LMFDB - Embedded newform 546.2.c.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(337,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	546.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:	$4.35983195036$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	546.337
Dual form			546.2.c.a.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} +(3.00000 + 2.00000i) q^{13} -1.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000i q^{18} -1.00000i q^{19} -1.00000i q^{20} +1.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} +4.00000 q^{25} +(2.00000 - 3.00000i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +9.00000 q^{29} -1.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -1.00000i q^{33} -1.00000i q^{34} +1.00000 q^{35} -1.00000 q^{36} -9.00000i q^{37} -1.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -1.00000 q^{40} +8.00000i q^{41} +1.00000 q^{42} +7.00000 q^{43} -1.00000i q^{44} +1.00000i q^{45} -3.00000i q^{46} +8.00000i q^{47} -1.00000 q^{48} -1.00000 q^{49} -4.00000i q^{50} -1.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -10.0000 q^{53} +1.00000i q^{54} -1.00000 q^{55} +1.00000 q^{56} +1.00000i q^{57} -9.00000i q^{58} -6.00000i q^{59} +1.00000i q^{60} +11.0000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} -1.00000 q^{66} -12.0000i q^{67} -1.00000 q^{68} -3.00000 q^{69} -1.00000i q^{70} +6.00000i q^{71} +1.00000i q^{72} +11.0000i q^{73} -9.00000 q^{74} -4.00000 q^{75} +1.00000i q^{76} +1.00000 q^{77} +(-2.00000 + 3.00000i) q^{78} -12.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +8.00000 q^{82} +6.00000i q^{83} -1.00000i q^{84} +1.00000i q^{85} -7.00000i q^{86} -9.00000 q^{87} -1.00000 q^{88} +12.0000i q^{89} +1.00000 q^{90} +(2.00000 - 3.00000i) q^{91} -3.00000 q^{92} +4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{95} +1.00000i q^{96} -2.00000i q^{97} +1.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 6 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{22} + 6 q^{23} + 8 q^{25} + 4 q^{26} - 2 q^{27} + 18 q^{29} - 2 q^{30} + 2 q^{35} - 2 q^{36} - 2 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{42} + 14 q^{43} - 2 q^{48} - 2 q^{49} - 2 q^{51} - 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^{66} - 2 q^{68} - 6 q^{69} - 18 q^{74} - 8 q^{75} + 2 q^{77} - 4 q^{78} - 24 q^{79} + 2 q^{81} + 16 q^{82} - 18 q^{87} - 2 q^{88} + 2 q^{90} + 4 q^{91} - 6 q^{92} + 16 q^{94} + 2 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 6 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^22 + 6 * q^23 + 8 * q^25 + 4 * q^26 - 2 * q^27 + 18 * q^29 - 2 * q^30 + 2 * q^35 - 2 * q^36 - 2 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^42 + 14 * q^43 - 2 * q^48 - 2 * q^49 - 2 * q^51 - 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^66 - 2 * q^68 - 6 * q^69 - 18 * q^74 - 8 * q^75 + 2 * q^77 - 4 * q^78 - 24 * q^79 + 2 * q^81 + 16 * q^82 - 18 * q^87 - 2 * q^88 + 2 * q^90 + 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 $\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$.

$n$	$157$	$365$	$379$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−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	$0.928217\pi$
$6$			1.00000i			0.408248i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.00000i			0.353553i
$9$	1.00000			0.333333
$10$	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	$0.951829\pi$
$12$	1.00000			0.288675
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$13$	3.00000	+	2.00000i	0.832050	+	0.554700i
$14$	−1.00000			−0.267261
$15$		−	1.00000i		−	0.258199i
$16$	1.00000			0.250000
$17$	1.00000			0.242536			0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000			0.242536			0.121268	+	0.992620i	$0.461304\pi$
$18$		−	1.00000i		−	0.235702i
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		−	1.00000i		−	0.223607i
$21$			1.00000i			0.218218i
$22$	1.00000			0.213201
$23$	3.00000			0.625543			0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000			0.625543			0.312772	+	0.949828i	$0.398743\pi$
$24$		−	1.00000i		−	0.204124i
$25$	4.00000			0.800000
$26$	2.00000	−	3.00000i	0.392232	−	0.588348i
$27$	−1.00000			−0.192450
$28$			1.00000i			0.188982i
$29$	9.00000			1.67126			0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000			1.67126			0.835629	+	0.549294i	$0.185103\pi$
$30$	−1.00000			−0.182574
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		−	1.00000i		−	0.176777i
