LMFDB - Embedded newform 135.2.b.b.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(109,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("135.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$135 = 3^{3} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	135.b (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:	$1.07798042729$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$3^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	135.109
Dual form			135.2.b.b.109.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +(-2.12132 - 0.707107i) q^{5} -3.00000i q^{7} -2.82843i q^{8} +(-1.00000 + 3.00000i) q^{10} +4.24264 q^{11} +3.00000i q^{13} -4.24264 q^{14} -4.00000 q^{16} +2.82843i q^{17} -1.00000 q^{19} -6.00000i q^{22} +7.07107i q^{23} +(4.00000 + 3.00000i) q^{25} +4.24264 q^{26} +4.24264 q^{29} +2.00000 q^{31} +4.00000 q^{34} +(-2.12132 + 6.36396i) q^{35} -9.00000i q^{37} +1.41421i q^{38} +(-2.00000 + 6.00000i) q^{40} -4.24264 q^{41} +6.00000i q^{43} +10.0000 q^{46} +2.82843i q^{47} -2.00000 q^{49} +(4.24264 - 5.65685i) q^{50} -9.89949i q^{53} +(-9.00000 - 3.00000i) q^{55} -8.48528 q^{56} -6.00000i q^{58} +8.48528 q^{59} -13.0000 q^{61} -2.82843i q^{62} -8.00000 q^{64} +(2.12132 - 6.36396i) q^{65} +3.00000i q^{67} +(9.00000 + 3.00000i) q^{70} -12.7279 q^{71} +9.00000i q^{73} -12.7279 q^{74} -12.7279i q^{77} +5.00000 q^{79} +(8.48528 + 2.82843i) q^{80} +6.00000i q^{82} -1.41421i q^{83} +(2.00000 - 6.00000i) q^{85} +8.48528 q^{86} -12.0000i q^{88} +9.00000 q^{91} +4.00000 q^{94} +(2.12132 + 0.707107i) q^{95} -3.00000i q^{97} +2.82843i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{10} - 16 q^{16} - 4 q^{19} + 16 q^{25} + 8 q^{31} + 16 q^{34} - 8 q^{40} + 40 q^{46} - 8 q^{49} - 36 q^{55} - 52 q^{61} - 32 q^{64} + 36 q^{70} + 20 q^{79} + 8 q^{85} + 36 q^{91} + 16 q^{94}+O(q^{100})$ 4 * q - 4 * q^10 - 16 * q^16 - 4 * q^19 + 16 * q^25 + 8 * q^31 + 16 * q^34 - 8 * q^40 + 40 * q^46 - 8 * q^49 - 36 * q^55 - 52 * q^61 - 32 * q^64 + 36 * q^70 + 20 * q^79 + 8 * q^85 + 36 * q^91 + 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$ .

$n$	$56$	$82$
$\chi(n)$	$1$	$-1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		−	1.41421i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$		−	1.41421i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$5$	−2.12132	−	0.707107i	−0.948683	−	0.316228i
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$		−	2.82843i		−	1.00000i
$9$		0			0
$10$	−1.00000	+	3.00000i	−0.316228	+	0.948683i
$11$	4.24264			1.27920			0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264			1.27920			0.639602	+	0.768706i	$0.279099\pi$
$12$		0			0
$13$			3.00000i			0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$			3.00000i			0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	−4.24264			−1.13389
$15$		0			0
$16$	−4.00000			−1.00000
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	−1.00000			−0.229416			−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000			−0.229416			−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		−	6.00000i		−	1.27920i
$23$			7.07107i			1.47442i	0.675664	+	0.737210i	$0.263857\pi$
$23$			7.07107i			1.47442i	−0.675664	+	0.737210i	$0.736143\pi$
$24$		0			0
$25$	4.00000	+	3.00000i	0.800000	+	0.600000i
$26$	4.24264			0.832050
$27$		0			0
$28$		0			0
$29$	4.24264			0.787839			0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264			0.787839			0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$		0			0
$34$	4.00000			0.685994
$35$	−2.12132	+	6.36396i	−0.358569	+	1.07571i
$36$		0			0
$37$		−	9.00000i		−	1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$		−	9.00000i		−	1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$			1.41421i			0.229416i
$39$		0			0
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$	−4.24264			−0.662589			−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264			−0.662589			−0.331295	+	0.943527i	$0.607485\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$		0			0
$46$	10.0000			1.47442
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$	4.24264	−	5.65685i	0.600000	−	0.800000i
$51$		0			0
$52$		0			0
$53$		−	9.89949i		−	1.35980i	−0.733305	−	0.679900i	$-0.762023\pi$
$53$		−	9.89949i		−	1.35980i	0.733305	−	0.679900i	$-0.237977\pi$
$54$		0			0
$55$	−9.00000	−	3.00000i	−1.21356	−	0.404520i
$56$	−8.48528			−1.13389
$57$		0			0
$58$		−	6.00000i		−	0.787839i
$59$	8.48528			1.10469			0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528			1.10469			0.552345	+	0.833616i	$0.313733\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$		−	2.82843i		−	0.359211i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.12132	−	6.36396i	0.263117	−	0.789352i
$66$		0			0
$67$			3.00000i			0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$			3.00000i			0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$		0			0
$69$		0			0
$70$	9.00000	+	3.00000i	1.07571	+	0.358569i
$71$	−12.7279			−1.51053			−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279			−1.51053			−0.755263	+	0.655422i	$0.772491\pi$
$72$		0			0
$73$			9.00000i			1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$			9.00000i			1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	−12.7279			−1.47959
$75$		0			0
$76$		0			0
$77$		−	12.7279i		−	1.45048i
$78$		0			0
$79$	5.00000			0.562544			0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000			0.562544			0.281272	+	0.959628i	$0.409244\pi$
