# Properties

 Label 115.5.i.a Level $115$ Weight $5$ Character orbit 115.i Analytic conductor $11.888$ Analytic rank $0$ Dimension $460$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,5,Mod(14,115)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 21]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.14");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 115.i (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$11.8875457546$$ Analytic rank: $$0$$ Dimension: $$460$$ Relative dimension: $$46$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9}+O(q^{10})$$ 460 * q + 306 * q^4 - 11 * q^5 - 182 * q^6 + 944 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$460 q + 306 q^{4} - 11 q^{5} - 182 q^{6} + 944 q^{9} - 11 q^{10} - 22 q^{11} - 22 q^{14} - 1474 q^{15} - 2670 q^{16} - 22 q^{19} + 3157 q^{20} - 22 q^{21} + 3896 q^{24} - 1427 q^{25} - 3166 q^{26} - 2808 q^{29} + 1397 q^{30} + 2684 q^{31} - 12848 q^{34} + 2279 q^{35} + 1114 q^{36} + 12866 q^{39} - 11 q^{40} - 2652 q^{41} + 11110 q^{44} - 5394 q^{46} - 27026 q^{49} - 10931 q^{50} - 22 q^{51} - 51290 q^{54} - 10545 q^{55} + 25322 q^{56} + 41460 q^{59} + 49269 q^{60} - 22506 q^{61} + 68582 q^{64} + 27940 q^{65} - 24332 q^{66} + 7608 q^{69} - 15618 q^{70} + 35838 q^{71} - 59158 q^{74} + 30594 q^{75} - 58520 q^{76} - 17930 q^{79} - 63602 q^{80} + 91888 q^{81} + 92906 q^{84} - 33815 q^{85} + 101486 q^{86} - 22 q^{89} - 37598 q^{90} - 46242 q^{94} + 43336 q^{95} - 210180 q^{96} + 44528 q^{99}+O(q^{100})$$ 460 * q + 306 * q^4 - 11 * q^5 - 182 * q^6 + 944 * q^9 - 11 * q^10 - 22 * q^11 - 22 * q^14 - 1474 * q^15 - 2670 * q^16 - 22 * q^19 + 3157 * q^20 - 22 * q^21 + 3896 * q^24 - 1427 * q^25 - 3166 * q^26 - 2808 * q^29 + 1397 * q^30 + 2684 * q^31 - 12848 * q^34 + 2279 * q^35 + 1114 * q^36 + 12866 * q^39 - 11 * q^40 - 2652 * q^41 + 11110 * q^44 - 5394 * q^46 - 27026 * q^49 - 10931 * q^50 - 22 * q^51 - 51290 * q^54 - 10545 * q^55 + 25322 * q^56 + 41460 * q^59 + 49269 * q^60 - 22506 * q^61 + 68582 * q^64 + 27940 * q^65 - 24332 * q^66 + 7608 * q^69 - 15618 * q^70 + 35838 * q^71 - 59158 * q^74 + 30594 * q^75 - 58520 * q^76 - 17930 * q^79 - 63602 * q^80 + 91888 * q^81 + 92906 * q^84 - 33815 * q^85 + 101486 * q^86 - 22 * q^89 - 37598 * q^90 - 46242 * q^94 + 43336 * q^95 - 210180 * q^96 + 44528 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −2.18300 7.43463i −2.64484 + 2.29177i −37.0481 + 23.8094i −13.3410 21.1428i 22.8121 + 14.6605i −5.36934 + 11.7572i 164.196 + 142.276i −9.78452 + 68.0528i −128.066 + 145.340i
14.2 −2.10144 7.15685i 10.8466 9.39863i −33.3445 + 21.4292i 23.7360 7.84872i −90.0581 57.8768i 32.6308 71.4516i 133.243 + 115.456i 17.7869 123.711i −106.052 153.381i
14.3 −2.01699 6.86925i −6.92280 + 5.99864i −29.6583 + 19.0602i 3.87662 + 24.6976i 55.1695 + 35.4553i 37.5707 82.2683i 104.181 + 90.2729i 0.413989 2.87936i 161.835 76.4444i
14.4 −1.98199 6.75004i 5.91077 5.12171i −28.1747 + 18.1068i −0.890702 + 24.9841i −46.2868 29.7467i −18.3089 + 40.0910i 92.9960 + 80.5815i −2.82222 + 19.6290i 170.409 43.5060i
14.5 −1.80372 6.14291i −12.3789 + 10.7264i −21.0219 + 13.5100i −20.7399 + 13.9592i 88.2193 + 56.6951i −35.9524 + 78.7248i 43.4924 + 37.6864i 26.6544 185.386i 123.159 + 102.225i
