# Properties

 Label 13.8.b.a Level $13$ Weight $8$ Character orbit 13.b Analytic conductor $4.061$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,8,Mod(12,13)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("13.12");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 13.b (of order $$2$$, degree $$1$$, 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: $$4.06100533129$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 449x^{4} + 37224x^{2} + 205776$$ x^6 + 449*x^4 + 37224*x^2 + 205776 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - 9) q^{3} + ( - \beta_{4} - 22) q^{4} + \beta_{5} q^{5} + ( - \beta_{3} - 10 \beta_1) q^{6} + (\beta_{5} - \beta_{3} - 11 \beta_1) q^{7} + ( - 2 \beta_{5} - 3 \beta_{3} - 26 \beta_1) q^{8} + ( - 6 \beta_{4} - 5 \beta_{2} - 192) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - 9) * q^3 + (-b4 - 22) * q^4 + b5 * q^5 + (-b3 - 10*b1) * q^6 + (b5 - b3 - 11*b1) * q^7 + (-2*b5 - 3*b3 - 26*b1) * q^8 + (-6*b4 - 5*b2 - 192) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - 9) q^{3} + ( - \beta_{4} - 22) q^{4} + \beta_{5} q^{5} + ( - \beta_{3} - 10 \beta_1) q^{6} + (\beta_{5} - \beta_{3} - 11 \beta_1) q^{7} + ( - 2 \beta_{5} - 3 \beta_{3} - 26 \beta_1) q^{8} + ( - 6 \beta_{4} - 5 \beta_{2} - 192) q^{9} + ( - 7 \beta_{4} + 6 \beta_{2} - 72) q^{10} + (12 \beta_{5} + 9 \beta_{3} + 133 \beta_1) q^{11} + (33 \beta_{4} - 44 \beta_{2} + 342) q^{12} + ( - 13 \beta_{5} - 26 \beta_{4} + \cdots - 832) q^{13}+ \cdots + (1428 \beta_{5} - 4139 \beta_{3} - 414983 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - 9) * q^3 + (-b4 - 22) * q^4 + b5 * q^5 + (-b3 - 10*b1) * q^6 + (b5 - b3 - 11*b1) * q^7 + (-2*b5 - 3*b3 - 26*b1) * q^8 + (-6*b4 - 5*b2 - 192) * q^9 + (-7*b4 + 6*b2 - 72) * q^10 + (12*b5 + 9*b3 + 133*b1) * q^11 + (33*b4 - 44*b2 + 342) * q^12 + (-13*b5 - 26*b4 + 13*b3 - 39*b2 + 65*b1 - 832) * q^13 + (27*b4 + 90*b2 + 1572) * q^14 + (-51*b5 + 90*b1) * q^15 + (-19*b4 + 240*b2 + 1210) * q^16 + (138*b4 - 45*b2 + 2253) * q^17 + (-12*b5 - 23*b3 - 989*b1) * q^18 + (50*b5 + 37*b3 + 245*b1) * q^19 + (114*b5 - 15*b3 - 990*b1) * q^20 + (-63*b5 - 22*b3 + 1298*b1) * q^21 + (-424*b4 - 684*b2 - 20760) * q^22 + (-312*b4 + 522*b2 + 4266) * q^23 + (66*b5 - 73*b3 + 3374*b1) * q^24 + (566*b4 - 1803*b2 - 2254) * q^25 + (-52*b5 - 273*b4 - 117*b3 - 1170*b2 - 4303*b1 - 8736) * q^26 + (168*b4 + 1913*b2 + 31845) * q^27 + (182*b5 + 43*b3 + 3818*b1) * q^28 + (12*b4 + 3150*b2 + 6108) * q^29 + (267*b4 - 306*b2 - 9828) * q^30 + (-44*b5 + 227*b3 - 10721*b1) * q^31 + (-294*b5 - 201*b3 - 