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Properties

Label 288.8.d.c
Level $288$
Weight $8$
Character orbit 288.d
Analytic conductor $89.967$
Analytic rank $0$
Dimension $12$
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(145,288)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{72}\cdot 3^{6}\cdot 5^{2}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 72)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + (\beta_{4} - 11) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + (\beta_{4} - 11) q^{7} + ( - \beta_{2} - 10 \beta_1) q^{11} - \beta_{7} q^{13} - \beta_{8} q^{17} - \beta_{10} q^{19} + ( - \beta_{9} - \beta_{8}) q^{23} + ( - \beta_{5} + 21 \beta_{4} + 19159) q^{25} + ( - 13 \beta_{3} + 24 \beta_{2} - 238 \beta_1) q^{29} + ( - 6 \beta_{5} + 33 \beta_{4} + 3115) q^{31} + (30 \beta_{3} - 55 \beta_{2} + 360 \beta_1) q^{35} + (\beta_{11} - 9 \beta_{10} + 3 \beta_{7}) q^{37} + (8 \beta_{9} + 3 \beta_{8} - \beta_{6}) q^{41} + ( - 2 \beta_{11} + 3 \beta_{10} - 40 \beta_{7}) q^{43} + ( - 3 \beta_{9} + 29 \beta_{8} - 2 \beta_{6}) q^{47} + (35 \beta_{5} + 161 \beta_{4} + 222391) q^{49} + (19 \beta_{3} + 152 \beta_{2} + 816 \beta_1) q^{53} + ( - 34 \beta_{5} - 366 \beta_{4} - 577524) q^{55} + (34 \beta_{3} + 50 \beta_{2} - 346 \beta_1) q^{59} + (9 \beta_{11} + 47 \beta_{10} + 61 \beta_{7}) q^{61} + ( - 24 \beta_{9} - 13 \beta_{8} - 13 \beta_{6}) q^{65} + ( - 14 \beta_{11} + \cdots + 232 \beta_{7}) q^{67}+ \cdots + (110 \beta_{5} + 8314 \beta_{4} + 2289606) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 136 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 136 q^{7} + 229820 q^{25} + 37224 q^{31} + 2668188 q^{49} - 6928960 q^{55} + 13619048 q^{73} - 20470552 q^{79} + 27442456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 206x^{10} + 24336x^{8} - 1510912x^{6} + 398721024x^{4} - 55297703936x^{2} + 4398046511104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 99 \nu^{11} + 32682 \nu^{9} - 746288 \nu^{7} + 121465344 \nu^{5} - 40792489984 \nu^{3} + 8391829225472 \nu ) / 85899345920 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39 \nu^{11} + 30562 \nu^{9} - 1395568 \nu^{7} + 280011264 \nu^{5} + \cdots + 6794638262272 \nu ) / 10737418240 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1973 \nu^{11} + 492454 \nu^{9} - 36374096 \nu^{7} + 2784224768 \nu^{5} + \cdots + 85543400505344 \nu ) / 85899345920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -23\nu^{10} + 2690\nu^{8} - 137840\nu^{6} + 18465280\nu^{4} - 6814433280\nu^{2} + 575659835392 ) / 134217728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{10} + 5114\nu^{8} - 1143984\nu^{6} + 41302528\nu^{4} + 12105023488\nu^{2} + 632299716608 ) / 134217728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 206\nu^{9} - 24336\nu^{7} + 1510912\nu^{5} - 398721024\nu^{3} + 89657442304\nu ) / 8388608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\nu^{10} + 26758\nu^{8} - 288592\nu^{6} + 199312896\nu^{4} + 2346450944\nu^{2} + 7499549769728 ) / 335544320 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 559 \nu^{11} + 41426 \nu^{9} - 2610160 \nu^{7} + 451252736 \nu^{5} + \cdots + 8272643883008 \nu ) / 2147483648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 975 \nu^{11} + 94354 \nu^{9} - 5983728 \nu^{7} + 282350080 \nu^{5} + \cdots + 18348637159424 \nu ) / 2147483648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -11\nu^{10} + 12506\nu^{8} - 279984\nu^{6} + 1579520\nu^{4} - 6335234048\nu^{2} + 3106871967744 ) / 33554432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 467 \nu^{10} - 122826 \nu^{8} + 14752304 \nu^{6} - 1357711872 \nu^{4} + 390611075072 \nu^{2} - 34855270219776 ) / 167772160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 8\beta_{3} + 56\beta_1 ) / 8192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + 4\beta_{7} + 8\beta_{5} + 24\beta_{4} + 70320 ) / 2048 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64\beta_{9} - 64\beta_{8} - \beta_{6} - 440\beta_{3} + 1024\beta_{2} - 5112\beta_1 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{11} - 395\beta_{10} + 1324\beta_{7} + 88\beta_{5} + 2312\beta_{4} - 1063536 ) / 1024 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2624\beta_{9} + 6720\beta_{8} - 155\beta_{6} - 18536\beta_{3} + 80896\beta_{2} - 353576\beta_1 ) / 2048 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3113\beta_{11} - 3753\beta_{10} + 41892\beta_{7} - 39608\beta_{5} + 92120\beta_{4} - 150577616 ) / 512 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -69824\beta_{9} + 229568\beta_{8} - 207065\beta_{6} - 2143992\beta_{3} + 529408\beta_{2} + 38133960\beta_1 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 65747\beta_{11} + 631725\beta_{10} + 750668\beta_{7} - 89448\beta_{5} + 610248\beta_{4} - 63807199472 ) / 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4082624 \beta_{9} - 12549056 \beta_{8} - 18230627 \beta_{6} + 108865496 \beta_{3} + \cdots + 865831192 \beta_1 ) / 512 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18232895 \beta_{11} + 21442495 \beta_{10} + 39932420 \beta_{7} - 85196280 \beta_{5} + \cdots - 1710937618768 ) / 128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1245143744 \beta_{9} + 174805696 \beta_{8} + 31986447 \beta_{6} + 9306877896 \beta_{3} + \cdots + 74687713672 \beta_1 ) / 256 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
10.2927 − 4.69692i
−10.2927 − 4.69692i
10.6405 + 3.84452i
−10.6405 + 3.84452i
−4.93370 + 10.1813i
4.93370 + 10.1813i
4.93370 − 10.1813i
−4.93370 − 10.1813i
−10.6405 − 3.84452i
10.6405 − 3.84452i
−10.2927 + 4.69692i
10.2927 + 4.69692i
0 0 0 − 316.260i 0 −1193.16 0 0 0
145.2 0 0 0 − 316.260i 0 −1193.16 0 0 0
145.3 0 0 0 − 209.469i 0 1301.46 0 0 0
145.4 0 0 0 − 209.469i 0 1301.46 0 0 0
145.5 0 0 0 − 181.720i 0 −142.301 0 0 0
145.6 0 0 0 − 181.720i 0 −142.301 0 0 0
145.7 0 0 0 181.720i 0 −142.301 0 0 0
145.8 0 0 0 181.720i 0 −142.301 0 0 0
145.9 0 0 0 209.469i 0 1301.46 0 0 0
145.10 0 0 0 209.469i 0 1301.46 0 0 0
145.11 0 0 0 316.260i 0 −1193.16 0 0 0
145.12 0 0 0 316.260i 0 −1193.16 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.8.d.c 12
3.b odd 2 1 inner 288.8.d.c 12
4.b odd 2 1 72.8.d.c ✓ 12
8.b even 2 1 inner 288.8.d.c 12
8.d odd 2 1 72.8.d.c ✓ 12
12.b even 2 1 72.8.d.c ✓ 12
24.f even 2 1 72.8.d.c ✓ 12
24.h odd 2 1 inner 288.8.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.d.c ✓ 12 4.b odd 2 1
72.8.d.c ✓ 12 8.d odd 2 1
72.8.d.c ✓ 12 12.b even 2 1
72.8.d.c ✓ 12 24.f even 2 1
288.8.d.c 12 1.a even 1 1 trivial
288.8.d.c 12 3.b odd 2 1 inner
288.8.d.c 12 8.b even 2 1 inner
288.8.d.c 12 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 176920T_{5}^{4} + 9140446400T_{5}^{2} + 144921907520000 \) acting on \(S_{8}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 144921907520000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} + 34 T^{2} + \cdots - 220971400)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 77\!\cdots\!40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 24\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 647461203489880)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 34\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 54\!\cdots\!20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 16\!\cdots\!00)^{4} \) Copy content Toggle raw display
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