↑

Properties

Label 490.2.k.g
Level $490$
Weight $2$
Character orbit 490.k
Analytic conductor $3.913$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [490,2,Mod(71,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.k (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: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{2} + q^{3} - 5 q^{4} + 5 q^{5} + q^{6} + q^{7} - 5 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 5 q^{2} + q^{3} - 5 q^{4} + 5 q^{5} + q^{6} + q^{7} - 5 q^{8} - 16 q^{9} + 5 q^{10} - 5 q^{11} + q^{12} - 5 q^{13} + 8 q^{14} - q^{15} - 5 q^{16} + 9 q^{17} + 40 q^{18} - 20 q^{19} + 5 q^{20} + 9 q^{22} - 23 q^{23} + q^{24} - 5 q^{25} + 9 q^{26} + 25 q^{27} - 6 q^{28} + 12 q^{29} + 6 q^{30} - 32 q^{31} - 5 q^{32} + 25 q^{33} - 5 q^{34} - q^{35} - 16 q^{36} - 47 q^{37} - 20 q^{38} - 57 q^{39} + 5 q^{40} + 11 q^{41} - 35 q^{42} + 10 q^{43} - 12 q^{44} + 2 q^{45} - 9 q^{46} + 12 q^{47} - 6 q^{48} - 15 q^{49} + 30 q^{50} - 13 q^{51} - 5 q^{52} - 14 q^{53} + 4 q^{54} + 12 q^{55} + q^{56} + 42 q^{57} - 16 q^{58} + 13 q^{59} - q^{60} - 28 q^{61} + 10 q^{62} + 25 q^{63} - 5 q^{64} + 5 q^{65} - 10 q^{66} + 76 q^{67} - 40 q^{68} + 52 q^{69} - q^{70} + 41 q^{71} - 2 q^{72} + 15 q^{73} + 9 q^{74} + q^{75} + 15 q^{76} - 17 q^{77} + 27 q^{78} + 96 q^{79} - 30 q^{80} - 75 q^{81} + 11 q^{82} + q^{83} + 56 q^{84} + 5 q^{85} - 11 q^{86} - 58 q^{87} - 12 q^{88} - 47 q^{89} + 2 q^{90} + 8 q^{91} - 9 q^{92} + 66 q^{93} - 16 q^{94} - 15 q^{95} + q^{96} + 24 q^{97} - 15 q^{98} - 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −0.900969 + 0.433884i −0.569762 + 2.49629i 0.623490 − 0.781831i 0.222521 − 0.974928i −0.569762 − 2.49629i 2.25534 − 1.38327i −0.222521 + 0.974928i −3.20393 − 1.54293i 0.222521 + 0.974928i
71.2 −0.900969 + 0.433884i −0.461392 + 2.02149i 0.623490 − 0.781831i 0.222521 − 0.974928i −0.461392 − 2.02149i −1.44858 − 2.21396i −0.222521 + 0.974928i −1.17063 − 0.563747i 0.222521 + 0.974928i
71.3 −0.900969 + 0.433884i 0.143685 − 0.629525i 0.623490 − 0.781831i 0.222521 − 0.974928i 0.143685 + 0.629525i 1.55562 + 2.14010i −0.222521 + 0.974928i 2.32725 + 1.12074i 0.222521 + 0.974928i
71.4 −0.900969 + 0.433884i 0.406537 − 1.78115i 0.623490 − 0.781831i 0.222521 − 0.974928i 0.406537 + 1.78115i 0.964708 − 2.46360i −0.222521 + 0.974928i −0.304331 − 0.146558i 0.222521 + 0.974928i
71.5 −0.900969 + 0.433884i 0.703453 − 3.08203i 0.623490 − 0.781831i 0.222521 − 0.974928i 0.703453 + 3.08203i −2.42613 + 1.05542i −0.222521 + 0.974928i −6.30115 − 3.03447i 0.222521 + 0.974928i
141.1 0.623490 − 0.781831i −1.79255 − 0.863245i −0.222521 − 0.974928i 0.900969 + 0.433884i −1.79255 + 0.863245i 2.10503 − 1.60277i −0.900969 − 0.433884i 0.597563 + 0.749320i 0.900969 − 0.433884i
141.2 0.623490 − 0.781831i −1.43984 − 0.693388i −0.222521 − 0.974928i 0.900969 + 0.433884i −1.43984 + 0.693388i −2.63179 − 0.271420i −0.900969 − 0.433884i −0.278130 − 0.348763i 0.900969 − 0.433884i
141.3 0.623490 − 0.781831i −0.252998 − 0.121838i −0.222521 − 0.974928i 0.900969 + 0.433884i −0.252998 + 0.121838i −1.06287 + 2.42287i −0.900969 − 0.433884i −1.82131 − 2.28385i 0.900969 − 0.433884i
