LMFDB - Embedded newform 289.2.b.a.288.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,2,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not 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:	$2.30767661842$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	289.288
Dual form			289.2.b.a.288.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} -2.00000i q^{5} -4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} -2.00000i q^{5} -4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} -2.00000i q^{10} -2.00000 q^{13} -4.00000i q^{14} -1.00000 q^{16} +3.00000 q^{18} +4.00000 q^{19} +2.00000i q^{20} -4.00000i q^{23} +1.00000 q^{25} -2.00000 q^{26} +4.00000i q^{28} +6.00000i q^{29} +4.00000i q^{31} +5.00000 q^{32} -8.00000 q^{35} -3.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +6.00000i q^{40} +6.00000i q^{41} -4.00000 q^{43} -6.00000i q^{45} -4.00000i q^{46} -9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +12.0000i q^{56} +6.00000i q^{58} +12.0000 q^{59} +10.0000i q^{61} +4.00000i q^{62} -12.0000i q^{63} +7.00000 q^{64} +4.00000i q^{65} +4.00000 q^{67} -8.00000 q^{70} -4.00000i q^{71} -9.00000 q^{72} -6.00000i q^{73} -2.00000i q^{74} -4.00000 q^{76} -12.0000i q^{79} +2.00000i q^{80} +9.00000 q^{81} +6.00000i q^{82} +4.00000 q^{83} -4.00000 q^{86} +10.0000 q^{89} -6.00000i q^{90} +8.00000i q^{91} +4.00000i q^{92} -8.00000i q^{95} +2.00000i q^{97} -9.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 4 q^{13} - 2 q^{16} + 6 q^{18} + 8 q^{19} + 2 q^{25} - 4 q^{26} + 10 q^{32} - 16 q^{35} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 18 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 14 q^{64} + 8 q^{67} - 16 q^{70} - 18 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} - 8 q^{86} + 20 q^{89} - 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 - 4 * q^13 - 2 * q^16 + 6 * q^18 + 8 * q^19 + 2 * q^25 - 4 * q^26 + 10 * q^32 - 16 * q^35 - 6 * q^36 + 8 * q^38 - 8 * q^43 - 18 * q^49 + 2 * q^50 + 4 * q^52 - 12 * q^53 + 24 * q^59 + 14 * q^64 + 8 * q^67 - 16 * q^70 - 18 * q^72 - 8 * q^76 + 18 * q^81 + 8 * q^83 - 8 * q^86 + 20 * q^89 - 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	−3.00000	−1.06066
$9$	3.00000	1.00000
$10$	− 2.00000i	− 0.632456i
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	− 4.00000i	− 1.06904i
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	3.00000	0.707107
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	2.00000i	0.447214i
$21$	0	0
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	4.00000i	0.755929i
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	−8.00000	−1.35225
$36$	−3.00000	−0.500000
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	6.00000i	0.948683i
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	− 4.00000i	− 0.589768i
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	12.0000i	1.60357i
$57$	0	0
$58$	6.00000i	0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	10.0000i	1.28037i	0.768221	+	0.640184i	$0.221142\pi$
$61$	10.0000i	1.28037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$	4.00000i	0.508001i
$63$	− 12.0000i	− 1.51186i
$64$	7.00000	0.875000
$65$	4.00000i	0.496139i
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	−8.00000	−0.956183
$71$	− 4.00000i	− 0.474713i	−0.971423	−	0.237356i	$-0.923719\pi$
$71$	− 4.00000i	− 0.474713i	0.971423	−	0.237356i	$-0.0762809\pi$
$72$	−9.00000	−1.06066
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	− 2.00000i	− 0.232495i
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	− 12.0000i	− 1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$	− 12.0000i	− 1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$	2.00000i	0.223607i
