Properties

Label 975.2.i.g
Level $975$
Weight $2$
Character orbit 975.i
Analytic conductor $7.785$
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] = [975,2,Mod(451,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.i (of order \(3\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} + 3 q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 \zeta_{6} q^{7} + 3 q^{8} - \zeta_{6} q^{9} + (5 \zeta_{6} - 5) q^{11} - q^{12} + (3 \zeta_{6} - 4) q^{13} + 4 q^{14} + ( - \zeta_{6} + 1) q^{16} - 2 \zeta_{6} q^{17} - q^{18} - 6 \zeta_{6} q^{19} - 4 q^{21} + 5 \zeta_{6} q^{22} + (3 \zeta_{6} - 3) q^{23} + (3 \zeta_{6} - 3) q^{24} + (4 \zeta_{6} - 1) q^{26} + q^{27} + (4 \zeta_{6} - 4) q^{28} + ( - 10 \zeta_{6} + 10) q^{29} - 10 q^{31} + 5 \zeta_{6} q^{32} - 5 \zeta_{6} q^{33} - 2 q^{34} + ( - \zeta_{6} + 1) q^{36} + (\zeta_{6} - 1) q^{37} - 6 q^{38} + ( - 4 \zeta_{6} + 1) q^{39} + ( - 6 \zeta_{6} + 6) q^{41} + (4 \zeta_{6} - 4) q^{42} + 8 \zeta_{6} q^{43} - 5 q^{44} + 3 \zeta_{6} q^{46} + 4 q^{47} + \zeta_{6} q^{48} + (9 \zeta_{6} - 9) q^{49} + 2 q^{51} + ( - \zeta_{6} - 3) q^{52} + 8 q^{53} + ( - \zeta_{6} + 1) q^{54} + 12 \zeta_{6} q^{56} + 6 q^{57} - 10 \zeta_{6} q^{58} + 7 \zeta_{6} q^{61} + (10 \zeta_{6} - 10) q^{62} + ( - 4 \zeta_{6} + 4) q^{63} + 7 q^{64} - 5 q^{66} + (8 \zeta_{6} - 8) q^{67} + ( - 2 \zeta_{6} + 2) q^{68} - 3 \zeta_{6} q^{69} + 15 \zeta_{6} q^{71} - 3 \zeta_{6} q^{72} + 9 q^{73} + \zeta_{6} q^{74} + ( - 6 \zeta_{6} + 6) q^{76} - 20 q^{77} + ( - \zeta_{6} - 3) q^{78} - 6 q^{79} + (\zeta_{6} - 1) q^{81} - 6 \zeta_{6} q^{82} + 5 q^{83} - 4 \zeta_{6} q^{84} + 8 q^{86} + 10 \zeta_{6} q^{87} + (15 \zeta_{6} - 15) q^{88} + (2 \zeta_{6} - 2) q^{89} + ( - 4 \zeta_{6} - 12) q^{91} - 3 q^{92} + ( - 10 \zeta_{6} + 10) q^{93} + ( - 4 \zeta_{6} + 4) q^{94} - 5 q^{96} + 11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{98} + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} + 6 q^{8} - q^{9} - 5 q^{11} - 2 q^{12} - 5 q^{13} + 8 q^{14} + q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} - 8 q^{21} + 5 q^{22} - 3 q^{23} - 3 q^{24} + 2 q^{26} + 2 q^{27} - 4 q^{28} + 10 q^{29} - 20 q^{31} + 5 q^{32} - 5 q^{33} - 4 q^{34} + q^{36} - q^{37} - 12 q^{38} - 2 q^{39} + 6 q^{41} - 4 q^{42} + 8 q^{43} - 10 q^{44} + 3 q^{46} + 8 q^{47} + q^{48} - 9 q^{49} + 4 q^{51} - 7 q^{52} + 16 q^{53} + q^{54} + 12 q^{56} + 12 q^{57} - 10 q^{58} + 7 q^{61} - 10 q^{62} + 4 q^{63} + 14 q^{64} - 10 q^{66} - 8 q^{67} + 2 q^{68} - 3 q^{69} + 15 q^{71} - 3 q^{72} + 18 q^{73} + q^{74} + 6 q^{76} - 40 q^{77} - 7 q^{78} - 12 q^{79} - q^{81} - 6 q^{82} + 10 q^{83} - 4 q^{84} + 16 q^{86} + 10 q^{87} - 15 q^{88} - 2 q^{89} - 28 q^{91} - 6 q^{92} + 10 q^{93} + 4 q^{94} - 10 q^{96} + 11 q^{97} + 9 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 2.00000 + 3.46410i 3.00000 −0.500000 0.866025i 0
601.1 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 0 0.500000 0.866025i 2.00000 3.46410i 3.00000 −0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.i.g yes 2
5.b even 2 1 975.2.i.b 2
5.c odd 4 2 975.2.bb.c 4
13.c even 3 1 inner 975.2.i.g yes 2
65.n even 6 1 975.2.i.b 2
65.q odd 12 2 975.2.bb.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.i.b 2 5.b even 2 1
975.2.i.b 2 65.n even 6 1
975.2.i.g yes 2 1.a even 1 1 trivial
975.2.i.g yes 2 13.c even 3 1 inner
975.2.bb.c 4 5.c odd 4 2
975.2.bb.c 4 65.q odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( (T - 4)^{2} \) Copy content Toggle raw display
$53$ \( (T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( (T - 9)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6)^{2} \) Copy content Toggle raw display
$83$ \( (T - 5)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
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