LMFDB - Embedded newform 490.2.c.g.99.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,2,Mod(99,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("490.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	490.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.91266969904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	490.99
Dual form			490.2.c.g.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -1.00000i q^{8} +3.00000 q^{9} +(-2.12132 - 0.707107i) q^{10} -4.00000 q^{11} +4.24264i q^{13} +1.00000 q^{16} +4.24264i q^{17} +3.00000i q^{18} -5.65685 q^{19} +(0.707107 - 2.12132i) q^{20} -4.00000i q^{22} +(-4.00000 - 3.00000i) q^{25} -4.24264 q^{26} +4.00000 q^{29} -5.65685 q^{31} +1.00000i q^{32} -4.24264 q^{34} -3.00000 q^{36} -6.00000i q^{37} -5.65685i q^{38} +(2.12132 + 0.707107i) q^{40} -1.41421 q^{41} +12.0000i q^{43} +4.00000 q^{44} +(-2.12132 + 6.36396i) q^{45} +(3.00000 - 4.00000i) q^{50} -4.24264i q^{52} +12.0000i q^{53} +(2.82843 - 8.48528i) q^{55} +4.00000i q^{58} +11.3137 q^{59} +7.07107 q^{61} -5.65685i q^{62} -1.00000 q^{64} +(-9.00000 - 3.00000i) q^{65} -12.0000i q^{67} -4.24264i q^{68} +8.00000 q^{71} -3.00000i q^{72} -4.24264i q^{73} +6.00000 q^{74} +5.65685 q^{76} +(-0.707107 + 2.12132i) q^{80} +9.00000 q^{81} -1.41421i q^{82} +16.9706i q^{83} +(-9.00000 - 3.00000i) q^{85} -12.0000 q^{86} +4.00000i q^{88} +4.24264 q^{89} +(-6.36396 - 2.12132i) q^{90} +(4.00000 - 12.0000i) q^{95} -4.24264i q^{97} -12.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 12 * q^9	$ 4 q - 4 q^{4} + 12 q^{9} - 16 q^{11} + 4 q^{16} - 16 q^{25} + 16 q^{29} - 12 q^{36} + 16 q^{44} + 12 q^{50} - 4 q^{64} - 36 q^{65} + 32 q^{71} + 24 q^{74} + 36 q^{81} - 36 q^{85} - 48 q^{86} + 16 q^{95} - 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^9 - 16 * q^11 + 4 * q^16 - 16 * q^25 + 16 * q^29 - 12 * q^36 + 16 * q^44 + 12 * q^50 - 4 * q^64 - 36 * q^65 + 32 * q^71 + 24 * q^74 + 36 * q^81 - 36 * q^85 - 48 * q^86 + 16 * q^95 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−1.00000			−0.500000
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	3.00000			1.00000
$10$	−2.12132	−	0.707107i	−0.670820	−	0.223607i
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$			3.00000i			0.707107i
$19$	−5.65685			−1.29777			−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685			−1.29777			−0.648886	+	0.760886i	$0.724765\pi$
$20$	0.707107	−	2.12132i	0.158114	−	0.474342i
$21$		0			0
$22$		−	4.00000i		−	0.852803i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$	−4.24264			−0.832050
$27$		0			0
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	−5.65685			−1.01600			−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685			−1.01600			−0.508001	+	0.861357i	$0.669615\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−4.24264			−0.727607
$35$		0			0
$36$	−3.00000			−0.500000
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		−	5.65685i		−	0.917663i
$39$		0			0
$40$	2.12132	+	0.707107i	0.335410	+	0.111803i
$41$	−1.41421			−0.220863			−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421			−0.220863			−0.110432	+	0.993884i	$0.535223\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	4.00000			0.603023
$45$	−2.12132	+	6.36396i	−0.316228	+	0.948683i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$		0			0
$50$	3.00000	−	4.00000i	0.424264	−	0.565685i
$51$		0			0
$52$		−	4.24264i		−	0.588348i
$53$			12.0000i			1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$			12.0000i			1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$		0			0
$55$	2.82843	−	8.48528i	0.381385	−	1.14416i
$56$		0			0
$57$		0			0
$58$			4.00000i			0.525226i
$59$	11.3137			1.47292			0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137			1.47292			0.736460	+	0.676481i	$0.236496\pi$
$60$		0			0
$61$	7.07107			0.905357			0.452679	−	0.891674i	$-0.350468\pi$
$61$	7.07107			0.905357			0.452679	+	0.891674i	$0.350468\pi$
$62$		−	5.65685i		−	0.718421i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−9.00000	−	3.00000i	−1.11631	−	0.372104i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		−	4.24264i		−	0.514496i
$69$		0			0
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		−	3.00000i		−	0.353553i
$73$		−	4.24264i		−	0.496564i	−0.968688	−	0.248282i	$-0.920134\pi$
$73$		−	4.24264i		−	0.496564i	0.968688	−	0.248282i	$-0.0798659\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$	5.65685			0.648886
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−0.707107	+	2.12132i	−0.0790569	+	0.237171i
$81$	9.00000			1.00000
$82$		−	1.41421i		−	0.156174i
$83$			16.9706i			1.86276i	0.364047	+	0.931381i	$0.381395\pi$
$83$			16.9706i			1.86276i	−0.364047	+	0.931381i	$0.618605\pi$
$84$		0			0
$85$	−9.00000	−	3.00000i	−0.976187	−	0.325396i
$86$	−12.0000			−1.29399
$87$		0			0
$88$			4.00000i			0.426401i
$89$	4.24264			0.449719			0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264			0.449719			0.224860	+	0.974391i	$0.427808\pi$
