LMFDB - Embedded newform 775.2.b.a.249.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [775,2,Mod(249,775)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("775.249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,4,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$6.18840615665$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.a.249.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} -4.00000 q^{16} -7.00000i q^{17} -4.00000i q^{18} +5.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} +12.0000 q^{26} +5.00000i q^{27} +4.00000i q^{28} +1.00000 q^{31} +8.00000i q^{32} +2.00000i q^{33} -14.0000 q^{34} -4.00000 q^{36} -7.00000i q^{37} -10.0000i q^{38} -6.00000 q^{39} -3.00000 q^{41} -4.00000i q^{42} -9.00000i q^{43} -4.00000 q^{44} -8.00000 q^{46} -2.00000i q^{47} -4.00000i q^{48} +3.00000 q^{49} +7.00000 q^{51} -12.0000i q^{52} -9.00000i q^{53} +10.0000 q^{54} +5.00000i q^{57} +5.00000 q^{59} -8.00000 q^{61} -2.00000i q^{62} -4.00000i q^{63} +8.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +14.0000i q^{68} +4.00000 q^{69} -3.00000 q^{71} +1.00000i q^{73} -14.0000 q^{74} -10.0000 q^{76} -4.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} +6.00000i q^{82} +11.0000i q^{83} -4.00000 q^{84} -18.0000 q^{86} -10.0000 q^{89} +12.0000 q^{91} +8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -8.00000 q^{96} +18.0000i q^{97} -6.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 4 q^{11} - 8 q^{14} - 8 q^{16} + 10 q^{19} + 4 q^{21} + 24 q^{26} + 2 q^{31} - 28 q^{34} - 8 q^{36} - 12 q^{39} - 6 q^{41} - 8 q^{44} - 16 q^{46} + 6 q^{49} + 14 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 + 4 * q^11 - 8 * q^14 - 8 * q^16 + 10 * q^19 + 4 * q^21 + 24 * q^26 + 2 * q^31 - 28 * q^34 - 8 * q^36 - 12 * q^39 - 6 * q^41 - 8 * q^44 - 16 * q^46 + 6 * q^49 + 14 * q^51 + 20 * q^54 + 10 * q^59 - 16 * q^61 + 16 * q^64 + 8 * q^66 + 8 * q^69 - 6 * q^71 - 28 * q^74 - 20 * q^76 + 2 * q^81 - 8 * q^84 - 36 * q^86 - 20 * q^89 + 24 * q^91 - 8 * q^94 - 16 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	− 2.00000i	− 0.577350i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	− 4.00000i	− 0.942809i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	− 4.00000i	− 0.852803i
$23$	− 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
$25$	0	0
$26$	12.0000	2.35339
$27$	5.00000i	0.962250i
$28$	4.00000i	0.755929i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	8.00000i	1.41421i
$33$	2.00000i	0.348155i
$34$	−14.0000	−2.40098
$35$	0	0
$36$	−4.00000	−0.666667
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 10.0000i	− 1.62221i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−8.00000	−1.17954
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	− 4.00000i	− 0.577350i
$49$	3.00000	0.428571
$50$	0	0
$51$	7.00000	0.980196
$52$	− 12.0000i	− 1.66410i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	10.0000	1.36083
$55$	0	0
$56$	0	0
$57$	5.00000i	0.662266i
$58$	0	0
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	− 2.00000i	− 0.254000i
$63$	− 4.00000i	− 0.503953i
$64$	8.00000	1.00000
$65$	0	0
$66$	4.00000	0.492366
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	14.0000i	1.69775i
$69$	4.00000	0.481543
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	1.00000i	0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$	1.00000i	0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	−14.0000	−1.62747
$75$	0	0
$76$	−10.0000	−1.14708
$77$	− 4.00000i	− 0.455842i
$78$	12.0000i	1.35873i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000i	0.662589i
$83$	11.0000i	1.20741i	0.797209	+	0.603703i	$0.206309\pi$
$83$	11.0000i	1.20741i	−0.797209	+	0.603703i	$0.793691\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−18.0000	−1.94099
