Properties

Label 30.3.f.a
Level $30$
Weight $3$
Character orbit 30.f
Analytic conductor $0.817$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 30.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.817440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 4 \beta_{3} + 4 \beta_{2} - 4) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 4 \beta_{3} + 4 \beta_{2} - 4) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9} + (4 \beta_{3} + \beta_{2} + 1) q^{10} + (4 \beta_{3} - 4 \beta_1 - 4) q^{11} - 2 \beta_{3} q^{12} + (3 \beta_{2} + 8 \beta_1 + 3) q^{13} + ( - 4 \beta_{3} + 8 \beta_{2} - 4 \beta_1) q^{14} + (\beta_{3} - 6 \beta_{2} - 6) q^{15} - 4 q^{16} + ( - 4 \beta_{3} - 11 \beta_{2} + 11) q^{17} + (3 \beta_{2} + 3) q^{18} + ( - 4 \beta_{3} - 16 \beta_{2} - 4 \beta_1) q^{19} + (4 \beta_{3} + 4 \beta_1 + 2) q^{20} + (4 \beta_{3} - 4 \beta_1 + 12) q^{21} + (8 \beta_{3} + 4 \beta_{2} - 4) q^{22} + ( - 4 \beta_{2} + 12 \beta_1 - 4) q^{23} + ( - 2 \beta_{3} - 2 \beta_1) q^{24} + ( - 4 \beta_{3} - 4 \beta_1 + 23) q^{25} + ( - 8 \beta_{3} + 8 \beta_1 + 6) q^{26} + 3 \beta_{3} q^{27} + (8 \beta_{2} - 8 \beta_1 + 8) q^{28} + (4 \beta_{3} + 16 \beta_{2} + 4 \beta_1) q^{29} + (\beta_{3} + \beta_1 - 12) q^{30} + (8 \beta_{3} - 8 \beta_1 - 20) q^{31} + (4 \beta_{2} - 4) q^{32} + ( - 12 \beta_{2} - 4 \beta_1 - 12) q^{33} + ( - 4 \beta_{3} - 22 \beta_{2} - 4 \beta_1) q^{34} + ( - 16 \beta_{3} + 20 \beta_{2} + 4 \beta_1 - 28) q^{35} + 6 q^{36} + (27 \beta_{2} - 27) q^{37} + ( - 16 \beta_{2} - 8 \beta_1 - 16) q^{38} + (3 \beta_{3} + 24 \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{2} + 8 \beta_1 + 2) q^{40} + ( - 4 \beta_{3} + 4 \beta_1 + 8) q^{41} + (8 \beta_{3} - 12 \beta_{2} + 12) q^{42} + (12 \beta_{2} - 20 \beta_1 + 12) q^{43} + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1) q^{44} + ( - 6 \beta_{3} - 6 \beta_1 - 3) q^{45} + ( - 12 \beta_{3} + 12 \beta_1 - 8) q^{46} + (12 \beta_{3} - 24 \beta_{2} + 24) q^{47} - 4 \beta_1 q^{48} + (32 \beta_{3} - 31 \beta_{2} + 32 \beta_1) q^{49} + ( - 23 \beta_{2} - 8 \beta_1 + 23) q^{50} + ( - 11 \beta_{3} + 11 \beta_1 + 12) q^{51} + ( - 16 \beta_{3} - 6 \beta_{2} + 6) q^{52} + (25 \beta_{2} - 36 \beta_1 + 25) q^{53} + (3 \beta_{3} + 3 \beta_1) q^{54} + ( - 12 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 48) q^{55} + (8 \beta_{3} - 8 \beta_1 + 16) q^{56} + ( - 16 \beta_{3} - 12 \beta_{2} + 12) q^{57} + (16 \beta_{2} + 8 \beta_1 + 16) q^{58} - 20 \beta_{2} q^{59} + (12 \beta_{2} + 2 \beta_1 - 12) q^{60} + ( - 16 \beta_{3} + 16 \beta_1 - 24) q^{61} + (16 \beta_{3} + 20 \beta_{2} - 20) q^{62} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{63} + 8 \beta_{2} q^{64} + (8 \beta_{3} - 45 \beta_{2} - 12 \beta_1 - 51) q^{65} + (4 \beta_{3} - 4 \beta_1 - 24) q^{66} + (36 \beta_{3} + 4 \beta_{2} - 