Properties

Label 30.3.b.a
Level 30
Weight 3
Character orbit 30.b
Analytic conductor 0.817
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 30.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.817440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-17})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -\beta_{1} - \beta_{2} ) q^{3} \) \( + 2 q^{4} \) \( + ( -2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -1 + \beta_{3} ) q^{6} \) \( + ( 2 \beta_{1} + \beta_{2} ) q^{7} \) \( + 2 \beta_{2} q^{8} \) \( + ( -8 - \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -\beta_{1} - \beta_{2} ) q^{3} \) \( + 2 q^{4} \) \( + ( -2 \beta_{2} - \beta_{3} ) q^{5} \) \( + ( -1 + \beta_{3} ) q^{6} \) \( + ( 2 \beta_{1} + \beta_{2} ) q^{7} \) \( + 2 \beta_{2} q^{8} \) \( + ( -8 - \beta_{3} ) q^{9} \) \( + ( -4 + 2 \beta_{1} + \beta_{2} ) q^{10} \) \( + 4 \beta_{3} q^{11} \) \( + ( -2 \beta_{1} - 2 \beta_{2} ) q^{12} \) \( -2 \beta_{3} q^{14} \) \( + ( 2 - \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{15} \) \( + 4 q^{16} \) \( -8 \beta_{2} q^{17} \) \( + ( 2 \beta_{1} - 7 \beta_{2} ) q^{18} \) \( + 12 q^{19} \) \( + ( -4 \beta_{2} - 2 \beta_{3} ) q^{20} \) \( + ( 17 + \beta_{3} ) q^{21} \) \( + ( -8 \beta_{1} - 4 \beta_{2} ) q^{22} \) \( + 17 \beta_{2} q^{23} \) \( + ( -2 + 2 \beta_{3} ) q^{24} \) \( + ( -9 - 8 \beta_{1} - 4 \beta_{2} ) q^{25} \) \( + ( 7 \beta_{1} + 16 \beta_{2} ) q^{27} \) \( + ( 4 \beta_{1} + 2 \beta_{2} ) q^{28} \) \( + ( 17 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{30} \) \( -32 q^{31} \) \( + 4 \beta_{2} q^{32} \) \( + ( 4 \beta_{1} - 32 \beta_{2} ) q^{33} \) \( -16 q^{34} \) \( + ( -17 \beta_{2} + 4 \beta_{3} ) q^{35} \) \( + ( -16 - 2 \beta_{3} ) q^{36} \) \( + ( 8 \beta_{1} + 4 \beta_{2} ) q^{37} \) \( + 12 \beta_{2} q^{38} \) \( + ( -8 + 4 \beta_{1} + 2 \beta_{2} ) q^{40} \) \( -14 \beta_{3} q^{41} \) \( + ( -2 \beta_{1} + 16 \beta_{2} ) q^{42} \) \( + ( -14 \beta_{1} - 7 \beta_{2} ) q^{43} \) \( + 8 \beta_{3} q^{44} \) \( + ( -17 - 4 \beta_{1} + 14 \beta_{2} + 8 \beta_{3} ) q^{45} \) \( + 34 q^{46} \) \( -25 \beta_{2} q^{47} \) \( + ( -4 \beta_{1} - 4 \beta_{2} ) q^{48} \) \( + 15 q^{49} \) \( + ( -9 \beta_{2} + 8 \beta_{3} ) q^{50} \) \( + ( 8 - 8 \beta_{3} ) q^{51} \) \( + 48 \beta_{2} q^{53} \) \( + ( 25 - 7 \beta_{3} ) q^{54} \) \( + ( 68 + 16 \beta_{1} + 8 \beta_{2} ) q^{55} \) \( -4 \beta_{3} q^{56} \) \( + ( -12 \beta_{1} - 12 \beta_{2} ) q^{57} \) \( + 4 \beta_{3} q^{59} \) \( + ( 4 - 2 \beta_{1} + 16 \beta_{2} - 4 \beta_{3} ) q^{60} \) \( -16 q^{61} \) \( -32 \beta_{2} q^{62} \) \( + ( -16 \beta_{1} - 25 \beta_{2} ) q^{63} \) \( + 8 q^{64} \) \( + ( -68 - 4 \beta_{3} ) q^{66} \) \( + ( -2 \beta_{1} - \beta_{2} ) q^{67} \) \( -16 \beta_{2} q^{68} \) \( + ( -17 + 17 \beta_{3} ) q^{69} \) \( + ( -34 - 8 \beta_{1} - 4 \beta_{2} ) q^{70} \) \( + ( 4 \beta_{1} - 14 \beta_{2} ) q^{72} \) \( + ( 40 \beta_{1} + 20 \beta_{2} ) q^{73} \) \( -8 \beta_{3} q^{74} \) \( + ( -68 + 9 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{75} \) \( + 24 q^{76} \) \( + 68 \beta_{2} q^{77} \) \( -72 q^{79} \) \( + ( -8 \beta_{2} - 4 \beta_{3} ) q^{80} \) \( + ( 47 + 16 \beta_{3} ) q^{81} \) \( + ( 28 \beta_{1} + 14 \beta_{2} ) q^{82} \) \( -31 \beta_{2} q^{83} \) \( + ( 34 + 2 \beta_{3} ) q^{84} \) \( + ( 32 - 16 \beta_{1} - 8 \beta_{2} ) q^{85} \) \( + 14 \beta_{3} q^{86} \) \( + ( -16 \beta_{1} - 8 \beta_{2} ) q^{88} \) \( -16 \beta_{3} q^{89} \) \( + ( 32 - 16 \beta_{1} - 25 \beta_{2} + 4 \beta_{3} ) q^{90} \) \( + 34 \beta_{2} q^{92} \) \( + ( 32 \beta_{1} + 32 \beta_{2} ) q^{93} \) \( -50 q^{94} \) \( + ( -24 \beta_{2} - 12 \beta_{3} ) q^{95} \) \( + ( -4 + 4 \beta_{3} ) q^{96} \) \( + ( -56 \beta_{1} - 28 \beta_{2} ) q^{97} \) \( + 15 \beta_{2} q^{98} \) \( + ( 68 - 32 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 32q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut -\mathstrut 32q^{9} \) \(\mathstrut -\mathstrut 16q^{10} \) \(\mathstrut +\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 48q^{19} \) \(\mathstrut +\mathstrut 68q^{21} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut -\mathstrut 36q^{25} \) \(\mathstrut +\mathstrut 68q^{30} \) \(\mathstrut -\mathstrut 128q^{31} \) \(\mathstrut -\mathstrut 64q^{34} \) \(\mathstrut -\mathstrut 64q^{36} \) \(\mathstrut -\mathstrut 32q^{40} \) \(\mathstrut -\mathstrut 68q^{45} \) \(\mathstrut +\mathstrut 136q^{46} \) \(\mathstrut +\mathstrut 60q^{49} \) \(\mathstrut +\mathstrut 32q^{51} \) \(\mathstrut +\mathstrut 100q^{54} \) \(\mathstrut +\mathstrut 272q^{55} \) \(\mathstrut +\mathstrut 16q^{60} \) \(\mathstrut -\mathstrut 64q^{61} \) \(\mathstrut +\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 272q^{66} \) \(\mathstrut -\mathstrut 68q^{69} \) \(\mathstrut -\mathstrut 136q^{70} \) \(\mathstrut -\mathstrut 272q^{75} \) \(\mathstrut +\mathstrut 96q^{76} \) \(\mathstrut -\mathstrut 288q^{79} \) \(\mathstrut +\mathstrut 188q^{81} \) \(\mathstrut +\mathstrut 136q^{84} \) \(\mathstrut +\mathstrut 128q^{85} \) \(\mathstrut +\mathstrut 128q^{90} \) \(\mathstrut -\mathstrut 200q^{94} \) \(\mathstrut -\mathstrut 16q^{96} \) \(\mathstrut +\mathstrut 272q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(16\) \(x^{2}\mathstrut +\mathstrut \) \(81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 7 \nu \)\()/9\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut -\mathstrut \) \(8\)
\(\nu^{3}\)\(=\)\(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.707107 + 2.91548i
0.707107 2.91548i
−0.707107 + 2.91548i
−0.707107 2.91548i
−1.41421 0.707107 2.91548i 2.00000 2.82843 4.12311i −1.00000 + 4.12311i 5.83095i −2.82843 −8.00000 4.12311i −4.00000 + 5.83095i
29.2 −1.41421 0.707107 + 2.91548i 2.00000 2.82843 + 4.12311i −1.00000 4.12311i 5.83095i −2.82843 −8.00000 + 4.12311i −4.00000 5.83095i
29.3 1.41421 −0.707107 2.91548i 2.00000 −2.82843 + 4.12311i −1.00000 4.12311i 5.83095i 2.82843 −8.00000 + 4.12311i −4.00000 + 5.83095i
29.4 1.41421 −0.707107 + 2.91548i 2.00000 −2.82843 4.12311i −1.00000 + 4.12311i 5.83095i 2.82843 −8.00000 4.12311i −4.00000 5.83095i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
5.b Even 1 yes
15.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(30, [\chi])\).