# Properties

 Label 637.2.r.a Level $637$ Weight $2$ Character orbit 637.r Analytic conductor $5.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{2} - 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} - \zeta_{12} q^{5} - 4 \zeta_{12}^{3} q^{6} + (\zeta_{12}^{2} - 1) q^{9} +O(q^{10})$$ q + 2*z * q^2 - 2*z^2 * q^3 + 2*z^2 * q^4 - z * q^5 - 4*z^3 * q^6 + (z^2 - 1) * q^9 $$q + 2 \zeta_{12} q^{2} - 2 \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{2} q^{4} - \zeta_{12} q^{5} - 4 \zeta_{12}^{3} q^{6} + (\zeta_{12}^{2} - 1) q^{9} - 2 \zeta_{12}^{2} q^{10} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 4) q^{12} + ( - 3 \zeta_{12}^{3} + 2) q^{13} + 2 \zeta_{12}^{3} q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} - 6 \zeta_{12}^{2} q^{17} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{18} + 3 \zeta_{12} q^{19} - 2 \zeta_{12}^{3} q^{20} - 4 q^{22} + ( - 3 \zeta_{12}^{2} + 3) q^{23} - 4 \zeta_{12}^{2} q^{25} + ( - 6 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{26} - 4 q^{27} + 3 q^{29} + (4 \zeta_{12}^{2} - 4) q^{30} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{31} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{32} + 4 \zeta_{12} q^{33} - 12 \zeta_{12}^{3} q^{34} - 2 q^{36} - 6 \zeta_{12} q^{37} + 6 \zeta_{12}^{2} q^{38} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 6 \zeta_{12}) q^{39} + 10 \zeta_{12}^{3} q^{41} + q^{43} - 4 \zeta_{12} q^{44} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{45} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{46} + 11 \zeta_{12} q^{47} - 8 q^{48} - 8 \zeta_{12}^{3} q^{50} + (12 \zeta_{12}^{2} - 12) q^{51} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 6 \zeta_{12}) q^{52} + 9 \zeta_{12}^{2} q^{53} - 8 \zeta_{12} q^{54} + 2 q^{55} - 6 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{59} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{60} + (8 \zeta_{12}^{2} - 8) q^{61} - 6 q^{62} + 8 q^{64} + (3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{65} + 8 \zeta_{12}^{2} q^{66} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{67} + ( - 12 \zeta_{12}^{2} + 12) q^{68} - 6 q^{69} + 14 \zeta_{12}^{3} q^{71} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{73} - 12 \zeta_{12}^{2} q^{74} + (8 \zeta_{12}^{2} - 8) q^{75} + 6 \zeta_{12}^{3} q^{76} + ( - 8 \zeta_{12}^{3} - 12) q^{78} + ( - 9 \zeta_{12}^{2} + 9) q^{79} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{80} + 11 \zeta_{12}^{2} q^{81} + (20 \zeta_{12}^{2} - 20) q^{82} + 11 \zeta_{12}^{3} q^{83} + 6 \zeta_{12}^{3} q^{85} + 2 \zeta_{12} q^{86} - 6 \zeta_{12}^{2} q^{87} - 5 \zeta_{12} q^{89} + 2 q^{90} + 6 q^{92} + 6 \zeta_{12} q^{93} + 22 \zeta_{12}^{2} q^{94} - 3 \zeta_{12}^{2} q^{95} - 16 \zeta_{12} q^{96} - 9 \zeta_{12}^{3} q^{97} - 2 \zeta_{12}^{3} q^{99} +O(q^{100})$$ q + 2*z * q^2 - 2*z^2 * q^3 + 2*z^2 * q^4 - z * q^5 - 4*z^3 * q^6 + (z^2 - 1) * q^9 - 2*z^2 * q^10 + (2*z^3 - 2*z) * q^11 + (-4*z^2 + 4) * q^12 + (-3*z^3 + 2) * q^13 + 2*z^3 * q^15 + (-4*z^2 + 4) * q^16 - 6*z^2 * q^17 + (2*z^3 - 2*z) * q^18 + 3*z * q^19 - 2*z^3 * q^20 - 4 * q^22 + (-3*z^2 + 3) * q^23 - 4*z^2 * q^25 + (-6*z^2 + 4*z + 6) * q^26 - 4 * q^27 + 3 * q^29 + (4*z^2 - 4) * q^30 + (3*z^3 - 3*z) * q^31 + (-8*z^3 + 8*z) * q^32 + 4*z * q^33 - 12*z^3 * q^34 - 2 * q^36 - 6*z * q^37 + 6*z^2 * q^38 + (6*z^3 - 4*z^2 - 6*z) * q^39 + 10*z^3 * q^41 + q^43 - 4*z * q^44 + (-z^3 + z) * q^45 + (-6*z^3 + 6*z) * q^46 + 11*z * q^47 - 8 * q^48 - 8*z^3 * q^50 + (12*z^2 - 12) * q^51 + (-6*z^3 + 4*z^2 + 6*z) * q^52 + 9*z^2 * q^53 - 8*z * q^54 + 2 * q^55 - 6*z^3 * q^57 + 6*z * q^58 + (-8*z^3 + 8*z) * q^59 + (4*z^3 - 4*z) * q^60 + (8*z^2 - 8) * q^61 - 6 * q^62 + 8 * q^64 + (3*z^2 - 2*z - 3) * q^65 + 8*z^2 * q^66 + (-12*z^3 + 12*z) * q^67 + (-12*z^2 + 12) * q^68 - 6 * q^69 + 14*z^3 * q^71 + (-9*z^3 + 9*z) * q^73 - 12*z^2 * q^74 + (8*z^2 - 8) * q^75 + 6*z^3 * q^76 + (-8*z^3 - 12) * q^78 + (-9*z^2 + 9) * q^79 + (4*z^3 - 4*z) * q^80 + 11*z^2 * q^81 + (20*z^2 - 20) * q^82 + 11*z^3 * q^83 + 6*z^3 * q^85 + 2*z * q^86 - 6*z^2 * q^87 - 5*z * q^89 + 2 * q^90 + 6 * q^92 + 6*z * q^93 + 22*z^2 * q^94 - 3*z^2 * q^95 - 16*z * q^96 - 9*z^3 * q^97 - 2*z^3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} + 4 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 $$4 q - 4 q^{3} + 4 q^{4} - 2 q^{9} - 4 q^{10} + 8 q^{12} + 8 q^{13} + 8 q^{16} - 12 q^{17} - 16 q^{22} + 6 q^{23} - 8 q^{25} + 12 q^{26} - 16 q^{27} + 12 q^{29} - 8 q^{30} - 8 q^{36} + 12 q^{38} - 8 q^{39} + 4 q^{43} - 32 q^{48} - 24 q^{51} + 8 q^{52} + 18 q^{53} + 8 q^{55} - 16 q^{61} - 24 q^{62} + 32 q^{64} - 6 q^{65} + 16 q^{66} + 24 q^{68} - 24 q^{69} - 24 q^{74} - 16 q^{75} - 48 q^{78} + 18 q^{79} + 22 q^{81} - 40 q^{82} - 12 q^{87} + 8 q^{90} + 24 q^{92} + 44 q^{94} - 6 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 + 4 * q^4 - 2 * q^9 - 4 * q^10 + 8 * q^12 + 8 * q^13 + 8 * q^16 - 12 * q^17 - 16 * q^22 + 6 * q^23 - 8 * q^25 + 12 * q^26 - 16 * q^27 + 12 * q^29 - 8 * q^30 - 8 * q^36 + 12 * q^38 - 8 * q^39 + 4 * q^43 - 32 * q^48 - 24 * q^51 + 8 * q^52 + 18 * q^53 + 8 * q^55 - 16 * q^61 - 24 * q^62 + 32 * q^64 - 6 * q^65 + 16 * q^66 + 24 * q^68 - 24 * q^69 - 24 * q^74 - 16 * q^75 - 48 * q^78 + 18 * q^79 + 22 * q^81 - 40 * q^82 - 12 * q^87 + 8 * q^90 + 24 * q^92 + 44 * q^94 - 6 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−1.73205 + 1.00000i −1.00000 + 1.73205i 1.00000 1.73205i 0.866025 0.500000i 4.00000i 0 0 −0.500000 0.866025i −1.00000 + 1.73205i
