Properties

Label 47.8.a.b
Level $47$
Weight $8$
Character orbit 47.a
Self dual yes
Analytic conductor $14.682$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,8,Mod(1,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, 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("47.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 47.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: \(14.6820961978\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 1579 x^{14} + 411 x^{13} + 976985 x^{12} + 344128 x^{11} - 300557483 x^{10} - 241372726 x^{9} + 48208527759 x^{8} + 51058288103 x^{7} + \cdots - 870476371764660 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 3) q^{3} + (\beta_{2} - \beta_1 + 70) q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 28) q^{5} + ( - \beta_{5} + 6 \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{6} + ( - \beta_{10} - \beta_{7} + 5 \beta_{3} + 3 \beta_1 + 118) q^{7} + (\beta_{12} + \beta_{11} + 2 \beta_{10} + 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 142) q^{8}+ \cdots + (\beta_{14} - 2 \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \cdots + 1070) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 3) q^{3} + (\beta_{2} - \beta_1 + 70) q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 28) q^{5} + ( - \beta_{5} + 6 \beta_{3} + \beta_{2} + 4 \beta_1 - 6) q^{6} + ( - \beta_{10} - \beta_{7} + 5 \beta_{3} + 3 \beta_1 + 118) q^{7} + (\beta_{12} + \beta_{11} + 2 \beta_{10} + 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots + 142) q^{8}+ \cdots + ( - 14161 \beta_{15} - 1711 \beta_{14} + 4637 \beta_{13} + \cdots + 3477441) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 15 q^{2} + 40 q^{3} + 1125 q^{4} + 444 q^{5} - 128 q^{6} + 1860 q^{7} + 2145 q^{8} + 16986 q^{9} + 7680 q^{10} + 4776 q^{11} - 20035 q^{12} + 16074 q^{13} - 7749 q^{14} + 22614 q^{15} + 131873 q^{16} + 67106 q^{17} + 243090 q^{18} + 69730 q^{19} + 300392 q^{20} + 247700 q^{21} + 357976 q^{22} + 101110 q^{23} + 167915 q^{24} + 542284 q^{25} + 424864 q^{26} + 253084 q^{27} + 383052 q^{28} + 199976 q^{29} + 252626 q^{30} + 373036 q^{31} + 463950 q^{32} + 64236 q^{33} + 17022 q^{34} - 140956 q^{35} + 694529 q^{36} + 596106 q^{37} - 488694 q^{38} - 352018 q^{39} - 188562 q^{40} - 66678 q^{41} - 776155 q^{42} + 1037964 q^{43} - 1598298 q^{44} + 905532 q^{45} - 2460646 q^{46} - 1661168 q^{47} - 9341196 q^{48} - 1580054 q^{49} - 7616723 q^{50} - 4897564 q^{51} + 1045432 q^{52} - 1253454 q^{53} - 11836385 q^{54} - 1489932 q^{55} - 9175912 q^{56} + 1630346 q^{57} + 1640408 q^{58} - 1977256 q^{59} - 12723470 q^{60} - 2381646 q^{61} - 5329480 q^{62} + 320028 q^{63} + 10719545 q^{64} + 9219376 q^{65} - 13727472 q^{66} + 11455752 q^{67} - 6720409 q^{68} + 9814322 q^{69} - 6175334 q^{70} + 5467788 q^{71} + 19454169 q^{72} + 12652462 q^{73} + 3377814 q^{74} + 14522488 q^{75} - 4047624 q^{76} + 12951326 q^{77} - 8107404 q^{78} + 34366848 q^{79} + 19400408 q^{80} + 24384080 q^{81} - 16612272 q^{82} + 27856008 q^{83} + 18267309 q^{84} + 8672002 q^{85} - 34325926 q^{86} + 41013236 q^{87} + 36623538 q^{88} + 14703638 q^{89} + 20243114 q^{90} + 11783202 q^{91} + 4020484 q^{92} + 35185978 q^{93} - 1557345 q^{94} + 14274548 q^{95} - 56487828 q^{96} + 51072842 q^{97} - 22140217 q^{98} + 55509018 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} - 1579 x^{14} + 411 x^{13} + 976985 x^{12} + 344128 x^{11} - 300557483 x^{10} - 241372726 x^{9} + 48208527759 x^{8} + 51058288103 x^{7} + \cdots - 870476371764660 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 197 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 51\!\cdots\!79 \nu^{15} + \cdots + 46\!\cdots\!44 ) / 71\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!55 \nu^{15} + \cdots - 12\!\cdots\!40 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70\!\cdots\!13 \nu^{15} + \cdots + 53\!\cdots\!92 ) / 22\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 66\!\cdots\!65 \nu^{15} + \cdots + 84\!