LMFDB - Embedded newform 1638.2.c.g.883.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1638.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{10} +5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000i q^{19} -3.00000i q^{20} +5.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000i q^{28} -5.00000 q^{29} -1.00000i q^{32} +3.00000i q^{34} -3.00000 q^{35} -7.00000i q^{37} -1.00000 q^{38} -3.00000 q^{40} -1.00000 q^{43} -5.00000i q^{44} -1.00000i q^{46} -8.00000i q^{47} -1.00000 q^{49} +4.00000i q^{50} +(3.00000 + 2.00000i) q^{52} -14.0000 q^{53} -15.0000 q^{55} -1.00000 q^{56} +5.00000i q^{58} -14.0000i q^{59} -3.00000 q^{61} -1.00000 q^{64} +(6.00000 - 9.00000i) q^{65} +8.00000i q^{67} +3.00000 q^{68} +3.00000i q^{70} +10.0000i q^{71} +11.0000i q^{73} -7.00000 q^{74} +1.00000i q^{76} -5.00000 q^{77} +3.00000i q^{80} -6.00000i q^{83} -9.00000i q^{85} +1.00000i q^{86} -5.00000 q^{88} +16.0000i q^{89} +(2.00000 - 3.00000i) q^{91} -1.00000 q^{92} -8.00000 q^{94} +3.00000 q^{95} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{10} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 6 q^{17} + 10 q^{22} + 2 q^{23} - 8 q^{25} - 4 q^{26} - 10 q^{29} - 6 q^{35} - 2 q^{38} - 6 q^{40} - 2 q^{43} - 2 q^{49} + 6 q^{52} - 28 q^{53}+ \cdots + 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^10 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 6 * q^17 + 10 * q^22 + 2 * q^23 - 8 * q^25 - 4 * q^26 - 10 * q^29 - 6 * q^35 - 2 * q^38 - 6 * q^40 - 2 * q^43 - 2 * q^49 + 6 * q^52 - 28 * q^53 - 30 * q^55 - 2 * q^56 - 6 * q^61 - 2 * q^64 + 12 * q^65 + 6 * q^68 - 14 * q^74 - 10 * q^77 - 10 * q^88 + 4 * q^91 - 2 * q^92 - 16 * q^94 + 6 * 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
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	3.00000			0.948683
$11$			5.00000i			1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$			5.00000i			1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$		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
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$		−	1.00000i		−	0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$		−	3.00000i		−	0.670820i
$21$		0			0
$22$	5.00000			1.06600
$23$	1.00000			0.208514			0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000			0.208514			0.104257	+	0.994550i	$0.466753\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$	−2.00000	+	3.00000i	−0.392232	+	0.588348i
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$			3.00000i			0.514496i
$35$	−3.00000			−0.507093
$36$		0			0
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	−1.00000			−0.162221
$39$		0			0
$40$	−3.00000			−0.474342
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		−	5.00000i		−	0.753778i
$45$		0			0
$46$		−	1.00000i		−	0.147442i
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$			4.00000i			0.565685i
$51$		0			0
$52$	3.00000	+	2.00000i	0.416025	+	0.277350i
$53$	−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
$55$	−15.0000			−2.02260
$56$	−1.00000			−0.133631
$57$		0			0
$58$			5.00000i			0.656532i
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−0.125000
$65$	6.00000	−	9.00000i	0.744208	−	1.11631i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	3.00000			0.363803
$69$		0			0
$70$			3.00000i			0.358569i
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	−7.00000			−0.813733
$75$		0			0
$76$			1.00000i			0.114708i
$77$	−5.00000			−0.569803
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			3.00000i			0.335410i
$81$		0			0
$82$		0			0
$83$		−	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
$85$		−	9.00000i		−	0.976187i
$86$			1.00000i			0.107833i
$87$		0			0
$88$	−5.00000			−0.533002
$89$			16.0000i			1.69600i	0.529999	+	0.847998i	$0.322192\pi$
$89$			16.0000i			1.69600i	−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	2.00000	−	3.00000i	0.209657	−	0.314485i
$92$	−1.00000			−0.104257
$93$		0			0
$94$	−8.00000			−0.825137
$95$	3.00000			0.307794
$96$		0			0
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	4.00000			0.400000
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	−11.0000			−1.08386			−0.541931	−	0.840423i	$-0.682307\pi$
$103$	−11.0000			−1.08386			−0.541931	+	0.840423i	$0.682307\pi$
$104$	2.00000	−	3.00000i	0.196116	−	0.294174i
$105$		0			0
$106$			14.0000i			1.35980i
