LMFDB - Embedded newform 152.4.o.a.107.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [152,4,Mod(27,152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(152, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("152.27");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	152.o (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:	$8.96829032087$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			107.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	152.107
Dual form			152.4.o.a.27.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 + 1.41421i) q^{2} +(-8.72474 + 5.03723i) q^{3} +(4.00000 - 6.92820i) q^{4} +(14.2474 - 24.6773i) q^{6} +22.6274i q^{8} +(37.2474 - 64.5145i) q^{9} +O(q^{10})$	\(q+(-2.44949 + 1.41421i) q^{2} +(-8.72474 + 5.03723i) q^{3} +(4.00000 - 6.92820i) q^{4} +(14.2474 - 24.6773i) q^{6} +22.6274i q^{8} +(37.2474 - 64.5145i) q^{9} +52.2372 q^{11} +80.5957i q^{12} +(-32.0000 - 55.4256i) q^{16} +(-45.0000 - 77.9423i) q^{17} +210.703i q^{18} +(28.6135 + 77.7192i) q^{19} +(-127.955 + 73.8746i) q^{22} +(-113.980 - 197.418i) q^{24} +(-62.5000 + 108.253i) q^{25} +478.486i q^{27} +(156.767 + 90.5097i) q^{32} +(-455.757 + 263.131i) q^{33} +(220.454 + 127.279i) q^{34} +(-297.980 - 516.116i) q^{36} +(-180.000 - 149.907i) q^{38} +(367.005 - 211.890i) q^{41} +(145.000 + 251.147i) q^{43} +(208.949 - 361.910i) q^{44} +(558.384 + 322.383i) q^{48} +343.000 q^{49} -353.553i q^{50} +(785.227 + 453.351i) q^{51} +(-676.681 - 1172.05i) q^{54} +(-641.135 - 533.947i) q^{57} +(-775.346 + 447.646i) q^{59} -512.000 q^{64} +(744.247 - 1289.07i) q^{66} +(526.476 + 303.961i) q^{67} -720.000 q^{68} +(1459.80 + 842.814i) q^{72} +(614.544 + 1064.42i) q^{73} -1259.31i q^{75} +(652.908 + 112.636i) q^{76} +(-1404.56 - 2432.78i) q^{81} +(-599.317 + 1038.05i) q^{82} +84.6730 q^{83} +(-710.352 - 410.122i) q^{86} +1181.99i q^{88} +(1151.26 + 664.680i) q^{89} -1823.67 q^{96} +(-1410.45 + 814.326i) q^{97} +(-840.175 + 485.075i) q^{98} +(1945.70 - 3370.06i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 30 q^{3} + 16 q^{4} + 8 q^{6} + 100 q^{9}+O(q^{10}) $ 4 * q - 30 * q^3 + 16 * q^4 + 8 * q^6 + 100 * q^9	$ 4 q - 30 q^{3} + 16 q^{4} + 8 q^{6} + 100 q^{9} - 36 q^{11} - 128 q^{16} - 180 q^{17} - 106 q^{19} - 600 q^{22} - 64 q^{24} - 250 q^{25} - 30 q^{33} - 800 q^{36} - 720 q^{38} + 1566 q^{41} + 580 q^{43} - 144 q^{44} + 1920 q^{48} + 1372 q^{49} + 2700 q^{51} - 1384 q^{54} - 360 q^{57} - 2538 q^{59} - 2048 q^{64} + 2928 q^{66} + 210 q^{67} - 2880 q^{68} + 1920 q^{72} + 430 q^{73} + 848 q^{76} - 2042 q^{81} + 160 q^{82} + 2700 q^{83} - 1024 q^{96} - 5730 q^{97} + 2100 q^{99}+O(q^{100}) $ 4 * q - 30 * q^3 + 16 * q^4 + 8 * q^6 + 100 * q^9 - 36 * q^11 - 128 * q^16 - 180 * q^17 - 106 * q^19 - 600 * q^22 - 64 * q^24 - 250 * q^25 - 30 * q^33 - 800 * q^36 - 720 * q^38 + 1566 * q^41 + 580 * q^43 - 144 * q^44 + 1920 * q^48 + 1372 * q^49 + 2700 * q^51 - 1384 * q^54 - 360 * q^57 - 2538 * q^59 - 2048 * q^64 + 2928 * q^66 + 210 * q^67 - 2880 * q^68 + 1920 * q^72 + 430 * q^73 + 848 * q^76 - 2042 * q^81 + 160 * q^82 + 2700 * q^83 - 1024 * q^96 - 5730 * q^97 + 2100 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/152\mathbb{Z}\right)^\times$.

