−0.143496	+	1.23682i
$567$		0			0
$568$	17101.8	+	6175.12i	1.26333	+	0.456166i
$569$	−383.132			−0.0282280			−0.0141140	−	0.999900i	$-0.504493\pi$
$569$	−383.132			−0.0282280			−0.0141140	+	0.999900i	$0.504493\pi$
$570$		0			0
$571$		−	21917.8i		−	1.60636i	−0.595739	−	0.803178i	$-0.703141\pi$
$571$		−	21917.8i		−	1.60636i	0.595739	−	0.803178i	$-0.296859\pi$
$572$	−10197.6	−	2398.56i	−0.745428	−	0.175330i
$573$		0			0
$574$	−5131.47	−	595.356i	−0.373142	−	0.0432921i
$575$	9922.44			0.719642
$576$		0			0
$577$	1570.50			0.113311			0.0566556	−	0.998394i	$-0.481956\pi$
$577$	1570.50			0.113311			0.0566556	+	0.998394i	$0.481956\pi$
$578$	9258.48	+	1074.17i	0.666267	+	0.0773006i
$579$		0			0
$580$	20701.4	+	4869.12i	1.48203	+	0.348585i
$581$		−	1632.76i		−	0.116589i
$582$		0			0
$583$	9588.84			0.681182
$584$	9572.57	+	3456.47i	0.678280	+	0.244914i
$585$		0			0
$586$	−1762.19	+	15188.6i	−0.124224	+	1.07071i
$587$		−	14387.7i		−	1.01166i	−0.862633	−	0.505830i	$-0.831186\pi$
$587$		−	14387.7i		−	1.01166i	0.862633	−	0.505830i	$-0.168814\pi$
$588$		0			0
$589$		−	6569.30i		−	0.459564i
$590$	7800.00	+	904.960i	0.544272	+	0.0631468i
$591$		0			0
$592$	−50.7739	+	101.963i	−0.00352499	+	0.00707882i
$593$	−17903.0			−1.23978			−0.619889	−	0.784690i	$-0.712822\pi$
$593$	−17903.0			−1.23978			−0.619889	+	0.784690i	$0.712822\pi$
$594$		0			0
$595$			15668.1i			1.07955i
$596$	−786.624	+	3344.39i	−0.0540627	+	0.229852i
$597$		0			0
$598$	487.102	−	4198.41i	0.0333095	−	0.287100i
$599$	−9474.88			−0.646299			−0.323150	−	0.946348i	$-0.604742\pi$
$599$	−9474.88			−0.646299			−0.323150	+	0.946348i	$0.604742\pi$
$600$		0			0
$601$	−16945.6			−1.15012			−0.575061	−	0.818110i	$-0.695022\pi$
$601$	−16945.6			−1.15012			−0.575061	+	0.818110i	$0.695022\pi$
$602$	−1237.45	+	10665.8i	−0.0837784	+	0.722100i
$603$		0			0
$604$	−212.695	−	50.0275i	−0.0143286	−	0.00337018i
$605$			4600.16i			0.309129i
$606$		0			0
$607$	18736.2			1.25285			0.626423	−	0.779483i	$-0.284518\pi$
$607$	18736.2			1.25285			0.626423	+	0.779483i	$0.284518\pi$
$608$	7174.84	−	11010.6i	0.478582	−	0.734438i
$609$		0			0
$610$	−26481.0	−	3072.34i	−1.75768	−	0.203927i
$611$			9176.39i			0.607589i
$612$		0			0
$613$			27850.0i			1.83499i	0.397742	+	0.917497i	$0.369794\pi$
$613$			27850.0i			1.83499i	−0.397742	+	0.917497i	$0.630206\pi$
$614$	−508.043	+	4378.90i	−0.0333924	+	0.287815i
$615$		0			0
$616$	−2848.13	+	7887.77i	−0.186289	+	0.515921i
$617$	12836.3			0.837551			0.418776	−	0.908090i	$-0.362459\pi$
$617$	12836.3			0.837551			0.418776	+	0.908090i	$0.362459\pi$
$618$		0			0
$619$		−	18030.3i		−	1.17076i	−0.810760	−	0.585378i	$-0.800946\pi$
