# Properties

 Label 507.4.b.d Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $2$ 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(337,507)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 3 q^{3} + 7 q^{4} - 7 i q^{5} + 3 i q^{6} - 10 i q^{7} + 15 i q^{8} + 9 q^{9} +O(q^{10})$$ q + i * q^2 + 3 * q^3 + 7 * q^4 - 7*i * q^5 + 3*i * q^6 - 10*i * q^7 + 15*i * q^8 + 9 * q^9 $$q + i q^{2} + 3 q^{3} + 7 q^{4} - 7 i q^{5} + 3 i q^{6} - 10 i q^{7} + 15 i q^{8} + 9 q^{9} + 7 q^{10} - 22 i q^{11} + 21 q^{12} + 10 q^{14} - 21 i q^{15} + 41 q^{16} - 37 q^{17} + 9 i q^{18} - 30 i q^{19} - 49 i q^{20} - 30 i q^{21} + 22 q^{22} + 162 q^{23} + 45 i q^{24} + 76 q^{25} + 27 q^{27} - 70 i q^{28} - 113 q^{29} + 21 q^{30} - 196 i q^{31} + 161 i q^{32} - 66 i q^{33} - 37 i q^{34} - 70 q^{35} + 63 q^{36} + 13 i q^{37} + 30 q^{38} + 105 q^{40} - 285 i q^{41} + 30 q^{42} + 246 q^{43} - 154 i q^{44} - 63 i q^{45} + 162 i q^{46} - 462 i q^{47} + 123 q^{48} + 243 q^{49} + 76 i q^{50} - 111 q^{51} - 537 q^{53} + 27 i q^{54} - 154 q^{55} + 150 q^{56} - 90 i q^{57} - 113 i q^{58} + 576 i q^{59} - 147 i q^{60} - 635 q^{61} + 196 q^{62} - 90 i q^{63} + 167 q^{64} + 66 q^{66} - 202 i q^{67} - 259 q^{68} + 486 q^{69} - 70 i q^{70} + 1086 i q^{71} + 135 i q^{72} - 805 i q^{73} - 13 q^{74} + 228 q^{75} - 210 i q^{76} - 220 q^{77} + 884 q^{79} - 287 i q^{80} + 81 q^{81} + 285 q^{82} - 518 i q^{83} - 210 i q^{84} + 259 i q^{85} + 246 i q^{86} - 339 q^{87} + 330 q^{88} + 194 i q^{89} + 63 q^{90} + 1134 q^{92} - 588 i q^{93} + 462 q^{94} - 210 q^{95} + 483 i q^{96} + 1202 i q^{97} + 243 i q^{98} - 198 i q^{99} +O(q^{100})$$ q + i * q^2 + 3 * q^3 + 7 * q^4 - 7*i * q^5 + 3*i * q^6 - 10*i * q^7 + 15*i * q^8 + 9 * q^9 + 7 * q^10 - 22*i * q^11 + 21 * q^12 + 10 * q^14 - 21*i * q^15 + 41 * q^16 - 37 * q^17 + 9*i * q^18 - 30*i * q^19 - 49*i * q^20 - 30*i * q^21 + 22 * q^22 + 162 * q^23 + 45*i * q^24 + 76 * q^25 + 27 * q^27 - 70*i * q^28 - 113 * q^29 + 21 * q^30 - 196*i * q^31 + 161*i * q^32 - 66*i * q^33 - 37*i * q^34 - 70 * q^35 + 63 * q^36 + 13*i * q^37 + 30 * q^38 + 105 * q^40 - 285*i * q^41 + 30 * q^42 + 246 * q^43 - 154*i * q^44 - 63*i * q^45 + 162*i * q^46 - 462*i * q^47 + 123 * q^48 + 243 * q^49 + 76*i * q^50 - 111 * q^51 - 537 * q^53 + 27*i * q^54 - 154 * q^55 + 150 * q^56 - 90*i * q^57 - 113*i * q^58 + 576*i * q^59 - 147*i * q^60 - 635 * q^61 + 196 * q^62 - 90*i * q^63 + 167 * q^64 + 66 * q^66 - 202*i * q^67 - 259 * q^68 + 486 * q^69 - 70*i * q^70 + 1086*i * q^71 + 135*i * q^72 - 805*i * q^73 - 13 * q^74 + 228 * q^75 - 210*i * q^76 - 220 * q^77 + 884 * q^79 - 287*i * q^80 + 81 * q^81 + 285 * q^82 - 518*i * q^83 - 210*i * q^84 + 259*i * q^85 + 246*i * q^86 - 339 * q^87 + 330 * q^88 + 194*i * q^89 + 63 * q^90 + 1134 * q^92 - 588*i * q^93 + 462 * q^94 - 210 * q^95 + 483*i * q^96 + 1202*i * q^97 + 243*i * q^98 - 198*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 14 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 $$2 q + 6 q^{3} + 14 q^{4} + 18 q^{9} + 14 q^{10} + 42 q^{12} + 20 q^{14} + 82 q^{16} - 74 q^{17} + 44 q^{22} + 324 q^{23} + 152 q^{25} + 54 q^{27} - 226 q^{29} + 42 q^{30} - 140 q^{35} + 126 q^{36} + 60 q^{38} + 210 q^{40} + 60 q^{42} + 492 q^{43} + 246 q^{48} + 486 q^{49} - 222 q^{51} - 1074 q^{53} - 308 q^{55} + 300 q^{56} - 1270 q^{61} + 392 q^{62} + 334 q^{64} + 132 q^{66} - 518 q^{68} + 972 q^{69} - 26 q^{74} + 456 q^{75} - 440 q^{77} + 1768 q^{79} + 162 q^{81} + 570 q^{82} - 678 q^{87} + 660 q^{88} + 126 q^{90} + 2268 q^{92} + 924 q^{94} - 420 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 + 14 * q^4 + 18 * q^9 + 14 * q^10 + 42 * q^12 + 20 * q^14 + 82 * q^16 - 74 * q^17 + 44 * q^22 + 324 * q^23 + 152 * q^25 + 54 * q^27 - 226 * q^29 + 42 * q^30 - 140 * q^35 + 126 * q^36 + 60 * q^38 + 210 * q^40 + 60 * q^42 + 492 * q^43 + 246 * q^48 + 486 * q^49 - 222 * q^51 - 1074 * q^53 - 308 * q^55 + 300 * q^56 - 1270 * q^61 + 392 * q^62 + 334 * q^64 + 132 * q^66 - 518 * q^68 + 972 * q^69 - 26 * q^74 + 456 * q^75 - 440 * q^77 + 1768 * q^79 + 162 * q^81 + 570 * q^82 - 678 * q^87 + 660 * q^88 + 126 * q^90 + 2268 * q^92 + 924 * q^94 - 420 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i 3.00000 7.00000 7.00000i 3.00000i 10.0000i 15.0000i 9.00000 7.00000
337.2 1.00000i 3.00000 7.00000 7.00000i 3.00000i 10.0000i 15.0000i 9.00000 7.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.d 2
13.b even 2 1 inner 507.4.b.d 2
13.d odd 4 1 507.4.a.b 1
13.d odd 4 1 507.4.a.d 1
13.f odd 12 2 39.4.e.b 2
39.f even 4 1 1521.4.a.e 1
39.f even 4 1 1521.4.a.h 1
39.k even 12 2 117.4.g.a 2
52.l even 12 2 624.4.q.c 2

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

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{5}^{2} + 49$$ T5^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 49$$
$7$ $$T^{2} + 100$$
$11$ $$T^{2} + 484$$
$13$ $$T^{2}$$
$17$ $$(T + 37)^{2}$$
$19$ $$T^{2} + 900$$
$23$ $$(T - 162)^{2}$$
$29$ $$(T + 113)^{2}$$
$31$ $$T^{2} + 38416$$
$37$ $$T^{2} + 169$$
$41$ $$T^{2} + 81225$$
$43$ $$(T - 246)^{2}$$
$47$ $$T^{2} + 213444$$
$53$ $$(T + 537)^{2}$$
$59$ $$T^{2} + 331776$$
$61$ $$(T + 635)^{2}$$
$67$ $$T^{2} + 40804$$
$71$ $$T^{2} + 1179396$$
$73$ $$T^{2} + 648025$$
$79$ $$(T - 884)^{2}$$
$83$ $$T^{2} + 268324$$
$89$ $$T^{2} + 37636$$
$97$ $$T^{2} + 1444804$$