LMFDB - Embedded newform 152.4.o.a.27.2

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			27.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	152.27
Dual form			152.4.o.a.107.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.44949 + 1.41421i) q^{2} +(-6.27526 - 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(-10.2474 - 17.7491i) q^{6} +22.6274i q^{8} +(12.7526 + 22.0881i) q^{9} +O(q^{10})$	\(q+(2.44949 + 1.41421i) q^{2} +(-6.27526 - 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(-10.2474 - 17.7491i) q^{6} +22.6274i q^{8} +(12.7526 + 22.0881i) q^{9} -70.2372 q^{11} -57.9683i q^{12} +(-32.0000 + 55.4256i) q^{16} +(-45.0000 + 77.9423i) q^{17} +72.1393i q^{18} +(-81.6135 - 14.0795i) q^{19} +(-172.045 - 99.3305i) q^{22} +(81.9796 - 141.993i) q^{24} +(-62.5000 - 108.253i) q^{25} +10.8321i q^{27} +(-156.767 + 90.5097i) q^{32} +(440.757 + 254.471i) q^{33} +(-220.454 + 127.279i) q^{34} +(-102.020 + 176.705i) q^{36} +(-180.000 - 149.907i) q^{38} +(415.995 + 240.175i) q^{41} +(145.000 - 251.147i) q^{43} +(-280.949 - 486.618i) q^{44} +(401.616 - 231.873i) q^{48} +343.000 q^{49} -353.553i q^{50} +(564.773 - 326.072i) q^{51} +(-15.3189 + 26.5331i) q^{54} +(461.135 + 384.040i) q^{57} +(-493.654 - 285.011i) q^{59} -512.000 q^{64} +(719.753 + 1246.65i) q^{66} +(-421.476 + 243.339i) q^{67} -720.000 q^{68} +(-499.796 + 288.557i) q^{72} +(-399.544 + 692.031i) q^{73} +905.755i q^{75} +(-228.908 - 621.753i) q^{76} +(383.564 - 664.352i) q^{81} +(679.317 + 1176.61i) q^{82} +1265.33 q^{83} +(710.352 - 410.122i) q^{86} -1589.29i q^{88} +(-1151.26 + 664.680i) q^{89} +1311.67 q^{96} +(-1454.55 - 839.782i) q^{97} +(840.175 + 485.075i) q^{98} +(-895.704 - 1551.40i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/152\mathbb{Z}\right)^\times$.

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

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
$3$	−6.27526	−	3.62302i	−1.20767	−	0.697251i	−0.245423	−	0.969416i	$-0.578927\pi$
$3$	−6.27526	−	3.62302i	−1.20767	−	0.697251i	−0.962250	+	0.272166i	$0.912260\pi$
$4$	4.00000	+	6.92820i	0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	−10.2474	−	17.7491i	−0.697251	−	1.20767i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			22.6274i			1.00000i
$9$	12.7526	+	22.0881i	0.472317	+	0.818077i
$10$		0			0
$11$	−70.2372			−1.92521			−0.962606	−	0.270906i	$-0.912677\pi$
$11$	−70.2372			−1.92521			−0.962606	+	0.270906i	$0.912677\pi$
$12$		−	57.9683i		−	1.39450i
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$	−45.0000	+	77.9423i	−0.642006	+	1.11199i	0.342978	+	0.939343i	$0.388564\pi$
$17$	−45.0000	+	77.9423i	−0.642006	+	1.11199i	−0.984984	+	0.172644i	$0.944769\pi$
$18$			72.1393i			0.944633i
$19$	−81.6135	−	14.0795i	−0.985443	−	0.170004i
$19$	−81.6135	−	14.0795i	−0.985443	−	0.170004i
$20$		0			0
$21$		0			0
$22$	−172.045	−	99.3305i	−1.66728	−	0.962606i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	81.9796	−	141.993i	0.697251	−	1.20767i
$25$	−62.5000	−	108.253i	−0.500000	−	0.866025i
$26$		0			0
$27$			10.8321i			0.0772087i
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		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
$33$	440.757	+	254.471i	2.32503	+	1.34235i
$34$	−220.454	+	127.279i	−1.11199	+	0.642006i
$35$		0			0
$36$	−102.020	+	176.705i	−0.472317	+	0.818077i
$37$		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
$41$	415.995	+	240.175i	1.58457	+	0.914854i	0.994179	+	0.107738i	$0.0343608\pi$
$41$	415.995	+	240.175i	1.58457	+	0.914854i	0.590394	+	0.807116i	$0.298973\pi$
$42$		0			0
$43$	145.000	−	251.147i	0.514239	−	0.890689i	−0.485624	−	0.874168i	$-0.661408\pi$
$43$	145.000	−	251.147i	0.514239	−	0.890689i	0.999864	−	0.0165210i	$-0.00525904\pi$
$44$	−280.949	−	486.618i	−0.962606	−	1.66728i
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	401.616	−	231.873i	1.20767	−	0.697251i
$49$	343.000			1.00000
$50$		−	353.553i		−	1.00000i
$51$	564.773	−	326.072i	1.55067	−	0.895278i
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$	−15.3189	+	26.5331i	−0.0386044	+	0.0668647i
$55$		0			0
$56$		0			0
$57$	461.135	+	384.040i	1.07156	+	0.892410i
$58$		0			0
$59$	−493.654	−	285.011i	−1.08929	−	0.628904i	−0.155905	−	0.987772i	$-0.549829\pi$
$59$	−493.654	−	285.011i	−1.08929	−	0.628904i	−0.933388	+	0.358868i	$0.883163\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$	719.753	+	1246.65i	1.34235	+	2.32503i
