LMFDB - Embedded newform 1040.2.f.c.129.1

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.c.129.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -2.37228 q^{7} -3.37228 q^{9} +O(q^{10})$	$q-2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -2.37228 q^{7} -3.37228 q^{9} +1.58457i q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.18614 + 5.51856i) q^{15} -5.98844i q^{17} +3.46410i q^{19} +5.98844i q^{21} +6.63325i q^{23} +(4.55842 + 2.05446i) q^{25} +0.939764i q^{27} +2.74456 q^{29} -3.46410i q^{31} +4.00000 q^{33} +(5.18614 + 1.11469i) q^{35} -9.11684 q^{37} +(8.74456 + 2.52434i) q^{39} +10.0974i q^{41} -0.644810i q^{43} +(7.37228 + 1.58457i) q^{45} +10.3723 q^{47} -1.37228 q^{49} -15.1168 q^{51} +5.04868i q^{53} +(0.744563 - 3.46410i) q^{55} +8.74456 q^{57} -6.63325i q^{59} -6.74456 q^{61} +8.00000 q^{63} +(3.81386 - 7.10313i) q^{65} -4.00000 q^{67} +16.7446 q^{69} +12.6217i q^{71} +10.0000 q^{73} +(5.18614 - 11.5070i) q^{75} -3.75906i q^{77} -4.74456 q^{79} -7.74456 q^{81} -8.74456 q^{83} +(-2.81386 + 13.0916i) q^{85} -6.92820i q^{87} +1.87953i q^{89} +(2.37228 - 8.21782i) q^{91} -8.74456 q^{93} +(1.62772 - 7.57301i) q^{95} +6.74456 q^{97} -5.34363i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) $ 4 * q - 3 * q^5 + 2 * q^7 - 2 * q^9	$ 4 q - 3 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{13} + q^{15} + q^{25} - 12 q^{29} + 16 q^{33} + 15 q^{35} - 2 q^{37} + 12 q^{39} + 18 q^{45} + 30 q^{47} + 6 q^{49} - 26 q^{51} - 20 q^{55} + 12 q^{57} - 4 q^{61} + 32 q^{63} + 21 q^{65} - 16 q^{67} + 44 q^{69} + 40 q^{73} + 15 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} - 17 q^{85} - 2 q^{91} - 12 q^{93} + 18 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 + 2 * q^7 - 2 * q^9 - 4 * q^13 + q^15 + q^25 - 12 * q^29 + 16 * q^33 + 15 * q^35 - 2 * q^37 + 12 * q^39 + 18 * q^45 + 30 * q^47 + 6 * q^49 - 26 * q^51 - 20 * q^55 + 12 * q^57 - 4 * q^61 + 32 * q^63 + 21 * q^65 - 16 * q^67 + 44 * q^69 + 40 * q^73 + 15 * q^75 + 4 * q^79 - 8 * q^81 - 12 * q^83 - 17 * q^85 - 2 * q^91 - 12 * q^93 + 18 * q^95 + 4 * 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
$3$		−	2.52434i		−	1.45743i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$		−	2.52434i		−	1.45743i	0.684819	−	0.728714i	$-0.259881\pi$
$4$		0			0
$5$	−2.18614	−	0.469882i	−0.977672	−	0.210138i
$5$	−2.18614	−	0.469882i	−0.977672	−	0.210138i
$6$		0			0
$7$	−2.37228			−0.896638			−0.448319	−	0.893874i	$-0.647977\pi$
$7$	−2.37228			−0.896638			−0.448319	+	0.893874i	$0.647977\pi$
$8$		0			0
$9$	−3.37228			−1.12409
$10$		0			0
$11$			1.58457i			0.477767i	0.971048	+	0.238884i	$0.0767814\pi$
$11$			1.58457i			0.477767i	−0.971048	+	0.238884i	$0.923219\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$	−1.18614	+	5.51856i	−0.306260	+	1.42489i
$16$		0			0
$17$		−	5.98844i		−	1.45241i	−0.687478	−	0.726205i	$-0.741282\pi$
$17$		−	5.98844i		−	1.45241i	0.687478	−	0.726205i	$-0.258718\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$			5.98844i			1.30678i
$22$		0			0
$23$			6.63325i			1.38313i	0.722315	+	0.691564i	$0.243078\pi$
$23$			6.63325i			1.38313i	−0.722315	+	0.691564i	$0.756922\pi$
$24$		0			0
$25$	4.55842	+	2.05446i	0.911684	+	0.410891i
$26$		0			0
$27$			0.939764i			0.180858i
$28$		0			0
$29$	2.74456			0.509652			0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456			0.509652			0.254826	+	0.966987i	$0.417982\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	4.00000			0.696311
$34$		0			0
$35$	5.18614	+	1.11469i	0.876618	+	0.188417i
$36$		0			0
$37$	−9.11684			−1.49880			−0.749400	−	0.662118i	$-0.769658\pi$
$37$	−9.11684			−1.49880			−0.749400	+	0.662118i	$0.769658\pi$
$38$		0			0
$39$	8.74456	+	2.52434i	1.40025	+	0.404218i
$40$		0			0
$41$			10.0974i			1.57694i	0.615072	+	0.788471i	$0.289127\pi$
$41$			10.0974i			1.57694i	−0.615072	+	0.788471i	$0.710873\pi$
$42$		0			0
$43$		−	0.644810i		−	0.0983326i	−0.998791	−	0.0491663i	$-0.984344\pi$
$43$		−	0.644810i		−	0.0983326i	0.998791	−	0.0491663i	$-0.0156564\pi$
$44$		0			0
$45$	7.37228	+	1.58457i	1.09899	+	0.236214i
$46$		0			0
$47$	10.3723			1.51295			0.756476	−	0.654021i	$-0.226919\pi$
$47$	10.3723			1.51295			0.756476	+	0.654021i	$0.226919\pi$
$48$		0			0
$49$	−1.37228			−0.196040
$50$		0			0
$51$	−15.1168			−2.11678
$52$		0			0
$53$			5.04868i			0.693489i	0.937960	+	0.346744i	$0.112713\pi$
$53$			5.04868i			0.693489i	−0.937960	+	0.346744i	$0.887287\pi$
$54$		0			0
$55$	0.744563	−	3.46410i	0.100397	−	0.467099i
$56$		0			0
$57$	8.74456			1.15825
$58$		0			0
$59$		−	6.63325i		−	0.863576i	−0.901975	−	0.431788i	$-0.857883\pi$
$59$		−	6.63325i		−	0.863576i	0.901975	−	0.431788i	$-0.142117\pi$
$60$		0			0
$61$	−6.74456			−0.863553			−0.431776	−	0.901981i	$-0.642113\pi$
$61$	−6.74456			−0.863553			−0.431776	+	0.901981i	$0.642113\pi$
$62$		0			0
$63$	8.00000			1.00791
$64$		0			0
$65$	3.81386	−	7.10313i	0.473051	−	0.881035i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	16.7446			2.01581
$70$		0			0
$71$			12.6217i			1.49792i	0.662616	+	0.748959i	$0.269446\pi$
$71$			12.6217i			1.49792i	−0.662616	+	0.748959i	$0.730554\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	5.18614	−	11.5070i	0.598844	−	1.32871i
$76$		0			0
$77$		−	3.75906i		−	0.428384i
$78$		0			0
$79$	−4.74456			−0.533805			−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456			−0.533805			−0.266903	+	0.963724i	$0.586000\pi$
$80$		0			0
$81$	−7.74456			−0.860507
$82$		0			0
$83$	−8.74456			−0.959840			−0.479920	−	0.877312i	$-0.659334\pi$
$83$	−8.74456			−0.959840			−0.479920	+	0.877312i	$0.659334\pi$
