LMFDB - Embedded newform 1525.2.d.c.1524.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1525,2,Mod(1524,1525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1525.1524"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1525 = 5^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$12.1771863082$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.897122304.10
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 61)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1524.7
Root			$-2.07341 + 1.19709i$ of defining polynomial
Character	$\chi$	$=$	1525.1524
Dual form			1525.2.d.c.1524.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.39417 q^{2} -2.73205i q^{3} +3.73205 q^{4} -6.54099i q^{6} +4.14682 q^{7} +4.14682 q^{8} -4.46410 q^{9} -2.39417i q^{11} -10.1962i q^{12} +3.00000i q^{13} +9.92820 q^{14} +2.46410 q^{16} +1.75265 q^{17} -10.6878 q^{18} -4.73205 q^{19} -11.3293i q^{21} -5.73205i q^{22} +5.89948 q^{23} -11.3293i q^{24} +7.18251i q^{26} +4.00000i q^{27} +15.4762 q^{28} +1.75265i q^{29} +3.03569i q^{31} -2.39417 q^{32} -6.54099 q^{33} +4.19615 q^{34} -16.6603 q^{36} -3.03569 q^{37} -11.3293 q^{38} +8.19615 q^{39} +0.464102 q^{41} -27.1244i q^{42} -11.3293 q^{43} -8.93516i q^{44} +14.1244 q^{46} +0.928203i q^{47} -6.73205i q^{48} +10.1962 q^{49} -4.78834i q^{51} +11.1962i q^{52} +1.28303 q^{53} +9.57668i q^{54} +17.1962 q^{56} +12.9282i q^{57} +4.19615i q^{58} +8.93516i q^{59} +(7.19615 + 3.03569i) q^{61} +7.26795i q^{62} -18.5118 q^{63} -10.6603 q^{64} -15.6603 q^{66} -10.2182 q^{67} +6.54099 q^{68} -16.1177i q^{69} -6.54099i q^{71} -18.5118 q^{72} -1.19615i q^{73} -7.26795 q^{74} -17.6603 q^{76} -9.92820i q^{77} +19.6230 q^{78} +9.40479i q^{79} -2.46410 q^{81} +1.11114 q^{82} +9.46410i q^{83} -42.2817i q^{84} -27.1244 q^{86} +4.78834 q^{87} -9.92820i q^{88} -11.7990i q^{89} +12.4405i q^{91} +22.0172 q^{92} +8.29365 q^{93} +2.22228i q^{94} +6.54099i q^{96} +3.46410i q^{97} +24.4113 q^{98} +10.6878i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 8 q^{9} + 24 q^{14} - 8 q^{16} - 24 q^{19} - 8 q^{34} - 64 q^{36} + 24 q^{39} - 24 q^{41} + 16 q^{46} + 40 q^{49} + 96 q^{56} + 16 q^{61} - 16 q^{64} - 56 q^{66} - 72 q^{74} - 72 q^{76}+ \cdots - 120 q^{86}+O(q^{100}) $ 8 * q + 16 * q^4 - 8 * q^9 + 24 * q^14 - 8 * q^16 - 24 * q^19 - 8 * q^34 - 64 * q^36 + 24 * q^39 - 24 * q^41 + 16 * q^46 + 40 * q^49 + 96 * q^56 + 16 * q^61 - 16 * q^64 - 56 * q^66 - 72 * q^74 - 72 * q^76 + 8 * q^81 - 120 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1525\mathbb{Z}\right)^\times$.

$n$	$551$	$977$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.39417			1.69293			0.846467	−	0.532441i	$-0.178725\pi$
$2$	2.39417			1.69293			0.846467	+	0.532441i	$0.178725\pi$
$3$		−	2.73205i		−	1.57735i	−0.614810	−	0.788675i	$-0.710767\pi$
$3$		−	2.73205i		−	1.57735i	0.614810	−	0.788675i	$-0.289233\pi$
$4$	3.73205			1.86603
$5$		0			0
$5$		0			0
$6$		−	6.54099i		−	2.67035i
$7$	4.14682			1.56735			0.783676	−	0.621170i	$-0.213342\pi$
$7$	4.14682			1.56735			0.783676	+	0.621170i	$0.213342\pi$
$8$	4.14682			1.46612
$9$	−4.46410			−1.48803
$10$		0			0
$11$		−	2.39417i		−	0.721869i	−0.932591	−	0.360935i	$-0.882458\pi$
$11$		−	2.39417i		−	0.721869i	0.932591	−	0.360935i	$-0.117542\pi$
$12$		−	10.1962i		−	2.94338i
$13$			3.00000i			0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$			3.00000i			0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	9.92820			2.65342
$15$		0			0
$16$	2.46410			0.616025
$17$	1.75265			0.425081			0.212541	−	0.977152i	$-0.431826\pi$
$17$	1.75265			0.425081			0.212541	+	0.977152i	$0.431826\pi$
$18$	−10.6878			−2.51914
$19$	−4.73205			−1.08561			−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205			−1.08561			−0.542803	+	0.839860i	$0.682637\pi$
$20$		0			0
$21$		−	11.3293i		−	2.47226i
$22$		−	5.73205i		−	1.22208i
$23$	5.89948			1.23013			0.615063	−	0.788478i	$-0.289131\pi$
$23$	5.89948			1.23013			0.615063	+	0.788478i	$0.289131\pi$
$24$		−	11.3293i		−	2.31259i
$25$		0			0
$26$			7.18251i			1.40861i
$27$			4.00000i			0.769800i
$28$	15.4762			2.92472
$29$			1.75265i			0.325460i	0.986671	+	0.162730i	$0.0520299\pi$
$29$			1.75265i			0.325460i	−0.986671	+	0.162730i	$0.947970\pi$
$30$		0			0
$31$			3.03569i			0.545225i	0.962124	+	0.272613i	$0.0878877\pi$
$31$			3.03569i			0.545225i	−0.962124	+	0.272613i	$0.912112\pi$
$32$	−2.39417			−0.423233
$33$	−6.54099			−1.13864
$34$	4.19615			0.719634
$35$		0			0
$36$	−16.6603			−2.77671
$37$	−3.03569			−0.499064			−0.249532	−	0.968367i	$-0.580277\pi$
$37$	−3.03569			−0.499064			−0.249532	+	0.968367i	$0.580277\pi$
$38$	−11.3293			−1.83786
$39$	8.19615			1.31243
$40$		0			0
$41$	0.464102			0.0724805			0.0362402	−	0.999343i	$-0.488462\pi$
$41$	0.464102			0.0724805			0.0362402	+	0.999343i	$0.488462\pi$
$42$		−	27.1244i		−	4.18538i
$43$	−11.3293			−1.72771			−0.863854	−	0.503743i	$-0.831956\pi$
$43$	−11.3293			−1.72771			−0.863854	+	0.503743i	$0.831956\pi$
$44$		−	8.93516i		−	1.34703i
$45$		0			0
$46$	14.1244			2.08252
$47$			0.928203i			0.135392i	0.997706	+	0.0676962i	$0.0215649\pi$
$47$			0.928203i			0.135392i	−0.997706	+	0.0676962i	$0.978435\pi$
$48$		−	6.73205i		−	0.971688i
$49$	10.1962			1.45659
$50$		0			0
$51$		−	4.78834i		−	0.670502i
$52$			11.1962i			1.55263i
$53$	1.28303			0.176238			0.0881190	−	0.996110i	$-0.471914\pi$
$53$	1.28303			0.176238			0.0881190	+	0.996110i	$0.471914\pi$
$54$			9.57668i			1.30322i
$55$		0			0
$56$	17.1962			2.29793
$57$			12.9282i			1.71238i
$58$			4.19615i			0.550982i
$59$			8.93516i			1.16326i	0.813454	+	0.581630i	$0.197585\pi$
$59$			8.93516i			1.16326i	−0.813454	+	0.581630i	$0.802415\pi$
$60$		0			0
$61$	7.19615	+	3.03569i	0.921373	+	0.388680i
$61$	7.19615	+	3.03569i	0.921373	+	0.388680i
$62$			7.26795i			0.923030i
$63$	−18.5118			−2.33227
$64$	−10.6603			−1.33253
$65$		0			0
$66$	−15.6603			−1.92764
$67$	−10.2182			−1.24835			−0.624176	−	0.781284i	$-0.714565\pi$
$67$	−10.2182			−1.24835			−0.624176	+	0.781284i	$0.714565\pi$
