LMFDB - Embedded newform 731.2.d.a.560.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [731,2,Mod(560,731)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(731, 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("731.560");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 731 = 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	731.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:	$5.83706438776$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			560.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	731.560
Dual form			731.2.d.a.560.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} -2.82843i q^{5} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{4} -2.82843i q^{5} -3.00000 q^{8} +3.00000 q^{9} -2.82843i q^{10} -2.82843i q^{11} -6.00000 q^{13} -1.00000 q^{16} +(-3.00000 + 2.82843i) q^{17} +3.00000 q^{18} -4.00000 q^{19} +2.82843i q^{20} -2.82843i q^{22} -2.82843i q^{23} -3.00000 q^{25} -6.00000 q^{26} +2.82843i q^{29} -8.48528i q^{31} +5.00000 q^{32} +(-3.00000 + 2.82843i) q^{34} -3.00000 q^{36} -8.48528i q^{37} -4.00000 q^{38} +8.48528i q^{40} -5.65685i q^{41} +1.00000 q^{43} +2.82843i q^{44} -8.48528i q^{45} -2.82843i q^{46} +7.00000 q^{49} -3.00000 q^{50} +6.00000 q^{52} -10.0000 q^{53} -8.00000 q^{55} +2.82843i q^{58} -4.00000 q^{59} +8.48528i q^{61} -8.48528i q^{62} +7.00000 q^{64} +16.9706i q^{65} +12.0000 q^{67} +(3.00000 - 2.82843i) q^{68} +11.3137i q^{71} -9.00000 q^{72} -14.1421i q^{73} -8.48528i q^{74} +4.00000 q^{76} +8.48528i q^{79} +2.82843i q^{80} +9.00000 q^{81} -5.65685i q^{82} +4.00000 q^{83} +(8.00000 + 8.48528i) q^{85} +1.00000 q^{86} +8.48528i q^{88} +6.00000 q^{89} -8.48528i q^{90} +2.82843i q^{92} +11.3137i q^{95} -11.3137i q^{97} +7.00000 q^{98} -8.48528i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9	$ 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 12 q^{13} - 2 q^{16} - 6 q^{17} + 6 q^{18} - 8 q^{19} - 6 q^{25} - 12 q^{26} + 10 q^{32} - 6 q^{34} - 6 q^{36} - 8 q^{38} + 2 q^{43} + 14 q^{49} - 6 q^{50} + 12 q^{52} - 20 q^{53} - 16 q^{55} - 8 q^{59} + 14 q^{64} + 24 q^{67} + 6 q^{68} - 18 q^{72} + 8 q^{76} + 18 q^{81} + 8 q^{83} + 16 q^{85} + 2 q^{86} + 12 q^{89} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 - 12 * q^13 - 2 * q^16 - 6 * q^17 + 6 * q^18 - 8 * q^19 - 6 * q^25 - 12 * q^26 + 10 * q^32 - 6 * q^34 - 6 * q^36 - 8 * q^38 + 2 * q^43 + 14 * q^49 - 6 * q^50 + 12 * q^52 - 20 * q^53 - 16 * q^55 - 8 * q^59 + 14 * q^64 + 24 * q^67 + 6 * q^68 - 18 * q^72 + 8 * q^76 + 18 * q^81 + 8 * q^83 + 16 * q^85 + 2 * q^86 + 12 * q^89 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000			0.707107			0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000			0.707107			0.353553	+	0.935414i	$0.384973\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−1.00000			−0.500000
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−3.00000			−1.06066
$9$	3.00000			1.00000
$10$		−	2.82843i		−	0.894427i
$11$		−	2.82843i		−	0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$		−	2.82843i		−	0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$		0			0
$13$	−6.00000			−1.66410			−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000			−1.66410			−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$18$	3.00000			0.707107
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$			2.82843i			0.632456i
$21$		0			0
$22$		−	2.82843i		−	0.603023i
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$	−6.00000			−1.17670
$27$		0			0
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	5.00000			0.883883
$33$		0			0
$34$	−3.00000	+	2.82843i	−0.514496	+	0.485071i
$35$		0			0
$36$	−3.00000			−0.500000
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	−4.00000			−0.648886
$39$		0			0
$40$			8.48528i			1.34164i
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	1.00000			0.152499
$43$	1.00000			0.152499
$44$			2.82843i			0.426401i
$45$		−	8.48528i		−	1.26491i
$46$		−	2.82843i		−	0.417029i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	−3.00000			−0.424264
$51$		0			0
$52$	6.00000			0.832050
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$			2.82843i			0.371391i
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$			8.48528i			1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		−	8.48528i		−	1.07763i
$63$		0			0
$64$	7.00000			0.875000
$65$			16.9706i			2.10494i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$	3.00000	−	2.82843i	0.363803	−	0.342997i
$69$		0			0
$70$		0			0
$71$			11.3137i			1.34269i	0.741145	+	0.671345i	$0.234283\pi$
$71$			11.3137i			1.34269i	−0.741145	+	0.671345i	$0.765717\pi$
$72$	−9.00000			−1.06066
