# Properties

 Label 507.4.b.f Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49$$ x^4 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_{3} - 7) q^{4} + ( - 2 \beta_{2} + 7 \beta_1) q^{5} - 3 \beta_{2} q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 13 \beta_1) q^{8} + 9 q^{9}+O(q^{10})$$ q + b2 * q^2 - 3 * q^3 + (-b3 - 7) * q^4 + (-2*b2 + 7*b1) * q^5 - 3*b2 * q^6 + (2*b2 - b1) * q^7 + (-b2 - 13*b1) * q^8 + 9 * q^9 $$q + \beta_{2} q^{2} - 3 q^{3} + ( - \beta_{3} - 7) q^{4} + ( - 2 \beta_{2} + 7 \beta_1) q^{5} - 3 \beta_{2} q^{6} + (2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 13 \beta_1) q^{8} + 9 q^{9} + ( - 5 \beta_{3} + 16) q^{10} + (12 \beta_{2} + 5 \beta_1) q^{11} + (3 \beta_{3} + 21) q^{12} + ( - \beta_{3} - 28) q^{14} + (6 \beta_{2} - 21 \beta_1) q^{15} + (6 \beta_{3} - 15) q^{16} + ( - 2 \beta_{3} - 82) q^{17} + 9 \beta_{2} q^{18} + (2 \beta_{2} + 11 \beta_1) q^{19} + ( - 10 \beta_{2} - 9 \beta_1) q^{20} + ( - 6 \beta_{2} + 3 \beta_1) q^{21} + ( - 17 \beta_{3} - 190) q^{22} + ( - 24 \beta_{3} - 4) q^{23} + (3 \beta_{2} + 39 \beta_1) q^{24} + (24 \beta_{3} - 75) q^{25} - 27 q^{27} + ( - 14 \beta_{2} - 21 \beta_1) q^{28} + ( - 12 \beta_{3} + 202) q^{29} + (15 \beta_{3} - 48) q^{30} + ( - 26 \beta_{2} + 23 \beta_1) q^{31} + ( - 11 \beta_{2} - 26 \beta_1) q^{32} + ( - 36 \beta_{2} - 15 \beta_1) q^{33} + ( - 86 \beta_{2} - 26 \beta_1) q^{34} + ( - 12 \beta_{3} + 56) q^{35} + ( - 9 \beta_{3} - 63) q^{36} + ( - 28 \beta_{2} + 39 \beta_1) q^{37} + ( - 13 \beta_{3} - 52) q^{38} + ( - 21 \beta_{3} + 296) q^{40} + (94 \beta_{2} + 3 \beta_1) q^{41} + (3 \beta_{3} + 84) q^{42} + ( - 26 \beta_{3} + 308) q^{43} + ( - 128 \beta_{2} - 181 \beta_1) q^{44} + ( - 18 \beta_{2} + 63 \beta_1) q^{45} + ( - 52 \beta_{2} - 312 \beta_1) q^{46} + ( - 32 \beta_{2} + 97 \beta_1) q^{47} + ( - 18 \beta_{3} + 45) q^{48} + 287 q^{49} + ( - 27 \beta_{2} + 312 \beta_1) q^{50} + (6 \beta_{3} + 246) q^{51} + ( - 60 \beta_{3} - 82) q^{53} - 27 \beta_{2} q^{54} + ( - 50 \beta_{3} + 72) q^{55} + (27 \beta_{3} + 28) q^{56} + ( - 6 \beta_{2} - 33 \beta_1) q^{57} + (178 \beta_{2} - 156 \beta_1) q^{58} + ( - 40 \beta_{2} - 15 \beta_1) q^{59} + (30 \beta_{2} + 27 \beta_1) q^{60} + (68 \beta_{3} + 314) q^{61} + (3 \beta_{3} + 344) q^{62} + (18 \beta_{2} - 9 \beta_1) q^{63} + (85 \beta_{3} + 97) q^{64} + (51 \beta_{3} + 570) q^{66} + ( - 170 \beta_{2} - 33 \beta_1) q^{67} + (96 \beta_{3} + 686) q^{68} + (72 \beta_{3} + 12) q^{69} + (32 \beta_{2} - 156 \beta_1) q^{70} + ( - 84 \beta_{2} + 149 \beta_1) q^{71} + ( - 9 \beta_{2} - 117 \beta_1) q^{72} + ( - 76 \beta_{2} + 263 \beta_1) q^{73} + ( - 11 \beta_{3} + 342) q^{74} + ( - 72 \beta_{3} + 225) q^{75} + ( - 62 \beta_{2} - 81 \beta_1) q^{76} + ( - 22 \beta_{3} - 336) q^{77} + ( - 44 \beta_{3} - 216) q^{79} + (174 \beta_{2} - 345 \beta_1) q^{80} + 81 q^{81} + ( - 97 \beta_{3} - 1416) q^{82} + (64 \beta_{2} - 379 \beta_1) q^{83} + (42 \beta_{2} + 63 \beta_1) q^{84} + (116 \beta_{2} - 494 \beta_1) q^{85} + (256 \beta_{2} - 338 \beta_1) q^{86} + (36 \beta_{3} - 606) q^{87} + (173 \beta_{3} + 762) q^{88} + (190 \beta_{2} - 335 \beta_1) q^{89} + ( - 45 \beta_{3} + 144) q^{90} + (172 \beta_{3} + 1372) q^{92} + (78 \beta_{2} - 69 \beta_1) q^{93} + ( - 65 \beta_{3} + 286) q^{94} + (12 \beta_{3} - 232) q^{95} + (33 \beta_{2} + 78 \beta_1) q^{96} + ( - 220 \beta_{2} - 23 \beta_1) q^{97} + 287 \beta_{2} q^{98} + (108 \beta_{2} + 45 \beta_1) q^{99}+O(q^{100})$$ q + b2 * q^2 - 3 * q^3 + (-b3 - 7) * q^4 + (-2*b2 + 7*b1) * q^5 - 3*b2 * q^6 + (2*b2 - b1) * q^7 + (-b2 - 13*b1) * q^8 + 9 * q^9 + (-5*b3 + 16) * q^10 + (12*b2 + 5*b1) * q^11 + (3*b3 + 21) * q^12 + (-b3 - 28) * q^14 + (6*b2 - 21*b1) * q^15 + (6*b3 - 15) * q^16 + (-2*b3 - 82) * q^17 + 9*b2 * q^18 + (2*b2 + 11*b1) * q^19 + (-10*b2 - 9*b1) * q^20 + (-6*b2 + 3*b1) * q^21 + (-17*b3 - 190) * q^22 + (-24*b3 - 4) * q^23 + (3*b2 + 39*b1) * q^24 + (24*b3 - 75) * q^25 - 27 * q^27 + (-14*b2 - 21*b1) * q^28 + (-12*b3 + 202) * q^29 + (15*b3 - 48) * q^30 + (-26*b2 + 23*b1) * q^31 + (-11*b2 - 26*b1) * q^32 + (-36*b2 - 15*b1) * q^33 + (-86*b2 - 26*b1) * q^34 + (-12*b3 + 56) * q^35 + (-9*b3 - 63) * q^36 + (-28*b2 + 39*b1) * q^37 + (-13*b3 - 52) * q^38 + (-21*b3 + 296) * q^40 + (94*b2 + 3*b1) * q^41 + (3*b3 + 84) * q^42 + (-26*b3 + 308) * q^43 + (-128*b2 - 181*b1) * q^44 + (-18*b2 + 63*b1) * q^45 + (-52*b2 - 312*b1) * q^46 + (-32*b2 + 97*b1) * q^47 + (-18*b3 + 45) * q^48 + 287 * q^49 + (-27*b2 + 312*b1) * q^50 + (6*b3 + 246) * q^51 + (-60*b3 - 82) * q^53 - 27*b2 * q^54 + (-50*b3 + 72) * q^55 + (27*b3 + 28) * q^56 + (-6*b2 - 33*b1) * q^57 + (178*b2 - 156*b1) * q^58 + (-40*b2 - 15*b1) * q^59 + (30*b2 + 27*b1) * q^60 + (68*b3 + 314) * q^61 + (3*b3 + 344) * q^62 + (18*b2 - 9*b1) * q^63 + (85*b3 + 97) * q^64 + (51*b3 + 570) * q^66 + (-170*b2 - 33*b1) * q^67 + (96*b3 + 686) * q^68 + (72*b3 + 12) * q^69 + (32*b2 - 156*b1) * q^70 + (-84*b2 + 149*b1) * q^71 + (-9*b2 - 117*b1) * q^72 + (-76*b2 + 263*b1) * q^73 + (-11*b3 + 342) * q^74 + (-72*b3 + 225) * q^75 + (-62*b2 - 81*b1) * q^76 + (-22*b3 - 336) * q^77 + (-44*b3 - 216) * q^79 + (174*b2 - 345*b1) * q^80 + 81 * q^81 + (-97*b3 - 1416) * q^82 + (64*b2 - 379*b1) * q^83 + (42*b2 + 63*b1) * q^84 + (116*b2 - 494*b1) * q^85 + (256*b2 - 338*b1) * q^86 + (36*b3 - 606) * q^87 + (173*b3 + 762) * q^88 + (190*b2 - 335*b1) * q^89 + (-45*b3 + 144) * q^90 + (172*b3 + 1372) * q^92 + (78*b2 - 69*b1) * q^93 + (-65*b3 + 286) * q^94 + (12*b3 - 232) * q^95 + (33*b2 + 78*b1) * q^96 + (-220*b2 - 23*b1) * q^97 + 287*b2 * q^98 + (108*b2 + 45*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 $$4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{2} ) / 7$$ (2*v^2) / 7 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} + 7\nu ) / 7$$ (v^3 + v^2 + 7*v) / 7 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 14\nu ) / 7$$ (-2*v^3 + 14*v) / 7