$68$	6.54099			0.793212
$69$		−	16.1177i		−	1.94034i
$70$		0			0
$71$		−	6.54099i		−	0.776273i	−0.921602	−	0.388137i	$-0.873119\pi$
$71$		−	6.54099i		−	0.776273i	0.921602	−	0.388137i	$-0.126881\pi$
$72$	−18.5118			−2.18164
$73$		−	1.19615i		−	0.139999i	−0.997547	−	0.0699995i	$-0.977700\pi$
$73$		−	1.19615i		−	0.139999i	0.997547	−	0.0699995i	$-0.0222998\pi$
$74$	−7.26795			−0.844882
$75$		0			0
$76$	−17.6603			−2.02577
$77$		−	9.92820i		−	1.13142i
$78$	19.6230			2.22187
$79$			9.40479i			1.05812i	0.848584	+	0.529061i	$0.177456\pi$
$79$			9.40479i			1.05812i	−0.848584	+	0.529061i	$0.822544\pi$
$80$		0			0
$81$	−2.46410			−0.273789
$82$	1.11114			0.122705
$83$			9.46410i			1.03882i	0.854525	+	0.519410i	$0.173848\pi$
$83$			9.46410i			1.03882i	−0.854525	+	0.519410i	$0.826152\pi$
$84$		−	42.2817i		−	4.61331i
$85$		0			0
$86$	−27.1244			−2.92489
$87$	4.78834			0.513364
$88$		−	9.92820i		−	1.05835i
$89$		−	11.7990i		−	1.25069i	−0.780350	−	0.625343i	$-0.784959\pi$
$89$		−	11.7990i		−	1.25069i	0.780350	−	0.625343i	$-0.215041\pi$
$90$		0			0
$91$			12.4405i			1.30412i
$92$	22.0172			2.29545
$93$	8.29365			0.860011
$94$			2.22228i			0.229210i
$95$		0			0
$96$			6.54099i			0.667587i
$97$			3.46410i			0.351726i	0.984415	+	0.175863i	$0.0562716\pi$
$97$			3.46410i			0.351726i	−0.984415	+	0.175863i	$0.943728\pi$
$98$	24.4113			2.46592
$99$			10.6878i			1.07417i
$100$		0			0
$101$		−	7.82403i		−	0.778520i	−0.921128	−	0.389260i	$-0.872731\pi$
$101$		−	7.82403i		−	0.778520i	0.921128	−	0.389260i	$-0.127269\pi$
$102$		−	11.4641i		−	1.13512i
$103$		−	9.12436i		−	0.899049i	−0.893268	−	0.449525i	$-0.851593\pi$
$103$		−	9.12436i		−	0.899049i	0.893268	−	0.449525i	$-0.148407\pi$
$104$			12.4405i			1.21989i
$105$		0			0
$106$	3.07180			0.298359
$107$			15.4641i			1.49497i	0.664278	+	0.747486i	$0.268739\pi$
$107$			15.4641i			1.49497i	−0.664278	+	0.747486i	$0.731261\pi$
$108$			14.9282i			1.43647i
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$			8.29365i			0.787198i
$112$	10.2182			0.965529
$113$		−	2.07180i		−	0.194898i	−0.995241	−	0.0974491i	$-0.968932\pi$
$113$		−	2.07180i		−	0.194898i	0.995241	−	0.0974491i	$-0.0310683\pi$
$114$			30.9523i			2.89895i
$115$		0			0
$116$			6.54099i			0.607316i
$117$		−	13.3923i		−	1.23812i
$118$			21.3923i			1.96932i
$119$	7.26795			0.666252
$120$		0			0
$121$	5.26795			0.478904
$122$	17.2288	+	7.26795i	1.55982	+	0.658009i
$123$		−	1.26795i		−	0.114327i
$124$			11.3293i			1.01740i
$125$		0			0
$126$	−44.3205			−3.94838
$127$		−	12.5885i		−	1.11704i	−0.829489	−	0.558522i	$-0.811368\pi$
$127$		−	12.5885i		−	1.11704i	0.829489	−	0.558522i	$-0.188632\pi$
$128$	−20.7341			−1.83265
$129$			30.9523i			2.72520i
$130$		0			0
$131$	11.6603			1.01876			0.509381	−	0.860541i	$-0.329875\pi$
$131$	11.6603			1.01876			0.509381	+	0.860541i	$0.329875\pi$
$132$	−24.4113			−2.12473
$133$	−19.6230			−1.70153
$134$	−24.4641			−2.11338
$135$		0			0
$136$	7.26795			0.623222
$137$		−	1.73205i		−	0.147979i	−0.997259	−	0.0739895i	$-0.976427\pi$
$137$		−	1.73205i		−	0.147979i	0.997259	−	0.0739895i	$-0.0235731\pi$
$138$		−	38.5885i		−	3.28487i
$139$		−	4.14682i		−	0.351729i	−0.984414	−	0.175865i	$-0.943728\pi$
$139$		−	4.14682i		−	0.351729i	0.984414	−	0.175865i	$-0.0562721\pi$
$140$		0			0
$141$	2.53590			0.213561
$142$		−	15.6603i		−	1.31418i
$143$	7.18251			0.600632
$144$	−11.0000			−0.916667
$145$		0			0
$146$		−	2.86379i		−	0.237009i
$147$		−	27.8564i		−	2.29756i
$148$	−11.3293			−0.931266
$149$	6.46410			0.529560			0.264780	−	0.964309i	$-0.414701\pi$
$149$	6.46410			0.529560			0.264780	+	0.964309i	$0.414701\pi$
$150$		0			0
$151$			12.4405i			1.01239i	0.862419	+	0.506196i	$0.168949\pi$
$151$			12.4405i			1.01239i	−0.862419	+	0.506196i	$0.831051\pi$
$152$	−19.6230			−1.59163
$153$	−7.82403			−0.632535
$154$		−	23.7698i		−	1.91543i
$155$		0			0
$156$	30.5885			2.44904
$157$	9.10706			0.726822			0.363411	−	0.931629i	$-0.381612\pi$
$157$	9.10706			0.726822			0.363411	+	0.931629i	$0.381612\pi$
$158$			22.5167i			1.79133i
$159$		−	3.50531i		−	0.277989i
$160$		0			0
$161$	24.4641			1.92804
$162$	−5.89948			−0.463507
$163$		−	15.1244i		−	1.18463i	−0.805706	−	0.592315i	$-0.798214\pi$
$163$		−	15.1244i		−	1.18463i	0.805706	−	0.592315i	$-0.201786\pi$
$164$	1.73205			0.135250
$165$		0			0
$166$			22.6587i			1.75865i
$167$			24.5885i			1.90271i	0.308094	+	0.951356i	$0.400309\pi$
$167$			24.5885i			1.90271i	−0.308094	+	0.951356i	$0.599691\pi$
$168$		−	46.9808i		−	3.62464i
$169$	4.00000			0.307692
$170$		0			0
$171$	21.1244			1.61542
$172$	−42.2817			−3.22395
$173$	9.57668			0.728102			0.364051	−	0.931379i	$-0.381393\pi$
$173$	9.57668			0.728102			0.364051	+	0.931379i	$0.381393\pi$
$174$	11.4641			0.869091
$175$		0			0
$176$		−	5.89948i		−	0.444690i
$177$	24.4113			1.83487
$178$		−	28.2487i		−	2.11733i
$179$	−22.3923			−1.67368			−0.836840	−	0.547448i	$-0.815599\pi$
$179$	−22.3923			−1.67368			−0.836840	+	0.547448i	$0.815599\pi$
$180$		0			0
$181$		−	22.6587i		−	1.68421i	−0.539317	−	0.842103i	$-0.681318\pi$
$181$		−	22.6587i		−	1.68421i	0.539317	−	0.842103i	$-0.318682\pi$
$182$			29.7846i			2.20778i
$183$	8.29365	−	19.6603i	0.613084	−	1.45333i
$184$	24.4641			1.80352
$185$		0			0
$186$	19.8564			1.45594
$187$		−	4.19615i		−	0.306853i
$188$			3.46410i			0.252646i
$189$			16.5873i			1.20655i
$190$		0			0
$191$			23.3002i			1.68594i	0.537959	+	0.842971i	$0.319196\pi$
$191$			23.3002i			1.68594i	−0.537959	+	0.842971i	$0.680804\pi$
$192$			29.1244i			2.10187i
$193$	−6.07137			−0.437027			−0.218513	−	0.975834i	$-0.570121\pi$
$193$	−6.07137			−0.437027			−0.218513	+	0.975834i	$0.570121\pi$
$194$			8.29365i			0.595449i
$195$		0			0
$196$	38.0526			2.71804
$197$			4.26795i			0.304079i	0.988374	+	0.152039i	$0.0485841\pi$
$197$			4.26795i			0.304079i	−0.988374	+	0.152039i	$0.951416\pi$
$198$			25.5885i			1.81849i
$199$	−2.00000			−0.141776			−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000			−0.141776			−0.0708881	+	0.997484i	$0.522583\pi$
