Properties

Label 8281.2.a.bt
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.27004.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 - 2) q^{5} + ( - \beta_1 - 1) q^{6} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 - 2) q^{5} + ( - \beta_1 - 1) q^{6} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{8} + (\beta_{3} - \beta_{2} + 2) q^{9} + (\beta_{2} - \beta_1 + 3) q^{10} - \beta_{3} q^{11} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{12} + ( - 2 \beta_{3} - \beta_1 - 1) q^{15} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 4) q^{16} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} + ( - \beta_{3} - 2 \beta_1 + 1) q^{18} + (2 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{20} + (\beta_1 + 1) q^{22} + (\beta_{3} + \beta_{2} - \beta_1) q^{23} + ( - \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{24} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + (3 \beta_{3} + \beta_1 + 7) q^{27} + ( - \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{2} - 1) q^{30} + ( - \beta_{2} - 1) q^{31} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 7) q^{32} + ( - \beta_{3} + \beta_{2} - 5) q^{33} + (\beta_{3} + \beta_{2} + 4 \beta_1) q^{34} + ( - 2 \beta_{3} - 9) q^{36} + ( - \beta_{3} + \beta_1 + 2) q^{37} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 5) q^{38} + ( - \beta_{3} + \beta_{2} + 4) q^{40} + (\beta_{2} + 2 \beta_1 - 6) q^{41} + ( - \beta_1 + 1) q^{43} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{44} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{45} + (\beta_{3} - \beta_{2} + \beta_1 - 6) q^{46} + ( - \beta_{3} - 3 \beta_1 + 1) q^{47} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 1) q^{48} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 11) q^{50} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{51} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{53} + (\beta_{2} + 5 \beta_1) q^{54} + (2 \beta_{3} + \beta_1 + 1) q^{55} + ( - \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 5) q^{57} + ( - \beta_{3} - 2 \beta_1 + 3) q^{58} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{59} + (3 \beta_{3} - 2 \beta_1 + 4) q^{60} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{61} + ( - \beta_{3} - 4 \beta_1 + 2) q^{62} + (2 \beta_{2} + 10 \beta_1 + 1) q^{64} + (\beta_{3} - \beta_1 - 1) q^{66} + (4 \beta_{3} + \beta_{2} + 6 \beta_1 + 1) q^{67} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 7) q^{68} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{69} + (2 \beta_{3} + 2 \beta_{2} + 4) q^{71} + (2 \beta_{3} - 3 \beta_1) q^{72} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{73} + (\beta_{2} + 4 \beta_1 + 4) q^{74} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{75} + ( - \beta_{3} + 2 \beta_{2} + 11 \beta_1 + 5) q^{76} + (3 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{79} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{80} + (7 \beta_{3} - \beta_1 + 8) q^{81} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{82} + (\beta_{2} + 4 \beta_1 - 1) q^{83} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{85} + ( - \beta_{2} - 3) q^{86} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{87} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{88} + (2 \beta_{3} - \beta_{2} + \beta_1) q^{89} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 7) q^{90} + ( - 3 \beta_{3} - \beta_{2} - 7 \beta_1 + 4) q^{92} + (2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{93} + ( - 3 \beta_{2} - \beta_1 - 8) q^{94} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 5) q^{95} + ( - \beta_{2} - 6 \beta_1 - 13) q^{96} + ( - 5 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{97} + ( - 6 \beta_{3} - \beta_1 - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} + 5 q^{4} - 7 q^{5} - 5 q^{6} + 6 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{3} + 5 q^{4} - 7 q^{5} - 5 q^{6} + 6 q^{8} + 7 q^{9} + 11 q^{10} + q^{11} - 12 q^{12} - 3 q^{15} + 19 q^{16} + 4 q^{17} + 3 q^{18} + q^{19} - 2 q^{20} + 5 q^{22} - 2 q^{23} - 3 q^{24} + 5 q^{25} + 26 q^{27} + q^{29} - 4 q^{30} - 4 q^{31} + 33 q^{32} - 19 q^{33} + 3 q^{34} - 34 q^{36} + 10 q^{37} + 23 q^{38} + 17 q^{40} - 22 q^{41} + 3 q^{43} + 12 