# Properties

 Label 245.4.a.p Level $245$ Weight $4$ Character orbit 245.a Self dual yes Analytic conductor $14.455$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.1163891200.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ x^6 - 2*x^5 - 23*x^4 + 12*x^3 + 154*x^2 + 152*x + 28 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 7$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + ( - 6 \beta_{5} + 3 \beta_{4} + \cdots + 16) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b5 + 3) * q^3 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 - 5 * q^5 + (b3 + 2*b2 - 4*b1 + 6) * q^6 + (2*b5 + 3*b4 + 5*b3 - b2 - 9*b1 - 8) * q^8 + (-6*b5 + 3*b4 + b3 - b2 - 5*b1 + 16) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 2) q^{4}+ \cdots + (26 \beta_{5} + 244 \beta_{4} + \cdots - 536) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b5 + 3) * q^3 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 - 5 * q^5 + (b3 + 2*b2 - 4*b1 + 6) * q^6 + (2*b5 + 3*b4 + 5*b3 - b2 - 9*b1 - 8) * q^8 + (-6*b5 + 3*b4 + b3 - b2 - 5*b1 + 16) * q^9 + 5*b1 * q^10 + (-2*b5 - 5*b4 - b3 - 5*b2 + 5*b1 - 7) * q^11 + (-2*b5 - 5*b4 - 8*b3 + 13*b2 + 6*b1 + 28) * q^12 + (-5*b4 - 8*b3 + 4*b2 + 2*b1 + 27) * q^13 + (5*b5 - 15) * q^15 + (-12*b5 - 3*b4 - 15*b3 + b2 + 27*b1 + 44) * q^16 + (-b5 - 2*b4 + 4*b3 - 10*b2 - 16*b1 + 1) * q^17 + (-8*b5 - 3*b4 - 3*b3 + 21*b2 - b1 + 68) * q^18 + (-3*b5 + 3*b4 + 4*b3 - 10*b2 - 6*b1 + 52) * q^19 + (5*b5 + 5*b4 + 5*b3 - 10*b2 - 15*b1 - 10) * q^20 + (22*b5 + b4 + 19*b3 - 17*b2 - 12*b1 - 48) * q^22 + (-6*b5 + 8*b4 - 9*b3 - 11*b2 + 3*b1 - 56) * q^23 + (b5 + 9*b4 + 8*b3 - 6*b2 - 50*b1 + 2) * q^24 + 25 * q^25 + (15*b5 + 5*b4 + 19*b3 - 4*b2 - 58*b1 + 24) * q^26 + (-12*b5 + 21*b4 + 16*b3 - 10*b2 - 50*b1 + 185) * q^27 + (-8*b5 - b4 + 13*b3 + 7*b2 + 39*b1 + 21) * q^29 + (-5*b3 - 10*b2 + 20*b1 - 30) * q^30 + (10*b5 - 17*b3 + 29*b2 + 11*b1 + 68) * q^31 + (42*b5 + 15*b4 + 31*b3 - 35*b2 - 99*b1 - 116) * q^32 + (8*b5 + 23*b4 + 8*b3 - 44*b2 + 26*b1 + 13) * q^33 + (-2*b5 - 22*b4 - 11*b3 + 18*b2 + 62*b1 + 114) * q^34 + (14*b5 - 25*b4 - 13*b3 + 65*b2 - 63*b1 + 68) * q^36 + (4*b5 + 28*b4 - 9*b3 - 11*b2 - 33*b1 - 8) * q^37 + (3*b5 - 7*b4 - 4*b3 + 2*b2 - 16*b1 + 6) * q^38 + (-6*b5 - 31*b4 - 11*b3 + 57*b2 + 39*b1 + 21) * q^39 + (-10*b5 - 15*b4 - 25*b3 + 5*b2 + 45*b1 + 40) * q^40 + (37*b5 - 15*b4 - 3*b3 - 7*b2 + 11*b1 + 86) * q^41 + (2*b5 - 34*b4 - 16*b3 + 8*b2 - 74) * q^43 + (-b5 + 10*b4 - 48*b3 + 5*b2 + 122*b1 - 62) * q^44 + (30*b5 - 15*b4 - 5*b3 + 5*b2 + 25*b1 - 80) * q^45 + (35*b5 + 20*b4 + 30*b3 - 25*b2 + 42*b1 - 92) * q^46 + (-2*b5 - 9*b4 + 32*b3 - 14*b2 + 50*b1 + 81) * q^47 + (-47*b5 - 9*b4 - 6*b3 - 10*b2 + 132*b1 + 198) * q^48 - 25*b1 * q^50 + (-10*b5 + 47*b4 + 41*b3 - 59*b2 - 65*b1 + 145) * q^51 + (-93*b5 - 32*b4 - 48*b3 + 65*b2 + 196*b1 + 230) * q^52 + (2*b5 - 60*b4 - 19*b3 + 35*b2 + 73*b1 - 146) * q^53 + (-83*b5 - 45*b4 - 81*b3 + 120*b2 + 16*b1 + 428) * q^54 + (10*b5 + 25*b4 + 5*b3 + 25*b2 - 25*b1 + 35) * q^55 + (-72*b5 + 48*b4 + 38*b3 - 86*b2 - 30*b1 + 266) * q^57 + (25*b4 + 15*b3 - 35*b2 - 128*b1 - 296) * q^58 + (-11*b5 + 59*b4 + 47*b3 - 33*b2 + 49*b1 + 162) * q^59 + (10*b5 + 25*b4 + 40*b3 - 65*b2 - 30*b1 - 140) * q^60 + (24*b5 - 10*b4 - 4*b3 + 48*b2 + 132*b1 + 84) * q^61 + (-13*b5 + 28*b4 + 6*b3 - b2 - 154*b1 + 4) * q^62 + (-10*b5 - 91*b4 - 63*b3 + 67*b2 + 351*b1 + 116) * q^64 + (25*b4 + 40*b3 - 20*b2 - 10*b1 - 135) * q^65 + (75*b5 + 41*b4 + 23*b3 - 148*b2 - 664) * q^66 + (-12*b5 + 12*b4 - 35*b3 + 31*b2 - 51*b1 + 330) * q^67 + (78*b5 + 67*b4 + 58*b3 - 15*b2 - 236*b1 - 420) * q^68 + (48*b5 + 16*b4 + 53*b3 - 81*b2 + 77*b1 - 238) * q^69 + (-14*b5 - 92*b3 + 60*b2 - 116*b1 + 26) * q^71 + (-78*b5 - 51*b4 - 67*b3 + 47*b2 + 27*b1 + 492) * q^72 + (-11*b5 - 85*b4 + 46*b3 + 60*b2 + 64*b1 + 414) * q^73 + (-21*b5 + 4*b4 - 36*b3 + 27*b2 + 132*b1 + 128) * q^74 + (-25*b5 + 75) * q^75 + (19*b5 - 43*b4 - 38*b3 + 106*b2 + 76*b1 - 234) * q^76 + (-22*b5 + 19*b4 + 41*b3 + 37*b2 - 254*b1 + 112) * q^78 + (10*b5 - 55*b4 + 65*b3 + 25*b2 + 67*b1 + 237) * q^79 + (60*b5 + 15*b4 + 75*b3 - 5*b2 - 135*b1 - 220) * q^80 + (-112*b5 + 12*b4 + 70*b3 - 34*b2 - 170*b1 + 827) * q^81 + (46*b5 - b4 + 2*b3 - 113*b2 - 96*b1 - 314) * q^82 + (-69*b5 - 41*b4 - 154*b3 + 80*b2 - 32*b1 + 70) * q^83 + (5*b5 + 10*b4 - 20*b3 + 50*b2 + 80*b1 - 5) * q^85 + (50*b5 - 18*b4 + 56*b3 - 4*b2 + 2*b1 + 172) * q^86 + (-70*b5 + 43*b4 - 66*b3 - 68*b2 + 16*b1 + 333) * q^87 + (22*b5 + 172*b4 + 52*b3 - 144*b2 - 300*b1 - 840) * q^88 + (-33*b5 - 79*b4 - 80*b3 - 62*b2 + 74*b1 - 378) * q^89 + (40*b5 + 15*b4 + 15*b3 - 105*b2 + 5*b1 - 340) * q^90 + (60*b5 - 32*b4 + 24*b3 - 86*b2 + 82*b1 - 412) * q^92 + (-4*b5 - 168*b4 - 91*b3 + 271*b2 + 109*b1 - 266) * q^93 + (23*b5 + 9*b4 + 11*b3 - 92*b2 - 164*b1 - 536) * q^94 + (15*b5 - 15*b4 - 20*b3 + 50*b2 + 30*b1 - 260) * q^95 + (165*b5 + 57*b4 + 146*b3 - 148*b2 - 252*b1 - 1078) * q^96 + (-77*b5 - 84*b4 + 14*b3 + 72*b2 + 134*b1 + 63) * q^97 + (26*b5 + 244*b4 + 140*b3 - 308*b2 + 116*b1 - 536) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 + 16 * q^3 + 14 * q^4 - 30 * q^5 + 24 * q^6 - 66 * q^8 + 70 * q^9 $$6 q - 2 q^{2} + 16 q^{3} + 14 q^{4} - 30 q^{5} + 24 q^{6} - 66 q^{8} + 70 q^{9} + 10 q^{10} - 16 q^{11} + 160 q^{12} + 168 q^{13} - 80 q^{15} + 298 q^{16} - 4 q^{17} + 354 q^{18} + 308 q^{19} - 70 q^{20} - 236 q^{22} - 336 q^{23} - 92 q^{24} + 150 q^{25} + 56 q^{26} + 964 q^{27} + 176 q^{29} - 120 q^{30} + 392 q^{31} - 770 q^{32} + 