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Properties

Label 1365.2.f.d
Level $1365$
Weight $2$
Character orbit 1365.f
Analytic conductor $10.900$
Analytic rank $0$
Dimension $16$
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] = [1365,2,Mod(274,1365)] 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("1365.274"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1365, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1365 = 3 \cdot 5 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1365.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-32,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8995798759\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 416x^{12} + 2802x^{10} + 10280x^{8} + 19568x^{6} + 16273x^{4} + 4072x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{9} q^{5} + \beta_{5} q^{6} + \beta_{3} q^{7} + ( - \beta_{10} + \beta_{9} + \cdots - 2 \beta_1) q^{8} - q^{9} + (\beta_{13} - \beta_{12} + \cdots - \beta_{2}) q^{10}+ \cdots + ( - \beta_{5} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 2 q^{5} + 4 q^{6} - 16 q^{9} - 2 q^{10} + 4 q^{11} + 4 q^{14} + 64 q^{16} - 4 q^{19} - 2 q^{20} - 16 q^{21} - 24 q^{24} + 2 q^{25} + 4 q^{26} + 24 q^{29} + 2 q^{30} + 4 q^{31} - 4 q^{34}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 416x^{12} + 2802x^{10} + 10280x^{8} + 19568x^{6} + 16273x^{4} + 4072x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{15} - 8\nu^{13} + 292\nu^{11} + 5154\nu^{9} + 31660\nu^{7} + 81796\nu^{5} + 72755\nu^{3} + 12980\nu ) / 2064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{14} - 244\nu^{12} - 2532\nu^{10} - 12266\nu^{8} - 25864\nu^{6} - 12036\nu^{4} + 14095\nu^{2} + 3644 ) / 688 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{14} + 59\nu^{12} + 663\nu^{10} + 3495\nu^{8} + 8447\nu^{6} + 7419\nu^{4} + 1421\nu^{2} + 12 ) / 172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} - 9 \nu^{14} - 8 \nu^{13} - 72 \nu^{12} + 292 \nu^{11} + 1596 \nu^{10} + 5154 \nu^{9} + \cdots + 18780 ) / 4128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{15} + 168 \nu^{14} + 668 \nu^{13} + 4440 \nu^{12} + 9416 \nu^{11} + 44856 \nu^{10} + \cdots - 1056 ) / 8256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19 \nu^{15} + 168 \nu^{14} - 668 \nu^{13} + 4440 \nu^{12} - 9416 \nu^{11} + 44856 \nu^{10} + \cdots - 1056 ) / 8256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 165 \nu^{14} + 8 \nu^{13} - 4416 \nu^{12} - 292 \nu^{11} - 45732 \nu^{10} - 5154 \nu^{9} + \cdots - 16212 ) / 4128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{15} - 165 \nu^{14} - 8 \nu^{13} - 4416 \nu^{12} + 292 \nu^{11} - 45732 \nu^{10} + \cdots - 16212 ) / 4128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 83 \nu^{15} - 2212 \nu^{13} - 22720 \nu^{11} - 111438 \nu^{9} - 258208 \nu^{7} - 230080 \nu^{5} + \cdots - 15548 \nu ) / 4128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{15} - 169 \nu^{13} - 2281 \nu^{11} - 15639 \nu^{9} - 57001 \nu^{7} - 104053 \nu^{5} + \cdots - 13016 \nu ) / 516 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 227 \nu^{15} - 210 \nu^{14} - 6460 \nu^{13} - 5808 \nu^{12} - 72520 \nu^{11} - 62520 \nu^{10} + \cdots - 744 ) / 8256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 139 \nu^{15} - 3692 \nu^{13} - 37328 \nu^{11} - 174726 \nu^{9} - 350072 \nu^{7} - 134576 \nu^{5} + \cdots + 98324 \nu ) / 4128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 227 \nu^{15} - 210 \nu^{14} + 6460 \nu^{13} - 5808 \nu^{12} + 72520 \nu^{11} - 62520 \nu^{10} + \cdots - 744 ) / 8256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{13} - \beta_{12} - 2 \beta_{11} + 11 \beta_{10} - 11 \beta_{9} - \beta_{8} + \cdots + 38 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{15} - 11 \beta_{13} + 11 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + 14 \beta_{7} + \cdots - 155 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{15} - 14 \beta_{13} + 16 \beta_{12} + 28 \beta_{11} - 100 \beta_{10} + 100 \beta_{9} + \cdots - 249 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{15} + 96 \beta_{13} - 96 \beta_{10} - 132 \beta_{9} - 148 \beta_{8} - 148 \beta_{7} + \cdots + 1040 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 140 \beta_{15} + 8 \beta_{14} + 140 \beta_{13} - 176 \beta_{12} - 292 \beta_{11} + 845 \beta_{10} + \cdots + 1674 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 773 \beta_{15} - 773 \beta_{13} + 769 \beta_{10} + 1217 \beta_{9} + 1397 \beta_{8} + 1397 \beta_{7} + \cdots - 7180 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1237 \beta_{15} - 164 \beta_{14} - 1237 \beta_{13} + 1685 \beta_{12} + 2718 \beta_{11} + \cdots - 11498 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5999 \beta_{15} + 5999 \beta_{13} - 5915 \beta_{10} - 10657 \beta_{9} - 12426 \beta_{8} - 12426 \beta_{7} + \cdots + 50723 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10322 \beta_{15} + 2188 \beta_{14} + 10322 \beta_{13} - 15100 \beta_{12} - 23868 \beta_{11} + \cdots + 80441 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45732 \beta_{15} - 45732 \beta_{13} + 44580 \beta_{10} + 90420 \beta_{9} + 106672 \beta_{8} + \cdots - 365108 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 83664 \beta_{15} - 24160 \beta_{14} - 83664 \beta_{13} + 130480 \beta_{12} + 202520 \beta_{11} + \cdots - 571790 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1365\mathbb{Z}\right)^\times\).