$n$	$39$	$77$	$97$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−2.44949	+	1.41421i	−0.866025	+	0.500000i
$2$	−2.44949	+	1.41421i	−0.866025	+	0.500000i
$3$	−8.72474	+	5.03723i	−1.67908	+	0.969416i	−0.716827	+	0.697251i	$0.754406\pi$
$3$	−8.72474	+	5.03723i	−1.67908	+	0.969416i	−0.962250	+	0.272166i	$0.912260\pi$
$4$	4.00000	−	6.92820i	0.500000	−	0.866025i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	14.2474	−	24.6773i	0.969416	−	1.67908i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			22.6274i			1.00000i
$9$	37.2474	−	64.5145i	1.37954	−	2.38942i
$10$		0			0
$11$	52.2372			1.43183			0.715915	−	0.698188i	$-0.246010\pi$
$11$	52.2372			1.43183			0.715915	+	0.698188i	$0.246010\pi$
$12$			80.5957i			1.93883i
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	−	55.4256i	−0.500000	−	0.866025i
$17$	−45.0000	−	77.9423i	−0.642006	−	1.11199i	−0.984984	−	0.172644i	$-0.944769\pi$
$17$	−45.0000	−	77.9423i	−0.642006	−	1.11199i	0.342978	−	0.939343i	$-0.388564\pi$
$18$			210.703i			2.75907i
$19$	28.6135	+	77.7192i	0.345494	+	0.938421i
$19$	28.6135	+	77.7192i	0.345494	+	0.938421i
$20$		0			0
$21$		0			0
$22$	−127.955	+	73.8746i	−1.24000	+	0.715915i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−113.980	−	197.418i	−0.969416	−	1.67908i
$25$	−62.5000	+	108.253i	−0.500000	+	0.866025i
$26$		0			0
$27$			478.486i			3.41054i
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	156.767	+	90.5097i	0.866025	+	0.500000i
$33$	−455.757	+	263.131i	−2.40415	+	1.38804i
$34$	220.454	+	127.279i	1.11199	+	0.642006i
$35$		0			0
$36$	−297.980	−	516.116i	−1.37954	−	2.38942i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−180.000	−	149.907i	−0.768417	−	0.639949i
$39$		0			0
$40$		0			0
$41$	367.005	−	211.890i	1.39797	−	0.807116i	0.403786	−	0.914854i	$-0.367694\pi$
$41$	367.005	−	211.890i	1.39797	−	0.807116i	0.994179	+	0.107738i	$0.0343608\pi$
$42$		0			0
$43$	145.000	+	251.147i	0.514239	+	0.890689i	0.999864	+	0.0165210i	$0.00525904\pi$
$43$	145.000	+	251.147i	0.514239	+	0.890689i	−0.485624	+	0.874168i	$0.661408\pi$
$44$	208.949	−	361.910i	0.715915	−	1.24000i
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	558.384	+	322.383i	1.67908	+	0.969416i
$49$	343.000			1.00000
$50$		−	353.553i		−	1.00000i
$51$	785.227	+	453.351i	2.15596	+	1.24474i
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$	−676.681	−	1172.05i	−1.70527	−	2.95362i
$55$		0			0
$56$		0			0
$57$	−641.135	−	533.947i	−1.48983	−	1.24075i
$58$		0			0
$59$	−775.346	+	447.646i	−1.71087	+	0.987772i	−0.777483	+	0.628904i	$0.783504\pi$
$59$	−775.346	+	447.646i	−1.71087	+	0.987772i	−0.933388	+	0.358868i	$0.883163\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$	744.247	−	1289.07i	1.38804	−	2.40415i
$67$	526.476	+	303.961i	0.959990	+	0.554250i	0.896170	−	0.443711i	$-0.146338\pi$
$67$	526.476	+	303.961i	0.959990	+	0.554250i	0.0638199	+	0.997961i	$0.479672\pi$
$68$	−720.000			−1.28401
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$	1459.80	+	842.814i	2.38942	+	1.37954i
$73$	614.544	+	1064.42i	0.985301	+	1.70659i	0.640591	+	0.767882i	$0.278689\pi$
$73$	614.544	+	1064.42i	0.985301	+	1.70659i	0.344710	+	0.938709i	$0.387977\pi$
$74$		0			0
$75$		−	1259.31i		−	1.93883i
$76$	652.908	+	112.636i	0.985443	+	0.170004i
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	−1404.56	−	2432.78i	−1.92670	−	3.33714i
$82$	−599.317	+	1038.05i	−0.807116	+	1.39797i
$83$	84.6730			0.111977			0.0559884	−	0.998431i	$-0.482169\pi$
$83$	84.6730			0.111977			0.0559884	+	0.998431i	$0.482169\pi$
$84$		0			0
$85$		0			0
$86$	−710.352	−	410.122i	−0.890689	−	0.514239i
$87$		0			0
$88$			1181.99i			1.43183i
$89$	1151.26	+	664.680i	1.37116	+	0.791640i	0.991074	−	0.133311i	$-0.0425608\pi$
$89$	1151.26	+	664.680i	1.37116	+	0.791640i	0.380087	+	0.924951i	$0.375894\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1823.67			−1.93883
$97$	−1410.45	+	814.326i	−1.47639	+	0.852395i	−0.999645	−	0.0266459i	$-0.991517\pi$
$97$	−1410.45	+	814.326i	−1.47639	+	0.852395i	−0.476746	+	0.879041i	$0.658184\pi$
$98$	−840.175	+	485.075i	−0.866025	+	0.500000i
$99$	1945.70	−	3370.06i	1.97526	−	3.42125i
$100$	500.000	+	866.025i	0.500000	+	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	−2564.54			−2.48948
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1405.73i		−	1.27006i	−0.772486	−	0.635032i	$-0.780987\pi$
$107$		−	1405.73i		−	1.27006i	0.772486	−	0.635032i	$-0.219013\pi$
$108$	3315.05	+	1913.94i	2.95362	+	1.70527i
$109$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$109$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1427.42i			1.18832i	0.804345	+	0.594162i	$0.202516\pi$
$113$			1427.42i			1.18832i	−0.804345	+	0.594162i	$0.797484\pi$
$114$	2325.57	+	401.195i	1.91061	+	0.329608i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1266.13	−	2193.01i	0.987772	−	1.71087i
$119$		0			0
$120$		0			0
$121$	1397.73			1.05013
$122$		0			0
$123$	−2134.68	+	3697.38i	−1.56486	+	2.71042i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$	1254.14	−	724.077i	0.866025	−	0.500000i
$129$	−2530.18	−	1460.80i	−1.72690	−	0.997024i
$130$		0			0
