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Properties

Label 1872.2.by.o
Level $1872$
Weight $2$
Character orbit 1872.by
Analytic conductor $14.948$
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] = [1872,2,Mod(433,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1872.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.by (of order \(6\), degree \(2\), not 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: \(14.9479952584\)
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: \( x^{16} + 9x^{14} + 58x^{12} + 171x^{10} + 366x^{8} + 396x^{6} + 301x^{4} + 18x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 936)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{5} - \beta_{2} q^{7} + ( - \beta_{12} + \beta_{8} - \beta_{5}) q^{11} + (\beta_{10} - \beta_{4}) q^{13} + (\beta_{13} - \beta_{11}) q^{17} + (\beta_{10} - \beta_{7} - \beta_{6} + \cdots + 1) q^{19}+ \cdots + ( - \beta_{10} + \beta_{7} + \beta_{6} + \cdots + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{7} + 12 q^{19} - 28 q^{25} + 6 q^{37} + 18 q^{43} - 2 q^{49} + 16 q^{55} - 20 q^{61} - 42 q^{67} + 52 q^{79} + 30 q^{85} - 34 q^{91} + 78 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 9x^{14} + 58x^{12} + 171x^{10} + 366x^{8} + 396x^{6} + 301x^{4} + 18x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -36\nu^{14} - 248\nu^{12} - 1507\nu^{10} - 2656\nu^{8} - 5903\nu^{6} - 2567\nu^{4} - 9229\nu^{2} - 1902 ) / 3243 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -68\nu^{14} - 599\nu^{12} - 3938\nu^{10} - 11717\nu^{8} - 27856\nu^{6} - 33007\nu^{4} - 33517\nu^{2} + 1781 ) / 3243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 104 \nu^{14} - 1129 \nu^{12} - 7607 \nu^{10} - 27345 \nu^{8} - 60925 \nu^{6} - 80976 \nu^{4} + \cdots - 13234 ) / 3243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 201\nu^{14} + 1792\nu^{12} + 11520\nu^{10} + 33504\nu^{8} + 71424\nu^{6} + 75264\nu^{4} + 58512\nu^{2} + 256 ) / 3243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 199 \nu^{15} - 2008 \nu^{13} - 13439 \nu^{11} - 46036 \nu^{9} - 106474 \nu^{7} - 147833 \nu^{5} + \cdots - 54968 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 483 \nu^{14} - 4377 \nu^{12} - 28487 \nu^{10} - 85815 \nu^{8} - 190381 \nu^{6} - 217392 \nu^{4} + \cdots - 13698 ) / 3243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{14} + 118\nu^{12} + 762\nu^{10} + 2276\nu^{8} + 4902\nu^{6} + 5512\nu^{4} + 4270\nu^{2} + 423 ) / 69 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 506 \nu^{15} - 4572 \nu^{13} - 29425 \nu^{11} - 86598 \nu^{9} - 182012 \nu^{7} - 185397 \nu^{5} + \cdots + 15864 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 711 \nu^{15} + 6214 \nu^{13} + 39551 \nu^{11} + 110548 \nu^{9} + 226858 \nu^{7} + 207737 \nu^{5} + \cdots - 46354 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1053 \nu^{14} + 9510 \nu^{12} + 61129 \nu^{10} + 180054 \nu^{8} + 379733 \nu^{6} + 404637 \nu^{4} + \cdots + 11148 ) / 3243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1210 \nu^{15} - 11072 \nu^{13} - 71768 \nu^{11} - 217038 \nu^{9} - 471244 \nu^{7} + \cdots - 61124 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1381 \nu^{15} + 12438 \nu^{13} + 80160 \nu^{11} + 236704 \nu^{9} + 507426 \nu^{7} + 556142 \nu^{5} + \cdots + 49690 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1497 \nu^{15} - 13446 \nu^{13} - 86405 \nu^{11} - 253200 \nu^{9} - 536134 \nu^{7} + \cdots - 18438 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1606 \nu^{15} - 14270 \nu^{13} - 91635 \nu^{11} - 265054 \nu^{9} - 562920 \nu^{7} + \cdots + 21918 \nu ) / 3243 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 840 \nu^{15} - 7604 \nu^{13} - 49091 \nu^{11} - 145884 \nu^{9} - 312984 \nu^{7} - 341411 \nu^{5} + \cdots - 15428 \nu ) / 1081 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} + \beta_{7} + \beta_{6} + 10\beta_{4} - \beta_{3} + 2\beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{15} + 2\beta_{14} + 5\beta_{11} - 2\beta_{9} + 3\beta_{8} + 2\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{10} - \beta_{7} - 2\beta_{6} - 13\beta_{4} - \beta_{3} + 3\beta_{2} - 7\beta _1 - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 26\beta_{15} - 24\beta_{14} - 11\beta_{13} - 11\beta_{9} - 26\beta_{8} - 26\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 19\beta_{10} - 3\beta_{7} + 16\beta_{6} - 66\beta_{4} + 19\beta_{3} - 97\beta_{2} + 50\beta _1 + 186 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 65\beta_{14} + 47\beta_{13} - 24\beta_{12} - 136\beta_{11} + 138\beta_{9} + 89\beta_{8} + 49\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -194\beta_{10} + 97\beta_{7} - \beta_{6} + 868\beta_{4} + \beta_{3} + 354\beta_{2} + 160\beta _1 - 97 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 