Properties

Label 1690.2.d.a
Level $1690$
Weight $2$
Character orbit 1690.d
Analytic conductor $13.495$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - 2 q^{3} - q^{4} + i q^{5} - 2 i q^{6} - i q^{7} - i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - 2 q^{3} - q^{4} + i q^{5} - 2 i q^{6} - i q^{7} - i q^{8} + q^{9} - q^{10} + 3 i q^{11} + 2 q^{12} + q^{14} - 2 i q^{15} + q^{16} + 6 q^{17} + i q^{18} - 5 i q^{19} - i q^{20} + 2 i q^{21} - 3 q^{22} + 2 i q^{24} - q^{25} + 4 q^{27} + i q^{28} + 2 q^{30} + 4 i q^{31} + i q^{32} - 6 i q^{33} + 6 i q^{34} + q^{35} - q^{36} + 11 i q^{37} + 5 q^{38} + q^{40} - 6 i q^{41} - 2 q^{42} - 2 q^{43} - 3 i q^{44} + i q^{45} - 3 i q^{47} - 2 q^{48} + 6 q^{49} - i q^{50} - 12 q^{51} - 9 q^{53} + 4 i q^{54} - 3 q^{55} - q^{56} + 10 i q^{57} + 2 i q^{60} + 8 q^{61} - 4 q^{62} - i q^{63} - q^{64} + 6 q^{66} + 16 i q^{67} - 6 q^{68} + i q^{70} - 6 i q^{71} - i q^{72} + 14 i q^{73} - 11 q^{74} + 2 q^{75} + 5 i q^{76} + 3 q^{77} - 16 q^{79} + i q^{80} - 11 q^{81} + 6 q^{82} + 6 i q^{83} - 2 i q^{84} + 6 i q^{85} - 2 i q^{86} + 3 q^{88} + 9 i q^{89} - q^{90} - 8 i q^{93} + 3 q^{94} + 5 q^{95} - 2 i q^{96} + 10 i q^{97} + 6 i q^{98} + 3 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 4 q^{12} + 2 q^{14} + 2 q^{16} + 12 q^{17} - 6 q^{22} - 2 q^{25} + 8 q^{27} + 4 q^{30} + 2 q^{35} - 2 q^{36} + 10 q^{38} + 2 q^{40} - 4 q^{42} - 4 q^{43} - 4 q^{48} + 12 q^{49} - 24 q^{51} - 18 q^{53} - 6 q^{55} - 2 q^{56} + 16 q^{61} - 8 q^{62} - 2 q^{64} + 12 q^{66} - 12 q^{68} - 22 q^{74} + 4 q^{75} + 6 q^{77} - 32 q^{79} - 22 q^{81} + 12 q^{82} + 6 q^{88} - 2 q^{90} + 6 q^{94} + 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
1.00000i
1.00000i
1.00000i −2.00000 −1.00000 1.00000i 2.00000i 1.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −2.00000 −1.00000 1.00000i 2.00000i 1.00000i 1.00000i 1.00000 −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.a 2
13.b even 2 1 inner 1690.2.d.a 2
13.c even 3 2 1690.2.l.i 4
13.d odd 4 1 1690.2.a.a 1
13.d odd 4 1 1690.2.a.g 1
13.e even 6 2 1690.2.l.i 4
13.f odd 12 2 130.2.e.b 2
13.f odd 12 2 1690.2.e.e 2
39.k even 12 2 1170.2.i.f 2
52.l even 12 2 1040.2.q.c 2
65.g odd 4 1 8450.2.a.k 1
65.g odd 4 1 8450.2.a.w 1
65.o even 12 2 650.2.o.b 4
65.s odd 12 2 650.2.e.a 2
65.t even 12 2 650.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 13.f odd 12 2
650.2.e.a 2 65.s odd 12 2
650.2.o.b 4 65.o even 12 2
650.2.o.b 4 65.t even 12 2
1040.2.q.c 2 52.l even 12 2
1170.2.i.f 2 39.k even 12 2
1690.2.a.a 1 13.d odd 4 1
1690.2.a.g 1 13.d odd 4 1
1690.2.d.a 2 1.a even 1 1 trivial
1690.2.d.a 2 13.b even 2 1 inner
1690.2.e.e 2 13.f odd 12 2
1690.2.l.i 4 13.c even 3 2
1690.2.l.i 4 13.e even 6 2
8450.2.a.k 1 65.g odd 4 1
8450.2.a.w 1 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 \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 25 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 121 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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