$81$	9.00000	1.00000
$82$	6.00000i	0.662589i
$83$	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
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	− 6.00000i	− 0.632456i
$91$	8.00000i	0.838628i
$92$	4.00000i	0.417029i
$93$	0	0
$94$	0	0
$95$	− 8.00000i	− 0.820783i
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	−9.00000	−0.909137
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	− 6.00000i	− 0.557086i
$117$	−6.00000	−0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	11.0000	1.00000
$122$	10.0000i	0.905357i
$123$	0	0
$124$	− 4.00000i	− 0.359211i
$125$	− 12.0000i	− 1.07331i
$126$	− 12.0000i	− 1.06904i
$127$	−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
$130$	4.00000i	0.350823i
$131$	16.0000i	1.39793i	0.715158	+	0.698963i	$0.246355\pi$
$131$	16.0000i	1.39793i	−0.715158	+	0.698963i	$0.753645\pi$
$132$	0	0
$133$	− 16.0000i	− 1.38738i
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	8.00000	0.676123
$141$	0	0
$142$	− 4.00000i	− 0.335673i
$143$	0	0
$144$	−3.00000	−0.250000
$145$	12.0000	0.996546
$146$	− 6.00000i	− 0.496564i
$147$	0	0
$148$	2.00000i	0.164399i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−12.0000	−0.973329
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	− 12.0000i	− 0.954669i
$159$	0	0
$160$	− 10.0000i	− 0.790569i
$161$	−16.0000	−1.26098
$162$	9.00000	0.707107
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	− 6.00000i	− 0.468521i
$165$	0	0
$166$	4.00000	0.310460
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	12.0000	0.917663
$172$	4.00000	0.304997
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	0	0
$178$	10.0000	0.749532
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	6.00000i	0.447214i
$181$	2.00000i	0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$	2.00000i	0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$	8.00000i	0.592999i
$183$	0	0
$184$	12.0000i	0.884652i
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	− 8.00000i	− 0.580381i
$191$	−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
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	2.00000i	0.143592i
$195$	0	0
$196$	9.00000	0.642857
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	− 20.0000i	− 1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$	− 20.0000i	− 1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−10.0000	−0.703598
$203$	24.0000	1.68447
$204$	0	0
$205$	12.0000	0.838116
$206$	8.00000	0.557386
$207$	− 12.0000i	− 0.834058i
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	− 8.00000i	− 0.550743i	−0.961338	−	0.275371i	$-0.911199\pi$
$211$	− 8.00000i	− 0.550743i	0.961338	−	0.275371i	$-0.0888008\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	8.00000i	0.546869i
$215$	8.00000i	0.545595i
$216$	0	0
$217$	16.0000	1.08615
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	− 20.0000i	− 1.33631i
$225$	3.00000	0.200000
$226$	14.0000i	0.931266i
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	− 18.0000i	− 1.18176i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000i	1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$	18.0000i	1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	− 10.0000i	− 0.640184i
$245$	18.0000i	1.14998i
$246$	0	0
$247$	−8.00000	−0.509028
$248$	− 12.0000i	− 0.762001i
$249$	0	0
$250$	− 12.0000i	− 0.758947i
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	12.0000i	0.755929i
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	− 4.00000i	− 0.248069i
$261$	18.0000i	1.11417i
$262$	16.0000i	0.988483i
$263$	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
