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Properties

Label 2376.2.q.g
Level $2376$
Weight $2$
Character orbit 2376.q
Analytic conductor $18.972$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2376,2,Mod(793,2376)] 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("2376.793"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2376.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,1] 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: no
Analytic conductor: \(18.9724555203\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
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: \( x^{16} - 2 x^{15} + 3 x^{14} - 5 x^{13} + 13 x^{12} + 3 x^{11} - 48 x^{10} + 111 x^{9} - 114 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 792)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} + ( - \beta_{14} + \beta_{8} - 1) q^{7} + (\beta_{8} - 1) q^{11} + (\beta_{13} + \beta_{7}) q^{13} + (\beta_{11} - \beta_{10} + \cdots - \beta_1) q^{17} + \beta_{3} q^{19} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{23}+ \cdots + (\beta_{14} - \beta_{13} + 2 \beta_{12} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{5} - 5 q^{7} - 8 q^{11} + 3 q^{13} + 6 q^{17} - 2 q^{19} - 2 q^{23} - 9 q^{25} - 8 q^{29} + 7 q^{31} + 40 q^{35} - 32 q^{37} - 12 q^{41} + 11 q^{43} - 17 q^{47} - 9 q^{49} + 56 q^{53} - 2 q^{55}+ \cdots - 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 5 x^{13} + 13 x^{12} + 3 x^{11} - 48 x^{10} + 111 x^{9} - 114 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{15} + 86 \nu^{14} + 93 \nu^{13} + 317 \nu^{12} + 260 \nu^{11} + 1047 \nu^{10} + 2289 \nu^{9} + \cdots + 183708 ) / 6561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 101 \nu^{15} + 1437 \nu^{14} + 1330 \nu^{13} + 5549 \nu^{12} + 4008 \nu^{11} + 16177 \nu^{10} + \cdots + 3289977 ) / 100602 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3 \nu^{15} + 5 \nu^{14} - \nu^{13} + 11 \nu^{11} - 25 \nu^{10} + 138 \nu^{9} - 159 \nu^{8} + \cdots - 2916 ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} - 94 \nu^{14} - 60 \nu^{13} - 196 \nu^{12} - 244 \nu^{11} - 672 \nu^{10} - 2355 \nu^{9} + \cdots - 107163 ) / 4374 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13 \nu^{15} - 17 \nu^{14} + 66 \nu^{13} + 88 \nu^{12} + 70 \nu^{11} + 336 \nu^{10} - 120 \nu^{9} + \cdots + 56862 ) / 6561 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16 \nu^{15} + 160 \nu^{14} + 258 \nu^{13} + 577 \nu^{12} + 598 \nu^{11} + 2085 \nu^{10} + \cdots + 341172 ) / 6561 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{15} - 20 \nu^{14} + 66 \nu^{13} + 85 \nu^{12} + 103 \nu^{11} + 471 \nu^{10} - 381 \nu^{9} + \cdots + 50301 ) / 6561 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1456 \nu^{15} + 431 \nu^{14} - 1980 \nu^{13} + 3959 \nu^{12} - 9358 \nu^{11} - 16947 \nu^{10} + \cdots + 5174442 ) / 301806 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1507 \nu^{15} - 62 \nu^{14} - 312 \nu^{13} - 3764 \nu^{12} + 9070 \nu^{11} + 10974 \nu^{10} + \cdots - 5500305 ) / 301806 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 950 \nu^{15} - 1420 \nu^{14} + 945 \nu^{13} - 6604 \nu^{12} + 5333 \nu^{11} + 1422 \nu^{10} + \cdots - 5526549 ) / 100602 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3379 \nu^{15} + 340 \nu^{14} + 9045 \nu^{13} + 520 \nu^{12} + 26257 \nu^{11} + 66780 \nu^{10} + \cdots - 3954096 ) / 301806 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1168 \nu^{15} + 574 \nu^{14} + 2085 \nu^{13} - 80 \nu^{12} + 9787 \nu^{11} + 22326 \nu^{10} + \cdots - 2289789 ) / 100602 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1786 \nu^{15} - 1400 \nu^{14} + 4578 \nu^{13} - 9650 \nu^{12} + 13465 \nu^{11} + 16098 \nu^{10} + \cdots - 8522739 ) / 150903 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1284 \nu^{15} + 79 \nu^{14} - 3416 \nu^{13} + 1881 \nu^{12} - 10286 \nu^{11} - 22355 \nu^{10} + \cdots + 2911626 ) / 100602 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2197 \nu^{15} + 476 \nu^{14} - 3345 \nu^{13} + 5765 \nu^{12} - 14452 \nu^{11} - 26637 \nu^{10} + \cdots + 8719569 ) / 150903 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \cdots - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 12 \beta_{14} - 8 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} + 5 \beta_{10} + 4 \beta_{9} + 9 \beta_{8} + \cdots - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9 \beta_{15} - 6 \beta_{14} - 10 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 7 \beta_{10} + 2 \beta_{9} + \cdots + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3 \beta_{15} + 24 \beta_{14} + 28 \beta_{13} - 9 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 19 \beta_{9} + \cdots - 45 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9 \beta_{15} + 21 \beta_{14} - 4 \beta_{13} + 33 \beta_{12} + 10 \beta_{11} + 19 \beta_{10} - 22 \beta_{9} + \cdots - 108 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 27 \beta_{15} + 33 \beta_{14} - 17 \beta_{13} + 29 \beta_{11} - 25 \beta_{10} - 29 \beta_{9} - 228 \beta_{8} + \cdots + 66 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 48 \beta_{15} + 24 \beta_{14} + 95 \beta_{13} + 111 \beta_{12} - 47 \beta_{11} - 92 \beta_{10} + \cdots - 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 126 \beta_{15} - 57 \beta_{14} + 13 \beta_{13} + 111 \beta_{12} - 199 \beta_{11} + 98 \beta_{10} + \cdots - 504 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 297 \beta_{15} + 102 \beta_{14} - 244 \beta_{13} - 117 \beta_{12} - 119 \beta_{11} + 196 \beta_{10} + \cdots + 1032 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 255 \beta_{15} + 453 \beta_{14} + 625 \beta_{13} - 561 \beta_{12} + 299 \beta_{11} - 484 \beta_{10} + \cdots + 765 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 45 \beta_{15} + 102 \beta_{14} + 113 \beta_{13} + 978 \beta_{12} + 496 \beta_{11} - 80 \beta_{10} + \cdots - 297 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 540 \beta_{15} + 123 \beta_{14} - 953 \beta_{13} - 657 \beta_{12} + 947 \beta_{11} - 574 \beta_{10} + \cdots + 6834 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2376\mathbb{Z}\right)^\times\).

