LMFDB - Embedded newform 600.2.b.c.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(251,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("600.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.b (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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	600.251
Dual form			600.2.b.c.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.11803 - 0.866025i) q^{2} -1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +(-1.50000 + 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +O(q^{10})$	$q+(-1.11803 - 0.866025i) q^{2} -1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +(-1.50000 + 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +(3.35410 - 0.866025i) q^{12} +(-3.50000 + 1.93649i) q^{16} -6.92820i q^{17} +(3.35410 + 2.59808i) q^{18} -4.00000 q^{19} -8.94427 q^{23} +(-4.50000 - 1.93649i) q^{24} +5.19615i q^{27} -7.74597i q^{31} +(5.59017 + 0.866025i) q^{32} +(-6.00000 + 7.74597i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(4.47214 + 3.46410i) q^{38} +(10.0000 + 7.74597i) q^{46} -8.94427 q^{47} +(3.35410 + 6.06218i) q^{48} +7.00000 q^{49} -12.0000 q^{51} -4.47214 q^{53} +(4.50000 - 5.80948i) q^{54} +6.92820i q^{57} +15.4919i q^{61} +(-6.70820 + 8.66025i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(13.4164 - 3.46410i) q^{68} +15.4919i q^{69} +(-3.35410 + 7.79423i) q^{72} +(-2.00000 - 7.74597i) q^{76} -7.74597i q^{79} +9.00000 q^{81} +3.46410i q^{83} +(-4.47214 - 17.3205i) q^{92} -13.4164 q^{93} +(10.0000 + 7.74597i) q^{94} +(1.50000 - 9.68246i) q^{96} +(-7.82624 - 6.06218i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 6 * q^6 - 12 * q^9	$ 4 q + 2 q^{4} - 6 q^{6} - 12 q^{9} - 14 q^{16} - 16 q^{19} - 18 q^{24} - 24 q^{34} - 6 q^{36} + 40 q^{46} + 28 q^{49} - 48 q^{51} + 18 q^{54} - 22 q^{64} - 8 q^{76} + 36 q^{81} + 40 q^{94} + 6 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 6 * q^6 - 12 * q^9 - 14 * q^16 - 16 * q^19 - 18 * q^24 - 24 * q^34 - 6 * q^36 + 40 * q^46 + 28 * q^49 - 48 * q^51 + 18 * q^54 - 22 * q^64 - 8 * q^76 + 36 * q^81 + 40 * q^94 + 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$	−1.11803	−	0.866025i	−0.790569	−	0.612372i
$2$	−1.11803	−	0.866025i	−0.790569	−	0.612372i
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	0.500000	+	1.93649i	0.250000	+	0.968246i
$5$		0			0
$5$		0			0
$6$	−1.50000	+	1.93649i	−0.612372	+	0.790569i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	1.11803	−	2.59808i	0.395285	−	0.918559i
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	3.35410	−	0.866025i	0.968246	−	0.250000i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$		−	6.92820i		−	1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$17$		−	6.92820i		−	1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$18$	3.35410	+	2.59808i	0.790569	+	0.612372i
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−8.94427			−1.86501			−0.932505	−	0.361158i	$-0.882382\pi$
$23$	−8.94427			−1.86501			−0.932505	+	0.361158i	$0.882382\pi$
$24$	−4.50000	−	1.93649i	−0.918559	−	0.395285i
$25$		0			0
$26$		0			0
$27$			5.19615i			1.00000i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	7.74597i		−	1.39122i	−0.718421	−	0.695608i	$-0.755135\pi$
$31$		−	7.74597i		−	1.39122i	0.718421	−	0.695608i	$-0.244865\pi$
$32$	5.59017	+	0.866025i	0.988212	+	0.153093i
$33$		0			0
$34$	−6.00000	+	7.74597i	−1.02899	+	1.32842i
$35$		0			0
$36$	−1.50000	−	5.80948i	−0.250000	−	0.968246i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	4.47214	+	3.46410i	0.725476	+	0.561951i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	10.0000	+	7.74597i	1.47442	+	1.14208i
$47$	−8.94427			−1.30466			−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427			−1.30466			−0.652328	+	0.757937i	$0.726208\pi$
$48$	3.35410	+	6.06218i	0.484123	+	0.875000i
$49$	7.00000			1.00000
$50$		0			0
$51$	−12.0000			−1.68034
$52$		0			0
$53$	−4.47214			−0.614295			−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214			−0.614295			−0.307148	+	0.951662i	$0.599375\pi$
$54$	4.50000	−	5.80948i	0.612372	−	0.790569i
$55$		0			0
