LMFDB - Embedded newform 1040.2.f.g.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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 $ x^10 + 11x^8 + 36x^6 + 42x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.35685i$ of defining polynomial
Character	$\chi$	$=$	1040.129
Dual form			1040.2.f.g.129.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22659i q^{3} +(-0.589653 + 2.15692i) q^{5} +1.65488 q^{7} -7.41088 q^{9} +4.44624i q^{11} +(2.04103 + 2.97224i) q^{13} +(6.95950 + 1.90257i) q^{15} +5.33962i q^{17} -0.472457i q^{19} -5.33962i q^{21} +1.08725i q^{23} +(-4.30462 - 2.54367i) q^{25} +14.2321i q^{27} +5.50812 q^{29} +5.47201i q^{31} +14.3462 q^{33} +(-0.975805 + 3.56945i) q^{35} -3.65488 q^{37} +(9.59018 - 6.58557i) q^{39} +10.4270i q^{41} -8.14528i q^{43} +(4.36985 - 15.9847i) q^{45} -1.65488 q^{47} -4.26137 q^{49} +17.2288 q^{51} -4.14520i q^{53} +(-9.59018 - 2.62174i) q^{55} -1.52443 q^{57} -2.52401i q^{59} +0.160253 q^{61} -12.2641 q^{63} +(-7.61438 + 2.64976i) q^{65} +8.26412 q^{67} +3.50812 q^{69} -12.9861i q^{71} -10.2641 q^{73} +(-8.20738 + 13.8892i) q^{75} +7.35799i q^{77} -8.65763 q^{79} +23.6885 q^{81} +14.0696 q^{83} +(-11.5171 - 3.14852i) q^{85} -17.7724i q^{87} +7.61027i q^{89} +(3.37766 + 4.91869i) q^{91} +17.6559 q^{93} +(1.01905 + 0.278586i) q^{95} +16.7397 q^{97} -32.9505i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61}+ \cdots + 72 q^{97}+O(q^{100}) $ 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

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
$3$		−	3.22659i		−	1.86287i	−0.363905	−	0.931436i	$-0.618557\pi$
$3$		−	3.22659i		−	1.86287i	0.363905	−	0.931436i	$-0.381443\pi$
$4$		0			0
$5$	−0.589653	+	2.15692i	−0.263701	+	0.964605i
$5$	−0.589653	+	2.15692i	−0.263701	+	0.964605i
$6$		0			0
$7$	1.65488			0.625486			0.312743	−	0.949838i	$-0.398752\pi$
$7$	1.65488			0.625486			0.312743	+	0.949838i	$0.398752\pi$
$8$		0			0
$9$	−7.41088			−2.47029
$10$		0			0
$11$			4.44624i			1.34059i	0.742094	+	0.670296i	$0.233833\pi$
$11$			4.44624i			1.34059i	−0.742094	+	0.670296i	$0.766167\pi$
$12$		0			0
$13$	2.04103	+	2.97224i	0.566081	+	0.824350i
$13$	2.04103	+	2.97224i	0.566081	+	0.824350i
$14$		0			0
$15$	6.95950	+	1.90257i	1.79693	+	0.491241i
$16$		0			0
$17$			5.33962i			1.29505i	0.762045	+	0.647524i	$0.224195\pi$
$17$			5.33962i			1.29505i	−0.762045	+	0.647524i	$0.775805\pi$
$18$		0			0
$19$		−	0.472457i		−	0.108389i	−0.998530	−	0.0541946i	$-0.982741\pi$
$19$		−	0.472457i		−	0.108389i	0.998530	−	0.0541946i	$-0.0172591\pi$
$20$		0			0
$21$		−	5.33962i		−	1.16520i
$22$		0			0
$23$			1.08725i			0.226708i	0.993555	+	0.113354i	$0.0361594\pi$
$23$			1.08725i			0.226708i	−0.993555	+	0.113354i	$0.963841\pi$
$24$		0			0
$25$	−4.30462	−	2.54367i	−0.860924	−	0.508734i
$26$		0			0
$27$			14.2321i			2.73897i
$28$		0			0
$29$	5.50812			1.02283			0.511416	−	0.859333i	$-0.329121\pi$
$29$	5.50812			1.02283			0.511416	+	0.859333i	$0.329121\pi$
$30$		0			0
$31$			5.47201i			0.982803i	0.870933	+	0.491401i	$0.163515\pi$
$31$			5.47201i			0.982803i	−0.870933	+	0.491401i	$0.836485\pi$
$32$		0			0
$33$	14.3462			2.49735
$34$		0			0
$35$	−0.975805	+	3.56945i	−0.164941	+	0.603347i
$36$		0			0
$37$	−3.65488			−0.600859			−0.300429	−	0.953804i	$-0.597130\pi$
$37$	−3.65488			−0.600859			−0.300429	+	0.953804i	$0.597130\pi$
$38$		0			0
$39$	9.59018	−	6.58557i	1.53566	−	1.05454i
$40$		0			0
$41$			10.4270i			1.62842i	0.580572	+	0.814209i	$0.302829\pi$
$41$			10.4270i			1.62842i	−0.580572	+	0.814209i	$0.697171\pi$
$42$		0			0
$43$		−	8.14528i		−	1.24214i	−0.783753	−	0.621072i	$-0.786697\pi$
$43$		−	8.14528i		−	1.24214i	0.783753	−	0.621072i	$-0.213303\pi$
$44$		0			0
$45$	4.36985	−	15.9847i	0.651418	−	2.38286i
$46$		0			0
$47$	−1.65488			−0.241389			−0.120695	−	0.992690i	$-0.538512\pi$
$47$	−1.65488			−0.241389			−0.120695	+	0.992690i	$0.538512\pi$
$48$		0			0
$49$	−4.26137			−0.608767
$50$		0			0
$51$	17.2288			2.41251
$52$		0			0
$53$		−	4.14520i		−	0.569387i	−0.958619	−	0.284693i	$-0.908108\pi$
$53$		−	4.14520i		−	0.569387i	0.958619	−	0.284693i	$-0.0918918\pi$
$54$		0			0
$55$	−9.59018	−	2.62174i	−1.29314	−	0.353515i
$56$		0			0
$57$	−1.52443			−0.201915
$58$		0			0
$59$		−	2.52401i		−	0.328598i	−0.986411	−	0.164299i	$-0.947464\pi$
$59$		−	2.52401i		−	0.328598i	0.986411	−	0.164299i	$-0.0525362\pi$
$60$		0			0
$61$	0.160253			0.0205183			0.0102591	−	0.999947i	$-0.496734\pi$
$61$	0.160253			0.0205183			0.0102591	+	0.999947i	$0.496734\pi$
$62$		0			0
$63$	−12.2641			−1.54513
$64$		0			0
$65$	−7.61438	+	2.64976i	−0.944448	+	0.328662i
$66$		0			0
$67$	8.26412			1.00962			0.504811	−	0.863230i	$-0.331562\pi$
$67$	8.26412			1.00962			0.504811	+	0.863230i	$0.331562\pi$
$68$		0			0
$69$	3.50812			0.422328
$70$		0			0
$71$		−	12.9861i		−	1.54117i	−0.637336	−	0.770586i	$-0.719964\pi$
$71$		−	12.9861i		−	1.54117i	0.637336	−	0.770586i	$-0.280036\pi$
$72$		0			0
$73$	−10.2641			−1.20132			−0.600662	−	0.799503i	$-0.705096\pi$
$73$	−10.2641			−1.20132			−0.600662	+	0.799503i	$0.705096\pi$
$74$		0			0
$75$	−8.20738	+	13.8892i	−0.947706	+	1.60379i
$76$		0			0
$77$			7.35799i			0.838521i
$78$		0			0
$79$	−8.65763			−0.974059			−0.487030	−	0.873385i	$-0.661920\pi$
$79$	−8.65763			−0.974059			−0.487030	+	0.873385i	$0.661920\pi$
$80$		0			0
$81$	23.6885			2.63205
$82$		0			0
$83$	14.0696			1.54434			0.772172	−	0.635414i	$-0.219171\pi$
$83$	14.0696			1.54434			0.772172	+	0.635414i	$0.219171\pi$
$84$		0			0
$85$	−11.5171	−	3.14852i	−1.24921	−	0.341505i
$86$		0			0
$87$		−	17.7724i		−	1.90541i
