# Properties

 Label 45.4.b.b Level $45$ Weight $4$ Character orbit 45.b Analytic conductor $2.655$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.65508595026$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Defining polynomial: $$x^{4} + 21 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -5 + \beta_{3} ) q^{4} + ( -2 - \beta_{2} + \beta_{3} ) q^{5} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -5 + \beta_{3} ) q^{4} + ( -2 - \beta_{2} + \beta_{3} ) q^{5} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{7} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{8} + ( 3 + 10 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{10} + ( 22 - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{13} + ( -58 + 2 \beta_{3} ) q^{14} + ( 13 - \beta_{3} ) q^{16} + ( -16 \beta_{1} - 6 \beta_{2} ) q^{17} + ( -24 - 8 \beta_{3} ) q^{19} + ( 102 + 5 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{20} + ( -38 \beta_{1} - 8 \beta_{2} ) q^{22} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{23} + ( 67 + 10 \beta_{1} - 14 \beta_{2} - 6 \beta_{3} ) q^{25} + ( -2 + 10 \beta_{3} ) q^{26} + ( 42 \beta_{1} + 24 \beta_{2} ) q^{28} + ( -152 - 14 \beta_{3} ) q^{29} + ( 24 + 4 \beta_{3} ) q^{31} + ( 19 \beta_{1} + 28 \beta_{2} ) q^{32} + ( -190 + 22 \beta_{3} ) q^{34} + ( 70 + 30 \beta_{1} + 30 \beta_{2} + 10 \beta_{3} ) q^{35} + ( 66 \beta_{1} + 12 \beta_{2} ) q^{37} + ( -40 \beta_{1} - 32 \beta_{2} ) q^{38} + ( 101 - 70 \beta_{1} + 8 \beta_{2} + 7 \beta_{3} ) q^{40} + ( 206 - 4 \beta_{3} ) q^{41} + ( -82 \beta_{1} + 14 \beta_{2} ) q^{43} + ( -294 + 30 \beta_{3} ) q^{44} + ( 44 + 4 \beta_{3} ) q^{46} + ( 54 \beta_{1} - 38 \beta_{2} ) q^{47} + ( -29 - 12 \beta_{3} ) q^{49} + ( 172 - 115 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{50} + ( 66 \beta_{1} - 24 \beta_{2} ) q^{52} + ( -28 \beta_{1} + 54 \beta_{2} ) q^{53} + ( -228 - 10 \beta_{1} - 4 \beta_{2} + 24 \beta_{3} ) q^{55} + ( 10 - 50 \beta_{3} ) q^{56} + ( 40 \beta_{1} - 56 \beta_{2} ) q^{58} + ( -94 + 2 \beta_{3} ) q^{59} + ( 186 - 32 \beta_{3} ) q^{61} + ( 8 \beta_{1} + 16 \beta_{2} ) q^{62} + ( 267 - 55 \beta_{3} ) q^{64} + ( -226 + 60 \beta_{1} - 48 \beta_{2} - 22 \beta_{3} ) q^{65} + ( 106 \beta_{1} + 46 \beta_{2} ) q^{67} + ( 238 \beta_{1} + 40 \beta_{2} ) q^{68} + ( 300 + 10 \beta_{1} + 40 \beta_{2} - 60 \beta_{3} ) q^{70} + ( -12 + 60 \beta_{3} ) q^{71} + ( -192 \beta_{1} - 84 \beta_{2} ) q^{73} + ( 822 - 78 \beta_{3} ) q^{74} + ( -616 + 8 \beta_{3} ) q^{76} + ( -132 \beta_{1} - 24 \beta_{2} ) q^{77} + ( -240 + 100 \beta_{3} ) q^{79} + ( -118 - 5 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} ) q^{80} + ( -238 \beta_{1} - 16 \beta_{2} ) q^{82} + ( -238 \beta_{1} - 30 \beta_{2} ) q^{83} + ( -126 + 190 \beta_{1} + 22 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -1108 + 68 \beta_{3} ) q^{86} + ( 230 \beta_{1} + 56 \beta_{2} ) q^{88} + ( 534 + 48 \beta_{3} ) q^{89} + ( 444 + 84 \beta_{3} ) q^{91} + ( 4 \beta_{1} - 32 \beta_{2} ) q^{92} + ( 816 - 16 \beta_{3} ) q^{94} + ( -688 - 40 \beta_{1} + 96 \beta_{2} - 16 \beta_{3} ) q^{95} + ( -244 \beta_{1} - 4 \beta_{2} ) q^{97} + ( -67 \beta_{1} - 48 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 18 q^{4} - 6 q^{5} + O(q^{10})$$ $$4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 21 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 13 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$3 \nu^{2} + 32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 5 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 32$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 65 \beta_{1}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 3.70156i − 2.70156i 2.70156i 3.70156i
4.70156i 0 −14.1047 −11.1047 1.29844i 0 16.2094i 28.7016i 0 −6.10469 + 52.2094i
19.2 1.70156i 0 5.10469 8.10469 + 7.70156i 0 22.2094i 22.2984i 0 13.1047 13.7906i
19.3 1.70156i 0 5.10469 8.10469 7.70156i 0 22.2094i 22.2984i 0 13.1047 + 13.7906i
19.4 4.70156i 0 −14.1047 −11.1047 + 1.29844i 0 16.2094i 28.7016i 0 −6.10469 52.2094i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.b.b 4
3.b odd 2 1 15.4.b.a 4
4.b odd 2 1 720.4.f.j 4
5.b even 2 1 inner 45.4.b.b 4
5.c odd 4 1 225.4.a.i 2
5.c odd 4 1 225.4.a.o 2
12.b even 2 1 240.4.f.f 4
15.d odd 2 1 15.4.b.a 4
15.e even 4 1 75.4.a.c 2
15.e even 4 1 75.4.a.f 2
20.d odd 2 1 720.4.f.j 4
24.f even 2 1 960.4.f.p 4
24.h odd 2 1 960.4.f.q 4
60.h even 2 1 240.4.f.f 4
60.l odd 4 1 1200.4.a.bn 2
60.l odd 4 1 1200.4.a.bt 2
120.i odd 2 1 960.4.f.q 4
120.m even 2 1 960.4.f.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 3.b odd 2 1
15.4.b.a 4 15.d odd 2 1
45.4.b.b 4 1.a even 1 1 trivial
45.4.b.b 4 5.b even 2 1 inner
75.4.a.c 2 15.e even 4 1
75.4.a.f 2 15.e even 4 1
225.4.a.i 2 5.c odd 4 1
225.4.a.o 2 5.c odd 4 1
240.4.f.f 4 12.b even 2 1
240.4.f.f 4 60.h even 2 1
720.4.f.j 4 4.b odd 2 1
720.4.f.j 4 20.d odd 2 1
960.4.f.p 4 24.f even 2 1
960.4.f.p 4 120.m even 2 1
960.4.f.q 4 24.h odd 2 1
960.4.f.q 4 120.i odd 2 1
1200.4.a.bn 2 60.l odd 4 1
1200.4.a.bt 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25 T_{2}^{2} + 64$$ acting on $$S_{4}^{\mathrm{new}}(45, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 25 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$15625 + 750 T - 110 T^{2} + 6 T^{3} + T^{4}$$
$7$ $$129600 + 756 T^{2} + T^{4}$$
$11$ $$( 72 - 42 T + T^{2} )^{2}$$
$13$ $$1327104 + 3780 T^{2} + T^{4}$$
$17$ $$2483776 + 7252 T^{2} + T^{4}$$
$19$ $$( -5120 + 56 T + T^{2} )^{2}$$
$23$ $$6400 + 2464 T^{2} + T^{4}$$
$29$ $$( 7200 + 318 T + T^{2} )^{2}$$
$31$ $$( -800 - 52 T + T^{2} )^{2}$$
$37$ $$41990400 + 106596 T^{2} + T^{4}$$
$41$ $$( 40140 - 408 T + T^{2} )^{2}$$
$43$ $$8256266496 + 196128 T^{2} + T^{4}$$
$47$ $$6186766336 + 189712 T^{2} + T^{4}$$
$53$ $$813390400 + 218644 T^{2} + T^{4}$$
$59$ $$( 8280 + 186 T + T^{2} )^{2}$$
$61$ $$( -65564 - 340 T + T^{2} )^{2}$$
$67$ $$9419867136 + 341712 T^{2} + T^{4}$$
$71$ $$( -331776 - 36 T + T^{2} )^{2}$$
$73$ $$104976000000 + 1126224 T^{2} + T^{4}$$
$79$ $$( -886400 + 380 T + T^{2} )^{2}$$
$83$ $$40558737664 + 1371040 T^{2} + T^{4}$$
$89$ $$( 98820 - 1116 T + T^{2} )^{2}$$
$97$ $$196199387136 + 1475712 T^{2} + T^{4}$$