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Properties

Label 128.8.b.g
Level $128$
Weight $8$
Character orbit 128.b
Analytic conductor $39.985$
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] = [128,8,Mod(65,128)] 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("128.65"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-7080] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.9852832620\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.194307987513600.13
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 22x^{6} - 52x^{5} + 659x^{4} - 1236x^{3} + 4978x^{2} - 4368x + 59105 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{52} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_{5} q^{7} + ( - 3 \beta_{4} - 885) q^{9} + (\beta_{6} - 59 \beta_{3}) q^{11} + ( - 41 \beta_{2} + 48 \beta_1) q^{13} + ( - 5 \beta_{7} - 3 \beta_{5}) q^{15}+ \cdots + (1476 \beta_{6} + 280683 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 7080 q^{9} + 34032 q^{17} + 85224 q^{25} + 1449984 q^{33} + 2337232 q^{41} + 3307592 q^{49} + 15507456 q^{57} + 22403200 q^{65} + 20737200 q^{73} + 23591880 q^{81} + 14086448 q^{89} + 7869808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 22x^{6} - 52x^{5} + 659x^{4} - 1236x^{3} + 4978x^{2} - 4368x + 59105 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{6} + 24\nu^{5} - 120\nu^{4} + 200\nu^{3} - 2792\nu^{2} + 2696\nu - 9760 ) / 225 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\nu^{6} - 57\nu^{5} + 285\nu^{4} - 475\nu^{3} + 13831\nu^{2} - 13603\nu + 51980 ) / 450 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 166 \nu^{7} - 581 \nu^{6} + 5289 \nu^{5} - 11770 \nu^{4} + 103509 \nu^{3} - 143784 \nu^{2} + \cdots - 675535 ) / 38550 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\nu^{4} - 16\nu^{3} + 72\nu^{2} - 64\nu + 2184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\nu^{7} - 42\nu^{6} - 14\nu^{5} + 140\nu^{4} - 246\nu^{3} + 208\nu^{2} - 21886\nu + 10914 ) / 257 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4412 \nu^{7} + 15442 \nu^{6} - 170298 \nu^{5} + 387140 \nu^{4} - 2097138 \nu^{3} + \cdots + 10716470 ) / 19275 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4928 \nu^{7} + 17248 \nu^{6} - 38112 \nu^{5} + 52160 \nu^{4} - 1741152 \nu^{3} + 2568192 \nu^{2} + \cdots - 4284640 ) / 11565 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + 4\beta_{6} + 16\beta_{5} + 336\beta_{3} + 4096 ) / 8192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 4\beta_{6} + 16\beta_{5} + 336\beta_{3} + 512\beta_{2} + 608\beta _1 - 28672 ) / 8192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -27\beta_{7} + 28\beta_{6} - 208\beta_{5} + 1072\beta_{3} + 384\beta_{2} + 456\beta _1 - 22528 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 111 \beta_{7} + 108 \beta_{6} - 848 \beta_{5} + 1024 \beta_{4} + 3952 \beta_{3} - 3072 \beta_{2} + \cdots - 2035712 ) / 8192 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 75 \beta_{7} - 1956 \beta_{6} - 3856 \beta_{5} + 2560 \beta_{4} - 69584 \beta_{3} - 8960 \beta_{2} + \cdots - 5013504 ) / 8192 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27 \beta_{7} - 3068 \beta_{6} - 4720 \beta_{5} - 3840 \beta_{4} - 109232 \beta_{3} - 70144 \beta_{2} + \cdots + 7880704 ) / 4096 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5709 \beta_{7} - 16644 \beta_{6} + 168176 \beta_{5} - 35840 \beta_{4} - 240976 \beta_{3} + \cdots + 72658944 ) / 8192 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
2.54560 − 2.81682i
−1.54560 − 2.81682i
3.33999 − 3.43737i
−2.33999 − 3.43737i
−2.33999 + 3.43737i
3.33999 + 3.43737i
−1.54560 + 2.81682i
2.54560 + 2.81682i
0 − 77.7987i 0 − 324.387i 0 113.762 0 −3865.64 0
65.2 0 − 77.7987i 0 324.387i 0 −113.762 0 −3865.64 0
65.3 0 − 9.55816i 0 − 172.387i 0 1568.77 0 2095.64 0
65.4 0 − 9.55816i 0 172.387i 0 −1568.77 0 2095.64 0
65.5 0 9.55816i 0 − 172.387i 0 −1568.77 0 2095.64 0
65.6 0 9.55816i 0 172.387i 0 1568.77 0 2095.64 0
65.7 0 77.7987i 0 − 324.387i 0 −113.762 0 −3865.64 0
65.8 0 77.7987i 0 324.387i 0 113.762 0 −3865.64 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.8.b.g ✓ 8
4.b odd 2 1 inner 128.8.b.g ✓ 8
8.b even 2 1 inner 128.8.b.g ✓ 8
8.d odd 2 1 inner 128.8.b.g ✓ 8
16.e even 4 1 256.8.a.k 4
16.e even 4 1 256.8.a.o 4
16.f odd 4 1 256.8.a.k 4
16.f odd 4 1 256.8.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.8.b.g ✓ 8 1.a even 1 1 trivial
128.8.b.g ✓ 8 4.b odd 2 1 inner
128.8.b.g ✓ 8 8.b even 2 1 inner
128.8.b.g ✓ 8 8.d odd 2 1 inner
256.8.a.k 4 16.e even 4 1
256.8.a.k 4 16.f odd 4 1
256.8.a.o 4 16.e even 4 1
256.8.a.o 4 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6144T_{3}^{2} + 552960 \) acting on \(S_{8}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 6144 T^{2} + 552960)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 134944 T^{2} + 3127046400)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2473984 T^{2} + 31850496000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 923324185866240)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8508 T - 30273148)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 47\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 97\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 15\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 584308 T + 61943042340)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 28\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 48\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 10\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 5509556798436)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 47\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 10640851706460)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1967452 T - 708251541948)^{4} \) Copy content Toggle raw display
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