LMFDB - Embedded newform 1040.2.f.d.129.2

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			129.2
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.d.129.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.792287i q^{3} +(-0.686141 - 2.12819i) q^{5} -3.37228 q^{7} +2.37228 q^{9} +O(q^{10})$	$q-0.792287i q^{3} +(-0.686141 - 2.12819i) q^{5} -3.37228 q^{7} +2.37228 q^{9} -5.04868i q^{11} +(1.00000 - 3.46410i) q^{13} +(-1.68614 + 0.543620i) q^{15} +2.67181i q^{17} +3.46410i q^{19} +2.67181i q^{21} +6.63325i q^{23} +(-4.05842 + 2.92048i) q^{25} -4.25639i q^{27} -8.74456 q^{29} -3.46410i q^{31} -4.00000 q^{33} +(2.31386 + 7.17687i) q^{35} -8.11684 q^{37} +(-2.74456 - 0.792287i) q^{39} -3.16915i q^{41} -9.30506i q^{43} +(-1.62772 - 5.04868i) q^{45} -4.62772 q^{47} +4.37228 q^{49} +2.11684 q^{51} +1.58457i q^{53} +(-10.7446 + 3.46410i) q^{55} +2.74456 q^{57} +6.63325i q^{59} +4.74456 q^{61} -8.00000 q^{63} +(-8.05842 + 0.248667i) q^{65} +4.00000 q^{67} +5.25544 q^{69} -3.96143i q^{71} -10.0000 q^{73} +(2.31386 + 3.21543i) q^{75} +17.0256i q^{77} +6.74456 q^{79} +3.74456 q^{81} -2.74456 q^{83} +(5.68614 - 1.83324i) q^{85} +6.92820i q^{87} +8.51278i q^{89} +(-3.37228 + 11.6819i) q^{91} -2.74456 q^{93} +(7.37228 - 2.37686i) q^{95} +4.74456 q^{97} -11.9769i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 15 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 - 15 * 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
$3$		−	0.792287i		−	0.457427i	−0.973494	−	0.228714i	$-0.926548\pi$
$3$		−	0.792287i		−	0.457427i	0.973494	−	0.228714i	$-0.0734519\pi$
$4$		0			0
$5$	−0.686141	−	2.12819i	−0.306851	−	0.951757i
$5$	−0.686141	−	2.12819i	−0.306851	−	0.951757i
$6$		0			0
$7$	−3.37228			−1.27460			−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228			−1.27460			−0.637301	+	0.770615i	$0.719949\pi$
$8$		0			0
$9$	2.37228			0.790760
$10$		0			0
$11$		−	5.04868i		−	1.52223i	−0.648615	−	0.761116i	$-0.724652\pi$
$11$		−	5.04868i		−	1.52223i	0.648615	−	0.761116i	$-0.275348\pi$
$12$		0			0
$13$	1.00000	−	3.46410i	0.277350	−	0.960769i
$13$	1.00000	−	3.46410i	0.277350	−	0.960769i
$14$		0			0
$15$	−1.68614	+	0.543620i	−0.435360	+	0.140362i
$16$		0			0
$17$			2.67181i			0.648010i	0.946055	+	0.324005i	$0.105030\pi$
$17$			2.67181i			0.648010i	−0.946055	+	0.324005i	$0.894970\pi$
$18$		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
$21$			2.67181i			0.583038i
$22$		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
$25$	−4.05842	+	2.92048i	−0.811684	+	0.584096i
$26$		0			0
$27$		−	4.25639i		−	0.819142i
$28$		0			0
$29$	−8.74456			−1.62382			−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456			−1.62382			−0.811912	+	0.583779i	$0.801573\pi$
$30$		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
$35$	2.31386	+	7.17687i	0.391114	+	1.21311i
$36$		0			0
$37$	−8.11684			−1.33440			−0.667200	−	0.744878i	$-0.732508\pi$
$37$	−8.11684			−1.33440			−0.667200	+	0.744878i	$0.732508\pi$
$38$		0			0
$39$	−2.74456	−	0.792287i	−0.439482	−	0.126867i
$40$		0			0
$41$		−	3.16915i		−	0.494938i	−0.968896	−	0.247469i	$-0.920401\pi$
$41$		−	3.16915i		−	0.494938i	0.968896	−	0.247469i	$-0.0795988\pi$
$42$		0			0
$43$		−	9.30506i		−	1.41901i	−0.704701	−	0.709504i	$-0.748919\pi$
$43$		−	9.30506i		−	1.41901i	0.704701	−	0.709504i	$-0.251081\pi$
$44$		0			0
$45$	−1.62772	−	5.04868i	−0.242646	−	0.752612i
$46$		0			0
$47$	−4.62772			−0.675022			−0.337511	−	0.941322i	$-0.609585\pi$
$47$	−4.62772			−0.675022			−0.337511	+	0.941322i	$0.609585\pi$
$48$		0			0
$49$	4.37228			0.624612
$50$		0			0
$51$	2.11684			0.296417
$52$		0			0
$53$			1.58457i			0.217658i	0.994060	+	0.108829i	$0.0347101\pi$
$53$			1.58457i			0.217658i	−0.994060	+	0.108829i	$0.965290\pi$
$54$		0			0
$55$	−10.7446	+	3.46410i	−1.44880	+	0.467099i
$56$		0			0
$57$	2.74456			0.363526
$58$		0			0
$59$			6.63325i			0.863576i	0.901975	+	0.431788i	$0.142117\pi$
$59$			6.63325i			0.863576i	−0.901975	+	0.431788i	$0.857883\pi$
$60$		0			0
$61$	4.74456			0.607479			0.303739	−	0.952755i	$-0.401765\pi$
$61$	4.74456			0.607479			0.303739	+	0.952755i	$0.401765\pi$
$62$		0			0
$63$	−8.00000			−1.00791
$64$		0			0
$65$	−8.05842	+	0.248667i	−0.999524	+	0.0308433i
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$	5.25544			0.632680
$70$		0			0
$71$		−	3.96143i		−	0.470136i	−0.971979	−	0.235068i	$-0.924469\pi$
$71$		−	3.96143i		−	0.470136i	0.971979	−	0.235068i	$-0.0755313\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$	2.31386	+	3.21543i	0.267181	+	0.371286i
$76$		0			0
$77$			17.0256i			1.94024i
$78$		0			0
$79$	6.74456			0.758823			0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456			0.758823			0.379411	+	0.925228i	$0.376127\pi$
$80$		0			0
$81$	3.74456			0.416063
$82$		0			0
$83$	−2.74456			−0.301255			−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456			−0.301255			−0.150627	+	0.988591i	$0.548129\pi$
$84$		0			0
$85$	5.68614	−	1.83324i	0.616749	−	0.198843i
$86$		0			0
$87$			6.92820i			0.742781i
$88$		0			0
$89$			8.51278i			0.902353i	0.892435	+	0.451176i	$0.148995\pi$
$89$			8.51278i			0.902353i	−0.892435	+	0.451176i	$0.851005\pi$
$90$		0			0
$91$	−3.37228	+	11.6819i	−0.353511	+	1.22460i
$92$		0			0
$93$	−2.74456			−0.284598
$94$		0			0
$95$	7.37228	−	2.37686i	0.756380	−	0.243861i
$96$		0			0
$97$	4.74456			0.481737			0.240869	−	0.970558i	$-0.422568\pi$
$97$	4.74456			0.481737			0.240869	+	0.970558i	$0.422568\pi$