$33$		−	1.00000i		−	0.174078i
$34$		−	1.00000i		−	0.171499i
$35$	1.00000			0.169031
$36$	−1.00000			−0.166667
$37$		−	9.00000i		−	1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$		−	9.00000i		−	1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	−1.00000			−0.162221
$39$	−3.00000	−	2.00000i	−0.480384	−	0.320256i
$40$	−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	$0.785223\pi$
$42$	1.00000			0.154303
$43$	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
$45$			1.00000i			0.149071i
$46$		−	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	$0.801699\pi$
$48$	−1.00000			−0.144338
$49$	−1.00000			−0.142857
$50$		−	4.00000i		−	0.565685i
$51$	−1.00000			−0.140028
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$			1.00000i			0.136083i
$55$	−1.00000			−0.134840
$56$	1.00000			0.133631
$57$			1.00000i			0.132453i
$58$		−	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	$-0.127721\pi$
$60$			1.00000i			0.129099i
$61$	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
$63$		−	1.00000i		−	0.125988i
$64$	−1.00000			−0.125000
$65$	−2.00000	+	3.00000i	−0.248069	+	0.372104i
$66$	−1.00000			−0.123091
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	−1.00000			−0.121268
$69$	−3.00000			−0.361158
$70$		−	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	$0.884129\pi$
$72$			1.00000i			0.117851i
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	−9.00000			−1.04623
$75$	−4.00000			−0.461880
$76$			1.00000i			0.114708i
$77$	1.00000			0.113961
$78$	−2.00000	+	3.00000i	−0.226455	+	0.339683i
$79$	−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
$81$	1.00000			0.111111
$82$	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	$0.893190\pi$
$84$		−	1.00000i		−	0.109109i
$85$			1.00000i			0.108465i
$86$		−	7.00000i		−	0.754829i
$87$	−9.00000			−0.964901
$88$	−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	$0.780588\pi$
$90$	1.00000			0.105409
$91$	2.00000	−	3.00000i	0.209657	−	0.314485i
$92$	−3.00000			−0.312772
$93$			4.00000i			0.414781i
$94$	8.00000			0.825137
$95$	1.00000			0.102598
$96$			1.00000i			0.102062i
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$			1.00000i			0.101015i
$99$			1.00000i			0.100504i
$100$	−4.00000			−0.400000
$101$	−2.00000			−0.199007			−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000			−0.199007			−0.0995037	+	0.995037i	$0.531726\pi$
$102$			1.00000i			0.0990148i
$103$	−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
$105$	−1.00000			−0.0975900
$106$			10.0000i			0.971286i
$107$	−2.00000			−0.193347			−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000			−0.193347			−0.0966736	+	0.995316i	$0.530820\pi$
$108$	1.00000			0.0962250
$109$			7.00000i			0.670478i	0.942133	+	0.335239i	$0.108817\pi$
$109$			7.00000i			0.670478i	−0.942133	+	0.335239i	$0.891183\pi$
$110$			1.00000i			0.0953463i
$111$			9.00000i			0.854242i
$112$		−	1.00000i		−	0.0944911i
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$	1.00000			0.0936586
$115$			3.00000i			0.279751i
$116$	−9.00000			−0.835629
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$	−6.00000			−0.552345
$119$		−	1.00000i		−	0.0916698i
$120$	1.00000			0.0912871
$121$	10.0000			0.909091
$122$		−	11.0000i		−	0.995893i
$123$		−	8.00000i		−	0.721336i
$124$			4.00000i			0.359211i
$125$			9.00000i			0.804984i
$126$	−1.00000			−0.0890871
$127$	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
$129$	−7.00000			−0.616316
$130$	3.00000	+	2.00000i	0.263117	+	0.175412i
$131$	−3.00000			−0.262111			−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000			−0.262111			−0.131056	+	0.991375i	$0.541837\pi$
$132$			1.00000i			0.0870388i
$133$	−1.00000			−0.0867110
$134$	−12.0000			−1.03664
$135$		−	1.00000i		−	0.0860663i
$136$			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	$-0.221384\pi$
$138$			3.00000i			0.255377i
$139$	−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
$141$		−	8.00000i		−	0.673722i
$142$	6.00000			0.503509
$143$	−2.00000	+	3.00000i	−0.167248	+	0.250873i
$144$	1.00000			0.0833333
$145$			9.00000i			0.747409i
$146$	11.0000			0.910366
$147$	1.00000			0.0824786
$148$			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	$0.920959\pi$
$150$			4.00000i			0.326599i
$151$			1.00000i			0.0813788i	0.999172	+	0.0406894i	$0.0129554\pi$
$151$			1.00000i			0.0813788i	−0.999172	+	0.0406894i	$0.987045\pi$
$152$	1.00000			0.0811107
