Properties

Label 975.2.o.a
Level $975$
Weight $2$
Character orbit 975.o
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(476,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("975.476");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.o (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{2} + (\zeta_{12}^{2} - 2) q^{3} + 4 \zeta_{12}^{3} q^{4} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{6} + ( - \zeta_{12}^{3} - 1) q^{7} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{8} + ( - 3 \zeta_{12}^{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{2} + (\zeta_{12}^{2} - 2) q^{3} + 4 \zeta_{12}^{3} q^{4} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{6} + ( - \zeta_{12}^{3} - 1) q^{7} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{8} + ( - 3 \zeta_{12}^{2} + 3) q^{9} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{11} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 3) q^{13} + (4 \zeta_{12}^{2} - 2) q^{14} - 4 q^{16} + (6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{18} + (4 \zeta_{12}^{3} - 4) q^{19} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{21} - 6 q^{22} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{23} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{24} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 10 \zeta_{12} - 1) q^{26} + (6 \zeta_{12}^{2} - 3) q^{27} + ( - 4 \zeta_{12}^{3} + 4) q^{28} + (2 \zeta_{12}^{2} - 1) q^{29} + (5 \zeta_{12}^{3} - 5) q^{31} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{33} + 12 \zeta_{12} q^{36} + (4 \zeta_{12}^{3} + 4) q^{37} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{38} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{39} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{41} - 6 \zeta_{12}^{2} q^{42} - 9 \zeta_{12}^{3} q^{43} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12} - 4) q^{44} + ( - 9 \zeta_{12}^{3} - 9) q^{46} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{47} + ( - 4 \zeta_{12}^{2} + 8) q^{48} - 5 \zeta_{12}^{3} q^{49} + ( - 12 \zeta_{12}^{3} - 8) q^{52} + (14 \zeta_{12}^{2} - 7) q^{53} + ( - 9 \zeta_{12}^{3} + 9) q^{54} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{56} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 8) q^{57} + ( - 3 \zeta_{12}^{3} + 3) q^{58} + ( - 6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 12 \zeta_{12} + 6) q^{59} + 3 q^{61} + ( - 10 \zeta_{12}^{3} + 20 \zeta_{12}) q^{62} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 6 \zeta_{12}^{2} + 12) q^{66} + ( - 5 \zeta_{12}^{3} + 5) q^{67} + (9 \zeta_{12}^{3} - 9 \zeta_{12}) q^{69} + (6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 12 \zeta_{12} + 6) q^{71} + ( - 12 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{72} + ( - \zeta_{12}^{3} - 1) q^{73} + ( - 16 \zeta_{12}^{2} + 8) q^{74} + ( - 16 \zeta_{12}^{3} - 16) q^{76} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{77} + (15 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 15 \zeta_{12}) q^{78} + 15 q^{79} - 9 \zeta_{12}^{2} q^{81} + 12 \zeta_{12}^{3} q^{82} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{83} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{84} + (9 \zeta_{12}^{3} + 18 \zeta_{12}^{2} - 18 \zeta_{12} - 9) q^{86} - 3 \zeta_{12}^{2} q^{87} - 12 \zeta_{12}^{3} q^{88} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{89} + (\zeta_{12}^{3} + 5) q^{91} + (24 \zeta_{12}^{2} - 12) q^{92} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 10) q^{93} - 24 q^{94} + (2 \zeta_{12}^{3} - 2) q^{97} + (5 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} - 5) q^{98} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} + 6 q^{6} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} + 6 q^{6} - 4 q^{7} + 6 q^{9} - 12 q^{13} - 16 q^{16} - 18 q^{18} - 16 q^{19} + 6 q^{21} - 24 q^{22} + 12 q^{24} + 16 q^{28} - 20 q^{31} + 6 q^{33} + 16 q^{37} + 18 q^{39} - 12 q^{42} - 36 q^{46} + 24 q^{48} - 32 q^{52} + 36 q^{54} + 24 q^{57} + 12 q^{58} + 12 q^{61} - 6 q^{63} + 36 q^{66} + 20 q^{67} - 36 q^{72} - 4 q^{73} - 64 q^{76} - 6 q^{78} + 60 q^{79} - 18 q^{81} - 24 q^{84} - 6 q^{87} + 20 q^{91} + 30 q^{93} - 96 q^{94} - 8 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\zeta_{12}^{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
476.1
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.73205 + 1.73205i −1.50000 0.866025i 4.00000i 0 4.09808 1.09808i −1.00000 + 1.00000i 3.46410 + 3.46410i 1.50000 + 2.59808i 0
476.2 1.73205 1.73205i −1.50000 + 0.866025i 4.00000i 0 −1.09808 + 4.09808i −1.00000 + 1.00000i −3.46410 3.46410i 1.50000 2.59808i 0
551.1 −1.73205 1.73205i −1.50000 + 0.866025i 4.00000i 0 4.09808 + 1.09808i −1.00000 1.00000i 3.46410 3.46410i 1.50000 2.59808i 0
551.2 1.73205 + 1.73205i −1.50000 0.866025i 4.00000i 0 −1.09808 4.09808i −1.00000 1.00000i −3.46410 + 3.46410i 1.50000 + 2.59808i 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
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.o.a 4
3.b odd 2 1 inner 975.2.o.a 4
5.b even 2 1 975.2.o.k yes 4
5.c odd 4 1 975.2.n.k 4
5.c odd 4 1 975.2.n.l 4
13.d odd 4 1 inner 975.2.o.a 4
15.d odd 2 1 975.2.o.k yes 4
15.e even 4 1 975.2.n.k 4
15.e even 4 1 975.2.n.l 4
39.f even 4 1 inner 975.2.o.a 4
65.f even 4 1 975.2.n.k 4
65.g odd 4 1 975.2.o.k yes 4
65.k even 4 1 975.2.n.l 4
195.j odd 4 1 975.2.n.l 4
195.n even 4 1 975.2.o.k yes 4
195.u odd 4 1 975.2.n.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.n.k 4 5.c odd 4 1
975.2.n.k 4 15.e even 4 1
975.2.n.k 4 65.f even 4 1
975.2.n.k 4 195.u odd 4 1
975.2.n.l 4 5.c odd 4 1
975.2.n.l 4 15.e even 4 1
975.2.n.l 4 65.k even 4 1
975.2.n.l 4 195.j odd 4 1
975.2.o.a 4 1.a even 1 1 trivial
975.2.o.a 4 3.b odd 2 1 inner
975.2.o.a 4 13.d odd 4 1 inner
975.2.o.a 4 39.f even 4 1 inner
975.2.o.k yes 4 5.b even 2 1
975.2.o.k yes 4 15.d odd 2 1
975.2.o.k yes 4 65.g odd 4 1
975.2.o.k yes 4 195.n even 4 1

Hecke kernels

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

\( T_{2}^{4} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 36 \) Copy content Toggle raw display
\( T_{37}^{2} - 8T_{37} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 36 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 576 \) Copy content Toggle raw display
$43$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 9216 \) Copy content Toggle raw display
$53$ \( (T^{2} + 147)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 46656 \) Copy content Toggle raw display
$61$ \( (T - 3)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 46656 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 15)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 576 \) Copy content Toggle raw display
$89$ \( T^{4} + 36 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
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