Properties

Label 103.8.a.b
Level $103$
Weight $8$
Character orbit 103.a
Self dual yes
Analytic conductor $32.176$
Analytic rank $0$
Dimension $32$
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] = [103,8,Mod(1,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, 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("103.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(32.1756576249\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 39 q^{2} + 94 q^{3} + 2111 q^{4} + 945 q^{5} + 271 q^{6} + 1226 q^{7} + 8052 q^{8} + 31566 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 39 q^{2} + 94 q^{3} + 2111 q^{4} + 945 q^{5} + 271 q^{6} + 1226 q^{7} + 8052 q^{8} + 31566 q^{9} + 2638 q^{10} + 8169 q^{11} + 12039 q^{12} + 14171 q^{13} + 10695 q^{14} + 10759 q^{15} + 143515 q^{16} + 120237 q^{17} + 102288 q^{18} + 45057 q^{19} + 140613 q^{20} + 115762 q^{21} + 242049 q^{22} + 203820 q^{23} + 74610 q^{24} + 686055 q^{25} - 127251 q^{26} + 132508 q^{27} - 769784 q^{28} + 310416 q^{29} - 576370 q^{30} - 101176 q^{31} + 1585740 q^{32} + 715539 q^{33} + 760438 q^{34} + 1016613 q^{35} + 4351624 q^{36} + 1548134 q^{37} + 2850210 q^{38} + 2195909 q^{39} + 4470331 q^{40} + 2031714 q^{41} + 6479435 q^{42} + 1565984 q^{43} + 4069392 q^{44} + 5790546 q^{45} + 7892113 q^{46} + 4034469 q^{47} + 7138233 q^{48} + 7627000 q^{49} + 5213658 q^{50} + 2565519 q^{51} + 8129871 q^{52} + 4799985 q^{53} + 9982795 q^{54} + 3583642 q^{55} + 4552923 q^{56} + 8733123 q^{57} + 2903090 q^{58} + 1725891 q^{59} + 7361351 q^{60} + 4299641 q^{61} + 6367509 q^{62} + 4202312 q^{63} + 13772046 q^{64} + 13614927 q^{65} + 9746103 q^{66} + 2271650 q^{67} + 17935092 q^{68} - 364650 q^{69} + 454144 q^{70} + 11201481 q^{71} + 9438726 q^{72} + 9409961 q^{73} + 8665539 q^{74} + 939165 q^{75} + 3560132 q^{76} + 14167347 q^{77} - 7436704 q^{78} + 1165551 q^{79} + 8385198 q^{80} + 29742120 q^{81} - 35007935 q^{82} + 2497407 q^{83} - 34226991 q^{84} + 607507 q^{85} - 19425795 q^{86} - 7760456 q^{87} + 29754765 q^{88} + 6604398 q^{89} - 49873976 q^{90} - 14775477 q^{91} + 27875580 q^{92} + 7773096 q^{93} - 36766264 q^{94} + 19069674 q^{95} - 40704833 q^{96} + 7841444 q^{97} - 21724491 q^{98} - 9334584 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −20.7827 33.4298 303.921 −243.724 −694.762 −1694.08 −3656.12 −1069.45 5065.25
1.2 −20.6574 46.4673 298.728 75.4363 −959.894 112.969 −3526.80 −27.7873 −1558.32
1.3 −18.2028 −40.9590 203.341 −132.100 745.566 −340.437 −1371.41 −509.364 2404.59
1.4 −18.1848 −92.4197 202.686 −198.302 1680.63 538.310 −1358.14 6354.40 3606.08
1.5 −17.3474 87.7597 172.931 467.264 −1522.40 −1002.57 −779.432 5514.76 −8105.79
1.6 −13.8714 24.6737 64.4145 −100.756 −342.258 1781.02 882.017 −1578.21 1397.62
1.7 −13.3268 −73.6815 49.6030 444.430 981.937 751.237 1044.78 3241.96 −5922.82
1.8 −12.8065 16.6711 36.0070 −184.175 −213.499 −819.277 1178.11 −1909.07 2358.64
1.9 −11.2327 −47.2601 −1.82614 33.9435 530.860 1327.58 1458.30 46.5205 −381.278
1.10 −8.28916 10.1290 −59.2899 336.004 −83.9610 −1536.90 1552.48 −2084.40 −2785.19
1.11 −7.05130 −68.8783 −78.2791 300.510 485.682 −499.751 1454.54 2557.23 −2118.98
1.12 −6.62833 67.4256 −84.0652 484.648 −446.919 1346.24 1405.64 2359.21 −3212.40
1.13 −5.95705 72.0720 −92.5135 16.6373 −429.337 408.502 1313.61 3007.37 −99.1094
1.14 −4.47080 5.61726 −108.012 −478.616 −25.1136 370.899 1055.16 −2155.45 2139.80
1.15 0.682485 33.7116 −127.534 −330.528 23.0076 −1374.53 −174.398 −1050.53 −225.580
1.16 2.60267 76.1686 −121.226 −469.972 198.242 1258.56 −648.654 3614.66 −1223.18
1.17 3.05039 −16.1277 −118.695 221.532 −49.1956 531.922 −752.515 −1926.90 675.757
1.18 5.04695 −31.1930 −102.528 523.347 −157.430 1709.09 −1163.46 −1213.99 2641.31
1.19 5.56903 −39.5094 −96.9859 −140.033 −220.029 −1518.37 −1252.95 −626.003 −779.849
1.20 6.13722 −29.4700 −90.3346 −100.733 −180.863 −488.558 −1339.97 −1318.52 −618.221
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(103\) \(1\)

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 103.8.a.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.8.a.b 32 1.a even 1 1 trivial