$56$		0			0
$57$			6.92820i			0.917663i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$			15.4919i			1.98354i	0.128037	+	0.991769i	$0.459132\pi$
$61$			15.4919i			1.98354i	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−6.70820	+	8.66025i	−0.851943	+	1.09985i
$63$		0			0
$64$	−5.50000	−	5.80948i	−0.687500	−	0.726184i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	13.4164	−	3.46410i	1.62698	−	0.420084i
$69$			15.4919i			1.86501i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−3.35410	+	7.79423i	−0.395285	+	0.918559i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	−2.00000	−	7.74597i	−0.229416	−	0.888523i
$77$		0			0
$78$		0			0
$79$		−	7.74597i		−	0.871489i	−0.900070	−	0.435745i	$-0.856485\pi$
$79$		−	7.74597i		−	0.871489i	0.900070	−	0.435745i	$-0.143515\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$			3.46410i			0.380235i	0.981761	+	0.190117i	$0.0608868\pi$
$83$			3.46410i			0.380235i	−0.981761	+	0.190117i	$0.939113\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−4.47214	−	17.3205i	−0.466252	−	1.80579i
$93$	−13.4164			−1.39122
$94$	10.0000	+	7.74597i	1.03142	+	0.798935i
$95$		0			0
$96$	1.50000	−	9.68246i	0.153093	−	0.988212i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−7.82624	−	6.06218i	−0.790569	−	0.612372i
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	13.4164	+	10.3923i	1.32842	+	1.02899i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	5.00000	+	3.87298i	0.485643	+	0.376177i
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$	−10.0623	+	2.59808i	−0.968246	+	0.250000i
$109$		−	15.4919i		−	1.48386i	−0.670478	−	0.741929i	$-0.733911\pi$
$109$		−	15.4919i		−	1.48386i	0.670478	−	0.741929i	$-0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	20.7846i		−	1.95525i	−0.210352	−	0.977626i	$-0.567461\pi$
$113$		−	20.7846i		−	1.95525i	0.210352	−	0.977626i	$-0.432539\pi$
$114$	6.00000	−	7.74597i	0.561951	−	0.725476i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	13.4164	−	17.3205i	1.21466	−	1.56813i
$123$		0			0
$124$	15.0000	−	3.87298i	1.34704	−	0.347804i
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	1.11803	+	11.2583i	0.0988212	+	0.995105i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−18.0000	−	7.74597i	−1.54349	−	0.664211i
$137$			6.92820i			0.591916i	0.955201	+	0.295958i	$0.0956389\pi$
$137$			6.92820i			0.591916i	−0.955201	+	0.295958i	$0.904361\pi$
$138$	13.4164	−	17.3205i	1.14208	−	1.47442i
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			15.4919i			1.30466i
$142$		0			0
$143$		0			0
$144$	10.5000	−	5.80948i	0.875000	−	0.484123i
$145$		0			0
$146$		0			0
$147$		−	12.1244i		−	1.00000i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		−	23.2379i		−	1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$		−	23.2379i		−	1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$	−4.47214	+	10.3923i	−0.362738	+	0.842927i
$153$			20.7846i			1.68034i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−6.70820	+	8.66025i	−0.533676	+	0.688973i
$159$			7.74597i			0.614295i
$160$		0			0
$161$		0			0
$162$	−10.0623	−	7.79423i	−0.790569	−	0.612372i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	3.00000	−	3.87298i	0.232845	−	0.300602i
$167$	8.94427			0.692129			0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427			0.692129			0.346064	+	0.938211i	$0.387518\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	12.0000			0.917663
$172$		0			0
$173$	22.3607			1.70005			0.850026	−	0.526742i	$-0.176586\pi$
$173$	22.3607			1.70005			0.850026	+	0.526742i	$0.176586\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	15.4919i		−	1.15151i	−0.817624	−	0.575753i	$-0.804709\pi$
$181$		−	15.4919i		−	1.15151i	0.817624	−	0.575753i	$-0.195291\pi$
$182$		0			0
$183$	26.8328			1.98354
$184$	−10.0000	+	23.2379i	−0.737210	+	1.71312i
$185$		0			0
$186$	15.0000	+	11.6190i	1.09985	+	0.851943i
$187$		0			0
$188$	−4.47214	−	17.3205i	−0.326164	−	1.26323i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−10.0623	+	9.52628i	−0.726184	+	0.687500i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	3.50000	+	13.5554i	0.250000	+	0.968246i
