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Properties

Label 387.2.d.b
Level $387$
Weight $2$
Character orbit 387.d
Analytic conductor $3.090$
Analytic rank $0$
Dimension $12$
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] = [387,2,Mod(386,387)]
 
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([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.386");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.d (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: \(3.09021055822\)
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: \( x^{12} - 4 x^{11} + 22 x^{10} - 60 x^{9} + 190 x^{8} - 460 x^{7} + 1088 x^{6} - 2108 x^{5} + 2849 x^{4} + \cdots + 1688 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{6} + \beta_{3} + 2) q^{4} + \beta_1 q^{5} - \beta_{8} q^{7} + (\beta_{7} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_{6} + \beta_{3} + 2) q^{4} + \beta_1 q^{5} - \beta_{8} q^{7} + (\beta_{7} - \beta_{4}) q^{8} + (2 \beta_{6} - \beta_{3} - 2) q^{10} + (\beta_{10} + \beta_{5}) q^{11} + ( - \beta_{3} - 2) q^{13} + ( - \beta_{10} - 2 \beta_{5} + 4 \beta_{2}) q^{14} + (2 \beta_{6} + 4 \beta_{3} + 1) q^{16} + ( - \beta_{10} + \beta_{5} - 3 \beta_{2}) q^{17} + ( - \beta_{9} + \beta_{8}) q^{19} + ( - \beta_{7} + \beta_1) q^{20} + ( - \beta_{11} - \beta_{9} + \beta_{8}) q^{22} + (\beta_{5} + 2 \beta_{2}) q^{23} + ( - 3 \beta_{6} - 2 \beta_{3} + 4) q^{25} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{26} + (2 \beta_{11} + \beta_{9} - 3 \beta_{8}) q^{28} + (\beta_{7} + \beta_1) q^{29} - 2 \beta_{6} q^{31} + (2 \beta_{7} - \beta_{4} - 2 \beta_1) q^{32} + ( - \beta_{11} + \beta_{9} + 2 \beta_{8}) q^{34} + ( - 2 \beta_{10} + 2 \beta_{5} - 5 \beta_{2}) q^{35} + (2 \beta_{11} - \beta_{9}) q^{37} + (4 \beta_{10} + 2 \beta_{5} - 3 \beta_{2}) q^{38} + ( - 5 \beta_{6} - 4 \beta_{3} + 1) q^{40} + ( - \beta_{10} - \beta_{5} - \beta_{2}) q^{41} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - 2) q^{43}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{4} + 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 28 q^{4} - 16 q^{10} - 24 q^{13} + 20 q^{16} + 36 q^{25} - 8 q^{31} - 8 q^{40} - 28 q^{43} - 36 q^{49} - 96 q^{52} + 8 q^{58} + 92 q^{64} + 16 q^{67} - 64 q^{79} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 22 x^{10} - 60 x^{9} + 190 x^{8} - 460 x^{7} + 1088 x^{6} - 2108 x^{5} + 2849 x^{4} + \cdots + 1688 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1905412138160 \nu^{11} + 6055986948655 \nu^{10} - 42764285674970 \nu^{9} + \cdots + 18\!\cdots\!84 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2596581584507 \nu^{11} - 13668496444297 \nu^{10} + 56499203391654 \nu^{9} + \cdots - 14\!\cdots\!28 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1306951942335 \nu^{11} + 3180734964163 \nu^{10} - 21852312583641 \nu^{9} + \cdots - 22\!\cdots\!62 ) / 14\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7133920415090 \nu^{11} + 17872989574889 \nu^{10} - 126887477211610 \nu^{9} + \cdots - 20\!\cdots\!20 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5879399 \nu^{11} - 23390026 \nu^{10} + 124851066 \nu^{9} - 344190040 \nu^{8} + \cdots - 18885986876 ) / 3715746616 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4768643163681 \nu^{11} - 12366172905905 \nu^{10} + 85200387992407 \nu^{9} + \cdots + 12\!\cdots\!34 ) / 28\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7802079153913 \nu^{11} + 21383446858213 \nu^{10} - 140529292785756 \nu^{9} + \cdots + 16\!\cdots\!68 ) / 28\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16594852763437 \nu^{11} + 49988064420676 \nu^{10} - 318413963288486 \nu^{9} + \cdots + 38\!\cdots\!00 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19553320334713 \nu^{11} - 80824432481518 \nu^{10} + 415661920688776 \nu^{9} + \cdots - 72\!\cdots\!88 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23500212398376 \nu^{11} + 76571904050549 \nu^{10} - 461190751128314 \nu^{9} + \cdots + 59\!\cdots\!28 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50998009459923 \nu^{11} + 172640391750634 \nu^{10} + \cdots + 12\!