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Properties

Label 959.2.o.b
Level $959$
Weight $2$
Character orbit 959.o
Analytic conductor $7.658$
Analytic rank $0$
Dimension $576$
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] = [959,2,Mod(50,959)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(959, base_ring=CyclotomicField(34))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("959.50");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 959 = 7 \cdot 137 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 959.o (of order \(17\), 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: \(7.65765355384\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(36\) over \(\Q(\zeta_{17})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{17}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{2} - 4 q^{3} - 14 q^{4} - 6 q^{5} - 16 q^{6} - 36 q^{7} - 24 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{2} - 4 q^{3} - 14 q^{4} - 6 q^{5} - 16 q^{6} - 36 q^{7} - 24 q^{8} - 54 q^{9} - 18 q^{10} + 14 q^{11} + 29 q^{12} - 8 q^{13} - 6 q^{14} - 34 q^{15} - 46 q^{16} + 42 q^{17} - 58 q^{18} + 14 q^{19} + 77 q^{20} - 4 q^{21} - 58 q^{22} + 65 q^{23} - 38 q^{24} - 86 q^{25} - 14 q^{26} + 68 q^{27} - 48 q^{28} - 32 q^{29} - 34 q^{30} + 19 q^{31} - 94 q^{32} - 36 q^{33} + 58 q^{34} + 11 q^{35} - 152 q^{36} + 104 q^{37} + 69 q^{38} - 14 q^{39} - 110 q^{40} - 52 q^{41} - 16 q^{42} - 68 q^{43} + 93 q^{44} + 60 q^{45} - 12 q^{46} + 6 q^{47} - 76 q^{48} - 36 q^{49} - 48 q^{50} - 84 q^{51} - 70 q^{52} + 110 q^{53} + 225 q^{54} - 48 q^{55} - 24 q^{56} + 36 q^{57} + 31 q^{58} - 9 q^{59} - 37 q^{60} - 66 q^{61} - 26 q^{62} - 54 q^{63} + 24 q^{64} - 114 q^{65} - 188 q^{66} + 160 q^{67} - 160 q^{68} - 4 q^{69} - 18 q^{70} + 199 q^{71} + 254 q^{72} - 86 q^{73} - 35 q^{74} + 366 q^{75} + 44 q^{76} - 3 q^{77} - 144 q^{78} - 68 q^{79} - 200 q^{80} - 188 q^{81} - 34 q^{82} - 54 q^{83} - 22 q^{84} - 124 q^{85} - 86 q^{86} - 16 q^{87} + 210 q^{88} - 43 q^{89} - 116 q^{90} - 8 q^{91} - 106 q^{92} - 140 q^{93} + 172 q^{94} - 62 q^{95} - 224 q^{96} + 133 q^{97} - 6 q^{98} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
50.1 −1.99343 + 1.81725i −0.0961418 − 0.0595285i 0.486824 − 5.25366i 2.25345 + 2.05429i 0.299830 − 0.0560480i −0.273663 − 0.961826i 5.32564 + 7.05229i −1.33152 − 2.67404i −8.22526
50.2 −1.84352 + 1.68059i 1.67460 + 1.03687i 0.389647 − 4.20497i −1.08482 − 0.988942i −4.82973 + 0.902833i −0.273663 − 0.961826i 3.34187 + 4.42535i 0.391979 + 0.787199i 3.66190
50.3 −1.74799 + 1.59350i 2.62440 + 1.62496i 0.331677 − 3.57937i 1.59363 + 1.45279i −7.17680 + 1.34158i −0.273663 − 0.961826i 2.27311 + 3.01009i 2.90978 + 5.84363i −5.10067
50.4 −1.67611 + 1.52797i −2.05454 − 1.27211i 0.290096 − 3.13063i −0.514451 − 0.468984i 5.38738 − 1.00708i −0.273663 − 0.961826i 1.56369 + 2.07065i 1.26563 + 2.54172i 1.57887
50.5 −1.64153 + 1.49646i 0.268788 + 0.166427i 0.270716 − 2.92150i −2.56189 − 2.33547i −0.690275 + 0.129035i −0.273663 − 0.961826i 1.25028 + 1.65563i −1.29267 − 2.59602i 7.70036
50.6 −1.50550 + 1.37245i −0.972880 − 0.602382i 0.198391 − 2.14098i 1.60221 + 1.46061i 2.29141 − 0.428339i −0.273663 − 0.961826i 0.184335 + 0.244099i −0.753583 − 1.51340i −4.41674