$84$		0			0
$85$	−2.81386	+	13.0916i	−0.305206	+	1.41998i
$86$		0			0
$87$		−	6.92820i		−	0.742781i
$88$		0			0
$89$			1.87953i			0.199230i	0.995026	+	0.0996148i	$0.0317610\pi$
$89$			1.87953i			0.199230i	−0.995026	+	0.0996148i	$0.968239\pi$
$90$		0			0
$91$	2.37228	−	8.21782i	0.248683	−	0.861462i
$92$		0			0
$93$	−8.74456			−0.906769
$94$		0			0
$95$	1.62772	−	7.57301i	0.167000	−	0.776975i
$96$		0			0
$97$	6.74456			0.684807			0.342403	−	0.939553i	$-0.388759\pi$
$97$	6.74456			0.684807			0.342403	+	0.939553i	$0.388759\pi$
$98$		0			0
$99$		−	5.34363i		−	0.537055i
$100$		0			0
$101$	−14.7446			−1.46714			−0.733569	−	0.679615i	$-0.762147\pi$
$101$	−14.7446			−1.46714			−0.733569	+	0.679615i	$0.762147\pi$
$102$		0			0
$103$			10.3923i			1.02398i	0.858990	+	0.511992i	$0.171092\pi$
$103$			10.3923i			1.02398i	−0.858990	+	0.511992i	$0.828908\pi$
$104$		0			0
$105$	2.81386	−	13.0916i	0.274605	−	1.27761i
$106$		0			0
$107$			13.5615i			1.31104i	0.755180	+	0.655518i	$0.227549\pi$
$107$			13.5615i			1.31104i	−0.755180	+	0.655518i	$0.772451\pi$
$108$		0			0
$109$		−	2.81929i		−	0.270039i	−0.990843	−	0.135020i	$-0.956890\pi$
$109$		−	2.81929i		−	0.270039i	0.990843	−	0.135020i	$-0.0431097\pi$
$110$		0			0
$111$			23.0140i			2.18439i
$112$		0			0
$113$			3.75906i			0.353622i	0.984245	+	0.176811i	$0.0565782\pi$
$113$			3.75906i			0.353622i	−0.984245	+	0.176811i	$0.943422\pi$
$114$		0			0
$115$	3.11684	−	14.5012i	0.290647	−	1.35225i
$116$		0			0
$117$	3.37228	−	11.6819i	0.311768	−	1.07999i
$118$		0			0
$119$			14.2063i			1.30229i
$120$		0			0
$121$	8.48913			0.771739
$122$		0			0
$123$	25.4891			2.29828
$124$		0			0
$125$	−9.00000	−	6.63325i	−0.804984	−	0.593296i
$126$		0			0
$127$		−	3.46410i		−	0.307389i	−0.988118	−	0.153695i	$-0.950883\pi$
$127$		−	3.46410i		−	0.307389i	0.988118	−	0.153695i	$-0.0491172\pi$
$128$		0			0
$129$	−1.62772			−0.143313
$130$		0			0
$131$	1.62772			0.142214			0.0711072	−	0.997469i	$-0.477347\pi$
$131$	1.62772			0.142214			0.0711072	+	0.997469i	$0.477347\pi$
$132$		0			0
$133$		−	8.21782i		−	0.712576i
$134$		0			0
$135$	0.441578	−	2.05446i	0.0380050	−	0.176819i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	−6.37228			−0.540490			−0.270245	−	0.962792i	$-0.587105\pi$
$139$	−6.37228			−0.540490			−0.270245	+	0.962792i	$0.587105\pi$
$140$		0			0
$141$		−	26.1831i		−	2.20502i
$142$		0			0
$143$	−5.48913	−	1.58457i	−0.459024	−	0.132509i
$144$		0			0
$145$	−6.00000	−	1.28962i	−0.498273	−	0.107097i
$146$		0			0
$147$			3.46410i			0.285714i
$148$		0			0
$149$			17.0256i			1.39479i	0.716688	+	0.697394i	$0.245657\pi$
$149$			17.0256i			1.39479i	−0.716688	+	0.697394i	$0.754343\pi$
$150$		0			0
$151$		−	7.57301i		−	0.616283i	−0.951341	−	0.308142i	$-0.900293\pi$
$151$		−	7.57301i		−	0.616283i	0.951341	−	0.308142i	$-0.0997070\pi$
$152$		0			0
$153$			20.1947i			1.63264i
$154$		0			0
$155$	−1.62772	+	7.57301i	−0.130742	+	0.608279i
$156$		0			0
$157$			15.1460i			1.20878i	0.796687	+	0.604392i	$0.206584\pi$
$157$			15.1460i			1.20878i	−0.796687	+	0.604392i	$0.793416\pi$
$158$		0			0
$159$	12.7446			1.01071
$160$		0			0
$161$		−	15.7359i		−	1.24017i
$162$		0			0
$163$	−24.7446			−1.93814			−0.969072	−	0.246779i	$-0.920628\pi$
$163$	−24.7446			−1.93814			−0.969072	+	0.246779i	$0.920628\pi$
$164$		0			0
$165$	−8.74456	−	1.87953i	−0.680763	−	0.146321i
$166$		0			0
$167$	−17.4891			−1.35335			−0.676675	−	0.736282i	$-0.736580\pi$
$167$	−17.4891			−1.35335			−0.676675	+	0.736282i	$0.736580\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$		−	11.6819i		−	0.893339i
$172$		0			0
$173$			18.9051i			1.43733i	0.695358	+	0.718663i	$0.255246\pi$
$173$			18.9051i			1.43733i	−0.695358	+	0.718663i	$0.744754\pi$
$174$		0			0
$175$	−10.8139	−	4.87375i	−0.817451	−	0.368421i
$176$		0			0
$177$	−16.7446			−1.25860
$178$		0			0
$179$	−4.88316			−0.364984			−0.182492	−	0.983207i	$-0.558416\pi$
$179$	−4.88316			−0.364984			−0.182492	+	0.983207i	$0.558416\pi$
$180$		0			0
$181$	−3.48913			−0.259345			−0.129672	−	0.991557i	$-0.541393\pi$
$181$	−3.48913			−0.259345			−0.129672	+	0.991557i	$0.541393\pi$
$182$		0			0
$183$			17.0256i			1.25857i
$184$		0			0
$185$	19.9307	+	4.28384i	1.46533	+	0.314954i
$186$		0			0
$187$	9.48913			0.693914
$188$		0			0
$189$		−	2.22938i		−	0.162164i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	−17.9307	−	9.62747i	−1.28404	−	0.689437i
$196$		0			0
$197$	−4.37228			−0.311512			−0.155756	−	0.987796i	$-0.549781\pi$
$197$	−4.37228			−0.311512			−0.155756	+	0.987796i	$0.549781\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$			10.0974i			0.712212i
$202$		0			0
$203$	−6.51087			−0.456974
$204$		0			0
$205$	4.74456	−	22.0742i	0.331375	−	1.54173i
$206$		0			0
$207$		−	22.3692i		−	1.55477i
$208$		0			0
$209$	−5.48913			−0.379691
$210$		0			0
$211$	−3.11684			−0.214572			−0.107286	−	0.994228i	$-0.534216\pi$
$211$	−3.11684			−0.214572			−0.107286	+	0.994228i	$0.534216\pi$
$212$		0			0
$213$	31.8614			2.18311
$214$		0			0
$215$	−0.302985	+	1.40965i	−0.0206634	+	0.0961370i
$216$		0			0
$217$			8.21782i			0.557862i
$218$		0			0
$219$		−	25.2434i		−	1.70579i
$220$		0			0
$221$	20.7446	+	5.98844i	1.39543	+	0.402826i
$222$		0			0
$223$	15.1168			1.01230			0.506149	−	0.862446i	$-0.331068\pi$
$223$	15.1168			1.01230			0.506149	+	0.862446i	$0.331068\pi$
$224$		0			0
$225$	−15.3723	−	6.92820i	−1.02482	−	0.461880i
$226$		0			0
$227$	5.48913			0.364326			0.182163	−	0.983268i	$-0.441690\pi$
$227$	5.48913			0.364326			0.182163	+	0.983268i	$0.441690\pi$
$228$		0			0
