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Properties

Label 1764.4.f.a
Level $1764$
Weight $4$
Character orbit 1764.f
Analytic conductor $104.079$
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] = [1764,4,Mod(881,1764)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.881");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.f (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: \(104.079369250\)
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: \( x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{18}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + \beta_{8} q^{11} - \beta_{4} q^{13} + ( - \beta_{10} - 3 \beta_{3}) q^{17} + (\beta_{9} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{19} + ( - 3 \beta_{8} - \beta_{5}) q^{23} + ( - \beta_{12} + 2 \beta_{6} + 27) q^{25} + ( - \beta_{15} - 2 \beta_{8} + \beta_{5}) q^{29} + (4 \beta_{9} + 6 \beta_{4} + \cdots - 5 \beta_1) q^{31}+ \cdots + (\beta_{9} + 33 \beta_{4} + \cdots + 48 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8606933517 \nu^{15} + 17491457046 \nu^{14} + 2762073108256 \nu^{13} + \cdots - 12\!\cdots\!68 ) / 67\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!75 \nu^{15} + \cdots - 22\!\cdots\!68 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!96 \nu^{15} + \cdots - 88\!\cdots\!72 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!29 \nu^{15} + \cdots + 29\!\cdots\!84 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!01 \nu^{15} + \cdots + 14\!\cdots\!24 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!49 \nu^{15} + \cdots + 12\!\cdots\!04 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!99 \nu^{15} + \cdots - 52\!\cdots\!96 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!95 \nu^{15} + \cdots - 10\!\cdots\!84 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!67 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 36\!\cdots\!39 \nu^{15} + \cdots + 12\!\cdots\!28 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!95 \nu^{15} + \cdots - 54\!\cdots\!76 ) / 23\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!07 \nu^{15} + \cdots - 17\!\cdots\!76 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30\!\cdots\!29 \nu^{15} + \cdots - 15\!\cdots\!12 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 65\!\cdots\!01 \nu^{15} + \cdots - 18\!\cdots\!56 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 17\!\cdots\!73 \nu^{15} + \cdots - 22\!\cdots\!64 ) / 70\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{14} + \beta_{13} - 21\beta_{12} + 7\beta_{11} - \beta_{10} - 15\beta_{7} - 12\beta_{6} + 7\beta_{3} + 78 ) / 378 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 15 \beta_{15} + 3 \beta_{14} - 20 \beta_{13} + 21 \beta_{12} + \beta_{11} - 43 \beta_{10} + \cdots + 14106 ) / 378 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 207 \beta_{15} + 801 \beta_{14} + 204 \beta_{13} - 2520 \beta_{12} + 1655 \beta_{11} + 1119 \beta_{10} + \cdots - 79956 ) / 756 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1890 \beta_{15} - 966 \beta_{14} - 3304 \beta_{13} + 4851 \beta_{12} - 3458 \beta_{11} + \cdots + 1002852 ) / 378 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 38925 \beta_{15} + 89691 \beta_{14} + 36932 \beta_{13} - 194082 \beta_{12} + 201581 \beta_{11} + \cdots - 10523976 ) / 756 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 247251 \beta_{15} - 218997 \beta_{14} - 375588 \beta_{13} + 575946 \beta_{12} - 671387 \beta_{11} + \cdots + 82934064 ) / 378 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5457375 \beta_{15} + 9167565 \beta_{14} + 5256004 \beta_{13} - 16598400 \beta_{12} + \cdots - 1143777696 ) / 756 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 31761492 \beta_{15} - 30732828 \beta_{14} - 38908016 \beta_{13} + 61087341 \beta_{12} + \cdots + 7342936392 ) / 378 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 705685329 \beta_{15} + 905768487 \beta_{14} + 630291996 \beta_{13} - 1491982674 \beta_{12} + \cdots - 117465566928 ) / 756 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3983508633 \beta_{15} - 