↑

Properties

Label 539.2.i.c
Level $539$
Weight $2$
Character orbit 539.i
Analytic conductor $4.304$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(362,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.362");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.i (of order \(6\), degree \(2\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_1) q^{2} + (\beta_{9} + \beta_{8} - \beta_{2} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + (\beta_{9} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_1) q^{2} + (\beta_{9} + \beta_{8} - \beta_{2} + 1) q^{3} + ( - \beta_{9} - \beta_{8} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{11} - \beta_{9} - \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 4 q^{4} - 4 q^{9} - 4 q^{11} + 18 q^{12} - 20 q^{15} + 12 q^{16} - 4 q^{22} - 20 q^{23} + 14 q^{25} - 18 q^{26} - 6 q^{31} - 18 q^{33} - 12 q^{36} + 16 q^{37} + 48 q^{38} + 20 q^{44} - 54 q^{45} + 18 q^{47} - 2 q^{53} - 6 q^{58} + 12 q^{59} + 28 q^{64} + 42 q^{66} - 24 q^{67} + 20 q^{71} + 78 q^{75} + 8 q^{78} - 30 q^{80} + 14 q^{81} - 54 q^{82} - 38 q^{86} - 4 q^{88} + 66 q^{89} - 20 q^{92} + 12 q^{93} + 12 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{10} + 113\nu^{8} - 929\nu^{6} + 6401\nu^{4} - 11510\nu^{2} + 16793 ) / 6363 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -148\nu^{10} - 797\nu^{8} + 1766\nu^{6} - 27353\nu^{4} + 2798\nu^{2} - 55601 ) / 69993 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 305\nu^{10} - 3875\nu^{8} + 25682\nu^{6} - 83459\nu^{4} + 133274\nu^{2} - 13895 ) / 69993 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 64\nu^{11} - 376\nu^{9} + 2209\nu^{7} - 1864\nu^{5} + 952\nu^{3} + 34403\nu ) / 9999 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 656\nu^{11} - 7187\nu^{9} + 45140\nu^{7} - 152426\nu^{5} + 299729\nu^{3} - 203147\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -799\nu^{11} + 5944\nu^{9} - 34921\nu^{7} + 82015\nu^{5} - 173119\nu^{3} + 18424\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 799\nu^{10} - 5944\nu^{8} + 34921\nu^{6} - 82015\nu^{4} + 173119\nu^{2} - 88417 ) / 69993 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2249\nu^{10} + 18629\nu^{8} - 106529\nu^{6} + 273398\nu^{4} - 452162\nu^{2} + 180866 ) / 69993 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2392\nu^{11} + 17386\nu^{9} - 96310\nu^{7} + 202987\nu^{5} - 325552\nu^{3} - 3857\nu ) / 69993 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -916\nu^{11} + 7048\nu^{9} - 41407\nu^{7} + 105004\nu^{5} - 205273\nu^{3} + 104839\nu ) / 23331 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{8} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 4\beta_{7} - \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{9} + 12\beta_{8} + 6\beta_{4} + 6\beta_{3} + 5\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{11} - \beta_{10} - 16\beta_{7} + 2\beta_{6} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{9} + 8\beta_{8} + 31\beta_{4} + 15\beta_{2} - 44 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{10} + 8\beta_{6} + 31\beta_{5} - 83\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -59\beta_{9} - 186\beta_{8} - 153\beta_{3} - 47\beta_{2} - 80 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -153\beta_{11} + 94\beta_{10} + 292\beta_{7} - 47\beta_{6} + 153\beta_{5} - 47\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -492\beta_{9} - 1064\beta_{8} - 739\beta_{4} - 739\beta_{3} - 492\beta_{2} + 492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -739\beta_{11} + 247\beta_{10} + 1309\beta_{7} - 494\beta_{6} + 1556\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(199\) \(442\)
\(\chi(n)\) \(1 + \beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1
−1.87742 + 1.08393i
−1.43898 + 0.830794i
−0.636099 + 0.367252i
0.636099 − 0.367252i
1.43898 − 0.830794i
1.87742 − 1.08393i
−1.87742 − 1.08393i
−1.43898 − 0.830794i
−0.636099 − 0.367252i
0.636099 + 0.367252i
1.43898 + 0.830794i
1.87742 + 1.08393i
−1.87742 − 1.08393i 0.555632 − 0.320794i 1.34981 + 2.33795i 2.93818 + 1.69636i −1.39088 0 − 1.51670i −1.29418 + 2.24159i −3.67747 − 6.36957i
362.2 −1.43898 − 0.830794i 1.97141 − 1.13819i 0.380438 + 0.658939i −2.80150 − 1.61745i −3.78242 0 2.05891i 1.09097 − 1.88962i 2.68754 + 4.65495i
362.3 −0.636099 − 0.367252i −1.02704 + 0.592963i −0.730252 − 1.26483i −0.136673 − 0.0789082i 0.871067 0 2.54175i −0.796790 + 1.38008i 0.0579584 + 0.100387i
