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Properties

Label 241.2.l.a
Level $241$
Weight $2$
Character orbit 241.l
Analytic conductor $1.924$
Analytic rank $0$
Dimension $144$
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(6,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.l (of order \(20\), 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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 10 q^{3} - 120 q^{4} - 10 q^{5} + 6 q^{6} - 10 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 10 q^{3} - 120 q^{4} - 10 q^{5} + 6 q^{6} - 10 q^{7} + 14 q^{9} - 2 q^{10} - 14 q^{11} - 20 q^{12} - 26 q^{13} + 6 q^{14} - 60 q^{15} + 56 q^{16} - 38 q^{17} + 10 q^{18} + 12 q^{19} - 30 q^{20} - 16 q^{21} - 4 q^{23} + 10 q^{25} + 16 q^{26} - 10 q^{27} - 4 q^{28} - 10 q^{29} - 44 q^{31} + 20 q^{33} + 136 q^{34} - 14 q^{35} + 30 q^{36} - 12 q^{37} + 22 q^{38} + 64 q^{40} - 10 q^{41} + 26 q^{42} + 6 q^{43} + 8 q^{44} - 40 q^{45} - 56 q^{46} - 10 q^{47} - 10 q^{49} - 170 q^{50} + 26 q^{51} + 2 q^{52} + 50 q^{53} + 32 q^{55} + 42 q^{56} + 78 q^{57} + 38 q^{58} + 10 q^{59} - 72 q^{60} + 50 q^{61} - 44 q^{62} + 150 q^{63} + 104 q^{64} + 58 q^{65} - 30 q^{66} + 20 q^{67} + 52 q^{68} - 42 q^{69} + 58 q^{70} - 26 q^{71} + 100 q^{72} - 32 q^{73} - 98 q^{74} - 140 q^{75} - 128 q^{76} - 130 q^{77} - 130 q^{78} + 50 q^{79} - 80 q^{80} + 60 q^{81} - 26 q^{82} + 2 q^{83} - 98 q^{84} + 6 q^{85} - 8 q^{86} + 62 q^{87} + 52 q^{88} + 84 q^{89} - 24 q^{90} + 22 q^{91} + 10 q^{92} + 20 q^{93} + 130 q^{94} - 104 q^{95} + 76 q^{96} + 52 q^{97} + 10 q^{98} - 74 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 − 2.45426i 0.00731224 + 0.00237589i −4.02339 2.76684 − 0.899001i 0.00583105 − 0.0179461i 0.223410 − 1.41055i 4.96592i −2.42700 − 1.76332i −2.20638 − 6.79055i
6.2 − 2.26591i −2.01017 − 0.653145i −3.13433 −0.919661 + 0.298816i −1.47996 + 4.55486i −0.558386 + 3.52551i 2.57029i 1.18714 + 0.862511i 0.677089 + 2.08387i
6.3 − 2.12343i 2.53741 + 0.824455i −2.50896 1.30733 − 0.424776i 1.75067 − 5.38802i −0.159330 + 1.00597i 1.08073i 3.33169 + 2.42061i −0.901982 − 2.77602i
6.4 − 2.06163i 0.487344 + 0.158348i −2.25033 −3.22438 + 1.04767i 0.326455 − 1.00472i 0.590185 − 3.72628i 0.516082i −2.21462 − 1.60902i 2.15990 + 6.64749i
6.5 − 1.37169i −3.16210 − 1.02743i 0.118476 4.03305 − 1.31042i −1.40931 + 4.33741i 0.385709 − 2.43527i − 2.90589i 6.51623 + 4.73432i −1.79748 − 5.53208i
6.6 − 1.15783i −0.867138 − 0.281750i 0.659429 −0.0490953 + 0.0159520i −0.326219 + 1.00400i 0.110876 − 0.700045i − 3.07917i −1.75451 − 1.27472i 0.0184697 + 0.0568440i
6.7 − 1.10511i 2.11886 + 0.688459i 0.778736 −1.35885 + 0.441517i 0.760822 − 2.34157i −0.218915 + 1.38217i − 3.07080i 1.58854 + 1.15414i 0.487924 + 1.50168i
6.8 − 0.400056i 0.252540 + 0.0820551i 1.83995 3.41552 − 1.10977i 0.0328266 − 0.101030i −0.786559 + 4.96614i − 1.53620i −2.37001 − 1.72191i −0.443971 − 1.36640i
6.9 − 0.273296i −2.11663 − 0.687736i 1.92531 −3.20357 + 1.04090i −0.187955 + 0.578468i 0.0587466 − 0.370911i − 1.07277i 1.58011 + 1.14802i 0.284475 + 0.875523i
6.10 0.531224i −1.43464 − 0.466142i 1.71780 0.939984 − 0.305419i 0.247626 − 0.762114i 0.517947 − 3.27019i 1.97499i −0.586156 − 0.425867i 0.162246 + 0.499342i
6.11 0.701751i 2.43038 + 0.789680i 1.50755 −2.01585 + 0.654989i −0.554158 + 1.70552i 0.454170 − 2.86752i 2.46142i 2.85612 + 2.07509i −0.459639 − 1.41462i
6.12 0.724418i 1.00171 + 0.325474i 1.47522 −2.36176 + 0.767384i −0.235779 + 0.725654i −0.594250 + 3.75195i 2.51751i −1.52957 − 1.11130i −0.555907 − 1.71090i
6.13 0.968747i 0.249445 + 0.0810497i 1.06153 2.12994 − 0.692060i −0.0785167 + 0.241649i 0.146415 − 0.924426i 2.96585i −2.37140 − 1.72292i 0.670431 + 2.06337i
6.14 1.62967i −0.600382 − 0.195076i −0.655838 −2.89402 + 0.940325i 0.317910 − 0.978426i −0.415190 + 2.62140i 2.19055i −2.10465 − 1.52912i −1.53242 − 4.71632i
6.15 1.81678i −2.24713 − 0.730135i −1.30067 2.15538 − 0.700325i 1.32649 − 4.08252i −0.458090 + 2.89226i 1.27052i 2.08942 + 1.51805i 1.27233 + 3.91584i
6.16 1.93203i −2.99023 − 0.971584i −1.73276 −2.36656 + 0.768943i 1.87713 − 5.77722i 0.401702 − 2.53625i 0.516326i 5.57044 + 4.04716i −1.48562 − 4.57228i
6.17 2.14238i 1.81045 + 0.588250i −2.58980 1.78548 − 0.580137i −1.26026 + 3.87867i 0.0328580 − 0.207457i − 1.26358i 0.504627 + 0.366633i 1.24287 + 3.82518i
6.18 2.76620i −0.717001 − 0.232968i −5.65186 −0.773207 + 0.251230i 0.644435 − 1.98337i −0.174086 + 1.09914i − 10.1018i −1.96723 − 1.42928i −0.694953 − 2.13885i
25.1 − 2.76145i 0.788005 + 1.08460i −5.62561 1.43592 − 1.97637i 2.99506 − 2.17604i −3.45658 − 1.76122i 10.0119i 0.371655 − 1.14384i −5.45766 − 3.96522i
25.2 − 2.39700i −0.661252 − 0.910136i −3.74561 −2.59396 + 3.57029i −2.18160 + 1.58502i −0.580275 − 0.295665i 4.18423i 0.535959 − 1.64951i 8.55798 + 6.21773i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
241.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 241.2.l.a ✓ 144
241.l even 20 1 inner 241.2.l.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.l.a ✓ 144 1.a even 1 1 trivial
241.2.l.a ✓ 144 241.l even 20 1 inner

Hecke kernels

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