LMFDB - Embedded newform 1710.2.d.b.1369.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(1369,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1710.1369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1710.d (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:	$13.6544187456$
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:	no (minimal twist has level 570)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1710.1369
Dual form			1710.2.d.b.1369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +4.00000i q^{22} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -2.00000 q^{29} -6.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +10.0000i q^{37} +1.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -6.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -2.00000i q^{52} -10.0000i q^{53} +(4.00000 - 8.00000i) q^{55} -2.00000i q^{58} +2.00000 q^{59} -6.00000 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(4.00000 + 2.00000i) q^{65} -8.00000i q^{67} +6.00000i q^{68} +12.0000 q^{71} -16.0000i q^{73} -10.0000 q^{74} -1.00000 q^{76} +14.0000 q^{79} +(1.00000 - 2.00000i) q^{80} -12.0000i q^{83} +(-12.0000 - 6.00000i) q^{85} +6.00000 q^{86} -4.00000i q^{88} +4.00000 q^{89} +6.00000i q^{92} -6.00000 q^{94} +(1.00000 - 2.00000i) q^{95} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^5	$ 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 8 q^{11} + 2 q^{16} + 2 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 4 q^{29} - 12 q^{31} + 12 q^{34} - 4 q^{40} - 8 q^{44} + 12 q^{46} + 14 q^{49} + 8 q^{50} + 8 q^{55} + 4 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} + 24 q^{71} - 20 q^{74} - 2 q^{76} + 28 q^{79} + 2 q^{80} - 24 q^{85} + 12 q^{86} + 8 q^{89} - 12 q^{94} + 2 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 8 * q^11 + 2 * q^16 + 2 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 4 * q^29 - 12 * q^31 + 12 * q^34 - 4 * q^40 - 8 * q^44 + 12 * q^46 + 14 * q^49 + 8 * q^50 + 8 * q^55 + 4 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 + 24 * q^71 - 20 * q^74 - 2 * q^76 + 28 * q^79 + 2 * q^80 - 24 * q^85 + 12 * q^86 + 8 * q^89 - 12 * q^94 + 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$			4.00000i			0.852803i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	−2.00000			−0.392232
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−6.00000			−1.07763			−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000			−1.07763			−0.538816	+	0.842424i	$0.681128\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	6.00000			1.02899
$35$		0			0
$36$		0			0
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$			1.00000i			0.162221i
$39$		0			0
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	−4.00000			−0.603023
$45$		0			0
$46$	6.00000			0.884652
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$		−	2.00000i		−	0.277350i
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$	4.00000	−	8.00000i	0.539360	−	1.07872i
$56$		0			0
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	2.00000			0.260378			0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000			0.260378			0.130189	+	0.991489i	$0.458442\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		−	6.00000i		−	0.762001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$			6.00000i			0.727607i
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$		−	16.0000i		−	1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$		−	16.0000i		−	1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	−10.0000			−1.16248
$75$		0			0
$76$	−1.00000			−0.114708
$77$		0			0
$78$		0			0
$79$	14.0000			1.57512			0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000			1.57512			0.787562	+	0.616236i	$0.211343\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−12.0000	−	6.00000i	−1.30158	−	0.650791i
$86$	6.00000			0.646997
$87$		0			0
$88$		−	4.00000i		−	0.426401i
$89$	4.00000			0.423999			0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000			0.423999			0.212000	+	0.977270i	$0.432002\pi$
$90$		0			0
$91$		0			0
$92$			6.00000i			0.625543i
$93$		0			0
$94$	−6.00000			−0.618853
$95$	1.00000	−	2.00000i	0.102598	−	0.205196i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$			7.00000i			0.707107i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	10.0000			0.971286