\cdots\!04 ) / 14\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 66\!\cdots\!39 \nu^{15} + \cdots - 42\!\cdots\!24 ) / 13\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!89 \nu^{15} + \cdots + 25\!\cdots\!72 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 79\!\cdots\!49 \nu^{15} + \cdots - 69\!\cdots\!56 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!87 \nu^{15} + \cdots - 12\!\cdots\!40 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91\!\cdots\!85 \nu^{15} + \cdots + 89\!\cdots\!60 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!37 \nu^{15} + \cdots + 20\!\cdots\!40 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 62\!\cdots\!97 \nu^{15} + \cdots - 46\!\cdots\!68 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 82\!\cdots\!15 \nu^{15} + \cdots - 64\!\cdots\!36 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 42\!\cdots\!09 \nu^{15} + \cdots - 34\!\cdots\!12 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 197 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 343 \beta _1 + 194 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{15} - 11 \beta_{14} + 8 \beta_{13} - 9 \beta_{12} + 13 \beta_{11} - \beta_{10} - 10 \beta_{9} + 7 \beta_{8} + 16 \beta_{7} - 11 \beta_{6} + 5 \beta_{5} + 11 \beta_{4} - 191 \beta_{3} + 446 \beta_{2} + 798 \beta _1 + 67384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 137 \beta_{15} - 7 \beta_{14} + 204 \beta_{13} - 610 \beta_{12} - 551 \beta_{11} - 1141 \beta_{10} - 117 \beta_{9} + 100 \beta_{8} - 1167 \beta_{7} - 112 \beta_{6} - 1168 \beta_{5} - 861 \beta_{4} - 828 \beta_{3} + \cdots + 153351 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3675 \beta_{15} - 8233 \beta_{14} + 4652 \beta_{13} - 7867 \beta_{12} + 6606 \beta_{11} - 740 \beta_{10} - 6653 \beta_{9} + 4629 \beta_{8} + 9681 \beta_{7} - 6837 \beta_{6} + 2481 \beta_{5} + \cdots + 26592616 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 95320 \beta_{15} - 24243 \beta_{14} + 154338 \beta_{13} - 307700 \beta_{12} - 247795 \beta_{11} - 547875 \beta_{10} - 84312 \beta_{9} + 99747 \beta_{8} - 544562 \beta_{7} - 92024 \beta_{6} + \cdots + 95462952 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2198038 \beta_{15} - 4922180 \beta_{14} + 2240982 \beta_{13} - 5037271 \beta_{12} + 2852707 \beta_{11} - 488760 \beta_{10} - 3353110 \beta_{9} + 2630954 \beta_{8} + \cdots + 11243083195 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 48465264 \beta_{15} - 24263747 \beta_{14} + 83973954 \beta_{13} - 150308524 \beta_{12} - 106982876 \beta_{11} - 252146121 \beta_{10} - 47179838 \beta_{9} + \cdots + 54657697539 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1114663319 \beta_{15} - 2713658816 \beta_{14} + 1032793818 \beta_{13} - 2873204490 \beta_{12} + 1187719515 \beta_{11} - 313806502 \beta_{10} + \cdots + 4935099838229 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21530818306 \beta_{15} - 17938017941 \beta_{14} + 40192188782 \beta_{13} - 73432453466 \beta_{12} - 45804933478 \beta_{11} - 114835351620 \beta_{10} + \cdots + 29927939935759 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 516738472708 \beta_{15} - 1437448896495 \beta_{14} + 466925765448 \beta_{13} - 1550826132063 \beta_{12} + 487178731102 \beta_{11} + \cdots + 22\!\cdots\!39 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8748313500683 \beta_{15} - 11593830754080 \beta_{14} + 18014994452058 \beta_{13} - 36099132564578 \beta_{12} - 19610882527088 \beta_{11} + \cdots + 15\!\cdots\!53 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 225132991337410 \beta_{15} - 744478986575148 \beta_{14} + 207689132822540 \beta_{13} - 812254199967303 \beta_{12} + 197733087895086 \beta_{11} + \cdots + 10\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 32\!\cdots\!93 \beta_{15} + \cdots + 83\!\cdots\!47 \) 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
22.1645
21.3441
16.8796
14.0126
12.0759
5.93023
4.73488
−0.948867
−1.15943
−2.06656
−9.90793
