LMFDB - Embedded newform 60.4.h.a.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,4,Mod(59,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	60.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.54011460034$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.1
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.4.h.a.59.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +(-1.58114 + 4.94975i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +(14.0000 + 4.47214i) q^{6} -34.7851 q^{7} +22.6274i q^{8} +(-22.0000 - 15.6525i) q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +(-1.58114 + 4.94975i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +(14.0000 + 4.47214i) q^{6} -34.7851 q^{7} +22.6274i q^{8} +(-22.0000 - 15.6525i) q^{9} -31.6228 q^{10} +(12.6491 - 39.5980i) q^{12} +98.3870i q^{14} +(55.3399 + 17.6777i) q^{15} +64.0000 q^{16} +(-44.2719 + 62.2254i) q^{18} +89.4427i q^{20} +(55.0000 - 172.177i) q^{21} -94.7523i q^{23} +(-112.000 - 35.7771i) q^{24} -125.000 q^{25} +(112.261 - 84.1457i) q^{27} +278.280 q^{28} -62.6099i q^{29} +(50.0000 - 156.525i) q^{30} -181.019i q^{32} +388.909i q^{35} +(176.000 + 125.220i) q^{36} +252.982 q^{40} +460.630i q^{41} +(-486.991 - 155.563i) q^{42} -376.311 q^{43} +(-175.000 + 245.967i) q^{45} -268.000 q^{46} -425.678i q^{47} +(-101.193 + 316.784i) q^{48} +867.000 q^{49} +353.553i q^{50} +(-238.000 - 317.522i) q^{54} -787.096i q^{56} -177.088 q^{58} +(-442.719 - 141.421i) q^{60} -952.000 q^{61} +(765.271 + 544.472i) q^{63} -512.000 q^{64} -774.758 q^{67} +(469.000 + 149.817i) q^{69} +1100.00 q^{70} +(354.175 - 497.803i) q^{72} +(197.642 - 618.718i) q^{75} -715.542i q^{80} +(239.000 + 688.709i) q^{81} +1302.86 q^{82} +108.894i q^{83} +(-440.000 + 1377.42i) q^{84} +1064.37i q^{86} +(309.903 + 98.9949i) q^{87} -948.093i q^{89} +(695.701 + 494.975i) q^{90} +758.018i q^{92} -1204.00 q^{94} +(896.000 + 286.217i) q^{96} -2452.25i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} + 56 q^{6} - 88 q^{9}+O(q^{10}) $ 4 * q - 32 * q^4 + 56 * q^6 - 88 * q^9	$ 4 q - 32 q^{4} + 56 q^{6} - 88 q^{9} + 256 q^{16} + 220 q^{21} - 448 q^{24} - 500 q^{25} + 200 q^{30} + 704 q^{36} - 700 q^{45} - 1072 q^{46} + 3468 q^{49} - 952 q^{54} - 3808 q^{61} - 2048 q^{64} + 1876 q^{69} + 4400 q^{70} + 956 q^{81} - 1760 q^{84} - 4816 q^{94} + 3584 q^{96}+O(q^{100}) $ 4 * q - 32 * q^4 + 56 * q^6 - 88 * q^9 + 256 * q^16 + 220 * q^21 - 448 * q^24 - 500 * q^25 + 200 * q^30 + 704 * q^36 - 700 * q^45 - 1072 * q^46 + 3468 * q^49 - 952 * q^54 - 3808 * q^61 - 2048 * q^64 + 1876 * q^69 + 4400 * q^70 + 956 * q^81 - 1760 * q^84 - 4816 * q^94 + 3584 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.82843i		−	1.00000i
$2$		−	2.82843i		−	1.00000i
$3$	−1.58114	+	4.94975i	−0.304290	+	0.952579i
$3$	−1.58114	+	4.94975i	−0.304290	+	0.952579i
$4$	−8.00000			−1.00000
$5$		−	11.1803i		−	1.00000i
$5$		−	11.1803i		−	1.00000i
$6$	14.0000	+	4.47214i	0.952579	+	0.304290i
$7$	−34.7851			−1.87822			−0.939108	−	0.343622i	$-0.888346\pi$
$7$	−34.7851			−1.87822			−0.939108	+	0.343622i	$0.888346\pi$
$8$			22.6274i			1.00000i
$9$	−22.0000	−	15.6525i	−0.814815	−	0.579721i
$10$	−31.6228			−1.00000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	12.6491	−	39.5980i	0.304290	−	0.952579i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			98.3870i			1.87822i
$15$	55.3399	+	17.6777i	0.952579	+	0.304290i
$16$	64.0000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−44.2719	+	62.2254i	−0.579721	+	0.814815i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			89.4427i			1.00000i
$21$	55.0000	−	172.177i	0.571523	−	1.78915i
$22$		0			0
$23$		−	94.7523i		−	0.859010i	−0.903065	−	0.429505i	$-0.858688\pi$
$23$		−	94.7523i		−	0.859010i	0.903065	−	0.429505i	$-0.141312\pi$
$24$	−112.000	−	35.7771i	−0.952579	−	0.304290i
$25$	−125.000			−1.00000
$26$		0			0
$27$	112.261	−	84.1457i	0.800171	−	0.599772i
$28$	278.280			1.87822
$29$		−	62.6099i		−	0.400909i	−0.979703	−	0.200455i	$-0.935758\pi$
$29$		−	62.6099i		−	0.400909i	0.979703	−	0.200455i	$-0.0642419\pi$
$30$	50.0000	−	156.525i	0.304290	−	0.952579i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	181.019i		−	1.00000i
$33$		0			0
$34$		0			0
$35$			388.909i			1.87822i
$36$	176.000	+	125.220i	0.814815	+	0.579721i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	252.982			1.00000
$41$			460.630i			1.75459i	0.479949	+	0.877297i	$0.340655\pi$
$41$			460.630i			1.75459i	−0.479949	+	0.877297i	$0.659345\pi$
$42$	−486.991	−	155.563i	−1.78915	−	0.571523i
$43$	−376.311			−1.33458			−0.667289	−	0.744798i	$-0.732546\pi$
$43$	−376.311			−1.33458			−0.667289	+	0.744798i	$0.732546\pi$
$44$		0			0
$45$	−175.000	+	245.967i	−0.579721	+	0.814815i