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$			15.0000i			1.43019i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$			3.00000i			0.279751i
$116$	5.00000			0.464238
$117$		0			0
$118$	−14.0000			−1.28880
$119$		−	3.00000i		−	0.275010i
$120$		0			0
$121$	−14.0000			−1.27273
$122$			3.00000i			0.271607i
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−9.00000	−	6.00000i	−0.789352	−	0.526235i
$131$	−7.00000			−0.611593			−0.305796	−	0.952097i	$-0.598923\pi$
$131$	−7.00000			−0.611593			−0.305796	+	0.952097i	$0.598923\pi$
$132$		0			0
$133$	1.00000			0.0867110
$134$	8.00000			0.691095
$135$		0			0
$136$		−	3.00000i		−	0.257248i
$137$		−	3.00000i		−	0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$		−	3.00000i		−	0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	3.00000			0.253546
$141$		0			0
$142$	10.0000			0.839181
$143$	10.0000	−	15.0000i	0.836242	−	1.25436i
$144$		0			0
$145$		−	15.0000i		−	1.24568i
$146$	11.0000			0.910366
$147$		0			0
$148$			7.00000i			0.575396i
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$			15.0000i			1.22068i	0.792139	+	0.610341i	$0.208968\pi$
$151$			15.0000i			1.22068i	−0.792139	+	0.610341i	$0.791032\pi$
$152$	1.00000			0.0811107
$153$		0			0
$154$			5.00000i			0.402911i
$155$		0			0
$156$		0			0
$157$	13.0000			1.03751			0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000			1.03751			0.518756	+	0.854922i	$0.326395\pi$
$158$		0			0
$159$		0			0
$160$	3.00000			0.237171
$161$			1.00000i			0.0788110i
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$	−6.00000			−0.465690
$167$			7.00000i			0.541676i	0.962625	+	0.270838i	$0.0873008\pi$
$167$			7.00000i			0.541676i	−0.962625	+	0.270838i	$0.912699\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$	−9.00000			−0.690268
$171$		0			0
$172$	1.00000			0.0762493
$173$	26.0000			1.97674			0.988372	−	0.152057i	$-0.0485898\pi$
$173$	26.0000			1.97674			0.988372	+	0.152057i	$0.0485898\pi$
$174$		0			0
$175$		−	4.00000i		−	0.302372i
$176$			5.00000i			0.376889i
$177$		0			0
$178$	16.0000			1.19925
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	−3.00000	−	2.00000i	−0.222375	−	0.148250i
$183$		0			0
$184$			1.00000i			0.0737210i
$185$	21.0000			1.54395
$186$		0			0
$187$		−	15.0000i		−	1.09691i
$188$			8.00000i			0.583460i
$189$		0			0
$190$		−	3.00000i		−	0.217643i
$191$	−17.0000			−1.23008			−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000			−1.23008			−0.615038	+	0.788497i	$0.710860\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−2.00000			−0.143592
$195$		0			0
$196$	1.00000			0.0714286
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−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
$201$		0			0
$202$		−	18.0000i		−	1.26648i
$203$		−	5.00000i		−	0.350931i
$204$		0			0
$205$		0			0
$206$			11.0000i			0.766406i
$207$		0			0
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$	5.00000			0.345857
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$	14.0000			0.961524
$213$		0			0
$214$			18.0000i			1.23045i
$215$		−	3.00000i		−	0.204598i
$216$		0			0
$217$		0			0
$218$	9.00000			0.609557
$219$		0			0
$220$	15.0000			1.01130
$221$	9.00000	+	6.00000i	0.605406	+	0.403604i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$		−	6.00000i		−	0.399114i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	26.0000i		−	1.71813i	−0.511868	−	0.859064i	$-0.671046\pi$
$229$		−	26.0000i		−	1.71813i	0.511868	−	0.859064i	$-0.328954\pi$
$230$	3.00000			0.197814
$231$		0			0
$232$		−	5.00000i		−	0.328266i
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$	24.0000			1.56559
$236$			14.0000i			0.911322i
$237$		0			0
$238$	−3.00000			−0.194461
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$			14.0000i			0.899954i
$243$		0			0
$244$	3.00000			0.192055
$245$		−	3.00000i		−	0.191663i
$246$		0			0
$247$	−2.00000	+	3.00000i	−0.127257	+	0.190885i
$248$		0			0
$249$		0			0
$250$	3.00000			0.189737
$251$	−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
$253$			5.00000i			0.314347i
$254$			12.0000i			0.752947i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		0			0
$259$	7.00000			0.434959
$260$	−6.00000	+	9.00000i	−0.372104	+	0.558156i
$261$		0			0
$262$			7.00000i			0.432461i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		−	42.0000i		−	2.58004i
$266$		−	1.00000i		−	0.0613139i
$267$		0			0
