LMFDB - Embedded newform 1638.2.c.e.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(883,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1638.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.c (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:	$13.0794958511$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1638.883
Dual form			1638.2.c.e.883.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{10} -3.00000i q^{11} +(3.00000 - 2.00000i) q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +3.00000i q^{19} -1.00000i q^{20} -3.00000 q^{22} -1.00000 q^{23} +4.00000 q^{25} +(-2.00000 - 3.00000i) q^{26} +1.00000i q^{28} +1.00000 q^{29} -8.00000i q^{31} -1.00000i q^{32} +7.00000i q^{34} +1.00000 q^{35} -1.00000i q^{37} +3.00000 q^{38} -1.00000 q^{40} -4.00000i q^{41} -5.00000 q^{43} +3.00000i q^{44} +1.00000i q^{46} -1.00000 q^{49} -4.00000i q^{50} +(-3.00000 + 2.00000i) q^{52} +6.00000 q^{53} +3.00000 q^{55} +1.00000 q^{56} -1.00000i q^{58} -10.0000i q^{59} -13.0000 q^{61} -8.00000 q^{62} -1.00000 q^{64} +(2.00000 + 3.00000i) q^{65} -8.00000i q^{67} +7.00000 q^{68} -1.00000i q^{70} -6.00000i q^{71} -13.0000i q^{73} -1.00000 q^{74} -3.00000i q^{76} -3.00000 q^{77} -12.0000 q^{79} +1.00000i q^{80} -4.00000 q^{82} -2.00000i q^{83} -7.00000i q^{85} +5.00000i q^{86} +3.00000 q^{88} -12.0000i q^{89} +(-2.00000 - 3.00000i) q^{91} +1.00000 q^{92} -3.00000 q^{95} +6.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 2 q^{10} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 14 q^{17} - 6 q^{22} - 2 q^{23} + 8 q^{25} - 4 q^{26} + 2 q^{29} + 2 q^{35} + 6 q^{38} - 2 q^{40} - 10 q^{43} - 2 q^{49} - 6 q^{52} + 12 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 16 q^{62} - 2 q^{64} + 4 q^{65} + 14 q^{68} - 2 q^{74} - 6 q^{77} - 24 q^{79} - 8 q^{82} + 6 q^{88} - 4 q^{91} + 2 q^{92} - 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^10 + 6 * q^13 - 2 * q^14 + 2 * q^16 - 14 * q^17 - 6 * q^22 - 2 * q^23 + 8 * q^25 - 4 * q^26 + 2 * q^29 + 2 * q^35 + 6 * q^38 - 2 * q^40 - 10 * q^43 - 2 * q^49 - 6 * q^52 + 12 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 16 * q^62 - 2 * q^64 + 4 * q^65 + 14 * q^68 - 2 * q^74 - 6 * q^77 - 24 * q^79 - 8 * q^82 + 6 * q^88 - 4 * q^91 + 2 * q^92 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.00000			0.316228
$11$		−	3.00000i		−	0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$		−	3.00000i		−	0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$		0			0
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$13$	3.00000	−	2.00000i	0.832050	−	0.554700i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	−7.00000			−1.69775			−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000			−1.69775			−0.848875	+	0.528594i	$0.822719\pi$
$18$		0			0
$19$			3.00000i			0.688247i	0.938924	+	0.344124i	$0.111824\pi$
$19$			3.00000i			0.688247i	−0.938924	+	0.344124i	$0.888176\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$	−3.00000			−0.639602
$23$	−1.00000			−0.208514			−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000			−0.208514			−0.104257	+	0.994550i	$0.533247\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$	−2.00000	−	3.00000i	−0.392232	−	0.588348i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			7.00000i			1.20049i
$35$	1.00000			0.169031
$36$		0			0
$37$		−	1.00000i		−	0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$		−	1.00000i		−	0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	3.00000			0.486664
$39$		0			0
$40$	−1.00000			−0.158114
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$			3.00000i			0.452267i
$45$		0			0
$46$			1.00000i			0.147442i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	4.00000i		−	0.565685i
$51$		0			0
$52$	−3.00000	+	2.00000i	−0.416025	+	0.277350i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$	1.00000			0.133631
$57$		0			0
$58$		−	1.00000i		−	0.131306i
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$	−8.00000			−1.01600
$63$		0			0
$64$	−1.00000			−0.125000
$65$	2.00000	+	3.00000i	0.248069	+	0.372104i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	7.00000			0.848875
$69$		0			0
$70$		−	1.00000i		−	0.119523i
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$		−	13.0000i		−	1.52153i	−0.649025	−	0.760767i	$-0.724823\pi$
$73$		−	13.0000i		−	1.52153i	0.649025	−	0.760767i	$-0.275177\pi$
$74$	−1.00000			−0.116248
$75$		0			0
$76$		−	3.00000i		−	0.344124i
$77$	−3.00000			−0.341882
$78$		0			0
$79$	−12.0000			−1.35011			−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000			−1.35011			−0.675053	+	0.737769i	$0.735879\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$	−4.00000			−0.441726
$83$		−	2.00000i		−	0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$		−	2.00000i		−	0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$		0			0
$85$		−	7.00000i		−	0.759257i
$86$			5.00000i			0.539164i
$87$		0			0
$88$	3.00000			0.319801
