↑

Properties

Label 456.2.j.c
Level $456$
Weight $2$
Character orbit 456.j
Analytic conductor $3.641$
Analytic rank $0$
Dimension $12$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [456,2,Mod(419,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.j (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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.3493441689358336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 10 x^{10} - 11 x^{9} + 13 x^{8} - 28 x^{7} + 50 x^{6} - 56 x^{5} + 52 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{8} q^{3} + (\beta_{5} - \beta_{4}) q^{4} + (\beta_{7} + \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{2}) q^{6} + (\beta_{8} + 2 \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{11} - \beta_{6} - \beta_{3}) q^{8}+ \cdots + (\beta_{8} + 3 \beta_{7} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 2 q^{4} + 3 q^{6} + 6 q^{9} - 20 q^{10} + 21 q^{12} - 6 q^{16} - q^{18} - 12 q^{19} - 20 q^{22} + 5 q^{24} + 44 q^{25} + 4 q^{27} - 54 q^{28} - 38 q^{30} - 24 q^{33} - 10 q^{34} - q^{36}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 10 x^{10} - 11 x^{9} + 13 x^{8} - 28 x^{7} + 50 x^{6} - 56 x^{5} + 52 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} - 3 \nu^{8} + 7 \nu^{7} - 14 \nu^{6} + 22 \nu^{5} - 12 \nu^{4} + \cdots - 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3 \nu^{11} + 15 \nu^{10} - 26 \nu^{9} + 21 \nu^{8} - 23 \nu^{7} + 72 \nu^{6} - 122 \nu^{5} + \cdots + 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{11} + 9 \nu^{10} - 15 \nu^{9} + 12 \nu^{8} - 15 \nu^{7} + 43 \nu^{6} - 72 \nu^{5} + \cdots + 160 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{11} + 27 \nu^{10} - 32 \nu^{9} + 23 \nu^{8} - 55 \nu^{7} + 130 \nu^{6} - 162 \nu^{5} + \cdots + 224 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{11} - 31 \nu^{10} + 44 \nu^{9} - 39 \nu^{8} + 67 \nu^{7} - 158 \nu^{6} + 218 \nu^{5} + \cdots - 416 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{11} + 37 \nu^{10} - 52 \nu^{9} + 45 \nu^{8} - 81 \nu^{7} + 186 \nu^{6} - 262 \nu^{5} + \cdots + 512 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13 \nu^{11} + 50 \nu^{10} - 71 \nu^{9} + 57 \nu^{8} - 100 \nu^{7} + 249 \nu^{6} - 358 \nu^{5} + \cdots + 672 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15 \nu^{11} - 52 \nu^{10} + 69 \nu^{9} - 57 \nu^{8} + 106 \nu^{7} - 255 \nu^{6} + 356 \nu^{5} + \cdots - 608 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{11} + 41 \nu^{10} - 57 \nu^{9} + 46 \nu^{8} - 83 \nu^{7} + 201 \nu^{6} - 287 \nu^{5} + \cdots + 520 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36 \nu^{11} + 131 \nu^{10} - 183 \nu^{9} + 150 \nu^{8} - 265 \nu^{7} + 647 \nu^{6} - 928 \nu^{5} + \cdots + 1728 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85 \nu^{11} - 305 \nu^{10} + 418 \nu^{9} - 343 \nu^{8} + 625 \nu^{7} - 1500 \nu^{6} + 2122 \nu^{5} + \cdots - 3840 ) / 32 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{6} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{8} + \beta_{6} + 2\beta_{3} - 2\beta_{2} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - \beta_{8} + \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{3} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{11} - 4 \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{11} + \beta_{10} - 3 \beta_{8} + 5 \beta_{7} + \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{11} - 2 \beta_{10} - 5 \beta_{9} - 5 \beta_{8} + 4 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6 \beta_{11} - 5 \beta_{10} - 4 \beta_{9} - 5 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 10 \beta_{5} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15 \beta_{11} - 2 \beta_{10} - 19 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} - 17 \beta_{6} + 8 \beta_{5} + \cdots + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 34 \beta_{11} - 21 \beta_{10} - 10 \beta_{9} + 5 \beta_{8} - 11 \beta_{7} - 27 \beta_{6} + 14 \beta_{5} + \cdots + 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23 \beta_{11} - 30 \beta_{10} - \beta_{9} - \beta_{8} + 24 \beta_{7} - 13 \beta_{6} + 6 \beta_{5} + \cdots + 17 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
1.41283 + 0.0625922i
1.41283 − 0.0625922i
−0.726098 + 1.21358i
−0.726098 − 1.21358i
0.531721 − 1.31045i
0.531721 + 1.31045i
−1.18026 + 0.779099i
−1.18026 − 0.779099i
1.33270 + 0.473200i
