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Properties

Label 286.2.j.b
Level $286$
Weight $2$
Character orbit 286.j
Analytic conductor $2.284$
Analytic rank $0$
Dimension $16$
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(23,286)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(286, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("286.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 286 = 2 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 286.j (of order \(6\), 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: \(2.28372149781\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 44x^{14} + 784x^{12} + 7200x^{10} + 35724x^{8} + 90936x^{6} + 101124x^{4} + 42336x^{2} + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + ( - \beta_{13} - \beta_{2}) q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{7}) q^{5} + \beta_{4} q^{6} + (\beta_{15} + \beta_{13} + \cdots + \beta_{5}) q^{7}+ \cdots + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + ( - \beta_{13} - \beta_{2}) q^{3} + ( - \beta_{2} + 1) q^{4} + (\beta_{8} - \beta_{7}) q^{5} + \beta_{4} q^{6} + (\beta_{15} + \beta_{13} + \cdots + \beta_{5}) q^{7}+ \cdots + ( - \beta_{15} - \beta_{14} + 2 \beta_{11} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} + 8 q^{4} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} + 8 q^{4} - 20 q^{9} - 2 q^{10} - 8 q^{12} + 10 q^{13} + 16 q^{14} - 30 q^{15} - 8 q^{16} + 14 q^{17} + 6 q^{19} - 8 q^{22} + 14 q^{23} - 12 q^{25} + 4 q^{26} + 20 q^{27} - 2 q^{29} - 6 q^{30} - 10 q^{35} + 20 q^{36} - 24 q^{38} - 14 q^{39} - 4 q^{40} - 48 q^{41} + 6 q^{42} + 2 q^{43} - 12 q^{45} + 12 q^{46} - 4 q^{48} + 40 q^{49} + 60 q^{51} + 2 q^{52} + 24 q^{53} - 18 q^{54} + 2 q^{55} + 8 q^{56} - 18 q^{58} + 54 q^{59} - 22 q^{61} + 14 q^{62} - 54 q^{63} - 16 q^{64} + 54 q^{65} + 8 q^{66} - 12 q^{67} - 14 q^{68} - 36 q^{71} - 12 q^{72} + 6 q^{74} + 40 q^{75} + 6 q^{76} - 16 q^{77} + 6 q^{78} - 24 q^{79} - 44 q^{81} + 6 q^{82} - 54 q^{84} + 30 q^{85} - 48 q^{87} + 8 q^{88} - 30 q^{89} + 108 q^{90} + 46 q^{91} + 28 q^{92} - 72 q^{93} + 6 q^{94} - 8 q^{95} - 90 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 44x^{14} + 784x^{12} + 7200x^{10} + 35724x^{8} + 90936x^{6} + 101124x^{4} + 42336x^{2} + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{14} + 88 \nu^{12} - 1024 \nu^{10} - 33936 \nu^{8} - 283044 \nu^{6} - 962712 \nu^{4} + \cdots - 277776 ) / 78624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{15} - 88 \nu^{13} + 1024 \nu^{11} + 33936 \nu^{9} + 283044 \nu^{7} + 962712 \nu^{5} + \cdots + 157248 ) / 314496 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11 \nu^{14} + 412 \nu^{12} + 5828 \nu^{10} + 38472 \nu^{8} + 118116 \nu^{6} + 150336 \nu^{4} + \cdots + 2160 ) / 26208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11 \nu^{14} + 412 \nu^{12} + 5828 \nu^{10} + 38472 \nu^{8} + 118116 \nu^{6} + 150336 \nu^{4} + \cdots + 2160 ) / 26208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{14} + 248 \nu^{12} + 3496 \nu^{10} + 25248 \nu^{8} + 99444 \nu^{6} + 201096 \nu^{4} + \cdots + 28944 ) / 11232 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107 \nu^{15} + 366 \nu^{14} - 4504 \nu^{13} + 14304 \nu^{12} - 75056 \nu^{11} + 215952 \nu^{10} + \cdots + 4103136 ) / 943488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47 \nu^{15} + 222 \nu^{14} + 1264 \nu^{13} + 6528 \nu^{12} + 8720 \nu^{11} + 64608 \nu^{10} + \cdots - 256608 ) / 314496 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47 \nu^{15} + 222 \nu^{14} - 1264 \nu^{13} + 6528 \nu^{12} - 8720 \nu^{11} + 64608 \nu^{10} + \cdots - 256608 ) / 314496 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 469 \nu^{15} + 1950 \nu^{14} + 20048 \nu^{13} + 73632 \nu^{12} + 346864 \nu^{11} + 1094496 \nu^{10} + \cdots + 7716384 ) / 1886976 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 85 \nu^{15} - 10 \nu^{14} - 3680 \nu^{13} - 176 \nu^{12} - 65584 \nu^{11} + 2048 \nu^{10} + \cdots + 555552 ) / 314496 