LMFDB - Embedded newform 60.8.h.a.59.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,8,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, 8, names="a")

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

chi := DirichletCharacter("60.59");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
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:	$18.7431015290$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			59.3
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	60.59
Dual form			60.8.h.a.59.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.3137i q^{2} +(-45.8530 - 9.19239i) q^{3} -128.000 q^{4} -279.508i q^{5} +(104.000 - 518.768i) q^{6} -1767.71 q^{7} -1448.15i q^{8} +(2018.00 + 842.998i) q^{9} +O(q^{10})$	\(q+11.3137i q^{2} +(-45.8530 - 9.19239i) q^{3} -128.000 q^{4} -279.508i q^{5} +(104.000 - 518.768i) q^{6} -1767.71 q^{7} -1448.15i q^{8} +(2018.00 + 842.998i) q^{9} +3162.28 q^{10} +(5869.19 + 1176.63i) q^{12} -19999.4i q^{14} +(-2569.35 + 12816.3i) q^{15} +16384.0 q^{16} +(-9537.43 + 22831.1i) q^{18} +35777.1i q^{20} +(81055.0 + 16249.5i) q^{21} +100399. i q^{23} +(-13312.0 + 66402.3i) q^{24} -78125.0 q^{25} +(-84782.2 - 57204.2i) q^{27} +226267. q^{28} -143135. i q^{29} +(-145000. - 29068.9i) q^{30} +185364. i q^{32} +494091. i q^{35} +(-258304. - 107904. i) q^{36} -404772. q^{40} +530248. i q^{41} +(-183842. + 917033. i) q^{42} +1.03340e6 q^{43} +(235625. - 564048. i) q^{45} -1.13589e6 q^{46} -846041. i q^{47} +(-751256. - 150608. i) q^{48} +2.30127e6 q^{49} -883883. i q^{50} +(647192. - 959202. i) q^{54} +2.55992e6i q^{56} +1.61939e6 q^{58} +(328877. - 1.64049e6i) q^{60} +2.06641e6 q^{61} +(-3.56725e6 - 1.49018e6i) q^{63} -2.09715e6 q^{64} -1.28611e6 q^{67} +(922909. - 4.60361e6i) q^{69} -5.59000e6 q^{70} +(1.22079e6 - 2.92238e6i) q^{72} +(3.58227e6 + 718155. i) q^{75} -4.57947e6i q^{80} +(3.36168e6 + 3.40234e6i) q^{81} -5.99907e6 q^{82} -8.02354e6i q^{83} +(-1.03750e7 - 2.07994e6i) q^{84} +1.16916e7i q^{86} +(-1.31575e6 + 6.56318e6i) q^{87} +8.51321e6i q^{89} +(6.38148e6 + 2.66579e6i) q^{90} -1.28511e7i q^{92} +9.57186e6 q^{94} +(1.70394e6 - 8.49949e6i) q^{96} +2.60359e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 512 q^{4} + 416 q^{6} + 8072 q^{9}+O(q^{10}) $ 4 * q - 512 * q^4 + 416 * q^6 + 8072 * q^9	$ 4 q - 512 q^{4} + 416 q^{6} + 8072 q^{9} + 65536 q^{16} + 324220 q^{21} - 53248 q^{24} - 312500 q^{25} - 580000 q^{30} - 1033216 q^{36} + 942500 q^{45} - 4543552 q^{46} + 9205068 q^{49} + 2588768 q^{54} + 8265632 q^{61} - 8388608 q^{64} + 3691636 q^{69} - 22360000 q^{70} + 13446716 q^{81} - 41500160 q^{84} + 38287424 q^{94} + 6815744 q^{96}+O(q^{100}) $ 4 * q - 512 * q^4 + 416 * q^6 + 8072 * q^9 + 65536 * q^16 + 324220 * q^21 - 53248 * q^24 - 312500 * q^25 - 580000 * q^30 - 1033216 * q^36 + 942500 * q^45 - 4543552 * q^46 + 9205068 * q^49 + 2588768 * q^54 + 8265632 * q^61 - 8388608 * q^64 + 3691636 * q^69 - 22360000 * q^70 + 13446716 * q^81 - 41500160 * q^84 + 38287424 * q^94 + 6815744 * 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 $

$2$			11.3137i			1.00000i
$2$			11.3137i			1.00000i
$3$	−45.8530	−	9.19239i	−0.980491	−	0.196564i
$3$	−45.8530	−	9.19239i	−0.980491	−	0.196564i
$4$	−128.000			−1.00000
$5$		−	279.508i		−	1.00000i
$5$		−	279.508i		−	1.00000i
$6$	104.000	−	518.768i	0.196564	−	0.980491i
$7$	−1767.71			−1.94791			−0.973955	−	0.226744i	$-0.927192\pi$
$7$	−1767.71			−1.94791			−0.973955	+	0.226744i	$0.927192\pi$
$8$		−	1448.15i		−	1.00000i
$9$	2018.00	+	842.998i	0.922725	+	0.385458i
$10$	3162.28			1.00000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	5869.19	+	1176.63i	0.980491	+	0.196564i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		−	19999.4i		−	1.94791i
$15$	−2569.35	+	12816.3i	−0.196564	+	0.980491i
$16$	16384.0			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−9537.43	+	22831.1i	−0.385458	+	0.922725i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$			35777.1i			1.00000i
$21$	81055.0	+	16249.5i	1.90991	+	0.382889i
$22$		0			0
$23$			100399.i			1.72061i	0.509777	+	0.860306i	$0.329728\pi$
$23$			100399.i			1.72061i	−0.509777	+	0.860306i	$0.670272\pi$
$24$	−13312.0	+	66402.3i	−0.196564	+	0.980491i
$25$	−78125.0			−1.00000
$26$		0			0
$27$	−84782.2	−	57204.2i	−0.828956	−	0.559313i
$28$	226267.			1.94791
$29$		−	143135.i		−	1.08982i	−0.838496	−	0.544908i	$-0.816564\pi$
$29$		−	143135.i		−	1.08982i	0.838496	−	0.544908i	$-0.183436\pi$
$30$	−145000.	−	29068.9i	−0.980491	−	0.196564i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			185364.i			1.00000i
$33$		0			0
$34$		0			0
$35$			494091.i			1.94791i
$36$	−258304.	−	107904.i	−0.922725	−	0.385458i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−404772.			−1.00000
$41$			530248.i			1.20153i	0.799425	+	0.600766i	$0.205138\pi$
$41$			530248.i			1.20153i	−0.799425	+	0.600766i	$0.794862\pi$
$42$	−183842.	+	917033.i	−0.382889	+	1.90991i
$43$	1.03340e6			1.98211			0.991055	−	0.133453i	$-0.0426064\pi$
