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Properties

Label 1728.4.c.i
Level $1728$
Weight $4$
Character orbit 1728.c
Analytic conductor $101.955$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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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: \( x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{36}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} - \beta_1 q^{7} + \beta_{5} q^{11} + ( - \beta_{2} - 3) q^{13} + ( - \beta_{9} - \beta_{6}) q^{17} + (\beta_{8} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{10} - \beta_{5}) q^{23} + (\beta_{4} - 11) q^{25} + (\beta_{11} + 2 \beta_{9} - 3 \beta_{6}) q^{29} + (\beta_{8} + 2 \beta_{3} - \beta_1) q^{31} + ( - \beta_{7} + 7 \beta_{5}) q^{35} + (3 \beta_{4} + \beta_{2} - 43) q^{37} + (\beta_{11} - 2 \beta_{9} + 9 \beta_{6}) q^{41} + ( - 3 \beta_{8} + 2 \beta_{3} - 5 \beta_1) q^{43} + ( - \beta_{10} - 3 \beta_{7} - 3 \beta_{5}) q^{47} + (2 \beta_{4} - 6 \beta_{2} - 60) q^{49} + ( - \beta_{11} - 13 \beta_{6}) q^{53} + ( - 3 \beta_{8} + 8 \beta_{3} + 19 \beta_1) q^{55} + ( - 2 \beta_{10} - 4 \beta_{7} + 3 \beta_{5}) q^{59} + ( - 2 \beta_{4} + \beta_{2} + 81) q^{61} + ( - 5 \beta_{11} + 3 \beta_{9} + 2 \beta_{6}) q^{65} + (15 \beta_{3} - 4 \beta_1) q^{67} + ( - 2 \beta_{10} + 20 \beta_{5}) q^{71} + ( - 4 \beta_{4} + 6 \beta_{2} + 55) q^{73} + ( - 5 \beta_{11} - 2 \beta_{9} + 62 \beta_{6}) q^{77} + (\beta_{8} + 18 \beta_{3} - 24 \beta_1) q^{79} + ( - 2 \beta_{10} + 7 \beta_{7} - 14 \beta_{5}) q^{83} + ( - \beta_{4} - 8 \beta_{2} - 88) q^{85} + (4 \beta_{11} + 7 \beta_{9} - 45 \beta_{6}) q^{89} + (\beta_{8} + 27 \beta_{3} + 45 \beta_1) q^{91} + (5 \beta_{10} + 6 \beta_{7} + 7 \beta_{5}) q^{95} + ( - \beta_{4} + 6 \beta_{2} + 211) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{13} - 132 q^{25} - 516 q^{37} - 720 q^{49} + 972 q^{61} + 660 q^{73} - 1056 q^{85} + 2532 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 12x^{10} + 112x^{8} - 368x^{6} + 928x^{4} - 256x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 67\nu^{10} - 691\nu^{8} + 6712\nu^{6} - 18840\nu^{4} + 63944\nu^{2} - 9488 ) / 3152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -153\nu^{10} + 1428\nu^{8} - 12540\nu^{6} + 11832\nu^{4} - 3264\nu^{2} - 240608 ) / 3152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -252\nu^{10} + 2943\nu^{8} - 27468\nu^{6} + 85680\nu^{4} - 227592\nu^{2} + 34416 ) / 3152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -18\nu^{10} + 168\nu^{8} - 1371\nu^{6} + 1392\nu^{4} - 384\nu^{2} - 7448 ) / 197 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -287\nu^{11} + 3598\nu^{9} - 33844\nu^{7} + 122008\nu^{5} - 310816\nu^{3} + 162912\nu ) / 3152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -190\nu^{11} + 2233\nu^{9} - 20710\nu^{7} + 64600\nu^{5} - 159552\nu^{3} + 4560\nu ) / 1576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -73\nu^{11} + 944\nu^{9} - 8942\nu^{7} + 33488\nu^{5} - 84560\nu^{3} + 44304\nu ) / 197 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -1909\nu^{10} + 23005\nu^{8} - 213400\nu^{6} + 697128\nu^{4} - 1650872\nu^{2} + 252272 ) / 3152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 106\nu^{11} - 1252\nu^{9} + 11554\nu^{7} - 36040\nu^{5} + 84870\nu^{3} - 2544\nu ) / 197 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1219\nu^{11} + 14792\nu^{9} - 138584\nu^{7} + 470408\nu^{5} - 1218512\nu^{3} + 639168\nu ) / 1576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2382\nu^{11} + 27945\nu^{9} - 259638\nu^{7} + 809880\nu^{5} - 2005056\nu^{3} + 57168\nu ) / 1576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 4\beta_{10} - 4\beta_{9} + \beta_{7} - 41\beta_{6} - 24\beta_{5} ) / 576 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{8} - \beta_{4} - 28\beta_{3} + 4\beta_{2} - 15\beta _1 + 288 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - \beta_{9} - 17\beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{8} + 7\beta_{4} - 88\beta_{3} - 16\beta_{2} - 81\beta _1 - 960 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 37\beta_{11} - 24\beta_{10} - 16\beta_{9} - 45\beta_{7} - 533\beta_{6} + 