↑

Properties

Label 624.2.bf.g
Level $624$
Weight $2$
Character orbit 624.bf
Analytic conductor $4.983$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(161,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bf (of order \(4\), 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: \(4.98266508613\)
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: \( x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 312)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{4}) q^{5}+ \cdots + ( - \beta_{15} - \beta_{13} - \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_{4}) q^{5}+ \cdots + (4 \beta_{13} + \beta_{11} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 24 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 24 q^{7} + 12 q^{9} + 8 q^{19} + 4 q^{21} + 8 q^{27} - 8 q^{31} - 20 q^{33} + 48 q^{37} + 8 q^{39} - 28 q^{45} - 48 q^{55} + 16 q^{57} - 48 q^{61} - 8 q^{63} + 16 q^{67} - 8 q^{73} + 80 q^{79} + 36 q^{81} - 24 q^{85} - 16 q^{87} + 48 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{13} + 5 x^{12} - 4 x^{11} + 8 x^{10} - 16 x^{9} + 28 x^{8} - 32 x^{7} + 32 x^{6} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{15} - 11 \nu^{14} + 32 \nu^{13} - 52 \nu^{12} + 103 \nu^{11} - 155 \nu^{10} + 276 \nu^{9} + \cdots - 768 ) / 1088 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{15} - 4 \nu^{14} - 10 \nu^{13} - 22 \nu^{12} + 39 \nu^{11} - 44 \nu^{10} + 54 \nu^{9} + \cdots + 1600 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{15} + 4 \nu^{14} - 24 \nu^{13} + 22 \nu^{12} - 39 \nu^{11} + 44 \nu^{10} - 88 \nu^{9} + \cdots + 576 ) / 1088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{15} + 2 \nu^{14} + 6 \nu^{12} - 21 \nu^{11} + 30 \nu^{10} - 48 \nu^{9} + 66 \nu^{8} + \cdots + 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29 \nu^{15} + 13 \nu^{14} - 44 \nu^{13} - 56 \nu^{12} + 73 \nu^{11} + 109 \nu^{10} - 320 \nu^{9} + \cdots + 9216 ) / 3264 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15 \nu^{15} + 22 \nu^{14} - 64 \nu^{13} + 155 \nu^{12} - 325 \nu^{11} + 650 \nu^{10} - 892 \nu^{9} + \cdots + 4256 ) / 1632 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{15} - 4 \nu^{14} + 4 \nu^{13} - 8 \nu^{12} + 19 \nu^{11} - 20 \nu^{10} + 28 \nu^{9} + \cdots + 448 ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47 \nu^{15} - 214 \nu^{14} + 366 \nu^{13} - 480 \nu^{12} + 939 \nu^{11} - 1878 \nu^{10} + 2838 \nu^{9} + \cdots - 8512 ) / 3264 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56 \nu^{15} - 47 \nu^{14} + 10 \nu^{13} - 80 \nu^{12} + 352 \nu^{11} - 347 \nu^{10} + 286 \nu^{9} + \cdots + 3840 ) / 3264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67 \nu^{15} - 62 \nu^{14} - 36 \nu^{13} - 120 \nu^{12} + 375 \nu^{11} - 138 \nu^{10} - 132 \nu^{9} + \cdots + 14464 ) / 3264 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37 \nu^{15} + 69 \nu^{14} - 40 \nu^{13} + 116 \nu^{12} - 235 \nu^{11} + 317 \nu^{10} - 328 \nu^{9} + \cdots - 5024 ) / 1632 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79 \nu^{15} + 66 \nu^{14} - 22 \nu^{13} + 176 \nu^{12} - 295 \nu^{11} + 182 \nu^{10} + \cdots - 16064 ) / 3264 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60 \nu^{15} - 37 \nu^{14} - 50 \nu^{13} - 110 \nu^{12} + 280 \nu^{11} - 101 \nu^{10} - 206 \nu^{9} + \cdots + 12352 ) / 1632 