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Properties

Label 403.2.bi.a
Level $403$
Weight $2$
Character orbit 403.bi
Analytic conductor $3.218$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

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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: \(120\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9} + 3 q^{10} + 3 q^{11} - 12 q^{12} - 15 q^{13} - 3 q^{14} - 39 q^{15} - 2 q^{16} + 6 q^{17} - 34 q^{18} - 25 q^{19} - 10 q^{20} + 40 q^{21} + 18 q^{22} + q^{23} + 46 q^{24} - 45 q^{25} - 12 q^{26} + 26 q^{27} - 13 q^{28} + 14 q^{29} - 28 q^{30} - 14 q^{31} - 76 q^{32} + 63 q^{33} + 36 q^{34} + 50 q^{35} - 42 q^{36} + 14 q^{37} + 44 q^{38} - 6 q^{39} + 27 q^{40} - 17 q^{41} - 159 q^{42} - 15 q^{43} + 70 q^{44} + 40 q^{45} - 62 q^{46} + 9 q^{47} + 19 q^{48} - 151 q^{49} - 22 q^{50} - 25 q^{51} - q^{52} - 69 q^{53} + 51 q^{54} + 27 q^{55} + 51 q^{56} - 28 q^{57} + 156 q^{58} - 27 q^{59} + 87 q^{60} - 28 q^{61} - 25 q^{62} - 88 q^{63} + 10 q^{64} + 7 q^{65} + 30 q^{66} + 61 q^{67} + 124 q^{68} + 3 q^{69} - 263 q^{70} + 31 q^{71} + 31 q^{72} - 50 q^{73} - 90 q^{74} + 78 q^{75} + 66 q^{76} - 38 q^{77} + 31 q^{79} - 145 q^{80} + 36 q^{81} + 3 q^{82} + 23 q^{83} - 228 q^{84} - 58 q^{85} - 12 q^{86} + 68 q^{87} + 28 q^{88} + 39 q^{89} - 7 q^{90} - 12 q^{91} - 32 q^{92} - 17 q^{93} + 14 q^{94} + 123 q^{95} + 165 q^{96} - 51 q^{97} + 45 q^{98} - 45 q^{99}+O(q^{100}) \) 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 −0.745649 + 2.29487i −1.56275 + 1.73561i −3.09241 − 2.24677i −1.53204 + 2.65358i −2.81775 − 4.88048i 0.477992 − 4.54779i 3.55762 − 2.58476i −0.256571 − 2.44111i −4.94725 − 5.49448i
14.2 −0.717728 + 2.20894i 1.96095 − 2.17786i −2.74625 − 1.99527i 1.43599 − 2.48720i 3.40333 + 5.89474i −0.518982 + 4.93778i 2.62042 − 1.90385i −0.584148 − 5.55780i 4.46343 + 4.95714i
14.3 −0.629077 + 1.93610i −0.654890 + 0.727329i −1.73471 − 1.26034i 0.778108 − 1.34772i −0.996205 − 1.72548i −0.152927 + 1.45500i 0.237526 − 0.172573i 0.213459 + 2.03093i 2.11984 + 2.35432i
14.4 −0.540256 + 1.66274i −0.217770 + 0.241858i −0.854781 − 0.621035i 1.00253 − 1.73643i −0.284495 − 0.492760i 0.416075 − 3.95868i −1.33440 + 0.969498i 0.302514 + 2.87823i 2.34560 + 2.60506i
14.5 −0.527412 + 1.62321i 1.32464 − 1.47116i −0.738608 − 0.536630i −1.29201 + 2.23783i 1.68936 + 2.92607i −0.115558 + 1.09946i −1.50095 + 1.09051i −0.0960576 − 0.913927i −2.95105 − 3.27747i
14.6 −0.269588 + 0.829708i −1.42637 + 1.58415i 1.00230 + 0.728211i −1.58260 + 2.74115i −0.929847 − 1.61054i −0.112042 + 1.06601i −2.28599 + 1.66087i −0.161400 − 1.53561i −1.84770 − 2.05208i
14.7 −0.162742 + 0.500869i −1.66779 + 1.85227i 1.39365 + 1.01255i 1.93093 − 3.34447i −0.656324 − 1.13679i −0.465155 + 4.42565i −1.58609 + 1.15236i −0.335789 − 3.19482i 1.36090 + 1.51143i