$90$	−6.36396	−	2.12132i	−0.670820	−	0.223607i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	12.0000i	0.410391	−	1.23117i
$96$		0			0
$97$		−	4.24264i		−	0.430775i	−0.976529	−	0.215387i	$-0.930899\pi$
$97$		−	4.24264i		−	0.430775i	0.976529	−	0.215387i	$-0.0691014\pi$
$98$		0			0
$99$	−12.0000			−1.20605
$100$	4.00000	+	3.00000i	0.400000	+	0.300000i
$101$	9.89949			0.985037			0.492518	−	0.870302i	$-0.336076\pi$
$101$	9.89949			0.985037			0.492518	+	0.870302i	$0.336076\pi$
$102$		0			0
$103$		−	16.9706i		−	1.67216i	−0.548608	−	0.836080i	$-0.684842\pi$
$103$		−	16.9706i		−	1.67216i	0.548608	−	0.836080i	$-0.315158\pi$
$104$	4.24264			0.416025
$105$		0			0
$106$	−12.0000			−1.16554
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	12.0000			1.14939			0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000			1.14939			0.574696	+	0.818367i	$0.305120\pi$
$110$	8.48528	+	2.82843i	0.809040	+	0.269680i
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	−4.00000			−0.371391
$117$			12.7279i			1.17670i
$118$			11.3137i			1.04151i
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$			7.07107i			0.640184i
$123$		0			0
$124$	5.65685			0.508001
$125$	9.19239	−	6.36396i	0.822192	−	0.569210i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	3.00000	−	9.00000i	0.263117	−	0.789352i
$131$	16.9706			1.48272			0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706			1.48272			0.741362	+	0.671105i	$0.234180\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		0			0
$136$	4.24264			0.363803
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−5.65685			−0.479808			−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685			−0.479808			−0.239904	+	0.970797i	$0.577116\pi$
$140$		0			0
$141$		0			0
$142$			8.00000i			0.671345i
$143$		−	16.9706i		−	1.41915i
$144$	3.00000			0.250000
$145$	−2.82843	+	8.48528i	−0.234888	+	0.704664i
$146$	4.24264			0.351123
$147$		0			0
$148$			6.00000i			0.493197i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$			5.65685i			0.458831i
$153$			12.7279i			1.02899i
$154$		0			0
$155$	4.00000	−	12.0000i	0.321288	−	0.963863i
$156$		0			0
$157$			4.24264i			0.338600i	0.985565	+	0.169300i	$0.0541506\pi$
$157$			4.24264i			0.338600i	−0.985565	+	0.169300i	$0.945849\pi$
$158$		0			0
$159$		0			0
$160$	−2.12132	−	0.707107i	−0.167705	−	0.0559017i
$161$		0			0
$162$			9.00000i			0.707107i
$163$			12.0000i			0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$			12.0000i			0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	1.41421			0.110432
$165$		0			0
$166$	−16.9706			−1.31717
$167$		−	16.9706i		−	1.31322i	−0.754230	−	0.656611i	$-0.771989\pi$
$167$		−	16.9706i		−	1.31322i	0.754230	−	0.656611i	$-0.228011\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$	3.00000	−	9.00000i	0.230089	−	0.690268i
$171$	−16.9706			−1.29777
$172$		−	12.0000i		−	0.914991i
$173$			4.24264i			0.322562i	0.986909	+	0.161281i	$0.0515625\pi$
$173$			4.24264i			0.322562i	−0.986909	+	0.161281i	$0.948437\pi$
$174$		0			0
$175$		0			0
$176$	−4.00000			−0.301511
$177$		0			0
$178$			4.24264i			0.317999i
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$	2.12132	−	6.36396i	0.158114	−	0.474342i
$181$	−15.5563			−1.15629			−0.578147	−	0.815933i	$-0.696224\pi$
$181$	−15.5563			−1.15629			−0.578147	+	0.815933i	$0.696224\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.7279	+	4.24264i	0.935775	+	0.311925i
$186$		0			0
$187$		−	16.9706i		−	1.24101i
$188$		0			0
$189$		0			0
$190$	12.0000	+	4.00000i	0.870572	+	0.290191i
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$	4.24264			0.304604
$195$		0			0
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		−	12.0000i		−	0.852803i
$199$	−11.3137			−0.802008			−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137			−0.802008			−0.401004	+	0.916076i	$0.631339\pi$
$200$	−3.00000	+	4.00000i	−0.212132	+	0.282843i
$201$		0			0
$202$			9.89949i			0.696526i
$203$		0			0
$204$		0			0
$205$	1.00000	−	3.00000i	0.0698430	−	0.209529i
$206$	16.9706			1.18240
$207$		0			0
$208$			4.24264i			0.294174i
$209$	22.6274			1.56517
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		−	12.0000i		−	0.824163i
$213$		0			0
$214$	−12.0000			−0.820303
$215$	−25.4558	−	8.48528i	−1.73607	−	0.578691i
$216$		0			0
$217$		0			0
$218$			12.0000i			0.812743i
$219$		0			0
$220$	−2.82843	+	8.48528i	−0.190693	+	0.572078i
$221$	−18.0000			−1.21081
$222$		0			0
$223$		−	16.9706i		−	1.13643i	−0.822879	−	0.568216i	$-0.807634\pi$
$223$		−	16.9706i		−	1.13643i	0.822879	−	0.568216i	$-0.192366\pi$
$224$		0			0
$225$	−12.0000	−	9.00000i	−0.800000	−	0.600000i
$226$		0			0
$227$		−	16.9706i		−	1.12638i	−0.826329	−	0.563188i	$-0.809575\pi$
$227$		−	16.9706i		−	1.12638i	0.826329	−	0.563188i	$-0.190425\pi$
$228$		0			0
$229$	−15.5563			−1.02799			−0.513996	−	0.857792i	$-0.671835\pi$
$229$	−15.5563			−1.02799			−0.513996	+	0.857792i	$0.671835\pi$