$43$	1.03340e6			1.98211			0.991055	+	0.133453i	$0.0426064\pi$
$44$		0			0
$45$	235625.	−	564048.i	0.385458	−	0.922725i
$46$	−1.13589e6			−1.72061
$47$		−	846041.i		−	1.18864i	−0.804230	−	0.594318i	$-0.797422\pi$
$47$		−	846041.i		−	1.18864i	0.804230	−	0.594318i	$-0.202578\pi$
$48$	−751256.	−	150608.i	−0.980491	−	0.196564i
$49$	2.30127e6			2.79435
$50$		−	883883.i		−	1.00000i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	647192.	−	959202.i	0.559313	−	0.828956i
$55$		0			0
$56$			2.55992e6i			1.94791i
$57$		0			0
$58$	1.61939e6			1.08982
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	328877.	−	1.64049e6i	0.196564	−	0.980491i
$61$	2.06641e6			1.16563			0.582816	−	0.812604i	$-0.301951\pi$
$61$	2.06641e6			1.16563			0.582816	+	0.812604i	$0.301951\pi$
$62$		0			0
$63$	−3.56725e6	−	1.49018e6i	−1.79738	−	0.750838i
$64$	−2.09715e6			−1.00000
$65$		0			0
$66$		0			0
$67$	−1.28611e6			−0.522417			−0.261209	−	0.965282i	$-0.584121\pi$
$67$	−1.28611e6			−0.522417			−0.261209	+	0.965282i	$0.584121\pi$
$68$		0			0
$69$	922909.	−	4.60361e6i	0.338210	−	1.68705i
$70$	−5.59000e6			−1.94791
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	1.22079e6	−	2.92238e6i	0.385458	−	0.922725i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	3.58227e6	+	718155.i	0.980491	+	0.196564i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	4.57947e6i		−	1.00000i
$81$	3.36168e6	+	3.40234e6i	0.702844	+	0.711344i
$82$	−5.99907e6			−1.20153
$83$		−	8.02354e6i		−	1.54026i	−0.637889	−	0.770128i	$-0.720192\pi$
$83$		−	8.02354e6i		−	1.54026i	0.637889	−	0.770128i	$-0.279808\pi$
$84$	−1.03750e7	−	2.07994e6i	−1.90991	−	0.382889i
$85$		0			0
$86$			1.16916e7i			1.98211i
$87$	−1.31575e6	+	6.56318e6i	−0.214219	+	1.06856i
$88$		0			0
$89$			8.51321e6i			1.28005i	0.768352	+	0.640027i	$0.221077\pi$
$89$			8.51321e6i			1.28005i	−0.768352	+	0.640027i	$0.778923\pi$
$90$	6.38148e6	+	2.66579e6i	0.922725	+	0.385458i
$91$		0			0
$92$		−	1.28511e7i		−	1.72061i
$93$		0			0
$94$	9.57186e6			1.18864
$95$		0			0
$96$	1.70394e6	−	8.49949e6i	0.196564	−	0.980491i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$			2.60359e7i			2.79435i
$99$		0			0
$100$	1.00000e7			1.00000
$101$			1.79076e6i			0.172947i	0.996254	+	0.0864734i	$0.0275598\pi$
$101$			1.79076e6i			0.172947i	−0.996254	+	0.0864734i	$0.972440\pi$
$102$		0			0
$103$	710105.			0.0640313			0.0320157	−	0.999487i	$-0.489807\pi$
$103$	710105.			0.0640313			0.0320157	+	0.999487i	$0.489807\pi$
$104$		0			0
$105$	4.54188e6	−	2.26556e7i	0.382889	−	1.90991i
$106$		0			0
$107$			2.48731e7i			1.96285i	0.191856	+	0.981423i	$0.438549\pi$
$107$			2.48731e7i			1.96285i	−0.191856	+	0.981423i	$0.561451\pi$
$108$	1.08521e7	+	7.32214e6i	0.828956	+	0.559313i
$109$	−4.63278e6			−0.342649			−0.171324	−	0.985215i	$-0.554805\pi$
$109$	−4.63278e6			−0.342649			−0.171324	+	0.985215i	$0.554805\pi$
$110$		0			0
$111$		0			0
$112$	−2.89622e7			−1.94791
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	2.80624e7			1.72061
$116$			1.83213e7i			1.08982i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	1.85600e7	+	3.72082e6i	0.980491	+	0.196564i
$121$	−1.94872e7			−1.00000
$122$			2.33787e7i			1.16563i
$123$	4.87424e6	−	2.43135e7i	0.236178	−	1.17809i
$124$		0			0
$125$			2.18366e7i			1.00000i
$126$	1.68594e7	−	4.03588e7i	0.750838	−	1.79738i
$127$	−1.12087e7			−0.485561			−0.242780	−	0.970081i	$-0.578059\pi$
$127$	−1.12087e7			−0.485561			−0.242780	+	0.970081i	$0.578059\pi$
$128$		−	2.37266e7i		−	1.00000i
$129$	−4.73844e7	−	9.49939e6i	−1.94344	−	0.389612i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		−	1.45507e7i		−	0.522417i
$135$	−1.59891e7	+	2.36974e7i	−0.559313	+	0.828956i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	5.20839e7	+	1.04415e7i	1.68705	+	0.338210i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		−	6.32436e7i		−	1.94791i
$141$	−7.77713e6	+	3.87935e7i	−0.233643	+	1.16545i
$142$		0			0
$143$		0			0
$144$	3.30629e7	+	1.38117e7i	0.922725	+	0.385458i
$145$	−4.00075e7			−1.08982
$146$		0			0
$147$	−1.05520e8	−	2.11541e7i	−2.73983	−	0.549268i
$148$		0			0
$149$			7.75851e7i			1.92144i	0.277522	+	0.960719i	$0.410487\pi$
$149$			7.75851e7i			1.92144i	−0.277522	+	0.960719i	$0.589513\pi$
$150$	−8.12500e6	+	4.05287e7i	−0.196564	+	0.980491i
$151$		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
$160$	5.18108e7			1.00000
$161$		−	1.77477e8i		−	3.35160i
$162$	−3.84931e7	+	3.80331e7i	−0.711344	+	0.702844i
$163$	9.64461e7			1.74433			0.872163	−	0.489215i	$-0.162716\pi$
$163$	9.64461e7			1.74433			0.872163	+	0.489215i	$0.162716\pi$
$164$		−	6.78717e7i		−	1.20153i
$165$		0			0
$166$	9.07760e7			1.54026
$167$		−	1.05016e8i		−	1.74481i	−0.488782	−	0.872406i	$-0.662559\pi$
$167$		−	1.05016e8i		−	1.74481i	0.488782	−	0.872406i	$-0.337441\pi$
$168$	2.35318e7	−	1.17380e8i	0.382889	−	1.90991i
$169$	6.27485e7			1.00000
$170$		0			0
$171$		0			0
$172$	−1.32275e8			−1.98211
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	−7.42539e7	−	1.48861e7i	−1.06856	−	0.214219i
$175$	1.38103e8			1.94791
