# Properties

 Label 23.15.d.a Level $23$ Weight $15$ Character orbit 23.d Analytic conductor $28.596$ Analytic rank $0$ Dimension $270$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 23.d (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.5956626749$$ Analytic rank: $$0$$ Dimension: $$270$$ Relative dimension: $$27$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9}+O(q^{10})$$ 270 * q + 173 * q^2 + 2149 * q^3 - 184115 * q^4 - 11 * q^5 + 44722 * q^6 - 11 * q^7 + 7241658 * q^8 - 39199166 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$270 q + 173 q^{2} + 2149 q^{3} - 184115 q^{4} - 11 q^{5} + 44722 q^{6} - 11 q^{7} + 7241658 q^{8} - 39199166 q^{9} - 11 q^{10} - 11 q^{11} + 123246346 q^{12} + 75796021 q^{13} - 11 q^{14} - 100263174 q^{15} - 2293744667 q^{16} + 412743804 q^{17} + 3858902346 q^{18} - 2351744978 q^{19} - 201490443 q^{20} + 13037325862 q^{21} - 16459019108 q^{23} - 11985092422 q^{24} + 53500912048 q^{25} - 15367300238 q^{26} - 59088035798 q^{27} - 113542201355 q^{28} - 13335164408 q^{29} + 229640542197 q^{30} + 13668629386 q^{31} - 19386720331 q^{32} + 231293232588 q^{33} - 596485598577 q^{34} + 178461474729 q^{35} - 509341530479 q^{36} - 531055292779 q^{37} + 1024048005354 q^{38} + 483189758359 q^{39} - 1388798125011 q^{40} - 93386524595 q^{41} - 515159244281 q^{42} + 1551162694301 q^{43} + 3235619676322 q^{44} - 4650794794275 q^{46} - 3470215856414 q^{47} - 1781351369326 q^{48} + 5936274217858 q^{49} + 7261092475944 q^{50} + 1291880338045 q^{51} - 8918366421904 q^{52} - 4071540211035 q^{53} - 23203855111662 q^{54} + 21106594800555 q^{55} + 27086747877288 q^{56} - 16763164950752 q^{57} - 36011600663437 q^{58} + 14392567819783 q^{59} + 57162393175019 q^{60} + 4364821463145 q^{61} - 22783954069470 q^{62} - 28024226583936 q^{63} - 24724647234718 q^{64} + 23743133734540 q^{65} + 58812349391400 q^{66} + 12322307029307 q^{67} - 35019783038946 q^{69} - 97427918060118 q^{70} - 50835887726728 q^{71} + 57733468922911 q^{72} + 7745762862791 q^{73} + 211572998537340 q^{74} + 12297756473794 q^{75} + 105996063920846 q^{76} - 117855211954146 q^{77} - 329907597606864 q^{78} - 114924580693791 q^{79} + 310536504942598 q^{80} + 308801320702838 q^{81} + 97168250777412 q^{82} - 131361125041938 q^{83} - 712029570237205 q^{84} - 103660002899365 q^{85} - 178873746859533 q^{86} + 284460279511999 q^{87} + 302606695327101 q^{88} + 310097614758481 q^{89} + 710096816884202 q^{90} - 350219490315254 q^{92} - 846851840300236 q^{93} - 599024596919251 q^{94} + 48176519704686 q^{95} - 228166812129947 q^{96} + 10\!