Properties

Label 30.3.d.a
Level $30$
Weight $3$
Character orbit 30.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.817440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{6} + (3 \beta_{2} - 6 \beta_1 + 2) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{6} + (3 \beta_{2} - 6 \beta_1 + 2) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + 6 \beta_{2} q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{12} - 10 q^{13} + (6 \beta_{3} - 2 \beta_{2}) q^{14} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 5) q^{15} + 4 q^{16} + ( - 6 \beta_{3} - 12 \beta_{2}) q^{17} + ( - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{18} + ( - 6 \beta_{2} + 12 \beta_1 + 8) q^{19} + 2 \beta_{3} q^{20} + (5 \beta_{3} - 14 \beta_{2} - 4 \beta_1 - 13) q^{21} + 12 q^{22} + (6 \beta_{3} - 3 \beta_{2}) q^{23} + (2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{24} - 5 q^{25} + 10 \beta_{2} q^{26} + (\beta_{3} + 20 \beta_{2} - 5 \beta_1 + 7) q^{27} + ( - 6 \beta_{2} + 12 \beta_1 - 4) q^{28} - 12 \beta_{3} q^{29} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 - 5) q^{30} + 8 q^{31} - 4 \beta_{2} q^{32} + (6 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 6) q^{33} + (6 \beta_{2} - 12 \beta_1 - 24) q^{34} + ( - 2 \beta_{3} + 15 \beta_{2}) q^{35} + ( - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{36} + ( - 12 \beta_{2} + 24 \beta_1 - 22) q^{37} + ( - 12 \beta_{3} - 8 \beta_{2}) q^{38} + ( - 10 \beta_{3} + 10 \beta_{2} - 10 \beta_1 - 10) q^{39} + ( - 2 \beta_{2} + 4 \beta_1) q^{40} + (6 \beta_{3} + 24 \beta_{2}) q^{41} + (4 \beta_{3} + 8 \beta_{2} + 10 \beta_1 - 32) q^{42} + (9 \beta_{2} - 18 \beta_1 + 14) q^{43} - 12 \beta_{2} q^{44} + (2 \beta_{3} - 14 \beta_{2} + 8 \beta_1 + 5) q^{45} + ( - 6 \beta_{2} + 12 \beta_1 - 6) q^{46} + (30 \beta_{3} - 15 \beta_{2}) q^{47} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 4) q^{48} + (12 \beta_{2} - 24 \beta_1 + 45) q^{49} + 5 \beta_{2} q^{50} + ( - 18 \beta_{3} - 36 \beta_{2} + 18 \beta_1 + 18) q^{51} + 20 q^{52} + ( - 6 \beta_{3} - 12 \beta_{2}) q^{53} + (5 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 35) q^{54} + ( - 6 \beta_{2} + 12 \beta_1) q^{55} + ( - 12 \beta_{3} + 4 \beta_{2}) q^{56} + (2 \beta_{3} + 16 \beta_{2} + 20 \beta_1 + 38) q^{57} + (12 \beta_{2} - 24 \beta_1) q^{58} + (12 \beta_{3} - 36 \beta_{2}) q^{59} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 10) q^{60} + (12 \beta_{2} - 24 \beta_1 - 16) q^{61} - 8 \beta_{2} q^{62} + ( - 22 \beta_{3} - 17 \beta_{2} + 20 \beta_1 - 64) q^{63} - 8 q^{64} + 10 \beta_{3} q^{65} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 12) q^{66} + (9 \beta_{2} - 18 \beta_1 - 82) q^{67} + (12 \beta_{3} + 24 \beta_{2}) q^{68} + (3 \beta_{3} + 6 \beta_{2} + 12 \beta_1 - 33) q^{69} + (2 \beta_{2} - 4 \beta_1 + 30) q^{70} + ( - 12 \beta_{3} + 30 \beta_{2}) q^{71} + (8 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 16) q^{72} + ( - 12 \beta_{2} + 24 \beta_1 + 50) q^{73} + ( - 24 \beta_{3} + 22 \beta_{2}) q^{74} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 5) q^{75} + (12 \beta_{2} - 24 \beta_1 - 16) q^{76} + ( - 36 \beta_{3} + 12 \beta_{2}) q^{77} + (10 \beta_{3} + 20 \beta_{2} - 20 \beta_1 + 10) q^{78} + (6 \beta_{2} - 12 \beta_1 - 28) q^{79} - 4 \beta_{3} q^{80} + (28 \beta_{3} + 20 \beta_{2} - 32 \beta_1 + 7) q^{81} + ( - 6 \beta_{2} + 12 \beta_1 + 48) q^{82} + (6 \beta_{3} + 9 \beta_{2}) q^{83} + ( - 10 \beta_{3} + 28 \beta_{2} + 8 \beta_1 + 26) q^{84} + (12 \beta_{2} - 24 \beta_1 - 30) q^{85} + (18 \beta_{3} - 14 \beta_{2}) q^{86} + ( - 12 \beta_{3} - 24 \beta_{2} - 12 \beta_1 + 60) q^{87} - 24 q^{88} + (12 \beta_{3} + 24 \beta_{2}) q^{89} + ( - 8 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 20) q^{90} + ( - 30 \beta_{2} + 60 \beta_1 - 20) q^{91} + ( - 12 \beta_{3} + 6 \beta_{2}) q^{92} + (8 \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 8) q^{93} + ( - 30 \beta_{2} + 60 \beta_1 - 30) q^{94} + ( - 8 \beta_{3} - 30 \beta_{2}) q^{95} + ( - 4 \beta_{3} - 8 \beta_{2} + 8 \beta_1 - 4) q^{96} + (12 \beta_{2} - 24 \beta_1 + 74) q^{97} + (24 \beta_{3} - 45 \beta_{2}) q^{98} + (24 \beta_{3} - 6 \beta_{2} - 12 \beta_1 - 48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{4} - 4 q^{6} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{4} - 4 q^{6} + 8 q^{7} - 8 q^{9} - 8 q^{12} - 40 q^{13} + 20 q^{15} + 16 q^{16} + 32 q^{18} + 32 q^{19} - 52 q^{21} + 48 q^{22} + 8 q^{24} - 20 q^{25} + 28 q^{27} - 16 q^{28} - 20 q^{30} + 32 q^{31} + 24 q^{33} - 96 q^{34} + 16 q^{36} - 88 q^{37} - 40 q^{39} - 128 q^{42} + 56 q^{43} + 20 q^{45} - 24 q^{46} + 16 q^{48} + 180 q^{49} + 72 q^{51} + 80 q^{52} + 140 q^{54} + 152 q^{57} - 40 q^{60} - 64 q^{61} - 256 q^{63} - 32 q^{64} + 48 q^{66} - 328 q^{67} - 132 q^{69} + 120 q^{70} - 64 q^{72} + 200 q^{73} - 20 q^{75} - 64 q^{76} + 40 q^{78} - 112 q^{79} + 28 q^{81} + 192 q^{82} + 104 q^{84} - 120 q^{85} + 240 q^{87} - 96 q^{88} - 80 q^{90} - 80 q^{91} + 32 q^{93} - 120 q^{94} - 16 q^{96} + 296 q^{97} - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.58114 + 0.707107i
1.58114 + 0.707107i
−1.58114 0.707107i
1.58114 0.707107i
1.41421i −0.581139 2.94317i −2.00000 2.23607i −4.16228 + 0.821854i 11.4868 2.82843i −8.32456 + 3.42079i 3.16228
11.2 1.41421i 2.58114 + 1.52896i −2.00000 2.23607i 2.16228 3.65028i −7.48683 2.82843i 4.32456 + 7.89292i −3.16228
11.3 1.41421i −0.581139 + 2.94317i −2.00000 2.23607i −4.16228 0.821854i 11.4868 2.82843i −8.32456 3.42079i 3.16228
11.4 1.41421i 2.58114 1.52896i −2.00000 2.23607i 2.16228 + 3.65028i −7.48683 2.82843i 4.32456 7.89292i −3.16228
\(n\): 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

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + 12 T^{2} - 36 T + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T - 86)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T + 10)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 296)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 396 T^{2} + 26244 \) Copy content Toggle raw display
$29$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 44 T - 956)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2664 T^{2} + 944784 \) Copy content Toggle raw display
$43$ \( (T^{2} - 28 T - 614)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 9900 T^{2} + \cdots + 16402500 \) Copy content Toggle raw display
$53$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 6624 T^{2} + \cdots + 3504384 \) Copy content Toggle raw display
$61$ \( (T^{2} + 32 T - 1184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 164 T + 5914)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 5040 T^{2} + \cdots + 1166400 \) Copy content Toggle raw display
$73$ \( (T^{2} - 100 T + 1060)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 56 T + 424)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 684T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 3744 T^{2} + 186624 \) Copy content Toggle raw display
$97$ \( (T^{2} - 148 T + 4036)^{2} \) Copy content Toggle raw display
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