LMFDB - Embedded newform 850.2.b.a.101.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(101,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, 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("850.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$850 = 2 \cdot 5^{2} \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	850.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:	$6.78728417181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	850.101
Dual form			850.2.b.a.101.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.00000i q^{3} +1.00000 q^{4} +3.00000i q^{6} +4.00000i q^{7} -1.00000 q^{8} -6.00000 q^{9} +2.00000i q^{11} -3.00000i q^{12} -1.00000 q^{13} -4.00000i q^{14} +1.00000 q^{16} +(4.00000 + 1.00000i) q^{17} +6.00000 q^{18} +7.00000 q^{19} +12.0000 q^{21} -2.00000i q^{22} +6.00000i q^{23} +3.00000i q^{24} +1.00000 q^{26} +9.00000i q^{27} +4.00000i q^{28} +3.00000i q^{29} -7.00000i q^{31} -1.00000 q^{32} +6.00000 q^{33} +(-4.00000 - 1.00000i) q^{34} -6.00000 q^{36} -2.00000i q^{37} -7.00000 q^{38} +3.00000i q^{39} +8.00000i q^{41} -12.0000 q^{42} +8.00000 q^{43} +2.00000i q^{44} -6.00000i q^{46} -9.00000 q^{47} -3.00000i q^{48} -9.00000 q^{49} +(3.00000 - 12.0000i) q^{51} -1.00000 q^{52} +11.0000 q^{53} -9.00000i q^{54} -4.00000i q^{56} -21.0000i q^{57} -3.00000i q^{58} -5.00000 q^{59} -1.00000i q^{61} +7.00000i q^{62} -24.0000i q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.0000 q^{67} +(4.00000 + 1.00000i) q^{68} +18.0000 q^{69} +1.00000i q^{71} +6.00000 q^{72} +9.00000i q^{73} +2.00000i q^{74} +7.00000 q^{76} -8.00000 q^{77} -3.00000i q^{78} +9.00000 q^{81} -8.00000i q^{82} -6.00000 q^{83} +12.0000 q^{84} -8.00000 q^{86} +9.00000 q^{87} -2.00000i q^{88} -1.00000 q^{89} -4.00000i q^{91} +6.00000i q^{92} -21.0000 q^{93} +9.00000 q^{94} +3.00000i q^{96} -1.00000i q^{97} +9.00000 q^{98} -12.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 12 q^{9} - 2 q^{13} + 2 q^{16} + 8 q^{17} + 12 q^{18} + 14 q^{19} + 24 q^{21} + 2 q^{26} - 2 q^{32} + 12 q^{33} - 8 q^{34} - 12 q^{36} - 14 q^{38} - 24 q^{42} + 16 q^{43}+ \cdots + 18 q^{98}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 12 * q^9 - 2 * q^13 + 2 * q^16 + 8 * q^17 + 12 * q^18 + 14 * q^19 + 24 * q^21 + 2 * q^26 - 2 * q^32 + 12 * q^33 - 8 * q^34 - 12 * q^36 - 14 * q^38 - 24 * q^42 + 16 * q^43 - 18 * q^47 - 18 * q^49 + 6 * q^51 - 2 * q^52 + 22 * q^53 - 10 * q^59 + 2 * q^64 - 12 * q^66 + 20 * q^67 + 8 * q^68 + 36 * q^69 + 12 * q^72 + 14 * q^76 - 16 * q^77 + 18 * q^81 - 12 * q^83 + 24 * q^84 - 16 * q^86 + 18 * q^87 - 2 * q^89 - 42 * q^93 + 18 * q^94 + 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$ .

$n$	$477$	$751$
$\chi(n)$	$1$	$-1$

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

)

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

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	3.00000i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$		−	3.00000i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			3.00000i			1.22474i
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	−1.00000			−0.353553
$9$	−6.00000			−2.00000
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		−	3.00000i		−	0.866025i
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		−	4.00000i		−	1.06904i
$15$		0			0
$16$	1.00000			0.250000
$17$	4.00000	+	1.00000i	0.970143	+	0.242536i
$17$	4.00000	+	1.00000i	0.970143	+	0.242536i
$18$	6.00000			1.41421
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$		0			0
$21$	12.0000			2.61861
$22$		−	2.00000i		−	0.426401i
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$			3.00000i			0.612372i
$25$		0			0
$26$	1.00000			0.196116
$27$			9.00000i			1.73205i
$28$			4.00000i			0.755929i
$29$			3.00000i			0.557086i	0.960424	+	0.278543i	$0.0898515\pi$
$29$			3.00000i			0.557086i	−0.960424	+	0.278543i	$0.910149\pi$
$30$		0			0
$31$		−	7.00000i		−	1.25724i	−0.777714	−	0.628619i	$-0.783621\pi$
$31$		−	7.00000i		−	1.25724i	0.777714	−	0.628619i	$-0.216379\pi$
$32$	−1.00000			−0.176777
$33$	6.00000			1.04447
$34$	−4.00000	−	1.00000i	−0.685994	−	0.171499i
$35$		0			0
$36$	−6.00000			−1.00000
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	−7.00000			−1.13555
$39$			3.00000i			0.480384i
$40$		0			0
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−12.0000			−1.85164
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$			2.00000i			0.301511i
$45$		0			0
$46$		−	6.00000i		−	0.884652i
$47$	−9.00000			−1.31278			−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000			−1.31278			−0.656392	+	0.754420i	$0.727918\pi$
$48$		−	3.00000i		−	0.433013i
$49$	−9.00000			−1.28571
$50$		0			0
$51$	3.00000	−	12.0000i	0.420084	−	1.68034i
$52$	−1.00000			−0.138675
$53$	11.0000			1.51097			0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000			1.51097			0.755483	+	0.655168i	$0.227402\pi$
$54$		−	9.00000i		−	1.22474i
$55$		0			0
$56$		−	4.00000i		−	0.534522i
$57$		−	21.0000i		−	2.78152i
$58$		−	3.00000i		−	0.393919i
$59$	−5.00000			−0.650945			−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000			−0.650945			−0.325472	+	0.945552i	$0.605523\pi$
$60$		0			0
$61$		−	1.00000i		−	0.128037i	−0.997949	−	0.0640184i	$-0.979608\pi$
$61$		−	1.00000i		−	0.128037i	0.997949	−	0.0640184i	$-0.0203916\pi$
