Properties

 Label 784.6.a.u 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: $$\Q(\sqrt{345})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 86$$ x^2 - x - 86 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) 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 $$\beta = \sqrt{345}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 3) q^{3} + (3 \beta - 41) q^{5} + (6 \beta + 111) q^{9}+O(q^{10})$$ q + (-b - 3) * q^3 + (3*b - 41) * q^5 + (6*b + 111) * q^9 $$q + ( - \beta - 3) q^{3} + (3 \beta - 41) q^{5} + (6 \beta + 111) q^{9} + (6 \beta - 170) q^{11} + ( - 39 \beta - 455) q^{13} + (32 \beta - 912) q^{15} + (6 \beta - 1608) q^{17} + (45 \beta - 337) q^{19} + (72 \beta + 552) q^{23} + ( - 246 \beta + 1661) q^{25} + (114 \beta - 1674) q^{27} + ( - 114 \beta + 4032) q^{29} + (210 \beta - 3106) q^{31} + (152 \beta - 1560) q^{33} + (390 \beta - 4256) q^{37} + (572 \beta + 14820) q^{39} + ( - 678 \beta + 652) q^{41} + ( - 798 \beta + 5002) q^{43} + (87 \beta + 1659) q^{45} + ( - 714 \beta - 6374) q^{47} + (1590 \beta + 2754) q^{51} + ( - 768 \beta - 5610) q^{53} + ( - 756 \beta + 13180) q^{55} + (202 \beta - 14514) q^{57} + (2253 \beta - 6009) q^{59} + (75 \beta - 51369) q^{61} + (234 \beta - 21710) q^{65} + ( - 1944 \beta - 12068) q^{67} + ( - 768 \beta - 26496) q^{69} + ( - 1740 \beta - 44860) q^{71} + (1716 \beta + 27794) q^{73} + ( - 923 \beta + 79887) q^{75} + ( - 1956 \beta - 24412) q^{79} + ( - 126 \beta - 61281) q^{81} + ( - 3447 \beta + 17891) q^{83} + ( - 5070 \beta + 72138) q^{85} + ( - 3690 \beta + 27234) q^{87} + ( - 4728 \beta + 9150) q^{89} + (2476 \beta - 63132) q^{93} + ( - 2856 \beta + 60392) q^{95} + (462 \beta + 34992) q^{97} + ( - 354 \beta - 6450) q^{99}+O(q^{100})$$ q + (-b - 3) * q^3 + (3*b - 41) * q^5 + (6*b + 111) * q^9 + (6*b - 170) * q^11 + (-39*b - 455) * q^13 + (32*b - 912) * q^15 + (6*b - 1608) * q^17 + (45*b - 337) * q^19 + (72*b + 552) * q^23 + (-246*b + 1661) * q^25 + (114*b - 1674) * q^27 + (-114*b + 4032) * q^29 + (210*b - 3106) * q^31 + (152*b - 1560) * q^33 + (390*b - 4256) * q^37 + (572*b + 14820) * q^39 + (-678*b + 652) * q^41 + (-798*b + 5002) * q^43 + (87*b + 1659) * q^45 + (-714*b - 6374) * q^47 + (1590*b + 2754) * q^51 + (-768*b - 5610) * q^53 + (-756*b + 13180) * q^55 + (202*b - 14514) * q^57 + (2253*b - 6009) * q^59 + (75*b - 51369) * q^61 + (234*b - 21710) * q^65 + (-1944*b - 12068) * q^67 + (-768*b - 26496) * q^69 + (-1740*b - 44860) * q^71 + (1716*b + 27794) * q^73 + (-923*b + 79887) * q^75 + (-1956*b - 24412) * q^79 + (-126*b - 61281) * q^81 + (-3447*b + 17891) * q^83 + (-5070*b + 72138) * q^85 + (-3690*b + 27234) * q^87 + (-4728*b + 9150) * q^89 + (2476*b - 63132) * q^93 + (-2856*b + 60392) * q^95 + (462*b + 34992) * q^97 + (-354*b - 6450) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 82 q^{5} + 222 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 82 * q^5 + 222 * q^9 $$2 