$131$	871.379	+	1509.27i	0.581166	+	1.00661i	0.995342	+	0.0964118i	$0.0307366\pi$
$131$	871.379	+	1509.27i	0.581166	+	1.00661i	−0.414176	+	0.910197i	$0.635930\pi$
$132$			4210.10i			2.77608i
$133$		0			0
$134$	−1719.46			−1.10850
$135$		0			0
$136$	1763.63	−	1018.23i	1.11199	−	0.642006i
$137$	1552.09	−	2688.31i	0.967915	−	1.67648i	0.266344	−	0.963878i	$-0.414184\pi$
$137$	1552.09	−	2688.31i	0.967915	−	1.67648i	0.701571	−	0.712599i	$-0.252482\pi$
$138$		0			0
$139$	−1636.11	+	2833.83i	−0.998368	+	1.72922i	−0.449723	+	0.893168i	$0.648477\pi$
$139$	−1636.11	+	2833.83i	−0.998368	+	1.72922i	−0.548645	+	0.836056i	$0.684856\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−4767.67			−2.75907
$145$		0			0
$146$	−3010.64	−	1738.19i	−1.70659	−	0.985301i
$147$	−2992.59	+	1727.77i	−1.67908	+	0.969416i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	1780.93	+	3084.66i	0.969416	+	1.67908i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−1758.58	+	647.450i	−0.938421	+	0.345494i
$153$	−6704.54			−3.54268
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	6880.93	+	3972.71i	3.33714	+	1.92670i
$163$	3990.22			1.91741			0.958706	−	0.284399i	$-0.0917941\pi$
$163$	3990.22			1.91741			0.958706	+	0.284399i	$0.0917941\pi$
$164$		−	3390.25i		−	1.61423i
$165$		0			0
$166$	−207.406	+	119.746i	−0.0969747	+	0.0559884i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	1098.50	+	1902.66i	0.500000	+	0.866025i
$170$		0			0
$171$	6079.79	+	1048.85i	2.71891	+	0.469052i
$172$	2320.00			1.02848
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	−1671.59	−	2895.28i	−0.715915	−	1.24000i
$177$	4509.80	−	7811.19i	1.91512	−	3.31709i
$178$	−3760.00			−1.58328
$179$		−	1884.91i		−	0.787068i	−0.919310	−	0.393534i	$-0.871252\pi$
$179$		−	1884.91i		−	0.787068i	0.919310	−	0.393534i	$-0.128748\pi$
$180$		0			0
$181$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$181$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2350.68	−	4071.49i	−0.919243	−	1.59218i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	4467.07	−	2579.06i	1.67908	−	0.969416i
$193$	4276.81	−	2469.22i	1.59509	−	0.920923i	0.602670	−	0.797990i	$-0.294103\pi$
$193$	4276.81	−	2469.22i	1.59509	−	0.920923i	0.992415	−	0.122933i	$-0.0392299\pi$
$194$	2303.26	−	3989.37i	0.852395	−	1.47639i
$195$		0			0
$196$	1372.00	−	2376.37i	0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$			11006.6i			3.95052i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−2449.49	−	1414.21i	−0.866025	−	0.500000i
$201$	−6124.49			−2.14920
$202$		0			0
$203$		0			0
$204$	6281.82	−	3626.81i	2.15596	−	1.24474i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1494.69	+	4059.83i	0.494689	+	1.34366i
$210$		0			0
$211$	−330.681	+	190.919i	−0.107891	+	0.0622910i	−0.552975	−	0.833198i	$-0.686507\pi$
$211$	−330.681	+	190.919i	−0.107891	+	0.0622910i	0.445083	+	0.895489i	$0.353174\pi$
$212$		0			0
$213$		0			0
$214$	1988.00	+	3443.32i	0.635032	+	1.09991i
$215$		0			0
$216$	−10826.9			−3.41054
$217$		0			0
$218$		0			0
$219$	−10723.5	−	6191.21i	−3.30879	−	1.91033i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$	4655.93	+	8064.31i	1.37954	+	2.38942i
$226$	−2018.68	−	3496.46i	−0.594162	−	1.02912i
$227$		−	4738.02i		−	1.38535i	−0.721252	−	0.692673i	$-0.756433\pi$
$227$		−	4738.02i		−	1.38535i	0.721252	−	0.692673i	$-0.243567\pi$
$228$	−6263.83	+	2306.13i	−1.81944	+	0.669855i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3141.31	−	5440.91i	−0.883236	−	1.52981i	−0.847722	−	0.530441i	$-0.822026\pi$
$233$	−3141.31	−	5440.91i	−0.883236	−	1.52981i	−0.0355143	−	0.999369i	$-0.511307\pi$
$234$		0			0
$235$		0			0
$236$			7162.34i			1.97554i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	2280.08	+	1316.41i	0.609432	+	0.351856i	0.772743	−	0.634719i	$-0.218884\pi$
$241$	2280.08	+	1316.41i	0.609432	+	0.351856i	−0.163311	+	0.986575i	$0.552217\pi$
$242$	−3423.72	+	1976.69i	−0.909444	+	0.525067i
$243$	13320.6	+	7690.67i	3.51654	+	2.03028i
$244$		0			0
$245$		0			0
$246$		−	12075.6i		−	3.12972i
$247$		0			0
$248$		0			0
$249$	−738.750	+	426.518i	−0.188018	+	0.108552i
$250$		0			0
$251$	−1821.02	+	3154.10i	−0.457936	+	0.793168i	−0.998852	−	0.0479087i	$-0.984744\pi$
$251$	−1821.02	+	3154.10i	−0.457936	+	0.793168i	0.540916	+	0.841077i	$0.318078\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2048.00	+	3547.24i	−0.500000	+	0.866025i
$257$	6052.54	+	3494.44i	1.46906	+	0.848160i	0.999398	−	0.0346890i	$-0.0110441\pi$
$257$	6052.54	+	3494.44i	1.46906	+	0.848160i	0.469658	+	0.882849i	$0.344377\pi$
$258$	8263.52			1.99405
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−4268.87	−	2464.63i	−1.00661	−	0.581166i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−5953.98	−	10312.6i	−1.38804	−	2.40415i
$265$		0			0
$266$		0			0
$267$	−13392.6			−3.06972
$268$	4211.81	−	2431.69i	0.959990	−	0.554250i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$	−2880.00	+	4988.31i	−0.642006	+	1.11199i
$273$		0			0
$274$			8779.97i			1.93583i
$275$	−3264.83	+	5654.85i	−0.715915	+	1.24000i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	9255.24i		−	1.99674i
