Properties

Label 3800.1.bn.b
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 $8$

Related objects

Downloads

Learn more

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: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
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.

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

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 −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0 0
1299.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0 0
2899.1 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0 0
2899.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 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
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.b 4
5.b even 2 1 inner 3800.1.bn.b 4
5.c odd 4 1 152.1.k.a 2
5.c odd 4 1 3800.1.bd.c 2
8.d odd 2 1 CM 3800.1.bn.b 4
15.e even 4 1 1368.1.bz.a 2
19.c even 3 1 inner 3800.1.bn.b 4
20.e even 4 1 608.1.o.a 2
40.e odd 2 1 inner 3800.1.bn.b 4
40.i odd 4 1 608.1.o.a 2
40.k even 4 1 152.1.k.a 2
40.k even 4 1 3800.1.bd.c 2
95.g even 4 1 2888.1.k.a 2
95.i even 6 1 inner 3800.1.bn.b 4
95.l even 12 1 2888.1.f.a 1
95.l even 12 1 2888.1.k.a 2
95.m odd 12 1 152.1.k.a 2
95.m odd 12 1 2888.1.f.b 1
95.m odd 12 1 3800.1.bd.c 2
95.q odd 36 6 2888.1.u.c 6
95.r even 36 6 2888.1.u.d 6
120.q odd 4 1 1368.1.bz.a 2
152.k odd 6 1 inner 3800.1.bn.b 4
285.v even 12 1 1368.1.bz.a 2
380.v even 12 1 608.1.o.a 2
760.y odd 4 1 2888.1.k.a 2
760.bm odd 6 1 inner 3800.1.bn.b 4
760.br odd 12 1 608.1.o.a 2
760.bu odd 12 1 2888.1.f.a 1
760.bu odd 12 1 2888.1.k.a 2
760.bw even 12 1 152.1.k.a 2
760.bw even 12 1 2888.1.f.b 1
760.bw even 12 1 3800.1.bd.c 2
760.cn odd 36 6 2888.1.u.d 6
760.cp even 36 6 2888.1.u.c 6
2280.dj odd 12 1 1368.1.bz.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 5.c odd 4 1
152.1.k.a 2 40.k even 4 1
152.1.k.a 2 95.m odd 12 1
152.1.k.a 2 760.bw even 12 1
608.1.o.a 2 20.e even 4 1
608.1.o.a 2 40.i odd 4 1
608.1.o.a 2 380.v even 12 1
608.1.o.a 2 760.br odd 12 1
1368.1.bz.a 2 15.e even 4 1
1368.1.bz.a 2 120.q odd 4 1
1368.1.bz.a 2 285.v even 12 1
1368.1.bz.a 2 2280.dj odd 12 1
2888.1.f.a 1 95.l even 12 1
2888.1.f.a 1 760.bu odd 12 1
2888.1.f.b 1 95.m odd 12 1
2888.1.f.b 1 760.bw even 12 1
2888.1.k.a 2 95.g even 4 1
2888.1.k.a 2 95.l even 12 1
2888.1.k.a 2 760.y odd 4 1
2888.1.k.a 2 760.bu odd 12 1
2888.1.u.c 6 95.q odd 36 6
2888.1.u.c 6 760.cp even 36 6
2888.1.u.d 6 95.r even 36 6
2888.1.u.d 6 760.cn odd 36 6
3800.1.bd.c 2 5.c odd 4 1
3800.1.bd.c 2 40.k even 4 1
3800.1.bd.c 2 95.m odd 12 1
3800.1.bd.c 2 760.bw even 12 1
3800.1.bn.b 4 1.a even 1 1 trivial
3800.1.bn.b 4 5.b even 2 1 inner
3800.1.bn.b 4 8.d odd 2 1 CM
3800.1.bn.b 4 19.c even 3 1 inner
3800.1.bn.b 4 40.e odd 2 1 inner
3800.1.bn.b 4 95.i even 6 1 inner
3800.1.bn.b 4 152.k odd 6 1 inner
3800.1.bn.b 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 \(S_{1}^{\mathrm{new}}(3800, [\chi])\):

\( T_{3}^{4} - T_{3}^{2} + 1 \)
\( T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( T^{4} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( 16 - 4 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 1 + T + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 1 - T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 4 - 2 T + T^{2} )^{2} \)
$97$ \( 1 - T^{2} + T^{4} \)
show more
show less