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	8.00000i	0.834058i
$93$	1.00000i	0.103695i
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−8.00000	−0.816497
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 6.00000i	− 0.606092i
$99$	4.00000	0.402015
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	− 14.0000i	− 1.38621i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	−18.0000	−1.74831
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	− 10.0000i	− 0.962250i
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	8.00000i	0.755929i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	10.0000	0.936586
$115$	0	0
$116$	0	0
$117$	12.0000i	1.10940i
$118$	− 10.0000i	− 0.920575i
$119$	−14.0000	−1.28338
$120$	0	0
$121$	−7.00000	−0.636364
$122$	16.0000i	1.44857i
$123$	− 3.00000i	− 0.270501i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−8.00000	−0.712697
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	0	0
$129$	9.00000	0.792406
$130$	0	0
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	− 4.00000i	− 0.348155i
$133$	− 10.0000i	− 0.867110i
$134$	16.0000	1.38219
$135$	0	0
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	− 8.00000i	− 0.681005i
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	6.00000i	0.503509i
$143$	12.0000i	1.00349i
$144$	−8.00000	−0.666667
$145$	0	0
$146$	2.00000	0.165521
$147$	3.00000i	0.247436i
$148$	14.0000i	1.15079i
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	− 14.0000i	− 1.13183i
$154$	−8.00000	−0.644658
$155$	0	0
$156$	12.0000	0.960769
$157$	− 22.0000i	− 1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$	− 22.0000i	− 1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	−8.00000	−0.630488
$162$	− 2.00000i	− 0.157135i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	22.0000	1.70753
$167$	13.0000i	1.00597i	0.864295	+	0.502985i	$0.167765\pi$
$167$	13.0000i	1.00597i	−0.864295	+	0.502985i	$0.832235\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	10.0000	0.764719
$172$	18.0000i	1.37249i
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	−8.00000	−0.603023
$177$	5.00000i	0.375823i
$178$	20.0000i	1.49906i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	− 24.0000i	− 1.77900i
$183$	− 8.00000i	− 0.591377i
$184$	0	0
$185$	0	0
$186$	2.00000	0.146647
$187$	− 14.0000i	− 1.02378i
$188$	4.00000i	0.291730i
$189$	10.0000	0.727393
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	8.00000i	0.577350i
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	36.0000	2.58465
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	− 8.00000i	− 0.568535i
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	− 34.0000i	− 2.39223i
$203$	0	0
$204$	−14.0000	−0.980196
$205$	0	0
$206$	32.0000	2.22955
$207$	− 8.00000i	− 0.556038i
$208$	− 24.0000i	− 1.66410i
$209$	10.0000	0.691714
$210$	0	0
$211$	−28.0000	−1.92760	−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000	−1.92760	−0.963800	+	0.266627i	$0.914091\pi$
$212$	18.0000i	1.23625i
$213$	− 3.00000i	− 0.205557i
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	− 10.0000i	− 0.677285i
$219$	−1.00000	−0.0675737
$220$	0	0
$221$	42.0000	2.82523
$222$	− 14.0000i	− 0.939618i
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	16.0000	1.06904
$225$	0	0
$226$	12.0000	0.798228
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	− 10.0000i	− 0.662266i
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	24.0000	1.56893
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	28.0000i	1.81497i
$239$	−10.0000	−0.646846	−0.323423	−	0.946254i	$-0.604834\pi$
$239$	−10.0000	−0.646846	−0.323423	+	0.946254i	$0.604834\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	14.0000i	0.899954i
$243$	16.0000i	1.02640i
$244$	16.0000	1.02430
$245$	0	0
$246$	−6.00000	−0.382546
