Properties

Label 23.11.b.a
Level $23$
Weight $11$
Character orbit 23.b
Self dual yes
Analytic conductor $14.613$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6132168115\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 23 \beta_1) q^{2} + (119 \beta_{2} - 66 \beta_1) q^{3} + (433 \beta_{2} + 613 \beta_1 + 1024) q^{4} + (4361 \beta_{2} - 611 \beta_1 + 7249) q^{6} + ( - 2048 \beta_{2} - 23552 \beta_1 - 50407) q^{8} + (5903 \beta_{2} - 39674 \beta_1 + 59049) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 23 \beta_1) q^{2} + (119 \beta_{2} - 66 \beta_1) q^{3} + (433 \beta_{2} + 613 \beta_1 + 1024) q^{4} + (4361 \beta_{2} - 611 \beta_1 + 7249) q^{6} + ( - 2048 \beta_{2} - 23552 \beta_1 - 50407) q^{8} + (5903 \beta_{2} - 39674 \beta_1 + 59049) q^{9} + ( - 14498 \beta_{2} - 166727 \beta_1 + 102961) q^{12} + ( - 218321 \beta_{2} - 74882 \beta_1) q^{13} + (100814 \beta_{2} + 1159361 \beta_1 + 1048576) q^{16} + (862631 \beta_{2} - 478434 \beta_1 + 3635593) q^{18} - 6436343 q^{23} + ( - 1532769 \beta_{2} + 2701198 \beta_1 + 7422976) q^{24} + 9765625 q^{25} + ( - 3885503 \beta_{2} + 6020149 \beta_1 + 4337849) q^{26} + (7026831 \beta_{2} - 3897234 \beta_1 + 19799482) q^{27} + (1170439 \beta_{2} - 14142914 \beta_1) q^{29} + (997111 \beta_{2} + 20457814 \beta_1) q^{31} + ( - 21826231 \beta_{2} - 30899491 \beta_1 - 51616768) q^{32} + (18297031 \beta_{2} - 47421602 \beta_1 - 7918175) q^{36} + (25424183 \beta_{2} + 62400838 \beta_1 - 141610574) q^{39} + (23460967 \beta_{2} + 82807462 \beta_1) q^{41} + (12872686 \beta_{2} + 148035889 \beta_1) q^{46} + ( - 110682929 \beta_{2} + 65544982 \beta_1) q^{47} + ( - 95044383 \beta_{2} - 38407339 \beta_1 - 365400343) q^{48} + 282475249 q^{49} + ( - 19531250 \beta_{2} - 224609375 \beta_1) q^{50} + ( - 8675698 \beta_{2} - 99770527 \beta_1 - 584553943) q^{52} + (217913725 \beta_{2} - 491467025 \beta_1 + 428046201) q^{54} + (326262169 \beta_{2} + 331334509 \beta_1 + 1285737089) q^{58} + 1281228026 q^{59} + ( - 404686319 \beta_{2} - 530390459 \beta_1 - 1830235039) q^{62} + (103233536 \beta_{2} + 1187185664 \beta_1 + 1467123825) q^{64} + ( - 765924817 \beta_{2} + 424798638 \beta_1) q^{69} + (97551247 \beta_{2} - 1246022426 \beta_1) q^{71} + (585781623 \beta_{2} + 1509930902 \beta_1 + 746364289) q^{72} + ( - 1110128873 \beta_{2} + 336680998 \beta_1) q^{73} + (1162109375 \beta_{2} - 644531250 \beta_1) q^{75} + ( - 391591875 \beta_{2} + 1213962775 \beta_1 - 5336409407) q^{78} + (2356138358 \beta_{2} - 1306765812 \beta_1 + 3486784401) q^{81} + ( - 1152432527 \beta_{2} - 2515944923 \beta_1 - 7194600943) q^{82} + (2554405823 \beta_{2} - 1105387898 \beta_1 + 6329678482) q^{87} + ( - 2786936519 \beta_{2} - 3945478259 \beta_1 - 6590815232) q^{92} + ( - 3837542473 \beta_{2} + 781142398 \beta_1 - 7057596182) q^{93} + ( - 4143517847 \beta_{2} + 464351101 \beta_1 - 7116560599) q^{94} + (730800686 \beta_{2} + 8404207889 \beta_1 - 5189955127) q^{96} + ( - 564950498 \beta_{2} - 6496930727 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3072 q^{4} + 21747 q^{6} - 151221 q^{8} + 177147 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3072 q^{4} + 21747 q^{6} - 151221 q^{8} + 177147 q^{9} + 308883 q^{12} + 3145728 q^{16} + 10906779 q^{18} - 19309029 q^{23} + 22268928 q^{24} + 29296875 q^{25} + 13013547 q^{26} + 59398446 q^{27} - 154850304 q^{32} - 23754525 q^{36} - 424831722 q^{39} - 1096201029 q^{48} + 847425747 q^{49} - 1753661829 q^{52} + 1284138603 q^{54} + 3857211267 q^{58} + 3843684078 q^{59} - 5490705117 q^{62} + 4401371475 q^{64} + 2239092867 q^{72} - 16009228221 q^{78} + 10460353203 q^{81} - 21583802829 q^{82} + 18989035446 q^{87} - 19772445696 q^{92} - 21172788546 q^{93} - 21349681797 q^{94} - 15569865381 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display

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
2.66908
−0.523976
−2.14510
−62.2986 −122.026 2857.12 0 7602.03 0 −114201. −44158.7 0
22.2 18.4544 −346.393 −683.435 0 −6392.47 0 −31509.7 60938.9 0
22.3 43.8442 468.418 898.316 0 20537.4 0 −5510.51 160367. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 CM by \(\Q(\sqrt{-23}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.11.b.a 3
3.b odd 2 1 207.11.d.a 3
23.b odd 2 1 CM 23.11.b.a 3
69.c even 2 1 207.11.d.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.11.b.a 3 1.a even 1 1 trivial
23.11.b.a 3 23.b odd 2 1 CM
207.11.d.a 3 3.b odd 2 1
207.11.d.a 3 69.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3072T_{2} + 50407 \) acting on \(S_{11}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3072T + 50407 \) Copy content Toggle raw display
$3$ \( T^{3} - 177147 T - 19799482 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 413575475547 T - 96\!\cdots\!82 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T + 6436343)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 45\!\cdots\!74 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots - 31\!\cdots\!74 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 31\!\cdots\!74 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 17\!\cdots\!82 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T - 1281228026)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots + 32\!\cdots\!26 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots + 50\!\cdots\!18 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} \) Copy content Toggle raw display
$89$ \( T^{3} \) Copy content Toggle raw display
$97$ \( T^{3} \) Copy content Toggle raw display
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