Properties

Label 289.2.d.d
Level $289$
Weight $2$
Character orbit 289.d
Analytic conductor $2.308$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 289.d (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.30767661842\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{4} q^{4} - \beta_{7} q^{5} - 2 \beta_1 q^{7} - 3 \beta_{6} q^{8} - 3 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{4} q^{4} - \beta_{7} q^{5} - 2 \beta_1 q^{7} - 3 \beta_{6} q^{8} - 3 \beta_{6} q^{9} + \beta_1 q^{10} + 2 \beta_{4} q^{13} - 2 \beta_{3} q^{14} + q^{16} + 3 q^{18} + 4 \beta_{2} q^{19} - \beta_{3} q^{20} + 2 \beta_{5} q^{23} + \beta_{6} q^{25} + 2 \beta_{6} q^{26} + 2 \beta_{5} q^{28} - 3 \beta_{7} q^{29} + 2 \beta_{3} q^{31} - 5 \beta_{2} q^{32} - 8 q^{35} - 3 \beta_{2} q^{36} + \beta_{3} q^{37} + 4 \beta_{4} q^{38} - 3 \beta_{5} q^{40} + 3 \beta_1 q^{41} + 4 \beta_{6} q^{43} - 3 \beta_{5} q^{45} + 2 \beta_{7} q^{46} + 9 \beta_{2} q^{49} - q^{50} + 2 q^{52} - 6 \beta_{2} q^{53} + 6 \beta_{7} q^{56} + 3 \beta_1 q^{58} + 12 \beta_{6} q^{59} - 5 \beta_1 q^{61} + 2 \beta_{5} q^{62} + 6 \beta_{7} q^{63} - 7 \beta_{4} q^{64} + 2 \beta_{3} q^{65} - 4 q^{67} - 8 \beta_{2} q^{70} + 2 \beta_{3} q^{71} - 9 \beta_{4} q^{72} - 3 \beta_{7} q^{73} + \beta_{5} q^{74} - 4 \beta_{6} q^{76} - 6 \beta_{5} q^{79} - \beta_{7} q^{80} - 9 \beta_{4} q^{81} + 3 \beta_{3} q^{82} - 4 \beta_{2} q^{83} - 4 q^{86} + 10 \beta_{4} q^{89} - 3 \beta_{7} q^{90} - 4 \beta_{5} q^{91} + 2 \beta_1 q^{92} + 4 \beta_1 q^{95} - \beta_{7} q^{97} + 9 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{16} + 24 q^{18} - 64 q^{35} - 8 q^{50} + 16 q^{52} - 32 q^{67} - 32 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{16}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{6} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{16}^{7} \) Copy content Toggle raw display
\(\zeta_{16}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( \beta_{7} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
110.1
−0.923880 + 0.382683i
0.923880 0.382683i
−0.923880 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
0.707107 0.707107i 0 1.00000i −1.84776 0.765367i 0 3.69552 1.53073i 2.12132 + 2.12132i 2.12132 + 2.12132i −1.84776 + 0.765367i
110.2 0.707107 0.707107i 0 1.00000i 1.84776 + 0.765367i 0 −3.69552 + 1.53073i 2.12132 + 2.12132i 2.12132 + 2.12132i 1.84776 0.765367i
134.1 0.707107 + 0.707107i 0 1.00000i −1.84776 + 0.765367i 0 3.69552 + 1.53073i 2.12132 2.12132i 2.12132 2.12132i −1.84776 0.765367i
134.2 0.707107 + 0.707107i 0 1.00000i 1.84776 0.765367i 0 −3.69552 1.53073i 2.12132 2.12132i 2.12132 2.12132i 1.84776 + 0.765367i
155.1 −0.707107 0.707107i 0 1.00000i −0.765367 1.84776i 0 1.53073 3.69552i −2.12132 + 2.12132i −2.12132 + 2.12132i −0.765367 + 1.84776i
155.2 −0.707107 0.707107i 0 1.00000i 0.765367 + 1.84776i 0 −1.53073 + 3.69552i −2.12132 + 2.12132i −2.12132 + 2.12132i 0.765367 1.84776i
