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Properties

Label 250.4.d.b
Level $250$
Weight $4$
Character orbit 250.d
Analytic conductor $14.750$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [250,4,Mod(51,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 250.d (of order \(5\), degree \(4\), not minimal)

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: no
Analytic conductor: \(14.7504775014\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 5^{6} \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{8} q^{2} + (\beta_{7} + \beta_{4} - 1) q^{3} + (4 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} - 4) q^{4} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{6} + ( - \beta_{14} - \beta_{13} + \beta_{8} + \cdots + 3) q^{7}+ \cdots + (30 \beta_{14} + 30 \beta_{13} + \cdots + 219) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 7 q^{3} - 16 q^{4} + 14 q^{6} + 54 q^{7} + 32 q^{8} - 83 q^{9} + 152 q^{11} + 12 q^{12} - 77 q^{13} + 52 q^{14} - 64 q^{16} - 76 q^{17} - 304 q^{18} - 20 q^{19} + 257 q^{21} + 226 q^{22}+ \cdots - 1786 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 135 x^{14} + 7296 x^{12} + 200295 x^{10} + 2912021 x^{8} + 20937420 x^{6} + 57578496 x^{4} + \cdots + 952576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2606260817 \nu^{14} - 299720681931 \nu^{12} - 13455196227932 \nu^{10} + \cdots + 704732477177408 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8091599105 \nu^{14} + 1099677668619 \nu^{12} + 59976834922988 \nu^{10} + \cdots + 35\!\cdots\!84 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 892560297 \nu^{14} + 112376768612 \nu^{12} + 5630820410899 \nu^{10} + \cdots + 62\!\cdots\!28 ) / 146090958472080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 630304983 \nu^{15} - 533360406 \nu^{14} - 73305733293 \nu^{13} - 13349395378 \nu^{12} + \cdots + 18\!\cdots\!04 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 630304983 \nu^{15} + 721297714 \nu^{14} + 73305733293 \nu^{13} + 106532776630 \nu^{12} + \cdots + 24\!\cdots\!72 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 630304983 \nu^{15} - 721297714 \nu^{14} + 73305733293 \nu^{13} - 106532776630 \nu^{12} + \cdots - 24\!\cdots\!72 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 236761278263 \nu^{15} - 76897207926 \nu^{14} - 32027842535037 \nu^{13} + \cdots + 45\!\cdots\!56 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 236761278263 \nu^{15} - 76897207926 \nu^{14} + 32027842535037 \nu^{13} + \cdots + 45\!\cdots\!56 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 388969833455 \nu^{15} + 76897207926 \nu^{14} + 52269860904677 \nu^{13} + \cdots - 17\!\cdots\!96 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1632378220845 \nu^{15} + 987175090810 \nu^{14} - 211453688342671 \nu^{13} + \cdots + 45\!\cdots\!08 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1861751406679 \nu^{15} + 610272512254 \nu^{14} + 258143900192877 \nu^{13} + \cdots + 73\!\cdots\!80 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1049256342471 \nu^{15} + 343584860090 \nu^{14} - 145085871363957 \nu^{13} + \cdots + 34\!\cdots\!12 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 386689991580 \nu^{15} + 301308049115 \nu^{14} - 52777807690166 \nu^{13} + \cdots + 24\!\cdots\!40 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3330281210903 \nu^{15} + 2333567184994 \nu^{14} + 454250304056365 \nu^{13} + \cdots + 20\!\cdots\!76 ) / 57\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1266846637783 \nu^{15} - 167046673294 \nu^{14} + 170998057181529 \nu^{13} + \cdots - 18\!\cdots\!16 ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{8} + 2\beta_{7} + 3\beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 22 \beta_{8} + 26 \beta_{7} + 3 \beta_{6} + \cdots - 95 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{15} + 5 \beta_{14} - 9 \beta_{13} + \beta_{12} - \beta_{11} - 8 \beta_{10} - 50 \beta_{9} + \cdots + 29 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 54 \beta_{14} - 54 \beta_{13} + 78 \beta_{12} + 78 \beta_{11} - 680 \beta_{8} - 812 \beta_{7} + \cdots + 2594 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 186 \beta_{15} - 206 \beta_{14} + 392 \beta_{13} - 34 \beta_{12} + 34 \beta_{11} + 414 \beta_{10} + \cdots - 1495 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1421 \beta_{14} + 1421 \beta_{13} - 2795 \beta_{12} - 2795 \beta_{11} + 22947 \beta_{8} + 27163 \beta_{7} + \cdots - 77349 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 7036 \beta_{15} + 7172 \beta_{14} - 14208 \beta_{13} + 1348 \beta_{12} - 1348 \beta_{11} + \cdots + 55910 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 38778 \beta_{14} - 38778 \beta_{13} + 96006 \beta_{12} + 96006 \beta_{11} - 787830 \beta_{8} + \cdots + 2400613 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 252516 \beta_{15} - 235731 \beta_{14} + 488247 \beta_{13} - 54495 \beta_{12} + 54495 \beta_{11} + \cdots - 1852479 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1097410 \beta_{14} + 1097410 \beta_{13} - 3226978 \beta_{12} - 3226978 \beta_{11} + 27089852 \beta_{8} + \cdots - 76078198 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 8848142 \beta_{15} + 7559638 \beta_{14} - 16407780 \beta_{13} + 2145002 \beta_{12} - 2145002 \beta_{11} + \cdots + 57973345 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 31856307 \beta_{14} - 31856307 \beta_{13} + 107278749 \beta_{12} + 107278749 \beta_{11} + \cdots + 2439096967 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 305600820 \beta_{15} - 239777260 \beta_{14} + 545378080 \beta_{13} - 81805508 \beta_{12} + \cdots - 1759493642 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 939132274 \beta_{14} + 939132274 \beta_{13} - 3547108246 \beta_{12} - 3547108246 \beta_{11} + \cdots - 78719495907 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10452419252 \beta_{15} + 7571112493 \beta_{14} - 18023531745 \beta_{13} + 3037954073 \beta_{12} + \cdots + 52420041109 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/250\mathbb{Z}\right)^\times\).

