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Properties

Label 560.2.bs.c
Level $560$
Weight $2$
Character orbit 560.bs
Analytic conductor $4.472$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,2,Mod(31,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.31"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.bs (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{4} - \beta_{3}) q^{3} - \beta_{4} q^{5} + (\beta_{11} + \beta_{9} + \cdots + \beta_{3}) q^{7} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{9} + (\beta_{11} - \beta_{7} - \beta_{6} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 2 q^{7} - 10 q^{9} + 6 q^{11} + 2 q^{19} + 12 q^{21} - 18 q^{23} + 6 q^{25} + 20 q^{27} + 16 q^{29} + 4 q^{31} - 6 q^{35} - 4 q^{37} + 24 q^{39} - 12 q^{45} + 12 q^{47} - 8 q^{49} - 12 q^{51}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8 \nu^{10} - 40 \nu^{9} + 295 \nu^{8} - 940 \nu^{7} + 3336 \nu^{6} - 6886 \nu^{5} + 14489 \nu^{4} + \cdots + 6831 ) / 306 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 845 \nu^{11} - 24323 \nu^{10} + 130623 \nu^{9} - 826134 \nu^{8} + 2641205 \nu^{7} - 8701717 \nu^{6} + \cdots - 14895288 ) / 617508 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 845 \nu^{11} - 15028 \nu^{10} + 66132 \nu^{9} - 605637 \nu^{8} + 1905349 \nu^{7} + \cdots - 13900563 ) / 617508 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 100 \nu^{11} - 550 \nu^{10} + 4025 \nu^{9} - 14492 \nu^{8} + 51972 \nu^{7} - 127540 \nu^{6} + \cdots - 125346 ) / 6054 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11603 \nu^{11} + 94591 \nu^{10} - 620641 \nu^{9} + 2753428 \nu^{8} - 9402147 \nu^{7} + \cdots + 26543106 ) / 617508 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11603 \nu^{11} - 33042 \nu^{10} + 312896 \nu^{9} - 490241 \nu^{8} + 2195869 \nu^{7} + \cdots + 13912749 ) / 617508 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20098 \nu^{11} - 118611 \nu^{10} + 821557 \nu^{9} - 2983999 \nu^{8} + 10164992 \nu^{7} + \cdots - 14340123 ) / 617508 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 200 \nu^{11} - 1100 \nu^{10} + 8050 \nu^{9} - 27975 \nu^{8} + 99908 \nu^{7} - 226828 \nu^{6} + \cdots - 90261 ) / 6054 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 24686 \nu^{11} - 143845 \nu^{10} + 1007233 \nu^{9} - 3681745 \nu^{8} + 12651336 \nu^{7} + \cdots - 23982993 ) / 617508 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24686 \nu^{11} + 127701 \nu^{10} - 926513 \nu^{9} + 2983517 \nu^{8} - 10342744 \nu^{7} + \cdots - 9253467 ) / 617508 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} + 2\beta_{5} - \beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} - \beta_{10} - 4\beta_{9} - \beta_{8} + 2\beta_{7} - 2\beta_{6} + 3\beta_{5} - 2\beta_{2} - 6\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{11} + 11 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 22 \beta_{5} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35 \beta_{11} + 20 \beta_{10} + 64 \beta_{9} + 12 \beta_{8} - 31 \beta_{7} + 21 \beta_{6} - 60 \beta_{5} + \cdots + 68 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 66 \beta_{11} - 116 \beta_{10} + 10 \beta_{9} + 41 \beta_{8} - 64 \beta_{7} + 112 \beta_{6} + \cdots - 333 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 487 \beta_{11} - 346 \beta_{10} - 810 \beta_{9} - 100 \beta_{8} + 358 \beta_{7} - 155 \beta_{6} + \cdots - 1020 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 311 \beta_{11} + 1113 \beta_{10} - 1058 \beta_{9} - 596 \beta_{8} + 1248 \beta_{7} - 1644 \beta_{6} + \cdots + 2949 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6301 \beta_{11} + 5342 \beta_{10} + 9154 \beta_{9} + 608 \beta_{8} - 3341 \beta_{7} + 299 \beta_{6} + \cdots + 14706 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2353 \beta_{11} - 8817 \beta_{10} + 23221 \beta_{9} + 7807 \beta_{8} - 19648 \beta_{7} + 21514 \beta_{6} + \cdots - 22508 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 76477 \beta_{11} - 75587 \beta_{10} - 91956 \beta_{9} + 71 \beta_{8} + 22398 \beta_{7} + \cdots - 202462 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1 - \beta_{9}\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 2.35365i
