LMFDB - Embedded newform 740.2.k.b.339.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [740,2,Mod(179,740)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(740, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("740.179"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 740 = 2^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	740.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$5.90892974957$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			339.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	740.339
Dual form			740.2.k.b.179.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} +3.00000 q^{9} +(3.00000 + 1.00000i) q^{10} +(5.00000 - 5.00000i) q^{13} -4.00000 q^{16} +(-3.00000 + 3.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} +(2.00000 + 4.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +10.0000 q^{26} +(-7.00000 + 7.00000i) q^{29} +(-4.00000 - 4.00000i) q^{32} -6.00000 q^{34} +6.00000i q^{36} +(6.00000 + 1.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} +10.0000i q^{41} +(6.00000 - 3.00000i) q^{45} -7.00000 q^{49} +(7.00000 - 1.00000i) q^{50} +(10.0000 + 10.0000i) q^{52} -14.0000i q^{53} -14.0000 q^{58} +(-1.00000 + 1.00000i) q^{61} -8.00000i q^{64} +(5.00000 - 15.0000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-6.00000 + 6.00000i) q^{72} -16.0000 q^{73} +(5.00000 + 7.00000i) q^{74} +(-8.00000 + 4.00000i) q^{80} +9.00000 q^{81} +(-10.0000 + 10.0000i) q^{82} +(-3.00000 + 9.00000i) q^{85} +(-3.00000 + 3.00000i) q^{89} +(9.00000 + 3.00000i) q^{90} +(5.00000 - 5.00000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{5} - 4 q^{8} + 6 q^{9} + 6 q^{10} + 10 q^{13} - 8 q^{16} - 6 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} + 20 q^{26} - 14 q^{29} - 8 q^{32} - 12 q^{34} + 12 q^{37} - 4 q^{40} + 12 q^{45}+ \cdots - 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^5 - 4 * q^8 + 6 * q^9 + 6 * q^10 + 10 * q^13 - 8 * q^16 - 6 * q^17 + 6 * q^18 + 4 * q^20 + 6 * q^25 + 20 * q^26 - 14 * q^29 - 8 * q^32 - 12 * q^34 + 12 * q^37 - 4 * q^40 + 12 * q^45 - 14 * q^49 + 14 * q^50 + 20 * q^52 - 28 * q^58 - 2 * q^61 + 10 * q^65 - 12 * q^68 - 12 * q^72 - 32 * q^73 + 10 * q^74 - 16 * q^80 + 18 * q^81 - 20 * q^82 - 6 * q^85 - 6 * q^89 + 18 * q^90 + 10 * q^97 - 14 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$.

$n$	$261$	$297$	$371$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$2$	1.00000	+	1.00000i	0.707107	+	0.707107i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$			2.00000i			1.00000i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$5$	2.00000	−	1.00000i	0.894427	−	0.447214i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$	3.00000			1.00000
$10$	3.00000	+	1.00000i	0.948683	+	0.316228i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	5.00000	−	5.00000i	1.38675	−	1.38675i	0.554700	−	0.832050i	$-0.312833\pi$
$13$	5.00000	−	5.00000i	1.38675	−	1.38675i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	0.242536	+	0.970143i	$0.422021\pi$
$18$	3.00000	+	3.00000i	0.707107	+	0.707107i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$	2.00000	+	4.00000i	0.447214	+	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	10.0000			1.96116
$27$		0			0
$28$		0			0
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.371391	+	0.928477i	$0.621119\pi$
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$	−4.00000	−	4.00000i	−0.707107	−	0.707107i
$33$		0			0
$34$	−6.00000			−1.02899
$35$		0			0
$36$			6.00000i			1.00000i
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$37$	6.00000	+	1.00000i	0.986394	+	0.164399i
$38$		0			0
$39$		0			0
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	6.00000	−	3.00000i	0.894427	−	0.447214i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	7.00000	−	1.00000i	0.989949	−	0.141421i
$51$		0			0
$52$	10.0000	+	10.0000i	1.38675	+	1.38675i
$53$		−	14.0000i		−	1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$		−	14.0000i		−	1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−14.0000			−1.83829