$265$	12.0000i	0.737154i
$266$	− 16.0000i	− 0.981023i
$267$	0	0
$268$	−4.00000	−0.244339
$269$	22.0000i	1.34136i	0.741745	+	0.670682i	$0.233998\pi$
$269$	22.0000i	1.34136i	−0.741745	+	0.670682i	$0.766002\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	− 8.00000i	− 0.479808i
$279$	12.0000i	0.718421i
$280$	24.0000	1.43427
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	4.00000i	0.237356i
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	15.0000	0.883883
$289$	0	0
$290$	12.0000	0.704664
$291$	0	0
$292$	6.00000i	0.351123i
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	− 24.0000i	− 1.39733i
$296$	6.00000i	0.348743i
$297$	0	0
$298$	−10.0000	−0.579284
$299$	8.00000i	0.462652i
$300$	0	0
$301$	16.0000i	0.922225i
$302$	16.0000	0.920697
$303$	0	0
$304$	−4.00000	−0.229416
$305$	20.0000	1.14520
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	28.0000i	1.58773i	0.608091	+	0.793867i	$0.291935\pi$
$311$	28.0000i	1.58773i	−0.608091	+	0.793867i	$0.708065\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−2.00000	−0.112867
$315$	−24.0000	−1.35225
$316$	12.0000i	0.675053i
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	0	0
$319$	0	0
$320$	− 14.0000i	− 0.782624i
$321$	0	0
$322$	−16.0000	−0.891645
$323$	0	0
$324$	−9.00000	−0.500000
$325$	−2.00000	−0.110940
$326$	− 24.0000i	− 1.32924i
$327$	0	0
$328$	− 18.0000i	− 0.993884i
$329$	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	$0.535063\pi$
$332$	−4.00000	−0.219529
$333$	− 6.00000i	− 0.328798i
$334$	− 4.00000i	− 0.218870i
$335$	− 8.00000i	− 0.437087i
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	0	0
$341$	0	0
$342$	12.0000	0.648886
$343$	8.00000i	0.431959i
$344$	12.0000	0.646997
$345$	0	0
$346$	22.0000i	1.18273i
$347$	− 32.0000i	− 1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$	− 32.0000i	− 1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	− 4.00000i	− 0.213809i
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−10.0000	−0.529999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	18.0000i	0.948683i
$361$	−3.00000	−0.157895
$362$	2.00000i	0.105118i
$363$	0	0
$364$	− 8.00000i	− 0.419314i
$365$	−12.0000	−0.628109
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	4.00000i	0.208514i
$369$	18.0000i	0.937043i
$370$	−4.00000	−0.207950
$371$	24.0000i	1.24602i
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	8.00000i	0.410391i
$381$	0	0
$382$	−16.0000	−0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	− 2.00000i	− 0.101797i
$387$	−12.0000	−0.609994
$388$	− 2.00000i	− 0.101535i
$389$	−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
$391$	0	0
$392$	27.0000	1.36371
$393$	0	0
$394$	18.0000i	0.906827i
$395$	−24.0000	−1.20757
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	14.0000i	0.699127i	0.936913	+	0.349563i	$0.113670\pi$
$401$	14.0000i	0.699127i	−0.936913	+	0.349563i	$0.886330\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	10.0000	0.497519
$405$	− 18.0000i	− 0.894427i
$406$	24.0000	1.19110
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	−8.00000	−0.394132
$413$	− 48.0000i	− 2.36193i
$414$	− 12.0000i	− 0.589768i
$415$	− 8.00000i	− 0.392705i
$416$	−10.0000	−0.490290
$417$	0	0
$418$	0	0
$419$	− 8.00000i	− 0.390826i	−0.980721	−	0.195413i	$-0.937395\pi$
$419$	− 8.00000i	− 0.390826i	0.980721	−	0.195413i	$-0.0626047\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	40.0000	1.93574
$428$	− 8.00000i	− 0.386695i
$429$	0	0
$430$	8.00000i	0.385794i
$431$	− 12.0000i	− 0.578020i	−0.957326	−	0.289010i	$-0.906674\pi$
$431$	− 12.0000i	− 0.578020i	0.957326	−	0.289010i	$-0.0933260\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	6.00000i	0.287348i
