Properties

 Label 245.2.e.h 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: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + \beta_{2} q^{5} + 4 q^{6} + ( -4 + \beta_{3} ) q^{8} + ( \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + \beta_{2} q^{5} + 4 q^{6} + ( -4 + \beta_{3} ) q^{8} + ( \beta_{1} - 2 \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{3} ) q^{10} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{12} + ( -3 - \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} -3 \beta_{1} q^{16} + ( -3 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{17} + ( -4 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{18} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{19} + ( -2 + \beta_{3} ) q^{20} + 4 q^{22} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{24} + ( -1 + \beta_{2} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{26} + ( 3 - \beta_{3} ) q^{27} + ( -1 - 3 \beta_{3} ) q^{29} + 4 \beta_{2} q^{30} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{32} + ( \beta_{1} - 5 \beta_{2} ) q^{33} + ( 4 + 2 \beta_{3} ) q^{34} + \beta_{3} q^{36} -6 \beta_{2} q^{37} + ( 8 - 4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{38} + ( 7 + 3 \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -\beta_{1} - 4 \beta_{2} ) q^{40} + 2 \beta_{3} q^{41} + ( 6 + 2 \beta_{3} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 2 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{45} + ( 8 - 8 \beta_{2} ) q^{46} + ( -3 \beta_{1} - \beta_{2} ) q^{47} -12 q^{48} + \beta_{3} q^{50} + ( 3 \beta_{1} - 7 \beta_{2} ) q^{51} + ( 2 - 2 \beta_{2} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{54} + ( -1 - \beta_{3} ) q^{55} + ( -6 + 2 \beta_{3} ) q^{57} + ( 2 \beta_{1} + 12 \beta_{2} ) q^{58} + ( -4 + 4 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{60} + 6 \beta_{1} q^{61} + ( 4 + \beta_{3} ) q^{64} + ( \beta_{1} - 3 \beta_{2} ) q^{65} + ( -4 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{66} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{67} -2 \beta_{2} q^{68} + ( -10 - 2 \beta_{3} ) q^{69} + 8 q^{71} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{72} + ( -2 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{73} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{74} + ( \beta_{1} - \beta_{2} ) q^{75} + ( 12 - 8 \beta_{3} ) q^{76} + ( -12 - 4 \beta_{3} ) q^{78} + ( -\beta_{1} + 5 \beta_{2} ) q^{79} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{80} + ( 7 - 7 \beta_{2} ) q^{81} + ( -2 \beta_{1} - 8 \beta_{2} ) q^{82} -4 q^{83} + ( -3 - \beta_{3} ) q^{85} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{86} + ( 13 + \beta_{1} - 13 \beta_{2} + \beta_{3} ) q^{87} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{88} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -4 - \beta_{3} ) q^{90} + ( 4 - 4 \beta_{3} ) q^{92} + ( 12 - 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{94} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{95} -4 \beta_{1} q^{96} + ( 7 + 5 \beta_{3} ) q^{97} + ( 6 + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{3} - 5q^{4} + 2q^{5} + 16q^{6} - 18q^{8} - 3q^{9} + O(q^{10})$$ $$4q + q^{2} - q^{3} - 5q^{4} + 2q^{5} + 16q^{6} - 18q^{8} - 3q^{9} - q^{10} - q^{11} + 6q^{12} - 10q^{13} - 2q^{15} - 3q^{16} - 5q^{17} - 7q^{18} - 6q^{19} - 10q^{20} + 16q^{22} + 2q^{23} - 4q^{24} - 2q^{25} + 6q^{26} + 14q^{27} + 2q^{29} + 8q^{30} + 9q^{32} - 9q^{33} + 12q^{34} - 2q^{36} - 12q^{37} + 20q^{38} + 11q^{39} - 9q^{40} - 4q^{41} + 20q^{43} + 6q^{44} + 3q^{45} + 16q^{46} - 5q^{47} - 48q^{48} - 2q^{50} - 11q^{51} + 4q^{52} + 2q^{53} + 12q^{54} - 2q^{55} - 28q^{57} + 26q^{58} - 8q^{59} - 6q^{60} + 6q^{61} + 14q^{64} - 5q^{65} - 4q^{66} - 4q^{67} - 4q^{68} - 36q^{69} + 32q^{71} + 5q^{72} - 8q^{73} + 6q^{74} - q^{75} + 64q^{76} - 40q^{78} + 9q^{79} + 3q^{80} + 14q^{81} - 18q^{82} - 16q^{83} - 10q^{85} - 12q^{86} + 25q^{87} - 4q^{88} + 6q^{89} - 14q^{90} + 24q^{92} + 28q^{94} + 6q^{95} - 4q^{96} + 18q^{97} + 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

