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Properties

Label 275.2.l.d
Level $275$
Weight $2$
Character orbit 275.l
Analytic conductor $2.196$
Analytic rank $0$
Dimension $100$
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] = [275,2,Mod(31,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.l (of order \(5\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(25\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} - 2 q^{3} + 106 q^{4} + 5 q^{5} + 5 q^{6} + 5 q^{7} - 30 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q + 2 q^{2} - 2 q^{3} + 106 q^{4} + 5 q^{5} + 5 q^{6} + 5 q^{7} - 30 q^{8} - 25 q^{9} - 22 q^{10} - 53 q^{12} - 18 q^{13} - 10 q^{14} + 16 q^{15} + 62 q^{16} + q^{17} - 21 q^{18} + 6 q^{19} + q^{20} - 22 q^{21} + 6 q^{22} + 20 q^{23} - 18 q^{24} + q^{25} - 37 q^{26} + 10 q^{27} - 64 q^{28} - 28 q^{29} + 15 q^{30} + 9 q^{31} - 98 q^{32} + 13 q^{33} - 14 q^{34} + 13 q^{35} - 30 q^{36} - 21 q^{37} + 18 q^{38} + 44 q^{39} + 9 q^{40} + 28 q^{41} + 31 q^{42} + 82 q^{43} - 4 q^{44} - 25 q^{45} - 16 q^{46} - 10 q^{47} - 72 q^{48} - 26 q^{49} - 46 q^{50} - 14 q^{51} - 23 q^{52} - 5 q^{53} - 22 q^{54} + 32 q^{55} + q^{56} - 85 q^{57} + 80 q^{58} - 3 q^{59} + 61 q^{60} + 35 q^{61} + 43 q^{62} - 47 q^{63} - 54 q^{64} - 64 q^{65} - 13 q^{66} - 20 q^{67} + 5 q^{68} + 76 q^{69} - 23 q^{70} + 28 q^{71} - 53 q^{72} + 33 q^{73} - 3 q^{74} - 34 q^{75} - 156 q^{76} + 45 q^{77} + 30 q^{78} + 4 q^{79} - 144 q^{80} - 27 q^{81} - 59 q^{82} - 92 q^{83} + 9 q^{84} - 26 q^{85} - 56 q^{86} + 38 q^{87} + 18 q^{88} + 39 q^{89} + 114 q^{90} - 11 q^{91} - 60 q^{92} - 19 q^{93} + 46 q^{94} - 46 q^{95} + 9 q^{96} + 20 q^{97} + 122 q^{98} - 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.71083 −0.438651 + 1.35003i 5.34858 −2.23591 + 0.0264998i 1.18911 − 3.65969i 1.37359 + 0.997974i −9.07740 0.796888 + 0.578973i 6.06116 − 0.0718363i
31.2 −2.57726 0.590903 − 1.81861i 4.64225 −0.573113 − 2.16138i −1.52291 + 4.68703i −3.15069 − 2.28911i −6.80976 −0.531139 − 0.385895i 1.47706 + 5.57042i
31.3 −2.44032 −0.915135 + 2.81650i 3.95515 2.18577 − 0.471625i 2.23322 − 6.87314i −2.76256 − 2.00712i −4.77118 −4.66813 − 3.39159i −5.33396 + 1.15091i
31.4 −2.28652 0.204707 − 0.630023i 3.22819 1.97653 + 1.04562i −0.468067 + 1.44056i −0.442376 − 0.321405i −2.80829 2.07203 + 1.50542i −4.51938 − 2.39084i
31.5 −1.91520 −0.482800 + 1.48591i 1.66800 −0.0235589 + 2.23594i 0.924660 − 2.84581i 2.10304 + 1.52795i 0.635849 0.452229 + 0.328564i 0.0451200 − 4.28228i
31.6 −1.83159 0.699896 − 2.15406i 1.35471 1.42132 − 1.72623i −1.28192 + 3.94534i 2.79400 + 2.02996i 1.18190 −1.72306 − 1.25188i −2.60327 + 3.16173i
31.7 −1.49148 0.0955568 − 0.294094i 0.224502 −1.68887 − 1.46551i −0.142521 + 0.438634i 2.02744 + 1.47302i 2.64811 2.34969 + 1.70715i 2.51891 + 2.18577i
