LMFDB - Embedded newform 1638.2.c.f.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.f.883.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{10} +1.00000i q^{11} +(3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000i q^{19} -1.00000i q^{20} +1.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +(-2.00000 - 3.00000i) q^{26} -1.00000i q^{28} -9.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} +1.00000i q^{34} -1.00000 q^{35} +9.00000i q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000i q^{41} +7.00000 q^{43} -1.00000i q^{44} +3.00000i q^{46} +8.00000i q^{47} -1.00000 q^{49} -4.00000i q^{50} +(-3.00000 + 2.00000i) q^{52} +10.0000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +9.00000i q^{58} -6.00000i q^{59} +11.0000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +(2.00000 + 3.00000i) q^{65} +12.0000i q^{67} +1.00000 q^{68} +1.00000i q^{70} +6.00000i q^{71} -11.0000i q^{73} +9.00000 q^{74} -1.00000i q^{76} -1.00000 q^{77} -12.0000 q^{79} +1.00000i q^{80} +8.00000 q^{82} +6.00000i q^{83} -1.00000i q^{85} -7.00000i q^{86} -1.00000 q^{88} +12.0000i q^{89} +(2.00000 + 3.00000i) q^{91} +3.00000 q^{92} +8.00000 q^{94} -1.00000 q^{95} +2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{17} + 2 q^{22} - 6 q^{23} + 8 q^{25} - 4 q^{26} - 18 q^{29} - 2 q^{35} + 2 q^{38} - 2 q^{40} + 14 q^{43} - 2 q^{49} - 6 q^{52} + 20 q^{53} - 2 q^{55} - 2 q^{56} + 22 q^{61} + 8 q^{62} - 2 q^{64} + 4 q^{65} + 2 q^{68} + 18 q^{74} - 2 q^{77} - 24 q^{79} + 16 q^{82} - 2 q^{88} + 4 q^{91} + 6 q^{92} + 16 q^{94} - 2 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^10 + 6 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^22 - 6 * q^23 + 8 * q^25 - 4 * q^26 - 18 * q^29 - 2 * q^35 + 2 * q^38 - 2 * q^40 + 14 * q^43 - 2 * q^49 - 6 * q^52 + 20 * q^53 - 2 * q^55 - 2 * q^56 + 22 * q^61 + 8 * q^62 - 2 * q^64 + 4 * q^65 + 2 * q^68 + 18 * q^74 - 2 * q^77 - 24 * q^79 + 16 * q^82 - 2 * q^88 + 4 * q^91 + 6 * q^92 + 16 * q^94 - 2 * 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
$11$			1.00000i			0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$			1.00000i			0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$		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
$17$	−1.00000			−0.242536			−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000			−0.242536			−0.121268	+	0.992620i	$0.538696\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$	1.00000			0.213201
$23$	−3.00000			−0.625543			−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000			−0.625543			−0.312772	+	0.949828i	$0.601257\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$	−2.00000	−	3.00000i	−0.392232	−	0.588348i
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			1.00000i			0.171499i
$35$	−1.00000			−0.169031
$36$		0			0
$37$			9.00000i			1.47959i	0.672832	+	0.739795i	$0.265078\pi$
$37$			9.00000i			1.47959i	−0.672832	+	0.739795i	$0.734922\pi$
$38$	1.00000			0.162221
$39$		0			0
$40$	−1.00000			−0.158114
$41$			8.00000i			1.24939i	0.780869	+	0.624695i	$0.214777\pi$
$41$			8.00000i			1.24939i	−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$		−	1.00000i		−	0.150756i
$45$		0			0
$46$			3.00000i			0.442326i
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	4.00000i		−	0.565685i
$51$		0			0
$52$	−3.00000	+	2.00000i	−0.416025	+	0.277350i
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	−1.00000			−0.134840
$56$	−1.00000			−0.133631
$57$		0			0
$58$			9.00000i			1.18176i
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	11.0000			1.40841			0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000			1.40841			0.704203	+	0.709999i	$0.251305\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	−1.00000			−0.125000
$65$	2.00000	+	3.00000i	0.248069	+	0.372104i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	1.00000			0.121268
$69$		0			0
$70$			1.00000i			0.119523i
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$		−	11.0000i		−	1.28745i	−0.765256	−	0.643726i	$-0.777388\pi$
$73$		−	11.0000i		−	1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	9.00000			1.04623
$75$		0			0
$76$		−	1.00000i		−	0.114708i
$77$	−1.00000			−0.113961
$78$		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
$82$	8.00000			0.883452
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$		−	1.00000i		−	0.108465i
$86$		−	7.00000i		−	0.754829i
$87$		0			0
$88$	−1.00000			−0.106600
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$	2.00000	+	3.00000i	0.209657	+	0.314485i
$92$	3.00000			0.312772
$93$		0			0
$94$	8.00000			0.825137
$95$	−1.00000			−0.102598
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	−4.00000			−0.400000
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$	−5.00000			−0.492665			−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000			−0.492665			−0.246332	+	0.969185i	$0.579225\pi$
$104$	2.00000	+	3.00000i	0.196116	+	0.294174i
$105$		0			0
$106$		−	10.0000i		−	0.971286i
