Properties

Label 975.2.b.b
Level $975$
Weight $2$
Character orbit 975.b
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(376,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("975.376");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.b (of order \(2\), degree \(1\), 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: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
376.1
1.00000i
1.00000i
1.00000i −1.00000 1.00000 0 1.00000i 2.00000i 3.00000i 1.00000 0
376.2 1.00000i −1.00000 1.00000 0 1.00000i 2.00000i 3.00000i 1.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.b.b 2
5.b even 2 1 195.2.b.b 2
5.c odd 4 1 975.2.h.a 2
5.c odd 4 1 975.2.h.d 2
13.b even 2 1 inner 975.2.b.b 2
15.d odd 2 1 585.2.b.a 2
20.d odd 2 1 3120.2.g.a 2
65.d even 2 1 195.2.b.b 2
65.g odd 4 1 2535.2.a.e 1
65.g odd 4 1 2535.2.a.l 1
65.h odd 4 1 975.2.h.a 2
65.h odd 4 1 975.2.h.d 2
195.e odd 2 1 585.2.b.a 2
195.n even 4 1 7605.2.a.d 1
195.n even 4 1 7605.2.a.p 1
260.g odd 2 1 3120.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.b 2 5.b even 2 1
195.2.b.b 2 65.d even 2 1
585.2.b.a 2 15.d odd 2 1
585.2.b.a 2 195.e odd 2 1
975.2.b.b 2 1.a even 1 1 trivial
975.2.b.b 2 13.b even 2 1 inner
975.2.h.a 2 5.c odd 4 1
975.2.h.a 2 65.h odd 4 1
975.2.h.d 2 5.c odd 4 1
975.2.h.d 2 65.h odd 4 1
2535.2.a.e 1 65.g odd 4 1
2535.2.a.l 1 65.g odd 4 1
3120.2.g.a 2 20.d odd 2 1
3120.2.g.a 2 260.g odd 2 1
7605.2.a.d 1 195.n even 4 1
7605.2.a.p 1 195.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 \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 13 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 36 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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