Properties

Label 25.5.c.c
Level $25$
Weight $5$
Character orbit 25.c
Analytic conductor $2.584$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,5,Mod(7,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), 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: \(2.58424907710\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{2} q^{3} - 26 \beta_1 q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} - 10 \beta_{2} q^{8} - 39 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{2} q^{3} - 26 \beta_1 q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} - 10 \beta_{2} q^{8} - 39 \beta_1 q^{9} - 108 q^{11} - 26 \beta_{3} q^{12} + 36 \beta_{2} q^{13} + 378 \beta_1 q^{14} - 4 q^{16} + 4 \beta_{3} q^{17} - 39 \beta_{2} q^{18} - 140 \beta_1 q^{19} - 378 q^{21} + 108 \beta_{3} q^{22} - 79 \beta_{2} q^{23} - 420 \beta_1 q^{24} + 1512 q^{26} - 120 \beta_{3} q^{27} + 234 \beta_{2} q^{28} - 810 \beta_1 q^{29} - 728 q^{31} - 156 \beta_{3} q^{32} - 108 \beta_{2} q^{33} + 168 \beta_1 q^{34} - 1014 q^{36} + 144 \beta_{3} q^{37} - 140 \beta_{2} q^{38} + 1512 \beta_1 q^{39} + 1512 q^{41} + 378 \beta_{3} q^{42} - 9 \beta_{2} q^{43} + 2808 \beta_1 q^{44} - 3318 q^{46} + 39 \beta_{3} q^{47} - 4 \beta_{2} q^{48} - 1001 \beta_1 q^{49} - 168 q^{51} - 936 \beta_{3} q^{52} + 676 \beta_{2} q^{53} - 5040 \beta_1 q^{54} + 3780 q^{56} - 140 \beta_{3} q^{57} - 810 \beta_{2} q^{58} + 3780 \beta_1 q^{59} + 4592 q^{61} + 728 \beta_{3} q^{62} + 351 \beta_{2} q^{63} - 6616 \beta_1 q^{64} - 4536 q^{66} + 729 \beta_{3} q^{67} + 104 \beta_{2} q^{68} - 3318 \beta_1 q^{69} + 432 q^{71} + 390 \beta_{3} q^{72} - 1404 \beta_{2} q^{73} + 6048 \beta_1 q^{74} - 3640 q^{76} - 972 \beta_{3} q^{77} + 1512 \beta_{2} q^{78} + 8840 \beta_1 q^{79} + 1881 q^{81} - 1512 \beta_{3} q^{82} - 169 \beta_{2} q^{83} + 9828 \beta_1 q^{84} - 378 q^{86} - 810 \beta_{3} q^{87} + 1080 \beta_{2} q^{88} - 13230 \beta_1 q^{89} - 13608 q^{91} + 2054 \beta_{3} q^{92} - 728 \beta_{2} q^{93} + 1638 \beta_1 q^{94} + 6552 q^{96} + 1764 \beta_{3} q^{97} - 1001 \beta_{2} q^{98} + 4212 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 168 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 168 q^{6} - 432 q^{11} - 16 q^{16} - 1512 q^{21} + 6048 q^{26} - 2912 q^{31} - 4056 q^{36} + 6048 q^{41} - 13272 q^{46} - 672 q^{51} + 15120 q^{56} + 18368 q^{61} - 18144 q^{66} + 1728 q^{71} - 14560 q^{76} + 7524 q^{81} - 1512 q^{86} - 54432 q^{91} + 26208 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 11x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 6\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu^{2} + 16\nu + 55 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 10\nu^{2} + 16\nu - 55 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - 3\beta_{2} + 16\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.79129i
1.79129i
2.79129i
1.79129i
−4.58258 4.58258i −4.58258 + 4.58258i 26.0000i 0 42.0000 41.2432 + 41.2432i 45.8258 45.8258i 39.0000i 0
7.2 4.58258 + 4.58258i 4.58258 4.58258i 26.0000i 0 42.0000 −41.2432 41.2432i −45.8258 + 45.8258i 39.0000i 0
18.1 −4.58258 + 4.58258i −4.58258 4.58258i 26.0000i 0 42.0000 41.2432 41.2432i 45.8258 + 45.8258i 39.0000i 0
18.2 4.58258 4.58258i 4.58258 + 4.58258i 26.0000i 0 42.0000 −41.2432 + 41.2432i −45.8258 45.8258i 39.0000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.5.c.c 4
3.b odd 2 1 225.5.g.j 4
4.b odd 2 1 400.5.p.l 4
5.b even 2 1 inner 25.5.c.c 4
5.c odd 4 2 inner 25.5.c.c 4
15.d odd 2 1 225.5.g.j 4
15.e even 4 2 225.5.g.j 4
20.d odd 2 1 400.5.p.l 4
20.e even 4 2 400.5.p.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.c.c 4 1.a even 1 1 trivial
25.5.c.c 4 5.b even 2 1 inner
25.5.c.c 4 5.c odd 4 2 inner
225.5.g.j 4 3.b odd 2 1
225.5.g.j 4 15.d odd 2 1
225.5.g.j 4 15.e even 4 2
400.5.p.l 4 4.b odd 2 1
400.5.p.l 4 20.d odd 2 1
400.5.p.l 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1764 \) acting on \(S_{5}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1764 \) Copy content Toggle raw display
$3$ \( T^{4} + 1764 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11573604 \) Copy content Toggle raw display
$11$ \( (T + 108)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 2962842624 \) Copy content Toggle raw display
$17$ \( T^{4} + 451584 \) Copy content Toggle raw display
$19$ \( (T^{2} + 19600)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 68707942884 \) Copy content Toggle raw display
$29$ \( (T^{2} + 656100)^{2} \) Copy content Toggle raw display
$31$ \( (T + 728)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 758487711744 \) Copy content Toggle raw display
$41$ \( (T - 1512)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 11573604 \) Copy content Toggle raw display
$47$ \( T^{4} + 4080909924 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 368370941912064 \) Copy content Toggle raw display
$59$ \( (T^{2} + 14288400)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4592)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 498205702352484 \) Copy content Toggle raw display
$71$ \( (T - 432)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 68\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{2} + 78145600)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1438948991844 \) Copy content Toggle raw display
$89$ \( (T^{2} + 175032900)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 17\!\cdots\!24 \) Copy content Toggle raw display
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