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$		−	7.00000i		−	0.670478i	−0.942133	−	0.335239i	$-0.891183\pi$
$109$		−	7.00000i		−	0.670478i	0.942133	−	0.335239i	$-0.108817\pi$
$110$			1.00000i			0.0953463i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$		−	3.00000i		−	0.279751i
$116$	9.00000			0.835629
$117$		0			0
$118$	−6.00000			−0.552345
$119$		−	1.00000i		−	0.0916698i
$120$		0			0
$121$	10.0000			0.909091
$122$		−	11.0000i		−	0.995893i
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$			9.00000i			0.804984i
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	3.00000	−	2.00000i	0.263117	−	0.175412i
$131$	3.00000			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$132$		0			0
$133$	−1.00000			−0.0867110
$134$	12.0000			1.03664
$135$		0			0
$136$		−	1.00000i		−	0.0857493i
$137$		−	15.0000i		−	1.28154i	−0.767734	−	0.640768i	$-0.778616\pi$
$137$		−	15.0000i		−	1.28154i	0.767734	−	0.640768i	$-0.221384\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$	1.00000			0.0845154
$141$		0			0
$142$	6.00000			0.503509
$143$	2.00000	+	3.00000i	0.167248	+	0.250873i
$144$		0			0
$145$		−	9.00000i		−	0.747409i
$146$	−11.0000			−0.910366
$147$		0			0
$148$		−	9.00000i		−	0.739795i
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		−	1.00000i		−	0.0813788i	−0.999172	−	0.0406894i	$-0.987045\pi$
$151$		−	1.00000i		−	0.0813788i	0.999172	−	0.0406894i	$-0.0129554\pi$
$152$	−1.00000			−0.0811107
$153$		0			0
$154$			1.00000i			0.0805823i
$155$	−4.00000			−0.321288
$156$		0			0
$157$	−21.0000			−1.67598			−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000			−1.67598			−0.837991	+	0.545684i	$0.816270\pi$
$158$			12.0000i			0.954669i
$159$		0			0
$160$	1.00000			0.0790569
$161$		−	3.00000i		−	0.236433i
$162$		0			0
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$		−	8.00000i		−	0.624695i
$165$		0			0
$166$	6.00000			0.465690
$167$		−	11.0000i		−	0.851206i	−0.904910	−	0.425603i	$-0.860062\pi$
$167$		−	11.0000i		−	0.851206i	0.904910	−	0.425603i	$-0.139938\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$	−1.00000			−0.0766965
$171$		0			0
$172$	−7.00000			−0.533745
$173$	−14.0000			−1.06440			−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$			1.00000i			0.0753778i
$177$		0			0
$178$	12.0000			0.899438
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$	3.00000	−	2.00000i	0.222375	−	0.148250i
$183$		0			0
$184$		−	3.00000i		−	0.221163i
$185$	−9.00000			−0.661693
$186$		0			0
$187$		−	1.00000i		−	0.0731272i
$188$		−	8.00000i		−	0.583460i
$189$		0			0
$190$			1.00000i			0.0725476i
$191$	11.0000			0.795932			0.397966	−	0.917400i	$-0.369716\pi$
$191$	11.0000			0.795932			0.397966	+	0.917400i	$0.369716\pi$
$192$		0			0
$193$			12.0000i			0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$			12.0000i			0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	2.00000			0.143592
$195$		0			0
$196$	1.00000			0.0714286
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	5.00000			0.354441			0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000			0.354441			0.177220	+	0.984171i	$0.443289\pi$
$200$			4.00000i			0.282843i
$201$		0			0
$202$		−	2.00000i		−	0.140720i
$203$		−	9.00000i		−	0.631676i
$204$		0			0
$205$	−8.00000			−0.558744
$206$			5.00000i			0.348367i
$207$		0			0
$208$	3.00000	−	2.00000i	0.208013	−	0.138675i
$209$	−1.00000			−0.0691714
$210$		0			0
$211$	−5.00000			−0.344214			−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000			−0.344214			−0.172107	+	0.985078i	$0.555058\pi$
$212$	−10.0000			−0.686803
$213$		0			0
$214$		−	2.00000i		−	0.136717i
$215$			7.00000i			0.477396i
$216$		0			0
$217$	−4.00000			−0.271538
$218$	−7.00000			−0.474100
$219$		0			0
$220$	1.00000			0.0674200
$221$	−3.00000	+	2.00000i	−0.201802	+	0.134535i
$222$		0			0
$223$			8.00000i			0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$			8.00000i			0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$		−	2.00000i		−	0.133038i
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$		−	2.00000i		−	0.132164i	−0.997814	−	0.0660819i	$-0.978950\pi$
$229$		−	2.00000i		−	0.132164i	0.997814	−	0.0660819i	$-0.0210498\pi$
$230$	−3.00000			−0.197814
$231$		0			0
$232$		−	9.00000i		−	0.590879i
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$			6.00000i			0.390567i
$237$		0			0
$238$	−1.00000			−0.0648204
$239$		−	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
$241$		−	10.0000i		−	0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$		−	10.0000i		−	0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$		−	10.0000i		−	0.642824i
$243$		0			0
$244$	−11.0000			−0.704203
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	2.00000	+	3.00000i	0.127257	+	0.190885i
$248$	−4.00000			−0.254000
$249$		0			0
$250$	9.00000			0.569210
$251$	−11.0000			−0.694314			−0.347157	−	0.937807i	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$		−	3.00000i		−	0.188608i
