# 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 $2$

# Related objects

## 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: $$x^{3} - 6x - 3$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} - 23 \beta_1) q^{2} + (119 \beta_{2} - 66 \beta_1) q^{3} + (433 \beta_{2} + 613 \beta_1 + 1024) q^{4} + (4361 \beta_{2} - 611 \beta_1 + 7249) q^{6} + ( - 2048 \beta_{2} - 23552 \beta_1 - 50407) q^{8} + (5903 \beta_{2} - 39674 \beta_1 + 59049) q^{9}+O(q^{10})$$ q + (-2*b2 - 23*b1) * q^2 + (119*b2 - 66*b1) * q^3 + (433*b2 + 613*b1 + 1024) * q^4 + (4361*b2 - 611*b1 + 7249) * q^6 + (-2048*b2 - 23552*b1 - 50407) * q^8 + (5903*b2 - 39674*b1 + 59049) * q^9 $$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})$$ q + (-2*b2 - 23*b1) * q^2 + (119*b2 - 66*b1) * q^3 + (433*b2 + 613*b1 + 1024) * q^4 + (4361*b2 - 611*b1 + 7249) * q^6 + (-2048*b2 - 23552*b1 - 50407) * q^8 + (5903*b2 - 39674*b1 + 59049) * q^9 + (-14498*b2 - 166727*b1 + 102961) * q^12 + (-218321*b2 - 74882*b1) * q^13 + (100814*b2 + 1159361*b1 + 1048576) * q^16 + (862631*b2 - 478434*b1 + 3635593) * q^18 - 6436343 * q^23 + (-1532769*b2 + 2701198*b1 + 7422976) * q^24 + 9765625 * q^25 + (-3885503*b2 + 6020149*b1 + 4337849) * q^26 + (7026831*b2 - 3897234*b1 + 19799482) * q^27 + (1170439*b2 - 14142914*b1) * q^29 + (997111*b2 + 20457814*b1) * q^31 + (-21826231*b2 - 30899491*b1 - 51616768) * q^32 + (18297031*b2 - 47421602*b1 - 7918175) * q^36 + (25424183*b2 + 62400838*b1 - 141610574) * q^39 + (23460967*b2 + 82807462*b1) * q^41 + (12872686*b2 + 148035889*b1) * q^46 + (-110682929*b2 + 65544982*b1) * q^47 + (-95044383*b2 - 38407339*b1 - 365400343) * q^48 + 282475249 * q^49 + (-19531250*b2 - 224609375*b1) * q^50 + (-8675698*b2 - 99770527*b1 - 584553943) * q^52 + (217913725*b2 - 491467025*b1 + 428046201) * q^54 + (326262169*b2 + 331334509*b1 + 1285737089) * q^58 + 1281228026 * q^59 + (-404686319*b2 - 530390459*b1 - 1830235039) * q^62 + (103233536*b2 + 1187185664*b1 + 1467123825) * q^64 + (-765924817*b2 + 424798638*b1) * q^69 + (97551247*b2 - 1246022426*b1) * q^71 + (585781623*b2 + 1509930902*b1 + 746364289) * q^72 + (-1110128873*b2 + 336680998*b1) * q^73 + (1162109375*b2 - 644531250*b1) * q^75 + (-391591875*b2 + 1213962775*b1 - 5336409407) * q^78 + (2356138358*b2 - 1306765812*b1 + 3486784401) * q^81 + (-1152432527*b2 - 2515944923*b1 - 7194600943) * q^82 + (2554405823*b2 - 1105387898*b1 + 6329678482) * q^87 + (-2786936519*b2 - 3945478259*b1 - 6590815232) * q^92 + (-3837542473*b2 + 781142398*b1 - 7057596182) * q^93 + (-4143517847*b2 + 464351101*b1 - 7116560599) * q^94 + (730800686*b2 + 8404207889*b1 - 5189955127) * q^96 + (-564950498*b2 - 6496930727*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3072 q^{4} + 21747 q^{6} - 151221 q^{8} + 177147 q^{9}+O(q^{10})$$ 3 * q + 3072 * q^4 + 21747 * q^6 - 151221 * q^8 + 177147 * q^9 $$3 q + 3072 q^{4} + 21747 q^{6} - 151221 q^{8} + 177147 q^{9} + 308883 q^{12} + 3145728 q^{16} + 10906779 q^{18} - 19309029 q^{23} + 22268928 q^{24} + 29296875 q^{25} + 13013547 q^{26} + 59398446 q^{27} - 154850304 q^{32} - 23754525 q^{36} - 424831722 q^{39} - 1096201029 q^{48} + 847425747 q^{49} - 1753661829 q^{52} + 1284138603 q^{54} + 3857211267 q^{58} + 3843684078 q^{59} - 5490705117 q^{62} + 4401371475 q^{64} + 2239092867 q^{72} - 16009228221 q^{78} + 10460353203 q^{81} - 21583802829 q^{82} + 18989035446 q^{87} - 19772445696 q^{92} - 21172788546 q^{93} - 21349681797 q^{94} - 15569865381 q^{96}+O(q^{100})$$ 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

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
 2.66908 −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: 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 23.11.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 $$S_{11}^{\mathrm{new}}(23, [\chi])$$.

## Hecke characteristic polynomials

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