# Properties

 Label 245.4.a.b Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $0$ 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] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.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: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 6 q^{3} - 7 q^{4} + 5 q^{5} - 6 q^{6} - 15 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 - 6 * q^3 - 7 * q^4 + 5 * q^5 - 6 * q^6 - 15 * q^8 + 9 * q^9 $$q + q^{2} - 6 q^{3} - 7 q^{4} + 5 q^{5} - 6 q^{6} - 15 q^{8} + 9 q^{9} + 5 q^{10} - 44 q^{11} + 42 q^{12} - 6 q^{13} - 30 q^{15} + 41 q^{16} + 24 q^{17} + 9 q^{18} + 114 q^{19} - 35 q^{20} - 44 q^{22} - 52 q^{23} + 90 q^{24} + 25 q^{25} - 6 q^{26} + 108 q^{27} + 146 q^{29} - 30 q^{30} + 276 q^{31} + 161 q^{32} + 264 q^{33} + 24 q^{34} - 63 q^{36} - 210 q^{37} + 114 q^{38} + 36 q^{39} - 75 q^{40} - 444 q^{41} + 492 q^{43} + 308 q^{44} + 45 q^{45} - 52 q^{46} + 612 q^{47} - 246 q^{48} + 25 q^{50} - 144 q^{51} + 42 q^{52} + 50 q^{53} + 108 q^{54} - 220 q^{55} - 684 q^{57} + 146 q^{58} - 294 q^{59} + 210 q^{60} - 450 q^{61} + 276 q^{62} - 167 q^{64} - 30 q^{65} + 264 q^{66} - 668 q^{67} - 168 q^{68} + 312 q^{69} - 308 q^{71} - 135 q^{72} - 12 q^{73} - 210 q^{74} - 150 q^{75} - 798 q^{76} + 36 q^{78} + 596 q^{79} + 205 q^{80} - 891 q^{81} - 444 q^{82} + 966 q^{83} + 120 q^{85} + 492 q^{86} - 876 q^{87} + 660 q^{88} + 408 q^{89} + 45 q^{90} + 364 q^{92} - 1656 q^{93} + 612 q^{94} + 570 q^{95} - 966 q^{96} + 1200 q^{97} - 396 q^{99}+O(q^{100})$$ q + q^2 - 6 * q^3 - 7 * q^4 + 5 * q^5 - 6 * q^6 - 15 * q^8 + 9 * q^9 + 5 * q^10 - 44 * q^11 + 42 * q^12 - 6 * q^13 - 30 * q^15 + 41 * q^16 + 24 * q^17 + 9 * q^18 + 114 * q^19 - 35 * q^20 - 44 * q^22 - 52 * q^23 + 90 * q^24 + 25 * q^25 - 6 * q^26 + 108 * q^27 + 146 * q^29 - 30 * q^30 + 276 * q^31 + 161 * q^32 + 264 * q^33 + 24 * q^34 - 63 * q^36 - 210 * q^37 + 114 * q^38 + 36 * q^39 - 75 * q^40 - 444 * q^41 + 492 * q^43 + 308 * q^44 + 45 * q^45 - 52 * q^46 + 612 * q^47 - 246 * q^48 + 25 * q^50 - 144 * q^51 + 42 * q^52 + 50 * q^53 + 108 * q^54 - 220 * q^55 - 684 * q^57 + 146 * q^58 - 294 * q^59 + 210 * q^60 - 450 * q^61 + 276 * q^62 - 167 * q^64 - 30 * q^65 + 264 * q^66 - 668 * q^67 - 168 * q^68 + 312 * q^69 - 308 * q^71 - 135 * q^72 - 12 * q^73 - 210 * q^74 - 150 * q^75 - 798 * q^76 + 36 * q^78 + 596 * q^79 + 205 * q^80 - 891 * q^81 - 444 * q^82 + 966 * q^83 + 120 * q^85 + 492 * q^86 - 876 * q^87 + 660 * q^88 + 408 * q^89 + 45 * q^90 + 364 * q^92 - 1656 * q^93 + 612 * q^94 + 570 * q^95 - 966 * q^96 + 1200 * q^97 - 396 * 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
1.00000 −6.00000 −7.00000 5.00000 −6.00000 0 −15.0000 9.00000 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
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 245.4.a.b 1
3.b odd 2 1 2205.4.a.k 1
5.b even 2 1 1225.4.a.g 1
7.b odd 2 1 245.4.a.c yes 1
7.c even 3 2 245.4.e.d 2
7.d odd 6 2 245.4.e.c 2
21.c even 2 1 2205.4.a.n 1
35.c odd 2 1 1225.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.b 1 1.a even 1 1 trivial
245.4.a.c yes 1 7.b odd 2 1
245.4.e.c 2 7.d odd 6 2
245.4.e.d 2 7.c even 3 2
1225.4.a.f 1 35.c odd 2 1
1225.4.a.g 1 5.b even 2 1
2205.4.a.k 1 3.b odd 2 1
2205.4.a.n 1 21.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{3} + 6$$ T3 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 6$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T + 44$$
$13$ $$T + 6$$
$17$ $$T - 24$$
$19$ $$T - 114$$
$23$ $$T + 52$$
$29$ $$T - 146$$
$31$ $$T - 276$$
$37$ $$T + 210$$
$41$ $$T + 444$$
$43$ $$T - 492$$
$47$ $$T - 612$$
$53$ $$T - 50$$
$59$ $$T + 294$$
$61$ $$T + 450$$
$67$ $$T + 668$$
$71$ $$T + 308$$
$73$ $$T + 12$$
$79$ $$T - 596$$
$83$ $$T - 966$$
$89$ $$T - 408$$
$97$ $$T - 1200$$