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Properties

Label 288.8.c.b
Level $288$
Weight $8$
Character orbit 288.c
Analytic conductor $89.967$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,8,Mod(287,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.287");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 1360 x^{14} + 710372 x^{12} - 180776904 x^{10} + 23779111124 x^{8} - 1575523372272 x^{6} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{98}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} - 7 \beta_{8}) q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} - 7 \beta_{8}) q^{5} + \beta_{4} q^{7} + ( - \beta_{14} + 2 \beta_{10} - 3 \beta_{9}) q^{11} + (\beta_{3} - 7 \beta_{2} + 2080) q^{13} + (\beta_{15} - 43 \beta_{12} + \cdots - 447 \beta_{8}) q^{17}+ \cdots + (325 \beta_{5} - 489 \beta_{3} + \cdots + 5133856) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 33280 q^{13} - 55248 q^{25} + 869664 q^{37} + 1042032 q^{49} + 5525344 q^{61} + 29037312 q^{73} + 56734176 q^{85} + 82141696 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 1360 x^{14} + 710372 x^{12} - 180776904 x^{10} + 23779111124 x^{8} - 1575523372272 x^{6} + \cdots + 18\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5068953983 \nu^{14} + 4018896861114 \nu^{12} + 3362828952452 \nu^{10} + \cdots - 61\!\cdots\!96 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!61 \nu^{14} + \cdots - 56\!\cdots\!32 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!93 \nu^{14} + \cdots + 26\!\cdots\!16 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{14} + \cdots + 65\!\cdots\!84 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 36\!\cdots\!51 \nu^{14} + \cdots - 61\!\cdots\!12 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26\!\cdots\!09 \nu^{14} + \cdots - 73\!\cdots\!92 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\!\cdots\!43 \nu^{14} + \cdots - 65\!\cdots\!84 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!76 \nu^{15} + \cdots - 25\!\cdots\!88 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 87053080108379 \nu^{15} + \cdots - 48\!\cdots\!68 \nu ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!59 \nu^{15} + \cdots + 96\!\cdots\!92 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43\!\cdots\!31 \nu^{15} + \cdots - 12\!\cdots\!72 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!59 \nu^{15} + \cdots + 19\!\cdots\!92 \nu ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{15} + \cdots + 45\!\cdots\!48 \nu ) / 78\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 66\!\cdots\!27 \nu^{15} + \cdots - 79\!\cdots\!76 \nu ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 35\!\cdots\!27 \nu^{15} + \cdots - 83\!\cdots\!76 \nu ) / 70\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -16\beta_{14} - 8\beta_{13} + 16\beta_{10} - 15\beta_{9} + 4608\beta_{8} ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{7} - 16\beta_{6} - 26\beta_{5} - 32\beta_{4} - 34\beta_{3} + 3476\beta_{2} + 31\beta _1 + 1566720 ) / 9216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{15} - 2544 \beta_{14} - 2326 \beta_{13} + 6944 \beta_{12} + 60 \beta_{11} + \cdots + 1569792 \beta_{8} ) / 6144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7664 \beta_{7} - 13988 \beta_{6} - 18200 \beta_{5} - 7456 \beta_{4} - 1528 \beta_{3} + \cdots + 494042112 ) / 9216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10410 \beta_{15} - 300832 \beta_{14} - 356348 \beta_{13} + 1957560 \beta_{12} + \cdots + 310739328 \beta_{8} ) / 2304 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1216112 \beta_{7} - 2874108 \beta_{6} - 2714668 \beta_{5} - 2350752 \beta_{4} + 797028 \beta_{3} + \cdots + 61236077568 ) / 3072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 37164232 \beta_{15} - 432277296 \beta_{14} - 586602222 \beta_{13} + 4687646432 \beta_{12} + \cdots + 651690510336 \beta_{8} ) / 9216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 440807984 \beta_{7} - 1193355860 \beta_{6} - 855007040 \beta_{5} - 1280361376 \beta_{4} + \cdots + 18135540066816 ) / 2304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3825962592 \beta_{15} - 27773469808 \beta_{14} - 40247567134 \beta_{13} + \cdots + 55713487784448 \beta_{8} ) / 1536 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 429877516816 \beta_{7} - 1243348682308 \beta_{6} - 707199669788 \beta_{5} - 1499770230368 \beta_{4} + \cdots + 14\!