Properties

Label 363.4.a.k
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,4,Mod(1,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

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: yes
Analytic conductor: \(21.4176933321\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} - 3 q^{3} - 3 q^{4} - 2 q^{5} + 3 \beta q^{6} + 10 \beta q^{7} + 11 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 3 q^{3} - 3 q^{4} - 2 q^{5} + 3 \beta q^{6} + 10 \beta q^{7} + 11 \beta q^{8} + 9 q^{9} + 2 \beta q^{10} + 9 q^{12} - 12 \beta q^{13} - 50 q^{14} + 6 q^{15} - 31 q^{16} - 38 \beta q^{17} - 9 \beta q^{18} - 50 \beta q^{19} + 6 q^{20} - 30 \beta q^{21} + 112 q^{23} - 33 \beta q^{24} - 121 q^{25} + 60 q^{26} - 27 q^{27} - 30 \beta q^{28} + 82 \beta q^{29} - 6 \beta q^{30} + 120 q^{31} - 57 \beta q^{32} + 190 q^{34} - 20 \beta q^{35} - 27 q^{36} + 386 q^{37} + 250 q^{38} + 36 \beta q^{39} - 22 \beta q^{40} + 62 \beta q^{41} + 150 q^{42} + 74 \beta q^{43} - 18 q^{45} - 112 \beta q^{46} - 236 q^{47} + 93 q^{48} + 157 q^{49} + 121 \beta q^{50} + 114 \beta q^{51} + 36 \beta q^{52} - 78 q^{53} + 27 \beta q^{54} + 550 q^{56} + 150 \beta q^{57} - 410 q^{58} + 840 q^{59} - 18 q^{60} - 320 \beta q^{61} - 120 \beta q^{62} + 90 \beta q^{63} + 533 q^{64} + 24 \beta q^{65} - 276 q^{67} + 114 \beta q^{68} - 336 q^{69} + 100 q^{70} + 572 q^{71} + 99 \beta q^{72} - 484 \beta q^{73} - 386 \beta q^{74} + 363 q^{75} + 150 \beta q^{76} - 180 q^{78} + 330 \beta q^{79} + 62 q^{80} + 81 q^{81} - 310 q^{82} + 460 \beta q^{83} + 90 \beta q^{84} + 76 \beta q^{85} - 370 q^{86} - 246 \beta q^{87} + 914 q^{89} + 18 \beta q^{90} - 600 q^{91} - 336 q^{92} - 360 q^{93} + 236 \beta q^{94} + 100 \beta q^{95} + 171 \beta q^{96} + 386 q^{97} - 157 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 6 q^{4} - 4 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 6 q^{4} - 4 q^{5} + 18 q^{9} + 18 q^{12} - 100 q^{14} + 12 q^{15} - 62 q^{16} + 12 q^{20} + 224 q^{23} - 242 q^{25} + 120 q^{26} - 54 q^{27} + 240 q^{31} + 380 q^{34} - 54 q^{36} + 772 q^{37} + 500 q^{38} + 300 q^{42} - 36 q^{45} - 472 q^{47} + 186 q^{48} + 314 q^{49} - 156 q^{53} + 1100 q^{56} - 820 q^{58} + 1680 q^{59} - 36 q^{60} + 1066 q^{64} - 552 q^{67} - 672 q^{69} + 200 q^{70} + 1144 q^{71} + 726 q^{75} - 360 q^{78} + 124 q^{80} + 162 q^{81} - 620 q^{82} - 740 q^{86} + 1828 q^{89} - 1200 q^{91} - 672 q^{92} - 720 q^{93} + 772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−2.23607 −3.00000 −3.00000 −2.00000 6.70820 22.3607 24.5967 9.00000 4.47214
1.2 2.23607 −3.00000 −3.00000 −2.00000 −6.70820 −22.3607 −24.5967 9.00000 −4.47214
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(1\)

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 363.4.a.k 2
3.b odd 2 1 1089.4.a.o 2
11.b odd 2 1 inner 363.4.a.k 2
33.d even 2 1 1089.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.k 2 1.a even 1 1 trivial
363.4.a.k 2 11.b odd 2 1 inner
1089.4.a.o 2 3.b odd 2 1
1089.4.a.o 2 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 500 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 720 \) Copy content Toggle raw display
$17$ \( T^{2} - 7220 \) Copy content Toggle raw display
$19$ \( T^{2} - 12500 \) Copy content Toggle raw display
$23$ \( (T - 112)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 33620 \) Copy content Toggle raw display
$31$ \( (T - 120)^{2} \) Copy content Toggle raw display
$37$ \( (T - 386)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 19220 \) Copy content Toggle raw display
$43$ \( T^{2} - 27380 \) Copy content Toggle raw display
$47$ \( (T + 236)^{2} \) Copy content Toggle raw display
$53$ \( (T + 78)^{2} \) Copy content Toggle raw display
$59$ \( (T - 840)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 512000 \) Copy content Toggle raw display
$67$ \( (T + 276)^{2} \) Copy content Toggle raw display
$71$ \( (T - 572)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1171280 \) Copy content Toggle raw display
$79$ \( T^{2} - 544500 \) Copy content Toggle raw display
$83$ \( T^{2} - 1058000 \) Copy content Toggle raw display
$89$ \( (T - 914)^{2} \) Copy content Toggle raw display
$97$ \( (T - 386)^{2} \) Copy content Toggle raw display
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