$279$		0			0
$280$		0			0
$281$	6479.86	+	3741.15i	1.37564	+	0.794229i	0.991632	−	0.129099i	$-0.0412086\pi$
$281$	6479.86	+	3741.15i	1.37564	+	0.794229i	0.384013	+	0.923328i	$0.374542\pi$
$282$		0			0
$283$	−208.063	−	360.377i	−0.0437035	−	0.0756967i	0.843346	−	0.537371i	$-0.180582\pi$
$283$	−208.063	−	360.377i	−0.0437035	−	0.0756967i	−0.887050	+	0.461674i	$0.847249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	11678.4	−	6742.51i	2.38942	−	1.37954i
$289$	−1593.50	+	2760.02i	−0.324344	+	0.561780i
$290$		0			0
$291$	8203.90	−	14209.6i	1.65265	−	2.86248i
$292$	9832.71			1.97060
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	4886.87	−	8464.32i	0.969416	−	1.67908i
$295$		0			0
$296$		0			0
$297$			24994.8i			4.88331i
$298$		0			0
$299$		0			0
$300$	−8724.74	−	5037.23i	−1.67908	−	0.969416i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	3392.00	−	4072.94i	0.639949	−	0.768417i
$305$		0			0
$306$	16422.7	−	9481.65i	3.06805	−	1.77134i
$307$	−9111.93	+	5260.77i	−1.69396	+	0.978007i	−0.742693	+	0.669632i	$0.766452\pi$
$307$	−9111.93	+	5260.77i	−1.69396	+	0.978007i	−0.951265	+	0.308375i	$0.900215\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	1032.95	−	1789.12i	0.186536	−	0.323089i	−0.757557	−	0.652769i	$-0.773607\pi$
$313$	1032.95	−	1789.12i	0.186536	−	0.323089i	0.944093	+	0.329679i	$0.106941\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7080.98	+	12264.6i	1.23122	+	2.13254i
$322$		0			0
$323$	4770.00	−	5727.56i	0.821702	−	0.986657i
$324$	−22473.0			−3.85340
$325$		0			0
$326$	−9774.00	+	5643.02i	−1.66053	+	0.958706i
$327$		0			0
$328$	4794.53	+	8304.38i	0.807116	+	1.39797i
$329$		0			0
$330$		0			0
$331$		−	2746.65i		−	0.456101i	−0.973649	−	0.228051i	$-0.926765\pi$
$331$		−	2746.65i		−	0.456101i	0.973649	−	0.228051i	$-0.0732351\pi$
$332$	338.692	−	586.632i	0.0559884	−	0.0969747i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−10629.8	+	6137.10i	−1.71822	+	0.992015i	−0.796051	+	0.605229i	$0.793081\pi$
$337$	−10629.8	+	6137.10i	−1.71822	+	0.992015i	−0.922170	+	0.386786i	$0.873585\pi$
$338$	−5381.53	−	3107.03i	−0.866025	−	0.500000i
$339$	−7190.26	−	12453.9i	−1.15198	−	1.99529i
$340$		0			0
$341$		0			0
$342$	−16375.7	+	6028.97i	−2.58917	+	0.953243i
$343$		0			0
$344$	−5682.82	+	3280.98i	−0.890689	+	0.514239i
$345$		0			0
$346$		0			0
$347$	3444.14	+	5965.43i	0.532828	+	0.922885i	0.999265	+	0.0383308i	$0.0122041\pi$
$347$	3444.14	+	5965.43i	0.532828	+	0.922885i	−0.466437	+	0.884554i	$0.654463\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	8189.09	+	4727.98i	1.24000	+	0.715915i
$353$	13104.0			1.97579			0.987895	−	0.155125i	$-0.0495779\pi$
$353$	13104.0			1.97579			0.987895	+	0.155125i	$0.0495779\pi$
$354$			25511.3i			3.83025i
$355$		0			0
$356$	9210.08	−	5317.44i	1.37116	−	0.791640i
$357$		0			0
$358$	2665.67	+	4617.08i	0.393534	+	0.681621i
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−5221.53	+	4447.64i	−0.761267	+	0.648438i
$362$		0			0
$363$	−12194.8	+	7040.69i	−1.76326	+	1.01802i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		−	31569.5i		−	4.45378i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	11515.9	+	6648.72i	1.59218	+	0.919243i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	9036.82i		−	1.22478i	−0.790557	−	0.612389i	$-0.790209\pi$
$379$		−	9036.82i		−	1.22478i	0.790557	−	0.612389i	$-0.209791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$	−7294.69	+	12634.8i	−0.969416	+	1.67908i
$385$		0			0
$386$	−6984.00	+	12096.6i	−0.920923	+	1.59509i
$387$	21603.5			2.83765
$388$			13029.2i			1.70479i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$			7761.20i			1.00000i
$393$	−15205.1	−	8778.68i	−1.95165	−	1.12678i
$394$		0			0
$395$		0			0
$396$	−15565.6	−	26960.5i	−1.97526	−	3.42125i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$		0			0
$400$	8000.00			1.00000
$401$	1006.95	−	581.361i	0.125398	−	0.0723984i	−0.435989	−	0.899952i	$-0.643601\pi$
$401$	1006.95	−	581.361i	0.125398	−	0.0723984i	0.561387	+	0.827554i	$0.310268\pi$
$402$	15001.9	−	8661.34i	1.86126	−	1.07460i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−10258.2	+	17767.7i	−1.24474	+	2.15596i
$409$	−11157.2	−	6441.63i	−1.34887	−	0.778773i	−0.360784	−	0.932649i	$-0.617491\pi$
$409$	−11157.2	−	6441.63i	−1.34887	−	0.778773i	−0.988090	+	0.153877i	$0.950824\pi$
$410$		0			0
$411$			31273.0i			3.75325i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	32965.9i		−	3.87134i
$418$	−9402.70	−	7830.71i	−1.10024	−	0.916298i
$419$	−16794.0			−1.95809			−0.979046	−	0.203639i	$-0.934723\pi$
$419$	−16794.0			−1.95809			−0.979046	+	0.203639i	$0.934723\pi$
$420$		0			0
$421$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$421$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$422$	540.000	−	935.307i	0.0622910	−	0.107891i
$423$		0			0
$424$		0			0
$425$	11250.0			1.28401
$426$		0			0
$427$		0			0
$428$	−9739.17	−	5622.91i	−1.09991	−	0.635032i
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$	26520.4	−	15311.5i	2.95362	−	1.70527i
$433$	−14858.6	−	8578.62i	−1.64910	−	0.952107i	−0.977432	−	0.211252i	$-0.932246\pi$
