LMFDB - Embedded newform 240.12.f.b.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,12,Mod(49,240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 12, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("240.49"); S:= CuspForms(chi, 12); N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	240.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$184.402363334$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 350078x^{4} + 30638651521x^{2} + 173683668788100 $ x^6 + 350078x^4 + 30638651521x^2 + 173683668788100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$78.0027i$ of defining polynomial
Character	$\chi$	$=$	240.49
Dual form			240.12.f.b.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-243.000i q^{3} +(-6709.81 + 1951.05i) q^{5} +52907.3i q^{7} -59049.0 q^{9} -531122. q^{11} +373795. i q^{13} +(474106. + 1.63048e6i) q^{15} -4.81399e6i q^{17} +5.80222e6 q^{19} +1.28565e7 q^{21} -3.32148e7i q^{23} +(4.12149e7 - 2.61824e7i) q^{25} +1.43489e7i q^{27} +1.30307e8 q^{29} -6.18143e7 q^{31} +1.29063e8i q^{33} +(-1.03225e8 - 3.54997e8i) q^{35} -1.37358e8i q^{37} +9.08322e7 q^{39} +1.46886e9 q^{41} +5.30160e8i q^{43} +(3.96207e8 - 1.15208e8i) q^{45} +1.34319e9i q^{47} -8.21851e8 q^{49} -1.16980e9 q^{51} -4.74535e9i q^{53} +(3.56372e9 - 1.03625e9i) q^{55} -1.40994e9i q^{57} +4.37861e9 q^{59} -7.78414e9 q^{61} -3.12412e9i q^{63} +(-7.29295e8 - 2.50809e9i) q^{65} +4.98481e9i q^{67} -8.07119e9 q^{69} -9.09417e9 q^{71} -1.88613e10i q^{73} +(-6.36232e9 - 1.00152e10i) q^{75} -2.81002e10i q^{77} -4.50312e10 q^{79} +3.48678e9 q^{81} +6.88856e10i q^{83} +(9.39236e9 + 3.23009e10i) q^{85} -3.16645e10i q^{87} +3.49597e9 q^{89} -1.97765e10 q^{91} +1.50209e10i q^{93} +(-3.89318e10 + 1.13205e10i) q^{95} +1.62848e11i q^{97} +3.13622e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 9926 q^{5} - 354294 q^{9} - 1753400 q^{11} - 551124 q^{15} + 4069824 q^{19} - 32748624 q^{21} + 169978326 q^{25} + 71715564 q^{29} + 122243352 q^{31} - 825571904 q^{35} - 153183312 q^{39} + 2006768564 q^{41}+ \cdots + 103536516600 q^{99}+O(q^{100}) $ 6 * q - 9926 * q^5 - 354294 * q^9 - 1753400 * q^11 - 551124 * q^15 + 4069824 * q^19 - 32748624 * q^21 + 169978326 * q^25 + 71715564 * q^29 + 122243352 * q^31 - 825571904 * q^35 - 153183312 * q^39 + 2006768564 * q^41 + 586120374 * q^45 - 9590905446 * q^49 + 1396921464 * q^51 - 9162341400 * q^55 + 4069460200 * q^59 - 12386466660 * q^61 - 30310342152 * q^65 + 24235021800 * q^69 - 13703505600 * q^71 + 22735606824 * q^75 + 13438327752 * q^79 + 20920706406 * q^81 + 50595843144 * q^85 - 127436589092 * q^89 - 271843383552 * q^91 + 87931077696 * q^95 + 103536516600 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	243.000i		−	0.577350i
$3$		−	243.000i		−	0.577350i
$4$		0			0
$5$	−6709.81	+	1951.05i	−0.960229	+	0.279212i
$5$	−6709.81	+	1951.05i	−0.960229	+	0.279212i
$6$		0			0
$7$			52907.3i			1.18981i	0.803798	+	0.594903i	$0.202809\pi$
$7$			52907.3i			1.18981i	−0.803798	+	0.594903i	$0.797191\pi$
$8$		0			0
$9$	−59049.0			−0.333333
$10$		0			0
$11$	−531122.			−0.994339			−0.497169	−	0.867654i	$-0.665627\pi$
$11$	−531122.			−0.994339			−0.497169	+	0.867654i	$0.665627\pi$
$12$		0			0
$13$			373795.i			0.279219i	0.990207	+	0.139610i	$0.0445848\pi$
$13$			373795.i			0.279219i	−0.990207	+	0.139610i	$0.955415\pi$
$14$		0			0
$15$	474106.	+	1.63048e6i	0.161203	+	0.554389i
$16$		0			0
$17$		−	4.81399e6i		−	0.822311i	−0.911565	−	0.411156i	$-0.865125\pi$
$17$		−	4.81399e6i		−	0.822311i	0.911565	−	0.411156i	$-0.134875\pi$
$18$		0			0
$19$	5.80222e6			0.537588			0.268794	−	0.963198i	$-0.413375\pi$
$19$	5.80222e6			0.537588			0.268794	+	0.963198i	$0.413375\pi$
$20$		0			0
$21$	1.28565e7			0.686935
$22$		0			0
$23$		−	3.32148e7i		−	1.07604i	−0.842932	−	0.538020i	$-0.819173\pi$
$23$		−	3.32148e7i		−	1.07604i	0.842932	−	0.538020i	$-0.180827\pi$
$24$		0			0
$25$	4.12149e7	−	2.61824e7i	0.844081	−	0.536216i
$26$		0			0
$27$			1.43489e7i			0.192450i
$28$		0			0
$29$	1.30307e8			1.17972			0.589859	−	0.807507i	$-0.299184\pi$
$29$	1.30307e8			1.17972			0.589859	+	0.807507i	$0.299184\pi$
$30$		0			0
$31$	−6.18143e7			−0.387793			−0.193896	−	0.981022i	$-0.562113\pi$
$31$	−6.18143e7			−0.387793			−0.193896	+	0.981022i	$0.562113\pi$
$32$		0			0
$33$			1.29063e8i			0.574082i
$34$		0			0
$35$	−1.03225e8	−	3.54997e8i	−0.332208	−	1.14249i
$36$		0			0
$37$		−	1.37358e8i		−	0.325645i	−0.986655	−	0.162822i	$-0.947940\pi$
$37$		−	1.37358e8i		−	0.325645i	0.986655	−	0.162822i	$-0.0520598\pi$
$38$		0			0
$39$	9.08322e7			0.161207
$40$		0			0
$41$	1.46886e9			1.98002			0.990009	−	0.141003i	$-0.0450326\pi$
$41$	1.46886e9			1.98002			0.990009	+	0.141003i	$0.0450326\pi$
$42$		0			0
$43$			5.30160e8i			0.549960i	0.961450	+	0.274980i	$0.0886712\pi$
$43$			5.30160e8i			0.549960i	−0.961450	+	0.274980i	$0.911329\pi$
$44$		0			0
$45$	3.96207e8	−	1.15208e8i	0.320076	−	0.0930708i
$46$		0			0
$47$			1.34319e9i			0.854277i	0.904186	+	0.427138i	$0.140478\pi$
$47$			1.34319e9i			0.854277i	−0.904186	+	0.427138i	$0.859522\pi$
$48$		0			0
$49$	−8.21851e8			−0.415637
$50$		0			0
$51$	−1.16980e9			−0.474762
$52$		0			0
$53$		−	4.74535e9i		−	1.55866i	−0.626615	−	0.779329i	$-0.715560\pi$
$53$		−	4.74535e9i		−	1.55866i	0.626615	−	0.779329i	$-0.284440\pi$
$54$		0			0
$55$	3.56372e9	−	1.03625e9i	0.954793	−	0.277632i
$56$		0			0
$57$		−	1.40994e9i		−	0.310377i
$58$		0			0
$59$	4.37861e9			0.797352			0.398676	−	0.917092i	$-0.369470\pi$
$59$	4.37861e9			0.797352			0.398676	+	0.917092i	$0.369470\pi$
$60$		0			0
$61$	−7.78414e9			−1.18004			−0.590020	−	0.807389i	$-0.700880\pi$
$61$	−7.78414e9			−1.18004			−0.590020	+	0.807389i	$0.700880\pi$
$62$		0			0
$63$		−	3.12412e9i		−	0.396602i
$64$		0			0
$65$	−7.29295e8	−	2.50809e9i	−0.0779614	−	0.268114i
$66$		0			0
$67$			4.98481e9i			0.451063i	0.974236	+	0.225531i	$0.0724118\pi$
$67$			4.98481e9i			0.451063i	−0.974236	+	0.225531i	$0.927588\pi$