$88$		0			0
$89$			7.61027i			0.806687i	0.915049	+	0.403344i	$0.132152\pi$
$89$			7.61027i			0.806687i	−0.915049	+	0.403344i	$0.867848\pi$
$90$		0			0
$91$	3.37766	+	4.91869i	0.354075	+	0.515619i
$92$		0			0
$93$	17.6559			1.83084
$94$		0			0
$95$	1.01905	+	0.278586i	0.104553	+	0.0285823i
$96$		0			0
$97$	16.7397			1.69966			0.849829	−	0.527058i	$-0.176705\pi$
$97$	16.7397			1.69966			0.849829	+	0.527058i	$0.176705\pi$
$98$		0			0
$99$		−	32.9505i		−	3.31165i
$100$		0			0
$101$	15.7729			1.56946			0.784731	−	0.619836i	$-0.212801\pi$
$101$	15.7729			1.56946			0.784731	+	0.619836i	$0.212801\pi$
$102$		0			0
$103$		−	1.60432i		−	0.158079i	−0.996871	−	0.0790393i	$-0.974815\pi$
$103$		−	1.60432i		−	0.158079i	0.996871	−	0.0790393i	$-0.0251853\pi$
$104$		0			0
$105$	11.5171	+	3.14852i	1.12396	+	0.307264i
$106$		0			0
$107$		−	1.34368i		−	0.129898i	−0.997889	−	0.0649492i	$-0.979311\pi$
$107$		−	1.34368i		−	0.129898i	0.997889	−	0.0649492i	$-0.0206885\pi$
$108$		0			0
$109$		−	5.33126i		−	0.510642i	−0.966856	−	0.255321i	$-0.917819\pi$
$109$		−	5.33126i		−	0.510642i	0.966856	−	0.255321i	$-0.0821811\pi$
$110$		0			0
$111$			11.7928i			1.11932i
$112$		0			0
$113$			14.7760i			1.39001i	0.719007	+	0.695003i	$0.244597\pi$
$113$			14.7760i			1.39001i	−0.719007	+	0.695003i	$0.755403\pi$
$114$		0			0
$115$	−2.34512	−	0.641102i	−0.218684	−	0.0597831i
$116$		0			0
$117$	−15.1258	−	22.0269i	−1.39838	−	2.03639i
$118$		0			0
$119$			8.83643i			0.810034i
$120$		0			0
$121$	−8.76903			−0.797185
$122$		0			0
$123$	33.6435			3.03353
$124$		0			0
$125$	8.02473	−	7.78484i	0.717753	−	0.696297i
$126$		0			0
$127$		−	3.15132i		−	0.279634i	−0.990177	−	0.139817i	$-0.955349\pi$
$127$		−	3.15132i		−	0.279634i	0.990177	−	0.139817i	$-0.0446515\pi$
$128$		0			0
$129$	−26.2815			−2.31396
$130$		0			0
$131$	−21.0724			−1.84110			−0.920551	−	0.390622i	$-0.872260\pi$
$131$	−21.0724			−1.84110			−0.920551	+	0.390622i	$0.872260\pi$
$132$		0			0
$133$		−	0.781860i		−	0.0677959i
$134$		0			0
$135$	−30.6975	−	8.39200i	−2.64202	−	0.722268i
$136$		0			0
$137$	5.58631			0.477270			0.238635	−	0.971109i	$-0.423300\pi$
$137$	5.58631			0.477270			0.238635	+	0.971109i	$0.423300\pi$
$138$		0			0
$139$	7.85391			0.666160			0.333080	−	0.942899i	$-0.391912\pi$
$139$	7.85391			0.666160			0.333080	+	0.942899i	$0.391912\pi$
$140$		0			0
$141$			5.33962i			0.449677i
$142$		0			0
$143$	−13.2153	+	9.07492i	−1.10512	+	0.758883i
$144$		0			0
$145$	−3.24788	+	11.8806i	−0.269722	+	0.986629i
$146$		0			0
$147$			13.7497i			1.13406i
$148$		0			0
$149$			15.7314i			1.28877i	0.764702	+	0.644384i	$0.222886\pi$
$149$			15.7314i			1.28877i	−0.764702	+	0.644384i	$0.777114\pi$
$150$		0			0
$151$			18.2097i			1.48189i	0.671568	+	0.740943i	$0.265621\pi$
$151$			18.2097i			1.48189i	−0.671568	+	0.740943i	$0.734379\pi$
$152$		0			0
$153$		−	39.5713i		−	3.19915i
$154$		0			0
$155$	−11.8027	−	3.22659i	−0.948016	−	0.259166i
$156$		0			0
$157$			0.385754i			0.0307865i	0.999882	+	0.0153933i	$0.00490002\pi$
$157$			0.385754i			0.0307865i	−0.999882	+	0.0153933i	$0.995100\pi$
$158$		0			0
$159$	−13.3749			−1.06069
$160$		0			0
$161$			1.79927i			0.141803i
$162$		0			0
$163$	−6.85666			−0.537055			−0.268527	−	0.963272i	$-0.586537\pi$
$163$	−6.85666			−0.537055			−0.268527	+	0.963272i	$0.586537\pi$
$164$		0			0
$165$	−8.45927	+	30.9436i	−0.658553	+	2.40896i
$166$		0			0
$167$	−10.8836			−0.842201			−0.421101	−	0.907014i	$-0.638356\pi$
$167$	−10.8836			−0.842201			−0.421101	+	0.907014i	$0.638356\pi$
$168$		0			0
$169$	−4.66837	+	12.1329i	−0.359106	+	0.933297i
$170$		0			0
$171$			3.50132i			0.267753i
$172$		0			0
$173$		−	20.6501i		−	1.57000i	−0.619497	−	0.784999i	$-0.712663\pi$
$173$		−	20.6501i		−	1.57000i	0.619497	−	0.784999i	$-0.287337\pi$
$174$		0			0
$175$	−7.12363	−	4.20947i	−0.538496	−	0.318206i
$176$		0			0
$177$	−8.14395			−0.612137
$178$		0			0
$179$	−1.89201			−0.141416			−0.0707079	−	0.997497i	$-0.522526\pi$
$179$	−1.89201			−0.141416			−0.0707079	+	0.997497i	$0.522526\pi$
$180$		0			0
$181$	4.69412			0.348911			0.174455	−	0.984665i	$-0.444183\pi$
$181$	4.69412			0.348911			0.174455	+	0.984665i	$0.444183\pi$
$182$		0			0
$183$		−	0.517070i		−	0.0382229i
$184$		0			0
$185$	2.15511	−	7.88329i	0.158447	−	0.579591i
$186$		0			0
$187$	−23.7412			−1.73613
$188$		0			0
$189$			23.5524i			1.71319i
$190$		0			0
$191$	−11.3367			−0.820298			−0.410149	−	0.912019i	$-0.634523\pi$
$191$	−11.3367			−0.820298			−0.410149	+	0.912019i	$0.634523\pi$
$192$		0			0
$193$	−16.4462			−1.18382			−0.591911	−	0.806004i	$-0.701626\pi$
$193$	−16.4462			−1.18382			−0.591911	+	0.806004i	$0.701626\pi$
$194$		0			0
$195$	8.54968	+	24.5685i	0.612256	+	1.75939i
$196$		0			0
$197$	16.8858			1.20306			0.601531	−	0.798849i	$-0.294558\pi$
$197$	16.8858			1.20306			0.601531	+	0.798849i	$0.294558\pi$
$198$		0			0
$199$	19.0240			1.34858			0.674288	−	0.738469i	$-0.264451\pi$
$199$	19.0240			1.34858			0.674288	+	0.738469i	$0.264451\pi$
$200$		0			0
$201$		−	26.6649i		−	1.88080i
$202$		0			0
$203$	9.11528			0.639767
$204$		0			0
$205$	−22.4901	−	6.14829i	−1.57078	−	0.429415i
$206$		0			0
$207$		−	8.05750i		−	0.560035i
$208$		0			0
$209$	2.10066			0.145305
$210$		0			0
$211$	1.89201			0.130252			0.0651258	−	0.997877i	$-0.479255\pi$
$211$	1.89201			0.130252			0.0651258	+	0.997877i	$0.479255\pi$
$212$		0			0
$213$	−41.9009			−2.87101
$214$		0			0
$215$	17.5687	+	4.80289i	1.19818	+	0.327554i
$216$		0			0
$217$			9.05553i			0.614729i
$218$		0			0
$219$			33.1181i			2.23791i
$220$		0			0