$67$	−421.476	+	243.339i	−0.768530	+	0.443711i	−0.832350	−	0.554250i	$-0.813005\pi$
$67$	−421.476	+	243.339i	−0.768530	+	0.443711i	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	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	−499.796	+	288.557i	−0.818077	+	0.472317i
$73$	−399.544	+	692.031i	−0.640591	+	1.10954i	0.344710	+	0.938709i	$0.387977\pi$
$73$	−399.544	+	692.031i	−0.640591	+	1.10954i	−0.985301	+	0.170827i	$0.945356\pi$
$74$		0			0
$75$			905.755i			1.39450i
$76$	−228.908	−	621.753i	−0.345494	−	0.938421i
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$	383.564	−	664.352i	0.526151	−	0.911319i
$82$	679.317	+	1176.61i	0.914854	+	1.58457i
$83$	1265.33			1.67335			0.836673	−	0.547703i	$-0.184498\pi$
$83$	1265.33			1.67335			0.836673	+	0.547703i	$0.184498\pi$
$84$		0			0
$85$		0			0
$86$	710.352	−	410.122i	0.890689	−	0.514239i
$87$		0			0
$88$		−	1589.29i		−	1.92521i
$89$	−1151.26	+	664.680i	−1.37116	+	0.791640i	−0.991074	−	0.133311i	$-0.957439\pi$
$89$	−1151.26	+	664.680i	−1.37116	+	0.791640i	−0.380087	+	0.924951i	$0.624106\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1311.67			1.39450
$97$	−1454.55	−	839.782i	−1.52254	−	0.879041i	−0.999645	−	0.0266459i	$-0.991517\pi$
$97$	−1454.55	−	839.782i	−1.52254	−	0.879041i	−0.522898	−	0.852395i	$-0.675149\pi$
$98$	840.175	+	485.075i	0.866025	+	0.500000i
$99$	−895.704	−	1551.40i	−0.909310	−	1.57497i
$100$	500.000	−	866.025i	0.500000	−	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	1844.54			1.79056
$103$		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	$-0.219013\pi$
$108$	−75.0469	+	43.3283i	−0.0668647	+	0.0386044i
$109$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$109$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			959.769i			0.799005i	0.916732	+	0.399502i	$0.130817\pi$
$113$			959.769i			0.799005i	−0.916732	+	0.399502i	$0.869183\pi$
$114$	586.431	+	1592.85i	0.481792	+	1.30863i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−806.134	−	1396.27i	−0.628904	−	1.08929i
$119$		0			0
$120$		0			0
$121$	3602.27			2.70644
$122$		0			0
$123$	−1740.32	−	3014.32i	−1.27576	−	2.20969i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$	−1254.14	−	724.077i	−0.866025	−	0.500000i
$129$	−1819.82	+	1050.68i	−1.24207	+	0.717107i
$130$		0			0
$131$	−1492.38	+	2584.88i	−0.995342	+	1.72398i	−0.414176	+	0.910197i	$0.635930\pi$
$131$	−1492.38	+	2584.88i	−0.995342	+	1.72398i	−0.581166	+	0.813785i	$0.697403\pi$
$132$			4071.54i			2.68471i
$133$		0			0
$134$	−1376.54			−0.887422
$135$		0			0
$136$	−1763.63	−	1018.23i	−1.11199	−	0.642006i
$137$	−427.094	−	739.748i	−0.266344	−	0.461321i	0.701571	−	0.712599i	$-0.252482\pi$
$137$	−427.094	−	739.748i	−0.266344	−	0.461321i	−0.967915	+	0.251279i	$0.919149\pi$
$138$		0			0
$139$	899.111	+	1557.31i	0.548645	+	0.950280i	0.998368	+	0.0571123i	$0.0181893\pi$
$139$	899.111	+	1557.31i	0.548645	+	0.950280i	−0.449723	+	0.893168i	$0.648477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1632.33			−0.944633
$145$		0			0
$146$	−1957.36	+	1130.08i	−1.10954	+	0.640591i
$147$	−2152.41	−	1242.70i	−1.20767	−	0.697251i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−1280.93	+	2218.64i	−0.697251	+	1.20767i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	318.584	−	1846.70i	0.170004	−	0.985443i
$153$	−2295.46			−1.21292
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1879.07	−	1084.88i	0.911319	−	0.526151i
$163$	−3020.22			−1.45130			−0.725650	−	0.688064i	$-0.758461\pi$
$163$	−3020.22			−1.45130			−0.725650	+	0.688064i	$0.758461\pi$
$164$			3842.80i			1.82971i
$165$		0			0
$166$	3099.41	+	1789.44i	1.44916	+	0.836673i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	1098.50	−	1902.66i	0.500000	−	0.866025i
$170$		0			0
$171$	−729.791	−	1982.23i	−0.326365	−	0.886464i
$172$	2320.00			1.02848
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	2247.59	−	3892.94i	0.962606	−	1.66728i
$177$	2065.20	+	3577.04i	0.877007	+	1.51902i
$178$	−3760.00			−1.58328
$179$			4755.77i			1.98583i	0.118845	+	0.992913i	$0.462081\pi$
$179$			4755.77i			1.98583i	−0.118845	+	0.992913i	$0.537919\pi$
$180$		0			0
$181$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$181$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	3160.68	−	5474.45i	1.23600	−	2.14081i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	3212.93	+	1854.99i	1.20767	+	0.697251i