$62$			7.00000i			0.889001i
$63$		−	24.0000i		−	3.02372i
$64$	1.00000			0.125000
$65$		0			0
$66$	−6.00000			−0.738549
$67$	10.0000			1.22169			0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000			1.22169			0.610847	+	0.791748i	$0.290829\pi$
$68$	4.00000	+	1.00000i	0.485071	+	0.121268i
$69$	18.0000			2.16695
$70$		0			0
$71$			1.00000i			0.118678i	0.998238	+	0.0593391i	$0.0188993\pi$
$71$			1.00000i			0.118678i	−0.998238	+	0.0593391i	$0.981101\pi$
$72$	6.00000			0.707107
$73$			9.00000i			1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$			9.00000i			1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$			2.00000i			0.232495i
$75$		0			0
$76$	7.00000			0.802955
$77$	−8.00000			−0.911685
$78$		−	3.00000i		−	0.339683i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		−	8.00000i		−	0.883452i
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$	12.0000			1.30931
$85$		0			0
$86$	−8.00000			−0.862662
$87$	9.00000			0.964901
$88$		−	2.00000i		−	0.213201i
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$			6.00000i			0.625543i
$93$	−21.0000			−2.17760
$94$	9.00000			0.928279
$95$		0			0
$96$			3.00000i			0.306186i
$97$		−	1.00000i		−	0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$		−	1.00000i		−	0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$	9.00000			0.909137
$99$		−	12.0000i		−	1.20605i
$100$		0			0
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$	−3.00000	+	12.0000i	−0.297044	+	1.18818i
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	−11.0000			−1.06841
$107$		−	16.0000i		−	1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$		−	16.0000i		−	1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$			9.00000i			0.866025i
$109$			19.0000i			1.81987i	0.414751	+	0.909935i	$0.363869\pi$
$109$			19.0000i			1.81987i	−0.414751	+	0.909935i	$0.636131\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$			4.00000i			0.377964i
$113$			1.00000i			0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$			1.00000i			0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$			21.0000i			1.96683i
$115$		0			0
$116$			3.00000i			0.278543i
$117$	6.00000			0.554700
$118$	5.00000			0.460287
$119$	−4.00000	+	16.0000i	−0.366679	+	1.46672i
$120$		0			0
$121$	7.00000			0.636364
$122$			1.00000i			0.0905357i
$123$	24.0000			2.16401
$124$		−	7.00000i		−	0.628619i
$125$		0			0
$126$			24.0000i			2.13809i
$127$	15.0000			1.33103			0.665517	−	0.746382i	$-0.268211\pi$
$127$	15.0000			1.33103			0.665517	+	0.746382i	$0.268211\pi$
$128$	−1.00000			−0.0883883
$129$		−	24.0000i		−	2.11308i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	6.00000			0.522233
$133$			28.0000i			2.42791i
$134$	−10.0000			−0.863868
$135$		0			0
$136$	−4.00000	−	1.00000i	−0.342997	−	0.0857493i
$137$	−8.00000			−0.683486			−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000			−0.683486			−0.341743	+	0.939793i	$0.611017\pi$
$138$	−18.0000			−1.53226
$139$		−	14.0000i		−	1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$		−	14.0000i		−	1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$		0			0
$141$			27.0000i			2.27381i
$142$		−	1.00000i		−	0.0839181i
$143$		−	2.00000i		−	0.167248i
$144$	−6.00000			−0.500000
$145$		0			0
$146$		−	9.00000i		−	0.744845i
$147$			27.0000i			2.22692i
$148$		−	2.00000i		−	0.164399i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	22.0000			1.79033			0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000			1.79033			0.895167	+	0.445730i	$0.147056\pi$
$152$	−7.00000			−0.567775
$153$	−24.0000	−	6.00000i	−1.94029	−	0.485071i
$154$	8.00000			0.644658
$155$		0			0
$156$			3.00000i			0.240192i
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$		−	33.0000i		−	2.61707i
$160$		0			0
$161$	−24.0000			−1.89146
$162$	−9.00000			−0.707107
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$			8.00000i			0.624695i
$165$		0			0
$166$	6.00000			0.465690
$167$		−	2.00000i		−	0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$		−	2.00000i		−	0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	−12.0000			−0.925820
$169$	−12.0000			−0.923077
$170$		0			0
$171$	−42.0000			−3.21182
$172$	8.00000			0.609994
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	−9.00000			−0.682288
$175$		0			0
$176$			2.00000i			0.150756i
$177$			15.0000i			1.12747i
$178$	1.00000			0.0749532
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$			4.00000i			0.296500i
$183$	−3.00000			−0.221766
$184$		−	6.00000i		−	0.442326i
$185$		0			0
$186$	21.0000			1.53979
$187$	−2.00000	+	8.00000i	−0.146254	+	0.585018i
$188$	−9.00000			−0.656392
$189$	−36.0000			−2.61861
$190$		0			0
$191$	10.0000			0.723575			0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000			0.723575			0.361787	+	0.932261i	$0.382167\pi$
$192$		−	3.00000i		−	0.216506i
$193$		−	2.00000i		−	0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$		−	2.00000i		−	0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$			1.00000i			0.0717958i
$195$		0			0
$196$	−9.00000			−0.642857
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$			12.0000i			0.852803i
$199$			11.0000i			0.779769i	0.920864	+	0.389885i	$0.127485\pi$
$199$			11.0000i			0.779769i	−0.920864	+	0.389885i	$0.872515\pi$
$200$		0			0
$201$		−	30.0000i		−	2.11604i
$202$	2.00000			0.140720
