Properties

Label 8034.2.a.t
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 11
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} - 1494 x^{2} - 1161 x - 162\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-5953 \nu^{10} + 71878 \nu^{9} - 597285 \nu^{8} + 825299 \nu^{7} + 11387240 \nu^{6} - 30641537 \nu^{5} - 54431851 \nu^{4} + 176099749 \nu^{3} + 73406730 \nu^{2} - 265050639 \nu - 16896519\)\()/16688889\)
\(\beta_{3}\)\(=\)\((\)\(-32168 \nu^{10} - 265732 \nu^{9} + 2755953 \nu^{8} + 3489646 \nu^{7} - 42680480 \nu^{6} - 10039690 \nu^{5} + 212224462 \nu^{4} + 23110703 \nu^{3} - 312848580 \nu^{2} - 42135030 \nu + 42233265\)\()/16688889\)
\(\beta_{4}\)\(=\)\((\)\(53471 \nu^{10} - 338801 \nu^{9} - 262824 \nu^{8} + 4406315 \nu^{7} - 4924159 \nu^{6} - 2522459 \nu^{5} + 23092565 \nu^{4} - 84254744 \nu^{3} - 26952456 \nu^{2} + 115529085 \nu + 33384069\)\()/16688889\)
\(\beta_{5}\)\(=\)\((\)\(35999 \nu^{10} - 355853 \nu^{9} + 249954 \nu^{8} + 6151373 \nu^{7} - 12119914 \nu^{6} - 30684362 \nu^{5} + 69103229 \nu^{4} + 50477308 \nu^{3} - 98979894 \nu^{2} - 8715126 \nu + 10853631\)\()/5562963\)
\(\beta_{6}\)\(=\)\((\)\(-57370 \nu^{10} + 225044 \nu^{9} + 1255382 \nu^{8} - 4321489 \nu^{7} - 10803735 \nu^{6} + 25564583 \nu^{5} + 48171502 \nu^{4} - 55233325 \nu^{3} - 83365948 \nu^{2} + 46532766 \nu + 18345483\)\()/5562963\)
\(\beta_{7}\)\(=\)\((\)\(-176675 \nu^{10} + 631196 \nu^{9} + 4961220 \nu^{8} - 15848045 \nu^{7} - 51857339 \nu^{6} + 134095505 \nu^{5} + 242900272 \nu^{4} - 436355941 \nu^{3} - 452083029 \nu^{2} + 474682383 \nu + 184701816\)\()/16688889\)
\(\beta_{8}\)\(=\)\((\)\(-75437 \nu^{10} + 263873 \nu^{9} + 1986210 \nu^{8} - 6019115 \nu^{7} - 19487438 \nu^{6} + 44806799 \nu^{5} + 83571139 \nu^{4} - 121626778 \nu^{3} - 120826686 \nu^{2} + 108695202 \nu + 22884885\)\()/5562963\)
\(\beta_{9}\)\(=\)\((\)\(-168806 \nu^{10} + 844770 \nu^{9} + 2991638 \nu^{8} - 16491977 \nu^{7} - 18171259 \nu^{6} + 101055744 \nu^{5} + 62639412 \nu^{4} - 227337411 \nu^{3} - 110775703 \nu^{2} + 169506057 \nu + 58191552\)\()/5562963\)
\(\beta_{10}\)\(=\)\((\)\(77942 \nu^{10} - 390059 \nu^{9} - 1408296 \nu^{8} + 7828925 \nu^{7} + 8470799 \nu^{6} - 49927739 \nu^{5} - 26369470 \nu^{4} + 117061726 \nu^{3} + 44577900 \nu^{2} - 84578166 \nu - 31061394\)\()/2384127\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 9 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-\beta_{10} - 13 \beta_{9} + 9 \beta_{8} + \beta_{7} + 14 \beta_{6} - 13 \beta_{5} + \beta_{4} + 2 \beta_{3} + 17 \beta_{1} + 47\)
\(\nu^{5}\)\(=\)\(12 \beta_{10} - 6 \beta_{9} + 13 \beta_{8} - 2 \beta_{7} + 26 \beta_{6} - 33 \beta_{5} + 30 \beta_{4} + 16 \beta_{3} + 8 \beta_{2} + 102 \beta_{1} + 74\)
\(\nu^{6}\)\(=\)\(-24 \beta_{10} - 168 \beta_{9} + 85 \beta_{8} + 6 \beta_{7} + 194 \beta_{6} - 168 \beta_{5} + 23 \beta_{4} + 46 \beta_{3} - 16 \beta_{2} + 253 \beta_{1} + 538\)
\(\nu^{7}\)\(=\)\(111 \beta_{10} - 166 \beta_{9} + 137 \beta_{8} - 48 \beta_{7} + 470 \beta_{6} - 489 \beta_{5} + 373 \beta_{4} + 257 \beta_{3} + 13 \beta_{2} + 1260 \beta_{1} + 1171\)
\(\nu^{8}\)\(=\)\(-379 \beta_{10} - 2174 \beta_{9} + 823 \beta_{8} - 26 \beta_{7} + 2689 \beta_{6} - 2268 \beta_{5} + 386 \beta_{4} + 876 \beta_{3} - 465 \beta_{2} + 3565 \beta_{1} + 6803\)
\(\nu^{9}\)\(=\)\(852 \beta_{10} - 3280 \beta_{9} + 1337 \beta_{8} - 824 \beta_{7} + 7508 \beta_{6} - 7242 \beta_{5} + 4382 \beta_{4} + 4211 \beta_{3} - 1103 \beta_{2} + 16113 \beta_{1} + 17861\)
\(\nu^{10}\)\(=\)\(-5248 \beta_{10} - 28433 \beta_{9} + 7861 \beta_{8} - 1367 \beta_{7} + 37184 \beta_{6} - 31807 \beta_{5} + 5490 \beta_{4} + 15512 \beta_{3} - 9866 \beta_{2} + 49080 \beta_{1} + 91048\)

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
−3.18653
−2.18067
−2.06747
−1.52965
−0.387444
−0.203778
1.30889
1.64076
3.03614
3.71131
3.85844
−1.00000 1.00000 1.00000 −3.18653 −1.00000 1.47944 −1.00000 1.00000 3.18653
1.2 −1.00000 1.00000 1.00000 −2.18067 −1.00000 −1.20699 −1.00000 1.00000 2.18067
1.3 −1.00000 1.00000 1.00000 −2.06747 −1.00000 −2.54251 −1.00000 1.00000 2.06747
1.4 −1.00000 1.00000 1.00000 −1.52965 −1.00000 −0.354974 −1.00000 1.00000 1.52965
1.5 −1.00000 1.00000 1.00000 −0.387444 −1.00000 5.18739 −1.00000 1.00000 0.387444
1.6 −1.00000 1.00000 1.00000 −0.203778 −1.00000 0.561930 −1.00000 1.00000 0.203778
1.7 −1.00000 1.00000 1.00000 1.30889 −1.00000 2.83911 −1.00000 1.00000 −1.30889
1.8 −1.00000 1.00000 1.00000 1.64076 −1.00000 −3.72013 −1.00000 1.00000 −1.64076
1.9 −1.00000 1.00000 1.00000 3.03614 −1.00000 −4.00525 −1.00000 1.00000 −3.03614
1.10 −1.00000 1.00000 1.00000 3.71131 −1.00000 1.88167 −1.00000 1.00000 −3.71131
1.11 −1.00000 1.00000 1.00000 3.85844 −1.00000 3.88029 −1.00000 1.00000 −3.85844
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)