Properties

Label 6005.2.a.c
Level 6005
Weight 2
Character orbit 6005.a
Self dual Yes
Analytic conductor 47.950
Analytic rank 1
Dimension 4
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9501664138\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(4\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut +\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)

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
−0.679643
2.36234
−1.50848
0.825785
−1.26308 2.12152 −0.404635 −1.00000 −2.67964 −2.67964 3.03724 1.50084 1.26308
1.2 −0.515722 −0.702588 −1.73403 −1.00000 0.362340 0.362340 1.92572 −2.50637 0.515722
1.3 1.18264 −2.96664 −0.601352 −1.00000 −3.50848 −3.50848 −3.07647 5.80096 −1.18264
1.4 2.59615 −0.452290 4.74002 −1.00000 −1.17422 −1.17422 7.11351 −2.79543 −2.59615
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(1201\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 2 T_{2}^{3} \) \(\mathstrut -\mathstrut 3 T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\).