# 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 discriminant -23 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.626704608029$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1 + 4) q^{4} + (2 \beta_{2} - 3 \beta_1 - 11) q^{6} + (4 \beta_{2} + 4 \beta_1 - 7) q^{8} + ( - \beta_{2} + 6 \beta_1 + 9) q^{9}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-b2 - 2*b1) * q^3 + (-2*b2 + b1 + 4) * q^4 + (2*b2 - 3*b1 - 11) * q^6 + (4*b2 + 4*b1 - 7) * q^8 + (-b2 + 6*b1 + 9) * q^9 $$q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{2} - 2 \beta_1) q^{3} + ( - 2 \beta_{2} + \beta_1 + 4) q^{4} + (2 \beta_{2} - 3 \beta_1 - 11) q^{6} + (4 \beta_{2} + 4 \beta_1 - 7) q^{8} + ( - \beta_{2} + 6 \beta_1 + 9) q^{9} + ( - 11 \beta_{2} - 11 \beta_1 + 1) q^{12} + (7 \beta_{2} - 2 \beta_1) q^{13} + ( - 7 \beta_{2} - 7 \beta_1 + 16) q^{16} + (11 \beta_{2} + 22 \beta_1 + 13) q^{18} - 23 q^{23} + (15 \beta_{2} + 2 \beta_1 - 44) q^{24} + 25 q^{25} + ( - 14 \beta_{2} - 11 \beta_1 + 29) q^{26} + ( - 9 \beta_{2} - 18 \beta_1 - 38) q^{27} + ( - 17 \beta_{2} - 2 \beta_1) q^{29} + (7 \beta_{2} + 22 \beta_1) q^{31} + (14 \beta_{2} - 7 \beta_1 - 28) q^{32} + ( - 5 \beta_{2} + 22 \beta_1 + 85) q^{36} + (23 \beta_{2} + 6 \beta_1 - 14) q^{39} + (7 \beta_{2} - 26 \beta_1) q^{41} + ( - 23 \beta_{2} - 23 \beta_1) q^{46} + ( - 17 \beta_{2} + 22 \beta_1) q^{47} + ( - 30 \beta_{2} - 11 \beta_1 + 77) q^{48} + 49 q^{49} + (25 \beta_{2} + 25 \beta_1) q^{50} + (29 \beta_{2} + 29 \beta_1 - 103) q^{52} + ( - 20 \beta_{2} - 65 \beta_1 - 99) q^{54} + (34 \beta_{2} + 13 \beta_1 - 91) q^{58} + 26 q^{59} + ( - 14 \beta_{2} + 37 \beta_1 + 101) q^{62} + ( - 28 \beta_{2} - 28 \beta_1 - 15) q^{64} + (23 \beta_{2} + 46 \beta_1) q^{69} + (31 \beta_{2} - 26 \beta_1) q^{71} + (51 \beta_{2} + 46 \beta_1 - 11) q^{72} + ( - 41 \beta_{2} - 26 \beta_1) q^{73} + ( - 25 \beta_{2} - 50 \beta_1) q^{75} + ( - 60 \beta_{2} - 25 \beta_1 + 133) q^{78} + (38 \beta_{2} + 76 \beta_1 + 81) q^{81} + ( - 14 \beta_{2} - 59 \beta_1 - 43) q^{82} + ( - 49 \beta_{2} + 6 \beta_1 + 82) q^{87} + (46 \beta_{2} - 23 \beta_1 - 92) q^{92} + ( - \beta_{2} - 66 \beta_1 - 182) q^{93} + (34 \beta_{2} + 61 \beta_1 - 19) q^{94} + (77 \beta_{2} + 77 \beta_1 - 7) q^{96} + (49 \beta_{2} + 49 \beta_1) q^{98}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-b2 - 2*b1) * q^3 + (-2*b2 + b1 + 4) * q^4 + (2*b2 - 3*b1 - 11) * q^6 + (4*b2 + 4*b1 - 7) * q^8 + (-b2 + 6*b1 + 9) * q^9 + (-11*b2 - 11*b1 + 1) * q^12 + (7*b2 - 2*b1) * q^13 + (-7*b2 - 7*b1 + 16) * q^16 + (11*b2 + 22*b1 + 13) * q^18 - 23 * q^23 + (15*b2 + 2*b1 - 44) * q^24 + 25 * q^25 + (-14*b2 - 11*b1 + 29) * q^26 + (-9*b2 - 18*b1 - 38) * q^27 + (-17*b2 - 2*b1) * q^29 + (7*b2 + 22*b1) * q^31 + (14*b2 - 7*b1 - 28) * q^32 + (-5*b2 + 22*b1 + 85) * q^36 + (23*b2 + 6*b1 - 14) * q^39 + (7*b2 - 26*b1) * q^41 + (-23*b2 - 23*b1) * q^46 + (-17*b2 + 22*b1) * q^47 + (-30*b2 - 