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Properties

Label 288.8.c.a
Level $288$
Weight $8$
Character orbit 288.c
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(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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 156x^{10} + 9341x^{8} - 257316x^{6} + 2917940x^{4} - 5804352x^{2} + 9684544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{53}\cdot 3^{12} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - 37 \beta_{4}) q^{5} + (\beta_{3} + 5 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - 37 \beta_{4}) q^{5} + (\beta_{3} + 5 \beta_1) q^{7} + ( - \beta_{10} + \beta_{9} + 7 \beta_{7}) q^{11} + ( - \beta_{6} + 5 \beta_{2} - 1596) q^{13} + ( - \beta_{11} + 37 \beta_{8} - 131 \beta_{4}) q^{17} + (5 \beta_{5} + 24 \beta_{3} + 95 \beta_1) q^{19} + ( - 34 \beta_{10} + \cdots + 178 \beta_{7}) q^{23}+ \cdots + (1490 \beta_{6} + 14278 \beta_{2} - 5333480) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 19152 q^{13} - 423228 q^{25} + 563928 q^{37} - 1281324 q^{49} - 5265528 q^{61} - 16672512 q^{73} - 49733736 q^{85} - 64001760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 156x^{10} + 9341x^{8} - 257316x^{6} + 2917940x^{4} - 5804352x^{2} + 9684544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 149\nu^{10} - 20506\nu^{8} + 1002957\nu^{6} - 19982130\nu^{4} + 126843040\nu^{2} - 159941408 ) / 2353995 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1841\nu^{10} + 195722\nu^{8} - 6879273\nu^{6} + 79126554\nu^{4} - 157826944\nu^{2} + 3651907176 ) / 6364505 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 764543 \nu^{10} - 98921806 \nu^{8} + 5003333799 \nu^{6} - 114888644502 \nu^{4} + \cdots - 1237666422368 ) / 1963904400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 126767 \nu^{11} - 18141074 \nu^{9} + 982689231 \nu^{7} - 23707893018 \nu^{5} + \cdots - 89111827072 \nu ) / 339537249600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 168128 \nu^{10} + 18487360 \nu^{8} - 572061984 \nu^{6} - 1017420096 \nu^{4} + \cdots - 218058584320 ) / 171841635 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 117761 \nu^{10} - 13266242 \nu^{8} + 515706873 \nu^{6} - 7300591194 \nu^{4} + 14603791744 \nu^{2} + 68237759424 ) / 31822525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 878\nu^{11} - 56056\nu^{9} - 1016346\nu^{7} + 137302968\nu^{5} - 2738637368\nu^{3} + 10454206432\nu ) / 134175825 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9659737 \nu^{11} - 2334546814 \nu^{9} + 195620150841 \nu^{7} - 7001975079798 \nu^{5} + \cdots - 27335154699392 \nu ) / 594190186800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58659691 \nu^{11} + 7818105992 \nu^{9} - 398783992863 \nu^{7} + 9427443738384 \nu^{5} + \cdots + 349731817251376 \nu ) / 381979405800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62615206 \nu^{11} + 8276874452 \nu^{9} - 424026561558 \nu^{7} + 10418129546484 \nu^{5} + \cdots + 443077534608256 \nu ) / 334231980075 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 252107323 \nu^{11} + 36773036506 \nu^{9} - 1982281826139 \nu^{7} + \cdots + 175369650973568 \nu ) / 198063395600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -8\beta_{10} + 16\beta_{9} + 15\beta_{7} + 2304\beta_{4} ) / 4608 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{6} - 4\beta_{5} + 16\beta_{3} - 64\beta_{2} + 15\beta _1 + 59904 ) / 2304 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - 28\beta_{10} + 48\beta_{9} - 30\beta_{8} - 77\beta_{7} + 15168\beta_{4} ) / 384 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -176\beta_{6} - 172\beta_{5} + 592\beta_{3} - 