$268$		−	8.00000i		−	0.488678i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−3.00000			−0.181902
$273$		0			0
$274$	−3.00000			−0.181237
$275$		−	20.0000i		−	1.20605i
$276$		0			0
$277$	28.0000			1.68236			0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000			1.68236			0.841178	+	0.540758i	$0.181862\pi$
$278$		−	20.0000i		−	1.19952i
$279$		0			0
$280$		−	3.00000i		−	0.179284i
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$	14.0000			0.832214			0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000			0.832214			0.416107	+	0.909316i	$0.363394\pi$
$284$		−	10.0000i		−	0.593391i
$285$		0			0
$286$	−15.0000	−	10.0000i	−0.886969	−	0.591312i
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$	−15.0000			−0.880830
$291$		0			0
$292$		−	11.0000i		−	0.643726i
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	42.0000			2.44533
$296$	7.00000			0.406867
$297$		0			0
$298$	6.00000			0.347571
$299$	−3.00000	−	2.00000i	−0.173494	−	0.115663i
$300$		0			0
$301$		−	1.00000i		−	0.0576390i
$302$	15.0000			0.863153
$303$		0			0
$304$		−	1.00000i		−	0.0573539i
$305$		−	9.00000i		−	0.515339i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	5.00000			0.284901
$309$		0			0
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$		−	13.0000i		−	0.733632i
$315$		0			0
$316$		0			0
$317$		−	8.00000i		−	0.449325i	−0.974437	−	0.224662i	$-0.927872\pi$
$317$		−	8.00000i		−	0.449325i	0.974437	−	0.224662i	$-0.0721279\pi$
$318$		0			0
$319$		−	25.0000i		−	1.39973i
$320$		−	3.00000i		−	0.167705i
$321$		0			0
$322$	1.00000			0.0557278
$323$			3.00000i			0.166924i
$324$		0			0
$325$	12.0000	+	8.00000i	0.665640	+	0.443760i
$326$	−4.00000			−0.221540
$327$		0			0
$328$		0			0
$329$	8.00000			0.441054
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$			6.00000i			0.329293i
$333$		0			0
$334$	7.00000			0.383023
$335$	−24.0000			−1.31126
$336$		0			0
$337$	33.0000			1.79762			0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000			1.79762			0.898812	+	0.438334i	$0.144431\pi$
$338$	12.0000	−	5.00000i	0.652714	−	0.271964i
$339$		0			0
$340$			9.00000i			0.488094i
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		−	1.00000i		−	0.0539164i
$345$		0			0
$346$		−	26.0000i		−	1.39777i
$347$	−28.0000			−1.50312			−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000			−1.50312			−0.751559	+	0.659665i	$0.770698\pi$
$348$		0			0
$349$		−	16.0000i		−	0.856460i	−0.903670	−	0.428230i	$-0.859137\pi$
$349$		−	16.0000i		−	0.856460i	0.903670	−	0.428230i	$-0.140863\pi$
$350$	−4.00000			−0.213809
$351$		0			0
$352$	5.00000			0.266501
$353$			24.0000i			1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$			24.0000i			1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$		0			0
$355$	−30.0000			−1.59223
$356$		−	16.0000i		−	0.847998i
$357$		0			0
$358$			10.0000i			0.528516i
$359$		−	34.0000i		−	1.79445i	−0.441572	−	0.897226i	$-0.645579\pi$
$359$		−	34.0000i		−	1.79445i	0.441572	−	0.897226i	$-0.354421\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		−	2.00000i		−	0.105118i
$363$		0			0
$364$	−2.00000	+	3.00000i	−0.104828	+	0.157243i
$365$	−33.0000			−1.72730
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$	1.00000			0.0521286
$369$		0			0
$370$		−	21.0000i		−	1.09174i
$371$		−	14.0000i		−	0.726844i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$	−15.0000			−0.775632
$375$		0			0
$376$	8.00000			0.412568
$377$	15.0000	+	10.0000i	0.772539	+	0.515026i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	−3.00000			−0.153897
$381$		0			0
$382$			17.0000i			0.869796i
$383$		−	31.0000i		−	1.58403i	−0.610504	−	0.792013i	$-0.709033\pi$
$383$		−	31.0000i		−	1.58403i	0.610504	−	0.792013i	$-0.290967\pi$
$384$		0			0
$385$		−	15.0000i		−	0.764471i
$386$	16.0000			0.814379
$387$		0			0
$388$			2.00000i			0.101535i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−3.00000			−0.151717
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	2.00000			0.100759
$395$		0			0
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$			5.00000i			0.250627i
$399$		0			0
$400$	−4.00000			−0.200000
$401$			30.0000i			1.49813i	0.662497	+	0.749064i	$0.269497\pi$
$401$			30.0000i			1.49813i	−0.662497	+	0.749064i	$0.730503\pi$
$402$		0			0
$403$		0			0
$404$	−18.0000			−0.895533
$405$		0			0
$406$	−5.00000			−0.248146
$407$	35.0000			1.73489
$408$		0			0