$200$		0			0
$201$			27.9166i			1.96909i
$202$		−	18.7321i		−	1.31798i
$203$			7.26795i			0.510110i
$204$		−	17.8703i		−	1.25117i
$205$		0			0
$206$		−	21.8453i		−	1.52203i
$207$	−26.3359			−1.83047
$208$			7.39230i			0.512564i
$209$			11.3293i			0.783666i
$210$		0			0
$211$		−	19.6230i		−	1.35090i	−0.737405	−	0.675451i	$-0.763949\pi$
$211$		−	19.6230i		−	1.35090i	0.737405	−	0.675451i	$-0.236051\pi$
$212$	4.78834			0.328865
$213$	−17.8703			−1.22445
$214$			37.0237i			2.53089i
$215$		0			0
$216$			16.5873i			1.12862i
$217$			12.5885i			0.854560i
$218$	26.3359			1.78369
$219$	−3.26795			−0.220828
$220$		0			0
$221$			5.25796i			0.353689i
$222$			19.8564i			1.33267i
$223$	1.11114			0.0744073			0.0372037	−	0.999308i	$-0.488155\pi$
$223$	1.11114			0.0744073			0.0372037	+	0.999308i	$0.488155\pi$
$224$	−9.92820			−0.663356
$225$		0			0
$226$		−	4.96023i		−	0.329950i
$227$	−11.9709			−0.794533			−0.397267	−	0.917703i	$-0.630041\pi$
$227$	−11.9709			−0.794533			−0.397267	+	0.917703i	$0.630041\pi$
$228$			48.2487i			3.19535i
$229$	16.2679			1.07502			0.537508	−	0.843259i	$-0.319366\pi$
$229$	16.2679			1.07502			0.537508	+	0.843259i	$0.319366\pi$
$230$		0			0
$231$	−27.1244			−1.78465
$232$			7.26795i			0.477164i
$233$	−12.6124			−0.826264			−0.413132	−	0.910671i	$-0.635565\pi$
$233$	−12.6124			−0.826264			−0.413132	+	0.910671i	$0.635565\pi$
$234$		−	32.0635i		−	2.09605i
$235$		0			0
$236$			33.3465i			2.17067i
$237$	25.6944			1.66903
$238$	17.4007			1.12792
$239$	−13.2679			−0.858232			−0.429116	−	0.903249i	$-0.641175\pi$
$239$	−13.2679			−0.858232			−0.429116	+	0.903249i	$0.641175\pi$
$240$		0			0
$241$	19.0526			1.22728			0.613642	−	0.789585i	$-0.289704\pi$
$241$	19.0526			1.22728			0.613642	+	0.789585i	$0.289704\pi$
$242$	12.6124			0.810754
$243$			18.7321i			1.20166i
$244$	26.8564	+	11.3293i	1.71931	+	0.725286i
$245$		0			0
$246$		−	3.03569i		−	0.193548i
$247$		−	14.1962i		−	0.903280i
$248$			12.5885i			0.799368i
$249$	25.8564			1.63858
$250$		0			0
$251$			14.8346i			0.936354i	0.883635	+	0.468177i	$0.155089\pi$
$251$			14.8346i			0.936354i	−0.883635	+	0.468177i	$0.844911\pi$
$252$	−69.0871			−4.35208
$253$		−	14.1244i		−	0.887991i
$254$		−	30.1389i		−	1.89108i
$255$		0			0
$256$	−28.3205			−1.77003
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$			74.1051i			4.61358i
$259$	−12.5885			−0.782209
$260$		0			0
$261$		−	7.82403i		−	0.484295i
$262$	27.9166			1.72470
$263$		−	20.1962i		−	1.24535i	−0.782481	−	0.622674i	$-0.786046\pi$
$263$		−	20.1962i		−	1.24535i	0.782481	−	0.622674i	$-0.213954\pi$
$264$	−27.1244			−1.66939
$265$		0			0
$266$	−46.9808			−2.88058
$267$	−32.2354			−1.97277
$268$	−38.1348			−2.32946
$269$	2.53590			0.154616			0.0773082	−	0.997007i	$-0.475367\pi$
$269$	2.53590			0.154616			0.0773082	+	0.997007i	$0.475367\pi$
$270$		0			0
$271$	6.19615			0.376389			0.188195	−	0.982132i	$-0.439736\pi$
$271$	6.19615			0.376389			0.188195	+	0.982132i	$0.439736\pi$
$272$	4.31872			0.261861
$273$	33.9880			2.05705
$274$		−	4.14682i		−	0.250519i
$275$		0			0
$276$		−	60.1520i		−	3.62072i
$277$	−19.6230			−1.17903			−0.589515	−	0.807757i	$-0.700681\pi$
$277$	−19.6230			−1.17903			−0.589515	+	0.807757i	$0.700681\pi$
$278$		−	9.92820i		−	0.595454i
$279$		−	13.5516i		−	0.811314i
$280$		0			0
$281$		−	5.72758i		−	0.341679i	−0.985299	−	0.170840i	$-0.945352\pi$
$281$		−	5.72758i		−	0.341679i	0.985299	−	0.170840i	$-0.0546480\pi$
$282$	6.07137			0.361545
$283$			14.5885i			0.867194i	0.901107	+	0.433597i	$0.142756\pi$
$283$			14.5885i			0.867194i	−0.901107	+	0.433597i	$0.857244\pi$
$284$		−	24.4113i		−	1.44855i
$285$		0			0
$286$	17.1962			1.01683
$287$	1.92455			0.113602
$288$	10.6878			0.629786
$289$	−13.9282			−0.819306
$290$		0			0
$291$	9.46410			0.554795
$292$		−	4.46410i		−	0.261242i
$293$		−	1.85641i		−	0.108452i	−0.998529	−	0.0542262i	$-0.982731\pi$
$293$		−	1.85641i		−	0.108452i	0.998529	−	0.0542262i	$-0.0172692\pi$
$294$		−	66.6930i		−	3.88961i
$295$		0			0
$296$	−12.5885			−0.731689
$297$	9.57668			0.555695
$298$	15.4762			0.896510
$299$			17.6984i			1.02353i
$300$		0			0
$301$	−46.9808			−2.70793
$302$			29.7846i			1.71391i
$303$	−21.3756			−1.22800
$304$	−11.6603			−0.668761
$305$		0			0
$306$	−18.7321			−1.07084
$307$	1.11114			0.0634160			0.0317080	−	0.999497i	$-0.489905\pi$
$307$	1.11114			0.0634160			0.0317080	+	0.999497i	$0.489905\pi$
$308$		−	37.0526i		−	2.11127i
$309$	−24.9282			−1.41812
$310$		0			0
$311$			31.1242i			1.76489i	0.470414	+	0.882446i	$0.344105\pi$
$311$			31.1242i			1.76489i	−0.470414	+	0.882446i	$0.655895\pi$
$312$	33.9880			1.92419
$313$	22.6587			1.28074			0.640372	−	0.768065i	$-0.278780\pi$
$313$	22.6587			1.28074			0.640372	+	0.768065i	$0.278780\pi$
$314$	21.8038			1.23046
$315$		0			0
$316$			35.0991i			1.97448i
$317$		−	5.07180i		−	0.284860i	−0.989805	−	0.142430i	$-0.954508\pi$
$317$		−	5.07180i		−	0.284860i	0.989805	−	0.142430i	$-0.0454917\pi$
$318$		−	8.39230i		−	0.470617i
$319$	4.19615			0.234939
$320$		0			0
$321$	42.2487			2.35809
$322$	58.5712			3.26405
$323$	−8.29365			−0.461471
$324$	−9.19615			−0.510897
$325$		0			0
$326$		−	36.2103i		−	2.00550i
$327$		−	30.0526i		−	1.66191i
$328$	1.92455			0.106265
$329$			3.84910i			0.212208i
$330$		0			0
$331$		−	30.6546i		−	1.68493i	−0.538752	−	0.842464i	$-0.681104\pi$
$331$		−	30.6546i		−	1.68493i	0.538752	−	0.842464i	$-0.318896\pi$
$332$			35.3205i			1.93846i
$333$	13.5516			0.742624
$334$			58.8690i			3.22117i
$335$		0			0
$336$		−	27.9166i		−	1.52298i
$337$	14.3650			0.782513			0.391256	−	0.920282i	$-0.372041\pi$
$337$	14.3650			0.782513			0.391256	+	0.920282i	$0.372041\pi$
$338$	9.57668			0.520903
$339$	−5.66025			−0.307423
$340$		0			0
$341$	7.26795			0.393582
$342$	50.5753			2.73480
$343$	13.2539			0.715642
$344$	−46.9808			−2.53303
$345$		0			0
$346$	22.9282			1.23263
$347$			25.5167i			1.36981i	0.728634	+	0.684903i	$0.240155\pi$
