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Properties

Label 93.2.f.b
Level $93$
Weight $2$
Character orbit 93.f
Analytic conductor $0.743$
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] = [93,2,Mod(4,93)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(93, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("93.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: no
Analytic conductor: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
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: \( x^{16} - 3 x^{15} + 13 x^{14} - 28 x^{13} + 90 x^{12} - 119 x^{11} + 382 x^{10} - 356 x^{9} + 1869 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{8} q^{3} + ( - \beta_{11} - \beta_{9} - \beta_{8} - 1) q^{4} + \beta_{7} q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{6} - \beta_{12} q^{7} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots - 1) q^{8}+ \cdots + (\beta_{15} - \beta_{13} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{2} - 4 q^{3} - 9 q^{4} + 6 q^{5} + 2 q^{6} - 7 q^{7} - 2 q^{8} - 4 q^{9} - 3 q^{10} + 10 q^{11} - 4 q^{12} - 3 q^{13} - 4 q^{14} - 4 q^{15} - 17 q^{16} - q^{17} - 3 q^{18} - 4 q^{19} + 7 q^{20}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 13 x^{14} - 28 x^{13} + 90 x^{12} - 119 x^{11} + 382 x^{10} - 356 x^{9} + 1869 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\!\cdots\!17 \nu^{15} + \cdots + 18\!\cdots\!99 ) / 69\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!11 \nu^{15} + \cdots + 27\!\cdots\!87 ) / 69\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!99 \nu^{15} + \cdots + 12\!\cdots\!28 ) / 62\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!40 \nu^{15} + \cdots + 84\!\cdots\!34 ) / 62\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!42 \nu^{15} + \cdots + 40\!\cdots\!01 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36\!\cdots\!26 \nu^{15} + \cdots + 82\!\cdots\!20 ) / 20\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 44\!\cdots\!11 \nu^{15} + \cdots - 27\!\cdots\!37 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!64 \nu^{15} + \cdots + 13\!\cdots\!82 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32\!\cdots\!10 \nu^{15} + \cdots - 40\!\cdots\!06 ) / 62\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 98\!\cdots\!81 \nu^{15} + \cdots - 31\!\cdots\!66 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!72 \nu^{15} + \cdots + 30\!\cdots\!93 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{15} + \cdots + 15\!\cdots\!62 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!68 \nu^{15} + \cdots + 71\!\cdots\!26 ) / 18\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17\!\cdots\!37 \nu^{15} + \cdots - 11\!\cdots\!91 ) / 18\!\cdots\!57 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} - 3\beta_{9} - \beta_{8} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} + \beta_{8} - 2\beta_{6} + 5\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{15} - \beta_{14} - \beta_{9} + 23\beta_{8} - \beta_{7} - \beta_{6} + 2\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} + \beta_{13} + \beta_{8} - 8 \beta_{7} - \beta_{6} + 29 \beta_{5} - 29 \beta_{4} + \cdots - 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46 \beta_{13} + 9 \beta_{12} + \beta_{11} + \beta_{10} + 9 \beta_{9} - 8 \beta_{8} - 9 \beta_{7} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{15} + 2 \beta_{14} + 54 \beta_{12} + 83 \beta_{11} + 2 \beta_{10} + 134 \beta_{9} + 9 \beta_{8} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 320 \beta_{15} + 85 \beta_{14} - 304 \beta_{13} + 320 \beta_{11} + 66 \beta_{10} + 628 \beta_{9} + \cdots - 606 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 649 \beta_{15} + 386 \beta_{14} - 112 \beta_{13} - 351 \beta_{12} + 112 \beta_{11} + 351 \beta_{10} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 185 \beta_{15} + 463 \beta_{14} - 185 \beta_{13} - 463 \beta_{12} - 299 \beta_{8} + 684 \beta_{7} + \cdots + 5468 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3970 \beta_{13} - 406 \beta_{12} - 962 \beta_{11} - 2275 \beta_{10} - 1348 \beta_{9} - 556 \beta_{8} + \cdots + 1348 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1852 \beta_{15} - 3237 \beta_{14} - 2101 \beta_{12} - 15598 \beta_{11} - 3237 \beta_{10} + \cdots - 6956 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 36890 \beta_{15} - 18835 \beta_{14} + 28935 \beta_{13} - 36890 \beta_{11} - 3953 \beta_{10} + \cdots + 50144 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 110876 \beta_{15} - 40843 \beta_{14} + 17064 \beta_{13} + 18006 \beta_{12} - 17064 \beta_{11} + \cdots + 5773 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 64409 \beta_{15} - 35070 \beta_{14} + 64409 \beta_{13} + 35070 \beta_{12} + 137933 \beta_{8} + \cdots - 500354 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).

