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Properties

Label 361.10.a.e
Level $361$
Weight $10$
Character orbit 361.a
Self dual yes
Analytic conductor $185.928$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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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] = [361,10,Mod(1,361)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("361.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-15] 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: \(185.927936855\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 5069 x^{12} + 6049 x^{11} + 9806858 x^{10} - 13799702 x^{9} - 9054174058 x^{8} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 19 \)
Twist minimal: no (minimal twist has level 19)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + 5) q^{3} + (\beta_{2} + 2 \beta_1 + 213) q^{4} + (\beta_{7} + \beta_{3} + 4 \beta_1 + 20) q^{5} + (\beta_{6} - 15 \beta_1 + 39) q^{6} + (\beta_{8} + 2 \beta_{3} - 15 \beta_1 - 94) q^{7}+ \cdots + ( - 401 \beta_{13} + 738 \beta_{12} + \cdots + 53002865) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 15 q^{2} + 74 q^{3} + 2987 q^{4} + 285 q^{5} + 535 q^{6} - 1338 q^{7} - 12135 q^{8} + 57928 q^{9} - 41180 q^{10} - 57405 q^{11} + 117729 q^{12} - 98671 q^{13} + 148290 q^{14} - 428251 q^{15} + 279203 q^{16}+ \cdots + 736622698 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 5069 x^{12} + 6049 x^{11} + 9806858 x^{10} - 13799702 x^{9} - 9054174058 x^{8} + \cdots - 17\!\cdots\!44 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 724 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 47\!\cdots\!09 \nu^{13} + \cdots - 10\!\cdots\!52 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!23 \nu^{13} + \cdots - 77\!\cdots\!88 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!09 \nu^{13} + \cdots - 50\!\cdots\!52 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74\!\cdots\!39 \nu^{13} + \cdots + 18\!\cdots\!36 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!91 \nu^{13} + \cdots - 63\!\cdots\!48 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\!\cdots\!73 \nu^{13} + \cdots - 23\!\cdots\!96 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27\!\cdots\!27 \nu^{13} + \cdots - 21\!\cdots\!80 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42\!\cdots\!73 \nu^{13} + \cdots + 65\!\cdots\!12 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!29 \nu^{13} + \cdots - 68\!\cdots\!84 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 43\!\cdots\!49 \nu^{13} + \cdots + 12\!\cdots\!08 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!93 \nu^{13} + \cdots + 82\!\cdots\!72 ) / 73\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 724 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} + 2\beta_{3} + 1199\beta _1 - 295 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 6 \beta_{7} + \cdots + 868135 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{13} - 11 \beta_{12} - 7 \beta_{11} - 93 \beta_{10} + 2 \beta_{9} - 586 \beta_{8} + \cdots - 466170 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2652 \beta_{13} + 2028 \beta_{12} + 2524 \beta_{11} - 748 \beta_{10} + 5472 \beta_{9} + \cdots + 1181810126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 69006 \beta_{13} - 30942 \beta_{12} - 19534 \beta_{11} - 239490 \beta_{10} + 14044 \beta_{9} + \cdots - 803679329 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5454315 \beta_{13} + 3395371 \beta_{12} + 4895187 \beta_{11} + 594853 \beta_{10} + 11195118 \beta_{9} + \cdots + 1691797203689 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 168440853 \beta_{13} - 65872765 \beta_{12} - 41432681 \beta_{11} - 475697675 \beta_{10} + \cdots - 1942131235628 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10201331558 \beta_{13} + 5490946294 \beta_{12} + 8693131678 \beta_{11} + 3629764410 \beta_{10} + \cdots + 24\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 352312045704 \beta_{13} - 126360856328 \beta_{12} - 80614690008 \beta_{11} - 864709289048 \beta_{10} + \cdots - 49\!\cdots\!15 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18195958915517 \beta_{13} + 8873847314301 \beta_{12} + 14845822981133 \beta_{11} + 9697587201059 \beta_{10} + \cdots + 37\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 681458149769847 \beta_{13} - 230584720711631 \beta_{12} - 151438982774283 \beta_{11} + \cdots - 11\!\cdots\!02 \) 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
39.3897
37.9671
33.1564
20.9383
17.4150
13.7186
5.65163
−7.92656
−8.67559
−14.6476
−26.2811
−33.6436
−35.1331
−40.9292
−40.3897 130.181 1119.32 −1773.63 −5257.96 −9612.95 −24529.6 −2735.95 71636.1
1.2 −38.9671 −178.567 1006.44 498.643 6958.24 698.529 −19266.8 12203.2 −19430.7
1.3 −34.1564 124.489 654.659 1904.13 −4252.10 11028.1 −4872.73 −4185.46 −65038.3
1.4 −21.9383 71.9034 −30.7120 −484.837 −1577.44 1688.25 11906.2 −14512.9 10636.5
1.5 −18.4150 −170.770 −172.887 −1591.63 3144.74 2900.36 12612.2 9479.50 29310.0
1.6 −14.7186 −85.4741 −295.361 2214.52 1258.06 −12185.9 11883.3 −12377.2 −32594.7
1.7 −6.65163 262.515 −467.756 155.085 −1746.15 −3034.24 6516.98 49231.1 −1031.57
1.8 6.92656 17.1250 −464.023 −1562.76 118.617 −83.0972 −6760.48 −19389.7 −10824.6
1.9 7.67559 −223.497 −453.085 1492.24 −1715.47 8355.84 −7407.60 30268.1 11453.8
1.10 13.6476 70.8395 −325.744 1144.07 966.787 5106.36 −11433.2 −14664.8 15613.8
1.11 25.2811 −176.997 127.135 −997.756 −4474.69 −9657.65 −9729.83 11645.0 −25224.4
1.12 32.6436 130.829 553.602 1611.76 4270.71 −7030.62 1358.05 −2566.90 52613.6
1.13 34.1331 208.441 653.070 −2506.72 7114.73 4067.64 4815.16 23764.5 −85562.3
1.14 39.9292 −107.016 1082.34 181.893 −4273.07 6421.38 22773.3 −8230.53 7262.85
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.10.a.e 14
19.b odd 2 1 361.10.a.f 14
19.c even 3 2 19.10.c.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.10.c.a ✓ 28 19.c even 3 2
361.10.a.e 14 1.a even 1 1 trivial
361.10.a.f 14 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 15 T_{2}^{13} - 4965 T_{2}^{12} - 66435 T_{2}^{11} + 9407052 T_{2}^{10} + \cdots - 17\!\cdots\!20 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(361))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 11\!\cdots\!35 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 84\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 42\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 99\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 26\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 49\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 17\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 79\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 92\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 50\!\cdots\!55 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 27\!\cdots\!45 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 37\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 67\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 38\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 37\!\cdots\!51 \) Copy content Toggle raw display
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