$347$			25.5167i			1.36981i	−0.728634	+	0.684903i	$0.759845\pi$
$348$	17.8703			0.957950
$349$		−	30.1389i		−	1.61330i	−0.591030	−	0.806649i	$-0.701279\pi$
$349$		−	30.1389i		−	1.61330i	0.591030	−	0.806649i	$-0.298721\pi$
$350$		0			0
$351$	−12.0000			−0.640513
$352$			5.73205i			0.305519i
$353$			25.9808i			1.38282i	0.722464	+	0.691408i	$0.243009\pi$
$353$			25.9808i			1.38282i	−0.722464	+	0.691408i	$0.756991\pi$
$354$	58.4449			3.10631
$355$		0			0
$356$		−	44.0343i		−	2.33381i
$357$		−	19.8564i		−	1.05091i
$358$	−53.6110			−2.83343
$359$		−	18.6837i		−	0.986090i	−0.870004	−	0.493045i	$-0.835884\pi$
$359$		−	18.6837i		−	0.986090i	0.870004	−	0.493045i	$-0.164116\pi$
$360$		0			0
$361$	3.39230			0.178542
$362$		−	54.2487i		−	2.85125i
$363$		−	14.3923i		−	0.755400i
$364$			46.4285i			2.43351i
$365$		0			0
$366$	19.8564	−	47.0700i	1.03791	−	2.46039i
$367$		−	11.0718i		−	0.577943i	−0.957338	−	0.288972i	$-0.906687\pi$
$367$		−	11.0718i		−	0.577943i	0.957338	−	0.288972i	$-0.0933133\pi$
$368$	14.5369			0.757789
$369$	−2.07180			−0.107853
$370$		0			0
$371$	5.32051			0.276227
$372$	30.9523			1.60480
$373$	−11.3293			−0.586611			−0.293305	−	0.956019i	$-0.594755\pi$
$373$	−11.3293			−0.586611			−0.293305	+	0.956019i	$0.594755\pi$
$374$		−	10.0463i		−	0.519482i
$375$		0			0
$376$			3.84910i			0.198502i
$377$	−5.25796			−0.270799
$378$			39.7128i			2.04261i
$379$	10.5885			0.543893			0.271946	−	0.962312i	$-0.412333\pi$
$379$	10.5885			0.543893			0.271946	+	0.962312i	$0.412333\pi$
$380$		0			0
$381$	−34.3923			−1.76197
$382$			55.7846i			2.85419i
$383$	18.9815			0.969908			0.484954	−	0.874540i	$-0.338836\pi$
$383$	18.9815			0.969908			0.484954	+	0.874540i	$0.338836\pi$
$384$			56.6467i			2.89074i
$385$		0			0
$386$	−14.5359			−0.739858
$387$	50.5753			2.57089
$388$			12.9282i			0.656330i
$389$			7.35440i			0.372883i	0.982466	+	0.186442i	$0.0596955\pi$
$389$			7.35440i			0.372883i	−0.982466	+	0.186442i	$0.940305\pi$
$390$		0			0
$391$	10.3397			0.522903
$392$	42.2817			2.13555
$393$		−	31.8564i		−	1.60694i
$394$			10.2182i			0.514785i
$395$		0			0
$396$			39.8875i			2.00442i
$397$	7.48024			0.375422			0.187711	−	0.982224i	$-0.439893\pi$
$397$	7.48024			0.375422			0.187711	+	0.982224i	$0.439893\pi$
$398$	−4.78834			−0.240018
$399$			53.6110i			2.68391i
$400$		0			0
$401$		−	10.8597i		−	0.542308i	−0.962536	−	0.271154i	$-0.912595\pi$
$401$		−	10.8597i		−	0.542308i	0.962536	−	0.271154i	$-0.0874053\pi$
$402$			66.8372i			3.33353i
$403$	−9.10706			−0.453655
$404$		−	29.1997i		−	1.45274i
$405$		0			0
$406$			17.4007i			0.863583i
$407$			7.26795i			0.360259i
$408$		−	19.8564i		−	0.983039i
$409$		−	8.29365i		−	0.410095i	−0.978752	−	0.205047i	$-0.934265\pi$
$409$		−	8.29365i		−	0.410095i	0.978752	−	0.205047i	$-0.0657348\pi$
$410$		0			0
$411$	−4.73205			−0.233415
$412$		−	34.0526i		−	1.67765i
$413$			37.0526i			1.82324i
$414$	−63.0526			−3.09886
$415$		0			0
$416$		−	7.18251i		−	0.352152i
$417$	−11.3293			−0.554800
$418$			27.1244i			1.32670i
$419$		−	16.1177i		−	0.787400i	−0.919239	−	0.393700i	$-0.871195\pi$
$419$		−	16.1177i		−	0.787400i	0.919239	−	0.393700i	$-0.128805\pi$
$420$		0			0
$421$		−	11.3293i		−	0.552158i	−0.961135	−	0.276079i	$-0.910965\pi$
$421$		−	11.3293i		−	0.552158i	0.961135	−	0.276079i	$-0.0890351\pi$
$422$		−	46.9808i		−	2.28699i
$423$		−	4.14359i		−	0.201468i
$424$	5.32051			0.258387
$425$		0			0
$426$	−42.7846			−2.07292
$427$	29.8412	+	12.5885i	1.44412	+	0.609198i
$428$			57.7128i			2.78965i
$429$		−	19.6230i		−	0.947407i
$430$		0			0
$431$	−36.2487			−1.74604			−0.873019	−	0.487685i	$-0.837841\pi$
$431$	−36.2487			−1.74604			−0.873019	+	0.487685i	$0.837841\pi$
$432$			9.85641i			0.474217i
$433$	−24.8809			−1.19570			−0.597851	−	0.801607i	$-0.703979\pi$
$433$	−24.8809			−1.19570			−0.597851	+	0.801607i	$0.703979\pi$
$434$			30.1389i			1.44671i
$435$		0			0
$436$	41.0526			1.96606
$437$	−27.9166			−1.33543
$438$	−7.82403			−0.373846
$439$	1.80385			0.0860929			0.0430465	−	0.999073i	$-0.486294\pi$
$439$	1.80385			0.0860929			0.0430465	+	0.999073i	$0.486294\pi$
$440$		0			0
$441$	−45.5167			−2.16746
$442$			12.5885i			0.598772i
$443$		−	24.2487i		−	1.15209i	−0.817418	−	0.576046i	$-0.804595\pi$
$443$		−	24.2487i		−	1.15209i	0.817418	−	0.576046i	$-0.195405\pi$
$444$			30.9523i			1.46893i
$445$		0			0
$446$	2.66025			0.125967
$447$		−	17.6603i		−	0.835301i
$448$	−44.2062			−2.08855
$449$	−0.803848			−0.0379359			−0.0189680	−	0.999820i	$-0.506038\pi$
$449$	−0.803848			−0.0379359			−0.0189680	+	0.999820i	$0.506038\pi$
$450$		0			0
$451$		−	1.11114i		−	0.0523215i
$452$		−	7.73205i		−	0.363685i
$453$	33.9880			1.59690
$454$	−28.6603			−1.34509
$455$		0			0
$456$			53.6110i			2.51056i
$457$	−21.8453			−1.02188			−0.510939	−	0.859617i	$-0.670702\pi$
$457$	−21.8453			−1.02188			−0.510939	+	0.859617i	$0.670702\pi$
$458$	38.9482			1.81993
$459$			7.01062i			0.327228i
$460$		0			0
$461$	−8.53590			−0.397556			−0.198778	−	0.980045i	$-0.563697\pi$
$461$	−8.53590			−0.397556			−0.198778	+	0.980045i	$0.563697\pi$
$462$	−64.9403			−3.02130
$463$			6.92820i			0.321981i	0.986956	+	0.160990i	$0.0514688\pi$
$463$			6.92820i			0.321981i	−0.986956	+	0.160990i	$0.948531\pi$
$464$			4.31872i			0.200491i
$465$		0			0
$466$	−30.1962			−1.39881
$467$	−24.2394			−1.12167			−0.560834	−	0.827929i	$-0.689519\pi$
$467$	−24.2394			−1.12167			−0.560834	+	0.827929i	$0.689519\pi$
$468$		−	49.9808i		−	2.31036i
$469$	−42.3731			−1.95661
$470$		0			0
$471$		−	24.8809i		−	1.14645i
$472$			37.0526i			1.70548i
$473$			27.1244i			1.24718i
$474$	61.5167			2.82555
$475$		0			0
$476$	27.1244			1.24324
$477$	−5.72758			−0.262248
$478$	−31.7657			−1.45293
$479$	37.1769			1.69866			0.849328	−	0.527865i	$-0.177007\pi$
$479$	37.1769			1.69866			0.849328	+	0.527865i	$0.177007\pi$
$480$		0			0
$481$		−	9.10706i		−	0.415246i
$482$	45.6151			2.07771
$483$		−	66.8372i		−	3.04120i
$484$	19.6603			0.893648
$485$		0			0
$486$			44.8477i			2.03433i