$73$		−	14.1421i		−	1.65521i	−0.561310	−	0.827606i	$-0.689702\pi$
$73$		−	14.1421i		−	1.65521i	0.561310	−	0.827606i	$-0.310298\pi$
$74$		−	8.48528i		−	0.986394i
$75$		0			0
$76$	4.00000			0.458831
$77$		0			0
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$			2.82843i			0.316228i
$81$	9.00000			1.00000
$82$		−	5.65685i		−	0.624695i
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	8.00000	+	8.48528i	0.867722	+	0.920358i
$86$	1.00000			0.107833
$87$		0			0
$88$			8.48528i			0.904534i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		−	8.48528i		−	0.894427i
$91$		0			0
$92$			2.82843i			0.294884i
$93$		0			0
$94$		0			0
$95$			11.3137i			1.16076i
$96$		0			0
$97$		−	11.3137i		−	1.14873i	−0.818598	−	0.574367i	$-0.805248\pi$
$97$		−	11.3137i		−	1.14873i	0.818598	−	0.574367i	$-0.194752\pi$
$98$	7.00000			0.707107
$99$		−	8.48528i		−	0.852803i
$100$	3.00000			0.300000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$	18.0000			1.76505
$105$		0			0
$106$	−10.0000			−0.971286
$107$			8.48528i			0.820303i	0.912017	+	0.410152i	$0.134524\pi$
$107$			8.48528i			0.820303i	−0.912017	+	0.410152i	$0.865476\pi$
$108$		0			0
$109$			5.65685i			0.541828i	0.962604	+	0.270914i	$0.0873260\pi$
$109$			5.65685i			0.541828i	−0.962604	+	0.270914i	$0.912674\pi$
$110$	−8.00000			−0.762770
$111$		0			0
$112$		0			0
$113$		−	19.7990i		−	1.86253i	−0.364340	−	0.931266i	$-0.618705\pi$
$113$		−	19.7990i		−	1.86253i	0.364340	−	0.931266i	$-0.381295\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$		−	2.82843i		−	0.262613i
$117$	−18.0000			−1.66410
$118$	−4.00000			−0.368230
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$			8.48528i			0.768221i
$123$		0			0
$124$			8.48528i			0.762001i
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000			−0.265165
$129$		0			0
$130$			16.9706i			1.48842i
$131$			16.9706i			1.48272i	0.671105	+	0.741362i	$0.265820\pi$
$131$			16.9706i			1.48272i	−0.671105	+	0.741362i	$0.734180\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		0			0
$136$	9.00000	−	8.48528i	0.771744	−	0.727607i
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$			14.1421i			1.19952i	0.800180	+	0.599760i	$0.204737\pi$
$139$			14.1421i			1.19952i	−0.800180	+	0.599760i	$0.795263\pi$
$140$		0			0
$141$		0			0
$142$			11.3137i			0.949425i
$143$			16.9706i			1.41915i
$144$	−3.00000			−0.250000
$145$	8.00000			0.664364
$146$		−	14.1421i		−	1.17041i
$147$		0			0
$148$			8.48528i			0.697486i
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	12.0000			0.973329
$153$	−9.00000	+	8.48528i	−0.727607	+	0.685994i
$154$		0			0
$155$	−24.0000			−1.92773
$156$		0			0
$157$	−18.0000			−1.43656			−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000			−1.43656			−0.718278	+	0.695756i	$0.755069\pi$
$158$			8.48528i			0.675053i
$159$		0			0
$160$		−	14.1421i		−	1.11803i
$161$		0			0
$162$	9.00000			0.707107
$163$		−	16.9706i		−	1.32924i	−0.747183	−	0.664619i	$-0.768594\pi$
$163$		−	16.9706i		−	1.32924i	0.747183	−	0.664619i	$-0.231406\pi$
$164$			5.65685i			0.441726i
$165$		0			0
$166$	4.00000			0.310460
$167$		−	2.82843i		−	0.218870i	−0.993994	−	0.109435i	$-0.965096\pi$
$167$		−	2.82843i		−	0.218870i	0.993994	−	0.109435i	$-0.0349042\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$	8.00000	+	8.48528i	0.613572	+	0.650791i
$171$	−12.0000			−0.917663
$172$	−1.00000			−0.0762493
$173$		−	22.6274i		−	1.72033i	−0.510015	−	0.860165i	$-0.670360\pi$
$173$		−	22.6274i		−	1.72033i	0.510015	−	0.860165i	$-0.329640\pi$
$174$		0			0
$175$		0			0
$176$			2.82843i			0.213201i
$177$		0			0
$178$	6.00000			0.449719
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$			8.48528i			0.632456i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$			8.48528i			0.625543i
$185$	−24.0000			−1.76452
$186$		0			0
$187$	8.00000	+	8.48528i	0.585018	+	0.620505i
$188$		0			0
$189$		0			0
$190$			11.3137i			0.820783i
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$		−	22.6274i		−	1.62876i	−0.580334	−	0.814379i	$-0.697078\pi$
$193$		−	22.6274i		−	1.62876i	0.580334	−	0.814379i	$-0.302922\pi$
$194$		−	11.3137i		−	0.812277i
$195$		0			0
$196$	−7.00000			−0.500000
$197$		−	16.9706i		−	1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$		−	16.9706i		−	1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$		−	8.48528i		−	0.603023i
$199$			16.9706i			1.20301i	0.798869	+	0.601506i	$0.205432\pi$
$199$			16.9706i			1.20301i	−0.798869	+	0.601506i	$0.794568\pi$
$200$	9.00000			0.636396
$201$		0			0
$202$	−6.00000			−0.422159
$203$		0			0
$204$		0			0
$205$	−16.0000			−1.11749
$206$	16.0000			1.11477
$207$		−	8.48528i		−	0.589768i
$208$	6.00000			0.416025
$209$			11.3137i			0.782586i
$210$		0			0
