# Properties

 Label 637.4.a.b Level $637$ Weight $4$ Character orbit 637.a Self dual yes Analytic conductor $37.584$ 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] = [637,4,Mod(1,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("637.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 637.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: $$37.5842166737$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 4) q^{8} + ( - 15 \beta + 25) q^{9}+O(q^{10})$$ q + b * q^2 + (3*b - 4) * q^3 + (b - 4) * q^4 + (-b + 2) * q^5 + (-b + 12) * q^6 + (-11*b + 4) * q^8 + (-15*b + 25) * q^9 $$q + \beta q^{2} + (3 \beta - 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 4) q^{8} + ( - 15 \beta + 25) q^{9} + (\beta - 4) q^{10} + (12 \beta + 34) q^{11} + ( - 13 \beta + 28) q^{12} + 13 q^{13} + (7 \beta - 20) q^{15} + ( - 15 \beta - 12) q^{16} + (17 \beta - 18) q^{17} + (10 \beta - 60) q^{18} + (32 \beta + 26) q^{19} + (5 \beta - 12) q^{20} + (46 \beta + 48) q^{22} + ( - 12 \beta + 104) q^{23} + (23 \beta - 148) q^{24} + ( - 3 \beta - 117) q^{25} + 13 \beta q^{26} + (9 \beta - 172) q^{27} + (96 \beta - 70) q^{29} + ( - 13 \beta + 28) q^{30} + (34 \beta + 26) q^{31} + (61 \beta - 92) q^{32} + (90 \beta + 8) q^{33} + ( - \beta + 68) q^{34} + (70 \beta - 160) q^{36} + (5 \beta + 102) q^{37} + (58 \beta + 128) q^{38} + (39 \beta - 52) q^{39} + ( - 15 \beta + 52) q^{40} + ( - 22 \beta + 126) q^{41} + (143 \beta + 72) q^{43} + ( - 2 \beta - 88) q^{44} + ( - 40 \beta + 110) q^{45} + (92 \beta - 48) q^{46} + (121 \beta - 278) q^{47} + ( - 21 \beta - 132) q^{48} + ( - 120 \beta - 12) q^{50} + ( - 71 \beta + 276) q^{51} + (13 \beta - 52) q^{52} + (30 \beta - 74) q^{53} + ( - 163 \beta + 36) q^{54} + ( - 22 \beta + 20) q^{55} + (46 \beta + 280) q^{57} + (26 \beta + 384) q^{58} + ( - 124 \beta + 246) q^{59} + ( - 41 \beta + 108) q^{60} + (190 \beta + 434) q^{61} + (60 \beta + 136) q^{62} + (89 \beta + 340) q^{64} + ( - 13 \beta + 26) q^{65} + (98 \beta + 360) q^{66} + ( - 232 \beta + 150) q^{67} + ( - 69 \beta + 140) q^{68} + (324 \beta - 560) q^{69} + ( - 231 \beta + 50) q^{71} + ( - 170 \beta + 760) q^{72} + ( - 260 \beta - 98) q^{73} + (107 \beta + 20) q^{74} + ( - 348 \beta + 432) q^{75} + ( - 70 \beta + 24) q^{76} + ( - 13 \beta + 156) q^{78} + (40 \beta - 524) q^{79} + ( - 3 \beta + 36) q^{80} + ( - 120 \beta + 121) q^{81} + (104 \beta - 88) q^{82} + (182 \beta - 1070) q^{83} + (35 \beta - 104) q^{85} + (215 \beta + 572) q^{86} + ( - 306 \beta + 1432) q^{87} + ( - 458 \beta - 392) q^{88} + (388 \beta + 166) q^{89} + (70 \beta - 160) q^{90} + (140 \beta - 464) q^{92} + (44 \beta + 304) q^{93} + ( - 157 \beta + 484) q^{94} + (6 \beta - 76) q^{95} + ( - 337 \beta + 1100) q^{96} + ( - 508 \beta + 718) q^{97} + ( - 390 \beta + 130) q^{99}+O(q^{100})$$ q + b * q^2 + (3*b - 4) * q^3 + (b - 4) * q^4 + (-b + 2) * q^5 + (-b + 12) * q^6 + (-11*b + 4) * q^8 + (-15*b + 25) * q^9 + (b - 4) * q^10 + (12*b + 