↑

Properties

Label 536.4.a.a
Level $536$
Weight $4$
Character orbit 536.a
Self dual yes
Analytic conductor $31.625$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [536,4,Mod(1,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 536.a (trivial)

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: yes
Analytic conductor: \(31.6250237631\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 147x^{8} + 7506x^{6} - 617x^{5} - 157469x^{4} + 31339x^{3} + 1285235x^{2} - 143141x - 3421846 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{7} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} + (\beta_{7} - \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} + 2) q^{9} + (\beta_{9} - \beta_{7} - \beta_{4} + \cdots - 4) q^{11}+ \cdots + ( - 20 \beta_{9} + 15 \beta_{8} + \cdots + 47) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{5} + 3 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{5} + 3 q^{7} + 24 q^{9} - 35 q^{11} - 74 q^{13} - 77 q^{15} - 238 q^{17} - 92 q^{19} - 168 q^{21} + 74 q^{23} - 67 q^{25} - 99 q^{29} - 278 q^{31} - 453 q^{33} - 61 q^{35} - 616 q^{37} - 414 q^{39} - 1107 q^{41} - 558 q^{43} - 1190 q^{45} - 651 q^{47} - 881 q^{49} - 948 q^{51} - 471 q^{53} - 1116 q^{55} - 1877 q^{57} - 95 q^{59} - 1684 q^{61} - 580 q^{63} - 1923 q^{65} - 670 q^{67} - 1504 q^{69} - 141 q^{71} - 2630 q^{73} - 84 q^{75} - 2531 q^{77} + 119 q^{79} - 3330 q^{81} - 43 q^{83} - 2793 q^{85} + 105 q^{87} - 4058 q^{89} - 1179 q^{91} - 2431 q^{93} + 633 q^{95} - 3052 q^{97} + 441 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 147x^{8} + 7506x^{6} - 617x^{5} - 157469x^{4} + 31339x^{3} + 1285235x^{2} - 143141x - 3421846 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13993607 \nu^{9} + 589845344 \nu^{8} - 2245395040 \nu^{7} - 75886543660 \nu^{6} + \cdots + 111862647767699 ) / 3649680339489 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13993607 \nu^{9} + 589845344 \nu^{8} - 2245395040 \nu^{7} - 75886543660 \nu^{6} + \cdots + 217703377612880 ) / 3649680339489 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1217402 \nu^{9} + 4858177 \nu^{8} + 162456556 \nu^{7} - 868060379 \nu^{6} + \cdots + 4389439132051 ) / 43972052283 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36015876 \nu^{9} + 182847428 \nu^{8} - 4115998121 \nu^{7} - 31124467818 \nu^{6} + \cdots + 158628117777028 ) / 1216560113163 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 129958215 \nu^{9} + 166051132 \nu^{8} + 17265577793 \nu^{7} - 25247887008 \nu^{6} + \cdots + 168276266634656 ) / 1216560113163 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 934453948 \nu^{9} + 1543095134 \nu^{8} + 129348151997 \nu^{7} - 215694038962 \nu^{6} + \cdots + 10\!\cdots\!32 ) / 3649680339489 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 940155926 \nu^{9} + 1592462950 \nu^{8} + 134177550496 \nu^{7} - 243924278813 \nu^{6} + \cdots + 12\!\cdots\!31 ) / 3649680339489 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1043284628 \nu^{9} - 3221720947 \nu^{8} - 146521783459 \nu^{7} + 434287941377 \nu^{6} + \cdots - 14\!\cdots\!23 ) / 3649680339489 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} + 5\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_{3} + 3\beta_{2} + 46\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{9} + 12\beta_{7} + \beta_{6} + 8\beta_{5} - 11\beta_{4} - 56\beta_{3} + 73\beta_{2} + 15\beta _1 + 1291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 201 \beta_{9} - 42 \beta_{8} + 391 \beta_{7} - 293 \beta_{6} + 39 \beta_{5} + 71 \beta_{4} + 136 \beta_{3} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 825 \beta_{9} - 99 \beta_{8} + 1252 \beta_{7} + 21 \beta_{6} + 796 \beta_{5} - 1157 \beta_{4} + \cdots + 65935 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12069 \beta_{9} - 4014 \beta_{8} + 25836 \beta_{7} - 18943 \beta_{6} + 4489 \beta_{5} + 3497 \beta_{4} + \cdots + 30617 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 57522 \beta_{9} - 15303 \beta_{8} + 100062 \beta_{7} - 4923 \beta_{6} + 61665 \beta_{5} - 89217 \beta_{4} + \cdots + 3543240 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 720447 \beta_{9} - 291060 \beta_{8} + 1633135 \beta_{7} - 1185098 \beta_{6} + 384756 \beta_{5} + \cdots + 3023992 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.41710
−7.07351
−5.48325
−2.56001
−2.42382
2.31677
3.45662
4.40333
6.88349
7.89747
0 −7.41710 0 2.59196 0 2.88599 0 28.0133 0
1.2 0 −7.07351 0 −15.5576 0 −9.59598 0 23.0346 0
1.3 0 −5.48325 0 −3.11474 0 10.4956 0 3.06603 0
1.4 0 −2.56001 0 13.2720 0 −9.91255 0 −20.4464 0
1.5 0 −2.42382 0 16.4341 0 27.7229 0 −21.1251 0
1.6 0 2.31677 0 1.03258 0 7.79764 0 −21.6326 0
1.7 0 3.45662 0 −13.8154 0 10.6633 0 −15.0518 0
1.8 0 4.40333 0 3.96772 0 −30.1981 0 −7.61065 0
1.9 0 6.88349 0 5.64267 0 −16.8766 0 20.3825 0
1.10 0 7.89747 0 −15.4533 0 10.0178 0 35.3701 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.4.a.a ✓ 10
4.b odd 2 1 1072.4.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.4.a.a ✓ 10 1.a even 1 1 trivial
1072.4.a.j 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 147 T_{3}^{8} + 7506 T_{3}^{6} - 617 T_{3}^{5} - 157469 T_{3}^{4} + 31339 T_{3}^{3} + \cdots - 3421846 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(536))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 147 T^{8} + \cdots - 3421846 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 135209366 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 33908472016 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots - 18060172185200 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 57\!\cdots\!78 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 24\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 50\!\cdots\!71 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 27\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 11\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 20\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 28\!\cdots\!58 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 31\!\cdots\!67 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 11\!\cdots\!86 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 45\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 17\!\cdots\!94 \) Copy content Toggle raw display
$67$ \( (T + 67)^{10} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 10\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 50\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 39\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 51\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
show more
show less