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Properties

Label 304.4.i.e
Level $304$
Weight $4$
Character orbit 304.i
Analytic conductor $17.937$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [304,4,Mod(49,304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("304.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.9365806417\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 38)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{4} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{2} - 9) q^{7} + ( - 2 \beta_{5} + 20 \beta_{4} + \cdots - 20) q^{9} + ( - 3 \beta_{3} + \beta_{2} - 2) q^{11}+ \cdots + ( - 20 \beta_{5} - 1060 \beta_{4} + \cdots + 1060) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{3} - q^{5} - 52 q^{7} - 54 q^{9} - 8 q^{11} + 129 q^{13} + 77 q^{15} - 51 q^{17} - 40 q^{19} - 170 q^{21} - 47 q^{23} - 338 q^{25} - 718 q^{27} - 125 q^{29} + 100 q^{31} + 274 q^{33} + 84 q^{35}+ \cdots + 3184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 64x^{4} + 33x^{3} + 3984x^{2} - 945x + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 64\nu^{4} + 4096\nu^{3} - 3984\nu^{2} + 945\nu - 60480 ) / 254031 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{5} - 1344\nu^{4} + 1339\nu^{3} - 83664\nu^{2} + 19845\nu - 3641036 ) / 169354 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1344\nu^{5} - 1339\nu^{4} + 85696\nu^{3} + 64832\nu^{2} + 5334576\nu + 4800 ) / 1270155 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57792\nu^{5} - 57577\nu^{4} + 3684928\nu^{3} + 1517621\nu^{2} + 229386768\nu - 54410265 ) / 2540310 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{5} + 43\beta_{4} - 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 63\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 128\beta_{5} - 2737\beta_{4} - 128\beta_{3} - 48\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 224\beta_{5} - 3856\beta_{4} - 4017\beta_{2} - 4017\beta _1 + 3856 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
4.16954 − 7.22186i
0.118706 − 0.205606i
−3.78825 + 6.56144i
4.16954 + 7.22186i
0.118706 + 0.205606i
−3.78825 − 6.56144i
0 −3.16954 + 5.48981i 0 −2.96554 + 5.13646i 0 −0.660916 0 −6.59199 − 11.4177i 0
49.2 0 0.881294 − 1.52645i 0 10.3546 − 17.9347i 0 −8.76259 0 11.9466 + 20.6922i 0
49.3 0 4.78825 − 8.29349i 0 −7.88908 + 13.6643i 0 −16.5765 0 −32.3546 − 56.0399i 0
273.1 0 −3.16954 − 5.48981i 0 −2.96554 − 5.13646i 0 −0.660916 0 −6.59199 + 11.4177i 0
273.2 0 0.881294 + 1.52645i 0 10.3546 + 17.9347i 0 −8.76259 0 11.9466 − 20.6922i 0
273.3 0 4.78825 + 8.29349i 0 −7.88908 − 13.6643i 0 −16.5765 0 −32.3546 + 56.0399i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.4.i.e 6
4.b odd 2 1 38.4.c.c ✓ 6
12.b even 2 1 342.4.g.f 6
19.c even 3 1 inner 304.4.i.e 6
76.f even 6 1 722.4.a.k 3
76.g odd 6 1 38.4.c.c ✓ 6
76.g odd 6 1 722.4.a.j 3
228.m even 6 1 342.4.g.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.c ✓ 6 4.b odd 2 1
38.4.c.c ✓ 6 76.g odd 6 1
304.4.i.e 6 1.a even 1 1 trivial
304.4.i.e 6 19.c even 3 1 inner
342.4.g.f 6 12.b even 2 1
342.4.g.f 6 228.m even 6 1
722.4.a.j 3 76.g odd 6 1
722.4.a.k 3 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 5T_{3}^{5} + 80T_{3}^{4} + 61T_{3}^{3} + 3560T_{3}^{2} - 5885T_{3} + 11449 \) acting on \(S_{4}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 5 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + \cdots + 3755844 \) Copy content Toggle raw display
$7$ \( (T^{3} + 26 T^{2} + \cdots + 96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 4 T^{2} + \cdots - 49980)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 129 T^{5} + \cdots + 129322384 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 11293737984 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 4555440036 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 5937750562500 \) Copy content Toggle raw display
$31$ \( (T^{3} - 50 T^{2} + \cdots - 3809848)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 188 T^{2} + \cdots - 88004)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 81183541856481 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 259289089231936 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 25671752892900 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 10857156800400 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 356206057990084 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 964239549849 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 713681971209 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{3} + 588 T^{2} + \cdots + 162474984)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 68\!\cdots\!25 \) Copy content Toggle raw display
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