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$	8.00000	+	4.00000i	0.762770	+	0.381385i
$111$		0			0
$112$		0			0
$113$		−	2.00000i		−	0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$		−	2.00000i		−	0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$		0			0
$115$	−12.0000	−	6.00000i	−1.11901	−	0.559503i
$116$	2.00000			0.185695
$117$		0			0
$118$			2.00000i			0.184115i
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		−	6.00000i		−	0.543214i
$123$		0			0
$124$	6.00000			0.538816
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−2.00000	+	4.00000i	−0.175412	+	0.350823i
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		0			0
$134$	8.00000			0.691095
$135$		0			0
$136$	−6.00000			−0.514496
$137$			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
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$			12.0000i			1.00702i
$143$			8.00000i			0.668994i
$144$		0			0
$145$	−2.00000	+	4.00000i	−0.166091	+	0.332182i
$146$	16.0000			1.32417
$147$		0			0
$148$		−	10.0000i		−	0.821995i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	10.0000			0.813788			0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000			0.813788			0.406894	+	0.913475i	$0.366612\pi$
$152$		−	1.00000i		−	0.0811107i
$153$		0			0
$154$		0			0
$155$	−6.00000	+	12.0000i	−0.481932	+	0.963863i
$156$		0			0
$157$			18.0000i			1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$			18.0000i			1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$			14.0000i			1.11378i
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$		0			0
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$		0			0
$165$		0			0
$166$	12.0000			0.931381
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	6.00000	−	12.0000i	0.460179	−	0.920358i
$171$		0			0
$172$			6.00000i			0.457496i
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$		0			0
$176$	4.00000			0.301511
$177$		0			0
$178$			4.00000i			0.299813i
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	−4.00000			−0.297318			−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000			−0.297318			−0.148659	+	0.988889i	$0.547496\pi$
$182$		0			0
$183$		0			0
$184$	−6.00000			−0.442326
$185$	20.0000	+	10.0000i	1.47043	+	0.735215i
$186$		0			0
$187$		−	24.0000i		−	1.75505i
$188$		−	6.00000i		−	0.437595i
$189$		0			0
$190$	2.00000	+	1.00000i	0.145095	+	0.0725476i
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	−7.00000			−0.500000
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$			18.0000i			1.26648i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−8.00000			−0.557386
$207$		0			0
$208$			2.00000i			0.138675i
$209$	4.00000			0.276686
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$			10.0000i			0.686803i
$213$		0			0
$214$	4.00000			0.273434
$215$	−12.0000	−	6.00000i	−0.818393	−	0.409197i
$216$		0			0
$217$		0			0
$218$		−	4.00000i		−	0.270914i
$219$		0			0
$220$	−4.00000	+	8.00000i	−0.269680	+	0.539360i
$221$	12.0000			0.807207
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$		0			0
$226$	2.00000			0.133038
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$	6.00000	−	12.0000i	0.395628	−	0.791257i
$231$		0			0
$232$			2.00000i			0.131306i
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	12.0000	+	6.00000i	0.782794	+	0.391397i
$236$	−2.00000			−0.130189
$237$		0			0
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$			5.00000i			0.321412i
$243$		0			0
$244$	6.00000			0.384111
$245$	7.00000	−	14.0000i	0.447214	−	0.894427i
$246$		0			0
$247$			2.00000i			0.127257i
$248$			6.00000i			0.381000i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		−	24.0000i		−	1.50887i
$254$	8.00000			0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$			22.0000i			1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$			22.0000i			1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$		0			0
$259$		0			0
$260$	−4.00000	−	2.00000i	−0.248069	−	0.124035i
$261$		0			0
$262$			8.00000i			0.494242i
$263$			2.00000i			0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$			2.00000i			0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$		0			0
$265$	−20.0000	−	10.0000i	−1.22859	−	0.614295i
$266$		0			0
$267$		0			0
$268$			8.00000i			0.488678i
