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Properties

Label 286.2.h.d
Level $286$
Weight $2$
Character orbit 286.h
Analytic conductor $2.284$
Analytic rank $0$
Dimension $12$
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] = [286,2,Mod(27,286)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(286, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("286.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 286.h (of order \(5\), 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: \(2.28372149781\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 17x^{10} + 99x^{8} + 233x^{6} + 226x^{4} + 80x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 17x^{10} + 99x^{8} + 233x^{6} + 226x^{4} + 80x^{2} + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 16\nu^{6} + 79\nu^{4} + 115\nu^{2} - 5\nu + 30 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4 \nu^{11} - \nu^{10} + 61 \nu^{9} - 14 \nu^{8} + 283 \nu^{7} - 57 \nu^{6} + 343 \nu^{5} - 37 \nu^{4} + \cdots + 115 ) / 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} - 80 \nu^{9} + 33 \nu^{8} - 420 \nu^{7} + 169 \nu^{6} - 800 \nu^{5} + \cdots + 20 ) / 50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4 \nu^{11} - \nu^{10} - 61 \nu^{9} - 14 \nu^{8} - 283 \nu^{7} - 57 \nu^{6} - 343 \nu^{5} + \cdots + 115 ) / 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5 \nu^{11} + 2 \nu^{10} + 80 \nu^{9} + 33 \nu^{8} + 420 \nu^{7} + 169 \nu^{6} + 800 \nu^{5} + 269 \nu^{4} + \cdots + 20 ) / 50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4 \nu^{11} + 5 \nu^{10} - 66 \nu^{9} + 80 \nu^{8} - 363 \nu^{7} + 420 \nu^{6} - 763 \nu^{5} + \cdots + 50 ) / 50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4 \nu^{11} - 5 \nu^{10} - 66 \nu^{9} - 80 \nu^{8} - 363 \nu^{7} - 420 \nu^{6} - 763 \nu^{5} + \cdots - 50 ) / 50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{11} + 4 \nu^{10} + 80 \nu^{9} + 66 \nu^{8} + 420 \nu^{7} + 363 \nu^{6} + 800 \nu^{5} + 738 \nu^{4} + \cdots + 40 ) / 50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8 \nu^{11} + 5 \nu^{10} + 132 \nu^{9} + 80 \nu^{8} + 726 \nu^{7} + 420 \nu^{6} + 1501 \nu^{5} + \cdots + 75 ) / 50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8 \nu^{11} + 5 \nu^{10} + 132 \nu^{9} + 80 \nu^{8} + 726 \nu^{7} + 420 \nu^{6} + 1501 \nu^{5} + \cdots + 75 ) / 50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8 \nu^{11} + 6 \nu^{10} - 137 \nu^{9} + 99 \nu^{8} - 806 \nu^{7} + 532 \nu^{6} - 1921 \nu^{5} + \cdots + 10 ) / 50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{10} + \beta_{9} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + 7\beta_{10} - 5\beta_{9} - \beta_{8} - 2\beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{10} + 8 \beta_{9} - 9 \beta_{8} - \beta_{7} + \beta_{6} + 8 \beta_{5} - 6 \beta_{4} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{11} - 48 \beta_{10} + 30 \beta_{9} + 8 \beta_{8} - 3 \beta_{7} + 15 \beta_{6} + 8 \beta_{5} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55 \beta_{10} - 55 \beta_{9} + 64 \beta_{8} + 8 \beta_{7} - 8 \beta_{6} - 57 \beta_{5} + 38 \beta_{4} + \cdots - 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 110 \beta_{11} + 318 \beta_{10} - 190 \beta_{9} - 55 \beta_{8} + 30 \beta_{7} - 98 \beta_{6} - 51 \beta_{5} + \cdots - 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 368 \beta_{10} + 368 \beta_{9} - 428 \beta_{8} - 49 \beta_{7} + 49 \beta_{6} + 395 \beta_{5} + \cdots + 321 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 736 \beta_{11} - 2076 \beta_{10} + 1220 \beta_{9} + 368 \beta_{8} - 233 \beta_{7} + 623 \beta_{6} + \cdots + 428 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2448 \beta_{10} - 2448 \beta_{9} + 2812 \beta_{8} + 267 \beta_{7} - 267 \beta_{6} - 2712 \beta_{5} + \cdots - 2018 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4896 \beta_{11} + 13494 \beta_{10} - 7870 \beta_{9} - 2448 \beta_{8} + 1688 \beta_{7} - 3936 \beta_{6} + \cdots - 2812 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/286\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\)
\(\chi(n)\) \(1\) \(-\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
2.49480i
− 1.03892i
− 0.280313i
− 2.49480i
1.03892i
0.280313i
− 0.793295i
− 2.60062i
1.49180i
0.793295i
2.60062i
− 1.49180i
