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Properties

Label 3006.2.a.w
Level $3006$
Weight $2$
Character orbit 3006.a
Self dual yes
Analytic conductor $24.003$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3006,2,Mod(1,3006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3006 = 2 \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3006.a (trivial)

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: yes
Analytic conductor: \(24.0030308476\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 29x^{8} + 118x^{7} + 283x^{6} - 1228x^{5} - 1029x^{4} + 5258x^{3} + 874x^{2} - 7932x + 1188 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{5} + ( - \beta_{4} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{5} + ( - \beta_{4} + 1) q^{7} + q^{8} + \beta_1 q^{10} + (\beta_{7} + 1) q^{11} + ( - \beta_{8} - \beta_1 + 1) q^{13} + ( - \beta_{4} + 1) q^{14} + q^{16} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_{2}) q^{17}+ \cdots + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + 4 q^{5} + 6 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 10 q^{4} + 4 q^{5} + 6 q^{7} + 10 q^{8} + 4 q^{10} + 8 q^{11} + 6 q^{13} + 6 q^{14} + 10 q^{16} + 12 q^{19} + 4 q^{20} + 8 q^{22} + 16 q^{23} + 24 q^{25} + 6 q^{26} + 6 q^{28} + 4 q^{29} + 10 q^{31} + 10 q^{32} + 10 q^{35} + 20 q^{37} + 12 q^{38} + 4 q^{40} + 8 q^{41} + 22 q^{43} + 8 q^{44} + 16 q^{46} + 2 q^{47} + 36 q^{49} + 24 q^{50} + 6 q^{52} + 4 q^{53} + 12 q^{55} + 6 q^{56} + 4 q^{58} + 8 q^{59} + 24 q^{61} + 10 q^{62} + 10 q^{64} - 36 q^{65} + 24 q^{67} + 10 q^{70} + 16 q^{71} + 12 q^{73} + 20 q^{74} + 12 q^{76} - 8 q^{77} + 56 q^{79} + 4 q^{80} + 8 q^{82} - 8 q^{83} + 16 q^{85} + 22 q^{86} + 8 q^{88} - 6 q^{89} + 20 q^{91} + 16 q^{92} + 2 q^{94} - 4 q^{95} + 14 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 29x^{8} + 118x^{7} + 283x^{6} - 1228x^{5} - 1029x^{4} + 5258x^{3} + 874x^{2} - 7932x + 1188 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 29 \nu^{9} + 37 \nu^{8} + 828 \nu^{7} - 644 \nu^{6} - 7851 \nu^{5} + 3761 \nu^{4} + 29302 \nu^{3} + \cdots + 7902 ) / 450 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7 \nu^{9} - 11 \nu^{8} - 204 \nu^{7} + 202 \nu^{6} + 2073 \nu^{5} - 1183 \nu^{4} - 8816 \nu^{3} + \cdots - 1926 ) / 90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{9} + 33 \nu^{8} + 302 \nu^{7} - 746 \nu^{6} - 2959 \nu^{5} + 5399 \nu^{4} + 12518 \nu^{3} + \cdots + 3018 ) / 150 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17 \nu^{9} - \nu^{8} - 669 \nu^{7} + 137 \nu^{6} + 8673 \nu^{5} - 2378 \nu^{4} - 44746 \nu^{3} + \cdots - 12321 ) / 225 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16 \nu^{9} + 23 \nu^{8} + 512 \nu^{7} - 526 \nu^{6} - 5679 \nu^{5} + 4119 \nu^{4} + 26058 \nu^{3} + \cdots + 6408 ) / 75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 113 \nu^{9} - 139 \nu^{8} - 3666 \nu^{7} + 3068 \nu^{6} + 41397 \nu^{5} - 23867 \nu^{4} - 195094 \nu^{3} + \cdots - 50994 ) / 450 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 127 \nu^{9} + 206 \nu^{8} + 4089 \nu^{7} - 4822 \nu^{6} - 46263 \nu^{5} + 38218 \nu^{4} + \cdots + 61326 ) / 450 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 253 \nu^{9} + 359 \nu^{8} + 8196 \nu^{7} - 8458 \nu^{6} - 91857 \nu^{5} + 68227 \nu^{4} + \cdots + 114264 ) / 450 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 3\beta_{5} - 3\beta_{4} - \beta_{3} - 3\beta_{2} + 11\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + 4 \beta_{8} + 3 \beta_{7} + 15 \beta_{6} + 20 \beta_{5} - 23 \beta_{4} - 2 \beta_{3} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} + 9 \beta_{6} + 72 \beta_{5} - 78 \beta_{4} - 25 \beta_{3} + \cdots + 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 39 \beta_{9} + 84 \beta_{8} + 70 \beta_{7} + 218 \beta_{6} + 369 \beta_{5} - 437 \beta_{4} - 72 \beta_{3} + \cdots + 928 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 372 \beta_{9} + 188 \beta_{8} + 213 \beta_{7} + 306 \beta_{6} + 1429 \beta_{5} - 1615 \beta_{4} + \cdots + 1525 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 964 \beta_{9} + 1498 \beta_{8} + 1352 \beta_{7} + 3317 \beta_{6} + 6694 \beta_{5} - 7969 \beta_{4} + \cdots + 13560 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6440 \beta_{9} + 4308 \beta_{8} + 4746 \beta_{7} + 7339 \beta_{6} + 26982 \beta_{5} - 31174 \beta_{4} + \cdots + 32856 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.29223
−3.27639
−2.06898
−1.95701
0.154783
2.22033
2.50890
2.66297
2.76829
4.27932
1.00000 0 1.00000 −3.29223 0 −3.52778 1.00000 0 −3.29223
1.2 1.00000 0 1.00000 −3.27639 0 5.10437 1.00000 0 −3.27639
1.3 1.00000 0 1.00000 −2.06898 0 −3.76266 1.00000 0 −2.06898
1.4 1.00000 0 1.00000 −1.95701 0 2.55597 1.00000 0 −1.95701
1.5 1.00000 0 1.00000 0.154783 0 3.22855 1.00000 0 0.154783
1.6 1.00000 0 1.00000 2.22033 0 2.42075 1.00000 0 2.22033
1.7 1.00000 0 1.00000 2.50890 0 −3.84795 1.00000 0 2.50890
1.8 1.00000 0 1.00000 2.66297 0 −1.08399 1.00000 0 2.66297
1.9 1.00000 0 1.00000 2.76829 0 1.34352 1.00000 0 2.76829
1.10 1.00000 0 1.00000 4.27932 0 3.56921 1.00000 0 4.27932
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3006.2.a.w yes 10
3.b odd 2 1 3006.2.a.v ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3006.2.a.v ✓ 10 3.b odd 2 1
3006.2.a.w yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3006))\):