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$		−	6.00000i		−	0.363803i
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−12.0000	−	16.0000i	−0.723627	−	0.964836i
$276$		0			0
$277$			26.0000i			1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$			26.0000i			1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$		−	12.0000i		−	0.719712i
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$			22.0000i			1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$			22.0000i			1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	−12.0000			−0.712069
$285$		0			0
$286$	−8.00000			−0.473050
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$	−4.00000	−	2.00000i	−0.234888	−	0.117444i
$291$		0			0
$292$			16.0000i			0.936329i
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$		0			0
$295$	2.00000	−	4.00000i	0.116445	−	0.232889i
$296$	10.0000			0.581238
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$	12.0000			0.693978
$300$		0			0
$301$		0			0
$302$			10.0000i			0.575435i
$303$		0			0
$304$	1.00000			0.0573539
$305$	−6.00000	+	12.0000i	−0.343559	+	0.687118i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$		0			0
$310$	−12.0000	−	6.00000i	−0.681554	−	0.340777i
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$			28.0000i			1.58265i	0.611393	+	0.791327i	$0.290609\pi$
$313$			28.0000i			1.58265i	−0.611393	+	0.791327i	$0.709391\pi$
$314$	−18.0000			−1.01580
$315$		0			0
$316$	−14.0000			−0.787562
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$		0			0
$322$		0			0
$323$		−	6.00000i		−	0.333849i
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$	10.0000			0.553849
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$			12.0000i			0.658586i
$333$		0			0
$334$		0			0
$335$	−16.0000	−	8.00000i	−0.874173	−	0.437087i
$336$		0			0
$337$		−	6.00000i		−	0.326841i	−0.986557	−	0.163420i	$-0.947747\pi$
$337$		−	6.00000i		−	0.326841i	0.986557	−	0.163420i	$-0.0522527\pi$
$338$			9.00000i			0.489535i
$339$		0			0
$340$	12.0000	+	6.00000i	0.650791	+	0.325396i
$341$	−24.0000			−1.29967
$342$		0			0
$343$		0			0
$344$	−6.00000			−0.323498
$345$		0			0
$346$	−14.0000			−0.752645
$347$			20.0000i			1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$			20.0000i			1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$			4.00000i			0.213201i
$353$			2.00000i			0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$			2.00000i			0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$		0			0
$355$	12.0000	−	24.0000i	0.636894	−	1.27379i
$356$	−4.00000			−0.212000
$357$		0			0
$358$		−	18.0000i		−	0.951330i
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		−	4.00000i		−	0.210235i
$363$		0			0
$364$		0			0
$365$	−32.0000	−	16.0000i	−1.67496	−	0.837478i
$366$		0			0
$367$		−	12.0000i		−	0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$		−	12.0000i		−	0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$	−10.0000	+	20.0000i	−0.519875	+	1.03975i
$371$		0			0
$372$		0			0
$373$			6.00000i			0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$			6.00000i			0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	24.0000			1.24101
$375$		0			0
$376$	6.00000			0.309426
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$	−1.00000	+	2.00000i	−0.0512989	+	0.102598i
$381$		0			0
$382$		−	24.0000i		−	1.22795i
$383$		−	36.0000i		−	1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$		−	36.0000i		−	1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$		0			0
$385$		0			0
$386$	−14.0000			−0.712581
$387$		0			0
$388$		−	10.0000i		−	0.507673i
$389$	38.0000			1.92668			0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000			1.92668			0.963338	+	0.268290i	$0.0864585\pi$
$390$		0			0
$391$	−36.0000			−1.82060
$392$		−	7.00000i		−	0.353553i
$393$		0			0
$394$	−8.00000			−0.403034
$395$	14.0000	−	28.0000i	0.704416	−	1.40883i
$396$		0			0
$397$			30.0000i			1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$			30.0000i			1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$			16.0000i			0.802008i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	−16.0000			−0.799002			−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000			−0.799002			−0.399501	+	0.916733i	$0.630817\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$	−18.0000			−0.895533
