Properties

Label 2736.2.dc.d
Level $2736$
Weight $2$
Character orbit 2736.dc
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
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] = [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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 684)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{4}) q^{5} + ( - \beta_{8} - \beta_{7}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{4}) q^{5} + ( - \beta_{8} - \beta_{7}) q^{7} + ( - \beta_{13} + \beta_{2}) q^{11} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{3} + 1) q^{13} + (\beta_{11} - \beta_{10} + \beta_{4}) q^{17} + (\beta_{5} + \beta_{3} - \beta_1 + 2) q^{19} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11}) q^{23} + (\beta_{8} + \beta_{6}) q^{25} + (\beta_{15} - 2 \beta_{2}) q^{29} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 2) q^{31} + ( - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{4}) q^{35} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_1) q^{37} + ( - \beta_{15} + 2 \beta_{13} + \beta_{11} + \beta_{10} + \beta_{4} - \beta_{2}) q^{41} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{43} + ( - \beta_{15} - \beta_{14} + \beta_{12}) q^{47} + (2 \beta_{9} - \beta_{8} + \beta_{6} - \beta_1 + 1) q^{49} + (\beta_{13} - \beta_{11} - 2 \beta_{10} + 4 \beta_{4}) q^{53} + (\beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{3} - 3) q^{55} + (2 \beta_{13} + \beta_{11} + 2 \beta_{10} + 2 \beta_{4}) q^{59} + (3 \beta_{8} - \beta_{6} + 2 \beta_{5} + 2) q^{61} + ( - 2 \beta_{15} + \beta_{13} + 2 \beta_{11} + \beta_{2}) q^{65} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} - 1) q^{67} + (2 \beta_{14} - \beta_{12}) q^{71} + ( - \beta_{9} - 5 \beta_{5} - \beta_1) q^{73} + (\beta_{14} + \beta_{4}) q^{77} + (2 \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_1 + 3) q^{79} + (\beta_{13} + 3 \beta_{4} - 2 \beta_{2}) q^{83} + (2 \beta_{6} + 6 \beta_{5} + 6) q^{85} + ( - \beta_{14} - 2 \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{4}) q^{89} + (2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + 3) q^{91} + ( - \beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + 3 \beta_{4} + \beta_{2}) q^{95} + (\beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 523\nu^{12} + 2584\nu^{10} + 5175\nu^{8} + 8496\nu^{6} + 57834\nu^{4} + 214407\nu^{2} + 282852 ) / 12393 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4 \nu^{15} + 152 \nu^{13} + 1088 \nu^{11} + 2658 \nu^{9} + 3654 \nu^{7} + 18360 \nu^{5} + 84240 \nu^{3} + 165726 \nu ) / 12393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 10\nu^{12} - 46\nu^{10} - 126\nu^{8} - 315\nu^{6} - 1134\nu^{4} - 2997\nu^{2} - 5832 ) / 729 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -23\nu^{15} + 58\nu^{13} + 850\nu^{11} + 1845\nu^{9} + 126\nu^{7} + 8262\nu^{5} + 63018\nu^{3} + 87480\nu ) / 37179 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50 \nu^{14} - 446 \nu^{12} - 1598 \nu^{10} - 3087 \nu^{8} - 8946 \nu^{6} - 41310 \nu^{4} - 111942 \nu^{2} - 126117 ) / 12393 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\nu^{14} + 158\nu^{12} + 527\nu^{10} + 939\nu^{8} + 3303\nu^{6} + 15606\nu^{4} + 41472\nu^{2} + 38880 ) / 4131 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\nu^{14} + 218\nu^{12} - 34\nu^{10} - 900\nu^{8} + 3465\nu^{6} + 13770\nu^{4} - 26811\nu^{2} - 122472 ) / 12393 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{14} + 40\nu^{12} + 85\nu^{10} + 168\nu^{8} + 873\nu^{6} + 3060\nu^{4} + 5427\nu^{2} + 2592 ) / 1377 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -22\nu^{14} - 133\nu^{12} - 493\nu^{10} - 1038\nu^{8} - 2853\nu^{6} - 13311\nu^{4} - 36531\nu^{2} - 46899 ) / 4131 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 