$254$		−	16.0000i		−	1.00393i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−14.0000			−0.873296			−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000			−0.873296			−0.436648	+	0.899632i	$0.643834\pi$
$258$		0			0
$259$	−9.00000			−0.559233
$260$	−2.00000	−	3.00000i	−0.124035	−	0.186052i
$261$		0			0
$262$		−	3.00000i		−	0.185341i
$263$	−16.0000			−0.986602			−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000			−0.986602			−0.493301	+	0.869859i	$0.664210\pi$
$264$		0			0
$265$			10.0000i			0.614295i
$266$			1.00000i			0.0613139i
$267$		0			0
$268$		−	12.0000i		−	0.733017i
$269$	20.0000			1.21942			0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000			1.21942			0.609711	+	0.792624i	$0.291286\pi$
$270$		0			0
$271$			4.00000i			0.242983i	0.992592	+	0.121491i	$0.0387677\pi$
$271$			4.00000i			0.242983i	−0.992592	+	0.121491i	$0.961232\pi$
$272$	−1.00000			−0.0606339
$273$		0			0
$274$	−15.0000			−0.906183
$275$			4.00000i			0.241209i
$276$		0			0
$277$	16.0000			0.961347			0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000			0.961347			0.480673	+	0.876900i	$0.340392\pi$
$278$			12.0000i			0.719712i
$279$		0			0
$280$		−	1.00000i		−	0.0597614i
$281$		−	22.0000i		−	1.31241i	−0.754583	−	0.656205i	$-0.772161\pi$
$281$		−	22.0000i		−	1.31241i	0.754583	−	0.656205i	$-0.227839\pi$
$282$		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
$286$	3.00000	−	2.00000i	0.177394	−	0.118262i
$287$	−8.00000			−0.472225
$288$		0			0
$289$	−16.0000			−0.941176
$290$	−9.00000			−0.528498
$291$		0			0
$292$			11.0000i			0.643726i
$293$		−	26.0000i		−	1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$		−	26.0000i		−	1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$		0			0
$295$	6.00000			0.349334
$296$	−9.00000			−0.523114
$297$		0			0
$298$	6.00000			0.347571
$299$	−9.00000	+	6.00000i	−0.520483	+	0.346989i
$300$		0			0
$301$			7.00000i			0.403473i
$302$	−1.00000			−0.0575435
$303$		0			0
$304$			1.00000i			0.0573539i
$305$			11.0000i			0.629858i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	1.00000			0.0569803
$309$		0			0
$310$			4.00000i			0.227185i
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$			21.0000i			1.18510i
$315$		0			0
$316$	12.0000			0.675053
$317$		−	12.0000i		−	0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$		−	12.0000i		−	0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$		0			0
$319$		−	9.00000i		−	0.503903i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	−3.00000			−0.167183
$323$		−	1.00000i		−	0.0556415i
$324$		0			0
$325$	12.0000	−	8.00000i	0.665640	−	0.443760i
$326$	20.0000			1.10770
$327$		0			0
$328$	−8.00000			−0.441726
$329$	−8.00000			−0.441054
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		−	6.00000i		−	0.329293i
$333$		0			0
$334$	−11.0000			−0.601893
$335$	−12.0000			−0.655630
$336$		0			0
$337$	17.0000			0.926049			0.463025	−	0.886345i	$-0.346764\pi$
$337$	17.0000			0.926049			0.463025	+	0.886345i	$0.346764\pi$
$338$	−12.0000	−	5.00000i	−0.652714	−	0.271964i
$339$		0			0
$340$			1.00000i			0.0542326i
$341$	−4.00000			−0.216612
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$			7.00000i			0.377415i
$345$		0			0
$346$			14.0000i			0.752645i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$			4.00000i			0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$			4.00000i			0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$	4.00000			0.213809
$351$		0			0
$352$	1.00000			0.0533002
$353$		−	16.0000i		−	0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$		−	16.0000i		−	0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$		0			0
$355$	−6.00000			−0.318447
$356$		−	12.0000i		−	0.635999i
$357$		0			0
$358$		−	18.0000i		−	0.951330i
$359$			30.0000i			1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$			30.0000i			1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$			2.00000i			0.105118i
$363$		0			0
$364$	−2.00000	−	3.00000i	−0.104828	−	0.157243i
$365$	11.0000			0.575766
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$	−3.00000			−0.156386
$369$		0			0
$370$			9.00000i			0.467888i
$371$			10.0000i			0.519174i
$372$		0			0
$373$	−14.0000			−0.724893			−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000			−0.724893			−0.362446	+	0.932005i	$0.618058\pi$
$374$	−1.00000			−0.0517088
$375$		0			0
$376$	−8.00000			−0.412568
$377$	−27.0000	+	18.0000i	−1.39057	+	0.927047i
$378$		0			0
$379$		−	16.0000i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$		−	16.0000i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	1.00000			0.0512989
$381$		0			0
$382$		−	11.0000i		−	0.562809i
$383$			35.0000i			1.78842i	0.447651	+	0.894208i	$0.352261\pi$
$383$			35.0000i			1.78842i	−0.447651	+	0.894208i	$0.647739\pi$
$384$		0			0
$385$		−	1.00000i		−	0.0509647i
$386$	12.0000			0.610784
$387$		0			0
$388$		−	2.00000i		−	0.101535i
$389$	−2.00000			−0.101404			−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000			−0.101404			−0.0507020	+	0.998714i	$0.516146\pi$