384\beta_{5} ) / 144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 17\beta_{4} - 32\beta_{2} - 1800 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -53\beta_{11} - 28\beta_{10} + 12\beta_{9} - 69\beta_{7} + 717\beta_{6} + 520\beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -111\beta_{8} + 149\beta_{4} + 1244\beta_{3} - 260\beta_{2} + 1563\beta _1 - 14208 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -111\beta_{11} + 16\beta_{9} + 1463\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -435\beta_{8} - 629\beta_{4} + 4992\beta_{3} + 1064\beta_{2} + 6531\beta _1 + 57408 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -2761\beta_{11} + 1272\beta_{10} + 304\beta_{9} + 3729\beta_{7} + 35993\beta_{6} - 26208\beta_{5} ) / 18 \) 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
2.48442 − 1.43438i
−2.48442 − 1.43438i
−0.456937 − 0.263813i
0.456937 − 0.263813i
−1.61829 − 0.934317i
1.61829 − 0.934317i
1.61829 + 0.934317i
−1.61829 + 0.934317i
0.456937 + 0.263813i
−0.456937 + 0.263813i
−2.48442 + 1.43438i
2.48442 + 1.43438i
0 0 0 − 14.9230i 0 − 30.0528i 0 0 0
1727.2 0 0 0 − 14.9230i 0 30.0528i 0 0 0
1727.3 0 0 0 − 13.1987i 0 − 4.49091i 0 0 0
1727.4 0 0 0 − 13.1987i 0 4.49091i 0 0 0
1727.5 0 0 0 − 3.33155i 0 − 16.9016i 0 0 0
1727.6 0 0 0 − 3.33155i 0 16.9016i 0 0 0
1727.7 0 0 0 3.33155i 0 − 16.9016i 0 0 0
1727.8 0 0 0 3.33155i 0 16.9016i 0 0 0
1727.9 0 0 0 13.1987i 0 − 4.49091i 0 0 0
1727.10 0 0 0 13.1987i 0 4.49091i 0 0 0
1727.11 0 0 0 14.9230i 0 − 30.0528i 0 0 0
1727.12 0 0 0 14.9230i 0 30.0528i 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.i 12
3.b odd 2 1 inner 1728.4.c.i 12
4.b odd 2 1 inner 1728.4.c.i 12
8.b even 2 1 108.4.b.a ✓ 12
8.d odd 2 1 108.4.b.a ✓ 12
12.b even 2 1 inner 1728.4.c.i 12
24.f even 2 1 108.4.b.a ✓ 12
24.h odd 2 1 108.4.b.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.a ✓ 12 8.b even 2 1
108.4.b.a ✓ 12 8.d odd 2 1
108.4.b.a ✓ 12 24.f even 2 1
108.4.b.a ✓ 12 24.h odd 2 1
1728.4.c.i 12 1.a even 1 1 trivial
1728.4.c.i 12 3.b odd 2 1 inner
1728.4.c.i 12 4.b odd 2 1 inner
1728.4.c.i 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{6} + 408T_{5}^{4} + 43200T_{5}^{2} + 430592 \) Copy content Toggle raw display
\( T_{7}^{6} + 1209T_{7}^{4} + 281979T_{7}^{2} + 5203467 \) Copy content Toggle raw display
\( T_{11}^{6} - 3912T_{11}^{4} + 2591424T_{11}^{2} - 442934784 \) 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
$5$ \( (T^{6} + 408 T^{4} + \cdots + 430592)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1209 T^{4} + \cdots + 5203467)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3912 T^{4} + \cdots - 442934784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 9 T^{2} + \cdots - 105973)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 16344 T^{4} + \cdots + 21293875712)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 22329 T^{4} + \cdots + 55289547)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 2278747067904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 103008 T^{4} + \cdots + 469937979392)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 24996 T^{4} + \cdots + 57669803712)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 129 T^{2} + \cdots - 20595901)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 8632577589248)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 4259903272128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 5276542957056)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 1446033784832)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 66\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 243 T^{2} + \cdots + 3664703)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 74359881698067)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 55\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 165 T^{2} + \cdots - 57176231)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 21\!\cdots\!47)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 632941911834624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 633 T^{2} + \cdots + 28667317)^{4} \) Copy content Toggle raw display
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