4) q^{67} + ( - 22 \beta_{2} - 8 \beta_1 - 22) q^{68} + ( - 4 \beta_{3} + 36 \beta_{2} - 4 \beta_1) q^{69} + ( - 20 \beta_{3} + 48 \beta_{2} - 12 \beta_1 - 8) q^{70} + (4 \beta_{3} - 4 \beta_1 + 16) q^{71} + ( - 6 \beta_{2} + 6) q^{72} + ( - 47 \beta_{2} + 8 \beta_1 - 47) q^{73} + 54 \beta_{2} q^{74} + ( - 12 \beta_{2} + 23 \beta_1 + 12) q^{75} + (8 \beta_{3} - 8 \beta_1 - 32) q^{76} + ( - 16 \beta_{3} + 32 \beta_{2} - 32) q^{77} + (24 \beta_{2} + 6 \beta_1 + 24) q^{78} + ( - 52 \beta_{3} + 12 \beta_{2} - 52 \beta_1) q^{79} + ( - 8 \beta_{3} - 4 \beta_{2} + 8 \beta_1) q^{80} - 9 q^{81} + ( - 8 \beta_{3} - 8 \beta_{2} + 8) q^{82} + (48 \beta_{2} + 28 \beta_1 + 48) q^{83} + (8 \beta_{3} - 24 \beta_{2} + 8 \beta_1) q^{84} + (44 \beta_{3} + 35 \beta_{2} + 4 \beta_1 - 13) q^{85} + (20 \beta_{3} - 20 \beta_1 + 24) q^{86} + (16 \beta_{3} + 12 \beta_{2} - 12) q^{87} + (8 \beta_{2} + 16 \beta_1 + 8) q^{88} + ( - 12 \beta_{3} - 88 \beta_{2} - 12 \beta_1) q^{89} + (3 \beta_{2} - 12 \beta_1 - 3) q^{90} + (20 \beta_{3} - 20 \beta_1 + 72) q^{91} + ( - 24 \beta_{3} + 8 \beta_{2} - 8) q^{92} + ( - 24 \beta_{2} - 20 \beta_1 - 24) q^{93} + (12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{94} + (28 \beta_{3} + 48 \beta_{2} + 36 \beta_1 + 16) q^{95} + (4 \beta_{3} - 4 \beta_1) q^{96} + ( - 40 \beta_{3} - 33 \beta_{2} + 33) q^{97} + ( - 31 \beta_{2} + 64 \beta_1 - 31) q^{98} + ( - 12 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 16 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 16 q^{7} - 8 q^{8} + 4 q^{10} - 16 q^{11} + 12 q^{13} - 24 q^{15} - 16 q^{16} + 44 q^{17} + 12 q^{18} + 8 q^{20} + 48 q^{21} - 16 q^{22} - 16 q^{23} + 92 q^{25} + 24 q^{26} + 32 q^{28} - 48 q^{30} - 80 q^{31} - 16 q^{32} - 48 q^{33} - 112 q^{35} + 24 q^{36} - 108 q^{37} - 64 q^{38} + 8 q^{40} + 32 q^{41} + 48 q^{42} + 48 q^{43} - 12 q^{45} - 32 q^{46} + 96 q^{47} + 92 q^{50} + 48 q^{51} + 24 q^{52} + 100 q^{53} + 192 q^{55} + 64 q^{56} + 48 q^{57} + 64 q^{58} - 48 q^{60} - 96 q^{61} - 80 q^{62} - 48 q^{63} - 204 q^{65} - 96 q^{66} - 16 q^{67} - 88 q^{68} - 32 q^{70} + 64 q^{71} + 24 q^{72} - 188 q^{73} + 48 q^{75} - 128 q^{76} - 128 q^{77} + 96 q^{78} - 36 q^{81} + 32 q^{82} + 192 q^{83} - 52 q^{85} + 96 q^{86} - 48 q^{87} + 32 q^{88} - 12 q^{90} + 288 q^{91} - 32 q^{92} - 96 q^{93} + 64 q^{95} + 132 q^{97} - 124 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 4.89898 1.00000i −2.44949 −8.89898 8.89898i −2.00000 + 2.00000i 3.00000i 5.89898 + 3.89898i
7.2 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −4.89898 1.00000i 2.44949 0.898979 + 0.898979i −2.00000 + 2.00000i 3.00000i −3.89898 5.89898i
13.1 1.00000 1.00000i −1.22474 1.22474i 2.00000i 4.89898 + 1.00000i −2.44949 −8.89898 + 8.89898i −2.00000 2.00000i 3.00000i 5.89898 3.89898i