1072\beta_{2} - 909\beta _1 + 1085568 ) / 1152 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 230\beta_{11} - 1378\beta_{10} + 2672\beta_{9} - 2610\beta_{8} - 7281\beta_{7} + 1432992\beta_{4} ) / 576 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3520\beta_{6} - 4566\beta_{5} + 17520\beta_{3} - 16064\beta_{2} - 45951\beta _1 + 18890496 ) / 576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13944\beta_{11} - 41854\beta_{10} + 93200\beta_{9} - 118608\beta_{8} - 354717\beta_{7} + 76254240\beta_{4} ) / 576 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -21572\beta_{6} - 35140\beta_{5} + 151984\beta_{3} - 71860\beta_{2} - 545139\beta _1 + 101641248 ) / 96 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 753684 \beta_{11} - 1133974 \beta_{10} + 2967824 \beta_{9} - 4765860 \beta_{8} + \cdots + 3683804256 \beta_{4} ) / 576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2108944 \beta_{6} - 4481891 \beta_{5} + 21929648 \beta_{3} - 4734224 \beta_{2} - 97941219 \beta _1 + 8717056128 ) / 288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12549988 \beta_{11} - 8150822 \beta_{10} + 26760720 \beta_{9} - 57927012 \beta_{8} + \cdots + 55635776160 \beta_{4} ) / 192 \) 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
−5.97864 + 0.707107i
5.97864 + 0.707107i
1.22825 − 0.707107i
−1.22825 − 0.707107i
6.49979 − 0.707107i
−6.49979 − 0.707107i
−6.49979 + 0.707107i
6.49979 + 0.707107i
−1.22825 + 0.707107i
1.22825 + 0.707107i
5.97864 − 0.707107i
−5.97864 − 0.707107i
0 0 0 − 484.866i 0 − 857.827i 0 0 0
287.2 0 0 0 − 484.866i 0 857.827i 0 0 0
287.3 0 0 0 − 324.150i 0 − 1034.55i 0 0 0
287.4 0 0 0 − 324.150i 0 1034.55i 0 0 0
287.5 0 0 0 − 3.73823i 0 − 992.374i 0 0 0
287.6 0 0 0 − 3.73823i 0 992.374i 0 0 0
287.7 0 0 0 3.73823i 0 − 992.374i 0 0 0
287.8 0 0 0 3.73823i 0 992.374i 0 0 0
287.9 0 0 0 324.150i 0 − 1034.55i 0 0 0
287.10 0 0 0 324.150i 0 1034.55i 0 0 0
287.11 0 0 0 484.866i 0 − 857.827i 0 0 0
287.12 0 0 0 484.866i 0 857.827i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.12
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.a ✓ 12
3.b odd 2 1 inner 288.8.c.a ✓ 12
4.b odd 2 1 inner 288.8.c.a ✓ 12
8.b even 2 1 576.8.c.f 12
8.d odd 2 1 576.8.c.f 12
12.b even 2 1 inner 288.8.c.a ✓ 12
24.f even 2 1 576.8.c.f 12
24.h odd 2 1 576.8.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.8.c.a ✓ 12 1.a even 1 1 trivial
288.8.c.a ✓ 12 3.b odd 2 1 inner
288.8.c.a ✓ 12 4.b odd 2 1 inner
288.8.c.a ✓ 12 12.b even 2 1 inner
576.8.c.f 12 8.b even 2 1
576.8.c.f 12 8.d odd 2 1
576.8.c.f 12 24.f even 2 1
576.8.c.f 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 340182T_{5}^{4} + 24706913100T_{5}^{2} + 345197405000 \) 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
$5$ \( (T^{6} + 340182 T^{4} + \cdots + 345197405000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 77\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 51\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 4788 T^{2} + \cdots - 51176189248)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 28\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 84\!\cdots\!44)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 16\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 17\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 17\!\cdots\!28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 49\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 30\!\cdots\!52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 64\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 84\!\cdots\!24)^{4} \) Copy content Toggle raw display
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