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Properties

Label 9016.2.a.bs
Level $9016$
Weight $2$
Character orbit 9016.a
Self dual yes
Analytic conductor $71.993$
Analytic rank $1$
Dimension $14$
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] = [9016,2,Mod(1,9016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("9016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9016.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: \(71.9931224624\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 28x^{12} + 293x^{10} - 1394x^{8} + 2848x^{6} - 1722x^{4} + 332x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{7} - \beta_1) q^{5} + (\beta_{8} + \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{7} - \beta_1) q^{5} + (\beta_{8} + \beta_{5} + \beta_{4} + 1) q^{9} + ( - \beta_{4} - 1) q^{11} + (\beta_{7} + \beta_{3}) q^{13} + ( - \beta_{8} - \beta_{4} - 2) q^{15} + \beta_{12} q^{17} + (\beta_{13} - \beta_{3}) q^{19} - q^{23} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \cdots + 1) q^{25}+ \cdots + ( - \beta_{9} - 3 \beta_{8} + \beta_{5} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{9} - 8 q^{11} - 28 q^{15} - 14 q^{23} + 10 q^{25} + 4 q^{29} + 12 q^{37} - 24 q^{39} - 44 q^{43} - 28 q^{51} - 4 q^{53} - 8 q^{57} - 44 q^{65} - 28 q^{67} - 28 q^{71} - 36 q^{79} + 18 q^{81} + 8 q^{85} + 44 q^{93} - 8 q^{95} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 28x^{12} + 293x^{10} - 1394x^{8} + 2848x^{6} - 1722x^{4} + 332x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -49\nu^{12} + 1204\nu^{10} - 10229\nu^{8} + 35044\nu^{6} - 46534\nu^{4} + 36982\nu^{2} + 1996 ) / 12662 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 499\nu^{13} - 14070\nu^{11} + 148615\nu^{9} - 716064\nu^{7} + 1491240\nu^{5} - 952346\nu^{3} + 239632\nu ) / 25324 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{12} + 561\nu^{10} - 5815\nu^{8} + 26588\nu^{6} - 46991\nu^{4} + 9201\nu^{2} + 765 ) / 487 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -695\nu^{12} + 18886\nu^{10} - 189531\nu^{8} + 843578\nu^{6} - 1500108\nu^{4} + 492498\nu^{2} - 29056 ) / 12662 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -933\nu^{13} + 24734\nu^{11} - 235597\nu^{9} + 921540\nu^{7} - 970028\nu^{5} - 1368266\nu^{3} + 472648\nu ) / 25324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 933\nu^{13} - 24734\nu^{11} + 235597\nu^{9} - 921540\nu^{7} + 970028\nu^{5} + 1393590\nu^{3} - 675240\nu ) / 25324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1215\nu^{12} - 33472\nu^{10} + 340721\nu^{8} - 1534866\nu^{6} + 2721874\nu^{4} - 719062\nu^{2} - 41482 ) / 12662 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -1763\nu^{12} + 48746\nu^{10} - 499565\nu^{8} + 2285974\nu^{6} - 4259648\nu^{4} + 1695728\nu^{2} - 94858 ) / 12662 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2409\nu^{12} + 66428\nu^{10} - 678867\nu^{8} + 3094508\nu^{6} - 5700560\nu^{4} + 2037286\nu^{2} - 24614 ) / 12662 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1763 \nu^{13} + 48746 \nu^{11} - 499565 \nu^{9} + 2285974 \nu^{7} - 4259648 \nu^{5} + \cdots - 170830 \nu ) / 12662 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5219 \nu^{13} - 144518 \nu^{11} + 1485891 \nu^{9} - 6847654 \nu^{7} + 12989222 \nu^{5} + \cdots + 558754 \nu ) / 12662 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5739 \nu^{13} + 159104 \nu^{11} - 1637081 \nu^{9} + 7538942 \nu^{7} - 14210988 \nu^{5} + \cdots - 462892 \nu ) / 12662 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{5} + \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + 9\beta_{8} + 8\beta_{5} + 9\beta_{4} + \beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - 2\beta_{11} + 13\beta_{7} + 12\beta_{6} + 7\beta_{3} + 66\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{10} - 13\beta_{9} + 80\beta_{8} + 64\beta_{5} + 79\beta_{4} + 17\beta_{2} + 212 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15\beta_{13} + \beta_{12} - 24\beta_{11} + 144\beta_{7} + 125\beta_{6} + 119\beta_{3} + 561\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 148\beta_{10} - 133\beta_{9} + 709\beta_{8} + 524\beta_{5} + 701\beta_{4} + 218\beta_{2} + 1681 