# Properties

 Label 245.2.e.b Level $245$ Weight $2$ Character orbit 245.e Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + (-2*z + 2) * q^4 - z * q^5 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - 2 \zeta_{6} q^{12} - 5 q^{13} - q^{15} - 4 \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + 2 \zeta_{6} q^{19} - 2 q^{20} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 q^{27} + 3 q^{29} + (4 \zeta_{6} - 4) q^{31} - 3 \zeta_{6} q^{33} + 4 q^{36} - 2 \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{39} + 12 q^{41} - 10 q^{43} - 6 \zeta_{6} q^{44} + ( - 2 \zeta_{6} + 2) q^{45} + 9 \zeta_{6} q^{47} - 4 q^{48} - 3 \zeta_{6} q^{51} + (10 \zeta_{6} - 10) q^{52} + (12 \zeta_{6} - 12) q^{53} - 3 q^{55} + 2 q^{57} + (2 \zeta_{6} - 2) q^{60} + 8 \zeta_{6} q^{61} - 8 q^{64} + 5 \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 6 \zeta_{6} q^{68} + 6 q^{69} + ( - 2 \zeta_{6} + 2) q^{73} + \zeta_{6} q^{75} + 4 q^{76} + \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + (\zeta_{6} - 1) q^{81} - 12 q^{83} - 3 q^{85} + ( - 3 \zeta_{6} + 3) q^{87} - 12 \zeta_{6} q^{89} + 12 q^{92} + 4 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{95} + q^{97} + 6 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (-2*z + 2) * q^4 - z * q^5 + 2*z * q^9 + (-3*z + 3) * q^11 - 2*z * q^12 - 5 * q^13 - q^15 - 4*z * q^16 + (-3*z + 3) * q^17 + 2*z * q^19 - 2 * q^20 + 6*z * q^23 + (z - 1) * q^25 + 5 * q^27 + 3 * q^29 + (4*z - 4) * q^31 - 3*z * q^33 + 4 * q^36 - 2*z * q^37 + (5*z - 5) * q^39 + 12 * q^41 - 10 * q^43 - 6*z * q^44 + (-2*z + 2) * q^45 + 9*z * q^47 - 4 * q^48 - 3*z * q^51 + (10*z - 10) * q^52 + (12*z - 12) * q^53 - 3 * q^55 + 2 * q^57 + (2*z - 2) * q^60 + 8*z * q^61 - 8 * q^64 + 5*z * q^65 + (-4*z + 4) * q^67 - 6*z * q^68 + 6 * q^69 + (-2*z + 2) * q^73 + z * q^75 + 4 * q^76 + z * q^79 + (4*z - 4) * q^80 + (z - 1) * q^81 - 12 * q^83 - 3 * q^85 + (-3*z + 3) * q^87 - 12*z * q^89 + 12 * q^92 + 4*z * q^93 + (-2*z + 2) * q^95 + q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{4} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^4 - q^5 + 2 * q^9 $$2 q + q^{3} + 2 q^{4} - q^{5} + 2 q^{9} + 3 q^{11} - 2 q^{12} - 10 q^{13} - 2 q^{15} - 4 q^{16} + 3 q^{17} + 2 q^{19} - 4 q^{20} + 6 q^{23} - q^{25} + 10 q^{27} + 6 q^{29} - 4 q^{31} - 3 q^{33} + 8 q^{36} - 2 q^{37} - 5 q^{39} + 24 q^{41} - 20 q^{43} - 6 q^{44} + 2 q^{45} + 9 q^{47} - 8 q^{48} - 3 q^{51} - 10 q^{52} - 12 q^{53} - 6 q^{55} + 4 q^{57} - 2 q^{60} + 8 q^{61} - 16 q^{64} + 5 q^{65} + 4 q^{67} - 6 q^{68} + 12 q^{69} + 2 q^{73} + q^{75} + 8 q^{76} + q^{79} - 4 q^{80} - q^{81} - 24 q^{83} - 6 q^{85} + 3 q^{87} - 12 q^{89} + 24 q^{92} + 4 q^{93} + 2 q^{95} + 2 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^4 - q^5 + 2 * q^9 + 3 * q^11 - 2 * q^12 - 10 * q^13 - 2 * q^15 - 4 * q^16 + 3 * q^17 + 2 * q^19 - 4 * q^20 + 6 * q^23 - q^25 + 10 * q^27 + 6 * q^29 - 4 * q^31 - 3 * q^33 + 8 * q^36 - 2 * q^37 - 5 * q^39 + 24 * q^41 - 20 * q^43 - 6 * q^44 + 2 * q^45 + 9 * q^47 - 8 * q^48 - 3 * q^51 - 10 * q^52 - 12 * q^53 - 6 * q^55 + 4 * q^57 - 2 * q^60 + 8 * q^61 - 16 * q^64 + 5 * q^65 + 4 * q^67 - 6 * q^68 + 12 * q^69 + 2 * q^73 + q^75 + 8 * q^76 + q^79 - 4 * q^80 - q^81 - 24 * q^83 - 6 * q^85 + 3 * q^87 - 12 * q^89 + 24 * q^92 + 4 * q^93 + 2 * q^95 + 2 * q^97 + 12 * q^99

## 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
0 0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
226.1 0 0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 0
 $$n$$: 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.b 2
7.b odd 2 1 245.2.e.a 2
7.c even 3 1 245.2.a.c 1
7.c even 3 1 inner 245.2.e.b 2
7.d odd 6 1 35.2.a.a 1
7.d odd 6 1 245.2.e.a 2
21.g even 6 1 315.2.a.b 1
21.h odd 6 1 2205.2.a.e 1
28.f even 6 1 560.2.a.b 1
28.g odd 6 1 3920.2.a.ba 1
35.i odd 6 1 175.2.a.b 1
35.j even 6 1 1225.2.a.e 1
35.k even 12 2 175.2.b.a 2
35.l odd 12 2 1225.2.b.d 2
56.j odd 6 1 2240.2.a.k 1
56.m even 6 1 2240.2.a.u 1
77.i even 6 1 4235.2.a.c 1
84.j odd 6 1 5040.2.a.v 1
91.s odd 6 1 5915.2.a.f 1
105.p even 6 1 1575.2.a.f 1
105.w odd 12 2 1575.2.d.c 2
140.s even 6 1 2800.2.a.z 1
140.x odd 12 2 2800.2.g.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 7.d odd 6 1
175.2.a.b 1 35.i odd 6 1
175.2.b.a 2 35.k even 12 2
245.2.a.c 1 7.c even 3 1
245.2.e.a 2 7.b odd 2 1
245.2.e.a 2 7.d odd 6 1
245.2.e.b 2 1.a even 1 1 trivial
245.2.e.b 2 7.c even 3 1 inner
315.2.a.b 1 21.g even 6 1
560.2.a.b 1 28.f even 6 1
1225.2.a.e 1 35.j even 6 1
1225.2.b.d 2 35.l odd 12 2
1575.2.a.f 1 105.p even 6 1
1575.2.d.c 2 105.w odd 12 2
2205.2.a.e 1 21.h odd 6 1
2240.2.a.k 1 56.j odd 6 1
2240.2.a.u 1 56.m even 6 1
2800.2.a.z 1 140.s even 6 1
2800.2.g.l 2 140.x odd 12 2
3920.2.a.ba 1 28.g odd 6 1
4235.2.a.c 1 77.i even 6 1
5040.2.a.v 1 84.j odd 6 1
5915.2.a.f 1 91.s odd 6 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 3)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T - 12)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 8T + 64$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$(T - 1)^{2}$$