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Properties

Label 1764.2.e.j
Level $1764$
Weight $2$
Character orbit 1764.e
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + ( - \beta_{8} + 1) q^{4} + \beta_{2} q^{5} + (\beta_{14} + \beta_{12} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + ( - \beta_{8} + 1) q^{4} + \beta_{2} q^{5} + (\beta_{14} + \beta_{12} - \beta_{3}) q^{8} + (\beta_{11} + \beta_{10} - \beta_{5}) q^{10} + (\beta_{14} + 2 \beta_{12} + \beta_{3}) q^{11} - \beta_{10} q^{13} + ( - 2 \beta_{8} + \beta_{6} + \beta_{4}) q^{16} + ( - \beta_{9} - 2 \beta_{2}) q^{17} + (2 \beta_{15} - \beta_{10} - \beta_{5}) q^{19} + ( - \beta_{7} + \beta_{2} - \beta_1) q^{20} + (\beta_{6} + 2 \beta_{4} - 1) q^{22} + (\beta_{14} + 2 \beta_{12} + \beta_{3}) q^{23} + ( - \beta_{8} + \beta_{4} + 1) q^{25} + \beta_{7} q^{26} + (\beta_{14} + 2 \beta_{13} - \beta_{3}) q^{29} + ( - 2 \beta_{15} + 4 \beta_{11} + \cdots - \beta_{5}) q^{31}+ \cdots + ( - 2 \beta_{10} - 5 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 8 q^{16} - 24 q^{22} + 16 q^{25} - 24 q^{46} - 8 q^{58} + 16 q^{64} + 160 q^{85} + 120 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -52\nu^{14} + 655\nu^{12} - 4512\nu^{10} + 17312\nu^{8} - 33992\nu^{6} + 38784\nu^{4} - 17044\nu^{2} - 305 ) / 1584 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 116\nu^{14} - 791\nu^{12} + 4704\nu^{10} - 6208\nu^{8} + 9544\nu^{6} + 480\nu^{4} + 7316\nu^{2} - 599 ) / 1584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -13\nu^{15} + 227\nu^{13} - 1568\nu^{11} + 6968\nu^{9} - 12392\nu^{7} + 16208\nu^{5} - 6637\nu^{3} + 2451\nu ) / 528 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{14} - 32\nu^{12} + 184\nu^{10} - 152\nu^{8} + 112\nu^{6} + 464\nu^{4} - 195\nu^{2} + 88 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} + 13\nu^{13} - 88\nu^{11} + 340\nu^{9} - 624\nu^{7} + 760\nu^{5} - 357\nu^{3} + 101\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{14} - 56\nu^{12} + 344\nu^{10} - 736\nu^{8} + 1168\nu^{6} - 824\nu^{4} + 447\nu^{2} - 80 ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 284\nu^{14} - 2105\nu^{12} + 12600\nu^{10} - 21544\nu^{8} + 29320\nu^{6} - 6408\nu^{4} + 2900\nu^{2} + 1351 ) / 792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{14} + 40\nu^{12} - 244\nu^{10} + 512\nu^{8} - 752\nu^{6} + 424\nu^{4} - 129\nu^{2} + 4 ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1060 \nu^{14} - 8407 \nu^{12} + 51360 \nu^{10} - 106688 \nu^{8} + 162152 \nu^{6} - 98592 \nu^{4} + \cdots - 4759 ) / 1584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -41\nu^{15} + 329\nu^{13} - 2016\nu^{11} + 4312\nu^{9} - 6664\nu^{7} + 4608\nu^{5} - 2345\nu^{3} + 665\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -47\nu^{15} + 383\nu^{13} - 2352\nu^{11} + 5176\nu^{9} - 7912\nu^{7} + 5328\nu^{5} - 2015\nu^{3} + 287\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( \nu^{15} - 8\nu^{13} + 49\nu^{11} - 104\nu^{9} + 160\nu^{7} - 104\nu^{5} + 49\nu^{3} - 9\nu \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 183 \nu^{15} + 1449 \nu^{13} - 8832 \nu^{11} + 18184 \nu^{9} - 27048 \nu^{7} + 15456 \nu^{5} + \cdots + 345 \nu ) / 176 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 689 \nu^{15} + 5519 \nu^{13} - 33824 \nu^{11} + 72040 \nu^{9} - 111176 \nu^{7} + 72656 \nu^{5} + \cdots + 3711 \nu ) / 528 