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Properties

Label 828.2.i.b
Level $828$
Weight $2$
Character orbit 828.i
Analytic conductor $6.612$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [828,2,Mod(277,828)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("828.277"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.288778218147.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{7} + \beta_{3} - \beta_{2}) q^{5} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - 1) q^{7} + ( - \beta_{8} - \beta_{3} - \beta_1) q^{9} + ( - \beta_{8} - \beta_{2} - \beta_1) q^{11}+ \cdots + ( - 2 \beta_{8} + 9 \beta_{7} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} + 6 q^{5} - 3 q^{7} - 3 q^{9} - 2 q^{11} + 7 q^{13} + 9 q^{15} - 4 q^{17} - 22 q^{19} + 12 q^{21} - 5 q^{23} - 7 q^{25} - 27 q^{27} - 5 q^{29} + 13 q^{31} - 18 q^{33} + 20 q^{35} + 34 q^{37}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 4x^{7} + 34x^{6} - 19x^{5} + 64x^{4} - x^{3} + 64x^{2} - 21x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 658 \nu^{9} - 2394 \nu^{8} - 4352 \nu^{7} - 10326 \nu^{6} - 25351 \nu^{5} - 51907 \nu^{4} + \cdots - 98232 ) / 72795 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 999 \nu^{9} - 537 \nu^{8} - 4321 \nu^{7} - 4688 \nu^{6} - 34543 \nu^{5} - 20431 \nu^{4} + \cdots - 9081 ) / 72795 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 339 \nu^{9} - 1348 \nu^{8} + 4381 \nu^{7} - 7882 \nu^{6} + 19883 \nu^{5} - 36059 \nu^{4} + \cdots - 29709 ) / 24265 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1348 \nu^{9} + 3356 \nu^{8} - 10907 \nu^{7} + 16239 \nu^{6} - 49501 \nu^{5} + 89773 \nu^{4} + \cdots + 245043 ) / 72795 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2533 \nu^{9} - 1626 \nu^{8} + 17417 \nu^{7} + 3021 \nu^{6} + 84646 \nu^{5} + 17167 \nu^{4} + \cdots + 59532 ) / 72795 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2694 \nu^{9} - 6203 \nu^{8} + 26226 \nu^{7} - 34722 \nu^{6} + 133958 \nu^{5} - 159864 \nu^{4} + \cdots - 139464 ) / 72795 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + \cdots - 54156 ) / 72795 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} + \cdots + 11514 ) / 14559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6042 \nu^{9} - 8994 \nu^{8} + 41363 \nu^{7} - 39341 \nu^{6} + 193324 \nu^{5} - 179927 \nu^{4} + \cdots - 134607 ) / 72795 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{8} - \beta_{6} + 2\beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} + 2\beta_{8} + 9\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{9} - 2 \beta_{8} - 36 \beta_{7} - 5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + 5 \beta_{3} + \cdots - 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{9} - 8\beta_{8} - 11\beta_{6} - 8\beta_{5} + 11\beta_{4} + 29\beta_{3} - 29\beta_{2} + 14\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{9} - 8\beta_{8} - 7\beta_{6} - \beta_{5} - 15\beta_{4} + 8\beta_{3} - 7\beta_{2} + 8\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{9} + 65\beta_{8} - 43\beta_{6} + 86\beta_{5} - 89\beta_{4} + 43\beta_{3} + 47\beta_{2} - 110\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 198 \beta_{9} + 86 \beta_{8} + 666 \beta_{7} + 200 \beta_{6} - 226 \beta_{5} + 112 \beta_{4} + \cdots - 2 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 47\beta_{9} - 58\beta_{8} + 126\beta_{6} - 105\beta_{5} + 68\beta_{4} - 268\beta_{3} + 126\beta_{2} + 58\beta _1 - 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(\beta_{7}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
