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Properties

Label 637.2.k.g
Level $637$
Weight $2$
Character orbit 637.k
Analytic conductor $5.086$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(459,637)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.459");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.k (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: 12.0.58891012706304.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{4}) q^{2} + (\beta_{9} + \beta_{2}) q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} + \cdots - 1) q^{4}+ \cdots + (\beta_{11} + \beta_{10} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{4}) q^{2} + (\beta_{9} + \beta_{2}) q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} + \cdots - 1) q^{4}+ \cdots + ( - 4 \beta_{11} - 4 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{4} - 6 q^{5} + 18 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{4} - 6 q^{5} + 18 q^{6} - 4 q^{9} - 12 q^{10} - 6 q^{11} - 2 q^{12} - 4 q^{13} + 6 q^{15} + 16 q^{16} - 8 q^{17} + 12 q^{18} + 12 q^{20} + 6 q^{22} + 24 q^{23} + 12 q^{24} + 10 q^{25} - 18 q^{26} - 12 q^{27} + 8 q^{29} + 8 q^{30} + 18 q^{31} - 30 q^{33} - 10 q^{36} + 2 q^{38} + 14 q^{39} + 46 q^{40} - 30 q^{41} + 2 q^{43} - 24 q^{44} + 42 q^{47} + 2 q^{48} - 18 q^{50} - 26 q^{51} + 28 q^{52} + 22 q^{53} + 6 q^{55} + 12 q^{58} - 66 q^{60} - 14 q^{61} + 4 q^{62} - 52 q^{64} - 18 q^{65} - 26 q^{66} + 24 q^{67} - 16 q^{68} - 4 q^{69} - 24 q^{71} - 60 q^{72} + 30 q^{73} - 12 q^{74} + 92 q^{75} + 18 q^{76} - 10 q^{78} + 28 q^{79} - 72 q^{80} + 2 q^{81} - 14 q^{82} - 48 q^{85} + 60 q^{86} - 4 q^{87} - 14 q^{88} - 24 q^{90} + 24 q^{92} - 4 q^{94} + 44 q^{95} + 6 q^{96} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} - 2x^{9} + 15x^{8} + 2x^{7} - 30x^{6} + 4x^{5} + 60x^{4} - 16x^{3} - 80x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{10} - 2\nu^{9} - 7\nu^{8} + 4\nu^{7} + 17\nu^{6} - 24\nu^{5} - 14\nu^{4} + 40\nu^{3} + 36\nu^{2} - 40\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} - 9 \nu^{7} + 24 \nu^{6} + 4 \nu^{5} - 44 \nu^{4} + \cdots - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} + \nu^{10} - 6 \nu^{9} - 5 \nu^{8} + 16 \nu^{7} - \nu^{6} - 30 \nu^{5} + 6 \nu^{4} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 3 \nu^{9} - 10 \nu^{8} + 9 \nu^{7} + 26 \nu^{6} - 42 \nu^{5} - 12 \nu^{4} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 6 \nu^{10} - 9 \nu^{9} + 20 \nu^{8} + 31 \nu^{7} - 56 \nu^{6} - 38 \nu^{5} + 136 \nu^{4} + \cdots + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} - 5 \nu^{9} + 13 \nu^{8} + 13 \nu^{7} - 35 \nu^{6} - 12 \nu^{5} + 70 \nu^{4} + \cdots + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 20 \nu^{8} - 27 \nu^{7} - 32 \nu^{6} + 46 \nu^{5} + \cdots + 224 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} + 3 \nu^{10} - 14 \nu^{9} - 7 \nu^{8} + 36 \nu^{7} + 5 \nu^{6} - 82 \nu^{5} + 34 \nu^{4} + \cdots + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{11} - 8 \nu^{10} - 19 \nu^{9} + 26 \nu^{8} + 41 \nu^{7} - 90 \nu^{6} - 18 \nu^{5} + 156 \nu^{4} + \cdots + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} - 12 \nu^{10} + 13 \nu^{9} + 46 \nu^{8} - 31 \nu^{7} - 102 \nu^{6} + 126 \nu^{5} + \cdots + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{10} + 8 \nu^{9} + 19 \nu^{8} - 26 \nu^{7} - 41 \nu^{6} + 90 \nu^{5} + 18 \nu^{4} - 140 \nu^{3} + \cdots - 80 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{9} - 3\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{5} - 3\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{10} + \beta_{9} + 2\beta_{8} - 4\beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} - \beta_{10} + 2\beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - 4\beta_{3} - 3\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + 7 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} + 3 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 3 \beta_{8} - 3 \beta_{7} - \beta_{5} - 7 \beta_{4} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{11} + 6 \beta_{10} - \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 3 \beta_{6} + 5 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6 \beta_{11} + 14 \beta_{10} + 5 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} - 13 \beta_{6} - 3 \beta_{5} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29 \beta_{11} - 11 \beta_{10} + 24 \beta_{9} - 20 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} - 10 \beta_{5} + \cdots - 7 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-1 - \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
459.1
−1.30089 − 0.554694i
1.40744 − 0.138282i
−1.08105 − 0.911778i
1.34408 − 0.439820i
0.759479 + 1.19298i
−1.12906 + 0.851598i
−1.12906 − 0.851598i
0.759479 − 1.19298i
1.34408 + 0.439820i
−1.08105 + 0.911778i
1.40744 + 0.138282i
−1.30089 + 0.554694i
− 2.10939i 1.13082 + 1.95864i −2.44952 3.11923 − 1.80089i 4.13154 − 2.38535i 0 0.948212i −1.05753 + 1.83169i −3.79878 − 6.57967i
459.2 − 1.27656i −0.583963 − 1.01145i 0.370384 −1.57173 + 0.907437i −1.29118 + 0.745466i 0 − 3.02595i 0.817975 − 1.41677i 1.15840 + 2.00641i
459.3 − 0.823556i 1.33015 + 2.30388i 1.32176 −2.73845 + 1.58105i 1.89737 − 1.09545i 0 − 2.73565i −2.03858 + 3.53092i 1.30208 + 2.25527i
459.4 0.120360i −0.291146 − 0.504280i 1.98551 1.46199 − 0.844083i 0.0606950 − 0.0350423i 0 0.479696i 1.33047 − 2.30444i 0.101594 + 0.175965i
459.5 1.38595i −1.41289 − 2.44719i 0.0791355 −0.449430 + 0.259479i 3.39169 − 1.95819i 0 2.88158i −2.49250 + 4.31714i −0.359625 − 0.622889i
459.6 2.70320i −0.172975 − 0.299601i −5.30727 −2.82162 + 1.62906i 0.809880 − 0.467584i 0 − 8.94020i 1.44016 − 2.49443i −4.40367 − 7.62739i
569.1 − 2.70320i −0.172975 + 0.299601i −5.30727 −2.82162 − 1.62906i 0.809880 + 0.467584i 0 8.94020i 1.44016 + 2.49443i −4.40367 + 7.62739i
569.2 − 1.38595i −1.41289 + 2.44719i 0.0791355 −0.449430 − 0.259479i 3.39169 + 1.95819i 0 − 2.88158i −2.49250 − 4.31714i −0.359625 + 0.622889i
569.3 − 0.120360i −0.291146 + 0.504280i 1.98551 1.46199 + 0.844083i 0.0606950 + 0.0350423i 0 − 0.479696i 1.33047 + 2.30444i 0.101594 − 0.175965i
569.4 0.823556i 1.33015 − 2.30388i 1.32176 −2.73845 − 1.58105i 1.89737 + 1.09545i 0 2.73565i −2.03858 − 3.53092i 1.30208 − 2.25527i
569.5 1.27656i −0.583963 + 1.01145i 0.370384 −1.57173 − 0.907437i −1.29118 − 0.745466i 0 3.02595i 0.817975 + 1.41677i 1.15840 − 2.00641i
