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Properties

Label 468.2.p.a
Level $468$
Weight $2$
Character orbit 468.p
Analytic conductor $3.737$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(125,468)]
 
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([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.p (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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 12x^{10} - 8x^{9} + 72x^{8} - 12x^{7} - 198x^{6} + 72x^{5} + 276x^{4} - 212x^{3} + 108x^{2} - 24x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} - \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{5} - \beta_{4} q^{7} + (\beta_{9} + \beta_{6}) q^{11} + ( - \beta_{11} + \beta_{3} - 1) q^{13} + (\beta_{6} + \beta_{5} - \beta_{2}) q^{17} + ( - \beta_{11} - \beta_{3} + \beta_1) q^{19} + (\beta_{9} + \beta_{8}) q^{23} + ( - 2 \beta_{11} - 3 \beta_{7} + \cdots + \beta_{3}) q^{25}+ \cdots + ( - \beta_{11} - \beta_{7} + 2 \beta_{3} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{13} + 24 q^{31} + 12 q^{37} + 48 q^{55} + 24 q^{61} - 72 q^{67} - 36 q^{73} - 48 q^{85} - 72 q^{91} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12x^{10} - 8x^{9} + 72x^{8} - 12x^{7} - 198x^{6} + 72x^{5} + 276x^{4} - 212x^{3} + 108x^{2} - 24x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37696274124 \nu^{11} - 97451157618 \nu^{10} - 520658173008 \nu^{9} + \cdots - 24457237127768 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 87189858 \nu^{11} - 202155894 \nu^{10} - 1097969265 \nu^{9} - 1707013710 \nu^{8} + \cdots - 6711331692 ) / 1169428591 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 711004074960 \nu^{11} + 90707194879 \nu^{10} + 8492188116312 \nu^{9} - 4616105248806 \nu^{8} + \cdots - 4752644879844 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 762684585636 \nu^{11} - 130722788001 \nu^{10} - 9211264620768 \nu^{9} + \cdots + 9072541350468 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1413886022520 \nu^{11} + 16303555550 \nu^{10} + 16910655924470 \nu^{9} + \cdots - 25690765469950 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1694466639060 \nu^{11} - 553238337604 \nu^{10} - 20537561042216 \nu^{9} + \cdots + 9784433910654 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84513528 \nu^{11} + 13220778 \nu^{10} + 1014074018 \nu^{9} - 517514454 \nu^{8} + \cdots - 777617133 ) / 240422869 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4191493001621 \nu^{11} + 45770646942 \nu^{10} + 50326203309850 \nu^{9} + \cdots - 75556851078224 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4854520681203 \nu^{11} - 1597015481442 \nu^{10} - 58739183628520 \nu^{9} + \cdots + 28736335763664 ) / 6313744962809 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 156084138 \nu^{11} - 27991314 \nu^{10} - 1881972543 \nu^{9} + 908700930 \nu^{8} + \cdots + 1806167010 ) / 141783139 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8653889733354 \nu^{11} + 1368115375578 \nu^{10} + 103834693065400 \nu^{9} + \cdots - 79431018499776 ) / 6313744962809 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{10} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{11} - 4\beta_{10} + 6\beta_{9} - 12\beta_{7} - 4\beta_{2} - 3\beta _1 - 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{10} - 6\beta_{7} + 9\beta_{6} + 18\beta_{5} - 27\beta_{4} - 9\beta_{3} - 6\beta_{2} + 12 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 54 \beta_{11} + 44 \beta_{10} - 36 \beta_{9} + 36 \beta_{8} + 216 \beta_{7} + 6 \beta_{6} + \cdots - 9 \beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9 \beta_{11} + 16 \beta_{10} + 21 \beta_{9} + 9 \beta_{8} - 90 \beta_{7} - 189 \beta_{6} - 81 \beta_{5} + \cdots - 210 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 189 \beta_{11} - 155 \beta_{10} - 270 \beta_{8} - 777 \beta_{7} + 90 \beta_{5} - 126 \beta_{4} + \cdots + 777 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 432 \beta_{11} - 221 \beta_{10} - 432 \beta_{9} + 180 \beta_{8} + 3024 \beta_{7} + 1980 \beta_{6} + \cdots + 1260 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 660 \beta_{9} + 660 \beta_{8} - 324 \beta_{6} - 324 \beta_{5} + 459 \beta_{4} + 459 \beta_{3} + \cdots - 3944 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6642 \beta_{11} + 464 \beta_{10} + 2754 \beta_{9} - 6642 \beta_{8} - 40098 \beta_{7} - 8721 \beta_{6} + \cdots + 16626 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21033 \beta_{11} + 15109 \beta_{10} - 29754 \beta_{9} + 90774 \beta_{7} + 18879 \beta_{6} + \cdots + 90774 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18879 \beta_{11} + 2665 \beta_{10} + 18879 \beta_{9} + 45570 \beta_{8} + 105270 \beta_{7} + \cdots - 254100 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\beta_{7}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
0.169938 + 0.0703908i
−1.30084 + 3.14050i
1.13090 − 2.73024i
1.13090 + 0.468435i
−1.30084 − 0.538825i
0.169938 − 0.410268i
0.169938 − 0.0703908i
−1.30084 − 3.14050i
1.13090 + 2.73024i
1.13090 − 0.468435i
−1.30084 + 0.538825i
0.169938 + 0.410268i
0 0 0 −2.74674 + 2.74674i 0 −0.339877 + 0.339877i 0 0 0
125.2 0 0 0 −1.95779 + 1.95779i 0 2.60168 − 2.60168i 0 0 0
125.3 0 0 0 −0.788954 + 0.788954i 0 −2.26180 + 2.26180i 0 0 0
125.4 0 0 0 0.788954 − 0.788954i 0 −2.26180 + 2.26180i 0 0 0
125.5 0 0 0 1.95779 − 1.95779i 0 2.60168 − 2.60168i 0 0 0
125.6 0 0 0 2.74674 − 2.74674i 0 −0.339877 + 0.339877i 0 0 0
161.1 0 0 0 −2.74674 − 2.74674i 0 −0.339877 − 0.339877i 0 0 0
161.2 0 0 0 −1.95779 − 1.95779i 0 2.60168 + 2.60168i 0 0 0
161.3 0 0 0 −0.788954 − 0.788954i 0 −2.26180 − 2.26180i 0 0 0
161.4 0 0 0 0.788954 + 0.788954i 0 −2.26180 − 2.26180i 0 0 0
161.5 0 0 0 1.95779 + 1.95779i 0 2.60168 + 2.60168i 0 0 0
161.6 0 0 0 2.74674 + 2.74674i 0 −0.339877 − 0.339877i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.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.p.a ✓ 12
3.b odd 2 1 inner 468.2.p.a ✓ 12
4.b odd 2 1 1872.2.bi.d 12
12.b even 2 1 1872.2.bi.d 12
13.b even 2 1 6084.2.p.c 12
13.d odd 4 1 inner 468.2.p.a ✓ 12
13.d odd 4 1 6084.2.p.c 12
39.d odd 2 1 6084.2.p.c 12
39.f even 4 1 inner 468.2.p.a ✓ 12
39.f even 4 1 6084.2.p.c 12
52.f even 4 1 1872.2.bi.d 12
156.l odd 4 1 1872.2.bi.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.p.a ✓ 12 1.a even 1 1 trivial
468.2.p.a ✓ 12 3.b odd 2 1 inner
468.2.p.a ✓ 12 13.d odd 4 1 inner
468.2.p.a ✓ 12 39.f even 4 1 inner
1872.2.bi.d 12 4.b odd 2 1
1872.2.bi.d 12 12.b even 2 1
1872.2.bi.d 12 52.f even 4 1
1872.2.bi.d 12 156.l odd 4 1
6084.2.p.c 12 13.b even 2 1
6084.2.p.c 12 13.d odd 4 1
6084.2.p.c 12 39.d odd 2 1
6084.2.p.c 12 39.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(468, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 288 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( (T^{6} + 8 T^{3} + 144 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 1296 T^{8} + \cdots + 49787136 \) Copy content Toggle raw display
$13$ \( (T^{6} + 6 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 102 T^{4} + \cdots - 12168)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 104 T^{3} + \cdots + 5408)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 48 T^{4} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 102 T^{4} + \cdots + 3528)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + \cdots + 1568)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 6 T^{5} + \cdots + 163592)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 592240896 \) Copy content Toggle raw display
$43$ \( (T^{6} + 144 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 9936 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( (T^{6} + 102 T^{4} + \cdots + 12168)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 303595776 \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 36 T^{5} + \cdots + 253472)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 47664 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$73$ \( (T^{6} + 18 T^{5} + \cdots + 63368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 48 T + 96)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 592240896 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1421970391296 \) Copy content Toggle raw display
$97$ \( (T^{6} + 6 T^{5} + \cdots + 35912)^{2} \) Copy content Toggle raw display
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