LMFDB - Embedded newform 520.2.f.a.129.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(129,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, 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("520.129");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.15222090511$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.10
Root			$1.35685i$ of defining polynomial
Character	$\chi$	$=$	520.129
Dual form			520.2.f.a.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22659i q^{3} +(0.589653 - 2.15692i) q^{5} +1.65488 q^{7} -7.41088 q^{9} +4.44624i q^{11} +(-2.04103 + 2.97224i) q^{13} +(6.95950 + 1.90257i) q^{15} +5.33962i q^{17} -0.472457i q^{19} +5.33962i q^{21} -1.08725i q^{23} +(-4.30462 - 2.54367i) q^{25} -14.2321i q^{27} +5.50812 q^{29} +5.47201i q^{31} -14.3462 q^{33} +(0.975805 - 3.56945i) q^{35} +3.65488 q^{37} +(-9.59018 - 6.58557i) q^{39} -10.4270i q^{41} +8.14528i q^{43} +(-4.36985 + 15.9847i) q^{45} -1.65488 q^{47} -4.26137 q^{49} -17.2288 q^{51} -4.14520i q^{53} +(9.59018 + 2.62174i) q^{55} +1.52443 q^{57} -2.52401i q^{59} +0.160253 q^{61} -12.2641 q^{63} +(5.20738 + 6.15493i) q^{65} +8.26412 q^{67} +3.50812 q^{69} -12.9861i q^{71} +10.2641 q^{73} +(8.20738 - 13.8892i) q^{75} +7.35799i q^{77} +8.65763 q^{79} +23.6885 q^{81} +14.0696 q^{83} +(11.5171 + 3.14852i) q^{85} +17.7724i q^{87} -7.61027i q^{89} +(-3.37766 + 4.91869i) q^{91} -17.6559 q^{93} +(-1.01905 - 0.278586i) q^{95} -16.7397 q^{97} -32.9505i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 - 9 * 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 $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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	$0.381443\pi$
$3$			3.22659i			1.86287i	−0.363905	+	0.931436i	$0.618557\pi$
$4$		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	$-0.963841\pi$
$23$		−	1.08725i		−	0.226708i	0.993555	−	0.113354i	$-0.0361594\pi$
$24$		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	$-0.402870\pi$
$37$	3.65488			0.600859			0.300429	+	0.953804i	$0.402870\pi$
$38$		0			0
$39$	−9.59018	−	6.58557i	−1.53566	−	1.05454i
$40$		0			0
$41$		−	10.4270i		−	1.62842i	−0.580572	−	0.814209i	$-0.697171\pi$
$41$		−	10.4270i		−	1.62842i	0.580572	−	0.814209i	$-0.302829\pi$
$42$		0			0
$43$			8.14528i			1.24214i	0.783753	+	0.621072i	$0.213303\pi$
$43$			8.14528i			1.24214i	−0.783753	+	0.621072i	$0.786697\pi$
$44$		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
$65$	5.20738	+	6.15493i	0.645896	+	0.763426i
$66$		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	$-0.294904\pi$
$73$	10.2641			1.20132			0.600662	+	0.799503i	$0.294904\pi$
$74$		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	$-0.338080\pi$
$79$	8.65763			0.974059			0.487030	+	0.873385i	$0.338080\pi$
$80$		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	$-0.867848\pi$
$89$		−	7.61027i		−	0.806687i	0.915049	−	0.403344i	$-0.132152\pi$
$90$		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	$-0.823295\pi$
$97$	−16.7397			−1.69966			−0.849829	+	0.527058i	$0.823295\pi$
$98$		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	$0.0251853\pi$
$103$			1.60432i			0.158079i	−0.996871	+	0.0790393i	$0.974815\pi$
$104$		0			0
$105$	11.5171	+	3.14852i	1.12396	+	0.307264i
$106$		0			0
$107$			1.34368i			0.129898i	0.997889	+	0.0649492i	$0.0206885\pi$
$107$			1.34368i			0.129898i	−0.997889	+	0.0649492i	$0.979311\pi$
$108$		0			0
$109$			5.33126i			0.510642i	0.966856	+	0.255321i	$0.0821811\pi$
$109$			5.33126i			0.510642i	−0.966856	+	0.255321i	$0.917819\pi$
$110$		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	$0.0446515\pi$
