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Properties

Label 468.2.n.j
Level $468$
Weight $2$
Character orbit 468.n
Analytic conductor $3.737$
Analytic rank $0$
Dimension $10$
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] = [468,2,Mod(307,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.n (of order \(4\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: 10.0.578281160704.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{7} - \beta_{5}) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{7} - \beta_{5}) q^{4} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{5}+ \cdots + (2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} + 4 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} + 4 q^{4} + 2 q^{8} + 12 q^{10} + 10 q^{11} + 2 q^{13} - 16 q^{14} + 20 q^{16} - 20 q^{19} + 32 q^{20} - 20 q^{22} - 20 q^{23} + 8 q^{26} - 8 q^{28} + 20 q^{29} + 4 q^{31} - 14 q^{32} - 12 q^{34} - 10 q^{37} - 8 q^{38} - 8 q^{40} - 4 q^{41} - 24 q^{43} + 8 q^{44} + 20 q^{46} - 6 q^{47} - 32 q^{50} - 28 q^{52} - 12 q^{53} - 24 q^{56} - 20 q^{58} + 10 q^{59} + 16 q^{61} + 44 q^{62} + 4 q^{64} + 4 q^{68} + 12 q^{70} - 34 q^{71} + 10 q^{73} - 8 q^{74} - 4 q^{76} + 20 q^{80} - 60 q^{82} + 10 q^{83} + 16 q^{85} - 32 q^{86} + 4 q^{88} + 16 q^{89} + 16 q^{91} + 24 q^{92} + 36 q^{94} - 40 q^{95} + 10 q^{97} - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} + 2\nu^{8} - 6\nu^{7} - 2\nu^{6} - 9\nu^{5} - 8\nu^{4} + 6\nu^{3} + 28\nu^{2} + 8\nu + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + \nu^{8} - 2\nu^{7} + \nu^{5} - \nu^{4} + 10\nu^{2} - 12\nu + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{9} + 2\nu^{8} + 2\nu^{7} + 6\nu^{6} + 19\nu^{5} + 16\nu^{4} - 2\nu^{3} + 12\nu^{2} - 88\nu - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 6\nu^{8} - 2\nu^{7} + 2\nu^{6} - 7\nu^{5} - 16\nu^{4} - 14\nu^{3} + 4\nu^{2} - 24\nu + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{9} + 8\nu^{8} - 2\nu^{7} + 6\nu^{6} + \nu^{5} - 14\nu^{4} - 26\nu^{3} + 16\nu^{2} - 72\nu + 96 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{9} - 6\nu^{8} + 2\nu^{7} - 2\nu^{6} + 7\nu^{5} + 16\nu^{4} + 14\nu^{3} + 12\nu^{2} + 24\nu - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 4\nu^{8} + 2\nu^{7} + 2\nu^{6} - 9\nu^{5} - 22\nu^{4} - 14\nu^{3} - 24\nu^{2} + 8\nu + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{9} + 2\nu^{7} + 10\nu^{6} + 23\nu^{5} + 22\nu^{4} - 6\nu^{3} - 88\nu - 64 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} + \beta_{4} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2\beta_{3} - 2\beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{6} + 3\beta_{4} - 2\beta_{3} + \beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{9} - 2\beta_{8} - 3\beta_{7} - 2\beta_{6} + 3\beta_{5} + 4\beta_{4} - \beta_{3} + 9\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2\beta_{9} - \beta_{8} + 5\beta_{7} - 5\beta_{6} + 13\beta_{5} - 2\beta_{4} + 2\beta_{3} - 4\beta_{2} - 2\beta _1 + 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\beta_{8}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.239536 + 1.39378i
0.437403 + 1.34487i
−1.28245 + 0.596081i
1.41363 + 0.0406696i
−0.329042 − 1.37540i
−0.239536 − 1.39378i
0.437403 − 1.34487i
−1.28245 − 0.596081i
1.41363 − 0.0406696i
−0.329042 + 1.37540i
−1.39378 − 0.239536i 0 1.88525 + 0.667720i −0.856369 − 0.856369i 0 −3.17474 − 3.17474i −2.46767 − 1.38224i 0 0.988459 + 1.39872i
307.2 −1.34487 + 0.437403i 0 1.61736 − 1.17650i 1.47820 + 1.47820i 0 3.28622 + 3.28622i −1.66053 + 2.28968i 0 −2.63455 − 1.34142i
307.3 −0.596081 − 1.28245i 0 −1.28937 + 1.52889i −0.492880 − 0.492880i 0 0.220245 + 0.220245i 2.72931 + 0.742217i 0 −0.338299 + 0.925891i
307.4 −0.0406696 + 1.41363i 0 −1.99669 − 0.114983i −2.59729 − 2.59729i 0 1.70740 + 1.70740i 0.243748 − 2.81790i 0 3.77724 − 3.56597i
307.5 1.37540 − 0.329042i 0 1.78346 − 0.905130i 2.46834 + 2.46834i 0 −2.03913 − 2.03913i 2.15515 − 1.83175i 0 4.20715 + 2.58278i
343.1 −1.39378 + 0.239536i 0 1.88525 − 0.667720i −0.856369 + 0.856369i 0 −3.17474 + 3.17474i −2.46767 + 1.38224i 0 0.988459 − 1.39872i
343.2 −1.34487 − 0.437403i 0 1.61736 + 1.17650i 1.47820 − 1.47820i 0 3.28622 − 3.28622i −1.66053 − 2.28968i 0 −2.63455 + 1.34142i
343.3 −0.596081 + 1.28245i 0 −1.28937 − 1.52889i −0.492880 + 0.492880i 0 0.220245 − 0.220245i 2.72931 − 0.742217i 0 −0.338299 − 0.925891i
343.4 −0.0406696 − 1.41363i 0 −1.99669 + 0.114983i −2.59729 + 2.59729i 0 1.70740 − 1.70740i 0.243748 + 2.81790i 0 3.77724 + 3.56597i
343.5 1.37540 + 0.329042i 0 1.78346 + 0.905130i 2.46834 − 2.46834i 0 −2.03913 + 2.03913i 2.15515 + 1.83175i 0 4.20715 − 2.58278i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.n.j 10
3.b odd 2 1 156.2.k.f yes 10
4.b odd 2 1 468.2.n.k 10
12.b even 2 1 156.2.k.e ✓ 10
13.d odd 4 1 468.2.n.k 10
39.f even 4 1 156.2.k.e ✓ 10
52.f even 4 1 inner 468.2.n.j 10
156.l odd 4 1 156.2.k.f yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.k.e ✓ 10 12.b even 2 1
156.2.k.e ✓ 10 39.f even 4 1
156.2.k.f yes 10 3.b odd 2 1
156.2.k.f yes 10 156.l odd 4 1
468.2.n.j 10 1.a even 1 1 trivial
468.2.n.j 10 52.f even 4 1 inner
468.2.n.k 10 4.b odd 2 1
468.2.n.k 10 13.d odd 4 1

