Properties

Label 1800.1.dg.a
Level 1800
Weight 1
Character orbit 1800.dg
Analytic conductor 0.898
Analytic rank 0
Dimension 16
Projective image \(A_{5}\)
CM/RM No
Inner twists 8

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.dg (of order \(30\) and degree \(8\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: \(\Q(\zeta_{60})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(A_{5}\)
Projective field Galois closure of 5.1.2025000000.9

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \zeta_{60}^{29} q^{2} \) \( + \zeta_{60}^{18} q^{3} \) \( -\zeta_{60}^{28} q^{4} \) \( + \zeta_{60}^{13} q^{5} \) \( -\zeta_{60}^{17} q^{6} \) \( + \zeta_{60}^{5} q^{7} \) \( + \zeta_{60}^{27} q^{8} \) \( -\zeta_{60}^{6} q^{9} \) \(+O(q^{10})\) \( q\) \( + \zeta_{60}^{29} q^{2} \) \( + \zeta_{60}^{18} q^{3} \) \( -\zeta_{60}^{28} q^{4} \) \( + \zeta_{60}^{13} q^{5} \) \( -\zeta_{60}^{17} q^{6} \) \( + \zeta_{60}^{5} q^{7} \) \( + \zeta_{60}^{27} q^{8} \) \( -\zeta_{60}^{6} q^{9} \) \( -\zeta_{60}^{12} q^{10} \) \( + ( -\zeta_{60}^{8} - \zeta_{60}^{20} ) q^{11} \) \( + \zeta_{60}^{16} q^{12} \) \( + ( -\zeta_{60}^{13} + \zeta_{60}^{19} ) q^{13} \) \( -\zeta_{60}^{4} q^{14} \) \( -\zeta_{60} q^{15} \) \( -\zeta_{60}^{26} q^{16} \) \( + \zeta_{60}^{12} q^{17} \) \( + \zeta_{60}^{5} q^{18} \) \( + \zeta_{60}^{12} q^{19} \) \( + \zeta_{60}^{11} q^{20} \) \( + \zeta_{60}^{23} q^{21} \) \( + ( \zeta_{60}^{7} + \zeta_{60}^{19} ) q^{22} \) \( + \zeta_{60}^{19} q^{23} \) \( -\zeta_{60}^{15} q^{24} \) \( + \zeta_{60}^{26} q^{25} \) \( + ( \zeta_{60}^{12} - \zeta_{60}^{18} ) q^{26} \) \( -\zeta_{60}^{24} q^{27} \) \( + \zeta_{60}^{3} q^{28} \) \( + ( \zeta_{60}^{17} + \zeta_{60}^{29} ) q^{29} \) \(+ q^{30}\) \( + \zeta_{60}^{7} q^{31} \) \( + \zeta_{60}^{25} q^{32} \) \( + ( \zeta_{60}^{8} - \zeta_{60}^{26} ) q^{33} \) \( -\zeta_{60}^{11} q^{34} \) \( + \zeta_{60}^{18} q^{35} \) \( -\zeta_{60}^{4} q^{36} \) \( -\zeta_{60}^{11} q^{38} \) \( + ( \zeta_{60} - \zeta_{60}^{7} ) q^{39} \) \( -\zeta_{60}^{10} q^{40} \) \( + ( \zeta_{60}^{4} + \zeta_{60}^{28} ) q^{41} \) \( -\zeta_{60}^{22} q^{42} \) \( + \zeta_{60}^{20} q^{43} \) \( + ( -\zeta_{60}^{6} - \zeta_{60}^{18} ) q^{44} \) \( -\zeta_{60}^{19} q^{45} \) \( -\zeta_{60}^{18} q^{46} \) \( + ( -\zeta_{60}^{17} - \zeta_{60}^{29} ) q^{47} \) \( + \zeta_{60}^{14} q^{48} \) \( -\zeta_{60}^{25} q^{50} \) \(- q^{51}\) \( + ( -\zeta_{60}^{11} + \zeta_{60}^{17} ) q^{52} \) \( + ( -\zeta_{60}^{15} + \zeta_{60}^{21} ) q^{53} \) \( + \zeta_{60}^{23} q^{54} \) \( + ( \zeta_{60}^{3} - \zeta_{60}^{21} ) q^{55} \) \( -\zeta_{60}^{2} q^{56} \) \(- q^{57}\) \( + ( -\zeta_{60}^{16} - \zeta_{60}^{28} ) q^{58} \) \( + \zeta_{60}^{29} q^{60} \) \( + \zeta_{60}^{29} q^{61} \) \( -\zeta_{60}^{6} q^{62} \) \( -\zeta_{60}^{11} q^{63} \) \( -\zeta_{60}^{24} q^{64} \) \( + ( -\zeta_{60}^{2} - \zeta_{60}^{26} ) q^{65} \) \( + ( -\zeta_{60}^{7} + \zeta_{60}^{25} ) q^{66} \) \( + ( -\zeta_{60}^{4} + \zeta_{60}^{10} ) q^{67} \) \( + \zeta_{60}^{10} q^{68} \) \( -\zeta_{60}^{7} q^{69} \) \( -\zeta_{60}^{17} q^{70} \) \( + \zeta_{60}^{3} q^{72} \) \( -\zeta_{60}^{14} q^{75} \) \( + \zeta_{60}^{10} q^{76} \) \( + ( -\zeta_{60}^{13} - \zeta_{60}^{25} ) q^{77} \) \( + ( -1 + \zeta_{60}^{6} ) q^{78} \) \( + ( -\zeta_{60}^{17} - \zeta_{60}^{29} ) q^{79} \) \( + \zeta_{60}^{9} q^{80} \) \( + \zeta_{60}^{12} q^{81} \) \( + ( -\zeta_{60}^{3} - \zeta_{60}^{27} ) q^{82} \) \( + ( \zeta_{60}^{8} - \zeta_{60}^{26} ) q^{83} \) \( + \zeta_{60}^{21} q^{84} \) \( + \zeta_{60}^{25} q^{85} \) \( -\zeta_{60}^{19} q^{86} \) \( + ( -\zeta_{60}^{5} - \zeta_{60}^{17} ) q^{87} \) \( + ( \zeta_{60}^{5} + \zeta_{60}^{17} ) q^{88} \) \( + \zeta_{60}^{24} q^{89} \) \( + \zeta_{60}^{18} q^{90} \) \( + ( -\zeta_{60}^{18} + \zeta_{60}^{24} ) q^{91} \) \( + \zeta_{60}^{17} q^{92} \) \( + \zeta_{60}^{25} q^{93} \) \( + ( \zeta_{60}^{16} + \zeta_{60}^{28} ) q^{94} \) \( + \zeta_{60}^{25} q^{95} \) \( -\zeta_{60}^{13} q^{96} \) \( + ( \zeta_{60}^{14} + \zeta_{60}^{26} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut +\mathstrut 2q^{12} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 16q^{30} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 8q^{44} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 2q^{48} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 2q^{56} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 4q^{62} \) \(\mathstrut +\mathstrut 4q^{64} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut +\mathstrut 6q^{67} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut +\mathstrut 2q^{75} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 12q^{78} \) \(\mathstrut -\mathstrut 4q^{81} \) \(\mathstrut +\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 4q^{89} \) \(\mathstrut +\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 4q^{94} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-\zeta_{60}^{18}\) \(-1\) \(-\zeta_{60}^{10}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
211.1
0.406737 + 0.913545i
−0.406737 0.913545i
0.207912 + 0.978148i
−0.207912 0.978148i
0.207912 0.978148i
−0.207912 + 0.978148i
0.406737 0.913545i
−0.406737 + 0.913545i
0.743145 0.669131i
−0.743145 + 0.669131i
0.994522 0.104528i
−0.994522 + 0.104528i
0.994522 + 0.104528i
−0.994522 0.104528i
0.743145 + 0.669131i
−0.743145 0.669131i
−0.406737 + 0.913545i −0.309017 + 0.951057i −0.669131 0.743145i −0.743145 + 0.669131i −0.743145 0.669131i 0.866025 0.500000i 0.951057 0.309017i −0.809017 0.587785i −0.309017 0.951057i
211.2 0.406737 0.913545i −0.309017 + 0.951057i −0.669131 0.743145i 0.743145 0.669131i 0.743145 + 0.669131i −0.866025 + 0.500000i −0.951057 + 0.309017i −0.809017 0.587785i −0.309017 0.951057i
