# Properties

 Label 432.2.be.c Level $432$ Weight $2$ Character orbit 432.be Analytic conductor $3.450$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(47,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 7]))

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

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

chi := DirichletCharacter("432.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.be (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: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{5} + 6 q^{9}+O(q^{10})$$ 36 * q + 3 * q^5 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 3 q^{5} + 6 q^{9} + 18 q^{11} - 9 q^{15} + 18 q^{21} - 9 q^{25} + 30 q^{29} - 27 q^{31} + 27 q^{33} - 27 q^{35} + 45 q^{39} + 18 q^{41} + 27 q^{45} - 45 q^{47} + 63 q^{51} - 9 q^{57} - 54 q^{59} + 63 q^{63} - 57 q^{65} - 63 q^{69} - 36 q^{71} + 9 q^{73} + 45 q^{75} - 81 q^{77} - 54 q^{81} + 27 q^{83} - 36 q^{85} - 45 q^{87} - 63 q^{89} - 27 q^{91} - 63 q^{93} + 72 q^{95} - 99 q^{99}+O(q^{100})$$ 36 * q + 3 * q^5 + 6 * q^9 + 18 * q^11 - 9 * q^15 + 18 * q^21 - 9 * q^25 + 30 * q^29 - 27 * q^31 + 27 * q^33 - 27 * q^35 + 45 * q^39 + 18 * q^41 + 27 * q^45 - 45 * q^47 + 63 * q^51 - 9 * q^57 - 54 * q^59 + 63 * q^63 - 57 * q^65 - 63 * q^69 - 36 * q^71 + 9 * q^73 + 45 * q^75 - 81 * q^77 - 54 * q^81 + 27 * q^83 - 36 * q^85 - 45 * q^87 - 63 * q^89 - 27 * q^91 - 63 * q^93 + 72 * q^95 - 99 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 0 −1.50778 0.852402i 0 −2.12767 + 0.375166i 0 2.50142 2.98107i 0 1.54682 + 2.57047i 0
47.2 0 −1.36628 + 1.06456i 0 1.83017 0.322709i 0 0.441545 0.526213i 0 0.733433 2.90896i 0
47.3 0 −0.721706 1.57453i 0 0.976679 0.172215i 0 −3.12803 + 3.72784i 0 −1.95828 + 2.27269i 0
47.4 0 0.939092 1.45537i 0 −1.07086 + 0.188822i 0 0.0466102 0.0555478i 0 −1.23621 2.73346i 0
47.5 0 1.59623 + 0.672336i 0 4.18690 0.738263i 0 −1.26944 + 1.51285i 0 2.09593 + 2.14641i 0
47.6 0 1.65284 + 0.517813i 0 −2.52917 + 0.445961i 0 1.40789 1.67786i 0 2.46374 + 1.71172i 0
95.1 0 −1.70424 + 0.309120i 0 −1.45395 + 1.73275i 0 1.49276 4.10134i 0 2.80889 1.05363i 0
95.2 0 −1.63553 0.570126i 0 2.34697 2.79701i 0 −0.550750 + 1.51317i 0 2.34991 + 1.86492i 0
95.3 0 −0.536320 + 1.64692i 0 −1.32159 + 1.57501i 0 −1.34711 + 3.70114i 0 −2.42472 1.76656i 0
95.4 0 −0.249148 1.71404i 0 −1.47612 + 1.75917i 0 −0.590654 + 1.62281i 0 −2.87585 + 0.854099i 0
95.5 0 1.19574 + 1.25308i 0 0.311559 0.371302i 0 0.958275 2.63284i 0 −0.140409 + 2.99671i 0
95.6 0 1.22376 1.22573i 0 1.15344 1.37462i 0 0.0374696 0.102947i 0 −0.00480931 3.00000i 0
191.1 0 −1.70424 0.309120i 0 −1.45395 1.73275i 0 1.49276 + 4.10134i 0 2.80889 + 1.05363i 0
191.2 0 −1.63553 + 0.570126i 0 2.34697 + 2.79701i 0 −0.550750 1.51317i 0 2.34991 1.86492i 0
191.3 0 −0.536320 1.64692i 0 −1.32159 1.57501i 0 −1.34711 3.70114i 0 −2.42472 + 1.76656i 0
191.4 0 −0.249148 + 1.71404i 0 −1.47612 1.75917i 0 −0.590654 1.62281i 0 −2.87585 0.854099i 0
191.5 0 1.19574 1.25308i 0 0.311559 + 0.371302i 0 0.958275 + 2.63284i 0 −0.140409 2.99671i 0
191.6 0 1.22376 + 1.22573i 0 1.15344 + 1.37462i 0 0.0374696 + 0.102947i 0 −0.00480931 + 3.00000i 0
239.1 0 −1.50778 + 0.852402i 0 −2.12767 0.375166i 0 2.50142 + 2.98107i 0 1.54682 2.57047i 0
239.2 0 −1.36628 1.06456i 0 1.83017 + 0.322709i 0 0.441545 + 0.526213i 0 0.733433 + 2.90896i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 383.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 432.2.be.c yes 36
4.b odd 2 1 432.2.be.b 36
27.f odd 18 1 432.2.be.b 36
108.l even 18 1 inner 432.2.be.c yes 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.be.b 36 4.b odd 2 1
432.2.be.b 36 27.f odd 18 1
432.2.be.c yes 36 1.a even 1 1 trivial
432.2.be.c yes 36 108.l even 18 1 inner

