Properties

Label 370.2.g.a
Level $370$
Weight $2$
Character orbit 370.g
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(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: \(2\)
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: 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.

\(f(q)\) \(=\) \( q + i q^{2} + ( - i - 1) q^{3} - q^{4} + (i + 2) q^{5} + ( - i + 1) q^{6} + (i + 1) q^{7} - i q^{8} - i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + ( - i - 1) q^{3} - q^{4} + (i + 2) q^{5} + ( - i + 1) q^{6} + (i + 1) q^{7} - i q^{8} - i q^{9} + (2 i - 1) q^{10} + 2 i q^{11} + (i + 1) q^{12} - 2 i q^{13} + (i - 1) q^{14} + ( - 3 i - 1) q^{15} + q^{16} + 4 q^{17} + q^{18} + (5 i + 5) q^{19} + ( - i - 2) q^{20} - 2 i q^{21} - 2 q^{22} + (i - 1) q^{24} + (4 i + 3) q^{25} + 2 q^{26} + (4 i - 4) q^{27} + ( - i - 1) q^{28} + ( - 3 i + 3) q^{29} + ( - i + 3) q^{30} + (7 i + 7) q^{31} + i q^{32} + ( - 2 i + 2) q^{33} + 4 i q^{34} + (3 i + 1) q^{35} + i q^{36} + ( - i - 6) q^{37} + (5 i - 5) q^{38} + (2 i - 2) q^{39} + ( - 2 i + 1) q^{40} + 2 q^{42} - 4 i q^{43} - 2 i q^{44} + ( - 2 i + 1) q^{45} + ( - 7 i - 7) q^{47} + ( - i - 1) q^{48} - 5 i q^{49} + (3 i - 4) q^{50} + ( - 4 i - 4) q^{51} + 2 i q^{52} + ( - i + 1) q^{53} + ( - 4 i - 4) q^{54} + (4 i - 2) q^{55} + ( - i + 1) q^{56} - 10 i q^{57} + (3 i + 3) q^{58} + (i + 1) q^{59} + (3 i + 1) q^{60} + ( - 3 i - 3) q^{61} + (7 i - 7) q^{62} + ( - i + 1) q^{63} - q^{64} + ( - 4 i + 2) q^{65} + (2 i + 2) q^{66} + (3 i - 3) q^{67} - 4 q^{68} + (i - 3) q^{70} - 8 q^{71} - q^{72} + ( - 9 i - 9) q^{73} + ( - 6 i + 1) q^{74} + ( - 7 i + 1) q^{75} + ( - 5 i - 5) q^{76} + (2 i - 2) q^{77} + ( - 2 i - 2) q^{78} + (i + 1) q^{79} + (i + 2) q^{80} + 5 q^{81} + (5 i - 5) q^{83} + 2 i q^{84} + (4 i + 8) q^{85} + 4 q^{86} - 6 q^{87} + 2 q^{88} + (5 i - 5) q^{89} + (i + 2) q^{90} + ( - 2 i + 2) q^{91} - 14 i q^{93} + ( - 7 i + 7) q^{94} + (15 i + 5) q^{95} + ( - i + 1) q^{96} + 8 q^{97} + 5 q^{98} + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{10} + 2 q^{12} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} + 2 q^{18} + 10 q^{19} - 4 q^{20} - 4 q^{22} - 2 q^{24} + 6 q^{25} + 4 q^{26} - 8 q^{27} - 2 q^{28} + 6 q^{29} + 6 q^{30} + 14 q^{31} + 4 q^{33} + 2 q^{35} - 12 q^{37} - 10 q^{38} - 4 q^{39} + 2 q^{40} + 4 q^{42} + 2 q^{45} - 14 q^{47} - 2 q^{48} - 8 q^{50} - 8 q^{51} + 2 q^{53} - 8 q^{54} - 4 q^{55} + 2 q^{56} + 6 q^{58} + 2 q^{59} + 2 q^{60} - 6 q^{61} - 14 q^{62} + 2 q^{63} - 2 q^{64} + 4 q^{65} + 4 q^{66} - 6 q^{67} - 8 q^{68} - 6 q^{70} - 16 q^{71} - 2 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 10 q^{76} - 4 q^{77} - 4 q^{78} + 2 q^{79} + 4 q^{80} + 10 q^{81} - 10 q^{83} + 16 q^{85} + 8 q^{86} - 12 q^{87} + 4 q^{88} - 10 q^{89} + 4 q^{90} + 4 q^{91} + 14 q^{94} + 10 q^{95} + 2 q^{96} + 16 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.00000i
1.00000i
1.00000i −1.00000 + 1.00000i −1.00000 2.00000 1.00000i 1.00000 + 1.00000i 1.00000 1.00000i 1.00000i 1.00000i −1.00000 2.00000i
327.1 1.00000i −1.00000 1.00000i −1.00000 2.00000 + 1.00000i 1.00000 1.00000i 1.00000 + 1.00000i 1.00000i 1.00000i −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.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.a 2
5.c odd 4 1 370.2.h.b yes 2
37.d odd 4 1 370.2.h.b yes 2
185.f even 4 1 inner 370.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.a 2 1.a even 1 1 trivial
370.2.g.a 2 185.f even 4 1 inner
370.2.h.b yes 2 5.c odd 4 1
370.2.h.b yes 2 37.d odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$79$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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