Properties

Label 861.2.a.l
Level $861$
Weight $2$
Character orbit 861.a
Self dual yes
Analytic conductor $6.875$
Analytic rank $0$
Dimension $5$
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] = [861,2,Mod(1,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("861.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.626512.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_1 + 1) q^{6} - q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} + 1) q^{5} + ( - \beta_1 + 1) q^{6} - q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{8} + q^{9} + ( - \beta_{4} - \beta_1) q^{10} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{12} + ( - \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_1 - 1) q^{14} + (\beta_{3} + 1) q^{15} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{16} + ( - \beta_{3} + 2) q^{17} + ( - \beta_1 + 1) q^{18} + (\beta_{4} + 2 \beta_{3} + 3) q^{19} + ( - 2 \beta_{2} + 1) q^{20} - q^{21} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{22} + ( - 2 \beta_{4} + \beta_1 + 1) q^{23} + ( - \beta_{2} - 2 \beta_1 + 1) q^{24} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{25} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{26} + q^{27} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{28} + (\beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{29} + ( - \beta_{4} - \beta_1) q^{30} + ( - \beta_{4} - 2 \beta_{3} + 1) q^{31} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{32} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{33} + (\beta_{4} - 2 \beta_1 + 3) q^{34} + ( - \beta_{3} - 1) q^{35} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{36} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{37} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 1) q^{38} + ( - \beta_{3} + \beta_{2} + 1) q^{39} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{40} - q^{41} + (\beta_1 - 1) q^{42} + ( - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{44} + (\beta_{3} + 1) q^{45} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{47} + ( - \beta_{4} - \beta_{2} - \beta_1 + 2) q^{48} + q^{49} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 3) q^{50} + ( - \beta_{3} + 2) q^{51} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{52} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{53} + ( - \beta_1 + 1) q^{54} + ( - \beta_{4} + \beta_{2} + 3 \beta_1 - 1) q^{55} + (\beta_{2} + 2 \beta_1 - 1) q^{56} + (\beta_{4} + 2 \beta_{3} + 3) q^{57} + ( - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 6) q^{58} + ( - \beta_{3} - 2 \beta_1 + 2) q^{59} + ( - 2 \beta_{2} + 1) q^{60} + (2 \beta_{3} - 5 \beta_{2} - \beta_1 - 2) q^{61} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{62} - q^{63} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_1 - 5) q^{64} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{65} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{66} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{67} + (3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 5) q^{68} + ( - 2 \beta_{4} + \beta_1 + 1) q^{69} + (\beta_{4} + \beta_1) q^{70} + (2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 2) q^{71} + ( - \beta_{2} - 2 \beta_1 + 1) q^{72} + ( - \beta_{4} + 2 \beta_1 - 3) q^{73} + ( - 2 \beta_{4} + \beta_{2} + 4 \beta_1 - 5) q^{74} + ( - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{75} + (2 \beta_{4} - 3 \beta_{2} + \beta_1 + 6) q^{76} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{77} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{78} + (\beta_{4} + 4 \beta_{3} - 2 \beta_1 + 4) q^{79} + (\beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{80} + q^{81} + (\beta_1 - 1) q^{82} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 8) q^{83} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{84} + (\beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{85} + (\beta_{4} + 2 \beta_{2} + 6 \beta_1 - 3) q^{86} + (\beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{87} + (\beta_{4} - 4 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 9) q^{88} + (\beta_{4} + \beta_{2} + 3 \beta_1 + 1) q^{89} + ( - \beta_{4} - \beta_1) q^{90} + (\beta_{3} - \beta_{2} - 1) q^{91} + (4 \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 3) q^{92} + ( - \beta_{4} - 2 \beta_{3} + 1) q^{93} + ( - \beta_{3} + \beta_1 - 5) q^{94} + ( - 2 \beta_{4} + \beta_{2} + 3 \beta_1 + 6) q^{95} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2) q^{96} + (5 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{97} + ( - \beta_1 + 1) q^{98} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{2} + 5 q^{3} + 7 q^{4} + 3 q^{5} + 3 