# Properties

 Label 115.4.b.a Level $115$ Weight $4$ Character orbit 115.b Analytic conductor $6.785$ Analytic rank $0$ Dimension $34$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(24,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("115.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 115.b (of order $$2$$, degree $$1$$, 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: $$6.78521965066$$ Analytic rank: $$0$$ Dimension: $$34$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$34 q - 144 q^{4} + 14 q^{5} + 24 q^{6} - 310 q^{9}+O(q^{10})$$ 34 * q - 144 * q^4 + 14 * q^5 + 24 * q^6 - 310 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$34 q - 144 q^{4} + 14 q^{5} + 24 q^{6} - 310 q^{9} + 14 q^{10} - 8 q^{11} + 236 q^{14} + 440 q^{16} + 144 q^{19} - 180 q^{20} - 32 q^{21} + 108 q^{24} + 134 q^{25} - 144 q^{26} + 56 q^{29} - 294 q^{30} - 80 q^{31} + 264 q^{34} + 116 q^{35} + 1864 q^{36} - 1200 q^{39} + 650 q^{40} + 268 q^{41} - 1612 q^{44} - 1346 q^{45} + 184 q^{46} - 1474 q^{49} + 120 q^{50} - 1104 q^{51} + 1564 q^{54} + 1160 q^{55} - 2300 q^{56} - 708 q^{59} - 516 q^{60} + 1100 q^{61} + 100 q^{64} + 1164 q^{65} - 1416 q^{66} - 552 q^{69} + 1144 q^{70} + 1360 q^{71} + 1588 q^{74} - 2064 q^{75} + 108 q^{76} + 3968 q^{79} + 2542 q^{80} + 4914 q^{81} - 1948 q^{84} + 124 q^{85} - 6148 q^{86} + 1196 q^{89} + 2760 q^{90} - 544 q^{91} - 2340 q^{94} + 3920 q^{95} + 2960 q^{96} - 3816 q^{99}+O(q^{100})$$ 34 * q - 144 * q^4 + 14 * q^5 + 24 * q^6 - 310 * q^9 + 14 * q^10 - 8 * q^11 + 236 * q^14 + 440 * q^16 + 144 * q^19 - 180 * q^20 - 32 * q^21 + 108 * q^24 + 134 * q^25 - 144 * q^26 + 56 * q^29 - 294 * q^30 - 80 * q^31 + 264 * q^34 + 116 * q^35 + 1864 * q^36 - 1200 * q^39 + 650 * q^40 + 268 * q^41 - 1612 * q^44 - 1346 * q^45 + 184 * q^46 - 1474 * q^49 + 120 * q^50 - 1104 * q^51 + 1564 * q^54 + 1160 * q^55 - 2300 * q^56 - 708 * q^59 - 516 * q^60 + 1100 * q^61 + 100 * q^64 + 1164 * q^65 - 1416 * q^66 - 552 * q^69 + 1144 * q^70 + 1360 * q^71 + 1588 * q^74 - 2064 * q^75 + 108 * q^76 + 3968 * q^79 + 2542 * q^80 + 4914 * q^81 - 1948 * q^84 + 124 * q^85 - 6148 * q^86 + 1196 * q^89 + 2760 * q^90 - 544 * q^91 - 2340 * q^94 + 3920 * q^95 + 2960 * q^96 - 3816 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
24.1 5.19037i 7.60701i −18.9399 −0.939979 11.1408i 39.4832 25.9715i 56.7822i −30.8666 −57.8246 + 4.87884i
24.2 5.07225i 3.81839i −17.7277 −11.0224 1.87263i −19.3678 14.6908i 49.3412i 12.4199 −9.49845 + 55.9083i
24.3 5.04242i 0.0620551i −17.4260 1.76431 + 11.0403i −0.312908 30.6525i 47.5296i 26.9961 55.6696 8.89637i
24.4 4.94561i 5.42743i −16.4591 9.51102 + 5.87712i 26.8419 28.7748i 41.8353i −2.45696 29.0660 47.0378i
24.5 4.93784i 9.92307i −16.3822 10.8531 2.68512i −48.9985 14.6304i 41.3902i −71.4673 −13.2587 53.5909i
24.6 4.24824i 2.36809i −10.0475 4.85714 10.0702i −10.0602 12.8082i 8.69841i 21.3922 −42.7805 20.6343i
24.7 3.60232i 8.84330i −4.97671 −8.81390 + 6.87860i 31.8564 13.0981i 10.8909i −51.2040 24.7789 + 31.7505i
24.8 3.17951i 4.30546i −2.10929 −7.05197 8.67581i 13.6892 0.115569i 18.7296i 8.46305 −27.5848 + 22.4218i
24.9 3.13266i 8.19196i −1.81356 −9.71296 + 5.53701i −25.6626 13.4923i 19.3800i −40.1082 17.3456 + 30.4274i
24.10 3.00148i 1.85196i −1.00888 −6.22111 + 9.28966i 5.55862 4.63746i 20.9837i 23.5702 27.8827 + 18.6725i
24.11 2.81381i 0.460867i 0.0824876 11.0215 1.87796i −1.29679 16.1192i 22.7426i 26.7876 −5.28422 31.0124i
24.12 2.51792i 6.03797i 1.66008 6.02099 + 9.42060i −15.2031 24.0921i 24.3233i −9.45712 23.7203 15.1604i
24.13 1.98882i 9.68598i 4.04459 10.8383 + 2.74438i 19.2637 25.5696i 23.9545i −66.8182 5.45807 21.5554i
24.14 1.45307i 7.24162i 5.88858 −4.02716 10.4299i −10.5226 0.356556i 20.1811i −25.4411 −15.1553 + 5.85176i
24.15 0.752118i 6.31519i 7.43432 2.89657 10.7986i 4.74977 25.4579i 11.6084i −12.8816 −8.12183 2.17857i
24.16 0.407036i 3.62432i 7.83432 7.84032 + 7.97054i 1.47523 11.1245i 6.44514i 13.8643 3.24429 3.19129i
24.17 0.231361i 2.18914i 7.94647 −10.8138 2.83950i 0.506482 29.8467i 3.68940i 22.2077 −0.656949 + 2.50188i
24.18 0.231361i 2.18914i 7.94647 −10.8138 + 2.83950i 0.506482 29.8467i 3.68940i 22.2077 −0.656949 2.50188i
24.19 0.407036i 3.62432i 7.83432 7.84032 7.97054i 1.47523 11.1245i 6.44514i 13.8643 3.24429 + 3.19129i
24.20 0.752118i 6.31519i 7.43432 2.89657 + 10.7986i 4.74977 25.4579i 11.6084i −12.8816 −8.12183 + 2.17857i
See all 34 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.34 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.4.b.a 34
5.b even 2 1 inner 115.4.b.a 34
5.c odd 4 1 575.4.a.q 17
5.c odd 4 1 575.4.a.r 17

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.b.a 34 1.a even 1 1 trivial
115.4.b.a 34 5.b even 2 1 inner
575.4.a.q 17 5.c odd 4 1
575.4.a.r 17 5.c odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(115, [\chi])$$.