Properties

Label 4030.2.a.q
Level 4030
Weight 2
Character orbit 4030.a
Self dual Yes
Analytic conductor 32.180
Analytic rank 0
Dimension 9
CM No
Inner twists 1

Related objects

Downloads

Learn more about

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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(3\) \(x^{8}\mathstrut -\mathstrut \) \(16\) \(x^{7}\mathstrut +\mathstrut \) \(48\) \(x^{6}\mathstrut +\mathstrut \) \(66\) \(x^{5}\mathstrut -\mathstrut \) \(202\) \(x^{4}\mathstrut -\mathstrut \) \(75\) \(x^{3}\mathstrut +\mathstrut \) \(210\) \(x^{2}\mathstrut +\mathstrut \) \(68\) \(x\mathstrut -\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -125 \nu^{8} - 623 \nu^{7} + 3654 \nu^{6} + 11052 \nu^{5} - 31314 \nu^{4} - 52462 \nu^{3} + 78455 \nu^{2} + 59144 \nu - 10736 \)\()/3732\)
\(\beta_{3}\)\(=\)\((\)\( -377 \nu^{8} + 823 \nu^{7} + 6348 \nu^{6} - 12108 \nu^{5} - 27894 \nu^{4} + 40526 \nu^{3} + 23747 \nu^{2} - 9610 \nu + 10792 \)\()/3732\)
\(\beta_{4}\)\(=\)\((\)\( 202 \nu^{8} - 389 \nu^{7} - 3285 \nu^{6} + 5562 \nu^{5} + 13164 \nu^{4} - 17314 \nu^{3} - 7210 \nu^{2} + 791 \nu - 8282 \)\()/1866\)
\(\beta_{5}\)\(=\)\((\)\( 541 \nu^{8} - 1379 \nu^{7} - 8664 \nu^{6} + 21612 \nu^{5} + 34554 \nu^{4} - 87358 \nu^{3} - 28603 \nu^{2} + 79442 \nu + 16072 \)\()/3732\)
\(\beta_{6}\)\(=\)\((\)\( -99 \nu^{8} + 146 \nu^{7} + 1747 \nu^{6} - 1990 \nu^{5} - 8564 \nu^{4} + 5742 \nu^{3} + 11829 \nu^{2} - 731 \nu - 2472 \)\()/622\)
\(\beta_{7}\)\(=\)\((\)\( 99 \nu^{8} - 146 \nu^{7} - 1747 \nu^{6} + 1990 \nu^{5} + 8564 \nu^{4} - 5742 \nu^{3} - 11207 \nu^{2} + 731 \nu - 16 \)\()/622\)
\(\beta_{8}\)\(=\)\((\)\( 659 \nu^{8} - 1597 \nu^{7} - 10740 \nu^{6} + 23808 \nu^{5} + 45126 \nu^{4} - 83006 \nu^{3} - 45341 \nu^{2} + 36478 \nu + 16832 \)\()/3732\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(74\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{6}\)\(=\)\(45\) \(\beta_{8}\mathstrut +\mathstrut \) \(146\) \(\beta_{7}\mathstrut +\mathstrut \) \(163\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(69\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut -\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(246\)
\(\nu^{7}\)\(=\)\(149\) \(\beta_{8}\mathstrut -\mathstrut \) \(134\) \(\beta_{7}\mathstrut +\mathstrut \) \(245\) \(\beta_{6}\mathstrut +\mathstrut \) \(69\) \(\beta_{5}\mathstrut +\mathstrut \) \(196\) \(\beta_{4}\mathstrut +\mathstrut \) \(39\) \(\beta_{3}\mathstrut -\mathstrut \) \(201\) \(\beta_{2}\mathstrut +\mathstrut \) \(723\) \(\beta_{1}\mathstrut +\mathstrut \) \(180\)
\(\nu^{8}\)\(=\)\(551\) \(\beta_{8}\mathstrut +\mathstrut \) \(1577\) \(\beta_{7}\mathstrut +\mathstrut \) \(1836\) \(\beta_{6}\mathstrut +\mathstrut \) \(39\) \(\beta_{5}\mathstrut +\mathstrut \) \(222\) \(\beta_{4}\mathstrut +\mathstrut \) \(909\) \(\beta_{3}\mathstrut -\mathstrut \) \(211\) \(\beta_{2}\mathstrut -\mathstrut \) \(321\) \(\beta_{1}\mathstrut +\mathstrut \) \(2434\)

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
−3.17012
−2.15368
−0.959789
−0.522486
0.161390
1.68895
1.90335
2.63839
3.41400
1.00000 −3.17012 1.00000 −1.00000 −3.17012 −0.334308 1.00000 7.04967 −1.00000
1.2 1.00000 −2.15368 1.00000 −1.00000 −2.15368 1.26985 1.00000 1.63833 −1.00000
1.3 1.00000 −0.959789 1.00000 −1.00000 −0.959789 5.04132 1.00000 −2.07880 −1.00000
1.4 1.00000 −0.522486 1.00000 −1.00000 −0.522486 −4.13215 1.00000 −2.72701 −1.00000
1.5 1.00000 0.161390 1.00000 −1.00000 0.161390 0.164045 1.00000 −2.97395 −1.00000
1.6 1.00000 1.68895 1.00000 −1.00000 1.68895 2.51858 1.00000 −0.147452 −1.00000
1.7 1.00000 1.90335 1.00000 −1.00000 1.90335 −3.95405 1.00000 0.622734 −1.00000
1.8 1.00000 2.63839 1.00000 −1.00000 2.63839 2.17174 1.00000 3.96109 −1.00000
1.9 1.00000 3.41400 1.00000 −1.00000 3.41400 0.254978 1.00000 8.65540 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4030))\).