# Properties

 Label 23.5.b.a Level $23$ Weight $5$ Character orbit 23.b Self dual yes Analytic conductor $2.378$ Analytic rank $0$ Dimension $3$ CM discriminant -23 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 23.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.37750915093$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} + 6 \beta_1) q^{3} + (\beta_{2} - 11 \beta_1 + 16) q^{4} + (17 \beta_{2} - 11 \beta_1 + 49) q^{6} + ( - 32 \beta_{2} + 16 \beta_1 - 79) q^{8} + (47 \beta_{2} + 22 \beta_1 + 81) q^{9}+O(q^{10})$$ q + (-2*b2 + b1) * q^2 + (-b2 + 6*b1) * q^3 + (b2 - 11*b1 + 16) * q^4 + (17*b2 - 11*b1 + 49) * q^6 + (-32*b2 + 16*b1 - 79) * q^8 + (47*b2 + 22*b1 + 81) * q^9 $$q + ( - 2 \beta_{2} + \beta_1) q^{2} + ( - \beta_{2} + 6 \beta_1) q^{3} + (\beta_{2} - 11 \beta_1 + 16) q^{4} + (17 \beta_{2} - 11 \beta_1 + 49) q^{6} + ( - 32 \beta_{2} + 16 \beta_1 - 79) q^{8} + (47 \beta_{2} + 22 \beta_1 + 81) q^{9} + ( - 98 \beta_{2} + 49 \beta_1 - 287) q^{12} + ( - 17 \beta_{2} - 122 \beta_1) q^{13} + (158 \beta_{2} - 79 \beta_1 + 256) q^{16} + ( - 49 \beta_{2} + 294 \beta_1 - 479) q^{18} + 529 q^{23} + (351 \beta_{2} - 650 \beta_1 + 784) q^{24} + 625 q^{25} + ( - 383 \beta_{2} + 37 \beta_1 - 511) q^{26} + ( - 81 \beta_{2} + 486 \beta_1 - 14) q^{27} + ( - 353 \beta_{2} - 506 \beta_1) q^{29} + (127 \beta_{2} + 694 \beta_1) q^{31} + ( - 79 \beta_{2} + 869 \beta_1 - 1264) q^{32} + (1039 \beta_{2} - 1370 \beta_1 + 1105) q^{36} + ( - 769 \beta_{2} - 746 \beta_1 - 2846) q^{39} + (991 \beta_{2} + 214 \beta_1) q^{41} + ( - 1058 \beta_{2} + 529 \beta_1) q^{46} + (943 \beta_{2} - 842 \beta_1) q^{47} + ( - 1599 \beta_{2} + 2405 \beta_1 - 3871) q^{48} + 2401 q^{49} + ( - 1250 \beta_{2} + 625 \beta_1) q^{50} + (1022 \beta_{2} - 511 \beta_1 + 5201) q^{52} + (1405 \beta_{2} - 905 \beta_1 + 3969) q^{54} + ( - 1871 \beta_{2} - 1259 \beta_1 + 1553) q^{58} - 6286 q^{59} + (2209 \beta_{2} - 59 \beta_1 + 2513) q^{62} + (2528 \beta_{2} - 1264 \beta_1 + 2145) q^{64} + ( - 529 \beta_{2} + 3174 \beta_1) q^{69} + (1327 \beta_{2} - 2858 \beta_1) q^{71} + ( - 4497 \beta_{2} + 2966 \beta_1 - 14063) q^{72} + ( - 3137 \beta_{2} - 554 \beta_1) q^{73} + ( - 625 \beta_{2} + 3750 \beta_1) q^{75} + (2685 \beta_{2} - 5945 \beta_1 + 5521) q^{78} + (14 \beta_{2} - 84 \beta_1 + 6561) q^{81} + (1633 \beta_{2} + 4741 \beta_1 - 11599) q^{82} + ( - 1777 \beta_{2} - 5354 \beta_1 - 8414) q^{87} + (529 \beta_{2} - 5819 \beta_1 + 8464) q^{92} + (4223 \beta_{2} + 4486 \beta_1 + 15826) q^{93} + ( - 1583 \beta_{2} + 5557 \beta_1 - 17311) q^{94} + (7742 \beta_{2} - 3871 \beta_1 + 22673) q^{96} + ( - 4802 \beta_{2} + 2401 \beta_1) q^{98}+O(q^{100})$$ q + (-2*b2 + b1) * q^2 + (-b2 + 6*b1) * q^3 + (b2 - 11*b1 + 16) * q^4 + (17*b2 - 11*b1 + 49) * q^6 + (-32*b2 + 16*b1 - 79) * q^8 + (47*b2 + 22*b1 + 81) * q^9 + (-98*b2 + 49*b1 - 287) * q^12 + (-17*b2 - 122*b1) * q^13 + (158*b2 - 79*b1 + 256) * q^16 + (-49*b2 + 294*b1 - 479) * q^18 + 529 * q^23 + (351*b2 - 650*b1 + 784) * q^24 + 625 * q^25 + (-383*b2 + 37*b1 - 511) * q^26 + (-81*b2 + 486*b1 - 14) * q^27 + (-353*b2 - 506*b1) * q^29 + (127*b2 + 694*b1) * q^31 + (-79*b2 + 869*b1 - 1264) * q^32 + (1039*b2 - 1370*b1 + 1105) * q^36 + (-769*b2 - 746*b1 - 2846) * q^39 + (991*b2 + 