Properties

Label 260.4.a.c
Level $260$
Weight $4$
Character orbit 260.a
Self dual yes
Analytic conductor $15.340$
Analytic rank $0$
Dimension $4$
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] = [260,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(15.3404966015\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2785296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 30x^{2} + 2x + 65 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} - 5 q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + ( - 4 \beta_{2} + 2 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} - 5 q^{5} + (\beta_{3} - \beta_{2} - 2) q^{7} + ( - 4 \beta_{2} + 2 \beta_1 + 9) q^{9} + ( - \beta_{3} - \beta_{2} - 3 \beta_1 + 7) q^{11} + 13 q^{13} + (5 \beta_{2} - 5) q^{15} + ( - 4 \beta_{3} - 5 \beta_{2} + \beta_1 + 50) q^{17} + (3 \beta_{3} - 7 \beta_{2} + \beta_1 + 33) q^{19} + (\beta_{2} - \beta_1 + 40) q^{21} + ( - 2 \beta_{3} - 7 \beta_1 + 61) q^{23} + 25 q^{25} + ( - 4 \beta_{3} - 20 \beta_{2} + 8 \beta_1 + 124) q^{27} + (6 \beta_{3} + 16 \beta_{2} + 4 \beta_1 - 2) q^{29} + (9 \beta_{3} - 6 \beta_{2} + 2 \beta_1 + 73) q^{31} + (6 \beta_{3} + 27 \beta_{2} + 5 \beta_1 + 32) q^{33} + ( - 5 \beta_{3} + 5 \beta_{2} + 10) q^{35} + ( - 4 \beta_{3} + 33 \beta_{2} - 3 \beta_1 + 42) q^{37} + ( - 13 \beta_{2} + 13) q^{39} + ( - 8 \beta_{3} + 5 \beta_{2} - 13 \beta_1 + 70) q^{41} + (6 \beta_{3} + 43 \beta_{2} + 6 \beta_1 + 113) q^{43} + (20 \beta_{2} - 10 \beta_1 - 45) q^{45} + (3 \beta_{3} + 29 \beta_{2} - 16 \beta_1 - 70) q^{47} + ( - 8 \beta_{3} + 12 \beta_{2} - 28 \beta_1 + 85) q^{49} + ( - 2 \beta_{3} - 86 \beta_{2} + 22 \beta_1 + 198) q^{51} + (39 \beta_{2} + 17 \beta_1 + 146) q^{53} + (5 \beta_{3} + 5 \beta_{2} + 15 \beta_1 - 35) q^{55} + ( - 2 \beta_{3} - 61 \beta_{2} + 5 \beta_1 + 300) q^{57} + ( - 11 \beta_{3} - 27 \beta_{2} + \beta_1 - 177) q^{59} + ( - 10 \beta_{3} - 26 \beta_{2} + 18 \beta_1 + 282) q^{61} + ( - 25 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 58) q^{63} - 65 q^{65} + (11 \beta_{3} - 59 \beta_{2} - 10 \beta_1 + 44) q^{67} + (14 \beta_{3} + 26 \beta_{2} + 6 \beta_1 + 40) q^{69} + (23 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 257) q^{71} + (26 \beta_{3} + 85 \beta_{2} - 17 \beta_1 + 166) q^{73} + ( - 25 \beta_{2} + 25) q^{75} + (56 \beta_{3} - 19 \beta_{2} + 23 \beta_1 - 216) q^{77} + ( - 28 \beta_{3} - 110 \beta_{2} - 8 \beta_1 - 182) q^{79} + ( - 16 \beta_{3} - 188 \beta_{2} - 2 \beta_1 + 561) q^{81} + ( - 23 \beta_{3} + 25 \beta_{2} - 14 \beta_1 + 138) q^{83} + (20 \beta_{3} + 25 \beta_{2} - 5 \beta_1 - 250) q^{85} + ( - 8 \beta_{3} + 10 \beta_{2} - 50 \beta_1 - 516) q^{87} + ( - 24 \beta_{3} + 118 \beta_{2} + 2 \beta_1 - 706) q^{89} + (13 \beta_{3} - 13 \beta_{2} - 26) q^{91} + ( - 4 \beta_{3} - 99 \beta_{2} - 15 \beta_1 + 348) q^{93} + ( - 15 \beta_{3} + 35 \beta_{2} - 5 \beta_1 - 165) q^{95} + ( - 32 \beta_{3} + 16 \beta_1 + 290) q^{97} + (17 \beta_{3} + 23 \beta_{2} + 9 \beta_1 - 1055) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 20 q^{5} - 8 q^{7} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 20 q^{5} - 8 q^{7} + 36 q^{9} + 28 q^{11} + 52 q^{13} - 20 q^{15} + 200 q^{17} + 132 q^{19} + 160 q^{21} + 244 q^{23} + 100 q^{25} + 496 q^{27} - 8 q^{29} + 292 q^{31} + 128 q^{33} + 40 q^{35} + 168 q^{37} + 52 q^{39} + 280 q^{41} + 452 q^{43} - 180 q^{45} - 280 q^{47} + 340 q^{49} + 792 q^{51} + 584 q^{53} - 140 q^{55} + 1200 q^{57} - 708 q^{59} + 1128 