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Properties

Label 465.2.bj.a
Level $465$
Weight $2$
Character orbit 465.bj
Analytic conductor $3.713$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(2,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 5, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.bj (of order \(20\), degree \(8\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{3} - 32 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 10 q^{3} - 32 q^{6} - 4 q^{10} + 10 q^{12} - 12 q^{13} - 22 q^{15} + 40 q^{16} + 2 q^{18} - 12 q^{21} - 56 q^{22} - 40 q^{25} - 58 q^{27} - 52 q^{28} + 80 q^{30} - 40 q^{31} - 50 q^{33} - 64 q^{36} - 112 q^{37} - 44 q^{40} - 102 q^{42} - 12 q^{43} - 16 q^{45} + 92 q^{46} - 26 q^{48} - 4 q^{51} - 20 q^{52} + 64 q^{55} - 48 q^{57} - 52 q^{58} - 212 q^{60} - 16 q^{61} + 68 q^{63} + 84 q^{66} - 56 q^{67} - 144 q^{70} + 52 q^{72} - 12 q^{73} + 4 q^{75} - 172 q^{76} + 60 q^{78} + 52 q^{81} + 20 q^{82} - 12 q^{85} + 56 q^{87} + 104 q^{88} - 118 q^{90} - 112 q^{91} + 70 q^{93} - 32 q^{96} - 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −2.74265 + 0.434392i −0.912887 + 1.47195i 5.43130 − 1.76474i 2.23435 + 0.0875720i 1.86432 − 4.43359i −0.284293 + 0.557957i −9.18119 + 4.67805i −1.33328 − 2.68745i −6.16608 + 0.730407i
2.2 −2.59619 + 0.411195i 1.55633 + 0.760162i 4.66899 − 1.51705i −0.216887 − 2.22552i −4.35309 − 1.33357i 1.18822 − 2.33200i −6.81365 + 3.47173i 1.84431 + 2.36612i 1.47821 + 5.68869i
2.3 −2.56075 + 0.405583i 1.25838 − 1.19016i 4.49082 − 1.45916i −2.05645 + 0.878072i −2.73969 + 3.55807i −0.256976 + 0.504343i −6.28787 + 3.20383i 0.167041 − 2.99535i 4.90992 − 3.08258i
2.4 −2.43986 + 0.386435i −0.748743 − 1.56185i 3.90145 − 1.26766i 1.07197 + 1.96237i 2.43038 + 3.52136i −1.83949 + 3.61020i −4.62706 + 2.35760i −1.87877 + 2.33885i −3.37377 − 4.37365i
2.5 −2.32591 + 0.368387i −1.53017 − 0.811530i 3.37201 − 1.09563i 0.881741 − 2.05488i 3.85799 + 1.32385i 0.0380771 − 0.0747306i −3.24291 + 1.65234i 1.68284 + 2.48356i −1.29385 + 5.10428i
2.6 −2.28905 + 0.362549i 0.305002 + 1.70499i 3.20617 − 1.04175i −2.15191 − 0.607698i −1.31630 − 3.79221i −2.32040 + 4.55405i −2.83144 + 1.44269i −2.81395 + 1.04005i 5.14613 + 0.610877i
2.7 −2.24595 + 0.355724i 1.50539 − 0.856628i 3.01565 − 0.979843i 1.82911 + 1.28622i −3.07630 + 2.45945i 1.04746 − 2.05576i −2.37224 + 1.20871i 1.53238 − 2.57911i −4.56563 − 2.23813i
2.8 −2.17908 + 0.345132i 0.522157 + 1.65147i 2.72716 − 0.886108i −0.925332 + 2.03562i −1.70780 − 3.41847i 1.64427 − 3.22707i −1.70532 + 0.868906i −2.45470 + 1.72465i 1.31381 − 4.75515i
2.9 −2.17802 + 0.344965i 0.191639 − 1.72142i 2.72266 − 0.884647i −1.02042 − 1.98966i 0.176433 + 3.81539i −0.288292 + 0.565805i −1.69520 + 0.863749i −2.92655 − 0.659782i 2.90885 + 3.98152i
