# Properties

 Label 72.22.f.a Level $72$ Weight $22$ Character orbit 72.f Analytic conductor $201.224$ Analytic rank $0$ Dimension $84$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,Mod(35,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 72.f (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: $$201.223687887$$ Analytic rank: $$0$$ Dimension: $$84$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84 q + 2424084 q^{4}+O(q^{10})$$ 84 * q + 2424084 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$84 q + 2424084 q^{4} + 17057181612 q^{10} - 4099708064904 q^{16} + 92015527242864 q^{19} - 236011369239528 q^{22} + 80\!\cdots\!00 q^{25}+ \cdots - 16\!\cdots\!12 q^{97}+O(q^{100})$$ 84 * q + 2424084 * q^4 + 17057181612 * q^10 - 4099708064904 * q^16 + 92015527242864 * q^19 - 236011369239528 * q^22 + 8010864257812500 * q^25 - 3042238700022024 * q^28 + 74414158327329924 * q^34 + 75018327243317184 * q^40 + 707788847336454432 * q^43 - 162639476543943288 * q^46 - 5423745715136249124 * q^49 - 70566629446202808 * q^52 - 14491558288019904108 * q^58 - 28749968749508108064 * q^64 - 13653811098426494352 * q^67 + 140256684870035374392 * q^70 - 71781309378863778384 * q^73 + 376055720386305092400 * q^76 - 368362114062816367548 * q^82 - 592572120184082106720 * q^88 - 522206955652189957200 * q^91 - 528772551789914644488 * q^94 - 1686050813935078492512 * q^97

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1443.33 118.056i 0 2.06928e6 + 340787.i −2.14944e6 0 2.68546e8i −2.94643e9 7.36160e8i 0 3.10237e9 + 2.53754e8i
35.2 −1443.33 + 118.056i 0 2.06928e6 340787.i −2.14944e6 0 2.68546e8i −2.94643e9 + 7.36160e8i 0 3.10237e9 2.53754e8i
35.3 −1434.58 197.828i 0 2.01888e6 + 567599.i −3.03198e7 0 9.80063e8i −2.78396e9 1.21366e9i 0 4.34961e10 + 5.99809e9i
35.4 −1434.58 + 197.828i 0 2.01888e6 567599.i −3.03198e7 0 9.80063e8i −2.78396e9 + 1.21366e9i 0 4.34961e10 5.99809e9i
35.5 −1420.72 280.555i 0 1.93973e6 + 797180.i 1.26307e7 0 2.88458e8i −2.53216e9 1.67677e9i 0 −1.79447e10 3.54361e9i
35.6 −1420.72 + 280.555i 0 1.93973e6 797180.i 1.26307e7 0 2.88458e8i −2.53216e9 + 1.67677e9i 0 −1.79447e10 + 3.54361e9i
35.7 −1383.40 428.211i 0 1.73042e6 + 1.18477e6i 4.14728e7 0 2.70660e8i −1.88653e9 2.38000e9i 0 −5.73733e10 1.77591e10i
35.8 −1383.40 + 428.211i 0 1.73042e6 1.18477e6i 4.14728e7 0 2.70660e8i −1.88653e9 + 2.38000e9i 0 −5.73733e10 + 1.77591e10i
35.9 −1366.89 478.294i 0 1.63962e6 + 1.30755e6i −1.38142e7 0 1.09260e9i −1.61579e9 2.57150e9i 0 1.88825e10 + 6.60726e9i
35.10 −1366.89 + 478.294i 0 1.63962e6 1.30755e6i −1.38142e7 0 1.09260e9i −1.61579e9 + 2.57150e9i 0 1.88825e10 6.60726e9i
35.11 −1331.80 568.730i 0 1.45025e6 + 1.51487e6i −2.65834e7 0 1.06432e9i −1.06989e9 2.84231e9i 0 3.54039e10 + 1.51188e10i
35.12 −1331.80 + 568.730i 0 1.45025e6 1.51487e6i −2.65834e7 0 1.06432e9i −1.06989e9 + 2.84231e9i 0 3.54039e10 1.51188e10i
35.13 −1301.95 634.092i 0 1.29301e6 + 1.65112e6i 2.45910e7 0 1.11543e9i −6.36472e8 2.96956e9i 0 −3.20163e10 1.55929e10i
35.14 −1301.95 + 634.092i 0 1.29301e6 1.65112e6i 2.45910e7 0 1.11543e9i −6.36472e8 + 2.96956e9i 0 −3.20163e10 + 1.55929e10i
35.15 −1225.13 772.149i 0 904725. + 1.89196e6i −3.55045e7 0 7.25119e8i 3.52472e8 3.01648e9i 0 4.34975e10 + 2.74147e10i
35.16 −1225.13 + 772.149i 0 904725. 1.89196e6i −3.55045e7 0 7.25119e8i 3.52472e8 + 3.01648e9i 0 4.34975e10 2.74147e10i
35.17 −1203.79 805.009i 0 801072. + 1.93813e6i 2.79565e7 0 1.16976e9i 5.95886e8 2.97797e9i 0 −3.36537e10 2.25052e10i
35.18 −1203.79 + 805.009i 0 801072. 1.93813e6i 2.79565e7 0 1.16976e9i 5.95886e8 + 2.97797e9i 0 −3.36537e10 + 2.25052e10i
35.19 −1075.57 969.687i 0 216565. + 2.08594e6i 2.58036e6 0 9.12625e7i 1.78978e9 2.45358e9i 0 −2.77537e9 2.50215e9i
35.20 −1075.57 + 969.687i 0 216565. 2.08594e6i 2.58036e6 0 9.12625e7i 1.78978e9 + 2.45358e9i 0 −2.77537e9 + 2.50215e9i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 35.84 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.22.f.a 84
3.b odd 2 1 inner 72.22.f.a 84
8.d odd 2 1 inner 72.22.f.a 84
24.f even 2 1 inner 72.22.f.a 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.22.f.a 84 1.a even 1 1 trivial
72.22.f.a 84 3.b odd 2 1 inner
72.22.f.a 84 8.d odd 2 1 inner
72.22.f.a 84 24.f even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{22}^{\mathrm{new}}(72, [\chi])$$.