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Properties

Label 3136.2.e.d
Level $3136$
Weight $2$
Character orbit 3136.e
Analytic conductor $25.041$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,2,Mod(1567,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.e (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: \(25.0410860739\)
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: \( x^{12} - 4x^{9} - 16x^{8} + 8x^{7} + 8x^{6} + 32x^{5} + 240x^{4} + 120x^{3} + 32x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 448)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{9} + \beta_{7} q^{11} + (\beta_{2} - 1) q^{13} + (2 \beta_{10} + \beta_{8} + \beta_{6}) q^{15} + (\beta_{5} + \beta_{3}) q^{17} + (2 \beta_{10} + \beta_{8} - \beta_{6}) q^{19} - 3 \beta_{10} q^{23} + ( - \beta_{2} - 3 \beta_1 + 1) q^{25} + ( - \beta_{8} - 2 \beta_{6}) q^{27} + ( - 2 \beta_{5} + \beta_{3}) q^{29} + (\beta_{7} - \beta_{4}) q^{31} + ( - \beta_{11} - 2 \beta_{5} + \beta_{3}) q^{33} - \beta_{5} q^{37} + ( - \beta_{10} + \beta_{8} + 2 \beta_{6}) q^{39} + ( - 2 \beta_{5} + \beta_{3}) q^{41} + 2 \beta_{4} q^{43} + (\beta_1 + 2) q^{45} + ( - 2 \beta_{9} + \beta_{7} + \beta_{4}) q^{47} + ( - 2 \beta_{9} - \beta_{7} + \beta_{4}) q^{51} + ( - \beta_{11} - 2 \beta_{5} + \beta_{3}) q^{53} + ( - \beta_{9} + 2 \beta_{4}) q^{55} + (2 \beta_{2} + 7) q^{57} + (\beta_{10} - 2 \beta_{8} + 2 \beta_{6}) q^{59} + (2 \beta_{2} - 5) q^{61} - 3 \beta_1 q^{65} + ( - \beta_{9} - 2 \beta_{7} + \beta_{4}) q^{67} + ( - 3 \beta_{2} - 12) q^{69} + 3 \beta_{6} q^{71} + ( - \beta_{11} - \beta_{3}) q^{73} + ( - 2 \beta_{10} - 4 \beta_{8} - 5 \beta_{6}) q^{75} + ( - \beta_{10} - 2 \beta_{8}) q^{79} + (3 \beta_1 + 1) q^{81} + (2 \beta_{10} + 2 \beta_{8} - 5 \beta_{6}) q^{83} + (\beta_{11} - 6 \beta_{5} - 3 \beta_{3}) q^{85} + (\beta_{9} - \beta_{7} + \beta_{4}) q^{87} - 3 \beta_{5} q^{89} + ( - 2 \beta_{11} - 3 \beta_{5} + 2 \beta_{3}) q^{93} + ( - 3 \beta_{10} - 3 \beta_{6}) q^{95} + ( - \beta_{11} - 5 \beta_{5} - 2 \beta_{3}) q^{97} + (3 \beta_{9} + \beta_{7} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 8 q^{9} - 16 q^{13} + 16 q^{25} + 24 q^{45} + 76 q^{57} - 68 q^{61} - 132 q^{69} + 12 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} - 16x^{8} + 8x^{7} + 8x^{6} + 32x^{5} + 240x^{4} + 120x^{3} + 32x^{2} + 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 76 \nu^{11} - 38 \nu^{10} + 241 \nu^{9} - 480 \nu^{8} - 976 \nu^{7} + 430 \nu^{6} + 504 \nu^{5} + \cdots + 29024 ) / 55944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 80 \nu^{11} + 40 \nu^{10} - 131 \nu^{9} + 996 \nu^{8} + 782 \nu^{7} - 698 \nu^{6} - 1512 \nu^{5} + \cdots + 78392 ) / 18648 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2612 \nu^{11} - 714 \nu^{10} + 283 \nu^{9} - 10608 \nu^{8} - 38264 \nu^{7} + 30778 \nu^{6} + \cdots + 30368 ) / 18648 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65 \nu^{11} + 8 \nu^{9} + 258 \nu^{8} + 1040 \nu^{7} - 544 \nu^{6} - 642 \nu^{5} - 1984 \nu^{4} + \cdots - 536 ) / 252 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 239 \nu^{11} + 64 \nu^{10} - 32 \nu^{9} + 972 \nu^{8} + 3560 \nu^{7} - 2804 \nu^{6} - 954 \nu^{5} + \cdots - 2800 ) / 888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1940 \nu^{11} - 970 \nu^{10} + 263 \nu^{9} - 7836 \nu^{8} - 27122 \nu^{7} + 29747 \nu^{6} + \cdots + 15076 ) / 6993 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5611 \nu^{11} + 210 \nu^{10} - 919 \nu^{9} - 22188 \nu^{8} - 91225 \nu^{7} + 45860 \nu^{6} + \cdots + 46516 ) / 18648 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18316 \nu^{11} - 9158 \nu^{10} + 2137 \nu^{9} - 73722 \nu^{8} - 256195 \nu^{7} + 285448 \nu^{6} + \cdots + 141644 ) / 55944 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2293 \nu^{11} - 56 \nu^{10} + 324 \nu^{9} + 9084 \nu^{8} + 36919 \nu^{7} - 18840 \nu^{6} + \cdots - 18936 ) / 6216 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7424 \nu^{11} + 3712 \nu^{10} - 968 \nu^{9} + 29958 \nu^{8} + 103805 \nu^{7} - 115124 \nu^{6} + \cdots - 57616 ) / 18648 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7045 \nu^{11} + 1876 \nu^{10} - 975 \nu^{9} + 28464 \nu^{8} + 105370 \nu^{7} - 82590 \nu^{6} + \cdots - 82704 ) / 6216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} + 2\beta_{9} - 3\beta_{6} - 3\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{10} - 4\beta_{8} - \beta_{6} - 4\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 10\beta_{5} + 3\beta_{3} - 2\beta_{2} - 3\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} - 18 \beta_{10} + 18 \beta_{9} - 