LMFDB - Embedded newform 1040.2.f.g.129.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,2,Mod(129,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1040.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1040.f (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:	$8.30444181021$
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} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 $ x^10 + 11x^8 + 36x^6 + 42x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$0.549054i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.g.129.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.57201i q^{3} +(2.14575 + 0.629081i) q^{5} +4.19743 q^{7} +0.528799 q^{9} +O(q^{10})$	$q-1.57201i q^{3} +(2.14575 + 0.629081i) q^{5} +4.19743 q^{7} +0.528799 q^{9} -6.07284i q^{11} +(-2.66347 - 2.43021i) q^{13} +(0.988918 - 3.37314i) q^{15} +6.59839i q^{17} +4.12781i q^{19} -6.59839i q^{21} -0.313844i q^{23} +(4.20852 + 2.69970i) q^{25} -5.54729i q^{27} +1.50663 q^{29} -0.732612i q^{31} -9.54654 q^{33} +(9.00666 + 2.64052i) q^{35} -6.19743 q^{37} +(-3.82031 + 4.18699i) q^{39} +1.19898i q^{41} +8.62864i q^{43} +(1.13467 + 0.332657i) q^{45} -4.19743 q^{47} +10.6184 q^{49} +10.3727 q^{51} +3.54308i q^{53} +(3.82031 - 13.0308i) q^{55} +6.48894 q^{57} -6.55264i q^{59} -1.69478 q^{61} +2.21960 q^{63} +(-4.18635 - 6.89017i) q^{65} -6.21960 q^{67} -0.493365 q^{69} -5.23809i q^{71} +4.21960 q^{73} +(4.24395 - 6.61581i) q^{75} -25.4903i q^{77} -11.5963 q^{79} -7.13398 q^{81} -8.29046 q^{83} +(-4.15092 + 14.1585i) q^{85} -2.36844i q^{87} -13.4925i q^{89} +(-11.1797 - 10.2006i) q^{91} -1.15167 q^{93} +(-2.59673 + 8.85726i) q^{95} +10.2693 q^{97} -3.21131i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$.

$n$	$261$	$417$	$561$	$911$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.57201i		−	0.907598i	−0.891104	−	0.453799i	$-0.850068\pi$
$3$		−	1.57201i		−	0.907598i	0.891104	−	0.453799i	$-0.149932\pi$
$4$		0			0
$5$	2.14575	+	0.629081i	0.959610	+	0.281333i
$5$	2.14575	+	0.629081i	0.959610	+	0.281333i
$6$		0			0
$7$	4.19743			1.58648			0.793240	−	0.608909i	$-0.208392\pi$
$7$	4.19743			1.58648			0.793240	+	0.608909i	$0.208392\pi$
$8$		0			0
$9$	0.528799			0.176266
$10$		0			0
$11$		−	6.07284i		−	1.83103i	−0.402284	−	0.915515i	$-0.631783\pi$
$11$		−	6.07284i		−	1.83103i	0.402284	−	0.915515i	$-0.368217\pi$
$12$		0			0
$13$	−2.66347	−	2.43021i	−0.738714	−	0.674019i
$13$	−2.66347	−	2.43021i	−0.738714	−	0.674019i
$14$		0			0
$15$	0.988918	−	3.37314i	0.255338	−	0.870940i
$16$		0			0
$17$			6.59839i			1.60034i	0.599770	+	0.800172i	$0.295259\pi$
$17$			6.59839i			1.60034i	−0.599770	+	0.800172i	$0.704741\pi$
$18$		0			0
$19$			4.12781i			0.946985i	0.880798	+	0.473492i	$0.157007\pi$
$19$			4.12781i			0.946985i	−0.880798	+	0.473492i	$0.842993\pi$
$20$		0			0
$21$		−	6.59839i		−	1.43989i
$22$		0			0
$23$		−	0.313844i		−	0.0654411i	−0.999465	−	0.0327205i	$-0.989583\pi$
$23$		−	0.313844i		−	0.0654411i	0.999465	−	0.0327205i	$-0.0104171\pi$
$24$		0			0
$25$	4.20852	+	2.69970i	0.841703	+	0.539941i
$26$		0			0
$27$		−	5.54729i		−	1.06758i
$28$		0			0
$29$	1.50663			0.279775			0.139888	−	0.990167i	$-0.455326\pi$
$29$	1.50663			0.279775			0.139888	+	0.990167i	$0.455326\pi$
$30$		0			0
$31$		−	0.732612i		−	0.131581i	−0.997833	−	0.0657905i	$-0.979043\pi$
$31$		−	0.732612i		−	0.131581i	0.997833	−	0.0657905i	$-0.0209569\pi$
$32$		0			0
$33$	−9.54654			−1.66184
$34$		0			0
$35$	9.00666	+	2.64052i	1.52240	+	0.446330i
$36$		0			0
$37$	−6.19743			−1.01885			−0.509426	−	0.860515i	$-0.670142\pi$
$37$	−6.19743			−1.01885			−0.509426	+	0.860515i	$0.670142\pi$
$38$		0			0
$39$	−3.82031	+	4.18699i	−0.611738	+	0.670455i
$40$		0			0
$41$			1.19898i			0.187249i	0.995608	+	0.0936247i	$0.0298454\pi$
$41$			1.19898i			0.187249i	−0.995608	+	0.0936247i	$0.970155\pi$
$42$		0			0
$43$			8.62864i			1.31586i	0.753081	+	0.657928i	$0.228567\pi$
$43$			8.62864i			1.31586i	−0.753081	+	0.657928i	$0.771433\pi$
$44$		0			0
$45$	1.13467	+	0.332657i	0.169147	+	0.0495896i
$46$		0			0
$47$	−4.19743			−0.612259			−0.306129	−	0.951990i	$-0.599034\pi$
$47$	−4.19743			−0.612259			−0.306129	+	0.951990i	$0.599034\pi$
$48$		0			0
$49$	10.6184			1.51692
$50$		0			0
$51$	10.3727			1.45247
$52$		0			0
$53$			3.54308i			0.486680i	0.969941	+	0.243340i	$0.0782430\pi$
$53$			3.54308i			0.486680i	−0.969941	+	0.243340i	$0.921757\pi$
$54$		0			0
$55$	3.82031	−	13.0308i	0.515130	−	1.75707i
$56$		0			0
$57$	6.48894			0.859481
$58$		0			0
$59$		−	6.55264i		−	0.853082i	−0.904468	−	0.426541i	$-0.859732\pi$
$59$		−	6.55264i		−	0.853082i	0.904468	−	0.426541i	$-0.140268\pi$
$60$		0			0
$61$	−1.69478			−0.216995			−0.108497	−	0.994097i	$-0.534604\pi$
$61$	−1.69478			−0.216995			−0.108497	+	0.994097i	$0.534604\pi$
$62$		0			0
$63$	2.21960			0.279643
$64$		0			0
$65$	−4.18635	−	6.89017i	−0.519253	−	0.854621i
$66$		0			0
$67$	−6.21960			−0.759845			−0.379922	−	0.925018i	$-0.624049\pi$
$67$	−6.21960			−0.759845			−0.379922	+	0.925018i	$0.624049\pi$
$68$		0			0
$69$	−0.493365			−0.0593942
$70$		0			0
$71$		−	5.23809i		−	0.621647i	−0.950468	−	0.310823i	$-0.899395\pi$
$71$		−	5.23809i		−	0.621647i	0.950468	−	0.310823i	$-0.100605\pi$
$72$		0			0
$73$	4.21960			0.493866			0.246933	−	0.969032i	$-0.420577\pi$
$73$	4.21960			0.493866			0.246933	+	0.969032i	$0.420577\pi$
$74$		0			0
$75$	4.24395	−	6.61581i	0.490049	−	0.763928i
$76$		0			0
$77$		−	25.4903i		−	2.90489i
$78$		0			0
$79$	−11.5963			−1.30468			−0.652342	−	0.757925i	$-0.726213\pi$
$79$	−11.5963			−1.30468			−0.652342	+	0.757925i	$0.726213\pi$
