Properties

Label 975.2.o.g
Level $975$
Weight $2$
Character orbit 975.o
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(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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + 3 \beta_{2} q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} - \beta_{3} q^{8} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + 3 \beta_{2} q^{6} + (\beta_{2} + \beta_1 + 1) q^{7} - \beta_{3} q^{8} + 3 \beta_{2} q^{9} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{11} + \beta_{3} q^{12} + ( - 2 \beta_{2} - 3) q^{13} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{14} + 5 q^{16} - 3 q^{17} + 3 \beta_{3} q^{18} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{19} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{21} + (3 \beta_{3} - 3 \beta_1 + 3) q^{22} + 6 q^{23} + 3 q^{24} + ( - 2 \beta_{3} - 3 \beta_1) q^{26} + 3 \beta_{3} q^{27} + (\beta_{3} + \beta_{2} - 1) q^{28} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{29} + ( - 5 \beta_{3} - \beta_{2} + 1) q^{31} + 3 \beta_1 q^{32} + (3 \beta_{3} - 3 \beta_1 + 3) q^{33} - 3 \beta_1 q^{34} - 3 q^{36} + (2 \beta_{2} - 2 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{38} + ( - 2 \beta_{3} - 3 \beta_1) q^{39} + (3 \beta_{2} - 2 \beta_1 + 3) q^{41} + (3 \beta_{3} + 3 \beta_{2} - 3) q^{42} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{43} + ( - 3 \beta_{2} + \beta_1 - 3) q^{44} + 6 \beta_1 q^{46} + (\beta_{3} + 3 \beta_{2} - 3) q^{47} + 5 \beta_1 q^{48} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{49} - 3 \beta_1 q^{51} + ( - 3 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{53} - 9 q^{54} + ( - \beta_{3} + \beta_1 + 3) q^{56} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{57} + (3 \beta_{3} + 6 \beta_{2} - 6) q^{58} + (3 \beta_{3} - 3 \beta_{2} + 3) q^{59} + (4 \beta_{3} - 4 \beta_1 + 3) q^{61} + ( - \beta_{3} + \beta_1 + 15) q^{62} + (3 \beta_{3} + 3 \beta_{2} - 3) q^{63} - \beta_{2} q^{64} + ( - 9 \beta_{2} + 3 \beta_1 - 9) q^{66} + (3 \beta_{3} - \beta_{2} + 1) q^{67} - 3 \beta_{2} q^{68} + 6 \beta_1 q^{69} + ( - 6 \beta_{2} - 6) q^{71} + 3 \beta_1 q^{72} + (4 \beta_{2} - 2 \beta_1 + 4) q^{73} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{74} + (2 \beta_{2} - 2 \beta_1 + 2) q^{76} + (2 \beta_{3} - 2 \beta_1 - 3) q^{77} + ( - 9 \beta_{2} + 6) q^{78} + ( - \beta_{3} + \beta_1 + 6) q^{79} - 9 q^{81} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1) q^{82} + ( - 3 \beta_{2} - \beta_1 - 3) q^{83} + (\beta_{3} - \beta_1 - 3) q^{84} + ( - 6 \beta_{3} - 6 \beta_{2} + 6) q^{86} + (3 \beta_{3} + 6 \beta_{2} - 6) q^{87} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{88} + ( - 2 \beta_{3} - 3 \beta_{2} + 3) q^{89} + ( - 2 \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 1) q^{91} + 6 \beta_{2} q^{92} + ( - \beta_{3} + \beta_1 + 15) q^{93} + (3 \beta_{3} - 3 \beta_1 - 3) q^{94} + 9 \beta_{2} q^{96} + ( - 2 \beta_{3} + 4 \beta_{2} - 4) q^{97} + ( - 2 \beta_{3} + 6 \beta_{2} - 6) q^{98} + ( - 9 \beta_{2} + 3 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 12 q^{11} - 12 q^{13} + 20 q^{16} - 12 q^{17} + 8 q^{19} + 12 q^{22} + 24 q^{23} + 12 q^{24} - 4 q^{28} + 4 q^{31} + 12 q^{33} - 12 q^{36} + 8 q^{37} - 24 q^{38} + 12 q^{41} - 12 q^{42} - 12 q^{44} - 12 q^{47} + 8 q^{52} - 36 q^{54} + 12 q^{56} - 24 q^{57} - 24 q^{58} + 12 q^{59} + 12 q^{61} + 60 q^{62} - 12 q^{63} - 36 q^{66} + 4 q^{67} - 24 q^{71} + 16 q^{73} + 8 q^{76} - 12 q^{77} + 24 q^{78} + 24 q^{79} - 36 q^{81} - 12 q^{83} - 12 q^{84} + 24 q^{86} - 24 q^{87} + 12 q^{89} - 4 q^{91} + 60 q^{93} - 12 q^{94} - 16 q^{97} - 24 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\)

