# Properties

 Label 1445.4.a.a Level $1445$ Weight $4$ Character orbit 1445.a Self dual yes Analytic conductor $85.258$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,4,Mod(1,1445)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1445.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1445 = 5 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1445.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$85.2577599583$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9}+O(q^{10})$$ q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 - 6 * q^7 - 23 * q^9 $$q - 4 q^{2} - 2 q^{3} + 8 q^{4} + 5 q^{5} + 8 q^{6} - 6 q^{7} - 23 q^{9} - 20 q^{10} - 32 q^{11} - 16 q^{12} - 38 q^{13} + 24 q^{14} - 10 q^{15} - 64 q^{16} + 92 q^{18} + 100 q^{19} + 40 q^{20} + 12 q^{21} + 128 q^{22} + 78 q^{23} + 25 q^{25} + 152 q^{26} + 100 q^{27} - 48 q^{28} + 50 q^{29} + 40 q^{30} + 108 q^{31} + 256 q^{32} + 64 q^{33} - 30 q^{35} - 184 q^{36} - 266 q^{37} - 400 q^{38} + 76 q^{39} - 22 q^{41} - 48 q^{42} + 442 q^{43} - 256 q^{44} - 115 q^{45} - 312 q^{46} - 514 q^{47} + 128 q^{48} - 307 q^{49} - 100 q^{50} - 304 q^{52} + 2 q^{53} - 400 q^{54} - 160 q^{55} - 200 q^{57} - 200 q^{58} + 500 q^{59} - 80 q^{60} + 518 q^{61} - 432 q^{62} + 138 q^{63} - 512 q^{64} - 190 q^{65} - 256 q^{66} + 126 q^{67} - 156 q^{69} + 120 q^{70} - 412 q^{71} + 878 q^{73} + 1064 q^{74} - 50 q^{75} + 800 q^{76} + 192 q^{77} - 304 q^{78} - 600 q^{79} - 320 q^{80} + 421 q^{81} + 88 q^{82} + 282 q^{83} + 96 q^{84} - 1768 q^{86} - 100 q^{87} - 150 q^{89} + 460 q^{90} + 228 q^{91} + 624 q^{92} - 216 q^{93} + 2056 q^{94} + 500 q^{95} - 512 q^{96} - 386 q^{97} + 1228 q^{98} + 736 q^{99}+O(q^{100})$$ q - 4 * q^2 - 2 * q^3 + 8 * q^4 + 5 * q^5 + 8 * q^6 - 6 * q^7 - 23 * q^9 - 20 * q^10 - 32 * q^11 - 16 * q^12 - 38 * q^13 + 24 * q^14 - 10 * q^15 - 64 * q^16 + 92 * q^18 + 100 * q^19 + 40 * q^20 + 12 * q^21 + 128 * q^22 + 78 * q^23 + 25 * q^25 + 152 * q^26 + 100 * q^27 - 48 * q^28 + 50 * q^29 + 40 * q^30 + 108 * q^31 + 256 * q^32 + 64 * q^33 - 30 * q^35 - 184 * q^36 - 266 * q^37 - 400 * q^38 + 76 * q^39 - 22 * q^41 - 48 * q^42 + 442 * q^43 - 256 * q^44 - 115 * q^45 - 312 * q^46 - 514 * q^47 + 128 * q^48 - 307 * q^49 - 100 * q^50 - 304 * q^52 + 2 * q^53 - 400 * q^54 - 160 * q^55 - 200 * q^57 - 200 * q^58 + 500 * q^59 - 80 * q^60 + 518 * q^61 - 432 * q^62 + 138 * q^63 - 512 * q^64 - 190 * q^65 - 256 * q^66 + 126 * q^67 - 156 * q^69 + 120 * q^70 - 412 * q^71 + 878 * q^73 + 1064 * q^74 - 50 * q^75 + 800 * q^76 + 192 * q^77 - 304 * q^78 - 600 * q^79 - 320 * q^80 + 421 * q^81 + 88 * q^82 + 282 * q^83 + 96 * q^84 - 1768 * q^86 - 100 * q^87 - 150 * q^89 + 460 * q^90 + 228 * q^91 + 624 * q^92 - 216 * q^93 + 2056 * q^94 + 500 * q^95 - 512 * q^96 - 386 * q^97 + 1228 * q^98 + 736 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
−4.00000 −2.00000 8.00000 5.00000 8.00000 −6.00000 0 −23.0000 −20.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$17$$ $$1$$

