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Properties

Label 288.7.b.c
Level $288$
Weight $7$
Character orbit 288.b
Analytic conductor $66.256$
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] = [288,7,Mod(271,288)]
 
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, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.271");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.b (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: \(66.2555760825\)
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: \( x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{66}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_1 q^{7} - \beta_{3} q^{11} - \beta_{4} q^{13} + \beta_{5} q^{17} + (\beta_{8} - 328) q^{19} + ( - \beta_{7} - \beta_{6} + 17 \beta_{2}) q^{23} + (\beta_{9} + 2 \beta_{8} - 3983) q^{25} + (2 \beta_{7} + \beta_{6} - 35 \beta_{2}) q^{29} + ( - \beta_{11} - 4 \beta_{4} - 4 \beta_1) q^{31} + (\beta_{10} - 8 \beta_{5} - 4 \beta_{3}) q^{35} + (2 \beta_{11} - 5 \beta_{4} - 22 \beta_1) q^{37} + (2 \beta_{10} + \beta_{5} + 30 \beta_{3}) q^{41} + ( - 8 \beta_{9} - 5 \beta_{8} + 28392) q^{43} + (\beta_{7} - 7 \beta_{6} - 193 \beta_{2}) q^{47} + (9 \beta_{9} - 14 \beta_{8} - 25387) q^{49} + ( - 6 \beta_{7} - 19 \beta_{6} + 11 \beta_{2}) q^{53} + ( - 7 \beta_{11} + 100 \beta_{4} - 11 \beta_1) q^{55} + (11 \beta_{10} + 40 \beta_{5} + 7 \beta_{3}) q^{59} + (10 \beta_{11} - 23 \beta_{4} + 402 \beta_1) q^{61} + (18 \beta_{10} - 19 \beta_{5} - 242 \beta_{3}) q^{65} + ( - 24 \beta_{9} - 6 \beta_{8} + 80176) q^{67} + (16 \beta_{7} - 40 \beta_{6} + 544 \beta_{2}) q^{71} + (25 \beta_{9} + 18 \beta_{8} - 89130) q^{73} + ( - 18 \beta_{7} + \cdots - 220 \beta_{2}) q^{77}+ \cdots + ( - 6 \beta_{9} + 84 \beta_{8} - 7246) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3936 q^{19} - 47796 q^{25} + 340704 q^{43} - 304644 q^{49} + 962112 q^{67} - 1069560 q^{73} - 775008 q^{91} - 86952 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\nu^{10} - 1390\nu^{8} + 48048\nu^{6} - 2776576\nu^{4} + 125763584\nu^{2} - 8908701696 ) / 92274688 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 75\nu^{11} - 1318\nu^{9} - 41360\nu^{7} + 18287104\nu^{5} + 845086720\nu^{3} + 8422162432\nu ) / 5905580032 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{11} + 2974\nu^{9} - 97072\nu^{7} + 1650176\nu^{5} + 570753024\nu^{3} + 37211865088\nu ) / 268435456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -63\nu^{10} - 5426\nu^{8} + 73040\nu^{6} + 17104384\nu^{4} - 1020723200\nu^{2} - 77594624000 ) / 46137344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51\nu^{11} - 1142\nu^{9} + 36592\nu^{7} - 10044928\nu^{5} + 1263992832\nu^{3} + 7214202880\nu ) / 134217728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1633 \nu^{11} - 105870 \nu^{9} - 4674384 \nu^{7} - 157949440 \nu^{5} + \cdots - 3379635945472 \nu ) / 2952790016 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3451 \nu^{11} - 16250 \nu^{9} + 11899536 \nu^{7} + 1167676928 \nu^{5} + \cdots + 1626349764608 \nu ) / 5905580032 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{10} + 134\nu^{8} - 2928\nu^{6} + 542208\nu^{4} + 48955392\nu^{2} + 3170893824 ) / 524288 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} - 78\nu^{8} - 3408\nu^{6} - 73216\nu^{4} + 2818048\nu^{2} - 872415232 ) / 131072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -231\nu^{11} - 16994\nu^{9} - 969520\nu^{7} - 17093120\nu^{5} - 3002793984\nu^{3} - 22917677056\nu ) / 268435456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 829\nu^{10} + 73366\nu^{8} + 5142544\nu^{6} + 253806080\nu^{4} + 34482225152\nu^{2} + 1266327486464 ) / 92274688 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 2\beta_{6} - \beta_{3} - 12\beta_{2} ) / 2048 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{9} - 4\beta_{4} + 5\beta _1 - 6656 ) / 512 