Properties

Label 370.2.q.c
Level $370$
Weight $2$
Character orbit 370.q
Analytic conductor $2.954$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(97,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.q (of order \(12\), degree \(4\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{4} q^{2} + (\zeta_{24}^{6} + \cdots + \zeta_{24}^{4}) q^{3}+ \cdots + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{4} q^{2} + (\zeta_{24}^{6} + \cdots + \zeta_{24}^{4}) q^{3}+ \cdots + ( - 4 \zeta_{24}^{7} + \cdots - 4 \zeta_{24}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{7} - 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 4 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 12 q^{7} - 8 q^{8} - 12 q^{9} - 4 q^{10} - 8 q^{12} - 4 q^{13} + 12 q^{14} - 4 q^{16} + 12 q^{17} - 12 q^{18} + 4 q^{19} - 8 q^{20} + 12 q^{21} - 12 q^{22} - 8 q^{23} - 4 q^{24} - 8 q^{26} - 8 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{31} + 4 q^{32} - 12 q^{33} + 12 q^{34} + 8 q^{38} + 16 q^{39} - 4 q^{40} + 12 q^{41} + 12 q^{42} + 16 q^{43} - 12 q^{44} - 12 q^{45} - 4 q^{46} - 8 q^{47} + 4 q^{48} + 36 q^{49} + 20 q^{51} - 4 q^{52} + 16 q^{53} - 4 q^{54} + 16 q^{55} - 12 q^{56} + 4 q^{57} - 24 q^{58} - 8 q^{59} - 12 q^{60} + 20 q^{62} - 4 q^{63} + 8 q^{64} + 16 q^{65} - 16 q^{67} + 16 q^{69} - 12 q^{70} - 4 q^{71} + 12 q^{72} + 16 q^{73} - 20 q^{75} + 4 q^{76} + 4 q^{77} + 8 q^{78} - 32 q^{79} + 4 q^{80} + 8 q^{81} + 16 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 24 q^{90} - 24 q^{91} + 4 q^{92} + 20 q^{93} - 16 q^{94} + 16 q^{95} + 8 q^{96} + 36 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(-\zeta_{24}^{2}\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
0.500000 0.866025i 0.241181 0.900100i −0.500000 0.866025i 1.81431 + 1.30701i −0.658919 0.658919i 0.0267682 0.0999004i −1.00000 1.84607 + 1.06583i 2.03906 0.917738i
97.2 0.500000 0.866025i 0.758819 2.83195i −0.500000 0.866025i 0.917738 2.03906i −2.07313 2.07313i −0.490870 + 1.83195i −1.00000 −4.84607 2.79788i −1.30701 1.81431i
103.1 0.500000 + 0.866025i 0.241181 + 0.900100i −0.500000 + 0.866025i 1.81431 1.30701i −0.658919 + 0.658919i 0.0267682 + 0.0999004i −1.00000 1.84607 1.06583i 2.03906 + 0.917738i
103.2 0.500000 + 0.866025i 0.758819 + 2.83195i −0.500000 + 0.866025i 0.917738 + 2.03906i −2.07313 + 2.07313i −0.490870 1.83195i −1.00000 −4.84607 + 2.79788i −1.30701 + 1.81431i
267.1 0.500000 + 0.866025i −0.465926 + 0.124844i −0.500000 + 0.866025i −2.03906 + 0.917738i −0.341081 0.341081i 4.19798 1.12484i −1.00000 −2.39658 + 1.38366i −1.81431 1.30701i
267.2 0.500000 + 0.866025i 1.46593 0.392794i −0.500000 + 0.866025i 1.30701 + 1.81431i 1.07313 + 1.07313i 2.26612 0.607206i −1.00000 −0.603425 + 0.348387i −0.917738 + 2.03906i
273.1 0.500000 0.866025i −0.465926 0.124844i −0.500000 0.866025i −2.03906 0.917738i −0.341081 + 0.341081i 4.19798 + 1.12484i −1.00000 −2.39658 1.38366i −1.81431 + 1.30701i
273.2 0.500000 0.866025i 1.46593 + 0.392794i −0.500000 0.866025i 1.30701 1.81431i 1.07313 1.07313i 2.26612 + 0.607206i −1.00000 −0.603425 0.348387i −0.917738 2.03906i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.p even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.q.c 8
5.c odd 4 1 370.2.r.c yes 8
37.g odd 12 1 370.2.r.c yes 8
185.p even 12 1 inner 370.2.q.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.q.c 8 1.a even 1 1 trivial
370.2.q.c 8 185.p even 12 1 inner
370.2.r.c yes 8 5.c odd 4 1
370.2.r.c yes 8 37.g odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 4T_{3}^{7} + 14T_{3}^{6} - 24T_{3}^{5} + 14T_{3}^{4} - 4T_{3}^{2} + 8T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} + 48 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} + \cdots + 142)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 24 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + \cdots + 85264 \) Copy content Toggle raw display
$37$ \( T^{8} - 529 T^{4} + 1874161 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 1270129 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots - 386)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} + \cdots + 20394256 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + \cdots + 160000 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + \cdots + 2262016 \) Copy content Toggle raw display
$61$ \( T^{8} - 48 T^{6} + \cdots + 8427409 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} + \cdots + 35344 \) Copy content Toggle raw display
$71$ \( T^{8} + 4 T^{7} + \cdots + 21316 \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + \cdots + 341056 \) Copy content Toggle raw display
$79$ \( T^{8} + 32 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$83$ \( T^{8} - 126 T^{6} + \cdots + 240436036 \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} + \cdots + 332929 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 1120843441 \) Copy content Toggle raw display
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