# 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 $4$

# Related objects

## 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: $$x^{4} + 1$$ x^4 + 1 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + q^{4} + \beta_1 q^{5} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{8} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + q^4 + b1 * q^5 + 2*b3 * q^7 + 3*b2 * q^8 - 3*b2 * q^9 $$q + \beta_{2} q^{2} + q^{4} + \beta_1 q^{5} + 2 \beta_{3} q^{7} + 3 \beta_{2} q^{8} - 3 \beta_{2} q^{9} + \beta_{3} q^{10} + 2 q^{13} - 2 \beta_1 q^{14} - q^{16} + 3 q^{18} + 4 \beta_{2} q^{19} + \beta_1 q^{20} - 2 \beta_{3} q^{23} - \beta_{2} q^{25} + 2 \beta_{2} q^{26} + 2 \beta_{3} q^{28} - 3 \beta_1 q^{29} + 2 \beta_1 q^{31} + 5 \beta_{2} q^{32} - 8 q^{35} - 3 \beta_{2} q^{36} - \beta_1 q^{37} - 4 q^{38} + 3 \beta_{3} q^{40} - 3 \beta_{3} q^{41} + 4 \beta_{2} q^{43} - 3 \beta_{3} q^{45} + 2 \beta_1 q^{46} - 9 \beta_{2} q^{49} + q^{50} + 2 q^{52} - 6 \beta_{2} q^{53} - 6 \beta_1 q^{56} - 3 \beta_{3} q^{58} - 12 \beta_{2} q^{59} - 5 \beta_{3} q^{61} + 2 \beta_{3} q^{62} + 6 \beta_1 q^{63} - 7 q^{64} + 2 \beta_1 q^{65} + 4 q^{67} - 8 \beta_{2} q^{70} - 2 \beta_1 q^{71} + 9 q^{72} + 3 \beta_1 q^{73} - \beta_{3} q^{74} + 4 \beta_{2} q^{76} - 6 \beta_{3} q^{79} - \beta_1 q^{80} - 9 q^{81} + 3 \beta_1 q^{82} + 4 \beta_{2} q^{83} - 4 q^{86} - 10 q^{89} + 3 \beta_1 q^{90} + 4 \beta_{3} q^{91} - 2 \beta_{3} q^{92} + 4 \beta_{3} q^{95} - \beta_1 q^{97} + 9 q^{98}+O(q^{100})$$ q + b2 * q^2 + q^4 + b1 * q^5 + 2*b3 * q^7 + 3*b2 * q^8 - 3*b2 * q^9 + b3 * q^10 + 2 * q^13 - 2*b1 * q^14 - q^16 + 3 * q^18 + 4*b2 * q^19 + b1 * q^20 - 2*b3 * q^23 - b2 * q^25 + 2*b2 * q^26 + 2*b3 * q^28 - 3*b1 * q^29 + 2*b1 * q^31 + 5*b2 * q^32 - 8 * q^35 - 3*b2 * q^36 - b1 * q^37 - 4 * q^38 + 3*b3 * q^40 - 3*b3 * q^41 + 4*b2 * q^43 - 3*b3 * q^45 + 2*b1 * q^46 - 9*b2 * q^49 + q^50 + 2 * q^52 - 6*b2 * q^53 - 6*b1 * q^56 - 3*b3 * q^58 - 12*b2 * q^59 - 5*b3 * q^61 + 2*b3 * q^62 + 6*b1 * q^63 - 7 * q^64 + 2*b1 * q^65 + 4 * q^67 - 8*b2 * q^70 - 2*b1 * q^71 + 9 * q^72 + 3*b1 * q^73 - b3 * q^74 + 4*b2 * q^76 - 6*b3 * q^79 - b1 * q^80 - 9 * q^81 + 3*b1 * q^82 + 4*b2 * q^83 - 4 * q^86 - 10 * q^89 + 3*b1 * q^90 + 4*b3 * q^91 - 2*b3 * q^92 + 4*b3 * q^95 - b1 * q^97 + 9 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4}+O(q^{10})$$ 4 * q + 4 * q^4 $$4 q + 4 q^{4} + 8 q^{13} - 4 q^{16} + 12 q^{18} - 32 q^{35} - 16 q^{38} + 4 q^{50} + 8 q^{52} - 28 q^{64} + 16 q^{67} + 36 q^{72} - 36 q^{81} - 16 q^{86} - 40 q^{89} + 36 q^{98}+O(q^{100})$$ 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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3}$$ 2*v^3
 $$\zeta_{8}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(289, [\chi])$$.

## Hecke characteristic polynomials

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