188 q^{33} + 812 q^{34} + 230 q^{36} - 140 q^{37} + 20 q^{38} + 140 q^{39} + 330 q^{40} + 656 q^{41} - 388 q^{43} - 160 q^{44} - 350 q^{45} - 388 q^{46} + 628 q^{47} + 1396 q^{48} - 50 q^{50} + 744 q^{51} + 1520 q^{52} - 676 q^{53} + 2284 q^{54} + 80 q^{55} + 1468 q^{57} - 2012 q^{58} + 996 q^{59} - 800 q^{60} + 740 q^{61} - 364 q^{62} + 1426 q^{64} - 840 q^{65} - 3620 q^{66} + 1768 q^{67} - 2940 q^{68} - 1048 q^{69} - 224 q^{71} + 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 400 q^{75} - 1340 q^{76} + 8 q^{78} + 1636 q^{79} - 1490 q^{80} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 20 q^{85} + 1180 q^{86} + 1940 q^{87} - 5652 q^{88} - 1904 q^{89} - 1770 q^{90} - 1952 q^{92} - 1592 q^{93} - 3332 q^{94} - 1540 q^{95} - 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 + 16 * q^3 + 14 * q^4 - 30 * q^5 + 24 * q^6 - 66 * q^8 + 70 * q^9 + 10 * q^10 - 16 * q^11 + 160 * q^12 + 168 * q^13 - 80 * q^15 + 298 * q^16 - 4 * q^17 + 354 * q^18 + 308 * q^19 - 70 * q^20 - 236 * q^22 - 336 * q^23 - 92 * q^24 + 150 * q^25 + 56 * q^26 + 964 * q^27 + 176 * q^29 - 120 * q^30 + 392 * q^31 - 770 * q^32 + 188 * q^33 + 812 * q^34 + 230 * q^36 - 140 * q^37 + 20 * q^38 + 140 * q^39 + 330 * q^40 + 656 * q^41 - 388 * q^43 - 160 * q^44 - 350 * q^45 - 388 * q^46 + 628 * q^47 + 1396 * q^48 - 50 * q^50 + 744 * q^51 + 1520 * q^52 - 676 * q^53 + 2284 * q^54 + 80 * q^55 + 1468 * q^57 - 2012 * q^58 + 996 * q^59 - 800 * q^60 + 740 * q^61 - 364 * q^62 + 1426 * q^64 - 840 * q^65 - 3620 * q^66 + 1768 * q^67 - 2940 * q^68 - 1048 * q^69 - 224 * q^71 + 2858 * q^72 + 2640 * q^73 + 928 * q^74 + 400 * q^75 - 1340 * q^76 + 8 * q^78 + 1636 * q^79 - 1490 * q^80 + 4442 * q^81 - 1756 * q^82 + 140 * q^83 + 20 * q^85 + 1180 * q^86 + 1940 * q^87 - 5652 * q^88 - 1904 * q^89 - 1770 * q^90 - 1952 * q^92 - 1592 * q^93 - 3332 * q^94 - 1540 * q^95 - 6460 * q^96 + 516 * q^97 - 2804 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26$$ (-v^5 + 23*v^3 + 8*v^2 - 86*v - 64) / 26 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26$$ (v^5 - 23*v^3 + 18*v^2 + 60*v - 144) / 26 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26$$ (-5*v^5 + 26*v^4 + 89*v^3 - 298*v^2 - 638*v - 164) / 26 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26$$ (-v^5 + 49*v^3 + 8*v^2 - 476*v - 428) / 26 $$\beta_{5}$$ $$=$$ $$( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26$$ (9*v^5 - 26*v^4 - 155*v^3 + 240*v^2 + 800*v + 264) / 26