$197$	−4.47214			−0.318626			−0.159313	−	0.987228i	$-0.550928\pi$
$197$	−4.47214			−0.318626			−0.159313	+	0.987228i	$0.550928\pi$
$198$		0			0
$199$		−	23.2379i		−	1.64729i	−0.567105	−	0.823646i	$-0.691937\pi$
$199$		−	23.2379i		−	1.64729i	0.567105	−	0.823646i	$-0.308063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−6.00000	−	23.2379i	−0.420084	−	1.62698i
$205$		0			0
$206$		0			0
$207$	26.8328			1.86501
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$	−2.23607	−	8.66025i	−0.153574	−	0.594789i
$213$		0			0
$214$	−9.00000	+	11.6190i	−0.615227	+	0.794255i
$215$		0			0
$216$	13.5000	+	5.80948i	0.918559	+	0.395285i
$217$		0			0
$218$	−13.4164	+	17.3205i	−0.908674	+	1.17309i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	−18.0000	+	23.2379i	−1.19734	+	1.54576i
$227$			24.2487i			1.60944i	0.593652	+	0.804722i	$0.297686\pi$
$227$			24.2487i			1.60944i	−0.593652	+	0.804722i	$0.702314\pi$
$228$	−13.4164	+	3.46410i	−0.888523	+	0.229416i
$229$			15.4919i			1.02374i	0.859064	+	0.511868i	$0.171046\pi$
$229$			15.4919i			1.02374i	−0.859064	+	0.511868i	$0.828954\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.7846i			1.36165i	0.732448	+	0.680823i	$0.238378\pi$
$233$			20.7846i			1.36165i	−0.732448	+	0.680823i	$0.761622\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−13.4164			−0.871489
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$	−12.2984	−	9.52628i	−0.790569	−	0.612372i
$243$		−	15.5885i		−	1.00000i
$244$	−30.0000	+	7.74597i	−1.92055	+	0.495885i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−20.1246	−	8.66025i	−1.27791	−	0.549927i
$249$	6.00000			0.380235
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$		−	6.92820i		−	0.432169i	−0.976375	−	0.216085i	$-0.930671\pi$
$257$		−	6.92820i		−	0.432169i	0.976375	−	0.216085i	$-0.0693287\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.94427			−0.551527			−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427			−0.551527			−0.275764	+	0.961225i	$0.588931\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			7.74597i			0.470534i	0.971931	+	0.235267i	$0.0755965\pi$
$271$			7.74597i			0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$272$	13.4164	+	24.2487i	0.813489	+	1.47029i
$273$		0			0
$274$	6.00000	−	7.74597i	0.362473	−	0.467951i
$275$		0			0
$276$	−30.0000	+	7.74597i	−1.80579	+	0.466252i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−4.47214	−	3.46410i	−0.268221	−	0.207763i
$279$			23.2379i			1.39122i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	13.4164	−	17.3205i	0.798935	−	1.03142i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.7705	−	2.59808i	−0.988212	−	0.153093i
$289$	−31.0000			−1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	31.3050			1.82885			0.914427	−	0.404750i	$-0.132641\pi$
$293$	31.3050			1.82885			0.914427	+	0.404750i	$0.132641\pi$
$294$	−10.5000	+	13.5554i	−0.612372	+	0.790569i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−20.1246	+	25.9808i	−1.15804	+	1.49502i
$303$		0			0
$304$	14.0000	−	7.74597i	0.802955	−	0.444262i
$305$		0			0
$306$	18.0000	−	23.2379i	1.02899	−	1.32842i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	15.0000	−	3.87298i	0.843816	−	0.217872i
$317$	22.3607			1.25590			0.627950	−	0.778253i	$-0.283894\pi$
$317$	22.3607			1.25590			0.627950	+	0.778253i	$0.283894\pi$
$318$	6.70820	−	8.66025i	0.376177	−	0.485643i
$319$		0			0
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$			27.7128i			1.54198i
$324$	4.50000	+	17.4284i	0.250000	+	0.968246i
$325$		0			0
$326$		0			0
$327$	−26.8328			−1.48386
$328$		0			0
$329$		0			0
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$	−6.70820	+	1.73205i	−0.368161	+	0.0950586i
$333$		0			0
$334$	−10.0000	−	7.74597i	−0.547176	−	0.423840i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−14.5344	−	11.2583i	−0.790569	−	0.612372i
$339$	−36.0000			−1.95525
$340$		0			0
$341$		0			0
$342$	−13.4164	−	10.3923i	−0.725476	−	0.561951i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−25.0000	−	19.3649i	−1.34401	−	1.04106i
$347$			10.3923i			0.557888i	0.960307	+	0.278944i	$0.0899844\pi$
