# Properties

 Label 3479.1.g.e Level $3479$ Weight $1$ Character orbit 3479.g Analytic conductor $1.736$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -71 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3479 = 7^{2} \cdot 71$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3479.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.73624717895$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 71) Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.357911.1 Artin image: $C_3\times D_7$ Artin field: Galois closure of 21.3.31095511042786085990206459319.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{4} + \beta_1 q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{2} - 1) q^{8} + (\beta_{5} + \beta_{4} - 1) q^{9}+O(q^{10})$$ q + b4 * q^2 + (b5 + b4 + b3 - b2 + b1) * q^3 + (-b5 + b2 - b1) * q^4 + b1 * q^5 + (-b3 - 1) * q^6 + (b2 - 1) * q^8 + (b5 + b4 - 1) * q^9 $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{4} + \beta_1 q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{2} - 1) q^{8} + (\beta_{5} + \beta_{4} - 1) q^{9} + (\beta_{5} - \beta_{2} + \beta_1) q^{10} + ( - \beta_{5} + 1) q^{12} + (\beta_{3} - \beta_{2}) q^{15} - \beta_{4} q^{16} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{18} + \beta_{4} q^{19} + ( - \beta_{3} + 1) q^{20} - \beta_{5} q^{24} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{25} + ( - \beta_{3} - 1) q^{27} + ( - \beta_{3} + \beta_{2} - 1) q^{29} + (\beta_{5} - 1) q^{30} + \beta_{5} q^{32} + \beta_{3} q^{36} + \beta_{4} q^{37} + ( - 2 \beta_{5} + \beta_{2} - \beta_1) q^{38} + ( - \beta_{5} + \beta_{4} + 1) q^{40} - \beta_{2} q^{43} + \beta_{5} q^{45} + (\beta_{3} + 1) q^{48} - q^{50} + ( - 2 \beta_{5} - \beta_{4} - \beta_1 + 2) q^{54} + ( - \beta_{3} - 1) q^{57} + ( - \beta_{5} - \beta_{4} + 1) q^{58} + ( - \beta_{2} + \beta_1) q^{60} + q^{71} - \beta_{4} q^{72} + (\beta_{2} - \beta_1) q^{73} + ( - 2 \beta_{5} + \beta_{2} - \beta_1) q^{74} + (\beta_{5} - \beta_1 - 1) q^{75} + (\beta_{3} + \beta_{2} - 1) q^{76} + \beta_1 q^{79} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{80} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{81} + \beta_{3} q^{83} + ( - \beta_{5} - \beta_1 + 1) q^{86} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{87} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{89} - \beta_{3} q^{90} + (\beta_{5} - \beta_{2} + \beta_1) q^{95} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{96}+O(q^{100})$$ q + b4 * q^2 + (b5 + b4 + b3 - b2 + b1) * q^3 + (-b5 + b2 - b1) * q^4 + b1 * q^5 + (-b3 - 1) * q^6 + (b2 - 1) * q^8 + (b5 + b4 - 1) * q^9 + (b5 - b2 + b1) * q^10 + (-b5 + 1) * q^12 + (b3 - b2) * q^15 - b4 * q^16 + (-2*b5 - b4 - b3 + b2 - b1) * q^18 + b4 * q^19 + (-b3 + 1) * q^20 - b5 * q^24 + (b4 + b3 - b2 + b1) * q^25 + (-b3 - 1) * q^27 + (-b3 + b2 - 1) * q^29 + (b5 - 1) * q^30 + b5 * q^32 + b3 * q^36 + b4 * q^37 + (-2*b5 + b2 - b1) * q^38 + (-b5 + b4 + 1) * q^40 - b2 * q^43 + b5 * q^45 + (b3 + 1) * q^48 - q^50 + (-2*b5 - b4 - b1 + 2) * q^54 + (-b3 - 1) * q^57 + (-b5 - b4 + 1) * q^58 + (-b2 + b1) * q^60 + q^71 - b4 * q^72 + (b2 - b1) * q^73 + (-2*b5 + b2 - b1) * q^74 + (b5 - b1 - 1) * q^75 + (b3 + b2 - 1) * q^76 + b1 * q^79 + (-b5 + b2 - b1) * q^80 + (-b5 - b4 - b3 + b2 - b1) * q^81 + b3 * q^83 + (-b5 - b1 + 1) * q^86 + (-2*b5 - b4 - b3) * q^87 + (-b5 - b4 - b1 + 1) * q^89 - b3 * q^90 + (b5 - b2 + b1) * q^95 + (b5 + b4 + b1 - 1) * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} + q^{3} - 2 q^{4} + q^{5} - 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 6 * q + q^2 + q^3 - 2 * q^4 + q^5 - 4 * q^6 - 4 * q^8 - 2 * q^9 $$6 q + q^{2} + q^{3} - 2 q^{4} + q^{5} - 4 q^{6} - 4 q^{8} - 2 q^{9} + 2 q^{10} + 3 q^{12} - 4 q^{15} - q^{16} - 4 q^{18} + q^{19} + 8 q^{20} - 3 q^{24} - 2 q^{25} - 4 q^{27} - 2 q^{29} - 3 q^{30} + 3 q^{32} - 2 q^{36} + q^{37} - 5 q^{38} + 4 q^{40} - 2 q^{43} + 3 q^{45} + 4 q^{48} - 6 q^{50} + 4 q^{54} - 4 q^{57} + 2 q^{58} - q^{60} + 6 q^{71} - q^{72} + q^{73} - 5 q^{74} - 4 q^{75} - 6 q^{76} + q^{79} - 2 q^{80} - q^{81} - 2 q^{83} + 2 q^{86} - 5 q^{87} + q^{89} + 2 q^{90} + 2 q^{95} - q^{96}+O(q^{100})$$ 6 * q + q^2 + q^3 - 2 * q^4 + q^5 - 4 * q^6 - 4 * q^8 - 2 * q^9 + 2 * q^10 + 3 * q^12 - 4 * q^15 - q^16 - 4 * q^18 + q^19 + 8 * q^20 - 3 * q^24 - 2 * q^25 - 4 * q^27 - 2 * q^29 - 3 * q^30 + 3 * q^32 - 2 * q^36 + q^37 - 5 * q^38 + 4 * q^40 - 2 * q^43 + 3 * q^45 + 4 * q^48 - 6 * q^50 + 4 * q^54 - 4 * q^57 + 2 * q^58 - q^60 + 6 * q^71 - q^72 + q^73 - 5 * q^74 - 4 * q^75 - 6 * q^76 + q^79 - 2 * q^80 - q^81 - 2 * q^83 + 2 * q^86 - 5 * q^87 + q^89 + 2 * q^90 + 2 * q^95 - q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3479\mathbb{Z}\right)^\times$$.

