# Properties

 Label 624.2.cn.c Level $624$ Weight $2$ Character orbit 624.cn Analytic conductor $4.983$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{3} + ( -\beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( \beta_{5} + \beta_{6} ) q^{7} + ( 1 + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} + ( 1 - 2 \beta_{1} - \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{11} + ( 2 - 3 \beta_{4} - 2 \beta_{6} ) q^{13} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{15} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{17} + ( 2 - 2 \beta_{4} ) q^{19} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{21} + ( 1 + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{25} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{4} - 4 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{27} + ( -2 + 5 \beta_{1} + 5 \beta_{2} - \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{29} + ( 1 - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{31} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{33} + ( 2 \beta_{2} - \beta_{5} - \beta_{6} ) q^{35} + ( -2 - 3 \beta_{4} + 5 \beta_{5} - 3 \beta_{6} ) q^{37} + ( 3 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{39} + ( -1 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} + ( -6 - 3 \beta_{5} + 3 \beta_{6} ) q^{43} + ( -5 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{45} + ( -4 + 4 \beta_{1} + 4 \beta_{3} ) q^{47} -5 \beta_{4} q^{49} + ( -2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{51} + ( \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( -1 - 3 \beta_{4} + 6 \beta_{5} + \beta_{6} ) q^{55} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{57} + ( -2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{59} + 7 \beta_{6} q^{61} + ( -1 + 2 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{63} + ( 3 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( 4 - 2 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{67} + ( -2 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{71} + ( -5 + 2 \beta_{4} - 5 \beta_{5} + 3 \beta_{6} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{77} -2 q^{79} + ( 3 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{81} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -1 - \beta_{4} + 5 \beta_{5} + 5 \beta_{6} ) q^{85} + ( 3 + 4 \beta_{2} - 6 \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -1 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{89} + ( -1 - 5 \beta_{4} + 5 \beta_{5} ) q^{91} + ( 4 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 5 \beta_{6} + 3 \beta_{7} ) q^{93} + ( 2 - 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{95} + ( 6 - 6 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{97} + ( -5 + 2 \beta_{1} + 2 \beta_{3} + 7 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} + 4q^{7} + 4q^{9} + O(q^{10})$$ $$8q + 2q^{3} + 4q^{7} + 4q^{9} + 8q^{13} + 14q^{15} + 16q^{19} + 4q^{21} - 4q^{27} - 8q^{31} + 16q^{33} - 28q^{37} + 14q^{39} - 36q^{43} - 20q^{45} - 4q^{55} + 16q^{57} + 28q^{61} + 8q^{63} + 40q^{67} - 28q^{73} - 12q^{75} - 16q^{79} + 4q^{81} + 12q^{85} + 34q^{87} - 8q^{91} + 4q^{93} + 20q^{97} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 15 \nu^{6} + 32 \nu^{5} - 172 \nu^{4} + 221 \nu^{3} - 426 \nu^{2} + 235 \nu - 159$$$$)/37$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 390 \nu^{2} + 298 \nu - 70$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 427 \nu^{2} + 335 \nu - 181$$$$)/37$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - 29 \nu^{6} + 89 \nu^{5} - 261 \nu^{4} + 373 \nu^{3} - 498 \nu^{2} + 294 \nu - 152$$$$)/37$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 28 \nu^{6} - 114 \nu^{5} + 215 \nu^{4} - 378 \nu^{3} + 366 \nu^{2} - 266 \nu + 97$$$$)/37$$ $$\beta_{7}$$ $$=$$ $$($$$$17 \nu^{7} - 41 \nu^{6} + 159 \nu^{5} - 184 \nu^{4} + 276 \nu^{3} - 84 \nu^{2} + 38 \nu + 39$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{1} - 4$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} + 3 \beta_{6} + 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{7} - 3 \beta_{6} + 7 \beta_{5} + 6 \beta_{4} - 12 \beta_{3} - 5 \beta_{2} + \beta_{1} + 26$$ $$\nu^{6}$$ $$=$$ $$-17 \beta_{7} - 25 \beta_{6} + 3 \beta_{5} - 24 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} + 27 \beta_{1} - 1$$ $$\nu^{7}$$ $$=$$ $$4 \beta_{7} - 16 \beta_{6} - 42 \beta_{5} - 54 \beta_{4} + 51 \beta_{3} + 42 \beta_{2} + 26 \beta_{1} - 122$$

