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Properties

Label 538.2.b.b
Level $538$
Weight $2$
Character orbit 538.b
Analytic conductor $4.296$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(537,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.537");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 40 x^{16} + 656 x^{14} + 5672 x^{12} + 27720 x^{10} + 76589 x^{8} + 114310 x^{6} + 83081 x^{4} + \cdots + 784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_1 q^{3} - q^{4} - \beta_{11} q^{5} - \beta_{3} q^{6} + (\beta_{14} - \beta_{8}) q^{7} + \beta_{8} q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} - \beta_1 q^{3} - q^{4} - \beta_{11} q^{5} - \beta_{3} q^{6} + (\beta_{14} - \beta_{8}) q^{7} + \beta_{8} q^{8} + (\beta_{2} - 1) q^{9} - \beta_{16} q^{10} + (\beta_{7} + \beta_{3}) q^{11} + \beta_1 q^{12} + ( - \beta_{5} - \beta_{4} + 1) q^{13} + ( - \beta_{5} - 1) q^{14} + ( - \beta_{17} - \beta_{16} + \cdots + \beta_{12}) q^{15}+ \cdots + (\beta_{11} - \beta_{9} - 2 \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{4} - 4 q^{5} - 4 q^{6} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{4} - 4 q^{5} - 4 q^{6} - 26 q^{9} - 2 q^{11} + 10 q^{13} - 24 q^{14} + 18 q^{16} + 4 q^{20} - 6 q^{21} - 20 q^{23} + 4 q^{24} + 30 q^{25} + 26 q^{36} - 10 q^{37} + 14 q^{38} - 12 q^{41} - 22 q^{43} + 2 q^{44} + 38 q^{45} - 18 q^{47} - 14 q^{49} - 6 q^{51} - 10 q^{52} - 14 q^{53} + 4 q^{54} - 6 q^{55} + 24 q^{56} - 12 q^{57} - 4 q^{58} + 6 q^{61} + 20 q^{62} - 18 q^{64} + 14 q^{65} - 50 q^{66} - 14 q^{67} - 18 q^{70} + 8 q^{73} - 10 q^{78} - 10 q^{79} - 4 q^{80} + 18 q^{81} + 6 q^{84} + 96 q^{87} - 18 q^{89} + 20 q^{92} + 70 q^{93} - 4 q^{96} + 72 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 40 x^{16} + 656 x^{14} + 5672 x^{12} + 27720 x^{10} + 76589 x^{8} + 114310 x^{6} + 83081 x^{4} + \cdots + 784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8223 \nu^{16} + 256202 \nu^{14} + 3092206 \nu^{12} + 18202410 \nu^{10} + 54665450 \nu^{8} + \cdots + 1873452 ) / 4608050 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 221171 \nu^{16} + 9177904 \nu^{14} + 153798112 \nu^{12} + 1331370520 \nu^{10} + 6317434200 \nu^{8} + \cdots + 411730604 ) / 82944900 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 262333 \nu^{16} - 9840592 \nu^{14} - 150815476 \nu^{12} - 1212073060 \nu^{10} + \cdots - 253735992 ) / 82944900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4674 \nu^{17} + 326351 \nu^{15} + 7536603 \nu^{13} + 82754605 \nu^{11} + 481195925 \nu^{9} + \cdots + 392160726 \nu ) / 16128175 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 103193 \nu^{16} + 4338732 \nu^{14} + 73947396 \nu^{12} + 653445960 \nu^{10} + 3174539600 \nu^{8} + \cdots + 269258632 ) / 27648300 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 66909 \nu^{17} + 2446116 \nu^{15} + 36718648 \nu^{13} + 292926080 \nu^{11} + \cdots + 797435616 \nu ) / 129025400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 541243 \nu^{16} + 21509532 \nu^{14} + 347098896 \nu^{12} + 2905101660 \nu^{10} + 13345216900 \nu^{8} + \cdots + 182199332 ) / 82944900 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 159013 \nu^{16} + 6143187 \nu^{14} + 96507786 \nu^{12} + 787973460 \nu^{10} + 3544555150 \nu^{8} + \cdots + 375139112 ) / 20736225 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 178792 \nu^{16} + 6877958 \nu^{14} + 107467574 \nu^{12} + 