$98$		0			0
$99$		−	11.9769i		−	1.20372i
$100$		0			0
$101$	−3.25544			−0.323928			−0.161964	−	0.986797i	$-0.551783\pi$
$101$	−3.25544			−0.323928			−0.161964	+	0.986797i	$0.551783\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$		0			0
$105$	5.68614	−	1.83324i	0.554911	−	0.178906i
$106$		0			0
$107$		−	0.294954i		−	0.0285142i	−0.999898	−	0.0142571i	$-0.995462\pi$
$107$		−	0.294954i		−	0.0285142i	0.999898	−	0.0142571i	$-0.00453834\pi$
$108$		0			0
$109$		−	12.7692i		−	1.22306i	−0.791220	−	0.611532i	$-0.790554\pi$
$109$		−	12.7692i		−	1.22306i	0.791220	−	0.611532i	$-0.209446\pi$
$110$		0			0
$111$			6.43087i			0.610391i
$112$		0			0
$113$		−	17.0256i		−	1.60163i	−0.598912	−	0.800815i	$-0.704400\pi$
$113$		−	17.0256i		−	1.60163i	0.598912	−	0.800815i	$-0.295600\pi$
$114$		0			0
$115$	14.1168	−	4.55134i	1.31640	−	0.424415i
$116$		0			0
$117$	2.37228	−	8.21782i	0.219317	−	0.759738i
$118$		0			0
$119$		−	9.01011i		−	0.825955i
$120$		0			0
$121$	−14.4891			−1.31719
$122$		0			0
$123$	−2.51087			−0.226398
$124$		0			0
$125$	9.00000	+	6.63325i	0.804984	+	0.593296i
$126$		0			0
$127$			3.46410i			0.307389i	0.988118	+	0.153695i	$0.0491172\pi$
$127$			3.46410i			0.307389i	−0.988118	+	0.153695i	$0.950883\pi$
$128$		0			0
$129$	−7.37228			−0.649093
$130$		0			0
$131$	7.37228			0.644119			0.322060	−	0.946719i	$-0.395625\pi$
$131$	7.37228			0.644119			0.322060	+	0.946719i	$0.395625\pi$
$132$		0			0
$133$		−	11.6819i		−	1.01295i
$134$		0			0
$135$	−9.05842	+	2.92048i	−0.779625	+	0.251355i
$136$		0			0
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−0.627719			−0.0532424			−0.0266212	−	0.999646i	$-0.508475\pi$
$139$	−0.627719			−0.0532424			−0.0266212	+	0.999646i	$0.508475\pi$
$140$		0			0
$141$			3.66648i			0.308773i
$142$		0			0
$143$	−17.4891	−	5.04868i	−1.46251	−	0.422191i
$144$		0			0
$145$	6.00000	+	18.6101i	0.498273	+	1.54549i
$146$		0			0
$147$		−	3.46410i		−	0.285714i
$148$		0			0
$149$			3.75906i			0.307954i	0.988074	+	0.153977i	$0.0492081\pi$
$149$			3.75906i			0.307954i	−0.988074	+	0.153977i	$0.950792\pi$
$150$		0			0
$151$			2.37686i			0.193426i	0.995312	+	0.0967131i	$0.0308329\pi$
$151$			2.37686i			0.193426i	−0.995312	+	0.0967131i	$0.969167\pi$
$152$		0			0
$153$			6.33830i			0.512421i
$154$		0			0
$155$	−7.37228	+	2.37686i	−0.592156	+	0.190914i
$156$		0			0
$157$			4.75372i			0.379388i	0.981843	+	0.189694i	$0.0607496\pi$
$157$			4.75372i			0.379388i	−0.981843	+	0.189694i	$0.939250\pi$
$158$		0			0
$159$	1.25544			0.0995627
$160$		0			0
$161$		−	22.3692i		−	1.76294i
$162$		0			0
$163$	13.2554			1.03825			0.519123	−	0.854700i	$-0.326259\pi$
$163$	13.2554			1.03825			0.519123	+	0.854700i	$0.326259\pi$
$164$		0			0
$165$	2.74456	+	8.51278i	0.213664	+	0.662719i
$166$		0			0
$167$	−5.48913			−0.424761			−0.212381	−	0.977187i	$-0.568122\pi$
$167$	−5.48913			−0.424761			−0.212381	+	0.977187i	$0.568122\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$			8.21782i			0.628433i
$172$		0			0
$173$		−	12.2718i		−	0.933010i	−0.884519	−	0.466505i	$-0.845513\pi$
$173$		−	12.2718i		−	0.933010i	0.884519	−	0.466505i	$-0.154487\pi$
$174$		0			0
$175$	13.6861	−	9.84868i	1.03457	−	0.744491i
$176$		0			0
$177$	5.25544			0.395023
$178$		0			0
$179$	−22.1168			−1.65309			−0.826545	−	0.562870i	$-0.809697\pi$
$179$	−22.1168			−1.65309			−0.826545	+	0.562870i	$0.809697\pi$
$180$		0			0
$181$	19.4891			1.44862			0.724308	−	0.689477i	$-0.242160\pi$
$181$	19.4891			1.44862			0.724308	+	0.689477i	$0.242160\pi$
$182$		0			0
$183$		−	3.75906i		−	0.277877i
$184$		0			0
$185$	5.56930	+	17.2742i	0.409463	+	1.27003i
$186$		0			0
$187$	13.4891			0.986423
$188$		0			0
$189$			14.3537i			1.04408i
$190$		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	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$	0.197015	+	6.38458i	0.0141086	+	0.457209i
$196$		0			0
$197$	−1.37228			−0.0977710			−0.0488855	−	0.998804i	$-0.515567\pi$
$197$	−1.37228			−0.0977710			−0.0488855	+	0.998804i	$0.515567\pi$
$198$		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
$201$		−	3.16915i		−	0.223534i
$202$		0			0
$203$	29.4891			2.06973
$204$		0			0
$205$	−6.74456	+	2.17448i	−0.471061	+	0.151872i
$206$		0			0
$207$			15.7359i			1.09372i
$208$		0			0
$209$	17.4891			1.20975
$210$		0			0
$211$	14.1168			0.971844			0.485922	−	0.874002i	$-0.338484\pi$
$211$	14.1168			0.971844			0.485922	+	0.874002i	$0.338484\pi$
$212$		0			0
$213$	−3.13859			−0.215053
$214$		0			0
$215$	−19.8030	+	6.38458i	−1.35055	+	0.435425i
$216$		0			0
$217$			11.6819i			0.793021i
$218$		0			0
$219$			7.92287i			0.535378i
$220$		0			0
$221$	9.25544	+	2.67181i	0.622588	+	0.179726i
$222$		0			0
$223$	2.11684			0.141754			0.0708772	−	0.997485i	$-0.477420\pi$
$223$	2.11684			0.141754			0.0708772	+	0.997485i	$0.477420\pi$
$224$		0			0
$225$	−9.62772	+	6.92820i	−0.641848	+	0.461880i
$226$		0			0
$227$	17.4891			1.16079			0.580397	−	0.814334i	$-0.302897\pi$
$227$	17.4891			1.16079			0.580397	+	0.814334i	$0.302897\pi$
$228$		0			0
$229$		−	14.9436i		−	0.987504i	−0.869603	−	0.493752i	$-0.835625\pi$
$229$		−	14.9436i		−	0.987504i	0.869603	−	0.493752i	$-0.164375\pi$
$230$		0			0
$231$	13.4891			0.887519
$232$		0			0
$233$		−	11.1846i		−	0.732727i	−0.930472	−	0.366363i	$-0.880603\pi$
$233$		−	11.1846i		−	0.732727i	0.930472	−	0.366363i	$-0.119397\pi$
$234$		0			0
$235$	3.17527	+	9.84868i	0.207132	+	0.642457i