q - 6 q^{3} - 82 q^{5} + 222 q^{9} - 340 q^{11} - 910 q^{13} - 1824 q^{15} - 3216 q^{17} - 674 q^{19} + 1104 q^{23} + 3322 q^{25} - 3348 q^{27} + 8064 q^{29} - 6212 q^{31} - 3120 q^{33} - 8512 q^{37} + 29640 q^{39} + 1304 q^{41} + 10004 q^{43} + 3318 q^{45} - 12748 q^{47} + 5508 q^{51} - 11220 q^{53} + 26360 q^{55} - 29028 q^{57} - 12018 q^{59} - 102738 q^{61} - 43420 q^{65} - 24136 q^{67} - 52992 q^{69} - 89720 q^{71} + 55588 q^{73} + 159774 q^{75} - 48824 q^{79} - 122562 q^{81} + 35782 q^{83} + 144276 q^{85} + 54468 q^{87} + 18300 q^{89} - 126264 q^{93} + 120784 q^{95} + 69984 q^{97} - 12900 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 82 * q^5 + 222 * q^9 - 340 * q^11 - 910 * q^13 - 1824 * q^15 - 3216 * q^17 - 674 * q^19 + 1104 * q^23 + 3322 * q^25 - 3348 * q^27 + 8064 * q^29 - 6212 * q^31 - 3120 * q^33 - 8512 * q^37 + 29640 * q^39 + 1304 * q^41 + 10004 * q^43 + 3318 * q^45 - 12748 * q^47 + 5508 * q^51 - 11220 * q^53 + 26360 * q^55 - 29028 * q^57 - 12018 * q^59 - 102738 * q^61 - 43420 * q^65 - 24136 * q^67 - 52992 * q^69 - 89720 * q^71 + 55588 * q^73 + 159774 * q^75 - 48824 * q^79 - 122562 * q^81 + 35782 * q^83 + 144276 * q^85 + 54468 * q^87 + 18300 * q^89 - 126264 * q^93 + 120784 * q^95 + 69984 * q^97 - 12900 * 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
 9.78709 −8.78709
0 −21.5742 0 14.7225 0 0 0 222.445 0
1.2 0 15.5742 0 −96.7225 0 0 0 −0.445054 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.u 2
4.b odd 2 1 392.6.a.d 2
7.b odd 2 1 112.6.a.i 2
21.c even 2 1 1008.6.a.bd 2
28.d even 2 1 56.6.a.e 2
28.f even 6 2 392.6.i.j 4
28.g odd 6 2 392.6.i.i 4
56.e even 2 1 448.6.a.v 2
56.h odd 2 1 448.6.a.t 2
84.h odd 2 1 504.6.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 28.d even 2 1
112.6.a.i 2 7.b odd 2 1
392.6.a.d 2 4.b odd 2 1
392.6.i.i 4 28.g odd 6 2
392.6.i.j 4 28.f even 6 2
448.6.a.t 2 56.h odd 2 1
448.6.a.v 2 56.e even 2 1
504.6.a.i 2 84.h odd 2 1
784.6.a.u 2 1.a even 1 1 trivial
1008.6.a.bd 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 6T_{3} - 336$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 6T - 336$$
$5$ $$T^{2} + 82T - 1424$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 340T + 16480$$
$13$ $$T^{2} + 910T - 317720$$
$17$ $$T^{2} + 3216 T + 2573244$$
$19$ $$T^{2} + 674T - 585056$$
$23$ $$T^{2} - 1104 T - 1483776$$
$29$ $$T^{2} - 8064 T + 11773404$$
$31$ $$T^{2} + 6212 T - 5567264$$
$37$ $$T^{2} + 8512 T - 34360964$$
$41$ $$T^{2} - 1304 T - 158165876$$
$43$ $$T^{2} - 10004 T - 194677376$$
$47$ $$T^{2} + 12748 T - 135251744$$
$53$ $$T^{2} + 11220 T - 172017180$$
$59$ $$T^{2} + \cdots - 1715115024$$
$61$ $$T^{2} + \cdots + 2636833536$$
$67$ $$T^{2} + \cdots - 1158165296$$
$71$ $$T^{2} + 89720 T + 967897600$$
$73$ $$T^{2} - 55588 T - 243399884$$
$79$ $$T^{2} + 48824 T - 724002176$$
$83$ $$T^{2} + \cdots - 3779136224$$
$89$ $$T^{2} + \cdots - 7628401980$$
$97$ $$T^{2} + \cdots + 1150801884$$