Properties

Label 260.6.a.d
Level $260$
Weight $6$
Character orbit 260.a
Self dual yes
Analytic conductor $41.700$
Analytic rank $1$
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] = [260,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(41.6997931514\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 335x^{2} - 928x + 4264 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 5) q^{3} + 25 q^{5} + ( - 2 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{3} + \beta_{2} - \beta_1 - 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 5) q^{3} + 25 q^{5} + ( - 2 \beta_{2} + \beta_1 - 6) q^{7} + (9 \beta_{3} + \beta_{2} - \beta_1 - 21) q^{9} + (7 \beta_{2} - 2 \beta_1 - 3) q^{11} - 169 q^{13} + ( - 25 \beta_{3} - 125) q^{15} + (23 \beta_{3} + 5 \beta_{2} - 7 \beta_1 - 104) q^{17} + (76 \beta_{3} - 11 \beta_{2} - 6 \beta_1 - 161) q^{19} + (41 \beta_{3} + 11 \beta_{2} + 15 \beta_1 - 94) q^{21} + (154 \beta_{3} + 25 \beta_{2} + 5 \beta_1 + 437) q^{23} + 625 q^{25} + (206 \beta_{3} - 6 \beta_{2} - 416) q^{27} + ( - \beta_{3} - 101 \beta_{2} + 31 \beta_1 - 1020) q^{29} + ( - 33 \beta_{3} - 38 \beta_{2} - 13 \beta_1 - 375) q^{31} + ( - 106 \beta_{3} - 64 \beta_{2} - 48 \beta_1 + 524) q^{33} + ( - 50 \beta_{2} + 25 \beta_1 - 150) q^{35} + (116 \beta_{3} + 178 \beta_{2} + 36 \beta_1 - 962) q^{37} + (169 \beta_{3} + 845) q^{39} + ( - 529 \beta_{3} - 35 \beta_{2} + 81 \beta_1 - 4492) q^{41} + (379 \beta_{3} + 240 \beta_{2} + 56 \beta_1 + 755) q^{43} + (225 \beta_{3} + 25 \beta_{2} - 25 \beta_1 - 525) q^{45} + ( - 332 \beta_{3} - 10 \beta_{2} - 341 \beta_1 - 3082) q^{47} + ( - 164 \beta_{3} + 20 \beta_{2} + 4 \beta_1 - 7027) q^{49} + ( - 116 \beta_{3} + 26 \beta_{2} - 28 \beta_1 - 3926) q^{51} + (265 \beta_{3} - 157 \beta_{2} + 35 \beta_1 - 13168) q^{53} + (175 \beta_{2} - 50 \beta_1 - 75) q^{55} + ( - 54 \beta_{3} + 180 \beta_{2} + 124 \beta_1 - 15424) q^{57} + ( - 130 \beta_{3} - 263 \beta_{2} + 484 \beta_1 - 14747) q^{59} + ( - 1189 \beta_{3} + 3 \beta_{2} - 525 \beta_1 - 14184) q^{61} + ( - 78 \beta_{3} + 36 \beta_{2} - 223 \beta_1 - 4442) q^{63} - 4225 q^{65} + ( - 1710 \beta_{3} + 258 \beta_{2} - 157 \beta_1 - 14120) q^{67} + ( - 1333 \beta_{3} - 589 \beta_{2} + 19 \beta_1 - 30098) q^{69} + (491 \beta_{3} - 84 \beta_{2} + 781 \beta_1 - 17065) q^{71} + ( - 414 \beta_{3} - 700 \beta_{2} + 748 \beta_1 - 28250) q^{73} + ( - 625 \beta_{3} - 3125) q^{75} + ( - 53 \beta_{3} - 43 \beta_{2} + 13 \beta_1 - 31650) q^{77} + (414 \beta_{3} + 468 \beta_{2} - 836 \beta_1 - 10910) q^{79} + ( - 2517 \beta_{3} - 365 \beta_{2} + 485 \beta_1 - 33921) q^{81} + (1148 \beta_{3} - 790 \beta_{2} - 607 \beta_1 - 13062) q^{83} + (575 \beta_{3} + 125 \beta_{2} - 175 \beta_1 - 2600) q^{85} + (2616 \beta_{3} + 888 \beta_{2} + 698 \beta_1 - 1940) q^{87} + ( - 1012 \beta_{3} + 716 \beta_{2} - 496 \beta_1 - 28594) q^{89} + (338 \beta_{2} - 169 \beta_1 + 1014) q^{91} + (884 \beta_{3} + 786 \beta_{2} + 156 \beta_1 + 4420) q^{93} + (1900 \beta_{3} - 275 \beta_{2} - 150 \beta_1 - 4025) q^{95} + ( - 2834 \beta_{3} - 814 \beta_{2} - 1138 \beta_1 - 5358) q^{97} + (300 \beta_{3} + 117 \beta_{2} + 620 \beta_1 + 11023) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{3} + 100 q^{5} - 24 q^{7} - 84 q^{9} - 12 q^{11} - 676 q^{13} - 500 q^{15} - 416 q^{17} - 644 q^{19} - 376 q^{21} + 1748 q^{23} + 2500 q^{25} - 1664 q^{27} - 4080 q^{29} - 1500 q^{31} + 2096 q^{33} - 600 q^{35} - 3848 q^{37} + 3380 q^{39} - 17968 q^{41} + 3020 q^{43} - 2100 