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Properties

Label 403.2.cd.a
Level $403$
Weight $2$
Character orbit 403.cd
Analytic conductor $3.218$
Analytic rank $0$
Dimension $576$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [403,2,Mod(21,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([15, 58]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.cd (of order \(60\), degree \(16\), 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: \(576\)
Relative dimension: \(36\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 12 q^{2} - 28 q^{3} - 8 q^{5} + 6 q^{6} - 16 q^{7} - 46 q^{8} - 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 12 q^{2} - 28 q^{3} - 8 q^{5} + 6 q^{6} - 16 q^{7} - 46 q^{8} - 104 q^{9} - 18 q^{11} - 24 q^{13} - 44 q^{14} + 10 q^{15} + 120 q^{16} - 10 q^{18} - 16 q^{19} - 32 q^{20} - 28 q^{21} - 28 q^{24} - 72 q^{26} - 100 q^{27} + 38 q^{28} - 40 q^{29} - 66 q^{31} - 124 q^{32} + 52 q^{33} + 76 q^{34} - 32 q^{35} - 36 q^{37} + 10 q^{39} - 52 q^{40} + 2 q^{41} + 8 q^{42} + 12 q^{44} + 30 q^{45} + 20 q^{46} - 20 q^{47} - 40 q^{48} + 50 q^{52} - 84 q^{53} - 80 q^{54} - 28 q^{55} - 126 q^{57} + 70 q^{58} - 74 q^{59} - 50 q^{60} + 212 q^{63} + 84 q^{65} + 136 q^{66} - 20 q^{67} - 60 q^{68} + 18 q^{70} - 62 q^{71} - 54 q^{72} - 32 q^{73} - 112 q^{74} - 64 q^{76} + 6 q^{78} - 56 q^{79} + 60 q^{80} - 168 q^{81} + 238 q^{83} - 206 q^{84} - 30 q^{85} - 158 q^{86} + 76 q^{87} + 140 q^{89} - 50 q^{91} + 70 q^{93} - 48 q^{94} + 8 q^{96} - 92 q^{97} + 50 q^{98} - 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −0.438402 − 2.76796i −2.21379 + 0.232679i −5.56730 + 1.80893i 0.467617 − 1.74517i 1.61458 + 6.02569i −2.46919 + 1.60351i 4.90317 + 9.62301i 1.91230 − 0.406472i −5.03557 − 0.529260i
21.2 −0.420851 − 2.65715i 0.222716 − 0.0234084i −4.98120 + 1.61849i −0.667546 + 2.49132i −0.155930 − 0.581938i 0.821932 − 0.533769i 3.95420 + 7.76055i −2.88539 + 0.613308i 6.90073 + 0.725296i
21.3 −0.375607 − 2.37149i 2.23445 − 0.234850i −3.58075 + 1.16346i 0.997640 − 3.72324i −1.39622 − 5.21075i 0.525344 − 0.341162i 1.92397 + 3.77601i 2.00316 − 0.425785i −9.20434 − 0.967415i
21.4 −0.342253 − 2.16090i 1.06144 − 0.111561i −2.65023 + 0.861112i −0.672966 + 2.51154i −0.604352 − 2.25547i −3.40576 + 2.21173i 0.781311 + 1.53341i −1.82024 + 0.386904i 5.65752 + 0.594629i
21.5 −0.339072 − 2.14081i −2.52783 + 0.265685i −2.56601 + 0.833746i −1.03003 + 3.84414i 1.42590 + 5.32152i 2.45967 − 1.59733i 0.686907 + 1.34813i 3.38488 − 0.719478i 8.57885 + 0.901674i
21.6 −0.336779 − 2.12634i 2.51608 − 0.264450i −2.50579 + 0.814180i −0.295700 + 1.10357i −1.40967 − 5.26097i 0.467445 − 0.303562i 0.620378 + 1.21756i 3.32626 − 0.707019i 2.44614 + 0.257100i
21.7 −0.331707 − 2.09431i −1.13363 + 0.119149i −2.37401 + 0.771361i 0.0970019 − 0.362016i 0.625567 + 2.33465i 0.682237 − 0.443050i 0.477646 + 0.937433i −1.66353 + 0.353594i −0.790351 − 0.0830693i