4386*b1) * q^32 + (-504*b5 + 164*b3 - 10132*b1) * q^33 + (276*b5 + 369*b3 + 20424*b1) * q^34 + (840*b4 - 2115*b2 - 90135) * q^35 + (834*b4 + 1220*b2 + 124500) * q^36 + (-713*b5 + 188*b3 + 32332*b1) * q^37 + (-1446*b4 - 2808*b2 - 40128) * q^38 + (819*b5 + 624*b4 + 364*b3 - 1430*b2 - 16094*b1 + 85878) * q^39 + (-359*b4 + 2712*b2 + 130986) * q^40 + (152*b5 - 516*b3 + 10148*b1) * q^41 + (-351*b4 + 1470*b2 - 190296) * q^42 + (-2256*b4 - 1257*b2 - 167929) * q^43 + (688*b5 - 804*b3 - 60388*b1) * q^44 + (414*b5 - 90*b3 - 5490*b1) * q^45 + (-624*b5 - 414*b3 - 36396*b1) * q^46 + (833*b5 - 1293*b3 + 37865*b1) * q^47 + (2067*b4 + 896*b2 - 467514) * q^48 + (462*b4 - 735*b2 + 541540) * q^49 + (1132*b5 - 105*b3 + 70655*b1) * q^50 + (-4824*b4 + 6225*b2 + 45981) * q^51 + (-2210*b5 + 4030*b4 - 325*b3 + 4524*b2 - 37622*b1 + 541996) * q^52 + (-3648*b4 - 14364*b2 + 287826) * q^53 + (336*b5 + 2417*b3 + 55934*b1) * q^54 + (4088*b4 - 18624*b2 - 879192) * q^55 + (-2625*b4 + 9000*b2 - 384330) * q^56 + (-2106*b5 + 976*b3 - 38576*b1) * q^57 + (24*b5 + 3186*b3 + 10842*b1) * q^58 + (4858*b5 - 87*b3 - 49503*b1) * q^59 + (-5994*b5 + 495*b3 + 36630*b1) * q^60 + (4268*b4 + 28350*b2 - 1380956) * q^61 + (5808*b4 - 19332*b2 + 1612680) * q^62 + (762*b5 + 163*b3 + 29563*b1) * q^63 + (8635*b4 + 45840*b2 + 832742) * q^64 + (1066*b5 - 10374*b4 - 390*b3 + 27027*b2 - 22230*b1 + 1177371) * q^65 + (9888*b4 - 16800*b2 + 1557072) * q^66 + (-7872*b5 - 4803*b3 - 75975*b1) * q^67 + (-13179*b4 - 35100*b2 - 2792874) * q^68 + (13428*b4 - 17550*b2 - 992574) * q^69 + (1680*b5 + 405*b3 + 18630*b1) * q^70 + (2213*b5 - 7209*b3 - 20203*b1) * q^71 + (132*b5 + 778*b3 + 109216*b1) * q^72 + (9822*b5 + 5448*b3 - 4896*b1) * q^73 + (-31665*b4 - 20070*b2 - 4797336) * q^74 + (-29496*b4 + 14368*b2 + 3389724) * q^75 + (3508*b5 - 2410*b3 - 202448*b1) * q^76 + (14400*b4 - 34740*b2 + 698640) * q^77 + (1248*b5 + 1989*b4 + 442*b3 - 25662*b2 + 166816*b1 + 2357316) * q^78 + (-4240*b4 - 17670*b2 + 2529010) * q^79 + (13874*b5 - 285*b3 - 40410*b1) * q^80 + (19056*b4 + 16960*b2 - 3552375) * q^81 + (656*b4 + 44256*b2 - 1536240) * q^82 + (-186*b5 - 99*b3 + 428245*b1) * q^83 + (-8766*b5 - 2399*b3 - 69014*b1) * q^84 + (-18405*b5 + 2070*b3 + 140670*b1) * q^85 + (-4512*b5 - 8025*b3 - 466978*b1) * q^86 + (18504*b4 + 38784*b2 - 6085800) * q^87 + (19792*b4 - 15888*b2 + 6346560) * q^88 + (-17062*b5 + 7008*b3 - 425488*b1) * q^89 + (4662*b4 + 10044*b2 + 793152) * q^90 + (2366*b5 - 8112*b4 + 169*b3 + 2535*b2 + 148889*b1 + 3543423) * q^91 + (10350*b4 + 97848*b2 + 6047892) * q^92 + (4968*b5 + 