$203$	−12.0000			−0.842235
$204$	3.00000	−	12.0000i	0.210042	−	0.840168i
$205$		0			0
$206$		0			0
$207$		−	36.0000i		−	2.50217i
$208$	−1.00000			−0.0693375
$209$			14.0000i			0.968400i
$210$		0			0
$211$		−	10.0000i		−	0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$		−	10.0000i		−	0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$	11.0000			0.755483
$213$	3.00000			0.205557
$214$			16.0000i			1.09374i
$215$		0			0
$216$		−	9.00000i		−	0.612372i
$217$	28.0000			1.90076
$218$		−	19.0000i		−	1.28684i
$219$	27.0000			1.82449
$220$		0			0
$221$	−4.00000	−	1.00000i	−0.269069	−	0.0672673i
$222$	6.00000			0.402694
$223$	−29.0000			−1.94198			−0.970992	−	0.239113i	$-0.923143\pi$
$223$	−29.0000			−1.94198			−0.970992	+	0.239113i	$0.923143\pi$
$224$		−	4.00000i		−	0.267261i
$225$		0			0
$226$		−	1.00000i		−	0.0665190i
$227$		−	27.0000i		−	1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$		−	27.0000i		−	1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$		−	21.0000i		−	1.39076i
$229$	−4.00000			−0.264327			−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000			−0.264327			−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$			24.0000i			1.57908i
$232$		−	3.00000i		−	0.196960i
$233$			3.00000i			0.196537i	0.995160	+	0.0982683i	$0.0313303\pi$
$233$			3.00000i			0.196537i	−0.995160	+	0.0982683i	$0.968670\pi$
$234$	−6.00000			−0.392232
$235$		0			0
$236$	−5.00000			−0.325472
$237$		0			0
$238$	4.00000	−	16.0000i	0.259281	−	1.03713i
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$		−	4.00000i		−	0.257663i	−0.991667	−	0.128831i	$-0.958877\pi$
$241$		−	4.00000i		−	0.257663i	0.991667	−	0.128831i	$-0.0411226\pi$
$242$	−7.00000			−0.449977
$243$		0			0
$244$		−	1.00000i		−	0.0640184i
$245$		0			0
$246$	−24.0000			−1.53018
$247$	−7.00000			−0.445399
$248$			7.00000i			0.444500i
$249$			18.0000i			1.14070i
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		−	24.0000i		−	1.51186i
$253$	−12.0000			−0.754434
$254$	−15.0000			−0.941184
$255$		0			0
$256$	1.00000			0.0625000
$257$	−24.0000			−1.49708			−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000			−1.49708			−0.748539	+	0.663090i	$0.769245\pi$
$258$			24.0000i			1.49417i
$259$	8.00000			0.497096
$260$		0			0
$261$		−	18.0000i		−	1.11417i
$262$		0			0
$263$	9.00000			0.554964			0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000			0.554964			0.277482	+	0.960731i	$0.410500\pi$
$264$	−6.00000			−0.369274
$265$		0			0
$266$		−	28.0000i		−	1.71679i
$267$			3.00000i			0.183597i
$268$	10.0000			0.610847
$269$			25.0000i			1.52428i	0.647414	+	0.762138i	$0.275850\pi$
$269$			25.0000i			1.52428i	−0.647414	+	0.762138i	$0.724150\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$	4.00000	+	1.00000i	0.242536	+	0.0606339i
$273$	−12.0000			−0.726273
$274$	8.00000			0.483298
$275$		0			0
$276$	18.0000			1.08347
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$			14.0000i			0.839664i
$279$			42.0000i			2.51447i
$280$		0			0
$281$	9.00000			0.536895			0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000			0.536895			0.268447	+	0.963294i	$0.413489\pi$
$282$		−	27.0000i		−	1.60783i
$283$		−	7.00000i		−	0.416107i	−0.978117	−	0.208053i	$-0.933287\pi$
$283$		−	7.00000i		−	0.416107i	0.978117	−	0.208053i	$-0.0667128\pi$
$284$			1.00000i			0.0593391i
$285$		0			0
$286$			2.00000i			0.118262i
$287$	−32.0000			−1.88890
$288$	6.00000			0.353553
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$	−3.00000			−0.175863
$292$			9.00000i			0.526685i
$293$	5.00000			0.292103			0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000			0.292103			0.146052	+	0.989277i	$0.453343\pi$
$294$		−	27.0000i		−	1.57467i
$295$		0			0
$296$			2.00000i			0.116248i
$297$	−18.0000			−1.04447
$298$	6.00000			0.347571
$299$		−	6.00000i		−	0.346989i
$300$		0			0
$301$			32.0000i			1.84445i
$302$	−22.0000			−1.26596
$303$			6.00000i			0.344691i
$304$	7.00000			0.401478
$305$		0			0
$306$	24.0000	+	6.00000i	1.37199	+	0.342997i
$307$	−22.0000			−1.25561			−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000			−1.25561			−0.627803	+	0.778372i	$0.716046\pi$
$308$	−8.00000			−0.455842
$309$		0			0
$310$		0			0
$311$		−	16.0000i		−	0.907277i	−0.891186	−	0.453638i	$-0.850126\pi$
$311$		−	16.0000i		−	0.907277i	0.891186	−	0.453638i	$-0.149874\pi$
$312$		−	3.00000i		−	0.169842i
$313$		−	26.0000i		−	1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$		−	26.0000i		−	1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	2.00000			0.112867
$315$		0			0
$316$		0			0
$317$			20.0000i			1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$			20.0000i			1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$			33.0000i			1.85055i
$319$	−6.00000			−0.335936
$320$		0			0
$321$	−48.0000			−2.67910
$322$	24.0000			1.33747
$323$	28.0000	+	7.00000i	1.55796	+	0.389490i
$324$	9.00000			0.500000
$325$		0			0
$326$		−	4.00000i		−	0.221540i
$327$	57.0000			3.15211
$328$		−	8.00000i		−	0.441726i
$329$		−	36.0000i		−	1.98474i
$330$		0			0
$331$	−19.0000			−1.04433			−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000			−1.04433			−0.522167	+	0.852843i	$0.674876\pi$
$332$	−6.00000			−0.329293
$333$			12.0000i			0.657596i
$334$			2.00000i			0.109435i
$335$		0			0
$336$	12.0000			0.654654
