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Properties

Label 468.2.n.l
Level $468$
Weight $2$
Character orbit 468.n
Analytic conductor $3.737$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(307,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("468.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.n (of order \(4\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.29960650073923649536.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + \beta_{7} q^{4} - \beta_{2} q^{5} + (\beta_{10} - \beta_{9} + \beta_{7}) q^{7} + (\beta_{15} + \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + \beta_{7} q^{4} - \beta_{2} q^{5} + (\beta_{10} - \beta_{9} + \beta_{7}) q^{7} + (\beta_{15} + \beta_{5}) q^{8} + ( - \beta_{13} + \beta_{8}) q^{10} + ( - \beta_{6} - \beta_{4}) q^{11} + (\beta_{10} + \beta_{8} + 2 \beta_1 + 1) q^{13} + (\beta_{15} - \beta_{14}) q^{14} + ( - \beta_{13} + \beta_{12} + \beta_{10} + \cdots + 2) q^{16}+ \cdots + ( - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{13} + 28 q^{16} - 24 q^{22} + 36 q^{28} + 8 q^{34} - 16 q^{37} + 56 q^{40} - 16 q^{46} + 28 q^{52} - 72 q^{58} - 32 q^{61} - 8 q^{70} + 32 q^{73} + 4 q^{76} - 64 q^{85} - 152 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 3\nu^{6} + 12\nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{15} + 3\nu^{11} + 4\nu^{7} + 32\nu^{3} ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{15} - \nu^{14} - 10 \nu^{12} + 33 \nu^{11} + 11 \nu^{10} - 18 \nu^{8} - 124 \nu^{7} + \cdots - 640 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 6 \nu^{13} - 10 \nu^{12} - 11 \nu^{10} + 2 \nu^{9} - 18 \nu^{8} + 84 \nu^{6} - 56 \nu^{5} + \cdots - 640 ) / 512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{13} - 17\nu^{9} + 76\nu^{5} - 128\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{14} + 6 \nu^{13} + 2 \nu^{12} + 23 \nu^{10} - 2 \nu^{9} - 22 \nu^{8} - 68 \nu^{6} + \cdots - 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7 \nu^{15} + 4 \nu^{14} - 6 \nu^{13} + 29 \nu^{11} - 12 \nu^{10} + 34 \nu^{9} - 124 \nu^{7} + \cdots + 128 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{14} - \nu^{13} - 4 \nu^{12} + 3 \nu^{10} - 5 \nu^{9} + 28 \nu^{8} - 28 \nu^{6} + 28 \nu^{5} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{15} + 6 \nu^{13} - 16 \nu^{12} + 29 \nu^{11} - 34 \nu^{9} + 112 \nu^{8} - 124 \nu^{7} + \cdots + 1024 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{14} + \nu^{13} - 4 \nu^{12} + 3 \nu^{10} + 5 \nu^{9} + 28 \nu^{8} - 28 \nu^{6} - 28 \nu^{5} + \cdots + 256 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{15} + 5 \nu^{14} + 2 \nu^{12} - 33 \nu^{11} - 23 \nu^{10} - 22 \nu^{8} + 124 \nu^{7} + \cdots - 384 ) / 512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5 \nu^{15} + 4 \nu^{14} - 6 \nu^{13} - 7 \nu^{11} - 12 \nu^{10} + 34 \nu^{9} + 84 \nu^{7} + \cdots + 128 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + 2\nu^{14} - 8\nu^{12} - 11\nu^{11} - 6\nu^{10} + 56\nu^{8} + 20\nu^{7} + 56\nu^{6} - 192\nu^{4} + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3 \nu^{15} - 11 \nu^{14} - 14 \nu^{12} - 33 \nu^{11} + 57 \nu^{10} + 26 \nu^{8} + 124 \nu^{7} + \cdots + 128 ) / 512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11 \nu^{14} - 6 \nu^{13} + 14 \nu^{12} + 57 \nu^{10} + 2 \nu^{9} - 26 \nu^{8} - 220 \nu^{6} + \cdots - 128 ) / 512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \cdots - 2 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} + 