$193$	−4276.81	−	2469.22i	−1.59509	−	0.920923i	−0.992415	−	0.122933i	$-0.960770\pi$
$193$	−4276.81	−	2469.22i	−1.59509	−	0.920923i	−0.602670	−	0.797990i	$-0.705897\pi$
$194$	−2375.26	−	4114.08i	−0.879041	−	1.52254i
$195$		0			0
$196$	1372.00	+	2376.37i	0.500000	+	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		−	5066.87i		−	1.81862i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	2449.49	−	1414.21i	0.866025	−	0.500000i
$201$	3526.49			1.23751
$202$		0			0
$203$		0			0
$204$	4518.18	+	2608.57i	1.55067	+	0.895278i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5732.31	+	988.908i	1.89719	+	0.327293i
$210$		0			0
$211$	330.681	+	190.919i	0.107891	+	0.0622910i	0.552975	−	0.833198i	$-0.313493\pi$
$211$	330.681	+	190.919i	0.107891	+	0.0622910i	−0.445083	+	0.895489i	$0.646826\pi$
$212$		0			0
$213$		0			0
$214$	1988.00	−	3443.32i	0.635032	−	1.09991i
$215$		0			0
$216$	−245.102			−0.0772087
$217$		0			0
$218$		0			0
$219$	5014.49	−	2895.11i	1.54725	−	0.893305i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$	1594.07	−	2761.01i	0.472317	−	0.818077i
$226$	−1357.32	+	2350.95i	−0.399502	+	0.691958i
$227$			6641.55i			1.94192i	0.239246	+	0.970959i	$0.423100\pi$
$227$			6641.55i			1.94192i	−0.239246	+	0.970959i	$0.576900\pi$
$228$	−816.167	+	4731.00i	−0.237070	+	1.37420i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	126.310	−	218.775i	0.0355143	−	0.0615125i	−0.847722	−	0.530441i	$-0.822026\pi$
$233$	126.310	−	218.775i	0.0355143	−	0.0615125i	0.883236	+	0.468928i	$0.155360\pi$
$234$		0			0
$235$		0			0
$236$		−	4560.18i		−	1.25781i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−4113.08	+	2374.69i	−1.09937	+	0.634719i	−0.936054	−	0.351856i	$-0.885551\pi$
$241$	−4113.08	+	2374.69i	−1.09937	+	0.634719i	−0.163311	+	0.986575i	$0.552217\pi$
$242$	8823.72	+	5094.38i	2.34384	+	1.35322i
$243$	−4560.64	+	2633.09i	−1.20397	+	0.695113i
$244$		0			0
$245$		0			0
$246$		−	9844.71i		−	2.55153i
$247$		0			0
$248$		0			0
$249$	−7940.25	−	4584.31i	−2.02085	−	1.16674i
$250$		0			0
$251$	3972.02	+	6879.74i	0.998852	+	1.73006i	0.540916	+	0.841077i	$0.318078\pi$
$251$	3972.02	+	6879.74i	0.998852	+	1.73006i	0.457936	+	0.888985i	$0.348589\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$257$	−247.544	+	142.919i	−0.0600831	+	0.0346890i	−0.529741	−	0.848160i	$-0.677711\pi$
$257$	−247.544	+	142.919i	−0.0600831	+	0.0346890i	0.469658	+	0.882849i	$0.344377\pi$
$258$	−5943.52			−1.43421
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−7311.13	+	4221.08i	−1.72398	+	0.995342i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	−5758.02	+	9973.18i	−1.34235	+	2.32503i
$265$		0			0
$266$		0			0
$267$	9632.60			2.20789
$268$	−3371.81	−	1946.72i	−0.768530	−	0.443711i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	−2880.00	−	4988.31i	−0.642006	−	1.11199i
$273$		0			0
$274$		−	2416.01i		−	0.532687i
$275$	4389.83	+	7603.40i	0.962606	+	1.66728i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			5086.14i			1.09729i
$279$		0			0
$280$		0			0
$281$	7533.14	−	4349.26i	1.59925	−	0.923328i	0.607619	−	0.794229i	$-0.292125\pi$
$281$	7533.14	−	4349.26i	1.59925	−	0.923328i	0.991632	−	0.129099i	$-0.0412086\pi$
$282$		0			0
$283$	4223.06	−	7314.56i	0.887050	−	1.53642i	0.0437035	−	0.999045i	$-0.486084\pi$
$283$	4223.06	−	7314.56i	0.887050	−	1.53642i	0.843346	−	0.537371i	$-0.180582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−3998.37	−	2308.46i	−0.818077	−	0.472317i
$289$	−1593.50	−	2760.02i	−0.324344	−	0.561780i
$290$		0			0
$291$	6085.10	+	10539.7i	1.22582	+	2.12319i
$292$	−6392.71			−1.28118
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−3514.87	−	6087.94i	−0.697251	−	1.20767i
$295$		0			0
$296$		0			0
$297$		−	760.816i		−	0.148643i
$298$		0			0
$299$		0			0
$300$	−6275.26	+	3623.02i	−1.20767	+	0.697251i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	3392.00	−	4072.94i	0.639949	−	0.768417i
$305$		0			0
$306$	−5622.70	−	3246.27i	−1.05042	−	0.606460i
$307$	−2873.07	−	1658.77i	−0.534121	−	0.308375i	0.208572	−	0.978007i	$-0.433118\pi$
$307$	−2873.07	−	1658.77i	−0.534121	−	0.308375i	−0.742693	+	0.669632i	$0.766452\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−5227.95	−	9055.07i	−0.944093	−	1.63522i	−0.757557	−	0.652769i	$-0.773607\pi$
$313$	−5227.95	−	9055.07i	−0.944093	−	1.63522i	−0.186536	−	0.982448i	$-0.559726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−5092.98	+	8821.30i	−0.885553	+	1.53382i
