Properties

Label 975.2.i.j
Level $975$
Weight $2$
Character orbit 975.i
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 195)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2}) q^{7} + ( - 2 \beta_{3} + 6) q^{8} + (\beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + \beta_1 q^{3} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{6} + ( - \beta_{3} + \beta_{2}) q^{7} + ( - 2 \beta_{3} + 6) q^{8} + (\beta_1 - 1) q^{9} + 2 \beta_1 q^{11} + (2 \beta_{3} - 2) q^{12} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{13} + (\beta_{3} - 3) q^{14} + (4 \beta_{2} - 8 \beta_1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{3} + 1) q^{18} + ( - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{19} - \beta_{3} q^{21} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{22} + 2 \beta_1 q^{23} + ( - 2 \beta_{2} + 6 \beta_1) q^{24} + ( - \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 1) q^{26} - q^{27} + ( - 2 \beta_{2} + 6 \beta_1) q^{28} + ( - \beta_{2} + 5 \beta_1) q^{29} + (2 \beta_{3} + 1) q^{31} + (8 \beta_{3} - 8 \beta_{2} + 8 \beta_1 - 8) q^{32} + (2 \beta_1 - 2) q^{33} + ( - 2 \beta_{3} + 4) q^{34} + (2 \beta_{2} - 2 \beta_1) q^{36} - 6 \beta_{2} q^{37} + (6 \beta_{3} - 10) q^{38} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{39} + (\beta_{2} + 9 \beta_1) q^{41} + (\beta_{2} - 3 \beta_1) q^{42} + ( - 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 4) q^{43} + (4 \beta_{3} - 4) q^{44} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{46} + ( - 3 \beta_{3} + 5) q^{47} + ( - 4 \beta_{3} + 4 \beta_{2} - 8 \beta_1 + 8) q^{48} + 4 \beta_1 q^{49} + (\beta_{3} - 1) q^{51} + ( - 4 \beta_{3} + 6 \beta_{2} - 10 \beta_1 + 12) q^{52} + (2 \beta_{3} + 6) q^{53} + ( - \beta_{2} + \beta_1) q^{54} + ( - 6 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 6) q^{56} + ( - 2 \beta_{3} + 4) q^{57} + ( - 6 \beta_{3} + 6 \beta_{2} - 8 \beta_1 + 8) q^{58} + (5 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{59} + ( - 6 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 5) q^{61} + ( - \beta_{2} + 5 \beta_1) q^{62} - \beta_{2} q^{63} + ( - 8 \beta_{3} + 16) q^{64} + ( - 2 \beta_{3} + 2) q^{66} + ( - 3 \beta_{2} - 4 \beta_1) q^{67} + (4 \beta_{2} - 8 \beta_1) q^{68} + (2 \beta_1 - 2) q^{69} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 3) q^{71} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{72} - \beta_{3} q^{73} + ( - 6 \beta_{3} + 6 \beta_{2} - 18 \beta_1 + 18) q^{74} + ( - 12 \beta_{2} + 20 \beta_1) q^{76} - 2 \beta_{3} q^{77} + (2 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 6) q^{78} - 11 q^{79} - \beta_1 q^{81} + ( - 8 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 6) q^{82} + ( - 4 \beta_{3} + 4) q^{83} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{84} + ( - \beta_{3} - 5) q^{86} + (\beta_{3} - \beta_{2} + 5 \beta_1 - 5) q^{87} + ( - 4 \beta_{2} + 12 \beta_1) q^{88} + (\beta_{2} - 7 \beta_1) q^{89} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{91} + (4 \beta_{3} - 4) q^{92} + (2 \beta_{2} + \beta_1) q^{93} + (8 \beta_{2} - 14 \beta_1) q^{94} + (8 \beta_{3} - 8) q^{96} + (3 \beta_{3} - 3 \beta_{2}) q^{97} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 