31.8 −1.47120 −0.500346 + 1.53991i 0.164422 −0.844616 − 2.07042i 0.736108 − 2.26551i −1.36475 − 0.991546i 2.70050 0.306083 + 0.222382i 1.24260 + 3.04599i
31.9 −1.07861 0.938461 − 2.88828i −0.836610 1.23116 + 1.86661i −1.01223 + 3.11532i −3.20906 − 2.33152i 3.05958 −5.03443 − 3.65773i −1.32794 − 2.01334i
31.10 −0.806214 −0.840984 + 2.58828i −1.35002 2.14075 − 0.645889i 0.678013 − 2.08671i 4.08974 + 2.97137i 2.70083 −3.56490 − 2.59005i −1.72591 + 0.520725i
31.11 −0.787013 0.316633 − 0.974495i −1.38061 −2.16322 + 0.566116i −0.249194 + 0.766940i −0.577283 − 0.419421i 2.66059 1.57767 + 1.14624i 1.70248 − 0.445541i
31.12 −0.462607 −0.145469 + 0.447707i −1.78599 2.02304 − 0.952527i 0.0672950 − 0.207113i −0.788024 − 0.572533i 1.75143 2.24777 + 1.63310i −0.935873 + 0.440646i
31.13 −0.205041 0.469534 − 1.44508i −1.95796 −0.320281 + 2.21301i −0.0962738 + 0.296300i 2.75899 + 2.00452i 0.811544 0.559262 + 0.406328i 0.0656708 − 0.453758i
31.14 −0.0196155 0.735377 − 2.26326i −1.99962 1.04148 − 1.97871i −0.0144248 + 0.0443950i −1.31439 − 0.954957i 0.0784545 −2.15451 − 1.56534i −0.0204292 + 0.0388135i
31.15 0.796321 0.147790 − 0.454851i −1.36587 −2.20171 − 0.390463i 0.117688 − 0.362208i −3.47097 − 2.52180i −2.68032 2.24200 + 1.62891i −1.75327 − 0.310934i
31.16 0.888309 0.964052 − 2.96705i −1.21091 −1.97035 − 1.05722i 0.856376 − 2.63565i 1.96632 + 1.42862i −2.85228 −5.44692 − 3.95742i −1.75028 − 0.939138i
31.17 1.30750 −0.414130 + 1.27456i −0.290451 −1.95087 + 1.09275i −0.541474 + 1.66648i 2.41591 + 1.75526i −2.99476 0.974049 + 0.707688i −2.55076 + 1.42877i
31.18 1.47809 −0.996172 + 3.06590i 0.184753 1.40497 + 1.73955i −1.47243 + 4.53168i −0.230114 − 0.167188i −2.68310 −5.98034 − 4.34497i 2.07668 + 2.57122i
31.19 1.57454 0.586933 − 1.80639i 0.479162 2.11673 + 0.720732i 0.924146 − 2.84423i 2.53222 + 1.83977i −2.39461 −0.491515 − 0.357107i 3.33287 + 1.13482i
31.20 1.70637 −0.215860 + 0.664349i 0.911711 0.843606 − 2.07083i −0.368338 + 1.13363i 2.29042 + 1.66409i −1.85703 2.03229 + 1.47654i 1.43951 − 3.53361i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.l even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.l.d yes 100
11.c even 5 1 275.2.k.d ✓ 100
25.d even 5 1 275.2.k.d ✓ 100
275.l even 5 1 inner 275.2.l.d yes 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.k.d ✓ 100 11.c even 5 1
275.2.k.d ✓ 100 25.d even 5 1
275.2.l.d yes 100 1.a even 1 1 trivial
275.2.l.d yes 100 275.l even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\):

\( T_{2}^{50} - T_{2}^{49} - 76 T_{2}^{48} + 80 T_{2}^{47} + 2697 T_{2}^{46} - 2975 T_{2}^{45} + \cdots + 2299 \) Copy content Toggle raw display
\( T_{3}^{100} + 2 T_{3}^{99} + 52 T_{3}^{98} + 102 T_{3}^{97} + 1505 T_{3}^{96} + 2930 T_{3}^{95} + \cdots + 202471413270025 \) Copy content Toggle raw display