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Properties

Label 152.3.u.a
Level $152$
Weight $3$
Character orbit 152.u
Analytic conductor $4.142$
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] = [152,3,Mod(35,152)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 152.u (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: \(4.14170001828\)
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: \( 2^{6}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_{5} - 2 \beta_1) q^{2} + (\beta_{11} - \beta_{9} + \cdots + \beta_{3}) q^{3}+ \cdots + (2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{5} - 2 \beta_1) q^{2} + (\beta_{11} - \beta_{9} + \cdots + \beta_{3}) q^{3}+ \cdots + ( - 14 \beta_{11} + 27 \beta_{10} + \cdots - 64) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 12 q^{6} + 48 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 12 q^{6} + 48 q^{8} + 42 q^{9} - 216 q^{18} - 168 q^{22} - 48 q^{24} + 138 q^{27} - 102 q^{33} + 168 q^{36} - 204 q^{38} + 138 q^{41} + 336 q^{44} - 192 q^{48} - 294 q^{49} + 300 q^{50} + 858 q^{51} - 108 q^{54} + 246 q^{59} - 384 q^{64} + 204 q^{66} - 186 q^{67} + 24 q^{68} + 672 q^{72} - 852 q^{73} - 858 q^{81} - 276 q^{82} + 282 q^{97} - 390 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{10} - 4\beta_{9} - 4\beta_{8} + 6\beta_{7} - 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12\beta_{11} - 4\beta_{9} - 4\beta_{8} + 12\beta_{7} - 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{5} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8\beta_{9} - 16\beta_{8} + 8\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{6} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -16\beta_{9} - 16\beta_{8} + 48\beta_{7} - 16\beta_{2} ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−1.39273 + 0.245576i
1.39273 − 0.245576i
−0.909039 − 1.08335i
0.909039 + 1.08335i
−0.909039 + 1.08335i
0.909039 − 1.08335i
0.483690 + 1.32893i
−0.483690 − 1.32893i
0.483690 − 1.32893i
−0.483690 + 1.32893i
−1.39273 − 0.245576i
1.39273 + 0.245576i
−1.53209 − 1.28558i −1.97718 + 0.719633i 0.694593 + 3.93923i 0 3.95435 + 1.43927i 0 4.00000 − 6.92820i −3.50305 + 2.93940i 0
35.2 −1.53209 − 1.28558i 4.85656 − 1.76764i 0.694593 + 3.93923i 0 −9.71312 − 3.53529i 0 4.00000 − 6.92820i 13.5672 − 11.3843i 0
43.1 1.87939 − 0.684040i −0.00961540 + 0.0545316i 3.06418 − 2.57115i 0 0.0192308 + 0.109063i 0 4.00000 − 6.92820i 8.45435 + 3.07713i 0
43.2 1.87939 − 0.684040i 0.662319 − 3.75620i 3.06418 − 2.57115i 0 −1.32464 − 7.51240i 0 4.00000 − 6.92820i −5.21312 − 1.89742i 0
99.1 1.87939 + 0.684040i −0.00961540 − 0.0545316i 3.06418 + 2.57115i 0 0.0192308 − 0.109063i 0 4.00000 + 6.92820i 8.45435 − 3.07713i 0
99.2 1.87939 + 0.684040i 0.662319 + 3.75620i 3.06418 + 2.57115i 0 −1.32464 + 7.51240i 0 4.00000 + 6.92820i −5.21312 + 1.89742i 0
123.1 −0.347296 − 1.96962i −4.53361 + 3.80415i −3.75877 + 1.36808i 0 9.06722 + 7.60830i 0 4.00000 + 6.92820i 4.51923 − 25.6298i 0
123.2 −0.347296 − 1.96962i 4.00152 − 3.35768i −3.75877 + 1.36808i 0 −8.00305 − 6.71535i 0 4.00000 + 6.92820i 3.17536 − 18.0084i 0
131.1 −0.347296 + 1.96962i −4.53361 − 3.80415i −3.75877 − 1.36808i 0 9.06722 − 7.60830i 0 4.00000 − 6.92820i 4.51923 + 25.6298i 0
131.2 −0.347296 + 1.96962i 4.00152 + 3.35768i −3.75877 − 1.36808i 0 −8.00305 + 6.71535i 0 4.00000 − 6.92820i 3.17536 + 18.0084i 0
139.1 −1.53209 + 1.28558i −1.97718 − 0.719633i 0.694593 − 3.93923i 0 3.95435 − 1.43927i 0 4.00000 + 6.92820i −3.50305 − 2.93940i 0
139.2 −1.53209 + 1.28558i 4.85656 + 1.76764i 0.694593 − 3.93923i 0 −9.71312 + 3.53529i 0 4.00000 + 6.92820i 13.5672 + 11.3843i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.2
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.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.3.u.a ✓ 12
8.d odd 2 1 CM 152.3.u.a ✓ 12
19.e even 9 1 inner 152.3.u.a ✓ 12
152.u odd 18 1 inner 152.3.u.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.u.a ✓ 12 1.a even 1 1 trivial
152.3.u.a ✓ 12 8.d odd 2 1 CM
152.3.u.a ✓ 12 19.e even 9 1 inner
152.3.u.a ✓ 12 152.u odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 6 T_{3}^{11} - 3 T_{3}^{10} + 44 T_{3}^{9} + 954 T_{3}^{8} - 6444 T_{3}^{7} + 17748 T_{3}^{6} + \cdots + 5041 \) acting on \(S_{3}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 5041 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 22970736721 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 41\!\cdots\!09 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) 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
$41$ \( T^{12} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 23\!\cdots\!01 \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 71\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 67\!\cdots\!41 \) Copy content Toggle raw display
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