$409$			19.0000i			0.939490i	0.882802	+	0.469745i	$0.155654\pi$
$409$			19.0000i			0.939490i	−0.882802	+	0.469745i	$0.844346\pi$
$410$		0			0
$411$		0			0
$412$	11.0000			0.541931
$413$	14.0000			0.688895
$414$		0			0
$415$	18.0000			0.883585
$416$	−2.00000	+	3.00000i	−0.0980581	+	0.147087i
$417$		0			0
$418$		−	5.00000i		−	0.244558i
$419$	35.0000			1.70986			0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000			1.70986			0.854931	+	0.518742i	$0.173599\pi$
$420$		0			0
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$			13.0000i			0.632830i
$423$		0			0
$424$		−	14.0000i		−	0.679900i
$425$	12.0000			0.582086
$426$		0			0
$427$		−	3.00000i		−	0.145180i
$428$	18.0000			0.870063
$429$		0			0
$430$	−3.00000			−0.144673
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$		0			0
$435$		0			0
$436$		−	9.00000i		−	0.431022i
$437$		−	1.00000i		−	0.0478365i
$438$		0			0
$439$	−15.0000			−0.715911			−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000			−0.715911			−0.357955	+	0.933739i	$0.616526\pi$
$440$		−	15.0000i		−	0.715097i
$441$		0			0
$442$	6.00000	−	9.00000i	0.285391	−	0.428086i
$443$	6.00000			0.285069			0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000			0.285069			0.142534	+	0.989790i	$0.454475\pi$
$444$		0			0
$445$	−48.0000			−2.27542
$446$	16.0000			0.757622
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$			21.0000i			0.991051i	0.868593	+	0.495526i	$0.165025\pi$
$449$			21.0000i			0.991051i	−0.868593	+	0.495526i	$0.834975\pi$
$450$		0			0
$451$		0			0
$452$	−6.00000			−0.282216
$453$		0			0
$454$	−8.00000			−0.375459
$455$	9.00000	+	6.00000i	0.421927	+	0.281284i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−26.0000			−1.21490
$459$		0			0
$460$		−	3.00000i		−	0.139876i
$461$		−	15.0000i		−	0.698620i	−0.937007	−	0.349310i	$-0.886416\pi$
$461$		−	15.0000i		−	0.698620i	0.937007	−	0.349310i	$-0.113584\pi$
$462$		0			0
$463$			11.0000i			0.511213i	0.966781	+	0.255607i	$0.0822752\pi$
$463$			11.0000i			0.511213i	−0.966781	+	0.255607i	$0.917725\pi$
$464$	−5.00000			−0.232119
$465$		0			0
$466$			14.0000i			0.648537i
$467$	7.00000			0.323921			0.161961	−	0.986797i	$-0.448218\pi$
$467$	7.00000			0.323921			0.161961	+	0.986797i	$0.448218\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		−	24.0000i		−	1.10704i
$471$		0			0
$472$	14.0000			0.644402
$473$		−	5.00000i		−	0.229900i
$474$		0			0
$475$			4.00000i			0.183533i
$476$			3.00000i			0.137505i
$477$		0			0
$478$	−4.00000			−0.182956
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$		0			0
$481$	−14.0000	+	21.0000i	−0.638345	+	0.957518i
$482$	10.0000			0.455488
$483$		0			0
$484$	14.0000			0.636364
$485$	6.00000			0.272446
$486$		0			0
$487$		−	32.0000i		−	1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$		−	32.0000i		−	1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$		−	3.00000i		−	0.135804i
$489$		0			0
$490$	−3.00000			−0.135526
$491$	28.0000			1.26362			0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000			1.26362			0.631811	+	0.775122i	$0.282312\pi$
$492$		0			0
$493$	15.0000			0.675566
$494$	3.00000	+	2.00000i	0.134976	+	0.0899843i
$495$		0			0
$496$		0			0
$497$	−10.0000			−0.448561
$498$		0			0
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		−	3.00000i		−	0.134164i
$501$		0			0
$502$			17.0000i			0.758747i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$			54.0000i			2.40297i
$506$	5.00000			0.222277
$507$		0			0
$508$	12.0000			0.532414
$509$			1.00000i			0.0443242i	0.999754	+	0.0221621i	$0.00705500\pi$
$509$			1.00000i			0.0443242i	−0.999754	+	0.0221621i	$0.992945\pi$
$510$		0			0
$511$	−11.0000			−0.486611
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			18.0000i			0.793946i
$515$		−	33.0000i		−	1.45415i
$516$		0			0
$517$	40.0000			1.75920
$518$		−	7.00000i		−	0.307562i
$519$		0			0
$520$	9.00000	+	6.00000i	0.394676	+	0.263117i
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$	7.00000			0.305796
$525$		0			0
$526$			24.0000i			1.04645i
$527$		0			0
$528$		0			0
$529$	−22.0000			−0.956522
$530$	−42.0000			−1.82436
$531$		0			0
$532$	−1.00000			−0.0433555
$533$		0			0
$534$		0			0
$535$		−	54.0000i		−	2.33462i
$536$	−8.00000			−0.345547
$537$		0			0
$538$		0			0
$539$		−	5.00000i		−	0.215365i
$540$		0			0
$541$			15.0000i			0.644900i	0.946586	+	0.322450i	$0.104506\pi$