$390$		0			0
$391$	3.00000			0.151717
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	−2.00000			−0.100759
$395$		−	12.0000i		−	0.603786i
$396$		0			0
$397$		−	4.00000i		−	0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$		−	4.00000i		−	0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$		−	5.00000i		−	0.250627i
$399$		0			0
$400$	4.00000			0.200000
$401$			14.0000i			0.699127i	0.936913	+	0.349563i	$0.113670\pi$
$401$			14.0000i			0.699127i	−0.936913	+	0.349563i	$0.886330\pi$
$402$		0			0
$403$	8.00000	+	12.0000i	0.398508	+	0.597763i
$404$	−2.00000			−0.0995037
$405$		0			0
$406$	−9.00000			−0.446663
$407$	−9.00000			−0.446113
$408$		0			0
$409$		−	27.0000i		−	1.33506i	−0.744581	−	0.667532i	$-0.767351\pi$
$409$		−	27.0000i		−	1.33506i	0.744581	−	0.667532i	$-0.232649\pi$
$410$			8.00000i			0.395092i
$411$		0			0
$412$	5.00000			0.246332
$413$	6.00000			0.295241
$414$		0			0
$415$	−6.00000			−0.294528
$416$	−2.00000	−	3.00000i	−0.0980581	−	0.147087i
$417$		0			0
$418$			1.00000i			0.0489116i
$419$	−23.0000			−1.12362			−0.561812	−	0.827265i	$-0.689895\pi$
$419$	−23.0000			−1.12362			−0.561812	+	0.827265i	$0.689895\pi$
$420$		0			0
$421$		−	22.0000i		−	1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$		−	22.0000i		−	1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$			5.00000i			0.243396i
$423$		0			0
$424$			10.0000i			0.485643i
$425$	−4.00000			−0.194029
$426$		0			0
$427$			11.0000i			0.532327i
$428$	−2.00000			−0.0966736
$429$		0			0
$430$	7.00000			0.337570
$431$		−	18.0000i		−	0.867029i	−0.901146	−	0.433515i	$-0.857273\pi$
$431$		−	18.0000i		−	0.867029i	0.901146	−	0.433515i	$-0.142727\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$			4.00000i			0.192006i
$435$		0			0
$436$			7.00000i			0.335239i
$437$		−	3.00000i		−	0.143509i
$438$		0			0
$439$	−33.0000			−1.57500			−0.787502	−	0.616312i	$-0.788626\pi$
$439$	−33.0000			−1.57500			−0.787502	+	0.616312i	$0.788626\pi$
$440$		−	1.00000i		−	0.0476731i
$441$		0			0
$442$	2.00000	+	3.00000i	0.0951303	+	0.142695i
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$	8.00000			0.378811
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$		−	15.0000i		−	0.707894i	−0.935266	−	0.353947i	$-0.884839\pi$
$449$		−	15.0000i		−	0.707894i	0.935266	−	0.353947i	$-0.115161\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$	−2.00000			−0.0940721
$453$		0			0
$454$	−12.0000			−0.563188
$455$	−3.00000	+	2.00000i	−0.140642	+	0.0937614i
$456$		0			0
$457$		−	4.00000i		−	0.187112i	−0.995614	−	0.0935561i	$-0.970177\pi$
$457$		−	4.00000i		−	0.187112i	0.995614	−	0.0935561i	$-0.0298234\pi$
$458$	−2.00000			−0.0934539
$459$		0			0
$460$			3.00000i			0.139876i
$461$			3.00000i			0.139724i	0.997557	+	0.0698620i	$0.0222559\pi$
$461$			3.00000i			0.139724i	−0.997557	+	0.0698620i	$0.977744\pi$
$462$		0			0
$463$		−	29.0000i		−	1.34774i	−0.738848	−	0.673872i	$-0.764630\pi$
$463$		−	29.0000i		−	1.34774i	0.738848	−	0.673872i	$-0.235370\pi$
$464$	−9.00000			−0.417815
$465$		0			0
$466$		−	18.0000i		−	0.833834i
$467$	37.0000			1.71216			0.856078	−	0.516847i	$-0.172894\pi$
$467$	37.0000			1.71216			0.856078	+	0.516847i	$0.172894\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$			8.00000i			0.369012i
$471$		0			0
$472$	6.00000			0.276172
$473$			7.00000i			0.321860i
$474$		0			0
$475$			4.00000i			0.183533i
$476$			1.00000i			0.0458349i
$477$		0			0
$478$	−8.00000			−0.365911
$479$			39.0000i			1.78196i	0.454047	+	0.890978i	$0.349980\pi$
$479$			39.0000i			1.78196i	−0.454047	+	0.890978i	$0.650020\pi$
$480$		0			0
$481$	18.0000	+	27.0000i	0.820729	+	1.23109i
$482$	−10.0000			−0.455488
$483$		0			0
$484$	−10.0000			−0.454545
$485$	−2.00000			−0.0908153
$486$		0			0
$487$		−	32.0000i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$		−	32.0000i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$			11.0000i			0.497947i
$489$		0			0
$490$	−1.00000			−0.0451754
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	9.00000			0.405340
$494$	3.00000	−	2.00000i	0.134976	−	0.0899843i
$495$		0			0
$496$			4.00000i			0.179605i
$497$	−6.00000			−0.269137
$498$		0			0
$499$			10.0000i			0.447661i	0.974628	+	0.223831i	$0.0718563\pi$
$499$			10.0000i			0.447661i	−0.974628	+	0.223831i	$0.928144\pi$
$500$		−	9.00000i		−	0.402492i
$501$		0			0
$502$			11.0000i			0.490954i
$503$	38.0000			1.69434			0.847168	−	0.531325i	$-0.178306\pi$
$503$	38.0000			1.69434			0.847168	+	0.531325i	$0.178306\pi$
$504$		0			0
$505$			2.00000i			0.0889988i
$506$	−3.00000			−0.133366
$507$		0			0
$508$	−16.0000			−0.709885
$509$		−	37.0000i		−	1.64000i	−0.572366	−	0.819998i	$-0.693974\pi$
$509$		−	37.0000i		−	1.64000i	0.572366	−	0.819998i	$-0.306026\pi$
$510$		0			0
$511$	11.0000			0.486611
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			14.0000i			0.617514i
$515$		−	5.00000i		−	0.220326i
$516$		0			0