$405$		0			0
$406$		0			0
$407$			40.0000i			1.98273i
$408$		0			0
$409$	−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
$411$		0			0
$412$		−	8.00000i		−	0.394132i
$413$		0			0
$414$		0			0
$415$	−24.0000	−	12.0000i	−1.17811	−	0.589057i
$416$	−2.00000			−0.0980581
$417$		0			0
$418$			4.00000i			0.195646i
$419$	8.00000			0.390826			0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000			0.390826			0.195413	+	0.980721i	$0.437395\pi$
$420$		0			0
$421$	16.0000			0.779792			0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000			0.779792			0.389896	+	0.920859i	$0.372511\pi$
$422$		−	8.00000i		−	0.389434i
$423$		0			0
$424$	−10.0000			−0.485643
$425$	−24.0000	+	18.0000i	−1.16417	+	0.873128i
$426$		0			0
$427$		0			0
$428$			4.00000i			0.193347i
$429$		0			0
$430$	6.00000	−	12.0000i	0.289346	−	0.578691i
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$			6.00000i			0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$			6.00000i			0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$		0			0
$435$		0			0
$436$	4.00000			0.191565
$437$		−	6.00000i		−	0.287019i
$438$		0			0
$439$	14.0000			0.668184			0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000			0.668184			0.334092	+	0.942541i	$0.391570\pi$
$440$	−8.00000	−	4.00000i	−0.381385	−	0.190693i
$441$		0			0
$442$			12.0000i			0.570782i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	4.00000	−	8.00000i	0.189618	−	0.379236i
$446$	8.00000			0.378811
$447$		0			0
$448$		0			0
$449$	4.00000			0.188772			0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000			0.188772			0.0943858	+	0.995536i	$0.469911\pi$
$450$		0			0
$451$		0			0
$452$			2.00000i			0.0940721i
$453$		0			0
$454$	12.0000			0.563188
$455$		0			0
$456$		0			0
$457$		−	16.0000i		−	0.748448i	−0.927338	−	0.374224i	$-0.877909\pi$
$457$		−	16.0000i		−	0.748448i	0.927338	−	0.374224i	$-0.122091\pi$
$458$			10.0000i			0.467269i
$459$		0			0
$460$	12.0000	+	6.00000i	0.559503	+	0.279751i
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$	14.0000			0.648537
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$	−6.00000	+	12.0000i	−0.276759	+	0.553519i
$471$		0			0
$472$		−	2.00000i		−	0.0920575i
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$	−3.00000	−	4.00000i	−0.137649	−	0.183533i
$476$		0			0
$477$		0			0
$478$		−	8.00000i		−	0.365911i
$479$	−16.0000			−0.731059			−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000			−0.731059			−0.365529	+	0.930800i	$0.619112\pi$
$480$		0			0
$481$	−20.0000			−0.911922
$482$		−	10.0000i		−	0.455488i
$483$		0			0
$484$	−5.00000			−0.227273
$485$	20.0000	+	10.0000i	0.908153	+	0.454077i
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$			6.00000i			0.271607i
$489$		0			0
$490$	14.0000	+	7.00000i	0.632456	+	0.316228i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$	−2.00000			−0.0899843
$495$		0			0
$496$	−6.00000			−0.269408
$497$		0			0
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$		0			0
$502$		−	12.0000i		−	0.535586i
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$	18.0000	−	36.0000i	0.800989	−	1.60198i
$506$	24.0000			1.06693
$507$		0			0
$508$			8.00000i			0.354943i
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−22.0000			−0.970378
$515$	16.0000	+	8.00000i	0.705044	+	0.352522i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		0			0
$519$		0			0
$520$	2.00000	−	4.00000i	0.0877058	−	0.175412i
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$			36.0000i			1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$			36.0000i			1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	−8.00000			−0.349482
$525$		0			0
$526$	−2.00000			−0.0872041
$527$			36.0000i			1.56818i
$528$		0			0
$529$	−13.0000			−0.565217
$530$	10.0000	−	20.0000i	0.434372	−	0.868744i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−8.00000	−	4.00000i	−0.345870	−	0.172935i
$536$	−8.00000			−0.345547
$537$		0			0
$538$		−	14.0000i		−	0.603583i
$539$	28.0000			1.20605
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$			32.0000i			1.37452i
$543$		0			0
$544$	6.00000			0.257248
$545$	−4.00000	+	8.00000i	−0.171341	+	0.342682i
$546$		0			0
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$		−	6.00000i		−	0.256307i
