Properties

Label 1452.4.a.u
Level $1452$
Weight $4$
Character orbit 1452.a
Self dual yes
Analytic conductor $85.671$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1452,4,Mod(1,1452)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1452, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1452.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1452.a (trivial)

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: yes
Analytic conductor: \(85.6707733283\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 11 \)
Twist minimal: no (minimal twist has level 132)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + ( - \beta_{2} + 2) q^{5} + (\beta_{3} - \beta_{2} + 4) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + ( - \beta_{2} + 2) q^{5} + (\beta_{3} - \beta_{2} + 4) q^{7} + 9 q^{9} + (\beta_{4} - 3 \beta_{2} - 3 \beta_1 + 10) q^{13} + ( - 3 \beta_{2} + 6) q^{15} + ( - 3 \beta_{5} - \beta_{4} - 4 \beta_{3} + \cdots + 5) q^{17}+ \cdots + ( - 26 \beta_{5} + 12 \beta_{4} + \cdots - 669) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 13 q^{5} + 23 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 13 q^{5} + 23 q^{7} + 54 q^{9} + 66 q^{13} + 39 q^{15} + 44 q^{17} + 270 q^{19} + 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} + 141 q^{29} - 253 q^{31} + 884 q^{35} + 288 q^{37} + 198 q^{39} + 428 q^{41} + 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} + 132 q^{51} - 773 q^{53} + 810 q^{57} - 17 q^{59} + 1016 q^{61} + 207 q^{63} + 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} + 1521 q^{73} + 393 q^{75} + 1425 q^{79} + 486 q^{81} + 3065 q^{83} + 1304 q^{85} + 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} + 1760 q^{95} - 3887 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 9\nu^{4} - 151\nu^{3} + 781\nu^{2} - 545\nu - 725 ) / 3234 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 58\nu^{4} - 641\nu^{3} + 8572\nu^{2} + 57471\nu - 187954 ) / 12936 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 67\nu^{4} + 286\nu^{3} + 6119\nu^{2} - 36860\nu - 43149 ) / 4312 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 9\nu^{4} - 151\nu^{3} + 781\nu^{2} + 5923\nu - 1019 ) / 588 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -31\nu^{5} - 15\nu^{4} + 4975\nu^{3} - 103\nu^{2} - 188023\nu + 98915 ) / 12936 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{4} - 11\beta _1 + 1 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -22\beta_{5} - 42\beta_{4} + 22\beta_{3} + 66\beta_{2} + 11\beta _1 + 1277 ) / 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -33\beta_{5} + 24\beta_{4} + 77\beta_{3} - 33\beta_{2} - 495\beta _1 + 474 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2838\beta_{5} - 4790\beta_{4} + 1958\beta_{3} + 5346\beta_{2} + 77\beta _1 + 110685 ) / 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18326\beta_{5} - 1970\beta_{4} + 23694\beta_{3} - 13398\beta_{2} - 92235\beta _1 + 158471 ) / 22 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
8.85455
10.2999
−9.47700
−8.43670
1.20018
−1.44098
0 3.00000 0 −17.0090 0 −13.6696 0 9.00000 0
1.2 0 3.00000 0 −3.88318 0 1.97618 0 9.00000 0
1.3 0 3.00000 0 −0.981524 0 −17.7745 0 9.00000 0
1.4 0 3.00000 0 3.10899 0 29.8524 0 9.00000 0
1.5 0 3.00000 0 10.3377 0 −5.80080 0 9.00000 0
1.6 0 3.00000 0 21.4270 0 28.4163 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.4.a.u 6
11.b odd 2 1 1452.4.a.t 6
11.c even 5 2 132.4.i.c 12
33.h odd 10 2 396.4.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.c 12 11.c even 5 2
396.4.j.c 12 33.h odd 10 2
1452.4.a.t 6 11.b odd 2 1
1452.4.a.u 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1452))\):

\( T_{5}^{6} - 13T_{5}^{5} - 356T_{5}^{4} + 3363T_{5}^{3} + 10396T_{5}^{2} - 38845T_{5} - 44645 \) Copy content Toggle raw display
\( T_{7}^{6} - 23T_{7}^{5} - 855T_{7}^{4} + 9990T_{7}^{3} + 262475T_{7}^{2} + 644817T_{7} - 2362741 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 13 T^{5} + \cdots - 44645 \) Copy content Toggle raw display
$7$ \( T^{6} - 23 T^{5} + \cdots - 2362741 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 2715277100 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 33421191664 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 61426596100 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 18587649476 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 3236332781296 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 2389629513941 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 33044892890544 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 6483755263220 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 27670010268564 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 40658968741616 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 28\!\cdots\!79 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 672659313542861 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 12926954748556 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots - 362260302137744 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 583597535659220 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 769184681261380 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 56\!\cdots\!59 \) Copy content Toggle raw display
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