$176$		0			0
$177$		0			0
$178$	−9.63160e7			−1.28005
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−3.01600e7	+	7.21982e7i	−0.385458	+	0.922725i
$181$	−7.93876e7			−0.995125			−0.497562	−	0.867428i	$-0.665771\pi$
$181$	−7.93876e7			−0.995125			−0.497562	+	0.867428i	$0.665771\pi$
$182$		0			0
$183$	−9.47511e7	−	1.89952e7i	−1.14289	−	0.229121i
$184$	1.45394e8			1.72061
$185$		0			0
$186$		0			0
$187$		0			0
$188$			1.08293e8i			1.18864i
$189$	1.49871e8	+	1.01121e8i	1.61473	+	1.08949i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	9.61608e7	+	1.92778e7i	0.980491	+	0.196564i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−2.94562e8			−2.79435
$197$		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			$\pi$
$200$			1.13137e8i			1.00000i
$201$	5.89722e7	+	1.18225e7i	0.512226	+	0.102688i
$202$	−2.02601e7			−0.172947
$203$			2.53022e8i			2.12286i
$204$		0			0
$205$	1.48209e8			1.20153
$206$			8.03392e6i			0.0640313i
$207$	−8.46363e7	+	2.02606e8i	−0.663225	+	1.58765i
$208$		0			0
$209$		0			0
$210$	2.56318e8	+	5.13854e7i	1.90991	+	0.382889i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	−2.81407e8			−1.96285
$215$		−	2.88843e8i		−	1.98211i
$216$	−8.28406e7	+	1.22778e8i	−0.559313	+	0.828956i
$217$		0			0
$218$		−	5.24140e7i		−	0.342649i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.23953e8			1.35235			0.676175	−	0.736741i	$-0.263636\pi$
$223$	2.23953e8			1.35235			0.676175	+	0.736741i	$0.263636\pi$
$224$		−	3.27670e8i		−	1.94791i
$225$	−1.57656e8	−	6.58592e7i	−0.922725	−	0.385458i
$226$		0			0
$227$			2.52451e8i			1.43247i	0.697859	+	0.716235i	$0.254136\pi$
$227$			2.52451e8i			1.43247i	−0.697859	+	0.716235i	$0.745864\pi$
$228$		0			0
$229$	8.09742e7			0.445577			0.222788	−	0.974867i	$-0.428484\pi$
$229$	8.09742e7			0.445577			0.222788	+	0.974867i	$0.428484\pi$
$230$			3.17490e8i			1.72061i
$231$		0			0
$232$	−2.07282e8			−1.08982
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−2.36476e8			−1.18864
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−4.20962e7	+	2.09982e8i	−0.196564	+	0.980491i
$241$	4.34559e8			1.99981			0.999905	−	0.0137476i	$-0.00437612\pi$
$241$	4.34559e8			1.99981			0.999905	+	0.0137476i	$0.00437612\pi$
$242$		−	2.20472e8i		−	1.00000i
$243$	−1.22868e8	−	1.86909e8i	−0.549307	−	0.835621i
$244$	−2.64500e8			−1.16563
$245$		−	6.43224e8i		−	2.79435i
$246$	2.75075e8	+	5.51458e7i	1.17809	+	0.236178i
$247$		0			0
$248$		0			0
$249$	−7.37555e7	+	3.67904e8i	−0.302759	+	1.51021i
$250$	−2.47053e8			−1.00000
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	4.56607e8	+	1.90743e8i	1.79738	+	0.750838i
$253$		0			0
$254$		−	1.26812e8i		−	0.485561i
$255$		0			0
$256$	2.68435e8			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$	1.07473e8	−	5.36093e8i	0.389612	−	1.94344i
$259$		0			0
$260$		0			0
$261$	1.20663e8	−	2.88847e8i	0.420079	−	1.00560i
$262$		0			0
$263$			2.07842e8i			0.704513i	0.935904	+	0.352256i	$0.114585\pi$
$263$			2.07842e8i			0.704513i	−0.935904	+	0.352256i	$0.885415\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	7.82567e7	−	3.90357e8i	0.251613	−	1.25508i
$268$	1.64623e8			0.522417
$269$			5.48312e8i			1.71749i	0.512403	+	0.858745i	$0.328756\pi$
$269$			5.48312e8i			1.71749i	−0.512403	+	0.858745i	$0.671244\pi$
$270$	−2.68105e8	−	1.80896e8i	−0.828956	−	0.559313i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−1.18132e8	+	5.89262e8i	−0.338210	+	1.68705i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	7.15520e8			1.94791
$281$			6.22535e8i			1.67376i	0.547390	+	0.836878i	$0.315621\pi$
$281$			6.22535e8i			1.67376i	−0.547390	+	0.836878i	$0.684379\pi$
$282$	−4.38899e8	−	8.79882e7i	−1.16545	−	0.233643i
$283$	2.12943e8			0.558484			0.279242	−	0.960221i	$-0.409917\pi$
$283$	2.12943e8			0.558484			0.279242	+	0.960221i	$0.409917\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	9.37326e8i		−	2.34047i
$288$	−1.56261e8	+	3.74064e8i	−0.385458	+	0.922725i
$289$	−4.10339e8			−1.00000
$290$		−	4.52633e8i		−	1.08982i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	2.39332e8	−	1.19382e9i	0.549268	−	2.73983i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−8.77776e8			−1.92144
$299$		0			0
$300$	−4.58530e8	−	9.19239e7i	−0.980491	−	0.196564i
$301$	−1.82675e9			−3.86097
$302$		0			0
$303$	1.64614e7	−	8.21117e7i	0.0339951	−	0.169573i
$304$		0			0
$305$		−	5.77579e8i		−	1.16563i
$306$		0			0
$307$	6.69510e8			1.32060			0.660302	−	0.751000i	$-0.270428\pi$
$307$	6.69510e8			1.32060			0.660302	+	0.751000i	$0.270428\pi$
$308$		0			0
$309$	−3.25605e7	−	6.52756e6i	−0.0627821	−	0.0125863i
$310$		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
$315$	−4.16517e8	+	9.97075e8i	−0.750838	+	1.79738i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$			5.86172e8i			1.00000i
$321$	2.28643e8	−	1.14051e9i	0.385825	−	1.92455i
$322$	2.00792e9			3.35160
$323$		0			0
$324$	−4.30295e8	−	4.35499e8i	−0.702844	−	0.711344i
$325$		0			0
$326$			1.09116e9i			1.74433i
$327$	2.12427e8	+	4.25863e7i	0.335964	+	0.0673524i