$211$		−	16.9706i		−	1.16830i	−0.811645	−	0.584151i	$-0.801428\pi$
$211$		−	16.9706i		−	1.16830i	0.811645	−	0.584151i	$-0.198572\pi$
$212$	10.0000			0.686803
$213$		0			0
$214$			8.48528i			0.580042i
$215$		−	2.82843i		−	0.192897i
$216$		0			0
$217$		0			0
$218$			5.65685i			0.383131i
$219$		0			0
$220$	8.00000			0.539360
$221$	18.0000	−	16.9706i	1.21081	−	1.14156i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−9.00000			−0.600000
$226$		−	19.7990i		−	1.31701i
$227$		−	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
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$	−8.00000			−0.527504
$231$		0			0
$232$		−	8.48528i		−	0.557086i
$233$			19.7990i			1.29707i	0.761183	+	0.648537i	$0.224619\pi$
$233$			19.7990i			1.29707i	−0.761183	+	0.648537i	$0.775381\pi$
$234$	−18.0000			−1.17670
$235$		0			0
$236$	4.00000			0.260378
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$		−	8.48528i		−	0.546585i	−0.961931	−	0.273293i	$-0.911887\pi$
$241$		−	8.48528i		−	0.546585i	0.961931	−	0.273293i	$-0.0881127\pi$
$242$	3.00000			0.192847
$243$		0			0
$244$		−	8.48528i		−	0.543214i
$245$		−	19.7990i		−	1.26491i
$246$		0			0
$247$	24.0000			1.52708
$248$			25.4558i			1.61645i
$249$		0			0
$250$		−	5.65685i		−	0.357771i
$251$	−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
$253$	−8.00000			−0.502956
$254$	−8.00000			−0.501965
$255$		0			0
$256$	−17.0000			−1.06250
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$		0			0
$260$		−	16.9706i		−	1.05247i
$261$			8.48528i			0.525226i
$262$			16.9706i			1.04844i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$			28.2843i			1.73749i
$266$		0			0
$267$		0			0
$268$	−12.0000			−0.733017
$269$			16.9706i			1.03471i	0.855770	+	0.517357i	$0.173084\pi$
$269$			16.9706i			1.03471i	−0.855770	+	0.517357i	$0.826916\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$	3.00000	−	2.82843i	0.181902	−	0.171499i
$273$		0			0
$274$	6.00000			0.362473
$275$			8.48528i			0.511682i
$276$		0			0
$277$			2.82843i			0.169944i	0.996383	+	0.0849719i	$0.0270800\pi$
$277$			2.82843i			0.169944i	−0.996383	+	0.0849719i	$0.972920\pi$
$278$			14.1421i			0.848189i
$279$		−	25.4558i		−	1.52400i
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$			31.1127i			1.84946i	0.380626	+	0.924729i	$0.375708\pi$
$283$			31.1127i			1.84946i	−0.380626	+	0.924729i	$0.624292\pi$
$284$		−	11.3137i		−	0.671345i
$285$		0			0
$286$			16.9706i			1.00349i
$287$		0			0
$288$	15.0000			0.883883
$289$	1.00000	−	16.9706i	0.0588235	−	0.998268i
$290$	8.00000			0.469776
$291$		0			0
$292$			14.1421i			0.827606i
$293$	−26.0000			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$294$		0			0
$295$			11.3137i			0.658710i
$296$			25.4558i			1.47959i
$297$		0			0
$298$	6.00000			0.347571
$299$			16.9706i			0.981433i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	4.00000			0.229416
$305$	24.0000			1.37424
$306$	−9.00000	+	8.48528i	−0.514496	+	0.485071i
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$		0			0
$309$		0			0
$310$	−24.0000			−1.36311
$311$		−	25.4558i		−	1.44347i	−0.692170	−	0.721734i	$-0.743345\pi$
$311$		−	25.4558i		−	1.44347i	0.692170	−	0.721734i	$-0.256655\pi$
$312$		0			0
$313$			8.48528i			0.479616i	0.970820	+	0.239808i	$0.0770846\pi$
$313$			8.48528i			0.479616i	−0.970820	+	0.239808i	$0.922915\pi$
$314$	−18.0000			−1.01580
$315$		0			0
$316$		−	8.48528i		−	0.477334i
$317$			11.3137i			0.635441i	0.948184	+	0.317721i	$0.102917\pi$
$317$			11.3137i			0.635441i	−0.948184	+	0.317721i	$0.897083\pi$
$318$		0			0
$319$	8.00000			0.447914
$320$		−	19.7990i		−	1.10680i
$321$		0			0
$322$		0			0
$323$	12.0000	−	11.3137i	0.667698	−	0.629512i
$324$	−9.00000			−0.500000
$325$	18.0000			0.998460
$326$		−	16.9706i		−	0.939913i
$327$		0			0
$328$			16.9706i			0.937043i
$329$		0			0
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$	−4.00000			−0.219529
$333$		−	25.4558i		−	1.39497i
$334$		−	2.82843i		−	0.154765i
$335$		−	33.9411i		−	1.85440i
$336$		0			0
$337$			16.9706i			0.924445i	0.886764	+	0.462223i	$0.152948\pi$
$337$			16.9706i			0.924445i	−0.886764	+	0.462223i	$0.847052\pi$
$338$	23.0000			1.25104
$339$		0			0
$340$	−8.00000	−	8.48528i	−0.433861	−	0.460179i
$341$	−24.0000			−1.29967
$342$	−12.0000			−0.648886
$343$		0			0
$344$	−3.00000			−0.161749
$345$		0			0
$346$		−	22.6274i		−	1.21646i
$347$			5.65685i			0.303676i	0.988405	+	0.151838i	$0.0485192\pi$
$347$			5.65685i			0.303676i	−0.988405	+	0.151838i	$0.951481\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$		−	14.1421i		−	0.753778i
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	32.0000			1.69838
$356$	−6.00000			−0.317999
$357$		0			0
$358$	20.0000			1.05703
