# Properties

 Label 92.2.h.a Level $92$ Weight $2$ Character orbit 92.h Analytic conductor $0.735$ Analytic rank $0$ Dimension $100$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [92,2,Mod(7,92)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(92, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 19]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("92.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$92 = 2^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 92.h (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: $$0.734623698596$$ Analytic rank: $$0$$ Dimension: $$100$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$100 q - 7 q^{2} - 11 q^{4} - 22 q^{5} - 12 q^{6} - 10 q^{8} - 12 q^{9}+O(q^{10})$$ 100 * q - 7 * q^2 - 11 * q^4 - 22 * q^5 - 12 * q^6 - 10 * q^8 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$100 q - 7 q^{2} - 11 q^{4} - 22 q^{5} - 12 q^{6} - 10 q^{8} - 12 q^{9} - 11 q^{10} - 18 q^{12} - 18 q^{13} - 11 q^{14} + 5 q^{16} - 22 q^{17} - 24 q^{18} - 11 q^{20} - 22 q^{21} - 30 q^{24} - 16 q^{25} + 12 q^{26} - 11 q^{28} - 42 q^{29} - 11 q^{30} - 27 q^{32} - 22 q^{33} + 11 q^{34} + 77 q^{36} - 22 q^{37} + 44 q^{38} + 77 q^{40} - 10 q^{41} + 99 q^{42} + 66 q^{44} + 65 q^{46} + 38 q^{48} - 8 q^{49} + 30 q^{50} + 96 q^{52} - 22 q^{53} + 48 q^{54} + 44 q^{56} - 22 q^{57} + 79 q^{58} + 11 q^{60} - 22 q^{61} - 36 q^{62} + 10 q^{64} - 22 q^{65} - 44 q^{66} - 10 q^{69} + 34 q^{70} - 21 q^{72} - 18 q^{73} - 22 q^{74} - 66 q^{76} + 122 q^{77} - 66 q^{78} - 110 q^{80} + 8 q^{81} - 122 q^{82} - 165 q^{84} + 54 q^{85} - 121 q^{86} - 99 q^{88} + 22 q^{89} - 198 q^{90} - 86 q^{92} + 212 q^{93} - 61 q^{94} - 147 q^{96} + 22 q^{97} - 71 q^{98}+O(q^{100})$$ 100 * q - 7 * q^2 - 11 * q^4 - 22 * q^5 - 12 * q^6 - 10 * q^8 - 12 * q^9 - 11 * q^10 - 18 * q^12 - 18 * q^13 - 11 * q^14 + 5 * q^16 - 22 * q^17 - 24 * q^18 - 11 * q^20 - 22 * q^21 - 30 * q^24 - 16 * q^25 + 12 * q^26 - 11 * q^28 - 42 * q^29 - 11 * q^30 - 27 * q^32 - 22 * q^33 + 11 * q^34 + 77 * q^36 - 22 * q^37 + 44 * q^38 + 77 * q^40 - 10 * q^41 + 99 * q^42 + 66 * q^44 + 65 * q^46 + 38 * q^48 - 8 * q^49 + 30 * q^50 + 96 * q^52 - 22 * q^53 + 48 * q^54 + 44 * q^56 - 22 * q^57 + 79 * q^58 + 11 * q^60 - 22 * q^61 - 36 * q^62 + 10 * q^64 - 22 * q^65 - 44 * q^66 - 10 * q^69 + 34 * q^70 - 21 * q^72 - 18 * q^73 - 22 * q^74 - 66 * q^76 + 122 * q^77 - 66 * q^78 - 110 * q^80 + 8 * q^81 - 122 * q^82 - 165 * q^84 + 54 * q^85 - 121 * q^86 - 99 * q^88 + 22 * q^89 - 198 * q^90 - 86 * q^92 + 212 * q^93 - 61 * q^94 - 147 * q^96 + 22 * q^97 - 71 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.32329 + 0.498911i −0.998425 1.55358i 1.50217 1.32041i −3.48007 + 1.58929i 2.09630 + 1.55770i −2.92167 + 0.857881i −1.32904 + 2.49673i −0.170511 + 0.373368i 3.81221 3.83934i
7.2 −1.29560 0.566948i −0.145312 0.226109i 1.35714 + 1.46907i 1.05084 0.479903i 0.0600729 + 0.375331i 1.55538 0.456702i −0.925416 2.67275i 1.21624 2.66318i −1.63355 + 0.0259878i
7.3 −1.25282 + 0.656083i 1.27476 + 1.98356i 1.13911 1.64391i 2.16405 0.988288i −2.89842 1.64870i −2.09343 + 0.614686i −0.348560 + 2.80687i −1.06327 + 2.32823i −2.06276 + 2.65794i
7.4 −0.471110 + 1.33344i −1.27476 1.98356i −1.55611 1.25639i 2.16405 0.988288i 3.24551 0.765333i 2.09343 0.614686i 2.40842 1.48308i −1.06327 + 2.32823i 0.298314 + 3.35122i
7.5 −0.312716 1.37921i 1.44811 + 2.25330i −1.80442 + 0.862599i 0.189929 0.0867378i 2.65492 2.70189i 2.09970 0.616528i 1.75397 + 2.21892i −1.73411 + 3.79717i −0.179023 0.234827i
