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Properties

Label 468.2.i.b
Level $468$
Weight $2$
Character orbit 468.i
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(157,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.i (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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.224744991207075.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} + 7x^{8} - 15x^{7} + 31x^{6} - 51x^{5} + 93x^{4} - 135x^{3} + 189x^{2} - 243x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots + 2) q^{5}+ \cdots + (\beta_{8} - \beta_{7} + \beta_{5} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots + 2) q^{5}+ \cdots + (\beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} + 7 q^{5} + 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} + 7 q^{5} + 2 q^{7} - 5 q^{9} - 5 q^{11} - 5 q^{13} + 14 q^{15} - 26 q^{17} + 2 q^{19} - 4 q^{21} - 8 q^{25} - 9 q^{27} + 12 q^{29} + 2 q^{31} + 23 q^{33} + 10 q^{35} + 26 q^{37} + 4 q^{41} + 8 q^{43} - 2 q^{45} + 7 q^{47} - 15 q^{49} + 17 q^{51} - 38 q^{53} - 42 q^{55} + 8 q^{57} + 11 q^{59} - 7 q^{61} - 6 q^{63} + 7 q^{65} + 2 q^{67} + 24 q^{69} - 58 q^{71} + 38 q^{73} - 20 q^{75} + 7 q^{77} + 5 q^{79} - 17 q^{81} + 24 q^{83} - 9 q^{85} + 27 q^{87} - 38 q^{89} - 4 q^{91} - 30 q^{93} + 10 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} + 7x^{8} - 15x^{7} + 31x^{6} - 51x^{5} + 93x^{4} - 135x^{3} + 189x^{2} - 243x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + 9 \nu^{8} + 115 \nu^{7} + 96 \nu^{6} - 86 \nu^{5} + 186 \nu^{4} - 213 \nu^{3} + 1116 \nu^{2} + \cdots + 2835 ) / 3807 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{9} + 387 \nu^{8} - 460 \nu^{7} + 1308 \nu^{6} - 2194 \nu^{5} + 6024 \nu^{4} - 2955 \nu^{3} + \cdots + 22923 ) / 11421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35 \nu^{9} + 108 \nu^{8} - 218 \nu^{7} + 870 \nu^{6} - 797 \nu^{5} + 1527 \nu^{4} - 2697 \nu^{3} + \cdots + 3564 ) / 11421 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44 \nu^{9} + 27 \nu^{8} + 16 \nu^{7} + 6 \nu^{6} + 1246 \nu^{5} - 147 \nu^{4} + 489 \nu^{3} + \cdots + 891 ) / 11421 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46 \nu^{9} + 9 \nu^{8} - 214 \nu^{7} + 1083 \nu^{6} - 1120 \nu^{5} + 5826 \nu^{4} - 4161 \nu^{3} + \cdots + 18063 ) / 11421 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 3\nu^{8} + 7\nu^{7} - 15\nu^{6} + 31\nu^{5} - 51\nu^{4} + 93\nu^{3} - 135\nu^{2} + 189\nu - 243 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50 \nu^{9} - 114 \nu^{8} + 251 \nu^{7} - 417 \nu^{6} + 776 \nu^{5} - 1839 \nu^{4} + 3309 \nu^{3} + \cdots - 2916 ) / 3807 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 287 \nu^{9} + 378 \nu^{8} - 1280 \nu^{7} + 2058 \nu^{6} - 4505 \nu^{5} + 6684 \nu^{4} + \cdots + 20088 ) / 11421 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + \beta_{7} + 2\beta_{6} - 3\beta_{4} + \beta_{3} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - 3\beta_{7} + \beta_{6} + 9\beta_{5} - 3\beta_{2} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{7} + \beta_{6} - 4\beta_{5} + 28\beta_{4} - \beta_{3} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} - 5\beta_{8} + 4\beta_{7} - \beta_{6} - 2\beta_{5} - 7\beta_{4} - \beta_{3} + 30\beta_{2} + 5\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{9} - 20 \beta_{8} + 8 \beta_{7} - 30 \beta_{6} + 13 \beta_{5} - 7 \beta_{4} + 24 \beta_{3} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 47 \beta_{9} - 3 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 58 \beta_{5} + 67 \beta_{4} + 22 \beta_{3} + \cdots + 32 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(1\) \(-1 - \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