$230$		0			0
$231$		0			0
$232$		−	4.00000i		−	0.262613i
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−12.7279			−0.832050
$235$		0			0
$236$	−11.3137			−0.736460
$237$		0			0
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	−15.5563			−1.00207			−0.501036	−	0.865426i	$-0.667048\pi$
$241$	−15.5563			−1.00207			−0.501036	+	0.865426i	$0.667048\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$	−7.07107			−0.452679
$245$		0			0
$246$		0			0
$247$		−	24.0000i		−	1.52708i
$248$			5.65685i			0.359211i
$249$		0			0
$250$	6.36396	+	9.19239i	0.402492	+	0.581378i
$251$	−5.65685			−0.357057			−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685			−0.357057			−0.178529	+	0.983935i	$0.557134\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			0.0625000
$257$			12.7279i			0.793946i	0.917830	+	0.396973i	$0.129939\pi$
$257$			12.7279i			0.793946i	−0.917830	+	0.396973i	$0.870061\pi$
$258$		0			0
$259$		0			0
$260$	9.00000	+	3.00000i	0.558156	+	0.186052i
$261$	12.0000			0.742781
$262$			16.9706i			1.04844i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−25.4558	−	8.48528i	−1.56374	−	0.521247i
$266$		0			0
$267$		0			0
$268$			12.0000i			0.733017i
$269$	−21.2132			−1.29339			−0.646696	−	0.762748i	$-0.723850\pi$
$269$	−21.2132			−1.29339			−0.646696	+	0.762748i	$0.723850\pi$
$270$		0			0
$271$	−11.3137			−0.687259			−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137			−0.687259			−0.343629	+	0.939105i	$0.611656\pi$
$272$			4.24264i			0.257248i
$273$		0			0
$274$	−6.00000			−0.362473
$275$	16.0000	+	12.0000i	0.964836	+	0.723627i
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		−	5.65685i		−	0.339276i
$279$	−16.9706			−1.01600
$280$		0			0
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−8.00000			−0.474713
$285$		0			0
$286$	16.9706			1.00349
$287$		0			0
$288$			3.00000i			0.176777i
$289$	−1.00000			−0.0588235
$290$	−8.48528	−	2.82843i	−0.498273	−	0.166091i
$291$		0			0
$292$			4.24264i			0.248282i
$293$		−	4.24264i		−	0.247858i	−0.992291	−	0.123929i	$-0.960451\pi$
$293$		−	4.24264i		−	0.247858i	0.992291	−	0.123929i	$-0.0395495\pi$
$294$		0			0
$295$	−8.00000	+	24.0000i	−0.465778	+	1.39733i
$296$	−6.00000			−0.348743
$297$		0			0
$298$		−	10.0000i		−	0.579284i
$299$		0			0
$300$		0			0
$301$		0			0
$302$			8.00000i			0.460348i
$303$		0			0
$304$	−5.65685			−0.324443
$305$	−5.00000	+	15.0000i	−0.286299	+	0.858898i
$306$	−12.7279			−0.727607
$307$			16.9706i			0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$			16.9706i			0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$		0			0
$309$		0			0
$310$	12.0000	+	4.00000i	0.681554	+	0.227185i
$311$	16.9706			0.962312			0.481156	−	0.876635i	$-0.340217\pi$
$311$	16.9706			0.962312			0.481156	+	0.876635i	$0.340217\pi$
$312$		0			0
$313$			12.7279i			0.719425i	0.933063	+	0.359712i	$0.117125\pi$
$313$			12.7279i			0.719425i	−0.933063	+	0.359712i	$0.882875\pi$
$314$	−4.24264			−0.239426
$315$		0			0
$316$		0			0
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$	−16.0000			−0.895828
$320$	0.707107	−	2.12132i	0.0395285	−	0.118585i
$321$		0			0
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$	−9.00000			−0.500000
$325$	12.7279	−	16.9706i	0.706018	−	0.941357i
$326$	−12.0000			−0.664619
$327$		0			0
$328$			1.41421i			0.0780869i
$329$		0			0
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		−	16.9706i		−	0.931381i
$333$		−	18.0000i		−	0.986394i
$334$	16.9706			0.928588
$335$	25.4558	+	8.48528i	1.39080	+	0.463600i
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$		−	5.00000i		−	0.271964i
$339$		0			0
$340$	9.00000	+	3.00000i	0.488094	+	0.162698i
$341$	22.6274			1.22534
$342$		−	16.9706i		−	0.917663i
$343$		0			0
$344$	12.0000			0.646997
$345$		0			0
$346$	−4.24264			−0.228086
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	24.0416			1.28692			0.643459	−	0.765480i	$-0.277498\pi$
$349$	24.0416			1.28692			0.643459	+	0.765480i	$0.277498\pi$
$350$		0			0
$351$		0			0
$352$		−	4.00000i		−	0.213201i
$353$			21.2132i			1.12906i	0.825411	+	0.564532i	$0.190943\pi$
$353$			21.2132i			1.12906i	−0.825411	+	0.564532i	$0.809057\pi$
$354$		0			0
$355$	−5.65685	+	16.9706i	−0.300235	+	0.900704i
$356$	−4.24264			−0.224860
$357$		0			0
$358$		−	20.0000i		−	1.05703i
$359$	−16.0000			−0.844448			−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000			−0.844448			−0.422224	+	0.906492i	$0.638750\pi$
$360$	6.36396	+	2.12132i	0.335410	+	0.111803i
$361$	13.0000			0.684211
$362$		−	15.5563i		−	0.817624i
$363$		0			0
$364$		0			0
$365$	9.00000	+	3.00000i	0.471082	+	0.157027i
$366$		0			0
$367$			16.9706i			0.885856i	0.896557	+	0.442928i	$0.146060\pi$
$367$			16.9706i			0.885856i	−0.896557	+	0.442928i	$0.853940\pi$
$368$		0			0
$369$	−4.24264			−0.220863
$370$	−4.24264	+	12.7279i	−0.220564	+	0.661693i
$371$		0			0
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	16.9706			0.877527
