Properties

Label 23.9.b.a
Level $23$
Weight $9$
Character orbit 23.b
Self dual yes
Analytic conductor $9.370$
Analytic rank $0$
Dimension $3$
CM discriminant -23
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 23.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(9.36970803141\)
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: \( 7 \)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 3 \beta_1) q^{2} + ( - 17 \beta_{2} + 13 \beta_1) q^{3} + ( - 47 \beta_{2} + 48 \beta_1 + 256) q^{4} + ( - 207 \beta_{2} - 208 \beta_1 - 191) q^{6} + ( - 512 \beta_{2} - 768 \beta_1 - 1951) q^{8} + (1359 \beta_{2} - 1075 \beta_1 + 6561) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 3 \beta_1) q^{2} + ( - 17 \beta_{2} + 13 \beta_1) q^{3} + ( - 47 \beta_{2} + 48 \beta_1 + 256) q^{4} + ( - 207 \beta_{2} - 208 \beta_1 - 191) q^{6} + ( - 512 \beta_{2} - 768 \beta_1 - 1951) q^{8} + (1359 \beta_{2} - 1075 \beta_1 + 6561) q^{9} + (382 \beta_{2} + 573 \beta_1 + 40897) q^{12} + ( - 1841 \beta_{2} + 6765 \beta_1) q^{13} + (3902 \beta_{2} + 5853 \beta_1 + 65536) q^{16} + (3247 \beta_{2} - 2483 \beta_1 + 19489) q^{18} + 279841 q^{23} + ( - 19825 \beta_{2} - 78611 \beta_1 - 48896) q^{24} + 390625 q^{25} + (4369 \beta_{2} - 108240 \beta_1 - 652831) q^{26} + ( - 111537 \beta_{2} + 85293 \beta_1 - 1062686) q^{27} + (148783 \beta_{2} + 71757 \beta_1) q^{29} + ( - 34577 \beta_{2} + 220077 \beta_1) q^{31} + (91697 \beta_{2} - 93648 \beta_1 - 499456) q^{32} + ( - 347345 \beta_{2} + 256461 \beta_1 - 1643135) q^{36} + (581103 \beta_{2} - 19987 \beta_1 + 3472834) q^{39} + (369583 \beta_{2} - 621267 \beta_1) q^{41} + ( - 559682 \beta_{2} - 839523 \beta_1) q^{46} + ( - 982961 \beta_{2} - 558195 \beta_1) q^{47} + ( - 710255 \beta_{2} + 1257776 \beta_1 + 372641) q^{48} + 5764801 q^{49} + ( - 781250 \beta_{2} - 1171875 \beta_1) q^{50} + (1305662 \beta_{2} + 1958493 \beta_1 + 12427169) q^{52} + (767245 \beta_{2} + 1823370 \beta_1 - 1253151) q^{54} + (2739313 \beta_{2} - 1148112 \beta_1 - 20221183) q^{58} + 15279074 q^{59} + (547153 \beta_{2} - 3521232 \beta_1 - 23237503) q^{62} + (998912 \beta_{2} + 1498368 \beta_1 - 12970815) q^{64} + ( - 4757297 \beta_{2} + 3637933 \beta_1) q^{69} + ( - 6416561 \beta_{2} + 1159245 \beta_1) q^{71} + ( - 1820177 \beta_{2} + 1461677 \beta_1 - 7811327) q^{72} + (3304303 \beta_{2} - 6401043 \beta_1) q^{73} + ( - 6640625 \beta_{2} + 5078125 \beta_1) q^{75} + (2252045 \beta_{2} - 10098710 \beta_1 - 43548671) q^{78} + (18065662 \beta_{2} - 13814918 \beta_1 + 43046721) q^{81} + (2806993 \beta_{2} + 9940272 \beta_1 + 44112449) q^{82} + (3134223 \beta_{2} + 12747917 \beta_1 - 43784126) q^{87} + ( - 13152527 \beta_{2} + 13432368 \beta_1 + 71639296) q^{92} + (18448623 \beta_{2} + 1298957 \beta_1 + 100851874) q^{93} + ( - 18518351 \beta_{2} + 8931120 \beta_1 + 143520929) q^{94} + ( - 745282 \beta_{2} - 1117923 \beta_1 - 79790047) q^{96} + ( - 11529602 \beta_{2} - 17294403 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 768 q^{4} - 573 q^{6} - 5853 q^{8} + 19683 q^{9} + 122691 q^{12} + 196608 q^{16} + 58467 q^{18} + 839523 q^{23} - 146688 q^{24} + 1171875 q^{25} - 1958493 q^{26} - 3188058 q^{27} - 1498368 q^{32} - 4929405 q^{36} + 10418502 q^{39} + 1117923 q^{48} + 17294403 q^{49} + 37281507 q^{52} - 3759453 q^{54} - 60663549 q^{58} + 45837222 q^{59} - 69712509 q^{62} - 38912445 q^{64} - 23433981 q^{72} - 130646013 q^{78} + 129140163 q^{81} + 132337347 q^{82} - 131352378 q^{87} + 214917888 q^{92} + 302555622 q^{93} + 430562787 q^{94} - 239370141 q^{96}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu^{2} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{2} + 3\nu + 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{2} + 3\beta _1 + 28 ) / 7 \) 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
−28.9050 80.1002 579.497 0 −2315.29 0 −9350.67 −144.953 0
22.2 2.56227 −161.997 −249.435 0 −415.079 0 −1295.06 19681.9 0
22.3 26.3427 81.8964 437.938 0 2157.37 0 4792.73 146.028 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 23.9.b.a 3
3.b odd 2 1 207.9.d.a 3
4.b odd 2 1 368.9.f.a 3
23.b odd 2 1 CM 23.9.b.a 3
69.c even 2 1 207.9.d.a 3
92.b even 2 1 368.9.f.a 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.9.b.a 3 1.a even 1 1 trivial
23.9.b.a 3 23.b odd 2 1 CM
207.9.d.a 3 3.b odd 2 1
207.9.d.a 3 69.c even 2 1
368.9.f.a 3 4.b odd 2 1
368.9.f.a 3 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 768T_{2} + 1951 \) acting on \(S_{9}^{\mathrm{new}}(23, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 768T + 1951 \) Copy content Toggle raw display
$3$ \( T^{3} - 19683 T + 1062686 \) 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
$13$ \( T^{3} - 2447192163 T - 25363320370274 \) Copy content Toggle raw display
$17$ \( T^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( (T - 279841)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 1500739238883 T + 64\!\cdots\!06 \) Copy content Toggle raw display
$31$ \( T^{3} - 2558673112323 T - 12\!\cdots\!94 \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 23954775687363 T - 12\!\cdots\!14 \) Copy content Toggle raw display
$43$ \( T^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 71433859985283 T - 19\!\cdots\!54 \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( (T - 15279074)^{3} \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 28\!\cdots\!94 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 39\!\cdots\!54 \) 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
show more
show less