Properties

Label 370.2.g.c
Level $370$
Weight $2$
Character orbit 370.g
Analytic conductor $2.954$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(43,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.g (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: \(2.95446487479\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} + 2 \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + 2 \zeta_{8}^{3} q^{6} + (2 \zeta_{8}^{2} + \zeta_{8} + 2) q^{7} - \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8}^{2} q^{2} + 2 \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + 2 \zeta_{8}^{3} q^{6} + (2 \zeta_{8}^{2} + \zeta_{8} + 2) q^{7} - \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} + (2 \zeta_{8}^{3} + \zeta_{8}) q^{10} + ( - 2 \zeta_{8}^{3} + \cdots - 2 \zeta_{8}) q^{11} + \cdots + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} - 8 q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} + 4 q^{22} + 16 q^{25} - 8 q^{26} - 8 q^{28} - 24 q^{29} - 16 q^{30} - 8 q^{31} + 16 q^{33} + 4 q^{35} + 12 q^{38} + 24 q^{39} - 8 q^{42} - 16 q^{46} - 16 q^{47} - 12 q^{50} - 16 q^{51} + 8 q^{55} + 8 q^{56} - 24 q^{58} - 16 q^{59} - 8 q^{60} + 40 q^{61} + 8 q^{62} - 8 q^{63} - 4 q^{64} + 12 q^{65} + 16 q^{66} - 12 q^{67} - 4 q^{68} + 8 q^{69} - 8 q^{70} - 24 q^{71} + 4 q^{72} + 24 q^{73} + 12 q^{76} + 16 q^{77} + 24 q^{78} + 16 q^{79} + 44 q^{81} + 28 q^{82} + 8 q^{83} - 24 q^{85} - 28 q^{86} - 8 q^{87} - 4 q^{88} + 24 q^{89} - 4 q^{91} + 16 q^{94} - 16 q^{95} - 28 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i −1.41421 + 1.41421i −1.00000 −2.12132 + 0.707107i 1.41421 + 1.41421i 1.29289 1.29289i 1.00000i 1.00000i 0.707107 + 2.12132i
43.2 1.00000i 1.41421 1.41421i −1.00000 2.12132 0.707107i −1.41421 1.41421i 2.70711 2.70711i 1.00000i 1.00000i −0.707107 2.12132i
327.1 1.00000i −1.41421 1.41421i −1.00000 −2.12132 0.707107i 1.41421 1.41421i 1.29289 + 1.29289i 1.00000i 1.00000i 0.707107 2.12132i
327.2 1.00000i 1.41421 + 1.41421i −1.00000 2.12132 + 0.707107i −1.41421 + 1.41421i 2.70711 + 2.70711i 1.00000i 1.00000i −0.707107 + 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.g.c 4
5.c odd 4 1 370.2.h.c yes 4
37.d odd 4 1 370.2.h.c yes 4
185.f even 4 1 inner 370.2.g.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.c 4 1.a even 1 1 trivial
370.2.g.c 4 185.f even 4 1 inner
370.2.h.c yes 4 5.c odd 4 1
370.2.h.c yes 4 37.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 16 \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 49 \) Copy content Toggle raw display
$13$ \( T^{4} + 44T^{2} + 196 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$29$ \( T^{4} + 24 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 24T^{2} + 1369 \) Copy content Toggle raw display
$41$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 16 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{4} + 81 \) Copy content Toggle raw display
$59$ \( T^{4} + 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{4} - 40 T^{3} + \cdots + 39601 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 72)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( T^{4} - 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$89$ \( T^{4} - 24 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$97$ \( (T + 7)^{4} \) Copy content Toggle raw display
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