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Properties

Label 127.4.a.b
Level $127$
Weight $4$
Character orbit 127.a
Self dual yes
Analytic conductor $7.493$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [127,4,Mod(1,127)] 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("127.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 127 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 127.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.49324257073\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + \cdots - 130048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{8} - 1) q^{3} + (\beta_{9} + \beta_{8} - \beta_1 + 3) q^{4} + (\beta_{12} - \beta_{9} - \beta_1 - 3) q^{5} + ( - \beta_{12} - \beta_{9} + \cdots - 3 \beta_1) q^{6}+ \cdots + ( - 85 \beta_{12} + 51 \beta_{11} + \cdots + 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 8 q^{2} - 16 q^{3} + 34 q^{4} - 46 q^{5} - 2 q^{6} - 26 q^{7} - 117 q^{8} + 3 q^{9} - 25 q^{10} - 53 q^{11} - 255 q^{12} - 75 q^{13} - 152 q^{14} - 62 q^{15} + 82 q^{16} - 479 q^{17} - 292 q^{18}+ \cdots + 1769 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 5 x^{12} - 55 x^{11} + 264 x^{10} + 1126 x^{9} - 5085 x^{8} - 10823 x^{7} + 44242 x^{6} + \cdots - 130048 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1002737 \nu^{12} + 723467 \nu^{11} + 81924597 \nu^{10} - 45464246 \nu^{9} + \cdots - 379711735296 ) / 12616105984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60894341 \nu^{12} - 421156103 \nu^{11} - 2418602233 \nu^{10} + 20568960398 \nu^{9} + \cdots - 6445760160256 ) / 227089907712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 66841603 \nu^{12} - 437650993 \nu^{11} - 2909259215 \nu^{10} + 21534874690 \nu^{9} + \cdots - 1992359022080 ) / 227089907712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10515013 \nu^{12} - 85125187 \nu^{11} - 357280253 \nu^{10} + 4134258130 \nu^{9} + \cdots - 585397755392 ) / 28386238464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13771177 \nu^{12} - 116883187 \nu^{11} - 491339021 \nu^{10} + 5838860294 \nu^{9} + \cdots - 1040420694528 ) / 25232211968 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31415069 \nu^{12} + 251048519 \nu^{11} + 1092161593 \nu^{10} - 12184503782 \nu^{9} + \cdots + 2433094191616 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52572899 \nu^{12} - 363365393 \nu^{11} - 2167612399 \nu^{10} + 17815473986 \nu^{9} + \cdots - 2608004468224 ) / 75696635904 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 52572899 \nu^{12} + 363365393 \nu^{11} + 2167612399 \nu^{10} - 17815473986 \nu^{9} + \cdots + 1851038109184 ) / 75696635904 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46441789 \nu^{12} + 285190399 \nu^{11} + 2043676673 \nu^{10} - 13889131726 \nu^{9} + \cdots + 2901811569152 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 56602447 \nu^{12} - 379328077 \nu^{11} - 2373552179 \nu^{10} + 18649757362 \nu^{9} + \cdots - 2727849532928 ) / 56772476928 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 227904101 \nu^{12} + 1762596455 \nu^{11} + 8164090777 \nu^{10} - 85107637262 \nu^{9} + \cdots + 19413978915328 ) / 227089907712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} + \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + 3\beta_{8} + 2\beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} + 17\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 27 \beta_{9} + 29 \beta_{8} + 7 \beta_{7} - \beta_{6} + \cdots + 169 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{12} - 18 \beta_{11} + 2 \beta_{10} + 54 \beta_{9} + 110 \beta_{8} + 74 \beta_{7} + \cdots + 224 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 83 \beta_{12} + 101 \beta_{11} + 87 \beta_{10} + 704 \beta_{9} + 818 \beta_{8} + 307 \beta_{7} + \cdots + 3491 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 