Properties

Label 186.2.m.b
Level $186$
Weight $2$
Character orbit 186.m
Analytic conductor $1.485$
Analytic rank $0$
Dimension $8$
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] = [186,2,Mod(7,186)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(186, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("186.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 186 = 2 \cdot 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 186.m (of order \(15\), degree \(8\), 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: \(1.48521747760\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{15}^{6} q^{2} - \zeta_{15}^{4} q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{4} + (\zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{4}) q^{5} + ( - \zeta_{15}^{5} - 1) q^{6} + ( - \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + \zeta_{15} + 2) q^{7} - \zeta_{15}^{3} q^{8} + (\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{15}^{6} q^{2} - \zeta_{15}^{4} q^{3} + ( - \zeta_{15}^{7} - \zeta_{15}^{2}) q^{4} + (\zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{4}) q^{5} + ( - \zeta_{15}^{5} - 1) q^{6} + ( - \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + \zeta_{15} + 2) q^{7} - \zeta_{15}^{3} q^{8} + (\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1) q^{9} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + \zeta_{15}^{5} + \zeta_{15}^{2} - 2 \zeta_{15} + 1) q^{10} + (2 \zeta_{15}^{7} - 2 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{4} + 2 \zeta_{15} - 3) q^{11} - \zeta_{15} q^{12} + (\zeta_{15}^{7} - 3 \zeta_{15}^{6} + \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + 2 \zeta_{15}^{2} - 1) q^{13} + (2 \zeta_{15}^{7} - 3 \zeta_{15}^{6} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15}) q^{14} + (\zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15}) q^{15} + (\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{16} + (\zeta_{15}^{6} + 3 \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15}) q^{17} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{18} + ( - 2 \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + 2 \zeta_{15}^{3} - 3 \zeta_{15} + 3) q^{19} + (\zeta_{15}^{3} - 2 \zeta_{15}^{2} + \zeta_{15}) q^{20} + (\zeta_{15}^{7} - \zeta_{15}^{6} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{4} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{2} - \zeta_{15} + 1) q^{21} + ( - 2 \zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{5} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{2} - 1) q^{22} + ( - \zeta_{15}^{4} + 4 \zeta_{15}^{3} - \zeta_{15}^{2}) q^{23} + \zeta_{15}^{7} q^{24} + ( - 4 \zeta_{15}^{7} + 8 \zeta_{15}^{6} - 2 \zeta_{15}^{5} + \zeta_{15}^{4} + 3 \zeta_{15}^{3} - 5 \zeta_{15}^{2} + \cdots + 2) q^{25} + \cdots + (2 \zeta_{15}^{7} - 4 \zeta_{15}^{6} + \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + 3 \zeta_{15}^{2} - \zeta_{15} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - q^{3} - 2 q^{4} + 7 q^{5} - 4 q^{6} + 14 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - q^{3} - 2 q^{4} + 7 q^{5} - 4 q^{6} + 14 q^{7} + 2 q^{8} + q^{9} + 8 q^{10} - 7 q^{11} - q^{12} + 5 q^{13} + 16 q^{14} - q^{15} - 2 q^{16} - 12 q^{17} - q^{18} + 8 q^{19} - 3 q^{20} + 6 q^{21} - 8 q^{22} - 10 q^{23} + q^{24} - q^{25} - 10 q^{26} + 2 q^{27} - 6 q^{28} - 27 q^{29} - 14 q^{30} + 4 q^{31} - 8 q^{32} - 4 q^{33} - 8 q^{34} - 26 q^{35} - 4 q^{36} + 18 q^{37} + 12 q^{38} - 5 q^{39} + 3 q^{40} + 3 q^{41} - 16 q^{42} - 5 q^{43} + 13 q^{44} - 3 q^{45} + 5 q^{46} - 16 q^{47} - q^{48} - 21 q^{49} + 36 q^{50} + 12 q^{51} - 10 q^{52} - 16 q^{53} - 2 q^{54} - 4 q^{55} + q^{56} - 8 q^{57} - 18 q^{58} - 13 q^{59} - q^{60} + 30 q^{61} + 16 q^{62} + 2 q^{63} - 2 q^{64} + 35 q^{65} - q^{66} + 27 q^{67} - 7 q^{68} - 5 q^{69} + 26 q^{70} - 8 q^{71} - q^{72} - 53 q^{73} - 18 q^{74} + 36 q^{75} + 3 q^{76} + 26 q^{77} + 5 q^{78} - 31 q^{79} + 7 q^{80} + q^{81} - 23 q^{82} - 4 q^{83} - 14 q^{84} + 8 q^{85} + 25 q^{86} + 9 q^{87} - 3 q^{88} - 32 q^{89} + 8 q^{90} + 30 q^{91} + 30 q^{92} + 13 q^{93} - 4 q^{94} - 2 q^{95} + q^{96} + 51 q^{97} + 11 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).