$247$	30.0000i	1.90885i
$248$	0	0
$249$	−11.0000	−0.697097
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	8.00000i	0.503953i
$253$	− 8.00000i	− 0.502956i
$254$	16.0000	1.00393
$255$	0	0
$256$	16.0000	1.00000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	− 18.0000i	− 1.12063i
$259$	−14.0000	−0.869918
$260$	0	0
$261$	0	0
$262$	− 14.0000i	− 0.864923i
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	0	0
$265$	0	0
$266$	−20.0000	−1.22628
$267$	− 10.0000i	− 0.611990i
$268$	− 16.0000i	− 0.977356i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	28.0000i	1.69775i
$273$	12.0000i	0.726273i
$274$	6.00000	0.362473
$275$	0	0
$276$	−8.00000	−0.481543
$277$	− 7.00000i	− 0.420589i	−0.977638	−	0.210295i	$-0.932558\pi$
$277$	− 7.00000i	− 0.420589i	0.977638	−	0.210295i	$-0.0674423\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	− 4.00000i	− 0.238197i
$283$	26.0000i	1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$	26.0000i	1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	24.0000	1.41915
$287$	6.00000i	0.354169i
$288$	16.0000i	0.942809i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	−18.0000	−1.05518
$292$	− 2.00000i	− 0.117041i
$293$	16.0000i	0.934730i	0.884064	+	0.467365i	$0.154797\pi$
$293$	16.0000i	0.934730i	−0.884064	+	0.467365i	$0.845203\pi$
$294$	6.00000	0.349927
$295$	0	0
$296$	0	0
$297$	10.0000i	0.580259i
$298$	− 30.0000i	− 1.73785i
$299$	24.0000	1.38796
$300$	0	0
$301$	−18.0000	−1.03750
$302$	− 24.0000i	− 1.38104i
$303$	17.0000i	0.976624i
$304$	−20.0000	−1.14708
$305$	0	0
$306$	−28.0000	−1.60065
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	8.00000i	0.455842i
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	11.0000i	0.621757i	0.950450	+	0.310878i	$0.100623\pi$
$313$	11.0000i	0.621757i	−0.950450	+	0.310878i	$0.899377\pi$
$314$	−44.0000	−2.48306
$315$	0	0
$316$	0	0
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	− 18.0000i	− 1.00939i
$319$	0	0
$320$	0	0
$321$	2.00000	0.111629
$322$	16.0000i	0.891645i
$323$	− 35.0000i	− 1.94745i
$324$	−2.00000	−0.111111
$325$	0	0
$326$	32.0000	1.77232
$327$	5.00000i	0.276501i
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	− 22.0000i	− 1.20741i
$333$	− 14.0000i	− 0.767195i
$334$	26.0000	1.42266
$335$	0	0
$336$	−8.00000	−0.436436
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	46.0000i	2.50207i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	2.00000	0.108306
$342$	− 20.0000i	− 1.08148i
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	16.0000i	0.852803i
$353$	− 34.0000i	− 1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$	− 34.0000i	− 1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	20.0000	1.06000
$357$	− 14.0000i	− 0.740959i
$358$	− 20.0000i	− 1.05703i
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	36.0000i	1.89212i
$363$	− 7.00000i	− 0.367405i
$364$	−24.0000	−1.25794
$365$	0	0
$366$	−16.0000	−0.836333
$367$	13.0000i	0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$	13.0000i	0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$	16.0000i	0.834058i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−18.0000	−0.934513
$372$	− 2.00000i	− 0.103695i
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	−28.0000	−1.44785
$375$	0	0
$376$	0	0
$377$	0	0
$378$	− 20.0000i	− 1.02869i
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	16.0000i	0.818631i
$383$	− 9.00000i	− 0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$	− 9.00000i	− 0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$	0	0
$385$	0	0
$386$	12.0000	0.610784
$387$	− 18.0000i	− 0.914991i
$388$	− 36.0000i	− 1.82762i
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	0	0
$393$	7.00000i	0.353103i
$394$	−44.0000	−2.21669
$395$	0	0
$396$	−8.00000	−0.402015