$127$			3.15132i			0.279634i	−0.990177	+	0.139817i	$0.955349\pi$
$128$		0			0
$129$	−26.2815			−2.31396
$130$		0			0
$131$	21.0724			1.84110			0.920551	−	0.390622i	$-0.127740\pi$
$131$	21.0724			1.84110			0.920551	+	0.390622i	$0.127740\pi$
$132$		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	$-0.576700\pi$
$137$	−5.58631			−0.477270			−0.238635	+	0.971109i	$0.576700\pi$
$138$		0			0
$139$	−7.85391			−0.666160			−0.333080	−	0.942899i	$-0.608088\pi$
$139$	−7.85391			−0.666160			−0.333080	+	0.942899i	$0.608088\pi$
$140$		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	$-0.777114\pi$
$149$		−	15.7314i		−	1.28877i	0.764702	−	0.644384i	$-0.222886\pi$
$150$		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	$-0.477474\pi$
$179$	1.89201			0.141416			0.0707079	+	0.997497i	$0.477474\pi$
$180$		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	$-0.365477\pi$
$191$	11.3367			0.820298			0.410149	+	0.912019i	$0.365477\pi$
$192$		0			0
$193$	16.4462			1.18382			0.591911	−	0.806004i	$-0.298374\pi$
$193$	16.4462			1.18382			0.591911	+	0.806004i	$0.298374\pi$
$194$		0			0
$195$	−19.8594	+	16.8021i	−1.42216	+	1.20322i
$196$		0			0
$197$	−16.8858			−1.20306			−0.601531	−	0.798849i	$-0.705442\pi$
$197$	−16.8858			−1.20306			−0.601531	+	0.798849i	$0.705442\pi$
$198$		0			0
$199$	−19.0240			−1.34858			−0.674288	−	0.738469i	$-0.735549\pi$
$199$	−19.0240			−1.34858			−0.674288	+	0.738469i	$0.735549\pi$
$200$		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	$-0.520745\pi$
$211$	−1.89201			−0.130252			−0.0651258	+	0.997877i	$0.520745\pi$
$212$		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	$-0.830681\pi$
$229$		−	15.3506i		−	1.01440i	0.861830	−	0.507198i	$-0.169319\pi$
$230$		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	$0.00801653\pi$
$241$			0.781860i			0.0503640i	−0.999683	+	0.0251820i	$0.991983\pi$
$242$		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	$-0.412917\pi$
$251$	8.56085			0.540356			0.270178	+	0.962810i	$0.412917\pi$
$252$		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	$-0.973724\pi$
$263$		−	2.67435i		−	0.164907i	0.996595	−	0.0824537i	$-0.0262757\pi$
$264$		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	$0.00726511\pi$
$281$			0.765134i			0.0456441i	−0.999740	+	0.0228220i	$0.992735\pi$
$282$		0			0
$283$			9.38817i			0.558069i	0.960281	+	0.279034i	$0.0900144\pi$
$283$			9.38817i			0.558069i	−0.960281	+	0.279034i	$0.909986\pi$
$284$		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	$-0.628899\pi$
$293$	−13.4874			−0.787941			−0.393971	+	0.919123i	$0.628899\pi$
$294$		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	$-0.614593\pi$
$311$	−12.4250			−0.704559			−0.352280	+	0.935895i	$0.614593\pi$
$312$		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	$-0.306239\pi$
$317$	20.3618			1.14363			0.571817	+	0.820381i	$0.306239\pi$
$318$		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
$325$	16.3463	−	7.60263i	0.906727	−	0.421718i
$326$		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	$-0.896301\pi$
$347$		−	11.9236i		−	0.640093i	0.947402	−	0.320047i	$-0.103699\pi$
$348$		0			0
$349$		−	3.38144i		−	0.181004i	−0.995896	−	0.0905021i	$-0.971153\pi$
$349$		−	3.38144i		−	0.181004i	0.995896	−	0.0905021i	$-0.0288472\pi$
$350$		0			0
$351$	42.3011	+	29.0482i	2.25787	+	1.55048i
$352$		0			0
$353$	−13.4771			−0.717314			−0.358657	−	0.933469i	$-0.616765\pi$
$353$	−13.4771			−0.717314			−0.358657	+	0.933469i	$0.616765\pi$
$354$		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	$0.306185\pi$
$367$			31.4288i			1.64057i	−0.571956	+	0.820284i	$0.693815\pi$