## 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 1445.4.a.a 1
17.b even 2 1 5.4.a.a 1
51.c odd 2 1 45.4.a.d 1
68.d odd 2 1 80.4.a.d 1
85.c even 2 1 25.4.a.c 1
85.g odd 4 2 25.4.b.a 2
119.d odd 2 1 245.4.a.a 1
119.h odd 6 2 245.4.e.g 2
119.j even 6 2 245.4.e.f 2
136.e odd 2 1 320.4.a.h 1
136.h even 2 1 320.4.a.g 1
153.h even 6 2 405.4.e.l 2
153.i odd 6 2 405.4.e.c 2
187.b odd 2 1 605.4.a.d 1
204.h even 2 1 720.4.a.u 1
221.b even 2 1 845.4.a.b 1
255.h odd 2 1 225.4.a.b 1
255.o even 4 2 225.4.b.c 2
272.k odd 4 2 1280.4.d.l 2
272.r even 4 2 1280.4.d.e 2
323.c odd 2 1 1805.4.a.h 1
340.d odd 2 1 400.4.a.m 1
340.r even 4 2 400.4.c.k 2
357.c even 2 1 2205.4.a.q 1
595.b odd 2 1 1225.4.a.k 1
680.h even 2 1 1600.4.a.bi 1
680.k odd 2 1 1600.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 17.b even 2 1
25.4.a.c 1 85.c even 2 1
25.4.b.a 2 85.g odd 4 2
45.4.a.d 1 51.c odd 2 1
80.4.a.d 1 68.d odd 2 1
225.4.a.b 1 255.h odd 2 1
225.4.b.c 2 255.o even 4 2
245.4.a.a 1 119.d odd 2 1
245.4.e.f 2 119.j even 6 2
245.4.e.g 2 119.h odd 6 2
320.4.a.g 1 136.h even 2 1
320.4.a.h 1 136.e odd 2 1
400.4.a.m 1 340.d odd 2 1
400.4.c.k 2 340.r even 4 2
405.4.e.c 2 153.i odd 6 2
405.4.e.l 2 153.h even 6 2
605.4.a.d 1 187.b odd 2 1
720.4.a.u 1 204.h even 2 1
845.4.a.b 1 221.b even 2 1
1225.4.a.k 1 595.b odd 2 1
1280.4.d.e 2 272.r even 4 2
1280.4.d.l 2 272.k odd 4 2
1445.4.a.a 1 1.a even 1 1 trivial
1600.4.a.s 1 680.k odd 2 1
1600.4.a.bi 1 680.h even 2 1
1805.4.a.h 1 323.c odd 2 1
2205.4.a.q 1 357.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1445))$$:

 $$T_{2} + 4$$ T2 + 4 $$T_{3} + 2$$ T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T + 2$$
$5$ $$T - 5$$
$7$ $$T + 6$$
$11$ $$T + 32$$
$13$ $$T + 38$$
$17$ $$T$$
$19$ $$T - 100$$
$23$ $$T - 78$$
$29$ $$T - 50$$
$31$ $$T - 108$$
$37$ $$T + 266$$
$41$ $$T + 22$$
$43$ $$T - 442$$
$47$ $$T + 514$$
$53$ $$T - 2$$
$59$ $$T - 500$$
$61$ $$T - 518$$
$67$ $$T - 126$$
$71$ $$T + 412$$
$73$ $$T - 878$$
$79$ $$T + 600$$
$83$ $$T - 282$$
$89$ $$T + 150$$
$97$ $$T + 386$$