# Properties

 Label 245.4.a.f Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 2 q^{3} + q^{4} - 5 q^{5} + 6 q^{6} - 21 q^{8} - 23 q^{9}+O(q^{10})$$ q + 3 * q^2 + 2 * q^3 + q^4 - 5 * q^5 + 6 * q^6 - 21 * q^8 - 23 * q^9 $$q + 3 q^{2} + 2 q^{3} + q^{4} - 5 q^{5} + 6 q^{6} - 21 q^{8} - 23 q^{9} - 15 q^{10} - 45 q^{11} + 2 q^{12} - 59 q^{13} - 10 q^{15} - 71 q^{16} + 54 q^{17} - 69 q^{18} + 121 q^{19} - 5 q^{20} - 135 q^{22} + 69 q^{23} - 42 q^{24} + 25 q^{25} - 177 q^{26} - 100 q^{27} - 162 q^{29} - 30 q^{30} + 88 q^{31} - 45 q^{32} - 90 q^{33} + 162 q^{34} - 23 q^{36} - 259 q^{37} + 363 q^{38} - 118 q^{39} + 105 q^{40} - 195 q^{41} - 286 q^{43} - 45 q^{44} + 115 q^{45} + 207 q^{46} - 45 q^{47} - 142 q^{48} + 75 q^{50} + 108 q^{51} - 59 q^{52} + 597 q^{53} - 300 q^{54} + 225 q^{55} + 242 q^{57} - 486 q^{58} + 360 q^{59} - 10 q^{60} - 392 q^{61} + 264 q^{62} + 433 q^{64} + 295 q^{65} - 270 q^{66} - 280 q^{67} + 54 q^{68} + 138 q^{69} + 48 q^{71} + 483 q^{72} - 668 q^{73} - 777 q^{74} + 50 q^{75} + 121 q^{76} - 354 q^{78} + 782 q^{79} + 355 q^{80} + 421 q^{81} - 585 q^{82} - 768 q^{83} - 270 q^{85} - 858 q^{86} - 324 q^{87} + 945 q^{88} + 1194 q^{89} + 345 q^{90} + 69 q^{92} + 176 q^{93} - 135 q^{94} - 605 q^{95} - 90 q^{96} - 902 q^{97} + 1035 q^{99}+O(q^{100})$$ q + 3 * q^2 + 2 * q^3 + q^4 - 5 * q^5 + 6 * q^6 - 21 * q^8 - 23 * q^9 - 15 * q^10 - 45 * q^11 + 2 * q^12 - 59 * q^13 - 10 * q^15 - 71 * q^16 + 54 * q^17 - 69 * q^18 + 121 * q^19 - 5 * q^20 - 135 * q^22 + 69 * q^23 - 42 * q^24 + 25 * q^25 - 177 * q^26 - 100 * q^27 - 162 * q^29 - 30 * q^30 + 88 * q^31 - 45 * q^32 - 90 * q^33 + 162 * q^34 - 23 * q^36 - 259 * q^37 + 363 * q^38 - 118 * q^39 + 105 * q^40 - 195 * q^41 - 286 * q^43 - 45 * q^44 + 115 * q^45 + 207 * q^46 - 45 * q^47 - 142 * q^48 + 75 * q^50 + 108 * q^51 - 59 * q^52 + 597 * q^53 - 300 * q^54 + 225 * q^55 + 242 * q^57 - 486 * q^58 + 360 * q^59 - 10 * q^60 - 392 * q^61 + 264 * q^62 + 433 * q^64 + 295 * q^65 - 270 * q^66 - 280 * q^67 + 54 * q^68 + 138 * q^69 + 48 * q^71 + 483 * q^72 - 668 * q^73 - 777 * q^74 + 50 * q^75 + 121 * q^76 - 354 * q^78 + 782 * q^79 + 355 * q^80 + 421 * q^81 - 585 * q^82 - 768 * q^83 - 270 * q^85 - 858 * q^86 - 324 * q^87 + 945 * q^88 + 1194 * q^89 + 345 * q^90 + 69 * q^92 + 176 * q^93 - 135 * q^94 - 605 * q^95 - 90 * q^96 - 902 * q^97 + 1035 * 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
3.00000 2.00000 1.00000 −5.00000 6.00000 0 −21.0000 −23.0000 −15.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$7$$ $$-1$$

## 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.f 1
3.b odd 2 1 2205.4.a.g 1
5.b even 2 1 1225.4.a.a 1
7.b odd 2 1 245.4.a.e 1
7.c even 3 2 245.4.e.a 2
7.d odd 6 2 35.4.e.a 2
21.c even 2 1 2205.4.a.e 1
21.g even 6 2 315.4.j.b 2
28.f even 6 2 560.4.q.b 2
35.c odd 2 1 1225.4.a.b 1
35.i odd 6 2 175.4.e.b 2
35.k even 12 4 175.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 7.d odd 6 2
175.4.e.b 2 35.i odd 6 2
175.4.k.b 4 35.k even 12 4
245.4.a.e 1 7.b odd 2 1
245.4.a.f 1 1.a even 1 1 trivial
245.4.e.a 2 7.c even 3 2
315.4.j.b 2 21.g even 6 2
560.4.q.b 2 28.f even 6 2
1225.4.a.a 1 5.b even 2 1
1225.4.a.b 1 35.c odd 2 1
2205.4.a.e 1 21.c even 2 1
2205.4.a.g 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2} - 3$$ T2 - 3 $$T_{3} - 2$$ T3 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T - 2$$
$5$ $$T + 5$$
$7$ $$T$$
$11$ $$T + 45$$
$13$ $$T + 59$$
$17$ $$T - 54$$
$19$ $$T - 121$$
$23$ $$T - 69$$
$29$ $$T + 162$$
$31$ $$T - 88$$
$37$ $$T + 259$$
$41$ $$T + 195$$
$43$ $$T + 286$$
$47$ $$T + 45$$
$53$ $$T - 597$$
$59$ $$T - 360$$
$61$ $$T + 392$$
$67$ $$T + 280$$
$71$ $$T - 48$$
$73$ $$T + 668$$
$79$ $$T - 782$$
$83$ $$T + 768$$
$89$ $$T - 1194$$
$97$ $$T + 902$$