Properties

Label 3100.1.t.b
Level $3100$
Weight $1$
Character orbit 3100.t
Analytic conductor $1.547$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3100 = 2^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3100.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.54710153916\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + q^{2} - \zeta_{12}^{2} q^{3} + q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \zeta_{12}^{2} q^{3} + q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} + q^{8} - \zeta_{12} q^{11} - \zeta_{12}^{2} q^{12} - \zeta_{12} q^{13} - \zeta_{12}^{2} q^{14} + q^{16} - \zeta_{12}^{5} q^{17} + \zeta_{12}^{5} q^{19} + \zeta_{12}^{4} q^{21} - \zeta_{12} q^{22} - \zeta_{12}^{2} q^{24} - \zeta_{12} q^{26} - q^{27} - \zeta_{12}^{2} q^{28} + \zeta_{12}^{3} q^{31} + q^{32} + \zeta_{12}^{3} q^{33} - \zeta_{12}^{5} q^{34} - \zeta_{12}^{5} q^{37} + \zeta_{12}^{5} q^{38} + \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{4} q^{42} - \zeta_{12}^{2} q^{43} - \zeta_{12} q^{44} - \zeta_{12}^{2} q^{48} - \zeta_{12} q^{51} - \zeta_{12} q^{52} - \zeta_{12} q^{53} - q^{54} - \zeta_{12}^{2} q^{56} + \zeta_{12} q^{57} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{3} q^{62} + q^{64} + \zeta_{12}^{3} q^{66} - \zeta_{12}^{4} q^{67} - \zeta_{12}^{5} q^{68} + \zeta_{12} q^{71} + \zeta_{12} q^{73} - \zeta_{12}^{5} q^{74} + \zeta_{12}^{5} q^{76} + \zeta_{12}^{3} q^{77} + \zeta_{12}^{3} q^{78} - \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{81} + \zeta_{12}^{4} q^{82} - \zeta_{12}^{4} q^{83} + \zeta_{12}^{4} q^{84} - \zeta_{12}^{2} q^{86} - \zeta_{12} q^{88} + \zeta_{12}^{3} q^{91} - \zeta_{12}^{5} q^{93} - \zeta_{12}^{2} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{12} - 2 q^{14} + 4 q^{16} - 2 q^{21} - 2 q^{24} - 4 q^{27} - 2 q^{28} + 4 q^{32} - 2 q^{41} - 2 q^{42} - 2 q^{43} - 2 q^{48} - 4 q^{54} - 2 q^{56} + 4 q^{64} + 2 q^{67} + 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times\).

\(n\) \(1551\) \(1801\) \(2977\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i −0.500000 0.866025i 1.00000 0 0
1699.2 1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i −0.500000 0.866025i 1.00000 0 0
2299.1 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 0
2299.2 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner
31.c even 3 1 inner
620.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3100.1.t.b 4
4.b odd 2 1 3100.1.t.a 4
5.b even 2 1 3100.1.t.a 4
5.c odd 4 1 124.1.i.a 4
5.c odd 4 1 3100.1.z.a 4
15.e even 4 1 1116.1.x.a 4
20.d odd 2 1 inner 3100.1.t.b 4
20.e even 4 1 124.1.i.a 4
20.e even 4 1 3100.1.z.a 4
31.c even 3 1 inner 3100.1.t.b 4
40.i odd 4 1 1984.1.s.a 4
40.k even 4 1 1984.1.s.a 4
60.l odd 4 1 1116.1.x.a 4
124.i odd 6 1 3100.1.t.a 4
155.f even 4 1 3844.1.i.d 4
155.j even 6 1 3100.1.t.a 4
155.o odd 12 1 124.1.i.a 4
155.o odd 12 1 3100.1.z.a 4
155.o odd 12 1 3844.1.b.d 2
155.p even 12 1 3844.1.b.c 2
155.p even 12 1 3844.1.i.d 4
155.r even 20 4 3844.1.n.f 16
155.s odd 20 4 3844.1.n.e 16
155.w odd 60 4 3844.1.l.d 8
155.w odd 60 4 3844.1.n.e 16
155.x even 60 4 3844.1.l.c 8
155.x even 60 4 3844.1.n.f 16
465.be even 12 1 1116.1.x.a 4
620.m odd 4 1 3844.1.i.d 4
620.o odd 6 1 inner 3100.1.t.b 4
620.bc odd 12 1 3844.1.b.c 2
620.bc odd 12 1 3844.1.i.d 4
620.be even 12 1 124.1.i.a 4
620.be even 12 1 3100.1.z.a 4
620.be even 12 1 3844.1.b.d 2
620.bh odd 20 4 3844.1.n.f 16
620.bj even 20 4 3844.1.n.e 16
620.bt even 60 4 3844.1.l.d 8
620.bt even 60 4 3844.1.n.e 16
620.bv odd 60 4 3844.1.l.c 8
620.bv odd 60 4 3844.1.n.f 16
1240.ch even 12 1 1984.1.s.a 4
1240.cj odd 12 1 1984.1.s.a 4
1860.cf odd 12 1 1116.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 5.c odd 4 1
124.1.i.a 4 20.e even 4 1
124.1.i.a 4 155.o odd 12 1
124.1.i.a 4 620.be even 12 1
1116.1.x.a 4 15.e even 4 1
1116.1.x.a 4 60.l odd 4 1
1116.1.x.a 4 465.be even 12 1
1116.1.x.a 4 1860.cf odd 12 1
1984.1.s.a 4 40.i odd 4 1
1984.1.s.a 4 40.k even 4 1
1984.1.s.a 4 1240.ch even 12 1
1984.1.s.a 4 1240.cj odd 12 1
3100.1.t.a 4 4.b odd 2 1
3100.1.t.a 4 5.b even 2 1
3100.1.t.a 4 124.i odd 6 1
3100.1.t.a 4 155.j even 6 1
3100.1.t.b 4 1.a even 1 1 trivial
3100.1.t.b 4 20.d odd 2 1 inner
3100.1.t.b 4 31.c even 3 1 inner
3100.1.t.b 4 620.o odd 6 1 inner
3100.1.z.a 4 5.c odd 4 1
3100.1.z.a 4 20.e even 4 1
3100.1.z.a 4 155.o odd 12 1
3100.1.z.a 4 620.be even 12 1
3844.1.b.c 2 155.p even 12 1
3844.1.b.c 2 620.bc odd 12 1
3844.1.b.d 2 155.o odd 12 1
3844.1.b.d 2 620.be even 12 1
3844.1.i.d 4 155.f even 4 1
3844.1.i.d 4 155.p even 12 1
3844.1.i.d 4 620.m odd 4 1
3844.1.i.d 4 620.bc odd 12 1
3844.1.l.c 8 155.x even 60 4
3844.1.l.c 8 620.bv odd 60 4
3844.1.l.d 8 155.w odd 60 4
3844.1.l.d 8 620.bt even 60 4
3844.1.n.e 16 155.s odd 20 4
3844.1.n.e 16 155.w odd 60 4
3844.1.n.e 16 620.bj even 20 4
3844.1.n.e 16 620.bt even 60 4
3844.1.n.f 16 155.r even 20 4
3844.1.n.f 16 155.x even 60 4
3844.1.n.f 16 620.bh odd 20 4
3844.1.n.f 16 620.bv odd 60 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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