$221$	−15.8706	+	10.8983i	−1.06757	+	0.733101i
$222$		0			0
$223$	14.3721			0.962427			0.481214	−	0.876603i	$-0.340196\pi$
$223$	14.3721			0.962427			0.481214	+	0.876603i	$0.340196\pi$
$224$		0			0
$225$	31.9010	+	18.8508i	2.12673	+	1.25672i
$226$		0			0
$227$	−6.94886			−0.461212			−0.230606	−	0.973047i	$-0.574071\pi$
$227$	−6.94886			−0.461212			−0.230606	+	0.973047i	$0.574071\pi$
$228$		0			0
$229$			15.3506i			1.01440i	0.861830	+	0.507198i	$0.169319\pi$
$229$			15.3506i			1.01440i	−0.861830	+	0.507198i	$0.830681\pi$
$230$		0			0
$231$	23.7412			1.56206
$232$		0			0
$233$		−	4.06993i		−	0.266630i	−0.991074	−	0.133315i	$-0.957438\pi$
$233$		−	4.06993i		−	0.266630i	0.991074	−	0.133315i	$-0.0425622\pi$
$234$		0			0
$235$	0.975805	−	3.56945i	0.0636545	−	0.232845i
$236$		0			0
$237$			27.9346i			1.81455i
$238$		0			0
$239$			6.89047i			0.445707i	0.974852	+	0.222854i	$0.0715372\pi$
$239$			6.89047i			0.445707i	−0.974852	+	0.222854i	$0.928463\pi$
$240$		0			0
$241$		−	0.781860i		−	0.0503640i	−0.999683	−	0.0251820i	$-0.991983\pi$
$241$		−	0.781860i		−	0.0503640i	0.999683	−	0.0251820i	$-0.00801653\pi$
$242$		0			0
$243$		−	33.7367i		−	2.16421i
$244$		0			0
$245$	2.51273	−	9.19144i	0.160532	−	0.587220i
$246$		0			0
$247$	1.40425	−	0.964301i	0.0893506	−	0.0613570i
$248$		0			0
$249$		−	45.3969i		−	2.87691i
$250$		0			0
$251$	−8.56085			−0.540356			−0.270178	−	0.962810i	$-0.587083\pi$
$251$	−8.56085			−0.540356			−0.270178	+	0.962810i	$0.587083\pi$
$252$		0			0
$253$	−4.83419			−0.303923
$254$		0			0
$255$	−10.1590	+	37.1611i	−0.636181	+	2.32712i
$256$		0			0
$257$		−	0.279087i		−	0.0174090i	−0.999962	−	0.00870448i	$-0.997229\pi$
$257$		−	0.279087i		−	0.0174090i	0.999962	−	0.00870448i	$-0.00277076\pi$
$258$		0			0
$259$	−6.04839			−0.375829
$260$		0			0
$261$	−40.8200			−2.52670
$262$		0			0
$263$			2.67435i			0.164907i	0.996595	+	0.0824537i	$0.0262757\pi$
$263$			2.67435i			0.164907i	−0.996595	+	0.0824537i	$0.973724\pi$
$264$		0			0
$265$	8.94087	+	2.44423i	0.549233	+	0.150148i
$266$		0			0
$267$	24.5552			1.50276
$268$		0			0
$269$	26.5878			1.62109			0.810544	−	0.585677i	$-0.199171\pi$
$269$	26.5878			1.62109			0.810544	+	0.585677i	$0.199171\pi$
$270$		0			0
$271$		−	25.3879i		−	1.54221i	−0.636710	−	0.771104i	$-0.719705\pi$
$271$		−	25.3879i		−	1.54221i	0.636710	−	0.771104i	$-0.280295\pi$
$272$		0			0
$273$	15.8706	−	10.8983i	0.960533	−	0.659597i
$274$		0			0
$275$	11.3098	−	19.1394i	0.682004	−	1.15415i
$276$		0			0
$277$			6.37232i			0.382875i	0.981505	+	0.191438i	$0.0613150\pi$
$277$			6.37232i			0.382875i	−0.981505	+	0.191438i	$0.938685\pi$
$278$		0			0
$279$		−	40.5524i		−	2.42781i
$280$		0			0
$281$		−	0.765134i		−	0.0456441i	−0.999740	−	0.0228220i	$-0.992735\pi$
$281$		−	0.765134i		−	0.0456441i	0.999740	−	0.0228220i	$-0.00726511\pi$
$282$		0			0
$283$		−	9.38817i		−	0.558069i	−0.960281	−	0.279034i	$-0.909986\pi$
$283$		−	9.38817i		−	0.558069i	0.960281	−	0.279034i	$-0.0900144\pi$
$284$		0			0
$285$	0.898882	−	3.28807i	0.0532452	−	0.194768i
$286$		0			0
$287$			17.2554i			1.01855i
$288$		0			0
$289$	−11.5115			−0.677149
$290$		0			0
$291$		−	54.0121i		−	3.16625i
$292$		0			0
$293$	13.4874			0.787941			0.393971	−	0.919123i	$-0.371101\pi$
$293$	13.4874			0.787941			0.393971	+	0.919123i	$0.371101\pi$
$294$		0			0
$295$	5.44409	+	1.48829i	0.316967	+	0.0866516i
$296$		0			0
$297$	−63.2793			−3.67184
$298$		0			0
$299$	−3.23157	+	2.21912i	−0.186887	+	0.128335i
$300$		0			0
$301$		−	13.4795i		−	0.776944i
$302$		0			0
$303$		−	50.8927i		−	2.92371i
$304$		0			0
$305$	−0.0944935	+	0.345653i	−0.00541068	+	0.0197920i
$306$		0			0
$307$	−25.4064			−1.45002			−0.725009	−	0.688739i	$-0.758165\pi$
$307$	−25.4064			−1.45002			−0.725009	+	0.688739i	$0.758165\pi$
$308$		0			0
$309$	−5.17649			−0.294480
$310$		0			0
$311$	12.4250			0.704559			0.352280	−	0.935895i	$-0.385407\pi$
$311$	12.4250			0.704559			0.352280	+	0.935895i	$0.385407\pi$
$312$		0			0
$313$		−	23.9664i		−	1.35466i	−0.735678	−	0.677331i	$-0.763136\pi$
$313$		−	23.9664i		−	1.35466i	0.735678	−	0.677331i	$-0.236864\pi$
$314$		0			0
$315$	7.23157	−	26.4527i	0.407453	−	1.49044i
$316$		0			0
$317$	−20.3618			−1.14363			−0.571817	−	0.820381i	$-0.693761\pi$
$317$	−20.3618			−1.14363			−0.571817	+	0.820381i	$0.693761\pi$
$318$		0			0
$319$			24.4904i			1.37120i
$320$		0			0
$321$	−4.33550			−0.241984
$322$		0			0
$323$	2.52274			0.140369
$324$		0			0
$325$	−1.22548	−	17.9861i	−0.0679774	−	0.997687i
$326$		0			0
$327$	−17.2018			−0.951260
$328$		0			0
$329$	−2.73863			−0.150986
$330$		0			0
$331$			30.0834i			1.65353i	0.562546	+	0.826766i	$0.309822\pi$
$331$			30.0834i			1.65353i	−0.562546	+	0.826766i	$0.690178\pi$
$332$		0			0
$333$	27.0859			1.48430
$334$		0			0
$335$	−4.87296	+	17.8251i	−0.266238	+	0.973887i
$336$		0			0
$337$		−	10.8983i		−	0.593670i	−0.954929	−	0.296835i	$-0.904069\pi$
$337$		−	10.8983i		−	0.593670i	0.954929	−	0.296835i	$-0.0959312\pi$
$338$		0			0
$339$	47.6760			2.58940
$340$		0			0
$341$	−24.3299			−1.31754
$342$		0			0
$343$	−18.6362			−1.00626
$344$		0			0
$345$	−2.06857	+	7.56674i	−0.111368	+	0.407379i
$346$		0			0
$347$			11.9236i			0.640093i	0.947402	+	0.320047i	$0.103699\pi$
$347$			11.9236i			0.640093i	−0.947402	+	0.320047i	$0.896301\pi$
$348$		0			0
$349$			3.38144i			0.181004i	0.995896	+	0.0905021i	$0.0288472\pi$
$349$			3.38144i			0.181004i	−0.995896	+	0.0905021i	$0.971153\pi$
$350$		0			0
$351$	−42.3011	+	29.0482i	−2.25787	+	1.55048i
$352$		0			0
$353$	13.4771			0.717314			0.358657	−	0.933469i	$-0.383235\pi$
