LMFDB - Embedded newform 3822.2.c.e.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(883,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, 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("3822.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.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:	$30.5188236525$
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	$\chi$	$=$	3822.883
Dual form			3822.2.c.e.883.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000i q^{11} -1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} -1.00000i q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000i q^{18} +1.00000i q^{19} +1.00000i q^{20} +1.00000 q^{22} +3.00000 q^{23} +1.00000i q^{24} +4.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} +1.00000 q^{27} +9.00000 q^{29} -1.00000 q^{30} +4.00000i q^{31} -1.00000i q^{32} +1.00000i q^{33} +1.00000i q^{34} -1.00000 q^{36} -9.00000i q^{37} +1.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} +1.00000 q^{40} -8.00000i q^{41} +7.00000 q^{43} -1.00000i q^{44} -1.00000i q^{45} -3.00000i q^{46} -8.00000i q^{47} +1.00000 q^{48} -4.00000i q^{50} -1.00000 q^{51} +(3.00000 + 2.00000i) q^{52} -10.0000 q^{53} -1.00000i q^{54} +1.00000 q^{55} +1.00000i q^{57} -9.00000i q^{58} +6.00000i q^{59} +1.00000i q^{60} -11.0000 q^{61} +4.00000 q^{62} -1.00000 q^{64} +(-2.00000 + 3.00000i) q^{65} +1.00000 q^{66} -12.0000i q^{67} +1.00000 q^{68} +3.00000 q^{69} +6.00000i q^{71} +1.00000i q^{72} -11.0000i q^{73} -9.00000 q^{74} +4.00000 q^{75} -1.00000i q^{76} +(-2.00000 + 3.00000i) q^{78} -12.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -8.00000 q^{82} -6.00000i q^{83} +1.00000i q^{85} -7.00000i q^{86} +9.00000 q^{87} -1.00000 q^{88} -12.0000i q^{89} -1.00000 q^{90} -3.00000 q^{92} +4.00000i q^{93} -8.00000 q^{94} +1.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{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^{36} + 2 q^{38} - 6 q^{39} + 2 q^{40} + 14 q^{43} + 2 q^{48} - 2 q^{51} + 6 q^{52} - 20 q^{53} + 2 q^{55} - 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} - 4 q^{78} - 24 q^{79} + 2 q^{81} - 16 q^{82} + 18 q^{87} - 2 q^{88} - 2 q^{90} - 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^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^36 + 2 * q^38 - 6 * q^39 + 2 * q^40 + 14 * q^43 + 2 * q^48 - 2 * q^51 + 6 * q^52 - 20 * q^53 + 2 * q^55 - 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 - 4 * q^78 - 24 * q^79 + 2 * q^81 - 16 * q^82 + 18 * q^87 - 2 * q^88 - 2 * q^90 - 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 $\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$.

$n$	$1471$	$2549$	$3433$
$\chi(n)$	$-1$	$1$	$1$

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	$-0.928217\pi$
$5$		−	1.00000i		−	0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$		−	1.00000i		−	0.408248i
$7$		0			0
$7$		0			0
$8$			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
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$	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	$0.538696\pi$
$18$		−	1.00000i		−	0.235702i
$19$			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	$-0.398743\pi$
$23$	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
$28$		0			0
$29$	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	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		−	1.00000i		−	0.176777i
$33$			1.00000i			0.174078i
$34$			1.00000i			0.171499i
$35$		0			0
$36$	−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	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		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
$45$		−	1.00000i		−	0.149071i
$46$		−	3.00000i		−	0.442326i
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000			0.144338
$49$		0			0
$50$		−	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
$56$		0			0
$57$			1.00000i			0.132453i
$58$		−	9.00000i		−	1.18176i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$			1.00000i			0.129099i
$61$	−11.0000			−1.40841			−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000			−1.40841			−0.704203	+	0.709999i	$0.748695\pi$
$62$	4.00000			0.508001
$63$		0			0
$64$	−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
$70$		0			0
$71$			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	$-0.777388\pi$
$73$		−	11.0000i		−	1.28745i	0.765256	−	0.643726i	$-0.222612\pi$
$74$	−9.00000			−1.04623
$75$	4.00000			0.461880
$76$		−	1.00000i		−	0.114708i
$77$		0			0
