# Properties

 Label 624.2.c.a Level $624$ Weight $2$ Character orbit 624.c Analytic conductor $4.983$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + b * q^7 + q^9 $$q - q^{3} + \beta q^{5} + \beta q^{7} + q^{9} + ( - \beta - 3) q^{13} - \beta q^{15} - 2 q^{17} + 3 \beta q^{19} - \beta q^{21} - 4 q^{23} + q^{25} - q^{27} - 10 q^{29} - 5 \beta q^{31} - 4 q^{35} + 4 \beta q^{37} + (\beta + 3) q^{39} + 5 \beta q^{41} - 4 q^{43} + \beta q^{45} + 6 \beta q^{47} + 3 q^{49} + 2 q^{51} - 6 q^{53} - 3 \beta q^{57} - 2 \beta q^{59} + 2 q^{61} + \beta q^{63} + ( - 3 \beta + 4) q^{65} + \beta q^{67} + 4 q^{69} - 2 \beta q^{73} - q^{75} + q^{81} + 2 \beta q^{83} - 2 \beta q^{85} + 10 q^{87} - 3 \beta q^{89} + ( - 3 \beta + 4) q^{91} + 5 \beta q^{93} - 12 q^{95} - 6 \beta q^{97} +O(q^{100})$$ q - q^3 + b * q^5 + b * q^7 + q^9 + (-b - 3) * q^13 - b * q^15 - 2 * q^17 + 3*b * q^19 - b * q^21 - 4 * q^23 + q^25 - q^27 - 10 * q^29 - 5*b * q^31 - 4 * q^35 + 4*b * q^37 + (b + 3) * q^39 + 5*b * q^41 - 4 * q^43 + b * q^45 + 6*b * q^47 + 3 * q^49 + 2 * q^51 - 6 * q^53 - 3*b * q^57 - 2*b * q^59 + 2 * q^61 + b * q^63 + (-3*b + 4) * q^65 + b * q^67 + 4 * q^69 - 2*b * q^73 - q^75 + q^81 + 2*b * q^83 - 2*b * q^85 + 10 * q^87 - 3*b * q^89 + (-3*b + 4) * q^91 + 5*b * q^93 - 12 * q^95 - 6*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 20 q^{29} - 8 q^{35} + 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} - 12 q^{53} + 4 q^{61} + 8 q^{65} + 8 q^{69} - 2 q^{75} + 2 q^{81} + 20 q^{87} + 8 q^{91} - 24 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 6 * q^13 - 4 * q^17 - 8 * q^23 + 2 * q^25 - 2 * q^27 - 20 * q^29 - 8 * q^35 + 6 * q^39 - 8 * q^43 + 6 * q^49 + 4 * q^51 - 12 * q^53 + 4 * q^61 + 8 * q^65 + 8 * q^69 - 2 * q^75 + 2 * q^81 + 20 * q^87 + 8 * q^91 - 24 * q^95

## 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
 − 1.00000i 1.00000i
0 −1.00000 0 2.00000i 0 2.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 0 2.00000i 0 1.00000 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 624.2.c.a 2
3.b odd 2 1 1872.2.c.b 2
4.b odd 2 1 78.2.b.a 2
8.b even 2 1 2496.2.c.m 2
8.d odd 2 1 2496.2.c.f 2
12.b even 2 1 234.2.b.a 2
13.b even 2 1 inner 624.2.c.a 2
13.d odd 4 1 8112.2.a.g 1
13.d odd 4 1 8112.2.a.j 1
20.d odd 2 1 1950.2.b.c 2
20.e even 4 1 1950.2.f.d 2
20.e even 4 1 1950.2.f.g 2
28.d even 2 1 3822.2.c.d 2
39.d odd 2 1 1872.2.c.b 2
52.b odd 2 1 78.2.b.a 2
52.f even 4 1 1014.2.a.b 1
52.f even 4 1 1014.2.a.g 1
52.i odd 6 2 1014.2.i.c 4
52.j odd 6 2 1014.2.i.c 4
52.l even 12 2 1014.2.e.b 2
52.l even 12 2 1014.2.e.e 2
104.e even 2 1 2496.2.c.m 2
104.h odd 2 1 2496.2.c.f 2
156.h even 2 1 234.2.b.a 2
156.l odd 4 1 3042.2.a.c 1
156.l odd 4 1 3042.2.a.n 1
260.g odd 2 1 1950.2.b.c 2
260.p even 4 1 1950.2.f.d 2
260.p even 4 1 1950.2.f.g 2
364.h even 2 1 3822.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 4.b odd 2 1
78.2.b.a 2 52.b odd 2 1
234.2.b.a 2 12.b even 2 1
234.2.b.a 2 156.h even 2 1
624.2.c.a 2 1.a even 1 1 trivial
624.2.c.a 2 13.b even 2 1 inner
1014.2.a.b 1 52.f even 4 1
1014.2.a.g 1 52.f even 4 1
1014.2.e.b 2 52.l even 12 2
1014.2.e.e 2 52.l even 12 2
1014.2.i.c 4 52.i odd 6 2
1014.2.i.c 4 52.j odd 6 2
1872.2.c.b 2 3.b odd 2 1
1872.2.c.b 2 39.d odd 2 1
1950.2.b.c 2 20.d odd 2 1
1950.2.b.c 2 260.g odd 2 1
1950.2.f.d 2 20.e even 4 1
1950.2.f.d 2 260.p even 4 1
1950.2.f.g 2 20.e even 4 1
1950.2.f.g 2 260.p even 4 1
2496.2.c.f 2 8.d odd 2 1
2496.2.c.f 2 104.h odd 2 1
2496.2.c.m 2 8.b even 2 1
2496.2.c.m 2 104.e even 2 1
3042.2.a.c 1 156.l odd 4 1
3042.2.a.n 1 156.l odd 4 1
3822.2.c.d 2 28.d even 2 1
3822.2.c.d 2 364.h even 2 1
8112.2.a.g 1 13.d odd 4 1
8112.2.a.j 1 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$