# Properties

 Label 507.2.k.g Level $507$ Weight $2$ Character orbit 507.k Analytic conductor $4.048$ Analytic rank $0$ Dimension $8$ CM discriminant -39 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.k (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} + ( -\beta_{2} - 2 \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{5} + ( -2 \beta_{1} + \beta_{7} ) q^{6} + ( \beta_{5} + \beta_{6} ) q^{8} + 3 \beta_{4} q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{2} + ( -\beta_{2} - 2 \beta_{3} ) q^{3} + ( -2 + 2 \beta_{3} - \beta_{4} ) q^{4} + ( -\beta_{1} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{5} + ( -2 \beta_{1} + \beta_{7} ) q^{6} + ( \beta_{5} + \beta_{6} ) q^{8} + 3 \beta_{4} q^{9} + ( 3 - \beta_{2} - 3 \beta_{4} ) q^{10} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( -2 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} ) q^{12} + ( -\beta_{1} + 3 \beta_{6} - \beta_{7} ) q^{15} + ( -1 - 2 \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{16} + ( -3 \beta_{5} + 3 \beta_{6} ) q^{18} + ( \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( 10 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} ) q^{22} + ( 3 \beta_{1} - 3 \beta_{7} ) q^{24} + ( -4 - 5 \beta_{2} - 5 \beta_{3} - 8 \beta_{4} ) q^{25} + ( -3 \beta_{2} + 3 \beta_{3} ) q^{27} + ( -2 - 9 \beta_{3} - \beta_{4} ) q^{30} + \beta_{5} q^{32} + ( -\beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{33} + ( 3 + 6 \beta_{2} - 3 \beta_{4} ) q^{36} + ( -9 + \beta_{2} - \beta_{3} ) q^{40} + ( \beta_{1} - 5 \beta_{6} + \beta_{7} ) q^{41} + 4 \beta_{3} q^{43} + ( 3 \beta_{1} + 5 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} ) q^{44} + ( 6 \beta_{1} - 3 \beta_{5} - 3 \beta_{7} ) q^{45} + ( \beta_{1} - 3 \beta_{5} - 3 \beta_{6} ) q^{47} + ( 2 \beta_{2} + \beta_{3} + 6 \beta_{4} ) q^{48} -7 \beta_{2} q^{49} + ( -5 \beta_{1} + 8 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} ) q^{50} + ( 3 \beta_{1} + 3 \beta_{7} ) q^{54} + ( 8 - 2 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} ) q^{55} + ( -3 \beta_{5} + 6 \beta_{6} - 5 \beta_{7} ) q^{59} + ( -3 \beta_{1} - \beta_{5} - \beta_{6} ) q^{60} + ( -16 \beta_{2} - 8 \beta_{3} ) q^{61} + ( 5 - 2 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} ) q^{64} + ( 15 + 7 \beta_{2} - 7 \beta_{3} ) q^{66} + ( 2 \beta_{1} + 5 \beta_{5} - \beta_{7} ) q^{71} + ( 3 \beta_{5} - 6 \beta_{6} ) q^{72} + ( -5 + 12 \beta_{2} + 5 \beta_{4} ) q^{75} + ( -6 \beta_{2} + 6 \beta_{3} ) q^{79} + ( -3 \beta_{1} + 7 \beta_{6} - 3 \beta_{7} ) q^{80} + ( -9 - 9 \beta_{4} ) q^{81} + ( -2 + 17 \beta_{3} - \beta_{4} ) q^{82} + ( \beta_{1} - 7 \beta_{5} + 7 \beta_{6} - 2 \beta_{7} ) q^{83} + 4 \beta_{1} q^{86} + ( -7 - 15 \beta_{2} + 7 \beta_{4} ) q^{88} + ( 7 \beta_{1} - 6 \beta_{5} + 3 \beta_{6} - 7 \beta_{7} ) q^{89} + ( 9 + 3 \beta_{2} + 3 \beta_{3} + 18 \beta_{4} ) q^{90} + ( -5 + 11 \beta_{2} + 22 \beta_{3} - 5 \beta_{4} ) q^{94} + ( \beta_{1} - 2 \beta_{7} ) q^{96} + 7 \beta_{7} q^{98} + ( 3 \beta_{1} + 3 \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{4} - 12q^{9} + O(q^{10})$$ $$8q - 12q^{4} - 12q^{9} + 36q^{10} - 4q^{16} + 28q^{22} - 12q^{30} + 36q^{36} - 72q^{40} - 24q^{48} + 32q^{55} + 120q^{66} - 60q^{75} - 36q^{81} - 12q^{82} - 84q^{88} - 20q^{94} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 390 \nu^{2} + 335 \nu - 107$$$$)/37$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{7} - 8 \nu^{6} + 22 \nu^{5} - 146 \nu^{4} + 256 \nu^{3} - 427 \nu^{2} + 335 \nu - 181$$$$)/37$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{7} + 29 \nu^{6} - 89 \nu^{5} + 261 \nu^{4} - 373 \nu^{3} + 498 \nu^{2} - 294 \nu + 152$$$$)/37$$ $$\beta_{4}$$ $$=$$ $$($$$$8 \nu^{7} - 28 \nu^{6} + 114 \nu^{5} - 215 \nu^{4} + 378 \nu^{3} - 366 \nu^{2} + 266 \nu - 97$$$$)/37$$ $$\beta_{5}$$ $$=$$ $$($$$$-21 \nu^{7} + 55 \nu^{6} - 216 \nu^{5} + 273 \nu^{4} - 428 \nu^{3} + 156 \nu^{2} - 97 \nu - 46$$$$)/37$$ $$\beta_{6}$$ $$=$$ $$($$$$-24 \nu^{7} + 84 \nu^{6} - 305 \nu^{5} + 534 \nu^{4} - 801 \nu^{3} + 617 \nu^{2} - 317 \nu - 5$$$$)/37$$ $$\beta_{7}$$ $$=$$ $$($$$$-24 \nu^{7} + 84 \nu^{6} - 305 \nu^{5} + 571 \nu^{4} - 875 \nu^{3} + 876 \nu^{2} - 539 \nu + 217$$$$)/37$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} + 4 \beta_{2} - 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{6} - 3 \beta_{4} + 4 \beta_{3} + 8 \beta_{2} - 4 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$($$$$9 \beta_{7} + 13 \beta_{6} - 14 \beta_{5} + 15 \beta_{4} - 26 \beta_{3} - \beta_{2} - 11 \beta_{1} + 41$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$5 \beta_{7} + 16 \beta_{6} - 4 \beta_{5} + 30 \beta_{4} - 31 \beta_{3} - 40 \beta_{2} + 11 \beta_{1} + 10$$ $$\nu^{7}$$ $$=$$ $$($$$$-46 \beta_{7} - 25 \beta_{6} + 63 \beta_{5} - 14 \beta_{4} + 71 \beta_{3} - 83 \beta_{2} + 77 \beta_{1} - 167$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
80.1
 0.5 − 1.19293i 0.5 + 2.19293i 0.5 − 1.56488i 0.5 + 0.564882i 0.5 + 1.56488i 0.5 − 0.564882i 0.5 + 1.19293i 0.5 − 2.19293i
−0.619657 + 2.31259i 0.866025 + 1.50000i −3.23205 1.86603i −1.23931 1.23931i −4.00552 + 1.07328i 0 2.93225 2.93225i −1.50000 + 2.59808i 3.63397 2.09808i
80.2 0.619657 2.31259i 0.866025 + 1.50000i −3.23205 1.86603i 1.23931 + 1.23931i 4.00552 1.07328i 0 −2.93225 + 2.93225i −1.50000 + 2.59808i 3.63397 2.09808i
89.1 −1.45466 0.389774i −0.866025 1.50000i 0.232051 + 0.133975i −2.90931 + 2.90931i 0.675108 + 2.51954i 0 1.84443 + 1.84443i −1.50000 + 2.59808i 5.36603 3.09808i
89.2 1.45466 + 0.389774i −0.866025 1.50000i 0.232051 + 0.133975i 2.90931 2.90931i −0.675108 2.51954i 0 −1.84443 1.84443i −1.50000 + 2.59808i 5.36603 3.09808i
