Properties

Label 630.3.e.a
Level $630$
Weight $3$
Character orbit 630.e
Analytic conductor $17.166$
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] = [630,3,Mod(71,630)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 630.e (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: \(17.1662566547\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.98344960000.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - 2 q^{4} - \beta_{6} q^{5} - \beta_{2} q^{7} - 2 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - 2 q^{4} - \beta_{6} q^{5} - \beta_{2} q^{7} - 2 \beta_{4} q^{8} + \beta_1 q^{10} + ( - \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{7} - 2 \beta_1 - 4) q^{13} + \beta_{3} q^{14} + 4 q^{16} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{17} + ( - 4 \beta_{2} + 3 \beta_1 + 10) q^{19} + 2 \beta_{6} q^{20} + (2 \beta_{2} + 2) q^{22} + ( - 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{23} - 5 q^{25} + ( - 4 \beta_{6} + 2 \beta_{5} - 4 \beta_{4}) q^{26} + 2 \beta_{2} q^{28} + (4 \beta_{6} + 4 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{29} + ( - 2 \beta_{7} + 8 \beta_{2} - \beta_1 - 14) q^{31} + 4 \beta_{4} q^{32} + (2 \beta_{7} - 4 \beta_{2} - 2 \beta_1 - 4) q^{34} + \beta_{5} q^{35} + (6 \beta_{2} - 6 \beta_1 - 4) q^{37} + (6 \beta_{6} + 10 \beta_{4} + 4 \beta_{3}) q^{38} - 2 \beta_1 q^{40} + (6 \beta_{5} - 8 \beta_{4} + 2 \beta_{3}) q^{41} + (2 \beta_{7} - 10 \beta_{2} + 4 \beta_1 + 16) q^{43} + (2 \beta_{4} - 2 \beta_{3}) q^{44} + (4 \beta_{7} + 2 \beta_{2} + 2 \beta_1 - 4) q^{46} + ( - 6 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - 10 \beta_{3}) q^{47} + 7 q^{49} - 5 \beta_{4} q^{50} + (2 \beta_{7} + 4 \beta_1 + 8) q^{52} + ( - 10 \beta_{6} + 8 \beta_{5} - 9 \beta_{4} + 3 \beta_{3}) q^{53} + (\beta_{7} - \beta_1) q^{55} - 2 \beta_{3} q^{56} + (4 \beta_{7} - 4 \beta_{2} - 4 \beta_1 + 2) q^{58} + (2 \beta_{6} + 8 \beta_{5} - 6 \beta_{4}) q^{59} + (4 \beta_{7} + 12 \beta_{2} + 2 \beta_1 - 10) q^{61} + ( - 2 \beta_{6} + 4 \beta_{5} - 14 \beta_{4} - 8 \beta_{3}) q^{62} - 8 q^{64} + (4 \beta_{6} + 10 \beta_{4} + 5 \beta_{3}) q^{65} + (2 \beta_{2} + 18 \beta_1 + 16) q^{67} + ( - 4 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 4 \beta_{3}) q^{68} + \beta_{7} q^{70} + (20 \beta_{6} + 8 \beta_{5} + 23 \beta_{4} + 7 \beta_{3}) q^{71} + (5 \beta_{7} - 20 \beta_{2} + 10 \beta_1) q^{73} + ( - 12 \beta_{6} - 4 \beta_{4} - 6 \beta_{3}) q^{74} + (8 \beta_{2} - 6 \beta_1 - 20) q^{76} + (7 \beta_{4} - \beta_{3}) q^{77} + ( - 8 \beta_{2} - 18 \beta_1) q^{79} - 4 \beta_{6} q^{80} + (6 \beta_{7} + 4 \beta_{2} + 16) q^{82} + (10 \beta_{6} - 2 \beta_{5} - 12 \beta_{4} + 6 \beta_{3}) q^{83} + ( - 2 \beta_{7} + 10 \beta_{2} + 2 \beta_1 + 10) q^{85} + (8 \beta_{6} - 4 \beta_{5} + 16 \beta_{4} + 10 \beta_{3}) q^{86} + ( - 4 \beta_{2} - 4) q^{88} + ( - 18 \beta_{6} + 26 \beta_{4} + 16 \beta_{3}) q^{89} + ( - 2 \beta_{7} + 4 \beta_{2} - 7 \beta_1) q^{91} + (4 \beta_{6} - 8 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{92} + (6 \beta_{7} - 20 \beta_{2} + 6 \beta_1 - 8) q^{94} + ( - 10 \beta_{6} + 4 \beta_{5} - 15 \beta_{4}) q^{95} + (3 \beta_{7} + 12 \beta_{2} - 30 \beta_1 - 28) q^{97} + 7 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} - 32 q^{13} + 32 q^{16} + 80 q^{19} + 16 q^{22} - 40 q^{25} - 112 q^{31} - 32 q^{34} - 32 q^{37} + 128 q^{43} - 32 q^{46} + 56 q^{49} + 64 q^{52} + 16 q^{58} - 80 q^{61} - 64 q^{64} + 128 q^{67} - 160 q^{76} + 128 q^{82} + 80 q^{85} - 32 q^{88} - 64 q^{94} - 224 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 6x^{6} + 39x^{4} - 190x^{2} + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - \nu^{4} + 11\nu^{2} + 270 ) / 90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{7} + 19\nu^{5} + 151\nu^{3} - 1155\nu ) / 450 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 55\nu^{2} - 108 ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} + 52\nu^{5} + 358\nu^{3} - 615\nu ) / 450 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{6} - 34\nu^{4} - 196\nu^{2} + 405 ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 17\nu^{5} + 113\nu^{3} - 195\nu ) / 90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 6\nu^{5} - 24\nu^{3} + 235\nu ) / 30 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + 2\beta_{4} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 2\beta_{6} - \beta_{4} + 8\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{5} - 16\beta_{3} + 8\beta _1 - 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -15\beta_{7} + 65\beta_{6} - 97\beta_{4} - 49\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{5} + 19\beta_{3} - 83\beta _1 + 264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -42\beta_{7} - 673\beta_{6} + 1076\beta_{4} - 133\beta_{2} ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−1.32288 2.53225i
1.32288 2.53225i
−1.32288 0.296180i
1.32288 0.296180i
−1.32288 + 0.296180i
1.32288 + 0.296180i
−1.32288 + 2.53225i
1.32288 + 2.53225i
1.41421i 0 −2.00000 2.23607i 0 −2.64575 2.82843i 0 −3.16228
71.2 1.41421i 0 −2.00000 2.23607i 0 2.64575 2.82843i 0 −3.16228
71.3 1.41421i 0 −2.00000 2.23607i 0 −2.64575 2.82843i 0 3.16228
71.4 1.41421i 0 −2.00000 2.23607i 0 2.64575 2.82843i 0 3.16228
71.5 1.41421i 0 −2.00000 2.23607i 0 −2.64575 2.82843i 0 3.16228
71.6 1.41421i 0 −2.00000 2.23607i 0 2.64575 2.82843i 0 3.16228
71.7 1.41421i 0 −2.00000 2.23607i 0 −2.64575 2.82843i 0 −3.16228
71.8 1.41421i 0 −2.00000 2.23607i 0 2.64575 2.82843i 0 −3.16228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.3.e.a 8
3.b odd 2 1 inner 630.3.e.a 8
5.b even 2 1 3150.3.e.i 8
5.c odd 4 2 3150.3.c.d 16
15.d odd 2 1 3150.3.e.i 8
15.e even 4 2 3150.3.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.3.e.a 8 1.a even 1 1 trivial
630.3.e.a 8 3.b odd 2 1 inner
3150.3.c.d 16 5.c odd 4 2
3150.3.c.d 16 15.e even 4 2
3150.3.e.i 8 5.b even 2 1
3150.3.e.i 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 32T_{11}^{2} + 144 \) acting on \(S_{3}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16 T^{3} - 124 T^{2} - 1504 T - 2364)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 896 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{3} + 196 T^{2} + \cdots - 29916)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2408 T^{6} + \cdots + 75666805776 \) Copy content Toggle raw display
$29$ \( T^{8} + 2792 T^{6} + \cdots + 19925016336 \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{3} - 300 T^{2} + \cdots - 111676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} - 1128 T^{2} + \cdots - 7664)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 5776 T^{6} + \cdots + 1274875842816 \) Copy content Toggle raw display
$43$ \( (T^{4} - 64 T^{3} - 744 T^{2} + \cdots + 87056)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 11488 T^{6} + \cdots + 3582449565696 \) Copy content Toggle raw display
$53$ \( T^{8} + 12112 T^{6} + \cdots + 58107995136 \) Copy content Toggle raw display
$59$ \( T^{8} + 9328 T^{6} + \cdots + 21234991124736 \) Copy content Toggle raw display
$61$ \( (T^{4} + 40 T^{3} - 3736 T^{2} + \cdots - 42096)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 64 T^{3} - 5000 T^{2} + \cdots + 8709264)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 842087039037696 \) Copy content Toggle raw display
$73$ \( (T^{4} - 11100 T^{2} - 560000 T - 6997500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 7376 T^{2} + 7795264)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 5728 T^{6} + \cdots + 2702420361216 \) Copy content Toggle raw display
$89$ \( T^{8} + 26224 T^{6} + \cdots + 70331973325056 \) Copy content Toggle raw display
$97$ \( (T^{4} + 112 T^{3} - 16572 T^{2} + \cdots + 18658756)^{2} \) Copy content Toggle raw display
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