$236$		0			0
$237$		−	5.34363i		−	0.347106i
$238$		0			0
$239$		−	6.13592i		−	0.396899i	−0.980111	−	0.198450i	$-0.936409\pi$
$239$		−	6.13592i		−	0.396899i	0.980111	−	0.198450i	$-0.0635907\pi$
$240$		0			0
$241$			11.6819i			0.752499i	0.926518	+	0.376249i	$0.122786\pi$
$241$			11.6819i			0.752499i	−0.926518	+	0.376249i	$0.877214\pi$
$242$		0			0
$243$		−	15.7359i		−	1.00946i
$244$		0			0
$245$	−3.00000	−	9.30506i	−0.191663	−	0.594479i
$246$		0			0
$247$	12.0000	+	3.46410i	0.763542	+	0.220416i
$248$		0			0
$249$			2.17448i			0.137802i
$250$		0			0
$251$	−22.9783			−1.45037			−0.725187	−	0.688552i	$-0.758247\pi$
$251$	−22.9783			−1.45037			−0.725187	+	0.688552i	$0.758247\pi$
$252$		0			0
$253$	33.4891			2.10544
$254$		0			0
$255$	−1.45245	−	4.50506i	−0.0909561	−	0.282118i
$256$		0			0
$257$		−	9.01011i		−	0.562035i	−0.959703	−	0.281018i	$-0.909328\pi$
$257$		−	9.01011i		−	0.562035i	0.959703	−	0.281018i	$-0.0906719\pi$
$258$		0			0
$259$	27.3723			1.70083
$260$		0			0
$261$	−20.7446			−1.28406
$262$		0			0
$263$		−	28.4125i		−	1.75199i	−0.482319	−	0.875996i	$-0.660205\pi$
$263$		−	28.4125i		−	1.75199i	0.482319	−	0.875996i	$-0.339795\pi$
$264$		0			0
$265$	3.37228	−	1.08724i	0.207158	−	0.0667887i
$266$		0			0
$267$	6.74456			0.412761
$268$		0			0
$269$	−8.74456			−0.533165			−0.266583	−	0.963812i	$-0.585895\pi$
$269$	−8.74456			−0.533165			−0.266583	+	0.963812i	$0.585895\pi$
$270$		0			0
$271$		−	11.4795i		−	0.697333i	−0.937247	−	0.348666i	$-0.886635\pi$
$271$		−	11.4795i		−	0.697333i	0.937247	−	0.348666i	$-0.113365\pi$
$272$		0			0
$273$	9.25544	+	2.67181i	0.560165	+	0.161706i
$274$		0			0
$275$	14.7446	+	20.4897i	0.889131	+	1.23557i
$276$		0			0
$277$		−	16.4356i		−	0.987522i	−0.869598	−	0.493761i	$-0.835622\pi$
$277$		−	16.4356i		−	0.987522i	0.869598	−	0.493761i	$-0.164378\pi$
$278$		0			0
$279$		−	8.21782i		−	0.491988i
$280$		0			0
$281$			8.51278i			0.507830i	0.967227	+	0.253915i	$0.0817183\pi$
$281$			8.51278i			0.507830i	−0.967227	+	0.253915i	$0.918282\pi$
$282$		0			0
$283$		−	17.3205i		−	1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$		−	17.3205i		−	1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$		0			0
$285$	−1.88316	−	5.84096i	−0.111549	−	0.345989i
$286$		0			0
$287$			10.6873i			0.630849i
$288$		0			0
$289$	9.86141			0.580083
$290$		0			0
$291$		−	3.75906i		−	0.220360i
$292$		0			0
$293$	−30.8614			−1.80294			−0.901471	−	0.432839i	$-0.857512\pi$
$293$	−30.8614			−1.80294			−0.901471	+	0.432839i	$0.857512\pi$
$294$		0			0
$295$	14.1168	−	4.55134i	0.821914	−	0.264989i
$296$		0			0
$297$	−21.4891			−1.24693
$298$		0			0
$299$	22.9783	+	6.63325i	1.32887	+	0.383611i
$300$		0			0
$301$			31.3793i			1.80867i
$302$		0			0
$303$			2.57924i			0.148174i
$304$		0			0
$305$	−3.25544	−	10.0974i	−0.186406	−	0.578173i
$306$		0			0
$307$	−16.2337			−0.926506			−0.463253	−	0.886226i	$-0.653318\pi$
$307$	−16.2337			−0.926506			−0.463253	+	0.886226i	$0.653318\pi$
$308$		0			0
$309$	−8.23369			−0.468398
$310$		0			0
$311$	−14.7446			−0.836087			−0.418044	−	0.908427i	$-0.637284\pi$
$311$	−14.7446			−0.836087			−0.418044	+	0.908427i	$0.637284\pi$
$312$		0			0
$313$		−	17.5229i		−	0.990452i	−0.868764	−	0.495226i	$-0.835085\pi$
$313$		−	17.5229i		−	0.990452i	0.868764	−	0.495226i	$-0.164915\pi$
$314$		0			0
$315$	5.48913	+	17.0256i	0.309277	+	0.959281i
$316$		0			0
$317$	28.9783			1.62758			0.813790	−	0.581159i	$-0.197400\pi$
$317$	28.9783			1.62758			0.813790	+	0.581159i	$0.197400\pi$
$318$		0			0
$319$			44.1485i			2.47184i
$320$		0			0
$321$	−0.233688			−0.0130432
$322$		0			0
$323$	−9.25544			−0.514986
$324$		0			0
$325$	6.05842	+	16.9793i	0.336061	+	0.941840i
$326$		0			0
$327$	−10.1168			−0.559463
$328$		0			0
$329$	15.6060			0.860385
$330$		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
$333$	−19.2554			−1.05519
$334$		0			0
$335$	−2.74456	−	8.51278i	−0.149951	−	0.465103i
$336$		0			0
$337$		−	31.3793i		−	1.70934i	−0.519172	−	0.854670i	$-0.673760\pi$
$337$		−	31.3793i		−	1.70934i	0.519172	−	0.854670i	$-0.326240\pi$
$338$		0			0
$339$	−13.4891			−0.732629
$340$		0			0
$341$	−17.4891			−0.947089
$342$		0			0
$343$	8.86141			0.478471
$344$		0			0
$345$	−3.60597	−	11.1846i	−0.194139	−	0.602158i
$346$		0			0
$347$		−	15.6434i		−	0.839780i	−0.907575	−	0.419890i	$-0.862069\pi$
$347$		−	15.6434i		−	0.839780i	0.907575	−	0.419890i	$-0.137931\pi$
$348$		0			0
$349$			22.2766i			1.19244i	0.802821	+	0.596220i	$0.203331\pi$
$349$			22.2766i			1.19244i	−0.802821	+	0.596220i	$0.796669\pi$
$350$		0			0
$351$	−14.7446	−	4.25639i	−0.787007	−	0.227189i
$352$		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
$355$	−8.43070	+	2.71810i	−0.447455	+	0.144262i
$356$		0			0
$357$	−7.13859			−0.377814
$358$		0			0
$359$		−	23.6588i		−	1.24866i	−0.781159	−	0.624332i	$-0.785371\pi$
$359$		−	23.6588i		−	1.24866i	0.781159	−	0.624332i	$-0.214629\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$			11.4795i			0.602520i
$364$		0			0
$365$	6.86141	+	21.2819i	0.359142	+	1.11395i
$366$		0			0
$367$			15.1460i			0.790616i	0.918549	+	0.395308i	$0.129362\pi$
$367$			15.1460i			0.790616i	−0.918549	+	0.395308i	$0.870638\pi$
$368$		0			0
$369$		−	7.51811i		−	0.391377i
$370$		0			0
$371$		−	5.34363i		−	0.277427i
$372$		0			0
$373$			30.2921i			1.56846i	0.620468	+	0.784232i	$0.286943\pi$
$373$			30.2921i			1.56846i	−0.620468	+	0.784232i	$0.713057\pi$
$374$		0			0
