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Properties

Label 456.2.bg.d
Level $456$
Weight $2$
Character orbit 456.bg
Analytic conductor $3.641$
Analytic rank $0$
Dimension $18$
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] = [456,2,Mod(25,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bg (of order \(9\), degree \(6\), 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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 42 x^{16} + 621 x^{14} + 4302 x^{12} + 15174 x^{10} + 27540 x^{8} + 25929 x^{6} + 12204 x^{4} + \cdots + 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + (\beta_{16} + \beta_{12} - \beta_{6} + \cdots + 1) q^{5}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} + (\beta_{16} + \beta_{12} - \beta_{6} + \cdots + 1) q^{5}+ \cdots + (\beta_{13} + \beta_{11} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{5} + 6 q^{7} + 9 q^{11} - 18 q^{13} - 3 q^{15} - 3 q^{17} + 3 q^{19} + 3 q^{21} - 24 q^{23} - 3 q^{25} + 9 q^{27} + 15 q^{29} + 9 q^{31} + 6 q^{35} + 30 q^{37} - 18 q^{39} - 3 q^{41} + 30 q^{43} + 3 q^{45} + 54 q^{47} - 9 q^{49} - 6 q^{51} - 24 q^{53} + 12 q^{55} + 15 q^{57} + 9 q^{59} + 6 q^{61} + 6 q^{63} + 3 q^{65} - 18 q^{67} - 30 q^{69} - 63 q^{71} - 24 q^{73} - 60 q^{75} - 66 q^{77} - 15 q^{79} + 36 q^{83} + 39 q^{85} + 3 q^{87} - 63 q^{89} - 66 q^{91} + 27 q^{93} + 42 q^{95} - 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 42 x^{16} + 621 x^{14} + 4302 x^{12} + 15174 x^{10} + 27540 x^{8} + 25929 x^{6} + 12204 x^{4} + \cdots + 192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 66511 \nu^{17} - 86548 \nu^{16} + 2684418 \nu^{15} - 3563622 \nu^{14} + 36883691 \nu^{13} + \cdots - 30139040 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 98577 \nu^{17} - 4103358 \nu^{15} - 59688981 \nu^{13} - 402050898 \nu^{11} - 1349392622 \nu^{9} + \cdots + 1824224 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70409 \nu^{16} + 2858258 \nu^{14} + 39692000 \nu^{12} + 246473858 \nu^{10} + 712958533 \nu^{8} + \cdots - 578832 ) / 912112 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97502 \nu^{17} + 70409 \nu^{16} + 3963222 \nu^{15} + 2858258 \nu^{14} + 55187677 \nu^{13} + \cdots - 578832 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86743 \nu^{17} + 491154 \nu^{16} + 3524876 \nu^{15} + 19969672 \nu^{14} + 49039483 \nu^{13} + \cdots + 57199712 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 98282 \nu^{17} - 134621 \nu^{16} - 4036266 \nu^{15} - 5484372 \nu^{14} - 57287880 \nu^{13} + \cdots - 22960944 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 86743 \nu^{17} + 491154 \nu^{16} - 3524876 \nu^{15} + 19969672 \nu^{14} - 49039483 \nu^{13} + \cdots + 57199712 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 174572 \nu^{17} - 150512 \nu^{16} + 7102338 \nu^{15} - 6116521 \nu^{14} + 99056525 \nu^{13} + \cdots - 18955088 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 174572 \nu^{17} - 168950 \nu^{16} + 7102338 \nu^{15} - 6880189 \nu^{14} + 99056525 \nu^{13} + \cdots - 28418480 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 435887 \nu^{17} - 153254 \nu^{16} + 17729552 \nu^{15} - 6209294 \nu^{14} + 247152533 \nu^{13} + \cdots - 362752 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 435887 \nu^{17} - 190130 \nu^{16} + 17729552 \nu^{15} - 7736630 \nu^{14} + 247152533 \nu^{13} + \cdots - 19289536 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 404411 \nu^{17} - 372824 \nu^{16} + 16444796 \nu^{15} - 15141752 \nu^{14} + 229149445 \nu^{13} + \cdots - 40545056 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 245577 \nu^{17} + 59165 \nu^{16} - 9984836 \nu^{15} + 2413960 \nu^{14} - 139094464 \nu^{13} + \cdots + 10151552 