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	5.00000	−	15.0000i	0.620174	−	1.86052i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−6.00000	−	6.00000i	−0.727607	−	0.727607i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$	−16.0000			−1.87266			−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000			−1.87266			−0.936329	+	0.351123i	$0.885800\pi$
$74$	5.00000	+	7.00000i	0.581238	+	0.813733i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$	−8.00000	+	4.00000i	−0.894427	+	0.447214i
$81$	9.00000			1.00000
$82$	−10.0000	+	10.0000i	−1.10432	+	1.10432i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−3.00000	+	9.00000i	−0.325396	+	0.976187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	+	3.00000i	−0.317999	+	0.317999i	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−3.00000	+	3.00000i	−0.317999	+	0.317999i	0.529999	+	0.847998i	$0.322192\pi$
$90$	9.00000	+	3.00000i	0.948683	+	0.316228i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	0.913812	+	0.406138i	$0.133125\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$		0			0
$100$	8.00000	+	6.00000i	0.800000	+	0.600000i
$101$		−	2.00000i		−	0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$		−	2.00000i		−	0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			20.0000i			1.96116i
$105$		0			0
$106$	14.0000	−	14.0000i	1.35980	−	1.35980i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−13.0000	−	13.0000i	−1.24517	−	1.24517i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−13.0000	−	13.0000i	−1.24517	−	1.24517i	−0.287348	−	0.957826i	$-0.592774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	0.752577	−	0.658505i	$-0.228811\pi$
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$		0			0
$116$	−14.0000	−	14.0000i	−1.29987	−	1.29987i
$117$	15.0000	−	15.0000i	1.38675	−	1.38675i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$	−2.00000			−0.181071
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	8.00000	−	8.00000i	0.707107	−	0.707107i
$129$		0			0
$130$	20.0000	−	10.0000i	1.75412	−	0.877058i
$131$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$131$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		−	12.0000i		−	1.02899i
$137$		−	22.0000i		−	1.87959i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$		−	22.0000i		−	1.87959i	0.341743	−	0.939793i	$-0.388983\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$	−7.00000	+	21.0000i	−0.581318	+	1.74396i
$146$	−16.0000	−	16.0000i	−1.32417	−	1.32417i
$147$		0			0
$148$	−2.00000	+	12.0000i	−0.164399	+	0.986394i
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−9.00000	+	9.00000i	−0.727607	+	0.727607i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$		0			0
$160$	−12.0000	−	4.00000i	−0.948683	−	0.316228i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$	−20.0000			−1.56174
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		−	37.0000i		−	2.84615i
$170$	−12.0000	+	6.00000i	−0.920358	+	0.460179i
$171$		0			0
$172$		0			0
$173$	−26.0000			−1.97674			−0.988372	−	0.152057i	$-0.951410\pi$
$173$	−26.0000			−1.97674			−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−6.00000			−0.449719
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$	6.00000	+	12.0000i	0.447214	+	0.894427i
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	13.0000	−	4.00000i	0.955779	−	0.294086i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$191$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$192$		0			0
$193$	19.0000	−	19.0000i	1.36765	−	1.36765i	0.503871	−	0.863779i	$-0.331909\pi$
$193$	19.0000	−	19.0000i	1.36765	−	1.36765i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$		−	14.0000i		−	1.00000i
$197$			28.0000i			1.99492i	0.0712470	+	0.997459i	$0.477302\pi$
$197$			28.0000i			1.99492i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$	2.00000	+	14.0000i	0.141421	+	0.989949i
$201$		0			0
$202$	2.00000	−	2.00000i	0.140720	−	0.140720i
$203$		0			0
$204$		0			0
