Properties

Label 468.2.h.d
Level $468$
Weight $2$
Character orbit 468.h
Analytic conductor $3.737$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(467,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.467");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.h (of order \(2\), degree \(1\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 10x^{12} + 49x^{8} - 160x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{8} - \beta_{3}) q^{4} + ( - \beta_{12} - \beta_{6}) q^{5} + ( - \beta_{14} - \beta_{9}) q^{7} - \beta_{15} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{8} - \beta_{3}) q^{4} + ( - \beta_{12} - \beta_{6}) q^{5} + ( - \beta_{14} - \beta_{9}) q^{7} - \beta_{15} q^{8} + ( - \beta_{11} + 2 \beta_{3} - 1) q^{10} + ( - \beta_{15} - \beta_{12} + \beta_{7}) q^{11} + (\beta_{13} - \beta_{10} + \beta_{3} + 1) q^{13} + (\beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{14} + (\beta_{11} - \beta_{8} - \beta_{3} - 3) q^{16} + ( - \beta_{5} + \beta_{4}) q^{17} + (\beta_{14} + 2 \beta_{13} + 2 \beta_{10} + \beta_{9}) q^{19} + (2 \beta_{12} - 2 \beta_{7} - 3 \beta_{6}) q^{20} + (\beta_{11} - 2 \beta_{8} - 3) q^{22} + ( - \beta_{4} - 2 \beta_{2}) q^{23} + ( - 2 \beta_{3} + 5) q^{25} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{26} + ( - 2 \beta_{14} + 2 \beta_{13} - \beta_{10}) q^{28} + ( - 3 \beta_{5} - 2 \beta_{4}) q^{29} + (2 \beta_{14} + \beta_{13} + \beta_{10} + 2 \beta_{9}) q^{31} + ( - \beta_{15} - 2 \beta_{12} - 2 \beta_{7} + \beta_{6}) q^{32} + ( - \beta_{13} + \beta_{10} - \beta_{9}) q^{34} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{35} + (\beta_{14} - \beta_{13} + \beta_{10} - \beta_{9}) q^{37} + (3 \beta_{5} + 3 \beta_{4} + \beta_{2} + \beta_1) q^{38} + ( - 3 \beta_{11} + 2 \beta_{8} - 2 \beta_{3} - 3) q^{40} + (\beta_{12} + \beta_{6}) q^{41} + (4 \beta_{8} + 2 \beta_{3} + 2) q^{43} + ( - 2 \beta_{15} - 2 \beta_{12} - 2 \beta_{7} - \beta_{6}) q^{44} + ( - 4 \beta_{14} - \beta_{13} - 2 \beta_{10} - \beta_{9}) q^{46} + ( - \beta_{15} - \beta_{12} + \beta_{7}) q^{47} + ( - 6 \beta_{3} + 3) q^{49} + (5 \beta_{7} + 2 \beta_{6}) q^{50} + ( - \beta_{13} - \beta_{11} - 3 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} - 1) q^{52} + ( - 3 \beta_{5} + 3 \beta_{4}) q^{53} + ( - 4 \beta_{11} + 4 \beta_{8} + 2) q^{55} + (4 \beta_{5} - 3 \beta_{4} + \beta_1) q^{56} + (2 \beta_{13} + 3 \beta_{10} + 2 \beta_{9}) q^{58} + (2 \beta_{15} + 3 \beta_{12} - 2 \beta_{7} - \beta_{6}) q^{59} + (4 \beta_{3} + 2) q^{61} + (3 \beta_{4} + 2 \beta_{2} - \beta_1) q^{62} + (2 \beta_{11} + \beta_{8} + 5 \beta_{3} - 2) q^{64} + (\beta_{15} - \beta_{12} + 3 \beta_{7} - 2 \beta_{5} + 5 \beta_{4}) q^{65} + ( - 2 \beta_{14} + \beta_{13} + \beta_{10} - 2 \beta_{9}) q^{67} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{68} + (4 \beta_{14} - 5 \beta_{13} + 3 \beta_{9}) q^{70} + (2 \beta_{15} - \beta_{12} - 2 \beta_{7} + 3 \beta_{6}) q^{71} + (\beta_{14} - 3 \beta_{13} + 3 \beta_{10} - \beta_{9}) q^{73} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{74} + (2 \beta_{14} - \beta_{10} - 2 \beta_{9}) q^{76} + (6 \beta_{5} - 5 \beta_{4}) q^{77} + (4 \beta_{11} + 2 \beta_{3}) q^{79} + (2 \beta_{15} + 6 \beta_{12} - 6 \beta_{7} + \beta_{6}) q^{80} + (\beta_{11} - 2 \beta_{3} + 1) q^{82} + ( - 3 \beta_{12} + 3 \beta_{6}) q^{83} + ( - \beta_{14} + 3 \beta_{13} - 3 \beta_{10} + \beta_{9}) q^{85} + (4 \beta_{15} + 2 \beta_{7} + 2 \beta_{6}) q^{86} + (\beta_{11} + 4 \beta_{3} - 7) q^{88} + ( - \beta_{12} - \beta_{6}) q^{89} + ( - \beta_{13} + 4 \beta_{11} - \beta_{10} - 4 \beta_{8} - 2) q^{91} + ( - \beta_{5} - 6 \beta_{4} - \beta_{2} + 2 \beta_1) q^{92} + (\beta_{11} - 2 \beta_{8} - 3) q^{94} + (2 \beta_{5} + 3 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{95} + ( - \beta_{14} - \beta_{13} + \beta_{10} + \beta_{9}) q^{97} + (3 \beta_{7} + 6 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 16 q^{10} + 16 q^{13} - 40 q^{16} - 