Properties

Label 1690.2.d.f
Level $1690$
Weight $2$
Character orbit 1690.d
Analytic conductor $13.495$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{6} - 3 \beta_1 q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{6} - 3 \beta_1 q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} + 1) q^{9} + q^{10} + 3 \beta_1 q^{11} + ( - \beta_{3} + 1) q^{12} - 3 q^{14} + (\beta_{2} - \beta_1) q^{15} + q^{16} + (3 \beta_{3} + 3) q^{17} + (2 \beta_{2} - \beta_1) q^{18} + (2 \beta_{2} - 3 \beta_1) q^{19} - \beta_1 q^{20} + ( - 3 \beta_{2} + 3 \beta_1) q^{21} + 3 q^{22} + (2 \beta_{3} + 6) q^{23} + (\beta_{2} - \beta_1) q^{24} - q^{25} - 4 q^{27} + 3 \beta_1 q^{28} + (2 \beta_{3} - 6) q^{29} + (\beta_{3} - 1) q^{30} + ( - \beta_{2} - 3 \beta_1) q^{31} - \beta_1 q^{32} + (3 \beta_{2} - 3 \beta_1) q^{33} + ( - 3 \beta_{2} - 3 \beta_1) q^{34} + 3 q^{35} + (2 \beta_{3} - 1) q^{36} + (3 \beta_{2} - 6 \beta_1) q^{37} + (2 \beta_{3} - 3) q^{38} - q^{40} - 6 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3) q^{42} + 2 q^{43} - 3 \beta_1 q^{44} + ( - 2 \beta_{2} + \beta_1) q^{45} + ( - 2 \beta_{2} - 6 \beta_1) q^{46} - 3 \beta_1 q^{47} + (\beta_{3} - 1) q^{48} - 2 q^{49} + \beta_1 q^{50} + 6 q^{51} + (2 \beta_{3} - 3) q^{53} + 4 \beta_1 q^{54} - 3 q^{55} + 3 q^{56} + ( - 5 \beta_{2} + 9 \beta_1) q^{57} + ( - 2 \beta_{2} + 6 \beta_1) q^{58} - 6 \beta_{2} q^{59} + ( - \beta_{2} + \beta_1) q^{60} + (3 \beta_{3} + 1) q^{61} + ( - \beta_{3} - 3) q^{62} + (6 \beta_{2} - 3 \beta_1) q^{63} - q^{64} + (3 \beta_{3} - 3) q^{66} + ( - 3 \beta_{3} - 3) q^{68} + 4 \beta_{3} q^{69} - 3 \beta_1 q^{70} + 6 \beta_1 q^{71} + ( - 2 \beta_{2} + \beta_1) q^{72} + ( - 5 \beta_{2} - 3 \beta_1) q^{73} + (3 \beta_{3} - 6) q^{74} + ( - \beta_{3} + 1) q^{75} + ( - 2 \beta_{2} + 3 \beta_1) q^{76} + 9 q^{77} + ( - 3 \beta_{3} + 1) q^{79} + \beta_1 q^{80} + (2 \beta_{3} + 1) q^{81} - 6 \beta_{3} q^{82} + (3 \beta_{2} + 3 \beta_1) q^{83} + (3 \beta_{2} - 3 \beta_1) q^{84} + (3 \beta_{2} + 3 \beta_1) q^{85} - 2 \beta_1 q^{86} + ( - 8 \beta_{3} + 12) q^{87} - 3 q^{88} + ( - 3 \beta_{2} + 12 \beta_1) q^{89} + ( - 2 \beta_{3} + 1) q^{90} + ( - 2 \beta_{3} - 6) q^{92} - 2 \beta_{2} q^{93} - 3 q^{94} + ( - 2 \beta_{3} + 3) q^{95} + ( - \beta_{2} + \beta_1) q^{96} + ( - 7 \beta_{2} + 3 \beta_1) q^{97} + 2 \beta_1 q^{98} + ( - 6 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} - 12 q^{14} + 4 q^{16} + 12 q^{17} + 12 q^{22} + 24 q^{23} - 4 q^{25} - 16 q^{27} - 24 q^{29} - 4 q^{30} + 12 q^{35} - 4 q^{36} - 12 q^{38} - 4 q^{40} + 12 q^{42} + 8 q^{43} - 4 q^{48} - 8 q^{49} + 24 q^{51} - 12 q^{53} - 12 q^{55} + 12 q^{56} + 4 q^{61} - 12 q^{62} - 4 q^{64} - 12 q^{66} - 12 q^{68} - 24 q^{74} + 4 q^{75} + 36 q^{77} + 4 q^{79} + 4 q^{81} + 48 q^{87} - 12 q^{88} + 4 q^{90} - 24 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.00000i −2.73205 −1.00000 1.00000i 2.73205i 3.00000i 1.00000i 4.46410 1.00000
1351.2 1.00000i 0.732051 −1.00000 1.00000i 0.732051i 3.00000i 1.00000i −2.46410 1.00000
1351.3 1.00000i −2.73205 −1.00000 1.00000i 2.73205i 3.00000i 1.00000i 4.46410 1.00000
1351.4 1.00000i 0.732051 −1.00000 1.00000i 0.732051i 3.00000i 1.00000i −2.46410 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.d.f 4
13.b even 2 1 inner 1690.2.d.f 4
13.c even 3 1 130.2.l.a 4
13.c even 3 1 1690.2.l.g 4
13.d odd 4 1 1690.2.a.j 2
13.d odd 4 1 1690.2.a.m 2
13.e even 6 1 130.2.l.a 4
13.e even 6 1 1690.2.l.g 4
13.f odd 12 2 1690.2.e.l 4
13.f odd 12 2 1690.2.e.n 4
39.h odd 6 1 1170.2.bs.c 4
39.i odd 6 1 1170.2.bs.c 4
52.i odd 6 1 1040.2.da.a 4
52.j odd 6 1 1040.2.da.a 4
65.g odd 4 1 8450.2.a.bf 2
65.g odd 4 1 8450.2.a.bm 2
65.l even 6 1 650.2.m.a 4
65.n even 6 1 650.2.m.a 4
65.q odd 12 1 650.2.n.a 4
65.q odd 12 1 650.2.n.b 4
65.r odd 12 1 650.2.n.a 4
65.r odd 12 1 650.2.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 13.c even 3 1
130.2.l.a 4 13.e even 6 1
650.2.m.a 4 65.l even 6 1
650.2.m.a 4 65.n even 6 1
650.2.n.a 4 65.q odd 12 1
650.2.n.a 4 65.r odd 12 1
650.2.n.b 4 65.q odd 12 1
650.2.n.b 4 65.r odd 12 1
1040.2.da.a 4 52.i odd 6 1
1040.2.da.a 4 52.j odd 6 1
1170.2.bs.c 4 39.h odd 6 1
1170.2.bs.c 4 39.i odd 6 1
1690.2.a.j 2 13.d odd 4 1
1690.2.a.m 2 13.d odd 4 1
1690.2.d.f 4 1.a even 1 1 trivial
1690.2.d.f 4 13.b even 2 1 inner
1690.2.e.l 4 13.f odd 12 2
1690.2.e.n 4 13.f odd 12 2
1690.2.l.g 4 13.c even 3 1
1690.2.l.g 4 13.e even 6 1
8450.2.a.bf 2 65.g odd 4 1
8450.2.a.bm 2 65.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3}^{2} + 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$23$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} + 126T^{2} + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T - 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{4} + 342 T^{2} + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} + 312 T^{2} + 19044 \) Copy content Toggle raw display
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