$80$		0			0
$81$	−7.13398			−0.792664
$82$		0			0
$83$	−8.29046			−0.909997			−0.454998	−	0.890492i	$-0.650360\pi$
$83$	−8.29046			−0.909997			−0.454998	+	0.890492i	$0.650360\pi$
$84$		0			0
$85$	−4.15092	+	14.1585i	−0.450230	+	1.53571i
$86$		0			0
$87$		−	2.36844i		−	0.253923i
$88$		0			0
$89$		−	13.4925i		−	1.43021i	−0.699019	−	0.715103i	$-0.746380\pi$
$89$		−	13.4925i		−	1.43021i	0.699019	−	0.715103i	$-0.253620\pi$
$90$		0			0
$91$	−11.1797	−	10.2006i	−1.17196	−	1.06932i
$92$		0			0
$93$	−1.15167			−0.119423
$94$		0			0
$95$	−2.59673	+	8.85726i	−0.266418	+	0.908736i
$96$		0			0
$97$	10.2693			1.04269			0.521347	−	0.853345i	$-0.325430\pi$
$97$	10.2693			1.04269			0.521347	+	0.853345i	$0.325430\pi$
$98$		0			0
$99$		−	3.21131i		−	0.322749i
$100$		0			0
$101$	15.9203			1.58413			0.792064	−	0.610438i	$-0.209007\pi$
$101$	15.9203			1.58413			0.792064	+	0.610438i	$0.209007\pi$
$102$		0			0
$103$			2.97805i			0.293436i	0.989178	+	0.146718i	$0.0468710\pi$
$103$			2.97805i			0.293436i	−0.989178	+	0.146718i	$0.953129\pi$
$104$		0			0
$105$	4.15092	−	14.1585i	0.405088	−	1.38173i
$106$		0			0
$107$			4.30721i			0.416393i	0.978087	+	0.208197i	$0.0667594\pi$
$107$			4.30721i			0.416393i	−0.978087	+	0.208197i	$0.933241\pi$
$108$		0			0
$109$		−	17.2670i		−	1.65388i	−0.562289	−	0.826941i	$-0.690079\pi$
$109$		−	17.2670i		−	1.65388i	0.562289	−	0.826941i	$-0.309921\pi$
$110$		0			0
$111$			9.74240i			0.924708i
$112$		0			0
$113$		−	0.0563961i		−	0.00530530i	−0.999996	−	0.00265265i	$-0.999156\pi$
$113$		−	0.0563961i		−	0.00530530i	0.999996	−	0.00265265i	$-0.000844366\pi$
$114$		0			0
$115$	0.197433	−	0.673433i	0.0184108	−	0.0627979i
$116$		0			0
$117$	−1.40844	−	1.28509i	−0.130210	−	0.118807i
$118$		0			0
$119$			27.6963i			2.53892i
$120$		0			0
$121$	−25.8794			−2.35267
$122$		0			0
$123$	1.88481			0.169947
$124$		0			0
$125$	7.33210	+	8.44039i	0.655803	+	0.754932i
$126$		0			0
$127$			16.2932i			1.44579i	0.690960	+	0.722893i	$0.257188\pi$
$127$			16.2932i			1.44579i	−0.690960	+	0.722893i	$0.742812\pi$
$128$		0			0
$129$	13.5643			1.19427
$130$		0			0
$131$	0.891614			0.0779007			0.0389504	−	0.999241i	$-0.487599\pi$
$131$	0.891614			0.0779007			0.0389504	+	0.999241i	$0.487599\pi$
$132$		0			0
$133$			17.3262i			1.50237i
$134$		0			0
$135$	3.48969	−	11.9031i	0.300345	−	1.02446i
$136$		0			0
$137$	9.13879			0.780780			0.390390	−	0.920650i	$-0.372340\pi$
$137$	9.13879			0.780780			0.390390	+	0.920650i	$0.372340\pi$
$138$		0			0
$139$	19.9424			1.69150			0.845748	−	0.533582i	$-0.179155\pi$
$139$	19.9424			1.69150			0.845748	+	0.533582i	$0.179155\pi$
$140$		0			0
$141$			6.59839i			0.555685i
$142$		0			0
$143$	−14.7583	+	16.1748i	−1.23415	+	1.35261i
$144$		0			0
$145$	3.23287	+	0.947795i	0.268475	+	0.0787101i
$146$		0			0
$147$		−	16.6923i		−	1.37675i
$148$		0			0
$149$			10.3109i			0.844703i	0.906432	+	0.422351i	$0.138795\pi$
$149$			10.3109i			0.844703i	−0.906432	+	0.422351i	$0.861205\pi$
$150$		0			0
$151$			0.754167i			0.0613732i	0.999529	+	0.0306866i	$0.00976939\pi$
$151$			0.754167i			0.0613732i	−0.999529	+	0.0306866i	$0.990231\pi$
$152$		0			0
$153$			3.48922i			0.282087i
$154$		0			0
$155$	0.460872	−	1.57201i	0.0370181	−	0.126267i
$156$		0			0
$157$			19.3126i			1.54131i	0.637252	+	0.770656i	$0.280071\pi$
$157$			19.3126i			1.54131i	−0.637252	+	0.770656i	$0.719929\pi$
$158$		0			0
$159$	5.56974			0.441709
$160$		0			0
$161$		−	1.31734i		−	0.103821i
$162$		0			0
$163$	−19.3413			−1.51493			−0.757464	−	0.652877i	$-0.773562\pi$
$163$	−19.3413			−1.51493			−0.757464	+	0.652877i	$0.773562\pi$
$164$		0			0
$165$	−20.4845	−	6.00554i	−1.59472	−	0.467531i
$166$		0			0
$167$	−6.57014			−0.508412			−0.254206	−	0.967150i	$-0.581814\pi$
$167$	−6.57014			−0.508412			−0.254206	+	0.967150i	$0.581814\pi$
$168$		0			0
$169$	1.18815	+	12.9456i	0.0913959	+	0.995815i
$170$		0			0
$171$			2.18278i			0.166921i
$172$		0			0
$173$		−	0.110259i		−	0.00838282i	−0.999991	−	0.00419141i	$-0.998666\pi$
$173$		−	0.110259i		−	0.00838282i	0.999991	−	0.00419141i	$-0.00133417\pi$
$174$		0			0
$175$	17.6650	+	11.3318i	1.33535	+	0.856606i
$176$		0			0
$177$	−10.3008			−0.774255
$178$		0			0
$179$	−6.74900			−0.504444			−0.252222	−	0.967669i	$-0.581161\pi$
$179$	−6.74900			−0.504444			−0.252222	+	0.967669i	$0.581161\pi$
$180$		0			0
$181$	−17.3540			−1.28991			−0.644955	−	0.764220i	$-0.723124\pi$
$181$	−17.3540			−1.28991			−0.644955	+	0.764220i	$0.723124\pi$
$182$		0			0
$183$			2.66421i			0.196944i
$184$		0			0
$185$	−13.2982	−	3.89868i	−0.977700	−	0.286637i
$186$		0			0
$187$	40.0710			2.93028
$188$		0			0
$189$		−	23.2844i		−	1.69369i
$190$		0			0
$191$	0.376294			0.0272276			0.0136138	−	0.999907i	$-0.495666\pi$
$191$	0.376294			0.0272276			0.0136138	+	0.999907i	$0.495666\pi$
$192$		0			0
$193$	3.11225			0.224025			0.112012	−	0.993707i	$-0.464270\pi$
$193$	3.11225			0.224025			0.112012	+	0.993707i	$0.464270\pi$
$194$		0			0
$195$	−10.8314	+	6.58097i	−0.775652	+	0.471273i
$196$		0			0
$197$	−1.67310			−0.119204			−0.0596019	−	0.998222i	$-0.518983\pi$
$197$	−1.67310			−0.119204			−0.0596019	+	0.998222i	$0.518983\pi$
$198$		0			0
$199$	−22.9049			−1.62369			−0.811844	−	0.583875i	$-0.801536\pi$
$199$	−22.9049			−1.62369			−0.811844	+	0.583875i	$0.801536\pi$
$200$		0			0
$201$			9.77724i			0.689633i
$202$		0			0
$203$	6.32400			0.443858
$204$		0			0
$205$	−0.754256	+	2.57272i	−0.0526795	+	0.179686i
$206$		0			0
$207$		−	0.165960i		−	0.0115351i
$208$		0			0
$209$	25.0675			1.73396
$210$		0			0
$211$	6.74900			0.464620			0.232310	−	0.972642i	$-0.425372\pi$