$153$	1.00000			0.0808452
$154$		−	1.00000i		−	0.0805823i
$155$	4.00000			0.321288
$156$	3.00000	+	2.00000i	0.240192	+	0.160128i
$157$	−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
$159$	10.0000			0.793052
$160$	1.00000			0.0790569
$161$		−	3.00000i		−	0.236433i
$162$		−	1.00000i		−	0.0785674i
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		−	8.00000i		−	0.624695i
$165$	1.00000			0.0778499
$166$	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	$-0.139938\pi$
$168$	−1.00000			−0.0771517
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	1.00000			0.0766965
$171$		−	1.00000i		−	0.0764719i
$172$	−7.00000			−0.533745
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$			9.00000i			0.682288i
$175$		−	4.00000i		−	0.302372i
$176$			1.00000i			0.0753778i
$177$			6.00000i			0.450988i
$178$	12.0000			0.899438
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		−	1.00000i		−	0.0745356i
$181$	−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
$183$	−11.0000			−0.813143
$184$			3.00000i			0.221163i
$185$	9.00000			0.661693
$186$	4.00000			0.293294
$187$			1.00000i			0.0731272i
$188$		−	8.00000i		−	0.583460i
$189$			1.00000i			0.0727393i
$190$		−	1.00000i		−	0.0725476i
$191$	−11.0000			−0.795932			−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000			−0.795932			−0.397966	+	0.917400i	$0.630284\pi$
$192$	1.00000			0.0721688
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$	−2.00000			−0.143592
$195$	2.00000	−	3.00000i	0.143223	−	0.214834i
$196$	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	$-0.0226979\pi$
$198$	1.00000			0.0710669
$199$	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
$201$			12.0000i			0.846415i
$202$			2.00000i			0.140720i
$203$		−	9.00000i		−	0.631676i
$204$	1.00000			0.0700140
$205$	−8.00000			−0.558744
$206$			5.00000i			0.348367i
$207$	3.00000			0.208514
$208$	3.00000	+	2.00000i	0.208013	+	0.138675i
$209$	1.00000			0.0691714
$210$			1.00000i			0.0690066i
$211$	−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
$213$		−	6.00000i		−	0.411113i
$214$			2.00000i			0.136717i
$215$			7.00000i			0.477396i
$216$		−	1.00000i		−	0.0680414i
$217$	−4.00000			−0.271538
$218$	7.00000			0.474100
$219$		−	11.0000i		−	0.743311i
$220$	1.00000			0.0674200
$221$	3.00000	+	2.00000i	0.201802	+	0.134535i
$222$	9.00000			0.604040
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	−1.00000			−0.0668153
$225$	4.00000			0.266667
$226$			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	$-0.130377\pi$
$228$		−	1.00000i		−	0.0662266i
$229$			2.00000i			0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$			2.00000i			0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	3.00000			0.197814
$231$	−1.00000			−0.0657952
$232$			9.00000i			0.590879i
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$	2.00000	−	3.00000i	0.130744	−	0.196116i
$235$	−8.00000			−0.521862
$236$			6.00000i			0.390567i
$237$	12.0000			0.779484
$238$	−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	$-0.0833068\pi$
$240$		−	1.00000i		−	0.0645497i
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$		−	10.0000i		−	0.642824i
$243$	−1.00000			−0.0641500
$244$	−11.0000			−0.704203
$245$		−	1.00000i		−	0.0638877i
$246$	−8.00000			−0.510061
$247$	2.00000	−	3.00000i	0.127257	−	0.190885i
$248$	4.00000			0.254000
$249$		−	6.00000i		−	0.380235i
$250$	9.00000			0.569210
$251$	11.0000			0.694314			0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000			0.694314			0.347157	+	0.937807i	$0.387147\pi$
$252$			1.00000i			0.0629941i
$253$			3.00000i			0.188608i
$254$		−	16.0000i		−	1.00393i
$255$		−	1.00000i		−	0.0626224i
$256$	1.00000			0.0625000
$257$	14.0000			0.873296			0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000			0.873296			0.436648	+	0.899632i	$0.356166\pi$
$258$			7.00000i			0.435801i
$259$	−9.00000			−0.559233
$260$	2.00000	−	3.00000i	0.124035	−	0.186052i
$261$	9.00000			0.557086
$262$			3.00000i			0.185341i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$	1.00000			0.0615457
$265$		−	10.0000i		−	0.614295i
$266$			1.00000i			0.0613139i
$267$		−	12.0000i		−	0.734388i
$268$			12.0000i			0.733017i
$269$	−20.0000			−1.21942			−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000			−1.21942			−0.609711	+	0.792624i	$0.708714\pi$
$270$	−1.00000			−0.0608581
$271$		−	4.00000i		−	0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$		−	4.00000i		−	0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$	1.00000			0.0606339
