Properties

Label 75.2.g.a
Level $75$
Weight $2$
Character orbit 75.g
Analytic conductor $0.599$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,2,Mod(16,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\), degree \(4\), 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: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots - \zeta_{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{2}+ \cdots + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - q^{3} + q^{4} + 5 q^{5} - q^{6} + 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - q^{3} + q^{4} + 5 q^{5} - q^{6} + 3 q^{8} - q^{9} - 5 q^{10} - 6 q^{11} + q^{12} - 2 q^{13} - 10 q^{14} + q^{16} - 8 q^{17} + 4 q^{18} - 4 q^{19} + 10 q^{21} + 4 q^{22} + 10 q^{23} - 12 q^{24} - 5 q^{25} + 18 q^{26} - q^{27} - 10 q^{28} - 8 q^{29} - 5 q^{30} + 20 q^{32} + 4 q^{33} + 7 q^{34} - 10 q^{35} + q^{36} - 15 q^{37} + 6 q^{38} - 2 q^{39} + 15 q^{40} + 10 q^{42} + 4 q^{43} - 4 q^{44} + 5 q^{45} - 10 q^{46} - 2 q^{47} + q^{48} + 52 q^{49} - 5 q^{50} + 2 q^{51} + 2 q^{52} - 5 q^{53} - q^{54} - 10 q^{55} - 30 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 5 q^{60} + 2 q^{61} - 10 q^{62} - 10 q^{63} - 7 q^{64} + 25 q^{65} - 6 q^{66} + 2 q^{67} - 2 q^{68} - 10 q^{69} - 10 q^{70} + 8 q^{71} + 3 q^{72} - 10 q^{73} + 10 q^{74} - 5 q^{75} + 4 q^{76} - 20 q^{77} - 7 q^{78} - 5 q^{80} - q^{81} - 10 q^{82} - 18 q^{83} + 10 q^{84} + 5 q^{85} + 14 q^{86} - 3 q^{87} + 18 q^{88} - 9 q^{89} + 5 q^{90} - 40 q^{91} + 10 q^{92} + 20 q^{93} - 2 q^{94} + 10 q^{95} - 5 q^{96} + 2 q^{97} - 13 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.809017 0.587785i
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
−0.809017 0.587785i 0.309017 + 0.951057i −0.309017 0.951057i 0.690983 2.12663i 0.309017 0.951057i 4.47214 −0.927051 + 2.85317i −0.809017 + 0.587785i −1.80902 + 1.31433i
31.1 0.309017 + 0.951057i −0.809017 + 0.587785i 0.809017 0.587785i 1.80902 + 1.31433i −0.809017 0.587785i −4.47214 2.42705 + 1.76336i 0.309017 0.951057i −0.690983 + 2.12663i
46.1 0.309017 0.951057i −0.809017 0.587785i 0.809017 + 0.587785i 1.80902 1.31433i −0.809017 + 0.587785i −4.47214 2.42705 1.76336i 0.309017 + 0.951057i −0.690983 2.12663i
61.1 −0.809017 + 0.587785i 0.309017 0.951057i −0.309017 + 0.951057i 0.690983 + 2.12663i 0.309017 + 0.951057i 4.47214 −0.927051 2.85317i −0.809017 0.587785i −1.80902 1.31433i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.g.a 4
3.b odd 2 1 225.2.h.a 4
5.b even 2 1 375.2.g.a 4
5.c odd 4 2 375.2.i.a 8
25.d even 5 1 inner 75.2.g.a 4
25.d even 5 1 1875.2.a.d 2
25.e even 10 1 375.2.g.a 4
25.e even 10 1 1875.2.a.a 2
25.f odd 20 2 375.2.i.a 8
25.f odd 20 2 1875.2.b.b 4
75.h odd 10 1 5625.2.a.h 2
75.j odd 10 1 225.2.h.a 4
75.j odd 10 1 5625.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 1.a even 1 1 trivial
75.2.g.a 4 25.d even 5 1 inner
225.2.h.a 4 3.b odd 2 1
225.2.h.a 4 75.j odd 10 1
375.2.g.a 4 5.b even 2 1
375.2.g.a 4 25.e even 10 1
375.2.i.a 8 5.c odd 4 2
375.2.i.a 8 25.f odd 20 2
1875.2.a.a 2 25.e even 10 1
1875.2.a.d 2 25.d even 5 1
1875.2.b.b 4 25.f odd 20 2
5625.2.a.a 2 75.j odd 10 1
5625.2.a.h 2 75.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} + 40 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$37$ \( T^{4} + 15 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 18 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( T^{4} + 9 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + \cdots + 361 \) Copy content Toggle raw display
show more
show less