$337$		−	23.0000i		−	1.25289i	−0.779466	−	0.626445i	$-0.784509\pi$
$337$		−	23.0000i		−	1.25289i	0.779466	−	0.626445i	$-0.215491\pi$
$338$	12.0000			0.652714
$339$	3.00000			0.162938
$340$		0			0
$341$	14.0000			0.758143
$342$	42.0000			2.27110
$343$		−	8.00000i		−	0.431959i
$344$	−8.00000			−0.431331
$345$		0			0
$346$		−	2.00000i		−	0.107521i
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$	9.00000			0.482451
$349$	12.0000			0.642345			0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000			0.642345			0.321173	+	0.947021i	$0.395923\pi$
$350$		0			0
$351$		−	9.00000i		−	0.480384i
$352$		−	2.00000i		−	0.106600i
$353$	10.0000			0.532246			0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000			0.532246			0.266123	+	0.963939i	$0.414257\pi$
$354$		−	15.0000i		−	0.797241i
$355$		0			0
$356$	−1.00000			−0.0529999
$357$	48.0000	+	12.0000i	2.54043	+	0.635107i
$358$	−24.0000			−1.26844
$359$	−4.00000			−0.211112			−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000			−0.211112			−0.105556	+	0.994413i	$0.533662\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		−	10.0000i		−	0.525588i
$363$		−	21.0000i		−	1.10221i
$364$		−	4.00000i		−	0.209657i
$365$		0			0
$366$	3.00000			0.156813
$367$			10.0000i			0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$			10.0000i			0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$			6.00000i			0.312772i
$369$		−	48.0000i		−	2.49878i
$370$		0			0
$371$			44.0000i			2.28437i
$372$	−21.0000			−1.08880
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$	2.00000	−	8.00000i	0.103418	−	0.413670i
$375$		0			0
$376$	9.00000			0.464140
$377$		−	3.00000i		−	0.154508i
$378$	36.0000			1.85164
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$		−	45.0000i		−	2.30542i
$382$	−10.0000			−0.511645
$383$	19.0000			0.970855			0.485427	−	0.874277i	$-0.338664\pi$
$383$	19.0000			0.970855			0.485427	+	0.874277i	$0.338664\pi$
$384$			3.00000i			0.153093i
$385$		0			0
$386$			2.00000i			0.101797i
$387$	−48.0000			−2.43998
$388$		−	1.00000i		−	0.0507673i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	−6.00000	+	24.0000i	−0.303433	+	1.21373i
$392$	9.00000			0.454569
$393$		0			0
$394$		−	18.0000i		−	0.906827i
$395$		0			0
$396$		−	12.0000i		−	0.603023i
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$		−	11.0000i		−	0.551380i
$399$	84.0000			4.20526
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$			30.0000i			1.49626i
$403$			7.00000i			0.348695i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	12.0000			0.595550
$407$	4.00000			0.198273
$408$	−3.00000	+	12.0000i	−0.148522	+	0.594089i
$409$	−19.0000			−0.939490			−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000			−0.939490			−0.469745	+	0.882802i	$0.655654\pi$
$410$		0			0
$411$			24.0000i			1.18383i
$412$		0			0
$413$		−	20.0000i		−	0.984136i
$414$			36.0000i			1.76930i
$415$		0			0
$416$	1.00000			0.0490290
$417$	−42.0000			−2.05675
$418$		−	14.0000i		−	0.684762i
$419$		−	10.0000i		−	0.488532i	−0.969708	−	0.244266i	$-0.921453\pi$
$419$		−	10.0000i		−	0.488532i	0.969708	−	0.244266i	$-0.0785470\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$			10.0000i			0.486792i
$423$	54.0000			2.62557
$424$	−11.0000			−0.534207
$425$		0			0
$426$	−3.00000			−0.145350
$427$	4.00000			0.193574
$428$		−	16.0000i		−	0.773389i
$429$	−6.00000			−0.289683
$430$		0			0
$431$			8.00000i			0.385346i	0.981263	+	0.192673i	$0.0617157\pi$
$431$			8.00000i			0.385346i	−0.981263	+	0.192673i	$0.938284\pi$
$432$			9.00000i			0.433013i
$433$	8.00000			0.384455			0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000			0.384455			0.192228	+	0.981350i	$0.438429\pi$
$434$	−28.0000			−1.34404
$435$		0			0
$436$			19.0000i			0.909935i
$437$			42.0000i			2.00913i
$438$	−27.0000			−1.29011
$439$		−	32.0000i		−	1.52728i	−0.645644	−	0.763638i	$-0.723411\pi$
$439$		−	32.0000i		−	1.52728i	0.645644	−	0.763638i	$-0.276589\pi$
$440$		0			0
$441$	54.0000			2.57143
$442$	4.00000	+	1.00000i	0.190261	+	0.0475651i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$	−6.00000			−0.284747
$445$		0			0
$446$	29.0000			1.37319
$447$			18.0000i			0.851371i
$448$			4.00000i			0.188982i
$449$		−	30.0000i		−	1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$		−	30.0000i		−	1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$			1.00000i			0.0470360i
$453$		−	66.0000i		−	3.10095i
$454$			27.0000i			1.26717i
$455$		0			0
$456$			21.0000i			0.983415i
$457$	36.0000			1.68401			0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000			1.68401			0.842004	+	0.539471i	$0.181376\pi$
$458$	4.00000			0.186908
$459$	−9.00000	+	36.0000i	−0.420084	+	1.68034i
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		−	24.0000i		−	1.11658i
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$			3.00000i			0.139272i
$465$		0			0
$466$		−	3.00000i		−	0.138972i
$467$	−26.0000			−1.20314			−0.601568	−	0.798821i	$-0.705457\pi$
$467$	−26.0000			−1.20314			−0.601568	+	0.798821i	$0.705457\pi$
$468$	6.00000			0.277350
$469$			40.0000i			1.84703i
$470$		0			0
$471$			6.00000i			0.276465i
$472$	5.00000			0.230144
$473$			16.0000i			0.735681i
$474$		0			0
$475$		0			0
$476$	−4.00000	+	16.0000i	−0.183340	+	0.733359i
$477$	−66.0000			−3.02193
$478$	20.0000			0.914779
$479$			15.0000i			0.685367i	0.939451	+	0.342684i	$0.111336\pi$