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	−2.00000	−	3.00000i	−0.209657	−	0.314485i
$92$	1.00000			0.104257
$93$		0			0
$94$		0			0
$95$	−3.00000			−0.307794
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	−4.00000			−0.400000
$101$	−14.0000			−1.39305			−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000			−1.39305			−0.696526	+	0.717532i	$0.745272\pi$
$102$		0			0
$103$	−1.00000			−0.0985329			−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000			−0.0985329			−0.0492665	+	0.998786i	$0.515688\pi$
$104$	2.00000	+	3.00000i	0.196116	+	0.294174i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	−2.00000			−0.193347			−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000			−0.193347			−0.0966736	+	0.995316i	$0.530820\pi$
$108$		0			0
$109$			7.00000i			0.670478i	0.942133	+	0.335239i	$0.108817\pi$
$109$			7.00000i			0.670478i	−0.942133	+	0.335239i	$0.891183\pi$
$110$		−	3.00000i		−	0.286039i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$		−	1.00000i		−	0.0932505i
$116$	−1.00000			−0.0928477
$117$		0			0
$118$	−10.0000			−0.920575
$119$			7.00000i			0.641689i
$120$		0			0
$121$	2.00000			0.181818
$122$			13.0000i			1.17696i
$123$		0			0
$124$			8.00000i			0.718421i
$125$			9.00000i			0.804984i
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	3.00000	−	2.00000i	0.263117	−	0.175412i
$131$	17.0000			1.48530			0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000			1.48530			0.742648	+	0.669681i	$0.233569\pi$
$132$		0			0
$133$	3.00000			0.260133
$134$	−8.00000			−0.691095
$135$		0			0
$136$		−	7.00000i		−	0.600245i
$137$			17.0000i			1.45241i	0.687479	+	0.726204i	$0.258717\pi$
$137$			17.0000i			1.45241i	−0.687479	+	0.726204i	$0.741283\pi$
$138$		0			0
$139$	−8.00000			−0.678551			−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000			−0.678551			−0.339276	+	0.940687i	$0.610182\pi$
$140$	−1.00000			−0.0845154
$141$		0			0
$142$	−6.00000			−0.503509
$143$	−6.00000	−	9.00000i	−0.501745	−	0.752618i
$144$		0			0
$145$			1.00000i			0.0830455i
$146$	−13.0000			−1.07589
$147$		0			0
$148$			1.00000i			0.0821995i
$149$			22.0000i			1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$			22.0000i			1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$		0			0
$151$		−	11.0000i		−	0.895167i	−0.894242	−	0.447584i	$-0.852285\pi$
$151$		−	11.0000i		−	0.895167i	0.894242	−	0.447584i	$-0.147715\pi$
$152$	−3.00000			−0.243332
$153$		0			0
$154$			3.00000i			0.241747i
$155$	8.00000			0.642575
$156$		0			0
$157$	19.0000			1.51637			0.758183	−	0.652042i	$-0.226088\pi$
$157$	19.0000			1.51637			0.758183	+	0.652042i	$0.226088\pi$
$158$			12.0000i			0.954669i
$159$		0			0
$160$	1.00000			0.0790569
$161$			1.00000i			0.0788110i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$			4.00000i			0.312348i
$165$		0			0
$166$	−2.00000			−0.155230
$167$		−	15.0000i		−	1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$		−	15.0000i		−	1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$	−7.00000			−0.536875
$171$		0			0
$172$	5.00000			0.381246
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		−	4.00000i		−	0.302372i
$176$		−	3.00000i		−	0.226134i
$177$		0			0
$178$	−12.0000			−0.899438
$179$	6.00000			0.448461			0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000			0.448461			0.224231	+	0.974536i	$0.428013\pi$
$180$		0			0
$181$	−26.0000			−1.93256			−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000			−1.93256			−0.966282	+	0.257485i	$0.917106\pi$
$182$	−3.00000	+	2.00000i	−0.222375	+	0.148250i
$183$		0			0
$184$		−	1.00000i		−	0.0737210i
$185$	1.00000			0.0735215
$186$		0			0
$187$			21.0000i			1.53567i
$188$		0			0
$189$		0			0
$190$			3.00000i			0.217643i
$191$	9.00000			0.651217			0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000			0.651217			0.325609	+	0.945505i	$0.394431\pi$
$192$		0			0
$193$		−	8.00000i		−	0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$		−	8.00000i		−	0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	6.00000			0.430775
$195$		0			0
$196$	1.00000			0.0714286
$197$		−	14.0000i		−	0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$		−	14.0000i		−	0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$		0			0
$199$	9.00000			0.637993			0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000			0.637993			0.318997	+	0.947756i	$0.396654\pi$
$200$			4.00000i			0.282843i
$201$		0			0
$202$			14.0000i			0.985037i
$203$		−	1.00000i		−	0.0701862i
$204$		0			0
$205$	4.00000			0.279372
$206$			1.00000i			0.0696733i
$207$		0			0
$208$	3.00000	−	2.00000i	0.208013	−	0.138675i
$209$	9.00000			0.622543
$210$		0			0
$211$	15.0000			1.03264			0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000			1.03264			0.516321	+	0.856395i	$0.327301\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$			2.00000i			0.136717i
$215$		−	5.00000i		−	0.340997i
$216$		0			0
$217$	−8.00000			−0.543075