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	20.0000i	1.00251i
$399$	10.0000	0.500626
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	16.0000i	0.798007i
$403$	6.00000i	0.298881i
$404$	−34.0000	−1.69156
$405$	0	0
$406$	0	0
$407$	− 14.0000i	− 0.693954i
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	− 32.0000i	− 1.57653i
$413$	− 10.0000i	− 0.492068i
$414$	−16.0000	−0.786357
$415$	0	0
$416$	−48.0000	−2.35339
$417$	0	0
$418$	− 20.0000i	− 0.978232i
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	0	0
$421$	−33.0000	−1.60832	−0.804161	−	0.594412i	$-0.797385\pi$
$421$	−33.0000	−1.60832	−0.804161	+	0.594412i	$0.797385\pi$
$422$	56.0000i	2.72604i
$423$	− 4.00000i	− 0.194487i
$424$	0	0
$425$	0	0
$426$	−6.00000	−0.290701
$427$	16.0000i	0.774294i
$428$	4.00000i	0.193347i
$429$	−12.0000	−0.579365
$430$	0	0
$431$	−13.0000	−0.626188	−0.313094	−	0.949722i	$-0.601365\pi$
$431$	−13.0000	−0.626188	−0.313094	+	0.949722i	$0.601365\pi$
$432$	− 20.0000i	− 0.962250i
$433$	6.00000i	0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$	6.00000i	0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 20.0000i	− 0.956730i
$438$	2.00000i	0.0955637i
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	− 84.0000i	− 3.99547i
$443$	26.0000i	1.23530i	0.786454	+	0.617649i	$0.211915\pi$
$443$	26.0000i	1.23530i	−0.786454	+	0.617649i	$0.788085\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	−38.0000	−1.79935
$447$	15.0000i	0.709476i
$448$	− 16.0000i	− 0.755929i
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 12.0000i	− 0.564433i
$453$	12.0000i	0.563809i
$454$	56.0000	2.62821
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	0	0
$459$	35.0000	1.63366
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	− 8.00000i	− 0.372194i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	0	0
$465$	0	0
$466$	12.0000	0.555889
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	− 24.0000i	− 1.10940i
$469$	16.0000	0.738811
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	− 18.0000i	− 0.827641i
$474$	0	0
$475$	0	0
$476$	28.0000	1.28338
$477$	− 18.0000i	− 0.824163i
$478$	20.0000i	0.914779i
$479$	40.0000	1.82765	0.913823	−	0.406112i	$-0.133116\pi$
$479$	40.0000	1.82765	0.913823	+	0.406112i	$0.133116\pi$
$480$	0	0
$481$	42.0000	1.91504
$482$	− 4.00000i	− 0.182195i
$483$	− 8.00000i	− 0.364013i
$484$	14.0000	0.636364
$485$	0	0
$486$	32.0000	1.45155
$487$	− 7.00000i	− 0.317200i	−0.987343	−	0.158600i	$-0.949302\pi$
$487$	− 7.00000i	− 0.317200i	0.987343	−	0.158600i	$-0.0506981\pi$
$488$	0	0
$489$	−16.0000	−0.723545
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	6.00000i	0.270501i
$493$	0	0
$494$	60.0000	2.69953
$495$	0	0
$496$	−4.00000	−0.179605
$497$	6.00000i	0.269137i
$498$	22.0000i	0.985844i
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	−13.0000	−0.580797
$502$	− 4.00000i	− 0.178529i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	−16.0000	−0.711287
$507$	− 23.0000i	− 1.02147i
$508$	− 16.0000i	− 0.709885i
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	− 32.0000i	− 1.41421i
$513$	25.0000i	1.10378i
$514$	−24.0000	−1.05859
$515$	0	0
$516$	−18.0000	−0.792406
$517$	− 4.00000i	− 0.175920i
$518$	28.0000i	1.23025i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	− 19.0000i	− 0.830812i	−0.909636	−	0.415406i	$-0.863640\pi$
$523$	− 19.0000i	− 0.830812i	0.909636	−	0.415406i	$-0.136360\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	42.0000	1.83129
$527$	− 7.00000i	− 0.304925i
$528$	− 8.00000i	− 0.348155i
$529$	7.00000	0.304348
$530$	0	0
$531$	10.0000	0.433963
$532$	20.0000i	0.867110i
$533$	− 18.0000i	− 0.779667i
$534$	−20.0000	−0.865485
$535$	0	0
$536$	0	0
$537$	10.0000i	0.431532i
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	56.0000i	2.40541i
