# Properties

 Label 704.4.a.p Level $704$ Weight $4$ Character orbit 704.a Self dual yes Analytic conductor $41.537$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,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, 4, names="a")

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

chi := DirichletCharacter("704.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$41.5373446440$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 4\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + ( - 2 \beta - 1) q^{5} + ( - \beta + 10) q^{7} + (2 \beta + 22) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (-2*b - 1) * q^5 + (-b + 10) * q^7 + (2*b + 22) * q^9 $$q + (\beta + 1) q^{3} + ( - 2 \beta - 1) q^{5} + ( - \beta + 10) q^{7} + (2 \beta + 22) q^{9} + 11 q^{11} + (5 \beta - 40) q^{13} + ( - 3 \beta - 97) q^{15} + (3 \beta - 62) q^{17} + ( - 15 \beta - 36) q^{19} + (9 \beta - 38) q^{21} + ( - 9 \beta - 49) q^{23} + (4 \beta + 68) q^{25} + ( - 3 \beta + 91) q^{27} + (14 \beta - 72) q^{29} + (7 \beta - 17) q^{31} + (11 \beta + 11) q^{33} + ( - 19 \beta + 86) q^{35} + (2 \beta - 27) q^{37} + ( - 35 \beta + 200) q^{39} + ( - \beta + 268) q^{41} + (4 \beta + 30) q^{43} + ( - 46 \beta - 214) q^{45} + ( - 30 \beta - 136) q^{47} + ( - 20 \beta - 195) q^{49} + ( - 59 \beta + 82) q^{51} + (14 \beta + 246) q^{53} + ( - 22 \beta - 11) q^{55} + ( - 51 \beta - 756) q^{57} + (33 \beta - 317) q^{59} + ( - 46 \beta - 420) q^{61} + ( - 2 \beta + 124) q^{63} + (75 \beta - 440) q^{65} + (5 \beta - 377) q^{67} + ( - 58 \beta - 481) q^{69} + (19 \beta - 339) q^{71} + ( - 117 \beta - 200) q^{73} + (72 \beta + 260) q^{75} + ( - 11 \beta + 110) q^{77} + (164 \beta + 158) q^{79} + (34 \beta - 647) q^{81} + ( - 30 \beta - 234) q^{83} + (121 \beta - 226) q^{85} + ( - 58 \beta + 600) q^{87} + ( - 82 \beta - 921) q^{89} + (90 \beta - 640) q^{91} + ( - 10 \beta + 319) q^{93} + (87 \beta + 1476) q^{95} + (36 \beta + 1097) q^{97} + (22 \beta + 242) q^{99}+O(q^{100})$$ q + (b + 1) * q^3 + (-2*b - 1) * q^5 + (-b + 10) * q^7 + (2*b + 22) * q^9 + 11 * q^11 + (5*b - 40) * q^13 + (-3*b - 97) * q^15 + (3*b - 62) * q^17 + (-15*b - 36) * q^19 + (9*b - 38) * q^21 + (-9*b - 49) * q^23 + (4*b + 68) * q^25 + (-3*b + 91) * q^27 + (14*b - 72) * q^29 + (7*b - 17) * q^31 + (11*b + 11) * q^33 + (-19*b + 86) * q^35 + (2*b - 27) * q^37 + (-35*b + 200) * q^39 + (-b + 268) * q^41 + (4*b + 30) * q^43 + (-46*b - 214) * q^45 + (-30*b - 136) * q^47 + (-20*b - 195) * q^49 + (-59*b + 82) * q^51 + (14*b + 246) * q^53 + (-22*b - 11) * q^55 + (-51*b - 756) * q^57 + (33*b - 317) * q^59 + (-46*b - 420) * q^61 + (-2*b + 124) * q^63 + (75*b - 440) * q^65 + (5*b - 377) * q^67 + (-58*b - 481) * q^69 + (19*b - 339) * q^71 + (-117*b - 200) * q^73 + (72*b + 260) * q^75 + (-11*b + 110) * q^77 + (164*b + 158) * q^79 + (34*b - 647) * q^81 + (-30*b - 234) * q^83 + (121*b - 226) * q^85 + (-58*b + 600) * q^87 + (-82*b - 921) * q^89 + (90*b - 640) * q^91 + (-10*b + 319) * q^93 + (87*b + 1476) * q^95 + (36*b + 1097) * q^97 + (22*b + 242) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^5 + 20 * q^7 + 44 * q^9 $$2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9} + 22 q^{11} - 80 q^{13} - 194 q^{15} - 124 q^{17} - 72 q^{19} - 76 q^{21} - 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} - 34 q^{31} + 22 q^{33} + 172 q^{35} - 54 q^{37} + 400 q^{39} + 536 q^{41} + 60 q^{43} - 428 q^{45} - 272 q^{47} - 390 q^{49} + 164 q^{51} + 492 q^{53} - 22 q^{55} - 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} - 880 q^{65} - 754 q^{67} - 962 q^{69} - 678 q^{71} - 400 q^{73} + 520 q^{75} + 220 q^{77} + 316 q^{79} - 1294 q^{81} - 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} + 638 q^{93} + 2952 q^{95} + 2194 q^{97} + 484 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^5 + 20 * q^7 + 44 * q^9 + 22 * q^11 - 80 * q^13 - 194 * q^15 - 124 * q^17 - 72 * q^19 - 76 * q^21 - 98 * q^23 + 136 * q^25 + 182 * q^27 - 144 * q^29 - 34 * q^31 + 22 * q^33 + 172 * q^35 - 54 * q^37 + 400 * q^39 + 536 * q^41 + 60 * q^43 - 428 * q^45 - 272 * q^47 - 390 * q^49 + 164 * q^51 + 492 * q^53 - 22 * q^55 - 1512 * q^57 - 634 * q^59 - 840 * q^61 + 248 * q^63 - 880 * q^65 - 754 * q^67 - 962 * q^69 - 678 * q^71 - 400 * q^73 + 520 * q^75 + 220 * q^77 + 316 * q^79 - 1294 * q^81 - 468 * q^83 - 452 * q^85 + 1200 * q^87 - 1842 * q^89 - 1280 * q^91 + 638 * q^93 + 2952 * q^95 + 2194 * q^97 + 484 * q^99