$517$	−8.00000			−0.351840
$518$			9.00000i			0.395437i
$519$		0			0
$520$	−3.00000	+	2.00000i	−0.131559	+	0.0877058i
$521$	31.0000			1.35813			0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000			1.35813			0.679067	+	0.734076i	$0.262384\pi$
$522$		0			0
$523$	−36.0000			−1.57417			−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000			−1.57417			−0.787085	+	0.616844i	$0.788411\pi$
$524$	−3.00000			−0.131056
$525$		0			0
$526$			16.0000i			0.697633i
$527$		−	4.00000i		−	0.174243i
$528$		0			0
$529$	−14.0000			−0.608696
$530$	10.0000			0.434372
$531$		0			0
$532$	1.00000			0.0433555
$533$	16.0000	+	24.0000i	0.693037	+	1.03956i
$534$		0			0
$535$			2.00000i			0.0864675i
$536$	−12.0000			−0.518321
$537$		0			0
$538$		−	20.0000i		−	0.862261i
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$		−	41.0000i		−	1.76273i	−0.472438	−	0.881364i	$-0.656626\pi$
$541$		−	41.0000i		−	1.76273i	0.472438	−	0.881364i	$-0.343374\pi$
$542$	4.00000			0.171815
$543$		0			0
$544$			1.00000i			0.0428746i
$545$	7.00000			0.299847
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$			15.0000i			0.640768i
$549$		0			0
$550$	4.00000			0.170561
$551$		−	9.00000i		−	0.383413i
$552$		0			0
$553$		−	12.0000i		−	0.510292i
$554$		−	16.0000i		−	0.679775i
$555$		0			0
$556$	12.0000			0.508913
$557$		−	2.00000i		−	0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$		−	2.00000i		−	0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$		0			0
$559$	21.0000	−	14.0000i	0.888205	−	0.592137i
$560$	−1.00000			−0.0422577
$561$		0			0
$562$	−22.0000			−0.928014
$563$	−11.0000			−0.463595			−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000			−0.463595			−0.231797	+	0.972764i	$0.574461\pi$
$564$		0			0
$565$			2.00000i			0.0841406i
$566$			14.0000i			0.588464i
$567$		0			0
$568$	−6.00000			−0.251754
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	−2.00000	−	3.00000i	−0.0836242	−	0.125436i
$573$		0			0
$574$			8.00000i			0.333914i
$575$	−12.0000			−0.500435
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$			16.0000i			0.665512i
$579$		0			0
$580$			9.00000i			0.373705i
$581$	−6.00000			−0.248922
$582$		0			0
$583$			10.0000i			0.414158i
$584$	11.0000			0.455183
$585$		0			0
$586$	−26.0000			−1.07405
$587$		−	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
$589$	−4.00000			−0.164817
$590$		−	6.00000i		−	0.247016i
$591$		0			0
$592$			9.00000i			0.369898i
$593$		−	28.0000i		−	1.14982i	−0.818216	−	0.574911i	$-0.805037\pi$
$593$		−	28.0000i		−	1.14982i	0.818216	−	0.574911i	$-0.194963\pi$
$594$		0			0
$595$	1.00000			0.0409960
$596$		−	6.00000i		−	0.245770i
$597$		0			0
$598$	6.00000	+	9.00000i	0.245358	+	0.368037i
$599$	−39.0000			−1.59350			−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000			−1.59350			−0.796748	+	0.604311i	$0.793448\pi$
$600$		0			0
$601$	−16.0000			−0.652654			−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000			−0.652654			−0.326327	+	0.945257i	$0.605811\pi$
$602$	7.00000			0.285299
$603$		0			0
$604$			1.00000i			0.0406894i
$605$			10.0000i			0.406558i
$606$		0			0
$607$	1.00000			0.0405887			0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000			0.0405887			0.0202944	+	0.999794i	$0.493540\pi$
$608$	1.00000			0.0405554
$609$		0			0
$610$	11.0000			0.445377
$611$	16.0000	+	24.0000i	0.647291	+	0.970936i
$612$		0			0
$613$			17.0000i			0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$			17.0000i			0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$	8.00000			0.322854
$615$		0			0
$616$		−	1.00000i		−	0.0402911i
$617$		−	19.0000i		−	0.764911i	−0.923974	−	0.382456i	$-0.875078\pi$
$617$		−	19.0000i		−	0.764911i	0.923974	−	0.382456i	$-0.124922\pi$
$618$		0			0
$619$		−	21.0000i		−	0.844061i	−0.906582	−	0.422031i	$-0.861317\pi$
$619$		−	21.0000i		−	0.844061i	0.906582	−	0.422031i	$-0.138683\pi$
$620$	4.00000			0.160644
$621$		0			0
$622$			8.00000i			0.320771i
$623$	−12.0000			−0.480770
$624$		0			0
$625$	11.0000			0.440000
$626$			14.0000i			0.559553i
$627$		0			0
$628$	21.0000			0.837991
$629$		−	9.00000i		−	0.358854i
$630$		0			0
$631$		−	29.0000i		−	1.15447i	−0.816577	−	0.577236i	$-0.804131\pi$
$631$		−	29.0000i		−	1.15447i	0.816577	−	0.577236i	$-0.195869\pi$
$632$		−	12.0000i		−	0.477334i
$633$		0			0
$634$	−12.0000			−0.476581
$635$			16.0000i			0.634941i
$636$		0			0
$637$	−3.00000	+	2.00000i	−0.118864	+	0.0792429i
$638$	−9.00000			−0.356313
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	44.0000			1.73790			0.868948	−	0.494904i	$-0.164797\pi$
$641$	44.0000			1.73790			0.868948	+	0.494904i	$0.164797\pi$
$642$		0			0
$643$			19.0000i			0.749287i	0.927169	+	0.374643i	$0.122235\pi$
$643$			19.0000i			0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$			3.00000i			0.118217i
$645$		0			0
$646$	−1.00000			−0.0393445
$647$	−42.0000			−1.65119			−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000			−1.65119			−0.825595	+	0.564263i	$0.809160\pi$
