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Properties

Label 490.2.q.d
Level $490$
Weight $2$
Character orbit 490.q
Analytic conductor $3.913$
Analytic rank $0$
Dimension $72$
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] = [490,2,Mod(11,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.q (of order \(21\), degree \(12\), 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: \(72\)
Relative dimension: \(6\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} + 6 q^{4} - 6 q^{5} + 3 q^{7} - 12 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{2} + 6 q^{4} - 6 q^{5} + 3 q^{7} - 12 q^{8} - 16 q^{9} - 6 q^{10} + 2 q^{11} + 10 q^{13} - 10 q^{14} + 6 q^{16} - 9 q^{17} - 58 q^{18} + 13 q^{19} + 12 q^{20} + 25 q^{21} - 18 q^{22} + 29 q^{23} + 6 q^{25} + 2 q^{26} + 24 q^{27} - 14 q^{28} - 33 q^{29} + 16 q^{31} + 6 q^{32} + 68 q^{33} + 11 q^{34} + 3 q^{35} - 10 q^{36} - 6 q^{37} - 15 q^{38} - 44 q^{39} - 6 q^{40} - 8 q^{41} - 18 q^{42} - 5 q^{43} + 9 q^{44} - 12 q^{45} - 6 q^{46} + 12 q^{47} - 59 q^{49} + 72 q^{50} - 53 q^{51} + 9 q^{52} + 38 q^{53} + 9 q^{54} - 3 q^{55} + 3 q^{56} - 52 q^{57} - 50 q^{58} - 58 q^{59} + 32 q^{61} - 11 q^{62} + 26 q^{63} - 12 q^{64} - 9 q^{65} - 44 q^{66} + 7 q^{67} - 2 q^{68} - 28 q^{69} - 41 q^{71} + 12 q^{72} + 101 q^{73} + 8 q^{74} + 2 q^{76} - 4 q^{77} - 31 q^{78} - 25 q^{79} + 36 q^{80} + 22 q^{81} + 25 q^{82} + 56 q^{83} + 14 q^{84} - 11 q^{85} - 8 q^{86} - 134 q^{87} + 9 q^{88} - 40 q^{89} + 24 q^{90} - 36 q^{91} - 9 q^{92} + 77 q^{93} + 19 q^{94} - 20 q^{95} + 80 q^{97} - 8 q^{98} + 236 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −0.733052 + 0.680173i −2.93420 + 0.442260i 0.0747301 − 0.997204i −0.365341 + 0.930874i 1.85011 − 2.31997i 0.581581 + 2.58104i 0.623490 + 0.781831i 5.54724 − 1.71110i −0.365341 − 0.930874i
11.2 −0.733052 + 0.680173i −1.77232 + 0.267134i 0.0747301 − 0.997204i −0.365341 + 0.930874i 1.11750 − 1.40131i 2.31412 − 1.28252i 0.623490 + 0.781831i 0.203038 − 0.0626291i −0.365341 − 0.930874i
11.3 −0.733052 + 0.680173i −0.243553 + 0.0367097i 0.0747301 − 0.997204i −0.365341 + 0.930874i 0.153568 − 0.192568i −2.42476 − 1.05856i 0.623490 + 0.781831i −2.80875 + 0.866384i −0.365341 − 0.930874i
11.4 −0.733052 + 0.680173i 0.0912265 − 0.0137502i 0.0747301 − 0.997204i −0.365341 + 0.930874i −0.0575213 + 0.0721294i −0.150643 − 2.64146i 0.623490 + 0.781831i −2.85859 + 0.881757i −0.365341 − 0.930874i
11.5 −0.733052 + 0.680173i 2.33429 − 0.351837i 0.0747301 − 0.997204i −0.365341 + 0.930874i −1.47184 + 1.84563i −1.57584 + 2.12526i 0.623490 + 0.781831i 2.45839 − 0.758312i −0.365341 − 0.930874i
11.6 −0.733052 + 0.680173i 2.52456 − 0.380517i 0.0747301 − 0.997204i −0.365341 + 0.930874i −1.59182 + 1.99608i 2.43363 − 1.03800i 0.623490 + 0.781831i 3.36192 − 1.03701i −0.365341 − 0.930874i
51.1 0.955573 + 0.294755i −1.12170 + 2.85805i 0.826239 + 0.563320i 0.988831 − 0.149042i −1.91429 + 2.40045i −2.54935 − 0.707689i 0.623490 + 0.781831i −4.71109 − 4.37125i 0.988831 + 0.149042i