$375$	5.25544	−	7.13058i	0.271390	−	0.368222i
$376$		0			0
$377$	−8.74456	+	30.2921i	−0.450368	+	1.56012i
$378$		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
$381$	2.74456			0.140608
$382$		0			0
$383$	−34.1168			−1.74329			−0.871645	−	0.490138i	$-0.836946\pi$
$383$	−34.1168			−1.74329			−0.871645	+	0.490138i	$0.836946\pi$
$384$		0			0
$385$	36.2337	−	11.6819i	1.84664	−	0.595366i
$386$		0			0
$387$		−	22.0742i		−	1.12210i
$388$		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
$391$	−17.7228			−0.896281
$392$		0			0
$393$		−	5.84096i		−	0.294638i
$394$		0			0
$395$	−4.62772	−	14.3537i	−0.232846	−	0.722215i
$396$		0			0
$397$	31.4891			1.58039			0.790197	−	0.612853i	$-0.209978\pi$
$397$	31.4891			1.58039			0.790197	+	0.612853i	$0.209978\pi$
$398$		0			0
$399$	−9.25544			−0.463351
$400$		0			0
$401$		−	0.994667i		−	0.0496713i	−0.999692	−	0.0248356i	$-0.992094\pi$
$401$		−	0.994667i		−	0.0496713i	0.999692	−	0.0248356i	$-0.00790624\pi$
$402$		0			0
$403$	−12.0000	−	3.46410i	−0.597763	−	0.172559i
$404$		0			0
$405$	−2.56930	−	7.96916i	−0.127669	−	0.395991i
$406$		0			0
$407$			40.9793i			2.03127i
$408$		0			0
$409$		−	25.5383i		−	1.26279i	−0.775462	−	0.631395i	$-0.782483\pi$
$409$		−	25.5383i		−	1.26279i	0.775462	−	0.631395i	$-0.217517\pi$
$410$		0			0
$411$		−	4.75372i		−	0.234484i
$412$		0			0
$413$		−	22.3692i		−	1.10072i
$414$		0			0
$415$	1.88316	+	5.84096i	0.0924405	+	0.286722i
$416$		0			0
$417$			0.497333i			0.0243545i
$418$		0			0
$419$	−1.88316			−0.0919982			−0.0459991	−	0.998941i	$-0.514647\pi$
$419$	−1.88316			−0.0919982			−0.0459991	+	0.998941i	$0.514647\pi$
$420$		0			0
$421$			1.08724i			0.0529889i	0.999649	+	0.0264944i	$0.00843443\pi$
$421$			1.08724i			0.0529889i	−0.999649	+	0.0264944i	$0.991566\pi$
$422$		0			0
$423$	−10.9783			−0.533781
$424$		0			0
$425$	−7.80298	−	10.8434i	−0.378500	−	0.525980i
$426$		0			0
$427$	−16.0000			−0.774294
$428$		0			0
$429$	−4.00000	+	13.8564i	−0.193122	+	0.668994i
$430$		0			0
$431$			9.89497i			0.476624i	0.971189	+	0.238312i	$0.0765941\pi$
$431$			9.89497i			0.476624i	−0.971189	+	0.238312i	$0.923406\pi$
$432$		0			0
$433$			15.3484i			0.737597i	0.929509	+	0.368799i	$0.120231\pi$
$433$			15.3484i			0.737597i	−0.929509	+	0.368799i	$0.879769\pi$
$434$		0			0
$435$	14.7446	−	4.75372i	0.706948	−	0.227924i
$436$		0			0
$437$	−22.9783			−1.09920
$438$		0			0
$439$	12.2337			0.583882			0.291941	−	0.956436i	$-0.405699\pi$
$439$	12.2337			0.583882			0.291941	+	0.956436i	$0.405699\pi$
$440$		0			0
$441$	10.3723			0.493918
$442$		0			0
$443$			7.72049i			0.366812i	0.983037	+	0.183406i	$0.0587122\pi$
$443$			7.72049i			0.366812i	−0.983037	+	0.183406i	$0.941288\pi$
$444$		0			0
$445$	18.1168	−	5.84096i	0.858821	−	0.276888i
$446$		0			0
$447$	2.97825			0.140866
$448$		0			0
$449$			34.0511i			1.60697i	0.595324	+	0.803486i	$0.297024\pi$
$449$			34.0511i			1.60697i	−0.595324	+	0.803486i	$0.702976\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	1.88316			0.0884784
$454$		0			0
$455$	27.1753	−	0.838574i	1.27400	−	0.0393130i
$456$		0			0
$457$	−20.9783			−0.981321			−0.490661	−	0.871351i	$-0.663244\pi$
$457$	−20.9783			−0.981321			−0.490661	+	0.871351i	$0.663244\pi$
$458$		0			0
$459$	11.3723			0.530813
$460$		0			0
$461$		−	41.4766i		−	1.93176i	−0.258990	−	0.965880i	$-0.583390\pi$
$461$		−	41.4766i		−	1.93176i	0.258990	−	0.965880i	$-0.416610\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$	1.88316	+	5.84096i	0.0873293	+	0.270868i
$466$		0			0
$467$		−	14.1514i		−	0.654847i	−0.944878	−	0.327423i	$-0.893820\pi$
$467$		−	14.1514i		−	0.654847i	0.944878	−	0.327423i	$-0.106180\pi$
$468$		0			0
$469$	−13.4891			−0.622870
$470$		0			0
$471$	3.76631			0.173542
$472$		0			0
$473$	−46.9783			−2.16006
$474$		0			0
$475$	−10.1168	−	14.0588i	−0.464193	−	0.645061i
$476$		0			0
$477$			3.75906i			0.172115i
$478$		0			0
$479$			21.5769i			0.985874i	0.870065	+	0.492937i	$0.164077\pi$
$479$			21.5769i			0.985874i	−0.870065	+	0.492937i	$0.835923\pi$
$480$		0			0
$481$	−8.11684	+	28.1176i	−0.370096	+	1.28205i
$482$		0			0
$483$	−17.7228			−0.806416
$484$		0			0
$485$	−3.25544	−	10.0974i	−0.147822	−	0.458497i
$486$		0			0
$487$	21.4891			0.973765			0.486883	−	0.873467i	$-0.338134\pi$
$487$	21.4891			0.973765			0.486883	+	0.873467i	$0.338134\pi$
$488$		0			0
$489$		−	10.5021i		−	0.474922i
$490$		0			0
$491$	31.3723			1.41581			0.707906	−	0.706307i	$-0.249640\pi$
$491$	31.3723			1.41581			0.707906	+	0.706307i	$0.249640\pi$
$492$		0			0
$493$		−	23.3639i		−	1.05225i
$494$		0			0
$495$	−25.4891	+	8.21782i	−1.14565	+	0.369364i
$496$		0			0
$497$			13.3591i			0.599236i
$498$		0			0
$499$		−	35.9306i		−	1.60848i	−0.594307	−	0.804238i	$-0.702574\pi$
$499$		−	35.9306i		−	1.60848i	0.594307	−	0.804238i	$-0.297426\pi$
$500$		0			0
$501$			4.34896i			0.194297i
$502$		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
$505$	2.23369	+	6.92820i	0.0993978	+	0.308301i
$506$		0			0
$507$	−5.48913	+	8.71516i	−0.243781	+	0.387054i
$508$		0			0
$509$		−	10.0974i		−	0.447557i	−0.974640	−	0.223779i	$-0.928161\pi$
$509$		−	10.0974i		−	0.447557i	0.974640	−	0.223779i	$-0.0718393\pi$
$510$		0			0
$511$	33.7228			1.49181
$512$		0			0
$513$	14.7446			0.650988
$514$		0			0
$515$	−22.1168	+	7.13058i	−0.974585	+	0.314211i
$516$		0			0
$517$			23.3639i			1.02754i
$518$		0			0
$519$	−9.72281			−0.426784
$520$		0			0
$521$	−21.6060			−0.946575			−0.473287	−	0.880908i	$-0.656933\pi$