$648$		0			0
$649$	6.00000			0.235521
$650$	−8.00000	−	12.0000i	−0.313786	−	0.470679i
$651$		0			0
$652$		−	20.0000i		−	0.783260i
$653$	13.0000			0.508729			0.254365	−	0.967108i	$-0.418134\pi$
$653$	13.0000			0.508729			0.254365	+	0.967108i	$0.418134\pi$
$654$		0			0
$655$			3.00000i			0.117220i
$656$			8.00000i			0.312348i
$657$		0			0
$658$			8.00000i			0.311872i
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$		−	26.0000i		−	1.01128i	−0.862744	−	0.505641i	$-0.831256\pi$
$661$		−	26.0000i		−	1.01128i	0.862744	−	0.505641i	$-0.168744\pi$
$662$	8.00000			0.310929
$663$		0			0
$664$	−6.00000			−0.232845
$665$		−	1.00000i		−	0.0387783i
$666$		0			0
$667$	27.0000			1.04544
$668$			11.0000i			0.425603i
$669$		0			0
$670$			12.0000i			0.463600i
$671$			11.0000i			0.424650i
$672$		0			0
$673$	29.0000			1.11787			0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000			1.11787			0.558934	+	0.829212i	$0.311211\pi$
$674$		−	17.0000i		−	0.654816i
$675$		0			0
$676$	−5.00000	+	12.0000i	−0.192308	+	0.461538i
$677$	−32.0000			−1.22986			−0.614930	−	0.788582i	$-0.710816\pi$
$677$	−32.0000			−1.22986			−0.614930	+	0.788582i	$0.710816\pi$
$678$		0			0
$679$	−2.00000			−0.0767530
$680$	1.00000			0.0383482
$681$		0			0
$682$			4.00000i			0.153168i
$683$			19.0000i			0.727015i	0.931591	+	0.363507i	$0.118421\pi$
$683$			19.0000i			0.727015i	−0.931591	+	0.363507i	$0.881579\pi$
$684$		0			0
$685$	15.0000			0.573121
$686$	−1.00000			−0.0381802
$687$		0			0
$688$	7.00000			0.266872
$689$	30.0000	−	20.0000i	1.14291	−	0.761939i
$690$		0			0
$691$			40.0000i			1.52167i	0.648944	+	0.760836i	$0.275211\pi$
$691$			40.0000i			1.52167i	−0.648944	+	0.760836i	$0.724789\pi$
$692$	14.0000			0.532200
$693$		0			0
$694$		0			0
$695$		−	12.0000i		−	0.455186i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$	4.00000			0.151402
$699$		0			0
$700$		−	4.00000i		−	0.151186i
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−9.00000			−0.339441
$704$		−	1.00000i		−	0.0376889i
$705$		0			0
$706$	−16.0000			−0.602168
$707$			2.00000i			0.0752177i
$708$		0			0
$709$			14.0000i			0.525781i	0.964826	+	0.262891i	$0.0846758\pi$
$709$			14.0000i			0.525781i	−0.964826	+	0.262891i	$0.915324\pi$
$710$			6.00000i			0.225176i
$711$		0			0
$712$	−12.0000			−0.449719
$713$		−	12.0000i		−	0.449404i
$714$		0			0
$715$	−3.00000	+	2.00000i	−0.112194	+	0.0747958i
$716$	−18.0000			−0.672692
$717$		0			0
$718$	30.0000			1.11959
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$		−	5.00000i		−	0.186210i
$722$		−	18.0000i		−	0.669891i
$723$		0			0
$724$	2.00000			0.0743294
$725$	−36.0000			−1.33701
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$	−3.00000	+	2.00000i	−0.111187	+	0.0741249i
$729$		0			0
$730$		−	11.0000i		−	0.407128i
$731$	−7.00000			−0.258904
$732$		0			0
$733$			40.0000i			1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$			40.0000i			1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$			8.00000i			0.295285i
$735$		0			0
$736$			3.00000i			0.110581i
$737$	−12.0000			−0.442026
$738$		0			0
$739$		−	8.00000i		−	0.294285i	−0.989115	−	0.147142i	$-0.952992\pi$
$739$		−	8.00000i		−	0.294285i	0.989115	−	0.147142i	$-0.0470076\pi$
$740$	9.00000			0.330847
$741$		0			0
$742$	10.0000			0.367112
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$			14.0000i			0.512576i
$747$		0			0
$748$			1.00000i			0.0365636i
$749$			2.00000i			0.0730784i
$750$		0			0
$751$	20.0000			0.729810			0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000			0.729810			0.364905	+	0.931045i	$0.381101\pi$
$752$			8.00000i			0.291730i
$753$		0			0
$754$	18.0000	+	27.0000i	0.655521	+	0.983282i
$755$	1.00000			0.0363937
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$	−16.0000			−0.581146
$759$		0			0
$760$		−	1.00000i		−	0.0362738i
$761$		−	42.0000i		−	1.52250i	−0.648459	−	0.761249i	$-0.724586\pi$
$761$		−	42.0000i		−	1.52250i	0.648459	−	0.761249i	$-0.275414\pi$
$762$		0			0
$763$	7.00000			0.253417
$764$	−11.0000			−0.397966
$765$		0			0
$766$	35.0000			1.26460
$767$	−12.0000	−	18.0000i	−0.433295	−	0.649942i
$768$		0			0
$769$		−	31.0000i		−	1.11789i	−0.829205	−	0.558944i	$-0.811207\pi$
$769$		−	31.0000i		−	1.11789i	0.829205	−	0.558944i	$-0.188793\pi$
$770$	−1.00000			−0.0360375
$771$		0			0
$772$		−	12.0000i		−	0.431889i
$773$			19.0000i			0.683383i	0.939812	+	0.341691i	$0.111000\pi$
$773$			19.0000i			0.683383i	−0.939812	+	0.341691i	$0.889000\pi$
$774$		0			0
$775$			16.0000i			0.574737i
$776$	−2.00000			−0.0717958
$777$		0			0
$778$			2.00000i			0.0717035i
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−6.00000			−0.214697
$782$		−	3.00000i		−	0.107280i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		−	21.0000i		−	0.749522i
$786$		0			0
$787$		−	53.0000i		−	1.88925i	−0.328158	−	0.944623i	$-0.606428\pi$
$787$		−	53.0000i		−	1.88925i	0.328158	−	0.944623i	$-0.393572\pi$
$788$			2.00000i			0.0712470i
$789$		0			0