$433$	−14858.6	−	8578.62i	−1.64910	−	0.952107i	−0.671665	−	0.740855i	$-0.734421\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	35022.8			3.82067
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	12775.9	−	22128.5i	1.37954	−	2.38942i
$442$		0			0
$443$	−2949.61	+	5108.88i	−0.316344	+	0.547924i	−0.979722	−	0.200361i	$-0.935789\pi$
$443$	−2949.61	+	5108.88i	−0.316344	+	0.547924i	0.663378	+	0.748284i	$0.269122\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	11448.2i		−	1.20328i	−0.798767	−	0.601641i	$-0.794514\pi$
$449$		−	11448.2i		−	1.20328i	0.798767	−	0.601641i	$-0.205486\pi$
$450$	−22809.3	−	13169.0i	−2.38942	−	1.37954i
$451$	19171.3	−	11068.6i	2.00165	−	1.15565i
$452$	9889.48	+	5709.69i	1.02912	+	0.594162i
$453$		0			0
$454$	6700.57	+	11605.7i	0.692673	+	1.19974i
$455$		0			0
$456$	12081.8	−	14507.2i	1.24075	−	1.48983i
$457$	15472.3			1.58372			0.791862	−	0.610700i	$-0.209112\pi$
$457$	15472.3			1.58372			0.791862	+	0.610700i	$0.209112\pi$
$458$		0			0
$459$	37294.3	−	21531.9i	3.79248	−	2.18959i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	15389.2	+	8884.97i	1.52981	+	0.883236i
$467$	−4151.92			−0.411409			−0.205704	−	0.978614i	$-0.565949\pi$
$467$	−4151.92			−0.411409			−0.205704	+	0.978614i	$0.565949\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−10129.1	−	17544.1i	−0.987772	−	1.71087i
$473$	7574.40	+	13119.2i	0.736303	+	1.27531i
$474$		0			0
$475$	−10201.7	−	1759.94i	−0.985443	−	0.170004i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$	−7446.72			−0.703712
$483$		0			0
$484$	5590.92	−	9683.75i	0.525067	−	0.909444i
$485$		0			0
$486$	−43505.0			−4.06055
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	−34813.6	+	20099.7i	−3.21948	+	1.85877i
$490$		0			0
$491$	6111.00	+	10584.6i	0.561681	+	0.972861i	0.997350	+	0.0727536i	$0.0231787\pi$
$491$	6111.00	+	10584.6i	0.561681	+	0.972861i	−0.435669	+	0.900107i	$0.643488\pi$
$492$	17077.5	+	29579.0i	1.56486	+	2.71042i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	1206.37	−	2089.50i	0.108552	−	0.188018i
$499$	−1012.97	−	1754.51i	−0.0908749	−	0.157400i	0.817005	−	0.576631i	$-0.195633\pi$
$499$	−1012.97	−	1754.51i	−0.0908749	−	0.157400i	−0.907880	+	0.419231i	$0.862300\pi$
$500$		0			0
$501$		0			0
$502$		−	10301.3i		−	0.915871i
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19168.3	−	11066.8i	−1.67908	−	0.969416i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		−	11585.2i		−	1.00000i
$513$	−37187.5	+	13691.2i	−3.20052	+	1.17832i
$514$	−19767.5			−1.69632
$515$		0			0
$516$	−20241.4	+	11686.4i	−1.72690	+	0.997024i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18908.8i			1.59004i	0.606584	+	0.795020i	$0.292540\pi$
$521$			18908.8i			1.59004i	−0.606584	+	0.795020i	$0.707460\pi$
$522$		0			0
$523$	−20303.8	−	11722.4i	−1.69756	−	0.980087i	−0.948065	−	0.318078i	$-0.896963\pi$
$523$	−20303.8	−	11722.4i	−1.69756	−	0.980087i	−0.749496	−	0.662009i	$-0.769704\pi$
$524$	13942.1			1.16233
$525$		0			0
$526$		0			0
$527$		0			0
$528$	29168.4	+	16840.4i	2.40415	+	1.38804i
$529$	−6083.50	−	10536.9i	−0.500000	−	0.866025i
$530$		0			0
$531$			66694.7i			5.45066i
$532$		0			0
$533$		0			0
$534$	32805.0	−	18940.0i	2.65845	−	1.53486i
$535$		0			0
$536$	−6877.86	+	11912.8i	−0.554250	+	0.959990i
$537$	9494.76	+	16445.4i	0.762996	+	1.32155i
$538$		0			0
$539$	17917.4			1.43183
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		−	16291.7i		−	1.28401i
$545$		0			0
$546$		0			0
$547$	11529.7	+	6656.70i	0.901237	+	0.520329i	0.877601	−	0.479391i	$-0.159143\pi$
$547$	11529.7	+	6656.70i	0.901237	+	0.520329i	0.0236354	+	0.999721i	$0.492476\pi$
$548$	−12416.8	−	21506.4i	−0.967915	−	1.67648i
$549$		0			0
$550$		−	18468.7i		−	1.43183i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	13088.9	+	22670.6i	0.998368	+	1.72922i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	41018.1	+	23681.8i	3.08696	+	1.78226i
$562$	−21163.1			−1.58846
$563$		−	26694.5i		−	1.99829i	−0.0413135	−	0.999146i	$-0.513154\pi$
$563$		−	26694.5i		−	1.99829i	0.0413135	−	0.999146i	$-0.486846\pi$
$564$		0			0
$565$		0			0
$566$	1019.30	+	588.492i	0.0756967	+	0.0437035i
$567$		0			0
$568$		0			0
$569$		−	27039.8i		−	1.99221i	−0.0881913	−	0.996104i	$-0.528109\pi$
$569$		−	27039.8i		−	1.99221i	0.0881913	−	0.996104i	$-0.471891\pi$
$570$		0			0
$571$	10322.4			0.756532			0.378266	−	0.925697i	$-0.376521\pi$
$571$	10322.4			0.756532			0.378266	+	0.925697i	$0.376521\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−19070.7	+	33031.4i	−1.37954	+	2.38942i
$577$	−26794.1			−1.93319			−0.966595	−	0.256307i	$-0.917494\pi$
$577$	−26794.1			−1.93319			−0.966595	+	0.256307i	$0.917494\pi$
$578$		−	9014.20i		−	0.648687i
$579$	−24876.0	+	43086.6i	−1.78551	+	3.09260i
$580$		0			0
$581$		0			0
$582$			46408.3i			3.30530i
$583$		0			0
$584$	−24085.1	+	13905.6i	−1.70659	+	0.985301i
$585$		0			0
$586$		0			0
$587$	−5175.00	−	8963.36i	−0.363876	−	0.630251i	0.624719	−	0.780849i	$-0.285213\pi$
$587$	−5175.00	−	8963.36i	−0.363876	−	0.630251i	−0.988595	+	0.150598i	$0.951880\pi$
