# Properties

 Label 245.2.e.e 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{6} + ( 3 + \beta_{3} ) q^{8} -2 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{4} + ( 1 + \beta_{2} ) q^{5} + q^{6} + ( 3 + \beta_{3} ) q^{8} -2 \beta_{1} q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{12} + ( 2 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{3} ) q^{15} + ( -3 - 3 \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} -2 \beta_{1} q^{19} + ( -1 - 2 \beta_{3} ) q^{20} -2 q^{22} + ( 1 + \beta_{1} + \beta_{2} ) q^{23} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{24} + \beta_{2} q^{25} + ( -6 + 4 \beta_{1} - 6 \beta_{2} ) q^{26} + ( -1 - \beta_{3} ) q^{27} - q^{29} + ( 1 + \beta_{2} ) q^{30} + 6 \beta_{2} q^{31} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{32} + ( 6 + 4 \beta_{1} + 6 \beta_{2} ) q^{33} + ( -6 - 4 \beta_{3} ) q^{34} + ( -8 - 2 \beta_{3} ) q^{36} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{38} + 2 \beta_{2} q^{39} + ( 3 - \beta_{1} + 3 \beta_{2} ) q^{40} + ( 5 + 2 \beta_{3} ) q^{41} + ( 5 - \beta_{3} ) q^{43} + ( 6 + 2 \beta_{1} + 6 \beta_{2} ) q^{44} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{45} + \beta_{2} q^{46} + ( 2 + 2 \beta_{2} ) q^{47} + ( -3 + 3 \beta_{3} ) q^{48} + ( 1 + \beta_{3} ) q^{50} + ( 2 + 2 \beta_{2} ) q^{51} + ( -6 \beta_{1} + 10 \beta_{2} - 6 \beta_{3} ) q^{52} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{54} + ( -2 + 2 \beta_{3} ) q^{55} + ( -4 + 2 \beta_{3} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} ) q^{58} + ( 6 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{59} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{60} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{61} + ( 6 + 6 \beta_{3} ) q^{62} + ( -7 - 2 \beta_{3} ) q^{64} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( \beta_{1} + 11 \beta_{2} + \beta_{3} ) q^{67} + ( 10 - 6 \beta_{1} + 10 \beta_{2} ) q^{68} + ( 3 - 2 \beta_{3} ) q^{69} + ( -4 + 6 \beta_{3} ) q^{71} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{72} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 1 + \beta_{1} + \beta_{2} ) q^{75} + ( -8 - 2 \beta_{3} ) q^{76} + ( 2 + 2 \beta_{3} ) q^{78} + ( -12 + 2 \beta_{1} - 12 \beta_{2} ) q^{79} -3 \beta_{2} q^{80} + ( -6 \beta_{1} - \beta_{2} - 6 \beta_{3} ) q^{81} + ( -9 + 7 \beta_{1} - 9 \beta_{2} ) q^{82} + ( -1 - 9 \beta_{3} ) q^{83} + ( 2 + 2 \beta_{3} ) q^{85} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{88} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{89} + ( 4 + 2 \beta_{3} ) q^{90} + ( 3 - \beta_{3} ) q^{92} + ( 6 + 6 \beta_{1} + 6 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -5 - 4 \beta_{1} - 5 \beta_{2} ) q^{96} + ( -6 - 4 \beta_{3} ) q^{97} + ( 8 - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} + 4q^{6} + 12q^{8} + O(q^{10})$$ $$4q - 2q^{2} + 2q^{3} - 2q^{4} + 2q^{5} + 4q^{6} + 12q^{8} + 2q^{10} - 4q^{11} - 6q^{12} + 8q^{13} + 4q^{15} - 6q^{16} + 4q^{17} + 8q^{18} - 4q^{20} - 8q^{22} + 2q^{23} + 2q^{24} - 2q^{25} - 12q^{26} - 4q^{27} - 4q^{29} + 2q^{30} - 12q^{31} + 6q^{32} + 12q^{33} - 24q^{34} - 32q^{36} + 8q^{38} - 4q^{39} + 6q^{40} + 20q^{41} + 20q^{43} + 12q^{44} - 2q^{46} + 4q^{47} - 12q^{48} + 4q^{50} + 4q^{51} - 20q^{52} + 8q^{53} + 6q^{54} - 8q^{55} - 16q^{57} + 2q^{58} - 8q^{59} + 6q^{60} - 6q^{61} + 24q^{62} - 28q^{64} + 4q^{65} - 4q^{66} - 22q^{67} + 20q^{68} + 12q^{69} - 16q^{71} + 8q^{72} + 4q^{73} + 2q^{75} - 32q^{76} + 8q^{78} - 24q^{79} + 6q^{80} + 2q^{81} - 18q^{82} - 4q^{83} + 8q^{85} - 6q^{86} - 2q^{87} - 4q^{88} - 6q^{89} + 16q^{90} + 12q^{92} + 12q^{93} + 4q^{94} - 10q^{96} - 24q^{97} + 32q^{99} + O(q^{100})$$

