Properties

Label 975.2.n.c
Level $975$
Weight $2$
Character orbit 975.n
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(749,975)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.n (of order \(4\), degree \(2\), not 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: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 1.41421 1.00000i 1.00000i 0 −1.70711 0.292893i −1.00000 1.00000i −2.12132 + 2.12132i 1.00000 2.82843i 0
749.2 0.707107 + 0.707107i −1.41421 1.00000i 1.00000i 0 −0.292893 1.70711i −1.00000 1.00000i 2.12132 2.12132i 1.00000 + 2.82843i 0
824.1 −0.707107 + 0.707107i 1.41421 + 1.00000i 1.00000i 0 −1.70711 + 0.292893i −1.00000 + 1.00000i −2.12132 2.12132i 1.00000 + 2.82843i 0
824.2 0.707107 0.707107i −1.41421 + 1.00000i 1.00000i 0 −0.292893 + 1.70711i −1.00000 + 1.00000i 2.12132 + 2.12132i 1.00000 2.82843i 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
65.g odd 4 1 inner
195.n 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.n.c 4
3.b odd 2 1 inner 975.2.n.c 4
5.b even 2 1 975.2.n.d 4
5.c odd 4 1 39.2.f.a 4
5.c odd 4 1 975.2.o.j 4
13.d odd 4 1 975.2.n.d 4
15.d odd 2 1 975.2.n.d 4
15.e even 4 1 39.2.f.a 4
15.e even 4 1 975.2.o.j 4
20.e even 4 1 624.2.bf.d 4
39.f even 4 1 975.2.n.d 4
60.l odd 4 1 624.2.bf.d 4
65.f even 4 1 507.2.f.a 4
65.f even 4 1 975.2.o.j 4
65.g odd 4 1 inner 975.2.n.c 4
65.h odd 4 1 507.2.f.a 4
65.k even 4 1 39.2.f.a 4
65.o even 12 2 507.2.k.j 8
65.q odd 12 2 507.2.k.j 8
65.r odd 12 2 507.2.k.i 8
65.t even 12 2 507.2.k.i 8
195.j odd 4 1 39.2.f.a 4
195.n even 4 1 inner 975.2.n.c 4
195.s even 4 1 507.2.f.a 4
195.u odd 4 1 507.2.f.a 4
195.u odd 4 1 975.2.o.j 4
195.bc odd 12 2 507.2.k.i 8
195.bf even 12 2 507.2.k.i 8
195.bl even 12 2 507.2.k.j 8
195.bn odd 12 2 507.2.k.j 8
260.s odd 4 1 624.2.bf.d 4
780.bn even 4 1 624.2.bf.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 5.c odd 4 1
39.2.f.a 4 15.e even 4 1
39.2.f.a 4 65.k even 4 1
39.2.f.a 4 195.j odd 4 1
507.2.f.a 4 65.f even 4 1
507.2.f.a 4 65.h odd 4 1
507.2.f.a 4 195.s even 4 1
507.2.f.a 4 195.u odd 4 1
507.2.k.i 8 65.r odd 12 2
507.2.k.i 8 65.t even 12 2
507.2.k.i 8 195.bc odd 12 2
507.2.k.i 8 195.bf even 12 2
507.2.k.j 8 65.o even 12 2
507.2.k.j 8 65.q odd 12 2
507.2.k.j 8 195.bl even 12 2
507.2.k.j 8 195.bn odd 12 2
624.2.bf.d 4 20.e even 4 1
624.2.bf.d 4 60.l odd 4 1
624.2.bf.d 4 260.s odd 4 1
624.2.bf.d 4 780.bn even 4 1
975.2.n.c 4 1.a even 1 1 trivial
975.2.n.c 4 3.b odd 2 1 inner
975.2.n.c 4 65.g odd 4 1 inner
975.2.n.c 4 195.n even 4 1 inner
975.2.n.d 4 5.b even 2 1
975.2.n.d 4 13.d odd 4 1
975.2.n.d 4 15.d odd 2 1
975.2.n.d 4 39.f even 4 1
975.2.o.j 4 5.c odd 4 1
975.2.o.j 4 15.e even 4 1
975.2.o.j 4 65.f even 4 1
975.2.o.j 4 195.u odd 4 1

Hecke kernels

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

\( T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{4} + 256 \) Copy content Toggle raw display
\( T_{37}^{2} + 2T_{37} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) 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
$11$ \( T^{4} + 256 \) 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
$19$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 16 \) Copy content Toggle raw display
$43$ \( (T + 6)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 256 \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 256 \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} + 38416 \) Copy content Toggle raw display
$97$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
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