$229$		−	24.8935i		−	1.64501i	−0.568758	−	0.822505i	$-0.692576\pi$
$229$		−	24.8935i		−	1.64501i	0.568758	−	0.822505i	$-0.307424\pi$
$230$		0			0
$231$	−9.48913			−0.624339
$232$		0			0
$233$			7.86797i			0.515448i	0.966219	+	0.257724i	$0.0829725\pi$
$233$			7.86797i			0.515448i	−0.966219	+	0.257724i	$0.917028\pi$
$234$		0			0
$235$	−22.6753	−	4.87375i	−1.47917	−	0.317928i
$236$		0			0
$237$			11.9769i			0.777982i
$238$		0			0
$239$		−	9.45254i		−	0.611434i	−0.952122	−	0.305717i	$-0.901104\pi$
$239$		−	9.45254i		−	0.611434i	0.952122	−	0.305717i	$-0.0988962\pi$
$240$		0			0
$241$		−	8.21782i		−	0.529357i	−0.964337	−	0.264678i	$-0.914734\pi$
$241$		−	8.21782i		−	0.529357i	0.964337	−	0.264678i	$-0.0852658\pi$
$242$		0			0
$243$			22.3692i			1.43498i
$244$		0			0
$245$	3.00000	+	0.644810i	0.191663	+	0.0411954i
$246$		0			0
$247$	−12.0000	−	3.46410i	−0.763542	−	0.220416i
$248$		0			0
$249$			22.0742i			1.39890i
$250$		0			0
$251$	22.9783			1.45037			0.725187	−	0.688552i	$-0.241753\pi$
$251$	22.9783			1.45037			0.725187	+	0.688552i	$0.241753\pi$
$252$		0			0
$253$	−10.5109			−0.660813
$254$		0			0
$255$	33.0475	+	7.10313i	2.06952	+	0.444815i
$256$		0			0
$257$		−	14.2063i		−	0.886162i	−0.896481	−	0.443081i	$-0.853885\pi$
$257$		−	14.2063i		−	0.886162i	0.896481	−	0.443081i	$-0.146115\pi$
$258$		0			0
$259$	21.6277			1.34388
$260$		0			0
$261$	−9.25544			−0.572897
$262$		0			0
$263$		−	18.0202i		−	1.11117i	−0.831458	−	0.555587i	$-0.812493\pi$
$263$		−	18.0202i		−	1.11117i	0.831458	−	0.555587i	$-0.187507\pi$
$264$		0			0
$265$	2.37228	−	11.0371i	0.145728	−	0.678005i
$266$		0			0
$267$	4.74456			0.290363
$268$		0			0
$269$	2.74456			0.167339			0.0836695	−	0.996494i	$-0.473336\pi$
$269$	2.74456			0.167339			0.0836695	+	0.996494i	$0.473336\pi$
$270$		0			0
$271$		−	21.4294i		−	1.30174i	−0.759187	−	0.650872i	$-0.774403\pi$
$271$		−	21.4294i		−	1.30174i	0.759187	−	0.650872i	$-0.225597\pi$
$272$		0			0
$273$	−20.7446	−	5.98844i	−1.25552	−	0.362437i
$274$		0			0
$275$	−3.25544	+	7.22316i	−0.196310	+	0.435573i
$276$		0			0
$277$		−	23.3639i		−	1.40380i	−0.712277	−	0.701899i	$-0.752336\pi$
$277$		−	23.3639i		−	1.40380i	0.712277	−	0.701899i	$-0.247664\pi$
$278$		0			0
$279$			11.6819i			0.699379i
$280$		0			0
$281$			1.87953i			0.112123i	0.998427	+	0.0560616i	$0.0178543\pi$
$281$			1.87953i			0.112123i	−0.998427	+	0.0560616i	$0.982146\pi$
$282$		0			0
$283$			17.3205i			1.02960i	0.857311	+	0.514799i	$0.172133\pi$
$283$			17.3205i			1.02960i	−0.857311	+	0.514799i	$0.827867\pi$
$284$		0			0
$285$	−19.1168	−	4.10891i	−1.13238	−	0.243391i
$286$		0			0
$287$		−	23.9538i		−	1.41395i
$288$		0			0
$289$	−18.8614			−1.10949
$290$		0			0
$291$		−	17.0256i		−	0.998056i
$292$		0			0
$293$	2.13859			0.124938			0.0624690	−	0.998047i	$-0.480103\pi$
$293$	2.13859			0.124938			0.0624690	+	0.998047i	$0.480103\pi$
$294$		0			0
$295$	−3.11684	+	14.5012i	−0.181470	+	0.844293i
$296$		0			0
$297$	−1.48913			−0.0864078
$298$		0			0
$299$	−22.9783	−	6.63325i	−1.32887	−	0.383611i
$300$		0			0
$301$			1.52967i			0.0881688i
$302$		0			0
$303$			37.2203i			2.13825i
$304$		0			0
$305$	14.7446	+	3.16915i	0.844271	+	0.181465i
$306$		0			0
$307$	−18.2337			−1.04065			−0.520326	−	0.853968i	$-0.674189\pi$
$307$	−18.2337			−1.04065			−0.520326	+	0.853968i	$0.674189\pi$
$308$		0			0
$309$	26.2337			1.49238
$310$		0			0
$311$	−3.25544			−0.184599			−0.0922995	−	0.995731i	$-0.529422\pi$
$311$	−3.25544			−0.184599			−0.0922995	+	0.995731i	$0.529422\pi$
$312$		0			0
$313$		−	12.3267i		−	0.696748i	−0.937356	−	0.348374i	$-0.886734\pi$
$313$		−	12.3267i		−	0.696748i	0.937356	−	0.348374i	$-0.113266\pi$
$314$		0			0
$315$	−17.4891	−	3.75906i	−0.985401	−	0.211799i
$316$		0			0
$317$	16.9783			0.953594			0.476797	−	0.879014i	$-0.341798\pi$
$317$	16.9783			0.953594			0.476797	+	0.879014i	$0.341798\pi$
$318$		0			0
$319$			4.34896i			0.243495i
$320$		0			0
$321$	34.2337			1.91074
$322$		0			0
$323$	20.7446			1.15426
$324$		0			0
$325$	−11.6753	+	13.7364i	−0.647627	+	0.761957i
$326$		0			0
$327$	−7.11684			−0.393562
$328$		0			0
$329$	−24.6060			−1.35657
$330$		0			0
$331$		−	10.3923i		−	0.571213i	−0.958347	−	0.285606i	$-0.907805\pi$
$331$		−	10.3923i		−	0.571213i	0.958347	−	0.285606i	$-0.0921950\pi$
$332$		0			0
$333$	30.7446			1.68479
$334$		0			0
$335$	8.74456	+	1.87953i	0.477766	+	0.102690i
$336$		0			0
$337$			1.52967i			0.0833265i	0.999132	+	0.0416632i	$0.0132657\pi$
$337$			1.52967i			0.0833265i	−0.999132	+	0.0416632i	$0.986734\pi$
$338$		0			0
$339$	9.48913			0.515379
$340$		0			0
$341$	5.48913			0.297253
$342$		0			0
$343$	19.8614			1.07242
$344$		0			0
$345$	−36.6060	−	7.86797i	−1.97080	−	0.423597i
$346$		0			0
$347$		−	20.8395i		−	1.11872i	−0.828924	−	0.559362i	$-0.811046\pi$
$347$		−	20.8395i		−	1.11872i	0.828924	−	0.559362i	$-0.188954\pi$
$348$		0			0
$349$		−	27.4728i		−	1.47058i	−0.677751	−	0.735292i	$-0.737045\pi$
$349$		−	27.4728i		−	1.47058i	0.677751	−	0.735292i	$-0.262955\pi$
$350$		0			0
$351$	−3.25544	−	0.939764i	−0.173762	−	0.0501609i
$352$		0			0
$353$	11.4891			0.611504			0.305752	−	0.952111i	$-0.401092\pi$
$353$	11.4891			0.611504			0.305752	+	0.952111i	$0.401092\pi$
$354$		0			0
$355$	5.93070	−	27.5928i	0.314769	−	1.46447i
$356$		0			0
$357$	35.8614			1.89799
$358$		0			0
$359$			2.87419i			0.151694i	0.997119	+	0.0758471i	$0.0241661\pi$
$359$			2.87419i			0.151694i	−0.997119	+	0.0758471i	$0.975834\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$		−	21.4294i		−	1.12475i
$364$		0			0
$365$	−21.8614	−	4.69882i	−1.14428	−	0.245947i
$366$		0			0
