# Properties

 Label 23.13.d.a Level $23$ Weight $13$ Character orbit 23.d Analytic conductor $21.022$ Analytic rank $0$ Dimension $230$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$23$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 23.d (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.0218577974$$ Analytic rank: $$0$$ Dimension: $$230$$ Relative dimension: $$23$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9}+O(q^{10})$$ 230 * q - 99 * q^2 - 329 * q^3 - 48371 * q^4 - 11 * q^5 - 45662 * q^6 - 11 * q^7 - 665422 * q^8 - 3565760 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$230 q - 99 q^{2} - 329 q^{3} - 48371 q^{4} - 11 q^{5} - 45662 q^{6} - 11 q^{7} - 665422 q^{8} - 3565760 q^{9} - 11 q^{10} - 11 q^{11} - 2474870 q^{12} - 8049129 q^{13} - 11 q^{14} - 62475963 q^{15} - 36781851 q^{16} - 80926571 q^{17} - 6061470 q^{18} + 111165109 q^{19} - 272498699 q^{20} - 600623771 q^{21} + 571777550 q^{23} + 2007958826 q^{24} + 537533342 q^{25} - 673402534 q^{26} + 976861513 q^{27} + 3957719029 q^{28} + 1704049911 q^{29} - 3340699659 q^{30} - 1272203913 q^{31} - 3285092075 q^{32} + 3117947349 q^{33} - 13971708113 q^{34} + 1229011837 q^{35} + 7602651625 q^{36} - 17898788171 q^{37} - 26779027286 q^{38} + 15514291129 q^{39} + 61210874989 q^{40} + 10806018039 q^{41} - 31821396761 q^{42} - 41404506011 q^{43} - 66452112342 q^{44} + 32831809005 q^{46} + 113777724940 q^{47} + 106596462218 q^{48} - 20816559970 q^{49} - 111742524936 q^{50} - 105406226651 q^{51} - 128565172520 q^{52} + 51469276309 q^{53} + 374890654986 q^{54} + 85037505685 q^{55} + 293616662416 q^{56} - 250324213931 q^{57} - 318984378637 q^{58} + 161752468527 q^{59} - 233131567141 q^{60} + 210924022837 q^{61} + 248275451378 q^{62} + 295298022789 q^{63} - 577532028430 q^{64} - 572376553067 q^{65} - 898520009904 q^{66} - 163067150411 q^{67} + 465583967511 q^{69} + 644411595290 q^{70} + 752742985959 q^{71} - 95647246505 q^{72} + 282922065111 q^{73} - 311901254588 q^{74} - 4026285059321 q^{75} - 845476021610 q^{76} + 2342664521989 q^{77} + 5018685734112 q^{78} + 686225579413 q^{79} - 1205174134802 q^{80} - 2186195321662 q^{81} - 871064811532 q^{82} - 415316705771 q^{83} + 7244741619995 q^{84} + 4797841175197 q^{85} - 200733189437 q^{86} - 3589216626407 q^{87} - 4135942853955 q^{88} - 3344097505331 q^{89} - 6933278864446 q^{90} - 84300700958 q^{92} + 7319743188062 q^{93} + 6803696836893 q^{94} + 1953813559237 q^{95} - 3263490949643 q^{96} - 2782724911571 q^{97} - 5026611935979 q^{98} - 7415031217451 q^{99}+O(q^{100})$$ 230 * q - 99 * q^2 - 329 * q^3 - 48371 * q^4 - 11 * q^5 - 45662 * q^6 - 11 * q^7 - 665422 * q^8 - 3565760 * q^9 - 11 * q^10 - 11 * q^11 - 2474870 * q^12 - 8049129 * q^13 - 11 * q^14 - 62475963 * q^15 - 36781851 * q^16 - 80926571 * q^17 - 