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Properties

Label 560.3.br.a
Level $560$
Weight $3$
Character orbit 560.br
Analytic conductor $15.259$
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] = [560,3,Mod(129,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.129");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.br (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: \(15.2588948042\)
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: \( x^{12} - 15x^{10} + 180x^{8} - 669x^{6} + 1980x^{4} - 135x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + (\beta_{11} + \beta_{10}) q^{3} + ( - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} + \beta_{10}) q^{3} + ( - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{5}+ \cdots + ( - 3 \beta_{6} - 3 \beta_{5} + \cdots - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{9} - 6 q^{11} - 48 q^{15} - 18 q^{19} + 12 q^{21} + 18 q^{25} + 108 q^{31} - 222 q^{35} + 240 q^{39} - 216 q^{45} - 324 q^{49} - 180 q^{51} - 396 q^{59} - 108 q^{61} - 54 q^{65} - 192 q^{71} + 216 q^{75} + 192 q^{79} + 294 q^{81} - 192 q^{85} + 684 q^{89} + 72 q^{91} - 288 q^{95} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 180x^{8} - 669x^{6} + 1980x^{4} - 135x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 12\nu^{8} - 144\nu^{6} + 132\nu^{4} - 9\nu^{2} - 7767 ) / 1575 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\nu^{10} - 179\nu^{8} + 2148\nu^{6} - 7884\nu^{4} + 23628\nu^{2} - 1611 ) / 1575 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{11} + 179\nu^{9} - 2148\nu^{7} + 7884\nu^{5} - 23628\nu^{3} + 1611\nu ) / 1575 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\nu^{10} - 811\nu^{8} + 9767\nu^{6} - 37971\nu^{4} + 111777\nu^{2} - 15069 ) / 3150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -62\nu^{10} + 919\nu^{8} - 10958\nu^{6} + 39159\nu^{4} - 111858\nu^{2} - 7269 ) / 3150 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -12\nu^{10} + 179\nu^{8} - 2148\nu^{6} + 7884\nu^{4} - 23313\nu^{2} + 36 ) / 315 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\nu^{11} - 801\nu^{9} + 9627\nu^{7} - 36471\nu^{5} + 108207\nu^{3} - 15939\nu ) / 1350 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -404\nu^{11} + 6003\nu^{9} - 71826\nu^{7} + 259653\nu^{5} - 757746\nu^{3} - 58743\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -583\nu^{11} + 8781\nu^{9} - 105477\nu^{7} + 396681\nu^{5} - 1179567\nu^{3} + 168489\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -622\nu^{11} + 9249\nu^{9} - 110778\nu^{7} + 401829\nu^{5} - 1179918\nu^{3} - 86229\nu ) / 9450 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 5\beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{10} - 2\beta_{9} + \beta_{8} - 9\beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} - 2\beta_{6} + 4\beta_{5} + 45\beta_{3} - 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{11} + 28\beta_{10} - 16\beta_{9} + 32\beta_{8} - 93\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{6} + 30\beta_{5} - 135\beta_{2} - 453 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{11} + 165\beta_{10} + 195\beta_{9} + 195\beta_{8} - 933\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1488\beta_{7} + 720\beta_{6} - 360\beta_{5} - 4785\beta_{3} - 4785 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3696\beta_{11} - 1848\beta_{10} + 4416\beta_{9} - 2208\beta_{8} + 10737\beta_{4} - 10377\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -16281\beta_{7} + 4056\beta_{6} - 8112\beta_{5} - 51525\beta_{3} + 16281\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -20337\beta_{11} - 40674\beta_{10} + 24393\beta_{9} - 48786\beta_{8} + 116649\beta_{4} - 4056\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1 + \beta_{3}\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
2.85853 + 1.65037i
−0.226181 − 0.130586i
−1.74002 − 1.00460i
1.74002 + 1.00460i
0.226181 + 0.130586i
−2.85853 − 1.65037i
2.85853 − 1.65037i
−0.226181 + 0.130586i
−1.74002 + 1.00460i
1.74002 − 1.00460i
0.226181 − 0.130586i
−2.85853 + 1.65037i