$367$			4.75372i			0.248142i	0.992273	+	0.124071i	$0.0395951\pi$
$367$			4.75372i			0.248142i	−0.992273	+	0.124071i	$0.960405\pi$
$368$		0			0
$369$		−	34.0511i		−	1.77263i
$370$		0			0
$371$		−	11.9769i		−	0.621809i
$372$		0			0
$373$			9.50744i			0.492277i	0.969235	+	0.246138i	$0.0791618\pi$
$373$			9.50744i			0.492277i	−0.969235	+	0.246138i	$0.920838\pi$
$374$		0			0
$375$	−16.7446	+	22.7190i	−0.864685	+	1.17321i
$376$		0			0
$377$	−2.74456	+	9.50744i	−0.141352	+	0.489658i
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$	−8.74456			−0.447998
$382$		0			0
$383$	16.8832			0.862689			0.431344	−	0.902187i	$-0.358039\pi$
$383$	16.8832			0.862689			0.431344	+	0.902187i	$0.358039\pi$
$384$		0			0
$385$	−1.76631	+	8.21782i	−0.0900196	+	0.418819i
$386$		0			0
$387$			2.17448i			0.110535i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	39.7228			2.00887
$392$		0			0
$393$		−	4.10891i		−	0.207267i
$394$		0			0
$395$	10.3723	+	2.22938i	0.521886	+	0.112172i
$396$		0			0
$397$	−8.51087			−0.427149			−0.213574	−	0.976927i	$-0.568511\pi$
$397$	−8.51087			−0.427149			−0.213574	+	0.976927i	$0.568511\pi$
$398$		0			0
$399$	−20.7446			−1.03853
$400$		0			0
$401$			32.1716i			1.60657i	0.595593	+	0.803286i	$0.296917\pi$
$401$			32.1716i			1.60657i	−0.595593	+	0.803286i	$0.703083\pi$
$402$		0			0
$403$	12.0000	+	3.46410i	0.597763	+	0.172559i
$404$		0			0
$405$	16.9307	+	3.63903i	0.841293	+	0.180825i
$406$		0			0
$407$		−	14.4463i		−	0.716077i
$408$		0			0
$409$		−	5.63858i		−	0.278810i	−0.990235	−	0.139405i	$-0.955481\pi$
$409$		−	5.63858i		−	0.278810i	0.990235	−	0.139405i	$-0.0445190\pi$
$410$		0			0
$411$			15.1460i			0.747098i
$412$		0			0
$413$			15.7359i			0.774315i
$414$		0			0
$415$	19.1168	+	4.10891i	0.938409	+	0.201699i
$416$		0			0
$417$			16.0858i			0.787725i
$418$		0			0
$419$	−19.1168			−0.933919			−0.466959	−	0.884279i	$-0.654651\pi$
$419$	−19.1168			−0.933919			−0.466959	+	0.884279i	$0.654651\pi$
$420$		0			0
$421$			11.0371i			0.537916i	0.963152	+	0.268958i	$0.0866793\pi$
$421$			11.0371i			0.537916i	−0.963152	+	0.268958i	$0.913321\pi$
$422$		0			0
$423$	−34.9783			−1.70070
$424$		0			0
$425$	12.3030	−	27.2978i	0.596782	−	1.32414i
$426$		0			0
$427$	16.0000			0.774294
$428$		0			0
$429$	−4.00000	+	13.8564i	−0.193122	+	0.668994i
$430$		0			0
$431$			26.4781i			1.27540i	0.770283	+	0.637702i	$0.220115\pi$
$431$			26.4781i			1.27540i	−0.770283	+	0.637702i	$0.779885\pi$
$432$		0			0
$433$			34.4010i			1.65320i	0.562786	+	0.826602i	$0.309729\pi$
$433$			34.4010i			1.65320i	−0.562786	+	0.826602i	$0.690271\pi$
$434$		0			0
$435$	−3.25544	+	15.1460i	−0.156086	+	0.726196i
$436$		0			0
$437$	−22.9783			−1.09920
$438$		0			0
$439$	−22.2337			−1.06116			−0.530578	−	0.847636i	$-0.678025\pi$
$439$	−22.2337			−1.06116			−0.530578	+	0.847636i	$0.678025\pi$
$440$		0			0
$441$	4.62772			0.220368
$442$		0			0
$443$		−	4.40387i		−	0.209234i	−0.994513	−	0.104617i	$-0.966638\pi$
$443$		−	4.40387i		−	0.209234i	0.994513	−	0.104617i	$-0.0333617\pi$
$444$		0			0
$445$	0.883156	−	4.10891i	0.0418656	−	0.194781i
$446$		0			0
$447$	42.9783			2.03280
$448$		0			0
$449$			7.51811i			0.354802i	0.984139	+	0.177401i	$0.0567689\pi$
$449$			7.51811i			0.354802i	−0.984139	+	0.177401i	$0.943231\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	−19.1168			−0.898188
$454$		0			0
$455$	−9.04755	+	16.8506i	−0.424156	+	0.789970i
$456$		0			0
$457$	−24.9783			−1.16843			−0.584217	−	0.811598i	$-0.698598\pi$
$457$	−24.9783			−1.16843			−0.584217	+	0.811598i	$0.698598\pi$
$458$		0			0
$459$	5.62772			0.262679
$460$		0			0
$461$			1.63948i			0.0763580i	0.999271	+	0.0381790i	$0.0121557\pi$
$461$			1.63948i			0.0763580i	−0.999271	+	0.0381790i	$0.987844\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	19.1168	+	4.10891i	0.886522	+	0.190546i
$466$		0			0
$467$			27.4179i			1.26875i	0.773027	+	0.634374i	$0.218742\pi$
$467$			27.4179i			1.26875i	−0.773027	+	0.634374i	$0.781258\pi$
$468$		0			0
$469$	9.48913			0.438167
$470$		0			0
$471$	38.2337			1.76172
$472$		0			0
$473$	1.02175			0.0469801
$474$		0			0
$475$	−7.11684	+	15.7908i	−0.326543	+	0.724533i
$476$		0			0
$477$		−	17.0256i		−	0.779547i
$478$		0			0
$479$			18.2603i			0.834333i	0.908830	+	0.417167i	$0.136977\pi$
$479$			18.2603i			0.834333i	−0.908830	+	0.417167i	$0.863023\pi$
$480$		0			0
$481$	9.11684	−	31.5817i	0.415692	−	1.44000i
$482$		0			0
$483$	−39.7228			−1.80745
$484$		0			0
$485$	−14.7446	−	3.16915i	−0.669516	−	0.143904i
$486$		0			0
$487$	1.48913			0.0674787			0.0337394	−	0.999431i	$-0.489258\pi$
$487$	1.48913			0.0674787			0.0337394	+	0.999431i	$0.489258\pi$
$488$		0			0
$489$			62.4636i			2.82470i
$490$		0			0
$491$	25.6277			1.15656			0.578281	−	0.815837i	$-0.303724\pi$
$491$	25.6277			1.15656			0.578281	+	0.815837i	$0.303724\pi$
$492$		0			0
$493$		−	16.4356i		−	0.740224i
$494$		0			0
$495$	−2.51087	+	11.6819i	−0.112855	+	0.525063i
$496$		0			0
$497$		−	29.9422i		−	1.34309i
$498$		0			0
$499$		−	16.0309i		−	0.717641i	−0.933407	−	0.358821i	$-0.883179\pi$
$499$		−	16.0309i		−	0.717641i	0.933407	−	0.358821i	$-0.116821\pi$
$500$		0			0
$501$			44.1485i			1.97241i
$502$		0			0
$503$			6.63325i			0.295762i	0.989005	+	0.147881i	$0.0472453\pi$
$503$			6.63325i			0.295762i	−0.989005	+	0.147881i	$0.952755\pi$
$504$		0			0
$505$	32.2337	+	6.92820i	1.43438	+	0.308301i
$506$		0			0
$507$	−17.4891	+	27.7677i	−0.776719	+	1.23321i
$508$		0			0
$509$			3.16915i			0.140470i	0.997530	+	0.0702350i	$0.0223749\pi$
$509$			3.16915i			0.140470i	−0.997530	+	0.0702350i	$0.977625\pi$
