Properties

Label 775.2.k.f
Level $775$
Weight $2$
Character orbit 775.k
Analytic conductor $6.188$
Analytic rank $0$
Dimension $36$
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] = [775,2,Mod(101,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.k (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + q^{3} - 10 q^{4} - 6 q^{6} - 6 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{2} + q^{3} - 10 q^{4} - 6 q^{6} - 6 q^{7} + 4 q^{8} - q^{11} - 16 q^{12} - 6 q^{13} - 15 q^{14} + 2 q^{16} + 11 q^{17} - 15 q^{18} + 11 q^{19} + 7 q^{21} - 4 q^{22} + 4 q^{23} + 10 q^{24} - 28 q^{26} - 17 q^{27} + 14 q^{28} + 25 q^{29} - 9 q^{31} + 46 q^{32} + 5 q^{33} + 18 q^{34} - 24 q^{36} - 24 q^{37} + 35 q^{38} + 7 q^{39} + q^{41} + 53 q^{42} - 2 q^{43} + 11 q^{44} + 36 q^{46} - 7 q^{47} + 64 q^{48} + 7 q^{49} - 34 q^{51} + 38 q^{52} - 13 q^{53} - 2 q^{54} - 36 q^{56} + 24 q^{57} - 34 q^{58} - 25 q^{59} + 54 q^{61} - 15 q^{62} + 18 q^{63} - 60 q^{64} - 8 q^{66} + 44 q^{67} - 32 q^{68} - 44 q^{69} + 32 q^{71} + 26 q^{72} + 21 q^{73} - 56 q^{74} - 15 q^{76} - 18 q^{77} + 8 q^{78} + 14 q^{79} - 46 q^{81} + 10 q^{82} + 21 q^{83} + 27 q^{84} + 25 q^{86} - 156 q^{87} + 8 q^{88} - 6 q^{89} - 64 q^{91} - 92 q^{92} - 50 q^{93} + 22 q^{94} + 36 q^{96} - 13 q^{97} + 88 q^{98} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.756851 + 2.32935i −0.554304 1.70597i −3.23501 2.35037i 0 4.39333 2.00183 + 1.45441i 3.96032 2.87734i −0.176040 + 0.127900i 0
101.2 −0.679709 + 2.09193i 0.127997 + 0.393934i −2.29613 1.66823i 0 −0.911081 −2.45457 1.78335i 1.49152 1.08366i 2.28825 1.66251i 0
101.3 −0.593461 + 1.82648i 0.885070 + 2.72397i −1.36582 0.992323i 0 −5.50054 0.299634 + 0.217697i −0.484381 + 0.351923i −4.20959 + 3.05845i 0
101.4 −0.216539 + 0.666437i −0.898636 2.76572i 1.22078 + 0.886952i 0 2.03777 0.447464 + 0.325102i −1.98926 + 1.44528i −4.41459 + 3.20739i 0
101.5 −0.140176 + 0.431417i 0.352463 + 1.08477i 1.45156 + 1.05462i 0 −0.517395 1.66302 + 1.20825i −1.39243 + 1.01166i 1.37455 0.998672i 0
101.6 0.237416 0.730690i 0.201777 + 0.621004i 1.14049 + 0.828616i 0 0.501667 −3.69321 2.68327i 2.11936 1.53980i 2.08212 1.51275i 0
101.7 0.262890 0.809091i −0.441915 1.36007i 1.03252 + 0.750167i 0 −1.21660 0.894593 + 0.649960i 2.25490 1.63828i 0.772536 0.561280i 0
101.8 0.664374 2.04473i −0.331864 1.02137i −2.12151 1.54137i 0 −2.30891 3.55777 + 2.58487i −1.08245 + 0.786446i 1.49398 1.08544i 0
101.9 0.722056 2.22226i −0.767639 2.36255i −2.79904 2.03362i 0 −5.80447 −4.21653 3.06349i −2.75955 + 2.00493i −2.56532 + 1.86381i 0
126.1 −2.10630 1.53032i −1.44142 + 1.04725i 1.47660 + 4.54452i 0 4.63870 −0.566161 1.74246i 2.23532 6.87960i 0.0539012 0.165891i 0
126.2 −1.45154 1.05460i 0.436799 0.317353i 0.376735 + 1.15947i 0 −0.968711 −0.374177 1.15160i −0.432937 + 1.33244i −0.836970 + 2.57593i 0
126.3 −0.930073 0.675738i 2.66102 1.93334i −0.209619 0.645141i 0 −3.78137 −0.813453 2.50355i −0.951498 + 2.92841i 2.41615 7.43615i 0
126.4 −0.750933 0.545585i −2.30185 + 1.67239i −0.351796 1.08272i 0 2.64097 0.289318 + 0.890428i −0.900201 + 2.77053i 1.57457 4.84604i 0
126.5 −0.470650 0.341947i 0.758887 0.551364i −0.513450 1.58024i 0 −0.545708 0.739195 + 2.27501i −0.658247 + 2.02588i −0.655143 + 2.01632i 0
126.6 0.601276 + 0.436853i −1.34036 + 0.973831i −0.447341 1.37678i 0 −1.23135 −1.14480 3.52332i 0.791806 2.43693i −0.0788236 + 0.242594i 0
126.7 1.19002 + 0.864600i 2.40953 1.75063i 0.0505793 + 0.155667i 0 4.38098 1.26381 + 3.88961i 0.834694 2.56893i 1.81409 5.58319i 0
126.8 1.44025 + 1.04641i 1.28666 0.934816i 0.361331 + 1.11206i 0 2.83132 −1.30585 4.01899i 0.456995 1.40649i −0.145428 + 0.447583i 0
126.9 1.97795 + 1.43706i −0.542210 + 0.393939i 1.22909 + 3.78276i 0 −1.63858 0.412110 + 1.26835i −1.49396 + 4.59795i −0.788247 + 2.42597i 0
326.1 −2.10630 + 1.53032i −1.44142 1.04725i 1.47660 4.54452i 0 4.63870 −0.566161 + 1.74246i 2.23532 + 6.87960i 0.0539012 + 0.165891i 0
326.2 −1.45154 + 1.05460i 0.436799 + 0.317353i 0.376735 1.15947i 0 −0.968711 −0.374177 + 1.15160i −0.432937 1.33244i −0.836970 2.57593i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.k.f 36
5.b even 2 1 775.2.k.g yes 36
5.c odd 4 2 775.2.bf.e 72
31.d even 5 1 inner 775.2.k.f 36
155.n even 10 1 775.2.k.g yes 36
155.s odd 20 2 775.2.bf.e 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.k.f 36 1.a even 1 1 trivial
775.2.k.f 36 31.d even 5 1 inner
775.2.k.g yes 36 5.b even 2 1
775.2.k.g yes 36 155.n even 10 1
775.2.bf.e 72 5.c odd 4 2
775.2.bf.e 72 155.s odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 2 T_{2}^{35} + 16 T_{2}^{34} + 24 T_{2}^{33} + 141 T_{2}^{32} + 188 T_{2}^{31} + 1034 T_{2}^{30} + 1240 T_{2}^{29} + 6847 T_{2}^{28} + 7973 T_{2}^{27} + 34561 T_{2}^{26} + 36292 T_{2}^{25} + 128399 T_{2}^{24} + \cdots + 22201 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display