↑

Properties

Label 1728.4.c.j
Level $1728$
Weight $4$
Character orbit 1728.c
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
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] = [1728,4,Mod(1727,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1727");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), 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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 108)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} - \beta_1 q^{7} - \beta_{6} q^{11} + ( - \beta_{5} + 6) q^{13} + \beta_{10} q^{17} + (\beta_{9} - \beta_1) q^{19} + ( - \beta_{8} + \beta_{6} + 2 \beta_{3}) q^{23} + (2 \beta_{5} - \beta_{4} - 32) q^{25} + (\beta_{10} - \beta_{7} - 3 \beta_{2}) q^{29} + ( - \beta_{11} - \beta_{9} - 4 \beta_1) q^{31} + ( - 2 \beta_{8} + 2 \beta_{6} + 5 \beta_{3}) q^{35} + ( - 5 \beta_{5} + 20) q^{37} + ( - \beta_{10} - 4 \beta_{7} - 2 \beta_{2}) q^{41} + ( - 2 \beta_{11} + 2 \beta_{9} + 8 \beta_1) q^{43} + (2 \beta_{8} - 6 \beta_{6} + 2 \beta_{3}) q^{47} + (4 \beta_{5} + \beta_{4} + 24) q^{49} + (3 \beta_{10} - 5 \beta_{7} + 18 \beta_{2}) q^{53} + ( - 2 \beta_{11} + 2 \beta_{9} - 13 \beta_1) q^{55} + (8 \beta_{8} + 6 \beta_{6}) q^{59} + (6 \beta_{5} + 2 \beta_{4} - 12) q^{61} + (3 \beta_{10} + 2 \beta_{7} - 26 \beta_{2}) q^{65} - 22 \beta_1 q^{67} + (7 \beta_{8} + 7 \beta_{6} + 10 \beta_{3}) q^{71} + ( - 8 \beta_{5} + \beta_{4} + 13) q^{73} + ( - 4 \beta_{10} + 14 \beta_{7} - 27 \beta_{2}) q^{77} + (\beta_{11} + 3 \beta_{9} + 29 \beta_1) q^{79} + ( - 6 \beta_{8} - 13 \beta_{6} + 32 \beta_{3}) q^{83} + (11 \beta_{5} - 2 \beta_{4} + 14) q^{85} + ( - 4 \beta_{10} + 14 \beta_{7} + 64 \beta_{2}) q^{89} + ( - 2 \beta_{11} - 3 \beta_{9} - 31 \beta_1) q^{91} + ( - 19 \beta_{8} + 11 \beta_{6} + 50 \beta_{3}) q^{95} + ( - 8 \beta_{5} + 4 \beta_{4} + 43) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 72 q^{13} - 384 q^{25} + 240 q^{37} + 288 q^{49} - 144 q^{61} + 156 q^{73} + 168 q^{85} + 516 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + \cdots + 9496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + \cdots - 3774567143 ) / 3046053151 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 454920367261 \nu^{11} + 699408466494 \nu^{10} + 909094946747 \nu^{9} + \cdots - 873663409442004 ) / 449256287134688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2201987655 \nu^{11} + 7037103366 \nu^{10} + 321745311 \nu^{9} + 50820234894 \nu^{8} + \cdots + 33640324703676 ) / 1072210709152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3668351532 \nu^{11} - 35323691472 \nu^{10} - 36152667432 \nu^{9} - 188138665632 \nu^{8} + \cdots - 171807057509552 ) / 1276296270269 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9151812708 \nu^{11} + 12155753232 \nu^{10} + 35817493464 \nu^{9} + 135939078600 \nu^{8} + \cdots + 120796778623888 ) / 1276296270269 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3222436257455 \nu^{11} + 5388050787734 \nu^{10} + 2723330832455 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 449256287134688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18642750728255 \nu^{11} - 54280349116842 \nu^{10} - 38930579155849 \nu^{9} + \cdots - 45\!\cdots\!68 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4893277942309 \nu^{11} + 8469535734322 \nu^{10} - 13632883283795 \nu^{9} + \cdots + 41\!\cdots\!12 ) / 449256287134688 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 82140767042 \nu^{11} + 128260169172 \nu^{10} + 66239186292 \nu^{9} - 613299928464 \nu^{8} + \cdots + 579514515914887 ) / 6381481351345 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24096087148205 \nu^{11} - 30785315288286 \nu^{10} - 65693209087147 \nu^{9} + \cdots + 49\!\cdots\!76 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 426968239932 \nu^{11} + 457010627112 \nu^{10} + 737602108632 \nu^{9} + \cdots + 749667835341342 ) / 6381481351345 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{10} + 2\beta_{7} - \beta_{5} + \beta_{4} + 8\beta_{3} - 14\beta_{2} ) / 144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 6 \beta_{6} - \beta_{5} + \cdots - 72 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 9 \beta_{8} + 17 \beta_{7} + 18 \beta_{6} - 8 \beta_{5} + \cdots + 864 ) / 288 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{11} + 8 \beta_{10} - 28 \beta_{9} + 21 \beta_{8} + 7 \beta_{7} - 96 \beta_{6} + 41 \beta_{5} + \cdots - 3288 ) / 144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 55 \beta_{11} + 122 \beta_{10} - 130 \beta_{9} + 240 \beta_{8} - 152 \beta_{7} + 150 \beta_{6} + \cdots - 2880 ) / 288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 93 \beta_{11} - 480 \beta_{10} + 408 \beta_{9} + 711 \beta_{8} + 795 \beta_{7} - 936 \beta_{6} + \cdots - 44208 ) / 288 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 469 \beta_{11} + 714 \beta_{10} + 154 \beta_{9} + 1554 \beta_{8} - 2454 \beta_{7} - 4242 \beta_{6} + \cdots - 149184 ) / 288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 634 \beta_{11} - 800 \beta_{10} + 1114 \beta_{9} + 654 \beta_{8} + 470 \beta_{7} + 2256 \beta_{6} + \cdots + 50628 ) / 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 8343 \beta_{11} - 13446 \beta_{10} + 17118 \beta_{9} - 6588 \beta_{8} + 11856 \beta_{7} + \cdots - 632448 ) / 288 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14603 \beta_{11} + 22552 \beta_{10} + 3248 \beta_{9} - 42141 \beta_{8} - 81157 \beta_{7} + \cdots + 3303696 ) / 288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 24805 \beta_{11} - 117994 \beta_{10} + 96910 \beta_{9} - 176418 \beta_{8} + 208474 \beta_{7} + \cdots + 15174720 ) / 288 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1727.1
0.886307 − 1.60260i
2.61836 − 1.60260i
0.433705 − 1.36719i
−1.29835 − 1.36719i
−0.453986 − 2.07664i
−2.18604 − 2.07664i
−2.18604 + 2.07664i
−0.453986 + 2.07664i
−1.29835 + 1.36719i
0.433705 + 1.36719i
2.61836 + 1.60260i
0.886307 + 1.60260i
0 0 0 − 20.8488i 0 − 13.9048i 0 0 0
1727.2 0 0 0 − 20.8488i 0 13.9048i 0 0 0
1727.3 0 0 0 − 5.83890i 0 − 8.83113i 0 0 0
1727.4 0 0 0 − 5.83890i 0 8.83113i 0 0 0
1727.5 0 0 0 − 1.49508i 0 − 26.1852i 0 0 0
1727.6 0 0 0 − 1.49508i 0 26.1852i 0 0 0
1727.7 0 0 0 1.49508i 0 − 26.1852i 0 0 0
1727.8 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.9 0 0 0 5.83890i 0 − 8.83113i 0 0 0
1727.10 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.11 0 0 0 20.8488i 0 − 13.9048i 0 0 0
1727.12 0 0 0 20.8488i 0 13.9048i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.c.j 12
3.b odd 2 1 inner 1728.4.c.j 12
4.b odd 2 1 inner 1728.4.c.j 12
8.b even 2 1 108.4.b.b ✓ 12
8.d odd 2 1 108.4.b.b ✓ 12
12.b even 2 1 inner 1728.4.c.j 12
24.f even 2 1 108.4.b.b ✓ 12
24.h odd 2 1 108.4.b.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b ✓ 12 8.b even 2 1
108.4.b.b ✓ 12 8.d odd 2 1
108.4.b.b ✓ 12 24.f even 2 1
108.4.b.b ✓ 12 24.h odd 2 1
1728.4.c.j 12 1.a even 1 1 trivial
1728.4.c.j 12 3.b odd 2 1 inner
1728.4.c.j 12 4.b odd 2 1 inner
1728.4.c.j 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{6} + 471T_{5}^{4} + 15867T_{5}^{2} + 33125 \) Copy content Toggle raw display
\( T_{7}^{6} + 957T_{7}^{4} + 201123T_{7}^{2} + 10338975 \) Copy content Toggle raw display
\( T_{11}^{6} - 4929T_{11}^{4} + 6225579T_{11}^{2} - 2112318675 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 471 T^{4} + \cdots + 33125)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 957 T^{4} + \cdots + 10338975)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4929 T^{4} + \cdots - 2112318675)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 18 T^{2} + \cdots + 70760)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 17532 T^{4} + \cdots + 42531205952)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 1093873434624)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 20688 T^{4} + \cdots - 15885545472)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 28380 T^{4} + \cdots + 251748219200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 37183780589679)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 60 T^{2} + \cdots + 8168000)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 223920 T^{4} + \cdots + 855375564800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 62306621744832)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 820610118691973)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 36 T^{2} + \cdots + 68226752)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 467529795115200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 39 T^{2} + \cdots + 14711755)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 49\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 39\!\cdots\!43)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 129 T^{2} + \cdots - 298155155)^{4} \) Copy content Toggle raw display
show more
show less