# Properties

 Label 704.6.a.q Level $704$ Weight $6$ Character orbit 704.a Self dual yes Analytic conductor $112.910$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 704.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: $$112.910209148$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.54492.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 52x - 38$$ x^3 - x^2 - 52*x - 38 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 11) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 11) q^{3} + ( - \beta_{2} - 4 \beta_1 - 7) q^{5} + ( - 5 \beta_{2} - 5 \beta_1 + 28) q^{7} + ( - \beta_{2} + 16 \beta_1 - 8) q^{9}+O(q^{10})$$ q + (-b1 - 11) * q^3 + (-b2 - 4*b1 - 7) * q^5 + (-5*b2 - 5*b1 + 28) * q^7 + (-b2 + 16*b1 - 8) * q^9 $$q + ( - \beta_1 - 11) q^{3} + ( - \beta_{2} - 4 \beta_1 - 7) q^{5} + ( - 5 \beta_{2} - 5 \beta_1 + 28) q^{7} + ( - \beta_{2} + 16 \beta_1 - 8) q^{9} - 121 q^{11} + ( - 11 \beta_{2} + 37 \beta_1 - 178) q^{13} + (14 \beta_{2} + 15 \beta_1 + 551) q^{15} + (9 \beta_{2} - 105 \beta_1 + 400) q^{17} + (15 \beta_{2} - 15 \beta_1 - 450) q^{19} + (85 \beta_{2} - 63 \beta_1 + 352) q^{21} + (60 \beta_{2} + 201 \beta_1 - 1069) q^{23} + (65 \beta_{2} - 70 \beta_1 + 26) q^{25} + (34 \beta_{2} + 159 \beta_1 + 955) q^{27} + (40 \beta_{2} - 62 \beta_1 + 1176) q^{29} + ( - 4 \beta_{2} + 89 \beta_1 - 1397) q^{31} + (121 \beta_1 + 1331) q^{33} + (217 \beta_{2} - 167 \beta_1 + 8204) q^{35} + ( - 191 \beta_{2} + 364 \beta_1 - 6093) q^{37} + (235 \beta_{2} - 139 \beta_1 - 2062) q^{39} + ( - 137 \beta_{2} + 967 \beta_1 + 1630) q^{41} + ( - 10 \beta_{2} + 1160 \beta_1 + 8346) q^{43} + (6 \beta_{2} + 514 \beta_1 - 6322) q^{45} + ( - 234 \beta_{2} - 102 \beta_1 - 5788) q^{47} + (320 \beta_{2} + 620 \beta_1 + 16077) q^{49} + ( - 267 \beta_{2} + 233 \beta_1 + 7408) q^{51} + (262 \beta_{2} + 310 \beta_1 - 16878) q^{53} + (121 \beta_{2} + 484 \beta_1 + 847) q^{55} + ( - 285 \beta_{2} + 705 \beta_1 + 6390) q^{57} + ( - 486 \beta_{2} + 1683 \beta_1 + 523) q^{59} + (688 \beta_{2} + 598 \beta_1 - 6132) q^{61} + ( - 378 \beta_{2} + 2198 \beta_1 - 5024) q^{63} + (573 \beta_{2} + 1983 \beta_1 - 3026) q^{65} + (1162 \beta_{2} - 3089 \beta_1 + 17335) q^{67} + ( - 879 \beta_{2} + 784 \beta_1 - 12235) q^{69} + ( - 826 \beta_{2} + 1457 \beta_1 + 12333) q^{71} + (231 \beta_{2} - 4677 \beta_1 + 6778) q^{73} + ( - 1240 \beta_{2} + 1104 \beta_1 + 6524) q^{75} + (605 \beta_{2} + 605 \beta_1 - 3388) q^{77} + (658 \beta_{2} - 1928 \beta_1 + 42578) q^{79} + ( - 210 \beta_{2} - 5230 \beta_1 - 27299) q^{81} + ( - 1800 \beta_{2} - 2250 \beta_1 + 48126) q^{83} + ( - 499 \beta_{2} - 4807 \beta_1 + 36116) q^{85} + ( - 782 \beta_{2} - 386 \beta_1 - 6588) q^{87} + ( - 1217 \beta_{2} + 1516 \beta_1 - 36519) q^{89} + (462 \beta_{2} + 8226 \beta_1 + 33956) q^{91} + (161 \beta_{2} + 904 \beta_1 + 5293) q^{93} + ( - 195 \beta_{2} + 1095 \beta_1 - 7830) q^{95} + (819 \beta_{2} + 2334 \beta_1 + 2723) q^{97} + (121 \beta_{2} - 1936 \beta_1 + 968) q^{99}+O(q^{100})$$ q + (-b1 - 11) * q^3 + (-b2 - 4*b1 - 7) * q^5 + (-5*b2 - 5*b1 + 28) * q^7 + (-b2 + 16*b1 - 8) * q^9 - 121 * q^11 + (-11*b2 + 37*b1 - 178) * q^13 + (14*b2 + 15*b1 + 551) * q^15 + (9*b2 - 105*b1 + 400) * q^17 + (15*b2 - 15*b1 - 450) * q^19 + (85*b2 - 63*b1 + 352) * q^21 + (60*b2 + 201*b1 - 1069) * q^23 + (65*b2 - 70*b1 + 26) * q^25 + (34*b2 + 159*b1 + 955) * q^27 + (40*b2 - 62*b1 + 1176) * q^29 + (-4*b2 + 89*b1 - 1397) * q^31 + (121*b1 + 1331) * q^33 + (217*b2 - 167*b1 + 8204) * q^35 + (-191*b2 + 364*b1 - 6093) * q^37 + (235*b2 - 139*b1 - 2062) * q^39 + (-137*b2 + 967*b1 + 1630) * q^41 + (-10*b2 + 1160*b1 + 8346) * q^43 + (6*b2 + 514*b1 - 6322) * q^45 + (-234*b2 - 102*b1 - 5788) * q^47 + (320*b2 + 620*b1 + 16077) * q^49 + (-267*b2 + 233*b1 + 7408) * q^51 + (262*b2 + 310*b1 - 16878) * q^53 + (121*b2 + 484*b1 + 847) * q^55 + (-285*b2 + 705*b1 + 6390) * q^57 + (-486*b2 + 1683*b1 + 523) * q^59 + (688*b2 + 598*b1 - 6132) * q^61 + (-378*b2 + 2198*b1 - 5024) * q^63 + (573*b2 + 1983*b1 - 3026) * q^65 + (1162*b2 - 3089*b1 + 17335) * q^67 + (-879*b2 + 784*b1 - 12235) * q^69 + (-826*b2 + 1457*b1 + 12333) * q^71 + (231*b2 - 4677*b1 + 6778) * q^73 + (-1240*b2 + 1104*b1 + 6524) * q^75 + (605*b2 + 605*b1 - 3388) * q^77 + (658*b2 - 1928*b1 + 42578) * q^79 + (-210*b2 - 5230*b1 - 27299) * q^81 + (-1800*b2 - 2250*b1 + 48126) * q^83 + (-499*b2 - 4807*b1 + 36116) * q^85 + (-782*b2 - 386*b1 - 6588) * q^87 + (-1217*b2 + 1516*b1 - 36519) * q^89 + (462*b2 + 8226*b1 + 33956) * q^91 + (161*b2 + 904*b1 + 5293) * q^93 + (-195*b2 + 1095*b1 - 7830) * q^95 + (819*b2 + 2334*b1 + 2723) * q^97 + (121*b2 - 1936*b1 + 968) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 34 q^{3} - 24 q^{5} + 84 q^{7} - 7 q^{9}+O(q^{10})$$ 3 * q - 34 * q^3 - 24 * q^5 + 84 * q^7 - 7 * q^9 $$3 q - 34 q^{3} - 24 q^{5} + 84 q^{7} - 7 q^{9} - 363 q^{11} - 486 q^{13} + 1654 q^{15} + 1086 q^{17} - 1380 q^{19} + 908 q^{21} - 3066 q^{23} - 57 q^{25} + 2990 q^{27} + 3426 q^{29} - 4098 q^{31} + 4114 q^{33} + 24228 q^{35} - 17724 q^{37} - 6560 q^{39} + 5994 q^{41} + 26208 q^{43} - 18458 q^{45} - 17232 q^{47} + 48531 q^{49} + 22724 q^{51} - 50586 q^{53} + 2904 q^{55} + 20160 q^{57} + 3738 q^{59} - 18486 q^{61} - 12496 q^{63} - 7668 q^{65} + 47754 q^{67} - 35042 q^{69} + 39282 q^{71} + 15426 q^{73} + 21916 q^{75} - 10164 q^{77} + 125148 q^{79} - 86917 q^{81} + 143928 q^{83} + 104040 q^{85} - 19368 q^{87} - 106824 q^{89} + 109632 q^{91} + 16622 q^{93} - 22200 q^{95} + 9684 q^{97} + 847 q^{99}+O(q^{100})$$ 3 * q - 34 * q^3 - 24 * q^5 + 84 * q^7 - 7 * q^9 - 363 * q^11 - 486 * q^13 + 1654 * q^15 + 1086 * q^17 - 1380 * q^19 + 908 * q^21 - 3066 * q^23 - 57 * q^25 + 2990 * q^27 + 3426 * q^29 - 4098 * q^31 + 4114 * q^33 + 24228 * q^35 - 17724 * q^37 - 6560 * q^39 + 5994 * q^41 + 26208 * q^43 - 