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Properties

Label 2376.2.q.h
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 = [16,0,0,0,3] 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: \( x^{16} - x^{15} - x^{14} + x^{13} - 8 x^{12} - 18 x^{11} + 24 x^{10} + 54 x^{9} - 45 x^{8} + 162 x^{7} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_1 - 1) q^{11} + (\beta_{14} + \beta_{6} + \beta_{4} + \cdots - 1) q^{13} + ( - \beta_{12} + \beta_{11} + \beta_{2}) q^{17}+ \cdots + (2 \beta_{14} + 2 \beta_{4} - 2 \beta_{3} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{5} + 5 q^{7} - 8 q^{11} + 5 q^{13} - 2 q^{17} - 34 q^{19} + 9 q^{23} - 21 q^{25} - 2 q^{29} + 10 q^{31} - 4 q^{35} + 14 q^{37} + 3 q^{43} + q^{47} - 25 q^{49} - 52 q^{53} - 6 q^{55} + 4 q^{59}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - x^{14} + x^{13} - 8 x^{12} - 18 x^{11} + 24 x^{10} + 54 x^{9} - 45 x^{8} + 162 x^{7} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 523 \nu^{15} - 551 \nu^{14} - 1364 \nu^{13} - 1591 \nu^{12} + 6638 \nu^{11} + 19581 \nu^{10} + \cdots + 2687823 ) / 883548 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 377 \nu^{15} - 319 \nu^{14} - 322 \nu^{13} - 203 \nu^{12} + 2554 \nu^{11} + 16035 \nu^{10} + \cdots + 1856763 ) / 441774 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 256 \nu^{15} + 133 \nu^{14} + 442 \nu^{13} - 1816 \nu^{12} + 1457 \nu^{11} + 9138 \nu^{10} + \cdots + 1587762 ) / 220887 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 274 \nu^{15} + 61 \nu^{14} + 1108 \nu^{13} + 479 \nu^{12} - 532 \nu^{11} + 4152 \nu^{10} + \cdots + 242757 ) / 220887 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1163 \nu^{15} + 145 \nu^{14} - 680 \nu^{13} - 73 \nu^{12} - 7276 \nu^{11} - 29931 \nu^{10} + \cdots - 4056885 ) / 883548 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 284 \nu^{15} + 86 \nu^{14} + 140 \nu^{13} - 689 \nu^{12} + 1498 \nu^{11} + 7704 \nu^{10} + \cdots + 1257525 ) / 220887 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 444 \nu^{15} + 821 \nu^{14} + 76 \nu^{13} - 2156 \nu^{12} - 3307 \nu^{11} - 12511 \nu^{10} + \cdots - 1161297 ) / 294516 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1541 \nu^{15} - 530 \nu^{14} - 1868 \nu^{13} + 1709 \nu^{12} - 6169 \nu^{11} - 36114 \nu^{10} + \cdots - 5113206 ) / 883548 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 547 \nu^{15} + 685 \nu^{14} + 912 \nu^{13} + 2159 \nu^{12} - 4080 \nu^{11} - 16363 \nu^{10} + \cdots - 3167505 ) / 294516 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 409 \nu^{15} - 800 \nu^{14} - 1838 \nu^{13} - 1306 \nu^{12} + 4601 \nu^{11} + 13146 \nu^{10} + \cdots + 2375082 ) / 220887 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2081 \nu^{15} + 851 \nu^{14} + 80 \nu^{13} - 13469 \nu^{12} + 6958 \nu^{11} + 60159 \nu^{10} + \cdots + 12157533 ) / 883548 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 707 \nu^{15} - 1095 \nu^{14} - 752 \nu^{13} + 693 \nu^{12} + 6350 \nu^{11} + 17581 \nu^{10} + \cdots + 2317491 ) / 294516 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 919 \nu^{15} - 259 \nu^{14} - 2568 \nu^{13} - 5111 \nu^{12} + 8346 \nu^{11} + 28501 \nu^{10} + \cdots + 5621319 ) / 294516 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3289 \nu^{15} + 2573 \nu^{14} + 1604 \nu^{13} - 1151 \nu^{12} - 11306 \nu^{11} - 80871 \nu^{10} + \cdots - 9498141 ) / 883548 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1629 \nu^{15} - 890 \nu^{14} + 1184 \nu^{13} - 1321 \nu^{12} + 6529 \nu^{11} + 39778 \nu^{10} + \cdots + 4448358 ) / 294516 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{3} - 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{12} - \beta_{10} - \beta_{9} + 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{3} + 2\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{14} + 4 \beta_{12} - \beta_{11} - \beta_{10} + 4 \beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots + 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8 \beta_{15} + 4 \beta_{14} + \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + \cdots + 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{15} - 8 \beta_{14} - 2 \beta_{13} - 3 \beta_{12} - 12 \beta_{11} - 8 \beta_{9} + \cdots - 9 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8 \beta_{15} + 5 \beta_{14} - 4 \beta_{13} + 10 \beta_{12} + 5 \beta_{11} + 18 \beta_{10} + 4 \beta_{9} + \cdots - 71 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{15} - 13 \beta_{14} + 11 \beta_{13} + 15 \beta_{12} - 4 \beta_{11} - 15 \beta_{10} + 19 \beta_{9} + \cdots + 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 13 \beta_{15} - 4 \beta_{14} + 35 \beta_{13} + 