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Properties

Label 588.2.e.e
Level $588$
Weight $2$
Character orbit 588.e
Analytic conductor $4.695$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(491,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("588.491");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.e (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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.312013725601644544.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} - 2x^{8} + 8x^{6} - 8x^{4} - 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{8} q^{3} + \beta_{2} q^{4} + \beta_{3} q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{6}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{8} q^{3} + \beta_{2} q^{4} + \beta_{3} q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{6}+ \cdots + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} + 2 q^{9} + 10 q^{10} + 12 q^{12} - 12 q^{13} + 10 q^{16} + 10 q^{18} + 14 q^{22} + 14 q^{24} + 12 q^{25} + 14 q^{30} - 10 q^{33} - 4 q^{34} + 22 q^{36} + 8 q^{37} - 34 q^{40} + 18 q^{45} - 24 q^{46} + 4 q^{48} - 16 q^{52} - 38 q^{54} - 2 q^{57} - 14 q^{58} - 14 q^{60} - 4 q^{61} - 34 q^{64} - 30 q^{66} + 18 q^{69} - 20 q^{72} - 12 q^{76} - 52 q^{78} - 26 q^{81} + 68 q^{82} - 20 q^{85} + 34 q^{88} - 20 q^{90} + 6 q^{93} + 24 q^{94} + 62 q^{96} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{10} - 2x^{8} + 8x^{6} - 8x^{4} - 16x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + \nu^{7} + 8\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - \nu^{7} + 8\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} - \nu^{6} + 2\nu^{4} + 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 2\nu^{10} - \nu^{9} - 2\nu^{8} + 4\nu^{6} + 8\nu^{5} + 8\nu^{4} + 8\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - \nu^{10} - \nu^{9} - \nu^{8} + 2\nu^{7} + 4\nu^{5} + 8\nu^{4} + 8\nu^{2} ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} - 2\nu^{10} - \nu^{9} + 2\nu^{8} - 4\nu^{6} + 8\nu^{5} - 8\nu^{4} + 8\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + \nu^{9} - 8\nu^{7} + 8\nu^{3} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + \nu^{8} + 2\nu^{6} - 8\nu^{4} + 8\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} - \nu^{10} + \nu^{9} - \nu^{8} - 2\nu^{7} - 4\nu^{5} + 8\nu^{4} + 8\nu^{2} ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} + \beta_{9} + 3\beta_{8} + \beta_{7} + 3\beta_{6} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + 4\beta_{10} - 5\beta_{8} + \beta_{7} + 5\beta_{6} + \beta_{5} - 3\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{11} - 3\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -\beta_{11} + 4\beta_{10} - 3\beta_{8} - \beta_{7} + 3\beta_{6} + 7\beta_{5} - 5\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{11} + 3\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 8\beta_{4} + 6\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -7\beta_{11} - 4\beta_{10} - 5\beta_{8} - 7\beta_{7} + 5\beta_{6} + \beta_{5} + 5\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -9\beta_{11} + 5\beta_{9} - 9\beta_{8} + 9\beta_{7} - 9\beta_{6} + 2\beta_{3} + 6\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
−1.38193 − 0.300427i
−1.38193 + 0.300427i
−1.13556 − 0.842913i
−1.13556 + 0.842913i
−0.225298 − 1.39615i
−0.225298 + 1.39615i
0.225298 − 1.39615i
0.225298 + 1.39615i
1.13556 − 0.842913i
1.13556 + 0.842913i
1.38193 − 0.300427i
1.38193 + 0.300427i
−1.38193 − 0.300427i −1.24483 − 1.20432i 1.81949 + 0.830342i − 2.72774i 1.35846 + 2.03828i 0 −2.26495 − 1.69410i 0.0992110 + 2.99836i −0.819487 + 3.76955i
491.2 −1.38193 + 0.300427i −1.24483 + 1.20432i 1.81949 − 0.830342i 2.72774i 1.35846 − 2.03828i 0 −2.26495 + 1.69410i 0.0992110 − 2.99836i −0.819487 − 3.76955i
491.3 −1.13556 − 0.842913i 1.65140 − 0.522368i 0.578995 + 1.91436i 0.499464i −2.31558 − 0.798808i 0 0.956154 − 2.66191i 2.45426 − 1.72528i 0.421005 − 0.567172i
491.4 −1.13556 + 0.842913i 1.65140 + 0.522368i 0.578995 − 1.91436i − 0.499464i −2.31558 + 0.798808i 0 0.956154 + 2.66191i 2.45426 + 1.72528i 0.421005 + 0.567172i
