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Properties

Label 2736.2.dc.f
Level $2736$
Weight $2$
Character orbit 2736.dc
Analytic conductor $21.847$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (of order \(6\), 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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + \cdots + 414864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{14} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{14} q^{5} + ( - \beta_{10} - \beta_{7}) q^{7} + ( - \beta_{18} - \beta_{6}) q^{11} + ( - \beta_{18} + \beta_{10} + \beta_{9} + \cdots + 1) q^{13}+ \cdots + (\beta_{19} - \beta_{18} + 2 \beta_{17} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{7} + 6 q^{13} + 12 q^{17} - 4 q^{19} + 14 q^{25} + 12 q^{35} + 8 q^{41} + 2 q^{43} + 36 q^{47} + 32 q^{49} - 8 q^{53} - 12 q^{55} - 8 q^{59} - 2 q^{61} + 8 q^{65} - 30 q^{67} + 4 q^{71} - 22 q^{73} - 54 q^{79} - 4 q^{85} - 32 q^{89} - 18 q^{91} + 32 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + \cdots + 414864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\!\cdots\!17 \nu^{19} + \cdots + 88\!\cdots\!20 ) / 13\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!15 \nu^{19} + \cdots + 15\!\cdots\!44 ) / 73\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!53 \nu^{19} + \cdots - 15\!\cdots\!36 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!24 \nu^{19} + \cdots - 29\!\cdots\!00 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!15 \nu^{19} + \cdots - 35\!\cdots\!84 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15\!\cdots\!08 \nu^{19} + \cdots + 44\!\cdots\!08 ) / 34\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 66\!\cdots\!42 \nu^{19} + \cdots + 48\!\cdots\!00 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!12 \nu^{19} + \cdots + 22\!\cdots\!60 ) / 73\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!12 \nu^{19} + \cdots - 96\!\cdots\!60 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 74\!\cdots\!97 \nu^{19} + \cdots + 27\!\cdots\!20 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!81 \nu^{19} + \cdots - 33\!\cdots\!28 ) / 73\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 99\!\cdots\!07 \nu^{19} + \cdots + 91\!\cdots\!84 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!67 \nu^{19} + \cdots + 68\!\cdots\!36 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!40 \nu^{19} + \cdots - 22\!\cdots\!12 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!81 \nu^{19} + \cdots + 63\!\cdots\!16 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 89\!\cdots\!29 \nu^{19} + \cdots - 56\!\cdots\!72 ) / 73\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 70\!\cdots\!85 \nu^{19} + \cdots + 43\!\cdots\!72 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 75\!\cdots\!00 \nu^{19} + \cdots - 41\!\cdots\!00 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 82\!\cdots\!92 \nu^{19} + \cdots - 30\!\cdots\!20 ) / 49\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{18} + \beta_{17} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 2 \beta_{18} + \beta_{16} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - 9 \beta_{18} + 3 \beta_{17} + 2 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + 8 \beta_{13} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{19} - 45 \beta_{18} + \beta_{17} + 24 \beta_{16} + 8 \beta_{15} - 18 \beta_{14} + 34 \beta_{13} + \cdots + 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{19} - 183 \beta_{18} + 19 \beta_{17} + 108 \beta_{16} + 85 \beta_{15} - 109 \beta_{14} + \cdots + 201 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} - 859 \beta_{18} + 11 \beta_{17} + 664 \beta_{16} + 388 \beta_{15} - 545 \beta_{14} + \cdots + 687 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 288 \beta_{19} - 3480 \beta_{18} - 4 \beta_{17} + 3375 \beta_{16} + 2366 \beta_{15} - 2696 \beta_{14} + \cdots + 1742 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2780 \beta_{19} - 15022 \beta_{18} - 705 \beta_{17} + 17713 \beta_{16} + 11856 \beta_{15} + \cdots + 2001 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21092 \beta_{19} - 58465 \beta_{18} - 6619 \beta_{17} + 88838 \beta_{16} + 61746 \beta_{15} + \cdots - 21606 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 135802 \beta_{19} - 223950 \beta_{18} - 43510 \beta_{17} + 438744 \beta_{16} + 304234 \beta_{15} + \cdots - 247596 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 820442 \beta_{19} - 752876 \beta_{18} - 278560 \beta_{17} + 2121960 \beta_{16} + 1478444 \beta_{15} + \cdots - 1854886 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4655336 \beta_{19} - 2085544 \beta_{18} - 1584368 \beta_{17} + 10040992 \beta_{16} + 6999828 \beta_{15} + \cdots - 11948222 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 