LMFDB - Embedded newform 1053.2.b.j.649.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1053,2,Mod(649,1053)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1053, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1053.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 $ x^10 + 16x^8 + 91x^6 + 222x^4 + 228x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.10
Root			$2.47728i$ of defining polynomial
Character	$\chi$	$=$	1053.649
Dual form			1053.2.b.j.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.47728i q^{2} -4.13693 q^{4} -0.890732i q^{5} -0.982330i q^{7} -5.29379i q^{8} +2.20660 q^{10} -2.88577i q^{11} +(2.70342 + 2.38569i) q^{13} +2.43351 q^{14} +4.84036 q^{16} +5.34353 q^{17} +7.19723i q^{19} +3.68490i q^{20} +7.14886 q^{22} -4.63376 q^{23} +4.20660 q^{25} +(-5.91002 + 6.69715i) q^{26} +4.06384i q^{28} +1.94227 q^{29} +10.0875i q^{31} +1.40336i q^{32} +13.2374i q^{34} -0.874993 q^{35} -4.82809i q^{37} -17.8296 q^{38} -4.71535 q^{40} +2.76379i q^{41} +4.91002 q^{43} +11.9382i q^{44} -11.4791i q^{46} +4.41560i q^{47} +6.03503 q^{49} +10.4209i q^{50} +(-11.1839 - 9.86943i) q^{52} +6.30850 q^{53} -2.57044 q^{55} -5.20025 q^{56} +4.81154i q^{58} -3.17228i q^{59} -5.52068 q^{61} -24.9896 q^{62} +6.20421 q^{64} +(2.12501 - 2.40803i) q^{65} -3.63799i q^{67} -22.1058 q^{68} -2.16761i q^{70} +6.69715i q^{71} +9.33980i q^{73} +11.9605 q^{74} -29.7745i q^{76} -2.83478 q^{77} +4.74760 q^{79} -4.31146i q^{80} -6.84670 q^{82} +5.52308i q^{83} -4.75965i q^{85} +12.1635i q^{86} -15.2767 q^{88} -17.5838i q^{89} +(2.34353 - 2.65566i) q^{91} +19.1696 q^{92} -10.9387 q^{94} +6.41080 q^{95} -0.246338i q^{97} +14.9505i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) $ 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$1$	$-1$

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 $

$2$			2.47728i			1.75170i	0.482580	+	0.875852i	$0.339700\pi$
$2$			2.47728i			1.75170i	−0.482580	+	0.875852i	$0.660300\pi$
$3$		0			0
$3$		0			0
$4$	−4.13693			−2.06847
$5$		−	0.890732i		−	0.398348i	−0.979964	−	0.199174i	$-0.936174\pi$
$5$		−	0.890732i		−	0.398348i	0.979964	−	0.199174i	$-0.0638258\pi$
$6$		0			0
$7$		−	0.982330i		−	0.371286i	−0.982617	−	0.185643i	$-0.940563\pi$
$7$		−	0.982330i		−	0.371286i	0.982617	−	0.185643i	$-0.0594368\pi$
$8$		−	5.29379i		−	1.87164i
$9$		0			0
$10$	2.20660			0.697787
$11$		−	2.88577i		−	0.870091i	−0.900408	−	0.435046i	$-0.856732\pi$
$11$		−	2.88577i		−	0.870091i	0.900408	−	0.435046i	$-0.143268\pi$
$12$		0			0
$13$	2.70342	+	2.38569i	0.749795	+	0.661670i
$13$	2.70342	+	2.38569i	0.749795	+	0.661670i
$14$	2.43351			0.650383
$15$		0			0
$16$	4.84036			1.21009
$17$	5.34353			1.29600			0.647998	−	0.761642i	$-0.275606\pi$
$17$	5.34353			1.29600			0.647998	+	0.761642i	$0.275606\pi$
$18$		0			0
$19$			7.19723i			1.65116i	0.564287	+	0.825579i	$0.309151\pi$
$19$			7.19723i			1.65116i	−0.564287	+	0.825579i	$0.690849\pi$
$20$			3.68490i			0.823969i
$21$		0			0
$22$	7.14886			1.52414
$23$	−4.63376			−0.966206			−0.483103	−	0.875563i	$-0.660490\pi$
$23$	−4.63376			−0.966206			−0.483103	+	0.875563i	$0.660490\pi$
$24$		0			0
$25$	4.20660			0.841319
$26$	−5.91002	+	6.69715i	−1.15905	+	1.31342i
$27$		0			0
$28$			4.06384i			0.767993i
$29$	1.94227			0.360670			0.180335	−	0.983605i	$-0.442282\pi$
$29$	1.94227			0.360670			0.180335	+	0.983605i	$0.442282\pi$
$30$		0			0
$31$			10.0875i			1.81177i	0.423524	+	0.905885i	$0.360793\pi$
$31$			10.0875i			1.81177i	−0.423524	+	0.905885i	$0.639207\pi$
$32$			1.40336i			0.248081i
$33$		0			0
$34$			13.2374i			2.27020i
$35$	−0.874993			−0.147901
$36$		0			0
$37$		−	4.82809i		−	0.793732i	−0.917876	−	0.396866i	$-0.870098\pi$
$37$		−	4.82809i		−	0.793732i	0.917876	−	0.396866i	$-0.129902\pi$
$38$	−17.8296			−2.89234
$39$		0			0
$40$	−4.71535			−0.745563
$41$			2.76379i			0.431632i	0.976434	+	0.215816i	$0.0692412\pi$
$41$			2.76379i			0.431632i	−0.976434	+	0.215816i	$0.930759\pi$
$42$		0			0
$43$	4.91002			0.748771			0.374386	−	0.927273i	$-0.377854\pi$
$43$	4.91002			0.748771			0.374386	+	0.927273i	$0.377854\pi$
$44$			11.9382i			1.79976i
$45$		0			0
$46$		−	11.4791i		−	1.69251i
$47$			4.41560i			0.644082i	0.946726	+	0.322041i	$0.104369\pi$
$47$			4.41560i			0.644082i	−0.946726	+	0.322041i	$0.895631\pi$
$48$		0			0
$49$	6.03503			0.862147
$50$			10.4209i			1.47374i
$51$		0			0
$52$	−11.1839	−	9.86943i	−1.55093	−	1.36864i
$53$	6.30850			0.866540			0.433270	−	0.901264i	$-0.357360\pi$
$53$	6.30850			0.866540			0.433270	+	0.901264i	$0.357360\pi$
$54$		0			0
$55$	−2.57044			−0.346599
$56$	−5.20025			−0.694913
$57$		0			0
$58$			4.81154i			0.631787i
$59$		−	3.17228i		−	0.412995i	−0.978447	−	0.206498i	$-0.933793\pi$
$59$		−	3.17228i		−	0.412995i	0.978447	−	0.206498i	$-0.0662066\pi$
$60$		0			0
$61$	−5.52068			−0.706851			−0.353426	−	0.935463i	$-0.614983\pi$
$61$	−5.52068			−0.706851			−0.353426	+	0.935463i	$0.614983\pi$
$62$	−24.9896			−3.17368
$63$		0			0
$64$	6.20421			0.775526
$65$	2.12501	−	2.40803i	0.263575	−	0.298679i
$66$		0			0
$67$		−	3.63799i		−	0.444451i	−0.974995	−	0.222225i	$-0.928668\pi$
$67$		−	3.63799i		−	0.444451i	0.974995	−	0.222225i	$-0.0713321\pi$
$68$	−22.1058			−2.68073
$69$		0			0
$70$		−	2.16761i		−	0.259078i
$71$			6.69715i			0.794805i	0.917644	+	0.397403i	$0.130088\pi$
$71$			6.69715i			0.794805i	−0.917644	+	0.397403i	$0.869912\pi$
$72$		0			0
$73$			9.33980i			1.09314i	0.837413	+	0.546570i	$0.184067\pi$
$73$			9.33980i			1.09314i	−0.837413	+	0.546570i	$0.815933\pi$
$74$	11.9605			1.39038
$75$		0			0
$76$		−	29.7745i		−	3.41537i
$77$	−2.83478			−0.323053
$78$		0			0
$79$	4.74760			0.534146			0.267073	−	0.963676i	$-0.413943\pi$