$218$	7.00000			0.474100
$219$		0			0
$220$	−3.00000			−0.202260
$221$	−21.0000	+	14.0000i	−1.41261	+	0.941742i
$222$		0			0
$223$			4.00000i			0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$			4.00000i			0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$		−	10.0000i		−	0.665190i
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	−1.00000			−0.0659380
$231$		0			0
$232$			1.00000i			0.0656532i
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$		0			0
$236$			10.0000i			0.650945i
$237$		0			0
$238$	7.00000			0.453743
$239$		−	8.00000i		−	0.517477i	−0.965947	−	0.258738i	$-0.916693\pi$
$239$		−	8.00000i		−	0.517477i	0.965947	−	0.258738i	$-0.0833068\pi$
$240$		0			0
$241$			26.0000i			1.67481i	0.546585	+	0.837404i	$0.315928\pi$
$241$			26.0000i			1.67481i	−0.546585	+	0.837404i	$0.684072\pi$
$242$		−	2.00000i		−	0.128565i
$243$		0			0
$244$	13.0000			0.832240
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	6.00000	+	9.00000i	0.381771	+	0.572656i
$248$	8.00000			0.508001
$249$		0			0
$250$	9.00000			0.569210
$251$	−17.0000			−1.07303			−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000			−1.07303			−0.536515	+	0.843891i	$0.680260\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$			4.00000i			0.250982i
$255$		0			0
$256$	1.00000			0.0625000
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	−1.00000			−0.0621370
$260$	−2.00000	−	3.00000i	−0.124035	−	0.186052i
$261$		0			0
$262$		−	17.0000i		−	1.05026i
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		−	3.00000i		−	0.183942i
$267$		0			0
$268$			8.00000i			0.488678i
$269$	4.00000			0.243884			0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000			0.243884			0.121942	+	0.992537i	$0.461088\pi$
$270$		0			0
$271$			20.0000i			1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$			20.0000i			1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$	−7.00000			−0.424437
$273$		0			0
$274$	17.0000			1.02701
$275$		−	12.0000i		−	0.723627i
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$			8.00000i			0.479808i
$279$		0			0
$280$			1.00000i			0.0597614i
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$			6.00000i			0.356034i
$285$		0			0
$286$	−9.00000	+	6.00000i	−0.532181	+	0.354787i
$287$	−4.00000			−0.236113
$288$		0			0
$289$	32.0000			1.88235
$290$	1.00000			0.0587220
$291$		0			0
$292$			13.0000i			0.760767i
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	10.0000			0.582223
$296$	1.00000			0.0581238
$297$		0			0
$298$	22.0000			1.27443
$299$	−3.00000	+	2.00000i	−0.173494	+	0.115663i
$300$		0			0
$301$			5.00000i			0.288195i
$302$	−11.0000			−0.632979
$303$		0			0
$304$			3.00000i			0.172062i
$305$		−	13.0000i		−	0.744378i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	3.00000			0.170941
$309$		0			0
$310$		−	8.00000i		−	0.454369i
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$	−26.0000			−1.46961			−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000			−1.46961			−0.734803	+	0.678280i	$0.762726\pi$
$314$		−	19.0000i		−	1.07223i
$315$		0			0
$316$	12.0000			0.675053
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$		−	3.00000i		−	0.167968i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	1.00000			0.0557278
$323$		−	21.0000i		−	1.16847i
$324$		0			0
$325$	12.0000	−	8.00000i	0.665640	−	0.443760i
$326$	4.00000			0.221540
$327$		0			0
$328$	4.00000			0.220863
$329$		0			0
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$			2.00000i			0.109764i
$333$		0			0
$334$	−15.0000			−0.820763
$335$	8.00000			0.437087
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$	−12.0000	−	5.00000i	−0.652714	−	0.271964i
$339$		0			0
$340$			7.00000i			0.379628i
$341$	−24.0000			−1.29967
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		−	5.00000i		−	0.269582i
$345$		0			0
$346$		−	6.00000i		−	0.322562i
$347$	24.0000			1.28839			0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000			1.28839			0.644194	+	0.764862i	$0.277193\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	−4.00000			−0.213809
$351$		0			0
$352$	−3.00000			−0.159901
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	6.00000			0.318447
$356$			12.0000i			0.635999i
$357$		0			0
$358$		−	6.00000i		−	0.317110i
$359$			26.0000i			1.37223i	0.727494	+	0.686114i	$0.240685\pi$
$359$			26.0000i			1.37223i	−0.727494	+	0.686114i	$0.759315\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$			26.0000i			1.36653i
$363$		0			0
$364$	2.00000	+	3.00000i	0.104828	+	0.157243i
$365$	13.0000			0.680451
$366$		0			0
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$	−1.00000			−0.0521286
$369$		0			0
$370$		−	1.00000i		−	0.0519875i
$371$		−	6.00000i		−	0.311504i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$	21.0000			1.08588
$375$		0			0
$376$		0			0
$377$	3.00000	−	2.00000i	0.154508	−	0.103005i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	3.00000			0.153897
$381$		0			0
