Properties

Label 1800.1.cr.a
Level 1800
Weight 1
Character orbit 1800.cr
Analytic conductor 0.898
Analytic rank 0
Dimension 16
Projective image \(D_{20}\)
CM disc. -24
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.cr (of order \(20\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{20}\)
Projective field Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\zeta_{40}^{19} q^{2} \) \( -\zeta_{40}^{18} q^{4} \) \( + \zeta_{40}^{9} q^{5} \) \( + ( -\zeta_{40}^{2} - \zeta_{40}^{8} ) q^{7} \) \( -\zeta_{40}^{17} q^{8} \) \(+O(q^{10})\) \( q\) \( -\zeta_{40}^{19} q^{2} \) \( -\zeta_{40}^{18} q^{4} \) \( + \zeta_{40}^{9} q^{5} \) \( + ( -\zeta_{40}^{2} - \zeta_{40}^{8} ) q^{7} \) \( -\zeta_{40}^{17} q^{8} \) \( + \zeta_{40}^{8} q^{10} \) \( + ( -\zeta_{40}^{13} - \zeta_{40}^{15} ) q^{11} \) \( + ( -\zeta_{40} - \zeta_{40}^{7} ) q^{14} \) \( -\zeta_{40}^{16} q^{16} \) \( + \zeta_{40}^{7} q^{20} \) \( + ( -\zeta_{40}^{12} - \zeta_{40}^{14} ) q^{22} \) \( + \zeta_{40}^{18} q^{25} \) \( + ( -1 - \zeta_{40}^{6} ) q^{28} \) \( + ( \zeta_{40}^{3} + \zeta_{40}^{13} ) q^{29} \) \( + ( \zeta_{40}^{10} + \zeta_{40}^{14} ) q^{31} \) \( -\zeta_{40}^{15} q^{32} \) \( + ( -\zeta_{40}^{11} - \zeta_{40}^{17} ) q^{35} \) \( + \zeta_{40}^{6} q^{40} \) \( + ( -\zeta_{40}^{11} - \zeta_{40}^{13} ) q^{44} \) \( + ( \zeta_{40}^{4} + \zeta_{40}^{10} + \zeta_{40}^{16} ) q^{49} \) \( + \zeta_{40}^{17} q^{50} \) \( + ( -\zeta_{40}^{7} - \zeta_{40}^{19} ) q^{53} \) \( + ( \zeta_{40}^{2} + \zeta_{40}^{4} ) q^{55} \) \( + ( -\zeta_{40}^{5} + \zeta_{40}^{19} ) q^{56} \) \( + ( \zeta_{40}^{2} + \zeta_{40}^{12} ) q^{58} \) \( + ( -\zeta_{40}^{5} + \zeta_{40}^{7} ) q^{59} \) \( + ( \zeta_{40}^{9} + \zeta_{40}^{13} ) q^{62} \) \( -\zeta_{40}^{14} q^{64} \) \( + ( -\zeta_{40}^{10} - \zeta_{40}^{16} ) q^{70} \) \( + ( -\zeta_{40}^{4} + \zeta_{40}^{14} ) q^{73} \) \( + ( -\zeta_{40} - \zeta_{40}^{3} + \zeta_{40}^{15} + \zeta_{40}^{17} ) q^{77} \) \( + ( -\zeta_{40}^{4} + \zeta_{40}^{12} ) q^{79} \) \( + \zeta_{40}^{5} q^{80} \) \( + ( \zeta_{40}^{5} - \zeta_{40}^{9} ) q^{83} \) \( + ( -\zeta_{40}^{10} - \zeta_{40}^{12} ) q^{88} \) \( + ( -1 + \zeta_{40}^{6} ) q^{97} \) \( + ( \zeta_{40}^{3} + \zeta_{40}^{9} + \zeta_{40}^{15} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 16q^{28} \) \(\mathstrut +\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(\zeta_{40}^{18}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.987688 0.156434i
0.987688 + 0.156434i
−0.891007 0.453990i
0.891007 + 0.453990i
−0.453990 + 0.891007i
0.453990 0.891007i
−0.156434 + 0.987688i
0.156434 0.987688i