$479$			15.0000i			0.685367i	−0.939451	+	0.342684i	$0.888664\pi$
$480$		0			0
$481$			2.00000i			0.0911922i
$482$			4.00000i			0.182195i
$483$			72.0000i			3.27611i
$484$	7.00000			0.318182
$485$		0			0
$486$		0			0
$487$		−	40.0000i		−	1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$		−	40.0000i		−	1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$			1.00000i			0.0452679i
$489$	12.0000			0.542659
$490$		0			0
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$	24.0000			1.08200
$493$	−3.00000	+	12.0000i	−0.135113	+	0.540453i
$494$	7.00000			0.314945
$495$		0			0
$496$		−	7.00000i		−	0.314309i
$497$	−4.00000			−0.179425
$498$		−	18.0000i		−	0.806599i
$499$		−	20.0000i		−	0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$		−	20.0000i		−	0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$		0			0
$501$	−6.00000			−0.268060
$502$	8.00000			0.357057
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$			24.0000i			1.06904i
$505$		0			0
$506$	12.0000			0.533465
$507$			36.0000i			1.59882i
$508$	15.0000			0.665517
$509$	−36.0000			−1.59567			−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000			−1.59567			−0.797836	+	0.602875i	$0.794022\pi$
$510$		0			0
$511$	−36.0000			−1.59255
$512$	−1.00000			−0.0441942
$513$			63.0000i			2.78152i
$514$	24.0000			1.05859
$515$		0			0
$516$		−	24.0000i		−	1.05654i
$517$		−	18.0000i		−	0.791639i
$518$	−8.00000			−0.351500
$519$	6.00000			0.263371
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$			18.0000i			0.787839i
$523$	22.0000			0.961993			0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000			0.961993			0.480996	+	0.876723i	$0.340275\pi$
$524$		0			0
$525$		0			0
$526$	−9.00000			−0.392419
$527$	7.00000	−	28.0000i	0.304925	−	1.21970i
$528$	6.00000			0.261116
$529$	−13.0000			−0.565217
$530$		0			0
$531$	30.0000			1.30189
$532$			28.0000i			1.21395i
$533$		−	8.00000i		−	0.346518i
$534$		−	3.00000i		−	0.129823i
$535$		0			0
$536$	−10.0000			−0.431934
$537$		−	72.0000i		−	3.10703i
$538$		−	25.0000i		−	1.07783i
$539$		−	18.0000i		−	0.775315i
$540$		0			0
$541$		−	22.0000i		−	0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$		−	22.0000i		−	0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$	−12.0000			−0.515444
$543$	30.0000			1.28742
$544$	−4.00000	−	1.00000i	−0.171499	−	0.0428746i
$545$		0			0
$546$	12.0000			0.513553
$547$		−	39.0000i		−	1.66752i	−0.552127	−	0.833760i	$-0.686184\pi$
$547$		−	39.0000i		−	1.66752i	0.552127	−	0.833760i	$-0.313816\pi$
$548$	−8.00000			−0.341743
$549$			6.00000i			0.256074i
$550$		0			0
$551$			21.0000i			0.894630i
$552$	−18.0000			−0.766131
$553$		0			0
$554$		−	2.00000i		−	0.0849719i
$555$		0			0
$556$		−	14.0000i		−	0.593732i
$557$	27.0000			1.14403			0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000			1.14403			0.572013	+	0.820244i	$0.306163\pi$
$558$		−	42.0000i		−	1.77800i
$559$	−8.00000			−0.338364
$560$		0			0
$561$	24.0000	+	6.00000i	1.01328	+	0.253320i
$562$	−9.00000			−0.379642
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$			27.0000i			1.13691i
$565$		0			0
$566$			7.00000i			0.294232i
$567$			36.0000i			1.51186i
$568$		−	1.00000i		−	0.0419591i
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		0			0
$571$		−	16.0000i		−	0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		−	2.00000i		−	0.0836242i
$573$		−	30.0000i		−	1.25327i
$574$	32.0000			1.33565
$575$		0			0
$576$	−6.00000			−0.250000
$577$	−32.0000			−1.33218			−0.666089	−	0.745873i	$-0.732033\pi$
$577$	−32.0000			−1.33218			−0.666089	+	0.745873i	$0.732033\pi$
$578$	−15.0000	−	8.00000i	−0.623918	−	0.332756i
$579$	−6.00000			−0.249351
$580$		0			0
$581$		−	24.0000i		−	0.995688i
$582$	3.00000			0.124354
$583$			22.0000i			0.911147i
$584$		−	9.00000i		−	0.372423i
$585$		0			0
$586$	−5.00000			−0.206548
$587$	10.0000			0.412744			0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000			0.412744			0.206372	+	0.978474i	$0.433834\pi$
$588$			27.0000i			1.11346i
$589$		−	49.0000i		−	2.01901i
$590$		0			0
$591$	54.0000			2.22126
$592$		−	2.00000i		−	0.0821995i
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$	18.0000			0.738549
$595$		0			0
$596$	−6.00000			−0.245770
$597$	33.0000			1.35060
$598$			6.00000i			0.245358i
$599$	2.00000			0.0817178			0.0408589	−	0.999165i	$-0.486991\pi$
$599$	2.00000			0.0817178			0.0408589	+	0.999165i	$0.486991\pi$
$600$		0			0
$601$			16.0000i			0.652654i	0.945257	+	0.326327i	$0.105811\pi$
$601$			16.0000i			0.652654i	−0.945257	+	0.326327i	$0.894189\pi$
$602$		−	32.0000i		−	1.30422i
$603$	−60.0000			−2.44339
$604$	22.0000			0.895167
$605$		0			0
$606$		−	6.00000i		−	0.243733i
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$	−7.00000			−0.283887
$609$			36.0000i			1.45879i
$610$		0			0
$611$	9.00000			0.364101
$612$	−24.0000	−	6.00000i	−0.970143	−	0.242536i
$613$	−29.0000			−1.17130			−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000			−1.17130			−0.585649	+	0.810564i	$0.699160\pi$
$614$	22.0000			0.887848
$615$		0			0
$616$	8.00000			0.322329
$617$		−	1.00000i		−	0.0402585i	−0.999797	−	0.0201292i	$-0.993592\pi$
$617$		−	1.00000i		−	0.0402585i	0.999797	−	0.0201292i	$-0.00640777\pi$
$618$		0			0
$619$		−	22.0000i		−	0.884255i	−0.896952	−	0.442127i	$-0.854224\pi$