$510$		0			0
$511$	−23.7228			−1.04944
$512$		0			0
$513$	−3.25544			−0.143731
$514$		0			0
$515$	4.88316	−	22.7190i	0.215178	−	1.00112i
$516$		0			0
$517$			16.4356i			0.722839i
$518$		0			0
$519$	47.7228			2.09480
$520$		0			0
$521$	18.6060			0.815142			0.407571	−	0.913173i	$-0.366376\pi$
$521$	18.6060			0.815142			0.407571	+	0.913173i	$0.366376\pi$
$522$		0			0
$523$		−	10.3923i		−	0.454424i	−0.973845	−	0.227212i	$-0.927039\pi$
$523$		−	10.3923i		−	0.454424i	0.973845	−	0.227212i	$-0.0729610\pi$
$524$		0			0
$525$	−12.3030	+	27.2978i	−0.536946	+	1.19138i
$526$		0			0
$527$	−20.7446			−0.903647
$528$		0			0
$529$	−21.0000			−0.913043
$530$		0			0
$531$			22.3692i			0.970740i
$532$		0			0
$533$	−34.9783	−	10.0974i	−1.51508	−	0.437365i
$534$		0			0
$535$	6.37228	−	29.6472i	0.275498	−	1.28176i
$536$		0			0
$537$			12.3267i			0.531938i
$538$		0			0
$539$		−	2.17448i		−	0.0936615i
$540$		0			0
$541$		−	30.5321i		−	1.31268i	−0.754466	−	0.656339i	$-0.772104\pi$
$541$		−	30.5321i		−	1.31268i	0.754466	−	0.656339i	$-0.227896\pi$
$542$		0			0
$543$			8.80773i			0.377976i
$544$		0			0
$545$	−1.32473	+	6.16337i	−0.0567454	+	0.264010i
$546$		0			0
$547$			21.4294i			0.916256i	0.888886	+	0.458128i	$0.151480\pi$
$547$			21.4294i			0.916256i	−0.888886	+	0.458128i	$0.848520\pi$
$548$		0			0
$549$	22.7446			0.970714
$550$		0			0
$551$			9.50744i			0.405031i
$552$		0			0
$553$	11.2554			0.478630
$554$		0			0
$555$	10.8139	−	50.3118i	0.459023	−	2.13562i
$556$		0			0
$557$	−1.11684			−0.0473222			−0.0236611	−	0.999720i	$-0.507532\pi$
$557$	−1.11684			−0.0473222			−0.0236611	+	0.999720i	$0.507532\pi$
$558$		0			0
$559$	2.23369	+	0.644810i	0.0944749	+	0.0272726i
$560$		0			0
$561$		−	23.9538i		−	1.01133i
$562$		0			0
$563$		−	1.82462i		−	0.0768988i	−0.999261	−	0.0384494i	$-0.987758\pi$
$563$		−	1.82462i		−	0.0768988i	0.999261	−	0.0384494i	$-0.0122418\pi$
$564$		0			0
$565$	1.76631	−	8.21782i	0.0743093	−	0.345726i
$566$		0			0
$567$	18.3723			0.771563
$568$		0			0
$569$	−16.3723			−0.686362			−0.343181	−	0.939269i	$-0.611504\pi$
$569$	−16.3723			−0.686362			−0.343181	+	0.939269i	$0.611504\pi$
$570$		0			0
$571$	−12.8832			−0.539143			−0.269572	−	0.962980i	$-0.586882\pi$
$571$	−12.8832			−0.539143			−0.269572	+	0.962980i	$0.586882\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−13.6277	+	30.2372i	−0.568315	+	1.26098i
$576$		0			0
$577$	20.9783			0.873336			0.436668	−	0.899623i	$-0.356158\pi$
$577$	20.9783			0.873336			0.436668	+	0.899623i	$0.356158\pi$
$578$		0			0
$579$			35.3407i			1.46871i
$580$		0			0
$581$	20.7446			0.860629
$582$		0			0
$583$	−8.00000			−0.331326
$584$		0			0
$585$	−12.8614	+	23.9538i	−0.531754	+	0.990366i
$586$		0			0
$587$	−19.7228			−0.814048			−0.407024	−	0.913418i	$-0.633433\pi$
$587$	−19.7228			−0.814048			−0.407024	+	0.913418i	$0.633433\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$			11.0371i			0.454006i
$592$		0			0
$593$	32.2337			1.32368			0.661839	−	0.749646i	$-0.269776\pi$
$593$	32.2337			1.32368			0.661839	+	0.749646i	$0.269776\pi$
$594$		0			0
$595$	6.67527	−	31.0569i	0.273659	−	1.27321i
$596$		0			0
$597$			20.1947i			0.826514i
$598$		0			0
$599$	−31.7228			−1.29616			−0.648080	−	0.761573i	$-0.724428\pi$
$599$	−31.7228			−1.29616			−0.648080	+	0.761573i	$0.724428\pi$
$600$		0			0
$601$	−32.3723			−1.32049			−0.660246	−	0.751049i	$-0.729548\pi$
$601$	−32.3723			−1.32049			−0.660246	+	0.751049i	$0.729548\pi$
$602$		0			0
$603$	13.4891			0.549320
$604$		0			0
$605$	−18.5584	−	3.98889i	−0.754507	−	0.162171i
$606$		0			0
$607$		−	3.46410i		−	0.140604i	−0.997526	−	0.0703018i	$-0.977604\pi$
$607$		−	3.46410i		−	0.140604i	0.997526	−	0.0703018i	$-0.0223962\pi$
$608$		0			0
$609$			16.4356i			0.666006i
$610$		0			0
$611$	−10.3723	+	35.9306i	−0.419618	+	1.45360i
$612$		0			0
$613$	−19.4891			−0.787158			−0.393579	−	0.919291i	$-0.628763\pi$
$613$	−19.4891			−0.787158			−0.393579	+	0.919291i	$0.628763\pi$
$614$		0			0
$615$	−55.7228	−	11.9769i	−2.24696	−	0.482954i
$616$		0			0
$617$	14.7446			0.593594			0.296797	−	0.954941i	$-0.404082\pi$
$617$	14.7446			0.593594			0.296797	+	0.954941i	$0.404082\pi$
$618$		0			0
$619$			9.10268i			0.365868i	0.983125	+	0.182934i	$0.0585594\pi$
$619$			9.10268i			0.365868i	−0.983125	+	0.182934i	$0.941441\pi$
$620$		0			0
$621$	−6.23369			−0.250149
$622$		0			0
$623$		−	4.45877i		−	0.178637i
$624$		0			0
$625$	16.5584	+	18.7302i	0.662337	+	0.749206i
$626$		0			0
$627$			13.8564i			0.553372i
$628$		0			0
$629$			54.5957i			2.17687i
$630$		0			0
$631$		−	40.4443i		−	1.61006i	−0.593232	−	0.805031i	$-0.702148\pi$
$631$		−	40.4443i		−	1.61006i	0.593232	−	0.805031i	$-0.297852\pi$
$632$		0			0
$633$			7.86797i			0.312724i
$634$		0			0
$635$	−1.62772	+	7.57301i	−0.0645940	+	0.300526i
$636$		0			0
$637$	1.37228	−	4.75372i	0.0543718	−	0.188349i
$638$		0			0
$639$		−	42.5639i		−	1.68380i
$640$		0			0
$641$	−16.9783			−0.670601			−0.335300	−	0.942111i	$-0.608838\pi$
$641$	−16.9783			−0.670601			−0.335300	+	0.942111i	$0.608838\pi$
$642$		0			0
$643$	37.4891			1.47843			0.739213	−	0.673471i	$-0.235197\pi$
$643$	37.4891			1.47843			0.739213	+	0.673471i	$0.235197\pi$
$644$		0			0
$645$	3.55842	+	0.764836i	0.140113	+	0.0301154i
$646$		0			0
$647$		−	21.0796i		−	0.828723i	−0.910112	−	0.414362i	$-0.864005\pi$
$647$		−	21.0796i		−	0.828723i	0.910112	−	0.414362i	$-0.135995\pi$
$648$		0			0
$649$	10.5109			0.412588
$650$		0			0
$651$	20.7446			0.813044
$652$		0			0
$653$		−	17.0256i		−	0.666261i	−0.942881	−	0.333131i	$-0.891895\pi$
$653$		−	17.0256i		−	0.666261i	0.942881	−	0.333131i	$-0.108105\pi$
