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Properties

Label 644.2.i.b
Level $644$
Weight $2$
Character orbit 644.i
Analytic conductor $5.142$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [644,2,Mod(93,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.i (of order \(3\), 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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 13 x^{12} - 4 x^{11} + 80 x^{10} - 16 x^{9} + 292 x^{8} + 112 x^{7} + 535 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_1 + 1) q^{3} + ( - \beta_{5} - \beta_{3}) q^{5} - \beta_{12} q^{7} + ( - \beta_{11} - \beta_{7} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_1 + 1) q^{3} + ( - \beta_{5} - \beta_{3}) q^{5} - \beta_{12} q^{7} + ( - \beta_{11} - \beta_{7} + \cdots - \beta_1) q^{9}+ \cdots + ( - 2 \beta_{10} - \beta_{9} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 5 q^{3} + 2 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 5 q^{3} + 2 q^{5} + 4 q^{7} - 4 q^{9} - 4 q^{11} - 20 q^{13} + 2 q^{15} + 6 q^{17} + 9 q^{19} - 14 q^{21} - 7 q^{23} - 11 q^{25} - 16 q^{27} + 14 q^{29} + 8 q^{31} - 7 q^{33} - 35 q^{35} + 9 q^{37} - 8 q^{39} - 18 q^{41} + 18 q^{43} + 8 q^{45} - 16 q^{49} + 15 q^{51} + 6 q^{53} - 46 q^{55} + 48 q^{57} + 19 q^{59} + 10 q^{61} - 38 q^{63} + 31 q^{65} - 10 q^{67} - 10 q^{69} + 16 q^{71} + q^{73} + 34 q^{75} - 26 q^{77} - 2 q^{79} + 5 q^{81} - 38 q^{83} + 66 q^{85} + 29 q^{87} - 15 q^{89} - 28 q^{91} + 29 q^{93} - 24 q^{95} - 34 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 13 x^{12} - 4 x^{11} + 80 x^{10} - 16 x^{9} + 292 x^{8} + 112 x^{7} + 535 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2298936 \nu^{13} - 54125220 \nu^{12} + 172831728 \nu^{11} - 712252188 \nu^{10} + \cdots - 3726614697 ) / 12024825729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 55717733 \nu^{13} - 1667790750 \nu^{12} + 3592180188 \nu^{11} - 19178672435 \nu^{10} + \cdots - 9522222246 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 631991039 \nu^{13} + 983555507 \nu^{12} + 5079658985 \nu^{11} + 23424484924 \nu^{10} + \cdots + 121442749038 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 908925513 \nu^{13} + 3364479626 \nu^{12} - 15351112964 \nu^{11} + 24643779763 \nu^{10} + \cdots + 59766109146 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 147535743 \nu^{13} - 330600910 \nu^{12} + 1830790024 \nu^{11} - 794372900 \nu^{10} + \cdots - 6794023518 ) / 5153496741 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1242204899 \nu^{13} + 2477512990 \nu^{12} - 15986288027 \nu^{11} + 4450324412 \nu^{10} + \cdots + 27666677133 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2452861130 \nu^{13} + 3738364095 \nu^{12} - 29800169145 \nu^{11} - 4917001732 \nu^{10} + \cdots - 24711042249 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2770945854 \nu^{13} - 7690329137 \nu^{12} + 38703091892 \nu^{11} - 35862162877 \nu^{10} + \cdots - 207954305388 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3672513749 \nu^{13} + 6914944350 \nu^{12} - 45927108805 \nu^{11} + 8065002195 \nu^{10} + \cdots + 149891469003 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4365502544 \nu^{13} - 6611359123 \nu^{12} + 53557018250 \nu^{11} + 7936755124 \nu^{10} + \cdots + 27262873548 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6696050275 \nu^{13} - 12125368893 \nu^{12} + 84462354700 \nu^{11} - 10632663281 \nu^{10} + \cdots - 21217413375 ) / 36074477187 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8439307977 \nu^{13} + 15855592437 \nu^{12} - 106729565695 \nu^{11} + 18889425532 \nu^{10} + \cdots + 147760300404 ) / 36074477187 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - 3\beta_{7} + \beta_{4} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{12} - 2\beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{7} - \beta_{4} - 2\beta_{3} + 6\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} + 7 \beta_{12} - 2 \beta_{11} + 9 \beta_{10} - \beta_{9} - 2 \beta_{8} + 9 \beta_{7} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{13} - 12 \beta_{12} + 12 \beta_{11} + 12 \beta_{10} - 6 \beta_{8} + 28 \beta_{7} + \cdots - 42 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 85 \beta_{12} + 85 \beta_{11} - 54 \beta_{10} + 19 \beta_{9} + 85 \beta_{7} + 29 \beta_{6} + \cdots + 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 94 \beta_{13} - 125 \beta_{12} + 125 \beta_{11} - 250 \beta_{10} + 94 \beta_{9} + 90 \beta_{8} + \cdots + 175 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 235 \beta_{13} + 360 \beta_{12} - 461 \beta_{11} - 360 \beta_{10} + 329 \beta_{8} - 1129 \beta_{7} + \cdots + 886 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2501 \beta_{12} - 2501 \beta_{11} + 1246 \beta_{10} - 895 \beta_{9} - 2501 \beta_{7} - 1029 \beta_{6} + \cdots - 1765 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2530 \beta_{13} + 4192 \beta_{12} - 3785 \beta_{11} + 7977 \beta_{10} - 2530 \beta_{9} - 3434 \beta_{8} + \cdots - 6017 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8626 \beta_{13} - 12411 \beta_{12} + 12225 \beta_{11} + 12411 \beta_{10} - 10749 \beta_{8} + \cdots - 27033 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 77614 \beta_{12} + 77614 \beta_{11} - 39444 \beta_{10} + 25759 \beta_{9} + 77614 \beta_{7} + \cdots + 56375 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 83641 \beta_{13} - 119255 \beta_{12} + 121811 \beta_{11} - 241066 \beta_{10} + 83641 \beta_{9} + \cdots + 169708 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/644\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\) \(323\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
1.55959 + 2.70129i
1.10426 + 1.91264i
0.300709 + 0.520843i
0.162655 + 0.281726i
−0.378805 − 0.656110i
−0.701808 − 1.21557i
−1.04660 − 1.81277i
1.55959 − 2.70129i
1.10426 − 1.91264i
0.300709 − 0.520843i
0.162655 − 0.281726i
−0.378805 + 0.656110i
−0.701808 + 1.21557i
−1.04660 + 1.81277i
0 −1.05959 − 1.83526i 0 1.13321 − 1.96278i 0 1.17184 − 2.37209i 0 −0.745456 + 1.29117i 0
93.2 0 −0.604263 − 1.04661i 0 −1.05231 + 1.82265i 0 1.38684 + 2.25315i 0 0.769732 − 1.33322i 0
93.3 0 0.199291 + 0.345183i 0 0.230146 − 0.398624i 0 −1.35000 + 2.27541i 0 1.42057 − 2.46049i 0
93.4 0 0.337345 + 0.584299i 0 −0.432215 + 0.748619i 0 −0.677526 − 2.55753i 0 1.27240 − 2.20386i 0
93.5 0 0.878805 + 1.52214i 0 2.18670 − 3.78748i 0 −2.52091 − 0.803133i 0 −0.0445972 + 0.0772446i 0
93.6 0 1.20181 + 2.08159i 0 −1.85930 + 3.22040i 0 2.54617 + 0.719026i 0 −1.38868 + 2.40527i 0
93.7 0 1.54660 + 2.67879i 0 0.793766 − 1.37484i 0 1.44359 + 2.21722i 0 −3.28396 + 5.68798i 0
277.1 0 −1.05959 + 1.83526i 0 1.13321 + 1.96278i 0 1.17184 + 2.37209i 0 −0.745456 − 1.29117i 0
277.2 0 −0.604263 + 1.04661i 0 −1.05231 − 1.82265i 0 1.38684 − 2.25315i 0 0.769732 + 1.33322i 0
277.3 0 0.199291 − 0.345183i 0 0.230146 + 0.398624i 0 −1.35000 − 2.27541i 0 1.42057 + 2.46049i 0
277.4 0 0.337345 − 0.584299i 0 −0.432215 − 0.748619i 0 −0.677526 + 2.55753i 0 1.27240 + 2.20386i 0
277.5 0 0.878805 − 1.52214i 0 2.18670 + 3.78748i 0 −2.52091 + 0.803133i 0 −0.0445972 − 0.0772446i 0
277.6 0 1.20181 − 2.08159i 0 −1.85930 − 3.22040i 0 2.54617 − 0.719026i 0 −1.38868 − 2.40527i 0
277.7 0 1.54660 − 2.67879i 0 0.793766 + 1.37484i 0 1.44359 − 2.21722i 0 −3.28396 − 5.68798i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.i.b ✓ 14
7.c even 3 1 inner 644.2.i.b ✓ 14
7.c even 3 1 4508.2.a.k 7
7.d odd 6 1 4508.2.a.n 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.i.b ✓ 14 1.a even 1 1 trivial
644.2.i.b ✓ 14 7.c even 3 1 inner
4508.2.a.k 7 7.c even 3 1
4508.2.a.n 7 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 5 T_{3}^{13} + 25 T_{3}^{12} - 58 T_{3}^{11} + 173 T_{3}^{10} - 307 T_{3}^{9} + 778 T_{3}^{8} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(644, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 5 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} + 4 T^{13} + \cdots + 2601 \) Copy content Toggle raw display
$13$ \( (T^{7} + 10 T^{6} + \cdots - 163)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 6 T^{13} + \cdots + 1929321 \) Copy content Toggle raw display
$19$ \( T^{14} - 9 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} - 7 T^{6} + \cdots + 29813)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 23481578169 \) Copy content Toggle raw display
$37$ \( T^{14} - 9 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$41$ \( (T^{7} + 9 T^{6} + \cdots - 452493)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 9 T^{6} + \cdots - 6507)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 817559649 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 141348321 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 7901965449 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 89545176081 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 32861109116521 \) Copy content Toggle raw display
$71$ \( (T^{7} - 8 T^{6} + \cdots - 244041)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 320194144449 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 46302862761 \) Copy content Toggle raw display
$83$ \( (T^{7} + 19 T^{6} + \cdots + 114671)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 51070847153769 \) Copy content Toggle raw display
$97$ \( (T^{7} + 17 T^{6} + \cdots + 290073)^{2} \) Copy content Toggle raw display
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