# Properties

 Label 507.4.a.r Level $507$ Weight $4$ Character orbit 507.a Self dual yes Analytic conductor $29.914$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# 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: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648$$ x^10 - 70*x^8 + 1645*x^6 - 14700*x^4 + 44100*x^2 - 27648 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \cdots + 7 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 + 3 * q^3 + (b4 + 6) * q^4 + (-b8 + b2 + b1) * q^5 + 3*b1 * q^6 + (-b9 + 2*b2) * q^7 + (b9 - b8 + b6 + 7*b1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{4} + 6) q^{4} + ( - \beta_{8} + \beta_{2} + \beta_1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{9} + 2 \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \cdots + 7 \beta_1) q^{8}+ \cdots + ( - 9 \beta_{9} - 9 \beta_{8} + \cdots + 45 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + 3 * q^3 + (b4 + 6) * q^4 + (-b8 + b2 + b1) * q^5 + 3*b1 * q^6 + (-b9 + 2*b2) * q^7 + (b9 - b8 + b6 + 7*b1) * q^8 + 9 * q^9 + (-b7 + b5 + 2*b4 + 3*b3 + 8) * q^10 + (-b9 - b8 + 2*b6 - 4*b2 + 5*b1) * q^11 + (3*b4 + 18) * q^12 + (-2*b5 - 5*b4 + b3 - 6) * q^14 + (-3*b8 + 3*b2 + 3*b1) * q^15 + (2*b5 + 7*b4 - 6*b3 + 50) * q^16 + (b7 + 4*b3 + 21) * q^17 + 9*b1 * q^18 + (-b9 + 3*b8 - 4*b6 - 12*b2 + 9*b1) * q^19 + (4*b9 - 2*b8 - 5*b6 + 34*b2 + 18*b1) * q^20 + (-3*b9 + 6*b2) * q^21 + (b7 - 3*b5 + 5*b4 - 21*b3 + 58) * q^22 + (3*b7 - b5 - b4 - b3 - 12) * q^23 + (3*b9 - 3*b8 + 3*b6 + 21*b1) * q^24 + (b5 - b4 + 21*b3 + 96) * q^25 + 27 * q^27 + (-b9 + 9*b8 - 4*b6 + 10*b2 - 45*b1) * q^28 + (-3*b5 + 3*b4 - 11*b3 + 99) * q^29 + (-3*b7 + 3*b5 + 6*b4 + 9*b3 + 24) * q^30 + (-2*b9 - 3*b8 - 6*b6 - 22*b2 - 9*b1) * q^31 + (3*b9 - 3*b8 + 3*b6 - 96*b2 + 51*b1) * q^32 + (-3*b9 - 3*b8 + 6*b6 - 12*b2 + 15*b1) * q^33 + (6*b8 - b6 + 50*b2 + 18*b1) * q^34 + (-3*b7 + 5*b5 - 13*b4 - 35*b3 - 12) * q^35 + (9*b4 + 54) * q^36 + (-b9 + 12*b8 + 27*b2 - 36*b1) * q^37 + (-b7 - b5 - 7*b4 + 17*b3 + 138) * q^38 + (b7 + 7*b5 + 14*b4 + 63*b3 + 200) * q^40 + (3*b9 - 8*b8 + 6*b6 + 71*b2 - 32*b1) * q^41 + (-6*b5 - 15*b4 + 3*b3 - 18) * q^42 + (-b7 - 10*b5 - 4*b4 - 18*b3 - 74) * q^43 + (7*b9 + 15*b8 + 16*b6 - 250*b2 + 69*b1) * q^44 + (-9*b8 + 9*b2 + 9*b1) * q^45 + (-3*b9 + 21*b8 + 10*b6 - 26*b2 - 27*b1) * q^46 + (b9 - 3*b8 + 4*b6 - 4*b2 + 47*b1) * q^47 + (6*b5 + 21*b4 - 18*b3 + 150) * q^48 + (-2*b7 - 7*b5 - 19*b4 - 3*b3 + 155) * q^49 + (b9 - b8 - 23*b6 + 288*b2 + 84*b1) * q^50 + (3*b7 + 12*b3 + 63) * q^51 + (-4*b7 + 9*b5 + 3*b4 - 3*b3 + 33) * q^53 + 27*b1 * q^54 + (b7 + 11*b5 - 35*b4 - 33*b3 + 52) * q^55 + (5*b7 + 9*b5 - 27*b4 + 19*b3 - 534) * q^56 + (-3*b9 + 9*b8 - 12*b6 - 36*b2 + 27*b1) * q^57 + (-3*b9 + 3*b8 + 17*b6 - 136*b2 + 135*b1) * q^58 + (-4*b9 - 6*b8 - 16*b6 + 52*b2 + 26*b1) * q^59 + (12*b9 - 6*b8 - 15*b6 + 102*b2 + 54*b1) * q^60 + (b7 - 3*b5 - 7*b4 + 9*b3 + 275) * q^61 + (-9*b7 + 5*b5 - 28*b4 + 36*b3 - 156) * q^62 + (-9*b9 + 18*b2) * q^63 + (-10*b5 + 19*b4 - 66*b3 + 314) * q^64 + (3*b7 - 9*b5 + 15*b4 - 63*b3 + 174) * q^66 + (-3*b9 + 12*b8 + 14*b6 + 106*b2 - 36*b1) * q^67 + (-3*b7 - 5*b5 + 10*b4 + 15*b3 + 120) * q^68 + (9*b7 - 3*b5 - 3*b4 - 3*b3 - 36) * q^69 + (-3*b9 - 15*b8 + 8*b6 - 502*b2 - 135*b1) * q^70 + (-5*b9 - b8 + 4*b6 - 108*b2 - 95*b1) * q^71 + (9*b9 - 9*b8 + 9*b6 + 63*b1) * q^72 + (-10*b6 + 57*b2 - 72*b1) * q^73 + (12*b7 - 14*b5 - 53*b4 + 2*b3 - 438) * q^74 + (3*b5 - 3*b4 + 63*b3 + 288) * q^75 + (-b9 - 21*b8 + 6*b6 + 346*b2 + 9*b1) * q^76 + (-2*b7 + 8*b5 - 58*b4 + 56*b3 + 432) * q^77 + (2*b7 - 13*b5 - 11*b4 + 63*b3 + 110) * q^79 + (-4*b9 - 6*b8 - 13*b6 + 562*b2 + 158*b1) * q^80 + 81 * q^81 + (-2*b7 + 8*b5 + 3*b4 + 36*b3 - 478) * q^82 + (-19*b9 - 37*b8 - 10*b6 - 172*b2 - 35*b1) * q^83 + (-3*b9 + 27*b8 - 12*b6 + 30*b2 - 135*b1) * q^84 + (21*b9 - 42*b8 + 14*b6 + 35*b2 - 54*b1) * q^85 + (-24*b9 + 18*b8 + 21*b6 - 186*b2 - 77*b1) * q^86 + (-9*b5 + 9*b4 - 33*b3 + 297) * q^87 + (23*b7 + 7*b5 + 81*b4 - 249*b3 + 634) * q^88 + (10*b9 - 6*b8 + 4*b6 - 136*b2 - 118*b1) * q^89 + (-9*b7 + 9*b5 + 18*b4 + 27*b3 + 72) * q^90 + (7*b7 - 29*b5 - 35*b4 - 153*b3 - 174) * q^92 + (-6*b9 - 9*b8 - 18*b6 - 66*b2 - 27*b1) * q^93 + (b7 + b5 + 63*b4 - 33*b3 + 646) * q^94 + (-31*b7 - 13*b5 + 47*b4 - 25*b3 - 276) * q^95 + (9*b9 - 9*b8 + 9*b6 - 288*b2 + 153*b1) * q^96 + (22*b9 - 3*b8 - 32*b6 + 250*b2 - 81*b1) * q^97 + (-33*b9 + 21*b8 - 15*b6 + 12*b2 + 11*b1) * q^98 + (-9*b9 - 9*b8 + 18*b6 - 36*b2 + 45*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 30 q^{3} + 60 q^{4} + 90 q^{9}+O(q^{10})$$ 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 $$10 q + 30 q^{3} + 60 q^{4} + 90 q^{9} + 80 q^{10} + 180 q^{12} - 60 q^{14} + 500 q^{16} + 210 q^{17} + 580 q^{22} - 120 q^{23} + 960 q^{25} + 270 q^{27} + 990 q^{29} + 240 q^{30} - 120 q^{35} + 540 q^{36} + 1380 q^{38} + 2000 q^{40} - 180 q^{42} - 740 q^{43} + 1500 q^{48} + 1550 q^{49} + 630 q^{51} + 330 q^{53} + 520 q^{55} - 5340 q^{56} + 2750 q^{61} - 1560 q^{62} + 3140 q^{64} + 1740 q^{66} + 1200 q^{68} - 360 q^{69} - 4380 q^{74} + 2880 q^{75} + 4320 q^{77} + 1100 q^{79} + 810 q^{81} - 4780 q^{82} + 2970 q^{87} + 6340 q^{88} + 720 q^{90} - 1740 q^{92} + 6460 q^{94} - 2760 q^{95}+O(q^{100})$$ 10 * q + 30 * q^3 + 60 * q^4 + 90 * q^9 + 80 * q^10 + 180 * q^12 - 60 * q^14 + 500 * q^16 + 210 * q^17 + 580 * q^22 - 120 * q^23 + 