277.1
−1.04536 + 1.81062i
1.07065 − 1.85442i
−0.539982 + 0.935277i
0.827154 − 1.43267i
0.187540 − 0.324828i
−1.04536 − 1.81062i
1.07065 + 1.85442i
−0.539982 − 0.935277i
0.827154 + 1.43267i
0.187540 + 0.324828i
0 −1.73091 + 0.0627999i 0 −0.933298 − 1.61652i 0 −1.57349 + 2.72537i 0 2.99211 − 0.217402i 0
277.2 0 0.278072 − 1.70958i 0 2.19714 + 3.80556i 0 −0.666084 + 1.15369i 0 −2.84535 − 0.950775i 0
277.3 0 0.376855 + 1.69056i 0 1.49015 + 2.58102i 0 1.94697 − 3.37226i 0 −2.71596 + 1.27419i 0
277.4 0 0.958787 + 1.44247i 0 0.282239 + 0.488851i 0 −1.41328 + 2.44788i 0 −1.16146 + 2.76605i 0
277.5 0 1.61720 − 0.620220i 0 −0.0362347 − 0.0627604i 0 0.205884 − 0.356601i 0 2.23065 − 2.00604i 0
553.1 0 −1.73091 − 0.0627999i 0 −0.933298 + 1.61652i 0 −1.57349 − 2.72537i 0 2.99211 + 0.217402i 0
553.2 0 0.278072 + 1.70958i 0 2.19714 − 3.80556i 0 −0.666084 − 1.15369i 0 −2.84535 + 0.950775i 0
553.3 0 0.376855 − 1.69056i 0 1.49015 − 2.58102i 0 1.94697 + 3.37226i 0 −2.71596 − 1.27419i 0
553.4 0 0.958787 − 1.44247i 0 0.282239 − 0.488851i 0 −1.41328 − 2.44788i 0 −1.16146 − 2.76605i 0
553.5 0 1.61720 + 0.620220i 0 −0.0362347 + 0.0627604i 0 0.205884 + 0.356601i 0 2.23065 + 2.00604i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.i.b ✓ 10
3.b odd 2 1 2484.2.i.b 10
9.c even 3 1 inner 828.2.i.b ✓ 10
9.c even 3 1 7452.2.a.c 5
9.d odd 6 1 2484.2.i.b 10
9.d odd 6 1 7452.2.a.h 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
828.2.i.b ✓ 10 1.a even 1 1 trivial
828.2.i.b ✓ 10 9.c even 3 1 inner
2484.2.i.b 10 3.b odd 2 1
2484.2.i.b 10 9.d odd 6 1
7452.2.a.c 5 9.c even 3 1
7452.2.a.h 5 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 6T_{5}^{9} + 34T_{5}^{8} - 62T_{5}^{7} + 166T_{5}^{6} - 95T_{5}^{5} + 643T_{5}^{4} - 296T_{5}^{3} + 169T_{5}^{2} + 12T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} - 6 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 361 \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + \cdots + 53361 \) Copy content Toggle raw display
$13$ \( T^{10} - 7 T^{9} + \cdots + 85849 \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} - 21 T^{3} + \cdots - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 11 T^{4} + \cdots - 1029)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} + \cdots + 720801 \) Copy content Toggle raw display
$31$ \( T^{10} - 13 T^{9} + \cdots + 2259009 \) Copy content Toggle raw display
$37$ \( (T^{5} - 17 T^{4} + \cdots - 2601)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 742181049 \) Copy content Toggle raw display
$43$ \( T^{10} - 26 T^{9} + \cdots + 63489024 \) Copy content Toggle raw display
$47$ \( T^{10} + 11 T^{9} + \cdots + 408321 \) Copy content Toggle raw display
$53$ \( (T^{5} - 15 T^{4} + \cdots + 5567)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} - 15 T^{9} + \cdots + 59049 \) Copy content Toggle raw display
$61$ \( T^{10} - 7 T^{9} + \cdots + 1423249 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 175271121 \) Copy content Toggle raw display
$71$ \( (T^{5} + 12 T^{4} + \cdots + 69471)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 19 T^{4} + \cdots + 195075)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} - 18 T^{9} + \cdots + 6561 \) Copy content Toggle raw display
$83$ \( T^{10} + 18 T^{9} + \cdots + 22686169 \) Copy content Toggle raw display
$89$ \( (T^{5} + 18 T^{4} + \cdots + 81)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1170255681 \) Copy content Toggle raw display
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