11*b1 + 77) * q^48 + 49 * q^49 + (25*b2 + 25*b1) * q^50 + (29*b2 + 29*b1 - 103) * q^52 + (-20*b2 - 65*b1 - 99) * q^54 + (34*b2 + 13*b1 - 91) * q^58 + 26 * q^59 + (-14*b2 + 37*b1 + 101) * q^62 + (-28*b2 - 28*b1 - 15) * q^64 + (23*b2 + 46*b1) * q^69 + (31*b2 - 26*b1) * q^71 + (51*b2 + 46*b1 - 11) * q^72 + (-41*b2 - 26*b1) * q^73 + (-25*b2 - 50*b1) * q^75 + (-60*b2 - 25*b1 + 133) * q^78 + (38*b2 + 76*b1 + 81) * q^81 + (-14*b2 - 59*b1 - 43) * q^82 + (-49*b2 + 6*b1 + 82) * q^87 + (46*b2 - 23*b1 - 92) * q^92 + (-b2 - 66*b1 - 182) * q^93 + (34*b2 + 61*b1 - 19) * q^94 + (77*b2 + 77*b1 - 7) * q^96 + (49*b2 + 49*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9 $$3 q + 12 q^{4} - 33 q^{6} - 21 q^{8} + 27 q^{9} + 3 q^{12} + 48 q^{16} + 39 q^{18} - 69 q^{23} - 132 q^{24} + 75 q^{25} + 87 q^{26} - 114 q^{27} - 84 q^{32} + 255 q^{36} - 42 q^{39} + 231 q^{48} + 147 q^{49} - 309 q^{52} - 297 q^{54} - 273 q^{58} + 78 q^{59} + 303 q^{62} - 45 q^{64} - 33 q^{72} + 399 q^{78} + 243 q^{81} - 129 q^{82} + 246 q^{87} - 276 q^{92} - 546 q^{93} - 57 q^{94} - 21 q^{96}+O(q^{100})$$ 3 * q + 12 * q^4 - 33 * q^6 - 21 * q^8 + 27 * q^9 + 3 * q^12 + 48 * q^16 + 39 * q^18 - 69 * q^23 - 132 * q^24 + 75 * q^25 + 87 * q^26 - 114 * q^27 - 84 * q^32 + 255 * q^36 - 42 * q^39 + 231 * q^48 + 147 * q^49 - 309 * q^52 - 297 * q^54 - 273 * q^58 + 78 * q^59 + 303 * q^62 - 45 * q^64 - 33 * q^72 + 399 * q^78 + 243 * q^81 - 129 * q^82 + 246 * q^87 - 276 * q^92 - 546 * q^93 - 57 * q^94 - 21 * q^96

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 23.3.b.a 3
3.b odd 2 1 207.3.d.a 3
4.b odd 2 1 368.3.f.a 3
5.b even 2 1 575.3.d.b 3
5.c odd 4 2 575.3.c.a 6
8.b even 2 1 1472.3.f.a 3
8.d odd 2 1 1472.3.f.b 3
23.b odd 2 1 CM 23.3.b.a 3
69.c even 2 1 207.3.d.a 3
92.b even 2 1 368.3.f.a 3
115.c odd 2 1 575.3.d.b 3
115.e even 4 2 575.3.c.a 6
184.e odd 2 1 1472.3.f.a 3
184.h even 2 1 1472.3.f.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.3.b.a 3 1.a even 1 1 trivial
23.3.b.a 3 23.b odd 2 1 CM
207.3.d.a 3 3.b odd 2 1
207.3.d.a 3 69.c even 2 1
368.3.f.a 3 4.b odd 2 1
368.3.f.a 3 92.b even 2 1
575.3.c.a 6 5.c odd 4 2
575.3.c.a 6 115.e even 4 2
575.3.d.b 3 5.b even 2 1
575.3.d.b 3 115.c odd 2 1
1472.3.f.a 3 8.b even 2 1
1472.3.f.a 3 184.e odd 2 1
1472.3.f.b 3 8.d odd 2 1
1472.3.f.b 3 184.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(23, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 12T + 7$$
$3$ $$T^{3} - 27T + 38$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 507T - 1082$$
$17$ $$T^{3}$$
$19$ $$T^{3}$$
$23$ $$(T + 23)^{3}$$
$29$ $$T^{3} - 2523T - 30746$$
$31$ $$T^{3} - 2883T - 58754$$
$37$ $$T^{3}$$
$41$ $$T^{3} - 5043T - 43634$$
$43$ $$T^{3}$$
$47$ $$T^{3} - 6627 T + 205342$$
$53$ $$T^{3}$$
$59$ $$(T - 26)^{3}$$
$61$ $$T^{3}$$
$67$ $$T^{3}$$
$71$ $$T^{3} - 15123 T - 667154$$
$73$ $$T^{3} - 15987 T - 725042$$
$79$ $$T^{3}$$
$83$ $$T^{3}$$
$89$ $$T^{3}$$
$97$ $$T^{3}$$