Properties

Label 3844.1.l.c
Level $3844$
Weight $1$
Character orbit 3844.l
Analytic conductor $1.918$
Analytic rank $0$
Dimension $8$
Projective image $A_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3844 = 2^{2} \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3844.l (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91840590856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 124)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.15376.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{20}^{7} q^{2} - \zeta_{20}^{3} q^{3} - \zeta_{20}^{4} q^{4} + q^{5} - q^{6} + \zeta_{20}^{9} q^{7} - \zeta_{20} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{20}^{7} q^{2} - \zeta_{20}^{3} q^{3} - \zeta_{20}^{4} q^{4} + q^{5} - q^{6} + \zeta_{20}^{9} q^{7} - \zeta_{20} q^{8} - \zeta_{20}^{7} q^{10} + \zeta_{20}^{9} q^{11} + \zeta_{20}^{7} q^{12} + \zeta_{20}^{8} q^{13} + \zeta_{20}^{6} q^{14} - \zeta_{20}^{3} q^{15} + \zeta_{20}^{8} q^{16} + \zeta_{20}^{6} q^{17} + \zeta_{20}^{7} q^{19} - \zeta_{20}^{4} q^{20} + \zeta_{20}^{2} q^{21} + \zeta_{20}^{6} q^{22} + \zeta_{20}^{4} q^{24} + \zeta_{20}^{5} q^{26} + \zeta_{20}^{9} q^{27} + \zeta_{20}^{3} q^{28} - q^{30} + \zeta_{20}^{5} q^{32} + \zeta_{20}^{2} q^{33} + \zeta_{20}^{3} q^{34} + \zeta_{20}^{9} q^{35} - q^{37} + \zeta_{20}^{4} q^{38} + \zeta_{20} q^{39} - \zeta_{20} q^{40} - \zeta_{20}^{2} q^{41} - \zeta_{20}^{9} q^{42} - \zeta_{20}^{7} q^{43} + \zeta_{20}^{3} q^{44} + \zeta_{20} q^{48} - \zeta_{20}^{9} q^{51} + \zeta_{20}^{2} q^{52} - \zeta_{20}^{6} q^{53} + \zeta_{20}^{6} q^{54} + \zeta_{20}^{9} q^{55} + q^{56} + q^{57} - \zeta_{20}^{3} q^{59} + \zeta_{20}^{7} q^{60} + \zeta_{20}^{2} q^{64} + \zeta_{20}^{8} q^{65} - \zeta_{20}^{9} q^{66} - \zeta_{20}^{5} q^{67} + q^{68} + \zeta_{20}^{6} q^{70} + \zeta_{20} q^{71} - \zeta_{20}^{4} q^{73} + \zeta_{20}^{7} q^{74} + \zeta_{20} q^{76} - \zeta_{20}^{8} q^{77} - \zeta_{20}^{8} q^{78} - \zeta_{20} q^{79} + \zeta_{20}^{8} q^{80} + \zeta_{20}^{2} q^{81} + \zeta_{20}^{9} q^{82} + \zeta_{20}^{7} q^{83} - \zeta_{20}^{6} q^{84} + \zeta_{20}^{6} q^{85} - \zeta_{20}^{4} q^{86} + q^{88} - \zeta_{20}^{7} q^{91} + \zeta_{20}^{7} q^{95} - \zeta_{20}^{8} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} + 8 q^{5} - 8 q^{6} - 2 q^{13} + 2 q^{14} - 2 q^{16} + 2 q^{17} + 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{24} - 8 q^{30} + 2 q^{33} - 8 q^{37} - 2 q^{38} - 2 q^{41} + 2 q^{52} - 2 q^{53} + 2 q^{54} + 8 q^{56} + 8 q^{57} + 2 q^{64} - 2 q^{65} + 8 q^{68} + 2 q^{70} + 2 q^{73} + 2 q^{77} + 2 q^{78} - 2 q^{80} + 2 q^{81} - 2 q^{84} + 2 q^{85} + 2 q^{86} + 8 q^{88} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3844\mathbb{Z}\right)^\times\).