$79$	4.74760			0.534146			0.267073	+	0.963676i	$0.413943\pi$
$80$		−	4.31146i		−	0.482036i
$81$		0			0
$82$	−6.84670			−0.756092
$83$			5.52308i			0.606237i	0.952953	+	0.303118i	$0.0980277\pi$
$83$			5.52308i			0.606237i	−0.952953	+	0.303118i	$0.901972\pi$
$84$		0			0
$85$		−	4.75965i		−	0.516257i
$86$			12.1635i			1.31163i
$87$		0			0
$88$	−15.2767			−1.62850
$89$		−	17.5838i		−	1.86388i	−0.362615	−	0.931939i	$-0.618116\pi$
$89$		−	17.5838i		−	1.86388i	0.362615	−	0.931939i	$-0.381884\pi$
$90$		0			0
$91$	2.34353	−	2.65566i	0.245669	−	0.278388i
$92$	19.1696			1.99857
$93$		0			0
$94$	−10.9387			−1.12824
$95$	6.41080			0.657735
$96$		0			0
$97$		−	0.246338i		−	0.0250119i	−0.999922	−	0.0125059i	$-0.996019\pi$
$97$		−	0.246338i		−	0.0250119i	0.999922	−	0.0125059i	$-0.00398086\pi$
$98$			14.9505i			1.51023i
$99$		0			0
$100$	−17.4024			−1.74024
$101$	−4.77426			−0.475057			−0.237528	−	0.971381i	$-0.576337\pi$
$101$	−4.77426			−0.475057			−0.237528	+	0.971381i	$0.576337\pi$
$102$		0			0
$103$	−3.43351			−0.338314			−0.169157	−	0.985589i	$-0.554104\pi$
$103$	−3.43351			−0.338314			−0.169157	+	0.985589i	$0.554104\pi$
$104$	12.6293	−	14.3114i	1.23841	−	1.40335i
$105$		0			0
$106$			15.6280i			1.51792i
$107$	−2.34075			−0.226289			−0.113144	−	0.993579i	$-0.536092\pi$
$107$	−2.34075			−0.226289			−0.113144	+	0.993579i	$0.536092\pi$
$108$		0			0
$109$			10.2127i			0.978195i	0.872229	+	0.489097i	$0.162674\pi$
$109$			10.2127i			0.978195i	−0.872229	+	0.489097i	$0.837326\pi$
$110$		−	6.36772i		−	0.607138i
$111$		0			0
$112$		−	4.75483i		−	0.449289i
$113$	−1.21055			−0.113879			−0.0569396	−	0.998378i	$-0.518134\pi$
$113$	−1.21055			−0.113879			−0.0569396	+	0.998378i	$0.518134\pi$
$114$		0			0
$115$			4.12744i			0.384886i
$116$	−8.03503			−0.746034
$117$		0			0
$118$	7.85863			0.723446
$119$		−	5.24911i		−	0.481185i
$120$		0			0
$121$	2.67235			0.242941
$122$		−	13.6763i		−	1.23819i
$123$		0			0
$124$		−	41.7314i		−	3.74759i
$125$		−	8.20061i		−	0.733485i
$126$		0			0
$127$	7.58237			0.672827			0.336413	−	0.941714i	$-0.390786\pi$
$127$	7.58237			0.672827			0.336413	+	0.941714i	$0.390786\pi$
$128$			18.1763i			1.60657i
$129$		0			0
$130$	5.96537	+	5.26425i	0.523197	+	0.461705i
$131$	18.8121			1.64362			0.821809	−	0.569763i	$-0.192965\pi$
$131$	18.8121			1.64362			0.821809	+	0.569763i	$0.192965\pi$
$132$		0			0
$133$	7.07006			0.613052
$134$	9.01232			0.778546
$135$		0			0
$136$		−	28.2875i		−	2.42564i
$137$		−	18.0615i		−	1.54310i	−0.636168	−	0.771550i	$-0.719482\pi$
$137$		−	18.0615i		−	1.54310i	0.636168	−	0.771550i	$-0.280518\pi$
$138$		0			0
$139$	−4.85153			−0.411502			−0.205751	−	0.978604i	$-0.565964\pi$
$139$	−4.85153			−0.411502			−0.205751	+	0.978604i	$0.565964\pi$
$140$	3.61979			0.305928
$141$		0			0
$142$	−16.5907			−1.39226
$143$	6.88453	−	7.80145i	0.575713	−	0.652390i
$144$		0			0
$145$		−	1.73004i		−	0.143672i
$146$	−23.1373			−1.91486
$147$		0			0
$148$			19.9735i			1.64181i
$149$		−	12.2677i		−	1.00501i	−0.864576	−	0.502503i	$-0.832413\pi$
$149$		−	12.2677i		−	1.00501i	0.864576	−	0.502503i	$-0.167587\pi$
$150$		0			0
$151$			5.56730i			0.453060i	0.974004	+	0.226530i	$0.0727382\pi$
$151$			5.56730i			0.453060i	−0.974004	+	0.226530i	$0.927262\pi$
$152$	38.1006			3.09037
$153$		0			0
$154$		−	7.02254i		−	0.565893i
$155$	8.98527			0.721714
$156$		0			0
$157$	−4.05052			−0.323266			−0.161633	−	0.986851i	$-0.551676\pi$
$157$	−4.05052			−0.323266			−0.161633	+	0.986851i	$0.551676\pi$
$158$			11.7611i			0.935667i
$159$		0			0
$160$	1.25001			0.0988223
$161$			4.55188i			0.358739i
$162$		0			0
$163$		−	10.3387i		−	0.809787i	−0.914364	−	0.404894i	$-0.867309\pi$
$163$		−	10.3387i		−	0.809787i	0.914364	−	0.404894i	$-0.132691\pi$
$164$		−	11.4336i		−	0.892817i
$165$		0			0
$166$	−13.6822			−1.06195
$167$			13.2084i			1.02209i	0.859553	+	0.511047i	$0.170742\pi$
$167$			13.2084i			1.02209i	−0.859553	+	0.511047i	$0.829258\pi$
$168$		0			0
$169$	1.61701	+	12.8990i	0.124385	+	0.992234i
$170$	11.7910			0.904330
$171$		0			0
$172$	−20.3124			−1.54881
$173$	11.4283			0.868879			0.434439	−	0.900701i	$-0.356947\pi$
$173$	11.4283			0.868879			0.434439	+	0.900701i	$0.356947\pi$
$174$		0			0
$175$		−	4.13227i		−	0.312370i
$176$		−	13.9681i		−	1.05289i
$177$		0			0
$178$	43.5600			3.26496
$179$	0.150491			0.0112482			0.00562411	−	0.999984i	$-0.498210\pi$
$179$	0.150491			0.0112482			0.00562411	+	0.999984i	$0.498210\pi$
$180$		0			0
$181$	−22.0219			−1.63687			−0.818436	−	0.574598i	$-0.805159\pi$
$181$	−22.0219			−1.63687			−0.818436	+	0.574598i	$0.805159\pi$
$182$	6.57881	+	5.80559i	0.487654	+	0.430339i
$183$		0			0
$184$			24.5302i			1.80839i
$185$	−4.30053			−0.316181
$186$		0			0
$187$		−	15.4202i		−	1.12764i
$188$		−	18.2671i		−	1.33226i
$189$		0			0
$190$			15.8814i			1.15216i
$191$	−19.9574			−1.44406			−0.722032	−	0.691859i	$-0.756792\pi$
$191$	−19.9574			−1.44406			−0.722032	+	0.691859i	$0.756792\pi$
$192$		0			0
$193$		−	10.0522i		−	0.723570i	−0.932262	−	0.361785i	$-0.882167\pi$
$193$		−	10.0522i		−	0.723570i	0.932262	−	0.361785i	$-0.117833\pi$
$194$	0.610250			0.0438134
$195$		0			0
$196$	−24.9665			−1.78332
$197$			3.94715i			0.281223i	0.990065	+	0.140611i	$0.0449068\pi$
$197$			3.94715i			0.281223i	−0.990065	+	0.140611i	$0.955093\pi$
$198$		0			0
$199$	11.4749			0.813433			0.406716	−	0.913554i	$-0.366674\pi$
$199$	11.4749			0.813433			0.406716	+	0.913554i	$0.366674\pi$
$200$		−	22.2689i		−	1.57465i
$201$		0			0
$202$		−	11.8272i		−	0.832158i
$203$		−	1.90795i		−	0.133912i
$204$		0			0
$205$	2.46180			0.171940
$206$		−	8.50578i		−	0.592626i
$207$		0			0
