# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.h (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ 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: $$3$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{6} + \beta_{3} q^{8} + (2 \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-b3 + b2 - b1 + 1) * q^3 + q^4 + (-b3 + b2) * q^5 + (-b3 + b2 - 3*b1 + 3) * q^6 + b3 * q^8 + (2*b2 - b1) * q^9 $$q - \beta_{3} q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{6} + \beta_{3} q^{8} + (2 \beta_{2} - \beta_1) q^{9} + ( - 3 \beta_1 + 3) q^{10} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{12} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{13} + (\beta_{2} - 3 \beta_1) q^{15} - 5 q^{16} + ( - \beta_{3} + 6) q^{17} + (\beta_{2} - 6 \beta_1) q^{18} + 2 \beta_1 q^{19} + ( - \beta_{3} + \beta_{2}) q^{20} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - \beta_{3} + 3) q^{23} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 3) q^{24} + 2 \beta_1 q^{25} + (2 \beta_{2} + 3 \beta_1 - 6) q^{26} - 4 q^{27} + 3 \beta_1 q^{29} + (3 \beta_{2} - 3 \beta_1) q^{30} + ( - 3 \beta_{2} - \beta_1) q^{31} + 3 \beta_{3} q^{32} - 2 \beta_{2} q^{33} + ( - 6 \beta_{3} + 3) q^{34} + (2 \beta_{2} - \beta_1) q^{36} - 7 q^{37} - 2 \beta_{2} q^{38} + (3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 5) q^{39} + (3 \beta_1 - 3) q^{40} + 3 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{43} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{44} + (\beta_{3} - 6) q^{45} + ( - 3 \beta_{3} + 3) q^{46} + ( - 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 6) q^{47} + (5 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 5) q^{48} - 2 \beta_{2} q^{50} + ( - 7 \beta_{3} + 7 \beta_{2} - 9 \beta_1 + 9) q^{51} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{52} + ( - 4 \beta_{2} + 3 \beta_1) q^{53} + 4 \beta_{3} q^{54} + ( - 3 \beta_{2} - 3 \beta_1) q^{55} + ( - 2 \beta_{3} + 2) q^{57} - 3 \beta_{2} q^{58} + ( - \beta_{3} - 9) q^{59} + (\beta_{2} - 3 \beta_1) q^{60} + ( - 3 \beta_{2} - 10 \beta_1) q^{61} + (\beta_{2} + 9 \beta_1) q^{62} + q^{64} + (2 \beta_{3} + 6 \beta_1 - 3) q^{65} + 6 \beta_1 q^{66} + ( - 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{67} + ( - \beta_{3} + 6) q^{68} + ( - 4 \beta_{3} + 4 \beta_{2} - 6 \beta_1 + 6) q^{69} + (6 \beta_1 - 6) q^{71} + ( - \beta_{2} + 6 \beta_1) q^{72} + (3 \beta_{2} + 2 \beta_1) q^{73} + 7 \beta_{3} q^{74} + ( - 2 \beta_{3} + 2) q^{75} + 2 \beta_1 q^{76} + (5 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 9) q^{78} + (3 \beta_{3} - 3 \beta_{2} + 11 \beta_1 - 11) q^{79} + (5 \beta_{3} - 5 \beta_{2}) q^{80} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{81} - 9 \beta_1 q^{82} + ( - 3 \beta_{3} - 3) q^{83} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 3) q^{85} + (5 \beta_{3} - 5 \beta_{2} - 9 \beta_1 + 9) q^{86} + ( - 3 \beta_{3} + 3) q^{87} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{88} + (4 \beta_{3} - 6) q^{89} + (6 \beta_{3} - 3) q^{90} + ( - \beta_{3} + 3) q^{92} + ( - 2 \beta_{3} + 8) q^{93} + ( - 6 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 12) q^{94} - 2 \beta_{3} q^{95} + (3 \beta_{3} - 3 \beta_{2} + 9 \beta_1 - 9) q^{96} + ( - 6 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 