7.6 −0.305510 + 1.38082i 0.998425 + 1.55358i −1.81333 0.843709i −3.48007 + 1.58929i −2.45024 + 0.904011i 2.92167 0.857881i 1.71900 2.24612i −0.170511 + 0.373368i −1.13133 5.29089i
7.7 0.550867 1.30252i −0.874133 1.36018i −1.39309 1.43503i −1.57961 + 0.721385i −2.25318 + 0.389295i 3.24497 0.952810i −2.63655 + 1.02401i 0.160270 0.350942i 0.0694584 + 2.45486i
7.8 0.745560 + 1.20172i 0.145312 + 0.226109i −0.888280 + 1.79191i 1.05084 0.479903i −0.163382 + 0.343203i −1.55538 + 0.456702i −2.81565 + 0.268514i 1.21624 2.66318i 1.36018 + 0.905024i
7.9 1.21086 0.730627i 0.874133 + 1.36018i 0.932368 1.76938i −1.57961 + 0.721385i 2.05224 + 1.00832i −3.24497 + 0.952810i −0.163786 2.82368i 0.160270 0.350942i −1.38563 + 2.02761i
7.10 1.40967 + 0.113251i −1.44811 2.25330i 1.97435 + 0.319294i 0.189929 0.0867378i −1.78617 3.34042i −2.09970 + 0.616528i 2.74702 + 0.673698i −1.73411 + 3.79717i 0.277561 0.100762i
11.1 −1.39348 + 0.241286i 2.01686 + 0.289980i 1.88356 0.672454i −0.680282 2.31683i −2.88042 + 0.0825585i 0.800569 + 0.923906i −2.46245 + 1.39153i 1.10515 + 0.324501i 1.50698 + 3.06431i
11.2 −1.35313 0.411128i −2.71502 0.390360i 1.66195 + 1.11262i 0.232350 + 0.791311i 3.51329 + 1.64443i 2.90308 + 3.35033i −1.79141 2.18880i 4.34045 + 1.27447i 0.0109292 1.16628i
11.3 −1.04738 0.950264i 1.65123 + 0.237411i 0.193997 + 1.99057i 0.934364 + 3.18215i −1.50386 1.81776i −0.148303 0.171151i 1.68838 2.26922i −0.208284 0.0611576i 2.04525 4.22080i
11.4 −0.798353 + 1.16732i −2.01686 0.289980i −0.725264 1.86386i −0.680282 2.31683i 1.94866 2.12281i −0.800569 0.923906i 2.75474 + 0.641408i 1.10515 + 0.324501i 3.24758 + 1.05554i
11.5 −0.665666 1.24775i −0.837775 0.120454i −1.11378 + 1.66117i −0.870000 2.96295i 0.407382 + 1.12552i −2.18927 2.52655i 2.81414 + 0.283931i −2.19112 0.643371i −3.11790 + 3.05788i
11.6 −0.188138 + 1.40164i 2.71502 + 0.390360i −1.92921 0.527404i 0.232350 + 0.791311i −1.05794 + 3.73204i −2.90308 3.35033i 1.10219 2.60484i 4.34045 + 1.27447i −1.15285 + 0.176796i
11.7 0.429294 + 1.34748i −1.65123 0.237411i −1.63141 + 1.15693i 0.934364 + 3.18215i −0.388957 2.32692i 0.148303 + 0.171151i −2.25930 1.70163i −0.208284 0.0611576i −3.88677 + 2.62512i
11.8 0.630624 1.26582i 0.942266 + 0.135477i −1.20463 1.59652i 0.224821 + 0.765671i 0.765707 1.10731i 0.326270 + 0.376536i −2.78058 + 0.518040i −2.00897 0.589886i 1.11098 + 0.198266i
11.9 0.858469 + 1.12385i 0.837775 + 0.120454i −0.526063 + 1.92957i −0.870000 2.96295i 0.583832 + 1.04494i 2.18927 + 2.52655i −2.62015 + 1.06526i −2.19112 0.643371i 2.58303 3.52134i
11.10 1.41341 0.0477935i −0.942266 0.135477i 1.99543 0.135103i 0.224821 + 0.765671i −1.33828 0.146450i −0.326270 0.376536i 2.81390 0.286325i −2.00897 0.589886i 0.354358 + 1.07146i
See all 100 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.10 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
23.d odd 22 1 inner
92.h even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.2.h.a 100
3.b odd 2 1 828.2.u.a 100
4.b odd 2 1 inner 92.2.h.a 100
12.b even 2 1 828.2.u.a 100
23.d odd 22 1 inner 92.2.h.a 100
69.g even 22 1 828.2.u.a 100
92.h even 22 1 inner 92.2.h.a 100
276.j odd 22 1 828.2.u.a 100

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.2.h.a 100 1.a even 1 1 trivial
92.2.h.a 100 4.b odd 2 1 inner
92.2.h.a 100 23.d odd 22 1 inner
92.2.h.a 100 92.h even 22 1 inner
828.2.u.a 100 3.b odd 2 1
828.2.u.a 100 12.b even 2 1
828.2.u.a 100 69.g even 22 1
828.2.u.a 100 276.j odd 22 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(92, [\chi])$$.