$68$		0			0
$69$	−8.07119e9			−0.621252
$70$		0			0
$71$	−9.09417e9			−0.598194			−0.299097	−	0.954223i	$-0.596685\pi$
$71$	−9.09417e9			−0.598194			−0.299097	+	0.954223i	$0.596685\pi$
$72$		0			0
$73$		−	1.88613e10i		−	1.06487i	−0.846471	−	0.532435i	$-0.821277\pi$
$73$		−	1.88613e10i		−	1.06487i	0.846471	−	0.532435i	$-0.178723\pi$
$74$		0			0
$75$	−6.36232e9	−	1.00152e10i	−0.309584	−	0.487330i
$76$		0			0
$77$		−	2.81002e10i		−	1.18307i
$78$		0			0
$79$	−4.50312e10			−1.64651			−0.823255	−	0.567672i	$-0.807844\pi$
$79$	−4.50312e10			−1.64651			−0.823255	+	0.567672i	$0.807844\pi$
$80$		0			0
$81$	3.48678e9			0.111111
$82$		0			0
$83$			6.88856e10i			1.91955i	0.280773	+	0.959774i	$0.409409\pi$
$83$			6.88856e10i			1.91955i	−0.280773	+	0.959774i	$0.590591\pi$
$84$		0			0
$85$	9.39236e9	+	3.23009e10i	0.229599	+	0.789607i
$86$		0			0
$87$		−	3.16645e10i		−	0.681110i
$88$		0			0
$89$	3.49597e9			0.0663624			0.0331812	−	0.999449i	$-0.489436\pi$
$89$	3.49597e9			0.0663624			0.0331812	+	0.999449i	$0.489436\pi$
$90$		0			0
$91$	−1.97765e10			−0.332217
$92$		0			0
$93$			1.50209e10i			0.223892i
$94$		0			0
$95$	−3.89318e10	+	1.13205e10i	−0.516208	+	0.150101i
$96$		0			0
$97$			1.62848e11i			1.92548i	0.270437	+	0.962738i	$0.412832\pi$
$97$			1.62848e11i			1.92548i	−0.270437	+	0.962738i	$0.587168\pi$
$98$		0			0
$99$	3.13622e10			0.331446
$100$		0			0
$101$	1.33281e11			1.26183			0.630915	−	0.775852i	$-0.282680\pi$
$101$	1.33281e11			1.26183			0.630915	+	0.775852i	$0.282680\pi$
$102$		0			0
$103$		−	8.49052e10i		−	0.721655i	−0.932633	−	0.360827i	$-0.882494\pi$
$103$		−	8.49052e10i		−	0.721655i	0.932633	−	0.360827i	$-0.117506\pi$
$104$		0			0
$105$	−8.62644e10	+	2.50837e10i	−0.659615	+	0.191801i
$106$		0			0
$107$			4.19151e10i			0.288908i	0.989512	+	0.144454i	$0.0461425\pi$
$107$			4.19151e10i			0.288908i	−0.989512	+	0.144454i	$0.953857\pi$
$108$		0			0
$109$	−2.67212e11			−1.66345			−0.831727	−	0.555185i	$-0.812647\pi$
$109$	−2.67212e11			−1.66345			−0.831727	+	0.555185i	$0.812647\pi$
$110$		0			0
$111$	−3.33780e10			−0.188011
$112$		0			0
$113$		−	2.37749e11i		−	1.21391i	−0.794735	−	0.606957i	$-0.792390\pi$
$113$		−	2.37749e11i		−	1.21391i	0.794735	−	0.606957i	$-0.207610\pi$
$114$		0			0
$115$	6.48038e10	+	2.22865e11i	0.300443	+	1.03324i
$116$		0			0
$117$		−	2.20722e10i		−	0.0930731i
$118$		0			0
$119$	2.54695e11			0.978390
$120$		0			0
$121$	−3.22132e9			−0.0112905
$122$		0			0
$123$		−	3.56933e11i		−	1.14316i
$124$		0			0
$125$	−2.25461e11	+	2.56091e11i	−0.660793	+	0.750568i
$126$		0			0
$127$		−	7.15504e10i		−	0.192173i	−0.995373	−	0.0960863i	$-0.969368\pi$
$127$		−	7.15504e10i		−	0.192173i	0.995373	−	0.0960863i	$-0.0306325\pi$
$128$		0			0
$129$	1.28829e11			0.317519
$130$		0			0
$131$	−4.56783e11			−1.03447			−0.517235	−	0.855844i	$-0.673039\pi$
$131$	−4.56783e11			−1.03447			−0.517235	+	0.855844i	$0.673039\pi$
$132$		0			0
$133$			3.06980e11i			0.639626i
$134$		0			0
$135$	−2.79955e10	−	9.62784e10i	−0.0537344	−	0.184796i
$136$		0			0
$137$		−	2.86655e10i		−	0.0507454i	−0.999678	−	0.0253727i	$-0.991923\pi$
$137$		−	2.86655e10i		−	0.0507454i	0.999678	−	0.0253727i	$-0.00807725\pi$
$138$		0			0
$139$	1.20093e12			1.96307			0.981536	−	0.191280i	$-0.0612638\pi$
$139$	1.20093e12			1.96307			0.981536	+	0.191280i	$0.0612638\pi$
$140$		0			0
$141$	3.26395e11			0.493217
$142$		0			0
$143$		−	1.98531e11i		−	0.277638i
$144$		0			0
$145$	−8.74333e11	+	2.54236e11i	−1.13280	+	0.329391i
$146$		0			0
$147$			1.99710e11i			0.239968i
$148$		0			0
$149$	6.64637e11			0.741412			0.370706	−	0.928750i	$-0.379116\pi$
$149$	6.64637e11			0.741412			0.370706	+	0.928750i	$0.379116\pi$
$150$		0			0
$151$	−1.70963e12			−1.77226			−0.886130	−	0.463436i	$-0.846616\pi$
$151$	−1.70963e12			−1.77226			−0.886130	+	0.463436i	$0.846616\pi$
$152$		0			0
$153$			2.84261e11i			0.274104i
$154$		0			0
$155$	4.14762e11	−	1.20603e11i	0.372370	−	0.108276i
$156$		0			0
$157$		−	9.96231e11i		−	0.833512i	−0.909018	−	0.416756i	$-0.863167\pi$
$157$		−	9.96231e11i		−	0.833512i	0.909018	−	0.416756i	$-0.136833\pi$
$158$		0			0
$159$	−1.15312e12			−0.899892
$160$		0			0
$161$	1.75730e12			1.28028
$162$		0			0
$163$			2.56482e11i			0.174592i	0.996182	+	0.0872960i	$0.0278226\pi$
$163$			2.56482e11i			0.174592i	−0.996182	+	0.0872960i	$0.972177\pi$
$164$		0			0
$165$	−2.51808e11	−	8.65985e11i	−0.160291	−	0.551250i
$166$		0			0
$167$			1.02735e12i			0.612040i	0.952025	+	0.306020i	$0.0989974\pi$
$167$			1.02735e12i			0.612040i	−0.952025	+	0.306020i	$0.901003\pi$
$168$		0			0
$169$	1.65244e12			0.922037
$170$		0			0
$171$	−3.42616e11			−0.179196
$172$		0			0
$173$			3.51424e12i			1.72416i	0.506769	+	0.862082i	$0.330840\pi$
$173$			3.51424e12i			1.72416i	−0.506769	+	0.862082i	$0.669160\pi$
$174$		0			0
$175$	1.38524e12	+	2.18057e12i	0.637992	+	1.00429i
$176$		0			0
$177$		−	1.06400e12i		−	0.460352i
$178$		0			0
$179$	4.59480e12			1.86885			0.934427	−	0.356156i	$-0.115913\pi$
$179$	4.59480e12			1.86885			0.934427	+	0.356156i	$0.115913\pi$
$180$		0			0
$181$	−9.54953e11			−0.365384			−0.182692	−	0.983170i	$-0.558481\pi$
$181$	−9.54953e11			−0.365384			−0.182692	+	0.983170i	$0.558481\pi$
$182$		0			0
$183$			1.89155e12i			0.681297i
$184$		0			0
$185$	2.67993e11	+	9.21645e11i	0.0909240	+	0.312694i
$186$		0			0
$187$			2.55682e12i			0.817656i
$188$		0			0
$189$	−7.59161e11			−0.228978
$190$		0			0
$191$	−4.18247e12			−1.19055			−0.595277	−	0.803520i	$-0.702958\pi$
$191$	−4.18247e12			−1.19055			−0.595277	+	0.803520i	$0.702958\pi$
$192$		0			0
$193$		−	3.35128e12i		−	0.900836i	−0.892818	−	0.450418i	$-0.851275\pi$
$193$		−	3.35128e12i		−	0.900836i	0.892818	−	0.450418i	$-0.148725\pi$
$194$		0			0
$195$	−6.09467e11	+	1.77219e11i	−0.154796	+	0.0450110i
$196$		0			0
$197$			7.22875e12i			1.73580i	0.496742	+	0.867899i	$0.334530\pi$
$197$			7.22875e12i			1.73580i	−0.496742	+	0.867899i	$0.665470\pi$
