Properties

Label 3100.1.z.a
Level $3100$
Weight $1$
Character orbit 3100.z
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.z (of order \(6\), degree \(2\), 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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} + \zeta_{12}^{5} q^{3} - q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{3} q^{2} + \zeta_{12}^{5} q^{3} - q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12} q^{11} - \zeta_{12}^{5} q^{12} + \zeta_{12}^{4} q^{13} + \zeta_{12}^{2} q^{14} + q^{16} + \zeta_{12}^{2} q^{17} - \zeta_{12}^{5} q^{19} + \zeta_{12}^{4} q^{21} - \zeta_{12}^{4} q^{22} + \zeta_{12}^{2} q^{24} - \zeta_{12} q^{26} - \zeta_{12}^{3} q^{27} + \zeta_{12}^{5} q^{28} + \zeta_{12}^{3} q^{31} + \zeta_{12}^{3} q^{32} + q^{33} + \zeta_{12}^{5} q^{34} + \zeta_{12}^{2} q^{37} + \zeta_{12}^{2} q^{38} - \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} - \zeta_{12} q^{42} + \zeta_{12}^{5} q^{43} + \zeta_{12} q^{44} + \zeta_{12}^{5} q^{48} - \zeta_{12} q^{51} - \zeta_{12}^{4} q^{52} + \zeta_{12}^{4} q^{53} + q^{54} - \zeta_{12}^{2} q^{56} + \zeta_{12}^{4} q^{57} + \zeta_{12}^{5} q^{59} - q^{62} - q^{64} + \zeta_{12}^{3} q^{66} + \zeta_{12} q^{67} - \zeta_{12}^{2} q^{68} + \zeta_{12} q^{71} - \zeta_{12}^{4} q^{73} + \zeta_{12}^{5} q^{74} + \zeta_{12}^{5} q^{76} - q^{77} + q^{78} + \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{81} - \zeta_{12} q^{82} - \zeta_{12} q^{83} - \zeta_{12}^{4} q^{84} - \zeta_{12}^{2} q^{86} + \zeta_{12}^{4} q^{88} + \zeta_{12}^{3} q^{91} - \zeta_{12}^{2} q^{93} - \zeta_{12}^{2} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{6} - 2 q^{13} + 2 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{21} + 2 q^{22} + 2 q^{24} + 4 q^{33} + 2 q^{37} + 2 q^{38} - 2 q^{41} + 2 q^{52} - 2 q^{53} + 4 q^{54} - 2 q^{56} - 2 q^{57} - 4 q^{62} - 4 q^{64} - 2 q^{68} + 2 q^{73} - 4 q^{77} + 4 q^{78} + 2 q^{81} + 2 q^{84} - 2 q^{86} - 2 q^{88} - 2 q^{93} - 2 q^{96}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1451.1
−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 −0.500000 0.866025i −0.866025 + 0.500000i 1.00000i 0 0
1451.2 1.00000i −0.866025 + 0.500000i −1.00000 0 −0.500000 0.866025i 0.866025 0.500000i 1.00000i 0 0
2051.1 1.00000i −0.866025 0.500000i −1.00000 0 −0.500000 + 0.866025i 0.866025 + 0.500000i 1.00000i 0 0
2051.2 1.00000i 0.866025 + 0.500000i −1.00000 0 −0.500000 + 0.866025i −0.866025 0.500000i 1.00000i 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
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 3100.1.z.a 4
4.b odd 2 1 inner 3100.1.z.a 4
5.b even 2 1 124.1.i.a 4
5.c odd 4 1 3100.1.t.a 4
5.c odd 4 1 3100.1.t.b 4
15.d odd 2 1 1116.1.x.a 4