\cdots\!66 q^{97}+ \cdots + 185682966518038 q^{99}+O(q^{100})$$ 270 * q + 173 * q^2 + 2149 * q^3 - 184115 * q^4 - 11 * q^5 + 44722 * q^6 - 11 * q^7 + 7241658 * q^8 - 39199166 * q^9 - 11 * q^10 - 11 * q^11 + 123246346 * q^12 + 75796021 * q^13 - 11 * q^14 - 100263174 * q^15 - 2293744667 * q^16 + 412743804 * q^17 + 3858902346 * q^18 - 2351744978 * q^19 - 201490443 * q^20 + 13037325862 * q^21 - 16459019108 * q^23 - 11985092422 * q^24 + 53500912048 * q^25 - 15367300238 * q^26 - 59088035798 * q^27 - 113542201355 * q^28 - 13335164408 * q^29 + 229640542197 * q^30 + 13668629386 * q^31 - 19386720331 * q^32 + 231293232588 * q^33 - 596485598577 * q^34 + 178461474729 * q^35 - 509341530479 * q^36 - 531055292779 * q^37 + 1024048005354 * q^38 + 483189758359 * q^39 - 1388798125011 * q^40 - 93386524595 * q^41 - 515159244281 * q^42 + 1551162694301 * q^43 + 3235619676322 * q^44 - 4650794794275 * q^46 - 3470215856414 * q^47 - 1781351369326 * q^48 + 5936274217858 * q^49 + 7261092475944 * q^50 + 1291880338045 * q^51 - 8918366421904 * q^52 - 4071540211035 * q^53 - 23203855111662 * q^54 + 21106594800555 * q^55 + 27086747877288 * q^56 - 16763164950752 * q^57 - 36011600663437 * q^58 + 14392567819783 * q^59 + 57162393175019 * q^60 + 4364821463145 * q^61 - 22783954069470 * q^62 - 28024226583936 * q^63 - 24724647234718 * q^64 + 23743133734540 * q^65 + 58812349391400 * q^66 + 12322307029307 * q^67 - 35019783038946 * q^69 - 97427918060118 * q^70 - 50835887726728 * q^71 + 57733468922911 * q^72 + 7745762862791 * q^73 + 211572998537340 * q^74 + 12297756473794 * q^75 + 105996063920846 * q^76 - 117855211954146 * q^77 - 329907597606864 * q^78 - 114924580693791 * q^79 + 310536504942598 * q^80 + 308801320702838 * q^81 + 97168250777412 * q^82 - 131361125041938 * q^83 - 712029570237205 * q^84 - 103660002899365 * q^85 - 178873746859533 * q^86 + 284460279511999 * q^87 + 302606695327101 * q^88 + 310097614758481 * q^89 + 710096816884202 * q^90 - 350219490315254 * q^92 - 846851840300236 * q^93 - 599024596919251 * q^94 + 48176519704686 * q^95 - 228166812129947 * q^96 + 1012889741736466 * q^97 + 1686581493323797 * q^98 + 185682966518038 * 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 −239.810 70.4144i 1474.41 1701.56i 38767.3 + 24914.3i 40294.5 + 5793.47i −473392. + 304231.i 341082. 155767.i −4.86086e6 5.60973e6i −40735.0 283318.i −9.25506e6 4.22664e6i
5.2 −215.886 63.3897i −2125.96 + 2453.49i 28805.2 + 18512.0i 7903.43 + 1136.34i 614491. 394909.i 96970.1 44284.8i −2.63109e6 3.03644e6i −819216. 5.69777e6i −1.63420e6 746316.i
5.3 −192.648 56.5664i −516.411 + 595.970i 20130.2 + 12936.9i −96544.3 13881.0i 133197. 85600.6i 133595. 61010.7i −992020. 1.14485e6i 592187. + 4.11875e6i 1.78138e7 + 8.13530e6i
5.4 −182.537 53.5978i 43.2855 49.9541i 16664.0 + 10709.3i 88332.7 + 12700.3i −10578.6 + 6798.48i −1.44754e6 + 661071.i −426644. 492373.i 680066. + 4.72996e6i −1.54433e7 7.05272e6i
5.5 −174.358 51.1962i 2314.56 2671.14i 13996.7 + 8995.12i −126019. 18118.8i −540315. + 347240.i −496327. + 226665.i −30212.8 34867.4i −1.09714e6 7.63076e6i 2.10449e7 + 9.61087e6i
5.6 −162.162 47.6150i 1111.61 1282.87i 10246.1 + 6584.79i 32770.0 + 4711.62i −241345. + 155103.i 862654. 393961.i 465328. + 537017.i 270616. + 1.88217e6i −5.08970e6 2.32439e6i