\cdots\!00 ) / 11\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{11} + 4\beta_{10} + 3\beta_{9} - 2\beta_{8} + 5\beta_{6} - 10\beta_{5} + 9\beta_{3} + 6\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{11} + 10 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{11} - 20 \beta_{10} - 37 \beta_{9} + 10 \beta_{8} + 20 \beta_{7} + 23 \beta_{6} + 56 \beta_{5} + \cdots + 53 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54 \beta_{11} - 77 \beta_{10} - 54 \beta_{9} + 29 \beta_{8} - 35 \beta_{7} - 73 \beta_{6} + 118 \beta_{5} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10 \beta_{11} + 122 \beta_{10} + 149 \beta_{9} - 90 \beta_{8} - 252 \beta_{7} - 779 \beta_{6} + \cdots - 337 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 632 \beta_{11} + 660 \beta_{10} + 596 \beta_{9} - 160 \beta_{8} + 68 \beta_{7} - 66 \beta_{6} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 230 \beta_{11} - 360 \beta_{10} + 795 \beta_{9} + 550 \beta_{8} + 1944 \beta_{7} + 6911 \beta_{6} + \cdots + 5421 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6622 \beta_{11} - 7535 \beta_{10} - 4398 \beta_{9} + 2685 \beta_{8} + 1583 \beta_{7} + 5397 \beta_{6} + \cdots - 1773 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13230 \beta_{11} - 17058 \beta_{10} - 18871 \beta_{9} + 5426 \beta_{8} - 11044 \beta_{7} - 34907 \beta_{6} + \cdots - 77625 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
386.1
−1.12457 − 2.23665i
−1.12457 + 2.23665i
0.573183 + 3.04480i
0.573183 − 3.04480i
1.55139 − 0.308446i
1.55139 + 0.308446i
1.55139 + 2.25638i
1.55139 − 2.25638i
0.573183 − 0.268598i
0.573183 + 0.268598i
−1.12457 − 1.57101i
−1.12457 + 1.57101i
−2.69242 0 5.24914 1.42514 0 − 4.28198i −8.74806 0 −3.83709
386.2 −2.69242 0 5.24914 1.42514 0 4.28198i −8.74806 0 −3.83709
386.3 −1.96307 0 1.85363 −2.63625 0 − 1.59127i 0.287325 0 5.17513
386.4 −1.96307 0 1.85363 −2.63625 0 1.59127i 0.287325 0 5.17513
386.5 −1.37740 0 −0.102775 3.87546 0 − 3.02200i 2.89636 0 −5.33804
386.6 −1.37740 0 −0.102775 3.87546 0 3.02200i 2.89636 0 −5.33804
386.7 1.37740 0 −0.102775 −3.87546 0 − 3.02200i −2.89636 0 −5.33804
386.8 1.37740 0 −0.102775 −3.87546 0 3.02200i −2.89636 0 −5.33804
386.9 1.96307 0 1.85363 2.63625 0 − 1.59127i −0.287325 0 5.17513
386.10 1.96307 0 1.85363 2.63625 0 1.59127i −0.287325 0 5.17513
386.11 2.69242 0 5.24914 −1.42514 0 − 4.28198i 8.74806 0 −3.83709
386.12 2.69242 0 5.24914 −1.42514 0 4.28198i 8.74806 0 −3.83709
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 386.12
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 387.2.d.b ✓ 12
3.b odd 2 1 inner 387.2.d.b ✓ 12
4.b odd 2 1 6192.2.l.j 12
12.b even 2 1 6192.2.l.j 12
43.b odd 2 1 inner 387.2.d.b ✓ 12
129.d even 2 1 inner 387.2.d.b ✓ 12
172.d even 2 1 6192.2.l.j 12
516.h odd 2 1 6192.2.l.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.d.b ✓ 12 1.a even 1 1 trivial
387.2.d.b ✓ 12 3.b odd 2 1 inner
387.2.d.b ✓ 12 43.b odd 2 1 inner
387.2.d.b ✓ 12 129.d even 2 1 inner
6192.2.l.j 12 4.b odd 2 1
6192.2.l.j 12 12.b even 2 1
6192.2.l.j 12 172.d even 2 1
6192.2.l.j 12 516.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 13T_{2}^{4} + 49T_{2}^{2} - 53 \) acting on \(S_{2}^{\mathrm{new}}(387, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 13 T^{4} + \cdots - 53)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 24 T^{4} + \cdots - 212)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 30 T^{4} + \cdots + 424)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 26 T^{4} + \cdots + 242)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} + 5 T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 66 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 74 T^{4} + \cdots + 6784)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 46 T^{4} + \cdots + 968)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 88 T^{4} + \cdots - 212)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 2 T^{2} - 16 T - 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 120 T^{4} + \cdots + 6784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 14 T^{5} + \cdots + 79507)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 162 T^{4} + \cdots + 1458)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 146 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 286 T^{4} + \cdots + 587528)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 184 T^{4} + \cdots + 27136)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 4 T^{2} + \cdots - 316)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 208 T^{4} + \cdots - 217088)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 176 T^{4} + \cdots + 6784)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + \cdots - 172)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 466 T^{4} + \cdots + 2789522)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 224 T^{4} + \cdots - 848)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 14 T^{2} + \cdots + 146)^{4} \) Copy content Toggle raw display
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