214*b1) * q^41 + (-1058*b2 + 529*b1) * q^46 + (943*b2 - 842*b1) * q^47 + (-1599*b2 + 2405*b1 - 3871) * q^48 + 2401 * q^49 + (-1250*b2 + 625*b1) * q^50 + (1022*b2 - 511*b1 + 5201) * q^52 + (1405*b2 - 905*b1 + 3969) * q^54 + (-1871*b2 - 1259*b1 + 1553) * q^58 - 6286 * q^59 + (2209*b2 - 59*b1 + 2513) * q^62 + (2528*b2 - 1264*b1 + 2145) * q^64 + (-529*b2 + 3174*b1) * q^69 + (1327*b2 - 2858*b1) * q^71 + (-4497*b2 + 2966*b1 - 14063) * q^72 + (-3137*b2 - 554*b1) * q^73 + (-625*b2 + 3750*b1) * q^75 + (2685*b2 - 5945*b1 + 5521) * q^78 + (14*b2 - 84*b1 + 6561) * q^81 + (1633*b2 + 4741*b1 - 11599) * q^82 + (-1777*b2 - 5354*b1 - 8414) * q^87 + (529*b2 - 5819*b1 + 8464) * q^92 + (4223*b2 + 4486*b1 + 15826) * q^93 + (-1583*b2 + 5557*b1 - 17311) * q^94 + (7742*b2 - 3871*b1 + 22673) * q^96 + (-4802*b2 + 2401*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9}+O(q^{10})$$ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 $$3 q + 48 q^{4} + 147 q^{6} - 237 q^{8} + 243 q^{9} - 861 q^{12} + 768 q^{16} - 1437 q^{18} + 1587 q^{23} + 2352 q^{24} + 1875 q^{25} - 1533 q^{26} - 42 q^{27} - 3792 q^{32} + 3315 q^{36} - 8538 q^{39} - 11613 q^{48} + 7203 q^{49} + 15603 q^{52} + 11907 q^{54} + 4659 q^{58} - 18858 q^{59} + 7539 q^{62} + 6435 q^{64} - 42189 q^{72} + 16563 q^{78} + 19683 q^{81} - 34797 q^{82} - 25242 q^{87} + 25392 q^{92} + 47478 q^{93} - 51933 q^{94} + 68019 q^{96}+O(q^{100})$$ 3 * q + 48 * q^4 + 147 * q^6 - 237 * q^8 + 243 * q^9 - 861 * q^12 + 768 * q^16 - 1437 * q^18 + 1587 * q^23 + 2352 * q^24 + 1875 * q^25 - 1533 * q^26 - 42 * q^27 - 3792 * q^32 + 3315 * q^36 - 8538 * q^39 - 11613 * q^48 + 7203 * q^49 + 15603 * q^52 + 11907 * q^54 + 4659 * q^58 - 18858 * q^59 + 7539 * q^62 + 6435 * q^64 - 42189 * q^72 + 16563 * q^78 + 19683 * q^81 - 34797 * q^82 - 25242 * q^87 + 25392 * q^92 + 47478 * q^93 - 51933 * q^94 + 68019 * 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
 −2.14510 2.66908 −0.523976
−7.63824 −15.6172 42.3427 0 119.288 0 −201.212 162.896 0
22.2 1.75927 15.5596 −12.9050 0 27.3735 0 −50.8517 161.100 0
22.3 5.87897 0.0576140 18.5623 0 0.338711 0 15.0635 −80.9967 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 23.5.b.a 3
3.b odd 2 1 207.5.d.a 3
4.b odd 2 1 368.5.f.a 3
23.b odd 2 1 CM 23.5.b.a 3
69.c even 2 1 207.5.d.a 3
92.b even 2 1 368.5.f.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.5.b.a 3 1.a even 1 1 trivial
23.5.b.a 3 23.b odd 2 1 CM
207.5.d.a 3 3.b odd 2 1
207.5.d.a 3 69.c even 2 1
368.5.f.a 3 4.b odd 2 1
368.5.f.a 3 92.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 48T_{2} + 79$$ acting on $$S_{5}^{\mathrm{new}}(23, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 48T + 79$$
$3$ $$T^{3} - 243T + 14$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3}$$
$13$ $$T^{3} - 85683 T + 8482894$$
$17$ $$T^{3}$$
$19$ $$T^{3}$$
$23$ $$(T - 529)^{3}$$
$29$ $$T^{3} - 2121843 T + 244330126$$
$31$ $$T^{3} - 2770563 T - 1677025154$$
$37$ $$T^{3}$$
$41$ $$T^{3} - 8477283 T + 7596282526$$
$43$ $$T^{3}$$
$47$ $$T^{3} - 14639043 T - 20606906306$$
$53$ $$T^{3}$$
$59$ $$(T + 6286)^{3}$$
$61$ $$T^{3}$$
$67$ $$T^{3}$$
$71$ $$T^{3} - 76235043 T - 188893891874$$
$73$ $$T^{3} - 85194723 T - 223017449186$$
$79$ $$T^{3}$$
$83$ $$T^{3}$$
$89$ $$T^{3}$$
$97$ $$T^{3}$$