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Properties

Label 417.2.i.b
Level $417$
Weight $2$
Character orbit 417.i
Analytic conductor $3.330$
Analytic rank $0$
Dimension $286$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [417,2,Mod(34,417)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(417, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([0, 36]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("417.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 417 = 3 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 417.i (of order \(23\), degree \(22\), 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: \(3.32976176429\)
Analytic rank: \(0\)
Dimension: \(286\)
Relative dimension: \(13\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 286 q - 2 q^{2} - 13 q^{3} - 22 q^{4} - 4 q^{5} - 2 q^{6} - 12 q^{7} + 34 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 286 q - 2 q^{2} - 13 q^{3} - 22 q^{4} - 4 q^{5} - 2 q^{6} - 12 q^{7} + 34 q^{8} - 13 q^{9} - 24 q^{10} + 34 q^{11} - 22 q^{12} - 16 q^{13} + 32 q^{14} - 4 q^{15} - 48 q^{16} + 32 q^{17} - 2 q^{18} - 30 q^{19} - 26 q^{20} - 12 q^{21} - 20 q^{22} - 32 q^{23} - 12 q^{24} - 35 q^{25} + 102 q^{26} - 13 q^{27} - 70 q^{28} - 20 q^{29} + 68 q^{30} + 2 q^{31} + 92 q^{32} - 12 q^{33} + 60 q^{34} - 41 q^{35} - 22 q^{36} - 49 q^{37} + 68 q^{38} - 16 q^{39} + 6 q^{40} - 30 q^{41} - 14 q^{42} + 50 q^{43} + 12 q^{44} - 4 q^{45} - 4 q^{46} + 27 q^{47} - 48 q^{48} - 103 q^{49} - 78 q^{50} + 9 q^{51} + 122 q^{52} - 15 q^{53} - 2 q^{54} + 5 q^{55} - 90 q^{56} + 16 q^{57} + 132 q^{58} + 94 q^{59} - 26 q^{60} - 62 q^{61} - 106 q^{62} - 12 q^{63} - 42 q^{64} - 65 q^{65} + 26 q^{66} + 78 q^{67} - 156 q^{68} + 60 q^{69} - 28 q^{70} - 23 q^{71} - 12 q^{72} - 62 q^{73} + 140 q^{74} - 12 q^{75} - 116 q^{76} - 93 q^{77} + 10 q^{78} - 98 q^{79} - 174 q^{80} - 13 q^{81} + 128 q^{82} - 70 q^{84} - 29 q^{85} + 70 q^{86} + 49 q^{87} - 2 q^{88} - 16 q^{89} + 22 q^{90} - 123 q^{91} - 14 q^{92} - 44 q^{93} + 8 q^{94} - 86 q^{95} - 46 q^{96} - 10 q^{97} - 34 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.17768 + 1.77168i 0.962917 + 0.269797i 1.19656 − 5.75814i −1.93301 − 2.73845i −2.57492 + 1.11845i −2.19624 − 0.953961i 5.01274 + 9.67414i 0.854419 + 0.519584i 9.06113 + 2.53881i
34.2 −1.79663 + 1.46167i 0.962917 + 0.269797i 0.684503 − 3.29401i 1.85194 + 2.62360i −2.12436 + 0.922742i −0.494269 − 0.214691i 1.45383 + 2.80577i 0.854419 + 0.519584i −7.16210 − 2.00673i
34.3 −1.41965 + 1.15497i 0.962917 + 0.269797i 0.274534 − 1.32113i −0.304698 − 0.431658i −1.67861 + 0.729124i −2.66196 − 1.15625i −0.547828 − 1.05726i 0.854419 + 0.519584i 0.931116 + 0.260886i
34.4 −1.24388 + 1.01197i 0.962917 + 0.269797i 0.116238 − 0.559369i −1.97192 − 2.79358i −1.47078 + 0.638849i 2.36284 + 1.02633i −1.05398 − 2.03409i 0.854419 + 0.519584i 5.27985 + 1.47934i
34.5 −0.499238 + 0.406161i 0.962917 + 0.269797i −0.322639 + 1.55263i −0.996782 − 1.41212i −0.590306 + 0.256406i −3.37772 − 1.46715i −1.06173 − 2.04904i 0.854419 + 0.519584i 1.07118 + 0.300130i
34.6 −0.306480 + 0.249340i 0.962917 + 0.269797i −0.375153 + 1.80533i 1.95653 + 2.77177i −0.362386 + 0.157406i 2.75282 + 1.19572i −0.698703 − 1.34844i 0.854419 + 0.519584i −1.29075 − 0.361651i