179.1 −0.707107 + 0.707107i 0 1.00000i −0.765367 + 1.84776i 0 1.53073 + 3.69552i −2.12132 2.12132i −2.12132 2.12132i −0.765367 1.84776i
179.2 −0.707107 + 0.707107i 0 1.00000i 0.765367 1.84776i 0 −1.53073 3.69552i −2.12132 2.12132i −2.12132 2.12132i 0.765367 + 1.84776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.2
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
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.2.d.d 8
17.b even 2 1 inner 289.2.d.d 8
17.c even 4 2 inner 289.2.d.d 8
17.d even 8 4 inner 289.2.d.d 8
17.e odd 16 1 17.2.a.a 1
17.e odd 16 1 289.2.a.a 1
17.e odd 16 2 289.2.b.a 2
17.e odd 16 4 289.2.c.a 4
51.i even 16 1 153.2.a.c 1
51.i even 16 1 2601.2.a.g 1
68.i even 16 1 272.2.a.b 1
68.i even 16 1 4624.2.a.d 1
85.o even 16 1 425.2.b.b 2
85.p odd 16 1 425.2.a.d 1
85.p odd 16 1 7225.2.a.g 1
85.r even 16 1 425.2.b.b 2
119.p even 16 1 833.2.a.a 1
119.s even 48 2 833.2.e.a 2
119.t odd 48 2 833.2.e.b 2
136.q odd 16 1 1088.2.a.i 1
136.s even 16 1 1088.2.a.h 1
187.m even 16 1 2057.2.a.e 1
204.t odd 16 1 2448.2.a.o 1
221.y odd 16 1 2873.2.a.c 1
255.be even 16 1 3825.2.a.d 1
323.p even 16 1 6137.2.a.b 1
340.bg even 16 1 6800.2.a.n 1
357.be odd 16 1 7497.2.a.l 1
391.k even 16 1 8993.2.a.a 1
408.bg odd 16 1 9792.2.a.i 1
408.bm even 16 1 9792.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 17.e odd 16 1
153.2.a.c 1 51.i even 16 1
272.2.a.b 1 68.i even 16 1
289.2.a.a 1 17.e odd 16 1
289.2.b.a 2 17.e odd 16 2
289.2.c.a 4 17.e odd 16 4
289.2.d.d 8 1.a even 1 1 trivial
289.2.d.d 8 17.b even 2 1 inner
289.2.d.d 8 17.c even 4 2 inner
289.2.d.d 8 17.d even 8 4 inner
425.2.a.d 1 85.p odd 16 1
425.2.b.b 2 85.o even 16 1
425.2.b.b 2 85.r even 16 1
833.2.a.a 1 119.p even 16 1
833.2.e.a 2 119.s even 48 2
833.2.e.b 2 119.t odd 48 2
1088.2.a.h 1 136.s even 16 1
1088.2.a.i 1 136.q odd 16 1
2057.2.a.e 1 187.m even 16 1
2448.2.a.o 1 204.t odd 16 1
2601.2.a.g 1 51.i even 16 1
2873.2.a.c 1 221.y odd 16 1
3825.2.a.d 1 255.be even 16 1
4624.2.a.d 1 68.i even 16 1
6137.2.a.b 1 323.p even 16 1
6800.2.a.n 1 340.bg even 16 1
7225.2.a.g 1 85.p odd 16 1
7497.2.a.l 1 357.be odd 16 1
8993.2.a.a 1 391.k even 16 1
9792.2.a.i 1 408.bg odd 16 1
9792.2.a.n 1 408.bm even 16 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + 65536 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 65536 \) Copy content Toggle raw display
$29$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$31$ \( T^{8} + 65536 \) Copy content Toggle raw display
$37$ \( T^{8} + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$43$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 100000000 \) Copy content Toggle raw display
$67$ \( (T + 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 65536 \) Copy content Toggle raw display
$73$ \( T^{8} + 1679616 \) Copy content Toggle raw display
$79$ \( T^{8} + 429981696 \) Copy content Toggle raw display
$83$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 256 \) Copy content Toggle raw display
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