Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1386,2,Mod(881,1386)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1386.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-16,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - q^{4} + \beta_1 q^{5} - \beta_{10} q^{7} - \beta_{5} q^{8} + \beta_{2} q^{10} + \beta_{5} q^{11} + (\beta_{15} + \beta_{14} + 1) q^{13} - \beta_{8} q^{14} + q^{16} + ( - \beta_{8} - \beta_{7} - \beta_1) q^{17}+ \cdots + (\beta_{11} + \beta_{8} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{22} + 48 q^{25} - 16 q^{37} - 80 q^{43} + 24 q^{49} + 16 q^{58} - 16 q^{64} + 16 q^{67} + 24 q^{70} + 96 q^{79} - 80 q^{85} + 16 q^{88} - 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 72x^{12} + 1296x^{10} + 8245x^{8} + 18780x^{6} + 20808x^{4} - 3672x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1381965647 \nu^{15} - 16053899943 \nu^{13} + 91281494448 \nu^{11} + 1854613004958 \nu^{9} + \cdots - 7903986405426 \nu ) / 8313550996212 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3221232346 \nu^{15} - 39189954813 \nu^{13} + 239020561026 \nu^{11} + \cdots - 2619108000162 \nu ) / 8313550996212 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3221232346 \nu^{15} + 39189954813 \nu^{13} - 239020561026 \nu^{11} + \cdots + 19246209992586 \nu ) / 8313550996212 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 133007 \nu^{14} - 1743582 \nu^{12} + 11322414 \nu^{10} + 162109944 \nu^{8} + 903956675 \nu^{6} + \cdots - 2021921136 ) / 481024764 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12219569 \nu^{14} + 143946111 \nu^{12} - 844183530 \nu^{10} - 16075743516 \nu^{8} + \cdots + 22892859036 ) / 26476277058 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2097307895 \nu^{15} - 24781419255 \nu^{13} + 146288587464 \nu^{11} + \cdots - 2157576356394 \nu ) / 2771183665404 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19480188302 \nu^{15} + 13882041737 \nu^{14} - 234062963571 \nu^{13} - 161840906400 \nu^{12} + \cdots - 26508114581148 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19480188302 \nu^{15} - 13882041737 \nu^{14} - 234062963571 \nu^{13} + 161840906400 \nu^{12} + \cdots + 26508114581148 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39894432671 \nu^{15} + 384169521 \nu^{14} - 475048585479 \nu^{13} - 4764275946 \nu^{12} + \cdots + 30719356487784 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39894432671 \nu^{15} + 384169521 \nu^{14} + 475048585479 \nu^{13} + \cdots + 30719356487784 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27614604307 \nu^{15} + 17650113884 \nu^{14} - 335527493352 \nu^{13} - 212774422656 \nu^{12} + \cdots - 31653237281040 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27614604307 \nu^{15} + 17650113884 \nu^{14} + 335527493352 \nu^{13} + \cdots - 31653237281040 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 66695 \nu^{14} - 802932 \nu^{12} + 4850514 \nu^{10} + 86012388 \nu^{8} + 548291507 \nu^{6} + \cdots - 120049236 ) / 32561172 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 54470935301 \nu^{15} - 1838672364 \nu^{14} + 645220229322 \nu^{13} + \cdots + 19609438426344 ) / 16627101992424 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 54470935301 \nu^{15} + 1838672364 \nu^{14} + 645220229322 \nu^{13} + \cdots - 36236540418768 ) / 16627101992424 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 13 \beta_{6} + 4 \beta_{3} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{13} - 17\beta_{12} - 17\beta_{11} + 28\beta_{8} - 28\beta_{7} - 144\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{15} + 14 \beta_{14} + 3 \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 17 \beta_{9} - 11 \beta_{8} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 202 \beta_{15} + 202 \beta_{14} + 117 \beta_{13} - 202 \beta_{12} - 202 \beta_{11} + 271 \beta_{10} + \cdots - 1188 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69 \beta_{15} + 69 \beta_{14} + 202 \beta_{12} - 202 \beta_{11} + 133 \beta_{10} - 133 \beta_{9} + \cdots + 69 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -4175\beta_{15} + 4175\beta_{14} + 6004\beta_{10} + 6004\beta_{9} + 2336\beta_{4} - 26389 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1173 \beta_{15} - 1173 \beta_{14} + 3002 \beta_{12} - 3002 \beta_{11} - 1829 \beta_{10} + 1829 \beta_{9} + \cdots - 1173 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 45148 \beta_{15} + 45148 \beta_{14} - 25461 \beta_{13} + 