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Properties

Label 1386.2.bk.c
Level $1386$
Weight $2$
Character orbit 1386.bk
Analytic conductor $11.067$
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] = [1386,2,Mod(703,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.bk (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: \(11.0672657201\)
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: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 154)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{10} q^{4} + (\beta_{10} + \beta_{8}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_1) q^{7}+ \cdots - \beta_{11} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{10} q^{4} + (\beta_{10} + \beta_{8}) q^{5} + ( - \beta_{15} - \beta_{14} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{15} - 4 \beta_{14} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 12 q^{5} - 8 q^{11} + 8 q^{14} - 8 q^{16} - 8 q^{22} - 16 q^{23} + 36 q^{26} - 12 q^{31} - 16 q^{37} - 12 q^{38} + 8 q^{44} - 24 q^{47} + 8 q^{49} + 28 q^{53} + 4 q^{56} - 12 q^{58} - 60 q^{59} - 16 q^{64} + 12 q^{67} + 60 q^{70} - 8 q^{71} - 44 q^{77} - 12 q^{80} - 20 q^{86} - 4 q^{88} - 96 q^{89} - 36 q^{91} - 32 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 115452774644 \nu^{15} + 886173093256 \nu^{14} - 3342656190846 \nu^{13} + \cdots - 1622048373702 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 229876192869 \nu^{15} - 1434169055319 \nu^{14} + 4481995762636 \nu^{13} + \cdots + 7097366139528 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 234407101203 \nu^{15} - 1463062841541 \nu^{14} + 4568245377402 \nu^{13} + \cdots - 5702240414038 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 323659513794 \nu^{15} + 2126436132890 \nu^{14} - 7021940502570 \nu^{13} + \cdots - 2750714974023 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 394538957168 \nu^{15} + 2231811901383 \nu^{14} - 6199106801734 \nu^{13} + \cdots - 1505816495286 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 401980321088 \nu^{15} - 2507531613006 \nu^{14} + 7838812215514 \nu^{13} + \cdots + 6420259797322 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 403027374657 \nu^{15} + 2521501247748 \nu^{14} - 7885517844470 \nu^{13} + \cdots + 4012225523467 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 556158552240 \nu^{15} - 3247720931080 \nu^{14} + 9404820278882 \nu^{13} + \cdots + 2653688275785 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 1370541375 ) / 1973128213 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2432602484212 \nu^{15} - 15210362430228 \nu^{14} + 47566230428388 \nu^{13} + \cdots + 3660982645558 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2667083154356 \nu^{15} + 16670596841871 \nu^{14} - 52123769377554 \nu^{13} + \cdots - 5233663502032 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2737320095452 \nu^{15} - 16635640723011 \nu^{14} + 50529646182960 \nu^{13} + \cdots + 16350688060416 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2891771622084 \nu^{15} + 17710086389784 \nu^{14} - 54266401205842 \nu^{13} + \cdots - 17956801097484 ) / 3707507912227 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{6} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 4 \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + \beta_{9} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{12} + 9\beta_{10} + 2\beta_{8} - \beta_{7} + 5\beta_{4} + 5\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 19 \beta_{12} + 28 \beta_{11} + 14 \beta_{10} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{15} + \beta_{14} + 46\beta_{11} - 8\beta_{9} + \beta_{6} + 16\beta_{5} - 26\beta_{2} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 96 \beta_{15} - 47 \beta_{14} - 96 \beta_{13} - 47 \beta_{12} + 82 \beta_{11} + 82 \beta_{10} + \cdots - 113 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -138\beta_{13} + 10\beta_{12} + 245\beta_{10} + 50\beta_{8} - 100\beta_{7} - 10\beta_{4} + 138\beta_{3} - 195 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 223 \beta_{15} + 501 \beta_{14} - 278 \beta_{13} + 278 \beta_{12} + 456 \beta_{11} + 912 \beta_{10} + \cdots - 456 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 70 \beta_{15} + 739 \beta_{14} + 1320 \beta_{11} - 576 \beta_{9} - 669 \beta_{6} + 288 \beta_{5} + \cdots + 962 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1533 \beta_{15} + 1533 \beta_{14} - 1125 \beta_{13} - 2658 \beta_{12} + 4980 \beta_{11} - 2490 \beta_{10} + \cdots - 817 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -3545\beta_{13} - 3545\beta_{12} - 1603\beta_{8} - 1603\beta_{7} - 3973\beta_{4} + 428\beta_{3} - 3923 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14219 \beta_{15} + 5865 \beta_{14} - 14219 \beta_{13} - 5865 \beta_{12} - 13502 \beta_{11} + \cdots - 17794 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18939\beta_{15} + 18939\beta_{14} - 8782\beta_{9} - 21398\beta_{6} - 8782\beta_{5} + 2459\beta_{2} + 39879\beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 31097 \beta_{15} + 76382 \beta_{14} + 45285 \beta_{13} - 45285 \beta_{12} + 73006 \beta_{11} + \cdots + 73006 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(1 - \beta_{10}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.60599 − 0.430324i
−1.29724 + 0.347596i
−0.186243 + 0.0499037i
2.24352 − 0.601150i
0.430324 + 1.60599i
−0.347596 − 1.29724i
