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Properties

Label 1638.2.bj.g
Level $1638$
Weight $2$
Character orbit 1638.bj
Analytic conductor $13.079$
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] = [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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + \beta_{7} q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{6} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + \beta_{7} q^{4} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{5}+ \cdots + \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - 2 q^{10} + 18 q^{11} - 8 q^{13} - 12 q^{14} - 6 q^{16} - 4 q^{17} + 12 q^{19} - 2 q^{22} + 6 q^{23} - 24 q^{25} + 14 q^{26} + 10 q^{29} - 2 q^{35} - 6 q^{37} - 8 q^{38} - 4 q^{40} + 24 q^{41} + 26 q^{43} - 6 q^{46} + 6 q^{49} + 12 q^{50} - 4 q^{52} - 36 q^{53} - 6 q^{55} - 6 q^{56} + 24 q^{58} - 6 q^{59} - 28 q^{61} + 2 q^{62} - 12 q^{64} + 34 q^{65} - 42 q^{67} + 4 q^{68} - 48 q^{71} + 12 q^{76} + 4 q^{77} + 44 q^{79} + 6 q^{82} + 54 q^{85} + 2 q^{88} - 12 q^{89} - 16 q^{91} + 12 q^{92} + 8 q^{94} - 32 q^{95} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24 \nu^{10} - 120 \nu^{9} + 751 \nu^{8} - 2284 \nu^{7} + 6728 \nu^{6} - 12694 \nu^{5} + 20323 \nu^{4} + \cdots + 2041 ) / 286 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31 \nu^{10} + 155 \nu^{9} - 976 \nu^{8} + 2974 \nu^{7} - 8881 \nu^{6} + 16885 \nu^{5} - 28044 \nu^{4} + \cdots - 5252 ) / 286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 162 \nu^{11} + 186 \nu^{10} - 51 \nu^{9} + 15887 \nu^{8} - 51264 \nu^{7} + 192268 \nu^{6} + \cdots + 66227 ) / 7898 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2323 \nu^{11} - 22649 \nu^{10} + 132943 \nu^{9} - 590285 \nu^{8} + 1852053 \nu^{7} - 4762085 \nu^{6} + \cdots - 1517893 ) / 102674 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2323 \nu^{11} + 2904 \nu^{10} - 34218 \nu^{9} - 29708 \nu^{8} + 35569 \nu^{7} - 841546 \nu^{6} + \cdots - 689598 ) / 102674 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 324 \nu^{11} + 1782 \nu^{10} - 10668 \nu^{9} + 34641 \nu^{8} - 98512 \nu^{7} + 195608 \nu^{6} + \cdots - 3573 ) / 7898 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5977 \nu^{11} + 33053 \nu^{10} - 223218 \nu^{9} + 741326 \nu^{8} - 2485739 \nu^{7} + \cdots - 527956 ) / 102674 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5977 \nu^{11} - 32694 \nu^{10} + 221423 \nu^{9} - 766456 \nu^{8} + 2597029 \nu^{7} - 6035626 \nu^{6} + \cdots - 3597672 ) / 102674 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6388 \nu^{11} + 30826 \nu^{10} - 187767 \nu^{9} + 543572 \nu^{8} - 1501380 \nu^{7} + \cdots - 435708 ) / 102674 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9240 \nu^{11} + 55128 \nu^{10} - 344643 \nu^{9} + 1207618 \nu^{8} - 3759676 \nu^{7} + \cdots + 1667016 ) / 102674 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 4 \beta_{5} - \beta_{3} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - \beta_{8} - 6 \beta_{7} - \beta_{6} - 21 \beta_{5} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{11} - 15 \beta_{10} - 8 \beta_{9} + 13 \beta_{8} + 25 \beta_{7} - 73 \beta_{6} + 26 \beta_{5} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 29 \beta_{11} - 50 \beta_{10} - 48 \beta_{9} + 25 \beta_{8} + 124 \beta_{7} - 81 \beta_{6} + 266 \beta_{5} + \cdots - 247 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 60 \beta_{11} + 135 \beta_{10} + 17 \beta_{9} - 115 \beta_{8} - 145 \beta_{7} + 668 \beta_{6} + \cdots - 526 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 380 \beta_{11} + 778 \beta_{10} + 515 \beta_{9} - 363 \beta_{8} - 1632 \beta_{7} + 1517 \beta_{6} + \cdots + 2008 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 264 \beta_{11} - 673 \beta_{10} + 454 \beta_{9} + 839 \beta_{8} - 287 \beta_{7} - 5265 \beta_{6} + \cdots + 7116 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4383 \beta_{11} - 9560 \beta_{10} - 4630 \beta_{9} + 4411 \beta_{8} + 17550 \beta_{7} - 20512 \beta_{6} + \cdots - 13713 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1759 \beta_{11} - 3186 \beta_{10} - 9684 \beta_{9} - 4506 \beta_{8} + 22413 \beta_{7} + 33458 \beta_{6} + \cdots - 85518 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\)

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
0.500000 − 0.399480i
0.500000 − 3.15681i
0.500000 + 1.69027i
0.500000 − 1.73154i
0.500000 − 0.613147i
0.500000 + 2.47866i
0.500000 − 1.69027i
0.500000 + 3.15681i
0.500000 + 0.399480i
0.500000 − 2.47866i