$375$		0			0
$376$		0			0
$377$			16.9706i			0.874028i
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$	−4.00000	+	12.0000i	−0.205196	+	0.615587i
$381$		0			0
$382$			8.00000i			0.409316i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$	−24.0000			−1.22157
$387$			36.0000i			1.82998i
$388$			4.24264i			0.215387i
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	12.0000			0.604551
$395$		0			0
$396$	12.0000			0.603023
$397$		−	29.6985i		−	1.49052i	−0.666772	−	0.745262i	$-0.732324\pi$
$397$		−	29.6985i		−	1.49052i	0.666772	−	0.745262i	$-0.267676\pi$
$398$		−	11.3137i		−	0.567105i
$399$		0			0
$400$	−4.00000	−	3.00000i	−0.200000	−	0.150000i
$401$	−8.00000			−0.399501			−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000			−0.399501			−0.199750	+	0.979847i	$0.564013\pi$
$402$		0			0
$403$		−	24.0000i		−	1.19553i
$404$	−9.89949			−0.492518
$405$	−6.36396	+	19.0919i	−0.316228	+	0.948683i
$406$		0			0
$407$			24.0000i			1.18964i
$408$		0			0
$409$	9.89949			0.489499			0.244749	−	0.969586i	$-0.421294\pi$
$409$	9.89949			0.489499			0.244749	+	0.969586i	$0.421294\pi$
$410$	3.00000	+	1.00000i	0.148159	+	0.0493865i
$411$		0			0
$412$			16.9706i			0.836080i
$413$		0			0
$414$		0			0
$415$	−36.0000	−	12.0000i	−1.76717	−	0.589057i
$416$	−4.24264			−0.208013
$417$		0			0
$418$			22.6274i			1.10674i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$			12.0000i			0.584151i
$423$		0			0
$424$	12.0000			0.582772
$425$	12.7279	−	16.9706i	0.617395	−	0.823193i
$426$		0			0
$427$		0			0
$428$		−	12.0000i		−	0.580042i
$429$		0			0
$430$	8.48528	−	25.4558i	0.409197	−	1.22759i
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$		−	29.6985i		−	1.42722i	−0.700544	−	0.713609i	$-0.747059\pi$
$433$		−	29.6985i		−	1.42722i	0.700544	−	0.713609i	$-0.252941\pi$
$434$		0			0
$435$		0			0
$436$	−12.0000			−0.574696
$437$		0			0
$438$		0			0
$439$	39.5980			1.88991			0.944954	−	0.327203i	$-0.106106\pi$
$439$	39.5980			1.88991			0.944954	+	0.327203i	$0.106106\pi$
$440$	−8.48528	−	2.82843i	−0.404520	−	0.134840i
$441$		0			0
$442$		−	18.0000i		−	0.856173i
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	−3.00000	+	9.00000i	−0.142214	+	0.426641i
$446$	16.9706			0.803579
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$	9.00000	−	12.0000i	0.424264	−	0.565685i
$451$	5.65685			0.266371
$452$		0			0
$453$		0			0
$454$	16.9706			0.796468
$455$		0			0
$456$		0			0
$457$		−	24.0000i		−	1.12267i	−0.827588	−	0.561336i	$-0.810287\pi$
$457$		−	24.0000i		−	1.12267i	0.827588	−	0.561336i	$-0.189713\pi$
$458$		−	15.5563i		−	0.726900i
$459$		0			0
$460$		0			0
$461$	−21.2132			−0.987997			−0.493999	−	0.869463i	$-0.664465\pi$
$461$	−21.2132			−0.987997			−0.493999	+	0.869463i	$0.664465\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	4.00000			0.185695
$465$		0			0
$466$	6.00000			0.277945
$467$			16.9706i			0.785304i	0.919687	+	0.392652i	$0.128442\pi$
$467$			16.9706i			0.785304i	−0.919687	+	0.392652i	$0.871558\pi$
$468$		−	12.7279i		−	0.588348i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		−	11.3137i		−	0.520756i
$473$		−	48.0000i		−	2.20704i
$474$		0			0
$475$	22.6274	+	16.9706i	1.03822	+	0.778663i
$476$		0			0
$477$			36.0000i			1.64833i
$478$			8.00000i			0.365911i
$479$	16.9706			0.775405			0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706			0.775405			0.387702	+	0.921785i	$0.373269\pi$
$480$		0			0
$481$	25.4558			1.16069
$482$		−	15.5563i		−	0.708572i
$483$		0			0
$484$	−5.00000			−0.227273
$485$	9.00000	+	3.00000i	0.408669	+	0.136223i
$486$		0			0
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$		−	7.07107i		−	0.320092i
$489$		0			0
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$			16.9706i			0.764316i
$494$	24.0000			1.07981
$495$	8.48528	−	25.4558i	0.381385	−	1.14416i
$496$	−5.65685			−0.254000
$497$		0			0
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−9.19239	+	6.36396i	−0.411096	+	0.284605i
$501$		0			0
$502$		−	5.65685i		−	0.252478i
$503$		−	33.9411i		−	1.51336i	−0.653785	−	0.756680i	$-0.726820\pi$
$503$		−	33.9411i		−	1.51336i	0.653785	−	0.756680i	$-0.273180\pi$
$504$		0			0
$505$	−7.00000	+	21.0000i	−0.311496	+	0.934488i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12.7279			−0.564155			−0.282078	−	0.959392i	$-0.591024\pi$
$509$	−12.7279			−0.564155			−0.282078	+	0.959392i	$0.591024\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−12.7279			−0.561405
$515$	36.0000	+	12.0000i	1.58635	+	0.528783i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−3.00000	+	9.00000i	−0.131559	+	0.394676i
$521$	12.7279			0.557620			0.278810	−	0.960346i	$-0.410060\pi$
$521$	12.7279			0.557620			0.278810	+	0.960346i	$0.410060\pi$
$522$			12.0000i			0.525226i
$523$			33.9411i			1.48414i	0.670321	+	0.742071i	$0.266156\pi$
$523$			33.9411i			1.48414i	−0.670321	+	0.742071i	$0.733844\pi$
$524$	−16.9706			−0.741362
$525$		0			0
$526$		0			0