$328$	7.67881e8			1.20153
$329$			1.49556e9i			2.31535i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			1.02701e9i			1.54026i
$333$		0			0
$334$	1.18812e9			1.74481
$335$			3.59480e8i			0.522417i
$336$	1.32801e9	+	2.66232e8i	1.90991	+	0.382889i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			7.09918e8i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.61219e9			−3.49523
$344$		−	1.49652e9i		−	1.98211i
$345$	−1.28675e9	−	2.57961e8i	−1.68705	−	0.338210i
$346$		0			0
$347$			1.52632e9i			1.96107i	0.196341	+	0.980536i	$0.437094\pi$
$347$			1.52632e9i			1.96107i	−0.196341	+	0.980536i	$0.562906\pi$
$348$	1.68417e8	−	8.40087e8i	0.214219	−	1.06856i
$349$	1.23989e9			1.56133			0.780667	−	0.624948i	$-0.214880\pi$
$349$	1.23989e9			1.56133			0.780667	+	0.624948i	$0.214880\pi$
$350$			1.56245e9i			1.94791i
$351$		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
$356$		−	1.08969e9i		−	1.28005i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	−8.16829e8	−	3.41221e8i	−0.922725	−	0.385458i
$361$	8.93872e8			1.00000
$362$		−	8.98168e8i		−	0.995125i
$363$	8.93546e8	+	1.79134e8i	0.980491	+	0.196564i
$364$		0			0
$365$		0			0
$366$	2.14906e8	−	1.07199e9i	0.229121	−	1.14289i
$367$	1.03538e9			1.09337			0.546686	−	0.837338i	$-0.315889\pi$
$367$	1.03538e9			1.09337			0.546686	+	0.837338i	$0.315889\pi$
$368$			1.64494e9i			1.72061i
$369$	−4.46998e8	+	1.07004e9i	−0.463141	+	1.10868i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	2.00731e8	−	1.00127e9i	0.196564	−	0.980491i
$376$	−1.22520e9			−1.18864
$377$		0			0
$378$	−1.14405e9	+	1.69559e9i	−1.08949	+	1.61473i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	5.13954e8	+	1.03035e8i	0.476088	+	0.0954438i
$382$		0			0
$383$		−	1.83425e9i		−	1.66826i	−0.551568	−	0.834130i	$-0.685970\pi$
$383$		−	1.83425e9i		−	1.66826i	0.551568	−	0.834130i	$-0.314030\pi$
$384$	−2.18104e8	+	1.08793e9i	−0.196564	+	0.980491i
$385$		0			0
$386$		0			0
$387$	2.08540e9	+	8.71152e8i	1.82894	+	0.764021i
$388$		0			0
$389$			3.09924e8i			0.266951i	0.991052	+	0.133476i	$0.0426138\pi$
$389$			3.09924e8i			0.266951i	−0.991052	+	0.133476i	$0.957386\pi$
$390$		0			0
$391$		0			0
$392$		−	3.33259e9i		−	2.79435i
$393$		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
$400$	−1.28000e9			−1.00000
$401$		−	2.56347e9i		−	1.98529i	−0.121067	−	0.992644i	$-0.538632\pi$
$401$		−	2.56347e9i		−	1.98529i	0.121067	−	0.992644i	$-0.461368\pi$
$402$	−1.33756e8	+	6.67195e8i	−0.102688	+	0.512226i
$403$		0			0
$404$		−	2.29217e8i		−	0.172947i
$405$	9.50982e8	−	9.39618e8i	0.711344	−	0.702844i
$406$	−2.86262e9			−2.12286
$407$		0			0
$408$		0			0
$409$	1.76873e9			1.27829			0.639147	−	0.769084i	$-0.279287\pi$
$409$	1.76873e9			1.27829			0.639147	+	0.769084i	$0.279287\pi$
$410$			1.67679e9i			1.20153i
$411$		0			0
$412$	−9.08935e7			−0.0640313
$413$		0			0
$414$	−2.29222e9	−	9.57551e8i	−1.58765	−	0.663225i
$415$	−2.24265e9			−1.54026
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−5.81360e8	+	2.89991e9i	−0.382889	+	1.90991i
$421$	−3.00264e9			−1.96117			−0.980587	−	0.196086i	$-0.937177\pi$
$421$	−3.00264e9			−1.96117			−0.980587	+	0.196086i	$0.937177\pi$
$422$		0			0
$423$	7.13210e8	−	1.70731e9i	0.458170	−	1.09678i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.65282e9			−2.27055
$428$		−	3.18375e9i		−	1.96285i
$429$		0			0
$430$	3.26789e9			1.98211
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−1.38907e9	−	9.37234e8i	−0.828956	−	0.559313i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	1.83446e9	+	3.67764e8i	1.06856	+	0.214219i
$436$	5.92996e8			0.342649
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	4.64396e9	+	1.93996e9i	2.57842	+	1.07711i
$442$		0			0
$443$		−	7.47920e7i		−	0.0408735i	−0.999791	−	0.0204368i	$-0.993494\pi$
$443$		−	7.47920e7i		−	0.0408735i	0.999791	−	0.0204368i	$-0.00650568\pi$
$444$		0			0
$445$	2.37952e9			1.28005
$446$			2.53373e9i			1.35235i
$447$	7.13193e8	−	3.55751e9i	0.377686	−	1.88395i
$448$	3.70716e9			1.94791
$449$			2.61229e9i			1.36194i	0.732310	+	0.680971i	$0.238442\pi$
$449$			2.61229e9i			1.36194i	−0.732310	+	0.680971i	$0.761558\pi$
$450$	7.45112e8	−	1.78368e9i	0.385458	−	0.922725i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−2.85615e9			−1.43247
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$			9.16118e8i			0.445577i
$459$		0			0
$460$	−3.59199e9			−1.72061
$461$		−	4.18523e9i		−	1.98960i	−0.101849	−	0.994800i	$-0.532476\pi$
$461$		−	4.18523e9i		−	1.98960i	0.101849	−	0.994800i	$-0.467524\pi$
$462$		0			0
$463$	2.16280e9			1.01270			0.506352	−	0.862327i	$-0.330994\pi$
$463$	2.16280e9			1.01270			0.506352	+	0.862327i	$0.330994\pi$
$464$		−	2.34513e9i		−	1.08982i
$465$		0			0
$466$		0			0
$467$			1.38199e9i			0.627907i	0.949438	+	0.313953i	$0.101654\pi$
$467$			1.38199e9i			0.627907i	−0.949438	+	0.313953i	$0.898346\pi$
$468$		0			0
$469$	2.27348e9			1.01762
$470$		−	2.67542e9i		−	1.18864i
$471$		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	$0.5\pi$
