# Properties

 Label 23.5.d.a Level $23$ Weight $5$ Character orbit 23.d Analytic conductor $2.378$ Analytic rank $0$ Dimension $70$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 23.d (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.37750915093$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$7$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$70 q - 3 q^{2} + q^{3} - 35 q^{4} - 11 q^{5} - 110 q^{6} - 11 q^{7} + 146 q^{8} - 182 q^{9}+O(q^{10})$$ 70 * q - 3 * q^2 + q^3 - 35 * q^4 - 11 * q^5 - 110 * q^6 - 11 * q^7 + 146 * q^8 - 182 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$70 q - 3 q^{2} + q^{3} - 35 q^{4} - 11 q^{5} - 110 q^{6} - 11 q^{7} + 146 q^{8} - 182 q^{9} - 11 q^{10} - 11 q^{11} + 490 q^{12} - 7 q^{13} - 11 q^{14} + 1452 q^{15} - 139 q^{16} - 506 q^{17} - 2670 q^{18} - 968 q^{19} - 3179 q^{20} - 1364 q^{21} + 356 q^{23} + 2666 q^{24} + 3432 q^{25} + 5642 q^{26} + 4924 q^{27} + 5269 q^{28} - 378 q^{29} - 1419 q^{30} - 2800 q^{31} - 5771 q^{32} - 6666 q^{33} + 12815 q^{34} + 5797 q^{35} + 2425 q^{36} - 3531 q^{37} - 10406 q^{38} - 5453 q^{39} - 24211 q^{40} - 6591 q^{41} - 24761 q^{42} - 7227 q^{43} - 6534 q^{44} + 8301 q^{46} - 4706 q^{47} + 42122 q^{48} + 28806 q^{49} + 29544 q^{50} + 17413 q^{51} + 22872 q^{52} + 7381 q^{53} + 30522 q^{54} + 22055 q^{55} + 19888 q^{56} - 13310 q^{57} - 45421 q^{58} - 17613 q^{59} - 83941 q^{60} - 22495 q^{61} - 30910 q^{62} - 27786 q^{63} - 43726 q^{64} - 27962 q^{65} - 21648 q^{66} - 2937 q^{67} - 2064 q^{69} + 730 q^{70} + 42522 q^{71} + 116359 q^{72} + 5383 q^{73} + 85492 q^{74} + 103504 q^{75} + 122166 q^{76} + 38020 q^{77} + 41808 q^{78} + 10505 q^{79} - 21362 q^{80} - 94198 q^{81} - 19388 q^{82} - 65120 q^{83} - 221221 q^{84} - 105813 q^{85} - 141053 q^{86} - 61205 q^{87} - 41283 q^{88} - 22979 q^{89} - 37246 q^{90} - 21998 q^{92} - 25744 q^{93} + 128365 q^{94} + 146512 q^{95} + 211093 q^{96} + 115852 q^{97} + 84021 q^{98} + 163768 q^{99}+O(q^{100})$$ 70 * q - 3 * q^2 + q^3 - 35 * q^4 - 11 * q^5 - 110 * q^6 - 11 * q^7 + 146 * q^8 - 182 * q^9 - 11 * q^10 - 11 * q^11 + 490 * q^12 - 7 * q^13 - 11 * q^14 + 1452 * q^15 - 139 * q^16 - 506 * q^17 - 2670 * q^18 - 968 * q^19 - 3179 * q^20 - 1364 * q^21 + 356 * q^23 + 2666 * q^24 + 3432 * q^25 + 5642 * q^26 + 4924 * q^27 + 5269 * q^28 - 378 * q^29 - 1419 * q^30 - 2800 * q^31 - 5771 * q^32 - 6666 * q^33 + 12815 * q^34 + 5797 * q^35 + 2425 * q^36 - 3531 * q^37 - 10406 * q^38 - 5453 * q^39 - 24211 * q^40 - 6591 * q^41 - 24761 * q^42 - 7227 * q^43 - 6534 * q^44 + 8301 * q^46 - 4706 * q^47 + 42122 * q^48 + 28806 * q^49 + 29544 * q^50 + 17413 * q^51 + 22872 * q^52 + 7381 * q^53 + 30522 * q^54 + 22055 * q^55 + 19888 * q^56 - 13310 * q^57 - 45421 * q^58 - 17613 * q^59 - 83941 * q^60 - 22495 * q^61 - 30910 * q^62 - 27786 * q^63 - 43726 * q^64 - 27962 * q^65 - 21648 * q^66 - 2937 * q^67 - 2064 * q^69 + 730 * q^70 + 42522 * q^71 + 116359 * q^72 + 5383 * q^73 + 85492 * q^74 + 103504 * q^75 + 122166 * q^76 + 38020 * q^77 + 41808 * q^78 + 10505 * q^79 - 21362 * q^80 - 94198 * q^81 - 19388 * q^82 - 65120 * q^83 - 221221 * q^84 - 105813 * q^85 - 141053 * q^86 - 61205 * q^87 - 41283 * q^88 - 22979 * q^89 - 37246 * q^90 - 21998 * q^92 - 25744 * q^93 + 128365 * q^94 + 146512 * q^95 + 211093 * q^96 + 115852 * q^97 + 84021 * q^98 + 163768 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −6.23539 1.83088i 6.55829 7.56867i 22.0680 + 14.1822i −37.2451 5.35504i −54.7508 + 35.1862i −43.4159 + 19.8274i −43.5454 50.2541i −2.74610 19.0996i 222.434 + 101.582i
