Properties

Label 19.8.a.b
Level $19$
Weight $8$
Character orbit 19.a
Self dual yes
Analytic conductor $5.935$
Analytic rank $0$
Dimension $6$
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] = [19,8,Mod(1,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 19.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: \(5.93531548420\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 540x^{4} + 610x^{3} + 80412x^{2} + 7680x - 2267712 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{2} + (\beta_{4} + \beta_1 + 6) q^{3} + (\beta_{2} + 6 \beta_1 + 57) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_1 + 32) q^{5} + (4 \beta_{5} + \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 12 \beta_1 + 148) q^{6} + ( - \beta_{5} - 8 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 352) q^{7} + ( - 8 \beta_{5} - 15 \beta_{4} + \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 972) q^{8} + ( - 10 \beta_{5} + 15 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 57 \beta_1 + 1011) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{2} + (\beta_{4} + \beta_1 + 6) q^{3} + (\beta_{2} + 6 \beta_1 + 57) q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 7 \beta_1 + 32) q^{5} + (4 \beta_{5} + \beta_{4} - 5 \beta_{3} - 6 \beta_{2} + 12 \beta_1 + 148) q^{6} + ( - \beta_{5} - 8 \beta_{4} + 7 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 352) q^{7} + ( - 8 \beta_{5} - 15 \beta_{4} + \beta_{3} + 14 \beta_{2} + 14 \beta_1 + 972) q^{8} + ( - 10 \beta_{5} + 15 \beta_{4} - 10 \beta_{3} - 2 \beta_{2} - 57 \beta_1 + 1011) q^{9} + (22 \beta_{5} - \beta_{4} + 11 \beta_{3} + 4 \beta_{2} - 67 \beta_1 + 1408) q^{10} + (41 \beta_{5} + 49 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} - 223 \beta_1 + 1314) q^{11} + ( - 28 \beta_{5} + 35 \beta_{4} - 49 \beta_{3} - 36 \beta_{2} + \cdots + 1154) q^{12}+ \cdots + ( - 20611 \beta_{5} + 21631 \beta_{4} + 33575 \beta_{3} + \cdots + 1899718) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 15 q^{2} + 40 q^{3} + 357 q^{4} + 219 q^{5} + 925 q^{6} + 2105 q^{7} + 5835 q^{8} + 5916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 15 q^{2} + 40 q^{3} + 357 q^{4} + 219 q^{5} + 925 q^{6} + 2105 q^{7} + 5835 q^{8} + 5916 q^{9} + 8212 q^{10} + 7257 q^{11} + 7025 q^{12} + 6850 q^{13} + 8859 q^{14} + 3650 q^{15} - 9159 q^{16} + 5415 q^{17} - 58980 q^{18} - 41154 q^{19} - 67620 q^{20} - 83290 q^{21} - 223870 q^{22} - 720 q^{23} - 151113 q^{24} - 53567 q^{25} - 106527 q^{26} + 199450 q^{27} + 91615 q^{28} + 381624 q^{29} - 137776 q^{30} + 264080 q^{31} + 259155 q^{32} + 496430 q^{33} + 297463 q^{34} + 739767 q^{35} - 147282 q^{36} + 1082300 q^{37} - 102885 q^{38} + 1129528 q^{39} - 524232 q^{40} + 485232 q^{41} - 1753105 q^{42} + 198705 q^{43} - 1729290 q^{44} - 478705 q^{45} - 1565713 q^{46} - 247125 q^{47} - 2937955 q^{48} - 538861 q^{49} - 2396859 q^{50} - 72176 q^{51} - 2647795 q^{52} + 3226770 q^{53} - 1217249 q^{54} - 1490553 q^{55} + 3718965 q^{56} - 274360 q^{57} + 1048405 q^{58} + 2305380 q^{59} + 647440 q^{60} + 585731 q^{61} + 2583780 q^{62} - 3209015 q^{63} + 2380137 q^{64} + 4809420 q^{65} + 2420402 q^{66} - 3264030 q^{67} + 8276595 q^{68} - 1867056 q^{69} + 5936880 q^{70} + 6833682 q^{71} + 3040530 q^{72} - 4160625 q^{73} + 20750550 q^{74} - 11237814 q^{75} - 2448663 q^{76} + 1659195 q^{77} - 1839095 q^{78} - 8680576 q^{79} + 14904048 q^{80} - 16541142 q^{81} - 9240140 q^{82} - 3785040 q^{83} - 18290321 q^{84} - 16108227 q^{85} + 11192544 q^{86} - 25742220 q^{87} - 24666570 q^{88} + 12473466 q^{89} - 4688908 q^{90} - 14289854 q^{91} + 9179655 q^{92} + 5742820 q^{93} - 10757712 q^{94} - 1502121 q^{95} - 8195689 q^{96} + 882830 q^{97} + 47239200 q^{98} + 10726225 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 540x^{4} + 610x^{3} + 80412x^{2} + 7680x - 2267712 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 181 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{5} - 249\nu^{4} - 7416\nu^{3} + 52718\nu^{2} + 454620\nu - 510288 ) / 11712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\nu^{5} - 805\nu^{4} - 13016\nu^{3} + 219414\nu^{2} + 775596\nu - 7796880 ) / 35136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 59\nu^{4} + 718\nu^{3} - 14854\nu^{2} - 11400\nu + 407544 ) / 1464 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 181 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{5} - 15\beta_{4} + \beta_{3} + 8\beta_{2} + 246\beta _1 + 390 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -20\beta_{5} - 141\beta_{4} + 67\beta_{3} + 376\beta_{2} + 1108\beta _1 + 45254 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2796\beta_{5} - 6363\beta_{4} + 1557\beta_{3} + 4358\beta_{2} + 66964\beta _1 + 222992 \) Copy content Toggle raw display

