# Properties

 Label 425.4.a.d Level $425$ Weight $4$ Character orbit 425.a Self dual yes Analytic conductor $25.076$ 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] = [425,4,Mod(1,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, 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("425.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 425.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: $$25.0758117524$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 8 q^{3} + q^{4} + 24 q^{6} + 28 q^{7} - 21 q^{8} + 37 q^{9}+O(q^{10})$$ q + 3 * q^2 + 8 * q^3 + q^4 + 24 * q^6 + 28 * q^7 - 21 * q^8 + 37 * q^9 $$q + 3 q^{2} + 8 q^{3} + q^{4} + 24 q^{6} + 28 q^{7} - 21 q^{8} + 37 q^{9} - 24 q^{11} + 8 q^{12} + 58 q^{13} + 84 q^{14} - 71 q^{16} - 17 q^{17} + 111 q^{18} + 116 q^{19} + 224 q^{21} - 72 q^{22} + 60 q^{23} - 168 q^{24} + 174 q^{26} + 80 q^{27} + 28 q^{28} + 30 q^{29} - 172 q^{31} - 45 q^{32} - 192 q^{33} - 51 q^{34} + 37 q^{36} + 58 q^{37} + 348 q^{38} + 464 q^{39} - 342 q^{41} + 672 q^{42} + 148 q^{43} - 24 q^{44} + 180 q^{46} - 288 q^{47} - 568 q^{48} + 441 q^{49} - 136 q^{51} + 58 q^{52} - 318 q^{53} + 240 q^{54} - 588 q^{56} + 928 q^{57} + 90 q^{58} + 252 q^{59} + 110 q^{61} - 516 q^{62} + 1036 q^{63} + 433 q^{64} - 576 q^{66} + 484 q^{67} - 17 q^{68} + 480 q^{69} - 708 q^{71} - 777 q^{72} - 362 q^{73} + 174 q^{74} + 116 q^{76} - 672 q^{77} + 1392 q^{78} - 484 q^{79} - 359 q^{81} - 1026 q^{82} - 756 q^{83} + 224 q^{84} + 444 q^{86} + 240 q^{87} + 504 q^{88} - 774 q^{89} + 1624 q^{91} + 60 q^{92} - 1376 q^{93} - 864 q^{94} - 360 q^{96} + 382 q^{97} + 1323 q^{98} - 888 q^{99}+O(q^{100})$$ q + 3 * q^2 + 8 * q^3 + q^4 + 24 * q^6 + 28 * q^7 - 21 * q^8 + 37 * q^9 - 24 * q^11 + 8 * q^12 + 58 * q^13 + 84 * q^14 - 71 * q^16 - 17 * q^17 + 111 * q^18 + 116 * q^19 + 224 * q^21 - 72 * q^22 + 60 * q^23 - 168 * q^24 + 174 * q^26 + 80 * q^27 + 28 * q^28 + 30 * q^29 - 172 * q^31 - 45 * q^32 - 192 * q^33 - 51 * q^34 + 37 * q^36 + 58 * q^37 + 348 * q^38 + 464 * q^39 - 342 * q^41 + 672 * q^42 + 148 * q^43 - 24 * q^44 + 180 * q^46 - 288 * q^47 - 568 * q^48 + 441 * q^49 - 136 * q^51 + 58 * q^52 - 318 * q^53 + 240 * q^54 - 588 * q^56 + 928 * q^57 + 90 * q^58 + 252 * q^59 + 110 * q^61 - 516 * q^62 + 1036 * q^63 + 433 * q^64 - 576 * q^66 + 484 * q^67 - 17 * q^68 + 480 * q^69 - 708 * q^71 - 777 * q^72 - 362 * q^73 + 174 * q^74 + 116 * q^76 - 672 * q^77 + 1392 * q^78 - 484 * q^79 - 359 * q^81 - 1026 * q^82 - 756 * q^83 + 224 * q^84 + 444 * q^86 + 240 * q^87 + 504 * q^88 - 774 * q^89 + 1624 * q^91 + 60 * q^92 - 1376 * q^93 - 864 * q^94 - 360 * q^96 + 382 * q^97 + 1323 * q^98 - 888 * 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 8.00000 1.00000 0 24.0000 28.0000 −21.0000 37.0000 0
 $$n$$: 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 425.4.a.d 1
5.b even 2 1 17.4.a.a 1
5.c odd 4 2 425.4.b.c 2
15.d odd 2 1 153.4.a.d 1
20.d odd 2 1 272.4.a.d 1
35.c odd 2 1 833.4.a.a 1
40.e odd 2 1 1088.4.a.a 1
40.f even 2 1 1088.4.a.l 1
55.d odd 2 1 2057.4.a.d 1
60.h even 2 1 2448.4.a.f 1
85.c even 2 1 289.4.a.a 1
85.j even 4 2 289.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 5.b even 2 1
153.4.a.d 1 15.d odd 2 1
272.4.a.d 1 20.d odd 2 1
289.4.a.a 1 85.c even 2 1
289.4.b.a 2 85.j even 4 2
425.4.a.d 1 1.a even 1 1 trivial
425.4.b.c 2 5.c odd 4 2
833.4.a.a 1 35.c odd 2 1
1088.4.a.a 1 40.e odd 2 1
1088.4.a.l 1 40.f even 2 1
2057.4.a.d 1 55.d odd 2 1
2448.4.a.f 1 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(425))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T - 8$$
$5$ $$T$$
$7$ $$T - 28$$
$11$ $$T + 24$$
$13$ $$T - 58$$
$17$ $$T + 17$$
$19$ $$T - 116$$
$23$ $$T - 60$$
$29$ $$T - 30$$
$31$ $$T + 172$$
$37$ $$T - 58$$
$41$ $$T + 342$$
$43$ $$T - 148$$
$47$ $$T + 288$$
$53$ $$T + 318$$
$59$ $$T - 252$$
$61$ $$T - 110$$
$67$ $$T - 484$$
$71$ $$T + 708$$
$73$ $$T + 362$$
$79$ $$T + 484$$
$83$ $$T + 756$$
$89$ $$T + 774$$
$97$ $$T - 382$$