\cdots\!20 ) / 4608 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6193484304920 \beta_{15} - 33188607056880 \beta_{14} - 49557725873094 \beta_{13} + \cdots + 83\!\cdots\!32 \beta_{8} ) / 4608 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 8693644017984 \beta_{7} - 25925194378068 \beta_{6} - 12094342447770 \beta_{5} - 32879955179136 \beta_{4} + \cdots + 24\!\cdots\!24 ) / 192 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 781544692835868 \beta_{15} + \cdots + 10\!\cdots\!48 \beta_{8} ) / 1152 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 50\!\cdots\!56 \beta_{7} + \cdots + 12\!\cdots\!04 ) / 2304 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 25\!\cdots\!08 \beta_{15} + \cdots + 32\!\cdots\!96 \beta_{8} ) / 768 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−20.5167 − 0.707107i
20.5167 − 0.707107i
2.74559 + 0.707107i
−2.74559 + 0.707107i
−13.6495 + 0.707107i
13.6495 + 0.707107i
8.19858 + 0.707107i
−8.19858 + 0.707107i
−8.19858 − 0.707107i
8.19858 − 0.707107i
13.6495 − 0.707107i
−13.6495 − 0.707107i
−2.74559 − 0.707107i
2.74559 − 0.707107i
20.5167 + 0.707107i
−20.5167 + 0.707107i
0 0 0 − 445.722i 0 − 201.952i 0 0 0
287.2 0 0 0 − 445.722i 0 201.952i 0 0 0
287.3 0 0 0 − 342.753i 0 − 274.982i 0 0 0
287.4 0 0 0 − 342.753i 0 274.982i 0 0 0
287.5 0 0 0 − 72.1819i 0 − 653.220i 0 0 0
287.6 0 0 0 − 72.1819i 0 653.220i 0 0 0
287.7 0 0 0 − 70.3851i 0 − 1578.15i 0 0 0
287.8 0 0 0 − 70.3851i 0 1578.15i 0 0 0
287.9 0 0 0 70.3851i 0 − 1578.15i 0 0 0
287.10 0 0 0 70.3851i 0 1578.15i 0 0 0
287.11 0 0 0 72.1819i 0 − 653.220i 0 0 0
287.12 0 0 0 72.1819i 0 653.220i 0 0 0
287.13 0 0 0 342.753i 0 − 274.982i 0 0 0
287.14 0 0 0 342.753i 0 274.982i 0 0 0
287.15 0 0 0 445.722i 0 − 201.952i 0 0 0
287.16 0 0 0 445.722i 0 201.952i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.8.c.b ✓ 16
3.b odd 2 1 inner 288.8.c.b ✓ 16
4.b odd 2 1 inner 288.8.c.b ✓ 16
8.b even 2 1 576.8.c.g 16
8.d odd 2 1 576.8.c.g 16
12.b even 2 1 inner 288.8.c.b ✓ 16
24.f even 2 1 576.8.c.g 16
24.h odd 2 1 576.8.c.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.8.c.b ✓ 16 1.a even 1 1 trivial
288.8.c.b ✓ 16 3.b odd 2 1 inner
288.8.c.b ✓ 16 4.b odd 2 1 inner
288.8.c.b ✓ 16 12.b even 2 1 inner
576.8.c.g 16 8.b even 2 1
576.8.c.g 16 8.d odd 2 1
576.8.c.g 16 24.f even 2 1
576.8.c.g 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 326312T_{5}^{6} + 26578680024T_{5}^{4} + 245389005956000T_{5}^{2} + 602431641526090000 \) acting on \(S_{8}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 32\!\cdots\!24)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 16\!\cdots\!96)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 41\!\cdots\!64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 65\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 53\!\cdots\!12)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 26\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 69\!\cdots\!56)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 13\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 59\!\cdots\!60)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 33\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 27\!\cdots\!24)^{4} \) Copy content Toggle raw display
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