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Properties

Label 1386.2.c.a
Level $1386$
Weight $2$
Character orbit 1386.c
Analytic conductor $11.067$
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] = [1386,2,Mod(197,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.197");
 
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.c (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: \(11.0672657201\)
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: \( x^{12} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{5} + \beta_1) q^{5} - \beta_1 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + ( - \beta_{5} + \beta_1) q^{5} - \beta_1 q^{7} - q^{8} + (\beta_{5} - \beta_1) q^{10} + ( - \beta_{9} - \beta_1) q^{11} + ( - \beta_{11} + \beta_{7}) q^{13} + \beta_1 q^{14} + q^{16} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 1) q^{17}+ \cdots + q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{11} + 12 q^{16} - 16 q^{17} - 4 q^{22} - 4 q^{25} - 16 q^{29} - 12 q^{32} + 16 q^{34} + 8 q^{35} + 24 q^{37} - 16 q^{41} + 4 q^{44} - 12 q^{49} + 4 q^{50} - 8 q^{55} + 16 q^{58} + 12 q^{64} - 48 q^{67} - 16 q^{68} - 8 q^{70} - 24 q^{74} - 8 q^{77} + 16 q^{82} - 16 q^{83} - 4 q^{88} + 48 q^{95} + 48 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4x^{10} - 8x^{9} + 17x^{8} - 16x^{7} + 88x^{6} - 48x^{5} + 153x^{4} - 216x^{3} + 324x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6 \nu^{11} - 97 \nu^{10} + 42 \nu^{9} - 301 \nu^{8} + 1004 \nu^{7} - 944 \nu^{6} + 1684 \nu^{5} + \cdots + 567 ) / 11502 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20 \nu^{11} - 1767 \nu^{10} + 3122 \nu^{9} - 7417 \nu^{8} + 16600 \nu^{7} - 44516 \nu^{6} + \cdots - 502281 ) / 34506 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 89 \nu^{11} - 206 \nu^{10} + 655 \nu^{9} - 1156 \nu^{8} + 2100 \nu^{7} - 9806 \nu^{6} + \cdots - 145476 ) / 11502 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 335 \nu^{11} + 489 \nu^{10} - 2341 \nu^{9} - 535 \nu^{8} - 8648 \nu^{7} + 22222 \nu^{6} + \cdots + 364257 ) / 34506 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 421 \nu^{11} - 495 \nu^{10} + 35 \nu^{9} - 97 \nu^{8} + 1168 \nu^{7} - 14324 \nu^{6} + \cdots - 328293 ) / 34506 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{11} - 12 \nu^{10} + 37 \nu^{9} - 50 \nu^{8} + 170 \nu^{7} - 172 \nu^{6} + 448 \nu^{5} + \cdots + 1863 \nu ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 707 \nu^{11} - 510 \nu^{10} - 263 \nu^{9} + 808 \nu^{8} - 1486 \nu^{7} - 14530 \nu^{6} + \cdots - 324648 ) / 34506 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 761 \nu^{11} - 489 \nu^{10} - 641 \nu^{9} + 109 \nu^{8} - 3706 \nu^{7} - 20518 \nu^{6} + \cdots - 398763 ) / 34506 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 267 \nu^{11} + 1015 \nu^{10} - 1869 \nu^{9} + 4981 \nu^{8} - 10598 \nu^{7} + 19430 \nu^{6} + \cdots + 155925 ) / 11502 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1471 \nu^{11} + 2067 \nu^{10} - 3481 \nu^{9} + 12179 \nu^{8} - 24722 \nu^{7} + 18958 \nu^{6} + \cdots - 157221 ) / 34506 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 305 \nu^{11} - 264 \nu^{10} + 208 \nu^{9} - 47 \nu^{8} + 533 \nu^{7} - 9547 \nu^{6} + \cdots - 194643 ) / 5751 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 2\beta_{10} + \beta_{9} + 2\beta_{8} - 2\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - 3\beta_{9} + \beta_{8} - 3\beta_{6} - 3\beta_{5} - 2\beta_{4} - 3\beta_{3} - \beta_{2} - 3\beta _1 - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 9 \beta_{10} - 8 \beta_{9} - 5 \beta_{8} - 5 \beta_{7} + 5 \beta_{5} + 4 \beta_{4} + \cdots + 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} + 2 \beta_{10} + \beta_{9} + 3 \beta_{8} - 5 \beta_{7} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + \cdots - 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{11} - 13 \beta_{10} + 20 \beta_{9} - 5 \beta_{8} + 11 \beta_{7} - 6 \beta_{6} - 29 \beta_{5} + \cdots - 42 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12 \beta_{11} + 23 \beta_{10} - 