$211$	6.74900			0.464620			0.232310	+	0.972642i	$0.425372\pi$
$212$		0			0
$213$	−8.23430			−0.564205
$214$		0			0
$215$	−5.42811	+	18.5149i	−0.370194	+	1.26271i
$216$		0			0
$217$		−	3.07509i		−	0.208751i
$218$		0			0
$219$		−	6.63323i		−	0.448232i
$220$		0			0
$221$	16.0355	−	17.5746i	1.07866	−	1.18220i
$222$		0			0
$223$	−4.96859			−0.332722			−0.166361	−	0.986065i	$-0.553202\pi$
$223$	−4.96859			−0.332722			−0.166361	+	0.986065i	$0.553202\pi$
$224$		0			0
$225$	2.22546	+	1.42760i	0.148364	+	0.0951733i
$226$		0			0
$227$	13.4122			0.890197			0.445098	−	0.895482i	$-0.353169\pi$
$227$	13.4122			0.890197			0.445098	+	0.895482i	$0.353169\pi$
$228$		0			0
$229$			13.8906i			0.917917i	0.888458	+	0.458958i	$0.151777\pi$
$229$			13.8906i			0.917917i	−0.888458	+	0.458958i	$0.848223\pi$
$230$		0			0
$231$	−40.0710			−2.63648
$232$		0			0
$233$			21.4083i			1.40250i	0.712914	+	0.701251i	$0.247375\pi$
$233$			21.4083i			1.40250i	−0.712914	+	0.701251i	$0.752625\pi$
$234$		0			0
$235$	−9.00666	−	2.64052i	−0.587530	−	0.172249i
$236$		0			0
$237$			18.2294i			1.18413i
$238$		0			0
$239$			1.52974i			0.0989506i	0.998775	+	0.0494753i	$0.0157549\pi$
$239$			1.52974i			0.0989506i	−0.998775	+	0.0494753i	$0.984245\pi$
$240$		0			0
$241$			17.3262i			1.11608i	0.829814	+	0.558040i	$0.188446\pi$
$241$			17.3262i			1.11608i	−0.829814	+	0.558040i	$0.811554\pi$
$242$		0			0
$243$		−	5.42722i		−	0.348157i
$244$		0			0
$245$	22.7846	+	6.67986i	1.45565	+	0.426761i
$246$		0			0
$247$	10.0315	−	10.9943i	0.638286	−	0.699550i
$248$		0			0
$249$			13.0327i			0.825911i
$250$		0			0
$251$	28.4303			1.79451			0.897254	−	0.441515i	$-0.145559\pi$
$251$	28.4303			1.79451			0.897254	+	0.441515i	$0.145559\pi$
$252$		0			0
$253$	−1.90593			−0.119825
$254$		0			0
$255$	22.2573	+	6.52527i	1.39380	+	0.408628i
$256$		0			0
$257$		−	20.1529i		−	1.25710i	−0.777768	−	0.628552i	$-0.783648\pi$
$257$		−	20.1529i		−	1.25710i	0.777768	−	0.628552i	$-0.216352\pi$
$258$		0			0
$259$	−26.0133			−1.61639
$260$		0			0
$261$	0.796707			0.0493149
$262$		0			0
$263$			6.74975i			0.416207i	0.978107	+	0.208104i	$0.0667292\pi$
$263$			6.74975i			0.416207i	−0.978107	+	0.208104i	$0.933271\pi$
$264$		0			0
$265$	−2.22888	+	7.60258i	−0.136919	+	0.467023i
$266$		0			0
$267$	−21.2104			−1.29805
$268$		0			0
$269$	−27.2015			−1.65851			−0.829253	−	0.558874i	$-0.811233\pi$
$269$	−27.2015			−1.65851			−0.829253	+	0.558874i	$0.811233\pi$
$270$		0			0
$271$			12.4695i			0.757467i	0.925506	+	0.378734i	$0.123640\pi$
$271$			12.4695i			0.757467i	−0.925506	+	0.378734i	$0.876360\pi$
$272$		0			0
$273$	−16.0355	+	17.5746i	−0.970511	+	1.06366i
$274$		0			0
$275$	16.3949	−	25.5576i	0.988648	−	1.54118i
$276$		0			0
$277$		−	10.4518i		−	0.627990i	−0.949425	−	0.313995i	$-0.898332\pi$
$277$		−	10.4518i		−	0.627990i	0.949425	−	0.313995i	$-0.101668\pi$
$278$		0			0
$279$		−	0.387404i		−	0.0231933i
$280$		0			0
$281$		−	4.01107i		−	0.239280i	−0.992817	−	0.119640i	$-0.961826\pi$
$281$		−	4.01107i		−	0.239280i	0.992817	−	0.119640i	$-0.0381741\pi$
$282$		0			0
$283$		−	11.2229i		−	0.667133i	−0.942727	−	0.333567i	$-0.891748\pi$
$283$		−	11.2229i		−	0.667133i	0.942727	−	0.333567i	$-0.108252\pi$
$284$		0			0
$285$	13.9237	+	4.08207i	0.824767	+	0.241801i
$286$		0			0
$287$			5.03264i			0.297068i
$288$		0			0
$289$	−26.5387			−1.56110
$290$		0			0
$291$		−	16.1435i		−	0.946347i
$292$		0			0
$293$	−8.64459			−0.505023			−0.252511	−	0.967594i	$-0.581256\pi$
$293$	−8.64459			−0.505023			−0.252511	+	0.967594i	$0.581256\pi$
$294$		0			0
$295$	4.12214	−	14.0604i	0.240000	−	0.818626i
$296$		0			0
$297$	−33.6878			−1.95476
$298$		0			0
$299$	−0.762708	+	0.835915i	−0.0441086	+	0.0483422i
$300$		0			0
$301$			36.2182i			2.08758i
$302$		0			0
$303$		−	25.0268i		−	1.43775i
$304$		0			0
$305$	−3.63658	−	1.06615i	−0.208230	−	0.0610478i
$306$		0			0
$307$	8.66676			0.494638			0.247319	−	0.968934i	$-0.420450\pi$
$307$	8.66676			0.494638			0.247319	+	0.968934i	$0.420450\pi$
$308$		0			0
$309$	4.68151			0.266322
$310$		0			0
$311$	14.7189			0.834630			0.417315	−	0.908762i	$-0.362971\pi$
$311$	14.7189			0.834630			0.417315	+	0.908762i	$0.362971\pi$
$312$		0			0
$313$		−	15.9051i		−	0.899010i	−0.893278	−	0.449505i	$-0.851600\pi$
$313$		−	15.9051i		−	0.899010i	0.893278	−	0.449505i	$-0.148400\pi$
$314$		0			0
$315$	4.76271	+	1.39631i	0.268348	+	0.0786729i
$316$		0			0
$317$	26.1754			1.47015			0.735077	−	0.677983i	$-0.237146\pi$
$317$	26.1754			1.47015			0.735077	+	0.677983i	$0.237146\pi$
$318$		0			0
$319$		−	9.14955i		−	0.512277i
$320$		0			0
$321$	6.77095			0.377918
$322$		0			0
$323$	−27.2369			−1.51550
$324$		0			0
$325$	−4.64841	−	17.4182i	−0.257847	−	0.966186i
$326$		0			0
$327$	−27.1439			−1.50106
$328$		0			0
$329$	−17.6184			−0.971336
$330$		0			0
$331$			24.0924i			1.32424i	0.749398	+	0.662119i	$0.230343\pi$
$331$			24.0924i			1.32424i	−0.749398	+	0.662119i	$0.769657\pi$
$332$		0			0
$333$	−3.27719			−0.179589
$334$		0			0
$335$	−13.3457	−	3.91263i	−0.729155	−	0.213770i
$336$		0			0
$337$			17.5746i			0.957350i	0.877992	+	0.478675i	$0.158883\pi$
$337$			17.5746i			0.957350i	−0.877992	+	0.478675i	$0.841117\pi$
$338$		0			0
$339$	−0.0886550			−0.00481508
$340$		0			0
$341$	−4.44904			−0.240929
$342$		0			0
$343$	15.1882			0.820085
$344$		0			0
$345$	−1.05864	−	0.310366i	−0.0569953	−	0.0167096i
$346$		0			0
$347$		−	23.5300i		−	1.26316i	−0.775312	−	0.631579i	$-0.782407\pi$
$347$		−	23.5300i		−	1.26316i	0.775312	−	0.631579i	$-0.217593\pi$
$348$		0			0