$368$		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	$-0.487641\pi$
$397$	1.54689			0.0776364			0.0388182	+	0.999246i	$0.487641\pi$
$398$		0			0
$399$	2.52274			0.126295
$400$		0			0
$401$		−	5.55872i		−	0.277589i	−0.990321	−	0.138795i	$-0.955677\pi$
$401$		−	5.55872i		−	0.277589i	0.990321	−	0.138795i	$-0.0443228\pi$
$402$		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	$-0.962877\pi$
$409$		−	4.70652i		−	0.232722i	0.993207	−	0.116361i	$-0.0371230\pi$
$410$		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	$-0.640261\pi$
$419$	−17.4613			−0.853043			−0.426521	+	0.904478i	$0.640261\pi$
$420$		0			0
$421$			23.1162i			1.12662i	0.826247	+	0.563308i	$0.190471\pi$
$421$			23.1162i			1.12662i	−0.826247	+	0.563308i	$0.809529\pi$
$422$		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	$-0.614538\pi$
$439$	−14.7553			−0.704233			−0.352117	+	0.935956i	$0.614538\pi$
$440$		0			0
$441$	31.5805			1.50383
$442$		0			0
$443$		−	12.8191i		−	0.609053i	−0.952504	−	0.304526i	$-0.901502\pi$
$443$		−	12.8191i		−	0.609053i	0.952504	−	0.304526i	$-0.0984982\pi$
$444$		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	$-0.791727\pi$
$449$		−	25.7925i		−	1.21722i	0.793468	−	0.608612i	$-0.208273\pi$
$450$		0			0
$451$	46.3607			2.18304
$452$		0			0
$453$	−58.7553			−2.76057
$454$		0			0
$455$	8.61759	+	10.1857i	0.403999	+	0.477512i
$456$		0			0
$457$	−9.61892			−0.449954			−0.224977	−	0.974364i	$-0.572231\pi$
$457$	−9.61892			−0.449954			−0.224977	+	0.974364i	$0.572231\pi$
$458$		0			0
$459$	75.9940			3.54709
$460$		0			0
$461$			18.4899i			0.861160i	0.902552	+	0.430580i	$0.141691\pi$
$461$			18.4899i			0.861160i	−0.902552	+	0.430580i	$0.858309\pi$
$462$		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	$0.0601721\pi$
$467$			8.12164i			0.375825i	−0.982186	+	0.187912i	$0.939828\pi$
$468$		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	$-0.754240\pi$
$491$	−31.7515			−1.43293			−0.716463	+	0.697626i	$0.754240\pi$
$492$		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	$0.201345\pi$
$503$			26.5184i			1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$504$		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	$0.338060\pi$
$509$			39.4076i			1.74671i	−0.487085	+	0.873355i	$0.661940\pi$
$510$		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	$-0.961354\pi$
$523$		−	5.53950i		−	0.242225i	0.992639	−	0.121113i	$-0.0386463\pi$
$524$		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	$0.0741208\pi$
$541$			10.7346i			0.461518i	−0.973011	+	0.230759i	$0.925879\pi$
$542$		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	$0.246101\pi$
$547$			32.6681i			1.39679i	−0.715714	+	0.698393i	$0.753899\pi$
$548$		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	$-0.649792\pi$
$557$	−21.4016			−0.906815			−0.453407	+	0.891303i	$0.649792\pi$
$558$		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	$-0.968552\pi$
$563$		−	4.68087i		−	0.197275i	0.995123	−	0.0986376i	$-0.0314485\pi$
$564$		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	$-0.495551\pi$
$571$	0.667911			0.0279512			0.0139756	+	0.999902i	$0.495551\pi$
$572$		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	$-0.671846\pi$
$577$	−24.6947			−1.02805			−0.514026	+	0.857775i	$0.671846\pi$
$578$		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
$585$	−38.5912	−	45.6135i	−1.59555	−	1.88588i
$586$		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	$-0.456248\pi$
$593$	6.67326			0.274038			0.137019	+	0.990568i	$0.456248\pi$
$594$		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	$-0.201670\pi$
$599$	39.4490			1.61184			0.805922	+	0.592022i	$0.201670\pi$
