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Properties

Label 756.2.t.e
Level $756$
Weight $2$
Character orbit 756.t
Analytic conductor $6.037$
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] = [756,2,Mod(269,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.t (of order \(6\), degree \(2\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + \beta_{10} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + \beta_{10} q^{7} + ( - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{10} + \beta_{9} + \cdots + \beta_{4}) q^{13}+ \cdots + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{19} - 24 q^{37} - 12 q^{43} + 18 q^{61} + 54 q^{73} + 24 q^{79} - 12 q^{85} + 42 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -99\nu^{11} + 488\nu^{9} - 1519\nu^{7} + 1061\nu^{5} + 1670\nu^{3} - 3551\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} + 464\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -114\nu^{11} + 545\nu^{9} - 2071\nu^{7} + 2831\nu^{5} - 3379\nu^{3} + 1213\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\nu^{10} - 298\nu^{8} + 1356\nu^{6} - 2987\nu^{4} + 4419\nu^{2} - 1811 ) / 559 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -171\nu^{11} + 538\nu^{9} - 1709\nu^{7} - 1064\nu^{5} + 3037\nu^{3} - 7963\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 654 ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 402\nu^{11} - 2422\nu^{9} + 9539\nu^{7} - 17809\nu^{5} + 19712\nu^{3} - 7043\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -187\nu^{10} + 1046\nu^{8} - 3863\nu^{6} + 6414\nu^{4} - 6038\nu^{2} + 1926 ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 259\nu^{10} - 1096\nu^{8} + 4053\nu^{6} - 4289\nu^{4} + 4671\nu^{2} + 809 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -262\nu^{10} + 1331\nu^{8} - 4946\nu^{6} + 6879\nu^{4} - 6128\nu^{2} - 527 ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1449\nu^{11} + 7295\nu^{9} - 27721\nu^{7} + 41294\nu^{5} - 45229\nu^{3} + 9313\nu ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{9} + 5\beta_{6} - \beta_{4} + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{7} + 3\beta_{5} - 5\beta_{3} + 14\beta_{2} - 2\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - 4\beta_{9} + 3\beta_{8} + 13\beta_{6} - 3\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{11} + 5\beta_{7} - 47\beta_{3} + 20\beta_{2} - 5\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -14\beta_{10} - 5\beta_{9} + 14\beta_{8} + 5\beta_{4} - 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19\beta_{11} + 38\beta_{7} - 42\beta_{5} - 122\beta_{3} - 61\beta_{2} + 19\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -28\beta_{10} + 28\beta_{9} + 19\beta_{8} - 117\beta_{6} + 47\beta_{4} - 117 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3\beta_{11} + 22\beta_{7} - 47\beta_{5} + 20\beta_{3} - 128\beta_{2} + 44\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 66\beta_{10} + 155\beta_{9} - 89\beta_{8} - 370\beta_{6} + 89\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -244\beta_{11} - 221\beta_{7} + 1472\beta_{3} - 614\beta_{2} + 221\beta_1 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.56052 + 0.900969i
1.07992 + 0.623490i
0.385418 + 0.222521i
−0.385418 − 0.222521i
−1.07992 − 0.623490i
−1.56052 − 0.900969i
1.56052 − 0.900969i
1.07992 − 0.623490i
0.385418 − 0.222521i
−0.385418 + 0.222521i
−1.07992 + 0.623490i
−1.56052 + 0.900969i
0 0 0 −1.56052 − 2.70291i 0 2.37047 − 1.17511i 0 0 0
269.2 0 0 0 −1.07992 − 1.87047i 0 −0.167563 + 2.64044i 0 0 0
269.3 0 0 0 −0.385418 − 0.667563i 0 −2.20291 − 1.46533i 0 0 0
269.4 0 0 0 0.385418 + 0.667563i 0 −2.20291 − 1.46533i 0 0 0
269.5 0 0 0 1.07992 + 1.87047i 0 −0.167563 + 2.64044i 0 0 0
269.6 0 0 0 1.56052 + 2.70291i 0 2.37047 − 1.17511i 0 0 0
593.1 0 0 0 −1.56052 + 2.70291i 0 2.37047 + 1.17511i 0 0 0
593.2 0 0 0 −1.07992 + 1.87047i 0 −0.167563 − 2.64044i 0 0 0
593.3 0 0 0 −0.385418 + 0.667563i 0 −2.20291 + 1.46533i 0 0 0
593.4 0 0 0 0.385418 − 0.667563i 0 −2.20291 + 1.46533i 0 0 0
593.5 0 0 0 1.07992 − 1.87047i 0 −0.167563 − 2.64044i 0 0 0
593.6 0 0 0 1.56052 − 2.70291i 0 2.37047 + 1.17511i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.t.e ✓ 12
3.b odd 2 1 inner 756.2.t.e ✓ 12
7.c even 3 1 5292.2.f.e 12
7.d odd 6 1 inner 756.2.t.e ✓ 12
7.d odd 6 1 5292.2.f.e 12
9.c even 3 1 2268.2.w.i 12
9.c even 3 1 2268.2.bm.i 12
9.d odd 6 1 2268.2.w.i 12
9.d odd 6 1 2268.2.bm.i 12
21.g even 6 1 inner 756.2.t.e ✓ 12
21.g even 6 1 5292.2.f.e 12
21.h odd 6 1 5292.2.f.e 12
63.i even 6 1 2268.2.bm.i 12
63.k odd 6 1 2268.2.w.i 12
63.s even 6 1 2268.2.w.i 12
63.t odd 6 1 2268.2.bm.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.e ✓ 12 1.a even 1 1 trivial
756.2.t.e ✓ 12 3.b odd 2 1 inner
756.2.t.e ✓ 12 7.d odd 6 1 inner
756.2.t.e ✓ 12 21.g even 6 1 inner
2268.2.w.i 12 9.c even 3 1
2268.2.w.i 12 9.d odd 6 1
2268.2.w.i 12 63.k odd 6 1
2268.2.w.i 12 63.s even 6 1
2268.2.bm.i 12 9.c even 3 1
2268.2.bm.i 12 9.d odd 6 1
2268.2.bm.i 12 63.i even 6 1
2268.2.bm.i 12 63.t odd 6 1
5292.2.f.e 12 7.c even 3 1
5292.2.f.e 12 7.d odd 6 1
5292.2.f.e 12 21.g even 6 1
5292.2.f.e 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{12} + 15T_{5}^{10} + 171T_{5}^{8} + 756T_{5}^{6} + 2511T_{5}^{4} + 1458T_{5}^{2} + 729 \) Copy content Toggle raw display
\( T_{13}^{6} + 42T_{13}^{4} + 441T_{13}^{2} + 1323 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( (T^{6} - 7 T^{3} + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 45 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$13$ \( (T^{6} + 42 T^{4} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 2492305929 \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{5} + 15 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 54 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$29$ \( (T^{6} + 153 T^{4} + \cdots + 123201)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 63 T^{4} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 12 T^{5} + \cdots + 142129)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 186 T^{4} + \cdots - 136107)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 3 T^{2} - 18 T - 13)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15178486401 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 118861526169 \) Copy content Toggle raw display
$61$ \( (T^{6} - 9 T^{5} + \cdots + 22707)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 147 T^{4} + \cdots + 247009)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 189 T^{4} + \cdots + 35721)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 27 T^{5} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots + 142129)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 375 T^{4} + \cdots - 421875)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 27 T^{2} + 729)^{3} \) Copy content Toggle raw display
$97$ \( (T^{6} + 204 T^{4} + \cdots + 1728)^{2} \) Copy content Toggle raw display
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