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7$$ (b5 - b4 + b3 + b2 + 6*b1) / 7 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7$$ (b5 - b4 + b3 + 8*b2 + 13*b1 + 56) / 7 $$\nu^{3}$$ $$=$$ $$( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7$$ (15*b5 - 8*b4 + 15*b3 + 15*b2 + 83*b1 + 98) / 7 $$\nu^{4}$$ $$=$$ $$( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7$$ (36*b5 - 29*b4 + 43*b3 + 127*b2 + 265*b1 + 784) / 7 $$\nu^{5}$$ $$=$$ $$( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7$$ (267*b5 - 106*b4 + 267*b3 + 323*b2 + 1315*b1 + 2254) / 7

## 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
 4.10376 4.29508 −1.04490 −2.05886 −0.241849 −3.05323
−5.51797 3.86039 22.4480 −5.00000 −21.3015 0 −79.7239 −12.0974 27.5899
1.2 −2.88087 −2.89052 0.299392 −5.00000 8.32721 0 22.1844 −18.6449 14.4043
1.3 −0.369315 9.74070 −7.86361 −5.00000 −3.59738 0 5.85867 67.8812 1.84657
1.4 0.644648 −4.18687 −7.58443 −5.00000 −2.69906 0 −10.0465 −9.47008 −3.22324
1.5 1.65606 −0.332888 −5.25746 −5.00000 −0.551283 0 −21.9552 −26.8892 −8.28031
1.6 4.46745 9.80920 11.9581 −5.00000 43.8221 0 17.6824 69.2204 −22.3372
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.p yes 6
3.b odd 2 1 2205.4.a.ca 6
5.b even 2 1 1225.4.a.bi 6
7.b odd 2 1 245.4.a.o 6
7.c even 3 2 245.4.e.p 12
7.d odd 6 2 245.4.e.q 12
21.c even 2 1 2205.4.a.bz 6
35.c odd 2 1 1225.4.a.bj 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 7.b odd 2 1
245.4.a.p yes 6 1.a even 1 1 trivial
245.4.e.p 12 7.c even 3 2
245.4.e.q 12 7.d odd 6 2
1225.4.a.bi 6 5.b even 2 1
1225.4.a.bj 6 35.c odd 2 1
2205.4.a.bz 6 21.c even 2 1
2205.4.a.ca 6 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{6} + 2T_{2}^{5} - 29T_{2}^{4} - 28T_{2}^{3} + 134T_{2}^{2} - 24T_{2} - 28$$ T2^6 + 2*T2^5 - 29*T2^4 - 28*T2^3 + 134*T2^2 - 24*T2 - 28 $$T_{3}^{6} - 16T_{3}^{5} + 12T_{3}^{4} + 564T_{3}^{3} - 355T_{3}^{2} - 4644T_{3} - 1486$$ T3^6 - 16*T3^5 + 12*T3^4 + 564*T3^3 - 355*T3^2 - 4644*T3 - 1486

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + \cdots - 28$$
$3$ $$T^{6} - 16 T^{5} + \cdots - 1486$$
$5$ $$(T + 5)^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + \cdots - 9225436100$$
$13$ $$T^{6} + \cdots + 12513937372$$
$17$ $$T^{6} + \cdots - 3807091762$$
$19$ $$T^{6} + \cdots + 1617325344$$
$23$ $$T^{6} + 336 T^{5} + \cdots + 298897696$$
$29$ $$T^{6} + \cdots - 544215793700$$
$31$ $$T^{6} + \cdots + 16539893268192$$
$37$ $$T^{6} + \cdots + 271258136464$$
$41$ $$T^{6} + \cdots + 144691772208184$$
$43$ $$T^{6} + \cdots + 440374360000$$
$47$ $$T^{6} + \cdots + 251564448569400$$
$53$ $$T^{6} + \cdots + 590408333736048$$
$59$ $$T^{6} + \cdots + 14\!\cdots\!04$$
$61$ $$T^{6} + \cdots - 842000334839552$$
$67$ $$T^{6} + \cdots + 885207397312$$
$71$ $$T^{6} + \cdots + 12\!\cdots\!88$$
$73$ $$T^{6} + \cdots - 85\!\cdots\!92$$
$79$ $$T^{6} + \cdots - 12\!\cdots\!68$$
$83$ $$T^{6} + \cdots + 22\!\cdots\!00$$
$89$ $$T^{6} + \cdots + 45\!\cdots\!76$$
$97$ $$T^{6} + \cdots - 510868966648482$$