$80$	8.48528	+	2.82843i	0.948683	+	0.316228i
$81$		0			0
$82$			6.00000i			0.662589i
$83$		−	1.41421i		−	0.155230i	−0.996983	−	0.0776151i	$-0.975269\pi$
$83$		−	1.41421i		−	0.155230i	0.996983	−	0.0776151i	$-0.0247305\pi$
$84$		0			0
$85$	2.00000	−	6.00000i	0.216930	−	0.650791i
$86$	8.48528			0.914991
$87$		0			0
$88$		−	12.0000i		−	1.27920i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	9.00000			0.943456
$92$		0			0
$93$		0			0
$94$	4.00000			0.412568
$95$	2.12132	+	0.707107i	0.217643	+	0.0725476i
$96$		0			0
$97$		−	3.00000i		−	0.304604i	−0.988334	−	0.152302i	$-0.951331\pi$
$97$		−	3.00000i		−	0.304604i	0.988334	−	0.152302i	$-0.0486686\pi$
$98$			2.82843i			0.285714i
$99$		0			0
$100$		0			0
$101$	−8.48528			−0.844317			−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528			−0.844317			−0.422159	+	0.906522i	$0.638727\pi$
$102$		0			0
$103$			3.00000i			0.295599i	0.989017	+	0.147799i	$0.0472190\pi$
$103$			3.00000i			0.295599i	−0.989017	+	0.147799i	$0.952781\pi$
$104$	8.48528			0.832050
$105$		0			0
$106$	−14.0000			−1.35980
$107$			7.07107i			0.683586i	0.939775	+	0.341793i	$0.111034\pi$
$107$			7.07107i			0.683586i	−0.939775	+	0.341793i	$0.888966\pi$
$108$		0			0
$109$	8.00000			0.766261			0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000			0.766261			0.383131	+	0.923694i	$0.374846\pi$
$110$	−4.24264	+	12.7279i	−0.404520	+	1.21356i
$111$		0			0
$112$			12.0000i			1.13389i
$113$		−	1.41421i		−	0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$		−	1.41421i		−	0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$		0			0
$115$	5.00000	−	15.0000i	0.466252	−	1.39876i
$116$		0			0
$117$		0			0
$118$		−	12.0000i		−	1.10469i
$119$	8.48528			0.777844
$120$		0			0
$121$	7.00000			0.636364
$122$			18.3848i			1.66448i
$123$		0			0
$124$		0			0
$125$	−6.36396	−	9.19239i	−0.569210	−	0.822192i
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$	−9.00000	−	3.00000i	−0.789352	−	0.263117i
$131$	8.48528			0.741362			0.370681	−	0.928760i	$-0.379124\pi$
$131$	8.48528			0.741362			0.370681	+	0.928760i	$0.379124\pi$
$132$		0			0
$133$			3.00000i			0.260133i
$134$	4.24264			0.366508
$135$		0			0
$136$	8.00000			0.685994
$137$		−	1.41421i		−	0.120824i	−0.998174	−	0.0604122i	$-0.980758\pi$
$137$		−	1.41421i		−	0.120824i	0.998174	−	0.0604122i	$-0.0192415\pi$
$138$		0			0
$139$	−13.0000			−1.10265			−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000			−1.10265			−0.551323	+	0.834292i	$0.685877\pi$
$140$		0			0
$141$		0			0
$142$			18.0000i			1.51053i
$143$			12.7279i			1.06436i
$144$		0			0
$145$	−9.00000	−	3.00000i	−0.747409	−	0.249136i
$146$	12.7279			1.05337
$147$		0			0
$148$		0			0
$149$	−16.9706			−1.39028			−0.695141	−	0.718873i	$-0.744658\pi$
$149$	−16.9706			−1.39028			−0.695141	+	0.718873i	$0.744658\pi$
$150$		0			0
$151$	−1.00000			−0.0813788			−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000			−0.0813788			−0.0406894	+	0.999172i	$0.512955\pi$
$152$			2.82843i			0.229416i
$153$		0			0
$154$	−18.0000			−1.45048
$155$	−4.24264	−	1.41421i	−0.340777	−	0.113592i
$156$		0			0
$157$		−	6.00000i		−	0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$		−	6.00000i		−	0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$		−	7.07107i		−	0.562544i
$159$		0			0
$160$		0			0
$161$	21.2132			1.67183
$162$		0			0
$163$		−	9.00000i		−	0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$		−	9.00000i		−	0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$		0			0
$165$		0			0
$166$	−2.00000			−0.155230
$167$		−	5.65685i		−	0.437741i	−0.975754	−	0.218870i	$-0.929763\pi$
$167$		−	5.65685i		−	0.437741i	0.975754	−	0.218870i	$-0.0702371\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$	−8.48528	−	2.82843i	−0.650791	−	0.216930i
$171$		0			0
$172$		0			0
$173$		−	14.1421i		−	1.07521i	−0.843198	−	0.537603i	$-0.819330\pi$
$173$		−	14.1421i		−	1.07521i	0.843198	−	0.537603i	$-0.180670\pi$
$174$		0			0
$175$	9.00000	−	12.0000i	0.680336	−	0.907115i
$176$	−16.9706			−1.27920
$177$		0			0
$178$		0			0
$179$	−12.7279			−0.951330			−0.475665	−	0.879627i	$-0.657792\pi$
$179$	−12.7279			−0.951330			−0.475665	+	0.879627i	$0.657792\pi$
$180$		0			0
$181$	−1.00000			−0.0743294			−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000			−0.0743294			−0.0371647	+	0.999309i	$0.511833\pi$
$182$		−	12.7279i		−	0.943456i
$183$		0			0
$184$	20.0000			1.47442
$185$	−6.36396	+	19.0919i	−0.467888	+	1.40366i
$186$		0			0
$187$			12.0000i			0.877527i
$188$		0			0
$189$		0			0
$190$	1.00000	−	3.00000i	0.0725476	−	0.217643i
$191$	16.9706			1.22795			0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706			1.22795			0.613973	+	0.789327i	$0.289570\pi$
$192$		0			0
$193$			3.00000i			0.215945i	0.994154	+	0.107972i	$0.0344358\pi$
$193$			3.00000i			0.215945i	−0.994154	+	0.107972i	$0.965564\pi$
$194$	−4.24264			−0.304604
$195$		0			0
$196$		0			0
$197$		−	22.6274i		−	1.61214i	−0.591822	−	0.806068i	$-0.701591\pi$
$197$		−	22.6274i		−	1.61214i	0.591822	−	0.806068i	$-0.298409\pi$
$198$		0			0
$199$	11.0000			0.779769			0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000			0.779769			0.389885	+	0.920864i	$0.372515\pi$
$200$	8.48528	−	11.3137i	0.600000	−	0.800000i
$201$		0			0
$202$			12.0000i			0.844317i
$203$		−	12.7279i		−	0.893325i
$204$		0			0
$205$	9.00000	+	3.00000i	0.628587	+	0.209529i
$206$	4.24264			0.295599