$322$		0			0
$323$	4770.00	−	5727.56i	0.821702	−	0.986657i
$324$	6137.02			1.05230
$325$		0			0
$326$	−7398.00	−	4271.24i	−1.25686	−	0.725650i
$327$		0			0
$328$	−5434.53	+	9412.89i	−0.914854	+	1.58457i
$329$		0			0
$330$		0			0
$331$			11528.9i			1.91446i	0.289327	+	0.957230i	$0.406568\pi$
$331$			11528.9i			1.91446i	−0.289327	+	0.957230i	$0.593432\pi$
$332$	5061.31	+	8766.44i	0.836673	+	1.44916i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6485.23	−	3744.25i	−1.04829	−	0.605229i	−0.126118	−	0.992015i	$-0.540252\pi$
$337$	−6485.23	−	3744.25i	−1.04829	−	0.605229i	−0.922170	+	0.386786i	$0.873585\pi$
$338$	5381.53	−	3107.03i	0.866025	−	0.500000i
$339$	3477.26	−	6022.80i	0.557106	−	0.964936i
$340$		0			0
$341$		0			0
$342$	1015.69	−	5887.54i	0.160591	−	0.930883i
$343$		0			0
$344$	5682.82	+	3280.98i	0.890689	+	0.514239i
$345$		0			0
$346$		0			0
$347$	−6459.14	+	11187.6i	−0.999265	+	1.73078i	−0.466437	+	0.884554i	$0.654463\pi$
$347$	−6459.14	+	11187.6i	−0.999265	+	1.73078i	−0.532828	+	0.846224i	$0.678871\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	11010.9	−	6357.15i	1.66728	−	0.962606i
$353$	−8333.97			−1.25658			−0.628289	−	0.777980i	$-0.716245\pi$
$353$	−8333.97			−1.25658			−0.628289	+	0.777980i	$0.716245\pi$
$354$			11682.6i			1.75401i
$355$		0			0
$356$	−9210.08	−	5317.44i	−1.37116	−	0.791640i
$357$		0			0
$358$	−6725.67	+	11649.2i	−0.992913	+	1.71978i
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	6462.53	+	2298.16i	0.942198	+	0.335058i
$362$		0			0
$363$	−22605.2	−	13051.1i	−3.26849	−	1.88707i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$			12251.4i			1.72840i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	15484.1	−	8939.74i	2.14081	−	1.23600i
$375$		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	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	5246.69	+	9087.54i	0.697251	+	1.20767i
$385$		0			0
$386$	−6984.00	−	12096.6i	−0.920923	−	1.59509i
$387$	7396.48			0.971535
$388$		−	13436.5i		−	1.75808i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$			7761.20i			1.00000i
$393$	18730.1	−	10813.8i	2.40410	−	1.38800i
$394$		0			0
$395$		0			0
$396$	7165.63	−	12411.2i	0.909310	−	1.57497i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	8000.00			1.00000
$401$	−11509.9	−	6645.27i	−1.43336	−	0.827554i	−0.435989	−	0.899952i	$-0.643601\pi$
$401$	−11509.9	−	6645.27i	−1.43336	−	0.827554i	−0.997376	+	0.0723984i	$0.976935\pi$
$402$	8638.11	+	4987.22i	1.07172	+	0.618756i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	7378.16	+	12779.4i	0.895278	+	1.55067i
$409$	−13361.8	+	7714.42i	−1.61540	+	0.932649i	−0.627306	+	0.778773i	$0.715842\pi$
$409$	−13361.8	+	7714.42i	−1.61540	+	0.932649i	−0.988090	+	0.153877i	$0.950824\pi$
$410$		0			0
$411$			6189.48i			0.742833i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	13030.0i		−	1.53017i
$418$	12642.7	+	10529.0i	1.47937	+	1.23204i
$419$	−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	$-0.833333\pi$
$421$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$422$	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	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	−600.375	−	346.627i	−0.0668647	−	0.0386044i
$433$	14858.6	−	8578.62i	1.64910	−	0.952107i	0.671665	−	0.740855i	$-0.265579\pi$
$433$	14858.6	−	8578.62i	1.64910	−	0.952107i	0.977432	−	0.211252i	$-0.0677540\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	16377.2			1.78661
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	4374.13	+	7576.21i	0.472317	+	0.818077i
$442$		0			0
$443$	−6185.39	−	10713.4i	−0.663378	−	1.14901i	−0.979722	−	0.200361i	$-0.935789\pi$
$443$	−6185.39	−	10713.4i	−0.663378	−	1.14901i	0.316344	−	0.948645i	$-0.397545\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			18887.0i			1.98515i	0.121653	+	0.992573i	$0.461181\pi$
$449$			18887.0i			1.98515i	−0.121653	+	0.992573i	$0.538819\pi$
$450$	7809.31	−	4508.71i	0.818077	−	0.472317i
$451$	−29218.3	−	16869.2i	−3.05064	−	1.76129i
$452$	−6649.48	+	3839.08i	−0.691958	+	0.399502i
$453$		0			0
$454$	−9392.57	+	16268.4i	−0.970959	+	1.68175i
$455$		0			0
$456$	−8689.84	+	10434.3i	−0.892410	+	1.07156i
$457$	2597.74			0.265902			0.132951	−	0.991123i	$-0.457555\pi$
$457$	2597.74			0.265902			0.132951	+	0.991123i	$0.457555\pi$
$458$		0			0