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 184251 \nu^{17} + 1197742 \nu^{16} - 7378154 \nu^{15} + 48840556 \nu^{14} - 99770897 \nu^{13} + \cdots + 200805888 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 816927 \nu^{17} + 945504 \nu^{16} - 33389870 \nu^{15} + 38574862 \nu^{14} - 469771363 \nu^{13} + \cdots + 177904800 ) / 3648448 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 500589 \nu^{17} + 126119 \nu^{16} + 20384012 \nu^{15} + 5132847 \nu^{14} + 284771130 \nu^{13} + \cdots + 11450544 ) / 1824224 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 816927 \nu^{17} + 945504 \nu^{16} + 33389870 \nu^{15} + 38574862 \nu^{14} + 469771363 \nu^{13} + \cdots + 177904800 ) / 3648448 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{14} + \beta_{12} - \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - \beta_{6} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 2 \beta_{12} - 9 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{17} + 13 \beta_{16} - 3 \beta_{15} + 13 \beta_{14} - 10 \beta_{13} - 16 \beta_{12} + \cdots + 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{17} + 60 \beta_{16} + 67 \beta_{15} - 60 \beta_{14} - 8 \beta_{13} - 43 \beta_{12} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 311 \beta_{17} - 223 \beta_{16} + 88 \beta_{15} - 223 \beta_{14} + 250 \beta_{13} + 263 \beta_{12} + \cdots - 826 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 201 \beta_{17} - 1152 \beta_{16} - 1353 \beta_{15} + 1152 \beta_{14} + 290 \beta_{13} + 837 \beta_{12} + \cdots - 541 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6232 \beta_{17} + 4230 \beta_{16} - 2002 \beta_{15} + 4230 \beta_{14} - 5190 \beta_{13} + \cdots + 15480 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4545 \beta_{17} + 22401 \beta_{16} + 26946 \beta_{15} - 22401 \beta_{14} - 7206 \beta_{13} + \cdots + 11188 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 124842 \beta_{17} - 82827 \beta_{16} + 42015 \beta_{15} - 82827 \beta_{14} + 103956 \beta_{13} + \cdots - 301215 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 95218 \beta_{17} - 440539 \beta_{16} - 535757 \beta_{15} + 440539 \beta_{14} + 157214 \beta_{13} + \cdots - 226645 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2493411 \beta_{17} + 1638678 \beta_{16} - 854733 \beta_{15} + 1638678 \beta_{14} - 2068510 \beta_{13} + \cdots + 5941991 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1936111 \beta_{17} + 8717899 \beta_{16} + 10654010 \beta_{15} - 8717899 \beta_{14} - 3253996 \beta_{13} + \cdots + 4546778 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 49703113 \beta_{17} - 32535830 \beta_{16} + 17167283 \beta_{15} - 32535830 \beta_{14} + 41129404 \beta_{13} + \cdots - 117825083 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 38881344 \beta_{17} - 173040882 \beta_{16} - 211922226 \beta_{15} + 173040882 \beta_{14} + \cdots - 90802376 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 989790567 \beta_{17} + 646837959 \beta_{16} - 342952608 \beta_{15} + 646837959 \beta_{14} + \cdots + 2341134580 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 776707592 \beta_{17} + 3439394657 \beta_{16} + 4216102249 \beta_{15} - 3439394657 \beta_{14} + \cdots + 1809673851 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(-\beta_{10}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
− 2.91656i
0.784748i
− 1.12338i
2.91656i
− 0.784748i
1.12338i
− 0.400115i
− 4.46066i
2.20712i
2.26471i
− 0.535872i
− 1.12730i
0.400115i
4.46066i
− 2.20712i
− 2.26471i
0.535872i
1.12730i
0 0.939693 + 0.342020i 0 −0.549114 − 3.11418i 0 1.34897 − 2.33648i 0 0.766044 + 0.642788i 0
25.2 0 0.939693 + 0.342020i 0 0.0502528 + 0.284998i 0 −1.83622 + 3.18043i 0 0.766044 + 0.642788i 0
