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Properties

Label 1932.2.q.h
Level $1932$
Weight $2$
Character orbit 1932.q
Analytic conductor $15.427$
Analytic rank $0$
Dimension $20$
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] = [1932,2,Mod(277,1932)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1932, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1932.277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1932.q (of order \(3\), degree \(2\), 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: \(15.4270976705\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 34 x^{18} + 11 x^{17} + 775 x^{16} + 458 x^{15} + 9071 x^{14} + 14821 x^{13} + \cdots + 8100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - 1) q^{3} - \beta_1 q^{5} + \beta_{16} q^{7} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - 1) q^{3} - \beta_1 q^{5} + \beta_{16} q^{7} + \beta_{7} q^{9} - \beta_{11} q^{11} + (\beta_{4} + \beta_{3}) q^{13} + \beta_{3} q^{15} + ( - \beta_{15} - \beta_{12} + \cdots + \beta_{2}) q^{17}+ \cdots - \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{3} - q^{5} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 10 q^{3} - q^{5} - 10 q^{9} + 2 q^{11} + 2 q^{15} + q^{17} + 3 q^{19} - 10 q^{23} - 17 q^{25} + 20 q^{27} + 16 q^{29} - q^{31} + 2 q^{33} - 7 q^{35} - 23 q^{37} + 12 q^{41} - 10 q^{43} - q^{45} + q^{47} + 6 q^{49} + q^{51} - 13 q^{53} + 32 q^{55} - 6 q^{57} - 12 q^{59} + 16 q^{61} - 47 q^{65} + 18 q^{67} + 20 q^{69} + 6 q^{71} - 8 q^{73} - 17 q^{75} - 24 q^{77} - 33 q^{79} - 10 q^{81} + 76 q^{83} - 26 q^{85} - 8 q^{87} + 20 q^{89} + 2 q^{91} - q^{93} - 6 q^{95} + 56 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 34 x^{18} + 11 x^{17} + 775 x^{16} + 458 x^{15} + 9071 x^{14} + 14821 x^{13} + \cdots + 8100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!00 \nu^{19} + \cdots + 54\!\cdots\!20 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 68\!\cdots\!49 \nu^{19} + \cdots - 29\!\cdots\!00 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 49\!\cdots\!73 \nu^{19} + \cdots + 56\!\cdots\!80 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18\!\cdots\!64 \nu^{19} + \cdots + 44\!\cdots\!78 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!51 \nu^{19} + \cdots - 35\!\cdots\!50 ) / 16\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!99 \nu^{19} + \cdots + 20\!\cdots\!80 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 90\!\cdots\!85 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!34 \nu^{19} + \cdots - 85\!\cdots\!20 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!93 \nu^{19} + \cdots - 14\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!54 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 16\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!55 \nu^{19} + \cdots - 25\!\cdots\!20 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!82 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 83\!\cdots\!52 \nu^{19} + \cdots - 25\!\cdots\!00 ) / 66\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25\!\cdots\!39 \nu^{19} + \cdots - 82\!\cdots\!70 ) / 19\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 55\!\cdots\!47 \nu^{19} + \cdots - 14\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 61\!\cdots\!98 \nu^{19} + \cdots + 37\!\cdots\!20 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15\!\cdots\!67 \nu^{19} + \cdots + 57\!\cdots\!60 ) / 79\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 26\!\cdots\!41 \nu^{19} + \cdots + 35\!