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Properties

Label 25.4.b.a
Level $25$
Weight $4$
Character orbit 25.b
Analytic conductor $1.475$
Analytic rank $0$
Dimension $2$
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] = [25,4,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), 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: \(1.47504775014\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} + \beta q^{3} - 8 q^{4} - 8 q^{6} - 3 \beta q^{7} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{2} + \beta q^{3} - 8 q^{4} - 8 q^{6} - 3 \beta q^{7} + 23 q^{9} + 32 q^{11} - 8 \beta q^{12} - 19 \beta q^{13} + 24 q^{14} - 64 q^{16} - 13 \beta q^{17} + 46 \beta q^{18} - 100 q^{19} + 12 q^{21} + 64 \beta q^{22} - 39 \beta q^{23} + 152 q^{26} + 50 \beta q^{27} + 24 \beta q^{28} + 50 q^{29} - 108 q^{31} - 128 \beta q^{32} + 32 \beta q^{33} + 104 q^{34} - 184 q^{36} - 133 \beta q^{37} - 200 \beta q^{38} + 76 q^{39} + 22 q^{41} + 24 \beta q^{42} + 221 \beta q^{43} - 256 q^{44} + 312 q^{46} + 257 \beta q^{47} - 64 \beta q^{48} + 307 q^{49} + 52 q^{51} + 152 \beta q^{52} + \beta q^{53} - 400 q^{54} - 100 \beta q^{57} + 100 \beta q^{58} - 500 q^{59} - 518 q^{61} - 216 \beta q^{62} - 69 \beta q^{63} + 512 q^{64} - 256 q^{66} - 63 \beta q^{67} + 104 \beta q^{68} + 156 q^{69} + 412 q^{71} - 439 \beta q^{73} + 1064 q^{74} + 800 q^{76} - 96 \beta q^{77} + 152 \beta q^{78} - 600 q^{79} + 421 q^{81} + 44 \beta q^{82} + 141 \beta q^{83} - 96 q^{84} - 1768 q^{86} + 50 \beta q^{87} + 150 q^{89} - 228 q^{91} + 312 \beta q^{92} - 108 \beta q^{93} - 2056 q^{94} + 512 q^{96} - 193 \beta q^{97} + 614 \beta q^{98} + 736 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 16 q^{6} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} - 16 q^{6} + 46 q^{9} + 64 q^{11} + 48 q^{14} - 128 q^{16} - 200 q^{19} + 24 q^{21} + 304 q^{26} + 100 q^{29} - 216 q^{31} + 208 q^{34} - 368 q^{36} + 152 q^{39} + 44 q^{41} - 512 q^{44} + 624 q^{46} + 614 q^{49} + 104 q^{51} - 800 q^{54} - 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 512 q^{66} + 312 q^{69} + 824 q^{71} + 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 842 q^{81} - 192 q^{84} - 3536 q^{86} + 300 q^{89} - 456 q^{91} - 4112 q^{94} + 1024 q^{96} + 1472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 1.00000i
1.00000i
− 4.00000i − 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 0
24.2 4.00000i 2.00000i −8.00000 0 −8.00000 − 6.00000i 0 23.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.4.b.a 2
3.b odd 2 1 225.4.b.c 2
4.b odd 2 1 400.4.c.k 2
5.b even 2 1 inner 25.4.b.a 2
5.c odd 4 1 5.4.a.a ✓ 1
5.c odd 4 1 25.4.a.c 1
15.d odd 2 1 225.4.b.c 2
15.e even 4 1 45.4.a.d 1
15.e even 4 1 225.4.a.b 1
20.d odd 2 1 400.4.c.k 2
20.e even 4 1 80.4.a.d 1
20.e even 4 1 400.4.a.m 1
35.f even 4 1 245.4.a.a 1
35.f even 4 1 1225.4.a.k 1
35.k even 12 2 245.4.e.g 2
35.l odd 12 2 245.4.e.f 2
40.i odd 4 1 320.4.a.g 1
40.i odd 4 1 1600.4.a.bi 1
40.k even 4 1 320.4.a.h 1
40.k even 4 1 1600.4.a.s 1
45.k odd 12 2 405.4.e.l 2
45.l even 12 2 405.4.e.c 2
55.e even 4 1 605.4.a.d 1
60.l odd 4 1 720.4.a.u 1
65.h odd 4 1 845.4.a.b 1
80.i odd 4 1 1280.4.d.e 2
80.j even 4 1 1280.4.d.l 2
80.s even 4 1 1280.4.d.l 2
80.t odd 4 1 1280.4.d.e 2
85.g odd 4 1 1445.4.a.a 1
95.g even 4 1 1805.4.a.h 1
105.k odd 4 1 2205.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a ✓ 1 5.c odd 4 1
25.4.a.c 1 5.c odd 4 1
25.4.b.a 2 1.a even 1 1 trivial
25.4.b.a 2 5.b even 2 1 inner
45.4.a.d 1 15.e even 4 1
80.4.a.d 1 20.e even 4 1
225.4.a.b 1 15.e even 4 1
225.4.b.c 2 3.b odd 2 1
225.4.b.c 2 15.d odd 2 1
245.4.a.a 1 35.f even 4 1
245.4.e.f 2 35.l odd 12 2
245.4.e.g 2 35.k even 12 2
320.4.a.g 1 40.i odd 4 1
320.4.a.h 1 40.k even 4 1
400.4.a.m 1 20.e even 4 1
400.4.c.k 2 4.b odd 2 1
400.4.c.k 2 20.d odd 2 1
405.4.e.c 2 45.l even 12 2
405.4.e.l 2 45.k odd 12 2
605.4.a.d 1 55.e even 4 1
720.4.a.u 1 60.l odd 4 1
845.4.a.b 1 65.h odd 4 1
1225.4.a.k 1 35.f even 4 1
1280.4.d.e 2 80.i odd 4 1
1280.4.d.e 2 80.t odd 4 1
1280.4.d.l 2 80.j even 4 1
1280.4.d.l 2 80.s even 4 1
1445.4.a.a 1 85.g odd 4 1
1600.4.a.s 1 40.k even 4 1
1600.4.a.bi 1 40.i odd 4 1
1805.4.a.h 1 95.g even 4 1
2205.4.a.q 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{2} + 676 \) Copy content Toggle raw display
$19$ \( (T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6084 \) Copy content Toggle raw display
$29$ \( (T - 50)^{2} \) Copy content Toggle raw display
$31$ \( (T + 108)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 70756 \) Copy content Toggle raw display
$41$ \( (T - 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 195364 \) Copy content Toggle raw display
$47$ \( T^{2} + 264196 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 500)^{2} \) Copy content Toggle raw display
$61$ \( (T + 518)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15876 \) Copy content Toggle raw display
$71$ \( (T - 412)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 770884 \) Copy content Toggle raw display
$79$ \( (T + 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 79524 \) Copy content Toggle raw display
$89$ \( (T - 150)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 148996 \) Copy content Toggle raw display
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