$46$	−268.000			−0.859010
$47$		−	425.678i		−	1.32110i	−0.750783	−	0.660549i	$-0.770324\pi$
$47$		−	425.678i		−	1.32110i	0.750783	−	0.660549i	$-0.229676\pi$
$48$	−101.193	+	316.784i	−0.304290	+	0.952579i
$49$	867.000			2.52770
$50$			353.553i			1.00000i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−238.000	−	317.522i	−0.599772	−	0.800171i
$55$		0			0
$56$		−	787.096i		−	1.87822i
$57$		0			0
$58$	−177.088			−0.400909
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−442.719	−	141.421i	−0.952579	−	0.304290i
$61$	−952.000			−1.99821			−0.999107	−	0.0422409i	$-0.986550\pi$
$61$	−952.000			−1.99821			−0.999107	+	0.0422409i	$0.986550\pi$
$62$		0			0
$63$	765.271	+	544.472i	1.53040	+	1.08884i
$64$	−512.000			−1.00000
$65$		0			0
$66$		0			0
$67$	−774.758			−1.41271			−0.706356	−	0.707856i	$-0.749662\pi$
$67$	−774.758			−1.41271			−0.706356	+	0.707856i	$0.749662\pi$
$68$		0			0
$69$	469.000	+	149.817i	0.818275	+	0.261388i
$70$	1100.00			1.87822
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	354.175	−	497.803i	0.579721	−	0.814815i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	197.642	−	618.718i	0.304290	−	0.952579i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	715.542i		−	1.00000i
$81$	239.000	+	688.709i	0.327846	+	0.944731i
$82$	1302.86			1.75459
$83$			108.894i			0.144009i	0.997404	+	0.0720043i	$0.0229395\pi$
$83$			108.894i			0.144009i	−0.997404	+	0.0720043i	$0.977060\pi$
$84$	−440.000	+	1377.42i	−0.571523	+	1.78915i
$85$		0			0
$86$			1064.37i			1.33458i
$87$	309.903	+	98.9949i	0.381898	+	0.121993i
$88$		0			0
$89$		−	948.093i		−	1.12919i	−0.825369	−	0.564593i	$-0.809033\pi$
$89$		−	948.093i		−	1.12919i	0.825369	−	0.564593i	$-0.190967\pi$
$90$	695.701	+	494.975i	0.814815	+	0.579721i
$91$		0			0
$92$			758.018i			0.859010i
$93$		0			0
$94$	−1204.00			−1.32110
$95$		0			0
$96$	896.000	+	286.217i	0.952579	+	0.304290i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	2452.25i		−	2.52770i
$99$		0			0
$100$	1000.00			1.00000
$101$		−	1994.57i		−	1.96502i	−0.186200	−	0.982512i	$-0.559617\pi$
$101$		−	1994.57i		−	1.96502i	0.186200	−	0.982512i	$-0.440383\pi$
$102$		0			0
$103$	932.872			0.892414			0.446207	−	0.894930i	$-0.352775\pi$
$103$	932.872			0.892414			0.446207	+	0.894930i	$0.352775\pi$
$104$		0			0
$105$	−1925.00	−	614.919i	−1.78915	−	0.571523i
$106$		0			0
$107$			312.541i			0.282378i	0.989983	+	0.141189i	$0.0450926\pi$
$107$			312.541i			0.282378i	−0.989983	+	0.141189i	$0.954907\pi$
$108$	−898.087	+	673.166i	−0.800171	+	0.599772i
$109$	1136.00			0.998248			0.499124	−	0.866530i	$-0.333655\pi$
$109$	1136.00			0.998248			0.499124	+	0.866530i	$0.333655\pi$
$110$		0			0
$111$		0			0
$112$	−2226.24			−1.87822
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−1059.36			−0.859010
$116$			500.879i			0.400909i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−400.000	+	1252.20i	−0.304290	+	0.952579i
$121$	−1331.00			−1.00000
$122$			2692.66i			1.99821i
$123$	−2280.00	−	728.320i	−1.67139	−	0.533906i
$124$		0			0
$125$			1397.54i			1.00000i
$126$	1540.00	−	2164.51i	1.08884	−	1.53040i
$127$	2412.82			1.68585			0.842925	−	0.538031i	$-0.180832\pi$
$127$	2412.82			1.68585			0.842925	+	0.538031i	$0.180832\pi$
$128$			1448.15i			1.00000i
$129$	595.000	−	1862.64i	0.406099	−	1.27129i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$			2191.35i			1.41271i
$135$	−940.778	−	1255.11i	−0.599772	−	0.800171i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	423.745	−	1326.53i	0.261388	−	0.818275i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		−	3111.27i		−	1.87822i
$141$	2107.00	+	673.056i	1.25845	+	0.401997i
$142$		0			0
$143$		0			0
$144$	−1408.00	−	1001.76i	−0.814815	−	0.579721i
$145$	−700.000			−0.400909
$146$		0			0
$147$	−1370.85	+	4291.43i	−0.769154	+	2.40783i
$148$		0			0
$149$			1909.60i			1.04994i	0.851121	+	0.524969i	$0.175923\pi$
$149$			1909.60i			1.04994i	−0.851121	+	0.524969i	$0.824077\pi$
$150$	−1750.00	−	559.017i	−0.952579	−	0.304290i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	−2023.86			−1.00000
$161$			3295.96i			1.61341i
$162$	1947.96	−	675.994i	0.944731	−	0.327846i
$163$	22.1359			0.0106369			0.00531847	−	0.999986i	$-0.498307\pi$
$163$	22.1359			0.0106369			0.00531847	+	0.999986i	$0.498307\pi$
$164$		−	3685.04i		−	1.75459i
$165$		0			0
$166$	308.000			0.144009
$167$		−	1851.21i		−	0.857788i	−0.903355	−	0.428894i	$-0.858903\pi$
$167$		−	1851.21i		−	0.857788i	0.903355	−	0.428894i	$-0.141097\pi$
$168$	3895.93	+	1244.51i	1.78915	+	0.571523i
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$	3010.49			1.33458
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	280.000	−	876.539i	0.121993	−	0.381898i
$175$	4348.13			1.87822
$176$		0			0
