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Properties

Label 630.2.p.d
Level $630$
Weight $2$
Character orbit 630.p
Analytic conductor $5.031$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(307,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.307");
 
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.p (of order \(4\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 44 x^{14} - 160 x^{13} + 468 x^{12} - 1060 x^{11} + 2038 x^{10} - 3208 x^{9} + \cdots + 2468 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{9} q^{4} - \beta_{12} q^{5} - \beta_1 q^{7} - \beta_{7} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_{9} q^{4} - \beta_{12} q^{5} - \beta_1 q^{7} - \beta_{7} q^{8} - \beta_{5} q^{10} + (\beta_{7} - \beta_{2}) q^{11} + (\beta_{14} + \beta_{13} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - \beta_{15} + \beta_{12} + \cdots - 2 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 16 q^{16} + 16 q^{22} + 24 q^{25} + 4 q^{28} + 40 q^{37} - 32 q^{43} + 16 q^{46} + 40 q^{58} - 64 q^{67} - 12 q^{70} + 24 q^{85} - 16 q^{88} + 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 44 x^{14} - 160 x^{13} + 468 x^{12} - 1060 x^{11} + 2038 x^{10} - 3208 x^{9} + \cdots + 2468 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 56\!\cdots\!91 \nu^{15} + \cdots + 16\!\cdots\!34 ) / 76\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!67 \nu^{15} + \cdots + 25\!\cdots\!22 ) / 36\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!01 \nu^{15} + \cdots - 17\!\cdots\!90 ) / 15\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57\!\cdots\!56 \nu^{15} + \cdots - 32\!\cdots\!70 ) / 41\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!99 \nu^{15} + \cdots - 32\!\cdots\!34 ) / 15\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79\!\cdots\!58 \nu^{15} + \cdots - 65\!\cdots\!70 ) / 41\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 72\!\cdots\!89 \nu^{15} + \cdots + 49\!\cdots\!86 ) / 36\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 45\!\cdots\!41 \nu^{15} + \cdots - 42\!\cdots\!86 ) / 20\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 359914193422 \nu^{15} - 2785448219070 \nu^{14} + 15345912920792 \nu^{13} + \cdots - 10\!\cdots\!10 ) / 16\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!37 \nu^{15} + \cdots + 21\!\cdots\!58 ) / 76\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 82\!\cdots\!27 \nu^{15} + \cdots - 27\!\cdots\!78 ) / 27\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 21\!\cdots\!32 \nu^{15} + \cdots + 37\!\cdots\!10 ) / 41\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!07 \nu^{15} + \cdots + 19\!\cdots\!38 ) / 76\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!01 \nu^{15} + \cdots + 18\!\cdots\!66 ) / 76\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!66 \nu^{15} + \cdots + 83\!\cdots\!30 ) / 41\!\cdots\!30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{6} + \beta_{4} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{13} - 2\beta_{11} + 3\beta_{9} - 2\beta_{8} + 2\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + 3 \beta_{14} - 5 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 11 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} - 3 \beta_{12} + 8 \beta_{11} + 10 \beta_{10} - 18 \beta_{9} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{15} - 10 \beta_{14} - 13 \beta_{13} + 20 \beta_{12} + 35 \beta_{11} - 15 \beta_{10} - 80 \beta_{9} + \cdots + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 64 \beta_{15} + 45 \beta_{14} - 31 \beta_{13} + 48 \beta_{12} + 20 \beta_{11} - 90 \beta_{10} + \cdots + 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 79 \beta_{15} + 110 \beta_{14} + 177 \beta_{13} + 12 \beta_{12} - 203 \beta_{11} - 21 \beta_{10} + \cdots + 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 286 \beta_{15} - 49 \beta_{14} + 673 \beta_{13} - 130 \beta_{12} - 448 \beta_{11} + 458 \beta_{10} + \cdots + 296 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 747 \beta_{15} - 494 \beta_{14} - 435 \beta_{13} - 478 \beta_{12} + 759 \beta_{11} + 975 \beta_{10} + \cdots - 998 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2042 \beta_{15} - 567 \beta_{14} - 6885 \beta_{13} - 1694 \beta_{12} + 3512 \beta_{11} + 120 \beta_{10} + \cdots - 6278 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7501 \beta_{15} - 1362 \beta_{14} - 9667 \beta_{13} + 428 \beta_{12} - 77 \beta_{11} - 4081 \beta_{10} + \cdots + 1596 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 10644 \beta_{15} - 10401 \beta_{14} + 33503 \beta_{13} + 27812 \beta_{12} - 16520 \beta_{11} + \cdots + 59104 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 68339 \beta_{15} + 3432 \beta_{14} + 128955 \beta_{13} + 56836 \beta_{12} - 23075 \beta_{11} + \cdots + 49882 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 31182 \beta_{15} + 195267 \beta_{14} + 44459 \beta_{13} - 195574 \beta_{12} + 13676 \beta_{11} + \cdots - 423790 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 302753 \beta_{15} + 401944 \beta_{14} - 629213 \beta_{13} - 939738 \beta_{12} + 132793 \beta_{11} + \cdots - 890088 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-\beta_{9}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
