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Properties

Label 1472.4.a.bk
Level $1472$
Weight $4$
Character orbit 1472.a
Self dual yes
Analytic conductor $86.851$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1472,4,Mod(1,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1472.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,0,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(86.8508115285\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 147x^{7} - 97x^{6} + 4561x^{5} + 7383x^{4} - 31427x^{3} - 43981x^{2} + 17596x + 12306 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 736)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + ( - \beta_{3} + 5) q^{7} + (\beta_{6} - \beta_{3} + 8) q^{9} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 7) q^{11} + (\beta_{7} + \beta_{6} - \beta_{4} + \cdots - 5) q^{13}+ \cdots + ( - 10 \beta_{8} + 10 \beta_{7} + \cdots - 61) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 42 q^{7} + 71 q^{9} - 66 q^{11} - 48 q^{13} + 90 q^{15} - 32 q^{17} - 6 q^{21} + 207 q^{23} + 135 q^{25} + 88 q^{29} + 556 q^{31} - 230 q^{33} - 368 q^{35} + 368 q^{37} + 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 147x^{7} - 97x^{6} + 4561x^{5} + 7383x^{4} - 31427x^{3} - 43981x^{2} + 17596x + 12306 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6731296043 \nu^{8} + 39410357444 \nu^{7} + 847773567986 \nu^{6} - 1690872679616 \nu^{5} + \cdots + 282337619064722 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10715933589 \nu^{8} - 144859711294 \nu^{7} - 950683976041 \nu^{6} + 13007152563976 \nu^{5} + \cdots + 203043051006328 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15796856925 \nu^{8} + 176790102375 \nu^{7} + 1712896624496 \nu^{6} + \cdots - 27\!\cdots\!04 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28522385174 \nu^{8} - 113788242967 \nu^{7} - 3884854083669 \nu^{6} - 230688766580 \nu^{5} + \cdots + 172208871473286 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 50540637472 \nu^{8} + 236799808312 \nu^{7} + 7232085928707 \nu^{6} + \cdots - 16\!\cdots\!28 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 93317435601 \nu^{8} - 410361525767 \nu^{7} - 13158325691937 \nu^{6} + 9217093974918 \nu^{5} + \cdots + 27\!\cdots\!56 ) / 78046956578582 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 110741263130 \nu^{8} + 315951004683 \nu^{7} + 16418943936491 \nu^{6} + \cdots - 12\!\cdots\!92 ) / 78046956578582 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{8} - 4\beta_{7} - \beta_{6} - 5\beta_{5} + 3\beta_{3} - 15\beta_{2} + 4\beta _1 + 132 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 25 \beta_{8} - 46 \beta_{7} - 20 \beta_{6} - 14 \beta_{5} - 13 \beta_{4} + 17 \beta_{3} - 78 \beta_{2} + \cdots + 689 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 438 \beta_{8} - 648 \beta_{7} - 173 \beta_{6} - 559 \beta_{5} - 197 \beta_{4} + 260 \beta_{3} + \cdots + 13568 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4789 \beta_{8} - 8292 \beta_{7} - 3202 \beta_{6} - 3988 \beta_{5} - 2657 \beta_{4} + 2647 \beta_{3} + \cdots + 133885 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 67143 \beta_{8} - 106264 \beta_{7} - 32713 \beta_{6} - 72885 \beta_{5} - 36384 \beta_{4} + \cdots + 1936840 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 812827 \beta_{8} - 1362970 \beta_{7} - 487124 \beta_{6} - 756986 \beta_{5} - 456959 \beta_{4} + \cdots + 22858459 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10687358 \beta_{8} - 17344444 \beta_{7} - 5675555 \beta_{6} - 10874601 \beta_{5} - 5982835 \beta_{4} + \cdots + 303437676 ) / 4 \) Copy content Toggle raw display

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−7.44967
2.75786
12.7496
−4.12339
0.625016
5.55395
−5.15057
−0.410550
−1.55219
0 −8.55290 0 −17.0239 0 8.01120 0 46.1522 0
1.2 0 −7.38682 0 10.2647 0 31.6457 0 27.5651 0
1.3 0 −5.48281 0 −1.44246 0 −5.95524 0 3.06121 0
1.4 0 −2.13547 0 18.3172 0 −12.7076 0 −22.4398 0
1.5 0 1.66678 0 −9.23864 0 −3.16263 0 −24.2218 0
1.6 0 3.02965 0 0.275790 0 −36.2914 0 −17.8212 0
1.7 0 3.60124 0 −17.5031 0 33.3352 0 −14.0311 0
1.8 0 5.36118 0 7.23489 0 8.11944 0 1.74224 0
1.9 0 9.89915 0 9.11549 0 19.0053 0 70.9931 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(23\) \( -1 \)

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 1472.4.a.bk 9
4.b odd 2 1 1472.4.a.bj 9
8.b even 2 1 736.4.a.i yes 9
8.d odd 2 1 736.4.a.h ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
736.4.a.h ✓ 9 8.d odd 2 1
736.4.a.i yes 9 8.b even 2 1
1472.4.a.bj 9 4.b odd 2 1
1472.4.a.bk 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 157T_{3}^{7} + 7339T_{3}^{5} - 5520T_{3}^{4} - 120407T_{3}^{3} + 187312T_{3}^{2} + 422848T_{3} - 713920 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 157 T^{7} + \cdots - 713920 \) Copy content Toggle raw display
$5$ \( T^{9} - 630 T^{7} + \cdots - 13579392 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 11327322624 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 586363744320 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots - 3501854757672 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 29038425785664 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( (T - 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 99\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 25\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 38\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 32\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 48\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 76\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 59\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 49\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 22\!\cdots\!60 \) Copy content Toggle raw display
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