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Properties

Label 6416.2.a.k
Level $6416$
Weight $2$
Character orbit 6416.a
Self dual yes
Analytic conductor $51.232$
Analytic rank $0$
Dimension $12$
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] = [6416,2,Mod(1,6416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6416.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6416 = 2^{4} \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6416.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: \(51.2320179369\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 401)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{10} - \beta_{8} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + (\beta_{10} + 3 \beta_{9} + 2 \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{3} - 7 q^{5} + 20 q^{7} + 3 q^{9} + 11 q^{11} - 11 q^{13} + 11 q^{15} + q^{17} + 34 q^{19} - 3 q^{21} + 7 q^{23} + 7 q^{25} + 2 q^{27} - 6 q^{29} + 52 q^{31} + 4 q^{33} - 12 q^{35} + 3 q^{37} + 24 q^{39} - 16 q^{41} + 2 q^{43} - 23 q^{45} + 3 q^{47} + 6 q^{49} + 16 q^{51} + 19 q^{53} + 43 q^{55} + 11 q^{57} + q^{59} - 24 q^{61} + 11 q^{63} + 13 q^{65} - 6 q^{67} + 29 q^{69} + 15 q^{71} - 20 q^{73} - 31 q^{75} + 38 q^{77} + 53 q^{79} - 8 q^{81} - 17 q^{83} + 7 q^{85} + 5 q^{87} - q^{89} + 6 q^{91} + 44 q^{93} - 34 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} - \nu^{10} - 10 \nu^{9} + 10 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} + 6 \nu^{5} + 59 \nu^{4} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{10} - \nu^{9} - 11\nu^{8} + 9\nu^{7} + 38\nu^{6} - 23\nu^{5} - 49\nu^{4} + 16\nu^{3} + 22\nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{11} + \nu^{10} + 11\nu^{9} - 9\nu^{8} - 38\nu^{7} + 23\nu^{6} + 49\nu^{5} - 16\nu^{4} - 23\nu^{3} + 5\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{11} - \nu^{10} - 13 \nu^{9} + 9 \nu^{8} + 61 \nu^{7} - 19 \nu^{6} - 132 \nu^{5} - 11 \nu^{4} + \cdots - 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( - 2 \nu^{11} + 4 \nu^{10} + 21 \nu^{9} - 43 \nu^{8} - 66 \nu^{7} + 151 \nu^{6} + 63 \nu^{5} - 211 \nu^{4} + \cdots - 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2 \nu^{11} - 4 \nu^{10} - 23 \nu^{9} + 43 \nu^{8} + 89 \nu^{7} - 147 \nu^{6} - 145 \nu^{5} + 183 \nu^{4} + \cdots + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{11} - 9 \nu^{10} - 56 \nu^{9} + 94 \nu^{8} + 205 \nu^{7} - 309 \nu^{6} - 302 \nu^{5} + 369 \nu^{4} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} - 5 \nu^{10} + 22 \nu^{9} + 60 \nu^{8} - 153 \nu^{7} - 245 \nu^{6} + 424 \nu^{5} + 431 \nu^{4} + \cdots + 48 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} + 9 \nu^{10} + 58 \nu^{9} - 98 \nu^{8} - 225 \nu^{7} + 347 \nu^{6} + 360 \nu^{5} + \cdots - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{11} - \nu^{10} - 68 \nu^{9} + 2 \nu^{8} + 329 \nu^{7} + 43 \nu^{6} - 706 \nu^{5} - 193 \nu^{4} + \cdots - 54 ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7 \nu^{11} - 7 \nu^{10} - 90 \nu^{9} + 68 \nu^{8} + 411 \nu^{7} - 189 \nu^{6} - 842 \nu^{5} + 111 \nu^{4} + \cdots - 42 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 2\beta_{9} + \beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} - 4 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \cdots - 6 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} - 3 \beta_{10} - 13 \beta_{9} - 4 \beta_{8} + 5 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} + \cdots + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{11} - 11 \beta_{10} - 11 \beta_{9} - 9 \beta_{8} + 6 \beta_{7} - 6 \beta_{6} + 3 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38 \beta_{11} - 29 \beta_{10} - 81 \beta_{9} - 40 \beta_{8} + 23 \beta_{7} - 41 \beta_{6} + 41 \beta_{5} + \cdots + 78 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 53 \beta_{11} - 