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Properties

Label 185.2.e.b
Level $185$
Weight $2$
Character orbit 185.e
Analytic conductor $1.477$
Analytic rank $0$
Dimension $14$
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(26,185)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(185, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("185.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.e (of order \(3\), 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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 13 x^{12} - 16 x^{11} + 98 x^{10} - 116 x^{9} + 378 x^{8} - 264 x^{7} + 795 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{6} q^{3} + ( - \beta_{7} + \beta_{5} - 1) q^{4} + ( - \beta_{5} + 1) q^{5} + ( - \beta_{12} + \beta_{11} - \beta_{10} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{12} + \beta_{11} - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + \beta_{6} q^{3} + ( - \beta_{7} + \beta_{5} - 1) q^{4} + ( - \beta_{5} + 1) q^{5} + ( - \beta_{12} + \beta_{11} - \beta_{10} + \cdots + 1) q^{6}+ \cdots + (3 \beta_{12} - \beta_{11} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} - 2 q^{3} - 8 q^{4} + 7 q^{5} + 4 q^{6} + 2 q^{7} - 6 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} - 2 q^{3} - 8 q^{4} + 7 q^{5} + 4 q^{6} + 2 q^{7} - 6 q^{8} - 5 q^{9} + 4 q^{10} - 10 q^{11} - 8 q^{12} + 6 q^{13} - 36 q^{14} + 2 q^{15} - 14 q^{16} - q^{17} - 4 q^{18} + 6 q^{19} + 8 q^{20} + 13 q^{21} - q^{22} + 12 q^{23} - 21 q^{24} - 7 q^{25} + 2 q^{26} + 22 q^{27} + 13 q^{28} - 12 q^{29} + 2 q^{30} - 8 q^{31} + 18 q^{32} + q^{33} - 11 q^{34} - 2 q^{35} - 8 q^{36} + 12 q^{37} + 16 q^{38} + 23 q^{39} - 3 q^{40} - 3 q^{41} + 29 q^{42} - 38 q^{43} + 25 q^{44} - 10 q^{45} + 10 q^{46} + 4 q^{47} - 20 q^{48} - 7 q^{49} + 2 q^{50} - 14 q^{51} + 46 q^{52} - 2 q^{53} + 23 q^{54} - 5 q^{55} + 19 q^{56} + 22 q^{57} - 12 q^{58} - 18 q^{59} - 16 q^{60} - 20 q^{61} - 21 q^{62} + 46 q^{63} + 50 q^{64} - 6 q^{65} - 42 q^{66} - 20 q^{67} + 110 q^{68} + 17 q^{69} - 18 q^{70} - 11 q^{71} - 29 q^{72} - 36 q^{73} - 66 q^{74} + 4 q^{75} + 40 q^{76} - q^{77} + 6 q^{78} + 23 q^{79} - 28 q^{80} + 29 q^{81} - 24 q^{82} - 9 q^{83} + 8 q^{84} - 2 q^{85} - 3 q^{86} - 43 q^{87} - 116 q^{88} - 16 q^{89} + 4 q^{90} + 12 q^{91} - 33 q^{92} + 25 q^{93} + 22 q^{94} - 6 q^{95} - 67 q^{96} - 62 q^{97} - 24 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 13 x^{12} - 16 x^{11} + 98 x^{10} - 116 x^{9} + 378 x^{8} - 264 x^{7} + 795 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9478348504 \nu^{13} + 5939251426 \nu^{12} - 80268303612 \nu^{11} - 36288739259 \nu^{10} + \cdots + 36649787622939 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1783045984 \nu^{13} - 60819893197 \nu^{12} + 104613638997 \nu^{11} - 717256889278 \nu^{10} + \cdots - 4054063958955 ) / 1807607208897 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11870550653 \nu^{13} - 68330049472 \nu^{12} - 10160356080 \nu^{11} - 953424895018 \nu^{10} + \cdots - 7664107792896 ) / 4217750154093 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50193883907 \nu^{13} - 45163761508 \nu^{12} - 351939083643 \nu^{11} + \cdots - 819032579649 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53204394598 \nu^{13} + 184672214059 \nu^{12} - 758931579168 \nu^{11} + \cdots + 1563889592313 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 141103303217 \nu^{13} - 141430535950 \nu^{12} - 975548947317 \nu^{11} + \cdots - 39106885361886 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 145551529322 \nu^{13} + 300581407148 \nu^{12} - 1898109132612 \nu^{11} + \cdots + 451744955163 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 207325143397 \nu^{13} + 86191905049 \nu^{12} - 