$541$			15.0000i			0.644900i	−0.946586	+	0.322450i	$0.895494\pi$
$542$		0			0
$543$		0			0
$544$			3.00000i			0.128624i
$545$	−27.0000			−1.15655
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$			3.00000i			0.128154i
$549$		0			0
$550$	−20.0000			−0.852803
$551$			5.00000i			0.213007i
$552$		0			0
$553$		0			0
$554$		−	28.0000i		−	1.18961i
$555$		0			0
$556$	−20.0000			−0.848189
$557$			22.0000i			0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$			22.0000i			0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$		0			0
$559$	3.00000	+	2.00000i	0.126886	+	0.0845910i
$560$	−3.00000			−0.126773
$561$		0			0
$562$	10.0000			0.421825
$563$	−9.00000			−0.379305			−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000			−0.379305			−0.189652	+	0.981851i	$0.560736\pi$
$564$		0			0
$565$			18.0000i			0.757266i
$566$		−	14.0000i		−	0.588464i
$567$		0			0
$568$	−10.0000			−0.419591
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	−10.0000	+	15.0000i	−0.418121	+	0.627182i
$573$		0			0
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$			8.00000i			0.332756i
$579$		0			0
$580$			15.0000i			0.622841i
$581$	6.00000			0.248922
$582$		0			0
$583$		−	70.0000i		−	2.89910i
$584$	−11.0000			−0.455183
$585$		0			0
$586$	−6.00000			−0.247858
$587$		−	8.00000i		−	0.330195i	−0.986277	−	0.165098i	$-0.947206\pi$
$587$		−	8.00000i		−	0.330195i	0.986277	−	0.165098i	$-0.0527939\pi$
$588$		0			0
$589$		0			0
$590$		−	42.0000i		−	1.72911i
$591$		0			0
$592$		−	7.00000i		−	0.287698i
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	9.00000			0.368964
$596$		−	6.00000i		−	0.245770i
$597$		0			0
$598$	−2.00000	+	3.00000i	−0.0817861	+	0.122679i
$599$	45.0000			1.83865			0.919325	−	0.393499i	$-0.128735\pi$
$599$	45.0000			1.83865			0.919325	+	0.393499i	$0.128735\pi$
$600$		0			0
$601$	−48.0000			−1.95796			−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−48.0000			−1.95796			−0.978980	+	0.203954i	$0.934621\pi$
$602$	−1.00000			−0.0407570
$603$		0			0
$604$		−	15.0000i		−	0.610341i
$605$		−	42.0000i		−	1.70754i
$606$		0			0
$607$	23.0000			0.933541			0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000			0.933541			0.466771	+	0.884378i	$0.345417\pi$
$608$	−1.00000			−0.0405554
$609$		0			0
$610$	−9.00000			−0.364399
$611$	−16.0000	+	24.0000i	−0.647291	+	0.970936i
$612$		0			0
$613$		−	39.0000i		−	1.57520i	−0.616190	−	0.787598i	$-0.711325\pi$
$613$		−	39.0000i		−	1.57520i	0.616190	−	0.787598i	$-0.288675\pi$
$614$	8.00000			0.322854
$615$		0			0
$616$		−	5.00000i		−	0.201456i
$617$			17.0000i			0.684394i	0.939628	+	0.342197i	$0.111171\pi$
$617$			17.0000i			0.684394i	−0.939628	+	0.342197i	$0.888829\pi$
$618$		0			0
$619$		−	11.0000i		−	0.442127i	−0.975259	−	0.221064i	$-0.929047\pi$
$619$		−	11.0000i		−	0.442127i	0.975259	−	0.221064i	$-0.0709529\pi$
$620$		0			0
$621$		0			0
$622$		−	8.00000i		−	0.320771i
$623$	−16.0000			−0.641026
$624$		0			0
$625$	−29.0000			−1.16000
$626$			6.00000i			0.239808i
$627$		0			0
$628$	−13.0000			−0.518756
$629$			21.0000i			0.837325i
$630$		0			0
$631$			35.0000i			1.39333i	0.717398	+	0.696664i	$0.245333\pi$
$631$			35.0000i			1.39333i	−0.717398	+	0.696664i	$0.754667\pi$
$632$		0			0
$633$		0			0
$634$	−8.00000			−0.317721
$635$		−	36.0000i		−	1.42862i
$636$		0			0
$637$	3.00000	+	2.00000i	0.118864	+	0.0792429i
$638$	−25.0000			−0.989759
$639$		0			0
$640$	−3.00000			−0.118585
$641$	8.00000			0.315981			0.157991	−	0.987441i	$-0.449498\pi$
$641$	8.00000			0.315981			0.157991	+	0.987441i	$0.449498\pi$
$642$		0			0
$643$		−	19.0000i		−	0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$		−	19.0000i		−	0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$		−	1.00000i		−	0.0394055i
$645$		0			0
$646$	3.00000			0.118033
$647$	2.00000			0.0786281			0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000			0.0786281			0.0393141	+	0.999227i	$0.487483\pi$
$648$		0			0
$649$	70.0000			2.74774
$650$	8.00000	−	12.0000i	0.313786	−	0.470679i
$651$		0			0
$652$			4.00000i			0.156652i
$653$	1.00000			0.0391330			0.0195665	−	0.999809i	$-0.493771\pi$
$653$	1.00000			0.0391330			0.0195665	+	0.999809i	$0.493771\pi$
$654$		0			0
$655$		−	21.0000i		−	0.820538i
$656$		0			0
$657$		0			0
$658$		−	8.00000i		−	0.311872i
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$			50.0000i			1.94477i	0.233373	+	0.972387i	$0.425024\pi$