$549$		0			0
$550$	16.0000	−	12.0000i	0.682242	−	0.511682i
$551$	−2.00000			−0.0852029
$552$		0			0
$553$		0			0
$554$	−26.0000			−1.10463
$555$		0			0
$556$	12.0000			0.508913
$557$			24.0000i			1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$			24.0000i			1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$			12.0000i			0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$			12.0000i			0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$		0			0
$565$	−4.00000	−	2.00000i	−0.168281	−	0.0841406i
$566$	−22.0000			−0.924729
$567$		0			0
$568$		−	12.0000i		−	0.503509i
$569$	−28.0000			−1.17382			−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000			−1.17382			−0.586911	+	0.809652i	$0.699656\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		−	8.00000i		−	0.334497i
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$			8.00000i			0.333044i	0.986038	+	0.166522i	$0.0532537\pi$
$577$			8.00000i			0.333044i	−0.986038	+	0.166522i	$0.946746\pi$
$578$		−	19.0000i		−	0.790296i
$579$		0			0
$580$	2.00000	−	4.00000i	0.0830455	−	0.166091i
$581$		0			0
$582$		0			0
$583$		−	40.0000i		−	1.65663i
$584$	−16.0000			−0.662085
$585$		0			0
$586$	2.00000			0.0826192
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	−6.00000			−0.247226
$590$	4.00000	+	2.00000i	0.164677	+	0.0823387i
$591$		0			0
$592$			10.0000i			0.410997i
$593$			46.0000i			1.88899i	0.328521	+	0.944497i	$0.393450\pi$
$593$			46.0000i			1.88899i	−0.328521	+	0.944497i	$0.606550\pi$
$594$		0			0
$595$		0			0
$596$	6.00000			0.245770
$597$		0			0
$598$			12.0000i			0.490716i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$		0			0
$604$	−10.0000			−0.406894
$605$	5.00000	−	10.0000i	0.203279	−	0.406558i
$606$		0			0
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$			1.00000i			0.0405554i
$609$		0			0
$610$	−12.0000	−	6.00000i	−0.485866	−	0.242933i
$611$	−12.0000			−0.485468
$612$		0			0
$613$		−	22.0000i		−	0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$		−	22.0000i		−	0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$	−44.0000			−1.76851			−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000			−1.76851			−0.884255	+	0.467005i	$0.845333\pi$
$620$	6.00000	−	12.0000i	0.240966	−	0.481932i
$621$		0			0
$622$		−	24.0000i		−	0.962312i
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−28.0000			−1.11911
$627$		0			0
$628$		−	18.0000i		−	0.718278i
$629$	60.0000			2.39236
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		−	14.0000i		−	0.556890i
$633$		0			0
$634$	−6.00000			−0.238290
$635$	−16.0000	−	8.00000i	−0.634941	−	0.317470i
$636$		0			0
$637$			14.0000i			0.554700i
$638$		−	8.00000i		−	0.316723i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	−48.0000			−1.89589			−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000			−1.89589			−0.947943	+	0.318440i	$0.896841\pi$
$642$		0			0
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$		0			0
$646$	6.00000			0.236067
$647$			6.00000i			0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$			6.00000i			0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$	6.00000	+	8.00000i	0.235339	+	0.313786i
$651$		0			0
$652$			10.0000i			0.391630i
$653$		−	48.0000i		−	1.87839i	−0.343391	−	0.939193i	$-0.611576\pi$
$653$		−	48.0000i		−	1.87839i	0.343391	−	0.939193i	$-0.388424\pi$
$654$		0			0
$655$	8.00000	−	16.0000i	0.312586	−	0.625172i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	46.0000			1.79191			0.895953	−	0.444149i	$-0.146494\pi$
$659$	46.0000			1.79191			0.895953	+	0.444149i	$0.146494\pi$
$660$		0			0
$661$	−8.00000			−0.311164			−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000			−0.311164			−0.155582	+	0.987823i	$0.549725\pi$
$662$		−	4.00000i		−	0.155464i
$663$		0			0
$664$	−12.0000			−0.465690
$665$		0			0
$666$		0			0
$667$			12.0000i			0.464642i
$668$		0			0
$669$		0			0
$670$	8.00000	−	16.0000i	0.309067	−	0.618134i
$671$	−24.0000			−0.926510
$672$		0			0
$673$			26.0000i			1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$			26.0000i			1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	6.00000			0.231111
$675$		0			0
$676$	−9.00000			−0.346154
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$		0			0
$679$		0			0
$680$	−6.00000	+	12.0000i	−0.230089	+	0.460179i