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	23.0000			1.00000
$530$	8.48528	−	25.4558i	0.368577	−	1.10573i
$531$	33.9411			1.47292
$532$		0			0
$533$		−	6.00000i		−	0.259889i
$534$		0			0
$535$	−25.4558	−	8.48528i	−1.10055	−	0.366851i
$536$	−12.0000			−0.518321
$537$		0			0
$538$		−	21.2132i		−	0.914566i
$539$		0			0
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		−	11.3137i		−	0.485965i
$543$		0			0
$544$	−4.24264			−0.181902
$545$	−8.48528	+	25.4558i	−0.363470	+	1.09041i
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		−	6.00000i		−	0.256307i
$549$	21.2132			0.905357
$550$	−12.0000	+	16.0000i	−0.511682	+	0.682242i
$551$	−22.6274			−0.963960
$552$		0			0
$553$		0			0
$554$	−12.0000			−0.509831
$555$		0			0
$556$	5.65685			0.239904
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$		−	16.9706i		−	0.718421i
$559$	−50.9117			−2.15333
$560$		0			0
$561$		0			0
$562$			16.0000i			0.674919i
$563$			33.9411i			1.43045i	0.698895	+	0.715224i	$0.253675\pi$
$563$			33.9411i			1.43045i	−0.698895	+	0.715224i	$0.746325\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		−	8.00000i		−	0.335673i
$569$	8.00000			0.335377			0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000			0.335377			0.167689	+	0.985840i	$0.446370\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$			16.9706i			0.709575i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.00000			−0.125000
$577$		−	21.2132i		−	0.883117i	−0.897232	−	0.441559i	$-0.854426\pi$
$577$		−	21.2132i		−	0.883117i	0.897232	−	0.441559i	$-0.145574\pi$
$578$		−	1.00000i		−	0.0415945i
$579$		0			0
$580$	2.82843	−	8.48528i	0.117444	−	0.352332i
$581$		0			0
$582$		0			0
$583$		−	48.0000i		−	1.98796i
$584$	−4.24264			−0.175562
$585$	−27.0000	−	9.00000i	−1.11631	−	0.372104i
$586$	4.24264			0.175262
$587$			16.9706i			0.700450i	0.936666	+	0.350225i	$0.113895\pi$
$587$			16.9706i			0.700450i	−0.936666	+	0.350225i	$0.886105\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$	−24.0000	−	8.00000i	−0.988064	−	0.329355i
$591$		0			0
$592$		−	6.00000i		−	0.246598i
$593$		−	12.7279i		−	0.522673i	−0.965248	−	0.261337i	$-0.915837\pi$
$593$		−	12.7279i		−	0.522673i	0.965248	−	0.261337i	$-0.0841632\pi$
$594$		0			0
$595$		0			0
$596$	10.0000			0.409616
$597$		0			0
$598$		0			0
$599$	−16.0000			−0.653742			−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000			−0.653742			−0.326871	+	0.945069i	$0.605994\pi$
$600$		0			0
$601$	26.8701			1.09605			0.548026	−	0.836461i	$-0.315379\pi$
$601$	26.8701			1.09605			0.548026	+	0.836461i	$0.315379\pi$
$602$		0			0
$603$		−	36.0000i		−	1.46603i
$604$	−8.00000			−0.325515
$605$	−3.53553	+	10.6066i	−0.143740	+	0.431220i
$606$		0			0
$607$		−	16.9706i		−	0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$		−	16.9706i		−	0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$		−	5.65685i		−	0.229416i
$609$		0			0
$610$	−15.0000	−	5.00000i	−0.607332	−	0.202444i
$611$		0			0
$612$		−	12.7279i		−	0.514496i
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−16.9706			−0.684876
$615$		0			0
$616$		0			0
$617$			42.0000i			1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$			42.0000i			1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$		0			0
$619$	45.2548			1.81895			0.909473	−	0.415764i	$-0.136486\pi$
$619$	45.2548			1.81895			0.909473	+	0.415764i	$0.136486\pi$
$620$	−4.00000	+	12.0000i	−0.160644	+	0.481932i
$621$		0			0
$622$			16.9706i			0.680458i
$623$		0			0
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	−12.7279			−0.508710
$627$		0			0
$628$		−	4.24264i		−	0.169300i
$629$	25.4558			1.01499
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$		0			0
$633$		0			0
$634$	−12.0000			−0.476581
$635$		0			0
$636$		0			0
$637$		0			0
$638$		−	16.0000i		−	0.633446i
$639$	24.0000			0.949425
$640$	2.12132	+	0.707107i	0.0838525	+	0.0279508i
$641$	−40.0000			−1.57991			−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000			−1.57991			−0.789953	+	0.613168i	$0.789895\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	24.0000			0.944267
$647$		−	16.9706i		−	0.667182i	−0.942718	−	0.333591i	$-0.891740\pi$
$647$		−	16.9706i		−	0.667182i	0.942718	−	0.333591i	$-0.108260\pi$
$648$		−	9.00000i		−	0.353553i
$649$	−45.2548			−1.77641
$650$	16.9706	+	12.7279i	0.665640	+	0.499230i
$651$		0			0
$652$		−	12.0000i		−	0.469956i
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$	−12.0000	+	36.0000i	−0.468879	+	1.40664i
$656$	−1.41421			−0.0552158
$657$		−	12.7279i		−	0.496564i
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	−24.0416			−0.935111			−0.467556	−	0.883964i	$-0.654865\pi$
$661$	−24.0416			−0.935111			−0.467556	+	0.883964i	$0.654865\pi$
$662$			12.0000i			0.466393i
$663$		0			0
$664$	16.9706			0.658586
$665$		0			0
$666$	18.0000			0.697486
$667$		0			0
$668$			16.9706i			0.656611i
$669$		0			0
$670$	−8.48528	+	25.4558i	−0.327815	+	0.983445i
$671$	−28.2843			−1.09190
$672$		0			0
