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Properties

Label 624.4.bc.a
Level $624$
Weight $4$
Character orbit 624.bc
Analytic conductor $36.817$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(31,624)]
 
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([2, 0, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.bc (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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 557 x^{12} + 114776 x^{10} + 11098364 x^{8} + 523047796 x^{6} + 11529575940 x^{4} + \cdots + 338947524864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta_{5} q^{3} + \beta_{2} q^{5} + ( - \beta_{8} + \beta_{5} - 1) q^{7} - 9 q^{9} + ( - \beta_{10} + 4 \beta_{5} + \cdots - 4) q^{11} + ( - \beta_{12} - \beta_{9} + \beta_{7} + \cdots + 3) q^{13}+ \cdots + (9 \beta_{10} - 36 \beta_{5} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{5} - 12 q^{7} - 126 q^{9} - 64 q^{11} + 32 q^{13} - 6 q^{15} - 84 q^{19} + 36 q^{21} - 384 q^{23} + 32 q^{29} - 140 q^{31} + 192 q^{33} - 466 q^{37} + 270 q^{39} - 98 q^{41} - 104 q^{43} - 18 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 557 x^{12} + 114776 x^{10} + 11098364 x^{8} + 523047796 x^{6} + 11529575940 x^{4} + \cdots + 338947524864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23820957 \nu^{12} + 12418249133 \nu^{10} + 2297165746328 \nu^{8} + 185740132811984 \nu^{6} + \cdots + 21\!\cdots\!96 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23820957 \nu^{12} - 12418249133 \nu^{10} - 2297165746328 \nu^{8} - 185740132811984 \nu^{6} + \cdots - 21\!\cdots\!96 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24\!\cdots\!29 \nu^{12} + \cdots - 17\!\cdots\!12 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!41 \nu^{12} + \cdots - 62\!\cdots\!48 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20733591991 \nu^{13} + 10778145705779 \nu^{11} + \cdots - 33\!\cdots\!52 \nu ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!61 \nu^{13} + \cdots - 13\!\cdots\!08 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!61 \nu^{13} + \cdots - 14\!\cdots\!12 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!61 \nu^{13} + \cdots + 14\!\cdots\!12 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 60\!\cdots\!69 \nu^{13} + \cdots - 64\!\cdots\!24 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60\!\cdots\!69 \nu^{13} + \cdots - 64\!\cdots\!24 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18\!\cdots\!47 \nu^{13} + \cdots - 15\!\cdots\!24 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!47 \nu^{13} + \cdots - 15\!\cdots\!24 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!39 \nu^{13} + \cdots + 13\!\cdots\!92 \nu ) / 65\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + \beta_{11} + 2\beta_{10} + 2\beta_{9} - 2\beta_{4} - 3\beta_{3} - 2\beta_{2} + 2\beta _1 - 162 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24 \beta_{13} + 7 \beta_{10} - 7 \beta_{9} - 18 \beta_{8} - 18 \beta_{7} - 5 \beta_{6} + \cdots - 149 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 102 \beta_{12} - 102 \beta_{11} - 310 \beta_{10} - 310 \beta_{9} + 197 \beta_{8} - 197 \beta_{7} + \cdots + 23407 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6066 \beta_{13} + 90 \beta_{12} - 90 \beta_{11} - 1926 \beta_{10} + 1926 \beta_{9} + \cdots + 26639 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11636 \beta_{12} + 11636 \beta_{11} + 54212 \beta_{10} + 54212 \beta_{9} - 59305 \beta_{8} + \cdots - 4132881 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1418912 \beta_{13} + 5258 \beta_{12} - 5258 \beta_{11} + 472156 \beta_{10} - 472156 \beta_{9} + \cdots - 5227541 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1619790 \beta_{12} - 1619790 \beta_{11} - 10532544 \beta_{10} - 10532544 \beta_{9} + 14438927 \beta_{8} + \cdots + 805758119 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 323504552 \beta_{13} - 6177842 \beta_{12} + 6177842 \beta_{11} - 110818286 \beta_{10} + \cdots + 1081982861 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 267960706 \beta_{12} + 267960706 \beta_{11} + 2169490136 \beta_{10} + 2169490136 \beta_{9} + \cdots - 166174078035 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 72768080100 \beta_{13} + 2046011442 \beta_{12} - 2046011442 \beta_{11} + 25396419418 \beta_{10} + \cdots - 230867244761 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 50145889926 \beta_{12} - 50145889926 \beta_{11} - 461404567192 \beta_{10} - 461404567192 \beta_{9} + \cdots + 35386491191815 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 16249248303276 \beta_{13} - 534718372026 \beta_{12} + 534718372026 \beta_{11} + \cdots + 50106080971649 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
10.9706i
10.7227i
5.37090i
− 2.73810i
− 3.06419i
− 7.38006i
− 14.8819i
− 10.9706i
− 10.7227i
− 5.37090i
2.73810i
3.06419i
7.38006i
14.8819i
0 − 3.00000i 0 −10.9706 + 10.9706i 0 7.36979 − 7.36979i 0 −9.00000 0
31.2 0 − 3.00000i 0 −10.7227 + 10.7227i 0 −11.7124 + 11.7124i 0 −9.00000 0
31.3 0 − 3.00000i 0 −5.37090 + 5.37090i 0 −4.75457 + 4.75457i 0 −9.00000 0
31.4 0 − 3.00000i 0 2.73810 − 2.73810i 0 −11.6722 + 11.6722i 0 −9.00000 0
31.5 0 − 3.00000i 0 3.06419 − 3.06419i 0 18.6330 − 18.6330i 0 −9.00000 0
31.6 0 − 3.00000i 0 7.38006 − 7.38006i 0 15.7177 − 15.7177i 0 −9.00000 0
31.7 0 − 3.00000i 0 14.8819 − 14.8819i 0 −19.5813 + 19.5813i 0 −9.00000 0
463.1 0 3.00000i 0 −10.9706 − 10.9706i 0 7.36979 + 7.36979i 0 −9.00000 0
463.2 0 3.00000i 0 −10.7227 − 10.7227i 0 −11.7124 − 11.7124i 0 −9.00000 0
463.3 0 3.00000i 0 −5.37090 − 5.37090i 0 −4.75457 − 4.75457i 0 −9.00000 0
463.4 0 3.00000i 0 2.73810 + 2.73810i 0 −11.6722 − 11.6722i 0 −9.00000 0
463.5 0 3.00000i 0 3.06419 + 3.06419i 0 18.6330 + 18.6330i 0 −9.00000 0
463.6 0 3.00000i 0 7.38006 + 7.38006i 0 15.7177 + 15.7177i 0 −9.00000 0
463.7 0 3.00000i 0 14.8819 + 14.8819i 0 −19.5813 − 19.5813i 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.bc.a ✓ 14
4.b odd 2 1 624.4.bc.b yes 14
13.d odd 4 1 624.4.bc.b yes 14
52.f even 4 1 inner 624.4.bc.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.bc.a ✓ 14 1.a even 1 1 trivial
624.4.bc.a ✓ 14 52.f even 4 1 inner
624.4.bc.b yes 14 4.b odd 2 1
624.4.bc.b yes 14 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{14} - 2 T_{5}^{13} + 2 T_{5}^{12} + 1384 T_{5}^{11} + 158624 T_{5}^{10} + 395392 T_{5}^{9} + \cdots + 43385283182592 \) Copy content Toggle raw display
\( T_{7}^{14} + 12 T_{7}^{13} + 72 T_{7}^{12} + 208 T_{7}^{11} + 738760 T_{7}^{10} + 8716608 T_{7}^{9} + \cdots + 96\!\cdots\!48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 43385283182592 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 96\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 220399211520000 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 31\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 106375863638016)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 285938882099712)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 47\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 18\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 15\!\cdots\!92)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 46\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 39\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 33\!\cdots\!72 \) Copy content Toggle raw display
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