LMFDB - Newform orbit 63.6.e.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,6,Mod(37,63)] 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("63.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	63.e (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.1041806482$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8} + 22 \zeta_{6} q^{10} + 269 \zeta_{6} q^{11} - 308 q^{13} + (252 \zeta_{6} - 266) q^{14}+ \cdots + (31654 \zeta_{6} - 35280) q^{98}+O(q^{100}) $ q + (2z - 2) q^2 + 28z q^4 + (-11z + 11) q^5 + (7z + 126) q^7 - 120 * q^8 + 22z q^10 + 269z q^11 - 308 * q^13 + (252z - 266) q^14 + (656z - 656) q^16 + 1896z q^17 + (-164z + 164) q^19 + 308 * q^20 - 538 * q^22 + (3264z - 3264) q^23 + 3004z q^25 + (-616z + 616) q^26 + (3724z - 196) q^28 - 2417 * q^29 - 2841z q^31 - 5152z q^32 - 3792 * q^34 + (-1386z + 1463) q^35 + (-11328z + 11328) q^37 + 328z q^38 + (1320z - 1320) q^40 + 16856 * q^41 - 7894 * q^43 + (7532z - 7532) q^44 - 6528z q^46 + (-21102z + 21102) q^47 + (1813z + 15827) q^49 - 6008 * q^50 - 8624z q^52 - 29691z q^53 + 2959 * q^55 + (-840z - 15120) q^56 + (-4834z + 4834) q^58 - 8163z q^59 + (15166z - 15166) q^61 + 5682 * q^62 - 10688 * q^64 + (3388z - 3388) q^65 + 32078z q^67 + (53088z - 53088) q^68 + (2926z - 154) q^70 + 38274 * q^71 - 34866z q^73 + 22656z q^74 + 4592 * q^76 + (35777z - 1883) q^77 + (13529z - 13529) q^79 + 7216z q^80 + (33712z - 33712) q^82 + 68103 * q^83 + 20856 * q^85 + (-15788z + 15788) q^86 - 32280z q^88 + (114922z - 114922) q^89 + (-2156z - 38808) q^91 - 91392 * q^92 + 42204z q^94 - 1804z q^95 + 154959 * q^97 + (31654z - 35280) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8} + 22 q^{10} + 269 q^{11} - 616 q^{13} - 280 q^{14} - 656 q^{16} + 1896 q^{17} + 164 q^{19} + 616 q^{20} - 1076 q^{22} - 3264 q^{23} + 3004 q^{25}+ \cdots - 38906 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 28 * q^4 + 11 * q^5 + 259 * q^7 - 240 * q^8 + 22 * q^10 + 269 * q^11 - 616 * q^13 - 280 * q^14 - 656 * q^16 + 1896 * q^17 + 164 * q^19 + 616 * q^20 - 1076 * q^22 - 3264 * q^23 + 3004 * q^25 + 616 * q^26 + 3332 * q^28 - 4834 * q^29 - 2841 * q^31 - 5152 * q^32 - 7584 * q^34 + 1540 * q^35 + 11328 * q^37 + 328 * q^38 - 1320 * q^40 + 33712 * q^41 - 15788 * q^43 - 7532 * q^44 - 6528 * q^46 + 21102 * q^47 + 33467 * q^49 - 12016 * q^50 - 8624 * q^52 - 29691 * q^53 + 5918 * q^55 - 31080 * q^56 + 4834 * q^58 - 8163 * q^59 - 15166 * q^61 + 11364 * q^62 - 21376 * q^64 - 3388 * q^65 + 32078 * q^67 - 53088 * q^68 + 2618 * q^70 + 76548 * q^71 - 34866 * q^73 + 22656 * q^74 + 9184 * q^76 + 32011 * q^77 - 13529 * q^79 + 7216 * q^80 - 33712 * q^82 + 136206 * q^83 + 41712 * q^85 + 15788 * q^86 - 32280 * q^88 - 114922 * q^89 - 79772 * q^91 - 182784 * q^92 + 42204 * q^94 - 1804 * q^95 + 309918 * q^97 - 38906 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$-1 + \zeta_{6}$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

