Properties

 Label 45.4.a.c Level $45$ Weight $4$ Character orbit 45.a Self dual yes Analytic conductor $2.655$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 45.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.65508595026$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 −7.00000 −5.00000 0 −24.0000 15.0000 0 5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.a.c 1
3.b odd 2 1 15.4.a.a 1
4.b odd 2 1 720.4.a.n 1
5.b even 2 1 225.4.a.f 1
5.c odd 4 2 225.4.b.e 2
7.b odd 2 1 2205.4.a.l 1
9.c even 3 2 405.4.e.i 2
9.d odd 6 2 405.4.e.g 2
12.b even 2 1 240.4.a.e 1
15.d odd 2 1 75.4.a.b 1
15.e even 4 2 75.4.b.b 2
21.c even 2 1 735.4.a.e 1
24.f even 2 1 960.4.a.ba 1
24.h odd 2 1 960.4.a.b 1
33.d even 2 1 1815.4.a.e 1
60.h even 2 1 1200.4.a.t 1
60.l odd 4 2 1200.4.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 3.b odd 2 1
45.4.a.c 1 1.a even 1 1 trivial
75.4.a.b 1 15.d odd 2 1
75.4.b.b 2 15.e even 4 2
225.4.a.f 1 5.b even 2 1
225.4.b.e 2 5.c odd 4 2
240.4.a.e 1 12.b even 2 1
405.4.e.g 2 9.d odd 6 2
405.4.e.i 2 9.c even 3 2
720.4.a.n 1 4.b odd 2 1
735.4.a.e 1 21.c even 2 1
960.4.a.b 1 24.h odd 2 1
960.4.a.ba 1 24.f even 2 1
1200.4.a.t 1 60.h even 2 1
1200.4.f.b 2 60.l odd 4 2
1815.4.a.e 1 33.d even 2 1
2205.4.a.l 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(45))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$5 + T$$
$7$ $$24 + T$$
$11$ $$52 + T$$
$13$ $$-22 + T$$
$17$ $$-14 + T$$
$19$ $$20 + T$$
$23$ $$-168 + T$$
$29$ $$230 + T$$
$31$ $$288 + T$$
$37$ $$34 + T$$
$41$ $$122 + T$$
$43$ $$188 + T$$
$47$ $$256 + T$$
$53$ $$-338 + T$$
$59$ $$100 + T$$
$61$ $$-742 + T$$
$67$ $$84 + T$$
$71$ $$-328 + T$$
$73$ $$38 + T$$
$79$ $$240 + T$$
$83$ $$1212 + T$$
$89$ $$330 + T$$
$97$ $$-866 + T$$