# Properties

 Label 624.4.a.r Level $624$ Weight $4$ Character orbit 624.a Self dual yes Analytic conductor $36.817$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = 2\sqrt{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (\beta + 12) q^{5} - \beta q^{7} + 9 q^{9} +O(q^{10})$$ q + 3 * q^3 + (b + 12) * q^5 - b * q^7 + 9 * q^9 $$q + 3 q^{3} + (\beta + 12) q^{5} - \beta q^{7} + 9 q^{9} + ( - 6 \beta + 22) q^{11} - 13 q^{13} + (3 \beta + 36) q^{15} + ( - 2 \beta + 82) q^{17} + (\beta - 24) q^{19} - 3 \beta q^{21} + (24 \beta - 4) q^{23} + (24 \beta + 75) q^{25} + 27 q^{27} + (12 \beta + 202) q^{29} + ( - 13 \beta - 20) q^{31} + ( - 18 \beta + 66) q^{33} + ( - 12 \beta - 56) q^{35} + ( - 14 \beta - 50) q^{37} - 39 q^{39} + ( - 47 \beta + 100) q^{41} + (26 \beta + 308) q^{43} + (9 \beta + 108) q^{45} + (16 \beta + 162) q^{47} - 287 q^{49} + ( - 6 \beta + 246) q^{51} + (60 \beta - 82) q^{53} + ( - 50 \beta - 72) q^{55} + (3 \beta - 72) q^{57} + (20 \beta - 70) q^{59} + ( - 68 \beta + 314) q^{61} - 9 \beta q^{63} + ( - 13 \beta - 156) q^{65} + ( - 85 \beta + 236) q^{67} + (72 \beta - 12) q^{69} + ( - 42 \beta - 214) q^{71} + ( - 38 \beta - 450) q^{73} + (72 \beta + 225) q^{75} + ( - 22 \beta + 336) q^{77} + ( - 44 \beta + 216) q^{79} + 81 q^{81} + (32 \beta + 694) q^{83} + (58 \beta + 872) q^{85} + (36 \beta + 606) q^{87} + (95 \beta + 480) q^{89} + 13 \beta q^{91} + ( - 39 \beta - 60) q^{93} + ( - 12 \beta - 232) q^{95} + (110 \beta - 266) q^{97} + ( - 54 \beta + 198) q^{99} +O(q^{100})$$ q + 3 * q^3 + (b + 12) * q^5 - b * q^7 + 9 * q^9 + (-6*b + 22) * q^11 - 13 * q^13 + (3*b + 36) * q^15 + (-2*b + 82) * q^17 + (b - 24) * q^19 - 3*b * q^21 + (24*b - 4) * q^23 + (24*b + 75) * q^25 + 27 * q^27 + (12*b + 202) * q^29 + (-13*b - 20) * q^31 + (-18*b + 66) * q^33 + (-12*b - 56) * q^35 + (-14*b - 50) * q^37 - 39 * q^39 + (-47*b + 100) * q^41 + (26*b + 308) * q^43 + (9*b + 108) * q^45 + (16*b + 162) * q^47 - 287 * q^49 + (-6*b + 246) * q^51 + (60*b - 82) * q^53 + (-50*b - 72) * q^55 + (3*b - 72) * q^57 + (20*b - 70) * q^59 + (-68*b + 314) * q^61 - 9*b * q^63 + (-13*b - 156) * q^65 + (-85*b + 236) * q^67 + (72*b - 12) * q^69 + (-42*b - 214) * q^71 + (-38*b - 450) * q^73 + (72*b + 225) * q^75 + (-22*b + 336) * q^77 + (-44*b + 216) * q^79 + 81 * q^81 + (32*b + 694) * q^83 + (58*b + 872) * q^85 + (36*b + 606) * q^87 + (95*b + 480) * q^89 + 13*b * q^91 + (-39*b - 60) * q^93 + (-12*b - 232) * q^95 + (110*b - 266) * q^97 + (-54*b + 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 24 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 $$2 q + 6 q^{3} + 24 q^{5} + 18 q^{9} + 44 q^{11} - 26 q^{13} + 72 q^{15} + 164 q^{17} - 48 q^{19} - 8 q^{23} + 150 