5.2 −5.33512 1.56653i −2.92721 + 3.37818i 12.5494 + 8.06502i 31.6604 + 4.55208i 20.9090 13.4374i 36.8795 16.8423i 3.94161 + 4.54886i 8.68396 + 60.3983i −161.781 73.8829i
5.3 −0.880608 0.258570i −5.57964 + 6.43925i −12.7514 8.19486i −17.0417 2.45023i 6.57847 4.22773i −31.0622 + 14.1856i 18.7264 + 21.6114i 1.19597 + 8.31815i 14.3735 + 6.56417i
5.4 −0.152216 0.0446948i 7.85234 9.06208i −13.4389 8.63665i 0.171500 + 0.0246579i −1.60028 + 1.02844i 52.7309 24.0814i 3.32183 + 3.83359i −8.93460 62.1415i −0.0250030 0.0114185i
5.5 4.08377 + 1.19910i 2.59819 2.99847i 1.77931 + 1.14349i 35.9653 + 5.17103i 14.2059 9.12957i −63.1747 + 28.8509i −38.7002 44.6624i 9.28727 + 64.5944i 140.674 + 64.2434i
5.6 5.24881 + 1.54119i −10.2768 + 11.8601i 11.7147 + 7.52859i 13.8853 + 1.99640i −72.2198 + 46.4128i 72.8498 33.2694i −7.43228 8.57731i −23.5210 163.592i 69.8043 + 31.8786i
5.7 6.27256 + 1.84179i 3.63756 4.19797i 22.4927 + 14.4552i −41.1469 5.91603i 30.5486 19.6324i 0.548908 0.250678i 45.9665 + 53.0482i 7.13641 + 49.6348i −247.200 112.893i
7.1 −4.13729 + 4.77469i −5.59893 + 3.59821i −3.40343 23.6714i 0.838951 0.383136i 5.98405 41.6200i −15.6319 53.2373i 42.0661 + 27.0342i −15.2477 + 33.3879i −1.64163 + 5.59087i
7.2 −3.88218 + 4.48028i 14.3217 9.20399i −2.72451 18.9494i 14.9307 6.81860i −14.3630 + 99.8968i 18.1967 + 61.9723i 15.6809 + 10.0775i 86.7488 189.953i −27.4143 + 93.3646i
7.3 −1.65795 + 1.91338i −0.476447 + 0.306194i 1.36482 + 9.49254i −24.0936 + 11.0032i 0.204062 1.41928i 12.1157 + 41.2622i −54.5034 35.0272i −33.5154 + 73.3884i 18.8928 64.3430i
7.4 0.246852 0.284882i 2.49375 1.60264i 2.25682 + 15.6965i 37.9584 17.3350i 0.159024 1.10604i −7.04577 23.9957i 10.1026 + 6.49253i −29.9983 + 65.6870i 4.43167 15.0929i
7.5 1.24584 1.43777i −13.2108 + 8.49010i 1.76196 + 12.2547i −8.32630 + 3.80249i −4.25173 + 29.5714i −1.18231 4.02657i 45.4215 + 29.1907i 68.7961 150.642i −4.90608 + 16.7086i
7.6 2.42804 2.80211i 10.9682 7.04881i 0.320606 + 2.22986i −27.0695 + 12.3622i 6.87962 47.8488i −5.49864 18.7267i 56.9329 + 36.5886i 36.9662 80.9446i −31.0856 + 105.868i
7.7 4.33983 5.00844i −1.84102 + 1.18315i −3.97323 27.6344i 8.83387 4.03429i −2.06398 + 14.3553i 6.57566 + 22.3946i −66.4470 42.7029i −31.6591 + 69.3238i 18.1320 61.7520i
10.1 −4.13729 4.77469i −5.59893 3.59821i −3.40343 + 23.6714i 0.838951 + 0.383136i 5.98405 + 41.6200i −15.6319 + 53.2373i 42.0661 27.0342i −15.2477 33.3879i −1.64163 5.59087i
10.2 −3.88218 4.48028i 14.3217 + 9.20399i −2.72451 + 18.9494i 14.9307 + 6.81860i −14.3630 99.8968i 18.1967 61.9723i 15.6809 10.0775i 86.7488 + 189.953i −27.4143 93.3646i
10.3 −1.65795 1.91338i −0.476447 0.306194i 1.36482 9.49254i −24.0936 11.0032i 0.204062 + 1.41928i 12.1157 41.2622i −54.5034 + 35.0272i −33.5154 73.3884i 18.8928 + 64.3430i
10.4 0.246852 + 0.284882i 2.49375 + 1.60264i 2.25682 15.6965i 37.9584 + 17.3350i 0.159024 + 1.10604i −7.04577 + 23.9957i 10.1026 6.49253i −29.9983 65.6870i 4.43167 + 15.0929i
10.5 1.24584 + 1.43777i −13.2108 8.49010i 1.76196 12.2547i −8.32630 3.80249i −4.25173 29.5714i −1.18231 + 4.02657i 45.4215 29.1907i 68.7961 + 150.642i −4.90608 16.7086i
10.6 2.42804 + 2.80211i 10.9682 + 7.04881i 0.320606 2.22986i −27.0695 12.3622i 6.87962 + 47.8488i −5.49864 + 18.7267i 56.9329 36.5886i 36.9662 + 80.9446i −31.0856 105.868i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.7 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.d odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.5.d.a 70
23.d odd 22 1 inner 23.5.d.a 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.5.d.a 70 1.a even 1 1 trivial
23.5.d.a 70 23.d odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(23, [\chi])$$.