LMFDB - Embedded newform 1638.2.c.a.883.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(883,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.883"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,-4,0,0,-6,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{10} -5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000i q^{19} +2.00000i q^{20} -5.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000i q^{28} -5.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} +2.00000 q^{35} +3.00000i q^{37} +4.00000 q^{38} +2.00000 q^{40} +5.00000i q^{41} +4.00000 q^{43} +5.00000i q^{44} +9.00000i q^{46} -13.0000i q^{47} -1.00000 q^{49} -1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} -14.0000 q^{53} -10.0000 q^{55} -1.00000 q^{56} +6.00000i q^{59} -13.0000 q^{61} -5.00000 q^{62} -1.00000 q^{64} +(-4.00000 + 6.00000i) q^{65} +3.00000i q^{67} -2.00000 q^{68} -2.00000i q^{70} +1.00000i q^{73} +3.00000 q^{74} -4.00000i q^{76} +5.00000 q^{77} -15.0000 q^{79} -2.00000i q^{80} +5.00000 q^{82} -6.00000i q^{83} -4.00000i q^{85} -4.00000i q^{86} +5.00000 q^{88} +6.00000i q^{89} +(2.00000 - 3.00000i) q^{91} +9.00000 q^{92} -13.0000 q^{94} +8.00000 q^{95} -7.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{10} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{22} - 18 q^{23} + 2 q^{25} - 4 q^{26} + 4 q^{35} + 8 q^{38} + 4 q^{40} + 8 q^{43} - 2 q^{49} + 6 q^{52} - 28 q^{53} - 20 q^{55}+ \cdots + 16 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^10 - 6 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 10 * q^22 - 18 * q^23 + 2 * q^25 - 4 * q^26 + 4 * q^35 + 8 * q^38 + 4 * q^40 + 8 * q^43 - 2 * q^49 + 6 * q^52 - 28 * q^53 - 20 * q^55 - 2 * q^56 - 26 * q^61 - 10 * q^62 - 2 * q^64 - 8 * q^65 - 4 * q^68 + 6 * q^74 + 10 * q^77 - 30 * q^79 + 10 * q^82 + 10 * q^88 + 4 * q^91 + 18 * q^92 - 26 * q^94 + 16 * q^95

Character values

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

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

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−2.00000			−0.632456
$11$		−	5.00000i		−	1.50756i	−0.657129	−	0.753778i	$-0.728229\pi$
$11$		−	5.00000i		−	1.50756i	0.657129	−	0.753778i	$-0.271771\pi$
$12$		0			0
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$			2.00000i			0.447214i
$21$		0			0
$22$	−5.00000			−1.06600
$23$	−9.00000			−1.87663			−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000			−1.87663			−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$	−2.00000	+	3.00000i	−0.392232	+	0.588348i
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	5.00000i		−	0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$		−	5.00000i		−	0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	2.00000			0.338062
$36$		0			0
$37$			3.00000i			0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	4.00000			0.648886
$39$		0			0
$40$	2.00000			0.316228
$41$			5.00000i			0.780869i	0.920631	+	0.390434i	$0.127675\pi$
$41$			5.00000i			0.780869i	−0.920631	+	0.390434i	$0.872325\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$			5.00000i			0.753778i
$45$		0			0
$46$			9.00000i			1.32698i
$47$		−	13.0000i		−	1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$		−	13.0000i		−	1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	−14.0000			−1.92305			−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000			−1.92305			−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	−10.0000			−1.34840
$56$	−1.00000			−0.133631
$57$		0			0
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	−13.0000			−1.66448			−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000			−1.66448			−0.832240	+	0.554416i	$0.812942\pi$
$62$	−5.00000			−0.635001
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−4.00000	+	6.00000i	−0.496139	+	0.744208i
$66$		0			0
$67$			3.00000i			0.366508i	0.983066	+	0.183254i	$0.0586631\pi$
$67$			3.00000i			0.366508i	−0.983066	+	0.183254i	$0.941337\pi$
$68$	−2.00000			−0.242536
$69$		0			0
$70$		−	2.00000i		−	0.239046i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			1.00000i			0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$			1.00000i			0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	3.00000			0.348743
$75$		0			0
$76$		−	4.00000i		−	0.458831i
$77$	5.00000			0.569803
$78$		0			0
$79$	−15.0000			−1.68763			−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000			−1.68763			−0.843816	+	0.536633i	$0.819696\pi$
$80$		−	2.00000i		−	0.223607i
$81$		0			0
$82$	5.00000			0.552158
$83$		−	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
$85$		−	4.00000i		−	0.433861i
$86$		−	4.00000i		−	0.431331i
$87$		0			0
$88$	5.00000			0.533002
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	2.00000	−	3.00000i	0.209657	−	0.314485i
$92$	9.00000			0.938315
$93$		0			0
