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Properties

Label 1764.2.e.i
Level $1764$
Weight $2$
Character orbit 1764.e
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1079,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1079");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{12} - 10x^{10} + 4x^{8} - 40x^{6} + 16x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 252)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{5} q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{5} q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{3}) q^{8} + ( - \beta_{10} - \beta_{9}) q^{10} - \beta_{15} q^{11} + (\beta_{8} + \beta_{6} + \beta_{2} + 1) q^{13} + (\beta_{9} + \beta_{8} + \beta_{6}) q^{16} + ( - \beta_{14} - \beta_{13}) q^{17} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{4}) q^{19}+ \cdots + (2 \beta_{10} - \beta_{9} + 3 \beta_{8} + \cdots - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{13} - 4 q^{16} - 16 q^{22} - 24 q^{25} - 8 q^{34} - 8 q^{37} + 52 q^{40} + 24 q^{46} + 52 q^{52} + 12 q^{58} + 16 q^{61} + 60 q^{64} + 8 q^{73} + 36 q^{76} - 68 q^{82} - 16 q^{85} - 44 q^{88} - 60 q^{94} - 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{12} - 10x^{10} + 4x^{8} - 40x^{6} + 16x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + \nu^{9} - 10\nu^{7} + 4\nu^{5} - 40\nu^{3} + 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{12} - \nu^{8} + 10\nu^{6} - 4\nu^{4} + 24\nu^{2} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} + 4\nu^{13} + 17\nu^{11} - 6\nu^{9} - 20\nu^{7} - 56\nu^{5} + 48\nu^{3} - 320\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{14} - 4\nu^{12} + 15\nu^{10} + 6\nu^{8} + 52\nu^{6} - 8\nu^{4} + 80\nu^{2} - 448 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 4\nu^{13} + 17\nu^{11} - 14\nu^{9} + 60\nu^{7} - 88\nu^{5} + 112\nu^{3} - 448\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{14} - 4\nu^{12} + \nu^{10} - 14\nu^{8} + 12\nu^{6} - 24\nu^{4} + 112\nu^{2} + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} + 12\nu^{12} - 17\nu^{10} + 22\nu^{8} - 76\nu^{6} + 184\nu^{4} - 304\nu^{2} + 320 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{14} - 4\nu^{12} + 19\nu^{10} - 34\nu^{8} + 68\nu^{6} - 168\nu^{4} + 144\nu^{2} - 448 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5\nu^{14} - 4\nu^{12} - 11\nu^{10} + 10\nu^{8} - 20\nu^{6} - 56\nu^{4} - 80\nu^{2} + 704 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} + \nu^{11} - 10\nu^{9} + 4\nu^{7} - 40\nu^{5} + 16\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{15} + 4\nu^{13} - \nu^{11} + 14\nu^{9} - 12\nu^{7} + 24\nu^{5} - 112\nu^{3} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7\nu^{15} + 4\nu^{13} - 9\nu^{11} - 2\nu^{9} - 28\nu^{7} - 40\nu^{5} - 240\nu^{3} + 448\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{15} + \nu^{13} - \nu^{11} + 3\nu^{9} + 2\nu^{7} + 36\nu^{5} + 8\nu^{3} - 48\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{7} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{8} + 2\beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + 3\beta_{7} - 2\beta_{5} - \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{11} - \beta_{10} - \beta_{9} - 3\beta_{8} + 2\beta_{6} + 4\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3\beta_{15} + \beta_{14} + 3\beta_{13} - 5\beta_{12} - 3\beta_{7} + 4\beta_{5} - 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( \beta_{10} + \beta_{9} + 3\beta_{8} + 8\beta_{6} - 6\beta_{4} + 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( \beta_{15} + \beta_{14} - 3\beta_{13} - 7\beta_{12} + 11\beta_{7} + 6\beta_{5} + 7\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -2\beta_{11} + 11\beta_{10} + 7\beta_{9} - 11\beta_{8} - 6\beta_{6} + 4\beta_{4} + 20\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9\beta_{15} + 5\beta_{14} + 11\beta_{13} + 19\beta_{12} - 11\beta_{7} + 16\beta_{5} - 15\beta_{3} + 24\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20\beta_{11} + 17\beta_{10} + 25\beta_{9} + 11\beta_{8} + 20\beta_{6} - 2\beta_{4} + 24\beta_{2} - 56 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 5\beta_{15} + 45\beta_{14} - 11\beta_{13} - 23\beta_{12} + 3\beta_{7} + 26\beta_{5} - 17\beta_{3} - 36\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1079.1
−1.41135 − 0.0900240i
−1.41135 + 0.0900240i
−1.13008 − 0.850247i
−1.13008 + 0.850247i
−0.658334 − 1.25164i
−0.658334 + 1.25164i
−0.545545 − 1.30475i
−0.545545 + 1.30475i
0.545545 − 1.30475i
0.545545 + 1.30475i
0.658334 − 1.25164i
0.658334 + 1.25164i
