Properties

Label 8036.2.a.h
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta ) q^{3} + ( 4 + 2 \beta - 2 \beta^{2} ) q^{5} + ( -2 - 2 \beta + \beta^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta ) q^{3} + ( 4 + 2 \beta - 2 \beta^{2} ) q^{5} + ( -2 - 2 \beta + \beta^{2} ) q^{9} + ( 6 + 2 \beta - 4 \beta^{2} ) q^{11} + ( -5 - \beta + 3 \beta^{2} ) q^{13} + ( 6 + 4 \beta - 4 \beta^{2} ) q^{15} + ( 1 - \beta^{2} ) q^{17} + ( 5 - 2 \beta - \beta^{2} ) q^{19} + ( 5 - \beta - 2 \beta^{2} ) q^{23} + ( 3 - 4 \beta ) q^{25} + ( -6 + 3 \beta^{2} ) q^{27} + ( -4 + 2 \beta ) q^{29} + ( 12 + 4 \beta - 4 \beta^{2} ) q^{31} + ( 10 + 8 \beta - 6 \beta^{2} ) q^{33} + ( -9 - 3 \beta + \beta^{2} ) q^{37} + ( -8 - 5 \beta + 4 \beta^{2} ) q^{39} + q^{41} + ( 1 + 4 \beta - \beta^{2} ) q^{43} + ( -2 + 4 \beta - 2 \beta^{2} ) q^{45} + ( 3 - 5 \beta - \beta^{2} ) q^{47} + ( 2 + 2 \beta - \beta^{2} ) q^{51} + ( 2 + 2 \beta + 2 \beta^{2} ) q^{53} + ( 12 - 8 \beta ) q^{55} + ( 6 - 4 \beta + \beta^{2} ) q^{57} + ( 10 + 4 \beta - 4 \beta^{2} ) q^{59} + ( 10 - 2 \beta ) q^{61} + ( -12 + 4 \beta + 2 \beta^{2} ) q^{65} + ( -14 - 4 \beta + 4 \beta^{2} ) q^{67} + ( 7 - \beta^{2} ) q^{69} + ( 16 + 2 \beta - 6 \beta^{2} ) q^{71} + ( -4 - 4 \beta + 4 \beta^{2} ) q^{73} + ( 3 - 7 \beta + 4 \beta^{2} ) q^{75} + ( -8 - 6 \beta + 6 \beta^{2} ) q^{79} + ( -3 + 3 \beta ) q^{81} + ( 6 - 4 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + ( -4 + 6 \beta - 2 \beta^{2} ) q^{87} + ( -13 - 5 \beta + 7 \beta^{2} ) q^{89} + ( 16 + 4 \beta - 8 \beta^{2} ) q^{93} + ( 22 + 10 \beta - 12 \beta^{2} ) q^{95} + ( -9 - 5 \beta + 4 \beta^{2} ) q^{97} + ( -2 + 10 \beta - 2 \beta^{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + O(q^{10}) \) \( 3q + 3q^{3} - 6q^{11} + 3q^{13} - 6q^{15} - 3q^{17} + 9q^{19} + 3q^{23} + 9q^{25} - 12q^{29} + 12q^{31} - 6q^{33} - 21q^{37} + 3q^{41} - 3q^{43} - 18q^{45} + 3q^{47} + 18q^{53} + 36q^{55} + 24q^{57} + 6q^{59} + 30q^{61} - 24q^{65} - 18q^{67} + 15q^{69} + 12q^{71} + 12q^{73} + 33q^{75} + 12q^{79} - 9q^{81} + 18q^{83} + 6q^{85} - 24q^{87} + 3q^{89} - 6q^{95} - 3q^{97} - 18q^{99} + O(q^{100}) \)

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
1.87939
−0.347296
−1.53209
0 −0.879385 0 0.694593 0 0 0 −2.22668 0
1.2 0 1.34730 0 3.06418 0 0 0 −1.18479 0
1.3 0 2.53209 0 −3.75877 0 0 0 3.41147 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 8036.2.a.h 3
7.b odd 2 1 1148.2.a.b 3
28.d even 2 1 4592.2.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.a.b 3 7.b odd 2 1
4592.2.a.v 3 28.d even 2 1
8036.2.a.h 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3}^{3} - 3 T_{3}^{2} + 3 \)
\( T_{5}^{3} - 12 T_{5} + 8 \)
\( T_{11}^{3} + 6 T_{11}^{2} - 24 T_{11} - 136 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 9 T^{2} - 15 T^{3} + 27 T^{4} - 27 T^{5} + 27 T^{6} \)
$5$ \( 1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6} \)
$7$ 1
$11$ \( 1 + 6 T + 9 T^{2} - 4 T^{3} + 99 T^{4} + 726 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 3 T + 21 T^{2} - 21 T^{3} + 273 T^{4} - 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 3 T + 51 T^{2} + 99 T^{3} + 867 T^{4} + 867 T^{5} + 4913 T^{6} \)
$19$ \( 1 - 9 T + 63 T^{2} - 269 T^{3} + 1197 T^{4} - 3249 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 3 T + 51 T^{2} - 101 T^{3} + 1173 T^{4} - 1587 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 12 T + 123 T^{2} + 704 T^{3} + 3567 T^{4} + 10092 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 12 T + 93 T^{2} - 552 T^{3} + 2883 T^{4} - 11532 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 21 T + 237 T^{2} + 1733 T^{3} + 8769 T^{4} + 28749 T^{5} + 50653 T^{6} \)
$41$ \( ( 1 - T )^{3} \)
$43$ \( 1 + 3 T + 93 T^{2} + 239 T^{3} + 3999 T^{4} + 5547 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 3 T + 51 T^{2} + 99 T^{3} + 2397 T^{4} - 6627 T^{5} + 103823 T^{6} \)
$53$ \( 1 - 18 T + 231 T^{2} - 1980 T^{3} + 12243 T^{4} - 50562 T^{5} + 148877 T^{6} \)
$59$ \( 1 - 6 T + 141 T^{2} - 556 T^{3} + 8319 T^{4} - 20886 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 30 T + 471 T^{2} - 4532 T^{3} + 28731 T^{4} - 111630 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 18 T + 261 T^{2} + 2276 T^{3} + 17487 T^{4} + 80802 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 12 T + 177 T^{2} - 1728 T^{3} + 12567 T^{4} - 60492 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 12 T + 219 T^{2} - 1688 T^{3} + 15987 T^{4} - 63948 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 12 T + 177 T^{2} - 1744 T^{3} + 13983 T^{4} - 74892 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 18 T + 309 T^{2} - 2852 T^{3} + 25647 T^{4} - 124002 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 3 T + 153 T^{2} - 265 T^{3} + 13617 T^{4} - 23763 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 3 T + 231 T^{2} + 349 T^{3} + 22407 T^{4} + 28227 T^{5} + 912673 T^{6} \)
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