13.2 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i −4.89898 + 1.00000i 2.44949 0.898979 0.898979i −2.00000 2.00000i 3.00000i −3.89898 + 5.89898i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.f.a 4
3.b odd 2 1 90.3.g.d 4
4.b odd 2 1 240.3.bg.b 4
5.b even 2 1 150.3.f.b 4
5.c odd 4 1 inner 30.3.f.a 4
5.c odd 4 1 150.3.f.b 4
8.b even 2 1 960.3.bg.e 4
8.d odd 2 1 960.3.bg.g 4
12.b even 2 1 720.3.bh.i 4
15.d odd 2 1 450.3.g.j 4
15.e even 4 1 90.3.g.d 4
15.e even 4 1 450.3.g.j 4
20.d odd 2 1 1200.3.bg.d 4
20.e even 4 1 240.3.bg.b 4
20.e even 4 1 1200.3.bg.d 4
40.i odd 4 1 960.3.bg.e 4
40.k even 4 1 960.3.bg.g 4
60.l odd 4 1 720.3.bh.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 1.a even 1 1 trivial
30.3.f.a 4 5.c odd 4 1 inner
90.3.g.d 4 3.b odd 2 1
90.3.g.d 4 15.e even 4 1
150.3.f.b 4 5.b even 2 1
150.3.f.b 4 5.c odd 4 1
240.3.bg.b 4 4.b odd 2 1
240.3.bg.b 4 20.e even 4 1
450.3.g.j 4 15.d odd 2 1
450.3.g.j 4 15.e even 4 1
720.3.bh.i 4 12.b even 2 1
720.3.bh.i 4 60.l odd 4 1
960.3.bg.e 4 8.b even 2 1
960.3.bg.e 4 40.i odd 4 1
960.3.bg.g 4 8.d odd 2 1
960.3.bg.g 4 40.k even 4 1
1200.3.bg.d 4 20.d odd 2 1
1200.3.bg.d 4 20.e even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 46T^{2} + 625 \) Copy content Toggle raw display
$7$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 80)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 30276 \) Copy content Toggle raw display
$17$ \( T^{4} - 44 T^{3} + 968 T^{2} + \cdots + 37636 \) Copy content Toggle raw display
$19$ \( T^{4} + 704 T^{2} + 25600 \) Copy content Toggle raw display
$23$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 160000 \) Copy content Toggle raw display
$29$ \( T^{4} + 704 T^{2} + 25600 \) Copy content Toggle raw display
$31$ \( (T^{2} + 40 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 54 T + 1458)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 831744 \) Copy content Toggle raw display
$47$ \( T^{4} - 96 T^{3} + 4608 T^{2} + \cdots + 518400 \) Copy content Toggle raw display
$53$ \( T^{4} - 100 T^{3} + 5000 T^{2} + \cdots + 6959044 \) Copy content Toggle raw display
$59$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48 T - 960)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 14868736 \) Copy content Toggle raw display
$71$ \( (T^{2} - 32 T + 160)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 188 T^{3} + \cdots + 17859076 \) Copy content Toggle raw display
$79$ \( T^{4} + 32736 T^{2} + \cdots + 258566400 \) Copy content Toggle raw display
$83$ \( T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 5089536 \) Copy content Toggle raw display
$89$ \( T^{4} + 17216 T^{2} + \cdots + 47334400 \) Copy content Toggle raw display
$97$ \( T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 6874884 \) Copy content Toggle raw display
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