## 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
 −1.73205 1.73205
0 −5.92820 0 12.8564 0 16.9282 0 8.14359 0
1.2 0 7.92820 0 −14.8564 0 3.07180 0 35.8564 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 704.4.a.p 2
4.b odd 2 1 704.4.a.n 2
8.b even 2 1 11.4.a.a 2
8.d odd 2 1 176.4.a.i 2
24.f even 2 1 1584.4.a.bc 2
24.h odd 2 1 99.4.a.c 2
40.f even 2 1 275.4.a.b 2
40.i odd 4 2 275.4.b.c 4
56.h odd 2 1 539.4.a.e 2
88.b odd 2 1 121.4.a.c 2
88.g even 2 1 1936.4.a.w 2
88.o even 10 4 121.4.c.c 8
88.p odd 10 4 121.4.c.f 8
104.e even 2 1 1859.4.a.a 2
120.i odd 2 1 2475.4.a.q 2
264.m even 2 1 1089.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 8.b even 2 1
99.4.a.c 2 24.h odd 2 1
121.4.a.c 2 88.b odd 2 1
121.4.c.c 8 88.o even 10 4
121.4.c.f 8 88.p odd 10 4
176.4.a.i 2 8.d odd 2 1
275.4.a.b 2 40.f even 2 1
275.4.b.c 4 40.i odd 4 2
539.4.a.e 2 56.h odd 2 1
704.4.a.n 2 4.b odd 2 1
704.4.a.p 2 1.a even 1 1 trivial
1089.4.a.v 2 264.m even 2 1
1584.4.a.bc 2 24.f even 2 1
1859.4.a.a 2 104.e even 2 1
1936.4.a.w 2 88.g even 2 1
2475.4.a.q 2 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(704))$$:

 $$T_{3}^{2} - 2T_{3} - 47$$ T3^2 - 2*T3 - 47 $$T_{5}^{2} + 2T_{5} - 191$$ T5^2 + 2*T5 - 191

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 47$$
$5$ $$T^{2} + 2T - 191$$
$7$ $$T^{2} - 20T + 52$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} + 80T + 400$$
$17$ $$T^{2} + 124T + 3412$$
$19$ $$T^{2} + 72T - 9504$$
$23$ $$T^{2} + 98T - 1487$$
$29$ $$T^{2} + 144T - 4224$$
$31$ $$T^{2} + 34T - 2063$$
$37$ $$T^{2} + 54T + 537$$
$41$ $$T^{2} - 536T + 71776$$
$43$ $$T^{2} - 60T + 132$$
$47$ $$T^{2} + 272T - 24704$$
$53$ $$T^{2} - 492T + 51108$$
$59$ $$T^{2} + 634T + 48217$$
$61$ $$T^{2} + 840T + 74832$$
$67$ $$T^{2} + 754T + 140929$$
$71$ $$T^{2} + 678T + 97593$$
$73$ $$T^{2} + 400T - 617072$$
$79$ $$T^{2} - 316 T - 1266044$$
$83$ $$T^{2} + 468T + 11556$$
$89$ $$T^{2} + 1842 T + 525489$$
$97$ $$T^{2} - 2194 T + 1141201$$