↑

Properties

Label 637.2.r.f
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(116,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.r (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{4} + 1) q^{3} + ( - \beta_{11} - \beta_{7} + 1) q^{4} - \beta_{14} q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - \beta_{12} + \beta_{9} - 2 \beta_{7} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{4} + 1) q^{3} + ( - \beta_{11} - \beta_{7} + 1) q^{4} - \beta_{14} q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_{2}) q^{6}+ \cdots + (\beta_{15} - \beta_{14} + \cdots - \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{3} + 6 q^{4} - 12 q^{9} + 6 q^{10} - 18 q^{12} + 12 q^{13} + 2 q^{16} - 8 q^{17} - 36 q^{22} - 12 q^{23} + 6 q^{26} - 32 q^{27} - 16 q^{29} + 38 q^{30} - 56 q^{36} - 34 q^{38} + 18 q^{39} + 4 q^{40} + 16 q^{43} - 36 q^{48} + 16 q^{51} + 42 q^{52} - 20 q^{53} - 24 q^{55} + 12 q^{61} - 44 q^{62} + 88 q^{64} - 30 q^{65} - 2 q^{66} + 2 q^{68} + 56 q^{69} + 42 q^{74} - 8 q^{75} + 20 q^{78} + 20 q^{79} - 24 q^{81} + 16 q^{82} + 68 q^{87} + 4 q^{88} + 216 q^{90} + 12 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24498 \nu^{15} + 246060 \nu^{13} - 1852321 \nu^{11} + 6411671 \nu^{9} - 17193085 \nu^{7} + \cdots - 55035781 \nu ) / 14163622 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24498 \nu^{14} + 246060 \nu^{12} - 1852321 \nu^{10} + 6411671 \nu^{8} - 17193085 \nu^{6} + \cdots - 40872159 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 172099 \nu^{14} - 2170865 \nu^{12} + 17340370 \nu^{10} - 78484018 \nu^{8} + 236538400 \nu^{6} + \cdots - 23829231 ) / 42490866 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 99072 \nu^{14} + 1000291 \nu^{12} - 7490944 \nu^{10} + 25929344 \nu^{8} - 66564370 \nu^{6} + \cdots - 20454191 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 123570 \nu^{15} - 1246351 \nu^{13} + 9343265 \nu^{11} - 32341015 \nu^{9} + 83757455 \nu^{7} + \cdots + 61326350 \nu ) / 14163622 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 539569 \nu^{14} - 5861765 \nu^{12} + 45125185 \nu^{10} - 174659083 \nu^{8} + 494434675 \nu^{6} + \cdots - 5618970 ) / 42490866 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 539569 \nu^{15} + 5861765 \nu^{13} - 45125185 \nu^{11} + 174659083 \nu^{9} + \cdots + 5618970 \nu ) / 42490866 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 102312 \nu^{14} - 1048444 \nu^{12} + 7735924 \nu^{10} - 26777324 \nu^{8} + 67022271 \nu^{6} + \cdots + 23341327 ) / 7081811 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 476262 \nu^{15} - 4835650 \nu^{13} + 36010699 \nu^{11} - 124648349 \nu^{9} + 318752537 \nu^{7} + \cdots + 238534757 \nu ) / 14163622 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 515071 \nu^{14} + 5615705 \nu^{12} - 43272864 \nu^{10} + 168247412 \nu^{8} - 477241590 \nu^{6} + \cdots + 46491129 ) / 14163622 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3598 \nu^{14} + 40712 \nu^{12} - 317920 \nu^{10} + 1287475 \nu^{8} - 3722089 \nu^{6} + \cdots + 366555 ) / 95271 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2084782 \nu^{15} - 22708880 \nu^{13} + 174943777 \nu^{11} - 679401319 \nu^{9} + \cdots - 187583223 \nu ) / 42490866 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 205171 \nu^{15} - 2261795 \nu^{13} + 17480377 \nu^{11} - 68898667 \nu^{9} + 196608895 \nu^{7} + \cdots - 19219314 \nu ) / 3862806 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10171607 \nu^{15} - 110994637 \nu^{13} + 854461673 \nu^{11} - 3317989973 \nu^{9} + \cdots - 106397226 \nu ) / 42490866 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{7} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - 4\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + 14\beta_{7} + \beta_{4} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 6\beta_{13} - 19\beta_{8} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 8\beta_{5} - 24\beta_{3} - 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{10} - 10\beta_{6} - 