↑

Properties

Label 403.2.bi.b
Level $403$
Weight $2$
Character orbit 403.bi
Analytic conductor $3.218$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [403,2,Mod(14,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bi (of order \(15\), degree \(8\), minimal)

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: no
Analytic conductor: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(17\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9} + 7 q^{10} + 5 q^{11} + 28 q^{12} + 17 q^{13} - 3 q^{14} + 17 q^{15} - 78 q^{16} - 12 q^{17} + 34 q^{18} - 23 q^{19} + 8 q^{20} - 38 q^{21} + 8 q^{22} + 9 q^{23} + 46 q^{24} - 69 q^{25} - 12 q^{26} - 58 q^{27} + 7 q^{28} + 18 q^{29} + 60 q^{30} + 14 q^{31} + 212 q^{32} - 41 q^{33} - 30 q^{35} - 106 q^{36} + 6 q^{37} - 62 q^{38} + 6 q^{39} + 39 q^{40} - 5 q^{41} + 37 q^{42} + q^{43} - 88 q^{44} - 168 q^{45} + 6 q^{46} - 81 q^{47} - 101 q^{48} + 117 q^{49} - 20 q^{50} + 35 q^{51} + 7 q^{52} + 25 q^{53} + 57 q^{54} + 19 q^{55} - 63 q^{56} - 20 q^{57} - 44 q^{58} + 9 q^{59} + 113 q^{60} + 8 q^{61} + 57 q^{62} + 112 q^{63} - 146 q^{64} - 5 q^{65} + 70 q^{66} - 99 q^{67} - 56 q^{68} - 65 q^{69} + 169 q^{70} - q^{71} - 43 q^{72} + 50 q^{73} - 26 q^{74} - 18 q^{75} - 94 q^{76} + 114 q^{77} - 17 q^{79} + 217 q^{80} + 54 q^{81} - 25 q^{82} - 53 q^{83} - 22 q^{84} + 2 q^{85} - 54 q^{86} - 28 q^{87} + 70 q^{88} - 101 q^{89} + 165 q^{90} - 4 q^{91} + 64 q^{92} + 61 q^{93} - 90 q^{94} - 33 q^{95} - 217 q^{96} + 73 q^{97} + 33 q^{98} - 15 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −0.778445 + 2.39581i −0.897108 + 0.996339i −3.51588 − 2.55444i −1.29523 + 2.24341i −1.68869 − 2.92489i −0.499609 + 4.75346i 4.78087 − 3.47351i 0.125696 + 1.19592i −4.36650 − 4.84949i
14.2 −0.775728 + 2.38745i −1.68623 + 1.87274i −3.48011 − 2.52845i 1.72799 − 2.99297i −3.16302 − 5.47852i 0.0507028 − 0.482404i 4.67440 − 3.39615i −0.350226 − 3.33218i 5.80510 + 6.44721i
14.3 −0.577531 + 1.77746i 1.86722 − 2.07376i −1.20778 − 0.877505i −0.00232995 + 0.00403559i 2.60765 + 4.51658i 0.383684 − 3.65051i −0.766732 + 0.557063i −0.500379 − 4.76079i −0.00582747 − 0.00647206i
14.4 −0.516329 + 1.58910i −0.00193962 + 0.00215417i −0.640602 − 0.465424i −0.479120 + 0.829859i −0.00242170 − 0.00419451i −0.214858 + 2.04424i −1.63317 + 1.18657i 0.313585 + 2.98356i −1.07134 − 1.18985i
14.5 −0.292848 + 0.901292i −1.55469 + 1.72666i 0.891466 + 0.647688i 0.318197 − 0.551134i −1.10094 − 1.90688i 0.111899 − 1.06465i −2.37819 + 1.72786i −0.250704 − 2.38529i 0.403549 + 0.448187i
14.6 −0.256563 + 0.789619i 0.840376 − 0.933332i 1.06036 + 0.770397i 2.12781 − 3.68547i 0.521367 + 0.903035i 0.110851 − 1.05468i −2.22375 + 1.61565i 0.148708 + 1.41487i 2.36420 + 2.62571i
14.7 −0.215030 + 0.661795i 0.258564 − 0.287164i 1.22630 + 0.890959i −1.89053 + 3.27449i 0.134445 + 0.232865i 0.319980 − 3.04441i −1.97924 + 1.43800i 0.297977 + 2.83506i −1.76052 − 1.95526i
