# Properties

 Label 624.4.a.j Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta - 9) q^{5} + ( - 5 \beta + 5) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b - 9) * q^5 + (-5*b + 5) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta - 9) q^{5} + ( - 5 \beta + 5) q^{7} + 9 q^{9} + (14 \beta - 2) q^{11} + 13 q^{13} + (3 \beta + 27) q^{15} + (12 \beta - 26) q^{17} + ( - 11 \beta + 71) q^{19} + (15 \beta - 15) q^{21} + (12 \beta + 140) q^{23} + (18 \beta - 27) q^{25} - 27 q^{27} + ( - 6 \beta - 212) q^{29} + ( - 33 \beta + 89) q^{31} + ( - 42 \beta + 6) q^{33} + (40 \beta + 40) q^{35} + ( - 54 \beta + 68) q^{37} - 39 q^{39} + (19 \beta - 113) q^{41} + (88 \beta + 36) q^{43} + ( - 9 \beta - 81) q^{45} + ( - 120 \beta - 88) q^{47} + ( - 50 \beta + 107) q^{49} + ( - 36 \beta + 78) q^{51} + (8 \beta - 394) q^{53} + ( - 124 \beta - 220) q^{55} + (33 \beta - 213) q^{57} + ( - 8 \beta - 364) q^{59} + (38 \beta - 368) q^{61} + ( - 45 \beta + 45) q^{63} + ( - 13 \beta - 117) q^{65} + (107 \beta - 527) q^{67} + ( - 36 \beta - 420) q^{69} + (78 \beta + 330) q^{71} + (222 \beta + 12) q^{73} + ( - 54 \beta + 81) q^{75} + (80 \beta - 1200) q^{77} + (276 \beta - 20) q^{79} + 81 q^{81} + (68 \beta - 672) q^{83} + ( - 82 \beta + 30) q^{85} + (18 \beta + 636) q^{87} + ( - 219 \beta + 157) q^{89} + ( - 65 \beta + 65) q^{91} + (99 \beta - 267) q^{93} + (28 \beta - 452) q^{95} + ( - 10 \beta - 132) q^{97} + (126 \beta - 18) q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b - 9) * q^5 + (-5*b + 5) * q^7 + 9 * q^9 + (14*b - 2) * q^11 + 13 * q^13 + (3*b + 27) * q^15 + (12*b - 26) * q^17 + (-11*b + 71) * q^19 + (15*b - 15) * q^21 + (12*b + 140) * q^23 + (18*b - 27) * q^25 - 27 * q^27 + (-6*b - 212) * q^29 + (-33*b + 89) * q^31 + (-42*b + 6) * q^33 + (40*b + 40) * q^35 + (-54*b + 68) * q^37 - 39 * q^39 + (19*b - 113) * q^41 + (88*b + 36) * q^43 + (-9*b - 81) * q^45 + (-120*b - 88) * q^47 + (-50*b + 107) * q^49 + (-36*b + 78) * q^51 + (8*b - 394) * q^53 + (-124*b - 220) * q^55 + (33*b - 213) * q^57 + (-8*b - 364) * q^59 + (38*b - 368) * q^61 + (-45*b + 45) * q^63 + (-13*b - 117) * q^65 + (107*b - 527) * q^67 + (-36*b - 420) * q^69 + (78*b + 330) * q^71 + (222*b + 12) * q^73 + (-54*b + 81) * q^75 + (80*b - 1200) * q^77 + (276*b - 20) * q^79 + 81 * q^81 + (68*b - 672) * q^83 + (-82*b + 30) * q^85 + (18*b + 636) * q^87 + (-219*b + 157) * q^89 + (-65*b + 65) * q^91 + (99*b - 267) * q^93 + (28*b - 452) * q^95 + (-10*b - 132) * q^97 + (126*b - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 18 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 18 * q^5 + 10 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 