116.2 1.73205 1.00000i −1.00000 + 1.73205i 1.00000 1.73205i −0.866025 + 0.500000i 4.00000i 0 0 −0.500000 0.866025i −1.00000 + 1.73205i
324.1 −1.73205 1.00000i −1.00000 1.73205i 1.00000 + 1.73205i 0.866025 + 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i −1.00000 1.73205i
324.2 1.73205 + 1.00000i −1.00000 1.73205i 1.00000 + 1.73205i −0.866025 0.500000i 4.00000i 0 0 −0.500000 + 0.866025i −1.00000 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.a 4
7.b odd 2 1 637.2.r.c 4
7.c even 3 1 637.2.c.c yes 2
7.c even 3 1 inner 637.2.r.a 4
7.d odd 6 1 637.2.c.a 2
7.d odd 6 1 637.2.r.c 4
13.b even 2 1 inner 637.2.r.a 4
91.b odd 2 1 637.2.r.c 4
91.r even 6 1 637.2.c.c yes 2
91.r even 6 1 inner 637.2.r.a 4
91.s odd 6 1 637.2.c.a 2
91.s odd 6 1 637.2.r.c 4
91.z odd 12 1 8281.2.a.b 1
91.z odd 12 1 8281.2.a.m 1
91.bb even 12 1 8281.2.a.a 1
91.bb even 12 1 8281.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.a 2 7.d odd 6 1
637.2.c.a 2 91.s odd 6 1
637.2.c.c yes 2 7.c even 3 1
637.2.c.c yes 2 91.r even 6 1
637.2.r.a 4 1.a even 1 1 trivial
637.2.r.a 4 7.c even 3 1 inner
637.2.r.a 4 13.b even 2 1 inner
637.2.r.a 4 91.r even 6 1 inner
637.2.r.c 4 7.b odd 2 1
637.2.r.c 4 7.d odd 6 1
637.2.r.c 4 91.b odd 2 1
637.2.r.c 4 91.s odd 6 1
8281.2.a.a 1 91.bb even 12 1
8281.2.a.b 1 91.z odd 12 1
8281.2.a.k 1 91.bb even 12 1
8281.2.a.m 1 91.z odd 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 4T^{2} + 16$$
$3$ $$(T^{2} + 2 T + 4)^{2}$$
$5$ $$T^{4} - T^{2} + 1$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$(T^{2} - 4 T + 13)^{2}$$
$17$ $$(T^{2} + 6 T + 36)^{2}$$
$19$ $$T^{4} - 9T^{2} + 81$$
$23$ $$(T^{2} - 3 T + 9)^{2}$$
$29$ $$(T - 3)^{4}$$
$31$ $$T^{4} - 9T^{2} + 81$$
$37$ $$T^{4} - 36T^{2} + 1296$$
$41$ $$(T^{2} + 100)^{2}$$
$43$ $$(T - 1)^{4}$$
$47$ $$T^{4} - 121 T^{2} + 14641$$
$53$ $$(T^{2} - 9 T + 81)^{2}$$
$59$ $$T^{4} - 64T^{2} + 4096$$
$61$ $$(T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{4} - 144 T^{2} + 20736$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$T^{4} - 81T^{2} + 6561$$
$79$ $$(T^{2} - 9 T + 81)^{2}$$
$83$ $$(T^{2} + 121)^{2}$$
$89$ $$T^{4} - 25T^{2} + 625$$
$97$ $$(T^{2} + 81)^{2}$$