# Properties

 Label 3479.1.g.d 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: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ 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: $C_6\times D_7$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## $q$-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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## 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.d 6
7.b odd 2 1 3479.1.g.e 6
7.c even 3 1 3479.1.d.e 3
7.c even 3 1 inner 3479.1.g.d 6
7.d odd 6 1 71.1.b.a 3
7.d odd 6 1 3479.1.g.e 6
21.g even 6 1 639.1.d.a 3
28.f even 6 1 1136.1.h.a 3
35.i odd 6 1 1775.1.d.b 3
35.k even 12 2 1775.1.c.a 6
71.b odd 2 1 CM 3479.1.g.d 6
497.b even 2 1 3479.1.g.e 6
497.g odd 6 1 3479.1.d.e 3
497.g odd 6 1 inner 3479.1.g.d 6
497.i even 6 1 71.1.b.a 3
497.i even 6 1 3479.1.g.e 6
1491.n odd 6 1 639.1.d.a 3
1988.l odd 6 1 1136.1.h.a 3
2485.s even 6 1 1775.1.d.b 3
2485.be odd 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.d odd 6 1
71.1.b.a 3 497.i even 6 1
639.1.d.a 3 21.g even 6 1
639.1.d.a 3 1491.n odd 6 1
1136.1.h.a 3 28.f even 6 1
1136.1.h.a 3 1988.l odd 6 1
1775.1.c.a 6 35.k even 12 2
1775.1.c.a 6 2485.be odd 12 2
1775.1.d.b 3 35.i odd 6 1
1775.1.d.b 3 2485.s even 6 1
3479.1.d.e 3 7.c even 3 1
3479.1.d.e 3 497.g odd 6 1
3479.1.g.d 6 1.a even 1 1 trivial
3479.1.g.d 6 7.c even 3 1 inner
3479.1.g.d 6 71.b odd 2 1 CM
3479.1.g.d 6 497.g odd 6 1 inner
3479.1.g.e 6 7.b odd 2 1
3479.1.g.e 6 7.d odd 6 1
3479.1.g.e 6 497.b even 2 1
3479.1.g.e 6 497.i even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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