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$			25.4558i			1.34164i
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−40.0000			−2.09370
$366$		0			0
$367$		−	31.1127i		−	1.62407i	−0.583609	−	0.812035i	$-0.698360\pi$
$367$		−	31.1127i		−	1.62407i	0.583609	−	0.812035i	$-0.301640\pi$
$368$			2.82843i			0.147442i
$369$		−	16.9706i		−	0.883452i
$370$	−24.0000			−1.24770
$371$		0			0
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$	8.00000	+	8.48528i	0.413670	+	0.438763i
$375$		0			0
$376$		0			0
$377$		−	16.9706i		−	0.874028i
$378$		0			0
$379$			19.7990i			1.01701i	0.861060	+	0.508503i	$0.169801\pi$
$379$			19.7990i			1.01701i	−0.861060	+	0.508503i	$0.830199\pi$
$380$		−	11.3137i		−	0.580381i
$381$		0			0
$382$	−16.0000			−0.818631
$383$	8.00000			0.408781			0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000			0.408781			0.204390	+	0.978889i	$0.434479\pi$
$384$		0			0
$385$		0			0
$386$		−	22.6274i		−	1.15171i
$387$	3.00000			0.152499
$388$			11.3137i			0.574367i
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	8.00000	+	8.48528i	0.404577	+	0.429119i
$392$	−21.0000			−1.06066
$393$		0			0
$394$		−	16.9706i		−	0.854965i
$395$	24.0000			1.20757
$396$			8.48528i			0.426401i
$397$			16.9706i			0.851728i	0.904787	+	0.425864i	$0.140030\pi$
$397$			16.9706i			0.851728i	−0.904787	+	0.425864i	$0.859970\pi$
$398$			16.9706i			0.850657i
$399$		0			0
$400$	3.00000			0.150000
$401$		−	28.2843i		−	1.41245i	−0.707988	−	0.706225i	$-0.750397\pi$
$401$		−	28.2843i		−	1.41245i	0.707988	−	0.706225i	$-0.249603\pi$
$402$		0			0
$403$			50.9117i			2.53609i
$404$	6.00000			0.298511
$405$		−	25.4558i		−	1.26491i
$406$		0			0
$407$	−24.0000			−1.18964
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$	−16.0000			−0.790184
$411$		0			0
$412$	−16.0000			−0.788263
$413$		0			0
$414$		−	8.48528i		−	0.417029i
$415$		−	11.3137i		−	0.555368i
$416$	−30.0000			−1.47087
$417$		0			0
$418$			11.3137i			0.553372i
$419$		−	22.6274i		−	1.10542i	−0.833373	−	0.552711i	$-0.813593\pi$
$419$		−	22.6274i		−	1.10542i	0.833373	−	0.552711i	$-0.186407\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		−	16.9706i		−	0.826114i
$423$		0			0
$424$	30.0000			1.45693
$425$	9.00000	−	8.48528i	0.436564	−	0.411597i
$426$		0			0
$427$		0			0
$428$		−	8.48528i		−	0.410152i
$429$		0			0
$430$		−	2.82843i		−	0.136399i
$431$			14.1421i			0.681203i	0.940208	+	0.340601i	$0.110631\pi$
$431$			14.1421i			0.681203i	−0.940208	+	0.340601i	$0.889369\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$		0			0
$435$		0			0
$436$		−	5.65685i		−	0.270914i
$437$			11.3137i			0.541208i
$438$		0			0
$439$			8.48528i			0.404980i	0.979284	+	0.202490i	$0.0649034\pi$
$439$			8.48528i			0.404980i	−0.979284	+	0.202490i	$0.935097\pi$
$440$	24.0000			1.14416
$441$	21.0000			1.00000
$442$	18.0000	−	16.9706i	0.856173	−	0.807207i
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		−	16.9706i		−	0.804482i
$446$	16.0000			0.757622
$447$		0			0
$448$		0			0
$449$		−	2.82843i		−	0.133482i	−0.997770	−	0.0667409i	$-0.978740\pi$
$449$		−	2.82843i		−	0.133482i	0.997770	−	0.0667409i	$-0.0212601\pi$
$450$	−9.00000			−0.424264
$451$	−16.0000			−0.753411
$452$			19.7990i			0.931266i
$453$		0			0
$454$		−	16.9706i		−	0.796468i
$455$		0			0
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$	−14.0000			−0.654177
$459$		0			0
$460$	8.00000			0.373002
$461$	−10.0000			−0.465746			−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000			−0.465746			−0.232873	+	0.972507i	$0.574813\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		−	2.82843i		−	0.131306i
$465$		0			0
$466$			19.7990i			0.917170i
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$	18.0000			0.832050
$469$		0			0
$470$		0			0
$471$		0			0
$472$	12.0000			0.552345
$473$		−	2.82843i		−	0.130051i
$474$		0			0
$475$	12.0000			0.550598
$476$		0			0
$477$	−30.0000			−1.37361
$478$		0			0
$479$		−	25.4558i		−	1.16311i	−0.813508	−	0.581554i	$-0.802445\pi$
$479$		−	25.4558i		−	1.16311i	0.813508	−	0.581554i	$-0.197555\pi$
$480$		0			0
$481$			50.9117i			2.32137i
$482$		−	8.48528i		−	0.386494i
$483$		0			0
$484$	−3.00000			−0.136364
$485$	−32.0000			−1.45305
$486$		0			0
$487$			31.1127i			1.40985i	0.709281	+	0.704925i	$0.249020\pi$
$487$			31.1127i			1.40985i	−0.709281	+	0.704925i	$0.750980\pi$
$488$		−	25.4558i		−	1.15233i
$489$		0			0
$490$		−	19.7990i		−	0.894427i
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$	−8.00000	−	8.48528i	−0.360302	−	0.382158i
$494$	24.0000			1.07981
$495$	−24.0000			−1.07872
$496$			8.48528i			0.381000i
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			5.65685i			0.252982i
$501$		0			0
