# Properties

 Label 405.2.e.f Level $405$ Weight $2$ Character orbit 405.e Analytic conductor $3.234$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.23394128186$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -\zeta_{6} q^{5} + 3 q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -\zeta_{6} q^{5} + 3 q^{8} - q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{16} + 2 q^{17} + 4 q^{19} + ( 1 - \zeta_{6} ) q^{20} -4 \zeta_{6} q^{22} + ( -1 + \zeta_{6} ) q^{25} + 2 q^{26} + ( 2 - 2 \zeta_{6} ) q^{29} + 5 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{34} -10 q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} -10 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 4 q^{44} + ( -8 + 8 \zeta_{6} ) q^{47} + 7 \zeta_{6} q^{49} + \zeta_{6} q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} -10 q^{53} -4 q^{55} -2 \zeta_{6} q^{58} + 4 \zeta_{6} q^{59} + ( 2 - 2 \zeta_{6} ) q^{61} + 7 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} -12 \zeta_{6} q^{67} + 2 \zeta_{6} q^{68} -8 q^{71} + 10 q^{73} + ( -10 + 10 \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} - q^{80} -10 q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} -2 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( 12 - 12 \zeta_{6} ) q^{88} -6 q^{89} + 8 \zeta_{6} q^{94} -4 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} + 7 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{4} - q^{5} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + q^{4} - q^{5} + 6q^{8} - 2q^{10} + 4q^{11} + 2q^{13} + q^{16} + 4q^{17} + 8q^{19} + q^{20} - 4q^{22} - q^{25} + 4q^{26} + 2q^{29} + 5q^{32} + 2q^{34} - 20q^{37} + 4q^{38} - 3q^{40} - 10q^{41} - 4q^{43} + 8q^{44} - 8q^{47} + 7q^{49} + q^{50} - 2q^{52} - 20q^{53} - 8q^{55} - 2q^{58} + 4q^{59} + 2q^{61} + 14q^{64} + 2q^{65} - 12q^{67} + 2q^{68} - 16q^{71} + 20q^{73} - 10q^{74} + 4q^{76} - 2q^{80} - 20q^{82} - 12q^{83} - 2q^{85} + 4q^{86} + 12q^{88} - 12q^{89} + 8q^{94} - 4q^{95} - 2q^{97} + 14q^{98} + O(q^{100})$$

## 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.

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 0.500000 + 0.866025i −0.500000 0.866025i 0 0 3.00000 0 −1.00000
271.1 0.500000 + 0.866025i 0 0.500000 0.866025i −0.500000 + 0.866025i 0 0 3.00000 0 −1.00000
 $$n$$: 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 405.2.e.f 2
3.b odd 2 1 405.2.e.c 2
9.c even 3 1 15.2.a.a 1
9.c even 3 1 inner 405.2.e.f 2
9.d odd 6 1 45.2.a.a 1
9.d odd 6 1 405.2.e.c 2
36.f odd 6 1 240.2.a.d 1
36.h even 6 1 720.2.a.c 1
45.h odd 6 1 225.2.a.b 1
45.j even 6 1 75.2.a.b 1
45.k odd 12 2 75.2.b.b 2
45.l even 12 2 225.2.b.b 2
63.g even 3 1 735.2.i.e 2
63.h even 3 1 735.2.i.e 2
63.k odd 6 1 735.2.i.d 2
63.l odd 6 1 735.2.a.c 1
63.o even 6 1 2205.2.a.i 1
63.t odd 6 1 735.2.i.d 2
72.j odd 6 1 2880.2.a.y 1
72.l even 6 1 2880.2.a.bc 1
72.n even 6 1 960.2.a.l 1
72.p odd 6 1 960.2.a.a 1
99.g even 6 1 5445.2.a.c 1
99.h odd 6 1 1815.2.a.d 1
117.n odd 6 1 7605.2.a.g 1
117.t even 6 1 2535.2.a.j 1
144.v odd 12 2 3840.2.k.r 2
144.x even 12 2 3840.2.k.m 2
153.h even 6 1 4335.2.a.c 1
171.o odd 6 1 5415.2.a.j 1
180.n even 6 1 3600.2.a.u 1
180.p odd 6 1 1200.2.a.e 1
180.v odd 12 2 3600.2.f.e 2
180.x even 12 2 1200.2.f.h 2
207.f odd 6 1 7935.2.a.d 1
315.bg odd 6 1 3675.2.a.j 1
360.z odd 6 1 4800.2.a.bz 1
360.bk even 6 1 4800.2.a.t 1
360.bo even 12 2 4800.2.f.c 2
360.bu odd 12 2 4800.2.f.bf 2
495.o odd 6 1 9075.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 9.c even 3 1
45.2.a.a 1 9.d odd 6 1
75.2.a.b 1 45.j even 6 1
75.2.b.b 2 45.k odd 12 2
225.2.a.b 1 45.h odd 6 1
225.2.b.b 2 45.l even 12 2
240.2.a.d 1 36.f odd 6 1
405.2.e.c 2 3.b odd 2 1
405.2.e.c 2 9.d odd 6 1
405.2.e.f 2 1.a even 1 1 trivial
405.2.e.f 2 9.c even 3 1 inner
720.2.a.c 1 36.h even 6 1
735.2.a.c 1 63.l odd 6 1
735.2.i.d 2 63.k odd 6 1
735.2.i.d 2 63.t odd 6 1
735.2.i.e 2 63.g even 3 1
735.2.i.e 2 63.h even 3 1
960.2.a.a 1 72.p odd 6 1
960.2.a.l 1 72.n even 6 1
1200.2.a.e 1 180.p odd 6 1
1200.2.f.h 2 180.x even 12 2
1815.2.a.d 1 99.h odd 6 1
2205.2.a.i 1 63.o even 6 1
2535.2.a.j 1 117.t even 6 1
2880.2.a.y 1 72.j odd 6 1
2880.2.a.bc 1 72.l even 6 1
3600.2.a.u 1 180.n even 6 1
3600.2.f.e 2 180.v odd 12 2
3675.2.a.j 1 315.bg odd 6 1
3840.2.k.m 2 144.x even 12 2
3840.2.k.r 2 144.v odd 12 2
4335.2.a.c 1 153.h even 6 1
4800.2.a.t 1 360.bk even 6 1
4800.2.a.bz 1 360.z odd 6 1
4800.2.f.c 2 360.bo even 12 2
4800.2.f.bf 2 360.bu odd 12 2
5415.2.a.j 1 171.o odd 6 1
5445.2.a.c 1 99.g even 6 1
7605.2.a.g 1 117.n odd 6 1
7935.2.a.d 1 207.f odd 6 1
9075.2.a.g 1 495.o odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{7}$$ $$T_{11}^{2} - 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$4 - 2 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$100 + 10 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$64 + 8 T + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$144 + 12 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( -10 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$4 + 2 T + T^{2}$$