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Properties

Label 1638.2.p.m
Level $1638$
Weight $2$
Character orbit 1638.p
Analytic conductor $13.079$
Analytic rank $0$
Dimension $16$
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] = [1638,2,Mod(919,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.919");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.p (of order \(3\), degree \(2\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 28 x^{14} - 20 x^{13} + 463 x^{12} - 254 x^{11} + 4408 x^{10} - 566 x^{9} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{6} - 1) q^{4} + \beta_1 q^{5} + ( - \beta_{8} - \beta_{5}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{6} - 1) q^{4} + \beta_1 q^{5} + ( - \beta_{8} - \beta_{5}) q^{7} - q^{8} - \beta_{2} q^{10} + \beta_{11} q^{11} + (\beta_{15} - \beta_{5} + 1) q^{13} - \beta_{5} q^{14} - \beta_{6} q^{16} + (\beta_{14} + \beta_{9} - \beta_{6} + \cdots + 1) q^{17}+ \cdots + ( - \beta_{15} - \beta_{13} - \beta_{12} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} + 2 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 8 q^{4} + 2 q^{5} - 16 q^{8} + 4 q^{10} + 8 q^{13} - 6 q^{14} - 8 q^{16} + 4 q^{17} - 8 q^{19} + 2 q^{20} - 12 q^{25} + 4 q^{26} - 6 q^{28} - 20 q^{29} + 10 q^{31} + 8 q^{32} + 8 q^{34} + 16 q^{35} + 14 q^{37} - 4 q^{38} - 2 q^{40} - 6 q^{41} - 4 q^{43} - 4 q^{47} + 22 q^{49} + 12 q^{50} - 4 q^{52} - 14 q^{53} - 14 q^{55} - 40 q^{58} + 26 q^{59} + 16 q^{61} - 10 q^{62} + 16 q^{64} + 16 q^{65} + 4 q^{68} + 14 q^{70} - 2 q^{71} - 6 q^{73} - 14 q^{74} + 4 q^{76} + 16 q^{77} + 12 q^{79} - 4 q^{80} - 12 q^{82} - 64 q^{83} - 24 q^{85} + 4 q^{86} + 22 q^{89} + 12 q^{91} - 8 q^{94} + 6 q^{95} - 10 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 28 x^{14} - 20 x^{13} + 463 x^{12} - 254 x^{11} + 4408 x^{10} - 566 x^{9} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27\!\cdots\!40 \nu^{15} + \cdots + 91\!\cdots\!61 ) / 39\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!80 \nu^{15} + \cdots - 25\!\cdots\!43 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!03 \nu^{15} + \cdots - 21\!\cdots\!78 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 64\!\cdots\!97 \nu^{15} + \cdots - 12\!\cdots\!13 ) / 57\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\!\cdots\!09 \nu^{15} + \cdots + 25\!\cdots\!24 ) / 35\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67\!\cdots\!70 \nu^{15} + \cdots + 17\!\cdots\!31 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!78 \nu^{15} + \cdots + 24\!\cdots\!70 ) / 86\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50\!\cdots\!70 \nu^{15} + \cdots - 71\!\cdots\!50 ) / 86\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!30 \nu^{15} + \cdots + 73\!\cdots\!93 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!87 \nu^{15} + \cdots - 26\!\cdots\!33 ) / 57\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!05 \nu^{15} + \cdots + 65\!\cdots\!56 ) / 17\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!12 \nu^{15} + \cdots + 18\!\cdots\!93 ) / 19\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 26\!\cdots\!61 \nu^{15} + \cdots - 79\!\cdots\!67 ) / 28\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 76\!\cdots\!56 \nu^{15} + \cdots + 10\!\cdots\!55 ) / 57\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{9} - 6\beta_{6} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - \beta_{13} - 2\beta_{11} + 2\beta_{7} + 2\beta_{5} + 2\beta_{3} + 11\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{15} - 15 \beta_{14} - 13 \beta_{13} - 13 \beta_{12} - 2 \beta_{10} - 15 \beta_{9} + \cdots - 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 25 \beta_{14} - 23 \beta_{12} + 44 \beta_{11} + 9 \beta_{10} - 44 \beta_{9} + 34 \beta_{8} + \cdots - 140 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 162 \beta_{15} + 104 \beta_{14} + 170 \beta_{13} + 226 \beta_{11} + 34 \beta_{10} - 6 \beta_{8} + \cdots + 769 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 432 \beta_{15} + 476 \beta_{14} + 422 \beta_{13} + 422 \beta_{12} - 10 \beta_{10} + 780 \beta_{9} + \cdots + 1902 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 490 \beta_{15} + 1840 \beta_{14} + 2335 \beta_{12} - 3481 \beta_{11} + 280 \beta_{10} + \cdots + 5707 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6196 \beta_{15} + 880 \beta_{14} - 7057 \beta_{13} - 12944 \beta_{11} - 580 \beta_{10} - 426 \beta_{8} + \cdots - 30041 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 19687 \beta_{15} - 37985 \beta_{14} - 33385 \beta_{13} - 33385 \beta_{12} - 13698 \beta_{10} + \cdots - 139257 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12778 \beta_{15} - 152325 \beta_{14} - 112913 \beta_{12} + 208622 \beta_{11} - 27685 \beta_{10} + \cdots - 385838 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 351292 \beta_{15} - 23484 \beta_{14} + 490460 \beta_{13} + 848032 \beta_{11} + 99084 \beta_{10} + \cdots + 1976017 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 937324 \beta_{15} + 2021004 \beta_{14} + 1767920 \beta_{13} + 1767920 \beta_{12} + 830596 \beta_{10} + \cdots + 6957252 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1446888 \beta_{15} + 9697544 \beta_{14} + 7329721 \beta_{12} - 13255129 \beta_{11} + \cdots + 20732233 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 16419198 \beta_{15} + 9665300 \beta_{14} - 27388225 \beta_{13} - 52068410 \beta_{11} + \cdots - 104840105 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(-\beta_{6}\) \(-1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
