Properties

Label 8034.2.a.l
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
−1.00000 −1.00000 1.00000 −2.00000 1.00000 −0.561553 −1.00000 1.00000 2.00000
1.2 −1.00000 −1.00000 1.00000 −2.00000 1.00000 3.56155 −1.00000 1.00000 2.00000
\(n\): e.g. 2-40 or 990-1000
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.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.l 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5} + 2 \)
\( T_{7}^{2} - 3 T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( -2 - 3 T + T^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( 26 - 11 T + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( -16 + 2 T + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -8 + 6 T + T^{2} \)
$37$ \( 8 + 10 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -8 + 6 T + T^{2} \)
$47$ \( -64 + 4 T + T^{2} \)
$53$ \( 16 - 9 T + T^{2} \)
$59$ \( ( 8 + T )^{2} \)
$61$ \( 32 - 14 T + T^{2} \)
$67$ \( -8 - 11 T + T^{2} \)
$71$ \( -32 + 12 T + T^{2} \)
$73$ \( -32 + 5 T + T^{2} \)
$79$ \( -128 + 10 T + T^{2} \)
$83$ \( -152 + 2 T + T^{2} \)
$89$ \( ( -2 + T )^{2} \)
$97$ \( -52 + 8 T + T^{2} \)
show more
show less