# Properties

 Label 245.4.a.d Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

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: 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 8 q^{3} - 7 q^{4} + 5 q^{5} + 8 q^{6} - 15 q^{8} + 37 q^{9}+O(q^{10})$$ q + q^2 + 8 * q^3 - 7 * q^4 + 5 * q^5 + 8 * q^6 - 15 * q^8 + 37 * q^9 $$q + q^{2} + 8 q^{3} - 7 q^{4} + 5 q^{5} + 8 q^{6} - 15 q^{8} + 37 q^{9} + 5 q^{10} + 12 q^{11} - 56 q^{12} + 78 q^{13} + 40 q^{15} + 41 q^{16} + 94 q^{17} + 37 q^{18} - 40 q^{19} - 35 q^{20} + 12 q^{22} + 32 q^{23} - 120 q^{24} + 25 q^{25} + 78 q^{26} + 80 q^{27} - 50 q^{29} + 40 q^{30} + 248 q^{31} + 161 q^{32} + 96 q^{33} + 94 q^{34} - 259 q^{36} - 434 q^{37} - 40 q^{38} + 624 q^{39} - 75 q^{40} - 402 q^{41} - 68 q^{43} - 84 q^{44} + 185 q^{45} + 32 q^{46} - 536 q^{47} + 328 q^{48} + 25 q^{50} + 752 q^{51} - 546 q^{52} + 22 q^{53} + 80 q^{54} + 60 q^{55} - 320 q^{57} - 50 q^{58} + 560 q^{59} - 280 q^{60} + 278 q^{61} + 248 q^{62} - 167 q^{64} + 390 q^{65} + 96 q^{66} - 164 q^{67} - 658 q^{68} + 256 q^{69} + 672 q^{71} - 555 q^{72} - 82 q^{73} - 434 q^{74} + 200 q^{75} + 280 q^{76} + 624 q^{78} - 1000 q^{79} + 205 q^{80} - 359 q^{81} - 402 q^{82} + 448 q^{83} + 470 q^{85} - 68 q^{86} - 400 q^{87} - 180 q^{88} + 870 q^{89} + 185 q^{90} - 224 q^{92} + 1984 q^{93} - 536 q^{94} - 200 q^{95} + 1288 q^{96} - 1026 q^{97} + 444 q^{99}+O(q^{100})$$ q + q^2 + 8 * q^3 - 7 * q^4 + 5 * q^5 + 8 * q^6 - 15 * q^8 + 37 * q^9 + 5 * q^10 + 12 * q^11 - 56 * q^12 + 78 * q^13 + 40 * q^15 + 41 * q^16 + 94 * q^17 + 37 * q^18 - 40 * q^19 - 35 * q^20 + 12 * q^22 + 32 * q^23 - 120 * q^24 + 25 * q^25 + 78 * q^26 + 80 * q^27 - 50 * q^29 + 40 * q^30 + 248 * q^31 + 161 * q^32 + 96 * q^33 + 94 * q^34 - 259 * q^36 - 434 * q^37 - 40 * q^38 + 624 * q^39 - 75 * q^40 - 402 * q^41 - 68 * q^43 - 84 * q^44 + 185 * q^45 + 32 * q^46 - 536 * q^47 + 328 * q^48 + 25 * q^50 + 752 * q^51 - 546 * q^52 + 22 * q^53 + 80 * q^54 + 60 * q^55 - 320 * q^57 - 50 * q^58 + 560 * q^59 - 280 * q^60 + 278 * q^61 + 248 * q^62 - 167 * q^64 + 390 * q^65 + 96 * q^66 - 164 * q^67 - 658 * q^68 + 256 * q^69 + 672 * q^71 - 555 * q^72 - 82 * q^73 - 434 * q^74 + 200 * q^75 + 280 * q^76 + 624 * q^78 - 1000 * q^79 + 205 * q^80 - 359 * q^81 - 402 * q^82 + 448 * q^83 + 470 * q^85 - 68 * q^86 - 400 * q^87 - 180 * q^88 + 870 * q^89 + 185 * q^90 - 224 * q^92 + 1984 * q^93 - 536 * q^94 - 200 * q^95 + 1288 * q^96 - 1026 * q^97 + 444 * 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 8.00000 −7.00000 5.00000 8.00000 0 −15.0000 37.0000 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.d 1
3.b odd 2 1 2205.4.a.i 1
5.b even 2 1 1225.4.a.e 1
7.b odd 2 1 35.4.a.a 1
7.c even 3 2 245.4.e.b 2
7.d odd 6 2 245.4.e.e 2
21.c even 2 1 315.4.a.c 1
28.d even 2 1 560.4.a.p 1
35.c odd 2 1 175.4.a.a 1
35.f even 4 2 175.4.b.a 2
56.e even 2 1 2240.4.a.b 1
56.h odd 2 1 2240.4.a.bk 1
105.g even 2 1 1575.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.a 1 7.b odd 2 1
175.4.a.a 1 35.c odd 2 1
175.4.b.a 2 35.f even 4 2
245.4.a.d 1 1.a even 1 1 trivial
245.4.e.b 2 7.c even 3 2
245.4.e.e 2 7.d odd 6 2
315.4.a.c 1 21.c even 2 1
560.4.a.p 1 28.d even 2 1
1225.4.a.e 1 5.b even 2 1
1575.4.a.g 1 105.g even 2 1
2205.4.a.i 1 3.b odd 2 1
2240.4.a.b 1 56.e even 2 1
2240.4.a.bk 1 56.h odd 2 1

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{3} - 8$$ T3 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 8$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T - 12$$
$13$ $$T - 78$$
$17$ $$T - 94$$
$19$ $$T + 40$$
$23$ $$T - 32$$
$29$ $$T + 50$$
$31$ $$T - 248$$
$37$ $$T + 434$$
$41$ $$T + 402$$
$43$ $$T + 68$$
$47$ $$T + 536$$
$53$ $$T - 22$$
$59$ $$T - 560$$
$61$ $$T - 278$$
$67$ $$T + 164$$
$71$ $$T - 672$$
$73$ $$T + 82$$
$79$ $$T + 1000$$
$83$ $$T - 448$$
$89$ $$T - 870$$
$97$ $$T + 1026$$
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