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

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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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{3} - 2q^{4} - 4q^{9} + O(q^{10}) \) \( 16q + 4q^{3} - 2q^{4} - 4q^{9} + 4q^{10} + 6q^{11} + 2q^{12} - 2q^{14} + 2q^{16} - 4q^{17} - 4q^{19} - 2q^{25} - 8q^{26} + 4q^{27} + 16q^{30} + 4q^{33} + 4q^{35} - 2q^{36} - 8q^{40} + 4q^{41} + 2q^{42} - 8q^{43} - 8q^{44} - 4q^{46} - 2q^{48} - 16q^{51} + 2q^{56} - 16q^{57} - 4q^{58} - 4q^{62} + 4q^{64} + 4q^{65} + 6q^{67} + 8q^{68} + 2q^{75} + 8q^{76} - 12q^{78} - 4q^{81} + 4q^{83} - 4q^{89} + 4q^{90} - 8q^{91} + 4q^{94} - 4q^{99} + O(q^{100}) \)

Character Values

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

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

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 \(S_{1}^{\mathrm{new}}(1800, [\chi])\).