0 −2.20085 + 3.81198i 0 3.20324 + 3.83918i 0 −5.32175 + 4.54742i 0 −5.18747 − 8.98496i 0
129.2 0 −1.88157 + 3.25898i 0 2.11218 − 4.53196i 0 1.66053 − 6.80019i 0 −2.58064 − 4.46979i 0
129.3 0 −0.784824 + 1.35935i 0 −3.93275 + 3.08763i 0 1.38623 + 6.86137i 0 3.26810 + 5.66052i 0
129.4 0 0.784824 − 1.35935i 0 −4.64034 + 1.86205i 0 −1.38623 − 6.86137i 0 3.26810 + 5.66052i 0
129.5 0 1.88157 − 3.25898i 0 4.98089 + 0.436783i 0 −1.66053 + 6.80019i 0 −2.58064 − 4.46979i 0
129.6 0 2.20085 − 3.81198i 0 −1.72321 − 4.69367i 0 5.32175 − 4.54742i 0 −5.18747 − 8.98496i 0
369.1 0 −2.20085 − 3.81198i 0 3.20324 − 3.83918i 0 −5.32175 − 4.54742i 0 −5.18747 + 8.98496i 0
369.2 0 −1.88157 − 3.25898i 0 2.11218 + 4.53196i 0 1.66053 + 6.80019i 0 −2.58064 + 4.46979i 0
369.3 0 −0.784824 − 1.35935i 0 −3.93275 − 3.08763i 0 1.38623 − 6.86137i 0 3.26810 − 5.66052i 0
369.4 0 0.784824 + 1.35935i 0 −4.64034 − 1.86205i 0 −1.38623 + 6.86137i 0 3.26810 − 5.66052i 0
369.5 0 1.88157 + 3.25898i 0 4.98089 − 0.436783i 0 −1.66053 − 6.80019i 0 −2.58064 + 4.46979i 0
369.6 0 2.20085 + 3.81198i 0 −1.72321 + 4.69367i 0 5.32175 + 4.54742i 0 −5.18747 + 8.98496i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.br.a 12
4.b odd 2 1 35.3.i.a ✓ 12
5.b even 2 1 inner 560.3.br.a 12
7.d odd 6 1 inner 560.3.br.a 12
12.b even 2 1 315.3.bi.c 12
20.d odd 2 1 35.3.i.a ✓ 12
20.e even 4 2 175.3.i.c 12
28.d even 2 1 245.3.i.d 12
28.f even 6 1 35.3.i.a ✓ 12
28.f even 6 1 245.3.c.a 12
28.g odd 6 1 245.3.c.a 12
28.g odd 6 1 245.3.i.d 12
35.i odd 6 1 inner 560.3.br.a 12
60.h even 2 1 315.3.bi.c 12
84.j odd 6 1 315.3.bi.c 12
140.c even 2 1 245.3.i.d 12
140.p odd 6 1 245.3.c.a 12
140.p odd 6 1 245.3.i.d 12
140.s even 6 1 35.3.i.a ✓ 12
140.s even 6 1 245.3.c.a 12
140.x odd 12 2 175.3.i.c 12
420.be odd 6 1 315.3.bi.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.i.a ✓ 12 4.b odd 2 1
35.3.i.a ✓ 12 20.d odd 2 1
35.3.i.a ✓ 12 28.f even 6 1
35.3.i.a ✓ 12 140.s even 6 1
175.3.i.c 12 20.e even 4 2
175.3.i.c 12 140.x odd 12 2
245.3.c.a 12 28.f even 6 1
245.3.c.a 12 28.g odd 6 1
245.3.c.a 12 140.p odd 6 1
245.3.c.a 12 140.s even 6 1
245.3.i.d 12 28.d even 2 1
245.3.i.d 12 28.g odd 6 1
245.3.i.d 12 140.c even 2 1
245.3.i.d 12 140.p odd 6 1
315.3.bi.c 12 12.b even 2 1
315.3.bi.c 12 60.h even 2 1
315.3.bi.c 12 84.j odd 6 1
315.3.bi.c 12 420.be odd 6 1
560.3.br.a 12 1.a even 1 1 trivial
560.3.br.a 12 5.b even 2 1 inner
560.3.br.a 12 7.d odd 6 1 inner
560.3.br.a 12 35.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 36T_{3}^{10} + 939T_{3}^{8} + 11500T_{3}^{6} + 103113T_{3}^{4} + 241332T_{3}^{2} + 456976 \) acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 36 T^{10} + \cdots + 456976 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} + 3 T^{5} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 489 T^{4} + \cdots - 4104676)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 119168138285056 \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{5} + \cdots + 2333772)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59129994297609 \) Copy content Toggle raw display
$29$ \( (T^{3} - 543 T - 4508)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} - 54 T^{5} + \cdots + 12288)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + 1077 T^{4} + \cdots + 30432675)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6792 T^{4} + \cdots + 2881200)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + 198 T^{5} + \cdots + 139491478272)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 54 T^{5} + \cdots + 4280171952)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{3} + 48 T^{2} + \cdots - 15508)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} - 96 T^{5} + \cdots + 330803344)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 23520 T^{4} + \cdots - 118402057216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 342 T^{5} + \cdots + 285265069488)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 18228 T^{4} + \cdots - 207129664)^{2} \) Copy content Toggle raw display
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