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Properties

Label 836.2.j.a
Level $836$
Weight $2$
Character orbit 836.j
Analytic conductor $6.675$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [836,2,Mod(229,836)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(836, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("836.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 836 = 2^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 836.j (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: \(6.67549360898\)
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: \( x^{12} - x^{11} + 9 x^{10} - 11 x^{9} + 52 x^{8} + 45 x^{7} + 155 x^{6} + 472 x^{5} + 1093 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{11} - \beta_{6}) q^{5} + (\beta_{8} - \beta_{7} + \cdots - \beta_{2}) q^{7} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots - 1) q^{9} + ( - \beta_{10} - \beta_{8} - 3 \beta_{6} - 1) q^{11}+ \cdots + (2 \beta_{11} - 3 \beta_{10} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{3} + q^{5} - 2 q^{7} - 8 q^{9} - 3 q^{11} - 3 q^{13} - 6 q^{15} + 7 q^{17} - 3 q^{19} + 10 q^{21} - 24 q^{23} - 16 q^{25} + q^{27} - 11 q^{29} + 14 q^{31} + q^{33} + 8 q^{35} - 17 q^{37} + 41 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 9 x^{10} - 11 x^{9} + 52 x^{8} + 45 x^{7} + 155 x^{6} + 472 x^{5} + 1093 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2114388924 \nu^{11} - 820177395 \nu^{10} - 17716190783 \nu^{9} + 4091673330 \nu^{8} + \cdots - 1710624007848 ) / 2748484260796 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42858754393 \nu^{11} - 59888943358 \nu^{10} + 403665638004 \nu^{9} - 623265778721 \nu^{8} + \cdots + 1147713543840 ) / 2748484260796 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 87978105399 \nu^{11} + 18655085387 \nu^{10} + 646660453429 \nu^{9} - 14586819503 \nu^{8} + \cdots - 1342928205344 ) / 5496968521592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 52486080487 \nu^{11} + 58257349061 \nu^{10} - 472518263518 \nu^{9} + 621564465601 \nu^{8} + \cdots - 1574246306984 ) / 2748484260796 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 71732096490 \nu^{11} + 114590850883 \nu^{10} - 705477811768 \nu^{9} + \cdots - 1732729251804 ) / 2748484260796 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 146253926991 \nu^{11} + 261747295379 \nu^{10} - 1535092973019 \nu^{9} + \cdots + 2165713819192 ) / 5496968521592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 196780788373 \nu^{11} - 301752949347 \nu^{10} + 1887541793479 \nu^{9} - 3109625199139 \nu^{8} + \cdots - 2735334861176 ) / 5496968521592 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 201189241645 \nu^{11} - 29394873225 \nu^{10} - 1450033759123 \nu^{9} + \cdots - 38243484121976 ) / 5496968521592 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 213828000981 \nu^{11} + 218056778829 \nu^{10} - 1922811654039 \nu^{9} + \cdots - 14271461875488 ) / 5496968521592 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 428871497377 \nu^{11} + 527727290962 \nu^{10} - 3962882372755 \nu^{9} + \cdots - 12197591571336 ) / 2748484260796 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{7} + 4\beta_{6} - \beta_{5} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + 7\beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{11} + 19 \beta_{10} + \beta_{9} - 27 \beta_{8} + 8 \beta_{7} - 27 \beta_{6} - 7 \beta_{5} + \cdots - 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{10} - 3\beta_{6} - 52\beta_{5} - 11\beta_{4} - 52\beta_{3} - 42\beta_{2} - 42\beta _1 - 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 52 \beta_{11} - 52 \beta_{9} + 196 \beta_{8} - 62 \beta_{7} + 60 \beta_{6} + 52 \beta_{5} + 7 \beta_{3} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 7 \beta_{11} + 19 \beta_{10} - 19 \beta_{8} + 102 \beta_{7} - 291 \beta_{6} + 395 \beta_{5} + \cdots - 15 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 87 \beta_{11} - 1018 \beta_{10} + 402 \beta_{9} - 459 \beta_{8} + 402 \beta_{7} - 207 \beta_{5} + \cdots - 459 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 897 \beta_{11} - 2277 \beta_{10} - 120 \beta_{9} + 2730 \beta_{8} - 1017 \beta_{7} + 2730 \beta_{6} + \cdots + 2277 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3643 \beta_{10} - 753 \beta_{9} - 3643 \beta_{6} + 7702 \beta_{5} + 3178 \beta_{4} + 7702 \beta_{3} + \cdots + 7788 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 7702 \beta_{11} + 7702 \beta_{9} - 24392 \beta_{8} + 9148 \beta_{7} - 5684 \beta_{6} - 7702 \beta_{5} + \cdots - 5684 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/836\mathbb{Z}\right)^\times\).