0.131441 − 0.491850i
0.131441 + 2.64318i
1.57567 − 1.93608i
1.57567 + 1.19896i
1.25485 − 2.48318i
−0.961955 − 0.266374i
1.25485 − 0.440733i
−0.961955 + 1.77607i
0.131441 + 0.491850i
0.131441 − 2.64318i
1.57567 + 1.93608i
1.57567 − 1.19896i
1.25485 + 2.48318i
−0.961955 + 0.266374i
1.25485 + 0.440733i
−0.961955 − 1.77607i
−0.707107 − 0.707107i 0 1.00000i −2.20245 + 0.386289i 0 −1.57763 − 2.12393i 0.707107 − 0.707107i 0 1.83051 + 1.28422i
307.2 −0.707107 − 0.707107i 0 1.00000i −1.28422 − 1.83051i 0 0.0564123 + 2.64515i 0.707107 − 0.707107i 0 −0.386289 + 2.20245i
307.3 −0.707107 − 0.707107i 0 1.00000i 1.28422 + 1.83051i 0 2.64515 + 0.0564123i 0.707107 − 0.707107i 0 0.386289 − 2.20245i
307.4 −0.707107 − 0.707107i 0 1.00000i 2.20245 − 0.386289i 0 −2.12393 − 1.57763i 0.707107 − 0.707107i 0 −1.83051 − 1.28422i
307.5 0.707107 + 0.707107i 0 1.00000i −2.20245 + 0.386289i 0 −2.12393 − 1.57763i −0.707107 + 0.707107i 0 −1.83051 − 1.28422i
307.6 0.707107 + 0.707107i 0 1.00000i −1.28422 − 1.83051i 0 2.64515 + 0.0564123i −0.707107 + 0.707107i 0 0.386289 − 2.20245i
307.7 0.707107 + 0.707107i 0 1.00000i 1.28422 + 1.83051i 0 0.0564123 + 2.64515i −0.707107 + 0.707107i 0 −0.386289 + 2.20245i
307.8 0.707107 + 0.707107i 0 1.00000i 2.20245 − 0.386289i 0 −1.57763 − 2.12393i −0.707107 + 0.707107i 0 1.83051 + 1.28422i
433.1 −0.707107 + 0.707107i 0 − 1.00000i −2.20245 − 0.386289i 0 −1.57763 + 2.12393i 0.707107 + 0.707107i 0 1.83051 − 1.28422i
433.2 −0.707107 + 0.707107i 0 − 1.00000i −1.28422 + 1.83051i 0 0.0564123 − 2.64515i 0.707107 + 0.707107i 0 −0.386289 − 2.20245i
433.3 −0.707107 + 0.707107i 0 − 1.00000i 1.28422 − 1.83051i 0 2.64515 − 0.0564123i 0.707107 + 0.707107i 0 0.386289 + 2.20245i
433.4 −0.707107 + 0.707107i 0 − 1.00000i 2.20245 + 0.386289i 0 −2.12393 + 1.57763i 0.707107 + 0.707107i 0 −1.83051 + 1.28422i
433.5 0.707107 − 0.707107i 0 − 1.00000i −2.20245 − 0.386289i 0 −2.12393 + 1.57763i −0.707107 − 0.707107i 0 −1.83051 + 1.28422i
433.6 0.707107 − 0.707107i 0 − 1.00000i −1.28422 + 1.83051i 0 2.64515 − 0.0564123i −0.707107 − 0.707107i 0 0.386289 + 2.20245i
433.7 0.707107 − 0.707107i 0 − 1.00000i 1.28422 − 1.83051i 0 0.0564123 − 2.64515i −0.707107 − 0.707107i 0 −0.386289 − 2.20245i
433.8 0.707107 − 0.707107i 0 − 1.00000i 2.20245 + 0.386289i 0 −1.57763 + 2.12393i −0.707107 − 0.707107i 0 1.83051 − 1.28422i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.p.d ✓ 16
3.b odd 2 1 inner 630.2.p.d ✓ 16
5.c odd 4 1 inner 630.2.p.d ✓ 16
7.b odd 2 1 inner 630.2.p.d ✓ 16
15.e even 4 1 inner 630.2.p.d ✓ 16
21.c even 2 1 inner 630.2.p.d ✓ 16
35.f even 4 1 inner 630.2.p.d ✓ 16
105.k odd 4 1 inner 630.2.p.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.p.d ✓ 16 1.a even 1 1 trivial
630.2.p.d ✓ 16 3.b odd 2 1 inner
630.2.p.d ✓ 16 5.c odd 4 1 inner
630.2.p.d ✓ 16 7.b odd 2 1 inner
630.2.p.d ✓ 16 15.e even 4 1 inner
630.2.p.d ✓ 16 21.c even 2 1 inner
630.2.p.d ✓ 16 35.f even 4 1 inner
630.2.p.d ✓ 16 105.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{8} + 720T_{13}^{4} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 6 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + 720 T^{4} + 1024)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 180 T^{4} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 58 T^{2} + 800)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 3856 T^{4} + 2560000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 66 T^{2} + 64)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 106 T^{2} + 800)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots + 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 112 T^{2} + 512)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T + 8)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 11520 T^{4} + 262144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 7952 T^{4} + 1048576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 202 T^{2} + 8192)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 52 T^{2} + 512)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 32)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 66 T^{2} + 64)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 69300 T^{4} + 119946304)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 84 T^{2} + 1600)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 40656 T^{4} + 400000000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 284 T^{2} + 20000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 14580 T^{4} + 419904)^{2} \) Copy content Toggle raw display
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