569.6 2.10939i 1.13082 − 1.95864i −2.44952 3.11923 + 1.80089i 4.13154 + 2.38535i 0 − 0.948212i −1.05753 − 1.83169i −3.79878 + 6.57967i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k 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.k.g 12
7.b odd 2 1 637.2.k.h 12
7.c even 3 1 637.2.q.h 12
7.c even 3 1 637.2.u.i 12
7.d odd 6 1 91.2.q.a ✓ 12
7.d odd 6 1 637.2.u.h 12
13.e even 6 1 637.2.u.i 12
21.g even 6 1 819.2.ct.a 12
28.f even 6 1 1456.2.cc.c 12
91.k even 6 1 inner 637.2.k.g 12
91.l odd 6 1 637.2.k.h 12
91.l odd 6 1 1183.2.c.i 12
91.p odd 6 1 91.2.q.a ✓ 12
91.t odd 6 1 637.2.u.h 12
91.u even 6 1 637.2.q.h 12
91.v odd 6 1 1183.2.c.i 12
91.x odd 12 1 8281.2.a.by 6
91.x odd 12 1 8281.2.a.ch 6
91.ba even 12 1 1183.2.a.m 6
91.ba even 12 1 1183.2.a.p 6
273.y even 6 1 819.2.ct.a 12
364.bp even 6 1 1456.2.cc.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a ✓ 12 7.d odd 6 1
91.2.q.a ✓ 12 91.p odd 6 1
637.2.k.g 12 1.a even 1 1 trivial
637.2.k.g 12 91.k even 6 1 inner
637.2.k.h 12 7.b odd 2 1
637.2.k.h 12 91.l odd 6 1
637.2.q.h 12 7.c even 3 1
637.2.q.h 12 91.u even 6 1
637.2.u.h 12 7.d odd 6 1
637.2.u.h 12 91.t odd 6 1
637.2.u.i 12 7.c even 3 1
637.2.u.i 12 13.e even 6 1
819.2.ct.a 12 21.g even 6 1
819.2.ct.a 12 273.y even 6 1
1183.2.a.m 6 91.ba even 12 1
1183.2.a.p 6 91.ba even 12 1
1183.2.c.i 12 91.l odd 6 1
1183.2.c.i 12 91.v odd 6 1
1456.2.cc.c 12 28.f even 6 1
1456.2.cc.c 12 364.bp even 6 1
8281.2.a.by 6 91.x odd 12 1
8281.2.a.ch 6 91.x odd 12 1

Hecke kernels

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

\( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 206T_{2}^{6} + 208T_{2}^{4} + 72T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{12} + 11 T_{3}^{10} + 4 T_{3}^{9} + 96 T_{3}^{8} + 42 T_{3}^{7} + 287 T_{3}^{6} + 390 T_{3}^{5} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 11 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 3481 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 491)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 29 T^{10} + \cdots + 55696 \) Copy content Toggle raw display
$23$ \( (T^{6} - 12 T^{5} + \cdots + 6208)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 8 T^{11} + \cdots + 10042561 \) Copy content Toggle raw display
$31$ \( T^{12} - 18 T^{11} + \cdots + 913936 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1755945216 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 884705536 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + \cdots + 2408704 \) Copy content Toggle raw display
$47$ \( T^{12} - 42 T^{11} + \cdots + 9461776 \) Copy content Toggle raw display
$53$ \( T^{12} - 22 T^{11} + \cdots + 5470921 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4571923456 \) Copy content Toggle raw display
$61$ \( T^{12} + 14 T^{11} + \cdots + 5607424 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 613651984 \) Copy content Toggle raw display
$71$ \( T^{12} + 24 T^{11} + \cdots + 46895104 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1386221824 \) Copy content Toggle raw display
$79$ \( T^{12} - 28 T^{11} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 141324544 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1834580224 \) Copy content Toggle raw display
$97$ \( T^{12} + 6 T^{11} + \cdots + 53465344 \) Copy content Toggle raw display
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