$543$	− 18.0000i	− 0.772454i
$544$	56.0000	2.40098
$545$	0	0
$546$	24.0000	1.02711
$547$	38.0000i	1.62476i	0.583127	+	0.812381i	$0.301829\pi$
$547$	38.0000i	1.62476i	−0.583127	+	0.812381i	$0.698171\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−16.0000	−0.682863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−14.0000	−0.594803
$555$	0	0
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	− 4.00000i	− 0.169334i
$559$	54.0000	2.28396
$560$	0	0
$561$	14.0000	0.591080
$562$	− 54.0000i	− 2.27785i
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	52.0000	2.18572
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	− 24.0000i	− 1.00349i
$573$	− 8.00000i	− 0.334205i
$574$	12.0000	0.500870
$575$	0	0
$576$	16.0000	0.666667
$577$	− 32.0000i	− 1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$	− 32.0000i	− 1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	64.0000i	2.66205i
$579$	−6.00000	−0.249351
$580$	0	0
$581$	22.0000	0.912714
$582$	36.0000i	1.49225i
$583$	− 18.0000i	− 0.745484i
$584$	0	0
$585$	0	0
$586$	32.0000	1.32191
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 6.00000i	− 0.247436i
$589$	5.00000	0.206021
$590$	0	0
$591$	22.0000	0.904959
$592$	28.0000i	1.15079i
$593$	− 24.0000i	− 0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$	− 24.0000i	− 0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$	20.0000	0.820610
$595$	0	0
$596$	−30.0000	−1.22885
$597$	− 10.0000i	− 0.409273i
$598$	− 48.0000i	− 1.96287i
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	36.0000i	1.46725i
$603$	16.0000i	0.651570i
$604$	−24.0000	−0.976546
$605$	0	0
$606$	34.0000	1.38116
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	40.0000i	1.62221i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	28.0000i	1.13183i
$613$	− 19.0000i	− 0.767403i	−0.923457	−	0.383701i	$-0.874649\pi$
$613$	− 19.0000i	− 0.767403i	0.923457	−	0.383701i	$-0.125351\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	32.0000i	1.28723i
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	6.00000i	0.240578i
$623$	20.0000i	0.801283i
$624$	24.0000	0.960769
$625$	0	0
$626$	22.0000	0.879297
$627$	10.0000i	0.399362i
$628$	44.0000i	1.75579i
$629$	−49.0000	−1.95376
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	− 28.0000i	− 1.11290i
$634$	−24.0000	−0.953162
$635$	0	0
$636$	−18.0000	−0.713746
$637$	18.0000i	0.713186i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 19.0000i	− 0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$	− 19.0000i	− 0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	−70.0000	−2.75411
$647$	− 37.0000i	− 1.45462i	−0.686309	−	0.727310i	$-0.740770\pi$
$647$	− 37.0000i	− 1.45462i	0.686309	−	0.727310i	$-0.259230\pi$
$648$	0	0
$649$	10.0000	0.392534
$650$	0	0
$651$	2.00000	0.0783862
$652$	− 32.0000i	− 1.25322i
$653$	− 4.00000i	− 0.156532i	−0.996933	−	0.0782660i	$-0.975062\pi$
$653$	− 4.00000i	− 0.156532i	0.996933	−	0.0782660i	$-0.0249384\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	12.0000	0.468521
$657$	2.00000i	0.0780274i
$658$	8.00000i	0.311872i
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	−23.0000	−0.894596	−0.447298	−	0.894385i	$-0.647614\pi$
$661$	−23.0000	−0.894596	−0.447298	+	0.894385i	$0.647614\pi$
$662$	− 64.0000i	− 2.48743i
$663$	42.0000i	1.63114i
$664$	0	0
$665$	0	0
$666$	−28.0000	−1.08498
$667$	0	0
$668$	− 26.0000i	− 1.00597i
$669$	19.0000	0.734582
$670$	0	0
$671$	−16.0000	−0.617673
$672$	16.0000i	0.617213i
$673$	− 19.0000i	− 0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$	− 19.0000i	− 0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	46.0000	1.76923
$677$	43.0000i	1.65262i	0.563212	+	0.826312i	$0.309565\pi$
$677$	43.0000i	1.65262i	−0.563212	+	0.826312i	$0.690435\pi$
$678$	12.0000i	0.460857i
$679$	36.0000	1.38155
$680$	0	0
