# Properties

 Label 405.4.e.g Level $405$ Weight $4$ Character orbit 405.e Analytic conductor $23.896$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not 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: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 24 \zeta_{6} + 24) q^{7} - 15 q^{8} +O(q^{10})$$ q + (z - 1) * q^2 + 7*z * q^4 - 5*z * q^5 + (-24*z + 24) * q^7 - 15 * q^8 $$q + (\zeta_{6} - 1) q^{2} + 7 \zeta_{6} q^{4} - 5 \zeta_{6} q^{5} + ( - 24 \zeta_{6} + 24) q^{7} - 15 q^{8} + 5 q^{10} + (52 \zeta_{6} - 52) q^{11} - 22 \zeta_{6} q^{13} + 24 \zeta_{6} q^{14} + (41 \zeta_{6} - 41) q^{16} - 14 q^{17} - 20 q^{19} + ( - 35 \zeta_{6} + 35) q^{20} - 52 \zeta_{6} q^{22} + 168 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} + 22 q^{26} + 168 q^{28} + (230 \zeta_{6} - 230) q^{29} + 288 \zeta_{6} q^{31} - 161 \zeta_{6} q^{32} + ( - 14 \zeta_{6} + 14) q^{34} - 120 q^{35} - 34 q^{37} + ( - 20 \zeta_{6} + 20) q^{38} + 75 \zeta_{6} q^{40} - 122 \zeta_{6} q^{41} + ( - 188 \zeta_{6} + 188) q^{43} - 364 q^{44} - 168 q^{46} + (256 \zeta_{6} - 256) q^{47} - 233 \zeta_{6} q^{49} - 25 \zeta_{6} q^{50} + ( - 154 \zeta_{6} + 154) q^{52} - 338 q^{53} + 260 q^{55} + (360 \zeta_{6} - 360) q^{56} - 230 \zeta_{6} q^{58} - 100 \zeta_{6} q^{59} + (742 \zeta_{6} - 742) q^{61} - 288 q^{62} - 167 q^{64} + (110 \zeta_{6} - 110) q^{65} + 84 \zeta_{6} q^{67} - 98 \zeta_{6} q^{68} + ( - 120 \zeta_{6} + 120) q^{70} - 328 q^{71} - 38 q^{73} + ( - 34 \zeta_{6} + 34) q^{74} - 140 \zeta_{6} q^{76} + 1248 \zeta_{6} q^{77} + ( - 240 \zeta_{6} + 240) q^{79} + 205 q^{80} + 122 q^{82} + (1212 \zeta_{6} - 1212) q^{83} + 70 \zeta_{6} q^{85} + 188 \zeta_{6} q^{86} + ( - 780 \zeta_{6} + 780) q^{88} + 330 q^{89} - 528 q^{91} + (1176 \zeta_{6} - 1176) q^{92} - 256 \zeta_{6} q^{94} + 100 \zeta_{6} q^{95} + (866 \zeta_{6} - 866) q^{97} + 233 q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + 7*z * q^4 - 5*z * q^5 + (-24*z + 24) * q^7 - 15 * q^8 + 5 * q^10 + (52*z - 52) * q^11 - 22*z * q^13 + 24*z * q^14 + (41*z - 41) * q^16 - 14 * q^17 - 20 * q^19 + (-35*z + 35) * q^20 - 52*z * q^22 + 168*z * q^23 + (25*z - 25) * q^25 + 22 * q^26 + 168 * q^28 + (230*z - 230) * q^29 + 288*z * q^31 - 161*z * q^32 + (-14*z + 14) * q^34 - 120 * q^35 - 34 * q^37 + (-20*z + 20) * q^38 + 75*z * q^40 - 122*z * q^41 + (-188*z + 188) * q^43 - 364 * q^44 - 168 * q^46 + (256*z - 256) * q^47 - 233*z * q^49 - 25*z * q^50 + (-154*z + 154) * q^52 - 338 * q^53 + 260 * q^55 + (360*z - 360) * q^56 - 230*z * q^58 - 100*z * q^59 + (742*z - 742) * q^61 - 288 * q^62 - 167 * q^64 + (110*z - 110) * q^65 + 84*z * q^67 - 98*z * q^68 + (-120*z + 120) * q^70 - 328 * q^71 - 38 * q^73 + (-34*z + 34) * q^74 - 140*z * q^76 + 1248*z * q^77 + (-240*z + 240) * q^79 + 205 * q^80 + 122 * q^82 + (1212*z - 1212) * q^83 + 70*z * q^85 + 188*z * q^86 + (-780*z + 780) * q^88 + 330 * q^89 - 528 * q^91 + (1176*z - 1176) * q^92 - 256*z * q^94 + 100*z * q^95 + (866*z - 866) * q^97 + 233 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8}+O(q^{10})$$ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8 $$2 q - q^{2} + 7 q^{4} - 5 q^{5} + 24 q^{7} - 30 q^{8} + 10 q^{10} - 52 q^{11} - 22 q^{13} + 24 q^{14} - 41 q^{16} - 28 q^{17} - 40 q^{19} + 35 q^{20} - 52 q^{22} + 168 q^{23} - 25 q^{25} + 44 q^{26} + 336 q^{28} - 230 q^{29} + 288 q^{31} - 161 q^{32} + 14 q^{34} - 240 q^{35} - 68 q^{37} + 20 q^{38} + 75 q^{40} - 122 q^{41} + 188 q^{43} - 728 