↑

Properties

Label 1932.2.q.f
Level $1932$
Weight $2$
Character orbit 1932.q
Analytic conductor $15.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 22 x^{10} - 23 x^{9} + 212 x^{8} - 169 x^{7} + 1377 x^{6} - 112 x^{5} + 4312 x^{4} + \cdots + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + \beta_1 q^{5} + (\beta_{11} + \beta_{8}) q^{7} + (\beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + \beta_1 q^{5} + (\beta_{11} + \beta_{8}) q^{7} + (\beta_{5} - 1) q^{9} + (\beta_{6} - \beta_{2} + 1) q^{13} + (\beta_{2} - 1) q^{15} + (\beta_{6} - \beta_{3} + 1) q^{17} + ( - \beta_{4} + \beta_{3}) q^{19} - \beta_{11} q^{21} + ( - \beta_{5} + 1) q^{23} + (\beta_{11} + \beta_{10} + \beta_{2} + \cdots - 1) q^{25}+ \cdots + (2 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 3 q^{5} - 2 q^{7} - 6 q^{9} - 6 q^{15} + 3 q^{17} + q^{19} + 4 q^{21} + 6 q^{23} - 5 q^{25} + 12 q^{27} - 24 q^{29} + 7 q^{31} + q^{35} + 23 q^{37} - 12 q^{41} - 2 q^{43} + 3 q^{45} + q^{47} - 6 q^{49} + 3 q^{51} + 7 q^{53} - 2 q^{57} + 12 q^{59} - 18 q^{61} - 2 q^{63} + 23 q^{65} - 10 q^{67} - 12 q^{69} + 6 q^{71} - 5 q^{75} + 33 q^{79} - 6 q^{81} - 68 q^{83} - 18 q^{85} + 12 q^{87} - 4 q^{89} + 22 q^{91} + 7 q^{93} + 22 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 22 x^{10} - 23 x^{9} + 212 x^{8} - 169 x^{7} + 1377 x^{6} - 112 x^{5} + 4312 x^{4} + \cdots + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1415095 \nu^{11} - 145254 \nu^{10} + 21553936 \nu^{9} + 40787986 \nu^{8} + 292178081 \nu^{7} + \cdots + 22288751442 ) / 17821319508 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5335315 \nu^{11} - 4939021 \nu^{10} - 17411190 \nu^{9} - 418417275 \nu^{8} + \cdots - 1517360580 ) / 17821319508 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 74461669 \nu^{11} - 300405375 \nu^{10} + 1899797284 \nu^{9} - 3262027559 \nu^{8} + \cdots - 23715564804 ) / 106927917048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 82730221 \nu^{11} - 256681233 \nu^{10} + 1820936386 \nu^{9} - 2032118699 \nu^{8} + \cdots + 88431867828 ) / 106927917048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14776811 \nu^{11} + 28653118 \nu^{10} - 284481984 \nu^{9} - 52538742 \nu^{8} + \cdots - 37102670802 ) / 17821319508 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15420791 \nu^{11} - 10740186 \nu^{10} + 245766659 \nu^{9} + 318481829 \nu^{8} + \cdots + 53102284200 ) / 17821319508 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 102729425 \nu^{11} + 263059125 \nu^{10} - 2180460836 \nu^{9} + 1767370453 \nu^{8} + \cdots + 180920287800 ) / 106927917048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 115403875 \nu^{11} + 407487291 \nu^{10} - 2848787200 \nu^{9} + 4618329455 \nu^{8} + \cdots + 207843813864 ) / 106927917048 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 170917619 \nu^{11} - 555390825 \nu^{10} + 3635422868 \nu^{9} - 3727961233 \nu^{8} + \cdots + 299034800244 ) / 106927917048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39040486 \nu^{11} - 121190636 \nu^{10} + 889989241 \nu^{9} - 1112893363 \nu^{8} + \cdots + 19386657882 ) / 17821319508 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - 5\beta_{5} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + 9\beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 10\beta_{10} + 9\beta_{9} + 10\beta_{8} + 41\beta_{5} + 2\beta_{4} - 17\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 17 \beta_{11} - 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 13 \beta_{7} + 3 \beta_{6} + 68 \beta_{5} + \cdots + 93 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -106\beta_{11} + 19\beta_{10} - 106\beta_{9} - 87\beta_{8} + 34\beta_{7} + 9\beta_{6} - 228\beta_{2} + 618 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 52 \beta_{11} + 271 \beta_{10} - 219 \beta_{9} - 271 \beta_{8} - 886 \beta_{5} - 159 \beta_{4} + \cdots + 886 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 895 \beta_{11} + 895 \beta_{10} + 291 \beta_{9} - 291 \beta_{8} - 482 \beta_{7} - 222 \beta_{6} + \cdots - 3041 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3523 \beta_{11} - 926 \beta_{10} + 3523 \beta_{9} + 2597 \beta_{8} - 1959 \beta_{7} - 1095 \beta_{6} + \cdots - 22267 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4149 \beta_{11} - 13720 \beta_{10} + 9571 \beta_{9} + 13720 \beta_{8} + 39719 \beta_{5} + 6408 \beta_{4} + \cdots - 39719 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 29827 \beta_{11} - 29827 \beta_{10} - 14154 \beta_{9} + 14154 \beta_{8} + 24277 \beta_{7} + \cdots + 137307 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(829\) \(925\) \(967\) \(1289\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

