# Properties

 Label 245.3.h.a Level $245$ Weight $3$ Character orbit 245.h Analytic conductor $6.676$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,3,Mod(31,245)]

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, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("245.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 245.h (of order $$6$$, 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: $$6.67576647683$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 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

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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} - 8 q^{8} - 4 \beta_{2} q^{9}+O(q^{10})$$ q - 2*b2 * q^2 + b1 * q^3 + (-b3 + b1) * q^5 - 2*b3 * q^6 - 8 * q^8 - 4*b2 * q^9 $$q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} - 8 q^{8} - 4 \beta_{2} q^{9} - 2 \beta_1 q^{10} + ( - \beta_{2} + 1) q^{11} - 9 \beta_{3} q^{13} + 5 q^{15} + 16 \beta_{2} q^{16} + 3 \beta_1 q^{17} + (8 \beta_{2} - 8) q^{18} + ( - 6 \beta_{3} + 6 \beta_1) q^{19} - 2 q^{22} - 8 \beta_{2} q^{23} - 8 \beta_1 q^{24} + ( - 5 \beta_{2} + 5) q^{25} + (18 \beta_{3} - 18 \beta_1) q^{26} - 13 \beta_{3} q^{27} + 41 q^{29} - 10 \beta_{2} q^{30} - 18 \beta_1 q^{31} + ( - \beta_{3} + \beta_1) q^{33} - 6 \beta_{3} q^{34} + 28 \beta_{2} q^{37} - 12 \beta_1 q^{38} + ( - 45 \beta_{2} + 45) q^{39} + (8 \beta_{3} - 8 \beta_1) q^{40} + 6 \beta_{3} q^{41} - 82 q^{43} - 4 \beta_1 q^{45} + (16 \beta_{2} - 16) q^{46} + ( - 9 \beta_{3} + 9 \beta_1) q^{47} + 16 \beta_{3} q^{48} - 10 q^{50} + 15 \beta_{2} q^{51} + (74 \beta_{2} - 74) q^{53} + (26 \beta_{3} - 26 \beta_1) q^{54} - \beta_{3} q^{55} + 30 q^{57} - 82 \beta_{2} q^{58} + 42 \beta_1 q^{59} + ( - 36 \beta_{3} + 36 \beta_1) q^{61} + 36 \beta_{3} q^{62} + 64 q^{64} - 45 \beta_{2} q^{65} - 2 \beta_1 q^{66} + (2 \beta_{2} - 2) q^{67} - 8 \beta_{3} q^{69} + 14 q^{71} + 32 \beta_{2} q^{72} + 30 \beta_1 q^{73} + ( - 56 \beta_{2} + 56) q^{74} + ( - 5 \beta_{3} + 5 \beta_1) q^{75} - 90 q^{78} + 19 \beta_{2} q^{79} + 16 \beta_1 q^{80} + ( - 29 \beta_{2} + 29) q^{81} + ( - 12 \beta_{3} + 12 \beta_1) q^{82} + 42 \beta_{3} q^{83} + 15 q^{85} + 164 \beta_{2} q^{86} + 41 \beta_1 q^{87} + (8 \beta_{2} - 8) q^{88} + ( - 48 \beta_{3} + 48 \beta_1) q^{89} + 8 \beta_{3} q^{90} - 90 \beta_{2} q^{93} - 18 \beta_1 q^{94} + ( - 30 \beta_{2} + 30) q^{95} - 27 \beta_{3} q^{97} - 4 q^{99}+O(q^{100})$$ q - 2*b2 * q^2 + b1 * q^3 + (-b3 + b1) * q^5 - 2*b3 * q^6 - 8 * q^8 - 4*b2 * q^9 - 2*b1 * q^10 + (-b2 + 1) * q^11 - 9*b3 * q^13 + 5 * q^15 + 16*b2 * q^16 + 3*b1 * q^17 + (8*b2 - 8) * q^18 + (-6*b3 + 6*b1) * q^19 - 2 * q^22 - 8*b2 * q^23 - 8*b1 * q^24 + (-5*b2 + 5) * q^25 + (18*b3 - 18*b1) * q^26 - 13*b3 * q^27 + 41 * q^29 - 10*b2 * q^30 - 18*b1 * q^31 + (-b3 + b1) * q^33 - 6*b3 * q^34 + 28*b2 * q^37 - 12*b1 * q^38 + (-45*b2 + 45) * q^39 + (8*b3 - 8*b1) * q^40 + 6*b3 * q^41 - 82 * q^43 - 4*b1 * q^45 + (16*b2 - 16) * q^46 + (-9*b3 + 9*b1) * q^47 + 16*b3 * q^48 - 10 * q^50 + 15*b2 * q^51 + (74*b2 - 74) * q^53 + (26*b3 - 26*b1) * q^54 - b3 * q^55 + 30 * q^57 - 82*b2 * q^58 + 42*b1 * q^59 + (-36*b3 + 36*b1) * q^61 + 36*b3 * q^62 + 64 * q^64 - 45*b2 * q^65 - 2*b1 * q^66 + (2*b2 - 2) * q^67 - 8*b3 * q^69 + 14 * q^71 + 32*b2 * q^72 + 30*b1 * q^73 + (-56*b2 + 56) * q^74 + (-5*b3 + 5*b1) * q^75 - 90 * q^78 + 19*b2 * q^79 + 16*b1 * q^80 + (-29*b2 + 29) * q^81 + (-12*b3 + 12*b1) * q^82 + 42*b3 * q^83 + 15 * q^85 + 164*b2 * q^86 + 41*b1 * q^87 + (8*b2 - 8) * q^88 + (-48*b3 + 48*b1) * q^89 + 8*b3 * q^90 - 90*b2 * q^93 - 18*b1 * q^94 + (-30*b2 + 30) * q^95 - 27*b3 * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 32 q^{8} - 8 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 32 * q^8 - 8 * q^9 $$4 q - 4 q^{2} - 32 q^{8} - 8 q^{9} + 2 q^{11} + 20 q^{15} + 32 q^{16} - 16 q^{18} - 8 q^{22} - 16 q^{23} + 10 q^{25} + 164 q^{29} - 20 q^{30} + 56 q^{37} + 90 q^{39} - 328 q^{43} - 32 q^{46} - 40 q^{50} + 30 q^{51} - 148 q^{53} + 120 q^{57} - 164 q^{58} + 256 q^{64} - 90 q^{65} - 4 q^{67} + 56 q^{71} + 64 q^{72} + 112 q^{74} - 360 q^{78} + 38 q^{79} + 58 q^{81} + 60 q^{85} + 328 q^{86} - 16 q^{88} - 180 q^{93} + 60 q^{95} - 16 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 - 32 * q^8 - 8 * q^9 + 2 * q^11 + 20 * q^15 + 32 * q^16 - 16 * q^18 - 8 * q^22 - 16 * q^23 + 10 * q^25 + 164 * q^29 - 20 * q^30 + 56 * q^37 + 90 * q^39 - 328 * q^43 - 32 * q^46 - 40 * q^50 + 30 * q^51 - 148 * q^53 + 120 * q^57 - 164 * q^58 + 256 * q^64 - 90 * q^65 - 4 * q^67 + 56 * q^71 + 64 * q^72 + 112 * q^74 - 360 * q^78 + 38 * q^79 + 58 * q^81 + 60 * q^85 + 328 * q^86 - 16 * q^88 - 180 * q^93 + 60 * q^95 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 −1.93649 − 1.11803i 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i
−1.00000 1.73205i −1.93649 1.11803i 0 −1.93649 + 1.11803i 4.47214i 0 −8.00000 −2.00000 3.46410i 3.87298 + 2.23607i
31.2 −1.00000 1.73205i 1.93649 + 1.11803i 0 1.93649 1.11803i 4.47214i 0 −8.00000 −2.00000 3.46410i −3.87298 2.23607i
166.1 −1.00000 + 1.73205i −1.93649 + 1.11803i 0 −1.93649 1.11803i 4.47214i 0 −8.00000 −2.00000 + 3.46410i 3.87298 2.23607i
166.2 −1.00000 + 1.73205i 1.93649 1.11803i 0 1.93649 + 1.11803i 4.47214i 0 −8.00000 −2.00000 + 3.46410i −3.87298 + 2.23607i
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.3.h.a 4
7.b odd 2 1 inner 245.3.h.a 4
7.c even 3 1 35.3.d.b 2
7.c even 3 1 inner 245.3.h.a 4
7.d odd 6 1 35.3.d.b 2
7.d odd 6 1 inner 245.3.h.a 4
21.g even 6 1 315.3.h.a 2
21.h odd 6 1 315.3.h.a 2
28.f even 6 1 560.3.f.b 2
28.g odd 6 1 560.3.f.b 2
35.i odd 6 1 175.3.d.c 2
35.j even 6 1 175.3.d.c 2
35.k even 12 2 175.3.c.c 4
35.l odd 12 2 175.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 7.c even 3 1
35.3.d.b 2 7.d odd 6 1
175.3.c.c 4 35.k even 12 2
175.3.c.c 4 35.l odd 12 2
175.3.d.c 2 35.i odd 6 1
175.3.d.c 2 35.j even 6 1
245.3.h.a 4 1.a even 1 1 trivial
245.3.h.a 4 7.b odd 2 1 inner
245.3.h.a 4 7.c even 3 1 inner
245.3.h.a 4 7.d odd 6 1 inner
315.3.h.a 2 21.g even 6 1
315.3.h.a 2 21.h odd 6 1
560.3.f.b 2 28.f even 6 1
560.3.f.b 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} + 4$$ acting on $$S_{3}^{\mathrm{new}}(245, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4} - 5T^{2} + 25$$
$5$ $$T^{4} - 5T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - T + 1)^{2}$$
$13$ $$(T^{2} + 405)^{2}$$
$17$ $$T^{4} - 45T^{2} + 2025$$
$19$ $$T^{4} - 180 T^{2} + 32400$$
$23$ $$(T^{2} + 8 T + 64)^{2}$$
$29$ $$(T - 41)^{4}$$
$31$ $$T^{4} - 1620 T^{2} + 2624400$$
$37$ $$(T^{2} - 28 T + 784)^{2}$$
$41$ $$(T^{2} + 180)^{2}$$
$43$ $$(T + 82)^{4}$$
$47$ $$T^{4} - 405 T^{2} + 164025$$
$53$ $$(T^{2} + 74 T + 5476)^{2}$$
$59$ $$T^{4} - 8820 T^{2} + 77792400$$
$61$ $$T^{4} - 6480 T^{2} + 41990400$$
$67$ $$(T^{2} + 2 T + 4)^{2}$$
$71$ $$(T - 14)^{4}$$
$73$ $$T^{4} - 4500 T^{2} + 20250000$$
$79$ $$(T^{2} - 19 T + 361)^{2}$$
$83$ $$(T^{2} + 8820)^{2}$$
$89$ $$T^{4} - 11520 T^{2} + 132710400$$
$97$ $$(T^{2} + 3645)^{2}$$