$94$	−13.0000			−1.34085
$95$	8.00000			0.820783
$96$		0			0
$97$		−	7.00000i		−	0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$		−	7.00000i		−	0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−7.00000			−0.696526			−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000			−0.696526			−0.348263	+	0.937397i	$0.613228\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$	2.00000	−	3.00000i	0.196116	−	0.294174i
$105$		0			0
$106$			14.0000i			1.35980i
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$			10.0000i			0.953463i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$	11.0000			1.03479			0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000			1.03479			0.517396	+	0.855746i	$0.326901\pi$
$114$		0			0
$115$			18.0000i			1.67851i
$116$		0			0
$117$		0			0
$118$	6.00000			0.552345
$119$			2.00000i			0.183340i
$120$		0			0
$121$	−14.0000			−1.27273
$122$			13.0000i			1.17696i
$123$		0			0
$124$			5.00000i			0.449013i
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	6.00000	+	4.00000i	0.526235	+	0.350823i
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$	3.00000			0.259161
$135$		0			0
$136$			2.00000i			0.171499i
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−2.00000			−0.169031
$141$		0			0
$142$		0			0
$143$	−10.0000	+	15.0000i	−0.836242	+	1.25436i
$144$		0			0
$145$		0			0
$146$	1.00000			0.0827606
$147$		0			0
$148$		−	3.00000i		−	0.246598i
$149$		−	9.00000i		−	0.737309i	−0.929567	−	0.368654i	$-0.879819\pi$
$149$		−	9.00000i		−	0.737309i	0.929567	−	0.368654i	$-0.120181\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	−4.00000			−0.324443
$153$		0			0
$154$		−	5.00000i		−	0.402911i
$155$	−10.0000			−0.803219
$156$		0			0
$157$	3.00000			0.239426			0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000			0.239426			0.119713	+	0.992809i	$0.461803\pi$
$158$			15.0000i			1.19334i
$159$		0			0
$160$	−2.00000			−0.158114
$161$		−	9.00000i		−	0.709299i
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		−	5.00000i		−	0.390434i
$165$		0			0
$166$	−6.00000			−0.465690
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	−4.00000			−0.306786
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−14.0000			−1.06440			−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		−	5.00000i		−	0.376889i
$177$		0			0
$178$	6.00000			0.449719
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	7.00000			0.520306			0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000			0.520306			0.260153	+	0.965567i	$0.416227\pi$
$182$	−3.00000	−	2.00000i	−0.222375	−	0.148250i
$183$		0			0
$184$		−	9.00000i		−	0.663489i
$185$	6.00000			0.441129
$186$		0			0
$187$		−	10.0000i		−	0.731272i
$188$			13.0000i			0.948122i
$189$		0			0
$190$		−	8.00000i		−	0.580381i
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	24.0000i		−	1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$		−	24.0000i		−	1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$	−7.00000			−0.502571
$195$		0			0
$196$	1.00000			0.0714286
$197$			7.00000i			0.498729i	0.968410	+	0.249365i	$0.0802218\pi$
$197$			7.00000i			0.498729i	−0.968410	+	0.249365i	$0.919778\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$			7.00000i			0.492518i
$203$		0			0
$204$		0			0
$205$	10.0000			0.698430
$206$		−	4.00000i		−	0.278693i
$207$		0			0
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$	20.0000			1.38343
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$	14.0000			0.961524
$213$		0			0
$214$		−	2.00000i		−	0.136717i
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$	5.00000			0.339422
$218$	14.0000			0.948200
$219$		0			0
$220$	10.0000			0.674200
$221$	−6.00000	−	4.00000i	−0.403604	−	0.269069i
$222$		0			0
$223$		−	19.0000i		−	1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$		−	11.0000i		−	0.731709i
$227$		−	18.0000i		−	1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$		−	18.0000i		−	1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	18.0000			1.18688
$231$		0			0
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$	−26.0000			−1.69605
$236$		−	6.00000i		−	0.390567i
$237$		0			0
$238$	2.00000			0.129641
$239$		−	14.0000i		−	0.905585i	−0.891616	−	0.452792i	$-0.850428\pi$
$239$		−	14.0000i		−	0.905585i	0.891616	−	0.452792i	$-0.149572\pi$
$240$		0			0
$241$		−	10.0000i		−	0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$		−	10.0000i		−	0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$			14.0000i			0.899954i
$243$		0			0
$244$	13.0000			0.832240
$245$			2.00000i			0.127775i
$246$		0			0
$247$	8.00000	−	12.0000i	0.509028	−	0.763542i
$248$	5.00000			0.317500
$249$		0			0
$250$	−12.0000			−0.758947
$251$	−17.0000			−1.07303			−0.536515	−	0.843891i	$-0.680260\pi$
$251$	−17.0000			−1.07303			−0.536515	+	0.843891i	$0.680260\pi$
$252$		0			0