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
−1.20711 2.09077i −0.207107 + 0.358719i −1.91421 + 3.31552i 0.500000 + 0.866025i 1.00000 0 4.41421 1.41421 + 2.44949i 1.20711 2.09077i
116.2 0.207107 + 0.358719i 1.20711 2.09077i 0.914214 1.58346i 0.500000 + 0.866025i 1.00000 0 1.58579 −1.41421 2.44949i −0.207107 + 0.358719i
226.1 −1.20711 + 2.09077i −0.207107 0.358719i −1.91421 3.31552i 0.500000 0.866025i 1.00000 0 4.41421 1.41421 2.44949i 1.20711 + 2.09077i
226.2 0.207107 0.358719i 1.20711 + 2.09077i 0.914214 + 1.58346i 0.500000 0.866025i 1.00000 0 1.58579 −1.41421 + 2.44949i −0.207107 0.358719i
 $$n$$: 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.e 4
7.b odd 2 1 35.2.e.a 4
7.c even 3 1 245.2.a.g 2
7.c even 3 1 inner 245.2.e.e 4
7.d odd 6 1 35.2.e.a 4
7.d odd 6 1 245.2.a.h 2
21.c even 2 1 315.2.j.e 4
21.g even 6 1 315.2.j.e 4
21.g even 6 1 2205.2.a.n 2
21.h odd 6 1 2205.2.a.q 2
28.d even 2 1 560.2.q.k 4
28.f even 6 1 560.2.q.k 4
28.f even 6 1 3920.2.a.bq 2
28.g odd 6 1 3920.2.a.bv 2
35.c odd 2 1 175.2.e.c 4
35.f even 4 2 175.2.k.a 8
35.i odd 6 1 175.2.e.c 4
35.i odd 6 1 1225.2.a.k 2
35.j even 6 1 1225.2.a.m 2
35.k even 12 2 175.2.k.a 8
35.k even 12 2 1225.2.b.g 4
35.l odd 12 2 1225.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 7.b odd 2 1
35.2.e.a 4 7.d odd 6 1
175.2.e.c 4 35.c odd 2 1
175.2.e.c 4 35.i odd 6 1
175.2.k.a 8 35.f even 4 2
175.2.k.a 8 35.k even 12 2
245.2.a.g 2 7.c even 3 1
245.2.a.h 2 7.d odd 6 1
245.2.e.e 4 1.a even 1 1 trivial
245.2.e.e 4 7.c even 3 1 inner
315.2.j.e 4 21.c even 2 1
315.2.j.e 4 21.g even 6 1
560.2.q.k 4 28.d even 2 1
560.2.q.k 4 28.f even 6 1
1225.2.a.k 2 35.i odd 6 1
1225.2.a.m 2 35.j even 6 1
1225.2.b.g 4 35.k even 12 2
1225.2.b.h 4 35.l odd 12 2
2205.2.a.n 2 21.g even 6 1
2205.2.a.q 2 21.h odd 6 1
3920.2.a.bq 2 28.f even 6 1
3920.2.a.bv 2 28.g odd 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$3$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( -4 - 4 T + T^{2} )^{2}$$
$17$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$64 + 8 T^{2} + T^{4}$$
$23$ $$1 + 2 T + 5 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$( 36 + 6 T + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 17 - 10 T + T^{2} )^{2}$$
$43$ $$( 23 - 10 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$59$ $$3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$3969 - 378 T + 99 T^{2} + 6 T^{3} + T^{4}$$
$67$ $$14161 + 2618 T + 365 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$( -56 + 8 T + T^{2} )^{2}$$
$73$ $$16 + 16 T + 20 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4}$$
$83$ $$( -161 + 2 T + T^{2} )^{2}$$
$89$ $$529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 4 + 12 T + T^{2} )^{2}$$