$353$	13.4771			0.717314			0.358657	+	0.933469i	$0.383235\pi$
$354$		0			0
$355$	28.0101	+	7.65732i	1.48662	+	0.406408i
$356$		0			0
$357$	28.5115			1.50899
$358$		0			0
$359$		−	4.74277i		−	0.250314i	−0.992137	−	0.125157i	$-0.960057\pi$
$359$		−	4.74277i		−	0.250314i	0.992137	−	0.125157i	$-0.0399434\pi$
$360$		0			0
$361$	18.7768			0.988252
$362$		0			0
$363$			28.2941i			1.48505i
$364$		0			0
$365$	6.05227	−	22.1389i	0.316790	−	1.15880i
$366$		0			0
$367$		−	31.4288i		−	1.64057i	−0.571956	−	0.820284i	$-0.693815\pi$
$367$		−	31.4288i		−	1.64057i	0.571956	−	0.820284i	$-0.306185\pi$
$368$		0			0
$369$		−	77.2729i		−	4.02267i
$370$		0			0
$371$		−	6.85981i		−	0.356143i
$372$		0			0
$373$		−	25.7810i		−	1.33489i	−0.744660	−	0.667444i	$-0.767388\pi$
$373$		−	25.7810i		−	1.33489i	0.744660	−	0.667444i	$-0.232612\pi$
$374$		0			0
$375$	−25.1185	−	25.8925i	−1.29711	−	1.33708i
$376$		0			0
$377$	11.2423	+	16.3714i	0.579005	+	0.843172i
$378$		0			0
$379$			14.7011i			0.755146i	0.925980	+	0.377573i	$0.123241\pi$
$379$			14.7011i			0.755146i	−0.925980	+	0.377573i	$0.876759\pi$
$380$		0			0
$381$	−10.1680			−0.520923
$382$		0			0
$383$	1.62790			0.0831816			0.0415908	−	0.999135i	$-0.486757\pi$
$383$	1.62790			0.0831816			0.0415908	+	0.999135i	$0.486757\pi$
$384$		0			0
$385$	−15.8706	−	4.33866i	−0.808841	−	0.221119i
$386$		0			0
$387$			60.3637i			3.06846i
$388$		0			0
$389$	28.3607			1.43795			0.718973	−	0.695038i	$-0.244612\pi$
$389$	28.3607			1.43795			0.718973	+	0.695038i	$0.244612\pi$
$390$		0			0
$391$	−5.80552			−0.293598
$392$		0			0
$393$			67.9919i			3.42974i
$394$		0			0
$395$	5.10500	−	18.6738i	0.256860	−	0.939582i
$396$		0			0
$397$	−1.54689			−0.0776364			−0.0388182	−	0.999246i	$-0.512359\pi$
$397$	−1.54689			−0.0776364			−0.0388182	+	0.999246i	$0.512359\pi$
$398$		0			0
$399$	−2.52274			−0.126295
$400$		0			0
$401$			5.55872i			0.277589i	0.990321	+	0.138795i	$0.0443228\pi$
$401$			5.55872i			0.277589i	−0.990321	+	0.138795i	$0.955677\pi$
$402$		0			0
$403$	−16.2641	+	11.1686i	−0.810173	+	0.556346i
$404$		0			0
$405$	−13.9680	+	51.0942i	−0.694075	+	2.53889i
$406$		0			0
$407$		−	16.2505i		−	0.805506i
$408$		0			0
$409$			4.70652i			0.232722i	0.993207	+	0.116361i	$0.0371230\pi$
$409$			4.70652i			0.232722i	−0.993207	+	0.116361i	$0.962877\pi$
$410$		0			0
$411$		−	18.0247i		−	0.889094i
$412$		0			0
$413$		−	4.17694i		−	0.205534i
$414$		0			0
$415$	−8.29620	+	30.3471i	−0.407245	+	1.48968i
$416$		0			0
$417$		−	25.3413i		−	1.24097i
$418$		0			0
$419$	17.4613			0.853043			0.426521	−	0.904478i	$-0.359739\pi$
$419$	17.4613			0.853043			0.426521	+	0.904478i	$0.359739\pi$
$420$		0			0
$421$		−	23.1162i		−	1.12662i	−0.826247	−	0.563308i	$-0.809529\pi$
$421$		−	23.1162i		−	1.12662i	0.826247	−	0.563308i	$-0.190471\pi$
$422$		0			0
$423$	12.2641			0.596302
$424$		0			0
$425$	13.5822	−	22.9850i	0.658835	−	1.11494i
$426$		0			0
$427$	0.265199			0.0128339
$428$		0			0
$429$	29.2810	+	42.6402i	1.41370	+	2.05869i
$430$		0			0
$431$		−	19.8860i		−	0.957877i	−0.877848	−	0.478939i	$-0.841022\pi$
$431$		−	19.8860i		−	0.957877i	0.877848	−	0.478939i	$-0.158978\pi$
$432$		0			0
$433$		−	2.09858i		−	0.100851i	−0.998728	−	0.0504255i	$-0.983942\pi$
$433$		−	2.09858i		−	0.100851i	0.998728	−	0.0504255i	$-0.0160578\pi$
$434$		0			0
$435$	38.3338	+	10.4796i	1.83796	+	0.502457i
$436$		0			0
$437$	0.513681			0.0245727
$438$		0			0
$439$	14.7553			0.704233			0.352117	−	0.935956i	$-0.385462\pi$
$439$	14.7553			0.704233			0.352117	+	0.935956i	$0.385462\pi$
$440$		0			0
$441$	31.5805			1.50383
$442$		0			0
$443$			12.8191i			0.609053i	0.952504	+	0.304526i	$0.0984982\pi$
$443$			12.8191i			0.609053i	−0.952504	+	0.304526i	$0.901502\pi$
$444$		0			0
$445$	−16.4148	−	4.48742i	−0.778134	−	0.212724i
$446$		0			0
$447$	50.7588			2.40081
$448$		0			0
$449$			25.7925i			1.21722i	0.793468	+	0.608612i	$0.208273\pi$
$449$			25.7925i			1.21722i	−0.793468	+	0.608612i	$0.791727\pi$
$450$		0			0
$451$	−46.3607			−2.18304
$452$		0			0
$453$	58.7553			2.76057
$454$		0			0
$455$	−12.6009	+	4.38503i	−0.590739	+	0.205574i
$456$		0			0
$457$	9.61892			0.449954			0.224977	−	0.974364i	$-0.427769\pi$
$457$	9.61892			0.449954			0.224977	+	0.974364i	$0.427769\pi$
$458$		0			0
$459$	−75.9940			−3.54709
$460$		0			0
$461$		−	18.4899i		−	0.861160i	−0.902552	−	0.430580i	$-0.858309\pi$
$461$		−	18.4899i		−	0.861160i	0.902552	−	0.430580i	$-0.141691\pi$
$462$		0			0
$463$	4.26412			0.198170			0.0990852	−	0.995079i	$-0.468408\pi$
$463$	4.26412			0.198170			0.0990852	+	0.995079i	$0.468408\pi$
$464$		0			0
$465$	−10.4109	+	38.0825i	−0.482793	+	1.76603i
$466$		0			0
$467$		−	8.12164i		−	0.375825i	−0.982186	−	0.187912i	$-0.939828\pi$
$467$		−	8.12164i		−	0.375825i	0.982186	−	0.187912i	$-0.0601721\pi$
$468$		0			0
$469$	13.6761			0.631505
$470$		0			0
$471$	1.24467			0.0573514
$472$		0			0
$473$	36.2159			1.66521
$474$		0			0
$475$	−1.20178	+	2.03375i	−0.0551412	+	0.0933148i
$476$		0			0
$477$			30.7196i			1.40655i
$478$		0			0
$479$			9.07748i			0.414761i	0.978260	+	0.207380i	$0.0664938\pi$
$479$			9.07748i			0.414761i	−0.978260	+	0.207380i	$0.933506\pi$
$480$		0			0
$481$	−7.45973	−	10.8632i	−0.340134	−	0.495318i
$482$		0			0
$483$	5.80552			0.264160
$484$		0			0
$485$	−9.87061	+	36.1062i	−0.448201	+	1.63950i
$486$		0			0
$487$	−0.124845			−0.00565727			−0.00282863	−	0.999996i	$-0.500900\pi$
$487$	−0.124845			−0.00565727			−0.00282863	+	0.999996i	$0.500900\pi$
$488$		0			0
$489$			22.1236i			1.00046i
$490$		0			0
$491$	31.7515			1.43293			0.716463	−	0.697626i	$-0.245760\pi$
