# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 6 q^{6} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( -2 + 2 \zeta_{6} ) q^{4} -\zeta_{6} q^{5} + 6 q^{6} -6 \zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{10} + ( -1 + \zeta_{6} ) q^{11} + 6 \zeta_{6} q^{12} -3 q^{13} -3 q^{15} + 4 \zeta_{6} q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + ( 12 - 12 \zeta_{6} ) q^{18} + 6 \zeta_{6} q^{19} + 2 q^{20} -2 q^{22} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -6 \zeta_{6} q^{26} -9 q^{27} - q^{29} -6 \zeta_{6} q^{30} + ( 6 - 6 \zeta_{6} ) q^{31} + ( -8 + 8 \zeta_{6} ) q^{32} + 3 \zeta_{6} q^{33} -6 q^{34} + 12 q^{36} + ( -12 + 12 \zeta_{6} ) q^{38} + ( -9 + 9 \zeta_{6} ) q^{39} -6 q^{41} -6 q^{43} -2 \zeta_{6} q^{44} + ( -6 + 6 \zeta_{6} ) q^{45} + ( -8 + 8 \zeta_{6} ) q^{46} -9 \zeta_{6} q^{47} + 12 q^{48} -2 q^{50} + 9 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + ( 10 - 10 \zeta_{6} ) q^{53} -18 \zeta_{6} q^{54} + q^{55} + 18 q^{57} -2 \zeta_{6} q^{58} + ( -6 + 6 \zeta_{6} ) q^{59} + ( 6 - 6 \zeta_{6} ) q^{60} + 12 q^{62} -8 q^{64} + 3 \zeta_{6} q^{65} + ( -6 + 6 \zeta_{6} ) q^{66} + ( 14 - 14 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + 12 q^{69} -8 q^{71} + ( 6 - 6 \zeta_{6} ) q^{73} + 3 \zeta_{6} q^{75} -12 q^{76} -18 q^{78} + \zeta_{6} q^{79} + ( 4 - 4 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -12 \zeta_{6} q^{82} -12 q^{83} + 3 q^{85} -12 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + 12 \zeta_{6} q^{89} -12 q^{90} -8 q^{92} -18 \zeta_{6} q^{93} + ( 18 - 18 \zeta_{6} ) q^{94} + ( 6 - 6 \zeta_{6} ) q^{95} + 24 \zeta_{6} q^{96} + 15 q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} - 2q^{4} - q^{5} + 12q^{6} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} - 2q^{4} - q^{5} + 12q^{6} - 6q^{9} + 2q^{10} - q^{11} + 6q^{12} - 6q^{13} - 6q^{15} + 4q^{16} - 3q^{17} + 12q^{18} + 6q^{19} + 4q^{20} - 4q^{22} + 4q^{23} - q^{25} - 6q^{26} - 18q^{27} - 2q^{29} - 6q^{30} + 6q^{31} - 8q^{32} + 3q^{33} - 12q^{34} + 24q^{36} - 12q^{38} - 9q^{39} - 12q^{41} - 12q^{43} - 2q^{44} - 6q^{45} - 8q^{46} - 9q^{47} + 24q^{48} - 4q^{50} + 9q^{51} + 6q^{52} + 10q^{53} - 18q^{54} + 2q^{55} + 36q^{57} - 2q^{58} - 6q^{59} + 6q^{60} + 24q^{62} - 16q^{64} + 3q^{65} - 6q^{66} + 14q^{67} - 6q^{68} + 24q^{69} - 16q^{71} + 6q^{73} + 3q^{75} - 24q^{76} - 36q^{78} + q^{79} + 4q^{80} - 9q^{81} - 12q^{82} - 24q^{83} + 6q^{85} - 12q^{86} - 3q^{87} + 12q^{89} - 24q^{90} - 16q^{92} - 18q^{93} + 18q^{94} + 6q^{95} + 24q^{96} + 30q^{97} + 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 + 1.73205i 1.50000 2.59808i −1.00000 + 1.73205i −0.500000 0.866025i 6.00000 0 0 −3.00000 5.19615i 1.00000 1.73205i
226.1 1.00000 1.73205i 1.50000 + 2.59808i −1.00000 1.73205i −0.500000 + 0.866025i 6.00000 0 0 −3.00000 + 5.19615i 1.00000 + 1.73205i
 $$n$$: 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.d 2
7.b odd 2 1 245.2.e.c 2
7.c even 3 1 245.2.a.a 1
7.c even 3 1 inner 245.2.e.d 2
7.d odd 6 1 245.2.a.b yes 1
7.d odd 6 1 245.2.e.c 2
21.g even 6 1 2205.2.a.l 1
21.h odd 6 1 2205.2.a.j 1
28.f even 6 1 3920.2.a.a 1
28.g odd 6 1 3920.2.a.bj 1
35.i odd 6 1 1225.2.a.h 1
35.j even 6 1 1225.2.a.j 1
35.k even 12 2 1225.2.b.a 2
35.l odd 12 2 1225.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 7.c even 3 1
245.2.a.b yes 1 7.d odd 6 1
245.2.e.c 2 7.b odd 2 1
245.2.e.c 2 7.d odd 6 1
245.2.e.d 2 1.a even 1 1 trivial
245.2.e.d 2 7.c even 3 1 inner
1225.2.a.h 1 35.i odd 6 1
1225.2.a.j 1 35.j even 6 1
1225.2.b.a 2 35.k even 12 2
1225.2.b.b 2 35.l odd 12 2
2205.2.a.j 1 21.h odd 6 1
2205.2.a.l 1 21.g even 6 1
3920.2.a.a 1 28.f even 6 1
3920.2.a.bj 1 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 $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{2} - 2 T_{2} + 4$$ $$T_{3}^{2} - 3 T_{3} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T + T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$( 1 + T )^{2}$$
$31$ $$36 - 6 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$100 - 10 T + T^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$196 - 14 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$144 - 12 T + T^{2}$$
$97$ $$( -15 + T )^{2}$$
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