$177$		0			0
$178$	−2681.61			−1.12919
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	1400.00	−	1967.74i	0.579721	−	0.814815i
$181$	−1078.00			−0.442691			−0.221346	−	0.975195i	$-0.571045\pi$
$181$	−1078.00			−0.442691			−0.221346	+	0.975195i	$0.571045\pi$
$182$		0			0
$183$	1505.24	−	4712.16i	0.608037	−	1.90346i
$184$	2144.00			0.859010
$185$		0			0
$186$		0			0
$187$		0			0
$188$			3405.43i			1.32110i
$189$	−3905.00	+	2927.01i	−1.50289	+	1.12650i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	809.543	−	2534.27i	0.304290	−	0.952579i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−6936.00			−2.52770
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	2828.43i		−	1.00000i
$201$	1225.00	−	3834.86i	0.429875	−	1.34572i
$202$	−5641.50			−1.96502
$203$			2177.89i			0.752994i
$204$		0			0
$205$	5150.00			1.75459
$206$		−	2638.56i		−	0.892414i
$207$	−1483.11	+	2084.55i	−0.497986	+	0.699934i
$208$		0			0
$209$		0			0
$210$	−1739.25	+	5444.72i	−0.571523	+	1.78915i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	884.000			0.282378
$215$			4207.29i			1.33458i
$216$	1904.00	+	2540.17i	0.599772	+	0.800171i
$217$		0			0
$218$		−	3213.09i		−	0.998248i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2083.94			−0.625789			−0.312895	−	0.949788i	$-0.601299\pi$
$223$	−2083.94			−0.625789			−0.312895	+	0.949788i	$0.601299\pi$
$224$			6296.77i			1.87822i
$225$	2750.00	+	1956.56i	0.814815	+	0.579721i
$226$		0			0
$227$		−	5771.41i		−	1.68750i	−0.536739	−	0.843748i	$-0.680344\pi$
$227$		−	5771.41i		−	1.68750i	0.536739	−	0.843748i	$-0.319656\pi$
$228$		0			0
$229$	−6874.00			−1.98361			−0.991805	−	0.127761i	$-0.959221\pi$
$229$	−6874.00			−1.98361			−0.991805	+	0.127761i	$0.959221\pi$
$230$			2996.33i			0.859010i
$231$		0			0
$232$	1416.70			0.400909
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−4759.23			−1.32110
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	3541.75	+	1131.37i	0.952579	+	0.304290i
$241$	−1708.00			−0.456523			−0.228261	−	0.973600i	$-0.573304\pi$
$241$	−1708.00			−0.456523			−0.228261	+	0.973600i	$0.573304\pi$
$242$			3764.64i			1.00000i
$243$	−3786.83	+	94.0452i	−0.999692	+	0.0248272i
$244$	7616.00			1.99821
$245$		−	9693.35i		−	2.52770i
$246$	−2060.00	+	6448.82i	−0.533906	+	1.67139i
$247$		0			0
$248$		0			0
$249$	−539.000	−	172.177i	−0.137180	−	0.0438204i
$250$	3952.85			1.00000
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−6122.17	−	4355.78i	−1.53040	−	1.08884i
$253$		0			0
$254$		−	6824.48i		−	1.68585i
$255$		0			0
$256$	4096.00			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	−5268.35	−	1682.91i	−1.27129	−	0.406099i
$259$		0			0
$260$		0			0
$261$	−980.000	+	1377.42i	−0.232416	+	0.326667i
$262$		0			0
$263$			8509.32i			1.99508i	0.0700645	+	0.997542i	$0.477679\pi$
$263$			8509.32i			1.99508i	−0.0700645	+	0.997542i	$0.522321\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4692.82	+	1499.07i	1.07564	+	0.343601i
$268$	6198.06			1.41271
$269$		−	6864.73i		−	1.55595i	−0.628297	−	0.777974i	$-0.716248\pi$
$269$		−	6864.73i		−	1.55595i	0.628297	−	0.777974i	$-0.283752\pi$
$270$	−3550.00	+	2660.92i	−0.800171	+	0.599772i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−3752.00	−	1198.53i	−0.818275	−	0.261388i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	−8800.00			−1.87822
$281$		−	4288.78i		−	0.910488i	−0.890367	−	0.455244i	$-0.849552\pi$
$281$		−	4288.78i		−	0.910488i	0.890367	−	0.455244i	$-0.150448\pi$
$282$	1903.69	−	5959.50i	0.401997	−	1.25845i
$283$	9471.02			1.98938			0.994689	−	0.102929i	$-0.0328215\pi$
$283$	9471.02			1.98938			0.994689	+	0.102929i	$0.0328215\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16023.0i		−	3.29551i
$288$	−2833.40	+	3982.43i	−0.579721	+	0.814815i
$289$	−4913.00			−1.00000
$290$			1979.90i			0.400909i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	12138.0	+	3877.34i	2.40783	+	0.769154i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	5401.17			1.04994
$299$		0			0
$300$	−1581.14	+	4949.75i	−0.304290	+	0.952579i
$301$	13090.0			2.50663
$302$		0			0
$303$	9872.63	+	3153.70i	1.87184	+	0.597938i
$304$		0			0
$305$			10643.7i			1.99821i
$306$		0			0
$307$	−10052.9			−1.86889			−0.934443	−	0.356112i	$-0.884102\pi$
$307$	−10052.9			−1.86889			−0.934443	+	0.356112i	$0.884102\pi$
$308$		0			0
$309$	−1475.00	+	4617.48i	−0.271553	+	0.850095i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	6087.38	−	8555.99i	1.08884	−	1.53040i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$			5724.33i			1.00000i
$321$	−1547.00	−	494.171i	−0.268988	−	0.0859250i
$322$	9322.39			1.61341
$323$		0			0
$324$	−1912.00	−	5509.67i	−0.327846	−	0.944731i
$325$		0			0
$326$		−	62.6099i		−	0.0106369i
