# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} -2 q^{6} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + 2 \zeta_{12} q^{2} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{2} q^{4} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} -2 q^{6} + ( -2 + 2 \zeta_{12}^{2} ) q^{9} + ( -4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{10} + 3 \zeta_{12}^{2} q^{11} -2 \zeta_{12} q^{12} -\zeta_{12}^{3} q^{13} + ( -1 - 2 \zeta_{12}^{3} ) q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{17} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{18} + ( -4 + 2 \zeta_{12}^{3} ) q^{20} + 6 \zeta_{12}^{3} q^{22} -6 \zeta_{12} q^{23} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + ( 2 - 2 \zeta_{12}^{2} ) q^{26} -5 \zeta_{12}^{3} q^{27} + 5 q^{29} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{30} + 2 \zeta_{12}^{2} q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -3 \zeta_{12} q^{33} + 14 q^{34} -4 q^{36} + 2 \zeta_{12} q^{37} + \zeta_{12}^{2} q^{39} -2 q^{41} -4 \zeta_{12}^{3} q^{43} + ( -6 + 6 \zeta_{12}^{2} ) q^{44} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{45} -12 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} + 4 \zeta_{12}^{3} q^{48} + ( -8 - 6 \zeta_{12}^{3} ) q^{50} + ( -7 + 7 \zeta_{12}^{2} ) q^{51} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( 10 - 10 \zeta_{12}^{2} ) q^{54} + ( -6 + 3 \zeta_{12}^{3} ) q^{55} + 10 \zeta_{12} q^{58} -10 \zeta_{12}^{2} q^{59} + ( 4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{60} + ( -8 + 8 \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} + 8 q^{64} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} ) q^{65} -6 \zeta_{12}^{2} q^{66} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{67} + 14 \zeta_{12} q^{68} + 6 q^{69} -8 q^{71} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{73} + 4 \zeta_{12}^{2} q^{74} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{75} + 2 \zeta_{12}^{3} q^{78} + ( -5 + 5 \zeta_{12}^{2} ) q^{79} + ( 4 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{80} -\zeta_{12}^{2} q^{81} -4 \zeta_{12} q^{82} + 4 \zeta_{12}^{3} q^{83} + ( 7 + 14 \zeta_{12}^{3} ) q^{85} + ( 8 - 8 \zeta_{12}^{2} ) q^{86} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{87} + ( -4 - 8 \zeta_{12}^{3} ) q^{90} -12 \zeta_{12}^{3} q^{92} -2 \zeta_{12} q^{93} + 6 \zeta_{12}^{2} q^{94} + ( -8 + 8 \zeta_{12}^{2} ) q^{96} + 7 \zeta_{12}^{3} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{9} + 4 q^{10} + 6 q^{11} - 4 q^{15} + 8 q^{16} - 16 q^{20} - 6 q^{25} + 4 q^{26} + 20 q^{29} + 8 q^{30} + 4 q^{31} + 56 q^{34} - 16 q^{36} + 2 q^{39} - 8 q^{41} - 12 q^{44} - 8 q^{45} - 24 q^{46} - 32 q^{50} - 14 q^{51} + 20 q^{54} - 24 q^{55} - 20 q^{59} - 4 q^{60} - 16 q^{61} + 32 q^{64} + 2 q^{65} - 12 q^{66} + 24 q^{69} - 32 q^{71} + 8 q^{74} + 8 q^{75} - 10 q^{79} + 16 q^{80} - 2 q^{81} + 28 q^{85} + 16 q^{86} - 16 q^{90} + 12 q^{94} - 16 q^{96} - 24 q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$101$$ $$197$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.73205 1.00000i 0.866025 0.500000i 1.00000 + 1.73205i −1.86603 + 1.23205i −2.00000 0 0 −1.00000 + 1.73205i 4.46410 0.267949i