$205$	10.0000	+	20.0000i	0.698430	+	1.39686i
$206$		0			0
$207$		0			0
$208$	−20.0000	+	20.0000i	−1.38675	+	1.38675i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	28.0000			1.92305
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		−	26.0000i		−	1.76094i
$219$		0			0
$220$		0			0
$221$			30.0000i			2.01802i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	9.00000	−	12.0000i	0.600000	−	0.800000i
$226$			2.00000i			0.133038i
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	−30.0000			−1.98246			−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000			−1.98246			−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$		−	28.0000i		−	1.83829i
$233$	−16.0000			−1.04819			−0.524097	−	0.851658i	$-0.675597\pi$
$233$	−16.0000			−1.04819			−0.524097	+	0.851658i	$0.675597\pi$
$234$	30.0000			1.96116
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	11.0000	+	11.0000i	0.708572	+	0.708572i	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	11.0000	+	11.0000i	0.708572	+	0.708572i	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−11.0000	−	11.0000i	−0.707107	−	0.707107i
$243$		0			0
$244$	−2.00000	−	2.00000i	−0.128037	−	0.128037i
$245$	−14.0000	+	7.00000i	−0.894427	+	0.447214i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−17.0000	+	17.0000i	−1.06043	+	1.06043i	−0.0623783	+	0.998053i	$0.519869\pi$
$257$	−17.0000	+	17.0000i	−1.06043	+	1.06043i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$		0			0
$259$		0			0
$260$	30.0000	+	10.0000i	1.86052	+	0.620174i
$261$	−21.0000	+	21.0000i	−1.29987	+	1.29987i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−14.0000	−	28.0000i	−0.860013	−	1.72003i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	26.0000			1.58525			0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000			1.58525			0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	12.0000	−	12.0000i	0.727607	−	0.727607i
$273$		0			0
$274$	22.0000	−	22.0000i	1.32907	−	1.32907i
$275$		0			0
$276$		0			0
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	0.540758	−	0.841178i	$-0.318138\pi$
$277$	−5.00000	−	5.00000i	−0.300421	−	0.300421i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.0000	+	21.0000i	1.25275	+	1.25275i	0.954480	+	0.298275i	$0.0964112\pi$
$281$	21.0000	+	21.0000i	1.25275	+	1.25275i	0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	−	12.0000i	−0.707107	−	0.707107i
$289$		−	1.00000i		−	0.0588235i
$290$	−28.0000	+	14.0000i	−1.64422	+	0.822108i
$291$		0			0
$292$		−	32.0000i		−	1.87266i
$293$			34.0000i			1.98630i	0.116841	+	0.993151i	$0.462723\pi$
$293$			34.0000i			1.98630i	−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$		0			0
$296$	−14.0000	+	10.0000i	−0.813733	+	0.581238i
$297$		0			0
$298$	14.0000	+	14.0000i	0.810998	+	0.810998i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.00000	+	3.00000i	−0.0572598	+	0.171780i
$306$	−18.0000			−1.02899
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−1.00000	+	1.00000i	−0.0565233	+	0.0565233i	0.678280	+	0.734803i	$0.262726\pi$
$314$	−12.0000	+	12.0000i	−0.677199	+	0.677199i
$315$		0			0
$316$		0			0
$317$	22.0000			1.23564			0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000			1.23564			0.617822	+	0.786318i	$0.288015\pi$
$318$		0			0
$319$		0			0
$320$	−8.00000	−	16.0000i	−0.447214	−	0.894427i
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$	−5.00000	−	35.0000i	−0.277350	−	1.94145i
$326$		0			0
$327$		0			0
$328$	−20.0000	−	20.0000i	−1.10432	−	1.10432i
$329$		0			0
$330$		0			0
$331$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$331$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$332$		0			0
$333$	18.0000	+	3.00000i	0.986394	+	0.164399i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−32.0000			−1.74315			−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000			−1.74315			−0.871576	+	0.490261i	$0.836901\pi$
$338$	37.0000	−	37.0000i	2.01253	−	2.01253i
$339$		0			0
