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Properties

Label 464.6.e.c
Level $464$
Weight $6$
Character orbit 464.e
Analytic conductor $74.418$
Analytic rank $0$
Dimension $12$
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] = [464,6,Mod(289,464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("464.289");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 464.e (of order \(2\), degree \(1\), not 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: \(74.4180923932\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 5 \)
Twist minimal: no (minimal twist has level 29)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{6} + 4) q^{5} + ( - \beta_{2} - 2) q^{7} + (\beta_{10} + \beta_{6} - 131) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{6} + 4) q^{5} + ( - \beta_{2} - 2) q^{7} + (\beta_{10} + \beta_{6} - 131) q^{9} + (\beta_{9} - \beta_{7} + 5 \beta_{3} + \beta_1) q^{11} + ( - \beta_{10} - \beta_{8} + \cdots + 113) q^{13}+ \cdots + ( - 49 \beta_{11} - 111 \beta_{9} + \cdots - 674 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 46 q^{5} - 20 q^{7} - 1574 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 46 q^{5} - 20 q^{7} - 1574 q^{9} + 1362 q^{13} - 5852 q^{23} + 12678 q^{25} + 11328 q^{29} - 22694 q^{33} - 4532 q^{35} - 52816 q^{45} + 102836 q^{49} - 58540 q^{51} + 25650 q^{53} - 32824 q^{57} + 3900 q^{59} + 146616 q^{63} + 169574 q^{65} + 28264 q^{67} + 286832 q^{71} - 24084 q^{81} - 85692 q^{83} + 137716 q^{87} + 182372 q^{91} + 377966 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 278x^{10} + 28285x^{8} + 1260472x^{6} + 22944832x^{4} + 140087936x^{2} + 966400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 16\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{10} - 18002\nu^{8} - 3564589\nu^{6} - 193044948\nu^{4} - 2175673456\nu^{2} - 327591360 ) / 37604096 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -125\nu^{11} - 35271\nu^{9} - 3622703\nu^{7} - 161356045\nu^{5} - 2877987192\nu^{3} - 17080930576\nu ) / 56406144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -125\nu^{11} - 35271\nu^{9} - 3622703\nu^{7} - 161356045\nu^{5} - 2840383096\nu^{3} - 14260623376\nu ) / 18802048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 557\nu^{10} + 78434\nu^{8} + 741537\nu^{6} - 200200212\nu^{4} - 2433767248\nu^{2} + 48053446720 ) / 75208192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -543\nu^{10} - 114438\nu^{8} - 7870715\nu^{6} - 185889684\nu^{4} - 714248592\nu^{2} + 7095849024 ) / 75208192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4129 \nu^{11} - 1048734 \nu^{9} - 96066205 \nu^{7} - 3784789952 \nu^{5} + \cdots - 301378600064 \nu ) / 225624576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -521\nu^{10} - 87078\nu^{8} - 3797045\nu^{6} - 9883192\nu^{4} + 373655280\nu^{2} - 2389273408 ) / 28203072 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1489 \nu^{11} - 409572 \nu^{9} - 41435601 \nu^{7} - 1857670318 \nu^{5} + \cdots - 222823456096 \nu ) / 37604096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2129 \nu^{10} - 484398 \nu^{8} - 38102957 \nu^{6} - 1203093232 \nu^{4} - 13316257680 \nu^{2} - 19961226112 ) / 56406144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7485 \nu^{11} - 2070898 \nu^{9} - 209723921 \nu^{7} - 9293509212 \nu^{5} + \cdots - 1003074595648 \nu ) / 75208192 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{2} - 742 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{4} - 24\beta_{3} - 75\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{10} + 8\beta_{8} - 25\beta_{6} - 89\beta_{5} + 121\beta_{2} + 55294 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48\beta_{11} - 64\beta_{9} - 16\beta_{7} - 832\beta_{4} + 1616\beta_{3} + 6119\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1824\beta_{10} - 1776\beta_{8} + 2173\beta_{6} + 7293\beta_{5} - 12285\beta_{2} - 4476798 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8544\beta_{11} + 9984\beta_{9} + 3616\beta_{7} + 75544\beta_{4} - 51176\beta_{3} - 510219\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -182768\beta_{10} + 254008\beta_{8} - 271121\beta_{6} - 591825\beta_{5} + 1179441\beta_{2} + 370835070 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1095632 \beta_{11} - 1175232 \beta_{9} - 489968 \beta_{7} - 6721168 \beta_{4} - 3996128 \beta_{3} + 42912431 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17557376 \beta_{10} - 30528352 \beta_{8} + 30668773 \beta_{6} + 48169573 \beta_{5} - 110606885 \beta_{2} - 31007551966 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 123494016 \beta_{11} + 124874624 \beta_{9} + 54109312 \beta_{7} + 596905384 \beta_{4} + \cdots - 3630095859 \beta_1 ) / 16 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(175\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
− 9.02264i
0.0831044i
4.44887i
9.47123i
− 3.61683i
− 8.60273i
8.60273i
3.61683i
− 9.47123i
− 4.44887i
− 0.0831044i
9.02264i
0 − 26.5755i 0 71.1177 0 −131.830 0 −463.255 0
289.2 0 − 25.1364i 0 −90.4149 0 6.31225 0 −388.840 0
289.3 0 − 21.5070i 0 90.0922 0 211.678 0 −219.553 0
289.4 0 − 19.9099i 0 −4.84615 0 −219.131 0 −153.403 0
289.5 0 − 5.13082i 0 15.9022 0 −69.7674 0 216.675 0
289.6 0 − 4.65015i 0 −58.8511 0 192.738 0 221.376 0
289.7 0 4.65015i 0 −58.8511 0 192.738 0 221.376 0
289.8 0 5.13082i 0 15.9022 0 −69.7674 0 216.675 0
289.9 0 19.9099i 0 −4.84615 0 −219.131 0 −153.403 0
289.10 0 21.5070i 0 90.0922 0 211.678 0 −219.553 0
289.11 0 25.1364i 0 −90.4149 0 6.31225 0 −388.840 0
289.12 0 26.5755i 0 71.1177 0 −131.830 0 −463.255 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.6.e.c 12
4.b odd 2 1 29.6.b.a ✓ 12
12.b even 2 1 261.6.c.b 12
29.b even 2 1 inner 464.6.e.c 12
116.d odd 2 1 29.6.b.a ✓ 12
116.e even 4 2 841.6.a.d 12
348.b even 2 1 261.6.c.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.b.a ✓ 12 4.b odd 2 1
29.6.b.a ✓ 12 116.d odd 2 1
261.6.c.b 12 12.b even 2 1
261.6.c.b 12 348.b even 2 1
464.6.e.c 12 1.a even 1 1 trivial
464.6.e.c 12 29.b even 2 1 inner
841.6.a.d 12 116.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2245 T_{3}^{10} + 1884878 T_{3}^{8} + 715200530 T_{3}^{6} + 112977325989 T_{3}^{4} + \cdots + 46577165867100 \) acting on \(S_{6}^{\mathrm{new}}(464, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 46577165867100 \) Copy content Toggle raw display
$5$ \( (T^{6} - 23 T^{5} + \cdots - 2627317458)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 10 T^{5} + \cdots - 519034134784)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 15\!\cdots\!82)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 74\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 28\!\cdots\!50)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 16\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
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