LMFDB - Newform orbit 42.6.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,6,Mod(25,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("42.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$42 = 2 \cdot 3 \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	42.e (of order $3$ , degree $2$ , 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:	$6.73612043215$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + ( - 4 \zeta_{6} + 4) q^{2} + 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + (86 \zeta_{6} - 86) q^{5} + 36 q^{6} + (147 \zeta_{6} - 49) q^{7} - 64 q^{8} + (81 \zeta_{6} - 81) q^{9} + 344 \zeta_{6} q^{10} + \cdots + 2754 q^{99} +O(q^{100})$ q + (-4z + 4) q^2 + 9z q^3 - 16z q^4 + (86z - 86) q^5 + 36 * q^6 + (147z - 49) q^7 - 64 * q^8 + (81z - 81) q^9 + 344z q^10 - 34z q^11 + (-144z + 144) q^12 - 3 * q^13 + (196z + 392) q^14 - 774 * q^15 + (256z - 256) q^16 + 1904z q^17 + 324z q^18 + (-1489z + 1489) q^19 + 1376 * q^20 + (882z - 1323) q^21 - 136 * q^22 + (-224z + 224) q^23 - 576z q^24 - 4271z q^25 + (12z - 12) q^26 - 729 * q^27 + (-1568z + 2352) q^28 - 6508 * q^29 + (3096z - 3096) q^30 - 1731z q^31 + 1024z q^32 + (-306z + 306) q^33 + 7616 * q^34 + (-4214z - 8428) q^35 + 1296 * q^36 + (-7633z + 7633) q^37 - 5956z q^38 - 27z q^39 + (-5504z + 5504) q^40 + 15414 * q^41 + (5292z - 1764) q^42 + 18491 * q^43 + (544z - 544) q^44 - 6966z q^45 - 896z q^46 + (18462z - 18462) q^47 - 2304 * q^48 + (7203z - 19208) q^49 - 17084 * q^50 + (17136z - 17136) q^51 + 48z q^52 + 19956z q^53 + (2916z - 2916) q^54 + 2924 * q^55 + (-9408z + 3136) q^56 + 13401 * q^57 + (26032z - 26032) q^58 + 31828z q^59 + 12384z q^60 + (-57654z + 57654) q^61 - 6924 * q^62 + (-3969z - 7938) q^63 + 4096 * q^64 + (-258z + 258) q^65 - 1224z q^66 + 60563z q^67 + (-30464z + 30464) q^68 + 2016 * q^69 + (33712z - 50568) q^70 - 44834 * q^71 + (-5184z + 5184) q^72 - 20821z q^73 - 30532z q^74 + (-38439z + 38439) q^75 - 23824 * q^76 + (-3332z + 4998) q^77 - 108 * q^78 + (-30531z + 30531) q^79 - 22016z q^80 - 6561z q^81 + (-61656z + 61656) q^82 + 110602 * q^83 + (7056z + 14112) q^84 - 163744 * q^85 + (-73964z + 73964) q^86 - 58572z q^87 + 2176z q^88 + (-58992z + 58992) q^89 - 27864 * q^90 + (-441z + 147) q^91 - 3584 * q^92 + (-15579z + 15579) q^93 + 73848z q^94 + 128054z q^95 + (9216z - 9216) q^96 - 119846 * q^97 + (76832z - 48020) q^98 + 2754 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9} + 344 q^{10} - 34 q^{11} + 144 q^{12} - 6 q^{13} + 980 q^{14} - 1548 q^{15} - 256 q^{16} + 1904 q^{17} + 324 q^{18}+ \cdots + 5508 q^{99}+O(q^{100})$ 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 - 86 * q^5 + 72 * q^6 + 49 * q^7 - 128 * q^8 - 81 * q^9 + 344 * q^10 - 34 * q^11 + 144 * q^12 - 6 * q^13 + 980 * q^14 - 1548 * q^15 - 256 * q^16 + 1904 * q^17 + 324 * q^18 + 1489 * q^19 + 2752 * q^20 - 1764 * q^21 - 272 * q^22 + 224 * q^23 - 576 * q^24 - 4271 * q^25 - 12 * q^26 - 1458 * q^27 + 3136 * q^28 - 13016 * q^29 - 3096 * q^30 - 1731 * q^31 + 1024 * q^32 + 306 * q^33 + 15232 * q^34 - 21070 * q^35 + 2592 * q^36 + 7633 * q^37 - 5956 * q^38 - 27 * q^39 + 5504 * q^40 + 30828 * q^41 + 1764 * q^42 + 36982 * q^43 - 544 * q^44 - 6966 * q^45 - 896 * q^46 - 18462 * q^47 - 4608 * q^48 - 31213 * q^49 - 34168 * q^50 - 17136 * q^51 + 48 * q^52 + 19956 * q^53 - 2916 * q^54 + 5848 * q^55 - 3136 * q^56 + 26802 * q^57 - 26032 * q^58 + 31828 * q^59 + 12384 * q^60 + 57654 * q^61 - 13848 * q^62 - 19845 * q^63 + 8192 * q^64 + 258 * q^65 - 1224 * q^66 + 60563 * q^67 + 30464 * q^68 + 4032 * q^69 - 67424 * q^70 - 89668 * q^71 + 5184 * q^72 - 20821 * q^73 - 30532 * q^74 + 38439 * q^75 - 47648 * q^76 + 6664 * q^77 - 216 * q^78 + 30531 * q^79 - 22016 * q^80 - 6561 * q^81 + 61656 * q^82 + 221204 * q^83 + 35280 * q^84 - 327488 * q^85 + 73964 * q^86 - 58572 * q^87 + 2176 * q^88 + 58992 * q^89 - 55728 * q^90 - 147 * q^91 - 7168 * q^92 + 15579 * q^93 + 73848 * q^94 + 128054 * q^95 - 9216 * q^96 - 239692 * q^97 - 19208 * q^98 + 5508 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$ .