51.2 0.955573 + 0.294755i −0.916542 + 2.33531i 0.826239 + 0.563320i 0.988831 − 0.149042i −1.56417 + 1.96140i 1.77424 + 1.96267i 0.623490 + 0.781831i −2.41447 − 2.24030i 0.988831 + 0.149042i
51.3 0.955573 + 0.294755i −0.221927 + 0.565462i 0.826239 + 0.563320i 0.988831 − 0.149042i −0.378741 + 0.474926i 0.0548626 − 2.64518i 0.623490 + 0.781831i 1.92866 + 1.78954i 0.988831 + 0.149042i
51.4 0.955573 + 0.294755i 0.325051 − 0.828217i 0.826239 + 0.563320i 0.988831 − 0.149042i 0.554731 − 0.695611i 0.0298206 + 2.64558i 0.623490 + 0.781831i 1.61887 + 1.50209i 0.988831 + 0.149042i
51.5 0.955573 + 0.294755i 0.928033 − 2.36459i 0.826239 + 0.563320i 0.988831 − 0.149042i 1.58378 − 1.98600i −1.70240 + 2.02530i 0.623490 + 0.781831i −2.53089 − 2.34832i 0.988831 + 0.149042i
51.6 0.955573 + 0.294755i 1.00709 − 2.56602i 0.826239 + 0.563320i 0.988831 − 0.149042i 1.71869 − 2.15517i 1.88230 − 1.85929i 0.623490 + 0.781831i −3.37106 − 3.12789i 0.988831 + 0.149042i
81.1 0.0747301 + 0.997204i −2.47486 − 0.763393i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.576312 − 2.52499i −1.35483 − 2.27254i −0.222521 − 0.974928i 3.06344 + 2.08862i 0.733052 + 0.680173i
81.2 0.0747301 + 0.997204i −1.46786 − 0.452775i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.341815 − 1.49759i −1.74669 + 1.98723i −0.222521 − 0.974928i −0.529111 − 0.360742i 0.733052 + 0.680173i
81.3 0.0747301 + 0.997204i −1.39264 − 0.429572i −0.988831 + 0.149042i 0.733052 − 0.680173i 0.324299 − 1.42085i 2.45413 − 0.988548i −0.222521 − 0.974928i −0.723805 − 0.493482i 0.733052 + 0.680173i
81.4 0.0747301 + 0.997204i 1.08629 + 0.335075i −0.988831 + 0.149042i 0.733052 − 0.680173i −0.252959 + 1.10829i 0.894832 + 2.48983i −0.222521 − 0.974928i −1.41098 − 0.961987i 0.733052 + 0.680173i
81.5 0.0747301 + 0.997204i 1.09419 + 0.337513i −0.988831 + 0.149042i 0.733052 − 0.680173i −0.254800 + 1.11635i 0.875828 − 2.49658i −0.222521 − 0.974928i −1.39538 − 0.951354i 0.733052 + 0.680173i
81.6 0.0747301 + 0.997204i 3.15488 + 0.973152i −0.988831 + 0.149042i 0.733052 − 0.680173i −0.734666 + 3.21878i 0.603931 + 2.57590i −0.222521 − 0.974928i 6.52754 + 4.45040i 0.733052 + 0.680173i
121.1 0.0747301 − 0.997204i −2.47486 + 0.763393i −0.988831 − 0.149042i 0.733052 + 0.680173i 0.576312 + 2.52499i −1.35483 + 2.27254i −0.222521 + 0.974928i 3.06344 − 2.08862i 0.733052 − 0.680173i
121.2 0.0747301 − 0.997204i −1.46786 + 0.452775i −0.988831 − 0.149042i 0.733052 + 0.680173i 0.341815 + 1.49759i −1.74669 − 1.98723i −0.222521 + 0.974928i −0.529111 + 0.360742i 0.733052 − 0.680173i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.q.d ✓ 72
49.g even 21 1 inner 490.2.q.d ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.q.d ✓ 72 1.a even 1 1 trivial
490.2.q.d ✓ 72 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - T_{3}^{70} - 8 T_{3}^{69} - 59 T_{3}^{68} - 35 T_{3}^{67} + 1290 T_{3}^{66} + \cdots + 371101696 \) acting on \(S_{2}^{\mathrm{new}}(490, [\chi])\). Copy content Toggle raw display