$340$	−18.0000	−	6.00000i	−0.976187	−	0.325396i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−26.0000	−	26.0000i	−1.39777	−	1.39777i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	36.0000			1.92704			0.963518	−	0.267644i	$-0.0862451\pi$
$349$	36.0000			1.92704			0.963518	+	0.267644i	$0.0862451\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.0000	−	25.0000i	−1.33062	−	1.33062i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−25.0000	−	25.0000i	−1.33062	−	1.33062i	−0.425797	−	0.904819i	$-0.640006\pi$
$354$		0			0
$355$		0			0
$356$	−6.00000	−	6.00000i	−0.317999	−	0.317999i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$	−6.00000	+	18.0000i	−0.316228	+	0.948683i
$361$			19.0000i			1.00000i
$362$	20.0000	+	20.0000i	1.05118	+	1.05118i
$363$		0			0
$364$		0			0
$365$	−32.0000	+	16.0000i	−1.67496	+	0.837478i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$			30.0000i			1.56174i
$370$	17.0000	+	9.00000i	0.883788	+	0.467888i
$371$		0			0
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			70.0000i			3.60518i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	38.0000			1.93415
$387$		0			0
$388$	10.0000	+	10.0000i	0.507673	+	0.507673i
$389$	27.0000	+	27.0000i	1.36895	+	1.36895i	0.861934	+	0.507020i	$0.169253\pi$
$389$	27.0000	+	27.0000i	1.36895	+	1.36895i	0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$		0			0
$392$	14.0000	−	14.0000i	0.707107	−	0.707107i
$393$		0			0
$394$	−28.0000	+	28.0000i	−1.41062	+	1.41062i
$395$		0			0
$396$		0			0
$397$	−12.0000			−0.602263			−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000			−0.602263			−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	−19.0000	+	19.0000i	−0.948815	+	0.948815i	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−19.0000	+	19.0000i	−0.948815	+	0.948815i	0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$		0			0
$404$	4.00000			0.199007
$405$	18.0000	−	9.00000i	0.894427	−	0.447214i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	17.0000	−	17.0000i	0.840596	−	0.840596i	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	17.0000	−	17.0000i	0.840596	−	0.840596i	0.988936	+	0.148340i	$0.0473931\pi$
$410$	−10.0000	+	30.0000i	−0.493865	+	1.48159i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−40.0000			−1.96116
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	1.00000	−	1.00000i	0.0487370	−	0.0487370i	0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$		0			0
$424$	28.0000	+	28.0000i	1.35980	+	1.35980i
$425$	3.00000	+	21.0000i	0.145521	+	1.01865i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$		0			0
$436$	26.0000	−	26.0000i	1.24517	−	1.24517i
$437$		0			0
$438$		0			0
$439$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$439$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$	−30.0000	+	30.0000i	−1.42695	+	1.42695i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−3.00000	+	9.00000i	−0.142214	+	0.426641i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−27.0000	−	27.0000i	−1.27421	−	1.27421i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−27.0000	−	27.0000i	−1.27421	−	1.27421i	−0.330350	−	0.943858i	$-0.607167\pi$
$450$	21.0000	−	3.00000i	0.989949	−	0.141421i
$451$		0			0
$452$	−2.00000	+	2.00000i	−0.0940721	+	0.0940721i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.187112	−	0.982339i	$-0.440087\pi$
$457$	25.0000	−	25.0000i	1.16945	−	1.16945i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	−30.0000	−	30.0000i	−1.40181	−	1.40181i
$459$		0			0
$460$		0			0
$461$	9.00000	+	9.00000i	0.419172	+	0.419172i	0.884918	−	0.465746i	$-0.154214\pi$
$461$	9.00000	+	9.00000i	0.419172	+	0.419172i	−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	28.0000	−	28.0000i	1.29987	−	1.29987i
$465$		0			0
$466$	−16.0000	−	16.0000i	−0.741186	−	0.741186i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	30.0000	+	30.0000i	1.38675	+	1.38675i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	42.0000i		−	1.92305i
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	35.0000	−	25.0000i	1.59586	−	1.13990i
$482$			22.0000i			1.00207i