133 \nu^{15} + 1168 \nu^{13} + 4012 \nu^{11} + 7119 \nu^{9} + 21483 \nu^{7} + 104652 \nu^{5} + 272484 \nu^{3} + 254421 \nu ) / 37179 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 158 \nu^{15} - 2003 \nu^{13} - 8177 \nu^{11} - 16848 \nu^{9} - 39726 \nu^{7} - 195534 \nu^{5} - 625887 \nu^{3} - 794610 \nu ) / 37179 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -19\nu^{15} - 196\nu^{13} - 646\nu^{11} - 1221\nu^{9} - 3834\nu^{7} - 17901\nu^{5} - 44631\nu^{3} - 41067\nu ) / 4131 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 194 \nu^{15} - 635 \nu^{13} + 238 \nu^{11} + 1566 \nu^{9} - 12348 \nu^{7} - 34425 \nu^{5} + 82701 \nu^{3} + 325134 \nu ) / 37179 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73 \nu^{15} + 541 \nu^{13} + 1819 \nu^{11} + 3690 \nu^{9} + 11115 \nu^{7} + 52326 \nu^{5} + 131544 \nu^{3} + 137781 \nu ) / 12393 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 346 \nu^{15} - 2560 \nu^{13} - 7888 \nu^{11} - 14832 \nu^{9} - 53424 \nu^{7} - 231336 \nu^{5} - 545616 \nu^{3} - 433026 \nu ) / 37179 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{10} - 4\beta_{4} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} - 2\beta_{14} + 4\beta_{12} - 6\beta_{11} - 4\beta_{10} + 2\beta_{4} - 5\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{9} - 2\beta_{8} + 5\beta_{7} + \beta_{6} + 10\beta_{5} - 5\beta_{3} + 3\beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{15} + 22\beta_{14} + 12\beta_{13} - 14\beta_{12} + 18\beta_{11} + 2\beta_{10} - 4\beta_{4} - 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18\beta_{9} - 5\beta_{8} - 10\beta_{7} + 19\beta_{6} - 17\beta_{5} - 5\beta_{3} - 12\beta _1 - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 59 \beta_{15} - 44 \beta_{14} + 24 \beta_{13} + 4 \beta_{12} + 24 \beta_{11} - 16 \beta_{10} - 34 \beta_{4} + 25 \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -42\beta_{9} + 52\beta_{8} + 14\beta_{7} - 56\beta_{6} + 79\beta_{5} - 26\beta_{3} + 36\beta _1 - 135 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 70 \beta_{15} + 76 \beta_{14} - 144 \beta_{13} - 44 \beta_{12} - 96 \beta_{11} - 274 \beta_{10} + 290 \beta_{4} - 5 \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 78\beta_{9} - 245\beta_{8} - 22\beta_{7} + 28\beta_{6} - 422\beta_{5} + 49\beta_{3} - 132\beta _1 - 24 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 263 \beta_{15} - 122 \beta_{14} + 282 \beta_{13} + 58 \beta_{12} + 612 \beta_{11} + 758 \beta_{10} - 4 \beta_{4} + 322 \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 72\beta_{9} + 589\beta_{8} - 10\beta_{7} - 413\beta_{6} + 64\beta_{5} + 238\beta_{3} + 231\beta _1 + 780 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1237 \beta_{15} - 908 \beta_{14} - 1542 \beta_{13} - 158 \beta_{12} - 2136 \beta_{11} - 394 \beta_{10} - 466 \beta_{4} - 407 \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -1284\beta_{9} - 326\beta_{8} - 175\beta_{7} - 218\beta_{6} - 164\beta_{5} + 703\beta_{3} - 396\beta _1 - 648 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 470 \beta_{15} + 1480 \beta_{14} - 576 \beta_{13} + 4168 \beta_{12} - 312 \beta_{11} + 2318 \beta_{10} + 884 \beta_{4} - 5 \beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{5}\) \(-1\) \(1\) \(1\)

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
1.56478 0.742611i
0.750719 1.56090i
−0.243856 1.71480i
−0.654543 1.60361i
0.654543 + 1.60361i
0.243856 + 1.71480i
−0.750719 + 1.56090i
−1.56478 + 0.742611i
1.56478 + 0.742611i
0.750719 + 1.56090i
−0.243856 + 1.71480i
−0.654543 + 1.60361i
0.654543 1.60361i
0.243856 1.71480i
−0.750719 1.56090i
−1.56478 0.742611i
0 0 0 −2.99029 + 1.72644i 0 −0.128302 0 0 0
449.2 0 0 0 −2.47786 + 1.43059i 0 −3.93208 0 0 0