21.8 −0.310808 − 1.96236i 0.486514 − 0.0511347i −1.85216 + 0.601802i 0.642313 − 2.39714i −0.251557 − 0.938825i −3.43497 + 2.23070i −0.0473792 − 0.0929870i −2.70036 + 0.573980i −4.90370 − 0.515400i
21.9 −0.302034 − 1.90697i −2.73963 + 0.287946i −1.64319 + 0.533905i 0.850977 − 3.17589i 1.37656 + 5.13741i 3.33174 − 2.16366i −0.238636 − 0.468349i 4.48819 − 0.953995i −6.31334 − 0.663559i
21.10 −0.226478 − 1.42992i 2.63019 − 0.276444i −0.0912802 + 0.0296587i −0.411870 + 1.53712i −0.990974 − 3.69837i 2.42932 − 1.57762i −1.25145 − 2.45610i 3.90704 − 0.830467i 2.29124 + 0.240819i
21.11 −0.222256 − 1.40327i −3.26546 + 0.343213i −0.0176593 + 0.00573785i 0.213111 − 0.795340i 1.20739 + 4.50604i −2.75126 + 1.78669i −1.27805 − 2.50831i 7.61098 − 1.61776i −1.16344 − 0.122283i
21.12 −0.184795 − 1.16675i −0.0935769 + 0.00983533i 0.574955 − 0.186814i −0.0228269 + 0.0851910i 0.0287679 + 0.107363i 2.60956 − 1.69467i −1.39681 − 2.74139i −2.92578 + 0.621894i 0.103615 + 0.0108904i
21.13 −0.183398 − 1.15793i −1.43456 + 0.150779i 0.594952 − 0.193312i −0.478261 + 1.78489i 0.437686 + 1.63347i −1.94871 + 1.26551i −1.39744 − 2.74262i −0.899204 + 0.191132i 2.15449 + 0.226446i
21.14 −0.159860 − 1.00931i 1.15285 − 0.121170i 0.908952 − 0.295336i 0.515241 − 1.92291i −0.306593 − 1.14422i 0.690634 − 0.448503i −1.37125 − 2.69124i −1.62005 + 0.344353i −2.02318 − 0.212645i
21.15 −0.138227 − 0.872731i −1.52885 + 0.160688i 1.15956 − 0.376764i −0.666091 + 2.48588i 0.351566 + 1.31206i −0.721464 + 0.468524i −1.29140 − 2.53451i −0.622893 + 0.132400i 2.26158 + 0.237702i
21.16 −0.0719155 − 0.454056i 2.88221 − 0.302933i 1.70112 − 0.552727i 0.524460 − 1.95731i −0.344824 − 1.28690i −3.88603 + 2.52362i −0.790719 − 1.55187i 5.28094 − 1.12250i −0.926446 − 0.0973734i
21.17 −0.0567540 − 0.358331i 2.32789 − 0.244671i 1.77693 − 0.577361i −0.959878 + 3.58231i −0.219790 − 0.820268i −2.32142 + 1.50755i −0.637148 − 1.25047i 2.42476 − 0.515398i 1.33813 + 0.140643i
21.18 −0.0348740 − 0.220186i −1.73304 + 0.182150i 1.85485 − 0.602676i 0.939504 − 3.50628i 0.100545 + 0.375238i 3.04041 − 1.97447i −0.399803 − 0.784658i 0.0357975 − 0.00760900i −0.804796 − 0.0845874i
21.19 −0.0103892 − 0.0655950i 1.19020 − 0.125096i 1.89792 − 0.616671i 0.380352 − 1.41949i −0.0205710 − 0.0767719i 0.375204 − 0.243660i −0.120470 − 0.236436i −1.53350 + 0.325956i −0.0970632 − 0.0102018i
21.20 0.0347043 + 0.219114i −2.53826 + 0.266782i 1.85531 − 0.602826i 0.413236 − 1.54222i −0.146544 − 0.546910i −1.87187 + 1.21561i 0.397906 + 0.780935i 3.43713 − 0.730585i 0.352263 + 0.0370243i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.h odd 30 1 inner
403.cd even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.cd.a ✓ 576
13.d odd 4 1 inner 403.2.cd.a ✓ 576
31.h odd 30 1 inner 403.2.cd.a ✓ 576
403.cd even 60 1 inner 403.2.cd.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cd.a ✓ 576 1.a even 1 1 trivial
403.2.cd.a ✓ 576 13.d odd 4 1 inner
403.2.cd.a ✓ 576 31.h odd 30 1 inner
403.2.cd.a ✓ 576 403.cd even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).