Properties

Label 8034.2.a.q
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 11 x^{6} + 21 x^{5} + 23 x^{4} - 29 x^{3} - 27 x^{2} + x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 11 x^{6} + 21 x^{5} + 23 x^{4} - 29 x^{3} - 27 x^{2} + x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 9 \nu^{5} + 31 \nu^{4} + 3 \nu^{3} - 41 \nu^{2} - 6 \nu + 8 \)
\(\beta_{3}\)\(=\)\( 2 \nu^{7} - 6 \nu^{6} - 17 \nu^{5} + 60 \nu^{4} - 4 \nu^{3} - 63 \nu^{2} - 3 \nu + 6 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{7} + 6 \nu^{6} + 17 \nu^{5} - 60 \nu^{4} + 4 \nu^{3} + 64 \nu^{2} + 3 \nu - 9 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{7} + 5 \nu^{6} + 19 \nu^{5} - 51 \nu^{4} - 15 \nu^{3} + 61 \nu^{2} + 14 \nu - 9 \)
\(\beta_{6}\)\(=\)\( -6 \nu^{7} + 17 \nu^{6} + 53 \nu^{5} - 171 \nu^{4} - 8 \nu^{3} + 188 \nu^{2} + 26 \nu - 24 \)
\(\beta_{7}\)\(=\)\( 8 \nu^{7} - 22 \nu^{6} - 71 \nu^{5} + 221 \nu^{4} + 13 \nu^{3} - 240 \nu^{2} - 28 \nu + 33 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + 10 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(2 \beta_{7} - 8 \beta_{6} + 12 \beta_{5} + 11 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} + 45 \beta_{1} - 7\)
\(\nu^{6}\)\(=\)\(13 \beta_{7} + 12 \beta_{6} + 13 \beta_{5} + 91 \beta_{4} + 77 \beta_{3} + 22 \beta_{2} - 40 \beta_{1} + 157\)
\(\nu^{7}\)\(=\)\(26 \beta_{7} - 64 \beta_{6} + 113 \beta_{5} + 100 \beta_{4} - 94 \beta_{3} + 23 \beta_{2} + 366 \beta_{1} - 97\)

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
−2.96271
−0.811382
−0.749852
−0.501271
0.315468
1.79755
2.03483
2.87737
1.00000 1.00000 1.00000 −3.96271 1.00000 0.133176 1.00000 1.00000 −3.96271
1.2 1.00000 1.00000 1.00000 −1.81138 1.00000 4.37785 1.00000 1.00000 −1.81138
1.3 1.00000 1.00000 1.00000 −1.74985 1.00000 −2.02962 1.00000 1.00000 −1.74985
1.4 1.00000 1.00000 1.00000 −1.50127 1.00000 1.44399 1.00000 1.00000 −1.50127
1.5 1.00000 1.00000 1.00000 −0.684532 1.00000 −1.46314 1.00000 1.00000 −0.684532
1.6 1.00000 1.00000 1.00000 0.797548 1.00000 −4.52438 1.00000 1.00000 0.797548
1.7 1.00000 1.00000 1.00000 1.03483 1.00000 0.662784 1.00000 1.00000 1.03483
1.8 1.00000 1.00000 1.00000 1.87737 1.00000 −1.60066 1.00000 1.00000 1.87737
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.q 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( -20 - 15 T + 55 T^{2} + 39 T^{3} - 37 T^{4} - 31 T^{5} + 3 T^{6} + 6 T^{7} + T^{8} \)
