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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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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 4 x^{5} + 14 x^{4} + 3 x^{3} - 12 x^{2} - 3 x + 1\)
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 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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 3 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + \nu^{2} + 6 \nu \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 12 \nu^{3} + 2 \nu^{2} + 5 \nu - 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 3 \nu^{4} + 13 \nu^{3} - 2 \nu^{2} - 9 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{6} - 4 \nu^{5} - \nu^{4} + 17 \nu^{3} - 9 \nu^{2} - 10 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 6 \beta_{4} - 5 \beta_{3} + \beta_{2} + 6 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(-\beta_{6} + 7 \beta_{5} + 9 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 20 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(-3 \beta_{6} + 12 \beta_{5} + 34 \beta_{4} - 26 \beta_{3} + 9 \beta_{2} + 37 \beta_{1} + 47\)

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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{7} \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( 657 - 417 T - 279 T^{2} + 173 T^{3} + 41 T^{4} - 23 T^{5} - 2 T^{6} + T^{7} \)
$7$ \( 183 + 270 T - 32 T^{2} - 178 T^{3} - 51 T^{4} + 17 T^{5} + 9 T^{6} + T^{7} \)
$11$ \( -3 + 4 T + 108 T^{2} + 165 T^{3} - 5 T^{4} - 27 T^{5} + T^{7} \)
$13$ \( ( -1 + T )^{7} \)
$17$ \( -14643 - 11978 T + 365 T^{2} + 1694 T^{3} + 96 T^{4} - 71 T^{5} - 3 T^{6} + T^{7} \)
$19$ \( 219 - 386 T - 546 T^{2} + 3 T^{3} + 204 T^{4} + 92 T^{5} + 16 T^{6} + T^{7} \)
$23$ \( -11 + 12 T + 134 T^{2} - 339 T^{3} + 249 T^{4} - 37 T^{5} - 6 T^{6} + T^{7} \)
$29$ \( 657 + 741 T - 3081 T^{2} + 2125 T^{3} - 258 T^{4} - 101 T^{5} + 5 T^{6} + T^{7} \)
$31$ \( -4051 + 52340 T + 12488 T^{2} - 3069 T^{3} - 924 T^{4} + 8 T^{5} + 16 T^{6} + T^{7} \)
$37$ \( -4101 + 4745 T + 7465 T^{2} - 4573 T^{3} + 650 T^{4} + 47 T^{5} - 17 T^{6} + T^{7} \)
$41$ \( -419979 + 324275 T - 70528 T^{2} - 1513 T^{3} + 1988 T^{4} - 123 T^{5} - 12 T^{6} + T^{7} \)
$43$ \( 73433 - 12839 T - 51058 T^{2} - 19868 T^{3} - 2456 T^{4} + 24 T^{5} + 22 T^{6} + T^{7} \)
$47$ \( -415551 - 223656 T + 9962 T^{2} + 11829 T^{3} - 16 T^{4} - 198 T^{5} + T^{7} \)
$53$ \( 149421 + 86044 T - 85289 T^{2} + 12699 T^{3} + 1029 T^{4} - 234 T^{5} - 2 T^{6} + T^{7} \)
$59$ \( 113583 - 194038 T + 20202 T^{2} + 12076 T^{3} - 523 T^{4} - 203 T^{5} + 3 T^{6} + T^{7} \)
$61$ \( -5977 - 21639 T - 6810 T^{2} + 6210 T^{3} - 284 T^{4} - 148 T^{5} + 6 T^{6} + T^{7} \)
$67$ \( 218443 - 89273 T - 35244 T^{2} + 14037 T^{3} + 536 T^{4} - 258 T^{5} - T^{6} + T^{7} \)
$71$ \( -85937 + 98170 T - 18865 T^{2} - 4841 T^{3} + 1539 T^{4} - 45 T^{5} - 15 T^{6} + T^{7} \)
$73$ \( -1563 + 2535 T + 151 T^{2} - 1750 T^{3} + 627 T^{4} + 19 T^{5} - 17 T^{6} + T^{7} \)
$79$ \( 141111 + 52229 T - 17607 T^{2} - 8671 T^{3} - 434 T^{4} + 191 T^{5} + 27 T^{6} + T^{7} \)
$83$ \( -156427 - 179920 T - 29142 T^{2} + 7595 T^{3} + 1233 T^{4} - 135 T^{5} - 12 T^{6} + T^{7} \)
$89$ \( -4707 + 8295 T + 37845 T^{2} + 1376 T^{3} - 3117 T^{4} - 283 T^{5} + 9 T^{6} + T^{7} \)
$97$ \( -41263 - 199913 T - 50612 T^{2} + 27697 T^{3} + 36 T^{4} - 372 T^{5} + 3 T^{6} + T^{7} \)
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