Properties

Label 637.2.e.l
Level $637$
Weight $2$
Character orbit 637.e
Analytic conductor $5.086$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
Defining polynomial: \(x^{6} - x^{5} + 6 x^{4} + 7 x^{3} + 24 x^{2} + 5 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{2} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{3} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{5} + ( -3 \beta_{2} - \beta_{3} ) q^{6} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{2} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{3} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{4} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{5} + ( -3 \beta_{2} - \beta_{3} ) q^{6} + ( -2 + 2 \beta_{2} + 2 \beta_{3} ) q^{8} + ( \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{9} + ( -5 + 2 \beta_{1} + \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{10} + ( 1 + \beta_{1} - \beta_{4} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} ) q^{12} + q^{13} + ( 1 - 3 \beta_{2} - 2 \beta_{3} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{16} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{17} + ( -1 - 5 \beta_{1} + \beta_{4} ) q^{18} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( -6 + 5 \beta_{2} + 3 \beta_{3} ) q^{20} + ( -2 - \beta_{2} + \beta_{3} ) q^{22} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} ) q^{23} + ( -10 + 4 \beta_{1} + 2 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} ) q^{24} + ( -2 + 3 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{26} + ( -6 - 4 \beta_{2} ) q^{27} + ( -3 - 2 \beta_{3} ) q^{29} + ( 5 \beta_{1} - 5 \beta_{2} + 8 \beta_{4} - 5 \beta_{5} ) q^{30} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{31} + ( 8 - 2 \beta_{1} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{32} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( -3 \beta_{2} - \beta_{3} ) q^{34} + ( 8 - \beta_{2} - 3 \beta_{3} ) q^{36} + ( \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 3 - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{38} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{39} + ( -8 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} + 6 \beta_{5} ) q^{40} + ( -2 - 4 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 3 + 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 2 - 4 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{45} + ( 9 - 3 \beta_{1} - 4 \beta_{3} - 9 \beta_{4} + 4 \beta_{5} ) q^{46} + ( 6 \beta_{4} + \beta_{5} ) q^{47} + ( -8 + 10 \beta_{2} + 2 \beta_{3} ) q^{48} + ( -10 + 4 \beta_{2} + 4 \beta_{3} ) q^{50} + ( \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{51} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{52} + ( -5 + \beta_{1} - \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{53} + ( -6 \beta_{1} + 6 \beta_{2} + 6 \beta_{4} - 4 \beta_{5} ) q^{54} + ( -1 + \beta_{3} ) q^{55} + ( -1 + 5 \beta_{2} - 2 \beta_{3} ) q^{57} + ( \beta_{1} - \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 8 - \beta_{1} + \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{59} + ( -16 + 12 \beta_{1} + 6 \beta_{3} + 16 \beta_{4} - 6 \beta_{5} ) q^{60} + ( 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} - 2 \beta_{5} ) q^{61} + ( 5 + 4 \beta_{2} + \beta_{3} ) q^{62} + ( 8 - 4 \beta_{2} ) q^{64} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} ) q^{65} + ( -8 - 4 \beta_{1} + 2 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} ) q^{66} + ( 4 - 2 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} ) q^{68} + ( -3 + 8 \beta_{2} + \beta_{3} ) q^{69} + ( 