34.7 −0.291206 + 0.236913i 0.962917 + 0.269797i −0.378239 + 1.82019i −0.581539 − 0.823854i −0.344325 + 0.149562i 3.57860 + 1.55441i −0.666502 − 1.28629i 0.854419 + 0.519584i 0.364529 + 0.102136i
34.8 0.363887 − 0.296044i 0.962917 + 0.269797i −0.362140 + 1.74272i 2.19095 + 3.10388i 0.430265 − 0.186890i −4.73567 − 2.05699i 0.815776 + 1.57438i 0.854419 + 0.519584i 1.71614 + 0.480841i
34.9 0.575775 − 0.468428i 0.962917 + 0.269797i −0.294820 + 1.41875i −1.04715 − 1.48348i 0.680804 − 0.295715i 0.399389 + 0.173479i 1.17780 + 2.27306i 0.854419 + 0.519584i −1.29783 − 0.363634i
34.10 1.04875 − 0.853225i 0.962917 + 0.269797i −0.0350191 + 0.168521i 0.283080 + 0.401033i 1.24006 − 0.538635i 1.64324 + 0.713758i 1.35107 + 2.60744i 0.854419 + 0.519584i 0.639053 + 0.179054i
34.11 1.46590 − 1.19260i 0.962917 + 0.269797i 0.319661 − 1.53829i −2.49571 − 3.53562i 1.73330 − 0.752879i −3.58953 − 1.55915i 0.372840 + 0.719549i 0.854419 + 0.519584i −7.87505 − 2.20648i
34.12 1.50058 − 1.22082i 0.962917 + 0.269797i 0.354449 − 1.70570i 1.01558 + 1.43875i 1.77431 − 0.770692i −0.644163 − 0.279799i 0.229487 + 0.442891i 0.854419 + 0.519584i 3.28041 + 0.919128i
34.13 2.06611 − 1.68091i 0.962917 + 0.269797i 1.03646 − 4.98773i 0.348749 + 0.494065i 2.44300 − 1.06114i −1.51446 − 0.657823i −3.79169 − 7.31764i 0.854419 + 0.519584i 1.55103 + 0.434579i
52.1 −0.188913 − 2.76181i −0.334880 + 0.942261i −5.61055 + 0.771152i 1.57106 + 0.682407i 2.66561 + 0.746869i −1.36228 + 0.381692i 2.06324 + 9.92887i −0.775711 − 0.631088i 1.58789 − 4.46788i
52.2 −0.146306 − 2.13891i −0.334880 + 0.942261i −2.57217 + 0.353536i −3.65628 − 1.58815i 2.06441 + 0.578420i 0.716302 − 0.200698i 0.260123 + 1.25178i −0.775711 − 0.631088i −2.86197 + 8.05281i
52.3 −0.132667 − 1.93952i −0.334880 + 0.942261i −1.76278 + 0.242289i 0.499262 + 0.216860i 1.87197 + 0.524500i −3.52265 + 0.986999i −0.0872710 − 0.419971i −0.775711 − 0.631088i 0.354370 − 0.997102i
52.4 −0.0819845 − 1.19857i −0.334880 + 0.942261i 0.551522 − 0.0758050i −1.02736 − 0.446247i 1.15682 + 0.324126i 1.49940 − 0.420111i −0.624926 − 3.00731i −0.775711 − 0.631088i −0.450631 + 1.26795i
52.5 −0.0632136 − 0.924149i −0.334880 + 0.942261i 1.13132 − 0.155496i 3.66059 + 1.59002i 0.891959 + 0.249915i 1.25401 − 0.351358i −0.592142 − 2.84954i −0.775711 − 0.631088i 1.23802 − 3.48344i
52.6 −0.0500173 − 0.731227i −0.334880 + 0.942261i 1.44918 − 0.199185i −0.0983367 − 0.0427137i 0.705756 + 0.197744i 2.59357 − 0.726683i −0.516374 − 2.48493i −0.775711 − 0.631088i −0.0263148 + 0.0740428i
52.7 −0.00455489 − 0.0665902i −0.334880 + 0.942261i 1.97696 − 0.271727i −0.389256 − 0.169078i 0.0642707 + 0.0180078i −4.60685 + 1.29078i −0.0542588 − 0.261108i −0.775711 − 0.631088i −0.00948591 + 0.0266908i
See next 80 embeddings (of 286 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
139.e even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 417.2.i.b ✓ 286
139.e even 23 1 inner 417.2.i.b ✓ 286
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
417.2.i.b ✓ 286 1.a even 1 1 trivial
417.2.i.b ✓ 286 139.e even 23 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{286} + 2 T_{2}^{285} + 26 T_{2}^{284} + 10 T_{2}^{283} + 329 T_{2}^{282} - 374 T_{2}^{281} + \cdots + 92\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\). Copy content Toggle raw display