Properties

Label 8034.2.a.w
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 12
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(6\) \(x^{11}\mathstrut -\mathstrut \) \(8\) \(x^{10}\mathstrut +\mathstrut \) \(94\) \(x^{9}\mathstrut -\mathstrut \) \(7\) \(x^{8}\mathstrut -\mathstrut \) \(580\) \(x^{7}\mathstrut +\mathstrut \) \(180\) \(x^{6}\mathstrut +\mathstrut \) \(1787\) \(x^{5}\mathstrut -\mathstrut \) \(308\) \(x^{4}\mathstrut -\mathstrut \) \(2790\) \(x^{3}\mathstrut -\mathstrut \) \(352\) \(x^{2}\mathstrut +\mathstrut \) \(1768\) \(x\mathstrut +\mathstrut \) \(768\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 81 \nu^{11} - 874 \nu^{10} + 1400 \nu^{9} + 10622 \nu^{8} - 27271 \nu^{7} - 48976 \nu^{6} + 141564 \nu^{5} + 120931 \nu^{4} - 292064 \nu^{3} - 190830 \nu^{2} + 207480 \nu + 143896 \)\()/2936\)
\(\beta_{2}\)\(=\)\((\)\( 103 \nu^{11} - 835 \nu^{10} + 276 \nu^{9} + 11894 \nu^{8} - 17071 \nu^{7} - 64829 \nu^{6} + 108598 \nu^{5} + 175289 \nu^{4} - 254607 \nu^{3} - 253616 \nu^{2} + 199594 \nu + 165236 \)\()/1468\)
\(\beta_{3}\)\(=\)\((\)\( -75 \nu^{11} + 551 \nu^{10} - 5 \nu^{9} - 7606 \nu^{8} + 8967 \nu^{7} + 39381 \nu^{6} - 56903 \nu^{5} - 97565 \nu^{4} + 125519 \nu^{3} + 123561 \nu^{2} - 89922 \nu - 70548 \)\()/734\)
\(\beta_{4}\)\(=\)\((\)\( -268 \nu^{11} + 1827 \nu^{10} + 814 \nu^{9} - 26572 \nu^{8} + 21678 \nu^{7} + 146843 \nu^{6} - 158256 \nu^{5} - 389932 \nu^{4} + 369489 \nu^{3} + 514220 \nu^{2} - 276606 \nu - 281980 \)\()/1468\)
\(\beta_{5}\)\(=\)\((\)\( 176 \nu^{11} - 1156 \nu^{10} - 551 \nu^{9} + 16048 \nu^{8} - 13086 \nu^{7} - 82784 \nu^{6} + 93363 \nu^{5} + 199984 \nu^{4} - 211208 \nu^{3} - 240617 \nu^{2} + 152708 \nu + 131084 \)\()/734\)
\(\beta_{6}\)\(=\)\((\)\( 464 \nu^{11} - 3081 \nu^{10} - 1486 \nu^{9} + 43376 \nu^{8} - 34366 \nu^{7} - 228825 \nu^{6} + 247440 \nu^{5} + 569936 \nu^{4} - 561859 \nu^{3} - 702716 \nu^{2} + 402794 \nu + 376680 \)\()/1468\)
\(\beta_{7}\)\(=\)\((\)\( -927 \nu^{11} + 6414 \nu^{10} + 1920 \nu^{9} - 90898 \nu^{8} + 85377 \nu^{7} + 488408 \nu^{6} - 593500 \nu^{5} - 1267853 \nu^{4} + 1378000 \nu^{3} + 1679930 \nu^{2} - 1045464 \nu - 977728 \)\()/2936\)
\(\beta_{8}\)\(=\)\((\)\( -1151 \nu^{11} + 7952 \nu^{10} + 2688 \nu^{9} - 113458 \nu^{8} + 101765 \nu^{7} + 614922 \nu^{6} - 714928 \nu^{5} - 1610725 \nu^{4} + 1662758 \nu^{3} + 2140510 \nu^{2} - 1257836 \nu - 1227304 \)\()/2936\)
\(\beta_{9}\)\(=\)\((\)\( 1227 \nu^{11} - 8618 \nu^{10} - 1900 \nu^{9} + 121322 \nu^{8} - 121245 \nu^{7} - 645932 \nu^{6} + 821112 \nu^{5} + 1658113 \nu^{4} - 1877140 \nu^{3} - 2177110 \nu^{2} + 1387536 \nu + 1262856 \)\()/2936\)