491.5 −0.225298 − 1.39615i −0.687941 − 1.58957i −1.89848 + 0.629100i 2.07605i −2.06429 + 1.31860i 0 1.30604 + 2.50883i −2.05347 + 2.18706i 2.89848 − 0.467730i
491.6 −0.225298 + 1.39615i −0.687941 + 1.58957i −1.89848 − 0.629100i − 2.07605i −2.06429 − 1.31860i 0 1.30604 − 2.50883i −2.05347 − 2.18706i 2.89848 + 0.467730i
491.7 0.225298 − 1.39615i 0.687941 + 1.58957i −1.89848 − 0.629100i 2.07605i 2.37428 − 0.602344i 0 −1.30604 + 2.50883i −2.05347 + 2.18706i 2.89848 + 0.467730i
491.8 0.225298 + 1.39615i 0.687941 − 1.58957i −1.89848 + 0.629100i − 2.07605i 2.37428 + 0.602344i 0 −1.30604 − 2.50883i −2.05347 − 2.18706i 2.89848 − 0.467730i
491.9 1.13556 − 0.842913i −1.65140 + 0.522368i 0.578995 − 1.91436i 0.499464i −1.43496 + 1.98517i 0 −0.956154 − 2.66191i 2.45426 − 1.72528i 0.421005 + 0.567172i
491.10 1.13556 + 0.842913i −1.65140 − 0.522368i 0.578995 + 1.91436i − 0.499464i −1.43496 − 1.98517i 0 −0.956154 + 2.66191i 2.45426 + 1.72528i 0.421005 − 0.567172i
491.11 1.38193 − 0.300427i 1.24483 + 1.20432i 1.81949 − 0.830342i − 2.72774i 2.08209 + 1.29031i 0 2.26495 − 1.69410i 0.0992110 + 2.99836i −0.819487 − 3.76955i
491.12 1.38193 + 0.300427i 1.24483 − 1.20432i 1.81949 + 0.830342i 2.72774i 2.08209 − 1.29031i 0 2.26495 + 1.69410i 0.0992110 − 2.99836i −0.819487 + 3.76955i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.12
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.e.e 12
3.b odd 2 1 inner 588.2.e.e 12
4.b odd 2 1 inner 588.2.e.e 12
7.b odd 2 1 588.2.e.d 12
7.c even 3 2 84.2.n.a ✓ 24
7.d odd 6 2 588.2.n.e 24
12.b even 2 1 inner 588.2.e.e 12
21.c even 2 1 588.2.e.d 12
21.g even 6 2 588.2.n.e 24
21.h odd 6 2 84.2.n.a ✓ 24
28.d even 2 1 588.2.e.d 12
28.f even 6 2 588.2.n.e 24
28.g odd 6 2 84.2.n.a ✓ 24
84.h odd 2 1 588.2.e.d 12
84.j odd 6 2 588.2.n.e 24
84.n even 6 2 84.2.n.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.n.a ✓ 24 7.c even 3 2
84.2.n.a ✓ 24 21.h odd 6 2
84.2.n.a ✓ 24 28.g odd 6 2
84.2.n.a ✓ 24 84.n even 6 2
588.2.e.d 12 7.b odd 2 1
588.2.e.d 12 21.c even 2 1
588.2.e.d 12 28.d even 2 1
588.2.e.d 12 84.h odd 2 1
588.2.e.e 12 1.a even 1 1 trivial
588.2.e.e 12 3.b odd 2 1 inner
588.2.e.e 12 4.b odd 2 1 inner
588.2.e.e 12 12.b even 2 1 inner
588.2.n.e 24 7.d odd 6 2
588.2.n.e 24 21.g even 6 2
588.2.n.e 24 28.f even 6 2
588.2.n.e 24 84.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{6} + 12T_{5}^{4} + 35T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{6} - 22T_{11}^{4} + 105T_{11}^{2} - 128 \) Copy content Toggle raw display
\( T_{13}^{3} + 3T_{13}^{2} - 10T_{13} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{6} + 12 T^{4} + 35 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 22 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 3 T^{2} - 10 T - 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 43 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 16 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 83 T^{4} + \cdots - 1568)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 45 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 71 T^{4} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} + \cdots + 292)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 144 T^{4} + \cdots + 100352)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 31 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 83 T^{4} + \cdots - 1568)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 164 T^{4} + \cdots + 75272)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 46 T^{4} + \cdots - 2888)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 20 T - 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 144 T^{4} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 52 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 139 T - 394)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 231 T^{4} + \cdots + 58081)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 383 T^{4} + \cdots - 1874048)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 79 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 9 T^{2} - 102 T + 56)^{4} \) Copy content Toggle raw display
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