Properties

Label 185.2.k.c
Level $185$
Weight $2$
Character orbit 185.k
Analytic conductor $1.477$
Analytic rank $0$
Dimension $6$
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] = [185,2,Mod(68,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.k (of order \(4\), degree \(2\), 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: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{8} + (3 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{6} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{8} + (3 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1) q^{9} + (3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{10} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1) q^{11} + \beta_{2} q^{12} + (3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{13} + ( - 2 \beta_{4} - \beta_{2} + 2) q^{14} + ( - 2 \beta_{5} + 4 \beta_{4} - \beta_1) q^{15} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{16} + ( - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{17} + (2 \beta_{5} - 7 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{18} + (\beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_1 - 3) q^{19} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 1) q^{20} + (\beta_{5} - 3 \beta_{4}) q^{21} + ( - 2 \beta_{5} + 8 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{22} + (2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{23} + ( - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{24} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 1) q^{25} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{26} + (\beta_{5} - 4 \beta_{4} - \beta_{3} - 4) q^{27} + (\beta_{4} + 3 \beta_{2} - 1) q^{28} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 1) q^{29} + ( - 4 \beta_{5} + 5 \beta_{4} + 2 \beta_{2} - \beta_1 - 1) q^{30} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 5 \beta_1 - 2) q^{31} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{32} + ( - 5 \beta_{5} + 7 \beta_{4} + 5 \beta_{3} - 6 \beta_1 + 7) q^{33} + ( - \beta_{5} + 5 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{34} + (3 \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{36} + (\beta_{4} + 6) q^{37} + (3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_1 - 2) q^{38} + (\beta_{5} - 4 \beta_{4} + \beta_{3} + 4) q^{39} + (2 \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{40} + (2 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{41} + (3 \beta_{5} - 3 \beta_{4} - \beta_{2} + \beta_1) q^{42} + ( - 4 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{43} + ( - 4 \beta_{5} - 4 \beta_{4} + \beta_{2} - \beta_1) q^{44} + ( - 3 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{45} + ( - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 8) q^{46} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 1) q^{47} + ( - 3 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + \beta_{2} - 5) q^{48} + ( - \beta_{5} + 4 \beta_{4} + \beta_{2} - \beta_1) q^{49} + (2 \beta_{5} - 6 \beta_{4} - \beta_{3} - 4 \beta_{2} + 4) q^{50} + ( - 3 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} - 4 \beta_1 + 4) q^{51} + (\beta_{3} + 5 \beta_{2} + 5 \beta_1 - 5) q^{52} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} - 2) q^{53} + (4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 2 \beta_1 - 3) q^{54} + (6 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} + 7 \beta_1 - 8) q^{55} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{56} + ( - 7 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 9) q^{57} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2) q^{58} + (\beta_{5} - 6 \beta_{4} - \beta_{3} + 5 \beta_1 - 6) q^{59} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{60} + (5 \beta_{4} - 2 \beta_1 + 5) q^{61} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 2) q^{62} + (\beta_{5} - 5 \beta_{4} - \beta_{3} + \beta_1 - 5) q^{63} + (6 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{64} + (\beta_{5} + 7 \beta_{4} - \beta_{2} - 5 \beta_1 - 2) q^{65} + ( - 7 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} - 4 \beta_1 + 9) q^{66} + (4 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 3 \beta_1 - 6) q^{67} + ( - 3 \beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1) q^{68} + (\beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 3) q^{69} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{70} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{71} + ( - 6 \beta_{5} + 7 \beta_{4} + \beta_{2} - \beta_1) q^{72} + ( - 3 \beta_{5} - 4 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 4) q^{73} + ( - \beta_{5} + 6 \beta_{3}) q^{74} + (5 \beta_{5} - 7 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 5) q^{75} + ( - 2 \beta_{4} + 3 \beta_1 - 2) q^{76} + ( - 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} - 8 \beta_1 + 5) q^{77} + (4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 3) q^{78} + ( - 2 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} - \beta_1 + 5) q^{79} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 5) q^{80} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{81} + (4 \beta_{5} - 12 \beta_{4} - 5 \beta_{2} + 5 \beta_1) q^{82} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{83} + (\beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{84} + (4 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_1 - 5) q^{85} + ( - 2 \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 14) q^{86} + ( - 2 \beta_{5} + 6 \beta_{4} - \beta_{2} + \beta_1) q^{87} + (8 \beta_{5} - 6 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{88} + (3 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} + 