Character values

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

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

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
 −0.780776 − 1.35234i 1.28078 + 2.21837i −0.780776 + 1.35234i 1.28078 − 2.21837i
−0.780776 1.35234i −1.28078 + 2.21837i −0.219224 + 0.379706i 0.500000 + 0.866025i 4.00000 0 −2.43845 −1.78078 3.08440i 0.780776 1.35234i
116.2 1.28078 + 2.21837i 0.780776 1.35234i −2.28078 + 3.95042i 0.500000 + 0.866025i 4.00000 0 −6.56155 0.280776 + 0.486319i −1.28078 + 2.21837i
226.1 −0.780776 + 1.35234i −1.28078 2.21837i −0.219224 0.379706i 0.500000 0.866025i 4.00000 0 −2.43845 −1.78078 + 3.08440i 0.780776 + 1.35234i
226.2 1.28078 2.21837i 0.780776 + 1.35234i −2.28078 3.95042i 0.500000 0.866025i 4.00000 0 −6.56155 0.280776 0.486319i −1.28078 2.21837i
 $$n$$: 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.h 4
7.b odd 2 1 245.2.e.i 4
7.c even 3 1 245.2.a.d 2
7.c even 3 1 inner 245.2.e.h 4
7.d odd 6 1 35.2.a.b 2
7.d odd 6 1 245.2.e.i 4
21.g even 6 1 315.2.a.e 2
21.h odd 6 1 2205.2.a.x 2
28.f even 6 1 560.2.a.i 2
28.g odd 6 1 3920.2.a.bs 2
35.i odd 6 1 175.2.a.f 2
35.j even 6 1 1225.2.a.s 2
35.k even 12 2 175.2.b.b 4
35.l odd 12 2 1225.2.b.f 4
56.j odd 6 1 2240.2.a.bh 2
56.m even 6 1 2240.2.a.bd 2
77.i even 6 1 4235.2.a.m 2
84.j odd 6 1 5040.2.a.bt 2
91.s odd 6 1 5915.2.a.l 2
105.p even 6 1 1575.2.a.p 2
105.w odd 12 2 1575.2.d.e 4
140.s even 6 1 2800.2.a.bi 2
140.x odd 12 2 2800.2.g.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 7.d odd 6 1
175.2.a.f 2 35.i odd 6 1
175.2.b.b 4 35.k even 12 2
245.2.a.d 2 7.c even 3 1
245.2.e.h 4 1.a even 1 1 trivial
245.2.e.h 4 7.c even 3 1 inner
245.2.e.i 4 7.b odd 2 1
245.2.e.i 4 7.d odd 6 1
315.2.a.e 2 21.g even 6 1
560.2.a.i 2 28.f even 6 1
1225.2.a.s 2 35.j even 6 1
1225.2.b.f 4 35.l odd 12 2
1575.2.a.p 2 105.p even 6 1
1575.2.d.e 4 105.w odd 12 2
2205.2.a.x 2 21.h odd 6 1
2240.2.a.bd 2 56.m even 6 1
2240.2.a.bh 2 56.j odd 6 1
2800.2.a.bi 2 140.s even 6 1
2800.2.g.t 4 140.x odd 12 2
3920.2.a.bs 2 28.g odd 6 1
4235.2.a.m 2 77.i even 6 1
5040.2.a.bt 2 84.j odd 6 1
5915.2.a.l 2 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 $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 5 T_{2}^{2} + 4 T_{2} + 16$$ $$T_{3}^{4} + T_{3}^{3} + 5 T_{3}^{2} - 4 T_{3} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 4 T + 5 T^{2} - T^{3} + T^{4}$$
$3$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$13$ $$( 2 + 5 T + T^{2} )^{2}$$
$17$ $$4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4}$$
$19$ $$64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$( -38 - T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 36 + 6 T + T^{2} )^{2}$$
$41$ $$( -16 + 2 T + T^{2} )^{2}$$
$43$ $$( 8 - 10 T + T^{2} )^{2}$$
$47$ $$1024 - 160 T + 57 T^{2} + 5 T^{3} + T^{4}$$
$53$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$20736 + 864 T + 180 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$2704 - 416 T + 116 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4}$$
$83$ $$( 4 + T )^{4}$$
$89$ $$64 + 48 T + 44 T^{2} - 6 T^{3} + T^{4}$$
$97$ $$( -86 - 9 T + T^{2} )^{2}$$