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Properties

Label 425.3.t.d
Level $425$
Weight $3$
Character orbit 425.t
Analytic conductor $11.580$
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] = [425,3,Mod(24,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 425.t (of order \(16\), degree \(8\), 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: \(11.5804112353\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) 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: $\mathrm{SU}(2)[C_{16}]$

$q$-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 \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{16}^{6} + \zeta_{16}^{3} + 1) q^{2} + (\zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 1) q^{3}+ \cdots + ( - 2 \zeta_{16}^{7} + 7 \zeta_{16}^{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{16}^{6} + \zeta_{16}^{3} + 1) q^{2} + (\zeta_{16}^{7} + 2 \zeta_{16}^{6} + \cdots + 1) q^{3}+ \cdots + ( - 18 \zeta_{16}^{7} + 15 \zeta_{16}^{6} + \cdots - 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{9} + 40 q^{11} + 32 q^{12} - 16 q^{14} + 64 q^{17} + 32 q^{19} - 64 q^{21} - 8 q^{22} + 32 q^{23} - 24 q^{24} - 112 q^{27} + 8 q^{28} - 24 q^{29} + 32 q^{31} + 16 q^{32} - 16 q^{33} - 64 q^{34} - 104 q^{36} - 56 q^{37} + 8 q^{38} + 72 q^{39} - 88 q^{42} + 48 q^{43} + 96 q^{44} - 80 q^{47} - 16 q^{48} - 8 q^{49} - 176 q^{51} - 88 q^{53} - 208 q^{54} + 72 q^{56} + 168 q^{57} + 96 q^{58} - 8 q^{59} + 264 q^{61} + 96 q^{62} + 16 q^{63} + 120 q^{64} + 8 q^{66} + 32 q^{67} - 120 q^{68} + 208 q^{69} + 32 q^{71} - 24 q^{72} + 240 q^{73} - 176 q^{74} - 80 q^{76} - 120 q^{77} + 96 q^{79} - 224 q^{81} + 352 q^{82} - 504 q^{83} + 288 q^{86} + 216 q^{87} - 24 q^{88} - 288 q^{89} - 24 q^{91} + 72 q^{92} + 248 q^{93} + 8 q^{94} + 328 q^{96} - 280 q^{97} - 16 q^{98} - 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(\zeta_{16}\)