$347$			10.3923i			0.557888i	−0.960307	+	0.278944i	$0.910016\pi$
$348$		0			0
$349$		−	15.4919i		−	0.829264i	−0.909989	−	0.414632i	$-0.863910\pi$
$349$		−	15.4919i		−	0.829264i	0.909989	−	0.414632i	$-0.136090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	20.7846i		−	1.10625i	−0.833097	−	0.553127i	$-0.813435\pi$
$353$		−	20.7846i		−	1.10625i	0.833097	−	0.553127i	$-0.186565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−13.4164	+	17.3205i	−0.705151	+	0.910346i
$363$		−	19.0526i		−	1.00000i
$364$		0			0
$365$		0			0
$366$	−30.0000	−	23.2379i	−1.56813	−	1.21466i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	31.3050	−	17.3205i	1.63188	−	0.902894i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−6.70820	−	25.9808i	−0.347804	−	1.34704i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−10.0000	+	23.2379i	−0.515711	+	1.19840i
$377$		0			0
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.94427			0.457031			0.228515	−	0.973540i	$-0.426613\pi$
$383$	8.94427			0.457031			0.228515	+	0.973540i	$0.426613\pi$
$384$	19.5000	−	1.93649i	0.995105	−	0.0988212i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$			61.9677i			3.13384i
$392$	7.82624	−	18.1865i	0.395285	−	0.918559i
$393$		0			0
$394$	5.00000	+	3.87298i	0.251896	+	0.195118i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−20.1246	+	25.9808i	−1.00876	+	1.30230i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−13.4164	+	31.1769i	−0.664211	+	1.54349i
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		0			0
$414$	−30.0000	−	23.2379i	−1.47442	−	1.14208i
$415$		0			0
$416$		0			0
$417$		−	6.92820i		−	0.339276i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	15.4919i		−	0.755031i	−0.926003	−	0.377515i	$-0.876779\pi$
$421$		−	15.4919i		−	0.755031i	0.926003	−	0.377515i	$-0.123221\pi$
$422$	31.3050	+	24.2487i	1.52390	+	1.18041i
$423$	26.8328			1.30466
$424$	−5.00000	+	11.6190i	−0.242821	+	0.564266i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	20.1246	−	5.19615i	0.972760	−	0.251166i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−10.0623	−	18.1865i	−0.484123	−	0.875000i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	30.0000	−	7.74597i	1.43674	−	0.370965i
$437$	35.7771			1.71145
$438$		0			0
$439$			38.7298i			1.84847i	0.381819	+	0.924237i	$0.375298\pi$
$439$			38.7298i			1.84847i	−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$		0			0
$443$			38.1051i			1.81043i	0.424955	+	0.905214i	$0.360290\pi$
$443$			38.1051i			1.81043i	−0.424955	+	0.905214i	$0.639710\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	40.2492	−	10.3923i	1.89316	−	0.488813i
$453$	−40.2492			−1.89107
$454$	21.0000	−	27.1109i	0.985579	−	1.27238i
$455$		0			0
$456$	18.0000	+	7.74597i	0.842927	+	0.362738i
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	13.4164	−	17.3205i	0.626908	−	0.809334i
$459$	36.0000			1.68034
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	18.0000	−	23.2379i	0.833834	−	1.07647i
$467$		−	24.2487i		−	1.12210i	−0.827783	−	0.561048i	$-0.810398\pi$
$467$		−	24.2487i		−	1.12210i	0.827783	−	0.561048i	$-0.189602\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	15.0000	+	11.6190i	0.688973	+	0.533676i
$475$		0			0
$476$		0			0
$477$	13.4164			0.614295
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	2.23607	+	1.73205i	0.101850	+	0.0788928i
$483$		0			0
$484$	5.50000	+	21.3014i	0.250000	+	0.968246i
$485$		0			0
$486$	−13.5000	+	17.4284i	−0.612372	+	0.790569i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	40.2492	+	17.3205i	1.82200	+	0.784063i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	15.0000	+	27.1109i	0.673520	+	1.21731i
$497$		0			0
$498$	−6.70820	−	5.19615i	−0.300602	−	0.232845i
$499$	44.0000			1.96971			0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000			1.96971			0.984855	+	0.173379i	$0.0554684\pi$
$500$		0			0
$501$		−	15.4919i		−	0.692129i
$502$		0			0
$503$	−44.7214			−1.99403			−0.997013	−	0.0772283i	$-0.975393\pi$
