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Properties

Label 3276.2.cf.a
Level $3276$
Weight $2$
Character orbit 3276.cf
Analytic conductor $26.159$
Analytic rank $0$
Dimension $12$
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] = [3276,2,Mod(1765,3276)] 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("3276.1765"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3276.cf (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.1589917022\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 12 x^{9} + 65 x^{8} - 78 x^{7} + 98 x^{6} + 550 x^{5} + 850 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1092)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{4}) q^{5} + ( - \beta_{8} - \beta_{3}) q^{7} + ( - \beta_{10} + \beta_{5} + \cdots + \beta_{2}) q^{11} + (2 \beta_{11} - \beta_{9} + \beta_{8} + \cdots + 3) q^{13}+ \cdots + ( - 2 \beta_{11} - 3 \beta_{10} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{13} - 2 q^{23} + 4 q^{25} - 14 q^{29} + 2 q^{35} + 12 q^{37} - 12 q^{41} - 10 q^{43} + 6 q^{49} - 8 q^{53} - 2 q^{55} + 12 q^{59} - 12 q^{61} - 2 q^{65} - 42 q^{67} + 30 q^{71} - 8 q^{77} - 4 q^{79}+ \cdots + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 12 x^{9} + 65 x^{8} - 78 x^{7} + 98 x^{6} + 550 x^{5} + 850 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10143602 \nu^{11} - 25154217 \nu^{10} + 32442622 \nu^{9} + 105600176 \nu^{8} + 610755929 \nu^{7} + \cdots + 283495023 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3832204 \nu^{11} - 31313357 \nu^{10} + 94885587 \nu^{9} - 181535270 \nu^{8} + \cdots - 2120670780 ) / 121744506 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31499447 \nu^{11} + 73142496 \nu^{10} - 88153111 \nu^{9} - 345550742 \nu^{8} + \cdots - 985707675 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18048 \nu^{11} + 26570 \nu^{10} - 7140 \nu^{9} - 259652 \nu^{8} - 1259035 \nu^{7} + 899486 \nu^{6} + \cdots - 888672 ) / 96546 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 78688126 \nu^{11} - 169826406 \nu^{10} + 172625831 \nu^{9} + 948843514 \nu^{8} + \cdots + 1661260869 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 90630362 \nu^{11} - 281189754 \nu^{10} + 441551230 \nu^{9} + 740467427 \nu^{8} + \cdots - 1600333191 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31722591 \nu^{11} - 105862602 \nu^{10} + 172118209 \nu^{9} + 239977710 \nu^{8} + \cdots - 1066840374 ) / 121744506 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 153085094 \nu^{11} + 374714712 \nu^{10} - 450842377 \nu^{9} - 1696837919 \nu^{8} + \cdots - 829850499 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63338490 \nu^{11} + 128209671 \nu^{10} - 116009493 \nu^{9} - 792411727 \nu^{8} + \cdots - 1163505957 ) / 121744506 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 241363513 \nu^{11} - 642311589 \nu^{10} + 850662869 \nu^{9} + 2480046367 \nu^{8} + \cdots - 1184607990 ) / 365233518 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 344782835 \nu^{11} - 778867497 \nu^{10} + 862310893 \nu^{9} + 3995992562 \nu^{8} + \cdots + 7075728351 ) / 365233518 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 5\beta_{3} - \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2\beta_{3} + 7\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{11} - 11 \beta_{10} - 6 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 10 \beta_{5} - 11 \beta_{4} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{11} - 57 \beta_{10} + 9 \beta_{9} - 79 \beta_{8} - \beta_{7} - \beta_{6} - 10 \beta_{5} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{11} - 61 \beta_{10} - 46 \beta_{9} - 66 \beta_{8} + 46 \beta_{7} - 97 \beta_{6} - 92 \beta_{5} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15 \beta_{11} + 162 \beta_{10} - 15 \beta_{9} + 290 \beta_{8} + 247 \beta_{7} - 247 \beta_{6} + \cdots + 270 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 297 \beta_{11} + 1114 \beta_{10} - 85 \beta_{9} + 1311 \beta_{8} + 934 \beta_{7} - 552 \beta_{6} + \cdots + 1361 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 669 \beta_{11} + 4183 \beta_{10} - 669 \beta_{9} + 6338 \beta_{8} + 180 \beta_{7} + 180 \beta_{6} + \cdots + 1399 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1064 \beta_{11} + 3849 \beta_{10} + 1875 \beta_{9} + 4913 \beta_{8} - 6016 \beta_{7} + 8955 \beta_{6} + \cdots + 142 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1974 \beta_{11} - 17768 \beta_{10} + 1974 \beta_{9} - 30765 \beta_{8} - 29619 \beta_{7} + 29619 \beta_{6} + \cdots - 23733 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times\).

