Properties

Label 8034.2.a.o
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $7$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 4x^{5} + 14x^{4} + 3x^{3} - 12x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 4x^{5} + 14x^{4} + 3x^{3} - 12x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 3\nu^{4} - 12\nu^{3} + \nu^{2} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 3\nu^{4} - 12\nu^{3} + 2\nu^{2} + 5\nu - 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 13\nu^{3} - 2\nu^{2} - 9\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 4\nu^{5} - \nu^{4} + 17\nu^{3} - 9\nu^{2} - 10\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 6\beta_{4} - 5\beta_{3} + \beta_{2} + 6\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{6} + 7\beta_{5} + 9\beta_{4} - 3\beta_{3} + 2\beta_{2} + 20\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{6} + 12\beta_{5} + 34\beta_{4} - 26\beta_{3} + 9\beta_{2} + 37\beta _1 + 47 \) 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
−0.519850
2.16681
1.27539
−1.86678
2.51101
−0.761570
0.194986
−1.00000 −1.00000 1.00000 −2.94146 1.00000 −3.14882 −1.00000 1.00000 2.94146
1.2 −1.00000 −1.00000 1.00000 −2.61702 1.00000 −0.977840 −1.00000 1.00000 2.61702
1.3 −1.00000 −1.00000 1.00000 −2.55242 1.00000 1.37822 −1.00000 1.00000 2.55242
1.4 −1.00000 −1.00000 1.00000 1.72171 1.00000 2.39003 −1.00000 1.00000 −1.72171
1.5 −1.00000 −1.00000 1.00000 1.77686 1.00000 −3.99413 −1.00000 1.00000 −1.77686
1.6 −1.00000 −1.00000 1.00000 3.28276 1.00000 −3.26302 −1.00000 1.00000 −3.28276
1.7 −1.00000 −1.00000 1.00000 3.32958 1.00000 −1.38443 −1.00000 1.00000 −3.32958
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(-1\)
\(103\) \(-1\)

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 8034.2.a.o 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.o 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{7} - 2T_{5}^{6} - 23T_{5}^{5} + 41T_{5}^{4} + 173T_{5}^{3} - 279T_{5}^{2} - 417T_{5} + 657 \) Copy content Toggle raw display
\( T_{7}^{7} + 9T_{7}^{6} + 17T_{7}^{5} - 51T_{7}^{4} - 178T_{7}^{3} - 32T_{7}^{2} + 270T_{7} + 183 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 2 T^{6} - 23 T^{5} + 41 T^{4} + \cdots + 657 \) Copy content Toggle raw display
$7$ \( T^{7} + 9 T^{6} + 17 T^{5} - 51 T^{4} + \cdots + 183 \) Copy content Toggle raw display
$11$ \( T^{7} - 27 T^{5} - 5 T^{4} + 165 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$13$ \( (T - 1)^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 3 T^{6} - 71 T^{5} + \cdots - 14643 \) Copy content Toggle raw display
$19$ \( T^{7} + 16 T^{6} + 92 T^{5} + \cdots + 219 \) Copy content Toggle raw display
$23$ \( T^{7} - 6 T^{6} - 37 T^{5} + 249 T^{4} + \cdots - 11 \) Copy content Toggle raw display
$29$ \( T^{7} + 5 T^{6} - 101 T^{5} + \cdots + 657 \) Copy content Toggle raw display
$31$ \( T^{7} + 16 T^{6} + 8 T^{5} + \cdots - 4051 \) Copy content Toggle raw display
$37$ \( T^{7} - 17 T^{6} + 47 T^{5} + \cdots - 4101 \) Copy content Toggle raw display
$41$ \( T^{7} - 12 T^{6} - 123 T^{5} + \cdots - 419979 \) Copy content Toggle raw display
$43$ \( T^{7} + 22 T^{6} + 24 T^{5} + \cdots + 73433 \) Copy content Toggle raw display
$47$ \( T^{7} - 198 T^{5} - 16 T^{4} + \cdots - 415551 \) Copy content Toggle raw display
$53$ \( T^{7} - 2 T^{6} - 234 T^{5} + \cdots + 149421 \) Copy content Toggle raw display
$59$ \( T^{7} + 3 T^{6} - 203 T^{5} + \cdots + 113583 \) Copy content Toggle raw display
$61$ \( T^{7} + 6 T^{6} - 148 T^{5} + \cdots - 5977 \) Copy content Toggle raw display
$67$ \( T^{7} - T^{6} - 258 T^{5} + \cdots + 218443 \) Copy content Toggle raw display
$71$ \( T^{7} - 15 T^{6} - 45 T^{5} + \cdots - 85937 \) Copy content Toggle raw display
$73$ \( T^{7} - 17 T^{6} + 19 T^{5} + \cdots - 1563 \) Copy content Toggle raw display
$79$ \( T^{7} + 27 T^{6} + 191 T^{5} + \cdots + 141111 \) Copy content Toggle raw display
$83$ \( T^{7} - 12 T^{6} - 135 T^{5} + \cdots - 156427 \) Copy content Toggle raw display
$89$ \( T^{7} + 9 T^{6} - 283 T^{5} + \cdots - 4707 \) Copy content Toggle raw display
$97$ \( T^{7} + 3 T^{6} - 372 T^{5} + \cdots - 41263 \) Copy content Toggle raw display
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