Properties

Label 8036.2.a.g
Level 8036
Weight 2
Character orbit 8036.a
Self dual yes
Analytic conductor 64.168
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} + ( 1 + \beta ) q^{5} + ( 1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} + ( 1 + \beta ) q^{5} + ( 1 + 3 \beta ) q^{9} + q^{11} + 5 q^{13} + ( 4 + 3 \beta ) q^{15} + ( 3 - 4 \beta ) q^{17} + ( 3 - 3 \beta ) q^{19} + ( -1 - \beta ) q^{23} + ( -1 + 3 \beta ) q^{25} + ( 7 + 4 \beta ) q^{27} + ( 8 + \beta ) q^{29} + ( -7 + 3 \beta ) q^{31} + ( 1 + \beta ) q^{33} + ( 4 - 2 \beta ) q^{37} + ( 5 + 5 \beta ) q^{39} + q^{41} + ( -7 - 2 \beta ) q^{43} + ( 10 + 7 \beta ) q^{45} + ( 10 - 2 \beta ) q^{47} + ( -9 - 5 \beta ) q^{51} + ( -5 + \beta ) q^{53} + ( 1 + \beta ) q^{55} + ( -6 - 3 \beta ) q^{57} + ( 1 - 3 \beta ) q^{59} + ( 1 + 4 \beta ) q^{61} + ( 5 + 5 \beta ) q^{65} + ( -5 + 3 \beta ) q^{67} + ( -4 - 3 \beta ) q^{69} + ( -3 - 4 \beta ) q^{71} -7 q^{73} + ( 8 + 5 \beta ) q^{75} + ( -12 - 2 \beta ) q^{79} + ( 16 + 6 \beta ) q^{81} + ( -3 + 6 \beta ) q^{83} + ( -9 - 5 \beta ) q^{85} + ( 11 + 10 \beta ) q^{87} + ( -3 + 5 \beta ) q^{89} + ( 2 - \beta ) q^{93} + ( -6 - 3 \beta ) q^{95} + ( 7 - 7 \beta ) q^{97} + ( 1 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} + 3q^{5} + 5q^{9} + O(q^{10}) \) \( 2q + 3q^{3} + 3q^{5} + 5q^{9} + 2q^{11} + 10q^{13} + 11q^{15} + 2q^{17} + 3q^{19} - 3q^{23} + q^{25} + 18q^{27} + 17q^{29} - 11q^{31} + 3q^{33} + 6q^{37} + 15q^{39} + 2q^{41} - 16q^{43} + 27q^{45} + 18q^{47} - 23q^{51} - 9q^{53} + 3q^{55} - 15q^{57} - q^{59} + 6q^{61} + 15q^{65} - 7q^{67} - 11q^{69} - 10q^{71} - 14q^{73} + 21q^{75} - 26q^{79} + 38q^{81} - 23q^{85} + 32q^{87} - q^{89} + 3q^{93} - 15q^{95} + 7q^{97} + 5q^{99} + O(q^{100}) \)

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
−1.30278
2.30278
0 −0.302776 0 −0.302776 0 0 0 −2.90833 0
1.2 0 3.30278 0 3.30278 0 0 0 7.90833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 8036.2.a.g 2
7.b odd 2 1 1148.2.a.a 2
28.d even 2 1 4592.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.a 2 7.b odd 2 1
4592.2.a.o 2 28.d even 2 1
8036.2.a.g 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{2} - 3 T_{3} - 1 \)
\( T_{5}^{2} - 3 T_{5} - 1 \)
\( T_{11} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 3 T + 5 T^{2} - 9 T^{3} + 9 T^{4} \)
$5$ \( 1 - 3 T + 9 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ \( \)
$11$ \( ( 1 - T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T - 17 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 3 T + 11 T^{2} - 57 T^{3} + 361 T^{4} \)
$23$ \( 1 + 3 T + 45 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 - 17 T + 127 T^{2} - 493 T^{3} + 841 T^{4} \)
$31$ \( 1 + 11 T + 63 T^{2} + 341 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T + 70 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - T )^{2} \)
$43$ \( 1 + 16 T + 137 T^{2} + 688 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 18 T + 162 T^{2} - 846 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 123 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + T + 89 T^{2} + 59 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 6 T + 79 T^{2} - 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 7 T + 117 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 10 T + 115 T^{2} + 710 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 7 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 26 T + 314 T^{2} + 2054 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 49 T^{2} + 6889 T^{4} \)
$89$ \( 1 + T + 97 T^{2} + 89 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 7 T + 47 T^{2} - 679 T^{3} + 9409 T^{4} \)
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