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Properties

Label 490.2.s.a
Level $490$
Weight $2$
Character orbit 490.s
Analytic conductor $3.913$
Analytic rank $0$
Dimension $336$
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] = [490,2,Mod(13,490)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(490, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([21, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("490.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.s (of order \(28\), 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: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{6} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q + 28 q^{6} + 8 q^{7} + 8 q^{11} + 44 q^{15} + 56 q^{16} + 28 q^{17} - 16 q^{21} - 20 q^{22} + 8 q^{23} - 8 q^{25} - 28 q^{26} - 8 q^{28} - 24 q^{30} + 8 q^{35} + 76 q^{36} - 24 q^{37} - 56 q^{41} - 112 q^{42} - 24 q^{43} + 112 q^{45} - 68 q^{46} - 84 q^{47} - 32 q^{50} - 80 q^{51} - 100 q^{53} + 84 q^{55} - 20 q^{56} + 92 q^{57} - 80 q^{58} - 112 q^{61} - 32 q^{67} + 52 q^{70} + 16 q^{71} - 84 q^{75} + 16 q^{77} - 80 q^{78} + 12 q^{81} + 140 q^{83} + 40 q^{85} - 8 q^{86} - 28 q^{87} - 8 q^{88} - 84 q^{90} + 124 q^{91} + 8 q^{92} + 20 q^{93} - 56 q^{95} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.846724 − 0.532032i −3.24598 − 0.365734i 0.433884 + 0.900969i 2.09187 + 0.789982i 2.55386 + 2.03664i −0.465488 + 2.60448i 0.111964 − 0.993712i 7.47782 + 1.70676i −1.35094 − 1.78184i
13.2 −0.846724 − 0.532032i −2.37039 − 0.267078i 0.433884 + 0.900969i −1.20386 − 1.88433i 1.86497 + 1.48726i 0.295144 + 2.62924i 0.111964 − 0.993712i 2.62262 + 0.598596i 0.0168149 + 2.23600i
13.3 −0.846724 − 0.532032i −2.24605 − 0.253069i 0.433884 + 0.900969i 1.00029 + 1.99986i 1.76714 + 1.40925i −1.38836 − 2.25221i 0.111964 − 0.993712i 2.05591 + 0.469247i 0.217020 − 2.22551i
13.4 −0.846724 − 0.532032i −2.13880 − 0.240984i 0.433884 + 0.900969i −2.23606 + 0.00630103i 1.68276 + 1.34196i −1.22938 − 2.34278i 0.111964 − 0.993712i 1.59159 + 0.363270i 1.89668 + 1.18432i
13.5 −0.846724 − 0.532032i −1.38368 − 0.155903i 0.433884 + 0.900969i 2.08356 − 0.811650i 1.08865 + 0.868166i 2.63328 + 0.256628i 0.111964 − 0.993712i −1.03453 − 0.236125i −2.19602 − 0.421277i
13.6 −0.846724 − 0.532032i −1.01671 − 0.114556i 0.433884 + 0.900969i −1.43471 + 1.71511i 0.799929 + 0.637922i −1.90631 + 1.83466i 0.111964 − 0.993712i −1.90420 − 0.434621i 2.12730 − 0.688915i
13.7 −0.846724 − 0.532032i −0.224192 − 0.0252604i 0.433884 + 0.900969i −1.00878 + 1.99559i 0.176390 + 0.140666i 2.07246 + 1.64466i 0.111964 − 0.993712i −2.87516 − 0.656236i 1.91587 − 1.15301i
13.8 −0.846724 − 0.532032i 0.00984721 + 0.00110951i 0.433884 + 0.900969i −0.690291 − 2.12685i −0.00774758 − 0.00617849i −2.58964 + 0.541992i 0.111964 − 0.993712i −2.92469 − 0.667541i −0.547067 + 2.16811i
