# Properties

 Label 245.4.a.m Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## 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: yes Analytic conductor: $$14.4554679514$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64$$ x^5 - x^4 - 37*x^3 + 21*x^2 + 288*x + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{3} + 7) q^{4} - 5 q^{5} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 3) q^{6}+ \cdots + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots + 18) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 - 2) * q^3 + (b3 + 7) * q^4 - 5 * q^5 + (b4 + 2*b3 + b2 - 3*b1 + 3) * q^6 + (4*b2 + 5*b1 + 4) * q^8 + (-b4 + b3 - 3*b2 - b1 + 18) * q^9 $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{3} + (\beta_{3} + 7) q^{4} - 5 q^{5} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots + 3) q^{6}+ \cdots + ( - 15 \beta_{4} + \beta_{3} + \cdots + 28) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 - 2) * q^3 + (b3 + 7) * q^4 - 5 * q^5 + (b4 + 2*b3 + b2 - 3*b1 + 3) * q^6 + (4*b2 + 5*b1 + 4) * q^8 + (-b4 + b3 - 3*b2 - b1 + 18) * q^9 - 5*b1 * q^10 + (b4 + b3 - b2 + 7*b1 + 8) * q^11 + (b4 - b3 + 9*b2 + 12*b1 - 26) * q^12 + (-3*b4 - b3 - b2 - b1 + 2) * q^13 + (-5*b2 + 10) * q^15 + (4*b4 + 5*b3 + 4*b2 + 31) * q^16 + (-6*b3 + 2*b2 + 4*b1 - 2) * q^17 + (-3*b4 - 7*b3 - 7*b2 + 29*b1 - 12) * q^18 + (-3*b4 + b3 + 3*b2 + 9*b1 - 6) * q^19 + (-5*b3 - 35) * q^20 + (-b4 + 5*b3 + 11*b2 + 13*b1 + 98) * q^22 + (-3*b4 - 9*b3 + 7*b1 + 40) * q^23 + (b4 + 14*b3 + 5*b2 - 19*b1 + 171) * q^24 + 25 * q^25 + (-b4 - 3*b3 - 29*b2 + 3*b1 + 2) * q^26 + (4*b4 - 6*b3 + 5*b2 - 6*b1 - 106) * q^27 + (3*b4 - 13*b3 - b2 + 31*b1 + 31) * q^29 + (-5*b4 - 10*b3 - 5*b2 + 15*b1 - 15) * q^30 + (-4*b4 - 8*b3 - 12*b2 + 10*b1 + 82) * q^31 + (4*b4 + 8*b3 + 24*b2 + 9*b1 - 32) * q^32 + (8*b4 + 12*b3 + 8*b2 + 16*b1 - 62) * q^33 + (2*b4 + 8*b3 - 22*b2 - 40*b1 + 42) * q^34 + (b4 + 7*b3 - 35*b2 - 33*b1 + 266) * q^36 + (5*b4 - 11*b3 + b2 - 37*b1 - 24) * q^37 + (3*b4 + 15*b3 - 17*b2 + 3*b1 + 172) * q^38 + (2*b4 - 2*b3 + 30*b2 - 80*b1 + 6) * q^39 + (-20*b2 - 25*b1 - 20) * q^40 + (14*b3 + 26*b2 + 2*b1 - 67) * q^41 + (-8*b4 - 4*b3 - 15*b2 - 56*b1 + 134) * q^43 + (3*b4 + 27*b3 + 31*b2 + 63*b1 + 192) * q^44 + (5*b4 - 5*b3 + 15*b2 + 5*b1 - 90) * q^45 + (7*b3 - 60*b2 - 8*b1 + 93) * q^46 + (-9*b4 - 15*b3 + 9*b2 - 33*b1 - 88) * q^47 + (-3*b4 - b3 - 3*b2 + 152*b1 - 14) * q^48 + 25*b1 * q^50 + (-4*b4 + 16*b3 - 12*b2 - 86*b1 + 170) * q^51 + (-5*b4 - 47*b3 - 41*b2 + 23*b1 - 62) * q^52 + (-3*b4 - 13*b3 + 15*b2 + 7*b1 + 64) * q^53 + (5*b4 + 4*b3 + 13*b2 - 155*b1 - 131) * q^54 + (-5*b4 - 5*b3 + 5*b2 - 35*b1 - 40) * q^55 + (10*b4 + 20*b3 + 32*b2 - 90*b1 + 192) * q^57 + (-b4 + 29*b3 - 29*b2 - 52*b1 + 386) * q^58 + (-4*b4 + 8*b3 - 10*b2 + 46*b1 + 138) * q^59 + (-5*b4 + 5*b3 - 45*b2 - 60*b1 + 130) * q^60 + (3*b4 + b3 + b2 + 41*b1 + 379) * q^61 + (-12*b4 - 14*b3 - 76*b2 + 54*b1 + 114) * q^62 + (-8*b4 + 17*b3 + 56*b2 - 16*b1 - 41) * q^64 + (15*b4 + 5*b3 + 5*b2 + 5*b1 - 10) * q^65 + (8*b4 + 32*b3 + 120*b2 - 14*b1 + 248) * q^66 + (-12*b4 + 26*b3 + 59*b2 - 66*b1 + 218) * q^67 + (-22*b4 - 36*b3 + 10*b2 + 76*b1 - 634) * q^68 + (b4 + 23*b3 + 59*b2 - 201*b1 + 91) * q^69 + (-4*b4 - 32*b3 + 10*b2 + 62*b1 + 158) * q^71 + (-11*b4 - 47*b3 + 57*b2 + 109*b1 - 484) * q^72 + (-2*b4 + 8*b3 + 44*b2 + 74*b1 - 362) * q^73 + (b4 - 35*b3 - 3*b2 - 101*b1 - 636) * q^74 + (25*b2 - 50) * q^75 + (7*b4 - 39*b3 + 43*b2 + 201*b1 + 78) * q^76 + (30*b4 - 20*b3 + 38*b2 - 40*b1 - 1134) * q^78 + (-6*b4 - 22*b3 - 96*b2 - 8*b1 + 86) * q^79 + (-20*b4 - 25*b3 - 20*b2 - 155) * q^80 + (6*b4 - 28*b3 - 88*b2 + 64*b1 - 71) * q^81 + (26*b4 + 54*b3 + 82*b2 - 9*b1 + 164) * q^82 + (36*b4 + 22*b3 - 3*b2 + 22*b1 - 318) * q^83 + (30*b3 - 10*b2 - 20*b1 + 10) * q^85 + (-15*b4 - 86*b3 - 95*b2 + 141*b1 - 837) * q^86 + (16*b4 + 74*b3 + 7*b2 - 176*b1 + 104) * q^87 + (39*b4 + 85*b3 + 75*b2 + 213*b1 + 338) * q^88 + (23*b4 + 3*b3 + 63*b2 - 115*b1 + 453) * q^89 + (15*b4 + 35*b3 + 35*b2 - 145*b1 + 60) * q^90 + (-36*b4 - 56*b3 - 32*b2 + 139*b1 - 592) * q^92 + (18*b4 + 16*b3 + 128*b2 - 210*b1 - 474) * q^93 + (9*b4 - 15*b3 - 123*b2 - 169*b1 - 456) * q^94 + (15*b4 - 5*b3 - 15*b2 - 45*b1 + 30) * q^95 + (-11*b4 + 34*b3 - 71*b2 + 141*b1 + 923) * q^96 + (-8*b4 - 24*b3 - 8*b2 - 40*b1 + 134) * q^97 + (-15*b4 + b3 - 83*b2 + 91*b1 + 28) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9}+O(q^{10})$$ 5 * q + q^2 - 8 * q^3 + 35 * q^4 - 25 * q^5 + 16 * q^6 + 33 * q^8 + 81 * q^9 $$5 q + q^{2} - 8 q^{3} + 35 q^{4} - 25 q^{5} + 16 q^{6} + 33 q^{8} + 81 q^{9} - 5 q^{10} + 47 q^{11} - 98 q^{12} + q^{13} + 40 q^{15} + 171 q^{16} - 2 q^{17} - 51 q^{18} - 21 q^{19} - 175 q^{20} + 523 q^{22} + 201 q^{23} + 848 q^{24} + 125 q^{25} - 47 q^{26} - 518 q^{27} + 190 q^{29} - 80 q^{30} + 388 q^{31} - 95 q^{32} - 262 q^{33} + 130 q^{34} + 1229 q^{36} - 145 