↑

Properties

Label 387.3.b.d
Level $387$
Weight $3$
Character orbit 387.b
Analytic conductor $10.545$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,3,Mod(343,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.343");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.b (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: \(10.5449862307\)
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: \( x^{12} - 2 x^{11} - 41 x^{10} + 172 x^{9} + 185 x^{8} - 2594 x^{7} + 7839 x^{6} - 15468 x^{5} + \cdots + 45998 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{4} q^{5} + \beta_{8} q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{4} q^{5} + \beta_{8} q^{7} - \beta_{3} q^{8} + (3 \beta_{5} - \beta_{2} + 2) q^{10} + (\beta_{9} - \beta_{7}) q^{11} + ( - \beta_{5} - \beta_{2} - 4) q^{13} + (3 \beta_{7} + \beta_{6}) q^{14} + (2 \beta_{5} - 2 \beta_{2} - 3) q^{16} + ( - 2 \beta_{7} + \beta_{6}) q^{17} + \beta_{11} q^{19} + (\beta_{4} + \beta_{3} + \beta_1) q^{20} - \beta_{10} q^{22} + ( - \beta_{9} - 2 \beta_{7} + 2 \beta_{6}) q^{23} + (\beta_{5} - 5 \beta_{2} - 3) q^{25} + (\beta_{4} + \beta_{3} - \beta_1) q^{26} + (\beta_{11} - \beta_{10} - \beta_{8}) q^{28} + (\beta_{4} - \beta_{3} - \beta_1) q^{29} + (8 \beta_{5} + 3 \beta_{2} + 9) q^{31} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{32} + (\beta_{11} - \beta_{10}) q^{34} + ( - 3 \beta_{9} - 5 \beta_{7} + 2 \beta_{6}) q^{35} + ( - \beta_{10} + \beta_{8}) q^{37} + (3 \beta_{9} - \beta_{7} - 4 \beta_{6}) q^{38} + (13 \beta_{5} + 2 \beta_{2} - 1) q^{40} + (2 \beta_{9} + 2 \beta_{7}) q^{41} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \cdots - 5) q^{43}+ \cdots + (9 \beta_{4} - 15 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} + 8 q^{10} - 48 q^{13} - 52 q^{16} - 60 q^{25} + 88 q^{31} - 56 q^{40} - 100 q^{43} - 252 q^{49} - 144 q^{52} + 104 q^{58} - 100 q^{64} + 376 q^{67} + 8 q^{79} + 512 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 41 x^{10} + 172 x^{9} + 185 x^{8} - 2594 x^{7} + 7839 x^{6} - 15468 x^{5} + \cdots + 45998 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!33 \nu^{11} + \cdots + 47\!\cdots\!35 ) / 77\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 199548104474086 \nu^{11} - 54505469446973 \nu^{10} + \cdots + 52\!\cdots\!69 ) / 99\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!73 \nu^{11} + \cdots - 12\!\cdots\!89 ) / 70\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28\!\cdots\!49 \nu^{11} + \cdots + 59\!\cdots\!30 ) / 77\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 499713287381196 \nu^{11} + 491208559417812 \nu^{10} + \cdots + 11\!\cdots\!63 ) / 99\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\!\cdots\!28 \nu^{11} + \cdots - 70\!\cdots\!36 ) / 77\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!09 \nu^{11} + \cdots + 13\!\cdots\!38 ) / 77\!\cdots\!49 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 682251935928514 \nu^{11} + 502525320179081 \nu^{10} + \cdots + 17\!\cdots\!42 ) / 99\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!74 \nu^{11} + \cdots - 26\!\cdots\!56 ) / 70\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 829704490022920 \nu^{11} + \cdots - 86\!\cdots\!72 ) / 99\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!90 \nu^{11} - 577597497221889 \nu^{10} + \cdots - 76\!\cdots\!18 ) / 99\!\cdots\!29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - 2\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{2} - \beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + \beta_{9} - 3 \beta_{8} + 38 \beta_{7} - 7 \beta_{6} - 46 \beta_{5} + 6 \beta_{4} + \cdots - 98 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{11} - 5 \beta_{10} - 18 \beta_{9} + 30 \beta_{8} - 84 \beta_{7} + 22 \beta_{6} + 84 \beta_{5} + \cdots + 327 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 125 \beta_{11} + 20 \beta_{10} + 121 \beta_{9} - 285 \beta_{8} + 956 \beta_{7} - 203 \beta_{6} + \cdots - 3222 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 321 \beta_{11} - 156 \beta_{10} - 349 \beta_{9} + 1187 \beta_{8} - 2402 \beta_{7} + 537 \beta_{6} + \cdots + 8380 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 4767 \beta_{11} + 1330 \beta_{10} + 3801 \beta_{9} - 13321 \beta_{8} + 23380 \beta_{7} - 5481 \beta_{6} + \cdots - 84732 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 13558 \beta_{11} - 5203 \beta_{10} - 7364 \beta_{9} + 44392 \beta_{8} - 58136 \beta_{7} + 12292 \beta_{6} + \cdots + 193431 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 172197 \beta_{11} + 56436 \beta_{10} + 86527 \beta_{9} - 518109 \beta_{8} + 502694 \beta_{7} + \cdots - 1865666 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 501020 \beta_{11} - 177142 \beta_{10} - 135659 \beta_{9} + 1568559 \beta_{8} - 1111518 \beta_{7} + \cdots + 3650144 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5906637 \beta_{11} + 2028708 \beta_{10} + 1339353 \beta_{9} - 18212249 \beta_{8} + 7509904 \beta_{7} + \cdots - 28278582 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/387\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(173\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
−0.995169 + 1.66771i
1.48748 + 1.66771i
0.668304 + 0.991433i
3.62465 + 0.991433i
−5.60398 + 0.697006i
1.81872 + 0.697006i
1.81872 − 0.697006i
−5.60398 − 0.697006i
3.62465 − 0.991433i
0.668304 − 0.991433i
1.48748 − 1.66771i
−0.995169 − 1.66771i
− 3.33541i 0 −7.12497 1.69336i 0 − 8.28065i 10.4231i 0 5.64804
343.2 − 3.33541i 0 −7.12497 1.69336i 0 8.28065i 10.4231i 0 5.64804
343.3 − 1.98287i 0 0.0682401 − 6.52948i 0 − 5.86203i − 8.06678i 0 −12.9471
343.4 − 1.98287i 0 0.0682401 − 6.52948i 0 5.86203i − 8.06678i 0 −12.9471
343.5 − 1.39401i 0 2.05673 6.67071i 0 − 10.3473i − 8.44316i 0 9.29905
343.6 − 1.39401i 0 2.05673 6.67071i 0 10.3473i − 8.44316i 0 9.29905
343.7 1.39401i 0 2.05673 − 6.67071i 0 − 10.3473i 8.44316i 0 9.29905
343.8 1.39401i 0 2.05673 − 6.67071i 0 10.3473i 8.44316i 0 9.29905
343.9 1.98287i 0 0.0682401 6.52948i 0 − 5.86203i 8.06678i 0 −12.9471
343.10 1.98287i 0 0.0682401 6.52948i 0 5.86203i 8.06678i 0 −12.9471
343.11 3.33541i 0 −7.12497 − 1.69336i 0 − 8.28065i − 10.4231i 0 5.64804
343.12 3.33541i 0 −7.12497 − 1.69336i 0 8.28065i − 10.4231i 0 5.64804
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.b odd 2 1 inner
129.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.b.d ✓ 12
3.b odd 2 1 inner 387.3.b.d ✓ 12
43.b odd 2 1 inner 387.3.b.d ✓ 12
129.d even 2 1 inner 387.3.b.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.3.b.d ✓ 12 1.a even 1 1 trivial
387.3.b.d ✓ 12 3.b odd 2 1 inner
387.3.b.d ✓ 12 43.b odd 2 1 inner
387.3.b.d ✓ 12 129.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\):

