Properties

Label 361.3.d.b
Level $361$
Weight $3$
Character orbit 361.d
Analytic conductor $9.837$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [361,3,Mod(69,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.69");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.d (of order \(6\), degree \(2\), not 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: \(9.83653754341\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 13x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_1 q^{3} + 9 \beta_{2} q^{4} + (4 \beta_{2} - 4) q^{5} - 13 \beta_{2} q^{6} - 5 q^{7} + 5 \beta_{3} q^{8} + 4 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_1 q^{3} + 9 \beta_{2} q^{4} + (4 \beta_{2} - 4) q^{5} - 13 \beta_{2} q^{6} - 5 q^{7} + 5 \beta_{3} q^{8} + 4 \beta_{2} q^{9} + (4 \beta_{3} - 4 \beta_1) q^{10} - 10 q^{11} - 9 \beta_{3} q^{12} + (\beta_{3} - \beta_1) q^{13} - 5 \beta_1 q^{14} + ( - 4 \beta_{3} + 4 \beta_1) q^{15} + (29 \beta_{2} - 29) q^{16} + (15 \beta_{2} - 15) q^{17} + 4 \beta_{3} q^{18} - 36 q^{20} + 5 \beta_1 q^{21} - 10 \beta_1 q^{22} - 35 \beta_{2} q^{23} + ( - 65 \beta_{2} + 65) q^{24} + 9 \beta_{2} q^{25} - 13 q^{26} + 5 \beta_{3} q^{27} - 45 \beta_{2} q^{28} + (5 \beta_{3} - 5 \beta_1) q^{29} + 52 q^{30} + 10 \beta_{3} q^{31} + (9 \beta_{3} - 9 \beta_1) q^{32} + 10 \beta_1 q^{33} + (15 \beta_{3} - 15 \beta_1) q^{34} + ( - 20 \beta_{2} + 20) q^{35} + (36 \beta_{2} - 36) q^{36} + 6 \beta_{3} q^{37} + 13 q^{39} - 20 \beta_1 q^{40} + 10 \beta_1 q^{41} + 65 \beta_{2} q^{42} + ( - 20 \beta_{2} + 20) q^{43} - 90 \beta_{2} q^{44} - 16 q^{45} - 35 \beta_{3} q^{46} - 10 \beta_{2} q^{47} + ( - 29 \beta_{3} + 29 \beta_1) q^{48} - 24 q^{49} + 9 \beta_{3} q^{50} + ( - 15 \beta_{3} + 15 \beta_1) q^{51} - 9 \beta_1 q^{52} + ( - 21 \beta_{3} + 21 \beta_1) q^{53} + (65 \beta_{2} - 65) q^{54} + ( - 40 \beta_{2} + 40) q^{55} - 25 \beta_{3} q^{56} - 65 q^{58} + 5 \beta_1 q^{59} + 36 \beta_1 q^{60} + 40 \beta_{2} q^{61} + (130 \beta_{2} - 130) q^{62} - 20 \beta_{2} q^{63} - q^{64} - 4 \beta_{3} q^{65} + 130 \beta_{2} q^{66} + (11 \beta_{3} - 11 \beta_1) q^{67} - 135 q^{68} + 35 \beta_{3} q^{69} + ( - 20 \beta_{3} + 20 \beta_1) q^{70} + 30 \beta_1 q^{71} + (20 \beta_{3} - 20 \beta_1) q^{72} + (105 \beta_{2} - 105) q^{73} + (78 \beta_{2} - 78) q^{74} - 9 \beta_{3} q^{75} + 50 q^{77} + 13 \beta_1 q^{78} - 10 \beta_1 q^{79} - 116 \beta_{2} q^{80} + ( - 101 \beta_{2} + 101) q^{81} + 130 \beta_{2} q^{82} - 40 q^{83} + 45 \beta_{3} q^{84} - 60 \beta_{2} q^{85} + ( - 20 \beta_{3} + 20 \beta_1) q^{86} + 65 q^{87} - 50 \beta_{3} q^{88} - 16 \beta_1 q^{90} + ( - 5 \beta_{3} + 5 \beta_1) q^{91} + ( - 315 \beta_{2} + 315) q^{92} + ( - 130 \beta_{2} + 130) q^{93} - 10 \beta_{3} q^{94} + 117 q^{96} + 34 \beta_1 q^{97} - 24 \beta_1 q^{98} - 40 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{4} - 8 q^{5} - 26 q^{6} - 20 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{4} - 8 q^{5} - 26 q^{6} - 20 q^{7} + 8 q^{9} - 40 q^{11} - 58 q^{16} - 30 q^{17} - 144 q^{20} - 70 q^{23} + 130 q^{24} + 18 q^{25} - 52 q^{26} - 90 q^{28} + 208 q^{30} + 40 q^{35} - 72 q^{36} + 52 q^{39} + 130 q^{42} + 40 q^{43} - 180 q^{44} - 64 q^{45} - 20 q^{47} - 96 q^{49} - 130 q^{54} + 80 q^{55} - 260 q^{58} + 80 q^{61} - 260 q^{62} - 40 q^{63} - 4 q^{64} + 260 q^{66} - 540 q^{68} - 210 q^{73} - 156 q^{74} + 200 q^{77} - 232 q^{80} + 202 q^{81} + 260 q^{82} - 160 q^{83} - 120 q^{85} + 260 q^{87} + 630 q^{92} + 260 q^{93} + 468 q^{96} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 13\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 13\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