$349$		−	17.1460i		−	0.917805i	−0.888487	−	0.458903i	$-0.848243\pi$
$349$		−	17.1460i		−	0.917805i	0.888487	−	0.458903i	$-0.151757\pi$
$350$		0			0
$351$	−13.4811	+	14.7750i	−0.719567	+	0.788634i
$352$		0			0
$353$	−35.8514			−1.90818			−0.954088	−	0.299528i	$-0.903171\pi$
$353$	−35.8514			−1.90818			−0.954088	+	0.299528i	$0.903171\pi$
$354$		0			0
$355$	3.29518	−	11.2396i	0.174890	−	0.596538i
$356$		0			0
$357$	43.5387			2.30431
$358$		0			0
$359$		−	10.3125i		−	0.544274i	−0.962258	−	0.272137i	$-0.912270\pi$
$359$		−	10.3125i		−	0.544274i	0.962258	−	0.272137i	$-0.0877305\pi$
$360$		0			0
$361$	1.96118			0.103220
$362$		0			0
$363$			40.6825i			2.13528i
$364$		0			0
$365$	9.05421	+	2.65447i	0.473919	+	0.138941i
$366$		0			0
$367$			1.25987i			0.0657646i	0.999459	+	0.0328823i	$0.0104687\pi$
$367$			1.25987i			0.0657646i	−0.999459	+	0.0328823i	$0.989531\pi$
$368$		0			0
$369$			0.634020i			0.0330057i
$370$		0			0
$371$			14.8718i			0.772108i
$372$		0			0
$373$			18.4713i			0.956409i	0.878249	+	0.478205i	$0.158712\pi$
$373$			18.4713i			0.956409i	−0.878249	+	0.478205i	$0.841288\pi$
$374$		0			0
$375$	13.2683	−	11.5261i	0.685174	−	0.595206i
$376$		0			0
$377$	−4.01288	−	3.66144i	−0.206674	−	0.188574i
$378$		0			0
$379$			23.8858i			1.22693i	0.789721	+	0.613466i	$0.210225\pi$
$379$			23.8858i			1.22693i	−0.789721	+	0.613466i	$0.789775\pi$
$380$		0			0
$381$	25.6130			1.31219
$382$		0			0
$383$	20.9686			1.07145			0.535723	−	0.844394i	$-0.320039\pi$
$383$	20.9686			1.07145			0.535723	+	0.844394i	$0.320039\pi$
$384$		0			0
$385$	16.0355	−	54.6960i	0.817244	−	2.78757i
$386$		0			0
$387$			4.56282i			0.231941i
$388$		0			0
$389$	−25.2812			−1.28181			−0.640904	−	0.767621i	$-0.721441\pi$
$389$	−25.2812			−1.28181			−0.640904	+	0.767621i	$0.721441\pi$
$390$		0			0
$391$	2.07087			0.104728
$392$		0			0
$393$		−	1.40162i		−	0.0707025i
$394$		0			0
$395$	−24.8828	−	7.29500i	−1.25199	−	0.367051i
$396$		0			0
$397$	−8.94643			−0.449008			−0.224504	−	0.974473i	$-0.572076\pi$
$397$	−8.94643			−0.449008			−0.224504	+	0.974473i	$0.572076\pi$
$398$		0			0
$399$	27.2369			1.36355
$400$		0			0
$401$		−	24.1730i		−	1.20714i	−0.797309	−	0.603571i	$-0.793744\pi$
$401$		−	24.1730i		−	1.20714i	0.797309	−	0.603571i	$-0.206256\pi$
$402$		0			0
$403$	−1.78040	+	1.95129i	−0.0886882	+	0.0972007i
$404$		0			0
$405$	−15.3078	−	4.48785i	−0.760648	−	0.223003i
$406$		0			0
$407$			37.6360i			1.86555i
$408$		0			0
$409$		−	1.81971i		−	0.0899791i	−0.998987	−	0.0449895i	$-0.985675\pi$
$409$		−	1.81971i		−	0.0899791i	0.998987	−	0.0449895i	$-0.0143254\pi$
$410$		0			0
$411$		−	14.3662i		−	0.708634i
$412$		0			0
$413$		−	27.5043i		−	1.35340i
$414$		0			0
$415$	−17.7893	−	5.21537i	−0.873242	−	0.256012i
$416$		0			0
$417$		−	31.3496i		−	1.53520i
$418$		0			0
$419$	11.2501			0.549604			0.274802	−	0.961501i	$-0.411388\pi$
$419$	11.2501			0.549604			0.274802	+	0.961501i	$0.411388\pi$
$420$		0			0
$421$			7.02433i			0.342345i	0.985241	+	0.171172i	$0.0547555\pi$
$421$			7.02433i			0.342345i	−0.985241	+	0.171172i	$0.945244\pi$
$422$		0			0
$423$	−2.21960			−0.107921
$424$		0			0
$425$	−17.8137	+	27.7694i	−0.864091	+	1.34701i
$426$		0			0
$427$	−7.11373			−0.344258
$428$		0			0
$429$	25.4269	+	23.2001i	1.22762	+	1.12011i
$430$		0			0
$431$		−	10.7450i		−	0.517567i	−0.965935	−	0.258783i	$-0.916678\pi$
$431$		−	10.7450i		−	0.517567i	0.965935	−	0.258783i	$-0.0833216\pi$
$432$		0			0
$433$		−	8.66873i		−	0.416593i	−0.978066	−	0.208296i	$-0.933208\pi$
$433$		−	8.66873i		−	0.416593i	0.978066	−	0.208296i	$-0.0667919\pi$
$434$		0			0
$435$	1.48994	−	5.08208i	0.0714371	−	0.243667i
$436$		0			0
$437$	1.29549			0.0619717
$438$		0			0
$439$	−14.3595			−0.685340			−0.342670	−	0.939456i	$-0.611331\pi$
$439$	−14.3595			−0.685340			−0.342670	+	0.939456i	$0.611331\pi$
$440$		0			0
$441$	5.61502			0.267382
$442$		0			0
$443$			26.3783i			1.25327i	0.779313	+	0.626634i	$0.215568\pi$
$443$			26.3783i			1.25327i	−0.779313	+	0.626634i	$0.784432\pi$
$444$		0			0
$445$	8.48790	−	28.9517i	0.402365	−	1.37244i
$446$		0			0
$447$	16.2088			0.766650
$448$		0			0
$449$			22.7429i			1.07330i	0.843804	+	0.536651i	$0.180311\pi$
$449$			22.7429i			1.07330i	−0.843804	+	0.536651i	$0.819689\pi$
$450$		0			0
$451$	7.28122			0.342859
$452$		0			0
$453$	1.18555			0.0557022
$454$		0			0
$455$	−17.5719	−	28.9210i	−0.823785	−	1.35584i
$456$		0			0
$457$	5.14764			0.240797			0.120398	−	0.992726i	$-0.461583\pi$
$457$	5.14764			0.240797			0.120398	+	0.992726i	$0.461583\pi$
$458$		0			0
$459$	36.6032			1.70849
$460$		0			0
$461$		−	35.5528i		−	1.65586i	−0.560831	−	0.827931i	$-0.689518\pi$
$461$		−	35.5528i		−	1.65586i	0.560831	−	0.827931i	$-0.310482\pi$
$462$		0			0
$463$	−10.2196			−0.474945			−0.237472	−	0.971394i	$-0.576319\pi$
$463$	−10.2196			−0.474945			−0.237472	+	0.971394i	$0.576319\pi$
$464$		0			0
$465$	−2.47120	−	0.724494i	−0.114599	−	0.0335976i
$466$		0			0
$467$		−	35.0991i		−	1.62419i	−0.583524	−	0.812096i	$-0.698327\pi$
$467$		−	35.0991i		−	1.62419i	0.583524	−	0.812096i	$-0.301673\pi$
$468$		0			0
$469$	−26.1063			−1.20548
$470$		0			0
$471$	30.3595			1.39889
$472$		0			0
$473$	52.4004			2.40937
$474$		0			0
$475$	−11.1439	+	17.3720i	−0.511316	+	0.797080i
$476$		0			0
$477$			1.87358i			0.0857852i
$478$		0			0
$479$		−	25.7577i		−	1.17690i	−0.808533	−	0.588451i	$-0.799738\pi$
$479$		−	25.7577i		−	1.17690i	0.808533	−	0.588451i	$-0.200262\pi$
$480$		0			0
$481$	16.5067	+	15.0611i	0.752640	+	0.686726i
$482$		0			0
$483$	−2.07087			−0.0942277
$484$		0			0
$485$	22.0355	+	6.46024i	1.00058	+	0.293345i