79.2 1.73205 + 1.00000i −0.866025 + 0.500000i 1.00000 + 1.73205i −0.133975 + 2.23205i −2.00000 0 0 −1.00000 + 1.73205i −2.46410 + 3.73205i
214.1 −1.73205 + 1.00000i 0.866025 + 0.500000i 1.00000 1.73205i −1.86603 1.23205i −2.00000 0 0 −1.00000 1.73205i 4.46410 + 0.267949i
214.2 1.73205 1.00000i −0.866025 0.500000i 1.00000 1.73205i −0.133975 2.23205i −2.00000 0 0 −1.00000 1.73205i −2.46410 3.73205i
 $$n$$: 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.j.d 4
5.b even 2 1 inner 245.2.j.d 4
7.b odd 2 1 245.2.j.e 4
7.c even 3 1 245.2.b.a 2
7.c even 3 1 inner 245.2.j.d 4
7.d odd 6 1 35.2.b.a 2
7.d odd 6 1 245.2.j.e 4
21.g even 6 1 315.2.d.a 2
21.h odd 6 1 2205.2.d.b 2
28.f even 6 1 560.2.g.b 2
35.c odd 2 1 245.2.j.e 4
35.i odd 6 1 35.2.b.a 2
35.i odd 6 1 245.2.j.e 4
35.j even 6 1 245.2.b.a 2
35.j even 6 1 inner 245.2.j.d 4
35.k even 12 1 175.2.a.a 1
35.k even 12 1 175.2.a.c 1
35.l odd 12 1 1225.2.a.a 1
35.l odd 12 1 1225.2.a.i 1
56.j odd 6 1 2240.2.g.h 2
56.m even 6 1 2240.2.g.g 2
84.j odd 6 1 5040.2.t.p 2
105.o odd 6 1 2205.2.d.b 2
105.p even 6 1 315.2.d.a 2
105.w odd 12 1 1575.2.a.a 1
105.w odd 12 1 1575.2.a.k 1
140.s even 6 1 560.2.g.b 2
140.x odd 12 1 2800.2.a.l 1
140.x odd 12 1 2800.2.a.w 1
280.ba even 6 1 2240.2.g.g 2
280.bk odd 6 1 2240.2.g.h 2
420.be odd 6 1 5040.2.t.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 7.d odd 6 1
35.2.b.a 2 35.i odd 6 1
175.2.a.a 1 35.k even 12 1
175.2.a.c 1 35.k even 12 1
245.2.b.a 2 7.c even 3 1
245.2.b.a 2 35.j even 6 1
245.2.j.d 4 1.a even 1 1 trivial
245.2.j.d 4 5.b even 2 1 inner
245.2.j.d 4 7.c even 3 1 inner
245.2.j.d 4 35.j even 6 1 inner
245.2.j.e 4 7.b odd 2 1
245.2.j.e 4 7.d odd 6 1
245.2.j.e 4 35.c odd 2 1
245.2.j.e 4 35.i odd 6 1
315.2.d.a 2 21.g even 6 1
315.2.d.a 2 105.p even 6 1
560.2.g.b 2 28.f even 6 1
560.2.g.b 2 140.s even 6 1
1225.2.a.a 1 35.l odd 12 1
1225.2.a.i 1 35.l odd 12 1
1575.2.a.a 1 105.w odd 12 1
1575.2.a.k 1 105.w odd 12 1
2205.2.d.b 2 21.h odd 6 1
2205.2.d.b 2 105.o odd 6 1
2240.2.g.g 2 56.m even 6 1
2240.2.g.g 2 280.ba even 6 1
2240.2.g.h 2 56.j odd 6 1
2240.2.g.h 2 280.bk odd 6 1
2800.2.a.l 1 140.x odd 12 1
2800.2.a.w 1 140.x odd 12 1
5040.2.t.p 2 84.j odd 6 1
5040.2.t.p 2 420.be odd 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 4 T_{2}^{2} + 16$$ $$T_{3}^{4} - T_{3}^{2} + 1$$ $$T_{31}^{2} - 2 T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 9 - 3 T + T^{2} )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$2401 - 49 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$1296 - 36 T^{2} + T^{4}$$
$29$ $$( -5 + T )^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$16 - 4 T^{2} + T^{4}$$
$41$ $$( 2 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$81 - 9 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 100 + 10 T + T^{2} )^{2}$$
$61$ $$( 64 + 8 T + T^{2} )^{2}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$1296 - 36 T^{2} + T^{4}$$
$79$ $$( 25 + 5 T + T^{2} )^{2}$$
$83$ $$( 16 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 49 + T^{2} )^{2}$$