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Properties

Label 309.8.a.d
Level $309$
Weight $8$
Character orbit 309.a
Self dual yes
Analytic conductor $96.527$
Analytic rank $0$
Dimension $34$
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: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 8 q^{2} + 918 q^{3} + 2638 q^{4} + 305 q^{5} + 216 q^{6} + 2890 q^{7} + 1956 q^{8} + 24786 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 8 q^{2} + 918 q^{3} + 2638 q^{4} + 305 q^{5} + 216 q^{6} + 2890 q^{7} + 1956 q^{8} + 24786 q^{9} + 9557 q^{10} + 15970 q^{11} + 71226 q^{12} + 32451 q^{13} + 15296 q^{14} + 8235 q^{15} + 218014 q^{16} + 32628 q^{17} + 5832 q^{18} + 151848 q^{19} + 28135 q^{20} + 78030 q^{21} + 150965 q^{22} + 42883 q^{23} + 52812 q^{24} + 775955 q^{25} - 233740 q^{26} + 669222 q^{27} + 413883 q^{28} + 34480 q^{29} + 258039 q^{30} + 686025 q^{31} + 321721 q^{32} + 431190 q^{33} + 508474 q^{34} + 1351922 q^{35} + 1923102 q^{36} + 827942 q^{37} + 1540130 q^{38} + 876177 q^{39} + 4649673 q^{40} + 2123822 q^{41} + 412992 q^{42} + 2212547 q^{43} + 6856511 q^{44} + 222345 q^{45} + 1386813 q^{46} + 2516992 q^{47} + 5886378 q^{48} + 8771360 q^{49} + 3911701 q^{50} + 880956 q^{51} + 7599619 q^{52} + 2791918 q^{53} + 157464 q^{54} + 4993770 q^{55} + 7672809 q^{56} + 4099896 q^{57} + 5462829 q^{58} + 7285213 q^{59} + 759645 q^{60} + 6114699 q^{61} + 8417866 q^{62} + 2106810 q^{63} + 23866294 q^{64} + 2864765 q^{65} + 4076055 q^{66} + 6272423 q^{67} + 349155 q^{68} + 1157841 q^{69} + 3093632 q^{70} - 832790 q^{71} + 1425924 q^{72} + 11874710 q^{73} - 4539197 q^{74} + 20950785 q^{75} + 19489637 q^{76} + 580372 q^{77} - 6310980 q^{78} + 9977290 q^{79} - 38868623 q^{80} + 18068994 q^{81} - 35370837 q^{82} + 10189957 q^{83} + 11174841 q^{84} - 870648 q^{85} - 32101135 q^{86} + 930960 q^{87} + 4422897 q^{88} - 13662876 q^{89} + 6967053 q^{90} + 37580658 q^{91} - 33200830 q^{92} + 18522675 q^{93} - 7756936 q^{94} - 4522798 q^{95} + 8686467 q^{96} + 15501279 q^{97} + 4733343 q^{98} + 11642130 q^{99}+O(q^{100}) \) 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 −22.4259 27.0000 374.921 −446.031 −605.500 953.410 −5537.44 729.000 10002.7
1.2 −21.2797 27.0000 324.827 −335.965 −574.553 −1222.61 −4188.43 729.000 7149.25
1.3 −19.6818 27.0000 259.375 −37.2187 −531.410 −796.227 −2585.70 729.000 732.533
1.4 −19.6652 27.0000 258.721 418.429 −530.961 1513.71 −2570.66 729.000 −8228.49
1.5 −17.9196 27.0000 193.112 −202.120 −483.829 643.609 −1166.78 729.000 3621.91
1.6 −17.6394 27.0000 183.150 398.145 −476.265 −733.323 −972.818 729.000 −7023.05
1.7 −15.2902 27.0000 105.790 147.696 −412.835 1200.69 339.600 729.000 −2258.30
1.8 −14.5055 27.0000 82.4109 97.3383 −391.650 −918.129 661.295 729.000 −1411.94
1.9 −14.3189 27.0000 77.0320 −92.7912 −386.611 −1596.91 729.807 729.000 1328.67
1.10 −11.1790 27.0000 −3.03087 497.265 −301.832 1048.06 1464.79 729.000 −5558.91
1.11 −8.02886 27.0000 −63.5374 −400.814 −216.779 322.519 1537.83 729.000 3218.08
1.12 −7.62441 27.0000 −69.8683 −368.066 −205.859 1622.38 1508.63 729.000 2806.28
1.13 −7.58613 27.0000 −70.4506 376.559 −204.826 24.8384 1505.47 729.000 −2856.63
1.14 −6.78874 27.0000 −81.9130 −107.172 −183.296 125.382 1425.05 729.000 727.566
1.15 −5.35217 27.0000 −99.3543 −60.0238 −144.508 −1370.71 1216.84 729.000 321.257
1.16 −3.68654 27.0000 −114.409 −118.681 −99.5367 −103.117 893.653 729.000 437.523
1.17 0.169634 27.0000 −127.971 394.163 4.58012 −117.674 −43.4215 729.000 66.8636
1.18 0.822873 27.0000 −127.323 −32.1038 22.2176 1535.51 −210.098 729.000 −26.4174
1.19 3.23968 27.0000 −117.504 412.051 87.4714 578.150 −795.356 729.000 1334.91
1.20 6.00064 27.0000 −91.9923 −345.839 162.017 −518.628 −1320.09 729.000 −2075.25
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
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.d ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.8.a.d ✓ 34 1.a even 1 1 trivial