$521$	−21.6060			−0.946575			−0.473287	+	0.880908i	$0.656933\pi$
$522$		0			0
$523$			10.3923i			0.454424i	0.973845	+	0.227212i	$0.0729610\pi$
$523$			10.3923i			0.454424i	−0.973845	+	0.227212i	$0.927039\pi$
$524$		0			0
$525$	−7.80298	−	10.8434i	−0.340550	−	0.473243i
$526$		0			0
$527$	9.25544			0.403173
$528$		0			0
$529$	−21.0000			−0.913043
$530$		0			0
$531$			15.7359i			0.682881i
$532$		0			0
$533$	−10.9783	−	3.16915i	−0.475521	−	0.137271i
$534$		0			0
$535$	−0.627719	+	0.202380i	−0.0271386	+	0.00874964i
$536$		0			0
$537$			17.5229i			0.756168i
$538$		0			0
$539$		−	22.0742i		−	0.950804i
$540$		0			0
$541$		−	40.4820i		−	1.74046i	−0.492649	−	0.870228i	$-0.663971\pi$
$541$		−	40.4820i		−	1.74046i	0.492649	−	0.870228i	$-0.336029\pi$
$542$		0			0
$543$		−	15.4410i		−	0.662636i
$544$		0			0
$545$	−27.1753	+	8.76144i	−1.16406	+	0.375299i
$546$		0			0
$547$		−	11.4795i		−	0.490830i	−0.969418	−	0.245415i	$-0.921076\pi$
$547$		−	11.4795i		−	0.490830i	0.969418	−	0.245415i	$-0.0789242\pi$
$548$		0			0
$549$	11.2554			0.480370
$550$		0			0
$551$		−	30.2921i		−	1.29048i
$552$		0			0
$553$	−22.7446			−0.967197
$554$		0			0
$555$	13.6861	−	4.41248i	0.580944	−	0.187299i
$556$		0			0
$557$	−16.1168			−0.682893			−0.341446	−	0.939901i	$-0.610917\pi$
$557$	−16.1168			−0.682893			−0.341446	+	0.939901i	$0.610917\pi$
$558$		0			0
$559$	−32.2337	−	9.30506i	−1.36334	−	0.393562i
$560$		0			0
$561$		−	10.6873i		−	0.451216i
$562$		0			0
$563$			44.9407i			1.89403i	0.321194	+	0.947013i	$0.395916\pi$
$563$			44.9407i			1.89403i	−0.321194	+	0.947013i	$0.604084\pi$
$564$		0			0
$565$	−36.2337	+	11.6819i	−1.52436	+	0.491462i
$566$		0			0
$567$	−12.6277			−0.530314
$568$		0			0
$569$	−10.6277			−0.445537			−0.222769	−	0.974871i	$-0.571509\pi$
$569$	−10.6277			−0.445537			−0.222769	+	0.974871i	$0.571509\pi$
$570$		0			0
$571$	−30.1168			−1.26035			−0.630175	−	0.776453i	$-0.717017\pi$
$571$	−30.1168			−1.26035			−0.630175	+	0.776453i	$0.717017\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.3723	−	26.9205i	−0.807880	−	1.12266i
$576$		0			0
$577$	24.9783			1.03986			0.519929	−	0.854209i	$-0.325958\pi$
$577$	24.9783			1.03986			0.519929	+	0.854209i	$0.325958\pi$
$578$		0			0
$579$		−	11.0920i		−	0.460969i
$580$		0			0
$581$	9.25544			0.383980
$582$		0			0
$583$	8.00000			0.331326
$584$		0			0
$585$	−19.1168	+	0.589907i	−0.790384	+	0.0243897i
$586$		0			0
$587$	−37.7228			−1.55699			−0.778494	−	0.627653i	$-0.784016\pi$
$587$	−37.7228			−1.55699			−0.778494	+	0.627653i	$0.784016\pi$
$588$		0			0
$589$	12.0000			0.494451
$590$		0			0
$591$			1.08724i			0.0447231i
$592$		0			0
$593$	2.23369			0.0917266			0.0458633	−	0.998948i	$-0.485396\pi$
$593$	2.23369			0.0917266			0.0458633	+	0.998948i	$0.485396\pi$
$594$		0			0
$595$	−19.1753	+	6.18220i	−0.786109	+	0.253446i
$596$		0			0
$597$			6.33830i			0.259409i
$598$		0			0
$599$	25.7228			1.05101			0.525503	−	0.850792i	$-0.323877\pi$
$599$	25.7228			1.05101			0.525503	+	0.850792i	$0.323877\pi$
$600$		0			0
$601$	−26.6277			−1.08617			−0.543084	−	0.839679i	$-0.682743\pi$
$601$	−26.6277			−1.08617			−0.543084	+	0.839679i	$0.682743\pi$
$602$		0			0
$603$	9.48913			0.386427
$604$		0			0
$605$	9.94158	+	30.8357i	0.404183	+	1.25365i
$606$		0			0
$607$			3.46410i			0.140604i	0.997526	+	0.0703018i	$0.0223962\pi$
$607$			3.46410i			0.140604i	−0.997526	+	0.0703018i	$0.977604\pi$
$608$		0			0
$609$		−	23.3639i		−	0.946751i
$610$		0			0
$611$	−4.62772	+	16.0309i	−0.187217	+	0.648540i
$612$		0			0
$613$	−3.48913			−0.140924			−0.0704622	−	0.997514i	$-0.522447\pi$
$613$	−3.48913			−0.140924			−0.0704622	+	0.997514i	$0.522447\pi$
$614$		0			0
$615$	1.72281	+	5.34363i	0.0694705	+	0.215476i
$616$		0			0
$617$	−3.25544			−0.131059			−0.0655295	−	0.997851i	$-0.520874\pi$
$617$	−3.25544			−0.131059			−0.0655295	+	0.997851i	$0.520874\pi$
$618$		0			0
$619$			29.0024i			1.16571i	0.812578	+	0.582853i	$0.198064\pi$
$619$			29.0024i			1.16571i	−0.812578	+	0.582853i	$0.801936\pi$
$620$		0			0
$621$	28.2337			1.13298
$622$		0			0
$623$		−	28.7075i		−	1.15014i
$624$		0			0
$625$	7.94158	−	23.7051i	0.317663	−	0.948204i
$626$		0			0
$627$		−	13.8564i		−	0.553372i
$628$		0			0
$629$		−	21.6867i		−	0.864705i
$630$		0			0
$631$			49.1046i			1.95482i	0.211348	+	0.977411i	$0.432215\pi$
$631$			49.1046i			1.95482i	−0.211348	+	0.977411i	$0.567785\pi$
$632$		0			0
$633$		−	11.1846i		−	0.444548i
$634$		0			0
$635$	7.37228	−	2.37686i	0.292560	−	0.0943229i
$636$		0			0
$637$	4.37228	−	15.1460i	0.173236	−	0.600107i
$638$		0			0
$639$		−	9.39764i		−	0.371765i
$640$		0			0
$641$	28.9783			1.14457			0.572286	−	0.820054i	$-0.306057\pi$
$641$	28.9783			1.14457			0.572286	+	0.820054i	$0.306057\pi$
$642$		0			0
$643$	−14.5109			−0.572253			−0.286127	−	0.958192i	$-0.592368\pi$
$643$	−14.5109			−0.572253			−0.286127	+	0.958192i	$0.592368\pi$
$644$		0			0
$645$	5.05842	+	15.6896i	0.199175	+	0.617779i
$646$		0			0
$647$			34.3461i			1.35028i	0.737688	+	0.675142i	$0.235917\pi$
$647$			34.3461i			1.35028i	−0.737688	+	0.675142i	$0.764083\pi$
$648$		0			0
$649$	33.4891			1.31456
$650$		0			0
$651$	9.25544			0.362749
$652$		0			0
$653$			3.75906i			0.147103i	0.997291	+	0.0735516i	$0.0234334\pi$
$653$			3.75906i			0.147103i	−0.997291	+	0.0735516i	$0.976567\pi$
$654$		0			0
$655$	−5.05842	−	15.6896i	−0.197649	−	0.613045i
$656$		0			0
$657$	−23.7228			−0.925515
