# Properties

 Label 23.6.c.a Level $23$ Weight $6$ Character orbit 23.c Analytic conductor $3.689$ Analytic rank $0$ Dimension $90$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 23.c (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68882785570$$ Analytic rank: $$0$$ Dimension: $$90$$ Relative dimension: $$9$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9}+O(q^{10})$$ 90 * q - 11 * q^2 - 7 * q^3 - 139 * q^4 + 5 * q^5 - 148 * q^6 - 29 * q^7 - 44 * q^8 - 894 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$90 q - 11 q^{2} - 7 q^{3} - 139 q^{4} + 5 q^{5} - 148 q^{6} - 29 q^{7} - 44 q^{8} - 894 q^{9} + 155 q^{10} - 89 q^{11} + 276 q^{12} + 313 q^{13} + 957 q^{14} - 4232 q^{15} + 5637 q^{16} + 5256 q^{17} + 11184 q^{18} - 3622 q^{19} - 2135 q^{20} - 6000 q^{21} - 10050 q^{22} - 11740 q^{23} - 20042 q^{24} - 3312 q^{25} + 432 q^{26} + 18236 q^{27} + 42219 q^{28} + 820 q^{29} + 39879 q^{30} - 10618 q^{31} - 44595 q^{32} - 52342 q^{33} - 48583 q^{34} - 12737 q^{35} + 62095 q^{36} + 37741 q^{37} + 93044 q^{38} + 51241 q^{39} + 54519 q^{40} + 27731 q^{41} - 74509 q^{42} - 24555 q^{43} - 93314 q^{44} - 97906 q^{45} - 178395 q^{46} - 72522 q^{47} - 84106 q^{48} - 60450 q^{49} + 100394 q^{50} + 92167 q^{51} + 289884 q^{52} + 37321 q^{53} + 74406 q^{54} + 24327 q^{55} + 73828 q^{56} + 210858 q^{57} - 46295 q^{58} + 12675 q^{59} + 40595 q^{60} + 169431 q^{61} + 126424 q^{62} + 48782 q^{63} - 152118 q^{64} - 204392 q^{65} - 289970 q^{66} - 68547 q^{67} - 619298 q^{68} - 269580 q^{69} - 616314 q^{70} - 116130 q^{71} - 90563 q^{72} - 41005 q^{73} - 39958 q^{74} - 89428 q^{75} - 54510 q^{76} + 473172 q^{77} + 749704 q^{78} + 562545 q^{79} + 1784628 q^{80} + 1038036 q^{81} + 361674 q^{82} + 235154 q^{83} + 314877 q^{84} - 216047 q^{85} - 896543 q^{86} - 516271 q^{87} - 213359 q^{88} - 497007 q^{89} - 1378102 q^{90} - 630132 q^{91} - 861506 q^{92} - 1401408 q^{93} - 497781 q^{94} - 605790 q^{95} - 1213923 q^{96} - 292308 q^{97} + 943825 q^{98} + 525394 q^{99}+O(q^{100})$$ 90 * q - 11 * q^2 - 7 * q^3 - 139 * q^4 + 5 * q^5 - 148 * q^6 - 29 * q^7 - 44 * q^8 - 894 * q^9 + 155 * q^10 - 89 * q^11 + 276 * q^12 + 313 * q^13 + 957 * q^14 - 4232 * q^15 + 5637 * q^16 + 5256 * q^17 + 11184 * q^18 - 3622 * q^19 - 2135 * q^20 - 6000 * q^21 - 10050 * q^22 - 11740 * q^23 - 20042 * q^24 - 3312 * q^25 + 432 * q^26 + 18236 * q^27 + 42219 * q^28 + 820 * q^29 + 39879 * q^30 - 10618 * q^31 - 44595 * q^32 - 52342 * q^33 - 48583 * q^34 - 12737 * q^35 + 62095 * q^36 + 37741 * q^37 + 93044 * q^38 + 51241 * q^39 + 54519 * q^40 + 27731 * q^41 - 74509 * q^42 - 24555 * q^43 - 93314 * q^44 - 97906 * q^45 - 178395 * q^46 - 72522 * q^47 - 84106 * q^48 - 60450 * q^49 + 100394 * q^50 + 92167 * q^51 + 289884 * q^52 + 37321 * q^53 + 74406 * q^54 + 24327 * q^55 + 73828 * q^56 + 210858 * q^57 - 46295 * q^58 + 12675 * q^59 + 40595 * q^60 + 169431 * q^61 + 126424 * q^62 + 48782 * q^63 - 152118 * q^64 - 204392 * q^65 - 289970 * q^66 - 68547 * q^67 - 619298 * q^68 - 269580 * q^69 - 616314 * q^70 - 116130 * q^71 - 90563 * q^72 - 41005 * q^73 - 39958 * q^74 - 89428 * q^75 - 54510 * q^76 + 473172 * q^77 + 749704 * q^78 + 562545 * q^79 + 1784628 * q^80 + 1038036 * q^81 + 361674 * q^82 + 235154 * q^83 + 314877 * q^84 - 216047 * q^85 - 896543 * q^86 - 516271 * q^87 - 213359 * q^88 - 497007 * q^89 - 1378102 * q^90 - 630132 * q^91 - 861506 * q^92 - 1401408 * q^93 - 497781 * q^94 - 605790 * q^95 - 1213923 * q^96 - 292308 * q^97 + 943825 * q^98 + 525394 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −8.49747 5.46100i 1.17717 + 8.18741i 29.0913 + 63.7011i −4.12458 1.21109i 34.7084 76.0009i 66.5045 76.7503i 54.6681 380.225i 167.509 49.1850i 28.4348 + 32.8155i