37.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

1.73205i

14.0000

24.2487i

5.50000

−

9.52628i

129.500

6.06218i

−120.000

11.0000

19.0526i

46.1

−1.00000

−

1.73205i

14.0000

−

24.2487i

5.50000

9.52628i

129.500

−

6.06218i

−120.000

11.0000

−

19.0526i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.6.e.a		2
3.b	odd	2	1		21.6.e.a	✓	2
7.c	even	3	1	inner	63.6.e.a		2
7.c	even	3	1		441.6.a.g		1
7.d	odd	6	1		441.6.a.h		1
12.b	even	2	1		336.6.q.b		2
21.c	even	2	1		147.6.e.g		2
21.g	even	6	1		147.6.a.d		1
21.g	even	6	1		147.6.e.g		2
21.h	odd	6	1		21.6.e.a	✓	2
21.h	odd	6	1		147.6.a.c		1
84.n	even	6	1		336.6.q.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.e.a	✓	2	3.b	odd	2	1
21.6.e.a	✓	2	21.h	odd	6	1
63.6.e.a		2	1.a	even	1	1	trivial
63.6.e.a		2	7.c	even	3	1	inner
147.6.a.c		1	21.h	odd	6	1
147.6.a.d		1	21.g	even	6	1
147.6.e.g		2	21.c	even	2	1
147.6.e.g		2	21.g	even	6	1
336.6.q.b		2	12.b	even	2	1
336.6.q.b		2	84.n	even	6	1
441.6.a.g		1	7.c	even	3	1
441.6.a.h		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 2T_{2} + 4 $ T2^2 + 2*T2 + 4 acting on $S_{6}^{\mathrm{new}}(63, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$3$	$ T^{2} $ T^2
$5$	$ T^{2} - 11T + 121 $ T^2 - 11*T + 121
$7$	$ T^{2} - 259T + 16807 $ T^2 - 259*T + 16807
$11$	$ T^{2} - 269T + 72361 $ T^2 - 269*T + 72361
$13$	$ (T + 308)^{2} $ (T + 308)^2
$17$	$ T^{2} - 1896 T + 3594816 $ T^2 - 1896*T + 3594816
$19$	$ T^{2} - 164T + 26896 $ T^2 - 164*T + 26896
$23$	$ T^{2} + 3264 T + 10653696 $ T^2 + 3264*T + 10653696
$29$	$ (T + 2417)^{2} $ (T + 2417)^2
$31$	$ T^{2} + 2841 T + 8071281 $ T^2 + 2841*T + 8071281
$37$	$ T^{2} - 11328 T + 128323584 $ T^2 - 11328*T + 128323584
$41$	$ (T - 16856)^{2} $ (T - 16856)^2
$43$	$ (T + 7894)^{2} $ (T + 7894)^2
$47$	$ T^{2} - 21102 T + 445294404 $ T^2 - 21102*T + 445294404
$53$	$ T^{2} + 29691 T + 881555481 $ T^2 + 29691*T + 881555481
$59$	$ T^{2} + 8163 T + 66634569 $ T^2 + 8163*T + 66634569
$61$	$ T^{2} + 15166 T + 230007556 $ T^2 + 15166*T + 230007556
$67$	$ T^{2} + \cdots + 1028998084 $ T^2 - 32078*T + 1028998084
$71$	$ (T - 38274)^{2} $ (T - 38274)^2
$73$	$ T^{2} + \cdots + 1215637956 $ T^2 + 34866*T + 1215637956
$79$	$ T^{2} + 13529 T + 183033841 $ T^2 + 13529*T + 183033841
$83$	$ (T - 68103)^{2} $ (T - 68103)^2
$89$	$ T^{2} + \cdots + 13207066084 $ T^2 + 114922*T + 13207066084
$97$	$ (T - 154959)^{2} $ (T - 154959)^2
