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Properties

Label 378.2.h.b
Level $378$
Weight $2$
Character orbit 378.h
Analytic conductor $3.018$
Analytic rank $0$
Dimension $2$
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] = [378,2,Mod(289,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.h (of order \(3\), degree \(2\), 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 q^{5} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + 3 q^{5} + ( - 3 \zeta_{6} + 1) q^{7} - q^{8} + ( - 3 \zeta_{6} + 3) q^{10} + 3 q^{11} + (5 \zeta_{6} - 5) q^{13} + ( - \zeta_{6} - 2) q^{14} + (\zeta_{6} - 1) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - 5 \zeta_{6} q^{19} - 3 \zeta_{6} q^{20} + ( - 3 \zeta_{6} + 3) q^{22} + 3 q^{23} + 4 q^{25} + 5 \zeta_{6} q^{26} + (2 \zeta_{6} - 3) q^{28} - 3 \zeta_{6} q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} - 3 \zeta_{6} q^{34} + ( - 9 \zeta_{6} + 3) q^{35} + 7 \zeta_{6} q^{37} - 5 q^{38} - 3 q^{40} + (9 \zeta_{6} - 9) q^{41} - 11 \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{46} + (3 \zeta_{6} - 8) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} + 5 q^{52} + (3 \zeta_{6} - 3) q^{53} + 9 q^{55} + (3 \zeta_{6} - 1) q^{56} - 3 q^{58} + 12 \zeta_{6} q^{59} + (2 \zeta_{6} - 2) q^{61} + 4 q^{62} + q^{64} + (15 \zeta_{6} - 15) q^{65} + 4 \zeta_{6} q^{67} - 3 q^{68} + ( - 3 \zeta_{6} - 6) q^{70} + (11 \zeta_{6} - 11) q^{73} + 7 q^{74} + (5 \zeta_{6} - 5) q^{76} + ( - 9 \zeta_{6} + 3) q^{77} + (8 \zeta_{6} - 8) q^{79} + (3 \zeta_{6} - 3) q^{80} + 9 \zeta_{6} q^{82} + 3 \zeta_{6} q^{83} + ( - 9 \zeta_{6} + 9) q^{85} - 11 q^{86} - 3 q^{88} + 15 \zeta_{6} q^{89} + (5 \zeta_{6} + 10) q^{91} - 3 \zeta_{6} q^{92} - 15 \zeta_{6} q^{95} + \zeta_{6} q^{97} + (8 \zeta_{6} - 5) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 6 q^{5} - q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 6 q^{5} - q^{7} - 2 q^{8} + 3 q^{10} + 6 q^{11} - 5 q^{13} - 5 q^{14} - q^{16} + 3 q^{17} - 5 q^{19} - 3 q^{20} + 3 q^{22} + 6 q^{23} + 8 q^{25} + 5 q^{26} - 4 q^{28} - 3 q^{29} + 4 q^{31} + q^{32} - 3 q^{34} - 3 q^{35} + 7 q^{37} - 10 q^{38} - 6 q^{40} - 9 q^{41} - 11 q^{43} - 3 q^{44} + 3 q^{46} - 13 q^{49} + 4 q^{50} + 10 q^{52} - 3 q^{53} + 18 q^{55} + q^{56} - 6 q^{58} + 12 q^{59} - 2 q^{61} + 8 q^{62} + 2 q^{64} - 15 q^{65} + 4 q^{67} - 6 q^{68} - 15 q^{70} - 11 q^{73} + 14 q^{74} - 5 q^{76} - 3 q^{77} - 8 q^{79} - 3 q^{80} + 9 q^{82} + 3 q^{83} + 9 q^{85} - 22 q^{86} - 6 q^{88} + 15 q^{89} + 25 q^{91} - 3 q^{92} - 15 q^{95} + q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 3.00000 0 −0.500000 + 2.59808i −1.00000 0 1.50000 + 2.59808i