$790$	−12.0000			−0.426941
$791$			2.00000i			0.0711118i
$792$		0			0
$793$	33.0000	−	22.0000i	1.17186	−	0.781243i
$794$	−4.00000			−0.141955
$795$		0			0
$796$	−5.00000			−0.177220
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$		0			0
$799$		−	8.00000i		−	0.283020i
$800$		−	4.00000i		−	0.141421i
$801$		0			0
$802$	14.0000			0.494357
$803$	11.0000			0.388182
$804$		0			0
$805$	3.00000			0.105736
$806$	12.0000	−	8.00000i	0.422682	−	0.281788i
$807$		0			0
$808$			2.00000i			0.0703598i
$809$	4.00000			0.140633			0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000			0.140633			0.0703163	+	0.997525i	$0.477599\pi$
$810$		0			0
$811$			21.0000i			0.737410i	0.929547	+	0.368705i	$0.120199\pi$
$811$			21.0000i			0.737410i	−0.929547	+	0.368705i	$0.879801\pi$
$812$			9.00000i			0.315838i
$813$		0			0
$814$			9.00000i			0.315450i
$815$	−20.0000			−0.700569
$816$		0			0
$817$			7.00000i			0.244899i
$818$	−27.0000			−0.944033
$819$		0			0
$820$	8.00000			0.279372
$821$			36.0000i			1.25641i	0.778048	+	0.628204i	$0.216210\pi$
$821$			36.0000i			1.25641i	−0.778048	+	0.628204i	$0.783790\pi$
$822$		0			0
$823$	−8.00000			−0.278862			−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000			−0.278862			−0.139431	+	0.990232i	$0.544527\pi$
$824$		−	5.00000i		−	0.174183i
$825$		0			0
$826$		−	6.00000i		−	0.208767i
$827$		−	3.00000i		−	0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$		−	3.00000i		−	0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$		0			0
$829$	25.0000			0.868286			0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000			0.868286			0.434143	+	0.900844i	$0.357051\pi$
$830$			6.00000i			0.208263i
$831$		0			0
$832$	−3.00000	+	2.00000i	−0.104006	+	0.0693375i
$833$	1.00000			0.0346479
$834$		0			0
$835$	11.0000			0.380671
$836$	1.00000			0.0345857
$837$		0			0
$838$			23.0000i			0.794522i
$839$			52.0000i			1.79524i	0.440771	+	0.897620i	$0.354705\pi$
$839$			52.0000i			1.79524i	−0.440771	+	0.897620i	$0.645295\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$	−22.0000			−0.758170
$843$		0			0
$844$	5.00000			0.172107
$845$	12.0000	+	5.00000i	0.412813	+	0.172005i
$846$		0			0
$847$			10.0000i			0.343604i
$848$	10.0000			0.343401
$849$		0			0
$850$			4.00000i			0.137199i
$851$		−	27.0000i		−	0.925548i
$852$		0			0
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	11.0000			0.376412
$855$		0			0
$856$			2.00000i			0.0683586i
$857$	22.0000			0.751506			0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000			0.751506			0.375753	+	0.926720i	$0.377384\pi$
$858$		0			0
$859$	10.0000			0.341196			0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000			0.341196			0.170598	+	0.985341i	$0.445430\pi$
$860$		−	7.00000i		−	0.238698i
$861$		0			0
$862$	−18.0000			−0.613082
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$		0			0
$865$		−	14.0000i		−	0.476014i
$866$		−	4.00000i		−	0.135926i
$867$		0			0
$868$	4.00000			0.135769
$869$		−	12.0000i		−	0.407072i
$870$		0			0
$871$	24.0000	+	36.0000i	0.813209	+	1.21981i
$872$	7.00000			0.237050
$873$		0			0
$874$	−3.00000			−0.101477
$875$	−9.00000			−0.304256
$876$		0			0
$877$			22.0000i			0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$			22.0000i			0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$			33.0000i			1.11370i
$879$		0			0
$880$	−1.00000			−0.0337100
$881$	59.0000			1.98776			0.993880	−	0.110463i	$-0.0352333\pi$
$881$	59.0000			1.98776			0.993880	+	0.110463i	$0.0352333\pi$
$882$		0			0
$883$	31.0000			1.04323			0.521617	−	0.853180i	$-0.325329\pi$
$883$	31.0000			1.04323			0.521617	+	0.853180i	$0.325329\pi$
$884$	3.00000	−	2.00000i	0.100901	−	0.0672673i
$885$		0			0
$886$		−	6.00000i		−	0.201574i
$887$	−2.00000			−0.0671534			−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000			−0.0671534			−0.0335767	+	0.999436i	$0.510690\pi$
$888$		0			0
$889$			16.0000i			0.536623i
$890$			12.0000i			0.402241i
$891$		0			0
$892$		−	8.00000i		−	0.267860i
$893$	−8.00000			−0.267710
$894$		0			0
$895$			18.0000i			0.601674i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	−15.0000			−0.500556
$899$		−	36.0000i		−	1.20067i
$900$		0			0
$901$	−10.0000			−0.333148
$902$			8.00000i			0.266371i
$903$		0			0
$904$			2.00000i			0.0665190i
$905$		−	2.00000i		−	0.0664822i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$			12.0000i			0.398234i
$909$		0			0
$910$	2.00000	+	3.00000i	0.0662994	+	0.0994490i
$911$	29.0000			0.960813			0.480406	−	0.877046i	$-0.340489\pi$
$911$	29.0000			0.960813			0.480406	+	0.877046i	$0.340489\pi$
$912$		0			0
$913$	−6.00000			−0.198571
$914$	−4.00000			−0.132308
$915$		0			0
$916$			2.00000i			0.0660819i
$917$			3.00000i			0.0990687i
$918$		0			0
$919$	34.0000			1.12156			0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000			1.12156			0.560778	+	0.827966i	$0.310502\pi$
$920$	3.00000			0.0989071
$921$		0			0
$922$	3.00000			0.0987997
$923$	12.0000	+	18.0000i	0.394985	+	0.592477i
$924$		0			0
