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Properties

Label 4592.2.a.bk
Level $4592$
Weight $2$
Character orbit 4592.a
Self dual yes
Analytic conductor $36.667$
Analytic rank $1$
Dimension $9$
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] = [4592,2,Mod(1,4592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4592, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4592.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4592 = 2^{4} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4592.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: \(36.6673046082\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 77x^{5} - 164x^{4} - 165x^{3} + 182x^{2} + 116x - 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2296)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{4} q^{5} + q^{7} + ( - \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{4} q^{5} + q^{7} + ( - \beta_{3} + \beta_{2} + 2) q^{9} + (\beta_{6} - \beta_{2} - 1) q^{11} + (\beta_{8} + \beta_1 - 2) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \cdots - 1) q^{15}+ \cdots + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} - 2 q^{5} + 9 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} - 2 q^{5} + 9 q^{7} + 14 q^{9} - 8 q^{11} - 13 q^{13} - 4 q^{15} - 4 q^{17} - 14 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 21 q^{27} + 2 q^{29} - 14 q^{31} - 9 q^{33} - 2 q^{35} - q^{37} - 21 q^{39} + 9 q^{41} - 24 q^{43} - 7 q^{45} - 3 q^{47} + 9 q^{49} - 40 q^{51} + 8 q^{53} - 19 q^{55} + 25 q^{57} - 41 q^{59} - 25 q^{61} + 14 q^{63} + 23 q^{65} - 9 q^{67} + 2 q^{69} - 31 q^{71} - 6 q^{73} - 12 q^{75} - 8 q^{77} - 17 q^{79} + 45 q^{81} - 16 q^{83} - 30 q^{85} - 33 q^{87} + 7 q^{89} - 13 q^{91} + 32 q^{93} + q^{95} + 3 q^{97} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 77x^{5} - 164x^{4} - 165x^{3} + 182x^{2} + 116x - 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19 \nu^{8} + 117 \nu^{7} + 110 \nu^{6} - 1973 \nu^{5} + 1609 \nu^{4} + 9353 \nu^{3} - 8327 \nu^{2} + \cdots + 3019 ) / 1667 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 19 \nu^{8} + 117 \nu^{7} + 110 \nu^{6} - 1973 \nu^{5} + 1609 \nu^{4} + 9353 \nu^{3} - 9994 \nu^{2} + \cdots + 11354 ) / 1667 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 85 \nu^{8} - 3 \nu^{7} + 1194 \nu^{6} + 2316 \nu^{5} - 5085 \nu^{4} - 27382 \nu^{3} + 16969 \nu^{2} + \cdots - 11148 ) / 6668 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 275 \nu^{8} + 1167 \nu^{7} + 2294 \nu^{6} - 14080 \nu^{5} + 4337 \nu^{4} + 29474 \nu^{3} + \cdots - 17632 ) / 6668 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 579 \nu^{8} + 3039 \nu^{7} + 4054 \nu^{6} - 38980 \nu^{5} + 23413 \nu^{4} + 99106 \nu^{3} + \cdots + 37340 ) / 6668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 169 \nu^{8} - 602 \nu^{7} - 2119 \nu^{6} + 7986 \nu^{5} + 4815 \nu^{4} - 23005 \nu^{3} - 1124 \nu^{2} + \cdots - 1585 ) / 1667 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 771 \nu^{8} - 2993 \nu^{7} - 9026 \nu^{6} + 40142 \nu^{5} + 14549 \nu^{4} - 117114 \nu^{3} + \cdots - 13100 ) / 3334 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 2\beta_{7} + \beta_{5} - \beta_{4} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 2\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - 10\beta_{3} + 11\beta_{2} + \beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{8} - 26\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 2\beta_{3} - \beta_{2} + 63\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{8} - 31\beta_{7} + 15\beta_{6} - 32\beta_{5} + 7\beta_{4} - 105\beta_{3} + 107\beta_{2} + 15\beta _1 + 390 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 139 \beta_{8} - 282 \beta_{7} + 39 \beta_{6} + 63 \beta_{5} - 128 \beta_{4} - 52 \beta_{3} - 18 \beta_{2} + \cdots + 399 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 164 \beta_{8} - 370 \beta_{7} + 204 \beta_{6} - 409 \beta_{5} - 13 \beta_{4} - 1129 \beta_{3} + \cdots + 3893 \) 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.