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	− 20.0000i	− 0.954548i	−0.878755	−	0.477274i	$-0.841625\pi$
$439$	− 20.0000i	− 0.954548i	0.878755	−	0.477274i	$-0.158375\pi$
$440$	0	0
$441$	−27.0000	−1.28571
$442$	0	0
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	0	0
$445$	− 20.0000i	− 0.948091i
$446$	−24.0000	−1.13643
$447$	0	0
$448$	− 28.0000i	− 1.32288i
$449$	− 34.0000i	− 1.60456i	−0.596948	−	0.802280i	$-0.703620\pi$
$449$	− 34.0000i	− 1.60456i	0.596948	−	0.802280i	$-0.296380\pi$
$450$	3.00000	0.141421
$451$	0	0
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	24.0000i	1.12638i
$455$	16.0000	0.750092
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	8.00000	0.373002
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	− 6.00000i	− 0.278543i
$465$	0	0
$466$	− 6.00000i	− 0.277945i
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	6.00000	0.277350
$469$	− 16.0000i	− 0.738811i
$470$	0	0
$471$	0	0
$472$	−36.0000	−1.65703
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−18.0000	−0.824163
$478$	−16.0000	−0.731823
$479$	36.0000i	1.64488i	0.568850	+	0.822441i	$0.307388\pi$
$479$	36.0000i	1.64488i	−0.568850	+	0.822441i	$0.692612\pi$
$480$	0	0
$481$	4.00000i	0.182384i
$482$	18.0000i	0.819878i
$483$	0	0
$484$	−11.0000	−0.500000
$485$	4.00000	0.181631
$486$	0	0
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	− 30.0000i	− 1.35804i
$489$	0	0
$490$	18.0000i	0.813157i
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	−16.0000	−0.717698
$498$	0	0
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	12.0000i	0.536656i
$501$	0	0
$502$	12.0000	0.535586
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	36.0000i	1.60357i
$505$	20.0000i	0.889988i
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−18.0000	−0.793946
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	0	0
$518$	−8.00000	−0.351500
$519$	0	0
$520$	− 12.0000i	− 0.526235i
$521$	− 26.0000i	− 1.13908i	−0.821963	−	0.569540i	$-0.807121\pi$
$521$	− 26.0000i	− 1.13908i	0.821963	−	0.569540i	$-0.192879\pi$
$522$	18.0000i	0.787839i
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	− 16.0000i	− 0.698963i
$525$	0	0
$526$	16.0000	0.697633
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	12.0000i	0.521247i
$531$	36.0000	1.56227
$532$	16.0000i	0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	16.0000	0.691740
$536$	−12.0000	−0.518321
$537$	0	0
$538$	22.0000i	0.948487i
$539$	0	0
$540$	0	0
$541$	6.00000i	0.257960i	0.991647	+	0.128980i	$0.0411703\pi$
$541$	6.00000i	0.257960i	−0.991647	+	0.128980i	$0.958830\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	6.00000	0.256307
$549$	30.0000i	1.28037i
$550$	0	0
$551$	24.0000i	1.02243i
$552$	0	0
$553$	−48.0000	−2.04117
$554$	14.0000i	0.594803i
$555$	0	0
$556$	8.00000i	0.339276i
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	12.0000i	0.508001i
$559$	8.00000	0.338364
$560$	8.00000	0.338062
$561$	0	0
$562$	6.00000	0.253095
$563$	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
$565$	28.0000	1.17797
$566$	16.0000i	0.672530i
$567$	− 36.0000i	− 1.51186i
$568$	12.0000i	0.503509i
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	32.0000i	1.33916i	0.742741	+	0.669579i	$0.233526\pi$
$571$	32.0000i	1.33916i	−0.742741	+	0.669579i	$0.766474\pi$
$572$	0	0
$573$	0	0
$574$	24.0000	1.00174
$575$	− 4.00000i	− 0.166812i
$576$	21.0000	0.875000
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	0	0
$580$	−12.0000	−0.498273
$581$	− 16.0000i	− 0.663792i
$582$	0	0
$583$	0	0
$584$	18.0000i	0.744845i
$585$	12.0000i	0.496139i
$586$	6.00000	0.247858