$253$			45.0000i			2.82913i
$254$		−	13.0000i		−	0.815693i
$255$		0			0
$256$	1.00000			0.0625000
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$	4.00000	−	6.00000i	0.248069	−	0.372104i
$261$		0			0
$262$			12.0000i			0.741362i
$263$	−4.00000			−0.246651			−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000			−0.246651			−0.123325	+	0.992366i	$0.539356\pi$
$264$		0			0
$265$			28.0000i			1.72003i
$266$			4.00000i			0.245256i
$267$		0			0
$268$		−	3.00000i		−	0.183254i
$269$	5.00000			0.304855			0.152428	−	0.988315i	$-0.451291\pi$
$269$	5.00000			0.304855			0.152428	+	0.988315i	$0.451291\pi$
$270$		0			0
$271$		−	5.00000i		−	0.303728i	−0.988401	−	0.151864i	$-0.951472\pi$
$271$		−	5.00000i		−	0.303728i	0.988401	−	0.151864i	$-0.0485276\pi$
$272$	2.00000			0.121268
$273$		0			0
$274$	12.0000			0.724947
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$		0			0
$280$			2.00000i			0.119523i
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	−11.0000			−0.653882			−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000			−0.653882			−0.326941	+	0.945045i	$0.606018\pi$
$284$		0			0
$285$		0			0
$286$	15.0000	+	10.0000i	0.886969	+	0.591312i
$287$	−5.00000			−0.295141
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		−	1.00000i		−	0.0585206i
$293$		−	26.0000i		−	1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$		−	26.0000i		−	1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$		0			0
$295$	12.0000			0.698667
$296$	−3.00000			−0.174371
$297$		0			0
$298$	−9.00000			−0.521356
$299$	27.0000	+	18.0000i	1.56145	+	1.04097i
$300$		0			0
$301$			4.00000i			0.230556i
$302$	−10.0000			−0.575435
$303$		0			0
$304$			4.00000i			0.229416i
$305$			26.0000i			1.48876i
$306$		0			0
$307$		−	22.0000i		−	1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$		−	22.0000i		−	1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	−5.00000			−0.284901
$309$		0			0
$310$			10.0000i			0.567962i
$311$	28.0000			1.58773			0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000			1.58773			0.793867	+	0.608091i	$0.208065\pi$
$312$		0			0
$313$	4.00000			0.226093			0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000			0.226093			0.113047	+	0.993590i	$0.463939\pi$
$314$		−	3.00000i		−	0.169300i
$315$		0			0
$316$	15.0000			0.843816
$317$			27.0000i			1.51647i	0.651981	+	0.758236i	$0.273938\pi$
$317$			27.0000i			1.51647i	−0.651981	+	0.758236i	$0.726062\pi$
$318$		0			0
$319$		0			0
$320$			2.00000i			0.111803i
$321$		0			0
$322$	−9.00000			−0.501550
$323$			8.00000i			0.445132i
$324$		0			0
$325$	−3.00000	−	2.00000i	−0.166410	−	0.110940i
$326$	−4.00000			−0.221540
$327$		0			0
$328$	−5.00000			−0.276079
$329$	13.0000			0.716713
$330$		0			0
$331$		−	25.0000i		−	1.37412i	−0.726599	−	0.687062i	$-0.758900\pi$
$331$		−	25.0000i		−	1.37412i	0.726599	−	0.687062i	$-0.241100\pi$
$332$			6.00000i			0.329293i
$333$		0			0
$334$	−8.00000			−0.437741
$335$	6.00000			0.327815
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$	12.0000	−	5.00000i	0.652714	−	0.271964i
$339$		0			0
$340$			4.00000i			0.216930i
$341$	−25.0000			−1.35383
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$			4.00000i			0.215666i
$345$		0			0
$346$			14.0000i			0.752645i
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	26.0000i		−	1.39175i	−0.718164	−	0.695874i	$-0.755017\pi$
$349$		−	26.0000i		−	1.39175i	0.718164	−	0.695874i	$-0.244983\pi$
$350$	1.00000			0.0534522
$351$		0			0
$352$	−5.00000			−0.266501
$353$		−	31.0000i		−	1.64996i	−0.565159	−	0.824982i	$-0.691185\pi$
$353$		−	31.0000i		−	1.64996i	0.565159	−	0.824982i	$-0.308815\pi$
$354$		0			0
$355$		0			0
$356$		−	6.00000i		−	0.317999i
$357$		0			0
$358$		0			0
$359$		−	24.0000i		−	1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$		−	24.0000i		−	1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		−	7.00000i		−	0.367912i
$363$		0			0
$364$	−2.00000	+	3.00000i	−0.104828	+	0.157243i
$365$	2.00000			0.104685
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$	−9.00000			−0.469157
$369$		0			0
$370$		−	6.00000i		−	0.311925i
$371$		−	14.0000i		−	0.726844i
$372$		0			0
$373$	34.0000			1.76045			0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000			1.76045			0.880227	+	0.474554i	$0.157390\pi$
$374$	−10.0000			−0.517088
$375$		0			0
$376$	13.0000			0.670424
$377$		0			0
$378$		0			0
$379$		−	16.0000i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$		−	16.0000i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	−8.00000			−0.410391
$381$		0			0
$382$		−	8.00000i		−	0.409316i
$383$		−	1.00000i		−	0.0510976i	−0.999674	−	0.0255488i	$-0.991867\pi$
$383$		−	1.00000i		−	0.0510976i	0.999674	−	0.0255488i	$-0.00813332\pi$
$384$		0			0
$385$		−	10.0000i		−	0.509647i
$386$	−24.0000			−1.22157
$387$		0			0