32 q^{22} + 80 q^{25} - 64 q^{40} + 48 q^{49} - 8 q^{52} + 32 q^{61} - 40 q^{64} + 16 q^{82} - 112 q^{88} - 32 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 10x^{12} + 49x^{8} - 160x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 6\nu^{10} + 17\nu^{6} - 16\nu^{2} ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 6\nu^{10} - 47\nu^{6} + 304\nu^{2} ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{12} + 10\nu^{8} - 33\nu^{4} + 80 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{14} + 14\nu^{10} - 51\nu^{6} + 144\nu^{2} ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{14} + 34\nu^{10} - 149\nu^{6} + 336\nu^{2} ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} + 6\nu^{13} - 10\nu^{11} - 28\nu^{9} + 49\nu^{7} + 102\nu^{5} - 160\nu^{3} - 288\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{15} - 20\nu^{13} + 34\nu^{11} + 136\nu^{9} - 85\nu^{7} - 596\nu^{5} + 272\nu^{3} + 1344\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{12} - 6\nu^{8} + 21\nu^{4} - 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 20\nu^{13} + 6\nu^{11} - 136\nu^{9} + 17\nu^{7} + 596\nu^{5} - 16\nu^{3} - 1600\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{15} + 6\nu^{13} + 10\nu^{11} - 28\nu^{9} - 49\nu^{7} + 102\nu^{5} + 160\nu^{3} - 288\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5\nu^{12} - 34\nu^{8} + 149\nu^{4} - 368 ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -3\nu^{15} + 14\nu^{11} - 51\nu^{7} + 144\nu^{3} - 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{15} + 14\nu^{11} - 51\nu^{7} + 144\nu^{3} + 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -2\nu^{15} - 5\nu^{13} + 12\nu^{11} + 34\nu^{9} - 50\nu^{7} - 117\nu^{5} + 88\nu^{3} + 304\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\nu^{15} - 20\nu^{13} - 130\nu^{11} + 136\nu^{9} + 485\nu^{7} - 340\nu^{5} - 976\nu^{3} + 1088\nu ) / 512 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{8} + \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{15} + 2\beta_{14} + 3\beta_{13} - \beta_{12} + 2\beta_{9} - 2\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{5} + 5\beta_{4} + \beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{15} - 3\beta_{14} + 5\beta_{13} + \beta_{12} - \beta_{10} + 4\beta_{9} + 11\beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{11} - \beta_{8} + 11\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 12\beta_{15} + 12\beta_{14} + 3\beta_{13} + 3\beta_{12} + 10\beta_{10} + 6\beta_{9} + 10\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3\beta_{5} + \beta_{4} - 3\beta_{2} + 24\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -3\beta_{15} + 3\beta_{14} + \beta_{13} - 23\beta_{12} - 3\beta_{10} + 24\beta_{9} + 45\beta_{7} + 3\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -3\beta_{11} + 23\beta_{8} + 45\beta_{3} + 14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 22\beta_{15} + 22\beta_{14} + 11\beta_{13} - 17\beta_{12} + 68\beta_{10} - 6\beta_{9} + 34\beta_{7} + 68\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 51\beta_{5} - 75\beta_{4} + 17\beta_{2} + 44\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 17 \beta_{15} + 17 \beta_{14} - 75 \beta_{13} - 119 \beta_{12} + 51 \beta_{10} + 44 \beta_{9} + 71 \beta_{7} - 51 \beta_{6} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
467.1
0.450197 1.34064i
1.34064 + 0.450197i
0.450197 + 1.34064i
1.34064 0.450197i
0.0733219 + 1.41231i
−1.41231 + 0.0733219i
0.0733219 1.41231i
−1.41231 0.0733219i
1.41231 + 0.0733219i
−0.0733219 + 1.41231i
1.41231 0.0733219i
−0.0733219 1.41231i
−1.34064 + 0.450197i
−0.450197 1.34064i
−1.34064 0.450197i
−0.450197 + 1.34064i
−1.26631 0.629640i 0 1.20711 + 1.59465i 3.58168 0 −4.29945 −0.524525 2.77937i 0 −4.53553 2.25517i
467.2 −1.26631 0.629640i 0 1.20711 + 1.59465i 3.58168 0 4.29945 −0.524525 2.77937i 0 −4.53553 2.25517i
467.3 −1.26631 + 0.629640i 0 1.20711 1.59465i 3.58168 0 −4.29945 −0.524525 + 2.77937i 0 −4.53553 + 2.25517i
