Properties

Label 5.16.b.a
Level $5$
Weight $16$
Character orbit 5.b
Analytic conductor $7.135$
Analytic rank $0$
Dimension $6$
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] = [5,16,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(7.13467525500\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 29397x^{4} + 153469728x^{2} + 65015354624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + (\beta_{3} - 6428) q^{4} + ( - \beta_{4} + 4 \beta_{2} + 206 \beta_1 - 39725) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3} - 85908) q^{6} + (7 \beta_{5} + 7 \beta_{4} - 105 \beta_{2} - 2842 \beta_1) q^{7} + (26 \beta_{5} + 26 \beta_{4} - 76 \beta_{2} - 10300 \beta_1) q^{8} + (57 \beta_{5} - 57 \beta_{4} - 408 \beta_{3} - 1902777) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 2 \beta_1) q^{3} + (\beta_{3} - 6428) q^{4} + ( - \beta_{4} + 4 \beta_{2} + 206 \beta_1 - 39725) q^{5} + (\beta_{5} - \beta_{4} + \beta_{3} - 85908) q^{6} + (7 \beta_{5} + 7 \beta_{4} - 105 \beta_{2} - 2842 \beta_1) q^{7} + (26 \beta_{5} + 26 \beta_{4} - 76 \beta_{2} - 10300 \beta_1) q^{8} + (57 \beta_{5} - 57 \beta_{4} - 408 \beta_{3} - 1902777) q^{9} + (125 \beta_{5} + 23 \beta_{4} + 625 \beta_{3} + 1908 \beta_{2} + \cdots - 8045700) q^{10}+ \cdots + (447094278 \beta_{5} - 447094278 \beta_{4} + \cdots + 84499849944396) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 38568 q^{4} - 238350 q^{5} - 515448 q^{6} - 11416662 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 38568 q^{4} - 238350 q^{5} - 515448 q^{6} - 11416662 q^{9} - 48274200 q^{10} - 108590088 q^{11} + 663751704 q^{14} + 297743400 q^{15} + 1155522336 q^{16} - 3630995640 q^{19} + 2533753800 q^{20} - 8917537608 q^{21} - 2959765920 q^{24} + 18250878750 q^{25} - 81970953168 q^{26} + 286168468740 q^{29} + 203897251800 q^{30} - 276236748288 q^{31} - 127784939136 q^{34} + 1171274911800 q^{35} - 3326879331864 q^{36} + 2186980965936 q^{39} + 4214283852000 q^{40} - 6153278882388 q^{41} + 8250173021664 q^{44} + 10442765857950 q^{45} - 23334602656488 q^{46} + 11613390856242 q^{49} + 23694218070000 q^{50} - 43487373385728 q^{51} + 10162879468560 q^{54} + 33977390365800 q^{55} - 59280484297440 q^{56} + 14903258326680 q^{59} + 39248254864800 q^{60} - 11352061428588 q^{61} + 73265851251072 q^{64} - 50675287275600 q^{65} + 76208211455904 q^{66} - 150489671962824 q^{69} - 156447825521400 q^{70} + 131693145807312 q^{71} - 353606797863216 q^{74} - 360965926890000 q^{75} + 959127540575520 q^{76} - 26081853939360 q^{79} - 552945514077600 q^{80} + 11\!\cdots\!46 q^{81}+ \cdots + 506999099666376 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 29397x^{4} + 153469728x^{2} + 65015354624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{5} - 368353\nu^{3} - 1648953712\nu ) / 4805184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 39196 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -19\nu^{5} + 13808\nu^{4} + 200897\nu^{3} + 336542384\nu^{2} + 11605603568\nu + 732969732608 ) / 4805184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{5} - 13808\nu^{4} + 200897\nu^{3} - 336542384\nu^{2} + 11605603568\nu - 732969732608 ) / 4805184 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 39196 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{5} + 13\beta_{4} - 38\beta_{2} - 37918\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -696\beta_{5} + 696\beta_{4} - 24373\beta_{3} + 742992204 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -368353\beta_{5} - 368353\beta_{4} - 401794\beta_{2} + 820715510\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
150.955i
78.3925i
21.5470i
21.5470i
78.3925i
150.955i
301.910i 779.290i −58381.4 −88431.8 150657.i −235275. 2.08791e6i 7.73292e6i 1.37416e7 −4.54846e7 + 2.66984e7i
4.2 156.785i 1705.52i 8186.44 129622. + 117114.i 267401. 1.54725e6i 6.42105e6i 1.14401e7 1.83617e7 2.03228e7i
4.3 43.0940i 6725.99i 30910.9 −160365. + 69286.1i −289850. 1.29717e6i 2.74418e6i −3.08900e7 2.98581e6 + 6.91078e6i
4.4 43.0940i 6725.99i 30910.9 −160365. 69286.1i −289850. 1.29717e6i 2.74418e6i −3.08900e7 2.98581e6 6.91078e6i
4.5 156.785i 1705.52i 8186.44 129622. 117114.i 267401. 1.54725e6i 6.42105e6i 1.14401e7 1.83617e7 + 2.03228e7i
4.6 301.910i 779.290i −58381.4 −88431.8 + 150657.i −235275. 2.08791e6i 7.73292e6i 1.37416e7 −4.54846e7 2.66984e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.16.b.a 6
3.b odd 2 1 45.16.b.b 6
4.b odd 2 1 80.16.c.a 6
5.b even 2 1 inner 5.16.b.a 6
5.c odd 4 2 25.16.a.f 6
15.d odd 2 1 45.16.b.b 6
20.d odd 2 1 80.16.c.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.16.b.a 6 1.a even 1 1 trivial
5.16.b.a 6 5.b even 2 1 inner
25.16.a.f 6 5.c odd 4 2
45.16.b.b 6 3.b odd 2 1
45.16.b.b 6 15.d odd 2 1
80.16.c.a 6 4.b odd 2 1
80.16.c.a 6 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 117588 T^{4} + \cdots + 4160982695936 \) Copy content Toggle raw display
$3$ \( T^{6} + 48755052 T^{4} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{6} + 238350 T^{5} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{6} + 8435989101708 T^{4} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( (T^{3} + 54295044 T^{2} + \cdots - 91\!\cdots\!08)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{3} + 1815497820 T^{2} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 61\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{3} - 143084234370 T^{2} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 138118374144 T^{2} + \cdots + 49\!\cdots\!92)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{3} + 3076639441194 T^{2} + \cdots - 12\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 40\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( (T^{3} - 7451629163340 T^{2} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 5676030714294 T^{2} + \cdots + 78\!\cdots\!92)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} - 65846572903656 T^{2} + \cdots + 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{3} + 13040926969680 T^{2} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{3} - 29634845354910 T^{2} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
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