$619$		−	22.0000i		−	0.884255i	0.896952	−	0.442127i	$-0.145776\pi$
$620$		0			0
$621$	−54.0000			−2.16695
$622$			16.0000i			0.641542i
$623$		−	4.00000i		−	0.160257i
$624$			3.00000i			0.120096i
$625$		0			0
$626$			26.0000i			1.03917i
$627$	42.0000			1.67732
$628$	−2.00000			−0.0798087
$629$	2.00000	−	8.00000i	0.0797452	−	0.318981i
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$	−30.0000			−1.19239
$634$		−	20.0000i		−	0.794301i
$635$		0			0
$636$		−	33.0000i		−	1.30854i
$637$	9.00000			0.356593
$638$	6.00000			0.237542
$639$		−	6.00000i		−	0.237356i
$640$		0			0
$641$		−	2.00000i		−	0.0789953i	−0.999220	−	0.0394976i	$-0.987424\pi$
$641$		−	2.00000i		−	0.0789953i	0.999220	−	0.0394976i	$-0.0125758\pi$
$642$	48.0000			1.89441
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	−24.0000			−0.945732
$645$		0			0
$646$	−28.0000	−	7.00000i	−1.10165	−	0.275411i
$647$	−11.0000			−0.432455			−0.216227	−	0.976343i	$-0.569375\pi$
$647$	−11.0000			−0.432455			−0.216227	+	0.976343i	$0.569375\pi$
$648$	−9.00000			−0.353553
$649$		−	10.0000i		−	0.392534i
$650$		0			0
$651$		−	84.0000i		−	3.29222i
$652$			4.00000i			0.156652i
$653$		−	16.0000i		−	0.626128i	−0.949732	−	0.313064i	$-0.898644\pi$
$653$		−	16.0000i		−	0.626128i	0.949732	−	0.313064i	$-0.101356\pi$
$654$	−57.0000			−2.22888
$655$		0			0
$656$			8.00000i			0.312348i
$657$		−	54.0000i		−	2.10674i
$658$			36.0000i			1.40343i
$659$	−29.0000			−1.12968			−0.564840	−	0.825201i	$-0.691062\pi$
$659$	−29.0000			−1.12968			−0.564840	+	0.825201i	$0.691062\pi$
$660$		0			0
$661$	12.0000			0.466746			0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000			0.466746			0.233373	+	0.972387i	$0.425024\pi$
$662$	19.0000			0.738456
$663$	−3.00000	+	12.0000i	−0.116510	+	0.466041i
$664$	6.00000			0.232845
$665$		0			0
$666$		−	12.0000i		−	0.464991i
$667$	−18.0000			−0.696963
$668$		−	2.00000i		−	0.0773823i
$669$			87.0000i			3.36361i
$670$		0			0
$671$	2.00000			0.0772091
$672$	−12.0000			−0.462910
$673$		−	33.0000i		−	1.27206i	−0.771666	−	0.636028i	$-0.780576\pi$
$673$		−	33.0000i		−	1.27206i	0.771666	−	0.636028i	$-0.219424\pi$
$674$			23.0000i			0.885927i
$675$		0			0
$676$	−12.0000			−0.461538
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	−3.00000			−0.115214
$679$	4.00000			0.153506
$680$		0			0
$681$	−81.0000			−3.10393
$682$	−14.0000			−0.536088
$683$		−	35.0000i		−	1.33924i	−0.742705	−	0.669619i	$-0.766457\pi$
$683$		−	35.0000i		−	1.33924i	0.742705	−	0.669619i	$-0.233543\pi$
$684$	−42.0000			−1.60591
$685$		0			0
$686$			8.00000i			0.305441i
$687$			12.0000i			0.457829i
$688$	8.00000			0.304997
$689$	−11.0000			−0.419067
$690$		0			0
$691$		−	18.0000i		−	0.684752i	−0.939563	−	0.342376i	$-0.888768\pi$
$691$		−	18.0000i		−	0.684752i	0.939563	−	0.342376i	$-0.111232\pi$
$692$			2.00000i			0.0760286i
$693$	48.0000			1.82337
$694$		−	27.0000i		−	1.02491i
$695$		0			0
$696$	−9.00000			−0.341144
$697$	−8.00000	+	32.0000i	−0.303022	+	1.21209i
$698$	−12.0000			−0.454207
$699$	9.00000			0.340411
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$			9.00000i			0.339683i
$703$		−	14.0000i		−	0.528020i
$704$			2.00000i			0.0753778i
$705$		0			0
$706$	−10.0000			−0.376355
$707$		−	8.00000i		−	0.300871i
$708$			15.0000i			0.563735i
$709$		−	17.0000i		−	0.638448i	−0.947679	−	0.319224i	$-0.896578\pi$
$709$		−	17.0000i		−	0.638448i	0.947679	−	0.319224i	$-0.103422\pi$
$710$		0			0
$711$		0			0
$712$	1.00000			0.0374766
$713$	42.0000			1.57291
$714$	−48.0000	−	12.0000i	−1.79635	−	0.449089i
$715$		0			0
$716$	24.0000			0.896922
$717$			60.0000i			2.24074i
$718$	4.00000			0.149279
$719$		−	15.0000i		−	0.559406i	−0.960087	−	0.279703i	$-0.909764\pi$
$719$		−	15.0000i		−	0.559406i	0.960087	−	0.279703i	$-0.0902359\pi$
$720$		0			0
$721$		0			0
$722$	−30.0000			−1.11648
$723$	−12.0000			−0.446285
$724$			10.0000i			0.371647i
$725$		0			0
$726$			21.0000i			0.779383i
$727$	−13.0000			−0.482143			−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000			−0.482143			−0.241072	+	0.970507i	$0.577499\pi$
$728$			4.00000i			0.148250i
$729$	27.0000			1.00000
$730$		0			0
$731$	32.0000	+	8.00000i	1.18356	+	0.295891i
$732$	−3.00000			−0.110883
$733$	−6.00000			−0.221615			−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000			−0.221615			−0.110808	+	0.993842i	$0.535344\pi$
$734$		−	10.0000i		−	0.369107i
$735$		0			0
$736$		−	6.00000i		−	0.221163i
$737$			20.0000i			0.736709i
$738$			48.0000i			1.76690i
$739$	−27.0000			−0.993211			−0.496606	−	0.867976i	$-0.665420\pi$
$739$	−27.0000			−0.993211			−0.496606	+	0.867976i	$0.665420\pi$
$740$		0			0
$741$			21.0000i			0.771454i
$742$		−	44.0000i		−	1.61529i
$743$			44.0000i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$			44.0000i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$	21.0000			0.769897
$745$		0			0
$746$	10.0000			0.366126
$747$	36.0000			1.31717
$748$	−2.00000	+	8.00000i	−0.0731272	+	0.292509i
$749$	64.0000			2.33851
$750$		0			0
$751$		−	45.0000i		−	1.64207i	−0.570875	−	0.821037i	$-0.693396\pi$
$751$		−	45.0000i		−	1.64207i	0.570875	−	0.821037i	$-0.306604\pi$
$752$	−9.00000			−0.328196
$753$			24.0000i			0.874609i
$754$			3.00000i			0.109254i
$755$		0			0
$756$	−36.0000			−1.30931