q^{37} + 835 q^{38} + 14 q^{39} - 165 q^{40} - 281 q^{41} + 568 q^{43} + 1091 q^{44} - 405 q^{45} + 337 q^{46} - 473 q^{47} + 70 q^{48} + 25 q^{50} + 732 q^{51} - 379 q^{52} + 351 q^{53} - 774 q^{54} - 235 q^{55} + 954 q^{57} + 1818 q^{58} + 708 q^{59} + 490 q^{60} + 1944 q^{61} + 448 q^{62} - 125 q^{64} - 5 q^{65} + 1482 q^{66} + 1118 q^{67} - 3118 q^{68} + 374 q^{69} + 864 q^{71} - 2219 q^{72} - 1652 q^{73} - 3285 q^{74} - 200 q^{75} + 691 q^{76} - 5574 q^{78} + 218 q^{79} - 855 q^{80} - 455 q^{81} + 1027 q^{82} - 1502 q^{83} + 10 q^{85} - 4264 q^{86} + 390 q^{87} + 2131 q^{88} + 2322 q^{89} + 255 q^{90} - 2957 q^{92} - 2288 q^{93} - 2677 q^{94} + 105 q^{95} + 4592 q^{96} + 598 q^{97} + 35 q^{99}+O(q^{100})$$ 5 * q + q^2 - 8 * q^3 + 35 * q^4 - 25 * q^5 + 16 * q^6 + 33 * q^8 + 81 * q^9 - 5 * q^10 + 47 * q^11 - 98 * q^12 + q^13 + 40 * q^15 + 171 * q^16 - 2 * q^17 - 51 * q^18 - 21 * q^19 - 175 * q^20 + 523 * q^22 + 201 * q^23 + 848 * q^24 + 125 * q^25 - 47 * q^26 - 518 * q^27 + 190 * q^29 - 80 * q^30 + 388 * q^31 - 95 * q^32 - 262 * q^33 + 130 * q^34 + 1229 * q^36 - 145 * q^37 + 835 * q^38 + 14 * q^39 - 165 * q^40 - 281 * q^41 + 568 * q^43 + 1091 * q^44 - 405 * q^45 + 337 * q^46 - 473 * q^47 + 70 * q^48 + 25 * q^50 + 732 * q^51 - 379 * q^52 + 351 * q^53 - 774 * q^54 - 235 * q^55 + 954 * q^57 + 1818 * q^58 + 708 * q^59 + 490 * q^60 + 1944 * q^61 + 448 * q^62 - 125 * q^64 - 5 * q^65 + 1482 * q^66 + 1118 * q^67 - 3118 * q^68 + 374 * q^69 + 864 * q^71 - 2219 * q^72 - 1652 * q^73 - 3285 * q^74 - 200 * q^75 + 691 * q^76 - 5574 * q^78 + 218 * q^79 - 855 * q^80 - 455 * q^81 + 1027 * q^82 - 1502 * q^83 + 10 * q^85 - 4264 * q^86 + 390 * q^87 + 2131 * q^88 + 2322 * q^89 + 255 * q^90 - 2957 * q^92 - 2288 * q^93 - 2677 * q^94 + 105 * q^95 + 4592 * q^96 + 598 * q^97 + 35 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 37x^{3} + 21x^{2} + 288x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 21\nu - 4 ) / 4$$ (v^3 - 21*v - 4) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 15$$ v^2 - 15 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 29\nu^{2} + 21\nu + 112 ) / 4$$ (v^4 - v^3 - 29*v^2 + 21*v + 112) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 15$$ b3 + 15 $$\nu^{3}$$ $$=$$ $$4\beta_{2} + 21\beta _1 + 4$$ 4*b2 + 21*b1 + 4 $$\nu^{4}$$ $$=$$ $$4\beta_{4} + 29\beta_{3} + 4\beta_{2} + 327$$ 4*b4 + 29*b3 + 4*b2 + 327