$382$		−	9.00000i		−	0.460480i
$383$		−	17.0000i		−	0.868659i	−0.900754	−	0.434330i	$-0.856985\pi$
$383$		−	17.0000i		−	0.868659i	0.900754	−	0.434330i	$-0.143015\pi$
$384$		0			0
$385$		−	3.00000i		−	0.152894i
$386$	−8.00000			−0.407189
$387$		0			0
$388$		−	6.00000i		−	0.304604i
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	7.00000			0.354005
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	−14.0000			−0.705310
$395$		−	12.0000i		−	0.603786i
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		−	9.00000i		−	0.451129i
$399$		0			0
$400$	4.00000			0.200000
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	−16.0000	−	24.0000i	−0.797017	−	1.19553i
$404$	14.0000			0.696526
$405$		0			0
$406$	−1.00000			−0.0496292
$407$	−3.00000			−0.148704
$408$		0			0
$409$		−	29.0000i		−	1.43396i	−0.697095	−	0.716979i	$-0.745524\pi$
$409$		−	29.0000i		−	1.43396i	0.697095	−	0.716979i	$-0.254476\pi$
$410$		−	4.00000i		−	0.197546i
$411$		0			0
$412$	1.00000			0.0492665
$413$	−10.0000			−0.492068
$414$		0			0
$415$	2.00000			0.0981761
$416$	−2.00000	−	3.00000i	−0.0980581	−	0.147087i
$417$		0			0
$418$		−	9.00000i		−	0.440204i
$419$	11.0000			0.537385			0.268693	−	0.963226i	$-0.413408\pi$
$419$	11.0000			0.537385			0.268693	+	0.963226i	$0.413408\pi$
$420$		0			0
$421$		−	34.0000i		−	1.65706i	−0.559946	−	0.828529i	$-0.689178\pi$
$421$		−	34.0000i		−	1.65706i	0.559946	−	0.828529i	$-0.310822\pi$
$422$		−	15.0000i		−	0.730189i
$423$		0			0
$424$			6.00000i			0.291386i
$425$	−28.0000			−1.35820
$426$		0			0
$427$			13.0000i			0.629114i
$428$	2.00000			0.0966736
$429$		0			0
$430$	−5.00000			−0.241121
$431$			14.0000i			0.674356i	0.941441	+	0.337178i	$0.109472\pi$
$431$			14.0000i			0.674356i	−0.941441	+	0.337178i	$0.890528\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$			8.00000i			0.384012i
$435$		0			0
$436$		−	7.00000i		−	0.335239i
$437$		−	3.00000i		−	0.143509i
$438$		0			0
$439$	27.0000			1.28864			0.644320	−	0.764756i	$-0.277141\pi$
$439$	27.0000			1.28864			0.644320	+	0.764756i	$0.277141\pi$
$440$			3.00000i			0.143019i
$441$		0			0
$442$	14.0000	+	21.0000i	0.665912	+	0.998868i
$443$	34.0000			1.61539			0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000			1.61539			0.807694	+	0.589601i	$0.200715\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$	4.00000			0.189405
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			9.00000i			0.424736i	0.977190	+	0.212368i	$0.0681176\pi$
$449$			9.00000i			0.424736i	−0.977190	+	0.212368i	$0.931882\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$	−10.0000			−0.470360
$453$		0			0
$454$	−20.0000			−0.938647
$455$	3.00000	−	2.00000i	0.140642	−	0.0937614i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	10.0000			0.467269
$459$		0			0
$460$			1.00000i			0.0466252i
$461$			27.0000i			1.25752i	0.777601	+	0.628758i	$0.216436\pi$
$461$			27.0000i			1.25752i	−0.777601	+	0.628758i	$0.783564\pi$
$462$		0			0
$463$		−	15.0000i		−	0.697109i	−0.937288	−	0.348555i	$-0.886673\pi$
$463$		−	15.0000i		−	0.697109i	0.937288	−	0.348555i	$-0.113327\pi$
$464$	1.00000			0.0464238
$465$		0			0
$466$			18.0000i			0.833834i
$467$	−9.00000			−0.416470			−0.208235	−	0.978079i	$-0.566772\pi$
$467$	−9.00000			−0.416470			−0.208235	+	0.978079i	$0.566772\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$	10.0000			0.460287
$473$			15.0000i			0.689701i
$474$		0			0
$475$			12.0000i			0.550598i
$476$		−	7.00000i		−	0.320844i
$477$		0			0
$478$	−8.00000			−0.365911
$479$			3.00000i			0.137073i	0.997649	+	0.0685367i	$0.0218330\pi$
$479$			3.00000i			0.137073i	−0.997649	+	0.0685367i	$0.978167\pi$
$480$		0			0
$481$	−2.00000	−	3.00000i	−0.0911922	−	0.136788i
$482$	26.0000			1.18427
$483$		0			0
$484$	−2.00000			−0.0909091
$485$	−6.00000			−0.272446
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		−	13.0000i		−	0.588482i
$489$		0			0
$490$	−1.00000			−0.0451754
$491$	40.0000			1.80517			0.902587	−	0.430507i	$-0.141665\pi$
$491$	40.0000			1.80517			0.902587	+	0.430507i	$0.141665\pi$
$492$		0			0
$493$	−7.00000			−0.315264
$494$	9.00000	−	6.00000i	0.404929	−	0.269953i
$495$		0			0
$496$		−	8.00000i		−	0.359211i
$497$	−6.00000			−0.269137
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		−	9.00000i		−	0.402492i
$501$		0			0
$502$			17.0000i			0.758747i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		−	14.0000i		−	0.622992i
$506$	3.00000			0.133366
$507$		0			0
$508$	4.00000			0.177471
$509$			27.0000i			1.19675i	0.801215	+	0.598377i	$0.204187\pi$
$509$			27.0000i			1.19675i	−0.801215	+	0.598377i	$0.795813\pi$
$510$		0			0
$511$	−13.0000			−0.575086
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	6.00000i		−	0.264649i
$515$		−	1.00000i		−	0.0440653i
$516$		0			0
$517$		0			0
$518$			1.00000i			0.0439375i
$519$		0			0
$520$	−3.00000	+	2.00000i	−0.131559	+	0.0877058i
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$	−17.0000			−0.742648
