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Properties

Label 103.16.a.b
Level $103$
Weight $16$
Character orbit 103.a
Self dual yes
Analytic conductor $146.974$
Analytic rank $0$
Dimension $66$
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: \(0\)
Dimension: \(66\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 423 q^{2} + 9190 q^{3} + 1116607 q^{4} + 498360 q^{5} + 2423263 q^{6} + 3397991 q^{7} + 29820852 q^{8} + 353004600 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 423 q^{2} + 9190 q^{3} + 1116607 q^{4} + 498360 q^{5} + 2423263 q^{6} + 3397991 q^{7} + 29820852 q^{8} + 353004600 q^{9} - 3234482 q^{10} + 169072026 q^{11} + 313452231 q^{12} + 690811111 q^{13} + 509406231 q^{14} + 1375913974 q^{15} + 19881617307 q^{16} + 7612973517 q^{17} + 5197802016 q^{18} + 1789654793 q^{19} + 29094353253 q^{20} + 12356107780 q^{21} + 15957412513 q^{22} + 56405712045 q^{23} + 90359248626 q^{24} + 516857418570 q^{25} - 64210800531 q^{26} + 126530632414 q^{27} + 754686507176 q^{28} + 422641653615 q^{29} + 867124963310 q^{30} + 55148608036 q^{31} + 636916034604 q^{32} + 187772814138 q^{33} - 2840865820314 q^{34} + 880948726098 q^{35} + 2767232865256 q^{36} + 808966126180 q^{37} + 106764129186 q^{38} + 1575753089960 q^{39} + 100089635131 q^{40} + 2784639914577 q^{41} + 17919698865995 q^{42} + 3457297141074 q^{43} + 21819152697648 q^{44} + 21894528489366 q^{45} + 30381178081441 q^{46} + 15263569678140 q^{47} + 28400287341657 q^{48} + 69957624998945 q^{49} + 35582235370698 q^{50} + 26190040340208 q^{51} + 59036536240047 q^{52} + 54961738423104 q^{53} + 87543300758107 q^{54} + 9657901256182 q^{55} + 58059557004987 q^{56} + 88724869431330 q^{57} + 6823502620818 q^{58} + 4954961062581 q^{59} - 10243999797529 q^{60} - 7568142643595 q^{61} + 102930611243637 q^{62} + 90486305952809 q^{63} + 77979668138830 q^{64} + 193719625034712 q^{65} - 245386053084873 q^{66} + 21152819252064 q^{67} - 92276712110508 q^{68} - 167260316370522 q^{69} - 612453845355216 q^{70} + 54715234527708 q^{71} - 677797471884762 q^{72} + 948622986382 q^{73} - 132604588121805 q^{74} - 566385017880030 q^{75} - 600618564499196 q^{76} + 366017878851840 q^{77} - 10\!\cdots\!92 q^{78}+ \cdots + 419253554901342 q^{99}+O(q^{100}) \) 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 −353.998 4998.60 92546.3 −58013.4 −1.76949e6 −372847. −2.11614e7 1.06371e7 2.05366e7
1.2 −336.648 957.283 80563.7 −287999. −322267. −1.33792e6 −1.60903e7 −1.34325e7 9.69543e7
1.3 −336.326 −7242.57 80346.9 309128. 2.43586e6 4.11164e6 −1.60020e7 3.81058e7 −1.03968e8
1.4 −326.847 −3954.67 74060.7 −60594.1 1.29257e6 1.06875e6 −1.34964e7 1.29051e6 1.98050e7
1.5 −316.785 5949.24 67584.8 131144. −1.88463e6 −3.72132e6 −1.10294e7 2.10445e7 −4.15444e7
1.6 −306.739 −1991.76 61320.6 −134243. 610949. 4.24405e6 −8.75819e6 −1.03818e7 4.11775e7
1.7 −297.936 5840.61 55997.9 88085.1 −1.74013e6 3.38615e6 −6.92103e6 1.97638e7 −2.62437e7
1.8 −296.244 2353.65 54992.5 −52232.6 −697255. −2.09345e6 −6.58389e6 −8.80924e6 1.54736e7
1.9 −284.420 −4614.66 48126.6 143756. 1.31250e6 2598.81 −4.36830e6 6.94618e6 −4.08871e7
1.10 −277.989 444.751 44509.6 311412. −123636. 1.48043e6 −3.26403e6 −1.41511e7 −8.65689e7
1.11 −258.541 −3128.77 34075.3 −182900. 808916. −3.94303e6 −338002. −4.55968e6 4.72872e7
1.12 −257.289 −6184.66 33429.5 126748. 1.59124e6 −2.42809e6 −170184. 2.39012e7 −3.26108e7
1.13 −249.153 3654.45 29309.5 128359. −910519. −2.78623e6 861706. −993898. −3.19811e7
1.14 −233.818 −4457.69 21902.9 −301516. 1.04229e6 990496. 2.54046e6 5.52213e6 7.04998e7
1.15 −220.897 −2443.09 16027.5 197166. 539672. 2.02401e6 3.69793e6 −8.38020e6 −4.35534e7
1.16 −213.522 3344.35 12823.8 26781.2 −714093. 2.78768e6 4.25853e6 −3.16426e6 −5.71838e6
1.17 −195.627 4724.69 5501.80 −182120. −924275. −2.33218e6 5.33400e6 7.97378e6 3.56276e7
1.18 −193.119 895.072 4526.81 −186736. −172855. 800720. 5.45390e6 −1.35478e7 3.60622e7
1.19 −189.026 −2940.84 2962.69 297571. 555894. 2.23601e6 5.63397e6 −5.70037e6 −5.62485e7
1.20 −167.507 5168.95 −4709.39 260727. −865836. 2.03153e6 6.27773e6 1.23691e7 −4.36737e7
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
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.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.16.a.b ✓ 66 1.a even 1 1 trivial