Properties

Label 33.2.d.a
Level 33
Weight 2
Character orbit 33.d
Analytic conductor 0.264
Analytic rank 0
Dimension 2
CM disc. -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 33 = 3 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 33.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.26350632667\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{3} \) \( -2 q^{4} \) \( + ( 1 - 2 \beta ) q^{5} \) \( + ( -3 + \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{3} \) \( -2 q^{4} \) \( + ( 1 - 2 \beta ) q^{5} \) \( + ( -3 + \beta ) q^{9} \) \( + ( -1 + 2 \beta ) q^{11} \) \( -2 \beta q^{12} \) \( + ( 6 - \beta ) q^{15} \) \( + 4 q^{16} \) \( + ( -2 + 4 \beta ) q^{20} \) \( + ( 1 - 2 \beta ) q^{23} \) \( -6 q^{25} \) \( + ( -3 - 2 \beta ) q^{27} \) \( + 5 q^{31} \) \( + ( -6 + \beta ) q^{33} \) \( + ( 6 - 2 \beta ) q^{36} \) \( -7 q^{37} \) \( + ( 2 - 4 \beta ) q^{44} \) \( + ( 3 + 5 \beta ) q^{45} \) \( + ( -2 + 4 \beta ) q^{47} \) \( + 4 \beta q^{48} \) \( + 7 q^{49} \) \( + ( 4 - 8 \beta ) q^{53} \) \( + 11 q^{55} \) \( + ( 1 - 2 \beta ) q^{59} \) \( + ( -12 + 2 \beta ) q^{60} \) \( -8 q^{64} \) \( -13 q^{67} \) \( + ( 6 - \beta ) q^{69} \) \( + ( -5 + 10 \beta ) q^{71} \) \( -6 \beta q^{75} \) \( + ( 4 - 8 \beta ) q^{80} \) \( + ( 6 - 5 \beta ) q^{81} \) \( + ( -5 + 10 \beta ) q^{89} \) \( + ( -2 + 4 \beta ) q^{92} \) \( + 5 \beta q^{93} \) \( + 17 q^{97} \) \( + ( -3 - 5 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut q^{3} \) \(\mathstrut -\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut -\mathstrut 2q^{12} \) \(\mathstrut +\mathstrut 11q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut -\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 10q^{31} \) \(\mathstrut -\mathstrut 11q^{33} \) \(\mathstrut +\mathstrut 10q^{36} \) \(\mathstrut -\mathstrut 14q^{37} \) \(\mathstrut +\mathstrut 11q^{45} \) \(\mathstrut +\mathstrut 4q^{48} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 22q^{55} \) \(\mathstrut -\mathstrut 22q^{60} \) \(\mathstrut -\mathstrut 16q^{64} \) \(\mathstrut -\mathstrut 26q^{67} \) \(\mathstrut +\mathstrut 11q^{69} \) \(\mathstrut -\mathstrut 6q^{75} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut +\mathstrut 5q^{93} \) \(\mathstrut +\mathstrut 34q^{97} \) \(\mathstrut -\mathstrut 11q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
0.500000 1.65831i
0.500000 + 1.65831i
0 0.500000 1.65831i −2.00000 3.31662i 0 0 0 −2.50000 1.65831i 0
32.2 0 0.500000 + 1.65831i −2.00000 3.31662i 0 0 0 −2.50000 + 1.65831i 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
11.b Odd 1 CM by \(\Q(\sqrt{-11}) \) yes
3.b Odd 1 yes
33.d Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(33, [\chi])\).