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Properties

Label 152.4.v.a
Level $152$
Weight $4$
Character orbit 152.v
Analytic conductor $8.968$
Analytic rank $0$
Dimension $12$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [152,4,Mod(3,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 13]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 152.v (of order \(18\), degree \(6\), 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: \(8.96829032087\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.101559956668416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \beta_{11} q^{2} + ( - 5 \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{3} - 8 \beta_{4} q^{4} + ( - 10 \beta_{11} + 4 \beta_{8} + \cdots - 4) q^{6} + (16 \beta_{9} - 16 \beta_{3}) q^{8} + (10 \beta_{11} - 27 \beta_{10} + \cdots - 23) q^{9}+ \cdots + ( - 485 \beta_{11} + 485 \beta_{9} + \cdots + 950) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 30 q^{3} - 24 q^{6} - 138 q^{9} - 1200 q^{22} + 192 q^{24} - 1710 q^{27} + 30 q^{33} + 1104 q^{36} + 1080 q^{38} - 1566 q^{41} - 864 q^{44} + 3840 q^{48} - 2058 q^{49} + 2718 q^{51} + 648 q^{54}+ \cdots + 9942 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 16\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 32\beta_{11} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.483690 − 1.32893i
−0.483690 + 1.32893i
0.483690 + 1.32893i
−0.483690 − 1.32893i
−0.909039 − 1.08335i
0.909039 + 1.08335i
−0.909039 + 1.08335i
0.909039 − 1.08335i
−1.39273 − 0.245576i
1.39273 + 0.245576i
−1.39273 + 0.245576i
1.39273 − 0.245576i
−1.81808 + 2.16670i −1.45741 − 4.00419i −1.38919 − 7.87846i 0 11.3256 + 4.12217i 0 19.5959 + 11.3137i 6.77366 − 5.68378i 0
3.2 1.81808 − 2.16670i −2.93952 − 8.07626i −1.38919 − 7.87846i 0 −22.8431 − 8.31421i 0 −19.5959 − 11.3137i −35.9020 + 30.1254i 0
51.1 −1.81808 − 2.16670i −1.45741 + 4.00419i −1.38919 + 7.87846i 0 11.3256 − 4.12217i 0 19.5959 − 11.3137i 6.77366 + 5.68378i 0
51.2 1.81808 + 2.16670i −2.93952 + 8.07626i −1.38919 + 7.87846i 0 −22.8431 + 8.31421i 0 −19.5959 + 11.3137i −35.9020 − 30.1254i 0
59.1 −2.78546 − 0.491151i 6.01452 − 7.16782i 7.51754 + 2.73616i 0 −20.2737 + 17.0116i 0 −19.5959 − 11.3137i −10.5148 − 59.6321i 0
59.2 2.78546 + 0.491151i 6.64593 − 7.92031i 7.51754 + 2.73616i 0 22.4020 − 18.7975i 0 19.5959 + 11.3137i −13.8744 − 78.6858i 0
67.1 −2.78546 + 0.491151i 6.01452 + 7.16782i 7.51754 − 2.73616i 0 −20.2737 − 17.0116i 0 −19.5959 + 11.3137i −10.5148 + 59.6321i 0
67.2 2.78546 − 0.491151i 6.64593 + 7.92031i 7.51754 − 2.73616i 0 22.4020 + 18.7975i 0 19.5959 − 11.3137i −13.8744 + 78.6858i 0
91.1 −0.967379 + 2.65785i 5.98571 − 1.05544i −6.12836 − 5.14230i 0 −2.98524 + 16.9302i 0 19.5959 − 11.3137i 9.34311 − 3.40062i 0
91.2 0.967379 − 2.65785i 0.750768 − 0.132381i −6.12836 − 5.14230i 0 0.374429 − 2.12349i 0 −19.5959 + 11.3137i −24.8256 + 9.03577i 0
147.1 −0.967379 − 2.65785i 5.98571 + 1.05544i −6.12836 + 5.14230i 0 −2.98524 − 16.9302i 0 19.5959 + 11.3137i 9.34311 + 3.40062i 0
147.2 0.967379 + 2.65785i 0.750768 + 0.132381i −6.12836 + 5.14230i 0 0.374429 + 2.12349i 0 −19.5959 − 11.3137i −24.8256 − 9.03577i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.f odd 18 1 inner
152.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.4.v.a ✓ 12
8.d odd 2 1 CM 152.4.v.a ✓ 12
19.f odd 18 1 inner 152.4.v.a ✓ 12
152.v even 18 1 inner 152.4.v.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.v.a ✓ 12 1.a even 1 1 trivial
152.4.v.a ✓ 12 8.d odd 2 1 CM
152.4.v.a ✓ 12 19.f odd 18 1 inner
152.4.v.a ✓ 12 152.v even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 30 T_{3}^{11} + 519 T_{3}^{10} - 6000 T_{3}^{9} + 58026 T_{3}^{8} - 484080 T_{3}^{7} + \cdots + 269517889 \) acting on \(S_{4}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 512 T^{6} + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 269517889 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 73\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 85\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 72\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 54\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 29\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 17\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 51\!\cdots\!89 \) Copy content Toggle raw display
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