\(n\) \(125\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{15} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.669131 0.743145i
−0.104528 + 0.994522i
−0.104528 0.994522i
0.913545 0.406737i
0.913545 + 0.406737i
0.669131 + 0.743145i
−0.978148 + 0.207912i
−0.978148 0.207912i
−0.309017 0.951057i 0.978148 0.207912i −0.809017 + 0.587785i 0.330869 0.573083i −0.500000 0.866025i 3.08268 + 1.37250i 0.809017 + 0.587785i 0.913545 0.406737i −0.647278 0.137583i
19.1 0.809017 + 0.587785i −0.913545 0.406737i 0.309017 + 0.951057i 1.10453 1.91310i −0.500000 0.866025i 2.06460 2.29297i −0.309017 + 0.951057i 0.669131 + 0.743145i 2.01807 0.898504i
49.1 0.809017 0.587785i −0.913545 + 0.406737i 0.309017 0.951057i 1.10453 + 1.91310i −0.500000 + 0.866025i 2.06460 + 2.29297i −0.309017 0.951057i 0.669131 0.743145i 2.01807 + 0.898504i
103.1 0.809017 + 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i 0.0864545 + 0.149744i −0.500000 + 0.866025i 1.43540 + 0.305103i −0.309017 + 0.951057i −0.978148 + 0.207912i −0.0180739 + 0.171962i
121.1 0.809017 0.587785i 0.104528 0.994522i 0.309017 0.951057i 0.0864545 0.149744i −0.500000 0.866025i 1.43540 0.305103i −0.309017 0.951057i −0.978148 0.207912i −0.0180739 0.171962i
133.1 −0.309017 + 0.951057i 0.978148 + 0.207912i −0.809017 0.587785i 0.330869 + 0.573083i −0.500000 + 0.866025i 3.08268 1.37250i 0.809017 0.587785i 0.913545 + 0.406737i −0.647278 + 0.137583i
169.1 −0.309017 + 0.951057i −0.669131 + 0.743145i −0.809017 0.587785i 1.97815 3.42625i −0.500000 0.866025i 0.417324 3.97057i 0.809017 0.587785i −0.104528 0.994522i 2.64728 + 2.94010i
175.1 −0.309017 0.951057i −0.669131 0.743145i −0.809017 + 0.587785i 1.97815 + 3.42625i −0.500000 + 0.866025i 0.417324 + 3.97057i 0.809017 + 0.587785i −0.104528 + 0.994522i 2.64728 2.94010i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 186.2.m.b 8
3.b odd 2 1 558.2.ba.a 8
31.g even 15 1 inner 186.2.m.b 8
31.g even 15 1 5766.2.a.w 4
31.h odd 30 1 5766.2.a.z 4
93.o odd 30 1 558.2.ba.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
186.2.m.b 8 1.a even 1 1 trivial
186.2.m.b 8 31.g even 15 1 inner
558.2.ba.a 8 3.b odd 2 1
558.2.ba.a 8 93.o odd 30 1
5766.2.a.w 4 31.g even 15 1
5766.2.a.z 4 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 7T_{5}^{7} + 35T_{5}^{6} - 82T_{5}^{5} + 139T_{5}^{4} - 98T_{5}^{3} + 50T_{5}^{2} - 8T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(186, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + 35 T^{6} - 82 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - 14 T^{7} + 105 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + 25 T^{6} + 62 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + 30 T^{6} - 100 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} + 12 T^{7} + 50 T^{6} + \cdots + 7921 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + 30 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$23$ \( T^{8} + 10 T^{7} + 55 T^{6} + \cdots + 21025 \) Copy content Toggle raw display
$29$ \( T^{8} + 27 T^{7} + 348 T^{6} + \cdots + 68121 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} - 5 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( T^{8} - 18 T^{7} + 240 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} - 55 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} + 10 T^{6} + \cdots + 24025 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + 252 T^{6} + \cdots + 891136 \) Copy content Toggle raw display
$53$ \( T^{8} + 16 T^{7} + 70 T^{6} + \cdots + 201601 \) Copy content Toggle raw display
$59$ \( T^{8} + 13 T^{7} + 40 T^{6} + \cdots + 1515361 \) Copy content Toggle raw display
$61$ \( (T^{4} - 15 T^{3} + 65 T^{2} - 75 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 27 T^{7} + 625 T^{6} + \cdots + 79370281 \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} + 30 T^{6} + \cdots + 16394401 \) Copy content Toggle raw display
$73$ \( T^{8} + 53 T^{7} + 1365 T^{6} + \cdots + 82646281 \) Copy content Toggle raw display
$79$ \( T^{8} + 31 T^{7} + 445 T^{6} + \cdots + 1510441 \) Copy content Toggle raw display
$83$ \( T^{8} + 4 T^{7} + 150 T^{6} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( T^{8} + 32 T^{7} + 483 T^{6} + \cdots + 229441 \) Copy content Toggle raw display
$97$ \( T^{8} - 51 T^{7} + 1262 T^{6} + \cdots + 326041 \) Copy content Toggle raw display
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