141.4 0.623490 − 0.781831i 2.03225 + 0.978682i −0.222521 − 0.974928i 0.900969 + 0.433884i 2.03225 − 0.978682i 0.0762540 − 2.64465i −0.900969 − 0.433884i 1.30177 + 1.63237i 0.900969 − 0.433884i
141.5 0.623490 − 0.781831i 2.35410 + 1.13367i −0.222521 − 0.974928i 0.900969 + 0.433884i 2.35410 − 1.13367i 0.889889 + 2.49161i −0.900969 − 0.433884i 2.38608 + 2.99205i 0.900969 − 0.433884i
211.1 −0.222521 + 0.974928i −2.13841 + 2.68148i −0.900969 − 0.433884i −0.623490 + 0.781831i −2.13841 − 2.68148i 1.90698 − 1.83397i 0.623490 − 0.781831i −1.94997 − 8.54338i −0.623490 − 0.781831i
211.2 −0.222521 + 0.974928i −0.821191 + 1.02974i −0.900969 − 0.433884i −0.623490 + 0.781831i −0.821191 − 1.02974i 1.94374 + 1.79496i 0.623490 − 0.781831i 0.281551 + 1.23356i −0.623490 − 0.781831i
211.3 −0.222521 + 0.974928i 0.0279169 − 0.0350067i −0.900969 − 0.433884i −0.623490 + 0.781831i 0.0279169 + 0.0350067i −1.98180 − 1.75284i 0.623490 − 0.781831i 0.667117 + 2.92283i −0.623490 − 0.781831i
211.4 −0.222521 + 0.974928i 0.607396 − 0.761651i −0.900969 − 0.433884i −0.623490 + 0.781831i 0.607396 + 0.761651i 0.958089 − 2.46618i 0.623490 − 0.781831i 0.456381 + 1.99954i −0.623490 − 0.781831i
211.5 −0.222521 + 0.974928i 1.70079 − 2.13273i −0.900969 − 0.433884i −0.623490 + 0.781831i 1.70079 + 2.13273i −2.60448 + 0.465484i 0.623490 − 0.781831i −0.988266 − 4.32987i −0.623490 − 0.781831i
281.1 −0.222521 − 0.974928i −2.13841 − 2.68148i −0.900969 + 0.433884i −0.623490 − 0.781831i −2.13841 + 2.68148i 1.90698 + 1.83397i 0.623490 + 0.781831i −1.94997 + 8.54338i −0.623490 + 0.781831i
281.2 −0.222521 − 0.974928i −0.821191 − 1.02974i −0.900969 + 0.433884i −0.623490 − 0.781831i −0.821191 + 1.02974i 1.94374 − 1.79496i 0.623490 + 0.781831i 0.281551 − 1.23356i −0.623490 + 0.781831i
281.3 −0.222521 − 0.974928i 0.0279169 + 0.0350067i −0.900969 + 0.433884i −0.623490 − 0.781831i 0.0279169 − 0.0350067i −1.98180 + 1.75284i 0.623490 + 0.781831i 0.667117 − 2.92283i −0.623490 + 0.781831i
281.4 −0.222521 − 0.974928i 0.607396 + 0.761651i −0.900969 + 0.433884i −0.623490 − 0.781831i 0.607396 − 0.761651i 0.958089 + 2.46618i 0.623490 + 0.781831i 0.456381 − 1.99954i −0.623490 + 0.781831i
281.5 −0.222521 − 0.974928i 1.70079 + 2.13273i −0.900969 + 0.433884i −0.623490 − 0.781831i 1.70079 − 2.13273i −2.60448 − 0.465484i 0.623490 + 0.781831i −0.988266 + 4.32987i −0.623490 + 0.781831i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.5
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 490.2.k.g ✓ 30
49.e even 7 1 inner 490.2.k.g ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.k.g ✓ 30 1.a even 1 1 trivial
490.2.k.g ✓ 30 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\):

\( T_{3}^{30} - T_{3}^{29} + 16 T_{3}^{28} - 26 T_{3}^{27} + 184 T_{3}^{26} - 396 T_{3}^{25} + 1528 T_{3}^{24} + \cdots + 3136 \) Copy content Toggle raw display
\( T_{11}^{30} + 5 T_{11}^{29} + 64 T_{11}^{28} + 376 T_{11}^{27} + 3082 T_{11}^{26} + 16467 T_{11}^{25} + \cdots + 6594739264 \) Copy content Toggle raw display