3\beta_{13} - 2\beta_{12} - 2\beta_{11} - 3\beta_{9} - \beta_{7} + \beta_{3} + 3\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + \beta_{13} + \beta_{12} + 5 \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} + 6\beta_{10} - 9\beta_{9} + 3\beta_{8} + 9\beta_{7} - 2\beta_{6} + 5\beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{15} + 3 \beta_{14} + 7 \beta_{13} + 7 \beta_{12} - 5 \beta_{11} - 7 \beta_{10} + 7 \beta_{9} + \cdots - 6 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{14} - 3\beta_{13} + 10\beta_{12} + 2\beta_{11} + 3\beta_{9} - 7\beta_{7} - \beta_{3} + 21\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 13 \beta_{15} - 13 \beta_{14} + 7 \beta_{13} + 7 \beta_{12} + 11 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + \cdots - 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -17\beta_{15} + 10\beta_{10} - 23\beta_{9} + 13\beta_{8} + 23\beta_{7} + 34\beta_{6} - 5\beta_{5} - 17\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} - 27 \beta_{11} - 9 \beta_{10} + \cdots + 70 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -\beta_{14} - 61\beta_{13} + 86\beta_{12} - 2\beta_{11} + 61\beta_{9} - 25\beta_{7} + \beta_{3} + 11\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 19 \beta_{15} - 19 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} - 43 \beta_{11} - 7 \beta_{10} - 7 \beta_{9} + \cdots - 154 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -79\beta_{15} - 10\beta_{10} + 55\beta_{9} - 45\beta_{8} - 55\beta_{7} + 158\beta_{6} - 27\beta_{5} - 79\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 93 \beta_{15} - 93 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} + 59 \beta_{11} + 41 \beta_{10} + \cdots + 378 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 31 \beta_{14} - 99 \beta_{13} + 234 \beta_{12} - 62 \beta_{11} + 99 \beta_{9} - 135 \beta_{7} + \cdots - 299 \beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−1.32968 + 0.481610i
0.281691 + 1.38588i
−1.38588 − 0.281691i
1.32968 − 0.481610i
−0.481610 + 1.32968i
−0.281691 − 1.38588i
1.38588 + 0.281691i
0.481610 − 1.32968i
−1.32968 − 0.481610i
0.281691 − 1.38588i
−1.38588 + 0.281691i
1.32968 + 0.481610i
−0.481610 − 1.32968i
−0.281691 + 1.38588i
1.38588 − 0.281691i
0.481610 + 1.32968i
−1.38588 + 0.281691i 0 1.84130 − 0.780776i −1.41421 − 1.41421i 0 0.662153 + 0.662153i −2.33188 + 1.60074i 0 2.35829 + 1.56155i
307.2 −1.32968 − 0.481610i 0 1.53610 + 1.28078i 1.41421 + 1.41421i 0 2.13578 + 2.13578i −1.42569 − 2.44283i 0 −1.19935 − 2.56155i
307.3 −0.481610 − 1.32968i 0 −1.53610 + 1.28078i 1.41421 + 1.41421i 0 −2.13578 − 2.13578i 2.44283 + 1.42569i 0 1.19935 − 2.56155i
307.4 −0.281691 + 1.38588i 0 −1.84130 − 0.780776i 1.41421 + 1.41421i 0 −0.662153 − 0.662153i 1.60074 − 2.33188i 0 −2.35829 + 1.56155i
307.5 0.281691 − 1.38588i 0 −1.84130 − 0.780776i −1.41421 − 1.41421i 0 −0.662153 − 0.662153i −1.60074 + 2.33188i 0 −2.35829 + 1.56155i
307.6 0.481610 + 1.32968i 0 −1.53610 + 1.28078i −1.41421 − 1.41421i 0 −2.13578 − 2.13578i −2.44283 − 1.42569i 0 1.19935 − 2.56155i
307.7 1.32968 + 0.481610i 0 1.53610 + 1.28078i −1.41421 − 1.41421i 0 2.13578 + 2.13578i 1.42569 + 2.44283i 0 −1.19935 − 2.56155i
307.8 1.38588 − 0.281691i 0 1.84130 − 0.780776i 1.41421 + 1.41421i 0 0.662153 + 0.662153i 2.33188 − 1.60074i 0 2.35829 + 1.56155i
343.1 −1.38588 − 0.281691i 0 1.84130 + 0.780776i −1.41421 + 1.41421i 0 0.662153 − 0.662153i −2.33188 − 1.60074i 0 2.35829 − 1.56155i
