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Properties

Label 260.10.a.a
Level $260$
Weight $10$
Character orbit 260.a
Self dual yes
Analytic conductor $133.909$
Analytic rank $1$
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] = [260,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(133.909317403\)
Analytic rank: \(1\)
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: \( x^{8} - 101068 x^{6} - 2601248 x^{5} + 3121514358 x^{4} + 138107972288 x^{3} - 31648729098668 x^{2} + \cdots + 23\!\cdots\!37 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4}\cdot 5^{3} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - 625 q^{5} + ( - \beta_{3} - 11 \beta_1 - 601) q^{7} + ( - \beta_{3} + \beta_{2} + \cdots + 5585) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - 625 q^{5} + ( - \beta_{3} - 11 \beta_1 - 601) q^{7} + ( - \beta_{3} + \beta_{2} + \cdots + 5585) q^{9}+ \cdots + ( - 5085 \beta_{7} - 16659 \beta_{6} + \cdots - 215665902) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 5000 q^{5} - 4808 q^{7} + 44680 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 5000 q^{5} - 4808 q^{7} + 44680 q^{9} - 63088 q^{11} + 228488 q^{13} + 5000 q^{15} - 74512 q^{17} - 981248 q^{19} - 2204416 q^{21} - 1016136 q^{23} + 3125000 q^{25} + 7512256 q^{27} + 3957952 q^{29} + 11372736 q^{31} + 10969296 q^{33} + 3005000 q^{35} + 7674288 q^{37} - 228488 q^{39} + 5873200 q^{41} + 27394472 q^{43} - 27925000 q^{45} + 61633640 q^{47} - 26376216 q^{49} + 74412288 q^{51} + 106321168 q^{53} + 39430000 q^{55} + 45723248 q^{57} - 40587008 q^{59} - 239058464 q^{61} + 221501288 q^{63} - 142805000 q^{65} - 373408008 q^{67} - 484564464 q^{69} + 20321120 q^{71} - 359059744 q^{73} - 3125000 q^{75} - 741127872 q^{77} - 128740816 q^{79} - 923615480 q^{81} + 343750008 q^{83} + 46570000 q^{85} - 1481571168 q^{87} - 64531728 q^{89} - 137321288 q^{91} + 160356416 q^{93} + 613280000 q^{95} - 1380655920 q^{97} - 1725327216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 101068 x^{6} - 2601248 x^{5} + 3121514358 x^{4} + 138107972288 x^{3} - 31648729098668 x^{2} + \cdots + 23\!\cdots\!37 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2843429 \nu^{7} - 89495417 \nu^{6} + 220695648669 \nu^{5} + 13476185129905 \nu^{4} + \cdots - 63\!\cdots\!77 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2843429 \nu^{7} - 89495417 \nu^{6} + 220695648669 \nu^{5} + 13476185129905 \nu^{4} + \cdots + 56\!\cdots\!51 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2328269 \nu^{7} - 248766334 \nu^{6} - 198785622789 \nu^{5} + 14224082697326 \nu^{4} + \cdots - 22\!\cdots\!10 ) / 13\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4895545 \nu^{7} + 1167499991 \nu^{6} + 436136058513 \nu^{5} - 84126021644959 \nu^{4} + \cdots - 14\!\cdots\!65 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5040878 \nu^{7} + 57430753 \nu^{6} + 402630360030 \nu^{5} + 6789446875591 \nu^{4} + \cdots + 43\!\cdots\!09 ) / 13\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21835255 \nu^{7} + 5270436943 \nu^{6} - 1651401335295 \nu^{5} - 464378055423095 \nu^{4} + \cdots - 49\!\cdots\!69 ) / 27\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 39\beta _1 + 25267 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -108\beta_{6} + 9\beta_{5} - 54\beta_{4} + 215\beta_{3} + 64\beta_{2} + 39307\beta _1 + 975468 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 648 \beta_{7} - 5076 \beta_{6} - 7272 \beta_{5} - 13716 \beta_{4} - 46658 \beta_{3} + 59690 \beta_{2} + \cdots + 992927977 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 132192 \beta_{7} - 8469468 \beta_{6} + 757062 \beta_{5} - 5070168 \beta_{4} + 13144606 \beta_{3} + \cdots + 77996850360 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58242888 \beta_{7} - 637734708 \beta_{6} - 553264776 \beta_{5} - 1191147444 \beta_{4} + \cdots + 47755922503915 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9022241376 \beta_{7} - 532423623960 \beta_{6} + 30964868163 \beta_{5} - 356577485094 \beta_{4} + \cdots + 55\!\cdots\!36 \) 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
−213.115
−143.257
−130.504
−86.9673
10.9562
155.637
157.085
250.165
0 −214.115 0 −625.000 0 8529.82 0 26162.0 0
1.2 0 −144.257 0 −625.000 0 6788.28 0 1127.18 0
1.3 0 −131.504 0 −625.000 0 −6688.25 0 −2389.75 0
1.4 0 −87.9673 0 −625.000 0 −8802.37 0 −11944.7 0
1.5 0 9.95624 0 −625.000 0 691.933 0 −19583.9 0
1.6 0 154.637 0 −625.000 0 3272.65 0 4229.62 0
1.7 0 156.085 0 −625.000 0 −6201.38 0 4679.56 0
1.8 0 249.165 0 −625.000 0 −2398.69 0 42400.0 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 260.10.a.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.10.a.a ✓ 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 8 T_{3}^{7} - 101040 T_{3}^{6} - 3207600 T_{3}^{5} + 3106992168 T_{3}^{4} + \cdots + 21\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(260))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T + 625)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T - 28561)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 71\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 54\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 36\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 68\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 46\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 16\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 24\!\cdots\!48 \) Copy content Toggle raw display
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