# Properties

 Label 725.4.a.a Level $725$ Weight $4$ Character orbit 725.a Self dual yes Analytic conductor $42.776$ 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] = [725,4,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, 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("725.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$725 = 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 725.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: $$42.7763847542$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 145) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 8 q^{3} - 7 q^{4} - 8 q^{6} + 14 q^{7} + 15 q^{8} + 37 q^{9}+O(q^{10})$$ q - q^2 + 8 * q^3 - 7 * q^4 - 8 * q^6 + 14 * q^7 + 15 * q^8 + 37 * q^9 $$q - q^{2} + 8 q^{3} - 7 q^{4} - 8 q^{6} + 14 q^{7} + 15 q^{8} + 37 q^{9} + 62 q^{11} - 56 q^{12} - 42 q^{13} - 14 q^{14} + 41 q^{16} + 114 q^{17} - 37 q^{18} - 70 q^{19} + 112 q^{21} - 62 q^{22} - 62 q^{23} + 120 q^{24} + 42 q^{26} + 80 q^{27} - 98 q^{28} - 29 q^{29} + 142 q^{31} - 161 q^{32} + 496 q^{33} - 114 q^{34} - 259 q^{36} - 146 q^{37} + 70 q^{38} - 336 q^{39} + 162 q^{41} - 112 q^{42} - 352 q^{43} - 434 q^{44} + 62 q^{46} + 444 q^{47} + 328 q^{48} - 147 q^{49} + 912 q^{51} + 294 q^{52} + 238 q^{53} - 80 q^{54} + 210 q^{56} - 560 q^{57} + 29 q^{58} + 840 q^{59} + 2 q^{61} - 142 q^{62} + 518 q^{63} - 167 q^{64} - 496 q^{66} + 154 q^{67} - 798 q^{68} - 496 q^{69} + 892 q^{71} + 555 q^{72} + 38 q^{73} + 146 q^{74} + 490 q^{76} + 868 q^{77} + 336 q^{78} + 1050 q^{79} - 359 q^{81} - 162 q^{82} + 778 q^{83} - 784 q^{84} + 352 q^{86} - 232 q^{87} + 930 q^{88} + 1410 q^{89} - 588 q^{91} + 434 q^{92} + 1136 q^{93} - 444 q^{94} - 1288 q^{96} - 466 q^{97} + 147 q^{98} + 2294 q^{99}+O(q^{100})$$ q - q^2 + 8 * q^3 - 7 * q^4 - 8 * q^6 + 14 * q^7 + 15 * q^8 + 37 * q^9 + 62 * q^11 - 56 * q^12 - 42 * q^13 - 14 * q^14 + 41 * q^16 + 114 * q^17 - 37 * q^18 - 70 * q^19 + 112 * q^21 - 62 * q^22 - 62 * q^23 + 120 * q^24 + 42 * q^26 + 80 * q^27 - 98 * q^28 - 29 * q^29 + 142 * q^31 - 161 * q^32 + 496 * q^33 - 114 * q^34 - 259 * q^36 - 146 * q^37 + 70 * q^38 - 336 * q^39 + 162 * q^41 - 112 * q^42 - 352 * q^43 - 434 * q^44 + 62 * q^46 + 444 * q^47 + 328 * q^48 - 147 * q^49 + 912 * q^51 + 294 * q^52 + 238 * q^53 - 80 * q^54 + 210 * q^56 - 560 * q^57 + 29 * q^58 + 840 * q^59 + 2 * q^61 - 142 * q^62 + 518 * q^63 - 167 * q^64 - 496 * q^66 + 154 * q^67 - 798 * q^68 - 496 * q^69 + 892 * q^71 + 555 * q^72 + 38 * q^73 + 146 * q^74 + 490 * q^76 + 868 * q^77 + 336 * q^78 + 1050 * q^79 - 359 * q^81 - 162 * q^82 + 778 * q^83 - 784 * q^84 + 352 * q^86 - 232 * q^87 + 930 * q^88 + 1410 * q^89 - 588 * q^91 + 434 * q^92 + 1136 * q^93 - 444 * q^94 - 1288 * q^96 - 466 * q^97 + 147 * q^98 + 2294 * 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 0 −8.00000 14.0000 15.0000 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
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 725.4.a.a 1
5.b even 2 1 145.4.a.a 1
15.d odd 2 1 1305.4.a.b 1
20.d odd 2 1 2320.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.a.a 1 5.b even 2 1
725.4.a.a 1 1.a even 1 1 trivial
1305.4.a.b 1 15.d odd 2 1
2320.4.a.f 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(725))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 8$$
$5$ $$T$$
$7$ $$T - 14$$
$11$ $$T - 62$$
$13$ $$T + 42$$
$17$ $$T - 114$$
$19$ $$T + 70$$
$23$ $$T + 62$$
$29$ $$T + 29$$
$31$ $$T - 142$$
$37$ $$T + 146$$
$41$ $$T - 162$$
$43$ $$T + 352$$
$47$ $$T - 444$$
$53$ $$T - 238$$
$59$ $$T - 840$$
$61$ $$T - 2$$
$67$ $$T - 154$$
$71$ $$T - 892$$
$73$ $$T - 38$$
$79$ $$T - 1050$$
$83$ $$T - 778$$
$89$ $$T - 1410$$
$97$ $$T + 466$$