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
−0.382683 − 0.923880i
0.923880 − 0.382683i
−0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 − 0.923880i
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
2.63099 + 1.08979i 4.32023 + 2.88669i 2.90602 + 2.90602i 0 8.22059 + 12.3030i −3.23784 − 0.644047i 0.119585 + 0.288703i 6.88730 + 16.6274i 0
74.1 0.675577 − 1.63099i −0.789499 − 3.96908i 0.624715 + 0.624715i 0 −7.00688 − 1.39376i 3.40262 − 2.27356i 7.96489 − 3.29916i −6.81537 + 2.82302i 0
99.1 −0.0897902 − 0.216773i 1.37529 + 0.273561i 2.78950 − 2.78950i 0 −0.0641865 − 0.322688i −3.98841 + 5.96908i −1.72225 − 0.713379i −6.49834 − 2.69170i 0
124.1 2.63099 − 1.08979i 4.32023 − 2.88669i 2.90602 − 2.90602i 0 8.22059 − 12.3030i −3.23784 + 0.644047i 0.119585 − 0.288703i 6.88730 − 16.6274i 0
199.1 0.783227 − 0.324423i −0.906019 − 1.35595i −2.32023 + 2.32023i 0 −1.14952 − 0.768086i −0.176373 − 0.886687i −2.36223 + 5.70292i 2.42641 − 5.85788i 0
224.1 0.675577 + 1.63099i −0.789499 + 3.96908i 0.624715 − 0.624715i 0 −7.00688 + 1.39376i 3.40262 + 2.27356i 7.96489 + 3.29916i −6.81537 − 2.82302i 0
249.1 −0.0897902 + 0.216773i 1.37529 − 0.273561i 2.78950 + 2.78950i 0 −0.0641865 + 0.322688i −3.98841 − 5.96908i −1.72225 + 0.713379i −6.49834 + 2.69170i 0
299.1 0.783227 + 0.324423i −0.906019 + 1.35595i −2.32023 − 2.32023i 0 −1.14952 + 0.768086i −0.176373 + 0.886687i −2.36223 − 5.70292i 2.42641 + 5.85788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.p odd 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 425.3.t.d 8
5.b even 2 1 425.3.t.b 8
5.c odd 4 1 17.3.e.b ✓ 8
5.c odd 4 1 425.3.u.a 8
15.e even 4 1 153.3.p.a 8
17.e odd 16 1 425.3.t.b 8
20.e even 4 1 272.3.bh.b 8
85.f odd 4 1 289.3.e.f 8
85.g odd 4 1 289.3.e.g 8
85.i odd 4 1 289.3.e.h 8
85.k odd 8 1 289.3.e.a 8
85.k odd 8 1 289.3.e.e 8
85.n odd 8 1 289.3.e.j 8
85.n odd 8 1 289.3.e.n 8
85.o even 16 1 17.3.e.b ✓ 8
85.o even 16 1 289.3.e.f 8
85.o even 16 1 289.3.e.g 8
85.o even 16 1 289.3.e.h 8
85.p odd 16 1 inner 425.3.t.d 8
85.r even 16 1 289.3.e.a 8
85.r even 16 1 289.3.e.e 8
85.r even 16 1 289.3.e.j 8
85.r even 16 1 289.3.e.n 8
85.r even 16 1 425.3.u.a 8
255.bc odd 16 1 153.3.p.a 8
340.bc odd 16 1 272.3.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.3.e.b ✓ 8 5.c odd 4 1
17.3.e.b ✓ 8 85.o even 16 1
153.3.p.a 8 15.e even 4 1
153.3.p.a 8 255.bc odd 16 1
272.3.bh.b 8 20.e even 4 1
272.3.bh.b 8 340.bc odd 16 1
289.3.e.a 8 85.k odd 8 1
289.3.e.a 8 85.r even 16 1
289.3.e.e 8 85.k odd 8 1
289.3.e.e 8 85.r even 16 1
289.3.e.f 8 85.f odd 4 1
289.3.e.f 8 85.o even 16 1
289.3.e.g 8 85.g odd 4 1
289.3.e.g 8 85.o even 16 1
289.3.e.h 8 85.i odd 4 1
289.3.e.h 8 85.o even 16 1
289.3.e.j 8 85.n odd 8 1
289.3.e.j 8 85.r even 16 1
289.3.e.n 8 85.n odd 8 1
289.3.e.n 8 85.r even 16 1
425.3.t.b 8 5.b even 2 1
425.3.t.b 8 17.e odd 16 1
425.3.t.d 8 1.a even 1 1 trivial
425.3.t.d 8 85.p odd 16 1 inner
425.3.u.a 8 5.c odd 4 1
425.3.u.a 8 85.r even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 8T_{2}^{7} + 28T_{2}^{6} - 56T_{2}^{5} + 72T_{2}^{4} - 48T_{2}^{3} + 12T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 2312 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 7688 \) Copy content Toggle raw display
$11$ \( T^{8} - 40 T^{7} + \cdots + 2221832 \) Copy content Toggle raw display
$13$ \( T^{8} - 784 T^{5} + \cdots + 16273156 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 34090452496 \) Copy content Toggle raw display
$23$ \( T^{8} - 32 T^{7} + \cdots + 564614408 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 55846825218 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 182700453128 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 368317129538 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 126445152962 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 6626611216 \) Copy content Toggle raw display
$47$ \( T^{8} + 80 T^{7} + \cdots + 298321984 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 41458660996 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 2347596304 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22880163635522 \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots + 940168)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 53847249369608 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18825456624578 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 376288804594952 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 128947789048324 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 18\!\cdots\!42 \) Copy content Toggle raw display
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