$588$			27644.3i			1.93883i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	11818.8	−	20470.8i	0.818449	−	1.41759i	−0.0883763	−	0.996087i	$-0.528168\pi$
$593$	11818.8	−	20470.8i	0.818449	−	1.41759i	0.906825	−	0.421507i	$-0.138499\pi$
$594$	−35348.0	−	61224.5i	−2.44166	−	4.22907i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$	28494.9			1.93883
$601$			386.168i			0.0262098i	0.999914	+	0.0131049i	$0.00417154\pi$
$601$			386.168i			0.0262098i	−0.999914	+	0.0131049i	$0.995828\pi$
$602$		0			0
$603$	39219.8	−	22643.6i	2.64868	−	1.52922i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−2548.67	+	14773.6i	−0.170004	+	0.985443i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−26818.2	+	46450.4i	−1.77134	+	3.06805i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	14879.7	−	25772.4i	0.978007	−	1.69396i
$615$		0			0
$616$		0			0
$617$	−2280.06	+	3949.18i	−0.148771	+	0.257679i	−0.930774	−	0.365596i	$-0.880865\pi$
$617$	−2280.06	+	3949.18i	−0.148771	+	0.257679i	0.782002	+	0.623275i	$0.214198\pi$
$618$		0			0
$619$	30706.0			1.99383			0.996913	−	0.0785136i	$-0.0250174\pi$
$619$	30706.0			1.99383			0.996913	+	0.0785136i	$0.0250174\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$			5843.24i			0.373071i
$627$	−33491.1	−	27891.9i	−2.13319	−	1.77655i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	1923.41	−	3331.44i	0.120772	−	0.209183i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8745.38	+	5049.15i	−0.538880	+	0.311122i	−0.744625	−	0.667483i	$-0.767372\pi$
$641$	−8745.38	+	5049.15i	−0.538880	+	0.311122i	0.205745	+	0.978606i	$0.434038\pi$
$642$	−34689.6	−	20028.0i	−2.13254	−	1.23122i
$643$	314.446	+	544.637i	0.0192855	+	0.0334034i	0.875507	−	0.483205i	$-0.160528\pi$
$643$	314.446	+	544.637i	0.0192855	+	0.0334034i	−0.856222	+	0.516609i	$0.827194\pi$
$644$		0			0
$645$		0			0
$646$	−3584.07	+	20775.4i	−0.218287	+	1.26532i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	55047.4	−	31781.6i	3.33714	−	1.92670i
$649$	−40501.9	+	23383.8i	−2.44968	+	1.41432i
$650$		0			0
$651$		0			0
$652$	15960.9	−	27645.1i	0.958706	−	1.66053i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$	−23488.3	−	13561.0i	−1.39797	−	0.807116i
$657$	91560.8			5.43703
$658$		0			0
$659$	13949.8	+	8053.95i	0.824596	+	0.476081i	0.851999	−	0.523544i	$-0.175390\pi$
$659$	13949.8	+	8053.95i	0.824596	+	0.476081i	−0.0274028	+	0.999624i	$0.508724\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	3884.35	+	6727.89i	0.228051	+	0.394995i
$663$		0			0
$664$			1915.93i			0.111977i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$			29172.4i			1.67090i	0.549569	+	0.835448i	$0.314792\pi$
$673$			29172.4i			1.67090i	−0.549569	+	0.835448i	$0.685208\pi$
$674$	17358.3	−	30065.5i	0.992015	−	1.71822i
$675$	−51797.6	−	29905.4i	−2.95362	−	1.70527i
$676$	17576.0			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	35225.0	+	20337.1i	1.99529	+	1.15198i
$679$		0			0
$680$		0			0
$681$	23866.5	+	41338.0i	1.34298	+	2.32610i
$682$		0			0
$683$			33632.8i			1.88422i	0.335300	+	0.942112i	$0.391162\pi$
$683$			33632.8i			1.88422i	−0.335300	+	0.942112i	$0.608838\pi$
$684$	31585.8	−	37926.6i	1.76566	−	2.12012i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	9280.00	−	16073.4i	0.514239	−	0.890689i
$689$		0			0
$690$		0			0
$691$	1978.00			0.108895			0.0544477	−	0.998517i	$-0.482660\pi$
$691$	1978.00			0.108895			0.0544477	+	0.998517i	$0.482660\pi$
$692$		0			0
$693$		0			0
$694$	−16872.8	−	9741.51i	−0.922885	−	0.532828i
$695$		0			0
$696$		0			0
$697$	−33030.5	−	19070.1i	−1.79500	−	1.03635i
$698$		0			0
$699$	54814.3	+	31647.0i	2.96604	+	1.71245i
$700$		0			0
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$	−26745.5			−1.43183
$705$		0			0
$706$	−32098.0	+	18531.8i	−1.71108	+	0.987895i
$707$		0			0
$708$	−36078.4	−	62489.6i	−1.91512	−	3.31709i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−15040.0	+	26050.0i	−0.791640	+	1.37116i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−13059.1	−	7539.66i	−0.681621	−	0.393534i
$717$		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
$722$	6500.18	−	18278.8i	0.335058	−	0.942198i
$723$	−26524.2			−1.36438
$724$		0			0
$725$		0			0
$726$	19914.1	−	34492.2i	1.01802	−	1.76326i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−79112.4			−4.01933
$730$		0			0
$731$	13050.0	−	22603.3i	0.660290	−	1.14366i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	27501.7	+	15878.1i	1.37454	+	0.793592i
$738$	44646.0	+	77329.2i	2.22689	+	3.85708i
$739$	1357.72	+	2351.64i	0.0675840	+	0.117059i	0.897837	−	0.440327i	$-0.145138\pi$
$739$	1357.72	+	2351.64i	0.0675840	+	0.117059i	−0.830253	+	0.557386i	$0.811804\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3153.85	−	5462.63i	0.154476	−	0.267560i
$748$	−37610.8			−1.83849
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		−	36691.6i		−	1.77572i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$	12780.0	+	22135.6i	0.612389	+	1.06069i
$759$		0			0
$760$		0			0
$761$	−4023.60			−0.191663			−0.0958314	−	0.995398i	$-0.530551\pi$
$761$	−4023.60			−0.191663			−0.0958314	+	0.995398i	$0.530551\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	41265.0i		−	1.93883i