$503$	−44.7214			−1.99403			−0.997013	+	0.0772283i	$0.975393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	22.5167i		−	1.00000i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−21.2426	+	7.79423i	−0.938801	+	0.344459i
$513$		−	20.7846i		−	0.917663i
$514$	−6.00000	+	7.74597i	−0.264649	+	0.341660i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	38.7298i		−	1.70005i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	10.0000	+	7.74597i	0.436021	+	0.337740i
$527$	−53.6656			−2.33771
$528$		0			0
$529$	57.0000			2.47826
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	46.4758i		−	1.99815i	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$		−	46.4758i		−	1.99815i	0.0429934	−	0.999075i	$-0.486311\pi$
$542$	6.70820	−	8.66025i	0.288142	−	0.371990i
$543$	−26.8328			−1.15151
$544$	6.00000	−	38.7298i	0.257248	−	1.66053i
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−13.4164	+	3.46410i	−0.573121	+	0.147979i
$549$		−	46.4758i		−	1.98354i
$550$		0			0
$551$		0			0
$552$	40.2492	+	17.3205i	1.71312	+	0.737210i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	2.00000	+	7.74597i	0.0848189	+	0.328502i
$557$	−22.3607			−0.947452			−0.473726	−	0.880672i	$-0.657091\pi$
$557$	−22.3607			−0.947452			−0.473726	+	0.880672i	$0.657091\pi$
$558$	20.1246	−	25.9808i	0.851943	−	1.09985i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	31.1769i		−	1.31395i	−0.753912	−	0.656975i	$-0.771836\pi$
$563$		−	31.1769i		−	1.31395i	0.753912	−	0.656975i	$-0.228164\pi$
$564$	−30.0000	+	7.74597i	−1.26323	+	0.326164i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	16.5000	+	17.4284i	0.687500	+	0.726184i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	34.6591	+	26.8468i	1.44163	+	1.11668i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−35.0000	−	27.1109i	−1.44584	−	1.11994i
$587$			45.0333i			1.85872i	0.369170	+	0.929362i	$0.379642\pi$
$587$			45.0333i			1.85872i	−0.369170	+	0.929362i	$0.620358\pi$
$588$	23.4787	−	6.06218i	0.968246	−	0.250000i
$589$			30.9839i			1.27667i
$590$		0			0
$591$			7.74597i			0.318626i
$592$		0			0
$593$		−	48.4974i		−	1.99155i	−0.0918243	−	0.995775i	$-0.529270\pi$
$593$		−	48.4974i		−	1.99155i	0.0918243	−	0.995775i	$-0.470730\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−40.2492			−1.64729
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	45.0000	−	11.6190i	1.83102	−	0.472768i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−22.3607	−	3.46410i	−0.906845	−	0.140488i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−40.2492	+	10.3923i	−1.62698	+	0.420084i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6.92820i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$617$			6.92820i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$618$		0			0
$619$	−44.0000			−1.76851			−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000			−1.76851			−0.884255	+	0.467005i	$0.845333\pi$
$620$		0			0
$621$		−	46.4758i		−	1.86501i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	38.7298i		−	1.54181i	−0.636950	−	0.770905i	$-0.719804\pi$
$631$		−	38.7298i		−	1.54181i	0.636950	−	0.770905i	$-0.280196\pi$
$632$	−20.1246	−	8.66025i	−0.800514	−	0.344486i
$633$			48.4974i			1.92760i
$634$	−25.0000	−	19.3649i	−0.992877	−	0.769079i
$635$		0			0
$636$	−15.0000	+	3.87298i	−0.594789	+	0.153574i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	20.1246	+	15.5885i	0.794255	+	0.615227i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	24.0000	−	30.9839i	0.944267	−	1.21904i
$647$	44.7214			1.75818			0.879089	−	0.476658i	$-0.158152\pi$
$647$	44.7214			1.75818			0.879089	+	0.476658i	$0.158152\pi$
$648$	10.0623	−	23.3827i	0.395285	−	0.918559i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−49.1935			−1.92509			−0.962545	−	0.271122i	$-0.912605\pi$
$653$	−49.1935			−1.92509			−0.962545	+	0.271122i	$0.912605\pi$
$654$	30.0000	+	23.2379i	1.17309	+	0.908674i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			46.4758i			1.80770i	0.427850	+	0.903850i	$0.359271\pi$
$661$			46.4758i			1.80770i	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−31.3050	−	24.2487i	−1.21670	−	0.942453i
$663$		0			0