$661$			50.0000i			1.94477i	−0.233373	+	0.972387i	$0.574976\pi$
$662$	20.0000			0.777322
$663$		0			0
$664$	6.00000			0.232845
$665$			3.00000i			0.116335i
$666$		0			0
$667$	−5.00000			−0.193601
$668$		−	7.00000i		−	0.270838i
$669$		0			0
$670$			24.0000i			0.927201i
$671$		−	15.0000i		−	0.579069i
$672$		0			0
$673$	−11.0000			−0.424019			−0.212009	−	0.977268i	$-0.568001\pi$
$673$	−11.0000			−0.424019			−0.212009	+	0.977268i	$0.568001\pi$
$674$		−	33.0000i		−	1.27111i
$675$		0			0
$676$	−5.00000	−	12.0000i	−0.192308	−	0.461538i
$677$	−8.00000			−0.307465			−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000			−0.307465			−0.153732	+	0.988113i	$0.549129\pi$
$678$		0			0
$679$	2.00000			0.0767530
$680$	9.00000			0.345134
$681$		0			0
$682$		0			0
$683$		−	1.00000i		−	0.0382639i	−0.999817	−	0.0191320i	$-0.993910\pi$
$683$		−	1.00000i		−	0.0382639i	0.999817	−	0.0191320i	$-0.00609027\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$	−1.00000			−0.0381802
$687$		0			0
$688$	−1.00000			−0.0381246
$689$	42.0000	+	28.0000i	1.60007	+	1.06672i
$690$		0			0
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$	−26.0000			−0.988372
$693$		0			0
$694$			28.0000i			1.06287i
$695$			60.0000i			2.27593i
$696$		0			0
$697$		0			0
$698$	−16.0000			−0.605609
$699$		0			0
$700$			4.00000i			0.151186i
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	−7.00000			−0.264010
$704$		−	5.00000i		−	0.188445i
$705$		0			0
$706$	24.0000			0.903252
$707$			18.0000i			0.676960i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$			30.0000i			1.12588i
$711$		0			0
$712$	−16.0000			−0.599625
$713$		0			0
$714$		0			0
$715$	45.0000	+	30.0000i	1.68290	+	1.12194i
$716$	10.0000			0.373718
$717$		0			0
$718$	−34.0000			−1.26887
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	11.0000i		−	0.409661i
$722$		−	18.0000i		−	0.669891i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$	20.0000			0.742781
$726$		0			0
$727$	−7.00000			−0.259616			−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000			−0.259616			−0.129808	+	0.991539i	$0.541436\pi$
$728$	3.00000	+	2.00000i	0.111187	+	0.0741249i
$729$		0			0
$730$			33.0000i			1.22138i
$731$	3.00000			0.110959
$732$		0			0
$733$			36.0000i			1.32969i	0.746981	+	0.664845i	$0.231502\pi$
$733$			36.0000i			1.32969i	−0.746981	+	0.664845i	$0.768498\pi$
$734$			32.0000i			1.18114i
$735$		0			0
$736$		−	1.00000i		−	0.0368605i
$737$	−40.0000			−1.47342
$738$		0			0
$739$			24.0000i			0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$			24.0000i			0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$	−21.0000			−0.771975
$741$		0			0
$742$	−14.0000			−0.513956
$743$			4.00000i			0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$			4.00000i			0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$			26.0000i			0.951928i
$747$		0			0
$748$			15.0000i			0.548454i
$749$		−	18.0000i		−	0.657706i
$750$		0			0
$751$	−48.0000			−1.75154			−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000			−1.75154			−0.875772	+	0.482724i	$0.839647\pi$
$752$		−	8.00000i		−	0.291730i
$753$		0			0
$754$	10.0000	−	15.0000i	0.364179	−	0.546268i
$755$	−45.0000			−1.63772
$756$		0			0
$757$	18.0000			0.654221			0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000			0.654221			0.327111	+	0.944986i	$0.393925\pi$
$758$	4.00000			0.145287
$759$		0			0
$760$			3.00000i			0.108821i
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	−9.00000			−0.325822
$764$	17.0000			0.615038
$765$		0			0
$766$	−31.0000			−1.12008
$767$	−28.0000	+	42.0000i	−1.01102	+	1.51653i
$768$		0			0
$769$		−	1.00000i		−	0.0360609i	−0.999837	−	0.0180305i	$-0.994260\pi$
$769$		−	1.00000i		−	0.0360609i	0.999837	−	0.0180305i	$-0.00573959\pi$
$770$	−15.0000			−0.540562
$771$		0			0
$772$		−	16.0000i		−	0.575853i
$773$			9.00000i			0.323708i	0.986815	+	0.161854i	$0.0517473\pi$
$773$			9.00000i			0.323708i	−0.986815	+	0.161854i	$0.948253\pi$
$774$		0			0
$775$		0			0
$776$	2.00000			0.0717958
$777$		0			0
$778$		−	30.0000i		−	1.07555i
$779$		0			0
$780$		0			0
$781$	−50.0000			−1.78914
$782$			3.00000i			0.107280i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			39.0000i			1.39197i
$786$		0			0
$787$			13.0000i			0.463400i	0.972787	+	0.231700i	$0.0744288\pi$
$787$			13.0000i			0.463400i	−0.972787	+	0.231700i	$0.925571\pi$
$788$		−	2.00000i		−	0.0712470i