$486$		0			0
$487$	−30.3613			−1.37580			−0.687902	−	0.725804i	$-0.741468\pi$
$487$	−30.3613			−1.37580			−0.687902	+	0.725804i	$0.741468\pi$
$488$		0			0
$489$			30.4046i			1.37494i
$490$		0			0
$491$	−4.86419			−0.219518			−0.109759	−	0.993958i	$-0.535008\pi$
$491$	−4.86419			−0.219518			−0.109759	+	0.993958i	$0.535008\pi$
$492$		0			0
$493$			9.94136i			0.447736i
$494$		0			0
$495$	2.02017	−	6.89068i	0.0908000	−	0.309713i
$496$		0			0
$497$		−	21.9865i		−	0.986230i
$498$		0			0
$499$			15.4429i			0.691320i	0.938360	+	0.345660i	$0.112345\pi$
$499$			15.4429i			0.691320i	−0.938360	+	0.345660i	$0.887655\pi$
$500$		0			0
$501$			10.3283i			0.461434i
$502$		0			0
$503$			23.0415i			1.02737i	0.857978	+	0.513686i	$0.171720\pi$
$503$			23.0415i			1.02737i	−0.857978	+	0.513686i	$0.828280\pi$
$504$		0			0
$505$	34.1610	+	10.0151i	1.52014	+	0.445668i
$506$		0			0
$507$	20.3505	−	1.86777i	0.903799	−	0.0829507i
$508$		0			0
$509$		−	14.6018i		−	0.647214i	−0.946192	−	0.323607i	$-0.895104\pi$
$509$		−	14.6018i		−	0.647214i	0.946192	−	0.323607i	$-0.104896\pi$
$510$		0			0
$511$	17.7115			0.783510
$512$		0			0
$513$	22.8982			1.01098
$514$		0			0
$515$	−1.87343	+	6.39016i	−0.0825534	+	0.281584i
$516$		0			0
$517$			25.4903i			1.12106i
$518$		0			0
$519$	−0.173327			−0.00760823
$520$		0			0
$521$	38.8012			1.69991			0.849955	−	0.526855i	$-0.176629\pi$
$521$	38.8012			1.69991			0.849955	+	0.526855i	$0.176629\pi$
$522$		0			0
$523$			31.1514i			1.36215i	0.732212	+	0.681077i	$0.238488\pi$
$523$			31.1514i			1.36215i	−0.732212	+	0.681077i	$0.761512\pi$
$524$		0			0
$525$	17.8137	−	27.7694i	0.777453	−	1.21196i
$526$		0			0
$527$	4.83406			0.210575
$528$		0			0
$529$	22.9015			0.995717
$530$		0			0
$531$		−	3.46503i		−	0.150369i
$532$		0			0
$533$	2.91378	−	3.19345i	0.126210	−	0.138324i
$534$		0			0
$535$	−2.70958	+	9.24220i	−0.117145	+	0.399575i
$536$		0			0
$537$			10.6095i			0.457832i
$538$		0			0
$539$		−	64.4841i		−	2.77753i
$540$		0			0
$541$		−	0.516416i		−	0.0222025i	−0.999938	−	0.0111012i	$-0.996466\pi$
$541$		−	0.516416i		−	0.0222025i	0.999938	−	0.0111012i	$-0.00353370\pi$
$542$		0			0
$543$			27.2805i			1.17072i
$544$		0			0
$545$	10.8624	−	37.0508i	0.465292	−	1.58708i
$546$		0			0
$547$			13.8924i			0.593998i	0.954878	+	0.296999i	$0.0959858\pi$
$547$			13.8924i			0.593998i	−0.954878	+	0.296999i	$0.904014\pi$
$548$		0			0
$549$	−0.896198			−0.0382488
$550$		0			0
$551$			6.21910i			0.264943i
$552$		0			0
$553$	−48.6746			−2.06986
$554$		0			0
$555$	−6.12875	+	20.9048i	−0.260151	+	0.887359i
$556$		0			0
$557$	−39.0758			−1.65570			−0.827848	−	0.560953i	$-0.810435\pi$
$557$	−39.0758			−1.65570			−0.827848	+	0.560953i	$0.810435\pi$
$558$		0			0
$559$	20.9694	−	22.9821i	0.886912	−	0.972041i
$560$		0			0
$561$		−	62.9918i		−	2.65951i
$562$		0			0
$563$		−	25.8408i		−	1.08906i	−0.838741	−	0.544530i	$-0.816708\pi$
$563$		−	25.8408i		−	1.08906i	0.838741	−	0.544530i	$-0.183292\pi$
$564$		0			0
$565$	0.0354777	−	0.121012i	0.00149256	−	0.00509102i
$566$		0			0
$567$	−29.9444			−1.25755
$568$		0			0
$569$	5.50226			0.230667			0.115333	−	0.993327i	$-0.463206\pi$
$569$	5.50226			0.230667			0.115333	+	0.993327i	$0.463206\pi$
$570$		0			0
$571$	−30.8030			−1.28907			−0.644534	−	0.764576i	$-0.722948\pi$
$571$	−30.8030			−1.28907			−0.644534	+	0.764576i	$0.722948\pi$
$572$		0			0
$573$		−	0.591536i		−	0.0247118i
$574$		0			0
$575$	0.847287	−	1.32082i	0.0353343	−	0.0550820i
$576$		0			0
$577$	13.2970			0.553560			0.276780	−	0.960933i	$-0.410733\pi$
$577$	13.2970			0.553560			0.276780	+	0.960933i	$0.410733\pi$
$578$		0			0
$579$		−	4.89248i		−	0.203325i
$580$		0			0
$581$	−34.7987			−1.44369
$582$		0			0
$583$	21.5166			0.891125
$584$		0			0
$585$	−2.21374	−	3.64351i	−0.0915268	−	0.150641i
$586$		0			0
$587$	−2.98270			−0.123109			−0.0615547	−	0.998104i	$-0.519606\pi$
$587$	−2.98270			−0.123109			−0.0615547	+	0.998104i	$0.519606\pi$
$588$		0			0
$589$	3.02408			0.124605
$590$		0			0
$591$			2.63013i			0.108189i
$592$		0			0
$593$	13.0325			0.535182			0.267591	−	0.963533i	$-0.413772\pi$
$593$	13.0325			0.535182			0.267591	+	0.963533i	$0.413772\pi$
$594$		0			0
$595$	−17.4232	+	59.4294i	−0.714282	+	2.43637i
$596$		0			0
$597$			36.0067i			1.47366i
$598$		0			0
$599$	0.186062			0.00760228			0.00380114	−	0.999993i	$-0.498790\pi$
$599$	0.186062			0.00760228			0.00380114	+	0.999993i	$0.498790\pi$
$600$		0			0
$601$	−14.7755			−0.602707			−0.301353	−	0.953513i	$-0.597438\pi$
$601$	−14.7755			−0.602707			−0.301353	+	0.953513i	$0.597438\pi$
$602$		0			0
$603$	−3.28891			−0.133935
$604$		0			0
$605$	−55.5308	−	16.2802i	−2.25765	−	0.661885i
$606$		0			0
$607$			18.8321i			0.764370i	0.924086	+	0.382185i	$0.124828\pi$
$607$			18.8321i			0.764370i	−0.924086	+	0.382185i	$0.875172\pi$
$608$		0			0
$609$		−	9.94136i		−	0.402844i
$610$		0			0
$611$	11.1797	+	10.2006i	0.452284	+	0.412674i
$612$		0			0
$613$	4.40311			0.177840			0.0889200	−	0.996039i	$-0.471658\pi$
$613$	4.40311			0.177840			0.0889200	+	0.996039i	$0.471658\pi$
$614$		0			0
$615$	4.04433	+	1.18569i	0.163083	+	0.0478118i
$616$		0			0
$617$	29.5594			1.19002			0.595009	−	0.803719i	$-0.297149\pi$
$617$	29.5594			1.19002			0.595009	+	0.803719i	$0.297149\pi$
$618$		0			0
$619$		−	12.4029i		−	0.498515i	−0.968437	−	0.249258i	$-0.919813\pi$
$619$		−	12.4029i		−	0.498515i	0.968437	−	0.249258i	$-0.0801866\pi$
$620$		0			0
$621$	−1.74099			−0.0698634
$622$		0			0
$623$		−	56.6341i		−	2.26900i
$624$		0			0
$625$	10.4232	+	22.7235i	0.416928	+	0.908939i
$626$		0			0
