# Properties

 Label 49.4.a.e Level $49$ Weight $4$ Character orbit 49.a Self dual yes Analytic conductor $2.891$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,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, 4, names="a")

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

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$49 = 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$2.89109359028$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 35x^{2} + 36x + 194$$ x^4 - 2*x^3 - 35*x^2 + 36*x + 194 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$7$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 9) q^{4} - \beta_{3} q^{5} + (2 \beta_{3} - 3 \beta_{2}) q^{6} + (\beta_1 + 17) q^{8} + ( - 6 \beta_1 + 7) q^{9}+O(q^{10})$$ q + (b1 + 1) * q^2 + b2 * q^3 + (b1 + 9) * q^4 - b3 * q^5 + (2*b3 - 3*b2) * q^6 + (b1 + 17) * q^8 + (-6*b1 + 7) * q^9 $$q + (\beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 9) q^{4} - \beta_{3} q^{5} + (2 \beta_{3} - 3 \beta_{2}) q^{6} + (\beta_1 + 17) q^{8} + ( - 6 \beta_1 + 7) q^{9} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{10} + ( - 6 \beta_1 + 22) q^{11} + (2 \beta_{3} + 5 \beta_{2}) q^{12} + (\beta_{3} + 6 \beta_{2}) q^{13} + ( - 8 \beta_1 - 20) q^{15} + (9 \beta_1 - 39) q^{16} + (9 \beta_{3} - \beta_{2}) q^{17} + (7 \beta_1 - 89) q^{18} + ( - 4 \beta_{3} - 11 \beta_{2}) q^{19} + ( - 12 \beta_{3} - 2 \beta_{2}) q^{20} + (22 \beta_1 - 74) q^{22} + (8 \beta_1 + 92) q^{23} + (2 \beta_{3} + 13 \beta_{2}) q^{24} + (22 \beta_1 - 21) q^{25} + (16 \beta_{3} - 16 \beta_{2}) q^{26} + ( - 12 \beta_{3} + 4 \beta_{2}) q^{27} + ( - 14 \beta_1 + 58) q^{29} + ( - 20 \beta_1 - 148) q^{30} + (4 \beta_{3} - 38 \beta_{2}) q^{31} + ( - 47 \beta_1 - 31) q^{32} + ( - 12 \beta_{3} + 46 \beta_{2}) q^{33} + (34 \beta_{3} + 21 \beta_{2}) q^{34} + ( - 41 \beta_1 - 33) q^{36} + ( - 6 \beta_1 + 50) q^{37} + ( - 38 \beta_{3} + 25 \beta_{2}) q^{38} + ( - 28 \beta_1 + 224) q^{39} + ( - 20 \beta_{3} - 2 \beta_{2}) q^{40} + (3 \beta_{3} - 31 \beta_{2}) q^{41} + (70 \beta_1 + 170) q^{43} + ( - 26 \beta_1 + 102) q^{44} + (11 \beta_{3} + 12 \beta_{2}) q^{45} + (92 \beta_1 + 220) q^{46} + (32 \beta_{3} - 26 \beta_{2}) q^{47} + (18 \beta_{3} - 75 \beta_{2}) q^{48} + ( - 21 \beta_1 + 331) q^{50} + (78 \beta_1 + 146) q^{51} + (24 \beta_{3} + 32 \beta_{2}) q^{52} + (36 \beta_1 + 22) q^{53} + ( - 40 \beta_{3} - 36 \beta_{2}) q^{54} + ( - 4 \beta_{3} + 12 \beta_{2}) q^{55} + (34 \beta_1 - 454) q^{57} + (58 \beta_1 - 166) q^{58} + ( - 20 \beta_{3} + 49 \beta_{2}) q^{59} + ( - 84 \beta_1 - 308) q^{60} + ( - 27 \beta_{3} + 68 \beta_{2}) q^{61} + ( - 60 \beta_{3} + 122 \beta_{2}) q^{62} + ( - 103 \beta_1 - 471) q^{64} + ( - 70 \beta_1 - 224) q^{65} + (44 \beta_{3} - 162 \beta_{2}) q^{66} + ( - 76 \beta_1 - 524) q^{67} + (106 \beta_{3} + 13 \beta_{2}) q^{68} + (16 \beta_{3} + 60 \beta_{2}) q^{69} + ( - 28 \beta_1 + 548) q^{71} + ( - 89 \beta_1 + 23) q^{72} + ( - 25 \beta_{3} - 37 \beta_{2}) q^{73} + (50 \beta_1 - 46) q^{74} + (44 \beta_{3} - 109 \beta_{2}) q^{75} + ( - 70 \beta_{3} - 63 \beta_{2}) q^{76} + (224 \beta_1 - 224) q^{78} + ( - 188 \beta_1 - 356) q^{79} + (12 \beta_{3} - 18 \beta_{2}) q^{80} + (42 \beta_1 - 293) q^{81} + ( - 50 \beta_{3} + 99 \beta_{2}) q^{82} + ( - 8 \beta_{3} + \beta_{2}) q^{83} + ( - 190 \beta_1 - 916) q^{85} + (170 \beta_1 + 1290) q^{86} + ( - 28 \beta_{3} + 114 \beta_{2}) q^{87} + ( - 74 \beta_1 + 278) q^{88} + ( - 75 \beta_{3} - 157 \beta_{2}) q^{89} + (68 \beta_{3} - 14 \beta_{2}) q^{90} + (156 \beta_1 + 956) q^{92} + (260 \beta_1 - 1212) q^{93} + (76 \beta_{3} + 142 \beta_{2}) q^{94} + (176 \beta_1 + 636) q^{95} + ( - 94 \beta_{3} + 157 \beta_{2}) q^{96} + (91 \beta_{3} - 189 \beta_{2}) q^{97} + ( - 210 \beta_1 + 730) q^{99}+O(q^{100})$$ q + (b1 + 1) * q^2 + b2 * q^3 + (b1 + 9) * q^4 - b3 * q^5 + (2*b3 - 3*b2) * q^6 + (b1 + 17) * q^8 + (-6*b1 + 7) * q^9 + (-4*b3 - 2*b2) * q^10 + (-6*b1 + 22) * q^11 + (2*b3 + 5*b2) * q^12 + (b3 + 6*b2) * q^13 + (-8*b1 - 20) * q^15 + (9*b1 - 39) * q^16 + (9*b3 - b2) * q^17 + (7*b1 - 89) * q^18 + (-4*b3 - 11*b2) * q^19 + (-12*b3 - 2*b2) * q^20 + (22*b1 - 74) * q^22 + (8*b1 + 92) * q^23 + (2*b3 + 13*b2) * q^24 + (22*b1 - 21) * q^25 + (16*b3 - 16*b2) * q^26 + (-12*b3 + 4*b2) * q^27 + (-14*b1 + 58) * q^29 + (-20*b1 - 148) * q^30 + (4*b3 - 38*b2) * q^31 + (-47*b1 - 31) * q^32 + (-12*b3 + 46*b2) * q^33 + (34*b3 + 21*b2) * q^34 + (-41*b1 - 33) * q^36 + (-6*b1 + 50) * q^37 + (-38*b3 + 25*b2) * q^38 + (-28*b1 + 224) * q^39 + (-20*b3 - 2*b2) * q^40 + (3*b3 - 31*b2) * q^41 + (70*b1 + 170) * q^43 + (-26*b1 + 102) * q^44 + (11*b3 + 12*b2) * q^45 + (92*b1 + 220) * q^46 + (32*b3 - 26*b2) * q^47 + (18*b3 - 75*b2) * q^48 + (-21*b1 + 331) * q^50 + (78*b1 + 146) * q^51 + (24*b3 + 32*b2) * q^52 + (36*b1 + 22) * q^53 + (-40*b3 - 36*b2) * q^54 + (-4*b3 + 12*b2) * q^55 + (34*b1 - 454) * q^57 + (58*b1 - 166) * q^58 + (-20*b3 + 49*b2) * q^59 + (-84*b1 - 308) * q^60 + (-27*b3 + 68*b2) * q^61 + (-60*b3 + 122*b2) * q^62 + (-103*b1 - 471) * q^64 + (-70*b1 - 224) * q^65 + (44*b3 - 162*b2) * q^66 + (-76*b1 - 524) * q^67 + (106*b3 + 13*b2) * q^68 + (16*b3 + 60*b2) * q^69 + (-28*b1 + 548) * q^71 + (-89*b1 + 23) * q^72 + (-25*b3 - 37*b2) * q^73 + (50*b1 - 46) * q^74 + (44*b3 - 109*b2) * q^75 + (-70*b3 - 63*b2) * q^76 + (224*b1 - 224) * q^78 + (-188*b1 - 356) * q^79 + (12*b3 - 18*b2) * q^80 + (42*b1 - 293) * q^81 + (-50*b3 + 99*b2) * q^82 + (-8*b3 + b2) * q^83 + (-190*b1 - 916) * q^85 + (170*b1 + 1290) * q^86 + (-28*b3 + 114*b2) * q^87 + (-74*b1 + 278) * q^88 + (-75*b3 - 157*b2) * q^89 + (68*b3 - 14*b2) * q^90 + (156*b1 + 956) * q^92 + (260*b1 - 1212) * q^93 + (76*b3 + 142*b2) * q^94 + (176*b1 + 636) * q^95 + (-94*b3 + 157*b2) * q^96 + (91*b3 - 189*b2) * q^97 + (-210*b1 + 730) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 $$4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 + 100 * q^11 - 64 * q^15 - 174 * q^16 - 370 * q^18 - 340 * q^22 + 352 * q^23 - 128 * q^25 + 260 * q^29 - 552 * q^30 - 30 * q^32 - 50 * q^36 + 212 * q^37 + 952 * q^39 + 540 * q^43 + 460 * q^44 + 696 * q^46 + 1366 * q^50 + 428 * q^51 + 16 * q^53 - 1884 * q^57 - 780 * q^58 - 1064 * q^60 - 1678 * q^64 - 756 * q^65 - 1944 * q^67 + 2248 * q^71 + 270 * q^72 - 284 * q^74 - 1344 * q^78 - 1048 * q^79 - 1256 * q^81 - 3284 * q^85 + 4820 * q^86 + 1260 * q^88 + 3512 * q^92 - 5368 * q^93 + 2192 * q^95 + 3340 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 35x^{2} + 36x + 194$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} + 100\nu - 79 ) / 57$$ (-2*v^3 + 3*v^2 + 100*v - 79) / 57 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 11\nu^{2} + 12\nu - 182 ) / 19$$ (-v^3 + 11*v^2 + 12*v - 182) / 19 $$\beta_{3}$$ $$=$$ $$( 11\nu^{3} + 12\nu^{2} - 265\nu - 392 ) / 57$$ (11*v^3 + 12*v^2 - 265*v - 392) / 57
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + 7\beta _1 + 7 ) / 7$$ (b3 - b2 + 7*b1 + 7) / 7 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} + 10\beta_{2} + 7\beta _1 + 133 ) / 7$$ (4*b3 + 10*b2 + 7*b1 + 133) / 7 $$\nu^{3}$$ $$=$$ $$8\beta_{3} - 5\beta_{2} + 23\beta _1 + 39$$ 8*b3 - 5*b2 + 23*b1 + 39

