Properties

Label 150.3.f.b
Level $150$
Weight $3$
Character orbit 150.f
Analytic conductor $4.087$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,3,Mod(7,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 150.f (of order \(4\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\beta_{3} - \beta_1) q^{6} + ( - 4 \beta_{3} - 4 \beta_{2} + 4) q^{7} + (2 \beta_{2} + 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\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} + 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} - 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 + 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} + (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} + ( - 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} + 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} + (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} + (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} + (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} + ( - 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} + (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} + ( - 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} + ( - 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} + ( - 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} - 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} + ( - 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} + ( - 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} + ( - 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 \( 4 q - 4 q^{2} + 16 q^{7} + 8 q^{8} - 16 q^{11} - 12 q^{13} - 16 q^{16} - 44 q^{17} - 12 q^{18} + 48 q^{21} + 16 q^{22} + 16 q^{23} + 24 q^{26} - 32 q^{28} - 80 q^{31} + 16 q^{32} + 48 q^{33} + 24 q^{36} + 108 q^{37} + 64 q^{38} + 32 q^{41} - 48 q^{42} - 48 q^{43} - 32 q^{46} - 96 q^{47} + 48 q^{51} - 24 q^{52} - 100 q^{53} + 64 q^{56} - 48 q^{57} - 64 q^{58} - 96 q^{61} + 80 q^{62} + 48 q^{63} - 96 q^{66} + 16 q^{67} + 88 q^{68} + 64 q^{71} - 24 q^{72} + 188 q^{73} - 128 q^{76} + 128 q^{77} - 96 q^{78} - 36 q^{81} - 32 q^{82} - 192 q^{83} + 96 q^{86} + 48 q^{87} - 32 q^{88} + 288 q^{91} + 32 q^{92} + 96 q^{93} - 132 q^{97} + 124 q^{98}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 0 2.44949 −0.898979 0.898979i 2.00000 2.00000i 3.00000i 0
7.2 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 0 −2.44949 8.89898 + 8.89898i 2.00000 2.00000i 3.00000i 0
43.1 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 0 2.44949 −0.898979 + 0.898979i 2.00000 + 2.00000i 3.00000i 0
43.2 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 0 −2.44949 8.89898 8.89898i 2.00000 + 2.00000i 3.00000i 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
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 150.3.f.b 4
3.b odd 2 1 450.3.g.j 4
4.b odd 2 1 1200.3.bg.d 4
5.b even 2 1 30.3.f.a 4
5.c odd 4 1 30.3.f.a 4
5.c odd 4 1 inner 150.3.f.b 4
15.d odd 2 1 90.3.g.d 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 1 240.3.bg.b 4
20.e even 4 1 240.3.bg.b 4
20.e even 4 1 1200.3.bg.d 4
40.e odd 2 1 960.3.bg.g 4
40.f even 2 1 960.3.bg.e 4
40.i odd 4 1 960.3.bg.e 4
40.k even 4 1 960.3.bg.g 4
60.h even 2 1 720.3.bh.i 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 5.b even 2 1
30.3.f.a 4 5.c odd 4 1
90.3.g.d 4 15.d odd 2 1
90.3.g.d 4 15.e even 4 1
150.3.f.b 4 1.a even 1 1 trivial
150.3.f.b 4 5.c odd 4 1 inner
240.3.bg.b 4 20.d odd 2 1
240.3.bg.b 4 20.e even 4 1
450.3.g.j 4 3.b odd 2 1
450.3.g.j 4 15.e even 4 1
720.3.bh.i 4 60.h even 2 1
720.3.bh.i 4 60.l odd 4 1
960.3.bg.e 4 40.f even 2 1
960.3.bg.e 4 40.i odd 4 1
960.3.bg.g 4 40.e odd 2 1
960.3.bg.g 4 40.k even 4 1
1200.3.bg.d 4 4.b odd 2 1
1200.3.bg.d 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16T_{7}^{3} + 128T_{7}^{2} + 256T_{7} + 256 \) acting on \(S_{3}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 80)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + \cdots + 30276 \) Copy content Toggle raw display
$17$ \( T^{4} + 44 T^{3} + \cdots + 37636 \) Copy content Toggle raw display
$19$ \( T^{4} + 704 T^{2} + 25600 \) Copy content Toggle raw display
$23$ \( T^{4} - 16 T^{3} + \cdots + 160000 \) 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
$43$ \( T^{4} + 48 T^{3} + \cdots + 831744 \) Copy content Toggle raw display
$47$ \( T^{4} + 96 T^{3} + \cdots + 518400 \) Copy content Toggle raw display
$53$ \( T^{4} + 100 T^{3} + \cdots + 6959044 \) 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
$67$ \( T^{4} - 16 T^{3} + \cdots + 14868736 \) 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
$79$ \( T^{4} + 32736 T^{2} + 258566400 \) Copy content Toggle raw display
$83$ \( T^{4} + 192 T^{3} + \cdots + 5089536 \) Copy content Toggle raw display
$89$ \( T^{4} + 17216 T^{2} + 47334400 \) Copy content Toggle raw display
$97$ \( T^{4} + 132 T^{3} + \cdots + 6874884 \) Copy content Toggle raw display
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