18 q^{5} + 10 q^{7} + 18 q^{9} - 4 q^{11} + 26 q^{13} + 54 q^{15} - 52 q^{17} + 142 q^{19} - 30 q^{21} + 280 q^{23} - 54 q^{25} - 54 q^{27} - 424 q^{29} + 178 q^{31} + 12 q^{33} + 80 q^{35} + 136 q^{37} - 78 q^{39} - 226 q^{41} + 72 q^{43} - 162 q^{45} - 176 q^{47} + 214 q^{49} + 156 q^{51} - 788 q^{53} - 440 q^{55} - 426 q^{57} - 728 q^{59} - 736 q^{61} + 90 q^{63} - 234 q^{65} - 1054 q^{67} - 840 q^{69} + 660 q^{71} + 24 q^{73} + 162 q^{75} - 2400 q^{77} - 40 q^{79} + 162 q^{81} - 1344 q^{83} + 60 q^{85} + 1272 q^{87} + 314 q^{89} + 130 q^{91} - 534 q^{93} - 904 q^{95} - 264 q^{97} - 36 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 18 * q^5 + 10 * q^7 + 18 * q^9 - 4 * q^11 + 26 * q^13 + 54 * q^15 - 52 * q^17 + 142 * q^19 - 30 * q^21 + 280 * q^23 - 54 * q^25 - 54 * q^27 - 424 * q^29 + 178 * q^31 + 12 * q^33 + 80 * q^35 + 136 * q^37 - 78 * q^39 - 226 * q^41 + 72 * q^43 - 162 * q^45 - 176 * q^47 + 214 * q^49 + 156 * q^51 - 788 * q^53 - 440 * q^55 - 426 * q^57 - 728 * q^59 - 736 * q^61 + 90 * q^63 - 234 * q^65 - 1054 * q^67 - 840 * q^69 + 660 * q^71 + 24 * q^73 + 162 * q^75 - 2400 * q^77 - 40 * q^79 + 162 * q^81 - 1344 * q^83 + 60 * q^85 + 1272 * q^87 + 314 * q^89 + 130 * q^91 - 534 * q^93 - 904 * q^95 - 264 * q^97 - 36 * 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
 2.56155 −1.56155
0 −3.00000 0 −13.1231 0 −15.6155 0 9.00000 0
1.2 0 −3.00000 0 −4.87689 0 25.6155 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.j 2
3.b odd 2 1 1872.4.a.bi 2
4.b odd 2 1 312.4.a.d 2
8.b even 2 1 2496.4.a.bj 2
8.d odd 2 1 2496.4.a.ba 2
12.b even 2 1 936.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.d 2 4.b odd 2 1
624.4.a.j 2 1.a even 1 1 trivial
936.4.a.j 2 12.b even 2 1
1872.4.a.bi 2 3.b odd 2 1
2496.4.a.ba 2 8.d odd 2 1
2496.4.a.bj 2 8.b even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 18T_{5} + 64$$ T5^2 + 18*T5 + 64 $$T_{7}^{2} - 10T_{7} - 400$$ T7^2 - 10*T7 - 400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 18T + 64$$
$7$ $$T^{2} - 10T - 400$$
$11$ $$T^{2} + 4T - 3328$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 52T - 1772$$
$19$ $$T^{2} - 142T + 2984$$
$23$ $$T^{2} - 280T + 17152$$
$29$ $$T^{2} + 424T + 44332$$
$31$ $$T^{2} - 178T - 10592$$
$37$ $$T^{2} - 136T - 44948$$
$41$ $$T^{2} + 226T + 6632$$
$43$ $$T^{2} - 72T - 130352$$
$47$ $$T^{2} + 176T - 237056$$
$53$ $$T^{2} + 788T + 154148$$
$59$ $$T^{2} + 728T + 131408$$
$61$ $$T^{2} + 736T + 110876$$
$67$ $$T^{2} + 1054T + 83096$$
$71$ $$T^{2} - 660T + 5472$$
$73$ $$T^{2} - 24T - 837684$$
$79$ $$T^{2} + 40T - 1294592$$
$83$ $$T^{2} + 1344 T + 372976$$
$89$ $$T^{2} - 314T - 790688$$
$97$ $$T^{2} + 264T + 15724$$