$198$		0			0
$199$	1.05138e12			0.238818			0.119409	−	0.992845i	$-0.461900\pi$
$199$	1.05138e12			0.238818			0.119409	+	0.992845i	$0.461900\pi$
$200$		0			0
$201$	1.21131e12			0.260421
$202$		0			0
$203$			6.89417e12i			1.40363i
$204$		0			0
$205$	−9.85577e12	+	2.86583e12i	−1.90127	+	0.552845i
$206$		0			0
$207$			1.96130e12i			0.358680i
$208$		0			0
$209$	−3.08169e12			−0.534545
$210$		0			0
$211$	−3.80994e11			−0.0627141			−0.0313570	−	0.999508i	$-0.509983\pi$
$211$	−3.80994e11			−0.0627141			−0.0313570	+	0.999508i	$0.509983\pi$
$212$		0			0
$213$			2.20988e12i			0.345368i
$214$		0			0
$215$	−1.03437e12	−	3.55727e12i	−0.153555	−	0.528087i
$216$		0			0
$217$		−	3.27042e12i		−	0.461398i
$218$		0			0
$219$	−4.58331e12			−0.614803
$220$		0			0
$221$	1.79945e12			0.229605
$222$		0			0
$223$			3.50569e12i			0.425693i	0.977086	+	0.212846i	$0.0682734\pi$
$223$			3.50569e12i			0.425693i	−0.977086	+	0.212846i	$0.931727\pi$
$224$		0			0
$225$	−2.43370e12	+	1.54604e12i	−0.281360	+	0.178739i
$226$		0			0
$227$			4.84938e12i			0.534003i	0.963696	+	0.267002i	$0.0860329\pi$
$227$			4.84938e12i			0.534003i	−0.963696	+	0.267002i	$0.913967\pi$
$228$		0			0
$229$	−1.41260e12			−0.148225			−0.0741127	−	0.997250i	$-0.523612\pi$
$229$	−1.41260e12			−0.148225			−0.0741127	+	0.997250i	$0.523612\pi$
$230$		0			0
$231$	−6.82835e12			−0.683046
$232$		0			0
$233$			9.88438e12i			0.942957i	0.881878	+	0.471478i	$0.156279\pi$
$233$			9.88438e12i			0.942957i	−0.881878	+	0.471478i	$0.843721\pi$
$234$		0			0
$235$	−2.62063e12	−	9.01253e12i	−0.238525	−	0.820302i
$236$		0			0
$237$			1.09426e13i			0.950613i
$238$		0			0
$239$	−5.87747e12			−0.487531			−0.243765	−	0.969834i	$-0.578383\pi$
$239$	−5.87747e12			−0.487531			−0.243765	+	0.969834i	$0.578383\pi$
$240$		0			0
$241$	−9.32982e12			−0.739230			−0.369615	−	0.929185i	$-0.620510\pi$
$241$	−9.32982e12			−0.739230			−0.369615	+	0.929185i	$0.620510\pi$
$242$		0			0
$243$		−	8.47289e11i		−	0.0641500i
$244$		0			0
$245$	5.51446e12	−	1.60348e12i	0.399107	−	0.116051i
$246$		0			0
$247$			2.16884e12i			0.150105i
$248$		0			0
$249$	1.67392e13			1.10825
$250$		0			0
$251$	−1.76150e13			−1.11603			−0.558016	−	0.829830i	$-0.688437\pi$
$251$	−1.76150e13			−1.11603			−0.558016	+	0.829830i	$0.688437\pi$
$252$		0			0
$253$			1.76411e13i			1.06995i
$254$		0			0
$255$	7.84913e12	−	2.28234e12i	0.455880	−	0.132559i
$256$		0			0
$257$		−	1.61047e13i		−	0.896023i	−0.894028	−	0.448012i	$-0.852132\pi$
$257$		−	1.61047e13i		−	0.896023i	0.894028	−	0.448012i	$-0.147868\pi$
$258$		0			0
$259$	7.26723e12			0.387454
$260$		0			0
$261$	−7.69448e12			−0.393239
$262$		0			0
$263$		−	8.60423e12i		−	0.421653i	−0.977524	−	0.210826i	$-0.932384\pi$
$263$		−	8.60423e12i		−	0.421653i	0.977524	−	0.210826i	$-0.0676155\pi$
$264$		0			0
$265$	9.25844e12	+	3.18404e13i	0.435196	+	1.49667i
$266$		0			0
$267$		−	8.49520e11i		−	0.0383144i
$268$		0			0
$269$	−1.03462e13			−0.447861			−0.223931	−	0.974605i	$-0.571889\pi$
$269$	−1.03462e13			−0.447861			−0.223931	+	0.974605i	$0.571889\pi$
$270$		0			0
$271$	3.19603e13			1.32825			0.664124	−	0.747622i	$-0.268804\pi$
$271$	3.19603e13			1.32825			0.664124	+	0.747622i	$0.268804\pi$
$272$		0			0
$273$			4.80568e12i			0.191805i
$274$		0			0
$275$	−2.18901e13	+	1.39060e13i	−0.839302	+	0.533180i
$276$		0			0
$277$			7.21606e12i			0.265865i	0.991125	+	0.132933i	$0.0424394\pi$
$277$			7.21606e12i			0.265865i	−0.991125	+	0.132933i	$0.957561\pi$
$278$		0			0
$279$	3.65007e12			0.129264
$280$		0			0
$281$	−4.86472e13			−1.65643			−0.828216	−	0.560409i	$-0.810644\pi$
$281$	−4.86472e13			−1.65643			−0.828216	+	0.560409i	$0.810644\pi$
$282$		0			0
$283$			4.50061e13i			1.47382i	0.675988	+	0.736912i	$0.263717\pi$
$283$			4.50061e13i			1.47382i	−0.675988	+	0.736912i	$0.736283\pi$
$284$		0			0
$285$	2.75087e12	+	9.46043e12i	0.0866610	+	0.298033i
$286$		0			0
$287$			7.77134e13i			2.35584i
$288$		0			0
$289$	1.10974e13			0.323804
$290$		0			0
$291$	3.95721e13			1.11167
$292$		0			0
$293$			1.55228e13i			0.419949i	0.977707	+	0.209975i	$0.0673382\pi$
$293$			1.55228e13i			0.419949i	−0.977707	+	0.209975i	$0.932662\pi$
$294$		0			0
$295$	−2.93796e13	+	8.54291e12i	−0.765641	+	0.222631i
$296$		0			0
$297$		−	7.62102e12i		−	0.191361i
$298$		0			0
$299$	1.24155e13			0.300451
$300$		0			0
$301$	−2.80493e13			−0.654345
$302$		0			0
$303$		−	3.23873e13i		−	0.728518i
$304$		0			0
$305$	5.22301e13	−	1.51873e13i	1.13311	−	0.329482i
$306$		0			0
$307$			5.15314e13i			1.07848i	0.842153	+	0.539239i	$0.181288\pi$
$307$			5.15314e13i			1.07848i	−0.842153	+	0.539239i	$0.818712\pi$
$308$		0			0
$309$	−2.06320e13			−0.416648
$310$		0			0
$311$	1.21704e13			0.237204			0.118602	−	0.992942i	$-0.462159\pi$
$311$	1.21704e13			0.237204			0.118602	+	0.992942i	$0.462159\pi$
$312$		0			0
$313$			4.90592e13i			0.923053i	0.887126	+	0.461527i	$0.152698\pi$
$313$			4.90592e13i			0.923053i	−0.887126	+	0.461527i	$0.847302\pi$
$314$		0			0
$315$	6.09533e12	+	2.09622e13i	0.110736	+	0.380829i
$316$		0			0
$317$			2.65586e12i			0.0465992i	0.999729	+	0.0232996i	$0.00741717\pi$
$317$			2.65586e12i			0.0465992i	−0.999729	+	0.0232996i	$0.992583\pi$
$318$		0			0
$319$	−6.92087e13			−1.17304
$320$		0			0
$321$	1.01854e13			0.166801
$322$		0			0
$323$		−	2.79319e13i		−	0.442065i
$324$		0			0
$325$	9.78686e12	+	1.54059e13i	0.149722	+	0.235684i
$326$		0			0
$327$			6.49326e13i			0.960395i
$328$		0			0
$329$	−7.10644e13			−1.01642
$330$		0			0
$331$	−8.86699e13			−1.22665			−0.613327	−	0.789829i	$-0.710169\pi$
$331$	−8.86699e13			−1.22665			−0.613327	+	0.789829i	$0.710169\pi$
$332$		0			0
$333$			8.11085e12i			0.108548i
$334$		0			0
$335$	−9.72563e12	−	3.34471e13i	−0.125942	−	0.433124i
$336$		0			0
$337$			5.28648e13i			0.662525i	0.943539	+	0.331262i	$0.107475\pi$
$337$			5.28648e13i			0.662525i	−0.943539	+	0.331262i	$0.892525\pi$
$338$		0			0