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
3.34816
3.00282
2.13957
1.17241
0.266456
−0.905012
−1.35358
−1.56402
−3.10680
0 −3.34816 0 1.93489 0 1.00000 0 8.21019 0
1.2 0 −3.00282 0 −2.40652 0 1.00000 0 6.01693 0
1.3 0 −2.13957 0 2.41499 0 1.00000 0 1.57774 0
1.4 0 −1.17241 0 −4.13256 0 1.00000 0 −1.62546 0
1.5 0 −0.266456 0 −0.513584 0 1.00000 0 −2.92900 0
1.6 0 0.905012 0 4.25747 0 1.00000 0 −2.18095 0
1.7 0 1.35358 0 0.597517 0 1.00000 0 −1.16782 0
1.8 0 1.56402 0 −2.38226 0 1.00000 0 −0.553837 0
1.9 0 3.10680 0 −1.76994 0 1.00000 0 6.65221 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
\(2\) \(1\)
\(7\) \(-1\)
\(41\) \(-1\)

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 4592.2.a.bk 9
4.b odd 2 1 2296.2.a.m ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2296.2.a.m ✓ 9 4.b odd 2 1
4592.2.a.bk 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4592))\):

\( T_{3}^{9} + 3T_{3}^{8} - 16T_{3}^{7} - 44T_{3}^{6} + 77T_{3}^{5} + 164T_{3}^{4} - 165T_{3}^{3} - 182T_{3}^{2} + 116T_{3} + 40 \) Copy content Toggle raw display
\( T_{5}^{9} + 2T_{5}^{8} - 28T_{5}^{7} - 57T_{5}^{6} + 208T_{5}^{5} + 416T_{5}^{4} - 492T_{5}^{3} - 928T_{5}^{2} + 192T_{5} + 256 \) Copy content Toggle raw display
\( T_{11}^{9} + 8 T_{11}^{8} - 57 T_{11}^{7} - 560 T_{11}^{6} + 632 T_{11}^{5} + 12172 T_{11}^{4} + \cdots + 212096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots + 40 \) Copy content Toggle raw display
$5$ \( T^{9} + 2 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T - 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 8 T^{8} + \cdots + 212096 \) Copy content Toggle raw display
$13$ \( T^{9} + 13 T^{8} + \cdots + 864 \) Copy content Toggle raw display
$17$ \( T^{9} + 4 T^{8} + \cdots + 2328 \) Copy content Toggle raw display
$19$ \( T^{9} + 14 T^{8} + \cdots - 32 \) Copy content Toggle raw display
$23$ \( T^{9} + 10 T^{8} + \cdots + 69792 \) Copy content Toggle raw display
$29$ \( T^{9} - 2 T^{8} + \cdots + 431136 \) Copy content Toggle raw display
$31$ \( T^{9} + 14 T^{8} + \cdots - 64256 \) Copy content Toggle raw display
$37$ \( T^{9} + T^{8} + \cdots - 1276544 \) Copy content Toggle raw display
$41$ \( (T - 1)^{9} \) Copy content Toggle raw display
$43$ \( T^{9} + 24 T^{8} + \cdots - 8712672 \) Copy content Toggle raw display
$47$ \( T^{9} + 3 T^{8} + \cdots + 385920 \) Copy content Toggle raw display
$53$ \( T^{9} - 8 T^{8} + \cdots - 1276384 \) Copy content Toggle raw display
$59$ \( T^{9} + 41 T^{8} + \cdots - 7130176 \) Copy content Toggle raw display
$61$ \( T^{9} + 25 T^{8} + \cdots - 1354464 \) Copy content Toggle raw display
$67$ \( T^{9} + 9 T^{8} + \cdots + 1825792 \) Copy content Toggle raw display
$71$ \( T^{9} + 31 T^{8} + \cdots - 18758912 \) Copy content Toggle raw display
$73$ \( T^{9} + 6 T^{8} + \cdots - 1659776 \) Copy content Toggle raw display
$79$ \( T^{9} + 17 T^{8} + \cdots - 65904640 \) Copy content Toggle raw display
$83$ \( T^{9} + 16 T^{8} + \cdots - 461184 \) Copy content Toggle raw display
$89$ \( T^{9} - 7 T^{8} + \cdots - 36521288 \) Copy content Toggle raw display
$97$ \( T^{9} - 3 T^{8} + \cdots + 173549256 \) Copy content Toggle raw display
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