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Properties

Label 560.5.f.b
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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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: \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{5} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{4} q^{5} + ( - \beta_{5} + 4) q^{7} + (\beta_{11} + \beta_1 - 36) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{4} q^{5} + ( - \beta_{5} + 4) q^{7} + (\beta_{11} + \beta_1 - 36) q^{9} + ( - \beta_{11} - 2 \beta_{9} + \cdots - 12) q^{11}+ \cdots + (24 \beta_{11} + 128 \beta_{9} + \cdots - 1900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 50 q^{7} - 434 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 50 q^{7} - 434 q^{9} - 126 q^{11} - 50 q^{15} - 642 q^{21} + 756 q^{23} - 1500 q^{25} - 2190 q^{29} + 150 q^{35} + 5564 q^{37} - 8634 q^{39} - 3944 q^{43} - 8796 q^{49} - 7206 q^{51} + 11760 q^{53} - 12900 q^{57} + 4310 q^{63} - 750 q^{65} + 24096 q^{67} + 5664 q^{71} + 26904 q^{77} + 1590 q^{79} - 11912 q^{81} + 1050 q^{85} + 7182 q^{91} - 70980 q^{93} + 3000 q^{95} - 23084 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 109 x^{10} + 570 x^{9} + 5814 x^{8} - 22512 x^{7} - 151120 x^{6} + 300288 x^{5} + \cdots + 205833600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} + \cdots - 17\!\cdots\!80 ) / 80\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 230060779665 \nu^{11} - 4634987015638 \nu^{10} - 6235493671465 \nu^{9} + \cdots - 85\!\cdots\!60 ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22050672377839 \nu^{11} + 167967216649588 \nu^{10} + \cdots - 17\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 545049904483 \nu^{11} + 3342865644154 \nu^{10} + 54129391887267 \nu^{9} + \cdots - 93\!\cdots\!00 ) / 32\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10105263298801 \nu^{11} + 7668759064702 \nu^{10} + \cdots - 10\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7288216857073 \nu^{11} - 74184072541910 \nu^{10} + \cdots + 20\!\cdots\!40 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5267672206343 \nu^{11} + 38454150139532 \nu^{10} + 581950909205127 \nu^{9} + \cdots + 16\!\cdots\!80 ) / 80\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4210783908554 \nu^{11} + 39529944424377 \nu^{10} + 402269649089344 \nu^{9} + \cdots - 33\!\cdots\!80 ) / 53\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19651585016711 \nu^{11} - 155617378754012 \nu^{10} + \cdots + 53\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 143154543512359 \nu^{11} - 553353694360708 \nu^{10} + \cdots + 54\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15067388428199 \nu^{11} - 153255880811244 \nu^{10} + \cdots + 54\!\cdots\!20 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{4} + 5\beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{10} + 2\beta_{8} + 2\beta_{5} - 2\beta_{4} - 4\beta_{3} + 3\beta_{2} + 7\beta _1 + 210 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{9} + 12 \beta_{8} - 10 \beta_{7} + 6 \beta_{6} - 152 \beta_{4} + 42 \beta_{3} - 3 \beta_{2} + \cdots + 390 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 152 \beta_{10} - 104 \beta_{9} + 80 \beta_{8} - 10 \beta_{7} + 24 \beta_{6} + 152 \beta_{5} + \cdots + 3120 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16 \beta_{11} + 64 \beta_{10} - 106 \beta_{9} + 162 \beta_{8} - 20 \beta_{7} + 104 \beta_{6} + \cdots + 1112 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 240 \beta_{11} + 8936 \beta_{10} - 5080 \beta_{9} + 2816 \beta_{8} - 410 \beta_{7} + 2040 \beta_{6} + \cdots - 202440 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1760 \beta_{11} + 30100 \beta_{10} - 20014 \beta_{9} + 30198 \beta_{8} + 21680 \beta_{7} + \cdots - 916960 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6960 \beta_{11} + 372224 \beta_{10} - 10728 \beta_{9} + 77120 \beta_{8} + 52530 \beta_{7} + \cdots - 27583080 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 516480 \beta_{11} + 1474116 \beta_{10} + 448854 \beta_{9} + 513858 \beta_{8} + 2329640 \beta_{7} + \cdots - 115491840 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 537744 \beta_{11} + 1158592 \beta_{10} + 3436664 \beta_{9} + 52912 \beta_{8} + 1720606 \beta_{7} + \cdots - 392415288 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 41859520 \beta_{11} + 16646300 \beta_{10} + 129377938 \beta_{9} - 29727786 \beta_{8} + \cdots - 8777056160 ) / 10 \) 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
−2.23331 + 2.23607i
5.81769 + 2.23607i
−6.63119 − 2.23607i
2.17355 − 2.23607i
7.50299 − 2.23607i
−3.62973 + 2.23607i
−3.62973 − 2.23607i
7.50299 + 2.23607i
2.17355 + 2.23607i
−6.63119 + 2.23607i
5.81769 − 2.23607i
−2.23331 − 2.23607i
0 − 14.6970i 0 − 11.1803i 0 26.6804 + 41.0994i 0 −135.002 0
321.2 0 − 14.5704i 0 − 11.1803i 0 −44.8989 − 19.6236i 0 −131.296 0
321.3 0 − 12.6955i 0 11.1803i 0 −2.03056 − 48.9579i 0 −80.1757 0
321.4 0 − 9.10378i 0 11.1803i 0 40.5002 − 27.5814i 0 −1.87877 0
321.5 0 − 5.52807i 0 11.1803i 0 19.9894 + 44.7373i 0 50.4404 0
321.6 0 − 0.296012i 0 − 11.1803i 0 −15.2405 − 46.5696i 0 80.9124 0
321.7 0 0.296012i 0 11.1803i 0 −15.2405 + 46.5696i 0 80.9124 0
321.8 0 5.52807i 0 − 11.1803i 0 19.9894 − 44.7373i 0 50.4404 0
321.9 0 9.10378i 0 − 11.1803i 0 40.5002 + 27.5814i 0 −1.87877 0
321.10 0 12.6955i 0 − 11.1803i 0 −2.03056 + 48.9579i 0 −80.1757 0
321.11 0 14.5704i 0 11.1803i 0 −44.8989 + 19.6236i 0 −131.296 0
321.12 0 14.6970i 0 11.1803i 0 26.6804 − 41.0994i 0 −135.002 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.b 12
4.b odd 2 1 35.5.d.a ✓ 12
7.b odd 2 1 inner 560.5.f.b 12
12.b even 2 1 315.5.h.a 12
20.d odd 2 1 175.5.d.i 12
20.e even 4 2 175.5.c.d 24
28.d even 2 1 35.5.d.a ✓ 12
84.h odd 2 1 315.5.h.a 12
140.c even 2 1 175.5.d.i 12
140.j odd 4 2 175.5.c.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.d.a ✓ 12 4.b odd 2 1
35.5.d.a ✓ 12 28.d even 2 1
175.5.c.d 24 20.e even 4 2
175.5.c.d 24 140.j odd 4 2
175.5.d.i 12 20.d odd 2 1
175.5.d.i 12 140.c even 2 1
315.5.h.a 12 12.b even 2 1
315.5.h.a 12 84.h odd 2 1
560.5.f.b 12 1.a even 1 1 trivial
560.5.f.b 12 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 703T_{3}^{10} + 184351T_{3}^{8} + 21932745T_{3}^{6} + 1131317100T_{3}^{4} + 18818284500T_{3}^{2} + 1640250000 \) 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
$3$ \( T^{12} + \cdots + 1640250000 \) 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
$11$ \( (T^{6} + \cdots - 5924759078688)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 449297249971200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
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