q^{25} + 54 q^{27} + 404 q^{29} - 40 q^{31} + 132 q^{33} - 112 q^{35} - 100 q^{37} - 78 q^{39} + 200 q^{41} + 616 q^{43} + 216 q^{45} + 324 q^{47} - 574 q^{49} + 492 q^{51} - 164 q^{53} - 144 q^{55} - 144 q^{57} - 140 q^{59} + 628 q^{61} - 312 q^{65} + 472 q^{67} - 24 q^{69} - 428 q^{71} - 900 q^{73} + 450 q^{75} + 672 q^{77} + 432 q^{79} + 162 q^{81} + 1388 q^{83} + 1744 q^{85} + 1212 q^{87} + 960 q^{89} - 120 q^{93} - 464 q^{95} - 532 q^{97} + 396 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 24 * q^5 + 18 * q^9 + 44 * q^11 - 26 * q^13 + 72 * q^15 + 164 * q^17 - 48 * q^19 - 8 * q^23 + 150 * q^25 + 54 * q^27 + 404 * q^29 - 40 * q^31 + 132 * q^33 - 112 * q^35 - 100 * q^37 - 78 * q^39 + 200 * q^41 + 616 * q^43 + 216 * q^45 + 324 * q^47 - 574 * q^49 + 492 * q^51 - 164 * q^53 - 144 * q^55 - 144 * q^57 - 140 * q^59 + 628 * q^61 - 312 * q^65 + 472 * q^67 - 24 * q^69 - 428 * q^71 - 900 * q^73 + 450 * q^75 + 672 * q^77 + 432 * q^79 + 162 * q^81 + 1388 * q^83 + 1744 * q^85 + 1212 * q^87 + 960 * q^89 - 120 * q^93 - 464 * q^95 - 532 * q^97 + 396 * 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
 −3.74166 3.74166
0 3.00000 0 4.51669 0 7.48331 0 9.00000 0
1.2 0 3.00000 0 19.4833 0 −7.48331 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.r 2
3.b odd 2 1 1872.4.a.t 2
4.b odd 2 1 39.4.a.b 2
8.b even 2 1 2496.4.a.s 2
8.d odd 2 1 2496.4.a.bc 2
12.b even 2 1 117.4.a.c 2
20.d odd 2 1 975.4.a.j 2
28.d even 2 1 1911.4.a.h 2
52.b odd 2 1 507.4.a.f 2
52.f even 4 2 507.4.b.f 4
156.h even 2 1 1521.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 4.b odd 2 1
117.4.a.c 2 12.b even 2 1
507.4.a.f 2 52.b odd 2 1
507.4.b.f 4 52.f even 4 2
624.4.a.r 2 1.a even 1 1 trivial
975.4.a.j 2 20.d odd 2 1
1521.4.a.s 2 156.h even 2 1
1872.4.a.t 2 3.b odd 2 1
1911.4.a.h 2 28.d even 2 1
2496.4.a.s 2 8.b even 2 1
2496.4.a.bc 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(624))$$:

 $$T_{5}^{2} - 24T_{5} + 88$$ T5^2 - 24*T5 + 88 $$T_{7}^{2} - 56$$ T7^2 - 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 24T + 88$$
$7$ $$T^{2} - 56$$
$11$ $$T^{2} - 44T - 1532$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} - 164T + 6500$$
$19$ $$T^{2} + 48T + 520$$
$23$ $$T^{2} + 8T - 32240$$
$29$ $$T^{2} - 404T + 32740$$
$31$ $$T^{2} + 40T - 9064$$
$37$ $$T^{2} + 100T - 8476$$
$41$ $$T^{2} - 200T - 113704$$
$43$ $$T^{2} - 616T + 57008$$
$47$ $$T^{2} - 324T + 11908$$
$53$ $$T^{2} + 164T - 194876$$
$59$ $$T^{2} + 140T - 17500$$
$61$ $$T^{2} - 628T - 160348$$
$67$ $$T^{2} - 472T - 348904$$
$71$ $$T^{2} + 428T - 52988$$
$73$ $$T^{2} + 900T + 121636$$
$79$ $$T^{2} - 432T - 61760$$
$83$ $$T^{2} - 1388 T + 424292$$
$89$ $$T^{2} - 960T - 275000$$
$97$ $$T^{2} + 532T - 606844$$