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Properties

Label 448.3.r.f
Level $448$
Weight $3$
Character orbit 448.r
Analytic conductor $12.207$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,3,Mod(191,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.191"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.752609431977984.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{9} + \beta_{7}) q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_1) q^{5} + (\beta_{9} + 2 \beta_{8}) q^{7} + (\beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \cdots - 1) q^{9} + ( - 3 \beta_{10} + 4 \beta_{9} + \cdots + 4 \beta_{7}) q^{11}+ \cdots + ( - 11 \beta_{11} + \cdots + 40 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5} - 6 q^{17} - 18 q^{21} + 186 q^{33} + 114 q^{37} + 180 q^{49} + 18 q^{53} - 684 q^{57} - 318 q^{61} + 228 q^{69} + 342 q^{73} - 318 q^{77} - 186 q^{81} + 996 q^{85} + 150 q^{89} + 222 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{10} + 83\nu^{8} - 906\nu^{6} + 3874\nu^{4} - 5950\nu^{2} - 1505 ) / 2947 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{10} - 338\nu^{8} + 3304\nu^{6} - 10019\nu^{4} - 898\nu^{2} + 16177 ) / 2947 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\nu^{10} - 895\nu^{8} + 5032\nu^{6} - 8946\nu^{4} - 1912\nu^{2} - 2128 ) / 2947 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 58\nu^{10} - 723\nu^{8} + 3540\nu^{6} - 6675\nu^{4} - 902\nu^{2} + 14175 ) / 2947 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -89\nu^{10} + 1378\nu^{8} - 8321\nu^{6} + 23780\nu^{4} - 30699\nu^{2} + 8176 ) / 2947 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80\nu^{11} - 794\nu^{9} - 808\nu^{7} + 34490\nu^{5} - 122376\nu^{3} + 113246\nu ) / 20629 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 488\nu^{11} - 7201\nu^{9} + 43402\nu^{7} - 131463\nu^{5} + 222480\nu^{3} - 142611\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -97\nu^{11} + 1710\nu^{9} - 11945\nu^{7} + 39276\nu^{5} - 54499\nu^{3} + 8050\nu ) / 5894 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -151\nu^{11} + 2267\nu^{9} - 13673\nu^{7} + 38203\nu^{5} - 41697\nu^{3} - 14903\nu ) / 5894 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1137\nu^{11} + 16663\nu^{9} - 94903\nu^{7} + 232931\nu^{5} - 169503\nu^{3} - 135051\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1002\nu^{11} + 14218\nu^{9} - 80058\nu^{7} + 197934\nu^{5} - 177090\nu^{3} - 90930\nu ) / 20629 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{10} - 2\beta_{9} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 4\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} - \beta_{4} + 4\beta_{3} - 3\beta_{2} - 11\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -21\beta_{11} + 34\beta_{10} + 16\beta_{9} - 18\beta_{8} - 2\beta_{7} + 14\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 34\beta_{5} + 7\beta_{4} + 42\beta_{3} - 33\beta_{2} - 185\beta _1 - 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -76\beta_{11} + 69\beta_{10} + 85\beta_{9} - 54\beta_{8} - 44\beta_{7} + 5\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52\beta_{5} + 31\beta_{4} + 42\beta_{3} - 43\beta_{2} - 280\beta _1 - 170 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -893\beta_{11} + 480\beta_{10} + 1050\beta_{9} - 426\beta_{8} - 762\beta_{7} - 321\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 978\beta_{5} + 1076\beta_{4} + 467\beta_{3} - 836\beta_{2} - 5233\beta _1 - 5635 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -2202\beta_{11} + 453\beta_{10} + 2506\beta_{9} - 309\beta_{8} - 2265\beta_{7} - 1693\beta_{6} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.75780 + 0.500000i
0.385124 − 0.500000i
2.23871 − 0.500000i
−2.23871 + 0.500000i
−0.385124 + 0.500000i
−1.75780 − 0.500000i
