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Properties

Label 637.2.r.d
Level $637$
Weight $2$
Character orbit 637.r
Analytic conductor $5.086$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{6} + \cdots + \beta_{3}) q^{2} + \beta_{11} q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} + 1) q^{4} + ( - \beta_{9} - \beta_{5} + \cdots - \beta_{3}) q^{5} + ( - \beta_{9} + \beta_{7} + \beta_{4}) q^{6}+ \cdots + (2 \beta_{9} + \beta_{7} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} + 2 q^{9} - 20 q^{12} - 16 q^{13} - 8 q^{16} - 8 q^{17} + 48 q^{22} - 6 q^{23} - 8 q^{25} + 12 q^{26} + 24 q^{27} - 28 q^{29} - 16 q^{30} + 8 q^{36} - 4 q^{38} + 8 q^{39} - 20 q^{40} - 52 q^{43}+ \cdots - 58 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + \cdots + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} + \cdots + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + \cdots - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + \cdots - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} + \cdots + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + \cdots - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + \cdots - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + \cdots - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} + \cdots + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + \cdots - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} + \cdots + 612 ) / 230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(-\beta_{8}\)

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
0.550552 + 0.147520i
0.312819 − 1.16746i
1.98293 + 0.531325i
−0.531325 + 1.98293i
−1.16746 − 0.312819i
−0.147520 + 0.550552i
0.550552 − 0.147520i
0.312819 + 1.16746i
1.98293 − 0.531325i
−0.531325 − 1.98293i
−1.16746 + 0.312819i
−0.147520 − 0.550552i
−2.14878 + 1.24060i 0.837565 − 1.45071i 2.07816 − 3.59948i −0.584680 + 0.337565i 4.15633i 0 5.35026i 0.0969683 + 0.167954i 0.837565 − 1.45071i
116.2 −1.01332 + 0.585043i 0.269594 − 0.466951i −0.315449 + 0.546373i −0.399074 + 0.230406i 0.630898i 0 − 3.07838i 1.35464 + 2.34630i 0.269594 − 0.466951i
116.3 −0.596598 + 0.344446i −1.10716 + 1.91766i −0.762714 + 1.32106i 2.78368 − 1.60716i − 1.52543i 0 − 2.42864i −0.951606 − 1.64823i −1.10716 + 1.91766i
116.4 0.596598 − 0.344446i −1.10716 + 1.91766i −0.762714 + 1.32106i −2.78368 + 1.60716i 1.52543i 0 2.42864i −0.951606 − 1.64823i −1.10716 + 1.91766i
116.5 1.01332 − 0.585043i 0.269594 − 0.466951i −0.315449 + 0.546373i 0.399074 − 0.230406i − 0.630898i 0 3.07838i 1.35464 + 2.34630i 0.269594 − 0.466951i
116.6 2.14878 − 1.24060i 0.837565 − 1.45071i 2.07816 − 3.59948i 0.584680 − 0.337565i − 4.15633i 0 − 5.35026i 0.0969683 + 0.167954i 0.837565 − 1.45071i
324.1 −2.14878 − 1.24060i 0.837565 + 1.45071i 2.07816 + 3.59948i −0.584680 − 0.337565i − 4.15633i 0 − 5.35026i 0.0969683 − 0.167954i 0.837565 + 1.45071i
324.2 −1.01332 − 0.585043i 0.269594 + 0.466951i −0.315449 − 0.546373i −0.399074 − 0.230406i − 0.630898i 0 3.07838i 1.35464 − 2.34630i 0.269594 + 0.466951i
324.3 −0.596598 − 0.344446i −1.10716 − 1.91766i −0.762714 − 1.32106i 2.78368 + 1.60716i 1.52543i 0 2.42864i −0.951606 + 1.64823i −1.10716 − 1.91766i
324.4 0.596598 + 0.344446i −1.10716 − 1.91766i −0.762714 − 1.32106i −2.78368 − 1.60716i − 1.52543i 0 − 2.42864i −0.951606 + 1.64823i −1.10716 − 1.91766i
324.5 1.01332 + 0.585043i 0.269594 + 0.466951i −0.315449 − 0.546373i 0.399074 + 0.230406i 0.630898i 0 − 3.07838i 1.35464 − 2.34630i 0.269594 + 0.466951i