136 \beta_{10} - 136 \beta_{9} - 130 \beta_{8} + 55 \beta_{7} - 61 \beta_{6} + \cdots - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 239 \beta_{11} - 222 \beta_{10} - 506 \beta_{9} - 304 \beta_{8} + 111 \beta_{7} - 241 \beta_{6} + \cdots + 420 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 348 \beta_{11} - 865 \beta_{10} - 879 \beta_{9} - 880 \beta_{8} + 279 \beta_{7} - 357 \beta_{6} + \cdots - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 748 \beta_{11} - 785 \beta_{10} - 1589 \beta_{9} - 1056 \beta_{8} + 291 \beta_{7} - 725 \beta_{6} + \cdots + 1192 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2306 \beta_{11} - 5529 \beta_{10} - 5771 \beta_{9} - 5808 \beta_{8} + 1535 \beta_{7} - 2245 \beta_{6} + \cdots + 130 ) / 2 \) 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
1.82098
−0.257740
1.53244
−0.960737
2.54250
0.578957
−1.17976
−0.305046
−2.35127
1.44790
1.85869
−1.72691
0 −2.71334 0 −3.63372 0 2.90457 0 4.36224 0
1.2 0 −2.08548 0 −2.90849 0 2.64864 0 1.34921 0
1.3 0 −1.08207 0 1.17495 0 3.29109 0 −1.82913 0
1.4 0 −0.489694 0 −2.30095 0 1.58751 0 −2.76020 0
1.5 0 −0.462633 0 2.03931 0 1.72844 0 −2.78597 0
1.6 0 0.601622 0 −0.0489583 0 −2.79065 0 −2.63805 0
1.7 0 0.735736 0 −1.18930 0 0.108315 0 −2.45869 0
1.8 0 0.792698 0 3.43210 0 4.26743 0 −2.37163 0
1.9 0 1.88094 0 −4.09454 0 4.62990 0 0.537944 0
1.10 0 2.35797 0 −1.54450 0 −2.00193 0 2.56004 0
1.11 0 2.50269 0 0.985636 0 2.11223 0 3.26346 0
1.12 0 2.96155 0 1.08846 0 1.51444 0 5.77079 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(401\) \(1\)

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 6416.2.a.k 12
4.b odd 2 1 401.2.a.a ✓ 12
12.b even 2 1 3609.2.a.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
401.2.a.a ✓ 12 4.b odd 2 1
3609.2.a.b 12 12.b even 2 1
6416.2.a.k 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 5 T_{3}^{11} - 7 T_{3}^{10} + 66 T_{3}^{9} - 33 T_{3}^{8} - 249 T_{3}^{7} + 270 T_{3}^{6} + \cdots - 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6416))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 5 T^{11} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots - 79 \) Copy content Toggle raw display
$7$ \( T^{12} - 20 T^{11} + \cdots + 2657 \) Copy content Toggle raw display
$11$ \( T^{12} - 11 T^{11} + \cdots + 41849 \) Copy content Toggle raw display
$13$ \( T^{12} + 11 T^{11} + \cdots - 272 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + \cdots - 4208 \) Copy content Toggle raw display
$19$ \( T^{12} - 34 T^{11} + \cdots - 201104 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots + 23888 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 522688447 \) Copy content Toggle raw display
$31$ \( T^{12} - 52 T^{11} + \cdots + 69618448 \) Copy content Toggle raw display
$37$ \( T^{12} - 3 T^{11} + \cdots - 74838512 \) Copy content Toggle raw display
$41$ \( T^{12} + 16 T^{11} + \cdots - 24965557 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots - 32571323 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 120470771 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 3460081936 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 49673523056 \) Copy content Toggle raw display
$61$ \( T^{12} + 24 T^{11} + \cdots - 45635056 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 429818624 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2314277776 \) Copy content Toggle raw display
$73$ \( T^{12} + 20 T^{11} + \cdots - 10554352 \) Copy content Toggle raw display
$79$ \( T^{12} - 53 T^{11} + \cdots - 19238224 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 3709520128 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 291613788629 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 9379281712 \) Copy content Toggle raw display
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