q^{45} - 12328 q^{47} - 28108 q^{49} - 15704 q^{51} - 52672 q^{53} - 300 q^{55} - 61696 q^{57} - 58988 q^{59} - 56736 q^{61} - 17768 q^{63} - 16900 q^{65} - 56480 q^{67} - 120392 q^{69} - 68260 q^{71} - 113000 q^{73} - 12500 q^{75} - 126600 q^{77} - 43640 q^{79} - 135684 q^{81} - 52248 q^{83} - 10400 q^{85} - 7760 q^{87} - 114376 q^{89} + 4056 q^{91} + 17680 q^{93} - 16100 q^{95} - 21432 q^{97} + 44092 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 335x^{2} - 928x + 4264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 12\nu^{2} - 522\nu - 118 ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{2} + 45\nu + 146 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} - 5\nu - 166 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 45\beta_{3} + 5\beta_{2} + 674 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 531\beta_{3} + 291\beta_{2} + 50\beta _1 + 4802 ) / 4 \) 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
20.2718
−14.9677
−5.74868
2.44457
0 −19.3586 0 25.0000 0 −94.2579 0 131.757 0
1.2 0 −18.2870 0 25.0000 0 76.3271 0 91.4133 0
1.3 0 5.42092 0 25.0000 0 107.399 0 −213.614 0
1.4 0 12.2247 0 25.0000 0 −113.468 0 −93.5569 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 260.6.a.d 4
4.b odd 2 1 1040.6.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.6.a.d 4 1.a even 1 1 trivial
1040.6.a.p 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 20T_{3}^{3} - 244T_{3}^{2} - 3752T_{3} + 23460 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(260))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 20 T^{3} - 244 T^{2} + \cdots + 23460 \) Copy content Toggle raw display
$5$ \( (T - 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24 T^{3} - 19272 T^{2} + \cdots + 87674192 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + \cdots + 11345306516 \) Copy content Toggle raw display
$13$ \( (T + 169)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 416 T^{3} + \cdots - 2150418352 \) Copy content Toggle raw display
$19$ \( T^{4} + 644 T^{3} + \cdots - 109653949260 \) Copy content Toggle raw display
$23$ \( T^{4} - 1748 T^{3} + \cdots + 2466019063044 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 457208195704832 \) Copy content Toggle raw display
$31$ \( T^{4} + 1500 T^{3} + \cdots - 62395614060 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 209287701681600 \) Copy content Toggle raw display
$41$ \( T^{4} + 17968 T^{3} + \cdots - 12\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{4} - 3020 T^{3} + \cdots - 22\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{4} + 12328 T^{3} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{4} + 52672 T^{3} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{4} + 58988 T^{3} + \cdots - 59\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{4} + 56736 T^{3} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + 56480 T^{3} + \cdots - 53\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{4} + 68260 T^{3} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{4} + 113000 T^{3} + \cdots - 62\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} + 43640 T^{3} + \cdots + 57\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{4} + 52248 T^{3} + \cdots - 85\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{4} + 114376 T^{3} + \cdots - 26\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{4} + 21432 T^{3} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
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