$681$	−28.0000	−1.07296
$682$	− 4.00000i	− 0.153168i
$683$	− 14.0000i	− 0.535695i	−0.963461	−	0.267848i	$-0.913688\pi$
$683$	− 14.0000i	− 0.535695i	0.963461	−	0.267848i	$-0.0863124\pi$
$684$	−20.0000	−0.764719
$685$	0	0
$686$	−40.0000	−1.52721
$687$	0	0
$688$	36.0000i	1.37249i
$689$	54.0000	2.05724
$690$	0	0
$691$	27.0000	1.02713	0.513564	−	0.858051i	$-0.328325\pi$
$691$	27.0000	1.02713	0.513564	+	0.858051i	$0.328325\pi$
$692$	− 12.0000i	− 0.456172i
$693$	− 8.00000i	− 0.303895i
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	21.0000i	0.795432i
$698$	− 10.0000i	− 0.378506i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	60.0000i	2.26455i
$703$	− 35.0000i	− 1.32005i
$704$	16.0000	0.603023
$705$	0	0
$706$	−68.0000	−2.55921
$707$	− 34.0000i	− 1.27870i
$708$	− 10.0000i	− 0.375823i
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4.00000i	− 0.149801i
$714$	−28.0000	−1.04787
$715$	0	0
$716$	−20.0000	−0.747435
$717$	− 10.0000i	− 0.373457i
$718$	40.0000i	1.49279i
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	− 12.0000i	− 0.446594i
$723$	2.00000i	0.0743808i
$724$	36.0000	1.33793
$725$	0	0
$726$	−14.0000	−0.519589
$727$	− 12.0000i	− 0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$	− 12.0000i	− 0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−63.0000	−2.33014
$732$	16.0000i	0.591377i
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	32.0000	1.17954
$737$	16.0000i	0.589368i
$738$	12.0000i	0.441726i
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	0	0
$741$	−30.0000	−1.10208
$742$	36.0000i	1.32160i
$743$	− 9.00000i	− 0.330178i	−0.986279	−	0.165089i	$-0.947209\pi$
$743$	− 9.00000i	− 0.330178i	0.986279	−	0.165089i	$-0.0527911\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	22.0000i	0.804938i
$748$	28.0000i	1.02378i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	8.00000i	0.291730i
$753$	2.00000i	0.0728841i
$754$	0	0
$755$	0	0
$756$	−20.0000	−0.727393
$757$	43.0000i	1.56286i	0.623992	+	0.781431i	$0.285510\pi$
$757$	43.0000i	1.56286i	−0.623992	+	0.781431i	$0.714490\pi$
$758$	− 10.0000i	− 0.363216i
$759$	8.00000	0.290382
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	16.0000i	0.579619i
$763$	− 10.0000i	− 0.362024i
$764$	16.0000	0.578860
$765$	0	0
$766$	−18.0000	−0.650366
$767$	30.0000i	1.08324i
$768$	16.0000i	0.577350i
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	− 12.0000i	− 0.431889i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	−36.0000	−1.29399
$775$	0	0
$776$	0	0
$777$	− 14.0000i	− 0.502247i
$778$	20.0000i	0.717035i
$779$	−15.0000	−0.537431
$780$	0	0
$781$	−6.00000	−0.214697
$782$	56.0000i	2.00256i
$783$	0	0
$784$	−12.0000	−0.428571
$785$	0	0
$786$	14.0000	0.499363
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	44.0000i	1.56744i
$789$	−21.0000	−0.747620
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	− 48.0000i	− 1.70453i
$794$	−44.0000	−1.56150
$795$	0	0
$796$	20.0000	0.708881
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	− 20.0000i	− 0.707992i
$799$	−14.0000	−0.495284
$800$	0	0
$801$	−20.0000	−0.706665
$802$	− 44.0000i	− 1.55369i
$803$	2.00000i	0.0705785i
$804$	16.0000	0.564276
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	17.0000	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$811$	17.0000	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$812$	0	0
$813$	− 28.0000i	− 0.982003i
$814$	−28.0000	−0.981399
$815$	0	0
$816$	−28.0000	−0.980196
$817$	− 45.0000i	− 1.57435i
$818$	0	0
$819$	24.0000	0.838628
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	6.00000i	0.209274i
$823$	11.0000i	0.383436i	0.981450	+	0.191718i	$0.0614059\pi$
$823$	11.0000i	0.383436i	−0.981450	+	0.191718i	$0.938594\pi$
$824$	0	0
$825$	0	0
$826$	−20.0000	−0.695889