$339$	−5.77731e13			−0.700853
$340$		0			0
$341$	3.28309e13			0.385597
$342$		0			0
$343$			6.11331e13i			0.695278i
$344$		0			0
$345$	5.41561e13	−	1.57473e13i	0.596544	−	0.173461i
$346$		0			0
$347$			2.14116e13i			0.228474i	0.993454	+	0.114237i	$0.0364424\pi$
$347$			2.14116e13i			0.228474i	−0.993454	+	0.114237i	$0.963558\pi$
$348$		0			0
$349$	1.29282e14			1.33659			0.668295	−	0.743896i	$-0.267024\pi$
$349$	1.29282e14			1.33659			0.668295	+	0.743896i	$0.267024\pi$
$350$		0			0
$351$	−5.36355e12			−0.0537358
$352$		0			0
$353$			3.24173e13i			0.314786i	0.987536	+	0.157393i	$0.0503090\pi$
$353$			3.24173e13i			0.314786i	−0.987536	+	0.157393i	$0.949691\pi$
$354$		0			0
$355$	6.10201e13	−	1.77432e13i	0.574404	−	0.167023i
$356$		0			0
$357$		−	6.18909e13i		−	0.564874i
$358$		0			0
$359$	−1.01176e14			−0.895483			−0.447741	−	0.894163i	$-0.647771\pi$
$359$	−1.01176e14			−0.895483			−0.447741	+	0.894163i	$0.647771\pi$
$360$		0			0
$361$	−8.28244e13			−0.710999
$362$		0			0
$363$			7.82780e11i			0.00651858i
$364$		0			0
$365$	3.67995e13	+	1.26556e14i	0.297325	+	1.02252i
$366$		0			0
$367$		−	6.19096e12i		−	0.0485395i	−0.999705	−	0.0242697i	$-0.992274\pi$
$367$		−	6.19096e12i		−	0.0485395i	0.999705	−	0.0242697i	$-0.00772605\pi$
$368$		0			0
$369$	−8.67347e13			−0.660006
$370$		0			0
$371$	2.51063e14			1.85450
$372$		0			0
$373$		−	2.17159e14i		−	1.55733i	−0.627442	−	0.778664i	$-0.715898\pi$
$373$		−	2.17159e14i		−	1.55733i	0.627442	−	0.778664i	$-0.284102\pi$
$374$		0			0
$375$	6.22302e13	+	5.47869e13i	0.433341	+	0.381509i
$376$		0			0
$377$			4.87080e13i			0.329400i
$378$		0			0
$379$	1.35854e14			0.892397			0.446199	−	0.894934i	$-0.352778\pi$
$379$	1.35854e14			0.892397			0.446199	+	0.894934i	$0.352778\pi$
$380$		0			0
$381$	−1.73867e13			−0.110951
$382$		0			0
$383$		−	1.53210e14i		−	0.949932i	−0.880004	−	0.474966i	$-0.842460\pi$
$383$		−	1.53210e14i		−	0.949932i	0.880004	−	0.474966i	$-0.157540\pi$
$384$		0			0
$385$	5.48250e13	+	1.88547e14i	0.330328	+	1.13602i
$386$		0			0
$387$		−	3.13054e13i		−	0.183320i
$388$		0			0
$389$	−6.34461e13			−0.361145			−0.180573	−	0.983562i	$-0.557795\pi$
$389$	−6.34461e13			−0.361145			−0.180573	+	0.983562i	$0.557795\pi$
$390$		0			0
$391$	−1.59896e14			−0.884839
$392$		0			0
$393$			1.10998e14i			0.597251i
$394$		0			0
$395$	3.02151e14	−	8.78583e13i	1.58103	−	0.459726i
$396$		0			0
$397$		−	2.82212e14i		−	1.43624i	−0.695918	−	0.718121i	$-0.745002\pi$
$397$		−	2.82212e14i		−	1.43624i	0.695918	−	0.718121i	$-0.254998\pi$
$398$		0			0
$399$	7.45961e13			0.369288
$400$		0			0
$401$	−4.11588e13			−0.198230			−0.0991149	−	0.995076i	$-0.531601\pi$
$401$	−4.11588e13			−0.198230			−0.0991149	+	0.995076i	$0.531601\pi$
$402$		0			0
$403$		−	2.31059e13i		−	0.108279i
$404$		0			0
$405$	−2.33957e13	+	6.80291e12i	−0.106692	+	0.0310236i
$406$		0			0
$407$			7.29538e13i			0.323801i
$408$		0			0
$409$	−7.28620e13			−0.314792			−0.157396	−	0.987536i	$-0.550310\pi$
$409$	−7.28620e13			−0.314792			−0.157396	+	0.987536i	$0.550310\pi$
$410$		0			0
$411$	−6.96573e12			−0.0292979
$412$		0			0
$413$			2.31660e14i			0.948694i
$414$		0			0
$415$	−1.34400e14	−	4.62209e14i	−0.535961	−	1.84321i
$416$		0			0
$417$		−	2.91826e14i		−	1.13338i
$418$		0			0
$419$	−4.44807e13			−0.168265			−0.0841326	−	0.996455i	$-0.526812\pi$
$419$	−4.44807e13			−0.168265			−0.0841326	+	0.996455i	$0.526812\pi$
$420$		0			0
$421$	4.08282e13			0.150456			0.0752279	−	0.997166i	$-0.476032\pi$
$421$	4.08282e13			0.150456			0.0752279	+	0.997166i	$0.476032\pi$
$422$		0			0
$423$		−	7.93139e13i		−	0.284759i
$424$		0			0
$425$	−1.26042e14	−	1.98408e14i	−0.440936	−	0.694097i
$426$		0			0
$427$		−	4.11838e14i		−	1.40402i
$428$		0			0
$429$	−4.82430e13			−0.160295
$430$		0			0
$431$	−5.01831e14			−1.62530			−0.812648	−	0.582754i	$-0.801975\pi$
$431$	−5.01831e14			−1.62530			−0.812648	+	0.582754i	$0.801975\pi$
$432$		0			0
$433$		−	3.09744e14i		−	0.977957i	−0.872296	−	0.488979i	$-0.837370\pi$
$433$		−	3.09744e14i		−	0.977957i	0.872296	−	0.488979i	$-0.162630\pi$
$434$		0			0
$435$	6.17793e13	+	2.12463e14i	0.190174	+	0.654022i
$436$		0			0
$437$		−	1.92720e14i		−	0.578466i
$438$		0			0
$439$	−1.08027e13			−0.0316213			−0.0158106	−	0.999875i	$-0.505033\pi$
$439$	−1.08027e13			−0.0316213			−0.0158106	+	0.999875i	$0.505033\pi$
$440$		0			0
$441$	4.85295e13			0.138546
$442$		0			0
$443$			3.45097e14i			0.960993i	0.876997	+	0.480497i	$0.159544\pi$
$443$			3.45097e14i			0.960993i	−0.876997	+	0.480497i	$0.840456\pi$
$444$		0			0
$445$	−2.34573e13	+	6.82082e12i	−0.0637232	+	0.0185292i
$446$		0			0
$447$		−	1.61507e14i		−	0.428055i
$448$		0			0
$449$	5.12768e14			1.32607			0.663035	−	0.748589i	$-0.269268\pi$
$449$	5.12768e14			1.32607			0.663035	+	0.748589i	$0.269268\pi$
$450$		0			0
$451$	−7.80144e14			−1.96881
$452$		0			0
$453$			4.15439e14i			1.02322i
$454$		0			0
$455$	1.32696e14	−	3.85850e13i	0.319004	−	0.0927589i
$456$		0			0
$457$			2.33364e14i			0.547640i	0.961781	+	0.273820i	$0.0882873\pi$
$457$			2.33364e14i			0.547640i	−0.961781	+	0.273820i	$0.911713\pi$
$458$		0			0
$459$	6.90755e13			0.158254
$460$		0			0
$461$	1.62814e14			0.364197			0.182098	−	0.983280i	$-0.441711\pi$
$461$	1.62814e14			0.364197			0.182098	+	0.983280i	$0.441711\pi$
$462$		0			0
$463$			2.48926e14i			0.543718i	0.962337	+	0.271859i	$0.0876385\pi$
$463$			2.48926e14i			0.543718i	−0.962337	+	0.271859i	$0.912361\pi$
$464$		0			0
$465$	−2.93065e13	−	1.00787e14i	−0.0625134	−	0.214988i
$466$		0			0
$467$			3.50158e14i			0.729494i	0.931107	+	0.364747i	$0.118845\pi$
$467$			3.50158e14i			0.729494i	−0.931107	+	0.364747i	$0.881155\pi$
$468$		0			0
$469$	−2.63732e14			−0.536677
$470$		0			0
$471$	−2.42084e14			−0.481228
$472$		0			0
$473$		−	2.81580e14i		−	0.546846i
$474$		0			0
$475$	2.39138e14	−	1.51916e14i	0.453768	−	0.288263i
$476$		0			0
$477$			2.80208e14i			0.519553i
$478$		0			0