$600$		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	$0.0284688\pi$
$607$			4.40112i			0.178636i	−0.996003	+	0.0893181i	$0.971531\pi$
$608$		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	$-0.407035\pi$
$613$	14.2573			0.575848			0.287924	+	0.957653i	$0.407035\pi$
$614$		0			0
$615$	19.8380	−	72.5664i	0.799945	−	2.92616i
$616$		0			0
$617$	9.58843			0.386016			0.193008	−	0.981197i	$-0.438176\pi$
$617$	9.58843			0.386016			0.193008	+	0.981197i	$0.438176\pi$
$618$		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	$-0.842705\pi$
$647$		−	24.1282i		−	0.948577i	0.880369	−	0.474288i	$-0.157295\pi$
$648$		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	$-0.626316\pi$
$659$	−19.8436			−0.772998			−0.386499	+	0.922290i	$0.626316\pi$
$660$		0			0
$661$			40.1796i			1.56280i	0.624028	+	0.781402i	$0.285495\pi$
$661$			40.1796i			1.56280i	−0.624028	+	0.781402i	$0.714505\pi$
$662$		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	$-0.939894\pi$
$709$		−	9.99630i		−	0.375419i	0.982225	−	0.187709i	$-0.0601063\pi$
$710$		0			0
$711$	−64.1606			−2.40621
$712$		0			0
$713$	5.94947			0.222809
$714$		0			0
$715$	−27.3663	+	23.1532i	−1.02344	+	0.865882i
$716$		0			0
$717$	−22.2327			−0.830296
$718$		0			0
$719$	−52.7587			−1.96757			−0.983783	−	0.179363i	$-0.942596\pi$
$719$	−52.7587			−1.96757			−0.983783	+	0.179363i	$0.942596\pi$
$720$		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	$-0.915024\pi$
$727$		−	14.2257i		−	0.527601i	0.964577	−	0.263801i	$-0.0849761\pi$
$728$		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	$-0.570264\pi$
$733$	−11.8558			−0.437904			−0.218952	+	0.975736i	$0.570264\pi$
$734$		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	$-0.434222\pi$
$751$	11.2456			0.410358			0.205179	+	0.978725i	$0.434222\pi$
$752$		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	$0.0449891\pi$
$761$			7.77200i			0.281735i	−0.990028	+	0.140867i	$0.955011\pi$
$762$		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	$-0.827981\pi$
$769$		−	28.5345i		−	1.02898i	0.857496	−	0.514491i	$-0.172019\pi$
$770$		0			0
$771$	0.900499			0.0324307
$772$		0			0
$773$	31.8056			1.14397			0.571984	−	0.820264i	$-0.306174\pi$
$773$	31.8056			1.14397			0.571984	+	0.820264i	$0.306174\pi$
$774$		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	$-0.856856\pi$
$821$		−	24.9107i		−	0.869389i	0.900578	−	0.434694i	$-0.143144\pi$
$822$		0			0
$823$			3.47743i			0.121215i	0.998162	+	0.0606077i	$0.0193039\pi$
$823$			3.47743i			0.121215i	−0.998162	+	0.0606077i	$0.980696\pi$
$824$		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
$845$	−28.9223	+	2.91514i	−0.994959	+	0.100284i
$846$		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	$-0.509932\pi$
$853$	−1.82238			−0.0623970			−0.0311985	+	0.999513i	$0.509932\pi$
$854$		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	$-0.602981\pi$
$859$	−18.6350			−0.635819			−0.317910	+	0.948121i	$0.602981\pi$
$860$		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	$-0.389570\pi$
$877$	20.1381			0.680015			0.340007	+	0.940423i	$0.389570\pi$
$878$		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	$-0.920559\pi$
$883$		−	14.6787i		−	0.493976i	0.969018	−	0.246988i	$-0.0794409\pi$
$884$		0			0
$885$	4.80210	−	17.5659i	0.161421	−	0.590470i
$886$		0			0
$887$		−	12.6524i		−	0.424827i	−0.977180	−	0.212413i	$-0.931868\pi$
$887$		−	12.6524i		−	0.424827i	0.977180	−	0.212413i	$-0.0681323\pi$
$888$		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	$0.226060\pi$
$907$			39.2705i			1.30396i	−0.758238	+	0.651978i	$0.773940\pi$
$908$		0			0