0.500000 − 2.22465i
0.500000 − 0.694987i
0.500000 + 0.777299i
0.500000 + 1.58132i
0.500000 − 3.52469i
0.500000 − 2.35365i
0.500000 + 2.22465i
0.500000 + 0.694987i
0.500000 − 0.777299i
0.500000 − 1.58132i
0.500000 + 3.52469i
0 −1.60984 + 2.78832i 0 0.866025 − 0.500000i 0 −0.279599 − 2.63094i 0 −3.68317 − 6.37943i 0
31.2 0 −0.679312 + 1.17660i 0 −0.866025 + 0.500000i 0 1.26330 − 2.32467i 0 0.577070 + 0.999515i 0
31.3 0 −0.0855194 + 0.148124i 0 0.866025 − 0.500000i 0 −2.53891 + 0.744258i 0 1.48537 + 2.57274i 0
31.4 0 0.821662 − 1.42316i 0 −0.866025 + 0.500000i 0 1.59679 + 2.10956i 0 0.149742 + 0.259361i 0
31.5 0 1.22368 − 2.11947i 0 −0.866025 + 0.500000i 0 −2.49406 − 0.882973i 0 −1.49476 − 2.58900i 0
31.6 0 1.32933 − 2.30247i 0 0.866025 − 0.500000i 0 1.45249 − 2.21140i 0 −2.03426 − 3.52343i 0
271.1 0 −1.60984 − 2.78832i 0 0.866025 + 0.500000i 0 −0.279599 + 2.63094i 0 −3.68317 + 6.37943i 0
271.2 0 −0.679312 − 1.17660i 0 −0.866025 − 0.500000i 0 1.26330 + 2.32467i 0 0.577070 − 0.999515i 0
271.3 0 −0.0855194 − 0.148124i 0 0.866025 + 0.500000i 0 −2.53891 − 0.744258i 0 1.48537 − 2.57274i 0
271.4 0 0.821662 + 1.42316i 0 −0.866025 − 0.500000i 0 1.59679 − 2.10956i 0 0.149742 − 0.259361i 0
271.5 0 1.22368 + 2.11947i 0 −0.866025 − 0.500000i 0 −2.49406 + 0.882973i 0 −1.49476 + 2.58900i 0
271.6 0 1.32933 + 2.30247i 0 0.866025 + 0.500000i 0 1.45249 + 2.21140i 0 −2.03426 + 3.52343i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bs.c yes 12
4.b odd 2 1 560.2.bs.b ✓ 12
7.c even 3 1 3920.2.k.d 12
7.d odd 6 1 560.2.bs.b ✓ 12
7.d odd 6 1 3920.2.k.e 12
28.f even 6 1 inner 560.2.bs.c yes 12
28.f even 6 1 3920.2.k.d 12
28.g odd 6 1 3920.2.k.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bs.b ✓ 12 4.b odd 2 1
560.2.bs.b ✓ 12 7.d odd 6 1
560.2.bs.c yes 12 1.a even 1 1 trivial
560.2.bs.c yes 12 28.f even 6 1 inner
3920.2.k.d 12 7.c even 3 1
3920.2.k.d 12 28.f even 6 1
3920.2.k.e 12 7.d odd 6 1
3920.2.k.e 12 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 2 T_{3}^{11} + 16 T_{3}^{10} - 28 T_{3}^{9} + 175 T_{3}^{8} - 272 T_{3}^{7} + 824 T_{3}^{6} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + \cdots + 82944 \) Copy content Toggle raw display
$13$ \( T^{12} + 106 T^{10} + \cdots + 2108304 \) Copy content Toggle raw display
$17$ \( T^{12} - 56 T^{10} + \cdots + 11943936 \) Copy content Toggle raw display
$19$ \( T^{12} - 2 T^{11} + \cdots + 156816 \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} + \cdots + 9801 \) Copy content Toggle raw display
$29$ \( (T^{6} - 8 T^{5} + \cdots + 37476)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 107827456 \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{12} + 234 T^{10} + \cdots + 77951241 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 394896384 \) Copy content Toggle raw display
$47$ \( T^{12} - 12 T^{11} + \cdots + 1411344 \) Copy content Toggle raw display
$53$ \( T^{12} + 12 T^{11} + \cdots + 104976 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18951376896 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 742671504 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 75475473984 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 14561731584 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 12230590464 \) Copy content Toggle raw display
$79$ \( T^{12} - 12 T^{11} + \cdots + 22581504 \) Copy content Toggle raw display
$83$ \( (T^{6} - 18 T^{5} + \cdots - 3564)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 264722598144 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 642318336 \) Copy content Toggle raw display
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