# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$23.8957735523$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{4} + ( 5 - 5 \zeta_{12}^{2} ) q^{5} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{7} + ( 6 - 20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( 4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{4} + ( 5 - 5 \zeta_{12}^{2} ) q^{5} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{7} + ( 6 - 20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{8} + ( 5 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{10} + ( 28 \zeta_{12} - 11 \zeta_{12}^{2} + 28 \zeta_{12}^{3} ) q^{11} + ( 32 + 4 \zeta_{12} - 32 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{13} + ( 12 - 4 \zeta_{12} - 12 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{14} + ( -32 \zeta_{12} + 4 \zeta_{12}^{2} - 32 \zeta_{12}^{3} ) q^{16} + ( -16 + 112 \zeta_{12} - 56 \zeta_{12}^{3} ) q^{17} + ( -5 + 136 \zeta_{12} - 68 \zeta_{12}^{3} ) q^{19} + ( -10 \zeta_{12} - 20 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{20} + ( 95 - 39 \zeta_{12} - 95 \zeta_{12}^{2} + 78 \zeta_{12}^{3} ) q^{22} + ( -42 + 32 \zeta_{12} + 42 \zeta_{12}^{2} - 64 \zeta_{12}^{3} ) q^{23} -25 \zeta_{12}^{2} q^{25} + ( 20 - 56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{26} + ( 24 + 32 \zeta_{12} - 16 \zeta_{12}^{3} ) q^{28} + ( 24 \zeta_{12} - 85 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{29} + ( 129 + 12 \zeta_{12} - 129 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{31} + ( -52 - 44 \zeta_{12} + 52 \zeta_{12}^{2} + 88 \zeta_{12}^{3} ) q^{32} + ( 72 \zeta_{12} - 184 \zeta_{12}^{2} + 72 \zeta_{12}^{3} ) q^{34} + ( 40 \zeta_{12} - 20 \zeta_{12}^{3} ) q^{35} + ( 38 - 296 \zeta_{12} + 148 \zeta_{12}^{3} ) q^{37} + ( 73 \zeta_{12} - 209 \zeta_{12}^{2} + 73 \zeta_{12}^{3} ) q^{38} + ( 30 - 50 \zeta_{12} - 30 \zeta_{12}^{2} + 100 \zeta_{12}^{3} ) q^{40} + ( -289 - 48 \zeta_{12} + 289 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{41} + ( -156 \zeta_{12} - 190 \zeta_{12}^{2} - 156 \zeta_{12}^{3} ) q^{43} + ( 124 + 180 \zeta_{12} - 90 \zeta_{12}^{3} ) q^{44} + ( -138 + 148 \zeta_{12} - 74 \zeta_{12}^{3} ) q^{46} + ( 16 \zeta_{12} - 242 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{47} + ( 295 - 295 \zeta_{12}^{2} ) q^{49} + ( 25 - 25 \zeta_{12} - 25 \zeta_{12}^{2} + 50 \zeta_{12}^{3} ) q^{50} + ( -80 \zeta_{12} - 152 \zeta_{12}^{2} - 80 \zeta_{12}^{3} ) q^{52} + ( 272 + 200 \zeta_{12} - 100 \zeta_{12}^{3} ) q^{53} + ( -55 + 280 \zeta_{12} - 140 \zeta_{12}^{3} ) q^{55} + ( 24 \zeta_{12} - 120 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{56} + ( 