# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,2,Mod(116,245)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("245.116");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_{2} q^{5} + 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + b2 - b1 - 1) * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + b2 * q^5 + 4 * q^6 + (b3 - 4) * q^8 + (-2*b2 + b1) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_{2} q^{5} + 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9} + (\beta_{3} + \beta_1) q^{10} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{11} + (2 \beta_{2} + 2 \beta_1) q^{12} + ( - \beta_{3} - 3) q^{13} + ( - \beta_{3} - 1) q^{15} - 3 \beta_1 q^{16} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{17} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{18} + ( - 2 \beta_{2} - 2 \beta_1) q^{19} + (\beta_{3} - 2) q^{20} + 4 q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} + (4 \beta_{3} + 4 \beta_1) q^{24} + (\beta_{2} - 1) q^{25} + (4 \beta_{2} - 2 \beta_1) q^{26} + ( - \beta_{3} + 3) q^{27} + ( - 3 \beta_{3} - 1) q^{29} + 4 \beta_{2} q^{30} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{32} + ( - 5 \beta_{2} + \beta_1) q^{33} + (2 \beta_{3} + 4) q^{34} + \beta_{3} q^{36} - 6 \beta_{2} q^{37} + ( - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 8) q^{38} + (3 \beta_{3} - 7 \beta_{2} + 3 \beta_1 + 7) q^{39} + ( - 4 \beta_{2} - \beta_1) q^{40} + 2 \beta_{3} q^{41} + (2 \beta_{3} + 6) q^{43} + (2 \beta_{2} + 2 \beta_1) q^{44} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{45} + ( - 8 \beta_{2} + 8) q^{46} + ( - \beta_{2} - 3 \beta_1) q^{47} - 12 q^{48} + \beta_{3} q^{50} + ( - 7 \beta_{2} + 3 \beta_1) q^{51} + ( - 2 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} - 2 \beta_1) q^{53} + (4 \beta_{2} + 4 \beta_1) q^{54} + ( - \beta_{3} - 1) q^{55} + (2 \beta_{3} - 6) q^{57} + (12 \beta_{2} + 2 \beta_1) q^{58} + (4 \beta_{2} - 4) q^{59} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{60} + 6 \beta_1 q^{61} + (\beta_{3} + 4) q^{64} + ( - 3 \beta_{2} + \beta_1) q^{65} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 4) q^{66} + (4 \beta_{3} + 4 \beta_1) q^{67} - 2 \beta_{2} q^{68} + ( - 2 \beta_{3} - 10) q^{69} + 8 q^{71} + (4 \beta_{2} - 3 \beta_1) q^{72} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{73} + ( - 6 \beta_{3} - 6 \beta_1) q^{74} + ( - \beta_{2} + \beta_1) q^{75} + ( - 8 \beta_{3} + 12) q^{76} + ( - 4 \beta_{3} - 12) q^{78} + (5 \beta_{2} - \beta_1) q^{79} + ( - 3 \beta_{3} - 3 \beta_1) q^{80} + ( - 7 \beta_{2} + 7) q^{81} + ( - 8 \beta_{2} - 2 \beta_1) q^{82} - 4 q^{83} + ( - \beta_{3} - 3) q^{85} + ( - 8 \beta_{2} + 4 \beta_1) q^{86} + (\beta_{3} - 13 \beta_{2} + \beta_1 + 13) q^{87} + (4 \beta_{3} + 4 \beta_1) q^{88} + (4 \beta_{2} - 2 \beta_1) q^{89} + ( - \beta_{3} - 4) q^{90} + ( - 4 \beta_{3} + 4) q^{92} + ( - 4 \beta_{3} - 12 \beta_{2} - 4 \beta_1 + 12) q^{94} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{95} - 4 \beta_1 q^{96} + (5 \beta_{3} + 7) q^{97} + (2 \beta_{3} + 6) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 + b2 - b1 - 1) * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + b2 * q^5 + 4 * q^6 + (b3 - 4) * q^8 + (-2*b2 + b1) * q^9 + (b3 + b1) * q^10 + (-b3 + b2 - b1 - 1) * q^11 + (2*b2 + 2*b1) * q^12 + (-b3 - 3) * q^13 + (-b3 - 1) * q^15 - 3*b1 * q^16 + (-b3 + 3*b2 - b1 - 3) * q^17 + (-b3 + 4*b2 - b1 - 4) * q^18 + (-2*b2 - 2*b1) * q^19 + (b3 - 2) * q^20 + 4 * q^22 + (2*b2 - 2*b1) * q^23 + (4*b3 + 4*b1) * q^24 + (b2 - 1) * q^25 + (4*b2 - 2*b1) * q^26 + (-b3 + 3) * q^27 + (-3*b3 - 1) * q^29 + 4*b2 * q^30 + (-b3 - 4*b2 - b1 + 4) * q^32 + (-5*b2 + b1) * q^33 + (2*b3 + 4) * q^34 + b3 * q^36 - 6*b2 * q^37 + (-4*b3 - 8*b2 - 4*b1 + 8) * q^38 + (3*b3 - 7*b2 + 3*b1 + 7) * q^39 + (-4*b2 - b1) * q^40 + 2*b3 * q^41 + (2*b3 + 6) * q^43 + (2*b2 + 2*b1) * q^44 + (b3 - 2*b2 + b1 + 2) * q^45 + (-8*b2 + 8) * q^46 + (-b2 - 3*b1) * q^47 - 12 * q^48 + b3 * q^50 + (-7*b2 + 3*b1) * q^51 + (-2*b2 + 2) * q^52 + (-2*b3 - 2*b1) * q^53 + (4*b2 + 4*b1) * q^54 + (-b3 - 1) * q^55 + (2*b3 - 6) * q^57 + (12*b2 + 2*b1) * q^58 + (4*b2 - 4) * q^59 + (2*b3 + 2*b2 + 2*b1 - 2) * q^60 + 6*b1 * q^61 + (b3 + 4) * q^64 + (-3*b2 + b1) * q^65 + (-4*b3 + 4*b2 - 4*b1 - 4) * q^66 + (4*b3 + 4*b1) * q^67 - 2*b2 * q^68 + (-2*b3 - 10) * q^69 + 8 * q^71 + (4*b2 - 3*b1) * q^72 + (4*b3 + 2*b2 + 4*b1 - 2) * q^73 + (-6*b3 - 6*b1) * q^74 + (-b2 + b1) * q^75 + (-8*b3 + 12) * q^76 + (-4*b3 - 12) * q^78 + (5*b2 - b1) * q^79 + (-3*b3 - 3*b1) * q^80 + (-7*b2 + 7) * q^81 + (-8*b2 - 2*b1) * q^82 - 4 * q^83 + (-b3 - 3) * q^85 + (-8*b2 + 4*b1) * q^86 + (b3 - 13*b2 + b1 + 13) * q^87 + (4*b3 + 4*b1) * q^88 + (4*b2 - 2*b1) * q^89 + (-b3 - 4) * q^90 + (-4*b3 + 4) * q^92 + (-4*b3 - 12*b2 - 4*b1 + 12) * q^94 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^95 - 4*b1 * q^96 + (5*b3 + 7) * q^97 + (2*b3 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - q^{3} - 5 q^{4} + 2 q^{5} + 16 q^{6} - 18 q^{8} - 3 q^{9}+O(q^{10})$$ 4 * q + q^2 - q^3 - 5 * q^4 + 2 * q^5 + 16 * q^6 - 18 * q^8 - 3 * q^9 $$4 q + q^{2} - q^{3} - 5 q^{4} + 2 q^{5} + 16 q^{6} - 18 q^{8} - 3 q^{9} - q^{10} - q^{11} + 6 q^{12} - 10 q^{13} - 2 q^{15} - 3 q^{16} - 5 q^{17} - 7 q^{18} - 6 q^{19} - 10 q^{20} + 16 q^{22} + 2 q^{23} - 4 q^{24} - 2 q^{25} + 6 q^{26} + 14 q^{27} + 2 q^{29} + 8 q^{30} + 9 q^{32} - 9 q^{33} + 12 q^{34} - 2 q^{36} - 12 q^{37} + 20 q^{38} + 11 q^{39} - 9 q^{40} - 4 q^{41} + 20 q^{43} + 6 q^{44} + 3 q^{45} + 16 q^{46} - 5 q^{47} - 48 q^{48} - 2 q^{50} - 11 q^{51} + 4 q^{52} + 2 q^{53} + 12 q^{54} - 2 q^{55} - 28 q^{57} + 26 q^{58} - 8 q^{59} - 6 q^{60} + 6 q^{61} + 14 q^{64} - 5 q^{65} - 4 q^{66} - 4 q^{67} - 4 q^{68} - 36 q^{69} + 32 q^{71} + 5 q^{72} - 8 q^{73} + 6 q^{74} - q^{75} + 64 q^{76} - 40 q^{78} + 9 q^{79} + 3 q^{80} + 14 q^{81} - 18 q^{82} - 16 q^{83} - 10 q^{85} - 12 q^{86} + 25 q^{87} - 4 q^{88} + 6 q^{89} - 14 q^{90} + 24 q^{92} + 28 q^{94} + 6 q^{95} - 4 q^{96} + 18 q^{97} + 20 q^{99}+O(q^{100})$$ 4 * q + q^2 - q^3 - 5 * q^4 + 2 * q^5 + 16 * q^6 - 18 * q^8 - 3 * q^9 - q^10 - q^11 + 6 * q^12 - 10 * q^13 - 2 * q^15 - 3 * q^16 - 5 * q^17 - 7 * q^18 - 6 * q^19 - 10 * q^20 + 16 * q^22 + 2 * q^23 - 4 * q^24 - 2 * q^25 + 6 * q^26 + 14 * q^27 + 2 * q^29 + 8 * q^30 + 9 * q^32 - 9 * q^33 + 12 * q^34 - 2 * q^36 - 12 * q^37 + 20 * q^38 + 11 * q^39 - 9 * q^40 - 4 * q^41 + 20 * q^43 + 6 * q^44 + 3 * q^45 + 16 * q^46 - 5 * q^47 - 48 * q^48 - 2 * q^50 - 11 * q^51 + 4 * q^52 + 2 * q^53 + 12 * q^54 - 2 * q^55 - 28 * q^57 + 26 * q^58 - 8 * q^59 - 6 * q^60 + 6 * q^61 + 14 * q^64 - 5 * q^65 - 4 * q^66 - 4 * q^67 - 4 * q^68 - 36 * q^69 + 32 * q^71 + 5 * q^72 - 8 * q^73 + 6 * q^74 - q^75 + 64 * q^76 - 40 * q^78 + 9 * q^79 + 3 * q^80 + 14 * q^81 - 18 * q^82 - 16 * q^83 - 10 * q^85 - 12 * q^86 + 25 * q^87 - 4 * q^88 + 6 * q^89 - 14 * q^90 + 24 * q^92 + 28 * q^94 + 6 * q^95 - 4 * q^96 + 18 * q^97 + 20 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 $$S_{2}^{\mathrm{new}}(245, [\chi])$$:

 $$T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16$$ T2^4 - T2^3 + 5*T2^2 + 4*T2 + 16 $$T_{3}^{4} + T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 16$$ T3^4 + T3^3 + 5*T3^2 - 4*T3 + 16

## Hecke characteristic polynomials

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