$208$	13.0855	+	11.5476i	0.907319	+	0.800680i
$209$	20.7695			1.43666
$210$		0			0
$211$	10.9661			0.754939			0.377469	−	0.926022i	$-0.376794\pi$
$211$	10.9661			0.754939			0.377469	+	0.926022i	$0.376794\pi$
$212$	−26.0979			−1.79241
$213$		0			0
$214$		−	5.79870i		−	0.396391i
$215$		−	4.37351i		−	0.298271i
$216$		0			0
$217$	9.90926			0.672684
$218$	−25.2996			−1.71351
$219$		0			0
$220$	10.6338			0.716928
$221$	14.4458	+	12.7480i	0.971732	+	0.857522i
$222$		0			0
$223$			12.8423i			0.859986i	0.902832	+	0.429993i	$0.141484\pi$
$223$			12.8423i			0.859986i	−0.902832	+	0.429993i	$0.858516\pi$
$224$	1.37856			0.0921088
$225$		0			0
$226$		−	2.99888i		−	0.199482i
$227$			23.6176i			1.56755i	0.621043	+	0.783777i	$0.286709\pi$
$227$			23.6176i			1.56755i	−0.621043	+	0.783777i	$0.713291\pi$
$228$		0			0
$229$			12.4378i			0.821916i	0.911654	+	0.410958i	$0.134806\pi$
$229$			12.4378i			0.821916i	−0.911654	+	0.410958i	$0.865194\pi$
$230$	−10.2248			−0.674206
$231$		0			0
$232$		−	10.2820i		−	0.675043i
$233$	−18.7821			−1.23046			−0.615230	−	0.788348i	$-0.710937\pi$
$233$	−18.7821			−1.23046			−0.615230	+	0.788348i	$0.710937\pi$
$234$		0			0
$235$	3.93312			0.256568
$236$			13.1235i			0.854267i
$237$		0			0
$238$	13.0035			0.842894
$239$		−	19.7831i		−	1.27966i	−0.768516	−	0.639830i	$-0.779005\pi$
$239$		−	19.7831i		−	1.27966i	0.768516	−	0.639830i	$-0.220995\pi$
$240$		0			0
$241$			11.3917i			0.733804i	0.930260	+	0.366902i	$0.119581\pi$
$241$			11.3917i			0.733804i	−0.930260	+	0.366902i	$0.880419\pi$
$242$			6.62017i			0.425561i
$243$		0			0
$244$	22.8387			1.46210
$245$		−	5.37559i		−	0.343434i
$246$		0			0
$247$	−17.1703	+	19.4572i	−1.09252	+	1.23803i
$248$	53.4012			3.39098
$249$		0			0
$250$	20.3152			1.28485
$251$	−18.6950			−1.18002			−0.590010	−	0.807396i	$-0.700876\pi$
$251$	−18.6950			−1.18002			−0.590010	+	0.807396i	$0.700876\pi$
$252$		0			0
$253$			13.3720i			0.840688i
$254$			18.7837i			1.17859i
$255$		0			0
$256$	−32.6194			−2.03871
$257$	−8.75953			−0.546404			−0.273202	−	0.961957i	$-0.588083\pi$
$257$	−8.75953			−0.546404			−0.273202	+	0.961957i	$0.588083\pi$
$258$		0			0
$259$	−4.74277			−0.294702
$260$	−8.79102	+	9.96185i	−0.545196	+	0.617808i
$261$		0			0
$262$			46.6028i			2.87913i
$263$	7.72804			0.476531			0.238266	−	0.971200i	$-0.423421\pi$
$263$	7.72804			0.476531			0.238266	+	0.971200i	$0.423421\pi$
$264$		0			0
$265$		−	5.61919i		−	0.345184i
$266$			17.5145i			1.07388i
$267$		0			0
$268$			15.0501i			0.919332i
$269$	14.8448			0.905102			0.452551	−	0.891739i	$-0.350514\pi$
$269$	14.8448			0.905102			0.452551	+	0.891739i	$0.350514\pi$
$270$		0			0
$271$		−	19.3486i		−	1.17534i	−0.809100	−	0.587671i	$-0.800045\pi$
$271$		−	19.3486i		−	1.17534i	0.809100	−	0.587671i	$-0.199955\pi$
$272$	25.8646			1.56827
$273$		0			0
$274$	44.7435			2.70306
$275$		−	12.1393i		−	0.732025i
$276$		0			0
$277$	1.17552			0.0706304			0.0353152	−	0.999376i	$-0.488756\pi$
$277$	1.17552			0.0706304			0.0353152	+	0.999376i	$0.488756\pi$
$278$		−	12.0186i		−	0.720829i
$279$		0			0
$280$			4.63203i			0.276817i
$281$		−	4.58097i		−	0.273278i	−0.990621	−	0.136639i	$-0.956370\pi$
$281$		−	4.58097i		−	0.273278i	0.990621	−	0.136639i	$-0.0436300\pi$
$282$		0			0
$283$	0.0318323			0.00189224			0.000946118	−	1.00000i	$-0.499699\pi$
$283$	0.0318323			0.00189224			0.000946118		1.00000i	$0.499699\pi$
$284$		−	27.7057i		−	1.64403i
$285$		0			0
$286$	19.3264	+	17.0549i	1.14279	+	1.00848i
$287$	2.71496			0.160259
$288$		0			0
$289$	11.5533			0.679607
$290$	4.28580			0.251671
$291$		0			0
$292$		−	38.6381i		−	2.26113i
$293$			10.9082i			0.637261i	0.947879	+	0.318631i	$0.103223\pi$
$293$			10.9082i			0.637261i	−0.947879	+	0.318631i	$0.896777\pi$
$294$		0			0
$295$	−2.82565			−0.164516
$296$	−25.5589			−1.48558
$297$		0			0
$298$	30.3905			1.76047
$299$	−12.5270	−	11.0547i	−0.724457	−	0.639310i
$300$		0			0
$301$		−	4.82326i		−	0.278008i
$302$	−13.7918			−0.793627
$303$		0			0
$304$			34.8372i			1.99805i
$305$			4.91745i			0.281572i
$306$		0			0
$307$			6.90363i			0.394011i	0.980402	+	0.197005i	$0.0631217\pi$
$307$			6.90363i			0.394011i	−0.980402	+	0.197005i	$0.936878\pi$
$308$	11.7273			0.668224
$309$		0			0
$310$			22.2591i			1.26423i
$311$	−15.3961			−0.873031			−0.436516	−	0.899697i	$-0.643788\pi$
$311$	−15.3961			−0.873031			−0.436516	+	0.899697i	$0.643788\pi$
$312$		0			0
$313$	−17.3415			−0.980202			−0.490101	−	0.871666i	$-0.663040\pi$
$313$	−17.3415			−0.980202			−0.490101	+	0.871666i	$0.663040\pi$
$314$		−	10.0343i		−	0.566267i
$315$		0			0
$316$	−19.6405			−1.10486
$317$			10.6355i			0.597348i	0.954355	+	0.298674i	$0.0965443\pi$
$317$			10.6355i			0.597348i	−0.954355	+	0.298674i	$0.903456\pi$
$318$		0			0
$319$		−	5.60493i		−	0.313816i
$320$		−	5.52629i		−	0.308929i
$321$		0			0
$322$	−11.2763			−0.628404
$323$			38.4586i			2.13989i
$324$		0			0
$325$	11.3722	+	10.0356i	0.630817	+	0.556676i
$326$	25.6118			1.41851
$327$		0			0
$328$	14.6310			0.807859
$329$	4.33758			0.239139
$330$		0			0
$331$		−	31.6792i		−	1.74125i	−0.491951	−	0.870623i	$-0.663716\pi$
$331$		−	31.6792i		−	1.74125i	0.491951	−	0.870623i	$-0.336284\pi$
$332$		−	22.8486i		−	1.25398i
$333$		0			0
$334$	−32.7208			−1.79041
$335$	−3.24047			−0.177046
$336$		0			0
$337$	24.2965			1.32351			0.661757	−	0.749718i	$-0.269811\pi$
$337$	24.2965			1.32351			0.661757	+	0.749718i	$0.269811\pi$
$338$	−31.9546	+	4.00578i	−1.73810	+	0.217886i
$339$		0			0
$340$			19.6904i			1.06786i
$341$	29.1102			1.57641
$342$		0			0
$343$		−	12.8047i		−	0.691389i
$344$		−	25.9926i		−	1.40143i
$345$		0			0
$346$			28.3112i			1.52202i