31\beta_{2} - 94\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -11\beta_{12} - 115\beta_{11} + 11\beta_{9} - 345\beta_{7} + 52\beta_{5} - 52\beta_{4} - 115\beta_{3} + 52 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11\beta_{15} - 74\beta_{14} + 156\beta_{13} + 471\beta_{8} - 74\beta_{6} - 156\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -85\beta_{12} - 553\beta_{11} - 1736\beta_{7} - 315\beta_{4} + 1736 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 85\beta_{15} - 485\beta_{14} + 783\beta_{13} - 85\beta_{10} + 2374\beta_{8} + 2374\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -570\beta_{9} - 1838\beta_{5} + 2672\beta_{3} + 6935 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -570\beta_{10} + 2978\beta_{6} + 3940\beta_{2} + 12015\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3548 \beta_{12} + 12977 \beta_{11} - 3548 \beta_{9} + 44495 \beta_{7} - 10466 \beta_{5} + 10466 \beta_{4} + \cdots - 10466 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -3548\beta_{15} + 17562\beta_{14} - 19895\beta_{13} - 61020\beta_{8} + 17562\beta_{6} + 19895\beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
1.97871 + 1.14241i
1.84073 + 1.06275i
0.929293 + 0.536527i
0.287846 + 0.166188i
−0.287846 − 0.166188i
−0.929293 − 0.536527i
−1.84073 − 1.06275i
−1.97871 − 1.14241i
1.97871 − 1.14241i
1.84073 − 1.06275i
0.929293 − 0.536527i
0.287846 − 0.166188i
−0.287846 + 0.166188i
−0.929293 + 0.536527i
−1.84073 + 1.06275i
−1.97871 + 1.14241i
−1.97871 + 1.14241i 1.57521 − 2.72835i 1.61019 − 2.78892i 1.84030 − 1.06250i 7.19813i 0 2.78832i −3.46258 − 5.99736i −2.42760 + 4.20473i
116.2 −1.84073 + 1.06275i −0.0894272 + 0.154892i 1.25885 − 2.18040i −3.12291 + 1.80301i − 0.380153i 0 1.10038i 1.48401 + 2.57037i 3.83229 − 6.63772i
116.3 −0.929293 + 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i −0.541640 + 0.312716i − 2.60903i 0 − 3.05665i −1.45586 − 2.52163i 0.335561 − 0.581209i
116.4 −0.287846 + 0.166188i 0.729919 − 1.26426i −0.944763 + 1.63638i 1.25195 − 0.722811i 0.485214i 0 − 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.5 0.287846 − 0.166188i 0.729919 − 1.26426i −0.944763 + 1.63638i −1.25195 + 0.722811i − 0.485214i 0 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.6 0.929293 − 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i 0.541640 − 0.312716i 2.60903i 0 3.05665i −1.45586 − 2.52163i 0.335561 − 0.581209i
116.7 1.84073 − 1.06275i −0.0894272 + 0.154892i 1.25885 − 2.18040i 3.12291 − 1.80301i 0.380153i 0 − 1.10038i 1.48401 + 2.57037i 3.83229 − 6.63772i
116.8 1.97871 − 1.14241i 1.57521 − 2.72835i 1.61019 − 2.78892i −1.84030 + 1.06250i − 7.19813i 0 − 2.78832i −3.46258 − 5.99736i −2.42760 + 4.20473i
324.1 −1.97871 − 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i 1.84030 + 1.06250i − 7.19813i 0 − 2.78832i −3.46258 + 5.99736i −2.42760 − 4.20473i
324.2 −1.84073 − 1.06275i −0.0894272 − 0.154892i 1.25885 + 2.18040i −3.12291 − 1.80301i 0.380153i 0 − 1.10038i 1.48401 − 2.57037i 3.83229 + 6.63772i
324.3 −0.929293 − 0.536527i −1.21570 − 2.10566i −0.424277 − 0.734868i −0.541640 − 0.312716i 2.60903i 0 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.4 −0.287846 − 0.166188i 0.729919 + 1.26426i −0.944763 − 1.63638i 1.25195 + 0.722811i − 0.485214i 0 1.29278i 0.434437 − 0.752468i −0.240245 − 0.416116i
324.5 0.287846 + 0.166188i 0.729919 + 1.26426i −0.944763 − 1.63638i −1.25195 − 0.722811i 0.485214i 0 − 1.29278i 0.434437 − 0.752468i −0.240245 − 0.416116i
324.6 0.929293 + 0.536527i −1.21570 − 2.10566i −0.424277 − 0.734868i 0.541640 + 0.312716i − 2.60903i 0 − 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.7 1.84073 + 1.06275i −0.0894272 − 0.154892i 1.25885 + 2.18040i 3.12291 + 1.80301i − 0.380153i 0 1.10038i 1.48401 − 2.57037i 3.83229 + 6.63772i
324.8 1.97871 + 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i −1.84030 − 1.06250i 7.19813i 0 2.78832i −3.46258 + 5.99736i −2.42760 − 4.20473i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.8
Significant digits:
Format:

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.f 16
7.b odd 2 1 91.2.r.a ✓ 16
7.c even 3 1 637.2.c.e 8
7.c even 3 1 inner 637.2.r.f 16
7.d odd 6 1 91.2.r.a ✓ 16
7.d odd 6 1 637.2.c.f 8
13.b even 2 1 inner 637.2.r.f 16
21.c even 2 1 819.2.dl.e 16
21.g even 6 1 819.2.dl.e 16
91.b odd 2 1 91.2.r.a ✓ 16
91.i even 4 2 1183.2.e.i 16
91.r even 6 1 637.2.c.e 8
91.r even 6 1 inner 637.2.r.f 16
91.s odd 6 1 91.2.r.a ✓ 16
91.s odd 6 1 637.2.c.f 8
91.z odd 12 2 8281.2.a.cj 8
91.bb even 12 2 1183.2.e.i 16
91.bb even 12 2 8281.2.a.ck 8
273.g even 2 1 819.2.dl.e 16
273.ba even 6 1 819.2.dl.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a ✓ 16 7.b odd 2 1
91.2.r.a ✓ 16 7.d odd 6 1
91.2.r.a ✓ 16 91.b odd 2 1
91.2.r.a ✓ 16 91.s odd 6 1
637.2.c.e 8 7.c even 3 1
637.2.c.e 8 91.r even 6 1
637.2.c.f 8 7.d odd 6 1
637.2.c.f 8 91.s odd 6 1
637.2.r.f 16 1.a even 1 1 trivial
637.2.r.f 16 7.c even 3 1 inner
637.2.r.f 16 13.b even 2 1 inner
637.2.r.f 16 91.r even 6 1 inner
819.2.dl.e 16 21.c even 2 1
819.2.dl.e 16 21.g even 6 1
819.2.dl.e 16 273.g even 2 1
819.2.dl.e 16 273.ba even 6 1
1183.2.e.i 16 91.i even 4 2
1183.2.e.i 16 91.bb even 12 2
8281.2.a.cj 8 91.z odd 12 2
8281.2.a.ck 8 91.bb even 12 2

Hecke kernels

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

\( T_{2}^{16} - 11T_{2}^{14} + 85T_{2}^{12} - 334T_{2}^{10} + 952T_{2}^{8} - 1050T_{2}^{6} + 853T_{2}^{4} - 93T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 6T_{3}^{5} + 67T_{3}^{4} - 62T_{3}^{3} + 114T_{3}^{2} + 20T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 11 T^{14} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{8} - 2 T^{7} + 11 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 20 T^{14} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 52 T^{14} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{8} - 6 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 4 T^{7} + \cdots + 15129)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} - 44 T^{14} + \cdots + 10673289 \) Copy content Toggle raw display
$23$ \( (T^{8} + 6 T^{7} + 31 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 624)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1136229264 \) Copy content Toggle raw display
$37$ \( T^{16} - 120 T^{14} + \cdots + 76527504 \) Copy content Toggle raw display
$41$ \( (T^{8} + 132 T^{6} + \cdots + 292032)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots - 104)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 57728231289 \) Copy content Toggle raw display
$53$ \( (T^{8} + 10 T^{7} + \cdots + 7569)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 12487392009 \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots + 49729)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 66330457209 \) Copy content Toggle raw display
$71$ \( (T^{8} + 292 T^{6} + \cdots + 397488)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 8437677133824 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 296 T^{6} + \cdots + 5483712)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58102628210064 \) Copy content Toggle raw display
$97$ \( (T^{8} + 104 T^{6} + \cdots + 192)^{2} \) Copy content Toggle raw display
show more
show less