$479$	4.41157e14			0.799370			0.399685	−	0.916653i	$-0.369119\pi$
$479$	4.41157e14			0.799370			0.399685	+	0.916653i	$0.369119\pi$
$480$		0			0
$481$	5.13437e13			0.0909263
$482$		0			0
$483$		−	4.27024e14i		−	0.739169i
$484$		0			0
$485$	−3.17725e14	−	1.09268e15i	−0.537616	−	1.84890i
$486$		0			0
$487$		−	4.35803e12i		−	0.00720909i	−0.999994	−	0.00360455i	$-0.998853\pi$
$487$		−	4.35803e12i		−	0.00720909i	0.999994	−	0.00360455i	$-0.00114736\pi$
$488$		0			0
$489$	6.23250e13			0.100801
$490$		0			0
$491$	6.73844e13			0.106564			0.0532821	−	0.998580i	$-0.483032\pi$
$491$	6.73844e13			0.106564			0.0532821	+	0.998580i	$0.483032\pi$
$492$		0			0
$493$		−	6.27295e14i		−	0.970094i
$494$		0			0
$495$	−2.10434e14	+	6.11894e13i	−0.318264	+	0.0925439i
$496$		0			0
$497$		−	4.81148e14i		−	0.711735i
$498$		0			0
$499$	8.06287e14			1.16664			0.583320	−	0.812243i	$-0.301753\pi$
$499$	8.06287e14			1.16664			0.583320	+	0.812243i	$0.301753\pi$
$500$		0			0
$501$	2.49647e14			0.353361
$502$		0			0
$503$		−	1.30287e15i		−	1.80416i	−0.431565	−	0.902082i	$-0.642038\pi$
$503$		−	1.30287e15i		−	1.80416i	0.431565	−	0.902082i	$-0.357962\pi$
$504$		0			0
$505$	−8.94290e14	+	2.60039e14i	−1.21165	+	0.352318i
$506$		0			0
$507$		−	4.01542e14i		−	0.532338i
$508$		0			0
$509$	1.09216e15			1.41690			0.708450	−	0.705761i	$-0.249395\pi$
$509$	1.09216e15			1.41690			0.708450	+	0.705761i	$0.249395\pi$
$510$		0			0
$511$	9.97902e14			1.26699
$512$		0			0
$513$			8.32556e13i			0.103459i
$514$		0			0
$515$	1.65655e14	+	5.69697e14i	0.201495	+	0.692954i
$516$		0			0
$517$		−	7.13397e14i		−	0.849441i
$518$		0			0
$519$	8.53961e14			0.995446
$520$		0			0
$521$	5.50995e14			0.628840			0.314420	−	0.949284i	$-0.398190\pi$
$521$	5.50995e14			0.628840			0.314420	+	0.949284i	$0.398190\pi$
$522$		0			0
$523$			4.43612e14i			0.495729i	0.968795	+	0.247865i	$0.0797288\pi$
$523$			4.43612e14i			0.495729i	−0.968795	+	0.247865i	$0.920271\pi$
$524$		0			0
$525$	5.29878e14	−	3.36613e14i	0.579828	−	0.368345i
$526$		0			0
$527$			2.97573e14i			0.318886i
$528$		0			0
$529$	−1.50411e14			−0.157861
$530$		0			0
$531$	−2.58553e14			−0.265784
$532$		0			0
$533$			5.49053e14i			0.552859i
$534$		0			0
$535$	−8.17786e13	−	2.81242e14i	−0.0806666	−	0.277418i
$536$		0			0
$537$		−	1.11654e15i		−	1.07898i
$538$		0			0
$539$	4.36503e14			0.413284
$540$		0			0
$541$	−7.37013e14			−0.683739			−0.341869	−	0.939747i	$-0.611060\pi$
$541$	−7.37013e14			−0.683739			−0.341869	+	0.939747i	$0.611060\pi$
$542$		0			0
$543$			2.32054e14i			0.210955i
$544$		0			0
$545$	1.79294e15	−	5.21346e14i	1.59730	−	0.464457i
$546$		0			0
$547$		−	2.04180e15i		−	1.78272i	−0.453299	−	0.891358i	$-0.649753\pi$
$547$		−	2.04180e15i		−	1.78272i	0.453299	−	0.891358i	$-0.350247\pi$
$548$		0			0
$549$	4.59646e14			0.393347
$550$		0			0
$551$	7.56069e14			0.634202
$552$		0			0
$553$		−	2.38248e15i		−	1.95903i
$554$		0			0
$555$	2.23960e14	−	6.51223e13i	0.180534	−	0.0524950i
$556$		0			0
$557$			2.16374e15i			1.71002i	0.518611	+	0.855011i	$0.326449\pi$
$557$			2.16374e15i			1.71002i	−0.518611	+	0.855011i	$0.673551\pi$
$558$		0			0
$559$	−1.98171e14			−0.153559
$560$		0			0
$561$	6.21306e14			0.472074
$562$		0			0
$563$		−	2.38895e15i		−	1.77996i	−0.455995	−	0.889982i	$-0.650717\pi$
$563$		−	2.38895e15i		−	1.77996i	0.455995	−	0.889982i	$-0.349283\pi$
$564$		0			0
$565$	4.63862e14	+	1.59525e15i	0.338940	+	1.16564i
$566$		0			0
$567$			1.84476e14i			0.132201i
$568$		0			0
$569$	−8.76463e13			−0.0616050			−0.0308025	−	0.999525i	$-0.509806\pi$
$569$	−8.76463e13			−0.0616050			−0.0308025	+	0.999525i	$0.509806\pi$
$570$		0			0
$571$	−1.74097e15			−1.20031			−0.600155	−	0.799884i	$-0.704894\pi$
$571$	−1.74097e15			−1.20031			−0.600155	+	0.799884i	$0.704894\pi$
$572$		0			0
$573$			1.01634e15i			0.687367i
$574$		0			0
$575$	−8.69643e14	−	1.36894e15i	−0.576989	−	0.908264i
$576$		0			0
$577$			1.81953e15i			1.18438i	0.805797	+	0.592192i	$0.201737\pi$
$577$			1.81953e15i			1.18438i	−0.805797	+	0.592192i	$0.798263\pi$
$578$		0			0
$579$	−8.14362e14			−0.520098
$580$		0			0
$581$	−3.64455e15			−2.28389
$582$		0			0
$583$			2.52036e15i			1.54983i
$584$		0			0
$585$	4.30641e13	+	1.48100e14i	0.0259871	+	0.0893715i
$586$		0			0
$587$		−	7.68456e14i		−	0.455103i	−0.973766	−	0.227551i	$-0.926928\pi$
$587$		−	7.68456e14i		−	0.455103i	0.973766	−	0.227551i	$-0.0730719\pi$
$588$		0			0
$589$	−3.58660e14			−0.208473
$590$		0			0
$591$	1.75659e15			1.00216
$592$		0			0
$593$		−	1.21221e15i		−	0.678855i	−0.940632	−	0.339427i	$-0.889767\pi$
$593$		−	1.21221e15i		−	0.678855i	0.940632	−	0.339427i	$-0.110233\pi$
$594$		0			0
$595$	−1.70895e15	+	4.96924e14i	−0.939479	+	0.273179i
$596$		0			0
$597$		−	2.55485e14i		−	0.137882i
$598$		0			0
$599$	−1.29651e15			−0.686954			−0.343477	−	0.939161i	$-0.611605\pi$
$599$	−1.29651e15			−0.686954			−0.343477	+	0.939161i	$0.611605\pi$
$600$		0			0
$601$	−1.83742e15			−0.955871			−0.477935	−	0.878395i	$-0.658615\pi$
$601$	−1.83742e15			−0.955871			−0.477935	+	0.878395i	$0.658615\pi$
$602$		0			0
$603$		−	2.94348e14i		−	0.150354i
$604$		0			0
$605$	2.16144e13	−	6.28497e12i	0.0108415	−	0.00315245i
$606$		0			0
$607$		−	9.38310e14i		−	0.462177i	−0.972933	−	0.231089i	$-0.925771\pi$
$607$		−	9.38310e14i		−	0.462177i	0.972933	−	0.231089i	$-0.0742287\pi$
$608$		0			0
$609$	1.67528e15			0.810388
$610$		0			0
$611$	−5.02077e14			−0.238531
$612$		0			0
$613$			8.45726e13i			0.0394636i	0.999805	+	0.0197318i	$0.00628124\pi$
$613$			8.45726e13i			0.0394636i	−0.999805	+	0.0197318i	$0.993719\pi$
$614$		0			0
$615$	6.96396e14	+	2.39495e15i	0.319185	+	1.09770i
$616$		0			0
$617$			1.30490e15i			0.587501i	0.955882	+	0.293750i	$0.0949034\pi$
$617$			1.30490e15i			0.587501i	−0.955882	+	0.293750i	$0.905097\pi$
$618$		0			0
$619$	−1.79372e15			−0.793335			−0.396668	−	0.917962i	$-0.629833\pi$
$619$	−1.79372e15			−0.793335			−0.396668	+	0.917962i	$0.629833\pi$