$525$		0			0
$526$		−	24.0000i		−	1.04645i
$527$			56.0000i			2.43940i
$528$		0			0
$529$	−22.0000			−0.956522
$530$	6.00000			0.260623
$531$		0			0
$532$	−3.00000			−0.130066
$533$	−8.00000	−	12.0000i	−0.346518	−	0.519778i
$534$		0			0
$535$		−	2.00000i		−	0.0864675i
$536$	8.00000			0.345547
$537$		0			0
$538$		−	4.00000i		−	0.172452i
$539$			3.00000i			0.129219i
$540$		0			0
$541$			25.0000i			1.07483i	0.843317	+	0.537417i	$0.180600\pi$
$541$			25.0000i			1.07483i	−0.843317	+	0.537417i	$0.819400\pi$
$542$	20.0000			0.859074
$543$		0			0
$544$			7.00000i			0.300123i
$545$	−7.00000			−0.299847
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		−	17.0000i		−	0.726204i
$549$		0			0
$550$	−12.0000			−0.511682
$551$			3.00000i			0.127804i
$552$		0			0
$553$			12.0000i			0.510292i
$554$		−	28.0000i		−	1.18961i
$555$		0			0
$556$	8.00000			0.339276
$557$		−	30.0000i		−	1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$		−	30.0000i		−	1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$		0			0
$559$	−15.0000	+	10.0000i	−0.634432	+	0.422955i
$560$	1.00000			0.0422577
$561$		0			0
$562$	10.0000			0.421825
$563$	−41.0000			−1.72794			−0.863972	−	0.503540i	$-0.832031\pi$
$563$	−41.0000			−1.72794			−0.863972	+	0.503540i	$0.832031\pi$
$564$		0			0
$565$			10.0000i			0.420703i
$566$			14.0000i			0.588464i
$567$		0			0
$568$	6.00000			0.251754
$569$	2.00000			0.0838444			0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000			0.0838444			0.0419222	+	0.999121i	$0.486652\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$	6.00000	+	9.00000i	0.250873	+	0.376309i
$573$		0			0
$574$			4.00000i			0.166957i
$575$	−4.00000			−0.166812
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$		−	32.0000i		−	1.33102i
$579$		0			0
$580$		−	1.00000i		−	0.0415227i
$581$	−2.00000			−0.0829740
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$	13.0000			0.537944
$585$		0			0
$586$	14.0000			0.578335
$587$		−	16.0000i		−	0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$		−	16.0000i		−	0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$		−	10.0000i		−	0.411693i
$591$		0			0
$592$		−	1.00000i		−	0.0410997i
$593$			16.0000i			0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$			16.0000i			0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$	−7.00000			−0.286972
$596$		−	22.0000i		−	0.901155i
$597$		0			0
$598$	2.00000	+	3.00000i	0.0817861	+	0.122679i
$599$	3.00000			0.122577			0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000			0.122577			0.0612883	+	0.998120i	$0.480479\pi$
$600$		0			0
$601$	4.00000			0.163163			0.0815817	−	0.996667i	$-0.474003\pi$
$601$	4.00000			0.163163			0.0815817	+	0.996667i	$0.474003\pi$
$602$	5.00000			0.203785
$603$		0			0
$604$			11.0000i			0.447584i
$605$			2.00000i			0.0813116i
$606$		0			0
$607$	−11.0000			−0.446476			−0.223238	−	0.974764i	$-0.571663\pi$
$607$	−11.0000			−0.446476			−0.223238	+	0.974764i	$0.571663\pi$
$608$	3.00000			0.121666
$609$		0			0
$610$	−13.0000			−0.526355
$611$		0			0
$612$		0			0
$613$			31.0000i			1.25208i	0.779792	+	0.626039i	$0.215325\pi$
$613$			31.0000i			1.25208i	−0.779792	+	0.626039i	$0.784675\pi$
$614$	16.0000			0.645707
$615$		0			0
$616$		−	3.00000i		−	0.120873i
$617$			37.0000i			1.48956i	0.667308	+	0.744782i	$0.267447\pi$
$617$			37.0000i			1.48956i	−0.667308	+	0.744782i	$0.732553\pi$
$618$		0			0
$619$		−	23.0000i		−	0.924448i	−0.886763	−	0.462224i	$-0.847052\pi$
$619$		−	23.0000i		−	0.924448i	0.886763	−	0.462224i	$-0.152948\pi$
$620$	−8.00000			−0.321288
$621$		0			0
$622$		−	4.00000i		−	0.160385i
$623$	−12.0000			−0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$			26.0000i			1.03917i
$627$		0			0
$628$	−19.0000			−0.758183
$629$			7.00000i			0.279108i
$630$		0			0
$631$		−	47.0000i		−	1.87104i	−0.353273	−	0.935520i	$-0.614931\pi$
$631$		−	47.0000i		−	1.87104i	0.353273	−	0.935520i	$-0.385069\pi$
$632$		−	12.0000i		−	0.477334i
$633$		0			0
$634$	−24.0000			−0.953162
$635$		−	4.00000i		−	0.158735i
$636$		0			0
$637$	−3.00000	+	2.00000i	−0.118864	+	0.0792429i
$638$	−3.00000			−0.118771
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$		−	39.0000i		−	1.53801i	−0.639243	−	0.769005i	$-0.720752\pi$
$643$		−	39.0000i		−	1.53801i	0.639243	−	0.769005i	$-0.279248\pi$
$644$		−	1.00000i		−	0.0394055i
$645$		0			0
$646$	−21.0000			−0.826234
$647$	−22.0000			−0.864909			−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000			−0.864909			−0.432455	+	0.901656i	$0.642352\pi$
$648$		0			0
$649$	−30.0000			−1.17760
$650$	−8.00000	−	12.0000i	−0.313786	−	0.470679i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$	35.0000			1.36966			0.684828	−	0.728705i	$-0.259877\pi$
$653$	35.0000			1.36966			0.684828	+	0.728705i	$0.259877\pi$
$654$		0			0
$655$			17.0000i			0.664245i
$656$		−	4.00000i		−	0.156174i
$657$		0			0
$658$		0			0
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$	8.00000			0.310929