331.1 −0.207912 + 0.978148i 0.809017 0.587785i −0.913545 0.406737i 0.406737 0.913545i 0.406737 + 0.913545i 0.866025 + 0.500000i 0.587785 0.809017i 0.309017 0.951057i 0.809017 + 0.587785i
331.2 0.207912 0.978148i 0.809017 0.587785i −0.913545 0.406737i −0.406737 + 0.913545i −0.406737 0.913545i −0.866025 0.500000i −0.587785 + 0.809017i 0.309017 0.951057i 0.809017 + 0.587785i
571.1 −0.207912 0.978148i 0.809017 + 0.587785i −0.913545 + 0.406737i 0.406737 + 0.913545i 0.406737 0.913545i 0.866025 0.500000i 0.587785 + 0.809017i 0.309017 + 0.951057i 0.809017 0.587785i
571.2 0.207912 + 0.978148i 0.809017 + 0.587785i −0.913545 + 0.406737i −0.406737 0.913545i −0.406737 + 0.913545i −0.866025 + 0.500000i −0.587785 0.809017i 0.309017 + 0.951057i 0.809017 0.587785i
691.1 −0.406737 0.913545i −0.309017 0.951057i −0.669131 + 0.743145i −0.743145 0.669131i −0.743145 + 0.669131i 0.866025 + 0.500000i 0.951057 + 0.309017i −0.809017 + 0.587785i −0.309017 + 0.951057i
691.2 0.406737 + 0.913545i −0.309017 0.951057i −0.669131 + 0.743145i 0.743145 + 0.669131i 0.743145 0.669131i −0.866025 0.500000i −0.951057 0.309017i −0.809017 + 0.587785i −0.309017 + 0.951057i
931.1 −0.743145 0.669131i 0.809017 0.587785i 0.104528 + 0.994522i −0.994522 + 0.104528i −0.994522 0.104528i −0.866025 + 0.500000i 0.587785 0.809017i 0.309017 0.951057i 0.809017 + 0.587785i
931.2 0.743145 + 0.669131i 0.809017 0.587785i 0.104528 + 0.994522i 0.994522 0.104528i 0.994522 + 0.104528i 0.866025 0.500000i −0.587785 + 0.809017i 0.309017 0.951057i 0.809017 + 0.587785i
1291.1 −0.994522 0.104528i −0.309017 0.951057i 0.978148 + 0.207912i 0.207912 0.978148i 0.207912 + 0.978148i 0.866025 0.500000i −0.951057 0.309017i −0.809017 + 0.587785i −0.309017 + 0.951057i
1291.2 0.994522 + 0.104528i −0.309017 0.951057i 0.978148 + 0.207912i −0.207912 + 0.978148i −0.207912 0.978148i −0.866025 + 0.500000i 0.951057 + 0.309017i −0.809017 + 0.587785i −0.309017 + 0.951057i
1411.1 −0.994522 + 0.104528i −0.309017 + 0.951057i 0.978148 0.207912i 0.207912 + 0.978148i 0.207912 0.978148i 0.866025 + 0.500000i −0.951057 + 0.309017i −0.809017 0.587785i −0.309017 0.951057i
1411.2 0.994522 0.104528i −0.309017 + 0.951057i 0.978148 0.207912i −0.207912 0.978148i −0.207912 + 0.978148i −0.866025 0.500000i 0.951057 0.309017i −0.809017 0.587785i −0.309017 0.951057i
1771.1 −0.743145 + 0.669131i 0.809017 + 0.587785i 0.104528 0.994522i −0.994522 0.104528i −0.994522 + 0.104528i −0.866025 0.500000i 0.587785 + 0.809017i 0.309017 + 0.951057i 0.809017 0.587785i
1771.2 0.743145 0.669131i 0.809017 + 0.587785i 0.104528 0.994522i 0.994522 + 0.104528i 0.994522 0.104528i 0.866025 + 0.500000i −0.587785 0.809017i 0.309017 + 0.951057i 0.809017 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1771.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 yes
9.c Even 1 yes
25.d Even 1 yes
72.p Odd 1 yes
200.n Odd 1 yes
225.q Even 1 yes
1800.dg Odd 1 yes

Hecke kernels

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