Properties

Label 289.2.c.a
Level $289$
Weight $2$
Character orbit 289.c
Analytic conductor $2.308$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.c (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + q^{4} + \beta_1 q^{5} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + q^{4} + \beta_1 q^{5} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{8} - 3 \beta_{2} q^{9} + \beta_{3} q^{10} + 2 q^{13} - 2 \beta_1 q^{14} - q^{16} + 3 q^{18} + 4 \beta_{2} q^{19} + \beta_1 q^{20} - 2 \beta_{3} q^{23} - \beta_{2} q^{25} + 2 \beta_{2} q^{26} + 2 \beta_{3} q^{28} - 3 \beta_1 q^{29} + 2 \beta_1 q^{31} + 5 \beta_{2} q^{32} - 8 q^{35} - 3 \beta_{2} q^{36} - \beta_1 q^{37} - 4 q^{38} + 3 \beta_{3} q^{40} - 3 \beta_{3} q^{41} + 4 \beta_{2} q^{43} - 3 \beta_{3} q^{45} + 2 \beta_1 q^{46} - 9 \beta_{2} q^{49} + q^{50} + 2 q^{52} - 6 \beta_{2} q^{53} - 6 \beta_1 q^{56} - 3 \beta_{3} q^{58} - 12 \beta_{2} q^{59} - 5 \beta_{3} q^{61} + 2 \beta_{3} q^{62} + 6 \beta_1 q^{63} - 7 q^{64} + 2 \beta_1 q^{65} + 4 q^{67} - 8 \beta_{2} q^{70} - 2 \beta_1 q^{71} + 9 q^{72} + 3 \beta_1 q^{73} - \beta_{3} q^{74} + 4 \beta_{2} q^{76} - 6 \beta_{3} q^{79} - \beta_1 q^{80} - 9 q^{81} + 3 \beta_1 q^{82} + 4 \beta_{2} q^{83} - 4 q^{86} - 10 q^{89} + 3 \beta_1 q^{90} + 4 \beta_{3} q^{91} - 2 \beta_{3} q^{92} + 4 \beta_{3} q^{95} - \beta_1 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 8 q^{13} - 4 q^{16} + 12 q^{18} - 32 q^{35} - 16 q^{38} + 4 q^{50} + 8 q^{52} - 28 q^{64} + 16 q^{67} + 36 q^{72} - 36 q^{81} - 16 q^{86} - 40 q^{89} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i 0 1.00000 −1.41421 + 1.41421i 0 2.82843 + 2.82843i 3.00000i 3.00000i 1.41421 + 1.41421i
38.2 1.00000i 0 1.00000 1.41421 1.41421i 0 −2.82843 2.82843i 3.00000i 3.00000i −1.41421 1.41421i
251.1 1.00000i 0 1.00000 −1.41421 1.41421i 0 2.82843 2.82843i 3.00000i 3.00000i 1.41421 1.41421i
251.2 1.00000i 0 1.00000 1.41421 + 1.41421i 0 −2.82843 + 2.82843i 3.00000i 3.00000i −1.41421 + 1.41421i
\(n\): 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
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.c.a 4
17.b even 2 1 inner 289.2.c.a 4
17.c even 4 2 inner 289.2.c.a 4
17.d even 8 1 17.2.a.a 1
17.d even 8 1 289.2.a.a 1
17.d even 8 2 289.2.b.a 2
17.e odd 16 8 289.2.d.d 8
51.g odd 8 1 153.2.a.c 1
51.g odd 8 1 2601.2.a.g 1
68.g odd 8 1 272.2.a.b 1
68.g odd 8 1 4624.2.a.d 1
85.k odd 8 1 425.2.b.b 2
85.m even 8 1 425.2.a.d 1
85.m even 8 1 7225.2.a.g 1
85.n odd 8 1 425.2.b.b 2
119.l odd 8 1 833.2.a.a 1
119.q even 24 2 833.2.e.b 2
119.r odd 24 2 833.2.e.a 2
136.o even 8 1 1088.2.a.i 1
136.p odd 8 1 1088.2.a.h 1
187.i odd 8 1 2057.2.a.e 1
204.p even 8 1 2448.2.a.o 1
221.p even 8 1 2873.2.a.c 1
255.y odd 8 1 3825.2.a.d 1
323.l odd 8 1 6137.2.a.b 1
340.ba odd 8 1 6800.2.a.n 1
357.w even 8 1 7497.2.a.l 1
391.h odd 8 1 8993.2.a.a 1
408.bd even 8 1 9792.2.a.i 1
408.be odd 8 1 9792.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 17.d even 8 1
153.2.a.c 1 51.g odd 8 1
272.2.a.b 1 68.g odd 8 1
289.2.a.a 1 17.d even 8 1
289.2.b.a 2 17.d even 8 2
289.2.c.a 4 1.a even 1 1 trivial
289.2.c.a 4 17.b even 2 1 inner
289.2.c.a 4 17.c even 4 2 inner
289.2.d.d 8 17.e odd 16 8
425.2.a.d 1 85.m even 8 1
425.2.b.b 2 85.k odd 8 1
425.2.b.b 2 85.n odd 8 1
833.2.a.a 1 119.l odd 8 1
833.2.e.a 2 119.r odd 24 2
833.2.e.b 2 119.q even 24 2
1088.2.a.h 1 136.p odd 8 1
1088.2.a.i 1 136.o even 8 1
2057.2.a.e 1 187.i odd 8 1
2448.2.a.o 1 204.p even 8 1
2601.2.a.g 1 51.g odd 8 1
2873.2.a.c 1 221.p even 8 1
3825.2.a.d 1 255.y odd 8 1
4624.2.a.d 1 68.g odd 8 1
6137.2.a.b 1 323.l odd 8 1
6800.2.a.n 1 340.ba odd 8 1
7225.2.a.g 1 85.m even 8 1
7497.2.a.l 1 357.w even 8 1
8993.2.a.a 1 391.h odd 8 1
9792.2.a.i 1 408.bd even 8 1
9792.2.a.n 1 408.be odd 8 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 16 \) Copy content Toggle raw display
$7$ \( T^{4} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 256 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 1296 \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 10000 \) Copy content Toggle raw display
$67$ \( (T - 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 256 \) Copy content Toggle raw display
$73$ \( T^{4} + 1296 \) Copy content Toggle raw display
$79$ \( T^{4} + 20736 \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 16 \) Copy content Toggle raw display
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