−0.309017 − 0.951057i −1.04462 − 0.758962i −0.809017 + 0.587785i 0.336594 − 1.03593i −0.399010 + 1.22803i −4.14812 + 3.01379i 0.809017 + 0.587785i −0.411841 − 1.26752i −1.08924
27.2 −0.309017 − 0.951057i −0.243251 − 0.176732i −0.809017 + 0.587785i −0.158680 + 0.488366i −0.0929135 + 0.285958i 1.28971 − 0.937032i 0.809017 + 0.587785i −0.899114 − 2.76719i 0.513499
27.3 −0.309017 − 0.951057i 1.28787 + 0.935693i −0.809017 + 0.587785i −1.10497 + 3.40073i 0.491923 − 1.51398i 0.122340 − 0.0888855i 0.809017 + 0.587785i −0.143960 − 0.443063i 3.57574
53.1 −0.309017 + 0.951057i −1.04462 + 0.758962i −0.809017 − 0.587785i 0.336594 + 1.03593i −0.399010 − 1.22803i −4.14812 − 3.01379i 0.809017 − 0.587785i −0.411841 + 1.26752i −1.08924
53.2 −0.309017 + 0.951057i −0.243251 + 0.176732i −0.809017 − 0.587785i −0.158680 − 0.488366i −0.0929135 − 0.285958i 1.28971 + 0.937032i 0.809017 − 0.587785i −0.899114 + 2.76719i 0.513499
53.3 −0.309017 + 0.951057i 1.28787 − 0.935693i −0.809017 − 0.587785i −1.10497 − 3.40073i 0.491923 + 1.51398i 0.122340 + 0.0888855i 0.809017 − 0.587785i −0.143960 + 0.443063i 3.57574
157.1 0.809017 − 0.587785i −0.889851 − 2.73868i 0.309017 − 0.951057i −0.630793 − 0.458298i −2.32966 − 1.69260i 0.520836 − 1.60297i −0.309017 − 0.951057i −4.28149 + 3.11068i −0.779703
157.2 0.809017 − 0.587785i −0.0566864 − 0.174463i 0.309017 − 0.951057i 0.717296 + 0.521146i −0.148407 − 0.107824i −0.135714 + 0.417686i −0.309017 − 0.951057i 2.39983 − 1.74358i 0.886627
157.3 0.809017 − 0.587785i 0.946538 + 2.91314i 0.309017 − 0.951057i 2.34055 + 1.70051i 2.47807 + 1.80042i 1.35095 − 4.15779i −0.309017 − 0.951057i −5.16342 + 3.75145i 2.89308
235.1 0.809017 + 0.587785i −0.889851 + 2.73868i 0.309017 + 0.951057i −0.630793 + 0.458298i −2.32966 + 1.69260i 0.520836 + 1.60297i −0.309017 + 0.951057i −4.28149 − 3.11068i −0.779703
235.2 0.809017 + 0.587785i −0.0566864 + 0.174463i 0.309017 + 0.951057i 0.717296 − 0.521146i −0.148407 + 0.107824i −0.135714 − 0.417686i −0.309017 + 0.951057i 2.39983 + 1.74358i 0.886627
235.3 0.809017 + 0.587785i 0.946538 − 2.91314i 0.309017 + 0.951057i 2.34055 − 1.70051i 2.47807 − 1.80042i 1.35095 + 4.15779i −0.309017 + 0.951057i −5.16342 − 3.75145i 2.89308
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 286.2.h.d ✓ 12
11.c even 5 1 inner 286.2.h.d ✓ 12
11.c even 5 1 3146.2.a.bf 6
11.d odd 10 1 3146.2.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.h.d ✓ 12 1.a even 1 1 trivial
286.2.h.d ✓ 12 11.c even 5 1 inner
3146.2.a.bf 6 11.c even 5 1
3146.2.a.bi 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 13 T_{3}^{10} + 3 T_{3}^{9} + 64 T_{3}^{8} + 13 T_{3}^{7} - 34 T_{3}^{6} + 61 T_{3}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(286, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + 13 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{12} + 20 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + 10 T^{11} + \cdots + 28185481 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 326041 \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} - 36 T^{4} + \cdots - 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + \cdots + 2972176 \) Copy content Toggle raw display
$31$ \( T^{12} + 29 T^{11} + \cdots + 92236816 \) Copy content Toggle raw display
$37$ \( T^{12} + 16 T^{11} + \cdots + 6400 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 723879025 \) Copy content Toggle raw display
$43$ \( (T^{6} - 19 T^{5} + \cdots - 4729)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 102400 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1130304400 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 666620761 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1856231056 \) Copy content Toggle raw display
$67$ \( (T^{6} - 7 T^{5} + \cdots - 166291)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 18188298496 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 210115891456 \) Copy content Toggle raw display
$79$ \( T^{12} + 12 T^{11} + \cdots + 69488896 \) Copy content Toggle raw display
$83$ \( T^{12} - 16 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$89$ \( (T^{6} + 5 T^{5} + \cdots - 219781)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 16 T^{11} + \cdots + 17598025 \) Copy content Toggle raw display
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