$347$	3.82448			0.205309			0.102654	−	0.994717i	$-0.467266\pi$
$347$	3.82448			0.205309			0.102654	+	0.994717i	$0.467266\pi$
$348$		0			0
$349$		−	23.5327i		−	1.25968i	−0.776726	−	0.629839i	$-0.783121\pi$
$349$		−	23.5327i		−	1.25968i	0.776726	−	0.629839i	$-0.216879\pi$
$350$	10.2368			0.547180
$351$		0			0
$352$	4.04976			0.215853
$353$		−	23.5936i		−	1.25576i	−0.778309	−	0.627881i	$-0.783922\pi$
$353$		−	23.5936i		−	1.25576i	0.778309	−	0.627881i	$-0.216078\pi$
$354$		0			0
$355$	5.96537			0.316609
$356$			72.7430i			3.85537i
$357$		0			0
$358$			0.372809i			0.0197035i
$359$			0.220474i			0.0116362i	0.999983	+	0.00581809i	$0.00185196\pi$
$359$			0.220474i			0.0116362i	−0.999983	+	0.00581809i	$0.998148\pi$
$360$		0			0
$361$	−32.8001			−1.72632
$362$		−	54.5544i		−	2.86732i
$363$		0			0
$364$	−9.69503	+	10.9863i	−0.508158	+	0.575837i
$365$	8.31926			0.435450
$366$		0			0
$367$	−20.5139			−1.07081			−0.535407	−	0.844594i	$-0.679842\pi$
$367$	−20.5139			−1.07081			−0.535407	+	0.844594i	$0.679842\pi$
$368$	−22.4291			−1.16920
$369$		0			0
$370$		−	10.6536i		−	0.553856i
$371$		−	6.19703i		−	0.321734i
$372$		0			0
$373$	10.8551			0.562055			0.281028	−	0.959700i	$-0.409325\pi$
$373$	10.8551			0.562055			0.281028	+	0.959700i	$0.409325\pi$
$374$	38.2002			1.97528
$375$		0			0
$376$	23.3753			1.20549
$377$	5.25077	+	4.63364i	0.270428	+	0.238644i
$378$		0			0
$379$		−	19.6987i		−	1.01185i	−0.862577	−	0.505927i	$-0.831151\pi$
$379$		−	19.6987i		−	1.01185i	0.862577	−	0.505927i	$-0.168849\pi$
$380$	−26.5211			−1.36050
$381$		0			0
$382$		−	49.4401i		−	2.52957i
$383$		−	20.3417i		−	1.03941i	−0.854346	−	0.519705i	$-0.826042\pi$
$383$		−	20.3417i		−	1.03941i	0.854346	−	0.519705i	$-0.173958\pi$
$384$		0			0
$385$			2.52503i			0.128687i
$386$	24.9020			1.26748
$387$		0			0
$388$			1.01909i			0.0517362i
$389$	21.3961			1.08482			0.542412	−	0.840113i	$-0.317511\pi$
$389$	21.3961			1.08482			0.542412	+	0.840113i	$0.317511\pi$
$390$		0			0
$391$	−24.7607			−1.25220
$392$		−	31.9482i		−	1.61363i
$393$		0			0
$394$	−9.77821			−0.492619
$395$		−	4.22884i		−	0.212776i
$396$		0			0
$397$		−	27.7995i		−	1.39522i	−0.716479	−	0.697608i	$-0.754248\pi$
$397$		−	27.7995i		−	1.39522i	0.716479	−	0.697608i	$-0.245752\pi$
$398$			28.4265i			1.42489i
$399$		0			0
$400$	20.3614			1.01807
$401$		−	15.1860i		−	0.758354i	−0.925324	−	0.379177i	$-0.876207\pi$
$401$		−	15.1860i		−	0.758354i	0.925324	−	0.379177i	$-0.123793\pi$
$402$		0			0
$403$	−24.0656	+	27.2708i	−1.19879	+	1.35846i
$404$	19.7508			0.982639
$405$		0			0
$406$	4.72652			0.234573
$407$	−13.9327			−0.690620
$408$		0			0
$409$		−	2.38569i		−	0.117965i	−0.998259	−	0.0589823i	$-0.981214\pi$
$409$		−	2.38569i		−	0.117965i	0.998259	−	0.0589823i	$-0.0187855\pi$
$410$			6.09858i			0.301187i
$411$		0			0
$412$	14.2042			0.699791
$413$	−3.11622			−0.153339
$414$		0			0
$415$	4.91958			0.241493
$416$	−3.34797	+	3.79387i	−0.164148	+	0.186010i
$417$		0			0
$418$			51.4520i			2.51660i
$419$	33.2909			1.62637			0.813183	−	0.582007i	$-0.197733\pi$
$419$	33.2909			1.62637			0.813183	+	0.582007i	$0.197733\pi$
$420$		0			0
$421$		−	24.3049i		−	1.18455i	−0.805736	−	0.592275i	$-0.798230\pi$
$421$		−	24.3049i		−	1.18455i	0.805736	−	0.592275i	$-0.201770\pi$
$422$			27.1662i			1.32243i
$423$		0			0
$424$		−	33.3959i		−	1.62185i
$425$	22.4781			1.09035
$426$		0			0
$427$			5.42313i			0.262444i
$428$	9.68352			0.468071
$429$		0			0
$430$	10.8344			0.522483
$431$		−	12.1410i		−	0.584812i	−0.956294	−	0.292406i	$-0.905544\pi$
$431$		−	12.1410i		−	0.584812i	0.956294	−	0.292406i	$-0.0944559\pi$
$432$		0			0
$433$	−13.2730			−0.637859			−0.318930	−	0.947778i	$-0.603323\pi$
$433$	−13.2730			−0.637859			−0.318930	+	0.947778i	$0.603323\pi$
$434$			24.5481i			1.17834i
$435$		0			0
$436$		−	42.2491i		−	2.02336i
$437$		−	33.3503i		−	1.59536i
$438$		0			0
$439$	−13.1015			−0.625300			−0.312650	−	0.949868i	$-0.601217\pi$
$439$	−13.1015			−0.625300			−0.312650	+	0.949868i	$0.601217\pi$
$440$			13.6074i			0.648708i
$441$		0			0
$442$	−31.5804	+	35.7864i	−1.50213	+	1.70219i
$443$	−34.4390			−1.63625			−0.818123	−	0.575044i	$-0.804985\pi$
$443$	−34.4390			−1.63625			−0.818123	+	0.575044i	$0.804985\pi$
$444$		0			0
$445$	−15.6624			−0.742471
$446$	−31.8141			−1.50644
$447$		0			0
$448$		−	6.09458i		−	0.287942i
$449$		−	5.32385i		−	0.251248i	−0.992078	−	0.125624i	$-0.959907\pi$
$449$		−	5.32385i		−	0.251248i	0.992078	−	0.125624i	$-0.0400933\pi$
$450$		0			0
$451$	7.97567			0.375559
$452$	5.00797			0.235555
$453$		0			0
$454$	−58.5074			−2.74589
$455$	−2.36548	−	2.08746i	−0.110895	−	0.0978616i
$456$		0			0
$457$		−	12.3896i		−	0.579562i	−0.957093	−	0.289781i	$-0.906417\pi$
$457$		−	12.3896i		−	0.579562i	0.957093	−	0.289781i	$-0.0935825\pi$
$458$	−30.8121			−1.43975
$459$		0			0
$460$		−	17.0750i		−	0.796124i
$461$		−	13.0058i		−	0.605739i	−0.953032	−	0.302869i	$-0.902055\pi$
$461$		−	13.0058i		−	0.605739i	0.953032	−	0.302869i	$-0.0979446\pi$
$462$		0			0
$463$			6.42448i			0.298571i	0.988794	+	0.149286i	$0.0476974\pi$
$463$			6.42448i			0.298571i	−0.988794	+	0.149286i	$0.952303\pi$
$464$	9.40126			0.436443
$465$		0			0
$466$		−	46.5287i		−	2.15540i
$467$	13.9598			0.645982			0.322991	−	0.946402i	$-0.395312\pi$
$467$	13.9598			0.645982			0.322991	+	0.946402i	$0.395312\pi$
$468$		0			0
$469$	−3.57370			−0.165018
$470$			9.74346i			0.449432i
$471$		0			0
$472$	−16.7934			−0.772978
$473$		−	14.1692i		−	0.651499i
$474$		0			0
$475$			30.2758i			1.38915i
$476$			21.7152i			0.995316i
$477$		0			0
$478$	49.0083			2.24159
$479$			34.6252i			1.58207i	0.611773	+	0.791033i	$0.290456\pi$
$479$			34.6252i			1.58207i	−0.611773	+	0.791033i	$0.709544\pi$