$459$	−844.278	−	487.444i	−0.0858551	−	0.0495685i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	618.788	−	357.258i	0.0615125	−	0.0355143i
$467$	19181.9			1.90071			0.950357	−	0.311162i	$-0.100718\pi$
$467$	19181.9			1.90071			0.950357	+	0.311162i	$0.100718\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	6449.07	−	11170.1i	0.628904	−	1.08929i
$473$	−10184.4	+	17639.9i	−0.990020	+	1.71476i
$474$		0			0
$475$	3576.69	+	9714.89i	0.345494	+	0.938421i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$	−13433.3			−1.26944
$483$		0			0
$484$	14409.1	+	24957.3i	1.35322	+	2.34384i
$485$		0			0
$486$	−14895.0			−1.39023
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$	18952.6	+	10942.3i	1.75270	+	1.01192i
$490$		0			0
$491$	6111.00	−	10584.6i	0.561681	−	0.972861i	−0.435669	−	0.900107i	$-0.643488\pi$
$491$	6111.00	−	10584.6i	0.561681	−	0.972861i	0.997350	−	0.0727536i	$-0.0231787\pi$
$492$	13922.5	−	24114.5i	1.27576	−	2.20969i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−12966.4	−	22458.4i	−1.16674	−	2.02085i
$499$	10120.0	−	17528.3i	0.907880	−	1.57249i	0.0908749	−	0.995862i	$-0.471034\pi$
$499$	10120.0	−	17528.3i	0.907880	−	1.57249i	0.817005	−	0.576631i	$-0.195633\pi$
$500$		0			0
$501$		0			0
$502$			22469.1i			1.99770i
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13786.7	+	7959.78i	−1.20767	+	0.697251i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		−	11585.2i		−	1.00000i
$513$	152.511	−	884.045i	0.0131258	−	0.0760849i
$514$	−808.475			−0.0693780
$515$		0			0
$516$	−14558.6	−	8405.41i	−1.24207	−	0.717107i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			3039.77i			0.255614i	0.991799	+	0.127807i	$0.0407938\pi$
$521$			3039.77i			0.255614i	−0.991799	+	0.127807i	$0.959206\pi$
$522$		0			0
$523$	20303.8	−	11722.4i	1.69756	−	0.980087i	0.749496	−	0.662009i	$-0.230296\pi$
$523$	20303.8	−	11722.4i	1.69756	−	0.980087i	0.948065	−	0.318078i	$-0.103037\pi$
$524$	−23878.1			−1.99068
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−28208.4	+	16286.1i	−2.32503	+	1.34235i
$529$	−6083.50	+	10536.9i	−0.500000	+	0.866025i
$530$		0			0
$531$		−	14538.5i		−	1.18817i
$532$		0			0
$533$		0			0
$534$	23595.0	+	13622.6i	1.91209	+	1.10394i
$535$		0			0
$536$	−5506.14	−	9536.92i	−0.443711	−	0.768530i
$537$	17230.2	−	29843.7i	1.38462	−	2.39823i
$538$		0			0
$539$	−24091.4			−1.92521
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		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	$-0.840857\pi$
$547$	−11529.7	+	6656.70i	−0.901237	+	0.520329i	−0.0236354	+	0.999721i	$0.507524\pi$
$548$	3416.75	−	5917.99i	0.266344	−	0.461321i
$549$		0			0
$550$			24832.6i			1.92521i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−7192.89	+	12458.4i	−0.548645	+	0.950280i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−39668.1	+	22902.4i	−2.98536	+	1.72360i
$562$	24603.1			1.84666
$563$			14303.2i			1.07070i	0.844629	+	0.535352i	$0.179821\pi$
$563$			14303.2i			1.07070i	−0.844629	+	0.535352i	$0.820179\pi$
$564$		0			0
$565$		0			0
$566$	20688.7	−	11944.6i	1.53642	−	0.887050i
$567$		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
$571$	16715.6			1.22509			0.612544	−	0.790436i	$-0.290146\pi$
$571$	16715.6			1.22509			0.612544	+	0.790436i	$0.290146\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−6529.31	−	11309.1i	−0.472317	−	0.818077i
$577$	7244.05			0.522658			0.261329	−	0.965250i	$-0.415839\pi$
$577$	7244.05			0.522658			0.261329	+	0.965250i	$0.415839\pi$
$578$		−	9014.20i		−	0.648687i
$579$	17892.0	+	30989.9i	1.28423	+	2.22435i
$580$		0			0
$581$		0			0
$582$			34422.5i			2.45165i
$583$		0			0
$584$	−15658.9	−	9040.66i	−1.10954	−	0.640591i
$585$		0			0
$586$		0			0
$587$	−5175.00	+	8963.36i	−0.363876	+	0.630251i	−0.988595	−	0.150598i	$-0.951880\pi$
$587$	−5175.00	+	8963.36i	−0.363876	+	0.630251i	0.624719	+	0.780849i	$0.285213\pi$
$588$		−	19883.1i		−	1.39450i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1276.20	+	2210.44i	0.0883763	+	0.153072i	0.906825	−	0.421507i	$-0.138499\pi$
$593$	1276.20	+	2210.44i	0.0883763	+	0.153072i	−0.818449	+	0.574580i	$0.805166\pi$
$594$	1075.96	−	1863.61i	0.0743216	−	0.128729i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$	−20494.9			−1.39450
$601$			25324.2i			1.71880i	0.511306	+	0.859399i	$0.329162\pi$
$601$			25324.2i			1.71880i	−0.511306	+	0.859399i	$0.670838\pi$
$602$		0			0
