Properties

Label 4030.2.a.m
Level 4030
Weight 2
Character orbit 4030.a
Self dual Yes
Analytic conductor 32.180
Analytic rank 0
Dimension 8
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4030.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1797120146\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut -\mathstrut \) \(15\) \(x^{6}\mathstrut +\mathstrut \) \(31\) \(x^{5}\mathstrut +\mathstrut \) \(79\) \(x^{4}\mathstrut -\mathstrut \) \(85\) \(x^{3}\mathstrut -\mathstrut \) \(162\) \(x^{2}\mathstrut +\mathstrut \) \(45\) \(x\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 8 \nu^{6} + 63 \nu^{5} + 50 \nu^{4} - 447 \nu^{3} - 106 \nu^{2} + 662 \nu + 267 \)\()/102\)
\(\beta_{3}\)\(=\)\((\)\( -23 \nu^{7} + 54 \nu^{6} + 327 \nu^{5} - 380 \nu^{4} - 1373 \nu^{3} + 554 \nu^{2} + 1320 \nu - 387 \)\()/306\)
\(\beta_{4}\)\(=\)\((\)\( 13 \nu^{7} - 66 \nu^{6} - 105 \nu^{5} + 727 \nu^{4} + 235 \nu^{3} - 2311 \nu^{2} - 378 \nu + 1629 \)\()/153\)
\(\beta_{5}\)\(=\)\((\)\( 19 \nu^{7} - 86 \nu^{6} - 177 \nu^{5} + 886 \nu^{4} + 503 \nu^{3} - 2406 \nu^{2} - 610 \nu + 1353 \)\()/102\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} + 19 \nu^{6} + 31 \nu^{5} - 191 \nu^{4} - 37 \nu^{3} + 511 \nu^{2} - 21 \nu - 309 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( -77 \nu^{7} + 336 \nu^{6} + 669 \nu^{5} - 3188 \nu^{4} - 1541 \nu^{3} + 7784 \nu^{2} + 1470 \nu - 4023 \)\()/306\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(36\)
\(\nu^{5}\)\(=\)\(-\)\(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(71\) \(\beta_{5}\mathstrut +\mathstrut \) \(39\) \(\beta_{4}\mathstrut +\mathstrut \) \(40\) \(\beta_{3}\mathstrut -\mathstrut \) \(45\) \(\beta_{2}\mathstrut +\mathstrut \) \(81\) \(\beta_{1}\mathstrut +\mathstrut \) \(96\)
\(\nu^{6}\)\(=\)\(-\)\(245\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(332\) \(\beta_{5}\mathstrut +\mathstrut \) \(154\) \(\beta_{4}\mathstrut +\mathstrut \) \(186\) \(\beta_{3}\mathstrut -\mathstrut \) \(231\) \(\beta_{2}\mathstrut +\mathstrut \) \(275\) \(\beta_{1}\mathstrut +\mathstrut \) \(474\)
\(\nu^{7}\)\(=\)\(-\)\(997\) \(\beta_{7}\mathstrut +\mathstrut \) \(80\) \(\beta_{6}\mathstrut -\mathstrut \) \(1320\) \(\beta_{5}\mathstrut +\mathstrut \) \(681\) \(\beta_{4}\mathstrut +\mathstrut \) \(682\) \(\beta_{3}\mathstrut -\mathstrut \) \(839\) \(\beta_{2}\mathstrut +\mathstrut \) \(1180\) \(\beta_{1}\mathstrut +\mathstrut \) \(1664\)

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
4.09186
2.49715
1.94220
0.842713
−0.869333
−1.28275
−1.77519
−2.44665
−1.00000 −3.09186 1.00000 1.00000 3.09186 −0.0229325 −1.00000 6.55960 −1.00000
1.2 −1.00000 −1.49715 1.00000 1.00000 1.49715 2.81472 −1.00000 −0.758530 −1.00000
1.3 −1.00000 −0.942199 1.00000 1.00000 0.942199 2.99101 −1.00000 −2.11226 −1.00000
1.4 −1.00000 0.157287 1.00000 1.00000 −0.157287 −4.20500 −1.00000 −2.97526 −1.00000
1.5 −1.00000 1.86933 1.00000 1.00000 −1.86933 1.98758 −1.00000 0.494407 −1.00000
1.6 −1.00000 2.28275 1.00000 1.00000 −2.28275 0.279113 −1.00000 2.21094 −1.00000
1.7 −1.00000 2.77519 1.00000 1.00000 −2.77519 −3.40676 −1.00000 4.70170 −1.00000
1.8 −1.00000 3.44665 1.00000 1.00000 −3.44665 4.56226 −1.00000 8.87940 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).