$909$	−116.891			−3.87703
$910$		0			0
$911$	−8.32826			−0.275928			−0.137964	−	0.990437i	$-0.544056\pi$
$911$	−8.32826			−0.275928			−0.137964	+	0.990437i	$0.544056\pi$
$912$		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	$-0.232292\pi$
$919$	45.1893			1.49066			0.745329	+	0.666697i	$0.232292\pi$
$920$		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	$-0.899989\pi$
$929$		−	18.8393i		−	0.618098i	0.951046	−	0.309049i	$-0.100011\pi$
$930$		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	$0.0437745\pi$
$941$			8.41058i			0.274177i	−0.990559	+	0.137088i	$0.956226\pi$
$942$		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	$-0.539694\pi$
$971$	−7.75150			−0.248758			−0.124379	+	0.992235i	$0.539694\pi$
$972$		0			0
$973$	−12.9973			−0.416674
$974$		0			0
$975$	24.5306	+	52.7426i	0.785607	+	1.68912i
$976$		0			0
$977$	0.0696366			0.00222787			0.00111394	−	0.999999i	$-0.499645\pi$
$977$	0.0696366			0.00222787			0.00111394	+	0.999999i	$0.499645\pi$
$978$		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	$-0.263243\pi$
$991$	42.6294			1.35417			0.677084	+	0.735906i	$0.263243\pi$
$992$		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	520.2.f.a.129.10	yes	10
4.3	odd	2		1040.2.f.f.129.1		10
5.2	odd	4		2600.2.k.f.2001.20		20
5.3	odd	4		2600.2.k.f.2001.1		20
5.4	even	2		520.2.f.b.129.1	yes	10
13.12	even	2		520.2.f.b.129.10	yes	10
20.19	odd	2		1040.2.f.g.129.10		10
52.51	odd	2		1040.2.f.g.129.1		10
65.12	odd	4		2600.2.k.f.2001.19		20
65.38	odd	4		2600.2.k.f.2001.2		20
65.64	even	2	inner	520.2.f.a.129.1	✓	10
260.259	odd	2		1040.2.f.f.129.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.f.a.129.1	✓	10	65.64	even	2	inner
520.2.f.a.129.10	yes	10	1.1	even	1	trivial
520.2.f.b.129.1	yes	10	5.4	even	2
520.2.f.b.129.10	yes	10	13.12	even	2
1040.2.f.f.129.1		10	4.3	odd	2
1040.2.f.f.129.10		10	260.259	odd	2
1040.2.f.g.129.1		10	52.51	odd	2
1040.2.f.g.129.10		10	20.19	odd	2
2600.2.k.f.2001.1		20	5.3	odd	4
2600.2.k.f.2001.2		20	65.38	odd	4
2600.2.k.f.2001.19		20	65.12	odd	4
2600.2.k.f.2001.20		20	5.2	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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.22659i q^{3} +(0.589653 - 2.15692i) q^{5} +1.65488 q^{7} -7.41088 q^{9} +4.44624i q^{11} +(-2.04103 + 2.97224i) q^{13} +(6.95950 + 1.90257i) q^{15} +5.33962i q^{17} -0.472457i q^{19} +5.33962i q^{21} -1.08725i q^{23} +(-4.30462 - 2.54367i) q^{25} -14.2321i q^{27} +5.50812 q^{29} +5.47201i q^{31} -14.3462 q^{33} +(0.975805 - 3.56945i) q^{35} +3.65488 q^{37} +(-9.59018 - 6.58557i) q^{39} -10.4270i q^{41} +8.14528i q^{43} +(-4.36985 + 15.9847i) q^{45} -1.65488 q^{47} -4.26137 q^{49} -17.2288 q^{51} -4.14520i q^{53} +(9.59018 + 2.62174i) q^{55} +1.52443 q^{57} -2.52401i q^{59} +0.160253 q^{61} -12.2641 q^{63} +(5.20738 + 6.15493i) q^{65} +8.26412 q^{67} +3.50812 q^{69} -12.9861i q^{71} +10.2641 q^{73} +(8.20738 - 13.8892i) q^{75} +7.35799i q^{77} +8.65763 q^{79} +23.6885 q^{81} +14.0696 q^{83} +(11.5171 + 3.14852i) q^{85} +17.7724i q^{87} -7.61027i q^{89} +(-3.37766 + 4.91869i) q^{91} -17.6559 q^{93} +(-1.01905 - 0.278586i) q^{95} -16.7397 q^{97} -32.9505i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{5} + 2 q^{7} - 8 q^{9} - 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} - 8 q^{33} - 3 q^{35} + 22 q^{37} - 12 q^{39} + 4 q^{45} - 2 q^{47} + 12 q^{49} - 30 q^{51} + 12 q^{55} + 12 q^{57} - 16 q^{61}+ \cdots - 72 q^{97}+O(q^{100}) \) 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 - 9 * 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

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