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Properties

Label 256.12.a.o
Level $256$
Weight $12$
Character orbit 256.a
Self dual yes
Analytic conductor $196.696$
Analytic rank $1$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [256,12,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(196.695854223\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5011x^{8} + 8617108x^{6} - 5521716928x^{4} + 760691642368x^{2} - 20440645894144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{85}\cdot 3^{2}\cdot 7 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{4} + 2 \beta_1) q^{5} + ( - \beta_{2} - 3362) q^{7} + (\beta_{3} - \beta_{2} + 47239) q^{9} + ( - \beta_{7} + 11 \beta_{4} + 30 \beta_1) q^{11} + (\beta_{8} + \beta_{7} + \cdots + 191 \beta_1) q^{13}+ \cdots + (12519 \beta_{9} + \cdots + 36045633 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 33616 q^{7} + 472394 q^{9} - 3391792 q^{15} + 2639732 q^{17} - 45357808 q^{23} + 41502654 q^{25} - 442035392 q^{31} - 54732216 q^{33} - 431459312 q^{39} + 654907964 q^{41} + 560226528 q^{47} + 1143661722 q^{49}+ \cdots + 60995282900 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5011x^{8} + 8617108x^{6} - 5521716928x^{4} + 760691642368x^{2} - 20440645894144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 149 \nu^{9} - 9960207 \nu^{7} + 31671582116 \nu^{5} - 27251267768256 \nu^{3} + 26\!\cdots\!88 \nu ) / 16750327136256 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 463\nu^{8} - 1593437\nu^{6} + 1624201260\nu^{4} - 450735578944\nu^{2} + 29780898726656 ) / 412137600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1283029 \nu^{8} + 4129573071 \nu^{6} - 3210195094180 \nu^{4} - 183379598326848 \nu^{2} + 55\!\cdots\!52 ) / 233682019200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 220573 \nu^{9} + 932839527 \nu^{7} - 1317203062660 \nu^{5} + 654642405138624 \nu^{3} - 46\!\cdots\!76 \nu ) / 26172386150400 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1004951 \nu^{8} - 3552349749 \nu^{6} + 3286505129420 \nu^{4} - 593318829165888 \nu^{2} + 29\!\cdots\!12 ) / 77894006400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 642661 \nu^{8} - 2625370239 \nu^{6} + 3364324782820 \nu^{4} + \cdots + 87\!\cdots\!32 ) / 16691572800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 209202013 \nu^{9} - 892317094887 \nu^{7} + \cdots + 86\!\cdots\!56 \nu ) / 418758178406400 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15204743 \nu^{9} + 62583530757 \nu^{7} - 79676872043660 \nu^{5} + \cdots + 29\!\cdots\!84 \nu ) / 23264343244800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 474068719 \nu^{9} - 1650307867581 \nu^{7} + \cdots + 40\!\cdots\!28 \nu ) / 418758178406400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} + 1420\beta_1 ) / 131072 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} - 16\beta_{5} - 45\beta_{3} - 139\beta_{2} + 65680128 ) / 65536 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1553\beta_{9} + 1361\beta_{8} - 3409\beta_{7} - 96593\beta_{4} + 2290956\beta_1 ) / 131072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3873\beta_{6} - 28368\beta_{5} - 50511\beta_{3} - 53817\beta_{2} + 103231047712 ) / 65536 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2477263\beta_{9} + 2662543\beta_{8} - 6097039\beta_{7} - 231320719\beta_{4} + 3610559092\beta_1 ) / 131072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2452407\beta_{6} - 47878192\beta_{5} - 56106201\beta_{3} + 191584289\beta_{2} + 168440485682784 ) / 65536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3991931081 \beta_{9} + 5154901769 \beta_{8} - 10601277193 \beta_{7} - 486153544457 \beta_{4} + 5493305137836 \beta_1 ) / 131072 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2225850303 \beta_{6} - 80836523856 \beta_{5} - 59707921839 \beta_{3} + 771154942119 \beta_{2} + 27\!\cdots\!08 ) / 65536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6488027798911 \beta_{9} + 9730139486527 \beta_{8} - 18332159091007 \beta_{7} + \cdots + 81\!\cdots\!12 \beta_1 ) / 131072 \) 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.

comment: embeddings in the coefficient field
 
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
40.4634
5.96180
−11.7812
−42.0479
−37.8331
37.8331
42.0479
11.7812
−5.96180
−40.4634
0 −712.065 0 7122.63 0 46911.8 0 329890. 0
1.2 0 −614.710 0 −4397.69 0 −41556.6 0 200721. 0
1.3 0 −381.717 0 −8937.55 0 10175.0 0 −31439.0 0
1.4 0 −284.352 0 10467.0 0 −70503.2 0 −96290.8 0
1.5 0 −102.287 0 −2319.82 0 38165.0 0 −166684. 0
1.6 0 102.287 0 2319.82 0 38165.0 0 −166684. 0
1.7 0 284.352 0 −10467.0 0 −70503.2 0 −96290.8 0
1.8 0 381.717 0 8937.55 0 10175.0 0 −31439.0 0
1.9 0 614.710 0 4397.69 0 −41556.6 0 200721. 0
1.10 0 712.065 0 −7122.63 0 46911.8 0 329890. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 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.o 10
4.b odd 2 1 256.12.a.p 10
8.b even 2 1 inner 256.12.a.o 10
8.d odd 2 1 256.12.a.p 10
16.e even 4 2 32.12.b.a 10
16.f odd 4 2 8.12.b.a ✓ 10
48.k even 4 2 72.12.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.12.b.a ✓ 10 16.f odd 4 2
32.12.b.a 10 16.e even 4 2
72.12.d.b 10 48.k even 4 2
256.12.a.o 10 1.a even 1 1 trivial
256.12.a.o 10 8.b even 2 1 inner
256.12.a.p 10 4.b odd 2 1
256.12.a.p 10 8.d odd 2 1

Hecke kernels

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

\( T_{3}^{10} - 1121932 T_{3}^{8} + 415491272352 T_{3}^{6} + \cdots - 23\!\cdots\!64 \) Copy content Toggle raw display
\( T_{5}^{10} - 264891952 T_{5}^{8} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{5} + 16808 T_{7}^{4} - 5087977856 T_{7}^{3} + 342941398016 T_{7}^{2} + \cdots - 53\!\cdots\!92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots - 23\!\cdots\!64 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 53\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 18\!\cdots\!08)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 57\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 62\!\cdots\!32)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 52\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 12\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 19\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 45\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 95\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 58\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 61\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 91\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
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