Properties

Label 150.2.g.b
Level $150$
Weight $2$
Character orbit 150.g
Analytic conductor $1.198$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,2,Mod(31,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.g (of order \(5\), degree \(4\), 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: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i 1.80902 + 1.31433i −0.809017 0.587785i 2.00000 0.809017 + 0.587785i 0.309017 0.951057i 0.690983 2.12663i
61.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.690983 + 2.12663i 0.309017 + 0.951057i 2.00000 −0.309017 0.951057i −0.809017 0.587785i 1.80902 + 1.31433i
91.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.690983 2.12663i 0.309017 0.951057i 2.00000 −0.309017 + 0.951057i −0.809017 + 0.587785i 1.80902 1.31433i
121.1 −0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i 1.80902 1.31433i −0.809017 + 0.587785i 2.00000 0.809017 0.587785i 0.309017 + 0.951057i 0.690983 + 2.12663i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.g.b 4
3.b odd 2 1 450.2.h.b 4
5.b even 2 1 750.2.g.a 4
5.c odd 4 2 750.2.h.a 8
25.d even 5 1 inner 150.2.g.b 4
25.d even 5 1 3750.2.a.b 2
25.e even 10 1 750.2.g.a 4
25.e even 10 1 3750.2.a.g 2
25.f odd 20 2 750.2.h.a 8
25.f odd 20 2 3750.2.c.c 4
75.j odd 10 1 450.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.b 4 1.a even 1 1 trivial
150.2.g.b 4 25.d even 5 1 inner
450.2.h.b 4 3.b odd 2 1
450.2.h.b 4 75.j odd 10 1
750.2.g.a 4 5.b even 2 1
750.2.g.a 4 25.e even 10 1
750.2.h.a 8 5.c odd 4 2
750.2.h.a 8 25.f odd 20 2
3750.2.a.b 2 25.d even 5 1
3750.2.a.g 2 25.e even 10 1
3750.2.c.c 4 25.f odd 20 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$7$ \( (T - 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 36 T^{2} + 81 T + 81 \) Copy content Toggle raw display
$17$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 40 T^{2} - 200 T + 400 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( T^{4} - 13 T^{3} + 64 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$53$ \( T^{4} + 21 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 64 T^{2} - 247 T + 361 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 144 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( T^{4} - 28 T^{3} + 384 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$73$ \( T^{4} - 14 T^{3} + 76 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$89$ \( T^{4} + 5 T^{3} + 10 T^{2} + 25 \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + 144 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
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