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Properties

Label 560.5.f.c
Level $560$
Weight $5$
Character orbit 560.f
Analytic conductor $57.887$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,5,Mod(321,560)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.321");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 560.f (of order \(2\), degree \(1\), 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: \(57.8871793270\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 735x^{10} + 187999x^{8} + 20647401x^{6} + 1027443564x^{4} + 21031479252x^{2} + 110003662224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{6}\cdot 7 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + (\beta_{6} + \beta_{5} + 7) q^{7} + (\beta_{2} - 42) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} + (\beta_{6} + \beta_{5} + 7) q^{7} + (\beta_{2} - 42) q^{9} + (\beta_{3} + 6) q^{11} + ( - \beta_{6} - 3 \beta_{5} + \cdots - 2 \beta_1) q^{13}+ \cdots + (3 \beta_{10} + 6 \beta_{9} + \cdots + 1242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 82 q^{7} - 498 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 82 q^{7} - 498 q^{9} + 66 q^{11} - 50 q^{15} + 638 q^{21} + 1332 q^{23} - 1500 q^{25} + 1554 q^{29} - 1050 q^{35} - 7844 q^{37} + 1830 q^{39} - 3720 q^{43} + 3940 q^{49} + 9242 q^{51} - 9840 q^{53} + 2076 q^{57} - 10634 q^{63} + 4050 q^{65} + 4480 q^{67} - 20256 q^{71} - 8616 q^{77} - 3050 q^{79} + 49976 q^{81} + 1050 q^{85} + 27182 q^{91} + 55036 q^{93} + 10200 q^{95} + 14484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 735x^{10} + 187999x^{8} + 20647401x^{6} + 1027443564x^{4} + 21031479252x^{2} + 110003662224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 123 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39239 \nu^{10} + 28318746 \nu^{8} + 6972551399 \nu^{6} + 697625941008 \nu^{4} + \cdots + 253311042447216 ) / 410865609168 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43286624 \nu^{11} - 1363754325 \nu^{10} + 34130821836 \nu^{9} - 933880683168 \nu^{8} + \cdots - 23\!\cdots\!92 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 110659 \nu^{11} - 76165872 \nu^{9} - 17286939703 \nu^{7} - 1506133740066 \nu^{5} + \cdots - 439875657036720 \nu ) / 34568994130272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33969905 \nu^{11} - 454584775 \nu^{10} + 22609582500 \nu^{9} - 311293561056 \nu^{8} + \cdots - 77\!\cdots\!64 ) / 75\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33969905 \nu^{11} - 1586484742 \nu^{10} + 22609582500 \nu^{9} - 1081481879778 \nu^{8} + \cdots - 34\!\cdots\!40 ) / 75\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 162188599 \nu^{11} + 10470592926 \nu^{10} + 99987768690 \nu^{9} + 7014558304116 \nu^{8} + \cdots + 23\!\cdots\!08 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 243584881 \nu^{11} - 4091262975 \nu^{10} - 174743739312 \nu^{9} - 2801642049504 \nu^{8} + \cdots - 69\!\cdots\!72 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 469573537 \nu^{11} - 2727508650 \nu^{10} - 301935106716 \nu^{9} - 1867761366336 \nu^{8} + \cdots - 46\!\cdots\!84 ) / 22\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1855892393 \nu^{11} - 1284261715614 \nu^{9} - 293885594876117 \nu^{7} + \cdots - 59\!