$664$	9.00000	+	3.87298i	0.349268	+	0.150301i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	4.47214	+	17.3205i	0.173032	+	0.670151i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	6.50000	+	25.1744i	0.250000	+	0.968246i
$677$	−31.3050			−1.20315			−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050			−1.20315			−0.601574	+	0.798817i	$0.705459\pi$
$678$	40.2492	+	31.1769i	1.54576	+	1.19734i
$679$		0			0
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$		−	38.1051i		−	1.45805i	−0.684486	−	0.729026i	$-0.739973\pi$
$683$		−	38.1051i		−	1.45805i	0.684486	−	0.729026i	$-0.260027\pi$
$684$	6.00000	+	23.2379i	0.229416	+	0.888523i
$685$		0			0
$686$		0			0
$687$	26.8328			1.02374
$688$		0			0
$689$		0			0
$690$		0			0
$691$	52.0000			1.97817			0.989087	−	0.147335i	$-0.0470696\pi$
$691$	52.0000			1.97817			0.989087	+	0.147335i	$0.0470696\pi$
$692$	11.1803	+	43.3013i	0.425013	+	1.64607i
$693$		0			0
$694$	9.00000	−	11.6190i	0.341635	−	0.441049i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−13.4164	+	17.3205i	−0.507819	+	0.655591i
$699$	36.0000			1.36165
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−18.0000	+	23.2379i	−0.677439	+	0.874570i
$707$		0			0
$708$		0			0
$709$		−	46.4758i		−	1.74544i	−0.488225	−	0.872718i	$-0.662356\pi$
$709$		−	46.4758i		−	1.74544i	0.488225	−	0.872718i	$-0.337644\pi$
$710$		0			0
$711$			23.2379i			0.871489i
$712$		0			0
$713$			69.2820i			2.59463i
$714$		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	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	3.35410	+	2.59808i	0.124827	+	0.0966904i
$723$			3.46410i			0.128831i
$724$	30.0000	−	7.74597i	1.11494	−	0.287877i
$725$		0			0
$726$	−16.5000	+	21.3014i	−0.612372	+	0.790569i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	13.4164	+	51.9615i	0.495885	+	1.92055i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−50.0000	−	7.74597i	−1.84302	−	0.285520i
$737$		0			0
$738$		0			0
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−44.7214			−1.64067			−0.820334	−	0.571885i	$-0.806212\pi$
$743$	−44.7214			−1.64067			−0.820334	+	0.571885i	$0.806212\pi$
$744$	−15.0000	+	34.8569i	−0.549927	+	1.27791i
$745$		0			0
$746$		0			0
$747$		−	10.3923i		−	0.380235i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			54.2218i			1.97858i	0.145962	+	0.989290i	$0.453372\pi$
$751$			54.2218i			1.97858i	−0.145962	+	0.989290i	$0.546628\pi$
$752$	31.3050	−	17.3205i	1.14157	−	0.631614i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−4.47214	−	3.46410i	−0.162435	−	0.125822i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−10.0000	−	7.74597i	−0.361315	−	0.279873i
$767$		0			0
$768$	−23.4787	−	14.7224i	−0.847215	−	0.531250i
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$	4.47214			0.160852			0.0804258	−	0.996761i	$-0.474372\pi$
$773$	4.47214			0.160852			0.0804258	+	0.996761i	$0.474372\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	53.6656	−	69.2820i	1.91908	−	2.47752i
$783$		0			0
$784$	−24.5000	+	13.5554i	−0.875000	+	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−2.23607	−	8.66025i	−0.0796566	−	0.308509i
$789$			15.4919i			0.551527i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	45.0000	−	11.6190i	1.59498	−	0.411823i
$797$	−49.1935			−1.74252			−0.871262	−	0.490819i	$-0.836698\pi$
$797$	−49.1935			−1.74252			−0.871262	+	0.490819i	$0.836698\pi$
$798$		0			0
$799$			61.9677i			2.19226i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$	13.4164			0.470534
$814$		0			0
$815$		0			0
$816$	42.0000	−	23.2379i	1.47029	−	0.813489i
$817$		0			0
$818$	29.0689	+	22.5167i	1.01637	+	0.787277i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	−13.4164	−	10.3923i	−0.467951	−	0.362473i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	45.0333i		−	1.56596i	−0.622046	−	0.782981i	$-0.713698\pi$
$827$		−	45.0333i		−	1.56596i	0.622046	−	0.782981i	$-0.286302\pi$
$828$	13.4164	+	51.9615i	0.466252	+	1.80579i
$829$			46.4758i			1.61417i	0.590434	+	0.807086i	$0.298956\pi$
$829$			46.4758i			1.61417i	−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	48.4974i		−	1.68034i
$834$	−6.00000	+	7.74597i	−0.207763	+	0.268221i