1922422841808 \nu^{11} + \cdots - 37457690633037 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 196326636943 \nu^{13} - 380961431131 \nu^{12} + 2492979987918 \nu^{11} + \cdots - 7617632295159 ) / 4217750154093 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1072227588916 \nu^{13} + 2189537640988 \nu^{12} - 13929871784304 \nu^{11} + \cdots + 4398007127490 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1469993686828 \nu^{13} + 2529739953856 \nu^{12} - 18343976363517 \nu^{11} + \cdots + 1902753058536 ) / 12653250462279 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 527727247875 \nu^{13} - 1200104506751 \nu^{12} + 7139242100648 \nu^{11} + \cdots - 2488211637801 ) / 4217750154093 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 3\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{11} + \beta_{9} - 5\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 7\beta_{7} - 2\beta_{6} + 16\beta_{5} + 2\beta_{4} - \beta_{3} - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{13} - \beta_{12} + 12 \beta_{11} - \beta_{10} + 29 \beta_{8} - 10 \beta_{7} + \cdots - 10 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{13} - 12 \beta_{12} + 24 \beta_{11} - 20 \beta_{10} - 11 \beta_{9} + 3 \beta_{8} + 12 \beta_{3} + \cdots + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -69\beta_{9} + 83\beta_{7} + 105\beta_{6} - 45\beta_{5} - 15\beta_{4} + 16\beta_{3} + 183\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 98 \beta_{13} + 106 \beta_{12} - 219 \beta_{11} + 159 \beta_{10} - 52 \beta_{8} + \cdots + 350 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 509 \beta_{13} + 166 \beta_{12} - 834 \beta_{11} + 158 \beta_{10} + 509 \beta_{9} - 1212 \beta_{8} + \cdots - 481 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 817\beta_{9} - 2538\beta_{7} - 1817\beta_{6} + 4636\beta_{5} + 1185\beta_{4} - 842\beta_{3} - 614\beta _1 - 4636 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3723 \beta_{13} - 1474 \beta_{12} + 6382 \beta_{11} - 1449 \beta_{10} + 8263 \beta_{8} + \cdots - 5154 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6603 \beta_{13} - 6407 \beta_{12} + 14459 \beta_{11} - 8656 \beta_{10} - 6603 \beta_{9} + 6153 \beta_{8} + \cdots + 32645 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 27245 \beta_{9} + 40001 \beta_{7} + 48111 \beta_{6} - 39325 \beta_{5} - 12406 \beta_{4} + \cdots + 39325 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(76\) \(112\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.24062 + 2.14882i
−0.795055 + 1.37708i
−0.155125 + 0.268684i
0.167462 − 0.290053i
0.704536 − 1.22029i
0.945119 − 1.63699i
1.37369 − 2.37929i
−1.24062 − 2.14882i
−0.795055 − 1.37708i
−0.155125 − 0.268684i
0.167462 + 0.290053i
0.704536 + 1.22029i
0.945119 + 1.63699i
1.37369 + 2.37929i
−1.24062 − 2.14882i −1.10207 + 1.90884i −2.07829 + 3.59971i 0.500000 − 0.866025i 5.46902 1.91357 − 3.31440i 5.35102 −0.929125 − 1.60929i −2.48125
26.2 −0.795055 − 1.37708i 0.779300 − 1.34979i −0.264225 + 0.457651i 0.500000 − 0.866025i −2.47835 0.801138 − 1.38761i −2.33993 0.285383 + 0.494298i −1.59011
26.3 −0.155125 − 0.268684i −1.03095 + 1.78566i 0.951873 − 1.64869i 0.500000 − 0.866025i 0.639704 1.06035 − 1.83659i −1.21113 −0.625722 − 1.08378i −0.310249
26.4 0.167462 + 0.290053i 1.19442 − 2.06880i 0.943913 − 1.63491i 0.500000 − 0.866025i 0.800081 −2.21517 + 3.83679i 1.30213 −1.35329 − 2.34397i 0.334924
26.5 0.704536 + 1.22029i 0.0771688 − 0.133660i 0.00725754 − 0.0125704i 0.500000 − 0.866025i 0.217473 1.20638 − 2.08952i 2.83860 1.48809 + 2.57745i 1.40907
26.6 0.945119 + 1.63699i −1.39740 + 2.42036i −0.786500 + 1.36226i 0.500000 − 0.866025i −5.28283 −1.45316 + 2.51695i 0.807130 −2.40544 − 4.16634i 1.89024