$207$		0			0
$208$		−	12.0000i		−	0.832050i
$209$	−4.24264			−0.293470
$210$		0			0
$211$	−7.00000			−0.481900			−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000			−0.481900			−0.240950	+	0.970538i	$0.577459\pi$
$212$		0			0
$213$		0			0
$214$	10.0000			0.683586
$215$	4.24264	−	12.7279i	0.289346	−	0.868037i
$216$		0			0
$217$		−	6.00000i		−	0.407307i
$218$		−	11.3137i		−	0.766261i
$219$		0			0
$220$		0			0
$221$	−8.48528			−0.570782
$222$		0			0
$223$			6.00000i			0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$			6.00000i			0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$		0			0
$225$		0			0
$226$	−2.00000			−0.133038
$227$			24.0416i			1.59570i	0.602857	+	0.797850i	$0.294029\pi$
$227$			24.0416i			1.59570i	−0.602857	+	0.797850i	$0.705971\pi$
$228$		0			0
$229$	−16.0000			−1.05731			−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000			−1.05731			−0.528655	+	0.848837i	$0.677303\pi$
$230$	−21.2132	−	7.07107i	−1.39876	−	0.466252i
$231$		0			0
$232$		−	12.0000i		−	0.787839i
$233$			19.7990i			1.29707i	0.761183	+	0.648537i	$0.224619\pi$
$233$			19.7990i			1.29707i	−0.761183	+	0.648537i	$0.775381\pi$
$234$		0			0
$235$	2.00000	−	6.00000i	0.130466	−	0.391397i
$236$		0			0
$237$		0			0
$238$		−	12.0000i		−	0.777844i
$239$	−4.24264			−0.274434			−0.137217	−	0.990541i	$-0.543816\pi$
$239$	−4.24264			−0.274434			−0.137217	+	0.990541i	$0.543816\pi$
$240$		0			0
$241$	5.00000			0.322078			0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000			0.322078			0.161039	+	0.986948i	$0.448515\pi$
$242$		−	9.89949i		−	0.636364i
$243$		0			0
$244$		0			0
$245$	4.24264	+	1.41421i	0.271052	+	0.0903508i
$246$		0			0
$247$		−	3.00000i		−	0.190885i
$248$		−	5.65685i		−	0.359211i
$249$		0			0
$250$	−13.0000	+	9.00000i	−0.822192	+	0.569210i
$251$	−12.7279			−0.803379			−0.401690	−	0.915776i	$-0.631577\pi$
$251$	−12.7279			−0.803379			−0.401690	+	0.915776i	$0.631577\pi$
$252$		0			0
$253$			30.0000i			1.88608i
$254$	25.4558			1.59724
$255$		0			0
$256$		0			0
$257$			19.7990i			1.23503i	0.786560	+	0.617514i	$0.211860\pi$
$257$			19.7990i			1.23503i	−0.786560	+	0.617514i	$0.788140\pi$
$258$		0			0
$259$	−27.0000			−1.67770
$260$		0			0
$261$		0			0
$262$		−	12.0000i		−	0.741362i
$263$			2.82843i			0.174408i	0.996190	+	0.0872041i	$0.0277932\pi$
$263$			2.82843i			0.174408i	−0.996190	+	0.0872041i	$0.972207\pi$
$264$		0			0
$265$	−7.00000	+	21.0000i	−0.430007	+	1.29002i
$266$	4.24264			0.260133
$267$		0			0
$268$		0			0
$269$	25.4558			1.55207			0.776035	−	0.630690i	$-0.217228\pi$
$269$	25.4558			1.55207			0.776035	+	0.630690i	$0.217228\pi$
$270$		0			0
$271$	−19.0000			−1.15417			−0.577084	−	0.816685i	$-0.695809\pi$
$271$	−19.0000			−1.15417			−0.577084	+	0.816685i	$0.695809\pi$
$272$		−	11.3137i		−	0.685994i
$273$		0			0
$274$	−2.00000			−0.120824
$275$	16.9706	+	12.7279i	1.02336	+	0.767523i
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$			18.3848i			1.10265i
$279$		0			0
$280$	18.0000	+	6.00000i	1.07571	+	0.358569i
$281$	−8.48528			−0.506189			−0.253095	−	0.967442i	$-0.581448\pi$
$281$	−8.48528			−0.506189			−0.253095	+	0.967442i	$0.581448\pi$
$282$		0			0
$283$		−	6.00000i		−	0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$		−	6.00000i		−	0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$		0			0
$285$		0			0
$286$	18.0000			1.06436
$287$			12.7279i			0.751305i
$288$		0			0
$289$	9.00000			0.529412
$290$	−4.24264	+	12.7279i	−0.249136	+	0.747409i
$291$		0			0
$292$		0			0
$293$			7.07107i			0.413096i	0.978436	+	0.206548i	$0.0662230\pi$
$293$			7.07107i			0.413096i	−0.978436	+	0.206548i	$0.933777\pi$
$294$		0			0
$295$	−18.0000	−	6.00000i	−1.04800	−	0.349334i
$296$	−25.4558			−1.47959
$297$		0			0
$298$			24.0000i			1.39028i
$299$	−21.2132			−1.22679
$300$		0			0
$301$	18.0000			1.03750
$302$			1.41421i			0.0813788i
$303$		0			0
$304$	4.00000			0.229416
$305$	27.5772	+	9.19239i	1.57906	+	0.526355i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−2.00000	+	6.00000i	−0.113592	+	0.340777i
$311$	8.48528			0.481156			0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528			0.481156			0.240578	+	0.970630i	$0.422663\pi$
$312$		0			0
$313$		−	21.0000i		−	1.18699i	−0.804838	−	0.593495i	$-0.797748\pi$
$313$		−	21.0000i		−	1.18699i	0.804838	−	0.593495i	$-0.202252\pi$
$314$	−8.48528			−0.478852
$315$		0			0
$316$		0			0
$317$			11.3137i			0.635441i	0.948184	+	0.317721i	$0.102917\pi$
$317$			11.3137i			0.635441i	−0.948184	+	0.317721i	$0.897083\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$	16.9706	+	5.65685i	0.948683	+	0.316228i
$321$		0			0
$322$		−	30.0000i		−	1.67183i
$323$		−	2.82843i		−	0.157378i
$324$		0			0
$325$	−9.00000	+	12.0000i	−0.499230	+	0.665640i
$326$	−12.7279			−0.704934
$327$		0			0
$328$			12.0000i			0.662589i
$329$	8.48528			0.467809
$330$		0			0
$331$	17.0000			0.934405			0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000			0.934405			0.467202	+	0.884150i	$0.345262\pi$
$332$		0			0
$333$		0			0
$334$	−8.00000			−0.437741
$335$	2.12132	−	6.36396i	0.115900	−	0.347700i
$336$		0			0
$337$		−	33.0000i		−	1.79762i	−0.438334	−	0.898812i	$-0.644431\pi$
$337$		−	33.0000i		−	1.79762i	0.438334	−	0.898812i	$-0.355569\pi$
$338$		−	5.65685i		−	0.307692i
$339$		0			0
$340$		0			0
$341$	8.48528			0.459504
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$	16.9706			0.914991
$345$		0			0
$346$	−20.0000			−1.07521