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	16.0000i	0.659269i
$590$	− 24.0000i	− 0.988064i
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	8.00000i	0.327144i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	− 10.0000i	− 0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	− 10.0000i	− 0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$	16.0000i	0.652111i
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	− 22.0000i	− 0.894427i
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	20.0000	0.809776
$611$	0	0
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	48.0000i	1.92928i	0.263566	+	0.964641i	$0.415101\pi$
$619$	48.0000i	1.92928i	−0.263566	+	0.964641i	$0.584899\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	28.0000i	1.12270i
$623$	− 40.0000i	− 1.60257i
$624$	0	0
$625$	−19.0000	−0.760000
$626$	22.0000i	0.879297i
$627$	0	0
$628$	2.00000	0.0798087
$629$	0	0
$630$	−24.0000	−0.956183
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	36.0000i	1.43200i
$633$	0	0
$634$	10.0000i	0.397151i
$635$	16.0000i	0.634941i
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	− 12.0000i	− 0.474713i
$640$	6.00000i	0.237171i
$641$	− 30.0000i	− 1.18493i	−0.805597	−	0.592464i	$-0.798155\pi$
$641$	− 30.0000i	− 1.18493i	0.805597	−	0.592464i	$-0.201845\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	−27.0000	−1.06066
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	24.0000i	0.939913i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	32.0000	1.25034
$656$	− 6.00000i	− 0.234261i
$657$	− 18.0000i	− 0.702247i
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−32.0000	−1.24091
$666$	− 6.00000i	− 0.232495i
$667$	24.0000	0.929284
$668$	4.00000i	0.154765i
$669$	0	0
$670$	− 8.00000i	− 0.309067i
$671$	0	0
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	− 14.0000i	− 0.539260i
$675$	0	0
$676$	9.00000	0.346154
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 40.0000i	− 1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$683$	− 40.0000i	− 1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$684$	−12.0000	−0.458831
$685$	12.0000i	0.458496i
$686$	8.00000i	0.305441i
$687$	0	0
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	− 22.0000i	− 0.836315i
$693$	0	0
$694$	− 32.0000i	− 1.21470i
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	18.0000	0.681310
$699$	0	0
$700$	4.00000i	0.151186i
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	−30.0000	−1.12906
$707$	40.0000i	1.50435i
$708$	0	0
$709$	− 34.0000i	− 1.27690i	−0.769665	−	0.638448i	$-0.779577\pi$
$709$	− 34.0000i	− 1.27690i	0.769665	−	0.638448i	$-0.220423\pi$
$710$	−8.00000	−0.300235
$711$	− 36.0000i	− 1.35011i
$712$	−30.0000	−1.12430
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	4.00000i	0.149175i	0.997214	+	0.0745874i	$0.0237640\pi$
$719$	4.00000i	0.149175i	−0.997214	+	0.0745874i	$0.976236\pi$
$720$	6.00000i	0.223607i
$721$	− 32.0000i	− 1.19174i
$722$	−3.00000	−0.111648
$723$	0	0
$724$	− 2.00000i	− 0.0743294i
$725$	6.00000i	0.222834i
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	− 24.0000i	− 0.889499i
$729$	27.0000	1.00000
$730$	−12.0000	−0.444140
$731$	0	0
$732$	0	0
$733$	50.0000	1.84679	0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000	1.84679	0.923396	+	0.383849i	$0.125402\pi$
$734$	− 28.0000i	− 1.03350i
$735$	0	0
$736$	− 20.0000i	− 0.737210i
$737$	0	0
$738$	18.0000i	0.662589i
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	24.0000i	0.881068i
$743$	12.0000i	0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$	12.0000i	0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$	0	0