$769$	20053.0	−	34732.8i	0.940351	−	1.62874i	0.175548	−	0.984471i	$-0.443830\pi$
$769$	20053.0	−	34732.8i	0.940351	−	1.62874i	0.764803	−	0.644264i	$-0.222836\pi$
$770$		0			0
$771$	−70409.2			−3.28888
$772$		−	39507.5i		−	1.84185i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	−52917.6	+	30552.0i	−2.45747	+	1.41882i
$775$		0			0
$776$	−18426.1	−	31914.9i	−0.852395	−	1.47639i
$777$		0			0
$778$		0			0
$779$	26969.3	+	22460.4i	1.24040	+	1.03303i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−10976.0	−	19011.0i	−0.500000	−	0.866025i
$785$		0			0
$786$	49659.7			2.25357
$787$			15784.1i			0.714918i	0.933929	+	0.357459i	$0.116357\pi$
$787$			15784.1i			0.714918i	−0.933929	+	0.357459i	$0.883643\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	76255.7	+	44026.3i	3.42125	+	1.97526i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−19595.9	+	11313.7i	−0.866025	+	0.500000i
$801$	85763.0	−	49515.3i	3.78313	−	2.18419i
$802$	−1644.34	+	2848.07i	−0.0723984	+	0.125398i
$803$	32102.1	+	55602.5i	1.41078	+	2.44355i
$804$	−24498.0	+	42431.7i	−1.07460	+	1.86126i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30696.1			−1.33401			−0.667007	−	0.745051i	$-0.732425\pi$
$809$	−30696.1			−1.33401			−0.667007	+	0.745051i	$0.732425\pi$
$810$		0			0
$811$	23257.9	+	13428.0i	1.00702	+	0.581405i	0.910319	−	0.413908i	$-0.135837\pi$
$811$	23257.9	+	13428.0i	1.00702	+	0.581405i	0.0967042	+	0.995313i	$0.469170\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		−	58028.9i		−	2.48948i
$817$	−15370.0	+	18455.5i	−0.658174	+	0.790301i
$818$	36439.4			1.55755
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	−44226.8	−	76603.0i	−1.87662	−	3.25041i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		−	65782.8i		−	2.77608i
$826$		0			0
$827$	−35526.9	−	20511.5i	−1.49382	−	0.862460i	−0.493850	−	0.869547i	$-0.664411\pi$
$827$	−35526.9	−	20511.5i	−1.49382	−	0.862460i	−0.999975	+	0.00708730i	$0.997744\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−15435.0	−	26734.2i	−0.642006	−	1.11199i
$834$	46620.8	+	80749.6i	1.93567	+	3.35267i
$835$		0			0
$836$	34106.1	+	5883.81i	1.41099	+	0.243416i
$837$		0			0
$838$	41136.7	−	23750.3i	1.69576	−	0.979046i
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	12194.5	+	21121.5i	0.500000	+	0.866025i
$842$		0			0
$843$	−75380.2			−3.07975
$844$			3054.70i			0.124582i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	3630.60	+	2096.13i	0.146763	+	0.0847337i
$850$	−27556.8	+	15909.9i	−1.11199	+	0.642006i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	31808.0			1.27006
$857$	6201.50	−	3580.44i	0.247187	−	0.142713i	−0.371289	−	0.928518i	$-0.621084\pi$
$857$	6201.50	−	3580.44i	0.247187	−	0.142713i	0.618475	+	0.785804i	$0.287751\pi$
$858$		0			0
$859$	20615.5	−	35707.0i	0.818848	−	1.41829i	−0.0876838	−	0.996148i	$-0.527947\pi$
$859$	20615.5	−	35707.0i	0.818848	−	1.41829i	0.906532	−	0.422138i	$-0.138720\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−43307.6	+	75010.9i	−1.70527	+	2.95362i
$865$		0			0
$866$	48528.0			1.90421
$867$		−	32107.3i		−	1.25770i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$			121326.i			4.70364i
$874$		0			0
$875$		0			0
$876$	−85787.9	+	49529.7i	−3.30879	+	1.91033i
$877$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$877$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6416.32			0.245370			0.122685	−	0.992446i	$-0.460849\pi$
$881$	6416.32			0.245370			0.122685	+	0.992446i	$0.460849\pi$
$882$			72271.3i			2.75907i
$883$	23930.1	−	41448.1i	0.912017	−	1.57966i	0.100806	−	0.994906i	$-0.467858\pi$
$883$	23930.1	−	41448.1i	0.912017	−	1.57966i	0.811211	−	0.584753i	$-0.198809\pi$
$884$		0			0
$885$		0			0
$886$		−	16685.5i		−	0.632688i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−73370.5	−	127082.i	−2.75870	−	4.77822i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	16190.2	+	28042.2i	0.601641	+	1.04207i
$899$		0			0
$900$	74494.9			2.75907
$901$		0			0
$902$	−31306.7	+	54224.7i	−1.15565	+	2.00165i
$903$		0			0
$904$	−32298.9			−1.18832
$905$		0			0
$906$		0			0
$907$	40413.3	−	23332.6i	1.47949	−	0.854186i	0.479762	−	0.877399i	$-0.340723\pi$
$907$	40413.3	−	23332.6i	1.47949	−	0.854186i	0.999731	+	0.0232129i	$0.00738957\pi$
$908$	−32826.0	−	18952.1i	−1.19974	−	0.692673i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−9078.01	+	52621.6i	−0.329608	+	1.91061i
$913$	4423.08			0.160332
$914$	−37899.1	+	21881.1i	−1.37154	+	0.791862i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−60901.3	+	105484.i	−2.18959	+	3.79248i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	52999.5	−	91797.8i	1.89619	−	3.28430i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26077.2	+	45167.0i	0.920952	+	1.59514i	0.797945	+	0.602730i	$0.205920\pi$
$929$	26077.2	+	45167.0i	0.920952	+	1.59514i	0.123007	+	0.992406i	$0.460746\pi$
$930$		0			0
$931$	9814.44	+	26657.7i	0.345494	+	0.938421i
$932$	−50261.0			−1.76647
$933$		0			0
$934$	10170.1	−	5871.70i	0.356291	−	0.205704i
$935$		0			0
$936$		0			0
$937$	−18895.4	+	32727.9i	−0.658791	+	1.14106i	0.322138	+	0.946693i	$0.395599\pi$
