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Properties

Label 309.10.a.d
Level $309$
Weight $10$
Character orbit 309.a
Self dual yes
Analytic conductor $159.146$
Analytic rank $0$
Dimension $42$
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] = [309,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(159.146073375\)
Analytic rank: \(0\)
Dimension: \(42\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 14 q^{2} + 3402 q^{3} + 12118 q^{4} + 2672 q^{5} + 1134 q^{6} + 14901 q^{7} + 24654 q^{8} + 275562 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 14 q^{2} + 3402 q^{3} + 12118 q^{4} + 2672 q^{5} + 1134 q^{6} + 14901 q^{7} + 24654 q^{8} + 275562 q^{9} + 130125 q^{10} + 95073 q^{11} + 981558 q^{12} + 439613 q^{13} + 456966 q^{14} + 216432 q^{15} + 3809750 q^{16} + 1138535 q^{17} + 91854 q^{18} + 2787090 q^{19} + 252055 q^{20} + 1206981 q^{21} + 2992161 q^{22} + 3595810 q^{23} + 1996974 q^{24} + 25809942 q^{25} + 9183694 q^{26} + 22320522 q^{27} + 15171971 q^{28} + 11821494 q^{29} + 10540125 q^{30} + 13256239 q^{31} - 28227783 q^{32} + 7700913 q^{33} - 3969066 q^{34} + 405418 q^{35} + 79506198 q^{36} + 25303145 q^{37} - 25090464 q^{38} + 35608653 q^{39} - 93315 q^{40} + 34710165 q^{41} + 37014246 q^{42} + 113885488 q^{43} - 9473605 q^{44} + 17530992 q^{45} + 132708981 q^{46} + 65555466 q^{47} + 308589750 q^{48} + 412449235 q^{49} + 223625919 q^{50} + 92221335 q^{51} + 467165531 q^{52} + 25767818 q^{53} + 7440174 q^{54} + 313092854 q^{55} + 941522123 q^{56} + 225754290 q^{57} + 415932141 q^{58} + 588922973 q^{59} + 20416455 q^{60} + 569746681 q^{61} + 471730490 q^{62} + 97765461 q^{63} + 1070306030 q^{64} + 607680520 q^{65} + 242365041 q^{66} + 501695028 q^{67} + 1968868559 q^{68} + 291260610 q^{69} + 1754432504 q^{70} + 310492124 q^{71} + 161754894 q^{72} + 1154078104 q^{73} + 1516731979 q^{74} + 2090605302 q^{75} + 2086387217 q^{76} + 924426278 q^{77} + 743879214 q^{78} + 2614635597 q^{79} + 4761602843 q^{80} + 1807962282 q^{81} + 4834781383 q^{82} + 2321624851 q^{83} + 1228929651 q^{84} + 1774895066 q^{85} + 1165523049 q^{86} + 957541014 q^{87} + 2548505065 q^{88} + 1793953327 q^{89} + 853750125 q^{90} + 795442770 q^{91} - 209442180 q^{92} + 1073755359 q^{93} + 1859549440 q^{94} + 2086137408 q^{95} - 2286450423 q^{96} + 2320198146 q^{97} - 3219622683 q^{98} + 623773953 q^{99}+O(q^{100}) \) 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 −44.2316 81.0000 1444.43 1357.02 −3582.76 −10096.2 −41243.0 6561.00 −60023.1
1.2 −44.0342 81.0000 1427.01 757.363 −3566.77 −2702.81 −40291.6 6561.00 −33349.8
1.3 −42.8596 81.0000 1324.95 −413.559 −3471.63 10116.4 −34842.6 6561.00 17725.0
1.4 −39.8141 81.0000 1073.16 −1565.87 −3224.94 −8259.90 −22342.1 6561.00 62343.8
1.5 −36.5384 81.0000 823.053 −2664.33 −2959.61 10313.4 −11365.4 6561.00 97350.3
1.6 −34.9243 81.0000 707.705 306.394 −2828.87 −10991.2 −6834.85 6561.00 −10700.6
1.7 −34.7586 81.0000 696.162 1827.26 −2815.45 −215.319 −6401.20 6561.00 −63512.9
1.8 −28.9328 81.0000 325.105 1254.29 −2343.55 10222.3 5407.38 6561.00 −36290.1
1.9 −28.5533 81.0000 303.292 −1415.72 −2312.82 2845.23 5959.30 6561.00 40423.6
1.10 −28.3254 81.0000 290.329 1269.30 −2294.36 8901.19 6278.92 6561.00 −35953.4
1.11 −25.5895 81.0000 142.820 862.266 −2072.75 −4151.91 9447.11 6561.00 −22064.9
1.12 −24.5310 81.0000 89.7693 −973.830 −1987.01 1226.04 10357.7 6561.00 23889.0
1.13 −24.0203 81.0000 64.9737 −1867.89 −1945.64 −1878.60 10737.7 6561.00 44867.2
1.14 −19.7041 81.0000 −123.749 −2357.06 −1596.03 7256.26 12526.9 6561.00 46443.7
1.15 −13.4143 81.0000 −332.057 −1061.97 −1086.56 −6035.88 11322.4 6561.00 14245.6
1.16 −10.2302 81.0000 −407.343 1998.38 −828.645 5506.13 9405.05 6561.00 −20443.8
1.17 −9.61107 81.0000 −419.627 152.571 −778.497 −4627.62 8953.93 6561.00 −1466.37
1.18 −9.23927 81.0000 −426.636 −2730.07 −748.381 −2823.68 8672.31 6561.00 25223.9
1.19 −7.07462 81.0000 −461.950 1702.83 −573.044 −7470.20 6890.32 6561.00 −12046.9
1.20 −7.00698 81.0000 −462.902 2625.58 −567.565 9659.93 6831.12 6561.00 −18397.4
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.42
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 309.10.a.d ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.10.a.d ✓ 42 1.a even 1 1 trivial