−0.987688 + 0.156434i
0.987688 0.156434i
−0.891007 + 0.453990i
0.891007 0.453990i
−0.453990 0.891007i
0.453990 + 0.891007i
−0.156434 0.987688i
0.156434 + 0.987688i
−0.987688 + 0.156434i 0 0.951057 0.309017i −0.156434 0.987688i 0 −1.26007 1.26007i −0.891007 + 0.453990i 0 0.309017 + 0.951057i
37.2 0.987688 0.156434i 0 0.951057 0.309017i 0.156434 + 0.987688i 0 −1.26007 1.26007i 0.891007 0.453990i 0 0.309017 + 0.951057i
253.1 −0.891007 + 0.453990i 0 0.587785 0.809017i 0.453990 + 0.891007i 0 0.221232 0.221232i −0.156434 + 0.987688i 0 −0.809017 0.587785i
253.2 0.891007 0.453990i 0 0.587785 0.809017i −0.453990 0.891007i 0 0.221232 0.221232i 0.156434 0.987688i 0 −0.809017 0.587785i
397.1 −0.453990 0.891007i 0 −0.587785 + 0.809017i 0.891007 0.453990i 0 1.39680 + 1.39680i 0.987688 + 0.156434i 0 −0.809017 0.587785i
397.2 0.453990 + 0.891007i 0 −0.587785 + 0.809017i −0.891007 + 0.453990i 0 1.39680 + 1.39680i −0.987688 0.156434i 0 −0.809017 0.587785i
613.1 −0.156434 0.987688i 0 −0.951057 + 0.309017i −0.987688 + 0.156434i 0 0.642040 0.642040i 0.453990 + 0.891007i 0 0.309017 + 0.951057i
613.2 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.987688 0.156434i 0 0.642040 0.642040i −0.453990 0.891007i 0 0.309017 + 0.951057i
973.1 −0.987688 0.156434i 0 0.951057 + 0.309017i −0.156434 + 0.987688i 0 −1.26007 + 1.26007i −0.891007 0.453990i 0 0.309017 0.951057i
973.2 0.987688 + 0.156434i 0 0.951057 + 0.309017i 0.156434 0.987688i 0 −1.26007 + 1.26007i 0.891007 + 0.453990i 0 0.309017 0.951057i
1117.1 −0.891007 0.453990i 0 0.587785 + 0.809017i 0.453990 0.891007i 0 0.221232 + 0.221232i −0.156434 0.987688i 0 −0.809017 + 0.587785i
1117.2 0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.453990 + 0.891007i 0 0.221232 + 0.221232i 0.156434 + 0.987688i 0 −0.809017 + 0.587785i
1333.1 −0.453990 + 0.891007i 0 −0.587785 0.809017i 0.891007 + 0.453990i 0 1.39680 1.39680i 0.987688 0.156434i 0 −0.809017 + 0.587785i
1333.2 0.453990 0.891007i 0 −0.587785 0.809017i −0.891007 0.453990i 0 1.39680 1.39680i −0.987688 + 0.156434i 0 −0.809017 + 0.587785i
1477.1 −0.156434 + 0.987688i 0 −0.951057 0.309017i −0.987688 0.156434i 0 0.642040 + 0.642040i 0.453990 0.891007i 0 0.309017 0.951057i
1477.2 0.156434 0.987688i 0 −0.951057 0.309017i 0.987688 + 0.156434i 0 0.642040 + 0.642040i −0.453990 + 0.891007i 0 0.309017 0.951057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1477.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
3.b Odd 1 yes
8.b Even 1 yes
25.f Odd 1 yes
75.l Even 1 yes
200.x Odd 1 yes
600.bp Even 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(1800, [\chi])\).