1.59256 − 0.680993i
1.34437 + 1.09210i
0.330295 − 1.70027i
−0.540243 + 1.64564i
−1.22698 − 1.22251i
1.59256 + 0.680993i
1.34437 − 1.09210i
0.330295 + 1.70027i
−0.540243 − 1.64564i
−1.22698 + 1.22251i
0 −1.59256 + 0.680993i 0 −0.592561 + 1.02634i 0 1.47994 + 2.56333i 0 2.07250 − 2.16904i 0
157.2 0 −1.34437 − 1.09210i 0 −0.344366 + 0.596459i 0 0.270272 + 0.468125i 0 0.614638 + 2.93636i 0
157.3 0 −0.330295 + 1.70027i 0 0.669705 − 1.15996i 0 −2.11211 − 3.65827i 0 −2.78181 − 1.12318i 0
157.4 0 0.540243 − 1.64564i 0 1.54024 − 2.66778i 0 −0.876031 − 1.51733i 0 −2.41627 − 1.77809i 0
157.5 0 1.22698 + 1.22251i 0 2.22698 − 3.85724i 0 2.23793 + 3.87620i 0 0.0109492 + 2.99998i 0
313.1 0 −1.59256 − 0.680993i 0 −0.592561 − 1.02634i 0 1.47994 − 2.56333i 0 2.07250 + 2.16904i 0
313.2 0 −1.34437 + 1.09210i 0 −0.344366 − 0.596459i 0 0.270272 − 0.468125i 0 0.614638 − 2.93636i 0
313.3 0 −0.330295 − 1.70027i 0 0.669705 + 1.15996i 0 −2.11211 + 3.65827i 0 −2.78181 + 1.12318i 0
313.4 0 0.540243 + 1.64564i 0 1.54024 + 2.66778i 0 −0.876031 + 1.51733i 0 −2.41627 + 1.77809i 0
313.5 0 1.22698 − 1.22251i 0 2.22698 + 3.85724i 0 2.23793 − 3.87620i 0 0.0109492 − 2.99998i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.i.b ✓ 10
3.b odd 2 1 1404.2.i.b 10
9.c even 3 1 inner 468.2.i.b ✓ 10
9.c even 3 1 4212.2.a.i 5
9.d odd 6 1 1404.2.i.b 10
9.d odd 6 1 4212.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.i.b ✓ 10 1.a even 1 1 trivial
468.2.i.b ✓ 10 9.c even 3 1 inner
1404.2.i.b 10 3.b odd 2 1
1404.2.i.b 10 9.d odd 6 1
4212.2.a.i 5 9.c even 3 1
4212.2.a.j 5 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 7 T_{5}^{9} + 41 T_{5}^{8} - 94 T_{5}^{7} + 212 T_{5}^{6} - 73 T_{5}^{5} + 376 T_{5}^{4} + \cdots + 225 \) acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} - 7 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + \cdots + 2809 \) Copy content Toggle raw display
$11$ \( T^{10} + 5 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$17$ \( (T^{5} + 13 T^{4} + \cdots - 147)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - T^{4} - 71 T^{3} + \cdots + 1031)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 60 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{10} - 12 T^{9} + \cdots + 793881 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + \cdots + 1729225 \) Copy content Toggle raw display
$37$ \( (T^{5} - 13 T^{4} + 43 T^{3} + \cdots + 5)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 389075625 \) Copy content Toggle raw display
$43$ \( T^{10} - 8 T^{9} + \cdots + 5148361 \) Copy content Toggle raw display
$47$ \( T^{10} - 7 T^{9} + \cdots + 6985449 \) Copy content Toggle raw display
$53$ \( (T^{5} + 19 T^{4} + \cdots - 2763)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 11 T^{9} + \cdots + 245025 \) Copy content Toggle raw display
$61$ \( T^{10} + 7 T^{9} + \cdots + 3481 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 2148415201 \) Copy content Toggle raw display
$71$ \( (T^{5} + 29 T^{4} + \cdots + 633)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 19 T^{4} + \cdots + 1375)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 216648961 \) Copy content Toggle raw display
$83$ \( T^{10} - 24 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( (T^{5} + 19 T^{4} + \cdots + 19617)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 59291763001 \) Copy content Toggle raw display
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