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Properties

Label 4704.2.b.f
Level $4704$
Weight $2$
Character orbit 4704.b
Analytic conductor $37.562$
Analytic rank $0$
Dimension $16$
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] = [4704,2,Mod(1567,4704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4704.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.b (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: \(37.5616291108\)
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: \( x^{16} + 36x^{14} + 502x^{12} + 3460x^{10} + 12205x^{8} + 19864x^{6} + 10140x^{4} + 1600x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2^{19} \)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_{8} q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_{8} q^{5} + q^{9} + ( - \beta_{12} + \beta_{8} - \beta_{5}) q^{11} + (\beta_{12} + \beta_{4}) q^{13} - \beta_{8} q^{15} + (\beta_{15} + \beta_{13} + \cdots - 2 \beta_{4}) q^{17}+ \cdots + ( - \beta_{12} + \beta_{8} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} + 16 q^{9} - 32 q^{25} + 16 q^{27} + 16 q^{29} + 32 q^{31} - 16 q^{47} + 16 q^{53} + 64 q^{55} + 16 q^{59} + 48 q^{65} - 32 q^{75} + 16 q^{81} + 32 q^{83} + 64 q^{85} + 16 q^{87} + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 502x^{12} + 3460x^{10} + 12205x^{8} + 19864x^{6} + 10140x^{4} + 1600x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 103 \nu^{15} + 2760 \nu^{13} + 21812 \nu^{11} + 11826 \nu^{9} - 550473 \nu^{7} - 2176086 \nu^{5} + \cdots - 539116 \nu ) / 10696 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 193 \nu^{14} - 4938 \nu^{12} - 34874 \nu^{10} + 19690 \nu^{8} + 989905 \nu^{6} + 2715364 \nu^{4} + \cdots - 37864 ) / 5348 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\nu^{14} + 433\nu^{12} + 7245\nu^{10} + 58743\nu^{8} + 234597\nu^{6} + 394066\nu^{4} + 117322\nu^{2} + 1376 ) / 764 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 319 \nu^{15} + 11196 \nu^{13} + 150227 \nu^{11} + 973967 \nu^{9} + 3073588 \nu^{7} + \cdots - 392426 \nu ) / 10696 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 216 \nu^{15} - 8436 \nu^{13} - 128415 \nu^{11} - 962141 \nu^{9} - 3624061 \nu^{7} + \cdots - 168082 \nu ) / 10696 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 116 \nu^{14} + 2005 \nu^{12} - 8211 \nu^{10} - 327214 \nu^{8} - 2080071 \nu^{6} - 4407859 \nu^{4} + \cdots - 10702 ) / 5348 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36 \nu^{14} - 1215 \nu^{12} - 15577 \nu^{10} - 96722 \nu^{8} - 299397 \nu^{6} - 405459 \nu^{4} + \cdots - 446 ) / 764 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 459 \nu^{15} + 17258 \nu^{13} + 251657 \nu^{11} + 1796369 \nu^{9} + 6357946 \nu^{7} + \cdots - 373162 \nu ) / 10696 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 544 \nu^{14} + 17978 \nu^{12} + 222992 \nu^{10} + 1314995 \nu^{8} + 3753218 \nu^{6} + 4465045 \nu^{4} + \cdots + 20534 ) / 5348 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 839 \nu^{14} - 29128 \nu^{12} - 386582 \nu^{10} - 2492390 \nu^{8} - 7986937 \nu^{6} - 10969282 \nu^{4} + \cdots + 8780 ) / 5348 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 544 \nu^{14} - 17978 \nu^{12} - 222992 \nu^{10} - 1314995 \nu^{8} - 3753218 \nu^{6} - 4465045 \nu^{4} + \cdots + 3532 ) / 2674 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 830 \nu^{15} - 29445 \nu^{13} - 402122 \nu^{11} - 2688337 \nu^{9} - 9019458 \nu^{7} + \cdots - 208944 \nu ) / 5348 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1913 \nu^{15} + 65280 \nu^{13} + 846489 \nu^{11} + 5293635 \nu^{9} + 16272508 \nu^{7} + \cdots - 545618 \nu ) / 10696 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1089 \nu^{15} - 39189 \nu^{13} - 544978 \nu^{11} - 3725653 \nu^{9} - 12869549 \nu^{7} + \cdots - 817888 \nu ) / 5348 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{15} - 35\nu^{13} - 470\nu^{11} - 3082\nu^{9} - 10135\nu^{7} - 14609\nu^{5} - 4986\nu^{3} - 330\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{9} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{13} - 3\beta_{12} - 2\beta_{8} + 9\beta_{5} + 10\beta_{4} - 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{11} + 2\beta_{10} - 24\beta_{9} + 4\beta_{7} + 2\beta_{6} + 14\beta_{3} - 2\beta_{2} + 73 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -17\beta_{15} - 4\beta_{14} + 15\beta_{13} + 53\beta_{12} + 40\beta_{8} - 77\beta_{5} - 114\beta_{4} + 63\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 161\beta_{11} - 30\beta_{10} + 297\beta_{9} - 77\beta_{7} - 46\beta_{6} - 267\beta_{3} + 36\beta_{2} - 705 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 221 \beta_{15} + 83 \beta_{14} - 219 \beta_{13} - 750 \beta_{12} - 610 \beta_{8} + 741 \beta_{5} + \cdots - 653 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1981 \beta_{11} + 362 \beta_{10} - 3734 \beta_{9} + 1138 \beta_{7} + 754 \beta_{6} + 3964 \beta_{3} + \cdots + 7669 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2705 \beta_{15} - 1302 \beta_{14} + 3063 \beta_{13} + 9931 \beta_{12} + 8452 \beta_{8} + \cdots + 7421 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 24523 \beta_{11} - 4186 \beta_{10} + 47243 \beta_{9} - 15459 \beta_{7} - 10886 \beta_{6} - 54065 \beta_{3} + \cdots - 89739 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 32895 \beta_{15} + 18421 \beta_{14} - 41333 \beta_{13} - 128314 \beta_{12} - 112194 \beta_{8} + \cdots - 88915 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 305931 \beta_{11} + 48754 \beta_{10} - 599124 \beta_{9} + 202832 \beta_{7} + 148006 \beta_{6} + \cdots + 1092667 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 403439 \beta_{15} - 248112 \beta_{14} + 544497 \beta_{13} + 1641575 \beta_{12} + 1458152 \beta_{8} + \cdots + 1095591 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3840429 \beta_{11} - 579810 \beta_{10} + 7603915 \beta_{9} - 2618763 \beta_{7} - 1950642 \beta_{6} + \cdots - 13578861 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5000049 \beta_{15} + 3256045 \beta_{14} - 7065635 \beta_{13} - 20912956 \beta_{12} + \cdots - 13694797 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4704\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(1765\) \(3137\) \(4609\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
− 2.25307i
3.56272i
− 2.66925i
0.608250i
− 2.33546i
− 0.0504054i
− 0.586853i
2.22138i
− 2.22138i
0.586853i
0.0504054i
2.33546i
− 0.608250i
2.66925i
− 3.56272i
2.25307i
0 1.00000 0 − 3.95169i 0 0 0 1.00000 0
1567.2 0 1.00000 0 − 3.19069i 0 0 0 1.00000 0
1567.3 0 1.00000 0 − 3.00952i 0 0 0 1.00000 0
1567.4 0 1.00000 0 − 2.70796i 0 0 0 1.00000 0
1567.5 0 1.00000 0 − 2.53748i 0 0 0 1.00000 0
1567.6 0 1.00000 0 − 1.77648i 0 0 0 1.00000 0
1567.7 0 1.00000 0 − 1.59530i 0 0 0 1.00000 0
1567.8 0 1.00000 0 − 1.29374i 0 0 0 1.00000 0
1567.9 0 1.00000 0 1.29374i 0 0 0 1.00000 0
1567.10 0 1.00000 0 1.59530i 0 0 0 1.00000 0
1567.11 0 1.00000 0 1.77648i 0 0 0 1.00000 0
1567.12 0 1.00000 0 2.53748i 0 0 0 1.00000 0
1567.13 0 1.00000 0 2.70796i 0 0 0 1.00000 0
1567.14 0 1.00000 0 3.00952i 0 0 0 1.00000 0
1567.15 0 1.00000 0 3.19069i 0 0 0 1.00000 0
1567.16 0 1.00000 0 3.95169i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4704.2.b.f yes 16
4.b odd 2 1 4704.2.b.c ✓ 16
7.b odd 2 1 4704.2.b.c ✓ 16
28.d even 2 1 inner 4704.2.b.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4704.2.b.c ✓ 16 4.b odd 2 1
4704.2.b.c ✓ 16 7.b odd 2 1
4704.2.b.f yes 16 1.a even 1 1 trivial
4704.2.b.f yes 16 28.d 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 \(S_{2}^{\mathrm{new}}(4704, [\chi])\):

\( T_{5}^{16} + 56 T_{5}^{14} + 1296 T_{5}^{12} + 16144 T_{5}^{10} + 117816 T_{5}^{8} + 512544 T_{5}^{6} + \cdots + 913936 \) Copy content Toggle raw display
\( T_{19}^{8} - 56T_{19}^{6} - 96T_{19}^{5} + 648T_{19}^{4} + 1536T_{19}^{3} - 768T_{19}^{2} - 2048T_{19} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T - 1)^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 56 T^{14} + \cdots + 913936 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + 120 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( T^{16} + 112 T^{14} + \cdots + 583696 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 28808951824 \) Copy content Toggle raw display
$19$ \( (T^{8} - 56 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 5473632256 \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots + 529904)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 16 T^{7} + \cdots + 31744)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 136 T^{6} + \cdots + 65552)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 18340743184 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 19394461696 \) Copy content Toggle raw display
$47$ \( (T^{8} + 8 T^{7} + \cdots + 361472)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots - 69632)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{7} + \cdots + 31744)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 368 T^{14} + \cdots + 38416 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 6701222723584 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 14105853362176 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 585163260261904 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{8} - 16 T^{7} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 27076044304 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
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