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Properties

Label 103.16.a.a
Level $103$
Weight $16$
Character orbit 103.a
Self dual yes
Analytic conductor $146.974$
Analytic rank $1$
Dimension $61$
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] = [103,16,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, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 16 \)
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: \(146.974310253\)
Analytic rank: \(1\)
Dimension: \(61\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 61 q - 857 q^{2} - 3932 q^{3} + 952767 q^{4} - 439140 q^{5} + 743647 q^{6} - 3190353 q^{7} - 33093708 q^{8} + 233430375 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 61 q - 857 q^{2} - 3932 q^{3} + 952767 q^{4} - 439140 q^{5} + 743647 q^{6} - 3190353 q^{7} - 33093708 q^{8} + 233430375 q^{9} - 63234482 q^{10} - 103748368 q^{11} - 331520313 q^{12} - 313165161 q^{13} - 755555817 q^{14} - 674398526 q^{15} + 14512908187 q^{16} - 12083282787 q^{17} - 13168798944 q^{18} - 5361319119 q^{19} - 9305646747 q^{20} - 16461308876 q^{21} - 28941029471 q^{22} - 66168004047 q^{23} + 7802762994 q^{24} + 303234371695 q^{25} + 182749185334 q^{26} - 156925014338 q^{27} - 664627255237 q^{28} - 645743643657 q^{29} - 1069618618594 q^{30} - 122874478996 q^{31} - 269627394831 q^{32} + 46150978058 q^{33} + 969116824174 q^{34} - 478522677982 q^{35} + 4223214984684 q^{36} + 819999551780 q^{37} - 1018697698883 q^{38} - 991652931072 q^{39} - 6174778893965 q^{40} - 7325440107283 q^{41} - 9956139869801 q^{42} - 8612850695736 q^{43} - 13352142895690 q^{44} - 15559896379366 q^{45} - 32065613137770 q^{46} - 15984352430276 q^{47} - 41506324107117 q^{48} + 17231550044622 q^{49} - 33888000032918 q^{50} - 27907219770060 q^{51} - 57828879269473 q^{52} - 37355418884312 q^{53} - 49810074049935 q^{54} - 46090202313238 q^{55} - 31772217542854 q^{56} - 57782774436846 q^{57} - 36740035491030 q^{58} - 41266858463091 q^{59} + 44989673695855 q^{60} - 45509082837011 q^{61} + 9938607229349 q^{62} + 47160755658129 q^{63} + 422835893804094 q^{64} - 177125638146712 q^{65} + 310487514411959 q^{66} + 91080256620398 q^{67} - 256019865760105 q^{68} + 80891639918642 q^{69} + 384378270951236 q^{70} - 46207729769832 q^{71} - 56835727444502 q^{72} - 114864967341268 q^{73} + 673296051484809 q^{74} + 251412324545408 q^{75} - 92392796537324 q^{76} - 426424980478584 q^{77} + 860430952610076 q^{78} + 27257492827013 q^{79} + 12\!\cdots\!86 q^{80}+ \cdots - 24\!\cdots\!92 q^{99}+O(q^{100}) \) 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 −357.874 −3198.88 95305.6 241789. 1.14480e6 −4.15427e6 −2.23805e7 −4.11605e6 −8.65299e7
1.2 −351.800 2818.71 90995.4 154555. −991623. 3.22909e6 −2.04844e7 −6.40377e6 −5.43723e7
1.3 −345.892 −3498.66 86873.6 −122746. 1.21016e6 628594. −1.87147e7 −2.10830e6 4.24568e7
1.4 −328.715 −7253.99 75285.3 −224372. 2.38449e6 −2.05636e6 −1.39760e7 3.82714e7 7.37544e7
1.5 −320.560 606.943 69990.8 226189. −194562. −182632. −1.19322e7 −1.39805e7 −7.25073e7
1.6 −301.379 415.803 58061.0 −184105. −125314. −3.14221e6 −7.62277e6 −1.41760e7 5.54854e7
1.7 −297.870 491.993 55958.8 102232. −146550. 577839. −6.90785e6 −1.41068e7 −3.04520e7
1.8 −295.329 6750.06 54451.5 −214413. −1.99349e6 −86989.0 −6.40376e6 3.12145e7 6.33225e7
1.9 −292.392 6570.72 52725.0 342215. −1.92122e6 −128251. −5.83527e6 2.88254e7 −1.00061e8
1.10 −291.585 2903.99 52253.8 −298360. −846760. 2.81699e6 −5.68177e6 −5.91575e6 8.69974e7
1.11 −282.619 −4569.96 47105.8 130821. 1.29156e6 −978086. −4.05213e6 6.53567e6 −3.69726e7
1.12 −247.906 5764.71 28689.3 −76957.0 −1.42911e6 1.78874e6 1.01114e6 1.88830e7 1.90781e7
1.13 −247.435 −6588.35 28455.9 −189728. 1.63019e6 2.77055e6 1.06696e6 2.90574e7 4.69453e7
1.14 −225.744 −1444.65 18192.2 −79240.1 326120. −2.45038e6 3.29040e6 −1.22619e7 1.78879e7
1.15 −207.496 436.660 10286.5 −62369.8 −90605.1 1.36658e6 4.66482e6 −1.41582e7 1.29415e7
1.16 −201.213 −2962.46 7718.81 −305026. 596086. −813455. 5.04023e6 −5.57275e6 6.13753e7
1.17 −196.400 −5985.44 5804.96 −35779.2 1.17554e6 739982. 5.29554e6 2.14766e7 7.02704e6
1.18 −186.305 4057.35 1941.56 160280. −755905. −1.44011e6 5.74312e6 2.11318e6 −2.98610e7
1.19 −182.639 −3002.52 589.102 37324.2 548379. 4.15954e6 5.87713e6 −5.33376e6 −6.81686e6
1.20 −181.045 −608.814 9.40830 319508. 110223. −3.84703e6 5.93079e6 −1.39783e7 −5.78455e7
See all 61 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.61
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 103.16.a.a ✓ 61
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.16.a.a ✓ 61 1.a even 1 1 trivial