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Properties

Label 2500.4.a.f
Level $2500$
Weight $4$
Character orbit 2500.a
Self dual yes
Analytic conductor $147.505$
Analytic rank $1$
Dimension $20$
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] = [2500,4,Mod(1,2500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2500.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2500 = 2^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2500.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: \(147.504775014\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} - 357 x^{18} + 357 x^{17} + 53281 x^{16} - 53520 x^{15} - 4320806 x^{14} + \cdots - 3694919941079 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{9} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{6} - \beta_{5} + 2) q^{7} + (\beta_{3} + \beta_{2} + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{6} - \beta_{5} + 2) q^{7} + (\beta_{3} + \beta_{2} + 9) q^{9} + ( - \beta_{14} - \beta_{2} - \beta_1 - 5) q^{11} + ( - \beta_{15} - \beta_{10} + \beta_{9} + \cdots - 5) q^{13}+ \cdots + ( - 4 \beta_{19} + 2 \beta_{18} + \cdots - 244) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + q^{3} + 26 q^{7} + 175 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + q^{3} + 26 q^{7} + 175 q^{9} - 90 q^{11} - 86 q^{13} - 303 q^{17} + 15 q^{19} + 70 q^{21} - 18 q^{23} - 53 q^{27} + 110 q^{29} + 140 q^{31} - 375 q^{33} - 584 q^{37} - 240 q^{39} + 35 q^{41} - 110 q^{43} - 158 q^{47} + 880 q^{49} - 1455 q^{51} - 1912 q^{53} - 966 q^{57} - 1420 q^{59} - 130 q^{61} - 676 q^{63} - 1079 q^{67} - 70 q^{69} - 1020 q^{71} - 911 q^{73} - 3600 q^{77} + 690 q^{79} - 1560 q^{81} - 3278 q^{83} + 1164 q^{87} + 1955 q^{89} - 1834 q^{93} + 66 q^{97} - 4265 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} - 357 x^{18} + 357 x^{17} + 53281 x^{16} - 53520 x^{15} - 4320806 x^{14} + \cdots - 3694919941079 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23\!\cdots\!96 \nu^{19} + \cdots - 83\!\cdots\!15 ) / 66\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!96 \nu^{19} + \cdots - 15\!\cdots\!85 ) / 66\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\!\cdots\!28 \nu^{19} + \cdots + 59\!\cdots\!71 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47\!\cdots\!84 \nu^{19} + \cdots + 23\!\cdots\!89 ) / 66\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!27 \nu^{19} + \cdots - 16\!\cdots\!87 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 97\!\cdots\!41 \nu^{19} + \cdots + 23\!\cdots\!83 ) / 42\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!27 \nu^{19} + \cdots + 11\!\cdots\!17 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!88 \nu^{19} + \cdots - 94\!\cdots\!41 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!66 \nu^{19} + \cdots - 18\!\cdots\!78 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!26 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 11\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!27 \nu^{19} + \cdots - 48\!\cdots\!71 ) / 38\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!66 \nu^{19} + \cdots - 14\!\cdots\!04 ) / 66\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 62\!\cdots\!76 \nu^{19} + \cdots - 17\!\cdots\!69 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 36\!\cdots\!50 \nu^{19} + \cdots + 89\!\cdots\!07 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!52 \nu^{19} + \cdots - 28\!\cdots\!65 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 14\!\cdots\!57 \nu^{19} + \cdots + 55\!\cdots\!11 ) / 87\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 71\!\cdots\!68 \nu^{19} + \cdots - 17\!\cdots\!88 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 35\!\cdots\!97 \nu^{19} + \cdots + 12\!\cdots\!25 ) / 87\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{19} - \beta_{18} + \beta_{17} - \beta_{15} + \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} - \beta_{17} - 2 \beta_{16} - 9 \beta_{15} + 8 \beta_{14} - 5 \beta_{12} + 15 \beta_{11} + \cdots + 2133 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 97 \beta_{19} - 73 \beta_{18} + 85 \beta_{17} - 9 \beta_{16} - 99 \beta_{15} + 96 \beta_{14} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 101 \beta_{19} + 243 \beta_{18} - 164 \beta_{17} - 240 \beta_{16} - 774 \beta_{15} + 788 \beta_{14} + \cdots + 144689 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8471 \beta_{19} - 4427 \beta_{18} + 5689 \beta_{17} - 1087 \beta_{16} - 7749 \beta_{15} + \cdots + 33774 