5.7 −125.080 36.7268i −1316.56 + 1519.39i 513.046 + 329.714i 138756. + 19950.1i 220477. 141692.i 793580. 362416.i 1.34661e6 + 1.55407e6i 105471. + 733564.i −1.66229e7 7.59143e6i
5.8 −102.168 29.9991i 2406.75 2777.53i −4244.84 2727.99i 72015.7 + 10354.3i −329215. + 211573.i −51794.5 + 23653.7i 1.49430e6 + 1.72452e6i −1.24157e6 8.63534e6i −7.04705e6 3.21828e6i
5.9 −101.219 29.7205i −1577.18 + 1820.17i −4421.18 2841.32i −68025.4 9780.58i 213737. 137360.i 169076. 77214.4i 1.49491e6 + 1.72522e6i −144812. 1.00719e6i 6.59476e6 + 3.01173e6i
5.10 −76.7679 22.5411i 184.251 212.637i −8397.89 5397.00i −65621.6 9434.97i −18937.6 + 12170.5i −1.11550e6 + 509430.i 1.38147e6 + 1.59430e6i 669421. + 4.65593e6i 4.82496e6 + 2.20349e6i
5.11 −72.6742 21.3391i −2557.19 + 2951.16i −8956.91 5756.26i −5706.48 820.468i 248817. 159905.i −701947. + 320568.i 1.34076e6 + 1.54732e6i −1.48941e6 1.03591e7i 397206. + 181398.i
5.12 −53.9715 15.8475i 306.188 353.360i −11121.3 7147.24i −52369.4 7529.58i −22125.3 + 14219.1i 1.12010e6 511532.i 1.09049e6 + 1.25849e6i 649575. + 4.51790e6i 2.70713e6 + 1.23630e6i
5.13 −18.9656 5.56881i 1300.36 1500.69i −13454.4 8646.63i 40364.7 + 5803.57i −33019.1 + 21220.1i −455018. + 207800.i 419097. + 483664.i 119540. + 831420.i −733223. 334852.i
5.14 22.9608 + 6.74191i −1346.51 + 1553.96i −13301.4 8548.26i 81926.2 + 11779.2i −41393.6 + 26602.1i −122133. + 55776.2i −504531. 582260.i 78999.3 + 549452.i 1.80168e6 + 822799.i
5.15 41.2203 + 12.1034i 2006.50 2315.62i −12230.5 7860.05i −102785. 14778.2i 110735. 71165.2i 373207. 170438.i −869944. 1.00397e6i −655385. 4.55830e6i −4.05795e6 1.85320e6i
5.16 52.3513 + 15.3717i −198.734 + 229.351i −11278.7 7248.40i 76643.7 + 11019.7i −13929.5 + 8951.94i −297141. + 135700.i −1.06444e6 1.22843e6i 667581. + 4.64313e6i 3.84301e6 + 1.75504e6i
5.17 67.1845 + 19.7271i −2168.41 + 2502.47i −9658.50 6207.14i −59403.5 8540.93i −195050. + 125351.i 1.32939e6 607114.i −1.27772e6 1.47457e6i −879702. 6.11846e6i −3.82250e6 1.74568e6i
5.18 77.1586 + 22.6558i −1079.39 + 1245.68i −8342.93 5361.68i −146358. 21043.0i −111506. + 71660.5i −868722. + 396732.i −1.38506e6 1.59844e6i 294047. + 2.04514e6i −1.08160e7 4.93950e6i
5.19 110.343 + 32.3998i 1950.82 2251.36i −2657.17 1707.66i 132314. + 19023.9i 288204. 185217.i 1.09279e6 499061.i −1.47176e6 1.69850e6i −582262. 4.04972e6i 1.39836e7 + 6.38610e6i
5.20 133.250 + 39.1257i 2339.92 2700.41i 2441.62 + 1569.14i 9176.41 + 1319.37i 417450. 268279.i −1.39751e6 + 638220.i −1.22607e6 1.41496e6i −1.13631e6 7.90321e6i 1.17113e6 + 534839.i
See next 80 embeddings (of 270 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.27 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 23.15.d.a 270
23.d odd 22 1 inner 23.15.d.a 270

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.15.d.a 270 1.a even 1 1 trivial
23.15.d.a 270 23.d odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{15}^{\mathrm{new}}(23, [\chi])$$.