$n$	$29$	$31$
$\chi(n)$	$1$	$-1 + \zeta_{6}$

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}

25.1

0.500000	−	0.866025i
0.500000	+	0.866025i

2.00000

3.46410i

4.50000

−

7.79423i

−8.00000

13.8564i

−43.0000

−

74.4782i

36.0000

24.5000

−

127.306i

−64.0000

−40.5000

−

70.1481i

172.000

−

297.913i

37.1

2.00000

−

3.46410i

4.50000

7.79423i

−8.00000

−

13.8564i

−43.0000

74.4782i

36.0000

24.5000

127.306i

−64.0000

−40.5000

70.1481i

172.000

297.913i

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	42.6.e.b	✓	2
3.b	odd	2	1		126.6.g.b		2
4.b	odd	2	1		336.6.q.a		2
7.b	odd	2	1		294.6.e.m		2
7.c	even	3	1	inner	42.6.e.b	✓	2
7.c	even	3	1		294.6.a.d		1
7.d	odd	6	1		294.6.a.e		1
7.d	odd	6	1		294.6.e.m		2
21.g	even	6	1		882.6.a.y		1
21.h	odd	6	1		126.6.g.b		2
21.h	odd	6	1		882.6.a.m		1
28.g	odd	6	1		336.6.q.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.e.b	✓	2	1.a	even	1	1	trivial
42.6.e.b	✓	2	7.c	even	3	1	inner
126.6.g.b		2	3.b	odd	2	1
126.6.g.b		2	21.h	odd	6	1
294.6.a.d		1	7.c	even	3	1
294.6.a.e		1	7.d	odd	6	1
294.6.e.m		2	7.b	odd	2	1
294.6.e.m		2	7.d	odd	6	1
336.6.q.a		2	4.b	odd	2	1
336.6.q.a		2	28.g	odd	6	1
882.6.a.m		1	21.h	odd	6	1
882.6.a.y		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 86T_{5} + 7396$ T5^2 + 86*T5 + 7396 acting on $S_{6}^{\mathrm{new}}(42, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$3$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$5$	$T^{2} + 86T + 7396$ T^2 + 86*T + 7396
$7$	$T^{2} - 49T + 16807$ T^2 - 49*T + 16807
$11$	$T^{2} + 34T + 1156$ T^2 + 34*T + 1156
$13$	$(T + 3)^{2}$ (T + 3)^2
$17$	$T^{2} - 1904 T + 3625216$ T^2 - 1904*T + 3625216
$19$	$T^{2} - 1489 T + 2217121$ T^2 - 1489*T + 2217121
$23$	$T^{2} - 224T + 50176$ T^2 - 224*T + 50176
$29$	$(T + 6508)^{2}$ (T + 6508)^2
$31$	$T^{2} + 1731 T + 2996361$ T^2 + 1731*T + 2996361
$37$	$T^{2} - 7633 T + 58262689$ T^2 - 7633*T + 58262689
$41$	$(T - 15414)^{2}$ (T - 15414)^2
$43$	$(T - 18491)^{2}$ (T - 18491)^2
$47$	$T^{2} + 18462 T + 340845444$ T^2 + 18462*T + 340845444
$53$	$T^{2} - 19956 T + 398241936$ T^2 - 19956*T + 398241936
$59$	$T^{2} + \cdots + 1013021584$ T^2 - 31828*T + 1013021584
$61$	$T^{2} + \cdots + 3323983716$ T^2 - 57654*T + 3323983716
$67$	$T^{2} + \cdots + 3667876969$ T^2 - 60563*T + 3667876969
$71$	$(T + 44834)^{2}$ (T + 44834)^2
$73$	$T^{2} + 20821 T + 433514041$ T^2 + 20821*T + 433514041
$79$	$T^{2} - 30531 T + 932141961$ T^2 - 30531*T + 932141961
$83$	$(T - 110602)^{2}$ (T - 110602)^2
$89$	$T^{2} + \cdots + 3480056064$ T^2 - 58992*T + 3480056064
$97$	$(T + 119846)^{2}$ (T + 119846)^2
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