362.4 0.636099 + 0.367252i −1.02704 + 0.592963i −0.730252 − 1.26483i −0.136673 − 0.0789082i −0.871067 0 − 2.54175i −0.796790 + 1.38008i −0.0579584 − 0.100387i
362.5 1.43898 + 0.830794i 1.97141 − 1.13819i 0.380438 + 0.658939i −2.80150 − 1.61745i 3.78242 0 − 2.05891i 1.09097 − 1.88962i −2.68754 − 4.65495i
362.6 1.87742 + 1.08393i 0.555632 − 0.320794i 1.34981 + 2.33795i 2.93818 + 1.69636i 1.39088 0 1.51670i −1.29418 + 2.24159i 3.67747 + 6.36957i
472.1 −1.87742 + 1.08393i 0.555632 + 0.320794i 1.34981 − 2.33795i 2.93818 − 1.69636i −1.39088 0 1.51670i −1.29418 − 2.24159i −3.67747 + 6.36957i
472.2 −1.43898 + 0.830794i 1.97141 + 1.13819i 0.380438 − 0.658939i −2.80150 + 1.61745i −3.78242 0 − 2.05891i 1.09097 + 1.88962i 2.68754 − 4.65495i
472.3 −0.636099 + 0.367252i −1.02704 − 0.592963i −0.730252 + 1.26483i −0.136673 + 0.0789082i 0.871067 0 − 2.54175i −0.796790 − 1.38008i 0.0579584 − 0.100387i
472.4 0.636099 − 0.367252i −1.02704 − 0.592963i −0.730252 + 1.26483i −0.136673 + 0.0789082i −0.871067 0 2.54175i −0.796790 − 1.38008i −0.0579584 + 0.100387i
472.5 1.43898 − 0.830794i 1.97141 + 1.13819i 0.380438 − 0.658939i −2.80150 + 1.61745i 3.78242 0 2.05891i 1.09097 + 1.88962i −2.68754 + 4.65495i
472.6 1.87742 − 1.08393i 0.555632 + 0.320794i 1.34981 − 2.33795i 2.93818 − 1.69636i 1.39088 0 − 1.51670i −1.29418 − 2.24159i 3.67747 − 6.36957i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.i.c 12
7.b odd 2 1 77.2.i.a ✓ 12
7.c even 3 1 77.2.i.a ✓ 12
7.c even 3 1 539.2.b.b 12
7.d odd 6 1 539.2.b.b 12
7.d odd 6 1 inner 539.2.i.c 12
11.b odd 2 1 inner 539.2.i.c 12
21.c even 2 1 693.2.bg.a 12
21.h odd 6 1 693.2.bg.a 12
28.d even 2 1 1232.2.bn.a 12
28.g odd 6 1 1232.2.bn.a 12
77.b even 2 1 77.2.i.a ✓ 12
77.h odd 6 1 77.2.i.a ✓ 12
77.h odd 6 1 539.2.b.b 12
77.i even 6 1 539.2.b.b 12
77.i even 6 1 inner 539.2.i.c 12
77.j odd 10 4 847.2.r.b 48
77.l even 10 4 847.2.r.b 48
77.m even 15 4 847.2.r.b 48
77.o odd 30 4 847.2.r.b 48
231.h odd 2 1 693.2.bg.a 12
231.l even 6 1 693.2.bg.a 12
308.g odd 2 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.b odd 2 1
77.2.i.a ✓ 12 7.c even 3 1
77.2.i.a ✓ 12 77.b even 2 1
77.2.i.a ✓ 12 77.h odd 6 1
539.2.b.b 12 7.c even 3 1
539.2.b.b 12 7.d odd 6 1
539.2.b.b 12 77.h odd 6 1
539.2.b.b 12 77.i even 6 1
539.2.i.c 12 1.a even 1 1 trivial
539.2.i.c 12 7.d odd 6 1 inner
539.2.i.c 12 11.b odd 2 1 inner
539.2.i.c 12 77.i even 6 1 inner
693.2.bg.a 12 21.c even 2 1
693.2.bg.a 12 21.h odd 6 1
693.2.bg.a 12 231.h odd 2 1
693.2.bg.a 12 231.l even 6 1
847.2.r.b 48 77.j odd 10 4
847.2.r.b 48 77.l even 10 4
847.2.r.b 48 77.m even 15 4
847.2.r.b 48 77.o odd 30 4
1232.2.bn.a 12 28.d even 2 1
1232.2.bn.a 12 28.g odd 6 1
1232.2.bn.a 12 308.g odd 2 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 \( T_{2}^{12} - 8T_{2}^{10} + 47T_{2}^{8} - 122T_{2}^{6} + 233T_{2}^{4} - 119T_{2}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} + T^{4} + \cdots + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 11 T^{4} + 121 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{6} - 11 T^{4} + \cdots - 21)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 39 T^{10} + \cdots + 2893401 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 311910921 \) Copy content Toggle raw display
$23$ \( (T^{6} + 10 T^{5} + \cdots + 441)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 83 T^{4} + \cdots + 5887)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 3 T^{5} - 9 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 8 T^{5} + \cdots + 576)^{2} \) 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
$47$ \( (T^{6} - 9 T^{5} + \cdots + 5043)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + T^{5} + 67 T^{4} + \cdots + 441)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 6 T^{5} + \cdots + 10443)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11851370496 \) Copy content Toggle raw display
$67$ \( (T^{6} + 12 T^{5} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 5 T^{2} - 12 T + 63)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + 68 T^{10} + \cdots + 1058841 \) Copy content Toggle raw display
$79$ \( T^{12} - 51 T^{10} + \cdots + 321489 \) Copy content Toggle raw display
$83$ \( (T^{6} - 471 T^{4} + \cdots - 3145149)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 33 T^{5} + \cdots + 10443)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 63 T^{4} + \cdots + 2187)^{2} \) Copy content Toggle raw display
show more
show less