Properties

Label 153.2.d.a
Level 153
Weight 2
Character orbit 153.d
Analytic conductor 1.222
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 153.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.22171115093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(- q^{4}\) \( + \beta q^{7} \) \( + 3 q^{8} \) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(- q^{4}\) \( + \beta q^{7} \) \( + 3 q^{8} \) \( + \beta q^{11} \) \( + 2 q^{13} \) \( -\beta q^{14} \) \(- q^{16}\) \( + ( -1 + \beta ) q^{17} \) \( -4 q^{19} \) \( -\beta q^{22} \) \( -\beta q^{23} \) \( + 5 q^{25} \) \( -2 q^{26} \) \( -\beta q^{28} \) \( + \beta q^{31} \) \( -5 q^{32} \) \( + ( 1 - \beta ) q^{34} \) \( -2 \beta q^{37} \) \( + 4 q^{38} \) \( -2 \beta q^{41} \) \( -4 q^{43} \) \( -\beta q^{44} \) \( + \beta q^{46} \) \( + 8 q^{47} \) \( -9 q^{49} \) \( -5 q^{50} \) \( -2 q^{52} \) \( -6 q^{53} \) \( + 3 \beta q^{56} \) \( + 12 q^{59} \) \( + 2 \beta q^{61} \) \( -\beta q^{62} \) \( + 7 q^{64} \) \( + 12 q^{67} \) \( + ( 1 - \beta ) q^{68} \) \( + 3 \beta q^{71} \) \( + 2 \beta q^{74} \) \( + 4 q^{76} \) \( -16 q^{77} \) \( + \beta q^{79} \) \( + 2 \beta q^{82} \) \( -12 q^{83} \) \( + 4 q^{86} \) \( + 3 \beta q^{88} \) \( + 10 q^{89} \) \( + 2 \beta q^{91} \) \( + \beta q^{92} \) \( -8 q^{94} \) \( -4 \beta q^{97} \) \( + 9 q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 10q^{32} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 16q^{47} \) \(\mathstrut -\mathstrut 18q^{49} \) \(\mathstrut -\mathstrut 10q^{50} \) \(\mathstrut -\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 14q^{64} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut +\mathstrut 2q^{68} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 32q^{77} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 16q^{94} \) \(\mathstrut +\mathstrut 18q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
1.00000i
1.00000i
−1.00000 0 −1.00000 0 0 4.00000i 3.00000 0 0
118.2 −1.00000 0 −1.00000 0 0 4.00000i 3.00000 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
17.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(153, [\chi])\).