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 97\nu^{15} - 745\nu^{13} + 4512\nu^{11} - 8624\nu^{9} + 12632\nu^{7} - 5808\nu^{5} + 2569\nu^{3} + 263\nu ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{12} - 2\beta_{10} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{9} + 2\beta_{8} - 2\beta_{7} - 4\beta_{6} + \beta_{2} + \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} + 3\beta_{14} + 4\beta_{13} - 2\beta_{11} - 2\beta_{10} - \beta_{5} - 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{9} + 20\beta_{8} + 12\beta_{7} - 22\beta_{6} - 6\beta_{4} - 7\beta_{2} - 5\beta _1 - 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21 \beta_{15} + 12 \beta_{14} + 42 \beta_{13} + 28 \beta_{12} - 14 \beta_{11} + 36 \beta_{10} + \cdots - 54 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -13\beta_{9} + 24\beta_{8} + 68\beta_{7} - 24\beta_{4} - 21\beta_{2} - 26\beta _1 - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 115 \beta_{15} - 108 \beta_{14} - 42 \beta_{13} + 148 \beta_{12} + 82 \beta_{11} + \cdots - 138 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -718\beta_{9} - 388\beta_{8} + 380\beta_{7} + 662\beta_{6} - 90\beta_{4} + 49\beta_{2} - 141\beta _1 - 148 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -636\beta_{15} - 459\beta_{14} - 788\beta_{13} + 462\beta_{11} + 318\beta_{10} + 87\beta_{5} + 459\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3212 \beta_{9} - 3682 \beta_{8} - 2114 \beta_{7} + 3668 \beta_{6} + 1008 \beta_{4} + 1625 \beta_{2} + \cdots + 4478 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3527 \beta_{15} - 1728 \beta_{14} - 7440 \beta_{13} - 4478 \beta_{12} + 2576 \beta_{11} + \cdots + 9456 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1076\beta_{9} - 2112\beta_{8} - 5872\beta_{7} + 2112\beta_{4} + 1860\beta_{2} + 2152\beta _1 + 7327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 19575 \beta_{15} + 18624 \beta_{14} + 7440 \beta_{13} - 24830 \beta_{12} - 14320 \beta_{11} + \cdots + 24336 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 122988 \beta_{9} + 66782 \beta_{8} - 65214 \beta_{7} - 112972 \beta_{6} + 15888 \beta_{4} - 9065 \beta_{2} + \cdots + 24830 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 108668 \beta_{15} + 78213 \beta_{14} + 135436 \beta_{13} - 79534 \beta_{11} - 54334 \beta_{10} + \cdots - 78213 \beta_{3} ) / 2 \) 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
2.04058 − 1.17813i
0.367543 + 0.212201i
0.367543 − 0.212201i
2.04058 + 1.17813i
−0.670418 + 0.387066i
−1.11871 − 0.645885i
−1.11871 + 0.645885i
−0.670418 − 0.387066i
1.11871 − 0.645885i
0.670418 + 0.387066i
0.670418 − 0.387066i
1.11871 + 0.645885i
−0.367543 + 0.212201i
−2.04058 − 1.17813i
−2.04058 + 1.17813i
−0.367543 − 0.212201i
−1.39033 − 0.258819i 0 1.86603 + 0.719687i − 0.732051i 0 0 −2.40812 − 1.48356i 0 −0.189469 + 1.01779i
1079.2 −1.39033 − 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 −2.40812 − 1.48356i 0 0.189469 − 1.01779i
1079.3 −1.39033 + 0.258819i 0 1.86603 − 0.719687i − 0.732051i 0 0 −2.40812 + 1.48356i 0 0.189469 + 1.01779i
1079.4 −1.39033 + 0.258819i 0 1.86603 − 0.719687i 0.732051i 0 0 −2.40812 + 1.48356i 0 −0.189469 − 1.01779i
1079.5 −1.03295 − 0.965926i 0 0.133975 + 1.99551i − 2.73205i 0 0 1.78912 − 2.19067i 0 −2.63896 + 2.82207i
1079.6 −1.03295 − 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 1.78912 − 2.19067i 0 2.63896 − 2.82207i
