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Properties

Label 3330.2.d.o
Level $3330$
Weight $2$
Character orbit 3330.d
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(1999,3330)]
 
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, 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("3330.1999");
 
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.d (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: 8.0.57815240704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots + (\beta_{7} - 3 \beta_{6} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{5} - 2 q^{10} + 10 q^{11} - 6 q^{14} + 8 q^{16} + 24 q^{19} - 2 q^{20} - 8 q^{26} + 10 q^{29} - 38 q^{31} - 2 q^{34} - 12 q^{35} + 2 q^{40} - 26 q^{41} - 10 q^{44} - 12 q^{46} - 30 q^{49} + 4 q^{50} + 30 q^{55} + 6 q^{56} + 20 q^{59} - 6 q^{61} - 8 q^{64} - 24 q^{65} + 26 q^{70} + 12 q^{71} - 8 q^{74} - 24 q^{76} + 16 q^{79} + 2 q^{80} + 44 q^{85} + 2 q^{86} + 64 q^{89} - 44 q^{91} + 52 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 198\nu^{7} - 217\nu^{6} + 209\nu^{5} + 88\nu^{4} + 18711\nu^{3} - 17653\nu^{2} + 16897\nu - 7524 ) / 9085 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1881\nu^{7} + 2970\nu^{6} - 2894\nu^{5} - 836\nu^{4} - 167761\nu^{3} + 244926\nu^{2} - 234110\nu + 67844 ) / 36340 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1969\nu^{7} - 2057\nu^{6} + 968\nu^{5} + 2894\nu^{4} + 176077\nu^{3} - 166969\nu^{2} + 74052\nu + 56002 ) / 18170 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -358\nu^{7} + 374\nu^{6} - 176\nu^{5} - 361\nu^{4} - 32014\nu^{3} + 30358\nu^{2} - 13464\nu - 2749 ) / 1817 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} + \cdots + 328276 ) / 36340 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9261 \nu^{7} - 16344 \nu^{6} + 14318 \nu^{5} + 4116 \nu^{4} + 825197 \nu^{3} - 1388536 \nu^{2} + \cdots - 333748 ) / 36340 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{7} - \beta_{6} - 5\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 9\beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{5} + 20\beta_{4} - 45 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 11\beta_{6} + 11\beta_{5} + 11\beta_{4} - 18\beta_{3} - 96\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 192\beta_{7} + 107\beta_{6} + 425\beta_{3} - 4\beta_{2} - 111\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 115\beta_{7} + 111\beta_{6} - 111\beta_{5} - 115\beta_{4} - 150\beta_{3} - 809\beta_{2} - 111\beta _1 - 150 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.594137 + 0.594137i
2.20793 + 2.20793i
−2.15569 − 2.15569i
0.353624 + 0.353624i
0.594137 − 0.594137i
2.20793 − 2.20793i
−2.15569 + 2.15569i
0.353624 − 0.353624i
− 1.00000i 0 −1.00000 −2.15569 − 0.594137i 0 − 2.38360i 1.00000i 0 −0.594137 + 2.15569i
1999.2 − 1.00000i 0 −1.00000 0.353624 − 2.20793i 0 − 3.94848i 1.00000i 0 −2.20793 − 0.353624i
1999.3 − 1.00000i 0 −1.00000 0.594137 + 2.15569i 0 − 1.17795i 1.00000i 0 2.15569 − 0.594137i
1999.4 − 1.00000i 0 −1.00000 2.20793 − 0.353624i 0 4.51003i 1.00000i 0 −0.353624 − 2.20793i
1999.5 1.00000i 0 −1.00000 −2.15569 + 0.594137i 0 2.38360i − 1.00000i 0 −0.594137 − 2.15569i
1999.6 1.00000i 0 −1.00000 0.353624 + 2.20793i 0 3.94848i − 1.00000i 0 −2.20793 + 0.353624i
1999.7 1.00000i 0 −1.00000 0.594137 − 2.15569i 0 1.17795i − 1.00000i 0 2.15569 + 0.594137i
1999.8 1.00000i 0 −1.00000 2.20793 + 0.353624i 0 − 4.51003i − 1.00000i 0 −0.353624 + 2.20793i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1999.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.d.o 8
3.b odd 2 1 1110.2.d.j ✓ 8
5.b even 2 1 inner 3330.2.d.o 8
15.d odd 2 1 1110.2.d.j ✓ 8
15.e even 4 1 5550.2.a.cl 4
15.e even 4 1 5550.2.a.cm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.j ✓ 8 3.b odd 2 1
1110.2.d.j ✓ 8 15.d odd 2 1
3330.2.d.o 8 1.a even 1 1 trivial
3330.2.d.o 8 5.b even 2 1 inner
5550.2.a.cl 4 15.e even 4 1
5550.2.a.cm 4 15.e even 4 1

Hecke kernels

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

\( T_{7}^{8} + 43T_{7}^{6} + 579T_{7}^{4} + 2525T_{7}^{2} + 2500 \) Copy content Toggle raw display
\( T_{11}^{4} - 5T_{11}^{3} - 11T_{11}^{2} + 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{8} + 75T_{17}^{6} + 1411T_{17}^{4} + 8285T_{17}^{2} + 11236 \) Copy content Toggle raw display
\( T_{29}^{4} - 5T_{29}^{3} - 34T_{29}^{2} - 46T_{29} - 16 \) 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
$5$ \( T^{8} - 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 43 T^{6} + \cdots + 2500 \) Copy content Toggle raw display
$11$ \( (T^{4} - 5 T^{3} - 11 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 26 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 75 T^{6} + \cdots + 11236 \) Copy content Toggle raw display
$19$ \( (T^{4} - 12 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 81 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} - 34 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 19 T^{3} + \cdots - 3328)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 13 T^{3} + \cdots + 172)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 25 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{8} + 208 T^{6} + \cdots + 719104 \) Copy content Toggle raw display
$53$ \( T^{8} + 335 T^{6} + \cdots + 1747684 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 568 T^{6} + \cdots + 11505664 \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} - 250 T^{2} + \cdots - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 278 T^{6} + \cdots + 11075584 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 518 T^{6} + \cdots + 163840000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 32 T^{3} + \cdots + 1616)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 337 T^{6} + \cdots + 839056 \) Copy content Toggle raw display
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