21 \beta_{9} - 49 \beta_{8} + 12 \beta_{7} + 3 \beta_{6} + 9 \beta_{5} + \cdots - 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 31 \beta_{11} + 24 \beta_{10} + 5 \beta_{9} - 10 \beta_{8} + 8 \beta_{7} + 18 \beta_{6} + 49 \beta_{5} + \cdots - 78 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 16 \beta_{11} - 71 \beta_{10} + 89 \beta_{9} + 18 \beta_{8} + 44 \beta_{7} + 4 \beta_{5} - 42 \beta_{4} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61 \beta_{11} - 38 \beta_{10} - 35 \beta_{9} - 70 \beta_{8} + 112 \beta_{7} + 102 \beta_{6} - 67 \beta_{5} + \cdots + 246 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 156 \beta_{11} + 61 \beta_{10} - 123 \beta_{9} - 41 \beta_{8} - 108 \beta_{7} - 357 \beta_{6} + \cdots + 435 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 359 \beta_{11} - 201 \beta_{10} - 26 \beta_{9} + 529 \beta_{8} - 209 \beta_{7} - 210 \beta_{6} + \cdots + 210 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−1.26053 + 1.18788i
1.49380 − 0.876678i
−0.0383715 − 1.73163i
−0.748321 − 1.56205i
−0.833477 + 1.51833i
1.38690 − 1.03755i
1.38690 + 1.03755i
−0.833477 − 1.51833i
−0.748321 + 1.56205i
−0.0383715 + 1.73163i
1.49380 + 0.876678i
−1.26053 − 1.18788i
−1.00000 0 1.00000 − 3.46258i 0 1.00000i −1.00000 0 3.46258i
197.2 −1.00000 0 1.00000 − 3.35236i 0 1.00000i −1.00000 0 3.35236i
197.3 −1.00000 0 1.00000 − 2.50315i 0 − 1.00000i −1.00000 0 2.50315i
197.4 −1.00000 0 1.00000 − 1.15079i 0 1.00000i −1.00000 0 1.15079i
197.5 −1.00000 0 1.00000 − 0.968524i 0 − 1.00000i −1.00000 0 0.968524i
197.6 −1.00000 0 1.00000 − 0.494054i 0 − 1.00000i −1.00000 0 0.494054i
197.7 −1.00000 0 1.00000 0.494054i 0 1.00000i −1.00000 0 − 0.494054i
197.8 −1.00000 0 1.00000 0.968524i 0 1.00000i −1.00000 0 − 0.968524i
197.9 −1.00000 0 1.00000 1.15079i 0 − 1.00000i −1.00000 0 − 1.15079i
197.10 −1.00000 0 1.00000 2.50315i 0 1.00000i −1.00000 0 − 2.50315i
197.11 −1.00000 0 1.00000 3.35236i 0 − 1.00000i −1.00000 0 − 3.35236i
197.12 −1.00000 0 1.00000 3.46258i 0 − 1.00000i −1.00000 0 − 3.46258i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.c.a ✓ 12
3.b odd 2 1 1386.2.c.b yes 12
11.b odd 2 1 1386.2.c.b yes 12
33.d even 2 1 inner 1386.2.c.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.c.a ✓ 12 1.a even 1 1 trivial
1386.2.c.a ✓ 12 33.d even 2 1 inner
1386.2.c.b yes 12 3.b odd 2 1
1386.2.c.b yes 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{6} + 8T_{17}^{5} - 20T_{17}^{4} - 136T_{17}^{3} + 306T_{17}^{2} + 136T_{17} - 376 \) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 84 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{6} + 8 T^{5} + \cdots - 376)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 68 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{12} + 180 T^{10} + \cdots + 79709184 \) Copy content Toggle raw display
$29$ \( (T^{6} + 8 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 76 T^{4} + \cdots + 3656)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 12 T^{5} + \cdots + 10688)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} - 76 T^{4} + \cdots - 56)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 212 T^{10} + \cdots + 541696 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 7322567184 \) Copy content Toggle raw display
$53$ \( T^{12} + 352 T^{10} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 339738624 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 183223296 \) Copy content Toggle raw display
$67$ \( (T^{6} + 24 T^{5} + \cdots - 42752)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 244 T^{10} + \cdots + 16516096 \) Copy content Toggle raw display
$73$ \( T^{12} + 496 T^{10} + \cdots + 14868736 \) Copy content Toggle raw display
$79$ \( T^{12} + 424 T^{10} + \cdots + 14992384 \) Copy content Toggle raw display
$83$ \( (T^{6} + 8 T^{5} + \cdots + 94856)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 133476238336 \) Copy content Toggle raw display
$97$ \( (T^{6} - 24 T^{5} + \cdots - 112896)^{2} \) Copy content Toggle raw display
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