Properties

Label 1690.2.l.g
Level $1690$
Weight $2$
Character orbit 1690.l
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.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 1.36603 + 2.36603i 0.500000 0.866025i 1.00000i −2.36603 1.36603i 2.59808 + 1.50000i 1.00000i −2.23205 + 3.86603i −0.500000 0.866025i
361.2 0.866025 0.500000i −0.366025 0.633975i 0.500000 0.866025i 1.00000i −0.633975 0.366025i −2.59808 1.50000i 1.00000i 1.23205 2.13397i −0.500000 0.866025i
1161.1 −0.866025 0.500000i 1.36603 2.36603i 0.500000 + 0.866025i 1.00000i −2.36603 + 1.36603i 2.59808 1.50000i 1.00000i −2.23205 3.86603i −0.500000 + 0.866025i
1161.2 0.866025 + 0.500000i −0.366025 + 0.633975i 0.500000 + 0.866025i 1.00000i −0.633975 + 0.366025i −2.59808 + 1.50000i 1.00000i 1.23205 + 2.13397i −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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.l.g 4
13.b even 2 1 130.2.l.a 4
13.c even 3 1 130.2.l.a 4
13.c even 3 1 1690.2.d.f 4
13.d odd 4 1 1690.2.e.l 4
13.d odd 4 1 1690.2.e.n 4
13.e even 6 1 1690.2.d.f 4
13.e even 6 1 inner 1690.2.l.g 4
13.f odd 12 1 1690.2.a.j 2
13.f odd 12 1 1690.2.a.m 2
13.f odd 12 1 1690.2.e.l 4
13.f odd 12 1 1690.2.e.n 4
39.d odd 2 1 1170.2.bs.c 4
39.i odd 6 1 1170.2.bs.c 4
52.b odd 2 1 1040.2.da.a 4
52.j odd 6 1 1040.2.da.a 4
65.d even 2 1 650.2.m.a 4
65.h odd 4 1 650.2.n.a 4
65.h odd 4 1 650.2.n.b 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.s odd 12 1 8450.2.a.bf 2
65.s odd 12 1 8450.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 13.b even 2 1
130.2.l.a 4 13.c even 3 1
650.2.m.a 4 65.d even 2 1
650.2.m.a 4 65.n even 6 1
650.2.n.a 4 65.h odd 4 1
650.2.n.a 4 65.q odd 12 1
650.2.n.b 4 65.h odd 4 1
650.2.n.b 4 65.q odd 12 1
1040.2.da.a 4 52.b odd 2 1
1040.2.da.a 4 52.j odd 6 1
1170.2.bs.c 4 39.d odd 2 1
1170.2.bs.c 4 39.i odd 6 1
1690.2.a.j 2 13.f odd 12 1
1690.2.a.m 2 13.f odd 12 1
1690.2.d.f 4 13.c even 3 1
1690.2.d.f 4 13.e even 6 1
1690.2.e.l 4 13.d odd 4 1
1690.2.e.l 4 13.f odd 12 1
1690.2.e.n 4 13.d odd 4 1
1690.2.e.n 4 13.f odd 12 1
1690.2.l.g 4 1.a even 1 1 trivial
1690.2.l.g 4 13.e even 6 1 inner
8450.2.a.bf 2 65.s odd 12 1
8450.2.a.bm 2 65.s odd 12 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
\( T_{7}^{4} - 9T_{7}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) 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
$19$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) 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
$31$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$37$ \( T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18 T + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 4)^{2} \) 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
$59$ \( (T^{2} + 18 T + 108)^{2} \) 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
$71$ \( T^{4} - 36T^{2} + 1296 \) 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
$89$ \( T^{4} - 18 T^{3} - 9 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$97$ \( T^{4} + 42 T^{3} + 726 T^{2} + \cdots + 19044 \) Copy content Toggle raw display
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