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Properties

Label 241.2.q.a
Level $241$
Weight $2$
Character orbit 241.q
Analytic conductor $1.924$
Analytic rank $0$
Dimension $304$
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] = [241,2,Mod(9,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.q (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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(304\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 304 q - 24 q^{2} - 14 q^{3} + 140 q^{4} - 20 q^{5} - 24 q^{6} - 10 q^{7} - 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 304 q - 24 q^{2} - 14 q^{3} + 140 q^{4} - 20 q^{5} - 24 q^{6} - 10 q^{7} - 34 q^{9} - 52 q^{10} - 16 q^{11} + 8 q^{12} - 42 q^{13} + 18 q^{15} - 124 q^{16} + 14 q^{17} - 16 q^{18} - 56 q^{19} - 18 q^{20} - 40 q^{21} - 42 q^{22} - 2 q^{23} + 72 q^{24} + 48 q^{25} - 40 q^{26} - 20 q^{27} + 24 q^{28} - 26 q^{29} - 56 q^{31} + 102 q^{32} + 28 q^{33} + 32 q^{34} - 16 q^{35} - 4 q^{36} - 26 q^{37} - 46 q^{38} - 16 q^{39} - 28 q^{40} - 20 q^{41} - 26 q^{42} - 16 q^{43} + 10 q^{44} + 4 q^{45} - 4 q^{46} - 20 q^{47} + 150 q^{48} - 14 q^{49} - 214 q^{50} - 50 q^{51} - 18 q^{52} - 38 q^{53} - 36 q^{54} - 98 q^{55} + 84 q^{56} - 100 q^{57} - 140 q^{58} - 76 q^{59} - 18 q^{60} - 80 q^{61} + 56 q^{62} + 140 q^{63} - 168 q^{64} + 8 q^{65} + 24 q^{66} - 44 q^{67} - 112 q^{68} - 108 q^{69} - 112 q^{70} - 40 q^{71} + 146 q^{72} + 52 q^{73} + 26 q^{74} + 80 q^{75} + 112 q^{76} + 10 q^{77} + 154 q^{78} - 80 q^{79} - 28 q^{80} + 152 q^{81} + 44 q^{82} - 68 q^{83} - 236 q^{84} - 36 q^{85} - 28 q^{86} - 8 q^{87} + 56 q^{88} + 72 q^{89} + 144 q^{90} + 12 q^{91} + 122 q^{92} - 10 q^{93} + 110 q^{94} - 82 q^{95} + 362 q^{96} + 212 q^{97} - 28 q^{98} - 190 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.43622 − 1.40655i 1.90491 + 0.200214i 2.95679 + 5.12132i −0.437026 − 0.601515i −4.35917 − 3.16712i −0.103096 + 1.96720i − 11.0093i 0.654141 + 0.139042i 0.218630 + 2.08013i
9.2 −2.04025 − 1.17794i −0.863220 − 0.0907280i 1.77508 + 3.07452i 1.18142 + 1.62609i 1.65431 + 1.20193i 0.0840992 − 1.60471i − 3.65197i −2.19753 − 0.467099i −0.494964 − 4.70927i
9.3 −2.01188 − 1.16156i −2.98340 − 0.313568i 1.69844 + 2.94178i −1.65462 − 2.27739i 5.63801 + 4.09626i −0.247034 + 4.71369i − 3.24509i 5.86792 + 1.24727i 0.683573 + 6.50377i
9.4 −1.81953 − 1.05051i −1.26440 − 0.132894i 1.20714 + 2.09082i 0.323219 + 0.444873i 2.16101 + 1.57007i −0.000110851 0.00211515i − 0.870397i −1.35340 − 0.287674i −0.120765 − 1.14901i
9.5 −1.59182 − 0.919038i 1.67083 + 0.175611i 0.689262 + 1.19384i −1.65861 − 2.28288i −2.49826 − 1.81509i 0.130727 − 2.49442i 1.14232i −0.173619 − 0.0369039i 0.542156 + 5.15827i
9.6 −1.22085 − 0.704856i 3.28104 + 0.344851i −0.00635713 − 0.0110109i 0.472014 + 0.649672i −3.76257 − 2.73367i −0.194390 + 3.70918i 2.83735i 7.71184 + 1.63920i −0.118332 − 1.12585i