q^{44} - 11 q^{45} - 24 q^{46} + 2 q^{47} - 11 q^{48} - 43 q^{50} + 7 q^{51} + 2 q^{53} + 5 q^{54} + 3 q^{55} + 17 q^{57} + 11 q^{58} - 8 q^{59} + 11 q^{60} - 8 q^{61} + 5 q^{62} + 14 q^{64} - 6 q^{66} + 6 q^{67} + 33 q^{68} + 18 q^{69} + 14 q^{71} - 5 q^{72} - 8 q^{73} + 20 q^{74} + 7 q^{75} + 32 q^{76} - 26 q^{79} + 7 q^{80} + 24 q^{81} + 14 q^{82} - 5 q^{85} - 12 q^{86} - 13 q^{87} + 3 q^{88} - q^{89} - 26 q^{90} + 12 q^{92} + 7 q^{93} - 33 q^{94} + 21 q^{95} - 58 q^{96} + 3 q^{97} - 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 6x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 6\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 7\beta _1 + 1 \) 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
−2.22001
−0.231361
0.710287
2.74108
−2.22001 −0.549551 2.92843 −4.22001 1.22001 0 −2.06113 −2.69799 9.36845
1.2 −0.231361 3.32225 −1.94647 −2.23136 −0.768639 0 0.913059 8.03736 0.516249
1.3 0.710287 −2.40788 −1.49549 −1.28971 −1.71029 0 −2.48280 2.79790 −0.916066
1.4 2.74108 −1.36482 5.51353 0.741082 −3.74108 0 9.63087 −1.13727 2.03137
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

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 8281.2.a.bt 4
7.b odd 2 1 1183.2.a.l 4
13.b even 2 1 8281.2.a.bp 4
13.e even 6 2 637.2.f.i 8
91.b odd 2 1 1183.2.a.k 4
91.i even 4 2 1183.2.c.g 8
91.k even 6 2 637.2.h.i 8
91.l odd 6 2 637.2.h.h 8
91.p odd 6 2 637.2.g.k 8
91.t odd 6 2 91.2.f.c 8
91.u even 6 2 637.2.g.j 8
273.u even 6 2 819.2.o.h 8
364.bc even 6 2 1456.2.s.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 91.t odd 6 2
637.2.f.i 8 13.e even 6 2
637.2.g.j 8 91.u even 6 2
637.2.g.k 8 91.p odd 6 2
637.2.h.h 8 91.l odd 6 2
637.2.h.i 8 91.k even 6 2
819.2.o.h 8 273.u even 6 2
1183.2.a.k 4 91.b odd 2 1
1183.2.a.l 4 7.b odd 2 1
1183.2.c.g 8 91.i even 4 2
1456.2.s.q 8 364.bc even 6 2
8281.2.a.bp 4 13.b even 2 1
8281.2.a.bt 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{4} - T_{2}^{3} - 6T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} - 9T_{3}^{2} - 16T_{3} - 6 \) Copy content Toggle raw display
\( T_{5}^{4} + 7T_{5}^{3} + 12T_{5}^{2} - T_{5} - 9 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 9T_{11}^{2} + 16T_{11} - 6 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 12T_{17}^{2} + 60T_{17} - 53 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 6 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 9 T^{2} - 16 T - 6 \) Copy content Toggle raw display
$5$ \( T^{4} + 7 T^{3} + 12 T^{2} - T - 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} - 9 T^{2} + 16 T - 6 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 12 T^{2} + 60 T - 53 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{3} - 55 T^{2} + 500 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} - 38 T^{2} - 56 T + 72 \) Copy content Toggle raw display
$29$ \( T^{4} - T^{3} - 22 T^{2} - 21 T - 5 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 13 T^{2} - 26 T + 54 \) Copy content Toggle raw display
$37$ \( T^{4} - 10 T^{3} + 17 T^{2} + 44 T - 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 22 T^{3} + 149 T^{2} + \cdots - 564 \) Copy content Toggle raw display
$43$ \( T^{4} - 3 T^{3} - 3 T^{2} + 8 T - 2 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} - 51 T^{2} - 12 T + 100 \) Copy content Toggle raw display
$53$ \( T^{4} - 2 T^{3} - 140 T^{2} + \cdots - 1389 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} - 77 T^{2} - 550 T - 706 \) Copy content Toggle raw display
$61$ \( T^{4} + 8 T^{3} - 15 T^{2} - 176 T - 100 \) Copy content Toggle raw display
$67$ \( T^{4} - 6 T^{3} - 211 T^{2} + \cdots + 11010 \) Copy content Toggle raw display
$71$ \( T^{4} - 14 T^{3} - 16 T^{2} + \cdots - 2032 \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} - 119 T^{2} + \cdots + 1772 \) Copy content Toggle raw display
$79$ \( T^{4} + 26 T^{3} + 134 T^{2} + \cdots - 7680 \) Copy content Toggle raw display
$83$ \( T^{4} - 97 T^{2} - 442 T - 426 \) Copy content Toggle raw display
$89$ \( T^{4} + T^{3} - 71 T^{2} - 184 T - 108 \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} - 305 T^{2} + \cdots + 15692 \) Copy content Toggle raw display
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