Properties

Label 637.2.c.f
Level $637$
Weight $2$
Character orbit 637.c
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
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(246,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.246");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.c (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 31x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_{5} q^{5} + (\beta_{5} - \beta_{3} + \beta_1) q^{6} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} - 1) q^{4} + \beta_{5} q^{5} + (\beta_{5} - \beta_{3} + \beta_1) q^{6} + \beta_{3} q^{8} + ( - \beta_{6} + \beta_{4} + 2) q^{9} + (\beta_{6} - 2 \beta_{4} - \beta_{2}) q^{10} + (\beta_{7} + \beta_1) q^{11} + (\beta_{6} - \beta_{4} + \beta_{2} - 3) q^{12} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{5} + 2 \beta_1) q^{15} + (\beta_{4} + \beta_{2}) q^{16} + ( - \beta_{6} - 1) q^{17} + ( - \beta_{7} + 3 \beta_{5} + 3 \beta_1) q^{18} + (\beta_{3} + 2 \beta_1) q^{19} + (\beta_{7} - 2 \beta_{5}) q^{20} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 2) q^{22} + ( - \beta_{4} + 1) q^{23} + (\beta_{7} - \beta_{5} - \beta_{3} - 3 \beta_1) q^{24} + ( - \beta_{6} + \beta_{4} + 2 \beta_{2}) q^{25} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{26} + ( - \beta_{6} + \beta_{4} - 2 \beta_{2} + 3) q^{27} + (3 \beta_{4} + 2 \beta_{2}) q^{29} + (\beta_{6} - 2 \beta_{4} + \beta_{2} - 6) q^{30} + ( - \beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{31} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{32} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{33} + ( - \beta_{7} + 2 \beta_{5} + \beta_{3}) q^{34} + (2 \beta_{6} - 5 \beta_{4} - 6) q^{36} + ( - \beta_{7} - \beta_{5} - \beta_{3} + \beta_1) q^{37} + (\beta_{4} + \beta_{2} - 4) q^{38} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 3) q^{39} + ( - \beta_{6} + \beta_{4} + 1) q^{40} + ( - \beta_{7} - 2 \beta_{5} + \beta_{3} - \beta_1) q^{41} + ( - \beta_{6} - 2 \beta_{4}) q^{43} + (\beta_{7} + 3 \beta_{5} + \beta_{3}) q^{44} + ( - 2 \beta_{3} + 4 \beta_1) q^{45} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{46} + ( - \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{47} + (\beta_{2} + 2) q^{48} + ( - \beta_{7} + 3 \beta_{5} + 2 \beta_{3} - \beta_1) q^{50} + (\beta_{4} - 2 \beta_{2} - 1) q^{51} + ( - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 6) q^{52} + ( - 2 \beta_{2} + 3) q^{53} + ( - \beta_{7} + 3 \beta_{5} - 2 \beta_{3} + 6 \beta_1) q^{54} + (\beta_{6} - 3 \beta_{4} - 4 \beta_{2} + 1) q^{55} + (\beta_{7} + \beta_{5} - 3 \beta_{3} - \beta_1) q^{57} + (3 \beta_{5} - \beta_{3} - 2 \beta_1) q^{58} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{59} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{60} + (2 \beta_{2} + 1) q^{61} + (2 \beta_{4} - \beta_{2} + 4) q^{62} + (\beta_{6} + 2 \beta_{4} - 2 \beta_{2} + 7) q^{64} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{2} + 4) q^{65} - \beta_{2} q^{66} + (2 \beta_{5} - 3 \beta_{3}) q^{67} + (\beta_{6} - 4 \beta_{4} - 3 \beta_{2} - 1) q^{68} + (\beta_{6} + \beta_{4} - 3) q^{69} + ( - \beta_{5} + 3 \beta_{3} + 4 \beta_1) q^{71} + ( - 3 \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{72} + (2 \beta_{7} - \beta_{5} - 2 \beta_1) q^{73} + (3 \beta_{2} - 6) q^{74} + (\beta_{6} + 2 \beta_{4}) q^{75} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{76} + (\beta_{7} - 3 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 + 3) q^{78} + (\beta_{6} - \beta_{4} - 3) q^{79} + (\beta_{7} - \beta_{5} + 2 \beta_1) q^{80} + (2 \beta_{4} - 4 \beta_{2} + 5) q^{81} + ( - \beta_{6} + 4 \beta_{4} + 4) q^{82} + (\beta_{7} - \beta_{3} - 5 \beta_1) q^{83} + ( - 3 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{85} + ( - \beta_{7} + 3 \beta_{3} + \beta_1) q^{86} + ( - \beta_{6} + 2 \beta_{2} + 8) q^{87} + ( - 2 \beta_{4} - 2 \beta_{2} - 1) q^{88} + ( - \beta_{7} + \beta_{5} - 3 \beta_{3} - 5 \beta_1) q^{89} + ( - 2 \beta_{4} + 6 \beta_{2} - 16) q^{90} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 1) q^{92} + ( - 3 \beta_{5} - 2 \beta_{3} - 6 \beta_1) q^{93} + ( - \beta_{6} + 5 \beta_{4} - \beta_{2} + 6) q^{94} + (\beta_{6} - 3 \beta_{4} - 2 \beta_{2} + 1) q^{95} + (2 \beta_{7} - 2 \beta_{5} - \beta_{3} - 5 \beta_1) q^{96} + (\beta_{7} + \beta_{3} - \beta_1) q^{97} + ( - \beta_{7} + \beta_{5} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 6 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 6 q^{4} + 12 q^{9} + 6 q^{10} - 18 q^{12} - 6 q^{13} - 2 q^{16} - 8 q^{17} - 18 q^{22} + 12 q^{23} + 6 q^{26} + 16 q^{27} - 8 q^{29} - 38 q^{30} - 28 q^{36} - 34 q^{38} - 18 q^{39} + 4 q^{40} + 8 q^{43} + 18 q^{48} - 16 q^{51} + 42 q^{52} + 20 q^{53} + 12 q^{55} + 12 q^{61} + 22 q^{62} + 44 q^{64} + 30 q^{65} - 2 q^{66} + 2 q^{68} - 28 q^{69} - 42 q^{74} - 8 q^{75} + 10 q^{78} - 20 q^{79} + 24 q^{81} + 16 q^{82} + 68 q^{87} - 4 q^{88} - 108 q^{90} + 6 q^{92} + 26 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 31x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 5\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 6\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 10\nu^{5} + 29\nu^{3} + 20\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 6\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 8\beta_{4} + 24\beta_{2} - 69 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 10\beta_{5} + 31\beta_{3} - 94\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(248\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
246.1
2.28481i
2.12549i
1.07305i
0.332375i
0.332375i
1.07305i
2.12549i
2.28481i
2.28481i 3.15042 −3.22037 2.12499i 7.19813i 0 2.78832i 6.92516 −4.85521
246.2 2.12549i −0.178854 −2.51771 3.60603i 0.380153i 0 1.10038i −2.96801 7.66457
246.3 1.07305i −2.43140 0.848553 0.625432i 2.60903i 0 3.05665i 2.91173 0.671123
246.4 0.332375i 1.45984 1.88953 1.44562i 0.485214i 0 1.29278i −0.868875 −0.480489
246.5 0.332375i 1.45984 1.88953 1.44562i 0.485214i 0 1.29278i −0.868875 −0.480489
246.6 1.07305i −2.43140 0.848553 0.625432i 2.60903i 0 3.05665i 2.91173 0.671123
246.7 2.12549i −0.178854 −2.51771 3.60603i 0.380153i 0 1.10038i −2.96801 7.66457
246.8 2.28481i 3.15042 −3.22037 2.12499i 7.19813i 0 2.78832i 6.92516 −4.85521
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 246.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{8} + 11T_{2}^{6} + 36T_{2}^{4} + 31T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 10T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 11 T^{6} + 36 T^{4} + 31 T^{2} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} - 7 T^{2} + 10 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 20 T^{6} + 103 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + 596 T^{4} + \cdots + 27 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + 28 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 4 T^{3} - 20 T^{2} - 52 T + 123)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 44 T^{6} + 540 T^{4} + \cdots + 3267 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + 5 T^{2} + 10 T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 63 T^{2} - 208 T + 624)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 80 T^{6} + 2091 T^{4} + \cdots + 33708 \) Copy content Toggle raw display
$37$ \( T^{8} + 120 T^{6} + 4347 T^{4} + \cdots + 8748 \) Copy content Toggle raw display
$41$ \( T^{8} + 132 T^{6} + 5732 T^{4} + \cdots + 292032 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 66 T^{2} - 156 T - 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 196 T^{6} + 12212 T^{4} + \cdots + 240267 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + 130 T + 87)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 188 T^{6} + 10476 T^{4} + \cdots + 111747 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} - 24 T^{2} + 94 T + 223)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 284 T^{6} + 21652 T^{4} + \cdots + 257547 \) Copy content Toggle raw display
$71$ \( T^{8} + 292 T^{6} + 19829 T^{4} + \cdots + 397488 \) Copy content Toggle raw display
$73$ \( T^{8} + 260 T^{6} + 19975 T^{4} + \cdots + 2904768 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + 9 T^{2} - 60 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 296 T^{6} + 28692 T^{4} + \cdots + 5483712 \) Copy content Toggle raw display
$89$ \( T^{8} + 440 T^{6} + 62899 T^{4} + \cdots + 7622508 \) Copy content Toggle raw display
$97$ \( T^{8} + 104 T^{6} + 2740 T^{4} + \cdots + 192 \) Copy content Toggle raw display
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