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Properties

Label 536.4.a.d
Level $536$
Weight $4$
Character orbit 536.a
Self dual yes
Analytic conductor $31.625$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 291 x^{13} + 792 x^{12} + 32250 x^{11} - 73564 x^{10} - 1717592 x^{9} + \cdots + 843285072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + 1) q^{5} - \beta_{6} q^{7} + (\beta_{2} + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} + 1) q^{5} - \beta_{6} q^{7} + (\beta_{2} + 12) q^{9} + (\beta_{5} + \beta_1 - 3) q^{11} + (\beta_{4} + \beta_{3} + \beta_1 + 5) q^{13} + ( - \beta_{14} + \beta_{13} + \beta_{12} + \cdots - 2) q^{15}+ \cdots + (15 \beta_{14} + 2 \beta_{13} + \cdots - 94) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 3 q^{3} + 10 q^{5} + 3 q^{7} + 186 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 3 q^{3} + 10 q^{5} + 3 q^{7} + 186 q^{9} - 46 q^{11} + 69 q^{13} - 17 q^{15} + 272 q^{17} + 117 q^{19} + 42 q^{21} - 18 q^{23} + 633 q^{25} + 108 q^{27} + 162 q^{29} + 342 q^{31} + 603 q^{33} + 323 q^{35} + 641 q^{37} + 692 q^{39} + 1195 q^{41} + 653 q^{43} + 1087 q^{45} + 967 q^{47} + 2004 q^{49} + 1682 q^{51} + 978 q^{53} + 1090 q^{55} + 3181 q^{57} + 982 q^{59} + 1545 q^{61} + 1046 q^{63} + 3307 q^{65} + 1005 q^{67} + 3649 q^{69} + 1158 q^{71} + 3166 q^{73} + 732 q^{75} + 1819 q^{77} - 727 q^{79} + 4479 q^{81} - 977 q^{83} + 3658 q^{85} + 1543 q^{87} + 4154 q^{89} - 1493 q^{91} + 4361 q^{93} - 2027 q^{95} + 5690 q^{97} - 1143 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 291 x^{13} + 792 x^{12} + 32250 x^{11} - 73564 x^{10} - 1717592 x^{9} + \cdots + 843285072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 39 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\!\cdots\!03 \nu^{14} + \cdots - 34\!\cdots\!36 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 63\!\cdots\!17 \nu^{14} + \cdots + 28\!\cdots\!48 ) / 63\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35\!\cdots\!99 \nu^{14} + \cdots - 21\!\cdots\!12 ) / 14\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 56\!\cdots\!45 \nu^{14} + \cdots + 19\!\cdots\!52 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 99\!\cdots\!87 \nu^{14} + \cdots - 10\!\cdots\!88 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!39 \nu^{14} + \cdots + 95\!\cdots\!50 ) / 36\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!61 \nu^{14} + \cdots - 50\!\cdots\!96 ) / 21\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 18\!\cdots\!51 \nu^{14} + \cdots + 24\!\cdots\!24 ) / 49\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 90\!\cdots\!85 \nu^{14} + \cdots + 30\!\cdots\!40 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!05 \nu^{14} + \cdots - 10\!\cdots\!92 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{14} + \cdots - 44\!\cdots\!32 ) / 19\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 73\!\cdots\!97 \nu^{14} + \cdots - 16\!\cdots\!92 ) / 63\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 39 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} - 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{14} + \beta_{13} + 6 \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + \beta_{8} + \cdots + 2722 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{14} + 118 \beta_{13} - 107 \beta_{12} - 16 \beta_{11} - 255 \beta_{10} + 63 \beta_{9} + \cdots - 700 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 915 \beta_{14} + 156 \beta_{13} + 1010 \beta_{12} - 167 \beta_{11} + 68 \beta_{10} - 458 \beta_{9} + \cdots + 218202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 426 \beta_{14} + 11772 \beta_{13} - 10683 \beta_{12} - 2856 \beta_{11} - 27199 \beta_{10} + \cdots - 127803 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 107340 \beta_{14} + 19780 \beta_{13} + 125754 \beta_{12} - 19530 \beta_{11} + 3092 \beta_{10} + \cdots + 18464849 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68193 \beta_{14} + 1101510 \beta_{13} - 1063198 \beta_{12} - 382412 \beta_{11} - 2734454 \beta_{10} + \cdots - 15006324 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11568717 \beta_{14} + 2347578 \beta_{13} + 13954294 \beta_{12} - 2117601 \beta_{11} + \cdots + 1607904112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 19383237 \beta_{14} + 99621006 \beta_{13} - 106014088 \beta_{12} - 46098364 \beta_{11} + \cdots - 1563061925 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1200027465 \beta_{14} + 266036438 \beta_{13} + 1463175695 \beta_{12} - 224262523 \beta_{11} + \cdots + 142418523359 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3124804677 \beta_{14} + 8823568029 \beta_{13} - 10569783512 \beta_{12} - 5246578280 \beta_{11} + \cdots - 155673797095 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 121885467864 \beta_{14} + 29065331060 \beta_{13} + 148672450705 \beta_{12} - 23388540440 \beta_{11} + \cdots + 12750046900257 \) 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
−9.69844
−9.32305
−7.45953
−3.65433
−3.39172
−3.08546
−2.52842
0.224854
0.894676
2.68742
5.52053
6.73414
7.31028
9.13098
9.63806
0 −9.69844 0 −9.74884 0 12.6619 0 67.0598 0
1.2 0 −9.32305 0 13.0980 0 −22.5919 0 59.9193 0
1.3 0 −7.45953 0 12.4763 0 32.1613 0 28.6446 0
1.4 0 −3.65433 0 16.1732 0 −17.9816 0 −13.6459 0
1.5 0 −3.39172 0 −1.53484 0 −19.5014 0 −15.4962 0
1.6 0 −3.08546 0 −14.8519 0 31.4046 0 −17.4800 0
1.7 0 −2.52842 0 −7.97788 0 −30.4878 0 −20.6071 0
1.8 0 0.224854 0 −20.9841 0 −11.4641 0 −26.9494 0
1.9 0 0.894676 0 −0.788596 0 16.9850 0 −26.1996 0
1.10 0 2.68742 0 17.5041 0 8.92397 0 −19.7778 0
1.11 0 5.52053 0 −12.7719 0 −19.3874 0 3.47630 0
1.12 0 6.73414 0 20.5841 0 −9.51846 0 18.3486 0
1.13 0 7.31028 0 0.998323 0 35.1531 0 26.4403 0
1.14 0 9.13098 0 8.91323 0 15.8876 0 56.3749 0
1.15 0 9.63806 0 −11.0892 0 −19.2447 0 65.8922 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.d ✓ 15
4.b odd 2 1 1072.4.a.m 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.4.a.d ✓ 15 1.a even 1 1 trivial
1072.4.a.m 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 3 T_{3}^{14} - 291 T_{3}^{13} + 792 T_{3}^{12} + 32250 T_{3}^{11} - 73564 T_{3}^{10} + \cdots + 843285072 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(536))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 843285072 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 35208924443784 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 37\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 37\!\cdots\!48 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 57\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 23\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 25\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 58\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 11\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 38\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 48\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T - 67)^{15} \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 55\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 47\!\cdots\!32 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 54\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 30\!\cdots\!26 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 41\!\cdots\!44 \) Copy content Toggle raw display
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