Properties

Label 363.2.e.b
Level $363$
Weight $2$
Character orbit 363.e
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
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] = [363,2,Mod(124,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\), degree \(4\), 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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
−2.11803 + 1.53884i −0.309017 0.951057i 1.50000 4.61653i 0.500000 + 0.363271i 2.11803 + 1.53884i 0.309017 0.951057i 2.30902 + 7.10642i −0.809017 + 0.587785i −1.61803
130.1 0.118034 0.363271i 0.809017 0.587785i 1.50000 + 1.08981i 0.500000 + 1.53884i −0.118034 0.363271i −0.809017 0.587785i 1.19098 0.865300i 0.309017 0.951057i 0.618034
148.1 0.118034 + 0.363271i 0.809017 + 0.587785i 1.50000 1.08981i 0.500000 1.53884i −0.118034 + 0.363271i −0.809017 + 0.587785i 1.19098 + 0.865300i 0.309017 + 0.951057i 0.618034
202.1 −2.11803 1.53884i −0.309017 + 0.951057i 1.50000 + 4.61653i 0.500000 0.363271i 2.11803 1.53884i 0.309017 + 0.951057i 2.30902 7.10642i −0.809017 0.587785i −1.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.e.b 4
11.b odd 2 1 363.2.e.k 4
11.c even 5 1 363.2.a.i 2
11.c even 5 1 inner 363.2.e.b 4
11.c even 5 2 363.2.e.f 4
11.d odd 10 2 33.2.e.b 4
11.d odd 10 1 363.2.a.d 2
11.d odd 10 1 363.2.e.k 4
33.f even 10 2 99.2.f.a 4
33.f even 10 1 1089.2.a.t 2
33.h odd 10 1 1089.2.a.l 2
44.g even 10 2 528.2.y.b 4
44.g even 10 1 5808.2.a.cj 2
44.h odd 10 1 5808.2.a.ci 2
55.h odd 10 2 825.2.n.c 4
55.h odd 10 1 9075.2.a.cb 2
55.j even 10 1 9075.2.a.u 2
55.l even 20 4 825.2.bx.d 8
99.o odd 30 4 891.2.n.c 8
99.p even 30 4 891.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 11.d odd 10 2
99.2.f.a 4 33.f even 10 2
363.2.a.d 2 11.d odd 10 1
363.2.a.i 2 11.c even 5 1
363.2.e.b 4 1.a even 1 1 trivial
363.2.e.b 4 11.c even 5 1 inner
363.2.e.f 4 11.c even 5 2
363.2.e.k 4 11.b odd 2 1
363.2.e.k 4 11.d odd 10 1
528.2.y.b 4 44.g even 10 2
825.2.n.c 4 55.h odd 10 2
825.2.bx.d 8 55.l even 20 4
891.2.n.b 8 99.p even 30 4
891.2.n.c 8 99.o odd 30 4
1089.2.a.l 2 33.h odd 10 1
1089.2.a.t 2 33.f even 10 1
5808.2.a.ci 2 44.h odd 10 1
5808.2.a.cj 2 44.g even 10 1
9075.2.a.u 2 55.j even 10 1
9075.2.a.cb 2 55.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 6T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 6 T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + 19 T^{2} - 7 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( T^{4} + T^{3} + 31 T^{2} - 99 T + 121 \) Copy content Toggle raw display
$41$ \( T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1 \) Copy content Toggle raw display
$43$ \( (T^{2} - 45)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} + 6 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$59$ \( T^{4} + 11 T^{3} + 76 T^{2} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$67$ \( (T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} + 136 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} - 11 T^{3} + 121 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$83$ \( T^{4} - 7 T^{3} + 69 T^{2} - 143 T + 121 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T + 31)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
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