# Properties

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

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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: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 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

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 2) q^{6} - 2 \beta_{3} q^{8} + 2 \beta_1 q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + b2 - b1) * q^3 + (-b2 - 1) * q^5 + (b3 + 2) * q^6 - 2*b3 * q^8 + 2*b1 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + ( - \beta_{2} - 1) q^{5} + (\beta_{3} + 2) q^{6} - 2 \beta_{3} q^{8} + 2 \beta_1 q^{9} + ( - \beta_{3} - \beta_1) q^{10} + (2 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{3} + 3) q^{13} + (\beta_{3} + 1) q^{15} + (4 \beta_{2} + 4) q^{16} + (3 \beta_{3} - \beta_{2} + 3 \beta_1) q^{17} + 4 \beta_{2} q^{18} + ( - 6 \beta_{2} - 6) q^{19} + ( - 3 \beta_{3} - 4) q^{22} + ( - 6 \beta_{2} - \beta_1 - 6) q^{23} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{24} + \beta_{2} q^{25} + (2 \beta_{2} + 3 \beta_1 + 2) q^{26} + ( - \beta_{3} + 1) q^{27} + (4 \beta_{3} - 3) q^{29} + ( - 2 \beta_{2} + \beta_1 - 2) q^{30} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{31} + (7 \beta_{2} - 5 \beta_1 + 7) q^{33} + ( - \beta_{3} - 6) q^{34} + (2 \beta_{2} - 3 \beta_1 + 2) q^{37} + ( - 6 \beta_{3} - 6 \beta_1) q^{38} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{39} - 2 \beta_1 q^{40} + (3 \beta_{3} - 2) q^{41} + 2 q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{45} + ( - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{46} + (3 \beta_{2} + 3 \beta_1 + 3) q^{47} + ( - 4 \beta_{3} - 4) q^{48} + \beta_{3} q^{50} + (7 \beta_{2} - 4 \beta_1 + 7) q^{51} + (3 \beta_{3} + 3 \beta_1) q^{53} + (2 \beta_{2} + \beta_1 + 2) q^{54} + ( - 2 \beta_{3} - 3) q^{55} + (6 \beta_{3} + 6) q^{57} + ( - 8 \beta_{2} - 3 \beta_1 - 8) q^{58} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{59} + 2 \beta_1 q^{61} + (6 \beta_{3} - 6) q^{62} + 8 q^{64} + ( - 3 \beta_{2} - \beta_1 - 3) q^{65} + (7 \beta_{3} - 10 \beta_{2} + 7 \beta_1) q^{66} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{67} + (5 \beta_{3} + 4) q^{69} + ( - 2 \beta_{3} - 6) q^{71} + (8 \beta_{2} + 8) q^{72} + ( - 6 \beta_{3} - 6 \beta_1) q^{73} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{74} + ( - \beta_{2} + \beta_1 - 1) q^{75} + (\beta_{3} + 4) q^{78} + (7 \beta_{2} - 6 \beta_1 + 7) q^{79} - 4 \beta_{2} q^{80} + (6 \beta_{3} - \beta_{2} + 6 \beta_1) q^{81} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{82} + ( - 3 \beta_{3} - 1) q^{85} + 2 \beta_1 q^{86} + ( - \beta_{3} + 5 \beta_{2} - \beta_1) q^{87} + ( - 6 \beta_{3} + 8 \beta_{2} - 6 \beta_1) q^{88} + (8 \beta_{2} + 8) q^{89} + 4 q^{90} + 3 \beta_1 q^{93} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{94} + 6 \beta_{2} q^{95} + ( - 3 \beta_{3} + 9) q^{97} + ( - 6 \beta_{3} - 8) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 + b2 - b1) * q^3 + (-b2 - 1) * q^5 + (b3 + 2) * q^6 - 2*b3 * q^8 + 