$627$		−	39.4063i		−	1.57374i
$628$		0			0
$629$		−	40.8931i		−	1.63051i
$630$		0			0
$631$		−	32.5256i		−	1.29482i	−0.762141	−	0.647411i	$-0.775852\pi$
$631$		−	32.5256i		−	1.29482i	0.762141	−	0.647411i	$-0.224148\pi$
$632$		0			0
$633$		−	10.6095i		−	0.421688i
$634$		0			0
$635$	−10.2497	+	34.9612i	−0.406748	+	1.38739i
$636$		0			0
$637$	−28.2819	−	25.8051i	−1.12057	−	1.02243i
$638$		0			0
$639$		−	2.76989i		−	0.109575i
$640$		0			0
$641$	24.9049			0.983686			0.491843	−	0.870684i	$-0.336323\pi$
$641$	24.9049			0.983686			0.491843	+	0.870684i	$0.336323\pi$
$642$		0			0
$643$	12.3348			0.486437			0.243218	−	0.969972i	$-0.421797\pi$
$643$	12.3348			0.486437			0.243218	+	0.969972i	$0.421797\pi$
$644$		0			0
$645$	29.1056	+	8.53302i	1.14603	+	0.335987i
$646$		0			0
$647$		−	25.2034i		−	0.990847i	−0.868652	−	0.495423i	$-0.835013\pi$
$647$		−	25.2034i		−	0.990847i	0.868652	−	0.495423i	$-0.164987\pi$
$648$		0			0
$649$	−39.7932			−1.56202
$650$		0			0
$651$	−4.83406			−0.189462
$652$		0			0
$653$			0.285014i			0.0111535i	0.999984	+	0.00557674i	$0.00177514\pi$
$653$			0.285014i			0.0111535i	−0.999984	+	0.00557674i	$0.998225\pi$
$654$		0			0
$655$	1.91318	+	0.560897i	0.0747543	+	0.0219161i
$656$		0			0
$657$	2.23132			0.0870520
$658$		0			0
$659$	4.73568			0.184476			0.0922380	−	0.995737i	$-0.470598\pi$
$659$	4.73568			0.184476			0.0922380	+	0.995737i	$0.470598\pi$
$660$		0			0
$661$			48.5089i			1.88678i	0.331687	+	0.943390i	$0.392382\pi$
$661$			48.5089i			1.88678i	−0.331687	+	0.943390i	$0.607618\pi$
$662$		0			0
$663$	−27.6274	−	25.2079i	−1.07296	−	0.978992i
$664$		0			0
$665$	−10.8996	+	37.1778i	−0.422668	+	1.44169i
$666$		0			0
$667$		−	0.472849i		−	0.0183088i
$668$		0			0
$669$			7.81066i			0.301977i
$670$		0			0
$671$			10.2921i			0.397324i
$672$		0			0
$673$		−	3.34297i		−	0.128862i	−0.997922	−	0.0644311i	$-0.979477\pi$
$673$		−	3.34297i		−	0.128862i	0.997922	−	0.0644311i	$-0.0205233\pi$
$674$		0			0
$675$	14.9760	−	23.3459i	0.576428	−	0.898583i
$676$		0			0
$677$		−	29.7161i		−	1.14208i	−0.820921	−	0.571042i	$-0.806539\pi$
$677$		−	29.7161i		−	1.14208i	0.820921	−	0.571042i	$-0.193461\pi$
$678$		0			0
$679$	43.1049			1.65421
$680$		0			0
$681$		−	21.0840i		−	0.807940i
$682$		0			0
$683$	−17.8328			−0.682353			−0.341176	−	0.939999i	$-0.610825\pi$
$683$	−17.8328			−0.682353			−0.341176	+	0.939999i	$0.610825\pi$
$684$		0			0
$685$	19.6096	+	5.74904i	0.749244	+	0.219659i
$686$		0			0
$687$	21.8361			0.833099
$688$		0			0
$689$	8.61044	−	9.43689i	0.328032	−	0.359517i
$690$		0			0
$691$			5.80394i			0.220792i	0.993888	+	0.110396i	$0.0352120\pi$
$691$			5.80394i			0.220792i	−0.993888	+	0.110396i	$0.964788\pi$
$692$		0			0
$693$		−	13.4793i		−	0.512035i
$694$		0			0
$695$	42.7916	+	12.5454i	1.62318	+	0.475874i
$696$		0			0
$697$	−7.91134			−0.299664
$698$		0			0
$699$	33.6539			1.27291
$700$		0			0
$701$	25.6149			0.967460			0.483730	−	0.875217i	$-0.339282\pi$
$701$	25.6149			0.967460			0.483730	+	0.875217i	$0.339282\pi$
$702$		0			0
$703$		−	25.5818i		−	0.964837i
$704$		0			0
$705$	−4.15092	+	14.1585i	−0.156333	+	0.533241i
$706$		0			0
$707$	66.8243			2.51319
$708$		0			0
$709$			4.66045i			0.175027i	0.996163	+	0.0875134i	$0.0278920\pi$
$709$			4.66045i			0.175027i	−0.996163	+	0.0875134i	$0.972108\pi$
$710$		0			0
$711$	−6.13210			−0.229972
$712$		0			0
$713$	−0.229926			−0.00861081
$714$		0			0
$715$	−41.8429	+	25.4230i	−1.56484	+	0.950768i
$716$		0			0
$717$	2.40476			0.0898074
$718$		0			0
$719$	10.2265			0.381384			0.190692	−	0.981650i	$-0.438927\pi$
$719$	10.2265			0.381384			0.190692	+	0.981650i	$0.438927\pi$
$720$		0			0
$721$			12.5002i			0.465531i
$722$		0			0
$723$	27.2369			1.01295
$724$		0			0
$725$	6.34070	+	4.06747i	0.235488	+	0.151062i
$726$		0			0
$727$			28.1388i			1.04361i	0.853065	+	0.521805i	$0.174741\pi$
$727$			28.1388i			1.04361i	−0.853065	+	0.521805i	$0.825259\pi$
$728$		0			0
$729$	−29.9336			−1.10865
$730$		0			0
$731$	−56.9351			−2.10582
$732$		0			0
$733$	−15.0357			−0.555357			−0.277678	−	0.960674i	$-0.589565\pi$
$733$	−15.0357			−0.555357			−0.277678	+	0.960674i	$0.589565\pi$
$734$		0			0
$735$	10.5008	−	35.8175i	0.387327	−	1.32115i
$736$		0			0
$737$			37.7706i			1.39130i
$738$		0			0
$739$			30.7946i			1.13280i	0.824132	+	0.566398i	$0.191664\pi$
$739$			30.7946i			1.13280i	−0.824132	+	0.566398i	$0.808336\pi$
$740$		0			0
$741$	−17.2831	−	15.7695i	−0.634910	−	0.579307i
$742$		0			0
$743$	−17.4078			−0.638630			−0.319315	−	0.947649i	$-0.603453\pi$
$743$	−17.4078			−0.638630			−0.319315	+	0.947649i	$0.603453\pi$
$744$		0			0
$745$	−6.48639	+	22.1247i	−0.237643	+	0.810585i
$746$		0			0
$747$	−4.38399			−0.160402
$748$		0			0
$749$			18.0792i			0.660600i
$750$		0			0
$751$	31.6229			1.15394			0.576968	−	0.816767i	$-0.304236\pi$
$751$	31.6229			1.15394			0.576968	+	0.816767i	$0.304236\pi$
$752$		0			0
$753$		−	44.6927i		−	1.62869i
$754$		0			0
$755$	−0.474432	+	1.61826i	−0.0172663	+	0.0588944i
$756$		0			0
$757$		−	47.6442i		−	1.73166i	−0.500340	−	0.865829i	$-0.666792\pi$
$757$		−	47.6442i		−	1.73166i	0.500340	−	0.865829i	$-0.333208\pi$
$758$		0			0
$759$			2.99613i			0.108753i
$760$		0			0
$761$		−	13.6992i		−	0.496594i	−0.968684	−	0.248297i	$-0.920129\pi$
$761$		−	13.6992i		−	0.496594i	0.968684	−	0.248297i	$-0.0798709\pi$
$762$		0			0
$763$		−	72.4772i		−	2.62385i
$764$		0			0
$765$	−2.19500	+	7.48700i	−0.0793604	+	0.270693i
$766$		0			0
$767$	−15.9243	+	17.4528i	−0.574994	+	0.630183i
$768$		0			0
$769$		−	16.1217i		−	0.581362i	−0.956820	−	0.290681i	$-0.906118\pi$
