LMFDB - Embedded newform 578.2.b.a.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [578,2,Mod(577,578)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(578, 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("578.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 578 = 2 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	578.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:	$4.61535323683$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	578.577
Dual form			578.2.b.a.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} +4.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} +4.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} -6.00000i q^{11} -2.00000i q^{12} +2.00000 q^{13} -4.00000i q^{14} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{19} +8.00000 q^{21} +6.00000i q^{22} +2.00000i q^{24} +5.00000 q^{25} -2.00000 q^{26} -4.00000i q^{27} +4.00000i q^{28} -4.00000i q^{31} -1.00000 q^{32} -12.0000 q^{33} -1.00000 q^{36} -4.00000i q^{37} -4.00000 q^{38} -4.00000i q^{39} -6.00000i q^{41} -8.00000 q^{42} -8.00000 q^{43} -6.00000i q^{44} -2.00000i q^{48} -9.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} +4.00000i q^{54} -4.00000i q^{56} -8.00000i q^{57} +4.00000i q^{61} +4.00000i q^{62} -4.00000i q^{63} +1.00000 q^{64} +12.0000 q^{66} +8.00000 q^{67} +1.00000 q^{72} +2.00000i q^{73} +4.00000i q^{74} -10.0000i q^{75} +4.00000 q^{76} +24.0000 q^{77} +4.00000i q^{78} -8.00000i q^{79} -11.0000 q^{81} +6.00000i q^{82} +8.00000 q^{84} +8.00000 q^{86} +6.00000i q^{88} -6.00000 q^{89} +8.00000i q^{91} -8.00000 q^{93} +2.00000i q^{96} +14.0000i q^{97} +9.00000 q^{98} +6.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 4 q^{13} + 2 q^{16} + 2 q^{18} + 8 q^{19} + 16 q^{21} + 10 q^{25} - 4 q^{26} - 2 q^{32} - 24 q^{33} - 2 q^{36} - 8 q^{38} - 16 q^{42} - 16 q^{43} - 18 q^{49} - 10 q^{50} + 4 q^{52} + 12 q^{53} + 2 q^{64} + 24 q^{66} + 16 q^{67} + 2 q^{72} + 8 q^{76} + 48 q^{77} - 22 q^{81} + 16 q^{84} + 16 q^{86} - 12 q^{89} - 16 q^{93} + 18 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 4 * q^13 + 2 * q^16 + 2 * q^18 + 8 * q^19 + 16 * q^21 + 10 * q^25 - 4 * q^26 - 2 * q^32 - 24 * q^33 - 2 * q^36 - 8 * q^38 - 16 * q^42 - 16 * q^43 - 18 * q^49 - 10 * q^50 + 4 * q^52 + 12 * q^53 + 2 * q^64 + 24 * q^66 + 16 * q^67 + 2 * q^72 + 8 * q^76 + 48 * q^77 - 22 * q^81 + 16 * q^84 + 16 * q^86 - 12 * q^89 - 16 * q^93 + 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/578\mathbb{Z}\right)^\times$.

