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Properties

Label 350.2.u.a
Level $350$
Weight $2$
Character orbit 350.u
Analytic conductor $2.795$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(9,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.u (of order \(30\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 20 q^{4} - 2 q^{5} + 8 q^{6} - 20 q^{9} - 2 q^{10} - 6 q^{11} - 16 q^{15} + 20 q^{16} - 20 q^{17} + 4 q^{19} - 4 q^{20} - 12 q^{21} + 40 q^{22} - 30 q^{23} - 16 q^{24} + 48 q^{26} - 120 q^{27} - 10 q^{28} + 24 q^{29} - 26 q^{30} - 6 q^{31} - 50 q^{33} - 16 q^{34} - 36 q^{35} - 40 q^{36} - 32 q^{39} + 2 q^{40} + 68 q^{41} + 10 q^{42} - 4 q^{44} + 16 q^{45} + 12 q^{46} + 24 q^{49} + 32 q^{50} + 12 q^{51} - 20 q^{53} + 16 q^{54} + 24 q^{55} + 24 q^{59} + 2 q^{60} - 8 q^{61} + 40 q^{64} + 2 q^{65} - 116 q^{69} - 118 q^{70} + 68 q^{71} - 40 q^{73} + 16 q^{74} + 196 q^{75} - 32 q^{76} - 80 q^{77} - 8 q^{79} - 2 q^{80} + 64 q^{81} - 160 q^{83} - 18 q^{84} + 12 q^{85} + 12 q^{86} - 10 q^{88} + 54 q^{89} + 44 q^{91} - 40 q^{92} - 16 q^{95} - 4 q^{96} - 20 q^{97} + 40 q^{98} - 24 q^{99}+O(q^{100}) \) 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 −0.406737 − 0.913545i −2.45852 − 2.21366i −0.669131 + 0.743145i 0.946275 + 2.02597i −1.02231 + 3.14634i 2.60820 − 0.444160i 0.951057 + 0.309017i 0.830434 + 7.90106i 1.46593 − 1.68850i
9.2 −0.406737 − 0.913545i −1.78149 − 1.60406i −0.669131 + 0.743145i −1.13845 − 1.92456i −0.740787 + 2.27991i −1.78876 − 1.94944i 0.951057 + 0.309017i 0.287113 + 2.73170i −1.29513 + 1.82281i
9.3 −0.406737 − 0.913545i −1.31575 − 1.18470i −0.669131 + 0.743145i 0.876426 − 2.05715i −0.547119 + 1.68386i 2.56906 + 0.632387i 0.951057 + 0.309017i 0.0140817 + 0.133978i −2.23578 + 0.0360644i
9.4 −0.406737 − 0.913545i −1.07672 − 0.969483i −0.669131 + 0.743145i −2.20911 + 0.346191i −0.447725 + 1.37796i 0.558372 + 2.58616i 0.951057 + 0.309017i −0.0941570 − 0.895844i 1.21479 + 1.87731i
9.5 −0.406737 − 0.913545i −0.895050 − 0.805906i −0.669131 + 0.743145i 2.05465 + 0.882269i −0.372183 + 1.14546i −2.64286 + 0.123674i 0.951057 + 0.309017i −0.161957 − 1.54091i −0.0297100 − 2.23587i
9.6 −0.406737 − 0.913545i −0.0285986 − 0.0257503i −0.669131 + 0.743145i −1.79621 + 1.33177i −0.0118920 + 0.0365997i 1.01591 − 2.44293i 0.951057 + 0.309017i −0.313431 − 2.98209i 1.94722 + 1.09924i
9.7 −0.406737 − 0.913545i 0.676976 + 0.609552i −0.669131 + 0.743145i 2.23601 + 0.0166621i 0.281502 − 0.866375i 0.728409 + 2.54351i 0.951057 + 0.309017i −0.226843 − 2.15826i −0.894244 − 2.04947i
9.8 −0.406737 − 0.913545i 1.34935 + 1.21496i −0.669131 + 0.743145i 1.21468 − 1.87738i 0.561093 − 1.72687i −0.495058 − 2.59902i 0.951057 + 0.309017i 0.0310342 + 0.295270i −2.20913 − 0.346063i