$502$	−12.0000			−0.535586
$503$			5.65685i			0.252227i	0.992016	+	0.126113i	$0.0402503\pi$
$503$			5.65685i			0.252227i	−0.992016	+	0.126113i	$0.959750\pi$
$504$		0			0
$505$			16.9706i			0.755180i
$506$	−8.00000			−0.355643
$507$		0			0
$508$	8.00000			0.354943
$509$	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
$512$	−11.0000			−0.486136
$513$		0			0
$514$	−2.00000			−0.0882162
$515$		−	45.2548i		−	1.99417i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		−	50.9117i		−	2.23263i
$521$		−	2.82843i		−	0.123916i	−0.998079	−	0.0619578i	$-0.980266\pi$
$521$		−	2.82843i		−	0.123916i	0.998079	−	0.0619578i	$-0.0197344\pi$
$522$			8.48528i			0.371391i
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		−	16.9706i		−	0.741362i
$525$		0			0
$526$	16.0000			0.697633
$527$	24.0000	+	25.4558i	1.04546	+	1.10887i
$528$		0			0
$529$	15.0000			0.652174
$530$			28.2843i			1.22859i
$531$	−12.0000			−0.520756
$532$		0			0
$533$			33.9411i			1.47015i
$534$		0			0
$535$	24.0000			1.03761
$536$	−36.0000			−1.55496
$537$		0			0
$538$			16.9706i			0.731653i
$539$		−	19.7990i		−	0.852803i
$540$		0			0
$541$		−	11.3137i		−	0.486414i	−0.969974	−	0.243207i	$-0.921801\pi$
$541$		−	11.3137i		−	0.486414i	0.969974	−	0.243207i	$-0.0781995\pi$
$542$	16.0000			0.687259
$543$		0			0
$544$	−15.0000	+	14.1421i	−0.643120	+	0.606339i
$545$	16.0000			0.685365
$546$		0			0
$547$		−	31.1127i		−	1.33028i	−0.746717	−	0.665141i	$-0.768371\pi$
$547$		−	31.1127i		−	1.33028i	0.746717	−	0.665141i	$-0.231629\pi$
$548$	−6.00000			−0.256307
$549$			25.4558i			1.08643i
$550$			8.48528i			0.361814i
$551$		−	11.3137i		−	0.481980i
$552$		0			0
$553$		0			0
$554$			2.82843i			0.120168i
$555$		0			0
$556$		−	14.1421i		−	0.599760i
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$		−	25.4558i		−	1.07763i
$559$	−6.00000			−0.253773
$560$		0			0
$561$		0			0
$562$	10.0000			0.421825
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$	−56.0000			−2.35594
$566$			31.1127i			1.30776i
$567$		0			0
$568$		−	33.9411i		−	1.42414i
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$		−	5.65685i		−	0.236732i	−0.992970	−	0.118366i	$-0.962234\pi$
$571$		−	5.65685i		−	0.236732i	0.992970	−	0.118366i	$-0.0377656\pi$
$572$		−	16.9706i		−	0.709575i
$573$		0			0
$574$		0			0
$575$			8.48528i			0.353861i
$576$	21.0000			0.875000
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$	1.00000	−	16.9706i	0.0415945	−	0.705882i
$579$		0			0
$580$	−8.00000			−0.332182
$581$		0			0
$582$		0			0
$583$			28.2843i			1.17141i
$584$			42.4264i			1.75562i
$585$			50.9117i			2.10494i
$586$	−26.0000			−1.07405
$587$	28.0000			1.15568			0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000			1.15568			0.577842	+	0.816149i	$0.303895\pi$
$588$		0			0
$589$			33.9411i			1.39852i
$590$			11.3137i			0.465778i
$591$		0			0
$592$			8.48528i			0.348743i
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$	−6.00000			−0.245770
$597$		0			0
$598$			16.9706i			0.693978i
$599$	−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
$601$		−	8.48528i		−	0.346122i	−0.984911	−	0.173061i	$-0.944634\pi$
$601$		−	8.48528i		−	0.346122i	0.984911	−	0.173061i	$-0.0553658\pi$
$602$		0			0
$603$	36.0000			1.46603
$604$		0			0
$605$		−	8.48528i		−	0.344976i
$606$		0			0
$607$		−	39.5980i		−	1.60723i	−0.595148	−	0.803616i	$-0.702907\pi$
$607$		−	39.5980i		−	1.60723i	0.595148	−	0.803616i	$-0.297093\pi$
$608$	−20.0000			−0.811107
$609$		0			0
$610$	24.0000			0.971732
$611$		0			0
$612$	9.00000	−	8.48528i	0.363803	−	0.342997i
$613$	46.0000			1.85792			0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000			1.85792			0.928961	+	0.370177i	$0.120703\pi$
$614$	−28.0000			−1.12999
$615$		0			0
$616$		0			0
$617$		−	11.3137i		−	0.455473i	−0.973723	−	0.227736i	$-0.926868\pi$
$617$		−	11.3137i		−	0.455473i	0.973723	−	0.227736i	$-0.0731324\pi$
$618$		0			0
$619$			31.1127i			1.25052i	0.780415	+	0.625262i	$0.215008\pi$
$619$			31.1127i			1.25052i	−0.780415	+	0.625262i	$0.784992\pi$
$620$	24.0000			0.963863
$621$		0			0
$622$		−	25.4558i		−	1.02069i
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$			8.48528i			0.339140i
$627$		0			0
$628$	18.0000			0.718278
$629$	24.0000	+	25.4558i	0.956943	+	1.01499i
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		−	25.4558i		−	1.01258i
$633$		0			0
$634$			11.3137i			0.449325i
$635$			22.6274i			0.897942i
$636$		0			0
$637$	−42.0000			−1.66410
$638$	8.00000			0.316723
$639$			33.9411i			1.34269i
$640$			8.48528i			0.335410i
$641$			14.1421i			0.558581i	0.960207	+	0.279290i	$0.0900992\pi$
$641$			14.1421i			0.558581i	−0.960207	+	0.279290i	$0.909901\pi$
$642$		0			0