$663$		0			0
$664$	2.00000			0.0776151
$665$			3.00000i			0.116335i
$666$		0			0
$667$	−1.00000			−0.0387202
$668$			15.0000i			0.580367i
$669$		0			0
$670$		−	8.00000i		−	0.309067i
$671$			39.0000i			1.50558i
$672$		0			0
$673$	−3.00000			−0.115642			−0.0578208	−	0.998327i	$-0.518415\pi$
$673$	−3.00000			−0.115642			−0.0578208	+	0.998327i	$0.518415\pi$
$674$			23.0000i			0.885927i
$675$		0			0
$676$	−5.00000	+	12.0000i	−0.192308	+	0.461538i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	6.00000			0.230259
$680$	7.00000			0.268438
$681$		0			0
$682$			24.0000i			0.919007i
$683$		−	9.00000i		−	0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$		−	9.00000i		−	0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$		0			0
$685$	−17.0000			−0.649537
$686$	1.00000			0.0381802
$687$		0			0
$688$	−5.00000			−0.190623
$689$	18.0000	−	12.0000i	0.685745	−	0.457164i
$690$		0			0
$691$		−	16.0000i		−	0.608669i	−0.952565	−	0.304334i	$-0.901566\pi$
$691$		−	16.0000i		−	0.608669i	0.952565	−	0.304334i	$-0.0984340\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$		−	24.0000i		−	0.911028i
$695$		−	8.00000i		−	0.303457i
$696$		0			0
$697$			28.0000i			1.06058i
$698$	−16.0000			−0.605609
$699$		0			0
$700$			4.00000i			0.151186i
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$	3.00000			0.113147
$704$			3.00000i			0.113067i
$705$		0			0
$706$	16.0000			0.602168
$707$			14.0000i			0.526524i
$708$		0			0
$709$		−	30.0000i		−	1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$		−	30.0000i		−	1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$		−	6.00000i		−	0.225176i
$711$		0			0
$712$	12.0000			0.449719
$713$			8.00000i			0.299602i
$714$		0			0
$715$	9.00000	−	6.00000i	0.336581	−	0.224387i
$716$	−6.00000			−0.224231
$717$		0			0
$718$	26.0000			0.970311
$719$	−20.0000			−0.745874			−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000			−0.745874			−0.372937	+	0.927857i	$0.621649\pi$
$720$		0			0
$721$			1.00000i			0.0372419i
$722$		−	10.0000i		−	0.372161i
$723$		0			0
$724$	26.0000			0.966282
$725$	4.00000			0.148556
$726$		0			0
$727$	35.0000			1.29808			0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000			1.29808			0.649039	+	0.760755i	$0.275171\pi$
$728$	3.00000	−	2.00000i	0.111187	−	0.0741249i
$729$		0			0
$730$		−	13.0000i		−	0.481152i
$731$	35.0000			1.29452
$732$		0			0
$733$		−	8.00000i		−	0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$		−	8.00000i		−	0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$		−	32.0000i		−	1.18114i
$735$		0			0
$736$			1.00000i			0.0368605i
$737$	−24.0000			−0.884051
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	−1.00000			−0.0367607
$741$		0			0
$742$	−6.00000			−0.220267
$743$		−	40.0000i		−	1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$		−	40.0000i		−	1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$		0			0
$745$	−22.0000			−0.806018
$746$		−	6.00000i		−	0.219676i
$747$		0			0
$748$		−	21.0000i		−	0.767836i
$749$			2.00000i			0.0730784i
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$		0			0
$754$	−2.00000	−	3.00000i	−0.0728357	−	0.109254i
$755$	11.0000			0.400331
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$	4.00000			0.145287
$759$		0			0
$760$		−	3.00000i		−	0.108821i
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	7.00000			0.253417
$764$	−9.00000			−0.325609
$765$		0			0
$766$	−17.0000			−0.614235
$767$	−20.0000	−	30.0000i	−0.722158	−	1.08324i
$768$		0			0
$769$			39.0000i			1.40638i	0.711004	+	0.703188i	$0.248241\pi$
$769$			39.0000i			1.40638i	−0.711004	+	0.703188i	$0.751759\pi$
$770$	−3.00000			−0.108112
$771$		0			0
$772$			8.00000i			0.287926i
$773$		−	21.0000i		−	0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$		−	21.0000i		−	0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$		0			0
$775$		−	32.0000i		−	1.14947i
$776$	−6.00000			−0.215387
$777$		0			0
$778$		−	18.0000i		−	0.645331i
$779$	12.0000			0.429945
$780$		0			0
$781$	−18.0000			−0.644091
$782$		−	7.00000i		−	0.250319i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			19.0000i			0.678139i
$786$		0			0
$787$			17.0000i			0.605985i	0.952993	+	0.302992i	$0.0979856\pi$
$787$			17.0000i			0.605985i	−0.952993	+	0.302992i	$0.902014\pi$
$788$			14.0000i			0.498729i
$789$		0			0
$790$	−12.0000			−0.426941
$791$		−	10.0000i		−	0.355559i
$792$		0			0
$793$	−39.0000	+	26.0000i	−1.38493	+	0.923287i
$794$	8.00000			0.283909
$795$		0			0
$796$	−9.00000			−0.318997
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$		0			0
$800$		−	4.00000i		−	0.141421i
$801$		0			0
$802$	6.00000			0.211867
$803$	−39.0000			−1.37628
$804$		0			0
$805$	−1.00000			−0.0352454
$806$	−24.0000	+	16.0000i	−0.845364	+	0.563576i
$807$		0			0
$808$		−	14.0000i		−	0.492518i
$809$	−52.0000			−1.82822			−0.914111	−	0.405463i	$-0.867110\pi$
$809$	−52.0000			−1.82822			−0.914111	+	0.405463i	$0.867110\pi$
