LMFDB - Newform orbit 26.8.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [26,8,Mod(1,26)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(26, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("26.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$26 = 2 \cdot 13$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	26.a (trivial)

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:	yes
Analytic conductor:	$8.12201066259$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q - 8 q^{2} - 39 q^{3} + 64 q^{4} + 385 q^{5} + 312 q^{6} - 293 q^{7} - 512 q^{8} - 666 q^{9} - 3080 q^{10} - 5402 q^{11} - 2496 q^{12} + 2197 q^{13} + 2344 q^{14} - 15015 q^{15} + 4096 q^{16} - 21011 q^{17}+ \cdots + 3597732 q^{99}+O(q^{100})

q - 8 * q^2 - 39 * q^3 + 64 * q^4 + 385 * q^5 + 312 * q^6 - 293 * q^7 - 512 * q^8 - 666 * q^9 - 3080 * q^10 - 5402 * q^11 - 2496 * q^12 + 2197 * q^13 + 2344 * q^14 - 15015 * q^15 + 4096 * q^16 - 21011 * q^17 + 5328 * q^18 - 27326 * q^19 + 24640 * q^20 + 11427 * q^21 + 43216 * q^22 - 63072 * q^23 + 19968 * q^24 + 70100 * q^25 - 17576 * q^26 + 111267 * q^27 - 18752 * q^28 + 122238 * q^29 + 120120 * q^30 - 208396 * q^31 - 32768 * q^32 + 210678 * q^33 + 168088 * q^34 - 112805 * q^35 - 42624 * q^36 - 442379 * q^37 + 218608 * q^38 - 85683 * q^39 - 197120 * q^40 + 58000 * q^41 - 91416 * q^42 - 202025 * q^43 - 345728 * q^44 - 256410 * q^45 + 504576 * q^46 + 588511 * q^47 - 159744 * q^48 - 737694 * q^49 - 560800 * q^50 + 819429 * q^51 + 140608 * q^52 + 1684336 * q^53 - 890136 * q^54 - 2079770 * q^55 + 150016 * q^56 + 1065714 * q^57 - 977904 * q^58 - 442630 * q^59 - 960960 * q^60 - 1083608 * q^61 + 1667168 * q^62 + 195138 * q^63 + 262144 * q^64 + 845845 * q^65 - 1685424 * q^66 + 3443486 * q^67 - 1344704 * q^68 + 2459808 * q^69 + 902440 * q^70 + 2084705 * q^71 + 340992 * q^72 + 5937890 * q^73 + 3539032 * q^74 - 2733900 * q^75 - 1748864 * q^76 + 1582786 * q^77 + 685464 * q^78 - 6609256 * q^79 + 1576960 * q^80 - 2882871 * q^81 - 464000 * q^82 - 142740 * q^83 + 731328 * q^84 - 8089235 * q^85 + 1616200 * q^86 - 4767282 * q^87 + 2765824 * q^88 - 6985286 * q^89 + 2051280 * q^90 - 643721 * q^91 - 4036608 * q^92 + 8127444 * q^93 - 4708088 * q^94 - 10520510 * q^95 + 1277952 * q^96 - 200762 * q^97 + 5901552 * q^98 + 3597732 * q^99

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}

1.1

−8.00000

−39.0000

64.0000

385.000

312.000

−293.000

−512.000

−666.000

−3080.00

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$13$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	26.8.a.a	✓	1
3.b	odd	2	1		234.8.a.d		1
4.b	odd	2	1		208.8.a.c		1
13.b	even	2	1		338.8.a.c		1
13.d	odd	4	2		338.8.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.8.a.a	✓	1	1.a	even	1	1	trivial
208.8.a.c		1	4.b	odd	2	1
234.8.a.d		1	3.b	odd	2	1
338.8.a.c		1	13.b	even	2	1
338.8.b.b		2	13.d	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 39$ T3 + 39 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(26))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 8$ T + 8
$3$	$T + 39$ T + 39
$5$	$T - 385$ T - 385
$7$	$T + 293$ T + 293
$11$	$T + 5402$ T + 5402
$13$	$T - 2197$ T - 2197
$17$	$T + 21011$ T + 21011
$19$	$T + 27326$ T + 27326
$23$	$T + 63072$ T + 63072
$29$	$T - 122238$ T - 122238
$31$	$T + 208396$ T + 208396
$37$	$T + 442379$ T + 442379
$41$	$T - 58000$ T - 58000
$43$	$T + 202025$ T + 202025
$47$	$T - 588511$ T - 588511
$53$	$T - 1684336$ T - 1684336
$59$	$T + 442630$ T + 442630
$61$	$T + 1083608$ T + 1083608
$67$	$T - 3443486$ T - 3443486
$71$	$T - 2084705$ T - 2084705
$73$	$T - 5937890$ T - 5937890
$79$	$T + 6609256$ T + 6609256
$83$	$T + 142740$ T + 142740
$89$	$T + 6985286$ T + 6985286
$97$	$T + 200762$ T + 200762
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