$673$		−	18.0000i		−	0.693849i	−0.937893	−	0.346925i	$-0.887226\pi$
$673$		−	18.0000i		−	0.693849i	0.937893	−	0.346925i	$-0.112774\pi$
$674$	18.0000			0.693334
$675$		0			0
$676$	5.00000			0.192308
$677$			21.2132i			0.815290i	0.913141	+	0.407645i	$0.133650\pi$
$677$			21.2132i			0.815290i	−0.913141	+	0.407645i	$0.866350\pi$
$678$		0			0
$679$		0			0
$680$	−3.00000	+	9.00000i	−0.115045	+	0.345134i
$681$		0			0
$682$			22.6274i			0.866449i
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	16.9706			0.648886
$685$	−12.7279	−	4.24264i	−0.486309	−	0.162103i
$686$		0			0
$687$		0			0
$688$			12.0000i			0.457496i
$689$	−50.9117			−1.93958
$690$		0			0
$691$	−5.65685			−0.215197			−0.107598	−	0.994194i	$-0.534316\pi$
$691$	−5.65685			−0.215197			−0.107598	+	0.994194i	$0.534316\pi$
$692$		−	4.24264i		−	0.161281i
$693$		0			0
$694$	12.0000			0.455514
$695$	4.00000	−	12.0000i	0.151729	−	0.455186i
$696$		0			0
$697$		−	6.00000i		−	0.227266i
$698$			24.0416i			0.909989i
$699$		0			0
$700$		0			0
$701$	−44.0000			−1.66186			−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000			−1.66186			−0.830929	+	0.556379i	$0.812190\pi$
$702$		0			0
$703$			33.9411i			1.28011i
$704$	4.00000			0.150756
$705$		0			0
$706$	−21.2132			−0.798369
$707$		0			0
$708$		0			0
$709$	44.0000			1.65245			0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000			1.65245			0.826227	+	0.563337i	$0.190483\pi$
$710$	−16.9706	−	5.65685i	−0.636894	−	0.212298i
$711$		0			0
$712$		−	4.24264i		−	0.159000i
$713$		0			0
$714$		0			0
$715$	36.0000	+	12.0000i	1.34632	+	0.448775i
$716$	20.0000			0.747435
$717$		0			0
$718$		−	16.0000i		−	0.597115i
$719$	5.65685			0.210965			0.105483	−	0.994421i	$-0.466361\pi$
$719$	5.65685			0.210965			0.105483	+	0.994421i	$0.466361\pi$
$720$	−2.12132	+	6.36396i	−0.0790569	+	0.237171i
$721$		0			0
$722$			13.0000i			0.483810i
$723$		0			0
$724$	15.5563			0.578147
$725$	−16.0000	−	12.0000i	−0.594225	−	0.445669i
$726$		0			0
$727$			16.9706i			0.629403i	0.949191	+	0.314702i	$0.101904\pi$
$727$			16.9706i			0.629403i	−0.949191	+	0.314702i	$0.898096\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	−3.00000	+	9.00000i	−0.111035	+	0.333105i
$731$	−50.9117			−1.88304
$732$		0			0
$733$			12.7279i			0.470117i	0.971981	+	0.235058i	$0.0755281\pi$
$733$			12.7279i			0.470117i	−0.971981	+	0.235058i	$0.924472\pi$
$734$	−16.9706			−0.626395
$735$		0			0
$736$		0			0
$737$			48.0000i			1.76810i
$738$		−	4.24264i		−	0.156174i
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	−12.7279	−	4.24264i	−0.467888	−	0.155963i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	7.07107	−	21.2132i	0.259064	−	0.777192i
$746$	12.0000			0.439351
$747$			50.9117i			1.86276i
$748$			16.9706i			0.620505i
$749$		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
$754$	−16.9706			−0.618031
$755$	−5.65685	+	16.9706i	−0.205874	+	0.617622i
$756$		0			0
$757$		−	42.0000i		−	1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$		−	42.0000i		−	1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$	−12.0000	−	4.00000i	−0.435286	−	0.145095i
$761$	−46.6690			−1.69175			−0.845876	−	0.533380i	$-0.820922\pi$
$761$	−46.6690			−1.69175			−0.845876	+	0.533380i	$0.820922\pi$
$762$		0			0
$763$		0			0
$764$	−8.00000			−0.289430
$765$	−27.0000	−	9.00000i	−0.976187	−	0.325396i
$766$		0			0
$767$			48.0000i			1.73318i
$768$		0			0
$769$	−7.07107			−0.254989			−0.127495	−	0.991839i	$-0.540694\pi$
$769$	−7.07107			−0.254989			−0.127495	+	0.991839i	$0.540694\pi$
$770$		0			0
$771$		0			0
$772$		−	24.0000i		−	0.863779i
$773$		−	38.1838i		−	1.37337i	−0.726953	−	0.686687i	$-0.759064\pi$
$773$		−	38.1838i		−	1.37337i	0.726953	−	0.686687i	$-0.240936\pi$
$774$	−36.0000			−1.29399
$775$	22.6274	+	16.9706i	0.812801	+	0.609601i
$776$	−4.24264			−0.152302
$777$		0			0
$778$			20.0000i			0.717035i
$779$	8.00000			0.286630
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.00000	−	3.00000i	−0.321224	−	0.107075i
$786$		0			0
$787$			50.9117i			1.81481i	0.420262	+	0.907403i	$0.361938\pi$
$787$			50.9117i			1.81481i	−0.420262	+	0.907403i	$0.638062\pi$
$788$			12.0000i			0.427482i
$789$		0			0
$790$		0			0
$791$		0			0
$792$			12.0000i			0.426401i
$793$			30.0000i			1.06533i
$794$	29.6985			1.05396
$795$		0			0
$796$	11.3137			0.401004
$797$			4.24264i			0.150282i	0.997173	+	0.0751410i	$0.0239407\pi$
$797$			4.24264i			0.150282i	−0.997173	+	0.0751410i	$0.976059\pi$
$798$		0			0
$799$		0			0
$800$	3.00000	−	4.00000i	0.106066	−	0.141421i
$801$	12.7279			0.449719
$802$		−	8.00000i		−	0.282490i
$803$			16.9706i			0.598878i
$804$		0			0
$805$		0			0
$806$	24.0000			0.845364
$807$		0			0
$808$		−	9.89949i		−	0.348263i
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$	−19.0919	−	6.36396i	−0.670820	−	0.223607i
$811$	45.2548			1.58911			0.794556	−	0.607191i	$-0.207704\pi$
$811$	45.2548			1.58911			0.794556	+	0.607191i	$0.207704\pi$
$812$		0			0
