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Properties

Label 187.4.a.e
Level $187$
Weight $4$
Character orbit 187.a
Self dual yes
Analytic conductor $11.033$
Analytic rank $0$
Dimension $10$
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] = [187,4,Mod(1,187)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(187, 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("187.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 187 = 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 187.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: \(11.0333571711\)
Analytic rank: \(0\)
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: \( x^{10} - 4 x^{9} - 52 x^{8} + 180 x^{7} + 933 x^{6} - 2524 x^{5} - 6654 x^{4} + 11196 x^{3} + 16044 x^{2} + \cdots - 88 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + (\beta_{8} + \beta_{6} - \beta_{5} + 5) q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1 + 10) q^{6} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \cdots - 2) q^{7}+ \cdots + ( - \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{5} + \beta_1 + 1) q^{3} + (\beta_{2} + \beta_1 + 4) q^{4} + (\beta_{8} + \beta_{6} - \beta_{5} + 5) q^{5} + ( - \beta_{4} + \beta_{2} + \beta_1 + 10) q^{6} + ( - \beta_{9} + \beta_{8} + \beta_{6} + \cdots - 2) q^{7}+ \cdots + (11 \beta_{8} - 11 \beta_{7} + \cdots - 132) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 11 q^{3} + 40 q^{4} + 47 q^{5} + 105 q^{6} - 17 q^{7} + 84 q^{8} + 107 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 11 q^{3} + 40 q^{4} + 47 q^{5} + 105 q^{6} - 17 q^{7} + 84 q^{8} + 107 q^{9} - 7 q^{10} - 110 q^{11} + 101 q^{12} + 15 q^{13} + 45 q^{14} + 224 q^{15} + 140 q^{16} - 170 q^{17} + 81 q^{18} + q^{19} + 289 q^{20} + 16 q^{21} - 44 q^{22} + 340 q^{23} + 663 q^{24} + 591 q^{25} - 277 q^{26} + 776 q^{27} - 127 q^{28} + 452 q^{29} + 594 q^{30} + 298 q^{31} + 536 q^{32} - 121 q^{33} - 68 q^{34} + 440 q^{35} - 15 q^{36} + 619 q^{37} + 349 q^{38} - 771 q^{39} + 379 q^{40} + 367 q^{41} + 292 q^{42} - 385 q^{43} - 440 q^{44} + 1542 q^{45} + 776 q^{46} + 365 q^{47} - 363 q^{48} + 985 q^{49} - 2167 q^{50} - 187 q^{51} + 133 q^{52} + 1624 q^{53} - 576 q^{54} - 517 q^{55} - 2217 q^{56} + 1498 q^{57} - 802 q^{58} - 414 q^{59} - 3420 q^{60} - 1162 q^{61} - 90 q^{62} - 1309 q^{63} - 1044 q^{64} + 2818 q^{65} - 1155 q^{66} - 1216 q^{67} - 680 q^{68} - 1048 q^{69} - 1556 q^{70} + 850 q^{71} - 117 q^{72} - 983 q^{73} - 555 q^{74} + 415 q^{75} - 1313 q^{76} + 187 q^{77} - 1939 q^{78} - 1165 q^{79} + 209 q^{80} + 1442 q^{81} + 481 q^{82} - 1391 q^{83} - 3728 q^{84} - 799 q^{85} + 2757 q^{86} - 1434 q^{87} - 924 q^{88} + 2225 q^{89} - 7526 q^{90} - 508 q^{91} + 358 q^{92} + 1266 q^{93} - 5233 q^{94} - 2768 q^{95} - 3209 q^{96} + 700 q^{97} + 3281 q^{98} - 1177 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 52 x^{8} + 180 x^{7} + 933 x^{6} - 2524 x^{5} - 6654 x^{4} + 11196 x^{3} + 16044 x^{2} + \cdots - 88 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83 \nu^{9} + 2717 \nu^{8} - 19573 \nu^{7} - 113171 \nu^{6} + 713942 \nu^{5} + 1274510 \nu^{4} + \cdots - 6454184 ) / 574464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 265 \nu^{9} - 3113 \nu^{8} - 10799 \nu^{7} + 138407 \nu^{6} + 164578 \nu^{5} - 1879382 \nu^{4} + \cdots - 1179640 ) / 574464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 349 \nu^{9} + 1661 \nu^{8} + 15035 \nu^{7} - 73619 \nu^{6} - 187210 \nu^{5} + 1045454 \nu^{4} + \cdots + 3970648 ) / 574464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 491 \nu^{9} - 1771 \nu^{8} - 27325 \nu^{7} + 86341 \nu^{6} + 505862 \nu^{5} - 1278754 \nu^{4} + \cdots - 1678952 ) / 287232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 109 \nu^{9} + 440 \nu^{8} + 5657 \nu^{7} - 19982 \nu^{6} - 101710 \nu^{5} + 281216 \nu^{4} + \cdots + 536272 ) / 35904 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2305 \nu^{9} + 7601 \nu^{8} + 121367 \nu^{7} - 328127 \nu^{6} - 2176018 \nu^{5} + 4269446 \nu^{4} + \cdots + 3585208 ) / 574464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 973 \nu^{9} + 4125 \nu^{8} + 49067 \nu^{7} - 180051 \nu^{6} - 842602 \nu^{5} + 2398094 \nu^{4} + \cdots + 3677784 ) / 191488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} + 20\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 3\beta_{8} - 2\beta_{7} + 5\beta_{5} + \beta_{4} - 2\beta_{3} + 25\beta_{2} + 32\beta _1 + 236 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{9} - 33 \beta_{8} - 14 \beta_{7} - 8 \beta_{6} - 21 \beta_{5} + 31 \beta_{4} - 2 \beta_{3} + \cdots + 312 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 135 \beta_{9} - 107 \beta_{8} - 106 \beta_{7} + 4 \beta_{6} + 133 \beta_{5} + 41 \beta_{4} - 70 \beta_{3} + \cdots + 5224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1137 \beta_{9} - 933 \beta_{8} - 678 \beta_{7} - 376 \beta_{6} - 457 \beta_{5} + 779 \beta_{4} + \cdots + 10128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4683 \beta_{9} - 3319 \beta_{8} - 3946 \beta_{7} - 44 \beta_{6} + 2753 \beta_{5} + 1165 \beta_{4} + \cdots + 122472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 33201 \beta_{9} - 25973 \beta_{8} - 24110 \beta_{7} - 12960 \beta_{6} - 11313 \beta_{5} + 18091 \beta_{4} + \cdots + 306944 \) 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
−4.58446
−4.10090
−2.62662
−1.58216
−0.0109263
0.417002
2.53711
3.74503
4.95094
5.25499
−4.58446 −7.60062 13.0173 20.5295 34.8447 7.60859 −23.0015 30.7694 −94.1165
1.2 −4.10090 −1.59161 8.81740 −13.7296 6.52702 −19.9578 −3.35210 −24.4668 56.3039
1.3 −2.62662 0.447567 −1.10085 9.81562 −1.17559 21.1571 23.9045 −26.7997 −25.7819
1.4 −1.58216 9.98623 −5.49676 18.3203 −15.7998 −20.6771 21.3541 72.7247 −28.9856
1.5 −0.0109263 −5.88845 −7.99988 −1.15833 0.0643390 −32.9609 0.174820 7.67385 0.0126563
1.6 0.417002 −5.36514 −7.82611 −14.2046 −2.23727 7.39855 −6.59952 1.78467 −5.92335
1.7 2.53711 5.15451 −1.56306 22.0279 13.0776 20.1356 −24.2626 −0.431059 55.8872
1.8 3.74503 9.00843 6.02526 −6.24362 33.7368 21.2202 −7.39547 54.1518 −23.3826
1.9 4.95094 0.0975223 16.5118 7.22801 0.482827 10.3683 42.1416 −26.9905 35.7855
1.10 5.25499 6.75156 19.6149 4.41497 35.4794 −31.2925 61.0362 18.5836 23.2006
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(17\) \(1\)

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 187.4.a.e ✓ 10
3.b odd 2 1 1683.4.a.j 10
11.b odd 2 1 2057.4.a.h 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.a.e ✓ 10 1.a even 1 1 trivial
1683.4.a.j 10 3.b odd 2 1
2057.4.a.h 10 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 4 T_{2}^{9} - 52 T_{2}^{8} + 180 T_{2}^{7} + 933 T_{2}^{6} - 2524 T_{2}^{5} - 6654 T_{2}^{4} + \cdots - 88 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(187))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots - 88 \) Copy content Toggle raw display
$3$ \( T^{10} - 11 T^{9} + \cdots + 52224 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 3660197440 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 2245822227552 \) Copy content Toggle raw display
$11$ \( (T + 11)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T + 17)^{10} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 37\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 10\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 23\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 76\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 50\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 75\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 76\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 35\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 92\!\cdots\!06 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
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