$78$	−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	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$			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	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$	−1.00000			−0.105409
$91$		0			0
$92$	−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	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$	−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	$0.468274\pi$
$102$			1.00000i			0.0990148i
$103$	5.00000			0.492665			0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000			0.492665			0.246332	+	0.969185i	$0.420775\pi$
$104$	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	$-0.530820\pi$
$107$	−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
$112$		0			0
$113$	−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
$119$		0			0
$120$	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
$126$		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
$129$	7.00000			0.616316
$130$	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	$0.458163\pi$
$132$		−	1.00000i		−	0.0870388i
$133$		0			0
$134$	−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	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		−	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
$147$		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	$0.920959\pi$
$150$		−	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
$154$		0			0
$155$	4.00000			0.321288
$156$	3.00000	+	2.00000i	0.240192	+	0.160128i
$157$	21.0000			1.67598			0.837991	−	0.545684i	$-0.183730\pi$
$157$	21.0000			1.67598			0.837991	+	0.545684i	$0.183730\pi$
$158$			12.0000i			0.954669i
$159$	−10.0000			−0.793052
$160$	−1.00000			−0.0790569
$161$		0			0
$162$		−	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	$0.139938\pi$
$167$			11.0000i			0.851206i	−0.904910	+	0.425603i	$0.860062\pi$
$168$		0			0
$169$	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	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		−	9.00000i		−	0.682288i
$175$		0			0
$176$			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	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	−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
$189$		0			0
$190$		−	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
$196$		0			0
$197$		−	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	$-0.556711\pi$
$199$	−5.00000			−0.354441			−0.177220	+	0.984171i	$0.556711\pi$
$200$			4.00000i			0.282843i
$201$		−	12.0000i		−	0.846415i
$202$		−	2.00000i		−	0.140720i
$203$		0			0
$204$	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
$210$		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
$213$			6.00000i			0.411113i
$214$			2.00000i			0.136717i
$215$		−	7.00000i		−	0.477396i
$216$			1.00000i			0.0680414i
$217$		0			0
$218$	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	$0.0863164\pi$
$223$			8.00000i			0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$			2.00000i			0.133038i
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		−	1.00000i		−	0.0662266i
$229$		−	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	$-0.700718\pi$
$233$	−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
$238$		0			0
$239$		−	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	$-0.895619\pi$
$241$		−	10.0000i		−	0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$		−	10.0000i		−	0.642824i
$243$	1.00000			0.0641500
$244$	11.0000			0.704203
$245$		0			0
$246$	−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	$-0.612853\pi$
$251$	−11.0000			−0.694314			−0.347157	+	0.937807i	$0.612853\pi$
$252$		0			0
$253$			3.00000i			0.188608i
$254$		−	16.0000i		−	1.00393i
$255$			1.00000i			0.0626224i
$256$	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	$0.643834\pi$
$258$		−	7.00000i		−	0.435801i
$259$		0			0
$260$	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
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$			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	$0.291286\pi$
$270$	−1.00000			−0.0608581
$271$			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
$276$	−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
$280$		0			0
$281$		−	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	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		−	6.00000i		−	0.356034i
$285$	1.00000			0.0592349
$286$	−3.00000	−	2.00000i	−0.177394	−	0.118262i
$287$		0			0
$288$		−	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	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	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
