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.

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

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 \(S_{2}^{\mathrm{new}}(150, [\chi])\).