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Properties

Label 1638.2.bj.i
Level $1638$
Weight $2$
Character orbit 1638.bj
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(127,1638)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.127");
 
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.bj (of order \(6\), 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 \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + \cdots + 97344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{5} q^{4} - \beta_{9} q^{5} - \beta_{4} q^{7} + (\beta_{4} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{5} q^{4} - \beta_{9} q^{5} - \beta_{4} q^{7} + (\beta_{4} - \beta_{3}) q^{8} - \beta_{13} q^{10} + (\beta_{15} - \beta_{14} + \beta_{8} + \cdots + 1) q^{11}+ \cdots - \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 2 q^{10} + 12 q^{11} + 10 q^{13} + 16 q^{14} - 8 q^{16} + 6 q^{17} - 4 q^{22} + 12 q^{23} - 20 q^{25} + 2 q^{26} - 16 q^{29} - 2 q^{35} - 6 q^{37} + 4 q^{40} - 12 q^{41} - 6 q^{43} + 6 q^{46} + 8 q^{49} + 24 q^{50} - 4 q^{52} + 40 q^{53} + 20 q^{55} + 8 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} + 14 q^{62} - 16 q^{64} + 52 q^{65} - 30 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{74} - 8 q^{77} - 16 q^{79} + 2 q^{82} + 6 q^{85} + 4 q^{88} - 30 q^{89} + 4 q^{91} + 24 q^{92} - 8 q^{94} + 40 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + \cdots + 97344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 282646183413977 \nu^{15} + \cdots - 56\!\cdots\!92 ) / 11\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{15} + \cdots - 67\!\cdots\!08 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\!\cdots\!57 \nu^{15} + \cdots - 66\!\cdots\!28 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 209664617 \nu^{15} - 2025754990 \nu^{14} + 4591800238 \nu^{13} - 476218986 \nu^{12} + \cdots + 19272755774688 ) / 35896513665792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!30 \nu^{15} + \cdots - 73\!\cdots\!04 ) / 11\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40\!\cdots\!40 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10297601 \nu^{15} - 26746609 \nu^{14} + 27244757 \nu^{13} + 236512383 \nu^{12} + \cdots + 130830721008 ) / 230105856832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!76 \nu^{15} + \cdots - 77\!\cdots\!28 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 88\!\cdots\!08 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!63 \nu^{15} + \cdots - 57\!\cdots\!04 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 46\!\cdots\!27 \nu^{15} + \cdots + 17\!\cdots\!64 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!15 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 59\!\cdots\!81 \nu^{15} + \cdots + 94\!\cdots\!84 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 80\!\cdots\!01 \nu^{15} + \cdots + 35\!\cdots\!68 ) / 57\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{9} - \beta_{8} - \beta_{5} - 5\beta_{4} + 5\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 8 \beta_{6} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - \beta_{14} - 14 \beta_{13} - 12 \beta_{11} + 14 \beta_{10} + \beta_{9} - 13 \beta_{8} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{15} - 30 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 26 \beta_{11} + 30 \beta_{10} - 2 \beta_{9} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 135 \beta_{14} + 24 \beta_{13} + 40 \beta_{12} + 20 \beta_{11} + 24 \beta_{10} - 163 \beta_{9} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 88 \beta_{15} - 290 \beta_{14} + 390 \beta_{13} + 88 \beta_{12} + 378 \beta_{11} - 56 \beta_{10} + \cdots + 760 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 622 \beta_{15} + 311 \beta_{14} + 1842 \beta_{13} + 1544 \beta_{11} - 1842 \beta_{10} - 411 \beta_{9} + \cdots + 3614 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1452 \beta_{15} + 4598 \beta_{14} + 1006 \beta_{13} - 1452 \beta_{12} + 3146 \beta_{11} - 4922 \beta_{10} + \cdots + 2004 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17921 \beta_{14} - 6136 \beta_{13} - 8896 \beta_{12} - 4448 \beta_{11} - 6136 \beta_{10} + 20953 \beta_{9} + \cdots - 11328 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 21488 \beta_{15} + 34122 \beta_{14} - 61474 \beta_{13} - 21488 \beta_{12} - 55610 \beta_{11} + \cdots - 111820 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 122050 \beta_{15} - 61025 \beta_{14} - 241902 \beta_{13} - 210020 \beta_{11} + 241902 \beta_{10} + \cdots - 418238 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 300388 \beta_{15} - 673430 \beta_{14} - 206178 \beta_{13} + 300388 \beta_{12} - 373042 \beta_{11} + \cdots - 273360 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2478759 \beta_{14} + 1138040 \beta_{13} + 1631704 \beta_{12} + 815852 \beta_{11} + 1138040 \beta_{10} + \cdots + 2226908 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4055624 \beta_{15} - 4121322 \beta_{14} + 9453366 \beta_{13} + 4055624 \beta_{12} + 8176946 \beta_{11} + \cdots + 16260016 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.15585 − 1.15585i