$603$	−10749.8	−	6206.40i	−0.725979	−	0.419144i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	14068.7	−	5179.60i	0.938421	−	0.345494i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−9181.84	−	15903.4i	−0.606460	−	1.05042i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−4691.71	−	8126.28i	−0.308375	−	0.534121i
$615$		0			0
$616$		0			0
$617$	−11984.9	−	20758.5i	−0.782002	−	1.35447i	−0.930774	−	0.365596i	$-0.880865\pi$
$617$	−11984.9	−	20758.5i	−0.782002	−	1.35447i	0.148771	−	0.988872i	$-0.452468\pi$
$618$		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
$626$		−	29573.7i		−	1.88819i
$627$	−32388.9	−	26973.9i	−2.06298	−	1.71808i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−1383.41	−	2396.13i	−0.0868648	−	0.150454i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18762.4	+	10832.5i	1.15611	+	0.667483i	0.950370	−	0.311122i	$-0.100705\pi$
$641$	18762.4	+	10832.5i	1.15611	+	0.667483i	0.205745	+	0.978606i	$0.434038\pi$
$642$	−24950.4	+	14405.1i	−1.53382	+	0.885553i
$643$	13960.6	−	24180.4i	0.856222	−	1.48302i	−0.0192855	−	0.999814i	$-0.506139\pi$
$643$	13960.6	−	24180.4i	0.856222	−	1.48302i	0.875507	−	0.483205i	$-0.160528\pi$
$644$		0			0
$645$		0			0
$646$	19784.1	−	7283.81i	1.20494	−	0.443619i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	15032.6	+	8679.06i	0.911319	+	0.526151i
$649$	34672.9	+	20018.4i	2.09712	+	1.21077i
$650$		0			0
$651$		0			0
$652$	−12080.9	−	20924.7i	−0.725650	−	1.25686i
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$	−26623.7	+	15371.2i	−1.58457	+	0.914854i
$657$	−20380.8			−1.21025
$658$		0			0
$659$	−13949.8	+	8053.95i	−0.824596	+	0.476081i	−0.851999	−	0.523544i	$-0.824610\pi$
$659$	−13949.8	+	8053.95i	−0.824596	+	0.476081i	0.0274028	+	0.999624i	$0.491276\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	−16304.3	+	28240.0i	−0.957230	+	1.65797i
$663$		0			0
$664$			28631.1i			1.67335i
$665$		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	$0.685208\pi$
$674$	−10590.3	−	18343.0i	−0.605229	−	1.04829i
$675$	1172.61	−	677.005i	0.0668647	−	0.0386044i
$676$	17576.0			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	17035.0	−	9835.19i	0.964936	−	0.557106i
$679$		0			0
$680$		0			0
$681$	24062.5	−	41677.4i	1.35400	−	2.34520i
$682$		0			0
$683$			33632.8i			1.88422i	0.335300	+	0.942112i	$0.391162\pi$
$683$			33632.8i			1.88422i	−0.335300	+	0.942112i	$0.608838\pi$
$684$	10814.2	−	12985.1i	0.604517	−	0.725873i
$685$		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
$694$	−31643.2	+	18269.2i	−1.73078	+	0.999265i
$695$		0			0
$696$		0			0
$697$	−37439.5	+	21615.7i	−2.03461	+	1.17468i
$698$		0			0
$699$	−1585.25	+	915.245i	−0.0857792	+	0.0495247i
$700$		0			0
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$	35961.5			1.92521
$705$		0			0
$706$	−20414.0	−	11786.0i	−1.08823	−	0.628289i
$707$		0			0
$708$	−16521.6	+	28616.3i	−0.877007	+	1.51902i
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	−15040.0	−	26050.0i	−0.791640	−	1.37116i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−32948.9	+	19023.1i	−1.71978	+	0.992913i
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$	12579.8	+	14768.7i	0.648438	+	0.761267i
$723$	34414.2			1.77023
$724$		0			0
$725$		0			0
$726$	−36914.1	−	63937.1i	−1.88707	−	3.26849i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	17446.4			0.886371
$730$		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
$737$	29603.3	−	17091.5i	1.47958	−	0.854238i
$738$	−17326.0	+	30009.6i	−0.864201	+	1.49684i
$739$	16679.3	−	28889.4i	0.830253	−	1.43804i	−0.0675840	−	0.997714i	$-0.521529\pi$
$739$	16679.3	−	28889.4i	0.830253	−	1.43804i	0.897837	−	0.440327i	$-0.145138\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	16136.1	+	27948.6i	0.790349	+	1.36892i
$748$	50570.8			2.47199
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		−	57562.9i		−	2.78580i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$	12780.0	−	22135.6i	0.612389	−	1.06069i
$759$		0			0
$760$		0			0
$761$	38205.6			1.81991			0.909955	−	0.414706i	$-0.136116\pi$
$761$	38205.6			1.81991			0.909955	+	0.414706i	$0.136116\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			29679.8i			1.39450i
$769$	20053.0	+	34732.8i	0.940351	+	1.62874i	0.764803	+	0.644264i	$0.222836\pi$
$769$	20053.0	+	34732.8i	0.940351	+	1.62874i	0.175548	+	0.984471i	$0.443830\pi$
$770$		0			0
$771$	2071.20			0.0967477
$772$		−	39507.5i		−	1.84185i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	18117.6	+	10460.2i	0.841374	+	0.485768i