q^{44} - 336 q^{46} - 256 q^{47} - 233 q^{49} - 25 q^{50} + 154 q^{52} - 676 q^{53} + 520 q^{55} - 360 q^{56} - 230 q^{58} - 100 q^{59} - 742 q^{61} - 576 q^{62} - 334 q^{64} - 110 q^{65} + 84 q^{67} - 98 q^{68} + 120 q^{70} - 656 q^{71} - 76 q^{73} + 34 q^{74} - 140 q^{76} + 1248 q^{77} + 240 q^{79} + 410 q^{80} + 244 q^{82} - 1212 q^{83} + 70 q^{85} + 188 q^{86} + 780 q^{88} + 660 q^{89} - 1056 q^{91} - 1176 q^{92} - 256 q^{94} + 100 q^{95} - 866 q^{97} + 466 q^{98}+O(q^{100})$$ 2 * q - q^2 + 7 * q^4 - 5 * q^5 + 24 * q^7 - 30 * q^8 + 10 * q^10 - 52 * q^11 - 22 * q^13 + 24 * q^14 - 41 * q^16 - 28 * q^17 - 40 * q^19 + 35 * q^20 - 52 * q^22 + 168 * q^23 - 25 * q^25 + 44 * q^26 + 336 * q^28 - 230 * q^29 + 288 * q^31 - 161 * q^32 + 14 * q^34 - 240 * q^35 - 68 * q^37 + 20 * q^38 + 75 * q^40 - 122 * q^41 + 188 * q^43 - 728 * q^44 - 336 * q^46 - 256 * q^47 - 233 * q^49 - 25 * q^50 + 154 * q^52 - 676 * q^53 + 520 * q^55 - 360 * q^56 - 230 * q^58 - 100 * q^59 - 742 * q^61 - 576 * q^62 - 334 * q^64 - 110 * q^65 + 84 * q^67 - 98 * q^68 + 120 * q^70 - 656 * q^71 - 76 * q^73 + 34 * q^74 - 140 * q^76 + 1248 * q^77 + 240 * q^79 + 410 * q^80 + 244 * q^82 - 1212 * q^83 + 70 * q^85 + 188 * q^86 + 780 * q^88 + 660 * q^89 - 1056 * q^91 - 1176 * q^92 - 256 * q^94 + 100 * q^95 - 866 * q^97 + 466 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$$.

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
136.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 3.50000 + 6.06218i −2.50000 4.33013i 0 12.0000 20.7846i −15.0000 0 5.00000
271.1 −0.500000 0.866025i 0 3.50000 6.06218i −2.50000 + 4.33013i 0 12.0000 + 20.7846i −15.0000 0 5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.e.g 2
3.b odd 2 1 405.4.e.i 2
9.c even 3 1 15.4.a.a 1
9.c even 3 1 inner 405.4.e.g 2
9.d odd 6 1 45.4.a.c 1
9.d odd 6 1 405.4.e.i 2
36.f odd 6 1 240.4.a.e 1
36.h even 6 1 720.4.a.n 1
45.h odd 6 1 225.4.a.f 1
45.j even 6 1 75.4.a.b 1
45.k odd 12 2 75.4.b.b 2
45.l even 12 2 225.4.b.e 2
63.l odd 6 1 735.4.a.e 1
63.o even 6 1 2205.4.a.l 1
72.n even 6 1 960.4.a.b 1
72.p odd 6 1 960.4.a.ba 1
99.h odd 6 1 1815.4.a.e 1
180.p odd 6 1 1200.4.a.t 1
180.x even 12 2 1200.4.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 9.c even 3 1
45.4.a.c 1 9.d odd 6 1
75.4.a.b 1 45.j even 6 1
75.4.b.b 2 45.k odd 12 2
225.4.a.f 1 45.h odd 6 1
225.4.b.e 2 45.l even 12 2
240.4.a.e 1 36.f odd 6 1
405.4.e.g 2 1.a even 1 1 trivial
405.4.e.g 2 9.c even 3 1 inner
405.4.e.i 2 3.b odd 2 1
405.4.e.i 2 9.d odd 6 1
720.4.a.n 1 36.h even 6 1
735.4.a.e 1 63.l odd 6 1
960.4.a.b 1 72.n even 6 1
960.4.a.ba 1 72.p odd 6 1
1200.4.a.t 1 180.p odd 6 1
1200.4.f.b 2 180.x even 12 2
1815.4.a.e 1 99.h odd 6 1
2205.4.a.l 1 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{7}^{2} - 24T_{7} + 576$$ T7^2 - 24*T7 + 576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 24T + 576$$
$11$ $$T^{2} + 52T + 2704$$
$13$ $$T^{2} + 22T + 484$$
$17$ $$(T + 14)^{2}$$
$19$ $$(T + 20)^{2}$$
$23$ $$T^{2} - 168T + 28224$$
$29$ $$T^{2} + 230T + 52900$$
$31$ $$T^{2} - 288T + 82944$$
$37$ $$(T + 34)^{2}$$
$41$ $$T^{2} + 122T + 14884$$
$43$ $$T^{2} - 188T + 35344$$
$47$ $$T^{2} + 256T + 65536$$
$53$ $$(T + 338)^{2}$$
$59$ $$T^{2} + 100T + 10000$$
$61$ $$T^{2} + 742T + 550564$$
$67$ $$T^{2} - 84T + 7056$$
$71$ $$(T + 328)^{2}$$
$73$ $$(T + 38)^{2}$$
$79$ $$T^{2} - 240T + 57600$$
$83$ $$T^{2} + 1212 T + 1468944$$
$89$ $$(T - 330)^{2}$$
$97$ $$T^{2} + 866T + 749956$$