25452540 \beta_{19} - 2320366 \beta_{18} - 8851794 \beta_{17} + 46569422 \beta_{16} + 32359158 \beta_{15} + \cdots - 70792962 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 134388246 \beta_{19} + 25660522 \beta_{18} - 47035156 \beta_{17} + 210719766 \beta_{16} + \cdots - 397964712 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 690653692 \beta_{19} + 297922790 \beta_{18} - 244628990 \beta_{17} + 929045294 \beta_{16} + \cdots - 2149996356 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3461541484 \beta_{19} + 2240581900 \beta_{18} - 1236318086 \beta_{17} + 3963909834 \beta_{16} + \cdots - 11261926130 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 16959549872 \beta_{19} + 14425981062 \beta_{18} - 6116742626 \beta_{17} + 16233773096 \beta_{16} + \cdots - 57441958996 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 81274368308 \beta_{19} + 85464416684 \beta_{18} - 29589942532 \beta_{17} + 62744460160 \beta_{16} + \cdots - 286157119204 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 380917114036 \beta_{19} + 479521118100 \beta_{18} - 140105517716 \beta_{17} + 221449454948 \beta_{16} + \cdots - 1394237620712 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.64948 − 0.707107i
−1.67971 + 0.707107i
4.76099 + 0.707107i
2.25091 − 0.707107i
−2.54394 + 0.707107i
1.85589 − 0.707107i
−0.561107 − 0.707107i
3.16384 + 0.707107i
−0.476437 + 0.707107i
−1.12095 − 0.707107i
−1.64948 + 0.707107i
−1.67971 − 0.707107i
4.76099 − 0.707107i
2.25091 + 0.707107i
−2.54394 − 0.707107i
1.85589 + 0.707107i
−0.561107 + 0.707107i
3.16384 − 0.707107i
−0.476437 − 0.707107i
−1.12095 + 0.707107i
0 0 0 −3.15998 + 1.82441i 0 −1.21776 0 0 0
449.2 0 0 0 −2.66570 + 1.53904i 0 −1.99260 0 0 0
449.3 0 0 0 −1.80341 + 1.04120i 0 4.37182 0 0 0
449.4 0 0 0 −0.967070 + 0.558338i 0 1.95123 0 0 0
449.5 0 0 0 −0.501439 + 0.289506i 0 0.198579 0 0 0
449.6 0 0 0 0.0242659 − 0.0140099i 0 1.95697 0 0 0
449.7 0 0 0 0.524572 − 0.302862i 0 −4.37361 0 0 0
449.8 0 0 0 1.76516 − 1.01912i 0 −3.69902 0 0 0
449.9 0 0 0 3.20539 − 1.85063i 0 4.57072 0 0 0
449.10 0 0 0 3.57821 − 2.06588i 0 0.233685 0 0 0
1889.1 0 0 0 −3.15998 − 1.82441i 0 −1.21776 0 0 0
1889.2 0 0 0 −2.66570 − 1.53904i 0 −1.99260 0 0 0
1889.3 0 0 0 −1.80341 − 1.04120i 0 4.37182 0 0 0
1889.4 0 0 0 −0.967070 − 0.558338i 0 1.95123 0 0 0
1889.5 0 0 0 −0.501439 − 0.289506i 0 0.198579 0 0 0
1889.6 0 0 0 0.0242659 + 0.0140099i 0 1.95697 0 0 0
1889.7 0 0 0 0.524572 + 0.302862i 0 −4.37361 0 0 0
1889.8 0 0 0 1.76516 + 1.01912i 0 −3.69902 0 0 0
1889.9 0 0 0 3.20539 + 1.85063i 0 4.57072 0 0 0
1889.10 0 0 0 3.57821 + 2.06588i 0 0.233685 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.dc.f 20
3.b odd 2 1 2736.2.dc.e 20
4.b odd 2 1 1368.2.cu.b yes 20
12.b even 2 1 1368.2.cu.a ✓ 20
19.d odd 6 1 2736.2.dc.e 20
57.f even 6 1 inner 2736.2.dc.f 20
76.f even 6 1 1368.2.cu.a ✓ 20
228.n odd 6 1 1368.2.cu.b yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.cu.a ✓ 20 12.b even 2 1
1368.2.cu.a ✓ 20 76.f even 6 1
1368.2.cu.b yes 20 4.b odd 2 1
1368.2.cu.b yes 20 228.n odd 6 1
2736.2.dc.e 20 3.b odd 2 1
2736.2.dc.e 20 19.d odd 6 1
2736.2.dc.f 20 1.a even 1 1 trivial
2736.2.dc.f 20 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{20} - 32 T_{5}^{18} + 708 T_{5}^{16} + 336 T_{5}^{15} - 8096 T_{5}^{14} - 6096 T_{5}^{13} + \cdots + 64 \) Copy content Toggle raw display
\( T_{17}^{20} - 12 T_{17}^{19} - 14 T_{17}^{18} + 744 T_{17}^{17} - 348 T_{17}^{16} - 37344 T_{17}^{15} + \cdots + 2064793600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 32 T^{18} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{10} - 2 T^{9} + \cdots + 139)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 100 T^{18} + \cdots + 4260096 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 1176147025 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2064793600 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 31021072384 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 34924134400 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 18420189841 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 266049536550625 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 11964640689 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2106445849600 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 831865578726400 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 2030625000000 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 175961790273481 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 32\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 796807954881 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 98425257984 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
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