Properties

Label 8034.2.a.u
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
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: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(4\) \(x^{10}\mathstrut -\mathstrut \) \(18\) \(x^{9}\mathstrut +\mathstrut \) \(64\) \(x^{8}\mathstrut +\mathstrut \) \(85\) \(x^{7}\mathstrut -\mathstrut \) \(249\) \(x^{6}\mathstrut -\mathstrut \) \(109\) \(x^{5}\mathstrut +\mathstrut \) \(230\) \(x^{4}\mathstrut +\mathstrut \) \(97\) \(x^{3}\mathstrut -\mathstrut \) \(53\) \(x^{2}\mathstrut -\mathstrut \) \(32\) \(x\mathstrut -\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25115 \nu^{10} + 123880 \nu^{9} + 347054 \nu^{8} - 1973096 \nu^{7} - 475719 \nu^{6} + 7344639 \nu^{5} - 3375623 \nu^{4} - 4857162 \nu^{3} + 1771243 \nu^{2} + 840961 \nu - 57834 \)\()/57098\)
\(\beta_{3}\)\(=\)\((\)\( -43620 \nu^{10} + 194155 \nu^{9} + 703885 \nu^{8} - 3133089 \nu^{7} - 2408751 \nu^{6} + 12312604 \nu^{5} - 266083 \nu^{4} - 11137378 \nu^{3} + 274806 \nu^{2} + 2744613 \nu + 326464 \)\()/57098\)
\(\beta_{4}\)\(=\)\((\)\(115421\) \(\nu^{10}\mathstrut -\mathstrut \) \(551264\) \(\nu^{9}\mathstrut -\mathstrut \) \(1667268\) \(\nu^{8}\mathstrut +\mathstrut \) \(8763870\) \(\nu^{7}\mathstrut +\mathstrut \) \(3266695\) \(\nu^{6}\mathstrut -\mathstrut \) \(32593715\) \(\nu^{5}\mathstrut +\mathstrut \) \(12149715\) \(\nu^{4}\mathstrut +\mathstrut \) \(22005140\) \(\nu^{3}\mathstrut -\mathstrut \) \(7530379\) \(\nu^{2}\mathstrut -\mathstrut \) \(3554529\) \(\nu\mathstrut +\mathstrut \) \(355594\)\()/114196\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(189461\) \(\nu^{10}\mathstrut +\mathstrut \) \(831576\) \(\nu^{9}\mathstrut +\mathstrut \) \(3083592\) \(\nu^{8}\mathstrut -\mathstrut \) \(13311542\) \(\nu^{7}\mathstrut -\mathstrut \) \(10877367\) \(\nu^{6}\mathstrut +\mathstrut \) \(51186895\) \(\nu^{5}\mathstrut +\mathstrut \) \(637757\) \(\nu^{4}\mathstrut -\mathstrut \) \(43024636\) \(\nu^{3}\mathstrut -\mathstrut \) \(2052417\) \(\nu^{2}\mathstrut +\mathstrut \) \(10501909\) \(\nu\mathstrut +\mathstrut \) \(2166734\)\()/114196\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(140379\) \(\nu^{10}\mathstrut +\mathstrut \) \(618522\) \(\nu^{9}\mathstrut +\mathstrut \) \(2271966\) \(\nu^{8}\mathstrut -\mathstrut \) \(9891326\) \(\nu^{7}\mathstrut -\mathstrut \) \(7853363\) \(\nu^{6}\mathstrut +\mathstrut \) \(37899765\) \(\nu^{5}\mathstrut -\mathstrut \) \(344375\) \(\nu^{4}\mathstrut -\mathstrut \) \(31274154\) \(\nu^{3}\mathstrut -\mathstrut \) \(665823\) \(\nu^{2}\mathstrut +\mathstrut \) \(7257953\) \(\nu\mathstrut +\mathstrut \) \(1342942\)\()/57098\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(366281\) \(\nu^{10}\mathstrut +\mathstrut \) \(1630196\) \(\nu^{9}\mathstrut +\mathstrut \) \(5868288\) \(\nu^{8}\mathstrut -\mathstrut \) \(26121810\) \(\nu^{7}\mathstrut -\mathstrut \) \(19548755\) \(\nu^{6}\mathstrut +\mathstrut \) \(100542707\) \(\nu^{5}\mathstrut -\mathstrut \) \(4436099\) \(\nu^{4}\mathstrut -\mathstrut \) \(84025608\) \(\nu^{3}\mathstrut +\mathstrut \) \(1059979\) \(\nu^{2}\mathstrut +\mathstrut \) \(20040201\) \(\nu\mathstrut +\mathstrut \) \(3395542\)\()/114196\)