1.75780 − 0.500000i
0.385124 + 0.500000i
2.23871 + 0.500000i
−2.23871 − 0.500000i
−0.385124 − 0.500000i
−1.75780 + 0.500000i
0 −4.23009 + 2.44224i 0 −1.16012 + 2.00939i 0 −4.65898 + 5.22436i 0 7.42909 − 12.8676i 0
191.2 0 −1.53177 + 0.884367i 0 −4.10168 + 7.10432i 0 5.46804 + 4.37041i 0 −2.93579 + 5.08494i 0
191.3 0 −0.100242 + 0.0578747i 0 0.761802 − 1.31948i 0 −6.66292 − 2.14605i 0 −4.49330 + 7.78263i 0
191.4 0 0.100242 − 0.0578747i 0 0.761802 − 1.31948i 0 6.66292 + 2.14605i 0 −4.49330 + 7.78263i 0
191.5 0 1.53177 − 0.884367i 0 −4.10168 + 7.10432i 0 −5.46804 − 4.37041i 0 −2.93579 + 5.08494i 0
191.6 0 4.23009 − 2.44224i 0 −1.16012 + 2.00939i 0 4.65898 − 5.22436i 0 7.42909 − 12.8676i 0
319.1 0 −4.23009 − 2.44224i 0 −1.16012 − 2.00939i 0 −4.65898 − 5.22436i 0 7.42909 + 12.8676i 0
319.2 0 −1.53177 − 0.884367i 0 −4.10168 − 7.10432i 0 5.46804 − 4.37041i 0 −2.93579 − 5.08494i 0
319.3 0 −0.100242 − 0.0578747i 0 0.761802 + 1.31948i 0 −6.66292 + 2.14605i 0 −4.49330 − 7.78263i 0
319.4 0 0.100242 + 0.0578747i 0 0.761802 + 1.31948i 0 6.66292 − 2.14605i 0 −4.49330 − 7.78263i 0
319.5 0 1.53177 + 0.884367i 0 −4.10168 − 7.10432i 0 −5.46804 + 4.37041i 0 −2.93579 − 5.08494i 0
319.6 0 4.23009 + 2.44224i 0 −1.16012 − 2.00939i 0 4.65898 + 5.22436i 0 7.42909 + 12.8676i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.r.f 12
4.b odd 2 1 inner 448.3.r.f 12
7.c even 3 1 inner 448.3.r.f 12
8.b even 2 1 224.3.r.d ✓ 12
8.d odd 2 1 224.3.r.d ✓ 12
28.g odd 6 1 inner 448.3.r.f 12
56.j odd 6 1 1568.3.d.l 6
56.k odd 6 1 224.3.r.d ✓ 12
56.k odd 6 1 1568.3.d.i 6
56.m even 6 1 1568.3.d.l 6
56.p even 6 1 224.3.r.d ✓ 12
56.p even 6 1 1568.3.d.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.d ✓ 12 8.b even 2 1
224.3.r.d ✓ 12 8.d odd 2 1
224.3.r.d ✓ 12 56.k odd 6 1
224.3.r.d ✓ 12 56.p even 6 1
448.3.r.f 12 1.a even 1 1 trivial
448.3.r.f 12 4.b odd 2 1 inner
448.3.r.f 12 7.c even 3 1 inner
448.3.r.f 12 28.g odd 6 1 inner
1568.3.d.i 6 56.k odd 6 1
1568.3.d.i 6 56.p even 6 1
1568.3.d.l 6 56.j odd 6 1
1568.3.d.l 6 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 27T_{3}^{10} + 654T_{3}^{8} - 2023T_{3}^{6} + 5598T_{3}^{4} - 75T_{3}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 27 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} + 9 T^{5} + \cdots + 841)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 205144679041 \) Copy content Toggle raw display
$13$ \( (T^{3} - 168 T + 784)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} + \cdots + 1962801)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 92\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{3} - 1176 T + 5488)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 3553711346641 \) Copy content Toggle raw display
$37$ \( (T^{6} - 57 T^{5} + \cdots + 131769)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 1848 T + 22736)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 5376 T^{4} + \cdots + 2517630976)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 65\!\cdots\!41 \) Copy content Toggle raw display
$53$ \( (T^{6} - 9 T^{5} + \cdots + 463583961)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 36\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( (T^{6} + 159 T^{5} + \cdots + 5396329)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 16550975387521 \) Copy content Toggle raw display
$71$ \( (T^{6} + 20160 T^{4} + \cdots + 37803802624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 171 T^{5} + \cdots + 17988442641)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{6} + 12096 T^{4} + \cdots + 14201012224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 75 T^{5} + \cdots + 127528266321)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 168 T - 784)^{4} \) Copy content Toggle raw display
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