$643$			14.1421i			0.557711i	0.960333	+	0.278856i	$0.0899551\pi$
$643$			14.1421i			0.557711i	−0.960333	+	0.278856i	$0.910045\pi$
$644$		0			0
$645$		0			0
$646$	12.0000	−	11.3137i	0.472134	−	0.445132i
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$	−27.0000			−1.06066
$649$			11.3137i			0.444102i
$650$	18.0000			0.706018
$651$		0			0
$652$			16.9706i			0.664619i
$653$		−	14.1421i		−	0.553425i	−0.960953	−	0.276712i	$-0.910755\pi$
$653$		−	14.1421i		−	0.553425i	0.960953	−	0.276712i	$-0.0892449\pi$
$654$		0			0
$655$	48.0000			1.87552
$656$			5.65685i			0.220863i
$657$		−	42.4264i		−	1.65521i
$658$		0			0
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$	−12.0000			−0.466393
$663$		0			0
$664$	−12.0000			−0.465690
$665$		0			0
$666$		−	25.4558i		−	0.986394i
$667$	8.00000			0.309761
$668$			2.82843i			0.109435i
$669$		0			0
$670$		−	33.9411i		−	1.31126i
$671$	24.0000			0.926510
$672$		0			0
$673$			2.82843i			0.109028i	0.998513	+	0.0545139i	$0.0173609\pi$
$673$			2.82843i			0.109028i	−0.998513	+	0.0545139i	$0.982639\pi$
$674$			16.9706i			0.653682i
$675$		0			0
$676$	−23.0000			−0.884615
$677$			2.82843i			0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$			2.82843i			0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$		0			0
$679$		0			0
$680$	−24.0000	−	25.4558i	−0.920358	−	0.976187i
$681$		0			0
$682$	−24.0000			−0.919007
$683$			8.48528i			0.324680i	0.986735	+	0.162340i	$0.0519042\pi$
$683$			8.48528i			0.324680i	−0.986735	+	0.162340i	$0.948096\pi$
$684$	12.0000			0.458831
$685$		−	16.9706i		−	0.648412i
$686$		0			0
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	60.0000			2.28582
$690$		0			0
$691$			22.6274i			0.860788i	0.902641	+	0.430394i	$0.141625\pi$
$691$			22.6274i			0.860788i	−0.902641	+	0.430394i	$0.858375\pi$
$692$			22.6274i			0.860165i
$693$		0			0
$694$			5.65685i			0.214731i
$695$	40.0000			1.51729
$696$		0			0
$697$	16.0000	+	16.9706i	0.606043	+	0.642806i
$698$	−2.00000			−0.0757011
$699$		0			0
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$			33.9411i			1.28011i
$704$		−	19.7990i		−	0.746203i
$705$		0			0
$706$	−14.0000			−0.526897
$707$		0			0
$708$		0			0
$709$		−	5.65685i		−	0.212448i	−0.994342	−	0.106224i	$-0.966124\pi$
$709$		−	5.65685i		−	0.212448i	0.994342	−	0.106224i	$-0.0338760\pi$
$710$	32.0000			1.20094
$711$			25.4558i			0.954669i
$712$	−18.0000			−0.674579
$713$	−24.0000			−0.898807
$714$		0			0
$715$	48.0000			1.79510
$716$	−20.0000			−0.747435
$717$		0			0
$718$	−24.0000			−0.895672
$719$			14.1421i			0.527413i	0.964603	+	0.263706i	$0.0849450\pi$
$719$			14.1421i			0.527413i	−0.964603	+	0.263706i	$0.915055\pi$
$720$			8.48528i			0.316228i
$721$		0			0
$722$	−3.00000			−0.111648
$723$		0			0
$724$		0			0
$725$		−	8.48528i		−	0.315135i
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$	−40.0000			−1.48047
$731$	−3.00000	+	2.82843i	−0.110959	+	0.104613i
$732$		0			0
$733$	30.0000			1.10808			0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000			1.10808			0.554038	+	0.832492i	$0.313086\pi$
$734$		−	31.1127i		−	1.14839i
$735$		0			0
$736$		−	14.1421i		−	0.521286i
$737$		−	33.9411i		−	1.25024i
$738$		−	16.9706i		−	0.624695i
$739$	52.0000			1.91285			0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000			1.91285			0.956425	+	0.291977i	$0.0943129\pi$
$740$	24.0000			0.882258
$741$		0			0
$742$		0			0
$743$			16.9706i			0.622590i	0.950313	+	0.311295i	$0.100763\pi$
$743$			16.9706i			0.622590i	−0.950313	+	0.311295i	$0.899237\pi$
$744$		0			0
$745$		−	16.9706i		−	0.621753i
$746$	26.0000			0.951928
$747$	12.0000			0.439057
$748$	−8.00000	−	8.48528i	−0.292509	−	0.310253i
$749$		0			0
$750$		0			0
$751$		−	16.9706i		−	0.619265i	−0.950856	−	0.309632i	$-0.899794\pi$
$751$		−	16.9706i		−	0.619265i	0.950856	−	0.309632i	$-0.100206\pi$
$752$		0			0
$753$		0			0
$754$		−	16.9706i		−	0.618031i
$755$		0			0
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$			19.7990i			0.719132i
$759$		0			0
$760$		−	33.9411i		−	1.23117i
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$		0			0
$764$	16.0000			0.578860
$765$	24.0000	+	25.4558i	0.867722	+	0.920358i
$766$	8.00000			0.289052
$767$	24.0000			0.866590
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$			22.6274i			0.814379i
$773$	−38.0000			−1.36677			−0.683383	−	0.730061i	$-0.739492\pi$
$773$	−38.0000			−1.36677			−0.683383	+	0.730061i	$0.739492\pi$
$774$	3.00000			0.107833
$775$			25.4558i			0.914401i
$776$			33.9411i			1.21842i
$777$		0			0
$778$	−6.00000			−0.215110
$779$			22.6274i			0.810711i
$780$		0			0
$781$	32.0000			1.14505
$782$	8.00000	+	8.48528i	0.286079	+	0.303433i
$783$		0			0
$784$	−7.00000			−0.250000
$785$			50.9117i			1.81712i
