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Properties

Label 468.3.m.d
Level $468$
Weight $3$
Character orbit 468.m
Analytic conductor $12.752$
Analytic rank $0$
Dimension $8$
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] = [468,3,Mod(73,468)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 468.m (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: \(12.7520763721\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1579585536.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{5} + (2 \beta_{6} - 2 \beta_{5} + \beta_{2} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{5} + (2 \beta_{6} - 2 \beta_{5} + \beta_{2} - 2) q^{7} + (2 \beta_{7} + 3 \beta_{5} - 2 \beta_{3}) q^{11} + ( - 3 \beta_{7} - 2 \beta_{6} + \cdots + 3) q^{13}+ \cdots + ( - 6 \beta_{7} + 39 \beta_{6} + \cdots + 39) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{5} - 8 q^{7} - 12 q^{11} + 24 q^{13} + 88 q^{19} - 24 q^{29} - 16 q^{31} + 216 q^{35} + 32 q^{37} + 180 q^{41} - 36 q^{47} + 72 q^{53} - 240 q^{55} + 228 q^{59} - 192 q^{61} - 132 q^{65} + 16 q^{67} - 36 q^{71} + 160 q^{73} + 48 q^{79} - 12 q^{83} + 24 q^{85} - 60 q^{89} + 112 q^{91} + 416 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 28x^{5} - 38x^{4} + 8x^{3} + 200x^{2} - 352x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 492 \nu^{7} - 43753 \nu^{6} + 126290 \nu^{5} - 25986 \nu^{4} - 676130 \nu^{3} + 1643424 \nu^{2} + \cdots + 510708 ) / 753610 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2104 \nu^{7} - 30399 \nu^{6} + 115510 \nu^{5} - 17538 \nu^{4} - 484510 \nu^{3} + 1206592 \nu^{2} + \cdots + 2521464 ) / 753610 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2373 \nu^{7} + 7243 \nu^{6} + 25530 \nu^{5} - 102359 \nu^{4} + 15275 \nu^{3} + 427176 \nu^{2} + \cdots + 707872 ) / 376805 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2373 \nu^{7} + 7243 \nu^{6} + 25530 \nu^{5} - 102359 \nu^{4} + 15275 \nu^{3} + 427176 \nu^{2} + \cdots + 707872 ) / 376805 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3374 \nu^{7} - 7486 \nu^{6} + 106325 \nu^{5} - 233347 \nu^{4} - 90545 \nu^{3} + 1234268 \nu^{2} + \cdots + 3796386 ) / 376805 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 86\nu^{7} - 227\nu^{6} + 162\nu^{5} + 758\nu^{4} - 1794\nu^{3} + 8124\nu^{2} - 8408\nu + 13662 ) / 8866 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2128\nu^{7} - 8213\nu^{6} + 7420\nu^{5} + 25504\nu^{4} - 112320\nu^{3} + 234124\nu^{2} - 185856\nu - 9912 ) / 68510 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} + 12\beta_{6} + 2\beta_{4} - 4\beta_{3} - \beta_{2} + 3\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -10\beta_{7} + 26\beta_{6} - 10\beta_{5} - 6\beta_{4} + 14\beta_{3} + 9\beta_{2} + 9\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{7} + 7\beta_{6} - 2\beta_{5} + \beta_{4} - 2\beta_{3} + 21\beta_{2} - 7\beta _1 - 55 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{6} - 22\beta_{5} - 22\beta_{4} + 70\beta_{3} + 46\beta_{2} - 24\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 59\beta_{7} - 99\beta_{6} + 37\beta_{5} - 11\beta_{4} - 11\beta_{3} + 79\beta_{2} - 153\beta _1 - 331 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
1.11361 + 1.42401i
2.22833 + 1.32913i
−2.59436 + 0.0368949i
0.252411 − 1.79004i
1.11361 − 1.42401i
2.22833 − 1.32913i
−2.59436 − 0.0368949i
0.252411 + 1.79004i
0 0 0 −5.58008 − 5.58008i 0 −7.42810 + 7.42810i 0 0 0
73.2 0 0 0 −1.92621 − 1.92621i 0 −3.58447 + 3.58447i 0 0 0
73.3 0 0 0 0.658261 + 0.658261i 0 1.58447 − 1.58447i 0 0 0
73.4 0 0 0 0.848026 + 0.848026i 0 5.42810 − 5.42810i 0 0 0
109.1 0 0 0 −5.58008 + 5.58008i 0 −7.42810 − 7.42810i 0 0 0
109.2 0 0 0 −1.92621 + 1.92621i 0 −3.58447 − 3.58447i 0 0 0
109.3 0 0 0 0.658261 − 0.658261i 0 1.58447 + 1.58447i 0 0 0
109.4 0 0 0 0.848026 − 0.848026i 0 5.42810 + 5.42810i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.3.m.d 8
3.b odd 2 1 156.3.j.a ✓ 8
12.b even 2 1 624.3.ba.d 8
13.d odd 4 1 inner 468.3.m.d 8
39.f even 4 1 156.3.j.a ✓ 8
156.l odd 4 1 624.3.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.j.a ✓ 8 3.b odd 2 1
156.3.j.a ✓ 8 39.f even 4 1
468.3.m.d 8 1.a even 1 1 trivial
468.3.m.d 8 13.d odd 4 1 inner
624.3.ba.d 8 12.b even 2 1
624.3.ba.d 8 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 12T_{5}^{7} + 72T_{5}^{6} + 48T_{5}^{5} - 48T_{5}^{4} - 288T_{5}^{3} + 1152T_{5}^{2} - 1152T_{5} + 576 \) acting on \(S_{3}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 839056 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots + 90782784 \) Copy content Toggle raw display
$13$ \( T^{8} - 24 T^{7} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{8} + 1296 T^{6} + \cdots + 427993344 \) Copy content Toggle raw display
$19$ \( T^{8} - 88 T^{7} + \cdots + 2972176 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 8422834176 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 75504)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 16 T^{7} + \cdots + 10061584 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 149561639824 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 28199813184 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 619986161664 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 4080820170816 \) Copy content Toggle raw display
$53$ \( (T^{4} - 36 T^{3} + \cdots - 976944)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 4746925417536 \) Copy content Toggle raw display
$61$ \( (T^{4} + 96 T^{3} + \cdots - 249024)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + \cdots + 929274256 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 448998131059776 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 601874565729424 \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots + 613248)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1057162337856 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 63914204772496 \) Copy content Toggle raw display
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