−1.52877 + 2.64791i
−1.40628 + 2.43575i
−0.954287 + 1.65287i
−0.111386 + 0.192926i
0.531084 − 0.919864i
0.760578 − 1.31736i
1.74722 − 3.02627i
1.96185 − 3.39802i
−1.52877 − 2.64791i
−1.40628 − 2.43575i
−0.954287 − 1.65287i
−0.111386 − 0.192926i
0.531084 + 0.919864i
0.760578 + 1.31736i
1.74722 + 3.02627i
1.96185 + 3.39802i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.52877 + 2.64791i 0 −0.824297 + 2.51407i −1.00000 0 −3.05755
919.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.40628 + 2.43575i 0 −2.40416 − 1.10454i −1.00000 0 −2.81256
919.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.954287 + 1.65287i 0 2.63899 − 0.189004i −1.00000 0 −1.90857
919.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.111386 + 0.192926i 0 0.263788 − 2.63257i −1.00000 0 −0.222772
919.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.531084 − 0.919864i 0 −1.79588 + 1.94289i −1.00000 0 1.06217
919.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.760578 − 1.31736i 0 1.93919 + 1.79987i −1.00000 0 1.52116
919.7 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.74722 − 3.02627i 0 −2.45876 + 0.976990i −1.00000 0 3.49444
919.8 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.96185 − 3.39802i 0 2.64112 + 0.156399i −1.00000 0 3.92370
991.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.52877 − 2.64791i 0 −0.824297 − 2.51407i −1.00000 0 −3.05755
991.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.40628 − 2.43575i 0 −2.40416 + 1.10454i −1.00000 0 −2.81256
991.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.954287 − 1.65287i 0 2.63899 + 0.189004i −1.00000 0 −1.90857
991.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.111386 − 0.192926i 0 0.263788 + 2.63257i −1.00000 0 −0.222772
991.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.531084 + 0.919864i 0 −1.79588 − 1.94289i −1.00000 0 1.06217
991.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.760578 + 1.31736i 0 1.93919 − 1.79987i −1.00000 0 1.52116
991.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.74722 + 3.02627i 0 −2.45876 − 0.976990i −1.00000 0 3.49444
991.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.96185 + 3.39802i 0 2.64112 − 0.156399i −1.00000 0 3.92370
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.p.m yes 16
3.b odd 2 1 1638.2.p.l yes 16
7.c even 3 1 1638.2.m.l ✓ 16
13.c even 3 1 1638.2.m.l ✓ 16
21.h odd 6 1 1638.2.m.m yes 16
39.i odd 6 1 1638.2.m.m yes 16
91.g even 3 1 inner 1638.2.p.m yes 16
273.bm odd 6 1 1638.2.p.l yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.m.l ✓ 16 7.c even 3 1
1638.2.m.l ✓ 16 13.c even 3 1
1638.2.m.m yes 16 21.h odd 6 1
1638.2.m.m yes 16 39.i odd 6 1
1638.2.p.l yes 16 3.b odd 2 1
1638.2.p.l yes 16 273.bm odd 6 1
1638.2.p.m yes 16 1.a even 1 1 trivial
1638.2.p.m yes 16 91.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 2 T_{5}^{15} + 28 T_{5}^{14} - 20 T_{5}^{13} + 463 T_{5}^{12} - 254 T_{5}^{11} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{16} - 11 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} - 52 T^{6} + \cdots - 3177)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + \cdots + 1108809 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots - 9123)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 140493609 \) Copy content Toggle raw display
$29$ \( T^{16} + 20 T^{15} + \cdots + 44475561 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 152201569 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 575856009 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 35101146609 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2802749481 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 184932801 \) Copy content Toggle raw display
$53$ \( T^{16} + 14 T^{15} + \cdots + 41641209 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 73235725641 \) Copy content Toggle raw display
$61$ \( (T^{8} - 8 T^{7} + \cdots - 2629888)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 193 T^{6} + \cdots + 306877)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 619745243121 \) Copy content Toggle raw display
$73$ \( T^{16} + 6 T^{15} + \cdots + 80089 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10500406027489 \) Copy content Toggle raw display
$83$ \( (T^{8} + 32 T^{7} + \cdots + 2749059)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 37313198053209 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 54\!\cdots\!89 \) Copy content Toggle raw display
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