18212*b3 - 145996*b1) * q^93 + (-13957*b4 + 113610*b2 - 5747484) * q^94 + (19296*b4 - 79578*b2 - 3646314) * q^95 + (12582*b5 - 2247*b3 + 238098*b1) * q^96 + (6460*b5 + 22760*b3 + 463456*b1) * q^97 + (924*b5 + 651*b3 + 601789*b1) * q^98 + (1428*b5 - 4139*b3 - 414983*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9}+O(q^{10})$$ 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 $$6 q - 56 q^{3} - 130 q^{4} - 1150 q^{9} - 406 q^{10} + 1898 q^{12} - 5018 q^{13} + 9558 q^{14} + 7778 q^{16} + 13152 q^{17} - 125080 q^{22} + 27264 q^{23} - 18262 q^{25} - 54210 q^{26} + 194560 q^{27} + 42924 q^{29} - 60114 q^{30} - 546720 q^{35} + 747772 q^{36} - 243492 q^{38} + 511160 q^{39} + 792058 q^{40} - 1138134 q^{42} - 1005576 q^{43} - 2807426 q^{48} + 3246846 q^{49} + 297984 q^{51} + 3252964 q^{52} + 1705524 q^{53} - 5320576 q^{55} - 2282730 q^{56} - 8237572 q^{61} + 9625800 q^{62} + 5070862 q^{64} + 7139028 q^{65} + 9289056 q^{66} - 16801086 q^{68} - 6017400 q^{69} - 28760826 q^{74} + 20426072 q^{75} + 4093560 q^{77} + 14088594 q^{78} + 15147200 q^{79} - 21318442 q^{81} - 9130240 q^{82} - 36474240 q^{87} + 38008000 q^{88} + 4769676 q^{90} + 21281832 q^{91} + 36462348 q^{92} - 34229770 q^{94} - 22075632 q^{95}+O(q^{100})$$ 6 * q - 56 * q^3 - 130 * q^4 - 1150 * q^9 - 406 * q^10 + 1898 * q^12 - 5018 * q^13 + 9558 * q^14 + 7778 * q^16 + 13152 * q^17 - 125080 * q^22 + 27264 * q^23 - 18262 * q^25 - 54210 * q^26 + 194560 * q^27 + 42924 * q^29 - 60114 * q^30 - 546720 * q^35 + 747772 * q^36 - 243492 * q^38 + 511160 * q^39 + 792058 * q^40 - 1138134 * q^42 - 1005576 * q^43 - 2807426 * q^48 + 3246846 * q^49 + 297984 * q^51 + 3252964 * q^52 + 1705524 * q^53 - 5320576 * q^55 - 2282730 * q^56 - 8237572 * q^61 + 9625800 * q^62 + 5070862 * q^64 + 7139028 * q^65 + 9289056 * q^66 - 16801086 * q^68 - 6017400 * q^69 - 28760826 * q^74 + 20426072 * q^75 + 4093560 * q^77 + 14088594 * q^78 + 15147200 * q^79 - 21318442 * q^81 - 9130240 * q^82 - 36474240 * q^87 + 38008000 * q^88 + 4769676 * q^90 + 21281832 * q^91 + 36462348 * q^92 - 34229770 * q^94 - 22075632 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 449x^{4} + 37224x^{2} + 205776$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 365\nu^{2} + 12324 ) / 240$$ (v^4 + 365*v^2 + 12324) / 240 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 365\nu^{3} + 12084\nu ) / 240$$ (v^5 + 365*v^3 + 12084*v) / 240 $$\beta_{4}$$ $$=$$ $$-\nu^{2} - 150$$ -v^2 - 150 $$\beta_{5}$$ $$=$$ $$( -\nu^{5} - 445\nu^{3} - 34644\nu ) / 160$$ (-v^5 - 445*v^3 - 34644*v) / 160
