# Properties

 Label 784.6.a.ba Level $784$ Weight $6$ Character orbit 784.a Self dual yes Analytic conductor $125.741$ Analytic rank $0$ 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] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$125.740914733$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 9$$ x^2 - x - 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 7) 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 $$\beta = \sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 4) q^{3} + ( - 10 \beta - 19) q^{5} + (8 \beta - 190) q^{9}+O(q^{10})$$ q + (b + 4) * q^3 + (-10*b - 19) * q^5 + (8*b - 190) * q^9 $$q + (\beta + 4) q^{3} + ( - 10 \beta - 19) q^{5} + (8 \beta - 190) q^{9} + ( - 23 \beta - 212) q^{11} + (28 \beta - 462) q^{13} + ( - 59 \beta - 446) q^{15} + (132 \beta - 1173) q^{17} + ( - 277 \beta + 180) q^{19} + (69 \beta + 6) q^{23} + (380 \beta + 936) q^{25} + ( - 401 \beta - 1436) q^{27} + ( - 700 \beta - 3526) q^{29} + (715 \beta - 1774) q^{31} + ( - 304 \beta - 1699) q^{33} + ( - 790 \beta + 5545) q^{37} + ( - 350 \beta - 812) q^{39} + (868 \beta + 1750) q^{41} + ( - 1344 \beta + 6340) q^{43} + (1748 \beta + 650) q^{45} + ( - 1635 \beta + 11478) q^{47} + ( - 645 \beta + 192) q^{51} + ( - 1818 \beta + 1521) q^{53} + (2557 \beta + 12538) q^{55} + ( - 928 \beta - 9529) q^{57} + ( - 531 \beta + 32904) q^{59} + ( - 4154 \beta - 21243) q^{61} + (4088 \beta - 1582) q^{65} + ( - 919 \beta - 21156) q^{67} + (282 \beta + 2577) q^{69} + (2184 \beta + 1104) q^{71} + ( - 7372 \beta - 25253) q^{73} + (2456 \beta + 17804) q^{75} + (5193 \beta - 4502) q^{79} + ( - 4984 \beta + 25589) q^{81} + (4536 \beta + 52164) q^{83} + (9222 \beta - 26553) q^{85} + ( - 6326 \beta - 40004) q^{87} + (9356 \beta - 13333) q^{89} + (1086 \beta + 19359) q^{93} + (3463 \beta + 99070) q^{95} + ( - 196 \beta + 104566) q^{97} + (2674 \beta + 33472) q^{99}+O(q^{100})$$ q + (b + 4) * q^3 + (-10*b - 19) * q^5 + (8*b - 190) * q^9 + (-23*b - 212) * q^11 + (28*b - 462) * q^13 + (-59*b - 446) * q^15 + (132*b - 1173) * q^17 + (-277*b + 180) * q^19 + (69*b + 6) * q^23 + (380*b + 936) * q^25 + (-401*b - 1436) * q^27 + (-700*b - 3526) * q^29 + (715*b - 1774) * q^31 + (-304*b - 1699) * q^33 + (-790*b + 5545) * q^37 + (-350*b - 812) * q^39 + (868*b + 1750) * q^41 + (-1344*b + 6340) * q^43 + (1748*b + 650) * q^45 + (-1635*b + 11478) * q^47 + (-645*b + 192) * q^51 + (-1818*b + 1521) * q^53 + (2557*b + 12538) * q^55 + (-928*b - 9529) * q^57 + (-531*b + 32904) * q^59 + (-4154*b - 21243) * q^61 + (4088*b - 1582) * q^65 + (-919*b - 21156) * q^67 + (282*b + 2577) * q^69 + (2184*b + 1104) * q^71 + (-7372*b - 25253) * q^73 + (2456*b + 17804) * q^75 + (5193*b - 4502) * q^79 + (-4984*b + 25589) * q^81 + (4536*b + 52164) * q^83 + (9222*b - 26553) * q^85 + (-6326*b - 40004) * q^87 + (9356*b - 13333) * q^89 + (1086*b + 19359) * q^93 + (3463*b + 99070) * q^95 + (-196*b + 104566) * q^97 + (2674*b + 33472) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{3} - 38 q^{5} - 380 