\(n\) \(1639\) \(2017\) \(2341\) \(2549\)
\(\chi(n)\) \(1\) \(1 - \beta_{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1765.1
−1.84370 + 1.84370i
−0.552679 − 0.552679i
1.82857 − 1.82857i
2.18161 + 2.18161i
−0.262901 − 0.262901i
−0.350904 + 0.350904i
−0.350904 − 0.350904i
−0.262901 + 0.262901i
2.18161 − 2.18161i
1.82857 + 1.82857i
−0.552679 + 0.552679i
−1.84370 − 1.84370i
0 0 0 − 2.87433i 0 0.866025 + 0.500000i 0 0 0
1765.2 0 0 0 − 2.84013i 0 −0.866025 − 0.500000i 0 0 0
1765.3 0 0 0 − 2.07271i 0 0.866025 + 0.500000i 0 0 0
1765.4 0 0 0 0.702222i 0 −0.866025 − 0.500000i 0 0 0
1765.5 0 0 0 1.40586i 0 −0.866025 − 0.500000i 0 0 0
1765.6 0 0 0 2.21499i 0 0.866025 + 0.500000i 0 0 0
2773.1 0 0 0 − 2.21499i 0 0.866025 − 0.500000i 0 0 0
2773.2 0 0 0 − 1.40586i 0 −0.866025 + 0.500000i 0 0 0
2773.3 0 0 0 − 0.702222i 0 −0.866025 + 0.500000i 0 0 0
2773.4 0 0 0 2.07271i 0 0.866025 − 0.500000i 0 0 0
2773.5 0 0 0 2.84013i 0 −0.866025 + 0.500000i 0 0 0
2773.6 0 0 0 2.87433i 0 0.866025 − 0.500000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1765.6
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 3276.2.cf.a 12
3.b odd 2 1 1092.2.bg.b ✓ 12
13.e even 6 1 inner 3276.2.cf.a 12
39.h odd 6 1 1092.2.bg.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1092.2.bg.b ✓ 12 3.b odd 2 1
1092.2.bg.b ✓ 12 39.h odd 6 1
3276.2.cf.a 12 1.a even 1 1 trivial
3276.2.cf.a 12 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 28T_{5}^{10} + 302T_{5}^{8} + 1570T_{5}^{6} + 4001T_{5}^{4} + 4402T_{5}^{2} + 1369 \) acting on \(S_{2}^{\mathrm{new}}(3276, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 28 T^{10} + \cdots + 1369 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} - 28 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 36 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{12} - 43 T^{10} + \cdots + 187489 \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + \cdots + 19881 \) Copy content Toggle raw display
$29$ \( T^{12} + 14 T^{11} + \cdots + 10673289 \) Copy content Toggle raw display
$31$ \( T^{12} + 158 T^{10} + \cdots + 38775529 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots + 45927729 \) Copy content Toggle raw display
$41$ \( T^{12} + 12 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( T^{12} + 10 T^{11} + \cdots + 9223369 \) Copy content Toggle raw display
$47$ \( T^{12} + 86 T^{10} + \cdots + 418609 \) Copy content Toggle raw display
$53$ \( (T^{6} + 4 T^{5} + \cdots + 7549)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 12 T^{11} + \cdots + 3767481 \) Copy content Toggle raw display
$61$ \( T^{12} + 12 T^{11} + \cdots + 625 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2826942561 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 7110031041 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 43392639481 \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots - 152787)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 303820337601 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 482109849 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 9630870769 \) Copy content Toggle raw display
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