1 + 3 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 4 \beta_{1} - 4 \beta_{2} + 10 \beta_{4} - 4 \beta_{5} ) q^{72} + ( 4 - 6 \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{73} + ( -4 + 7 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{74} + ( 8 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{75} + ( 4 - 5 \beta_{2} - \beta_{3} ) q^{76} + ( -3 \beta_{2} - \beta_{3} ) q^{78} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{79} + ( 14 - 10 \beta_{1} - 8 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} ) q^{80} + ( -1 - 9 \beta_{1} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{81} + ( 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{4} - 8 \beta_{5} ) q^{82} + ( -8 + 3 \beta_{2} + 2 \beta_{3} ) q^{83} + ( 1 - 3 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{86} + ( 7 + 2 \beta_{1} + 3 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} ) q^{87} + ( 2 - 2 \beta_{1} - 2 \beta_{4} ) q^{88} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + 3 \beta_{5} ) q^{89} + ( 13 - 4 \beta_{2} - 5 \beta_{3} ) q^{90} + ( 12 - 11 \beta_{2} - 5 \beta_{3} ) q^{92} + ( -5 \beta_{1} + 5 \beta_{2} + 11 \beta_{4} ) q^{93} + ( -7 + 4 \beta_{1} - \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{94} + ( 1 - 2 \beta_{1} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{95} + ( -4 \beta_{1} + 4 \beta_{2} - 16 \beta_{4} + 8 \beta_{5} ) q^{96} + ( 2 - 4 \beta_{2} - \beta_{3} ) q^{97} + ( -7 - 7 \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{3} - 6q^{4} + 5q^{5} - 4q^{6} - 12q^{8} - 11q^{9} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{3} - 6q^{4} + 5q^{5} - 4q^{6} - 12q^{8} - 11q^{9} - 14q^{10} + 4q^{11} + 18q^{12} + 6q^{13} + 4q^{15} - 4q^{16} + 4q^{17} - 8q^{18} + 7q^{19} - 32q^{20} - 16q^{22} - q^{23} - 28q^{24} - 4q^{25} + 2q^{26} - 44q^{27} - 14q^{29} + 24q^{30} - 3q^{31} + 24q^{32} - 10q^{33} - 4q^{34} + 52q^{36} + 10q^{37} + 12q^{38} + 4q^{39} - 22q^{40} - 12q^{41} + 18q^{43} + 2q^{44} + 3q^{45} + 28q^{46} + 17q^{47} - 32q^{48} - 60q^{50} - 20q^{51} - 6q^{52} - 13q^{53} + 28q^{54} - 8q^{55} + 8q^{57} - 14q^{58} + 22q^{59} - 42q^{60} - 24q^{61} + 36q^{62} + 40q^{64} + 5q^{65} - 30q^{66} + 14q^{67} + 18q^{68} - 4q^{69} + 8q^{71} + 30q^{72} + 5q^{73} - 8q^{74} + 6q^{75} + 16q^{76} - 4q^{78} - q^{79} + 40q^{80} - 15q^{81} + 20q^{82} - 46q^{83} + 4q^{85} - 6q^{86} + 20q^{87} + 4q^{88} - 11q^{89} + 80q^{90} + 60q^{92} + 38q^{93} - 16q^{94} + 5q^{95} - 52q^{96} + 6q^{97} - 60q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 6 x^{4} + 7 x^{3} + 24 x^{2} + 5 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 6 \nu^{4} - 36 \nu^{3} + 24 \nu^{2} + 5 \nu - 30 \)\()/149\)
\(\beta_{3}\)\(=\)\((\)\( -6 \nu^{5} + 36 \nu^{4} - 67 \nu^{3} + 144 \nu^{2} + 30 \nu + 416 \)\()/149\)
\(\beta_{4}\)\(=\)\((\)\( 30 \nu^{5} - 31 \nu^{4} + 186 \nu^{3} + 174 \nu^{2} + 744 \nu + 155 \)\()/149\)
\(\beta_{5}\)\(=\)\((\)\( 89 \nu^{5} - 87 \nu^{4} + 522 \nu^{3} + 695 \nu^{2} + 2088 \nu + 435 \)\()/149\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 6 \beta_{2} - 4\)
\(\nu^{4}\)\(=\)\(-6 \beta_{5} + 19 \beta_{4} + 6 \beta_{3} - 12 \beta_{1} - 19\)
\(\nu^{5}\)\(=\)\(-12 \beta_{5} + 42 \beta_{4} + 43 \beta_{2} - 43 \beta_{1}\)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.43310 + 2.48220i
−0.105378 0.182520i
−0.827721 1.43366i
1.43310 2.48220i
−0.105378 + 0.182520i
−0.827721 + 1.43366i
−0.933099 + 1.61618i 1.67445 + 2.90023i −0.741348 1.28405i −0.433099 + 0.750150i −6.24970 0 −0.965392 −4.10755 + 7.11448i −0.808249 1.39993i