157 - 109 \zeta_{12} - 157 \zeta_{12}^{2} + 218 \zeta_{12}^{3} ) q^{58} + ( -353 - 28 \zeta_{12} + 353 \zeta_{12}^{2} + 56 \zeta_{12}^{3} ) q^{59} + ( 192 \zeta_{12} - 334 \zeta_{12}^{2} + 192 \zeta_{12}^{3} ) q^{61} + ( 93 - 234 \zeta_{12} + 117 \zeta_{12}^{3} ) q^{62} + ( 112 - 496 \zeta_{12} + 248 \zeta_{12}^{3} ) q^{64} + ( -20 \zeta_{12} - 160 \zeta_{12}^{2} - 20 \zeta_{12}^{3} ) q^{65} + ( 726 + 52 \zeta_{12} - 726 \zeta_{12}^{2} - 104 \zeta_{12}^{3} ) q^{67} + ( 272 + 192 \zeta_{12} - 272 \zeta_{12}^{2} - 384 \zeta_{12}^{3} ) q^{68} + ( 20 \zeta_{12} - 60 \zeta_{12}^{2} + 20 \zeta_{12}^{3} ) q^{70} + ( 487 - 392 \zeta_{12} + 196 \zeta_{12}^{3} ) q^{71} + ( 592 - 184 \zeta_{12} + 92 \zeta_{12}^{3} ) q^{73} + ( -186 \zeta_{12} + 482 \zeta_{12}^{2} - 186 \zeta_{12}^{3} ) q^{74} + ( 388 + 262 \zeta_{12} - 388 \zeta_{12}^{2} - 524 \zeta_{12}^{3} ) q^{76} + ( -336 + 44 \zeta_{12} + 336 \zeta_{12}^{2} - 88 \zeta_{12}^{3} ) q^{77} + ( -104 \zeta_{12} - 204 \zeta_{12}^{2} - 104 \zeta_{12}^{3} ) q^{79} + ( 20 - 320 \zeta_{12} + 160 \zeta_{12}^{3} ) q^{80} + ( -145 + 482 \zeta_{12} - 241 \zeta_{12}^{3} ) q^{82} + ( -600 \zeta_{12} - 222 \zeta_{12}^{2} - 600 \zeta_{12}^{3} ) q^{83} + ( -80 + 280 \zeta_{12} + 80 \zeta_{12}^{2} - 560 \zeta_{12}^{3} ) q^{85} + ( -278 - 34 \zeta_{12} + 278 \zeta_{12}^{2} + 68 \zeta_{12}^{3} ) q^{86} + ( 278 \zeta_{12} - 906 \zeta_{12}^{2} + 278 \zeta_{12}^{3} ) q^{88} + 513 q^{89} + ( 48 + 256 \zeta_{12} - 128 \zeta_{12}^{3} ) q^{91} + ( -44 \zeta_{12} - 24 \zeta_{12}^{2} - 44 \zeta_{12}^{3} ) q^{92} + ( 290 - 258 \zeta_{12} - 290 \zeta_{12}^{2} + 516 \zeta_{12}^{3} ) q^{94} + ( -25 + 340 \zeta_{12} + 25 \zeta_{12}^{2} - 680 \zeta_{12}^{3} ) q^{95} + ( 440 \zeta_{12} + 334 \zeta_{12}^{2} + 440 \zeta_{12}^{3} ) q^{97} + ( 295 - 590 \zeta_{12} + 295 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 8 q^{4} + 10 q^{5} + 24 q^{8} + O(q^{10})$$ $$4 q + 2 q^{2} + 8 q^{4} + 10 q^{5} + 24 q^{8} + 20 q^{10} - 22 q^{11} + 64 q^{13} + 24 q^{14} + 8 q^{16} - 64 q^{17} - 20 q^{19} - 40 q^{20} + 190 q^{22} - 84 q^{23} - 50 q^{25} + 80 q^{26} + 96 q^{28} - 170 q^{29} + 258 q^{31} - 104 q^{32} - 368 q^{34} + 152 q^{37} - 418 q^{38} + 60 q^{40} - 578 q^{41} - 380 q^{43} + 496 q^{44} - 552 q^{46} - 484 q^{47} + 590 q^{49} + 50 q^{50} - 304 q^{52} + 1088 q^{53} - 220 q^{55} - 240 q^{56} + 314 q^{58} - 706 q^{59} - 668 q^{61} + 372 q^{62} + 448 q^{64} - 320 q^{65} + 1452 q^{67} + 544 q^{68} - 120 q^{70} + 1948 q^{71} + 2368 q^{73} + 964 q^{74} + 776 q^{76} - 672 q^{77} - 408 q^{79} + 80 q^{80} - 580 q^{82} - 444 q^{83} - 160 q^{85} - 556 q^{86} - 1812 q^{88} + 2052 q^{89} + 192 q^{91} - 48 q^{92} + 580 q^{94} - 50 q^{95} + 668 q^{97} + 1180 q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$82$$ $$326$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}$$