\cdots\!12 \nu ) / 22\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 123 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + 3\beta_{9} - 7\beta_{6} + 30\beta_{5} - 2\beta_{4} - 229\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{10} + 10 \beta_{9} + 2 \beta_{8} - 44 \beta_{7} + 125 \beta_{6} + 20 \beta_{5} + 11 \beta_{4} + \cdots + 27553 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 866\beta_{11} + 22\beta_{10} - 937\beta_{9} + 1979\beta_{6} - 15866\beta_{5} + 788\beta_{4} + 62379\beta _1 + 937 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3933 \beta_{10} - 4374 \beta_{9} + 882 \beta_{8} + 24780 \beta_{7} - 59331 \beta_{6} - 8748 \beta_{5} + \cdots - 7467789 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 310114 \beta_{11} - 6330 \beta_{10} + 255645 \beta_{9} - 464375 \beta_{6} + 6277134 \beta_{5} + \cdots - 255645 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1144021 \beta_{10} + 1563230 \beta_{9} - 838418 \beta_{8} - 10342276 \beta_{7} + 22428907 \beta_{6} + \cdots + 2163244733 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 104966482 \beta_{11} + 1951298 \beta_{10} - 69483125 \beta_{9} + 101902723 \beta_{6} - 2255337826 \beta_{5} + \cdots + 69483125 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 322334433 \beta_{10} - 528735006 \beta_{9} + 412801146 \beta_{8} + 3843013452 \beta_{7} + \cdots - 646091307429 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 34692914594 \beta_{11} - 653638002 \beta_{10} + 19297256661 \beta_{9} - 21393922927 \beta_{6} + \cdots - 19297256661 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
− 17.8437i
− 14.9748i
− 9.30742i
− 7.67564i
− 6.25214i
− 2.77897i
2.77897i
6.25214i
7.67564i
9.30742i
14.9748i
17.8437i
0 − 17.8437i 0 11.1803i 0 44.3596 + 20.8140i 0 −237.399 0
321.2 0 − 14.9748i 0 − 11.1803i 0 −12.4491 − 47.3922i 0 −143.246 0
321.3 0 − 9.30742i 0 − 11.1803i 0 −21.5650 + 43.9994i 0 −5.62807 0
321.4 0 − 7.67564i 0 11.1803i 0 −48.7488 + 4.95570i 0 22.0845 0
321.5 0 − 6.25214i 0 − 11.1803i 0 45.6730 + 17.7476i 0 41.9108 0
321.6 0 − 2.77897i 0 11.1803i 0 33.7302 + 35.5425i 0 73.2773 0
321.7 0 2.77897i 0 − 11.1803i 0 33.7302 − 35.5425i 0 73.2773 0
321.8 0 6.25214i 0 11.1803i 0 45.6730 − 17.7476i 0 41.9108 0
321.9 0 7.67564i 0 − 11.1803i 0 −48.7488 − 4.95570i 0 22.0845 0
321.10 0 9.30742i 0 11.1803i 0 −21.5650 − 43.9994i 0 −5.62807 0
321.11 0 14.9748i 0 11.1803i 0 −12.4491 + 47.3922i 0 −143.246 0
321.12 0 17.8437i 0 − 11.1803i 0 44.3596 − 20.8140i 0 −237.399 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.f.c 12
4.b odd 2 1 140.5.d.a ✓ 12
7.b odd 2 1 inner 560.5.f.c 12
12.b even 2 1 1260.5.j.a 12
20.d odd 2 1 700.5.d.f 12
20.e even 4 2 700.5.h.c 24
28.d even 2 1 140.5.d.a ✓ 12
84.h odd 2 1 1260.5.j.a 12
140.c even 2 1 700.5.d.f 12
140.j odd 4 2 700.5.h.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.5.d.a ✓ 12 4.b odd 2 1
140.5.d.a ✓ 12 28.d even 2 1
560.5.f.c 12 1.a even 1 1 trivial
560.5.f.c 12 7.b odd 2 1 inner
700.5.d.f 12 20.d odd 2 1
700.5.d.f 12 140.c even 2 1
700.5.h.c 24 20.e even 4 2
700.5.h.c 24 140.j odd 4 2
1260.5.j.a 12 12.b even 2 1
1260.5.j.a 12 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 735T_{3}^{10} + 187999T_{3}^{8} + 20647401T_{3}^{6} + 1027443564T_{3}^{4} + 21031479252T_{3}^{2} + 110003662224 \) acting on \(S_{5}^{\mathrm{new}}(560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 110003662224 \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{6} - 33 T^{5} + \cdots - 570564431136)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 11\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 82\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
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