# Properties

 Label 108.4.l.a Level $108$ Weight $4$ Character orbit 108.l Analytic conductor $6.372$ Analytic rank $0$ Dimension $312$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(11,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 13]))

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

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

chi := DirichletCharacter("108.11");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 108.l (of order $$18$$, degree $$6$$, 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: $$6.37220628062$$ Analytic rank: $$0$$ Dimension: $$312$$ Relative dimension: $$52$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10})$$ 312 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 6 * q^6 - 9 * q^8 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9} - 3 q^{10} - 123 q^{12} - 12 q^{13} + 69 q^{14} - 6 q^{16} - 18 q^{17} + 351 q^{18} + 225 q^{20} - 12 q^{21} - 6 q^{22} - 300 q^{24} - 12 q^{25} - 12 q^{28} - 96 q^{29} - 207 q^{30} - 696 q^{32} + 858 q^{33} - 30 q^{34} - 1056 q^{36} - 6 q^{37} - 900 q^{38} - 381 q^{40} + 138 q^{41} + 2574 q^{42} + 2655 q^{44} - 672 q^{45} - 3 q^{46} - 435 q^{48} - 12 q^{49} - 2829 q^{50} + 1371 q^{52} - 4458 q^{54} - 2925 q^{56} + 660 q^{57} + 885 q^{58} + 966 q^{60} - 12 q^{61} + 1872 q^{62} - 3 q^{64} - 708 q^{65} + 3093 q^{66} + 2211 q^{68} - 1572 q^{69} - 1011 q^{70} - 4524 q^{72} - 6 q^{73} - 5883 q^{74} - 198 q^{76} - 996 q^{77} - 2976 q^{78} + 444 q^{81} - 12 q^{82} + 6324 q^{84} - 762 q^{85} + 8322 q^{86} + 1530 q^{88} + 4212 q^{89} - 1104 q^{90} - 3255 q^{92} + 7404 q^{93} + 2019 q^{94} + 582 q^{96} - 66 q^{97} + 2898 q^{98}+O(q^{100})$$ 312 * q - 6 * q^2 - 6 * q^4 - 12 * q^5 - 6 * q^6 - 9 * q^8 - 12 * q^9 - 3 * q^10 - 123 * q^12 - 12 * q^13 + 69 * q^14 - 6 * q^16 - 18 * q^17 + 351 * q^18 + 225 * q^20 - 12 * q^21 - 6 * q^22 - 300 * q^24 - 12 * q^25 - 12 * q^28 - 96 * q^29 - 207 * q^30 - 696 * q^32 + 858 * q^33 - 30 * q^34 - 1056 * q^36 - 6 * q^37 - 900 * q^38 - 381 * q^40 + 138 * q^41 + 2574 * q^42 + 2655 * q^44 - 672 * q^45 - 3 * q^46 - 435 * q^48 - 12 * q^49 - 2829 * q^50 + 1371 * q^52 - 4458 * q^54 - 2925 * q^56 + 660 * q^57 + 885 * q^58 + 966 * q^60 - 12 * q^61 + 1872 * q^62 - 3 * q^64 - 708 * q^65 + 3093 * q^66 + 2211 * q^68 - 1572 * q^69 - 1011 * q^70 - 4524 * q^72 - 6 * q^73 - 5883 * q^74 - 198 * q^76 - 996 * q^77 - 2976 * q^78 + 444 * q^81 - 12 * q^82 + 6324 * q^84 - 762 * q^85 + 8322 * q^86 + 1530 * q^88 + 4212 * q^89 - 1104 * q^90 - 3255 * q^92 + 7404 * q^93 + 2019 * q^94 + 582 * q^96 - 66 * q^97 + 2898 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.82000 + 0.218140i −2.56535 + 4.51874i 7.90483 1.23031i 2.35563 6.47204i 6.24857 13.3025i −6.19890 1.09303i −22.0233 + 5.19383i −13.8380 23.1843i −5.23107 + 18.7650i
11.2 −2.81863 + 0.235239i −0.238051 5.19070i 7.88933 1.32610i 5.65263 15.5305i 1.89203 + 14.5746i 28.8083 + 5.07968i −21.9251 + 5.59366i −26.8867 + 2.47130i −12.2793 + 45.1044i
11.3 −2.76593 + 0.591311i −5.08011 + 1.09199i 7.30070 3.27105i −6.87889 + 18.8996i 13.4055 6.02429i 22.1047 + 3.89766i −18.2590 + 13.3645i 24.6151 11.0949i 7.85096 56.3424i
11.4 −2.75026 0.660376i 2.16188 4.72507i 7.12781 + 3.63240i −3.60998 + 9.91833i −9.06605 + 11.5675i −12.1874 2.14897i −17.2045 14.6971i −17.6525 20.4301i 16.4782 24.8940i