Hecke kernels

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

\( T_{5}^{10} + 176T_{5}^{6} - 32T_{5}^{5} + 512T_{5}^{3} + 1600T_{5}^{2} + 1280T_{5} + 512 \) Copy content Toggle raw display
\( T_{7}^{10} + 488T_{7}^{6} + 64T_{7}^{5} - 1792T_{7}^{3} + 21904T_{7}^{2} - 9472T_{7} + 2048 \) Copy content Toggle raw display
\( T_{11}^{10} - 10 T_{11}^{9} + 50 T_{11}^{8} - 88 T_{11}^{7} + 96 T_{11}^{6} - 224 T_{11}^{5} + \cdots + 128 \) Copy content Toggle raw display
\( T_{17}^{10} + 116T_{17}^{8} + 4704T_{17}^{6} + 86400T_{17}^{4} + 728320T_{17}^{2} + 2262016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 176 T^{6} + \cdots + 512 \) Copy content Toggle raw display
$7$ \( T^{10} + 488 T^{6} + \cdots + 2048 \) Copy content Toggle raw display
$11$ \( T^{10} - 10 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 116 T^{8} + \cdots + 2262016 \) Copy content Toggle raw display
$19$ \( T^{10} + 20 T^{9} + \cdots + 21632 \) Copy content Toggle raw display
$23$ \( (T^{5} + 10 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 10 T^{4} + \cdots - 224)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 4 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$37$ \( T^{10} + 10 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$41$ \( T^{10} + 4 T^{9} + \cdots + 7129088 \) Copy content Toggle raw display
$43$ \( (T^{5} + 12 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 6 T^{9} + \cdots + 2918528 \) Copy content Toggle raw display
$53$ \( (T^{5} + 6 T^{4} + \cdots - 1888)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 10 T^{9} + \cdots + 20173952 \) Copy content Toggle raw display
$61$ \( (T^{5} - 8 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 640 T^{7} + \cdots + 30482432 \) Copy content Toggle raw display
$71$ \( T^{10} + 34 T^{9} + \cdots + 128 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1303766048 \) Copy content Toggle raw display
$79$ \( T^{10} + 384 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$83$ \( T^{10} - 10 T^{9} + \cdots + 2918528 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 255922688 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 10203918368 \) Copy content Toggle raw display
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