\(\beta_{8}\)\(=\)\((\)\(510619\) \(\nu^{10}\mathstrut -\mathstrut \) \(2246898\) \(\nu^{9}\mathstrut -\mathstrut \) \(8303822\) \(\nu^{8}\mathstrut +\mathstrut \) \(36042260\) \(\nu^{7}\mathstrut +\mathstrut \) \(29219587\) \(\nu^{6}\mathstrut -\mathstrut \) \(139385317\) \(\nu^{5}\mathstrut -\mathstrut \) \(1255417\) \(\nu^{4}\mathstrut +\mathstrut \) \(119432904\) \(\nu^{3}\mathstrut +\mathstrut \) \(3928915\) \(\nu^{2}\mathstrut -\mathstrut \) \(28669817\) \(\nu\mathstrut -\mathstrut \) \(5761410\)\()/114196\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(671471\) \(\nu^{10}\mathstrut +\mathstrut \) \(2966642\) \(\nu^{9}\mathstrut +\mathstrut \) \(10849434\) \(\nu^{8}\mathstrut -\mathstrut \) \(47518076\) \(\nu^{7}\mathstrut -\mathstrut \) \(37292383\) \(\nu^{6}\mathstrut +\mathstrut \) \(182903005\) \(\nu^{5}\mathstrut -\mathstrut \) \(2609191\) \(\nu^{4}\mathstrut -\mathstrut \) \(153749580\) \(\nu^{3}\mathstrut -\mathstrut \) \(2584379\) \(\nu^{2}\mathstrut +\mathstrut \) \(36919609\) \(\nu\mathstrut +\mathstrut \) \(6971166\)\()/114196\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(460175\) \(\nu^{10}\mathstrut +\mathstrut \) \(2024699\) \(\nu^{9}\mathstrut +\mathstrut \) \(7470473\) \(\nu^{8}\mathstrut -\mathstrut \) \(32427051\) \(\nu^{7}\mathstrut -\mathstrut \) \(26090460\) \(\nu^{6}\mathstrut +\mathstrut \) \(124845689\) \(\nu^{5}\mathstrut -\mathstrut \) \(53294\) \(\nu^{4}\mathstrut -\mathstrut \) \(105229792\) \(\nu^{3}\mathstrut -\mathstrut \) \(2225443\) \(\nu^{2}\mathstrut +\mathstrut \) \(24837390\) \(\nu\mathstrut +\mathstrut \) \(4665936\)\()/57098\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(21\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(39\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{5}\)\(=\)\(81\) \(\beta_{10}\mathstrut -\mathstrut \) \(58\) \(\beta_{9}\mathstrut +\mathstrut \) \(42\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut -\mathstrut \) \(49\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(66\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(141\) \(\beta_{2}\mathstrut +\mathstrut \) \(173\) \(\beta_{1}\mathstrut +\mathstrut \) \(91\)
\(\nu^{6}\)\(=\)\(441\) \(\beta_{10}\mathstrut -\mathstrut \) \(383\) \(\beta_{9}\mathstrut +\mathstrut \) \(300\) \(\beta_{8}\mathstrut +\mathstrut \) \(49\) \(\beta_{7}\mathstrut -\mathstrut \) \(213\) \(\beta_{6}\mathstrut +\mathstrut \) \(222\) \(\beta_{5}\mathstrut +\mathstrut \) \(394\) \(\beta_{4}\mathstrut +\mathstrut \) \(112\) \(\beta_{3}\mathstrut +\mathstrut \) \(796\) \(\beta_{2}\mathstrut +\mathstrut \) \(525\) \(\beta_{1}\mathstrut +\mathstrut \) \(758\)