$388$			7.00000i			0.355371i
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	−18.0000			−0.910299
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	7.00000			0.352655
$395$			30.0000i			1.50946i
$396$		0			0
$397$		−	22.0000i		−	1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$		−	22.0000i		−	1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$		−	20.0000i		−	1.00251i
$399$		0			0
$400$	1.00000			0.0500000
$401$			20.0000i			0.998752i	0.866385	+	0.499376i	$0.166437\pi$
$401$			20.0000i			0.998752i	−0.866385	+	0.499376i	$0.833563\pi$
$402$		0			0
$403$	−10.0000	+	15.0000i	−0.498135	+	0.747203i
$404$	7.00000			0.348263
$405$		0			0
$406$		0			0
$407$	15.0000			0.743522
$408$		0			0
$409$		−	6.00000i		−	0.296681i	−0.988936	−	0.148340i	$-0.952607\pi$
$409$		−	6.00000i		−	0.296681i	0.988936	−	0.148340i	$-0.0473931\pi$
$410$		−	10.0000i		−	0.493865i
$411$		0			0
$412$	−4.00000			−0.197066
$413$	−6.00000			−0.295241
$414$		0			0
$415$	−12.0000			−0.589057
$416$	−2.00000	+	3.00000i	−0.0980581	+	0.147087i
$417$		0			0
$418$		−	20.0000i		−	0.978232i
$419$	−35.0000			−1.70986			−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000			−1.70986			−0.854931	+	0.518742i	$0.826401\pi$
$420$		0			0
$421$			35.0000i			1.70580i	0.522078	+	0.852898i	$0.325157\pi$
$421$			35.0000i			1.70580i	−0.522078	+	0.852898i	$0.674843\pi$
$422$			8.00000i			0.389434i
$423$		0			0
$424$		−	14.0000i		−	0.679900i
$425$	2.00000			0.0970143
$426$		0			0
$427$		−	13.0000i		−	0.629114i
$428$	−2.00000			−0.0966736
$429$		0			0
$430$	−8.00000			−0.385794
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		−	5.00000i		−	0.240008i
$435$		0			0
$436$		−	14.0000i		−	0.670478i
$437$		−	36.0000i		−	1.72211i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		−	10.0000i		−	0.476731i
$441$		0			0
$442$	−4.00000	+	6.00000i	−0.190261	+	0.285391i
$443$	−24.0000			−1.14027			−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000			−1.14027			−0.570137	+	0.821549i	$0.693110\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$	−19.0000			−0.899676
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$	25.0000			1.17720
$452$	−11.0000			−0.517396
$453$		0			0
$454$	−18.0000			−0.844782
$455$	−6.00000	−	4.00000i	−0.281284	−	0.187523i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	4.00000			0.186908
$459$		0			0
$460$		−	18.0000i		−	0.839254i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		−	14.0000i		−	0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$		−	14.0000i		−	0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$		0			0
$465$		0			0
$466$			9.00000i			0.416917i
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	−3.00000			−0.138527
$470$			26.0000i			1.19929i
$471$		0			0
$472$	−6.00000			−0.276172
$473$		−	20.0000i		−	0.919601i
$474$		0			0
$475$			4.00000i			0.183533i
$476$		−	2.00000i		−	0.0916698i
$477$		0			0
$478$	−14.0000			−0.640345
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	6.00000	−	9.00000i	0.273576	−	0.410365i
$482$	−10.0000			−0.455488
$483$		0			0
$484$	14.0000			0.636364
$485$	−14.0000			−0.635707
$486$		0			0
$487$			28.0000i			1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$			28.0000i			1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$		−	13.0000i		−	0.588482i
$489$		0			0
$490$	2.00000			0.0903508
$491$	18.0000			0.812329			0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000			0.812329			0.406164	+	0.913800i	$0.366866\pi$
$492$		0			0
$493$		0			0
$494$	−12.0000	−	8.00000i	−0.539906	−	0.359937i
$495$		0			0
$496$		−	5.00000i		−	0.224507i
$497$		0			0
$498$		0			0
$499$		−	11.0000i		−	0.492428i	−0.969216	−	0.246214i	$-0.920813\pi$
$499$		−	11.0000i		−	0.492428i	0.969216	−	0.246214i	$-0.0791865\pi$
$500$			12.0000i			0.536656i
$501$		0			0
$502$			17.0000i			0.758747i
$503$	−34.0000			−1.51599			−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000			−1.51599			−0.757993	+	0.652263i	$0.773820\pi$
$504$		0			0
$505$			14.0000i			0.622992i
$506$	45.0000			2.00049
$507$		0			0
$508$	−13.0000			−0.576782
$509$		−	34.0000i		−	1.50702i	−0.657434	−	0.753512i	$-0.728358\pi$
$509$		−	34.0000i		−	1.50702i	0.657434	−	0.753512i	$-0.271642\pi$
$510$		0			0
$511$	−1.00000			−0.0442374
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	22.0000i		−	0.970378i
$515$		−	8.00000i		−	0.352522i
$516$		0			0
$517$	−65.0000			−2.85870
$518$			3.00000i			0.131812i
$519$		0			0
$520$	−6.00000	−	4.00000i	−0.263117	−	0.175412i
$521$	28.0000			1.22670			0.613351	−	0.789810i	$-0.289821\pi$
$521$	28.0000			1.22670			0.613351	+	0.789810i	$0.289821\pi$
$522$		0			0
$523$	−21.0000			−0.918266			−0.459133	−	0.888368i	$-0.651840\pi$
$523$	−21.0000			−0.918266			−0.459133	+	0.888368i	$0.651840\pi$
$524$	12.0000			0.524222
$525$		0			0
