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Properties

Label 630.2.b.b
Level $630$
Weight $2$
Character orbit 630.b
Analytic conductor $5.031$
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] = [630,2,Mod(251,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("630.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.b (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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - q^{4} + q^{5} + \beta_{6} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - q^{4} + q^{5} + \beta_{6} q^{7} + \beta_{3} q^{8} - \beta_{3} q^{10} + (\beta_{6} + \beta_{5}) q^{11} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_{3} - \beta_1) q^{14} + q^{16} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{17} + (\beta_{7} - \beta_{2} + \beta_1) q^{19} - q^{20} + ( - \beta_{2} - \beta_1) q^{22} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{23} + q^{25} + (\beta_{6} - \beta_{5} + 2) q^{26} - \beta_{6} q^{28} + (\beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{29} + (\beta_{6} + \beta_{5} - 6 \beta_{3} + \cdots + \beta_1) q^{31}+ \cdots + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{5} - 4 q^{7} + 8 q^{16} - 8 q^{20} + 8 q^{25} + 8 q^{26} + 4 q^{28} - 4 q^{35} - 8 q^{37} + 8 q^{38} + 8 q^{41} - 16 q^{43} - 8 q^{46} + 40 q^{47} + 4 q^{49} - 8 q^{58} - 40 q^{62} - 8 q^{64} + 32 q^{67} - 52 q^{77} + 8 q^{79} + 8 q^{80} - 16 q^{83} + 8 q^{89} - 4 q^{91} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 205x^{4} + 540x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 12\nu^{4} + 73\nu^{3} + 156\nu^{2} - 198\nu + 216 ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 20\nu^{5} + 12\nu^{4} - 73\nu^{3} + 156\nu^{2} + 198\nu + 216 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 112\nu^{5} + 665\nu^{3} + 954\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 20\nu^{4} - 73\nu^{2} + 54 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 20\nu^{5} + 132\nu^{4} + 73\nu^{3} + 738\nu^{2} - 54\nu + 900 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} + 20\nu^{5} - 132\nu^{4} + 73\nu^{3} - 738\nu^{2} - 54\nu - 900 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 26\nu^{5} - 187\nu^{3} - 306\nu ) / 36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{2} - \beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 13\beta_{6} - 13\beta_{5} + 12\beta_{3} - 7\beta_{2} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{6} - 13\beta_{5} - 26\beta_{4} + 25\beta_{2} + 25\beta _1 + 146 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -50\beta_{7} + 163\beta_{6} + 163\beta_{5} - 228\beta_{3} + 73\beta_{2} - 73\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -187\beta_{6} + 187\beta_{5} + 326\beta_{4} - 427\beta_{2} - 427\beta _1 - 1790 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 854\beta_{7} - 2113\beta_{6} - 2113\beta_{5} + 3684\beta_{3} - 895\beta_{2} + 895\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0.916813i
2.73923i
− 3.73923i
− 1.91681i
− 0.916813i
− 2.73923i
3.73923i
1.91681i
− 1.00000i 0 −1.00000 1.00000 0 −2.56510 − 0.648285i 1.00000i 0 − 1.00000i
251.2 − 1.00000i 0 −1.00000 1.00000 0 −1.80230 + 1.93693i 1.00000i 0 − 1.00000i
251.3 − 1.00000i 0 −1.00000 1.00000 0 0.0951965 − 2.64404i 1.00000i 0 − 1.00000i
251.4 − 1.00000i 0 −1.00000 1.00000 0 2.27220 + 1.35539i 1.00000i 0 − 1.00000i
251.5 1.00000i 0 −1.00000 1.00000 0 −2.56510 + 0.648285i − 1.00000i 0 1.00000i
251.6 1.00000i 0 −1.00000 1.00000 0 −1.80230 − 1.93693i − 1.00000i 0 1.00000i
251.7 1.00000i 0 −1.00000 1.00000 0 0.0951965 + 2.64404i − 1.00000i 0 1.00000i
251.8 1.00000i 0 −1.00000 1.00000 0 2.27220 − 1.35539i − 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.b.b yes 8
3.b odd 2 1 630.2.b.a ✓ 8
4.b odd 2 1 5040.2.f.i 8
5.b even 2 1 3150.2.b.f 8
5.c odd 4 1 3150.2.d.a 8
5.c odd 4 1 3150.2.d.f 8
7.b odd 2 1 630.2.b.a ✓ 8
12.b even 2 1 5040.2.f.f 8
15.d odd 2 1 3150.2.b.e 8
15.e even 4 1 3150.2.d.c 8
15.e even 4 1 3150.2.d.d 8
21.c even 2 1 inner 630.2.b.b yes 8
28.d even 2 1 5040.2.f.f 8
35.c odd 2 1 3150.2.b.e 8
35.f even 4 1 3150.2.d.c 8
35.f even 4 1 3150.2.d.d 8
84.h odd 2 1 5040.2.f.i 8
105.g even 2 1 3150.2.b.f 8
105.k odd 4 1 3150.2.d.a 8
105.k odd 4 1 3150.2.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.b.a ✓ 8 3.b odd 2 1
630.2.b.a ✓ 8 7.b odd 2 1
630.2.b.b yes 8 1.a even 1 1 trivial
630.2.b.b yes 8 21.c even 2 1 inner
3150.2.b.e 8 15.d odd 2 1
3150.2.b.e 8 35.c odd 2 1
3150.2.b.f 8 5.b even 2 1
3150.2.b.f 8 105.g even 2 1
3150.2.d.a 8 5.c odd 4 1
3150.2.d.a 8 105.k odd 4 1
3150.2.d.c 8 15.e even 4 1
3150.2.d.c 8 35.f even 4 1
3150.2.d.d 8 15.e even 4 1
3150.2.d.d 8 35.f even 4 1
3150.2.d.f 8 5.c odd 4 1
3150.2.d.f 8 105.k odd 4 1
5040.2.f.f 8 12.b even 2 1
5040.2.f.f 8 28.d even 2 1
5040.2.f.i 8 4.b odd 2 1
5040.2.f.i 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{4} - 50T_{17}^{2} - 120T_{17} - 72 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 52 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$17$ \( (T^{4} - 50 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 108 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( T^{8} + 104 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( T^{8} + 104 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$31$ \( T^{8} + 216 T^{6} + \cdots + 1327104 \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 124)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 22 T^{2} + \cdots + 72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} - 26 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 20 T^{3} + \cdots - 12024)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 120 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$59$ \( (T^{4} - 194 T^{2} + \cdots + 5112)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 252 T^{6} + \cdots + 4981824 \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots - 4896)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 108 T^{6} + \cdots + 82944 \) Copy content Toggle raw display
$73$ \( T^{8} + 352 T^{6} + \cdots + 5992704 \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + \cdots + 14368)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{3} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + \cdots - 5832)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 568 T^{6} + \cdots + 199148544 \) Copy content Toggle raw display
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