# Properties

 Label 245.4.a.i Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 11$$ x^2 - 11 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 5 q^{3} + (2 \beta + 4) q^{4} - 5 q^{5} + ( - 5 \beta - 5) q^{6} + ( - 2 \beta + 18) q^{8} - 2 q^{9}+O(q^{10})$$ q + (b + 1) * q^2 - 5 * q^3 + (2*b + 4) * q^4 - 5 * q^5 + (-5*b - 5) * q^6 + (-2*b + 18) * q^8 - 2 * q^9 $$q + (\beta + 1) q^{2} - 5 q^{3} + (2 \beta + 4) q^{4} - 5 q^{5} + ( - 5 \beta - 5) q^{6} + ( - 2 \beta + 18) q^{8} - 2 q^{9} + ( - 5 \beta - 5) q^{10} + ( - 4 \beta + 33) q^{11} + ( - 10 \beta - 20) q^{12} + ( - 20 \beta - 5) q^{13} + 25 q^{15} - 36 q^{16} + (20 \beta - 35) q^{17} + ( - 2 \beta - 2) q^{18} + ( - 20 \beta - 70) q^{19} + ( - 10 \beta - 20) q^{20} + (29 \beta - 11) q^{22} + ( - 28 \beta - 8) q^{23} + (10 \beta - 90) q^{24} + 25 q^{25} + ( - 25 \beta - 225) q^{26} + 145 q^{27} + ( - 48 \beta - 129) q^{29} + (25 \beta + 25) q^{30} + (60 \beta + 10) q^{31} + ( - 20 \beta - 180) q^{32} + (20 \beta - 165) q^{33} + ( - 15 \beta + 185) q^{34} + ( - 4 \beta - 8) q^{36} + (44 \beta + 164) q^{37} + ( - 90 \beta - 290) q^{38} + (100 \beta + 25) q^{39} + (10 \beta - 90) q^{40} + (100 \beta - 150) q^{41} + (12 \beta - 58) q^{43} + (50 \beta + 44) q^{44} + 10 q^{45} + ( - 36 \beta - 316) q^{46} + (40 \beta + 15) q^{47} + 180 q^{48} + (25 \beta + 25) q^{50} + ( - 100 \beta + 175) q^{51} + ( - 90 \beta - 460) q^{52} + ( - 120 \beta + 270) q^{53} + (145 \beta + 145) q^{54} + (20 \beta - 165) q^{55} + (100 \beta + 350) q^{57} + ( - 177 \beta - 657) q^{58} + ( - 40 \beta - 190) q^{59} + (50 \beta + 100) q^{60} + (60 \beta - 540) q^{61} + (70 \beta + 670) q^{62} + ( - 200 \beta - 112) q^{64} + (100 \beta + 25) q^{65} + ( - 145 \beta + 55) q^{66} + ( - 96 \beta + 234) q^{67} + (10 \beta + 300) q^{68} + (140 \beta + 40) q^{69} + (64 \beta - 528) q^{71} + (4 \beta - 36) q^{72} + (200 \beta + 430) q^{73} + (208 \beta + 648) q^{74} - 125 q^{75} + ( - 220 \beta - 720) q^{76} + (125 \beta + 1125) q^{78} + (348 \beta + 79) q^{79} + 180 q^{80} - 671 q^{81} + ( - 50 \beta + 950) q^{82} + (200 \beta - 20) q^{83} + ( - 100 \beta + 175) q^{85} + ( - 46 \beta + 74) q^{86} + (240 \beta + 645) q^{87} + ( - 138 \beta + 682) q^{88} + ( - 380 \beta + 120) q^{89} + (10 \beta + 10) q^{90} + ( - 128 \beta - 648) q^{92} + ( - 300 \beta - 50) q^{93} + (55 \beta + 455) q^{94} + (100 \beta + 350) q^{95} + (100 \beta + 900) q^{96} + ( - 180 \beta - 815) q^{97} + (8 \beta - 66) q^{99}+O(q^{100})$$ q + (b + 1) * q^2 - 5 * q^3 + (2*b + 4) * q^4 - 5 * q^5 + (-5*b - 5) * q^6 + (-2*b + 18) * q^8 - 2 * q^9 + (-5*b - 5) * q^10 + (-4*b + 33) * q^11 + (-10*b - 20) * q^12 + (-20*b - 5) * q^13 + 25 * q^15 - 36 * q^16 + (20*b - 35) * q^17 + (-2*b - 2) * q^18 + (-20*b - 70) * q^19 + (-10*b - 20) * q^20 + (29*b - 11) * q^22 + (-28*b - 8) * q^23 + (10*b - 90) * q^24 + 25 * q^25 + (-25*b - 225) * q^26 + 145 * q^27 + (-48*b - 129) * q^29 + (25*b + 25) * q^30 + (60*b + 10) * q^31 + (-20*b - 180) * q^32 + (20*b - 165) * q^33 + (-15*b + 185) * q^34 + (-4*b - 8) * q^36 + (44*b + 164) * q^37 + (-90*b - 290) * q^38 + (100*b + 25) * q^39 + (10*b - 90) * q^40 + (100*b - 150) * q^41 + (12*b - 58) * q^43 + (50*b + 44) * q^44 + 10 * q^45 + (-36*b - 316) * q^46 + (40*b + 15) * q^47 + 180 * q^48 + (25*b + 25) * q^50 + (-100*b + 175) * q^51 + (-90*b - 460) * q^52 + (-120*b + 270) * q^53 + (145*b + 145) * q^54 + (20*b - 165) * q^55 + (100*b + 350) * q^57 + (-177*b - 657) * q^58 + (-40*b - 190) * q^59 + (50*b + 100) * q^60 + (60*b - 540) * q^61 + (70*b + 670) * q^62 + (-200*b - 112) * q^64 + (100*b + 25) * q^65 + (-145*b + 55) * q^66 + (-96*b + 234) * q^67 + (10*b + 300) * q^68 + (140*b + 40) * q^69 + (64*b - 528) * q^71 + (4*b - 36) * q^72 + (200*b + 430) * q^73 + (208*b + 648) * q^74 - 125 * q^75 + (-220*b - 720) * q^76 + (125*b + 1125) * q^78 + (348*b + 79) * q^79 + 180 * q^80 - 671 * q^81 + (-50*b + 950) * q^82 + (200*b - 20) * q^83 + (-100*b + 175) * q^85 + (-46*b + 74) * q^86 + (240*b + 645) * q^87 + (-138*b + 682) * q^88 + (-380*b + 120) * q^89 + (10*b + 10) * q^90 + (-128*b - 648) * q^92 + (-300*b - 50) * q^93 + (55*b + 455) * q^94 + (100*b + 350) * q^95 + (100*b + 900) * q^96 + (-180*b - 815) * q^97 + (8*b - 66) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 10 * q^3 + 8 * q^4 - 10 * q^5 - 10 * q^6 + 36 * q^8 - 4 * q^9 $$2 q + 2 q^{2} - 10 q^{3} + 8 q^{4} - 10 q^{5} - 10 q^{6} + 36 q^{8} - 4 q^{9} - 10 q^{10} + 66 q^{11} - 40 q^{12} - 10 q^{13} + 50 q^{15} - 72 q^{16} - 70 q^{17} - 4 q^{18} - 140 q^{19} - 40 q^{20} - 22 q^{22} - 16 q^{23} - 180 q^{24} + 50 q^{25} - 450 q^{26} + 290 q^{27} - 258 q^{29} + 50 q^{30} + 20 q^{31} - 360 q^{32} - 330 q^{33} + 370 q^{34} - 16 q^{36} + 328 q^{37} - 580 q^{38} + 50 q^{39} - 180 q^{40} - 300 q^{41} - 116 q^{43} + 88 q^{44} + 20 q^{45} - 632 q^{46} + 30 q^{47} + 360 q^{48} + 50 q^{50} + 350 q^{51} - 920 q^{52} + 540 q^{53} + 290 q^{54} - 330 q^{55} + 700 q^{57} - 1314 q^{58} - 380 q^{59} + 200 q^{60} - 1080 q^{61} + 1340 q^{62} - 224 q^{64} + 50 q^{65} + 110 q^{66} + 468 q^{67} + 600 q^{68} + 80 q^{69} - 1056 q^{71} - 72 q^{72} + 860 q^{73} + 1296 q^{74} - 250 q^{75} - 1440 q^{76} + 2250 q^{78} + 158 q^{79} + 360 q^{80} - 1342 q^{81} + 1900 q^{82} - 40 q^{83} + 350 q^{85} + 148 q^{86} + 1290 q^{87} + 1364 q^{88} + 240 q^{89} + 20 q^{90} - 1296 q^{92} - 100 q^{93} + 910 q^{94} + 700 q^{95} + 1800 q^{96} - 1630 q^{97} - 132 