$480$		0			0
$481$	11.5183	−	13.0524i	0.525189	−	0.595137i
$482$	−28.2205			−1.28541
$483$		0			0
$484$	−11.0553			−0.502516
$485$	−0.219421			−0.00996341
$486$		0			0
$487$		−	5.78811i		−	0.262284i	−0.991364	−	0.131142i	$-0.958136\pi$
$487$		−	5.78811i		−	0.262284i	0.991364	−	0.131142i	$-0.0418644\pi$
$488$			29.2254i			1.32297i
$489$		0			0
$490$	13.3169			0.601595
$491$	−24.9244			−1.12482			−0.562410	−	0.826858i	$-0.690126\pi$
$491$	−24.9244			−1.12482			−0.562410	+	0.826858i	$0.690126\pi$
$492$		0			0
$493$	10.3786			0.467427
$494$	−48.2009	−	42.5358i	−2.16866	−	1.91378i
$495$		0			0
$496$			48.8272i			2.19240i
$497$	6.57881			0.295100
$498$		0			0
$499$		−	39.8186i		−	1.78252i	−0.453488	−	0.891262i	$-0.649821\pi$
$499$		−	39.8186i		−	1.78252i	0.453488	−	0.891262i	$-0.350179\pi$
$500$			33.9254i			1.51719i
$501$		0			0
$502$		−	46.3129i		−	2.06705i
$503$	−12.9120			−0.575720			−0.287860	−	0.957673i	$-0.592944\pi$
$503$	−12.9120			−0.575720			−0.287860	+	0.957673i	$0.592944\pi$
$504$		0			0
$505$			4.25259i			0.189238i
$506$	−33.1261			−1.47264
$507$		0			0
$508$	−31.3678			−1.39172
$509$			37.1525i			1.64676i	0.567494	+	0.823378i	$0.307913\pi$
$509$			37.1525i			1.64676i	−0.567494	+	0.823378i	$0.692087\pi$
$510$		0			0
$511$	9.17476			0.405868
$512$		−	44.4550i		−	1.96465i
$513$		0			0
$514$		−	21.6998i		−	0.957139i
$515$			3.05834i			0.134766i
$516$		0			0
$517$	12.7424			0.560410
$518$		−	11.7492i		−	0.516230i
$519$		0			0
$520$	−12.7476	−	11.2493i	−0.559019	−	0.493317i
$521$	−24.6907			−1.08172			−0.540859	−	0.841114i	$-0.681901\pi$
$521$	−24.6907			−1.08172			−0.540859	+	0.841114i	$0.681901\pi$
$522$		0			0
$523$	−33.4434			−1.46238			−0.731189	−	0.682175i	$-0.761034\pi$
$523$	−33.4434			−1.46238			−0.731189	+	0.682175i	$0.761034\pi$
$524$	−77.8243			−3.39977
$525$		0			0
$526$			19.1445i			0.834742i
$527$			53.9029i			2.34805i
$528$		0			0
$529$	−1.52824			−0.0664453
$530$	13.9203			0.604660
$531$		0			0
$532$	−29.2484			−1.26808
$533$	−6.59354	+	7.47171i	−0.285598	+	0.323636i
$534$		0			0
$535$			2.08498i			0.0901416i
$536$	−19.2587			−0.831851
$537$		0			0
$538$			36.7747i			1.58547i
$539$		−	17.4157i		−	0.750146i
$540$		0			0
$541$			10.9418i			0.470423i	0.971944	+	0.235211i	$0.0755782\pi$
$541$			10.9418i			0.470423i	−0.971944	+	0.235211i	$0.924422\pi$
$542$	47.9319			2.05885
$543$		0			0
$544$			7.49887i			0.321512i
$545$	9.09674			0.389661
$546$		0			0
$547$	10.9481			0.468108			0.234054	−	0.972224i	$-0.424801\pi$
$547$	10.9481			0.468108			0.234054	+	0.972224i	$0.424801\pi$
$548$			74.7194i			3.19185i
$549$		0			0
$550$	30.0724			1.28229
$551$			13.9789i			0.595523i
$552$		0			0
$553$		−	4.66371i		−	0.198321i
$554$			2.91211i			0.123724i
$555$		0			0
$556$	20.0705			0.851177
$557$		−	12.5825i		−	0.533136i	−0.963816	−	0.266568i	$-0.914110\pi$
$557$		−	12.5825i		−	0.533136i	0.963816	−	0.266568i	$-0.0858898\pi$
$558$		0			0
$559$	13.2739	+	11.7138i	0.561425	+	0.495440i
$560$	−4.23528			−0.178973
$561$		0			0
$562$	11.3484			0.478702
$563$	−12.1557			−0.512301			−0.256151	−	0.966637i	$-0.582454\pi$
$563$	−12.1557			−0.512301			−0.256151	+	0.966637i	$0.582454\pi$
$564$		0			0
$565$			1.07828i			0.0453635i
$566$			0.0788577i			0.00331464i
$567$		0			0
$568$	35.4533			1.48759
$569$	35.3764			1.48306			0.741529	−	0.670921i	$-0.234101\pi$
$569$	35.3764			1.48306			0.741529	+	0.670921i	$0.234101\pi$
$570$		0			0
$571$	−12.4912			−0.522741			−0.261371	−	0.965239i	$-0.584174\pi$
$571$	−12.4912			−0.522741			−0.261371	+	0.965239i	$0.584174\pi$
$572$	−28.4809	+	32.2741i	−1.19084	+	1.34945i
$573$		0			0
$574$			6.72572i			0.280726i
$575$	−19.4924			−0.812888
$576$		0			0
$577$			25.1610i			1.04747i	0.851882	+	0.523734i	$0.175461\pi$
$577$			25.1610i			1.04747i	−0.851882	+	0.523734i	$0.824539\pi$
$578$			28.6209i			1.19047i
$579$		0			0
$580$			7.15706i			0.297181i
$581$	5.42549			0.225087
$582$		0			0
$583$		−	18.2049i		−	0.753969i
$584$	49.4430			2.04596
$585$		0			0
$586$	−27.0226			−1.11629
$587$			3.71270i			0.153239i	0.997060	+	0.0766197i	$0.0244127\pi$
$587$			3.71270i			0.153239i	−0.997060	+	0.0766197i	$0.975587\pi$
$588$		0			0
$589$	−72.6021			−2.99152
$590$		−	6.99993i		−	0.288183i
$591$		0			0
$592$		−	23.3697i		−	0.960487i
$593$			31.6710i			1.30057i	0.759689	+	0.650287i	$0.225351\pi$
$593$			31.6710i			1.30057i	−0.759689	+	0.650287i	$0.774649\pi$
$594$		0			0
$595$	−4.67555			−0.191679
$596$			50.7505i			2.07882i
$597$		0			0
$598$	27.3856	−	31.0330i	1.11988	−	1.26903i
$599$	−28.4597			−1.16283			−0.581415	−	0.813607i	$-0.697501\pi$
$599$	−28.4597			−1.16283			−0.581415	+	0.813607i	$0.697501\pi$
$600$		0			0
$601$	1.27109			0.0518487			0.0259244	−	0.999664i	$-0.491747\pi$
$601$	1.27109			0.0518487			0.0259244	+	0.999664i	$0.491747\pi$
$602$	11.9486			0.486988
$603$		0			0
$604$		−	23.0315i		−	0.937140i
$605$		−	2.38035i		−	0.0967750i
$606$		0			0
$607$	−0.747208			−0.0303283			−0.0151641	−	0.999885i	$-0.504827\pi$
$607$	−0.747208			−0.0303283			−0.0151641	+	0.999885i	$0.504827\pi$
$608$	−10.1003			−0.409620
$609$		0			0
$610$	−12.1819			−0.493231
$611$	−10.5342	+	11.9373i	−0.426170	+	0.482930i
$612$		0			0
$613$		−	41.1308i		−	1.66126i	−0.556826	−	0.830629i	$-0.687981\pi$
$613$		−	41.1308i		−	1.66126i	0.556826	−	0.830629i	$-0.312019\pi$
$614$	−17.1022			−0.690191
$615$		0			0
$616$			15.0067i			0.604638i
$617$			12.6582i			0.509598i	0.966994	+	0.254799i	$0.0820093\pi$
$617$			12.6582i			0.509598i	−0.966994	+	0.254799i	$0.917991\pi$
$618$		0			0
$619$		−	8.52968i		−	0.342837i	−0.985198	−	0.171418i	$-0.945165\pi$