$526$			4.00000i			0.174408i
$527$		−	10.0000i		−	0.435607i
$528$		0			0
$529$	58.0000			2.52174
$530$	28.0000			1.21624
$531$		0			0
$532$	4.00000			0.173422
$533$	10.0000	−	15.0000i	0.433148	−	0.649722i
$534$		0			0
$535$		−	4.00000i		−	0.172935i
$536$	−3.00000			−0.129580
$537$		0			0
$538$		−	5.00000i		−	0.215565i
$539$			5.00000i			0.215365i
$540$		0			0
$541$			10.0000i			0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$			10.0000i			0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	−5.00000			−0.214768
$543$		0			0
$544$		−	2.00000i		−	0.0857493i
$545$	28.0000			1.19939
$546$		0			0
$547$	−22.0000			−0.940652			−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000			−0.940652			−0.470326	+	0.882493i	$0.655864\pi$
$548$		−	12.0000i		−	0.512615i
$549$		0			0
$550$	−5.00000			−0.213201
$551$		0			0
$552$		0			0
$553$		−	15.0000i		−	0.637865i
$554$			2.00000i			0.0849719i
$555$		0			0
$556$		0			0
$557$			37.0000i			1.56774i	0.620925	+	0.783870i	$0.286757\pi$
$557$			37.0000i			1.56774i	−0.620925	+	0.783870i	$0.713243\pi$
$558$		0			0
$559$	−12.0000	−	8.00000i	−0.507546	−	0.338364i
$560$	2.00000			0.0845154
$561$		0			0
$562$	20.0000			0.843649
$563$	31.0000			1.30649			0.653247	−	0.757145i	$-0.273406\pi$
$563$	31.0000			1.30649			0.653247	+	0.757145i	$0.273406\pi$
$564$		0			0
$565$		−	22.0000i		−	0.925547i
$566$			11.0000i			0.462364i
$567$		0			0
$568$		0			0
$569$	−15.0000			−0.628833			−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000			−0.628833			−0.314416	+	0.949285i	$0.601809\pi$
$570$		0			0
$571$	2.00000			0.0836974			0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000			0.0836974			0.0418487	+	0.999124i	$0.486675\pi$
$572$	10.0000	−	15.0000i	0.418121	−	0.627182i
$573$		0			0
$574$			5.00000i			0.208696i
$575$	−9.00000			−0.375326
$576$		0			0
$577$		−	42.0000i		−	1.74848i	−0.485491	−	0.874241i	$-0.661359\pi$
$577$		−	42.0000i		−	1.74848i	0.485491	−	0.874241i	$-0.338641\pi$
$578$			13.0000i			0.540729i
$579$		0			0
$580$		0			0
$581$	6.00000			0.248922
$582$		0			0
$583$			70.0000i			2.89910i
$584$	−1.00000			−0.0413803
$585$		0			0
$586$	−26.0000			−1.07405
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$	20.0000			0.824086
$590$		−	12.0000i		−	0.494032i
$591$		0			0
$592$			3.00000i			0.123299i
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$			9.00000i			0.368654i
$597$		0			0
$598$	18.0000	−	27.0000i	0.736075	−	1.10411i
$599$	15.0000			0.612883			0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000			0.612883			0.306442	+	0.951889i	$0.400862\pi$
$600$		0			0
$601$	−28.0000			−1.14214			−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000			−1.14214			−0.571072	+	0.820900i	$0.693472\pi$
$602$	4.00000			0.163028
$603$		0			0
$604$			10.0000i			0.406894i
$605$			28.0000i			1.13836i
$606$		0			0
$607$	−2.00000			−0.0811775			−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000			−0.0811775			−0.0405887	+	0.999176i	$0.512923\pi$
$608$	4.00000			0.162221
$609$		0			0
$610$	26.0000			1.05271
$611$	−26.0000	+	39.0000i	−1.05185	+	1.57777i
$612$		0			0
$613$			11.0000i			0.444286i	0.975014	+	0.222143i	$0.0713052\pi$
$613$			11.0000i			0.444286i	−0.975014	+	0.222143i	$0.928695\pi$
$614$	−22.0000			−0.887848
$615$		0			0
$616$			5.00000i			0.201456i
$617$			32.0000i			1.28827i	0.764911	+	0.644136i	$0.222783\pi$
$617$			32.0000i			1.28827i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$	10.0000			0.401610
$621$		0			0
$622$		−	28.0000i		−	1.12270i
$623$	−6.00000			−0.240385
$624$		0			0
$625$	−19.0000			−0.760000
$626$		−	4.00000i		−	0.159872i
$627$		0			0
$628$	−3.00000			−0.119713
$629$			6.00000i			0.239236i
$630$		0			0
$631$			40.0000i			1.59237i	0.605050	+	0.796187i	$0.293153\pi$
$631$			40.0000i			1.59237i	−0.605050	+	0.796187i	$0.706847\pi$
$632$		−	15.0000i		−	0.596668i
$633$		0			0
$634$	27.0000			1.07231
$635$		−	26.0000i		−	1.03178i
$636$		0			0
$637$	3.00000	+	2.00000i	0.118864	+	0.0792429i
$638$		0			0
$639$		0			0
$640$	2.00000			0.0790569
$641$	−37.0000			−1.46141			−0.730706	−	0.682692i	$-0.760809\pi$
$641$	−37.0000			−1.46141			−0.730706	+	0.682692i	$0.760809\pi$
$642$		0			0
$643$		−	14.0000i		−	0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$		−	14.0000i		−	0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$			9.00000i			0.354650i
$645$		0			0
$646$	8.00000			0.314756
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$	−2.00000	+	3.00000i	−0.0784465	+	0.117670i
$651$		0			0
$652$			4.00000i			0.156652i
$653$	16.0000			0.626128			0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000			0.626128			0.313064	+	0.949732i	$0.398644\pi$
$654$		0			0
$655$			24.0000i			0.937758i
$656$			5.00000i			0.195217i
$657$		0			0