\(n\) \(127\)
\(\chi(n)\) \(-1 + \beta_{7} + \beta_{8} + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
4.04577i
0.493198i
− 0.276776i
− 5.43776i
− 5.68871i
− 2.36461i
4.19209i
5.76335i
5.68871i
2.36461i
− 4.19209i
− 5.76335i
− 4.04577i
− 0.493198i
0.276776i
5.43776i
1.61803 − 1.17557i −2.56903 + 7.90665i 1.23607 − 3.80423i 0 5.13805 + 15.8133i 26.4146 −2.47214 − 7.60845i −34.0718 − 24.7546i 0
51.2 1.61803 − 1.17557i −0.480877 + 1.47999i 1.23607 − 3.80423i 0 0.961755 + 2.95998i −19.9781 −2.47214 − 7.60845i 19.8843 + 14.4468i 0
51.3 1.61803 − 1.17557i −0.0282983 + 0.0870932i 1.23607 − 3.80423i 0 0.0565966 + 0.174186i −10.7092 −2.47214 − 7.60845i 21.8367 + 15.8653i 0
51.4 1.61803 − 1.17557i 3.00525 − 9.24922i 1.23607 − 3.80423i 0 −6.01051 − 18.4984i 23.3628 −2.47214 − 7.60845i −54.6730 − 39.7223i 0
101.1 −0.618034 + 1.90211i −6.71931 − 4.88186i −3.23607 − 2.35114i 0 13.4386 − 9.76372i 11.9615 6.47214 − 4.70228i 12.9730 + 39.9269i 0
101.2 −0.618034 + 1.90211i −3.55790 − 2.58496i −3.23607 − 2.35114i 0 7.11579 − 5.16993i −32.4633 6.47214 − 4.70228i −2.36687 − 7.28447i 0
101.3 −0.618034 + 1.90211i 2.67790 + 1.94561i −3.23607 − 2.35114i 0 −5.35580 + 3.89121i 25.0474 6.47214 − 4.70228i −4.95771 − 15.2583i 0
101.4 −0.618034 + 1.90211i 4.17225 + 3.03132i −3.23607 − 2.35114i 0 −8.34450 + 6.06264i 3.36421 6.47214 − 4.70228i −0.124662 − 0.383671i 0
151.1 −0.618034 − 1.90211i −6.71931 + 4.88186i −3.23607 + 2.35114i 0 13.4386 + 9.76372i 11.9615 6.47214 + 4.70228i 12.9730 − 39.9269i 0
151.2 −0.618034 − 1.90211i −3.55790 + 2.58496i −3.23607 + 2.35114i 0 7.11579 + 5.16993i −32.4633 6.47214 + 4.70228i −2.36687 + 7.28447i 0
151.3 −0.618034 − 1.90211i 2.67790 − 1.94561i −3.23607 + 2.35114i 0 −5.35580 − 3.89121i 25.0474 6.47214 + 4.70228i −4.95771 + 15.2583i 0
151.4 −0.618034 − 1.90211i 4.17225 − 3.03132i −3.23607 + 2.35114i 0 −8.34450 − 6.06264i 3.36421 6.47214 + 4.70228i −0.124662 + 0.383671i 0
201.1 1.61803 + 1.17557i −2.56903 − 7.90665i 1.23607 + 3.80423i 0 5.13805 − 15.8133i 26.4146 −2.47214 + 7.60845i −34.0718 + 24.7546i 0
201.2 1.61803 + 1.17557i −0.480877 − 1.47999i 1.23607 + 3.80423i 0 0.961755 − 2.95998i −19.9781 −2.47214 + 7.60845i 19.8843 − 14.4468i 0
201.3 1.61803 + 1.17557i −0.0282983 − 0.0870932i 1.23607 + 3.80423i 0 0.0565966 − 0.174186i −10.7092 −2.47214 + 7.60845i 21.8367 − 15.8653i 0
201.4 1.61803 + 1.17557i 3.00525 + 9.24922i 1.23607 + 3.80423i 0 −6.01051 + 18.4984i 23.3628 −2.47214 + 7.60845i −54.6730 + 39.7223i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 250.4.d.b 16
5.b even 2 1 50.4.d.b ✓ 16
5.c odd 4 2 250.4.e.c 32
25.d even 5 1 inner 250.4.d.b 16
25.d even 5 1 1250.4.a.g 8
25.e even 10 1 50.4.d.b ✓ 16
25.e even 10 1 1250.4.a.j 8
25.f odd 20 2 250.4.e.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.d.b ✓ 16 5.b even 2 1
50.4.d.b ✓ 16 25.e even 10 1
250.4.d.b 16 1.a even 1 1 trivial
250.4.d.b 16 25.d even 5 1 inner
250.4.e.c 32 5.c odd 4 2
250.4.e.c 32 25.f odd 20 2
1250.4.a.g 8 25.d even 5 1
1250.4.a.j 8 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 7 T_{3}^{15} + 120 T_{3}^{14} + 810 T_{3}^{13} + 7880 T_{3}^{12} + 15911 T_{3}^{11} + \cdots + 51609856 \) acting on \(S_{4}^{\mathrm{new}}(250, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 7 T^{15} + \cdots + 51609856 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 27 T^{7} + \cdots - 4320209664)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 96\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 78\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
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