# Properties

 Label 637.4.a.a Level $637$ Weight $4$ Character orbit 637.a Self dual yes Analytic conductor $37.584$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{2} + 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} - 45 q^{8} + 22 q^{9}+O(q^{10})$$ q - 5 * q^2 + 7 * q^3 + 17 * q^4 + 7 * q^5 - 35 * q^6 - 45 * q^8 + 22 * q^9 $$q - 5 q^{2} + 7 q^{3} + 17 q^{4} + 7 q^{5} - 35 q^{6} - 45 q^{8} + 22 q^{9} - 35 q^{10} - 26 q^{11} + 119 q^{12} - 13 q^{13} + 49 q^{15} + 89 q^{16} - 77 q^{17} - 110 q^{18} + 126 q^{19} + 119 q^{20} + 130 q^{22} - 96 q^{23} - 315 q^{24} - 76 q^{25} + 65 q^{26} - 35 q^{27} - 82 q^{29} - 245 q^{30} - 196 q^{31} - 85 q^{32} - 182 q^{33} + 385 q^{34} + 374 q^{36} - 131 q^{37} - 630 q^{38} - 91 q^{39} - 315 q^{40} - 336 q^{41} - 201 q^{43} - 442 q^{44} + 154 q^{45} + 480 q^{46} + 105 q^{47} + 623 q^{48} + 380 q^{50} - 539 q^{51} - 221 q^{52} - 432 q^{53} + 175 q^{54} - 182 q^{55} + 882 q^{57} + 410 q^{58} + 294 q^{59} + 833 q^{60} + 56 q^{61} + 980 q^{62} - 287 q^{64} - 91 q^{65} + 910 q^{66} + 478 q^{67} - 1309 q^{68} - 672 q^{69} + 9 q^{71} - 990 q^{72} - 98 q^{73} + 655 q^{74} - 532 q^{75} + 2142 q^{76} + 455 q^{78} + 1304 q^{79} + 623 q^{80} - 839 q^{81} + 1680 q^{82} + 308 q^{83} - 539 q^{85} + 1005 q^{86} - 574 q^{87} + 1170 q^{88} + 1190 q^{89} - 770 q^{90} - 1632 q^{92} - 1372 q^{93} - 525 q^{94} + 882 q^{95} - 595 q^{96} - 70 q^{97} - 572 q^{99}+O(q^{100})$$ q - 5 * q^2 + 7 * q^3 + 17 * q^4 + 7 * q^5 - 35 * q^6 - 45 * q^8 + 22 * q^9 - 35 * q^10 - 26 * q^11 + 119 * q^12 - 13 * q^13 + 49 * q^15 + 89 * q^16 - 77 * q^17 - 110 * q^18 + 126 * q^19 + 119 * q^20 + 130 * q^22 - 96 * q^23 - 315 * q^24 - 76 * q^25 + 65 * q^26 - 35 * q^27 - 82 * q^29 - 245 * q^30 - 196 * q^31 - 85 * q^32 - 182 * q^33 + 385 * q^34 + 374 * q^36 - 131 * q^37 - 630 * q^38 - 91 * q^39 - 315 * q^40 - 336 * q^41 - 201 * q^43 - 442 * q^44 + 154 * q^45 + 480 * q^46 + 105 * q^47 + 623 * q^48 + 380 * q^50 - 539 * q^51 - 221 * q^52 - 432 * q^53 + 175 * q^54 - 182 * q^55 + 882 * q^57 + 410 * q^58 + 294 * q^59 + 833 * q^60 + 56 * q^61 + 980 * q^62 - 287 * q^64 - 91 * q^65 + 910 * q^66 + 478 * q^67 - 1309 * q^68 - 672 * q^69 + 9 * q^71 - 990 * q^72 - 98 * q^73 + 655 * q^74 - 532 * q^75 + 2142 * q^76 + 455 * q^78 + 1304 * q^79 + 623 * q^80 - 839 * q^81 + 1680 * q^82 + 308 * q^83 - 539 * q^85 + 1005 * q^86 - 574 * q^87 + 1170 * q^88 + 1190 * q^89 - 770 * q^90 - 1632 * q^92 - 1372 * q^93 - 525 * q^94 + 882 * q^95 - 595 * q^96 - 70 * q^97 - 572 * 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
 0
−5.00000 7.00000 17.0000 7.00000 −35.0000 0 −45.0000 22.0000 −35.0000
 $$n$$: 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.a 1
7.b odd 2 1 13.4.a.a 1
21.c even 2 1 117.4.a.b 1
28.d even 2 1 208.4.a.g 1
35.c odd 2 1 325.4.a.d 1
35.f even 4 2 325.4.b.b 2
56.e even 2 1 832.4.a.a 1
56.h odd 2 1 832.4.a.r 1
77.b even 2 1 1573.4.a.a 1
84.h odd 2 1 1872.4.a.k 1
91.b odd 2 1 169.4.a.e 1
91.i even 4 2 169.4.b.a 2
91.n odd 6 2 169.4.c.e 2
91.t odd 6 2 169.4.c.a 2
91.bc even 12 4 169.4.e.e 4
273.g even 2 1 1521.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 7.b odd 2 1
117.4.a.b 1 21.c even 2 1
169.4.a.e 1 91.b odd 2 1
169.4.b.a 2 91.i even 4 2
169.4.c.a 2 91.t odd 6 2
169.4.c.e 2 91.n odd 6 2
169.4.e.e 4 91.bc even 12 4
208.4.a.g 1 28.d even 2 1
325.4.a.d 1 35.c odd 2 1
325.4.b.b 2 35.f even 4 2
637.4.a.a 1 1.a even 1 1 trivial
832.4.a.a 1 56.e even 2 1
832.4.a.r 1 56.h odd 2 1
1521.4.a.a 1 273.g even 2 1
1573.4.a.a 1 77.b even 2 1
1872.4.a.k 1 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(637))$$:

 $$T_{2} + 5$$ T2 + 5 $$T_{3} - 7$$ T3 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 5$$
$3$ $$T - 7$$
$5$ $$T - 7$$
$7$ $$T$$
$11$ $$T + 26$$
$13$ $$T + 13$$
$17$ $$T + 77$$
$19$ $$T - 126$$
$23$ $$T + 96$$
$29$ $$T + 82$$
$31$ $$T + 196$$
$37$ $$T + 131$$
$41$ $$T + 336$$
$43$ $$T + 201$$
$47$ $$T - 105$$
$53$ $$T + 432$$
$59$ $$T - 294$$
$61$ $$T - 56$$
$67$ $$T - 478$$
$71$ $$T - 9$$
$73$ $$T + 98$$
$79$ $$T - 1304$$
$83$ $$T - 308$$
$89$ $$T - 1190$$
$97$ $$T + 70$$