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Properties

Label 1931.2.a.b
Level $1931$
Weight $2$
Character orbit 1931.a
Self dual yes
Analytic conductor $15.419$
Analytic rank $0$
Dimension $101$
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] = [1931,2,Mod(1,1931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1931, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1931.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1931 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1931.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: \(15.4191126303\)
Analytic rank: \(0\)
Dimension: \(101\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 101 q + 5 q^{2} + 9 q^{3} + 131 q^{4} + 25 q^{5} + 15 q^{6} + 20 q^{7} + 12 q^{8} + 138 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 101 q + 5 q^{2} + 9 q^{3} + 131 q^{4} + 25 q^{5} + 15 q^{6} + 20 q^{7} + 12 q^{8} + 138 q^{9} + 14 q^{10} + 29 q^{11} + 15 q^{12} + 41 q^{13} + 4 q^{14} + 22 q^{15} + 187 q^{16} + 9 q^{17} + 11 q^{18} + 34 q^{19} + 42 q^{20} + 72 q^{21} + 17 q^{22} + 19 q^{23} + 33 q^{24} + 172 q^{25} + 28 q^{26} + 36 q^{27} + 47 q^{28} + 68 q^{29} + q^{30} + 76 q^{31} + 9 q^{32} + 8 q^{33} + 50 q^{34} + 7 q^{35} + 198 q^{36} + 141 q^{37} - 13 q^{38} + 42 q^{39} + 32 q^{40} + 35 q^{41} - 46 q^{42} + 45 q^{43} + 67 q^{44} + 106 q^{45} + 86 q^{46} - 5 q^{47} + 7 q^{48} + 203 q^{49} + 4 q^{50} + 5 q^{51} + 30 q^{52} + 47 q^{53} + 20 q^{54} + 11 q^{55} - 4 q^{56} + 19 q^{57} + 92 q^{58} + 30 q^{59} + 35 q^{60} + 153 q^{61} - 20 q^{62} + 19 q^{63} + 276 q^{64} - 7 q^{65} - 18 q^{66} + 39 q^{67} - 7 q^{68} + 55 q^{69} - 8 q^{70} + 47 q^{71} - 4 q^{72} + 86 q^{73} + 9 q^{74} + 11 q^{75} + 50 q^{76} + 15 q^{77} - 15 q^{78} + 89 q^{79} + 26 q^{80} + 201 q^{81} - 9 q^{82} - 15 q^{83} + 60 q^{84} + 239 q^{85} + 13 q^{86} - 22 q^{87} + 5 q^{88} + 14 q^{89} - 85 q^{90} + 58 q^{91} - 4 q^{92} + 86 q^{93} + 21 q^{94} - 10 q^{95} + 6 q^{96} + 46 q^{97} - 68 q^{98} + 100 q^{99}+O(q^{100}) \) 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 −2.82191 −2.51033 5.96320 −0.272521 7.08394 0.922658 −11.1838 3.30177 0.769032
1.2 −2.80220 3.00833 5.85234 3.47491 −8.42994 3.27757 −10.7951 6.05002 −9.73740
1.3 −2.75459 1.21931 5.58779 −2.23120 −3.35871 1.87951 −9.88291 −1.51327 6.14605
1.4 −2.74767 −1.13537 5.54967 −0.785684 3.11963 −5.06907 −9.75331 −1.71092 2.15880
1.5 −2.72890 1.92320 5.44689 0.976915 −5.24822 −4.25621 −9.40622 0.698698 −2.66590
1.6 −2.61328 −1.80746 4.82925 1.71424 4.72340 4.84622 −7.39365 0.266904 −4.47979
1.7 −2.58473 −3.05888 4.68083 −3.99775 7.90638 −0.257391 −6.92922 6.35675 10.3331
1.8 −2.58262 0.418355 4.66991 0.958842 −1.08045 3.24270 −6.89534 −2.82498 −2.47632
1.9 −2.52613 0.443336 4.38134 −3.54756 −1.11993 −1.45074 −6.01558 −2.80345 8.96161
1.10 −2.50752 −3.17004 4.28766 4.19591 7.94895 −2.58953 −5.73635 7.04918 −10.5213
1.11 −2.43657 2.88124 3.93686 −3.58348 −7.02035 4.97042 −4.71929 5.30157 8.73140
1.12 −2.37672 −1.17301 3.64881 −2.79721 2.78793 2.65701 −3.91875 −1.62404 6.64819
1.13 −2.30254 2.88733 3.30168 −0.276583 −6.64819 2.20621 −2.99717 5.33668 0.636843
1.14 −2.29286 −2.68651 3.25720 2.47279 6.15979 −1.89337 −2.88259 4.21734 −5.66976
1.15 −2.25413 −0.147613 3.08110 0.661141 0.332740 −2.11167 −2.43693 −2.97821 −1.49030
1.16 −2.23412 2.63573 2.99131 3.85671 −5.88855 −4.33741 −2.21471 3.94709 −8.61636
1.17 −2.11521 −1.74079 2.47410 3.90511 3.68213 4.66819 −1.00281 0.0303472 −8.26012
1.18 −2.08811 1.86927 2.36020 3.43441 −3.90323 3.28978 −0.752132 0.494160 −7.17142
1.19 −2.00825 0.545518 2.03308 2.04148 −1.09554 −1.36166 −0.0664267 −2.70241 −4.09980
1.20 −1.91212 1.11256 1.65620 −2.93704 −2.12735 −1.01726 0.657391 −1.76220 5.61596
See next 80 embeddings (of 101 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.101
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(1931\) \(-1\)

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 1931.2.a.b ✓ 101
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1931.2.a.b ✓ 101 1.a even 1 1 trivial