$327$	−1796.17	+	5622.91i	−0.303757	+	0.950911i
$328$	−10422.9			−1.75459
$329$			14807.2i			2.48131i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	871.156i		−	0.144009i
$333$		0			0
$334$	−5236.00			−0.857788
$335$			8662.06i			1.41271i
$336$	3520.00	−	11019.3i	0.571523	−	1.78915i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		−	6214.05i		−	1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18227.4			−2.86935
$344$		−	8514.95i		−	1.33458i
$345$	1675.00	−	5243.58i	0.261388	−	0.818275i
$346$		0			0
$347$			11131.3i			1.72207i	0.508546	+	0.861035i	$0.330183\pi$
$347$			11131.3i			1.72207i	−0.508546	+	0.861035i	$0.669817\pi$
$348$	−2479.23	−	791.960i	−0.381898	−	0.121993i
$349$	−9646.00			−1.47948			−0.739740	−	0.672893i	$-0.765052\pi$
$349$	−9646.00			−1.47948			−0.739740	+	0.672893i	$0.765052\pi$
$350$		−	12298.4i		−	1.87822i
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$			7584.74i			1.12919i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−5565.61	−	3959.80i	−0.814815	−	0.579721i
$361$	6859.00			1.00000
$362$			3049.04i			0.442691i
$363$	2104.50	−	6588.11i	0.304290	−	0.952579i
$364$		0			0
$365$		0			0
$366$	−13328.0	−	4257.47i	−1.90346	−	0.608037i
$367$	−3450.04			−0.490711			−0.245355	−	0.969433i	$-0.578905\pi$
$367$	−3450.04			−0.490711			−0.245355	+	0.969433i	$0.578905\pi$
$368$		−	6064.15i		−	0.859010i
$369$	7210.00	−	10133.9i	1.01718	−	1.42967i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−6917.48	−	2209.71i	−0.952579	−	0.304290i
$376$	9632.00			1.32110
$377$		0			0
$378$	8278.84	+	11045.0i	1.12650	+	1.50289i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−3815.00	+	11942.8i	−0.512988	+	1.60591i
$382$		0			0
$383$		−	11473.5i		−	1.53073i	−0.643597	−	0.765365i	$-0.722559\pi$
$383$		−	11473.5i		−	1.53073i	0.643597	−	0.765365i	$-0.277441\pi$
$384$	−7168.00	−	2289.73i	−0.952579	−	0.304290i
$385$		0			0
$386$		0			0
$387$	8278.84	+	5890.20i	1.08743	+	0.773684i
$388$		0			0
$389$			5854.03i			0.763010i	0.924367	+	0.381505i	$0.124594\pi$
$389$			5854.03i			0.763010i	−0.924367	+	0.381505i	$0.875406\pi$
$390$		0			0
$391$		0			0
$392$			19618.0i			2.52770i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−8000.00			−1.00000
$401$			2754.84i			0.343067i	0.985178	+	0.171534i	$0.0548722\pi$
$401$			2754.84i			0.343067i	−0.985178	+	0.171534i	$0.945128\pi$
$402$	−10846.6	−	3464.82i	−1.34572	−	0.429875i
$403$		0			0
$404$			15956.6i			1.96502i
$405$	7700.00	−	2672.10i	0.944731	−	0.327846i
$406$	6160.00			0.752994
$407$		0			0
$408$		0			0
$409$	4844.00			0.585624			0.292812	−	0.956170i	$-0.405409\pi$
$409$	4844.00			0.585624			0.292812	+	0.956170i	$0.405409\pi$
$410$		−	14566.4i		−	1.75459i
$411$		0			0
$412$	−7462.98			−0.892414
$413$		0			0
$414$	5896.00	+	4194.86i	0.699934	+	0.497986i
$415$	1217.48			0.144009
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	15400.0	+	4919.35i	1.78915	+	0.571523i
$421$	−9592.00			−1.11042			−0.555208	−	0.831711i	$-0.687362\pi$
$421$	−9592.00			−1.11042			−0.555208	+	0.831711i	$0.687362\pi$
$422$		0			0
$423$	−6662.92	+	9364.92i	−0.765868	+	1.07645i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	33115.4			3.75308
$428$		−	2500.33i		−	0.282378i
$429$		0			0
$430$	11900.0			1.33458
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	7184.69	−	5385.33i	0.800171	−	0.599772i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	1106.80	−	3464.82i	0.121993	−	0.381898i
$436$	−9088.00			−0.998248
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−19074.0	−	13570.7i	−2.05960	−	1.46536i
$442$		0			0
$443$		−	14477.3i		−	1.55268i	−0.630314	−	0.776340i	$-0.717074\pi$
$443$		−	14477.3i		−	1.55268i	0.630314	−	0.776340i	$-0.282926\pi$
$444$		0			0
$445$	−10600.0			−1.12919
$446$			5894.28i			0.625789i
$447$	−9452.05	−	3019.35i	−1.00015	−	0.319486i
$448$	17809.9			1.87822
$449$		−	18939.5i		−	1.99067i	−0.0964880	−	0.995334i	$-0.530761\pi$
$449$		−	18939.5i		−	1.99067i	0.0964880	−	0.995334i	$-0.469239\pi$
$450$	5533.99	−	7778.17i	0.579721	−	0.814815i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−16324.0			−1.68750
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$			19442.6i			1.98361i
$459$		0			0
$460$	8474.90			0.859010
$461$		−	11654.4i		−	1.17744i	−0.808338	−	0.588719i	$-0.799632\pi$
$461$		−	11654.4i		−	1.17744i	0.808338	−	0.588719i	$-0.200368\pi$
$462$		0			0
$463$	12374.0			1.24205			0.621024	−	0.783792i	$-0.286717\pi$
$463$	12374.0			1.24205			0.621024	+	0.783792i	$0.286717\pi$
$464$		−	4007.03i		−	0.400909i
$465$		0			0
$466$		0			0
$467$			11156.7i			1.10551i	0.833345	+	0.552754i	$0.186423\pi$
$467$			11156.7i			1.10551i	−0.833345	+	0.552754i	$0.813577\pi$
$468$		0			0
$469$	26950.0			2.65338