2*b1 * q^9 + (-b3 - b1) * q^10 + (2*b3 - 3*b2 + 2*b1) * q^11 + (-b3 + 3) * q^13 + (b3 + 1) * q^15 + (4*b2 + 4) * q^16 + (3*b3 - b2 + 3*b1) * q^17 + 4*b2 * q^18 + (-6*b2 - 6) * q^19 + (-3*b3 - 4) * q^22 + (-6*b2 - b1 - 6) * q^23 + (2*b3 - 4*b2 + 2*b1) * q^24 + b2 * q^25 + (2*b2 + 3*b1 + 2) * q^26 + (-b3 + 1) * q^27 + (4*b3 - 3) * q^29 + (-2*b2 + b1 - 2) * q^30 + (3*b3 + 6*b2 + 3*b1) * q^31 + (7*b2 - 5*b1 + 7) * q^33 + (-b3 - 6) * q^34 + (2*b2 - 3*b1 + 2) * q^37 + (-6*b3 - 6*b1) * q^38 + (-2*b3 + b2 - 2*b1) * q^39 - 2*b1 * q^40 + (3*b3 - 2) * q^41 + 2 * q^43 + (-2*b3 - 2*b1) * q^45 + (-6*b3 - 2*b2 - 6*b1) * q^46 + (3*b2 + 3*b1 + 3) * q^47 + (-4*b3 - 4) * q^48 + b3 * q^50 + (7*b2 - 4*b1 + 7) * q^51 + (3*b3 + 3*b1) * q^53 + (2*b2 + b1 + 2) * q^54 + (-2*b3 - 3) * q^55 + (6*b3 + 6) * q^57 + (-8*b2 - 3*b1 - 8) * q^58 + (-3*b3 + 2*b2 - 3*b1) * q^59 + 2*b1 * q^61 + (6*b3 - 6) * q^62 + 8 * q^64 + (-3*b2 - b1 - 3) * q^65 + (7*b3 - 10*b2 + 7*b1) * q^66 + (-3*b3 - 4*b2 - 3*b1) * q^67 + (5*b3 + 4) * q^69 + (-2*b3 - 6) * q^71 + (8*b2 + 8) * q^72 + (-6*b3 - 6*b1) * q^73 + (2*b3 - 6*b2 + 2*b1) * q^74 + (-b2 + b1 - 1) * q^75 + (b3 + 4) * q^78 + (7*b2 - 6*b1 + 7) * q^79 - 4*b2 * q^80 + (6*b3 - b2 + 6*b1) * q^81 + (-6*b2 - 2*b1 - 6) * q^82 + (-3*b3 - 1) * q^85 + 2*b1 * q^86 + (-b3 + 5*b2 - b1) * q^87 + (-6*b3 + 8*b2 - 6*b1) * q^88 + (8*b2 + 8) * q^89 + 4 * q^90 + 3*b1 * q^93 + (3*b3 + 6*b2 + 3*b1) * q^94 + 6*b2 * q^95 + (-3*b3 + 9) * q^97 + (-6*b3 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} + 8 q^{6}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 + 8 * q^6 $$4 q - 2 q^{3} - 2 q^{5} + 8 q^{6} + 6 q^{11} + 12 q^{13} + 4 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} - 12 q^{19} - 16 q^{22} - 12 q^{23} + 8 q^{24} - 2 q^{25} + 4 q^{26} + 4 q^{27} - 12 q^{29} - 4 q^{30} - 12 q^{31} + 14 q^{33} - 24 q^{34} + 4 q^{37} - 2 q^{39} - 8 q^{41} + 8 q^{43} + 4 q^{46} + 6 q^{47} - 16 q^{48} + 14 q^{51} + 4 q^{54} - 12 q^{55} + 24 q^{57} - 16 q^{58} - 4 q^{59} - 24 q^{62} + 32 q^{64} - 6 q^{65} + 20 q^{66} + 8 q^{67} + 16 q^{69} - 24 q^{71} + 16 q^{72} + 12 q^{74} - 2 q^{75} + 16 q^{78} + 14 q^{79} + 8 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{85} - 10 q^{87} - 16 q^{88} + 16 q^{89} + 16 q^{90} - 12 q^{94} - 12 q^{95} + 36 q^{97} - 32 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 + 8 * q^6 + 6 * q^11 + 12 * q^13 + 4 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 - 12 * q^19 - 16 * q^22 - 12 * q^23 + 8 * q^24 - 2 * q^25 + 4 * q^26 + 4 * q^27 - 12 * q^29 - 4 * q^30 - 12 * q^31 + 14 * q^33 - 24 * q^34 + 4 * q^37 - 2 * q^39 - 8 * q^41 + 8 * q^43 + 4 * q^46 + 6 * q^47 - 16 * q^48 + 14 * q^51 + 4 * q^54 - 12 * q^55 + 24 * q^57 - 16 * q^58 - 4 * q^59 - 24 * q^62 + 32 * q^64 - 6 * q^65 + 20 * q^66 + 8 * q^67 + 16 * q^69 - 24 * q^71 + 16 * q^72 + 12 * q^74 - 2 * q^75 + 16 * q^78 + 14 * q^79 + 8 * q^80 + 2 * q^81 - 12 * q^82 - 4 * q^85 - 10 * q^87 - 16 * q^88 + 16 * q^89 + 16 * q^90 - 12 * q^94 - 12 * q^95 + 36 * q^97 - 32 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