$835$		0			0
$836$		0			0
$837$	40.2492			1.39122
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	−13.4164	+	17.3205i	−0.462360	+	0.596904i
$843$		0			0
$844$	−14.0000	−	54.2218i	−0.481900	−	1.86639i
$845$		0			0
$846$	−30.0000	−	23.2379i	−1.03142	−	0.798935i
$847$		0			0
$848$	15.6525	−	8.66025i	0.537508	−	0.297394i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−27.0000	−	11.6190i	−0.922841	−	0.397128i
$857$			6.92820i			0.236663i	0.992974	+	0.118331i	$0.0377545\pi$
$857$			6.92820i			0.236663i	−0.992974	+	0.118331i	$0.962245\pi$
$858$		0			0
$859$	−44.0000			−1.50126			−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000			−1.50126			−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	44.7214			1.52233			0.761166	−	0.648557i	$-0.224627\pi$
$863$	44.7214			1.52233			0.761166	+	0.648557i	$0.224627\pi$
$864$	−4.50000	+	29.0474i	−0.153093	+	0.988212i
$865$		0			0
$866$		0			0
$867$			53.6936i			1.82353i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−40.2492	−	17.3205i	−1.36301	−	0.586546i
$873$		0			0
$874$	−40.0000	−	30.9839i	−1.35302	−	1.04804i
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$	33.5410	−	43.3013i	1.13195	−	1.46135i
$879$		−	54.2218i		−	1.82885i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	23.4787	+	18.1865i	0.790569	+	0.612372i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	33.0000	−	42.6028i	1.10866	−	1.43127i
$887$	8.94427			0.300319			0.150160	−	0.988662i	$-0.452021\pi$
$887$	8.94427			0.300319			0.150160	+	0.988662i	$0.452021\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	35.7771			1.19723
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			30.9839i			1.03222i
$902$		0			0
$903$		0			0
$904$	−54.0000	−	23.2379i	−1.79601	−	0.772881i
$905$		0			0
$906$	45.0000	+	34.8569i	1.49502	+	1.15804i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$	−46.9574	+	12.1244i	−1.55834	+	0.402361i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−13.4164	−	24.2487i	−0.444262	−	0.802955i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−30.0000	+	7.74597i	−0.991228	+	0.255934i
$917$		0			0
$918$	−40.2492	−	31.1769i	−1.32842	−	1.02899i
$919$			23.2379i			0.766548i	0.923635	+	0.383274i	$0.125203\pi$
$919$			23.2379i			0.766548i	−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$	−40.2492	+	10.3923i	−1.31841	+	0.340411i
$933$		0			0
$934$	−21.0000	+	27.1109i	−0.687141	+	0.887095i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	58.8897i		−	1.91366i	−0.290650	−	0.956830i	$-0.593871\pi$
$947$		−	58.8897i		−	1.91366i	0.290650	−	0.956830i	$-0.406129\pi$
$948$	−6.70820	−	25.9808i	−0.217872	−	0.843816i
$949$		0			0
$950$		0			0
$951$		−	38.7298i		−	1.25590i
$952$		0			0
$953$			20.7846i			0.673280i	0.941634	+	0.336640i	$0.109290\pi$
$953$			20.7846i			0.673280i	−0.941634	+	0.336640i	$0.890710\pi$
$954$	−15.0000	−	11.6190i	−0.485643	−	0.376177i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.0000			−0.935484
$962$		0			0
$963$			31.1769i			1.00466i
$964$	−1.00000	−	3.87298i	−0.0322078	−	0.124740i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	12.2984	−	28.5788i	0.395285	−	0.918559i
$969$	48.0000			1.54198
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	30.1869	−	7.79423i	0.968246	−	0.250000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−30.0000	−	54.2218i	−0.960277	−	1.73560i
$977$		−	62.3538i		−	1.99488i	−0.0715382	−	0.997438i	$-0.522791\pi$
$977$		−	62.3538i		−	1.99488i	0.0715382	−	0.997438i	$-0.477209\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			46.4758i			1.48386i
$982$		0			0
$983$	62.6099			1.99695			0.998473	−	0.0552438i	$-0.0175936\pi$
$983$	62.6099			1.99695			0.998473	+	0.0552438i	$0.0175936\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	54.2218i		−	1.72241i	−0.508257	−	0.861206i	$-0.669710\pi$
$991$		−	54.2218i		−	1.72241i	0.508257	−	0.861206i	$-0.330290\pi$
$992$	6.70820	−	43.3013i	0.212986	−	1.37482i
$993$		−	48.4974i		−	1.53902i
$994$		0			0
$995$		0			0
