Properties

Label 1800.2.b.a
Level 1800
Weight 2
Character orbit 1800.b
Analytic conductor 14.373
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{7} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -2 q^{4} + 3 \beta q^{7} -2 \beta q^{8} + \beta q^{11} + 3 \beta q^{13} -6 q^{14} + 4 q^{16} + 2 \beta q^{17} -4 q^{19} -2 q^{22} + 6 q^{23} -6 q^{26} -6 \beta q^{28} + 6 q^{29} -6 \beta q^{31} + 4 \beta q^{32} -4 q^{34} -3 \beta q^{37} -4 \beta q^{38} + 7 \beta q^{41} -8 q^{43} -2 \beta q^{44} + 6 \beta q^{46} -11 q^{49} -6 \beta q^{52} -12 q^{53} + 12 q^{56} + 6 \beta q^{58} + \beta q^{59} + 6 \beta q^{61} + 12 q^{62} -8 q^{64} -8 q^{67} -4 \beta q^{68} -14 q^{73} + 6 q^{74} + 8 q^{76} -6 q^{77} -6 \beta q^{79} -14 q^{82} + 2 \beta q^{83} -8 \beta q^{86} + 4 q^{88} -5 \beta q^{89} -18 q^{91} -12 q^{92} + 10 q^{97} -11 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} - 12q^{14} + 8q^{16} - 8q^{19} - 4q^{22} + 12q^{23} - 12q^{26} + 12q^{29} - 8q^{34} - 16q^{43} - 22q^{49} - 24q^{53} + 24q^{56} + 24q^{62} - 16q^{64} - 16q^{67} - 28q^{73} + 12q^{74} + 16q^{76} - 12q^{77} - 28q^{82} + 8q^{88} - 36q^{91} - 24q^{92} + 20q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 4.24264i 2.82843i 0 0
251.2 1.41421i 0 −2.00000 0 0 4.24264i 2.82843i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.f Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} + 18 \)
\( T_{23} - 6 \)
\( T_{43} + 8 \)