Properties

Label 286.2.h.c
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: \( x^{12} + 15x^{10} + 80x^{8} + 180x^{6} + 160x^{4} + 55x^{2} + 5 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 15x^{10} + 80x^{8} + 180x^{6} + 160x^{4} + 55x^{2} + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} + 5 \nu^{10} + 44 \nu^{9} + 72 \nu^{8} + 224 \nu^{7} + 356 \nu^{6} + 452 \nu^{5} + 680 \nu^{4} + 292 \nu^{3} + 380 \nu^{2} + 45 \nu + 47 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{11} - \nu^{10} + 72 \nu^{9} - 16 \nu^{8} + 356 \nu^{7} - 88 \nu^{6} + 680 \nu^{5} - 188 \nu^{4} + 380 \nu^{3} - 120 \nu^{2} + 47 \nu - 15 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{11} + 5 \nu^{10} - 44 \nu^{9} + 72 \nu^{8} - 224 \nu^{7} + 356 \nu^{6} - 452 \nu^{5} + 680 \nu^{4} - 292 \nu^{3} + 380 \nu^{2} - 45 \nu + 47 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{11} + 5 \nu^{10} + 72 \nu^{9} + 72 \nu^{8} + 356 \nu^{7} + 356 \nu^{6} + 680 \nu^{5} + 676 \nu^{4} + 380 \nu^{3} + 360 \nu^{2} + 43 \nu + 35 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} - 43\nu^{8} - 211\nu^{6} - 396\nu^{4} - \nu^{3} - 206\nu^{2} - 5\nu - 19 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{11} - 5 \nu^{10} + 72 \nu^{9} - 72 \nu^{8} + 356 \nu^{7} - 356 \nu^{6} + 680 \nu^{5} - 676 \nu^{4} + 380 \nu^{3} - 360 \nu^{2} + 43 \nu - 35 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{10} - 100\nu^{8} - 490\nu^{6} - 924\nu^{4} - 496\nu^{2} - 2\nu - 45 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} - 14 \nu^{9} + 86 \nu^{8} - 66 \nu^{7} + 424 \nu^{6} - 112 \nu^{5} + 810 \nu^{4} - 30 \nu^{3} + 454 \nu^{2} + 17 \nu + 50 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{11} - 5 \nu^{10} + 100 \nu^{9} - 72 \nu^{8} + 488 \nu^{7} - 356 \nu^{6} + 904 \nu^{5} - 680 \nu^{4} + 444 \nu^{3} - 380 \nu^{2} + 25 \nu - 43 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{11} - 5 \nu^{10} + 100 \nu^{9} - 72 \nu^{8} + 488 \nu^{7} - 356 \nu^{6} + 904 \nu^{5} - 680 \nu^{4} + 444 \nu^{3} - 380 \nu^{2} + 33 \nu - 43 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{11} + \nu^{10} + 158 \nu^{9} + 14 \nu^{8} + 780 \nu^{7} + 66 \nu^{6} + 1490 \nu^{5} + 114 \nu^{4} + 834 \nu^{3} + 44 \nu^{2} + 95 \nu + 1 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{10} - \beta_{9} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{10} + 5\beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} + 6\beta_{6} - \beta_{4} - 4\beta_{3} - 5\beta_{2} - 4\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{10} - 28\beta_{9} - 6\beta_{8} - 6\beta_{7} + 2\beta_{6} + 6\beta_{5} + 8\beta_{4} - 2\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + \beta_{9} + 2 \beta_{8} + 24 \beta_{7} - 34 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} + 16 \beta_{3} + 24 \beta_{2} + 16 \beta _1 - 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{11} - 104 \beta_{10} + 154 \beta_{9} + 34 \beta_{8} + 33 \beta_{7} - 21 \beta_{6} - 32 \beta_{5} - 54 \beta_{4} - \beta_{3} - \beta_{2} + 17 \beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 8 \beta_{10} - 13 \beta_{9} - 21 \beta_{8} - 117 \beta_{7} + 192 \beta_{6} - 21 \beta_{5} - 49 \beta_{4} - 66 \beta_{3} - 122 \beta_{2} - 66 \beta _1 + 235 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 26 \beta_{11} + 548 \beta_{10} - 838 \beta_{9} - 192 \beta_{8} - 179 \beta_{7} + 160 \beta_{6} + 166 \beta_{5} + 339 \beta_{4} + 8 \beta_{3} + 13 \beta_{2} - 106 \beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 44 \beta_{10} + 116 \beta_{9} + 160 \beta_{8} + 580 \beta_{7} - 1084 \beta_{6} + 160 \beta_{5} + 272 \beta_{4} + 280 \beta_{3} + 652 \beta_{2} + 280 \beta _1 - 1095 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 232 \beta_{11} - 2911 \beta_{10} + 4539 \beta_{9} + 1084 \beta_{8} + 968 \beta_{7} - 1080 \beta_{6} - 852 \beta_{5} - 2048 \beta_{4} - 44 \beta_{3} - 116 \beta_{2} + 588 \beta _1 - 1240 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.372297i
2.35758i
0.827762i
0.372297i
2.35758i
0.827762i
0.784128i
2.18652i
1.79508i
0.784128i
2.18652i
1.79508i
0.309017 + 0.951057i −1.24555 0.904942i −0.809017 + 0.587785i 0.590186 1.81641i 0.475756 1.46423i −1.16309 + 0.845036i −0.809017 0.587785i −0.194587 0.598877i 1.90988
27.2 0.309017 + 0.951057i 0.905613 + 0.657966i −0.809017 + 0.587785i −0.576732 + 1.77500i −0.345913 + 1.06461i −3.05121 + 2.21683i −0.809017 0.587785i −0.539836 1.66144i −1.86634
