Properties

Label 3800.1.bn.a
Level $3800$
Weight $1$
Character orbit 3800.bn
Analytic conductor $1.896$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.89644704801\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.72200.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} + 2 \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} + 2 \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} - 3 \zeta_{12}^{4} q^{9} - q^{11} + 2 \zeta_{12}^{3} q^{12} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} - 3 \zeta_{12}^{3} q^{18} - q^{19} + \zeta_{12}^{5} q^{22} + 2 \zeta_{12}^{2} q^{24} - 2 \zeta_{12}^{3} q^{27} - \zeta_{12} q^{32} - 2 \zeta_{12}^{5} q^{33} + \zeta_{12}^{4} q^{34} - 3 \zeta_{12}^{2} q^{36} + \zeta_{12}^{5} q^{38} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{43} + \zeta_{12}^{4} q^{44} + 2 \zeta_{12} q^{48} - q^{49} - 2 \zeta_{12}^{4} q^{51} + 4 \zeta_{12}^{2} q^{54} - 2 \zeta_{12}^{5} q^{57} - \zeta_{12}^{2} q^{59} - q^{64} - 2 \zeta_{12}^{4} q^{66} + \zeta_{12} q^{67} + \zeta_{12}^{3} q^{68} - 3 \zeta_{12} q^{72} + \zeta_{12}^{5} q^{73} + \zeta_{12}^{4} q^{76} + 3 \zeta_{12}^{2} q^{81} + \zeta_{12} q^{82} - \zeta_{12}^{3} q^{83} + 2 \zeta_{12}^{4} q^{86} + \zeta_{12}^{3} q^{88} + \zeta_{12}^{4} q^{89} + 2 q^{96} + \zeta_{12}^{5} q^{97} + \zeta_{12}^{5} q^{98} + 3 \zeta_{12}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{6} + 6 q^{9} - 4 q^{11} - 2 q^{16} - 4 q^{19} + 4 q^{24} - 2 q^{34} - 6 q^{36} + 2 q^{41} - 2 q^{44} - 4 q^{49} + 4 q^{51} + 8 q^{54} - 2 q^{59} - 4 q^{64} + 4 q^{66} - 2 q^{76} - 10 q^{81} - 4 q^{86} - 2 q^{89} + 8 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1299.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 1.73205 1.00000i 0.500000 0.866025i 0 −1.00000 + 1.73205i 0 1.00000i 1.50000 2.59808i 0
1299.2 0.866025 0.500000i −1.73205 + 1.00000i 0.500000 0.866025i 0 −1.00000 + 1.73205i 0 1.00000i 1.50000 2.59808i 0
2899.1 −0.866025 0.500000i 1.73205 + 1.00000i 0.500000 + 0.866025i 0 −1.00000 1.73205i 0 1.00000i 1.50000 + 2.59808i 0
2899.2 0.866025 + 0.500000i −1.73205 1.00000i 0.500000 + 0.866025i 0 −1.00000 1.73205i 0 1.00000i 1.50000 + 2.59808i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
19.c even 3 1 inner
40.e odd 2 1 inner
95.i even 6 1 inner
152.k odd 6 1 inner
760.bm odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.bn.a 4
5.b even 2 1 inner 3800.1.bn.a 4
5.c odd 4 1 3800.1.bd.a 2
5.c odd 4 1 3800.1.bd.e yes 2
8.d odd 2 1 CM 3800.1.bn.a 4
19.c even 3 1 inner 3800.1.bn.a 4
40.e odd 2 1 inner 3800.1.bn.a 4
40.k even 4 1 3800.1.bd.a 2
40.k even 4 1 3800.1.bd.e yes 2
95.i even 6 1 inner 3800.1.bn.a 4
95.m odd 12 1 3800.1.bd.a 2
95.m odd 12 1 3800.1.bd.e yes 2
152.k odd 6 1 inner 3800.1.bn.a 4
760.bm odd 6 1 inner 3800.1.bn.a 4
760.bw even 12 1 3800.1.bd.a 2
760.bw even 12 1 3800.1.bd.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.bd.a 2 5.c odd 4 1
3800.1.bd.a 2 40.k even 4 1
3800.1.bd.a 2 95.m odd 12 1
3800.1.bd.a 2 760.bw even 12 1
3800.1.bd.e yes 2 5.c odd 4 1
3800.1.bd.e yes 2 40.k even 4 1
3800.1.bd.e yes 2 95.m odd 12 1
3800.1.bd.e yes 2 760.bw even 12 1
3800.1.bn.a 4 1.a even 1 1 trivial
3800.1.bn.a 4 5.b even 2 1 inner
3800.1.bn.a 4 8.d odd 2 1 CM
3800.1.bn.a 4 19.c even 3 1 inner
3800.1.bn.a 4 40.e odd 2 1 inner
3800.1.bn.a 4 95.i even 6 1 inner
3800.1.bn.a 4 152.k odd 6 1 inner
3800.1.bn.a 4 760.bm odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{4} - 4T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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