Properties

Label 425.2.e.b
Level $425$
Weight $2$
Character orbit 425.e
Analytic conductor $3.394$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,2,Mod(251,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.e (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: \(3.39364208590\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} + (i + 1) q^{3} + q^{4} + ( - i + 1) q^{6} + ( - 3 i + 3) q^{7} - 3 i q^{8} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} + (i + 1) q^{3} + q^{4} + ( - i + 1) q^{6} + ( - 3 i + 3) q^{7} - 3 i q^{8} - i q^{9} + (3 i - 3) q^{11} + (i + 1) q^{12} + ( - 3 i - 3) q^{14} - q^{16} + ( - i - 4) q^{17} - q^{18} + 6 i q^{19} + 6 q^{21} + (3 i + 3) q^{22} + ( - i + 1) q^{23} + ( - 3 i + 3) q^{24} + ( - 4 i + 4) q^{27} + ( - 3 i + 3) q^{28} + (3 i + 3) q^{29} + ( - i - 1) q^{31} - 5 i q^{32} - 6 q^{33} + (4 i - 1) q^{34} - i q^{36} + (3 i + 3) q^{37} + 6 q^{38} + (3 i - 3) q^{41} - 6 i q^{42} + 12 i q^{43} + (3 i - 3) q^{44} + ( - i - 1) q^{46} - 2 q^{47} + ( - i - 1) q^{48} - 11 i q^{49} + ( - 5 i - 3) q^{51} - 2 i q^{53} + ( - 4 i - 4) q^{54} + ( - 9 i - 9) q^{56} + (6 i - 6) q^{57} + ( - 3 i + 3) q^{58} + 6 i q^{59} + ( - i + 1) q^{61} + (i - 1) q^{62} + ( - 3 i - 3) q^{63} - 7 q^{64} + 6 i q^{66} - 6 q^{67} + ( - i - 4) q^{68} + 2 q^{69} + (3 i + 3) q^{71} - 3 q^{72} + ( - 3 i - 3) q^{73} + ( - 3 i + 3) q^{74} + 6 i q^{76} + 18 i q^{77} + ( - 7 i + 7) q^{79} + 5 q^{81} + (3 i + 3) q^{82} + 4 i q^{83} + 6 q^{84} + 12 q^{86} + 6 i q^{87} + (9 i + 9) q^{88} - 6 q^{89} + ( - i + 1) q^{92} - 2 i q^{93} + 2 i q^{94} + ( - 5 i + 5) q^{96} + (3 i + 3) q^{97} - 11 q^{98} + (3 i + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 6 q^{11} + 2 q^{12} - 6 q^{14} - 2 q^{16} - 8 q^{17} - 2 q^{18} + 12 q^{21} + 6 q^{22} + 2 q^{23} + 6 q^{24} + 8 q^{27} + 6 q^{28} + 6 q^{29} - 2 q^{31} - 12 q^{33} - 2 q^{34} + 6 q^{37} + 12 q^{38} - 6 q^{41} - 6 q^{44} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 6 q^{51} - 8 q^{54} - 18 q^{56} - 12 q^{57} + 6 q^{58} + 2 q^{61} - 2 q^{62} - 6 q^{63} - 14 q^{64} - 12 q^{67} - 8 q^{68} + 4 q^{69} + 6 q^{71} - 6 q^{72} - 6 q^{73} + 6 q^{74} + 14 q^{79} + 10 q^{81} + 6 q^{82} + 12 q^{84} + 24 q^{86} + 18 q^{88} - 12 q^{89} + 2 q^{92} + 10 q^{96} + 6 q^{97} - 22 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.00000i
1.00000i
1.00000i 1.00000 + 1.00000i 1.00000 0 1.00000 1.00000i 3.00000 3.00000i 3.00000i 1.00000i 0
276.1 1.00000i 1.00000 1.00000i 1.00000 0 1.00000 + 1.00000i 3.00000 + 3.00000i 3.00000i 1.00000i 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
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.e.b 2
5.b even 2 1 425.2.e.a 2
5.c odd 4 1 85.2.j.a 2
5.c odd 4 1 85.2.j.b yes 2
15.e even 4 1 765.2.t.a 2
15.e even 4 1 765.2.t.b 2
17.c even 4 1 inner 425.2.e.b 2
17.d even 8 2 7225.2.a.p 2
85.f odd 4 1 85.2.j.a 2
85.i odd 4 1 85.2.j.b yes 2
85.j even 4 1 425.2.e.a 2
85.k odd 8 2 1445.2.b.a 4
85.m even 8 2 7225.2.a.i 2
85.n odd 8 2 1445.2.b.a 4
255.k even 4 1 765.2.t.b 2
255.r even 4 1 765.2.t.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.a 2 5.c odd 4 1
85.2.j.a 2 85.f odd 4 1
85.2.j.b yes 2 5.c odd 4 1
85.2.j.b yes 2 85.i odd 4 1
425.2.e.a 2 5.b even 2 1
425.2.e.a 2 85.j even 4 1
425.2.e.b 2 1.a even 1 1 trivial
425.2.e.b 2 17.c even 4 1 inner
765.2.t.a 2 15.e even 4 1
765.2.t.a 2 255.r even 4 1
765.2.t.b 2 15.e even 4 1
765.2.t.b 2 255.k even 4 1
1445.2.b.a 4 85.k odd 8 2
1445.2.b.a 4 85.n odd 8 2
7225.2.a.i 2 85.m even 8 2
7225.2.a.p 2 17.d even 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( (T + 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$67$ \( (T + 6)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$79$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
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