872051840 \nu^{10} + 3895497375 \nu^{8} + \cdots + 175862308 ) / 20736225 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1984467 \nu^{17} + 79047958 \nu^{15} + 1279581124 \nu^{13} + 10758709840 \nu^{11} + \cdots - 1314381242 \nu ) / 580614300 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1005379 \nu^{17} - 37572646 \nu^{15} - 569864938 \nu^{13} - 4472870830 \nu^{11} + \cdots + 15565106604 \nu ) / 290307150 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 4210649 \nu^{17} - 165052576 \nu^{15} - 2630690128 \nu^{13} - 21790338280 \nu^{11} + \cdots - 4227873476 \nu ) / 1161228600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 464701 \nu^{17} + 18003064 \nu^{15} + 282748552 \nu^{13} + 2298079600 \nu^{11} + \cdots - 1216231356 \nu ) / 77415240 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 1984467 \nu^{17} + 79047958 \nu^{15} + 1279581124 \nu^{13} + 10758709840 \nu^{11} + \cdots + 717768808 \nu ) / 290307150 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 80153 \nu^{17} - 3130487 \nu^{15} - 49762916 \nu^{13} - 411939050 \nu^{11} + \cdots - 270139077 \nu ) / 8294490 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} - 2\beta_{12} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_{4} - 9\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + 22\beta_{12} - 2\beta_{6} + 55\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{11} + \beta_{10} + 10\beta_{9} - 14\beta_{7} - 17\beta_{5} + 11\beta_{4} - 5\beta_{3} + 76\beta_{2} - 219 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{17} + 66 \beta_{16} + 14 \beta_{15} - 14 \beta_{14} - 17 \beta_{13} - 206 \beta_{12} + \cdots - 448 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 93 \beta_{11} - 14 \beta_{10} - 80 \beta_{9} + 154 \beta_{7} + 214 \beta_{5} - 92 \beta_{4} + \cdots + 1772 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 121 \beta_{17} - 445 \beta_{16} - 153 \beta_{15} + 137 \beta_{14} + 214 \beta_{13} + 1849 \beta_{12} + \cdots + 3706 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 712 \beta_{11} + 137 \beta_{10} + 582 \beta_{9} - 1557 \beta_{7} - 2381 \beta_{5} + 720 \beta_{4} + \cdots - 14550 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1669 \beta_{17} + 2774 \beta_{16} + 1565 \beta_{15} - 1122 \beta_{14} - 2381 \beta_{13} + \cdots - 30941 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5079 \beta_{11} - 1122 \beta_{10} - 3896 \beta_{9} + 15133 \beta_{7} + 24761 \beta_{5} - 5620 \beta_{4} + \cdots + 120530 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 19682 \beta_{17} - 15242 \beta_{16} - 15674 \beta_{15} + 7843 \beta_{14} + 24761 \beta_{13} + \cdots + 260135 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 33269 \beta_{11} + 7843 \beta_{10} + 23085 \beta_{9} - 143867 \beta_{7} - 246919 \beta_{5} + \cdots - 1005184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 213650 \beta_{17} + 61295 \beta_{16} + 155547 \beta_{15} - 42777 \beta_{14} - 246919 \beta_{13} + \cdots - 2200208 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 188001 \beta_{11} - 42777 \beta_{10} - 104072 \beta_{9} + 1348532 \beta_{7} + 2394311 \beta_{5} + \cdots + 8431635 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2206310 \beta_{17} + 39059 \beta_{16} - 1533360 \beta_{15} + 90352 \beta_{14} + 2394311 \beta_{13} + \cdots + 18709545 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/538\mathbb{Z}\right)^\times\).

