# Properties

 Label 507.4.b.h Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776$$ x^8 + 54*x^6 + 889*x^4 + 4584*x^2 + 5776 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 13^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_{2}) q^{8}+ \cdots + 9 q^{9}+O(q^{10})$$ q - b2 * q^2 - 3 * q^3 + (b1 - 6) * q^4 + (b5 + b2) * q^5 + 3*b2 * q^6 + (b3 + b2) * q^7 + (-b7 + 2*b5 + b3 + 5*b2) * q^8 + 9 * q^9 $$q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_{2}) q^{8}+ \cdots + ( - 9 \beta_{7} - 9 \beta_{5} + \cdots + 36 \beta_{2}) q^{99}+O(q^{100})$$ q - b2 * q^2 - 3 * q^3 + (b1 - 6) * q^4 + (b5 + b2) * q^5 + 3*b2 * q^6 + (b3 + b2) * q^7 + (-b7 + 2*b5 + b3 + 5*b2) * q^8 + 9 * q^9 + (-b4 - 3*b1 + 17) * q^10 + (-b7 - b5 + b3 + 4*b2) * q^11 + (-3*b1 + 18) * q^12 + (b6 + b4 - 5*b1 + 12) * q^14 + (-3*b5 - 3*b2) * q^15 + (-2*b4 - 7*b1 + 34) * q^16 + (2*b6 + b4 + 3*b1 - 27) * q^17 - 9*b2 * q^18 + (3*b7 + 7*b5 - b3 - 6*b2) * q^19 + (8*b7 - 9*b5 - 2*b3 - 23*b2) * q^20 + (-3*b3 - 3*b2) * q^21 + (b4 - 8*b1 + 59) * q^22 + (-2*b6 + b4 - 25) * q^23 + (3*b7 - 6*b5 - 3*b3 - 15*b2) * q^24 + (2*b4 + 7*b1 + 11) * q^25 - 27 * q^27 + (-5*b7 - b5 + b3 - 48*b2) * q^28 + (-4*b6 + 2*b4 - 3*b1 + 52) * q^29 + (3*b4 + 9*b1 - 51) * q^30 + (-b7 + 7*b5 - 53*b2) * q^31 + (9*b7 - 20*b5 + 3*b3 - 29*b2) * q^32 + (3*b7 + 3*b5 - 3*b3 - 12*b2) * q^33 + (-18*b7 + 13*b5 + 37*b2) * q^34 + (6*b6 + b4 + 8*b1 + 15) * q^35 + (9*b1 - 54) * q^36 + (-b7 - 2*b5 - b3 + 21*b2) * q^37 + (2*b6 - 5*b4 + 2*b1 - 85) * q^38 + (6*b6 + 7*b4 + 41*b1 - 273) * q^40 + (-21*b7 - 3*b3 - 43*b2) * q^41 + (-3*b6 - 3*b4 + 15*b1 - 36) * q^42 + (6*b6 + 5*b4 - 3*b1 - 114) * q^43 + (-5*b7 - 13*b5 - b3 - 90*b2) * q^44 + (9*b5 + 9*b2) * q^45 + (5*b7 + 15*b5 + b3 + 22*b2) * q^46 + (-3*b7 - 47*b5 + 3*b3 - 8*b2) * q^47 + (6*b4 + 21*b1 - 102) * q^48 + (-2*b4 - 31*b1 - 252) * q^49 + (-17*b7 + 36*b5 + 5*b3 + 24*b2) * q^50 + (-6*b6 - 3*b4 - 9*b1 + 81) * q^51 + (-2*b4 - 25*b1 + 78) * q^53 + 27*b2 * q^54 + (6*b6 + 3*b4 + 26*b1 + 35) * q^55 + (4*b6 + 5*b4 - 4*b1 - 541) * q^56 + (-9*b7 - 21*b5 + 3*b3 + 18*b2) * q^57 + (13*b7 + 24*b5 - b3 - 79*b2) * q^58 + (-8*b7 + 24*b5 + 14*b3 + 10*b2) * q^59 + (-24*b7 + 27*b5 + 6*b3 + 69*b2) * q^60 + (-6*b6 - 3*b4 + 22*b1 - 240) * q^61 + (-b6 - 8*b4 + 37*b1 - 713) * q^62 + (9*b3 + 9*b2) * q^63 + (12*b6 + 16*b4 + 19*b1 - 272) * q^64 + (-3*b4 + 24*b1 - 177) * q^66 + (-26*b7 + 14*b5 + b3 - 129*b2) * q^67 + (-2*b6 - 23*b4 - 75*b1 + 485) * q^68 + (6*b6 - 3*b4 + 75) * q^69 + (-43*b7 + 15*b5 + b3 + 22*b2) * q^70 + (23*b7 - 49*b5 - 11*b3 + 76*b2) * q^71 + (-9*b7 + 18*b5 + 9*b3 + 45*b2) * q^72 + (-27*b7 - 24*b5 - 2*b3 - 194*b2) * q^73 + (-2*b6 - 15*b1 + 298) * q^74 + (-6*b4 - 21*b1 - 33) * q^75 + (37*b7 + b5 - 3*b3 + 82*b2) * q^76 + (-16*b6 - 18*b1 - 610) * q^77 + (2*b4 + 9*b1 - 191) * q^79 + (-42*b7 + 75*b5 + 12*b3 + 315*b2) * q^80 + 81 * q^81 + (-24*b6 - 