$487$		−	38.9808i		−	1.76639i	−0.469009	−	0.883193i	$-0.655389\pi$
$487$		−	38.9808i		−	1.76639i	0.469009	−	0.883193i	$-0.344611\pi$
$488$	29.8412	+	12.5885i	1.35085	+	0.569853i
$489$	−41.3205			−1.86858
$490$		0			0
$491$	−15.8038			−0.713218			−0.356609	−	0.934254i	$-0.616067\pi$
$491$	−15.8038			−0.713218			−0.356609	+	0.934254i	$0.616067\pi$
$492$		−	4.73205i		−	0.213337i
$493$			3.07180i			0.138347i
$494$		−	33.9880i		−	1.52919i
$495$		0			0
$496$			7.48024i			0.335873i
$497$		−	27.1244i		−	1.21669i
$498$	61.9046			2.77401
$499$			0.297729i			0.0133282i	0.999978	+	0.00666408i	$0.00212126\pi$
$499$			0.297729i			0.0133282i	−0.999978	+	0.00666408i	$0.997879\pi$
$500$		0			0
$501$	67.1769			3.00124
$502$			35.5167i			1.58519i
$503$		−	18.5885i		−	0.828818i	−0.910091	−	0.414409i	$-0.863988\pi$
$503$		−	18.5885i		−	0.828818i	0.910091	−	0.414409i	$-0.136012\pi$
$504$	−76.7654			−3.41940
$505$		0			0
$506$		−	33.8161i		−	1.50331i
$507$		−	10.9282i		−	0.485339i
$508$		−	46.9808i		−	2.08443i
$509$		−	32.7050i		−	1.44962i	−0.688948	−	0.724811i	$-0.741927\pi$
$509$		−	32.7050i		−	1.44962i	0.688948	−	0.724811i	$-0.258073\pi$
$510$		0			0
$511$		−	4.96023i		−	0.219428i
$512$	−26.3359			−1.16389
$513$		−	18.9282i		−	0.835701i
$514$			14.3650i			0.633614i
$515$		0			0
$516$			115.516i			5.08529i
$517$	2.22228			0.0977356
$518$	−30.1389			−1.32423
$519$		−	26.1640i		−	1.14847i
$520$		0			0
$521$		−	35.2710i		−	1.54525i	−0.634861	−	0.772626i	$-0.718943\pi$
$521$		−	35.2710i		−	1.54525i	0.634861	−	0.772626i	$-0.281057\pi$
$522$		−	18.7321i		−	0.819880i
$523$	−44.2062			−1.93300			−0.966501	−	0.256662i	$-0.917377\pi$
$523$	−44.2062			−1.93300			−0.966501	+	0.256662i	$0.917377\pi$
$524$	43.5167			1.90103
$525$		0			0
$526$		−	48.3530i		−	2.10829i
$527$			5.32051i			0.231765i
$528$	−16.1177			−0.701432
$529$	11.8038			0.513211
$530$		0			0
$531$		−	39.8875i		−	1.73097i
$532$	−73.2340			−3.17510
$533$			1.39230i			0.0603074i
$534$	−77.1769			−3.33977
$535$		0			0
$536$	−42.3731			−1.83024
$537$			61.1769i			2.63998i
$538$	6.07137			0.261755
$539$		−	24.4113i		−	1.05147i
$540$		0			0
$541$		−	27.1032i		−	1.16526i	−0.812738	−	0.582629i	$-0.802024\pi$
$541$		−	27.1032i		−	1.16526i	0.812738	−	0.582629i	$-0.197976\pi$
$542$	14.8346			0.637202
$543$	−61.9046			−2.65658
$544$	−4.19615			−0.179909
$545$		0			0
$546$	81.3731			3.48245
$547$	19.9207			0.851748			0.425874	−	0.904782i	$-0.359967\pi$
$547$	19.9207			0.851748			0.425874	+	0.904782i	$0.359967\pi$
$548$		−	6.46410i		−	0.276133i
$549$	−32.1244	−	13.5516i	−1.37103	−	0.578369i
$550$		0			0
$551$		−	8.29365i		−	0.353321i
$552$		−	66.8372i		−	2.84478i
$553$			39.0000i			1.65845i
$554$	−46.9808			−1.99602
$555$		0			0
$556$		−	15.4762i		−	0.656335i
$557$	−29.6693			−1.25713			−0.628564	−	0.777758i	$-0.716357\pi$
$557$	−29.6693			−1.25713			−0.628564	+	0.777758i	$0.716357\pi$
$558$		−	32.4449i		−	1.37350i
$559$		−	33.9880i		−	1.43754i
$560$		0			0
$561$	−11.4641			−0.484015
$562$		−	13.7128i		−	0.578440i
$563$			22.0526i			0.929405i	0.885467	+	0.464702i	$0.153839\pi$
$563$			22.0526i			0.929405i	−0.885467	+	0.464702i	$0.846161\pi$
$564$	9.46410			0.398511
$565$		0			0
$566$			34.9272i			1.46810i
$567$	−10.2182			−0.429124
$568$		−	27.1244i		−	1.13811i
$569$	11.1962			0.469367			0.234684	−	0.972072i	$-0.424595\pi$
$569$	11.1962			0.469367			0.234684	+	0.972072i	$0.424595\pi$
$570$		0			0
$571$	−40.3923			−1.69037			−0.845183	−	0.534478i	$-0.820508\pi$
$571$	−40.3923			−1.69037			−0.845183	+	0.534478i	$0.820508\pi$
$572$	26.8055			1.12079
$573$	63.6573			2.65932
$574$	4.60770			0.192321
$575$		0			0
$576$	47.5885			1.98285
$577$	−15.1784			−0.631886			−0.315943	−	0.948778i	$-0.602321\pi$
$577$	−15.1784			−0.631886			−0.315943	+	0.948778i	$0.602321\pi$
$578$	−33.3465			−1.38703
$579$			16.5873i			0.689345i
$580$		0			0
$581$			39.2460i			1.62820i
$582$	22.6587			0.939232
$583$		−	3.07180i		−	0.127221i
$584$		−	4.96023i		−	0.205256i
$585$		0			0
$586$		−	4.44455i		−	0.183603i
$587$	−26.6336			−1.09929			−0.549643	−	0.835400i	$-0.685236\pi$
$587$	−26.6336			−1.09929			−0.549643	+	0.835400i	$0.685236\pi$
$588$		−	103.962i		−	4.28730i
$589$		−	14.3650i		−	0.591900i
$590$		0			0
$591$	11.6603			0.479639
$592$	−7.48024			−0.307436
$593$	−23.1283			−0.949765			−0.474883	−	0.880049i	$-0.657509\pi$
$593$	−23.1283			−0.949765			−0.474883	+	0.880049i	$0.657509\pi$
$594$	22.9282			0.940756
$595$		0			0
$596$	24.1244			0.988172
$597$			5.46410i			0.223631i
$598$			42.3731i			1.73276i
$599$		−	31.5938i		−	1.29089i	−0.763807	−	0.645445i	$-0.776672\pi$
$599$		−	31.5938i		−	1.29089i	0.763807	−	0.645445i	$-0.223328\pi$
$600$		0			0
$601$	−3.00000			−0.122373			−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000			−0.122373			−0.0611863	+	0.998126i	$0.519488\pi$
$602$	−112.480			−4.58434
$603$	45.6151			1.85759
$604$			46.4285i			1.88915i
$605$		0			0
$606$	−51.1769			−2.07892
$607$		−	6.78461i		−	0.275379i	−0.990475	−	0.137689i	$-0.956032\pi$
$607$		−	6.78461i		−	0.275379i	0.990475	−	0.137689i	$-0.0439676\pi$
$608$	11.3293			0.459465
$609$	19.8564			0.804622
$610$		0			0
$611$	−2.78461			−0.112653
$612$	−29.1997			−1.18033
$613$			24.9282i			1.00684i	0.864042	+	0.503420i	$0.167925\pi$
$613$			24.9282i			1.00684i	−0.864042	+	0.503420i	$0.832075\pi$
$614$	2.66025			0.107359
$615$		0			0
$616$		−	41.1705i		−	1.65881i
$617$	11.7990			0.475008			0.237504	−	0.971387i	$-0.423671\pi$
$617$	11.7990			0.475008			0.237504	+	0.971387i	$0.423671\pi$
$618$	−59.6824			−2.40078
$619$	−7.60770			−0.305779			−0.152890	−	0.988243i	$-0.548858\pi$
$619$	−7.60770			−0.305779			−0.152890	+	0.988243i	$0.548858\pi$
$620$		0			0
$621$			23.5979i			0.946952i
$622$			74.5167i			2.98785i
$623$		−	48.9282i		−	1.96027i
$624$	20.1962			0.808493
$625$		0			0
$626$	54.2487			2.16821
$627$	30.9523			1.23612
$628$	33.9880			1.35627
$629$	−5.32051			−0.212143
$630$		0			0
$631$			7.18251i			0.285931i	0.989728	+	0.142966i	$0.0456638\pi$