$937$	−18895.4	+	32727.9i	−0.658791	+	1.14106i	−0.980929	+	0.194367i	$0.937735\pi$
$938$		0			0
$939$			20812.8i			0.723323i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$	49622.1	+	28649.3i	1.71087	+	0.987772i
$945$		0			0
$946$	−37106.8	−	21423.6i	−1.27531	−	0.736303i
$947$	−29115.0	−	50428.7i	−0.999061	−	1.73042i	−0.537060	−	0.843544i	$-0.680465\pi$
$947$	−29115.0	−	50428.7i	−0.999061	−	1.73042i	−0.462001	−	0.886879i	$-0.652868\pi$
$948$		0			0
$949$		0			0
$950$	27477.9	−	10116.4i	0.938421	−	0.345494i
$951$		0			0
$952$		0			0
$953$	−18600.2	+	10738.8i	−0.632235	+	0.365021i	−0.781617	−	0.623759i	$-0.785605\pi$
$953$	−18600.2	+	10738.8i	−0.632235	+	0.365021i	0.149382	+	0.988780i	$0.452272\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29791.0			−1.00000
$962$		0			0
$963$	−90689.8	−	52359.8i	−3.03472	−	1.75210i
$964$	18240.7	−	10531.3i	0.609432	−	0.351856i
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$			31627.0i			1.05013i
$969$	−12765.9	+	73999.1i	−0.423221	+	2.45325i
$970$		0			0
$971$	−26324.9	+	15198.7i	−0.870038	+	0.502317i	−0.867361	−	0.497680i	$-0.834185\pi$
$971$	−26324.9	+	15198.7i	−0.870038	+	0.502317i	−0.00267705	+	0.999996i	$0.500852\pi$
$972$	106565.	−	61525.4i	3.51654	−	2.03028i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	14234.2i		−	0.466113i	−0.972463	−	0.233056i	$-0.925127\pi$
$977$		−	14234.2i		−	0.466113i	0.972463	−	0.233056i	$-0.0748726\pi$
$978$	56850.5	−	98467.9i	1.85877	−	3.21948i
$979$	60138.7	+	34721.1i	1.96327	+	1.13349i
$980$		0			0
$981$		0			0
$982$	−29937.7	−	17284.5i	−0.972861	−	0.561681i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$	−83662.2	−	48302.4i	−2.71042	−	1.56486i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$		0			0
$993$	13835.5	+	23963.8i	0.442152	+	0.765829i
$994$		0			0
$995$		0			0
$996$			6824.28i			0.217104i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	4962.50	+	2865.10i	0.157400	+	0.0908749i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.o.a.107.1	yes	4
8.3	odd	2	CM	152.4.o.a.107.1	yes	4
19.8	odd	6	inner	152.4.o.a.27.1	✓	4
152.27	even	6	inner	152.4.o.a.27.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.o.a.27.1	✓	4	19.8	odd	6	inner
152.4.o.a.27.1	✓	4	152.27	even	6	inner
152.4.o.a.107.1	yes	4	1.1	even	1	trivial
152.4.o.a.107.1	yes	4	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.o (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 + 1.41421i) q^{2} +(-8.72474 + 5.03723i) q^{3} +(4.00000 - 6.92820i) q^{4} +(14.2474 - 24.6773i) q^{6} +22.6274i q^{8} +(37.2474 - 64.5145i) q^{9} +O(q^{10})\)	\(q+(-2.44949 + 1.41421i) q^{2} +(-8.72474 + 5.03723i) q^{3} +(4.00000 - 6.92820i) q^{4} +(14.2474 - 24.6773i) q^{6} +22.6274i q^{8} +(37.2474 - 64.5145i) q^{9} +52.2372 q^{11} +80.5957i q^{12} +(-32.0000 - 55.4256i) q^{16} +(-45.0000 - 77.9423i) q^{17} +210.703i q^{18} +(28.6135 + 77.7192i) q^{19} +(-127.955 + 73.8746i) q^{22} +(-113.980 - 197.418i) q^{24} +(-62.5000 + 108.253i) q^{25} +478.486i q^{27} +(156.767 + 90.5097i) q^{32} +(-455.757 + 263.131i) q^{33} +(220.454 + 127.279i) q^{34} +(-297.980 - 516.116i) q^{36} +(-180.000 - 149.907i) q^{38} +(367.005 - 211.890i) q^{41} +(145.000 + 251.147i) q^{43} +(208.949 - 361.910i) q^{44} +(558.384 + 322.383i) q^{48} +343.000 q^{49} -353.553i q^{50} +(785.227 + 453.351i) q^{51} +(-676.681 - 1172.05i) q^{54} +(-641.135 - 533.947i) q^{57} +(-775.346 + 447.646i) q^{59} -512.000 q^{64} +(744.247 - 1289.07i) q^{66} +(526.476 + 303.961i) q^{67} -720.000 q^{68} +(1459.80 + 842.814i) q^{72} +(614.544 + 1064.42i) q^{73} -1259.31i q^{75} +(652.908 + 112.636i) q^{76} +(-1404.56 - 2432.78i) q^{81} +(-599.317 + 1038.05i) q^{82} +84.6730 q^{83} +(-710.352 - 410.122i) q^{86} +1181.99i q^{88} +(1151.26 + 664.680i) q^{89} -1823.67 q^{96} +(-1410.45 + 814.326i) q^{97} +(-840.175 + 485.075i) q^{98} +(1945.70 - 3370.06i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 30 q^{3} + 16 q^{4} + 8 q^{6} + 100 q^{9}+O(q^{10}) \) 4 * q - 30 * q^3 + 16 * q^4 + 8 * q^6 + 100 * q^9	\( 4 q - 30 q^{3} + 16 q^{4} + 8 q^{6} + 100 q^{9} - 36 q^{11} - 128 q^{16} - 180 q^{17} - 106 q^{19} - 600 q^{22} - 64 q^{24} - 250 q^{25} - 30 q^{33} - 800 q^{36} - 720 q^{38} + 1566 q^{41} + 580 q^{43} - 144 q^{44} + 1920 q^{48} + 1372 q^{49} + 2700 q^{51} - 1384 q^{54} - 360 q^{57} - 2538 q^{59} - 2048 q^{64} + 2928 q^{66} + 210 q^{67} - 2880 q^{68} + 1920 q^{72} + 430 q^{73} + 848 q^{76} - 2042 q^{81} + 160 q^{82} + 2700 q^{83} - 1024 q^{96} - 5730 q^{97} + 2100 q^{99}+O(q^{100}) \) 4 * q - 30 * q^3 + 16 * q^4 + 8 * q^6 + 100 * q^9 - 36 * q^11 - 128 * q^16 - 180 * q^17 - 106 * q^19 - 600 * q^22 - 64 * q^24 - 250 * q^25 - 30 * q^33 - 800 * q^36 - 720 * q^38 + 1566 * q^41 + 580 * q^43 - 144 * q^44 + 1920 * q^48 + 1372 * q^49 + 2700 * q^51 - 1384 * q^54 - 360 * q^57 - 2538 * q^59 - 2048 * q^64 + 2928 * q^66 + 210 * q^67 - 2880 * q^68 + 1920 * q^72 + 430 * q^73 + 848 * q^76 - 2042 * q^81 + 160 * q^82 + 2700 * q^83 - 1024 * q^96 - 5730 * q^97 + 2100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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