$7$ \( 12 - 95 T + 14 T^{2} + 168 T^{3} + 30 T^{4} - 65 T^{5} - 21 T^{6} + 3 T^{7} + T^{8} \)
$11$ \( -101 + 167 T + 198 T^{2} - 319 T^{3} - 136 T^{4} + 112 T^{5} + 75 T^{6} + 15 T^{7} + T^{8} \)
$13$ \( ( 1 + T )^{8} \)
$17$ \( 3114 - 13909 T + 6608 T^{2} + 5207 T^{3} - 434 T^{4} - 444 T^{5} - 15 T^{6} + 11 T^{7} + T^{8} \)
$19$ \( -9235 - 8185 T + 5880 T^{2} + 3649 T^{3} - 965 T^{4} - 418 T^{5} + 24 T^{6} + 15 T^{7} + T^{8} \)
$23$ \( 25017 + 37411 T - 30848 T^{2} - 3191 T^{3} + 3476 T^{4} + 82 T^{5} - 115 T^{6} - T^{7} + T^{8} \)
$29$ \( -25513 + 78970 T - 72836 T^{2} + 14782 T^{3} + 3943 T^{4} - 839 T^{5} - 100 T^{6} + 10 T^{7} + T^{8} \)
$31$ \( -5795 - 14955 T - 2260 T^{2} + 10481 T^{3} + 2929 T^{4} - 348 T^{5} - 108 T^{6} + 3 T^{7} + T^{8} \)
$37$ \( 421 + 164 T - 6266 T^{2} - 6322 T^{3} - 369 T^{4} + 793 T^{5} + 238 T^{6} + 26 T^{7} + T^{8} \)
$41$ \( -876 - 2423 T + 271 T^{2} + 3448 T^{3} + 835 T^{4} - 788 T^{5} - 65 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( -1893014 - 1893157 T - 209867 T^{2} + 92248 T^{3} + 14606 T^{4} - 1142 T^{5} - 222 T^{6} + 4 T^{7} + T^{8} \)
$47$ \( 2149042 - 419411 T - 228234 T^{2} + 30820 T^{3} + 9005 T^{4} - 748 T^{5} - 156 T^{6} + 6 T^{7} + T^{8} \)
$53$ \( 4 + 411 T + 1510 T^{2} + 1769 T^{3} + 563 T^{4} - 145 T^{5} - 64 T^{6} + 4 T^{7} + T^{8} \)
$59$ \( -153224 - 80937 T + 89596 T^{2} + 23816 T^{3} - 9822 T^{4} - 2455 T^{5} - 43 T^{6} + 19 T^{7} + T^{8} \)
$61$ \( -335576 + 320959 T + 447351 T^{2} + 95028 T^{3} - 7236 T^{4} - 3398 T^{5} - 174 T^{6} + 14 T^{7} + T^{8} \)
$67$ \( 693994 - 992681 T - 193 T^{2} + 97334 T^{3} + 2405 T^{4} - 2804 T^{5} - 190 T^{6} + 13 T^{7} + T^{8} \)
$71$ \( -39480 - 204325 T - 218810 T^{2} - 93441 T^{3} - 16035 T^{4} - 99 T^{5} + 287 T^{6} + 31 T^{7} + T^{8} \)
$73$ \( 372 + 2819 T - 6167 T^{2} - 2961 T^{3} + 1468 T^{4} + 1153 T^{5} + 269 T^{6} + 27 T^{7} + T^{8} \)
$79$ \( -230 + 1285 T - 1435 T^{2} - 1043 T^{3} + 2119 T^{4} - 642 T^{5} - 95 T^{6} + 13 T^{7} + T^{8} \)
$83$ \( -1706912 - 3112975 T - 1868666 T^{2} - 532510 T^{3} - 77053 T^{4} - 4645 T^{5} + 111 T^{6} + 28 T^{7} + T^{8} \)
$89$ \( -1021811 + 1275408 T - 420030 T^{2} - 9049 T^{3} + 18187 T^{4} - 480 T^{5} - 266 T^{6} + 2 T^{7} + T^{8} \)
$97$ \( 14708867 + 8918360 T + 1307927 T^{2} - 194557 T^{3} - 70829 T^{4} - 5664 T^{5} + 91 T^{6} + 30 T^{7} + T^{8} \)
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