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Properties

Label 309.8.a.b
Level $309$
Weight $8$
Character orbit 309.a
Self dual yes
Analytic conductor $96.527$
Analytic rank $0$
Dimension $29$
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] = [309,8,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(96.5269728746\)
Analytic rank: \(0\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q + 10 q^{2} - 783 q^{3} + 1998 q^{4} + 55 q^{5} - 270 q^{6} + 146 q^{7} + 2970 q^{8} + 21141 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q + 10 q^{2} - 783 q^{3} + 1998 q^{4} + 55 q^{5} - 270 q^{6} + 146 q^{7} + 2970 q^{8} + 21141 q^{9} + 4573 q^{10} + 13308 q^{11} - 53946 q^{12} + 6655 q^{13} + 29170 q^{14} - 1485 q^{15} + 126006 q^{16} + 3116 q^{17} + 7290 q^{18} + 92220 q^{19} + 74639 q^{20} - 3942 q^{21} - 174495 q^{22} + 105073 q^{23} - 80190 q^{24} + 390636 q^{25} + 471683 q^{26} - 570807 q^{27} + 122226 q^{28} + 421136 q^{29} - 123471 q^{30} + 437871 q^{31} - 39490 q^{32} - 359316 q^{33} + 189706 q^{34} + 753290 q^{35} + 1456542 q^{36} - 1161026 q^{37} - 887313 q^{38} - 179685 q^{39} - 1411175 q^{40} + 690208 q^{41} - 787590 q^{42} - 502371 q^{43} - 1391335 q^{44} + 40095 q^{45} - 148718 q^{46} - 878146 q^{47} - 3402162 q^{48} + 2436893 q^{49} - 1352093 q^{50} - 84132 q^{51} - 1988293 q^{52} - 242294 q^{53} - 196830 q^{54} + 2325162 q^{55} + 2573918 q^{56} - 2489940 q^{57} - 1209463 q^{58} + 5488145 q^{59} - 2015253 q^{60} + 556491 q^{61} + 2413770 q^{62} + 106434 q^{63} + 5280038 q^{64} + 4572757 q^{65} + 4711365 q^{66} + 432035 q^{67} + 3121842 q^{68} - 2836971 q^{69} + 3568986 q^{70} + 280604 q^{71} + 2165130 q^{72} - 9214716 q^{73} + 6119753 q^{74} - 10547172 q^{75} + 7559953 q^{76} + 11123600 q^{77} - 12735441 q^{78} + 7870230 q^{79} + 1332171 q^{80} + 15411789 q^{81} - 24700702 q^{82} + 12547021 q^{83} - 3300102 q^{84} - 24312336 q^{85} - 10338725 q^{86} - 11370672 q^{87} - 39398705 q^{88} + 22583902 q^{89} + 3333717 q^{90} + 8464878 q^{91} + 28620210 q^{92} - 11822517 q^{93} + 43426200 q^{94} + 29793692 q^{95} + 1066230 q^{96} + 18898895 q^{97} + 73332280 q^{98} + 9701532 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −22.0316 −27.0000 357.390 −107.936 594.852 171.200 −5053.82 729.000 2377.99
1.2 −19.4012 −27.0000 248.408 467.977 523.834 958.100 −2336.07 729.000 −9079.34
1.3 −18.5520 −27.0000 216.176 −469.540 500.904 −1027.04 −1635.85 729.000 8710.91
1.4 −18.5445 −27.0000 215.898 279.027 500.701 964.944 −1630.02 729.000 −5174.40
1.5 −17.8235 −27.0000 189.678 230.276 481.235 −1038.37 −1099.31 729.000 −4104.33
1.6 −17.4648 −27.0000 177.019 −120.994 471.549 −770.229 −856.102 729.000 2113.14
1.7 −13.1892 −27.0000 45.9541 −234.397 356.107 330.671 1082.12 729.000 3091.51
1.8 −9.75758 −27.0000 −32.7896 523.070 263.455 −713.895 1568.92 729.000 −5103.90
1.9 −9.41770 −27.0000 −39.3069 −514.212 254.278 968.319 1575.65 729.000 4842.70
1.10 −9.03677 −27.0000 −46.3369 −186.982 243.993 −1267.47 1575.44 729.000 1689.71
1.11 −7.82333 −27.0000 −66.7956 −144.780 211.230 −214.361 1523.95 729.000 1132.66
1.12 −5.25833 −27.0000 −100.350 −273.516 141.975 723.955 1200.74 729.000 1438.24
1.13 −4.00729 −27.0000 −111.942 172.991 108.197 −1328.09 961.515 729.000 −693.223
1.14 −1.13146 −27.0000 −126.720 307.147 30.5494 1215.46 288.205 729.000 −347.524
1.15 −0.128779 −27.0000 −127.983 107.179 3.47703 90.3308 32.9653 729.000 −13.8024
1.16 0.402078 −27.0000 −127.838 −350.097 −10.8561 639.437 −102.867 729.000 −140.767
1.17 2.62012 −27.0000 −121.135 126.815 −70.7432 −226.111 −652.763 729.000 332.270
1.18 7.38636 −27.0000 −73.4417 69.6564 −199.432 1363.21 −1487.92 729.000 514.507
1.19 7.70678 −27.0000 −68.6056 −54.6907 −208.083 807.357 −1515.20 729.000 −421.489
1.20 10.1794 −27.0000 −24.3802 40.5392 −274.843 −1566.51 −1551.14 729.000 412.664
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(103\) \(1\)

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 309.8.a.b ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.8.a.b ✓ 29 1.a even 1 1 trivial