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 469 \nu^{15} + 1950 \nu^{14} - 20048 \nu^{13} + 73632 \nu^{12} - 346864 \nu^{11} + \cdots + 7716384 ) / 1886976 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 469 \nu^{15} - 2070 \nu^{14} + 20048 \nu^{13} - 75744 \nu^{12} + 346864 \nu^{11} - 1069920 \nu^{10} + \cdots + 8385120 ) / 1886976 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 165 \nu^{15} + 49 \nu^{14} + 6180 \nu^{13} + 1736 \nu^{12} + 90696 \nu^{11} + 24472 \nu^{10} + \cdots + 123984 ) / 157248 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2069 \nu^{15} - 1086 \nu^{14} - 87520 \nu^{13} - 26976 \nu^{12} - 1486832 \nu^{11} + \cdots + 25989984 ) / 1886976 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2069 \nu^{15} + 1086 \nu^{14} - 87520 \nu^{13} + 26976 \nu^{12} - 1486832 \nu^{11} + \cdots - 25989984 ) / 1886976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{12} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 2 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{14} - 7 \beta_{12} - 8 \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + 3 \beta_{5} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{15} - 13 \beta_{14} - 28 \beta_{13} - 13 \beta_{12} - 6 \beta_{11} + 6 \beta_{10} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{15} + 18 \beta_{14} + 46 \beta_{12} + 64 \beta_{11} + 18 \beta_{9} + 20 \beta_{8} + \cdots - 344 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 142 \beta_{15} + 142 \beta_{14} + 328 \beta_{13} + 146 \beta_{12} + 128 \beta_{11} - 112 \beta_{10} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 236 \beta_{15} - 236 \beta_{14} - 286 \beta_{12} - 534 \beta_{11} - 248 \beta_{9} - 276 \beta_{8} + \cdots + 3054 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1468 \beta_{15} - 1468 \beta_{14} - 3628 \beta_{13} - 1548 \beta_{12} - 1924 \beta_{11} + 1532 \beta_{10} + \cdots - 1036 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2770 \beta_{15} + 2770 \beta_{14} + 1546 \beta_{12} + 4628 \beta_{11} + 3082 \beta_{9} + 3346 \beta_{8} + \cdots - 27772 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14866 \beta_{15} + 14866 \beta_{14} + 39064 \beta_{13} + 15958 \beta_{12} + 25032 \beta_{11} + \cdots + 10296 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 30912 \beta_{15} - 30912 \beta_{14} - 5068 \beta_{12} - 41308 \beta_{11} - 36240 \beta_{9} + \cdots + 257612 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 149740 \beta_{15} - 149740 \beta_{14} - 414784 \beta_{13} - 162212 \beta_{12} - 301472 \beta_{11} + \cdots - 95336 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 335984 \beta_{15} + 335984 \beta_{14} - 35156 \beta_{12} + 377052 \beta_{11} + 412208 \beta_{9} + \cdots - 2429532 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1511452 \beta_{15} + 1511452 \beta_{14} + 4372408 \beta_{13} + 1638060 \beta_{12} + 3464560 \beta_{11} + \cdots + 839200 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(67\) \(79\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
− 3.23409i
− 2.41891i
0.712633i
2.94037i
3.11119i
1.14262i
0.455963i
− 2.70978i
3.23409i
2.41891i
− 0.712633i
− 2.94037i
− 3.11119i
− 1.14262i
− 0.455963i
2.70978i
−0.866025 + 0.500000i −1.61705 − 2.80080i 0.500000 − 0.866025i 2.38822i 2.80080 + 1.61705i 3.39370 + 1.95936i 1.00000i −3.72967 + 6.45998i −1.19411 − 2.06826i
23.2 −0.866025 + 0.500000i −1.20946 − 2.09484i 0.500000 − 0.866025i − 2.58095i 2.09484 + 1.20946i −2.44511 − 1.41168i 1.00000i −1.42557 + 2.46916i 1.29048 + 2.23517i
23.3 −0.866025 + 0.500000i 0.356316 + 0.617158i 0.500000 − 0.866025i − 1.63116i −0.617158 − 0.356316i −0.869910 − 0.502243i 1.00000i 1.24608 − 2.15827i 0.815581 + 1.41263i
23.4 −0.866025 + 0.500000i 1.47019 + 2.54644i 0.500000 − 0.866025i 2.82390i −2.54644 − 1.47019i −3.54279 − 2.04543i 1.00000i −2.82289 + 4.88939i −1.41195 − 2.44556i
23.5 0.866025 − 0.500000i −1.55560 − 2.69437i 0.500000 − 0.866025i − 3.79295i −2.69437 − 1.55560i 3.01581 + 1.74118i − 1.00000i −3.33976 + 5.78464i −1.89648 − 3.28480i