$658$		−	13.0000i		−	0.506793i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			40.0000i			1.55582i	0.628376	+	0.777910i	$0.283720\pi$
$661$			40.0000i			1.55582i	−0.628376	+	0.777910i	$0.716280\pi$
$662$	−25.0000			−0.971653
$663$		0			0
$664$	6.00000			0.232845
$665$			8.00000i			0.310227i
$666$		0			0
$667$		0			0
$668$			8.00000i			0.309529i
$669$		0			0
$670$		−	6.00000i		−	0.231800i
$671$			65.0000i			2.50930i
$672$		0			0
$673$	−31.0000			−1.19496			−0.597481	−	0.801883i	$-0.703832\pi$
$673$	−31.0000			−1.19496			−0.597481	+	0.801883i	$0.703832\pi$
$674$		−	23.0000i		−	0.885927i
$675$		0			0
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	7.00000			0.269032			0.134516	−	0.990911i	$-0.457052\pi$
$677$	7.00000			0.269032			0.134516	+	0.990911i	$0.457052\pi$
$678$		0			0
$679$	7.00000			0.268635
$680$	4.00000			0.153393
$681$		0			0
$682$			25.0000i			0.957299i
$683$		−	41.0000i		−	1.56882i	−0.620242	−	0.784411i	$-0.712966\pi$
$683$		−	41.0000i		−	1.56882i	0.620242	−	0.784411i	$-0.287034\pi$
$684$		0			0
$685$	24.0000			0.916993
$686$	−1.00000			−0.0381802
$687$		0			0
$688$	4.00000			0.152499
$689$	42.0000	+	28.0000i	1.60007	+	1.06672i
$690$		0			0
$691$			10.0000i			0.380418i	0.981744	+	0.190209i	$0.0609166\pi$
$691$			10.0000i			0.380418i	−0.981744	+	0.190209i	$0.939083\pi$
$692$	14.0000			0.532200
$693$		0			0
$694$		−	12.0000i		−	0.455514i
$695$		0			0
$696$		0			0
$697$			10.0000i			0.378777i
$698$	−26.0000			−0.984115
$699$		0			0
$700$		−	1.00000i		−	0.0377964i
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−12.0000			−0.452589
$704$			5.00000i			0.188445i
$705$		0			0
$706$	−31.0000			−1.16670
$707$		−	7.00000i		−	0.263262i
$708$		0			0
$709$		−	31.0000i		−	1.16423i	−0.813107	−	0.582115i	$-0.802225\pi$
$709$		−	31.0000i		−	1.16423i	0.813107	−	0.582115i	$-0.197775\pi$
$710$		0			0
$711$		0			0
$712$	−6.00000			−0.224860
$713$			45.0000i			1.68526i
$714$		0			0
$715$	30.0000	+	20.0000i	1.12194	+	0.747958i
$716$		0			0
$717$		0			0
$718$	−24.0000			−0.895672
$719$	−40.0000			−1.49175			−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000			−1.49175			−0.745874	+	0.666087i	$0.767968\pi$
$720$		0			0
$721$			4.00000i			0.148968i
$722$		−	3.00000i		−	0.111648i
$723$		0			0
$724$	−7.00000			−0.260153
$725$		0			0
$726$		0			0
$727$	38.0000			1.40934			0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000			1.40934			0.704671	+	0.709534i	$0.251095\pi$
$728$	3.00000	+	2.00000i	0.111187	+	0.0741249i
$729$		0			0
$730$		−	2.00000i		−	0.0740233i
$731$	8.00000			0.295891
$732$		0			0
$733$		−	44.0000i		−	1.62518i	−0.582838	−	0.812589i	$-0.698058\pi$
$733$		−	44.0000i		−	1.62518i	0.582838	−	0.812589i	$-0.301942\pi$
$734$		−	28.0000i		−	1.03350i
$735$		0			0
$736$			9.00000i			0.331744i
$737$	15.0000			0.552532
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$	−6.00000			−0.220564
$741$		0			0
$742$	−14.0000			−0.513956
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		−	34.0000i		−	1.24483i
$747$		0			0
$748$			10.0000i			0.365636i
$749$			2.00000i			0.0730784i
$750$		0			0
$751$	−43.0000			−1.56909			−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000			−1.56909			−0.784546	+	0.620070i	$0.787104\pi$
$752$		−	13.0000i		−	0.474061i
$753$		0			0
$754$		0			0
$755$	−20.0000			−0.727875
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	−16.0000			−0.581146
$759$		0			0
$760$			8.00000i			0.290191i
$761$			15.0000i			0.543750i	0.962333	+	0.271875i	$0.0876437\pi$
$761$			15.0000i			0.543750i	−0.962333	+	0.271875i	$0.912356\pi$
$762$		0			0
$763$	−14.0000			−0.506834
$764$	−8.00000			−0.289430
$765$		0			0
$766$	−1.00000			−0.0361315
$767$	12.0000	−	18.0000i	0.433295	−	0.649942i
$768$		0			0
$769$		−	1.00000i		−	0.0360609i	−0.999837	−	0.0180305i	$-0.994260\pi$
$769$		−	1.00000i		−	0.0360609i	0.999837	−	0.0180305i	$-0.00573959\pi$
$770$	−10.0000			−0.360375
$771$		0			0
$772$			24.0000i			0.863779i
$773$			34.0000i			1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$			34.0000i			1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$		−	5.00000i		−	0.179605i
$776$	7.00000			0.251285
$777$		0			0
$778$		−	10.0000i		−	0.358517i
$779$	−20.0000			−0.716574
$780$		0			0
$781$		0			0
$782$			18.0000i			0.643679i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		−	6.00000i		−	0.214149i
$786$		0			0
$787$			28.0000i			0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$			28.0000i			0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$		−	7.00000i		−	0.249365i
$789$		0			0
$790$	30.0000			1.06735
$791$			11.0000i			0.391115i
$792$		0			0
$793$	39.0000	+	26.0000i	1.38493	+	0.923287i
$794$	−22.0000			−0.780751
$795$		0			0
$796$	−20.0000			−0.708881
$797$	37.0000			1.31061			0.655304	−	0.755366i	$-0.272541\pi$
$797$	37.0000			1.31061			0.655304	+	0.755366i	$0.272541\pi$
