Properties

Label 2736.3.m.f
Level $2736$
Weight $3$
Character orbit 2736.m
Analytic conductor $74.551$
Analytic rank $0$
Dimension $12$
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] = [2736,3,Mod(1711,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.1711");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.m (of order \(2\), degree \(1\), 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: \(74.5506003290\)
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: \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 912)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} - \beta_{11} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{5} - \beta_{11} q^{7} + (\beta_{10} + \beta_{9} - \beta_{7}) q^{11} + (\beta_1 + 3) q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{17} - \beta_{8} q^{19} + (\beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} - 2 \beta_{6}) q^{23} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{25} + ( - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2) q^{29} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8}) q^{31} + ( - 3 \beta_{11} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{35} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{37} + (\beta_{5} - 3 \beta_{3} + 3 \beta_{2} - 11) q^{41} + (3 \beta_{11} + 2 \beta_{10} - 8 \beta_{8} - 2 \beta_{7} + 3 \beta_{6}) q^{43} + (3 \beta_{11} - 2 \beta_{10} + \beta_{9} + 3 \beta_{7} + \beta_{6}) q^{47} + (2 \beta_{5} + 5 \beta_{2} - \beta_1 - 3) q^{49} + (3 \beta_{5} - 4 \beta_{4} + \beta_{3} - \beta_{2} + 17) q^{53} + (\beta_{11} + 4 \beta_{10} + 6 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} - 6 \beta_{6}) q^{55} + (4 \beta_{11} + 3 \beta_{10} + 4 \beta_{9} + 9 \beta_{8} - 3 \beta_{7} - 6 \beta_{6}) q^{59} + (4 \beta_{5} + 5 \beta_{4} + \beta_{3} - 3 \beta_1 - 16) q^{61} + (3 \beta_{5} - 7 \beta_{2} + 5) q^{65} + (4 \beta_{11} - \beta_{10} - 2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 6 \beta_{6}) q^{67} + ( - \beta_{10} - 4 \beta_{9} + 5 \beta_{8} + 5 \beta_{7} + 2 \beta_{6}) q^{71} + ( - 6 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + \beta_{2} + 3 \beta_1 - 12) q^{73} + ( - 3 \beta_{5} - 10 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 3) q^{77} + (9 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 5 \beta_{7}) q^{79} + ( - 7 \beta_{11} + 5 \beta_{9} + 9 \beta_{8} - 4 \beta_{7} + 2 \beta_{6}) q^{83} + ( - 7 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 + 16) q^{85} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 7 \beta_{2} - 8 \beta_1 - 37) q^{89} + ( - 4 \beta_{11} - \beta_{10} - 6 \beta_{9} - 17 \beta_{8} + \beta_{7} + 6 \beta_{6}) q^{91} + (2 \beta_{11} + \beta_{10} - 2 \beta_{7} - \beta_{6}) q^{95} + (3 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 13 \beta_{2} + 2 \beta_1 + 27) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{5} + 36 q^{13} + 10 q^{17} + 58 q^{25} - 12 q^{29} + 32 q^{37} - 136 q^{41} - 22 q^{49} + 236 q^{53} - 210 q^{61} + 52 q^{65} - 158 q^{73} + 70 q^{77} + 242 q^{85} - 444 q^{89} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1132830139545 \nu^{11} + 83463703932306 \nu^{10} + 82049035182874 \nu^{9} + \cdots - 63\!\cdots\!20 ) / 52\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38271754888553 \nu^{11} + 9342376003726 \nu^{10} + \cdots - 27\!\cdots\!72 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5763155937085 \nu^{11} + 13707431665502 \nu^{10} + 264163078401060 \nu^{9} + 394878866521450 \nu^{8} + \cdots + 11\!\cdots\!52 ) / 13\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79024470131873 \nu^{11} + 62863161452290 \nu^{10} + \cdots + 94\!\cdots\!84 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 104157093719227 \nu^{11} - 31799241318102 \nu^{10} + \cdots - 19\!\cdots\!44 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2248559103131 \nu^{11} + 4524708578702 \nu^{10} - 112234893734550 \nu^{9} + 307066972550728 \nu^{8} + \cdots - 98\!\cdots\!72 ) / 54\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 451147977297025 \nu^{11} - 972391254937206 \nu^{10} + \cdots + 15\!\cdots\!20 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5084441 \nu^{11} - 10363066 \nu^{10} + 255384002 \nu^{9} - 701038728 \nu^{8} + 11325739791 \nu^{7} - 25986026436 \nu^{6} + \cdots + 25239211968 ) / 10877179584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 563617568102453 \nu^{11} + \cdots + 24\!\cdots\!20 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8386307706881 \nu^{11} - 16626138955879 \nu^{10} + 416130916219964 \nu^{9} + \cdots + 33\!\cdots\!44 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 288508475434596 \nu^{11} + 596180148308589 \nu^{10} + \cdots - 13\!\cdots\!16 ) / 26\!