9.7 −0.945280 − 0.545758i 0.124558 + 0.0130915i −0.404297 − 0.700262i 1.81212 + 2.49417i −0.110597 − 0.0803535i −0.153001 + 2.91943i 3.06562i −2.91910 − 0.620474i −0.351750 − 3.34667i
9.8 −0.713012 − 0.411658i −2.75197 − 0.289244i −0.661076 − 1.14502i −0.465105 − 0.640162i 1.84312 + 1.33911i 0.216491 − 4.13089i 2.73518i 4.55525 + 0.968248i 0.0680978 + 0.647908i
9.9 −0.532365 − 0.307361i 1.57474 + 0.165512i −0.811058 − 1.40479i 1.15155 + 1.58497i −0.787464 − 0.572126i 0.205071 − 3.91299i 2.22660i −0.482037 − 0.102460i −0.125886 − 1.19772i
9.10 −0.108315 − 0.0625360i −0.269463 − 0.0283217i −0.992179 − 1.71850i −1.03912 − 1.43023i 0.0274159 + 0.0199188i −0.0838189 + 1.59936i 0.498331i −2.86263 − 0.608472i 0.0231123 + 0.219899i
9.11 0.310733 + 0.179402i −2.66649 − 0.280260i −0.935630 − 1.62056i 0.477120 + 0.656699i −0.778288 − 0.565460i −0.223404 + 4.26280i − 1.38902i 4.09719 + 0.870886i 0.0304439 + 0.289654i
9.12 0.459038 + 0.265026i 2.32971 + 0.244863i −0.859523 − 1.48874i −2.09916 − 2.88924i 1.00453 + 0.729835i −0.0615051 + 1.17359i − 1.97129i 2.43316 + 0.517183i −0.197869 − 1.88260i
9.13 0.794157 + 0.458507i 2.45833 + 0.258381i −0.579543 − 1.00380i 1.02316 + 1.40826i 1.83383 + 1.33236i 0.0402771 − 0.768533i − 2.89692i 3.04218 + 0.646636i 0.166854 + 1.58751i
9.14 0.840289 + 0.485141i −1.97751 − 0.207844i −0.529276 − 0.916733i 0.625741 + 0.861258i −1.56084 − 1.13402i 0.128748 − 2.45666i − 2.96766i 0.932885 + 0.198291i 0.107971 + 1.02728i
9.15 1.60829 + 0.928547i 0.150613 + 0.0158301i 0.724398 + 1.25469i 1.34771 + 1.85497i 0.227531 + 0.165311i −0.0684779 + 1.30664i − 1.02364i −2.91201 − 0.618967i 0.445089 + 4.23474i
9.16 1.67232 + 0.965512i −2.52592 − 0.265485i 0.864427 + 1.49723i −2.43407 − 3.35020i −3.96780 − 2.88278i 0.0127123 − 0.242565i − 0.523588i 3.37533 + 0.717449i −0.835864 − 7.95272i
9.17 1.86650 + 1.07762i 1.79608 + 0.188775i 1.32254 + 2.29070i −1.44031 − 1.98242i 3.14894 + 2.28784i −0.152314 + 2.90632i 1.39029i 0.255812 + 0.0543745i −0.552038 − 5.25229i
9.18 2.06561 + 1.19258i −2.79156 − 0.293405i 1.84449 + 3.19475i 1.52397 + 2.09757i −5.41636 − 3.93522i −0.0255690 + 0.487886i 4.02849i 4.77229 + 1.01438i 0.646414 + 6.15022i
9.19 2.30260 + 1.32941i 0.445113 + 0.0467833i 2.53464 + 4.39013i −0.525197 − 0.722872i 0.962723 + 0.699459i 0.183523 − 3.50183i 8.16064i −2.73851 − 0.582087i −0.248328 − 2.36269i
82.1 −2.16702 − 1.25113i 0.991120 − 2.22609i 2.13065 + 3.69039i 1.56916 − 2.15976i −4.93290 + 3.58396i −3.32812 + 2.16131i − 5.65833i −1.96578 − 2.18322i −6.10252 + 2.71702i
See next 80 embeddings (of 304 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.q even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.q.a ✓ 304
241.q even 60 1 inner 241.2.q.a ✓ 304
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.q.a ✓ 304 1.a even 1 1 trivial
241.2.q.a ✓ 304 241.q even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(241, [\chi])\).