Properties

Label 4025.2.a.x
Level 4025
Weight 2
Character orbit 4025.a
Self dual Yes
Analytic conductor 32.140
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(2\) \(x^{11}\mathstrut -\mathstrut \) \(14\) \(x^{10}\mathstrut +\mathstrut \) \(26\) \(x^{9}\mathstrut +\mathstrut \) \(71\) \(x^{8}\mathstrut -\mathstrut \) \(120\) \(x^{7}\mathstrut -\mathstrut \) \(162\) \(x^{6}\mathstrut +\mathstrut \) \(244\) \(x^{5}\mathstrut +\mathstrut \) \(170\) \(x^{4}\mathstrut -\mathstrut \) \(210\) \(x^{3}\mathstrut -\mathstrut \) \(81\) \(x^{2}\mathstrut +\mathstrut \) \(58\) \(x\mathstrut +\mathstrut \) \(17\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{11} - 24 \nu^{10} - 218 \nu^{9} + 237 \nu^{8} + 966 \nu^{7} - 670 \nu^{6} - 1780 \nu^{5} + 537 \nu^{4} + 1297 \nu^{3} + 81 \nu^{2} - 241 \nu - 94 \)\()/29\)
\(\beta_{4}\)\(=\)\((\)\( -12 \nu^{11} + 5 \nu^{10} + 188 \nu^{9} - 53 \nu^{8} - 1035 \nu^{7} + 171 \nu^{6} + 2367 \nu^{5} - 159 \nu^{4} - 2125 \nu^{3} - 151 \nu^{2} + 571 \nu + 138 \)\()/29\)
\(\beta_{5}\)\(=\)\((\)\( 18 \nu^{11} - 22 \nu^{10} - 253 \nu^{9} + 239 \nu^{8} + 1277 \nu^{7} - 822 \nu^{6} - 2840 \nu^{5} + 1036 \nu^{4} + 2767 \nu^{3} - 281 \nu^{2} - 958 \nu - 120 \)\()/29\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{11} - 18 \nu^{10} - 294 \nu^{9} + 185 \nu^{8} + 1551 \nu^{7} - 575 \nu^{6} - 3539 \nu^{5} + 613 \nu^{4} + 3329 \nu^{3} - 77 \nu^{2} - 913 \nu - 114 \)\()/29\)
\(\beta_{7}\)\(=\)\((\)\( -28 \nu^{11} + 31 \nu^{10} + 400 \nu^{9} - 317 \nu^{8} - 2067 \nu^{7} + 950 \nu^{6} + 4740 \nu^{5} - 806 \nu^{4} - 4736 \nu^{3} - 275 \nu^{2} + 1574 \nu + 322 \)\()/29\)
\(\beta_{8}\)\(=\)\((\)\( 24 \nu^{11} - 10 \nu^{10} - 376 \nu^{9} + 77 \nu^{8} + 2128 \nu^{7} - 52 \nu^{6} - 5256 \nu^{5} - 523 \nu^{4} + 5468 \nu^{3} + 1056 \nu^{2} - 1751 \nu - 508 \)\()/29\)
\(\beta_{9}\)\(=\)\((\)\( -33 \nu^{11} + 21 \nu^{10} + 488 \nu^{9} - 182 \nu^{8} - 2578 \nu^{7} + 347 \nu^{6} + 5864 \nu^{5} + 179 \nu^{4} - 5561 \nu^{3} - 785 \nu^{2} + 1592 \nu + 336 \)\()/29\)
\(\beta_{10}\)\(=\)\((\)\( 35 \nu^{11} - 17 \nu^{10} - 529 \nu^{9} + 128 \nu^{8} + 2852 \nu^{7} - 71 \nu^{6} - 6592 \nu^{5} - 863 \nu^{4} + 6355 \nu^{3} + 1569 \nu^{2} - 1953 \nu - 591 \)\()/29\)
\(\beta_{11}\)\(=\)\((\)\( -73 \nu^{11} + 57 \nu^{10} + 1076 \nu^{9} - 552 \nu^{8} - 5680 \nu^{7} + 1468 \nu^{6} + 12971 \nu^{5} - 757 \nu^{4} - 12451 \nu^{3} - 1414 \nu^{2} + 3824 \nu + 1028 \)\()/29\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(7\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(9\) \(\beta_{11}\mathstrut +\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(43\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(76\)
\(\nu^{7}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(43\) \(\beta_{10}\mathstrut +\mathstrut \) \(52\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(53\) \(\beta_{7}\mathstrut +\mathstrut \) \(52\) \(\beta_{6}\mathstrut +\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(43\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(57\) \(\beta_{1}\mathstrut +\mathstrut \) \(67\)