$347$		−	26.8701i		−	1.44246i	−0.692696	−	0.721230i	$-0.743577\pi$
$347$		−	26.8701i		−	1.44246i	0.692696	−	0.721230i	$-0.256423\pi$
$348$		0			0
$349$	5.00000			0.267644			0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000			0.267644			0.133822	+	0.991005i	$0.457275\pi$
$350$	−16.9706	−	12.7279i	−0.907115	−	0.680336i
$351$		0			0
$352$		0			0
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$		0			0
$355$	27.0000	+	9.00000i	1.43301	+	0.477670i
$356$		0			0
$357$		0			0
$358$			18.0000i			0.951330i
$359$	−12.7279			−0.671754			−0.335877	−	0.941906i	$-0.609033\pi$
$359$	−12.7279			−0.671754			−0.335877	+	0.941906i	$0.609033\pi$
$360$		0			0
$361$	−18.0000			−0.947368
$362$			1.41421i			0.0743294i
$363$		0			0
$364$		0			0
$365$	6.36396	−	19.0919i	0.333105	−	0.999315i
$366$		0			0
$367$			15.0000i			0.782994i	0.920179	+	0.391497i	$0.128043\pi$
$367$			15.0000i			0.782994i	−0.920179	+	0.391497i	$0.871957\pi$
$368$		−	28.2843i		−	1.47442i
$369$		0			0
$370$	27.0000	+	9.00000i	1.40366	+	0.467888i
$371$	−29.6985			−1.54187
$372$		0			0
$373$			21.0000i			1.08734i	0.839299	+	0.543669i	$0.182965\pi$
$373$			21.0000i			1.08734i	−0.839299	+	0.543669i	$0.817035\pi$
$374$	16.9706			0.877527
$375$		0			0
$376$	8.00000			0.412568
$377$			12.7279i			0.655521i
$378$		0			0
$379$	35.0000			1.79783			0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000			1.79783			0.898915	+	0.438124i	$0.144357\pi$
$380$		0			0
$381$		0			0
$382$		−	24.0000i		−	1.22795i
$383$		−	5.65685i		−	0.289052i	−0.989501	−	0.144526i	$-0.953834\pi$
$383$		−	5.65685i		−	0.289052i	0.989501	−	0.144526i	$-0.0461657\pi$
$384$		0			0
$385$	−9.00000	+	27.0000i	−0.458682	+	1.37605i
$386$	4.24264			0.215945
$387$		0			0
$388$		0			0
$389$	−33.9411			−1.72088			−0.860442	−	0.509549i	$-0.829812\pi$
$389$	−33.9411			−1.72088			−0.860442	+	0.509549i	$0.829812\pi$
$390$		0			0
$391$	−20.0000			−1.01144
$392$			5.65685i			0.285714i
$393$		0			0
$394$	−32.0000			−1.61214
$395$	−10.6066	−	3.53553i	−0.533676	−	0.177892i
$396$		0			0
$397$		−	36.0000i		−	1.80679i	−0.428811	−	0.903394i	$-0.641067\pi$
$397$		−	36.0000i		−	1.80679i	0.428811	−	0.903394i	$-0.358933\pi$
$398$		−	15.5563i		−	0.779769i
$399$		0			0
$400$	−16.0000	−	12.0000i	−0.800000	−	0.600000i
$401$	−4.24264			−0.211867			−0.105934	−	0.994373i	$-0.533783\pi$
$401$	−4.24264			−0.211867			−0.105934	+	0.994373i	$0.533783\pi$
$402$		0			0
$403$			6.00000i			0.298881i
$404$		0			0
$405$		0			0
$406$	−18.0000			−0.893325
$407$		−	38.1838i		−	1.89270i
$408$		0			0
$409$	−25.0000			−1.23617			−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000			−1.23617			−0.618085	+	0.786111i	$0.712091\pi$
$410$	4.24264	−	12.7279i	0.209529	−	0.628587i
$411$		0			0
$412$		0			0
$413$		−	25.4558i		−	1.25260i
$414$		0			0
$415$	−1.00000	+	3.00000i	−0.0490881	+	0.147264i
$416$		0			0
$417$		0			0
$418$			6.00000i			0.293470i
$419$	33.9411			1.65813			0.829066	−	0.559150i	$-0.188873\pi$
$419$	33.9411			1.65813			0.829066	+	0.559150i	$0.188873\pi$
$420$		0			0
$421$	23.0000			1.12095			0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000			1.12095			0.560476	+	0.828171i	$0.310618\pi$
$422$			9.89949i			0.481900i
$423$		0			0
$424$	−28.0000			−1.35980
$425$	−8.48528	+	11.3137i	−0.411597	+	0.548795i
$426$		0			0
$427$			39.0000i			1.88734i
$428$		0			0
$429$		0			0
$430$	−18.0000	−	6.00000i	−0.868037	−	0.289346i
$431$	12.7279			0.613082			0.306541	−	0.951857i	$-0.400828\pi$
$431$	12.7279			0.613082			0.306541	+	0.951857i	$0.400828\pi$
$432$		0			0
$433$		−	36.0000i		−	1.73005i	−0.501729	−	0.865025i	$-0.667303\pi$
$433$		−	36.0000i		−	1.73005i	0.501729	−	0.865025i	$-0.332697\pi$
$434$	−8.48528			−0.407307
$435$		0			0
$436$		0			0
$437$		−	7.07107i		−	0.338255i
$438$		0			0
$439$	14.0000			0.668184			0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000			0.668184			0.334092	+	0.942541i	$0.391570\pi$
$440$	−8.48528	+	25.4558i	−0.404520	+	1.21356i
$441$		0			0
$442$			12.0000i			0.570782i
$443$		−	14.1421i		−	0.671913i	−0.941877	−	0.335957i	$-0.890940\pi$
$443$		−	14.1421i		−	0.671913i	0.941877	−	0.335957i	$-0.109060\pi$
$444$		0			0
$445$		0			0
$446$	8.48528			0.401790
$447$		0			0
$448$			24.0000i			1.13389i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	−18.0000			−0.847587
$452$		0			0
$453$		0			0
$454$	34.0000			1.59570
$455$	−19.0919	−	6.36396i	−0.895041	−	0.298347i
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$			22.6274i			1.05731i
$459$		0			0
$460$		0			0
$461$	16.9706			0.790398			0.395199	−	0.918596i	$-0.370676\pi$
$461$	16.9706			0.790398			0.395199	+	0.918596i	$0.370676\pi$
$462$		0			0
$463$		−	15.0000i		−	0.697109i	−0.937288	−	0.348555i	$-0.886673\pi$
$463$		−	15.0000i		−	0.697109i	0.937288	−	0.348555i	$-0.113327\pi$
$464$	−16.9706			−0.787839
$465$		0			0
$466$	28.0000			1.29707
$467$		−	22.6274i		−	1.04707i	−0.852004	−	0.523536i	$-0.824613\pi$
$467$		−	22.6274i		−	1.04707i	0.852004	−	0.523536i	$-0.175387\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$	−8.48528	−	2.82843i	−0.391397	−	0.130466i
$471$		0			0
$472$		−	24.0000i		−	1.10469i
$473$			25.4558i			1.17046i
$474$		0			0
$475$	−4.00000	−	3.00000i	−0.183533	−	0.137649i
$476$		0			0
$477$		0			0
$478$			6.00000i			0.274434i
$479$	4.24264			0.193851			0.0969256	−	0.995292i	$-0.469099\pi$
$479$	4.24264			0.193851			0.0969256	+	0.995292i	$0.469099\pi$