3593813727 \beta_{14} - 3873978556 \beta_{13} + 6187643826 \beta_{12} + \cdots + 670379739216 ) / 378 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 88019325471 \beta_{15} + 87423864621 \beta_{14} + 68068170740 \beta_{13} - 136607395848 \beta_{12} + \cdots - 11644698741504 ) / 756 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 69826854858 \beta_{15} - 54148201326 \beta_{14} - 53369153208 \beta_{13} + 86322951231 \beta_{12} + \cdots + 8788001773704 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 10715369975505 \beta_{15} + 8205381038199 \beta_{14} + 6819476356444 \beta_{13} + \cdots - 11\!\cdots\!16 ) / 756 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 58781025379719 \beta_{15} - 37030892703297 \beta_{14} - 34684564933124 \beta_{13} + \cdots + 55\!\cdots\!92 ) / 378 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 12\!\cdots\!15 \beta_{15} + 739525552505109 \beta_{14} + 638478540273492 \beta_{13} + \cdots - 10\!\cdots\!76 ) / 756 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
4.65022 − 0.707107i
4.65022 + 0.707107i
−10.4548 − 0.707107i
−10.4548 + 0.707107i
−1.35249 − 0.707107i
−1.35249 + 0.707107i
8.15703 + 0.707107i
8.15703 − 0.707107i
5.70754 + 0.707107i
5.70754 − 0.707107i
1.09700 − 0.707107i
1.09700 + 0.707107i
−8.00527 − 0.707107i
−8.00527 + 0.707107i
2.20073 − 0.707107i
2.20073 + 0.707107i
0 0 0 −20.2518 0 0 0 0 0
881.2 0 0 0 −20.2518 0 0 0 0 0
881.3 0 0 0 −8.73625 0 0 0 0 0
881.4 0 0 0 −8.73625 0 0 0 0 0
881.5 0 0 0 −8.54212 0 0 0 0 0
881.6 0 0 0 −8.54212 0 0 0 0 0
881.7 0 0 0 −6.82452 0 0 0 0 0
881.8 0 0 0 −6.82452 0 0 0 0 0
881.9 0 0 0 6.82452 0 0 0 0 0
881.10 0 0 0 6.82452 0 0 0 0 0
881.11 0 0 0 8.54212 0 0 0 0 0
881.12 0 0 0 8.54212 0 0 0 0 0
881.13 0 0 0 8.73625 0 0 0 0 0
881.14 0 0 0 8.73625 0 0 0 0 0
881.15 0 0 0 20.2518 0 0 0 0 0
881.16 0 0 0 20.2518 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.f.a 16
3.b odd 2 1 inner 1764.4.f.a 16
7.b odd 2 1 inner 1764.4.f.a 16
7.c even 3 1 252.4.t.a ✓ 16
7.c even 3 1 1764.4.t.b 16
7.d odd 6 1 252.4.t.a ✓ 16
7.d odd 6 1 1764.4.t.b 16
21.c even 2 1 inner 1764.4.f.a 16
21.g even 6 1 252.4.t.a ✓ 16
21.g even 6 1 1764.4.t.b 16
21.h odd 6 1 252.4.t.a ✓ 16
21.h odd 6 1 1764.4.t.b 16
28.f even 6 1 1008.4.bt.b 16
28.g odd 6 1 1008.4.bt.b 16
84.j odd 6 1 1008.4.bt.b 16
84.n even 6 1 1008.4.bt.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a ✓ 16 7.c even 3 1
252.4.t.a ✓ 16 7.d odd 6 1
252.4.t.a ✓ 16 21.g even 6 1
252.4.t.a ✓ 16 21.h odd 6 1
1008.4.bt.b 16 28.f even 6 1
1008.4.bt.b 16 28.g odd 6 1
1008.4.bt.b 16 84.j odd 6 1
1008.4.bt.b 16 84.n even 6 1
1764.4.f.a 16 1.a even 1 1 trivial
1764.4.f.a 16 3.b odd 2 1 inner
1764.4.f.a 16 7.b odd 2 1 inner
1764.4.f.a 16 21.c even 2 1 inner
1764.4.t.b 16 7.c even 3 1
1764.4.t.b 16 7.d odd 6 1
1764.4.t.b 16 21.g even 6 1
1764.4.t.b 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 606T_{5}^{6} + 92853T_{5}^{4} - 5395140T_{5}^{2} + 106378596 \) acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\). 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
$5$ \( (T^{8} - 606 T^{6} + \cdots + 106378596)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4806 T^{6} + \cdots + 84182380164)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 2826 T^{6} + \cdots + 22573860516)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 70607585620224)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 46677071171844)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 17\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 38 T^{3} + \cdots - 424087754)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 352 T^{3} + \cdots - 6254809742)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 68\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 764 T^{3} + \cdots + 30097773916)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 182 T^{3} + \cdots + 1237158301)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 31\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 47\!\cdots\!36)^{2} \) Copy content Toggle raw display
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