Properties

Label 1800.1.bk.d
Level $1800$
Weight $1$
Character orbit 1800.bk
Analytic conductor $0.898$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1800.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.898317022739\)
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 72)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.648.1
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1051.1
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 −0.500000 0.866025i 0
1651.1 0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −1.00000 −0.500000 + 0.866025i 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 \(\Q(\sqrt{-2}) \)
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.bk.d 2
5.b even 2 1 72.1.p.a 2
5.c odd 4 2 1800.1.ba.b 4
8.d odd 2 1 CM 1800.1.bk.d 2
9.c even 3 1 inner 1800.1.bk.d 2
15.d odd 2 1 216.1.p.a 2
20.d odd 2 1 288.1.t.a 2
35.c odd 2 1 3528.1.cg.a 2
35.i odd 6 1 3528.1.ba.a 2
35.i odd 6 1 3528.1.ce.b 2
35.j even 6 1 3528.1.ba.b 2
35.j even 6 1 3528.1.ce.a 2
40.e odd 2 1 72.1.p.a 2
40.f even 2 1 288.1.t.a 2
40.k even 4 2 1800.1.ba.b 4
45.h odd 6 1 216.1.p.a 2
45.h odd 6 1 648.1.b.a 1
45.j even 6 1 72.1.p.a 2
45.j even 6 1 648.1.b.b 1
45.k odd 12 2 1800.1.ba.b 4
60.h even 2 1 864.1.t.a 2
72.p odd 6 1 inner 1800.1.bk.d 2
80.k odd 4 2 2304.1.o.c 4
80.q even 4 2 2304.1.o.c 4
120.i odd 2 1 864.1.t.a 2
120.m even 2 1 216.1.p.a 2
180.n even 6 1 864.1.t.a 2
180.n even 6 1 2592.1.b.a 1
180.p odd 6 1 288.1.t.a 2
180.p odd 6 1 2592.1.b.b 1
280.n even 2 1 3528.1.cg.a 2
280.ba even 6 1 3528.1.ba.a 2
280.ba even 6 1 3528.1.ce.b 2
280.bi odd 6 1 3528.1.ba.b 2
280.bi odd 6 1 3528.1.ce.a 2
315.q odd 6 1 3528.1.ba.a 2
315.r even 6 1 3528.1.ba.b 2
315.bg odd 6 1 3528.1.cg.a 2
315.bn odd 6 1 3528.1.ce.b 2
315.bo even 6 1 3528.1.ce.a 2
360.z odd 6 1 72.1.p.a 2
360.z odd 6 1 648.1.b.b 1
360.bd even 6 1 216.1.p.a 2
360.bd even 6 1 648.1.b.a 1
360.bh odd 6 1 864.1.t.a 2
360.bh odd 6 1 2592.1.b.a 1
360.bk even 6 1 288.1.t.a 2
360.bk even 6 1 2592.1.b.b 1
360.bo even 12 2 1800.1.ba.b 4
720.ce even 12 2 2304.1.o.c 4
720.cz odd 12 2 2304.1.o.c 4
2520.df odd 6 1 3528.1.ba.b 2
2520.ds even 6 1 3528.1.ba.a 2
2520.dy even 6 1 3528.1.cg.a 2
2520.gj odd 6 1 3528.1.ce.a 2
2520.gp even 6 1 3528.1.ce.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 5.b even 2 1
72.1.p.a 2 40.e odd 2 1
72.1.p.a 2 45.j even 6 1
72.1.p.a 2 360.z odd 6 1
216.1.p.a 2 15.d odd 2 1
216.1.p.a 2 45.h odd 6 1
216.1.p.a 2 120.m even 2 1
216.1.p.a 2 360.bd even 6 1
288.1.t.a 2 20.d odd 2 1
288.1.t.a 2 40.f even 2 1
288.1.t.a 2 180.p odd 6 1
288.1.t.a 2 360.bk even 6 1
648.1.b.a 1 45.h odd 6 1
648.1.b.a 1 360.bd even 6 1
648.1.b.b 1 45.j even 6 1
648.1.b.b 1 360.z odd 6 1
864.1.t.a 2 60.h even 2 1
864.1.t.a 2 120.i odd 2 1
864.1.t.a 2 180.n even 6 1
864.1.t.a 2 360.bh odd 6 1
1800.1.ba.b 4 5.c odd 4 2
1800.1.ba.b 4 40.k even 4 2
1800.1.ba.b 4 45.k odd 12 2
1800.1.ba.b 4 360.bo even 12 2
1800.1.bk.d 2 1.a even 1 1 trivial
1800.1.bk.d 2 8.d odd 2 1 CM
1800.1.bk.d 2 9.c even 3 1 inner
1800.1.bk.d 2 72.p odd 6 1 inner
2304.1.o.c 4 80.k odd 4 2
2304.1.o.c 4 80.q even 4 2
2304.1.o.c 4 720.ce even 12 2
2304.1.o.c 4 720.cz odd 12 2
2592.1.b.a 1 180.n even 6 1
2592.1.b.a 1 360.bh odd 6 1
2592.1.b.b 1 180.p odd 6 1
2592.1.b.b 1 360.bk even 6 1
3528.1.ba.a 2 35.i odd 6 1
3528.1.ba.a 2 280.ba even 6 1
3528.1.ba.a 2 315.q odd 6 1
3528.1.ba.a 2 2520.ds even 6 1
3528.1.ba.b 2 35.j even 6 1
3528.1.ba.b 2 280.bi odd 6 1
3528.1.ba.b 2 315.r even 6 1
3528.1.ba.b 2 2520.df odd 6 1
3528.1.ce.a 2 35.j even 6 1
3528.1.ce.a 2 280.bi odd 6 1
3528.1.ce.a 2 315.bo even 6 1
3528.1.ce.a 2 2520.gj odd 6 1
3528.1.ce.b 2 35.i odd 6 1
3528.1.ce.b 2 280.ba even 6 1
3528.1.ce.b 2 315.bn odd 6 1
3528.1.ce.b 2 2520.gp even 6 1
3528.1.cg.a 2 35.c odd 2 1
3528.1.cg.a 2 280.n even 2 1
3528.1.cg.a 2 315.bg odd 6 1
3528.1.cg.a 2 2520.dy even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \)
\( T_{17} - 1 \)
\( T_{19} + 1 \)

Hecke characteristic polynomials

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