$480$		0			0
$481$	27.0000			1.23109
$482$		−	7.07107i		−	0.322078i
$483$		0			0
$484$		0			0
$485$	−2.12132	+	6.36396i	−0.0963242	+	0.288973i
$486$		0			0
$487$			9.00000i			0.407829i	0.978989	+	0.203914i	$0.0653664\pi$
$487$			9.00000i			0.407829i	−0.978989	+	0.203914i	$0.934634\pi$
$488$			36.7696i			1.66448i
$489$		0			0
$490$	2.00000	−	6.00000i	0.0903508	−	0.271052i
$491$	21.2132			0.957338			0.478669	−	0.877995i	$-0.341119\pi$
$491$	21.2132			0.957338			0.478669	+	0.877995i	$0.341119\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$	−4.24264			−0.190885
$495$		0			0
$496$	−8.00000			−0.359211
$497$			38.1838i			1.71278i
$498$		0			0
$499$	−16.0000			−0.716258			−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000			−0.716258			−0.358129	+	0.933672i	$0.616585\pi$
$500$		0			0
$501$		0			0
$502$			18.0000i			0.803379i
$503$		−	35.3553i		−	1.57642i	−0.615409	−	0.788208i	$-0.711009\pi$
$503$		−	35.3553i		−	1.57642i	0.615409	−	0.788208i	$-0.288991\pi$
$504$		0			0
$505$	18.0000	+	6.00000i	0.800989	+	0.266996i
$506$	42.4264			1.88608
$507$		0			0
$508$		0			0
$509$	−29.6985			−1.31636			−0.658181	−	0.752860i	$-0.728674\pi$
$509$	−29.6985			−1.31636			−0.658181	+	0.752860i	$0.728674\pi$
$510$		0			0
$511$	27.0000			1.19441
$512$			22.6274i			1.00000i
$513$		0			0
$514$	28.0000			1.23503
$515$	2.12132	−	6.36396i	0.0934765	−	0.280430i
$516$		0			0
$517$			12.0000i			0.527759i
$518$			38.1838i			1.67770i
$519$		0			0
$520$	−18.0000	−	6.00000i	−0.789352	−	0.263117i
$521$	12.7279			0.557620			0.278810	−	0.960346i	$-0.410060\pi$
$521$	12.7279			0.557620			0.278810	+	0.960346i	$0.410060\pi$
$522$		0			0
$523$		−	9.00000i		−	0.393543i	−0.980449	−	0.196771i	$-0.936954\pi$
$523$		−	9.00000i		−	0.393543i	0.980449	−	0.196771i	$-0.0630456\pi$
$524$		0			0
$525$		0			0
$526$	4.00000			0.174408
$527$			5.65685i			0.246416i
$528$		0			0
$529$	−27.0000			−1.17391
$530$	29.6985	+	9.89949i	1.29002	+	0.430007i
$531$		0			0
$532$		0			0
$533$		−	12.7279i		−	0.551308i
$534$		0			0
$535$	5.00000	−	15.0000i	0.216169	−	0.648507i
$536$	8.48528			0.366508
$537$		0			0
$538$		−	36.0000i		−	1.55207i
$539$	−8.48528			−0.365487
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$			26.8701i			1.15417i
$543$		0			0
$544$		0			0
$545$	−16.9706	−	5.65685i	−0.726939	−	0.242313i
$546$		0			0
$547$		−	3.00000i		−	0.128271i	−0.997941	−	0.0641354i	$-0.979571\pi$
$547$		−	3.00000i		−	0.128271i	0.997941	−	0.0641354i	$-0.0204289\pi$
$548$		0			0
$549$		0			0
$550$	18.0000	−	24.0000i	0.767523	−	1.02336i
$551$	−4.24264			−0.180743
$552$		0			0
$553$		−	15.0000i		−	0.637865i
$554$	−16.9706			−0.721010
$555$		0			0
$556$		0			0
$557$			19.7990i			0.838910i	0.907776	+	0.419455i	$0.137779\pi$
$557$			19.7990i			0.838910i	−0.907776	+	0.419455i	$0.862221\pi$
$558$		0			0
$559$	−18.0000			−0.761319
$560$	8.48528	−	25.4558i	0.358569	−	1.07571i
$561$		0			0
$562$			12.0000i			0.506189i
$563$			11.3137i			0.476816i	0.971165	+	0.238408i	$0.0766255\pi$
$563$			11.3137i			0.476816i	−0.971165	+	0.238408i	$0.923374\pi$
$564$		0			0
$565$	−1.00000	+	3.00000i	−0.0420703	+	0.126211i
$566$	−8.48528			−0.356663
$567$		0			0
$568$			36.0000i			1.51053i
$569$	42.4264			1.77861			0.889304	−	0.457317i	$-0.151190\pi$
$569$	42.4264			1.77861			0.889304	+	0.457317i	$0.151190\pi$
$570$		0			0
$571$	−25.0000			−1.04622			−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000			−1.04622			−0.523109	+	0.852266i	$0.675228\pi$
$572$		0			0
$573$		0			0
$574$	18.0000			0.751305
$575$	−21.2132	+	28.2843i	−0.884652	+	1.17954i
$576$		0			0
$577$			45.0000i			1.87337i	0.350167	+	0.936687i	$0.386125\pi$
$577$			45.0000i			1.87337i	−0.350167	+	0.936687i	$0.613875\pi$
$578$		−	12.7279i		−	0.529412i
$579$		0			0
$580$		0			0
$581$	−4.24264			−0.176014
$582$		0			0
$583$		−	42.0000i		−	1.73946i
$584$	25.4558			1.05337
$585$		0			0
$586$	10.0000			0.413096
$587$		−	26.8701i		−	1.10905i	−0.832168	−	0.554523i	$-0.812901\pi$
$587$		−	26.8701i		−	1.10905i	0.832168	−	0.554523i	$-0.187099\pi$
$588$		0			0
$589$	−2.00000			−0.0824086
$590$	−8.48528	+	25.4558i	−0.349334	+	1.04800i
$591$		0			0
$592$			36.0000i			1.47959i
$593$		−	9.89949i		−	0.406524i	−0.979124	−	0.203262i	$-0.934846\pi$
$593$		−	9.89949i		−	0.406524i	0.979124	−	0.203262i	$-0.0651542\pi$
$594$		0			0
$595$	−18.0000	−	6.00000i	−0.737928	−	0.245976i
$596$		0			0
$597$		0			0
$598$			30.0000i			1.22679i
$599$	−29.6985			−1.21345			−0.606724	−	0.794913i	$-0.707517\pi$
$599$	−29.6985			−1.21345			−0.606724	+	0.794913i	$0.707517\pi$
$600$		0			0
$601$	−40.0000			−1.63163			−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000			−1.63163			−0.815817	+	0.578310i	$0.803712\pi$
$602$		−	25.4558i		−	1.03750i
$603$		0			0
$604$		0			0
$605$	−14.8492	−	4.94975i	−0.603708	−	0.201236i
$606$		0			0
$607$			39.0000i			1.58296i	0.611194	+	0.791481i	$0.290689\pi$
$607$			39.0000i			1.58296i	−0.611194	+	0.791481i	$0.709311\pi$
$608$		0			0
$609$		0			0
$610$	13.0000	−	39.0000i	0.526355	−	1.57906i
$611$	−8.48528			−0.343278
$612$		0			0
$613$			9.00000i			0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$			9.00000i			0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$		0			0
$615$		0			0
$616$	−36.0000			−1.45048
$617$			7.07107i			0.284670i	0.989819	+	0.142335i	$0.0454611\pi$
$617$			7.07107i			0.284670i	−0.989819	+	0.142335i	$0.954539\pi$
$618$		0			0