\cdots\!20 ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} - \beta_{16} + \beta_{12} - \beta_{11} - \beta_{10} - 7\beta_{7} - \beta_{6} - \beta_{3} + \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{17} + \beta_{16} + 2 \beta_{15} - \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{19} - 16 \beta_{18} + 23 \beta_{17} + \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - 16 \beta_{13} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 36 \beta_{19} - 19 \beta_{18} + 38 \beta_{17} + 21 \beta_{16} - 43 \beta_{15} - 19 \beta_{14} + \cdots + 95 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 402 \beta_{17} + 402 \beta_{16} + 21 \beta_{15} + 271 \beta_{13} - 292 \beta_{12} + 292 \beta_{11} + \cdots + 1724 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 626 \beta_{19} + 331 \beta_{18} - 383 \beta_{17} - 678 \beta_{16} + 78 \beta_{15} + 327 \beta_{14} + \cdots + 4802 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1231 \beta_{19} + 4693 \beta_{18} - 601 \beta_{17} - 7699 \beta_{16} + 847 \beta_{15} + 668 \beta_{14} + \cdots - 29982 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5265 \beta_{17} + 5265 \beta_{16} + 12434 \beta_{15} - 5692 \beta_{13} - 6742 \beta_{12} + \cdots - 28720 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21672 \beta_{19} - 81739 \beta_{18} + 135955 \beta_{17} + 11529 \beta_{16} - 23019 \beta_{15} + \cdots - 5733 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 190808 \beta_{19} - 97603 \beta_{18} + 215294 \beta_{17} + 120523 \beta_{16} - 248930 \beta_{15} + \cdots + 497879 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2177686 \beta_{17} + 2177686 \beta_{16} + 145621 \beta_{15} + 1425694 \beta_{13} - 1571315 \beta_{12} + \cdots + 9188190 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3335094 \beta_{19} + 1672195 \beta_{18} - 2129313 \beta_{17} - 3837842 \beta_{16} + \cdots + 25909697 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 6566761 \beta_{19} + 24876301 \beta_{18} - 4035179 \beta_{17} - 42135827 \beta_{16} + \cdots - 161012150 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 30790837 \beta_{17} + 30790837 \beta_{16} + 65323540 \beta_{15} - 28636405 \beta_{13} + \cdots - 149751793 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 114111229 \beta_{19} - 434109259 \beta_{18} + 741452731 \beta_{17} + 74878975 \beta_{16} + \cdots - 21765488 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1019244259 \beta_{19} - 490222534 \beta_{18} + 1218779016 \beta_{17} + 664301997 \beta_{16} + \cdots + 2597713114 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 11661984892 \beta_{17} + 11661984892 \beta_{16} + 958305930 \beta_{15} + 7576005025 \beta_{13} + \cdots + 49459699345 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 17819939000 \beta_{19} + 8389079155 \beta_{18} - 11734550179 \beta_{17} - 21711080204 \beta_{16} + \cdots + 140052126016 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times\).

\(n\) \(829\) \(925\) \(967\) \(1289\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
2.10113 + 3.63926i
2.08280 + 3.60752i
1.03387 + 1.79071i
0.439110 + 0.760561i
−0.0742788 − 0.128655i
−0.579613 − 1.00392i
−0.642882 − 1.11350i
−0.758143 − 1.31314i
−1.00568 − 1.74189i
−2.09631 − 3.63092i
2.10113 − 3.63926i
2.08280 − 3.60752i
1.03387 − 1.79071i
0.439110 − 0.760561i
−0.0742788 + 0.128655i
−0.579613 + 1.00392i
−0.642882 + 1.11350i
−0.758143 + 1.31314i
−1.00568 + 1.74189i
−2.09631 + 3.63092i
0 −0.500000 + 0.866025i 0 −2.10113 − 3.63926i 0 0.585236 − 2.58021i 0 −0.500000 − 0.866025i 0