\(\nu^{7}\)\(=\)\(1863\) \(\beta_{10}\mathstrut -\mathstrut \) \(1428\) \(\beta_{9}\mathstrut +\mathstrut \) \(1067\) \(\beta_{8}\mathstrut +\mathstrut \) \(109\) \(\beta_{7}\mathstrut -\mathstrut \) \(1067\) \(\beta_{6}\mathstrut +\mathstrut \) \(330\) \(\beta_{5}\mathstrut +\mathstrut \) \(1612\) \(\beta_{4}\mathstrut +\mathstrut \) \(82\) \(\beta_{3}\mathstrut +\mathstrut \) \(3287\) \(\beta_{2}\mathstrut +\mathstrut \) \(3131\) \(\beta_{1}\mathstrut +\mathstrut \) \(2199\)
\(\nu^{8}\)\(=\)\(9266\) \(\beta_{10}\mathstrut -\mathstrut \) \(7869\) \(\beta_{9}\mathstrut +\mathstrut \) \(5979\) \(\beta_{8}\mathstrut +\mathstrut \) \(1309\) \(\beta_{7}\mathstrut -\mathstrut \) \(4789\) \(\beta_{6}\mathstrut +\mathstrut \) \(3571\) \(\beta_{5}\mathstrut +\mathstrut \) \(8308\) \(\beta_{4}\mathstrut +\mathstrut \) \(1410\) \(\beta_{3}\mathstrut +\mathstrut \) \(16589\) \(\beta_{2}\mathstrut +\mathstrut \) \(11579\) \(\beta_{1}\mathstrut +\mathstrut \) \(13651\)
\(\nu^{9}\)\(=\)\(40813\) \(\beta_{10}\mathstrut -\mathstrut \) \(32453\) \(\beta_{9}\mathstrut +\mathstrut \) \(24224\) \(\beta_{8}\mathstrut +\mathstrut \) \(4636\) \(\beta_{7}\mathstrut -\mathstrut \) \(22775\) \(\beta_{6}\mathstrut +\mathstrut \) \(8756\) \(\beta_{5}\mathstrut +\mathstrut \) \(36143\) \(\beta_{4}\mathstrut +\mathstrut \) \(1962\) \(\beta_{3}\mathstrut +\mathstrut \) \(72387\) \(\beta_{2}\mathstrut +\mathstrut \) \(60981\) \(\beta_{1}\mathstrut +\mathstrut \) \(49344\)
\(\nu^{10}\)\(=\)\(194994\) \(\beta_{10}\mathstrut -\mathstrut \) \(163463\) \(\beta_{9}\mathstrut +\mathstrut \) \(122607\) \(\beta_{8}\mathstrut +\mathstrut \) \(30061\) \(\beta_{7}\mathstrut -\mathstrut \) \(103522\) \(\beta_{6}\mathstrut +\mathstrut \) \(63091\) \(\beta_{5}\mathstrut +\mathstrut \) \(175406\) \(\beta_{4}\mathstrut +\mathstrut \) \(20202\) \(\beta_{3}\mathstrut +\mathstrut \) \(348138\) \(\beta_{2}\mathstrut +\mathstrut \) \(250349\) \(\beta_{1}\mathstrut +\mathstrut \) \(264771\)

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
−0.740393
−0.403660
1.71784
1.09038
3.12575
−3.29880
4.59586
−2.14636
−0.444701
−0.218590
0.722678
1.00000 −1.00000 1.00000 −3.99294 −1.00000 −2.56934 1.00000 1.00000 −3.99294
1.2 1.00000 −1.00000 1.00000 −2.98995 −1.00000 3.36753 1.00000 1.00000 −2.98995
1.3 1.00000 −1.00000 1.00000 −2.58633 −1.00000 3.26383 1.00000 1.00000 −2.58633
1.4 1.00000 −1.00000 1.00000 −1.28102 −1.00000 −0.150820 1.00000 1.00000 −1.28102
1.5 1.00000 −1.00000 1.00000 −0.539721 −1.00000 0.708670 1.00000 1.00000 −0.539721
1.6 1.00000 −1.00000 1.00000 0.113923 −1.00000 2.65133 1.00000 1.00000 0.113923
1.7 1.00000 −1.00000 1.00000 0.819538 −1.00000 −3.89413 1.00000 1.00000 0.819538
1.8 1.00000 −1.00000 1.00000 1.41909 −1.00000 −0.663444 1.00000 1.00000 1.41909
1.9 1.00000 −1.00000 1.00000 1.48470 −1.00000 −3.17363 1.00000 1.00000 1.48470
1.10 1.00000 −1.00000 1.00000 2.47956 −1.00000 −2.89002 1.00000 1.00000 2.47956
1.11 1.00000 −1.00000 1.00000 3.07315 −1.00000 1.35002 1.00000 1.00000 3.07315
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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