50.7 −1.38759 + 1.26496i 0.555127 + 0.343720i 0.140759 − 1.51903i −0.329116 − 0.300029i −1.20508 + 0.225269i −0.273663 − 0.961826i −0.536865 − 0.710924i −1.14719 − 2.30387i 0.836204
50.8 −1.32484 + 1.20775i −2.54187 − 1.57386i 0.111999 − 1.20867i −3.19571 − 2.91328i 5.26838 − 0.984832i −0.273663 − 0.961826i −0.849328 − 1.12469i 2.64685 + 5.31560i 7.75230
50.9 −1.13138 + 1.03139i 2.54169 + 1.57375i 0.0317200 − 0.342313i −2.19289 − 1.99908i −4.49876 + 0.840964i −0.273663 − 0.961826i −1.52803 − 2.02343i 2.64629 + 5.31447i 4.54282
50.10 −1.08567 + 0.989720i 0.278556 + 0.172475i 0.0145993 − 0.157552i 1.68115 + 1.53257i −0.473122 + 0.0884418i −0.273663 − 0.961826i −1.63057 − 2.15922i −1.28937 − 2.58940i −3.34200
50.11 −0.898149 + 0.818771i 1.07242 + 0.664013i −0.0482511 + 0.520713i −0.732293 − 0.667574i −1.50686 + 0.281682i −0.273663 − 0.961826i −1.84782 − 2.44691i −0.628049 − 1.26129i 1.20430
50.12 −0.743406 + 0.677704i −1.38122 − 0.855217i −0.0911671 + 0.983849i 1.80991 + 1.64995i 1.60639 − 0.300287i −0.273663 − 0.961826i −1.81143 − 2.39872i −0.160836 − 0.323002i −2.46367
50.13 −0.621363 + 0.566448i −1.82349 − 1.12906i −0.119307 + 1.28753i −0.739555 − 0.674193i 1.77260 − 0.331357i −0.273663 − 0.961826i −1.66858 − 2.20956i 0.713131 + 1.43216i 0.841427
50.14 −0.317668 + 0.289592i 1.16499 + 0.721334i −0.167488 + 1.80748i −1.85964 − 1.69528i −0.578974 + 0.108229i −0.273663 − 0.961826i −0.988319 − 1.30875i −0.500326 − 1.00479i 1.08169
50.15 −0.291083 + 0.265358i 2.53433 + 1.56919i −0.170222 + 1.83699i 0.715567 + 0.652325i −1.15410 + 0.215738i −0.273663 − 0.961826i −0.912645 − 1.20854i 2.62325 + 5.26820i −0.381389
50.16 −0.186070 + 0.169625i 0.157810 + 0.0977116i −0.178687 + 1.92834i 0.754055 + 0.687412i −0.0459380 + 0.00858729i −0.273663 − 0.961826i −0.597314 − 0.790971i −1.32186 − 2.65465i −0.256909
50.17 0.0391285 − 0.0356703i 1.47775 + 0.914985i −0.184278 + 1.98868i 2.44859 + 2.23219i 0.0904599 − 0.0169099i −0.273663 − 0.961826i 0.127542 + 0.168893i 0.00933609 + 0.0187494i 0.175433
50.18 0.0445943 − 0.0406531i −0.826954 − 0.512028i −0.184201 + 1.98784i −1.66954 − 1.52199i −0.0576929 + 0.0107847i −0.273663 − 0.961826i 0.145328 + 0.192445i −0.915535 − 1.83864i −0.136325
50.19 0.315257 − 0.287395i −2.09289 − 1.29586i −0.167745 + 1.81026i 3.14650 + 2.86841i −1.03222 + 0.192956i −0.273663 − 0.961826i 0.981538 + 1.29977i 1.36371 + 2.73870i 1.81632
50.20 0.409450 − 0.373263i 1.24994 + 0.773928i −0.156213 + 1.68580i −2.39252 − 2.18107i 0.800664 − 0.149670i −0.273663 − 0.961826i 1.23307 + 1.63285i −0.373838 − 0.750769i −1.79373
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 50.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
137.e even 17 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 959.2.o.b ✓ 576
137.e even 17 1 inner 959.2.o.b ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
959.2.o.b ✓ 576 1.a even 1 1 trivial
959.2.o.b ✓ 576 137.e even 17 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{576} + 6 T_{2}^{575} + 61 T_{2}^{574} + 310 T_{2}^{573} + 2004 T_{2}^{572} + \cdots + 99\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(959, [\chi])\). Copy content Toggle raw display