20.d odd 2 1 124.1.i.a 4
20.e even 4 1 3100.1.t.a 4
20.e even 4 1 3100.1.t.b 4
31.c even 3 1 inner 3100.1.z.a 4
40.e odd 2 1 1984.1.s.a 4
40.f even 2 1 1984.1.s.a 4
60.h even 2 1 1116.1.x.a 4
124.i odd 6 1 inner 3100.1.z.a 4
155.c odd 2 1 3844.1.i.d 4
155.i odd 6 1 3844.1.b.c 2
155.i odd 6 1 3844.1.i.d 4
155.j even 6 1 124.1.i.a 4
155.j even 6 1 3844.1.b.d 2
155.m odd 10 4 3844.1.n.f 16
155.n even 10 4 3844.1.n.e 16
155.o odd 12 1 3100.1.t.a 4
155.o odd 12 1 3100.1.t.b 4
155.u even 30 4 3844.1.l.d 8
155.u even 30 4 3844.1.n.e 16
155.v odd 30 4 3844.1.l.c 8
155.v odd 30 4 3844.1.n.f 16
465.u odd 6 1 1116.1.x.a 4
620.e even 2 1 3844.1.i.d 4
620.o odd 6 1 124.1.i.a 4
620.o odd 6 1 3844.1.b.d 2
620.r even 6 1 3844.1.b.c 2
620.r even 6 1 3844.1.i.d 4
620.v odd 10 4 3844.1.n.e 16
620.y even 10 4 3844.1.n.f 16
620.be even 12 1 3100.1.t.a 4
620.be even 12 1 3100.1.t.b 4
620.bo even 30 4 3844.1.l.c 8
620.bo even 30 4 3844.1.n.f 16
620.br odd 30 4 3844.1.l.d 8
620.br odd 30 4 3844.1.n.e 16
1240.bg even 6 1 1984.1.s.a 4
1240.bi odd 6 1 1984.1.s.a 4
1860.bc even 6 1 1116.1.x.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 5.b even 2 1
124.1.i.a 4 20.d odd 2 1
124.1.i.a 4 155.j even 6 1
124.1.i.a 4 620.o odd 6 1
1116.1.x.a 4 15.d odd 2 1
1116.1.x.a 4 60.h even 2 1
1116.1.x.a 4 465.u odd 6 1
1116.1.x.a 4 1860.bc even 6 1
1984.1.s.a 4 40.e odd 2 1
1984.1.s.a 4 40.f even 2 1
1984.1.s.a 4 1240.bg even 6 1
1984.1.s.a 4 1240.bi odd 6 1
3100.1.t.a 4 5.c odd 4 1
3100.1.t.a 4 20.e even 4 1
3100.1.t.a 4 155.o odd 12 1
3100.1.t.a 4 620.be even 12 1
3100.1.t.b 4 5.c odd 4 1
3100.1.t.b 4 20.e even 4 1
3100.1.t.b 4 155.o odd 12 1
3100.1.t.b 4 620.be even 12 1
3100.1.z.a 4 1.a even 1 1 trivial
3100.1.z.a 4 4.b odd 2 1 inner
3100.1.z.a 4 31.c even 3 1 inner
3100.1.z.a 4 124.i odd 6 1 inner
3844.1.b.c 2 155.i odd 6 1
3844.1.b.c 2 620.r even 6 1
3844.1.b.d 2 155.j even 6 1
3844.1.b.d 2 620.o odd 6 1
3844.1.i.d 4 155.c odd 2 1
3844.1.i.d 4 155.i odd 6 1
3844.1.i.d 4 620.e even 2 1
3844.1.i.d 4 620.r even 6 1
3844.1.l.c 8 155.v odd 30 4
3844.1.l.c 8 620.bo even 30 4
3844.1.l.d 8 155.u even 30 4
3844.1.l.d 8 620.br odd 30 4
3844.1.n.e 16 155.n even 10 4
3844.1.n.e 16 155.u even 30 4
3844.1.n.e 16 620.v odd 10 4
3844.1.n.e 16 620.br odd 30 4
3844.1.n.f 16 155.m odd 10 4
3844.1.n.f 16 155.v odd 30 4
3844.1.n.f 16 620.y even 10 4
3844.1.n.f 16 620.bo even 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3100, [\chi])\).

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
$5$ \( T^{4} \) 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
$31$ \( (T^{2} + 1)^{2} \) 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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