Properties

Label 1682.2.b.a
Level $1682$
Weight $2$
Character orbit 1682.b
Analytic conductor $13.431$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4308376200\)
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 58)
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.

\(f(q)\) \(=\) \( q - i q^{2} + i q^{3} - q^{4} - q^{5} + q^{6} - 2 q^{7} + i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} + i q^{3} - q^{4} - q^{5} + q^{6} - 2 q^{7} + i q^{8} + 2 q^{9} + i q^{10} + 3 i q^{11} - i q^{12} + q^{13} + 2 i q^{14} - i q^{15} + q^{16} - 8 i q^{17} - 2 i q^{18} + q^{20} - 2 i q^{21} + 3 q^{22} + 4 q^{23} - q^{24} - 4 q^{25} - i q^{26} + 5 i q^{27} + 2 q^{28} - q^{30} + 3 i q^{31} - i q^{32} - 3 q^{33} - 8 q^{34} + 2 q^{35} - 2 q^{36} + 8 i q^{37} + i q^{39} - i q^{40} + 2 i q^{41} - 2 q^{42} + 11 i q^{43} - 3 i q^{44} - 2 q^{45} - 4 i q^{46} + 13 i q^{47} + i q^{48} - 3 q^{49} + 4 i q^{50} + 8 q^{51} - q^{52} - 11 q^{53} + 5 q^{54} - 3 i q^{55} - 2 i q^{56} + i q^{60} + 8 i q^{61} + 3 q^{62} - 4 q^{63} - q^{64} - q^{65} + 3 i q^{66} + 12 q^{67} + 8 i q^{68} + 4 i q^{69} - 2 i q^{70} - 2 q^{71} + 2 i q^{72} + 4 i q^{73} + 8 q^{74} - 4 i q^{75} - 6 i q^{77} + q^{78} - 15 i q^{79} - q^{80} + q^{81} + 2 q^{82} + 4 q^{83} + 2 i q^{84} + 8 i q^{85} + 11 q^{86} - 3 q^{88} + 10 i q^{89} + 2 i q^{90} - 2 q^{91} - 4 q^{92} - 3 q^{93} + 13 q^{94} + q^{96} - 2 i q^{97} + 3 i q^{98} + 6 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times\).

\(n\) \(843\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1681.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 −1.00000 1.00000 −2.00000 1.00000i 2.00000 1.00000i
1681.2 1.00000i 1.00000i −1.00000 −1.00000 1.00000 −2.00000 1.00000i 2.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1682.2.b.a 2
29.b even 2 1 inner 1682.2.b.a 2
29.c odd 4 1 58.2.a.b 1
29.c odd 4 1 1682.2.a.d 1
87.f even 4 1 522.2.a.b 1
116.e even 4 1 464.2.a.e 1
145.e even 4 1 1450.2.b.b 2
145.f odd 4 1 1450.2.a.c 1
145.j even 4 1 1450.2.b.b 2
203.g even 4 1 2842.2.a.e 1
232.k even 4 1 1856.2.a.f 1
232.l odd 4 1 1856.2.a.k 1
319.f even 4 1 7018.2.a.a 1
348.k odd 4 1 4176.2.a.n 1
377.i odd 4 1 9802.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 29.c odd 4 1
464.2.a.e 1 116.e even 4 1
522.2.a.b 1 87.f even 4 1
1450.2.a.c 1 145.f odd 4 1
1450.2.b.b 2 145.e even 4 1
1450.2.b.b 2 145.j even 4 1
1682.2.a.d 1 29.c odd 4 1
1682.2.b.a 2 1.a even 1 1 trivial
1682.2.b.a 2 29.b even 2 1 inner
1856.2.a.f 1 232.k even 4 1
1856.2.a.k 1 232.l odd 4 1
2842.2.a.e 1 203.g even 4 1
4176.2.a.n 1 348.k odd 4 1
7018.2.a.a 1 319.f even 4 1
9802.2.a.a 1 377.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1682, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 121 \) Copy content Toggle raw display
$47$ \( T^{2} + 169 \) Copy content Toggle raw display
$53$ \( (T + 11)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 64 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 225 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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