Properties

Label 3800.1.bd.c
Level $3800$
Weight $1$
Character orbit 3800.bd
Analytic conductor $1.896$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

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

Character values

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

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1451.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 0 −1.00000 0 0
3051.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 0 −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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.c even 3 1 inner
152.k 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.bd.c 2
5.b even 2 1 152.1.k.a 2
5.c odd 4 2 3800.1.bn.b 4
8.d odd 2 1 CM 3800.1.bd.c 2
15.d odd 2 1 1368.1.bz.a 2
19.c even 3 1 inner 3800.1.bd.c 2
20.d odd 2 1 608.1.o.a 2
40.e odd 2 1 152.1.k.a 2
40.f even 2 1 608.1.o.a 2
40.k even 4 2 3800.1.bn.b 4
95.d odd 2 1 2888.1.k.a 2
95.h odd 6 1 2888.1.f.a 1
95.h odd 6 1 2888.1.k.a 2
95.i even 6 1 152.1.k.a 2
95.i even 6 1 2888.1.f.b 1
95.m odd 12 2 3800.1.bn.b 4
95.o odd 18 6 2888.1.u.d 6
95.p even 18 6 2888.1.u.c 6
120.m even 2 1 1368.1.bz.a 2
152.k odd 6 1 inner 3800.1.bd.c 2
285.n odd 6 1 1368.1.bz.a 2
380.p odd 6 1 608.1.o.a 2
760.p even 2 1 2888.1.k.a 2
760.z even 6 1 608.1.o.a 2
760.bf even 6 1 2888.1.f.a 1
760.bf even 6 1 2888.1.k.a 2
760.bm odd 6 1 152.1.k.a 2
760.bm odd 6 1 2888.1.f.b 1
760.bw even 12 2 3800.1.bn.b 4
760.bx even 18 6 2888.1.u.d 6
760.bz odd 18 6 2888.1.u.c 6
2280.co even 6 1 1368.1.bz.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 5.b even 2 1
152.1.k.a 2 40.e odd 2 1
152.1.k.a 2 95.i even 6 1
152.1.k.a 2 760.bm odd 6 1
608.1.o.a 2 20.d odd 2 1
608.1.o.a 2 40.f even 2 1
608.1.o.a 2 380.p odd 6 1
608.1.o.a 2 760.z even 6 1
1368.1.bz.a 2 15.d odd 2 1
1368.1.bz.a 2 120.m even 2 1
1368.1.bz.a 2 285.n odd 6 1
1368.1.bz.a 2 2280.co even 6 1
2888.1.f.a 1 95.h odd 6 1
2888.1.f.a 1 760.bf even 6 1
2888.1.f.b 1 95.i even 6 1
2888.1.f.b 1 760.bm odd 6 1
2888.1.k.a 2 95.d odd 2 1
2888.1.k.a 2 95.h odd 6 1
2888.1.k.a 2 760.p even 2 1
2888.1.k.a 2 760.bf even 6 1
2888.1.u.c 6 95.p even 18 6
2888.1.u.c 6 760.bz odd 18 6
2888.1.u.d 6 95.o odd 18 6
2888.1.u.d 6 760.bx even 18 6
3800.1.bd.c 2 1.a even 1 1 trivial
3800.1.bd.c 2 8.d odd 2 1 CM
3800.1.bd.c 2 19.c even 3 1 inner
3800.1.bd.c 2 152.k odd 6 1 inner
3800.1.bn.b 4 5.c odd 4 2
3800.1.bn.b 4 40.k even 4 2
3800.1.bn.b 4 95.m odd 12 2
3800.1.bn.b 4 760.bw even 12 2

Hecke kernels

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

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

Hecke characteristic polynomials

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