$631$			7.18251i			0.285931i	−0.989728	+	0.142966i	$0.954336\pi$
$632$			39.0000i			1.55134i
$633$	−53.6110			−2.13085
$634$		−	12.1427i		−	0.482250i
$635$		0			0
$636$		−	13.0820i		−	0.518735i
$637$			30.5885i			1.21196i
$638$	10.0463			0.397737
$639$			29.1997i			1.15512i
$640$		0			0
$641$			13.4258i			0.530286i	0.964209	+	0.265143i	$0.0854192\pi$
$641$			13.4258i			0.530286i	−0.964209	+	0.265143i	$0.914581\pi$
$642$	101.151			3.99210
$643$	−25.6944			−1.01329			−0.506643	−	0.862156i	$-0.669114\pi$
$643$	−25.6944			−1.01329			−0.506643	+	0.862156i	$0.669114\pi$
$644$	91.3013			3.59777
$645$		0			0
$646$	−19.8564			−0.781240
$647$	−11.5012			−0.452160			−0.226080	−	0.974109i	$-0.572591\pi$
$647$	−11.5012			−0.452160			−0.226080	+	0.974109i	$0.572591\pi$
$648$	−10.2182			−0.401409
$649$	21.3923			0.839721
$650$		0			0
$651$	34.3923			1.34794
$652$		−	56.4449i		−	2.21055i
$653$	−1.28303			−0.0502089			−0.0251045	−	0.999685i	$-0.507992\pi$
$653$	−1.28303			−0.0502089			−0.0251045	+	0.999685i	$0.507992\pi$
$654$		−	71.9509i		−	2.81350i
$655$		0			0
$656$	1.14359			0.0446498
$657$			5.33975i			0.208323i
$658$			9.21539i			0.359253i
$659$	−28.7321			−1.11924			−0.559621	−	0.828749i	$-0.689053\pi$
$659$	−28.7321			−1.11924			−0.559621	+	0.828749i	$0.689053\pi$
$660$		0			0
$661$			27.9166i			1.08583i	0.839787	+	0.542916i	$0.182680\pi$
$661$			27.9166i			1.08583i	−0.839787	+	0.542916i	$0.817320\pi$
$662$		−	73.3923i		−	2.85247i
$663$	14.3650			0.557891
$664$			39.2460i			1.52304i
$665$		0			0
$666$	32.4449			1.25721
$667$			10.3397i			0.400357i
$668$			91.7654i			3.55051i
$669$		−	3.03569i		−	0.117366i
$670$		0			0
$671$	7.26795	−	17.2288i	0.280576	−	0.665111i
$672$			27.1244i			1.04634i
$673$	19.6230			0.756410			0.378205	−	0.925722i	$-0.376541\pi$
$673$	19.6230			0.756410			0.378205	+	0.925722i	$0.376541\pi$
$674$	34.3923			1.32474
$675$		0			0
$676$	14.9282			0.574162
$677$	−37.9629			−1.45903			−0.729517	−	0.683963i	$-0.760255\pi$
$677$	−37.9629			−1.45903			−0.729517	+	0.683963i	$0.760255\pi$
$678$	−13.5516			−0.520446
$679$			14.3650i			0.551279i
$680$		0			0
$681$			32.7050i			1.25326i
$682$	17.4007			0.666308
$683$		−	4.73205i		−	0.181067i	−0.995893	−	0.0905334i	$-0.971143\pi$
$683$		−	4.73205i		−	0.181067i	0.995893	−	0.0905334i	$-0.0288572\pi$
$684$	78.8372			3.01441
$685$		0			0
$686$	31.7321			1.21154
$687$		−	44.4449i		−	1.69568i
$688$	−27.9166			−1.06431
$689$			3.84910i			0.146639i
$690$		0			0
$691$	45.4641			1.72954			0.864768	−	0.502172i	$-0.167465\pi$
$691$	45.4641			1.72954			0.864768	+	0.502172i	$0.167465\pi$
$692$	35.7407			1.35866
$693$			44.3205i			1.68360i
$694$			61.0912i			2.31899i
$695$		0			0
$696$	19.8564			0.752655
$697$	0.813410			0.0308101
$698$		−	72.1577i		−	2.73121i
$699$			34.4576i			1.30331i
$700$		0			0
$701$			46.2566i			1.74709i	0.486746	+	0.873544i	$0.338184\pi$
$701$			46.2566i			1.74709i	−0.486746	+	0.873544i	$0.661816\pi$
$702$	−28.7300			−1.08435
$703$	14.3650			0.541787
$704$			25.5225i			0.961914i
$705$		0			0
$706$			62.2024i			2.34102i
$707$		−	32.4449i		−	1.22021i
$708$	91.1043			3.42391
$709$		−	5.25796i		−	0.197467i	−0.995114	−	0.0987335i	$-0.968521\pi$
$709$		−	5.25796i		−	0.197467i	0.995114	−	0.0987335i	$-0.0314791\pi$
$710$		0			0
$711$		−	41.9839i		−	1.57452i
$712$		−	48.9282i		−	1.83366i
$713$			17.9090i			0.670696i
$714$		−	47.5396i		−	1.77913i
$715$		0			0
$716$	−83.5692			−3.12313
$717$			36.2487i			1.35373i
$718$		−	44.7321i		−	1.66939i
$719$	17.6603			0.658616			0.329308	−	0.944222i	$-0.393185\pi$
$719$	17.6603			0.658616			0.329308	+	0.944222i	$0.393185\pi$
$720$		0			0
$721$		−	37.8371i		−	1.40913i
$722$	8.12176			0.302260
$723$		−	52.0526i		−	1.93586i
$724$		−	84.5633i		−	3.14277i
$725$		0			0
$726$		−	34.4576i		−	1.27884i
$727$		−	21.7128i		−	0.805284i	−0.915358	−	0.402642i	$-0.868092\pi$
$727$		−	21.7128i		−	0.805284i	0.915358	−	0.402642i	$-0.131908\pi$
$728$			51.5885i			1.91200i
$729$	43.7846			1.62165
$730$		0			0
$731$	−19.8564			−0.734416
$732$	30.9523	−	73.3731i	1.14403	−	2.71195i
$733$			15.3923i			0.568528i	0.958746	+	0.284264i	$0.0917492\pi$
$733$			15.3923i			0.568528i	−0.958746	+	0.284264i	$0.908251\pi$
$734$		−	26.5078i		−	0.978419i
$735$		0			0
$736$	−14.1244			−0.520631
$737$			24.4641i			0.901147i
$738$	−4.96023			−0.182589
$739$			9.40479i			0.345961i	0.984925	+	0.172980i	$0.0553397\pi$
$739$			9.40479i			0.345961i	−0.984925	+	0.172980i	$0.944660\pi$
$740$		0			0
$741$	−38.7846			−1.42479
$742$	12.7382			0.467634
$743$	12.7843			0.469009			0.234505	−	0.972115i	$-0.424653\pi$
$743$	12.7843			0.469009			0.234505	+	0.972115i	$0.424653\pi$
$744$	34.3923			1.26088
$745$		0			0
$746$	−27.1244			−0.993093
$747$		−	42.2487i		−	1.54580i
$748$		−	15.6603i		−	0.572596i
$749$			64.1269i			2.34315i
$750$		0			0
$751$	−36.3923			−1.32797			−0.663987	−	0.747744i	$-0.731137\pi$
$751$	−36.3923			−1.32797			−0.663987	+	0.747744i	$0.731137\pi$
$752$			2.28719i			0.0834051i
$753$	40.5290			1.47696
$754$	−12.5885			−0.458445
$755$		0			0
$756$			61.9046i			2.25145i
$757$		−	4.41154i		−	0.160340i	−0.996781	−	0.0801701i	$-0.974454\pi$
$757$		−	4.41154i		−	0.160340i	0.996781	−	0.0801701i	$-0.0255464\pi$
$758$	25.3506			0.920774
$759$	−38.5885			−1.40067
$760$		0			0
$761$		−	10.9855i		−	0.398226i	−0.979977	−	0.199113i	$-0.936194\pi$
$761$		−	10.9855i		−	0.398226i	0.979977	−	0.199113i	$-0.0638060\pi$
$762$	−82.3410			−2.98290
$763$	45.6151			1.65138
$764$			86.9575i			3.14601i
$765$		0			0
$766$	45.4449			1.64199
$767$	−26.8055			−0.967890
$768$			77.3731i			2.79196i
$769$			9.10706i			0.328409i	0.986426	+	0.164204i	$0.0525057\pi$
$769$			9.10706i			0.328409i	−0.986426	+	0.164204i	$0.947494\pi$
$770$		0			0
$771$	16.3923			0.590354
$772$	−22.6587			−0.815503
$773$		−	26.5359i		−	0.954430i	−0.878787	−	0.477215i	$-0.841646\pi$
$773$		−	26.5359i		−	0.954430i	0.878787	−	0.477215i	$-0.158354\pi$
$774$	121.086			4.35234
$775$		0			0
