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Properties

Label 363.8.a.n
Level $363$
Weight $8$
Character orbit 363.a
Self dual yes
Analytic conductor $113.396$
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] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(113.395764251\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \( x^{12} - 1139x^{10} + 474732x^{8} - 87245660x^{6} + 6439013152x^{4} - 100879962816x^{2} + 359539347456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 11^{6} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 27 q^{3} + (\beta_{2} + 62) q^{4} + (\beta_{4} + 4) q^{5} - 27 \beta_1 q^{6} + (\beta_{11} + \beta_{9} + \cdots + 11 \beta_1) q^{7} + ( - \beta_{11} - 3 \beta_{9} + \cdots + 49 \beta_1) q^{8}+ \cdots + (4664 \beta_{11} - 1117 \beta_{10} + \cdots - 424376 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 324 q^{3} + 742 q^{4} + 54 q^{5} + 8748 q^{9} - 20034 q^{12} + 25512 q^{14} - 1458 q^{15} + 17570 q^{16} - 14730 q^{20} - 108720 q^{23} + 395498 q^{25} + 498174 q^{26} - 236196 q^{27} - 261362 q^{31}+ \cdots - 51696396 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 1139x^{10} + 474732x^{8} - 87245660x^{6} + 6439013152x^{4} - 100879962816x^{2} + 359539347456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 190 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5411 \nu^{10} - 4752701 \nu^{8} + 1026215744 \nu^{6} + 58126844652 \nu^{4} + \cdots + 208704946065408 ) / 1792848837120 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2526503 \nu^{10} - 3354639753 \nu^{8} + 1573429286832 \nu^{6} - 304754961629764 \nu^{4} + \cdots - 12\!\cdots\!56 ) / 271616598823680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1922119 \nu^{10} + 2126602569 \nu^{8} - 855743089296 \nu^{6} + 150854012001668 \nu^{4} + \cdots + 94\!\cdots\!52 ) / 18107773254912 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10587871 \nu^{10} - 10097029681 \nu^{8} + 3257934211984 \nu^{6} - 404877142257828 \nu^{4} + \cdots - 61\!\cdots\!72 ) / 90538866274560 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6877591 \nu^{11} - 7901170361 \nu^{9} + 3324383271504 \nu^{7} - 612859453079108 \nu^{5} + \cdots - 39\!\cdots\!92 \nu ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11901350641 \nu^{11} - 14145937192751 \nu^{9} + \cdots - 49\!\cdots\!32 \nu ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62469841 \nu^{11} - 70936050431 \nu^{9} + 29293286757264 \nu^{7} + \cdots - 32\!\cdots\!32 \nu ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3538438009 \nu^{11} - 3715402218659 \nu^{9} + \cdots - 17\!\cdots\!48 \nu ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9851821393 \nu^{11} + 10829572755143 \nu^{9} + \cdots + 91\!\cdots\!16 \nu ) / 11\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 190 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - 3\beta_{9} + \beta_{8} + 9\beta_{7} + 305\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{6} + 7\beta_{5} + 30\beta_{4} - 40\beta_{3} + 385\beta_{2} + 58046 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -449\beta_{11} + 118\beta_{10} - 1941\beta_{9} + 587\beta_{8} + 10063\beta_{7} + 102607\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3153\beta_{6} + 4151\beta_{5} + 17394\beta_{4} - 29784\beta_{3} + 146667\beta_{2} + 19568698 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -186499\beta_{11} + 76198\beta_{10} - 977127\beta_{9} + 263533\beta_{8} + 6392809\beta_{7} + 36698801\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1577047\beta_{6} + 1954225\beta_{5} + 7754430\beta_{4} - 16273544\beta_{3} + 56775281\beta_{2} + 7013635022 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 76454201 \beta_{11} + 39179802 \beta_{10} - 450096333 \beta_{9} + 109158671 \beta_{8} + \cdots + 13703147803 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 733494805 \beta_{6} + 854026547 \beta_{5} + 3189925194 \beta_{4} - 7884042968 \beta_{3} + \cdots + 2623577339290 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 31362526099 \beta_{11} + 18694387214 \beta_{10} - 198736304079 \beta_{9} + 43995968917 \beta_{8} + \cdots + 5272841384057 \beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−20.3449
−17.5038
−15.3836
−12.7496
−3.79005
−2.26509
2.26509
3.79005
12.7496
15.3836
17.5038
20.3449
−20.3449 −27.0000 285.914 55.3567 549.312 480.919 −3212.74 729.000 −1126.23
1.2 −17.5038 −27.0000 178.382 395.788 472.602 −446.896 −881.876 729.000 −6927.79
1.3 −15.3836 −27.0000 108.656 −358.490 415.358 482.323 297.579 729.000 5514.88
1.4 −12.7496 −27.0000 34.5525 −425.277 344.239 −1135.81 1191.42 729.000 5422.11
1.5 −3.79005 −27.0000 −113.635 437.441 102.331 −1542.07 915.811 729.000 −1657.93
1.6 −2.26509 −27.0000 −122.869 −77.8195 61.1574 −799.987 568.242 729.000 176.268
1.7 2.26509 −27.0000 −122.869 −77.8195 −61.1574 799.987 −568.242 729.000 −176.268
1.8 3.79005 −27.0000 −113.635 437.441 −102.331 1542.07 −915.811 729.000 1657.93
1.9 12.7496 −27.0000 34.5525 −425.277 −344.239 1135.81 −1191.42 729.000 −5422.11
1.10 15.3836 −27.0000 108.656 −358.490 −415.358 −482.323 −297.579 729.000 −5514.88
1.11 17.5038 −27.0000 178.382 395.788 −472.602 446.896 881.876 729.000 6927.79
1.12 20.3449 −27.0000 285.914 55.3567 −549.312 −480.919 3212.74 729.000 1126.23
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.8.a.n ✓ 12
11.b odd 2 1 inner 363.8.a.n ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.8.a.n ✓ 12 1.a even 1 1 trivial
363.8.a.n ✓ 12 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 1139 T_{2}^{10} + 474732 T_{2}^{8} - 87245660 T_{2}^{6} + 6439013152 T_{2}^{4} + \cdots + 359539347456 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(363))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 359539347456 \) Copy content Toggle raw display
$3$ \( (T + 27)^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 113707929600000)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 71\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 78\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 76\!\cdots\!25)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 43\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 90\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 76\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 43\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 58\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 20\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 16\!\cdots\!25)^{2} \) Copy content Toggle raw display
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