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Properties

Label 260.8.a.d
Level $260$
Weight $8$
Character orbit 260.a
Self dual yes
Analytic conductor $81.220$
Analytic rank $0$
Dimension $8$
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] = [260,8,Mod(1,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("260.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 260.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: \(81.2201066259\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12244 x^{6} - 36352 x^{5} + 37696782 x^{4} + 651005440 x^{3} - 27470944932 x^{2} + \cdots - 6426971905095 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3 \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + (\beta_1 + 5) q^{3} + 125 q^{5} + (\beta_{2} + 6 \beta_1 - 77) q^{7} + (\beta_{4} + \beta_{3} + 14 \beta_1 + 899) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 5) q^{3} + 125 q^{5} + (\beta_{2} + 6 \beta_1 - 77) q^{7} + (\beta_{4} + \beta_{3} + 14 \beta_1 + 899) q^{9} + ( - \beta_{4} + 2 \beta_{3} + \cdots + 184) q^{11}+ \cdots + (34 \beta_{7} - 42 \beta_{6} + \cdots + 4857286) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{3} + 1000 q^{5} - 616 q^{7} + 7192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{3} + 1000 q^{5} - 616 q^{7} + 7192 q^{9} + 1472 q^{11} - 17576 q^{13} + 5000 q^{15} + 47968 q^{17} + 57056 q^{19} + 136080 q^{21} - 75688 q^{23} + 125000 q^{25} + 302416 q^{27} - 58000 q^{29} + 50608 q^{31} - 72752 q^{33} - 77000 q^{35} + 31760 q^{37} - 87880 q^{39} + 1315808 q^{41} - 820792 q^{43} + 899000 q^{45} - 425624 q^{47} + 2008168 q^{49} + 3861504 q^{51} - 69696 q^{53} + 184000 q^{55} + 436528 q^{57} + 173152 q^{59} + 1277520 q^{61} + 1784360 q^{63} - 2197000 q^{65} - 4853960 q^{67} - 1995648 q^{69} - 1457568 q^{71} + 7590592 q^{73} + 625000 q^{75} + 9171952 q^{77} - 1134192 q^{79} + 31189048 q^{81} + 11070840 q^{83} + 5996000 q^{85} + 1066160 q^{87} + 33632368 q^{89} + 1353352 q^{91} + 27799360 q^{93} + 7132000 q^{95} + 18398544 q^{97} + 38858288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12244 x^{6} - 36352 x^{5} + 37696782 x^{4} + 651005440 x^{3} - 27470944932 x^{2} + \cdots - 6426971905095 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 440575273 \nu^{7} + 9729335537 \nu^{6} + 5189003373621 \nu^{5} - 98142266604317 \nu^{4} + \cdots + 13\!\cdots\!89 ) / 22\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 714193349 \nu^{7} + 15399210865 \nu^{6} + 8391706662381 \nu^{5} - 154904740326313 \nu^{4} + \cdots + 21\!\cdots\!33 ) / 22\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 714193349 \nu^{7} - 15399210865 \nu^{6} - 8391706662381 \nu^{5} + 154904740326313 \nu^{4} + \cdots - 21\!\cdots\!97 ) / 22\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2823049907 \nu^{7} - 53647487587 \nu^{6} - 33526533332031 \nu^{5} + 538423542364927 \nu^{4} + \cdots - 91\!\cdots\!35 ) / 22\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1284180791 \nu^{7} + 20826858919 \nu^{6} + 15397215808443 \nu^{5} + \cdots + 49\!\cdots\!59 ) / 740276556293808 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 223750229 \nu^{7} + 4230510538 \nu^{6} + 2661493421103 \nu^{5} - 42271784379592 \nu^{4} + \cdots + 76\!\cdots\!48 ) / 123379426048968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 4\beta _1 + 3061 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{7} + 2\beta_{6} - 7\beta_{5} - 23\beta_{4} - 33\beta_{3} + 27\beta_{2} + 6110\beta _1 + 13632 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 72 \beta_{7} + 162 \beta_{6} + 330 \beta_{5} + 8230 \beta_{4} + 8578 \beta_{3} + 792 \beta_{2} + \cdots + 18630493 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 70016 \beta_{7} + 18926 \beta_{6} - 60376 \beta_{5} - 238082 \beta_{4} - 391134 \beta_{3} + \cdots - 128694720 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 283728 \beta_{7} + 1614090 \beta_{6} + 4123266 \beta_{5} + 65814907 \beta_{4} + 67926511 \beta_{3} + \cdots + 133820665753 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 551213264 \beta_{7} + 156551936 \beta_{6} - 462859795 \beta_{5} - 2356334213 \beta_{4} + \cdots - 2651130751584 \) Copy content Toggle raw display

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

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
−92.2519
−37.6110
−24.1963
−19.9766
−17.2845
37.9692
70.3594
82.9916
0 −87.2519 0 125.000 0 −732.025 0 5425.89 0
1.2 0 −32.6110 0 125.000 0 79.7819 0 −1123.52 0
1.3 0 −19.1963 0 125.000 0 1059.41 0 −1818.50 0
1.4 0 −14.9766 0 125.000 0 −1380.04 0 −1962.70 0
1.5 0 −12.2845 0 125.000 0 −560.280 0 −2036.09 0
1.6 0 42.9692 0 125.000 0 664.127 0 −340.644 0
1.7 0 75.3594 0 125.000 0 −1329.59 0 3492.05 0
1.8 0 87.9916 0 125.000 0 1582.61 0 5555.52 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)

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 260.8.a.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.8.a.d ✓ 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 40 T_{3}^{7} - 11544 T_{3}^{6} + 323968 T_{3}^{5} + 34057832 T_{3}^{4} + \cdots - 2863284312240 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(260))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots - 2863284312240 \) Copy content Toggle raw display
$5$ \( (T - 125)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 27\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T + 2197)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 11\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 54\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 20\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 87\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 60\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 54\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 18\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 10\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
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