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

## 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
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 1.22474i −1.20711 + 2.09077i 0 −0.500000 0.866025i 3.41421 0 −2.82843 −1.41421 2.44949i −0.707107 + 1.22474i
116.2 0.707107 + 1.22474i 0.207107 0.358719i 0 −0.500000 0.866025i 0.585786 0 2.82843 1.41421 + 2.44949i 0.707107 1.22474i
226.1 −0.707107 + 1.22474i −1.20711 2.09077i 0 −0.500000 + 0.866025i 3.41421 0 −2.82843 −1.41421 + 2.44949i −0.707107 1.22474i
226.2 0.707107 1.22474i 0.207107 + 0.358719i 0 −0.500000 + 0.866025i 0.585786 0 2.82843 1.41421 2.44949i 0.707107 + 1.22474i
 $$n$$: 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.f 4
7.b odd 2 1 245.2.e.g 4
7.c even 3 1 245.2.a.f yes 2
7.c even 3 1 inner 245.2.e.f 4
7.d odd 6 1 245.2.a.e 2
7.d odd 6 1 245.2.e.g 4
21.g even 6 1 2205.2.a.v 2
21.h odd 6 1 2205.2.a.t 2
28.f even 6 1 3920.2.a.bw 2
28.g odd 6 1 3920.2.a.br 2
35.i odd 6 1 1225.2.a.r 2
35.j even 6 1 1225.2.a.p 2
35.k even 12 2 1225.2.b.i 4
35.l odd 12 2 1225.2.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 7.d odd 6 1
245.2.a.f yes 2 7.c even 3 1
245.2.e.f 4 1.a even 1 1 trivial
245.2.e.f 4 7.c even 3 1 inner
245.2.e.g 4 7.b odd 2 1
245.2.e.g 4 7.d odd 6 1
1225.2.a.p 2 35.j even 6 1
1225.2.a.r 2 35.i odd 6 1
1225.2.b.i 4 35.k even 12 2
1225.2.b.j 4 35.l odd 12 2
2205.2.a.t 2 21.h odd 6 1
2205.2.a.v 2 21.g even 6 1
3920.2.a.br 2 28.g odd 6 1
3920.2.a.bw 2 28.f even 6 1

## Hecke kernels

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

 $$T_{2}^{4} + 2T_{2}^{2} + 4$$ T2^4 + 2*T2^2 + 4 $$T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1$$ T3^4 + 2*T3^3 + 5*T3^2 - 2*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 6 T^{3} + 35 T^{2} - 6 T + 1$$
$13$ $$(T^{2} - 6 T + 7)^{2}$$
$17$ $$T^{4} - 2 T^{3} + 21 T^{2} + 34 T + 289$$
$19$ $$(T^{2} + 6 T + 36)^{2}$$
$23$ $$T^{4} + 12 T^{3} + 110 T^{2} + \cdots + 1156$$
$29$ $$(T^{2} + 6 T - 23)^{2}$$
$31$ $$T^{4} + 12 T^{3} + 126 T^{2} + \cdots + 324$$
$37$ $$T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196$$
$41$ $$(T^{2} + 4 T - 14)^{2}$$
$43$ $$(T - 2)^{4}$$
$47$ $$T^{4} - 6 T^{3} + 45 T^{2} + 54 T + 81$$
$53$ $$T^{4} + 18T^{2} + 324$$
$59$ $$T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196$$
$61$ $$T^{4} + 8T^{2} + 64$$
$67$ $$T^{4} - 8 T^{3} + 66 T^{2} + 16 T + 4$$
$71$ $$(T^{2} + 12 T + 28)^{2}$$
$73$ $$T^{4} + 72T^{2} + 5184$$
$79$ $$T^{4} - 14 T^{3} + 219 T^{2} + \cdots + 529$$
$83$ $$T^{4}$$
$89$ $$(T^{2} - 8 T + 64)^{2}$$
$97$ $$(T^{2} - 18 T + 63)^{2}$$
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