1.33270 − 0.473200i
1.12911 − 0.851536i
1.12911 + 0.851536i
−1.33078 − 0.478573i 1.54194 − 0.788944i 1.54194 + 1.27375i 3.82023 −2.42954 + 0.311980i 2.54750i −1.44239 − 2.43300i 1.75513 − 2.43300i −5.08387 − 1.82826i
419.2 −1.33078 + 0.478573i 1.54194 + 0.788944i 1.54194 − 1.27375i 3.82023 −2.42954 − 0.311980i − 2.54750i −1.44239 + 2.43300i 1.75513 + 2.43300i −5.08387 + 1.82826i
419.3 −0.896270 − 1.09394i −0.393401 + 1.68678i −0.393401 + 1.96093i 1.35361 2.19783 − 1.08146i 3.92185i 2.49773 − 1.32716i −2.69047 − 1.32716i −1.21320 − 1.48076i
419.4 −0.896270 + 1.09394i −0.393401 − 1.68678i −0.393401 − 1.96093i 1.35361 2.19783 + 1.08146i − 3.92185i 2.49773 + 1.32716i −2.69047 + 1.32716i −1.21320 + 1.48076i
419.5 −0.419204 − 1.35065i −1.64854 − 0.531349i −1.64854 + 1.13240i −3.09412 −0.0265952 + 2.44935i 2.26480i 2.22056 + 1.75189i 2.43534 + 1.75189i 1.29707 + 4.17909i
419.6 −0.419204 + 1.35065i −1.64854 + 0.531349i −1.64854 − 1.13240i −3.09412 −0.0265952 − 2.44935i − 2.26480i 2.22056 − 1.75189i 2.43534 − 1.75189i 1.29707 − 4.17909i
419.7 0.419204 − 1.35065i −1.64854 − 0.531349i −1.64854 − 1.13240i 3.09412 −1.40874 + 2.00386i − 2.26480i −2.22056 + 1.75189i 2.43534 + 1.75189i 1.29707 − 4.17909i
419.8 0.419204 + 1.35065i −1.64854 + 0.531349i −1.64854 + 1.13240i 3.09412 −1.40874 − 2.00386i 2.26480i −2.22056 − 1.75189i 2.43534 − 1.75189i 1.29707 + 4.17909i
419.9 0.896270 − 1.09394i −0.393401 + 1.68678i −0.393401 − 1.96093i −1.35361 1.49264 + 1.94217i − 3.92185i −2.49773 − 1.32716i −2.69047 − 1.32716i −1.21320 + 1.48076i
419.10 0.896270 + 1.09394i −0.393401 − 1.68678i −0.393401 + 1.96093i −1.35361 1.49264 − 1.94217i 3.92185i −2.49773 + 1.32716i −2.69047 + 1.32716i −1.21320 − 1.48076i
419.11 1.33078 − 0.478573i 1.54194 − 0.788944i 1.54194 − 1.27375i −3.82023 1.67441 − 1.78784i − 2.54750i 1.44239 − 2.43300i 1.75513 − 2.43300i −5.08387 + 1.82826i
419.12 1.33078 + 0.478573i 1.54194 + 0.788944i 1.54194 + 1.27375i −3.82023 1.67441 + 1.78784i 2.54750i 1.44239 + 2.43300i 1.75513 + 2.43300i −5.08387 − 1.82826i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.j.c ✓ 12
3.b odd 2 1 inner 456.2.j.c ✓ 12
4.b odd 2 1 1824.2.j.c 12
8.b even 2 1 1824.2.j.c 12
8.d odd 2 1 inner 456.2.j.c ✓ 12
12.b even 2 1 1824.2.j.c 12
24.f even 2 1 inner 456.2.j.c ✓ 12
24.h odd 2 1 1824.2.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.j.c ✓ 12 1.a even 1 1 trivial
456.2.j.c ✓ 12 3.b odd 2 1 inner
456.2.j.c ✓ 12 8.d odd 2 1 inner
456.2.j.c ✓ 12 24.f even 2 1 inner
1824.2.j.c 12 4.b odd 2 1
1824.2.j.c 12 8.b even 2 1
1824.2.j.c 12 12.b even 2 1
1824.2.j.c 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\):

\( T_{5}^{6} - 26T_{5}^{4} + 184T_{5}^{2} - 256 \) Copy content Toggle raw display
\( T_{7}^{6} + 27T_{7}^{4} + 212T_{7}^{2} + 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} - T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 26 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 27 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 22 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 17 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 83 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} - 57 T^{4} + \cdots - 2116)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 75 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 114 T^{4} + \cdots + 16928)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 58 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 150 T^{4} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 14 T^{2} + \cdots + 128)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 68 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 99 T^{4} + \cdots - 11664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 91 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 240 T^{4} + \cdots + 476288)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 17 T^{2} + \cdots + 368)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 260 T^{4} + \cdots - 565504)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 11 T^{2} + \cdots - 1472)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 486 T^{4} + \cdots + 1013888)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 166 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 86 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$97$ \( (T - 6)^{12} \) Copy content Toggle raw display
show more
show less