Properties

Label 560.8.a.i
Level $560$
Weight $8$
Character orbit 560.a
Self dual yes
Analytic conductor $174.936$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,8,Mod(1,560)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 560.a (trivial)

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: yes
Analytic conductor: \(174.935614271\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{11}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \beta + 15) q^{3} + 125 q^{5} + 343 q^{7} + (90 \beta - 378) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \beta + 15) q^{3} + 125 q^{5} + 343 q^{7} + (90 \beta - 378) q^{9} + (190 \beta + 3953) q^{11} + ( - 209 \beta - 8909) q^{13} + (375 \beta + 1875) q^{15} + (1105 \beta - 1199) q^{17} + ( - 2771 \beta + 1806) q^{19} + (1029 \beta + 5145) q^{21} + ( - 5345 \beta - 6922) q^{23} + 15625 q^{25} + ( - 6345 \beta + 9045) q^{27} + ( - 11886 \beta - 63449) q^{29} + (11277 \beta - 126384) q^{31} + (14709 \beta + 159615) q^{33} + 42875 q^{35} + ( - 21819 \beta - 132930) q^{37} + ( - 29862 \beta - 243987) q^{39} + (18677 \beta - 55960) q^{41} + (12289 \beta - 473786) q^{43} + (11250 \beta - 47250) q^{45} + (28371 \beta - 135637) q^{47} + 117649 q^{49} + (12978 \beta + 565455) q^{51} + ( - 32612 \beta - 633896) q^{53} + (23750 \beta + 494125) q^{55} + ( - 36147 \beta - 1435998) q^{57} + ( - 85320 \beta + 680060) q^{59} + (5667 \beta - 906840) q^{61} + (30870 \beta - 129654) q^{63} + ( - 26125 \beta - 1113625) q^{65} + ( - 253172 \beta + 1094656) q^{67} + ( - 100941 \beta - 2925990) q^{69} + (111024 \beta + 747464) q^{71} + (106198 \beta + 3584894) q^{73} + (46875 \beta + 234375) q^{75} + (65170 \beta + 1355879) q^{77} + ( - 93752 \beta + 3971487) q^{79} + ( - 264870 \beta - 2387799) q^{81} + (469722 \beta + 152356) q^{83} + (138125 \beta - 149875) q^{85} + ( - 368637 \beta - 7227543) q^{87} + (123769 \beta - 8971764) q^{89} + ( - 71687 \beta - 3055787) q^{91} + ( - 209997 \beta + 4058496) q^{93} + ( - 346375 \beta + 225750) q^{95} + ( - 340391 \beta + 2129037) q^{97} + (283950 \beta + 1515366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 30 q^{3} + 250 q^{5} + 686 q^{7} - 756 q^{9} + 7906 q^{11} - 17818 q^{13} + 3750 q^{15} - 2398 q^{17} + 3612 q^{19} + 10290 q^{21} - 13844 q^{23} + 31250 q^{25} + 18090 q^{27} - 126898 q^{29} - 252768 q^{31} + 319230 q^{33} + 85750 q^{35} - 265860 q^{37} - 487974 q^{39} - 111920 q^{41} - 947572 q^{43} - 94500 q^{45} - 271274 q^{47} + 235298 q^{49} + 1130910 q^{51} - 1267792 q^{53} + 988250 q^{55} - 2871996 q^{57} + 1360120 q^{59} - 1813680 q^{61} - 259308 q^{63} - 2227250 q^{65} + 2189312 q^{67} - 5851980 q^{69} + 1494928 q^{71} + 7169788 q^{73} + 468750 q^{75} + 2711758 q^{77} + 7942974 q^{79} - 4775598 q^{81} + 304712 q^{83} - 299750 q^{85} - 14455086 q^{87} - 17943528 q^{89} - 6111574 q^{91} + 8116992 q^{93} + 451500 q^{95} + 4258074 q^{97} + 3030732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.31662
3.31662
0 −24.7995 0 125.000 0 343.000 0 −1571.98 0
1.2 0 54.7995 0 125.000 0 343.000 0 815.985 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.8.a.i 2
4.b odd 2 1 35.8.a.a 2
12.b even 2 1 315.8.a.c 2
20.d odd 2 1 175.8.a.b 2
20.e even 4 2 175.8.b.c 4
28.d even 2 1 245.8.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.8.a.a 2 4.b odd 2 1
175.8.a.b 2 20.d odd 2 1
175.8.b.c 4 20.e even 4 2
245.8.a.b 2 28.d even 2 1
315.8.a.c 2 12.b even 2 1
560.8.a.i 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 30T_{3} - 1359 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 30T - 1359 \) Copy content Toggle raw display
$5$ \( (T - 125)^{2} \) Copy content Toggle raw display
$7$ \( (T - 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7906 T + 9272609 \) Copy content Toggle raw display
$13$ \( T^{2} + 17818 T + 71682425 \) Copy content Toggle raw display
$17$ \( T^{2} + 2398 T - 213462799 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1348143980 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 4980234316 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 20838975695 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 6409132848 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 66117717036 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 58262616304 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 197893738100 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 123267405047 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 214640651072 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 818710818800 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 816706565136 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 10082635080448 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 1610731398080 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10866534315332 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14225767990465 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 38809208931248 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 77796446568160 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 15859623239687 \) Copy content Toggle raw display
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