$470$			13461.1i			1.32110i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	3200.00	−	10017.6i	0.304290	−	0.952579i
$481$		0			0
$482$			4830.95i			0.456523i
$483$	−16314.2	−	5211.38i	−1.53690	−	0.490944i
$484$	10648.0			1.00000
$485$		0			0
$486$	266.000	+	10710.8i	0.0248272	+	0.999692i
$487$	−21494.0			−1.99997			−0.999986	−	0.00532938i	$-0.998304\pi$
$487$	−21494.0			−1.99997			−0.999986	+	0.00532938i	$0.998304\pi$
$488$		−	21541.3i		−	1.99821i
$489$	−35.0000	+	109.567i	−0.00323672	+	0.0101325i
$490$	−27416.9			−2.52770
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	18240.0	+	5826.56i	1.67139	+	0.533906i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−486.991	+	1524.52i	−0.0438204	+	0.137180i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		−	11180.3i		−	1.00000i
$501$	9163.00	+	2927.01i	0.817111	+	0.261017i
$502$		0			0
$503$			18106.2i			1.60500i	0.596653	+	0.802499i	$0.296497\pi$
$503$			18106.2i			1.60500i	−0.596653	+	0.802499i	$0.703503\pi$
$504$	−12320.0	+	17316.1i	−1.08884	+	1.53040i
$505$	−22300.0			−1.96502
$506$		0			0
$507$	−3473.76	+	10874.6i	−0.304290	+	0.952579i
$508$	−19302.5			−1.68585
$509$		−	21153.2i		−	1.84204i	−0.389513	−	0.921021i	$-0.627357\pi$
$509$		−	21153.2i		−	1.84204i	0.389513	−	0.921021i	$-0.372643\pi$
$510$		0			0
$511$		0			0
$512$		−	11585.2i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		−	10429.8i		−	0.892414i
$516$	−4760.00	+	14901.2i	−0.406099	+	1.27129i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			22235.5i			1.86978i	0.354943	+	0.934888i	$0.384500\pi$
$521$			22235.5i			1.86978i	−0.354943	+	0.934888i	$0.615500\pi$
$522$	3895.93	+	2771.86i	0.326667	+	0.232416i
$523$	12487.8			1.04408			0.522041	−	0.852920i	$-0.325171\pi$
$523$	12487.8			1.04408			0.522041	+	0.852920i	$0.325171\pi$
$524$		0			0
$525$	−6875.00	+	21522.2i	−0.571523	+	1.78915i
$526$	24068.0			1.99508
$527$		0			0
$528$		0			0
$529$	3189.00			0.262102
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	4240.00	−	13273.3i	0.343601	−	1.07564i
$535$	3494.32			0.282378
$536$		−	17530.8i		−	1.41271i
$537$		0			0
$538$	−19416.4			−1.55595
$539$		0			0
$540$	7526.22	+	10040.9i	0.599772	+	0.800171i
$541$	−6802.00			−0.540556			−0.270278	−	0.962782i	$-0.587116\pi$
$541$	−6802.00			−0.540556			−0.270278	+	0.962782i	$0.587116\pi$
$542$		0			0
$543$	1704.47	−	5335.83i	0.134707	−	0.421699i
$544$		0			0
$545$		−	12700.9i		−	0.998248i
$546$		0			0
$547$	−5157.67			−0.403156			−0.201578	−	0.979472i	$-0.564607\pi$
$547$	−5157.67			−0.403156			−0.201578	+	0.979472i	$0.564607\pi$
$548$		0			0
$549$	20944.0	+	14901.2i	1.62818	+	1.15841i
$550$		0			0
$551$		0			0
$552$	−3389.96	+	10612.3i	−0.261388	+	0.818275i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$			24890.2i			1.87822i
$561$		0			0
$562$	−12130.5			−0.910488
$563$		−	2385.78i		−	0.178594i	−0.996005	−	0.0892971i	$-0.971538\pi$
$563$		−	2385.78i		−	0.178594i	0.996005	−	0.0892971i	$-0.0284621\pi$
$564$	−16856.0	−	5384.45i	−1.25845	−	0.401997i
$565$		0			0
$566$		−	26788.1i		−	1.98938i
$567$	−8313.63	−	23956.8i	−0.615766	−	1.77441i
$568$		0			0
$569$			22758.7i			1.67679i	0.545062	+	0.838396i	$0.316506\pi$
$569$			22758.7i			1.67679i	−0.545062	+	0.838396i	$0.683494\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	−45320.0			−3.29551
$575$			11844.0i			0.859010i
$576$	11264.0	+	8014.07i	0.814815	+	0.579721i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			13896.1i			1.00000i
$579$		0			0
$580$	5600.00			0.400909
$581$		−	3787.90i		−	0.270479i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	16462.9i		−	1.15757i	−0.815479	−	0.578786i	$-0.803527\pi$
$587$		−	16462.9i		−	1.15757i	0.815479	−	0.578786i	$-0.196473\pi$
$588$	10966.8	−	34331.4i	0.769154	−	2.40783i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		−	15276.8i		−	1.04994i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	14000.0	+	4472.14i	0.952579	+	0.304290i
$601$	28532.0			1.93651			0.968257	−	0.249958i	$-0.0804168\pi$
$601$	28532.0			1.93651			0.968257	+	0.249958i	$0.0804168\pi$
$602$		−	37024.1i		−	2.50663i
$603$	17044.7	+	12126.9i	1.15110	+	0.818980i
$604$		0			0
$605$			14881.0i			1.00000i
$606$	8920.00	−	27924.0i	0.597938	−	1.87184i
$607$	24839.7			1.66098			0.830488	−	0.557037i	$-0.188062\pi$
$607$	24839.7			1.66098			0.830488	+	0.557037i	$0.188062\pi$
$608$		0			0
$609$	−10780.0	−	3443.54i	−0.717287	−	0.229129i
$610$	30104.9			1.99821
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			28433.8i			1.86889i
$615$	−8142.86	+	25491.2i	−0.533906	+	1.67139i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	13060.2	+	4171.93i	0.850095	+	0.271553i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	−7973.00	−	10637.0i	−0.515210	−	0.687354i