$491$	31.7515			1.43293			0.716463	+	0.697626i	$0.245760\pi$
$492$		0			0
$493$			29.4113i			1.32462i
$494$		0			0
$495$	71.0717	+	19.4294i	3.19444	+	0.873285i
$496$		0			0
$497$		−	21.4905i		−	0.963981i
$498$		0			0
$499$			12.3636i			0.553469i	0.960946	+	0.276734i	$0.0892522\pi$
$499$			12.3636i			0.553469i	−0.960946	+	0.276734i	$0.910748\pi$
$500$		0			0
$501$			35.1170i			1.56891i
$502$		0			0
$503$		−	26.5184i		−	1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$503$		−	26.5184i		−	1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$504$		0			0
$505$	−9.30054	+	34.0209i	−0.413869	+	1.51391i
$506$		0			0
$507$	39.1478	+	15.0629i	1.73861	+	0.668968i
$508$		0			0
$509$		−	39.4076i		−	1.74671i	−0.487085	−	0.873355i	$-0.661940\pi$
$509$		−	39.4076i		−	1.74671i	0.487085	−	0.873355i	$-0.338060\pi$
$510$		0			0
$511$	−16.9859			−0.751411
$512$		0			0
$513$	6.72406			0.296874
$514$		0			0
$515$	3.46040	+	0.945994i	0.152483	+	0.0416855i
$516$		0			0
$517$		−	7.35799i		−	0.323604i
$518$		0			0
$519$	−66.6294			−2.92471
$520$		0			0
$521$	−30.8042			−1.34956			−0.674779	−	0.738020i	$-0.735761\pi$
$521$	−30.8042			−1.34956			−0.674779	+	0.738020i	$0.735761\pi$
$522$		0			0
$523$			5.53950i			0.242225i	0.992639	+	0.121113i	$0.0386463\pi$
$523$			5.53950i			0.242225i	−0.992639	+	0.121113i	$0.961354\pi$
$524$		0			0
$525$	−13.5822	+	22.9850i	−0.592777	+	1.00315i
$526$		0			0
$527$	−29.2185			−1.27278
$528$		0			0
$529$	21.8179			0.948603
$530$		0			0
$531$			18.7051i			0.811734i
$532$		0			0
$533$	−30.9914	+	21.2818i	−1.34239	+	0.921815i
$534$		0			0
$535$	2.89821	+	0.792305i	0.125301	+	0.0342543i
$536$		0			0
$537$			6.10475i			0.263440i
$538$		0			0
$539$		−	18.9471i		−	0.816108i
$540$		0			0
$541$		−	10.7346i		−	0.461518i	−0.973011	−	0.230759i	$-0.925879\pi$
$541$		−	10.7346i		−	0.461518i	0.973011	−	0.230759i	$-0.0741208\pi$
$542$		0			0
$543$		−	15.1460i		−	0.649976i
$544$		0			0
$545$	11.4991	+	3.14359i	0.492567	+	0.134657i
$546$		0			0
$547$		−	32.6681i		−	1.39679i	−0.715714	−	0.698393i	$-0.753899\pi$
$547$		−	32.6681i		−	1.39679i	0.715714	−	0.698393i	$-0.246101\pi$
$548$		0			0
$549$	−1.18761			−0.0506861
$550$		0			0
$551$		−	2.60235i		−	0.110864i
$552$		0			0
$553$	−14.3273			−0.609260
$554$		0			0
$555$	−25.4361	−	6.95366i	−1.07970	−	0.295166i
$556$		0			0
$557$	21.4016			0.906815			0.453407	−	0.891303i	$-0.350208\pi$
$557$	21.4016			0.906815			0.453407	+	0.891303i	$0.350208\pi$
$558$		0			0
$559$	24.2097	−	16.6248i	1.02396	−	0.703154i
$560$		0			0
$561$			76.6032i			3.23419i
$562$		0			0
$563$			4.68087i			0.197275i	0.995123	+	0.0986376i	$0.0314485\pi$
$563$			4.68087i			0.197275i	−0.995123	+	0.0986376i	$0.968552\pi$
$564$		0			0
$565$	−31.8706	−	8.71270i	−1.34081	−	0.366546i
$566$		0			0
$567$	39.2016			1.64631
$568$		0			0
$569$	−18.4434			−0.773186			−0.386593	−	0.922250i	$-0.626348\pi$
$569$	−18.4434			−0.773186			−0.386593	+	0.922250i	$0.626348\pi$
$570$		0			0
$571$	−0.667911			−0.0279512			−0.0139756	−	0.999902i	$-0.504449\pi$
$571$	−0.667911			−0.0279512			−0.0139756	+	0.999902i	$0.504449\pi$
$572$		0			0
$573$			36.5790i			1.52811i
$574$		0			0
$575$	2.76561	−	4.68021i	0.115334	−	0.195178i
$576$		0			0
$577$	24.6947			1.02805			0.514026	−	0.857775i	$-0.328154\pi$
$577$	24.6947			1.02805			0.514026	+	0.857775i	$0.328154\pi$
$578$		0			0
$579$			53.0650i			2.20531i
$580$		0			0
$581$	23.2836			0.965965
$582$		0			0
$583$	18.4305			0.763315
$584$		0			0
$585$	56.4292	−	19.6370i	2.33306	−	0.811892i
$586$		0			0
$587$	−18.2586			−0.753614			−0.376807	−	0.926292i	$-0.622978\pi$
$587$	−18.2586			−0.753614			−0.376807	+	0.926292i	$0.622978\pi$
$588$		0			0
$589$	2.58529			0.106525
$590$		0			0
$591$		−	54.4835i		−	2.24115i
$592$		0			0
$593$	−6.67326			−0.274038			−0.137019	−	0.990568i	$-0.543752\pi$
$593$	−6.67326			−0.274038			−0.137019	+	0.990568i	$0.543752\pi$
$594$		0			0
$595$	−19.0595	−	5.21043i	−0.781363	−	0.213607i
$596$		0			0
$597$		−	61.3826i		−	2.51222i
$598$		0			0
$599$	−39.4490			−1.61184			−0.805922	−	0.592022i	$-0.798330\pi$
$599$	−39.4490			−1.61184			−0.805922	+	0.592022i	$0.798330\pi$
$600$		0			0
$601$	−25.9245			−1.05748			−0.528741	−	0.848783i	$-0.677336\pi$
$601$	−25.9245			−1.05748			−0.528741	+	0.848783i	$0.677336\pi$
$602$		0			0
$603$	−61.2444			−2.49406
$604$		0			0
$605$	5.17068	−	18.9141i	0.210218	−	0.768968i
$606$		0			0
$607$		−	4.40112i		−	0.178636i	−0.996003	−	0.0893181i	$-0.971531\pi$
$607$		−	4.40112i		−	0.178636i	0.996003	−	0.0893181i	$-0.0284688\pi$
$608$		0			0
$609$		−	29.4113i		−	1.19180i
$610$		0			0
$611$	−3.37766	−	4.91869i	−0.136646	−	0.198989i
$612$		0			0
$613$	−14.2573			−0.575848			−0.287924	−	0.957653i	$-0.592965\pi$
$613$	−14.2573			−0.575848			−0.287924	+	0.957653i	$0.592965\pi$
$614$		0			0
$615$	−19.8380	+	72.5664i	−0.799945	+	2.92616i
$616$		0			0
$617$	−9.58843			−0.386016			−0.193008	−	0.981197i	$-0.561824\pi$
$617$	−9.58843			−0.386016			−0.193008	+	0.981197i	$0.561824\pi$
$618$		0			0
$619$		−	41.8263i		−	1.68114i	−0.541703	−	0.840570i	$-0.682220\pi$
$619$		−	41.8263i		−	1.68114i	0.541703	−	0.840570i	$-0.317780\pi$
$620$		0			0
$621$	−15.4739			−0.620946
$622$		0			0
$623$			12.5941i			0.504572i
$624$		0			0
$625$	12.0595	+	21.8991i	0.482379	+	0.875962i
$626$		0			0
$627$		−	6.77796i		−	0.270686i
$628$		0			0
$629$		−	19.5157i		−	0.778141i
$630$		0			0
$631$		−	10.7991i		−	0.429906i	−0.976624	−	0.214953i	$-0.931040\pi$
$631$		−	10.7991i		−	0.429906i	0.976624	−	0.214953i	$-0.0689599\pi$
$632$		0			0