361.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 3.00000 0 −0.500000 − 2.59808i −1.00000 0 1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.h.b 2
3.b odd 2 1 126.2.h.a yes 2
4.b odd 2 1 3024.2.t.f 2
7.b odd 2 1 2646.2.h.f 2
7.c even 3 1 378.2.e.a 2
7.c even 3 1 2646.2.f.e 2
7.d odd 6 1 2646.2.e.e 2
7.d odd 6 1 2646.2.f.i 2
9.c even 3 1 378.2.e.a 2
9.c even 3 1 1134.2.g.f 2
9.d odd 6 1 126.2.e.b ✓ 2
9.d odd 6 1 1134.2.g.d 2
12.b even 2 1 1008.2.t.c 2
21.c even 2 1 882.2.h.e 2
21.g even 6 1 882.2.e.h 2
21.g even 6 1 882.2.f.a 2
21.h odd 6 1 126.2.e.b ✓ 2
21.h odd 6 1 882.2.f.e 2
28.g odd 6 1 3024.2.q.a 2
36.f odd 6 1 3024.2.q.a 2
36.h even 6 1 1008.2.q.e 2
63.g even 3 1 inner 378.2.h.b 2
63.g even 3 1 7938.2.a.o 1
63.h even 3 1 1134.2.g.f 2
63.h even 3 1 2646.2.f.e 2
63.i even 6 1 882.2.f.a 2
63.j odd 6 1 882.2.f.e 2
63.j odd 6 1 1134.2.g.d 2
63.k odd 6 1 2646.2.h.f 2
63.k odd 6 1 7938.2.a.c 1
63.l odd 6 1 2646.2.e.e 2
63.n odd 6 1 126.2.h.a yes 2
63.n odd 6 1 7938.2.a.r 1
63.o even 6 1 882.2.e.h 2
63.s even 6 1 882.2.h.e 2
63.s even 6 1 7938.2.a.bd 1
63.t odd 6 1 2646.2.f.i 2
84.n even 6 1 1008.2.q.e 2
252.o even 6 1 1008.2.t.c 2
252.bl odd 6 1 3024.2.t.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b ✓ 2 9.d odd 6 1
126.2.e.b ✓ 2 21.h odd 6 1
126.2.h.a yes 2 3.b odd 2 1
126.2.h.a yes 2 63.n odd 6 1
378.2.e.a 2 7.c even 3 1
378.2.e.a 2 9.c even 3 1
378.2.h.b 2 1.a even 1 1 trivial
378.2.h.b 2 63.g even 3 1 inner
882.2.e.h 2 21.g even 6 1
882.2.e.h 2 63.o even 6 1
882.2.f.a 2 21.g even 6 1
882.2.f.a 2 63.i even 6 1
882.2.f.e 2 21.h odd 6 1
882.2.f.e 2 63.j odd 6 1
882.2.h.e 2 21.c even 2 1
882.2.h.e 2 63.s even 6 1
1008.2.q.e 2 36.h even 6 1
1008.2.q.e 2 84.n even 6 1
1008.2.t.c 2 12.b even 2 1
1008.2.t.c 2 252.o even 6 1
1134.2.g.d 2 9.d odd 6 1
1134.2.g.d 2 63.j odd 6 1
1134.2.g.f 2 9.c even 3 1
1134.2.g.f 2 63.h even 3 1
2646.2.e.e 2 7.d odd 6 1
2646.2.e.e 2 63.l odd 6 1
2646.2.f.e 2 7.c even 3 1
2646.2.f.e 2 63.h even 3 1
2646.2.f.i 2 7.d odd 6 1
2646.2.f.i 2 63.t odd 6 1
2646.2.h.f 2 7.b odd 2 1
2646.2.h.f 2 63.k odd 6 1
3024.2.q.a 2 28.g odd 6 1
3024.2.q.a 2 36.f odd 6 1
3024.2.t.f 2 4.b odd 2 1
3024.2.t.f 2 252.bl odd 6 1
7938.2.a.c 1 63.k odd 6 1
7938.2.a.o 1 63.g even 3 1
7938.2.a.r 1 63.n odd 6 1
7938.2.a.bd 1 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( (T - 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$89$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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