q^{9}+O(q^{10})$$ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 $$2 q + 8 q^{3} - 38 q^{5} - 380 q^{9} - 424 q^{11} - 924 q^{13} - 892 q^{15} - 2346 q^{17} + 360 q^{19} + 12 q^{23} + 1872 q^{25} - 2872 q^{27} - 7052 q^{29} - 3548 q^{31} - 3398 q^{33} + 11090 q^{37} - 1624 q^{39} + 3500 q^{41} + 12680 q^{43} + 1300 q^{45} + 22956 q^{47} + 384 q^{51} + 3042 q^{53} + 25076 q^{55} - 19058 q^{57} + 65808 q^{59} - 42486 q^{61} - 3164 q^{65} - 42312 q^{67} + 5154 q^{69} + 2208 q^{71} - 50506 q^{73} + 35608 q^{75} - 9004 q^{79} + 51178 q^{81} + 104328 q^{83} - 53106 q^{85} - 80008 q^{87} - 26666 q^{89} + 38718 q^{93} + 198140 q^{95} + 209132 q^{97} + 66944 q^{99}+O(q^{100})$$ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 - 424 * q^11 - 924 * q^13 - 892 * q^15 - 2346 * q^17 + 360 * q^19 + 12 * q^23 + 1872 * q^25 - 2872 * q^27 - 7052 * q^29 - 3548 * q^31 - 3398 * q^33 + 11090 * q^37 - 1624 * q^39 + 3500 * q^41 + 12680 * q^43 + 1300 * q^45 + 22956 * q^47 + 384 * q^51 + 3042 * q^53 + 25076 * q^55 - 19058 * q^57 + 65808 * q^59 - 42486 * q^61 - 3164 * q^65 - 42312 * q^67 + 5154 * q^69 + 2208 * q^71 - 50506 * q^73 + 35608 * q^75 - 9004 * q^79 + 51178 * q^81 + 104328 * q^83 - 53106 * q^85 - 80008 * q^87 - 26666 * q^89 + 38718 * q^93 + 198140 * q^95 + 209132 * q^97 + 66944 * 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
 −2.54138 3.54138
0 −2.08276 0 41.8276 0 0 0 −238.662 0
1.2 0 10.0828 0 −79.8276 0 0 0 −141.338 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
$$2$$ $$-1$$
$$7$$ $$+1$$

## 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 784.6.a.ba 2
4.b odd 2 1 49.6.a.d 2
7.b odd 2 1 784.6.a.t 2
7.c even 3 2 112.6.i.c 4
12.b even 2 1 441.6.a.n 2
28.d even 2 1 49.6.a.e 2
28.f even 6 2 49.6.c.f 4
28.g odd 6 2 7.6.c.a 4
84.h odd 2 1 441.6.a.m 2
84.n even 6 2 63.6.e.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.c.a 4 28.g odd 6 2
49.6.a.d 2 4.b odd 2 1
49.6.a.e 2 28.d even 2 1
49.6.c.f 4 28.f even 6 2
63.6.e.d 4 84.n even 6 2
112.6.i.c 4 7.c even 3 2
441.6.a.m 2 84.h odd 2 1
441.6.a.n 2 12.b even 2 1
784.6.a.t 2 7.b odd 2 1
784.6.a.ba 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 8T_{3} - 21$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 8T - 21$$
$5$ $$T^{2} + 38T - 3339$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 424T + 25371$$
$13$ $$T^{2} + 924T + 184436$$
$17$ $$T^{2} + 2346 T + 731241$$
$19$ $$T^{2} - 360 T - 2806573$$
$23$ $$T^{2} - 12T - 176121$$
$29$ $$T^{2} + 7052 T - 5697324$$
$31$ $$T^{2} + 3548 T - 15768249$$
$37$ $$T^{2} - 11090 T + 7655325$$
$41$ $$T^{2} - 3500 T - 24814188$$
$43$ $$T^{2} - 12680 T - 26638832$$
$47$ $$T^{2} - 22956 T + 32835159$$
$53$ $$T^{2} - 3042 T - 119976147$$
$59$ $$T^{2} + \cdots + 1072240659$$
$61$ $$T^{2} + 42486 T - 187196443$$
$67$ $$T^{2} + 42312 T + 416327579$$
$71$ $$T^{2} - 2208 T - 175265856$$
$73$ $$T^{2} + \cdots - 1373102199$$
$79$ $$T^{2} + 9004 T - 977520209$$
$83$ $$T^{2} + \cdots + 1959796944$$
$89$ $$T^{2} + \cdots - 3061016343$$
$97$ $$T^{2} + \cdots + 10932626964$$