$658$		0			0
$659$	−28.4674			−1.10893			−0.554466	−	0.832207i	$-0.687077\pi$
$659$	−28.4674			−1.10893			−0.554466	+	0.832207i	$0.687077\pi$
$660$		0			0
$661$		−	30.2921i		−	1.17822i	−0.808051	−	0.589112i	$-0.799478\pi$
$661$		−	30.2921i		−	1.17822i	0.808051	−	0.589112i	$-0.200522\pi$
$662$		0			0
$663$	2.11684	−	7.33296i	0.0822114	−	0.284789i
$664$		0			0
$665$	−24.8614	+	8.01544i	−0.964084	+	0.310826i
$666$		0			0
$667$		−	58.0049i		−	2.24596i
$668$		0			0
$669$		−	1.67715i		−	0.0648423i
$670$		0			0
$671$		−	23.9538i		−	0.924725i
$672$		0			0
$673$			19.6974i			0.759278i	0.925135	+	0.379639i	$0.123952\pi$
$673$			19.6974i			0.759278i	−0.925135	+	0.379639i	$0.876048\pi$
$674$		0			0
$675$	12.4307	+	17.2742i	0.478458	+	0.664885i
$676$		0			0
$677$			5.93354i			0.228044i	0.993478	+	0.114022i	$0.0363735\pi$
$677$			5.93354i			0.228044i	−0.993478	+	0.114022i	$0.963627\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$		−	13.8564i		−	0.530979i
$682$		0			0
$683$	2.74456			0.105018			0.0525089	−	0.998620i	$-0.483278\pi$
$683$	2.74456			0.105018			0.0525089	+	0.998620i	$0.483278\pi$
$684$		0			0
$685$	−4.11684	−	12.7692i	−0.157297	−	0.487885i
$686$		0			0
$687$	−11.8397			−0.451711
$688$		0			0
$689$	5.48913	+	1.58457i	0.209119	+	0.0603675i
$690$		0			0
$691$			19.4950i			0.741624i	0.928708	+	0.370812i	$0.120921\pi$
$691$			19.4950i			0.741624i	−0.928708	+	0.370812i	$0.879079\pi$
$692$		0			0
$693$			40.3894i			1.53427i
$694$		0			0
$695$	0.430703	+	1.33591i	0.0163375	+	0.0506739i
$696$		0			0
$697$	8.46738			0.320725
$698$		0			0
$699$	−8.86141			−0.335169
$700$		0			0
$701$	−28.9783			−1.09449			−0.547247	−	0.836971i	$-0.684324\pi$
$701$	−28.9783			−1.09449			−0.547247	+	0.836971i	$0.684324\pi$
$702$		0			0
$703$		−	28.1176i		−	1.06047i
$704$		0			0
$705$	7.80298	−	2.51572i	0.293877	−	0.0947476i
$706$		0			0
$707$	10.9783			0.412880
$708$		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
$715$	1.25544	+	40.6844i	0.0469507	+	1.52151i
$716$		0			0
$717$	−4.86141			−0.181553
$718$		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
$721$			35.0458i			1.30517i
$722$		0			0
$723$	9.25544			0.344213
$724$		0			0
$725$	35.4891	−	25.5383i	1.31803	−	0.948470i
$726$		0			0
$727$		−	22.0742i		−	0.818688i	−0.912380	−	0.409344i	$-0.865758\pi$
$727$		−	22.0742i		−	0.818688i	0.912380	−	0.409344i	$-0.134242\pi$
$728$		0			0
$729$	−1.23369			−0.0456921
$730$		0			0
$731$	24.8614			0.919532
$732$		0			0
$733$	47.0951			1.73950			0.869749	−	0.493495i	$-0.164281\pi$
$733$	47.0951			1.73950			0.869749	+	0.493495i	$0.164281\pi$
$734$		0			0
$735$	−7.37228	+	2.37686i	−0.271931	+	0.0876718i
$736$		0			0
$737$		−	20.1947i		−	0.743881i
$738$		0			0
$739$		−	31.5817i		−	1.16175i	−0.813993	−	0.580875i	$-0.802710\pi$
$739$		−	31.5817i		−	1.16175i	0.813993	−	0.580875i	$-0.197290\pi$
$740$		0			0
$741$	2.74456	−	9.50744i	0.100824	−	0.349265i
$742$		0			0
$743$	10.1168			0.371151			0.185576	−	0.982630i	$-0.440585\pi$
$743$	10.1168			0.371151			0.185576	+	0.982630i	$0.440585\pi$
$744$		0			0
$745$	8.00000	−	2.57924i	0.293097	−	0.0944961i
$746$		0			0
$747$	−6.51087			−0.238220
$748$		0			0
$749$			0.994667i			0.0363443i
$750$		0			0
$751$	−2.51087			−0.0916231			−0.0458116	−	0.998950i	$-0.514587\pi$
$751$	−2.51087			−0.0916231			−0.0458116	+	0.998950i	$0.514587\pi$
$752$		0			0
$753$			18.2054i			0.663441i
$754$		0			0
$755$	5.05842	−	1.63086i	0.184095	−	0.0593531i
$756$		0			0
$757$			14.2612i			0.518331i	0.965833	+	0.259165i	$0.0834475\pi$
$757$			14.2612i			0.518331i	−0.965833	+	0.259165i	$0.916553\pi$
$758$		0			0
$759$		−	26.5330i		−	0.963087i
$760$		0			0
$761$			8.51278i			0.308588i	0.988025	+	0.154294i	$0.0493103\pi$
$761$			8.51278i			0.308588i	−0.988025	+	0.154294i	$0.950690\pi$
$762$		0			0
$763$			43.0612i			1.55892i
$764$		0			0
$765$	13.4891	−	4.34896i	0.487700	−	0.157237i
$766$		0			0
$767$	22.9783	+	6.63325i	0.829697	+	0.239513i
$768$		0			0
$769$			23.3639i			0.842522i	0.906939	+	0.421261i	$0.138412\pi$
$769$			23.3639i			0.842522i	−0.906939	+	0.421261i	$0.861588\pi$
$770$		0			0
$771$	−7.13859			−0.257090
$772$		0			0
$773$	9.60597			0.345503			0.172751	−	0.984965i	$-0.444734\pi$
$773$	9.60597			0.345503			0.172751	+	0.984965i	$0.444734\pi$
$774$		0			0
$775$	10.1168	+	14.0588i	0.363408	+	0.505007i
$776$		0			0
$777$		−	21.6867i		−	0.778006i
$778$		0			0
$779$	10.9783			0.393337
$780$		0			0
$781$	−20.0000			−0.715656
$782$		0			0
$783$			37.2203i			1.33014i
$784$		0			0
$785$	10.1168	−	3.26172i	0.361086	−	0.116416i
$786$		0			0
$787$	−44.0000			−1.56843			−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000			−1.56843			−0.784215	+	0.620489i	$0.786934\pi$
$788$		0			0
$789$	−22.5109			−0.801408
$790$		0			0
$791$			57.4150i			2.04144i
$792$		0			0
$793$	4.74456	−	16.4356i	0.168484	−	0.583647i
$794$		0			0
$795$	−0.861407	−	2.67181i	−0.0305509	−	0.0947595i
$796$		0			0
$797$		−	7.92287i		−	0.280642i	−0.990106	−	0.140321i	$-0.955186\pi$
$797$		−	7.92287i		−	0.280642i	0.990106	−	0.140321i	$-0.0448135\pi$
$798$		0			0
$799$		−	12.3644i		−	0.437421i
$800$		0			0
$801$			20.1947i			0.713545i
$802$		0			0
$803$			50.4868i			1.78164i
$804$		0			0
$805$	−47.6060	+	15.3484i	−1.67789	+	0.540960i
$806$		0			0
$807$			6.92820i			0.243884i
$808$		0			0
$809$	−12.3505			−0.434222			−0.217111	−	0.976147i	$-0.569663\pi$
$809$	−12.3505			−0.434222			−0.217111	+	0.976147i	$0.569663\pi$