343.2 −1.32968 + 0.481610i 0 1.53610 − 1.28078i 1.41421 − 1.41421i 0 2.13578 − 2.13578i −1.42569 + 2.44283i 0 −1.19935 + 2.56155i
343.3 −0.481610 + 1.32968i 0 −1.53610 − 1.28078i 1.41421 − 1.41421i 0 −2.13578 + 2.13578i 2.44283 − 1.42569i 0 1.19935 + 2.56155i
343.4 −0.281691 − 1.38588i 0 −1.84130 + 0.780776i 1.41421 − 1.41421i 0 −0.662153 + 0.662153i 1.60074 + 2.33188i 0 −2.35829 − 1.56155i
343.5 0.281691 + 1.38588i 0 −1.84130 + 0.780776i −1.41421 + 1.41421i 0 −0.662153 + 0.662153i −1.60074 − 2.33188i 0 −2.35829 − 1.56155i
343.6 0.481610 − 1.32968i 0 −1.53610 − 1.28078i −1.41421 + 1.41421i 0 −2.13578 + 2.13578i −2.44283 + 1.42569i 0 1.19935 + 2.56155i
343.7 1.32968 − 0.481610i 0 1.53610 − 1.28078i −1.41421 + 1.41421i 0 2.13578 − 2.13578i 1.42569 − 2.44283i 0 −1.19935 + 2.56155i
343.8 1.38588 + 0.281691i 0 1.84130 + 0.780776i 1.41421 − 1.41421i 0 0.662153 − 0.662153i 2.33188 + 1.60074i 0 2.35829 − 1.56155i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner
52.f even 4 1 inner
156.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.n.l ✓ 16
3.b odd 2 1 inner 468.2.n.l ✓ 16
4.b odd 2 1 inner 468.2.n.l ✓ 16
12.b even 2 1 inner 468.2.n.l ✓ 16
13.d odd 4 1 inner 468.2.n.l ✓ 16
39.f even 4 1 inner 468.2.n.l ✓ 16
52.f even 4 1 inner 468.2.n.l ✓ 16
156.l odd 4 1 inner 468.2.n.l ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.n.l ✓ 16 1.a even 1 1 trivial
468.2.n.l ✓ 16 3.b odd 2 1 inner
468.2.n.l ✓ 16 4.b odd 2 1 inner
468.2.n.l ✓ 16 12.b even 2 1 inner
468.2.n.l ✓ 16 13.d odd 4 1 inner
468.2.n.l ✓ 16 39.f even 4 1 inner
468.2.n.l ✓ 16 52.f even 4 1 inner
468.2.n.l ✓ 16 156.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 84T_{7}^{4} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} + 84T_{11}^{4} + 64 \) Copy content Toggle raw display
\( T_{17}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 7 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 16)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + 84 T^{4} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 84 T^{4} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 6 T^{3} + \cdots + 169)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{8} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2772 T^{4} + 1827904)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 92 T^{2} + 2048)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 72 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 84 T^{4} + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1296)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 40 T^{2} + 128)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 32916 T^{4} + 218803264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 104 T^{2} + 256)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 84 T^{4} + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 64)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + 68052 T^{4} + 1827904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 41748 T^{4} + 8340544)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} + \cdots + 676)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 224 T^{2} + 8192)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 9108 T^{4} + 8340544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 4624)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{3} + \cdots + 17956)^{4} \) Copy content Toggle raw display
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