4) q^{89} + ( - 5 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 2) q^{90} + ( - 2 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 5) q^{91} + (4 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 8) q^{92} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{93} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{94} + (2 \beta_{4} + 6 \beta_{3} - \beta_{2} - 4 \beta_1 + 6) q^{95} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 2) q^{96} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{97} + ( - 4 \beta_{5} + \beta_{4}) q^{98} + (12 \beta_{3} - 9 \beta_{2} - 9 \beta_1 + 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{6} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 8 q^{6} - 4 q^{7} - 6 q^{8} - 4 q^{10} + 2 q^{12} - 16 q^{13} + 10 q^{14} - 2 q^{15} - 6 q^{16} - 14 q^{19} - 12 q^{20} - 12 q^{23} - 12 q^{24} + 2 q^{25} + 36 q^{26} - 22 q^{27} + 6 q^{29} - 4 q^{30} + 6 q^{32} + 20 q^{33} + 36 q^{37} - 4 q^{38} + 22 q^{39} + 16 q^{40} - 8 q^{43} + 30 q^{45} + 40 q^{46} - 8 q^{47} - 22 q^{48} + 18 q^{50} + 10 q^{51} - 12 q^{52} - 14 q^{53} - 6 q^{54} - 32 q^{55} - 6 q^{56} - 28 q^{57} - 10 q^{58} - 24 q^{59} + 10 q^{60} + 26 q^{61} + 10 q^{62} - 26 q^{63} - 22 q^{64} - 24 q^{65} + 32 q^{66} - 22 q^{67} + 12 q^{69} - 14 q^{70} + 4 q^{71} + 38 q^{73} - 12 q^{74} - 22 q^{75} - 6 q^{76} + 8 q^{77} + 6 q^{78} + 24 q^{79} + 32 q^{80} - 26 q^{81} + 18 q^{83} - 20 q^{85} - 60 q^{86} + 6 q^{89} - 2 q^{90} + 38 q^{91} - 44 q^{92} - 24 q^{93} + 6 q^{94} + 14 q^{95} - 4 q^{96} + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} - 15\nu + 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{3} - 5\beta_{2} - 5\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{5} - \beta_{4} + 5\beta_{3} - 16\beta _1 - 1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/185\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\)
\(\chi(n)\) \(\beta_{4}\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 0.403032i
−2.17009 −0.315449 + 0.315449i 2.70928 −0.539189 2.17009i 0.684551 0.684551i −1.31545 + 1.31545i −1.53919 2.80098i 1.17009 + 4.70928i
68.2 −0.311108 −0.762714 + 0.762714i −1.90321 2.21432 0.311108i 0.237286 0.237286i −1.76271 + 1.76271i 1.21432 1.83654i −0.688892 + 0.0967881i
68.3 1.48119 2.07816 2.07816i 0.193937 −1.67513 + 1.48119i 3.07816 3.07816i 1.07816 1.07816i −2.67513 5.63752i −2.48119 + 2.19394i
117.1 −2.17009 −0.315449 0.315449i 2.70928 −0.539189 + 2.17009i 0.684551 + 0.684551i −1.31545 1.31545i −1.53919 2.80098i 1.17009 4.70928i
117.2 −0.311108 −0.762714 0.762714i −1.90321 2.21432 + 0.311108i 0.237286 + 0.237286i −1.76271 1.76271i 1.21432 1.83654i −0.688892 0.0967881i
117.3 1.48119 2.07816 + 2.07816i 0.193937 −1.67513 1.48119i 3.07816 + 3.07816i 1.07816 + 1.07816i −2.67513 5.63752i −2.48119 2.19394i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.k.c yes 6
5.b even 2 1 925.2.k.c 6
5.c odd 4 1 185.2.f.c 6
5.c odd 4 1 925.2.f.c 6
37.d odd 4 1 185.2.f.c 6
185.f even 4 1 925.2.k.c 6
185.j odd 4 1 925.2.f.c 6
185.k even 4 1 inner 185.2.k.c yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.f.c 6 5.c odd 4 1
185.2.f.c 6 37.d odd 4 1
185.2.k.c yes 6 1.a even 1 1 trivial
185.2.k.c yes 6 185.k even 4 1 inner
925.2.f.c 6 5.c odd 4 1
925.2.f.c 6 185.j odd 4 1
925.2.k.c 6 5.b even 2 1
925.2.k.c 6 185.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\):

\( T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{6} - 2T_{3}^{5} + 2T_{3}^{4} + 10T_{3}^{3} + 16T_{3}^{2} + 8T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} + T^{2} - 3 T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + 2 T^{4} + 10 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} - 16 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + 8 T^{4} + 2 T^{3} + \cdots + 50 \) Copy content Toggle raw display
$11$ \( T^{6} + 68 T^{4} + 1456 T^{2} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( (T^{3} + 8 T^{2} - 74)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 32 T^{4} + 336 T^{2} + \cdots + 1156 \) Copy content Toggle raw display
$19$ \( T^{6} + 14 T^{5} + 98 T^{4} + \cdots + 578 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} - 16 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + 18 T^{4} + 160 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$31$ \( T^{6} + 134 T^{3} + 2500 T^{2} + \cdots + 8978 \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 37)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} + 264 T^{4} + 19856 T^{2} + \cdots + 384400 \) Copy content Toggle raw display
$43$ \( (T^{3} + 4 T^{2} - 72 T - 268)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 8 T^{5} + 32 T^{4} + \cdots + 1250 \) Copy content Toggle raw display
$53$ \( T^{6} + 14 T^{5} + 98 T^{4} + \cdots + 200 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{5} + 288 T^{4} + \cdots + 38642 \) Copy content Toggle raw display
$61$ \( T^{6} - 26 T^{5} + 338 T^{4} + \cdots + 27848 \) Copy content Toggle raw display
$67$ \( T^{6} + 22 T^{5} + 242 T^{4} + 134 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$71$ \( (T^{3} - 2 T^{2} - 52 T + 184)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 38 T^{5} + 722 T^{4} + \cdots + 63368 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 288 T^{4} + \cdots + 8978 \) Copy content Toggle raw display
$83$ \( T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 1250 \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + 18 T^{4} + \cdots + 63368 \) Copy content Toggle raw display
$97$ \( T^{6} + 20 T^{4} + 48 T^{2} + 16 \) Copy content Toggle raw display
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