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Properties

Label 3330.2.h.o
Level $3330$
Weight $2$
Character orbit 3330.h
Analytic conductor $26.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3330,2,Mod(2071,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3330.2071");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.h (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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1110)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} - \beta_{2} q^{5} - \beta_{5} q^{7} - \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - q^{4} - \beta_{2} q^{5} - \beta_{5} q^{7} - \beta_{2} q^{8} + q^{10} - \beta_{7} q^{11} + \beta_{3} q^{13} - \beta_{4} q^{14} + q^{16} + (\beta_{4} - 2 \beta_{2}) q^{17} + (\beta_{6} - \beta_{4} + \cdots + 2 \beta_{2}) q^{19}+ \cdots + (2 \beta_{6} - \beta_{4} + 5 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 4 q^{7} + 8 q^{10} + 4 q^{11} + 8 q^{16} - 8 q^{25} - 2 q^{26} + 4 q^{28} + 12 q^{34} + 18 q^{37} - 10 q^{38} - 8 q^{40} + 10 q^{41} - 4 q^{44} + 2 q^{46} - 28 q^{47} + 28 q^{49} - 12 q^{53} + 6 q^{58} + 10 q^{62} - 8 q^{64} + 2 q^{65} + 12 q^{67} - 4 q^{70} - 24 q^{71} + 10 q^{73} + 2 q^{74} + 32 q^{77} - 6 q^{83} - 12 q^{85} - 14 q^{86} + 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 18\nu^{2} - 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} + 44\nu^{5} + 180\nu^{3} + 155\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 44\nu^{5} - 124\nu^{3} + 181\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 527\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 27\nu^{4} - 85\nu^{2} + 11 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 639\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{6} + 27\nu^{4} + 99\nu^{2} + 59 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{6} + 3\beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{7} - 9\beta_{5} - 4\beta _1 + 68 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 41\beta_{6} - 35\beta_{4} - 24\beta_{3} - 22\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26\beta_{7} + 36\beta_{5} + 27\beta _1 - 241 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -293\beta_{6} + 205\beta_{4} + 232\beta_{3} + 240\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2071.1
− 0.490113i
2.20111i
2.87151i
− 2.58251i
0.490113i
− 2.20111i
− 2.87151i
2.58251i
− 1.00000i 0 −1.00000 1.00000i 0 −4.26968 1.00000i 0 1.00000
2071.2 − 1.00000i 0 −1.00000 1.00000i 0 −2.35622 1.00000i 0 1.00000
2071.3 − 1.00000i 0 −1.00000 1.00000i 0 0.374052 1.00000i 0 1.00000
2071.4 − 1.00000i 0 −1.00000 1.00000i 0 4.25184 1.00000i 0 1.00000
2071.5 1.00000i 0 −1.00000 − 1.00000i 0 −4.26968 − 1.00000i 0 1.00000
2071.6 1.00000i 0 −1.00000 − 1.00000i 0 −2.35622 − 1.00000i 0 1.00000
2071.7 1.00000i 0 −1.00000 − 1.00000i 0 0.374052 − 1.00000i 0 1.00000
2071.8 1.00000i 0 −1.00000 − 1.00000i 0 4.25184 − 1.00000i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2071.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.h.o 8
3.b odd 2 1 1110.2.h.g ✓ 8
37.b even 2 1 inner 3330.2.h.o 8
111.d odd 2 1 1110.2.h.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.g ✓ 8 3.b odd 2 1
1110.2.h.g ✓ 8 111.d odd 2 1
3330.2.h.o 8 1.a even 1 1 trivial
3330.2.h.o 8 37.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{4} + 2T_{7}^{3} - 19T_{7}^{2} - 36T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} - 33T_{11}^{2} + 38T_{11} + 60 \) Copy content Toggle raw display
\( T_{13}^{8} + 53T_{13}^{6} + 852T_{13}^{4} + 5056T_{13}^{2} + 9216 \) Copy content Toggle raw display
\( T_{41}^{4} - 5T_{41}^{3} - 46T_{41}^{2} + 52T_{41} + 360 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 19 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 33 T^{2} + \cdots + 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 53 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( T^{8} + 50 T^{6} + \cdots + 144 \) Copy content Toggle raw display
$19$ \( T^{8} + 81 T^{6} + \cdots + 46656 \) Copy content Toggle raw display
$23$ \( T^{8} + 53 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( T^{8} + 213 T^{6} + \cdots + 3594816 \) Copy content Toggle raw display
$31$ \( T^{8} + 117 T^{6} + \cdots + 129600 \) Copy content Toggle raw display
$37$ \( T^{8} - 18 T^{7} + \cdots + 1874161 \) Copy content Toggle raw display
$41$ \( (T^{4} - 5 T^{3} + \cdots + 360)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 89 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$47$ \( (T^{4} + 14 T^{3} + \cdots - 9216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} - 7 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 212 T^{6} + \cdots + 2359296 \) Copy content Toggle raw display
$61$ \( T^{8} + 177 T^{6} + \cdots + 484416 \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} + \cdots + 192)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 5 T^{3} + \cdots + 1880)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 492 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$83$ \( (T^{4} + 3 T^{3} + \cdots - 576)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 185 T^{6} + \cdots + 36864 \) Copy content Toggle raw display
$97$ \( T^{8} + 501 T^{6} + \cdots + 191102976 \) Copy content Toggle raw display
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