$273$	−2.00000	+	3.00000i	−0.121046	+	0.181568i
$274$	−15.0000			−0.906183
$275$			4.00000i			0.241209i
$276$	3.00000			0.180579
$277$	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
$279$		−	4.00000i		−	0.239474i
$280$			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	$-0.227839\pi$
$282$	−8.00000			−0.476393
$283$	−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
$285$	−1.00000			−0.0592349
$286$	3.00000	+	2.00000i	0.177394	+	0.118262i
$287$	8.00000			0.472225
$288$		−	1.00000i		−	0.0589256i
$289$	−16.0000			−0.941176
$290$	9.00000			0.528498
$291$			2.00000i			0.117242i
$292$		−	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	$-0.274541\pi$
$294$		−	1.00000i		−	0.0583212i
$295$	6.00000			0.349334
$296$	9.00000			0.523114
$297$		−	1.00000i		−	0.0580259i
$298$	6.00000			0.347571
$299$	9.00000	+	6.00000i	0.520483	+	0.346989i
$300$	4.00000			0.230940
$301$		−	7.00000i		−	0.403473i
$302$	1.00000			0.0575435
$303$	2.00000			0.114897
$304$		−	1.00000i		−	0.0573539i
$305$			11.0000i			0.629858i
$306$		−	1.00000i		−	0.0571662i
$307$		−	8.00000i		−	0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$		−	8.00000i		−	0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	−1.00000			−0.0569803
$309$	5.00000			0.284440
$310$		−	4.00000i		−	0.227185i
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$	2.00000	−	3.00000i	0.113228	−	0.169842i
$313$	−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
$315$	1.00000			0.0563436
$316$	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	$-0.109410\pi$
$318$		−	10.0000i		−	0.560772i
$319$			9.00000i			0.503903i
$320$		−	1.00000i		−	0.0559017i
$321$	2.00000			0.111629
$322$	−3.00000			−0.167183
$323$		−	1.00000i		−	0.0556415i
$324$	−1.00000			−0.0555556
$325$	12.0000	+	8.00000i	0.665640	+	0.443760i
$326$	−20.0000			−1.10770
$327$		−	7.00000i		−	0.387101i
$328$	−8.00000			−0.441726
$329$	8.00000			0.441054
$330$		−	1.00000i		−	0.0550482i
$331$		−	8.00000i		−	0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$331$		−	8.00000i		−	0.439720i	0.975531	−	0.219860i	$-0.0705600\pi$
$332$		−	6.00000i		−	0.329293i
$333$		−	9.00000i		−	0.493197i
$334$	−11.0000			−0.601893
$335$	12.0000			0.655630
$336$			1.00000i			0.0545545i
$337$	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
$339$	2.00000			0.108625
$340$		−	1.00000i		−	0.0542326i
$341$	4.00000			0.216612
$342$	−1.00000			−0.0540738
$343$			1.00000i			0.0539949i
$344$			7.00000i			0.377415i
$345$		−	3.00000i		−	0.161515i
$346$		−	14.0000i		−	0.752645i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	9.00000			0.482451
$349$		−	4.00000i		−	0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$		−	4.00000i		−	0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$	−4.00000			−0.213809
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$	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	$-0.140006\pi$
$354$	6.00000			0.318896
$355$	−6.00000			−0.318447
$356$		−	12.0000i		−	0.635999i
$357$			1.00000i			0.0529256i
$358$			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	$0.709212\pi$
$360$	−1.00000			−0.0527046
$361$	18.0000			0.947368
$362$			2.00000i			0.105118i
$363$	−10.0000			−0.524864
$364$	−2.00000	+	3.00000i	−0.104828	+	0.157243i
$365$	−11.0000			−0.575766
$366$			11.0000i			0.574979i
$367$	−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
$369$			8.00000i			0.416463i
$370$		−	9.00000i		−	0.467888i
$371$			10.0000i			0.519174i
$372$		−	4.00000i		−	0.207390i
$373$	−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
$375$		−	9.00000i		−	0.464758i
$376$	−8.00000			−0.412568
$377$	27.0000	+	18.0000i	1.39057	+	0.927047i
$378$	1.00000			0.0514344
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	−1.00000			−0.0512989
$381$	−16.0000			−0.819705
$382$			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	$0.647739\pi$
$384$		−	1.00000i		−	0.0510310i
$385$			1.00000i			0.0509647i
$386$	−12.0000			−0.610784
$387$	7.00000			0.355830
$388$			2.00000i			0.101535i
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$	−3.00000	−	2.00000i	−0.151911	−	0.101274i
$391$	3.00000			0.151717
$392$		−	1.00000i		−	0.0505076i
$393$	3.00000			0.151330
$394$	−2.00000			−0.100759
$395$		−	12.0000i		−	0.603786i
$396$		−	1.00000i		−	0.0502519i
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$		−	5.00000i		−	0.250627i