6061470 * q^18 + 111165109 * q^19 - 272498699 * q^20 - 600623771 * q^21 + 571777550 * q^23 + 2007958826 * q^24 + 537533342 * q^25 - 673402534 * q^26 + 976861513 * q^27 + 3957719029 * q^28 + 1704049911 * q^29 - 3340699659 * q^30 - 1272203913 * q^31 - 3285092075 * q^32 + 3117947349 * q^33 - 13971708113 * q^34 + 1229011837 * q^35 + 7602651625 * q^36 - 17898788171 * q^37 - 26779027286 * q^38 + 15514291129 * q^39 + 61210874989 * q^40 + 10806018039 * q^41 - 31821396761 * q^42 - 41404506011 * q^43 - 66452112342 * q^44 + 32831809005 * q^46 + 113777724940 * q^47 + 106596462218 * q^48 - 20816559970 * q^49 - 111742524936 * q^50 - 105406226651 * q^51 - 128565172520 * q^52 + 51469276309 * q^53 + 374890654986 * q^54 + 85037505685 * q^55 + 293616662416 * q^56 - 250324213931 * q^57 - 318984378637 * q^58 + 161752468527 * q^59 - 233131567141 * q^60 + 210924022837 * q^61 + 248275451378 * q^62 + 295298022789 * q^63 - 577532028430 * q^64 - 572376553067 * q^65 - 898520009904 * q^66 - 163067150411 * q^67 + 465583967511 * q^69 + 644411595290 * q^70 + 752742985959 * q^71 - 95647246505 * q^72 + 282922065111 * q^73 - 311901254588 * q^74 - 4026285059321 * q^75 - 845476021610 * q^76 + 2342664521989 * q^77 + 5018685734112 * q^78 + 686225579413 * q^79 - 1205174134802 * q^80 - 2186195321662 * q^81 - 871064811532 * q^82 - 415316705771 * q^83 + 7244741619995 * q^84 + 4797841175197 * q^85 - 200733189437 * q^86 - 3589216626407 * q^87 - 4135942853955 * q^88 - 3344097505331 * q^89 - 6933278864446 * q^90 - 84300700958 * q^92 + 7319743188062 * q^93 + 6803696836893 * q^94 + 1953813559237 * q^95 - 3263490949643 * q^96 - 2782724911571 * q^97 - 5026611935979 * q^98 - 7415031217451 * 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 −120.139 35.2760i −90.7690 + 104.753i 9743.22 + 6261.59i −23272.0 3346.01i 14600.2 9382.96i −140828. + 64314.1i −613803. 708366.i 72897.8 + 507015.i 2.67784e6 + 1.22293e6i
5.2 −107.972 31.7035i −294.207 + 339.533i 7207.11 + 4631.73i 15513.5 + 2230.51i 42530.6 27332.7i 176390. 80554.6i −329484. 380244.i 46907.1 + 326246.i −1.60431e6 732666.i
5.3 −102.129 29.9877i 821.786 948.392i 6085.27 + 3910.76i 12183.7 + 1751.75i −112368. + 72214.7i −67347.8 + 30756.7i −218700. 252393.i −148483. 1.03272e6i −1.19178e6 544267.i
5.4 −90.4098 26.5467i −761.907 + 879.288i 4023.43 + 2585.70i 13366.6 + 1921.82i 92226.1 59270.1i −116992. + 53428.4i −42370.4 48898.1i −117012. 813840.i −1.15745e6 528590.i
5.5 −87.7954 25.7791i 198.298 228.848i 3597.70 + 2312.10i 1515.00 + 217.824i −23309.1 + 14979.9i −31197.4 + 14247.4i −10821.5 12488.7i 62582.6 + 435272.i −127395. 58179.2i
5.6 −77.3817 22.7213i 495.427 571.753i 2025.90 + 1301.96i −13357.6 1920.53i −51328.0 + 32986.5i 130792. 59730.6i 89139.4 + 102872.i −5821.98 40492.8i 989994. + 452115.i
