# Properties

 Label 624.4.c.c Level $624$ Weight $4$ Character orbit 624.c Analytic conductor $36.817$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.1362828.1 Defining polynomial: $$x^{4} + 23x^{2} + 48$$ x^4 + 23*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + (\beta_{2} - \beta_1) q^{5} + (2 \beta_{2} - \beta_1) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (b2 - b1) * q^5 + (2*b2 - b1) * q^7 + 9 * q^9 $$q - 3 q^{3} + (\beta_{2} - \beta_1) q^{5} + (2 \beta_{2} - \beta_1) q^{7} + 9 q^{9} + ( - 3 \beta_{2} - 4 \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{13} + ( - 3 \beta_{2} + 3 \beta_1) q^{15} + ( - 3 \beta_{3} + 24) q^{17} + ( - 8 \beta_{2} - 5 \beta_1) q^{19} + ( - 6 \beta_{2} + 3 \beta_1) q^{21} + 72 q^{23} + (\beta_{3} - 5) q^{25} - 27 q^{27} + (3 \beta_{3} - 96) q^{29} + (20 \beta_{2} - 19 \beta_1) q^{31} + (9 \beta_{2} + 12 \beta_1) q^{33} - 216 q^{35} + (4 \beta_{2} + 34 \beta_1) q^{37} + (3 \beta_{3} + 3 \beta_{2} + 12 \beta_1 + 9) q^{39} + ( - 13 \beta_{2} - 33 \beta_1) q^{41} + ( - 3 \beta_{3} + 322) q^{43} + (9 \beta_{2} - 9 \beta_1) q^{45} + (\beta_{2} + 18 \beta_1) q^{47} + ( - 3 \beta_{3} - 47) q^{49} + (9 \beta_{3} - 72) q^{51} + (12 \beta_{3} + 246) q^{53} + (11 \beta_{3} + 82) q^{55} + (24 \beta_{2} + 15 \beta_1) q^{57} + ( - \beta_{2} - 26 \beta_1) q^{59} + (7 \beta_{3} + 72) q^{61} + (18 \beta_{2} - 9 \beta_1) q^{63} + (9 \beta_{3} - 17 \beta_{2} - 55 \beta_1 - 90) q^{65} + ( - 54 \beta_{2} - 27 \beta_1) q^{67} - 216 q^{69} + ( - 41 \beta_{2} + 84 \beta_1) q^{71} + ( - 42 \beta_{2} - 78 \beta_1) q^{73} + ( - 3 \beta_{3} + 15) q^{75} + (21 \beta_{3} + 354) q^{77} + ( - 4 \beta_{3} - 1080) q^{79} + 81 q^{81} + (63 \beta_{2} - 94 \beta_1) q^{83} + ( - 18 \beta_{2} - 198 \beta_1) q^{85} + ( - 9 \beta_{3} + 288) q^{87} + ( - 25 \beta_{2} - 69 \beta_1) q^{89} + (15 \beta_{3} - 50 \beta_{2} - 83 \beta_1 + 6) q^{91} + ( - 60 \beta_{2} + 57 \beta_1) q^{93} + (18 \beta_{3} + 468) q^{95} + ( - 14 \beta_{2} + 34 \beta_1) q^{97} + ( - 27 \beta_{2} - 36 \beta_1) q^{99}+O(q^{100})$$ q - 3 * q^3 + (b2 - b1) * q^5 + (2*b2 - b1) * q^7 + 9 * q^9 + (-3*b2 - 4*b1) * q^11 + (-b3 - b2 - 4*b1 - 3) * q^13 + (-3*b2 + 3*b1) * q^15 + (-3*b3 + 24) * q^17 + (-8*b2 - 5*b1) * q^19 + (-6*b2 + 3*b1) * q^21 + 72 * q^23 + (b3 - 5) * q^25 - 27 * q^27 + (3*b3 - 96) * q^29 + (20*b2 - 19*b1) * q^31 + (9*b2 + 12*b1) * q^33 - 216 * q^35 + (4*b2 + 34*b1) * q^37 + (3*b3 + 3*b2 + 12*b1 + 9) * q^39 + (-13*b2 - 33*b1) * q^41 + (-3*b3 + 322) * q^43 + (9*b2 - 9*b1) * q^45 + (b2 + 18*b1) * q^47 + (-3*b3 - 47) * q^49 + (9*b3 - 72) * q^51 + (12*b3 + 246) * q^53 + (11*b3 + 82) * q^55 + (24*b2 + 15*b1) * q^57 + (-b2 - 26*b1) * q^59 + (7*b3 + 72) * q^61 + (18*b2 - 9*b1) * q^63 + (9*b3 - 17*b2 - 55*b1 - 90) * q^65 + (-54*b2 - 27*b1) * q^67 - 216 * q^69 + (-41*b2 + 84*b1) * q^71 + (-42*b2 - 78*b1) * q^73 + (-3*b3 + 15) * q^75 + (21*b3 + 354) * q^77 + (-4*b3 - 1080) * q^79 + 81 * q^81 + (63*b2 - 94*b1) * q^83 + (-18*b2 - 198*b1) * q^85 + (-9*b3 + 288) * q^87 + (-25*b2 - 69*b1) * q^89 + (15*b3 - 50*b2 - 83*b1 + 6) * q^91 + (-60*b2 + 57*b1) * q^93 + (18*b3 + 468) * q^95 + (-14*b2 + 34*b1) * q^97 + (-27*b2 - 36*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 + 36 * q^9 $$4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 + 36 * q^9 - 12 * q^13 + 96 * q^17 + 288 * q^23 - 20 * q^25 - 108 * q^27 - 384 * q^29 - 864 * q^35 + 36 * q^39 + 1288 * q^43 - 188 * q^49 - 288 * q^51 + 984 * q^53 + 328 * q^55 + 288 * q^61 - 360 * q^65 - 864 * q^69 + 60 * q^75 + 1416 * q^77 - 4320 * q^79 + 324 * q^81 + 1152 * q^87 + 24 * q^91 + 1872 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 23x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 19\nu ) / 2$$ (v^3 + 19*v) / 2 $$\beta_{3}$$ $$=$$ $$4\nu^{2} + 46$$ 4*v^2 + 46
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 46 ) / 4$$ (b3 - 46) / 4 $$\nu^{3}$$ $$=$$ $$( 4\beta_{2} - 19\beta_1 ) / 2$$ (4*b2 - 19*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 4.54739i − 1.52356i 1.52356i − 4.54739i
0 −3.00000 0 12.9118i 0 16.7289i 0 9.00000 0
337.2 0 −3.00000 0 9.65841i 0 22.3639i 0 9.00000 0
337.3 0 −3.00000 0 9.65841i 0 22.3639i 0 9.00000 0
337.4 0 −3.00000 0 12.9118i 0 16.7289i 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.c.c 4
4.b odd 2 1 39.4.b.b 4
12.b even 2 1 117.4.b.e 4
13.b even 2 1 inner 624.4.c.c 4
52.b odd 2 1 39.4.b.b 4
52.f even 4 2 507.4.a.l 4
156.h even 2 1 117.4.b.e 4
156.l odd 4 2 1521.4.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b 4 4.b odd 2 1
39.4.b.b 4 52.b odd 2 1
117.4.b.e 4 12.b even 2 1
117.4.b.e 4 156.h even 2 1
507.4.a.l 4 52.f even 4 2
624.4.c.c 4 1.a even 1 1 trivial
624.4.c.c 4 13.b even 2 1 inner
1521.4.a.w 4 156.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 260T_{5}^{2} + 15552$$ acting on $$S_{4}^{\mathrm{new}}(624, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T + 3)^{4}$$
$5$ $$T^{4} + 260 T^{2} + 15552$$
$7$ $$T^{4} + 780 T^{2} + 139968$$
$11$ $$T^{4} + 3152 T^{2} + \cdots + 1572528$$
$13$ $$T^{4} + 12 T^{3} - 962 T^{2} + \cdots + 4826809$$
$17$ $$(T^{2} - 48 T - 11556)^{2}$$
$19$ $$T^{4} + 13884 T^{2} + \cdots + 3048192$$
$23$ $$(T - 72)^{4}$$
$29$ $$(T^{2} + 192 T - 2916)^{2}$$
$31$ $$T^{4} + 100572 T^{2} + \cdots + 2389782528$$
$37$ $$T^{4} + 110256 T^{2} + \cdots + 2060577792$$
$41$ $$T^{4} + 133364 T^{2} + \cdots + 4431055872$$
$43$ $$(T^{2} - 644 T + 91552)^{2}$$
$47$ $$T^{4} + 30128 T^{2} + \cdots + 116663088$$
$53$ $$(T^{2} - 492 T - 133596)^{2}$$
$59$ $$T^{4} + 62576 T^{2} + \cdots + 457419312$$
$61$ $$(T^{2} - 144 T - 60868)^{2}$$
$67$ $$T^{4} + 591948 T^{2} + \cdots + 918330048$$
$71$ $$T^{4} + 917456 T^{2} + \cdots + 59484058032$$
$73$ $$T^{4} + 896400 T^{2} + \cdots + 179358354432$$
$79$ $$(T^{2} + 2160 T + 1144832)^{2}$$
$83$ $$T^{4} + 1464080 T^{2} + \cdots + 317023217328$$
$89$ $$T^{4} + 561812 T^{2} + \cdots + 78903164928$$
$97$ $$T^{4} + 137040 T^{2} + \cdots + 725594112$$