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
−15.2573
−14.6171
−6.37480
5.78641
14.6629
18.7999
−13.2573 −51.9225 47.7563 −400.717 688.353 −4.24487 1063.82 508.948 5312.43
1.2 −12.6171 75.4805 31.1902 −68.7314 −952.342 325.908 1221.45 3510.30 867.188
1.3 −4.37480 −63.9788 −108.861 301.974 279.894 615.044 1036.22 1906.28 −1321.08
1.4 7.78641 37.1813 −67.3719 383.672 289.509 1025.33 −1521.24 −804.550 2987.43
1.5 16.6629 67.6316 149.652 −74.3884 1126.94 −1123.58 360.790 2387.04 −1239.53
1.6 20.7999 −24.3922 304.634 77.1906 −507.353 1266.54 3673.97 −1592.02 1605.55
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \(1\)

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 19.8.a.b 6
3.b odd 2 1 171.8.a.f 6
4.b odd 2 1 304.8.a.h 6
5.b even 2 1 475.8.a.b 6
19.b odd 2 1 361.8.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.8.a.b 6 1.a even 1 1 trivial
171.8.a.f 6 3.b odd 2 1
304.8.a.h 6 4.b odd 2 1
361.8.a.c 6 19.b odd 2 1
475.8.a.b 6 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 15T_{2}^{5} - 450T_{2}^{4} + 4650T_{2}^{3} + 64272T_{2}^{2} - 289800T_{2} - 1974784 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 15 T^{5} - 450 T^{4} + \cdots - 1974784 \) Copy content Toggle raw display
$3$ \( T^{6} - 40 T^{5} + \cdots - 15379814304 \) Copy content Toggle raw display
$5$ \( T^{6} - 219 T^{5} + \cdots - 18322855416000 \) Copy content Toggle raw display
$7$ \( T^{6} - 2105 T^{5} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{6} - 7257 T^{5} + \cdots - 86\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{6} - 6850 T^{5} + \cdots - 69\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{6} - 5415 T^{5} + \cdots + 17\!\cdots\!74 \) Copy content Toggle raw display
$19$ \( (T + 6859)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 720 T^{5} + \cdots - 52\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{6} - 381624 T^{5} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{6} - 264080 T^{5} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{6} - 1082300 T^{5} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{6} - 485232 T^{5} + \cdots - 50\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{6} - 198705 T^{5} + \cdots - 10\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + 247125 T^{5} + \cdots - 22\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} - 3226770 T^{5} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} - 2305380 T^{5} + \cdots - 16\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{6} - 585731 T^{5} + \cdots - 49\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{6} + 3264030 T^{5} + \cdots - 48\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{6} - 6833682 T^{5} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{6} + 4160625 T^{5} + \cdots + 42\!\cdots\!74 \) Copy content Toggle raw display
$79$ \( T^{6} + 8680576 T^{5} + \cdots - 74\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{6} + 3785040 T^{5} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{6} - 12473466 T^{5} + \cdots + 65\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{6} - 882830 T^{5} + \cdots - 44\!\cdots\!44 \) Copy content Toggle raw display
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