$798$		0			0
$799$		−	26.0000i		−	0.919814i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	20.0000			0.706225
$803$	5.00000			0.176446
$804$		0			0
$805$	−18.0000			−0.634417
$806$	15.0000	+	10.0000i	0.528352	+	0.352235i
$807$		0			0
$808$		−	7.00000i		−	0.246259i
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		−	15.0000i		−	0.525750i
$815$	−8.00000			−0.280228
$816$		0			0
$817$			16.0000i			0.559769i
$818$	−6.00000			−0.209785
$819$		0			0
$820$	−10.0000			−0.349215
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$		0			0
$823$	−11.0000			−0.383436			−0.191718	−	0.981450i	$-0.561406\pi$
$823$	−11.0000			−0.383436			−0.191718	+	0.981450i	$0.561406\pi$
$824$			4.00000i			0.139347i
$825$		0			0
$826$			6.00000i			0.208767i
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	10.0000			0.347314			0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000			0.347314			0.173657	+	0.984806i	$0.444442\pi$
$830$			12.0000i			0.416526i
$831$		0			0
$832$	3.00000	+	2.00000i	0.104006	+	0.0693375i
$833$	−2.00000			−0.0692959
$834$		0			0
$835$	−16.0000			−0.553703
$836$	−20.0000			−0.691714
$837$		0			0
$838$			35.0000i			1.20905i
$839$			1.00000i			0.0345238i	0.999851	+	0.0172619i	$0.00549491\pi$
$839$			1.00000i			0.0345238i	−0.999851	+	0.0172619i	$0.994505\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	35.0000			1.20618
$843$		0			0
$844$	8.00000			0.275371
$845$	24.0000	−	10.0000i	0.825625	−	0.344010i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$	−14.0000			−0.480762
$849$		0			0
$850$		−	2.00000i		−	0.0685994i
$851$		−	27.0000i		−	0.925548i
$852$		0			0
$853$			56.0000i			1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$			56.0000i			1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	−13.0000			−0.444851
$855$		0			0
$856$			2.00000i			0.0683586i
$857$	−8.00000			−0.273275			−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000			−0.273275			−0.136637	+	0.990621i	$0.543630\pi$
$858$		0			0
$859$	−35.0000			−1.19418			−0.597092	−	0.802173i	$-0.703677\pi$
$859$	−35.0000			−1.19418			−0.597092	+	0.802173i	$0.703677\pi$
$860$			8.00000i			0.272798i
$861$		0			0
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$			28.0000i			0.952029i
$866$		−	34.0000i		−	1.15537i
$867$		0			0
$868$	−5.00000			−0.169711
$869$			75.0000i			2.54420i
$870$		0			0
$871$	6.00000	−	9.00000i	0.203302	−	0.304953i
$872$	−14.0000			−0.474100
$873$		0			0
$874$	−36.0000			−1.21772
$875$	12.0000			0.405674
$876$		0			0
$877$		−	17.0000i		−	0.574049i	−0.957923	−	0.287025i	$-0.907334\pi$
$877$		−	17.0000i		−	0.574049i	0.957923	−	0.287025i	$-0.0926662\pi$
$878$		0			0
$879$		0			0
$880$	−10.0000			−0.337100
$881$	8.00000			0.269527			0.134763	−	0.990878i	$-0.456973\pi$
$881$	8.00000			0.269527			0.134763	+	0.990878i	$0.456973\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$	6.00000	+	4.00000i	0.201802	+	0.134535i
$885$		0			0
$886$			24.0000i			0.806296i
$887$	−8.00000			−0.268614			−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000			−0.268614			−0.134307	+	0.990940i	$0.542881\pi$
$888$		0			0
$889$			13.0000i			0.436006i
$890$		−	12.0000i		−	0.402241i
$891$		0			0
$892$			19.0000i			0.636167i
$893$	52.0000			1.74011
$894$		0			0
$895$		0			0
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	6.00000			0.200223
$899$		0			0
$900$		0			0
$901$	−28.0000			−0.932815
$902$		−	25.0000i		−	0.832409i
$903$		0			0
$904$			11.0000i			0.365855i
$905$		−	14.0000i		−	0.465376i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$			18.0000i			0.597351i
$909$		0			0
$910$	−4.00000	+	6.00000i	−0.132599	+	0.198898i
$911$	8.00000			0.265052			0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000			0.265052			0.132526	+	0.991180i	$0.457691\pi$
$912$		0			0
$913$	−30.0000			−0.992855
$914$	8.00000			0.264616
$915$		0			0
$916$		−	4.00000i		−	0.132164i
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	−35.0000			−1.15454			−0.577272	−	0.816552i	$-0.695883\pi$
$919$	−35.0000			−1.15454			−0.577272	+	0.816552i	$0.695883\pi$
$920$	−18.0000			−0.593442
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			3.00000i			0.0986394i
$926$	−14.0000			−0.460069
$927$		0			0
$928$		0			0
$929$			1.00000i			0.0328089i	0.999865	+	0.0164045i	$0.00522194\pi$
$929$			1.00000i			0.0328089i	−0.999865	+	0.0164045i	$0.994778\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$	9.00000			0.294805
$933$		0			0
$934$			8.00000i			0.261768i
$935$	−20.0000			−0.654070
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$			3.00000i			0.0979535i
$939$		0			0
$940$	26.0000			0.848026
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		−	45.0000i		−	1.46540i
$944$			6.00000i			0.195283i
$945$		0			0
$946$	−20.0000			−0.650256
