Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3844 = 2^{2} \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3844.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91840590856\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

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

\(n\) \(1923\) \(1925\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.00000i 0.866025 + 0.500000i −1.00000 −0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
439.2 1.00000i −0.866025 0.500000i −1.00000 −0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
2443.1 1.00000i −0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
2443.2 1.00000i 0.866025 0.500000i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.c even 3 1 inner
124.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3844.1.i.d 4
4.b odd 2 1 inner 3844.1.i.d 4
31.b odd 2 1 124.1.i.a 4
31.c even 3 1 3844.1.b.c 2
31.c even 3 1 inner 3844.1.i.d 4
31.d even 5 4 3844.1.n.f 16
31.e odd 6 1 124.1.i.a 4
31.e odd 6 1 3844.1.b.d 2
31.f odd 10 4 3844.1.n.e 16
31.g even 15 4 3844.1.l.c 8
31.g even 15 4 3844.1.n.f 16
31.h odd 30 4 3844.1.l.d 8
31.h odd 30 4 3844.1.n.e 16
93.c even 2 1 1116.1.x.a 4
93.g even 6 1 1116.1.x.a 4
124.d even 2 1 124.1.i.a 4
124.g even 6 1 124.1.i.a 4
124.g even 6 1 3844.1.b.d 2
124.i odd 6 1 3844.1.b.c 2
124.i odd 6 1 inner 3844.1.i.d 4
124.j even 10 4 3844.1.n.e 16
124.l odd 10 4 3844.1.n.f 16
124.n odd 30 4 3844.1.l.c 8
124.n odd 30 4 3844.1.n.f 16
124.p even 30 4 3844.1.l.d 8
124.p even 30 4 3844.1.n.e 16
155.c odd 2 1 3100.1.z.a 4
155.f even 4 1 3100.1.t.a 4
155.f even 4 1 3100.1.t.b 4
155.i odd 6 1 3100.1.z.a 4
155.p even 12 1 3100.1.t.a 4
155.p even 12 1 3100.1.t.b 4
248.b even 2 1 1984.1.s.a 4
248.g odd 2 1 1984.1.s.a 4
248.l odd 6 1 1984.1.s.a 4
248.q even 6 1 1984.1.s.a 4
372.b odd 2 1 1116.1.x.a 4
372.q odd 6 1 1116.1.x.a 4
620.e even 2 1 3100.1.z.a 4
620.m odd 4 1 3100.1.t.a 4
620.m odd 4 1 3100.1.t.b 4
620.r even 6 1 3100.1.z.a 4
620.bc odd 12 1 3100.1.t.a 4
620.bc odd 12 1 3100.1.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 31.b odd 2 1
124.1.i.a 4 31.e odd 6 1
124.1.i.a 4 124.d even 2 1
124.1.i.a 4 124.g even 6 1
1116.1.x.a 4 93.c even 2 1
1116.1.x.a 4 93.g even 6 1
1116.1.x.a 4 372.b odd 2 1
1116.1.x.a 4 372.q odd 6 1
1984.1.s.a 4 248.b even 2 1
1984.1.s.a 4 248.g odd 2 1
1984.1.s.a 4 248.l odd 6 1
1984.1.s.a 4 248.q even 6 1
3100.1.t.a 4 155.f even 4 1
3100.1.t.a 4 155.p even 12 1
3100.1.t.a 4 620.m odd 4 1
3100.1.t.a 4 620.bc odd 12 1
3100.1.t.b 4 155.f even 4 1
3100.1.t.b 4 155.p even 12 1
3100.1.t.b 4 620.m odd 4 1
3100.1.t.b 4 620.bc odd 12 1
3100.1.z.a 4 155.c odd 2 1
3100.1.z.a 4 155.i odd 6 1
3100.1.z.a 4 620.e even 2 1
3100.1.z.a 4 620.r even 6 1
3844.1.b.c 2 31.c even 3 1
3844.1.b.c 2 124.i odd 6 1
3844.1.b.d 2 31.e odd 6 1
3844.1.b.d 2 124.g even 6 1
3844.1.i.d 4 1.a even 1 1 trivial
3844.1.i.d 4 4.b odd 2 1 inner
3844.1.i.d 4 31.c even 3 1 inner
3844.1.i.d 4 124.i odd 6 1 inner
3844.1.l.c 8 31.g even 15 4
3844.1.l.c 8 124.n odd 30 4
3844.1.l.d 8 31.h odd 30 4
3844.1.l.d 8 124.p even 30 4
3844.1.n.e 16 31.f odd 10 4
3844.1.n.e 16 31.h odd 30 4
3844.1.n.e 16 124.j even 10 4
3844.1.n.e 16 124.p even 30 4
3844.1.n.f 16 31.d even 5 4
3844.1.n.f 16 31.g even 15 4
3844.1.n.f 16 124.l odd 10 4
3844.1.n.f 16 124.n odd 30 4

Hecke kernels

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

\( T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} - T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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