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 10 * q^3 + 8 * q^4 - 10 * q^5 - 10 * q^6 + 36 * q^8 - 4 * q^9 - 10 * q^10 + 66 * q^11 - 40 * q^12 - 10 * q^13 + 50 * q^15 - 72 * q^16 - 70 * q^17 - 4 * q^18 - 140 * q^19 - 40 * q^20 - 22 * q^22 - 16 * q^23 - 180 * q^24 + 50 * q^25 - 450 * q^26 + 290 * q^27 - 258 * q^29 + 50 * q^30 + 20 * q^31 - 360 * q^32 - 330 * q^33 + 370 * q^34 - 16 * q^36 + 328 * q^37 - 580 * q^38 + 50 * q^39 - 180 * q^40 - 300 * q^41 - 116 * q^43 + 88 * q^44 + 20 * q^45 - 632 * q^46 + 30 * q^47 + 360 * q^48 + 50 * q^50 + 350 * q^51 - 920 * q^52 + 540 * q^53 + 290 * q^54 - 330 * q^55 + 700 * q^57 - 1314 * q^58 - 380 * q^59 + 200 * q^60 - 1080 * q^61 + 1340 * q^62 - 224 * q^64 + 50 * q^65 + 110 * q^66 + 468 * q^67 + 600 * q^68 + 80 * q^69 - 1056 * q^71 - 72 * q^72 + 860 * q^73 + 1296 * q^74 - 250 * q^75 - 1440 * q^76 + 2250 * q^78 + 158 * q^79 + 360 * q^80 - 1342 * q^81 + 1900 * q^82 - 40 * q^83 + 350 * q^85 + 148 * q^86 + 1290 * q^87 + 1364 * q^88 + 240 * q^89 + 20 * q^90 - 1296 * q^92 - 100 * q^93 + 910 * q^94 + 700 * q^95 + 1800 * q^96 - 1630 * q^97 - 132 * q^99

## 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
 −3.31662 3.31662
−2.31662 −5.00000 −2.63325 −5.00000 11.5831 0 24.6332 −2.00000 11.5831
1.2 4.31662 −5.00000 10.6332 −5.00000 −21.5831 0 11.3668 −2.00000 −21.5831
 $$n$$: e.g. 2-40 or 990-1000 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.i 2
3.b odd 2 1 2205.4.a.x 2
5.b even 2 1 1225.4.a.q 2
7.b odd 2 1 245.4.a.j yes 2
7.c even 3 2 245.4.e.k 4
7.d odd 6 2 245.4.e.j 4
21.c even 2 1 2205.4.a.w 2
35.c odd 2 1 1225.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.i 2 1.a even 1 1 trivial
245.4.a.j yes 2 7.b odd 2 1
245.4.e.j 4 7.d odd 6 2
245.4.e.k 4 7.c even 3 2
1225.4.a.p 2 35.c odd 2 1
1225.4.a.q 2 5.b even 2 1
2205.4.a.w 2 21.c even 2 1
2205.4.a.x 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{2} - 2T_{2} - 10$$ T2^2 - 2*T2 - 10 $$T_{3} + 5$$ T3 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 10$$
$3$ $$(T + 5)^{2}$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 66T + 913$$
$13$ $$T^{2} + 10T - 4375$$
$17$ $$T^{2} + 70T - 3175$$
$19$ $$T^{2} + 140T + 500$$
$23$ $$T^{2} + 16T - 8560$$
$29$ $$T^{2} + 258T - 8703$$
$31$ $$T^{2} - 20T - 39500$$
$37$ $$T^{2} - 328T + 5600$$
$41$ $$T^{2} + 300T - 87500$$
$43$ $$T^{2} + 116T + 1780$$
$47$ $$T^{2} - 30T - 17375$$
$53$ $$T^{2} - 540T - 85500$$
$59$ $$T^{2} + 380T + 18500$$
$61$ $$T^{2} + 1080 T + 252000$$
$67$ $$T^{2} - 468T - 46620$$
$71$ $$T^{2} + 1056 T + 233728$$
$73$ $$T^{2} - 860T - 255100$$
$79$ $$T^{2} - 158 T - 1325903$$
$83$ $$T^{2} + 40T - 439600$$
$89$ $$T^{2} - 240 T - 1574000$$
$97$ $$T^{2} + 1630 T + 307825$$