$769$		−	16.1217i		−	0.581362i	0.956820	−	0.290681i	$-0.0938818\pi$
$770$		0			0
$771$	−31.6805			−1.14094
$772$		0			0
$773$	−40.6817			−1.46322			−0.731610	−	0.681723i	$-0.761231\pi$
$773$	−40.6817			−1.46322			−0.731610	+	0.681723i	$0.761231\pi$
$774$		0			0
$775$	1.97784	−	3.08321i	0.0710460	−	0.110752i
$776$		0			0
$777$			40.8931i			1.46703i
$778$		0			0
$779$	−4.94917			−0.177322
$780$		0			0
$781$	−31.8101			−1.13825
$782$		0			0
$783$		−	8.35774i		−	0.298681i
$784$		0			0
$785$	−12.1492	+	41.4400i	−0.433622	+	1.47906i
$786$		0			0
$787$	−21.2075			−0.755967			−0.377983	−	0.925812i	$-0.623382\pi$
$787$	−21.2075			−0.755967			−0.377983	+	0.925812i	$0.623382\pi$
$788$		0			0
$789$	10.6106			0.377749
$790$		0			0
$791$		−	0.236719i		−	0.00841676i
$792$		0			0
$793$	4.51400	+	4.11868i	0.160297	+	0.146259i
$794$		0			0
$795$	11.9513	+	3.50382i	0.423869	+	0.124268i
$796$		0			0
$797$			10.7343i			0.380229i	0.981762	+	0.190115i	$0.0608859\pi$
$797$			10.7343i			0.380229i	−0.981762	+	0.190115i	$0.939114\pi$
$798$		0			0
$799$		−	27.6963i		−	0.979825i
$800$		0			0
$801$		−	7.13484i		−	0.252097i
$802$		0			0
$803$		−	25.6249i		−	0.904284i
$804$		0			0
$805$	0.828714	−	2.82669i	0.0292083	−	0.0996277i
$806$		0			0
$807$			42.7609i			1.50526i
$808$		0			0
$809$	38.6151			1.35763			0.678817	−	0.734307i	$-0.262493\pi$
$809$	38.6151			1.35763			0.678817	+	0.734307i	$0.262493\pi$
$810$		0			0
$811$		−	44.3647i		−	1.55785i	−0.627114	−	0.778927i	$-0.715764\pi$
$811$		−	44.3647i		−	1.55785i	0.627114	−	0.778927i	$-0.284236\pi$
$812$		0			0
$813$	19.6021			0.687475
$814$		0			0
$815$	−41.5017	−	12.1672i	−1.45374	−	0.426200i
$816$		0			0
$817$	−35.6174			−1.24610
$818$		0			0
$819$	−5.91183	−	5.39409i	−0.206576	−	0.188485i
$820$		0			0
$821$			34.8111i			1.21492i	0.794352	+	0.607458i	$0.207811\pi$
$821$			34.8111i			1.21492i	−0.794352	+	0.607458i	$0.792189\pi$
$822$		0			0
$823$		−	1.84799i		−	0.0644168i	−0.999481	−	0.0322084i	$-0.989746\pi$
$823$		−	1.84799i		−	0.0644168i	0.999481	−	0.0322084i	$-0.0102540\pi$
$824$		0			0
$825$	−40.1767	−	25.7728i	−1.39877	−	0.897294i
$826$		0			0
$827$	36.2985			1.26222			0.631112	−	0.775692i	$-0.282599\pi$
$827$	36.2985			1.26222			0.631112	+	0.775692i	$0.282599\pi$
$828$		0			0
$829$	29.1215			1.01143			0.505716	−	0.862700i	$-0.331229\pi$
$829$	29.1215			1.01143			0.505716	+	0.862700i	$0.331229\pi$
$830$		0			0
$831$	−16.4303			−0.569962
$832$		0			0
$833$			70.0646i			2.42760i
$834$		0			0
$835$	−14.0979	−	4.13315i	−0.487878	−	0.143033i
$836$		0			0
$837$	−4.06401			−0.140473
$838$		0			0
$839$			3.95559i			0.136562i	0.997666	+	0.0682810i	$0.0217515\pi$
$839$			3.95559i			0.136562i	−0.997666	+	0.0682810i	$0.978249\pi$
$840$		0			0
$841$	−26.7301			−0.921726
$842$		0			0
$843$	−6.30543			−0.217170
$844$		0			0
$845$	−5.59435	+	28.5255i	−0.192451	+	0.981306i
$846$		0			0
$847$	−108.627			−3.73247
$848$		0			0
$849$	−17.6425			−0.605489
$850$		0			0
$851$			1.94503i			0.0666748i
$852$		0			0
$853$	29.0395			0.994292			0.497146	−	0.867667i	$-0.334381\pi$
$853$	29.0395			0.994292			0.497146	+	0.867667i	$0.334381\pi$
$854$		0			0
$855$	−1.37314	+	4.68371i	−0.0469606	+	0.160179i
$856$		0			0
$857$			5.48988i			0.187531i	0.995594	+	0.0937654i	$0.0298904\pi$
$857$			5.48988i			0.187531i	−0.995594	+	0.0937654i	$0.970110\pi$
$858$		0			0
$859$	−39.0467			−1.33225			−0.666127	−	0.745838i	$-0.732049\pi$
$859$	−39.0467			−1.33225			−0.666127	+	0.745838i	$0.732049\pi$
$860$		0			0
$861$	7.91134			0.269618
$862$		0			0
$863$	21.4609			0.730536			0.365268	−	0.930902i	$-0.380977\pi$
$863$	21.4609			0.730536			0.365268	+	0.930902i	$0.380977\pi$
$864$		0			0
$865$	0.0693616	−	0.236588i	0.00235837	−	0.00804423i
$866$		0			0
$867$			41.7190i			1.41685i
$868$		0			0
$869$			70.4224i			2.38892i
$870$		0			0
$871$	16.5657	+	15.1149i	0.561308	+	0.512150i
$872$		0			0
$873$	5.43041			0.183792
$874$		0			0
$875$	30.7760	+	35.4280i	1.04042	+	1.19768i
$876$		0			0
$877$	11.2141			0.378674			0.189337	−	0.981912i	$-0.439366\pi$
$877$	11.2141			0.378674			0.189337	+	0.981912i	$0.439366\pi$
$878$		0			0
$879$			13.5893i			0.458357i
$880$		0			0
$881$	5.59967			0.188658			0.0943288	−	0.995541i	$-0.469930\pi$
$881$	5.59967			0.188658			0.0943288	+	0.995541i	$0.469930\pi$
$882$		0			0
$883$			35.2728i			1.18703i	0.804825	+	0.593513i	$0.202259\pi$
$883$			35.2728i			1.18703i	−0.804825	+	0.593513i	$0.797741\pi$
$884$		0			0
$885$	−22.1030	−	6.48003i	−0.742983	−	0.217824i
$886$		0			0
$887$		−	23.7057i		−	0.795960i	−0.917394	−	0.397980i	$-0.869711\pi$
$887$		−	23.7057i		−	0.795960i	0.917394	−	0.397980i	$-0.130289\pi$
$888$		0			0
$889$			68.3896i			2.29371i
$890$		0			0
$891$			43.3235i			1.45139i
$892$		0			0
$893$		−	17.3262i		−	0.579799i
$894$		0			0
$895$	−14.4817	−	4.24566i	−0.484069	−	0.141917i
$896$		0			0
$897$	1.31406	+	1.19898i	0.0438753	+	0.0400328i
$898$		0			0
$899$		−	1.10378i		−	0.0368131i
$900$		0			0
$901$	−23.3786			−0.778855
$902$		0			0
$903$	56.9351			1.89468
$904$		0			0
$905$	−37.2373	−	10.9170i	−1.23781	−	0.362895i
$906$		0			0
$907$		−	43.1031i		−	1.43121i	−0.698503	−	0.715607i	$-0.746150\pi$
$907$		−	43.1031i		−	1.43121i	0.698503	−	0.715607i	$-0.253850\pi$
$908$		0			0
$909$	8.41862			0.279228
$910$		0			0
$911$	−29.3078			−0.971009			−0.485505	−	0.874234i	$-0.661364\pi$
$911$	−29.3078			−0.971009			−0.485505	+	0.874234i	$0.661364\pi$
$912$		0			0
$913$			50.3467i			1.66623i
$914$		0			0
$915$	−1.67600	+	5.71673i	−0.0554069	+	0.188989i
$916$		0			0
$917$	3.74249			0.123588
$918$		0			0