\(\beta_{10}\)\(=\)\((\)\( -1415 \nu^{11} + 9686 \nu^{10} + 2964 \nu^{9} - 134594 \nu^{8} + 124697 \nu^{7} + 704600 \nu^{6} - 850752 \nu^{5} - 1763901 \nu^{4} + 1927456 \nu^{3} + 2233342 \nu^{2} - 1408360 \nu - 1251440 \)\()/2936\)
\(\beta_{11}\)\(=\)\((\)\( 1555 \nu^{11} - 10372 \nu^{10} - 4912 \nu^{9} + 147226 \nu^{8} - 118241 \nu^{7} - 788534 \nu^{6} + 860768 \nu^{5} + 2017465 \nu^{4} - 1999642 \nu^{3} - 2588182 \nu^{2} + 1485492 \nu + 1437152 \)\()/2936\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{4}\)\(=\)\(8\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(34\)
\(\nu^{5}\)\(=\)\(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(24\) \(\beta_{9}\mathstrut +\mathstrut \) \(33\) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(37\) \(\beta_{4}\mathstrut +\mathstrut \) \(36\) \(\beta_{3}\mathstrut +\mathstrut \) \(37\) \(\beta_{2}\mathstrut +\mathstrut \) \(34\) \(\beta_{1}\mathstrut +\mathstrut \) \(71\)
\(\nu^{6}\)\(=\)\(55\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(75\) \(\beta_{9}\mathstrut +\mathstrut \) \(90\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(29\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(111\) \(\beta_{4}\mathstrut +\mathstrut \) \(80\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(102\) \(\beta_{1}\mathstrut +\mathstrut \) \(271\)
\(\nu^{7}\)\(=\)\(95\) \(\beta_{11}\mathstrut -\mathstrut \) \(34\) \(\beta_{10}\mathstrut -\mathstrut \) \(207\) \(\beta_{9}\mathstrut +\mathstrut \) \(284\) \(\beta_{8}\mathstrut -\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(151\) \(\beta_{6}\mathstrut +\mathstrut \) \(56\) \(\beta_{5}\mathstrut -\mathstrut \) \(360\) \(\beta_{4}\mathstrut +\mathstrut \) \(352\) \(\beta_{3}\mathstrut +\mathstrut \) \(350\) \(\beta_{2}\mathstrut +\mathstrut \) \(305\) \(\beta_{1}\mathstrut +\mathstrut \) \(692\)
\(\nu^{8}\)\(=\)\(380\) \(\beta_{11}\mathstrut -\mathstrut \) \(98\) \(\beta_{10}\mathstrut -\mathstrut \) \(641\) \(\beta_{9}\mathstrut +\mathstrut \) \(820\) \(\beta_{8}\mathstrut +\mathstrut \) \(64\) \(\beta_{7}\mathstrut +\mathstrut \) \(313\) \(\beta_{6}\mathstrut +\mathstrut \) \(235\) \(\beta_{5}\mathstrut -\mathstrut \) \(1117\) \(\beta_{4}\mathstrut +\mathstrut \) \(964\) \(\beta_{3}\mathstrut +\mathstrut \) \(1062\) \(\beta_{2}\mathstrut +\mathstrut \) \(937\) \(\beta_{1}\mathstrut +\mathstrut \) \(2375\)
