# Properties

 Label 23.15.b.b Level $23$ Weight $15$ Character orbit 23.b Analytic conductor $28.596$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 23.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.5956626749$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10})$$ 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100})$$ 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −241.960 570.908 42160.6 114189.i −138137. 270764.i −6.23690e6 −4.45703e6 2.76291e7i
22.2 −241.960 570.908 42160.6 114189.i −138137. 270764.i −6.23690e6 −4.45703e6 2.76291e7i
22.3 −181.785 3188.39 16661.8 67250.0i −579603. 311198.i −50505.8 5.38289e6 1.22251e7i
22.4 −181.785 3188.39 16661.8 67250.0i −579603. 311198.i −50505.8 5.38289e6 1.22251e7i
22.5 −176.800 −1555.37 14874.1 28617.6i 274989. 1.02277e6i 266944. −2.36379e6 5.05958e6i
22.6 −176.800 −1555.37 14874.1 28617.6i 274989. 1.02277e6i 266944. −2.36379e6 5.05958e6i
22.7 −115.892 −980.720 −2953.11 117312.i 113657. 1.26769e6i 2.24101e6 −3.82116e6 1.35955e7i
22.8 −115.892 −980.720 −2953.11 117312.i 113657. 1.26769e6i 2.24101e6 −3.82116e6 1.35955e7i
22.9 −73.1726 1838.22 −11029.8 72032.6i −134507. 1.03026e6i 2.00594e6 −1.40393e6 5.27081e6i
22.10 −73.1726 1838.22 −11029.8 72032.6i −134507. 1.03026e6i 2.00594e6 −1.40393e6 5.27081e6i
22.11 −54.1371 −3497.25 −13453.2 104018.i 189331. 363656.i 1.61530e6 7.44782e6 5.63123e6i
22.12 −54.1371 −3497.25 −13453.2 104018.i 189331. 363656.i 1.61530e6 7.44782e6 5.63123e6i
22.13 37.1439 4213.49 −15004.3 126427.i 156505. 1.01652e6i −1.16588e6 1.29705e7 4.69598e6i
22.14 37.1439 4213.49 −15004.3 126427.i 156505. 1.01652e6i −1.16588e6 1.29705e7 4.69598e6i
22.15 51.5135 −1848.44 −13730.4 58352.1i −95219.8 395426.i −1.55130e6 −1.36623e6 3.00593e6i
22.16 51.5135 −1848.44 −13730.4 58352.1i −95219.8 395426.i −1.55130e6 −1.36623e6 3.00593e6i
22.17 133.181 −181.571 1353.30 128567.i −24181.9 828509.i −2.00181e6 −4.75000e6 1.71228e7i
22.18 133.181 −181.571 1353.30 128567.i −24181.9 828509.i −2.00181e6 −4.75000e6 1.71228e7i
22.19 137.789 2037.01 2601.83 58362.5i 280678. 1.12900e6i −1.89903e6 −633549. 8.04171e6i
22.20 137.789 2037.01 2601.83 58362.5i 280678. 1.12900e6i −1.89903e6 −633549. 8.04171e6i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 22.24 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.15.b.b 24
23.b odd 2 1 inner 23.15.b.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.15.b.b 24 1.a even 1 1 trivial
23.15.b.b 24 23.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 92 T_{2}^{11} - 127810 T_{2}^{10} - 9865224 T_{2}^{9} + 5912332672 T_{2}^{8} + 366384186880 T_{2}^{7} - 123538605332480 T_{2}^{6} + \cdots + 47\!\cdots\!00$$ acting on $$S_{15}^{\mathrm{new}}(23, [\chi])$$.