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

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
$2$	−1.00000	−0.707107
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	1.00000	0.500000
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	2.00000i	0.816497i
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	− 6.00000i	− 1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$	− 2.00000i	− 0.577350i
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	− 4.00000i	− 1.06904i
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	8.00000	1.74574
$22$	6.00000i	1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.00000i	0.408248i
$25$	5.00000	1.00000
$26$	−2.00000	−0.392232
$27$	− 4.00000i	− 0.769800i
$28$	4.00000i	0.755929i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	−1.00000	−0.176777
$33$	−12.0000	−2.08893
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	−4.00000	−0.648886
$39$	− 4.00000i	− 0.640513i
$40$	0	0
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	−8.00000	−1.23443
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	− 6.00000i	− 0.904534i
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−9.00000	−1.28571
$50$	−5.00000	−0.707107
$51$	0	0
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	4.00000i	0.544331i
$55$	0	0
$56$	− 4.00000i	− 0.534522i
$57$	− 8.00000i	− 1.05963i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	4.00000i	0.512148i	0.966657	+	0.256074i	$0.0824290\pi$
$61$	4.00000i	0.512148i	−0.966657	+	0.256074i	$0.917571\pi$
$62$	4.00000i	0.508001i
$63$	− 4.00000i	− 0.503953i
$64$	1.00000	0.125000
$65$	0	0
$66$	12.0000	1.47710
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000	0.117851
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	4.00000i	0.464991i
$75$	− 10.0000i	− 1.15470i
$76$	4.00000	0.458831
$77$	24.0000	2.73505
$78$	4.00000i	0.452911i
$79$	− 8.00000i	− 0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$	− 8.00000i	− 0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	6.00000i	0.662589i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	8.00000	0.872872
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	6.00000i	0.639602i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	8.00000i	0.838628i
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	0	0
$96$	2.00000i	0.204124i
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	9.00000	0.909137
$99$	6.00000i	0.603023i
$100$	5.00000	0.500000
$101$	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
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	− 4.00000i	− 0.384900i
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	4.00000i	0.377964i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	8.00000i	0.749269i
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−25.0000	−2.27273
$122$	− 4.00000i	− 0.362143i
$123$	−12.0000	−1.08200
$124$	− 4.00000i	− 0.359211i
$125$	0	0
$126$	4.00000i	0.356348i
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	16.0000i	1.40872i
$130$	0	0
$131$	− 6.00000i	− 0.524222i	−0.965038	−	0.262111i	$-0.915581\pi$
$131$	− 6.00000i	− 0.524222i	0.965038	−	0.262111i	$-0.0844187\pi$
$132$	−12.0000	−1.04447
$133$	16.0000i	1.38738i
$134$	−8.00000	−0.691095
$135$	0	0
$136$	0	0
$137$	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
$139$	2.00000i	0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$	2.00000i	0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 12.0000i	− 1.00349i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	− 2.00000i	− 0.165521i
$147$	18.0000i	1.48461i
$148$	− 4.00000i	− 0.328798i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	10.0000i	0.816497i
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	0	0
$154$	−24.0000	−1.93398
$155$	0	0
$156$	− 4.00000i	− 0.320256i
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000i	0.636446i
$159$	− 12.0000i	− 0.951662i
$160$	0	0
$161$	0	0
$162$	11.0000	0.864242
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	− 6.00000i	− 0.468521i