\(n\) \(106\) \(547\) \(911\) \(976\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 2.80391i
− 2.57127i
− 2.49569i
− 2.21394i
− 2.15856i
− 1.16327i
− 0.584729i
− 0.205168i
0.205168i
0.584729i
1.16327i
2.15856i
2.21394i
2.49569i
2.57127i
2.80391i
− 2.80391i 1.00000i −5.86193 −0.739214 + 2.11035i 2.80391 1.00000i 10.8285i −1.00000 5.91723 + 2.07269i
274.2 − 2.57127i − 1.00000i −4.61144 −1.82920 + 1.28609i −2.57127 − 1.00000i 6.71472i −1.00000 3.30688 + 4.70337i
274.3 − 2.49569i 1.00000i −4.22847 2.22572 − 0.214900i 2.49569 1.00000i 5.56158i −1.00000 −0.536323 − 5.55470i
274.4 − 2.21394i 1.00000i −2.90155 −1.43370 − 1.71595i 2.21394 1.00000i 1.99598i −1.00000 −3.79903 + 3.17414i
274.5 − 2.15856i − 1.00000i −2.65940 2.02569 − 0.946887i −2.15856 − 1.00000i 1.42335i −1.00000 −2.04392 − 4.37257i
274.6 − 1.16327i − 1.00000i 0.646794 0.384586 − 2.20275i −1.16327 − 1.00000i − 3.07895i −1.00000 −2.56240 − 0.447379i
274.7 − 0.584729i 1.00000i 1.65809 0.599430 − 2.15422i 0.584729 1.00000i − 2.13899i −1.00000 −1.25964 − 0.350504i
274.8 − 0.205168i − 1.00000i 1.95791 −2.23330 − 0.111186i −0.205168 − 1.00000i − 0.812036i −1.00000 −0.0228118 + 0.458202i
274.9 0.205168i 1.00000i 1.95791 −2.23330 + 0.111186i −0.205168 1.00000i 0.812036i −1.00000 −0.0228118 − 0.458202i
274.10 0.584729i − 1.00000i 1.65809 0.599430 + 2.15422i 0.584729 − 1.00000i 2.13899i −1.00000 −1.25964 + 0.350504i
274.11 1.16327i 1.00000i 0.646794 0.384586 + 2.20275i −1.16327 1.00000i 3.07895i −1.00000 −2.56240 + 0.447379i
274.12 2.15856i 1.00000i −2.65940 2.02569 + 0.946887i −2.15856 1.00000i − 1.42335i −1.00000 −2.04392 + 4.37257i
274.13 2.21394i − 1.00000i −2.90155 −1.43370 + 1.71595i 2.21394 − 1.00000i − 1.99598i −1.00000 −3.79903 − 3.17414i
274.14 2.49569i − 1.00000i −4.22847 2.22572 + 0.214900i 2.49569 − 1.00000i − 5.56158i −1.00000 −0.536323 + 5.55470i
274.15 2.57127i 1.00000i −4.61144 −1.82920 − 1.28609i −2.57127 1.00000i − 6.71472i −1.00000 3.30688 − 4.70337i
274.16 2.80391i − 1.00000i −5.86193 −0.739214 − 2.11035i 2.80391 − 1.00000i − 10.8285i −1.00000 5.91723 − 2.07269i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1365.2.f.d ✓ 16
5.b even 2 1 inner 1365.2.f.d ✓ 16
5.c odd 4 1 6825.2.a.bv 8
5.c odd 4 1 6825.2.a.ca 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1365.2.f.d ✓ 16 1.a even 1 1 trivial
1365.2.f.d ✓ 16 5.b even 2 1 inner
6825.2.a.bv 8 5.c odd 4 1
6825.2.a.ca 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1365, [\chi])\):

\( T_{2}^{16} + 32T_{2}^{14} + 416T_{2}^{12} + 2802T_{2}^{10} + 10280T_{2}^{8} + 19568T_{2}^{6} + 16273T_{2}^{4} + 4072T_{2}^{2} + 144 \) Copy content Toggle raw display
\( T_{11}^{8} - 2T_{11}^{7} - 34T_{11}^{6} + 80T_{11}^{5} + 229T_{11}^{4} - 550T_{11}^{3} - 84T_{11}^{2} + 536T_{11} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 32 T^{14} + \cdots + 144 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} + \cdots - 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$17$ \( T^{16} + 124 T^{14} + \cdots + 8294400 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2 T^{7} + \cdots - 524)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1587225600 \) Copy content Toggle raw display
$29$ \( (T^{8} - 12 T^{7} + \cdots + 50396)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} - 65 T^{6} + \cdots - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 6156599296 \) Copy content Toggle raw display
$41$ \( (T^{8} - 10 T^{7} + \cdots - 615168)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 40613534784 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 107495424 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 15521267607616 \) Copy content Toggle raw display
$59$ \( (T^{8} + 2 T^{7} + \cdots - 217920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 28 T^{7} + \cdots + 23344)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5977145344 \) Copy content Toggle raw display
$71$ \( (T^{8} + 20 T^{7} + \cdots - 1536)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 253319056 \) Copy content Toggle raw display
$79$ \( (T^{8} - 12 T^{7} + \cdots - 221632)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 120712235295744 \) Copy content Toggle raw display
$89$ \( (T^{8} - 14 T^{7} + \cdots - 119867240)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 4472497469584 \) Copy content Toggle raw display
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