34) * q^11 + (-13*b + 28) * q^12 + 13 * q^13 + (7*b - 20) * q^15 + (-15*b - 12) * q^16 + (17*b - 18) * q^17 + (10*b - 60) * q^18 + (32*b + 26) * q^19 + (5*b - 12) * q^20 + (46*b + 48) * q^22 + (-12*b + 104) * q^23 + (23*b - 148) * q^24 + (-3*b - 117) * q^25 + 13*b * q^26 + (9*b - 172) * q^27 + (96*b - 70) * q^29 + (-13*b + 28) * q^30 + (34*b + 26) * q^31 + (61*b - 92) * q^32 + (90*b + 8) * q^33 + (-b + 68) * q^34 + (70*b - 160) * q^36 + (5*b + 102) * q^37 + (58*b + 128) * q^38 + (39*b - 52) * q^39 + (-15*b + 52) * q^40 + (-22*b + 126) * q^41 + (143*b + 72) * q^43 + (-2*b - 88) * q^44 + (-40*b + 110) * q^45 + (92*b - 48) * q^46 + (121*b - 278) * q^47 + (-21*b - 132) * q^48 + (-120*b - 12) * q^50 + (-71*b + 276) * q^51 + (13*b - 52) * q^52 + (30*b - 74) * q^53 + (-163*b + 36) * q^54 + (-22*b + 20) * q^55 + (46*b + 280) * q^57 + (26*b + 384) * q^58 + (-124*b + 246) * q^59 + (-41*b + 108) * q^60 + (190*b + 434) * q^61 + (60*b + 136) * q^62 + (89*b + 340) * q^64 + (-13*b + 26) * q^65 + (98*b + 360) * q^66 + (-232*b + 150) * q^67 + (-69*b + 140) * q^68 + (324*b - 560) * q^69 + (-231*b + 50) * q^71 + (-170*b + 760) * q^72 + (-260*b - 98) * q^73 + (107*b + 20) * q^74 + (-348*b + 432) * q^75 + (-70*b + 24) * q^76 + (-13*b + 156) * q^78 + (40*b - 524) * q^79 + (-3*b + 36) * q^80 + (-120*b + 121) * q^81 + (104*b - 88) * q^82 + (182*b - 1070) * q^83 + (35*b - 104) * q^85 + (215*b + 572) * q^86 + (-306*b + 1432) * q^87 + (-458*b - 392) * q^88 + (388*b + 166) * q^89 + (70*b - 160) * q^90 + (140*b - 464) * q^92 + (44*b + 304) * q^93 + (-157*b + 484) * q^94 + (6*b - 76) * q^95 + (-337*b + 1100) * q^96 + (-508*b + 718) * q^97 + (-390*b + 130) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9 $$2 q + q^{2} - 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} - 3 q^{8} + 35 q^{9} - 7 q^{10} + 80 q^{11} + 43 q^{12} + 26 q^{13} - 33 q^{15} - 39 q^{16} - 19 q^{17} - 110 q^{18} + 84 q^{19} - 19 q^{20} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 13 q^{26} - 335 q^{27} - 44 q^{29} + 43 q^{30} + 86 q^{31} - 123 q^{32} + 106 q^{33} + 135 q^{34} - 250 q^{36} + 209 q^{37} + 314 q^{38} - 65 q^{39} + 89 q^{40} + 230 q^{41} + 287 q^{43} - 178 q^{44} + 180 q^{45} - 4 q^{46} - 435 q^{47} - 285 q^{48} - 144 q^{50} + 481 q^{51} - 91 q^{52} - 118 q^{53} - 91 q^{54} + 18 q^{55} + 606 q^{57} + 794 q^{58} + 368 q^{59} + 175 q^{60} + 1058 q^{61} + 332 q^{62} + 769 q^{64} + 39 q^{65} + 818 q^{66} + 68 q^{67} + 211 q^{68} - 796 q^{69} - 131 q^{71} + 1350 q^{72} - 456 q^{73} + 147 q^{74} + 516 q^{75} - 22 q^{76} + 299 q^{78} - 1008 q^{79} + 69 q^{80} + 122 q^{81} - 72 q^{82} - 1958 q^{83} - 173 q^{85} + 1359 q^{86} + 2558 q^{87} - 1242 q^{88} + 720 q^{89} - 250 q^{90} - 788 q^{92} + 652 q^{93} + 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} - 130 q^{99}+O(q^{100})$$ 2 * q + q^2 - 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 - 3 * q^8 + 35 * q^9 - 7 * q^10 + 80 * q^11 + 43 * q^12 + 26 * q^13 - 33 * q^15 - 39 * q^16 - 19 * q^17 - 110 * q^18 + 84 * q^19 - 19 * q^20 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 13 * q^26 - 335 * q^27 - 44 * q^29 + 43 * q^30 + 86 * q^31 - 123 * q^32 + 106 * q^33 + 135 * q^34 - 250 * q^36 + 209 * q^37 + 314 * q^38 - 65 * q^39 + 89 * q^40 + 230 * q^41 + 287 * q^43 - 178 * q^44 + 180 * q^45 - 4 * q^46 - 435 * q^47 - 285 * q^48 - 144 * q^50 + 481 * q^51 - 91 * q^52 - 118 * q^53 - 91 * q^54 + 18 * q^55 + 606 * q^57 + 794 * q^58 + 368 * q^59 + 175 * q^60 + 1058 * q^61 + 332 * q^62 + 769 * q^64 + 39 * q^65 + 818 * q^66 + 68 * q^67 + 211 * q^68 - 796 * q^69 - 131 * q^71 + 1350 * q^72 - 456 * q^73 + 147 * q^74 + 516 * q^75 - 22 * q^76 + 299 * q^78 - 1008 * q^79 + 69 * q^80 + 122 * q^81 - 72 * q^82 - 1958 * q^83 - 173 * q^85 + 1359 * q^86 + 2558 * q^87 - 1242 * q^88 + 720 * q^89 - 250 * q^90 - 788 * q^92 + 652 * q^93 + 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 - 130 * 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
 −1.56155 2.56155
−1.56155 −8.68466 −5.56155 3.56155 13.5616 0 21.1771 48.4233 −5.56155
1.2 2.56155 3.68466 −1.43845 −0.561553 9.43845 0 −24.1771 −13.4233 −1.43845
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 637.4.a.b 2
7.b odd 2 1 13.4.a.b 2
21.c even 2 1 117.4.a.d 2
28.d even 2 1 208.4.a.h 2
35.c odd 2 1 325.4.a.f 2
35.f even 4 2 325.4.b.e 4
56.e even 2 1 832.4.a.z 2
56.h odd 2 1 832.4.a.s 2
77.b even 2 1 1573.4.a.b 2
84.h odd 2 1 1872.4.a.bb 2
91.b odd 2 1 169.4.a.g 2
91.i even 4 2 169.4.b.f 4
91.n odd 6 2 169.4.c.g 4
91.t odd 6 2 169.4.c.j 4
91.bc even 12 4 169.4.e.f 8
273.g even 2 1 1521.4.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b 2 7.b odd 2 1
117.4.a.d 2 21.c even 2 1
169.4.a.g 2 91.b odd 2 1
169.4.b.f 4 91.i even 4 2
169.4.c.g 4 91.n odd 6 2
169.4.c.j 4 91.t odd 6 2
169.4.e.f 8 91.bc even 12 4
208.4.a.h 2 28.d even 2 1
325.4.a.f 2 35.c odd 2 1
325.4.b.e 4 35.f even 4 2
637.4.a.b 2 1.a even 1 1 trivial
832.4.a.s 2 56.h odd 2 1
832.4.a.z 2 56.e even 2 1
1521.4.a.r 2 273.g even 2 1
1573.4.a.b 2 77.b even 2 1
1872.4.a.bb 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2}^{2} - T_{2} - 4$$ T2^2 - T2 - 4 $$T_{3}^{2} + 5T_{3} - 32$$ T3^2 + 5*T3 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 4$$
$3$ $$T^{2} + 5T - 32$$
$5$ $$T^{2} - 3T - 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 80T + 988$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 19T - 1138$$
$19$ $$T^{2} - 84T - 2588$$
$23$ $$T^{2} - 196T + 8992$$
$29$ $$T^{2} + 44T - 38684$$
$31$ $$T^{2} - 86T - 3064$$
$37$ $$T^{2} - 209T + 10814$$
$41$ $$T^{2} - 230T + 11168$$
$43$ $$T^{2} - 287T - 66316$$
$47$ $$T^{2} + 435T - 14918$$
$53$ $$T^{2} + 118T - 344$$
$59$ $$T^{2} - 368T - 31492$$
$61$ $$T^{2} - 1058 T + 126416$$
$67$ $$T^{2} - 68T - 227596$$
$71$ $$T^{2} + 131T - 222494$$
$73$ $$T^{2} + 456T - 235316$$
$79$ $$T^{2} + 1008 T + 247216$$
$83$ $$T^{2} + 1958 T + 817664$$
$89$ $$T^{2} - 720T - 510212$$
$97$ $$T^{2} - 928T - 881476$$