−1.29491 + 1.29491i
2.48294 − 2.48294i
−0.977855 + 0.977855i
−2.02798 − 2.02798i
2.24849 + 2.24849i
0.830471 + 0.830471i
−1.41701 − 1.41701i
−0.977855 − 0.977855i
2.48294 + 2.48294i
−1.29491 − 1.29491i
1.15585 + 1.15585i
−1.41701 + 1.41701i
0.830471 − 0.830471i
2.24849 − 2.24849i
−2.02798 + 2.02798i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i − 3.62374i 0 −0.866025 − 0.500000i 1.00000i 0 1.81187 + 3.13825i
127.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i − 1.34861i 0 −0.866025 − 0.500000i 1.00000i 0 0.674306 + 1.16793i
127.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i − 0.145508i 0 −0.866025 − 0.500000i 1.00000i 0 0.0727538 + 0.126013i
127.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 4.11786i 0 −0.866025 − 0.500000i 1.00000i 0 −2.05893 − 3.56617i
127.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i − 2.19147i 0 0.866025 + 0.500000i − 1.00000i 0 −1.09573 − 1.89787i
127.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i − 1.24768i 0 0.866025 + 0.500000i − 1.00000i 0 −0.623842 − 1.08053i
127.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.25534i 0 0.866025 + 0.500000i − 1.00000i 0 0.627670 + 1.08716i
127.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 3.18381i 0 0.866025 + 0.500000i − 1.00000i 0 1.59191 + 2.75726i
1135.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i − 4.11786i 0 −0.866025 + 0.500000i − 1.00000i 0 −2.05893 + 3.56617i
1135.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.145508i 0 −0.866025 + 0.500000i − 1.00000i 0 0.0727538 − 0.126013i
1135.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.34861i 0 −0.866025 + 0.500000i − 1.00000i 0 0.674306 − 1.16793i
1135.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.62374i 0 −0.866025 + 0.500000i − 1.00000i 0 1.81187 − 3.13825i
1135.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 3.18381i 0 0.866025 − 0.500000i 1.00000i 0 1.59191 − 2.75726i
1135.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.25534i 0 0.866025 − 0.500000i 1.00000i 0 0.627670 − 1.08716i
1135.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.24768i 0 0.866025 − 0.500000i 1.00000i 0 −0.623842 + 1.08053i
1135.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 2.19147i 0 0.866025 − 0.500000i 1.00000i 0 −1.09573 + 1.89787i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.i yes 16
3.b odd 2 1 1638.2.bj.h ✓ 16
13.e even 6 1 inner 1638.2.bj.i yes 16
39.h odd 6 1 1638.2.bj.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.bj.h ✓ 16 3.b odd 2 1
1638.2.bj.h ✓ 16 39.h odd 6 1
1638.2.bj.i yes 16 1.a even 1 1 trivial
1638.2.bj.i yes 16 13.e 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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{16} + 50 T_{5}^{14} + 953 T_{5}^{12} + 8752 T_{5}^{10} + 40824 T_{5}^{8} + 96800 T_{5}^{6} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{11}^{16} - 12 T_{11}^{15} + 18 T_{11}^{14} + 360 T_{11}^{13} - 881 T_{11}^{12} - 9276 T_{11}^{11} + \cdots + 11075584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 50 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} - 12 T^{15} + \cdots + 11075584 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 184226329 \) Copy content Toggle raw display
$19$ \( T^{16} - 62 T^{14} + \cdots + 11075584 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 112123183104 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 37903417344 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 645566464 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 897122304 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 887876521984 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4269838336 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 7653323063296 \) Copy content Toggle raw display
$53$ \( (T^{8} - 20 T^{7} + \cdots - 4239)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 6 T^{15} + \cdots + 37748736 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 3444868249 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 102072582144 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5464074551296 \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + \cdots + 975168)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 374 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
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