$757$	−43.0000			−1.56286			−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000			−1.56286			−0.781431	+	0.623992i	$0.785510\pi$
$758$		−	8.00000i		−	0.290573i
$759$			36.0000i			1.30672i
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$			45.0000i			1.63018i
$763$	−76.0000			−2.75138
$764$	10.0000			0.361787
$765$		0			0
$766$	−19.0000			−0.686498
$767$	5.00000			0.180540
$768$		−	3.00000i		−	0.108253i
$769$	−23.0000			−0.829401			−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000			−0.829401			−0.414701	+	0.909958i	$0.636114\pi$
$770$		0			0
$771$			72.0000i			2.59302i
$772$		−	2.00000i		−	0.0719816i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$	48.0000			1.72532
$775$		0			0
$776$			1.00000i			0.0358979i
$777$		−	24.0000i		−	0.860995i
$778$		0			0
$779$			56.0000i			2.00641i
$780$		0			0
$781$	−2.00000			−0.0715656
$782$	6.00000	−	24.0000i	0.214560	−	0.858238i
$783$	−27.0000			−0.964901
$784$	−9.00000			−0.321429
$785$		0			0
$786$		0			0
$787$			17.0000i			0.605985i	0.952993	+	0.302992i	$0.0979856\pi$
$787$			17.0000i			0.605985i	−0.952993	+	0.302992i	$0.902014\pi$
$788$			18.0000i			0.641223i
$789$		−	27.0000i		−	0.961225i
$790$		0			0
$791$	−4.00000			−0.142224
$792$			12.0000i			0.426401i
$793$			1.00000i			0.0355110i
$794$		−	4.00000i		−	0.141955i
$795$		0			0
$796$			11.0000i			0.389885i
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$	−84.0000			−2.97357
$799$	−36.0000	−	9.00000i	−1.27359	−	0.318397i
$800$		0			0
$801$	6.00000			0.212000
$802$			24.0000i			0.847469i
$803$	−18.0000			−0.635206
$804$		−	30.0000i		−	1.05802i
$805$		0			0
$806$		−	7.00000i		−	0.246564i
$807$	75.0000			2.64013
$808$	2.00000			0.0703598
$809$			16.0000i			0.562530i	0.959630	+	0.281265i	$0.0907540\pi$
$809$			16.0000i			0.562530i	−0.959630	+	0.281265i	$0.909246\pi$
$810$		0			0
$811$			32.0000i			1.12367i	0.827249	+	0.561836i	$0.189905\pi$
$811$			32.0000i			1.12367i	−0.827249	+	0.561836i	$0.810095\pi$
$812$	−12.0000			−0.421117
$813$		−	36.0000i		−	1.26258i
$814$	−4.00000			−0.140200
$815$		0			0
$816$	3.00000	−	12.0000i	0.105021	−	0.420084i
$817$	56.0000			1.95919
$818$	19.0000			0.664319
$819$			24.0000i			0.838628i
$820$		0			0
$821$			21.0000i			0.732905i	0.930437	+	0.366453i	$0.119428\pi$
$821$			21.0000i			0.732905i	−0.930437	+	0.366453i	$0.880572\pi$
$822$		−	24.0000i		−	0.837096i
$823$			2.00000i			0.0697156i	0.999392	+	0.0348578i	$0.0110978\pi$
$823$			2.00000i			0.0697156i	−0.999392	+	0.0348578i	$0.988902\pi$
$824$		0			0
$825$		0			0
$826$			20.0000i			0.695889i
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$		−	36.0000i		−	1.25109i
$829$	24.0000			0.833554			0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000			0.833554			0.416777	+	0.909009i	$0.363160\pi$
$830$		0			0
$831$	6.00000			0.208138
$832$	−1.00000			−0.0346688
$833$	−36.0000	−	9.00000i	−1.24733	−	0.311832i
$834$	42.0000			1.45434
$835$		0			0
$836$			14.0000i			0.484200i
$837$	63.0000			2.17760
$838$			10.0000i			0.345444i
$839$		−	43.0000i		−	1.48452i	−0.670109	−	0.742262i	$-0.733753\pi$
$839$		−	43.0000i		−	1.48452i	0.670109	−	0.742262i	$-0.266247\pi$
$840$		0			0
$841$	20.0000			0.689655
$842$	−6.00000			−0.206774
$843$		−	27.0000i		−	0.929929i
$844$		−	10.0000i		−	0.344214i
$845$		0			0
$846$	−54.0000			−1.85656
$847$			28.0000i			0.962091i
$848$	11.0000			0.377742
$849$	−21.0000			−0.720718
$850$		0			0
$851$	12.0000			0.411355
$852$	3.00000			0.102778
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$	−4.00000			−0.136877
$855$		0			0
$856$			16.0000i			0.546869i
$857$		−	23.0000i		−	0.785665i	−0.919610	−	0.392833i	$-0.871495\pi$
$857$		−	23.0000i		−	0.785665i	0.919610	−	0.392833i	$-0.128505\pi$
$858$	6.00000			0.204837
$859$	15.0000			0.511793			0.255897	−	0.966704i	$-0.417629\pi$
$859$	15.0000			0.511793			0.255897	+	0.966704i	$0.417629\pi$
$860$		0			0
$861$			96.0000i			3.27167i
$862$		−	8.00000i		−	0.272481i
$863$	20.0000			0.680808			0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000			0.680808			0.340404	+	0.940279i	$0.389436\pi$
$864$		−	9.00000i		−	0.306186i
$865$		0			0
$866$	−8.00000			−0.271851
$867$	24.0000	−	45.0000i	0.815083	−	1.52828i
$868$	28.0000			0.950382
$869$		0			0
$870$		0			0
$871$	−10.0000			−0.338837
$872$		−	19.0000i		−	0.643421i
$873$			6.00000i			0.203069i
$874$		−	42.0000i		−	1.42067i
$875$		0			0
$876$	27.0000			0.912245
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$			32.0000i			1.07995i
$879$		−	15.0000i		−	0.505937i
$880$		0			0
$881$		−	6.00000i		−	0.202145i	−0.994879	−	0.101073i	$-0.967773\pi$
$881$		−	6.00000i		−	0.202145i	0.994879	−	0.101073i	$-0.0322274\pi$
$882$	−54.0000			−1.81827
$883$	52.0000			1.74994			0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000			1.74994			0.874970	+	0.484178i	$0.160881\pi$
$884$	−4.00000	−	1.00000i	−0.134535	−	0.0336336i
$885$		0			0
$886$	−24.0000			−0.806296
$887$			36.0000i			1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$			36.0000i			1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	6.00000			0.201347
$889$			60.0000i			2.01234i
$890$		0			0
$891$			18.0000i			0.603023i
$892$	−29.0000			−0.970992
$893$	−63.0000			−2.10821
$894$		−	18.0000i		−	0.602010i
$895$		0			0
$896$		−	4.00000i		−	0.133631i
$897$	−18.0000			−0.601003
$898$			30.0000i			1.00111i