## 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
 −2.11692 −4.94534 3.11692 5.94534
−3.53113 −7.82220 4.46887 −2.07730 27.6212 0 12.4689 34.1868 7.33521
1.2 −3.53113 7.82220 4.46887 2.07730 −27.6212 0 12.4689 34.1868 −7.33521
1.3 4.53113 −3.57956 12.5311 13.4791 −16.2194 0 20.5311 −14.1868 61.0753
1.4 4.53113 3.57956 12.5311 −13.4791 16.2194 0 20.5311 −14.1868 −61.0753
 $$n$$: 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 49.4.a.e 4
3.b odd 2 1 441.4.a.u 4
4.b odd 2 1 784.4.a.bf 4
5.b even 2 1 1225.4.a.bb 4
7.b odd 2 1 inner 49.4.a.e 4
7.c even 3 2 49.4.c.e 8
7.d odd 6 2 49.4.c.e 8
21.c even 2 1 441.4.a.u 4
21.g even 6 2 441.4.e.y 8
21.h odd 6 2 441.4.e.y 8
28.d even 2 1 784.4.a.bf 4
35.c odd 2 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 1.a even 1 1 trivial
49.4.a.e 4 7.b odd 2 1 inner
49.4.c.e 8 7.c even 3 2
49.4.c.e 8 7.d odd 6 2
441.4.a.u 4 3.b odd 2 1
441.4.a.u 4 21.c even 2 1
441.4.e.y 8 21.g even 6 2
441.4.e.y 8 21.h odd 6 2
784.4.a.bf 4 4.b odd 2 1
784.4.a.bf 4 28.d even 2 1
1225.4.a.bb 4 5.b even 2 1
1225.4.a.bb 4 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(49))$$:

 $$T_{2}^{2} - T_{2} - 16$$ T2^2 - T2 - 16 $$T_{3}^{4} - 74T_{3}^{2} + 784$$ T3^4 - 74*T3^2 + 784

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T - 16)^{2}$$
$3$ $$T^{4} - 74T^{2} + 784$$
$5$ $$T^{4} - 186T^{2} + 784$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 50 T + 40)^{2}$$
$13$ $$T^{4} - 3234 T^{2} + \cdots + 2458624$$
$17$ $$T^{4} - 14564 T^{2} + \cdots + 9746884$$
$19$ $$T^{4} - 14746 T^{2} + \cdots + 52591504$$
$23$ $$(T^{2} - 176 T + 6704)^{2}$$
$29$ $$(T^{2} - 130 T + 1040)^{2}$$
$31$ $$T^{4} - 100104 T^{2} + \cdots + 629407744$$
$37$ $$(T^{2} - 106 T + 2224)^{2}$$
$41$ $$T^{4} - 66836 T^{2} + \cdots + 307721764$$
$43$ $$(T^{2} - 270 T - 61400)^{2}$$
$47$ $$T^{4} - 187240 T^{2} + \cdots + 8332038400$$
$53$ $$(T^{2} - 8 T - 21044)^{2}$$
$59$ $$T^{4} - 189354 T^{2} + \cdots + 1600960144$$
$61$ $$T^{4} - 360266 T^{2} + \cdots + 5022273424$$
$67$ $$(T^{2} + 972 T + 142336)^{2}$$
$71$ $$(T^{2} - 1124 T + 303104)^{2}$$
$73$ $$T^{4} - 276756 T^{2} + \cdots + 12428236324$$
$79$ $$(T^{2} + 524 T - 505696)^{2}$$
$83$ $$T^{4} - 11466 T^{2} + \cdots + 6492304$$
$89$ $$T^{4} - 3623876 T^{2} + \cdots + 2844693770884$$
$97$ $$T^{4} - 3082884 T^{2} + \cdots + 841222821124$$