960 * q^25 + 270 * q^27 + 990 * q^29 + 240 * q^30 - 120 * q^35 + 540 * q^36 + 1380 * q^38 + 2000 * q^40 - 180 * q^42 - 740 * q^43 + 1500 * q^48 + 1550 * q^49 + 630 * q^51 + 330 * q^53 + 520 * q^55 - 5340 * q^56 + 2750 * q^61 - 1560 * q^62 + 3140 * q^64 + 1740 * q^66 + 1200 * q^68 - 360 * q^69 - 4380 * q^74 + 2880 * q^75 + 4320 * q^77 + 1100 * q^79 + 810 * q^81 - 4780 * q^82 + 2970 * q^87 + 6340 * q^88 + 720 * q^90 - 1740 * q^92 + 6460 * q^94 - 2760 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 35\nu^{3} - 210\nu ) / 96$$ (-v^5 + 35*v^3 - 210*v) / 96 $$\beta_{3}$$ $$=$$ $$( -\nu^{6} + 35\nu^{4} - 210\nu^{2} ) / 96$$ (-v^6 + 35*v^4 - 210*v^2) / 96 $$\beta_{4}$$ $$=$$ $$\nu^{2} - 14$$ v^2 - 14 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 51\nu^{4} - 706\nu^{2} + 1792 ) / 32$$ (-v^6 + 51*v^4 - 706*v^2 + 1792) / 32 $$\beta_{6}$$ $$=$$ $$( \nu^{7} - 49\nu^{5} + 700\nu^{3} - 2940\nu ) / 96$$ (v^7 - 49*v^5 + 700*v^3 - 2940*v) / 96 $$\beta_{7}$$ $$=$$ $$( \nu^{8} - 61\nu^{6} + 1168\nu^{4} - 7140\nu^{2} + 8064 ) / 96$$ (v^8 - 61*v^6 + 1168*v^4 - 7140*v^2 + 8064) / 96 $$\beta_{8}$$ $$=$$ $$( \nu^{9} - 64\nu^{7} + 1309\nu^{5} - 9030\nu^{3} + 15912\nu ) / 576$$ (v^9 - 64*v^7 + 1309*v^5 - 9030*v^3 + 15912*v) / 576 $$\beta_{9}$$ $$=$$ $$( \nu^{9} - 70\nu^{7} + 1603\nu^{5} - 12654\nu^{3} + 20304\nu ) / 576$$ (v^9 - 70*v^7 + 1603*v^5 - 12654*v^3 + 20304*v) / 576
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + 14$$ b4 + 14 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_{6} + 23\beta_1$$ b9 - b8 + b6 + 23*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 31\beta_{4} - 6\beta_{3} + 322$$ 2*b5 + 31*b4 - 6*b3 + 322 $$\nu^{5}$$ $$=$$ $$35\beta_{9} - 35\beta_{8} + 35\beta_{6} - 96\beta_{2} + 595\beta_1$$ 35*b9 - 35*b8 + 35*b6 - 96*b2 + 595*b1 $$\nu^{6}$$ $$=$$ $$70\beta_{5} + 875\beta_{4} - 306\beta_{3} + 8330$$ 70*b5 + 875*b4 - 306*b3 + 8330 $$\nu^{7}$$ $$=$$ $$1015\beta_{9} - 1015\beta_{8} + 1111\beta_{6} - 4704\beta_{2} + 15995\beta_1$$ 1015*b9 - 1015*b8 + 1111*b6 - 4704*b2 + 15995*b1 $$\nu^{8}$$ $$=$$ $$96\beta_{7} + 1934\beta_{5} + 24307\beta_{4} - 11658\beta_{3} + 223930$$ 96*b7 + 1934*b5 + 24307*b4 - 11658*b3 + 223930 $$\nu^{9}$$ $$=$$ $$28175\beta_{9} - 27599\beta_{8} + 34319\beta_{6} - 175392\beta_{2} + 436603\beta_1$$ 28175*b9 - 27599*b8 + 34319*b6 - 175392*b2 + 436603*b1

## 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
 −5.36472 −5.04537 −3.27897 −2.04224 −0.917374 0.917374 2.04224 3.27897 5.04537 5.36472
−5.36472 3.00000 20.7803 2.69631 −16.0942 15.2025 −68.5626 9.00000 −14.4650
1.2 −5.04537 3.00000 17.4557 −20.1174 −15.1361 15.4279 −47.7076 9.00000 101.500
