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Properties

Label 2500.4.a.c
Level $2500$
Weight $4$
Character orbit 2500.a
Self dual yes
Analytic conductor $147.505$
Analytic rank $1$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 240 x^{12} + 242 x^{11} + 22134 x^{10} - 6820 x^{9} - 974680 x^{8} - 50130 x^{7} + \cdots - 43494224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 100)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{4} - 3 \beta_{2} + \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{4} - 3 \beta_{2} + \beta_1 + 9) q^{9} + (\beta_{9} - \beta_{6} + 2 \beta_{2} + \cdots - 3) q^{11}+ \cdots + (3 \beta_{13} + 6 \beta_{12} + \cdots - 143) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{3} + 8 q^{7} + 106 q^{9} - 30 q^{11} + 77 q^{13} + 111 q^{17} - 64 q^{19} - 156 q^{21} - 24 q^{23} - 614 q^{27} - 359 q^{29} - 156 q^{31} - 390 q^{33} - 307 q^{37} - 314 q^{39} - 791 q^{41} - 90 q^{43} + 276 q^{47} + 72 q^{49} - 832 q^{51} - 451 q^{53} + 192 q^{57} - 808 q^{59} - 1017 q^{61} + 2082 q^{63} + 1678 q^{67} - 1502 q^{69} - 148 q^{71} - 223 q^{73} - 1210 q^{77} - 92 q^{79} - 414 q^{81} - 474 q^{83} + 4262 q^{87} - 2111 q^{89} - 794 q^{91} - 2832 q^{93} + 353 q^{97} - 1050 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 240 x^{12} + 242 x^{11} + 22134 x^{10} - 6820 x^{9} - 974680 x^{8} - 50130 x^{7} + \cdots - 43494224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!66 \nu^{13} + \cdots - 52\!\cdots\!44 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!13 \nu^{13} + \cdots + 12\!\cdots\!04 ) / 33\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!66 \nu^{13} + \cdots - 41\!\cdots\!44 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\!\cdots\!81 \nu^{13} + \cdots - 75\!\cdots\!08 ) / 33\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!41 \nu^{13} + \cdots + 55\!\cdots\!84 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 32\!\cdots\!13 \nu^{13} + \cdots + 66\!\cdots\!72 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 72\!\cdots\!89 \nu^{13} + \cdots - 66\!\cdots\!76 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 80\!\cdots\!64 \nu^{13} + \cdots + 22\!\cdots\!96 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 87\!\cdots\!12 \nu^{13} + \cdots - 40\!\cdots\!88 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!07 \nu^{13} + \cdots - 68\!\cdots\!68 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!11 \nu^{13} + \cdots + 17\!\cdots\!64 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!96 \nu^{13} + \cdots + 10\!\cdots\!64 ) / 84\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 3\beta_{2} + \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} - 5 \beta_{6} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} + 23 \beta_{12} + 17 \beta_{11} + 10 \beta_{10} + 12 \beta_{9} + 22 \beta_{8} + \cdots + 2265 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 97 \beta_{13} + 333 \beta_{12} - 17 \beta_{11} + 43 \beta_{10} + 148 \beta_{9} + 337 \beta_{8} + \cdots + 8965 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 222 \beta_{13} + 2847 \beta_{12} + 1863 \beta_{11} + 839 \beta_{10} + 1603 \beta_{9} + 3180 \beta_{8} + \cdots + 173837 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8506 \beta_{13} + 32628 \beta_{12} + 3382 \beta_{11} + 1132 \beta_{10} + 16420 \beta_{9} + \cdots + 1074942 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 38488 \beta_{13} + 297738 \beta_{12} + 162590 \beta_{11} + 52366 \beta_{10} + 170818 \beta_{9} + \cdots + 15027088 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 795555 \beta_{13} + 3180113 \beta_{12} + 581631 \beta_{11} - 47401 \beta_{10} + 1688943 \beta_{9} + \cdots + 116423983 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5375875 \beta_{13} + 30148975 \beta_{12} + 13451775 \beta_{11} + 2489450 \beta_{10} + \cdots + 1390908931 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 79457265 \beta_{13} + 313053975 \beta_{12} + 67745165 \beta_{11} - 14667215 \beta_{10} + \cdots + 12126807685 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 662858740 \beta_{13} + 3036736405 \beta_{12} + 1113618315 \beta_{11} + 40686425 \beta_{10} + \cdots + 133747552031 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8219385416 \beta_{13} + 31108164588 \beta_{12} + 6996947064 \beta_{11} - 2202672264 \beta_{10} + \cdots + 1242523360028 \) 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
10.0772
8.60496
6.78160
5.30619
3.36500
1.63014
1.06104
−0.0952647
−2.25367
−5.75970
−5.88123
−5.93521
−7.28532
−7.61573
0 −10.0772 0 0 0 17.4845 0 74.5496 0
1.2 0 −8.60496 0 0 0 17.7526 0 47.0453 0
1.3 0 −6.78160 0 0 0 −19.6435 0 18.9900 0
1.4 0 −5.30619 0 0 0 5.90251 0 1.15568 0
1.5 0 −3.36500 0 0 0 −0.208499 0 −15.6768 0
1.6 0 −1.63014 0 0 0 −34.0141 0 −24.3426 0
1.7 0 −1.06104 0 0 0 3.00616 0 −25.8742 0
1.8 0 0.0952647 0 0 0 26.6009 0 −26.9909 0
1.9 0 2.25367 0 0 0 −29.6874 0 −21.9210 0
1.10 0 5.75970 0 0 0 12.5176 0 6.17410 0
1.11 0 5.88123 0 0 0 11.1886 0 7.58887 0
1.12 0 5.93521 0 0 0 21.1171 0 8.22666 0
1.13 0 7.28532 0 0 0 −6.47483 0 26.0760 0
1.14 0 7.61573 0 0 0 −17.5417 0 30.9993 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.c 14
5.b even 2 1 2500.4.a.d 14
25.d even 5 2 100.4.g.a ✓ 28
25.e even 10 2 500.4.g.a 28
25.f odd 20 4 500.4.i.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.g.a ✓ 28 25.d even 5 2
500.4.g.a 28 25.e even 10 2
500.4.i.b 56 25.f odd 20 4
2500.4.a.c 14 1.a even 1 1 trivial
2500.4.a.d 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 2 T_{3}^{13} - 240 T_{3}^{12} - 242 T_{3}^{11} + 22134 T_{3}^{10} + 6820 T_{3}^{9} + \cdots - 43494224 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2500))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots - 43494224 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 203538447138816 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 20\!\cdots\!39 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 49\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 28\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 50\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 16\!\cdots\!79 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 42\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 29\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 82\!\cdots\!51 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 74\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 44\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 40\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 83\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 20\!\cdots\!51 \) Copy content Toggle raw display
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