$776$			14.3650i			0.515674i
$777$			34.3923i			1.23382i
$778$			17.6077i			0.631266i
$779$	−2.19615			−0.0786853
$780$		0			0
$781$	−15.6603			−0.560368
$782$	24.7551			0.885241
$783$	−7.01062			−0.250539
$784$	25.1244			0.897298
$785$		0			0
$786$		−	76.2697i		−	2.72045i
$787$	−6.88478			−0.245416			−0.122708	−	0.992443i	$-0.539158\pi$
$787$	−6.88478			−0.245416			−0.122708	+	0.992443i	$0.539158\pi$
$788$			15.9282i			0.567419i
$789$	−55.1769			−1.96435
$790$		0			0
$791$		−	8.59138i		−	0.305474i
$792$			44.3205i			1.57486i
$793$	−9.10706	+	21.5885i	−0.323401	+	0.766629i
$794$	17.9090			0.635565
$795$		0			0
$796$	−7.46410			−0.264558
$797$			4.51666i			0.159988i	0.996795	+	0.0799942i	$0.0254902\pi$
$797$			4.51666i			0.159988i	−0.996795	+	0.0799942i	$0.974510\pi$
$798$			128.354i			4.54368i
$799$			1.62682i			0.0575527i
$800$		0			0
$801$			52.6717i			1.86106i
$802$		−	26.0000i		−	0.918092i
$803$	−2.86379			−0.101061
$804$			104.186i			3.67437i
$805$		0			0
$806$	−21.8038			−0.768008
$807$		−	6.92820i		−	0.243884i
$808$		−	32.4449i		−	1.14141i
$809$	−7.48334			−0.263100			−0.131550	−	0.991310i	$-0.541995\pi$
$809$	−7.48334			−0.263100			−0.131550	+	0.991310i	$0.541995\pi$
$810$		0			0
$811$			53.3133i			1.87208i	0.351891	+	0.936041i	$0.385539\pi$
$811$			53.3133i			1.87208i	−0.351891	+	0.936041i	$0.614461\pi$
$812$			27.1244i			0.951878i
$813$		−	16.9282i		−	0.593698i
$814$			17.4007i			0.609894i
$815$		0			0
$816$		−	11.7990i		−	0.413046i
$817$	53.6110			1.87561
$818$		−	19.8564i		−	0.694263i
$819$		−	55.5355i		−	1.94057i
$820$		0			0
$821$			2.56606i			0.0895562i	0.998997	+	0.0447781i	$0.0142581\pi$
$821$			2.56606i			0.0895562i	−0.998997	+	0.0447781i	$0.985742\pi$
$822$	−11.3293			−0.395156
$823$	17.4007			0.606551			0.303275	−	0.952903i	$-0.401920\pi$
$823$	17.4007			0.606551			0.303275	+	0.952903i	$0.401920\pi$
$824$		−	37.8371i		−	1.31812i
$825$		0			0
$826$			88.7101i			3.08662i
$827$		−	4.98076i		−	0.173198i	−0.996243	−	0.0865990i	$-0.972400\pi$
$827$		−	4.98076i		−	0.173198i	0.996243	−	0.0865990i	$-0.0275999\pi$
$828$	−98.2868			−3.41570
$829$	6.17691			0.214533			0.107267	−	0.994230i	$-0.465790\pi$
$829$	6.17691			0.214533			0.107267	+	0.994230i	$0.465790\pi$
$830$		0			0
$831$			53.6110i			1.85974i
$832$		−	31.9808i		−	1.10873i
$833$	17.8703			0.619170
$834$	−27.1244			−0.939240
$835$		0			0
$836$			42.2817i			1.46234i
$837$	−12.1427			−0.419715
$838$		−	38.5885i		−	1.33302i
$839$	18.0000			0.621429			0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000			0.621429			0.310715	+	0.950503i	$0.399432\pi$
$840$		0			0
$841$	25.9282			0.894076
$842$		−	27.1244i		−	0.934767i
$843$	−15.6481			−0.538948
$844$		−	73.2340i		−	2.52082i
$845$		0			0
$846$		−	9.92047i		−	0.341073i
$847$	21.8453			0.750612
$848$	3.16152			0.108567
$849$	39.8564			1.36787
$850$		0			0
$851$	−17.9090			−0.613911
$852$	−66.6930			−2.28486
$853$		−	31.5885i		−	1.08157i	−0.841161	−	0.540784i	$-0.818128\pi$
$853$		−	31.5885i		−	1.08157i	0.841161	−	0.540784i	$-0.181872\pi$
$854$	71.4449	+	30.1389i	2.44479	+	1.03133i
$855$		0			0
$856$			64.1269i			2.19181i
$857$		−	1.14359i		−	0.0390644i	−0.999809	−	0.0195322i	$-0.993782\pi$
$857$		−	1.14359i		−	0.0390644i	0.999809	−	0.0195322i	$-0.00621769\pi$
$858$		−	46.9808i		−	1.60390i
$859$	32.9808			1.12529			0.562645	−	0.826699i	$-0.309784\pi$
$859$	32.9808			1.12529			0.562645	+	0.826699i	$0.309784\pi$
$860$		0			0
$861$		−	5.25796i		−	0.179191i
$862$	−86.7856			−2.95593
$863$			23.0718i			0.785373i	0.919672	+	0.392687i	$0.128454\pi$
$863$			23.0718i			0.785373i	−0.919672	+	0.392687i	$0.871546\pi$
$864$		−	9.57668i		−	0.325805i
$865$		0			0
$866$	−59.5692			−2.02424
$867$			38.0526i			1.29233i
$868$			46.9808i			1.59463i
$869$	22.5167			0.763825
$870$		0			0
$871$		−	30.6546i		−	1.03869i
$872$	45.6151			1.54472
$873$		−	15.4641i		−	0.523381i
$874$	−66.8372			−2.26080
$875$		0			0
$876$	−12.1962			−0.412070
$877$	−12.7382			−0.430139			−0.215069	−	0.976599i	$-0.568998\pi$
$877$	−12.7382			−0.430139			−0.215069	+	0.976599i	$0.568998\pi$
$878$	4.31872			0.145750
$879$	−5.07180			−0.171067
$880$		0			0
$881$	43.0526			1.45048			0.725239	−	0.688497i	$-0.241729\pi$
$881$	43.0526			1.45048			0.725239	+	0.688497i	$0.241729\pi$
$882$	−108.975			−3.66937
$883$	8.59138			0.289123			0.144561	−	0.989496i	$-0.453823\pi$
$883$	8.59138			0.289123			0.144561	+	0.989496i	$0.453823\pi$
$884$			19.6230i			0.659992i
$885$		0			0
$886$		−	58.0555i		−	1.95041i
$887$	30.4827			1.02351			0.511754	−	0.859132i	$-0.328996\pi$
$887$	30.4827			1.02351			0.511754	+	0.859132i	$0.328996\pi$
$888$			34.3923i			1.15413i
$889$		−	52.2021i		−	1.75080i
$890$		0			0
$891$			5.89948i			0.197640i
$892$	4.14682			0.138846
$893$		−	4.39230i		−	0.146983i
$894$		−	42.2817i		−	1.41411i
$895$		0			0
$896$	−85.9808			−2.87242
$897$	48.3530			1.61446
$898$	−1.92455			−0.0642230
$899$	−5.32051			−0.177449
$900$		0			0
$901$	2.24871			0.0749154
$902$		−	2.66025i		−	0.0885768i
$903$			128.354i			4.27135i
$904$		−	8.59138i		−	0.285745i
$905$		0			0
$906$	81.3731			2.70344
$907$	−15.1784			−0.503992			−0.251996	−	0.967728i	$-0.581087\pi$
$907$	−15.1784			−0.503992			−0.251996	+	0.967728i	$0.581087\pi$
$908$	−44.6758			−1.48262
$909$			34.9272i			1.15846i
$910$		0			0
$911$	54.8372			1.81684			0.908418	−	0.418063i	$-0.137291\pi$
$911$	54.8372			1.81684			0.908418	+	0.418063i	$0.137291\pi$
$912$			31.8564i			1.05487i
$913$	22.6587			0.749893
$914$	−52.3013			−1.72997
$915$		0			0
$916$	60.7128			2.00601
$917$	48.3530			1.59676
$918$			16.7846i			0.553975i
$919$	26.0000			0.857661			0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000			0.857661			0.428830	+	0.903385i	$0.358926\pi$
$920$		0			0
$921$		−	3.03569i		−	0.100029i
$922$	−20.4364			−0.673037
$923$	19.6230			0.645898
$924$	−101.229			−3.33021
$925$		0			0
$926$			16.5873i			0.545092i
$927$			40.7321i			1.33782i
$928$		−	4.19615i		−	0.137745i