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} - \beta_1 ) / 4$$ (b3 + 2*b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( 7\beta_1 ) / 2$$ (7*b1) / 2 $$\nu^{3}$$ $$=$$ $$( -7\beta_{3} + 14\beta_{2} - 7\beta_1 ) / 4$$ (-7*b3 + 14*b2 - 7*b1) / 4

## 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
 1.87083 − 1.87083i −1.87083 − 1.87083i −1.87083 + 1.87083i 1.87083 + 1.87083i
4.74166i −3.00000 −14.4833 4.51669i 14.2250i 7.48331i 30.7417i 9.00000 −21.4166
337.2 2.74166i −3.00000 0.483315 19.4833i 8.22497i 7.48331i 23.2583i 9.00000 53.4166
337.3 2.74166i −3.00000 0.483315 19.4833i 8.22497i 7.48331i 23.2583i 9.00000 53.4166
337.4 4.74166i −3.00000 −14.4833 4.51669i 14.2250i 7.48331i 30.7417i 9.00000 −21.4166
 $$n$$: 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.f 4
13.b even 2 1 inner 507.4.b.f 4
13.d odd 4 1 39.4.a.b 2
13.d odd 4 1 507.4.a.f 2
39.f even 4 1 117.4.a.c 2
39.f even 4 1 1521.4.a.s 2
52.f even 4 1 624.4.a.r 2
65.g odd 4 1 975.4.a.j 2
91.i even 4 1 1911.4.a.h 2
104.j odd 4 1 2496.4.a.bc 2
104.m even 4 1 2496.4.a.s 2
156.l odd 4 1 1872.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 13.d odd 4 1
117.4.a.c 2 39.f even 4 1
507.4.a.f 2 13.d odd 4 1
507.4.b.f 4 1.a even 1 1 trivial
507.4.b.f 4 13.b even 2 1 inner
624.4.a.r 2 52.f even 4 1
975.4.a.j 2 65.g odd 4 1
1521.4.a.s 2 39.f even 4 1
1872.4.a.t 2 156.l odd 4 1
1911.4.a.h 2 91.i even 4 1
2496.4.a.s 2 104.m even 4 1
2496.4.a.bc 2 104.j odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 30T_{2}^{2} + 169$$ T2^4 + 30*T2^2 + 169 $$T_{5}^{4} + 400T_{5}^{2} + 7744$$ T5^4 + 400*T5^2 + 7744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 30T^{2} + 169$$
$3$ $$(T + 3)^{4}$$
$5$ $$T^{4} + 400T^{2} + 7744$$
$7$ $$(T^{2} + 56)^{2}$$
$11$ $$T^{4} + 5000 T^{2} + 2347024$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 164 T + 6500)^{2}$$
$19$ $$T^{4} + 1264 T^{2} + 270400$$
$23$ $$(T^{2} + 8 T - 32240)^{2}$$
$29$ $$(T^{2} - 404 T + 32740)^{2}$$
$31$ $$T^{4} + 19728 T^{2} + 82156096$$
$37$ $$T^{4} + 26952 T^{2} + 71842576$$
$41$ $$T^{4} + \cdots + 12928599616$$
$43$ $$(T^{2} - 616 T + 57008)^{2}$$
$47$ $$T^{4} + 81160 T^{2} + 141800464$$
$53$ $$(T^{2} + 164 T - 194876)^{2}$$
$59$ $$T^{4} + 54600 T^{2} + 306250000$$
$61$ $$(T^{2} - 628 T - 160348)^{2}$$
$67$ $$T^{4} + \cdots + 121734001216$$
$71$ $$T^{4} + \cdots + 2807728144$$
$73$ $$T^{4} + \cdots + 14795316496$$
$79$ $$(T^{2} + 432 T - 61760)^{2}$$
$83$ $$T^{4} + \cdots + 180023701264$$
$89$ $$T^{4} + \cdots + 75625000000$$
$97$ $$T^{4} + \cdots + 368259640336$$