# Properties

 Label 507.4.a.h Level $507$ Weight $4$ Character orbit 507.a Self dual yes Analytic conductor $29.914$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.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: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9}+O(q^{10})$$ q + (b1 - 1) * q^2 + 3 * q^3 + (b2 + 3) * q^4 + (2*b2 - 2) * q^5 + (3*b1 - 3) * q^6 + (-6*b1 - 8) * q^7 + (-2*b2 - b1 + 3) * q^8 + 9 * q^9 $$q + (\beta_1 - 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + (2 \beta_{2} - 2) q^{5} + (3 \beta_1 - 3) q^{6} + ( - 6 \beta_1 - 8) q^{7} + ( - 2 \beta_{2} - \beta_1 + 3) q^{8} + 9 q^{9} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{10} + ( - 6 \beta_{2} - 2 \beta_1 + 8) q^{11} + (3 \beta_{2} + 9) q^{12} + ( - 6 \beta_{2} - 14 \beta_1 - 52) q^{14} + (6 \beta_{2} - 6) q^{15} + ( - 5 \beta_{2} - 6 \beta_1 - 33) q^{16} + ( - 8 \beta_1 - 46) q^{17} + (9 \beta_1 - 9) q^{18} + ( - 16 \beta_{2} + 6 \beta_1 - 28) q^{19} + ( - 2 \beta_{2} - 12 \beta_1 + 86) q^{20} + ( - 18 \beta_1 - 24) q^{21} + (10 \beta_{2} - 18 \beta_1 - 16) q^{22} + ( - 8 \beta_{2} + 32 \beta_1 - 24) q^{23} + ( - 6 \beta_{2} - 3 \beta_1 + 9) q^{24} + ( - 20 \beta_{2} - 24 \beta_1 + 63) q^{25} + 27 q^{27} + ( - 2 \beta_{2} - 42 \beta_1 - 12) q^{28} + (8 \beta_{2} + 20 \beta_1 - 10) q^{29} + ( - 12 \beta_{2} + 18 \beta_1 - 6) q^{30} + (4 \beta_{2} + 54 \beta_1 - 120) q^{31} + (20 \beta_{2} - 51 \beta_1 - 41) q^{32} + ( - 18 \beta_{2} - 6 \beta_1 + 24) q^{33} + ( - 8 \beta_{2} - 54 \beta_1 - 34) q^{34} + ( - 4 \beta_{2} - 36 \beta_1 + 40) q^{35} + (9 \beta_{2} + 27) q^{36} + (28 \beta_{2} + 48 \beta_1 - 150) q^{37} + (38 \beta_{2} - 86 \beta_1 + 120) q^{38} + (24 \beta_{2} + 18 \beta_1 - 186) q^{40} + ( - 34 \beta_{2} + 4 \beta_1 - 150) q^{41} + ( - 18 \beta_{2} - 42 \beta_1 - 156) q^{42} + (4 \beta_{2} - 60 \beta_1 - 68) q^{43} + (10 \beta_{2} + 22 \beta_1 - 248) q^{44} + (18 \beta_{2} - 18) q^{45} + (48 \beta_{2} - 24 \beta_1 + 360) q^{46} + (42 \beta_{2} - 54 \beta_1 + 12) q^{47} + ( - 15 \beta_{2} - 18 \beta_1 - 99) q^{48} + (36 \beta_{2} + 168 \beta_1 + 81) q^{49} + (16 \beta_{2} - 41 \beta_1 - 263) q^{50} + ( - 24 \beta_1 - 138) q^{51} + ( - 12 \beta_{2} - 108 \beta_1 - 186) q^{53} + (27 \beta_1 - 27) q^{54} + (68 \beta_{2} + 60 \beta_1 - 560) q^{55} + (10 \beta_{2} + 50 \beta_1 + 12) q^{56} + ( - 48 \beta_{2} + 18 \beta_1 - 84) q^{57} + (4 \beta_{2} + 42 \beta_1 + 194) q^{58} + ( - 2 \beta_{2} - 82 \beta_1 + 624) q^{59} + ( - 6 \beta_{2} - 36 \beta_1 + 258) q^{60} + ( - 28 \beta_{2} + 24 \beta_1 + 78) q^{61} + (46 \beta_{2} - 50 \beta_1 + 652) q^{62} + ( - 54 \beta_1 - 72) q^{63} + ( - 51 \beta_{2} + 36 \beta_1 - 245) q^{64} + (30 \beta_{2} - 54 \beta_1 - 48) q^{66} + (76 \beta_{2} - 42 \beta_1 - 36) q^{67} + ( - 38 \beta_{2} - 56 \beta_1 - 122) q^{68} + ( - 24 \beta_{2} + 96 \beta_1 - 72) q^{69} + ( - 