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.02529 −2.79706 −0.227497 3.84623 5.20362
−5.02529 −8.34382 17.2535 −5.00000 41.9301 0 −46.5017 42.6193 25.1265
1.2 −2.79706 6.21383 −0.176445 −5.00000 −17.3805 0 22.8700 11.6117 13.9853
1.3 −0.227497 −1.80858 −7.94824 −5.00000 0.411448 0 3.62818 −23.7290 1.13749
1.4 3.84623 −8.96795 6.79345 −5.00000 −34.4927 0 −4.64067 53.4241 −19.2311
1.5 5.20362 4.90652 19.0777 −5.00000 25.5317 0 57.6442 −2.92609 −26.0181
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.a.m 5
3.b odd 2 1 2205.4.a.bu 5
5.b even 2 1 1225.4.a.bg 5
7.b odd 2 1 245.4.a.n 5
7.c even 3 2 35.4.e.c 10
7.d odd 6 2 245.4.e.o 10
21.c even 2 1 2205.4.a.bt 5
21.h odd 6 2 315.4.j.g 10
28.g odd 6 2 560.4.q.n 10
35.c odd 2 1 1225.4.a.bf 5
35.j even 6 2 175.4.e.d 10
35.l odd 12 4 175.4.k.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.c 10 7.c even 3 2
175.4.e.d 10 35.j even 6 2
175.4.k.d 20 35.l odd 12 4
245.4.a.m 5 1.a even 1 1 trivial
245.4.a.n 5 7.b odd 2 1
245.4.e.o 10 7.d odd 6 2
315.4.j.g 10 21.h odd 6 2
560.4.q.n 10 28.g odd 6 2
1225.4.a.bf 5 35.c odd 2 1
1225.4.a.bg 5 5.b even 2 1
2205.4.a.bt 5 21.c even 2 1
2205.4.a.bu 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{5} - T_{2}^{4} - 37T_{2}^{3} + 21T_{2}^{2} + 288T_{2} + 64$$ T2^5 - T2^4 - 37*T2^3 + 21*T2^2 + 288*T2 + 64 $$T_{3}^{5} + 8T_{3}^{4} - 76T_{3}^{3} - 462T_{3}^{2} + 1731T_{3} + 4126$$ T3^5 + 8*T3^4 - 76*T3^3 - 462*T3^2 + 1731*T3 + 4126

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} - T^{4} + \cdots + 64$$
$3$ $$T^{5} + 8 T^{4} + \cdots + 4126$$
$5$ $$(T + 5)^{5}$$
$7$ $$T^{5}$$
$11$ $$T^{5} - 47 T^{4} + \cdots - 1022160$$
$13$ $$T^{5} - T^{4} + \cdots + 307317696$$
$17$ $$T^{5} + 2 T^{4} + \cdots - 10657408$$
$19$ $$T^{5} + \cdots - 1626396544$$
$23$ $$T^{5} + \cdots - 11116980717$$
$29$ $$T^{5} + \cdots - 13190815450$$
$31$ $$T^{5} + \cdots - 2519500032$$
$37$ $$T^{5} + \cdots - 111213849600$$
$41$ $$T^{5} + \cdots + 77000100765$$
$43$ $$T^{5} + \cdots - 72928266842$$
$47$ $$T^{5} + \cdots - 376942584320$$
$53$ $$T^{5} + \cdots + 15657780928$$
$59$ $$T^{5} + \cdots - 9224535040$$
$61$ $$T^{5} + \cdots - 5292345093084$$
$67$ $$T^{5} + \cdots - 17139028321380$$
$71$ $$T^{5} + \cdots + 6439908260352$$
$73$ $$T^{5} + \cdots - 16172766031616$$
$79$ $$T^{5} + \cdots - 32568043234176$$
$83$ $$T^{5} + \cdots - 17832616128012$$
$89$ $$T^{5} + \cdots + 292828216813434$$
$97$ $$T^{5} + \cdots + 863391264288$$