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Properties

Label 539.2.b.b
Level $539$
Weight $2$
Character orbit 539.b
Analytic conductor $4.304$
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] = [539,2,Mod(538,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.538");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.b (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: \(4.30393666895\)
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: \( x^{12} - 8x^{10} + 47x^{8} - 122x^{6} + 233x^{4} - 119x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 77)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{7} + \beta_{5}) q^{5} + \beta_1 q^{6} + \beta_{10} q^{8} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{7} + \beta_{5}) q^{5} + \beta_1 q^{6} + \beta_{10} q^{8} + (\beta_{4} + 1) q^{9} + ( - \beta_{9} - 2 \beta_{3}) q^{10} + ( - \beta_{11} - \beta_{2} + 1) q^{11} - \beta_{8} q^{12} - \beta_{3} q^{13} + (2 \beta_{4} - 1) q^{15} + (\beta_{4} + \beta_{2} - 2) q^{16} + \beta_{9} q^{17} + ( - \beta_{11} - \beta_{10} + \beta_{6}) q^{18} + ( - \beta_{9} - 2 \beta_1) q^{19} + (\beta_{8} + \beta_{7} - 2 \beta_{5}) q^{20} + ( - \beta_{10} + 2 \beta_{6} + \cdots - 1) q^{22}+ \cdots + (\beta_{10} + 3 \beta_{6} + \beta_{4} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} + 8 q^{9} + 8 q^{11} - 20 q^{15} - 24 q^{16} - 4 q^{22} + 40 q^{23} - 28 q^{25} - 12 q^{36} - 32 q^{37} - 40 q^{44} + 4 q^{53} + 12 q^{58} + 28 q^{64} + 48 q^{67} + 20 q^{71} + 8 q^{78} - 28 q^{81} + 76 q^{86} + 8 q^{88} - 20 q^{92} - 24 q^{93} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{10} + 47x^{8} - 122x^{6} + 233x^{4} - 119x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -13\nu^{11} - 113\nu^{9} + 929\nu^{7} - 6401\nu^{5} + 11510\nu^{3} - 10430\nu ) / 6363 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64\nu^{10} - 376\nu^{8} + 2209\nu^{6} - 1864\nu^{4} + 952\nu^{2} + 24404 ) / 9999 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -656\nu^{11} + 7187\nu^{9} - 45140\nu^{7} + 152426\nu^{5} - 299729\nu^{3} + 273140\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 184\nu^{10} - 1081\nu^{8} + 5101\nu^{6} - 5359\nu^{4} + 2737\nu^{2} + 5168 ) / 9999 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 118\nu^{10} - 769\nu^{8} + 4783\nu^{6} - 11239\nu^{4} + 24253\nu^{2} - 7462 ) / 6363 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -799\nu^{11} + 5944\nu^{9} - 34921\nu^{7} + 82015\nu^{5} - 173119\nu^{3} + 18424\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1450\nu^{10} - 12685\nu^{8} + 71608\nu^{6} - 191383\nu^{4} + 279043\nu^{2} - 92449 ) / 69993 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1598\nu^{10} - 11888\nu^{8} + 69842\nu^{6} - 164030\nu^{4} + 346238\nu^{2} - 106841 ) / 69993 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2300\nu^{11} - 18512\nu^{9} + 108758\nu^{7} - 301964\nu^{5} + 609155\nu^{3} - 555338\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3196\nu^{11} + 23776\nu^{9} - 139684\nu^{7} + 328060\nu^{5} - 622483\nu^{3} + 73696\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 221\nu^{11} - 1715\nu^{9} + 9659\nu^{7} - 22685\nu^{5} + 37034\nu^{3} - 5096\nu ) / 3333 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} - \beta_{7} - \beta_{5} - \beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 4\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{8} - 3\beta_{7} - 6\beta_{5} - \beta_{4} + 5\beta_{2} - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{11} + 6\beta_{10} - 6\beta_{9} - 17\beta_{6} - 19\beta_{3} - 16\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{4} + 23\beta_{2} - 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{11} - 31\beta_{10} - 31\beta_{9} + 75\beta_{6} - 91\beta_{3} - 67\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -127\beta_{8} + 12\beta_{7} + 153\beta_{5} - 47\beta_{4} + 106\beta_{2} - 233 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} - 153\beta_{10} + 339\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -572\beta_{8} - 2\beta_{7} + 739\beta_{5} + 247\beta_{4} - 492\beta_{2} + 1064 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -247\beta_{11} - 739\beta_{10} + 739\beta_{9} + 1556\beta_{6} + 2050\beta_{3} + 1309\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(442\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
538.1
−1.87742 + 1.08393i
1.87742 + 1.08393i
−1.43898 + 0.830794i
1.43898 + 0.830794i
0.636099 + 0.367252i
−0.636099 + 0.367252i
−0.636099 − 0.367252i
0.636099 − 0.367252i
1.43898 − 0.830794i
