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Properties

Label 920.4.a.d
Level $920$
Weight $4$
Character orbit 920.a
Self dual yes
Analytic conductor $54.282$
Analytic rank $1$
Dimension $8$
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] = [920,4,Mod(1,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("920.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 920.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: \(54.2817572053\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 135x^{6} + 187x^{5} + 5854x^{4} - 3309x^{3} - 85092x^{2} - 22464x + 77760 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - 5 q^{5} + \beta_{5} q^{7} + (\beta_{2} - \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - 5 q^{5} + \beta_{5} q^{7} + (\beta_{2} - \beta_1 + 8) q^{9} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 7) q^{11}+ \cdots + ( - 3 \beta_{7} + 5 \beta_{6} + \cdots - 527) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{3} - 40 q^{5} - 3 q^{7} + 62 q^{9} - 55 q^{11} - 16 q^{13} + 30 q^{15} + 127 q^{17} + 61 q^{19} + 94 q^{21} + 184 q^{23} + 200 q^{25} - 243 q^{27} - 223 q^{29} + 102 q^{31} + 129 q^{33} + 15 q^{35} - 218 q^{37} - 403 q^{39} + 112 q^{41} - 560 q^{43} - 310 q^{45} + 379 q^{47} - 291 q^{49} - 453 q^{51} - 1006 q^{53} + 275 q^{55} + 1020 q^{57} - 860 q^{59} - 1477 q^{61} - 1206 q^{63} + 80 q^{65} - 740 q^{67} - 138 q^{69} - 1238 q^{71} - 2001 q^{73} - 150 q^{75} - 412 q^{77} - 1202 q^{79} - 2356 q^{81} - 1218 q^{83} - 635 q^{85} - 3419 q^{87} - 1318 q^{89} - 3123 q^{91} - 2445 q^{93} - 305 q^{95} + 151 q^{97} - 4075 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 135x^{6} + 187x^{5} + 5854x^{4} - 3309x^{3} - 85092x^{2} - 22464x + 77760 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 51\nu - 54 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 17\nu^{6} - 288\nu^{5} - 1527\nu^{4} + 6815\nu^{3} + 31428\nu^{2} - 15168\nu + 8640 ) / 2160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 13\nu^{6} - 258\nu^{5} + 1233\nu^{4} + 3845\nu^{3} - 25002\nu^{2} + 32352\nu + 38880 ) / 2160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 21\nu^{6} + 96\nu^{5} - 2011\nu^{4} - 2205\nu^{3} + 44484\nu^{2} + 18576\nu - 63720 ) / 1080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 9\nu^{6} + 108\nu^{5} - 835\nu^{4} - 3177\nu^{3} + 16872\nu^{2} + 27216\nu - 6912 ) / 432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} - \beta_{2} + 50\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} - 15\beta_{6} - 12\beta_{5} + 12\beta_{4} - 9\beta_{3} + 73\beta_{2} + 64\beta _1 + 1699 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{7} + 3\beta_{6} + 42\beta_{5} - 15\beta_{4} + 267\beta_{3} - 94\beta_{2} + 2714\beta _1 + 1331 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 567\beta_{7} - 1377\beta_{6} - 1134\beta_{5} + 1161\beta_{4} - 858\beta_{3} + 4840\beta_{2} + 3355\beta _1 + 92713 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1281 \beta_{7} + 456 \beta_{6} + 4350 \beta_{5} - 1191 \beta_{4} + 19098 \beta_{3} - 7498 \beta_{2} + \cdots + 62696 \) 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
−8.04302
−5.45758
−5.21946
−1.19376
0.838946
5.84508
7.52841
7.70138
0 −9.04302 0 −5.00000 0 −21.7314 0 54.7762 0
1.2 0 −6.45758 0 −5.00000 0 27.5388 0 14.7003 0
1.3 0 −6.21946 0 −5.00000 0 −10.5361 0 11.6816 0
1.4 0 −2.19376 0 −5.00000 0 −17.9765 0 −22.1874 0
1.5 0 −0.161054 0 −5.00000 0 23.7013 0 −26.9741 0
1.6 0 4.84508 0 −5.00000 0 0.673383 0 −3.52519 0
1.7 0 6.52841 0 −5.00000 0 −12.7051 0 15.6201 0
1.8 0 6.70138 0 −5.00000 0 8.03572 0 17.9085 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(23\) \(-1\)

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 920.4.a.d ✓ 8
4.b odd 2 1 1840.4.a.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.d ✓ 8 1.a even 1 1 trivial
1840.4.a.w 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 6T_{3}^{7} - 121T_{3}^{6} - 609T_{3}^{5} + 4764T_{3}^{4} + 19263T_{3}^{3} - 60064T_{3}^{2} - 179040T_{3} - 27200 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(920))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{7} + \cdots - 27200 \) Copy content Toggle raw display
$5$ \( (T + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + \cdots + 184695552 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 150383545600 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 211990650120 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 127561163391360 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 51555969091840 \) Copy content Toggle raw display
$23$ \( (T - 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 5965214878296 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 341560403089960 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 12\!\cdots\!14 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 50\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 94\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 39\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 70\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 69\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 19\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
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