Properties

Label 23.3.b.a
Level 23
Weight 3
Character orbit 23.b
Self dual Yes
Analytic conductor 0.627
Analytic rank 0
Dimension 3
CM disc. -23
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 23.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.626704608029\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( \beta_{1} + \beta_{2} ) q^{2} \) \( + ( -2 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{4} \) \( + ( -11 - 3 \beta_{1} + 2 \beta_{2} ) q^{6} \) \( + ( -7 + 4 \beta_{1} + 4 \beta_{2} ) q^{8} \) \( + ( 9 + 6 \beta_{1} - \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( \beta_{1} + \beta_{2} ) q^{2} \) \( + ( -2 \beta_{1} - \beta_{2} ) q^{3} \) \( + ( 4 + \beta_{1} - 2 \beta_{2} ) q^{4} \) \( + ( -11 - 3 \beta_{1} + 2 \beta_{2} ) q^{6} \) \( + ( -7 + 4 \beta_{1} + 4 \beta_{2} ) q^{8} \) \( + ( 9 + 6 \beta_{1} - \beta_{2} ) q^{9} \) \( + ( 1 - 11 \beta_{1} - 11 \beta_{2} ) q^{12} \) \( + ( -2 \beta_{1} + 7 \beta_{2} ) q^{13} \) \( + ( 16 - 7 \beta_{1} - 7 \beta_{2} ) q^{16} \) \( + ( 13 + 22 \beta_{1} + 11 \beta_{2} ) q^{18} \) \( -23 q^{23} \) \( + ( -44 + 2 \beta_{1} + 15 \beta_{2} ) q^{24} \) \( + 25 q^{25} \) \( + ( 29 - 11 \beta_{1} - 14 \beta_{2} ) q^{26} \) \( + ( -38 - 18 \beta_{1} - 9 \beta_{2} ) q^{27} \) \( + ( -2 \beta_{1} - 17 \beta_{2} ) q^{29} \) \( + ( 22 \beta_{1} + 7 \beta_{2} ) q^{31} \) \( + ( -28 - 7 \beta_{1} + 14 \beta_{2} ) q^{32} \) \( + ( 85 + 22 \beta_{1} - 5 \beta_{2} ) q^{36} \) \( + ( -14 + 6 \beta_{1} + 23 \beta_{2} ) q^{39} \) \( + ( -26 \beta_{1} + 7 \beta_{2} ) q^{41} \) \( + ( -23 \beta_{1} - 23 \beta_{2} ) q^{46} \) \( + ( 22 \beta_{1} - 17 \beta_{2} ) q^{47} \) \( + ( 77 - 11 \beta_{1} - 30 \beta_{2} ) q^{48} \) \( + 49 q^{49} \) \( + ( 25 \beta_{1} + 25 \beta_{2} ) q^{50} \) \( + ( -103 + 29 \beta_{1} + 29 \beta_{2} ) q^{52} \) \( + ( -99 - 65 \beta_{1} - 20 \beta_{2} ) q^{54} \) \( + ( -91 + 13 \beta_{1} + 34 \beta_{2} ) q^{58} \) \( + 26 q^{59} \) \( + ( 101 + 37 \beta_{1} - 14 \beta_{2} ) q^{62} \) \( + ( -15 - 28 \beta_{1} - 28 \beta_{2} ) q^{64} \) \( + ( 46 \beta_{1} + 23 \beta_{2} ) q^{69} \) \( + ( -26 \beta_{1} + 31 \beta_{2} ) q^{71} \) \( + ( -11 + 46 \beta_{1} + 51 \beta_{2} ) q^{72} \) \( + ( -26 \beta_{1} - 41 \beta_{2} ) q^{73} \) \( + ( -50 \beta_{1} - 25 \beta_{2} ) q^{75} \) \( + ( 133 - 25 \beta_{1} - 60 \beta_{2} ) q^{78} \) \( + ( 81 + 76 \beta_{1} + 38 \beta_{2} ) q^{81} \) \( + ( -43 - 59 \beta_{1} - 14 \beta_{2} ) q^{82} \) \( + ( 82 + 6 \beta_{1} - 49 \beta_{2} ) q^{87} \) \( + ( -92 - 23 \beta_{1} + 46 \beta_{2} ) q^{92} \) \( + ( -182 - 66 \beta_{1} - \beta_{2} ) q^{93} \) \( + ( -19 + 61 \beta_{1} + 34 \beta_{2} ) q^{94} \) \( + ( -7 + 77 \beta_{1} + 77 \beta_{2} ) q^{96} \) \( + ( 49 \beta_{1} + 49 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 33q^{6} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 33q^{6} \) \(\mathstrut -\mathstrut 21q^{8} \) \(\mathstrut +\mathstrut 27q^{9} \) \(\mathstrut +\mathstrut 3q^{12} \) \(\mathstrut +\mathstrut 48q^{16} \) \(\mathstrut +\mathstrut 39q^{18} \) \(\mathstrut -\mathstrut 69q^{23} \) \(\mathstrut -\mathstrut 132q^{24} \) \(\mathstrut +\mathstrut 75q^{25} \) \(\mathstrut +\mathstrut 87q^{26} \) \(\mathstrut -\mathstrut 114q^{27} \) \(\mathstrut -\mathstrut 84q^{32} \) \(\mathstrut +\mathstrut 255q^{36} \) \(\mathstrut -\mathstrut 42q^{39} \) \(\mathstrut +\mathstrut 231q^{48} \) \(\mathstrut +\mathstrut 147q^{49} \) \(\mathstrut -\mathstrut 309q^{52} \) \(\mathstrut -\mathstrut 297q^{54} \) \(\mathstrut -\mathstrut 273q^{58} \) \(\mathstrut +\mathstrut 78q^{59} \) \(\mathstrut +\mathstrut 303q^{62} \) \(\mathstrut -\mathstrut 45q^{64} \) \(\mathstrut -\mathstrut 33q^{72} \) \(\mathstrut +\mathstrut 399q^{78} \) \(\mathstrut +\mathstrut 243q^{81} \) \(\mathstrut -\mathstrut 129q^{82} \) \(\mathstrut +\mathstrut 246q^{87} \) \(\mathstrut -\mathstrut 276q^{92} \) \(\mathstrut -\mathstrut 546q^{93} \) \(\mathstrut -\mathstrut 57q^{94} \) \(\mathstrut -\mathstrut 21q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(6\) \(x\mathstrut -\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−0.523976
−2.14510
2.66908
−3.72545 4.24943 9.87897 0 −15.8310 0 −21.9018 9.05761 0
22.2 0.601466 1.54364 −3.63824 0 0.928445 0 −4.59414 −6.61718 0
22.3 3.12398 −5.79306 5.75927 0 −18.0974 0 5.49593 24.5596 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
23.b Odd 1 CM by \(\Q(\sqrt{-23}) \) yes

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(23, [\chi])\).