$619$	−19.0000			−0.763674			−0.381837	−	0.924230i	$-0.624709\pi$
$619$	−19.0000			−0.763674			−0.381837	+	0.924230i	$0.624709\pi$
$620$		0			0
$621$		0			0
$622$		−	12.0000i		−	0.481156i
$623$		0			0
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$	−29.6985			−1.18699
$627$		0			0
$628$		0			0
$629$	25.4558			1.01499
$630$		0			0
$631$	17.0000			0.676759			0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000			0.676759			0.338380	+	0.941010i	$0.390121\pi$
$632$		−	14.1421i		−	0.562544i
$633$		0			0
$634$	16.0000			0.635441
$635$	12.7279	−	38.1838i	0.505092	−	1.51528i
$636$		0			0
$637$		−	6.00000i		−	0.237729i
$638$		−	25.4558i		−	1.00781i
$639$		0			0
$640$	8.00000	−	24.0000i	0.316228	−	0.948683i
$641$	16.9706			0.670297			0.335148	−	0.942165i	$-0.391214\pi$
$641$	16.9706			0.670297			0.335148	+	0.942165i	$0.391214\pi$
$642$		0			0
$643$		−	42.0000i		−	1.65632i	−0.560493	−	0.828159i	$-0.689388\pi$
$643$		−	42.0000i		−	1.65632i	0.560493	−	0.828159i	$-0.310612\pi$
$644$		0			0
$645$		0			0
$646$	−4.00000			−0.157378
$647$		−	9.89949i		−	0.389189i	−0.980884	−	0.194595i	$-0.937661\pi$
$647$		−	9.89949i		−	0.389189i	0.980884	−	0.194595i	$-0.0623391\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$	16.9706	+	12.7279i	0.665640	+	0.499230i
$651$		0			0
$652$		0			0
$653$		−	5.65685i		−	0.221370i	−0.993856	−	0.110685i	$-0.964696\pi$
$653$		−	5.65685i		−	0.221370i	0.993856	−	0.110685i	$-0.0353044\pi$
$654$		0			0
$655$	−18.0000	−	6.00000i	−0.703318	−	0.234439i
$656$	16.9706			0.662589
$657$		0			0
$658$		−	12.0000i		−	0.467809i
$659$	16.9706			0.661079			0.330540	−	0.943792i	$-0.392769\pi$
$659$	16.9706			0.661079			0.330540	+	0.943792i	$0.392769\pi$
$660$		0			0
$661$	23.0000			0.894596			0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000			0.894596			0.447298	+	0.894385i	$0.352386\pi$
$662$		−	24.0416i		−	0.934405i
$663$		0			0
$664$	−4.00000			−0.155230
$665$	2.12132	−	6.36396i	0.0822613	−	0.246784i
$666$		0			0
$667$			30.0000i			1.16160i
$668$		0			0
$669$		0			0
$670$	−9.00000	−	3.00000i	−0.347700	−	0.115900i
$671$	−55.1543			−2.12921
$672$		0			0
$673$			15.0000i			0.578208i	0.957298	+	0.289104i	$0.0933573\pi$
$673$			15.0000i			0.578208i	−0.957298	+	0.289104i	$0.906643\pi$
$674$	−46.6690			−1.79762
$675$		0			0
$676$		0			0
$677$		−	48.0833i		−	1.84799i	−0.382405	−	0.923995i	$-0.624904\pi$
$677$		−	48.0833i		−	1.84799i	0.382405	−	0.923995i	$-0.375096\pi$
$678$		0			0
$679$	−9.00000			−0.345388
$680$	−16.9706	−	5.65685i	−0.650791	−	0.216930i
$681$		0			0
$682$		−	12.0000i		−	0.459504i
$683$			15.5563i			0.595247i	0.954683	+	0.297624i	$0.0961940\pi$
$683$			15.5563i			0.595247i	−0.954683	+	0.297624i	$0.903806\pi$
$684$		0			0
$685$	−1.00000	+	3.00000i	−0.0382080	+	0.114624i
$686$	−21.2132			−0.809924
$687$		0			0
$688$		−	24.0000i		−	0.914991i
$689$	29.6985			1.13142
$690$		0			0
$691$	50.0000			1.90209			0.951045	−	0.309053i	$-0.100012\pi$
$691$	50.0000			1.90209			0.951045	+	0.309053i	$0.100012\pi$
$692$		0			0
$693$		0			0
$694$	−38.0000			−1.44246
$695$	27.5772	+	9.19239i	1.04606	+	0.348687i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		−	7.07107i		−	0.267644i
$699$		0			0
$700$		0			0
$701$	−38.1838			−1.44218			−0.721090	−	0.692841i	$-0.756359\pi$
$701$	−38.1838			−1.44218			−0.721090	+	0.692841i	$0.756359\pi$
$702$		0			0
$703$			9.00000i			0.339441i
$704$	−33.9411			−1.27920
$705$		0			0
$706$	40.0000			1.50542
$707$			25.4558i			0.957366i
$708$		0			0
$709$	17.0000			0.638448			0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000			0.638448			0.319224	+	0.947679i	$0.396578\pi$
$710$	12.7279	−	38.1838i	0.477670	−	1.43301i
$711$		0			0
$712$		0			0
$713$			14.1421i			0.529627i
$714$		0			0
$715$	9.00000	−	27.0000i	0.336581	−	1.00974i
$716$		0			0
$717$		0			0
$718$			18.0000i			0.671754i
$719$	12.7279			0.474671			0.237336	−	0.971428i	$-0.423726\pi$
$719$	12.7279			0.474671			0.237336	+	0.971428i	$0.423726\pi$
$720$		0			0
$721$	9.00000			0.335178
$722$			25.4558i			0.947368i
$723$		0			0
$724$		0			0
$725$	16.9706	+	12.7279i	0.630271	+	0.472703i
$726$		0			0
$727$			42.0000i			1.55769i	0.627214	+	0.778847i	$0.284195\pi$
$727$			42.0000i			1.55769i	−0.627214	+	0.778847i	$0.715805\pi$
$728$		−	25.4558i		−	0.943456i
$729$		0			0
$730$	−27.0000	−	9.00000i	−0.999315	−	0.333105i
$731$	−16.9706			−0.627679
$732$		0			0
$733$		−	24.0000i		−	0.886460i	−0.896408	−	0.443230i	$-0.853832\pi$
$733$		−	24.0000i		−	0.886460i	0.896408	−	0.443230i	$-0.146168\pi$
$734$	21.2132			0.782994
$735$		0			0
$736$		0			0
$737$			12.7279i			0.468839i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$		0			0
$741$		0			0
$742$			42.0000i			1.54187i
$743$		−	14.1421i		−	0.518825i	−0.965767	−	0.259412i	$-0.916471\pi$
$743$		−	14.1421i		−	0.518825i	0.965767	−	0.259412i	$-0.0835289\pi$
$744$		0			0
$745$	36.0000	+	12.0000i	1.31894	+	0.439646i
$746$	29.6985			1.08734
$747$		0			0
$748$		0			0
$749$	21.2132			0.775114
$750$		0			0
$751$	−7.00000			−0.255434			−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000			−0.255434			−0.127717	+	0.991811i	$0.540765\pi$
$752$		−	11.3137i		−	0.412568i
$753$		0			0
$754$	18.0000			0.655521
$755$	2.12132	+	0.707107i	0.0772028	+	0.0257343i
$756$		0			0
$757$		−	27.0000i		−	0.981332i	−0.871348	−	0.490666i	$-0.836754\pi$
$757$		−	27.0000i		−	0.981332i	0.871348	−	0.490666i	$-0.163246\pi$