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	−8.00000	−0.617213
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−8.00000	−0.609994
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	20.0000i	1.51186i
$176$	− 6.00000i	− 0.452267i
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	4.00000i	0.297318i	0.988889	+	0.148659i	$0.0474956\pi$
$181$	4.00000i	0.297318i	−0.988889	+	0.148659i	$0.952504\pi$
$182$	− 8.00000i	− 0.592999i
$183$	8.00000	0.591377
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	− 2.00000i	− 0.144338i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	− 14.0000i	− 1.00514i
$195$	0	0
$196$	−9.00000	−0.642857
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	− 6.00000i	− 0.426401i
$199$	− 16.0000i	− 1.13421i	−0.823646	−	0.567105i	$-0.808063\pi$
$199$	− 16.0000i	− 1.13421i	0.823646	−	0.567105i	$-0.191937\pi$
$200$	−5.00000	−0.353553
$201$	− 16.0000i	− 1.12855i
$202$	−18.0000	−1.26648
$203$	0	0
$204$	0	0
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	2.00000	0.138675
$209$	− 24.0000i	− 1.66011i
$210$	0	0
$211$	10.0000i	0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$	10.0000i	0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	6.00000i	0.410152i
$215$	0	0
$216$	4.00000i	0.272166i
$217$	16.0000	1.08615
$218$	− 16.0000i	− 1.08366i
$219$	4.00000	0.270295
$220$	0	0
$221$	0	0
$222$	8.00000	0.536925
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	− 4.00000i	− 0.267261i
$225$	−5.00000	−0.333333
$226$	− 6.00000i	− 0.399114i
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	− 8.00000i	− 0.529813i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	− 48.0000i	− 3.15817i
$232$	0	0
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	25.0000	1.60706
$243$	10.0000i	0.641500i
$244$	4.00000i	0.256074i
$245$	0	0
$246$	12.0000	0.765092
$247$	8.00000	0.509028
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	− 16.0000i	− 0.996116i
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	− 16.0000i	− 0.981023i
$267$	12.0000i	0.734388i
$268$	8.00000	0.488678
$269$	− 24.0000i	− 1.46331i	−0.681677	−	0.731653i	$-0.738749\pi$
$269$	− 24.0000i	− 1.46331i	0.681677	−	0.731653i	$-0.261251\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	16.0000	0.968364
$274$	−6.00000	−0.362473
$275$	− 30.0000i	− 1.80907i
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	− 2.00000i	− 0.119952i
$279$	4.00000i	0.239474i
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	0	0
$286$	12.0000i	0.709575i
$287$	24.0000	1.41668
$288$	1.00000	0.0589256
$289$	0	0
$290$	0	0
$291$	28.0000	1.64139
$292$	2.00000i	0.117041i
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	− 18.0000i	− 1.04978i
$295$	0	0
$296$	4.00000i	0.232495i
$297$	−24.0000	−1.39262
$298$	−6.00000	−0.347571
$299$	0	0
$300$	− 10.0000i	− 0.577350i
$301$	− 32.0000i	− 1.84445i
$302$	−16.0000	−0.920697
$303$	− 36.0000i	− 2.06815i
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	24.0000	1.36753
$309$	32.0000i	1.82042i
$310$	0	0
$311$	12.0000i	0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$	12.0000i	0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$	4.00000i	0.226455i
$313$	34.0000i	1.92179i	0.276907	+	0.960897i	$0.410691\pi$
$313$	34.0000i	1.92179i	−0.276907	+	0.960897i	$0.589309\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	− 8.00000i	− 0.450035i
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	12.0000i	0.672927i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	10.0000	0.554700
$326$	2.00000i	0.110770i
$327$	32.0000	1.76960
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	− 12.0000i	− 0.656611i
$335$	0	0
$336$	8.00000	0.436436
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	9.00000	0.489535
$339$	12.0000	0.651751
$340$	0	0
$341$	−24.0000	−1.29967
$342$	4.00000	0.216295
$343$	− 8.00000i	− 0.431959i
$344$	8.00000	0.431331
$345$	0	0
$346$	− 24.0000i	− 1.29025i
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	− 20.0000i	− 1.06904i
$351$	− 8.00000i	− 0.427008i
