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Properties

Label 2366.4.a.bi
Level $2366$
Weight $4$
Character orbit 2366.a
Self dual yes
Analytic conductor $139.599$
Analytic rank $0$
Dimension $15$
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] = [2366,4,Mod(1,2366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,-30,-5,60,26] 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: yes
Analytic conductor: \(139.598519074\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 272 x^{13} + 1126 x^{12} + 29249 x^{11} - 95770 x^{10} - 1588299 x^{9} + \cdots - 13037372712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7\cdot 13^{4} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - \beta_1 q^{3} + 4 q^{4} + (\beta_{4} + 2) q^{5} + 2 \beta_1 q^{6} + 7 q^{7} - 8 q^{8} + (\beta_{4} + \beta_{3} + \beta_1 + 11) q^{9} + ( - 2 \beta_{4} - 4) q^{10} + (\beta_{14} + \beta_{12} + \beta_{10} + \cdots + 2) q^{11}+ \cdots + (22 \beta_{14} + 19 \beta_{13} + \cdots + 192) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 30 q^{2} - 5 q^{3} + 60 q^{4} + 26 q^{5} + 10 q^{6} + 105 q^{7} - 120 q^{8} + 164 q^{9} - 52 q^{10} + 7 q^{11} - 20 q^{12} - 210 q^{14} + 86 q^{15} + 240 q^{16} + 16 q^{17} - 328 q^{18} + 511 q^{19}+ \cdots + 2790 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 272 x^{13} + 1126 x^{12} + 29249 x^{11} - 95770 x^{10} - 1588299 x^{9} + \cdots - 13037372712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\!\cdots\!04 \nu^{14} + \cdots + 54\!\cdots\!44 ) / 40\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 94\!\cdots\!89 \nu^{14} + \cdots - 15\!\cdots\!80 ) / 40\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 94\!\cdots\!89 \nu^{14} + \cdots + 14\!\cdots\!88 ) / 40\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 84\!\cdots\!49 \nu^{14} + \cdots + 24\!\cdots\!32 ) / 31\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37\!\cdots\!39 \nu^{14} + \cdots + 13\!\cdots\!82 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\!\cdots\!31 \nu^{14} + \cdots + 10\!\cdots\!40 ) / 81\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!28 \nu^{14} + \cdots - 10\!\cdots\!62 ) / 40\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!11 \nu^{14} + \cdots + 21\!\cdots\!38 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51\!\cdots\!19 \nu^{14} + \cdots + 13\!\cdots\!12 ) / 81\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!67 \nu^{14} + \cdots - 46\!\cdots\!52 ) / 31\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!11 \nu^{14} + \cdots + 25\!\cdots\!52 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!71 \nu^{14} + \cdots + 85\!\cdots\!84 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 83\!\cdots\!55 \nu^{14} + \cdots - 17\!\cdots\!10 ) / 40\!\cdots\!34 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - 3 \beta_{11} - \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + 3 \beta_{5} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10 \beta_{14} + 5 \beta_{13} - 6 \beta_{12} - 17 \beta_{11} - 4 \beta_{10} - 7 \beta_{9} - 12 \beta_{8} + \cdots + 2414 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 153 \beta_{14} - 49 \beta_{13} - 30 \beta_{12} - 370 \beta_{11} - 53 \beta_{10} - 351 \beta_{9} + \cdots + 5027 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1583 \beta_{14} + 563 \beta_{13} - 381 \beta_{12} - 2726 \beta_{11} - 345 \beta_{10} - 1221 \beta_{9} + \cdots + 184571 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18475 \beta_{14} - 1298 \beta_{13} - 2787 \beta_{12} - 37463 \beta_{11} - 939 \beta_{10} - 33425 \beta_{9} + \cdots + 650386 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 193619 \beta_{14} + 52923 \beta_{13} - 8310 \beta_{12} - 324785 \beta_{11} - 11488 \beta_{10} + \cdots + 15402712 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2083660 \beta_{14} + 96763 \beta_{13} - 104784 \beta_{12} - 3675022 \beta_{11} + 253362 \beta_{10} + \cdots + 76205637 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21867813 \beta_{14} + 5012071 \beta_{13} + 1839534 \beta_{12} - 34929436 \beta_{11} + 1657673 \beta_{10} + \cdots + 1351373837 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 228469668 \beta_{14} + 24615658 \beta_{13} + 12011829 \beta_{12} - 361430676 \beta_{11} + \cdots + 8359432271 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2389557846 \beta_{14} + 494441835 \beta_{13} + 417254082 \beta_{12} - 3601278554 \beta_{11} + \cdots + 122479212814 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 24664327652 \beta_{14} + 3511822836 \beta_{13} + 3543790641 \beta_{12} - 35813912033 \beta_{11} + \cdots + 879954264359 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 256651914699 \beta_{14} + 50491080692 \beta_{13} + 62582570139 \beta_{12} - 364451073535 \beta_{11} + \cdots + 11354481655477 \) 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
10.0031
9.63589
5.92979
5.79056
5.74358
4.79740
2.09855
−1.05600
−2.45772
−2.93842
−3.64980
−4.62607
−7.81817
−8.17455
−8.27817
−2.00000 −10.0031 4.00000 18.0968 20.0062 7.00000 −8.00000 73.0624 −36.1936
1.2 −2.00000 −9.63589 4.00000 −2.86008 19.2718 7.00000 −8.00000 65.8504 5.72016
1.3 −2.00000 −5.92979 4.00000 4.19102 11.8596 7.00000 −8.00000 8.16242 −8.38205
1.4 −2.00000 −5.79056 4.00000 −16.6311 11.5811 7.00000 −8.00000 6.53058 33.2621
1.5 −2.00000 −5.74358 4.00000 −8.34699 11.4872 7.00000 −8.00000 5.98868 16.6940
1.6 −2.00000 −4.79740 4.00000 13.0998 9.59480 7.00000 −8.00000 −3.98494 −26.1995
1.7 −2.00000 −2.09855 4.00000 −4.70897 4.19711 7.00000 −8.00000 −22.5961 9.41794
1.8 −2.00000 1.05600 4.00000 9.53424 −2.11200 7.00000 −8.00000 −25.8849 −19.0685
1.9 −2.00000 2.45772 4.00000 21.4663 −4.91545 7.00000 −8.00000 −20.9596 −42.9327
1.10 −2.00000 2.93842 4.00000 −8.28127 −5.87684 7.00000 −8.00000 −18.3657 16.5625
1.11 −2.00000 3.64980 4.00000 −21.5157 −7.29960 7.00000 −8.00000 −13.6790 43.0314
1.12 −2.00000 4.62607 4.00000 −9.48128 −9.25213 7.00000 −8.00000 −5.59952 18.9626
1.13 −2.00000 7.81817 4.00000 3.05262 −15.6363 7.00000 −8.00000 34.1237 −6.10523
1.14 −2.00000 8.17455 4.00000 18.4026 −16.3491 7.00000 −8.00000 39.8233 −36.8052
1.15 −2.00000 8.27817 4.00000 9.98200 −16.5563 7.00000 −8.00000 41.5281 −19.9640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)
\(13\) \( -1 \)

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 2366.4.a.bi ✓ 15
13.b even 2 1 2366.4.a.bk yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.4.a.bi ✓ 15 1.a even 1 1 trivial
2366.4.a.bk yes 15 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{15} + 5 T_{3}^{14} - 272 T_{3}^{13} - 1126 T_{3}^{12} + 29249 T_{3}^{11} + 95770 T_{3}^{10} + \cdots + 13037372712 \) Copy content Toggle raw display
\( T_{5}^{15} - 26 T_{5}^{14} - 919 T_{5}^{13} + 26562 T_{5}^{12} + 269328 T_{5}^{11} + \cdots + 360139962867776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 13037372712 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 360139962867776 \) Copy content Toggle raw display
$7$ \( (T - 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 26\!\cdots\!17 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 64\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 52\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 29\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 28\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 98\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 43\!\cdots\!77 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 38\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 95\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 46\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 12\!\cdots\!72 \) Copy content Toggle raw display
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