Properties

Label 8034.2.a.p
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} - 12 x^{6} + 12 x^{5} + 43 x^{4} - 38 x^{3} - 49 x^{2} + 23 x + 20\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 12 x^{6} + 12 x^{5} + 43 x^{4} - 38 x^{3} - 49 x^{2} + 23 x + 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{6} - 11 \nu^{4} + \nu^{3} + 32 \nu^{2} - 5 \nu - 17 \)
\(\beta_{3}\)\(=\)\( 3 \nu^{7} + \nu^{6} - 32 \nu^{5} - 6 \nu^{4} + 92 \nu^{3} + 7 \nu^{2} - 53 \nu - 11 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 43 \nu^{2} - 37 \nu - 25 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 44 \nu^{2} - 37 \nu - 28 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 11 \nu^{5} + 33 \nu^{4} + 29 \nu^{3} - 95 \nu^{2} - 2 \nu + 46 \)
\(\beta_{7}\)\(=\)\( 2 \nu^{7} + 3 \nu^{6} - 21 \nu^{5} - 29 \nu^{4} + 62 \nu^{3} + 76 \nu^{2} - 47 \nu - 45 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{5} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - \beta_{6} + 6 \beta_{5} - 6 \beta_{4} + \beta_{3} - \beta_{2} - \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\(10 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 10 \beta_{2} + 27 \beta_{1} - 2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{7} - 11 \beta_{6} + 35 \beta_{5} - 34 \beta_{4} + 11 \beta_{3} - 9 \beta_{2} - 11 \beta_{1} + 64\)
\(\nu^{7}\)\(=\)\(78 \beta_{7} + 23 \beta_{6} - 46 \beta_{5} + 23 \beta_{4} - 44 \beta_{3} - 75 \beta_{2} + 154 \beta_{1} - 20\)

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
1.82602
−0.685988
0.965971
2.41269
−2.01106
−0.735807
−2.57192
1.80009
1.00000 1.00000 1.00000 −3.53629 1.00000 0.738654 1.00000 1.00000 −3.53629
1.2 1.00000 1.00000 1.00000 −3.08319 1.00000 1.60046 1.00000 1.00000 −3.08319
1.3 1.00000 1.00000 1.00000 −2.24957 1.00000 3.57309 1.00000 1.00000 −2.24957
1.4 1.00000 1.00000 1.00000 −1.15050 1.00000 −2.62035 1.00000 1.00000 −1.15050
1.5 1.00000 1.00000 1.00000 −0.612396 1.00000 −3.48353 1.00000 1.00000 −0.612396
1.6 1.00000 1.00000 1.00000 −0.216275 1.00000 −0.843680 1.00000 1.00000 −0.216275
1.7 1.00000 1.00000 1.00000 −0.176895 1.00000 −2.27129 1.00000 1.00000 −0.176895
1.8 1.00000 1.00000 1.00000 3.02511 1.00000 −2.69336 1.00000 1.00000 3.02511
\(n\): 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.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.p 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

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( -2 - 27 T - 125 T^{2} - 237 T^{3} - 189 T^{4} - 47 T^{5} + 13 T^{6} + 8 T^{7} + T^{8} \)
$7$ \( -199 - 81 T + 408 T^{2} + 228 T^{3} - 117 T^{4} - 94 T^{5} - 6 T^{6} + 6 T^{7} + T^{8} \)
$11$ \( 1565 + 5881 T + 5910 T^{2} + 1731 T^{3} - 418 T^{4} - 296 T^{5} - 27 T^{6} + 7 T^{7} + T^{8} \)
$13$ \( ( -1 + T )^{8} \)
$17$ \( -2827 - 10797 T - 13641 T^{2} - 7255 T^{3} - 1316 T^{4} + 239 T^{5} + 136 T^{6} + 20 T^{7} + T^{8} \)
$19$ \( 12034 - 27425 T + 5034 T^{2} + 6030 T^{3} - 543 T^{4} - 476 T^{5} - 8 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( -31420 - 9051 T + 17758 T^{2} + 5702 T^{3} - 1395 T^{4} - 571 T^{5} + 3 T^{6} + 14 T^{7} + T^{8} \)
$29$ \( 2650 + 11031 T + 4153 T^{2} - 7589 T^{3} - 2687 T^{4} + 276 T^{5} + 197 T^{6} + 25 T^{7} + T^{8} \)
$31$ \( -32146 + 220965 T + 186802 T^{2} + 39942 T^{3} - 2507 T^{4} - 1584 T^{5} - 88 T^{6} + 12 T^{7} + T^{8} \)
$37$ \( 9670 - 23027 T + 10243 T^{2} + 4761 T^{3} - 1851 T^{4} - 644 T^{5} + 9 T^{6} + 15 T^{7} + T^{8} \)
$41$ \( 21098 + 25457 T - 2291 T^{2} - 14100 T^{3} - 6561 T^{4} - 956 T^{5} + 37 T^{6} + 18 T^{7} + T^{8} \)
$43$ \( 17666 + 9185 T - 27297 T^{2} + 8080 T^{3} + 2090 T^{4} - 636 T^{5} - 92 T^{6} + 8 T^{7} + T^{8} \)
$47$ \( -2152628 - 727761 T + 320282 T^{2} + 129288 T^{3} + 3111 T^{4} - 2528 T^{5} - 172 T^{6} + 12 T^{7} + T^{8} \)
$53$ \( 266759 + 305475 T + 71675 T^{2} - 27568 T^{3} - 13804 T^{4} - 1409 T^{5} + 116 T^{6} + 25 T^{7} + T^{8} \)
$59$ \( 111752 - 47579 T - 80766 T^{2} + 20388 T^{3} + 8364 T^{4} - 1123 T^{5} - 175 T^{6} + 9 T^{7} + T^{8} \)
$61$ \( -73352 - 48627 T + 19169 T^{2} + 15686 T^{3} + 340 T^{4} - 1078 T^{5} - 160 T^{6} + 2 T^{7} + T^{8} \)
$67$ \( -183707 - 126806 T + 17847 T^{2} + 22807 T^{3} + 1087 T^{4} - 1100 T^{5} - 123 T^{6} + 8 T^{7} + T^{8} \)
$71$ \( 87092 - 211127 T + 12884 T^{2} + 33159 T^{3} - 377 T^{4} - 1623 T^{5} - 99 T^{6} + 13 T^{7} + T^{8} \)
$73$ \( -16304777 - 9145704 T - 575410 T^{2} + 355679 T^{3} + 48531 T^{4} - 1906 T^{5} - 426 T^{6} + 2 T^{7} + T^{8} \)
$79$ \( -551150 - 341209 T + 262711 T^{2} + 200921 T^{3} + 33301 T^{4} - 678 T^{5} - 377 T^{6} - T^{7} + T^{8} \)
$83$ \( -42334 + 473115 T + 1470670 T^{2} + 519038 T^{3} + 38373 T^{4} - 3839 T^{5} - 431 T^{6} + 6 T^{7} + T^{8} \)
$89$ \( -423490 - 804049 T - 181531 T^{2} + 94493 T^{3} + 7882 T^{4} - 2431 T^{5} - 135 T^{6} + 17 T^{7} + T^{8} \)
$97$ \( 118642 - 1337553 T + 2384907 T^{2} - 818056 T^{3} + 22643 T^{4} + 8186 T^{5} - 392 T^{6} - 19 T^{7} + T^{8} \)
show more
show less