$352$	6.00000i	0.319801i
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 4.00000i	− 0.210235i
$363$	50.0000i	2.62432i
$364$	8.00000i	0.419314i
$365$	0	0
$366$	−8.00000	−0.418167
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	6.00000i	0.312348i
$370$	0	0
$371$	24.0000i	1.24602i
$372$	−8.00000	−0.414781
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	−16.0000	−0.822951
$379$	14.0000i	0.719132i	0.933120	+	0.359566i	$0.117075\pi$
$379$	14.0000i	0.719132i	−0.933120	+	0.359566i	$0.882925\pi$
$380$	0	0
$381$	− 32.0000i	− 1.63941i
$382$	24.0000	1.22795
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	2.00000i	0.102062i
$385$	0	0
$386$	− 10.0000i	− 0.508987i
$387$	8.00000	0.406663
$388$	14.0000i	0.710742i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	−12.0000	−0.605320
$394$	− 12.0000i	− 0.604551i
$395$	0	0
$396$	6.00000i	0.301511i
$397$	− 20.0000i	− 1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$	− 20.0000i	− 1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$	16.0000i	0.802008i
$399$	32.0000	1.60200
$400$	5.00000	0.250000
$401$	− 30.0000i	− 1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$	− 30.0000i	− 1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	16.0000i	0.798007i
$403$	− 8.00000i	− 0.398508i
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	− 12.0000i	− 0.591916i
$412$	−16.0000	−0.788263
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	4.00000	0.195881
$418$	24.0000i	1.17388i
$419$	30.0000i	1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$419$	30.0000i	1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 10.0000i	− 0.486792i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−16.0000	−0.774294
$428$	− 6.00000i	− 0.290021i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	− 4.00000i	− 0.192450i
$433$	−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
$436$	16.0000i	0.766261i
$437$	0	0
$438$	−4.00000	−0.191127
$439$	8.00000i	0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$	8.00000i	0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	8.00000	0.378811
$447$	− 12.0000i	− 0.567581i
$448$	4.00000i	0.188982i
$449$	18.0000i	0.849473i	0.905317	+	0.424736i	$0.139633\pi$
$449$	18.0000i	0.849473i	−0.905317	+	0.424736i	$0.860367\pi$
$450$	5.00000	0.235702
$451$	−36.0000	−1.69517
$452$	6.00000i	0.282216i
$453$	− 32.0000i	− 1.50349i
$454$	− 6.00000i	− 0.281594i
$455$	0	0
$456$	8.00000i	0.374634i
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	48.0000i	2.23316i
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	− 18.0000i	− 0.833834i
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	32.0000i	1.47762i
$470$	0	0
$471$	− 28.0000i	− 1.29017i
$472$	0	0
$473$	48.0000i	2.20704i
$474$	16.0000	0.734904
$475$	20.0000	0.917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	0	0
$481$	− 8.00000i	− 0.364769i
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	− 10.0000i	− 0.453609i
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 4.00000i	− 0.181071i
$489$	−4.00000	−0.180886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−12.0000	−0.541002
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	− 4.00000i	− 0.179605i
$497$	0	0
$498$	0	0
$499$	− 14.0000i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	− 14.0000i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	24.0000	1.07117
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	4.00000i	0.178174i
$505$	0	0
$506$	0	0
$507$	18.0000i	0.799408i
$508$	16.0000	0.709885
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	−1.00000	−0.0441942
$513$	− 16.0000i	− 0.706417i
$514$	6.00000	0.264649
$515$	0	0
$516$	16.0000i	0.704361i
$517$	0	0
$518$	−16.0000	−0.703000
$519$	48.0000	2.10697
$520$	0	0
$521$	18.0000i	0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$521$	18.0000i	0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	− 6.00000i	− 0.262111i
$525$	40.0000	1.74574
$526$	24.0000	1.04645
$527$	0	0
$528$	−12.0000	−0.522233
