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Properties

Label 560.4.e.c
Level $560$
Weight $4$
Character orbit 560.e
Analytic conductor $33.041$
Analytic rank $0$
Dimension $8$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,4,Mod(559,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.559");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.e (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: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4}) q^{3} + 5 \beta_1 q^{5} + 7 \beta_{4} q^{7} + (2 \beta_{7} - 40) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{4}) q^{3} + 5 \beta_1 q^{5} + 7 \beta_{4} q^{7} + (2 \beta_{7} - 40) q^{9} + (\beta_{6} - 18 \beta_{2}) q^{11} + (8 \beta_{3} - 9 \beta_1) q^{13} + ( - 5 \beta_{6} + 25 \beta_{2}) q^{15} + (9 \beta_{3} - 23 \beta_1) q^{17} + ( - 7 \beta_{7} + 49) q^{21} + 125 q^{25} + (27 \beta_{5} + 133 \beta_{4}) q^{27} + ( - 13 \beta_{7} - 27) q^{29} + (23 \beta_{3} + 223 \beta_1) q^{33} + 35 \beta_{6} q^{35} + ( - 87 \beta_{6} + 11 \beta_{2}) q^{39} + ( - 50 \beta_{3} - 200 \beta_1) q^{45} + ( - 20 \beta_{5} + 117 \beta_{4}) q^{47} - 343 q^{49} + ( - 85 \beta_{6} - 52 \beta_{2}) q^{51} + (90 \beta_{5} + 25 \beta_{4}) q^{55} + ( - 98 \beta_{5} - 280 \beta_{4}) q^{63} + ( - 40 \beta_{7} - 225) q^{65} + 146 \beta_{6} q^{71} - 234 \beta_1 q^{73} + ( - 125 \beta_{5} - 125 \beta_{4}) q^{75} + ( - 126 \beta_{3} - 49 \beta_1) q^{77} + ( - 117 \beta_{6} + 59 \beta_{2}) q^{79} + ( - 106 \beta_{7} + 1471) q^{81} + 18 \beta_{4} q^{83} + ( - 45 \beta_{7} - 575) q^{85} + ( - 64 \beta_{5} - 753 \beta_{4}) q^{87} + ( - 63 \beta_{6} - 392 \beta_{2}) q^{91} + (91 \beta_{3} + 369 \beta_1) q^{97} + ( - 472 \beta_{6} + 790 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 320 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 320 q^{9} + 392 q^{21} + 1000 q^{25} - 216 q^{29} - 2744 q^{49} - 1800 q^{65} + 11768 q^{81} - 4600 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 25\nu^{7} - 3\nu^{6} - 70\nu^{5} - 205\nu^{4} - 95\nu^{3} + 40\nu^{2} - 120\nu + 2624 ) / 1224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -41\nu^{7} + 111\nu^{6} + 74\nu^{5} + 173\nu^{4} - 1517\nu^{3} + 1036\nu^{2} + 768\nu - 1072 ) / 204 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} - 14\nu^{5} - 15\nu^{4} + 95\nu^{3} + 60\nu^{2} - 128\nu + 16 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\nu^{7} - 66\nu^{6} - 112\nu^{5} - 328\nu^{4} + 1072\nu^{3} + 64\nu^{2} - 192\nu + 557 ) / 153 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 7\nu^{6} - 4\nu^{5} - 15\nu^{4} + 97\nu^{3} - 66\nu^{2} - 52\nu + 68 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 7\nu^{6} - 14\nu^{5} - 41\nu^{4} + 125\nu^{3} + 8\nu^{2} - 24\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -65\nu^{7} + 69\nu^{6} + 284\nu^{5} + 533\nu^{4} - 1691\nu^{3} - 1226\nu^{2} + 2556\nu - 172 ) / 102 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 + 2 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 3\beta_{5} + 6\beta_{4} + 3\beta_{3} + 9\beta_{2} - 6\beta _1 + 18 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - 2\beta_{4} - 5\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + 18\beta_{6} + 3\beta_{5} - 42\beta_{4} + 21\beta_{3} + 7\beta_{2} + 6\beta _1 - 14 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 10\beta_{6} + 55\beta_{5} + 22\beta_{4} + 11\beta_{3} + 121\beta_{2} - 110\beta _1 + 242 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 20\beta_{6} - 45\beta_{4} - 36\beta _1 + 81 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 91\beta_{7} + 182\beta_{6} + 169\beta_{5} - 406\beta_{4} + 203\beta_{3} + 377\beta_{2} + 338\beta _1 - 754 ) / 16 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.44918 − 1.77086i
−0.862555 + 0.141174i
2.25820 + 0.369600i
0.553538 − 0.676408i
2.25820 − 0.369600i
0.553538 + 0.676408i
−1.44918 + 1.77086i
−0.862555 − 0.141174i
0 − 10.3917i 0 −11.1803 0 18.5203i 0 −80.9878 0
559.2 0 − 10.3917i 0 11.1803 0 18.5203i 0 −80.9878 0
559.3 0 − 5.10022i 0 −11.1803 0 − 18.5203i 0 0.987803 0
559.4 0 − 5.10022i 0 11.1803 0 − 18.5203i 0 0.987803 0
559.5 0 5.10022i 0 −11.1803 0 18.5203i 0 0.987803 0
559.6 0 5.10022i 0 11.1803 0 18.5203i 0 0.987803 0
559.7 0 10.3917i 0 −11.1803 0 − 18.5203i 0 −80.9878 0
559.8 0 10.3917i 0 11.1803 0 − 18.5203i 0 −80.9878 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.4.e.c ✓ 8
4.b odd 2 1 inner 560.4.e.c ✓ 8
5.b even 2 1 inner 560.4.e.c ✓ 8
7.b odd 2 1 inner 560.4.e.c ✓ 8
20.d odd 2 1 inner 560.4.e.c ✓ 8
28.d even 2 1 inner 560.4.e.c ✓ 8
35.c odd 2 1 CM 560.4.e.c ✓ 8
140.c even 2 1 inner 560.4.e.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.4.e.c ✓ 8 1.a even 1 1 trivial
560.4.e.c ✓ 8 4.b odd 2 1 inner
560.4.e.c ✓ 8 5.b even 2 1 inner
560.4.e.c ✓ 8 7.b odd 2 1 inner
560.4.e.c ✓ 8 20.d odd 2 1 inner
560.4.e.c ✓ 8 28.d even 2 1 inner
560.4.e.c ✓ 8 35.c odd 2 1 CM
560.4.e.c ✓ 8 140.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 134T_{3}^{2} + 2809 \) acting on \(S_{4}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 134 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 343)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 7846 T^{2} + 14845609)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 11562 T^{2} + 24710841)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 18898 T^{2} + 17297281)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 54 T - 70251)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 239646 T^{2} + 5158543329)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 746060)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 273780)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1041774 T^{2} + 191268899649)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2268)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2752818 T^{2} + 219010401)^{2} \) Copy content Toggle raw display
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