$483$		0			0
$484$		−	22.0000i		−	1.00000i
$485$	5.00000	−	15.0000i	0.227038	−	0.681115i
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		−	4.00000i		−	0.181071i
$489$		0			0
$490$	−21.0000	−	7.00000i	−0.948683	−	0.316228i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		−	42.0000i		−	1.89158i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	22.0000	+	4.00000i	0.983870	+	0.178885i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−2.00000	−	4.00000i	−0.0889988	−	0.177998i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			44.0000i			1.95027i	0.221621	+	0.975133i	$0.428865\pi$
$509$			44.0000i			1.95027i	−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	−34.0000			−1.49968
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	20.0000	+	40.0000i	0.877058	+	1.75412i
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$	−42.0000			−1.83829
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	23.0000i		−	1.00000i
$530$	14.0000	−	42.0000i	0.608121	−	1.82436i
$531$		0			0
$532$		0			0
$533$	50.0000	+	50.0000i	2.16574	+	2.16574i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	26.0000	+	26.0000i	1.12094	+	1.12094i
$539$		0			0
$540$		0			0
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.902861	+	0.429934i	$0.141463\pi$
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.429934	+	0.902861i	$0.358537\pi$
$542$		0			0
$543$		0			0
$544$	24.0000			1.02899
$545$	−39.0000	−	13.0000i	−1.67058	−	0.556859i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	44.0000			1.87959
$549$	−3.00000	+	3.00000i	−0.128037	+	0.128037i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	10.0000i		−	0.424859i
$555$		0			0
$556$		0			0
$557$	−5.00000	−	5.00000i	−0.211857	−	0.211857i	0.593199	−	0.805056i	$-0.297865\pi$
$557$	−5.00000	−	5.00000i	−0.211857	−	0.211857i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$			42.0000i			1.77166i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	3.00000	+	1.00000i	0.126211	+	0.0420703i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	33.0000	−	33.0000i	1.38343	−	1.38343i	0.544988	−	0.838444i	$-0.316534\pi$
$569$	33.0000	−	33.0000i	1.38343	−	1.38343i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	24.0000i		−	1.00000i
$577$	23.0000	−	23.0000i	0.957503	−	0.957503i	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	23.0000	−	23.0000i	0.957503	−	0.957503i	0.999133	+	0.0416305i	$0.0132552\pi$
$578$	1.00000	−	1.00000i	0.0415945	−	0.0415945i
$579$		0			0
$580$	−42.0000	−	14.0000i	−1.74396	−	0.581318i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	32.0000	−	32.0000i	1.32417	−	1.32417i
$585$	15.0000	−	45.0000i	0.620174	−	1.86052i
$586$	−34.0000	+	34.0000i	−1.40453	+	1.40453i
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−24.0000	−	4.00000i	−0.986394	−	0.164399i
$593$		−	16.0000i		−	0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$		−	16.0000i		−	0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$		0			0
$595$		0			0
$596$			28.0000i			1.14692i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.0000	+	11.0000i	−0.894427	+	0.447214i
$606$		0			0
$607$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$607$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$	−4.00000	+	2.00000i	−0.161955	+	0.0809776i
$611$		0			0
$612$	−18.0000	−	18.0000i	−0.727607	−	0.727607i
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.0000			1.52982			0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000			1.52982			0.764911	+	0.644136i	$0.222783\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−2.00000			−0.0799361
$627$		0			0
$628$	−24.0000			−0.957704
$629$	−21.0000	+	15.0000i	−0.837325	+	0.598089i
$630$		0			0
$631$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$631$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$		0			0
$634$	22.0000	+	22.0000i	0.873732	+	0.873732i
$635$		0			0
$636$		0			0
$637$	−35.0000	+	35.0000i	−1.38675	+	1.38675i
$638$		0			0
