Properties

Label 1210.3.d.b
Level $1210$
Weight $3$
Character orbit 1210.d
Analytic conductor $32.970$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1210,3,Mod(241,1210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1210.241");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1210.d (of order \(2\), degree \(1\), 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: \(32.9701119876\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{6} + ( - \beta_{2} + 4 \beta_1) q^{7} + 2 \beta_1 q^{8} + ( - 2 \beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{6} + ( - \beta_{2} + 4 \beta_1) q^{7} + 2 \beta_1 q^{8} + ( - 2 \beta_{3} - 3) q^{9} + \beta_{2} q^{10} + (2 \beta_{3} - 2) q^{12} + ( - \beta_{2} + 3 \beta_1) q^{13} + ( - 2 \beta_{3} + 8) q^{14} + ( - \beta_{3} + 5) q^{15} + 4 q^{16} + (3 \beta_{2} + \beta_1) q^{17} + (2 \beta_{2} + 3 \beta_1) q^{18} + ( - 4 \beta_{2} + 14 \beta_1) q^{19} + 2 \beta_{3} q^{20} + ( - 5 \beta_{2} + 9 \beta_1) q^{21} + (13 \beta_{3} - 3) q^{23} + ( - 2 \beta_{2} + 2 \beta_1) q^{24} + 5 q^{25} + ( - 2 \beta_{3} + 6) q^{26} + (10 \beta_{3} - 2) q^{27} + (2 \beta_{2} - 8 \beta_1) q^{28} + (2 \beta_{2} + \beta_1) q^{29} + (\beta_{2} - 5 \beta_1) q^{30} + ( - 12 \beta_{3} - 12) q^{31} - 4 \beta_1 q^{32} + (6 \beta_{3} + 2) q^{34} + ( - 4 \beta_{2} + 5 \beta_1) q^{35} + (4 \beta_{3} + 6) q^{36} + (8 \beta_{3} - 50) q^{37} + ( - 8 \beta_{3} + 28) q^{38} + ( - 4 \beta_{2} + 8 \beta_1) q^{39} - 2 \beta_{2} q^{40} + ( - 3 \beta_{2} - 6 \beta_1) q^{41} + ( - 10 \beta_{3} + 18) q^{42} + ( - 16 \beta_{2} + 9 \beta_1) q^{43} + (3 \beta_{3} + 10) q^{45} + ( - 13 \beta_{2} + 3 \beta_1) q^{46} + ( - 11 \beta_{3} + 45) q^{47} + ( - 4 \beta_{3} + 4) q^{48} + (16 \beta_{3} + 7) q^{49} - 5 \beta_1 q^{50} + (2 \beta_{2} - 14 \beta_1) q^{51} + (2 \beta_{2} - 6 \beta_1) q^{52} + ( - 12 \beta_{3} - 18) q^{53} + ( - 10 \beta_{2} + 2 \beta_1) q^{54} + (4 \beta_{3} - 16) q^{56} + ( - 18 \beta_{2} + 34 \beta_1) q^{57} + (4 \beta_{3} + 2) q^{58} + ( - 2 \beta_{3} + 2) q^{59} + (2 \beta_{3} - 10) q^{60} + ( - 15 \beta_{2} + 6 \beta_1) q^{61} + (12 \beta_{2} + 12 \beta_1) q^{62} + ( - 5 \beta_{2} - 2 \beta_1) q^{63} - 8 q^{64} + ( - 3 \beta_{2} + 5 \beta_1) q^{65} + (35 \beta_{3} + 51) q^{67} + ( - 6 \beta_{2} - 2 \beta_1) q^{68} + (16 \beta_{3} - 68) q^{69} + ( - 8 \beta_{3} + 10) q^{70} + ( - 4 \beta_{3} - 40) q^{71} + ( - 4 \beta_{2} - 6 \beta_1) q^{72} + ( - 7 \beta_{2} - 37 \beta_1) q^{73} + ( - 8 \beta_{2} + 50 \beta_1) q^{74} + ( - 5 \beta_{3} + 5) q^{75} + (8 \beta_{2} - 28 \beta_1) q^{76} + ( - 8 \beta_{3} + 16) q^{78} + ( - 24 \beta_{2} + 6 \beta_1) q^{79} - 4 \beta_{3} q^{80} + (30 \beta_{3} - 25) q^{81} + ( - 6 \beta_{3} - 12) q^{82} + ( - 32 \beta_{2} - \beta_1) q^{83} + (10 \beta_{2} - 18 \beta_1) q^{84} + ( - \beta_{2} - 15 \beta_1) q^{85} + ( - 32 \beta_{3} + 18) q^{86} + (\beta_{2} - 9 \beta_1) q^{87} + (40 \beta_{3} + 24) q^{89} + ( - 3 \beta_{2} - 10 \beta_1) q^{90} + (14 \beta_{3} - 34) q^{91} + ( - 26 \beta_{3} + 6) q^{92} + 48 q^{93} + (11 \beta_{2} - 45 \beta_1) q^{94} + ( - 14 \beta_{2} + 20 \beta_1) q^{95} + (4 \beta_{2} - 4 \beta_1) q^{96} + ( - 66 \beta_{3} + 24) q^{97} + ( - 16 \beta_{2} - 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 8 q^{4} - 12 q^{9} - 8 q^{12} + 32 q^{14} + 20 q^{15} + 16 q^{16} - 12 q^{23} + 20 q^{25} + 24 q^{26} - 8 q^{27} - 48 q^{31} + 8 q^{34} + 24 q^{36} - 200 q^{37} + 112 q^{38} + 72 q^{42} + 40 q^{45} + 180 q^{47} + 16 q^{48} + 28 q^{49} - 72 q^{53} - 64 q^{56} + 8 q^{58} + 8 q^{59} - 40 q^{60} - 32 q^{64} + 204 q^{67} - 272 q^{69} + 40 q^{70} - 160 q^{71} + 20 q^{75} + 64 q^{78} - 100 q^{81} - 48 q^{82} + 72 q^{86} + 96 q^{89} - 136 q^{91} + 24 q^{92} + 192 q^{93} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{2} + 4\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1210\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1091\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
0.874032i
2.28825i
0.874032i
2.28825i
1.41421i −1.23607 −2.00000 −2.23607 1.74806i 2.49458i 2.82843i −7.47214 3.16228i
241.2 1.41421i 3.23607 −2.00000 2.23607 4.57649i 8.81913i 2.82843i 1.47214 3.16228i
241.3 1.41421i −1.23607 −2.00000 −2.23607 1.74806i 2.49458i 2.82843i −7.47214 3.16228i
241.4 1.41421i 3.23607 −2.00000 2.23607 4.57649i 8.81913i 2.82843i 1.47214 3.16228i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1210.3.d.b 4
11.b odd 2 1 inner 1210.3.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1210.3.d.b 4 1.a even 1 1 trivial
1210.3.d.b 4 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 4 \) acting on \(S_{3}^{\mathrm{new}}(1210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 84T^{2} + 484 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 56T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 184T^{2} + 7744 \) Copy content Toggle raw display
$19$ \( T^{4} + 1104 T^{2} + 53824 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T - 836)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 84T^{2} + 1444 \) Copy content Toggle raw display
$31$ \( (T^{2} + 24 T - 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 100 T + 2180)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 324T^{2} + 324 \) Copy content Toggle raw display
$43$ \( T^{4} + 5444 T^{2} + \cdots + 5750404 \) Copy content Toggle raw display
$47$ \( (T^{2} - 90 T + 1420)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36 T - 396)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 4644 T^{2} + \cdots + 4743684 \) Copy content Toggle raw display
$67$ \( (T^{2} - 102 T - 3524)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 80 T + 1520)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 6456 T^{2} + \cdots + 5053504 \) Copy content Toggle raw display
$79$ \( T^{4} + 11664 T^{2} + \cdots + 32353344 \) Copy content Toggle raw display
$83$ \( T^{4} + 20484 T^{2} + \cdots + 104816644 \) Copy content Toggle raw display
$89$ \( (T^{2} - 48 T - 7424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48 T - 21204)^{2} \) Copy content Toggle raw display
show more
show less