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 $4$

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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:

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

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