## 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
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.366025 + 0.633975i 0 3.73205 + 6.46410i 2.50000 + 4.33013i 0 3.46410 6.00000i −11.3205 0 −3.66025
136.2 1.36603 2.36603i 0 0.267949 + 0.464102i 2.50000 + 4.33013i 0 −3.46410 + 6.00000i 23.3205 0 13.6603
271.1 −0.366025 0.633975i 0 3.73205 6.46410i 2.50000 4.33013i 0 3.46410 + 6.00000i −11.3205 0 −3.66025
271.2 1.36603 + 2.36603i 0 0.267949 0.464102i 2.50000 4.33013i 0 −3.46410 6.00000i 23.3205 0 13.6603
 $$n$$: 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.p 4
3.b odd 2 1 405.4.e.o 4
9.c even 3 1 405.4.a.c 2
9.c even 3 1 inner 405.4.e.p 4
9.d odd 6 1 405.4.a.f yes 2
9.d odd 6 1 405.4.e.o 4
45.h odd 6 1 2025.4.a.i 2
45.j even 6 1 2025.4.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.c 2 9.c even 3 1
405.4.a.f yes 2 9.d odd 6 1
405.4.e.o 4 3.b odd 2 1
405.4.e.o 4 9.d odd 6 1
405.4.e.p 4 1.a even 1 1 trivial
405.4.e.p 4 9.c even 3 1 inner
2025.4.a.i 2 45.h odd 6 1
2025.4.a.m 2 45.j even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 4$$ $$T_{7}^{4} + 48 T_{7}^{2} + 2304$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 25 - 5 T + T^{2} )^{2}$$
$7$ $$2304 + 48 T^{2} + T^{4}$$
$11$ $$4977361 - 49082 T + 2715 T^{2} + 22 T^{3} + T^{4}$$
$13$ $$952576 - 62464 T + 3120 T^{2} - 64 T^{3} + T^{4}$$
$17$ $$( -9152 + 32 T + T^{2} )^{2}$$
$19$ $$( -13847 + 10 T + T^{2} )^{2}$$
$23$ $$1710864 - 109872 T + 8364 T^{2} + 84 T^{3} + T^{4}$$
$29$ $$30217009 + 934490 T + 23403 T^{2} + 170 T^{3} + T^{4}$$
$31$ $$262731681 - 4181922 T + 50355 T^{2} - 258 T^{3} + T^{4}$$
$37$ $$( -64268 - 76 T + T^{2} )^{2}$$
$41$ $$5868938881 + 44280002 T + 257475 T^{2} + 578 T^{3} + T^{4}$$
$43$ $$1362200464 - 14025040 T + 181308 T^{2} + 380 T^{3} + T^{4}$$
$47$ $$3340377616 + 27973264 T + 176460 T^{2} + 484 T^{3} + T^{4}$$
$53$ $$( 43984 - 544 T + T^{2} )^{2}$$
$59$ $$14946774049 + 86313442 T + 376179 T^{2} + 706 T^{3} + T^{4}$$
$61$ $$929296 + 643952 T + 445260 T^{2} + 668 T^{3} + T^{4}$$
$67$ $$269323633296 - 753535728 T + 1589340 T^{2} - 1452 T^{3} + T^{4}$$
$71$ $$( 121921 - 974 T + T^{2} )^{2}$$
$73$ $$( 325072 - 1184 T + T^{2} )^{2}$$
$79$ $$84052224 + 3740544 T + 157296 T^{2} + 408 T^{3} + T^{4}$$
$83$ $$1062375472656 - 457637904 T + 1227852 T^{2} + 444 T^{3} + T^{4}$$
$89$ $$( -513 + T )^{4}$$
$97$ $$220189931536 + 313454992 T + 915468 T^{2} - 668 T^{3} + T^{4}$$