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Properties

Label 286.2.h.e
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: \( x^{12} - 4 x^{11} + 10 x^{10} - 7 x^{9} + 52 x^{8} + 58 x^{7} + 213 x^{6} + 328 x^{5} + 402 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 10 x^{10} - 7 x^{9} + 52 x^{8} + 58 x^{7} + 213 x^{6} + 328 x^{5} + 402 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69292433938 \nu^{11} + 389280417649 \nu^{10} - 1327050520744 \nu^{9} + \cdots + 34154736915780 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 92341415778 \nu^{11} - 371569053467 \nu^{10} + 859571633717 \nu^{9} - 314637103573 \nu^{8} + \cdots - 9564367673240 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 93838406245 \nu^{11} - 443149068451 \nu^{10} + 1256084465115 \nu^{9} - 1587406533061 \nu^{8} + \cdots - 1539811285621 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 648006713347 \nu^{11} - 2591530485141 \nu^{10} + 6585630982992 \nu^{9} + \cdots + 1548508114907 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1003915254171 \nu^{11} + 4108002432462 \nu^{10} - 10410721595177 \nu^{9} + \cdots + 15421774031178 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1183406376550 \nu^{11} + 4654498638440 \nu^{10} - 11570468013162 \nu^{9} + \cdots + 6892552418188 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1183902744797 \nu^{11} + 4548934788918 \nu^{10} - 11129873175574 \nu^{9} + \cdots + 7540559131535 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1699323110240 \nu^{11} - 6379204503845 \nu^{10} + 15220563849734 \nu^{9} + \cdots - 28899354440725 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2039061314421 \nu^{11} + 7976754135305 \nu^{10} - 19844116938232 \nu^{9} + \cdots - 5093774301323 ) / 10568282927411 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3162368454566 \nu^{11} + 12913040943310 \nu^{10} - 32768587439514 \nu^{9} + \cdots + 49806275187853 ) / 10568282927411 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} - 4\beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 7 \beta_{7} - 5 \beta_{6} - \beta_{5} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{11} - 3\beta_{9} - 23\beta_{8} + 14\beta_{7} - 26\beta_{6} - 9\beta_{5} + 28\beta_{4} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17\beta_{11} - 17\beta_{10} - 54\beta_{6} - 54\beta_{5} + 57\beta_{4} - 51\beta_{3} + 11\beta_{2} - 57\beta _1 - 75 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 108 \beta_{10} + 57 \beta_{9} + 344 \beta_{8} - 179 \beta_{7} - 148 \beta_{5} - 236 \beta_{3} + \cdots - 401 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 236 \beta_{11} - 379 \beta_{10} + 379 \beta_{9} + 2122 \beta_{8} - 1036 \beta_{7} + 702 \beta_{6} + \cdots - 1539 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1415 \beta_{11} - 837 \beta_{10} + 1415 \beta_{9} + 7819 \beta_{8} - 3717 \beta_{7} + 4102 \beta_{6} + \cdots - 3626 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5132 \beta_{11} + 3190 \beta_{9} + 17698 \beta_{8} - 8387 \beta_{7} + 14818 \beta_{6} + \cdots + 8387 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11577 \beta_{11} + 11577 \beta_{10} + 33483 \beta_{6} + 33483 \beta_{5} - 43049 \beta_{4} + \cdots + 47908 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 69469 \beta_{10} - 43049 \beta_{9} - 239493 \beta_{8} + 114529 \beta_{7} + 76049 \beta_{5} + \cdots + 282542 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(67\) \(79\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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
−0.930199 + 0.675829i
2.97515 − 2.16157i
0.0730824 − 0.0530974i
−0.930199 − 0.675829i
2.97515 + 2.16157i
0.0730824 + 0.0530974i
−0.213690 + 0.657669i
0.650997 − 2.00356i
−0.555341 + 1.70916i
−0.213690 − 0.657669i
0.650997 + 2.00356i
−0.555341 − 1.70916i
−0.309017 − 0.951057i −1.65267 − 1.20074i −0.809017 + 0.587785i 0.754429 − 2.32189i −0.631264 + 1.94283i 2.05290 − 1.49152i 0.809017 + 0.587785i 0.362505 + 1.11567i −2.44138
