Properties

Label 1690.2.e.n
Level $1690$
Weight $2$
Character orbit 1690.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.500000 0.866025i −0.366025 + 0.633975i −0.500000 0.866025i 1.00000 0.366025 + 0.633975i −1.50000 2.59808i −1.00000 1.23205 + 2.13397i 0.500000 0.866025i
191.2 0.500000 0.866025i 1.36603 2.36603i −0.500000 0.866025i 1.00000 −1.36603 2.36603i −1.50000 2.59808i −1.00000 −2.23205 3.86603i 0.500000 0.866025i
991.1 0.500000 + 0.866025i −0.366025 0.633975i −0.500000 + 0.866025i 1.00000 0.366025 0.633975i −1.50000 + 2.59808i −1.00000 1.23205 2.13397i 0.500000 + 0.866025i
991.2 0.500000 + 0.866025i 1.36603 + 2.36603i −0.500000 + 0.866025i 1.00000 −1.36603 + 2.36603i −1.50000 + 2.59808i −1.00000 −2.23205 + 3.86603i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{19}^{4} - 6T_{19}^{3} + 39T_{19}^{2} + 18T_{19} + 9 \) Copy content Toggle raw display
\( T_{31}^{2} + 6T_{31} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 39 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 12 T^{3} + 135 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T - 3)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + 459 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} + 6 T^{3} + 174 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
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