26.7 1.37369 + 2.37929i 0.479530 − 0.830570i −2.77403 + 4.80475i 0.500000 − 0.866025i 2.63489 −0.313113 + 0.542327i −9.74781 1.04010 + 1.80151i 2.74737
121.1 −1.24062 + 2.14882i −1.10207 − 1.90884i −2.07829 − 3.59971i 0.500000 + 0.866025i 5.46902 1.91357 + 3.31440i 5.35102 −0.929125 + 1.60929i −2.48125
121.2 −0.795055 + 1.37708i 0.779300 + 1.34979i −0.264225 − 0.457651i 0.500000 + 0.866025i −2.47835 0.801138 + 1.38761i −2.33993 0.285383 − 0.494298i −1.59011
121.3 −0.155125 + 0.268684i −1.03095 − 1.78566i 0.951873 + 1.64869i 0.500000 + 0.866025i 0.639704 1.06035 + 1.83659i −1.21113 −0.625722 + 1.08378i −0.310249
121.4 0.167462 − 0.290053i 1.19442 + 2.06880i 0.943913 + 1.63491i 0.500000 + 0.866025i 0.800081 −2.21517 − 3.83679i 1.30213 −1.35329 + 2.34397i 0.334924
121.5 0.704536 − 1.22029i 0.0771688 + 0.133660i 0.00725754 + 0.0125704i 0.500000 + 0.866025i 0.217473 1.20638 + 2.08952i 2.83860 1.48809 − 2.57745i 1.40907
121.6 0.945119 − 1.63699i −1.39740 − 2.42036i −0.786500 − 1.36226i 0.500000 + 0.866025i −5.28283 −1.45316 − 2.51695i 0.807130 −2.40544 + 4.16634i 1.89024
121.7 1.37369 − 2.37929i 0.479530 + 0.830570i −2.77403 − 4.80475i 0.500000 + 0.866025i 2.63489 −0.313113 − 0.542327i −9.74781 1.04010 − 1.80151i 2.74737
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 185.2.e.b ✓ 14
5.b even 2 1 925.2.e.b 14
5.c odd 4 2 925.2.o.c 28
37.c even 3 1 inner 185.2.e.b ✓ 14
37.c even 3 1 6845.2.a.j 7
37.e even 6 1 6845.2.a.m 7
185.n even 6 1 925.2.e.b 14
185.s odd 12 2 925.2.o.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.e.b ✓ 14 1.a even 1 1 trivial
185.2.e.b ✓ 14 37.c even 3 1 inner
925.2.e.b 14 5.b even 2 1
925.2.e.b 14 185.n even 6 1
925.2.o.c 28 5.c odd 4 2
925.2.o.c 28 185.s odd 12 2
6845.2.a.j 7 37.c even 3 1
6845.2.a.m 7 37.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 2 T_{2}^{13} + 13 T_{2}^{12} - 16 T_{2}^{11} + 98 T_{2}^{10} - 116 T_{2}^{9} + 378 T_{2}^{8} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(185, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{14} + 2 T^{13} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots + 64009 \) Copy content Toggle raw display
$11$ \( (T^{7} + 5 T^{6} + \cdots - 333)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 6 T^{13} + \cdots + 6889 \) Copy content Toggle raw display
$17$ \( T^{14} + T^{13} + \cdots + 49070025 \) Copy content Toggle raw display
$19$ \( T^{14} - 6 T^{13} + \cdots + 116281 \) Copy content Toggle raw display
$23$ \( (T^{7} - 6 T^{6} - 81 T^{5} + \cdots + 75)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 6 T^{6} + \cdots - 2649)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 4 T^{6} + \cdots - 40581)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 94931877133 \) Copy content Toggle raw display
$41$ \( T^{14} + 3 T^{13} + \cdots + 106929 \) Copy content Toggle raw display
$43$ \( (T^{7} + 19 T^{6} + \cdots + 863977)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} - 2 T^{6} - 9 T^{5} + \cdots + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 3682911969 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 31784114961 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 4190620225 \) Copy content Toggle raw display
$67$ \( T^{14} + 20 T^{13} + \cdots + 885481 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 4179234609 \) Copy content Toggle raw display
$73$ \( (T^{7} + 18 T^{6} + \cdots - 23788741)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1637092521 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 48568666689 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 21599154005121 \) Copy content Toggle raw display
$97$ \( (T^{7} + 31 T^{6} + \cdots - 2647169)^{2} \) Copy content Toggle raw display
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