0.500000 + 0.613147i
0.500000 + 1.73154i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i − 3.38938i 0 0.866025 + 0.500000i 1.00000i 0 1.69469 + 2.93529i
127.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.901839i 0 0.866025 + 0.500000i 1.00000i 0 −0.450919 − 0.781015i
127.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 3.48754i 0 0.866025 + 0.500000i 1.00000i 0 −1.74377 − 3.02030i
127.4 0.866025 − 0.500000i 0 0.500000 − 0.866025i − 3.71131i 0 −0.866025 − 0.500000i − 1.00000i 0 −1.85566 − 3.21409i
127.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.14776i 0 −0.866025 − 0.500000i − 1.00000i 0 0.573878 + 0.993985i
127.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.56356i 0 −0.866025 − 0.500000i − 1.00000i 0 0.781779 + 1.35408i
1135.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i − 3.48754i 0 0.866025 − 0.500000i − 1.00000i 0 −1.74377 + 3.02030i
1135.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i − 0.901839i 0 0.866025 − 0.500000i − 1.00000i 0 −0.450919 + 0.781015i
1135.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.38938i 0 0.866025 − 0.500000i − 1.00000i 0 1.69469 − 2.93529i
1135.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.56356i 0 −0.866025 + 0.500000i 1.00000i 0 0.781779 − 1.35408i
1135.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i − 1.14776i 0 −0.866025 + 0.500000i 1.00000i 0 0.573878 − 0.993985i
1135.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.71131i 0 −0.866025 + 0.500000i 1.00000i 0 −1.85566 + 3.21409i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.6
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.g 12
3.b odd 2 1 182.2.m.b ✓ 12
12.b even 2 1 1456.2.cc.d 12
13.e even 6 1 inner 1638.2.bj.g 12
21.c even 2 1 1274.2.m.c 12
21.g even 6 1 1274.2.o.e 12
21.g even 6 1 1274.2.v.d 12
21.h odd 6 1 1274.2.o.d 12
21.h odd 6 1 1274.2.v.e 12
39.h odd 6 1 182.2.m.b ✓ 12
39.h odd 6 1 2366.2.d.r 12
39.i odd 6 1 2366.2.d.r 12
39.k even 12 1 2366.2.a.bf 6
39.k even 12 1 2366.2.a.bh 6
156.r even 6 1 1456.2.cc.d 12
273.u even 6 1 1274.2.m.c 12
273.x odd 6 1 1274.2.o.d 12
273.y even 6 1 1274.2.o.e 12
273.bp odd 6 1 1274.2.v.e 12
273.br even 6 1 1274.2.v.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b ✓ 12 3.b odd 2 1
182.2.m.b ✓ 12 39.h odd 6 1
1274.2.m.c 12 21.c even 2 1
1274.2.m.c 12 273.u even 6 1
1274.2.o.d 12 21.h odd 6 1
1274.2.o.d 12 273.x odd 6 1
1274.2.o.e 12 21.g even 6 1
1274.2.o.e 12 273.y even 6 1
1274.2.v.d 12 21.g even 6 1
1274.2.v.d 12 273.br even 6 1
1274.2.v.e 12 21.h odd 6 1
1274.2.v.e 12 273.bp odd 6 1
1456.2.cc.d 12 12.b even 2 1
1456.2.cc.d 12 156.r even 6 1
1638.2.bj.g 12 1.a even 1 1 trivial
1638.2.bj.g 12 13.e even 6 1 inner
2366.2.a.bf 6 39.k even 12 1
2366.2.a.bh 6 39.k even 12 1
2366.2.d.r 12 39.h odd 6 1
2366.2.d.r 12 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{12} + 42T_{5}^{10} + 643T_{5}^{8} + 4292T_{5}^{6} + 11827T_{5}^{4} + 13306T_{5}^{2} + 5041 \) Copy content Toggle raw display
\( T_{11}^{12} - 18 T_{11}^{11} + 118 T_{11}^{10} - 180 T_{11}^{9} - 1292 T_{11}^{8} + 2976 T_{11}^{7} + \cdots + 495616 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 42 T^{10} + \cdots + 5041 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - 18 T^{11} + \cdots + 495616 \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 369869824 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + \cdots + 9872164 \) Copy content Toggle raw display
$29$ \( T^{12} - 10 T^{11} + \cdots + 135424 \) Copy content Toggle raw display
$31$ \( T^{12} + 88 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{12} + 6 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$43$ \( T^{12} - 26 T^{11} + \cdots + 8667136 \) Copy content Toggle raw display
$47$ \( T^{12} + 272 T^{10} + \cdots + 31719424 \) Copy content Toggle raw display
$53$ \( (T^{6} + 18 T^{5} + \cdots + 44928)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 6 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 80364879169 \) Copy content Toggle raw display
$67$ \( T^{12} + 42 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 750321664 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2708994304 \) Copy content Toggle raw display
$79$ \( (T^{6} - 22 T^{5} + \cdots + 8032)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 879478336 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10303062016 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 6400000000 \) Copy content Toggle raw display
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