$633$		−	6.10475i		−	0.242642i
$634$		0			0
$635$	6.79714	+	1.85818i	0.269736	+	0.0737398i
$636$		0			0
$637$	−8.69760	−	12.6658i	−0.344611	−	0.501837i
$638$		0			0
$639$			96.2387i			3.80714i
$640$		0			0
$641$	−17.0240			−0.672407			−0.336204	−	0.941789i	$-0.609143\pi$
$641$	−17.0240			−0.672407			−0.336204	+	0.941789i	$0.609143\pi$
$642$		0			0
$643$	−33.9076			−1.33719			−0.668593	−	0.743628i	$-0.733103\pi$
$643$	−33.9076			−1.33719			−0.668593	+	0.743628i	$0.733103\pi$
$644$		0			0
$645$	15.4970	−	56.6871i	0.610192	−	2.23205i
$646$		0			0
$647$			24.1282i			0.948577i	0.880369	+	0.474288i	$0.157295\pi$
$647$			24.1282i			0.948577i	−0.880369	+	0.474288i	$0.842705\pi$
$648$		0			0
$649$	11.2224			0.440516
$650$		0			0
$651$	29.2185			1.14516
$652$		0			0
$653$		−	6.35210i		−	0.248577i	−0.992246	−	0.124288i	$-0.960335\pi$
$653$		−	6.35210i		−	0.248577i	0.992246	−	0.124288i	$-0.0396648\pi$
$654$		0			0
$655$	12.4254	−	45.4515i	0.485500	−	1.77594i
$656$		0			0
$657$	76.0661			2.96762
$658$		0			0
$659$	19.8436			0.772998			0.386499	−	0.922290i	$-0.373684\pi$
$659$	19.8436			0.772998			0.386499	+	0.922290i	$0.373684\pi$
$660$		0			0
$661$		−	40.1796i		−	1.56280i	−0.624028	−	0.781402i	$-0.714505\pi$
$661$		−	40.1796i		−	1.56280i	0.624028	−	0.781402i	$-0.285495\pi$
$662$		0			0
$663$	35.1645	+	51.2079i	1.36567	+	1.98875i
$664$		0			0
$665$	1.68641	+	0.461026i	0.0653962	+	0.0178778i
$666$		0			0
$667$			5.98872i			0.231884i
$668$		0			0
$669$		−	46.3729i		−	1.79288i
$670$		0			0
$671$			0.712522i			0.0275066i
$672$		0			0
$673$		−	24.0716i		−	0.927894i	−0.885863	−	0.463947i	$-0.846433\pi$
$673$		−	24.0716i		−	0.927894i	0.885863	−	0.463947i	$-0.153567\pi$
$674$		0			0
$675$	36.2018	−	61.2637i	1.39341	−	2.35804i
$676$		0			0
$677$			24.0869i			0.925736i	0.886427	+	0.462868i	$0.153180\pi$
$677$			24.0869i			0.925736i	−0.886427	+	0.462868i	$0.846820\pi$
$678$		0			0
$679$	27.7022			1.06311
$680$		0			0
$681$			22.4211i			0.859179i
$682$		0			0
$683$	38.1236			1.45876			0.729380	−	0.684109i	$-0.239809\pi$
$683$	38.1236			1.45876			0.729380	+	0.684109i	$0.239809\pi$
$684$		0			0
$685$	−3.29398	+	12.0492i	−0.125857	+	0.460377i
$686$		0			0
$687$	49.5300			1.88969
$688$		0			0
$689$	12.3205	−	8.46048i	0.469374	−	0.322319i
$690$		0			0
$691$		−	4.01418i		−	0.152707i	−0.997081	−	0.0763533i	$-0.975672\pi$
$691$		−	4.01418i		−	0.152707i	0.997081	−	0.0763533i	$-0.0243277\pi$
$692$		0			0
$693$		−	54.5292i		−	2.07139i
$694$		0			0
$695$	−4.63108	+	16.9403i	−0.175667	+	0.642581i
$696$		0			0
$697$	−55.6760			−2.10888
$698$		0			0
$699$	−13.1320			−0.496697
$700$		0			0
$701$	29.3041			1.10680			0.553401	−	0.832915i	$-0.313330\pi$
$701$	29.3041			1.10680			0.553401	+	0.832915i	$0.313330\pi$
$702$		0			0
$703$			1.72677i			0.0651265i
$704$		0			0
$705$	−11.5171	−	3.14852i	−0.433761	−	0.118580i
$706$		0			0
$707$	26.1023			0.981677
$708$		0			0
$709$			9.99630i			0.375419i	0.982225	+	0.187709i	$0.0601063\pi$
$709$			9.99630i			0.375419i	−0.982225	+	0.187709i	$0.939894\pi$
$710$		0			0
$711$	64.1606			2.40621
$712$		0			0
$713$	−5.94947			−0.222809
$714$		0			0
$715$	−11.7815	−	33.8553i	−0.440602	−	1.26612i
$716$		0			0
$717$	22.2327			0.830296
$718$		0			0
$719$	52.7587			1.96757			0.983783	−	0.179363i	$-0.0574035\pi$
$719$	52.7587			1.96757			0.983783	+	0.179363i	$0.0574035\pi$
$720$		0			0
$721$		−	2.65496i		−	0.0988760i
$722$		0			0
$723$	−2.52274			−0.0938218
$724$		0			0
$725$	−23.7104	−	14.0108i	−0.880581	−	0.520349i
$726$		0			0
$727$			14.2257i			0.527601i	0.964577	+	0.263801i	$0.0849761\pi$
$727$			14.2257i			0.527601i	−0.964577	+	0.263801i	$0.915024\pi$
$728$		0			0
$729$	−37.7891			−1.39960
$730$		0			0
$731$	43.4927			1.60864
$732$		0			0
$733$	11.8558			0.437904			0.218952	−	0.975736i	$-0.429736\pi$
$733$	11.8558			0.437904			0.218952	+	0.975736i	$0.429736\pi$
$734$		0			0
$735$	−29.6570	−	8.10755i	−1.09392	−	0.299051i
$736$		0			0
$737$			36.7442i			1.35349i
$738$		0			0
$739$			12.4740i			0.458864i	0.973325	+	0.229432i	$0.0736868\pi$
$739$			12.4740i			0.458864i	−0.973325	+	0.229432i	$0.926313\pi$
$740$		0			0
$741$	−3.11140	−	4.53095i	−0.114300	−	0.166449i
$742$		0			0
$743$	30.9003			1.13362			0.566812	−	0.823847i	$-0.308177\pi$
$743$	30.9003			1.13362			0.566812	+	0.823847i	$0.308177\pi$
$744$		0			0
$745$	−33.9314	−	9.27607i	−1.24315	−	0.339849i
$746$		0			0
$747$	−104.268			−3.81498
$748$		0			0
$749$		−	2.22363i		−	0.0812497i
$750$		0			0
$751$	−11.2456			−0.410358			−0.205179	−	0.978725i	$-0.565778\pi$
$751$	−11.2456			−0.410358			−0.205179	+	0.978725i	$0.565778\pi$
$752$		0			0
$753$			27.6223i			1.00661i
$754$		0			0
$755$	−39.2769	−	10.7374i	−1.42943	−	0.390775i
$756$		0			0
$757$		−	4.81148i		−	0.174876i	−0.996170	−	0.0874381i	$-0.972132\pi$
$757$		−	4.81148i		−	0.174876i	0.996170	−	0.0874381i	$-0.0278680\pi$
$758$		0			0
$759$			15.5979i			0.566169i
$760$		0			0
$761$		−	7.77200i		−	0.281735i	−0.990028	−	0.140867i	$-0.955011\pi$
$761$		−	7.77200i		−	0.281735i	0.990028	−	0.140867i	$-0.0449891\pi$
$762$		0			0
$763$		−	8.82259i		−	0.319399i
$764$		0			0
$765$	85.3521	+	23.3333i	3.08591	+	0.843618i
$766$		0			0
$767$	7.50196	−	5.15159i	0.270880	−	0.186013i
$768$		0			0
$769$			28.5345i			1.02898i	0.857496	+	0.514491i	$0.172019\pi$
$769$			28.5345i			1.02898i	−0.857496	+	0.514491i	$0.827981\pi$
$770$		0			0
$771$	−0.900499			−0.0324307
$772$		0			0
$773$	−31.8056			−1.14397			−0.571984	−	0.820264i	$-0.693826\pi$
$773$	−31.8056			−1.14397			−0.571984	+	0.820264i	$0.693826\pi$
$774$		0			0