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	16.0000i	0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	− 12.0000i	− 0.519291i
$535$	0	0
$536$	−8.00000	−0.345547
$537$	24.0000i	1.03568i
$538$	24.0000i	1.03471i
$539$	54.0000i	2.32594i
$540$	0	0
$541$	20.0000i	0.859867i	0.902861	+	0.429934i	$0.141463\pi$
$541$	20.0000i	0.859867i	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−8.00000	−0.343629
$543$	8.00000	0.343313
$544$	0	0
$545$	0	0
$546$	−16.0000	−0.684737
$547$	2.00000i	0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$	2.00000i	0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$	6.00000	0.256307
$549$	− 4.00000i	− 0.170716i
$550$	30.0000i	1.27920i
$551$	0	0
$552$	0	0
$553$	32.0000	1.36078
$554$	− 8.00000i	− 0.339887i
$555$	0	0
$556$	2.00000i	0.0848189i
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	14.0000i	0.588464i
$567$	− 44.0000i	− 1.84783i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	− 26.0000i	− 1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$571$	− 26.0000i	− 1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$572$	− 12.0000i	− 0.501745i
$573$	48.0000i	2.00523i
$574$	−24.0000	−1.00174
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	20.0000	0.831172
$580$	0	0
$581$	0	0
$582$	−28.0000	−1.16064
$583$	− 36.0000i	− 1.49097i
$584$	− 2.00000i	− 0.0827606i
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	18.0000i	0.742307i
$589$	− 16.0000i	− 0.659269i
$590$	0	0
$591$	24.0000	0.987228
$592$	− 4.00000i	− 0.164399i
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	24.0000	0.984732
$595$	0	0
$596$	6.00000	0.245770
$597$	−32.0000	−1.30967
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	10.0000i	0.408248i
$601$	46.0000i	1.87638i	0.346122	+	0.938190i	$0.387498\pi$
$601$	46.0000i	1.87638i	−0.346122	+	0.938190i	$0.612502\pi$
$602$	32.0000i	1.30422i
$603$	−8.00000	−0.325785
$604$	16.0000	0.651031
$605$	0	0
$606$	36.0000i	1.46240i
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	−24.0000	−0.966988
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	− 32.0000i	− 1.28723i
$619$	− 26.0000i	− 1.04503i	−0.852631	−	0.522514i	$-0.824994\pi$
$619$	− 26.0000i	− 1.04503i	0.852631	−	0.522514i	$-0.175006\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	− 24.0000i	− 0.961540i
$624$	− 4.00000i	− 0.160128i
$625$	25.0000	1.00000
$626$	− 34.0000i	− 1.35891i
$627$	−48.0000	−1.91694
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	8.00000i	0.318223i
$633$	20.0000	0.794929
$634$	− 12.0000i	− 0.476581i
$635$	0	0
$636$	− 12.0000i	− 0.475831i
$637$	−18.0000	−0.713186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 18.0000i	− 0.710957i	−0.934684	−	0.355479i	$-0.884318\pi$
$641$	− 18.0000i	− 0.710957i	0.934684	−	0.355479i	$-0.115682\pi$
$642$	12.0000	0.473602
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	−10.0000	−0.392232
$651$	− 32.0000i	− 1.25418i
$652$	− 2.00000i	− 0.0783260i
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	−32.0000	−1.25130
$655$	0	0
$656$	− 6.00000i	− 0.234261i
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−16.0000	−0.621858
$663$	0	0
$664$	0	0
$665$	0	0
$666$	− 4.00000i	− 0.154997i
$667$	0	0
$668$	12.0000i	0.464294i
$669$	16.0000i	0.618596i
$670$	0	0
$671$	24.0000	0.926510
$672$	−8.00000	−0.308607
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	22.0000i	0.847408i
$675$	− 20.0000i	− 0.769800i
$676$	−9.00000	−0.346154
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	−12.0000	−0.460857
$679$	−56.0000	−2.14908
$680$	0	0
$681$	12.0000	0.459841
$682$	24.0000	0.919007
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	8.00000i	0.305441i
$687$	28.0000i	1.06827i
$688$	−8.00000	−0.304997
$689$	12.0000	0.457164
$690$	0	0
$691$	10.0000i	0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$	10.0000i	0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	24.0000i	0.912343i
$693$	−24.0000	−0.911685
$694$	18.0000i	0.683271i
$695$	0	0
$696$	0	0
$697$	0	0
$698$	26.0000	0.984115
$699$	36.0000	1.36165
