Properties

Label 370.2.h.a
Level $370$
Weight $2$
Character orbit 370.h
Analytic conductor $2.954$
Analytic rank $0$
Dimension $2$
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(117,370)]
 
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([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.117");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.h (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: \(2\)
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: \(\Z[a_1, \ldots, a_{9}]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (2 i - 1) q^{5} + (2 i - 2) q^{7} + q^{8} + 3 i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (2 i - 1) q^{5} + (2 i - 2) q^{7} + q^{8} + 3 i q^{9} + (2 i - 1) q^{10} + 4 q^{13} + (2 i - 2) q^{14} + q^{16} + 2 i q^{17} + 3 i q^{18} + ( - 2 i - 2) q^{19} + (2 i - 1) q^{20} + 4 q^{23} + ( - 4 i - 3) q^{25} + 4 q^{26} + (2 i - 2) q^{28} + ( - 7 i + 7) q^{29} + ( - 4 i - 4) q^{31} + q^{32} + 2 i q^{34} + ( - 6 i - 2) q^{35} + 3 i q^{36} + (6 i - 1) q^{37} + ( - 2 i - 2) q^{38} + (2 i - 1) q^{40} + 4 q^{43} + ( - 3 i - 6) q^{45} + 4 q^{46} + (2 i - 2) q^{47} - i q^{49} + ( - 4 i - 3) q^{50} + 4 q^{52} + ( - i - 1) q^{53} + (2 i - 2) q^{56} + ( - 7 i + 7) q^{58} + ( - 2 i - 2) q^{59} + (i + 1) q^{61} + ( - 4 i - 4) q^{62} + ( - 6 i - 6) q^{63} + q^{64} + (8 i - 4) q^{65} + 2 i q^{68} + ( - 6 i - 2) q^{70} + 12 q^{71} + 3 i q^{72} + ( - 5 i + 5) q^{73} + (6 i - 1) q^{74} + ( - 2 i - 2) q^{76} + ( - 12 i - 12) q^{79} + (2 i - 1) q^{80} - 9 q^{81} + (4 i + 4) q^{83} + ( - 2 i - 4) q^{85} + 4 q^{86} + ( - 7 i + 7) q^{89} + ( - 3 i - 6) q^{90} + (8 i - 8) q^{91} + 4 q^{92} + (2 i - 2) q^{94} + ( - 2 i + 6) q^{95} + 12 i q^{97} - i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} + 8 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{19} - 2 q^{20} + 8 q^{23} - 6 q^{25} + 8 q^{26} - 4 q^{28} + 14 q^{29} - 8 q^{31} + 2 q^{32} - 4 q^{35} - 2 q^{37} - 4 q^{38} - 2 q^{40} + 8 q^{43} - 12 q^{45} + 8 q^{46} - 4 q^{47} - 6 q^{50} + 8 q^{52} - 2 q^{53} - 4 q^{56} + 14 q^{58} - 4 q^{59} + 2 q^{61} - 8 q^{62} - 12 q^{63} + 2 q^{64} - 8 q^{65} - 4 q^{70} + 24 q^{71} + 10 q^{73} - 2 q^{74} - 4 q^{76} - 24 q^{79} - 2 q^{80} - 18 q^{81} + 8 q^{83} - 8 q^{85} + 8 q^{86} + 14 q^{89} - 12 q^{90} - 16 q^{91} + 8 q^{92} - 4 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(261\) \(297\)
\(\chi(n)\) \(i\) \(-i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
1.00000i
1.00000i
1.00000 0 1.00000 −1.00000 2.00000i 0 −2.00000 2.00000i 1.00000 3.00000i −1.00000 2.00000i
253.1 1.00000 0 1.00000 −1.00000 + 2.00000i 0 −2.00000 + 2.00000i 1.00000 3.00000i −1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k 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.h.a yes 2
5.c odd 4 1 370.2.g.b 2
37.d odd 4 1 370.2.g.b 2
185.k even 4 1 inner 370.2.h.a yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.b 2 5.c odd 4 1
370.2.g.b 2 37.d odd 4 1
370.2.h.a yes 2 1.a even 1 1 trivial
370.2.h.a yes 2 185.k even 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$53$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( (T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} + 24T + 288 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$97$ \( T^{2} + 144 \) Copy content Toggle raw display
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