2.10 −2.15955 + 0.342038i −1.55053 + 0.771914i 2.64453 − 0.859261i −0.905851 + 2.04437i 3.08442 − 2.19732i −0.220064 + 0.431900i −1.52078 + 0.774878i 1.80830 − 2.39375i 1.25698 − 4.72474i
2.11 −2.02555 + 0.320816i 1.73024 + 0.0792596i 2.09783 − 0.681626i 1.82355 − 1.29409i −3.53011 + 0.394543i −1.92349 + 3.77505i −0.376033 + 0.191598i 2.98744 + 0.274276i −3.27852 + 3.20628i
2.12 −1.80538 + 0.285944i −1.69303 − 0.365570i 1.27552 − 0.414441i 2.00300 + 0.993974i 3.16110 + 0.175880i 1.66552 − 3.26878i 1.07303 − 0.546734i 2.73272 + 1.23784i −3.90040 − 1.22175i
2.13 −1.70896 + 0.270673i −1.10192 + 1.33632i 0.945180 − 0.307108i 0.234060 − 2.22378i 1.52144 − 2.58199i 0.918809 − 1.80326i 1.55120 − 0.790378i −0.571528 − 2.94506i 0.201919 + 3.86372i
2.14 −1.58415 + 0.250905i −0.829045 − 1.52075i 0.544468 − 0.176908i −2.23552 + 0.0495149i 1.69490 + 2.20109i 1.78627 − 3.50575i 2.04003 − 1.03945i −1.62537 + 2.52154i 3.52898 − 0.639342i
2.15 −1.57328 + 0.249184i −0.0101677 + 1.73202i 0.511014 − 0.166038i 1.57575 − 1.58651i −0.415594 − 2.72749i −0.248953 + 0.488597i 2.07596 − 1.05776i −2.99979 − 0.0352214i −2.08376 + 2.88869i
2.16 −1.46273 + 0.231674i 1.73089 + 0.0632977i 0.183806 − 0.0597220i −2.21983 + 0.268955i −2.54650 + 0.308416i 0.862581 − 1.69291i 2.38408 − 1.21475i 2.99199 + 0.219123i 3.18472 − 0.907688i
2.17 −1.36799 + 0.216669i 1.65708 + 0.504070i −0.0776502 + 0.0252301i −0.179540 + 2.22885i −2.37609 − 0.330527i −1.27496 + 2.50225i 2.56893 − 1.30893i 2.49183 + 1.67057i −0.237312 − 3.08795i
2.18 −1.24096 + 0.196548i −1.64920 + 0.529286i −0.400773 + 0.130219i −1.97360 − 1.05115i 1.94255 − 0.980967i 0.151021 − 0.296395i 2.71071 − 1.38118i 2.43971 − 1.74579i 2.65575 + 0.916525i
2.19 −1.22014 + 0.193251i 0.703493 − 1.58275i −0.450718 + 0.146447i 2.03867 − 0.918606i −0.552491 + 2.06713i −0.415656 + 0.815771i 2.72305 − 1.38746i −2.01020 − 2.22691i −2.30994 + 1.51480i
2.20 −1.19705 + 0.189594i 0.0565761 − 1.73113i −0.505132 + 0.164127i 0.118331 + 2.23293i 0.260487 + 2.08297i 0.563205 − 1.10535i 2.73330 − 1.39268i −2.99360 − 0.195881i −0.564999 − 2.65050i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
31.d even 5 1 inner
93.l odd 10 1 inner
155.s odd 20 1 inner
465.bj even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.bj.a ✓ 480
3.b odd 2 1 inner 465.2.bj.a ✓ 480
5.c odd 4 1 inner 465.2.bj.a ✓ 480
15.e even 4 1 inner 465.2.bj.a ✓ 480
31.d even 5 1 inner 465.2.bj.a ✓ 480
93.l odd 10 1 inner 465.2.bj.a ✓ 480
155.s odd 20 1 inner 465.2.bj.a ✓ 480
465.bj even 20 1 inner 465.2.bj.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bj.a ✓ 480 1.a even 1 1 trivial
465.2.bj.a ✓ 480 3.b odd 2 1 inner
465.2.bj.a ✓ 480 5.c odd 4 1 inner
465.2.bj.a ✓ 480 15.e even 4 1 inner
465.2.bj.a ✓ 480 31.d even 5 1 inner
465.2.bj.a ✓ 480 93.l odd 10 1 inner
465.2.bj.a ✓ 480 155.s odd 20 1 inner
465.2.bj.a ✓ 480 465.bj even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).