$654$		0			0
$655$	−3.55842	−	0.764836i	−0.139039	−	0.0298846i
$656$		0			0
$657$	−33.7228			−1.31565
$658$		0			0
$659$	40.4674			1.57639			0.788193	−	0.615429i	$-0.211017\pi$
$659$	40.4674			1.57639			0.788193	+	0.615429i	$0.211017\pi$
$660$		0			0
$661$			9.50744i			0.369797i	0.982758	+	0.184898i	$0.0591956\pi$
$661$			9.50744i			0.369797i	−0.982758	+	0.184898i	$0.940804\pi$
$662$		0			0
$663$	15.1168	−	52.3663i	0.587090	−	2.03374i
$664$		0			0
$665$	−3.86141	+	17.9653i	−0.149739	+	0.696665i
$666$		0			0
$667$			18.2054i			0.704915i
$668$		0			0
$669$		−	38.1600i		−	1.47535i
$670$		0			0
$671$		−	10.6873i		−	0.412577i
$672$		0			0
$673$		−	9.74749i		−	0.375738i	−0.982194	−	0.187869i	$-0.939842\pi$
$673$		−	9.74749i		−	0.375738i	0.982194	−	0.187869i	$-0.0601581\pi$
$674$		0			0
$675$	−1.93070	+	4.28384i	−0.0743128	+	0.164885i
$676$		0			0
$677$		−	39.0998i		−	1.50273i	−0.659889	−	0.751363i	$-0.729397\pi$
$677$		−	39.0998i		−	1.50273i	0.659889	−	0.751363i	$-0.270603\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$		−	13.8564i		−	0.530979i
$682$		0			0
$683$	8.74456			0.334601			0.167301	−	0.985906i	$-0.446495\pi$
$683$	8.74456			0.334601			0.167301	+	0.985906i	$0.446495\pi$
$684$		0			0
$685$	13.1168	+	2.81929i	0.501169	+	0.107720i
$686$		0			0
$687$	−62.8397			−2.39748
$688$		0			0
$689$	−17.4891	−	5.04868i	−0.666283	−	0.192339i
$690$		0			0
$691$			39.3947i			1.49865i	0.662204	+	0.749323i	$0.269621\pi$
$691$			39.3947i			1.49865i	−0.662204	+	0.749323i	$0.730379\pi$
$692$		0			0
$693$			12.6766i			0.481544i
$694$		0			0
$695$	13.9307	+	2.99422i	0.528422	+	0.113577i
$696$		0			0
$697$	60.4674			2.29037
$698$		0			0
$699$	19.8614			0.751227
$700$		0			0
$701$	16.9783			0.641260			0.320630	−	0.947205i	$-0.396105\pi$
$701$	16.9783			0.641260			0.320630	+	0.947205i	$0.396105\pi$
$702$		0			0
$703$		−	31.5817i		−	1.19113i
$704$		0			0
$705$	−12.3030	+	57.2400i	−0.463357	+	2.15578i
$706$		0			0
$707$	34.9783			1.31549
$708$		0			0
$709$			20.7846i			0.780582i	0.920691	+	0.390291i	$0.127626\pi$
$709$			20.7846i			0.780582i	−0.920691	+	0.390291i	$0.872374\pi$
$710$		0			0
$711$	16.0000			0.600047
$712$		0			0
$713$	22.9783			0.860542
$714$		0			0
$715$	11.2554	+	6.04334i	0.420929	+	0.226008i
$716$		0			0
$717$	−23.8614			−0.891121
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$		−	24.6535i		−	0.918143i
$722$		0			0
$723$	−20.7446			−0.771499
$724$		0			0
$725$	12.5109	+	5.63858i	0.464642	+	0.209412i
$726$		0			0
$727$			2.17448i			0.0806470i	0.999187	+	0.0403235i	$0.0128389\pi$
$727$			2.17448i			0.0806470i	−0.999187	+	0.0403235i	$0.987161\pi$
$728$		0			0
$729$	33.2337			1.23088
$730$		0			0
$731$	−3.86141			−0.142819
$732$		0			0
$733$	16.0951			0.594486			0.297243	−	0.954802i	$-0.403933\pi$
$733$	16.0951			0.594486			0.297243	+	0.954802i	$0.403933\pi$
$734$		0			0
$735$	1.62772	−	7.57301i	0.0600393	−	0.279335i
$736$		0			0
$737$		−	6.33830i		−	0.233474i
$738$		0			0
$739$			28.1176i			1.03432i	0.855888	+	0.517161i	$0.173011\pi$
$739$			28.1176i			1.03432i	−0.855888	+	0.517161i	$0.826989\pi$
$740$		0			0
$741$	−8.74456	+	30.2921i	−0.321240	+	1.11281i
$742$		0			0
$743$	7.11684			0.261092			0.130546	−	0.991442i	$-0.458327\pi$
$743$	7.11684			0.261092			0.130546	+	0.991442i	$0.458327\pi$
$744$		0			0
$745$	8.00000	−	37.2203i	0.293097	−	1.36364i
$746$		0			0
$747$	29.4891			1.07895
$748$		0			0
$749$		−	32.1716i		−	1.17552i
$750$		0			0
$751$	−25.4891			−0.930111			−0.465056	−	0.885281i	$-0.653966\pi$
$751$	−25.4891			−0.930111			−0.465056	+	0.885281i	$0.653966\pi$
$752$		0			0
$753$		−	58.0049i		−	2.11381i
$754$		0			0
$755$	−3.55842	+	16.5557i	−0.129504	+	0.602523i
$756$		0			0
$757$			45.4381i			1.65148i	0.564055	+	0.825738i	$0.309241\pi$
$757$			45.4381i			1.65148i	−0.564055	+	0.825738i	$0.690759\pi$
$758$		0			0
$759$			26.5330i			0.963087i
$760$		0			0
$761$			1.87953i			0.0681328i	0.999420	+	0.0340664i	$0.0108458\pi$
$761$			1.87953i			0.0681328i	−0.999420	+	0.0340664i	$0.989154\pi$
$762$		0			0
$763$			6.68815i			0.242127i
$764$		0			0
$765$	9.48913	−	44.1485i	0.343080	−	1.59619i
$766$		0			0
$767$	22.9783	+	6.63325i	0.829697	+	0.239513i
$768$		0			0
$769$		−	16.4356i		−	0.592685i	−0.955082	−	0.296342i	$-0.904233\pi$
$769$		−	16.4356i		−	0.592685i	0.955082	−	0.296342i	$-0.0957669\pi$
$770$		0			0
$771$	−35.8614			−1.29152
$772$		0			0
$773$	30.6060			1.10082			0.550410	−	0.834894i	$-0.314471\pi$
$773$	30.6060			1.10082			0.550410	+	0.834894i	$0.314471\pi$
$774$		0			0
$775$	7.11684	−	15.7908i	0.255645	−	0.567224i
$776$		0			0
$777$		−	54.5957i		−	1.95861i
$778$		0			0
$779$	−34.9783			−1.25323
$780$		0			0
$781$	−20.0000			−0.715656
$782$		0			0
$783$			2.57924i			0.0921745i
$784$		0			0
$785$	7.11684	−	33.1113i	0.254011	−	1.18179i
$786$		0			0
$787$	44.0000			1.56843			0.784215	−	0.620489i	$-0.213066\pi$
$787$	44.0000			1.56843			0.784215	+	0.620489i	$0.213066\pi$
$788$		0			0
$789$	−45.4891			−1.61946
$790$		0			0
$791$		−	8.91754i		−	0.317071i
$792$		0			0
$793$	6.74456	−	23.3639i	0.239506	−	0.829675i
$794$		0			0
$795$	−27.8614	−	5.98844i	−0.988142	−	0.212388i
$796$		0			0
$797$		−	25.2434i		−	0.894166i	−0.894492	−	0.447083i	$-0.852463\pi$
$797$		−	25.2434i		−	0.894166i	0.894492	−	0.447083i	$-0.147537\pi$
$798$		0			0
$799$		−	62.1138i		−	2.19743i
$800$		0			0
$801$		−	6.33830i		−	0.223953i
$802$		0			0
$803$			15.8457i			0.559184i
$804$		0			0
$805$	−7.39403	+	34.4010i	−0.260605	+	1.21247i
$806$		0			0
$807$		−	6.92820i		−	0.243884i
$808$		0			0