q^{61} + 232 q^{63} - 260 q^{65} + 176 q^{67} + 160 q^{69} - 1028 q^{71} + 664 q^{73} + 100 q^{75} - 864 q^{77} - 728 q^{79} + 2244 q^{81} + 552 q^{83} - 1000 q^{85} - 2064 q^{87} - 2824 q^{89} - 104 q^{91} + 1392 q^{93} - 660 q^{95} + 1160 q^{97} - 4220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 30x^{2} + 2x + 65 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 49\nu + 5 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 25\nu - 17 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} + 17\nu - 43 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 4\beta_{2} + \beta _1 + 32 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 57\beta_{2} + 27\beta _1 + 182 ) / 4 \) 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
6.39420
−1.59616
−4.28450
1.48645
0 −6.28190 0 −5.00000 0 −4.93274 0 12.4622 0
1.2 0 −1.71495 0 −5.00000 0 −31.3797 0 −24.0590 0
1.3 0 2.14910 0 −5.00000 0 26.4482 0 −22.3814 0
1.4 0 9.84775 0 −5.00000 0 1.86425 0 69.9781 0
\(n\): e.g. 2-40 or 990-1000
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 260.4.a.c 4
3.b odd 2 1 2340.4.a.m 4
4.b odd 2 1 1040.4.a.u 4
5.b even 2 1 1300.4.a.k 4
5.c odd 4 2 1300.4.c.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.4.a.c 4 1.a even 1 1 trivial
1040.4.a.u 4 4.b odd 2 1
1300.4.a.k 4 5.b even 2 1
1300.4.c.g 8 5.c odd 4 2
2340.4.a.m 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 64T_{3}^{2} + 40T_{3} + 228 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(260))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} - 64 T^{2} + 40 T + 228 \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} - 824 T^{2} + \cdots + 7632 \) Copy content Toggle raw display
$11$ \( T^{4} - 28 T^{3} - 3864 T^{2} + \cdots + 2100852 \) Copy content Toggle raw display
$13$ \( (T - 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 200 T^{3} + \cdots - 14951088 \) Copy content Toggle raw display
$19$ \( T^{4} - 132 T^{3} - 4136 T^{2} + \cdots - 8535020 \) Copy content Toggle raw display
$23$ \( T^{4} - 244 T^{3} + \cdots + 29171364 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} - 44448 T^{2} + \cdots + 293007168 \) Copy content Toggle raw display
$31$ \( T^{4} - 292 T^{3} + \cdots - 642276076 \) Copy content Toggle raw display
$37$ \( T^{4} - 168 T^{3} + \cdots + 805400832 \) Copy content Toggle raw display
$41$ \( T^{4} - 280 T^{3} + \cdots + 496536912 \) Copy content Toggle raw display
$43$ \( T^{4} - 452 T^{3} + \cdots - 45746748 \) Copy content Toggle raw display
$47$ \( T^{4} + 280 T^{3} + \cdots + 1647798480 \) Copy content Toggle raw display
$53$ \( T^{4} - 584 T^{3} + \cdots - 168468912 \) Copy content Toggle raw display
$59$ \( T^{4} + 708 T^{3} + \cdots + 777261396 \) Copy content Toggle raw display
$61$ \( T^{4} - 1128 T^{3} + \cdots - 21509627072 \) Copy content Toggle raw display
$67$ \( T^{4} - 176 T^{3} + \cdots + 33912154512 \) Copy content Toggle raw display
$71$ \( T^{4} + 1028 T^{3} + \cdots - 3387108492 \) Copy content Toggle raw display
$73$ \( T^{4} - 664 T^{3} + \cdots + 67723906304 \) Copy content Toggle raw display
$79$ \( T^{4} + 728 T^{3} + \cdots + 245418881600 \) Copy content Toggle raw display
$83$ \( T^{4} - 552 T^{3} + \cdots - 311448240 \) Copy content Toggle raw display
$89$ \( T^{4} + 2824 T^{3} + \cdots + 46519298064 \) Copy content Toggle raw display
$97$ \( T^{4} - 1160 T^{3} + \cdots - 226545046512 \) Copy content Toggle raw display
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