Properties

Label 560.6.a.l
Level $560$
Weight $6$
Character orbit 560.a
Self dual yes
Analytic conductor $89.815$
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] = [560,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(89.8149390953\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta q^{3} - 25 q^{5} + 49 q^{7} + (9 \beta - 99) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta q^{3} - 25 q^{5} + 49 q^{7} + (9 \beta - 99) q^{9} + (97 \beta + 252) q^{11} + ( - 53 \beta - 262) q^{13} + 75 \beta q^{15} + ( - 251 \beta + 146) q^{17} + ( - 86 \beta - 272) q^{19} - 147 \beta q^{21} + ( - 902 \beta + 672) q^{23} + 625 q^{25} + (999 \beta - 432) q^{27} + (945 \beta + 2470) q^{29} + (924 \beta - 264) q^{31} + ( - 1047 \beta - 4656) q^{33} - 1225 q^{35} + (1260 \beta - 5082) q^{37} + (945 \beta + 2544) q^{39} + (3818 \beta - 1022) q^{41} + (922 \beta + 13100) q^{43} + ( - 225 \beta + 2475) q^{45} + ( - 1575 \beta + 11432) q^{47} + 2401 q^{49} + (315 \beta + 12048) q^{51} + (454 \beta - 28018) q^{53} + ( - 2425 \beta - 6300) q^{55} + (1074 \beta + 4128) q^{57} + (5184 \beta - 32392) q^{59} + ( - 5706 \beta - 23070) q^{61} + (441 \beta - 4851) q^{63} + (1325 \beta + 6550) q^{65} + ( - 4568 \beta + 24956) q^{67} + (690 \beta + 43296) q^{69} + (5304 \beta - 43024) q^{71} + (4192 \beta - 8862) q^{73} - 1875 \beta q^{75} + (4753 \beta + 12348) q^{77} + (17635 \beta + 17080) q^{79} + ( - 3888 \beta - 23895) q^{81} + ( - 3924 \beta - 52952) q^{83} + (6275 \beta - 3650) q^{85} + ( - 10245 \beta - 45360) q^{87} + (5722 \beta - 21686) q^{89} + ( - 2597 \beta - 12838) q^{91} + ( - 1980 \beta - 44352) q^{93} + (2150 \beta + 6800) q^{95} + ( - 13943 \beta - 41198) q^{97} + ( - 6462 \beta - 10980) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9} + 601 q^{11} - 577 q^{13} + 75 q^{15} + 41 q^{17} - 630 q^{19} - 147 q^{21} + 442 q^{23} + 1250 q^{25} + 135 q^{27} + 5885 q^{29} + 396 q^{31} - 10359 q^{33} - 2450 q^{35} - 8904 q^{37} + 6033 q^{39} + 1774 q^{41} + 27122 q^{43} + 4725 q^{45} + 21289 q^{47} + 4802 q^{49} + 24411 q^{51} - 55582 q^{53} - 15025 q^{55} + 9330 q^{57} - 59600 q^{59} - 51846 q^{61} - 9261 q^{63} + 14425 q^{65} + 45344 q^{67} + 87282 q^{69} - 80744 q^{71} - 13532 q^{73} - 1875 q^{75} + 29449 q^{77} + 51795 q^{79} - 51678 q^{81} - 109828 q^{83} - 1025 q^{85} - 100965 q^{87} - 37650 q^{89} - 28273 q^{91} - 90684 q^{93} + 15750 q^{95} - 96339 q^{97} - 28422 q^{99}+O(q^{100}) \) 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
4.53113
−3.53113
0 −13.5934 0 −25.0000 0 49.0000 0 −58.2198 0
1.2 0 10.5934 0 −25.0000 0 49.0000 0 −130.780 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 560.6.a.l 2
4.b odd 2 1 35.6.a.b 2
12.b even 2 1 315.6.a.c 2
20.d odd 2 1 175.6.a.d 2
20.e even 4 2 175.6.b.d 4
28.d even 2 1 245.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.b 2 4.b odd 2 1
175.6.a.d 2 20.d odd 2 1
175.6.b.d 4 20.e even 4 2
245.6.a.c 2 28.d even 2 1
315.6.a.c 2 12.b even 2 1
560.6.a.l 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3T_{3} - 144 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 144 \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 601T - 62596 \) Copy content Toggle raw display
$13$ \( T^{2} + 577T + 37586 \) Copy content Toggle raw display
$17$ \( T^{2} - 41T - 1023346 \) Copy content Toggle raw display
$19$ \( T^{2} + 630T - 20960 \) Copy content Toggle raw display
$23$ \( T^{2} - 442 T - 13172224 \) Copy content Toggle raw display
$29$ \( T^{2} - 5885 T - 5853350 \) Copy content Toggle raw display
$31$ \( T^{2} - 396 T - 13834656 \) Copy content Toggle raw display
$37$ \( T^{2} + 8904 T - 5978196 \) Copy content Toggle raw display
$41$ \( T^{2} - 1774 T - 236091496 \) Copy content Toggle raw display
$43$ \( T^{2} - 27122 T + 170086856 \) Copy content Toggle raw display
$47$ \( T^{2} - 21289 T + 72995224 \) Copy content Toggle raw display
$53$ \( T^{2} + 55582 T + 768990296 \) Copy content Toggle raw display
$59$ \( T^{2} + 59600 T + 451339840 \) Copy content Toggle raw display
$61$ \( T^{2} + 51846 T + 142927344 \) Copy content Toggle raw display
$67$ \( T^{2} - 45344 T + 174936944 \) Copy content Toggle raw display
$71$ \( T^{2} + 80744 T + 1172746624 \) Copy content Toggle raw display
$73$ \( T^{2} + 13532 T - 239780284 \) Copy content Toggle raw display
$79$ \( T^{2} - 51795 T - 4382959400 \) Copy content Toggle raw display
$83$ \( T^{2} + 109828 T + 2765333536 \) Copy content Toggle raw display
$89$ \( T^{2} + 37650 T - 177665240 \) Copy content Toggle raw display
$97$ \( T^{2} + 96339 T - 838817066 \) Copy content Toggle raw display
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