$619$		−	18030.3i		−	1.17076i	0.810760	−	0.585378i	$-0.199054\pi$
$620$	3073.22	−	13066.0i	0.199070	−	0.846359i
$621$		0			0
$622$	23455.0	+	2721.26i	1.51199	+	0.175422i
$623$	−1186.35			−0.0762924
$624$		0			0
$625$	4902.56			0.313764
$626$	−14646.6	−	1699.31i	−0.935137	−	0.108495i
$627$		0			0
$628$	2291.58	−	9742.80i	0.145611	−	0.619077i
$629$		−	161.247i		−	0.0102215i
$630$		0			0
$631$	16460.1			1.03846			0.519229	−	0.854635i	$-0.326219\pi$
$631$	16460.1			1.03846			0.519229	+	0.854635i	$0.326219\pi$
$632$	−3346.03	−	1208.19i	−0.210598	−	0.0760429i
$633$		0			0
$634$	2956.73	−	25484.5i	0.185215	−	1.59640i
$635$			13680.3i			0.854940i
$636$		0			0
$637$		−	8436.46i		−	0.524749i
$638$	−16006.1	−	1857.04i	−0.993243	−	0.115237i
$639$		0			0
$640$	19421.3	−	18543.0i	1.19952	−	1.14527i
$641$	19443.3			1.19807			0.599035	−	0.800723i	$-0.295551\pi$
$641$	19443.3			1.19807			0.599035	+	0.800723i	$0.295551\pi$
$642$		0			0
$643$			7368.87i			0.451944i	0.974134	+	0.225972i	$0.0725558\pi$
$643$			7368.87i			0.451944i	−0.974134	+	0.225972i	$0.927444\pi$
$644$	−3293.59	−	774.677i	−0.201531	−	0.0474015i
$645$		0			0
$646$	−2144.07	+	18480.1i	−0.130584	+	1.12553i
$647$	11042.3			0.670969			0.335484	−	0.942046i	$-0.391100\pi$
$647$	11042.3			0.670969			0.335484	+	0.942046i	$0.391100\pi$
$648$		0			0
$649$	−5949.70			−0.359856
$650$	2350.44	−	20258.8i	0.141833	−	1.22249i
$651$		0			0
$652$	234.241	−	995.893i	0.0140699	−	0.0598193i
$653$			14174.3i			0.849440i	0.905325	+	0.424720i	$0.139627\pi$
$653$			14174.3i			0.849440i	−0.905325	+	0.424720i	$0.860373\pi$
$654$		0			0
$655$	2738.65			0.163371
$656$	−11218.9	−	5586.60i	−0.667721	−	0.332500i
$657$		0			0
$658$	7296.96	+	846.598i	0.432318	+	0.0501578i
$659$			1917.19i			0.113328i	0.998393	+	0.0566640i	$0.0180464\pi$
$659$			1917.19i			0.113328i	−0.998393	+	0.0566640i	$0.981954\pi$
$660$		0			0
$661$		−	28084.2i		−	1.65257i	−0.563254	−	0.826284i	$-0.690451\pi$
$661$		−	28084.2i		−	1.65257i	0.563254	−	0.826284i	$-0.309549\pi$
$662$	60.8052	−	524.090i	0.00356988	−	0.0307694i
$663$		0			0
$664$	1345.31	−	3725.77i	0.0786265	−	0.217753i
$665$	−12555.2			−0.732136
$666$		0			0
$667$		−	6501.09i		−	0.377396i
$668$	16195.3	+	3809.24i	0.938044	+	0.220635i
$669$		0			0
$670$	−49492.4	−	5742.13i	−2.85382	−	0.331101i
$671$	20199.3			1.16212
$672$		0			0
$673$	1756.20			0.100589			0.0502947	−	0.998734i	$-0.483984\pi$
$673$	1756.20			0.100589			0.0502947	+	0.998734i	$0.483984\pi$
$674$	2330.12	+	270.342i	0.133165	+	0.0154499i
$675$		0			0
$676$	8652.53	+	2035.14i	0.492292	+	0.115791i