$681$		0			0
$682$		−	24.0000i		−	0.919007i
$683$			36.0000i			1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$			36.0000i			1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$		0			0
$685$	12.0000	+	6.00000i	0.458496	+	0.229248i
$686$		0			0
$687$		0			0
$688$		−	6.00000i		−	0.228748i
$689$	20.0000			0.761939
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$		−	14.0000i		−	0.532200i
$693$		0			0
$694$	−20.0000			−0.759190
$695$	−12.0000	+	24.0000i	−0.455186	+	0.910372i
$696$		0			0
$697$		0			0
$698$			26.0000i			0.984115i
$699$		0			0
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$			10.0000i			0.377157i
$704$	−4.00000			−0.150756
$705$		0			0
$706$	−2.00000			−0.0752710
$707$		0			0
$708$		0			0
$709$	2.00000			0.0751116			0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000			0.0751116			0.0375558	+	0.999295i	$0.488043\pi$
$710$	24.0000	+	12.0000i	0.900704	+	0.450352i
$711$		0			0
$712$		−	4.00000i		−	0.149906i
$713$			36.0000i			1.34821i
$714$		0			0
$715$	16.0000	+	8.00000i	0.598366	+	0.299183i
$716$	18.0000			0.672692
$717$		0			0
$718$		−	8.00000i		−	0.298557i
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$		0			0
$722$			1.00000i			0.0372161i
$723$		0			0
$724$	4.00000			0.148659
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		0			0
$729$		0			0
$730$	16.0000	−	32.0000i	0.592187	−	1.18437i
$731$	−36.0000			−1.33151
$732$		0			0
$733$			46.0000i			1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$			46.0000i			1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	12.0000			0.442928
$735$		0			0
$736$	6.00000			0.221163
$737$		−	32.0000i		−	1.17874i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$	−20.0000	−	10.0000i	−0.735215	−	0.367607i
$741$		0			0
$742$		0			0
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$	−6.00000	+	12.0000i	−0.219823	+	0.439646i
$746$	−6.00000			−0.219676
$747$		0			0
$748$			24.0000i			0.877527i
$749$		0			0
$750$		0			0
$751$	−26.0000			−0.948753			−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000			−0.948753			−0.474377	+	0.880322i	$0.657327\pi$
$752$			6.00000i			0.218797i
$753$		0			0
$754$	4.00000			0.145671
$755$	10.0000	−	20.0000i	0.363937	−	0.727875i
$756$		0			0
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		−	24.0000i		−	0.871719i
$759$		0			0
$760$	−2.00000	−	1.00000i	−0.0725476	−	0.0362738i
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		0			0
$764$	24.0000			0.868290
$765$		0			0
$766$	36.0000			1.30073
$767$			4.00000i			0.144432i
$768$		0			0
$769$	34.0000			1.22607			0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000			1.22607			0.613036	+	0.790055i	$0.289948\pi$
$770$		0			0
$771$		0			0
$772$		−	14.0000i		−	0.503871i
$773$			50.0000i			1.79838i	0.437564	+	0.899188i	$0.355842\pi$
$773$			50.0000i			1.79838i	−0.437564	+	0.899188i	$0.644158\pi$
$774$		0			0
$775$	18.0000	+	24.0000i	0.646579	+	0.862105i
$776$	10.0000			0.358979
$777$		0			0
$778$			38.0000i			1.36237i
$779$		0			0
$780$		0			0
$781$	48.0000			1.71758
$782$		−	36.0000i		−	1.28736i
$783$		0			0
$784$	7.00000			0.250000
$785$	36.0000	+	18.0000i	1.28490	+	0.642448i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	8.00000i		−	0.284988i
$789$		0			0
$790$	28.0000	+	14.0000i	0.996195	+	0.498098i
$791$		0			0
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$	−30.0000			−1.06466
$795$		0			0
$796$	−16.0000			−0.567105
$797$		−	54.0000i		−	1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$		−	54.0000i		−	1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$		0			0
$799$	36.0000			1.27359
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$		−	16.0000i		−	0.564980i
$803$		−	64.0000i		−	2.25851i
$804$		0			0
$805$		0			0
$806$	12.0000			0.422682
$807$		0			0
$808$		−	18.0000i		−	0.633238i
$809$	−10.0000			−0.351581			−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000			−0.351581			−0.175791	+	0.984428i	$0.556248\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$	−40.0000			−1.40200
$815$	−20.0000	−	10.0000i	−0.700569	−	0.350285i
$816$		0			0
$817$		−	6.00000i		−	0.209913i
$818$		−	26.0000i		−	0.909069i
$819$		0			0
$820$		0			0