1.13008 − 0.850247i
1.13008 + 0.850247i
1.41135 − 0.0900240i
1.41135 + 0.0900240i
−1.41135 − 0.0900240i 0 1.98379 + 0.254110i 2.48866i 0 0 −2.77694 − 0.537226i 0 0.224040 − 3.51237i
1079.2 −1.41135 + 0.0900240i 0 1.98379 − 0.254110i − 2.48866i 0 0 −2.77694 + 0.537226i 0 0.224040 + 3.51237i
1079.3 −1.13008 − 0.850247i 0 0.554161 + 1.92169i 3.87147i 0 0 1.00767 − 2.64284i 0 3.29170 − 4.37507i
1079.4 −1.13008 + 0.850247i 0 0.554161 − 1.92169i − 3.87147i 0 0 1.00767 + 2.64284i 0 3.29170 + 4.37507i
1079.5 −0.658334 − 1.25164i 0 −1.13319 + 1.64799i − 2.08104i 0 0 2.80871 + 0.333415i 0 −2.60471 + 1.37002i
1079.6 −0.658334 + 1.25164i 0 −1.13319 − 1.64799i 2.08104i 0 0 2.80871 − 0.333415i 0 −2.60471 − 1.37002i
1079.7 −0.545545 − 1.30475i 0 −1.40476 + 1.42360i − 0.698240i 0 0 2.62381 + 1.05623i 0 −0.911031 + 0.380921i
1079.8 −0.545545 + 1.30475i 0 −1.40476 − 1.42360i 0.698240i 0 0 2.62381 − 1.05623i 0 −0.911031 − 0.380921i
1079.9 0.545545 − 1.30475i 0 −1.40476 − 1.42360i − 0.698240i 0 0 −2.62381 + 1.05623i 0 −0.911031 − 0.380921i
1079.10 0.545545 + 1.30475i 0 −1.40476 + 1.42360i 0.698240i 0 0 −2.62381 − 1.05623i 0 −0.911031 + 0.380921i
1079.11 0.658334 − 1.25164i 0 −1.13319 − 1.64799i − 2.08104i 0 0 −2.80871 + 0.333415i 0 −2.60471 − 1.37002i
1079.12 0.658334 + 1.25164i 0 −1.13319 + 1.64799i 2.08104i 0 0 −2.80871 − 0.333415i 0 −2.60471 + 1.37002i
1079.13 1.13008 − 0.850247i 0 0.554161 − 1.92169i 3.87147i 0 0 −1.00767 − 2.64284i 0 3.29170 + 4.37507i
1079.14 1.13008 + 0.850247i 0 0.554161 + 1.92169i − 3.87147i 0 0 −1.00767 + 2.64284i 0 3.29170 − 4.37507i
1079.15 1.41135 − 0.0900240i 0 1.98379 − 0.254110i 2.48866i 0 0 2.77694 − 0.537226i 0 0.224040 + 3.51237i
1079.16 1.41135 + 0.0900240i 0 1.98379 + 0.254110i − 2.48866i 0 0 2.77694 + 0.537226i 0 0.224040 − 3.51237i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1079.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.i 16
3.b odd 2 1 inner 1764.2.e.i 16
4.b odd 2 1 inner 1764.2.e.i 16
7.b odd 2 1 1764.2.e.h 16
7.c even 3 2 252.2.be.a ✓ 32
12.b even 2 1 inner 1764.2.e.i 16
21.c even 2 1 1764.2.e.h 16
21.h odd 6 2 252.2.be.a ✓ 32
28.d even 2 1 1764.2.e.h 16
28.g odd 6 2 252.2.be.a ✓ 32
84.h odd 2 1 1764.2.e.h 16
84.n even 6 2 252.2.be.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.be.a ✓ 32 7.c even 3 2
252.2.be.a ✓ 32 21.h odd 6 2
252.2.be.a ✓ 32 28.g odd 6 2
252.2.be.a ✓ 32 84.n even 6 2
1764.2.e.h 16 7.b odd 2 1
1764.2.e.h 16 21.c even 2 1
1764.2.e.h 16 28.d even 2 1
1764.2.e.h 16 84.h odd 2 1
1764.2.e.i 16 1.a even 1 1 trivial
1764.2.e.i 16 3.b odd 2 1 inner
1764.2.e.i 16 4.b odd 2 1 inner
1764.2.e.i 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} + 26T_{5}^{6} + 197T_{5}^{4} + 492T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{13}^{4} - 2T_{13}^{3} - 25T_{13}^{2} + 40T_{13} + 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 26 T^{6} + \cdots + 196)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 58 T^{6} + \cdots + 30324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 25 T^{2} + \cdots + 14)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 60 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 118 T^{6} + \cdots + 201684)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 68 T^{6} + \cdots + 5376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 86 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 116 T^{6} + \cdots + 1029)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 33 T^{2} + \cdots - 14)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 140 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 142 T^{6} + \cdots + 756)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 176 T^{6} + \cdots + 806736)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 142 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 286 T^{6} + \cdots + 3226944)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 224)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 282 T^{6} + \cdots + 263424)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 208 T^{6} + \cdots + 27216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} + \cdots + 2408)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + 300 T^{6} + \cdots + 371469)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 554 T^{6} + \cdots + 68494356)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 360 T^{6} + \cdots + 12845056)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 22 T^{3} + \cdots - 12208)^{4} \) Copy content Toggle raw display
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