18458 * q^45 - 17232 * q^47 + 48531 * q^49 + 22724 * q^51 - 50586 * q^53 + 2904 * q^55 + 20160 * q^57 + 3738 * q^59 - 18486 * q^61 - 12496 * q^63 - 7668 * q^65 + 47754 * q^67 - 35042 * q^69 + 39282 * q^71 + 15426 * q^73 + 21916 * q^75 - 10164 * q^77 + 125148 * q^79 - 86917 * q^81 + 143928 * q^83 + 104040 * q^85 - 19368 * q^87 - 106824 * q^89 + 109632 * q^91 + 16622 * q^93 - 22200 * q^95 + 9684 * q^97 + 847 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 52x - 38$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{2} + 6\nu + 34 ) / 3$$ (-v^2 + 6*v + 34) / 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2\nu - 36$$ v^2 + 2*v - 36
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta _1 + 2 ) / 8$$ (b2 + 3*b1 + 2) / 8 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} - 3\beta _1 + 142 ) / 4$$ (3*b2 - 3*b1 + 142) / 4

## 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
 −0.749680 8.04796 −6.29828
0 −20.6466 0 −8.64919 0 164.454 0 183.283 0
1.2 0 −16.8394 0 −75.2230 0 −225.525 0 40.5643 0
1.3 0 3.48600 0 59.8722 0 145.071 0 −230.848 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$1$$

## 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 704.6.a.q 3
4.b odd 2 1 704.6.a.t 3
8.b even 2 1 11.6.a.b 3
8.d odd 2 1 176.6.a.i 3
24.h odd 2 1 99.6.a.g 3
40.f even 2 1 275.6.a.b 3
40.i odd 4 2 275.6.b.b 6
56.h odd 2 1 539.6.a.e 3
88.b odd 2 1 121.6.a.d 3
264.m even 2 1 1089.6.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.b 3 8.b even 2 1
99.6.a.g 3 24.h odd 2 1
121.6.a.d 3 88.b odd 2 1
176.6.a.i 3 8.d odd 2 1
275.6.a.b 3 40.f even 2 1
275.6.b.b 6 40.i odd 4 2
539.6.a.e 3 56.h odd 2 1
704.6.a.q 3 1.a even 1 1 trivial
704.6.a.t 3 4.b odd 2 1
1089.6.a.r 3 264.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 34T_{3}^{2} + 217T_{3} - 1212$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(704))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} + 34 T^{2} + \cdots - 1212$$
$5$ $$T^{3} + 24 T^{2} + \cdots - 38954$$
$7$ $$T^{3} - 84 T^{2} + \cdots + 5380448$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} + 486 T^{2} + \cdots - 164136608$$
$17$ $$T^{3} - 1086 T^{2} + \cdots + 331752056$$
$19$ $$T^{3} + 1380 T^{2} + \cdots - 57024000$$
$23$ $$T^{3} + \cdots - 17004325928$$
$29$ $$T^{3} + \cdots + 4029189120$$
$31$ $$T^{3} + \cdots + 1094344400$$
$37$ $$T^{3} + \cdots - 541788167034$$
$41$ $$T^{3} + \cdots + 201929821568$$
$43$ $$T^{3} + \cdots + 2443875098544$$
$47$ $$T^{3} + \cdots - 70174939136$$
$53$ $$T^{3} + \cdots + 1850911309656$$
$59$ $$T^{3} + \cdots - 7759637437060$$
$61$ $$T^{3} + \cdots - 15233874751008$$
$67$ $$T^{3} + \cdots + 147288561330212$$
$71$ $$T^{3} + \cdots - 1290398551704$$
$73$ $$T^{3} + \cdots - 34539701265952$$
$79$ $$T^{3} + \cdots + 1279883216320$$
$83$ $$T^{3} + \cdots + 411597824719824$$
$89$ $$T^{3} + \cdots - 90320980174650$$
$97$ $$T^{3} + \cdots - 10221902527106$$