Properties

Label 150.3.b.a
Level $150$
Weight $3$
Character orbit 150.b
Analytic conductor $4.087$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,3,Mod(149,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 150.b (of order \(2\), degree \(1\), not 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: \(4.08720396540\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - 2 \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 4) q^{6} + 7 \beta_1 q^{7} - 2 \beta_{3} q^{8} + (4 \beta_{2} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - 2 \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 4) q^{6} + 7 \beta_1 q^{7} - 2 \beta_{3} q^{8} + (4 \beta_{2} + 7) q^{9} - 6 \beta_{2} q^{11} + ( - 4 \beta_{3} - 2 \beta_1) q^{12} - 25 \beta_1 q^{13} - 7 \beta_{2} q^{14} + 4 q^{16} + 18 \beta_{3} q^{17} + ( - 7 \beta_{3} - 8 \beta_1) q^{18} + 7 q^{19} + ( - 14 \beta_{2} + 7) q^{21} + 12 \beta_1 q^{22} + 18 \beta_{3} q^{23} + (2 \beta_{2} + 8) q^{24} + 25 \beta_{2} q^{26} + ( - 10 \beta_{3} - 23 \beta_1) q^{27} + 14 \beta_1 q^{28} - 30 \beta_{2} q^{29} - 7 q^{31} - 4 \beta_{3} q^{32} + ( - 6 \beta_{3} + 24 \beta_1) q^{33} - 36 q^{34} + (8 \beta_{2} + 14) q^{36} - 2 \beta_1 q^{37} - 7 \beta_{3} q^{38} + (50 \beta_{2} - 25) q^{39} + 6 \beta_{2} q^{41} + ( - 7 \beta_{3} + 28 \beta_1) q^{42} + 41 \beta_1 q^{43} - 12 \beta_{2} q^{44} - 36 q^{46} + ( - 8 \beta_{3} - 4 \beta_1) q^{48} + ( - 18 \beta_{2} - 72) q^{51} - 50 \beta_1 q^{52} + 42 \beta_{3} q^{53} + (23 \beta_{2} + 20) q^{54} - 14 \beta_{2} q^{56} + ( - 14 \beta_{3} - 7 \beta_1) q^{57} + 60 \beta_1 q^{58} + 24 \beta_{2} q^{59} - q^{61} + 7 \beta_{3} q^{62} + ( - 28 \beta_{3} + 49 \beta_1) q^{63} + 8 q^{64} + ( - 24 \beta_{2} + 12) q^{66} - 17 \beta_1 q^{67} + 36 \beta_{3} q^{68} + ( - 18 \beta_{2} - 72) q^{69} + 30 \beta_{2} q^{71} + ( - 14 \beta_{3} - 16 \beta_1) q^{72} - 70 \beta_1 q^{73} + 2 \beta_{2} q^{74} + 14 q^{76} + 42 \beta_{3} q^{77} + (25 \beta_{3} - 100 \beta_1) q^{78} + 58 q^{79} + (56 \beta_{2} + 17) q^{81} - 12 \beta_1 q^{82} - 84 \beta_{3} q^{83} + ( - 28 \beta_{2} + 14) q^{84} - 41 \beta_{2} q^{86} + ( - 30 \beta_{3} + 120 \beta_1) q^{87} + 24 \beta_1 q^{88} - 96 \beta_{2} q^{89} + 175 q^{91} + 36 \beta_{3} q^{92} + (14 \beta_{3} + 7 \beta_1) q^{93} + (4 \beta_{2} + 16) q^{96} + 49 \beta_1 q^{97} + ( - 42 \beta_{2} + 48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 16 q^{6} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 16 q^{6} + 28 q^{9} + 16 q^{16} + 28 q^{19} + 28 q^{21} + 32 q^{24} - 28 q^{31} - 144 q^{34} + 56 q^{36} - 100 q^{39} - 144 q^{46} - 288 q^{51} + 80 q^{54} - 4 q^{61} + 32 q^{64} + 48 q^{66} - 288 q^{69} + 56 q^{76} + 232 q^{79} + 68 q^{81} + 56 q^{84} + 700 q^{91} + 64 q^{96} + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−1.41421 −2.82843 1.00000i 2.00000 0 4.00000 + 1.41421i 7.00000i −2.82843 7.00000 + 5.65685i 0
149.2 −1.41421 −2.82843 + 1.00000i 2.00000 0 4.00000 1.41421i 7.00000i −2.82843 7.00000 5.65685i 0
149.3 1.41421 2.82843 1.00000i 2.00000 0 4.00000 1.41421i 7.00000i 2.82843 7.00000 5.65685i 0
149.4 1.41421 2.82843 + 1.00000i 2.00000 0 4.00000 + 1.41421i 7.00000i 2.82843 7.00000 + 5.65685i 0
\(n\): 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 150.3.b.a 4
3.b odd 2 1 inner 150.3.b.a 4
4.b odd 2 1 1200.3.c.h 4
5.b even 2 1 inner 150.3.b.a 4
5.c odd 4 1 150.3.d.a 2
5.c odd 4 1 150.3.d.b yes 2
12.b even 2 1 1200.3.c.h 4
15.d odd 2 1 inner 150.3.b.a 4
15.e even 4 1 150.3.d.a 2
15.e even 4 1 150.3.d.b yes 2
20.d odd 2 1 1200.3.c.h 4
20.e even 4 1 1200.3.l.i 2
20.e even 4 1 1200.3.l.p 2
60.h even 2 1 1200.3.c.h 4
60.l odd 4 1 1200.3.l.i 2
60.l odd 4 1 1200.3.l.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.b.a 4 1.a even 1 1 trivial
150.3.b.a 4 3.b odd 2 1 inner
150.3.b.a 4 5.b even 2 1 inner
150.3.b.a 4 15.d odd 2 1 inner
150.3.d.a 2 5.c odd 4 1
150.3.d.a 2 15.e even 4 1
150.3.d.b yes 2 5.c odd 4 1
150.3.d.b yes 2 15.e even 4 1
1200.3.c.h 4 4.b odd 2 1
1200.3.c.h 4 12.b even 2 1
1200.3.c.h 4 20.d odd 2 1
1200.3.c.h 4 60.h even 2 1
1200.3.l.i 2 20.e even 4 1
1200.3.l.i 2 60.l odd 4 1
1200.3.l.p 2 20.e even 4 1
1200.3.l.p 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 49 \) acting on \(S_{3}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 14T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$19$ \( (T - 7)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 648)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3528)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1152)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$79$ \( (T - 58)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 14112)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18432)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
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