Properties

Label 363.2.e.c
Level $363$
Weight $2$
Character orbit 363.e
Analytic conductor $2.899$
Analytic rank $0$
Dimension $4$
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + \zeta_{10}^{2} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (\zeta_{10}^{2} + \zeta_{10} + 1) q^{5} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{8}+ \cdots + (\zeta_{10}^{3} - \zeta_{10}^{2} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + \zeta_{10}^{2} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (\zeta_{10}^{2} + \zeta_{10} + 1) q^{5} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{6} + 3 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{8}+ \cdots + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 5 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} + 3 q^{7} - 5 q^{8} - q^{9} - 2 q^{10} - 2 q^{12} - q^{13} - 9 q^{14} - q^{15} - 6 q^{16} + 3 q^{17} + 3 q^{18} - 5 q^{19} - 7 q^{20} - 12 q^{21} - 4 q^{23} + 5 q^{24} + 9 q^{25} - 17 q^{26} - q^{27} - 9 q^{28} - 10 q^{29} - 2 q^{30} - 7 q^{31} + 18 q^{32} - 4 q^{34} - 12 q^{35} - 2 q^{36} + 7 q^{37} + 5 q^{38} - q^{39} + 5 q^{40} + 17 q^{41} + 6 q^{42} - 16 q^{43} - 6 q^{45} + 2 q^{46} + 2 q^{47} + 9 q^{48} - 2 q^{49} + 3 q^{50} - 2 q^{51} + 8 q^{52} + 11 q^{53} - 2 q^{54} + 10 q^{57} + 10 q^{58} + 15 q^{59} - 7 q^{60} + 12 q^{61} - 9 q^{62} + 3 q^{63} + 3 q^{64} + 14 q^{65} + 2 q^{67} - 4 q^{68} + 11 q^{69} + 6 q^{70} + 18 q^{71} + 5 q^{72} + 4 q^{73} - 11 q^{74} - 6 q^{75} + 20 q^{76} + 18 q^{78} + 15 q^{79} - 6 q^{80} - q^{81} - 6 q^{82} + 9 q^{83} + 6 q^{84} + 3 q^{85} + 8 q^{86} + 20 q^{89} + 3 q^{90} - 27 q^{91} + 2 q^{92} - 7 q^{93} - q^{94} + 10 q^{95} - 2 q^{96} + 32 q^{97} - 4 q^{98}+O(q^{100}) \) 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
−0.500000 + 0.363271i 0.309017 + 0.951057i −0.500000 + 1.53884i 2.11803 + 1.53884i −0.500000 0.363271i −0.927051 + 2.85317i −0.690983 2.12663i −0.809017 + 0.587785i −1.61803
130.1 −0.500000 + 1.53884i −0.809017 + 0.587785i −0.500000 0.363271i −0.118034 0.363271i −0.500000 1.53884i 2.42705 + 1.76336i −1.80902 + 1.31433i 0.309017 0.951057i 0.618034
148.1 −0.500000 1.53884i −0.809017 0.587785i −0.500000 + 0.363271i −0.118034 + 0.363271i −0.500000 + 1.53884i 2.42705 1.76336i −1.80902 1.31433i 0.309017 + 0.951057i 0.618034
202.1 −0.500000 0.363271i 0.309017 0.951057i −0.500000 1.53884i 2.11803 1.53884i −0.500000 + 0.363271i −0.927051 2.85317i −0.690983 + 2.12663i −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.c 4
11.b odd 2 1 363.2.e.h 4
11.c even 5 1 363.2.a.e 2
11.c even 5 1 inner 363.2.e.c 4
11.c even 5 2 363.2.e.j 4
11.d odd 10 2 33.2.e.a 4
11.d odd 10 1 363.2.a.h 2
11.d odd 10 1 363.2.e.h 4
33.f even 10 2 99.2.f.b 4
33.f even 10 1 1089.2.a.m 2
33.h odd 10 1 1089.2.a.s 2
44.g even 10 2 528.2.y.f 4
44.g even 10 1 5808.2.a.bl 2
44.h odd 10 1 5808.2.a.bm 2
55.h odd 10 2 825.2.n.f 4
55.h odd 10 1 9075.2.a.x 2
55.j even 10 1 9075.2.a.bv 2
55.l even 20 4 825.2.bx.b 8
99.o odd 30 4 891.2.n.d 8
99.p even 30 4 891.2.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 11.d odd 10 2
99.2.f.b 4 33.f even 10 2
363.2.a.e 2 11.c even 5 1
363.2.a.h 2 11.d odd 10 1
363.2.e.c 4 1.a even 1 1 trivial
363.2.e.c 4 11.c even 5 1 inner
363.2.e.h 4 11.b odd 2 1
363.2.e.h 4 11.d odd 10 1
363.2.e.j 4 11.c even 5 2
528.2.y.f 4 44.g even 10 2
825.2.n.f 4 55.h odd 10 2
825.2.bx.b 8 55.l even 20 4
891.2.n.a 8 99.p even 30 4
891.2.n.d 8 99.o odd 30 4
1089.2.a.m 2 33.f even 10 1
1089.2.a.s 2 33.h odd 10 1
5808.2.a.bl 2 44.g even 10 1
5808.2.a.bm 2 44.h odd 10 1
9075.2.a.x 2 55.h odd 10 1
9075.2.a.bv 2 55.j even 10 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 19)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 10 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$37$ \( T^{4} - 7 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} - 17 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$59$ \( T^{4} - 15 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$67$ \( (T^{2} - T - 101)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{4} - 15 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$83$ \( T^{4} - 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 32 T^{3} + \cdots + 44521 \) Copy content Toggle raw display
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