$745$	20.0000i	0.732743i
$746$	6.00000	0.219676
$747$	12.0000	0.439057
$748$	0	0
$749$	32.0000	1.16925
$750$	0	0
$751$	20.0000i	0.729810i	0.931045	+	0.364905i	$0.118899\pi$
$751$	20.0000i	0.729810i	−0.931045	+	0.364905i	$0.881101\pi$
$752$	0	0
$753$	0	0
$754$	− 12.0000i	− 0.437014i
$755$	− 32.0000i	− 1.16460i
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	− 8.00000i	− 0.290573i
$759$	0	0
$760$	24.0000i	0.870572i
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	16.0000	0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−12.0000	−0.431331
$775$	4.00000i	0.143684i
$776$	− 6.00000i	− 0.215387i
$777$	0	0
$778$	−6.00000	−0.215110
$779$	24.0000i	0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	4.00000i	0.142766i
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	−24.0000	−0.853882
$791$	56.0000	1.99113
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	− 6.00000i	− 0.212932i
$795$	0	0
$796$	20.0000i	0.708881i
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	30.0000	1.06000
$802$	14.0000i	0.494357i
$803$	0	0
$804$	0	0
$805$	32.0000i	1.12785i
$806$	− 8.00000i	− 0.281788i
$807$	0	0
$808$	30.0000	1.05540
$809$	− 26.0000i	− 0.914111i	−0.889438	−	0.457056i	$-0.848904\pi$
$809$	− 26.0000i	− 0.914111i	0.889438	−	0.457056i	$-0.151096\pi$
$810$	− 18.0000i	− 0.632456i
$811$	40.0000i	1.40459i	0.711886	+	0.702295i	$0.247841\pi$
$811$	40.0000i	1.40459i	−0.711886	+	0.702295i	$0.752159\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	0	0
$815$	−48.0000	−1.68137
$816$	0	0
$817$	−16.0000	−0.559769
$818$	26.0000	0.909069
$819$	24.0000i	0.838628i
$820$	−12.0000	−0.419058
$821$	− 18.0000i	− 0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$	− 18.0000i	− 0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$	0	0
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	− 48.0000i	− 1.67013i
$827$	48.0000i	1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$	48.0000i	1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$	12.0000i	0.417029i
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	− 8.00000i	− 0.277684i
$831$	0	0
$832$	−14.0000	−0.485363
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	− 8.00000i	− 0.276355i
$839$	− 20.0000i	− 0.690477i	−0.938515	−	0.345238i	$-0.887798\pi$
$839$	− 20.0000i	− 0.690477i	0.938515	−	0.345238i	$-0.112202\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	22.0000	0.758170
$843$	0	0
$844$	8.00000i	0.275371i
$845$	18.0000i	0.619219i
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	40.0000	1.36877
$855$	− 24.0000i	− 0.820783i
$856$	− 24.0000i	− 0.820303i
$857$	− 10.0000i	− 0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$	− 10.0000i	− 0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	− 8.00000i	− 0.272798i
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	44.0000	1.49604
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	−16.0000	−0.543075
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	18.0000i	0.609557i
$873$	6.00000i	0.203069i
$874$	− 16.0000i	− 0.541208i
$875$	−48.0000	−1.62270
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	− 20.0000i	− 0.674967i
$879$	0	0
$880$	0	0
$881$	− 46.0000i	− 1.54978i	−0.632096	−	0.774890i	$-0.717805\pi$
$881$	− 46.0000i	− 1.54978i	0.632096	−	0.774890i	$-0.282195\pi$
$882$	−27.0000	−0.909137
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	0	0
$886$	28.0000	0.940678
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	0	0
$889$	32.0000i	1.07325i
$890$	− 20.0000i	− 0.670402i
$891$	0	0
$892$	24.0000	0.803579
$893$	0	0
$894$	0	0
$895$	24.0000i	0.802232i
$896$	12.0000i	0.400892i
$897$	0	0
$898$	− 34.0000i	− 1.13459i