$758$		−	49.4975i		−	1.79783i
$759$		0			0
$760$	2.00000	−	6.00000i	0.0725476	−	0.217643i
$761$	−4.24264			−0.153796			−0.0768978	−	0.997039i	$-0.524502\pi$
$761$	−4.24264			−0.153796			−0.0768978	+	0.997039i	$0.524502\pi$
$762$		0			0
$763$		−	24.0000i		−	0.868858i
$764$		0			0
$765$		0			0
$766$	−8.00000			−0.289052
$767$			25.4558i			0.919157i
$768$		0			0
$769$	11.0000			0.396670			0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000			0.396670			0.198335	+	0.980134i	$0.436447\pi$
$770$	38.1838	+	12.7279i	1.37605	+	0.458682i
$771$		0			0
$772$		0			0
$773$			41.0122i			1.47511i	0.675289	+	0.737553i	$0.264019\pi$
$773$			41.0122i			1.47511i	−0.675289	+	0.737553i	$0.735981\pi$
$774$		0			0
$775$	8.00000	+	6.00000i	0.287368	+	0.215526i
$776$	−8.48528			−0.304604
$777$		0			0
$778$			48.0000i			1.72088i
$779$	4.24264			0.152008
$780$		0			0
$781$	−54.0000			−1.93227
$782$			28.2843i			1.01144i
$783$		0			0
$784$	8.00000			0.285714
$785$	−4.24264	+	12.7279i	−0.151426	+	0.454279i
$786$		0			0
$787$		−	15.0000i		−	0.534692i	−0.963601	−	0.267346i	$-0.913853\pi$
$787$		−	15.0000i		−	0.534692i	0.963601	−	0.267346i	$-0.0861467\pi$
$788$		0			0
$789$		0			0
$790$	−5.00000	+	15.0000i	−0.177892	+	0.533676i
$791$	−4.24264			−0.150851
$792$		0			0
$793$		−	39.0000i		−	1.38493i
$794$	−50.9117			−1.80679
$795$		0			0
$796$		0			0
$797$			19.7990i			0.701316i	0.936504	+	0.350658i	$0.114042\pi$
$797$			19.7990i			0.701316i	−0.936504	+	0.350658i	$0.885958\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$		0			0
$802$			6.00000i			0.211867i
$803$			38.1838i			1.34748i
$804$		0			0
$805$	−45.0000	−	15.0000i	−1.58604	−	0.528681i
$806$	8.48528			0.298881
$807$		0			0
$808$			24.0000i			0.844317i
$809$	−25.4558			−0.894980			−0.447490	−	0.894289i	$-0.647682\pi$
$809$	−25.4558			−0.894980			−0.447490	+	0.894289i	$0.647682\pi$
$810$		0			0
$811$	2.00000			0.0702295			0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000			0.0702295			0.0351147	+	0.999383i	$0.488820\pi$
$812$		0			0
$813$		0			0
$814$	−54.0000			−1.89270
$815$	−6.36396	+	19.0919i	−0.222920	+	0.668759i
$816$		0			0
$817$		−	6.00000i		−	0.209913i
$818$			35.3553i			1.23617i
$819$		0			0
$820$		0			0
$821$	−8.48528			−0.296138			−0.148069	−	0.988977i	$-0.547306\pi$
$821$	−8.48528			−0.296138			−0.148069	+	0.988977i	$0.547306\pi$
$822$		0			0
$823$			39.0000i			1.35945i	0.733465	+	0.679727i	$0.237902\pi$
$823$			39.0000i			1.35945i	−0.733465	+	0.679727i	$0.762098\pi$
$824$	8.48528			0.295599
$825$		0			0
$826$	−36.0000			−1.25260
$827$			2.82843i			0.0983540i	0.998790	+	0.0491770i	$0.0156598\pi$
$827$			2.82843i			0.0983540i	−0.998790	+	0.0491770i	$0.984340\pi$
$828$		0			0
$829$	−7.00000			−0.243120			−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000			−0.243120			−0.121560	+	0.992584i	$0.538790\pi$
$830$	4.24264	+	1.41421i	0.147264	+	0.0490881i
$831$		0			0
$832$		−	24.0000i		−	0.832050i
$833$		−	5.65685i		−	0.195998i
$834$		0			0
$835$	−4.00000	+	12.0000i	−0.138426	+	0.415277i
$836$		0			0
$837$		0			0
$838$		−	48.0000i		−	1.65813i
$839$	42.4264			1.46472			0.732361	−	0.680916i	$-0.238418\pi$
$839$	42.4264			1.46472			0.732361	+	0.680916i	$0.238418\pi$
$840$		0			0
$841$	−11.0000			−0.379310
$842$		−	32.5269i		−	1.12095i
$843$		0			0
$844$		0			0
$845$	−8.48528	−	2.82843i	−0.291903	−	0.0973009i
$846$		0			0
$847$		−	21.0000i		−	0.721569i
$848$			39.5980i			1.35980i
$849$		0			0
$850$	16.0000	+	12.0000i	0.548795	+	0.411597i
$851$	63.6396			2.18154
$852$		0			0
$853$			33.0000i			1.12990i	0.825126	+	0.564949i	$0.191104\pi$
$853$			33.0000i			1.12990i	−0.825126	+	0.564949i	$0.808896\pi$
$854$	55.1543			1.88734
$855$		0			0
$856$	20.0000			0.683586
$857$		−	26.8701i		−	0.917864i	−0.888471	−	0.458932i	$-0.848232\pi$
$857$		−	26.8701i		−	0.917864i	0.888471	−	0.458932i	$-0.151768\pi$
$858$		0			0
$859$	47.0000			1.60362			0.801810	−	0.597580i	$-0.203871\pi$
$859$	47.0000			1.60362			0.801810	+	0.597580i	$0.203871\pi$
$860$		0			0
$861$		0			0
$862$		−	18.0000i		−	0.613082i
$863$			28.2843i			0.962808i	0.876499	+	0.481404i	$0.159873\pi$
$863$			28.2843i			0.962808i	−0.876499	+	0.481404i	$0.840127\pi$
$864$		0			0
$865$	−10.0000	+	30.0000i	−0.340010	+	1.02003i
$866$	−50.9117			−1.73005
$867$		0			0
$868$		0			0
$869$	21.2132			0.719609
$870$		0			0
$871$	−9.00000			−0.304953
$872$		−	22.6274i		−	0.766261i
$873$		0			0
$874$	−10.0000			−0.338255
$875$	−27.5772	+	19.0919i	−0.932278	+	0.645423i
$876$		0			0
$877$		−	15.0000i		−	0.506514i	−0.967399	−	0.253257i	$-0.918498\pi$
$877$		−	15.0000i		−	0.506514i	0.967399	−	0.253257i	$-0.0815018\pi$
$878$		−	19.7990i		−	0.668184i
$879$		0			0
$880$	36.0000	+	12.0000i	1.21356	+	0.404520i
$881$	−38.1838			−1.28644			−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838			−1.28644			−0.643222	+	0.765680i	$0.722403\pi$
$882$		0			0
$883$			27.0000i			0.908622i	0.890843	+	0.454311i	$0.150115\pi$
$883$			27.0000i			0.908622i	−0.890843	+	0.454311i	$0.849885\pi$
$884$		0			0
$885$		0			0
$886$	−20.0000			−0.671913
$887$			45.2548i			1.51951i	0.650210	+	0.759754i	$0.274681\pi$
$887$			45.2548i			1.51951i	−0.650210	+	0.759754i	$0.725319\pi$
$888$		0			0
$889$	54.0000			1.81110
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.82843i		−	0.0946497i
$894$		0			0
$895$	27.0000	+	9.00000i	0.902510	+	0.300837i
$896$	33.9411			1.13389
$897$		0			0
$898$		0			0
