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Properties

Label 128.12.b.f
Level $128$
Weight $12$
Character orbit 128.b
Analytic conductor $98.348$
Analytic rank $0$
Dimension $12$
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] = [128,12,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, 12); 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, 12, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 128.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-316732] 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: \(98.3479271116\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3286 x^{10} + 3725205 x^{8} + 1773266980 x^{6} + 401838244180 x^{4} + 42969249696816 x^{2} + 17\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{2}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{8} + \beta_{7}) q^{5} - \beta_1 q^{7} + ( - \beta_{2} - 26394) q^{9} + ( - \beta_{9} - 152 \beta_{6}) q^{11} + (3 \beta_{10} - 23 \beta_{8} + 333 \beta_{7}) q^{13} + ( - \beta_{5} + \beta_{3} + 64 \beta_1) q^{15}+ \cdots + ( - 30739 \beta_{11} + \cdots + 37507796 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 316732 q^{9} + 11301736 q^{17} - 100402276 q^{25} - 370222336 q^{33} + 1458786872 q^{41} - 2425030420 q^{49} - 8422115584 q^{57} - 18057782080 q^{65} - 41779508088 q^{73} - 213089686484 q^{81} - 331639752632 q^{89}+ \cdots - 396775590616 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3286 x^{10} + 3725205 x^{8} + 1773266980 x^{6} + 401838244180 x^{4} + 42969249696816 x^{2} + 17\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 979001512 \nu^{10} - 3212689097920 \nu^{8} + \cdots - 18\!\cdots\!52 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38279392 \nu^{10} + 118646414752 \nu^{8} + 120417505751328 \nu^{6} + \cdots + 32\!\cdots\!79 ) / 78902062547625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 126998467432 \nu^{10} + 382835056189120 \nu^{8} + \cdots + 65\!\cdots\!72 ) / 96\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2297358848 \nu^{10} + 7125077298688 \nu^{8} + \cdots + 20\!\cdots\!51 ) / 12\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 850962980848 \nu^{10} + \cdots - 85\!\cdots\!08 ) / 32\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1361027478371 \nu^{11} + \cdots - 11\!\cdots\!16 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 688877728 \nu^{11} - 2137664017600 \nu^{9} + \cdots - 69\!\cdots\!48 \nu ) / 10\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1598843613077 \nu^{11} + \cdots + 72\!\cdots\!60 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 393508964860357 \nu^{11} + \cdots + 15\!\cdots\!72 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27577236650852 \nu^{11} + \cdots + 35\!\cdots\!60 \nu ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!01 \nu^{11} + \cdots + 91\!\cdots\!96 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} - 16\beta_{9} + 640\beta_{7} + 15621\beta_{6} ) / 655360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -16\beta_{5} + 880\beta_{4} - 55\beta_{3} - 3840\beta_{2} + 791\beta _1 - 717836080 ) / 1310720 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1927\beta_{11} - 720\beta_{10} + 4944\beta_{9} - 42240\beta_{8} - 505475\beta_{7} - 13057889\beta_{6} ) / 327680 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1232\beta_{5} - 85405\beta_{4} + 11985\beta_{3} + 329040\beta_{2} - 132657\beta _1 + 45702838705 ) / 81920 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 477439 \beta_{11} + 328080 \beta_{10} - 1053008 \beta_{9} + 21807360 \beta_{8} + \cdots + 3486344873 \beta_{6} ) / 65536 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 15696624 \beta_{5} + 900734600 \beta_{4} - 192537145 \beta_{3} - 3352732800 \beta_{2} + \cdots - 445456354011560 ) / 655360 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3010532073 \beta_{11} - 2910689040 \beta_{10} + 6477771056 \beta_{9} + \cdots - 22382441304111 \beta_{6} ) / 327680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6404612048 \beta_{5} - 288733280330 \beta_{4} + 82027175915 \beta_{3} + 1065944641440 \beta_{2} + \cdots + 14\!\cdots\!50 ) / 163840 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3829131381091 \beta_{11} + 4727004492240 \beta_{10} - 8206046971152 \beta_{9} + \cdots + 28\!\cdots\!37 \beta_{6} ) / 327680 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8052687256208 \beta_{5} + 295228471557160 \beta_{4} - 103957038733215 \beta_{3} + \cdots - 14\!\cdots\!92 ) / 131072 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48\!\cdots\!29 \beta_{11} + \cdots - 36\!\cdots\!03 \beta_{6} ) / 327680 \) 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
36.4873i
34.4873i
− 16.5733i
− 14.5733i
12.7527i
10.7527i
− 10.7527i
− 12.7527i
14.5733i
16.5733i
− 34.4873i
− 36.4873i
0 − 580.453i 0 − 6948.91i 0 45098.0 0 −159779. 0
65.2 0 − 580.453i 0 6948.91i 0 −45098.0 0 −159779. 0
65.3 0 − 501.129i 0 − 10865.5i 0 32440.9 0 −73983.7 0
65.4 0 − 501.129i 0 10865.5i 0 −32440.9 0 −73983.7 0
65.5 0 − 150.224i 0 − 2288.63i 0 −47323.2 0 154580. 0
65.6 0 − 150.224i 0 2288.63i 0 47323.2 0 154580. 0
65.7 0 150.224i 0 − 2288.63i 0 47323.2 0 154580. 0
65.8 0 150.224i 0 2288.63i 0 −47323.2 0 154580. 0
65.9 0 501.129i 0 − 10865.5i 0 −32440.9 0 −73983.7 0
65.10 0 501.129i 0 10865.5i 0 32440.9 0 −73983.7 0
65.11 0 580.453i 0 − 6948.91i 0 −45098.0 0 −159779. 0
65.12 0 580.453i 0 6948.91i 0 45098.0 0 −159779. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.12
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 128.12.b.f ✓ 12
4.b odd 2 1 inner 128.12.b.f ✓ 12
8.b even 2 1 inner 128.12.b.f ✓ 12
8.d odd 2 1 inner 128.12.b.f ✓ 12
16.e even 4 1 256.12.a.j 6
16.e even 4 1 256.12.a.k 6
16.f odd 4 1 256.12.a.j 6
16.f odd 4 1 256.12.a.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.12.b.f ✓ 12 1.a even 1 1 trivial
128.12.b.f ✓ 12 4.b odd 2 1 inner
128.12.b.f ✓ 12 8.b even 2 1 inner
128.12.b.f ✓ 12 8.d odd 2 1 inner
256.12.a.j 6 16.e even 4 1
256.12.a.j 6 16.f odd 4 1
256.12.a.k 6 16.e even 4 1
256.12.a.k 6 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 610624T_{3}^{4} + 97883320320T_{3}^{2} + 1909477048320000 \) acting on \(S_{12}^{\mathrm{new}}(128, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 94\!\cdots\!92)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 42\!\cdots\!84)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 37\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 44\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 93\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 13\!\cdots\!48)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 82\!\cdots\!00)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 11\!\cdots\!68)^{4} \) Copy content Toggle raw display
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