34 \beta_{12} - 30 \beta_{11} - 46 \beta_{10} - 11 \beta_{9} + \cdots - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 77 \beta_{15} - 23 \beta_{14} - 29 \beta_{13} - 108 \beta_{12} - 12 \beta_{11} + 35 \beta_{10} + \cdots + 159 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 31 \beta_{15} + 137 \beta_{14} + 11 \beta_{13} + 27 \beta_{12} - 112 \beta_{11} - 2 \beta_{10} + \cdots - 297 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 449 \beta_{15} - 298 \beta_{14} - 124 \beta_{13} + 275 \beta_{12} + 72 \beta_{11} + 130 \beta_{10} + \cdots - 700 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 233 \beta_{15} + 574 \beta_{14} + 157 \beta_{13} + 118 \beta_{12} - 145 \beta_{11} - 68 \beta_{10} + \cdots + 292 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 330 \beta_{15} - 435 \beta_{14} + 849 \beta_{13} - 84 \beta_{12} - 528 \beta_{11} - 880 \beta_{10} + \cdots - 468 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 306 \beta_{15} + 1116 \beta_{14} - 810 \beta_{13} - 1606 \beta_{12} + 465 \beta_{11} + 667 \beta_{10} + \cdots - 1417 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
0.229327 − 1.71680i
0.328371 + 1.70064i
−1.71147 − 0.266241i
1.68275 − 0.410311i
0.769857 − 1.55155i
−1.40349 + 1.01500i
1.73123 − 0.0532017i
−1.12658 − 1.31560i
0.229327 + 1.71680i
0.328371 − 1.70064i
−1.71147 + 0.266241i
1.68275 + 0.410311i
0.769857 + 1.55155i
−1.40349 − 1.01500i
1.73123 + 0.0532017i
−1.12658 + 1.31560i
0 0 0 −1.73110 + 2.99836i 0 1.55740 + 2.69749i 0 0 0
793.2 0 0 0 −1.32219 + 2.29010i 0 1.41483 + 2.45055i 0 0 0
793.3 0 0 0 −0.615861 + 1.06670i 0 −1.83259 − 3.17413i 0 0 0
793.4 0 0 0 −0.151414 + 0.262256i 0 −0.560811 − 0.971353i 0 0 0
793.5 0 0 0 0.253197 − 0.438549i 0 −0.837283 − 1.45022i 0 0 0
793.6 0 0 0 0.696028 − 1.20556i 0 1.98282 + 3.43435i 0 0 0
793.7 0 0 0 2.18298 − 3.78103i 0 2.28735 + 3.96181i 0 0 0
793.8 0 0 0 2.18837 − 3.79036i 0 −1.51172 − 2.61837i 0 0 0
1585.1 0 0 0 −1.73110 − 2.99836i 0 1.55740 − 2.69749i 0 0 0
1585.2 0 0 0 −1.32219 − 2.29010i 0 1.41483 − 2.45055i 0 0 0
1585.3 0 0 0 −0.615861 − 1.06670i 0 −1.83259 + 3.17413i 0 0 0
1585.4 0 0 0 −0.151414 − 0.262256i 0 −0.560811 + 0.971353i 0 0 0
1585.5 0 0 0 0.253197 + 0.438549i 0 −0.837283 + 1.45022i 0 0 0
1585.6 0 0 0 0.696028 + 1.20556i 0 1.98282 − 3.43435i 0 0 0
1585.7 0 0 0 2.18298 + 3.78103i 0 2.28735 − 3.96181i 0 0 0
1585.8 0 0 0 2.18837 + 3.79036i 0 −1.51172 + 2.61837i 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.h 16
3.b odd 2 1 792.2.q.g ✓ 16
9.c even 3 1 inner 2376.2.q.h 16
9.c even 3 1 7128.2.a.bc 8
9.d odd 6 1 792.2.q.g ✓ 16
9.d odd 6 1 7128.2.a.bf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
792.2.q.g ✓ 16 3.b odd 2 1
792.2.q.g ✓ 16 9.d odd 6 1
2376.2.q.h 16 1.a even 1 1 trivial
2376.2.q.h 16 9.c even 3 1 inner
7128.2.a.bc 8 9.c even 3 1
7128.2.a.bf 8 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 3 T_{5}^{15} + 35 T_{5}^{14} - 24 T_{5}^{13} + 622 T_{5}^{12} - 225 T_{5}^{11} + \cdots + 2116 \) 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
$5$ \( T^{16} - 3 T^{15} + \cdots + 2116 \) Copy content Toggle raw display
$7$ \( T^{16} - 5 T^{15} + \cdots + 11075584 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 101929216 \) Copy content Toggle raw display
$17$ \( (T^{8} + T^{7} - 62 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 17 T^{7} + \cdots - 77888)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 11722825984 \) Copy content Toggle raw display
$29$ \( T^{16} + 2 T^{15} + \cdots + 2611456 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 4420081555201 \) Copy content Toggle raw display
$37$ \( (T^{8} - 7 T^{7} + \cdots + 44971)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 93631104064 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 181666898176 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 841313867278921 \) Copy content Toggle raw display
$53$ \( (T^{8} + 26 T^{7} + \cdots + 57307)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 170015521 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 2908634742784 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 9526540816 \) Copy content Toggle raw display
$71$ \( (T^{8} + 13 T^{7} + \cdots + 1604629)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 2 T^{7} + \cdots + 1682176)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 249545967256576 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 480592789504 \) Copy content Toggle raw display
$89$ \( (T^{8} + 35 T^{7} + \cdots - 2515136)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 83192659242001 \) Copy content Toggle raw display
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