$925$			36.0000i			1.18367i
$926$	−29.0000			−0.952999
$927$		0			0
$928$			9.00000i			0.295439i
$929$		−	26.0000i		−	0.853032i	−0.904480	−	0.426516i	$-0.859741\pi$
$929$		−	26.0000i		−	0.853032i	0.904480	−	0.426516i	$-0.140259\pi$
$930$		0			0
$931$		−	1.00000i		−	0.0327737i
$932$	−18.0000			−0.589610
$933$		0			0
$934$		−	37.0000i		−	1.21068i
$935$	1.00000			0.0327035
$936$		0			0
$937$	−20.0000			−0.653372			−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000			−0.653372			−0.326686	+	0.945133i	$0.605932\pi$
$938$			12.0000i			0.391814i
$939$		0			0
$940$	8.00000			0.260931
$941$		−	30.0000i		−	0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$		−	30.0000i		−	0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$		0			0
$943$		−	24.0000i		−	0.781548i
$944$		−	6.00000i		−	0.195283i
$945$		0			0
$946$	7.00000			0.227590
$947$		−	35.0000i		−	1.13735i	−0.822563	−	0.568674i	$-0.807457\pi$
$947$		−	35.0000i		−	1.13735i	0.822563	−	0.568674i	$-0.192543\pi$
$948$		0			0
$949$	−22.0000	−	33.0000i	−0.714150	−	1.07123i
$950$	4.00000			0.129777
$951$		0			0
$952$	1.00000			0.0324102
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$			11.0000i			0.355952i
$956$			8.00000i			0.258738i
$957$		0			0
$958$	39.0000			1.26003
$959$	15.0000			0.484375
$960$		0			0
$961$	15.0000			0.483871
$962$	27.0000	−	18.0000i	0.870515	−	0.580343i
$963$		0			0
$964$			10.0000i			0.322078i
$965$	−12.0000			−0.386294
$966$		0			0
$967$		−	9.00000i		−	0.289420i	−0.989474	−	0.144710i	$-0.953775\pi$
$967$		−	9.00000i		−	0.289420i	0.989474	−	0.144710i	$-0.0462250\pi$
$968$			10.0000i			0.321412i
$969$		0			0
$970$			2.00000i			0.0642161i
$971$	4.00000			0.128366			0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000			0.128366			0.0641831	+	0.997938i	$0.479556\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$	−32.0000			−1.02535
$975$		0			0
$976$	11.0000			0.352101
$977$		−	39.0000i		−	1.24772i	−0.781536	−	0.623860i	$-0.785563\pi$
$977$		−	39.0000i		−	1.24772i	0.781536	−	0.623860i	$-0.214437\pi$
$978$		0			0
$979$	−12.0000			−0.383522
$980$			1.00000i			0.0319438i
$981$		0			0
$982$			12.0000i			0.382935i
$983$		−	9.00000i		−	0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$		−	9.00000i		−	0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$		0			0
$985$	2.00000			0.0637253
$986$		−	9.00000i		−	0.286618i
$987$		0			0
$988$	−2.00000	−	3.00000i	−0.0636285	−	0.0954427i
$989$	−21.0000			−0.667761
$990$		0			0
$991$	−12.0000			−0.381193			−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000			−0.381193			−0.190596	+	0.981669i	$0.561042\pi$
$992$	4.00000			0.127000
$993$		0			0
$994$			6.00000i			0.190308i
$995$			5.00000i			0.158511i
$996$		0			0
$997$	−14.0000			−0.443384			−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000			−0.443384			−0.221692	+	0.975117i	$0.571158\pi$
$998$	10.0000			0.316544
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{10} +1.00000i q^{11} +(3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000i q^{19} -1.00000i q^{20} +1.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +(-2.00000 - 3.00000i) q^{26} -1.00000i q^{28} -9.00000 q^{29} +4.00000i q^{31} -1.00000i q^{32} +1.00000i q^{34} -1.00000 q^{35} +9.00000i q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000i q^{41} +7.00000 q^{43} -1.00000i q^{44} +3.00000i q^{46} +8.00000i q^{47} -1.00000 q^{49} -4.00000i q^{50} +(-3.00000 + 2.00000i) q^{52} +10.0000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +9.00000i q^{58} -6.00000i q^{59} +11.0000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +(2.00000 + 3.00000i) q^{65} +12.0000i q^{67} +1.00000 q^{68} +1.00000i q^{70} +6.00000i q^{71} -11.0000i q^{73} +9.00000 q^{74} -1.00000i q^{76} -1.00000 q^{77} -12.0000 q^{79} +1.00000i q^{80} +8.00000 q^{82} +6.00000i q^{83} -1.00000i q^{85} -7.00000i q^{86} -1.00000 q^{88} +12.0000i q^{89} +(2.00000 + 3.00000i) q^{91} +3.00000 q^{92} +8.00000 q^{94} -1.00000 q^{95} +2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 2 q^{10} + 6 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{22} - 6 q^{23} + 8 q^{25} - 4 q^{26} - 18 q^{29} - 2 q^{35} + 2 q^{38} - 2 q^{40} + 14 q^{43} - 2 q^{49} - 6 q^{52} + 20 q^{53} - 2 q^{55} - 2 q^{56} + 22 q^{61} + 8 q^{62} - 2 q^{64} + 4 q^{65} + 2 q^{68} + 18 q^{74} - 2 q^{77} - 24 q^{79} + 16 q^{82} - 2 q^{88} + 4 q^{91} + 6 q^{92} + 16 q^{94} - 2 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^10 + 6 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^22 - 6 * q^23 + 8 * q^25 - 4 * q^26 - 18 * q^29 - 2 * q^35 + 2 * q^38 - 2 * q^40 + 14 * q^43 - 2 * q^49 - 6 * q^52 + 20 * q^53 - 2 * q^55 - 2 * q^56 + 22 * q^61 + 8 * q^62 - 2 * q^64 + 4 * q^65 + 2 * q^68 + 18 * q^74 - 2 * q^77 - 24 * q^79 + 16 * q^82 - 2 * q^88 + 4 * q^91 + 6 * q^92 + 16 * q^94 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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