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Properties

Label 288.11.b.c
Level $288$
Weight $11$
Character orbit 288.b
Analytic conductor $182.983$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.271");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(182.982888770\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 29481498 x^{18} + 358178748066573 x^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{185}\cdot 3^{42} \)
Twist minimal: no (minimal twist has level 24)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{11} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{11} q^{7} + ( - \beta_{4} - 4590) q^{11} + (\beta_{12} + 7 \beta_1) q^{13} + (\beta_{6} - 4 \beta_{4} + \cdots + 45290) q^{17}+ \cdots + ( - 379 \beta_{10} + 462 \beta_{9} + \cdots + 1163749905) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 91808 q^{11} + 905768 q^{17} + 5107040 q^{19} - 40539484 q^{25} - 68411424 q^{35} - 51872200 q^{41} + 392685920 q^{43} - 1130175964 q^{49} - 1499085440 q^{59} - 1717119456 q^{65} + 4570092800 q^{67} - 2312358680 q^{73} + 14192131360 q^{83} - 8963359000 q^{89} + 401968224 q^{91} + 23275134232 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 29481498 x^{18} + 358178748066573 x^{16} + \cdots + 59\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38\!\cdots\!87 \nu^{18} + \cdots + 12\!\cdots\!75 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22\!\cdots\!59 \nu^{18} + \cdots - 47\!\cdots\!75 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!13 \nu^{18} + \cdots - 18\!\cdots\!25 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!03 \nu^{18} + \cdots - 10\!\cdots\!75 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37\!\cdots\!67 \nu^{18} + \cdots + 14\!\cdots\!75 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48\!\cdots\!23 \nu^{18} + \cdots + 26\!\cdots\!75 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30\!\cdots\!81 \nu^{18} + \cdots - 15\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33\!\cdots\!19 \nu^{18} + \cdots - 12\!\cdots\!75 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!61 \nu^{18} + \cdots + 50\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 88\!\cdots\!99 \nu^{19} + \cdots + 51\!\cdots\!75 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!81 \nu^{19} + \cdots - 99\!\cdots\!25 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\!\cdots\!41 \nu^{19} + \cdots + 16\!\cdots\!25 \nu ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!83 \nu^{19} + \cdots + 50\!\cdots\!75 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 85\!\cdots\!07 \nu^{19} + \cdots + 38\!\cdots\!00 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 28\!\cdots\!88 \nu^{19} + \cdots - 90\!\cdots\!25 \nu ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{19} + \cdots - 25\!\cdots\!25 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 49\!\cdots\!78 \nu^{19} + \cdots + 19\!\cdots\!25 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 13\!\cdots\!17 \nu^{19} + \cdots + 55\!\cdots\!25 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 23\beta_{4} + \beta_{3} - 421\beta_{2} - 11792590 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2217 \beta_{19} - 5599 \beta_{18} - 5434 \beta_{17} - 1088 \beta_{16} + 7252 \beta_{15} + \cdots - 20701326 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2116087 \beta_{10} - 19765166 \beta_{9} + 16812540 \beta_{8} - 12918580 \beta_{7} + \cdots + 122240818014503 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 67895432853 \beta_{19} + 87679284452 \beta_{18} + 98279070091 \beta_{17} + 35065938996 \beta_{16} + \cdots + 272166230785031 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20170322972201 \beta_{10} + 181724226149372 \beta_{9} - 134547472966004 \beta_{8} + \cdots - 80\!\cdots\!59 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 68\!\cdots\!77 \beta_{19} + \cdots - 19\!\cdots\!37 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 84\!\cdots\!61 \beta_{10} + \cdots + 28\!\cdots\!69 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 31\!\cdots\!91 \beta_{19} + \cdots + 74\!\cdots\!89 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 71\!\cdots\!99 \beta_{10} + \cdots - 21\!\cdots\!51 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27\!\cdots\!50 \beta_{19} + \cdots - 58\!\cdots\!69 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 12\!\cdots\!93 \beta_{10} + \cdots + 34\!\cdots\!77 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 46\!\cdots\!79 \beta_{19} + \cdots + 93\!\cdots\!45 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 10\!\cdots\!75 \beta_{10} + \cdots - 27\!\cdots\!85 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 38\!\cdots\!07 \beta_{19} + \cdots - 75\!\cdots\!71 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 22\!\cdots\!49 \beta_{10} + \cdots + 54\!\cdots\!31 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 81\!\cdots\!11 \beta_{19} + \cdots + 15\!\cdots\!27 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 39\!\cdots\!34 \beta_{10} + \cdots - 88\!\cdots\!56 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 13\!\cdots\!03 \beta_{19} + \cdots - 25\!\cdots\!88 \beta_1 ) / 8 \) 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
− 2892.77i
− 2530.52i
− 2201.95i
− 1692.55i
− 1348.27i
− 1310.97i
− 1185.20i
− 1072.58i
− 815.992i
− 488.983i
488.983i
815.992i
1072.58i
1185.20i
1310.97i
1348.27i
1692.55i
2201.95i
2530.52i
2892.77i
0 0 0 − 5785.54i 0 14762.1i 0 0 0
271.2 0 0 0 − 5061.04i 0 − 31335.6i 0 0 0
271.3 0 0 0 − 4403.89i 0 − 7529.34i 0 0 0
271.4 0 0 0 − 3385.10i 0 2423.26i 0 0 0
271.5 0 0 0 − 2696.55i 0 18441.7i 0 0 0
271.6 0 0 0 − 2621.93i 0 22511.9i 0 0 0
271.7 0 0 0 − 2370.40i 0 10154.6i 0 0 0
271.8 0 0 0 − 2145.17i 0 − 29274.4i 0 0 0
271.9 0 0 0 − 1631.98i 0 − 11799.6i 0 0 0
271.10 0 0 0 − 977.966i 0 13461.7i 0 0 0
271.11 0 0 0 977.966i 0 − 13461.7i 0 0 0
271.12 0 0 0 1631.98i 0 11799.6i 0 0 0
271.13 0 0 0 2145.17i 0 29274.4i 0 0 0
271.14 0 0 0 2370.40i 0 − 10154.6i 0 0 0
271.15 0 0 0 2621.93i 0 − 22511.9i 0 0 0
271.16 0 0 0 2696.55i 0 − 18441.7i 0 0 0
271.17 0 0 0 3385.10i 0 − 2423.26i 0 0 0
271.18 0 0 0 4403.89i 0 7529.34i 0 0 0
271.19 0 0 0 5061.04i 0 31335.6i 0 0 0
271.20 0 0 0 5785.54i 0 − 14762.1i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.11.b.c 20
3.b odd 2 1 96.11.b.a 20
4.b odd 2 1 72.11.b.d 20
8.b even 2 1 72.11.b.d 20
8.d odd 2 1 inner 288.11.b.c 20
12.b even 2 1 24.11.b.a ✓ 20
24.f even 2 1 96.11.b.a 20
24.h odd 2 1 24.11.b.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.11.b.a ✓ 20 12.b even 2 1
24.11.b.a ✓ 20 24.h odd 2 1
72.11.b.d 20 4.b odd 2 1
72.11.b.d 20 8.b even 2 1
96.11.b.a 20 3.b odd 2 1
96.11.b.a 20 24.f even 2 1
288.11.b.c 20 1.a even 1 1 trivial
288.11.b.c 20 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 117925992 T_{5}^{18} + \cdots + 62\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 27\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 58\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 78\!\cdots\!12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 97\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 43\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 89\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 90\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 40\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 75\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 32\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 97\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
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