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Properties

Label 2178.3.d.l
Level $2178$
Weight $3$
Character orbit 2178.d
Analytic conductor $59.346$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2178,3,Mod(1693,2178)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2178, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2178.1693");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2178.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(59.3462015777\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} + (\beta_{6} + 4) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} + (\beta_{6} + 4) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{7} - 2 \beta_1 q^{8} + (2 \beta_{5} + \beta_{4} + 4 \beta_1) q^{10} + (3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3}) q^{13} + (\beta_{7} - \beta_{6} - \beta_{2} + 2) q^{14} + 4 q^{16} + ( - \beta_{5} - 2 \beta_{4} - 8 \beta_1) q^{17} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1) q^{19} + ( - 2 \beta_{6} - 8) q^{20} + ( - 2 \beta_{6} + 8 \beta_{2} + 16) q^{23} + ( - 2 \beta_{7} + 5 \beta_{6} + \cdots + 12) q^{25}+ \cdots + (2 \beta_{5} + 7 \beta_{4} + \cdots + 13 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 28 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 28 q^{5} + 24 q^{14} + 32 q^{16} - 56 q^{20} + 104 q^{23} + 68 q^{25} + 16 q^{26} - 124 q^{31} + 112 q^{34} + 36 q^{37} - 8 q^{38} - 76 q^{47} + 76 q^{49} + 44 q^{53} - 48 q^{56} - 16 q^{58} - 100 q^{59} - 64 q^{64} + 112 q^{67} - 136 q^{70} + 276 q^{71} + 112 q^{80} + 64 q^{82} - 128 q^{86} - 24 q^{89} + 176 q^{91} - 208 q^{92} + 92 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{4} + 4\nu^{3} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 4\nu^{3} - 4\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 4\nu^{4} - 4\nu^{3} + 16\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{4} - 8\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + \nu^{6} - 4\nu^{5} - 2\nu^{4} + 8\nu^{3} - 32\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{6} + \nu^{5} - 4\nu^{4} - 4\nu^{3} + 8\nu - 8 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 5\beta_{6} - 2\beta_{5} - 3\beta_{4} - 4\beta_{3} + \beta_{2} - 7\beta _1 - 2 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - \beta_{5} + 4\beta_{4} - 2\beta_{3} - 2\beta_{2} + 2\beta _1 + 4 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 2\beta_{5} - 3\beta_{4} - 4\beta_{3} + 9\beta_{2} + 4\beta _1 + 4 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 2\beta_{6} - 10\beta_{5} - 4\beta_{4} + 2\beta_{3} - 4\beta_{2} - 2\beta _1 - 14 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{7} + 4\beta_{6} - 20\beta_{5} - 8\beta_{4} + 4\beta_{3} + 8\beta_{2} - 4\beta _1 + 28 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 4\beta_{6} - 8\beta_{5} - 12\beta_{4} - 16\beta_{3} - 36\beta_{2} + 16\beta _1 - 16 ) / 11 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times\).

\(n\) \(1333\) \(1937\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1693.1
1.34500 − 0.437016i
0.831254 + 1.14412i
−1.34500 − 0.437016i
−0.831254 + 1.14412i
1.34500 + 0.437016i
0.831254 − 1.14412i
−1.34500 + 0.437016i
−0.831254 − 1.14412i
− 1.41421i 0 −2.00000 −2.99802 0 3.89538i 2.82843i 0 4.23984i
1693.2 − 1.41421i 0 −2.00000 1.29302 0 11.0334i 2.82843i 0 − 1.82860i
1693.3 − 1.41421i 0 −2.00000 7.76195 0 − 2.81502i 2.82843i 0 − 10.9771i
1693.4 − 1.41421i 0 −2.00000 7.94305 0 − 3.62848i 2.82843i 0 − 11.2332i
1693.5 1.41421i 0 −2.00000 −2.99802 0 − 3.89538i − 2.82843i 0 − 4.23984i
1693.6 1.41421i 0 −2.00000 1.29302 0 − 11.0334i − 2.82843i 0 1.82860i
1693.7 1.41421i 0 −2.00000 7.76195 0 2.81502i − 2.82843i 0 10.9771i
1693.8 1.41421i 0 −2.00000 7.94305 0 3.62848i − 2.82843i 0 11.2332i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1693.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.3.d.l 8
3.b odd 2 1 242.3.b.d 8
11.b odd 2 1 inner 2178.3.d.l 8
11.c even 5 1 198.3.j.a 8
11.d odd 10 1 198.3.j.a 8
33.d even 2 1 242.3.b.d 8
33.f even 10 1 22.3.d.a ✓ 8
33.f even 10 1 242.3.d.c 8
33.f even 10 1 242.3.d.d 8
33.f even 10 1 242.3.d.e 8
33.h odd 10 1 22.3.d.a ✓ 8
33.h odd 10 1 242.3.d.c 8
33.h odd 10 1 242.3.d.d 8
33.h odd 10 1 242.3.d.e 8
132.n odd 10 1 176.3.n.b 8
132.o even 10 1 176.3.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a ✓ 8 33.f even 10 1
22.3.d.a ✓ 8 33.h odd 10 1
176.3.n.b 8 132.n odd 10 1
176.3.n.b 8 132.o even 10 1
198.3.j.a 8 11.c even 5 1
198.3.j.a 8 11.d odd 10 1
242.3.b.d 8 3.b odd 2 1
242.3.b.d 8 33.d even 2 1
242.3.d.c 8 33.f even 10 1
242.3.d.c 8 33.h odd 10 1
242.3.d.d 8 33.f even 10 1
242.3.d.d 8 33.h odd 10 1
242.3.d.e 8 33.f even 10 1
242.3.d.e 8 33.h odd 10 1
2178.3.d.l 8 1.a even 1 1 trivial
2178.3.d.l 8 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 14T_{5}^{3} + 31T_{5}^{2} + 166T_{5} - 239 \) acting on \(S_{3}^{\mathrm{new}}(2178, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 14 T^{3} + \cdots - 239)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 158 T^{6} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2268521641 \) Copy content Toggle raw display
$17$ \( T^{8} + 562 T^{6} + \cdots + 22934521 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3189877441 \) Copy content Toggle raw display
$23$ \( (T^{4} - 52 T^{3} + \cdots - 90224)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 206760274681 \) Copy content Toggle raw display
$31$ \( (T^{4} + 62 T^{3} + \cdots + 31571)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 18 T^{3} + \cdots + 4481)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8458 T^{6} + \cdots + 405257161 \) Copy content Toggle raw display
$43$ \( T^{8} + 3632 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$47$ \( (T^{4} + 38 T^{3} + \cdots - 86189)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 22 T^{3} + \cdots + 13780441)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 50 T^{3} + \cdots + 6869995)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 4097365398025 \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{3} + \cdots + 112576)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 138 T^{3} + \cdots - 3768829)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36635874025 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 4768279282321 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots - 22808304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 46 T^{3} + \cdots + 430441)^{2} \) Copy content Toggle raw display
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