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{10} + 32\beta_{7} + 26\beta_{6} + 64\beta_{5} + 129\beta_{3} - 324\beta_{2} ) / 1024 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{11} + 2\beta_{9} + 64\beta_{8} + 348\beta_{4} - 1235\beta _1 - 31232 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -109\beta_{10} + 544\beta_{7} + 178\beta_{6} - 1984\beta_{5} + 237\beta_{3} + 83020\beta_{2} ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 717\beta_{11} - 1782\beta_{9} - 2496\beta_{8} + 2636\beta_{4} + 45313\beta _1 + 2203136 ) / 128 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -26577\beta_{10} + 8864\beta_{7} - 27174\beta_{6} + 6464\beta_{5} - 251311\beta_{3} - 926788\beta_{2} ) / 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4369\beta_{11} - 32430\beta_{9} - 22720\beta_{8} - 166084\beta_{4} - 1825707\beta _1 - 593988096 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 979109 \beta_{10} - 53472 \beta_{7} + 711138 \beta_{6} - 1244608 \beta_{5} + \cdots - 47105940 \beta_{2} ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -687515\beta_{11} - 1077318\beta_{9} + 2426944\beta_{8} + 341996\beta_{4} + 44779257\beta _1 - 7515296256 ) / 32 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11750025 \beta_{10} - 38363744 \beta_{7} - 14214774 \beta_{6} + 5353280 \beta_{5} + \cdots + 2237709532 \beta_{2} ) / 64 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−4.24448 + 6.78118i
4.24448 + 6.78118i
−1.98725 − 7.74925i
1.98725 − 7.74925i
7.38480 − 3.07648i
−7.38480 − 3.07648i
−7.38480 + 3.07648i
7.38480 + 3.07648i
1.98725 + 7.74925i
−1.98725 + 7.74925i
4.24448 − 6.78118i
−4.24448 − 6.78118i
0 0 0 − 206.098i 0 − 210.403i 0 0 0
271.2 0 0 0 − 206.098i 0 210.403i 0 0 0
271.3 0 0 0 − 106.706i 0 − 614.112i 0 0 0
271.4 0 0 0 − 106.706i 0 614.112i 0 0 0
271.5 0 0 0 − 70.4379i 0 − 87.7768i 0 0 0
271.6 0 0 0 − 70.4379i 0 87.7768i 0 0 0
271.7 0 0 0 70.4379i 0 − 87.7768i 0 0 0
271.8 0 0 0 70.4379i 0 87.7768i 0 0 0
271.9 0 0 0 106.706i 0 − 614.112i 0 0 0
271.10 0 0 0 106.706i 0 614.112i 0 0 0
271.11 0 0 0 206.098i 0 − 210.403i 0 0 0
271.12 0 0 0 206.098i 0 210.403i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.7.b.c 12
3.b odd 2 1 inner 288.7.b.c 12
4.b odd 2 1 72.7.b.d ✓ 12
8.b even 2 1 72.7.b.d ✓ 12
8.d odd 2 1 inner 288.7.b.c 12
12.b even 2 1 72.7.b.d ✓ 12
24.f even 2 1 inner 288.7.b.c 12
24.h odd 2 1 72.7.b.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.b.d ✓ 12 4.b odd 2 1
72.7.b.d ✓ 12 8.b even 2 1
72.7.b.d ✓ 12 12.b even 2 1
72.7.b.d ✓ 12 24.h odd 2 1
288.7.b.c 12 1.a even 1 1 trivial
288.7.b.c 12 3.b odd 2 1 inner
288.7.b.c 12 8.d odd 2 1 inner
288.7.b.c 12 24.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 58824T_{5}^{4} + 750878400T_{5}^{2} + 2399578560000 \) acting on \(S_{7}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( (T^{6} + \cdots + 2399578560000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 128634653983680)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 51\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 984 T^{2} + \cdots - 40086711808)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 28\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 91\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 657572380382720)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 81\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 44\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 35\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots + 18\!\cdots\!40)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots - 19\!\cdots\!00)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 38\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 11\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 62\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 71\!\cdots\!00)^{4} \) Copy content Toggle raw display
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