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Properties

Label 1000.4.a.e
Level $1000$
Weight $4$
Character orbit 1000.a
Self dual yes
Analytic conductor $59.002$
Analytic rank $1$
Dimension $10$
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] = [1000,4,Mod(1,1000)] 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("1000.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1000.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-1,0,0,0,-27] 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: \(59.0019100057\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 81x^{8} - 17x^{7} + 1728x^{6} + 1268x^{5} - 8606x^{4} - 1260x^{3} + 12092x^{2} - 5011x + 491 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{9} - 3) q^{7} + (\beta_{9} + \beta_{5} - \beta_{3} + 8) q^{9} + (\beta_{9} - \beta_{6} + 1) q^{11} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \cdots - 9) q^{13} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 16) q^{17}+ \cdots + (9 \beta_{9} - 13 \beta_{8} + \cdots - 85) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} - 27 q^{7} + 81 q^{9} + 2 q^{11} - 78 q^{13} - 166 q^{17} + 18 q^{19} - 26 q^{21} - 167 q^{23} - 172 q^{27} + 251 q^{29} - 302 q^{31} - 146 q^{33} - 436 q^{37} - 378 q^{39} + 293 q^{41} - 891 q^{43}+ \cdots - 518 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 81x^{8} - 17x^{7} + 1728x^{6} + 1268x^{5} - 8606x^{4} - 1260x^{3} + 12092x^{2} - 5011x + 491 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8448418261 \nu^{9} + 254880482293 \nu^{8} - 537238497589 \nu^{7} - 19796634603939 \nu^{6} + \cdots + 483990018995724 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 37662805 \nu^{9} + 8280650 \nu^{8} - 3102755360 \nu^{7} - 1202421465 \nu^{6} + \cdots - 116891554368 ) / 6447786301 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 119408235789 \nu^{9} - 13510170967 \nu^{8} + 9655201328695 \nu^{7} + \cdots + 324134012830974 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 121955932886 \nu^{9} + 337569845345 \nu^{8} - 10142321149135 \nu^{7} + \cdots + 172213359178273 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 144960268225 \nu^{9} - 3865770664 \nu^{8} - 11363090839802 \nu^{7} - 3244946477954 \nu^{6} + \cdots + 127136826453869 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 165549826026 \nu^{9} - 124798226137 \nu^{8} + 13200591029555 \nu^{7} + \cdots - 41240082538025 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 221717266752 \nu^{9} + 69211818739 \nu^{8} + 17790774353911 \nu^{7} + \cdots + 11\!\cdots\!41 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 280742230555 \nu^{9} + 409591047718 \nu^{8} - 23345448604056 \nu^{7} + \cdots - 954198766579727 ) / 13437186651284 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 689236045764 \nu^{9} - 104759826107 \nu^{8} + 55405108838657 \nu^{7} + \cdots + 13\!\cdots\!95 ) / 13437186651284 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} - 2\beta_{4} - 3\beta_{3} + 3\beta_{2} + 2\beta _1 + 1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 9 \beta_{9} + 5 \beta_{8} + \beta_{7} + 5 \beta_{6} - 8 \beta_{5} - 6 \beta_{4} + 26 \beta_{3} + \cdots + 314 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{9} - 9 \beta_{8} + 36 \beta_{7} - 17 \beta_{6} - 68 \beta_{5} - 79 \beta_{4} - 92 \beta_{3} + \cdots + 167 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 471 \beta_{9} + 243 \beta_{8} + 129 \beta_{7} + 314 \beta_{6} - 361 \beta_{5} - 236 \beta_{4} + \cdots + 11755 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 75 \beta_{9} - 291 \beta_{8} + 853 \beta_{7} - 583 \beta_{6} - 1700 \beta_{5} - 1732 \beta_{4} + \cdots + 2042 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 23082 \beta_{9} + 11690 \beta_{8} + 7040 \beta_{7} + 16770 \beta_{6} - 16309 \beta_{5} + \cdots + 513334 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2347 \beta_{9} - 34755 \beta_{8} + 83479 \beta_{7} - 65800 \beta_{6} - 163272 \beta_{5} - 155919 \beta_{4} + \cdots - 128705 ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1127177 \beta_{9} + 571881 \beta_{8} + 344092 \beta_{7} + 859588 \beta_{6} - 738067 \beta_{5} + \cdots + 23634776 ) / 20 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 582420 \beta_{9} - 1989984 \beta_{8} + 4068275 \beta_{7} - 3567787 \beta_{6} - 7752744 \beta_{5} + \cdots - 21749880 ) / 20 \) 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
1.57509
1.15222
0.319796
−3.53167
6.75695
−3.42748
−1.62802
−7.18832
5.81954
0.151875
0 −8.82719 0 0 0 −16.0975 0 50.9193 0
1.2 0 −8.36877 0 0 0 −23.5786 0 43.0364 0
1.3 0 −5.69743 0 0 0 34.3223 0 5.46071 0
1.4 0 −1.53284 0 0 0 21.8557 0 −24.6504 0
1.5 0 −1.24662 0 0 0 −13.3185 0 −25.4459 0
1.6 0 0.939484 0 0 0 −22.8799 0 −26.1174 0
1.7 0 1.93491 0 0 0 3.94792 0 −23.2561 0
1.8 0 5.87859 0 0 0 13.5532 0 7.55781 0
1.9 0 7.33771 0 0 0 4.75060 0 26.8419 0
1.10 0 8.58217 0 0 0 −29.5552 0 46.6536 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

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 1000.4.a.e ✓ 10
4.b odd 2 1 2000.4.a.x 10
5.b even 2 1 1000.4.a.h yes 10
5.c odd 4 2 1000.4.c.d 20
20.d odd 2 1 2000.4.a.u 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.4.a.e ✓ 10 1.a even 1 1 trivial
1000.4.a.h yes 10 5.b even 2 1
1000.4.c.d 20 5.c odd 4 2
2000.4.a.u 10 20.d odd 2 1
2000.4.a.x 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + T_{3}^{9} - 175 T_{3}^{8} - 100 T_{3}^{7} + 9995 T_{3}^{6} + 1951 T_{3}^{5} - 195999 T_{3}^{4} + \cdots - 541225 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1000))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots - 541225 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 651806993269 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 148169393652736 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 246638983617536 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 63\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 37\!\cdots\!91 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 41\!\cdots\!69 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 18\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 21\!\cdots\!79 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 41\!\cdots\!61 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 75\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 26\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 13\!\cdots\!75 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 35\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 47\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 40\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 40\!\cdots\!71 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 90\!\cdots\!56 \) Copy content Toggle raw display
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