449.3 0 0 0 −1.11928 + 0.646214i 0 0.567493 0 0 0
449.4 0 0 0 −0.406956 + 0.234956i 0 3.49289 0 0 0
449.5 0 0 0 0.406956 0.234956i 0 3.49289 0 0 0
449.6 0 0 0 1.11928 0.646214i 0 0.567493 0 0 0
449.7 0 0 0 2.47786 1.43059i 0 −3.93208 0 0 0
449.8 0 0 0 2.99029 1.72644i 0 −0.128302 0 0 0
1889.1 0 0 0 −2.99029 1.72644i 0 −0.128302 0 0 0
1889.2 0 0 0 −2.47786 1.43059i 0 −3.93208 0 0 0
1889.3 0 0 0 −1.11928 0.646214i 0 0.567493 0 0 0
1889.4 0 0 0 −0.406956 0.234956i 0 3.49289 0 0 0
1889.5 0 0 0 0.406956 + 0.234956i 0 3.49289 0 0 0
1889.6 0 0 0 1.11928 + 0.646214i 0 0.567493 0 0 0
1889.7 0 0 0 2.47786 + 1.43059i 0 −3.93208 0 0 0
1889.8 0 0 0 2.99029 + 1.72644i 0 −0.128302 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
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.d 16
3.b odd 2 1 inner 2736.2.dc.d 16
4.b odd 2 1 684.2.bk.a 16
12.b even 2 1 684.2.bk.a 16
19.d odd 6 1 inner 2736.2.dc.d 16
57.f even 6 1 inner 2736.2.dc.d 16
76.f even 6 1 684.2.bk.a 16
228.n odd 6 1 684.2.bk.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.bk.a 16 4.b odd 2 1
684.2.bk.a 16 12.b even 2 1
684.2.bk.a 16 76.f even 6 1
684.2.bk.a 16 228.n odd 6 1
2736.2.dc.d 16 1.a even 1 1 trivial
2736.2.dc.d 16 3.b odd 2 1 inner
2736.2.dc.d 16 19.d odd 6 1 inner
2736.2.dc.d 16 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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{16} - 22T_{5}^{14} + 348T_{5}^{12} - 2608T_{5}^{10} + 14236T_{5}^{8} - 24528T_{5}^{6} + 31968T_{5}^{4} - 6912T_{5}^{2} + 1296 \) Copy content Toggle raw display
\( T_{17}^{16} - 88 T_{17}^{14} + 5808 T_{17}^{12} - 161344 T_{17}^{10} + 3350464 T_{17}^{8} - 8633856 T_{17}^{6} + 19243008 T_{17}^{4} - 2598912 T_{17}^{2} + 331776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 22 T^{14} + 348 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T^{4} - 14 T^{2} + 6 T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 70 T^{6} + 1360 T^{4} + \cdots + 6084)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 42 T^{6} + 1647 T^{4} + \cdots + 13689)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 88 T^{14} + 5808 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + 44 T^{6} - 246 T^{5} + \cdots + 130321)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 154 T^{14} + \cdots + 358892053776 \) Copy content Toggle raw display
$29$ \( T^{16} + 192 T^{14} + \cdots + 34828517376 \) Copy content Toggle raw display
$31$ \( (T^{8} + 168 T^{6} + 7974 T^{4} + \cdots + 613089)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 156 T^{6} + 6102 T^{4} + \cdots + 23409)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 216 T^{14} + \cdots + 176319369216 \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + 94 T^{6} + \cdots + 2825761)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 88 T^{14} + 6600 T^{12} + \cdots + 20736 \) Copy content Toggle raw display
$53$ \( T^{16} + 354 T^{14} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} + 354 T^{14} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + 164 T^{6} + \cdots + 1990921)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 18 T^{7} + 84 T^{6} - 432 T^{5} + \cdots + 68121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 54 T^{2} + 2916)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 22 T^{7} + 376 T^{6} + \cdots + 85849)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} - 138 T^{6} + \cdots + 700569)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 328 T^{6} + 30352 T^{4} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 403540761128976 \) Copy content Toggle raw display
$97$ \( (T^{8} - 252 T^{6} + 66420 T^{4} + \cdots + 8503056)^{2} \) Copy content Toggle raw display
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