Properties

Label 425.2.n.a
Level $425$
Weight $2$
Character orbit 425.n
Analytic conductor $3.394$
Analytic rank $1$
Dimension $4$
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(49,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.n (of order \(8\), degree \(4\), 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: \(3.39364208590\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{2} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \cdots - 1) q^{3}+ \cdots + ( - 2 \zeta_{8}^{2} - \zeta_{8} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{8}^{2} + \zeta_{8} - 1) q^{2} + (\zeta_{8}^{3} + \zeta_{8}^{2} + \cdots - 1) q^{3}+ \cdots + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}^{2} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{6} - 4 q^{7} + 4 q^{8} - 8 q^{9} - 4 q^{11} + 12 q^{12} - 4 q^{14} - 12 q^{16} - 12 q^{17} - 8 q^{19} - 4 q^{22} - 12 q^{23} - 12 q^{24} - 4 q^{26} + 8 q^{27} + 20 q^{28} + 4 q^{29} - 12 q^{31} + 4 q^{32} + 20 q^{34} + 20 q^{37} + 8 q^{38} - 4 q^{41} + 8 q^{42} - 8 q^{43} + 20 q^{44} + 20 q^{46} - 32 q^{47} + 12 q^{48} - 8 q^{49} + 28 q^{51} + 4 q^{53} + 8 q^{54} - 20 q^{56} - 8 q^{57} - 4 q^{58} + 36 q^{62} + 4 q^{63} + 8 q^{66} - 32 q^{68} + 32 q^{69} + 20 q^{71} - 4 q^{72} - 20 q^{74} + 8 q^{76} - 8 q^{77} + 4 q^{79} - 8 q^{82} - 16 q^{83} - 32 q^{84} + 8 q^{86} + 16 q^{87} - 20 q^{88} - 28 q^{92} + 24 q^{93} + 24 q^{94} - 20 q^{96} + 4 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−1.70711 + 1.70711i −1.00000 + 0.414214i 3.82843i 0 1.00000 2.41421i −1.00000 + 2.41421i 3.12132 + 3.12132i −1.29289 + 1.29289i 0
274.1 −0.292893 + 0.292893i −1.00000 2.41421i 1.82843i 0 1.00000 + 0.414214i −1.00000 0.414214i −1.12132 1.12132i −2.70711 + 2.70711i 0
349.1 −0.292893 0.292893i −1.00000 + 2.41421i 1.82843i 0 1.00000 0.414214i −1.00000 + 0.414214i −1.12132 + 1.12132i −2.70711 2.70711i 0
399.1 −1.70711 1.70711i −1.00000 0.414214i 3.82843i 0 1.00000 + 2.41421i −1.00000 2.41421i 3.12132 3.12132i −1.29289 1.29289i 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
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.2.n.a 4
5.b even 2 1 425.2.n.b 4
5.c odd 4 1 17.2.d.a 4
5.c odd 4 1 425.2.m.a 4
15.e even 4 1 153.2.l.c 4
17.d even 8 1 425.2.n.b 4
20.e even 4 1 272.2.v.d 4
35.f even 4 1 833.2.l.a 4
35.k even 12 2 833.2.v.a 8
35.l odd 12 2 833.2.v.b 8
85.f odd 4 1 289.2.d.c 4
85.g odd 4 1 289.2.d.a 4
85.i odd 4 1 289.2.d.b 4
85.k odd 8 1 289.2.d.b 4
85.k odd 8 1 289.2.d.c 4
85.k odd 8 1 425.2.m.a 4
85.m even 8 1 inner 425.2.n.a 4
85.n odd 8 1 17.2.d.a 4
85.n odd 8 1 289.2.d.a 4
85.o even 16 4 289.2.c.c 8
85.o even 16 2 7225.2.a.u 4
85.r even 16 2 289.2.a.f 4
85.r even 16 2 289.2.b.b 4
255.v even 8 1 153.2.l.c 4
255.bj odd 16 2 2601.2.a.bb 4
340.w even 8 1 272.2.v.d 4
340.bj odd 16 2 4624.2.a.bp 4
595.bj even 8 1 833.2.l.a 4
595.ce odd 24 2 833.2.v.b 8
595.cf even 24 2 833.2.v.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 5.c odd 4 1
17.2.d.a 4 85.n odd 8 1
153.2.l.c 4 15.e even 4 1
153.2.l.c 4 255.v even 8 1
272.2.v.d 4 20.e even 4 1
272.2.v.d 4 340.w even 8 1
289.2.a.f 4 85.r even 16 2
289.2.b.b 4 85.r even 16 2
289.2.c.c 8 85.o even 16 4
289.2.d.a 4 85.g odd 4 1
289.2.d.a 4 85.n odd 8 1
289.2.d.b 4 85.i odd 4 1
289.2.d.b 4 85.k odd 8 1
289.2.d.c 4 85.f odd 4 1
289.2.d.c 4 85.k odd 8 1
425.2.m.a 4 5.c odd 4 1
425.2.m.a 4 85.k odd 8 1
425.2.n.a 4 1.a even 1 1 trivial
425.2.n.a 4 85.m even 8 1 inner
425.2.n.b 4 5.b even 2 1
425.2.n.b 4 17.d even 8 1
833.2.l.a 4 35.f even 4 1
833.2.l.a 4 595.bj even 8 1
833.2.v.a 8 35.k even 12 2
833.2.v.a 8 595.cf even 24 2
833.2.v.b 8 35.l odd 12 2
833.2.v.b 8 595.ce odd 24 2
2601.2.a.bb 4 255.bj odd 16 2
4624.2.a.bp 4 340.bj odd 16 2
7225.2.a.u 4 85.o even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots + 392 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 648 \) Copy content Toggle raw display
$37$ \( T^{4} - 20 T^{3} + \cdots + 1250 \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 16 T + 56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 1296 \) Copy content Toggle raw display
$61$ \( T^{4} + 50 T^{2} + \cdots + 1250 \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + \cdots + 5000 \) Copy content Toggle raw display
$73$ \( T^{4} + 98 T^{2} + \cdots + 4802 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} + \cdots + 392 \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{4} + 132T^{2} + 3844 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 4418 \) Copy content Toggle raw display
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