$622$		0			0
$623$			32979.5i			2.12086i
$624$		0			0
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−24200.0	−	17217.7i	−1.53040	−	1.08884i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	26976.1i		−	1.68585i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	16190.9			1.00000
$641$			24449.2i			1.50653i	0.657719	+	0.753264i	$0.271522\pi$
$641$			24449.2i			1.50653i	−0.657719	+	0.753264i	$0.728478\pi$
$642$	−1397.73	+	4375.58i	−0.0859250	+	0.268988i
$643$	−9825.20			−0.602594			−0.301297	−	0.953530i	$-0.597420\pi$
$643$	−9825.20			−0.602594			−0.301297	+	0.953530i	$0.597420\pi$
$644$		−	26367.7i		−	1.61341i
$645$	−20825.0	−	6652.30i	−1.27129	−	0.406099i
$646$		0			0
$647$		−	29470.8i		−	1.79075i	−0.445311	−	0.895376i	$-0.646907\pi$
$647$		−	29470.8i		−	1.79075i	0.445311	−	0.895376i	$-0.353093\pi$
$648$	−15583.7	+	5407.95i	−0.944731	+	0.327846i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−177.088			−0.0106369
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	15904.0	+	5080.35i	0.950911	+	0.303757i
$655$		0			0
$656$			29480.3i			1.75459i
$657$		0			0
$658$	41881.2			2.48131
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−30688.0			−1.80579			−0.902893	−	0.429865i	$-0.858561\pi$
$661$	−30688.0			−1.80579			−0.902893	+	0.429865i	$0.858561\pi$
$662$		0			0
$663$		0			0
$664$	−2464.00			−0.144009
$665$		0			0
$666$		0			0
$667$	−5932.43			−0.344385
$668$			14809.6i			0.857788i
$669$	3295.00	−	10315.0i	0.190422	−	0.596114i
$670$	24500.0			1.41271
$671$		0			0
$672$	−31167.4	−	9956.06i	−1.78915	−	0.571523i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−14032.6	+	10518.2i	−0.800171	+	0.599772i
$676$	−17576.0			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	28567.0	+	9125.39i	1.60747	+	0.513489i
$682$		0			0
$683$		−	5567.76i		−	0.311924i	−0.987763	−	0.155962i	$-0.950152\pi$
$683$		−	5567.76i		−	0.311924i	0.987763	−	0.155962i	$-0.0498478\pi$
$684$		0			0
$685$		0			0
$686$			51554.8i			2.86935i
$687$	10868.7	−	34024.6i	0.603593	−	1.88955i
$688$	−24083.9			−1.33458
$689$		0			0
$690$	−14831.1	−	4737.62i	−0.818275	−	0.261388i
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	31484.0			1.72207
$695$		0			0
$696$	−2240.00	+	7012.31i	−0.121993	+	0.381898i
$697$		0			0
$698$			27283.0i			1.47948i
$699$		0			0
$700$	−34785.1			−1.87822
$701$		−	35844.2i		−	1.93126i	−0.259914	−	0.965632i	$-0.583694\pi$
$701$		−	35844.2i		−	1.93126i	0.259914	−	0.965632i	$-0.416306\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	7525.00	−	23557.0i	0.401997	−	1.25845i
$706$		0			0
$707$			69381.3i			3.69074i
$708$		0			0
$709$	506.000			0.0268029			0.0134014	−	0.999910i	$-0.495734\pi$
$709$	506.000			0.0268029			0.0134014	+	0.999910i	$0.495734\pi$
$710$		0			0
$711$		0			0
$712$	21452.9			1.12919
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−11200.0	+	15741.9i	−0.579721	+	0.814815i
$721$	−32450.0			−1.67615
$722$		−	19400.2i		−	1.00000i
$723$	2700.59	−	8454.17i	0.138915	−	0.434874i
$724$	8624.00			0.442691
$725$			7826.24i			0.400909i
$726$	−18634.0	−	5952.41i	−0.952579	−	0.304290i
$727$	−38114.9			−1.94444			−0.972218	−	0.234078i	$-0.924793\pi$
$727$	−38114.9			−1.94444			−0.972218	+	0.234078i	$0.924793\pi$
$728$		0			0
$729$	5522.00	−	18892.5i	0.280547	−	0.959840i
$730$		0			0
$731$		0			0
$732$	−12042.0	+	37697.3i	−0.608037	+	1.90346i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$			9758.20i			0.490711i
$735$	47979.7	+	15326.5i	2.40783	+	0.769154i
$736$	−17152.0			−0.859010
$737$		0			0
$738$	−28662.9	−	20393.0i	−1.42967	−	1.01718i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	40493.2i		−	1.99940i	−0.0245922	−	0.999698i	$-0.507829\pi$
$743$		−	40493.2i		−	1.99940i	0.0245922	−	0.999698i	$-0.492171\pi$
$744$		0			0
$745$	21350.0			1.04994
$746$		0			0
$747$	1704.47	−	2395.68i	0.0834849	−	0.117340i
$748$		0			0
$749$		−	10871.8i		−	0.530368i
$750$	−6250.00	+	19565.6i	−0.304290	+	0.952579i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	27243.4i		−	1.32110i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	31240.0	−	23416.1i	1.50289	−	1.12650i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	35884.4i		−	1.70934i	−0.519170	−	0.854671i	$-0.673759\pi$
$761$		−	35884.4i		−	1.70934i	0.519170	−	0.854671i	$-0.326241\pi$
$762$	33779.4	+	10790.4i	1.60591	+	0.512988i
$763$	−39515.8			−1.87493
$764$		0			0
$765$		0			0
$766$	−32452.0			−1.53073
$767$		0			0
$768$	−6476.34	+	20274.2i	−0.304290	+	0.952579i
$769$	−29554.0			−1.38588			−0.692942	−	0.720994i	$-0.743686\pi$
$769$	−29554.0			−1.38588			−0.692942	+	0.720994i	$0.743686\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	16660.0	−	23416.1i	0.773684	−	1.08743i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	16557.7			0.763010