$929$	41.7846			1.37091			0.685454	−	0.728116i	$-0.259604\pi$
$929$	41.7846			1.37091			0.685454	+	0.728116i	$0.259604\pi$
$930$		0			0
$931$	−48.2487			−1.58129
$932$	−47.0700			−1.54183
$933$	85.0329			2.78385
$934$	−58.0333			−1.89891
$935$		0			0
$936$		−	55.5355i		−	1.81524i
$937$			35.1051i			1.14683i	0.819264	+	0.573417i	$0.194383\pi$
$937$			35.1051i			1.14683i	−0.819264	+	0.573417i	$0.805617\pi$
$938$	−101.448			−3.31241
$939$		−	61.9046i		−	2.02018i
$940$		0			0
$941$			1.75265i			0.0571349i	0.999592	+	0.0285674i	$0.00909454\pi$
$941$			1.75265i			0.0571349i	−0.999592	+	0.0285674i	$0.990905\pi$
$942$		−	59.5692i		−	1.94087i
$943$	2.73796			0.0891602
$944$			22.0172i			0.716597i
$945$		0			0
$946$			64.9403i			2.11139i
$947$	22.0172			0.715461			0.357731	−	0.933825i	$-0.383551\pi$
$947$	22.0172			0.715461			0.357731	+	0.933825i	$0.383551\pi$
$948$	95.8926			3.11445
$949$	3.58846			0.116486
$950$		0			0
$951$	−13.8564			−0.449325
$952$	30.1389			0.976808
$953$	25.3506			0.821185			0.410593	−	0.911819i	$-0.365322\pi$
$953$	25.3506			0.821185			0.410593	+	0.911819i	$0.365322\pi$
$954$	−13.7128			−0.443969
$955$		0			0
$956$	−49.5167			−1.60148
$957$		−	11.4641i		−	0.370582i
$958$	89.0079			2.87571
$959$		−	7.18251i		−	0.231935i
$960$		0			0
$961$	21.7846			0.702729
$962$		−	21.8038i		−	0.702984i
$963$		−	69.0333i		−	2.22457i
$964$	71.1051			2.29014
$965$		0			0
$966$		−	160.020i		−	5.14854i
$967$			31.3731i			1.00889i	0.863444	+	0.504445i	$0.168303\pi$
$967$			31.3731i			1.00889i	−0.863444	+	0.504445i	$0.831697\pi$
$968$	21.8453			0.702133
$969$			22.6587i			0.727901i
$970$		0			0
$971$	16.1436			0.518073			0.259036	−	0.965868i	$-0.416595\pi$
$971$	16.1436			0.518073			0.259036	+	0.965868i	$0.416595\pi$
$972$			69.9090i			2.24233i
$973$		−	17.1962i		−	0.551283i
$974$		−	93.3266i		−	2.99038i
$975$		0			0
$976$	17.7321	+	7.48024i	0.567589	+	0.239437i
$977$		−	34.6410i		−	1.10826i	−0.832429	−	0.554132i	$-0.813050\pi$
$977$		−	34.6410i		−	1.10826i	0.832429	−	0.554132i	$-0.186950\pi$
$978$	−98.9283			−3.16338
$979$	−28.2487			−0.902833
$980$		0			0
$981$	−49.1051			−1.56781
$982$	−37.8371			−1.20743
$983$	−20.5622			−0.655833			−0.327917	−	0.944707i	$-0.606347\pi$
$983$	−20.5622			−0.655833			−0.327917	+	0.944707i	$0.606347\pi$
$984$		−	5.25796i		−	0.167618i
$985$		0			0
$986$			7.35440i			0.234212i
$987$	10.5159			0.334726
$988$		−	52.9808i		−	1.68554i
$989$	−66.8372			−2.12530
$990$		0			0
$991$	−1.60770			−0.0510701			−0.0255351	−	0.999674i	$-0.508129\pi$
$991$	−1.60770			−0.0510701			−0.0255351	+	0.999674i	$0.508129\pi$
$992$		−	7.26795i		−	0.230758i
$993$	−83.7499			−2.65772
$994$		−	64.9403i		−	2.05978i
$995$		0			0
$996$	96.4974			3.05764
$997$	−13.5516			−0.429184			−0.214592	−	0.976704i	$-0.568842\pi$
$997$	−13.5516			−0.429184			−0.214592	+	0.976704i	$0.568842\pi$
$998$			0.712813i			0.0225637i
$999$		−	12.1427i		−	0.384179i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1525.2.d.c.1524.7		8
5.2	odd	4		61.2.b.a.60.4	yes	4
5.3	odd	4		1525.2.c.b.426.1		4
5.4	even	2	inner	1525.2.d.c.1524.2		8
15.2	even	4		549.2.c.c.487.1		4
20.7	even	4		976.2.h.d.609.3		4
61.60	even	2	inner	1525.2.d.c.1524.1		8
305.72	even	4		3721.2.a.f.1.1		4
305.172	even	4		3721.2.a.f.1.4		4
305.182	odd	4		61.2.b.a.60.1	✓	4
305.243	odd	4		1525.2.c.b.426.4		4
305.304	even	2	inner	1525.2.d.c.1524.8		8
915.182	even	4		549.2.c.c.487.4		4
1220.487	even	4		976.2.h.d.609.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.b.a.60.1	✓	4	305.182	odd	4
61.2.b.a.60.4	yes	4	5.2	odd	4
549.2.c.c.487.1		4	15.2	even	4
549.2.c.c.487.4		4	915.182	even	4
976.2.h.d.609.3		4	20.7	even	4
976.2.h.d.609.4		4	1220.487	even	4
1525.2.c.b.426.1		4	5.3	odd	4
1525.2.c.b.426.4		4	305.243	odd	4
1525.2.d.c.1524.1		8	61.60	even	2	inner
1525.2.d.c.1524.2		8	5.4	even	2	inner
1525.2.d.c.1524.7		8	1.1	even	1	trivial
1525.2.d.c.1524.8		8	305.304	even	2	inner
3721.2.a.f.1.1		4	305.72	even	4
3721.2.a.f.1.4		4	305.172	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1525 = 5^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.39417 q^{2} -2.73205i q^{3} +3.73205 q^{4} -6.54099i q^{6} +4.14682 q^{7} +4.14682 q^{8} -4.46410 q^{9} -2.39417i q^{11} -10.1962i q^{12} +3.00000i q^{13} +9.92820 q^{14} +2.46410 q^{16} +1.75265 q^{17} -10.6878 q^{18} -4.73205 q^{19} -11.3293i q^{21} -5.73205i q^{22} +5.89948 q^{23} -11.3293i q^{24} +7.18251i q^{26} +4.00000i q^{27} +15.4762 q^{28} +1.75265i q^{29} +3.03569i q^{31} -2.39417 q^{32} -6.54099 q^{33} +4.19615 q^{34} -16.6603 q^{36} -3.03569 q^{37} -11.3293 q^{38} +8.19615 q^{39} +0.464102 q^{41} -27.1244i q^{42} -11.3293 q^{43} -8.93516i q^{44} +14.1244 q^{46} +0.928203i q^{47} -6.73205i q^{48} +10.1962 q^{49} -4.78834i q^{51} +11.1962i q^{52} +1.28303 q^{53} +9.57668i q^{54} +17.1962 q^{56} +12.9282i q^{57} +4.19615i q^{58} +8.93516i q^{59} +(7.19615 + 3.03569i) q^{61} +7.26795i q^{62} -18.5118 q^{63} -10.6603 q^{64} -15.6603 q^{66} -10.2182 q^{67} +6.54099 q^{68} -16.1177i q^{69} -6.54099i q^{71} -18.5118 q^{72} -1.19615i q^{73} -7.26795 q^{74} -17.6603 q^{76} -9.92820i q^{77} +19.6230 q^{78} +9.40479i q^{79} -2.46410 q^{81} +1.11114 q^{82} +9.46410i q^{83} -42.2817i q^{84} -27.1244 q^{86} +4.78834 q^{87} -9.92820i q^{88} -11.7990i q^{89} +12.4405i q^{91} +22.0172 q^{92} +8.29365 q^{93} +2.22228i q^{94} +6.54099i q^{96} +3.46410i q^{97} +24.4113 q^{98} +10.6878i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 8 q^{9} + 24 q^{14} - 8 q^{16} - 24 q^{19} - 8 q^{34} - 64 q^{36} + 24 q^{39} - 24 q^{41} + 16 q^{46} + 40 q^{49} + 96 q^{56} + 16 q^{61} - 16 q^{64} - 56 q^{66} - 72 q^{74} - 72 q^{76}+ \cdots - 120 q^{86}+O(q^{100}) \) 8 * q + 16 * q^4 - 8 * q^9 + 24 * q^14 - 8 * q^16 - 24 * q^19 - 8 * q^34 - 64 * q^36 + 24 * q^39 - 24 * q^41 + 16 * q^46 + 40 * q^49 + 96 * q^56 + 16 * q^61 - 16 * q^64 - 56 * q^66 - 72 * q^74 - 72 * q^76 + 8 * q^81 - 120 * q^86

Introduction

L-functions

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