$619$		−	8.52968i		−	0.342837i	0.985198	−	0.171418i	$-0.0548350\pi$
$620$	−37.1715			−1.49284
$621$		0			0
$622$		−	38.1404i		−	1.52929i
$623$	−17.2731			−0.692032
$624$		0			0
$625$	13.7284			0.549137
$626$		−	42.9599i		−	1.71702i
$627$		0			0
$628$	16.7567			0.668666
$629$		−	25.7990i		−	1.02867i
$630$		0			0
$631$			2.50848i			0.0998612i	0.998753	+	0.0499306i	$0.0159000\pi$
$631$			2.50848i			0.0998612i	−0.998753	+	0.0499306i	$0.984100\pi$
$632$		−	25.1328i		−	0.999729i
$633$		0			0
$634$	−26.3471			−1.04638
$635$		−	6.75386i		−	0.268019i
$636$		0			0
$637$	16.3152	+	14.3977i	0.646433	+	0.570457i
$638$	13.8850			0.549712
$639$		0			0
$640$	16.1902			0.639974
$641$	−41.3383			−1.63277			−0.816383	−	0.577511i	$-0.804024\pi$
$641$	−41.3383			−1.63277			−0.816383	+	0.577511i	$0.804024\pi$
$642$		0			0
$643$			20.4206i			0.805310i	0.915352	+	0.402655i	$0.131913\pi$
$643$			20.4206i			0.805310i	−0.915352	+	0.402655i	$0.868087\pi$
$644$		−	18.8308i		−	0.742039i
$645$		0			0
$646$	−95.2729			−3.74846
$647$	15.7134			0.617756			0.308878	−	0.951102i	$-0.400047\pi$
$647$	15.7134			0.617756			0.308878	+	0.951102i	$0.400047\pi$
$648$		0			0
$649$	−9.15445			−0.359344
$650$	−24.8611	+	28.1722i	−0.975131	+	1.10500i
$651$		0			0
$652$			42.7704i			1.67502i
$653$	−32.2336			−1.26140			−0.630700	−	0.776027i	$-0.717232\pi$
$653$	−32.2336			−1.26140			−0.630700	+	0.776027i	$0.717232\pi$
$654$		0			0
$655$		−	16.7565i		−	0.654731i
$656$			13.3778i			0.522314i
$657$		0			0
$658$			10.7454i			0.418900i
$659$	−4.66757			−0.181823			−0.0909115	−	0.995859i	$-0.528978\pi$
$659$	−4.66757			−0.181823			−0.0909115	+	0.995859i	$0.528978\pi$
$660$		0			0
$661$		−	22.5788i		−	0.878215i	−0.898434	−	0.439108i	$-0.855295\pi$
$661$		−	22.5788i		−	0.878215i	0.898434	−	0.439108i	$-0.144705\pi$
$662$	78.4784			3.05015
$663$		0			0
$664$	29.2380			1.13466
$665$		−	6.29753i		−	0.244208i
$666$		0			0
$667$	−9.00000			−0.348481
$668$		−	54.6421i		−	2.11417i
$669$		0			0
$670$		−	8.02756i		−	0.310132i
$671$			15.9314i			0.615025i
$672$		0			0
$673$	−3.79929			−0.146452			−0.0732259	−	0.997315i	$-0.523329\pi$
$673$	−3.79929			−0.146452			−0.0732259	+	0.997315i	$0.523329\pi$
$674$			60.1893i			2.31841i
$675$		0			0
$676$	−6.68945	−	53.3625i	−0.257287	−	2.05240i
$677$	−29.8319			−1.14653			−0.573267	−	0.819369i	$-0.694324\pi$
$677$	−29.8319			−1.14653			−0.573267	+	0.819369i	$0.694324\pi$
$678$		0			0
$679$	−0.241985			−0.00928655
$680$	−25.1966			−0.966247
$681$		0			0
$682$			72.1142i			2.76140i
$683$			0.0316640i			0.00121159i	1.00000		0.000605795i	$0.000192831\pi$
$683$			0.0316640i			0.00121159i	−1.00000		0.000605795i	$0.999807\pi$
$684$		0			0
$685$	−16.0880			−0.614690
$686$	31.7209			1.21111
$687$		0			0
$688$	23.7663			0.906080
$689$	17.0546	+	15.0501i	0.649727	+	0.573363i
$690$		0			0
$691$		−	7.91140i		−	0.300964i	−0.988613	−	0.150482i	$-0.951917\pi$
$691$		−	7.91140i		−	0.300964i	0.988613	−	0.150482i	$-0.0480826\pi$
$692$	−47.2782			−1.79725
$693$		0			0
$694$			9.47431i			0.359640i
$695$			4.32141i			0.163921i
$696$		0			0
$697$			14.7684i			0.559394i
$698$	58.2972			2.20658
$699$		0			0
$700$			17.0949i			0.646127i
$701$	35.2396			1.33098			0.665491	−	0.746406i	$-0.268222\pi$
$701$	35.2396			1.33098			0.665491	+	0.746406i	$0.268222\pi$
$702$		0			0
$703$	34.7488			1.31058
$704$		−	17.9039i		−	0.674778i
$705$		0			0
$706$	58.4481			2.19972
$707$			4.68990i			0.176382i
$708$		0			0
$709$		−	22.4278i		−	0.842293i	−0.906993	−	0.421147i	$-0.861628\pi$
$709$		−	22.4278i		−	0.842293i	0.906993	−	0.421147i	$-0.138372\pi$
$710$			14.7779i			0.554605i
$711$		0			0
$712$	−93.0849			−3.48851
$713$		−	46.7431i		−	1.75054i
$714$		0			0
$715$	−6.94900	−	6.13227i	−0.259878	−	0.229334i
$716$	−0.622571			−0.0232666
$717$		0			0
$718$	−0.546177			−0.0203831
$719$	−37.6708			−1.40488			−0.702442	−	0.711741i	$-0.747907\pi$
$719$	−37.6708			−1.40488			−0.702442	+	0.711741i	$0.747907\pi$
$720$		0			0
$721$			3.37284i			0.125611i
$722$		−	81.2552i		−	3.02401i
$723$		0			0
$724$	91.1030			3.38582
$725$	8.17033			0.303438
$726$		0			0
$727$	44.0135			1.63237			0.816185	−	0.577791i	$-0.196085\pi$
$727$	44.0135			1.63237			0.816185	+	0.577791i	$0.196085\pi$
$728$	−14.0585	−	12.4062i	−0.521042	−	0.459803i
$729$		0			0
$730$			20.6092i			0.762779i
$731$	26.2368			0.970405
$732$		0			0
$733$			17.6349i			0.651358i	0.945480	+	0.325679i	$0.105593\pi$
$733$			17.6349i			0.651358i	−0.945480	+	0.325679i	$0.894407\pi$
$734$		−	50.8187i		−	1.87575i
$735$		0			0
$736$		−	6.50282i		−	0.239697i
$737$	−10.4984			−0.386713
$738$		0			0
$739$			13.4319i			0.494100i	0.969003	+	0.247050i	$0.0794612\pi$
$739$			13.4319i			0.494100i	−0.969003	+	0.247050i	$0.920539\pi$
$740$	17.7910			0.654011
$741$		0			0
$742$	15.3518			0.563583
$743$			32.7872i			1.20285i	0.798930	+	0.601423i	$0.205399\pi$
$743$			32.7872i			1.20285i	−0.798930	+	0.601423i	$0.794601\pi$
$744$		0			0
$745$	−10.9272			−0.400341
$746$			26.8911i			0.984555i
$747$		0			0
$748$			63.7923i			2.33248i
$749$			2.29939i			0.0840178i
$750$		0			0
$751$	24.9832			0.911648			0.455824	−	0.890070i	$-0.349345\pi$
$751$	24.9832			0.911648			0.455824	+	0.890070i	$0.349345\pi$
$752$			21.3731i			0.779397i
$753$		0			0
$754$	−11.4788	+	13.0076i	−0.418034	+	0.473711i
$755$	4.95897			0.180475
$756$		0			0
$757$	21.6436			0.786652			0.393326	−	0.919399i	$-0.371324\pi$
$757$	21.6436			0.786652			0.393326	+	0.919399i	$0.371324\pi$
$758$	48.7992			1.77247
$759$		0			0
$760$		−	33.9375i		−	1.23104i
$761$		−	19.9359i		−	0.722675i	−0.932435	−	0.361338i	$-0.882320\pi$
$761$		−	19.9359i		−	0.722675i	0.932435	−	0.361338i	$-0.117680\pi$