$677$		−	10424.0i		−	0.591766i	−0.955224	−	0.295883i	$-0.904386\pi$
$677$		−	10424.0i		−	0.591766i	0.955224	−	0.295883i	$-0.0956139\pi$
$678$		0			0
$679$	1481.32			0.0837230
$680$	−12909.7	+	35752.9i	−0.728036	+	2.01627i
$681$		0			0
$682$	−1172.10	+	10102.5i	−0.0658093	+	0.567221i
$683$			5828.41i			0.326527i	0.986582	+	0.163264i	$0.0522021\pi$
$683$			5828.41i			0.326527i	−0.986582	+	0.163264i	$0.947798\pi$
$684$		0			0
$685$		−	34873.9i		−	1.94520i
$686$	−15696.6	−	1821.13i	−0.873613	−	0.101357i
$687$		0			0
$688$	−11611.8	+	23318.5i	−0.643451	+	1.29217i
$689$	7951.72			0.439675
$690$		0			0
$691$			10673.5i			0.587611i	0.955865	+	0.293806i	$0.0949218\pi$
$691$			10673.5i			0.587611i	−0.955865	+	0.293806i	$0.905078\pi$
$692$	−1254.95	+	5335.52i	−0.0689395	+	0.293101i
$693$		0			0
$694$	−3913.47	+	33730.8i	−0.214054	+	1.84496i
$695$	11669.2			0.636890
$696$		0			0
$697$	17741.9			0.964163
$698$	−25.3784	+	218.740i	−0.00137620	+	0.0118617i
$699$		0			0
$700$	−15892.7	−	3738.08i	−0.858127	−	0.201838i
$701$			14367.0i			0.774084i	0.922062	+	0.387042i	$0.126503\pi$
$701$			14367.0i			0.774084i	−0.922062	+	0.387042i	$0.873497\pi$
$702$		0			0
$703$	129.211			0.00693214
$704$	−12998.2	+	15652.3i	−0.695865	+	0.837954i
$705$		0			0
$706$	13837.4	+	1605.42i	0.737643	+	0.0855818i
$707$		−	12742.7i		−	0.677848i
$708$		0			0
$709$			25026.4i			1.32565i	0.748774	+	0.662825i	$0.230643\pi$
$709$			25026.4i			1.32565i	−0.748774	+	0.662825i	$0.769357\pi$
$710$	4856.86	−	41862.1i	0.256725	−	2.21275i
$711$		0			0
$712$	−2707.13	−	977.493i	−0.142491	−	0.0514510i
$713$	−4103.26			−0.215523
$714$		0			0
$715$			24280.8i			1.27000i
$716$	786.562	−	3344.12i	0.0410547	−	0.174547i
$717$		0			0
$718$	34446.0	+	3996.44i	1.79041	+	0.207724i
$719$	−692.065			−0.0358966			−0.0179483	−	0.999839i	$-0.505713\pi$
$719$	−692.065			−0.0358966			−0.0179483	+	0.999839i	$0.505713\pi$
$720$		0			0
$721$	−16240.6			−0.838876
$722$	4462.37	+	517.726i	0.230017	+	0.0266867i
$723$		0			0
$724$	−5207.18	+	22138.7i	−0.267297	+	1.13643i
$725$		−	31370.0i		−	1.60697i
$726$		0			0
$727$	−23929.5			−1.22076			−0.610382	−	0.792107i	$-0.708984\pi$
$727$	−23929.5			−1.22076			−0.610382	+	0.792107i	$0.708984\pi$
$728$	−2361.86	+	6541.08i	−0.120242	+	0.333006i
$729$		0			0
$730$	2718.59	−	23431.9i	0.137835	−	1.18802i
$731$		−	36876.5i		−	1.86584i
$732$		0			0
$733$		−	5613.22i		−	0.282850i	−0.989949	−	0.141425i	$-0.954832\pi$
$733$		−	5613.22i		−	0.282850i	0.989949	−	0.141425i	$-0.0451684\pi$
$734$	−24164.8	−	2803.62i	−1.21518	−	0.140985i
$735$		0			0
$736$	−6877.33	−	4481.48i	−0.344432	−	0.224442i
$737$	37751.9			1.88685