$399$	1.00000			0.0500626
$400$	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	$0.886330\pi$
$402$	12.0000			0.598506
$403$	8.00000	−	12.0000i	0.398508	−	0.597763i
$404$	2.00000			0.0995037
$405$			1.00000i			0.0496904i
$406$	−9.00000			−0.446663
$407$	9.00000			0.446113
$408$		−	1.00000i		−	0.0495074i
$409$			27.0000i			1.33506i	0.744581	+	0.667532i	$0.232649\pi$
$409$			27.0000i			1.33506i	−0.744581	+	0.667532i	$0.767351\pi$
$410$			8.00000i			0.395092i
$411$			15.0000i			0.739895i
$412$	5.00000			0.246332
$413$	−6.00000			−0.295241
$414$		−	3.00000i		−	0.147442i
$415$	−6.00000			−0.294528
$416$	2.00000	−	3.00000i	0.0980581	−	0.147087i
$417$	12.0000			0.587643
$418$		−	1.00000i		−	0.0489116i
$419$	23.0000			1.12362			0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000			1.12362			0.561812	+	0.827265i	$0.310105\pi$
$420$	1.00000			0.0487950
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$			5.00000i			0.243396i
$423$			8.00000i			0.388973i
$424$		−	10.0000i		−	0.485643i
$425$	4.00000			0.194029
$426$	−6.00000			−0.290701
$427$		−	11.0000i		−	0.532327i
$428$	2.00000			0.0966736
$429$	2.00000	−	3.00000i	0.0965609	−	0.144841i
$430$	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	$-0.142727\pi$
$432$	−1.00000			−0.0481125
$433$	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
$435$		−	9.00000i		−	0.431517i
$436$		−	7.00000i		−	0.335239i
$437$		−	3.00000i		−	0.143509i
$438$	−11.0000			−0.525600
$439$	−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
$441$	−1.00000			−0.0476190
$442$	2.00000	−	3.00000i	0.0951303	−	0.142695i
$443$	−6.00000			−0.285069			−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000			−0.285069			−0.142534	+	0.989790i	$0.545525\pi$
$444$		−	9.00000i		−	0.427121i
$445$	−12.0000			−0.568855
$446$	−8.00000			−0.378811
$447$		−	6.00000i		−	0.283790i
$448$			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	$-0.115161\pi$
$450$		−	4.00000i		−	0.188562i
$451$	−8.00000			−0.376705
$452$	2.00000			0.0940721
$453$		−	1.00000i		−	0.0469841i
$454$	−12.0000			−0.563188
$455$	3.00000	+	2.00000i	0.140642	+	0.0937614i
$456$	−1.00000			−0.0468293
$457$			4.00000i			0.187112i	0.995614	+	0.0935561i	$0.0298234\pi$
$457$			4.00000i			0.187112i	−0.995614	+	0.0935561i	$0.970177\pi$
$458$	2.00000			0.0934539
$459$	−1.00000			−0.0466760
$460$		−	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	$0.977744\pi$
$462$			1.00000i			0.0465242i
$463$			29.0000i			1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$			29.0000i			1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$	9.00000			0.417815
$465$	−4.00000			−0.185496
$466$			18.0000i			0.833834i
$467$	−37.0000			−1.71216			−0.856078	−	0.516847i	$-0.827106\pi$
$467$	−37.0000			−1.71216			−0.856078	+	0.516847i	$0.827106\pi$
$468$	−3.00000	−	2.00000i	−0.138675	−	0.0924500i
$469$	−12.0000			−0.554109
$470$			8.00000i			0.369012i
$471$	21.0000			0.967629
$472$	6.00000			0.276172
$473$			7.00000i			0.321860i
$474$		−	12.0000i		−	0.551178i
$475$		−	4.00000i		−	0.183533i
$476$			1.00000i			0.0458349i
$477$	−10.0000			−0.457869
$478$	−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	$0.650020\pi$
$480$	−1.00000			−0.0456435
$481$	18.0000	−	27.0000i	0.820729	−	1.23109i
$482$	10.0000			0.455488
$483$			3.00000i			0.136505i
$484$	−10.0000			−0.454545
$485$	2.00000			0.0908153
$486$			1.00000i			0.0453609i
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$			11.0000i			0.497947i
$489$			20.0000i			0.904431i
$490$	−1.00000			−0.0451754
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$			8.00000i			0.360668i
$493$	9.00000			0.405340
$494$	−3.00000	−	2.00000i	−0.134976	−	0.0899843i
$495$	−1.00000			−0.0449467
$496$		−	4.00000i		−	0.179605i
$497$	6.00000			0.269137
$498$	−6.00000			−0.268866
$499$		−	10.0000i		−	0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$		−	10.0000i		−	0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$		−	9.00000i		−	0.402492i
$501$			11.0000i			0.491444i
$502$		−	11.0000i		−	0.490954i
$503$	−38.0000			−1.69434			−0.847168	−	0.531325i	$-0.821694\pi$
$503$	−38.0000			−1.69434			−0.847168	+	0.531325i	$0.821694\pi$
$504$	1.00000			0.0445435
$505$		−	2.00000i		−	0.0889988i
$506$	3.00000			0.133366
$507$	−5.00000	−	12.0000i	−0.222058	−	0.532939i
$508$	−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	$-0.306026\pi$
$510$	−1.00000			−0.0442807