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 120 \nu^{15} - 23 \nu^{14} - 100 \nu^{13} - 220 \nu^{12} + 356 \nu^{11} + 53 \nu^{10} - 616 \nu^{9} + \cdots + 24704 ) / 3264 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 159 \nu^{15} + 206 \nu^{14} - 80 \nu^{13} + 436 \nu^{12} - 1235 \nu^{11} + 1450 \nu^{10} + \cdots - 18752 ) / 3264 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - 2\beta_{14} + 3\beta_{13} + 3\beta_{7} - \beta_{6} + \beta_{3} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} + \beta_{13} - \beta_{9} + \beta_{5} + \beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{11} + 5 \beta_{10} + 3 \beta_{8} - 4 \beta_{5} - \beta_{4} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} - \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 4 \beta_{12} + 4 \beta_{9} - 9 \beta_{7} + 3 \beta_{6} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{14} + \beta_{13} + 2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10 \beta_{15} - 10 \beta_{14} - 16 \beta_{12} - 4 \beta_{11} - 13 \beta_{10} - 16 \beta_{9} + \cdots - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{15} - 10 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} + 11 \beta_{10} - 3 \beta_{8} - 3 \beta_{7} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 22 \beta_{15} + 22 \beta_{14} - 27 \beta_{13} + 28 \beta_{12} + 8 \beta_{11} - 28 \beta_{9} + \cdots - 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7 \beta_{14} - 11 \beta_{13} - 4 \beta_{10} + 3 \beta_{9} + 4 \beta_{8} - 4 \beta_{7} + 24 \beta_{6} + \cdots + 29 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14 \beta_{15} - 14 \beta_{14} + 24 \beta_{12} + 12 \beta_{11} + 45 \beta_{10} + 24 \beta_{9} + \cdots - 69 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 73 \beta_{15} - 36 \beta_{13} + 7 \beta_{12} + 19 \beta_{11} - 37 \beta_{10} - 19 \beta_{8} + \cdots + 51 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30 \beta_{15} - 30 \beta_{14} - 69 \beta_{13} - 148 \beta_{12} + 32 \beta_{11} + 148 \beta_{9} + \cdots + 83 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 47 \beta_{14} + 21 \beta_{13} - 26 \beta_{10} + 13 \beta_{9} - 54 \beta_{8} + 54 \beta_{7} + \cdots + 99 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 70 \beta_{15} + 70 \beta_{14} - 84 \beta_{11} - 45 \beta_{10} + 21 \beta_{8} + 84 \beta_{5} - 231 \beta_{4} + \cdots + 429 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.859879 − 1.12277i
−1.12277 + 0.859879i
1.39116 + 0.254324i
0.254324 + 1.39116i
−0.662534 + 1.24942i
1.24942 − 0.662534i
−1.15878 − 0.810702i
−0.810702 − 1.15878i
−1.12277 − 0.859879i
0.859879 + 1.12277i
0.254324 − 1.39116i
1.39116 − 0.254324i
1.24942 + 0.662534i
−0.662534 − 1.24942i
−0.810702 + 1.15878i
−1.15878 + 0.810702i
0 −1.63905 − 0.559912i 0 2.68975 + 2.68975i 0 1.79184 + 1.79184i 0 2.37300 + 1.83545i 0
161.2 0 −1.63905 + 0.559912i 0 −2.68975 − 2.68975i 0 1.79184 + 1.79184i 0 2.37300 − 1.83545i 0
161.3 0 −1.34162 − 1.09547i 0 0.429726 + 0.429726i 0 −0.549230 − 0.549230i 0 0.599886 + 2.93941i 0
161.4 0 −1.34162 + 1.09547i 0 −0.429726 − 0.429726i 0 −0.549230 − 0.549230i 0 0.599886 − 2.93941i 0
161.5 0 0.265070 − 1.71165i 0 −1.20485 − 1.20485i 0 3.42064 + 3.42064i 0 −2.85948 − 0.907412i 0
161.6 0 0.265070 + 1.71165i 0 1.20485 + 1.20485i 0 3.42064 + 3.42064i 0 −2.85948 + 0.907412i 0