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	−50.0000			−1.97488			−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000			−1.97488			−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−18.0000	+	18.0000i	−0.707107	+	0.707107i
$649$		0			0
$650$	30.0000	−	40.0000i	1.17670	−	1.56893i
$651$		0			0
$652$		0			0
$653$	35.0000	+	35.0000i	1.36966	+	1.36966i	0.860927	+	0.508729i	$0.169885\pi$
$653$	35.0000	+	35.0000i	1.36966	+	1.36966i	0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$		0			0
$656$		−	40.0000i		−	1.56174i
$657$	−48.0000			−1.87266
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−31.0000	+	31.0000i	−1.20576	+	1.20576i	−0.233373	+	0.972387i	$0.574976\pi$
$661$	−31.0000	+	31.0000i	−1.20576	+	1.20576i	−0.972387	+	0.233373i	$0.925024\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	15.0000	+	21.0000i	0.581238	+	0.813733i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$	−32.0000	−	32.0000i	−1.23259	−	1.23259i
$675$		0			0
$676$	74.0000			2.84615
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$		0			0
$680$	−12.0000	−	24.0000i	−0.460179	−	0.920358i
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−22.0000	−	44.0000i	−0.840577	−	1.68115i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−70.0000	−	70.0000i	−2.66679	−	2.66679i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		−	52.0000i		−	1.97674i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−30.0000	−	30.0000i	−1.13633	−	1.13633i
$698$	36.0000	+	36.0000i	1.36262	+	1.36262i
$699$		0			0
$700$		0			0
$701$	21.0000	+	21.0000i	0.793159	+	0.793159i	0.982006	−	0.188847i	$-0.0604752\pi$
$701$	21.0000	+	21.0000i	0.793159	+	0.793159i	−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		−	50.0000i		−	1.88177i
$707$		0			0
$708$		0			0
$709$	−37.0000	−	37.0000i	−1.38956	−	1.38956i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−37.0000	−	37.0000i	−1.38956	−	1.38956i	−0.563337	−	0.826227i	$-0.690483\pi$
$710$		0			0
$711$		0			0
$712$		−	12.0000i		−	0.449719i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	−24.0000	+	12.0000i	−0.894427	+	0.447214i
$721$		0			0
$722$	−19.0000	+	19.0000i	−0.707107	+	0.707107i
$723$		0			0
$724$			40.0000i			1.48659i
$725$	7.00000	+	49.0000i	0.259973	+	1.81981i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	−48.0000	−	16.0000i	−1.77656	−	0.592187i
$731$		0			0
$732$		0			0
$733$	−4.00000			−0.147743			−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000			−0.147743			−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−30.0000	+	30.0000i	−1.10432	+	1.10432i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	8.00000	+	26.0000i	0.294086	+	0.955779i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	28.0000	−	14.0000i	1.02584	−	0.512920i
$746$	−14.0000	−	14.0000i	−0.512576	−	0.512576i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$	−70.0000	+	70.0000i	−2.54925	+	2.54925i
$755$		0			0
$756$		0			0
$757$	35.0000	−	35.0000i	1.27210	−	1.27210i	0.327111	−	0.944986i	$-0.393925\pi$
$757$	35.0000	−	35.0000i	1.27210	−	1.27210i	0.944986	−	0.327111i	$-0.106075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	40.0000i		−	1.45000i	−0.688749	−	0.724999i	$-0.741840\pi$
$761$		−	40.0000i		−	1.45000i	0.688749	−	0.724999i	$-0.258160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−9.00000	+	27.0000i	−0.325396	+	0.976187i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−37.0000	+	37.0000i	−1.33425	+	1.33425i	−0.432731	+	0.901523i	$0.642450\pi$
$769$	−37.0000	+	37.0000i	−1.33425	+	1.33425i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$	38.0000	+	38.0000i	1.36765	+	1.36765i
$773$		−	44.0000i		−	1.58257i	−0.611448	−	0.791285i	$-0.709412\pi$
$773$		−	44.0000i		−	1.58257i	0.611448	−	0.791285i	$-0.290588\pi$
$774$		0			0
$775$		0			0
$776$			20.0000i			0.717958i
$777$		0			0
$778$			54.0000i			1.93599i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$	12.0000	+	24.0000i	0.428298	+	0.856597i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−56.0000			−1.99492
