Properties

Label 370.2.h.b
Level $370$
Weight $2$
Character orbit 370.h
Analytic conductor $2.954$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - i + 1) q^{3} + q^{4} + ( - 2 i - 1) q^{5} + ( - i + 1) q^{6} + ( - i + 1) q^{7} + q^{8} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - i + 1) q^{3} + q^{4} + ( - 2 i - 1) q^{5} + ( - i + 1) q^{6} + ( - i + 1) q^{7} + q^{8} + i q^{9} + ( - 2 i - 1) q^{10} + 2 i q^{11} + ( - i + 1) q^{12} + 2 q^{13} + ( - i + 1) q^{14} + ( - i - 3) q^{15} + q^{16} - 4 i q^{17} + i q^{18} + ( - 5 i - 5) q^{19} + ( - 2 i - 1) q^{20} - 2 i q^{21} + 2 i q^{22} + ( - i + 1) q^{24} + (4 i - 3) q^{25} + 2 q^{26} + (4 i + 4) q^{27} + ( - i + 1) q^{28} + (3 i - 3) q^{29} + ( - i - 3) q^{30} + (7 i + 7) q^{31} + q^{32} + (2 i + 2) q^{33} - 4 i q^{34} + ( - i - 3) q^{35} + i q^{36} + (6 i - 1) q^{37} + ( - 5 i - 5) q^{38} + ( - 2 i + 2) q^{39} + ( - 2 i - 1) q^{40} - 2 i q^{42} + 4 q^{43} + 2 i q^{44} + ( - i + 2) q^{45} + (7 i - 7) q^{47} + ( - i + 1) q^{48} + 5 i q^{49} + (4 i - 3) q^{50} + ( - 4 i - 4) q^{51} + 2 q^{52} + (i + 1) q^{53} + (4 i + 4) q^{54} + ( - 2 i + 4) q^{55} + ( - i + 1) q^{56} - 10 q^{57} + (3 i - 3) q^{58} + ( - i - 1) q^{59} + ( - i - 3) q^{60} + ( - 3 i - 3) q^{61} + (7 i + 7) q^{62} + (i + 1) q^{63} + q^{64} + ( - 4 i - 2) q^{65} + (2 i + 2) q^{66} + (3 i + 3) q^{67} - 4 i q^{68} + ( - i - 3) q^{70} - 8 q^{71} + i q^{72} + ( - 9 i + 9) q^{73} + (6 i - 1) q^{74} + (7 i + 1) q^{75} + ( - 5 i - 5) q^{76} + (2 i + 2) q^{77} + ( - 2 i + 2) q^{78} + ( - i - 1) q^{79} + ( - 2 i - 1) q^{80} + 5 q^{81} + ( - 5 i - 5) q^{83} - 2 i q^{84} + (4 i - 8) q^{85} + 4 q^{86} + 6 i q^{87} + 2 i q^{88} + ( - 5 i + 5) q^{89} + ( - i + 2) q^{90} + ( - 2 i + 2) q^{91} + 14 q^{93} + (7 i - 7) q^{94} + (15 i - 5) q^{95} + ( - i + 1) q^{96} - 8 i q^{97} + 5 i q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{12} + 4 q^{13} + 2 q^{14} - 6 q^{15} + 2 q^{16} - 10 q^{19} - 2 q^{20} + 2 q^{24} - 6 q^{25} + 4 q^{26} + 8 q^{27} + 2 q^{28} - 6 q^{29} - 6 q^{30} + 14 q^{31} + 2 q^{32} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 10 q^{38} + 4 q^{39} - 2 q^{40} + 8 q^{43} + 4 q^{45} - 14 q^{47} + 2 q^{48} - 6 q^{50} - 8 q^{51} + 4 q^{52} + 2 q^{53} + 8 q^{54} + 8 q^{55} + 2 q^{56} - 20 q^{57} - 6 q^{58} - 2 q^{59} - 6 q^{60} - 6 q^{61} + 14 q^{62} + 2 q^{63} + 2 q^{64} - 4 q^{65} + 4 q^{66} + 6 q^{67} - 6 q^{70} - 16 q^{71} + 18 q^{73} - 2 q^{74} + 2 q^{75} - 10 q^{76} + 4 q^{77} + 4 q^{78} - 2 q^{79} - 2 q^{80} + 10 q^{81} - 10 q^{83} - 16 q^{85} + 8 q^{86} + 10 q^{89} + 4 q^{90} + 4 q^{91} + 28 q^{93} - 14 q^{94} - 10 q^{95} + 2 q^{96} - 4 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
117.1
1.00000i
1.00000i
1.00000 1.00000 + 1.00000i 1.00000 −1.00000 + 2.00000i 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 1.00000i −1.00000 + 2.00000i
253.1 1.00000 1.00000 1.00000i 1.00000 −1.00000 2.00000i 1.00000 1.00000i 1.00000 1.00000i 1.00000 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.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.b yes 2
5.c odd 4 1 370.2.g.a 2
37.d odd 4 1 370.2.g.a 2
185.k even 4 1 inner 370.2.h.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.g.a 2 5.c odd 4 1
370.2.g.a 2 37.d odd 4 1
370.2.h.b yes 2 1.a even 1 1 trivial
370.2.h.b yes 2 185.k even 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) 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
$37$ \( T^{2} + 2T + 37 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 98 \) 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
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
show more
show less