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Properties

Label 756.4.f.d
Level $756$
Weight $4$
Character orbit 756.f
Analytic conductor $44.605$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [756,4,Mod(377,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.377"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.704310349824.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 23x^{6} + 50x^{5} + 501x^{4} - 908x^{3} - 644x^{2} - 6448x + 15376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - 2 \beta_{5} + \beta_{4} + \cdots - 10) q^{7} + \beta_{6} q^{11} + ( - 7 \beta_{5} - 3 \beta_{4}) q^{13} + ( - \beta_{3} + 2 \beta_{2}) q^{17} + ( - 5 \beta_{5} - 4 \beta_{4}) q^{19}+ \cdots + ( - 136 \beta_{5} + 32 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 80 q^{7} + 272 q^{25} - 1288 q^{37} - 1072 q^{43} + 8 q^{49} - 4304 q^{67} - 368 q^{79} - 2688 q^{85} - 2448 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 23x^{6} + 50x^{5} + 501x^{4} - 908x^{3} - 644x^{2} - 6448x + 15376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1539 \nu^{7} + 11994 \nu^{6} + 34257 \nu^{5} + 5358 \nu^{4} - 416019 \nu^{3} - 440952 \nu^{2} + \cdots + 78525288 ) / 8844304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 349379 \nu^{7} + 797462 \nu^{6} + 6560241 \nu^{5} - 22567334 \nu^{4} - 179371811 \nu^{3} + \cdots + 978034872 ) / 548346848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 165201 \nu^{7} - 310314 \nu^{6} - 5502267 \nu^{5} + 7812906 \nu^{4} + 112071489 \nu^{3} + \cdots - 773303928 ) / 19583816 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 654513 \nu^{7} - 200085 \nu^{6} + 14446137 \nu^{5} + 866043 \nu^{4} - 322657071 \nu^{3} + \cdots + 4256099430 ) / 34271678 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6652603 \nu^{7} + 1909842 \nu^{6} - 147192905 \nu^{5} - 3144242 \nu^{4} + 3283972491 \nu^{3} + \cdots - 43301993128 ) / 274173424 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2922399 \nu^{7} - 1164126 \nu^{6} + 65050437 \nu^{5} + 10174278 \nu^{4} - 1465867311 \nu^{3} + \cdots + 19344568872 ) / 17688608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2627235 \nu^{7} - 1070502 \nu^{6} + 58480305 \nu^{5} + 9146670 \nu^{4} - 1307876979 \nu^{3} + \cdots + 17266220520 ) / 8844304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 9\beta_{5} - 3\beta_{4} + 18\beta_{2} + 3\beta _1 + 18 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 2\beta_{6} + 126\beta_{5} + 192\beta_{4} - 3\beta_{3} - 42\beta _1 + 450 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{7} - 18\beta_{6} + 19\beta _1 - 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{7} + 50\beta_{6} + 3276\beta_{5} + 3978\beta_{4} - 33\beta_{3} - 252\beta_{2} + 1092\beta _1 - 8658 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 216\beta_{7} - 402\beta_{6} - 1167\beta_{5} - 1367\beta_{4} - 186\beta_{3} - 2502\beta_{2} + 389\beta _1 - 2934 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -847\beta_{7} + 1498\beta_{6} + 23646\beta _1 - 194634 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4862 \beta_{7} + 8754 \beta_{6} - 33933 \beta_{5} - 44241 \beta_{4} - 3892 \beta_{3} - 55434 \beta_{2} + \cdots + 98790 ) / 24 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
3.99910 + 2.54803i
3.99910 − 2.54803i
2.34803 + 0.0382094i
2.34803 − 0.0382094i
−1.14093 + 2.05256i
−1.14093 − 2.05256i
−4.20621 − 2.18931i
−4.20621 + 2.18931i
0 0 0 −16.4106 0 −18.4853 − 1.13770i 0 0 0
377.2 0 0 0 −16.4106 0 −18.4853 + 1.13770i 0 0 0
377.3 0 0 0 −6.97792 0 −1.51472 − 18.4582i 0 0 0
377.4 0 0 0 −6.97792 0 −1.51472 + 18.4582i 0 0 0
377.5 0 0 0 6.97792 0 −1.51472 − 18.4582i 0 0 0
377.6 0 0 0 6.97792 0 −1.51472 + 18.4582i 0 0 0
377.7 0 0 0 16.4106 0 −18.4853 − 1.13770i 0 0 0
377.8 0 0 0 16.4106 0 −18.4853 + 1.13770i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.4.f.d ✓ 8
3.b odd 2 1 inner 756.4.f.d ✓ 8
7.b odd 2 1 inner 756.4.f.d ✓ 8
21.c even 2 1 inner 756.4.f.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.4.f.d ✓ 8 1.a even 1 1 trivial
756.4.f.d ✓ 8 3.b odd 2 1 inner
756.4.f.d ✓ 8 7.b odd 2 1 inner
756.4.f.d ✓ 8 21.c even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} - 318T_{5}^{2} + 13113 \) Copy content Toggle raw display
\( T_{13}^{4} + 2376T_{13}^{2} + 1354896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 318 T^{2} + 13113)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 40 T^{3} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4842 T^{2} + 5782833)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2376 T^{2} + 1354896)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 13296 T^{2} + 839232)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1494 T^{2} + 205209)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28170 T^{2} + 198386577)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 18936 T^{2} + 1888272)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 101574 T^{2} + 493772841)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 322 T - 15551)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 91902 T^{2} + 1095210873)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 268 T - 42596)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 125544 T^{2} + 3937886352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 702432 T^{2} + 119912825088)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 891624 T^{2} + 154093275792)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 645408 T^{2} + 25995757824)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1076 T + 285916)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 289098 T^{2} + 2215061073)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 73800 T^{2} + 479610000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 92 T - 703556)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 362280 T^{2} + 28258829712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 1915751196297)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1210752 T^{2} + 79767834624)^{2} \) Copy content Toggle raw display
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