Properties

Label 150.3.d.a
Level $150$
Weight $3$
Character orbit 150.d
Analytic conductor $4.087$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,3,Mod(101,150)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(4.08720396540\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} + ( - \beta + 4) q^{6} - 7 q^{7} - 2 \beta q^{8} + (4 \beta - 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} + ( - \beta + 4) q^{6} - 7 q^{7} - 2 \beta q^{8} + (4 \beta - 7) q^{9} + 6 \beta q^{11} + (4 \beta + 2) q^{12} - 25 q^{13} - 7 \beta q^{14} + 4 q^{16} - 18 \beta q^{17} + ( - 7 \beta - 8) q^{18} - 7 q^{19} + (14 \beta + 7) q^{21} - 12 q^{22} + 18 \beta q^{23} + (2 \beta - 8) q^{24} - 25 \beta q^{26} + (10 \beta + 23) q^{27} + 14 q^{28} - 30 \beta q^{29} - 7 q^{31} + 4 \beta q^{32} + ( - 6 \beta + 24) q^{33} + 36 q^{34} + ( - 8 \beta + 14) q^{36} + 2 q^{37} - 7 \beta q^{38} + (50 \beta + 25) q^{39} - 6 \beta q^{41} + (7 \beta - 28) q^{42} + 41 q^{43} - 12 \beta q^{44} - 36 q^{46} + ( - 8 \beta - 4) q^{48} + (18 \beta - 72) q^{51} + 50 q^{52} + 42 \beta q^{53} + (23 \beta - 20) q^{54} + 14 \beta q^{56} + (14 \beta + 7) q^{57} + 60 q^{58} + 24 \beta q^{59} - q^{61} - 7 \beta q^{62} + ( - 28 \beta + 49) q^{63} - 8 q^{64} + (24 \beta + 12) q^{66} + 17 q^{67} + 36 \beta q^{68} + ( - 18 \beta + 72) q^{69} - 30 \beta q^{71} + (14 \beta + 16) q^{72} - 70 q^{73} + 2 \beta q^{74} + 14 q^{76} - 42 \beta q^{77} + (25 \beta - 100) q^{78} - 58 q^{79} + ( - 56 \beta + 17) q^{81} + 12 q^{82} - 84 \beta q^{83} + ( - 28 \beta - 14) q^{84} + 41 \beta q^{86} + (30 \beta - 120) q^{87} + 24 q^{88} - 96 \beta q^{89} + 175 q^{91} - 36 \beta q^{92} + (14 \beta + 7) q^{93} + ( - 4 \beta + 16) q^{96} - 49 q^{97} + ( - 42 \beta - 48) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{7} - 14 q^{9} + 4 q^{12} - 50 q^{13} + 8 q^{16} - 16 q^{18} - 14 q^{19} + 14 q^{21} - 24 q^{22} - 16 q^{24} + 46 q^{27} + 28 q^{28} - 14 q^{31} + 48 q^{33} + 72 q^{34} + 28 q^{36} + 4 q^{37} + 50 q^{39} - 56 q^{42} + 82 q^{43} - 72 q^{46} - 8 q^{48} - 144 q^{51} + 100 q^{52} - 40 q^{54} + 14 q^{57} + 120 q^{58} - 2 q^{61} + 98 q^{63} - 16 q^{64} + 24 q^{66} + 34 q^{67} + 144 q^{69} + 32 q^{72} - 140 q^{73} + 28 q^{76} - 200 q^{78} - 116 q^{79} + 34 q^{81} + 24 q^{82} - 28 q^{84} - 240 q^{87} + 48 q^{88} + 350 q^{91} + 14 q^{93} + 32 q^{96} - 98 q^{97} - 96 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.41421i
1.41421i
1.41421i −1.00000 + 2.82843i −2.00000 0 4.00000 + 1.41421i −7.00000 2.82843i −7.00000 5.65685i 0
101.2 1.41421i −1.00000 2.82843i −2.00000 0 4.00000 1.41421i −7.00000 2.82843i −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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.d.a 2
3.b odd 2 1 inner 150.3.d.a 2
4.b odd 2 1 1200.3.l.p 2
5.b even 2 1 150.3.d.b yes 2
5.c odd 4 2 150.3.b.a 4
12.b even 2 1 1200.3.l.p 2
15.d odd 2 1 150.3.d.b yes 2
15.e even 4 2 150.3.b.a 4
20.d odd 2 1 1200.3.l.i 2
20.e even 4 2 1200.3.c.h 4
60.h even 2 1 1200.3.l.i 2
60.l odd 4 2 1200.3.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.b.a 4 5.c odd 4 2
150.3.b.a 4 15.e even 4 2
150.3.d.a 2 1.a even 1 1 trivial
150.3.d.a 2 3.b odd 2 1 inner
150.3.d.b yes 2 5.b even 2 1
150.3.d.b yes 2 15.d odd 2 1
1200.3.c.h 4 20.e even 4 2
1200.3.c.h 4 60.l odd 4 2
1200.3.l.i 2 20.d odd 2 1
1200.3.l.i 2 60.h even 2 1
1200.3.l.p 2 4.b odd 2 1
1200.3.l.p 2 12.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 72 \) Copy content Toggle raw display
$13$ \( (T + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 648 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 648 \) Copy content Toggle raw display
$29$ \( T^{2} + 1800 \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 72 \) Copy content Toggle raw display
$43$ \( (T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3528 \) Copy content Toggle raw display
$59$ \( T^{2} + 1152 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T - 17)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 1800 \) Copy content Toggle raw display
$73$ \( (T + 70)^{2} \) Copy content Toggle raw display
$79$ \( (T + 58)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14112 \) Copy content Toggle raw display
$89$ \( T^{2} + 18432 \) Copy content Toggle raw display
$97$ \( (T + 49)^{2} \) Copy content Toggle raw display
show more
show less