45148 \beta_{12} + 45148 \beta_{11} + \cdots - 279342 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 45148 \beta_{15} - 45148 \beta_{14} + 18375 \beta_{12} - 18375 \beta_{11} - 63523 \beta_{10} + \cdots - 45148 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 541832 \beta_{13} + 962603 \beta_{12} + 962603 \beta_{11} - 1363420 \beta_{8} + \cdots + 6956172 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 681710 \beta_{15} - 681710 \beta_{14} - 280893 \beta_{12} + 280893 \beta_{11} - 962603 \beta_{10} + \cdots - 681710 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 10306300 \beta_{15} - 10306300 \beta_{14} - 5804109 \beta_{13} + 10306300 \beta_{12} + 10306300 \beta_{11} + \cdots + 64051830 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4262343 \beta_{15} - 4262343 \beta_{14} - 10306300 \beta_{12} + 10306300 \beta_{11} - 6043957 \beta_{10} + \cdots - 4262343 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.808328 + 1.95148i
3.59322 + 1.48836i
−0.579826 + 1.39982i
0.314903 + 0.130437i
−0.314903 − 0.130437i
0.579826 − 1.39982i
−3.59322 − 1.48836i
0.808328 − 1.95148i
−0.808328 − 1.95148i
3.59322 − 1.48836i
−0.579826 − 1.39982i
0.314903 − 0.130437i
−0.314903 + 0.130437i
0.579826 + 1.39982i
−3.59322 + 1.48836i
0.808328 + 1.95148i
− 1.00000i 0 −1.00000 −3.90295 0 1.49520 − 2.18274i 1.00000i 0 3.90295i
881.2 − 1.00000i 0 −1.00000 −2.97672 0 2.55176 − 0.698947i 1.00000i 0 2.97672i
881.3 − 1.00000i 0 −1.00000 −2.79965 0 −2.20231 + 1.46623i 1.00000i 0 2.79965i
881.4 − 1.00000i 0 −1.00000 −0.260874 0 −1.84465 + 1.89664i 1.00000i 0 0.260874i
881.5 − 1.00000i 0 −1.00000 0.260874 0 −1.84465 − 1.89664i 1.00000i 0 − 0.260874i
881.6 − 1.00000i 0 −1.00000 2.79965 0 −2.20231 − 1.46623i 1.00000i 0 − 2.79965i
881.7 − 1.00000i 0 −1.00000 2.97672 0 2.55176 + 0.698947i 1.00000i 0 − 2.97672i
881.8 − 1.00000i 0 −1.00000 3.90295 0 1.49520 + 2.18274i 1.00000i 0 − 3.90295i
881.9 1.00000i 0 −1.00000 −3.90295 0 1.49520 + 2.18274i − 1.00000i 0 − 3.90295i
881.10 1.00000i 0 −1.00000 −2.97672 0 2.55176 + 0.698947i − 1.00000i 0 − 2.97672i
881.11 1.00000i 0 −1.00000 −2.79965 0 −2.20231 − 1.46623i − 1.00000i 0 − 2.79965i
881.12 1.00000i 0 −1.00000 −0.260874 0 −1.84465 − 1.89664i − 1.00000i 0 − 0.260874i
881.13 1.00000i 0 −1.00000 0.260874 0 −1.84465 + 1.89664i − 1.00000i 0 0.260874i
881.14 1.00000i 0 −1.00000 2.79965 0 −2.20231 + 1.46623i − 1.00000i 0 2.79965i
881.15 1.00000i 0 −1.00000 2.97672 0 2.55176 − 0.698947i − 1.00000i 0 2.97672i
881.16 1.00000i 0 −1.00000 3.90295 0 1.49520 − 2.18274i − 1.00000i 0 3.90295i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.g.b ✓ 16
3.b odd 2 1 inner 1386.2.g.b ✓ 16
7.b odd 2 1 inner 1386.2.g.b ✓ 16
21.c even 2 1 inner 1386.2.g.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.g.b ✓ 16 1.a even 1 1 trivial
1386.2.g.b ✓ 16 3.b odd 2 1 inner
1386.2.g.b ✓ 16 7.b odd 2 1 inner
1386.2.g.b ✓ 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 32T_{5}^{6} + 326T_{5}^{4} - 1080T_{5}^{2} + 72 \) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 32 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 6 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + 88 T^{6} + \cdots + 73728)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 52 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 56 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} + 188 T^{6} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 92 T^{6} + \cdots + 73728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots + 448)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 116 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + \cdots - 7792)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 140 T^{6} + \cdots + 225792)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 284 T^{6} + \cdots + 11451456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 416 T^{6} + \cdots + 40716288)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 200 T^{6} + \cdots + 73728)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 144)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} + 484 T^{6} + \cdots + 206613792)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots - 1564)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 476 T^{6} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 500 T^{6} + \cdots + 12221568)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 256 T^{6} + \cdots + 294912)^{2} \) Copy content Toggle raw display
show more
show less