14.8 −0.0908434 + 0.279587i 1.96940 − 2.18724i 1.54812 + 1.12477i 0.739803 − 1.28138i 0.432618 + 0.749317i 0.0548395 − 0.521763i −0.930771 + 0.676245i −0.591901 − 5.63156i 0.291050 + 0.323244i
14.9 −0.0727036 + 0.223759i 0.337206 − 0.374505i 1.57325 + 1.14303i 0.254802 − 0.441330i 0.0592827 + 0.102681i −0.0906697 + 0.862665i −0.750826 + 0.545507i 0.287039 + 2.73099i 0.0802263 + 0.0891004i
14.10 0.138780 − 0.427121i −2.06376 + 2.29204i 1.45486 + 1.05702i 0.873401 − 1.51278i 0.692568 + 1.19956i 0.471332 − 4.48443i 1.38004 − 1.00266i −0.680746 − 6.47687i −0.524927 − 0.582991i
14.11 0.362743 − 1.11641i −0.166092 + 0.184464i 0.503249 + 0.365632i 2.00419 − 3.47136i 0.145688 + 0.252339i 0.284216 − 2.70413i 2.49009 − 1.80916i 0.307145 + 2.92229i −3.14845 − 3.49671i
14.12 0.395429 − 1.21700i 0.354263 − 0.393449i 0.293298 + 0.213093i −1.48333 + 2.56921i −0.338744 − 0.586721i −0.222430 + 2.11628i 2.44581 − 1.77698i 0.284286 + 2.70480i 2.54018 + 2.82116i
14.13 0.523105 − 1.60995i 0.991294 − 1.10094i −0.700275 − 0.508780i −1.03169 + 1.78693i −1.25392 − 2.17185i 0.334356 − 3.18118i 1.55359 − 1.12875i 0.0841724 + 0.800847i 2.33720 + 2.59572i
14.14 0.635252 − 1.95510i −1.26940 + 1.40981i −1.80085 − 1.30840i 0.148342 − 0.256935i 1.94994 + 3.37739i −0.266274 + 2.53343i −0.375822 + 0.273051i −0.0626061 − 0.595658i −0.408101 − 0.453242i
14.15 0.773640 − 2.38102i 1.00840 − 1.11994i −3.45270 − 2.50854i 1.04075 − 1.80263i −1.88646 − 3.26744i −0.0586254 + 0.557783i −4.59319 + 3.33715i 0.0761884 + 0.724884i −3.48694 − 3.87264i
40.1 −0.844756 + 2.59989i 1.33170 + 0.283062i −4.42779 − 3.21698i 1.99341 + 3.45269i −1.86090 + 3.22317i 3.32958 − 1.48242i 7.68099 − 5.58056i −1.04733 − 0.466300i −10.6606 + 2.26597i
40.2 −0.820567 + 2.52545i −1.82037 − 0.386931i −4.08652 − 2.96903i −1.79431 − 3.10783i 2.47091 − 4.27974i −1.93457 + 0.861327i 6.55484 − 4.76237i 0.423384 + 0.188503i 9.32102 − 1.98124i
40.3 −0.608805 + 1.87371i −1.89076 − 0.401893i −1.52211 − 1.10588i 0.912174 + 1.57993i 1.90413 − 3.29806i −0.756252 + 0.336705i −0.188985 + 0.137305i 0.672810 + 0.299554i −3.51567 + 0.747279i
40.4 −0.584417 + 1.79865i 3.21845 + 0.684102i −1.27557 − 0.926754i 1.13209 + 1.96083i −3.11138 + 5.38906i −1.46764 + 0.653435i −0.647676 + 0.470565i 7.14976 + 3.18328i −4.18846 + 0.890285i
40.5 −0.395989 + 1.21873i 2.04438 + 0.434547i 0.289543 + 0.210365i −0.128714 − 0.222939i −1.33915 + 2.31947i 1.64432 − 0.732098i −2.44446 + 1.77600i 1.25003 + 0.556550i 0.322672 − 0.0685860i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.15
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.a ✓ 120
31.g even 15 1 inner 403.2.bi.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bi.a ✓ 120 1.a even 1 1 trivial
403.2.bi.a ✓ 120 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} - 6 T_{2}^{119} + 59 T_{2}^{118} - 288 T_{2}^{117} + 1777 T_{2}^{116} - 7420 T_{2}^{115} + \cdots + 6507601 \) acting on \(S_{2}^{\mathrm{new}}(403, [\chi])\). Copy content Toggle raw display