−1.43898 − 0.830794i
1.87742 − 1.08393i
−1.87742 − 1.08393i
− 2.16786i − 0.641589i −2.69963 3.39272i −1.39088 0 1.51670i 2.58836 7.35494
538.2 − 2.16786i 0.641589i −2.69963 − 3.39272i 1.39088 0 1.51670i 2.58836 −7.35494
538.3 − 1.66159i − 2.27639i −0.760877 − 3.23490i −3.78242 0 − 2.05891i −2.18194 −5.37507
538.4 − 1.66159i 2.27639i −0.760877 3.23490i 3.78242 0 − 2.05891i −2.18194 5.37507
538.5 − 0.734503i − 1.18593i 1.46050 0.157816i −0.871067 0 − 2.54175i 1.59358 0.115917
538.6 − 0.734503i 1.18593i 1.46050 − 0.157816i 0.871067 0 − 2.54175i 1.59358 −0.115917
538.7 0.734503i − 1.18593i 1.46050 0.157816i 0.871067 0 2.54175i 1.59358 −0.115917
538.8 0.734503i 1.18593i 1.46050 − 0.157816i −0.871067 0 2.54175i 1.59358 0.115917
538.9 1.66159i − 2.27639i −0.760877 − 3.23490i 3.78242 0 2.05891i −2.18194 5.37507
538.10 1.66159i 2.27639i −0.760877 3.23490i −3.78242 0 2.05891i −2.18194 −5.37507
538.11 2.16786i − 0.641589i −2.69963 3.39272i 1.39088 0 − 1.51670i 2.58836 −7.35494
538.12 2.16786i 0.641589i −2.69963 − 3.39272i −1.39088 0 − 1.51670i 2.58836 7.35494
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 538.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.b.b 12
7.b odd 2 1 inner 539.2.b.b 12
7.c even 3 1 77.2.i.a ✓ 12
7.c even 3 1 539.2.i.c 12
7.d odd 6 1 77.2.i.a ✓ 12
7.d odd 6 1 539.2.i.c 12
11.b odd 2 1 inner 539.2.b.b 12
21.g even 6 1 693.2.bg.a 12
21.h odd 6 1 693.2.bg.a 12
28.f even 6 1 1232.2.bn.a 12
28.g odd 6 1 1232.2.bn.a 12
77.b even 2 1 inner 539.2.b.b 12
77.h odd 6 1 77.2.i.a ✓ 12
77.h odd 6 1 539.2.i.c 12
77.i even 6 1 77.2.i.a ✓ 12
77.i even 6 1 539.2.i.c 12
77.m even 15 4 847.2.r.b 48
77.n even 30 4 847.2.r.b 48
77.o odd 30 4 847.2.r.b 48
77.p odd 30 4 847.2.r.b 48
231.k odd 6 1 693.2.bg.a 12
231.l even 6 1 693.2.bg.a 12
308.m odd 6 1 1232.2.bn.a 12
308.n even 6 1 1232.2.bn.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.i.a ✓ 12 7.c even 3 1
77.2.i.a ✓ 12 7.d odd 6 1
77.2.i.a ✓ 12 77.h odd 6 1
77.2.i.a ✓ 12 77.i even 6 1
539.2.b.b 12 1.a even 1 1 trivial
539.2.b.b 12 7.b odd 2 1 inner
539.2.b.b 12 11.b odd 2 1 inner
539.2.b.b 12 77.b even 2 1 inner
539.2.i.c 12 7.c even 3 1
539.2.i.c 12 7.d odd 6 1
539.2.i.c 12 77.h odd 6 1
539.2.i.c 12 77.i even 6 1
693.2.bg.a 12 21.g even 6 1
693.2.bg.a 12 21.h odd 6 1
693.2.bg.a 12 231.k odd 6 1
693.2.bg.a 12 231.l even 6 1
847.2.r.b 48 77.m even 15 4
847.2.r.b 48 77.n even 30 4
847.2.r.b 48 77.o odd 30 4
847.2.r.b 48 77.p odd 30 4
1232.2.bn.a 12 28.f even 6 1
1232.2.bn.a 12 28.g odd 6 1
1232.2.bn.a 12 308.m odd 6 1
1232.2.bn.a 12 308.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 8T_{2}^{4} + 17T_{2}^{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 8 T^{4} + 17 T^{2} + 7)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 7 T^{4} + 10 T^{2} + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 22 T^{4} + 121 T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} + \cdots + 1331)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 11 T^{4} + \cdots - 21)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 39 T^{4} + \cdots - 1701)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 83 T^{4} + \cdots - 17661)^{2} \) Copy content Toggle raw display
$23$ \( (T^{3} - 10 T^{2} + \cdots - 21)^{4} \) Copy content Toggle raw display
$29$ \( (T^{6} + 83 T^{4} + \cdots + 5887)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 27 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 8 T^{2} + 4 T - 24)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 99 T^{4} + \cdots - 15309)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 76 T^{4} + \cdots + 5103)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 55 T^{4} + \cdots + 5043)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 66 T - 21)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 298 T^{4} + \cdots + 10443)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 212 T^{4} + \cdots - 108864)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 12 T^{2} + \cdots - 56)^{4} \) Copy content Toggle raw display
$71$ \( (T^{3} - 5 T^{2} - 12 T + 63)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 68 T^{4} + \cdots - 1029)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 51 T^{4} + \cdots + 567)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 471 T^{4} + \cdots - 3145149)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 199 T^{4} + \cdots + 10443)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 63 T^{4} + \cdots + 2187)^{2} \) Copy content Toggle raw display
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