$480$	−2.37568e9	−	4.76265e8i	−0.980491	−	0.196564i
$481$		0			0
$482$			4.91647e9i			1.99981i
$483$	−1.63144e9	+	8.13786e9i	−0.658803	+	3.28621i
$484$	2.49436e9			1.00000
$485$		0			0
$486$	2.11464e9	−	1.39009e9i	0.835621	−	0.549307i
$487$	−2.60359e9			−1.02146			−0.510730	−	0.859741i	$-0.670625\pi$
$487$	−2.60359e9			−1.02146			−0.510730	+	0.859741i	$0.670625\pi$
$488$		−	2.99248e9i		−	1.16563i
$489$	−4.42234e9	−	8.86570e8i	−1.71030	−	0.342872i
$490$	7.27725e9			2.79435
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	−6.23903e8	+	3.11212e9i	−0.236178	+	1.17809i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−4.16236e9	−	8.34449e8i	−1.51021	−	0.302759i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		−	2.79508e9i		−	1.00000i
$501$	−9.65349e8	+	4.81531e9i	−0.342967	+	1.71077i
$502$		0			0
$503$		−	4.50733e8i		−	0.157918i	−0.996878	−	0.0789589i	$-0.974840\pi$
$503$		−	4.50733e8i		−	0.157918i	0.996878	−	0.0789589i	$-0.0251596\pi$
$504$	−2.15801e9	+	5.16592e9i	−0.750838	+	1.79738i
$505$	5.00533e8			0.172947
$506$		0			0
$507$	−2.87721e9	−	5.76809e8i	−0.980491	−	0.196564i
$508$	1.43472e9			0.485561
$509$		−	3.54025e9i		−	1.18993i	−0.803751	−	0.594965i	$-0.797166\pi$
$509$		−	3.54025e9i		−	1.18993i	0.803751	−	0.594965i	$-0.202834\pi$
$510$		0			0
$511$		0			0
$512$			3.03700e9i			1.00000i
$513$		0			0
$514$		0			0
$515$		−	1.98480e8i		−	0.0640313i
$516$	6.06520e9	+	1.21592e9i	1.94344	+	0.389612i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			2.04957e9i			0.634937i	0.948269	+	0.317469i	$0.102833\pi$
$521$			2.04957e9i			0.634937i	−0.948269	+	0.317469i	$0.897167\pi$
$522$	3.26793e9	+	1.36514e9i	1.00560	+	0.420079i
$523$	1.51950e9			0.464456			0.232228	−	0.972661i	$-0.425399\pi$
$523$	1.51950e9			0.464456			0.232228	+	0.972661i	$0.425399\pi$
$524$		0			0
$525$	−6.33242e9	−	1.26949e9i	−1.90991	−	0.382889i
$526$	−2.35147e9			−0.704513
$527$		0			0
$528$		0			0
$529$	−6.67519e9			−1.96051
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	4.41638e9	+	8.85374e8i	1.25508	+	0.251613i
$535$	6.95224e9			1.96285
$536$			1.86249e9i			0.522417i
$537$		0			0
$538$	−6.20344e9			−1.71749
$539$		0			0
$540$	2.04660e9	−	3.03326e9i	0.559313	−	0.828956i
$541$	−2.92666e9			−0.794662			−0.397331	−	0.917675i	$-0.630064\pi$
$541$	−2.92666e9			−0.794662			−0.397331	+	0.917675i	$0.630064\pi$
$542$		0			0
$543$	3.64016e9	+	7.29761e8i	0.975711	+	0.195606i
$544$		0			0
$545$			1.29490e9i			0.342649i
$546$		0			0
$547$	3.49167e9			0.912174			0.456087	−	0.889935i	$-0.349251\pi$
$547$	3.49167e9			0.912174			0.456087	+	0.889935i	$0.349251\pi$
$548$		0			0
$549$	4.17001e9	+	1.74198e9i	1.07556	+	0.449303i
$550$		0			0
$551$		0			0
$552$	−6.66674e9	−	1.33651e9i	−1.68705	−	0.338210i
$553$		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
$560$			8.09518e9i			1.94791i
$561$		0			0
$562$	−7.04319e9			−1.67376
$563$			1.75407e9i			0.414255i	0.978314	+	0.207128i	$0.0664115\pi$
$563$			1.75407e9i			0.414255i	−0.978314	+	0.207128i	$0.933589\pi$
$564$	9.95473e8	−	4.96557e9i	0.233643	−	1.16545i
$565$		0			0
$566$			2.40917e9i			0.558484i
$567$	−5.94248e9	−	6.01436e9i	−1.36908	−	1.38563i
$568$		0			0
$569$		−	8.40161e9i		−	1.91192i	−0.293497	−	0.955960i	$-0.594819\pi$
$569$		−	8.40161e9i		−	1.91192i	0.293497	−	0.955960i	$-0.405181\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	1.06046e10			2.34047
$575$		−	7.84369e9i		−	1.72061i
$576$	−4.23205e9	−	1.76789e9i	−0.922725	−	0.385458i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		−	4.64245e9i		−	1.00000i
$579$		0			0
$580$	5.12096e9			1.08982
$581$			1.41833e10i			3.00028i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			9.71741e9i			1.98298i	0.130200	+	0.991488i	$0.458438\pi$
$587$			9.71741e9i			1.98298i	−0.130200	+	0.991488i	$0.541562\pi$
$588$	1.35066e10	+	2.70773e9i	2.73983	+	0.549268i
$589$		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
$596$		−	9.93090e9i		−	1.92144i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	1.04000e9	−	5.18768e9i	0.196564	−	0.980491i
$601$	−9.54806e9			−1.79413			−0.897066	−	0.441897i	$-0.854306\pi$
$601$	−9.54806e9			−1.79413			−0.897066	+	0.441897i	$0.854306\pi$
$602$		−	2.06673e10i		−	3.86097i
$603$	−2.59538e9	−	1.08419e9i	−0.482048	−	0.201370i
$604$		0			0
$605$			5.44683e9i			1.00000i
$606$	9.28988e8	+	1.86239e8i	0.169573	+	0.0339951i
$607$	−8.31530e9			−1.50910			−0.754549	−	0.656244i	$-0.772144\pi$
$607$	−8.31530e9			−1.50910			−0.754549	+	0.656244i	$0.772144\pi$
$608$		0			0
$609$	2.32588e9	−	1.16018e10i	0.417279	−	2.08145i
$610$	6.53456e9			1.16563
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			7.57464e9i			1.32060i
$615$	−6.79582e9	−	1.36239e9i	−1.17809	−	0.236178i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	7.38509e7	−	3.68380e8i	0.0125863	−	0.0627821i
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	5.74326e9	−	8.51207e9i	0.962361	−	1.42631i
$622$		0			0
$623$		−	1.50489e10i		−	2.49343i
$624$		0			0
