Properties

Label 42.2.e.b
Level $42$
Weight $2$
Character orbit 42.e
Analytic conductor $0.335$
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] = [42,2,Mod(25,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
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("42.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), 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: \(0.335371688489\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{6} + (\zeta_{6} + 2) q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{3} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{6} + (\zeta_{6} + 2) q^{7} - q^{8} + (\zeta_{6} - 1) q^{9} + 3 \zeta_{6} q^{10} - 3 \zeta_{6} q^{11} + (\zeta_{6} - 1) q^{12} - 4 q^{13} + ( - 2 \zeta_{6} + 3) q^{14} + 3 q^{15} + (\zeta_{6} - 1) q^{16} + \zeta_{6} q^{18} + ( - 4 \zeta_{6} + 4) q^{19} + 3 q^{20} + ( - 3 \zeta_{6} + 1) q^{21} - 3 q^{22} + \zeta_{6} q^{24} - 4 \zeta_{6} q^{25} + (4 \zeta_{6} - 4) q^{26} + q^{27} + ( - 3 \zeta_{6} + 1) q^{28} + 9 q^{29} + ( - 3 \zeta_{6} + 3) q^{30} + \zeta_{6} q^{31} + \zeta_{6} q^{32} + (3 \zeta_{6} - 3) q^{33} + (6 \zeta_{6} - 9) q^{35} + q^{36} + (8 \zeta_{6} - 8) q^{37} - 4 \zeta_{6} q^{38} + 4 \zeta_{6} q^{39} + ( - 3 \zeta_{6} + 3) q^{40} + ( - \zeta_{6} - 2) q^{42} - 10 q^{43} + (3 \zeta_{6} - 3) q^{44} - 3 \zeta_{6} q^{45} + ( - 6 \zeta_{6} + 6) q^{47} + q^{48} + (5 \zeta_{6} + 3) q^{49} - 4 q^{50} + 4 \zeta_{6} q^{52} + 3 \zeta_{6} q^{53} + ( - \zeta_{6} + 1) q^{54} + 9 q^{55} + ( - \zeta_{6} - 2) q^{56} - 4 q^{57} + ( - 9 \zeta_{6} + 9) q^{58} - 3 \zeta_{6} q^{59} - 3 \zeta_{6} q^{60} + ( - 10 \zeta_{6} + 10) q^{61} + q^{62} + (2 \zeta_{6} - 3) q^{63} + q^{64} + ( - 12 \zeta_{6} + 12) q^{65} + 3 \zeta_{6} q^{66} + 10 \zeta_{6} q^{67} + (9 \zeta_{6} - 3) q^{70} - 6 q^{71} + ( - \zeta_{6} + 1) q^{72} - 2 \zeta_{6} q^{73} + 8 \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} - 4 q^{76} + ( - 9 \zeta_{6} + 3) q^{77} + 4 q^{78} + ( - \zeta_{6} + 1) q^{79} - 3 \zeta_{6} q^{80} - \zeta_{6} q^{81} - 9 q^{83} + (2 \zeta_{6} - 3) q^{84} + (10 \zeta_{6} - 10) q^{86} - 9 \zeta_{6} q^{87} + 3 \zeta_{6} q^{88} + (6 \zeta_{6} - 6) q^{89} - 3 q^{90} + ( - 4 \zeta_{6} - 8) q^{91} + ( - \zeta_{6} + 1) q^{93} - 6 \zeta_{6} q^{94} + 12 \zeta_{6} q^{95} + ( - \zeta_{6} + 1) q^{96} - q^{97} + ( - 3 \zeta_{6} + 8) q^{98} + 3 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - 3 q^{5} - 2 q^{6} + 5 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} - 3 q^{5} - 2 q^{6} + 5 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} - 3 q^{11} - q^{12} - 8 q^{13} + 4 q^{14} + 6 q^{15} - q^{16} + q^{18} + 4 q^{19} + 6 q^{20} - q^{21} - 6 q^{22} + q^{24} - 4 q^{25} - 4 q^{26} + 2 q^{27} - q^{28} + 18 q^{29} + 3 q^{30} + q^{31} + q^{32} - 3 q^{33} - 12 q^{35} + 2 q^{36} - 8 q^{37} - 4 q^{38} + 4 q^{39} + 3 q^{40} - 5 q^{42} - 20 q^{43} - 3 q^{44} - 3 q^{45} + 6 q^{47} + 2 q^{48} + 11 q^{49} - 8 q^{50} + 4 q^{52} + 3 q^{53} + q^{54} + 18 q^{55} - 5 q^{56} - 8 q^{57} + 9 q^{58} - 3 q^{59} - 3 q^{60} + 10 q^{61} + 2 q^{62} - 4 q^{63} + 2 q^{64} + 12 q^{65} + 3 q^{66} + 10 q^{67} + 3 q^{70} - 12 q^{71} + q^{72} - 2 q^{73} + 8 q^{74} - 4 q^{75} - 8 q^{76} - 3 q^{77} + 8 q^{78} + q^{79} - 3 q^{80} - q^{81} - 18 q^{83} - 4 q^{84} - 10 q^{86} - 9 q^{87} + 3 q^{88} - 6 q^{89} - 6 q^{90} - 20 q^{91} + q^{93} - 6 q^{94} + 12 q^{95} + q^{96} - 2 q^{97} + 13 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.50000 2.59808i −1.00000 2.50000 0.866025i −1.00000 −0.500000 0.866025i 1.50000 2.59808i
