Properties

Label 150.12.a.f
Level $150$
Weight $12$
Character orbit 150.a
Self dual yes
Analytic conductor $115.251$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,12,Mod(1,150)]
 
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([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

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: yes
Analytic conductor: \(115.251477084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 32 q^{2} - 243 q^{3} + 1024 q^{4} - 7776 q^{6} + 50008 q^{7} + 32768 q^{8} + 59049 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 q^{2} - 243 q^{3} + 1024 q^{4} - 7776 q^{6} + 50008 q^{7} + 32768 q^{8} + 59049 q^{9} - 531420 q^{11} - 248832 q^{12} - 1332566 q^{13} + 1600256 q^{14} + 1048576 q^{16} + 5109678 q^{17} + 1889568 q^{18} + 2901404 q^{19} - 12151944 q^{21} - 17005440 q^{22} - 30597000 q^{23} - 7962624 q^{24} - 42642112 q^{26} - 14348907 q^{27} + 51208192 q^{28} - 77006634 q^{29} - 239418352 q^{31} + 33554432 q^{32} + 129135060 q^{33} + 163509696 q^{34} + 60466176 q^{36} + 785041666 q^{37} + 92844928 q^{38} + 323813538 q^{39} + 411252954 q^{41} - 388862208 q^{42} - 351233348 q^{43} - 544174080 q^{44} - 979104000 q^{46} - 95821680 q^{47} - 254803968 q^{48} + 523473321 q^{49} - 1241651754 q^{51} - 1364547584 q^{52} + 1465857378 q^{53} - 459165024 q^{54} + 1638662144 q^{56} - 705041172 q^{57} - 2464212288 q^{58} + 5621152020 q^{59} - 10473587770 q^{61} - 7661387264 q^{62} + 2952922392 q^{63} + 1073741824 q^{64} + 4132321920 q^{66} - 4515307532 q^{67} + 5232310272 q^{68} + 7435071000 q^{69} - 8509579560 q^{71} + 1934917632 q^{72} - 2012496986 q^{73} + 25121333312 q^{74} + 2971037696 q^{76} - 26575251360 q^{77} + 10362033216 q^{78} - 22238409568 q^{79} + 3486784401 q^{81} + 13160094528 q^{82} - 6328647516 q^{83} - 12443590656 q^{84} - 11239467136 q^{86} + 18712612062 q^{87} - 17413570560 q^{88} - 50123706678 q^{89} - 66638960528 q^{91} - 31331328000 q^{92} + 58178659536 q^{93} - 3066293760 q^{94} - 8153726976 q^{96} - 94805961314 q^{97} + 16751146272 q^{98} - 31379819580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
32.0000 −243.000 1024.00 0 −7776.00 50008.0 32768.0 59049.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.12.a.f 1
5.b even 2 1 6.12.a.b 1
5.c odd 4 2 150.12.c.b 2
15.d odd 2 1 18.12.a.e 1
20.d odd 2 1 48.12.a.a 1
40.e odd 2 1 192.12.a.t 1
40.f even 2 1 192.12.a.j 1
45.h odd 6 2 162.12.c.a 2
45.j even 6 2 162.12.c.j 2
60.h even 2 1 144.12.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.b 1 5.b even 2 1
18.12.a.e 1 15.d odd 2 1
48.12.a.a 1 20.d odd 2 1
144.12.a.o 1 60.h even 2 1
150.12.a.f 1 1.a even 1 1 trivial
150.12.c.b 2 5.c odd 4 2
162.12.c.a 2 45.h odd 6 2
162.12.c.j 2 45.j even 6 2
192.12.a.j 1 40.f even 2 1
192.12.a.t 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 50008 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(150))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 32 \) Copy content Toggle raw display
$3$ \( T + 243 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 50008 \) Copy content Toggle raw display
$11$ \( T + 531420 \) Copy content Toggle raw display
$13$ \( T + 1332566 \) Copy content Toggle raw display
$17$ \( T - 5109678 \) Copy content Toggle raw display
$19$ \( T - 2901404 \) Copy content Toggle raw display
$23$ \( T + 30597000 \) Copy content Toggle raw display
$29$ \( T + 77006634 \) Copy content Toggle raw display
$31$ \( T + 239418352 \) Copy content Toggle raw display
$37$ \( T - 785041666 \) Copy content Toggle raw display
$41$ \( T - 411252954 \) Copy content Toggle raw display
$43$ \( T + 351233348 \) Copy content Toggle raw display
$47$ \( T + 95821680 \) Copy content Toggle raw display
$53$ \( T - 1465857378 \) Copy content Toggle raw display
$59$ \( T - 5621152020 \) Copy content Toggle raw display
$61$ \( T + 10473587770 \) Copy content Toggle raw display
$67$ \( T + 4515307532 \) Copy content Toggle raw display
$71$ \( T + 8509579560 \) Copy content Toggle raw display
$73$ \( T + 2012496986 \) Copy content Toggle raw display
$79$ \( T + 22238409568 \) Copy content Toggle raw display
$83$ \( T + 6328647516 \) Copy content Toggle raw display
$89$ \( T + 50123706678 \) Copy content Toggle raw display
$97$ \( T + 94805961314 \) Copy content Toggle raw display
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