−10.3842
−14.0272
−16.6961
−19.7958
−21.1558
−21.1645 −29.5695 319.936 −285.628 625.824 689.565 −4062.24 −1312.64 6045.17
1.2 −20.3441 −40.9060 285.883 529.958 832.197 −226.949 −3212.00 −513.699 −10781.5
1.3 −15.8796 56.7286 124.163 377.285 −900.829 1657.42 60.9345 1031.13 −5991.15
1.4 −13.0126 −12.9197 41.3274 143.808 168.119 −1101.53 1127.83 −2020.08 −1871.32
1.5 −11.0759 54.8743 −5.32547 −526.586 −607.780 703.033 1476.69 824.193 5832.39
1.6 −4.93023 −23.0639 −103.693 −510.202 113.710 −930.373 1142.30 −1655.06 2515.42
1.7 −3.73488 90.8992 −114.051 −73.0157 −339.498 −138.398 904.031 6075.67 272.705
1.8 1.94887 12.3682 −124.202 380.678 24.1039 274.575 −491.508 −2034.03 741.892
1.9 2.15943 −69.5079 −123.337 −36.4103 −150.097 −1189.62 −542.744 2644.35 −78.6256
1.10 3.06656 −33.3484 −118.596 −215.229 −102.265 462.391 −756.203 −1074.89 −660.014
1.11 10.9079 81.4817 −9.01712 518.214 888.797 −142.434 −1494.57 4452.27 5652.64
1.12 11.3842 −85.1792 1.59931 −234.652 −969.694 761.301 −1438.97 5068.49 −2671.32
1.13 15.0272 67.4810 97.8164 −225.306 1014.05 1201.42 −453.575 2366.69 −3385.71
1.14 17.6961 0.554749 185.152 268.454 9.81689 1091.43 1011.36 −2186.69 4750.58
1.15 20.7958 53.0514 304.467 44.7319 1103.25 −563.442 3669.77 627.455 930.238
1.16 22.1558 −82.9447 362.877 287.899 −1837.70 −688.379 5203.88 4692.83 6378.63
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(47\) \(1\)

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 47.8.a.b 16
3.b odd 2 1 423.8.a.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.8.a.b 16 1.a even 1 1 trivial
423.8.a.f 16 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 15 T_{2}^{15} - 1474 T_{2}^{14} + 21240 T_{2}^{13} + 840004 T_{2}^{12} - 11528253 T_{2}^{11} - 233749093 T_{2}^{10} + 3015944674 T_{2}^{9} + 33047268840 T_{2}^{8} + \cdots - 35\!\cdots\!72 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(47))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 15 T^{15} + \cdots - 35\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( T^{16} - 40 T^{15} + \cdots + 33\!\cdots\!68 \) Copy content Toggle raw display
$5$ \( T^{16} - 444 T^{15} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} - 1860 T^{15} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{16} - 4776 T^{15} + \cdots - 51\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{16} - 16074 T^{15} + \cdots - 20\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{16} - 67106 T^{15} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{16} - 69730 T^{15} + \cdots - 68\!\cdots\!80 \) Copy content Toggle raw display
$23$ \( T^{16} - 101110 T^{15} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} - 199976 T^{15} + \cdots - 53\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{16} - 373036 T^{15} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{16} - 596106 T^{15} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{16} + 66678 T^{15} + \cdots - 78\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{16} - 1037964 T^{15} + \cdots + 15\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( (T + 103823)^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 1253454 T^{15} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + 1977256 T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + 2381646 T^{15} + \cdots - 42\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( T^{16} - 11455752 T^{15} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{16} - 5467788 T^{15} + \cdots - 41\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( T^{16} - 12652462 T^{15} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} - 34366848 T^{15} + \cdots - 32\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{16} - 27856008 T^{15} + \cdots + 20\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{16} - 14703638 T^{15} + \cdots + 97\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{16} - 51072842 T^{15} + \cdots - 52\!\cdots\!72 \) Copy content Toggle raw display
show more
show less