$301$		0			0
$302$	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	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$	5.00000			0.284440
$310$		−	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	$0.572833\pi$
$312$	2.00000	−	3.00000i	0.113228	−	0.169842i
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$		−	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	$-0.109410\pi$
$318$			10.0000i			0.560772i
$319$			9.00000i			0.503903i
$320$			1.00000i			0.0559017i
$321$	−2.00000			−0.111629
$322$		0			0
$323$		−	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
$329$		0			0
$330$		−	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
$336$		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
$339$	−2.00000			−0.108625
$340$		−	1.00000i		−	0.0542326i
$341$	−4.00000			−0.216612
$342$	1.00000			0.0540738
$343$		0			0
$344$			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	$0.0341429\pi$
$349$			4.00000i			0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$		0			0
$351$	−3.00000	−	2.00000i	−0.160128	−	0.106752i
$352$	1.00000			0.0533002
$353$			16.0000i			0.851594i	0.904819	+	0.425797i	$0.140006\pi$
$353$			16.0000i			0.851594i	−0.904819	+	0.425797i	$0.859994\pi$
$354$	6.00000			0.318896
$355$	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	$0.709212\pi$
$360$	1.00000			0.0527046
$361$	18.0000			0.947368
$362$		−	2.00000i		−	0.105118i
$363$	10.0000			0.524864
$364$		0			0
$365$	−11.0000			−0.575766
$366$			11.0000i			0.574979i
$367$	8.00000			0.417597			0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000			0.417597			0.208798	+	0.977959i	$0.433045\pi$
$368$	3.00000			0.156386
$369$		−	8.00000i		−	0.416463i
$370$			9.00000i			0.467888i
$371$		0			0
$372$		−	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
$378$		0			0
$379$			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	$-0.647739\pi$
$383$		−	35.0000i		−	1.78842i	0.447651	−	0.894208i	$-0.352261\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$	−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
$392$		0			0
$393$	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	$-0.967995\pi$
$397$		−	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	$0.886330\pi$
$402$	−12.0000			−0.598506
$403$	8.00000	−	12.0000i	0.398508	−	0.597763i
$404$	−2.00000			−0.0995037
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$	9.00000			0.446113
$408$		−	1.00000i		−	0.0495074i
$409$		−	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
$411$		−	15.0000i		−	0.739895i
$412$	−5.00000			−0.246332
$413$		0			0
$414$		−	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	$-0.689895\pi$
$419$	−23.0000			−1.12362			−0.561812	+	0.827265i	$0.689895\pi$
$420$		0			0
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			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
$427$		0			0
$428$	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	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$		−	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	$-0.211374\pi$
$439$	33.0000			1.57500			0.787502	+	0.616312i	$0.211374\pi$
$440$			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	$-0.545525\pi$
$443$	−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
$448$		0			0
$449$		−	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
$455$		0			0
$456$	−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	$-0.977744\pi$
$461$		−	3.00000i		−	0.139724i	0.997557	−	0.0698620i	$-0.0222559\pi$
$462$		0			0
$463$			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	$-0.172894\pi$
$467$	37.0000			1.71216			0.856078	+	0.516847i	$0.172894\pi$
$468$	3.00000	+	2.00000i	0.138675	+	0.0924500i
$469$		0			0
$470$			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
$476$		0			0
$477$	−10.0000			−0.457869
$478$	−8.00000			−0.365911
$479$		−	39.0000i		−	1.78196i	−0.454047	−	0.890978i	$-0.650020\pi$
$479$		−	39.0000i		−	1.78196i	0.454047	−	0.890978i	$-0.349980\pi$
$480$	−1.00000			−0.0456435
$481$	−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
$486$		−	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
$490$		0			0
$491$	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
$497$		0			0
$498$	−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	$-0.178306\pi$
$503$	38.0000			1.69434			0.847168	+	0.531325i	$0.178306\pi$
$504$		0			0
$505$		−	2.00000i		−	0.0889988i
$506$	3.00000			0.133366
