# Properties

 Label 624.2.cn.b Level $624$ Weight $2$ Character orbit 624.cn Analytic conductor $4.983$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 624.cn (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 + 2 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( 2 + 2 \zeta_{12} + \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 3 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{13} + ( -5 + 5 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( 4 + \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{21} -5 \zeta_{12}^{3} q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( 1 - 5 \zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{31} + ( -4 - 3 \zeta_{12} + 7 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{37} + ( 5 + 2 \zeta_{12}^{2} ) q^{39} + ( 1 + \zeta_{12}^{2} ) q^{43} + ( 6 + 7 \zeta_{12} - 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( -1 - 8 \zeta_{12} + 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{57} + ( -5 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{61} + ( -9 + 3 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{63} + ( -7 - 9 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( -9 + \zeta_{12} + \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{73} + ( 10 - 5 \zeta_{12}^{2} ) q^{75} + ( 14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 5 + 10 \zeta_{12} - 11 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{91} + ( 7 - 4 \zeta_{12} - 11 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{93} + ( -8 + 8 \zeta_{12} - 3 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 10q^{7} - 6q^{9} - 14q^{19} + 18q^{21} + 14q^{31} - 2q^{37} + 24q^{39} + 6q^{43} + 18q^{49} + 12q^{57} - 24q^{63} - 32q^{67} - 34q^{73} + 30q^{75} - 18q^{81} - 2q^{91} + 6q^{93} - 38q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
0 −0.866025 + 1.50000i 0 0 0 0.767949 2.86603i 0 −1.50000 2.59808i 0
353.1 0 0.866025 + 1.50000i 0 0 0 4.23205 1.13397i 0 −1.50000 + 2.59808i 0
401.1 0 −0.866025 1.50000i 0 0 0 0.767949 + 2.86603i 0 −1.50000 + 2.59808i 0
449.1 0 0.866025 1.50000i 0 0 0 4.23205 + 1.13397i 0 −1.50000 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cn.b 4
3.b odd 2 1 CM 624.2.cn.b 4
4.b odd 2 1 39.2.k.a 4
12.b even 2 1 39.2.k.a 4
13.f odd 12 1 inner 624.2.cn.b 4
20.d odd 2 1 975.2.bo.c 4
20.e even 4 1 975.2.bp.a 4
20.e even 4 1 975.2.bp.d 4
39.k even 12 1 inner 624.2.cn.b 4
52.b odd 2 1 507.2.k.c 4
52.f even 4 1 507.2.k.a 4
52.f even 4 1 507.2.k.b 4
52.i odd 6 1 507.2.f.b 4
52.i odd 6 1 507.2.k.a 4
52.j odd 6 1 507.2.f.c 4
52.j odd 6 1 507.2.k.b 4
52.l even 12 1 39.2.k.a 4
52.l even 12 1 507.2.f.b 4
52.l even 12 1 507.2.f.c 4
52.l even 12 1 507.2.k.c 4
60.h even 2 1 975.2.bo.c 4
60.l odd 4 1 975.2.bp.a 4
60.l odd 4 1 975.2.bp.d 4
156.h even 2 1 507.2.k.c 4
156.l odd 4 1 507.2.k.a 4
156.l odd 4 1 507.2.k.b 4
156.p even 6 1 507.2.f.c 4
156.p even 6 1 507.2.k.b 4
156.r even 6 1 507.2.f.b 4
156.r even 6 1 507.2.k.a 4
156.v odd 12 1 39.2.k.a 4
156.v odd 12 1 507.2.f.b 4
156.v odd 12 1 507.2.f.c 4
156.v odd 12 1 507.2.k.c 4
260.bc even 12 1 975.2.bo.c 4
260.be odd 12 1 975.2.bp.a 4
260.bl odd 12 1 975.2.bp.d 4
780.cf even 12 1 975.2.bp.a 4
780.cr odd 12 1 975.2.bo.c 4
780.cy even 12 1 975.2.bp.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 4.b odd 2 1
39.2.k.a 4 12.b even 2 1
39.2.k.a 4 52.l even 12 1
39.2.k.a 4 156.v odd 12 1
507.2.f.b 4 52.i odd 6 1
507.2.f.b 4 52.l even 12 1
507.2.f.b 4 156.r even 6 1
507.2.f.b 4 156.v odd 12 1
507.2.f.c 4 52.j odd 6 1
507.2.f.c 4 52.l even 12 1
507.2.f.c 4 156.p even 6 1
507.2.f.c 4 156.v odd 12 1
507.2.k.a 4 52.f even 4 1
507.2.k.a 4 52.i odd 6 1
507.2.k.a 4 156.l odd 4 1
507.2.k.a 4 156.r even 6 1
507.2.k.b 4 52.f even 4 1
507.2.k.b 4 52.j odd 6 1
507.2.k.b 4 156.l odd 4 1
507.2.k.b 4 156.p even 6 1
507.2.k.c 4 52.b odd 2 1
507.2.k.c 4 52.l even 12 1
507.2.k.c 4 156.h even 2 1
507.2.k.c 4 156.v odd 12 1
624.2.cn.b 4 1.a even 1 1 trivial
624.2.cn.b 4 3.b odd 2 1 CM
624.2.cn.b 4 13.f odd 12 1 inner
624.2.cn.b 4 39.k even 12 1 inner
975.2.bo.c 4 20.d odd 2 1
975.2.bo.c 4 60.h even 2 1
975.2.bo.c 4 260.bc even 12 1
975.2.bo.c 4 780.cr odd 12 1
975.2.bp.a 4 20.e even 4 1
975.2.bp.a 4 60.l odd 4 1
975.2.bp.a 4 260.be odd 12 1
975.2.bp.a 4 780.cf even 12 1
975.2.bp.d 4 20.e even 4 1
975.2.bp.d 4 60.l odd 4 1
975.2.bp.d 4 260.bl odd 12 1
975.2.bp.d 4 780.cy even 12 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{4} - 10 T_{7}^{3} + 41 T_{7}^{2} - 104 T_{7} + 169$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$169 - 104 T + 41 T^{2} - 10 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$169 - T^{2} + T^{4}$$
$17$ $$T^{4}$$
$19$ $$676 + 52 T + 50 T^{2} + 14 T^{3} + T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$169 + 182 T + 98 T^{2} - 14 T^{3} + T^{4}$$
$37$ $$676 - 572 T + 122 T^{2} + 2 T^{3} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 3 - 3 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$5625 + 75 T^{2} + T^{4}$$
$67$ $$169 + 130 T + 281 T^{2} + 32 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$20449 + 4862 T + 578 T^{2} + 34 T^{3} + T^{4}$$
$79$ $$( -147 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$28561 + 4732 T + 557 T^{2} + 38 T^{3} + T^{4}$$