Properties

Label 30.3.d.a
Level $30$
Weight $3$
Character orbit 30.d
Analytic conductor $0.817$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,3,Mod(11,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 30.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.817440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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 \( q - \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{6} + (3 \beta_{2} - 6 \beta_1 + 2) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + 6 \beta_{2} q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{12}+ \cdots + (24 \beta_{3} - 6 \beta_{2} + \cdots - 48) q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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
$3$ \( T^{4} - 4 T^{3} + \cdots + 81 \) 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
$47$ \( T^{4} + 9900 T^{2} + 16402500 \) Copy content Toggle raw display
$53$ \( T^{4} + 936 T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 6624 T^{2} + 3504384 \) 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
$71$ \( T^{4} + 5040 T^{2} + 1166400 \) 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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