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Properties

Label 186.2.m.e
Level $186$
Weight $2$
Character orbit 186.m
Analytic conductor $1.485$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 39x^{14} + 630x^{12} + 5436x^{10} + 26909x^{8} + 76236x^{6} + 116550x^{4} + 85119x^{2} + 22801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 39x^{14} + 630x^{12} + 5436x^{10} + 26909x^{8} + 76236x^{6} + 116550x^{4} + 85119x^{2} + 22801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13 \nu^{14} - 460 \nu^{12} - 6477 \nu^{10} - 46053 \nu^{8} - 173383 \nu^{6} - 332920 \nu^{4} + \cdots - 92412 ) / 2596 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13 \nu^{14} - 460 \nu^{12} - 6477 \nu^{10} - 46053 \nu^{8} - 173383 \nu^{6} - 332920 \nu^{4} + \cdots - 92412 ) / 2596 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 364222 \nu^{15} - 11000199 \nu^{14} + 11420067 \nu^{13} - 359137041 \nu^{12} + \cdots - 36170993755 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 189584 \nu^{15} - 19614145 \nu^{14} - 7776561 \nu^{13} - 637749859 \nu^{12} + \cdots - 85991967881 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 420 \nu^{15} + 3013 \nu^{14} + 14090 \nu^{13} + 97959 \nu^{12} + 181536 \nu^{11} + 1261121 \nu^{10} + \cdots + 13377543 ) / 524156 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1584824 \nu^{15} - 9604959 \nu^{14} + 54583541 \nu^{13} - 312330061 \nu^{12} + \cdots - 41551770035 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1687486 \nu^{15} - 28654515 \nu^{14} + 61915097 \nu^{13} - 940090515 \nu^{12} + \cdots - 114436188201 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 88593 \nu^{15} + 63420 \nu^{14} - 3000164 \nu^{13} + 2127590 \nu^{12} - 41021781 \nu^{11} + \cdots - 244580740 ) / 79147556 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 88593 \nu^{15} + 63420 \nu^{14} + 3000164 \nu^{13} + 2127590 \nu^{12} + 41021781 \nu^{11} + \cdots - 244580740 ) / 79147556 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2572785 \nu^{15} + 8082577 \nu^{14} - 82651079 \nu^{13} + 259373304 \nu^{12} + \cdots + 20592793214 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2572785 \nu^{15} - 8082577 \nu^{14} - 82651079 \nu^{13} - 259373304 \nu^{12} + \cdots - 20592793214 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 612 \nu^{15} - 21905 \nu^{13} - 316100 \nu^{11} - 2348805 \nu^{9} - 9514305 \nu^{7} + \cdots - 195998 ) / 391996 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6759313 \nu^{15} + 21967178 \nu^{14} - 208878274 \nu^{13} + 727524191 \nu^{12} + \cdots + 93686357403 ) / 1741246232 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4490610 \nu^{15} - 1126611 \nu^{14} + 147841295 \nu^{13} - 33555069 \nu^{12} + \cdots + 10461777545 ) / 870623116 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4490610 \nu^{15} + 1126611 \nu^{14} + 147841295 \nu^{13} + 33555069 \nu^{12} + \cdots - 10461777545 ) / 870623116 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{12} - 2\beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} - 2 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{15} + 7 \beta_{14} - 9 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 17 \beta_{9} + 16 \beta_{8} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{15} + 13 \beta_{14} + 24 \beta_{13} + 32 \beta_{12} - 46 \beta_{11} - 22 \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41 \beta_{15} - 41 \beta_{14} + 77 \beta_{12} - 60 \beta_{11} + 60 \beta_{10} - 130 \beta_{9} + \cdots - 193 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 109 \beta_{15} - 127 \beta_{14} - 236 \beta_{13} - 274 \beta_{12} + 458 \beta_{11} + 222 \beta_{10} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 214 \beta_{15} + 214 \beta_{14} - 663 \beta_{12} + 678 \beta_{11} - 678 \beta_{10} + 986 \beta_{9} + \cdots + 1507 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1046 \beta_{15} + 1134 \beta_{14} + 2180 \beta_{13} + 2224 \beta_{12} - 4338 \beta_{11} - 2158 \beta_{10} + \cdots - 523 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 889 \beta_{15} - 889 \beta_{14} + 5741 \beta_{12} - 6914 \beta_{11} + 6914 \beta_{10} - 7564 \beta_{9} + \cdots - 12124 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9850 \beta_{15} - 9744 \beta_{14} - 19594 \beta_{13} - 17803 \beta_{12} + 40032 \beta_{11} + 20438 \beta_{10} + \cdots + 5794 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1056 \beta_{15} + 1056 \beta_{14} - 49923 \beta_{12} + 67134 \beta_{11} - 67134 \beta_{10} + \cdots + 99683 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 91626 \beta_{15} + 82158 \beta_{14} + 173784 \beta_{13} + 142122 \beta_{12} - 363966 \beta_{11} + \cdots - 59277 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 37639 \beta_{15} + 37639 \beta_{14} + 435766 \beta_{12} - 634792 \beta_{11} + 634792 \beta_{10} + \cdots - 833688 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 845502 \beta_{15} - 686186 \beta_{14} - 1531688 \beta_{13} - 1135482 \beta_{12} + 3280672 \beta_{11} + \cdots + 581025 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(125\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{6} + \beta_{9}\)

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
2.47662i
− 0.990333i
0.805988i
− 2.79503i
− 0.805988i
2.79503i
2.32418i
− 1.51071i
− 2.32418i
1.51071i
− 2.47662i
0.990333i
2.58970i
− 3.00552i
− 2.58970i
3.00552i
0.309017 + 0.951057i 0.978148 − 0.207912i −0.809017 + 0.587785i −1.48491 + 2.57194i 0.500000 + 0.866025i 1.48643 + 0.661801i −0.809017 − 0.587785i 0.913545 − 0.406737i −2.90492 − 0.617459i
7.2 0.309017 + 0.951057i 0.978148 − 0.207912i −0.809017 + 0.587785i 1.96306 − 3.40011i 0.500000 + 0.866025i −0.105906 − 0.0471522i −0.809017 − 0.587785i 0.913545 − 0.406737i 3.84032 + 0.816284i
19.1 −0.809017 − 0.587785i −0.913545 − 0.406737i 0.309017 + 0.951057i −1.08112 + 1.87255i 0.500000 + 0.866025i 3.04878 − 3.38601i 0.309017 − 0.951057i 0.669131 + 0.743145i 1.97530 − 0.879462i
19.2 −0.809017 − 0.587785i −0.913545 − 0.406737i 0.309017 + 0.951057i −0.332426 + 0.575778i 0.500000 + 0.866025i −2.74588 + 3.04960i 0.309017 − 0.951057i 0.669131 + 0.743145i 0.607372 − 0.270419i
49.1 −0.809017 + 0.587785i −0.913545 + 0.406737i 0.309017 − 0.951057i −1.08112 − 1.87255i 0.500000 − 0.866025i 3.04878 + 3.38601i 0.309017 + 0.951057i 0.669131 − 0.743145i 1.97530 + 0.879462i
49.2 −0.809017 + 0.587785i −0.913545 + 0.406737i 0.309017 − 0.951057i −0.332426 − 0.575778i 0.500000 − 0.866025i −2.74588 − 3.04960i 0.309017 + 0.951057i 0.669131 − 0.743145i 0.607372 + 0.270419i
103.1 −0.809017 − 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i −1.62268 − 2.81056i 0.500000 − 0.866025i −4.09040 − 0.869442i 0.309017 − 0.951057i −0.978148 + 0.207912i −0.339232 + 3.22758i
103.2 −0.809017 − 0.587785i 0.104528 + 0.994522i 0.309017 + 0.951057i 1.22721 + 2.12558i 0.500000 − 0.866025i −1.56660 − 0.332992i 0.309017 − 0.951057i −0.978148 + 0.207912i 0.256556 − 2.44097i
121.1 −0.809017 + 0.587785i 0.104528 − 0.994522i 0.309017 − 0.951057i −1.62268 + 2.81056i 0.500000 + 0.866025i −4.09040 + 0.869442i 0.309017 + 0.951057i −0.978148 − 0.207912i −0.339232 − 3.22758i
121.2 −0.809017 + 0.587785i 0.104528 − 0.994522i 0.309017 − 0.951057i 1.22721 − 2.12558i 0.500000 + 0.866025i −1.56660 + 0.332992i 0.309017 + 0.951057i −0.978148 − 0.207912i 0.256556 + 2.44097i