$738$		0			0
$739$			4790.30i			0.238449i	0.992867	+	0.119225i	$0.0380408\pi$
$739$			4790.30i			0.238449i	−0.992867	+	0.119225i	$0.961959\pi$
$740$	256.994	+	60.4469i	0.0127666	+	0.00300280i
$741$		0			0
$742$	733.612	−	6323.12i	0.0362962	−	0.312842i
$743$	−16695.0			−0.824333			−0.412166	−	0.911109i	$-0.635228\pi$
$743$	−16695.0			−0.824333			−0.412166	+	0.911109i	$0.635228\pi$
$744$		0			0
$745$	7963.07			0.391603
$746$	−1302.73	+	11228.4i	−0.0639359	+	0.551074i
$747$		0			0
$748$	6594.46	−	28036.8i	0.322349	−	1.37049i
$749$			6056.55i			0.295463i
$750$		0			0
$751$	27366.8			1.32973			0.664866	−	0.746963i	$-0.268489\pi$
$751$	27366.8			1.32973			0.664866	+	0.746963i	$0.268489\pi$
$752$	15953.3	+	7944.17i	0.773615	+	0.385231i
$753$		0			0
$754$	−13273.4	−	1539.99i	−0.641098	−	0.0743806i
$755$			506.433i			0.0244119i
$756$		0			0
$757$		−	5712.23i		−	0.274260i	−0.990553	−	0.137130i	$-0.956212\pi$
$757$		−	5712.23i		−	0.274260i	0.990553	−	0.137130i	$-0.0437878\pi$
$758$	−3391.63	+	29233.0i	−0.162519	+	1.40078i
$759$		0			0
$760$	−28649.7	−	10344.9i	−1.36741	−	0.493747i
$761$	−14015.9			−0.667643			−0.333822	−	0.942636i	$-0.608338\pi$
$761$	−14015.9			−0.667643			−0.333822	+	0.942636i	$0.608338\pi$
$762$		0			0
$763$		−	13180.3i		−	0.625372i
$764$	−19833.0	−	4664.86i	−0.939179	−	0.220902i
$765$		0			0
$766$	−19145.0	−	2221.21i	−0.903051	−	0.104773i
$767$	−4933.90			−0.232272
$768$		0			0
$769$	−3430.70			−0.160877			−0.0804384	−	0.996760i	$-0.525632\pi$
$769$	−3430.70			−0.160877			−0.0804384	+	0.996760i	$0.525632\pi$
$770$	19307.9	+	2240.11i	0.903645	+	0.104841i
$771$		0			0
$772$	−27879.0	−	6557.34i	−1.29972	−	0.305704i
$773$			14821.8i			0.689654i	0.938666	+	0.344827i	$0.112062\pi$
$773$			14821.8i			0.689654i	−0.938666	+	0.344827i	$0.887938\pi$
$774$		0			0
$775$	−19799.6			−0.917708
$776$	3380.21	+	1220.53i	0.156369	+	0.0564621i
$777$		0			0
$778$	254.037	−	2189.59i	0.0117065	−	0.100900i
$779$			14217.0i			0.653885i
$780$		0			0
$781$			31931.7i			1.46300i
$782$	11542.9	+	1339.21i	0.527842	+	0.0612405i
$783$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.80958 + 0.325969i) q^{2} +(7.78749 + 1.83167i) q^{4} -18.5422i q^{5} +9.32669 q^{7} +(21.2825 + 7.68472i) q^{8} +O(q^{10})\)	\(q+(2.80958 + 0.325969i) q^{2} +(7.78749 + 1.83167i) q^{4} -18.5422i q^{5} +9.32669 q^{7} +(21.2825 + 7.68472i) q^{8} +(6.04419 - 52.0958i) q^{10} +39.7378i q^{11} +32.9533i q^{13} +(26.2041 + 3.04021i) q^{14} +(57.2899 + 28.5283i) q^{16} -90.5998 q^{17} -72.5998i q^{19} +(33.9633 - 144.397i) q^{20} +(-12.9533 + 111.647i) q^{22} -45.3466 q^{23} -218.813 q^{25} +(-10.7418 + 92.5849i) q^{26} +(72.6315 + 17.0835i) q^{28} +143.364i q^{29} +90.4865 q^{31} +(151.661 + 98.8272i) q^{32} +(-254.547 - 29.5327i) q^{34} -172.937i q^{35} +1.77977i q^{37} +(23.6653 - 203.975i) q^{38} +(142.492 - 394.625i) q^{40} -195.827 q^{41} +407.027i q^{43} +(-72.7866 + 309.458i) q^{44} +(-127.405 - 14.7816i) q^{46} +278.467 q^{47} -256.013 q^{49} +(-614.773 - 71.3263i) q^{50} +(-60.3597 + 256.623i) q^{52} -241.303i q^{53} +736.826 q^{55} +(198.495 + 71.6730i) q^{56} +(-46.7324 + 402.794i) q^{58} +149.724i q^{59} -508.314i q^{61} +(254.229 + 29.4958i) q^{62} +(393.890 + 327.100i) q^{64} +611.027 q^{65} -950.026i q^{67} +(-705.545 - 165.949i) q^{68} +(56.3723 - 485.882i) q^{70} +803.559 q^{71} +449.786 q^{73} +(-0.580152 + 5.00042i) q^{74} +(132.979 - 565.370i) q^{76} +370.622i q^{77} -157.220 q^{79} +(528.977 - 1062.28i) q^{80} +(-550.192 - 63.8335i) q^{82} -175.063i q^{83} +1679.92i q^{85} +(-132.678 + 1143.57i) q^{86} +(-305.374 + 845.720i) q^{88} -127.200 q^{89} +307.345i q^{91} +(-353.136 - 83.0602i) q^{92} +(782.374 + 90.7715i) q^{94} -1346.16 q^{95} +158.826 q^{97} +(-719.289 - 83.4523i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8	\( 6 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 76 q^{8} + 60 q^{10} + 100 q^{14} + 56 q^{16} - 52 q^{17} - 56 q^{20} + 224 q^{22} - 328 q^{23} - 106 q^{25} - 56 q^{26} - 352 q^{28} - 636 q^{31} + 248 q^{32} - 548 q^{34} + 776 q^{38} + 232 q^{40} - 236 q^{41} - 1152 q^{44} + 328 q^{46} + 408 q^{47} + 654 q^{49} - 1970 q^{50} - 368 q^{52} + 1024 q^{55} + 1864 q^{56} + 140 q^{58} + 2108 q^{62} + 832 q^{64} + 1744 q^{65} - 2976 q^{68} + 1352 q^{70} + 1704 q^{71} + 956 q^{73} - 1568 q^{74} - 1744 q^{76} - 44 q^{79} + 2112 q^{80} - 2236 q^{82} + 760 q^{86} + 1856 q^{88} + 220 q^{89} - 1728 q^{92} + 2088 q^{94} - 5104 q^{95} - 2444 q^{97} - 3354 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 16 * q^4 + 28 * q^7 + 76 * q^8 + 60 * q^10 + 100 * q^14 + 56 * q^16 - 52 * q^17 - 56 * q^20 + 224 * q^22 - 328 * q^23 - 106 * q^25 - 56 * q^26 - 352 * q^28 - 636 * q^31 + 248 * q^32 - 548 * q^34 + 776 * q^38 + 232 * q^40 - 236 * q^41 - 1152 * q^44 + 328 * q^46 + 408 * q^47 + 654 * q^49 - 1970 * q^50 - 368 * q^52 + 1024 * q^55 + 1864 * q^56 + 140 * q^58 + 2108 * q^62 + 832 * q^64 + 1744 * q^65 - 2976 * q^68 + 1352 * q^70 + 1704 * q^71 + 956 * q^73 - 1568 * q^74 - 1744 * q^76 - 44 * q^79 + 2112 * q^80 - 2236 * q^82 + 760 * q^86 + 1856 * q^88 + 220 * q^89 - 1728 * q^92 + 2088 * q^94 - 5104 * q^95 - 2444 * q^97 - 3354 * q^98

Introduction

L-functions

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