\(n\) \(32\) \(34\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6} - \beta_{8} + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.719646 + 2.21484i
0.516428 + 1.58940i
−0.239090 − 0.735844i
−0.806001 − 2.48062i
2.19791 − 1.59688i
0.968675 − 0.703784i
−0.133750 + 0.0971754i
−1.72382 + 1.25243i
2.19791 + 1.59688i
0.968675 + 0.703784i
−0.133750 − 0.0971754i
−1.72382 − 1.25243i
0.719646 − 2.21484i
0.516428 − 1.58940i
−0.239090 + 0.735844i
−0.806001 + 2.48062i
−0.719646 − 2.21484i 0.309017 − 0.951057i −2.76960 + 2.01223i −3.40947 −2.32882 2.25832 − 1.64077i 2.68180 + 1.94844i −0.809017 − 0.587785i 2.45361 + 7.55144i
4.2 −0.516428 − 1.58940i 0.309017 − 0.951057i −0.641469 + 0.466054i 3.89560 −1.67120 −3.65161 + 2.65305i −1.63203 − 1.18574i −0.809017 − 0.587785i −2.01180 − 6.19168i
4.3 0.239090 + 0.735844i 0.309017 − 0.951057i 1.13373 − 0.823704i −1.00402 0.773712 0.312272 − 0.226879i 2.12907 + 1.54686i −0.809017 − 0.587785i −0.240052 − 0.738805i
4.4 0.806001 + 2.48062i 0.309017 − 0.951057i −3.88578 + 2.82319i 0.899861 2.60827 −1.22800 + 0.892196i −5.91490 − 4.29743i −0.809017 − 0.587785i 0.725289 + 2.23221i
16.1 −2.19791 + 1.59688i −0.809017 − 0.587785i 1.66277 − 5.11749i 2.08737 2.71677 0.145034 − 0.446368i 2.83832 + 8.73545i 0.309017 + 0.951057i −4.58787 + 3.33328i
16.2 −0.968675 + 0.703784i −0.809017 − 0.587785i −0.175014 + 0.538638i −2.70381 1.19735 −1.33552 + 4.11032i −0.949555 − 2.92243i 0.309017 + 0.951057i 2.61912 − 1.90290i
16.3 0.133750 − 0.0971754i −0.809017 − 0.587785i −0.609588 + 1.87612i 3.79477 −0.165325 0.672649 − 2.07020i 0.202956 + 0.624635i 0.309017 + 0.951057i 0.507553 − 0.368759i
16.4 1.72382 − 1.25243i −0.809017 − 0.587785i 0.784947 − 2.41582i −0.560297 −2.13076 −0.673141 + 2.07172i −0.355652 − 1.09459i 0.309017 + 0.951057i −0.965853 + 0.701733i
64.1 −2.19791 − 1.59688i −0.809017 + 0.587785i 1.66277 + 5.11749i 2.08737 2.71677 0.145034 + 0.446368i 2.83832 − 8.73545i 0.309017 − 0.951057i −4.58787 − 3.33328i
64.2 −0.968675 − 0.703784i −0.809017 + 0.587785i −0.175014 − 0.538638i −2.70381 1.19735 −1.33552 − 4.11032i −0.949555 + 2.92243i 0.309017 − 0.951057i 2.61912 + 1.90290i
64.3 0.133750 + 0.0971754i −0.809017 + 0.587785i −0.609588 − 1.87612i 3.79477 −0.165325 0.672649 + 2.07020i 0.202956 − 0.624635i 0.309017 − 0.951057i 0.507553 + 0.368759i
64.4 1.72382 + 1.25243i −0.809017 + 0.587785i 0.784947 + 2.41582i −0.560297 −2.13076 −0.673141 − 2.07172i −0.355652 + 1.09459i 0.309017 − 0.951057i −0.965853 − 0.701733i
70.1 −0.719646 + 2.21484i 0.309017 + 0.951057i −2.76960 − 2.01223i −3.40947 −2.32882 2.25832 + 1.64077i 2.68180 − 1.94844i −0.809017 + 0.587785i 2.45361 − 7.55144i
70.2 −0.516428 + 1.58940i 0.309017 + 0.951057i −0.641469 − 0.466054i 3.89560 −1.67120 −3.65161 − 2.65305i −1.63203 + 1.18574i −0.809017 + 0.587785i −2.01180 + 6.19168i