$700$	20.0000i	0.755929i
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	8.00000i	0.301941i
$703$	− 16.0000i	− 0.603451i
$704$	− 6.00000i	− 0.226134i
$705$	0	0
$706$	−6.00000	−0.225813
$707$	72.0000i	2.70784i
$708$	0	0
$709$	− 16.0000i	− 0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$	− 16.0000i	− 0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$	0	0
$711$	8.00000i	0.300023i
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	− 48.0000i	− 1.79259i
$718$	24.0000	0.895672
$719$	48.0000i	1.79010i	0.445968	+	0.895049i	$0.352860\pi$
$719$	48.0000i	1.79010i	−0.445968	+	0.895049i	$0.647140\pi$
$720$	0	0
$721$	− 64.0000i	− 2.38348i
$722$	3.00000	0.111648
$723$	−20.0000	−0.743808
$724$	4.00000i	0.148659i
$725$	0	0
$726$	− 50.0000i	− 1.85567i
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	− 8.00000i	− 0.296500i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	8.00000	0.295689
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	− 16.0000i	− 0.590571i
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	− 6.00000i	− 0.220863i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	− 16.0000i	− 0.587775i
$742$	− 24.0000i	− 0.881068i
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	22.0000	0.805477
$747$	0	0
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	8.00000i	0.291924i	0.989290	+	0.145962i	$0.0466277\pi$
$751$	8.00000i	0.291924i	−0.989290	+	0.145962i	$0.953372\pi$
$752$	0	0
$753$	48.0000i	1.74922i
$754$	0	0
$755$	0	0
$756$	16.0000	0.581914
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	− 14.0000i	− 0.508503i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	32.0000i	1.15924i
$763$	−64.0000	−2.31696
$764$	−24.0000	−0.868290
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	− 2.00000i	− 0.0721688i
$769$	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
$771$	12.0000i	0.432169i
$772$	10.0000i	0.359908i
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−8.00000	−0.287554
$775$	− 20.0000i	− 0.718421i
$776$	− 14.0000i	− 0.502571i
$777$	− 32.0000i	− 1.14799i
$778$	30.0000	1.07555
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	− 46.0000i	− 1.63972i	−0.572562	−	0.819861i	$-0.694050\pi$
$787$	− 46.0000i	− 1.63972i	0.572562	−	0.819861i	$-0.305950\pi$
$788$	12.0000i	0.427482i
$789$	48.0000i	1.70885i
$790$	0	0
$791$	−24.0000	−0.853342
$792$	− 6.00000i	− 0.213201i
$793$	8.00000i	0.284088i
$794$	20.0000i	0.709773i
$795$	0	0
$796$	− 16.0000i	− 0.567105i
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	−32.0000	−1.13279
$799$	0	0
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	30.0000i	1.05934i
$803$	12.0000	0.423471
$804$	− 16.0000i	− 0.564276i
$805$	0	0
$806$	8.00000i	0.281788i
$807$	−48.0000	−1.68968
$808$	−18.0000	−0.633238
$809$	− 30.0000i	− 1.05474i	−0.849635	−	0.527372i	$-0.823177\pi$
$809$	− 30.0000i	− 1.05474i	0.849635	−	0.527372i	$-0.176823\pi$
$810$	0	0
$811$	38.0000i	1.33436i	0.744896	+	0.667180i	$0.232499\pi$
$811$	38.0000i	1.33436i	−0.744896	+	0.667180i	$0.767501\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	−32.0000	−1.11954
$818$	10.0000	0.349642
$819$	− 8.00000i	− 0.279543i
$820$	0	0
$821$	36.0000i	1.25641i	0.778048	+	0.628204i	$0.216210\pi$
$821$	36.0000i	1.25641i	−0.778048	+	0.628204i	$0.783790\pi$
$822$	12.0000i	0.418548i
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	16.0000	0.557386
$825$	−60.0000	−2.08893
$826$	0	0
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	0	0
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	16.0000	0.555034
$832$	2.00000	0.0693375
$833$	0	0
$834$	−4.00000	−0.138509
$835$	0	0
$836$	− 24.0000i	− 0.830057i
$837$	−16.0000	−0.553041
$838$	− 30.0000i	− 1.03633i
$839$	36.0000i	1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$	36.0000i	1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	−2.00000	−0.0689246
$843$	12.0000i	0.413302i
$844$	10.0000i	0.344214i
$845$	0	0
$846$	0	0
$847$	− 100.000i	− 3.43604i
$848$	6.00000	0.206041