$821$	−50.0000			−1.74501			−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000			−1.74501			−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$		−	12.0000i		−	0.418294i	−0.977884	−	0.209147i	$-0.932931\pi$
$823$		−	12.0000i		−	0.418294i	0.977884	−	0.209147i	$-0.0670687\pi$
$824$	8.00000			0.278693
$825$		0			0
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	28.0000			0.972480			0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000			0.972480			0.486240	+	0.873825i	$0.338368\pi$
$830$	12.0000	−	24.0000i	0.416526	−	0.833052i
$831$		0			0
$832$		−	2.00000i		−	0.0693375i
$833$		−	42.0000i		−	1.45521i
$834$		0			0
$835$		0			0
$836$	−4.00000			−0.138343
$837$		0			0
$838$			8.00000i			0.276355i
$839$	44.0000			1.51905			0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000			1.51905			0.759524	+	0.650479i	$0.225432\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$			16.0000i			0.551396i
$843$		0			0
$844$	8.00000			0.275371
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$		0			0
$848$		−	10.0000i		−	0.343401i
$849$		0			0
$850$	−18.0000	−	24.0000i	−0.617395	−	0.823193i
$851$	60.0000			2.05677
$852$		0			0
$853$		−	14.0000i		−	0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$		−	14.0000i		−	0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$		0			0
$855$		0			0
$856$	−4.00000			−0.136717
$857$		−	18.0000i		−	0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$		−	18.0000i		−	0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$	12.0000	+	6.00000i	0.409197	+	0.204598i
$861$		0			0
$862$			24.0000i			0.817443i
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$	28.0000	+	14.0000i	0.952029	+	0.476014i
$866$	−6.00000			−0.203888
$867$		0			0
$868$		0			0
$869$	56.0000			1.89967
$870$		0			0
$871$	16.0000			0.542139
$872$			4.00000i			0.135457i
$873$		0			0
$874$	6.00000			0.202953
$875$		0			0
$876$		0			0
$877$			46.0000i			1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$			46.0000i			1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$			14.0000i			0.472477i
$879$		0			0
$880$	4.00000	−	8.00000i	0.134840	−	0.269680i
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$			10.0000i			0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$			10.0000i			0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$	−12.0000			−0.403604
$885$		0			0
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$		0			0
$890$	8.00000	+	4.00000i	0.268161	+	0.134080i
$891$		0			0
$892$			8.00000i			0.267860i
$893$			6.00000i			0.200782i
$894$		0			0
$895$	−18.0000	+	36.0000i	−0.601674	+	1.20335i
$896$		0			0
$897$		0			0
$898$			4.00000i			0.133482i
$899$	12.0000			0.400222
$900$		0			0
$901$	−60.0000			−1.99889
$902$		0			0
$903$		0			0
$904$	−2.00000			−0.0665190
$905$	−4.00000	+	8.00000i	−0.132964	+	0.265929i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$		−	48.0000i		−	1.58857i
$914$	16.0000			0.529233
$915$		0			0
$916$	−10.0000			−0.330409
$917$		0			0
$918$		0			0
$919$	−28.0000			−0.923635			−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000			−0.923635			−0.461817	+	0.886975i	$0.652802\pi$
$920$	−6.00000	+	12.0000i	−0.197814	+	0.395628i
$921$		0			0
$922$			18.0000i			0.592798i
$923$			24.0000i			0.789970i
$924$		0			0
$925$	40.0000	−	30.0000i	1.31519	−	0.986394i
$926$	8.00000			0.262896
$927$		0			0
$928$		−	2.00000i		−	0.0656532i
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	7.00000			0.229416
$932$			14.0000i			0.458585i
$933$		0			0
$934$		0			0
$935$	−48.0000	−	24.0000i	−1.56977	−	0.784884i
$936$		0			0
$937$			44.0000i			1.43742i	0.695311	+	0.718709i	$0.255266\pi$
$937$			44.0000i			1.43742i	−0.695311	+	0.718709i	$0.744734\pi$
$938$		0			0
$939$		0			0
$940$	−12.0000	−	6.00000i	−0.391397	−	0.195698i
$941$	−22.0000			−0.717180			−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000			−0.717180			−0.358590	+	0.933495i	$0.616742\pi$
$942$		0			0
$943$		0			0
$944$	2.00000			0.0650945
$945$		0			0
$946$	24.0000			0.780307
$947$		−	8.00000i		−	0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$		−	8.00000i		−	0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$		0			0
$949$	32.0000			1.03876
$950$	4.00000	−	3.00000i	0.129777	−	0.0973329i
$951$		0			0
$952$		0			0
$953$			42.0000i			1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$			42.0000i			1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$		0			0