$620$		0			0
$621$	4.76596e14			0.207084
$622$		0			0
$623$			1.84962e14i			0.0789584i
$624$		0			0
$625$	1.01315e15	−	2.15821e15i	0.424946	−	0.905219i
$626$		0			0
$627$			7.48850e14i			0.308620i
$628$		0			0
$629$	−6.61240e14			−0.267781
$630$		0			0
$631$	1.89462e15			0.753981			0.376991	−	0.926217i	$-0.376959\pi$
$631$	1.89462e15			0.753981			0.376991	+	0.926217i	$0.376959\pi$
$632$		0			0
$633$			9.25816e13i			0.0362080i
$634$		0			0
$635$	1.39599e14	+	4.80089e14i	0.0536570	+	0.184530i
$636$		0			0
$637$		−	3.07204e14i		−	0.116054i
$638$		0			0
$639$	5.37002e14			0.199398
$640$		0			0
$641$	−2.48785e14			−0.0908040			−0.0454020	−	0.998969i	$-0.514457\pi$
$641$	−2.48785e14			−0.0908040			−0.0454020	+	0.998969i	$0.514457\pi$
$642$		0			0
$643$			4.34116e15i			1.55756i	0.627297	+	0.778780i	$0.284161\pi$
$643$			4.34116e15i			1.55756i	−0.627297	+	0.778780i	$0.715839\pi$
$644$		0			0
$645$	−8.64418e14	+	2.51352e14i	−0.304891	+	0.0886553i
$646$		0			0
$647$			1.76476e15i			0.611944i	0.952041	+	0.305972i	$0.0989814\pi$
$647$			1.76476e15i			0.611944i	−0.952041	+	0.305972i	$0.901019\pi$
$648$		0			0
$649$	−2.32558e15			−0.792838
$650$		0			0
$651$	−7.94713e14			−0.266388
$652$		0			0
$653$			8.57218e14i			0.282533i	0.989972	+	0.141266i	$0.0451174\pi$
$653$			8.57218e14i			0.282533i	−0.989972	+	0.141266i	$0.954883\pi$
$654$		0			0
$655$	3.06493e15	−	8.91209e14i	0.993328	−	0.288837i
$656$		0			0
$657$			1.11374e15i			0.354957i
$658$		0			0
$659$	2.32789e15			0.729613			0.364807	−	0.931083i	$-0.381135\pi$
$659$	2.32789e15			0.729613			0.364807	+	0.931083i	$0.381135\pi$
$660$		0			0
$661$	−3.06684e15			−0.945328			−0.472664	−	0.881243i	$-0.656708\pi$
$661$	−3.06684e15			−0.945328			−0.472664	+	0.881243i	$0.656708\pi$
$662$		0			0
$663$		−	4.37266e14i		−	0.132563i
$664$		0			0
$665$	−5.98934e14	−	2.05978e15i	−0.178591	−	0.614187i
$666$		0			0
$667$		−	4.32811e15i		−	1.26942i
$668$		0			0
$669$	8.51882e14			0.245774
$670$		0			0
$671$	4.13433e15			1.17336
$672$		0			0
$673$			2.69617e15i			0.752775i	0.926462	+	0.376387i	$0.122834\pi$
$673$			2.69617e15i			0.752775i	−0.926462	+	0.376387i	$0.877166\pi$
$674$		0			0
$675$	3.75689e14	+	5.91389e14i	0.103195	+	0.162443i
$676$		0			0
$677$			5.36073e15i			1.44872i	0.689419	+	0.724362i	$0.257866\pi$
$677$			5.36073e15i			1.44872i	−0.689419	+	0.724362i	$0.742134\pi$
$678$		0			0
$679$	−8.61584e15			−2.29094
$680$		0			0
$681$	1.17840e15			0.308307
$682$		0			0
$683$			7.26562e15i			1.87050i	0.353981	+	0.935252i	$0.384828\pi$
$683$			7.26562e15i			1.87050i	−0.353981	+	0.935252i	$0.615172\pi$
$684$		0			0
$685$	5.59280e13	+	1.92340e14i	0.0141687	+	0.0487273i
$686$		0			0
$687$			3.43261e14i			0.0855780i
$688$		0			0
$689$	1.77379e15			0.435207
$690$		0			0
$691$	−2.83285e15			−0.684061			−0.342030	−	0.939689i	$-0.611115\pi$
$691$	−2.83285e15			−0.684061			−0.342030	+	0.939689i	$0.611115\pi$
$692$		0			0
$693$			1.65929e15i			0.394357i
$694$		0			0
$695$	−8.05800e15	+	2.34308e15i	−1.88500	+	0.548114i
$696$		0			0
$697$		−	7.07108e15i		−	1.62819i
$698$		0			0
$699$	2.40190e15			0.544416
$700$		0			0
$701$	6.35487e15			1.41794			0.708969	−	0.705239i	$-0.249161\pi$
$701$	6.35487e15			1.41794			0.708969	+	0.705239i	$0.249161\pi$
$702$		0			0
$703$		−	7.96982e14i		−	0.175063i
$704$		0			0
$705$	−2.19005e15	+	6.36814e14i	−0.473601	+	0.137712i
$706$		0			0
$707$			7.05153e15i			1.50133i
$708$		0			0
$709$	−5.55781e15			−1.16506			−0.582531	−	0.812808i	$-0.697938\pi$
$709$	−5.55781e15			−1.16506			−0.582531	+	0.812808i	$0.697938\pi$
$710$		0			0
$711$	2.65905e15			0.548837
$712$		0			0
$713$			2.05315e15i			0.417280i
$714$		0			0
$715$	3.87344e14	+	1.33210e15i	0.0775201	+	0.266597i
$716$		0			0
$717$			1.42823e15i			0.281476i
$718$		0			0
$719$	1.05266e15			0.204305			0.102153	−	0.994769i	$-0.467427\pi$
$719$	1.05266e15			0.204305			0.102153	+	0.994769i	$0.467427\pi$
$720$		0			0
$721$	4.49210e15			0.858629
$722$		0			0
$723$			2.26715e15i			0.426795i
$724$		0			0
$725$	5.37058e15	−	3.41174e15i	0.995777	−	0.632583i
$726$		0			0
$727$		−	3.83121e15i		−	0.699675i	−0.936810	−	0.349838i	$-0.886237\pi$
$727$		−	3.83121e15i		−	0.699675i	0.936810	−	0.349838i	$-0.113763\pi$
$728$		0			0
$729$	−2.05891e14			−0.0370370
$730$		0			0
$731$	2.55219e15			0.452238
$732$		0			0
$733$			7.31689e15i			1.27719i	0.769544	+	0.638594i	$0.220483\pi$
$733$			7.31689e15i			1.27719i	−0.769544	+	0.638594i	$0.779517\pi$
$734$		0			0
$735$	−3.89645e14	−	1.34001e15i	−0.0670021	−	0.230425i
$736$		0			0
$737$		−	2.64754e15i		−	0.448509i
$738$		0			0
$739$	−1.04805e16			−1.74920			−0.874598	−	0.484848i	$-0.838875\pi$
$739$	−1.04805e16			−1.74920			−0.874598	+	0.484848i	$0.838875\pi$
$740$		0			0
$741$	5.27029e14			0.0866631
$742$		0			0
$743$			3.28089e15i			0.531561i	0.964034	+	0.265780i	$0.0856296\pi$
$743$			3.28089e15i			0.531561i	−0.964034	+	0.265780i	$0.914370\pi$
$744$		0			0
$745$	−4.45958e15	+	1.29674e15i	−0.711926	+	0.207011i
$746$		0			0
$747$		−	4.06763e15i		−	0.639850i
$748$		0			0
$749$	−2.21761e15			−0.343744
$750$		0			0
$751$	−9.30704e15			−1.42165			−0.710824	−	0.703370i	$-0.751678\pi$
$751$	−9.30704e15			−1.42165			−0.710824	+	0.703370i	$0.751678\pi$
$752$		0			0
$753$			4.28044e15i			0.644342i
$754$		0			0
$755$	1.14713e16	−	3.33557e15i	1.70178	−	0.494837i
$756$		0			0
$757$			6.20592e15i			0.907359i	0.891165	+	0.453680i	$0.149889\pi$
$757$			6.20592e15i			0.907359i	−0.891165	+	0.453680i	$0.850111\pi$
$758$		0			0
$759$	4.28678e15			0.617735
$760$		0			0
$761$	−6.29521e14			−0.0894118			−0.0447059	−	0.999000i	$-0.514235\pi$
$761$	−6.29521e14			−0.0894118			−0.0447059	+	0.999000i	$0.514235\pi$
$762$		0			0
$763$		−	1.41375e16i		−	1.97919i
$764$		0			0
$765$	−5.54609e14	−	1.90734e15i	−0.0765331	−	0.263202i
$766$		0			0
$767$			1.63670e15i			0.222636i
$768$		0			0
$769$	−4.37850e15			−0.587126			−0.293563	−	0.955940i	$-0.594841\pi$