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 40.0000i	− 1.36957i	−0.728743	−	0.684787i	$-0.759895\pi$
$853$	− 40.0000i	− 1.36957i	0.728743	−	0.684787i	$-0.240105\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	6.00000i	0.205076i
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	24.0000	0.819346
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	0	0
$861$	− 48.0000i	− 1.63584i
$862$	24.0000i	0.817443i
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	4.00000i	0.136083i
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	16.0000	0.543075
$869$	−48.0000	−1.62829
$870$	0	0
$871$	16.0000	0.542139
$872$	− 16.0000i	− 0.541828i
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	− 8.00000i	− 0.269987i
$879$	− 12.0000i	− 0.404750i
$880$	0	0
$881$	18.0000i	0.606435i	0.952921	+	0.303218i	$0.0980609\pi$
$881$	18.0000i	0.606435i	−0.952921	+	0.303218i	$0.901939\pi$
$882$	−9.00000	−0.303046
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	8.00000	0.268462
$889$	64.0000i	2.14649i
$890$	0	0
$891$	66.0000i	2.21108i
$892$	−8.00000	−0.267860
$893$	0	0
$894$	12.0000i	0.401340i
$895$	0	0
$896$	− 4.00000i	− 0.133631i
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	0	0
$900$	−5.00000	−0.166667
$901$	0	0
$902$	36.0000	1.19867
$903$	−64.0000	−2.12979
$904$	− 6.00000i	− 0.199557i
$905$	0	0
$906$	32.0000i	1.06313i
$907$	58.0000i	1.92586i	0.269754	+	0.962929i	$0.413058\pi$
$907$	58.0000i	1.92586i	−0.269754	+	0.962929i	$0.586942\pi$
$908$	6.00000i	0.199117i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	− 12.0000i	− 0.397578i	−0.980042	−	0.198789i	$-0.936299\pi$
$911$	− 12.0000i	− 0.397578i	0.980042	−	0.198789i	$-0.0637008\pi$
$912$	− 8.00000i	− 0.264906i
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	−14.0000	−0.462573
$917$	24.0000	0.792550
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	− 40.0000i	− 1.31804i
$922$	−6.00000	−0.197599
$923$	0	0
$924$	− 48.0000i	− 1.57908i
$925$	− 20.0000i	− 0.657596i
$926$	16.0000	0.525793
$927$	16.0000	0.525509
$928$	0	0
$929$	18.0000i	0.590561i	0.955411	+	0.295280i	$0.0954131\pi$
$929$	18.0000i	0.590561i	−0.955411	+	0.295280i	$0.904587\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	18.0000i	0.589610i
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	0	0
$936$	2.00000	0.0653720
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	− 32.0000i	− 1.04484i
$939$	68.0000	2.21910
$940$	0	0
$941$	36.0000i	1.17357i	0.809744	+	0.586783i	$0.199606\pi$
$941$	36.0000i	1.17357i	−0.809744	+	0.586783i	$0.800394\pi$
$942$	28.0000i	0.912289i
$943$	0	0
$944$	0	0
$945$	0	0
$946$	− 48.0000i	− 1.56061i
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	−16.0000	−0.519656
$949$	4.00000i	0.129845i
$950$	−20.0000	−0.648886
$951$	24.0000	0.778253
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	24.0000i	0.775000i
$960$	0	0
$961$	15.0000	0.483871
$962$	8.00000i	0.257930i
$963$	6.00000i	0.193347i
$964$	− 10.0000i	− 0.322078i
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	10.0000i	0.320750i
$973$	−8.00000	−0.256468
$974$	8.00000i	0.256337i
$975$	− 20.0000i	− 0.640513i
$976$	4.00000i	0.128037i
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	4.00000	0.127906
$979$	36.0000i	1.15056i
$980$	0	0
$981$	− 16.0000i	− 0.510841i
$982$	−12.0000	−0.382935
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	12.0000	0.382546
$985$	0	0
$986$	0	0
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	0	0
$991$	− 16.0000i	− 0.508257i	−0.967170	−	0.254128i	$-0.918211\pi$
$991$	− 16.0000i	− 0.508257i	0.967170	−	0.254128i	$-0.0817886\pi$
$992$	4.00000i	0.127000i
$993$	− 32.0000i	− 1.01549i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	14.0000i	0.443162i
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	578.2.b.a.577.1		2
17.2	even	8		578.2.c.e.251.1		4
17.3	odd	16		578.2.d.e.399.2		8
17.4	even	4		34.2.a.a.1.1	✓	1
17.5	odd	16		578.2.d.e.179.2		8
17.6	odd	16		578.2.d.e.423.1		8
17.7	odd	16		578.2.d.e.155.1		8
17.8	even	8		578.2.c.e.327.1		4
17.9	even	8		578.2.c.e.327.2		4