$775$		0			0
$776$	19002.1	−	32912.6i	0.879041	−	1.52254i
$777$		0			0
$778$		0			0
$779$	−30569.3	−	25458.5i	−1.40598	−	1.17092i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−10976.0	+	19011.0i	−0.500000	+	0.866025i
$785$		0			0
$786$	61172.3			2.77601
$787$			27821.8i			1.26015i	0.776533	+	0.630076i	$0.216976\pi$
$787$			27821.8i			1.26015i	−0.776533	+	0.630076i	$0.783024\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	35104.3	−	20267.5i	1.57497	−	0.909310i
$793$		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
$801$	−29363.0	−	16952.7i	−1.29524	−	0.747810i
$802$	−18795.7	−	32555.0i	−0.827554	−	1.43336i
$803$	28062.9	−	48606.4i	1.23327	−	2.13609i
$804$	14106.0	+	24432.3i	0.618756	+	1.07172i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45042.1			1.95747			0.978737	−	0.205120i	$-0.0657584\pi$
$809$	45042.1			1.95747			0.978737	+	0.205120i	$0.0657584\pi$
$810$		0			0
$811$	−23257.9	+	13428.0i	−1.00702	+	0.581405i	−0.910319	−	0.413908i	$-0.864163\pi$
$811$	−23257.9	+	13428.0i	−1.00702	+	0.581405i	−0.0967042	+	0.995313i	$0.530830\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$			41737.2i			1.79056i
$817$	−15370.0	+	18455.5i	−0.658174	+	0.790301i
$818$	−43639.4			−1.86530
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−8753.24	+	15161.1i	−0.371417	+	0.643312i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		−	63617.7i		−	2.68471i
$826$		0			0
$827$	291.944	−	168.554i	0.0122756	−	0.00708730i	−0.493850	−	0.869547i	$-0.664411\pi$
$827$	291.944	−	168.554i	0.0122756	−	0.00708730i	0.506125	+	0.862460i	$0.331077\pi$
$828$		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
$834$	18427.2	−	31916.8i	0.765085	−	1.32517i
$835$		0			0
$836$	16077.9	+	43670.2i	0.665150	+	1.80666i
$837$		0			0
$838$	−41136.7	−	23750.3i	−1.69576	−	0.979046i
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	12194.5	−	21121.5i	0.500000	−	0.866025i
$842$		0			0
$843$	−63029.8			−2.57516
$844$			3054.70i			0.124582i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−53001.6	+	30600.5i	−2.14253	+	1.23699i
$850$	27556.8	+	15909.9i	1.11199	+	0.642006i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	31808.0			1.27006
$857$	−34146.5	−	19714.5i	−1.36105	−	0.785804i	−0.371289	−	0.928518i	$-0.621084\pi$
$857$	−34146.5	−	19714.5i	−1.36105	−	0.785804i	−0.989764	+	0.142713i	$0.954417\pi$
$858$		0			0
$859$	2207.54	+	3823.58i	0.0876838	+	0.151873i	0.906532	−	0.422138i	$-0.138720\pi$
$859$	2207.54	+	3823.58i	0.0876838	+	0.151873i	−0.818848	+	0.574011i	$0.805387\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−980.409	−	1698.12i	−0.0386044	−	0.0668647i
$865$		0			0
$866$	48528.0			1.90421
$867$			23093.1i			0.904595i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	42837.5i		−	1.66074i
$874$		0			0
$875$		0			0
$876$	40115.9	+	23160.9i	1.54725	+	0.893305i
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−48158.3			−1.84165			−0.920826	−	0.389974i	$-0.872484\pi$
$881$	−48158.3			−1.84165			−0.920826	+	0.389974i	$0.872484\pi$
$882$			24743.8i			0.944633i
$883$	−21285.1	−	36866.8i	−0.811211	−	1.40506i	−0.912017	−	0.410153i	$-0.865475\pi$
$883$	−21285.1	−	36866.8i	−0.811211	−	1.40506i	0.100806	−	0.994906i	$-0.467858\pi$
$884$		0			0
$885$		0			0
$886$		−	34989.8i		−	1.32676i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−26940.5	+	46662.2i	−1.01295	+	1.75448i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−26710.2	+	46263.4i	−0.992573	+	1.71919i
$899$		0			0
$900$	25505.1			0.944633
$901$		0			0
$902$	−47713.3	−	82641.9i	−1.76129	−	3.05064i
$903$		0			0
$904$	−21717.1			−0.799005
$905$		0			0
$906$		0			0
$907$	−1098.25	−	634.076i	−0.0402060	−	0.0232129i	0.479762	−	0.877399i	$-0.340723\pi$
$907$	−1098.25	−	634.076i	−0.0402060	−	0.0232129i	−0.519968	+	0.854186i	$0.674056\pi$
$908$	−46014.0	+	26566.2i	−1.68175	+	0.970959i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−36042.0	+	13269.4i	−1.30863	+	0.481792i
$913$	−88873.1			−3.22154
$914$	6363.14	+	3673.76i	0.230278	+	0.132951i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1378.70	−	2387.98i	−0.0495685	−	0.0858551i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	12019.5	+	20818.4i	0.430029	+	0.744832i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22594.2	+	39134.3i	−0.797945	+	1.38208i	0.123007	+	0.992406i	$0.460746\pi$