$769$	−4.37850e15			−0.587126			−0.293563	+	0.955940i	$0.594841\pi$
$770$		0			0
$771$	−3.91343e15			−0.517319
$772$		0			0
$773$			6.24169e15i			0.813420i	0.913557	+	0.406710i	$0.133324\pi$
$773$			6.24169e15i			0.813420i	−0.913557	+	0.406710i	$0.866676\pi$
$774$		0			0
$775$	−2.54767e15	+	1.61845e15i	−0.327328	+	0.207940i
$776$		0			0
$777$		−	1.76594e15i		−	0.223697i
$778$		0			0
$779$	8.52266e15			1.06443
$780$		0			0
$781$	4.83011e15			0.594808
$782$		0			0
$783$			1.86976e15i			0.227037i
$784$		0			0
$785$	1.94370e15	+	6.68452e15i	0.232727	+	0.800363i
$786$		0			0
$787$			1.67198e16i			1.97411i	0.160397	+	0.987053i	$0.448722\pi$
$787$			1.67198e16i			1.97411i	−0.160397	+	0.987053i	$0.551278\pi$
$788$		0			0
$789$	−2.09083e15			−0.243441
$790$		0			0
$791$	1.25787e16			1.44432
$792$		0			0
$793$		−	2.90968e15i		−	0.329490i
$794$		0			0
$795$	7.73721e15	−	2.24980e15i	0.864102	−	0.251261i
$796$		0			0
$797$			4.36301e15i			0.480580i	0.970701	+	0.240290i	$0.0772425\pi$
$797$			4.36301e15i			0.480580i	−0.970701	+	0.240290i	$0.922758\pi$
$798$		0			0
$799$	6.46610e15			0.702481
$800$		0			0
$801$	−2.06433e14			−0.0221208
$802$		0			0
$803$			1.00177e16i			1.05884i
$804$		0			0
$805$	−1.17912e16	+	3.42859e15i	−1.22936	+	0.357469i
$806$		0			0
$807$			2.51413e15i			0.258573i
$808$		0			0
$809$	−3.55238e15			−0.360415			−0.180208	−	0.983629i	$-0.557677\pi$
$809$	−3.55238e15			−0.360415			−0.180208	+	0.983629i	$0.557677\pi$
$810$		0			0
$811$	−1.38794e16			−1.38917			−0.694585	−	0.719411i	$-0.744412\pi$
$811$	−1.38794e16			−1.38917			−0.694585	+	0.719411i	$0.744412\pi$
$812$		0			0
$813$		−	7.76635e15i		−	0.766865i
$814$		0			0
$815$	−5.00410e14	−	1.72094e15i	−0.0487482	−	0.167648i
$816$		0			0
$817$			3.07611e15i			0.295652i
$818$		0			0
$819$	1.16778e15			0.110739
$820$		0			0
$821$	9.65670e15			0.903528			0.451764	−	0.892138i	$-0.350795\pi$
$821$	9.65670e15			0.903528			0.451764	+	0.892138i	$0.350795\pi$
$822$		0			0
$823$			1.62599e16i			1.50114i	0.660793	+	0.750568i	$0.270220\pi$
$823$			1.62599e16i			1.50114i	−0.660793	+	0.750568i	$0.729780\pi$
$824$		0			0
$825$	3.37917e15	+	5.31930e15i	0.307832	+	0.484571i
$826$		0			0
$827$		−	9.52899e15i		−	0.856578i	−0.903642	−	0.428289i	$-0.859117\pi$
$827$		−	9.52899e15i		−	0.856578i	0.903642	−	0.428289i	$-0.140883\pi$
$828$		0			0
$829$	−1.66053e16			−1.47298			−0.736492	−	0.676446i	$-0.763519\pi$
$829$	−1.66053e16			−1.47298			−0.736492	+	0.676446i	$0.763519\pi$
$830$		0			0
$831$	1.75350e15			0.153497
$832$		0			0
$833$			3.95638e15i			0.341783i
$834$		0			0
$835$	−2.00442e15	−	6.89335e15i	−0.170889	−	0.587699i
$836$		0			0
$837$		−	8.86967e14i		−	0.0746307i
$838$		0			0
$839$	−1.01371e16			−0.841827			−0.420913	−	0.907101i	$-0.638290\pi$
$839$	−1.01371e16			−0.841827			−0.420913	+	0.907101i	$0.638290\pi$
$840$		0			0
$841$	4.77933e15			0.391732
$842$		0			0
$843$			1.18213e16i			0.956341i
$844$		0			0
$845$	−1.10875e16	+	3.22400e15i	−0.885367	+	0.257444i
$846$		0			0
$847$		−	1.70431e14i		−	0.0134335i
$848$		0			0
$849$	1.09365e16			0.850913
$850$		0			0
$851$	−4.56231e15			−0.350407
$852$		0			0
$853$			2.03075e16i			1.53970i	0.638224	+	0.769851i	$0.279669\pi$
$853$			2.03075e16i			1.53970i	−0.638224	+	0.769851i	$0.720331\pi$
$854$		0			0
$855$	2.29888e15	−	6.68462e14i	0.172069	−	0.0500337i
$856$		0			0
$857$		−	4.99342e15i		−	0.368981i	−0.982834	−	0.184490i	$-0.940937\pi$
$857$		−	4.99342e15i		−	0.368981i	0.982834	−	0.184490i	$-0.0590635\pi$
$858$		0			0
$859$	5.27939e15			0.385142			0.192571	−	0.981283i	$-0.438317\pi$
$859$	5.27939e15			0.385142			0.192571	+	0.981283i	$0.438317\pi$
$860$		0			0
$861$	1.88843e16			1.36014
$862$		0			0
$863$		−	6.94499e15i		−	0.493869i	−0.969032	−	0.246935i	$-0.920577\pi$
$863$		−	6.94499e15i		−	0.493869i	0.969032	−	0.246935i	$-0.0794233\pi$
$864$		0			0
$865$	−6.85648e15	−	2.35799e16i	−0.481408	−	1.65559i
$866$		0			0
$867$		−	2.69667e15i		−	0.186949i
$868$		0			0
$869$	2.39170e16			1.63719
$870$		0			0
$871$	−1.86330e15			−0.125945
$872$		0			0
$873$		−	9.61601e15i		−	0.641825i
$874$		0			0
$875$	−1.35491e16	−	1.19285e16i	−0.893030	−	0.786216i
$876$		0			0
$877$		−	1.66689e16i		−	1.08495i	−0.840073	−	0.542473i	$-0.817488\pi$
$877$		−	1.66689e16i		−	1.08495i	0.840073	−	0.542473i	$-0.182512\pi$
$878$		0			0
$879$	3.77203e15			0.242458
$880$		0			0
$881$	−7.30716e15			−0.463854			−0.231927	−	0.972733i	$-0.574503\pi$
$881$	−7.30716e15			−0.463854			−0.231927	+	0.972733i	$0.574503\pi$
$882$		0			0
$883$		−	1.41660e16i		−	0.888100i	−0.896002	−	0.444050i	$-0.853541\pi$
$883$		−	1.41660e16i		−	0.888100i	0.896002	−	0.444050i	$-0.146459\pi$
$884$		0			0
$885$	2.07593e15	+	7.13925e15i	0.128536	+	0.442043i
$886$		0			0
$887$		−	1.48432e16i		−	0.907713i	−0.891075	−	0.453857i	$-0.850048\pi$
$887$		−	1.48432e16i		−	0.907713i	0.891075	−	0.453857i	$-0.149952\pi$
$888$		0			0
$889$	3.78554e15			0.228648
$890$		0			0
$891$	−1.85191e15			−0.110482
$892$		0			0
$893$			7.79348e15i			0.459249i
$894$		0			0
$895$	−3.08302e16	+	8.96471e15i	−1.79453	+	0.521807i
$896$		0			0
$897$		−	3.01697e15i		−	0.173465i
$898$		0			0
$899$	−8.05482e15			−0.457486
$900$		0			0
$901$	−2.28441e16			−1.28170
$902$		0			0
$903$			6.81599e15i			0.377786i
$904$		0			0
$905$	6.40755e15	−	1.86317e15i	0.350853	−	0.102020i
$906$		0			0
$907$			1.13330e16i			0.613061i	0.951861	+	0.306530i	$0.0991681\pi$
$907$			1.13330e16i			0.613061i	−0.951861	+	0.306530i	$0.900832\pi$
$908$		0			0
$909$	−7.87011e15			−0.420610
$910$		0			0
$911$	1.91130e16			1.00920			0.504602	−	0.863352i	$-0.331639\pi$
$911$	1.91130e16			1.00920			0.504602	+	0.863352i	$0.331639\pi$
$912$		0			0
$913$		−	3.65867e16i		−	1.90868i
$914$		0			0
$915$	−3.69051e15	−	1.26919e16i	−0.190226	−	0.654201i
$916$		0			0
$917$		−	2.41671e16i		−	1.23082i
$918$		0			0
$919$	7.09706e15			0.357144			0.178572	−	0.983927i	$-0.442852\pi$