$511$	11.0000			0.486611
$512$		−	1.00000i		−	0.0441942i
$513$			1.00000i			0.0441511i
$514$		−	14.0000i		−	0.617514i
$515$		−	5.00000i		−	0.220326i
$516$	7.00000			0.308158
$517$	−8.00000			−0.351840
$518$			9.00000i			0.395437i
$519$	−14.0000			−0.614532
$520$	−3.00000	−	2.00000i	−0.131559	−	0.0877058i
$521$	−31.0000			−1.35813			−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000			−1.35813			−0.679067	+	0.734076i	$0.737616\pi$
$522$		−	9.00000i		−	0.393919i
$523$	−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
$525$			4.00000i			0.174574i
$526$		−	16.0000i		−	0.697633i
$527$		−	4.00000i		−	0.174243i
$528$		−	1.00000i		−	0.0435194i
$529$	−14.0000			−0.608696
$530$	−10.0000			−0.434372
$531$		−	6.00000i		−	0.260378i
$532$	1.00000			0.0433555
$533$	−16.0000	+	24.0000i	−0.693037	+	1.03956i
$534$	−12.0000			−0.519291
$535$		−	2.00000i		−	0.0864675i
$536$	12.0000			0.518321
$537$	18.0000			0.776757
$538$			20.0000i			0.862261i
$539$		−	1.00000i		−	0.0430730i
$540$			1.00000i			0.0430331i
$541$			41.0000i			1.76273i	0.472438	+	0.881364i	$0.343374\pi$
$541$			41.0000i			1.76273i	−0.472438	+	0.881364i	$0.656626\pi$
$542$	−4.00000			−0.171815
$543$	2.00000			0.0858282
$544$		−	1.00000i		−	0.0428746i
$545$	−7.00000			−0.299847
$546$	3.00000	+	2.00000i	0.128388	+	0.0855921i
$547$	−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
$549$	11.0000			0.469469
$550$	4.00000			0.170561
$551$		−	9.00000i		−	0.383413i
$552$		−	3.00000i		−	0.127688i
$553$			12.0000i			0.510292i
$554$		−	16.0000i		−	0.679775i
$555$	−9.00000			−0.382029
$556$	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	$-0.0134913\pi$
$558$	−4.00000			−0.169334
$559$	21.0000	+	14.0000i	0.888205	+	0.592137i
$560$	1.00000			0.0422577
$561$		−	1.00000i		−	0.0422200i
$562$	−22.0000			−0.928014
$563$	11.0000			0.463595			0.231797	−	0.972764i	$-0.425539\pi$
$563$	11.0000			0.463595			0.231797	+	0.972764i	$0.425539\pi$
$564$			8.00000i			0.336861i
$565$		−	2.00000i		−	0.0841406i
$566$			14.0000i			0.588464i
$567$		−	1.00000i		−	0.0419961i
$568$	−6.00000			−0.251754
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$			1.00000i			0.0418854i
$571$	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
$573$	11.0000			0.459532
$574$		−	8.00000i		−	0.333914i
$575$	12.0000			0.500435
$576$	−1.00000			−0.0416667
$577$		−	18.0000i		−	0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$		−	18.0000i		−	0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$			16.0000i			0.665512i
$579$			12.0000i			0.498703i
$580$		−	9.00000i		−	0.373705i
$581$	6.00000			0.248922
$582$	2.00000			0.0829027
$583$		−	10.0000i		−	0.414158i
$584$	−11.0000			−0.455183
$585$	−2.00000	+	3.00000i	−0.0826898	+	0.124035i
$586$	−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	$-0.107115\pi$
$588$	−1.00000			−0.0412393
$589$	−4.00000			−0.164817
$590$		−	6.00000i		−	0.247016i
$591$			2.00000i			0.0822690i
$592$		−	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	$-0.194963\pi$
$594$	−1.00000			−0.0410305
$595$	1.00000			0.0409960
$596$		−	6.00000i		−	0.245770i
$597$	−5.00000			−0.204636
$598$	6.00000	−	9.00000i	0.245358	−	0.368037i
$599$	39.0000			1.59350			0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000			1.59350			0.796748	+	0.604311i	$0.206552\pi$
$600$		−	4.00000i		−	0.163299i
$601$	−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
$603$		−	12.0000i		−	0.488678i
$604$		−	1.00000i		−	0.0406894i
$605$			10.0000i			0.406558i
$606$		−	2.00000i		−	0.0812444i
$607$	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
$609$			9.00000i			0.364698i
$610$	11.0000			0.445377
$611$	−16.0000	+	24.0000i	−0.647291	+	0.970936i
$612$	−1.00000			−0.0404226
$613$		−	17.0000i		−	0.686624i	−0.939222	−	0.343312i	$-0.888451\pi$
$613$		−	17.0000i		−	0.686624i	0.939222	−	0.343312i	$-0.111549\pi$
$614$	−8.00000			−0.322854
$615$	8.00000			0.322591
$616$			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	$-0.124922\pi$
$618$		−	5.00000i		−	0.201129i
$619$			21.0000i			0.844061i	0.906582	+	0.422031i	$0.138683\pi$
$619$			21.0000i			0.844061i	−0.906582	+	0.422031i	$0.861317\pi$
$620$	−4.00000			−0.160644
$621$	−3.00000			−0.120386
$622$		−	8.00000i		−	0.320771i
$623$	12.0000			0.480770
$624$	−3.00000	−	2.00000i	−0.120096	−	0.0800641i
$625$	11.0000			0.440000
$626$			14.0000i			0.559553i
$627$	−1.00000			−0.0399362
$628$	21.0000			0.837991