$762$		0			0
$763$	10.0322			0.363190
$764$	82.5623			2.98700
$765$		0			0
$766$	50.3921			1.82074
$767$	7.56806	−	8.57601i	0.273267	−	0.309662i
$768$		0			0
$769$		−	38.0071i		−	1.37057i	−0.728275	−	0.685285i	$-0.759678\pi$
$769$		−	38.0071i		−	1.37057i	0.728275	−	0.685285i	$-0.240322\pi$
$770$	−6.25520			−0.225422
$771$		0			0
$772$			41.5851i			1.49668i
$773$		−	36.5619i		−	1.31504i	−0.753437	−	0.657520i	$-0.771605\pi$
$773$		−	36.5619i		−	1.31504i	0.753437	−	0.657520i	$-0.228395\pi$
$774$		0			0
$775$			42.4341i			1.52428i
$776$	−1.30406			−0.0468132
$777$		0			0
$778$			53.0041i			1.90029i
$779$	−19.8917			−0.712693
$780$		0			0
$781$	19.3264			0.691553
$782$		−	61.3392i		−	2.19348i
$783$		0			0
$784$	29.2117			1.04328
$785$			3.60793i			0.128772i
$786$		0			0
$787$		−	1.61066i		−	0.0574138i	−0.999588	−	0.0287069i	$-0.990861\pi$
$787$		−	1.61066i		−	0.0574138i	0.999588	−	0.0287069i	$-0.00913894\pi$
$788$		−	16.3291i		−	0.581700i
$789$		0			0
$790$	10.4760			0.372720
$791$			1.18916i			0.0422817i
$792$		0			0
$793$	−14.9247	−	13.1706i	−0.529993	−	0.467702i
$794$	68.8672			2.44401
$795$		0			0
$796$	−47.4708			−1.68256
$797$	0.0983260			0.00348289			0.00174144	−	0.999998i	$-0.499446\pi$
$797$	0.0983260			0.00348289			0.00174144	+	0.999998i	$0.499446\pi$
$798$		0			0
$799$			23.5949i			0.834728i
$800$			5.90335i			0.208715i
$801$		0			0
$802$	37.6201			1.32841
$803$	26.9525			0.951132
$804$		0			0
$805$	4.05451			0.142903
$806$	−67.5575	−	59.6174i	−2.37961	−	2.09993i
$807$		0			0
$808$			25.2739i			0.889134i
$809$	33.5971			1.18121			0.590605	−	0.806961i	$-0.298889\pi$
$809$	33.5971			1.18121			0.590605	+	0.806961i	$0.298889\pi$
$810$		0			0
$811$			13.9011i			0.488134i	0.969758	+	0.244067i	$0.0784817\pi$
$811$			13.9011i			0.488134i	−0.969758	+	0.244067i	$0.921518\pi$
$812$			7.89305i			0.276992i
$813$		0			0
$814$		−	34.5153i		−	1.20976i
$815$	−9.20898			−0.322577
$816$		0			0
$817$			35.3385i			1.23634i
$818$	5.91002			0.206639
$819$		0			0
$820$	−10.1843			−0.355651
$821$			41.9370i			1.46361i	0.681513	+	0.731806i	$0.261322\pi$
$821$			41.9370i			1.46361i	−0.681513	+	0.731806i	$0.738678\pi$
$822$		0			0
$823$	44.9120			1.56553			0.782766	−	0.622316i	$-0.213808\pi$
$823$	44.9120			1.56553			0.782766	+	0.622316i	$0.213808\pi$
$824$			18.1763i			0.633201i
$825$		0			0
$826$		−	7.71977i		−	0.268605i
$827$		−	17.5703i		−	0.610980i	−0.952195	−	0.305490i	$-0.901180\pi$
$827$		−	17.5703i		−	0.610980i	0.952195	−	0.305490i	$-0.0988202\pi$
$828$		0			0
$829$	0.797445			0.0276964			0.0138482	−	0.999904i	$-0.495592\pi$
$829$	0.797445			0.0276964			0.0138482	+	0.999904i	$0.495592\pi$
$830$			12.1872i			0.423024i
$831$		0			0
$832$	16.7726	+	14.8013i	0.581485	+	0.513142i
$833$	32.2484			1.11734
$834$		0			0
$835$	11.7651			0.407148
$836$	−85.9222			−2.97168
$837$		0			0
$838$			82.4710i			2.84891i
$839$			16.7342i			0.577728i	0.957370	+	0.288864i	$0.0932775\pi$
$839$			16.7342i			0.577728i	−0.957370	+	0.288864i	$0.906722\pi$
$840$		0			0
$841$	−25.2276			−0.869917
$842$	60.2102			2.07498
$843$		0			0
$844$	−45.3661			−1.56157
$845$	11.4896	−	1.44032i	0.395254	−	0.0495485i
$846$		0			0
$847$		−	2.62513i		−	0.0902006i
$848$	30.5354			1.04859
$849$		0			0
$850$			55.6846i			1.90997i
$851$			22.3722i			0.766909i
$852$		0			0
$853$			22.7935i			0.780435i	0.920723	+	0.390217i	$0.127600\pi$
$853$			22.7935i			0.780435i	−0.920723	+	0.390217i	$0.872400\pi$
$854$	−13.4346			−0.459724
$855$		0			0
$856$			12.3914i			0.423531i
$857$	43.5246			1.48677			0.743386	−	0.668862i	$-0.233218\pi$
$857$	43.5246			1.48677			0.743386	+	0.668862i	$0.233218\pi$
$858$		0			0
$859$	−17.0382			−0.581337			−0.290668	−	0.956824i	$-0.593878\pi$
$859$	−17.0382			−0.581337			−0.290668	+	0.956824i	$0.593878\pi$
$860$			18.0929i			0.616964i
$861$		0			0
$862$	30.0768			1.02442
$863$			47.3664i			1.61237i	0.591663	+	0.806186i	$0.298472\pi$
$863$			47.3664i			1.61237i	−0.591663	+	0.806186i	$0.701528\pi$
$864$		0			0
$865$		−	10.1796i		−	0.346116i
$866$		−	32.8810i		−	1.11734i
$867$		0			0
$868$	−40.9940			−1.39143
$869$		−	13.7005i		−	0.464756i
$870$		0			0
$871$	8.67909	−	9.83502i	0.294080	−	0.333247i
$872$	54.0637			1.83083
$873$		0			0
$874$	82.6180			2.79460
$875$	−8.05571			−0.272333
$876$		0			0
$877$			46.7266i			1.57784i	0.614493	+	0.788922i	$0.289361\pi$
$877$			46.7266i			1.57784i	−0.614493	+	0.788922i	$0.710639\pi$
$878$		−	32.4561i		−	1.09534i
$879$		0			0
$880$	−12.4419			−0.419416
$881$	−15.7253			−0.529797			−0.264899	−	0.964276i	$-0.585338\pi$
$881$	−15.7253			−0.529797			−0.264899	+	0.964276i	$0.585338\pi$
$882$		0			0
$883$	−24.3745			−0.820269			−0.410134	−	0.912025i	$-0.634518\pi$
$883$	−24.3745			−0.820269			−0.410134	+	0.912025i	$0.634518\pi$
$884$	−59.7615	−	52.7376i	−2.01000	−	1.77376i
$885$		0			0
$886$		−	85.3151i		−	2.86622i
$887$	−3.92002			−0.131621			−0.0658107	−	0.997832i	$-0.520963\pi$
$887$	−3.92002			−0.131621			−0.0658107	+	0.997832i	$0.520963\pi$
$888$		0			0
$889$		−	7.44839i		−	0.249811i
$890$		−	38.8003i		−	1.30059i
$891$		0			0
$892$		−	53.1279i		−	1.77885i
$893$	−31.7801			−1.06348
$894$		0			0
$895$		−	0.134047i		−	0.00448070i
$896$	17.8551			0.596498
$897$		0			0
$898$	13.1887			0.440112
$899$			19.5926i			0.653450i
$900$		0			0
$901$	33.7097			1.12303
$902$			19.7580i			0.657869i
$903$		0			0
$904$			6.40841i			0.213141i
$905$			19.6156i			0.652044i
$906$		0			0
$907$	43.5593			1.44636			0.723181	−	0.690658i	$-0.242679\pi$
$907$	43.5593			1.44636			0.723181	+	0.690658i	$0.242679\pi$
$908$		−	97.7044i		−	3.24243i
$909$		0			0