27.2 −0.309017 − 0.951057i 1.11376 + 0.809192i −0.809017 + 0.587785i 0.671146 − 2.06558i 0.425417 − 1.30930i 1.48207 − 1.07679i 0.809017 + 0.587785i −0.341388 − 1.05068i −2.17188
27.3 −0.309017 − 0.951057i 2.65695 + 1.93039i −0.809017 + 0.587785i −0.116558 + 0.358728i 1.01486 − 3.12343i −3.91693 + 2.84582i 0.809017 + 0.587785i 2.40593 + 7.40470i 0.377189
53.1 −0.309017 + 0.951057i −1.65267 + 1.20074i −0.809017 − 0.587785i 0.754429 + 2.32189i −0.631264 − 1.94283i 2.05290 + 1.49152i 0.809017 − 0.587785i 0.362505 − 1.11567i −2.44138
53.2 −0.309017 + 0.951057i 1.11376 − 0.809192i −0.809017 − 0.587785i 0.671146 + 2.06558i 0.425417 + 1.30930i 1.48207 + 1.07679i 0.809017 − 0.587785i −0.341388 + 1.05068i −2.17188
53.3 −0.309017 + 0.951057i 2.65695 − 1.93039i −0.809017 − 0.587785i −0.116558 − 0.358728i 1.01486 + 3.12343i −3.91693 − 2.84582i 0.809017 − 0.587785i 2.40593 − 7.40470i 0.377189
157.1 0.809017 − 0.587785i −0.515146 − 1.58546i 0.309017 − 0.951057i −3.14685 − 2.28632i −1.34867 − 0.979867i −1.34108 + 4.12743i −0.309017 − 0.951057i 0.178750 − 0.129870i −3.88972
157.2 0.809017 − 0.587785i −0.360292 − 1.10886i 0.309017 − 0.951057i 3.43578 + 2.49624i −0.943256 − 0.685315i −0.380693 + 1.17165i −0.309017 − 0.951057i 1.32728 − 0.964327i 4.24686
157.3 0.809017 − 0.587785i 0.757404 + 2.33105i 0.309017 − 0.951057i −0.0979495 − 0.0711645i 1.98291 + 1.44067i −0.896257 + 2.75839i −0.309017 − 0.951057i −2.43308 + 1.76774i −0.121072
235.1 0.809017 + 0.587785i −0.515146 + 1.58546i 0.309017 + 0.951057i −3.14685 + 2.28632i −1.34867 + 0.979867i −1.34108 − 4.12743i −0.309017 + 0.951057i 0.178750 + 0.129870i −3.88972
235.2 0.809017 + 0.587785i −0.360292 + 1.10886i 0.309017 + 0.951057i 3.43578 − 2.49624i −0.943256 + 0.685315i −0.380693 − 1.17165i −0.309017 + 0.951057i 1.32728 + 0.964327i 4.24686
235.3 0.809017 + 0.587785i 0.757404 − 2.33105i 0.309017 + 0.951057i −0.0979495 + 0.0711645i 1.98291 − 1.44067i −0.896257 − 2.75839i −0.309017 + 0.951057i −2.43308 − 1.76774i −0.121072
\(n\): 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.e ✓ 12
11.c even 5 1 inner 286.2.h.e ✓ 12
11.c even 5 1 3146.2.a.bd 6
11.d odd 10 1 3146.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.h.e ✓ 12 1.a even 1 1 trivial
286.2.h.e ✓ 12 11.c even 5 1 inner
3146.2.a.bd 6 11.c even 5 1
3146.2.a.bh 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 4 T_{3}^{11} + 11 T_{3}^{10} - 12 T_{3}^{9} + 45 T_{3}^{8} - 24 T_{3}^{7} + 317 T_{3}^{6} + \cdots + 1936 \) 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
$3$ \( T^{12} - 4 T^{11} + \cdots + 1936 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 121801 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + 13 T^{11} + \cdots + 707281 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + 6 T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$23$ \( (T^{6} + 10 T^{5} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 338891281 \) Copy content Toggle raw display
$31$ \( T^{12} - 7 T^{11} + \cdots + 15437041 \) Copy content Toggle raw display
$37$ \( T^{12} - 18 T^{11} + \cdots + 7929856 \) Copy content Toggle raw display
$41$ \( T^{12} + 26 T^{11} + \cdots + 430336 \) Copy content Toggle raw display
$43$ \( (T^{6} - 174 T^{4} + \cdots - 176)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1611139321 \) Copy content Toggle raw display
$53$ \( T^{12} - 3 T^{11} + \cdots + 79727041 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 578931721 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23124980761 \) Copy content Toggle raw display
$67$ \( (T^{6} + 12 T^{5} + \cdots + 126736)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 8 T^{11} + \cdots + 81558961 \) Copy content Toggle raw display
$73$ \( T^{12} - 8 T^{11} + \cdots + 495616 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 193655056 \) Copy content Toggle raw display
$83$ \( T^{12} - 17 T^{11} + \cdots + 323761 \) Copy content Toggle raw display
$89$ \( (T^{6} + 8 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 95730835216 \) Copy content Toggle raw display
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