Properties

Label 1710.2.d.f
Level $1710$
Weight $2$
Character orbit 1710.d
Analytic conductor $13.654$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{4} - \beta_{4} q^{5} + ( - \beta_{5} - \beta_{2}) q^{7} - \beta_1 q^{8} + \beta_{2} q^{10} + (\beta_{5} - \beta_{2} - 2) q^{11} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{4} - \beta_{3}) q^{14} + q^{16} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{17} - q^{19} + \beta_{4} q^{20} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{22} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{23} + (\beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{25} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{26} + (\beta_{5} + \beta_{2}) q^{28} + (\beta_{5} - \beta_{2} + 4) q^{29} + ( - \beta_{4} - \beta_{3}) q^{31} + \beta_1 q^{32} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2}) q^{34} + (2 \beta_{5} + \beta_{4} - \beta_{3} - 6 \beta_1 + 2) q^{35} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{37} - \beta_1 q^{38} - \beta_{2} q^{40} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{41} + (2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{43} + ( - \beta_{5} + \beta_{2} + 2) q^{44} + (\beta_{5} - \beta_{2} - 2) q^{46} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{47} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5) q^{49} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{50} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{52} + 6 \beta_1 q^{53} + (2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 4 \beta_1 + 2) q^{55} + (\beta_{4} + \beta_{3}) q^{56} + ( - \beta_{4} + \beta_{3} + 4 \beta_1) q^{58} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 4) q^{59} + (2 \beta_{5} - 2 \beta_{2} + 6) q^{61} + (\beta_{5} + \beta_{2}) q^{62} - q^{64} + (3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 8 \beta_1) q^{67} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{68} + (\beta_{5} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 6) q^{70} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 8) q^{71} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{73} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{74} + q^{76} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{77} + ( - \beta_{4} - \beta_{3}) q^{79} - \beta_{4} q^{80} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{82} + (3 \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{83} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{85} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 4) q^{86} + (\beta_{4} - \beta_{3} + 2 \beta_1) q^{88} + (\beta_{4} + \beta_{3} + 8) q^{89} + ( - 4 \beta_{4} - 4 \beta_{3} - 8) q^{91} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{92} + (\beta_{5} - \beta_{2} - 2) q^{94} + \beta_{4} q^{95} + (\beta_{4} - \beta_{3} - 8 \beta_1) q^{97} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{10} - 16 q^{11} + 6 q^{16} - 6 q^{19} + 2 q^{25} - 8 q^{26} + 20 q^{29} - 4 q^{34} + 8 q^{35} - 2 q^{40} - 8 q^{41} + 16 q^{44} - 16 q^{46} - 22 q^{49} - 8 q^{50} + 8 q^{55} + 20 q^{59} + 28 q^{61} - 6 q^{64} - 16 q^{65} + 32 q^{70} - 40 q^{71} + 8 q^{74} + 6 q^{76} + 20 q^{85} + 12 q^{86} + 48 q^{89} - 48 q^{91} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 30\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{5} + 3\nu^{4} + 10\nu^{3} - 32\nu^{2} - 74\nu - 3 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -10\nu^{5} + 11\nu^{4} - 17\nu^{3} - 10\nu^{2} - 72\nu - 11 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 27\nu^{4} + 25\nu^{3} + 12\nu^{2} + 68\nu - 65 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} + 37\nu^{4} - 30\nu^{3} - 42\nu^{2} - 54\nu + 55 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{2} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + 3\beta_{2} - 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 5\beta_{4} - 5\beta_{3} + \beta_{2} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{5} - 11\beta_{4} - 5\beta_{3} - 5\beta_{2} + 18\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1027\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1369.1
1.45161 + 1.45161i
−0.854638 0.854638i
0.403032 + 0.403032i
1.45161 1.45161i
−0.854638 + 0.854638i
0.403032 0.403032i
1.00000i 0 −1.00000 −2.21432 + 0.311108i 0 4.42864i 1.00000i 0 0.311108 + 2.21432i
1369.2 1.00000i 0 −1.00000 0.539189 + 2.17009i 0 1.07838i 1.00000i 0 2.17009 0.539189i
1369.3 1.00000i 0 −1.00000 1.67513 1.48119i 0 3.35026i 1.00000i 0 −1.48119 1.67513i
1369.4 1.00000i 0 −1.00000 −2.21432 0.311108i 0 4.42864i 1.00000i 0 0.311108 2.21432i
1369.5 1.00000i 0 −1.00000 0.539189 2.17009i 0 1.07838i 1.00000i 0 2.17009 + 0.539189i
1369.6 1.00000i 0 −1.00000 1.67513 + 1.48119i 0 3.35026i 1.00000i 0 −1.48119 + 1.67513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1369.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1710.2.d.f 6
3.b odd 2 1 570.2.d.c 6
5.b even 2 1 inner 1710.2.d.f 6
5.c odd 4 1 8550.2.a.ce 3
5.c odd 4 1 8550.2.a.cq 3
15.d odd 2 1 570.2.d.c 6
15.e even 4 1 2850.2.a.bl 3
15.e even 4 1 2850.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.c 6 3.b odd 2 1
570.2.d.c 6 15.d odd 2 1
1710.2.d.f 6 1.a even 1 1 trivial
1710.2.d.f 6 5.b even 2 1 inner
2850.2.a.bl 3 15.e even 4 1
2850.2.a.bm 3 15.e even 4 1
8550.2.a.ce 3 5.c odd 4 1
8550.2.a.cq 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1710, [\chi])\):

\( T_{7}^{6} + 32T_{7}^{4} + 256T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{3} + 8T_{11}^{2} + 8T_{11} - 16 \) Copy content Toggle raw display
\( T_{13}^{6} + 48T_{13}^{4} + 512T_{13}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} + 8 T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{6} + 44 T^{4} + 432 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 10 T^{2} + 20 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 80 T - 400)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 284 T^{4} + 24496 T^{2} + \cdots + 577600 \) Copy content Toggle raw display
$47$ \( T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{3} \) Copy content Toggle raw display
$59$ \( (T^{3} - 10 T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 14 T^{2} + 12 T + 152)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 304 T^{4} + 15104 T^{2} + \cdots + 102400 \) Copy content Toggle raw display
$71$ \( (T^{3} + 20 T^{2} + 48 T - 320)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 192 T^{4} + 8192 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 368 T^{4} + 32256 T^{2} + \cdots + 25600 \) Copy content Toggle raw display
$89$ \( (T^{3} - 24 T^{2} + 176 T - 400)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 188 T^{4} + 8880 T^{2} + \cdots + 87616 \) Copy content Toggle raw display
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