1.3 −3.27897 3.00000 2.75167 17.5414 −9.83692 −26.6999 17.2091 9.00000 −57.5178
1.4 −2.04224 3.00000 −3.82924 −12.0825 −6.12673 −29.7373 24.1582 9.00000 24.6753
1.5 −0.917374 3.00000 −7.15843 15.4704 −2.75212 20.5833 13.9059 9.00000 −14.1922
1.6 0.917374 3.00000 −7.15843 −15.4704 2.75212 −20.5833 −13.9059 9.00000 −14.1922
1.7 2.04224 3.00000 −3.82924 12.0825 6.12673 29.7373 −24.1582 9.00000 24.6753
1.8 3.27897 3.00000 2.75167 −17.5414 9.83692 26.6999 −17.2091 9.00000 −57.5178
1.9 5.04537 3.00000 17.4557 20.1174 15.1361 −15.4279 47.7076 9.00000 101.500
1.10 5.36472 3.00000 20.7803 −2.69631 16.0942 −15.2025 68.5626 9.00000 −14.4650
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.a.r 10
3.b odd 2 1 1521.4.a.bk 10
13.b even 2 1 inner 507.4.a.r 10
13.d odd 4 2 507.4.b.i 10
13.f odd 12 2 39.4.j.c 10
39.d odd 2 1 1521.4.a.bk 10
39.k even 12 2 117.4.q.e 10
52.l even 12 2 624.4.bv.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.c 10 13.f odd 12 2
117.4.q.e 10 39.k even 12 2
507.4.a.r 10 1.a even 1 1 trivial
507.4.a.r 10 13.b even 2 1 inner
507.4.b.i 10 13.d odd 4 2
624.4.bv.h 10 52.l even 12 2
1521.4.a.bk 10 3.b odd 2 1
1521.4.a.bk 10 39.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{10} - 70T_{2}^{8} + 1645T_{2}^{6} - 14700T_{2}^{4} + 44100T_{2}^{2} - 27648$$ T2^10 - 70*T2^8 + 1645*T2^6 - 14700*T2^4 + 44100*T2^2 - 27648 $$T_{5}^{10} - 1105T_{5}^{8} + 441955T_{5}^{6} - 76029795T_{5}^{4} + 4880780280T_{5}^{2} - 31632011568$$ T5^10 - 1105*T5^8 + 441955*T5^6 - 76029795*T5^4 + 4880780280*T5^2 - 31632011568

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 70 T^{8} + \cdots - 27648$$
$3$ $$(T - 3)^{10}$$
$5$ $$T^{10} + \cdots - 31632011568$$
$7$ $$T^{10} + \cdots - 14692478786352$$
$11$ $$T^{10} + \cdots - 50\!\cdots\!32$$
$13$ $$T^{10}$$
$17$ $$(T^{5} - 105 T^{4} + \cdots + 18224352)^{2}$$
$19$ $$T^{10} + \cdots - 19\!\cdots\!68$$
$23$ $$(T^{5} + 60 T^{4} + \cdots - 8153671248)^{2}$$
$29$ $$(T^{5} - 495 T^{4} + \cdots + 427627836)^{2}$$
$31$ $$T^{10} + \cdots - 35\!\cdots\!00$$
$37$ $$T^{10} + \cdots - 48\!\cdots\!32$$
$41$ $$T^{10} + \cdots - 36\!\cdots\!52$$
$43$ $$(T^{5} + 370 T^{4} + \cdots - 227329236796)^{2}$$
$47$ $$T^{10} + \cdots - 21\!\cdots\!28$$
$53$ $$(T^{5} - 165 T^{4} + \cdots - 46733997168)^{2}$$
$59$ $$T^{10} + \cdots - 41\!\cdots\!68$$
$61$ $$(T^{5} - 1375 T^{4} + \cdots - 933851008945)^{2}$$
$67$ $$T^{10} + \cdots - 13\!\cdots\!88$$
$71$ $$T^{10} + \cdots - 71\!\cdots\!00$$
$73$ $$T^{10} + \cdots - 20\!\cdots\!75$$
$79$ $$(T^{5} - 550 T^{4} + \cdots - 920208867136)^{2}$$
$83$ $$T^{10} + \cdots - 16\!\cdots\!68$$
$89$ $$T^{10} + \cdots - 28\!\cdots\!28$$
$97$ $$T^{10} + \cdots - 15\!\cdots\!68$$