# Properties

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

# Learn more

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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 3 q^{3} - 7 q^{4} + 7 q^{5} - 3 q^{6} - 10 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10})$$ q - q^2 + 3 * q^3 - 7 * q^4 + 7 * q^5 - 3 * q^6 - 10 * q^7 + 15 * q^8 + 9 * q^9 $$q - q^{2} + 3 q^{3} - 7 q^{4} + 7 q^{5} - 3 q^{6} - 10 q^{7} + 15 q^{8} + 9 q^{9} - 7 q^{10} - 22 q^{11} - 21 q^{12} + 10 q^{14} + 21 q^{15} + 41 q^{16} + 37 q^{17} - 9 q^{18} + 30 q^{19} - 49 q^{20} - 30 q^{21} + 22 q^{22} - 162 q^{23} + 45 q^{24} - 76 q^{25} + 27 q^{27} + 70 q^{28} - 113 q^{29} - 21 q^{30} + 196 q^{31} - 161 q^{32} - 66 q^{33} - 37 q^{34} - 70 q^{35} - 63 q^{36} + 13 q^{37} - 30 q^{38} + 105 q^{40} + 285 q^{41} + 30 q^{42} - 246 q^{43} + 154 q^{44} + 63 q^{45} + 162 q^{46} - 462 q^{47} + 123 q^{48} - 243 q^{49} + 76 q^{50} + 111 q^{51} - 537 q^{53} - 27 q^{54} - 154 q^{55} - 150 q^{56} + 90 q^{57} + 113 q^{58} + 576 q^{59} - 147 q^{60} - 635 q^{61} - 196 q^{62} - 90 q^{63} - 167 q^{64} + 66 q^{66} + 202 q^{67} - 259 q^{68} - 486 q^{69} + 70 q^{70} - 1086 q^{71} + 135 q^{72} - 805 q^{73} - 13 q^{74} - 228 q^{75} - 210 q^{76} + 220 q^{77} + 884 q^{79} + 287 q^{80} + 81 q^{81} - 285 q^{82} + 518 q^{83} + 210 q^{84} + 259 q^{85} + 246 q^{86} - 339 q^{87} - 330 q^{88} + 194 q^{89} - 63 q^{90} + 1134 q^{92} + 588 q^{93} + 462 q^{94} + 210 q^{95} - 483 q^{96} - 1202 q^{97} + 243 q^{98} - 198 q^{99}+O(q^{100})$$ q - q^2 + 3 * q^3 - 7 * q^4 + 7 * q^5 - 3 * q^6 - 10 * q^7 + 15 * q^8 + 9 * q^9 - 7 * q^10 - 22 * q^11 - 21 * q^12 + 10 * q^14 + 21 * q^15 + 41 * q^16 + 37 * q^17 - 9 * q^18 + 30 * q^19 - 49 * q^20 - 30 * q^21 + 22 * q^22 - 162 * q^23 + 45 * q^24 - 76 * q^25 + 27 * q^27 + 70 * q^28 - 113 * q^29 - 21 * q^30 + 196 * q^31 - 161 * q^32 - 66 * q^33 - 37 * q^34 - 70 * q^35 - 63 * q^36 + 13 * q^37 - 30 * q^38 + 105 * q^40 + 285 * q^41 + 30 * q^42 - 246 * q^43 + 154 * q^44 + 63 * q^45 + 162 * q^46 - 462 * q^47 + 123 * q^48 - 243 * q^49 + 76 * q^50 + 111 * q^51 - 537 * q^53 - 27 * q^54 - 154 * q^55 - 150 * q^56 + 90 * q^57 + 113 * q^58 + 576 * q^59 - 147 * q^60 - 635 * q^61 - 196 * q^62 - 90 * q^63 - 167 * q^64 + 66 * q^66 + 202 * q^67 - 259 * q^68 - 486 * q^69 + 70 * q^70 - 1086 * q^71 + 135 * q^72 - 805 * q^73 - 13 * q^74 - 228 * q^75 - 210 * q^76 + 220 * q^77 + 884 * q^79 + 287 * q^80 + 81 * q^81 - 285 * q^82 + 518 * q^83 + 210 * q^84 + 259 * q^85 + 246 * q^86 - 339 * q^87 - 330 * q^88 + 194 * q^89 - 63 * q^90 + 1134 * q^92 + 588 * q^93 + 462 * q^94 + 210 * q^95 - 483 * q^96 - 1202 * q^97 + 243 * q^98 - 198 * 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
 0
−1.00000 3.00000 −7.00000 7.00000 −3.00000 −10.0000 15.0000 9.00000 −7.00000
 $$n$$: 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.b 1
3.b odd 2 1 1521.4.a.h 1
13.b even 2 1 507.4.a.d 1
13.c even 3 2 39.4.e.b 2
13.d odd 4 2 507.4.b.d 2
39.d odd 2 1 1521.4.a.e 1
39.i odd 6 2 117.4.g.a 2
52.j odd 6 2 624.4.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.b 2 13.c even 3 2
117.4.g.a 2 39.i odd 6 2
507.4.a.b 1 1.a even 1 1 trivial
507.4.a.d 1 13.b even 2 1
507.4.b.d 2 13.d odd 4 2
624.4.q.c 2 52.j odd 6 2
1521.4.a.e 1 39.d odd 2 1
1521.4.a.h 1 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 7$$ T5 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 3$$
$5$ $$T - 7$$
$7$ $$T + 10$$
$11$ $$T + 22$$
$13$ $$T$$
$17$ $$T - 37$$
$19$ $$T - 30$$
$23$ $$T + 162$$
$29$ $$T + 113$$
$31$ $$T - 196$$
$37$ $$T - 13$$
$41$ $$T - 285$$
$43$ $$T + 246$$
$47$ $$T + 462$$
$53$ $$T + 537$$
$59$ $$T - 576$$
$61$ $$T + 635$$
$67$ $$T - 202$$
$71$ $$T + 1086$$
$73$ $$T + 805$$
$79$ $$T - 884$$
$83$ $$T - 518$$
$89$ $$T - 194$$
$97$ $$T + 1202$$
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