28 \beta_{2} - 12 \beta_1 - 392) q^{70} + ( - 14 \beta_{2} + 134 \beta_1 + 276) q^{71} + ( - 18 \beta_{2} - 9 \beta_1 + 27) q^{72} + ( - 12 \beta_{2} + 240 \beta_1 - 2) q^{73} + ( - 8 \beta_{2} + 10 \beta_1 + 574) q^{74} + ( - 60 \beta_{2} - 72 \beta_1 + 189) q^{75} + ( - 34 \beta_{2} + 138 \beta_1 - 832) q^{76} + (24 \beta_{2} + 136 \beta_1 - 16) q^{77} + ( - 24 \beta_{2} + 48 \beta_1 - 16) q^{79} + ( - 14 \beta_{2} + 24 \beta_1 - 370) q^{80} + 81 q^{81} + (72 \beta_{2} - 282 \beta_1 + 258) q^{82} + (10 \beta_{2} - 30 \beta_1 + 272) q^{83} + ( - 6 \beta_{2} - 126 \beta_1 - 36) q^{84} + ( - 76 \beta_{2} - 48 \beta_1 + 124) q^{85} + ( - 68 \beta_{2} - 112 \beta_1 - 540) q^{86} + (24 \beta_{2} + 60 \beta_1 - 30) q^{87} + ( - 78 \beta_{2} - 42 \beta_1 + 576) q^{88} + ( - 30 \beta_{2} - 116 \beta_1 - 430) q^{89} + ( - 36 \beta_{2} + 54 \beta_1 - 18) q^{90} + ( - 56 \beta_{2} + 272 \beta_1 - 504) q^{92} + (12 \beta_{2} + 162 \beta_1 - 360) q^{93} + ( - 138 \beta_{2} + 126 \beta_1 - 636) q^{94} + (60 \beta_{2} + 228 \beta_1 - 1440) q^{95} + (60 \beta_{2} - 153 \beta_1 - 123) q^{96} + (4 \beta_{2} + 48 \beta_1 - 1098) q^{97} + (96 \beta_{2} + 393 \beta_1 + 1527) q^{98} + ( - 54 \beta_{2} - 18 \beta_1 + 72) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^2 + 3 * q^3 + (b2 + 3) * q^4 + (2*b2 - 2) * q^5 + (3*b1 - 3) * q^6 + (-6*b1 - 8) * q^7 + (-2*b2 - b1 + 3) * q^8 + 9 * q^9 + (-4*b2 + 6*b1 - 2) * q^10 + (-6*b2 - 2*b1 + 8) * q^11 + (3*b2 + 9) * q^12 + (-6*b2 - 14*b1 - 52) * q^14 + (6*b2 - 6) * q^15 + (-5*b2 - 6*b1 - 33) * q^16 + (-8*b1 - 46) * q^17 + (9*b1 - 9) * q^18 + (-16*b2 + 6*b1 - 28) * q^19 + (-2*b2 - 12*b1 + 86) * q^20 + (-18*b1 - 24) * q^21 + (10*b2 - 18*b1 - 16) * q^22 + (-8*b2 + 32*b1 - 24) * q^23 + (-6*b2 - 3*b1 + 9) * q^24 + (-20*b2 - 24*b1 + 63) * q^25 + 27 * q^27 + (-2*b2 - 42*b1 - 12) * q^28 + (8*b2 + 20*b1 - 10) * q^29 + (-12*b2 + 18*b1 - 6) * q^30 + (4*b2 + 54*b1 - 120) * q^31 + (20*b2 - 51*b1 - 41) * q^32 + (-18*b2 - 6*b1 + 24) * q^33 + (-8*b2 - 54*b1 - 34) * q^34 + (-4*b2 - 36*b1 + 40) * q^35 + (9*b2 + 27) * q^36 + (28*b2 + 48*b1 - 150) * q^37 + (38*b2 - 86*b1 + 120) * q^38 + (24*b2 + 18*b1 - 186) * q^40 + (-34*b2 + 4*b1 - 150) * q^41 + (-18*b2 - 42*b1 - 156) * q^42 + (4*b2 - 60*b1 - 68) * q^43 + (10*b2 + 22*b1 - 248) * q^44 + (18*b2 - 18) * q^45 + (48*b2 - 24*b1 + 360) * q^46 + (42*b2 - 54*b1 + 12) * q^47 + (-15*b2 - 18*b1 - 99) * q^48 + (36*b2 + 168*b1 + 81) * q^49 + (16*b2 - 41*b1 - 263) * q^50 + (-24*b1 - 138) * q^51 + (-12*b2 - 108*b1 - 186) * q^53 + (27*b1 - 27) * q^54 + (68*b2 + 60*b1 - 560) * q^55 + (10*b2 + 50*b1 + 12) * q^56 + (-48*b2 + 18*b1 - 84) * q^57 + (4*b2 + 42*b1 + 194) * q^58 + (-2*b2 - 82*b1 + 624) * q^59 + (-6*b2 - 36*b1 + 258) * q^60 + (-28*b2 + 24*b1 + 78) * q^61 + (46*b2 - 50*b1 + 652) * q^62 + (-54*b1 - 72) * q^63 + (-51*b2 + 36*b1 - 245) * q^64 + (30*b2 - 54*b1 - 