$789$		0			0
$790$		0			0
$791$			6.00000i			0.213335i
$792$		0			0
$793$	9.00000	+	6.00000i	0.319599	+	0.213066i
$794$	−12.0000			−0.425864
$795$		0			0
$796$	5.00000			0.177220
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$			4.00000i			0.141421i
$801$		0			0
$802$	30.0000			1.05934
$803$	−55.0000			−1.94091
$804$		0			0
$805$	−3.00000			−0.105736
$806$		0			0
$807$		0			0
$808$			18.0000i			0.633238i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$			35.0000i			1.22902i	0.788911	+	0.614508i	$0.210645\pi$
$811$			35.0000i			1.22902i	−0.788911	+	0.614508i	$0.789355\pi$
$812$			5.00000i			0.175466i
$813$		0			0
$814$		−	35.0000i		−	1.22675i
$815$	12.0000			0.420342
$816$		0			0
$817$			1.00000i			0.0349856i
$818$	19.0000			0.664319
$819$		0			0
$820$		0			0
$821$			20.0000i			0.698005i	0.937122	+	0.349002i	$0.113479\pi$
$821$			20.0000i			0.698005i	−0.937122	+	0.349002i	$0.886521\pi$
$822$		0			0
$823$	44.0000			1.53374			0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000			1.53374			0.766872	+	0.641800i	$0.221812\pi$
$824$		−	11.0000i		−	0.383203i
$825$		0			0
$826$		−	14.0000i		−	0.487122i
$827$		−	23.0000i		−	0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$		−	23.0000i		−	0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$		0			0
$829$	−25.0000			−0.868286			−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000			−0.868286			−0.434143	+	0.900844i	$0.642949\pi$
$830$		−	18.0000i		−	0.624789i
$831$		0			0
$832$	3.00000	+	2.00000i	0.104006	+	0.0693375i
$833$	3.00000			0.103944
$834$		0			0
$835$	−21.0000			−0.726735
$836$	−5.00000			−0.172929
$837$		0			0
$838$		−	35.0000i		−	1.20905i
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$	−30.0000			−1.03387
$843$		0			0
$844$	13.0000			0.447478
$845$	−36.0000	+	15.0000i	−1.23844	+	0.516016i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$	−14.0000			−0.480762
$849$		0			0
$850$		−	12.0000i		−	0.411597i
$851$		−	7.00000i		−	0.239957i
$852$		0			0
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$	−3.00000			−0.102658
$855$		0			0
$856$		−	18.0000i		−	0.615227i
$857$	42.0000			1.43469			0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000			1.43469			0.717346	+	0.696717i	$0.245357\pi$
$858$		0			0
$859$	−50.0000			−1.70598			−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000			−1.70598			−0.852989	+	0.521929i	$0.825213\pi$
$860$			3.00000i			0.102299i
$861$		0			0
$862$	30.0000			1.02180
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$			78.0000i			2.65208i
$866$		−	4.00000i		−	0.135926i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	16.0000	−	24.0000i	0.542139	−	0.813209i
$872$	−9.00000			−0.304778
$873$		0			0
$874$	−1.00000			−0.0338255
$875$	−3.00000			−0.101419
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$			15.0000i			0.506225i
$879$		0			0
$880$	−15.0000			−0.505650
$881$	−7.00000			−0.235836			−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000			−0.235836			−0.117918	+	0.993023i	$0.537622\pi$
$882$		0			0
$883$	−41.0000			−1.37976			−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000			−1.37976			−0.689880	+	0.723924i	$0.742337\pi$
$884$	−9.00000	−	6.00000i	−0.302703	−	0.201802i
$885$		0			0
$886$		−	6.00000i		−	0.201574i
$887$	2.00000			0.0671534			0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000			0.0671534			0.0335767	+	0.999436i	$0.489310\pi$
$888$		0			0
$889$		−	12.0000i		−	0.402467i
$890$			48.0000i			1.60896i
$891$		0			0
$892$		−	16.0000i		−	0.535720i
$893$	−8.00000			−0.267710
$894$		0			0
$895$		−	30.0000i		−	1.00279i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$	21.0000			0.700779
$899$		0			0
$900$		0			0
$901$	42.0000			1.39922
$902$		0			0
$903$		0			0
$904$			6.00000i			0.199557i
$905$			6.00000i			0.199447i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$			8.00000i			0.265489i
$909$		0			0
$910$	6.00000	−	9.00000i	0.198898	−	0.298347i
$911$	−47.0000			−1.55718			−0.778590	−	0.627533i	$-0.784065\pi$
$911$	−47.0000			−1.55718			−0.778590	+	0.627533i	$0.784065\pi$
$912$		0			0
$913$	30.0000			0.992855
$914$	8.00000			0.264616
$915$		0			0
$916$			26.0000i			0.859064i
$917$		−	7.00000i		−	0.231160i
$918$		0			0
$919$	10.0000			0.329870			0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000			0.329870			0.164935	+	0.986304i	$0.447259\pi$
