Properties

Label 289.2.b.a
Level $289$
Weight $2$
Character orbit 289.b
Analytic conductor $2.308$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + \beta q^{5} + 2 \beta q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + \beta q^{5} + 2 \beta q^{7} - 3 q^{8} + 3 q^{9} + \beta q^{10} - 2 q^{13} + 2 \beta q^{14} - q^{16} + 3 q^{18} + 4 q^{19} - \beta q^{20} + 2 \beta q^{23} + q^{25} - 2 q^{26} - 2 \beta q^{28} - 3 \beta q^{29} - 2 \beta q^{31} + 5 q^{32} - 8 q^{35} - 3 q^{36} + \beta q^{37} + 4 q^{38} - 3 \beta q^{40} - 3 \beta q^{41} - 4 q^{43} + 3 \beta q^{45} + 2 \beta q^{46} - 9 q^{49} + q^{50} + 2 q^{52} - 6 q^{53} - 6 \beta q^{56} - 3 \beta q^{58} + 12 q^{59} - 5 \beta q^{61} - 2 \beta q^{62} + 6 \beta q^{63} + 7 q^{64} - 2 \beta q^{65} + 4 q^{67} - 8 q^{70} + 2 \beta q^{71} - 9 q^{72} + 3 \beta q^{73} + \beta q^{74} - 4 q^{76} + 6 \beta q^{79} - \beta q^{80} + 9 q^{81} - 3 \beta q^{82} + 4 q^{83} - 4 q^{86} + 10 q^{89} + 3 \beta q^{90} - 4 \beta q^{91} - 2 \beta q^{92} + 4 \beta q^{95} - \beta q^{97} - 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 4 q^{13} - 2 q^{16} + 6 q^{18} + 8 q^{19} + 2 q^{25} - 4 q^{26} + 10 q^{32} - 16 q^{35} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 18 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 14 q^{64} + 8 q^{67} - 16 q^{70} - 18 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} - 8 q^{86} + 20 q^{89} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
288.1
1.00000i
1.00000i
1.00000 0 −1.00000 2.00000i 0 4.00000i −3.00000 3.00000 2.00000i
288.2 1.00000 0 −1.00000 2.00000i 0 4.00000i −3.00000 3.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.b.a 2
17.b even 2 1 inner 289.2.b.a 2
17.c even 4 1 17.2.a.a 1
17.c even 4 1 289.2.a.a 1
17.d even 8 4 289.2.c.a 4
17.e odd 16 8 289.2.d.d 8
51.f odd 4 1 153.2.a.c 1
51.f odd 4 1 2601.2.a.g 1
68.f odd 4 1 272.2.a.b 1
68.f odd 4 1 4624.2.a.d 1
85.f odd 4 1 425.2.b.b 2
85.i odd 4 1 425.2.b.b 2
85.j even 4 1 425.2.a.d 1
85.j even 4 1 7225.2.a.g 1
119.f odd 4 1 833.2.a.a 1
119.m odd 12 2 833.2.e.a 2
119.n even 12 2 833.2.e.b 2
136.i even 4 1 1088.2.a.i 1
136.j odd 4 1 1088.2.a.h 1
187.f odd 4 1 2057.2.a.e 1
204.l even 4 1 2448.2.a.o 1
221.k even 4 1 2873.2.a.c 1
255.i odd 4 1 3825.2.a.d 1
323.g odd 4 1 6137.2.a.b 1
340.n odd 4 1 6800.2.a.n 1
357.l even 4 1 7497.2.a.l 1
391.f odd 4 1 8993.2.a.a 1
408.q even 4 1 9792.2.a.i 1
408.t odd 4 1 9792.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 17.c even 4 1
153.2.a.c 1 51.f odd 4 1
272.2.a.b 1 68.f odd 4 1
289.2.a.a 1 17.c even 4 1
289.2.b.a 2 1.a even 1 1 trivial
289.2.b.a 2 17.b even 2 1 inner
289.2.c.a 4 17.d even 8 4
289.2.d.d 8 17.e odd 16 8
425.2.a.d 1 85.j even 4 1
425.2.b.b 2 85.f odd 4 1
425.2.b.b 2 85.i odd 4 1
833.2.a.a 1 119.f odd 4 1
833.2.e.a 2 119.m odd 12 2
833.2.e.b 2 119.n even 12 2
1088.2.a.h 1 136.j odd 4 1
1088.2.a.i 1 136.i even 4 1
2057.2.a.e 1 187.f odd 4 1
2448.2.a.o 1 204.l even 4 1
2601.2.a.g 1 51.f odd 4 1
2873.2.a.c 1 221.k even 4 1
3825.2.a.d 1 255.i odd 4 1
4624.2.a.d 1 68.f odd 4 1
6137.2.a.b 1 323.g odd 4 1
6800.2.a.n 1 340.n odd 4 1
7225.2.a.g 1 85.j even 4 1
7497.2.a.l 1 357.l even 4 1
8993.2.a.a 1 391.f odd 4 1
9792.2.a.i 1 408.q even 4 1
9792.2.a.n 1 408.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 100 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( T^{2} + 144 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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