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Properties

Label 201.4.a.e
Level $201$
Weight $4$
Character orbit 201.a
Self dual yes
Analytic conductor $11.859$
Analytic rank $0$
Dimension $11$
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] = [201,4,Mod(1,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("201.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 3 x^{10} - 74 x^{9} + 208 x^{8} + 1913 x^{7} - 4831 x^{6} - 20432 x^{5} + 42994 x^{4} + \cdots + 3072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} + 1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{7} + \beta_1 + 7) q^{7} + (\beta_{9} + \beta_{8} + \cdots + 7 \beta_1) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{3} + 1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{7} + \beta_1 + 7) q^{7} + (\beta_{9} + \beta_{8} + \cdots + 7 \beta_1) q^{8}+ \cdots + ( - 9 \beta_{10} - 9 \beta_{8} + \cdots + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 3 q^{2} + 33 q^{3} + 69 q^{4} + 8 q^{5} + 9 q^{6} + 78 q^{7} + 21 q^{8} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 3 q^{2} + 33 q^{3} + 69 q^{4} + 8 q^{5} + 9 q^{6} + 78 q^{7} + 21 q^{8} + 99 q^{9} + 29 q^{10} + 104 q^{11} + 207 q^{12} + 172 q^{13} + 143 q^{14} + 24 q^{15} + 485 q^{16} - 48 q^{17} + 27 q^{18} + 180 q^{19} - 539 q^{20} + 234 q^{21} - 144 q^{22} + 156 q^{23} + 63 q^{24} + 383 q^{25} - 252 q^{26} + 297 q^{27} + 1011 q^{28} - 4 q^{29} + 87 q^{30} + 514 q^{31} - 119 q^{32} + 312 q^{33} + 72 q^{34} - 338 q^{35} + 621 q^{36} + 854 q^{37} - 308 q^{38} + 516 q^{39} - 15 q^{40} + 674 q^{41} + 429 q^{42} + 738 q^{43} + 356 q^{44} + 72 q^{45} + 507 q^{46} + 54 q^{47} + 1455 q^{48} + 1465 q^{49} + 656 q^{50} - 144 q^{51} - 12 q^{52} - 190 q^{53} + 81 q^{54} + 262 q^{55} + 239 q^{56} + 540 q^{57} - 1466 q^{58} + 18 q^{59} - 1617 q^{60} + 328 q^{61} - 915 q^{62} + 702 q^{63} + 2253 q^{64} - 732 q^{65} - 432 q^{66} + 737 q^{67} - 5746 q^{68} + 468 q^{69} - 4451 q^{70} + 264 q^{71} + 189 q^{72} + 330 q^{73} - 5975 q^{74} + 1149 q^{75} - 178 q^{76} - 368 q^{77} - 756 q^{78} + 456 q^{79} - 8515 q^{80} + 891 q^{81} - 3629 q^{82} - 2432 q^{83} + 3033 q^{84} + 2882 q^{85} - 6225 q^{86} - 12 q^{87} - 5492 q^{88} - 2340 q^{89} + 261 q^{90} - 994 q^{91} - 2939 q^{92} + 1542 q^{93} - 3506 q^{94} - 2568 q^{95} - 357 q^{96} + 1892 q^{97} - 1078 q^{98} + 936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 3 x^{10} - 74 x^{9} + 208 x^{8} + 1913 x^{7} - 4831 x^{6} - 20432 x^{5} + 42994 x^{4} + \cdots + 3072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37 \nu^{10} - 10337 \nu^{9} - 910 \nu^{8} + 803028 \nu^{7} - 138167 \nu^{6} - 21118529 \nu^{5} + \cdots - 17164864 ) / 17885888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1563 \nu^{10} - 1415 \nu^{9} + 68654 \nu^{8} - 54128 \nu^{7} + 5581 \nu^{6} + 7020509 \nu^{5} + \cdots + 161452352 ) / 17885888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 999 \nu^{10} - 368 \nu^{9} + 24570 \nu^{8} + 116670 \nu^{7} + 935839 \nu^{6} - 5222270 \nu^{5} + \cdots - 260927136 ) / 8942944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2205 \nu^{10} - 41989 \nu^{9} - 190188 \nu^{8} + 2763750 \nu^{7} + 5709131 \nu^{6} + \cdots - 284052576 ) / 8942944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27763 \nu^{10} + 74816 \nu^{9} + 1906432 \nu^{8} - 5007380 \nu^{7} - 44730735 \nu^{6} + \cdots + 1141218944 ) / 35771776 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 28189 \nu^{10} + 47870 \nu^{9} - 1856484 \nu^{8} - 2968156 \nu^{7} + 40420809 \nu^{6} + \cdots - 713324160 ) / 35771776 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 28263 \nu^{10} - 27196 \nu^{9} + 1858304 \nu^{8} + 1362100 \nu^{7} - 40144475 \nu^{6} + \cdots + 246849024 ) / 35771776 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45089 \nu^{10} - 81302 \nu^{9} - 3359788 \nu^{8} + 5317020 \nu^{7} + 86621461 \nu^{6} + \cdots - 961174016 ) / 35771776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{8} + \beta_{3} - \beta_{2} + 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - 2\beta_{8} - \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + 4\beta_{3} + 28\beta_{2} + 313 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{10} + 29 \beta_{9} + 41 \beta_{8} + 9 \beta_{7} + 8 \beta_{6} - \beta_{5} + 9 \beta_{4} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34 \beta_{10} - 2 \beta_{9} - 108 \beta_{8} - 88 \beta_{7} - 106 \beta_{6} - 26 \beta_{5} + 54 \beta_{4} + \cdots + 7856 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 172 \beta_{10} + 755 \beta_{9} + 1395 \beta_{8} + 452 \beta_{7} + 412 \beta_{6} - 16 \beta_{5} + \cdots - 1548 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 911 \beta_{10} - 164 \beta_{9} - 4210 \beta_{8} - 4063 \beta_{7} - 4086 \beta_{6} - 631 \beta_{5} + \cdots + 208851 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7009 \beta_{10} + 19693 \beta_{9} + 44825 \beta_{8} + 17035 \beta_{7} + 15584 \beta_{6} + \cdots - 81047 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23316 \beta_{10} - 8654 \beta_{9} - 146472 \beta_{8} - 153782 \beta_{7} - 141002 \beta_{6} + \cdots + 5762126 \) 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
−5.56558
−4.47941
−3.89104
−2.12976
−0.638765
0.0344354
1.96751
2.85564
4.59496
4.82380
5.42821
−5.56558 3.00000 22.9757 −9.37616 −16.6967 18.3374 −83.3483 9.00000 52.1838
1.2 −4.47941 3.00000 12.0651 −12.5434 −13.4382 −24.1947 −18.2093 9.00000 56.1868
1.3 −3.89104 3.00000 7.14021 11.5975 −11.6731 32.2920 3.34547 9.00000 −45.1263
1.4 −2.12976 3.00000 −3.46413 17.7773 −6.38927 −9.52178 24.4158 9.00000 −37.8613
1.5 −0.638765 3.00000 −7.59198 −13.5481 −1.91629 17.9442 9.95961 9.00000 8.65405
1.6 0.0344354 3.00000 −7.99881 3.05553 0.103306 −26.7198 −0.550926 9.00000 0.105218
1.7 1.96751 3.00000 −4.12891 17.2496 5.90252 22.3999 −23.8637 9.00000 33.9387
1.8 2.85564 3.00000 0.154694 −2.88345 8.56693 19.3750 −22.4034 9.00000 −8.23409
1.9 4.59496 3.00000 13.1136 11.7140 13.7849 4.20913 23.4970 9.00000 53.8252
1.10 4.82380 3.00000 15.2690 4.99073 14.4714 −10.0496 35.0643 9.00000 24.0743
1.11 5.42821 3.00000 21.4655 −20.0336 16.2846 33.9282 73.0934 9.00000 −108.746
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(67\) \(-1\)

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 201.4.a.e ✓ 11
3.b odd 2 1 603.4.a.g 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.e ✓ 11 1.a even 1 1 trivial
603.4.a.g 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 3 T_{2}^{10} - 74 T_{2}^{9} + 208 T_{2}^{8} + 1913 T_{2}^{7} - 4831 T_{2}^{6} - 20432 T_{2}^{5} + \cdots + 3072 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 3 T^{10} + \cdots + 3072 \) Copy content Toggle raw display
$3$ \( (T - 3)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots + 58472299008 \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 40739453904896 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots - 43\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots - 64\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots - 78\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 52\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 97\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 22\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 22\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( (T - 67)^{11} \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 30\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots - 14\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 50\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 11\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
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