$910$	5.17123	−	5.85996i	0.171424	−	0.194256i
$911$	−28.1289			−0.931953			−0.465976	−	0.884797i	$-0.654297\pi$
$911$	−28.1289			−0.931953			−0.465976	+	0.884797i	$0.654297\pi$
$912$		0			0
$913$	15.9383			0.527481
$914$	30.6926			1.01522
$915$		0			0
$916$		−	51.4545i		−	1.70011i
$917$		−	18.4797i		−	0.610252i
$918$		0			0
$919$	−17.9402			−0.591793			−0.295897	−	0.955220i	$-0.595618\pi$
$919$	−17.9402			−0.591793			−0.295897	+	0.955220i	$0.595618\pi$
$920$	21.8498			0.720367
$921$		0			0
$922$	32.2190			1.06108
$923$	−15.9773	+	18.1052i	−0.525899	+	0.595941i
$924$		0			0
$925$		−	20.3098i		−	0.667782i
$926$	−15.9153			−0.523008
$927$		0			0
$928$			2.72569i			0.0894752i
$929$			43.0278i			1.41170i	0.708363	+	0.705849i	$0.249434\pi$
$929$			43.0278i			1.41170i	−0.708363	+	0.705849i	$0.750566\pi$
$930$		0			0
$931$			43.4355i			1.42354i
$932$	77.7005			2.54516
$933$		0			0
$934$			34.5823i			1.13157i
$935$	−13.7353			−0.449191
$936$		0			0
$937$	20.5616			0.671719			0.335860	−	0.941912i	$-0.390973\pi$
$937$	20.5616			0.671719			0.335860	+	0.941912i	$0.390973\pi$
$938$		−	8.85307i		−	0.289063i
$939$		0			0
$940$	−16.2711			−0.530704
$941$		−	36.0935i		−	1.17662i	−0.808637	−	0.588308i	$-0.799794\pi$
$941$		−	36.0935i		−	1.17662i	0.808637	−	0.588308i	$-0.200206\pi$
$942$		0			0
$943$		−	12.8068i		−	0.417046i
$944$		−	15.3550i		−	0.499761i
$945$		0			0
$946$	35.1011			1.14123
$947$		−	6.14337i		−	0.199633i	−0.995006	−	0.0998163i	$-0.968174\pi$
$947$		−	6.14337i		−	0.199633i	0.995006	−	0.0998163i	$-0.0318255\pi$
$948$		0			0
$949$	−22.2818	+	25.2494i	−0.723299	+	0.819631i
$950$	−75.0018			−2.43338
$951$		0			0
$952$	−27.7877			−0.900605
$953$	19.7667			0.640305			0.320152	−	0.947366i	$-0.396266\pi$
$953$	19.7667			0.640305			0.320152	+	0.947366i	$0.396266\pi$
$954$		0			0
$955$			17.7767i			0.575240i
$956$			81.8413i			2.64694i
$957$		0			0
$958$	−85.7765			−2.77131
$959$	−17.7424			−0.572931
$960$		0			0
$961$	−70.7578			−2.28251
$962$	32.3344	+	28.5341i	1.04250	+	0.919976i
$963$		0			0
$964$		−	47.1267i		−	1.51785i
$965$	−8.95378			−0.288232
$966$		0			0
$967$		−	30.5215i		−	0.981506i	−0.871299	−	0.490753i	$-0.836722\pi$
$967$		−	30.5215i		−	0.981506i	0.871299	−	0.490753i	$-0.163278\pi$
$968$		−	14.1469i		−	0.454698i
$969$		0			0
$970$		−	0.543569i		−	0.0174529i
$971$	−4.05102			−0.130004			−0.0650018	−	0.997885i	$-0.520705\pi$
$971$	−4.05102			−0.130004			−0.0650018	+	0.997885i	$0.520705\pi$
$972$		0			0
$973$			4.76581i			0.152785i
$974$	14.3388			0.459445
$975$		0			0
$976$	−26.7221			−0.855353
$977$		−	29.0337i		−	0.928872i	−0.885607	−	0.464436i	$-0.846257\pi$
$977$		−	29.0337i		−	0.928872i	0.885607	−	0.464436i	$-0.153743\pi$
$978$		0			0
$979$	−50.7427			−1.62174
$980$			22.2385i			0.710382i
$981$		0			0
$982$		−	61.7447i		−	1.97035i
$983$			6.03911i			0.192618i	0.995352	+	0.0963088i	$0.0307036\pi$
$983$			6.03911i			0.192618i	−0.995352	+	0.0963088i	$0.969296\pi$
$984$		0			0
$985$	3.51585			0.112024
$986$			25.7106i			0.818793i
$987$		0			0
$988$	71.0325	−	80.4930i	2.25985	−	2.56082i
$989$	−22.7519			−0.723467
$990$		0			0
$991$	−49.4199			−1.56987			−0.784937	−	0.619575i	$-0.787305\pi$
$991$	−49.4199			−1.56987			−0.784937	+	0.619575i	$0.787305\pi$
$992$	−14.1564			−0.449465
$993$		0			0
$994$			16.2976i			0.516928i
$995$		−	10.2210i		−	0.324029i
$996$		0			0
$997$	−21.2348			−0.672514			−0.336257	−	0.941770i	$-0.609161\pi$
$997$	−21.2348			−0.672514			−0.336257	+	0.941770i	$0.609161\pi$
$998$	98.6419			3.12246
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.b.j.649.10		10
3.2	odd	2		1053.2.b.i.649.1		10
9.2	odd	6		351.2.t.c.64.1		20
9.4	even	3		117.2.t.c.25.1	✓	20
9.5	odd	6		351.2.t.c.181.10		20
9.7	even	3		117.2.t.c.103.10	yes	20
13.12	even	2	inner	1053.2.b.j.649.1		10
39.38	odd	2		1053.2.b.i.649.10		10
117.25	even	6		117.2.t.c.103.1	yes	20
117.38	odd	6		351.2.t.c.64.10		20
117.77	odd	6		351.2.t.c.181.1		20
117.103	even	6		117.2.t.c.25.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.1	✓	20	9.4	even	3
117.2.t.c.25.10	yes	20	117.103	even	6
117.2.t.c.103.1	yes	20	117.25	even	6
117.2.t.c.103.10	yes	20	9.7	even	3
351.2.t.c.64.1		20	9.2	odd	6
351.2.t.c.64.10		20	117.38	odd	6
351.2.t.c.181.1		20	117.77	odd	6
351.2.t.c.181.10		20	9.5	odd	6
1053.2.b.i.649.1		10	3.2	odd	2
1053.2.b.i.649.10		10	39.38	odd	2
1053.2.b.j.649.1		10	13.12	even	2	inner
1053.2.b.j.649.10		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.47728i q^{2} -4.13693 q^{4} -0.890732i q^{5} -0.982330i q^{7} -5.29379i q^{8} +2.20660 q^{10} -2.88577i q^{11} +(2.70342 + 2.38569i) q^{13} +2.43351 q^{14} +4.84036 q^{16} +5.34353 q^{17} +7.19723i q^{19} +3.68490i q^{20} +7.14886 q^{22} -4.63376 q^{23} +4.20660 q^{25} +(-5.91002 + 6.69715i) q^{26} +4.06384i q^{28} +1.94227 q^{29} +10.0875i q^{31} +1.40336i q^{32} +13.2374i q^{34} -0.874993 q^{35} -4.82809i q^{37} -17.8296 q^{38} -4.71535 q^{40} +2.76379i q^{41} +4.91002 q^{43} +11.9382i q^{44} -11.4791i q^{46} +4.41560i q^{47} +6.03503 q^{49} +10.4209i q^{50} +(-11.1839 - 9.86943i) q^{52} +6.30850 q^{53} -2.57044 q^{55} -5.20025 q^{56} +4.81154i q^{58} -3.17228i q^{59} -5.52068 q^{61} -24.9896 q^{62} +6.20421 q^{64} +(2.12501 - 2.40803i) q^{65} -3.63799i q^{67} -22.1058 q^{68} -2.16761i q^{70} +6.69715i q^{71} +9.33980i q^{73} +11.9605 q^{74} -29.7745i q^{76} -2.83478 q^{77} +4.74760 q^{79} -4.31146i q^{80} -6.84670 q^{82} +5.52308i q^{83} -4.75965i q^{85} +12.1635i q^{86} -15.2767 q^{88} -17.5838i q^{89} +(2.34353 - 2.65566i) q^{91} +19.1696 q^{92} -10.9387 q^{94} +6.41080 q^{95} -0.246338i q^{97} +14.9505i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more