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$	−16.0000			−0.709885
$509$			37.0000i			1.64000i	0.572366	+	0.819998i	$0.306026\pi$
$509$			37.0000i			1.64000i	−0.572366	+	0.819998i	$0.693974\pi$
$510$	1.00000			0.0442807
$511$		0			0
$512$		−	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
$518$		0			0
$519$	−14.0000			−0.614532
$520$	−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	$0.262384\pi$
$522$		−	9.00000i		−	0.393919i
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$	−3.00000			−0.131056
$525$		0			0
$526$		−	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
$532$		0			0
$533$	−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
$539$		0			0
$540$			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
$546$		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
$549$	−11.0000			−0.469469
$550$	4.00000			0.170561
$551$			9.00000i			0.383413i
$552$			3.00000i			0.127688i
$553$		0			0
$554$		−	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
$560$		0			0
$561$		−	1.00000i		−	0.0422200i
$562$	−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	$0.574461\pi$
$564$			8.00000i			0.336861i
$565$			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	$-0.716467\pi$
$569$	−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
$574$		0			0
$575$	12.0000			0.500435
$576$	−1.00000			−0.0416667
$577$			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
$579$		−	12.0000i		−	0.498703i
$580$			9.00000i			0.373705i
$581$		0			0
$582$	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	$0.107115\pi$
$587$			16.0000i			0.660391i	−0.943913	+	0.330195i	$0.892885\pi$
$588$		0			0
$589$	−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	$0.194963\pi$
$593$			28.0000i			1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$	1.00000			0.0410305
$595$		0			0
$596$		−	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	$-0.394189\pi$
$601$	16.0000			0.652654			0.326327	+	0.945257i	$0.394189\pi$
$602$		0			0
$603$		−	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	$-0.506460\pi$
$607$	−1.00000			−0.0405887			−0.0202944	+	0.999794i	$0.506460\pi$
$608$	1.00000			0.0405554
$609$		0			0
$610$	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
$616$		0			0
$617$		−	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	$-0.861317\pi$
$619$		−	21.0000i		−	0.844061i	0.906582	−	0.422031i	$-0.138683\pi$
$620$	−4.00000			−0.160644
$621$	3.00000			0.120386
$622$			8.00000i			0.320771i
$623$		0			0
$624$	−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
$630$		0			0
$631$			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
$637$		0			0
$638$	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	$0.122235\pi$
$643$			19.0000i			0.749287i	−0.927169	+	0.374643i	$0.877765\pi$
$644$		0			0
$645$		−	7.00000i		−	0.275625i
$646$	−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	$0.809160\pi$
$648$			1.00000i			0.0392837i
$649$	−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	$-0.581866\pi$
$653$	−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
$658$		0			0
$659$	−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	$-0.831256\pi$
$661$		−	26.0000i		−	1.01128i	0.862744	−	0.505641i	$-0.168744\pi$
$662$	−8.00000			−0.310929
$663$	3.00000	+	2.00000i	0.116510	+	0.0776736i
$664$	6.00000			0.232845
$665$		0			0
$666$	−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
$672$		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
$675$	4.00000			0.153960
$676$	−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	$0.710816\pi$
$678$			2.00000i			0.0768095i
$679$		0			0
$680$	−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
$686$		0			0
$687$		−	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	$0.275211\pi$
$691$			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
$696$			9.00000i			0.341144i
$697$			8.00000i			0.303022i
$698$	4.00000			0.151402
$699$	−18.0000			−0.680823
$700$		0			0
$701$	−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
$707$		0			0
$708$		−	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
$714$		0			0
$715$	−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	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		−	1.00000i		−	0.0372678i
$721$		0			0
$722$		−	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	$-0.640258\pi$
$727$	−23.0000			−0.853023			−0.426511	+	0.904482i	$0.640258\pi$
$728$		0			0
$729$	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	$0.264568\pi$