\(n\) \(353\) \(1189\) \(1729\) \(1783\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
793.1
−0.620121 − 1.61724i
1.07799 + 1.35570i
−0.907666 + 1.47518i
0.810693 − 1.53061i
1.56837 − 0.734990i
1.56316 + 0.746010i
−1.72733 + 0.127803i
−0.765100 − 1.55391i
−0.620121 + 1.61724i
1.07799 − 1.35570i
−0.907666 − 1.47518i
0.810693 + 1.53061i
1.56837 + 0.734990i
1.56316 − 0.746010i
−1.72733 − 0.127803i
−0.765100 + 1.55391i
0 0 0 −1.99318 + 3.45229i 0 −0.239061 − 0.414066i 0 0 0
793.2 0 0 0 −1.37412 + 2.38004i 0 −1.52666 − 2.64424i 0 0 0
793.3 0 0 0 −0.576848 + 0.999130i 0 −2.38285 − 4.12721i 0 0 0
793.4 0 0 0 0.0982132 − 0.170110i 0 0.932476 + 1.61510i 0 0 0
793.5 0 0 0 0.181366 − 0.314134i 0 −0.856145 − 1.48289i 0 0 0
793.6 0 0 0 0.994481 − 1.72249i 0 2.32329 + 4.02405i 0 0 0
793.7 0 0 0 1.58120 − 2.73873i 0 0.296039 + 0.512755i 0 0 0
793.8 0 0 0 1.58888 − 2.75202i 0 −1.04710 − 1.81362i 0 0 0
1585.1 0 0 0 −1.99318 − 3.45229i 0 −0.239061 + 0.414066i 0 0 0
1585.2 0 0 0 −1.37412 − 2.38004i 0 −1.52666 + 2.64424i 0 0 0
1585.3 0 0 0 −0.576848 − 0.999130i 0 −2.38285 + 4.12721i 0 0 0
1585.4 0 0 0 0.0982132 + 0.170110i 0 0.932476 − 1.61510i 0 0 0
1585.5 0 0 0 0.181366 + 0.314134i 0 −0.856145 + 1.48289i 0 0 0
1585.6 0 0 0 0.994481 + 1.72249i 0 2.32329 − 4.02405i 0 0 0
1585.7 0 0 0 1.58120 + 2.73873i 0 0.296039 − 0.512755i 0 0 0
1585.8 0 0 0 1.58888 + 2.75202i 0 −1.04710 + 1.81362i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 793.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2376.2.q.g 16
3.b odd 2 1 792.2.q.h ✓ 16
9.c even 3 1 inner 2376.2.q.g 16
9.c even 3 1 7128.2.a.bd 8
9.d odd 6 1 792.2.q.h ✓ 16
9.d odd 6 1 7128.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.q.h ✓ 16 3.b odd 2 1
792.2.q.h ✓ 16 9.d odd 6 1
2376.2.q.g 16 1.a even 1 1 trivial
2376.2.q.g 16 9.c even 3 1 inner
7128.2.a.bd 8 9.c even 3 1
7128.2.a.be 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - T_{5}^{15} + 25 T_{5}^{14} - 34 T_{5}^{13} + 454 T_{5}^{12} - 571 T_{5}^{11} + 3904 T_{5}^{10} + \cdots + 324 \) acting on \(S_{2}^{\mathrm{new}}(2376, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{15} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{16} + 5 T^{15} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} - 3 T^{15} + \cdots + 186624 \) Copy content Toggle raw display
$17$ \( (T^{8} - 3 T^{7} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + T^{7} - 80 T^{6} + \cdots - 2432)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 17700705936 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 308904747264 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 9929723904 \) Copy content Toggle raw display
$37$ \( (T^{8} + 16 T^{7} + \cdots - 3352446)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 150270971904 \) Copy content Toggle raw display
$47$ \( T^{16} + 17 T^{15} + \cdots + 52577001 \) Copy content Toggle raw display
$53$ \( (T^{8} - 28 T^{7} + \cdots - 15273)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 144378872882176 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 4707057664 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 28224016266384 \) Copy content Toggle raw display
$71$ \( (T^{8} - 14 T^{7} + \cdots + 386352)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 42 T^{7} + \cdots + 86704)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 195342784576 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11372326354944 \) Copy content Toggle raw display
$89$ \( (T^{8} - 411 T^{6} + \cdots + 588384)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 7 T^{15} + \cdots + 42588676 \) Copy content Toggle raw display
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