$629$		−	9.00000i		−	0.358854i
$630$		−	1.00000i		−	0.0398410i
$631$			29.0000i			1.15447i	0.816577	+	0.577236i	$0.195869\pi$
$631$			29.0000i			1.15447i	−0.816577	+	0.577236i	$0.804131\pi$
$632$		−	12.0000i		−	0.477334i
$633$	5.00000			0.198732
$634$	−12.0000			−0.476581
$635$			16.0000i			0.634941i
$636$	−10.0000			−0.396526
$637$	−3.00000	−	2.00000i	−0.118864	−	0.0792429i
$638$	9.00000			0.356313
$639$			6.00000i			0.237356i
$640$	−1.00000			−0.0395285
$641$	−44.0000			−1.73790			−0.868948	−	0.494904i	$-0.835203\pi$
$641$	−44.0000			−1.73790			−0.868948	+	0.494904i	$0.835203\pi$
$642$		−	2.00000i		−	0.0789337i
$643$		−	19.0000i		−	0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$		−	19.0000i		−	0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$			3.00000i			0.118217i
$645$		−	7.00000i		−	0.275625i
$646$	−1.00000			−0.0393445
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$			1.00000i			0.0392837i
$649$	6.00000			0.235521
$650$	8.00000	−	12.0000i	0.313786	−	0.470679i
$651$	4.00000			0.156772
$652$			20.0000i			0.783260i
$653$	−13.0000			−0.508729			−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000			−0.508729			−0.254365	+	0.967108i	$0.581866\pi$
$654$	−7.00000			−0.273722
$655$		−	3.00000i		−	0.117220i
$656$			8.00000i			0.312348i
$657$			11.0000i			0.429151i
$658$		−	8.00000i		−	0.311872i
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$	−1.00000			−0.0389249
$661$			26.0000i			1.01128i	0.862744	+	0.505641i	$0.168744\pi$
$661$			26.0000i			1.01128i	−0.862744	+	0.505641i	$0.831256\pi$
$662$	−8.00000			−0.310929
$663$	−3.00000	−	2.00000i	−0.116510	−	0.0776736i
$664$	−6.00000			−0.232845
$665$		−	1.00000i		−	0.0387783i
$666$	−9.00000			−0.348743
$667$	27.0000			1.04544
$668$			11.0000i			0.425603i
$669$			8.00000i			0.309298i
$670$		−	12.0000i		−	0.463600i
$671$			11.0000i			0.424650i
$672$	1.00000			0.0385758
$673$	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
$675$	−4.00000			−0.153960
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	32.0000			1.22986			0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000			1.22986			0.614930	+	0.788582i	$0.289184\pi$
$678$		−	2.00000i		−	0.0768095i
$679$	−2.00000			−0.0767530
$680$	−1.00000			−0.0383482
$681$			12.0000i			0.459841i
$682$		−	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	$0.881579\pi$
$684$			1.00000i			0.0382360i
$685$	15.0000			0.573121
$686$	1.00000			0.0381802
$687$		−	2.00000i		−	0.0763048i
$688$	7.00000			0.266872
$689$	−30.0000	−	20.0000i	−1.14291	−	0.761939i
$690$	−3.00000			−0.114208
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$	−14.0000			−0.532200
$693$	1.00000			0.0379869
$694$		0			0
$695$		−	12.0000i		−	0.455186i
$696$		−	9.00000i		−	0.341144i
$697$			8.00000i			0.303022i
$698$	−4.00000			−0.151402
$699$	18.0000			0.680823
$700$			4.00000i			0.151186i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.00000	+	3.00000i	−0.0754851	+	0.113228i
$703$	−9.00000			−0.339441
$704$		−	1.00000i		−	0.0376889i
$705$	8.00000			0.301297
$706$	−16.0000			−0.602168
$707$			2.00000i			0.0752177i
$708$		−	6.00000i		−	0.225494i
$709$		−	14.0000i		−	0.525781i	−0.964826	−	0.262891i	$-0.915324\pi$
$709$		−	14.0000i		−	0.525781i	0.964826	−	0.262891i	$-0.0846758\pi$
$710$			6.00000i			0.225176i
$711$	−12.0000			−0.450035
$712$	−12.0000			−0.449719
$713$		−	12.0000i		−	0.449404i
$714$	1.00000			0.0374241
$715$	−3.00000	−	2.00000i	−0.112194	−	0.0747958i
$716$	18.0000			0.672692
$717$			8.00000i			0.298765i
$718$	30.0000			1.11959
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$			1.00000i			0.0372678i
$721$			5.00000i			0.186210i
$722$		−	18.0000i		−	0.669891i
$723$		−	10.0000i		−	0.371904i
$724$	2.00000			0.0743294
$725$	36.0000			1.33701
$726$			10.0000i			0.371135i
$727$	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
$729$	1.00000			0.0370370
$730$			11.0000i			0.407128i
$731$	7.00000			0.258904
$732$	11.0000			0.406572
$733$		−	40.0000i		−	1.47743i	−0.674016	−	0.738717i	$-0.735432\pi$
$733$		−	40.0000i		−	1.47743i	0.674016	−	0.738717i	$-0.264568\pi$
$734$			8.00000i			0.295285i
$735$			1.00000i			0.0368856i
$736$		−	3.00000i		−	0.110581i
$737$	12.0000			0.442026
$738$	8.00000			0.294484
$739$			8.00000i			0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$			8.00000i			0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$	−9.00000			−0.330847
$741$	−2.00000	+	3.00000i	−0.0734718	+	0.110208i