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$			10.0000i			0.355110i
$794$	−12.0000	−	12.0000i	−0.425864	−	0.425864i
$795$		0			0
$796$		0			0
$797$	−15.0000	−	15.0000i	−0.531327	−	0.531327i	0.389640	−	0.920967i	$-0.372599\pi$
$797$	−15.0000	−	15.0000i	−0.531327	−	0.531327i	−0.920967	+	0.389640i	$0.872599\pi$
$798$		0			0
$799$		0			0
$800$	−28.0000	+	4.00000i	−0.989949	+	0.141421i
$801$	−9.00000	+	9.00000i	−0.317999	+	0.317999i
$802$	−38.0000			−1.34183
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	4.00000	+	4.00000i	0.140720	+	0.140720i
$809$	23.0000	−	23.0000i	0.808637	−	0.808637i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	23.0000	−	23.0000i	0.808637	−	0.808637i	0.984428	+	0.175791i	$0.0562482\pi$
$810$	27.0000	+	9.00000i	0.948683	+	0.316228i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	34.0000			1.18878
$819$		0			0
$820$	−40.0000	+	20.0000i	−1.39686	+	0.698430i
$821$	28.0000			0.977207			0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000			0.977207			0.488603	+	0.872506i	$0.337507\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	37.0000	−	37.0000i	1.28506	−	1.28506i	0.347314	−	0.937749i	$-0.387094\pi$
$829$	37.0000	−	37.0000i	1.28506	−	1.28506i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		0			0
$832$	−40.0000	−	40.0000i	−1.38675	−	1.38675i
$833$	21.0000	−	21.0000i	0.727607	−	0.727607i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$	2.00000			0.0689246
$843$		0			0
$844$		0			0
$845$	−37.0000	−	74.0000i	−1.27284	−	2.54568i
$846$		0			0
$847$		0			0
$848$			56.0000i			1.92305i
$849$		0			0
$850$	−18.0000	+	24.0000i	−0.617395	+	0.823193i
$851$		0			0
$852$		0			0
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	0.787505	−	0.616308i	$-0.211372\pi$
$853$	5.00000	+	5.00000i	0.171197	+	0.171197i	−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	25.0000	−	25.0000i	0.853984	−	0.853984i	0.990621	+	0.136637i	$0.0436295\pi$
$858$		0			0
$859$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$859$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−52.0000	+	26.0000i	−1.76805	+	0.884027i
$866$	−34.0000	+	34.0000i	−1.15537	+	1.15537i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	52.0000			1.76094
$873$	15.0000	−	15.0000i	0.507673	−	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			12.0000i			0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$			12.0000i			0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	32.0000i		−	1.07811i	−0.842271	−	0.539054i	$-0.818782\pi$
$881$		−	32.0000i		−	1.07811i	0.842271	−	0.539054i	$-0.181218\pi$
$882$	−21.0000	−	21.0000i	−0.707107	−	0.707107i
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$	−60.0000			−2.01802
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	−12.0000	+	6.00000i	−0.402241	+	0.201120i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	54.0000i		−	1.80200i
$899$		0			0
$900$	24.0000	+	18.0000i	0.800000	+	0.600000i
$901$	42.0000	+	42.0000i	1.39922	+	1.39922i
$902$		0			0
$903$		0			0
$904$	−4.00000			−0.133038
$905$	40.0000	−	20.0000i	1.32964	−	0.664822i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		−	6.00000i		−	0.199007i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$	50.0000			1.65385
$915$		0			0
$916$		−	60.0000i		−	1.98246i
$917$		0			0
$918$		0			0
$919$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$919$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$920$		0			0
$921$		0			0
$922$			18.0000i			0.592798i
$923$		0			0
$924$		0			0
$925$	22.0000	−	21.0000i	0.723356	−	0.690476i
$926$		0			0
$927$		0			0
$928$	56.0000			1.83829
$929$			46.0000i			1.50921i	0.656179	+	0.754606i	$0.272172\pi$
$929$			46.0000i			1.50921i	−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$		0			0
$932$		−	32.0000i		−	1.04819i
$933$		0			0
$934$		0			0
$935$		0			0
$936$			60.0000i			1.96116i
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.0000			0.651981			0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000			0.651981			0.325991	+	0.945373i	$0.394302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$		0			0