1079.7 −1.03295 + 0.965926i 0 0.133975 − 1.99551i − 2.73205i 0 0 1.78912 + 2.19067i 0 2.63896 + 2.82207i
1079.8 −1.03295 + 0.965926i 0 0.133975 − 1.99551i 2.73205i 0 0 1.78912 + 2.19067i 0 −2.63896 − 2.82207i
1079.9 1.03295 − 0.965926i 0 0.133975 − 1.99551i − 2.73205i 0 0 −1.78912 − 2.19067i 0 −2.63896 − 2.82207i
1079.10 1.03295 − 0.965926i 0 0.133975 − 1.99551i 2.73205i 0 0 −1.78912 − 2.19067i 0 2.63896 + 2.82207i
1079.11 1.03295 + 0.965926i 0 0.133975 + 1.99551i − 2.73205i 0 0 −1.78912 + 2.19067i 0 2.63896 − 2.82207i
1079.12 1.03295 + 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 −1.78912 + 2.19067i 0 −2.63896 + 2.82207i
1079.13 1.39033 − 0.258819i 0 1.86603 − 0.719687i − 0.732051i 0 0 2.40812 − 1.48356i 0 −0.189469 − 1.01779i
1079.14 1.39033 − 0.258819i 0 1.86603 − 0.719687i 0.732051i 0 0 2.40812 − 1.48356i 0 0.189469 + 1.01779i
1079.15 1.39033 + 0.258819i 0 1.86603 + 0.719687i − 0.732051i 0 0 2.40812 + 1.48356i 0 0.189469 − 1.01779i
1079.16 1.39033 + 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 2.40812 + 1.48356i 0 −0.189469 + 1.01779i
\(n\): 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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 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.j ✓ 16
3.b odd 2 1 inner 1764.2.e.j ✓ 16
4.b odd 2 1 inner 1764.2.e.j ✓ 16
7.b odd 2 1 inner 1764.2.e.j ✓ 16
12.b even 2 1 inner 1764.2.e.j ✓ 16
21.c even 2 1 inner 1764.2.e.j ✓ 16
28.d even 2 1 inner 1764.2.e.j ✓ 16
84.h odd 2 1 inner 1764.2.e.j ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.j ✓ 16 1.a even 1 1 trivial
1764.2.e.j ✓ 16 3.b odd 2 1 inner
1764.2.e.j ✓ 16 4.b odd 2 1 inner
1764.2.e.j ✓ 16 7.b odd 2 1 inner
1764.2.e.j ✓ 16 12.b even 2 1 inner
1764.2.e.j ✓ 16 21.c even 2 1 inner
1764.2.e.j ✓ 16 28.d even 2 1 inner
1764.2.e.j ✓ 16 84.h 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 \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{6} + 9 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 4)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 36 T^{2} + 132)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 56 T^{2} + 676)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 48 T^{2} + 528)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 132)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 144 T^{2} + 4752)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 104 T^{2} + 2116)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 96 T^{2} + 2112)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 120 T^{2} + 528)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 84 T^{2} + 36)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 216 T^{2} + 4752)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 316 T^{2} + 20164)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 72 T^{2} + 528)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 276 T^{2} + 15972)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 172 T^{2} + 484)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 216 T^{2} + 4752)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 288 T^{2} + 19008)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 104 T^{2} + 2116)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 316 T^{2} + 20164)^{4} \) Copy content Toggle raw display
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