## Character values

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

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$ $$-1$$ $$1$$

## 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
 0.5 − 0.564882i 0.5 + 1.56488i 0.5 + 2.19293i 0.5 − 1.19293i 0.5 + 0.564882i 0.5 − 1.56488i 0.5 − 2.19293i 0.5 + 1.19293i
0 −0.239203 1.71545i 0 1.06488 + 1.06488i 0 −0.366025 + 1.36603i 0 −2.88556 + 0.820682i 0
305.2 0 1.60523 0.650571i 0 −1.06488 1.06488i 0 −0.366025 + 1.36603i 0 2.15351 2.08863i 0
353.1 0 −1.64914 + 0.529480i 0 −1.69293 1.69293i 0 1.36603 0.366025i 0 2.43930 1.74637i 0
353.2 0 1.28311 1.16345i 0 1.69293 + 1.69293i 0 1.36603 0.366025i 0 0.292748 2.98568i 0
401.1 0 −0.239203 + 1.71545i 0 1.06488 1.06488i 0 −0.366025 1.36603i 0 −2.88556 0.820682i 0
401.2 0 1.60523 + 0.650571i 0 −1.06488 + 1.06488i 0 −0.366025 1.36603i 0 2.15351 + 2.08863i 0
449.1 0 −1.64914 0.529480i 0 −1.69293 + 1.69293i 0 1.36603 + 0.366025i 0 2.43930 + 1.74637i 0
449.2 0 1.28311 + 1.16345i 0 1.69293 1.69293i 0 1.36603 + 0.366025i 0 0.292748 + 2.98568i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.2 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 inner
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.c 8
3.b odd 2 1 inner 624.2.cn.c 8
4.b odd 2 1 39.2.k.b 8
12.b even 2 1 39.2.k.b 8
13.f odd 12 1 inner 624.2.cn.c 8
20.d odd 2 1 975.2.bo.d 8
20.e even 4 1 975.2.bp.e 8
20.e even 4 1 975.2.bp.f 8
39.k even 12 1 inner 624.2.cn.c 8
52.b odd 2 1 507.2.k.d 8
52.f even 4 1 507.2.k.e 8
52.f even 4 1 507.2.k.f 8
52.i odd 6 1 507.2.f.e 8
52.i odd 6 1 507.2.k.f 8
52.j odd 6 1 507.2.f.f 8
52.j odd 6 1 507.2.k.e 8
52.l even 12 1 39.2.k.b 8
52.l even 12 1 507.2.f.e 8
52.l even 12 1 507.2.f.f 8
52.l even 12 1 507.2.k.d 8
60.h even 2 1 975.2.bo.d 8
60.l odd 4 1 975.2.bp.e 8
60.l odd 4 1 975.2.bp.f 8
156.h even 2 1 507.2.k.d 8
156.l odd 4 1 507.2.k.e 8
156.l odd 4 1 507.2.k.f 8
156.p even 6 1 507.2.f.f 8
156.p even 6 1 507.2.k.e 8
156.r even 6 1 507.2.f.e 8
156.r even 6 1 507.2.k.f 8
156.v odd 12 1 39.2.k.b 8
156.v odd 12 1 507.2.f.e 8
156.v odd 12 1 507.2.f.f 8
156.v odd 12 1 507.2.k.d 8
260.bc even 12 1 975.2.bo.d 8
260.be odd 12 1 975.2.bp.f 8
260.bl odd 12 1 975.2.bp.e 8
780.cf even 12 1 975.2.bp.f 8
780.cr odd 12 1 975.2.bo.d 8
780.cy even 12 1 975.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 4.b odd 2 1
39.2.k.b 8 12.b even 2 1
39.2.k.b 8 52.l even 12 1
39.2.k.b 8 156.v odd 12 1
507.2.f.e 8 52.i odd 6 1
507.2.f.e 8 52.l even 12 1
507.2.f.e 8 156.r even 6 1
507.2.f.e 8 156.v odd 12 1
507.2.f.f 8 52.j odd 6 1
507.2.f.f 8 52.l even 12 1
507.2.f.f 8 156.p even 6 1
507.2.f.f 8 156.v odd 12 1
507.2.k.d 8 52.b odd 2 1
507.2.k.d 8 52.l even 12 1
507.2.k.d 8 156.h even 2 1
507.2.k.d 8 156.v odd 12 1
507.2.k.e 8 52.f even 4 1
507.2.k.e 8 52.j odd 6 1
507.2.k.e 8 156.l odd 4 1
507.2.k.e 8 156.p even 6 1
507.2.k.f 8 52.f even 4 1
507.2.k.f 8 52.i odd 6 1
507.2.k.f 8 156.l odd 4 1
507.2.k.f 8 156.r even 6 1
624.2.cn.c 8 1.a even 1 1 trivial
624.2.cn.c 8 3.b odd 2 1 inner
624.2.cn.c 8 13.f odd 12 1 inner
624.2.cn.c 8 39.k even 12 1 inner
975.2.bo.d 8 20.d odd 2 1
975.2.bo.d 8 60.h even 2 1
975.2.bo.d 8 260.bc even 12 1
975.2.bo.d 8 780.cr odd 12 1
975.2.bp.e 8 20.e even 4 1
975.2.bp.e 8 60.l odd 4 1
975.2.bp.e 8 260.bl odd 12 1
975.2.bp.e 8 780.cy even 12 1
975.2.bp.f 8 20.e even 4 1
975.2.bp.f 8 60.l odd 4 1
975.2.bp.f 8 260.be odd 12 1
975.2.bp.f 8 780.cf even 12 1