23.6 0.866025 − 0.500000i −0.571310 − 0.989538i 0.500000 − 0.866025i 2.66270i −0.989538 − 0.571310i 3.99243 + 2.30503i − 1.00000i 0.847209 − 1.46741i 1.33135 + 2.30597i
23.7 0.866025 − 0.500000i −0.227982 − 0.394876i 0.500000 − 0.866025i − 0.805227i −0.394876 − 0.227982i −3.99480 − 2.30640i − 1.00000i 1.39605 − 2.41803i −0.402614 − 0.697347i
23.8 0.866025 − 0.500000i 1.35489 + 2.34674i 0.500000 − 0.866025i 0.935480i 2.34674 + 1.35489i 0.450662 + 0.260190i − 1.00000i −2.17145 + 3.76105i 0.467740 + 0.810150i
199.1 −0.866025 − 0.500000i −1.61705 + 2.80080i 0.500000 + 0.866025i − 2.38822i 2.80080 − 1.61705i 3.39370 − 1.95936i − 1.00000i −3.72967 − 6.45998i −1.19411 + 2.06826i
199.2 −0.866025 − 0.500000i −1.20946 + 2.09484i 0.500000 + 0.866025i 2.58095i 2.09484 − 1.20946i −2.44511 + 1.41168i − 1.00000i −1.42557 − 2.46916i 1.29048 − 2.23517i
199.3 −0.866025 − 0.500000i 0.356316 − 0.617158i 0.500000 + 0.866025i 1.63116i −0.617158 + 0.356316i −0.869910 + 0.502243i − 1.00000i 1.24608 + 2.15827i 0.815581 − 1.41263i
199.4 −0.866025 − 0.500000i 1.47019 − 2.54644i 0.500000 + 0.866025i − 2.82390i −2.54644 + 1.47019i −3.54279 + 2.04543i − 1.00000i −2.82289 − 4.88939i −1.41195 + 2.44556i
199.5 0.866025 + 0.500000i −1.55560 + 2.69437i 0.500000 + 0.866025i 3.79295i −2.69437 + 1.55560i 3.01581 − 1.74118i 1.00000i −3.33976 − 5.78464i −1.89648 + 3.28480i
199.6 0.866025 + 0.500000i −0.571310 + 0.989538i 0.500000 + 0.866025i − 2.66270i −0.989538 + 0.571310i 3.99243 − 2.30503i 1.00000i 0.847209 + 1.46741i 1.33135 − 2.30597i
199.7 0.866025 + 0.500000i −0.227982 + 0.394876i 0.500000 + 0.866025i 0.805227i −0.394876 + 0.227982i −3.99480 + 2.30640i 1.00000i 1.39605 + 2.41803i −0.402614 + 0.697347i
199.8 0.866025 + 0.500000i 1.35489 − 2.34674i 0.500000 + 0.866025i − 0.935480i 2.34674 − 1.35489i 0.450662 − 0.260190i 1.00000i −2.17145 − 3.76105i 0.467740 − 0.810150i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 286.2.j.b ✓ 16
13.e even 6 1 inner 286.2.j.b ✓ 16
13.f odd 12 1 3718.2.a.bn 8
13.f odd 12 1 3718.2.a.bo 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
286.2.j.b ✓ 16 1.a even 1 1 trivial
286.2.j.b ✓ 16 13.e even 6 1 inner
3718.2.a.bn 8 13.f odd 12 1
3718.2.a.bo 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 4 T_{3}^{15} + 30 T_{3}^{14} + 76 T_{3}^{13} + 430 T_{3}^{12} + 972 T_{3}^{11} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(286, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{15} + \cdots + 5184 \) Copy content Toggle raw display
$5$ \( T^{16} + 46 T^{14} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( T^{16} - 48 T^{14} + \cdots + 3069504 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 14 T^{15} + \cdots + 1557504 \) Copy content Toggle raw display
$19$ \( T^{16} - 6 T^{15} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{16} - 14 T^{15} + \cdots + 82944 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 7943622129 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 3328828416 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 33388060176 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1437016464 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 6843921984 \) Copy content Toggle raw display
$47$ \( T^{16} + 234 T^{14} + \cdots + 83283876 \) Copy content Toggle raw display
$53$ \( (T^{8} - 12 T^{7} + \cdots - 209628)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 54 T^{15} + \cdots + 26132544 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 128686336 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11423210270976 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 67509623347776 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 8243187264 \) Copy content Toggle raw display
$79$ \( (T^{8} + 12 T^{7} + \cdots + 3789376)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 12747556218384 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1065048768144 \) Copy content Toggle raw display
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