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 \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.817440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-17})\)
Defining polynomial: \(x^{4} + 16 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} - 4q^{6} - 32q^{9} + O(q^{10}) \) \( 4q + 8q^{4} - 4q^{6} - 32q^{9} - 16q^{10} + 8q^{15} + 16q^{16} + 48q^{19} + 68q^{21} - 8q^{24} - 36q^{25} + 68q^{30} - 128q^{31} - 64q^{34} - 64q^{36} - 32q^{40} - 68q^{45} + 136q^{46} + 60q^{49} + 32q^{51} + 100q^{54} + 272q^{55} + 16q^{60} - 64q^{61} + 32q^{64} - 272q^{66} - 68q^{69} - 136q^{70} - 272q^{75} + 96q^{76} - 288q^{79} + 188q^{81} + 136q^{84} + 128q^{85} + 128q^{90} - 200q^{94} - 16q^{96} + 272q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.b.a 4
3.b odd 2 1 inner 30.3.b.a 4
4.b odd 2 1 240.3.c.c 4
5.b even 2 1 inner 30.3.b.a 4
5.c odd 4 2 150.3.d.d 4
8.b even 2 1 960.3.c.f 4
8.d odd 2 1 960.3.c.e 4
9.c even 3 2 810.3.j.c 8
9.d odd 6 2 810.3.j.c 8
12.b even 2 1 240.3.c.c 4
15.d odd 2 1 inner 30.3.b.a 4
15.e even 4 2 150.3.d.d 4
20.d odd 2 1 240.3.c.c 4
20.e even 4 2 1200.3.l.t 4
24.f even 2 1 960.3.c.e 4
24.h odd 2 1 960.3.c.f 4
40.e odd 2 1 960.3.c.e 4
40.f even 2 1 960.3.c.f 4
45.h odd 6 2 810.3.j.c 8
45.j even 6 2 810.3.j.c 8
60.h even 2 1 240.3.c.c 4
60.l odd 4 2 1200.3.l.t 4
120.i odd 2 1 960.3.c.f 4
120.m even 2 1 960.3.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.b.a 4 1.a even 1 1 trivial
30.3.b.a 4 3.b odd 2 1 inner
30.3.b.a 4 5.b even 2 1 inner
30.3.b.a 4 15.d odd 2 1 inner
150.3.d.d 4 5.c odd 4 2
150.3.d.d 4 15.e even 4 2
240.3.c.c 4 4.b odd 2 1
240.3.c.c 4 12.b even 2 1
240.3.c.c 4 20.d odd 2 1
240.3.c.c 4 60.h even 2 1
810.3.j.c 8 9.c even 3 2
810.3.j.c 8 9.d odd 6 2
810.3.j.c 8 45.h odd 6 2
810.3.j.c 8 45.j even 6 2
960.3.c.e 4 8.d odd 2 1
960.3.c.e 4 24.f even 2 1
960.3.c.e 4 40.e odd 2 1
960.3.c.e 4 120.m even 2 1
960.3.c.f 4 8.b even 2 1
960.3.c.f 4 24.h odd 2 1
960.3.c.f 4 40.f even 2 1
960.3.c.f 4 120.i odd 2 1
1200.3.l.t 4 20.e even 4 2
1200.3.l.t 4 60.l odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( 1 + 16 T^{2} + 81 T^{4} \)
$5$ \( 1 + 18 T^{2} + 625 T^{4} \)
$7$ \( ( 1 - 64 T^{2} + 2401 T^{4} )^{2} \)
$11$ \( ( 1 + 30 T^{2} + 14641 T^{4} )^{2} \)
$13$ \( ( 1 - 13 T )^{4}( 1 + 13 T )^{4} \)
$17$ \( ( 1 + 450 T^{2} + 83521 T^{4} )^{2} \)
$19$ \( ( 1 - 12 T + 361 T^{2} )^{4} \)
$23$ \( ( 1 + 480 T^{2} + 279841 T^{4} )^{2} \)
$29$ \( ( 1 - 29 T )^{4}( 1 + 29 T )^{4} \)
$31$ \( ( 1 + 32 T + 961 T^{2} )^{4} \)
$37$ \( ( 1 - 2194 T^{2} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 - 30 T^{2} + 2825761 T^{4} )^{2} \)
$43$ \( ( 1 - 2032 T^{2} + 3418801 T^{4} )^{2} \)
$47$ \( ( 1 + 3168 T^{2} + 4879681 T^{4} )^{2} \)
$53$ \( ( 1 + 1010 T^{2} + 7890481 T^{4} )^{2} \)
$59$ \( ( 1 - 6690 T^{2} + 12117361 T^{4} )^{2} \)
$61$ \( ( 1 + 16 T + 3721 T^{2} )^{4} \)
$67$ \( ( 1 - 8944 T^{2} + 20151121 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T )^{4}( 1 + 71 T )^{4} \)
$73$ \( ( 1 + 2942 T^{2} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 + 72 T + 6241 T^{2} )^{4} \)
$83$ \( ( 1 + 11856 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( ( 1 - 11490 T^{2} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 + 7838 T^{2} + 88529281 T^{4} )^{2} \)
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