−3.12250 + 1.80278i
3.12250 1.80278i
−3.12250 1.80278i
3.12250 + 1.80278i
−3.12250 + 1.80278i 3.12250 1.80278i 4.50000 7.79423i −2.00000 3.46410i −6.50000 + 11.2583i −5.00000 18.0278i 2.00000 3.46410i 12.4900 + 7.21110i
69.2 3.12250 1.80278i −3.12250 + 1.80278i 4.50000 7.79423i −2.00000 3.46410i −6.50000 + 11.2583i −5.00000 18.0278i 2.00000 3.46410i −12.4900 7.21110i
293.1 −3.12250 1.80278i 3.12250 + 1.80278i 4.50000 + 7.79423i −2.00000 + 3.46410i −6.50000 11.2583i −5.00000 18.0278i 2.00000 + 3.46410i 12.4900 7.21110i
293.2 3.12250 + 1.80278i −3.12250 1.80278i 4.50000 + 7.79423i −2.00000 + 3.46410i −6.50000 11.2583i −5.00000 18.0278i 2.00000 + 3.46410i −12.4900 + 7.21110i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.d.b 4
19.b odd 2 1 inner 361.3.d.b 4
19.c even 3 1 19.3.b.b 2
19.c even 3 1 inner 361.3.d.b 4
19.d odd 6 1 19.3.b.b 2
19.d odd 6 1 inner 361.3.d.b 4
19.e even 9 6 361.3.f.d 12
19.f odd 18 6 361.3.f.d 12
57.f even 6 1 171.3.c.b 2
57.h odd 6 1 171.3.c.b 2
76.f even 6 1 304.3.e.d 2
76.g odd 6 1 304.3.e.d 2
95.h odd 6 1 475.3.c.b 2
95.i even 6 1 475.3.c.b 2
95.l even 12 2 475.3.d.b 4
95.m odd 12 2 475.3.d.b 4
152.k odd 6 1 1216.3.e.h 2
152.l odd 6 1 1216.3.e.g 2
152.o even 6 1 1216.3.e.h 2
152.p even 6 1 1216.3.e.g 2
228.m even 6 1 2736.3.o.d 2
228.n odd 6 1 2736.3.o.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 19.c even 3 1
19.3.b.b 2 19.d odd 6 1
171.3.c.b 2 57.f even 6 1
171.3.c.b 2 57.h odd 6 1
304.3.e.d 2 76.f even 6 1
304.3.e.d 2 76.g odd 6 1
361.3.d.b 4 1.a even 1 1 trivial
361.3.d.b 4 19.b odd 2 1 inner
361.3.d.b 4 19.c even 3 1 inner
361.3.d.b 4 19.d odd 6 1 inner
361.3.f.d 12 19.e even 9 6
361.3.f.d 12 19.f odd 18 6
475.3.c.b 2 95.h odd 6 1
475.3.c.b 2 95.i even 6 1
475.3.d.b 4 95.l even 12 2
475.3.d.b 4 95.m odd 12 2
1216.3.e.g 2 152.l odd 6 1
1216.3.e.g 2 152.p even 6 1
1216.3.e.h 2 152.k odd 6 1
1216.3.e.h 2 152.o even 6 1
2736.3.o.d 2 228.m even 6 1
2736.3.o.d 2 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 13T_{2}^{2} + 169 \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$3$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$5$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T + 5)^{4} \) Copy content Toggle raw display
$11$ \( (T + 10)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 13T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 15 T + 225)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 35 T + 1225)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 325 T^{2} + 105625 \) Copy content Toggle raw display
$31$ \( (T^{2} + 1300)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 468)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 1300 T^{2} + \cdots + 1690000 \) Copy content Toggle raw display
$43$ \( (T^{2} - 20 T + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 5733 T^{2} + \cdots + 32867289 \) Copy content Toggle raw display
$59$ \( T^{4} - 325 T^{2} + 105625 \) Copy content Toggle raw display
$61$ \( (T^{2} - 40 T + 1600)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 1573 T^{2} + \cdots + 2474329 \) Copy content Toggle raw display
$71$ \( T^{4} - 11700 T^{2} + \cdots + 136890000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 105 T + 11025)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1300 T^{2} + \cdots + 1690000 \) Copy content Toggle raw display
$83$ \( (T + 40)^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 15028 T^{2} + \cdots + 225840784 \) Copy content Toggle raw display
show more
show less