$810$		0			0
$811$			7.00000i			0.245803i	0.992419	+	0.122902i	$0.0392200\pi$
$811$			7.00000i			0.245803i	−0.992419	+	0.122902i	$0.960780\pi$
$812$			1.00000i			0.0350931i
$813$		0			0
$814$			3.00000i			0.105150i
$815$	−4.00000			−0.140114
$816$		0			0
$817$		−	15.0000i		−	0.524784i
$818$	−29.0000			−1.01396
$819$		0			0
$820$	−4.00000			−0.139686
$821$		−	36.0000i		−	1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$		−	36.0000i		−	1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$		0			0
$823$	12.0000			0.418294			0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000			0.418294			0.209147	+	0.977884i	$0.432931\pi$
$824$		−	1.00000i		−	0.0348367i
$825$		0			0
$826$			10.0000i			0.347945i
$827$			9.00000i			0.312961i	0.987681	+	0.156480i	$0.0500148\pi$
$827$			9.00000i			0.312961i	−0.987681	+	0.156480i	$0.949985\pi$
$828$		0			0
$829$	−15.0000			−0.520972			−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000			−0.520972			−0.260486	+	0.965478i	$0.583883\pi$
$830$		−	2.00000i		−	0.0694210i
$831$		0			0
$832$	−3.00000	+	2.00000i	−0.104006	+	0.0693375i
$833$	7.00000			0.242536
$834$		0			0
$835$	15.0000			0.519096
$836$	−9.00000			−0.311272
$837$		0			0
$838$		−	11.0000i		−	0.379989i
$839$			28.0000i			0.966667i	0.875436	+	0.483334i	$0.160574\pi$
$839$			28.0000i			0.966667i	−0.875436	+	0.483334i	$0.839426\pi$
$840$		0			0
$841$	−28.0000			−0.965517
$842$	−34.0000			−1.17172
$843$		0			0
$844$	−15.0000			−0.516321
$845$	12.0000	+	5.00000i	0.412813	+	0.172005i
$846$		0			0
$847$		−	2.00000i		−	0.0687208i
$848$	6.00000			0.206041
$849$		0			0
$850$			28.0000i			0.960392i
$851$			1.00000i			0.0342796i
$852$		0			0
$853$			32.0000i			1.09566i	0.836590	+	0.547830i	$0.184546\pi$
$853$			32.0000i			1.09566i	−0.836590	+	0.547830i	$0.815454\pi$
$854$	13.0000			0.444851
$855$		0			0
$856$		−	2.00000i		−	0.0683586i
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$	22.0000			0.750630			0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000			0.750630			0.375315	+	0.926897i	$0.377534\pi$
$860$			5.00000i			0.170499i
$861$		0			0
$862$	14.0000			0.476842
$863$		−	28.0000i		−	0.953131i	−0.879139	−	0.476566i	$-0.841881\pi$
$863$		−	28.0000i		−	0.953131i	0.879139	−	0.476566i	$-0.158119\pi$
$864$		0			0
$865$			6.00000i			0.204006i
$866$			8.00000i			0.271851i
$867$		0			0
$868$	8.00000			0.271538
$869$			36.0000i			1.22122i
$870$		0			0
$871$	−16.0000	−	24.0000i	−0.542139	−	0.813209i
$872$	−7.00000			−0.237050
$873$		0			0
$874$	−3.00000			−0.101477
$875$	9.00000			0.304256
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		−	27.0000i		−	0.911206i
$879$		0			0
$880$	3.00000			0.101130
$881$	−51.0000			−1.71823			−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000			−1.71823			−0.859117	+	0.511780i	$0.828986\pi$
$882$		0			0
$883$	3.00000			0.100958			0.0504790	−	0.998725i	$-0.483925\pi$
$883$	3.00000			0.100958			0.0504790	+	0.998725i	$0.483925\pi$
$884$	21.0000	−	14.0000i	0.706306	−	0.470871i
$885$		0			0
$886$		−	34.0000i		−	1.14225i
$887$	−42.0000			−1.41022			−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000			−1.41022			−0.705111	+	0.709097i	$0.749103\pi$
$888$		0			0
$889$			4.00000i			0.134156i
$890$		−	12.0000i		−	0.402241i
$891$		0			0
$892$		−	4.00000i		−	0.133930i
$893$		0			0
$894$		0			0
$895$			6.00000i			0.200558i
$896$	1.00000			0.0334077
$897$		0			0
$898$	9.00000			0.300334
$899$		−	8.00000i		−	0.266815i
$900$		0			0
$901$	−42.0000			−1.39922
$902$			12.0000i			0.399556i
$903$		0			0
$904$			10.0000i			0.332595i
$905$		−	26.0000i		−	0.864269i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$			20.0000i			0.663723i
$909$		0			0
$910$	−2.00000	−	3.00000i	−0.0662994	−	0.0994490i
$911$	−57.0000			−1.88849			−0.944247	−	0.329238i	$-0.893208\pi$
$911$	−57.0000			−1.88849			−0.944247	+	0.329238i	$0.893208\pi$
$912$		0			0
$913$	−6.00000			−0.198571
$914$	28.0000			0.926158
$915$		0			0
$916$		−	10.0000i		−	0.330409i
$917$		−	17.0000i		−	0.561389i
$918$		0			0
$919$	−30.0000			−0.989609			−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000			−0.989609			−0.494804	+	0.869004i	$0.664760\pi$
$920$	1.00000			0.0329690
$921$		0			0
$922$	27.0000			0.889198
$923$	−12.0000	−	18.0000i	−0.394985	−	0.592477i
$924$		0			0
$925$		−	4.00000i		−	0.131519i
$926$	−15.0000			−0.492931
$927$		0			0
$928$		−	1.00000i		−	0.0328266i
$929$			42.0000i			1.37798i	0.724773	+	0.688988i	$0.241945\pi$
$929$			42.0000i			1.37798i	−0.724773	+	0.688988i	$0.758055\pi$
$930$		0			0
$931$		−	3.00000i		−	0.0983210i
$932$	18.0000			0.589610
$933$		0			0
$934$			9.00000i			0.294489i
$935$	−21.0000			−0.686773
$936$		0			0
$937$	4.00000			0.130674			0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000			0.130674			0.0653372	+	0.997863i	$0.479188\pi$
$938$			8.00000i			0.261209i
$939$		0			0
$940$		0			0
$941$		−	38.0000i		−	1.23876i	−0.785090	−	0.619382i	$-0.787383\pi$