2.2 −5.98228 3.84458i −4.25176 29.5716i 7.71364 + 16.8905i 10.2972 + 3.02354i −88.2553 + 193.252i 60.9133 70.2977i −13.5930 + 94.5412i −623.247 + 183.002i −49.9767 57.6762i
2.3 −5.01311 3.22173i −0.293619 2.04217i 1.45843 + 3.19351i −10.1657 2.98491i −5.10736 + 11.1836i −159.381 + 183.936i −24.1608 + 168.042i 229.073 67.2618i 41.3450 + 47.7146i
2.4 −2.21730 1.42497i 4.20469 + 29.2443i −10.4074 22.7890i −83.1134 24.4043i 32.3492 70.8350i 22.1618 25.5761i −21.4006 + 148.845i −604.392 + 177.466i 149.512 + 172.546i
2.5 −1.85052 1.18926i 1.12337 + 7.81321i −11.2832 24.7067i 82.9428 + 24.3542i 7.21310 15.7945i 98.7062 113.913i −18.5206 + 128.813i 173.372 50.9067i −124.524 143.708i
2.6 1.87482 + 1.20487i −1.73793 12.0876i −11.2301 24.5904i −45.2299 13.2807i 11.3057 24.7560i 13.6774 15.7845i 18.7232 130.223i 90.0673 26.4461i −68.7963 79.3951i
2.7 5.39264 + 3.46564i 2.36907 + 16.4773i 3.77666 + 8.26972i 27.3965 + 8.04434i −44.3288 + 97.0664i −81.9154 + 94.5354i 20.8990 145.356i −32.7312 + 9.61074i 119.861 + 138.327i
2.8 6.83356 + 4.39166i −3.42127 23.7955i 14.1176 + 30.9132i 81.6604 + 23.9777i 81.1223 177.633i −26.6137 + 30.7138i −2.29389 + 15.9544i −321.363 + 94.3608i 452.730 + 522.478i
2.9 8.74318 + 5.61890i 0.603375 + 4.19657i 31.5779 + 69.1460i −63.3504 18.6013i −18.3047 + 40.0817i 143.409 165.503i −65.1024 + 452.797i 215.910 63.3968i −449.365 518.594i
3.1 −1.56617 10.8929i 3.45588 7.56732i −85.4995 + 25.1049i 43.9262 50.6935i −87.8428 25.7930i −42.6312 + 27.3974i 261.081 + 571.688i 113.810 + 131.344i −620.998 399.091i
3.2 −1.13164 7.87071i −6.81486 + 14.9225i −29.9637 + 8.79812i −54.8415 + 63.2904i 125.162 + 36.7510i −3.45670 + 2.22148i −2.54800 5.57934i −17.1066 19.7421i 560.201 + 360.019i
3.3 −0.696058 4.84119i 8.29077 18.1543i 7.75116 2.27595i −11.8362 + 13.6597i −93.6590 27.5008i −29.6881 + 19.0794i −81.4306 178.308i −101.709 117.378i 74.3679 + 47.7933i
3.4 −0.608938 4.23525i −6.59663 + 14.4446i 13.1372 3.85743i 52.7594 60.8875i 65.1935 + 19.1425i 55.4232 35.6183i −81.2163 177.839i −5.99980 6.92413i −290.001 186.373i
3.5 0.330150 + 2.29625i −5.46335 + 11.9631i 25.5400 7.49923i −32.3567 + 37.3416i −29.2739 8.59559i −167.445 + 107.610i 56.4907 + 123.697i 45.8643 + 52.9303i −96.4282 61.9706i
3.6 0.364275 + 2.53359i 3.17904 6.96113i 24.4174 7.16960i −2.80660 + 3.23899i 18.7947 + 5.51861i 130.259 83.7125i 61.0854 + 133.758i 120.780 + 139.388i −9.22864 5.93088i
3.7 0.991600 + 6.89673i 9.66731 21.1685i −15.8778 + 4.66215i 66.8592 77.1597i 155.579 + 45.6822i −163.317 + 104.958i 44.7249 + 97.9339i −195.515 225.637i 598.447 + 384.598i
3.8 1.13144 + 7.86930i −9.75312 + 21.3564i −29.9420 + 8.79177i 30.9167 35.6798i −179.095 52.5869i 34.5697 22.2166i 2.62201 + 5.74140i −201.839 232.935i 315.756 + 202.924i
3.9 1.41793 + 9.86195i 2.99089 6.54914i −64.5437 + 18.9517i −49.3449 + 56.9471i 68.8281 + 20.2098i 37.4214 24.0493i −145.974 319.639i 125.185 + 144.472i −631.577 405.890i
4.1 −4.47298 9.79445i −7.03817 + 2.06659i −54.9683 + 63.4367i 15.8872 + 10.2101i 51.7227 + 59.6912i −16.2855 113.268i 536.597 + 157.559i −159.160 + 102.286i 28.9391 201.276i
4.2 −2.60966 5.71435i −0.942040 + 0.276608i −4.88796 + 5.64100i −66.5354 42.7597i 4.03903 + 4.66129i 20.2510 + 140.849i −147.892 43.4250i −203.614 + 130.855i −70.7094 + 491.795i
See all 90 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.9 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.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.6.c.a 90
23.c even 11 1 inner 23.6.c.a 90
23.c even 11 1 529.6.a.j 45
23.d odd 22 1 529.6.a.k 45

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.c.a 90 1.a even 1 1 trivial
23.6.c.a 90 23.c even 11 1 inner
529.6.a.j 45 23.c even 11 1
529.6.a.k 45 23.d odd 22 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(23, [\chi])$$.