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
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i −1.22474 + 1.22474i 1.00000i 0 3.00000i −0.224745 + 0.224745i −1.22474 1.22474i 3.00000i 0
476.2 1.22474 1.22474i 1.22474 1.22474i 1.00000i 0 3.00000i 2.22474 2.22474i 1.22474 + 1.22474i 3.00000i 0
551.1 −1.22474 1.22474i −1.22474 1.22474i 1.00000i 0 3.00000i −0.224745 0.224745i −1.22474 + 1.22474i 3.00000i 0
551.2 1.22474 + 1.22474i 1.22474 + 1.22474i 1.00000i 0 3.00000i 2.22474 + 2.22474i 1.22474 1.22474i 3.00000i 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
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.g yes 4
3.b odd 2 1 975.2.o.d yes 4
5.b even 2 1 975.2.o.f yes 4
5.c odd 4 1 975.2.n.f 4
5.c odd 4 1 975.2.n.i 4
13.d odd 4 1 975.2.o.d yes 4
15.d odd 2 1 975.2.o.c 4
15.e even 4 1 975.2.n.a 4
15.e even 4 1 975.2.n.b 4
39.f even 4 1 inner 975.2.o.g yes 4
65.f even 4 1 975.2.n.b 4
65.g odd 4 1 975.2.o.c 4
65.k even 4 1 975.2.n.a 4
195.j odd 4 1 975.2.n.f 4
195.n even 4 1 975.2.o.f yes 4
195.u odd 4 1 975.2.n.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.n.a 4 15.e even 4 1
975.2.n.a 4 65.k even 4 1
975.2.n.b 4 15.e even 4 1
975.2.n.b 4 65.f even 4 1
975.2.n.f 4 5.c odd 4 1
975.2.n.f 4 195.j odd 4 1
975.2.n.i 4 5.c odd 4 1
975.2.n.i 4 195.u odd 4 1
975.2.o.c 4 15.d odd 2 1
975.2.o.c 4 65.g odd 4 1
975.2.o.d yes 4 3.b odd 2 1
975.2.o.d yes 4 13.d odd 4 1
975.2.o.f yes 4 5.b even 2 1
975.2.o.f yes 4 195.n even 4 1
975.2.o.g yes 4 1.a even 1 1 trivial
975.2.o.g yes 4 39.f even 4 1 inner

Hecke kernels

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

\( T_{2}^{4} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} + 8T_{7}^{2} + 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 12T_{11}^{3} + 72T_{11}^{2} + 180T_{11} + 225 \) Copy content Toggle raw display
\( T_{37}^{4} - 8T_{37}^{3} + 32T_{37}^{2} + 32T_{37} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 8 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$23$ \( (T - 6)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 66T^{2} + 225 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 8 T^{2} + 292 T + 5329 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 72 T^{2} - 72 T + 36 \) Copy content Toggle raw display
$43$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$53$ \( T^{4} + 66T^{2} + 225 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 72 T^{2} + 108 T + 81 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T - 87)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + 8 T^{2} + 100 T + 625 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 225 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 72 T^{2} - 72 T + 36 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 400 \) Copy content Toggle raw display
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