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18137 \beta_{19} + 33711 \beta_{18} - 18103 \beta_{17} - 26360 \beta_{16} - 52404 \beta_{15} + \cdots + 10472453 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 729511 \beta_{19} - 242438 \beta_{18} + 344101 \beta_{17} - 103472 \beta_{16} - 577008 \beta_{15} + \cdots + 3925229 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2282020 \beta_{19} + 3734660 \beta_{18} - 1773715 \beta_{17} - 2697665 \beta_{16} - 3271730 \beta_{15} + \cdots + 788020181 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 62836600 \beta_{19} - 11360060 \beta_{18} + 19334860 \beta_{17} - 9351740 \beta_{16} + \cdots + 411722800 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 248249080 \beta_{19} + 370585750 \beta_{18} - 165417670 \beta_{17} - 260880970 \beta_{16} + \cdots + 60849873941 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5420114796 \beta_{19} - 334303691 \beta_{18} + 993451316 \beta_{17} - 845444780 \beta_{16} + \cdots + 41926862518 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 25020111610 \beta_{19} + 34606568011 \beta_{18} - 15023177561 \beta_{17} - 24232998137 \beta_{16} + \cdots + 4785871627878 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 467670421472 \beta_{19} + 14956768907 \beta_{18} + 43097364240 \beta_{17} - 77280207969 \beta_{16} + \cdots + 4216115815323 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2409759486016 \beta_{19} + 3117359365833 \beta_{18} - 1341013132994 \beta_{17} + \cdots + 381583251221199 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 40322445880091 \beta_{19} + 4287404807843 \beta_{18} + 1054219956299 \beta_{17} + \cdots + 419452503763834 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 225411980234657 \beta_{19} + 274591395580141 \beta_{18} - 118217909866583 \beta_{17} + \cdots + 30\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 34\!\cdots\!01 \beta_{19} + 577156065998397 \beta_{18} - 75720819083609 \beta_{17} + \cdots + 41\!\cdots\!99 \) 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
−8.95450
−8.87117
−7.66876
−6.86511
−5.93972
−5.69300
−3.48426
−3.05333
−1.95603
0.590015
1.12036
1.27196
2.85937
3.97161
5.12650
5.74926
7.43775
7.95263
8.13833
9.26809
0 −8.95450 0 0 0 4.22788 0 53.1830 0
1.2 0 −8.87117 0 0 0 −22.3866 0 51.6976 0
1.3 0 −7.66876 0 0 0 −16.6863 0 31.8099 0
1.4 0 −6.86511 0 0 0 28.1745 0 20.1297 0
1.5 0 −5.93972 0 0 0 32.5517 0 8.28027 0
1.6 0 −5.69300 0 0 0 −22.9370 0 5.41024 0
1.7 0 −3.48426 0 0 0 −9.58418 0 −14.8599 0
1.8 0 −3.05333 0 0 0 19.9343 0 −17.6771 0
1.9 0 −1.95603 0 0 0 1.68911 0 −23.1739 0
1.10 0 0.590015 0 0 0 10.5566 0 −26.6519 0
1.11 0 1.12036 0 0 0 0.102035 0 −25.7448 0
1.12 0 1.27196 0 0 0 −11.7790 0 −25.3821 0
1.13 0 2.85937 0 0 0 22.9637 0 −18.8240 0
1.14 0 3.97161 0 0 0 −26.0120 0 −11.2263 0
1.15 0 5.12650 0 0 0 −10.4837 0 −0.718950 0
1.16 0 5.74926 0 0 0 21.0872 0 6.05400 0
1.17 0 7.43775 0 0 0 −31.0274 0 28.3201 0
1.18 0 7.95263 0 0 0 15.4463 0 36.2443 0
1.19 0 8.13833 0 0 0 28.2147 0 39.2325 0
1.20 0 9.26809 0 0 0 −8.05182 0 58.8975 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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 2500.4.a.f yes 20
5.b even 2 1 2500.4.a.e ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2500.4.a.e ✓ 20 5.b even 2 1
2500.4.a.f yes 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - T_{3}^{19} - 357 T_{3}^{18} + 357 T_{3}^{17} + 53281 T_{3}^{16} - 53520 T_{3}^{15} + \cdots - 3694919941079 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots - 3694919941079 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 19\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots - 82\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 17\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 16\!\cdots\!79 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 19\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 25\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots - 60\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots - 17\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots - 27\!\cdots\!39 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 47\!\cdots\!75 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 13\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 85\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 83\!\cdots\!71 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 12\!\cdots\!99 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
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