25.3 0 0.939693 + 0.342020i 0 0.651565 + 3.69521i 0 2.42694 − 4.20359i 0 0.766044 + 0.642788i 0
73.1 0 0.939693 − 0.342020i 0 −0.549114 + 3.11418i 0 1.34897 + 2.33648i 0 0.766044 − 0.642788i 0
73.2 0 0.939693 − 0.342020i 0 0.0502528 − 0.284998i 0 −1.83622 − 3.18043i 0 0.766044 − 0.642788i 0
73.3 0 0.939693 − 0.342020i 0 0.651565 − 3.69521i 0 2.42694 + 4.20359i 0 0.766044 − 0.642788i 0
169.1 0 −0.766044 − 0.642788i 0 −1.17314 + 0.426986i 0 1.26992 + 2.19956i 0 0.173648 + 0.984808i 0
169.2 0 −0.766044 − 0.642788i 0 −0.543584 + 0.197848i 0 −0.286343 − 0.495961i 0 0.173648 + 0.984808i 0
169.3 0 −0.766044 − 0.642788i 0 4.09611 − 1.49086i 0 −0.749616 − 1.29837i 0 0.173648 + 0.984808i 0
289.1 0 −0.173648 − 0.984808i 0 −3.10397 + 2.60454i 0 1.83370 − 3.17606i 0 −0.939693 + 0.342020i 0
289.2 0 −0.173648 − 0.984808i 0 −0.665625 + 0.558526i 0 −1.13299 + 1.96240i 0 −0.939693 + 0.342020i 0
289.3 0 −0.173648 − 0.984808i 0 2.73750 − 2.29704i 0 0.125641 − 0.217617i 0 −0.939693 + 0.342020i 0
313.1 0 −0.766044 + 0.642788i 0 −1.17314 − 0.426986i 0 1.26992 − 2.19956i 0 0.173648 − 0.984808i 0
313.2 0 −0.766044 + 0.642788i 0 −0.543584 − 0.197848i 0 −0.286343 + 0.495961i 0 0.173648 − 0.984808i 0
313.3 0 −0.766044 + 0.642788i 0 4.09611 + 1.49086i 0 −0.749616 + 1.29837i 0 0.173648 − 0.984808i 0
385.1 0 −0.173648 + 0.984808i 0 −3.10397 − 2.60454i 0 1.83370 + 3.17606i 0 −0.939693 − 0.342020i 0
385.2 0 −0.173648 + 0.984808i 0 −0.665625 − 0.558526i 0 −1.13299 − 1.96240i 0 −0.939693 − 0.342020i 0
385.3 0 −0.173648 + 0.984808i 0 2.73750 + 2.29704i 0 0.125641 + 0.217617i 0 −0.939693 − 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bg.d ✓ 18
4.b odd 2 1 912.2.bo.l 18
19.e even 9 1 inner 456.2.bg.d ✓ 18
19.e even 9 1 8664.2.a.bn 9
19.f odd 18 1 8664.2.a.bp 9
76.l odd 18 1 912.2.bo.l 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bg.d ✓ 18 1.a even 1 1 trivial
456.2.bg.d ✓ 18 19.e even 9 1 inner
912.2.bo.l 18 4.b odd 2 1
912.2.bo.l 18 76.l odd 18 1
8664.2.a.bn 9 19.e even 9 1
8664.2.a.bp 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 3 T_{5}^{17} + 6 T_{5}^{16} - 36 T_{5}^{15} + 150 T_{5}^{14} - 561 T_{5}^{13} + \cdots + 18496 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} - 3 T^{17} + \cdots + 18496 \) Copy content Toggle raw display
$7$ \( T^{18} - 6 T^{17} + \cdots + 47961 \) Copy content Toggle raw display
$11$ \( T^{18} - 9 T^{17} + \cdots + 23104 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 19808592049 \) Copy content Toggle raw display
$17$ \( T^{18} + 3 T^{17} + \cdots + 87616 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 1605410634304 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 3135704307264 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 830013924601 \) Copy content Toggle raw display
$37$ \( (T^{9} - 15 T^{8} + \cdots - 69933)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 350246177856 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 4805350636321 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 154373553216 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 2701504576 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 10307233198144 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 52415442502201 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 73203356362816 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 1280086325281 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 19990849321 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 69621134469184 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 20096239991689 \) Copy content Toggle raw display
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