188.1 −1.45466 + 0.389774i −0.866025 + 1.50000i 0.232051 0.133975i −2.90931 2.90931i 0.675108 2.51954i 0 1.84443 1.84443i −1.50000 2.59808i 5.36603 + 3.09808i
188.2 1.45466 0.389774i −0.866025 + 1.50000i 0.232051 0.133975i 2.90931 + 2.90931i −0.675108 + 2.51954i 0 −1.84443 + 1.84443i −1.50000 2.59808i 5.36603 + 3.09808i
488.1 −0.619657 2.31259i 0.866025 1.50000i −3.23205 + 1.86603i −1.23931 + 1.23931i −4.00552 1.07328i 0 2.93225 + 2.93225i −1.50000 2.59808i 3.63397 + 2.09808i
488.2 0.619657 + 2.31259i 0.866025 1.50000i −3.23205 + 1.86603i 1.23931 1.23931i 4.00552 + 1.07328i 0 −2.93225 2.93225i −1.50000 2.59808i 3.63397 + 2.09808i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 488.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
13.b even 2 1 inner
13.f odd 12 2 inner
39.k even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.k.g 8
3.b odd 2 1 inner 507.2.k.g 8
13.b even 2 1 inner 507.2.k.g 8
13.c even 3 1 507.2.f.d 8
13.c even 3 1 507.2.k.h 8
13.d odd 4 2 507.2.k.h 8
13.e even 6 1 507.2.f.d 8
13.e even 6 1 507.2.k.h 8
13.f odd 12 2 507.2.f.d 8
13.f odd 12 2 inner 507.2.k.g 8
39.d odd 2 1 CM 507.2.k.g 8
39.f even 4 2 507.2.k.h 8
39.h odd 6 1 507.2.f.d 8
39.h odd 6 1 507.2.k.h 8
39.i odd 6 1 507.2.f.d 8
39.i odd 6 1 507.2.k.h 8
39.k even 12 2 507.2.f.d 8
39.k even 12 2 inner 507.2.k.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.f.d 8 13.c even 3 1
507.2.f.d 8 13.e even 6 1
507.2.f.d 8 13.f odd 12 2
507.2.f.d 8 39.h odd 6 1
507.2.f.d 8 39.i odd 6 1
507.2.f.d 8 39.k even 12 2
507.2.k.g 8 1.a even 1 1 trivial
507.2.k.g 8 3.b odd 2 1 inner
507.2.k.g 8 13.b even 2 1 inner
507.2.k.g 8 13.f odd 12 2 inner
507.2.k.g 8 39.d odd 2 1 CM
507.2.k.g 8 39.k even 12 2 inner
507.2.k.h 8 13.c even 3 1
507.2.k.h 8 13.d odd 4 2
507.2.k.h 8 13.e even 6 1
507.2.k.h 8 39.f even 4 2
507.2.k.h 8 39.h odd 6 1
507.2.k.h 8 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{8} + 6 T_{2}^{6} - T_{2}^{4} - 78 T_{2}^{2} + 169$$ $$T_{5}^{8} + 296 T_{5}^{4} + 2704$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 - 78 T^{2} - T^{4} + 6 T^{6} + T^{8}$$
$3$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$5$ $$2704 + 296 T^{4} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$2704 + 3744 T^{2} + 1676 T^{4} - 72 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$39589264 + 453024 T^{2} - 4564 T^{4} - 72 T^{6} + T^{8}$$
$43$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$47$ $$77228944 + 17768 T^{4} + T^{8}$$
$53$ $$T^{8}$$
$59$ $$2704 + 21216 T^{2} + 55436 T^{4} - 408 T^{6} + T^{8}$$
$61$ $$( 36864 + 192 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$39589264 + 2567136 T^{2} + 49196 T^{4} - 408 T^{6} + T^{8}$$
$73$ $$T^{8}$$
$79$ $$( -108 + T^{2} )^{4}$$
$83$ $$756690064 + 55208 T^{4} + T^{8}$$
$89$ $$39589264 - 3473184 T^{2} + 95276 T^{4} + 552 T^{6} + T^{8}$$
$97$ $$T^{8}$$