$810$		0			0
$811$		−	17.7253i		−	0.622418i	−0.950341	−	0.311209i	$-0.899266\pi$
$811$		−	17.7253i		−	0.622418i	0.950341	−	0.311209i	$-0.100734\pi$
$812$		0			0
$813$	−9.09509			−0.318979
$814$		0			0
$815$	−9.09509	−	28.2101i	−0.318587	−	0.988158i
$816$		0			0
$817$	32.2337			1.12771
$818$		0			0
$819$	−8.00000	+	27.7128i	−0.279543	+	0.968364i
$820$		0			0
$821$			30.7894i			1.07456i	0.843405	+	0.537279i	$0.180548\pi$
$821$			30.7894i			1.07456i	−0.843405	+	0.537279i	$0.819452\pi$
$822$		0			0
$823$		−	14.7413i		−	0.513848i	−0.966432	−	0.256924i	$-0.917291\pi$
$823$		−	14.7413i		−	0.513848i	0.966432	−	0.256924i	$-0.0827091\pi$
$824$		0			0
$825$	16.2337	−	11.6819i	0.565184	−	0.406712i
$826$		0			0
$827$	−26.7446			−0.930000			−0.465000	−	0.885311i	$-0.653946\pi$
$827$	−26.7446			−0.930000			−0.465000	+	0.885311i	$0.653946\pi$
$828$		0			0
$829$	−13.7663			−0.478124			−0.239062	−	0.971004i	$-0.576840\pi$
$829$	−13.7663			−0.478124			−0.239062	+	0.971004i	$0.576840\pi$
$830$		0			0
$831$	−13.0217			−0.451719
$832$		0			0
$833$			11.6819i			0.404755i
$834$		0			0
$835$	3.76631	+	11.6819i	0.130339	+	0.404270i
$836$		0			0
$837$	−14.7446			−0.509647
$838$		0			0
$839$		−	0.294954i		−	0.0101829i	−0.999987	−	0.00509147i	$-0.998379\pi$
$839$		−	0.294954i		−	0.0101829i	0.999987	−	0.00509147i	$-0.00162067\pi$
$840$		0			0
$841$	47.4674			1.63681
$842$		0			0
$843$	6.74456			0.232295
$844$		0			0
$845$	−7.19702	+	28.1639i	−0.247585	+	0.968866i
$846$		0			0
$847$	48.8614			1.67890
$848$		0			0
$849$	−13.7228			−0.470966
$850$		0			0
$851$		−	53.8411i		−	1.84565i
$852$		0			0
$853$	12.1168			0.414873			0.207436	−	0.978249i	$-0.433488\pi$
$853$	12.1168			0.414873			0.207436	+	0.978249i	$0.433488\pi$
$854$		0			0
$855$	17.4891	−	5.63858i	0.598115	−	0.192835i
$856$		0			0
$857$			43.5586i			1.48793i	0.668218	+	0.743966i	$0.267058\pi$
$857$			43.5586i			1.48793i	−0.668218	+	0.743966i	$0.732942\pi$
$858$		0			0
$859$	−36.4674			−1.24425			−0.622125	−	0.782918i	$-0.713731\pi$
$859$	−36.4674			−1.24425			−0.622125	+	0.782918i	$0.713731\pi$
$860$		0			0
$861$	8.46738			0.288567
$862$		0			0
$863$	24.8614			0.846292			0.423146	−	0.906061i	$-0.360926\pi$
$863$	24.8614			0.846292			0.423146	+	0.906061i	$0.360926\pi$
$864$		0			0
$865$	−26.1168	+	8.42020i	−0.887999	+	0.286295i
$866$		0			0
$867$		−	7.81306i		−	0.265346i
$868$		0			0
$869$		−	34.0511i		−	1.15510i
$870$		0			0
$871$	4.00000	−	13.8564i	0.135535	−	0.469506i
$872$		0			0
$873$	11.2554			0.380939
$874$		0			0
$875$	−30.3505	−	22.3692i	−1.02604	−	0.756216i
$876$		0			0
$877$	−4.35053			−0.146907			−0.0734535	−	0.997299i	$-0.523402\pi$
$877$	−4.35053			−0.146907			−0.0734535	+	0.997299i	$0.523402\pi$
$878$		0			0
$879$			24.4511i			0.824715i
$880$		0			0
$881$	−14.3940			−0.484947			−0.242474	−	0.970158i	$-0.577959\pi$
$881$	−14.3940			−0.484947			−0.242474	+	0.970158i	$0.577959\pi$
$882$		0			0
$883$			8.90030i			0.299519i	0.988722	+	0.149760i	$0.0478500\pi$
$883$			8.90030i			0.299519i	−0.988722	+	0.149760i	$0.952150\pi$
$884$		0			0
$885$	−3.60597	−	11.1846i	−0.121213	−	0.375966i
$886$		0			0
$887$		−	0.699713i		−	0.0234941i	−0.999931	−	0.0117470i	$-0.996261\pi$
$887$		−	0.699713i		−	0.0234941i	0.999931	−	0.0117470i	$-0.00373928\pi$
$888$		0			0
$889$		−	11.6819i		−	0.391799i
$890$		0			0
$891$		−	18.9051i		−	0.633344i
$892$		0			0
$893$		−	16.0309i		−	0.536453i
$894$		0			0
$895$	15.1753	+	47.0689i	0.507253	+	1.57334i
$896$		0			0
$897$	5.25544	−	18.2054i	0.175474	−	0.607860i
$898$		0			0
$899$			30.2921i			1.01030i
$900$		0			0
$901$	−4.23369			−0.141045
$902$		0			0
$903$	24.8614			0.827336
$904$		0			0
$905$	−13.3723	−	41.4766i	−0.444510	−	1.37873i
$906$		0			0
$907$		−	9.30506i		−	0.308970i	−0.987995	−	0.154485i	$-0.950628\pi$
$907$		−	9.30506i		−	0.308970i	0.987995	−	0.154485i	$-0.0493718\pi$
$908$		0			0
$909$	−7.72281			−0.256150
$910$		0			0
$911$	10.9783			0.363726			0.181863	−	0.983324i	$-0.441787\pi$
$911$	10.9783			0.363726			0.181863	+	0.983324i	$0.441787\pi$
$912$		0			0
$913$			13.8564i			0.458580i
$914$		0			0
$915$	−8.00000	+	2.57924i	−0.264472	+	0.0852671i
$916$		0			0
$917$	−24.8614			−0.820996
$918$		0			0
$919$	2.97825			0.0982434			0.0491217	−	0.998793i	$-0.484358\pi$
$919$	2.97825			0.0982434			0.0491217	+	0.998793i	$0.484358\pi$
$920$		0			0
$921$			12.8617i			0.423809i
$922$		0			0
$923$	−13.7228	−	3.96143i	−0.451692	−	0.130392i
$924$		0			0
$925$	32.9416	−	23.7051i	1.08311	−	0.779419i
$926$		0			0
$927$		−	24.6535i		−	0.809726i
$928$		0			0
$929$		−	21.3745i		−	0.701275i	−0.936511	−	0.350638i	$-0.885965\pi$
$929$		−	21.3745i		−	0.701275i	0.936511	−	0.350638i	$-0.114035\pi$
$930$		0			0
$931$			15.1460i			0.496391i
$932$		0			0
$933$			11.6819i			0.382449i
$934$		0			0
$935$	−9.25544	−	28.7075i	−0.302685	−	0.938835i
$936$		0			0
$937$		−	9.50744i		−	0.310595i	−0.987868	−	0.155297i	$-0.950366\pi$
$937$		−	9.50744i		−	0.310595i	0.987868	−	0.155297i	$-0.0496336\pi$
$938$		0			0
$939$	−13.8832			−0.453060
$940$		0			0
$941$			37.3128i			1.21636i	0.793798	+	0.608182i	$0.208101\pi$
$941$			37.3128i			1.21636i	−0.793798	+	0.608182i	$0.791899\pi$
$942$		0			0
$943$	21.0217			0.684562
$944$		0			0
$945$	30.5475	−	9.84868i	0.993712	−	0.320378i
$946$		0			0
$947$	6.51087			0.211575			0.105787	−	0.994389i	$-0.466264\pi$
$947$	6.51087			0.211575			0.105787	+	0.994389i	$0.466264\pi$