$733$			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
$738$	−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
$742$		0			0
$743$		−	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
$749$		0			0
$750$	−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
$756$		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
$759$			3.00000i			0.108893i
$760$			1.00000i			0.0362738i
$761$			42.0000i			1.52250i	0.648459	+	0.761249i	$0.275414\pi$
$761$			42.0000i			1.52250i	−0.648459	+	0.761249i	$0.724586\pi$
$762$		−	16.0000i		−	0.579619i
$763$		0			0
$764$	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	$-0.811207\pi$
$769$		−	31.0000i		−	1.11789i	0.829205	−	0.558944i	$-0.188793\pi$
$770$		0			0
$771$	−14.0000			−0.504198
$772$			12.0000i			0.431889i
$773$		−	19.0000i		−	0.683383i	−0.939812	−	0.341691i	$-0.889000\pi$
$773$		−	19.0000i		−	0.683383i	0.939812	−	0.341691i	$-0.111000\pi$
$774$		−	7.00000i		−	0.251610i
$775$			16.0000i			0.574737i
$776$	−2.00000			−0.0717958
$777$		0			0
$778$		−	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
$784$		0			0
$785$		−	21.0000i		−	0.749522i
$786$		−	3.00000i		−	0.107006i
$787$		−	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
$789$	16.0000			0.569615
$790$	12.0000			0.426941
$791$		0			0
$792$	−1.00000			−0.0355335
$793$	33.0000	+	22.0000i	1.17186	+	0.781243i
$794$	−4.00000			−0.141955
$795$			10.0000i			0.354663i
$796$	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
$801$		−	12.0000i		−	0.423999i
$802$	14.0000			0.494357
$803$	11.0000			0.388182
$804$			12.0000i			0.423207i
$805$		0			0
$806$	−12.0000	−	8.00000i	−0.422682	−	0.281788i
$807$	20.0000			0.704033
$808$			2.00000i			0.0703598i
$809$	−4.00000			−0.140633			−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000			−0.140633			−0.0703163	+	0.997525i	$0.522401\pi$
$810$	−1.00000			−0.0351364
$811$			21.0000i			0.737410i	0.929547	+	0.368705i	$0.120199\pi$
$811$			21.0000i			0.737410i	−0.929547	+	0.368705i	$0.879801\pi$
$812$		0			0
$813$			4.00000i			0.140286i
$814$		−	9.00000i		−	0.315450i
$815$	−20.0000			−0.700569
$816$	−1.00000			−0.0350070
$817$			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	$0.783790\pi$
$822$	−15.0000			−0.523185
$823$	−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
$825$			4.00000i			0.139262i
$826$		0			0
$827$		−	3.00000i		−	0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$		−	3.00000i		−	0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$	−3.00000			−0.104257
$829$	−25.0000			−0.868286			−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000			−0.868286			−0.434143	+	0.900844i	$0.642949\pi$
$830$			6.00000i			0.208263i
$831$	16.0000			0.555034
$832$	3.00000	+	2.00000i	0.104006	+	0.0693375i
$833$		0			0
$834$		−	12.0000i		−	0.415526i
$835$	11.0000			0.380671
$836$	1.00000			0.0345857
$837$			4.00000i			0.138260i
$838$			23.0000i			0.794522i
$839$		−	52.0000i		−	1.79524i	−0.440771	−	0.897620i	$-0.645295\pi$
$839$		−	52.0000i		−	1.79524i	0.440771	−	0.897620i	$-0.354705\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$	22.0000			0.758170
$843$		−	22.0000i		−	0.757720i
$844$	5.00000			0.172107
$845$	12.0000	−	5.00000i	0.412813	−	0.172005i
$846$	−8.00000			−0.275046
$847$		0			0
$848$	−10.0000			−0.343401
$849$	14.0000			0.480479
$850$			4.00000i			0.137199i
$851$		−	27.0000i		−	0.925548i
$852$		−	6.00000i		−	0.205557i
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$		0			0
$855$	1.00000			0.0341993
$856$		−	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	$0.377384\pi$
$858$	−3.00000	−	2.00000i	−0.102418	−	0.0682789i
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$			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	$0.695652\pi$
$864$		−	1.00000i		−	0.0340207i
$865$			14.0000i			0.476014i
$866$			4.00000i			0.135926i
$867$	−16.0000			−0.543388
$868$		0			0
$869$		−	12.0000i		−	0.407072i
$870$	−9.00000			−0.305129
$871$	−24.0000	+	36.0000i	−0.813209	+	1.21981i
$872$	−7.00000			−0.237050
$873$			2.00000i			0.0676897i
$874$	3.00000			0.101477
$875$		0			0
$876$			11.0000i			0.371656i
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$		−	33.0000i		−	1.11370i
$879$			26.0000i			0.876958i
$880$	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
$885$	6.00000			0.201688
$886$			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	$0.510690\pi$
$888$	9.00000			0.302020
$889$		0			0
$890$			12.0000i			0.402241i
$891$			1.00000i			0.0335013i
