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Properties

Label 560.3.p.g
Level $560$
Weight $3$
Character orbit 560.p
Analytic conductor $15.259$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,3,Mod(209,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.209"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.211319484596224.6
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} - 48x^{5} - 23x^{4} - 48x^{3} + 1226x^{2} - 7512x + 24408 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{5} + 5) q^{9} + (\beta_{5} + 8) q^{11} + (2 \beta_{3} + 3 \beta_{2}) q^{13} + (\beta_{7} + \beta_{5} + \cdots + 2 \beta_1) q^{15}+ \cdots + (12 \beta_{5} + 124) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{9} + 60 q^{11} - 4 q^{15} + 56 q^{21} - 92 q^{29} - 52 q^{35} - 204 q^{39} - 284 q^{49} + 212 q^{51} - 292 q^{65} - 504 q^{71} - 692 q^{79} - 136 q^{81} + 92 q^{85} + 44 q^{91} + 176 q^{95}+ \cdots + 944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2x^{6} - 48x^{5} - 23x^{4} - 48x^{3} + 1226x^{2} - 7512x + 24408 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68583 \nu^{7} + 2881098 \nu^{6} + 14444162 \nu^{5} + 89546988 \nu^{4} - 50140069 \nu^{3} + \cdots - 148651332 ) / 1916061700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69474 \nu^{7} - 302427 \nu^{6} - 999414 \nu^{5} - 1549105 \nu^{4} + 25322976 \nu^{3} + \cdots - 1008257557 ) / 479015425 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20621 \nu^{7} + 33279 \nu^{6} + 247069 \nu^{5} - 1102127 \nu^{4} - 4098822 \nu^{3} + \cdots - 63934961 ) / 68430775 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18671 \nu^{7} + 44942 \nu^{6} + 242194 \nu^{5} - 392820 \nu^{4} + 271679 \nu^{3} + \cdots - 133926828 ) / 39103300 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -192\nu^{7} - 1258\nu^{6} - 480\nu^{5} + 6071\nu^{4} + 64560\nu^{3} + 236702\nu^{2} - 142728\nu + 18347 ) / 391033 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36245 \nu^{7} - 157626 \nu^{6} - 886970 \nu^{5} - 3306924 \nu^{4} - 7397227 \nu^{3} + \cdots - 127553316 ) / 39103300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 85343 \nu^{7} + 286266 \nu^{6} + 408874 \nu^{5} - 2207700 \nu^{4} - 8069593 \nu^{3} + \cdots - 427413804 ) / 76642468 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + 6\beta_{4} - 6\beta_{3} - 6\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 7\beta_{6} + 30\beta_{4} - 42\beta_{3} + 30\beta_{2} - 18\beta _1 + 108 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} + 4\beta_{6} - 3\beta_{5} - 12\beta_{3} + 14\beta_{2} + 24\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{7} - 17\beta_{6} + 180\beta_{5} + 168\beta_{4} + 297\beta_{3} - 333\beta_{2} + 15\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 192\beta_{7} - 85\beta_{6} - 128\beta_{5} - 53\beta_{4} - 480\beta_{3} - 148\beta_{2} - 180\beta _1 - 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -61\beta_{7} + 730\beta_{6} - 630\beta_{5} + 852\beta_{4} + 1608\beta_{3} + 7530\beta_{2} + 2751\beta _1 + 22554 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−3.29232 − 1.99575i
−3.29232 + 1.99575i
1.28865 − 2.58522i
1.28865 + 2.58522i
−1.28865 − 3.99943i
−1.28865 + 3.99943i
3.29232 − 0.581540i
3.29232 + 0.581540i
0 −4.76222 0 −1.82242 − 4.65605i 0 −1.46990 − 6.84393i 0 13.6788 0
209.2 0 −4.76222 0 −1.82242 + 4.65605i 0 −1.46990 + 6.84393i 0 13.6788 0
209.3 0 −2.07875 0 4.65605 − 1.82242i 0 −3.36740 − 6.13682i 0 −4.67878 0
209.4 0 −2.07875 0 4.65605 + 1.82242i 0 −3.36740 + 6.13682i 0 −4.67878 0
209.5 0 2.07875 0 −4.65605 − 1.82242i 0 3.36740 + 6.13682i 0 −4.67878 0
209.6 0 2.07875 0 −4.65605 + 1.82242i 0 3.36740 − 6.13682i 0 −4.67878 0
209.7 0 4.76222 0 1.82242 − 4.65605i 0 1.46990 + 6.84393i 0 13.6788 0
209.8 0 4.76222 0 1.82242 + 4.65605i 0 1.46990 − 6.84393i 0 13.6788 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.p.g 8
4.b odd 2 1 70.3.d.a ✓ 8
5.b even 2 1 inner 560.3.p.g 8
7.b odd 2 1 inner 560.3.p.g 8
12.b even 2 1 630.3.h.d 8
20.d odd 2 1 70.3.d.a ✓ 8
20.e even 4 2 350.3.b.c 8
28.d even 2 1 70.3.d.a ✓ 8
28.f even 6 2 490.3.h.a 16
28.g odd 6 2 490.3.h.a 16
35.c odd 2 1 inner 560.3.p.g 8
60.h even 2 1 630.3.h.d 8
84.h odd 2 1 630.3.h.d 8
140.c even 2 1 70.3.d.a ✓ 8
140.j odd 4 2 350.3.b.c 8
140.p odd 6 2 490.3.h.a 16
140.s even 6 2 490.3.h.a 16
420.o odd 2 1 630.3.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.d.a ✓ 8 4.b odd 2 1
70.3.d.a ✓ 8 20.d odd 2 1
70.3.d.a ✓ 8 28.d even 2 1
70.3.d.a ✓ 8 140.c even 2 1
350.3.b.c 8 20.e even 4 2
350.3.b.c 8 140.j odd 4 2
490.3.h.a 16 28.f even 6 2
490.3.h.a 16 28.g odd 6 2
490.3.h.a 16 140.p odd 6 2
490.3.h.a 16 140.s even 6 2
560.3.p.g 8 1.a even 1 1 trivial
560.3.p.g 8 5.b even 2 1 inner
560.3.p.g 8 7.b odd 2 1 inner
560.3.p.g 8 35.c odd 2 1 inner
630.3.h.d 8 12.b even 2 1
630.3.h.d 8 60.h even 2 1
630.3.h.d 8 84.h odd 2 1
630.3.h.d 8 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 27T_{3}^{2} + 98 \) acting on \(S_{3}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 27 T^{2} + 98)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 98 T^{4} + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 142 T^{6} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} - 15 T - 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 321 T^{2} + 21632)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 129 T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1048 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 506 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 23 T - 626)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1096 T^{2} + 294912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1544 T^{2} + 331776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 10636 T^{2} + 28215072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1562 T^{2} + 197136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 5227 T^{2} + 642978)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7328 T^{2} + 2985984)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 10000 T^{2} + 11520000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 4428 T^{2} + 4792608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 338 T^{2} + 28224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 126 T + 3632)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 12100 T^{2} + 16866432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 173 T + 5376)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1878 T^{2} + 137288)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5704 T^{2} + 8128512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 15361 T^{2} + 23722272)^{2} \) Copy content Toggle raw display
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