$813$		0			0
$814$	−24.0000			−0.841200
$815$	−25.4558	−	8.48528i	−0.891679	−	0.297226i
$816$		0			0
$817$		−	67.8823i		−	2.37490i
$818$			9.89949i			0.346128i
$819$		0			0
$820$	−1.00000	+	3.00000i	−0.0349215	+	0.104765i
$821$	10.0000			0.349002			0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000			0.349002			0.174501	+	0.984657i	$0.444169\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	−16.9706			−0.591198
$825$		0			0
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	−15.5563			−0.540294			−0.270147	−	0.962819i	$-0.587072\pi$
$829$	−15.5563			−0.540294			−0.270147	+	0.962819i	$0.587072\pi$
$830$	12.0000	−	36.0000i	0.416526	−	1.24958i
$831$		0			0
$832$		−	4.24264i		−	0.147087i
$833$		0			0
$834$		0			0
$835$	36.0000	+	12.0000i	1.24583	+	0.415277i
$836$	−22.6274			−0.782586
$837$		0			0
$838$		0			0
$839$	−22.6274			−0.781185			−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274			−0.781185			−0.390593	+	0.920564i	$0.627730\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		−	6.00000i		−	0.206774i
$843$		0			0
$844$	−12.0000			−0.413057
$845$	3.53553	−	10.6066i	0.121626	−	0.364878i
$846$		0			0
$847$		0			0
$848$			12.0000i			0.412082i
$849$		0			0
$850$	16.9706	+	12.7279i	0.582086	+	0.436564i
$851$		0			0
$852$		0			0
$853$		−	46.6690i		−	1.59792i	−0.601386	−	0.798959i	$-0.705384\pi$
$853$		−	46.6690i		−	1.59792i	0.601386	−	0.798959i	$-0.294616\pi$
$854$		0			0
$855$	12.0000	−	36.0000i	0.410391	−	1.23117i
$856$	12.0000			0.410152
$857$			12.7279i			0.434778i	0.976085	+	0.217389i	$0.0697539\pi$
$857$			12.7279i			0.434778i	−0.976085	+	0.217389i	$0.930246\pi$
$858$		0			0
$859$	−5.65685			−0.193009			−0.0965047	−	0.995333i	$-0.530766\pi$
$859$	−5.65685			−0.193009			−0.0965047	+	0.995333i	$0.530766\pi$
$860$	25.4558	+	8.48528i	0.868037	+	0.289346i
$861$		0			0
$862$			16.0000i			0.544962i
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−9.00000	−	3.00000i	−0.306009	−	0.102003i
$866$	29.6985			1.00920
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	50.9117			1.72508
$872$		−	12.0000i		−	0.406371i
$873$		−	12.7279i		−	0.430775i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	18.0000i		−	0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$		−	18.0000i		−	0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$			39.5980i			1.33637i
$879$		0			0
$880$	2.82843	−	8.48528i	0.0953463	−	0.286039i
$881$	−32.5269			−1.09586			−0.547930	−	0.836524i	$-0.684584\pi$
$881$	−32.5269			−1.09586			−0.547930	+	0.836524i	$0.684584\pi$
$882$		0			0
$883$			12.0000i			0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$			12.0000i			0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	18.0000			0.605406
$885$		0			0
$886$	−12.0000			−0.403148
$887$		−	16.9706i		−	0.569816i	−0.958555	−	0.284908i	$-0.908037\pi$
$887$		−	16.9706i		−	0.569816i	0.958555	−	0.284908i	$-0.0919630\pi$
$888$		0			0
$889$		0			0
$890$	−9.00000	−	3.00000i	−0.301681	−	0.100560i
$891$	−36.0000			−1.20605
$892$			16.9706i			0.568216i
$893$		0			0
$894$		0			0
$895$	14.1421	−	42.4264i	0.472719	−	1.41816i
$896$		0			0
$897$		0			0
$898$			2.00000i			0.0667409i
$899$	−22.6274			−0.754667
$900$	12.0000	+	9.00000i	0.400000	+	0.300000i
$901$	−50.9117			−1.69611
$902$			5.65685i			0.188353i
$903$		0			0
$904$		0			0
$905$	11.0000	−	33.0000i	0.365652	−	1.09696i
$906$		0			0
$907$		−	12.0000i		−	0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$		−	12.0000i		−	0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$			16.9706i			0.563188i
$909$	29.6985			0.985037
$910$		0			0
$911$	−40.0000			−1.32526			−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000			−1.32526			−0.662630	+	0.748947i	$0.730560\pi$
$912$		0			0
$913$		−	67.8823i		−	2.24657i
$914$	24.0000			0.793849
$915$		0			0
$916$	15.5563			0.513996
$917$		0			0
$918$		0			0
$919$	−48.0000			−1.58337			−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000			−1.58337			−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		−	21.2132i		−	0.698620i
$923$			33.9411i			1.11719i
$924$		0			0
$925$	−18.0000	+	24.0000i	−0.591836	+	0.789115i
$926$	−24.0000			−0.788689
$927$		−	50.9117i		−	1.67216i
$928$			4.00000i			0.131306i
$929$	29.6985			0.974376			0.487188	−	0.873297i	$-0.338023\pi$
$929$	29.6985			0.974376			0.487188	+	0.873297i	$0.338023\pi$
$930$		0			0
$931$		0			0
$932$			6.00000i			0.196537i
$933$		0			0
$934$	−16.9706			−0.555294
$935$	36.0000	+	12.0000i	1.17733	+	0.392442i
$936$	12.7279			0.416025
$937$		−	4.24264i		−	0.138601i	−0.997596	−	0.0693005i	$-0.977923\pi$
$937$		−	4.24264i		−	0.138601i	0.997596	−	0.0693005i	$-0.0220767\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	49.4975			1.61357			0.806786	−	0.590844i	$-0.201205\pi$
$941$	49.4975			1.61357			0.806786	+	0.590844i	$0.201205\pi$
$942$		0			0
$943$		0			0
$944$	11.3137			0.368230
$945$		0			0
$946$	48.0000			1.56061
