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Properties

Label 756.4.t.c
Level $756$
Weight $4$
Character orbit 756.t
Analytic conductor $44.605$
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] = [756,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.269");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 43x^{10} + 1500x^{8} - 13455x^{6} + 88433x^{4} - 270824x^{2} + 602176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{9}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{2}) q^{5} + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{2}) q^{5} + ( - 3 \beta_{5} + 2 \beta_{4} + \cdots - 8) q^{7}+ \cdots + ( - 76 \beta_{7} - 48 \beta_{5} + \cdots - 51) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 42 q^{7} + 186 q^{19} - 246 q^{25} + 738 q^{31} - 336 q^{37} + 756 q^{43} - 1218 q^{49} - 426 q^{61} - 1440 q^{67} + 2730 q^{73} + 1992 q^{79} + 3288 q^{85} + 1764 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 43x^{10} + 1500x^{8} - 13455x^{6} + 88433x^{4} - 270824x^{2} + 602176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 130875 \nu^{10} + 5268919 \nu^{8} - 183799500 \nu^{6} + 1324423125 \nu^{4} + \cdots + 9710088000 ) / 23474789192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 56860921 \nu^{11} + 224803741 \nu^{9} - 18214572236 \nu^{7} + 2749526254601 \nu^{5} + \cdots + 67054106448240 \nu ) / 2863924281424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 80805667 \nu^{10} - 4627531749 \nu^{8} + 162517376788 \nu^{6} - 2480206132813 \nu^{4} + \cdots - 43063637414912 ) / 1431962140712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129683213 \nu^{10} - 4081323949 \nu^{8} + 133091035052 \nu^{6} + 350109932413 \nu^{4} + \cdots + 27446708682752 ) / 1431962140712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 177933011 \nu^{10} + 8415289741 \nu^{8} - 292464768212 \nu^{6} + 3232399086893 \nu^{4} + \cdots + 39083085543408 ) / 1431962140712 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 308483869 \nu^{11} + 12495222335 \nu^{9} - 443522803516 \nu^{7} + \cdots - 42963827370696 \nu ) / 2863924281424 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 326887763 \nu^{10} + 13611385741 \nu^{8} - 465535051052 \nu^{6} + 3538736658605 \nu^{4} + \cdots + 32575836776704 ) / 1431962140712 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 411991405 \nu^{11} + 16105950335 \nu^{9} - 558013999492 \nu^{7} + \cdots + 162269842530768 \nu ) / 2863924281424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 561681695 \nu^{11} + 21801871741 \nu^{9} - 739785247028 \nu^{7} + \cdots + 6036778012584 \nu ) / 2863924281424 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 161302839 \nu^{11} - 7807026981 \nu^{9} + 272338150500 \nu^{7} - 3179461852353 \nu^{5} + \cdots - 49170471514008 \nu ) / 715981070356 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 989043 \nu^{11} - 34501500 \nu^{9} + 1152887832 \nu^{7} - 2034047433 \nu^{5} + \cdots + 118386512364 \nu ) / 3690624074 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9\beta_{11} + 11\beta_{9} + \beta_{8} - \beta_{6} - 11\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} + 5\beta_{5} + 7\beta_{4} + 12\beta_{3} - 81\beta _1 + 10 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 282\beta_{11} + 282\beta_{10} + 722\beta_{9} + 13\beta_{8} + 26\beta_{6} + 361\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 454\beta_{7} - 275\beta_{5} + 454\beta_{4} + 275\beta_{3} - 2302\beta _1 - 2302 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9333\beta_{10} + 11764\beta_{9} + 1328\beta_{8} + 664\beta_{6} + 23528\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4691\beta_{7} - 7667\beta_{5} + 