$929$	−22594.2	+	39134.3i	−0.797945	+	1.38208i	−0.920952	+	0.389676i	$0.872587\pi$
$930$		0			0
$931$	−27993.4	−	4829.28i	−0.985443	−	0.170004i
$932$	2020.95			0.0710285
$933$		0			0
$934$	46985.9	+	27127.3i	1.64607	+	0.950357i
$935$		0			0
$936$		0			0
$937$	−9239.56	−	16003.4i	−0.322138	−	0.557959i	0.658791	−	0.752326i	$-0.271068\pi$
$937$	−9239.56	−	16003.4i	−0.322138	−	0.557959i	−0.980929	+	0.194367i	$0.937735\pi$
$938$		0			0
$939$			75763.8i			2.63308i
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$	31593.9	−	18240.7i	1.08929	−	0.628904i
$945$		0			0
$946$	−49893.2	+	28805.8i	−1.71476	+	0.990020i
$947$	−29115.0	+	50428.7i	−0.999061	+	1.73042i	−0.462001	+	0.886879i	$0.652868\pi$
$947$	−29115.0	+	50428.7i	−0.999061	+	1.73042i	−0.537060	+	0.843544i	$0.680465\pi$
$948$		0			0
$949$		0			0
$950$	−4977.87	+	28854.7i	−0.170004	+	0.985443i
$951$		0			0
$952$		0			0
$953$	−50384.8	−	29089.7i	−1.71262	−	0.988780i	−0.930999	−	0.365021i	$-0.881062\pi$
$953$	−50384.8	−	29089.7i	−1.71262	−	0.988780i	−0.781617	−	0.623759i	$-0.785605\pi$
$954$		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
$963$	31049.8	−	17926.6i	1.03901	−	0.599873i
$964$	−32904.7	−	18997.5i	−1.09937	−	0.634719i
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$			81510.1i			2.70644i
$969$	−50684.1	+	18660.1i	−1.68030	+	0.618627i
$970$		0			0
$971$	26081.9	+	15058.4i	0.862007	+	0.497680i	0.864684	−	0.502317i	$-0.167519\pi$
$971$	26081.9	+	15058.4i	0.862007	+	0.497680i	−0.00267705	+	0.999996i	$0.500852\pi$
$972$	−36485.1	−	21064.7i	−1.20397	−	0.695113i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	44319.9i		−	1.45130i	−0.688064	−	0.725650i	$-0.741539\pi$
$977$		−	44319.9i		−	1.45130i	0.688064	−	0.725650i	$-0.258461\pi$
$978$	30949.5	+	53606.2i	1.01192	+	1.75270i
$979$	80861.3	−	46685.3i	2.63977	−	1.52407i
$980$		0			0
$981$		0			0
$982$	29937.7	−	17284.5i	0.972861	−	0.561681i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$	68206.2	−	39378.9i	2.20969	−	1.27576i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$991$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$992$		0			0
$993$	41769.5	−	72346.9i	1.33486	−	2.31204i
$994$		0			0
$995$		0			0
$996$		−	73348.9i		−	2.33348i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	49577.5	−	28623.6i	1.57249	−	0.907880i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.o.a.27.2	✓	4
8.3	odd	2	CM	152.4.o.a.27.2	✓	4
19.12	odd	6	inner	152.4.o.a.107.2	yes	4
152.107	even	6	inner	152.4.o.a.107.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.o.a.27.2	✓	4	1.1	even	1	trivial
152.4.o.a.27.2	✓	4	8.3	odd	2	CM
152.4.o.a.107.2	yes	4	19.12	odd	6	inner
152.4.o.a.107.2	yes	4	152.107	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.44949 + 1.41421i) q^{2} +(-6.27526 - 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(-10.2474 - 17.7491i) q^{6} +22.6274i q^{8} +(12.7526 + 22.0881i) q^{9} +O(q^{10})\)	\(q+(2.44949 + 1.41421i) q^{2} +(-6.27526 - 3.62302i) q^{3} +(4.00000 + 6.92820i) q^{4} +(-10.2474 - 17.7491i) q^{6} +22.6274i q^{8} +(12.7526 + 22.0881i) q^{9} -70.2372 q^{11} -57.9683i q^{12} +(-32.0000 + 55.4256i) q^{16} +(-45.0000 + 77.9423i) q^{17} +72.1393i q^{18} +(-81.6135 - 14.0795i) q^{19} +(-172.045 - 99.3305i) q^{22} +(81.9796 - 141.993i) q^{24} +(-62.5000 - 108.253i) q^{25} +10.8321i q^{27} +(-156.767 + 90.5097i) q^{32} +(440.757 + 254.471i) q^{33} +(-220.454 + 127.279i) q^{34} +(-102.020 + 176.705i) q^{36} +(-180.000 - 149.907i) q^{38} +(415.995 + 240.175i) q^{41} +(145.000 - 251.147i) q^{43} +(-280.949 - 486.618i) q^{44} +(401.616 - 231.873i) q^{48} +343.000 q^{49} -353.553i q^{50} +(564.773 - 326.072i) q^{51} +(-15.3189 + 26.5331i) q^{54} +(461.135 + 384.040i) q^{57} +(-493.654 - 285.011i) q^{59} -512.000 q^{64} +(719.753 + 1246.65i) q^{66} +(-421.476 + 243.339i) q^{67} -720.000 q^{68} +(-499.796 + 288.557i) q^{72} +(-399.544 + 692.031i) q^{73} +905.755i q^{75} +(-228.908 - 621.753i) q^{76} +(383.564 - 664.352i) q^{81} +(679.317 + 1176.61i) q^{82} +1265.33 q^{83} +(710.352 - 410.122i) q^{86} -1589.29i q^{88} +(-1151.26 + 664.680i) q^{89} +1311.67 q^{96} +(-1454.55 - 839.782i) q^{97} +(840.175 + 485.075i) q^{98} +(-895.704 - 1551.40i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 30 q^{3} + 16 q^{4} + 8 q^{6} + 100 q^{9}+O(q^{10}) \) 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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