24*b4 + 13*b1 - 428) * q^82 + (-b7 - b5 - 5*b3 + 138*b2) * q^83 + (15*b7 + 3*b5 - 3*b3 + 144*b2) * q^84 + (25*b7 - 50*b5 - 25*b3 - 147*b2) * q^85 + (-52*b7 + 37*b5 - 14*b3 + 46*b2) * q^86 + (12*b6 - 6*b4 + 9*b1 - 156) * q^87 + (-6*b6 + 15*b4 + 46*b1 - 785) * q^88 + (-32*b7 - 50*b5 - 28*b3 - 146*b2) * q^89 + (-9*b4 - 27*b1 + 153) * q^90 + (-10*b6 - b4 - 46*b1 + 111) * q^92 + (3*b7 - 21*b5 + 159*b2) * q^93 + (47*b4 + 84*b1 - 235) * q^94 + (-6*b6 + b4 - 34*b1 - 531) * q^95 + (-27*b7 + 60*b5 - 9*b3 + 87*b2) * q^96 + (37*b7 + 63*b5 + 22*b3 - 71*b2) * q^97 + (41*b7 - 84*b5 - 29*b3 + 49*b2) * q^98 + (-9*b7 - 9*b5 + 9*b3 + 36*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10})$$ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 $$8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100})$$ 8 * q - 24 * q^3 - 44 * q^4 + 72 * q^9 + 124 * q^10 + 132 * q^12 + 80 * q^14 + 244 * q^16 - 196 * q^17 + 440 * q^22 - 208 * q^23 + 116 * q^25 - 216 * q^27 + 388 * q^29 - 372 * q^30 + 176 * q^35 - 396 * q^36 - 664 * q^38 - 1996 * q^40 - 240 * q^42 - 900 * q^43 - 732 * q^48 - 2140 * q^49 + 588 * q^51 + 524 * q^53 + 408 * q^55 - 4328 * q^56 - 1856 * q^61 - 5560 * q^62 - 2052 * q^64 - 1320 * q^66 + 3572 * q^68 + 624 * q^69 + 2316 * q^74 - 348 * q^75 - 5016 * q^77 - 1492 * q^79 + 648 * q^81 - 3468 * q^82 - 1164 * q^87 - 6120 * q^88 + 1116 * q^90 + 664 * q^92 - 1544 * q^94 - 4408 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} - 40\nu^{4} - 329\nu^{2} + 178 ) / 78$$ (-v^6 - 40*v^4 - 329*v^2 + 178) / 78 $$\beta_{2}$$ $$=$$ $$( -11\nu^{7} - 518\nu^{5} - 6739\nu^{3} - 31348\nu ) / 11856$$ (-11*v^7 - 518*v^5 - 6739*v^3 - 31348*v) / 11856 $$\beta_{3}$$ $$=$$ $$( 23\nu^{7} + 1622\nu^{5} + 41575\nu^{3} + 349012\nu ) / 11856$$ (23*v^7 + 1622*v^5 + 41575*v^3 + 349012*v) / 11856 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 118\nu^{4} + 2357\nu^{2} + 5048 ) / 156$$ (v^6 + 118*v^4 + 2357*v^2 + 5048) / 156 $$\beta_{5}$$ $$=$$ $$( 7\nu^{7} + 397\nu^{5} + 6242\nu^{3} + 19814\nu ) / 1482$$ (7*v^7 + 397*v^5 + 6242*v^3 + 19814*v) / 1482 $$\beta_{6}$$ $$=$$ $$( -5\nu^{6} - 239\nu^{4} - 3166\nu^{2} - 8704 ) / 78$$ (-5*v^6 - 239*v^4 - 3166*v^2 - 8704) / 78 $$\beta_{7}$$ $$=$$ $$( 11\nu^{7} + 518\nu^{5} + 6739\nu^{3} + 19492\nu ) / 912$$ (11*v^7 + 518*v^5 + 6739*v^3 + 19492*v) / 912
 $$\nu$$ $$=$$ $$( -\beta_{7} - 13\beta_{2} ) / 13$$ (-b7 - 13*b2) / 13 $$\nu^{2}$$ $$=$$ $$( -2\beta_{6} - 2\beta_{4} + 9\beta _1 - 179 ) / 13$$ (-2*b6 - 2*b4 + 9*b1 - 179) / 13 $$\nu^{3}$$ $$=$$ $$( 24\beta_{7} - 13\beta_{5} + 13\beta_{3} + 273\beta_{2} ) / 13$$ (24*b7 - 13*b5 + 13*b3 + 273*b2) / 13 $$\nu^{4}$$ $$=$$ $$4\beta_{6} + 6\beta_{4} - 17\beta _1 + 291$$ 4*b6 + 6*b4 - 17*b1 + 291 $$\nu^{5}$$ $$=$$ $$( -698\beta_{7} + 663\beta_{5} - 377\beta_{3} - 6487\beta_{2} ) / 13$$ (-698*b7 + 663*b5 - 377*b3 - 6487*b2) / 13 $$\nu^{6}$$ $$=$$ $$( -1422\beta_{6} - 2462\beta_{4} + 4865\beta _1 - 90115 ) / 13$$ (-1422*b6 - 2462*b4 + 4865*b1 - 90115) / 13 $$\nu^{7}$$ $$=$$ $$( 21016\beta_{7} - 23257\beta_{5} + 9789\beta_{3} + 161265\beta_{2} ) / 13$$ (21016*b7 - 23257*b5 + 9789*b3 + 161265*b2) / 13