79.2 0.605378 1.04855i −0.872413 1.51106i 0.267035 + 0.462518i 1.10538 1.91457i −2.11256 0 3.06814 −0.0222090 + 0.0384672i −1.33834 2.31808i
79.3 1.32772 2.29968i 1.19797 + 2.07494i −2.52569 4.37462i 1.82772 3.16571i 6.36226 0 −8.10275 −1.37024 + 2.37333i −4.85341 8.40635i
508.1 −0.933099 1.61618i 1.67445 2.90023i −0.741348 + 1.28405i −0.433099 0.750150i −6.24970 0 −0.965392 −4.10755 7.11448i −0.808249 + 1.39993i
508.2 0.605378 + 1.04855i −0.872413 + 1.51106i 0.267035 0.462518i 1.10538 + 1.91457i −2.11256 0 3.06814 −0.0222090 0.0384672i −1.33834 + 2.31808i
508.3 1.32772 + 2.29968i 1.19797 2.07494i −2.52569 + 4.37462i 1.82772 + 3.16571i 6.36226 0 −8.10275 −1.37024 2.37333i −4.85341 + 8.40635i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 508.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c 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.e.l 6
7.b odd 2 1 637.2.e.k 6
7.c even 3 1 637.2.a.h 3
7.c even 3 1 inner 637.2.e.l 6
7.d odd 6 1 637.2.a.i yes 3
7.d odd 6 1 637.2.e.k 6
21.g even 6 1 5733.2.a.bd 3
21.h odd 6 1 5733.2.a.be 3
91.r even 6 1 8281.2.a.bh 3
91.s odd 6 1 8281.2.a.bk 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.h 3 7.c even 3 1
637.2.a.i yes 3 7.d odd 6 1
637.2.e.k 6 7.b odd 2 1
637.2.e.k 6 7.d odd 6 1
637.2.e.l 6 1.a even 1 1 trivial
637.2.e.l 6 7.c even 3 1 inner
5733.2.a.bd 3 21.g even 6 1
5733.2.a.be 3 21.h odd 6 1
8281.2.a.bh 3 91.r even 6 1
8281.2.a.bk 3 91.s odd 6 1

Hecke kernels

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

\( T_{2}^{6} - 2 T_{2}^{5} + 8 T_{2}^{4} - 4 T_{2}^{3} + 28 T_{2}^{2} - 24 T_{2} + 36 \)
\( T_{3}^{6} - 4 T_{3}^{5} + 18 T_{3}^{4} - 20 T_{3}^{3} + 60 T_{3}^{2} - 28 T_{3} + 196 \)
\( T_{5}^{6} - 5 T_{5}^{5} + 22 T_{5}^{4} - 29 T_{5}^{3} + 44 T_{5}^{2} + 21 T_{5} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 36 - 24 T + 28 T^{2} - 4 T^{3} + 8 T^{4} - 2 T^{5} + T^{6} \)
$3$ \( 196 - 28 T + 60 T^{2} - 20 T^{3} + 18 T^{4} - 4 T^{5} + T^{6} \)
$5$ \( 49 + 21 T + 44 T^{2} - 29 T^{3} + 22 T^{4} - 5 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 4 + 8 T^{2} - 4 T^{3} + 16 T^{4} - 4 T^{5} + T^{6} \)
$13$ \( ( -1 + T )^{6} \)
$17$ \( 196 - 28 T + 60 T^{2} - 20 T^{3} + 18 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( 3969 - 189 T + 450 T^{2} - 105 T^{3} + 52 T^{4} - 7 T^{5} + T^{6} \)
$23$ \( 1849 - 1763 T + 1638 T^{2} - 127 T^{3} + 42 T^{4} + T^{5} + T^{6} \)
$29$ \( ( -3 - 13 T + 7 T^{2} + T^{3} )^{2} \)
$31$ \( 2401 + 2009 T + 1828 T^{2} - 25 T^{3} + 50 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( 6724 + 656 T + 884 T^{2} - 244 T^{3} + 92 T^{4} - 10 T^{5} + T^{6} \)
$41$ \( ( -504 - 116 T + 6 T^{2} + T^{3} )^{2} \)
$43$ \( ( 101 - 5 T - 9 T^{2} + T^{3} )^{2} \)
$47$ \( 21609 - 13083 T + 5422 T^{2} - 1219 T^{3} + 200 T^{4} - 17 T^{5} + T^{6} \)
$53$ \( 81 - 351 T + 1638 T^{2} + 525 T^{3} + 130 T^{4} + 13 T^{5} + T^{6} \)
$59$ \( 63504 - 36288 T + 15192 T^{2} - 2664 T^{3} + 340 T^{4} - 22 T^{5} + T^{6} \)
$61$ \( 50176 + 35840 T + 20224 T^{2} + 3392 T^{3} + 416 T^{4} + 24 T^{5} + T^{6} \)
$67$ \( 419904 - 23328 T + 10368 T^{2} - 792 T^{3} + 232 T^{4} - 14 T^{5} + T^{6} \)
$71$ \( ( 194 - 44 T - 4 T^{2} + T^{3} )^{2} \)
$73$ \( 2436721 - 341859 T + 55766 T^{2} - 2027 T^{3} + 244 T^{4} - 5 T^{5} + T^{6} \)
$79$ \( 9801 - 6831 T + 4662 T^{2} - 267 T^{3} + 70 T^{4} + T^{5} + T^{6} \)
$83$ \( ( 203 + 127 T + 23 T^{2} + T^{3} )^{2} \)
$89$ \( 441 - 1155 T + 2794 T^{2} - 647 T^{3} + 176 T^{4} + 11 T^{5} + T^{6} \)
$97$ \( ( 7 - 71 T - 3 T^{2} + T^{3} )^{2} \)
show more
show less