14.6 −1.77123 6.03227i 11.0627 9.58586i −19.7910 + 12.7189i −22.7723 10.3161i −77.4191 49.7542i −22.6376 + 49.5694i 35.7565 + 30.9832i 18.9665 131.915i −21.8945 + 155.641i
14.7 −1.75464 5.97575i −7.06070 + 6.11814i −19.1708 + 12.3203i 24.2582 + 6.04485i 48.9494 + 31.4579i −12.9566 + 28.3710i 31.9516 + 27.6862i 0.894466 6.22115i −6.44183 155.567i
14.8 −1.72741 5.88302i −0.556901 + 0.482558i −18.1659 + 11.6745i 22.6244 10.6365i 3.80090 + 2.44269i −11.9218 + 26.1051i 25.9210 + 22.4607i −11.4502 + 79.6381i −101.657 114.726i
14.9 −1.60185 5.45541i 6.57117 5.69395i −13.7355 + 8.82725i −24.7386 + 3.60566i −41.5888 26.7275i 31.2269 68.3774i 1.40679 + 1.21899i −0.768314 + 5.34374i 59.2979 + 129.183i
14.10 −1.49289 5.08433i −10.7675 + 9.33009i −10.1616 + 6.53048i −3.03564 24.8150i 63.5120 + 40.8167i 25.1721 55.1192i −15.7018 13.6057i 17.3609 120.748i −121.636 + 52.4804i
14.11 −1.34542 4.58206i 4.34914 3.76855i −5.72511 + 3.67931i 4.19085 24.6462i −23.1191 14.8578i 7.38910 16.1799i −33.1839 28.7540i −6.81447 + 47.3957i −118.569 + 13.9567i
14.12 −1.33057 4.53150i −3.81868 + 3.30890i −5.30401 + 3.40868i −24.7617 + 3.44348i 20.0753 + 12.9016i 3.15920 6.91768i −34.6044 29.9848i −7.89404 + 54.9042i 48.5512 + 107.626i
14.13 −1.13341 3.86005i 2.81764 2.44150i −0.155323 + 0.0998199i 1.67039 + 24.9441i −12.6179 8.10901i −7.78355 + 17.0436i −48.0849 41.6658i −9.54933 + 66.4170i 94.3924 34.7198i
14.14 −1.04916 3.57310i 12.0617 10.4515i 1.79375 1.15277i 24.8811 2.43500i −49.9988 32.1323i −30.9341 + 67.7361i −51.0308 44.2185i 24.7226 171.949i −34.8047 86.3480i
14.15 −0.944999 3.21837i 6.93728 6.01119i 3.99518 2.56754i 17.7521 + 17.6030i −25.9019 16.6462i 19.4786 42.6523i −52.5982 45.5766i 0.463978 3.22704i 39.8771 73.7675i
14.16 −0.742368 2.52827i −9.67129 + 8.38022i 7.61900 4.89643i 19.4722 + 15.6790i 28.3671 + 18.2305i 8.49240 18.5958i −49.8982 43.2370i 11.7782 81.9194i 25.1853 60.8708i
14.17 −0.674293 2.29643i 0.167586 0.145214i 8.64114 5.55332i −14.8202 20.1336i −0.446476 0.286933i −39.0504 + 85.5085i −47.5202 41.1765i −11.5205 + 80.1269i −36.2423 + 47.6095i
14.18 −0.639671 2.17852i −10.2600 + 8.89037i 9.12329 5.86318i 13.0251 21.3388i 25.9309 + 16.6648i −26.3066 + 57.6035i −46.0637 39.9144i 14.7021 102.255i −54.8189 14.7256i
14.19 −0.617270 2.10223i −5.53313 + 4.79448i 9.42171 6.05496i −12.9412 + 21.3898i 13.4945 + 8.67241i −1.32728 + 2.90635i −45.0380 39.0256i −3.89907 + 27.1186i 52.9546 + 14.0021i
14.20 −0.488218 1.66272i 9.79680 8.48898i 10.9338 7.02672i −7.35574 23.8934i −18.8977 12.1448i 13.0669 28.6125i −37.9759 32.9063i 12.3871 86.1541i −36.1367 + 23.8957i
See next 80 embeddings (of 460 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.d odd 22 1 inner
115.i odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.5.i.a 460
5.b even 2 1 inner 115.5.i.a 460
23.d odd 22 1 inner 115.5.i.a 460
115.i odd 22 1 inner 115.5.i.a 460

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.5.i.a 460 1.a even 1 1 trivial
115.5.i.a 460 5.b even 2 1 inner
115.5.i.a 460 23.d odd 22 1 inner
115.5.i.a 460 115.i odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(115, [\chi])$$.