# Properties

 Label 245.2.e.f Level $245$ Weight $2$ Character orbit 245.e Analytic conductor $1.956$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 + \beta_{3} ) q^{6} -2 \beta_{3} q^{8} + 2 \beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -1 - \beta_{2} ) q^{5} + ( 2 + \beta_{3} ) q^{6} -2 \beta_{3} q^{8} + 2 \beta_{1} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{10} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{11} + ( 3 - \beta_{3} ) q^{13} + ( 1 + \beta_{3} ) q^{15} + ( 4 + 4 \beta_{2} ) q^{16} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{17} + 4 \beta_{2} q^{18} + ( -6 - 6 \beta_{2} ) q^{19} + ( -4 - 3 \beta_{3} ) q^{22} + ( -6 - \beta_{1} - 6 \beta_{2} ) q^{23} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{24} + \beta_{2} q^{25} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{26} + ( 1 - \beta_{3} ) q^{27} + ( -3 + 4 \beta_{3} ) q^{29} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{30} + ( 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{31} + ( 7 - 5 \beta_{1} + 7 \beta_{2} ) q^{33} + ( -6 - \beta_{3} ) q^{34} + ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{38} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{39} -2 \beta_{1} q^{40} + ( -2 + 3 \beta_{3} ) q^{41} + 2 q^{43} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{45} + ( -6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{46} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{47} + ( -4 - 4 \beta_{3} ) q^{48} + \beta_{3} q^{50} + ( 7 - 4 \beta_{1} + 7 \beta_{2} ) q^{51} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{53} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{54} + ( -3 - 2 \beta_{3} ) q^{55} + ( 6 + 6 \beta_{3} ) q^{57} + ( -8 - 3 \beta_{1} - 8 \beta_{2} ) q^{58} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{59} + 2 \beta_{1} q^{61} + ( -6 + 6 \beta_{3} ) q^{62} + 8 q^{64} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{65} + ( 7 \beta_{1} - 10 \beta_{2} + 7 \beta_{3} ) q^{66} + ( -3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 4 + 5 \beta_{3} ) q^{69} + ( -6 - 2 \beta_{3} ) q^{71} + ( 8 + 8 \beta_{2} ) q^{72} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{73} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} ) q^{75} + ( 4 + \beta_{3} ) q^{78} + ( 7 - 6 \beta_{1} + 7 \beta_{2} ) q^{79} -4 \beta_{2} q^{80} + ( 6 \beta_{1} - \beta_{2} + 6 \beta_{3} ) q^{81} + ( -6 - 2 \beta_{1} - 6 \beta_{2} ) q^{82} + ( -1 - 3 \beta_{3} ) q^{85} + 2 \beta_{1} q^{86} + ( -\beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{87} + ( -6 \beta_{1} + 8 \beta_{2} - 6 \beta_{3} ) q^{88} + ( 8 + 8 \beta_{2} ) q^{89} + 4 q^{90} + 3 \beta_{1} q^{93} + ( 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{94} + 6 \beta_{2} q^{95} + ( 9 - 3 \beta_{3} ) q^{97} + ( -8 - 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} + 8 q^{6} + O(q^{10})$$ $$4 q - 2 q^{3} - 2 q^{5} + 8 q^{6} + 6 q^{11} + 12 q^{13} + 4 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} - 12 q^{19} - 16 q^{22} - 12 q^{23} + 8 q^{24} - 2 q^{25} + 4 q^{26} + 4 q^{27} - 12 q^{29} - 4 q^{30} - 12 q^{31} + 14 q^{33} - 24 q^{34} + 4 q^{37} - 2 q^{39} - 8 q^{41} + 8 q^{43} + 4 q^{46} + 6 q^{47} - 16 q^{48} + 14 q^{51} + 4 q^{54} - 12 q^{55} + 24 q^{57} - 16 q^{58} - 4 q^{59} - 24 q^{62} + 32 q^{64} - 6 q^{65} + 20 q^{66} + 8 q^{67} + 16 q^{69} - 24 q^{71} + 16 q^{72} + 12 q^{74} - 2 q^{75} + 16 q^{78} + 14 q^{79} + 8 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{85} - 10 q^{87} - 16 q^{88} + 16 q^{89} + 16 q^{90} - 12 q^{94} - 12 q^{95} + 36 q^{97} - 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 1.22474i −1.20711 + 2.09077i 0 −0.500000 0.866025i 3.41421 0 −2.82843 −1.41421 2.44949i −0.707107 + 1.22474i
116.2 0.707107 + 1.22474i 0.207107 0.358719i 0 −0.500000 0.866025i 0.585786 0 2.82843 1.41421 + 2.44949i 0.707107 1.22474i
226.1 −0.707107 + 1.22474i −1.20711 2.09077i 0 −0.500000 + 0.866025i 3.41421 0 −2.82843 −1.41421 + 2.44949i −0.707107 1.22474i
226.2 0.707107 1.22474i 0.207107 + 0.358719i 0 −0.500000 + 0.866025i 0.585786 0 2.82843 1.41421 2.44949i 0.707107 + 1.22474i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.e.f 4
7.b odd 2 1 245.2.e.g 4
7.c even 3 1 245.2.a.f yes 2
7.c even 3 1 inner 245.2.e.f 4
7.d odd 6 1 245.2.a.e 2
7.d odd 6 1 245.2.e.g 4
21.g even 6 1 2205.2.a.v 2
21.h odd 6 1 2205.2.a.t 2
28.f even 6 1 3920.2.a.bw 2
28.g odd 6 1 3920.2.a.br 2
35.i odd 6 1 1225.2.a.r 2
35.j even 6 1 1225.2.a.p 2
35.k even 12 2 1225.2.b.i 4
35.l odd 12 2 1225.2.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 7.d odd 6 1
245.2.a.f yes 2 7.c even 3 1
245.2.e.f 4 1.a even 1 1 trivial
245.2.e.f 4 7.c even 3 1 inner
245.2.e.g 4 7.b odd 2 1
245.2.e.g 4 7.d odd 6 1
1225.2.a.p 2 35.j even 6 1
1225.2.a.r 2 35.i odd 6 1
1225.2.b.i 4 35.k even 12 2
1225.2.b.j 4 35.l odd 12 2
2205.2.a.t 2 21.h odd 6 1
2205.2.a.v 2 21.g even 6 1
3920.2.a.br 2 28.g odd 6 1
3920.2.a.bw 2 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} + 2 T_{2}^{2} + 4$$ $$T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$( 7 - 6 T + T^{2} )^{2}$$
$17$ $$289 + 34 T + 21 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$( 36 + 6 T + T^{2} )^{2}$$
$23$ $$1156 + 408 T + 110 T^{2} + 12 T^{3} + T^{4}$$
$29$ $$( -23 + 6 T + T^{2} )^{2}$$
$31$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$196 + 56 T + 30 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$( -14 + 4 T + T^{2} )^{2}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$81 + 54 T + 45 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$324 + 18 T^{2} + T^{4}$$
$59$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$64 + 8 T^{2} + T^{4}$$
$67$ $$4 + 16 T + 66 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( 28 + 12 T + T^{2} )^{2}$$
$73$ $$5184 + 72 T^{2} + T^{4}$$
$79$ $$529 + 322 T + 219 T^{2} - 14 T^{3} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 64 - 8 T + T^{2} )^{2}$$
$97$ $$( 63 - 18 T + T^{2} )^{2}$$