$949$	−80.0000	+	80.0000i	−2.59691	+	2.59691i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$	42.0000	−	42.0000i	1.35980	−	1.35980i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	31.0000i		−	1.00000i
$962$	60.0000	+	10.0000i	1.93448	+	0.322413i
$963$		0			0
$964$	−22.0000	+	22.0000i	−0.708572	+	0.708572i
$965$	19.0000	−	57.0000i	0.611632	−	1.83489i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	22.0000	−	22.0000i	0.707107	−	0.707107i
$969$		0			0
$970$	20.0000	−	10.0000i	0.642161	−	0.321081i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	4.00000	−	4.00000i	0.128037	−	0.128037i
$977$	27.0000	+	27.0000i	0.863807	+	0.863807i	0.991778	−	0.127971i	$-0.0408466\pi$
$977$	27.0000	+	27.0000i	0.863807	+	0.863807i	−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$		0			0
$980$	−14.0000	−	28.0000i	−0.447214	−	0.894427i
$981$	−39.0000	−	39.0000i	−1.24517	−	1.24517i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	28.0000	+	56.0000i	0.892154	+	1.78431i
$986$	42.0000	−	42.0000i	1.33755	−	1.33755i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$991$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−25.0000	+	25.0000i	−0.791758	+	0.791758i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−25.0000	+	25.0000i	−0.791758	+	0.791758i	0.190022	+	0.981780i	$0.439144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	740.2.k.b.339.1	yes	2
4.3	odd	2	CM	740.2.k.b.339.1	yes	2
5.4	even	2		740.2.k.a.339.1	yes	2
20.19	odd	2		740.2.k.a.339.1	yes	2
37.31	odd	4		740.2.k.a.179.1	✓	2
148.31	even	4		740.2.k.a.179.1	✓	2
185.179	odd	4	inner	740.2.k.b.179.1	yes	2
740.179	even	4	inner	740.2.k.b.179.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
740.2.k.a.179.1	✓	2	37.31	odd	4
740.2.k.a.179.1	✓	2	148.31	even	4
740.2.k.a.339.1	yes	2	5.4	even	2
740.2.k.a.339.1	yes	2	20.19	odd	2
740.2.k.b.179.1	yes	2	185.179	odd	4	inner
740.2.k.b.179.1	yes	2	740.179	even	4	inner
740.2.k.b.339.1	yes	2	1.1	even	1	trivial
740.2.k.b.339.1	yes	2	4.3	odd	2	CM

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:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{2} +2.00000i q^{4} +(2.00000 - 1.00000i) q^{5} +(-2.00000 + 2.00000i) q^{8} +3.00000 q^{9} +(3.00000 + 1.00000i) q^{10} +(5.00000 - 5.00000i) q^{13} -4.00000 q^{16} +(-3.00000 + 3.00000i) q^{17} +(3.00000 + 3.00000i) q^{18} +(2.00000 + 4.00000i) q^{20} +(3.00000 - 4.00000i) q^{25} +10.0000 q^{26} +(-7.00000 + 7.00000i) q^{29} +(-4.00000 - 4.00000i) q^{32} -6.00000 q^{34} +6.00000i q^{36} +(6.00000 + 1.00000i) q^{37} +(-2.00000 + 6.00000i) q^{40} +10.0000i q^{41} +(6.00000 - 3.00000i) q^{45} -7.00000 q^{49} +(7.00000 - 1.00000i) q^{50} +(10.0000 + 10.0000i) q^{52} -14.0000i q^{53} -14.0000 q^{58} +(-1.00000 + 1.00000i) q^{61} -8.00000i q^{64} +(5.00000 - 15.0000i) q^{65} +(-6.00000 - 6.00000i) q^{68} +(-6.00000 + 6.00000i) q^{72} -16.0000 q^{73} +(5.00000 + 7.00000i) q^{74} +(-8.00000 + 4.00000i) q^{80} +9.00000 q^{81} +(-10.0000 + 10.0000i) q^{82} +(-3.00000 + 9.00000i) q^{85} +(-3.00000 + 3.00000i) q^{89} +(9.00000 + 3.00000i) q^{90} +(5.00000 - 5.00000i) q^{97} +(-7.00000 - 7.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{5} - 4 q^{8} + 6 q^{9} + 6 q^{10} + 10 q^{13} - 8 q^{16} - 6 q^{17} + 6 q^{18} + 4 q^{20} + 6 q^{25} + 20 q^{26} - 14 q^{29} - 8 q^{32} - 12 q^{34} + 12 q^{37} - 4 q^{40} + 12 q^{45}+ \cdots - 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^5 - 4 * q^8 + 6 * q^9 + 6 * q^10 + 10 * q^13 - 8 * q^16 - 6 * q^17 + 6 * q^18 + 4 * q^20 + 6 * q^25 + 20 * q^26 - 14 * q^29 - 8 * q^32 - 12 * q^34 + 12 * q^37 - 4 * q^40 + 12 * q^45 - 14 * q^49 + 14 * q^50 + 20 * q^52 - 28 * q^58 - 2 * q^61 + 10 * q^65 - 12 * q^68 - 12 * q^72 - 32 * q^73 + 10 * q^74 - 16 * q^80 + 18 * q^81 - 20 * q^82 - 6 * q^85 - 6 * q^89 + 18 * q^90 + 10 * q^97 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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