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Properties

Label 735.2.u.d
Level $735$
Weight $2$
Character orbit 735.u
Analytic conductor $5.869$
Analytic rank $0$
Dimension $60$
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] = [735,2,Mod(106,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.u (of order \(7\), degree \(6\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + q^{2} + 10 q^{3} - 7 q^{4} - 10 q^{5} - q^{6} + 7 q^{7} - 9 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + q^{2} + 10 q^{3} - 7 q^{4} - 10 q^{5} - q^{6} + 7 q^{7} - 9 q^{8} - 10 q^{9} + q^{10} + 6 q^{11} + 7 q^{12} + 4 q^{13} + 21 q^{14} + 10 q^{15} - 29 q^{16} - 9 q^{17} - 6 q^{18} + 18 q^{19} - 7 q^{20} + 7 q^{21} + 41 q^{22} - 12 q^{24} - 10 q^{25} + 38 q^{26} + 10 q^{27} - 21 q^{28} - 12 q^{29} + 6 q^{30} - 30 q^{31} + 45 q^{32} + q^{33} - 12 q^{34} - 7 q^{35} - 7 q^{36} + 11 q^{37} + 37 q^{38} - 4 q^{39} + 12 q^{40} - 6 q^{41} - 14 q^{42} - 6 q^{43} - 39 q^{44} - 10 q^{45} + 4 q^{46} - 42 q^{47} - 90 q^{48} - 21 q^{49} - 6 q^{50} + 9 q^{51} + 2 q^{52} - 17 q^{53} - q^{54} - q^{55} - 35 q^{56} + 17 q^{57} - 40 q^{58} + 19 q^{59} + 7 q^{60} + 31 q^{61} + 30 q^{62} + 7 q^{63} + q^{64} + 4 q^{65} - 41 q^{66} - 66 q^{67} + 94 q^{68} + 14 q^{70} + 46 q^{71} + 12 q^{72} + 8 q^{73} + 30 q^{74} + 10 q^{75} - 46 q^{76} - 38 q^{78} + 42 q^{79} + 90 q^{80} - 10 q^{81} - 59 q^{82} - 35 q^{83} - 9 q^{85} - 47 q^{86} + 5 q^{87} + 41 q^{88} - 67 q^{89} + q^{90} - 105 q^{91} - 47 q^{92} + 2 q^{93} - 100 q^{94} - 17 q^{95} - 45 q^{96} + 150 q^{97} - 14 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −1.69498 − 2.12544i 0.900969 − 0.433884i −1.19948 + 5.25528i −0.900969 + 0.433884i −2.44931 − 1.17953i −0.584581 + 2.58036i 8.30423 − 3.99910i 0.623490 − 0.781831i 2.44931 + 1.17953i
106.2 −1.33255 − 1.67096i 0.900969 − 0.433884i −0.571392 + 2.50343i −0.900969 + 0.433884i −1.92559 − 0.927315i 2.62019 + 0.366893i 1.09337 − 0.526540i 0.623490 − 0.781831i 1.92559 + 0.927315i
106.3 −1.28937 − 1.61682i 0.900969 − 0.433884i −0.506587 + 2.21950i −0.900969 + 0.433884i −1.86319 − 0.897267i −2.63675 − 0.218029i 0.515325 − 0.248167i 0.623490 − 0.781831i 1.86319 + 0.897267i
106.4 −0.761139 − 0.954439i 0.900969 − 0.433884i 0.113422 − 0.496934i −0.900969 + 0.433884i −1.09988 − 0.529673i −0.302392 − 2.62841i −2.76038 + 1.32933i 0.623490 − 0.781831i 1.09988 + 0.529673i
106.5 −0.538574 − 0.675351i 0.900969 − 0.433884i 0.279006 − 1.22240i −0.900969 + 0.433884i −0.778262 − 0.374791i 1.97533 + 1.76013i −2.53234 + 1.21951i 0.623490 − 0.781831i 0.778262 + 0.374791i
106.6 0.261032 + 0.327324i 0.900969 − 0.433884i 0.406039 − 1.77897i −0.900969 + 0.433884i 0.377203 + 0.181651i −2.28980 − 1.32545i 1.44270 − 0.694766i 0.623490 − 0.781831i −0.377203 − 0.181651i
106.7 0.744805 + 0.933956i 0.900969 − 0.433884i 0.127502 − 0.558624i −0.900969 + 0.433884i 1.07628 + 0.518307i 2.50996 − 0.836717i 2.76925 − 1.33360i 0.623490 − 0.781831i −1.07628 − 0.518307i
