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Properties

Label 25.2.e.a
Level $25$
Weight $2$
Character orbit 25.e
Analytic conductor $0.200$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,2,Mod(4,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 25.e (of order \(10\), degree \(4\), 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: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{7} - \beta_{3} + \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 1) q^{4}+ \cdots + (2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} - 5 q^{3} - q^{4} - 9 q^{6} + 10 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} - 5 q^{3} - q^{4} - 9 q^{6} + 10 q^{8} + q^{9} - 5 q^{10} - 4 q^{11} + 15 q^{12} - 5 q^{13} + 13 q^{14} + 15 q^{15} + 3 q^{16} - 10 q^{17} - 5 q^{19} - 15 q^{20} - 4 q^{21} + 5 q^{23} - 20 q^{24} - 10 q^{25} + 6 q^{26} - 5 q^{27} - 15 q^{28} - 5 q^{29} + 15 q^{30} - 9 q^{31} + 10 q^{33} + 13 q^{34} + 15 q^{35} + 23 q^{36} + 30 q^{37} + 15 q^{38} - 3 q^{39} + 10 q^{40} - 4 q^{41} - 15 q^{42} - 2 q^{44} - 15 q^{45} - 19 q^{46} - 30 q^{48} + 14 q^{49} - 15 q^{50} - 4 q^{51} - 10 q^{52} - 10 q^{53} - 5 q^{54} - 10 q^{55} + 10 q^{56} + 20 q^{58} - 10 q^{60} - 9 q^{61} - 30 q^{62} + 10 q^{63} + 4 q^{64} + 5 q^{65} + 12 q^{66} + 20 q^{67} + 17 q^{69} + 30 q^{70} + 6 q^{71} + 5 q^{72} + 15 q^{73} - 12 q^{74} - 10 q^{75} - 20 q^{76} + 10 q^{77} + 25 q^{78} + 15 q^{79} + 20 q^{80} + 28 q^{81} - 45 q^{83} + 18 q^{84} - 15 q^{85} - 9 q^{86} - 20 q^{87} - 20 q^{88} - 25 q^{89} - 25 q^{90} + 6 q^{91} + 30 q^{92} - 27 q^{94} + 15 q^{95} + 16 q^{96} - 60 q^{97} - 10 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−0.983224 − 0.644389i
1.17421 + 0.0566033i
−0.357358 + 1.86824i
1.66637 − 0.917186i
−0.357358 − 1.86824i
1.66637 + 0.917186i
−0.983224 + 0.644389i
1.17421 − 0.0566033i
−1.98322 − 0.644389i 1.29224 − 1.77862i 1.89991 + 1.38036i −1.22570 + 1.87020i −3.70892 + 2.69469i 0.992398i −0.427051 − 0.587785i −0.566541 − 1.74363i 3.63597 − 2.91920i
4.2 0.174207 + 0.0566033i −0.865190 + 1.19083i −1.59089 − 1.15585i 0.107666 − 2.23347i −0.218127 + 0.158479i 3.26086i −0.427051 − 0.587785i 0.257524 + 0.792578i 0.145178 − 0.382993i
9.1 −1.35736 + 1.86824i −0.451659 + 0.146753i −1.02988 − 3.16963i 2.19625 + 0.420099i 0.338893 − 1.04301i − 3.03582i 2.92705 + 0.951057i −2.24459 + 1.63079i −3.76594 + 3.53290i
9.2 0.666375 − 0.917186i −2.47539 + 0.804303i 0.220859 + 0.679734i −1.07822 − 1.95894i −0.911842 + 2.80636i 0.407162i 2.92705 + 0.951057i 3.05361 − 2.21858i −2.51521 − 0.316463i
14.1 −1.35736 − 1.86824i −0.451659 − 0.146753i −1.02988 + 3.16963i 2.19625 − 0.420099i 0.338893 + 1.04301i 3.03582i 2.92705 − 0.951057i −2.24459 − 1.63079i −3.76594 − 3.53290i
14.2 0.666375 + 0.917186i −2.47539 − 0.804303i 0.220859 − 0.679734i −1.07822 + 1.95894i −0.911842 − 2.80636i − 0.407162i 2.92705 − 0.951057i 3.05361 + 2.21858i −2.51521 + 0.316463i
19.1 −1.98322 + 0.644389i 1.29224 + 1.77862i 1.89991 − 1.38036i −1.22570 − 1.87020i −3.70892 − 2.69469i − 0.992398i −0.427051 + 0.587785i −0.566541 + 1.74363i 3.63597 + 2.91920i
19.2 0.174207 − 0.0566033i −0.865190 − 1.19083i −1.59089 + 1.15585i 0.107666 + 2.23347i −0.218127 − 0.158479i − 3.26086i −0.427051 + 0.587785i 0.257524 − 0.792578i 0.145178 + 0.382993i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.2.e.a ✓ 8
3.b odd 2 1 225.2.m.a 8
4.b odd 2 1 400.2.y.c 8
5.b even 2 1 125.2.e.b 8
5.c odd 4 2 125.2.d.b 16
25.d even 5 1 125.2.e.b 8
25.d even 5 1 625.2.b.c 8
25.d even 5 1 625.2.e.a 8
25.d even 5 1 625.2.e.i 8
25.e even 10 1 inner 25.2.e.a ✓ 8
25.e even 10 1 625.2.b.c 8
25.e even 10 1 625.2.e.a 8
25.e even 10 1 625.2.e.i 8
25.f odd 20 2 125.2.d.b 16
25.f odd 20 2 625.2.a.f 8
25.f odd 20 4 625.2.d.o 16
75.h odd 10 1 225.2.m.a 8
75.l even 20 2 5625.2.a.x 8
100.h odd 10 1 400.2.y.c 8
100.l even 20 2 10000.2.a.bj 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a ✓ 8 1.a even 1 1 trivial
25.2.e.a ✓ 8 25.e even 10 1 inner
125.2.d.b 16 5.c odd 4 2
125.2.d.b 16 25.f odd 20 2
125.2.e.b 8 5.b even 2 1
125.2.e.b 8 25.d even 5 1
225.2.m.a 8 3.b odd 2 1
225.2.m.a 8 75.h odd 10 1
400.2.y.c 8 4.b odd 2 1
400.2.y.c 8 100.h odd 10 1
625.2.a.f 8 25.f odd 20 2
625.2.b.c 8 25.d even 5 1
625.2.b.c 8 25.e even 10 1
625.2.d.o 16 25.f odd 20 4
625.2.e.a 8 25.d even 5 1
625.2.e.a 8 25.e even 10 1
625.2.e.i 8 25.d even 5 1
625.2.e.i 8 25.e even 10 1
5625.2.a.x 8 75.l even 20 2
10000.2.a.bj 8 100.l even 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 21 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} + 5 T^{7} + \cdots + 483025 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{8} - 30 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 129 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( T^{8} + 15 T^{5} + \cdots + 4080400 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$67$ \( T^{8} - 20 T^{7} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 24245776 \) Copy content Toggle raw display
$73$ \( T^{8} - 15 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} - 15 T^{7} + \cdots + 33408400 \) Copy content Toggle raw display
$83$ \( T^{8} + 45 T^{7} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( T^{8} + 25 T^{7} + \cdots + 1392400 \) Copy content Toggle raw display
$97$ \( T^{8} + 60 T^{7} + \cdots + 301334881 \) Copy content Toggle raw display
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