# Properties

 Label 624.4.a.f Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ 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] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.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: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 16 q^{5} - 28 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 16 * q^5 - 28 * q^7 + 9 * q^9 $$q + 3 q^{3} - 16 q^{5} - 28 q^{7} + 9 q^{9} - 34 q^{11} - 13 q^{13} - 48 q^{15} + 138 q^{17} - 108 q^{19} - 84 q^{21} + 52 q^{23} + 131 q^{25} + 27 q^{27} - 190 q^{29} + 176 q^{31} - 102 q^{33} + 448 q^{35} + 342 q^{37} - 39 q^{39} + 240 q^{41} + 140 q^{43} - 144 q^{45} - 454 q^{47} + 441 q^{49} + 414 q^{51} + 198 q^{53} + 544 q^{55} - 324 q^{57} + 154 q^{59} + 34 q^{61} - 252 q^{63} + 208 q^{65} + 656 q^{67} + 156 q^{69} - 550 q^{71} + 614 q^{73} + 393 q^{75} + 952 q^{77} - 8 q^{79} + 81 q^{81} - 762 q^{83} - 2208 q^{85} - 570 q^{87} - 444 q^{89} + 364 q^{91} + 528 q^{93} + 1728 q^{95} + 1022 q^{97} - 306 q^{99}+O(q^{100})$$ q + 3 * q^3 - 16 * q^5 - 28 * q^7 + 9 * q^9 - 34 * q^11 - 13 * q^13 - 48 * q^15 + 138 * q^17 - 108 * q^19 - 84 * q^21 + 52 * q^23 + 131 * q^25 + 27 * q^27 - 190 * q^29 + 176 * q^31 - 102 * q^33 + 448 * q^35 + 342 * q^37 - 39 * q^39 + 240 * q^41 + 140 * q^43 - 144 * q^45 - 454 * q^47 + 441 * q^49 + 414 * q^51 + 198 * q^53 + 544 * q^55 - 324 * q^57 + 154 * q^59 + 34 * q^61 - 252 * q^63 + 208 * q^65 + 656 * q^67 + 156 * q^69 - 550 * q^71 + 614 * q^73 + 393 * q^75 + 952 * q^77 - 8 * q^79 + 81 * q^81 - 762 * q^83 - 2208 * q^85 - 570 * q^87 - 444 * q^89 + 364 * q^91 + 528 * q^93 + 1728 * q^95 + 1022 * q^97 - 306 * 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
0 3.00000 0 −16.0000 0 −28.0000 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$+1$$

## 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 624.4.a.f 1
3.b odd 2 1 1872.4.a.o 1
4.b odd 2 1 78.4.a.a 1
8.b even 2 1 2496.4.a.g 1
8.d odd 2 1 2496.4.a.q 1
12.b even 2 1 234.4.a.k 1
20.d odd 2 1 1950.4.a.o 1
52.b odd 2 1 1014.4.a.i 1
52.f even 4 2 1014.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.a 1 4.b odd 2 1
234.4.a.k 1 12.b even 2 1
624.4.a.f 1 1.a even 1 1 trivial
1014.4.a.i 1 52.b odd 2 1
1014.4.b.a 2 52.f even 4 2
1872.4.a.o 1 3.b odd 2 1
1950.4.a.o 1 20.d odd 2 1
2496.4.a.g 1 8.b even 2 1
2496.4.a.q 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5} + 16$$ T5 + 16 $$T_{7} + 28$$ T7 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 16$$
$7$ $$T + 28$$
$11$ $$T + 34$$
$13$ $$T + 13$$
$17$ $$T - 138$$
$19$ $$T + 108$$
$23$ $$T - 52$$
$29$ $$T + 190$$
$31$ $$T - 176$$
$37$ $$T - 342$$
$41$ $$T - 240$$
$43$ $$T - 140$$
$47$ $$T + 454$$
$53$ $$T - 198$$
$59$ $$T - 154$$
$61$ $$T - 34$$
$67$ $$T - 656$$
$71$ $$T + 550$$
$73$ $$T - 614$$
$79$ $$T + 8$$
$83$ $$T + 762$$
$89$ $$T + 444$$
$97$ $$T - 1022$$
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