106.8 0.908357 + 1.13904i 0.900969 − 0.433884i −0.0272663 + 0.119461i −0.900969 + 0.433884i 1.31261 + 0.632122i −1.54066 + 2.15090i 2.46439 − 1.18679i 0.623490 − 0.781831i −1.31261 − 0.632122i
106.9 1.33513 + 1.67421i 0.900969 − 0.433884i −0.575339 + 2.52073i −0.900969 + 0.433884i 1.92933 + 0.929114i 1.24066 − 2.33683i −1.12972 + 0.544044i 0.623490 − 0.781831i −1.92933 − 0.929114i
106.10 1.74379 + 2.18665i 0.900969 − 0.433884i −1.29557 + 5.67626i −0.900969 + 0.433884i 2.51985 + 1.21350i 0.909006 + 2.48469i −9.63147 + 4.63827i 0.623490 − 0.781831i −2.51985 − 1.21350i
211.1 −0.511495 + 2.24101i −0.623490 + 0.781831i −2.95855 − 1.42476i 0.623490 − 0.781831i −1.43318 − 1.79715i 2.59335 + 0.523976i 1.83983 − 2.30708i −0.222521 − 0.974928i 1.43318 + 1.79715i
211.2 −0.446564 + 1.95652i −0.623490 + 0.781831i −1.82663 − 0.879657i 0.623490 − 0.781831i −1.25124 − 1.56901i 1.19603 − 2.35998i 0.0342858 − 0.0429931i −0.222521 − 0.974928i 1.25124 + 1.56901i
211.3 −0.299530 + 1.31233i −0.623490 + 0.781831i 0.169451 + 0.0816035i 0.623490 − 0.781831i −0.839265 − 1.05241i −1.75004 + 1.98428i −1.83638 + 2.30274i −0.222521 − 0.974928i 0.839265 + 1.05241i
211.4 −0.223500 + 0.979215i −0.623490 + 0.781831i 0.893027 + 0.430059i 0.623490 − 0.781831i −0.626232 − 0.785270i −1.22283 − 2.34621i −1.87318 + 2.34889i −0.222521 − 0.974928i 0.626232 + 0.785270i
211.5 −0.0584752 + 0.256197i −0.623490 + 0.781831i 1.73972 + 0.837805i 0.623490 − 0.781831i −0.163844 − 0.205454i 2.43492 + 1.03496i −0.644061 + 0.807627i −0.222521 − 0.974928i 0.163844 + 0.205454i
211.6 0.0630617 − 0.276291i −0.623490 + 0.781831i 1.72958 + 0.832921i 0.623490 − 0.781831i 0.176695 + 0.221568i −2.48553 − 0.906726i 0.692589 − 0.868479i −0.222521 − 0.974928i −0.176695 − 0.221568i
211.7 0.237591 − 1.04095i −0.623490 + 0.781831i 0.774805 + 0.373127i 0.623490 − 0.781831i 0.665714 + 0.834779i 0.538866 − 2.59029i 1.90392 − 2.38744i −0.222521 − 0.974928i −0.665714 − 0.834779i
211.8 0.364276 − 1.59600i −0.623490 + 0.781831i −0.612571 − 0.294998i 0.623490 − 0.781831i 1.02068 + 1.27989i 0.179267 + 2.63967i 1.34740 − 1.68958i −0.222521 − 0.974928i −1.02068 − 1.27989i
211.9 0.469976 − 2.05910i −0.623490 + 0.781831i −2.21708 − 1.06769i 0.623490 − 0.781831i 1.31684 + 1.65127i −2.47858 + 0.925553i −0.606760 + 0.760853i −0.222521 − 0.974928i −1.31684 − 1.65127i
211.10 0.627181 − 2.74786i −0.623490 + 0.781831i −5.35544 − 2.57904i 0.623490 − 0.781831i 1.75732 + 2.20361i 1.37105 + 2.26279i −6.93103 + 8.69123i −0.222521 − 0.974928i −1.75732 − 2.20361i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.u.d ✓ 60
49.e even 7 1 inner 735.2.u.d ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.u.d ✓ 60 1.a even 1 1 trivial
735.2.u.d ✓ 60 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - T_{2}^{59} + 14 T_{2}^{58} - 12 T_{2}^{57} + 155 T_{2}^{56} - 161 T_{2}^{55} + \cdots + 6964321 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display