715 \beta_{15} + 296 \beta_{14} + 18 \beta_{13} + 98 \beta_{12} + 715 \beta_{11} + \cdots + 492 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 507\beta_{10} - 550\beta_{7} - 507\beta_{6} - 2602\beta_{4} - 550\beta_{3} + 825\beta_{2} - 2200\beta _1 - 4295 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3770 \beta_{15} - 3336 \beta_{14} - 1237 \beta_{13} + 312 \beta_{12} - 1237 \beta_{9} + \cdots - 3886 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1335 \beta_{10} + 156 \beta_{7} + 1491 \beta_{6} - 4986 \beta_{4} + 1335 \beta_{3} - 7146 \beta_{2} + \cdots + 12568 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 9443 \beta_{14} + 5795 \beta_{13} - 4669 \beta_{12} - 19907 \beta_{11} + 20573 \beta_{9} + \cdots + 6461 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7048 \beta_{10} + 3524 \beta_{7} - 460 \beta_{6} + 30923 \beta_{4} + 460 \beta_{3} + 13192 \beta_{2} + \cdots - 3524 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 105187 \beta_{15} + 42976 \beta_{14} + 3648 \beta_{13} + 15936 \beta_{12} + 105187 \beta_{11} + \cdots + 74848 \beta_{5} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1 + \beta_{4}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−0.590398 − 1.02260i
−0.122532 − 0.212232i
0.751486 + 1.30161i
1.14965 + 1.99125i
−1.14965 − 1.99125i
−0.751486 − 1.30161i
0.122532 + 0.212232i
0.590398 + 1.02260i
0.590398 − 1.02260i
0.122532 − 0.212232i
−0.751486 + 1.30161i
−1.14965 + 1.99125i
1.14965 − 1.99125i
0.751486 − 1.30161i
−0.122532 + 0.212232i
−0.590398 + 1.02260i
0 0 0 − 4.07257i 0 −3.14201 + 1.81404i 0 0 0
433.2 0 0 0 − 3.12779i 0 −0.876049 + 0.505787i 0 0 0
433.3 0 0 0 − 0.588659i 0 −0.535406 + 0.309117i 0 0 0
433.4 0 0 0 − 0.533445i 0 3.05347 − 1.76292i 0 0 0
433.5 0 0 0 0.533445i 0 3.05347 − 1.76292i 0 0 0
433.6 0 0 0 0.588659i 0 −0.535406 + 0.309117i 0 0 0
433.7 0 0 0 3.12779i 0 −0.876049 + 0.505787i 0 0 0
433.8 0 0 0 4.07257i 0 −3.14201 + 1.81404i 0 0 0
1297.1 0 0 0 − 4.07257i 0 −3.14201 − 1.81404i 0 0 0
1297.2 0 0 0 − 3.12779i 0 −0.876049 − 0.505787i 0 0 0
1297.3 0 0 0 − 0.588659i 0 −0.535406 − 0.309117i 0 0 0
1297.4 0 0 0 − 0.533445i 0 3.05347 + 1.76292i 0 0 0
1297.5 0 0 0 0.533445i 0 3.05347 + 1.76292i 0 0 0
1297.6 0 0 0 0.588659i 0 −0.535406 − 0.309117i 0 0 0
1297.7 0 0 0 3.12779i 0 −0.876049 − 0.505787i 0 0 0
1297.8 0 0 0 4.07257i 0 −3.14201 − 1.81404i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.by.o 16
3.b odd 2 1 inner 1872.2.by.o 16
4.b odd 2 1 936.2.bi.d ✓ 16
12.b even 2 1 936.2.bi.d ✓ 16
13.e even 6 1 inner 1872.2.by.o 16
39.h odd 6 1 inner 1872.2.by.o 16
52.i odd 6 1 936.2.bi.d ✓ 16
156.r even 6 1 936.2.bi.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.bi.d ✓ 16 4.b odd 2 1
936.2.bi.d ✓ 16 12.b even 2 1
936.2.bi.d ✓ 16 52.i odd 6 1
936.2.bi.d ✓ 16 156.r even 6 1
1872.2.by.o 16 1.a even 1 1 trivial
1872.2.by.o 16 3.b odd 2 1 inner
1872.2.by.o 16 13.e even 6 1 inner
1872.2.by.o 16 39.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{8} + 27T_{5}^{6} + 179T_{5}^{4} + 105T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} - 9T_{7}^{6} - 36T_{7}^{5} + 116T_{7}^{4} + 432T_{7}^{3} + 528T_{7}^{2} + 288T_{7} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 27 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} - 9 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 268435456 \) Copy content Toggle raw display
$13$ \( (T^{8} + 18 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12149330176 \) Copy content Toggle raw display
$19$ \( (T^{8} - 6 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 293434556416 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 26639462656 \) Copy content Toggle raw display
$31$ \( (T^{8} + 195 T^{6} + \cdots + 2166784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 3 T^{7} + \cdots + 13162384)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 159539531776 \) Copy content Toggle raw display
$43$ \( (T^{8} - 9 T^{7} + 97 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 264 T^{6} + \cdots + 802816)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 177 T^{6} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 16 T^{2} + 256)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 10 T^{7} + \cdots + 22801)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 21 T^{7} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{8} + 324 T^{6} + \cdots + 29241)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 13 T^{3} + \cdots - 8384)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 104 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} - 348 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$97$ \( (T^{8} - 39 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
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