11.5 −2.73813 0.708980i −4.61237 2.39291i 6.99469 + 3.88256i 0.577330 1.58620i 10.9327 + 9.82218i −27.0016 4.76111i −16.3997 15.5900i 15.5480 + 22.0740i −2.70539 + 3.93391i
11.6 −2.66507 + 0.947300i 5.14914 0.697370i 6.20524 5.04925i −1.80167 + 4.95006i −13.0622 + 6.73633i 14.7819 + 2.60644i −11.7543 + 19.3349i 26.0273 7.18172i 0.112404 14.8990i
11.7 −2.63308 1.03289i 5.16641 + 0.555160i 5.86626 + 5.43939i 6.86537 18.8624i −13.0302 6.79813i −18.0159 3.17669i −9.82807 20.3816i 26.3836 + 5.73637i −37.5600 + 42.5752i
11.8 −2.61271 + 1.08339i 2.79011 + 4.38352i 5.65252 5.66118i −1.04436 + 2.86937i −12.0388 8.43010i −28.5377 5.03197i −8.63512 + 20.9149i −11.4306 + 24.4610i −0.380029 8.62829i
11.9 −2.60460 1.10274i 2.16828 + 4.72213i 5.56791 + 5.74442i −0.836765 + 2.29899i −0.440216 14.6903i 22.6981 + 4.00229i −8.16758 21.1019i −17.5971 + 20.4779i 4.71464 5.06522i
11.10 −2.33178 + 1.60087i −4.92086 1.66888i 2.87442 7.46577i 3.67143 10.0872i 14.1460 3.98618i −6.04634 1.06613i 5.24922 + 22.0101i 21.4296 + 16.4247i 7.58728 + 29.3985i
11.11 −2.10858 + 1.88517i −1.35776 5.01562i 0.892234 7.95009i −5.18482 + 14.2452i 12.3183 + 8.01624i −9.21752 1.62530i 13.1060 + 18.4454i −23.3130 + 13.6200i −15.9220 39.8114i
11.12 −2.10574 1.88834i −3.74172 3.60549i 0.868318 + 7.95274i −1.60571 + 4.41165i 1.07071 + 14.6579i 23.1356 + 4.07942i 13.1890 18.3861i 1.00093 + 26.9814i 11.7119 6.25768i
11.13 −2.04068 1.95848i −2.83464 + 4.35486i 0.328747 + 7.99324i −6.07198 + 16.6826i 14.3135 3.33532i −22.8171 4.02326i 14.9837 16.9555i −10.9297 24.6889i 45.0635 22.1521i
11.14 −1.88673 + 2.10719i 2.88986 4.31841i −0.880530 7.95139i 3.28651 9.02962i 3.64735 + 14.2372i −18.0956 3.19074i 18.4164 + 13.1467i −10.2974 24.9592i 12.8264 + 23.9617i
11.15 −1.88377 2.10984i −4.43306 + 2.71072i −0.902841 + 7.94889i 5.32629 14.6339i 14.0700 + 4.24668i 6.67869 + 1.17763i 18.4716 13.0690i 12.3040 24.0335i −40.9086 + 16.3292i
11.16 −1.76579 2.20953i 4.72349 2.16532i −1.76401 + 7.80309i −1.80584 + 4.96151i −13.1250 6.61318i 9.40209 + 1.65784i 20.3560 9.88097i 17.6228 20.4558i 14.1513 4.77091i
11.17 −1.74755 + 2.22397i −2.88986 + 4.31841i −1.89211 7.77303i 3.28651 9.02962i −4.55384 13.9736i 18.0956 + 3.19074i 20.5936 + 9.37578i −10.2974 24.9592i 14.3383 + 23.0889i
11.18 −1.49038 + 2.40390i 1.35776 + 5.01562i −3.55752 7.16548i −5.18482 + 14.2452i −14.0807 4.21127i 9.21752 + 1.62530i 22.5272 + 2.12738i −23.3130 + 13.6200i −26.5167 33.6946i
11.19 −1.17164 + 2.57435i 4.92086 + 1.66888i −5.25451 6.03242i 3.67143 10.0872i −10.0618 + 10.7127i 6.04634 + 1.06613i 21.6859 6.45911i 21.4296 + 16.4247i 21.6662 + 21.2701i
11.20 −1.11352 2.60001i −0.629751 5.15785i −5.52015 + 5.79033i 4.68740 12.8785i −12.7092 + 7.38072i −21.3342 3.76180i 21.2017 + 7.90483i −26.2068 + 6.49632i −38.7038 + 2.15317i
See next 80 embeddings (of 312 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.52 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.4.l.a 312
4.b odd 2 1 inner 108.4.l.a 312
27.f odd 18 1 inner 108.4.l.a 312
108.l even 18 1 inner 108.4.l.a 312

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.l.a 312 1.a even 1 1 trivial
108.4.l.a 312 4.b odd 2 1 inner
108.4.l.a 312 27.f odd 18 1 inner
108.4.l.a 312 108.l even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(108, [\chi])$$.