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−5268.35	−	7028.64i	−0.240454	−	0.320796i
$784$	55488.0			2.52770
$785$		0			0
$786$		0			0
$787$	−27584.5			−1.24941			−0.624703	−	0.780862i	$-0.714780\pi$
$787$	−27584.5			−1.24941			−0.624703	+	0.780862i	$0.714780\pi$
$788$		0			0
$789$	−42119.0	−	13454.4i	−1.90048	−	0.607085i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		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
$800$			22627.4i			1.00000i
$801$	−14840.0	+	20858.0i	−0.654614	+	0.920078i
$802$	7791.85			0.343067
$803$		0			0
$804$	−9800.00	+	30678.9i	−0.429875	+	1.34572i
$805$	36850.0			1.61341
$806$		0			0
$807$	33978.7	+	10854.1i	1.48216	+	0.473460i
$808$	45132.0			1.96502
$809$			37691.2i			1.63801i	0.573786	+	0.819005i	$0.305474\pi$
$809$			37691.2i			1.63801i	−0.573786	+	0.819005i	$0.694526\pi$
$810$	−7557.84	−	21778.9i	−0.327846	−	0.944731i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	17423.1i		−	0.752994i
$813$		0			0
$814$		0			0
$815$		−	247.487i		−	0.0106369i
$816$		0			0
$817$		0			0
$818$		−	13700.9i		−	0.585624i
$819$		0			0
$820$	−41200.0			−1.75459
$821$			46425.2i			1.97351i	0.162216	+	0.986755i	$0.448136\pi$
$821$			46425.2i			1.97351i	−0.162216	+	0.986755i	$0.551864\pi$
$822$		0			0
$823$	35085.5			1.48603			0.743015	−	0.669275i	$-0.233395\pi$
$823$	35085.5			1.48603			0.743015	+	0.669275i	$0.233395\pi$
$824$			21108.5i			0.892414i
$825$		0			0
$826$		0			0
$827$			46285.8i			1.94621i	0.230364	+	0.973104i	$0.426008\pi$
$827$			46285.8i			1.94621i	−0.230364	+	0.973104i	$0.573992\pi$
$828$	11864.9	−	16676.4i	0.497986	−	0.699934i
$829$	36344.0			1.52265			0.761326	−	0.648369i	$-0.224548\pi$
$829$	36344.0			1.52265			0.761326	+	0.648369i	$0.224548\pi$
$830$		−	3443.54i		−	0.144009i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−20697.1			−0.857788
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	13914.0	−	43557.8i	0.571523	−	1.78915i
$841$	20469.0			0.839272
$842$			27130.3i			1.11042i
$843$	21228.4	+	6781.15i	0.867312	+	0.277053i
$844$		0			0
$845$		−	24563.2i		−	1.00000i
$846$	26488.0	+	18845.6i	1.07645	+	0.765868i
$847$	46298.9			1.87822
$848$		0			0
$849$	−14975.0	+	46879.2i	−0.605348	+	1.89504i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		−	93664.4i		−	3.75308i
$855$		0			0
$856$	−7072.00			−0.282378
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		−	33658.3i		−	1.33458i
$861$	79310.0	+	25334.7i	3.13923	+	1.00279i
$862$		0			0
$863$		−	44820.7i		−	1.76792i	−0.467564	−	0.883959i	$-0.654868\pi$
$863$		−	44820.7i		−	1.76792i	0.467564	−	0.883959i	$-0.345132\pi$
$864$	−15232.0	−	20321.4i	−0.599772	−	0.800171i
$865$		0			0
$866$		0			0
$867$	7768.14	−	24318.1i	0.304290	−	0.952579i
$868$		0			0
$869$		0			0
$870$	−9800.00	−	3130.50i	−0.381898	−	0.121993i
$871$		0			0
$872$			25704.7i			0.998248i
$873$		0			0
$874$		0			0
$875$		−	48613.6i		−	1.87822i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	42847.5i		−	1.63856i	−0.573395	−	0.819279i	$-0.694374\pi$
$881$		−	42847.5i		−	1.63856i	0.573395	−	0.819279i	$-0.305626\pi$
$882$	−38383.7	+	53949.4i	−1.46536	+	2.05960i
$883$	−47791.5			−1.82142			−0.910709	−	0.413049i	$-0.864464\pi$
$883$	−47791.5			−1.82142			−0.910709	+	0.413049i	$0.864464\pi$
$884$		0			0
$885$		0			0
$886$	−40948.0			−1.55268
$887$			4029.09i			0.152518i	0.997088	+	0.0762592i	$0.0242976\pi$
$887$			4029.09i			0.152518i	−0.997088	+	0.0762592i	$0.975702\pi$
$888$		0			0
$889$	−83930.0			−3.16639
$890$			29981.3i			1.12919i
$891$		0			0
$892$	16671.5			0.625789
$893$		0			0
$894$	−8540.00	+	26734.4i	−0.319486	+	1.00015i
$895$		0			0
$896$		−	50374.1i		−	1.87822i
$897$		0			0
$898$	−53569.0			−1.99067
$899$		0			0
$900$	−22000.0	−	15652.5i	−0.814815	−	0.579721i
$901$		0			0
$902$		0			0
$903$	−20697.1	+	64792.2i	−0.762743	+	2.38776i
$904$		0			0
$905$			12052.4i			0.442691i
$906$		0			0
$907$	53414.0			1.95544			0.977720	−	0.209914i	$-0.0673184\pi$
$907$	53414.0			1.95544			0.977720	+	0.209914i	$0.0673184\pi$
$908$			46171.2i			1.68750i
$909$	−31220.0	+	43880.6i	−1.13917	+	1.60113i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−52683.5	−	16829.1i	−1.90346	−	0.608037i
$916$	54992.0			1.98361
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		−	23970.6i		−	0.859010i
$921$	15895.0	−	49759.2i	0.568684	−	1.78026i
$922$	−32963.6			−1.17744
$923$		0			0
$924$		0			0
$925$		0			0
$926$		−	34998.9i		−	1.24205i
$927$	−20523.2	−	14601.8i	−0.727152	−	0.517351i
$928$	−11333.6			−0.400909
$929$		−	18054.0i		−	0.637603i	−0.947822	−	0.318801i	$-0.896720\pi$
$929$		−	18054.0i		−	0.637603i	0.947822	−	0.318801i	$-0.103280\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	31556.0			1.10551