\( T_{2}^{6} + 17T_{2}^{4} + 73T_{2}^{2} + 85 \) Copy content Toggle raw display
\( T_{11}^{6} - 490T_{11}^{4} + 48755T_{11}^{2} - 189952 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 17 T^{4} + \cdots + 85)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 90 T^{4} + \cdots + 5440)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 210 T^{4} + \cdots + 252280)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 490 T^{4} + \cdots - 189952)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 12 T^{2} + \cdots - 34)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} - 756 T^{4} + \cdots - 1187200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 1946 T^{4} + \cdots + 122103520)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 1988 T^{4} + \cdots - 141051232)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 398 T^{4} + \cdots + 5440)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 22 T^{2} + \cdots - 10156)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 2940 T^{4} + \cdots + 291635680)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 2408 T^{4} + \cdots - 474880000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 50 T^{5} + \cdots + 6321363049)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 5670 T^{4} + \cdots - 1577316888)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 4984 T^{4} + \cdots - 1308579328)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 12572 T^{4} + \cdots - 66234410368)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 22624 T^{4} + \cdots + 365709124480)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 94 T^{2} + \cdots + 213968)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + 21692 T^{4} + \cdots + 128702218240)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 8708 T^{4} + \cdots + 100912000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 2 T^{2} + \cdots - 505252)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 11438 T^{4} + \cdots - 48586504288)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 18172 T^{4} + \cdots + 15610819840)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 128 T^{2} + \cdots + 85058)^{4} \) Copy content Toggle raw display
show more
show less