324.6 2.14878 + 1.24060i 0.837565 + 1.45071i 2.07816 + 3.59948i 0.584680 + 0.337565i 4.15633i 0 5.35026i 0.0969683 − 0.167954i 0.837565 + 1.45071i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.6
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.d 12
7.b odd 2 1 637.2.r.e 12
7.c even 3 1 637.2.c.d 6
7.c even 3 1 inner 637.2.r.d 12
7.d odd 6 1 91.2.c.a ✓ 6
7.d odd 6 1 637.2.r.e 12
13.b even 2 1 inner 637.2.r.d 12
21.g even 6 1 819.2.c.b 6
28.f even 6 1 1456.2.k.c 6
91.b odd 2 1 637.2.r.e 12
91.r even 6 1 637.2.c.d 6
91.r even 6 1 inner 637.2.r.d 12
91.s odd 6 1 91.2.c.a ✓ 6
91.s odd 6 1 637.2.r.e 12
91.z odd 12 1 8281.2.a.be 3
91.z odd 12 1 8281.2.a.bi 3
91.bb even 12 1 1183.2.a.h 3
91.bb even 12 1 1183.2.a.j 3
273.ba even 6 1 819.2.c.b 6
364.x even 6 1 1456.2.k.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a ✓ 6 7.d odd 6 1
91.2.c.a ✓ 6 91.s odd 6 1
637.2.c.d 6 7.c even 3 1
637.2.c.d 6 91.r even 6 1
637.2.r.d 12 1.a even 1 1 trivial
637.2.r.d 12 7.c even 3 1 inner
637.2.r.d 12 13.b even 2 1 inner
637.2.r.d 12 91.r even 6 1 inner
637.2.r.e 12 7.b odd 2 1
637.2.r.e 12 7.d odd 6 1
637.2.r.e 12 91.b odd 2 1
637.2.r.e 12 91.s odd 6 1
819.2.c.b 6 21.g even 6 1
819.2.c.b 6 273.ba even 6 1
1183.2.a.h 3 91.bb even 12 1
1183.2.a.j 3 91.bb even 12 1
1456.2.k.c 6 28.f even 6 1
1456.2.k.c 6 364.x even 6 1
8281.2.a.be 3 91.z odd 12 1
8281.2.a.bi 3 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 \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{12} - 8T_{2}^{10} + 52T_{2}^{8} - 88T_{2}^{6} + 112T_{2}^{4} - 48T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{3}^{6} + 4T_{3}^{4} - 4T_{3}^{3} + 16T_{3}^{2} - 8T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{6} + 4 T^{4} - 4 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 11 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 28 T^{10} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( (T^{6} + 8 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 1387488001 \) Copy content Toggle raw display
$23$ \( (T^{6} + 3 T^{5} + \cdots + 6241)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 7 T^{2} - 21 T + 5)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} - 83 T^{10} + \cdots + 17850625 \) Copy content Toggle raw display
$37$ \( T^{12} - 108 T^{10} + \cdots + 8503056 \) Copy content Toggle raw display
$41$ \( (T^{6} + 108 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 13 T^{2} + \cdots - 17)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 352275361 \) Copy content Toggle raw display
$53$ \( (T^{6} + T^{5} + 10 T^{4} + \cdots + 169)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 68 T^{10} + \cdots + 7311616 \) Copy content Toggle raw display
$61$ \( (T^{6} - 14 T^{5} + \cdots + 23104)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 80 T^{10} + \cdots + 65536 \) Copy content Toggle raw display
$71$ \( (T^{6} + 304 T^{4} + \cdots + 792100)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 263 T^{10} + \cdots + 923521 \) Copy content Toggle raw display
$79$ \( (T^{6} + 13 T^{5} + \cdots + 34225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 227 T^{4} + \cdots + 26569)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2655237841 \) Copy content Toggle raw display
$97$ \( (T^{6} + 575 T^{4} + \cdots + 4765489)^{2} \) Copy content Toggle raw display
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