$919$	7.09706e15			0.357144			0.178572	+	0.983927i	$0.442852\pi$
$920$		0			0
$921$	1.25221e16			0.622660
$922$		0			0
$923$		−	3.39936e15i		−	0.167027i
$924$		0			0
$925$	−3.59636e15	−	5.66119e15i	−0.174616	−	0.274871i
$926$		0			0
$927$			5.01357e15i			0.240552i
$928$		0			0
$929$	1.87622e16			0.889604			0.444802	−	0.895629i	$-0.353274\pi$
$929$	1.87622e16			0.889604			0.444802	+	0.895629i	$0.353274\pi$
$930$		0			0
$931$	−4.76856e15			−0.223442
$932$		0			0
$933$		−	2.95741e15i		−	0.136950i
$934$		0			0
$935$	−4.98849e15	−	1.71557e16i	−0.228300	−	0.785137i
$936$		0			0
$937$		−	3.08165e16i		−	1.39385i	−0.717145	−	0.696924i	$-0.754552\pi$
$937$		−	3.08165e16i		−	1.39385i	0.717145	−	0.696924i	$-0.245448\pi$
$938$		0			0
$939$	1.19214e16			0.532925
$940$		0			0
$941$	−4.26974e16			−1.88651			−0.943255	−	0.332070i	$-0.892253\pi$
$941$	−4.26974e16			−1.88651			−0.943255	+	0.332070i	$0.892253\pi$
$942$		0			0
$943$		−	4.87879e16i		−	2.13058i
$944$		0			0
$945$	5.09383e15	−	1.48117e15i	0.219872	−	0.0639335i
$946$		0			0
$947$			1.63317e16i			0.696799i	0.937346	+	0.348399i	$0.113275\pi$
$947$			1.63317e16i			0.696799i	−0.937346	+	0.348399i	$0.886725\pi$
$948$		0			0
$949$	7.05028e15			0.297332
$950$		0			0
$951$	6.45373e14			0.0269041
$952$		0			0
$953$		−	2.78821e16i		−	1.14899i	−0.818509	−	0.574494i	$-0.805199\pi$
$953$		−	2.78821e16i		−	1.14899i	0.818509	−	0.574494i	$-0.194801\pi$
$954$		0			0
$955$	2.80636e16	−	8.16023e15i	1.14321	−	0.332417i
$956$		0			0
$957$			1.68177e16i			0.677254i
$958$		0			0
$959$	1.51662e15			0.0603772
$960$		0			0
$961$	−2.15875e16			−0.849617
$962$		0			0
$963$		−	2.47504e15i		−	0.0963026i
$964$		0			0
$965$	6.53854e15	+	2.24865e16i	0.251525	+	0.865010i
$966$		0			0
$967$		−	7.21160e15i		−	0.274275i	−0.990552	−	0.137137i	$-0.956210\pi$
$967$		−	7.21160e15i		−	0.274275i	0.990552	−	0.137137i	$-0.0437902\pi$
$968$		0			0
$969$	−6.78744e15			−0.255226
$970$		0			0
$971$	4.56053e16			1.69555			0.847773	−	0.530359i	$-0.177943\pi$
$971$	4.56053e16			1.69555			0.847773	+	0.530359i	$0.177943\pi$
$972$		0			0
$973$			6.35379e16i			2.33567i
$974$		0			0
$975$	3.74364e15	−	2.37821e15i	0.136072	−	0.0864419i
$976$		0			0
$977$		−	4.85178e16i		−	1.74374i	−0.489740	−	0.871868i	$-0.662908\pi$
$977$		−	4.85178e16i		−	1.74374i	0.489740	−	0.871868i	$-0.337092\pi$
$978$		0			0
$979$	−1.85678e15			−0.0659867
$980$		0			0
$981$	1.57786e16			0.554485
$982$		0			0
$983$			4.37954e16i			1.52189i	0.648814	+	0.760947i	$0.275265\pi$
$983$			4.37954e16i			1.52189i	−0.648814	+	0.760947i	$0.724735\pi$
$984$		0			0
$985$	−1.41037e16	−	4.85035e16i	−0.484656	−	1.66676i
$986$		0			0
$987$			1.72687e16i			0.586832i
$988$		0			0
$989$	1.76092e16			0.591778
$990$		0			0
$991$	2.55075e16			0.847740			0.423870	−	0.905723i	$-0.360671\pi$
$991$	2.55075e16			0.847740			0.423870	+	0.905723i	$0.360671\pi$
$992$		0			0
$993$			2.15468e16i			0.708209i
$994$		0			0
$995$	−7.05456e15	+	2.05130e15i	−0.229321	+	0.0666810i
$996$		0			0
$997$			8.08618e15i			0.259968i	0.991516	+	0.129984i	$0.0414926\pi$
$997$			8.08618e15i			0.259968i	−0.991516	+	0.129984i	$0.958507\pi$
$998$		0			0
$999$	1.97094e15			0.0626704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.12.f.b.49.1		6
4.3	odd	2		30.12.c.b.19.4	yes	6
5.4	even	2	inner	240.12.f.b.49.4		6
12.11	even	2		90.12.c.c.19.3		6
20.3	even	4		150.12.a.u.1.1		3
20.7	even	4		150.12.a.t.1.3		3
20.19	odd	2		30.12.c.b.19.1	✓	6
60.59	even	2		90.12.c.c.19.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.12.c.b.19.1	✓	6	20.19	odd	2
30.12.c.b.19.4	yes	6	4.3	odd	2
90.12.c.c.19.3		6	12.11	even	2
90.12.c.c.19.6		6	60.59	even	2
150.12.a.t.1.3		3	20.7	even	4
150.12.a.u.1.1		3	20.3	even	4
240.12.f.b.49.1		6	1.1	even	1	trivial
240.12.f.b.49.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	240.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-243.000i q^{3} +(-6709.81 + 1951.05i) q^{5} +52907.3i q^{7} -59049.0 q^{9} -531122. q^{11} +373795. i q^{13} +(474106. + 1.63048e6i) q^{15} -4.81399e6i q^{17} +5.80222e6 q^{19} +1.28565e7 q^{21} -3.32148e7i q^{23} +(4.12149e7 - 2.61824e7i) q^{25} +1.43489e7i q^{27} +1.30307e8 q^{29} -6.18143e7 q^{31} +1.29063e8i q^{33} +(-1.03225e8 - 3.54997e8i) q^{35} -1.37358e8i q^{37} +9.08322e7 q^{39} +1.46886e9 q^{41} +5.30160e8i q^{43} +(3.96207e8 - 1.15208e8i) q^{45} +1.34319e9i q^{47} -8.21851e8 q^{49} -1.16980e9 q^{51} -4.74535e9i q^{53} +(3.56372e9 - 1.03625e9i) q^{55} -1.40994e9i q^{57} +4.37861e9 q^{59} -7.78414e9 q^{61} -3.12412e9i q^{63} +(-7.29295e8 - 2.50809e9i) q^{65} +4.98481e9i q^{67} -8.07119e9 q^{69} -9.09417e9 q^{71} -1.88613e10i q^{73} +(-6.36232e9 - 1.00152e10i) q^{75} -2.81002e10i q^{77} -4.50312e10 q^{79} +3.48678e9 q^{81} +6.88856e10i q^{83} +(9.39236e9 + 3.23009e10i) q^{85} -3.16645e10i q^{87} +3.49597e9 q^{89} -1.97765e10 q^{91} +1.50209e10i q^{93} +(-3.89318e10 + 1.13205e10i) q^{95} +1.62848e11i q^{97} +3.13622e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 9926 q^{5} - 354294 q^{9} - 1753400 q^{11} - 551124 q^{15} + 4069824 q^{19} - 32748624 q^{21} + 169978326 q^{25} + 71715564 q^{29} + 122243352 q^{31} - 825571904 q^{35} - 153183312 q^{39} + 2006768564 q^{41}+ \cdots + 103536516600 q^{99}+O(q^{100}) \) 6 * q - 9926 * q^5 - 354294 * q^9 - 1753400 * q^11 - 551124 * q^15 + 4069824 * q^19 - 32748624 * q^21 + 169978326 * q^25 + 71715564 * q^29 + 122243352 * q^31 - 825571904 * q^35 - 153183312 * q^39 + 2006768564 * q^41 + 586120374 * q^45 - 9590905446 * q^49 + 1396921464 * q^51 - 9162341400 * q^55 + 4069460200 * q^59 - 12386466660 * q^61 - 30310342152 * q^65 + 24235021800 * q^69 - 13703505600 * q^71 + 22735606824 * q^75 + 13438327752 * q^79 + 20920706406 * q^81 + 50595843144 * q^85 - 127436589092 * q^89 - 271843383552 * q^91 + 87931077696 * q^95 + 103536516600 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more