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Properties

Label 2178.3.c.p
Level $2178$
Weight $3$
Character orbit 2178.c
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(485,2178)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2178.485");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2178.c (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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 102x^{6} + 3599x^{4} + 51708x^{2} + 249001 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 198)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - 2 q^{4} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{5} + ( - \beta_{4} + 2 \beta_{2} + 2) q^{7} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - 2 q^{4} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{5} + ( - \beta_{4} + 2 \beta_{2} + 2) q^{7} + 2 \beta_{6} q^{8} + (\beta_{3} - 3 \beta_{2} - 3) q^{10} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - 3) q^{13} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{14} + 4 q^{16} + (2 \beta_{6} - 10 \beta_{5} - \beta_1) q^{17} + (2 \beta_{4} - \beta_{3} + 9) q^{19} + ( - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5}) q^{20} + (3 \beta_{7} - 7 \beta_{6} + \cdots - \beta_1) q^{23}+ \cdots + (8 \beta_{7} - 4 \beta_{6} - 10 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 8 q^{7} - 12 q^{10} - 36 q^{13} + 32 q^{16} + 72 q^{19} - 64 q^{25} - 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} + 24 q^{40} - 220 q^{43} - 68 q^{46} + 56 q^{49} + 72 q^{52} + 112 q^{58} - 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} - 488 q^{73} - 144 q^{76} - 368 q^{79} - 388 q^{82} + 248 q^{85} - 356 q^{91} - 120 q^{94} - 832 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 102x^{6} + 3599x^{4} + 51708x^{2} + 249001 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 155\nu^{4} - 3296\nu^{2} - 18275 ) / 1881 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{6} + 647\nu^{4} + 17806\nu^{2} + 136799 ) / 1881 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} + 155\nu^{4} + 3505\nu^{2} + 22664 ) / 209 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -98\nu^{7} - 5505\nu^{5} - 56795\nu^{3} + 143673\nu ) / 938619 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -45\nu^{7} - 3592\nu^{5} - 84610\nu^{3} - 577865\nu ) / 104291 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 691\nu^{7} + 50522\nu^{5} + 940009\nu^{3} + 2836148\nu ) / 938619 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 9\beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -9\beta_{7} - 11\beta_{6} - 18\beta_{5} - 31\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -60\beta_{4} + 18\beta_{3} - 477\beta_{2} + 563 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 495\beta_{7} + 507\beta_{6} + 1395\beta_{5} + 1100\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3002\beta_{4} - 1395\beta_{3} + 21195\beta_{2} - 18162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -22590\beta_{7} - 22105\beta_{6} - 77508\beta_{5} - 42359\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
4.57204i
− 5.21424i
− 3.15782i
6.62845i
− 6.62845i
3.15782i
5.21424i
− 4.57204i
− 1.41421i 0 −2.00000 − 9.68596i 0 8.70191 2.82843i 0 −13.6980
485.2 − 1.41421i 0 −2.00000 − 2.34854i 0 −9.61011 2.82843i 0 −3.32134
485.3 − 1.41421i 0 −2.00000 2.82122i 0 −2.22977 2.82843i 0 3.98981
485.4 − 1.41421i 0 −2.00000 4.97064i 0 7.13798 2.82843i 0 7.02955
485.5 1.41421i 0 −2.00000 − 4.97064i 0 7.13798 − 2.82843i 0 7.02955
485.6 1.41421i 0 −2.00000 − 2.82122i 0 −2.22977 − 2.82843i 0 3.98981
485.7 1.41421i 0 −2.00000 2.34854i 0 −9.61011 − 2.82843i 0 −3.32134
485.8 1.41421i 0 −2.00000 9.68596i 0 8.70191 − 2.82843i 0 −13.6980
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 485.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.c.p 8
3.b odd 2 1 inner 2178.3.c.p 8
11.b odd 2 1 2178.3.c.m 8
11.c even 5 2 198.3.k.a ✓ 16
33.d even 2 1 2178.3.c.m 8
33.h odd 10 2 198.3.k.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
198.3.k.a ✓ 16 11.c even 5 2
198.3.k.a ✓ 16 33.h odd 10 2
2178.3.c.m 8 11.b odd 2 1
2178.3.c.m 8 33.d even 2 1
2178.3.c.p 8 1.a even 1 1 trivial
2178.3.c.p 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2178, [\chi])\):

\( T_{5}^{8} + 132T_{5}^{6} + 3959T_{5}^{4} + 36438T_{5}^{2} + 101761 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} - 104T_{7}^{2} + 396T_{7} + 1331 \) Copy content Toggle raw display
\( T_{13}^{4} + 18T_{13}^{3} - 96T_{13}^{2} - 438T_{13} + 1231 \) 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
$5$ \( T^{8} + 132 T^{6} + \cdots + 101761 \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + \cdots + 1331)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 18 T^{3} + \cdots + 1231)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1664558401 \) Copy content Toggle raw display
$19$ \( (T^{4} - 36 T^{3} + \cdots - 30629)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1342 T^{6} + \cdots + 92140801 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 685789359376 \) Copy content Toggle raw display
$31$ \( (T^{4} + 44 T^{3} + \cdots - 159599)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 54 T^{3} + \cdots - 95049)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 28245621544921 \) Copy content Toggle raw display
$43$ \( (T^{4} + 110 T^{3} + \cdots - 2538225)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32777292025 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10672498400641 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 33566832228721 \) Copy content Toggle raw display
$61$ \( (T^{4} + 80 T^{3} + \cdots + 121995)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 138 T^{3} + \cdots - 15268559)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 67740993441 \) Copy content Toggle raw display
$73$ \( (T^{4} + 244 T^{3} + \cdots - 17459244)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 184 T^{3} + \cdots - 4318479)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 184401217872481 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 301784681546401 \) Copy content Toggle raw display
$97$ \( (T^{4} + 416 T^{3} + \cdots + 43576551)^{2} \) Copy content Toggle raw display
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