$899$	−24.0000	−0.800445
$900$	−3.00000	−0.100000
$901$	0	0
$902$	0	0
$903$	0	0
$904$	− 42.0000i	− 1.39690i
$905$	4.00000	0.132964
$906$	0	0
$907$	− 32.0000i	− 1.06254i	−0.847202	−	0.531271i	$-0.821714\pi$
$907$	− 32.0000i	− 1.06254i	0.847202	−	0.531271i	$-0.178286\pi$
$908$	− 24.0000i	− 0.796468i
$909$	−30.0000	−0.995037
$910$	16.0000	0.530395
$911$	4.00000i	0.132526i	0.997802	+	0.0662630i	$0.0211076\pi$
$911$	4.00000i	0.132526i	−0.997802	+	0.0662630i	$0.978892\pi$
$912$	0	0
$913$	0	0
$914$	6.00000	0.198462
$915$	0	0
$916$	6.00000	0.198246
$917$	64.0000	2.11347
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	24.0000	0.791257
$921$	0	0
$922$	2.00000	0.0658665
$923$	8.00000i	0.263323i
$924$	0	0
$925$	− 2.00000i	− 0.0657596i
$926$	32.0000	1.05159
$927$	24.0000	0.788263
$928$	30.0000i	0.984798i
$929$	30.0000i	0.984268i	0.870519	+	0.492134i	$0.163783\pi$
$929$	30.0000i	0.984268i	−0.870519	+	0.492134i	$0.836217\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	6.00000i	0.196537i
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	18.0000	0.588348
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	− 6.00000i	− 0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$	− 6.00000i	− 0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	32.0000i	1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$	32.0000i	1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	4.00000	0.129777
$951$	0	0
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−18.0000	−0.582772
$955$	32.0000i	1.03550i
$956$	16.0000	0.517477
$957$	0	0
$958$	36.0000i	1.16311i
$959$	24.0000i	0.775000i
$960$	0	0
$961$	15.0000	0.483871
$962$	4.00000i	0.128965i
$963$	24.0000i	0.773389i
$964$	− 18.0000i	− 0.579741i
$965$	−4.00000	−0.128765
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−33.0000	−1.06066
$969$	0	0
$970$	4.00000	0.128432
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	− 20.0000i	− 0.640841i
$975$	0	0
$976$	− 10.0000i	− 0.320092i
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	0	0
$980$	− 18.0000i	− 0.574989i
$981$	− 18.0000i	− 0.574696i
$982$	−20.0000	−0.638226
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	36.0000	1.14706
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	16.0000i	0.508770i
$990$	0	0
$991$	− 12.0000i	− 0.381193i	−0.981669	−	0.190596i	$-0.938958\pi$
$991$	− 12.0000i	− 0.381193i	0.981669	−	0.190596i	$-0.0610421\pi$
$992$	20.0000i	0.635001i
$993$	0	0
$994$	−16.0000	−0.507489
$995$	−40.0000	−1.26809
$996$	0	0
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	40.0000i	1.26618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.2.b.a.288.1		2
17.2	even	8		289.2.c.a.251.2		4
17.3	odd	16		289.2.d.d.110.2		8
17.4	even	4		17.2.a.a.1.1	✓	1
17.5	odd	16		289.2.d.d.179.1		8
17.6	odd	16		289.2.d.d.134.1		8
17.7	odd	16		289.2.d.d.155.2		8
17.8	even	8		289.2.c.a.38.2		4
17.9	even	8		289.2.c.a.38.1		4
17.10	odd	16		289.2.d.d.155.1		8
17.11	odd	16		289.2.d.d.134.2		8
17.12	odd	16		289.2.d.d.179.2		8
17.13	even	4		289.2.a.a.1.1		1
17.14	odd	16		289.2.d.d.110.1		8
17.15	even	8		289.2.c.a.251.1		4
17.16	even	2	inner	289.2.b.a.288.2		2
51.38	odd	4		153.2.a.c.1.1		1
51.47	odd	4		2601.2.a.g.1.1		1
68.47	odd	4		4624.2.a.d.1.1		1
68.55	odd	4		272.2.a.b.1.1		1
85.4	even	4		425.2.a.d.1.1		1
85.38	odd	4		425.2.b.b.324.2		2
85.64	even	4		7225.2.a.g.1.1		1
85.72	odd	4		425.2.b.b.324.1		2
119.4	even	12		833.2.e.b.324.1		2
119.38	odd	12		833.2.e.a.324.1		2
119.55	odd	4		833.2.a.a.1.1		1
119.72	even	12		833.2.e.b.18.1		2
119.89	odd	12		833.2.e.a.18.1		2
136.21	even	4		1088.2.a.i.1.1		1
136.123	odd	4		1088.2.a.h.1.1		1
