# Properties

 Label 405.3.i.b Level $405$ Weight $3$ Character orbit 405.i Analytic conductor $11.035$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 405.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0354507066$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 6 - 6 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 6 - 6 \beta_{2} ) q^{7} -3 \beta_{3} q^{8} + 5 q^{10} -2 \beta_{1} q^{11} -16 \beta_{2} q^{13} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{14} + ( 19 - 19 \beta_{2} ) q^{16} + 2 \beta_{3} q^{17} -2 q^{19} + \beta_{1} q^{20} -10 \beta_{2} q^{22} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} -16 \beta_{3} q^{26} + 6 q^{28} + 14 \beta_{1} q^{29} + 18 \beta_{2} q^{31} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{32} + ( -10 + 10 \beta_{2} ) q^{34} -6 \beta_{3} q^{35} -16 q^{37} -2 \beta_{1} q^{38} -15 \beta_{2} q^{40} + ( 28 \beta_{1} - 28 \beta_{3} ) q^{41} + ( -16 + 16 \beta_{2} ) q^{43} -2 \beta_{3} q^{44} + 30 q^{46} + 22 \beta_{1} q^{47} + 13 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + ( 16 - 16 \beta_{2} ) q^{52} + 2 \beta_{3} q^{53} -10 q^{55} -18 \beta_{1} q^{56} + 70 \beta_{2} q^{58} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -82 + 82 \beta_{2} ) q^{61} + 18 \beta_{3} q^{62} -41 q^{64} -16 \beta_{1} q^{65} -24 \beta_{2} q^{67} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( 30 - 30 \beta_{2} ) q^{70} -56 \beta_{3} q^{71} -74 q^{73} -16 \beta_{1} q^{74} -2 \beta_{2} q^{76} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{77} + ( -138 + 138 \beta_{2} ) q^{79} -19 \beta_{3} q^{80} + 140 q^{82} + 42 \beta_{1} q^{83} + 10 \beta_{2} q^{85} + ( -16 \beta_{1} + 16 \beta_{3} ) q^{86} + ( -30 + 30 \beta_{2} ) q^{88} + 48 \beta_{3} q^{89} -96 q^{91} + 6 \beta_{1} q^{92} + 110 \beta_{2} q^{94} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{95} + ( 166 - 166 \beta_{2} ) q^{97} + 13 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} + 12q^{7} + O(q^{10})$$ $$4q + 2q^{4} + 12q^{7} + 20q^{10} - 32q^{13} + 38q^{16} - 8q^{19} - 20q^{22} + 10q^{25} + 24q^{28} + 36q^{31} - 20q^{34} - 64q^{37} - 30q^{40} - 32q^{43} + 120q^{46} + 26q^{49} + 32q^{52} - 40q^{55} + 140q^{58} - 164q^{61} - 164q^{64} - 48q^{67} + 60q^{70} - 296q^{73} - 4q^{76} - 276q^{79} + 560q^{82} + 20q^{85} - 60q^{88} - 384q^{91} + 220q^{94} + 332q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −1.93649 − 1.11803i 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i
−1.93649 1.11803i 0 0.500000 + 0.866025i −1.93649 + 1.11803i 0 3.00000 5.19615i 6.70820i 0 5.00000
26.2 1.93649 + 1.11803i 0 0.500000 + 0.866025i 1.93649 1.11803i 0 3.00000 5.19615i 6.70820i 0 5.00000
296.1 −1.93649 + 1.11803i 0 0.500000 0.866025i −1.93649 1.11803i 0 3.00000 + 5.19615i 6.70820i 0 5.00000
296.2 1.93649 1.11803i 0 0.500000 0.866025i 1.93649 + 1.11803i 0 3.00000 + 5.19615i 6.70820i 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
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.i.b 4
3.b odd 2 1 inner 405.3.i.b 4
9.c even 3 1 15.3.c.a 2
9.c even 3 1 inner 405.3.i.b 4
9.d odd 6 1 15.3.c.a 2
9.d odd 6 1 inner 405.3.i.b 4
36.f odd 6 1 240.3.l.b 2
36.h even 6 1 240.3.l.b 2
45.h odd 6 1 75.3.c.e 2
45.j even 6 1 75.3.c.e 2
45.k odd 12 2 75.3.d.b 4
45.l even 12 2 75.3.d.b 4
72.j odd 6 1 960.3.l.c 2
72.l even 6 1 960.3.l.b 2
72.n even 6 1 960.3.l.c 2
72.p odd 6 1 960.3.l.b 2
180.n even 6 1 1200.3.l.g 2
180.p odd 6 1 1200.3.l.g 2
180.v odd 12 2 1200.3.c.f 4
180.x even 12 2 1200.3.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 9.c even 3 1
15.3.c.a 2 9.d odd 6 1
75.3.c.e 2 45.h odd 6 1
75.3.c.e 2 45.j even 6 1
75.3.d.b 4 45.k odd 12 2
75.3.d.b 4 45.l even 12 2
240.3.l.b 2 36.f odd 6 1
240.3.l.b 2 36.h even 6 1
405.3.i.b 4 1.a even 1 1 trivial
405.3.i.b 4 3.b odd 2 1 inner
405.3.i.b 4 9.c even 3 1 inner
405.3.i.b 4 9.d odd 6 1 inner
960.3.l.b 2 72.l even 6 1
960.3.l.b 2 72.p odd 6 1
960.3.l.c 2 72.j odd 6 1
960.3.l.c 2 72.n even 6 1
1200.3.c.f 4 180.v odd 12 2
1200.3.c.f 4 180.x even 12 2
1200.3.l.g 2 180.n even 6 1
1200.3.l.g 2 180.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(405, [\chi])$$:

 $$T_{2}^{4} - 5 T_{2}^{2} + 25$$ $$T_{7}^{2} - 6 T_{7} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$25 - 5 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$25 - 5 T^{2} + T^{4}$$
$7$ $$( 36 - 6 T + T^{2} )^{2}$$
$11$ $$400 - 20 T^{2} + T^{4}$$
$13$ $$( 256 + 16 T + T^{2} )^{2}$$
$17$ $$( 20 + T^{2} )^{2}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$32400 - 180 T^{2} + T^{4}$$
$29$ $$960400 - 980 T^{2} + T^{4}$$
$31$ $$( 324 - 18 T + T^{2} )^{2}$$
$37$ $$( 16 + T )^{4}$$
$41$ $$15366400 - 3920 T^{2} + T^{4}$$
$43$ $$( 256 + 16 T + T^{2} )^{2}$$
$47$ $$5856400 - 2420 T^{2} + T^{4}$$
$53$ $$( 20 + T^{2} )^{2}$$
$59$ $$400 - 20 T^{2} + T^{4}$$
$61$ $$( 6724 + 82 T + T^{2} )^{2}$$
$67$ $$( 576 + 24 T + T^{2} )^{2}$$
$71$ $$( 15680 + T^{2} )^{2}$$
$73$ $$( 74 + T )^{4}$$
$79$ $$( 19044 + 138 T + T^{2} )^{2}$$
$83$ $$77792400 - 8820 T^{2} + T^{4}$$
$89$ $$( 11520 + T^{2} )^{2}$$
$97$ $$( 27556 - 166 T + T^{2} )^{2}$$
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