$775$	13.9190	−	23.5549i	0.499985	−	0.846118i
$776$		0			0
$777$			19.5157i			0.700121i
$778$		0			0
$779$	4.92629			0.176503
$780$		0			0
$781$	57.7395			2.06608
$782$		0			0
$783$			78.3921i			2.80150i
$784$		0			0
$785$	−0.832042	−	0.227461i	−0.0296968	−	0.00811844i
$786$		0			0
$787$	−33.8537			−1.20675			−0.603376	−	0.797457i	$-0.706178\pi$
$787$	−33.8537			−1.20675			−0.603376	+	0.797457i	$0.706178\pi$
$788$		0			0
$789$	8.62903			0.307201
$790$		0			0
$791$			24.4525i			0.869430i
$792$		0			0
$793$	0.327081	+	0.476309i	0.0116150	+	0.0169142i
$794$		0			0
$795$	7.88652	−	28.8485i	0.279706	−	1.02315i
$796$		0			0
$797$			37.4776i			1.32753i	0.747943	+	0.663763i	$0.231042\pi$
$797$			37.4776i			1.32753i	−0.747943	+	0.663763i	$0.768958\pi$
$798$		0			0
$799$		−	8.83643i		−	0.312610i
$800$		0			0
$801$		−	56.3988i		−	1.99275i
$802$		0			0
$803$		−	45.6367i		−	1.61048i
$804$		0			0
$805$	−3.88089	−	1.06095i	−0.136783	−	0.0373935i
$806$		0			0
$807$		−	85.7880i		−	3.01988i
$808$		0			0
$809$	8.64480			0.303935			0.151968	−	0.988385i	$-0.451439\pi$
$809$	8.64480			0.303935			0.151968	+	0.988385i	$0.451439\pi$
$810$		0			0
$811$			15.4230i			0.541574i	0.962639	+	0.270787i	$0.0872840\pi$
$811$			15.4230i			0.541574i	−0.962639	+	0.270787i	$0.912716\pi$
$812$		0			0
$813$	−81.9165			−2.87294
$814$		0			0
$815$	4.04305	−	14.7893i	0.141622	−	0.518045i
$816$		0			0
$817$	−3.84830			−0.134635
$818$		0			0
$819$	−25.0315	−	36.4519i	−0.874670	−	1.27373i
$820$		0			0
$821$			24.9107i			0.869389i	0.900578	+	0.434694i	$0.143144\pi$
$821$			24.9107i			0.869389i	−0.900578	+	0.434694i	$0.856856\pi$
$822$		0			0
$823$		−	3.47743i		−	0.121215i	−0.998162	−	0.0606077i	$-0.980696\pi$
$823$		−	3.47743i		−	0.121215i	0.998162	−	0.0606077i	$-0.0193039\pi$
$824$		0			0
$825$	−61.7549	−	36.4920i	−2.15003	−	1.27049i
$826$		0			0
$827$	−32.6194			−1.13429			−0.567143	−	0.823619i	$-0.691951\pi$
$827$	−32.6194			−1.13429			−0.567143	+	0.823619i	$0.691951\pi$
$828$		0			0
$829$	36.8123			1.27854			0.639271	−	0.768981i	$-0.279236\pi$
$829$	36.8123			1.27854			0.639271	+	0.768981i	$0.279236\pi$
$830$		0			0
$831$	20.5608			0.713248
$832$		0			0
$833$		−	22.7541i		−	0.788383i
$834$		0			0
$835$	6.41757	−	23.4752i	0.222089	−	0.812391i
$836$		0			0
$837$	−77.8782			−2.69187
$838$		0			0
$839$		−	23.6723i		−	0.817259i	−0.912700	−	0.408629i	$-0.866007\pi$
$839$		−	23.6723i		−	0.817259i	0.912700	−	0.408629i	$-0.133993\pi$
$840$		0			0
$841$	1.33938			0.0461856
$842$		0			0
$843$	−2.46877			−0.0850291
$844$		0			0
$845$	−23.4169	−	17.2235i	−0.805566	−	0.592506i
$846$		0			0
$847$	−14.5117			−0.498628
$848$		0			0
$849$	−30.2918			−1.03961
$850$		0			0
$851$		−	3.97378i		−	0.136219i
$852$		0			0
$853$	1.82238			0.0623970			0.0311985	−	0.999513i	$-0.490068\pi$
$853$	1.82238			0.0623970			0.0311985	+	0.999513i	$0.490068\pi$
$854$		0			0
$855$	−7.55208	−	2.06457i	−0.258276	−	0.0706066i
$856$		0			0
$857$			34.1031i			1.16494i	0.812852	+	0.582470i	$0.197914\pi$
$857$			34.1031i			1.16494i	−0.812852	+	0.582470i	$0.802086\pi$
$858$		0			0
$859$	18.6350			0.635819			0.317910	−	0.948121i	$-0.397019\pi$
$859$	18.6350			0.635819			0.317910	+	0.948121i	$0.397019\pi$
$860$		0			0
$861$	55.6760			1.89743
$862$		0			0
$863$	5.16462			0.175806			0.0879028	−	0.996129i	$-0.471984\pi$
$863$	5.16462			0.175806			0.0879028	+	0.996129i	$0.471984\pi$
$864$		0			0
$865$	44.5406	+	12.1764i	1.51443	+	0.414010i
$866$		0			0
$867$			37.1430i			1.26144i
$868$		0			0
$869$		−	38.4939i		−	1.30582i
$870$		0			0
$871$	16.8673	+	24.5629i	0.571528	+	0.832282i
$872$		0			0
$873$	−124.056			−4.19865
$874$		0			0
$875$	13.2800	−	12.8830i	0.448945	−	0.435524i
$876$		0			0
$877$	−20.1381			−0.680015			−0.340007	−	0.940423i	$-0.610430\pi$
$877$	−20.1381			−0.680015			−0.340007	+	0.940423i	$0.610430\pi$
$878$		0			0
$879$		−	43.5182i		−	1.46783i
$880$		0			0
$881$	−10.2164			−0.344199			−0.172100	−	0.985080i	$-0.555055\pi$
$881$	−10.2164			−0.344199			−0.172100	+	0.985080i	$0.555055\pi$
$882$		0			0
$883$			14.6787i			0.493976i	0.969018	+	0.246988i	$0.0794409\pi$
$883$			14.6787i			0.493976i	−0.969018	+	0.246988i	$0.920559\pi$
$884$		0			0
$885$	4.80210	−	17.5659i	0.161421	−	0.590470i
$886$		0			0
$887$			12.6524i			0.424827i	0.977180	+	0.212413i	$0.0681323\pi$
$887$			12.6524i			0.424827i	−0.977180	+	0.212413i	$0.931868\pi$
$888$		0			0
$889$		−	5.21505i		−	0.174907i
$890$		0			0
$891$			105.325i			3.52851i
$892$		0			0
$893$			0.781860i			0.0261640i
$894$		0			0
$895$	1.11563	−	4.08093i	0.0372915	−	0.136410i
$896$		0			0
$897$	7.16019	+	10.4270i	0.239072	+	0.348146i
$898$		0			0
$899$			30.1405i			1.00524i
$900$		0			0
$901$	22.1338			0.737383
$902$		0			0
$903$	−43.4927			−1.44735
$904$		0			0
$905$	−2.76790	+	10.1248i	−0.0920081	+	0.336561i
$906$		0			0
$907$		−	39.2705i		−	1.30396i	−0.758238	−	0.651978i	$-0.773940\pi$
$907$		−	39.2705i		−	1.30396i	0.758238	−	0.651978i	$-0.226060\pi$
$908$		0			0
$909$	−116.891			−3.87703
$910$		0			0
$911$	8.32826			0.275928			0.137964	−	0.990437i	$-0.455944\pi$
$911$	8.32826			0.275928			0.137964	+	0.990437i	$0.455944\pi$
$912$		0			0
$913$			62.5569i			2.07033i
$914$		0			0
$915$	1.11528	+	0.304892i	0.0368700	+	0.0100794i
$916$		0			0
$917$	−34.8723			−1.15158
$918$		0			0
$919$	−45.1893			−1.49066			−0.745329	−	0.666697i	$-0.767708\pi$
$919$	−45.1893			−1.49066			−0.745329	+	0.666697i	$0.767708\pi$
$920$		0			0
$921$			81.9760i			2.70120i
$922$		0			0
$923$	38.5979	−	26.5051i	1.27046	−	0.872427i
$924$		0			0