$996$	3.00000	+	11.6190i	0.0950586	+	0.368161i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−49.1935	−	38.1051i	−1.55719	−	1.20620i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.b.c.251.1		4
3.2	odd	2	inner	600.2.b.c.251.4		4
4.3	odd	2		2400.2.b.c.2351.4		4
5.2	odd	4		120.2.m.a.59.3	yes	4
5.3	odd	4		120.2.m.a.59.2	yes	4
5.4	even	2	inner	600.2.b.c.251.4		4
8.3	odd	2	inner	600.2.b.c.251.3		4
8.5	even	2		2400.2.b.c.2351.3		4
12.11	even	2		2400.2.b.c.2351.1		4
15.2	even	4		120.2.m.a.59.2	yes	4
15.8	even	4		120.2.m.a.59.3	yes	4
15.14	odd	2	CM	600.2.b.c.251.1		4
20.3	even	4		480.2.m.a.239.2		4
20.7	even	4		480.2.m.a.239.3		4
20.19	odd	2		2400.2.b.c.2351.1		4
24.5	odd	2		2400.2.b.c.2351.2		4
24.11	even	2	inner	600.2.b.c.251.2		4
40.3	even	4		120.2.m.a.59.1	✓	4
40.13	odd	4		480.2.m.a.239.1		4
40.19	odd	2	inner	600.2.b.c.251.2		4
40.27	even	4		120.2.m.a.59.4	yes	4
40.29	even	2		2400.2.b.c.2351.2		4
40.37	odd	4		480.2.m.a.239.4		4
60.23	odd	4		480.2.m.a.239.3		4
60.47	odd	4		480.2.m.a.239.2		4
60.59	even	2		2400.2.b.c.2351.4		4
120.29	odd	2		2400.2.b.c.2351.3		4
120.53	even	4		480.2.m.a.239.4		4
120.59	even	2	inner	600.2.b.c.251.3		4
120.77	even	4		480.2.m.a.239.1		4
120.83	odd	4		120.2.m.a.59.4	yes	4
120.107	odd	4		120.2.m.a.59.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.m.a.59.1	✓	4	40.3	even	4
120.2.m.a.59.1	✓	4	120.107	odd	4
120.2.m.a.59.2	yes	4	5.3	odd	4
120.2.m.a.59.2	yes	4	15.2	even	4
120.2.m.a.59.3	yes	4	5.2	odd	4
120.2.m.a.59.3	yes	4	15.8	even	4
120.2.m.a.59.4	yes	4	40.27	even	4
120.2.m.a.59.4	yes	4	120.83	odd	4
480.2.m.a.239.1		4	40.13	odd	4
480.2.m.a.239.1		4	120.77	even	4
480.2.m.a.239.2		4	20.3	even	4
480.2.m.a.239.2		4	60.47	odd	4
480.2.m.a.239.3		4	20.7	even	4
480.2.m.a.239.3		4	60.23	odd	4
480.2.m.a.239.4		4	40.37	odd	4
480.2.m.a.239.4		4	120.53	even	4
600.2.b.c.251.1		4	1.1	even	1	trivial
600.2.b.c.251.1		4	15.14	odd	2	CM
600.2.b.c.251.2		4	24.11	even	2	inner
600.2.b.c.251.2		4	40.19	odd	2	inner
600.2.b.c.251.3		4	8.3	odd	2	inner
600.2.b.c.251.3		4	120.59	even	2	inner
600.2.b.c.251.4		4	3.2	odd	2	inner
600.2.b.c.251.4		4	5.4	even	2	inner
2400.2.b.c.2351.1		4	12.11	even	2
2400.2.b.c.2351.1		4	20.19	odd	2
2400.2.b.c.2351.2		4	24.5	odd	2
2400.2.b.c.2351.2		4	40.29	even	2
2400.2.b.c.2351.3		4	8.5	even	2
2400.2.b.c.2351.3		4	120.29	odd	2
2400.2.b.c.2351.4		4	4.3	odd	2
2400.2.b.c.2351.4		4	60.59	even	2

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.11803 - 0.866025i) q^{2} -1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +(-1.50000 + 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+(-1.11803 - 0.866025i) q^{2} -1.73205i q^{3} +(0.500000 + 1.93649i) q^{4} +(-1.50000 + 1.93649i) q^{6} +(1.11803 - 2.59808i) q^{8} -3.00000 q^{9} +(3.35410 - 0.866025i) q^{12} +(-3.50000 + 1.93649i) q^{16} -6.92820i q^{17} +(3.35410 + 2.59808i) q^{18} -4.00000 q^{19} -8.94427 q^{23} +(-4.50000 - 1.93649i) q^{24} +5.19615i q^{27} -7.74597i q^{31} +(5.59017 + 0.866025i) q^{32} +(-6.00000 + 7.74597i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(4.47214 + 3.46410i) q^{38} +(10.0000 + 7.74597i) q^{46} -8.94427 q^{47} +(3.35410 + 6.06218i) q^{48} +7.00000 q^{49} -12.0000 q^{51} -4.47214 q^{53} +(4.50000 - 5.80948i) q^{54} +6.92820i q^{57} +15.4919i q^{61} +(-6.70820 + 8.66025i) q^{62} +(-5.50000 - 5.80948i) q^{64} +(13.4164 - 3.46410i) q^{68} +15.4919i q^{69} +(-3.35410 + 7.79423i) q^{72} +(-2.00000 - 7.74597i) q^{76} -7.74597i q^{79} +9.00000 q^{81} +3.46410i q^{83} +(-4.47214 - 17.3205i) q^{92} -13.4164 q^{93} +(10.0000 + 7.74597i) q^{94} +(1.50000 - 9.68246i) q^{96} +(-7.82624 - 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 6 * q^6 - 12 * q^9	\( 4 q + 2 q^{4} - 6 q^{6} - 12 q^{9} - 14 q^{16} - 16 q^{19} - 18 q^{24} - 24 q^{34} - 6 q^{36} + 40 q^{46} + 28 q^{49} - 48 q^{51} + 18 q^{54} - 22 q^{64} - 8 q^{76} + 36 q^{81} + 40 q^{94} + 6 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 6 * q^6 - 12 * q^9 - 14 * q^16 - 16 * q^19 - 18 * q^24 - 24 * q^34 - 6 * q^36 + 40 * q^46 + 28 * q^49 - 48 * q^51 + 18 * q^54 - 22 * q^64 - 8 * q^76 + 36 * q^81 + 40 * q^94 + 6 * q^96

Introduction

L-functions

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