$625$	6.10352e9			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−1.12806e10	−	4.71236e9i	−1.79738	−	0.750838i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			3.13294e9i			0.485561i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−6.63178e9			−1.00000
$641$		−	1.38497e9i		−	0.207701i	−0.994593	−	0.103850i	$-0.966884\pi$
$641$		−	1.38497e9i		−	0.207701i	0.994593	−	0.103850i	$-0.0331163\pi$
$642$	1.29034e10	+	2.58680e9i	1.92455	+	0.385825i
$643$	4.40809e9			0.653901			0.326950	−	0.945041i	$-0.393979\pi$
$643$	4.40809e9			0.653901			0.326950	+	0.945041i	$0.393979\pi$
$644$			2.27171e10i			3.35160i
$645$	−2.65516e9	+	1.32443e10i	−0.389612	+	1.94344i
$646$		0			0
$647$			7.24027e9i			1.05097i	0.850803	+	0.525485i	$0.176116\pi$
$647$			7.24027e9i			1.05097i	−0.850803	+	0.525485i	$0.823884\pi$
$648$	4.92711e9	−	4.86823e9i	0.711344	−	0.702844i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.23451e10			−1.74433
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$	−4.81810e8	+	2.40334e9i	−0.0673524	+	0.335964i
$655$		0			0
$656$			8.68758e9i			1.20153i
$657$		0			0
$658$	−1.69203e10			−2.31535
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.48495e10			1.99989			0.999946	−	0.0103968i	$-0.00330946\pi$
$661$	1.48495e10			1.99989			0.999946	+	0.0103968i	$0.00330946\pi$
$662$		0			0
$663$		0			0
$664$	−1.16193e10			−1.54026
$665$		0			0
$666$		0			0
$667$	1.43707e10			1.87515
$668$			1.34421e10i			1.74481i
$669$	−1.02689e10	−	2.05866e9i	−1.32597	−	0.265823i
$670$	−4.06705e9			−0.522417
$671$		0			0
$672$	−3.01207e9	+	1.50247e10i	−0.382889	+	1.90991i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	6.62361e9	+	4.46908e9i	0.828956	+	0.559313i
$676$	−8.03181e9			−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
$681$	2.32062e9	−	1.15756e10i	0.281572	−	1.40452i
$682$		0			0
$683$			1.04947e10i			1.26037i	0.776443	+	0.630187i	$0.217022\pi$
$683$			1.04947e10i			1.26037i	−0.776443	+	0.630187i	$0.782978\pi$
$684$		0			0
$685$		0			0
$686$		−	2.95536e10i		−	3.49523i
$687$	−3.71291e9	−	7.44346e8i	−0.436884	−	0.0875843i
$688$	1.69312e10			1.98211
$689$		0			0
$690$	2.91849e9	−	1.45579e10i	0.338210	−	1.68705i
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−1.72684e10			−1.96107
$695$		0			0
$696$	9.50450e9	+	1.90542e9i	1.06856	+	0.214219i
$697$		0			0
$698$			1.40278e10i			1.56133i
$699$		0			0
$700$	−1.76771e10			−1.94791
$701$		−	1.63787e9i		−	0.179583i	−0.995961	−	0.0897917i	$-0.971380\pi$
$701$		−	1.63787e9i		−	0.179583i	0.995961	−	0.0897917i	$-0.0286201\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	1.08431e10	+	2.17377e9i	1.16545	+	0.233643i
$706$		0			0
$707$		−	3.16555e9i		−	0.336885i
$708$		0			0
$709$	1.61323e10			1.69994			0.849969	−	0.526833i	$-0.176621\pi$
$709$	1.61323e10			1.69994			0.849969	+	0.526833i	$0.176621\pi$
$710$		0			0
$711$		0			0
$712$	1.23284e10			1.28005
$713$		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	$0.5\pi$
$720$	3.86048e9	−	9.24136e9i	0.385458	−	0.922725i
$721$	−1.25526e9			−0.124727
$722$			1.01130e10i			1.00000i
$723$	−1.99258e10	−	3.99463e9i	−1.96080	−	0.393091i
$724$	1.01616e10			0.995125
$725$			1.11824e10i			1.08982i
$726$	−2.02667e9	+	1.01093e10i	−0.196564	+	0.980491i
$727$	1.76507e10			1.70369			0.851846	−	0.523793i	$-0.175483\pi$
$727$	1.76507e10			1.70369			0.851846	+	0.523793i	$0.175483\pi$
$728$		0			0
$729$	3.91571e9	+	9.69981e9i	0.374338	+	0.927292i
$730$		0			0
$731$		0			0
$732$	1.21281e10	+	2.43139e9i	1.14289	+	0.229121i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$			1.17140e10i			1.09337i
$735$	−5.91276e9	+	2.94938e10i	−0.549268	+	2.73983i
$736$	−1.86104e10			−1.72061
$737$		0			0
$738$	−1.21061e10	−	5.05720e9i	−1.10868	−	0.463141i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.00514e10i		−	0.899010i	−0.893278	−	0.449505i	$-0.851600\pi$
$743$		−	1.00514e10i		−	0.899010i	0.893278	−	0.449505i	$-0.148400\pi$
$744$		0			0
$745$	2.16857e10			1.92144
$746$		0			0
$747$	6.76383e9	−	1.61915e10i	0.593705	−	1.42123i
$748$		0			0
$749$		−	4.39685e10i		−	3.82345i
$750$	1.13281e10	+	2.27101e9i	0.980491	+	0.196564i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	1.38615e10i		−	1.18864i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.91834e10	−	1.29434e10i	−1.61473	−	1.08949i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			2.36939e10i			1.94891i	0.224589	+	0.974453i	$0.427896\pi$
$761$			2.36939e10i			1.94891i	−0.224589	+	0.974453i	$0.572104\pi$
$762$	−1.16571e9	+	5.81473e9i	−0.0954438	+	0.476088i
$763$	8.18943e9			0.667449
$764$		0			0
$765$		0			0
$766$	2.07522e10			1.66826
$767$		0			0
$768$	−1.23086e10	−	2.46756e9i	−0.980491	−	0.196564i
$769$	−2.46381e10			−1.95373			−0.976867	−	0.213849i	$-0.931400\pi$
$769$	−2.46381e10			−1.95373			−0.976867	+	0.213849i	$0.931400\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	−9.85596e9	+	2.35936e10i	−0.764021	+	1.82894i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−3.50639e9			−0.266951
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−8.18794e9	+	1.21353e10i	−0.609549	+	0.903411i