$809$	39.3505			1.38349			0.691746	−	0.722141i	$-0.256842\pi$
$809$	39.3505			1.38349			0.691746	+	0.722141i	$0.256842\pi$
$810$		0			0
$811$			41.9740i			1.47391i	0.675944	+	0.736953i	$0.263736\pi$
$811$			41.9740i			1.47391i	−0.675944	+	0.736953i	$0.736264\pi$
$812$		0			0
$813$	−54.0951			−1.89720
$814$		0			0
$815$	54.0951	+	11.6270i	1.89487	+	0.407277i
$816$		0			0
$817$	2.23369			0.0781468
$818$		0			0
$819$	−8.00000	+	27.7128i	−0.279543	+	0.968364i
$820$		0			0
$821$		−	25.5932i		−	0.893210i	−0.894731	−	0.446605i	$-0.852633\pi$
$821$		−	25.5932i		−	0.893210i	0.894731	−	0.446605i	$-0.147367\pi$
$822$		0			0
$823$			54.5408i			1.90117i	0.310462	+	0.950586i	$0.399516\pi$
$823$			54.5408i			1.90117i	−0.310462	+	0.950586i	$0.600484\pi$
$824$		0			0
$825$	18.2337	+	8.21782i	0.634816	+	0.286108i
$826$		0			0
$827$	15.2554			0.530484			0.265242	−	0.964182i	$-0.414548\pi$
$827$	15.2554			0.530484			0.265242	+	0.964182i	$0.414548\pi$
$828$		0			0
$829$	−48.2337			−1.67523			−0.837613	−	0.546265i	$-0.816049\pi$
$829$	−48.2337			−1.67523			−0.837613	+	0.546265i	$0.816049\pi$
$830$		0			0
$831$	−58.9783			−2.04593
$832$		0			0
$833$			8.21782i			0.284731i
$834$		0			0
$835$	38.2337	+	8.21782i	1.32313	+	0.284390i
$836$		0			0
$837$	3.25544			0.112524
$838$		0			0
$839$		−	13.5615i		−	0.468193i	−0.972213	−	0.234097i	$-0.924787\pi$
$839$		−	13.5615i		−	0.468193i	0.972213	−	0.234097i	$-0.0752132\pi$
$840$		0			0
$841$	−21.4674			−0.740254
$842$		0			0
$843$	4.74456			0.163411
$844$		0			0
$845$	20.7921	+	20.3147i	0.715270	+	0.698848i
$846$		0			0
$847$	−20.1386			−0.691970
$848$		0			0
$849$	43.7228			1.50056
$850$		0			0
$851$		−	60.4743i		−	2.07303i
$852$		0			0
$853$	5.11684			0.175197			0.0875987	−	0.996156i	$-0.472081\pi$
$853$	5.11684			0.175197			0.0875987	+	0.996156i	$0.472081\pi$
$854$		0			0
$855$	−5.48913	+	25.5383i	−0.187724	+	0.873393i
$856$		0			0
$857$			22.7739i			0.777943i	0.921250	+	0.388972i	$0.127170\pi$
$857$			22.7739i			0.777943i	−0.921250	+	0.388972i	$0.872830\pi$
$858$		0			0
$859$	32.4674			1.10777			0.553886	−	0.832592i	$-0.313144\pi$
$859$	32.4674			1.10777			0.553886	+	0.832592i	$0.313144\pi$
$860$		0			0
$861$	−60.4674			−2.06072
$862$		0			0
$863$	3.86141			0.131444			0.0657219	−	0.997838i	$-0.479065\pi$
$863$	3.86141			0.131444			0.0657219	+	0.997838i	$0.479065\pi$
$864$		0			0
$865$	8.88316	−	41.3292i	0.302036	−	1.40523i
$866$		0			0
$867$			47.6126i			1.61701i
$868$		0			0
$869$		−	7.51811i		−	0.255034i
$870$		0			0
$871$	4.00000	−	13.8564i	0.135535	−	0.469506i
$872$		0			0
$873$	−22.7446			−0.769787
$874$		0			0
$875$	21.3505	+	15.7359i	0.721780	+	0.531972i
$876$		0			0
$877$	−47.3505			−1.59891			−0.799457	−	0.600723i	$-0.794879\pi$
$877$	−47.3505			−1.59891			−0.799457	+	0.600723i	$0.794879\pi$
$878$		0			0
$879$		−	5.39853i		−	0.182088i
$880$		0			0
$881$	−54.6060			−1.83972			−0.919861	−	0.392245i	$-0.871699\pi$
$881$	−54.6060			−1.83972			−0.919861	+	0.392245i	$0.871699\pi$
$882$		0			0
$883$		−	58.6497i		−	1.97372i	−0.161581	−	0.986859i	$-0.551659\pi$
$883$		−	58.6497i		−	1.97372i	0.161581	−	0.986859i	$-0.448341\pi$
$884$		0			0
$885$	36.6060	+	7.86797i	1.23050	+	0.264479i
$886$		0			0
$887$		−	45.7330i		−	1.53556i	−0.640711	−	0.767782i	$-0.721360\pi$
$887$		−	45.7330i		−	1.53556i	0.640711	−	0.767782i	$-0.278640\pi$
$888$		0			0
$889$			8.21782i			0.275617i
$890$		0			0
$891$		−	12.2718i		−	0.411122i
$892$		0			0
$893$			35.9306i			1.20237i
$894$		0			0
$895$	10.6753	+	2.29451i	0.356835	+	0.0766969i
$896$		0			0
$897$	−16.7446	+	58.0049i	−0.559085	+	1.93673i
$898$		0			0
$899$		−	9.50744i		−	0.317091i
$900$		0			0
$901$	30.2337			1.00723
$902$		0			0
$903$	3.86141			0.128500
$904$		0			0
$905$	7.62772	+	1.63948i	0.253554	+	0.0544981i
$906$		0			0
$907$		−	0.644810i		−	0.0214106i	−0.999943	−	0.0107053i	$-0.996592\pi$
$907$		−	0.644810i		−	0.0214106i	0.999943	−	0.0107053i	$-0.00340766\pi$
$908$		0			0
$909$	49.7228			1.64920
$910$		0			0
$911$	−34.9783			−1.15888			−0.579441	−	0.815014i	$-0.696729\pi$
$911$	−34.9783			−1.15888			−0.579441	+	0.815014i	$0.696729\pi$
$912$		0			0
$913$		−	13.8564i		−	0.458580i
$914$		0			0
$915$	8.00000	−	37.2203i	0.264472	−	1.23046i
$916$		0			0
$917$	−3.86141			−0.127515
$918$		0			0
$919$	−42.9783			−1.41772			−0.708861	−	0.705348i	$-0.750791\pi$
$919$	−42.9783			−1.41772			−0.708861	+	0.705348i	$0.750791\pi$
$920$		0			0
$921$			46.0280i			1.51667i
$922$		0			0
$923$	−43.7228	−	12.6217i	−1.43915	−	0.415448i
$924$		0			0
$925$	−41.5584	−	18.7302i	−1.36643	−	0.615844i
$926$		0			0
$927$		−	35.0458i		−	1.15105i
$928$		0			0
$929$		−	47.9075i		−	1.57179i	−0.618357	−	0.785897i	$-0.712201\pi$
$929$		−	47.9075i		−	1.57179i	0.618357	−	0.785897i	$-0.287799\pi$
$930$		0			0
$931$		−	4.75372i		−	0.155797i
$932$		0			0
$933$			8.21782i			0.269039i
$934$		0			0
$935$	−20.7446	−	4.45877i	−0.678420	−	0.145817i
$936$		0			0
$937$		−	30.2921i		−	0.989598i	−0.869007	−	0.494799i	$-0.835242\pi$
$937$		−	30.2921i		−	0.989598i	0.869007	−	0.494799i	$-0.164758\pi$
$938$		0			0
$939$	−31.1168			−1.01546
$940$		0			0
$941$			40.6295i			1.32448i	0.749291	+	0.662241i	$0.230395\pi$
$941$			40.6295i			1.32448i	−0.749291	+	0.662241i	$0.769605\pi$
$942$		0			0
$943$	−66.9783			−2.18111
$944$		0			0
$945$	−1.04755	+	4.87375i	−0.0340767	+	0.158543i
$946$		0			0
$947$	−29.4891			−0.958268			−0.479134	−	0.877742i	$-0.659049\pi$
$947$	−29.4891			−0.958268			−0.479134	+	0.877742i	$0.659049\pi$
$948$		0			0
$949$	−10.0000	+	34.6410i	−0.324614	+	1.12449i
$950$		0			0