16 \beta_{8} + 16 \beta_{7} - 7 \beta_{6} + 13 \beta_{5} + \cdots - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{10} - 8\beta_{8} - 25\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4 \beta_{11} - 41 \beta_{10} - 41 \beta_{9} - 33 \beta_{8} - 33 \beta_{7} - 20 \beta_{6} + 36 \beta_{5} + \cdots + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -33\beta_{11} + 181\beta_{5} + 81\beta_{3} + 33\beta_{2} + 81\beta _1 - 181 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -188\beta_{10} - 140\beta_{8} - 105\beta_{6} + 24\beta_{2} + 164\beta _1 - 186 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -269\beta_{10} - 269\beta_{9} - 129\beta_{8} - 129\beta_{7} - 234\beta_{6} + 234\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 129 \beta_{11} + 866 \beta_{10} - 866 \beta_{9} + 608 \beta_{8} - 608 \beta_{7} + 527 \beta_{6} + \cdots - 925 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.0946732 + 0.353325i
−0.353325 − 0.0946732i
−0.558440 + 2.08413i
2.08413 − 0.558440i
−1.73080 + 0.463767i
0.463767 − 1.73080i
−1.73080 − 0.463767i
0.463767 + 1.73080i
−0.558440 − 2.08413i
2.08413 + 0.558440i
0.0946732 − 0.353325i
−0.353325 + 0.0946732i
0 − 2.86620i 0 −0.482696 0 0 0 −5.21509 0
1567.2 0 − 2.86620i 0 −0.482696 0 0 0 −5.21509 0
1567.3 0 − 1.65544i 0 −4.05137 0 0 0 0.259511 0
1567.4 0 − 1.65544i 0 −4.05137 0 0 0 0.259511 0
1567.5 0 − 0.210756i 0 1.53407 0 0 0 2.95558 0
1567.6 0 − 0.210756i 0 1.53407 0 0 0 2.95558 0
1567.7 0 0.210756i 0 1.53407 0 0 0 2.95558 0
1567.8 0 0.210756i 0 1.53407 0 0 0 2.95558 0
1567.9 0 1.65544i 0 −4.05137 0 0 0 0.259511 0
1567.10 0 1.65544i 0 −4.05137 0 0 0 0.259511 0
1567.11 0 2.86620i 0 −0.482696 0 0 0 −5.21509 0
1567.12 0 2.86620i 0 −0.482696 0 0 0 −5.21509 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.e.d 12
4.b odd 2 1 inner 3136.2.e.d 12
7.b odd 2 1 3136.2.e.e 12
7.c even 3 1 448.2.q.c yes 12
7.d odd 6 1 448.2.q.b ✓ 12
8.b even 2 1 3136.2.e.e 12
8.d odd 2 1 3136.2.e.e 12
28.d even 2 1 3136.2.e.e 12
28.f even 6 1 448.2.q.b ✓ 12
28.g odd 6 1 448.2.q.c yes 12
56.e even 2 1 inner 3136.2.e.d 12
56.h odd 2 1 inner 3136.2.e.d 12
56.j odd 6 1 448.2.q.c yes 12
56.k odd 6 1 448.2.q.b ✓ 12
56.m even 6 1 448.2.q.c yes 12
56.p even 6 1 448.2.q.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.q.b ✓ 12 7.d odd 6 1
448.2.q.b ✓ 12 28.f even 6 1
448.2.q.b ✓ 12 56.k odd 6 1
448.2.q.b ✓ 12 56.p even 6 1
448.2.q.c yes 12 7.c even 3 1
448.2.q.c yes 12 28.g odd 6 1
448.2.q.c yes 12 56.j odd 6 1
448.2.q.c yes 12 56.m even 6 1
3136.2.e.d 12 1.a even 1 1 trivial
3136.2.e.d 12 4.b odd 2 1 inner
3136.2.e.d 12 56.e even 2 1 inner
3136.2.e.d 12 56.h odd 2 1 inner
3136.2.e.e 12 7.b odd 2 1
3136.2.e.e 12 8.b even 2 1
3136.2.e.e 12 8.d odd 2 1
3136.2.e.e 12 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{6} + 11T_{3}^{4} + 23T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 3T_{5}^{2} - 5T_{5} - 3 \) Copy content Toggle raw display
\( T_{11}^{6} - 45T_{11}^{4} + 603T_{11}^{2} - 2187 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 11 T^{4} + 23 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{3} + 3 T^{2} - 5 T - 3)^{4} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 45 T^{4} + \cdots - 2187)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 4 T^{2} - 12 T - 36)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 57 T^{4} + \cdots + 243)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 55 T^{4} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 99 T^{4} + \cdots + 729)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 84 T^{4} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 93 T^{4} + \cdots - 3267)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$41$ \( (T^{6} + 84 T^{4} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 48)^{6} \) Copy content Toggle raw display
$47$ \( (T^{6} - 237 T^{4} + \cdots - 408483)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 153 T^{4} + \cdots + 2187)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 115 T^{4} + \cdots + 9801)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 17 T^{2} + \cdots - 333)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 201 T^{4} + \cdots - 177147)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$73$ \( (T^{6} + 177 T^{4} + \cdots + 49923)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 51 T^{4} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 364 T^{4} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} + 540 T^{4} + \cdots + 2834352)^{2} \) Copy content Toggle raw display
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