$784$	3.77040e10			2.79435
$785$		0			0
$786$		0			0
$787$	−1.37602e10			−1.00627			−0.503135	−	0.864208i	$-0.667820\pi$
$787$	−1.37602e10			−1.00627			−0.503135	+	0.864208i	$0.667820\pi$
$788$		0			0
$789$	1.91057e9	−	9.53020e9i	0.138482	−	0.690768i
$790$		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
$800$		−	1.44815e10i		−	1.00000i
$801$	−7.17662e9	+	1.71797e10i	−0.493408	+	1.18114i
$802$	2.90024e10			1.98529
$803$		0			0
$804$	−7.54844e9	−	1.51327e9i	−0.512226	−	0.102688i
$805$	−4.96064e10			−3.35160
$806$		0			0
$807$	5.04029e9	−	2.51417e10i	0.337597	−	1.68398i
$808$	2.59330e9			0.172947
$809$			2.36375e10i			1.56957i	0.619767	+	0.784786i	$0.287227\pi$
$809$			2.36375e10i			1.56957i	−0.619767	+	0.784786i	$0.712773\pi$
$810$	1.06306e10	+	1.07591e10i	0.702844	+	0.711344i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	3.23868e10i		−	2.12286i
$813$		0			0
$814$		0			0
$815$		−	2.69575e10i		−	1.74433i
$816$		0			0
$817$		0			0
$818$			2.00109e10i			1.27829i
$819$		0			0
$820$	−1.89707e10			−1.20153
$821$		−	2.49157e10i		−	1.57135i	−0.618642	−	0.785673i	$-0.712317\pi$
$821$		−	2.49157e10i		−	1.57135i	0.618642	−	0.785673i	$-0.287683\pi$
$822$		0			0
$823$	−2.96602e10			−1.85471			−0.927353	−	0.374187i	$-0.877922\pi$
$823$	−2.96602e10			−1.85471			−0.927353	+	0.374187i	$0.877922\pi$
$824$		−	1.02834e9i		−	0.0640313i
$825$		0			0
$826$		0			0
$827$			2.84734e10i			1.75053i	0.483641	+	0.875267i	$0.339314\pi$
$827$			2.84734e10i			1.75053i	−0.483641	+	0.875267i	$0.660686\pi$
$828$	1.08335e10	−	2.59335e10i	0.663225	−	1.58765i
$829$	−2.46576e9			−0.150318			−0.0751589	−	0.997172i	$-0.523946\pi$
$829$	−2.46576e9			−0.150318			−0.0751589	+	0.997172i	$0.523946\pi$
$830$		−	2.53727e10i		−	1.54026i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−2.93529e10			−1.74481
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	−3.28088e10	−	6.57734e9i	−1.90991	−	0.382889i
$841$	−3.23780e9			−0.187700
$842$		−	3.39710e10i		−	1.96117i
$843$	5.72259e9	−	2.85451e10i	0.329000	−	1.64110i
$844$		0			0
$845$		−	1.75387e10i		−	1.00000i
$846$	1.93160e10	+	8.06905e9i	1.09678	+	0.458170i
$847$	3.44477e10			1.94791
$848$		0			0
$849$	−9.76408e9	−	1.95745e9i	−0.547588	−	0.109778i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		−	4.13269e10i		−	2.27055i
$855$		0			0
$856$	3.60201e10			1.96285
$857$		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			$\pi$
$860$			3.69720e10i			1.98211i
$861$	−8.61626e9	+	4.29792e10i	−0.460053	+	2.29481i
$862$		0			0
$863$			1.59327e10i			0.843822i	0.906637	+	0.421911i	$0.138641\pi$
$863$			1.59327e10i			0.843822i	−0.906637	+	0.421911i	$0.861359\pi$
$864$	1.06036e10	−	1.57156e10i	0.559313	−	0.828956i
$865$		0			0
$866$		0			0
$867$	1.88153e10	+	3.77199e9i	0.980491	+	0.196564i
$868$		0			0
$869$		0			0
$870$	−4.16078e9	+	2.07546e10i	−0.214219	+	1.06856i
$871$		0			0
$872$			6.70899e9i			0.342649i
$873$		0			0
$874$		0			0
$875$		−	3.86008e10i		−	1.94791i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.90394e9i		−	0.290888i	−0.989366	−	0.145444i	$-0.953539\pi$
$881$		−	5.90394e9i		−	0.290888i	0.989366	−	0.145444i	$-0.0464611\pi$
$882$	−2.19482e10	+	5.25404e10i	−1.07711	+	2.57842i
$883$	1.85278e10			0.905652			0.452826	−	0.891599i	$-0.350416\pi$
$883$	1.85278e10			0.905652			0.452826	+	0.891599i	$0.350416\pi$
$884$		0			0
$885$		0			0
$886$	8.46175e8			0.0408735
$887$			3.91121e10i			1.88182i	0.338656	+	0.940910i	$0.390028\pi$
$887$			3.91121e10i			1.88182i	−0.338656	+	0.940910i	$0.609972\pi$
$888$		0			0
$889$	1.98138e10			0.945828
$890$			2.69211e10i			1.28005i
$891$		0			0
$892$	−2.86659e10			−1.35235
$893$		0			0
$894$	4.02487e10	+	8.06885e9i	1.88395	+	0.377686i
$895$		0			0
$896$			4.19418e10i			1.94791i
$897$		0			0
$898$	−2.95547e10			−1.36194
$899$		0			0
$900$	2.01800e10	+	8.42998e9i	0.922725	+	0.385458i
$901$		0			0
$902$		0			0
$903$	8.37620e10	+	1.67922e10i	3.78565	+	0.758928i
$904$		0			0
$905$			2.21895e10i			0.995125i
$906$		0			0
$907$	−3.95805e10			−1.76139			−0.880695	−	0.473684i	$-0.842924\pi$
$907$	−3.95805e10			−1.76139			−0.880695	+	0.473684i	$0.842924\pi$
$908$		−	3.23137e10i		−	1.43247i
$909$	−1.50961e9	+	3.61375e9i	−0.0666638	+	0.159582i
$910$		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
$915$	−5.30933e9	+	2.64837e10i	−0.229121	+	1.14289i
$916$	−1.03647e10			−0.445577
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		−	4.06388e10i		−	1.72061i
$921$	−3.06991e10	−	6.15440e9i	−1.29484	−	0.259583i
$922$	4.73504e10			1.98960
$923$		0			0
$924$		0			0
$925$		0			0
$926$			2.44693e10i			1.01270i
$927$	1.43299e9	+	5.98617e8i	0.0590833	+	0.0246814i
$928$	2.65321e10			1.08982
$929$		−	1.39812e10i		−	0.572122i	−0.958211	−	0.286061i	$-0.907654\pi$
$929$		−	1.39812e10i		−	0.572122i	0.958211	−	0.286061i	$-0.0923460\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−1.56354e10			−0.627907
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$			2.57215e10i			1.01762i
$939$		0			0