13.9 −0.846724 − 0.532032i 0.275176 + 0.0310049i 0.433884 + 0.900969i 2.16282 − 0.567618i −0.216503 − 0.172655i −1.51360 − 2.17003i 0.111964 − 0.993712i −2.85002 − 0.650499i −2.13331 − 0.670076i
13.10 −0.846724 − 0.532032i 1.01523 + 0.114389i 0.433884 + 0.900969i −2.18655 − 0.467951i −0.798762 − 0.636992i 1.76089 − 1.97465i 0.111964 − 0.993712i −1.90717 − 0.435300i 1.60244 + 1.55954i
13.11 −0.846724 − 0.532032i 1.64716 + 0.185590i 0.433884 + 0.900969i 1.74399 + 1.39947i −1.29595 − 1.03349i −1.48002 + 2.19307i 0.111964 − 0.993712i −0.246094 − 0.0561694i −0.732113 − 2.11282i
13.12 −0.846724 − 0.532032i 1.85754 + 0.209295i 0.433884 + 0.900969i 0.807489 − 2.08518i −1.46148 − 1.16549i 1.88226 + 1.85933i 0.111964 − 0.993712i 0.481883 + 0.109987i −1.79310 + 1.33596i
13.13 −0.846724 − 0.532032i 2.96487 + 0.334061i 0.433884 + 0.900969i 1.14001 + 1.92364i −2.33270 − 1.86026i 2.03821 − 1.68692i 0.111964 − 0.993712i 5.75408 + 1.31333i 0.0581638 − 2.23531i
13.14 −0.846724 − 0.532032i 3.30213 + 0.372061i 0.433884 + 0.900969i 0.346271 − 2.20909i −2.59804 − 2.07187i −1.79867 − 1.94031i 0.111964 − 0.993712i 7.84084 + 1.78962i −1.46850 + 1.68627i
13.15 0.846724 + 0.532032i −2.67756 − 0.301689i 0.433884 + 0.900969i −2.10481 + 0.754840i −2.10665 − 1.67999i 1.62687 + 2.08646i −0.111964 + 0.993712i 4.15353 + 0.948016i −2.18379 − 0.480684i
13.16 0.846724 + 0.532032i −2.36144 − 0.266071i 0.433884 + 0.900969i 1.12920 − 1.93000i −1.85793 − 1.48165i −2.63369 − 0.252315i −0.111964 + 0.993712i 2.58084 + 0.589061i 1.98294 − 1.03341i
13.17 0.846724 + 0.532032i −2.17451 − 0.245008i 0.433884 + 0.900969i 2.23510 + 0.0657981i −1.71086 − 1.36436i −0.0302883 − 2.64558i −0.111964 + 0.993712i 1.74368 + 0.397983i 1.85751 + 1.24486i
13.18 0.846724 + 0.532032i −2.05457 − 0.231494i 0.433884 + 0.900969i −2.08089 − 0.818483i −1.61649 − 1.28911i 1.38568 − 2.25386i −0.111964 + 0.993712i 1.24288 + 0.283678i −1.32648 − 1.80013i
13.19 0.846724 + 0.532032i −1.43065 − 0.161195i 0.433884 + 0.900969i 2.00522 + 0.989483i −1.12560 − 0.897638i 2.38096 + 1.15371i −0.111964 + 0.993712i −0.904016 − 0.206336i 1.17143 + 1.90466i
13.20 0.846724 + 0.532032i −0.819320 − 0.0923152i 0.433884 + 0.900969i 0.200391 + 2.22707i −0.644623 − 0.514070i −1.94256 + 1.79623i −0.111964 + 0.993712i −2.26202 − 0.516291i −1.01520 + 1.99233i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
49.f odd 14 1 inner
245.s even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 490.2.s.a ✓ 336
5.c odd 4 1 inner 490.2.s.a ✓ 336
49.f odd 14 1 inner 490.2.s.a ✓ 336
245.s even 28 1 inner 490.2.s.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.s.a ✓ 336 1.a even 1 1 trivial
490.2.s.a ✓ 336 5.c odd 4 1 inner
490.2.s.a ✓ 336 49.f odd 14 1 inner
490.2.s.a ✓ 336 245.s even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(490, [\chi])\).