27.3 0.309017 + 0.951057i 2.45797 + 1.78582i −0.809017 + 0.587785i 1.29556 3.98733i −0.938860 + 2.88951i −0.0217689 + 0.0158160i −0.809017 0.587785i 1.92541 + 5.92579i 4.19253
53.1 0.309017 0.951057i −1.24555 + 0.904942i −0.809017 0.587785i 0.590186 + 1.81641i 0.475756 + 1.46423i −1.16309 0.845036i −0.809017 + 0.587785i −0.194587 + 0.598877i 1.90988
53.2 0.309017 0.951057i 0.905613 0.657966i −0.809017 0.587785i −0.576732 1.77500i −0.345913 1.06461i −3.05121 2.21683i −0.809017 + 0.587785i −0.539836 + 1.66144i −1.86634
53.3 0.309017 0.951057i 2.45797 1.78582i −0.809017 0.587785i 1.29556 + 3.98733i −0.938860 2.88951i −0.0217689 0.0158160i −0.809017 + 0.587785i 1.92541 5.92579i 4.19253
157.1 −0.809017 + 0.587785i −0.558621 1.71926i 0.309017 0.951057i 0.436733 + 0.317305i 1.46249 + 1.06256i −0.151882 + 0.467445i 0.309017 + 0.951057i −0.216737 + 0.157469i −0.539832
157.2 −0.809017 + 0.587785i −0.0778392 0.239564i 0.309017 0.951057i 1.77049 + 1.28633i 0.203786 + 0.148059i −0.976187 + 3.00440i 0.309017 + 0.951057i 2.37572 1.72606i −2.18844
157.3 −0.809017 + 0.587785i 0.518426 + 1.59555i 0.309017 0.951057i −2.01624 1.46488i −1.35726 0.986104i 1.36414 4.19838i 0.309017 + 0.951057i 0.150035 0.109007i 2.49221
235.1 −0.809017 0.587785i −0.558621 + 1.71926i 0.309017 + 0.951057i 0.436733 0.317305i 1.46249 1.06256i −0.151882 0.467445i 0.309017 0.951057i −0.216737 0.157469i −0.539832
235.2 −0.809017 0.587785i −0.0778392 + 0.239564i 0.309017 + 0.951057i 1.77049 1.28633i 0.203786 0.148059i −0.976187 3.00440i 0.309017 0.951057i 2.37572 + 1.72606i −2.18844
235.3 −0.809017 0.587785i 0.518426 1.59555i 0.309017 + 0.951057i −2.01624 + 1.46488i −1.35726 + 0.986104i 1.36414 + 4.19838i 0.309017 0.951057i 0.150035 + 0.109007i 2.49221
\(n\): 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.c 12
11.c even 5 1 inner 286.2.h.c 12
11.c even 5 1 3146.2.a.bg 6
11.d odd 10 1 3146.2.a.be 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.h.c 12 1.a even 1 1 trivial
286.2.h.c 12 11.c even 5 1 inner
3146.2.a.be 6 11.d odd 10 1
3146.2.a.bg 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 4 T_{3}^{11} + 9 T_{3}^{10} - 10 T_{3}^{9} + 35 T_{3}^{8} - 14 T_{3}^{7} + 71 T_{3}^{6} - 26 T_{3}^{5} + 145 T_{3}^{4} - 220 T_{3}^{3} + 224 T_{3}^{2} + 24 T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(286, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + 9 T^{10} - 10 T^{9} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + 21 T^{10} + \cdots + 1936 \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + 51 T^{10} + 255 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + 16 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + 9 T^{10} + 55 T^{9} + \cdots + 3481 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} + 81 T^{10} + \cdots + 25411681 \) Copy content Toggle raw display
$23$ \( (T^{6} - 16 T^{5} + 50 T^{4} + 400 T^{3} + \cdots - 6064)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 31 T^{11} + 519 T^{10} + \cdots + 2193361 \) Copy content Toggle raw display
$31$ \( T^{12} - 3 T^{11} + 16 T^{10} + \cdots + 19321 \) Copy content Toggle raw display
$37$ \( T^{12} - 16 T^{11} + 209 T^{10} + \cdots + 18800896 \) Copy content Toggle raw display
$41$ \( T^{12} + 10 T^{11} + \cdots + 403206400 \) Copy content Toggle raw display
$43$ \( (T^{6} - 30 T^{5} + 210 T^{4} + \cdots + 34880)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 11 T^{11} + \cdots + 5032341721 \) Copy content Toggle raw display
$53$ \( T^{12} + 11 T^{11} + 64 T^{10} + \cdots + 477481 \) Copy content Toggle raw display
$59$ \( T^{12} - 5 T^{11} + \cdots + 9205443025 \) Copy content Toggle raw display
$61$ \( T^{12} + 16 T^{11} + 209 T^{10} + \cdots + 11881 \) Copy content Toggle raw display
$67$ \( (T^{6} - 18 T^{5} - 25 T^{4} + 1170 T^{3} + \cdots + 14384)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 15 T^{10} - 95 T^{9} + \cdots + 6477025 \) Copy content Toggle raw display
$73$ \( T^{12} + 120 T^{10} - 1280 T^{9} + \cdots + 102400 \) Copy content Toggle raw display
$79$ \( T^{12} + 18 T^{11} + 321 T^{10} + \cdots + 80209936 \) Copy content Toggle raw display
$83$ \( T^{12} - 15 T^{11} + \cdots + 5507822265625 \) Copy content Toggle raw display
$89$ \( (T^{6} - 6 T^{5} - 120 T^{4} + 525 T^{3} + \cdots + 176)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 13 T^{11} + \cdots + 3508903696 \) Copy content Toggle raw display
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