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Properties

Label 429.2.bs.b
Level $429$
Weight $2$
Character orbit 429.bs
Analytic conductor $3.426$
Analytic rank $0$
Dimension $224$
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] = [429,2,Mod(7,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([0, 42, 55]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.bs (of order \(60\), degree \(16\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(14\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{3} + 6 q^{5} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 28 q^{3} + 6 q^{5} + 28 q^{9} - 8 q^{11} - 10 q^{13} + 36 q^{14} + 8 q^{15} - 8 q^{16} - 22 q^{20} + 10 q^{22} - 48 q^{23} - 30 q^{24} - 2 q^{26} - 56 q^{27} + 20 q^{29} - 4 q^{31} + 28 q^{33} - 72 q^{34} - 20 q^{35} - 36 q^{37} - 20 q^{39} + 80 q^{40} - 70 q^{41} - 8 q^{42} - 24 q^{44} - 4 q^{45} - 40 q^{46} - 42 q^{47} - 28 q^{48} + 24 q^{49} + 180 q^{50} + 10 q^{52} - 4 q^{53} + 82 q^{55} - 72 q^{56} - 132 q^{58} - 66 q^{59} - 16 q^{60} + 60 q^{61} - 120 q^{62} - 60 q^{66} - 12 q^{67} - 60 q^{68} - 12 q^{69} - 44 q^{70} - 32 q^{71} + 30 q^{72} - 70 q^{73} + 60 q^{74} + 12 q^{75} + 52 q^{78} - 120 q^{79} - 44 q^{80} + 28 q^{81} + 6 q^{82} - 240 q^{83} + 60 q^{84} + 40 q^{85} + 10 q^{86} + 6 q^{88} - 24 q^{89} + 88 q^{91} + 48 q^{92} + 20 q^{93} - 40 q^{94} - 300 q^{95} - 46 q^{97} + 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.987158 + 2.57163i 0.669131 + 0.743145i −4.15253 − 3.73895i 3.32910 + 0.527277i −2.57163 + 0.987158i 0.866370 − 0.0454045i 8.80570 − 4.48673i −0.104528 + 0.994522i −4.64231 + 8.04071i
7.2 −0.893395 + 2.32737i 0.669131 + 0.743145i −3.13222 − 2.82026i −3.41301 − 0.540567i −2.32737 + 0.893395i 1.64776 − 0.0863554i 4.91964 − 2.50668i −0.104528 + 0.994522i 4.30726 − 7.46040i
7.3 −0.742514 + 1.93431i 0.669131 + 0.743145i −1.70395 − 1.53425i 0.399672 + 0.0633018i −1.93431 + 0.742514i −2.63195 + 0.137934i 0.540719 − 0.275510i −0.104528 + 0.994522i −0.419207 + 0.726088i
7.4 −0.473247 + 1.23285i 0.669131 + 0.743145i 0.190334 + 0.171377i 4.09856 + 0.649148i −1.23285 + 0.473247i −1.65326 + 0.0866439i −2.65461 + 1.35259i −0.104528 + 0.994522i −2.73993 + 4.74570i
7.5 −0.436789 + 1.13787i 0.669131 + 0.743145i 0.382318 + 0.344240i −0.0897712 − 0.0142184i −1.13787 + 0.436789i 4.09917 − 0.214828i −2.73066 + 1.39134i −0.104528 + 0.994522i 0.0553897 − 0.0959378i
7.6 −0.215576 + 0.561596i 0.669131 + 0.743145i 1.21737 + 1.09613i 0.0568875 + 0.00901009i −0.561596 + 0.215576i 3.00661 − 0.157570i −1.94999 + 0.993569i −0.104528 + 0.994522i −0.0173236 + 0.0300054i
7.7 −0.158810 + 0.413715i 0.669131 + 0.743145i 1.34035 + 1.20686i −3.17932 − 0.503554i −0.413715 + 0.158810i −0.944050 + 0.0494756i −1.50185 + 0.765232i −0.104528 + 0.994522i 0.713236 − 1.23536i
7.8 0.0361343 − 0.0941330i 0.669131 + 0.743145i 1.47873 + 1.33146i 2.04988 + 0.324669i 0.0941330 − 0.0361343i −4.15746 + 0.217883i 0.358448 − 0.182638i −0.104528 + 0.994522i 0.104633 − 0.181230i