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{4} - 150$$ -b4 - 150 $$\nu^{3}$$ $$=$$ $$-2\beta_{5} - 3\beta_{3} - 282\beta_1$$ -2*b5 - 3*b3 - 282*b1 $$\nu^{4}$$ $$=$$ $$365\beta_{4} + 240\beta_{2} + 42426$$ 365*b4 + 240*b2 + 42426 $$\nu^{5}$$ $$=$$ $$730\beta_{5} + 1335\beta_{3} + 90846\beta_1$$ 730*b5 + 1335*b3 + 90846*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
12.1
 − 18.4900i − 10.0583i − 2.43912i 2.43912i 10.0583i 18.4900i
18.4900i −27.4160 −213.880 70.5606i 506.922i 454.852i 1587.93i −1435.36 −1304.67
12.2 10.0583i 50.8656 26.8297 8.88672i 511.623i 510.452i 1557.33i 400.306 −89.3858
12.3 2.43912i −51.4496 122.051 488.312i 125.492i 616.243i 609.904i 460.056 1191.05
12.4 2.43912i −51.4496 122.051 488.312i 125.492i 616.243i 609.904i 460.056 1191.05
12.5 10.0583i 50.8656 26.8297 8.88672i 511.623i 510.452i 1557.33i 400.306 −89.3858
12.6 18.4900i −27.4160 −213.880 70.5606i 506.922i 454.852i 1587.93i −1435.36 −1304.67
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 12.6 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 13.8.b.a 6
3.b odd 2 1 117.8.b.b 6
4.b odd 2 1 208.8.f.a 6
13.b even 2 1 inner 13.8.b.a 6
13.d odd 4 2 169.8.a.d 6
39.d odd 2 1 117.8.b.b 6
52.b odd 2 1 208.8.f.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.b.a 6 1.a even 1 1 trivial
13.8.b.a 6 13.b even 2 1 inner
117.8.b.b 6 3.b odd 2 1
117.8.b.b 6 39.d odd 2 1
169.8.a.d 6 13.d odd 4 2
208.8.f.a 6 4.b odd 2 1
208.8.f.a 6 52.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 449 T^{4} + \cdots + 205776$$
$3$ $$(T^{3} + 28 T^{2} + \cdots - 71748)^{2}$$
$5$ $$T^{6} + \cdots + 93756690000$$
$7$ $$T^{6} + \cdots + 20\!\cdots\!00$$
$11$ $$T^{6} + \cdots + 13\!\cdots\!00$$
$13$ $$T^{6} + \cdots + 24\!\cdots\!13$$
$17$ $$(T^{3} - 6576 T^{2} + \cdots - 976631438250)^{2}$$
$19$ $$T^{6} + \cdots + 62\!\cdots\!36$$
$23$ $$(T^{3} + \cdots - 38560806996192)^{2}$$
$29$ $$(T^{3} + \cdots + 16\!\cdots\!80)^{2}$$
$31$ $$T^{6} + \cdots + 33\!\cdots\!00$$
$37$ $$T^{6} + \cdots + 30\!\cdots\!96$$
$41$ $$T^{6} + \cdots + 18\!\cdots\!00$$
$43$ $$(T^{3} + \cdots + 11\!\cdots\!88)^{2}$$
$47$ $$T^{6} + \cdots + 32\!\cdots\!96$$
$53$ $$(T^{3} + \cdots + 18\!\cdots\!92)^{2}$$
$59$ $$T^{6} + \cdots + 36\!\cdots\!96$$
$61$ $$(T^{3} + \cdots + 10\!\cdots\!88)^{2}$$
$67$ $$T^{6} + \cdots + 54\!\cdots\!96$$
$71$ $$T^{6} + \cdots + 39\!\cdots\!00$$
$73$ $$T^{6} + \cdots + 10\!\cdots\!04$$
$79$ $$(T^{3} + \cdots - 12\!\cdots\!00)^{2}$$
$83$ $$T^{6} + \cdots + 14\!\cdots\!84$$
$89$ $$T^{6} + \cdots + 22\!\cdots\!96$$
$97$ $$T^{6} + \cdots + 17\!\cdots\!56$$