5.7 −72.3634 21.2478i −762.412 + 879.870i 1339.21 + 860.659i −30802.3 4428.71i 73866.0 47470.8i 143265. 65427.1i 123673. + 142726.i −117268. 815615.i 2.13486e6 + 974958.i
5.8 −54.9700 16.1406i −407.479 + 470.255i −684.594 439.962i 1588.57 + 228.402i 29989.3 19273.0i −79757.6 + 36424.1i 184202. + 212581.i 20530.6 + 142794.i −83637.1 38195.8i
5.9 −35.7422 10.4949i 347.101 400.576i −2278.41 1464.25i 28946.6 + 4161.89i −16610.1 + 10674.7i −39526.9 + 18051.3i 165987. + 191560.i 35650.0 + 247951.i −990936. 452545.i
5.10 −32.4770 9.53610i 448.373 517.450i −2481.96 1595.06i −21831.2 3138.85i −19496.3 + 12529.5i −104777. + 47850.0i 156187. + 180249.i 8915.82 + 62010.9i 679079. + 310125.i
5.11 −15.4298 4.53059i −161.338 + 186.194i −3228.22 2074.65i 3924.46 + 564.252i 3332.97 2141.97i 124602. 56904.0i 83546.0 + 96417.2i 66993.7 + 465951.i −57997.1 26486.4i
5.12 −0.705443 0.207137i 890.238 1027.39i −3445.32 2214.17i −536.567 77.1467i −840.822 + 540.363i 114910. 52477.8i 3943.94 + 4551.55i −187373. 1.30321e6i 362.538 + 165.565i
5.13 8.53737 + 2.50680i −912.924 + 1053.57i −3379.17 2171.66i 21701.0 + 3120.14i −10435.0 + 6706.20i 94155.6 42999.4i −47271.9 54554.7i −200948. 1.39763e6i 177448. + 81037.8i
5.14 16.1031 + 4.72830i −479.913 + 553.849i −3208.82 2062.18i −13434.2 1931.55i −10346.9 + 6649.52i −82122.5 + 37504.1i −86938.4 100332.i −800.303 5566.23i −207200. 94625.2i
5.15 35.2606 + 10.3534i 563.490 650.302i −2309.66 1484.33i 2550.09 + 366.648i 26601.9 17096.0i −174986. + 79913.5i −164645. 190010.i −29740.0 206846.i 86121.7 + 39330.5i
5.16 47.4597 + 13.9354i 72.2874 83.4241i −1387.55 891.721i −16609.8 2388.12i 4593.29 2951.93i 138459. 63232.2i −186102. 214773.i 73897.8 + 513971.i −755015. 344804.i
5.17 53.6967 + 15.7668i 228.961 264.235i −811.028 521.216i 16209.3 + 2330.55i 16460.6 10578.6i 57943.8 26462.0i −185443. 214013.i 58234.9 + 405033.i 833643. + 380712.i
5.18 68.8275 + 20.2096i −392.302 + 452.741i 883.025 + 567.486i 18893.6 + 2716.49i −36150.9 + 23232.8i −145477. + 66436.9i −143103. 165150.i 24558.6 + 170809.i 1.24550e6 + 568801.i
5.19 85.4052 + 25.0772i 424.703 490.133i 3219.40 + 2068.98i −25609.1 3682.03i 48563.0 31209.5i −11513.2 + 5257.91i −15685.2 18101.7i 15774.0 + 109710.i −2.09481e6 956670.i
5.20 87.7164 + 25.7559i −741.301 + 855.507i 3585.04 + 2303.96i −13018.4 1871.76i −87058.6 + 55949.2i 29354.1 13405.6i 9910.44 + 11437.3i −106733. 742344.i −1.09372e6 499484.i
See next 80 embeddings (of 230 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.23 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 23.13.d.a 230
23.d odd 22 1 inner 23.13.d.a 230

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.13.d.a 230 1.a even 1 1 trivial
23.13.d.a 230 23.d odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{13}^{\mathrm{new}}(23, [\chi])$$.