$925$	15.7329	+	9.29681i	0.517293	+	0.305677i
$926$		0			0
$927$			11.8894i			0.390501i
$928$		0			0
$929$			18.8393i			0.618098i	0.951046	+	0.309049i	$0.100011\pi$
$929$			18.8393i			0.618098i	−0.951046	+	0.309049i	$0.899989\pi$
$930$		0			0
$931$			2.01332i			0.0659837i
$932$		0			0
$933$		−	40.0905i		−	1.31250i
$934$		0			0
$935$	13.9991	−	51.2079i	0.457819	−	1.67468i
$936$		0			0
$937$		−	15.8303i		−	0.517154i	−0.965991	−	0.258577i	$-0.916746\pi$
$937$		−	15.8303i		−	0.517154i	0.965991	−	0.258577i	$-0.0832536\pi$
$938$		0			0
$939$	−77.3298			−2.52356
$940$		0			0
$941$		−	8.41058i		−	0.274177i	−0.990559	−	0.137088i	$-0.956226\pi$
$941$		−	8.41058i		−	0.274177i	0.990559	−	0.137088i	$-0.0437745\pi$
$942$		0			0
$943$	−11.3367			−0.369175
$944$		0			0
$945$	−50.8007	−	13.8878i	−1.65255	−	0.451769i
$946$		0			0
$947$	47.9898			1.55946			0.779730	−	0.626116i	$-0.215356\pi$
$947$	47.9898			1.55946			0.779730	+	0.626116i	$0.215356\pi$
$948$		0			0
$949$	−20.9494	−	30.5074i	−0.680046	−	0.990311i
$950$		0			0
$951$			65.6992i			2.13044i
$952$		0			0
$953$		−	38.6553i		−	1.25217i	−0.779755	−	0.626084i	$-0.784657\pi$
$953$		−	38.6553i		−	1.25217i	0.779755	−	0.626084i	$-0.215343\pi$
$954$		0			0
$955$	6.68474	−	24.4525i	0.216313	−	0.791263i
$956$		0			0
$957$	79.0205			2.55437
$958$		0			0
$959$	9.24467			0.298526
$960$		0			0
$961$	1.05706			0.0340986
$962$		0			0
$963$			9.95785i			0.320887i
$964$		0			0
$965$	9.69753	−	35.4731i	0.312175	−	1.14192i
$966$		0			0
$967$	32.5757			1.04756			0.523782	−	0.851853i	$-0.324521\pi$
$967$	32.5757			1.04756			0.523782	+	0.851853i	$0.324521\pi$
$968$		0			0
$969$		−	8.13985i		−	0.261490i
$970$		0			0
$971$	7.75150			0.248758			0.124379	−	0.992235i	$-0.460306\pi$
$971$	7.75150			0.248758			0.124379	+	0.992235i	$0.460306\pi$
$972$		0			0
$973$	12.9973			0.416674
$974$		0			0
$975$	−58.0336	+	3.95412i	−1.85856	+	0.126633i
$976$		0			0
$977$	−0.0696366			−0.00222787			−0.00111394	−	0.999999i	$-0.500355\pi$
$977$	−0.0696366			−0.00222787			−0.00111394	+	0.999999i	$0.500355\pi$
$978$		0			0
$979$	−33.8371			−1.08144
$980$		0			0
$981$			39.5093i			1.26143i
$982$		0			0
$983$	−51.0126			−1.62705			−0.813525	−	0.581530i	$-0.802454\pi$
$983$	−51.0126			−1.62705			−0.813525	+	0.581530i	$0.802454\pi$
$984$		0			0
$985$	−9.95675	+	36.4213i	−0.317248	+	1.16048i
$986$		0			0
$987$			8.83643i			0.281267i
$988$		0			0
$989$	8.85599			0.281604
$990$		0			0
$991$	−42.6294			−1.35417			−0.677084	−	0.735906i	$-0.736757\pi$
$991$	−42.6294			−1.35417			−0.677084	+	0.735906i	$0.736757\pi$
$992$		0			0
$993$	97.0667			3.08032
$994$		0			0
$995$	−11.2176	+	41.0333i	−0.355620	+	1.30084i
$996$		0			0
$997$		−	56.5323i		−	1.79040i	−0.445670	−	0.895198i	$-0.647034\pi$
$997$		−	56.5323i		−	1.79040i	0.445670	−	0.895198i	$-0.352966\pi$
$998$		0			0
$999$		−	52.0166i		−	1.64573i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.2.f.g.129.1		10
4.3	odd	2		520.2.f.b.129.10	yes	10
5.4	even	2		1040.2.f.f.129.10		10
13.12	even	2		1040.2.f.f.129.1		10
20.3	even	4		2600.2.k.f.2001.2		20
20.7	even	4		2600.2.k.f.2001.19		20
20.19	odd	2		520.2.f.a.129.1	✓	10
52.51	odd	2		520.2.f.a.129.10	yes	10
65.64	even	2	inner	1040.2.f.g.129.10		10
260.103	even	4		2600.2.k.f.2001.1		20
260.207	even	4		2600.2.k.f.2001.20		20
260.259	odd	2		520.2.f.b.129.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.1	✓	10	20.19	odd	2
520.2.f.a.129.10	yes	10	52.51	odd	2
520.2.f.b.129.1	yes	10	260.259	odd	2
520.2.f.b.129.10	yes	10	4.3	odd	2
1040.2.f.f.129.1		10	13.12	even	2
1040.2.f.f.129.10		10	5.4	even	2
1040.2.f.g.129.1		10	1.1	even	1	trivial
1040.2.f.g.129.10		10	65.64	even	2	inner
2600.2.k.f.2001.1		20	260.103	even	4
2600.2.k.f.2001.2		20	20.3	even	4
2600.2.k.f.2001.19		20	20.7	even	4
2600.2.k.f.2001.20		20	260.207	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.22659i q^{3} +(-0.589653 + 2.15692i) q^{5} +1.65488 q^{7} -7.41088 q^{9} +4.44624i q^{11} +(2.04103 + 2.97224i) q^{13} +(6.95950 + 1.90257i) q^{15} +5.33962i q^{17} -0.472457i q^{19} -5.33962i q^{21} +1.08725i q^{23} +(-4.30462 - 2.54367i) q^{25} +14.2321i q^{27} +5.50812 q^{29} +5.47201i q^{31} +14.3462 q^{33} +(-0.975805 + 3.56945i) q^{35} -3.65488 q^{37} +(9.59018 - 6.58557i) q^{39} +10.4270i q^{41} -8.14528i q^{43} +(4.36985 - 15.9847i) q^{45} -1.65488 q^{47} -4.26137 q^{49} +17.2288 q^{51} -4.14520i q^{53} +(-9.59018 - 2.62174i) q^{55} -1.52443 q^{57} -2.52401i q^{59} +0.160253 q^{61} -12.2641 q^{63} +(-7.61438 + 2.64976i) q^{65} +8.26412 q^{67} +3.50812 q^{69} -12.9861i q^{71} -10.2641 q^{73} +(-8.20738 + 13.8892i) q^{75} +7.35799i q^{77} -8.65763 q^{79} +23.6885 q^{81} +14.0696 q^{83} +(-11.5171 - 3.14852i) q^{85} -17.7724i q^{87} +7.61027i q^{89} +(3.37766 + 4.91869i) q^{91} +17.6559 q^{93} +(1.01905 + 0.278586i) q^{95} +16.7397 q^{97} -32.9505i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{5} + 2 q^{7} - 8 q^{9} + 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} + 8 q^{33} + 3 q^{35} - 22 q^{37} + 12 q^{39} - 4 q^{45} - 2 q^{47} + 12 q^{49} + 30 q^{51} - 12 q^{55} - 12 q^{57} - 16 q^{61}+ \cdots + 72 q^{97}+O(q^{100}) \) 10 * q + 3 * q^5 + 2 * q^7 - 8 * q^9 + 2 * q^13 + 13 * q^15 - q^25 + 8 * q^29 + 8 * q^33 + 3 * q^35 - 22 * q^37 + 12 * q^39 - 4 * q^45 - 2 * q^47 + 12 * q^49 + 30 * q^51 - 12 * q^55 - 12 * q^57 - 16 * q^61 - 24 * q^63 - 5 * q^65 - 16 * q^67 - 12 * q^69 - 4 * q^73 - 21 * q^75 - 28 * q^79 + 22 * q^81 + 4 * q^83 - 25 * q^85 - 2 * q^91 + 12 * q^93 + 10 * q^95 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more