\( T_{5}^{10} - 4 T_{5}^{9} - 29 T_{5}^{8} + 118 T_{5}^{7} + 283 T_{5}^{6} - 1228 T_{5}^{5} - 1029 T_{5}^{4} + \cdots + 1188 \) Copy content Toggle raw display
\( T_{7}^{10} - 6 T_{7}^{9} - 35 T_{7}^{8} + 252 T_{7}^{7} + 283 T_{7}^{6} - 3614 T_{7}^{5} + 1751 T_{7}^{4} + \cdots + 27072 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + \cdots + 1188 \) Copy content Toggle raw display
$7$ \( T^{10} - 6 T^{9} + \cdots + 27072 \) Copy content Toggle raw display
$11$ \( T^{10} - 8 T^{9} + \cdots - 20736 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + \cdots + 265792 \) Copy content Toggle raw display
$17$ \( T^{10} - 106 T^{8} + \cdots - 7872 \) Copy content Toggle raw display
$19$ \( T^{10} - 12 T^{9} + \cdots - 256 \) Copy content Toggle raw display
$23$ \( T^{10} - 16 T^{9} + \cdots + 162816 \) Copy content Toggle raw display
$29$ \( T^{10} - 4 T^{9} + \cdots + 69888 \) Copy content Toggle raw display
$31$ \( T^{10} - 10 T^{9} + \cdots - 10968848 \) Copy content Toggle raw display
$37$ \( T^{10} - 20 T^{9} + \cdots + 10528164 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 299388672 \) Copy content Toggle raw display
$43$ \( T^{10} - 22 T^{9} + \cdots - 4486976 \) Copy content Toggle raw display
$47$ \( T^{10} - 2 T^{9} + \cdots + 2305584 \) Copy content Toggle raw display
$53$ \( T^{10} - 4 T^{9} + \cdots - 59949948 \) Copy content Toggle raw display
$59$ \( T^{10} - 8 T^{9} + \cdots - 51437700 \) Copy content Toggle raw display
$61$ \( T^{10} - 24 T^{9} + \cdots + 3244032 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 321086268 \) Copy content Toggle raw display
$71$ \( T^{10} - 16 T^{9} + \cdots + 56650752 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + \cdots + 22150400 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 958390272 \) Copy content Toggle raw display
$83$ \( T^{10} + 8 T^{9} + \cdots + 4422444 \) Copy content Toggle raw display
$89$ \( T^{10} + 6 T^{9} + \cdots + 26293248 \) Copy content Toggle raw display
$97$ \( T^{10} - 14 T^{9} + \cdots + 18967824 \) Copy content Toggle raw display
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