q^{6} - 5 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{2} + 5 q^{3} + 7 q^{4} + 3 q^{5} + 3 q^{6} - 5 q^{7} + 3 q^{8} + 5 q^{9} - q^{10} + 4 q^{11} + 7 q^{12} + 5 q^{13} - 3 q^{14} + 3 q^{15} + 11 q^{16} + 12 q^{17} + 3 q^{18} + 10 q^{19} + 9 q^{20} - 5 q^{21} - 6 q^{22} + 9 q^{23} + 3 q^{24} - 4 q^{25} + 13 q^{26} + 5 q^{27} - 7 q^{28} + 7 q^{29} - q^{30} + 10 q^{31} + 9 q^{32} + 4 q^{33} + 10 q^{34} - 3 q^{35} + 7 q^{36} - 11 q^{37} + 5 q^{39} - 7 q^{40} - 5 q^{41} - 3 q^{42} - 2 q^{43} + 3 q^{45} - 9 q^{46} - 7 q^{47} + 11 q^{48} + 5 q^{49} - 10 q^{50} + 12 q^{51} - 11 q^{52} - 9 q^{53} + 3 q^{54} - 3 q^{56} + 10 q^{57} - 31 q^{58} + 8 q^{59} + 9 q^{60} - 6 q^{61} + 12 q^{62} - 5 q^{63} - 29 q^{64} - 13 q^{65} - 6 q^{66} - 9 q^{67} + 12 q^{68} + 9 q^{69} + q^{70} + 4 q^{71} + 3 q^{72} - 10 q^{73} - 17 q^{74} - 4 q^{75} + 36 q^{76} - 4 q^{77} + 13 q^{78} + 7 q^{79} + 9 q^{80} + 5 q^{81} - 3 q^{82} + 50 q^{83} - 7 q^{84} - 12 q^{85} - 8 q^{86} + 7 q^{87} - 38 q^{88} + 8 q^{89} - q^{90} - 5 q^{91} - 17 q^{92} + 10 q^{93} - 21 q^{94} + 36 q^{95} + 9 q^{96} - 11 q^{97} + 3 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 6x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu^{2} - 3\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + 3\nu^{3} + 3\nu^{2} - 6\nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} + 3\beta_{3} - 2\beta_{2} + 6\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11\beta_{4} + 12\beta_{3} - 9\beta_{2} + 15\beta _1 + 23 \) 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
3.30065
1.11295
0.302390
−1.14904
−1.56695
−2.30065 1.00000 3.29299 1.89893 −2.30065 −1.00000 −2.97472 1.00000 −4.36878
1.2 −0.112947 1.00000 −1.98724 −2.19021 −0.112947 −1.00000 0.450348 1.00000 0.247378
1.3 0.697610 1.00000 −1.51334 3.10064 0.697610 −1.00000 −2.45094 1.00000 2.16304
1.4 2.14904 1.00000 2.61836 −1.12223 2.14904 −1.00000 1.32887 1.00000 −2.41172
1.5 2.56695 1.00000 4.58924 1.31287 2.56695 −1.00000 6.64645 1.00000 3.37008
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 861.2.a.l 5
3.b odd 2 1 2583.2.a.p 5
7.b odd 2 1 6027.2.a.w 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.a.l 5 1.a even 1 1 trivial
2583.2.a.p 5 3.b odd 2 1
6027.2.a.w 5 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(861))\):

\( T_{2}^{5} - 3T_{2}^{4} - 4T_{2}^{3} + 16T_{2}^{2} - 7T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{5} - 3T_{5}^{4} - 6T_{5}^{3} + 18T_{5}^{2} + 5T_{5} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 3 T^{4} - 4 T^{3} + 16 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 3 T^{4} - 6 T^{3} + 18 T^{2} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( (T + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 4 T^{4} - 24 T^{3} + 96 T^{2} + \cdots - 112 \) Copy content Toggle raw display
$13$ \( T^{5} - 5 T^{4} - 10 T^{3} + 48 T^{2} + \cdots - 103 \) Copy content Toggle raw display
$17$ \( T^{5} - 12 T^{4} + 48 T^{3} - 72 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{5} - 10 T^{4} + 6 T^{3} + 200 T^{2} + \cdots + 556 \) Copy content Toggle raw display
$23$ \( T^{5} - 9 T^{4} - 24 T^{3} + \cdots + 1441 \) Copy content Toggle raw display
$29$ \( T^{5} - 7 T^{4} - 64 T^{3} + \cdots - 1421 \) Copy content Toggle raw display
$31$ \( T^{5} - 10 T^{4} + 6 T^{3} + 48 T^{2} + \cdots - 44 \) Copy content Toggle raw display
$37$ \( T^{5} + 11 T^{4} + 2 T^{3} + \cdots - 1021 \) Copy content Toggle raw display
$41$ \( (T + 1)^{5} \) Copy content Toggle raw display
$43$ \( T^{5} + 2 T^{4} - 88 T^{3} + \cdots + 1228 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 48 T^{3} - 212 T^{2} + \cdots + 341 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} - 22 T^{3} + \cdots + 1043 \) Copy content Toggle raw display
$59$ \( T^{5} - 8 T^{4} - 20 T^{3} + 252 T^{2} + \cdots - 356 \) Copy content Toggle raw display
$61$ \( T^{5} + 6 T^{4} - 298 T^{3} + \cdots + 25172 \) Copy content Toggle raw display
$67$ \( T^{5} + 9 T^{4} - 56 T^{3} + \cdots + 7471 \) Copy content Toggle raw display
$71$ \( T^{5} - 4 T^{4} - 114 T^{3} + \cdots - 4708 \) Copy content Toggle raw display
$73$ \( T^{5} + 10 T^{4} - 2 T^{3} - 100 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$79$ \( T^{5} - 7 T^{4} - 120 T^{3} + \cdots + 2143 \) Copy content Toggle raw display
$83$ \( T^{5} - 50 T^{4} + 824 T^{3} + \cdots + 163808 \) Copy content Toggle raw display
$89$ \( T^{5} - 8 T^{4} - 52 T^{3} + 472 T^{2} + \cdots + 688 \) Copy content Toggle raw display
$97$ \( T^{5} + 11 T^{4} - 276 T^{3} + \cdots + 168041 \) Copy content Toggle raw display
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