# Properties

 Label 432.2.k.c Level 432 Weight 2 Character orbit 432.k Analytic conductor 3.450 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 432.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 16q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 16q^{4} - 4q^{10} + 16q^{13} - 20q^{16} - 16q^{19} - 12q^{22} - 12q^{28} + 32q^{31} + 28q^{34} - 8q^{37} - 36q^{40} - 64q^{46} - 16q^{49} - 36q^{52} - 32q^{58} - 16q^{61} + 16q^{64} + 48q^{67} - 24q^{70} + 16q^{76} - 48q^{79} - 16q^{82} - 16q^{85} - 60q^{88} + 96q^{91} + 84q^{94} + 8q^{97} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.40946 + 0.115854i 0 1.97316 0.326583i −1.78448 + 1.78448i 0 4.77575i −2.74325 + 0.688904i 0 2.30841 2.72189i
109.2 −1.37261 0.340511i 0 1.76810 + 0.934777i 2.75203 2.75203i 0 0.113492i −2.10861 1.88514i 0 −4.71455 + 2.84036i
109.3 −1.20402 + 0.741841i 0 0.899344 1.78639i −0.431497 + 0.431497i 0 0.145064i 0.242383 + 2.81802i 0 0.199430 0.839634i
109.4 −1.15725 0.812885i 0 0.678437 + 1.88142i −2.39401 + 2.39401i 0 2.42931i 0.744255 2.72875i 0 4.71651 0.824404i
109.5 −0.936391 1.05980i 0 −0.246343 + 1.98477i 1.12951 1.12951i 0 1.33035i 2.33413 1.59745i 0 −2.25471 0.139389i
109.6 −0.680919 + 1.23950i 0 −1.07270 1.68799i −2.24693 + 2.24693i 0 3.93534i 2.82268 0.180219i 0 −1.25508 4.31505i
109.7 0.680919 1.23950i 0 −1.07270 1.68799i 2.24693 2.24693i 0 3.93534i −2.82268 + 0.180219i 0 −1.25508 4.31505i
109.8 0.936391 + 1.05980i 0 −0.246343 + 1.98477i −1.12951 + 1.12951i 0 1.33035i −2.33413 + 1.59745i 0 −2.25471 0.139389i
109.9 1.15725 + 0.812885i 0 0.678437 + 1.88142i 2.39401 2.39401i 0 2.42931i −0.744255 + 2.72875i 0 4.71651 0.824404i
109.10 1.20402 0.741841i 0 0.899344 1.78639i 0.431497 0.431497i 0 0.145064i −0.242383 2.81802i 0 0.199430 0.839634i
109.11 1.37261 + 0.340511i 0 1.76810 + 0.934777i −2.75203 + 2.75203i 0 0.113492i 2.10861 + 1.88514i 0 −4.71455 + 2.84036i
109.12 1.40946 0.115854i 0 1.97316 0.326583i 1.78448 1.78448i 0 4.77575i 2.74325 0.688904i 0 2.30841 2.72189i
325.1 −1.40946 0.115854i 0 1.97316 + 0.326583i −1.78448 1.78448i 0 4.77575i −2.74325 0.688904i 0 2.30841 + 2.72189i
325.2 −1.37261 + 0.340511i 0 1.76810 0.934777i 2.75203 + 2.75203i 0 0.113492i −2.10861 + 1.88514i 0 −4.71455 2.84036i
325.3 −1.20402 0.741841i 0 0.899344 + 1.78639i −0.431497 0.431497i 0 0.145064i 0.242383 2.81802i 0 0.199430 + 0.839634i
325.4 −1.15725 + 0.812885i 0 0.678437 1.88142i −2.39401 2.39401i 0 2.42931i 0.744255 + 2.72875i 0 4.71651 + 0.824404i
325.5 −0.936391 + 1.05980i 0 −0.246343 1.98477i 1.12951 + 1.12951i 0 1.33035i 2.33413 + 1.59745i 0 −2.25471 + 0.139389i
325.6 −0.680919 1.23950i 0 −1.07270 + 1.68799i −2.24693 2.24693i 0 3.93534i 2.82268 + 0.180219i 0 −1.25508 + 4.31505i
325.7 0.680919 + 1.23950i 0 −1.07270 + 1.68799i 2.24693 + 2.24693i 0 3.93534i −2.82268 0.180219i 0 −1.25508 + 4.31505i
325.8 0.936391 1.05980i 0 −0.246343 1.98477i −1.12951 1.12951i 0 1.33035i −2.33413 1.59745i 0 −2.25471 + 0.139389i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 325.12 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.k.c 24
3.b odd 2 1 inner 432.2.k.c 24
4.b odd 2 1 1728.2.k.c 24
12.b even 2 1 1728.2.k.c 24
16.e even 4 1 inner 432.2.k.c 24
16.f odd 4 1 1728.2.k.c 24
48.i odd 4 1 inner 432.2.k.c 24
48.k even 4 1 1728.2.k.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.c 24 1.a even 1 1 trivial
432.2.k.c 24 3.b odd 2 1 inner
432.2.k.c 24 16.e even 4 1 inner
432.2.k.c 24 48.i odd 4 1 inner
1728.2.k.c 24 4.b odd 2 1
1728.2.k.c 24 12.b even 2 1
1728.2.k.c 24 16.f odd 4 1
1728.2.k.c 24 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{24} + 510 T_{5}^{20} + 89055 T_{5}^{16} + 6358980 T_{5}^{12} + 163237055 T_{5}^{8} + 834181822 T_{5}^{4} + 112550881$$ $$T_{11}^{24} + 1494 T_{11}^{20} + 659871 T_{11}^{16} + 78092340 T_{11}^{12} + 1777212719 T_{11}^{8} + 2891176726 T_{11}^{4} + 244140625$$

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database