−0.0499037 − 0.186243i
0.601150 + 2.24352i
1.60599 + 0.430324i
−1.29724 − 0.347596i
−0.186243 − 0.0499037i
2.24352 + 0.601150i
0.430324 − 1.60599i
−0.347596 + 1.29724i
−0.0499037 + 0.186243i
0.601150 − 2.24352i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.09005 + 0.629341i 0 2.11465 + 1.59005i 1.00000i 0 0.629341 − 1.09005i
703.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.882559 + 0.509546i 0 −2.25578 + 1.38256i 1.00000i 0 0.509546 − 0.882559i
703.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.90775 − 1.10144i 0 −2.24014 − 1.40775i 1.00000i 0 −1.10144 + 1.90775i
703.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 3.06486 − 1.76950i 0 0.649221 − 2.56486i 1.00000i 0 −1.76950 + 3.06486i
703.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.09005 + 0.629341i 0 −2.11465 − 1.59005i − 1.00000i 0 −0.629341 + 1.09005i
703.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.882559 + 0.509546i 0 2.25578 − 1.38256i − 1.00000i 0 −0.509546 + 0.882559i
703.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.90775 − 1.10144i 0 2.24014 + 1.40775i − 1.00000i 0 1.10144 − 1.90775i
703.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 3.06486 − 1.76950i 0 −0.649221 + 2.56486i − 1.00000i 0 1.76950 − 3.06486i
901.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.09005 − 0.629341i 0 2.11465 − 1.59005i − 1.00000i 0 0.629341 + 1.09005i
901.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.882559 − 0.509546i 0 −2.25578 − 1.38256i − 1.00000i 0 0.509546 + 0.882559i
901.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.90775 + 1.10144i 0 −2.24014 + 1.40775i − 1.00000i 0 −1.10144 − 1.90775i
901.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 3.06486 + 1.76950i 0 0.649221 + 2.56486i − 1.00000i 0 −1.76950 − 3.06486i
901.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.09005 − 0.629341i 0 −2.11465 + 1.59005i 1.00000i 0 −0.629341 − 1.09005i
901.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.882559 − 0.509546i 0 2.25578 + 1.38256i 1.00000i 0 −0.509546 − 0.882559i
901.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.90775 + 1.10144i 0 2.24014 − 1.40775i 1.00000i 0 1.10144 + 1.90775i
901.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.06486 + 1.76950i 0 −0.649221 − 2.56486i 1.00000i 0 1.76950 + 3.06486i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.bk.c 16
3.b odd 2 1 154.2.i.a ✓ 16
7.d odd 6 1 inner 1386.2.bk.c 16
11.b odd 2 1 inner 1386.2.bk.c 16
12.b even 2 1 1232.2.bn.b 16
21.c even 2 1 1078.2.i.c 16
21.g even 6 1 154.2.i.a ✓ 16
21.g even 6 1 1078.2.c.b 16
21.h odd 6 1 1078.2.c.b 16
21.h odd 6 1 1078.2.i.c 16
33.d even 2 1 154.2.i.a ✓ 16
77.i even 6 1 inner 1386.2.bk.c 16
84.j odd 6 1 1232.2.bn.b 16
132.d odd 2 1 1232.2.bn.b 16
231.h odd 2 1 1078.2.i.c 16
231.k odd 6 1 154.2.i.a ✓ 16
231.k odd 6 1 1078.2.c.b 16
231.l even 6 1 1078.2.c.b 16
231.l even 6 1 1078.2.i.c 16
924.y even 6 1 1232.2.bn.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.i.a ✓ 16 3.b odd 2 1
154.2.i.a ✓ 16 21.g even 6 1
154.2.i.a ✓ 16 33.d even 2 1
154.2.i.a ✓ 16 231.k odd 6 1
1078.2.c.b 16 21.g even 6 1
1078.2.c.b 16 21.h odd 6 1
1078.2.c.b 16 231.k odd 6 1
1078.2.c.b 16 231.l even 6 1
1078.2.i.c 16 21.c even 2 1
1078.2.i.c 16 21.h odd 6 1
1078.2.i.c 16 231.h odd 2 1
1078.2.i.c 16 231.l even 6 1
1232.2.bn.b 16 12.b even 2 1
1232.2.bn.b 16 84.j odd 6 1
1232.2.bn.b 16 132.d odd 2 1
1232.2.bn.b 16 924.y even 6 1
1386.2.bk.c 16 1.a even 1 1 trivial
1386.2.bk.c 16 7.d odd 6 1 inner
1386.2.bk.c 16 11.b odd 2 1 inner
1386.2.bk.c 16 77.i 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 \(S_{2}^{\mathrm{new}}(1386, [\chi])\):

\( T_{5}^{8} - 6T_{5}^{7} + 8T_{5}^{6} + 24T_{5}^{5} - 30T_{5}^{4} - 72T_{5}^{3} + 68T_{5}^{2} + 180T_{5} + 100 \) Copy content Toggle raw display
\( T_{13}^{8} - 44T_{13}^{6} + 510T_{13}^{4} - 908T_{13}^{2} + 25 \) 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
$5$ \( (T^{8} - 6 T^{7} + \cdots + 100)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 44 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6146560000 \) Copy content Toggle raw display
$19$ \( T^{16} + 80 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{7} + \cdots + 196)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 36 T^{6} + 366 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 6 T^{7} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 8 T^{7} + 52 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 200 T^{6} + \cdots + 1674436)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 244 T^{6} + \cdots + 1201216)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 12 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 14 T^{7} + \cdots + 1170724)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 30 T^{7} + \cdots + 2356225)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1706808989601 \) Copy content Toggle raw display
$67$ \( (T^{8} - 6 T^{7} + \cdots + 173889)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + \cdots + 3262)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 15704099856 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{8} - 260 T^{6} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 48 T^{7} + \cdots + 49617936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 644 T^{6} + \cdots + 496532089)^{2} \) Copy content Toggle raw display
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