$899$	21.0000			0.700389
$900$		0			0
$901$	44.0000	+	11.0000i	1.46585	+	0.366463i
$902$	16.0000			0.532742
$903$	96.0000			3.19468
$904$		−	1.00000i		−	0.0332595i
$905$		0			0
$906$			66.0000i			2.19270i
$907$			45.0000i			1.49420i	0.664711	+	0.747100i	$0.268555\pi$
$907$			45.0000i			1.49420i	−0.664711	+	0.747100i	$0.731445\pi$
$908$		−	27.0000i		−	0.896026i
$909$	12.0000			0.398015
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	21.0000i		−	0.695379i
$913$		−	12.0000i		−	0.397142i
$914$	−36.0000			−1.19077
$915$		0			0
$916$	−4.00000			−0.132164
$917$		0			0
$918$	9.00000	−	36.0000i	0.297044	−	1.18818i
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$			66.0000i			2.17477i
$922$	6.00000			0.197599
$923$		−	1.00000i		−	0.0329154i
$924$			24.0000i			0.789542i
$925$		0			0
$926$	9.00000			0.295758
$927$		0			0
$928$		−	3.00000i		−	0.0984798i
$929$		−	12.0000i		−	0.393707i	−0.980433	−	0.196854i	$-0.936928\pi$
$929$		−	12.0000i		−	0.393707i	0.980433	−	0.196854i	$-0.0630724\pi$
$930$		0			0
$931$	−63.0000			−2.06474
$932$			3.00000i			0.0982683i
$933$	−48.0000			−1.57145
$934$	26.0000			0.850746
$935$		0			0
$936$	−6.00000			−0.196116
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		−	40.0000i		−	1.30605i
$939$	−78.0000			−2.54543
$940$		0			0
$941$		−	7.00000i		−	0.228193i	−0.993470	−	0.114097i	$-0.963603\pi$
$941$		−	7.00000i		−	0.228193i	0.993470	−	0.114097i	$-0.0363974\pi$
$942$		−	6.00000i		−	0.195491i
$943$	−48.0000			−1.56310
$944$	−5.00000			−0.162736
$945$		0			0
$946$		−	16.0000i		−	0.520205i
$947$		−	41.0000i		−	1.33232i	−0.745808	−	0.666160i	$-0.767937\pi$
$947$		−	41.0000i		−	1.33232i	0.745808	−	0.666160i	$-0.232063\pi$
$948$		0			0
$949$		−	9.00000i		−	0.292152i
$950$		0			0
$951$	60.0000			1.94563
$952$	4.00000	−	16.0000i	0.129641	−	0.518563i
$953$	34.0000			1.10137			0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000			1.10137			0.550684	+	0.834714i	$0.314367\pi$
$954$	66.0000			2.13683
$955$		0			0
$956$	−20.0000			−0.646846
$957$			18.0000i			0.581857i
$958$		−	15.0000i		−	0.484628i
$959$		−	32.0000i		−	1.03333i
$960$		0			0
$961$	−18.0000			−0.580645
$962$		−	2.00000i		−	0.0644826i
$963$			96.0000i			3.09356i
$964$		−	4.00000i		−	0.128831i
$965$		0			0
$966$		−	72.0000i		−	2.31656i
$967$	−32.0000			−1.02905			−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000			−1.02905			−0.514525	+	0.857475i	$0.672032\pi$
$968$	−7.00000			−0.224989
$969$	21.0000	−	84.0000i	0.674617	−	2.69847i
$970$		0			0
$971$	−29.0000			−0.930654			−0.465327	−	0.885139i	$-0.654063\pi$
$971$	−29.0000			−0.930654			−0.465327	+	0.885139i	$0.654063\pi$
$972$		0			0
$973$	56.0000			1.79528
$974$			40.0000i			1.28168i
$975$		0			0
$976$		−	1.00000i		−	0.0320092i
$977$	10.0000			0.319928			0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000			0.319928			0.159964	+	0.987123i	$0.448862\pi$
$978$	−12.0000			−0.383718
$979$		−	2.00000i		−	0.0639203i
$980$		0			0
$981$		−	114.000i		−	3.63974i
$982$	15.0000			0.478669
$983$		−	4.00000i		−	0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$		−	4.00000i		−	0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$	−24.0000			−0.765092
$985$		0			0
$986$	3.00000	−	12.0000i	0.0955395	−	0.382158i
$987$	−108.000			−3.43768
$988$	−7.00000			−0.222700
$989$			48.0000i			1.52631i
$990$		0			0
$991$		−	23.0000i		−	0.730619i	−0.930886	−	0.365310i	$-0.880963\pi$
$991$		−	23.0000i		−	0.730619i	0.930886	−	0.365310i	$-0.119037\pi$
$992$			7.00000i			0.222250i
$993$			57.0000i			1.80884i
$994$	4.00000			0.126872
$995$		0			0
$996$			18.0000i			0.570352i
$997$		−	12.0000i		−	0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$		−	12.0000i		−	0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$			20.0000i			0.633089i
$999$	18.0000			0.569495

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

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a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.b.a.101.1		2
5.2	odd	4		850.2.d.a.849.1		2
5.3	odd	4		850.2.d.h.849.2		2
5.4	even	2		170.2.b.b.101.2	yes	2
15.14	odd	2		1530.2.c.b.271.1		2
17.16	even	2	inner	850.2.b.a.101.2		2
20.19	odd	2		1360.2.c.a.1121.1		2
85.4	even	4		2890.2.a.l.1.1		1
85.33	odd	4		850.2.d.a.849.2		2
85.64	even	4		2890.2.a.a.1.1		1
85.67	odd	4		850.2.d.h.849.1		2
85.84	even	2		170.2.b.b.101.1	✓	2
255.254	odd	2		1530.2.c.b.271.2		2
340.339	odd	2		1360.2.c.a.1121.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.b.b.101.1	✓	2	85.84	even	2
170.2.b.b.101.2	yes	2	5.4	even	2
850.2.b.a.101.1		2	1.1	even	1	trivial
850.2.b.a.101.2		2	17.16	even	2	inner
850.2.d.a.849.1		2	5.2	odd	4
850.2.d.a.849.2		2	85.33	odd	4
850.2.d.h.849.1		2	85.67	odd	4
850.2.d.h.849.2		2	5.3	odd	4
1360.2.c.a.1121.1		2	20.19	odd	2
1360.2.c.a.1121.2		2	340.339	odd	2
1530.2.c.b.271.1		2	15.14	odd	2
1530.2.c.b.271.2		2	255.254	odd	2
2890.2.a.a.1.1		1	85.64	even	4
2890.2.a.l.1.1		1	85.4	even	4

Overview	Random
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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

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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	850.2.b.a.101.1

Level	$850$
Weight	$2$
Character	850.101
Analytic conductor	$6.787$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$