4 q^{4} + 2 q^{6} + 24 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 2 q^{3} - 4 q^{4} + 2 q^{6} + 24 q^{8} - 2 q^{9} + 4 q^{11} - 8 q^{12} - 4 q^{13} - 12 q^{14} - 16 q^{16} - 2 q^{17} + 4 q^{18} + 8 q^{19} + 4 q^{22} + 4 q^{23} + 12 q^{24} + 8 q^{26} - 4 q^{27} + 12 q^{28} + 10 q^{29} + 4 q^{31} - 16 q^{32} - 4 q^{33} + 16 q^{34} - 4 q^{36} - 40 q^{38} - 8 q^{39} + 18 q^{41} - 6 q^{42} - 8 q^{43} - 16 q^{44} + 4 q^{46} + 20 q^{47} + 16 q^{48} + 8 q^{49} - 4 q^{51} + 28 q^{52} + 24 q^{53} + 2 q^{54} + 12 q^{56} + 16 q^{57} + 16 q^{58} + 6 q^{59} + 10 q^{61} + 10 q^{62} + 64 q^{64} + 8 q^{66} - 8 q^{67} - 16 q^{68} - 4 q^{69} + 6 q^{71} - 12 q^{72} + 36 q^{74} + 40 q^{76} - 14 q^{78} - 44 q^{79} - 2 q^{81} + 12 q^{82} + 16 q^{83} - 12 q^{84} - 20 q^{86} - 10 q^{87} + 24 q^{88} - 14 q^{89} - 18 q^{91} - 16 q^{92} + 2 q^{93} - 28 q^{94} - 32 q^{96} + 8 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(1\) \(1\)

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
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.36603 + 2.36603i 0.500000 0.866025i −2.73205 4.73205i 0 1.36603 + 2.36603i 0.866025 + 1.50000i 9.46410 −0.500000 0.866025i 0
451.2 0.366025 0.633975i 0.500000 0.866025i 0.732051 + 1.26795i 0 −0.366025 0.633975i −0.866025 1.50000i 2.53590 −0.500000 0.866025i 0
601.1 −1.36603 2.36603i 0.500000 + 0.866025i −2.73205 + 4.73205i 0 1.36603 2.36603i 0.866025 1.50000i 9.46410 −0.500000 + 0.866025i 0
601.2 0.366025 + 0.633975i 0.500000 + 0.866025i 0.732051 1.26795i 0 −0.366025 + 0.633975i −0.866025 + 1.50000i 2.53590 −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.j 4
5.b even 2 1 195.2.i.c 4
5.c odd 4 1 975.2.bb.a 4
5.c odd 4 1 975.2.bb.h 4
13.c even 3 1 inner 975.2.i.j 4
15.d odd 2 1 585.2.j.c 4
65.l even 6 1 2535.2.a.r 2
65.n even 6 1 195.2.i.c 4
65.n even 6 1 2535.2.a.o 2
65.q odd 12 1 975.2.bb.a 4
65.q odd 12 1 975.2.bb.h 4
195.x odd 6 1 585.2.j.c 4
195.x odd 6 1 7605.2.a.bj 2
195.y odd 6 1 7605.2.a.z 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.c 4 5.b even 2 1
195.2.i.c 4 65.n even 6 1
585.2.j.c 4 15.d odd 2 1
585.2.j.c 4 195.x odd 6 1
975.2.i.j 4 1.a even 1 1 trivial
975.2.i.j 4 13.c even 3 1 inner
975.2.bb.a 4 5.c odd 4 1
975.2.bb.a 4 65.q odd 12 1
975.2.bb.h 4 5.c odd 4 1
975.2.bb.h 4 65.q odd 12 1
2535.2.a.o 2 65.n even 6 1
2535.2.a.r 2 65.l even 6 1
7605.2.a.z 2 195.y odd 6 1
7605.2.a.bj 2 195.x odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 3 T^{2} + 52 T + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 11)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$41$ \( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$47$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} + 102 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$61$ \( T^{4} - 10 T^{3} + 183 T^{2} + \cdots + 6889 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + 30 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T + 11)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 32)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 14 T^{3} + 150 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
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