9.9 −0.406737 − 0.913545i 1.91099 + 1.72067i −0.669131 + 0.743145i −0.421866 + 2.19591i 0.794636 − 2.44564i −2.51031 + 0.835675i 0.951057 + 0.309017i 0.377618 + 3.59279i 2.17765 − 0.507765i
9.10 −0.406737 − 0.913545i 2.13251 + 1.92012i −0.669131 + 0.743145i −2.06461 − 0.858708i 0.886748 − 2.72913i 2.62238 + 0.350889i 0.951057 + 0.309017i 0.547152 + 5.20580i 0.0552844 + 2.23538i
9.11 0.406737 + 0.913545i −2.02781 − 1.82584i −0.669131 + 0.743145i 1.93081 + 1.12782i 0.843209 − 2.59513i −1.42835 + 2.22707i −0.951057 − 0.309017i 0.464703 + 4.42135i −0.244982 + 2.22261i
9.12 0.406737 + 0.913545i −1.54024 − 1.38684i −0.669131 + 0.743145i −2.08526 + 0.807273i 0.640467 − 1.97115i 2.52231 + 0.798718i −0.951057 − 0.309017i 0.135432 + 1.28855i −1.58563 − 1.57663i
9.13 0.406737 + 0.913545i −1.12000 − 1.00845i −0.669131 + 0.743145i 0.567199 − 2.16293i 0.465720 − 1.43334i −2.39266 + 1.12923i −0.951057 − 0.309017i −0.0761640 − 0.724652i 2.20664 − 0.361583i
9.14 0.406737 + 0.913545i −0.383437 − 0.345248i −0.669131 + 0.743145i 2.15736 − 0.588036i 0.159442 − 0.490712i 0.792205 − 2.52436i −0.951057 − 0.309017i −0.285758 − 2.71880i 1.41468 + 1.73167i
9.15 0.406737 + 0.913545i −0.339833 − 0.305987i −0.669131 + 0.743145i 0.421551 + 2.19597i 0.141311 − 0.434909i 2.63940 + 0.183264i −0.951057 − 0.309017i −0.291727 − 2.77560i −1.83466 + 1.27829i
9.16 0.406737 + 0.913545i −0.0530570 − 0.0477727i −0.669131 + 0.743145i −1.81178 − 1.31051i 0.0220623 − 0.0679009i −0.672893 − 2.55875i −0.951057 − 0.309017i −0.313053 − 2.97850i 0.460294 − 2.18818i
9.17 0.406737 + 0.913545i 1.13669 + 1.02348i −0.669131 + 0.743145i −2.18536 + 0.473503i −0.472661 + 1.45470i −1.72507 + 2.00602i −0.951057 − 0.309017i −0.0690344 − 0.656819i −1.32143 − 1.80383i
9.18 0.406737 + 0.913545i 1.55431 + 1.39950i −0.669131 + 0.743145i 1.71274 + 1.43754i −0.646317 + 1.98916i 0.734885 + 2.54164i −0.951057 − 0.309017i 0.143672 + 1.36695i −0.616628 + 2.14936i
9.19 0.406737 + 0.913545i 2.06743 + 1.86152i −0.669131 + 0.743145i 0.677494 − 2.13096i −0.859686 + 2.64584i 2.16518 − 1.52052i −0.951057 − 0.309017i 0.495416 + 4.71357i 2.22229 − 0.247819i
9.20 0.406737 + 0.913545i 2.19223 + 1.97389i −0.669131 + 0.743145i −0.873486 + 2.05840i −0.911582 + 2.80556i 0.0303467 − 2.64558i −0.951057 − 0.309017i 0.596036 + 5.67091i −2.23572 + 0.0392584i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.e even 10 1 inner
175.t even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.u.a ✓ 160
7.c even 3 1 inner 350.2.u.a ✓ 160
25.e even 10 1 inner 350.2.u.a ✓ 160
175.t even 30 1 inner 350.2.u.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.u.a ✓ 160 1.a even 1 1 trivial
350.2.u.a ✓ 160 7.c even 3 1 inner
350.2.u.a ✓ 160 25.e even 10 1 inner
350.2.u.a ✓ 160 175.t even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(350, [\chi])\).