Properties

Label 363.4.a.m
Level $363$
Weight $4$
Character orbit 363.a
Self dual yes
Analytic conductor $21.418$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} - 5 q^{4} - 3 q^{5} + 3 \beta q^{6} + 2 \beta q^{7} - 13 \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 q^{3} - 5 q^{4} - 3 q^{5} + 3 \beta q^{6} + 2 \beta q^{7} - 13 \beta q^{8} + 9 q^{9} - 3 \beta q^{10} - 15 q^{12} + \beta q^{13} + 6 q^{14} - 9 q^{15} + q^{16} + 7 \beta q^{17} + 9 \beta q^{18} - 76 \beta q^{19} + 15 q^{20} + 6 \beta q^{21} - 174 q^{23} - 39 \beta q^{24} - 116 q^{25} + 3 q^{26} + 27 q^{27} - 10 \beta q^{28} - 31 \beta q^{29} - 9 \beta q^{30} - 20 q^{31} + 105 \beta q^{32} + 21 q^{34} - 6 \beta q^{35} - 45 q^{36} - 299 q^{37} - 228 q^{38} + 3 \beta q^{39} + 39 \beta q^{40} - 175 \beta q^{41} + 18 q^{42} + 286 \beta q^{43} - 27 q^{45} - 174 \beta q^{46} - 384 q^{47} + 3 q^{48} - 331 q^{49} - 116 \beta q^{50} + 21 \beta q^{51} - 5 \beta q^{52} + 255 q^{53} + 27 \beta q^{54} - 78 q^{56} - 228 \beta q^{57} - 93 q^{58} + 570 q^{59} + 45 q^{60} + 216 \beta q^{61} - 20 \beta q^{62} + 18 \beta q^{63} + 307 q^{64} - 3 \beta q^{65} + 46 q^{67} - 35 \beta q^{68} - 522 q^{69} - 18 q^{70} - 630 q^{71} - 117 \beta q^{72} + 332 \beta q^{73} - 299 \beta q^{74} - 348 q^{75} + 380 \beta q^{76} + 9 q^{78} + 144 \beta q^{79} - 3 q^{80} + 81 q^{81} - 525 q^{82} + 744 \beta q^{83} - 30 \beta q^{84} - 21 \beta q^{85} + 858 q^{86} - 93 \beta q^{87} - 207 q^{89} - 27 \beta q^{90} + 6 q^{91} + 870 q^{92} - 60 q^{93} - 384 \beta q^{94} + 228 \beta q^{95} + 315 \beta q^{96} - 1615 q^{97} - 331 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 10 q^{4} - 6 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} - 10 q^{4} - 6 q^{5} + 18 q^{9} - 30 q^{12} + 12 q^{14} - 18 q^{15} + 2 q^{16} + 30 q^{20} - 348 q^{23} - 232 q^{25} + 6 q^{26} + 54 q^{27} - 40 q^{31} + 42 q^{34} - 90 q^{36} - 598 q^{37} - 456 q^{38} + 36 q^{42} - 54 q^{45} - 768 q^{47} + 6 q^{48} - 662 q^{49} + 510 q^{53} - 156 q^{56} - 186 q^{58} + 1140 q^{59} + 90 q^{60} + 614 q^{64} + 92 q^{67} - 1044 q^{69} - 36 q^{70} - 1260 q^{71} - 696 q^{75} + 18 q^{78} - 6 q^{80} + 162 q^{81} - 1050 q^{82} + 1716 q^{86} - 414 q^{89} + 12 q^{91} + 1740 q^{92} - 120 q^{93} - 3230 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 3.00000 −5.00000 −3.00000 −5.19615 −3.46410 22.5167 9.00000 5.19615
1.2 1.73205 3.00000 −5.00000 −3.00000 5.19615 3.46410 −22.5167 9.00000 −5.19615
\(n\): 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.m 2
3.b odd 2 1 1089.4.a.n 2
11.b odd 2 1 inner 363.4.a.m 2
33.d even 2 1 1089.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.4.a.m 2 1.a even 1 1 trivial
363.4.a.m 2 11.b odd 2 1 inner
1089.4.a.n 2 3.b odd 2 1
1089.4.a.n 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(363))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 12 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3 \) Copy content Toggle raw display
$17$ \( T^{2} - 147 \) Copy content Toggle raw display
$19$ \( T^{2} - 17328 \) Copy content Toggle raw display
$23$ \( (T + 174)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2883 \) Copy content Toggle raw display
$31$ \( (T + 20)^{2} \) Copy content Toggle raw display
$37$ \( (T + 299)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 91875 \) Copy content Toggle raw display
$43$ \( T^{2} - 245388 \) Copy content Toggle raw display
$47$ \( (T + 384)^{2} \) Copy content Toggle raw display
$53$ \( (T - 255)^{2} \) Copy content Toggle raw display
$59$ \( (T - 570)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 139968 \) Copy content Toggle raw display
$67$ \( (T - 46)^{2} \) Copy content Toggle raw display
$71$ \( (T + 630)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 330672 \) Copy content Toggle raw display
$79$ \( T^{2} - 62208 \) Copy content Toggle raw display
$83$ \( T^{2} - 1660608 \) Copy content Toggle raw display
$89$ \( (T + 207)^{2} \) Copy content Toggle raw display
$97$ \( (T + 1615)^{2} \) Copy content Toggle raw display
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