277.2 0 −0.500000 + 0.866025i 0 −2.08280 − 3.60752i 0 2.11694 + 1.58699i 0 −0.500000 − 0.866025i 0
277.3 0 −0.500000 + 0.866025i 0 −1.03387 − 1.79071i 0 −2.63943 − 0.182785i 0 −0.500000 − 0.866025i 0
277.4 0 −0.500000 + 0.866025i 0 −0.439110 − 0.760561i 0 2.34346 + 1.22809i 0 −0.500000 − 0.866025i 0
277.5 0 −0.500000 + 0.866025i 0 0.0742788 + 0.128655i 0 −2.30437 + 1.29994i 0 −0.500000 − 0.866025i 0
277.6 0 −0.500000 + 0.866025i 0 0.579613 + 1.00392i 0 −0.357624 − 2.62147i 0 −0.500000 − 0.866025i 0
277.7 0 −0.500000 + 0.866025i 0 0.642882 + 1.11350i 0 0.798901 + 2.52225i 0 −0.500000 − 0.866025i 0
277.8 0 −0.500000 + 0.866025i 0 0.758143 + 1.31314i 0 −0.867155 − 2.49961i 0 −0.500000 − 0.866025i 0
277.9 0 −0.500000 + 0.866025i 0 1.00568 + 1.74189i 0 −2.32162 + 1.26890i 0 −0.500000 − 0.866025i 0
277.10 0 −0.500000 + 0.866025i 0 2.09631 + 3.63092i 0 2.64566 − 0.0220967i 0 −0.500000 − 0.866025i 0
1381.1 0 −0.500000 − 0.866025i 0 −2.10113 + 3.63926i 0 0.585236 + 2.58021i 0 −0.500000 + 0.866025i 0
1381.2 0 −0.500000 − 0.866025i 0 −2.08280 + 3.60752i 0 2.11694 − 1.58699i 0 −0.500000 + 0.866025i 0
1381.3 0 −0.500000 − 0.866025i 0 −1.03387 + 1.79071i 0 −2.63943 + 0.182785i 0 −0.500000 + 0.866025i 0
1381.4 0 −0.500000 − 0.866025i 0 −0.439110 + 0.760561i 0 2.34346 − 1.22809i 0 −0.500000 + 0.866025i 0
1381.5 0 −0.500000 − 0.866025i 0 0.0742788 − 0.128655i 0 −2.30437 − 1.29994i 0 −0.500000 + 0.866025i 0
1381.6 0 −0.500000 − 0.866025i 0 0.579613 − 1.00392i 0 −0.357624 + 2.62147i 0 −0.500000 + 0.866025i 0
1381.7 0 −0.500000 − 0.866025i 0 0.642882 − 1.11350i 0 0.798901 − 2.52225i 0 −0.500000 + 0.866025i 0
1381.8 0 −0.500000 − 0.866025i 0 0.758143 − 1.31314i 0 −0.867155 + 2.49961i 0 −0.500000 + 0.866025i 0
1381.9 0 −0.500000 − 0.866025i 0 1.00568 − 1.74189i 0 −2.32162 − 1.26890i 0 −0.500000 + 0.866025i 0
1381.10 0 −0.500000 − 0.866025i 0 2.09631 − 3.63092i 0 2.64566 + 0.0220967i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1932.2.q.h ✓ 20
7.c even 3 1 inner 1932.2.q.h ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.q.h ✓ 20 1.a even 1 1 trivial
1932.2.q.h ✓ 20 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + T_{5}^{19} + 34 T_{5}^{18} - 11 T_{5}^{17} + 775 T_{5}^{16} - 458 T_{5}^{15} + 9071 T_{5}^{14} + \cdots + 8100 \) acting on \(S_{2}^{\mathrm{new}}(1932, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 8100 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 33973862400 \) Copy content Toggle raw display
$13$ \( (T^{10} - 77 T^{8} + \cdots - 30429)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 156915807876 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 212576400 \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$29$ \( (T^{10} - 8 T^{9} + \cdots - 7908480)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 21175311568896 \) Copy content Toggle raw display
$41$ \( (T^{10} - 6 T^{9} + \cdots - 199056096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 5 T^{9} + \cdots + 22194036)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 5267275776 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 450777960000 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{10} - 3 T^{9} + \cdots + 340200)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{10} - 38 T^{9} + \cdots - 79435008)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 348809411073600 \) Copy content Toggle raw display
$97$ \( (T^{10} - 28 T^{9} + \cdots - 142976)^{2} \) Copy content Toggle raw display
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