$899$	8.48528			0.283000
$900$		0			0
$901$	28.0000			0.932815
$902$			25.4558i			0.847587i
$903$		0			0
$904$	−4.00000			−0.133038
$905$	2.12132	+	0.707107i	0.0705151	+	0.0235050i
$906$		0			0
$907$		−	3.00000i		−	0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$		−	3.00000i		−	0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$		0			0
$909$		0			0
$910$	−9.00000	+	27.0000i	−0.298347	+	0.895041i
$911$	42.4264			1.40565			0.702825	−	0.711363i	$-0.251922\pi$
$911$	42.4264			1.40565			0.702825	+	0.711363i	$0.251922\pi$
$912$		0			0
$913$		−	6.00000i		−	0.198571i
$914$	−16.9706			−0.561336
$915$		0			0
$916$		0			0
$917$		−	25.4558i		−	0.840626i
$918$		0			0
$919$	2.00000			0.0659739			0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000			0.0659739			0.0329870	+	0.999456i	$0.489498\pi$
$920$	−42.4264	−	14.1421i	−1.39876	−	0.466252i
$921$		0			0
$922$		−	24.0000i		−	0.790398i
$923$		−	38.1838i		−	1.25683i
$924$		0			0
$925$	27.0000	−	36.0000i	0.887755	−	1.18367i
$926$	−21.2132			−0.697109
$927$		0			0
$928$		0			0
$929$	−46.6690			−1.53116			−0.765581	−	0.643340i	$-0.777548\pi$
$929$	−46.6690			−1.53116			−0.765581	+	0.643340i	$0.777548\pi$
$930$		0			0
$931$	2.00000			0.0655474
$932$		0			0
$933$		0			0
$934$	−32.0000			−1.04707
$935$	8.48528	−	25.4558i	0.277498	−	0.832495i
$936$		0			0
$937$			9.00000i			0.294017i	0.989135	+	0.147009i	$0.0469645\pi$
$937$			9.00000i			0.294017i	−0.989135	+	0.147009i	$0.953036\pi$
$938$		−	12.7279i		−	0.415581i
$939$		0			0
$940$		0			0
$941$	21.2132			0.691531			0.345765	−	0.938321i	$-0.387619\pi$
$941$	21.2132			0.691531			0.345765	+	0.938321i	$0.387619\pi$
$942$		0			0
$943$		−	30.0000i		−	0.976934i
$944$	−33.9411			−1.10469
$945$		0			0
$946$	36.0000			1.17046
$947$			36.7696i			1.19485i	0.801925	+	0.597425i	$0.203809\pi$
$947$			36.7696i			1.19485i	−0.801925	+	0.597425i	$0.796191\pi$
$948$		0			0
$949$	−27.0000			−0.876457
$950$	−4.24264	+	5.65685i	−0.137649	+	0.183533i
$951$		0			0
$952$		−	24.0000i		−	0.777844i
$953$		−	5.65685i		−	0.183243i	−0.995794	−	0.0916217i	$-0.970795\pi$
$953$		−	5.65685i		−	0.183243i	0.995794	−	0.0916217i	$-0.0292051\pi$
$954$		0			0
$955$	−36.0000	−	12.0000i	−1.16493	−	0.388311i
$956$		0			0
$957$		0			0
$958$		−	6.00000i		−	0.193851i
$959$	−4.24264			−0.137002
$960$		0			0
$961$	−27.0000			−0.870968
$962$		−	38.1838i		−	1.23109i
$963$		0			0
$964$		0			0
$965$	2.12132	−	6.36396i	0.0682877	−	0.204863i
$966$		0			0
$967$		−	33.0000i		−	1.06121i	−0.847620	−	0.530604i	$-0.821965\pi$
$967$		−	33.0000i		−	1.06121i	0.847620	−	0.530604i	$-0.178035\pi$
$968$		−	19.7990i		−	0.636364i
$969$		0			0
$970$	9.00000	+	3.00000i	0.288973	+	0.0963242i
$971$	25.4558			0.816917			0.408458	−	0.912777i	$-0.366066\pi$
$971$	25.4558			0.816917			0.408458	+	0.912777i	$0.366066\pi$
$972$		0			0
$973$			39.0000i			1.25028i
$974$	12.7279			0.407829
$975$		0			0
$976$	52.0000			1.66448
$977$		−	1.41421i		−	0.0452447i	−0.999744	−	0.0226224i	$-0.992798\pi$
$977$		−	1.41421i		−	0.0452447i	0.999744	−	0.0226224i	$-0.00720153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	30.0000i		−	0.957338i
$983$		−	52.3259i		−	1.66894i	−0.551056	−	0.834469i	$-0.685775\pi$
$983$		−	52.3259i		−	1.66894i	0.551056	−	0.834469i	$-0.314225\pi$
$984$		0			0
$985$	−16.0000	+	48.0000i	−0.509802	+	1.52941i
$986$	16.9706			0.540453
$987$		0			0
$988$		0			0
$989$	−42.4264			−1.34908
$990$		0			0
$991$	53.0000			1.68360			0.841800	−	0.539789i	$-0.181496\pi$
$991$	53.0000			1.68360			0.841800	+	0.539789i	$0.181496\pi$
$992$		0			0
$993$		0			0
$994$	54.0000			1.71278
$995$	−23.3345	−	7.77817i	−0.739754	−	0.246585i
$996$		0			0
$997$		−	48.0000i		−	1.52018i	−0.649821	−	0.760088i	$-0.725156\pi$
$997$		−	48.0000i		−	1.52018i	0.649821	−	0.760088i	$-0.274844\pi$
$998$			22.6274i			0.716258i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.2.b.b.109.1	✓	4
3.2	odd	2	inner	135.2.b.b.109.4	yes	4
4.3	odd	2		2160.2.f.k.1729.1		4
5.2	odd	4		675.2.a.m.1.2		2
5.3	odd	4		675.2.a.l.1.1		2
5.4	even	2	inner	135.2.b.b.109.3	yes	4
9.2	odd	6		405.2.j.f.109.3		8
9.4	even	3		405.2.j.f.379.4		8
9.5	odd	6		405.2.j.f.379.1		8
9.7	even	3		405.2.j.f.109.2		8
12.11	even	2		2160.2.f.k.1729.4		4
15.2	even	4		675.2.a.m.1.1		2
15.8	even	4		675.2.a.l.1.2		2
15.14	odd	2	inner	135.2.b.b.109.2	yes	4
20.19	odd	2		2160.2.f.k.1729.2		4
45.4	even	6		405.2.j.f.379.2		8
45.14	odd	6		405.2.j.f.379.3		8
45.29	odd	6		405.2.j.f.109.1		8
45.34	even	6		405.2.j.f.109.4		8
60.59	even	2		2160.2.f.k.1729.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.b.b.109.1	✓	4	1.1	even	1	trivial
135.2.b.b.109.2	yes	4	15.14	odd	2	inner
135.2.b.b.109.3	yes	4	5.4	even	2	inner
135.2.b.b.109.4	yes	4	3.2	odd	2	inner
405.2.j.f.109.1		8	45.29	odd	6
405.2.j.f.109.2		8	9.7	even	3
405.2.j.f.109.3		8	9.2	odd	6
405.2.j.f.109.4		8	45.34	even	6
405.2.j.f.379.1		8	9.5	odd	6
405.2.j.f.379.2		8	45.4	even	6
405.2.j.f.379.3		8	45.14	odd	6
405.2.j.f.379.4		8	9.4	even	3
675.2.a.l.1.1		2	5.3	odd	4
675.2.a.l.1.2		2	15.8	even	4
675.2.a.m.1.1		2	15.2	even	4
675.2.a.m.1.2		2	5.2	odd	4
2160.2.f.k.1729.1		4	4.3	odd	2
2160.2.f.k.1729.2		4	20.19	odd	2
2160.2.f.k.1729.3		4	60.59	even	2
2160.2.f.k.1729.4		4	12.11	even	2

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	135.2.b.b.109.1

Level	$135$
Weight	$2$
Character	135.109
Analytic conductor	$1.078$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$