$948$		0			0
$949$	−10.0000	+	34.6410i	−0.324614	+	1.12449i
$950$		0			0
$951$		−	22.9591i		−	0.744500i
$952$		0			0
$953$			0.497333i			0.0161102i	0.999968	+	0.00805510i	$0.00256405\pi$
$953$			0.497333i			0.0161102i	−0.999968	+	0.00805510i	$0.997436\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	34.9783			1.13069
$958$		0			0
$959$	−20.2337			−0.653380
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		−	0.699713i		−	0.0225479i
$964$		0			0
$965$	−9.60597	−	29.7947i	−0.309227	−	0.959126i
$966$		0			0
$967$	46.3505			1.49053			0.745266	−	0.666767i	$-0.232322\pi$
$967$	46.3505			1.49053			0.745266	+	0.666767i	$0.232322\pi$
$968$		0			0
$969$			7.33296i			0.235569i
$970$		0			0
$971$	−27.6060			−0.885918			−0.442959	−	0.896542i	$-0.646071\pi$
$971$	−27.6060			−0.885918			−0.442959	+	0.896542i	$0.646071\pi$
$972$		0			0
$973$	2.11684			0.0678629
$974$		0			0
$975$	13.4525	−	4.80001i	0.430823	−	0.153723i
$976$		0			0
$977$	31.7228			1.01490			0.507451	−	0.861680i	$-0.330588\pi$
$977$	31.7228			1.01490			0.507451	+	0.861680i	$0.330588\pi$
$978$		0			0
$979$	42.9783			1.37359
$980$		0			0
$981$		−	30.2921i		−	0.967151i
$982$		0			0
$983$	21.0951			0.672829			0.336415	−	0.941714i	$-0.390786\pi$
$983$	21.0951			0.672829			0.336415	+	0.941714i	$0.390786\pi$
$984$		0			0
$985$	0.941578	+	2.92048i	0.0300012	+	0.0930543i
$986$		0			0
$987$		−	12.3644i		−	0.393563i
$988$		0			0
$989$	61.7228			1.96267
$990$		0			0
$991$	36.2337			1.15100			0.575501	−	0.817801i	$-0.304807\pi$
$991$	36.2337			1.15100			0.575501	+	0.817801i	$0.304807\pi$
$992$		0			0
$993$	−8.23369			−0.261288
$994$		0			0
$995$	5.48913	+	17.0256i	0.174017	+	0.539746i
$996$		0			0
$997$		−	44.1485i		−	1.39820i	−0.715026	−	0.699098i	$-0.753585\pi$
$997$		−	44.1485i		−	1.39820i	0.715026	−	0.699098i	$-0.246415\pi$
$998$		0			0
$999$			34.5484i			1.09306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.d.129.2		4
4.3	odd	2		130.2.c.a.129.3	yes	4
5.4	even	2		1040.2.f.c.129.3		4
12.11	even	2		1170.2.f.b.649.4		4
13.12	even	2		1040.2.f.c.129.2		4
20.3	even	4		650.2.d.e.51.6		8
20.7	even	4		650.2.d.e.51.3		8
20.19	odd	2		130.2.c.b.129.2	yes	4
52.31	even	4		1690.2.b.d.339.7		8
52.47	even	4		1690.2.b.d.339.3		8
52.51	odd	2		130.2.c.b.129.3	yes	4
60.59	even	2		1170.2.f.a.649.2		4
65.64	even	2	inner	1040.2.f.d.129.3		4
156.155	even	2		1170.2.f.a.649.1		4
260.47	odd	4		8450.2.a.cn.1.3		4
260.83	odd	4		8450.2.a.cn.1.2		4
260.99	even	4		1690.2.b.d.339.6		8
260.103	even	4		650.2.d.e.51.2		8
260.187	odd	4		8450.2.a.cj.1.3		4
260.203	odd	4		8450.2.a.cj.1.2		4
260.207	even	4		650.2.d.e.51.7		8
260.239	even	4		1690.2.b.d.339.2		8
260.259	odd	2		130.2.c.a.129.2	✓	4
780.779	even	2		1170.2.f.b.649.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.c.a.129.2	✓	4	260.259	odd	2
130.2.c.a.129.3	yes	4	4.3	odd	2
130.2.c.b.129.2	yes	4	20.19	odd	2
130.2.c.b.129.3	yes	4	52.51	odd	2
650.2.d.e.51.2		8	260.103	even	4
650.2.d.e.51.3		8	20.7	even	4
650.2.d.e.51.6		8	20.3	even	4
650.2.d.e.51.7		8	260.207	even	4
1040.2.f.c.129.2		4	13.12	even	2
1040.2.f.c.129.3		4	5.4	even	2
1040.2.f.d.129.2		4	1.1	even	1	trivial
1040.2.f.d.129.3		4	65.64	even	2	inner
1170.2.f.a.649.1		4	156.155	even	2
1170.2.f.a.649.2		4	60.59	even	2
1170.2.f.b.649.3		4	780.779	even	2
1170.2.f.b.649.4		4	12.11	even	2
1690.2.b.d.339.2		8	260.239	even	4
1690.2.b.d.339.3		8	52.47	even	4
1690.2.b.d.339.6		8	260.99	even	4
1690.2.b.d.339.7		8	52.31	even	4
8450.2.a.cj.1.2		4	260.203	odd	4
8450.2.a.cj.1.3		4	260.187	odd	4
8450.2.a.cn.1.2		4	260.83	odd	4
8450.2.a.cn.1.3		4	260.47	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)

\(f(q)\)	\(=\)	\(q-0.792287i q^{3} +(-0.686141 - 2.12819i) q^{5} -3.37228 q^{7} +2.37228 q^{9} +O(q^{10})\)	\(q-0.792287i q^{3} +(-0.686141 - 2.12819i) q^{5} -3.37228 q^{7} +2.37228 q^{9} -5.04868i q^{11} +(1.00000 - 3.46410i) q^{13} +(-1.68614 + 0.543620i) q^{15} +2.67181i q^{17} +3.46410i q^{19} +2.67181i q^{21} +6.63325i q^{23} +(-4.05842 + 2.92048i) q^{25} -4.25639i q^{27} -8.74456 q^{29} -3.46410i q^{31} -4.00000 q^{33} +(2.31386 + 7.17687i) q^{35} -8.11684 q^{37} +(-2.74456 - 0.792287i) q^{39} -3.16915i q^{41} -9.30506i q^{43} +(-1.62772 - 5.04868i) q^{45} -4.62772 q^{47} +4.37228 q^{49} +2.11684 q^{51} +1.58457i q^{53} +(-10.7446 + 3.46410i) q^{55} +2.74456 q^{57} +6.63325i q^{59} +4.74456 q^{61} -8.00000 q^{63} +(-8.05842 + 0.248667i) q^{65} +4.00000 q^{67} +5.25544 q^{69} -3.96143i q^{71} -10.0000 q^{73} +(2.31386 + 3.21543i) q^{75} +17.0256i q^{77} +6.74456 q^{79} +3.74456 q^{81} -2.74456 q^{83} +(5.68614 - 1.83324i) q^{85} +6.92820i q^{87} +8.51278i q^{89} +(-3.37228 + 11.6819i) q^{91} -2.74456 q^{93} +(7.37228 - 2.37686i) q^{95} +4.74456 q^{97} -11.9769i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 4 * q + 3 * q^5 - 2 * q^7 - 2 * q^9	\( 4 q + 3 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{13} - q^{15} + q^{25} - 12 q^{29} - 16 q^{33} + 15 q^{35} + 2 q^{37} + 12 q^{39} - 18 q^{45} - 30 q^{47} + 6 q^{49} - 26 q^{51} - 20 q^{55} - 12 q^{57} - 4 q^{61} - 32 q^{63} - 15 q^{65} + 16 q^{67} + 44 q^{69} - 40 q^{73} + 15 q^{75} + 4 q^{79} - 8 q^{81} + 12 q^{83} + 17 q^{85} - 2 q^{91} + 12 q^{93} + 18 q^{95} - 4 q^{97}+O(q^{100}) \) 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 - 15 * 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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