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 177\beta_{13} + 29\beta_{12} - 211\beta_{11} + 1492\beta_{7} + 1237\beta_{6} + 1497\beta_{3} + 4874\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1419\beta_{10} - 1242\beta_{9} + 6293\beta_{8} + 4355\beta_{5} + 6288\beta_{4} + 2465\beta_{2} + 13798 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1933\beta_{13} + 514\beta_{12} - 1615\beta_{11} + 14917\beta_{7} + 11933\beta_{6} + 16720\beta_{3} + 43009\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13034\beta_{10} - 11101\beta_{9} + 56043\beta_{8} + 36550\beta_{5} + 56875\beta_{4} + 26010\beta_{2} + 116207 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20325 \beta_{13} + 7291 \beta_{12} - 11159 \beta_{11} + 145948 \beta_{7} + 113479 \beta_{6} + \cdots + 383680 \beta_1 \) 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
−3.03220
−2.78536
−2.38642
−2.12838
−0.667834
−0.590343
−0.167238
0.167238
0.590343
0.667834
2.12838
2.38642
2.78536
3.03220
0 −3.03220 0 3.31370 0 0 0 6.19425 0
1.2 0 −2.78536 0 0.431379 0 0 0 4.75825 0
1.3 0 −2.38642 0 1.68161 0 0 0 2.69501 0
1.4 0 −2.12838 0 −0.504697 0 0 0 1.53001 0
1.5 0 −0.667834 0 2.41880 0 0 0 −2.55400 0
1.6 0 −0.590343 0 −1.91351 0 0 0 −2.65149 0
1.7 0 −0.167238 0 −4.02972 0 0 0 −2.97203 0
1.8 0 0.167238 0 4.02972 0 0 0 −2.97203 0
1.9 0 0.590343 0 1.91351 0 0 0 −2.65149 0
1.10 0 0.667834 0 −2.41880 0 0 0 −2.55400 0
1.11 0 2.12838 0 0.504697 0 0 0 1.53001 0
1.12 0 2.38642 0 −1.68161 0 0 0 2.69501 0
1.13 0 2.78536 0 −0.431379 0 0 0 4.75825 0
1.14 0 3.03220 0 −3.31370 0 0 0 6.19425 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(23\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9016.2.a.bs ✓ 14
7.b odd 2 1 inner 9016.2.a.bs ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9016.2.a.bs ✓ 14 1.a even 1 1 trivial
9016.2.a.bs ✓ 14 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9016))\):

\( T_{3}^{14} - 28T_{3}^{12} + 293T_{3}^{10} - 1394T_{3}^{8} + 2848T_{3}^{6} - 1722T_{3}^{4} + 332T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{14} - 40T_{5}^{12} + 580T_{5}^{10} - 3826T_{5}^{8} + 11868T_{5}^{6} - 15496T_{5}^{4} + 5248T_{5}^{2} - 512 \) Copy content Toggle raw display
\( T_{11}^{7} + 4T_{11}^{6} - 22T_{11}^{5} - 82T_{11}^{4} + 124T_{11}^{3} + 352T_{11}^{2} - 240T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{14} - 42T_{13}^{12} + 685T_{13}^{10} - 5558T_{13}^{8} + 23932T_{13}^{6} - 54248T_{13}^{4} + 59840T_{13}^{2} - 25088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 28 T^{12} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{14} - 40 T^{12} + \cdots - 512 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( (T^{7} + 4 T^{6} - 22 T^{5} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 42 T^{12} + \cdots - 25088 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 768320000 \) Copy content Toggle raw display
$19$ \( T^{14} - 108 T^{12} + \cdots - 3276800 \) Copy content Toggle raw display
$23$ \( (T + 1)^{14} \) Copy content Toggle raw display
$29$ \( (T^{7} - 2 T^{6} + \cdots + 2528)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 38765530568 \) Copy content Toggle raw display
$37$ \( (T^{7} - 6 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 1507224608 \) Copy content Toggle raw display
$43$ \( (T^{7} + 22 T^{6} + \cdots + 62720)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 134367206408 \) Copy content Toggle raw display
$53$ \( (T^{7} + 2 T^{6} + \cdots - 10240)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 2206073888 \) Copy content Toggle raw display
$61$ \( T^{14} - 348 T^{12} + \cdots - 19971200 \) Copy content Toggle raw display
$67$ \( (T^{7} + 14 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + 14 T^{6} + \cdots - 176992)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 2933780000 \) Copy content Toggle raw display
$79$ \( (T^{7} + 18 T^{6} + \cdots - 166976)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 7867596800 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 250382845952 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 3044496512 \) Copy content Toggle raw display
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