161.7 0 1.71560 − 0.238125i 0 0.359033 + 0.359033i 0 1.33676 + 1.33676i 0 2.88659 − 0.817056i 0
161.8 0 1.71560 + 0.238125i 0 −0.359033 − 0.359033i 0 1.33676 + 1.33676i 0 2.88659 + 0.817056i 0
593.1 0 −1.63905 − 0.559912i 0 −2.68975 + 2.68975i 0 1.79184 − 1.79184i 0 2.37300 + 1.83545i 0
593.2 0 −1.63905 + 0.559912i 0 2.68975 − 2.68975i 0 1.79184 − 1.79184i 0 2.37300 − 1.83545i 0
593.3 0 −1.34162 − 1.09547i 0 −0.429726 + 0.429726i 0 −0.549230 + 0.549230i 0 0.599886 + 2.93941i 0
593.4 0 −1.34162 + 1.09547i 0 0.429726 − 0.429726i 0 −0.549230 + 0.549230i 0 0.599886 − 2.93941i 0
593.5 0 0.265070 − 1.71165i 0 1.20485 − 1.20485i 0 3.42064 − 3.42064i 0 −2.85948 − 0.907412i 0
593.6 0 0.265070 + 1.71165i 0 −1.20485 + 1.20485i 0 3.42064 − 3.42064i 0 −2.85948 + 0.907412i 0
593.7 0 1.71560 − 0.238125i 0 −0.359033 + 0.359033i 0 1.33676 − 1.33676i 0 2.88659 − 0.817056i 0
593.8 0 1.71560 + 0.238125i 0 0.359033 − 0.359033i 0 1.33676 − 1.33676i 0 2.88659 + 0.817056i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bf.g 16
3.b odd 2 1 inner 624.2.bf.g 16
4.b odd 2 1 312.2.x.c ✓ 16
12.b even 2 1 312.2.x.c ✓ 16
13.d odd 4 1 inner 624.2.bf.g 16
39.f even 4 1 inner 624.2.bf.g 16
52.f even 4 1 312.2.x.c ✓ 16
156.l odd 4 1 312.2.x.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.x.c ✓ 16 4.b odd 2 1
312.2.x.c ✓ 16 12.b even 2 1
312.2.x.c ✓ 16 52.f even 4 1
312.2.x.c ✓ 16 156.l odd 4 1
624.2.bf.g 16 1.a even 1 1 trivial
624.2.bf.g 16 3.b odd 2 1 inner
624.2.bf.g 16 13.d odd 4 1 inner
624.2.bf.g 16 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{16} + 218T_{5}^{12} + 1809T_{5}^{8} + 360T_{5}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 12T_{7}^{7} + 72T_{7}^{6} - 224T_{7}^{5} + 397T_{7}^{4} - 292T_{7}^{3} + 8T_{7}^{2} + 72T_{7} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 2 T^{7} - T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 218 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{8} - 12 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 1040 T^{12} + \cdots + 1679616 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 62 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + \cdots + 576)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 60 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 124 T^{6} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 24 T^{7} + \cdots + 114244)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7376134537216 \) Copy content Toggle raw display
$43$ \( (T^{8} + 134 T^{6} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 290258027536 \) Copy content Toggle raw display
$53$ \( (T^{8} + 132 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7676563456 \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + \cdots + 2528)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 8 T^{7} + \cdots + 6411024)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 982045424460816 \) Copy content Toggle raw display
$73$ \( (T^{8} + 4 T^{7} + \cdots + 7884864)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots - 1216)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 75391979776 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 340609761939456 \) Copy content Toggle raw display
$97$ \( (T^{8} - 24 T^{7} + \cdots + 4717584)^{2} \) Copy content Toggle raw display
show more
show less