70.3 0.239090 − 0.735844i 0.309017 + 0.951057i 1.13373 + 0.823704i −1.00402 0.773712 0.312272 + 0.226879i 2.12907 − 1.54686i −0.809017 + 0.587785i −0.240052 + 0.738805i
70.4 0.806001 − 2.48062i 0.309017 + 0.951057i −3.88578 − 2.82319i 0.899861 2.60827 −1.22800 − 0.892196i −5.91490 + 4.29743i −0.809017 + 0.587785i 0.725289 − 2.23221i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.f.b ✓ 16
3.b odd 2 1 279.2.i.c 16
31.d even 5 1 inner 93.2.f.b ✓ 16
31.d even 5 1 2883.2.a.p 8
31.f odd 10 1 2883.2.a.o 8
93.k even 10 1 8649.2.a.bg 8
93.l odd 10 1 279.2.i.c 16
93.l odd 10 1 8649.2.a.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.f.b ✓ 16 1.a even 1 1 trivial
93.2.f.b ✓ 16 31.d even 5 1 inner
279.2.i.c 16 3.b odd 2 1
279.2.i.c 16 93.l odd 10 1
2883.2.a.o 8 31.f odd 10 1
2883.2.a.p 8 31.d even 5 1
8649.2.a.bg 8 93.k even 10 1
8649.2.a.bh 8 93.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 3 T_{2}^{15} + 13 T_{2}^{14} + 28 T_{2}^{13} + 90 T_{2}^{12} + 119 T_{2}^{11} + 382 T_{2}^{10} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 3 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{15} + \cdots + 5041 \) Copy content Toggle raw display
$11$ \( T^{16} - 10 T^{15} + \cdots + 20736 \) Copy content Toggle raw display
$13$ \( T^{16} + 3 T^{15} + \cdots + 9801 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} + \cdots + 1679616 \) Copy content Toggle raw display
$19$ \( T^{16} + 4 T^{15} + \cdots + 76125625 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 104530176 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 671846400 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( (T^{8} + 14 T^{7} + \cdots + 23216)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 3 T^{15} + \cdots + 1679616 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 444745921 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 582694062336 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 93988843776 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 8880876806400 \) Copy content Toggle raw display
$61$ \( (T^{8} - 154 T^{6} + \cdots - 508939)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 30 T^{7} + \cdots + 417024)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 21531453696 \) Copy content Toggle raw display
$73$ \( T^{16} + 17 T^{15} + \cdots + 398161 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 271425625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1261254779136 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 9625382150400 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 4495836601 \) Copy content Toggle raw display
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