 $$n$$ $$640$$ $$1569$$ $$\chi(n)$$ $$-1 + \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
851.1
 0.222521 + 0.385418i 0.900969 + 1.56052i −0.623490 − 1.07992i 0.222521 − 0.385418i 0.900969 − 1.56052i −0.623490 + 1.07992i
−0.623490 1.07992i 0.900969 1.56052i −0.277479 + 0.480608i 0.222521 + 0.385418i −2.24698 0 −0.554958 −1.12349 1.94594i 0.277479 0.480608i
851.2 0.222521 + 0.385418i −0.623490 + 1.07992i 0.400969 0.694498i 0.900969 + 1.56052i −0.554958 0 0.801938 −0.277479 0.480608i −0.400969 + 0.694498i
851.3 0.900969 + 1.56052i 0.222521 0.385418i −1.12349 + 1.94594i −0.623490 1.07992i 0.801938 0 −2.24698 0.400969 + 0.694498i 1.12349 1.94594i
1206.1 −0.623490 + 1.07992i 0.900969 + 1.56052i −0.277479 0.480608i 0.222521 0.385418i −2.24698 0 −0.554958 −1.12349 + 1.94594i 0.277479 + 0.480608i
1206.2 0.222521 0.385418i −0.623490 1.07992i 0.400969 + 0.694498i 0.900969 1.56052i −0.554958 0 0.801938 −0.277479 + 0.480608i −0.400969 0.694498i
1206.3 0.900969 1.56052i 0.222521 + 0.385418i −1.12349 1.94594i −0.623490 + 1.07992i 0.801938 0 −2.24698 0.400969 0.694498i 1.12349 + 1.94594i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1206.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.b odd 2 1 CM by $$\Q(\sqrt{-71})$$
7.c even 3 1 inner
497.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3479.1.g.e 6
7.b odd 2 1 3479.1.g.d 6
7.c even 3 1 71.1.b.a 3
7.c even 3 1 inner 3479.1.g.e 6
7.d odd 6 1 3479.1.d.e 3
7.d odd 6 1 3479.1.g.d 6
21.h odd 6 1 639.1.d.a 3
28.g odd 6 1 1136.1.h.a 3
35.j even 6 1 1775.1.d.b 3
35.l odd 12 2 1775.1.c.a 6
71.b odd 2 1 CM 3479.1.g.e 6
497.b even 2 1 3479.1.g.d 6
497.g odd 6 1 71.1.b.a 3
497.g odd 6 1 inner 3479.1.g.e 6
497.i even 6 1 3479.1.d.e 3
497.i even 6 1 3479.1.g.d 6
1491.p even 6 1 639.1.d.a 3
1988.n even 6 1 1136.1.h.a 3
2485.u odd 6 1 1775.1.d.b 3
2485.bg even 12 2 1775.1.c.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.1.b.a 3 7.c even 3 1
71.1.b.a 3 497.g odd 6 1
639.1.d.a 3 21.h odd 6 1
639.1.d.a 3 1491.p even 6 1
1136.1.h.a 3 28.g odd 6 1
1136.1.h.a 3 1988.n even 6 1
1775.1.c.a 6 35.l odd 12 2
1775.1.c.a 6 2485.bg even 12 2
1775.1.d.b 3 35.j even 6 1
1775.1.d.b 3 2485.u odd 6 1
3479.1.d.e 3 7.d odd 6 1
3479.1.d.e 3 497.i even 6 1
3479.1.g.d 6 7.b odd 2 1
3479.1.g.d 6 7.d odd 6 1
3479.1.g.d 6 497.b even 2 1
3479.1.g.d 6 497.i even 6 1
3479.1.g.e 6 1.a even 1 1 trivial
3479.1.g.e 6 7.c even 3 1 inner
3479.1.g.e 6 71.b odd 2 1 CM
3479.1.g.e 6 497.g odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3479, [\chi])$$:

 $$T_{2}^{6} - T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} - 2T_{2} + 1$$ T2^6 - T2^5 + 3*T2^4 + 5*T2^2 - 2*T2 + 1 $$T_{3}^{6} - T_{3}^{5} + 3T_{3}^{4} + 5T_{3}^{2} - 2T_{3} + 1$$ T3^6 - T3^5 + 3*T3^4 + 5*T3^2 - 2*T3 + 1

## Hecke characteristic polynomials

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