37.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.50000 + 2.59808i −1.00000 2.50000 + 0.866025i −1.00000 −0.500000 + 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.2.e.b 2
3.b odd 2 1 126.2.g.b 2
4.b odd 2 1 336.2.q.d 2
5.b even 2 1 1050.2.i.e 2
5.c odd 4 2 1050.2.o.b 4
7.b odd 2 1 294.2.e.f 2
7.c even 3 1 inner 42.2.e.b 2
7.c even 3 1 294.2.a.d 1
7.d odd 6 1 294.2.a.a 1
7.d odd 6 1 294.2.e.f 2
8.b even 2 1 1344.2.q.v 2
8.d odd 2 1 1344.2.q.j 2
9.c even 3 1 1134.2.e.a 2
9.c even 3 1 1134.2.h.p 2
9.d odd 6 1 1134.2.e.p 2
9.d odd 6 1 1134.2.h.a 2
12.b even 2 1 1008.2.s.n 2
21.c even 2 1 882.2.g.b 2
21.g even 6 1 882.2.a.k 1
21.g even 6 1 882.2.g.b 2
21.h odd 6 1 126.2.g.b 2
21.h odd 6 1 882.2.a.g 1
28.d even 2 1 2352.2.q.m 2
28.f even 6 1 2352.2.a.n 1
28.f even 6 1 2352.2.q.m 2
28.g odd 6 1 336.2.q.d 2
28.g odd 6 1 2352.2.a.m 1
35.i odd 6 1 7350.2.a.cw 1
35.j even 6 1 1050.2.i.e 2
35.j even 6 1 7350.2.a.ce 1
35.l odd 12 2 1050.2.o.b 4
56.j odd 6 1 9408.2.a.db 1
56.k odd 6 1 1344.2.q.j 2
56.k odd 6 1 9408.2.a.bu 1
56.m even 6 1 9408.2.a.bm 1
56.p even 6 1 1344.2.q.v 2
56.p even 6 1 9408.2.a.d 1
63.g even 3 1 1134.2.e.a 2
63.h even 3 1 1134.2.h.p 2
63.j odd 6 1 1134.2.h.a 2
63.n odd 6 1 1134.2.e.p 2
84.j odd 6 1 7056.2.a.bz 1
84.n even 6 1 1008.2.s.n 2
84.n even 6 1 7056.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 1.a even 1 1 trivial
42.2.e.b 2 7.c even 3 1 inner
126.2.g.b 2 3.b odd 2 1
126.2.g.b 2 21.h odd 6 1
294.2.a.a 1 7.d odd 6 1
294.2.a.d 1 7.c even 3 1
294.2.e.f 2 7.b odd 2 1
294.2.e.f 2 7.d odd 6 1
336.2.q.d 2 4.b odd 2 1
336.2.q.d 2 28.g odd 6 1
882.2.a.g 1 21.h odd 6 1
882.2.a.k 1 21.g even 6 1
882.2.g.b 2 21.c even 2 1
882.2.g.b 2 21.g even 6 1
1008.2.s.n 2 12.b even 2 1
1008.2.s.n 2 84.n even 6 1
1050.2.i.e 2 5.b even 2 1
1050.2.i.e 2 35.j even 6 1
1050.2.o.b 4 5.c odd 4 2
1050.2.o.b 4 35.l odd 12 2
1134.2.e.a 2 9.c even 3 1
1134.2.e.a 2 63.g even 3 1
1134.2.e.p 2 9.d odd 6 1
1134.2.e.p 2 63.n odd 6 1
1134.2.h.a 2 9.d odd 6 1
1134.2.h.a 2 63.j odd 6 1
1134.2.h.p 2 9.c even 3 1
1134.2.h.p 2 63.h even 3 1
1344.2.q.j 2 8.d odd 2 1
1344.2.q.j 2 56.k odd 6 1
1344.2.q.v 2 8.b even 2 1
1344.2.q.v 2 56.p even 6 1
2352.2.a.m 1 28.g odd 6 1
2352.2.a.n 1 28.f even 6 1
2352.2.q.m 2 28.d even 2 1
2352.2.q.m 2 28.f even 6 1
7056.2.a.g 1 84.n even 6 1
7056.2.a.bz 1 84.j odd 6 1
7350.2.a.ce 1 35.j even 6 1
7350.2.a.cw 1 35.i odd 6 1
9408.2.a.d 1 56.p even 6 1
9408.2.a.bm 1 56.m even 6 1
9408.2.a.bu 1 56.k odd 6 1
9408.2.a.db 1 56.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T + 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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