14.8 0.0129642 − 0.0398998i 1.94913 − 2.16473i 1.61661 + 1.17454i −1.81601 + 3.14542i −0.0611034 − 0.105834i −0.401482 + 3.81985i 0.135703 − 0.0985943i −0.573359 − 5.45514i 0.101958 + 0.113236i
14.9 0.124680 − 0.383725i −0.653315 + 0.725580i 1.48633 + 1.07989i −0.174215 + 0.301749i 0.196968 + 0.341158i 0.256927 − 2.44450i 1.25253 − 0.910013i 0.213940 + 2.03550i 0.0940675 + 0.104473i
14.10 0.201597 − 0.620451i −0.733986 + 0.815174i 1.27372 + 0.925409i 0.554991 − 0.961272i 0.357806 + 0.619739i −0.401317 + 3.81828i 1.88652 − 1.37064i 0.187812 + 1.78691i −0.484538 − 0.538134i
14.11 0.249715 − 0.768544i 1.39968 − 1.55451i 1.08973 + 0.791737i −0.342316 + 0.592908i −0.845184 − 1.46390i 0.346639 − 3.29805i 2.18813 − 1.58977i −0.143790 − 1.36807i 0.370195 + 0.411143i
14.12 0.313343 − 0.964371i −2.18936 + 2.43153i 0.786207 + 0.571213i −1.31857 + 2.28384i 1.65888 + 2.87326i −0.310865 + 2.95768i 2.43790 − 1.77124i −0.805458 − 7.66342i 1.78930 + 1.98722i
14.13 0.477192 − 1.46864i 1.45858 − 1.61992i −0.311171 − 0.226079i 1.14752 − 1.98757i −1.68306 − 2.91515i −0.410673 + 3.90729i 2.01809 − 1.46623i −0.183091 − 1.74199i −2.37144 − 2.63375i
14.14 0.563363 − 1.73385i −0.719072 + 0.798611i −1.07083 − 0.778005i 1.03814 − 1.79811i 0.979574 + 1.69667i 0.0320232 − 0.304681i 0.997593 − 0.724794i 0.192871 + 1.83505i −2.53281 − 2.81297i
14.15 0.748518 − 2.30370i 1.94845 − 2.16398i −3.12873 − 2.27316i −1.19465 + 2.06919i −3.52671 − 6.10844i 0.119417 − 1.13617i −3.65930 + 2.65864i −0.572740 − 5.44925i 3.87259 + 4.30095i
14.16 0.798897 − 2.45875i −0.340489 + 0.378151i −3.78920 − 2.75301i −1.48464 + 2.57146i 0.657765 + 1.13928i −0.327920 + 3.11995i −5.61307 + 4.07814i 0.286520 + 2.72605i 5.13652 + 5.70469i
14.17 0.849257 − 2.61374i −2.02850 + 2.25288i −4.49238 − 3.26391i 1.75209 − 3.03471i 4.16574 + 7.21527i 0.0345203 − 0.328439i −7.89944 + 5.73928i −0.647064 − 6.15640i −6.44398 − 7.15676i
40.1 −0.779971 + 2.40050i 2.95884 + 0.628922i −3.53603 − 2.56908i −2.10392 − 3.64409i −3.81754 + 6.61218i 3.44993 − 1.53601i 4.84111 − 3.51727i 5.61858 + 2.50155i 10.3887 − 2.20818i
40.2 −0.756168 + 2.32725i −0.493906 − 0.104983i −3.22625 − 2.34401i 0.105129 + 0.182089i 0.617798 − 1.07006i 0.113244 − 0.0504194i 3.93532 − 2.85917i −2.50771 − 1.11651i −0.503261 + 0.106971i
40.3 −0.715991 + 2.20359i 1.66971 + 0.354909i −2.72515 − 1.97993i 0.00534395 + 0.00925600i −1.97758 + 3.42526i −3.74078 + 1.66550i 2.56517 − 1.86371i −0.0786495 − 0.0350170i −0.0242227 + 0.00514869i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bi.b ✓ 136
31.g even 15 1 inner 403.2.bi.b ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bi.b ✓ 136 1.a even 1 1 trivial
403.2.bi.b ✓ 136 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{136} + 6 T_{2}^{135} + 69 T_{2}^{134} + 352 T_{2}^{133} + 2486 T_{2}^{132} + 11042 T_{2}^{131} + \cdots + 1834683795025 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display