$742$	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	$-0.229595\pi$
$744$	−4.00000			−0.146647
$745$	−6.00000			−0.219823
$746$			14.0000i			0.512576i
$747$			6.00000i			0.219529i
$748$		−	1.00000i		−	0.0365636i
$749$			2.00000i			0.0730784i
$750$	−9.00000			−0.328634
$751$	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
$753$	−11.0000			−0.400862
$754$	18.0000	−	27.0000i	0.655521	−	0.983282i
$755$	−1.00000			−0.0363937
$756$		−	1.00000i		−	0.0363696i
$757$	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
$759$		−	3.00000i		−	0.108893i
$760$			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	$-0.275414\pi$
$762$			16.0000i			0.579619i
$763$	7.00000			0.253417
$764$	11.0000			0.397966
$765$			1.00000i			0.0361551i
$766$	35.0000			1.26460
$767$	12.0000	−	18.0000i	0.433295	−	0.649942i
$768$	−1.00000			−0.0360844
$769$			31.0000i			1.11789i	0.829205	+	0.558944i	$0.188793\pi$
$769$			31.0000i			1.11789i	−0.829205	+	0.558944i	$0.811207\pi$
$770$	1.00000			0.0360375
$771$	−14.0000			−0.504198
$772$			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	$0.889000\pi$
$774$		−	7.00000i		−	0.251610i
$775$		−	16.0000i		−	0.574737i
$776$	2.00000			0.0717958
$777$	9.00000			0.322873
$778$		−	2.00000i		−	0.0717035i
$779$	8.00000			0.286630
$780$	−2.00000	+	3.00000i	−0.0716115	+	0.107417i
$781$	−6.00000			−0.214697
$782$		−	3.00000i		−	0.107280i
$783$	−9.00000			−0.321634
$784$	−1.00000			−0.0357143
$785$		−	21.0000i		−	0.749522i

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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000i q^{11} +1.00000 q^{12} +(3.00000 + 2.00000i) q^{13} -1.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000i q^{18} -1.00000i q^{19} -1.00000i q^{20} +1.00000i q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000i q^{24} +4.00000 q^{25} +(2.00000 - 3.00000i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +9.00000 q^{29} -1.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -1.00000i q^{33} -1.00000i q^{34} +1.00000 q^{35} -1.00000 q^{36} -9.00000i q^{37} -1.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -1.00000 q^{40} +8.00000i q^{41} +1.00000 q^{42} +7.00000 q^{43} -1.00000i q^{44} +1.00000i q^{45} -3.00000i q^{46} +8.00000i q^{47} -1.00000 q^{48} -1.00000 q^{49} -4.00000i q^{50} -1.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -10.0000 q^{53} +1.00000i q^{54} -1.00000 q^{55} +1.00000 q^{56} +1.00000i q^{57} -9.00000i q^{58} -6.00000i q^{59} +1.00000i q^{60} +11.0000 q^{61} -4.00000 q^{62} -1.00000i q^{63} -1.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} -1.00000 q^{66} -12.0000i q^{67} -1.00000 q^{68} -3.00000 q^{69} -1.00000i q^{70} +6.00000i q^{71} +1.00000i q^{72} +11.0000i q^{73} -9.00000 q^{74} -4.00000 q^{75} +1.00000i q^{76} +1.00000 q^{77} +(-2.00000 + 3.00000i) q^{78} -12.0000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +8.00000 q^{82} +6.00000i q^{83} -1.00000i q^{84} +1.00000i q^{85} -7.00000i q^{86} -9.00000 q^{87} -1.00000 q^{88} +12.0000i q^{89} +1.00000 q^{90} +(2.00000 - 3.00000i) q^{91} -3.00000 q^{92} +4.00000i q^{93} +8.00000 q^{94} +1.00000 q^{95} +1.00000i q^{96} -2.00000i q^{97} +1.00000i q^{98} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} + 2 q^{12} + 6 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{22} + 6 q^{23} + 8 q^{25} + 4 q^{26} - 2 q^{27} + 18 q^{29} - 2 q^{30} + 2 q^{35} - 2 q^{36} - 2 q^{38} - 6 q^{39} - 2 q^{40} + 2 q^{42} + 14 q^{43} - 2 q^{48} - 2 q^{49} - 2 q^{51} - 6 q^{52} - 20 q^{53} - 2 q^{55} + 2 q^{56} + 22 q^{61} - 8 q^{62} - 2 q^{64} - 4 q^{65} - 2 q^{66} - 2 q^{68} - 6 q^{69} - 18 q^{74} - 8 q^{75} + 2 q^{77} - 4 q^{78} - 24 q^{79} + 2 q^{81} + 16 q^{82} - 18 q^{87} - 2 q^{88} + 2 q^{90} + 4 q^{91} - 6 q^{92} + 16 q^{94} + 2 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 2 * q^10 + 2 * q^12 + 6 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 + 2 * q^22 + 6 * q^23 + 8 * q^25 + 4 * q^26 - 2 * q^27 + 18 * q^29 - 2 * q^30 + 2 * q^35 - 2 * q^36 - 2 * q^38 - 6 * q^39 - 2 * q^40 + 2 * q^42 + 14 * q^43 - 2 * q^48 - 2 * q^49 - 2 * q^51 - 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^66 - 2 * q^68 - 6 * q^69 - 18 * q^74 - 8 * q^75 + 2 * q^77 - 4 * q^78 - 24 * q^79 + 2 * q^81 + 16 * q^82 - 18 * q^87 - 2 * q^88 + 2 * q^90 + 4 * q^91 - 6 * q^92 + 16 * q^94 + 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more