$786$		0			0
$787$		−	19.7990i		−	0.705758i	−0.935669	−	0.352879i	$-0.885203\pi$
$787$		−	19.7990i		−	0.705758i	0.935669	−	0.352879i	$-0.114797\pi$
$788$			16.9706i			0.604551i
$789$		0			0
$790$	24.0000			0.853882
$791$		0			0
$792$			25.4558i			0.904534i
$793$		−	50.9117i		−	1.80793i
$794$			16.9706i			0.602263i
$795$		0			0
$796$		−	16.9706i		−	0.601506i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$		0			0
$799$		0			0
$800$	−15.0000			−0.530330
$801$	18.0000			0.635999
$802$		−	28.2843i		−	0.998752i
$803$	−40.0000			−1.41157
$804$		0			0
$805$		0			0
$806$			50.9117i			1.79329i
$807$		0			0
$808$	18.0000			0.633238
$809$			5.65685i			0.198884i	0.995043	+	0.0994422i	$0.0317058\pi$
$809$			5.65685i			0.198884i	−0.995043	+	0.0994422i	$0.968294\pi$
$810$		−	25.4558i		−	0.894427i
$811$			11.3137i			0.397278i	0.980073	+	0.198639i	$0.0636521\pi$
$811$			11.3137i			0.397278i	−0.980073	+	0.198639i	$0.936348\pi$
$812$		0			0
$813$		0			0
$814$	−24.0000			−0.841200
$815$	−48.0000			−1.68137
$816$		0			0
$817$	−4.00000			−0.139942
$818$	10.0000			0.349642
$819$		0			0
$820$	16.0000			0.558744
$821$			11.3137i			0.394851i	0.980318	+	0.197426i	$0.0632581\pi$
$821$			11.3137i			0.394851i	−0.980318	+	0.197426i	$0.936742\pi$
$822$		0			0
$823$			14.1421i			0.492964i	0.969147	+	0.246482i	$0.0792746\pi$
$823$			14.1421i			0.492964i	−0.969147	+	0.246482i	$0.920725\pi$
$824$	−48.0000			−1.67216
$825$		0			0
$826$		0			0
$827$			42.4264i			1.47531i	0.675177	+	0.737655i	$0.264067\pi$
$827$			42.4264i			1.47531i	−0.675177	+	0.737655i	$0.735933\pi$
$828$			8.48528i			0.294884i
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		−	11.3137i		−	0.392705i
$831$		0			0
$832$	−42.0000			−1.45609
$833$	−21.0000	+	19.7990i	−0.727607	+	0.685994i
$834$		0			0
$835$	−8.00000			−0.276851
$836$		−	11.3137i		−	0.391293i
$837$		0			0
$838$		−	22.6274i		−	0.781651i
$839$			45.2548i			1.56237i	0.624299	+	0.781185i	$0.285385\pi$
$839$			45.2548i			1.56237i	−0.624299	+	0.781185i	$0.714615\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$	−22.0000			−0.758170
$843$		0			0
$844$			16.9706i			0.584151i
$845$		−	65.0538i		−	2.23792i
$846$		0			0
$847$		0			0
$848$	10.0000			0.343401
$849$		0			0
$850$	9.00000	−	8.48528i	0.308697	−	0.291043i
$851$	−24.0000			−0.822709
$852$		0			0
$853$			5.65685i			0.193687i	0.995300	+	0.0968435i	$0.0308746\pi$
$853$			5.65685i			0.193687i	−0.995300	+	0.0968435i	$0.969125\pi$
$854$		0			0
$855$			33.9411i			1.16076i
$856$		−	25.4558i		−	0.870063i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$			2.82843i			0.0964486i
$861$		0			0
$862$			14.1421i			0.481683i
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	−64.0000			−2.17607
$866$	30.0000			1.01944
$867$		0			0
$868$		0			0
$869$	24.0000			0.814144
$870$		0			0
$871$	−72.0000			−2.43963
$872$		−	16.9706i		−	0.574696i
$873$		−	33.9411i		−	1.14873i
$874$			11.3137i			0.382692i
$875$		0			0
$876$		0			0
$877$		−	33.9411i		−	1.14611i	−0.819517	−	0.573055i	$-0.805758\pi$
$877$		−	33.9411i		−	1.14611i	0.819517	−	0.573055i	$-0.194242\pi$
$878$			8.48528i			0.286364i
$879$		0			0
$880$	8.00000			0.269680
$881$		−	5.65685i		−	0.190584i	−0.995449	−	0.0952921i	$-0.969621\pi$
$881$		−	5.65685i		−	0.190584i	0.995449	−	0.0952921i	$-0.0303785\pi$
$882$	21.0000			0.707107
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$	−18.0000	+	16.9706i	−0.605406	+	0.570782i
$885$		0			0
$886$	−12.0000			−0.403148
$887$			22.6274i			0.759754i	0.925037	+	0.379877i	$0.124034\pi$
$887$			22.6274i			0.759754i	−0.925037	+	0.379877i	$0.875966\pi$
$888$		0			0
$889$		0			0
$890$		−	16.9706i		−	0.568855i
$891$		−	25.4558i		−	0.852803i
$892$	−16.0000			−0.535720
$893$		0			0
$894$		0			0
$895$		−	56.5685i		−	1.89088i
$896$		0			0
$897$		0			0
$898$		−	2.82843i		−	0.0943858i
$899$	24.0000			0.800445
$900$	9.00000			0.300000
$901$	30.0000	−	28.2843i	0.999445	−	0.942286i
$902$	−16.0000			−0.532742
$903$		0			0
$904$			59.3970i			1.97551i
$905$		0			0
$906$		0			0
$907$			8.48528i			0.281749i	0.990027	+	0.140875i	$0.0449914\pi$
$907$			8.48528i			0.281749i	−0.990027	+	0.140875i	$0.955009\pi$
$908$			16.9706i			0.563188i
$909$	−18.0000			−0.597022
$910$		0			0
$911$			16.9706i			0.562260i	0.959670	+	0.281130i	$0.0907092\pi$
$911$			16.9706i			0.562260i	−0.959670	+	0.281130i	$0.909291\pi$
$912$		0			0
$913$		−	11.3137i		−	0.374429i
$914$	6.00000			0.198462
$915$		0			0
$916$	14.0000			0.462573
$917$		0			0
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$	24.0000			0.791257
$921$		0			0
$922$	−10.0000			−0.329332
$923$		−	67.8823i		−	2.23437i
$924$		0			0
$925$			25.4558i			0.836983i