4) q^{97} + ( - 5 \beta_{3} - 3) q^{99}+O(q^{100})$$ q - b3 * q^2 + (-b3 + b2 - b1 + 1) * q^3 + q^4 + (-b3 + b2) * q^5 + (-b3 + b2 - 3*b1 + 3) * q^6 + b3 * q^8 + (2*b2 - b1) * q^9 + (-3*b1 + 3) * q^10 + (-b3 + b2 + 3*b1 - 3) * q^11 + (-b3 + b2 - b1 + 1) * q^12 + (2*b3 - b2 - 2*b1) * q^13 + (b2 - 3*b1) * q^15 - 5 * q^16 + (-b3 + 6) * q^17 + (b2 - 6*b1) * q^18 + 2*b1 * q^19 + (-b3 + b2) * q^20 + (3*b3 - 3*b2 - 3*b1 + 3) * q^22 + (-b3 + 3) * q^23 + (b3 - b2 + 3*b1 - 3) * q^24 + 2*b1 * q^25 + (2*b2 + 3*b1 - 6) * q^26 - 4 * q^27 + 3*b1 * q^29 + (3*b2 - 3*b1) * q^30 + (-3*b2 - b1) * q^31 + 3*b3 * q^32 - 2*b2 * q^33 + (-6*b3 + 3) * q^34 + (2*b2 - b1) * q^36 - 7 * q^37 - 2*b2 * q^38 + (3*b3 - 2*b2 + 6*b1 - 5) * q^39 + (3*b1 - 3) * q^40 + 3*b2 * q^41 + (-3*b3 + 3*b2 + 5*b1 - 5) * q^43 + (-b3 + b2 + 3*b1 - 3) * q^44 + (b3 - 6) * q^45 + (-3*b3 + 3) * q^46 + (-4*b3 + 4*b2 - 6*b1 + 6) * q^47 + (5*b3 - 5*b2 + 5*b1 - 5) * q^48 - 2*b2 * q^50 + (-7*b3 + 7*b2 - 9*b1 + 9) * q^51 + (2*b3 - b2 - 2*b1) * q^52 + (-4*b2 + 3*b1) * q^53 + 4*b3 * q^54 + (-3*b2 - 3*b1) * q^55 + (-2*b3 + 2) * q^57 - 3*b2 * q^58 + (-b3 - 9) * q^59 + (b2 - 3*b1) * q^60 + (-3*b2 - 10*b1) * q^61 + (b2 + 9*b1) * q^62 + q^64 + (2*b3 + 6*b1 - 3) * q^65 + 6*b1 * q^66 + (-3*b3 + 3*b2 - b1 + 1) * q^67 + (-b3 + 6) * q^68 + (-4*b3 + 4*b2 - 6*b1 + 6) * q^69 + (6*b1 - 6) * q^71 + (-b2 + 6*b1) * q^72 + (3*b2 + 2*b1) * q^73 + 7*b3 * q^74 + (-2*b3 + 2) * q^75 + 2*b1 * q^76 + (5*b3 - 6*b2 + 6*b1 - 9) * q^78 + (3*b3 - 3*b2 + 11*b1 - 11) * q^79 + (5*b3 - 5*b2) * q^80 + (-2*b3 + 2*b2 + b1 - 1) * q^81 - 9*b1 * q^82 + (-3*b3 - 3) * q^83 + (-6*b3 + 6*b2 - 3*b1 + 3) * q^85 + (5*b3 - 5*b2 - 9*b1 + 9) * q^86 + (-3*b3 + 3) * q^87 + (-3*b3 + 3*b2 + 3*b1 - 3) * q^88 + (4*b3 - 6) * q^89 + (6*b3 - 3) * q^90 + (-b3 + 3) * q^92 + (-2*b3 + 8) * q^93 + (-6*b3 + 6*b2 - 12*b1 + 12) * q^94 - 2*b3 * q^95 + (3*b3 - 3*b2 + 9*b1 - 9) * q^96 + (-6*b3 + 6*b2 + 4*b1 - 4) * q^97 + (-5*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{4} + 6 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^4 + 6 * q^6 - 2 * q^9 $$4 q + 2 q^{3} + 4 q^{4} + 6 q^{6} - 2 q^{9} + 6 q^{10} - 6 q^{11} + 2 q^{12} - 4 q^{13} - 6 q^{15} - 20 q^{16} + 24 q^{17} - 12 q^{18} + 4 q^{19} + 6 q^{22} + 12 q^{23} - 6 q^{24} + 4 q^{25} - 18 q^{26} - 16 q^{27} + 6 q^{29} - 6 q^{30} - 2 q^{31} + 12 q^{34} - 2 q^{36} - 28 q^{37} - 8 q^{39} - 6 q^{40} - 10 q^{43} - 6 q^{44} - 24 q^{45} + 12 q^{46} + 12 q^{47} - 10 q^{48} + 18 q^{51} - 4 q^{52} + 6 q^{53} - 6 q^{55} + 8 q^{57} - 36 q^{59} - 6 q^{60} - 20 q^{61} + 18 q^{62} + 4 q^{64} + 12 q^{66} + 2 q^{67} + 24 q^{68} + 12 q^{69} - 12 q^{71} + 12 q^{72} + 4 q^{73} + 8 q^{75} + 4 q^{76} - 24 q^{78} - 22 q^{79} - 2 q^{81} - 18 q^{82} - 12 q^{83} + 6 q^{85} + 18 q^{86} + 12 q^{87} - 6 q^{88} - 24 q^{89} - 12 q^{90} + 12 q^{92} + 32 q^{93} + 24 q^{94} - 18 q^{96} - 8 q^{97} - 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^4 + 6 * q^6 - 2 * q^9 + 6 * q^10 - 6 * q^11 + 2 * q^12 - 4 * q^13 - 6 * q^15 - 20 * q^16 + 24 * q^17 - 12 * q^18 + 4 * q^19 + 6 * q^22 + 12 * q^23 - 6 * q^24 + 4 * q^25 - 18 * q^26 - 16 * q^27 + 6 * q^29 - 6 * q^30 - 2 * q^31 + 12 * q^34 - 2 * q^36 - 28 * q^37 - 8 * q^39 - 6 * q^40 - 10 * q^43 - 6 * q^44 - 24 * q^45 + 12 * q^46 + 12 * q^47 - 10 * q^48 + 18 * q^51 - 4 * q^52 + 6 * q^53 - 6 * q^55 + 8 * q^57 - 36 * q^59 - 6 * q^60 - 20 * q^61 + 18 * q^62 + 4 * q^64 + 12 * q^66 + 2 * q^67 + 24 * q^68 + 12 * q^69 - 12 * q^71 + 12 * q^72 + 4 * q^73 + 8 * q^75 + 4 * q^76 - 24 * q^78 - 22 * q^79 - 2 * q^81 - 18 * q^82 - 12 * q^83 + 6 * q^85 + 18 * q^86 + 12 * q^87 - 6 * q^88 - 24 * q^89 - 12 * q^90 + 12 * q^92 + 32 * q^93 + 24 * q^94 - 18 * q^96 - 8 * q^97 - 12 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−1.73205 −0.366025 0.633975i 1.00000 −0.866025 1.50000i 0.633975 + 1.09808i 0 1.73205 1.23205 2.13397i 1.50000 + 2.59808i
165.2 1.73205 1.36603 + 2.36603i 1.00000 0.866025 + 1.50000i 2.36603 + 4.09808i 0 −1.73205 −2.23205 + 3.86603i 1.50000 + 2.59808i
471.1 −1.73205 −0.366025 + 0.633975i 1.00000 −0.866025 + 1.50000i 0.633975 1.09808i 0 1.73205 1.23205 + 2.13397i 1.50000 2.59808i
471.2 1.73205 1.36603 2.36603i 1.00000 0.866025 1.50000i 2.36603 4.09808i 0 −1.73205 −2.23205 3.86603i 1.50000 2.59808i
 $$n$$: 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
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.h.e 4
7.b odd 2 1 637.2.h.d 4
7.c even 3 1 637.2.f.d 4
7.c even 3 1 637.2.g.d 4
7.d odd 6 1 91.2.f.b 4
7.d odd 6 1 637.2.g.e 4
13.c even 3 1 637.2.g.d 4
21.g even 6 1 819.2.o.b 4
28.f even 6 1 1456.2.s.o 4
91.g even 3 1 637.2.f.d 4
91.h even 3 1 inner 637.2.h.e 4
91.h even 3 1 8281.2.a.r 2
91.k even 6 1 8281.2.a.t 2
91.l odd 6 1 1183.2.a.e 2
91.m odd 6 1 91.2.f.b 4
91.n odd 6 1 637.2.g.e 4
91.v odd 6 1 637.2.h.d 4
91.v odd 6 1 1183.2.a.f 2
91.ba even 12 2 1183.2.c.e 4
273.bf even 6 1 819.2.o.b 4
364.br even 6 1 1456.2.s.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 7.d odd 6 1
91.2.f.b 4 91.m odd 6 1
637.2.f.d 4 7.c even 3 1
637.2.f.d 4 91.g even 3 1
637.2.g.d 4 7.c even 3 1
637.2.g.d 4 13.c even 3 1
637.2.g.e 4 7.d odd 6 1
637.2.g.e 4 91.n odd 6 1
637.2.h.d 4 7.b odd 2 1
637.2.h.d 4 91.v odd 6 1
637.2.h.e 4 1.a even 1 1 trivial
637.2.h.e 4 91.h even 3 1 inner
819.2.o.b 4 21.g even 6 1
819.2.o.b 4 273.bf even 6 1
1183.2.a.e 2 91.l odd 6 1
1183.2.a.f 2 91.v odd 6 1
1183.2.c.e 4 91.ba even 12 2
1456.2.s.o 4 28.f even 6 1
1456.2.s.o 4 364.br even 6 1
8281.2.a.r 2 91.h even 3 1
8281.2.a.t 2 91.k even 6 1

## Hecke kernels

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

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4$$ T3^4 - 2*T3^3 + 6*T3^2 + 4*T3 + 4 $$T_{5}^{4} + 3T_{5}^{2} + 9$$ T5^4 + 3*T5^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 3)^{2}$$
$3$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$13$ $$T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169$$
$17$ $$(T^{2} - 12 T + 33)^{2}$$
$19$ $$(T^{2} - 2 T + 4)^{2}$$
$23$ $$(T^{2} - 6 T + 6)^{2}$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$37$ $$(T + 7)^{4}$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4$$
$47$ $$T^{4} - 12 T^{3} + 156 T^{2} + \cdots + 144$$
$53$ $$T^{4} - 6 T^{3} + 75 T^{2} + \cdots + 1521$$
$59$ $$(T^{2} + 18 T + 78)^{2}$$
$61$ $$T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$T^{4} - 4 T^{3} + 39 T^{2} + 92 T + 529$$
$79$ $$T^{4} + 22 T^{3} + 390 T^{2} + \cdots + 8836$$
$83$ $$(T^{2} + 6 T - 18)^{2}$$
$89$ $$(T^{2} + 12 T - 12)^{2}$$
$97$ $$T^{4} + 8 T^{3} + 156 T^{2} + \cdots + 8464$$