2976\beta_{4} - 2976\beta_{3} - 2976\beta _1 - 42766 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -309885\beta_{11} - 388399\beta_{9} + 23239\beta_{8} - 23239\beta_{6} + 388399\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -197345\beta_{7} - 197345\beta_{5} - 312883\beta_{4} - 510228\beta_{3} + 2231997\beta _1 - 394690 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10286670 \beta_{11} - 10286670 \beta_{10} - 25751314 \beta_{9} - 777629 \beta_{8} + \cdots - 12875657 \beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -16940542\beta_{7} + 10393051\beta_{5} - 16940542\beta_{4} - 10393051\beta_{3} + 80551886\beta _1 + 80551886 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -113806451\beta_{10} - 142410288\beta_{9} - 17228600\beta_{8} - 8614300\beta_{6} - 284820576\beta_{2} ) / 42 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
2.07339 − 1.19707i
−1.74910 + 1.00984i
4.98916 − 2.88049i
−4.98916 + 2.88049i
1.74910 − 1.00984i
−2.07339 + 1.19707i
2.07339 + 1.19707i
−1.74910 − 1.00984i
4.98916 + 2.88049i
−4.98916 − 2.88049i
1.74910 + 1.00984i
−2.07339 − 1.19707i
0 0 0 −7.05863 − 12.2259i 0 −2.66851 + 18.3270i 0 0 0
269.2 0 0 0 −6.90269 − 11.9558i 0 8.82127 − 16.2845i 0 0 0
269.3 0 0 0 −5.19891 − 9.00477i 0 −16.6528 − 8.10467i 0 0 0
269.4 0 0 0 5.19891 + 9.00477i 0 −16.6528 − 8.10467i 0 0 0
269.5 0 0 0 6.90269 + 11.9558i 0 8.82127 − 16.2845i 0 0 0
269.6 0 0 0 7.05863 + 12.2259i 0 −2.66851 + 18.3270i 0 0 0
593.1 0 0 0 −7.05863 + 12.2259i 0 −2.66851 − 18.3270i 0 0 0
593.2 0 0 0 −6.90269 + 11.9558i 0 8.82127 + 16.2845i 0 0 0
593.3 0 0 0 −5.19891 + 9.00477i 0 −16.6528 + 8.10467i 0 0 0
593.4 0 0 0 5.19891 − 9.00477i 0 −16.6528 + 8.10467i 0 0 0
593.5 0 0 0 6.90269 − 11.9558i 0 8.82127 + 16.2845i 0 0 0
593.6 0 0 0 7.05863 − 12.2259i 0 −2.66851 − 18.3270i 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 756.4.t.c ✓ 12
3.b odd 2 1 inner 756.4.t.c ✓ 12
7.d odd 6 1 inner 756.4.t.c ✓ 12
21.g even 6 1 inner 756.4.t.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.4.t.c ✓ 12 1.a even 1 1 trivial
756.4.t.c ✓ 12 3.b odd 2 1 inner
756.4.t.c ✓ 12 7.d odd 6 1 inner
756.4.t.c ✓ 12 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{12} + 498 T_{5}^{10} + 167868 T_{5}^{8} + 31694544 T_{5}^{6} + 4376695680 T_{5}^{4} + \cdots + 16864097854464 \) Copy content Toggle raw display
\( T_{19}^{6} - 93T_{19}^{5} - 213T_{19}^{4} + 287928T_{19}^{3} + 9851940T_{19}^{2} + 26637984T_{19} + 24676272 \) 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
$5$ \( T^{12} + \cdots + 16864097854464 \) Copy content Toggle raw display
$7$ \( (T^{6} + 21 T^{5} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{6} + 3930 T^{4} + \cdots + 253147788)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 83\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{6} - 93 T^{5} + \cdots + 24676272)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 3092276095776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 369 T^{5} + \cdots + 1449493083)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 369044539723024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 179914563273312)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 189 T^{2} + \cdots + 1242677)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 66\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 169348444633107)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 60\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 90\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 367064062295056)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 41\!\cdots\!07)^{2} \) Copy content Toggle raw display
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