$951$		−	42.8588i		−	1.38979i
$952$		0			0
$953$			16.0858i			0.521070i	0.965464	+	0.260535i	$0.0838989\pi$
$953$			16.0858i			0.521070i	−0.965464	+	0.260535i	$0.916101\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	10.9783			0.354876
$958$		0			0
$959$	14.2337			0.459630
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		−	45.7330i		−	1.47373i
$964$		0			0
$965$	30.6060	+	6.57835i	0.985241	+	0.211764i
$966$		0			0
$967$	5.35053			0.172062			0.0860308	−	0.996292i	$-0.472582\pi$
$967$	5.35053			0.172062			0.0860308	+	0.996292i	$0.472582\pi$
$968$		0			0
$969$		−	52.3663i		−	1.68225i
$970$		0			0
$971$	12.6060			0.404545			0.202272	−	0.979329i	$-0.435167\pi$
$971$	12.6060			0.404545			0.202272	+	0.979329i	$0.435167\pi$
$972$		0			0
$973$	15.1168			0.484624
$974$		0			0
$975$	34.6753	+	29.4723i	1.11050	+	0.943869i
$976$		0			0
$977$	25.7228			0.822946			0.411473	−	0.911422i	$-0.365014\pi$
$977$	25.7228			0.822946			0.411473	+	0.911422i	$0.365014\pi$
$978$		0			0
$979$	−2.97825			−0.0951853
$980$		0			0
$981$			9.50744i			0.303549i
$982$		0			0
$983$	42.0951			1.34263			0.671313	−	0.741174i	$-0.265731\pi$
$983$	42.0951			1.34263			0.671313	+	0.741174i	$0.265731\pi$
$984$		0			0
$985$	9.55842	+	2.05446i	0.304557	+	0.0654604i
$986$		0			0
$987$			62.1138i			1.97710i
$988$		0			0
$989$	4.27719			0.136007
$990$		0			0
$991$	1.76631			0.0561088			0.0280544	−	0.999606i	$-0.491069\pi$
$991$	1.76631			0.0561088			0.0280544	+	0.999606i	$0.491069\pi$
$992$		0			0
$993$	−26.2337			−0.832501
$994$		0			0
$995$	17.4891	+	3.75906i	0.554443	+	0.119170i
$996$		0			0
$997$			4.34896i			0.137733i	0.997626	+	0.0688665i	$0.0219383\pi$
$997$			4.34896i			0.137733i	−0.997626	+	0.0688665i	$0.978062\pi$
$998$		0			0
$999$		−	8.56768i		−	0.271069i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.c.129.1		4
4.3	odd	2		130.2.c.b.129.4	yes	4
5.4	even	2		1040.2.f.d.129.4		4
12.11	even	2		1170.2.f.a.649.4		4
13.12	even	2		1040.2.f.d.129.1		4
20.3	even	4		650.2.d.e.51.1		8
20.7	even	4		650.2.d.e.51.8		8
20.19	odd	2		130.2.c.a.129.1	✓	4
52.31	even	4		1690.2.b.d.339.4		8
52.47	even	4		1690.2.b.d.339.8		8
52.51	odd	2		130.2.c.a.129.4	yes	4
60.59	even	2		1170.2.f.b.649.2		4
65.64	even	2	inner	1040.2.f.c.129.4		4
156.155	even	2		1170.2.f.b.649.1		4
260.47	odd	4		8450.2.a.cj.1.4		4
260.83	odd	4		8450.2.a.cj.1.1		4
260.99	even	4		1690.2.b.d.339.1		8
260.103	even	4		650.2.d.e.51.5		8
260.187	odd	4		8450.2.a.cn.1.4		4
260.203	odd	4		8450.2.a.cn.1.1		4
260.207	even	4		650.2.d.e.51.4		8
260.239	even	4		1690.2.b.d.339.5		8
260.259	odd	2		130.2.c.b.129.1	yes	4
780.779	even	2		1170.2.f.a.649.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.c.a.129.1	✓	4	20.19	odd	2
130.2.c.a.129.4	yes	4	52.51	odd	2
130.2.c.b.129.1	yes	4	260.259	odd	2
130.2.c.b.129.4	yes	4	4.3	odd	2
650.2.d.e.51.1		8	20.3	even	4
650.2.d.e.51.4		8	260.207	even	4
650.2.d.e.51.5		8	260.103	even	4
650.2.d.e.51.8		8	20.7	even	4
1040.2.f.c.129.1		4	1.1	even	1	trivial
1040.2.f.c.129.4		4	65.64	even	2	inner
1040.2.f.d.129.1		4	13.12	even	2
1040.2.f.d.129.4		4	5.4	even	2
1170.2.f.a.649.3		4	780.779	even	2
1170.2.f.a.649.4		4	12.11	even	2
1170.2.f.b.649.1		4	156.155	even	2
1170.2.f.b.649.2		4	60.59	even	2
1690.2.b.d.339.1		8	260.99	even	4
1690.2.b.d.339.4		8	52.31	even	4
1690.2.b.d.339.5		8	260.239	even	4
1690.2.b.d.339.8		8	52.47	even	4
8450.2.a.cj.1.1		4	260.83	odd	4
8450.2.a.cj.1.4		4	260.47	odd	4
8450.2.a.cn.1.1		4	260.203	odd	4
8450.2.a.cn.1.4		4	260.187	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1040.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -2.37228 q^{7} -3.37228 q^{9} +O(q^{10})\)	\(q-2.52434i q^{3} +(-2.18614 - 0.469882i) q^{5} -2.37228 q^{7} -3.37228 q^{9} +1.58457i q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.18614 + 5.51856i) q^{15} -5.98844i q^{17} +3.46410i q^{19} +5.98844i q^{21} +6.63325i q^{23} +(4.55842 + 2.05446i) q^{25} +0.939764i q^{27} +2.74456 q^{29} -3.46410i q^{31} +4.00000 q^{33} +(5.18614 + 1.11469i) q^{35} -9.11684 q^{37} +(8.74456 + 2.52434i) q^{39} +10.0974i q^{41} -0.644810i q^{43} +(7.37228 + 1.58457i) q^{45} +10.3723 q^{47} -1.37228 q^{49} -15.1168 q^{51} +5.04868i q^{53} +(0.744563 - 3.46410i) q^{55} +8.74456 q^{57} -6.63325i q^{59} -6.74456 q^{61} +8.00000 q^{63} +(3.81386 - 7.10313i) q^{65} -4.00000 q^{67} +16.7446 q^{69} +12.6217i q^{71} +10.0000 q^{73} +(5.18614 - 11.5070i) q^{75} -3.75906i q^{77} -4.74456 q^{79} -7.74456 q^{81} -8.74456 q^{83} +(-2.81386 + 13.0916i) q^{85} -6.92820i q^{87} +1.87953i q^{89} +(2.37228 - 8.21782i) q^{91} -8.74456 q^{93} +(1.62772 - 7.57301i) q^{95} +6.74456 q^{97} -5.34363i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) 4 * q - 3 * q^5 + 2 * q^7 - 2 * q^9	\( 4 q - 3 q^{5} + 2 q^{7} - 2 q^{9} - 4 q^{13} + q^{15} + q^{25} - 12 q^{29} + 16 q^{33} + 15 q^{35} - 2 q^{37} + 12 q^{39} + 18 q^{45} + 30 q^{47} + 6 q^{49} - 26 q^{51} - 20 q^{55} + 12 q^{57} - 4 q^{61} + 32 q^{63} + 21 q^{65} - 16 q^{67} + 44 q^{69} + 40 q^{73} + 15 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} - 17 q^{85} - 2 q^{91} - 12 q^{93} + 18 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 + 2 * q^7 - 2 * q^9 - 4 * q^13 + q^15 + q^25 - 12 * q^29 + 16 * q^33 + 15 * q^35 - 2 * q^37 + 12 * q^39 + 18 * q^45 + 30 * q^47 + 6 * q^49 - 26 * q^51 - 20 * q^55 + 12 * q^57 - 4 * q^61 + 32 * q^63 + 21 * q^65 - 16 * q^67 + 44 * q^69 + 40 * q^73 + 15 * q^75 + 4 * q^79 - 8 * q^81 - 12 * q^83 - 17 * q^85 - 2 * q^91 - 12 * q^93 + 18 * q^95 + 4 * q^97

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