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Properties

Label 256.12.a.m
Level $256$
Weight $12$
Character orbit 256.a
Self dual yes
Analytic conductor $196.696$
Analytic rank $1$
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] = [256,12,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0] 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: yes
Analytic conductor: \(196.695854223\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 1198x^{6} + 3608x^{5} + 410719x^{4} - 827456x^{3} - 32836930x^{2} + 33251260x + 673579417 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{54}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{3} - \beta_{5} q^{5} - \beta_1 q^{7} + ( - 15 \beta_{2} - 23859) q^{9} + ( - 967 \beta_{4} - 306 \beta_{3}) q^{11} + (\beta_{7} - \beta_{5}) q^{13} + ( - \beta_{6} + 5 \beta_1) q^{15}+ \cdots + (45585573 \beta_{4} + 231606039 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 190872 q^{9} + 4065072 q^{17} - 24949480 q^{25} + 872983488 q^{33} - 2793492048 q^{41} + 6639991624 q^{49} - 15016294080 q^{57} + 496419840 q^{65} + 38648281744 q^{73} - 97953918840 q^{81} + 138916158480 q^{89}+ \cdots - 101372690000 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 1198x^{6} + 3608x^{5} + 410719x^{4} - 827456x^{3} - 32836930x^{2} + 33251260x + 673579417 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 356\nu^{6} - 1068\nu^{5} - 202092\nu^{4} + 405964\nu^{3} - 324828\nu^{2} + 121668\nu + 555675232 ) / 89799 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{6} - 24\nu^{5} - 7232\nu^{4} + 14504\nu^{3} + 673424\nu^{2} - 680680\nu + 224466536 ) / 29933 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 74048 \nu^{7} - 259168 \nu^{6} - 82154016 \nu^{5} + 206032960 \nu^{4} + 24472631904 \nu^{3} + \cdots + 549566150496 ) / 12506516261 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 248720 \nu^{7} - 870520 \nu^{6} - 297566186 \nu^{5} + 746091765 \nu^{4} + 95518404296 \nu^{3} + \cdots + 2179334428179 ) / 25013032522 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1316246 \nu^{7} - 4606861 \nu^{6} - 1719761559 \nu^{5} + 4310921050 \nu^{4} + \cdots + 41934275429557 ) / 37519548783 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32512 \nu^{6} - 97536 \nu^{5} - 37053696 \nu^{4} + 74269952 \nu^{3} + 11296196352 \nu^{2} + \cdots - 476881420288 ) / 89799 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43715200 \nu^{7} + 153003200 \nu^{6} + 31897399872 \nu^{5} - 80126007680 \nu^{4} + \cdots + 351022673098048 ) / 37519548783 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 64\beta_{5} + 576\beta_{3} + 36864 ) / 73728 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - 64\beta_{5} + 576\beta_{3} - 2304\beta_{2} + 64\beta _1 + 22228992 ) / 73728 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 376 \beta_{7} + \beta_{6} - 5632 \beta_{5} - 147456 \beta_{4} + 355008 \beta_{3} - 2304 \beta_{2} + \cdots + 22216704 ) / 49152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1127 \beta_{7} + 256 \beta_{6} - 16832 \beta_{5} - 442368 \beta_{4} + 1064448 \beta_{3} + \cdots + 11489882112 ) / 73728 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 637562 \beta_{7} + 1275 \beta_{6} - 6465152 \beta_{5} - 445317120 \beta_{4} + 911659968 \beta_{3} + \cdots + 57338339328 ) / 147456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 317842 \beta_{7} + 48813 \beta_{6} - 3218560 \beta_{5} - 222289920 \beta_{4} + 454943040 \beta_{3} + \cdots + 2158567833600 ) / 24576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 182654871 \beta_{7} + 510307 \beta_{6} - 1752612288 \beta_{5} - 175035506688 \beta_{4} + \cdots + 22564659032064 ) / 73728 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−24.3394
22.5109
−5.14004
8.96847
−7.96847
6.14004
−21.5109
25.3394
0 −522.534 0 −531.752 0 −74823.3 0 95895.0 0
1.2 0 −522.534 0 531.752 0 74823.3 0 95895.0 0
1.3 0 −183.123 0 −9546.52 0 −4016.43 0 −143613. 0
1.4 0 −183.123 0 9546.52 0 4016.43 0 −143613. 0
1.5 0 183.123 0 −9546.52 0 4016.43 0 −143613. 0
1.6 0 183.123 0 9546.52 0 −4016.43 0 −143613. 0
1.7 0 522.534 0 −531.752 0 74823.3 0 95895.0 0
1.8 0 522.534 0 531.752 0 −74823.3 0 95895.0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)

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 256.12.a.m 8
4.b odd 2 1 inner 256.12.a.m 8
8.b even 2 1 inner 256.12.a.m 8
8.d odd 2 1 inner 256.12.a.m 8
16.e even 4 2 128.12.b.d ✓ 8
16.f odd 4 2 128.12.b.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.12.b.d ✓ 8 16.e even 4 2
128.12.b.d ✓ 8 16.f odd 4 2
256.12.a.m 8 1.a even 1 1 trivial
256.12.a.m 8 4.b odd 2 1 inner
256.12.a.m 8 8.b even 2 1 inner
256.12.a.m 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(256))\):

\( T_{3}^{4} - 306576T_{3}^{2} + 9156193344 \) Copy content Toggle raw display
\( T_{5}^{4} - 91418880T_{5}^{2} + 25769705472000 \) Copy content Toggle raw display
\( T_{7}^{4} - 5614651392T_{7}^{2} + 90313540585390080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 306576 T^{2} + 9156193344)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + \cdots + 25769705472000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 90\!\cdots\!80)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 1016268 T - 391989707100)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 76\!\cdots\!00)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 95\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 91\!\cdots\!80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 77\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + \cdots + 92\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 59\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 69\!\cdots\!16)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 56\!\cdots\!00)^{4} \) Copy content Toggle raw display
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