$955$	−24.0000	+	48.0000i	−0.776622	+	1.55324i
$956$	8.00000			0.258738
$957$		0			0
$958$		−	16.0000i		−	0.516937i
$959$		0			0
$960$		0			0
$961$	5.00000			0.161290
$962$		−	20.0000i		−	0.644826i
$963$		0			0
$964$	10.0000			0.322078
$965$	28.0000	+	14.0000i	0.901352	+	0.450676i
$966$		0			0
$967$		−	28.0000i		−	0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$		−	28.0000i		−	0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$		−	5.00000i		−	0.160706i
$969$		0			0
$970$	−10.0000	+	20.0000i	−0.321081	+	0.642161i
$971$	58.0000			1.86131			0.930654	−	0.365900i	$-0.119239\pi$
$971$	58.0000			1.86131			0.930654	+	0.365900i	$0.119239\pi$
$972$		0			0
$973$		0			0
$974$	24.0000			0.769010
$975$		0			0
$976$	−6.00000			−0.192055
$977$		−	26.0000i		−	0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$		−	26.0000i		−	0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$		0			0
$979$	16.0000			0.511362
$980$	−7.00000	+	14.0000i	−0.223607	+	0.447214i
$981$		0			0
$982$		0			0
$983$		−	56.0000i		−	1.78612i	−0.449935	−	0.893061i	$-0.648553\pi$
$983$		−	56.0000i		−	1.78612i	0.449935	−	0.893061i	$-0.351447\pi$
$984$		0			0
$985$	16.0000	+	8.00000i	0.509802	+	0.254901i
$986$	−12.0000			−0.382158
$987$		0			0
$988$		−	2.00000i		−	0.0636285i
$989$	−36.0000			−1.14473
$990$		0			0
$991$	−6.00000			−0.190596			−0.0952981	−	0.995449i	$-0.530380\pi$
$991$	−6.00000			−0.190596			−0.0952981	+	0.995449i	$0.530380\pi$
$992$		−	6.00000i		−	0.190500i
$993$		0			0
$994$		0			0
$995$	16.0000	−	32.0000i	0.507234	−	1.01447i
$996$		0			0
$997$			2.00000i			0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$			2.00000i			0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$		−	20.0000i		−	0.633089i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.2.d.b.1369.2		2
3.2	odd	2		570.2.d.a.229.1	✓	2
5.2	odd	4		8550.2.a.k.1.1		1
5.3	odd	4		8550.2.a.bb.1.1		1
5.4	even	2	inner	1710.2.d.b.1369.1		2
15.2	even	4		2850.2.a.t.1.1		1
15.8	even	4		2850.2.a.i.1.1		1
15.14	odd	2		570.2.d.a.229.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
570.2.d.a.229.1	✓	2	3.2	odd	2
570.2.d.a.229.2	yes	2	15.14	odd	2
1710.2.d.b.1369.1		2	5.4	even	2	inner
1710.2.d.b.1369.2		2	1.1	even	1	trivial
2850.2.a.i.1.1		1	15.8	even	4
2850.2.a.t.1.1		1	15.2	even	4
8550.2.a.k.1.1		1	5.2	odd	4
8550.2.a.bb.1.1		1	5.3	odd	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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +4.00000i q^{22} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{26} -2.00000 q^{29} -6.00000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +10.0000i q^{37} +1.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -6.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} +6.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} -2.00000i q^{52} -10.0000i q^{53} +(4.00000 - 8.00000i) q^{55} -2.00000i q^{58} +2.00000 q^{59} -6.00000 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(4.00000 + 2.00000i) q^{65} -8.00000i q^{67} +6.00000i q^{68} +12.0000 q^{71} -16.0000i q^{73} -10.0000 q^{74} -1.00000 q^{76} +14.0000 q^{79} +(1.00000 - 2.00000i) q^{80} -12.0000i q^{83} +(-12.0000 - 6.00000i) q^{85} +6.00000 q^{86} -4.00000i q^{88} +4.00000 q^{89} +6.00000i q^{92} -6.00000 q^{94} +(1.00000 - 2.00000i) q^{95} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^5	\( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 8 q^{11} + 2 q^{16} + 2 q^{19} - 2 q^{20} - 6 q^{25} - 4 q^{26} - 4 q^{29} - 12 q^{31} + 12 q^{34} - 4 q^{40} - 8 q^{44} + 12 q^{46} + 14 q^{49} + 8 q^{50} + 8 q^{55} + 4 q^{59} - 12 q^{61} - 2 q^{64} + 8 q^{65} + 24 q^{71} - 20 q^{74} - 2 q^{76} + 28 q^{79} + 2 q^{80} - 24 q^{85} + 12 q^{86} + 8 q^{89} - 12 q^{94} + 2 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 8 * q^11 + 2 * q^16 + 2 * q^19 - 2 * q^20 - 6 * q^25 - 4 * q^26 - 4 * q^29 - 12 * q^31 + 12 * q^34 - 4 * q^40 - 8 * q^44 + 12 * q^46 + 14 * q^49 + 8 * q^50 + 8 * q^55 + 4 * q^59 - 12 * q^61 - 2 * q^64 + 8 * q^65 + 24 * q^71 - 20 * q^74 - 2 * q^76 + 28 * q^79 + 2 * q^80 - 24 * q^85 + 12 * q^86 + 8 * q^89 - 12 * q^94 + 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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