187.21	odd	4		2057.2.a.e.1.1		1
204.191	even	4		2448.2.a.o.1.1		1
221.38	even	4		2873.2.a.c.1.1		1
255.89	odd	4		3825.2.a.d.1.1		1
323.208	odd	4		6137.2.a.b.1.1		1
340.259	odd	4		6800.2.a.n.1.1		1
357.293	even	4		7497.2.a.l.1.1		1
391.344	odd	4		8993.2.a.a.1.1		1
408.293	odd	4		9792.2.a.n.1.1		1
408.395	even	4		9792.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.2.a.a.1.1	✓	1	17.4	even	4
153.2.a.c.1.1		1	51.38	odd	4
272.2.a.b.1.1		1	68.55	odd	4
289.2.a.a.1.1		1	17.13	even	4
289.2.b.a.288.1		2	1.1	even	1	trivial
289.2.b.a.288.2		2	17.16	even	2	inner
289.2.c.a.38.1		4	17.9	even	8
289.2.c.a.38.2		4	17.8	even	8
289.2.c.a.251.1		4	17.15	even	8
289.2.c.a.251.2		4	17.2	even	8
289.2.d.d.110.1		8	17.14	odd	16
289.2.d.d.110.2		8	17.3	odd	16
289.2.d.d.134.1		8	17.6	odd	16
289.2.d.d.134.2		8	17.11	odd	16
289.2.d.d.155.1		8	17.10	odd	16
289.2.d.d.155.2		8	17.7	odd	16
289.2.d.d.179.1		8	17.5	odd	16
289.2.d.d.179.2		8	17.12	odd	16
425.2.a.d.1.1		1	85.4	even	4
425.2.b.b.324.1		2	85.72	odd	4
425.2.b.b.324.2		2	85.38	odd	4
833.2.a.a.1.1		1	119.55	odd	4
833.2.e.a.18.1		2	119.89	odd	12
833.2.e.a.324.1		2	119.38	odd	12
833.2.e.b.18.1		2	119.72	even	12
833.2.e.b.324.1		2	119.4	even	12
1088.2.a.h.1.1		1	136.123	odd	4
1088.2.a.i.1.1		1	136.21	even	4
2057.2.a.e.1.1		1	187.21	odd	4
2448.2.a.o.1.1		1	204.191	even	4
2601.2.a.g.1.1		1	51.47	odd	4
2873.2.a.c.1.1		1	221.38	even	4
3825.2.a.d.1.1		1	255.89	odd	4
4624.2.a.d.1.1		1	68.47	odd	4
6137.2.a.b.1.1		1	323.208	odd	4
6800.2.a.n.1.1		1	340.259	odd	4
7225.2.a.g.1.1		1	85.64	even	4
7497.2.a.l.1.1		1	357.293	even	4
8993.2.a.a.1.1		1	391.344	odd	4
9792.2.a.i.1.1		1	408.395	even	4
9792.2.a.n.1.1		1	408.293	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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} -2.00000i q^{5} -4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} -2.00000i q^{5} -4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} -2.00000i q^{10} -2.00000 q^{13} -4.00000i q^{14} -1.00000 q^{16} +3.00000 q^{18} +4.00000 q^{19} +2.00000i q^{20} -4.00000i q^{23} +1.00000 q^{25} -2.00000 q^{26} +4.00000i q^{28} +6.00000i q^{29} +4.00000i q^{31} +5.00000 q^{32} -8.00000 q^{35} -3.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +6.00000i q^{40} +6.00000i q^{41} -4.00000 q^{43} -6.00000i q^{45} -4.00000i q^{46} -9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +12.0000i q^{56} +6.00000i q^{58} +12.0000 q^{59} +10.0000i q^{61} +4.00000i q^{62} -12.0000i q^{63} +7.00000 q^{64} +4.00000i q^{65} +4.00000 q^{67} -8.00000 q^{70} -4.00000i q^{71} -9.00000 q^{72} -6.00000i q^{73} -2.00000i q^{74} -4.00000 q^{76} -12.0000i q^{79} +2.00000i q^{80} +9.00000 q^{81} +6.00000i q^{82} +4.00000 q^{83} -4.00000 q^{86} +10.0000 q^{89} -6.00000i q^{90} +8.00000i q^{91} +4.00000i q^{92} -8.00000i q^{95} +2.00000i q^{97} -9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 4 q^{13} - 2 q^{16} + 6 q^{18} + 8 q^{19} + 2 q^{25} - 4 q^{26} + 10 q^{32} - 16 q^{35} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 18 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 14 q^{64} + 8 q^{67} - 16 q^{70} - 18 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} - 8 q^{86} + 20 q^{89} - 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 - 4 * q^13 - 2 * q^16 + 6 * q^18 + 8 * q^19 + 2 * q^25 - 4 * q^26 + 10 * q^32 - 16 * q^35 - 6 * q^36 + 8 * q^38 - 8 * q^43 - 18 * q^49 + 2 * q^50 + 4 * q^52 - 12 * q^53 + 24 * q^59 + 14 * q^64 + 8 * q^67 - 16 * q^70 - 18 * q^72 - 8 * q^76 + 18 * q^81 + 8 * q^83 - 8 * q^86 + 20 * q^89 - 18 * q^98

Introduction

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