$827$	− 27.0000i	− 0.938882i	−0.882964	−	0.469441i	$-0.844455\pi$
$827$	− 27.0000i	− 0.938882i	0.882964	−	0.469441i	$-0.155545\pi$
$828$	16.0000i	0.556038i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	7.00000	0.242827
$832$	48.0000i	1.66410i
$833$	− 21.0000i	− 0.727607i
$834$	0	0
$835$	0	0
$836$	−20.0000	−0.691714
$837$	5.00000i	0.172825i
$838$	10.0000i	0.345444i
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	66.0000i	2.27451i
$843$	27.0000i	0.929929i
$844$	56.0000	1.92760
$845$	0	0
$846$	−8.00000	−0.275046
$847$	14.0000i	0.481046i
$848$	36.0000i	1.23625i
$849$	−26.0000	−0.892318
$850$	0	0
$851$	−28.0000	−0.959828
$852$	6.00000i	0.205557i
$853$	− 4.00000i	− 0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$	− 4.00000i	− 0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$	32.0000	1.09502
$855$	0	0
$856$	0	0
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	24.0000i	0.819346i
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	26.0000i	0.885564i
$863$	1.00000i	0.0340404i	0.999855	+	0.0170202i	$0.00541796\pi$
$863$	1.00000i	0.0340404i	−0.999855	+	0.0170202i	$0.994582\pi$
$864$	−40.0000	−1.36083
$865$	0	0
$866$	12.0000	0.407777
$867$	− 32.0000i	− 1.08678i
$868$	4.00000i	0.135769i
$869$	0	0
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	36.0000i	1.21842i
$874$	−40.0000	−1.35302
$875$	0	0
$876$	2.00000	0.0675737
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	− 30.0000i	− 1.01245i
$879$	−16.0000	−0.539667
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	− 12.0000i	− 0.404061i
$883$	11.0000i	0.370179i	0.982722	+	0.185090i	$0.0592576\pi$
$883$	11.0000i	0.370179i	−0.982722	+	0.185090i	$0.940742\pi$
$884$	−84.0000	−2.82523
$885$	0	0
$886$	52.0000	1.74697
$887$	48.0000i	1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$	48.0000i	1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	2.00000	0.0670025
$892$	38.0000i	1.27233i
$893$	− 10.0000i	− 0.334637i
$894$	30.0000	1.00335
$895$	0	0
$896$	0	0
$897$	24.0000i	0.801337i
$898$	20.0000i	0.667409i
$899$	0	0
$900$	0	0
$901$	−63.0000	−2.09883
$902$	12.0000i	0.399556i
$903$	− 18.0000i	− 0.599002i
$904$	0	0
$905$	0	0
$906$	24.0000	0.797347
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	− 56.0000i	− 1.85843i
$909$	34.0000	1.12771
$910$	0	0
$911$	−38.0000	−1.25900	−0.629498	−	0.777002i	$-0.716739\pi$
$911$	−38.0000	−1.25900	−0.629498	+	0.777002i	$0.716739\pi$
$912$	− 20.0000i	− 0.662266i
$913$	22.0000i	0.728094i
$914$	−4.00000	−0.132308
$915$	0	0
$916$	0	0
$917$	− 14.0000i	− 0.462321i
$918$	− 70.0000i	− 2.31034i
$919$	25.0000	0.824674	0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000	0.824674	0.412337	+	0.911031i	$0.364713\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	36.0000i	1.18560i
$923$	− 18.0000i	− 0.592477i
$924$	−8.00000	−0.263181
$925$	0	0
$926$	32.0000	1.05159
$927$	32.0000i	1.05102i
$928$	0	0
$929$	−40.0000	−1.31236	−0.656179	−	0.754606i	$-0.727828\pi$
$929$	−40.0000	−1.31236	−0.656179	+	0.754606i	$0.727828\pi$
$930$	0	0
$931$	15.0000	0.491605
$932$	− 12.0000i	− 0.393073i
$933$	− 3.00000i	− 0.0982156i
$934$	56.0000	1.83238
$935$	0	0
$936$	0	0
$937$	18.0000i	0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$	18.0000i	0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	− 32.0000i	− 1.04484i
$939$	−11.0000	−0.358971
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	− 44.0000i	− 1.43360i
$943$	12.0000i	0.390774i
$944$	−20.0000	−0.650945
$945$	0	0
$946$	−36.0000	−1.17046
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	21.0000i	0.680257i	0.940379	+	0.340128i	$0.110471\pi$
$953$	21.0000i	0.680257i	−0.940379	+	0.340128i	$0.889529\pi$
$954$	−36.0000	−1.16554
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	− 80.0000i	− 2.58468i
$959$	6.00000	0.193750