48) * q^66 + (76*b2 - 42*b1 - 36) * q^67 + (-38*b2 - 56*b1 - 122) * q^68 + (-24*b2 + 96*b1 - 72) * q^69 + (-28*b2 - 12*b1 - 392) * q^70 + (-14*b2 + 134*b1 + 276) * q^71 + (-18*b2 - 9*b1 + 27) * q^72 + (-12*b2 + 240*b1 - 2) * q^73 + (-8*b2 + 10*b1 + 574) * q^74 + (-60*b2 - 72*b1 + 189) * q^75 + (-34*b2 + 138*b1 - 832) * q^76 + (24*b2 + 136*b1 - 16) * q^77 + (-24*b2 + 48*b1 - 16) * q^79 + (-14*b2 + 24*b1 - 370) * q^80 + 81 * q^81 + (72*b2 - 282*b1 + 258) * q^82 + (10*b2 - 30*b1 + 272) * q^83 + (-6*b2 - 126*b1 - 36) * q^84 + (-76*b2 - 48*b1 + 124) * q^85 + (-68*b2 - 112*b1 - 540) * q^86 + (24*b2 + 60*b1 - 30) * q^87 + (-78*b2 - 42*b1 + 576) * q^88 + (-30*b2 - 116*b1 - 430) * q^89 + (-36*b2 + 54*b1 - 18) * q^90 + (-56*b2 + 272*b1 - 504) * q^92 + (12*b2 + 162*b1 - 360) * q^93 + (-138*b2 + 126*b1 - 636) * q^94 + (60*b2 + 228*b1 - 1440) * q^95 + (60*b2 - 153*b1 - 123) * q^96 + (4*b2 + 48*b1 - 1098) * q^97 + (96*b2 + 393*b1 + 1527) * q^98 + (-54*b2 - 18*b1 + 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q - 2 * q^2 + 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 $$3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 4 q^{10} + 16 q^{11} + 30 q^{12} - 176 q^{14} - 12 q^{15} - 110 q^{16} - 146 q^{17} - 18 q^{18} - 94 q^{19} + 244 q^{20} - 90 q^{21} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} + 81 q^{27} - 80 q^{28} - 2 q^{29} - 12 q^{30} - 302 q^{31} - 154 q^{32} + 48 q^{33} - 164 q^{34} + 80 q^{35} + 90 q^{36} - 374 q^{37} + 312 q^{38} - 516 q^{40} - 480 q^{41} - 528 q^{42} - 260 q^{43} - 712 q^{44} - 36 q^{45} + 1104 q^{46} + 24 q^{47} - 330 q^{48} + 447 q^{49} - 814 q^{50} - 438 q^{51} - 678 q^{53} - 54 q^{54} - 1552 q^{55} + 96 q^{56} - 282 q^{57} + 628 q^{58} + 1788 q^{59} + 732 q^{60} + 230 q^{61} + 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} - 460 q^{68} - 144 q^{69} - 1216 q^{70} + 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 435 q^{75} - 2392 q^{76} + 112 q^{77} - 24 q^{79} - 1100 q^{80} + 243 q^{81} + 564 q^{82} + 796 q^{83} - 240 q^{84} + 248 q^{85} - 1800 q^{86} - 6 q^{87} + 1608 q^{88} - 1436 q^{89} - 36 q^{90} - 1296 q^{92} - 906 q^{93} - 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} + 5070 q^{98} + 144 q^{99}+O(q^{100})$$ 3 * q - 2 * q^2 + 9 * q^3 + 10 * q^4 - 4 * q^5 - 6 * q^6 - 30 * q^7 + 6 * q^8 + 27 * q^9 - 4 * q^10 + 16 * q^11 + 30 * q^12 - 176 * q^14 - 12 * q^15 - 110 * q^16 - 146 * q^17 - 18 * q^18 - 94 * q^19 + 244 * q^20 - 90 * q^21 - 56 * q^22 - 48 * q^23 + 18 * q^24 + 145 * q^25 + 81 * q^27 - 80 * q^28 - 2 * q^29 - 12 * q^30 - 302 * q^31 - 154 * q^32 + 48 * q^33 - 164 * q^34 + 80 * q^35 + 90 * q^36 - 374 * q^37 + 312 * q^38 - 516 * q^40 - 480 * q^41 - 528 * q^42 - 260 * q^43 - 712 * q^44 - 36 * q^45 + 1104 * q^46 + 24 * q^47 - 330 * q^48 + 447 * q^49 - 814 * q^50 - 438 * q^51 - 678 * q^53 - 54 * q^54 - 1552 * q^55 + 96 * q^56 - 282 * q^57 + 628 * q^58 + 1788 * q^59 + 732 * q^60 + 230 * q^61 + 1952 * q^62 - 270 * q^63 - 750 * q^64 - 168 * q^66 - 74 * q^67 - 460 * q^68 - 144 * q^69 - 1216 * q^70 + 948 * q^71 + 54 * q^72 + 222 * q^73 + 1724 * q^74 + 435 * q^75 - 2392 * q^76 + 112 * q^77 - 24 * q^79 - 1100 * q^80 + 243 * q^81 + 564 * q^82 + 796 * q^83 - 240 * q^84 + 248 * q^85 - 1800 * q^86 - 6 * q^87 + 1608 * q^88 - 1436 * q^89 - 36 * q^90 - 1296 * q^92 - 906 * q^93 - 1920 * q^94 - 4032 * q^95 - 462 * q^96 - 3242 * q^97 + 5070 * q^98 + 144 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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
 −3.20905 −0.526440 4.73549
−4.20905 3.00000 9.71610 11.4322 −12.6271 11.2543 −7.22315 9.00000 −48.1187
1.2 −1.52644 3.00000 −5.66998 −19.3400 −4.57932 −4.84136 20.8664 9.00000 29.5213
1.3 3.73549 3.00000 5.95388 3.90776 11.2065 −36.4129 −7.64325 9.00000 14.5974
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 507.4.a.h 3
3.b odd 2 1 1521.4.a.u 3
13.b even 2 1 39.4.a.c 3
13.d odd 4 2 507.4.b.g 6
39.d odd 2 1 117.4.a.f 3
52.b odd 2 1 624.4.a.t 3
65.d even 2 1 975.4.a.l 3
91.b odd 2 1 1911.4.a.k 3
104.e even 2 1 2496.4.a.bl 3
104.h odd 2 1 2496.4.a.bp 3
156.h even 2 1 1872.4.a.bk 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 13.b even 2 1
117.4.a.f 3 39.d odd 2 1
507.4.a.h 3 1.a even 1 1 trivial
507.4.b.g 6 13.d odd 4 2
624.4.a.t 3 52.b odd 2 1
975.4.a.l 3 65.d even 2 1
1521.4.a.u 3 3.b odd 2 1
1872.4.a.bk 3 156.h even 2 1
1911.4.a.k 3 91.b odd 2 1
2496.4.a.bl 3 104.e even 2 1
2496.4.a.bp 3 104.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}^{3} + 2T_{2}^{2} - 15T_{2} - 24$$ T2^3 + 2*T2^2 - 15*T2 - 24 $$T_{5}^{3} + 4T_{5}^{2} - 252T_{5} + 864$$ T5^3 + 4*T5^2 - 252*T5 + 864

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} - 15 T - 24$$
$3$ $$(T - 3)^{3}$$
$5$ $$T^{3} + 4 T^{2} - 252 T + 864$$
$7$ $$T^{3} + 30 T^{2} - 288 T - 1984$$
$11$ $$T^{3} - 16 T^{2} - 2256 T - 30336$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 146 T^{2} + 6060 T + 71256$$
$19$ $$T^{3} + 94 T^{2} - 14432 T - 779616$$
$23$ $$T^{3} + 48 T^{2} - 20928 T + 534528$$
$29$ $$T^{3} + 2 T^{2} - 10116 T - 199176$$
$31$ $$T^{3} + 302 T^{2} - 17536 T - 7197248$$
$37$ $$T^{3} + 374 T^{2} - 36964 T - 7758104$$
$41$ $$T^{3} + 480 T^{2} + \cdots - 12919824$$
$43$ $$T^{3} + 260 T^{2} - 38096 T - 3663168$$
$47$ $$T^{3} - 24 T^{2} - 168480 T - 18102528$$
$53$ $$T^{3} + 678 T^{2} - 42228 T - 1471608$$
$59$ $$T^{3} - 1788 T^{2} + \cdots - 137423808$$
$61$ $$T^{3} - 230 T^{2} - 44452 T + 6279512$$
$67$ $$T^{3} + 74 T^{2} - 409216 T - 4260896$$
$71$ $$T^{3} - 948 T^{2} + \cdots + 70464384$$
$73$ $$T^{3} - 222 T^{2} + \cdots - 22780552$$
$79$ $$T^{3} + 24 T^{2} - 78336 T + 7757824$$
$83$ $$T^{3} - 796 T^{2} + \cdots - 13963968$$
$89$ $$T^{3} + 1436 T^{2} + \cdots + 30129888$$
$97$ $$T^{3} + 3242 T^{2} + \cdots + 1218481048$$