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Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8} + 22 \zeta_{6} q^{10} + 269 \zeta_{6} q^{11} - 308 q^{13} + (252 \zeta_{6} - 266) q^{14}+ \cdots + (31654 \zeta_{6} - 35280) q^{98}+O(q^{100}) \) q + (2z - 2) q^2 + 28z q^4 + (-11z + 11) q^5 + (7z + 126) q^7 - 120 * q^8 + 22z q^10 + 269z q^11 - 308 * q^13 + (252z - 266) q^14 + (656z - 656) q^16 + 1896z q^17 + (-164z + 164) q^19 + 308 * q^20 - 538 * q^22 + (3264z - 3264) q^23 + 3004z q^25 + (-616z + 616) q^26 + (3724z - 196) q^28 - 2417 * q^29 - 2841z q^31 - 5152z q^32 - 3792 * q^34 + (-1386z + 1463) q^35 + (-11328z + 11328) q^37 + 328z q^38 + (1320z - 1320) q^40 + 16856 * q^41 - 7894 * q^43 + (7532z - 7532) q^44 - 6528z q^46 + (-21102z + 21102) q^47 + (1813z + 15827) q^49 - 6008 * q^50 - 8624z q^52 - 29691z q^53 + 2959 * q^55 + (-840z - 15120) q^56 + (-4834z + 4834) q^58 - 8163z q^59 + (15166z - 15166) q^61 + 5682 * q^62 - 10688 * q^64 + (3388z - 3388) q^65 + 32078z q^67 + (53088z - 53088) q^68 + (2926z - 154) q^70 + 38274 * q^71 - 34866z q^73 + 22656z q^74 + 4592 * q^76 + (35777z - 1883) q^77 + (13529z - 13529) q^79 + 7216z q^80 + (33712z - 33712) q^82 + 68103 * q^83 + 20856 * q^85 + (-15788z + 15788) q^86 - 32280z q^88 + (114922z - 114922) q^89 + (-2156z - 38808) q^91 - 91392 * q^92 + 42204z q^94 - 1804z q^95 + 154959 * q^97 + (31654z - 35280) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8} + 22 q^{10} + 269 q^{11} - 616 q^{13} - 280 q^{14} - 656 q^{16} + 1896 q^{17} + 164 q^{19} + 616 q^{20} - 1076 q^{22} - 3264 q^{23} + 3004 q^{25}+ \cdots - 38906 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 28 * q^4 + 11 * q^5 + 259 * q^7 - 240 * q^8 + 22 * q^10 + 269 * q^11 - 616 * q^13 - 280 * q^14 - 656 * q^16 + 1896 * q^17 + 164 * q^19 + 616 * q^20 - 1076 * q^22 - 3264 * q^23 + 3004 * q^25 + 616 * q^26 + 3332 * q^28 - 4834 * q^29 - 2841 * q^31 - 5152 * q^32 - 7584 * q^34 + 1540 * q^35 + 11328 * q^37 + 328 * q^38 - 1320 * q^40 + 33712 * q^41 - 15788 * q^43 - 7532 * q^44 - 6528 * q^46 + 21102 * q^47 + 33467 * q^49 - 12016 * q^50 - 8624 * q^52 - 29691 * q^53 + 5918 * q^55 - 31080 * q^56 + 4834 * q^58 - 8163 * q^59 - 15166 * q^61 + 11364 * q^62 - 21376 * q^64 - 3388 * q^65 + 32078 * q^67 - 53088 * q^68 + 2618 * q^70 + 76548 * q^71 - 34866 * q^73 + 22656 * q^74 + 9184 * q^76 + 32011 * q^77 - 13529 * q^79 + 7216 * q^80 - 33712 * q^82 + 136206 * q^83 + 41712 * q^85 + 15788 * q^86 - 32280 * q^88 - 114922 * q^89 - 79772 * q^91 - 182784 * q^92 + 42204 * q^94 - 1804 * q^95 + 309918 * q^97 - 38906 * q^98

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