$947$		−	8.00000i		−	0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$		−	8.00000i		−	0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$		0			0
$949$	2.00000	−	3.00000i	0.0649227	−	0.0973841i
$950$	4.00000			0.129777
$951$		0			0
$952$	−2.00000			−0.0648204
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$		−	16.0000i		−	0.517748i
$956$			14.0000i			0.452792i
$957$		0			0
$958$	−24.0000			−0.775405
$959$	−12.0000			−0.387500
$960$		0			0
$961$	6.00000			0.193548
$962$	−9.00000	−	6.00000i	−0.290172	−	0.193448i
$963$		0			0
$964$			10.0000i			0.322078i
$965$	−48.0000			−1.54517
$966$		0			0
$967$			48.0000i			1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$			48.0000i			1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$		−	14.0000i		−	0.449977i
$969$		0			0
$970$			14.0000i			0.449513i
$971$	43.0000			1.37994			0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000			1.37994			0.689968	+	0.723840i	$0.257625\pi$
$972$		0			0
$973$		0			0
$974$	28.0000			0.897178
$975$		0			0
$976$	−13.0000			−0.416120
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	30.0000			0.958804
$980$		−	2.00000i		−	0.0638877i
$981$		0			0
$982$		−	18.0000i		−	0.574403i
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$	14.0000			0.446077
$986$		0			0
$987$		0			0
$988$	−8.00000	+	12.0000i	−0.254514	+	0.381771i
$989$	−36.0000			−1.14473
$990$		0			0
$991$	−3.00000			−0.0952981			−0.0476491	−	0.998864i	$-0.515173\pi$
$991$	−3.00000			−0.0952981			−0.0476491	+	0.998864i	$0.515173\pi$
$992$	−5.00000			−0.158750
$993$		0			0
$994$		0			0
$995$		−	40.0000i		−	1.26809i
$996$		0			0
$997$	−17.0000			−0.538395			−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000			−0.538395			−0.269198	+	0.963085i	$0.586759\pi$
$998$	−11.0000			−0.348199
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.c.a.883.1		2
3.2	odd	2		182.2.d.a.155.2	yes	2
12.11	even	2		1456.2.k.a.337.2		2
13.12	even	2	inner	1638.2.c.a.883.2		2
21.2	odd	6		1274.2.n.e.753.2		4
21.5	even	6		1274.2.n.b.753.2		4
21.11	odd	6		1274.2.n.e.961.1		4
21.17	even	6		1274.2.n.b.961.1		4
21.20	even	2		1274.2.d.d.883.2		2
39.5	even	4		2366.2.a.l.1.1		1
39.8	even	4		2366.2.a.c.1.1		1
39.38	odd	2		182.2.d.a.155.1	✓	2
156.155	even	2		1456.2.k.a.337.1		2
273.38	even	6		1274.2.n.b.961.2		4
273.116	odd	6		1274.2.n.e.961.2		4
273.194	even	6		1274.2.n.b.753.1		4
273.233	odd	6		1274.2.n.e.753.1		4
273.272	even	2		1274.2.d.d.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.d.a.155.1	✓	2	39.38	odd	2
182.2.d.a.155.2	yes	2	3.2	odd	2
1274.2.d.d.883.1		2	273.272	even	2
1274.2.d.d.883.2		2	21.20	even	2
1274.2.n.b.753.1		4	273.194	even	6
1274.2.n.b.753.2		4	21.5	even	6
1274.2.n.b.961.1		4	21.17	even	6
1274.2.n.b.961.2		4	273.38	even	6
1274.2.n.e.753.1		4	273.233	odd	6
1274.2.n.e.753.2		4	21.2	odd	6
1274.2.n.e.961.1		4	21.11	odd	6
1274.2.n.e.961.2		4	273.116	odd	6
1456.2.k.a.337.1		2	156.155	even	2
1456.2.k.a.337.2		2	12.11	even	2
1638.2.c.a.883.1		2	1.1	even	1	trivial
1638.2.c.a.883.2		2	13.12	even	2	inner
2366.2.a.c.1.1		1	39.8	even	4
2366.2.a.l.1.1		1	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{10} -5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +4.00000i q^{19} +2.00000i q^{20} -5.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000i q^{28} -5.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} +2.00000 q^{35} +3.00000i q^{37} +4.00000 q^{38} +2.00000 q^{40} +5.00000i q^{41} +4.00000 q^{43} +5.00000i q^{44} +9.00000i q^{46} -13.0000i q^{47} -1.00000 q^{49} -1.00000i q^{50} +(3.00000 + 2.00000i) q^{52} -14.0000 q^{53} -10.0000 q^{55} -1.00000 q^{56} +6.00000i q^{59} -13.0000 q^{61} -5.00000 q^{62} -1.00000 q^{64} +(-4.00000 + 6.00000i) q^{65} +3.00000i q^{67} -2.00000 q^{68} -2.00000i q^{70} +1.00000i q^{73} +3.00000 q^{74} -4.00000i q^{76} +5.00000 q^{77} -15.0000 q^{79} -2.00000i q^{80} +5.00000 q^{82} -6.00000i q^{83} -4.00000i q^{85} -4.00000i q^{86} +5.00000 q^{88} +6.00000i q^{89} +(2.00000 - 3.00000i) q^{91} +9.00000 q^{92} -13.0000 q^{94} +8.00000 q^{95} -7.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{10} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{22} - 18 q^{23} + 2 q^{25} - 4 q^{26} + 4 q^{35} + 8 q^{38} + 4 q^{40} + 8 q^{43} - 2 q^{49} + 6 q^{52} - 28 q^{53} - 20 q^{55}+ \cdots + 16 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^10 - 6 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 10 * q^22 - 18 * q^23 + 2 * q^25 - 4 * q^26 + 4 * q^35 + 8 * q^38 + 4 * q^40 + 8 * q^43 - 2 * q^49 + 6 * q^52 - 28 * q^53 - 20 * q^55 - 2 * q^56 - 26 * q^61 - 10 * q^62 - 2 * q^64 - 8 * q^65 - 4 * q^68 + 6 * q^74 + 10 * q^77 - 30 * q^79 + 10 * q^82 + 10 * q^88 + 4 * q^91 + 18 * q^92 - 26 * q^94 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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