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Properties

Label 456.4.q.c
Level $456$
Weight $4$
Character orbit 456.q
Analytic conductor $26.905$
Analytic rank $0$
Dimension $16$
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] = [456,4,Mod(49,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 456.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: \(26.9048709626\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 415 x^{14} + 2482 x^{13} + 134557 x^{12} + 576929 x^{11} + 14567909 x^{10} + \cdots + 7068621881344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta_1 - 3) q^{3} + \beta_{2} q^{5} + ( - \beta_{6} - 1) q^{7} + 9 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_1 - 3) q^{3} + \beta_{2} q^{5} + ( - \beta_{6} - 1) q^{7} + 9 \beta_1 q^{9} + ( - \beta_{10} - \beta_{7} + \beta_1 + 1) q^{11} + ( - \beta_{15} + \beta_{5} - \beta_{2} + \cdots - 1) q^{13}+ \cdots + 9 \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{3} - 4 q^{5} - 16 q^{7} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{3} - 4 q^{5} - 16 q^{7} - 72 q^{9} + 8 q^{11} + 16 q^{13} - 12 q^{15} - 64 q^{17} + 48 q^{19} + 24 q^{21} - 148 q^{23} - 132 q^{25} + 432 q^{27} + 4 q^{29} + 384 q^{31} - 12 q^{33} - 120 q^{35} + 392 q^{37} - 96 q^{39} + 4 q^{41} + 52 q^{43} + 72 q^{45} + 4 q^{47} - 72 q^{49} - 192 q^{51} - 228 q^{53} - 136 q^{55} - 252 q^{57} - 56 q^{59} + 288 q^{61} + 72 q^{63} + 2480 q^{65} - 244 q^{67} + 888 q^{69} + 176 q^{71} + 272 q^{73} + 792 q^{75} - 344 q^{77} - 744 q^{79} - 648 q^{81} - 320 q^{83} - 2328 q^{85} - 24 q^{87} - 1152 q^{89} - 236 q^{91} - 576 q^{93} - 1448 q^{95} + 2372 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 415 x^{14} + 2482 x^{13} + 134557 x^{12} + 576929 x^{11} + 14567909 x^{10} + \cdots + 7068621881344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 93\!\cdots\!33 \nu^{15} + \cdots + 18\!\cdots\!12 ) / 91\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 27\!\cdots\!57 \nu^{15} + \cdots - 17\!\cdots\!32 ) / 68\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!21 \nu^{15} + \cdots - 11\!\cdots\!48 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!77 \nu^{15} + \cdots + 42\!\cdots\!88 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\!\cdots\!25 \nu^{15} + \cdots - 28\!\cdots\!68 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{15} + \cdots + 82\!\cdots\!32 ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81\!\cdots\!07 \nu^{15} + \cdots - 31\!\cdots\!00 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\!\cdots\!95 \nu^{15} + \cdots + 87\!\cdots\!44 ) / 68\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!47 \nu^{15} + \cdots + 10\!\cdots\!28 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!61 \nu^{15} + \cdots + 20\!\cdots\!16 ) / 13\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!49 \nu^{15} + \cdots - 10\!\cdots\!76 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 31\!\cdots\!73 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!93 \nu^{15} + \cdots + 39\!\cdots\!40 ) / 34\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 78\!\cdots\!15 \nu^{15} + \cdots - 35\!\cdots\!96 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13\!\cdots\!19 \nu^{15} + \cdots + 70\!\cdots\!32 ) / 34\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8 \beta_{15} - 3 \beta_{14} - 13 \beta_{11} - 5 \beta_{10} - 12 \beta_{9} - \beta_{8} + 8 \beta_{6} + \cdots + 33 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 88 \beta_{15} + 137 \beta_{14} - 88 \beta_{13} - 137 \beta_{12} - 305 \beta_{11} - 369 \beta_{10} + \cdots - 3919 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2984 \beta_{13} + 195 \beta_{12} + 2315 \beta_{9} - 1353 \beta_{8} - 4389 \beta_{7} - 1353 \beta_{6} + \cdots - 210364 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 45368 \beta_{15} - 28617 \beta_{14} + 106049 \beta_{11} + 129361 \beta_{10} + 54912 \beta_{9} + \cdots - 151561 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1083336 \beta_{15} - 132605 \beta_{14} + 1083336 \beta_{13} + 132605 \beta_{12} + 1964653 \beta_{11} + \cdots + 64040883 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19182744 \beta_{13} + 7700841 \beta_{12} - 5428447 \beta_{9} + 17229609 \beta_{8} + 47240849 \beta_{7} + \cdots + 1083706080 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 406561896 \beta_{15} + 86782813 \beta_{14} - 768980573 \beta_{11} - 854840581 \beta_{10} + \cdots + 1355049209 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7677542264 \beta_{15} + 2494947689 \beta_{14} - 7677542264 \beta_{13} - 2494947689 \beta_{12} + \cdots - 416795247759 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 155632165256 \beta_{13} - 39339239133 \beta_{12} + 65367497323 \beta_{9} - 117308655681 \beta_{8} + \cdots - 9179684961908 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3016449647064 \beta_{15} - 895644620105 \beta_{14} + 5942585073969 \beta_{11} + 6847800480689 \beta_{10} + \cdots - 10013324511393 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 60112869098280 \beta_{15} - 16161713855485 \beta_{14} + 60112869098280 \beta_{13} + \cdots + 33\!\cdots\!27 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!48 \beta_{13} + 336960989509257 \beta_{12} - 456093500111359 \beta_{9} + \cdots + 68\!\cdots\!64 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 23\!\cdots\!44 \beta_{15} + \cdots + 77\!\cdots\!97 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 45\!\cdots\!36 \beta_{15} + \cdots - 25\!\cdots\!31 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.08785 + 1.88422i
−6.21424 + 10.7634i
1.20353 − 2.08457i
−3.04305 + 5.27072i
4.36843 − 7.56634i
1.58034 − 2.73722i
9.85710 − 17.0730i
−6.16425 + 10.6768i
−1.08785 − 1.88422i
−6.21424 − 10.7634i
1.20353 + 2.08457i
−3.04305 − 5.27072i
4.36843 + 7.56634i
1.58034 + 2.73722i
9.85710 + 17.0730i
−6.16425 − 10.6768i
0 −1.50000 + 2.59808i 0 −10.1652 + 17.6066i 0 −4.94599 0 −4.50000 − 7.79423i 0
49.2 0 −1.50000 + 2.59808i 0 −4.92053 + 8.52261i 0 32.6050 0 −4.50000 − 7.79423i 0
49.3 0 −1.50000 + 2.59808i 0 −4.69796 + 8.13711i 0 0.191741 0 −4.50000 − 7.79423i 0
49.4 0 −1.50000 + 2.59808i 0 −1.97731 + 3.42480i 0 −29.7428 0 −4.50000 − 7.79423i 0
49.5 0 −1.50000 + 2.59808i 0 1.22085 − 2.11457i 0 −19.0289 0 −4.50000 − 7.79423i 0
49.6 0 −1.50000 + 2.59808i 0 4.40626 − 7.63187i 0 14.7912 0 −4.50000 − 7.79423i 0
49.7 0 −1.50000 + 2.59808i 0 4.98270 − 8.63029i 0 7.81563 0 −4.50000 − 7.79423i 0
49.8 0 −1.50000 + 2.59808i 0 9.15117 − 15.8503i 0 −9.68601 0 −4.50000 − 7.79423i 0
121.1 0 −1.50000 − 2.59808i 0 −10.1652 − 17.6066i 0 −4.94599 0 −4.50000 + 7.79423i 0
121.2 0 −1.50000 − 2.59808i 0 −4.92053 − 8.52261i 0 32.6050 0 −4.50000 + 7.79423i 0
121.3 0 −1.50000 − 2.59808i 0 −4.69796 − 8.13711i 0 0.191741 0 −4.50000 + 7.79423i 0
121.4 0 −1.50000 − 2.59808i 0 −1.97731 − 3.42480i 0 −29.7428 0 −4.50000 + 7.79423i 0
121.5 0 −1.50000 − 2.59808i 0 1.22085 + 2.11457i 0 −19.0289 0 −4.50000 + 7.79423i 0
121.6 0 −1.50000 − 2.59808i 0 4.40626 + 7.63187i 0 14.7912 0 −4.50000 + 7.79423i 0
121.7 0 −1.50000 − 2.59808i 0 4.98270 + 8.63029i 0 7.81563 0 −4.50000 + 7.79423i 0
121.8 0 −1.50000 − 2.59808i 0 9.15117 + 15.8503i 0 −9.68601 0 −4.50000 + 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.4.q.c ✓ 16
19.c even 3 1 inner 456.4.q.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.4.q.c ✓ 16 1.a even 1 1 trivial
456.4.q.c ✓ 16 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 4 T_{5}^{15} + 574 T_{5}^{14} + 656 T_{5}^{13} + 237340 T_{5}^{12} + 320928 T_{5}^{11} + \cdots + 851235241476096 \) acting on \(S_{4}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 851235241476096 \) Copy content Toggle raw display
$7$ \( (T^{8} + 8 T^{7} + \cdots + 19595655)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots + 326909701056)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 48\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 59\!\cdots\!39)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 10\!\cdots\!83)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 91\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 30\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 30\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
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