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 84.0000i	− 2.70827i
$963$	− 4.00000i	− 0.128898i
$964$	−4.00000	−0.128831
$965$	0	0
$966$	−16.0000	−0.514792
$967$	48.0000i	1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$	48.0000i	1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$	0	0
$969$	35.0000	1.12436
$970$	0	0
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	− 32.0000i	− 1.02640i
$973$	0	0
$974$	−14.0000	−0.448589
$975$	0	0
$976$	32.0000	1.02430
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	32.0000i	1.02325i
$979$	−20.0000	−0.639203
$980$	0	0
$981$	10.0000	0.319275
$982$	− 24.0000i	− 0.765871i
$983$	− 4.00000i	− 0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$	− 4.00000i	− 0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.00000i	− 0.127321i
$988$	− 60.0000i	− 1.90885i
$989$	−36.0000	−1.14473
$990$	0	0
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	8.00000i	0.254000i
$993$	32.0000i	1.01549i
$994$	12.0000	0.380617
$995$	0	0
$996$	22.0000	0.697097
$997$	− 12.0000i	− 0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	− 12.0000i	− 0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$	80.0000i	2.53236i
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.a.249.1		2
5.2	odd	4		775.2.a.c.1.1		1
5.3	odd	4		155.2.a.a.1.1	✓	1
5.4	even	2	inner	775.2.b.a.249.2		2
15.2	even	4		6975.2.a.b.1.1		1
15.8	even	4		1395.2.a.e.1.1		1
20.3	even	4		2480.2.a.m.1.1		1
35.13	even	4		7595.2.a.b.1.1		1
40.3	even	4		9920.2.a.j.1.1		1
40.13	odd	4		9920.2.a.x.1.1		1
155.123	even	4		4805.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.a.1.1	✓	1	5.3	odd	4
775.2.a.c.1.1		1	5.2	odd	4
775.2.b.a.249.1		2	1.1	even	1	trivial
775.2.b.a.249.2		2	5.4	even	2	inner
1395.2.a.e.1.1		1	15.8	even	4
2480.2.a.m.1.1		1	20.3	even	4
4805.2.a.b.1.1		1	155.123	even	4
6975.2.a.b.1.1		1	15.2	even	4
7595.2.a.b.1.1		1	35.13	even	4
9920.2.a.j.1.1		1	40.3	even	4
9920.2.a.x.1.1		1	40.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -2.00000i q^{7} +2.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} -4.00000 q^{16} -7.00000i q^{17} -4.00000i q^{18} +5.00000 q^{19} +2.00000 q^{21} -4.00000i q^{22} -4.00000i q^{23} +12.0000 q^{26} +5.00000i q^{27} +4.00000i q^{28} +1.00000 q^{31} +8.00000i q^{32} +2.00000i q^{33} -14.0000 q^{34} -4.00000 q^{36} -7.00000i q^{37} -10.0000i q^{38} -6.00000 q^{39} -3.00000 q^{41} -4.00000i q^{42} -9.00000i q^{43} -4.00000 q^{44} -8.00000 q^{46} -2.00000i q^{47} -4.00000i q^{48} +3.00000 q^{49} +7.00000 q^{51} -12.0000i q^{52} -9.00000i q^{53} +10.0000 q^{54} +5.00000i q^{57} +5.00000 q^{59} -8.00000 q^{61} -2.00000i q^{62} -4.00000i q^{63} +8.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +14.0000i q^{68} +4.00000 q^{69} -3.00000 q^{71} +1.00000i q^{73} -14.0000 q^{74} -10.0000 q^{76} -4.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} +6.00000i q^{82} +11.0000i q^{83} -4.00000 q^{84} -18.0000 q^{86} -10.0000 q^{89} +12.0000 q^{91} +8.00000i q^{92} +1.00000i q^{93} -4.00000 q^{94} -8.00000 q^{96} +18.0000i q^{97} -6.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 4 q^{11} - 8 q^{14} - 8 q^{16} + 10 q^{19} + 4 q^{21} + 24 q^{26} + 2 q^{31} - 28 q^{34} - 8 q^{36} - 12 q^{39} - 6 q^{41} - 8 q^{44} - 16 q^{46} + 6 q^{49} + 14 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 + 4 * q^11 - 8 * q^14 - 8 * q^16 + 10 * q^19 + 4 * q^21 + 24 * q^26 + 2 * q^31 - 28 * q^34 - 8 * q^36 - 12 * q^39 - 6 * q^41 - 8 * q^44 - 16 * q^46 + 6 * q^49 + 14 * q^51 + 20 * q^54 + 10 * q^59 - 16 * q^61 + 16 * q^64 + 8 * q^66 + 8 * q^69 - 6 * q^71 - 28 * q^74 - 20 * q^76 + 2 * q^81 - 8 * q^84 - 36 * q^86 - 20 * q^89 + 24 * q^91 - 8 * q^94 - 16 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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