133.1 0.309017 − 0.951057i 0.978148 + 0.207912i −0.809017 − 0.587785i −1.48491 − 2.57194i 0.500000 − 0.866025i 1.48643 − 0.661801i −0.809017 + 0.587785i 0.913545 + 0.406737i −2.90492 + 0.617459i
133.2 0.309017 − 0.951057i 0.978148 + 0.207912i −0.809017 − 0.587785i 1.96306 + 3.40011i 0.500000 − 0.866025i −0.105906 + 0.0471522i −0.809017 + 0.587785i 0.913545 + 0.406737i 3.84032 − 0.816284i
169.1 0.309017 − 0.951057i −0.669131 + 0.743145i −0.809017 − 0.587785i −1.72246 + 2.98338i 0.500000 + 0.866025i −0.372693 + 3.54594i −0.809017 + 0.587785i −0.104528 − 0.994522i 2.30509 + 2.56007i
169.2 0.309017 − 0.951057i −0.669131 + 0.743145i −0.809017 − 0.587785i 0.553325 − 0.958386i 0.500000 + 0.866025i 0.346272 − 3.29456i −0.809017 + 0.587785i −0.104528 − 0.994522i −0.740493 − 0.822401i
175.1 0.309017 + 0.951057i −0.669131 − 0.743145i −0.809017 + 0.587785i −1.72246 − 2.98338i 0.500000 − 0.866025i −0.372693 − 3.54594i −0.809017 − 0.587785i −0.104528 + 0.994522i 2.30509 − 2.56007i
175.2 0.309017 + 0.951057i −0.669131 − 0.743145i −0.809017 + 0.587785i 0.553325 + 0.958386i 0.500000 − 0.866025i 0.346272 + 3.29456i −0.809017 − 0.587785i −0.104528 + 0.994522i −0.740493 + 0.822401i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
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.e ✓ 16
3.b odd 2 1 558.2.ba.g 16
31.g even 15 1 inner 186.2.m.e ✓ 16
31.g even 15 1 5766.2.a.bp 8
31.h odd 30 1 5766.2.a.br 8
93.o odd 30 1 558.2.ba.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
186.2.m.e ✓ 16 1.a even 1 1 trivial
186.2.m.e ✓ 16 31.g even 15 1 inner
558.2.ba.g 16 3.b odd 2 1
558.2.ba.g 16 93.o odd 30 1
5766.2.a.bp 8 31.g even 15 1
5766.2.a.br 8 31.h odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 5 T_{5}^{15} + 42 T_{5}^{14} + 145 T_{5}^{13} + 870 T_{5}^{12} + 2640 T_{5}^{11} + \cdots + 259081 \) acting on \(S_{2}^{\mathrm{new}}(186, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} - T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + \cdots + 259081 \) Copy content Toggle raw display
$7$ \( T^{16} + 8 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$11$ \( T^{16} - 17 T^{15} + \cdots + 2627641 \) Copy content Toggle raw display
$13$ \( T^{16} + 5 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{16} - 6 T^{15} + \cdots + 891136 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 9497671936 \) Copy content Toggle raw display
$23$ \( T^{16} + 2 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{16} + 42 T^{15} + \cdots + 82609921 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + 80 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$41$ \( T^{16} - 41 T^{15} + \cdots + 5837056 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 3549516288256 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 6116797382656 \) Copy content Toggle raw display
$53$ \( T^{16} + 3 T^{15} + \cdots + 3031081 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 238242586201 \) Copy content Toggle raw display
$61$ \( (T^{8} - 5 T^{7} + \cdots - 111600)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 422530060247296 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1153957202176 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 584351696041 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 907154161 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45921643455601 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 1290895902976 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 18332860692721 \) Copy content Toggle raw display
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