\cdots\!96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} + 2\beta_{9} + 2\beta_{8} - 2\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 16 \beta_{8} + 8 \beta_{7} + 31 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 4 \beta _1 - 64 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12\beta_{5} - 35\beta_{4} + 35\beta_{3} - 63\beta_{2} - 4\beta _1 + 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 90 \beta_{11} - 96 \beta_{10} + 338 \beta_{9} - 672 \beta_{8} - 408 \beta_{7} - 1149 \beta_{6} + 106 \beta_{5} + 247 \beta_{4} + 113 \beta_{3} + 345 \beta_{2} - 168 \beta _1 - 2360 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3850 \beta_{11} - 2554 \beta_{10} - 3762 \beta_{9} - 1858 \beta_{8} + 2338 \beta_{7} + 1135 \beta_{6} - 504 \beta_{5} + 1537 \beta_{4} - 1377 \beta_{3} + 2889 \beta_{2} + 72 \beta _1 - 2702 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4258\beta_{5} - 11341\beta_{4} - 3867\beta_{3} - 16079\beta_{2} + 6900\beta _1 + 97664 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 163514 \beta_{11} + 103394 \beta_{10} + 165862 \beta_{9} + 60614 \beta_{8} - 107558 \beta_{7} - 78445 \beta_{6} - 18652 \beta_{5} + 70235 \beta_{4} - 54587 \beta_{3} + 129927 \beta_{2} + \cdots - 176930 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 343514 \beta_{11} + 17776 \beta_{10} - 786242 \beta_{9} + 1080640 \beta_{8} + 840344 \beta_{7} + 2009317 \beta_{6} + 165610 \beta_{5} + 514511 \beta_{4} + 125817 \beta_{3} + \cdots - 4139064 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 670120\beta_{5} - 3225385\beta_{4} + 2178281\beta_{3} - 5845345\beta_{2} + 238472\beta _1 + 9957246 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18600482 \beta_{11} + 1973948 \beta_{10} + 37156070 \beta_{9} - 43542928 \beta_{8} - 37900232 \beta_{7} - 85899087 \beta_{6} + 6433250 \beta_{5} + 23319637 \beta_{4} + \cdots - 177340352 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 300468378 \beta_{11} - 177177090 \beta_{10} - 327495670 \beta_{9} - 48227286 \beta_{8} + 229470870 \beta_{7} + 241426709 \beta_{6} - 23665836 \beta_{5} + \cdots - 523950514 ) / 8 \) Copy content Toggle raw display

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1711.1
−0.820808 + 1.42168i
−0.820808 1.42168i
3.06079 + 5.30145i
3.06079 5.30145i
0.816029 1.41340i
0.816029 + 1.41340i
−0.0525878 0.0910847i
−0.0525878 + 0.0910847i
1.37340 2.37879i
1.37340 + 2.37879i
−3.37682 + 5.84883i
−3.37682 5.84883i
0 0 0 −8.90000 0 2.78940i 0 0 0
1711.2 0 0 0 −8.90000 0 2.78940i 0 0 0
1711.3 0 0 0 −1.38490 0 7.59081i 0 0 0
1711.4 0 0 0 −1.38490 0 7.59081i 0 0 0
1711.5 0 0 0 −0.606930 0 2.85501i 0 0 0
1711.6 0 0 0 −0.606930 0 2.85501i 0 0 0
1711.7 0 0 0 3.59607 0 9.49604i 0 0 0
1711.8 0 0 0 3.59607 0 9.49604i 0 0 0
1711.9 0 0 0 4.02901 0 7.12527i 0 0 0
1711.10 0 0 0 4.02901 0 7.12527i 0 0 0
1711.11 0 0 0 8.26675 0 9.51333i 0 0 0
1711.12 0 0 0 8.26675 0 9.51333i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1711.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.m.f 12
3.b odd 2 1 912.3.m.a 12
4.b odd 2 1 inner 2736.3.m.f 12
12.b even 2 1 912.3.m.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.a 12 3.b odd 2 1
912.3.m.a 12 12.b even 2 1
2736.3.m.f 12 1.a even 1 1 trivial
2736.3.m.f 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 5T_{5}^{5} - 77T_{5}^{4} + 437T_{5}^{3} + 16T_{5}^{2} - 1644T_{5} - 896 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\). 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
$5$ \( (T^{6} - 5 T^{5} - 77 T^{4} + 437 T^{3} + \cdots - 896)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 305 T^{10} + \cdots + 1514143744 \) Copy content Toggle raw display
$11$ \( T^{12} + 885 T^{10} + \cdots + 187580416 \) Copy content Toggle raw display
$13$ \( (T^{6} - 18 T^{5} - 416 T^{4} + \cdots + 2630656)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 5 T^{5} - 915 T^{4} + \cdots + 8199352)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 2152 T^{10} + \cdots + 99230924406784 \) Copy content Toggle raw display
$29$ \( (T^{6} + 6 T^{5} - 2096 T^{4} + \cdots + 76842496)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 1312 T^{10} + \cdots + 1396965376 \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} - 2740 T^{4} + \cdots + 6873088)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 68 T^{5} - 3012 T^{4} + \cdots + 1137344512)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 15745 T^{10} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + 12849 T^{10} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} - 118 T^{5} - 1928 T^{4} + \cdots + 2520564736)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 21104 T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + 105 T^{5} - 6647 T^{4} + \cdots + 826249672)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 15472 T^{10} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{12} + 19408 T^{10} + \cdots + 80\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 79 T^{5} - 11631 T^{4} + \cdots + 1441081768)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 38988 T^{10} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + 31516 T^{10} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + 222 T^{5} + \cdots + 307161915264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 160 T^{5} + \cdots + 100074188608)^{2} \) Copy content Toggle raw display
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