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
−1.21914 − 2.11161i
−1.12598 − 1.95026i
−0.118505 − 0.205256i
0.618505 + 1.07128i
1.62598 + 2.81628i
1.71914 + 2.97764i
−1.21914 + 2.11161i
−1.12598 + 1.95026i
−0.118505 + 0.205256i
0.618505 − 1.07128i
1.62598 − 2.81628i
1.71914 − 2.97764i
0 −0.500000 + 0.866025i 0 −1.21914 − 2.11161i 0 −1.69175 − 2.03420i 0 −0.500000 − 0.866025i 0
277.2 0 −0.500000 + 0.866025i 0 −1.12598 − 1.95026i 0 −1.16166 + 2.37709i 0 −0.500000 − 0.866025i 0
277.3 0 −0.500000 + 0.866025i 0 −0.118505 − 0.205256i 0 2.35341 − 1.20891i 0 −0.500000 − 0.866025i 0
277.4 0 −0.500000 + 0.866025i 0 0.618505 + 1.07128i 0 2.35341 − 1.20891i 0 −0.500000 − 0.866025i 0
277.5 0 −0.500000 + 0.866025i 0 1.62598 + 2.81628i 0 −1.16166 + 2.37709i 0 −0.500000 − 0.866025i 0
277.6 0 −0.500000 + 0.866025i 0 1.71914 + 2.97764i 0 −1.69175 − 2.03420i 0 −0.500000 − 0.866025i 0
1381.1 0 −0.500000 − 0.866025i 0 −1.21914 + 2.11161i 0 −1.69175 + 2.03420i 0 −0.500000 + 0.866025i 0
1381.2 0 −0.500000 − 0.866025i 0 −1.12598 + 1.95026i 0 −1.16166 − 2.37709i 0 −0.500000 + 0.866025i 0
1381.3 0 −0.500000 − 0.866025i 0 −0.118505 + 0.205256i 0 2.35341 + 1.20891i 0 −0.500000 + 0.866025i 0
1381.4 0 −0.500000 − 0.866025i 0 0.618505 − 1.07128i 0 2.35341 + 1.20891i 0 −0.500000 + 0.866025i 0
1381.5 0 −0.500000 − 0.866025i 0 1.62598 − 2.81628i 0 −1.16166 − 2.37709i 0 −0.500000 + 0.866025i 0
1381.6 0 −0.500000 − 0.866025i 0 1.71914 − 2.97764i 0 −1.69175 + 2.03420i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.6
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.f ✓ 12
7.c even 3 1 inner 1932.2.q.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.q.f ✓ 12 1.a even 1 1 trivial
1932.2.q.f ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 3 T_{5}^{11} + 22 T_{5}^{10} - 23 T_{5}^{9} + 212 T_{5}^{8} - 169 T_{5}^{7} + 1377 T_{5}^{6} + \cdots + 324 \) acting on \(S_{2}^{\mathrm{new}}(1932, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} + 2 T^{4} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 39 T^{4} + \cdots - 121)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 26244 \) Copy content Toggle raw display
$19$ \( T^{12} - T^{11} + \cdots + 1157776 \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} + 12 T^{5} + \cdots + 288)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} + \cdots + 20736 \) Copy content Toggle raw display
$37$ \( T^{12} - 23 T^{11} + \cdots + 17505856 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots + 14256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + T^{5} - 53 T^{4} + \cdots - 1076)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1816805376 \) Copy content Toggle raw display
$53$ \( T^{12} - 7 T^{11} + \cdots + 4359744 \) Copy content Toggle raw display
$59$ \( T^{12} - 12 T^{11} + \cdots + 82944 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 661476409344 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2961427561 \) Copy content Toggle raw display
$71$ \( (T^{6} - 3 T^{5} + \cdots + 3528)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 848964769 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 559417104 \) Copy content Toggle raw display
$83$ \( (T^{6} + 34 T^{5} + \cdots - 40752)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 4 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$97$ \( (T^{6} - 8 T^{5} + \cdots - 593728)^{2} \) Copy content Toggle raw display
show more
show less