Properties

Label 150.2.c.a
Level 150
Weight 2
Character orbit 150.c
Analytic conductor 1.198
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 4 i q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} + 4 i q^{7} -i q^{8} - q^{9} -i q^{12} + 2 i q^{13} -4 q^{14} + q^{16} -6 i q^{17} -i q^{18} + 4 q^{19} -4 q^{21} + q^{24} -2 q^{26} -i q^{27} -4 i q^{28} + 6 q^{29} + 8 q^{31} + i q^{32} + 6 q^{34} + q^{36} -2 i q^{37} + 4 i q^{38} -2 q^{39} -6 q^{41} -4 i q^{42} -4 i q^{43} + i q^{48} -9 q^{49} + 6 q^{51} -2 i q^{52} -6 i q^{53} + q^{54} + 4 q^{56} + 4 i q^{57} + 6 i q^{58} -10 q^{61} + 8 i q^{62} -4 i q^{63} - q^{64} + 4 i q^{67} + 6 i q^{68} + i q^{72} + 2 i q^{73} + 2 q^{74} -4 q^{76} -2 i q^{78} -8 q^{79} + q^{81} -6 i q^{82} + 12 i q^{83} + 4 q^{84} + 4 q^{86} + 6 i q^{87} -18 q^{89} -8 q^{91} + 8 i q^{93} - q^{96} -2 i q^{97} -9 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{14} + 2q^{16} + 8q^{19} - 8q^{21} + 2q^{24} - 4q^{26} + 12q^{29} + 16q^{31} + 12q^{34} + 2q^{36} - 4q^{39} - 12q^{41} - 18q^{49} + 12q^{51} + 2q^{54} + 8q^{56} - 20q^{61} - 2q^{64} + 4q^{74} - 8q^{76} - 16q^{79} + 2q^{81} + 8q^{84} + 8q^{86} - 36q^{89} - 16q^{91} - 2q^{96} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 4.00000i 1.00000i −1.00000 0
49.2 1.00000i 1.00000i −1.00000 0 −1.00000 4.00000i 1.00000i −1.00000 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
5.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(150, [\chi])\).