\(n\) \(1923\) \(1925\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{6}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
531.1
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
−0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i 1.00000 −1.00000 −0.587785 0.809017i −0.587785 + 0.809017i 0 −0.951057 + 0.309017i
531.2 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i 1.00000 −1.00000 0.587785 + 0.809017i 0.587785 0.809017i 0 0.951057 0.309017i
1335.1 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 1.00000 −1.00000 0.951057 0.309017i 0.951057 + 0.309017i 0 −0.587785 + 0.809017i
1335.2 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 1.00000 −1.00000 −0.951057 + 0.309017i −0.951057 0.309017i 0 0.587785 0.809017i
3271.1 −0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 1.00000 −1.00000 0.951057 + 0.309017i 0.951057 0.309017i 0 −0.587785 0.809017i
3271.2 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.00000 −1.00000 −0.951057 0.309017i −0.951057 + 0.309017i 0 0.587785 + 0.809017i
3511.1 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i 1.00000 −1.00000 −0.587785 + 0.809017i −0.587785 0.809017i 0 −0.951057 0.309017i
3511.2 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i 1.00000 −1.00000 0.587785 0.809017i 0.587785 + 0.809017i 0 0.951057 + 0.309017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3511.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.d even 5 3 inner
124.l odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3844.1.l.c 8
4.b odd 2 1 inner 3844.1.l.c 8
31.b odd 2 1 3844.1.l.d 8
31.c even 3 2 3844.1.n.f 16
31.d even 5 1 3844.1.b.c 2
31.d even 5 3 inner 3844.1.l.c 8
31.e odd 6 2 3844.1.n.e 16
31.f odd 10 1 3844.1.b.d 2
31.f odd 10 3 3844.1.l.d 8
31.g even 15 2 3844.1.i.d 4
31.g even 15 6 3844.1.n.f 16
31.h odd 30 2 124.1.i.a 4
31.h odd 30 6 3844.1.n.e 16
93.p even 30 2 1116.1.x.a 4
124.d even 2 1 3844.1.l.d 8
124.g even 6 2 3844.1.n.e 16
124.i odd 6 2 3844.1.n.f 16
124.j even 10 1 3844.1.b.d 2
124.j even 10 3 3844.1.l.d 8
124.l odd 10 1 3844.1.b.c 2
124.l odd 10 3 inner 3844.1.l.c 8
124.n odd 30 2 3844.1.i.d 4
124.n odd 30 6 3844.1.n.f 16
124.p even 30 2 124.1.i.a 4
124.p even 30 6 3844.1.n.e 16
155.v odd 30 2 3100.1.z.a 4
155.x even 60 2 3100.1.t.a 4
155.x even 60 2 3100.1.t.b 4
248.bb even 30 2 1984.1.s.a 4
248.bf odd 30 2 1984.1.s.a 4
372.bc odd 30 2 1116.1.x.a 4
620.bo even 30 2 3100.1.z.a 4
620.bv odd 60 2 3100.1.t.a 4
620.bv odd 60 2 3100.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 31.h odd 30 2
124.1.i.a 4 124.p even 30 2
1116.1.x.a 4 93.p even 30 2
1116.1.x.a 4 372.bc odd 30 2
1984.1.s.a 4 248.bb even 30 2
1984.1.s.a 4 248.bf odd 30 2
3100.1.t.a 4 155.x even 60 2
3100.1.t.a 4 620.bv odd 60 2
3100.1.t.b 4 155.x even 60 2
3100.1.t.b 4 620.bv odd 60 2
3100.1.z.a 4 155.v odd 30 2
3100.1.z.a 4 620.bo even 30 2
3844.1.b.c 2 31.d even 5 1
3844.1.b.c 2 124.l odd 10 1
3844.1.b.d 2 31.f odd 10 1
3844.1.b.d 2 124.j even 10 1
3844.1.i.d 4 31.g even 15 2
3844.1.i.d 4 124.n odd 30 2
3844.1.l.c 8 1.a even 1 1 trivial
3844.1.l.c 8 4.b odd 2 1 inner
3844.1.l.c 8 31.d even 5 3 inner
3844.1.l.c 8 124.l odd 10 3 inner
3844.1.l.d 8 31.b odd 2 1
3844.1.l.d 8 31.f odd 10 3
3844.1.l.d 8 124.d even 2 1
3844.1.l.d 8 124.j even 10 3
3844.1.n.e 16 31.e odd 6 2
3844.1.n.e 16 31.h odd 30 6
3844.1.n.e 16 124.g even 6 2
3844.1.n.e 16 124.p even 30 6
3844.1.n.f 16 31.c even 3 2
3844.1.n.f 16 31.g even 15 6
3844.1.n.f 16 124.i odd 6 2
3844.1.n.f 16 124.n odd 30 6

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3844, [\chi])\):

\( T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{4} + T_{13}^{3} + T_{13}^{2} + T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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