17.10	odd	16		578.2.d.e.155.2		8
17.11	odd	16		578.2.d.e.423.2		8
17.12	odd	16		578.2.d.e.179.1		8
17.13	even	4		578.2.a.a.1.1		1
17.14	odd	16		578.2.d.e.399.1		8
17.15	even	8		578.2.c.e.251.2		4
17.16	even	2	inner	578.2.b.a.577.2		2
51.38	odd	4		306.2.a.a.1.1		1
51.47	odd	4		5202.2.a.d.1.1		1
68.47	odd	4		4624.2.a.a.1.1		1
68.55	odd	4		272.2.a.d.1.1		1
85.4	even	4		850.2.a.e.1.1		1
85.38	odd	4		850.2.c.b.749.1		2
85.72	odd	4		850.2.c.b.749.2		2
119.55	odd	4		1666.2.a.m.1.1		1
136.21	even	4		1088.2.a.l.1.1		1
136.123	odd	4		1088.2.a.d.1.1		1
187.21	odd	4		4114.2.a.a.1.1		1
204.191	even	4		2448.2.a.k.1.1		1
221.38	even	4		5746.2.a.b.1.1		1
255.89	odd	4		7650.2.a.ci.1.1		1
340.259	odd	4		6800.2.a.b.1.1		1
408.293	odd	4		9792.2.a.y.1.1		1
408.395	even	4		9792.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.a.a.1.1	✓	1	17.4	even	4
272.2.a.d.1.1		1	68.55	odd	4
306.2.a.a.1.1		1	51.38	odd	4
578.2.a.a.1.1		1	17.13	even	4
578.2.b.a.577.1		2	1.1	even	1	trivial
578.2.b.a.577.2		2	17.16	even	2	inner
578.2.c.e.251.1		4	17.2	even	8
578.2.c.e.251.2		4	17.15	even	8
578.2.c.e.327.1		4	17.8	even	8
578.2.c.e.327.2		4	17.9	even	8
578.2.d.e.155.1		8	17.7	odd	16
578.2.d.e.155.2		8	17.10	odd	16
578.2.d.e.179.1		8	17.12	odd	16
578.2.d.e.179.2		8	17.5	odd	16
578.2.d.e.399.1		8	17.14	odd	16
578.2.d.e.399.2		8	17.3	odd	16
578.2.d.e.423.1		8	17.6	odd	16
578.2.d.e.423.2		8	17.11	odd	16
850.2.a.e.1.1		1	85.4	even	4
850.2.c.b.749.1		2	85.38	odd	4
850.2.c.b.749.2		2	85.72	odd	4
1088.2.a.d.1.1		1	136.123	odd	4
1088.2.a.l.1.1		1	136.21	even	4
1666.2.a.m.1.1		1	119.55	odd	4
2448.2.a.k.1.1		1	204.191	even	4
4114.2.a.a.1.1		1	187.21	odd	4
4624.2.a.a.1.1		1	68.47	odd	4
5202.2.a.d.1.1		1	51.47	odd	4
5746.2.a.b.1.1		1	221.38	even	4
6800.2.a.b.1.1		1	340.259	odd	4
7650.2.a.ci.1.1		1	255.89	odd	4
9792.2.a.y.1.1		1	408.293	odd	4
9792.2.a.bj.1.1		1	408.395	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 578 = 2 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	578.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} +4.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} +4.00000i q^{7} -1.00000 q^{8} -1.00000 q^{9} -6.00000i q^{11} -2.00000i q^{12} +2.00000 q^{13} -4.00000i q^{14} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{19} +8.00000 q^{21} +6.00000i q^{22} +2.00000i q^{24} +5.00000 q^{25} -2.00000 q^{26} -4.00000i q^{27} +4.00000i q^{28} -4.00000i q^{31} -1.00000 q^{32} -12.0000 q^{33} -1.00000 q^{36} -4.00000i q^{37} -4.00000 q^{38} -4.00000i q^{39} -6.00000i q^{41} -8.00000 q^{42} -8.00000 q^{43} -6.00000i q^{44} -2.00000i q^{48} -9.00000 q^{49} -5.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} +4.00000i q^{54} -4.00000i q^{56} -8.00000i q^{57} +4.00000i q^{61} +4.00000i q^{62} -4.00000i q^{63} +1.00000 q^{64} +12.0000 q^{66} +8.00000 q^{67} +1.00000 q^{72} +2.00000i q^{73} +4.00000i q^{74} -10.0000i q^{75} +4.00000 q^{76} +24.0000 q^{77} +4.00000i q^{78} -8.00000i q^{79} -11.0000 q^{81} +6.00000i q^{82} +8.00000 q^{84} +8.00000 q^{86} +6.00000i q^{88} -6.00000 q^{89} +8.00000i q^{91} -8.00000 q^{93} +2.00000i q^{96} +14.0000i q^{97} +9.00000 q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 4 q^{13} + 2 q^{16} + 2 q^{18} + 8 q^{19} + 16 q^{21} + 10 q^{25} - 4 q^{26} - 2 q^{32} - 24 q^{33} - 2 q^{36} - 8 q^{38} - 16 q^{42} - 16 q^{43} - 18 q^{49} - 10 q^{50} + 4 q^{52} + 12 q^{53} + 2 q^{64} + 24 q^{66} + 16 q^{67} + 2 q^{72} + 8 q^{76} + 48 q^{77} - 22 q^{81} + 16 q^{84} + 16 q^{86} - 12 q^{89} - 16 q^{93} + 18 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 4 * q^13 + 2 * q^16 + 2 * q^18 + 8 * q^19 + 16 * q^21 + 10 * q^25 - 4 * q^26 - 2 * q^32 - 24 * q^33 - 2 * q^36 - 8 * q^38 - 16 * q^42 - 16 * q^43 - 18 * q^49 - 10 * q^50 + 4 * q^52 + 12 * q^53 + 2 * q^64 + 24 * q^66 + 16 * q^67 + 2 * q^72 + 8 * q^76 + 48 * q^77 - 22 * q^81 + 16 * q^84 + 16 * q^86 - 12 * q^89 - 16 * q^93 + 18 * q^98

Introduction

L-functions

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