$920$	−3.00000			−0.0989071
$921$		0			0
$922$	−15.0000			−0.493999
$923$	20.0000	−	30.0000i	0.658308	−	0.987462i
$924$		0			0
$925$			28.0000i			0.920634i
$926$	11.0000			0.361482
$927$		0			0
$928$			5.00000i			0.164133i
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$			1.00000i			0.0327737i
$932$	14.0000			0.458585
$933$		0			0
$934$		−	7.00000i		−	0.229047i
$935$	45.0000			1.47166
$936$		0			0
$937$	−32.0000			−1.04539			−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000			−1.04539			−0.522697	+	0.852518i	$0.675074\pi$
$938$			8.00000i			0.261209i
$939$		0			0
$940$	−24.0000			−0.782794
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$		0			0
$944$		−	14.0000i		−	0.455661i
$945$		0			0
$946$	−5.00000			−0.162564
$947$			57.0000i			1.85225i	0.377215	+	0.926126i	$0.376882\pi$
$947$			57.0000i			1.85225i	−0.377215	+	0.926126i	$0.623118\pi$
$948$		0			0
$949$	22.0000	−	33.0000i	0.714150	−	1.07123i
$950$	4.00000			0.129777
$951$		0			0
$952$	3.00000			0.0972306
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$		−	51.0000i		−	1.65032i
$956$			4.00000i			0.129369i
$957$		0			0
$958$	21.0000			0.678479
$959$	3.00000			0.0968751
$960$		0			0
$961$	31.0000			1.00000
$962$	21.0000	+	14.0000i	0.677067	+	0.451378i
$963$		0			0
$964$		−	10.0000i		−	0.322078i
$965$	−48.0000			−1.54517
$966$		0			0
$967$		−	17.0000i		−	0.546683i	−0.961917	−	0.273342i	$-0.911871\pi$
$967$		−	17.0000i		−	0.546683i	0.961917	−	0.273342i	$-0.0881289\pi$
$968$		−	14.0000i		−	0.449977i
$969$		0			0
$970$		−	6.00000i		−	0.192648i
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$			20.0000i			0.641171i
$974$	−32.0000			−1.02535
$975$		0			0
$976$	−3.00000			−0.0960277
$977$		−	3.00000i		−	0.0959785i	−0.998848	−	0.0479893i	$-0.984719\pi$
$977$		−	3.00000i		−	0.0959785i	0.998848	−	0.0479893i	$-0.0152813\pi$
$978$		0			0
$979$	−80.0000			−2.55681
$980$			3.00000i			0.0958315i
$981$		0			0
$982$		−	28.0000i		−	0.893516i
$983$		−	11.0000i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$		−	11.0000i		−	0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$		0			0
$985$	−6.00000			−0.191176
$986$		−	15.0000i		−	0.477697i
$987$		0			0
$988$	2.00000	−	3.00000i	0.0636285	−	0.0954427i
$989$	−1.00000			−0.0317982
$990$		0			0
$991$	−28.0000			−0.889449			−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000			−0.889449			−0.444725	+	0.895667i	$0.646698\pi$
$992$		0			0
$993$		0			0
$994$			10.0000i			0.317181i
$995$		−	15.0000i		−	0.475532i
$996$		0			0
$997$	38.0000			1.20347			0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000			1.20347			0.601736	+	0.798695i	$0.294476\pi$
$998$	14.0000			0.443162
$999$		0			0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{10} +5.00000i q^{11} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000i q^{19} -3.00000i q^{20} +5.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000i q^{28} -5.00000 q^{29} -1.00000i q^{32} +3.00000i q^{34} -3.00000 q^{35} -7.00000i q^{37} -1.00000 q^{38} -3.00000 q^{40} -1.00000 q^{43} -5.00000i q^{44} -1.00000i q^{46} -8.00000i q^{47} -1.00000 q^{49} +4.00000i q^{50} +(3.00000 + 2.00000i) q^{52} -14.0000 q^{53} -15.0000 q^{55} -1.00000 q^{56} +5.00000i q^{58} -14.0000i q^{59} -3.00000 q^{61} -1.00000 q^{64} +(6.00000 - 9.00000i) q^{65} +8.00000i q^{67} +3.00000 q^{68} +3.00000i q^{70} +10.0000i q^{71} +11.0000i q^{73} -7.00000 q^{74} +1.00000i q^{76} -5.00000 q^{77} +3.00000i q^{80} -6.00000i q^{83} -9.00000i q^{85} +1.00000i q^{86} -5.00000 q^{88} +16.0000i q^{89} +(2.00000 - 3.00000i) q^{91} -1.00000 q^{92} -8.00000 q^{94} +3.00000 q^{95} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{10} - 6 q^{13} + 2 q^{14} + 2 q^{16} - 6 q^{17} + 10 q^{22} + 2 q^{23} - 8 q^{25} - 4 q^{26} - 10 q^{29} - 6 q^{35} - 2 q^{38} - 6 q^{40} - 2 q^{43} - 2 q^{49} + 6 q^{52} - 28 q^{53}+ \cdots + 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^10 - 6 * q^13 + 2 * q^14 + 2 * q^16 - 6 * q^17 + 10 * q^22 + 2 * q^23 - 8 * q^25 - 4 * q^26 - 10 * q^29 - 6 * q^35 - 2 * q^38 - 6 * q^40 - 2 * q^43 - 2 * q^49 + 6 * q^52 - 28 * q^53 - 30 * q^55 - 2 * q^56 - 6 * q^61 - 2 * q^64 + 12 * q^65 + 6 * q^68 - 14 * q^74 - 10 * q^77 - 10 * q^88 + 4 * q^91 - 2 * q^92 - 16 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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