467.4 −1.26631 + 0.629640i 0 1.20711 1.59465i 3.58168 0 4.29945 −0.524525 + 2.77937i 0 −4.53553 + 2.25517i
467.5 −0.946809 1.05050i 0 −0.207107 + 1.98925i −2.67798 0 −1.23074 2.28580 1.66587i 0 2.53553 + 2.81322i
467.6 −0.946809 1.05050i 0 −0.207107 + 1.98925i −2.67798 0 1.23074 2.28580 1.66587i 0 2.53553 + 2.81322i
467.7 −0.946809 + 1.05050i 0 −0.207107 1.98925i −2.67798 0 −1.23074 2.28580 + 1.66587i 0 2.53553 2.81322i
467.8 −0.946809 + 1.05050i 0 −0.207107 1.98925i −2.67798 0 1.23074 2.28580 + 1.66587i 0 2.53553 2.81322i
467.9 0.946809 1.05050i 0 −0.207107 1.98925i 2.67798 0 −1.23074 −2.28580 1.66587i 0 2.53553 2.81322i
467.10 0.946809 1.05050i 0 −0.207107 1.98925i 2.67798 0 1.23074 −2.28580 1.66587i 0 2.53553 2.81322i
467.11 0.946809 + 1.05050i 0 −0.207107 + 1.98925i 2.67798 0 −1.23074 −2.28580 + 1.66587i 0 2.53553 + 2.81322i
467.12 0.946809 + 1.05050i 0 −0.207107 + 1.98925i 2.67798 0 1.23074 −2.28580 + 1.66587i 0 2.53553 + 2.81322i
467.13 1.26631 0.629640i 0 1.20711 1.59465i −3.58168 0 −4.29945 0.524525 2.77937i 0 −4.53553 + 2.25517i
467.14 1.26631 0.629640i 0 1.20711 1.59465i −3.58168 0 4.29945 0.524525 2.77937i 0 −4.53553 + 2.25517i
467.15 1.26631 + 0.629640i 0 1.20711 + 1.59465i −3.58168 0 −4.29945 0.524525 + 2.77937i 0 −4.53553 2.25517i
467.16 1.26631 + 0.629640i 0 1.20711 + 1.59465i −3.58168 0 4.29945 0.524525 + 2.77937i 0 −4.53553 2.25517i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 467.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner
52.b odd 2 1 inner
156.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.h.d 16
3.b odd 2 1 inner 468.2.h.d 16
4.b odd 2 1 inner 468.2.h.d 16
12.b even 2 1 inner 468.2.h.d 16
13.b even 2 1 inner 468.2.h.d 16
39.d odd 2 1 inner 468.2.h.d 16
52.b odd 2 1 inner 468.2.h.d 16
156.h even 2 1 inner 468.2.h.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.h.d 16 1.a even 1 1 trivial
468.2.h.d 16 3.b odd 2 1 inner
468.2.h.d 16 4.b odd 2 1 inner
468.2.h.d 16 12.b even 2 1 inner
468.2.h.d 16 13.b even 2 1 inner
468.2.h.d 16 39.d odd 2 1 inner
468.2.h.d 16 52.b odd 2 1 inner
468.2.h.d 16 156.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):

\( T_{5}^{4} - 20T_{5}^{2} + 92 \) Copy content Toggle raw display
\( T_{7}^{4} - 20T_{7}^{2} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{6} + 7 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 20 T^{2} + 92)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 20 T^{2} + 28)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + 22 T^{2} - 52 T + 169)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{2} + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 52 T^{2} + 28)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 104 T^{2} + 2576)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 68 T^{2} + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 76 T^{2} + 1372)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 40 T^{2} + 368)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 20 T^{2} + 92)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 104 T^{2} + 2576)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 20 T^{2} + 28)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 108 T^{2} + 324)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 108 T^{2} + 28)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 28)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 108 T^{2} + 28)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 140 T^{2} + 1372)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 152 T^{2} + 368)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 208 T^{2} + 10304)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 108 T^{2} + 2268)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 20 T^{2} + 92)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 88 T^{2} + 368)^{4} \) Copy content Toggle raw display
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