\(n\) \(271\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
537.1
3.00256i
2.84486i
1.79438i
1.53574i
0.193979i
− 0.851562i
− 0.919863i
− 2.69147i
− 2.90863i
2.90863i
2.69147i
0.919863i
0.851562i
− 0.193979i
− 1.53574i
− 1.79438i
− 2.84486i
− 3.00256i
− 1.00000i − 3.00256i −1.00000 1.23471 −3.00256 − 4.98662i 1.00000i −6.01538 − 1.23471i
537.2 − 1.00000i − 2.84486i −1.00000 −3.96643 −2.84486 1.38718i 1.00000i −5.09323 3.96643i
537.3 − 1.00000i − 1.79438i −1.00000 −0.681291 −1.79438 − 1.89795i 1.00000i −0.219801 0.681291i
537.4 − 1.00000i − 1.53574i −1.00000 3.51954 −1.53574 1.46450i 1.00000i 0.641495 − 3.51954i
537.5 − 1.00000i − 0.193979i −1.00000 −0.907831 −0.193979 − 0.975413i 1.00000i 2.96237 0.907831i
537.6 − 1.00000i 0.851562i −1.00000 3.59767 0.851562 − 4.28533i 1.00000i 2.27484 − 3.59767i
537.7 − 1.00000i 0.919863i −1.00000 −3.72117 0.919863 − 1.30781i 1.00000i 2.15385 3.72117i
537.8 − 1.00000i 2.69147i −1.00000 0.384107 2.69147 2.08088i 1.00000i −4.24401 − 0.384107i
537.9 − 1.00000i 2.90863i −1.00000 −1.45931 2.90863 − 3.47945i 1.00000i −5.46013 1.45931i
537.10 1.00000i − 2.90863i −1.00000 −1.45931 2.90863 3.47945i − 1.00000i −5.46013 − 1.45931i
537.11 1.00000i − 2.69147i −1.00000 0.384107 2.69147 − 2.08088i − 1.00000i −4.24401 0.384107i
537.12 1.00000i − 0.919863i −1.00000 −3.72117 0.919863 1.30781i − 1.00000i 2.15385 − 3.72117i
537.13 1.00000i − 0.851562i −1.00000 3.59767 0.851562 4.28533i − 1.00000i 2.27484 3.59767i
537.14 1.00000i 0.193979i −1.00000 −0.907831 −0.193979 0.975413i − 1.00000i 2.96237 − 0.907831i
537.15 1.00000i 1.53574i −1.00000 3.51954 −1.53574 − 1.46450i − 1.00000i 0.641495 3.51954i
537.16 1.00000i 1.79438i −1.00000 −0.681291 −1.79438 1.89795i − 1.00000i −0.219801 − 0.681291i
537.17 1.00000i 2.84486i −1.00000 −3.96643 −2.84486 − 1.38718i − 1.00000i −5.09323 − 3.96643i
537.18 1.00000i 3.00256i −1.00000 1.23471 −3.00256 4.98662i − 1.00000i −6.01538 1.23471i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 537.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 538.2.b.b ✓ 18
269.b even 2 1 inner 538.2.b.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
538.2.b.b ✓ 18 1.a even 1 1 trivial
538.2.b.b ✓ 18 269.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 40 T_{3}^{16} + 656 T_{3}^{14} + 5672 T_{3}^{12} + 27720 T_{3}^{10} + 76589 T_{3}^{8} + \cdots + 784 \) acting on \(S_{2}^{\mathrm{new}}(538, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{18} + 40 T^{16} + \cdots + 784 \) Copy content Toggle raw display
$5$ \( (T^{9} + 2 T^{8} - 28 T^{7} + \cdots + 80)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + 70 T^{16} + \cdots + 579121 \) Copy content Toggle raw display
$11$ \( (T^{9} + T^{8} - 52 T^{7} + \cdots + 416)^{2} \) Copy content Toggle raw display
$13$ \( (T^{9} - 5 T^{8} - 21 T^{7} + \cdots + 76)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + 126 T^{16} + \cdots + 39337984 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 542703616 \) Copy content Toggle raw display
$23$ \( (T^{9} + 10 T^{8} + \cdots + 400000)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 133448704 \) Copy content Toggle raw display
$31$ \( T^{18} + 146 T^{16} + \cdots + 207025 \) Copy content Toggle raw display
$37$ \( (T^{9} + 5 T^{8} + \cdots - 6330196)^{2} \) Copy content Toggle raw display
$41$ \( (T^{9} + 6 T^{8} + \cdots + 527660)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} + 11 T^{8} + \cdots - 138848)^{2} \) Copy content Toggle raw display
$47$ \( (T^{9} + 9 T^{8} + \cdots + 1110400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + 7 T^{8} + \cdots + 3904)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 2460556816 \) Copy content Toggle raw display
$61$ \( (T^{9} - 3 T^{8} + \cdots - 473252)^{2} \) Copy content Toggle raw display
$67$ \( (T^{9} + 7 T^{8} + \cdots - 28533152)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 383811982854400 \) Copy content Toggle raw display
$73$ \( (T^{9} - 4 T^{8} + \cdots - 1023841)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + 5 T^{8} + \cdots + 114849920)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{9} + 9 T^{8} + \cdots + 17095805)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} - 36 T^{8} + \cdots + 214448)^{2} \) Copy content Toggle raw display
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