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: \( x^{4} + 16x^{2} + 81 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + (\beta_{3} - 1) q^{6} + (\beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + ( - \beta_{3} - 8) q^{9} + (\beta_{2} + 2 \beta_1 - 4) q^{10} + 4 \beta_{3} q^{11} + ( - 2 \beta_{2} - 2 \beta_1) q^{12} - 2 \beta_{3} q^{14} + ( - 2 \beta_{3} + 8 \beta_{2} - \beta_1 + 2) q^{15} + 4 q^{16} - 8 \beta_{2} q^{17} + ( - 7 \beta_{2} + 2 \beta_1) q^{18} + 12 q^{19} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{20} + (\beta_{3} + 17) q^{21} + ( - 4 \beta_{2} - 8 \beta_1) q^{22} + 17 \beta_{2} q^{23} + (2 \beta_{3} - 2) q^{24} + ( - 4 \beta_{2} - 8 \beta_1 - 9) q^{25} + (16 \beta_{2} + 7 \beta_1) q^{27} + (2 \beta_{2} + 4 \beta_1) q^{28} + (\beta_{3} + 4 \beta_{2} + 4 \beta_1 + 17) q^{30} - 32 q^{31} + 4 \beta_{2} q^{32} + ( - 32 \beta_{2} + 4 \beta_1) q^{33} - 16 q^{34} + (4 \beta_{3} - 17 \beta_{2}) q^{35} + ( - 2 \beta_{3} - 16) q^{36} + (4 \beta_{2} + 8 \beta_1) q^{37} + 12 \beta_{2} q^{38} + (2 \beta_{2} + 4 \beta_1 - 8) q^{40} - 14 \beta_{3} q^{41} + (16 \beta_{2} - 2 \beta_1) q^{42} + ( - 7 \beta_{2} - 14 \beta_1) q^{43} + 8 \beta_{3} q^{44} + (8 \beta_{3} + 14 \beta_{2} - 4 \beta_1 - 17) q^{45} + 34 q^{46} - 25 \beta_{2} q^{47} + ( - 4 \beta_{2} - 4 \beta_1) q^{48} + 15 q^{49} + (8 \beta_{3} - 9 \beta_{2}) q^{50} + ( - 8 \beta_{3} + 8) q^{51} + 48 \beta_{2} q^{53} + ( - 7 \beta_{3} + 25) q^{54} + (8 \beta_{2} + 16 \beta_1 + 68) q^{55} - 4 \beta_{3} q^{56} + ( - 12 \beta_{2} - 12 \beta_1) q^{57} + 4 \beta_{3} q^{59} + ( - 4 \beta_{3} + 16 \beta_{2} - 2 \beta_1 + 4) q^{60} - 16 q^{61} - 32 \beta_{2} q^{62} + ( - 25 \beta_{2} - 16 \beta_1) q^{63} + 8 q^{64} + ( - 4 \beta_{3} - 68) q^{66} + ( - \beta_{2} - 2 \beta_1) q^{67} - 16 \beta_{2} q^{68} + (17 \beta_{3} - 17) q^{69} + ( - 4 \beta_{2} - 8 \beta_1 - 34) q^{70} + ( - 14 \beta_{2} + 4 \beta_1) q^{72} + (20 \beta_{2} + 40 \beta_1) q^{73} - 8 \beta_{3} q^{74} + ( - 4 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 68) q^{75} + 24 q^{76} + 68 \beta_{2} q^{77} - 72 q^{79} + ( - 4 \beta_{3} - 8 \beta_{2}) q^{80} + (16 \beta_{3} + 47) q^{81} + (14 \beta_{2} + 28 \beta_1) q^{82} - 31 \beta_{2} q^{83} + (2 \beta_{3} + 34) q^{84} + ( - 8 \beta_{2} - 16 \beta_1 + 32) q^{85} + 14 \beta_{3} q^{86} + ( - 8 \beta_{2} - 16 \beta_1) q^{88} - 16 \beta_{3} q^{89} + (4 \beta_{3} - 25 \beta_{2} - 16 \beta_1 + 32) q^{90} + 34 \beta_{2} q^{92} + (32 \beta_{2} + 32 \beta_1) q^{93} - 50 q^{94} + ( - 12 \beta_{3} - 24 \beta_{2}) q^{95} + (4 \beta_{3} - 4) q^{96} + ( - 28 \beta_{2} - 56 \beta_1) q^{97} + 15 \beta_{2} q^{98} + ( - 32 \beta_{3} + 68) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 4 q^{6} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 4 q^{6} - 32 q^{9} - 16 q^{10} + 8 q^{15} + 16 q^{16} + 48 q^{19} + 68 q^{21} - 8 q^{24} - 36 q^{25} + 68 q^{30} - 128 q^{31} - 64 q^{34} - 64 q^{36} - 32 q^{40} - 68 q^{45} + 136 q^{46} + 60 q^{49} + 32 q^{51} + 100 q^{54} + 272 q^{55} + 16 q^{60} - 64 q^{61} + 32 q^{64} - 272 q^{66} - 68 q^{69} - 136 q^{70} - 272 q^{75} + 96 q^{76} - 288 q^{79} + 188 q^{81} + 136 q^{84} + 128 q^{85} + 128 q^{90} - 200 q^{94} - 16 q^{96} + 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{2} - 7\beta_1 \) Copy content Toggle raw display

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

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 16T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} + 18T^{2} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$19$ \( (T - 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 578)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 544)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3332)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1666)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 1250)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4608)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 272)^{2} \) Copy content Toggle raw display
$61$ \( (T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 34)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 13600)^{2} \) Copy content Toggle raw display
$79$ \( (T + 72)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1922)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4352)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 26656)^{2} \) Copy content Toggle raw display
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