$892$		−	8.00000i		−	0.267860i
$893$	8.00000			0.267710
$894$	6.00000			0.200670
$895$			18.0000i			0.601674i
$896$		0			0
$897$	−9.00000	−	6.00000i	−0.300501	−	0.200334i
$898$	−15.0000			−0.500556
$899$			36.0000i			1.20067i
$900$	−4.00000			−0.133333
$901$	10.0000			0.333148
$902$		−	8.00000i		−	0.266371i
$903$		0			0
$904$		−	2.00000i		−	0.0665190i
$905$		−	2.00000i		−	0.0664822i
$906$	1.00000			0.0332228
$907$	−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
$909$	2.00000			0.0663358
$910$		0			0
$911$	−29.0000			−0.960813			−0.480406	−	0.877046i	$-0.659511\pi$
$911$	−29.0000			−0.960813			−0.480406	+	0.877046i	$0.659511\pi$
$912$			1.00000i			0.0331133i
$913$	6.00000			0.198571
$914$	4.00000			0.132308
$915$			11.0000i			0.363649i
$916$			2.00000i			0.0660819i
$917$		0			0
$918$			1.00000i			0.0330049i
$919$	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
$921$			8.00000i			0.263609i
$922$	−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
$927$	5.00000			0.164222
$928$		−	9.00000i		−	0.295439i
$929$			26.0000i			0.853032i	0.904480	+	0.426516i	$0.140259\pi$
$929$			26.0000i			0.853032i	−0.904480	+	0.426516i	$0.859741\pi$
$930$		−	4.00000i		−	0.131165i
$931$		0			0
$932$	18.0000			0.589610
$933$	−8.00000			−0.261908
$934$		−	37.0000i		−	1.21068i
$935$	−1.00000			−0.0327035
$936$	2.00000	−	3.00000i	0.0653720	−	0.0980581i
$937$	20.0000			0.653372			0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000			0.653372			0.326686	+	0.945133i	$0.394068\pi$
$938$		0			0
$939$	14.0000			0.456873
$940$	8.00000			0.260931
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		−	21.0000i		−	0.684217i
$943$		−	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	$-0.192543\pi$
$948$	12.0000			0.389742
$949$	−22.0000	+	33.0000i	−0.714150	+	1.07123i
$950$	4.00000			0.129777
$951$		−	12.0000i		−	0.389127i
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$			10.0000i			0.323762i
$955$			11.0000i			0.355952i
$956$			8.00000i			0.258738i
$957$			9.00000i			0.290929i
$958$	−39.0000			−1.26003
$959$		0			0
$960$			1.00000i			0.0322749i
$961$	15.0000			0.483871
$962$	27.0000	+	18.0000i	0.870515	+	0.580343i
$963$	−2.00000			−0.0644491
$964$			10.0000i			0.322078i
$965$	−12.0000			−0.386294
$966$		0			0
$967$			9.00000i			0.289420i	0.989474	+	0.144710i	$0.0462250\pi$
$967$			9.00000i			0.289420i	−0.989474	+	0.144710i	$0.953775\pi$
$968$			10.0000i			0.321412i
$969$		−	1.00000i		−	0.0321246i
$970$		−	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	$0.479556\pi$
$972$	−1.00000			−0.0320750
$973$		0			0
$974$	32.0000			1.02535
$975$	−12.0000	−	8.00000i	−0.384308	−	0.256205i
$976$	−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	$-0.214437\pi$
$978$	−20.0000			−0.639529
$979$	12.0000			0.383522
$980$		0			0
$981$			7.00000i			0.223493i
$982$		−	12.0000i		−	0.382935i
$983$			9.00000i			0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$			9.00000i			0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$	8.00000			0.255031
$985$	−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
$990$		−	1.00000i		−	0.0317821i
$991$	−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
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$			5.00000i			0.158511i
$996$			6.00000i			0.190117i
$997$	14.0000			0.443384			0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000			0.443384			0.221692	+	0.975117i	$0.428842\pi$
$998$	−10.0000			−0.316544
$999$		−	9.00000i		−	0.284747i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.c.e.883.1		2
7.6	odd	2		546.2.c.a.337.1	✓	2
13.12	even	2	inner	3822.2.c.e.883.2		2
21.20	even	2		1638.2.c.f.883.2		2
28.27	even	2		4368.2.h.k.337.2		2
91.34	even	4		7098.2.a.s.1.1		1
91.83	even	4		7098.2.a.d.1.1		1
91.90	odd	2		546.2.c.a.337.2	yes	2
273.272	even	2		1638.2.c.f.883.1		2
364.363	even	2		4368.2.h.k.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.a.337.1	✓	2	7.6	odd	2
546.2.c.a.337.2	yes	2	91.90	odd	2
1638.2.c.f.883.1		2	273.272	even	2
1638.2.c.f.883.2		2	21.20	even	2
3822.2.c.e.883.1		2	1.1	even	1	trivial
3822.2.c.e.883.2		2	13.12	even	2	inner
4368.2.h.k.337.1		2	364.363	even	2
4368.2.h.k.337.2		2	28.27	even	2
7098.2.a.d.1.1		1	91.83	even	4
7098.2.a.s.1.1		1	91.34	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:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.c (of order \(2\), degree \(1\), minimal)

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

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