# Properties

 Label 49.6.a.c Level $49$ Weight $6$ Character orbit 49.a Self dual yes Analytic conductor $7.859$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 49.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: $$7.85880717084$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 39$$ x^2 - 39 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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 $$\beta = 4\sqrt{39}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + \beta q^{3} - 28 q^{4} + 3 \beta q^{5} - 2 \beta q^{6} + 120 q^{8} + 381 q^{9} +O(q^{10})$$ q - 2 * q^2 + b * q^3 - 28 * q^4 + 3*b * q^5 - 2*b * q^6 + 120 * q^8 + 381 * q^9 $$q - 2 q^{2} + \beta q^{3} - 28 q^{4} + 3 \beta q^{5} - 2 \beta q^{6} + 120 q^{8} + 381 q^{9} - 6 \beta q^{10} - 284 q^{11} - 28 \beta q^{12} + 21 \beta q^{13} + 1872 q^{15} + 656 q^{16} - 6 \beta q^{17} - 762 q^{18} + 87 \beta q^{19} - 84 \beta q^{20} + 568 q^{22} + 1496 q^{23} + 120 \beta q^{24} + 2491 q^{25} - 42 \beta q^{26} + 138 \beta q^{27} - 4366 q^{29} - 3744 q^{30} - 258 \beta q^{31} - 5152 q^{32} - 284 \beta q^{33} + 12 \beta q^{34} - 10668 q^{36} - 12630 q^{37} - 174 \beta q^{38} + 13104 q^{39} + 360 \beta q^{40} - 378 \beta q^{41} - 1356 q^{43} + 7952 q^{44} + 1143 \beta q^{45} - 2992 q^{46} + 402 \beta q^{47} + 656 \beta q^{48} - 4982 q^{50} - 3744 q^{51} - 588 \beta q^{52} + 14150 q^{53} - 276 \beta q^{54} - 852 \beta q^{55} + 54288 q^{57} + 8732 q^{58} - 1497 \beta q^{59} - 52416 q^{60} - 1425 \beta q^{61} + 516 \beta q^{62} - 10688 q^{64} + 39312 q^{65} + 568 \beta q^{66} - 3644 q^{67} + 168 \beta q^{68} + 1496 \beta q^{69} + 35632 q^{71} + 45720 q^{72} + 1632 \beta q^{73} + 25260 q^{74} + 2491 \beta q^{75} - 2436 \beta q^{76} - 26208 q^{78} - 54616 q^{79} + 1968 \beta q^{80} - 6471 q^{81} + 756 \beta q^{82} - 21 \beta q^{83} - 11232 q^{85} + 2712 q^{86} - 4366 \beta q^{87} - 34080 q^{88} - 816 \beta q^{89} - 2286 \beta q^{90} - 41888 q^{92} - 160992 q^{93} - 804 \beta q^{94} + 162864 q^{95} - 5152 \beta q^{96} + 7350 \beta q^{97} - 108204 q^{99} +O(q^{100})$$ q - 2 * q^2 + b * q^3 - 28 * q^4 + 3*b * q^5 - 2*b * q^6 + 120 * q^8 + 381 * q^9 - 6*b * q^10 - 284 * q^11 - 28*b * q^12 + 21*b * q^13 + 1872 * q^15 + 656 * q^16 - 6*b * q^17 - 762 * q^18 + 87*b * q^19 - 84*b * q^20 + 568 * q^22 + 1496 * q^23 + 120*b * q^24 + 2491 * q^25 - 42*b * q^26 + 138*b * q^27 - 4366 * q^29 - 3744 * q^30 - 258*b * q^31 - 5152 * q^32 - 284*b * q^33 + 12*b * q^34 - 10668 * q^36 - 12630 * q^37 - 174*b * q^38 + 13104 * q^39 + 360*b * q^40 - 378*b * q^41 - 1356 * q^43 + 7952 * q^44 + 1143*b * q^45 - 2992 * q^46 + 402*b * q^47 + 656*b * q^48 - 4982 * q^50 - 3744 * q^51 - 588*b * q^52 + 14150 * q^53 - 276*b * q^54 - 852*b * q^55 + 54288 * q^57 + 8732 * q^58 - 1497*b * q^59 - 52416 * q^60 - 1425*b * q^61 + 516*b * q^62 - 10688 * q^64 + 39312 * q^65 + 568*b * q^66 - 3644 * q^67 + 168*b * q^68 + 1496*b * q^69 + 35632 * q^71 + 45720 * q^72 + 1632*b * q^73 + 25260 * q^74 + 2491*b * q^75 - 2436*b * q^76 - 26208 * q^78 - 54616 * q^79 + 1968*b * q^80 - 6471 * q^81 + 756*b * q^82 - 21*b * q^83 - 11232 * q^85 + 2712 * q^86 - 4366*b * q^87 - 34080 * q^88 - 816*b * q^89 - 2286*b * q^90 - 41888 * q^92 - 160992 * q^93 - 804*b * q^94 + 162864 * q^95 - 5152*b * q^96 + 7350*b * q^97 - 108204 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - 56 * q^4 + 240 * q^8 + 762 * q^9 $$2 q - 4 q^{2} - 56 q^{4} + 240 q^{8} + 762 q^{9} - 568 q^{11} + 3744 q^{15} + 1312 q^{16} - 1524 q^{18} + 1136 q^{22} + 2992 q^{23} + 4982 q^{25} - 8732 q^{29} - 7488 q^{30} - 10304 q^{32} - 21336 q^{36} - 25260 q^{37} + 26208 q^{39} - 2712 q^{43} + 15904 q^{44} - 5984 q^{46} - 9964 q^{50} - 7488 q^{51} + 28300 q^{53} + 108576 q^{57} + 17464 q^{58} - 104832 q^{60} - 21376 q^{64} + 78624 q^{65} - 7288 q^{67} + 71264 q^{71} + 91440 q^{72} + 50520 q^{74} - 52416 q^{78} - 109232 q^{79} - 12942 q^{81} - 22464 q^{85} + 5424 q^{86} - 68160 q^{88} - 83776 q^{92} - 321984 q^{93} + 325728 q^{95} - 216408 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - 56 * q^4 + 240 * q^8 + 762 * q^9 - 568 * q^11 + 3744 * q^15 + 1312 * q^16 - 1524 * q^18 + 1136 * q^22 + 2992 * q^23 + 4982 * q^25 - 8732 * q^29 - 7488 * q^30 - 10304 * q^32 - 21336 * q^36 - 25260 * q^37 + 26208 * q^39 - 2712 * q^43 + 15904 * q^44 - 5984 * q^46 - 9964 * q^50 - 7488 * q^51 + 28300 * q^53 + 108576 * q^57 + 17464 * q^58 - 104832 * q^60 - 21376 * q^64 + 78624 * q^65 - 7288 * q^67 + 71264 * q^71 + 91440 * q^72 + 50520 * q^74 - 52416 * q^78 - 109232 * q^79 - 12942 * q^81 - 22464 * q^85 + 5424 * q^86 - 68160 * q^88 - 83776 * q^92 - 321984 * q^93 + 325728 * q^95 - 216408 * 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
 −6.24500 6.24500
−2.00000 −24.9800 −28.0000 −74.9400 49.9600 0 120.000 381.000 149.880
1.2 −2.00000 24.9800 −28.0000 74.9400 −49.9600 0 120.000 381.000 −149.880
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.a.c 2
3.b odd 2 1 441.6.a.u 2
4.b odd 2 1 784.6.a.z 2
7.b odd 2 1 inner 49.6.a.c 2
7.c even 3 2 49.6.c.g 4
7.d odd 6 2 49.6.c.g 4
21.c even 2 1 441.6.a.u 2
28.d even 2 1 784.6.a.z 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.c 2 1.a even 1 1 trivial
49.6.a.c 2 7.b odd 2 1 inner
49.6.c.g 4 7.c even 3 2
49.6.c.g 4 7.d odd 6 2
441.6.a.u 2 3.b odd 2 1
441.6.a.u 2 21.c even 2 1
784.6.a.z 2 4.b odd 2 1
784.6.a.z 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{3}^{2} - 624$$ T3^2 - 624

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} - 624$$
$5$ $$T^{2} - 5616$$
$7$ $$T^{2}$$
$11$ $$(T + 284)^{2}$$
$13$ $$T^{2} - 275184$$
$17$ $$T^{2} - 22464$$
$19$ $$T^{2} - 4723056$$
$23$ $$(T - 1496)^{2}$$
$29$ $$(T + 4366)^{2}$$
$31$ $$T^{2} - 41535936$$
$37$ $$(T + 12630)^{2}$$
$41$ $$T^{2} - 89159616$$
$43$ $$(T + 1356)^{2}$$
$47$ $$T^{2} - 100840896$$
$53$ $$(T - 14150)^{2}$$
$59$ $$T^{2} - 1398389616$$
$61$ $$T^{2} - 1267110000$$
$67$ $$(T + 3644)^{2}$$
$71$ $$(T - 35632)^{2}$$
$73$ $$T^{2} - 1661976576$$
$79$ $$(T + 54616)^{2}$$
$83$ $$T^{2} - 275184$$
$89$ $$T^{2} - 415494144$$
$97$ $$T^{2} - 33710040000$$