## 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
 − 4.33039i − 5.22605i − 1.36176i − 2.46610i 2.46610i 1.36176i 5.22605i 4.33039i
5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
337.2 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.3 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.4 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.5 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.6 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.7 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.8 5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 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.h 8
13.b even 2 1 inner 507.4.b.h 8
13.d odd 4 1 507.4.a.i 4
13.d odd 4 1 507.4.a.m 4
13.f odd 12 2 39.4.e.c 8
39.f even 4 1 1521.4.a.v 4
39.f even 4 1 1521.4.a.bb 4
39.k even 12 2 117.4.g.e 8
52.l even 12 2 624.4.q.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 13.f odd 12 2
117.4.g.e 8 39.k even 12 2
507.4.a.i 4 13.d odd 4 1
507.4.a.m 4 13.d odd 4 1
507.4.b.h 8 1.a even 1 1 trivial
507.4.b.h 8 13.b even 2 1 inner
624.4.q.i 8 52.l even 12 2
1521.4.a.v 4 39.f even 4 1
1521.4.a.bb 4 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 $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{8} + 54T_{2}^{6} + 877T_{2}^{4} + 4476T_{2}^{2} + 6084$$ T2^8 + 54*T2^6 + 877*T2^4 + 4476*T2^2 + 6084 $$T_{5}^{8} + 442T_{5}^{6} + 55249T_{5}^{4} + 2494440T_{5}^{2} + 37015056$$ T5^8 + 442*T5^6 + 55249*T5^4 + 2494440*T5^2 + 37015056

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 54 T^{6} + \cdots + 6084$$
$3$ $$(T + 3)^{8}$$
$5$ $$T^{8} + 442 T^{6} + \cdots + 37015056$$
$7$ $$T^{8} + \cdots + 42523388944$$
$11$ $$T^{8} + 4112 T^{6} + \cdots + 751198464$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 98 T^{3} + \cdots + 22571952)^{2}$$
$19$ $$T^{8} + \cdots + 2959721107456$$
$23$ $$(T^{4} + 104 T^{3} + \cdots - 2571504)^{2}$$
$29$ $$(T^{4} - 194 T^{3} + \cdots - 274591068)^{2}$$
$31$ $$T^{8} + \cdots + 10\!\cdots\!64$$
$37$ $$T^{8} + \cdots + 738573457717264$$
$41$ $$T^{8} + \cdots + 10\!\cdots\!04$$
$43$ $$(T^{4} + 450 T^{3} + \cdots - 2362804828)^{2}$$
$47$ $$T^{8} + \cdots + 18\!\cdots\!04$$
$53$ $$(T^{4} - 262 T^{3} + \cdots + 744728256)^{2}$$
$59$ $$T^{8} + \cdots + 44\!\cdots\!56$$
$61$ $$(T^{4} + 928 T^{3} + \cdots - 5230543711)^{2}$$
$67$ $$T^{8} + \cdots + 10\!\cdots\!44$$
$71$ $$T^{8} + \cdots + 98\!\cdots\!16$$
$73$ $$T^{8} + \cdots + 14\!\cdots\!09$$
$79$ $$(T^{4} + 746 T^{3} + \cdots + 680937616)^{2}$$
$83$ $$T^{8} + \cdots + 34\!\cdots\!36$$
$89$ $$T^{8} + \cdots + 72\!\cdots\!96$$
$97$ $$T^{8} + \cdots + 19\!\cdots\!96$$