\(\nu^{8}\)\(=\)\(63\) \(\beta_{11}\mathstrut +\mathstrut \) \(63\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(53\) \(\beta_{6}\mathstrut +\mathstrut \) \(67\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(257\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\) \(\beta_{1}\mathstrut +\mathstrut \) \(438\)
\(\nu^{9}\)\(=\)\(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(259\) \(\beta_{10}\mathstrut +\mathstrut \) \(319\) \(\beta_{9}\mathstrut +\mathstrut \) \(90\) \(\beta_{8}\mathstrut +\mathstrut \) \(334\) \(\beta_{7}\mathstrut +\mathstrut \) \(318\) \(\beta_{6}\mathstrut +\mathstrut \) \(288\) \(\beta_{5}\mathstrut +\mathstrut \) \(254\) \(\beta_{4}\mathstrut +\mathstrut \) \(78\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(299\) \(\beta_{1}\mathstrut +\mathstrut \) \(465\)
\(\nu^{10}\)\(=\)\(410\) \(\beta_{11}\mathstrut +\mathstrut \) \(412\) \(\beta_{10}\mathstrut +\mathstrut \) \(106\) \(\beta_{9}\mathstrut +\mathstrut \) \(124\) \(\beta_{8}\mathstrut +\mathstrut \) \(68\) \(\beta_{7}\mathstrut +\mathstrut \) \(337\) \(\beta_{6}\mathstrut +\mathstrut \) \(474\) \(\beta_{5}\mathstrut -\mathstrut \) \(69\) \(\beta_{4}\mathstrut +\mathstrut \) \(108\) \(\beta_{3}\mathstrut +\mathstrut \) \(1526\) \(\beta_{2}\mathstrut +\mathstrut \) \(184\) \(\beta_{1}\mathstrut +\mathstrut \) \(2594\)
\(\nu^{11}\)\(=\)\(172\) \(\beta_{11}\mathstrut +\mathstrut \) \(1561\) \(\beta_{10}\mathstrut +\mathstrut \) \(1919\) \(\beta_{9}\mathstrut +\mathstrut \) \(667\) \(\beta_{8}\mathstrut +\mathstrut \) \(2073\) \(\beta_{7}\mathstrut +\mathstrut \) \(1905\) \(\beta_{6}\mathstrut +\mathstrut \) \(1851\) \(\beta_{5}\mathstrut +\mathstrut \) \(1473\) \(\beta_{4}\mathstrut +\mathstrut \) \(563\) \(\beta_{3}\mathstrut +\mathstrut \) \(337\) \(\beta_{2}\mathstrut +\mathstrut \) \(1655\) \(\beta_{1}\mathstrut +\mathstrut \) \(3122\)

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
2.55311
2.43455
1.53413
1.31562
1.23080
0.792462
−0.271475
−0.649823
−1.11343
−1.69164
−1.76402
−2.37028
−2.55311 0.556415 4.51835 0 −1.42059 1.00000 −6.42962 −2.69040 0
1.2 −2.43455 2.82471 3.92701 0 −6.87689 1.00000 −4.69139 4.97900 0
1.3 −1.53413 −2.66039 0.353561 0 4.08139 1.00000 2.52585 4.07768 0
1.4 −1.31562 −2.16083 −0.269134 0 2.84283 1.00000 2.98533 1.66917 0
1.5 −1.23080 0.767488 −0.485126 0 −0.944626 1.00000 3.05870 −2.41096 0
1.6 −0.792462 2.09947 −1.37200 0 −1.66375 1.00000 2.67218 1.40777 0
1.7 0.271475 2.40463 −1.92630 0 0.652799 1.00000 −1.06589 2.78227 0
1.8 0.649823 −1.89575 −1.57773 0 −1.23191 1.00000 −2.32489 0.593883 0
1.9 1.11343 −2.87976 −0.760266 0 −3.20642 1.00000 −3.07337 5.29300 0
1.10 1.69164 −0.285164 0.861643 0 −0.482394 1.00000 −1.92569 −2.91868 0
1.11 1.76402 1.09514 1.11176 0 1.93184 1.00000 −1.56687 −1.80068 0
1.12 2.37028 0.134036 3.61823 0 0.317703 1.00000 3.83566 −2.98203 0
\(n\): 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
\(5\) \(-1\)
\(7\) \(-1\)
\(23\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{12} + \cdots\)
\(T_{3}^{12} - \cdots\)
\(T_{11}^{12} + \cdots\)