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	18.0000			0.584305
$950$	−16.9706	+	22.6274i	−0.550598	+	0.734130i
$951$		0			0
$952$		0			0
$953$			24.0000i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$			24.0000i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	−36.0000			−1.16554
$955$	−5.65685	+	16.9706i	−0.183052	+	0.549155i
$956$	−8.00000			−0.258738
$957$		0			0
$958$			16.9706i			0.548294i
$959$		0			0
$960$		0			0
$961$	1.00000			0.0322581
$962$			25.4558i			0.820729i
$963$			36.0000i			1.16008i
$964$	15.5563			0.501036
$965$	−50.9117	−	16.9706i	−1.63891	−	0.546302i
$966$		0			0
$967$		−	24.0000i		−	0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$		−	24.0000i		−	0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$		−	5.00000i		−	0.160706i
$969$		0			0
$970$	−3.00000	+	9.00000i	−0.0963242	+	0.288973i
$971$	45.2548			1.45230			0.726148	−	0.687538i	$-0.241309\pi$
$971$	45.2548			1.45230			0.726148	+	0.687538i	$0.241309\pi$
$972$		0			0
$973$		0			0
$974$	−24.0000			−0.769010
$975$		0			0
$976$	7.07107			0.226339
$977$		−	30.0000i		−	0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$		−	30.0000i		−	0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$		0			0
$979$	−16.9706			−0.542382
$980$		0			0
$981$	36.0000			1.14939
$982$		−	4.00000i		−	0.127645i
$983$		−	16.9706i		−	0.541277i	−0.962681	−	0.270638i	$-0.912765\pi$
$983$		−	16.9706i		−	0.541277i	0.962681	−	0.270638i	$-0.0872348\pi$
$984$		0			0
$985$	25.4558	+	8.48528i	0.811091	+	0.270364i
$986$	−16.9706			−0.540453
$987$		0			0
$988$			24.0000i			0.763542i
$989$		0			0
$990$	25.4558	+	8.48528i	0.809040	+	0.269680i
$991$	48.0000			1.52477			0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000			1.52477			0.762385	+	0.647124i	$0.224028\pi$
$992$		−	5.65685i		−	0.179605i
$993$		0			0
$994$		0			0
$995$	8.00000	−	24.0000i	0.253617	−	0.760851i
$996$		0			0
$997$		−	38.1838i		−	1.20929i	−0.796494	−	0.604646i	$-0.793315\pi$
$997$		−	38.1838i		−	1.20929i	0.796494	−	0.604646i	$-0.206685\pi$
$998$		−	4.00000i		−	0.126618i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.c.g.99.3	yes	4
5.2	odd	4		2450.2.a.bi.1.2		2
5.3	odd	4		2450.2.a.bo.1.1		2
5.4	even	2	inner	490.2.c.g.99.1	✓	4
7.2	even	3		490.2.i.d.459.4		8
7.3	odd	6		490.2.i.d.79.2		8
7.4	even	3		490.2.i.d.79.1		8
7.5	odd	6		490.2.i.d.459.3		8
7.6	odd	2	inner	490.2.c.g.99.4	yes	4
35.4	even	6		490.2.i.d.79.4		8
35.9	even	6		490.2.i.d.459.1		8
35.13	even	4		2450.2.a.bo.1.2		2
35.19	odd	6		490.2.i.d.459.2		8
35.24	odd	6		490.2.i.d.79.3		8
35.27	even	4		2450.2.a.bi.1.1		2
35.34	odd	2	inner	490.2.c.g.99.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.c.g.99.1	✓	4	5.4	even	2	inner
490.2.c.g.99.2	yes	4	35.34	odd	2	inner
490.2.c.g.99.3	yes	4	1.1	even	1	trivial
490.2.c.g.99.4	yes	4	7.6	odd	2	inner
490.2.i.d.79.1		8	7.4	even	3
490.2.i.d.79.2		8	7.3	odd	6
490.2.i.d.79.3		8	35.24	odd	6
490.2.i.d.79.4		8	35.4	even	6
490.2.i.d.459.1		8	35.9	even	6
490.2.i.d.459.2		8	35.19	odd	6
490.2.i.d.459.3		8	7.5	odd	6
490.2.i.d.459.4		8	7.2	even	3
2450.2.a.bi.1.1		2	35.27	even	4
2450.2.a.bi.1.2		2	5.2	odd	4
2450.2.a.bo.1.1		2	5.3	odd	4
2450.2.a.bo.1.2		2	35.13	even	4

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:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} -1.00000i q^{8} +3.00000 q^{9} +(-2.12132 - 0.707107i) q^{10} -4.00000 q^{11} +4.24264i q^{13} +1.00000 q^{16} +4.24264i q^{17} +3.00000i q^{18} -5.65685 q^{19} +(0.707107 - 2.12132i) q^{20} -4.00000i q^{22} +(-4.00000 - 3.00000i) q^{25} -4.24264 q^{26} +4.00000 q^{29} -5.65685 q^{31} +1.00000i q^{32} -4.24264 q^{34} -3.00000 q^{36} -6.00000i q^{37} -5.65685i q^{38} +(2.12132 + 0.707107i) q^{40} -1.41421 q^{41} +12.0000i q^{43} +4.00000 q^{44} +(-2.12132 + 6.36396i) q^{45} +(3.00000 - 4.00000i) q^{50} -4.24264i q^{52} +12.0000i q^{53} +(2.82843 - 8.48528i) q^{55} +4.00000i q^{58} +11.3137 q^{59} +7.07107 q^{61} -5.65685i q^{62} -1.00000 q^{64} +(-9.00000 - 3.00000i) q^{65} -12.0000i q^{67} -4.24264i q^{68} +8.00000 q^{71} -3.00000i q^{72} -4.24264i q^{73} +6.00000 q^{74} +5.65685 q^{76} +(-0.707107 + 2.12132i) q^{80} +9.00000 q^{81} -1.41421i q^{82} +16.9706i q^{83} +(-9.00000 - 3.00000i) q^{85} -12.0000 q^{86} +4.00000i q^{88} +4.24264 q^{89} +(-6.36396 - 2.12132i) q^{90} +(4.00000 - 12.0000i) q^{95} -4.24264i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 12 * q^9	\( 4 q - 4 q^{4} + 12 q^{9} - 16 q^{11} + 4 q^{16} - 16 q^{25} + 16 q^{29} - 12 q^{36} + 16 q^{44} + 12 q^{50} - 4 q^{64} - 36 q^{65} + 32 q^{71} + 24 q^{74} + 36 q^{81} - 36 q^{85} - 48 q^{86} + 16 q^{95} - 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^9 - 16 * q^11 + 4 * q^16 - 16 * q^25 + 16 * q^29 - 12 * q^36 + 16 * q^44 + 12 * q^50 - 4 * q^64 - 36 * q^65 + 32 * q^71 + 24 * q^74 + 36 * q^81 - 36 * q^85 - 48 * q^86 + 16 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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