$926$	−8.00000			−0.262896
$927$	48.0000			1.57653
$928$			14.1421i			0.464238i
$929$		−	25.4558i		−	0.835179i	−0.908636	−	0.417590i	$-0.862875\pi$
$929$		−	25.4558i		−	0.835179i	0.908636	−	0.417590i	$-0.137125\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$		−	19.7990i		−	0.648537i
$933$		0			0
$934$	−20.0000			−0.654420
$935$	24.0000	−	22.6274i	0.784884	−	0.739996i
$936$	54.0000			1.76505
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	22.6274i		−	0.737633i	−0.929502	−	0.368816i	$-0.879763\pi$
$941$		−	22.6274i		−	0.737633i	0.929502	−	0.368816i	$-0.120237\pi$
$942$		0			0
$943$	−16.0000			−0.521032
$944$	4.00000			0.130189
$945$		0			0
$946$		−	2.82843i		−	0.0919601i
$947$			31.1127i			1.01103i	0.862819	+	0.505513i	$0.168697\pi$
$947$			31.1127i			1.01103i	−0.862819	+	0.505513i	$0.831303\pi$
$948$		0			0
$949$			84.8528i			2.75444i
$950$	12.0000			0.389331
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$	−30.0000			−0.971286
$955$			45.2548i			1.46441i
$956$		0			0
$957$		0			0
$958$		−	25.4558i		−	0.822441i
$959$		0			0
$960$		0			0
$961$	−41.0000			−1.32258
$962$			50.9117i			1.64146i
$963$			25.4558i			0.820303i
$964$			8.48528i			0.273293i
$965$	−64.0000			−2.06023
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$	−9.00000			−0.289271
$969$		0			0
$970$	−32.0000			−1.02746
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$		0			0
$974$			31.1127i			0.996915i
$975$		0			0
$976$		−	8.48528i		−	0.271607i
$977$	−6.00000			−0.191957			−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000			−0.191957			−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		−	16.9706i		−	0.542382i
$980$			19.7990i			0.632456i
$981$			16.9706i			0.541828i
$982$	20.0000			0.638226
$983$		−	28.2843i		−	0.902128i	−0.892492	−	0.451064i	$-0.851045\pi$
$983$		−	28.2843i		−	0.902128i	0.892492	−	0.451064i	$-0.148955\pi$
$984$		0			0
$985$	−48.0000			−1.52941
$986$	−8.00000	−	8.48528i	−0.254772	−	0.270226i
$987$		0			0
$988$	−24.0000			−0.763542
$989$		−	2.82843i		−	0.0899388i
$990$	−24.0000			−0.762770
$991$			33.9411i			1.07818i	0.842250	+	0.539088i	$0.181231\pi$
$991$			33.9411i			1.07818i	−0.842250	+	0.539088i	$0.818769\pi$
$992$		−	42.4264i		−	1.34704i
$993$		0			0
$994$		0			0
$995$	48.0000			1.52170
$996$		0			0
$997$			48.0833i			1.52281i	0.648275	+	0.761406i	$0.275491\pi$
$997$			48.0833i			1.52281i	−0.648275	+	0.761406i	$0.724509\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.d.a.560.1	✓	2
17.16	even	2	inner	731.2.d.a.560.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.d.a.560.1	✓	2	1.1	even	1	trivial
731.2.d.a.560.2	yes	2	17.16	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} -2.82843i q^{5} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} -2.82843i q^{5} -3.00000 q^{8} +3.00000 q^{9} -2.82843i q^{10} -2.82843i q^{11} -6.00000 q^{13} -1.00000 q^{16} +(-3.00000 + 2.82843i) q^{17} +3.00000 q^{18} -4.00000 q^{19} +2.82843i q^{20} -2.82843i q^{22} -2.82843i q^{23} -3.00000 q^{25} -6.00000 q^{26} +2.82843i q^{29} -8.48528i q^{31} +5.00000 q^{32} +(-3.00000 + 2.82843i) q^{34} -3.00000 q^{36} -8.48528i q^{37} -4.00000 q^{38} +8.48528i q^{40} -5.65685i q^{41} +1.00000 q^{43} +2.82843i q^{44} -8.48528i q^{45} -2.82843i q^{46} +7.00000 q^{49} -3.00000 q^{50} +6.00000 q^{52} -10.0000 q^{53} -8.00000 q^{55} +2.82843i q^{58} -4.00000 q^{59} +8.48528i q^{61} -8.48528i q^{62} +7.00000 q^{64} +16.9706i q^{65} +12.0000 q^{67} +(3.00000 - 2.82843i) q^{68} +11.3137i q^{71} -9.00000 q^{72} -14.1421i q^{73} -8.48528i q^{74} +4.00000 q^{76} +8.48528i q^{79} +2.82843i q^{80} +9.00000 q^{81} -5.65685i q^{82} +4.00000 q^{83} +(8.00000 + 8.48528i) q^{85} +1.00000 q^{86} +8.48528i q^{88} +6.00000 q^{89} -8.48528i q^{90} +2.82843i q^{92} +11.3137i q^{95} -11.3137i q^{97} +7.00000 q^{98} -8.48528i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 12 q^{13} - 2 q^{16} - 6 q^{17} + 6 q^{18} - 8 q^{19} - 6 q^{25} - 12 q^{26} + 10 q^{32} - 6 q^{34} - 6 q^{36} - 8 q^{38} + 2 q^{43} + 14 q^{49} - 6 q^{50} + 12 q^{52} - 20 q^{53} - 16 q^{55} - 8 q^{59} + 14 q^{64} + 24 q^{67} + 6 q^{68} - 18 q^{72} + 8 q^{76} + 18 q^{81} + 8 q^{83} + 16 q^{85} + 2 q^{86} + 12 q^{89} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 - 12 * q^13 - 2 * q^16 - 6 * q^17 + 6 * q^18 - 8 * q^19 - 6 * q^25 - 12 * q^26 + 10 * q^32 - 6 * q^34 - 6 * q^36 - 8 * q^38 + 2 * q^43 + 14 * q^49 - 6 * q^50 + 12 * q^52 - 20 * q^53 - 16 * q^55 - 8 * q^59 + 14 * q^64 + 24 * q^67 + 6 * q^68 - 18 * q^72 + 8 * q^76 + 18 * q^81 + 8 * q^83 + 16 * q^85 + 2 * q^86 + 12 * q^89 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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