## Hecke kernels

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

 $$T_{5}^{8} + 38 T_{5}^{4} + 169$$ $$T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} - 4 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$81 - 54 T + 12 T^{3} - 5 T^{4} + 4 T^{5} - 2 T^{7} + T^{8}$$
$5$ $$169 + 38 T^{4} + T^{8}$$
$7$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$2704 - 1248 T^{2} + 140 T^{4} + 24 T^{6} + T^{8}$$
$13$ $$( 169 - 52 T + 3 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$17$ $$13689 + 3510 T^{2} + 783 T^{4} + 30 T^{6} + T^{8}$$
$19$ $$( 16 - 16 T + 20 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$23$ $$T^{8}$$
$29$ $$2474329 - 128986 T^{2} + 5151 T^{4} - 82 T^{6} + T^{8}$$
$31$ $$( 484 - 88 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$( 1369 + 592 T + 113 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$41$ $$169 + 702 T^{2} + 959 T^{4} - 54 T^{6} + T^{8}$$
$43$ $$( 324 + 324 T + 126 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$47$ $$11075584 + 9728 T^{4} + T^{8}$$
$53$ $$( 13 + 22 T^{2} + T^{4} )^{2}$$
$59$ $$43264 - 4992 T^{2} - 16 T^{4} + 24 T^{6} + T^{8}$$
$61$ $$( 49 - 7 T + T^{2} )^{4}$$
$67$ $$( 2704 - 832 T + 164 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$71$ $$43264 + 4992 T^{2} - 16 T^{4} - 24 T^{6} + T^{8}$$
$73$ $$( 121 + 154 T + 98 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$79$ $$( 2 + T )^{8}$$
$83$ $$2704 + 296 T^{4} + T^{8}$$
$89$ $$77228944 + 210912 T^{2} - 8596 T^{4} - 24 T^{6} + T^{8}$$
$97$ $$( 484 - 572 T + 194 T^{2} - 10 T^{3} + T^{4} )^{2}$$
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