7.9 0.342749 − 0.892893i 0.669131 + 0.743145i 0.806509 + 0.726184i 2.10352 + 0.333164i 0.892893 − 0.342749i 0.180391 − 0.00945392i 2.62918 − 1.33964i −0.104528 + 0.994522i 1.01846 − 1.76402i
7.10 0.400304 − 1.04283i 0.669131 + 0.743145i 0.559044 + 0.503365i −0.769642 − 0.121899i 1.04283 − 0.400304i 0.172980 − 0.00906547i 2.73925 − 1.39572i −0.104528 + 0.994522i −0.435211 + 0.753807i
7.11 0.727798 − 1.89598i 0.669131 + 0.743145i −1.57876 − 1.42152i 1.62284 + 0.257032i 1.89598 − 0.727798i 1.47389 − 0.0772434i −0.225156 + 0.114723i −0.104528 + 0.994522i 1.66843 − 2.88980i
7.12 0.750080 − 1.95402i 0.669131 + 0.743145i −1.76930 − 1.59309i −2.73934 − 0.433869i 1.95402 − 0.750080i 4.52723 − 0.237262i −0.710221 + 0.361876i −0.104528 + 0.994522i −2.90252 + 5.02731i
7.13 0.758057 − 1.97481i 0.669131 + 0.743145i −1.83892 − 1.65577i −3.65267 − 0.578526i 1.97481 − 0.758057i −4.58954 + 0.240527i −0.894330 + 0.455684i −0.104528 + 0.994522i −3.91141 + 6.77476i
7.14 0.892365 − 2.32469i 0.669131 + 0.743145i −3.12158 − 2.81068i 3.27062 + 0.518016i 2.32469 − 0.892365i −1.99815 + 0.104719i −4.88221 + 2.48761i −0.104528 + 0.994522i 4.12282 − 7.14093i
19.1 −1.56477 + 1.93233i −0.978148 − 0.207912i −0.869572 − 4.09102i −0.513743 + 0.0813689i 1.93233 − 1.56477i 1.92549 − 2.96500i 4.83500 + 2.46356i 0.913545 + 0.406737i 0.646659 − 1.12005i
19.2 −1.48670 + 1.83592i −0.978148 − 0.207912i −0.744504 − 3.50262i 0.0628305 − 0.00995138i 1.83592 − 1.48670i −0.583961 + 0.899221i 3.32757 + 1.69548i 0.913545 + 0.406737i −0.0751401 + 0.130146i
19.3 −1.36075 + 1.68039i −0.978148 − 0.207912i −0.556241 − 2.61691i 4.09781 − 0.649030i 1.68039 − 1.36075i −0.0869532 + 0.133896i 1.30116 + 0.662975i 0.913545 + 0.406737i −4.48550 + 7.76911i
19.4 −1.18625 + 1.46489i −0.978148 − 0.207912i −0.322905 − 1.51915i −2.00150 + 0.317006i 1.46489 − 1.18625i −1.72749 + 2.66010i −0.750595 − 0.382447i 0.913545 + 0.406737i 1.90989 − 3.30802i
19.5 −0.821551 + 1.01453i −0.978148 − 0.207912i 0.0614966 + 0.289319i −3.20835 + 0.508153i 1.01453 − 0.821551i −0.510870 + 0.786671i −2.67039 − 1.36063i 0.913545 + 0.406737i 2.12029 − 3.67244i
19.6 −0.560625 + 0.692314i −0.978148 − 0.207912i 0.250825 + 1.18004i 1.71423 − 0.271507i 0.692314 − 0.560625i 0.284275 − 0.437746i −2.54507 − 1.29678i 0.913545 + 0.406737i −0.773071 + 1.33900i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
13.f odd 12 1 inner
143.w even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bs.b ✓ 224
11.d odd 10 1 inner 429.2.bs.b ✓ 224
13.f odd 12 1 inner 429.2.bs.b ✓ 224
143.w even 60 1 inner 429.2.bs.b ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bs.b ✓ 224 1.a even 1 1 trivial
429.2.bs.b ✓ 224 11.d odd 10 1 inner
429.2.bs.b ✓ 224 13.f odd 12 1 inner
429.2.bs.b ✓ 224 143.w even 60 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} + 112 T_{2}^{220} - 300 T_{2}^{217} + 4723 T_{2}^{216} + 5030 T_{2}^{215} + \cdots + 19485170468401 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display