\(\nu^{9}\)\(=\)\(786\) \(\beta_{11}\mathstrut -\mathstrut \) \(434\) \(\beta_{10}\mathstrut -\mathstrut \) \(1845\) \(\beta_{9}\mathstrut +\mathstrut \) \(2568\) \(\beta_{8}\mathstrut +\mathstrut \) \(72\) \(\beta_{7}\mathstrut +\mathstrut \) \(1232\) \(\beta_{6}\mathstrut +\mathstrut \) \(749\) \(\beta_{5}\mathstrut -\mathstrut \) \(3584\) \(\beta_{4}\mathstrut +\mathstrut \) \(3542\) \(\beta_{3}\mathstrut +\mathstrut \) \(3356\) \(\beta_{2}\mathstrut +\mathstrut \) \(2835\) \(\beta_{1}\mathstrut +\mathstrut \) \(6734\)
\(\nu^{10}\)\(=\)\(2788\) \(\beta_{11}\mathstrut -\mathstrut \) \(1331\) \(\beta_{10}\mathstrut -\mathstrut \) \(5677\) \(\beta_{9}\mathstrut +\mathstrut \) \(7715\) \(\beta_{8}\mathstrut +\mathstrut \) \(920\) \(\beta_{7}\mathstrut +\mathstrut \) \(3041\) \(\beta_{6}\mathstrut +\mathstrut \) \(2744\) \(\beta_{5}\mathstrut -\mathstrut \) \(11286\) \(\beta_{4}\mathstrut +\mathstrut \) \(10557\) \(\beta_{3}\mathstrut +\mathstrut \) \(10348\) \(\beta_{2}\mathstrut +\mathstrut \) \(8791\) \(\beta_{1}\mathstrut +\mathstrut \) \(22049\)
\(\nu^{11}\)\(=\)\(6699\) \(\beta_{11}\mathstrut -\mathstrut \) \(4984\) \(\beta_{10}\mathstrut -\mathstrut \) \(16860\) \(\beta_{9}\mathstrut +\mathstrut \) \(24258\) \(\beta_{8}\mathstrut +\mathstrut \) \(2411\) \(\beta_{7}\mathstrut +\mathstrut \) \(10443\) \(\beta_{6}\mathstrut +\mathstrut \) \(8847\) \(\beta_{5}\mathstrut -\mathstrut \) \(36083\) \(\beta_{4}\mathstrut +\mathstrut \) \(35919\) \(\beta_{3}\mathstrut +\mathstrut \) \(32539\) \(\beta_{2}\mathstrut +\mathstrut \) \(27032\) \(\beta_{1}\mathstrut +\mathstrut \) \(66028\)

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
2.37328
−0.829811
1.78941
−2.08335
1.38143
3.10048
−1.02386
−1.61063
3.23048
−2.20360
2.65870
−0.782524
−1.00000 1.00000 1.00000 −3.83399 −1.00000 −3.29790 −1.00000 1.00000 3.83399
1.2 −1.00000 1.00000 1.00000 −3.54947 −1.00000 1.31288 −1.00000 1.00000 3.54947
1.3 −1.00000 1.00000 1.00000 −2.74714 −1.00000 3.39290 −1.00000 1.00000 2.74714
1.4 −1.00000 1.00000 1.00000 −1.71945 −1.00000 2.07930 −1.00000 1.00000 1.71945
1.5 −1.00000 1.00000 1.00000 −1.60911 −1.00000 2.84255 −1.00000 1.00000 1.60911
1.6 −1.00000 1.00000 1.00000 −1.18783 −1.00000 0.107097 −1.00000 1.00000 1.18783
1.7 −1.00000 1.00000 1.00000 −0.812175 −1.00000 −2.80411 −1.00000 1.00000 0.812175
1.8 −1.00000 1.00000 1.00000 1.40749 −1.00000 −0.998879 −1.00000 1.00000 −1.40749
1.9 −1.00000 1.00000 1.00000 1.42840 −1.00000 −2.28162 −1.00000 1.00000 −1.42840
1.10 −1.00000 1.00000 1.00000 2.09722 −1.00000 −0.0373018 −1.00000 1.00000 −2.09722
1.11 −1.00000 1.00000 1.00000 2.30797 −1.00000 4.49849 −1.00000 1.00000 −2.30797
1.12 −1.00000 1.00000 1.00000 4.21809 −1.00000 −4.81342 −1.00000 1.00000 −4.21809
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{12} + \cdots\)
\(T_{7}^{12} - \cdots\)