$919$	−13.7254			−0.452758			−0.226379	−	0.974039i	$-0.572689\pi$
$919$	−13.7254			−0.452758			−0.226379	+	0.974039i	$0.572689\pi$
$920$		0			0
$921$		−	13.6242i		−	0.448932i
$922$		0			0
$923$	−12.7297	+	13.9515i	−0.419002	+	0.459219i
$924$		0			0
$925$	−26.0820	−	16.7312i	−0.857571	−	0.550120i
$926$		0			0
$927$			1.57479i			0.0517229i
$928$		0			0
$929$		−	39.7866i		−	1.30536i	−0.757636	−	0.652678i	$-0.773646\pi$
$929$		−	39.7866i		−	1.30536i	0.757636	−	0.652678i	$-0.226354\pi$
$930$		0			0
$931$			43.8309i			1.43650i
$932$		0			0
$933$		−	23.1381i		−	0.757509i
$934$		0			0
$935$	85.9824	+	25.2079i	2.81192	+	0.824385i
$936$		0			0
$937$			15.5516i			0.508050i	0.967198	+	0.254025i	$0.0817545\pi$
$937$			15.5516i			0.508050i	−0.967198	+	0.254025i	$0.918246\pi$
$938$		0			0
$939$	−25.0029			−0.815939
$940$		0			0
$941$			11.9950i			0.391026i	0.980701	+	0.195513i	$0.0626372\pi$
$941$			11.9950i			0.391026i	−0.980701	+	0.195513i	$0.937363\pi$
$942$		0			0
$943$	0.376294			0.0122538
$944$		0			0
$945$	14.6478	−	49.9625i	0.476491	−	1.62528i
$946$		0			0
$947$	37.5458			1.22008			0.610038	−	0.792372i	$-0.291154\pi$
$947$	37.5458			1.22008			0.610038	+	0.792372i	$0.291154\pi$
$948$		0			0
$949$	−11.2388	−	10.2545i	−0.364826	−	0.332876i
$950$		0			0
$951$		−	41.1478i		−	1.33431i
$952$		0			0
$953$		−	42.2131i		−	1.36742i	−0.729756	−	0.683708i	$-0.760366\pi$
$953$		−	42.2131i		−	1.36742i	0.729756	−	0.683708i	$-0.239634\pi$
$954$		0			0
$955$	0.807433	+	0.236719i	0.0261279	+	0.00766005i
$956$		0			0
$957$	−14.3831			−0.464941
$958$		0			0
$959$	38.3595			1.23869
$960$		0			0
$961$	30.4633			0.982686
$962$		0			0
$963$			2.27764i			0.0733961i
$964$		0			0
$965$	6.67813	+	1.95786i	0.214977	+	0.0630257i
$966$		0			0
$967$	24.4506			0.786277			0.393139	−	0.919479i	$-0.371389\pi$
$967$	24.4506			0.786277			0.393139	+	0.919479i	$0.371389\pi$
$968$		0			0
$969$			42.8165i			1.37547i
$970$		0			0
$971$	−28.8642			−0.926296			−0.463148	−	0.886281i	$-0.653280\pi$
$971$	−28.8642			−0.926296			−0.463148	+	0.886281i	$0.653280\pi$
$972$		0			0
$973$	83.7071			2.68353
$974$		0			0
$975$	−27.3814	+	7.30732i	−0.876908	+	0.234021i
$976$		0			0
$977$	22.2905			0.713135			0.356568	−	0.934270i	$-0.383947\pi$
$977$	22.2905			0.713135			0.356568	+	0.934270i	$0.383947\pi$
$978$		0			0
$979$	−81.9381			−2.61875
$980$		0			0
$981$		−	9.13078i		−	0.291523i
$982$		0			0
$983$	25.2176			0.804315			0.402158	−	0.915570i	$-0.368260\pi$
$983$	25.2176			0.804315			0.402158	+	0.915570i	$0.368260\pi$
$984$		0			0
$985$	−3.59007	−	1.05252i	−0.114389	−	0.0335360i
$986$		0			0
$987$			27.6963i			0.881583i
$988$		0			0
$989$	2.70805			0.0861110
$990$		0			0
$991$	23.8267			0.756879			0.378440	−	0.925626i	$-0.376461\pi$
$991$	23.8267			0.756879			0.378440	+	0.925626i	$0.376461\pi$
$992$		0			0
$993$	37.8734			1.20188
$994$		0			0
$995$	−49.1483	−	14.4090i	−1.55811	−	0.456797i
$996$		0			0
$997$		−	14.4496i		−	0.457624i	−0.973471	−	0.228812i	$-0.926516\pi$
$997$		−	14.4496i		−	0.457624i	0.973471	−	0.228812i	$-0.0734841\pi$
$998$		0			0
$999$			34.3790i			1.08770i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.g.129.3		10
4.3	odd	2		520.2.f.b.129.8	yes	10
5.4	even	2		1040.2.f.f.129.8		10
13.12	even	2		1040.2.f.f.129.3		10
20.3	even	4		2600.2.k.f.2001.6		20
20.7	even	4		2600.2.k.f.2001.15		20
20.19	odd	2		520.2.f.a.129.3	✓	10
52.51	odd	2		520.2.f.a.129.8	yes	10
65.64	even	2	inner	1040.2.f.g.129.8		10
260.103	even	4		2600.2.k.f.2001.5		20
260.207	even	4		2600.2.k.f.2001.16		20
260.259	odd	2		520.2.f.b.129.3	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.3	✓	10	20.19	odd	2
520.2.f.a.129.8	yes	10	52.51	odd	2
520.2.f.b.129.3	yes	10	260.259	odd	2
520.2.f.b.129.8	yes	10	4.3	odd	2
1040.2.f.f.129.3		10	13.12	even	2
1040.2.f.f.129.8		10	5.4	even	2
1040.2.f.g.129.3		10	1.1	even	1	trivial
1040.2.f.g.129.8		10	65.64	even	2	inner
2600.2.k.f.2001.5		20	260.103	even	4
2600.2.k.f.2001.6		20	20.3	even	4
2600.2.k.f.2001.15		20	20.7	even	4
2600.2.k.f.2001.16		20	260.207	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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.57201i q^{3} +(2.14575 + 0.629081i) q^{5} +4.19743 q^{7} +0.528799 q^{9} +O(q^{10})\)	\(q-1.57201i q^{3} +(2.14575 + 0.629081i) q^{5} +4.19743 q^{7} +0.528799 q^{9} -6.07284i q^{11} +(-2.66347 - 2.43021i) q^{13} +(0.988918 - 3.37314i) q^{15} +6.59839i q^{17} +4.12781i q^{19} -6.59839i q^{21} -0.313844i q^{23} +(4.20852 + 2.69970i) q^{25} -5.54729i q^{27} +1.50663 q^{29} -0.732612i q^{31} -9.54654 q^{33} +(9.00666 + 2.64052i) q^{35} -6.19743 q^{37} +(-3.82031 + 4.18699i) q^{39} +1.19898i q^{41} +8.62864i q^{43} +(1.13467 + 0.332657i) q^{45} -4.19743 q^{47} +10.6184 q^{49} +10.3727 q^{51} +3.54308i q^{53} +(3.82031 - 13.0308i) q^{55} +6.48894 q^{57} -6.55264i q^{59} -1.69478 q^{61} +2.21960 q^{63} +(-4.18635 - 6.89017i) q^{65} -6.21960 q^{67} -0.493365 q^{69} -5.23809i q^{71} +4.21960 q^{73} +(4.24395 - 6.61581i) q^{75} -25.4903i q^{77} -11.5963 q^{79} -7.13398 q^{81} -8.29046 q^{83} +(-4.15092 + 14.1585i) q^{85} -2.36844i q^{87} -13.4925i q^{89} +(-11.1797 - 10.2006i) q^{91} -1.15167 q^{93} +(-2.59673 + 8.85726i) q^{95} +10.2693 q^{97} -3.21131i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61} - 24 q^{63} - 5 q^{65} - 16 q^{67} - 12 q^{69} - 4 q^{73} - 21 q^{75} - 28 q^{79} + 22 q^{81} + 4 q^{83} - 25 q^{85} - 2 q^{91} + 12 q^{93} + 10 q^{95} + 72 q^{97}+O(q^{100}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

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