$941$		−	38.0000i		−	1.23876i	0.785090	−	0.619382i	$-0.212617\pi$
$942$		0			0
$943$			4.00000i			0.130258i
$944$		−	10.0000i		−	0.325472i
$945$		0			0
$946$	15.0000			0.487692
$947$			25.0000i			0.812391i	0.913786	+	0.406195i	$0.133145\pi$
$947$			25.0000i			0.812391i	−0.913786	+	0.406195i	$0.866855\pi$
$948$		0			0
$949$	−26.0000	−	39.0000i	−0.843996	−	1.26599i
$950$	12.0000			0.389331
$951$		0			0
$952$	−7.00000			−0.226871
$953$	30.0000			0.971795			0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000			0.971795			0.485898	+	0.874016i	$0.338493\pi$
$954$		0			0
$955$			9.00000i			0.291233i
$956$			8.00000i			0.258738i
$957$		0			0
$958$	3.00000			0.0969256
$959$	17.0000			0.548959
$960$		0			0
$961$	−33.0000			−1.06452
$962$	−3.00000	+	2.00000i	−0.0967239	+	0.0644826i
$963$		0			0
$964$		−	26.0000i		−	0.837404i
$965$	8.00000			0.257529
$966$		0			0
$967$		−	43.0000i		−	1.38279i	−0.722478	−	0.691393i	$-0.756997\pi$
$967$		−	43.0000i		−	1.38279i	0.722478	−	0.691393i	$-0.243003\pi$
$968$			2.00000i			0.0642824i
$969$		0			0
$970$			6.00000i			0.192648i
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$	−24.0000			−0.769010
$975$		0			0
$976$	−13.0000			−0.416120
$977$		−	15.0000i		−	0.479893i	−0.970786	−	0.239946i	$-0.922870\pi$
$977$		−	15.0000i		−	0.479893i	0.970786	−	0.239946i	$-0.0771298\pi$
$978$		0			0
$979$	−36.0000			−1.15056
$980$			1.00000i			0.0319438i
$981$		0			0
$982$		−	40.0000i		−	1.27645i
$983$			51.0000i			1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$			51.0000i			1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$		0			0
$985$	14.0000			0.446077
$986$			7.00000i			0.222925i
$987$		0			0
$988$	−6.00000	−	9.00000i	−0.190885	−	0.286328i
$989$	5.00000			0.158991
$990$		0			0
$991$	−20.0000			−0.635321			−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000			−0.635321			−0.317660	+	0.948205i	$0.602897\pi$
$992$	−8.00000			−0.254000
$993$		0			0
$994$			6.00000i			0.190308i
$995$			9.00000i			0.285319i
$996$		0			0
$997$	18.0000			0.570066			0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000			0.570066			0.285033	+	0.958518i	$0.407995\pi$
$998$	14.0000			0.443162
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.c.e.883.1		2
3.2	odd	2		546.2.c.c.337.2	yes	2
12.11	even	2		4368.2.h.h.337.1		2
13.12	even	2	inner	1638.2.c.e.883.2		2
21.20	even	2		3822.2.c.b.883.2		2
39.5	even	4		7098.2.a.y.1.1		1
39.8	even	4		7098.2.a.o.1.1		1
39.38	odd	2		546.2.c.c.337.1	✓	2
156.155	even	2		4368.2.h.h.337.2		2
273.272	even	2		3822.2.c.b.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.c.337.1	✓	2	39.38	odd	2
546.2.c.c.337.2	yes	2	3.2	odd	2
1638.2.c.e.883.1		2	1.1	even	1	trivial
1638.2.c.e.883.2		2	13.12	even	2	inner
3822.2.c.b.883.1		2	273.272	even	2
3822.2.c.b.883.2		2	21.20	even	2
4368.2.h.h.337.1		2	12.11	even	2
4368.2.h.h.337.2		2	156.155	even	2
7098.2.a.o.1.1		1	39.8	even	4
7098.2.a.y.1.1		1	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{10} -3.00000i q^{11} +(3.00000 - 2.00000i) q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.00000 q^{17} +3.00000i q^{19} -1.00000i q^{20} -3.00000 q^{22} -1.00000 q^{23} +4.00000 q^{25} +(-2.00000 - 3.00000i) q^{26} +1.00000i q^{28} +1.00000 q^{29} -8.00000i q^{31} -1.00000i q^{32} +7.00000i q^{34} +1.00000 q^{35} -1.00000i q^{37} +3.00000 q^{38} -1.00000 q^{40} -4.00000i q^{41} -5.00000 q^{43} +3.00000i q^{44} +1.00000i q^{46} -1.00000 q^{49} -4.00000i q^{50} +(-3.00000 + 2.00000i) q^{52} +6.00000 q^{53} +3.00000 q^{55} +1.00000 q^{56} -1.00000i q^{58} -10.0000i q^{59} -13.0000 q^{61} -8.00000 q^{62} -1.00000 q^{64} +(2.00000 + 3.00000i) q^{65} -8.00000i q^{67} +7.00000 q^{68} -1.00000i q^{70} -6.00000i q^{71} -13.0000i q^{73} -1.00000 q^{74} -3.00000i q^{76} -3.00000 q^{77} -12.0000 q^{79} +1.00000i q^{80} -4.00000 q^{82} -2.00000i q^{83} -7.00000i q^{85} +5.00000i q^{86} +3.00000 q^{88} -12.0000i q^{89} +(-2.00000 - 3.00000i) q^{91} +1.00000 q^{92} -3.00000 q^{95} +6.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{10} + 6 q^{13} - 2 q^{14} + 2 q^{16} - 14 q^{17} - 6 q^{22} - 2 q^{23} + 8 q^{25} - 4 q^{26} + 2 q^{29} + 2 q^{35} + 6 q^{38} - 2 q^{40} - 10 q^{43} - 2 q^{49} - 6 q^{52} + 12 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 16 q^{62} - 2 q^{64} + 4 q^{65} + 14 q^{68} - 2 q^{74} - 6 q^{77} - 24 q^{79} - 8 q^{82} + 6 q^{88} - 4 q^{91} + 2 q^{92} - 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^10 + 6 * q^13 - 2 * q^14 + 2 * q^16 - 14 * q^17 - 6 * q^22 - 2 * q^23 + 8 * q^25 - 4 * q^26 + 2 * q^29 + 2 * q^35 + 6 * q^38 - 2 * q^40 - 10 * q^43 - 2 * q^49 - 6 * q^52 + 12 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 16 * q^62 - 2 * q^64 + 4 * q^65 + 14 * q^68 - 2 * q^74 - 6 * q^77 - 24 * q^79 - 8 * q^82 + 6 * q^88 - 4 * q^91 + 2 * q^92 - 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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