\(n\) \(419\) \(705\) \(761\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
−1.16744 − 0.848196i
−0.251622 − 0.182814i
2.22808 + 1.61880i
−1.16744 + 0.848196i
−0.251622 + 0.182814i
2.22808 − 1.61880i
−0.895600 − 2.75637i
−0.145620 − 0.448171i
0.732202 + 2.25349i
−0.895600 + 2.75637i
−0.145620 + 0.448171i
0.732202 − 2.25349i
0 −1.16744 + 0.848196i 0 −1.32665 − 4.08302i 0 2.57671 + 1.87209i 0 −0.283568 + 0.872732i 0
229.2 0 −0.251622 + 0.182814i 0 0.581976 + 1.79114i 0 −2.98616 − 2.16957i 0 −0.897158 + 2.76117i 0
229.3 0 2.22808 − 1.61880i 0 −0.682374 − 2.10013i 0 −1.20858 − 0.878088i 0 1.41679 − 4.36044i 0
533.1 0 −1.16744 − 0.848196i 0 −1.32665 + 4.08302i 0 2.57671 − 1.87209i 0 −0.283568 − 0.872732i 0
533.2 0 −0.251622 − 0.182814i 0 0.581976 − 1.79114i 0 −2.98616 + 2.16957i 0 −0.897158 − 2.76117i 0
533.3 0 2.22808 + 1.61880i 0 −0.682374 + 2.10013i 0 −1.20858 + 0.878088i 0 1.41679 + 4.36044i 0
609.1 0 −0.895600 + 2.75637i 0 −1.97741 + 1.43667i 0 −0.0757739 − 0.233208i 0 −4.36844 − 3.17385i 0
609.2 0 −0.145620 + 0.448171i 0 2.86138 − 2.07892i 0 −0.710535 − 2.18680i 0 2.24740 + 1.63283i 0
609.3 0 0.732202 − 2.25349i 0 1.04308 − 0.757839i 0 1.40434 + 4.32212i 0 −2.11503 − 1.53666i 0
685.1 0 −0.895600 − 2.75637i 0 −1.97741 − 1.43667i 0 −0.0757739 + 0.233208i 0 −4.36844 + 3.17385i 0
685.2 0 −0.145620 − 0.448171i 0 2.86138 + 2.07892i 0 −0.710535 + 2.18680i 0 2.24740 − 1.63283i 0
685.3 0 0.732202 + 2.25349i 0 1.04308 + 0.757839i 0 1.40434 − 4.32212i 0 −2.11503 + 1.53666i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.3
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 836.2.j.a ✓ 12
11.c even 5 1 inner 836.2.j.a ✓ 12
11.c even 5 1 9196.2.a.j 6
11.d odd 10 1 9196.2.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
836.2.j.a ✓ 12 1.a even 1 1 trivial
836.2.j.a ✓ 12 11.c even 5 1 inner
9196.2.a.i 6 11.d odd 10 1
9196.2.a.j 6 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - T_{3}^{11} + 9 T_{3}^{10} - 11 T_{3}^{9} + 52 T_{3}^{8} + 45 T_{3}^{7} + 155 T_{3}^{6} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(836, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 39601 \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 2025 \) Copy content Toggle raw display
$11$ \( (T^{4} + T^{3} + 21 T^{2} + \cdots + 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{12} + 3 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{12} - 7 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$23$ \( (T^{6} + 12 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 11 T^{11} + \cdots + 2062096 \) Copy content Toggle raw display
$31$ \( T^{12} - 14 T^{11} + \cdots + 3610000 \) Copy content Toggle raw display
$37$ \( T^{12} + 17 T^{11} + \cdots + 20736 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 392198416 \) Copy content Toggle raw display
$43$ \( (T^{6} + 23 T^{5} + \cdots + 69975)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 5 T^{11} + \cdots + 5470921 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 44451818896 \) Copy content Toggle raw display
$59$ \( T^{12} + 19 T^{11} + \cdots + 72522256 \) Copy content Toggle raw display
$61$ \( T^{12} - 36 T^{11} + \cdots + 13549761 \) Copy content Toggle raw display
$67$ \( (T^{6} + 11 T^{5} - 127 T^{4} + \cdots - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 18474246400 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 517835536 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 429981696 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1292474401 \) Copy content Toggle raw display
$89$ \( (T^{6} - 13 T^{5} + \cdots - 35044)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 821166336 \) Copy content Toggle raw display
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