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		−	76226.1i		−	2.65338i
$939$		0			0
$940$	38073.8			1.32110
$941$			36805.7i			1.27506i	0.770426	+	0.637530i	$0.220044\pi$
$941$			36805.7i			1.27506i	−0.770426	+	0.637530i	$0.779956\pi$
$942$		0			0
$943$	43645.8			1.50721
$944$		0			0
$945$	32725.0	+	43659.2i	1.12650	+	1.50289i
$946$		0			0
$947$			52115.2i			1.78830i	0.447772	+	0.894148i	$0.352218\pi$
$947$			52115.2i			1.78830i	−0.447772	+	0.894148i	$0.647782\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−28334.0	−	9050.97i	−0.952579	−	0.304290i
$961$	29791.0			1.00000
$962$		0			0
$963$	4892.04	−	6875.91i	0.163701	−	0.230086i
$964$	13664.0			0.456523
$965$		0			0
$966$	−14740.0	+	46143.5i	−0.490944	+	1.53690i
$967$	−49783.7			−1.65557			−0.827785	−	0.561045i	$-0.810400\pi$
$967$	−49783.7			−1.65557			−0.827785	+	0.561045i	$0.810400\pi$
$968$		−	30117.1i		−	1.00000i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	30294.6	−	752.362i	0.999692	−	0.0248272i
$973$		0			0
$974$			60794.2i			1.99997i
$975$		0			0
$976$	−60928.0			−1.99821
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	309.903	+	98.9949i	0.0101325	+	0.00323672i
$979$		0			0
$980$			77546.8i			2.52770i
$981$	−24992.0	−	17781.2i	−0.813388	−	0.578706i
$982$		0			0
$983$			51962.4i			1.68601i	0.537908	+	0.843003i	$0.319215\pi$
$983$			51962.4i			1.68601i	−0.537908	+	0.843003i	$0.680785\pi$
$984$	16480.0	−	51590.6i	0.533906	−	1.67139i
$985$		0			0
$986$		0			0
$987$	−73292.1	−	23412.3i	−2.36364	−	0.755037i
$988$		0			0
$989$			35656.3i			1.14642i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	4312.00	+	1377.42i	0.137180	+	0.0438204i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.h.a.59.1	✓	4
3.2	odd	2	inner	60.4.h.a.59.3	yes	4
4.3	odd	2	inner	60.4.h.a.59.4	yes	4
5.4	even	2	inner	60.4.h.a.59.4	yes	4
12.11	even	2	inner	60.4.h.a.59.2	yes	4
15.14	odd	2	inner	60.4.h.a.59.2	yes	4
20.19	odd	2	CM	60.4.h.a.59.1	✓	4
60.59	even	2	inner	60.4.h.a.59.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.h.a.59.1	✓	4	1.1	even	1	trivial
60.4.h.a.59.1	✓	4	20.19	odd	2	CM
60.4.h.a.59.2	yes	4	12.11	even	2	inner
60.4.h.a.59.2	yes	4	15.14	odd	2	inner
60.4.h.a.59.3	yes	4	3.2	odd	2	inner
60.4.h.a.59.3	yes	4	60.59	even	2	inner
60.4.h.a.59.4	yes	4	4.3	odd	2	inner
60.4.h.a.59.4	yes	4	5.4	even	2	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +(-1.58114 + 4.94975i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +(14.0000 + 4.47214i) q^{6} -34.7851 q^{7} +22.6274i q^{8} +(-22.0000 - 15.6525i) q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +(-1.58114 + 4.94975i) q^{3} -8.00000 q^{4} -11.1803i q^{5} +(14.0000 + 4.47214i) q^{6} -34.7851 q^{7} +22.6274i q^{8} +(-22.0000 - 15.6525i) q^{9} -31.6228 q^{10} +(12.6491 - 39.5980i) q^{12} +98.3870i q^{14} +(55.3399 + 17.6777i) q^{15} +64.0000 q^{16} +(-44.2719 + 62.2254i) q^{18} +89.4427i q^{20} +(55.0000 - 172.177i) q^{21} -94.7523i q^{23} +(-112.000 - 35.7771i) q^{24} -125.000 q^{25} +(112.261 - 84.1457i) q^{27} +278.280 q^{28} -62.6099i q^{29} +(50.0000 - 156.525i) q^{30} -181.019i q^{32} +388.909i q^{35} +(176.000 + 125.220i) q^{36} +252.982 q^{40} +460.630i q^{41} +(-486.991 - 155.563i) q^{42} -376.311 q^{43} +(-175.000 + 245.967i) q^{45} -268.000 q^{46} -425.678i q^{47} +(-101.193 + 316.784i) q^{48} +867.000 q^{49} +353.553i q^{50} +(-238.000 - 317.522i) q^{54} -787.096i q^{56} -177.088 q^{58} +(-442.719 - 141.421i) q^{60} -952.000 q^{61} +(765.271 + 544.472i) q^{63} -512.000 q^{64} -774.758 q^{67} +(469.000 + 149.817i) q^{69} +1100.00 q^{70} +(354.175 - 497.803i) q^{72} +(197.642 - 618.718i) q^{75} -715.542i q^{80} +(239.000 + 688.709i) q^{81} +1302.86 q^{82} +108.894i q^{83} +(-440.000 + 1377.42i) q^{84} +1064.37i q^{86} +(309.903 + 98.9949i) q^{87} -948.093i q^{89} +(695.701 + 494.975i) q^{90} +758.018i q^{92} -1204.00 q^{94} +(896.000 + 286.217i) q^{96} -2452.25i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} + 56 q^{6} - 88 q^{9}+O(q^{10}) \) 4 * q - 32 * q^4 + 56 * q^6 - 88 * q^9	\( 4 q - 32 q^{4} + 56 q^{6} - 88 q^{9} + 256 q^{16} + 220 q^{21} - 448 q^{24} - 500 q^{25} + 200 q^{30} + 704 q^{36} - 700 q^{45} - 1072 q^{46} + 3468 q^{49} - 952 q^{54} - 3808 q^{61} - 2048 q^{64} + 1876 q^{69} + 4400 q^{70} + 956 q^{81} - 1760 q^{84} - 4816 q^{94} + 3584 q^{96}+O(q^{100}) \) 4 * q - 32 * q^4 + 56 * q^6 - 88 * q^9 + 256 * q^16 + 220 * q^21 - 448 * q^24 - 500 * q^25 + 200 * q^30 + 704 * q^36 - 700 * q^45 - 1072 * q^46 + 3468 * q^49 - 952 * q^54 - 3808 * q^61 - 2048 * q^64 + 1876 * q^69 + 4400 * q^70 + 956 * q^81 - 1760 * q^84 - 4816 * q^94 + 3584 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more