330 \beta_{12} - 178 \beta_{11} + 95 \beta_{10} + 2101 \beta_{9} + 3561 \beta_{8} + 2257 \beta_{7} + \cdots + 7297 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2629 \beta_{12} + 2916 \beta_{11} + 2081 \beta_{10} + 18992 \beta_{9} + 23556 \beta_{8} + \cdots + 80379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11356 \beta_{12} + 2104 \beta_{11} + 3276 \beta_{10} + 71883 \beta_{9} + 111675 \beta_{8} + \cdots + 229436 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 77280 \beta_{12} + 84015 \beta_{11} + 48318 \beta_{10} + 530049 \beta_{9} + 689571 \beta_{8} + \cdots + 1991626 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 355376 \beta_{12} + 206755 \beta_{11} + 101151 \beta_{10} + 2316705 \beta_{9} + 3448947 \beta_{8} + \cdots + 7068295 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2230037 \beta_{12} + 2467992 \beta_{11} + 1135346 \beta_{10} + 15172180 \beta_{9} + 20396200 \beta_{8} + \cdots + 52123626 \) 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
−4.33359
−4.22658
−2.58112
−2.19365
−2.11792
−0.842259
0.744781
1.40898
2.38890
3.15095
3.56231
4.55672
5.48248
−5.33359 −9.19310 20.4472 −15.2037 49.0322 2.88328 −66.3885 57.5130 81.0905
1.2 −5.22658 5.22259 19.3172 −5.57653 −27.2963 −8.92344 −59.1502 0.275458 29.1462
1.3 −3.58112 −1.01773 4.82445 −8.90410 3.64461 34.6915 11.3721 −25.9642 31.8867
1.4 −3.19365 0.513415 2.19938 11.5549 −1.63967 −29.5975 18.5251 −26.7364 −36.9023
1.5 −3.11792 −8.26910 1.72143 6.07736 25.7824 8.24382 19.5761 41.3781 −18.9487
1.6 −1.84226 5.15345 −4.60608 −0.253207 −9.49398 −11.4184 23.2237 −0.441989 0.466472
1.7 −0.255219 −4.33464 −7.93486 14.2191 1.10628 17.9778 4.06688 −8.21090 −3.62899
1.8 0.408981 5.44463 −7.83273 −18.6969 2.22675 22.7740 −6.47528 2.64397 −7.64665
1.9 1.38890 4.45464 −6.07096 −6.99532 6.18704 −31.6914 −19.5431 −7.15620 −9.71578
1.10 2.15095 −1.20869 −3.37340 1.85550 −2.59983 −13.5146 −24.4637 −25.5391 3.99109
1.11 2.56231 −3.70212 −1.43456 3.57484 −9.48599 −0.237956 −24.1743 −13.2943 9.15985
1.12 3.55672 −1.25597 4.65028 −21.6593 −4.46714 3.10281 −11.9140 −25.4225 −77.0363
1.13 4.48248 −7.80738 12.0927 −5.99266 −34.9964 −20.2899 18.3452 33.9551 −26.8620
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(127\) \( -1 \)

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 127.4.a.b ✓ 13
3.b odd 2 1 1143.4.a.d 13
4.b odd 2 1 2032.4.a.g 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
127.4.a.b ✓ 13 1.a even 1 1 trivial
1143.4.a.d 13 3.b odd 2 1
2032.4.a.g 13 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 8 T_{2}^{12} - 37 T_{2}^{11} - 385 T_{2}^{10} + 356 T_{2}^{9} + 6666 T_{2}^{8} + \cdots + 23328 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(127))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 8 T^{12} + \cdots + 23328 \) Copy content Toggle raw display
$3$ \( T^{13} + 16 T^{12} + \cdots - 4931584 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 21492717120 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 6532521113600 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots - 87\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots - 29\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 12\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 12\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 16\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 15\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 40\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 93\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 40\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 19\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 15\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 59\!\cdots\!68 \) Copy content Toggle raw display
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