$940$	3.02689e10			1.18864
$941$		−	2.74055e10i		−	1.07219i	−0.844156	−	0.536097i	$-0.819898\pi$
$941$		−	2.74055e10i		−	1.07219i	0.844156	−	0.536097i	$-0.180102\pi$
$942$		0			0
$943$	−5.32365e10			−2.06737
$944$		0			0
$945$	2.82641e10	−	4.18901e10i	1.08949	−	1.61473i
$946$		0			0
$947$		−	2.44841e10i		−	0.936828i	−0.883509	−	0.468414i	$-0.844825\pi$
$947$		−	2.44841e10i		−	0.936828i	0.883509	−	0.468414i	$-0.155175\pi$
$948$		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
$960$	5.38832e9	−	2.68778e10i	0.196564	−	0.980491i
$961$	2.75126e10			1.00000
$962$		0			0
$963$	−2.09679e10	+	5.01939e10i	−0.756596	+	1.81117i
$964$	−5.56235e10			−1.99981
$965$		0			0
$966$	−9.20694e10	−	1.84576e10i	−3.28621	−	0.658803i
$967$	−5.29679e10			−1.88374			−0.941869	−	0.335981i	$-0.890932\pi$
$967$	−5.29679e10			−1.88374			−0.941869	+	0.335981i	$0.890932\pi$
$968$			2.82204e10i			1.00000i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	1.57270e10	+	2.39244e10i	0.549307	+	0.835621i
$973$		0			0
$974$		−	2.94563e10i		−	1.02146i
$975$		0			0
$976$	3.38560e10			1.16563
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	1.00304e10	−	5.00331e10i	0.342872	−	1.71030i
$979$		0			0
$980$			8.23326e10i			2.79435i
$981$	−9.34896e9	−	3.90543e9i	−0.316171	−	0.132077i
$982$		0			0
$983$			5.72745e10i			1.92320i	0.274460	+	0.961599i	$0.411501\pi$
$983$			5.72745e10i			1.92320i	−0.274460	+	0.961599i	$0.588499\pi$
$984$	−3.52097e10	−	7.05866e9i	−1.17809	−	0.236178i
$985$		0			0
$986$		0			0
$987$	1.37477e10	−	6.85758e10i	0.455115	−	2.27018i
$988$		0			0
$989$			1.03752e11i			3.41044i
$990$		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
$996$	9.44071e9	−	4.70917e10i	0.302759	−	1.51021i
$997$		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	60.8.h.a.59.3	yes	4
3.2	odd	2	inner	60.8.h.a.59.1	✓	4
4.3	odd	2	inner	60.8.h.a.59.2	yes	4
5.4	even	2	inner	60.8.h.a.59.2	yes	4
12.11	even	2	inner	60.8.h.a.59.4	yes	4
15.14	odd	2	inner	60.8.h.a.59.4	yes	4
20.19	odd	2	CM	60.8.h.a.59.3	yes	4
60.59	even	2	inner	60.8.h.a.59.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.8.h.a.59.1	✓	4	3.2	odd	2	inner
60.8.h.a.59.1	✓	4	60.59	even	2	inner
60.8.h.a.59.2	yes	4	4.3	odd	2	inner
60.8.h.a.59.2	yes	4	5.4	even	2	inner
60.8.h.a.59.3	yes	4	1.1	even	1	trivial
60.8.h.a.59.3	yes	4	20.19	odd	2	CM
60.8.h.a.59.4	yes	4	12.11	even	2	inner
60.8.h.a.59.4	yes	4	15.14	odd	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	60.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.3137i q^{2} +(-45.8530 - 9.19239i) q^{3} -128.000 q^{4} -279.508i q^{5} +(104.000 - 518.768i) q^{6} -1767.71 q^{7} -1448.15i q^{8} +(2018.00 + 842.998i) q^{9} +O(q^{10})\)	\(q+11.3137i q^{2} +(-45.8530 - 9.19239i) q^{3} -128.000 q^{4} -279.508i q^{5} +(104.000 - 518.768i) q^{6} -1767.71 q^{7} -1448.15i q^{8} +(2018.00 + 842.998i) q^{9} +3162.28 q^{10} +(5869.19 + 1176.63i) q^{12} -19999.4i q^{14} +(-2569.35 + 12816.3i) q^{15} +16384.0 q^{16} +(-9537.43 + 22831.1i) q^{18} +35777.1i q^{20} +(81055.0 + 16249.5i) q^{21} +100399. i q^{23} +(-13312.0 + 66402.3i) q^{24} -78125.0 q^{25} +(-84782.2 - 57204.2i) q^{27} +226267. q^{28} -143135. i q^{29} +(-145000. - 29068.9i) q^{30} +185364. i q^{32} +494091. i q^{35} +(-258304. - 107904. i) q^{36} -404772. q^{40} +530248. i q^{41} +(-183842. + 917033. i) q^{42} +1.03340e6 q^{43} +(235625. - 564048. i) q^{45} -1.13589e6 q^{46} -846041. i q^{47} +(-751256. - 150608. i) q^{48} +2.30127e6 q^{49} -883883. i q^{50} +(647192. - 959202. i) q^{54} +2.55992e6i q^{56} +1.61939e6 q^{58} +(328877. - 1.64049e6i) q^{60} +2.06641e6 q^{61} +(-3.56725e6 - 1.49018e6i) q^{63} -2.09715e6 q^{64} -1.28611e6 q^{67} +(922909. - 4.60361e6i) q^{69} -5.59000e6 q^{70} +(1.22079e6 - 2.92238e6i) q^{72} +(3.58227e6 + 718155. i) q^{75} -4.57947e6i q^{80} +(3.36168e6 + 3.40234e6i) q^{81} -5.99907e6 q^{82} -8.02354e6i q^{83} +(-1.03750e7 - 2.07994e6i) q^{84} +1.16916e7i q^{86} +(-1.31575e6 + 6.56318e6i) q^{87} +8.51321e6i q^{89} +(6.38148e6 + 2.66579e6i) q^{90} -1.28511e7i q^{92} +9.57186e6 q^{94} +(1.70394e6 - 8.49949e6i) q^{96} +2.60359e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 512 q^{4} + 416 q^{6} + 8072 q^{9}+O(q^{10}) \) 4 * q - 512 * q^4 + 416 * q^6 + 8072 * q^9	\( 4 q - 512 q^{4} + 416 q^{6} + 8072 q^{9} + 65536 q^{16} + 324220 q^{21} - 53248 q^{24} - 312500 q^{25} - 580000 q^{30} - 1033216 q^{36} + 942500 q^{45} - 4543552 q^{46} + 9205068 q^{49} + 2588768 q^{54} + 8265632 q^{61} - 8388608 q^{64} + 3691636 q^{69} - 22360000 q^{70} + 13446716 q^{81} - 41500160 q^{84} + 38287424 q^{94} + 6815744 q^{96}+O(q^{100}) \) 4 * q - 512 * q^4 + 416 * q^6 + 8072 * q^9 + 65536 * q^16 + 324220 * q^21 - 53248 * q^24 - 312500 * q^25 - 580000 * q^30 - 1033216 * q^36 + 942500 * q^45 - 4543552 * q^46 + 9205068 * q^49 + 2588768 * q^54 + 8265632 * q^61 - 8388608 * q^64 + 3691636 * q^69 - 22360000 * q^70 + 13446716 * q^81 - 41500160 * q^84 + 38287424 * q^94 + 6815744 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more