## Hecke kernels

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

 $$T_{5}^{36} - 3 T_{5}^{35} + 9 T_{5}^{34} - 33 T_{5}^{33} - 108 T_{5}^{31} - 2493 T_{5}^{30} + 2277 T_{5}^{29} + 10935 T_{5}^{28} - 2952 T_{5}^{27} + 169857 T_{5}^{26} + 1473687 T_{5}^{25} + 3988251 T_{5}^{24} + \cdots + 1847624256$$ T5^36 - 3*T5^35 + 9*T5^34 - 33*T5^33 - 108*T5^31 - 2493*T5^30 + 2277*T5^29 + 10935*T5^28 - 2952*T5^27 + 169857*T5^26 + 1473687*T5^25 + 3988251*T5^24 + 5571072*T5^23 + 21695445*T5^22 + 52419744*T5^21 - 40406283*T5^20 - 395074827*T5^19 - 410772249*T5^18 + 728806734*T5^17 + 595177713*T5^16 - 2869259598*T5^15 - 2593039149*T5^14 + 9825163110*T5^13 + 20842273782*T5^12 + 6777318600*T5^11 - 618190542*T5^10 - 15193153395*T5^9 - 8579305755*T5^8 - 7846770834*T5^7 - 33302928141*T5^6 + 20676363102*T5^5 + 27541205928*T5^4 - 8394149232*T5^3 + 7486363440*T5^2 - 1490169312*T5 + 1847624256 $$T_{7}^{36} + 81 T_{7}^{32} - 351 T_{7}^{31} - 6399 T_{7}^{30} + 19521 T_{7}^{29} - 29322 T_{7}^{28} - 28431 T_{7}^{27} + 269487 T_{7}^{26} - 3623616 T_{7}^{25} + 36709524 T_{7}^{24} - 69289992 T_{7}^{23} + \cdots + 34012224$$ T7^36 + 81*T7^32 - 351*T7^31 - 6399*T7^30 + 19521*T7^29 - 29322*T7^28 - 28431*T7^27 + 269487*T7^26 - 3623616*T7^25 + 36709524*T7^24 - 69289992*T7^23 + 1750329*T7^22 + 165315330*T7^21 - 635767461*T7^20 - 1286292798*T7^19 - 2071352169*T7^18 + 11892921309*T7^17 + 12079949175*T7^16 - 1212866460*T7^15 + 56784825144*T7^14 + 193913530524*T7^13 + 635472642465*T7^12 + 173002704984*T7^11 - 377874332415*T7^10 - 897025641192*T7^9 - 386523593739*T7^8 + 1376567353161*T7^7 - 264127771323*T7^6 - 785115858294*T7^5 + 674560187064*T7^4 - 110454697440*T7^3 + 14310643248*T7^2 - 867311712*T7 + 34012224