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Properties

Label 945.2.bv.e
Level $945$
Weight $2$
Character orbit 945.bv
Analytic conductor $7.546$
Analytic rank $0$
Dimension $160$
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] = [945,2,Mod(73,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bv (of order \(12\), degree \(4\), not 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{2} + 6 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q - 4 q^{2} + 6 q^{5} + 16 q^{8} - 24 q^{10} + 16 q^{11} - 152 q^{16} + 6 q^{17} - 60 q^{20} + 8 q^{22} - 8 q^{23} + 2 q^{25} + 36 q^{26} + 22 q^{28} - 12 q^{32} + 36 q^{35} - 4 q^{37} + 18 q^{38} - 6 q^{40} + 12 q^{41} - 4 q^{43} - 16 q^{46} + 44 q^{50} + 54 q^{52} - 8 q^{53} - 148 q^{56} + 28 q^{58} + 124 q^{65} - 24 q^{67} - 42 q^{68} - 34 q^{70} + 40 q^{71} + 36 q^{73} + 96 q^{76} - 58 q^{77} - 36 q^{80} - 66 q^{82} + 138 q^{83} - 20 q^{85} + 16 q^{86} + 46 q^{88} - 48 q^{91} + 26 q^{92} - 188 q^{95} + 48 q^{97} - 102 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −1.89944 + 1.89944i 0 − 5.21578i 2.17053 + 0.537411i 0 −2.39282 − 1.12891i 6.10819 + 6.10819i 0 −5.14358 + 3.10201i
73.2 −1.88738 + 1.88738i 0 − 5.12443i 0.859858 − 2.06413i 0 2.44492 + 1.01112i 5.89699 + 5.89699i 0 2.27293 + 5.51869i
73.3 −1.88327 + 1.88327i 0 − 5.09343i −2.13610 − 0.661134i 0 −1.23324 + 2.34075i 5.82578 + 5.82578i 0 5.26794 − 2.77775i
73.4 −1.70572 + 1.70572i 0 − 3.81894i −1.19668 + 1.88890i 0 −2.60688 − 0.451887i 3.10261 + 3.10261i 0 −1.18073 − 5.26314i
73.5 −1.66610 + 1.66610i 0 − 3.55179i 1.68710 + 1.46755i 0 2.64329 + 0.114057i 2.58544 + 2.58544i 0 −5.25596 + 0.365799i
73.6 −1.59360 + 1.59360i 0 − 3.07913i 1.46808 − 1.68663i 0 −0.324829 − 2.62574i 1.71970 + 1.71970i 0 0.348280 + 5.02736i
73.7 −1.44715 + 1.44715i 0 − 2.18846i −2.19749 + 0.413548i 0 −0.575284 − 2.58245i 0.272736 + 0.272736i 0 2.58163 − 3.77856i
73.8 −1.38705 + 1.38705i 0 − 1.84779i −0.0285272 − 2.23589i 0 0.385956 + 2.61745i −0.211119 − 0.211119i 0 3.14084 + 3.06171i
73.9 −1.35218 + 1.35218i 0 − 1.65676i −0.823809 + 2.07878i 0 2.09405 + 1.61708i −0.464119 − 0.464119i 0 −1.69695 − 3.92482i
73.10 −1.15985 + 1.15985i 0 − 0.690520i −1.38328 − 1.75685i 0 −2.48248 + 0.915044i −1.51881 − 1.51881i 0 3.64210 + 0.433282i
73.11 −0.994069 + 0.994069i 0 0.0236551i 0.292236 + 2.21689i 0 −2.63236 − 0.265907i −2.01165 − 2.01165i 0 −2.49424 − 1.91324i
73.12 −0.974511 + 0.974511i 0 0.100655i 1.63908 + 1.52100i 0 1.06331 − 2.42268i −2.04711 − 2.04711i 0 −3.07953 + 0.115069i
73.13 −0.955716 + 0.955716i 0 0.173213i 2.23581 − 0.0342301i 0 −1.21501 + 2.35027i −2.07698 − 2.07698i 0 −2.10408 + 2.16951i
73.14 −0.888914 + 0.888914i 0 0.419665i −1.45181 + 1.70066i 0 2.13716 + 1.55967i −2.15087 − 2.15087i 0 −0.221199 − 2.80227i
73.15 −0.747603 + 0.747603i 0 0.882180i 2.19474 − 0.427907i 0 −2.63599 − 0.227075i −2.15473 − 2.15473i 0 −1.32089 + 1.96070i
73.16 −0.742042 + 0.742042i 0 0.898747i −0.642138 − 2.14188i 0 2.48255 − 0.914856i −2.15099 − 2.15099i 0 2.06586 + 1.11287i
73.17 −0.475340 + 0.475340i 0 1.54810i 0.171432 + 2.22949i 0 0.999832 − 2.44956i −1.68656 − 1.68656i 0 −1.14125 − 0.978275i
73.18 −0.300078 + 0.300078i 0 1.81991i −2.21779 − 0.285321i 0 2.31030 − 1.28938i −1.14627 − 1.14627i 0 0.751129 − 0.579892i
73.19 −0.0133941 + 0.0133941i 0 1.99964i 1.57464 + 1.58761i 0 0.664445 + 2.56096i −0.0535716 − 0.0535716i 0 −0.0423555 0.000173729i
73.20 0.0825718 − 0.0825718i 0 1.98636i −2.17426 − 0.522126i 0 −0.400384 − 2.61528i 0.329161 + 0.329161i 0 −0.222645 + 0.136419i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.t odd 6 1 inner
315.bs even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bv.e 160
3.b odd 2 1 315.2.bs.e ✓ 160
5.c odd 4 1 inner 945.2.bv.e 160
7.d odd 6 1 945.2.cj.e 160
9.c even 3 1 945.2.cj.e 160
9.d odd 6 1 315.2.cg.e yes 160
15.e even 4 1 315.2.bs.e ✓ 160
21.g even 6 1 315.2.cg.e yes 160
35.k even 12 1 945.2.cj.e 160
45.k odd 12 1 945.2.cj.e 160
45.l even 12 1 315.2.cg.e yes 160
63.i even 6 1 315.2.bs.e ✓ 160
63.t odd 6 1 inner 945.2.bv.e 160
105.w odd 12 1 315.2.cg.e yes 160
315.bs even 12 1 inner 945.2.bv.e 160
315.bu odd 12 1 315.2.bs.e ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.e ✓ 160 3.b odd 2 1
315.2.bs.e ✓ 160 15.e even 4 1
315.2.bs.e ✓ 160 63.i even 6 1
315.2.bs.e ✓ 160 315.bu odd 12 1
315.2.cg.e yes 160 9.d odd 6 1
315.2.cg.e yes 160 21.g even 6 1
315.2.cg.e yes 160 45.l even 12 1
315.2.cg.e yes 160 105.w odd 12 1
945.2.bv.e 160 1.a even 1 1 trivial
945.2.bv.e 160 5.c odd 4 1 inner
945.2.bv.e 160 63.t odd 6 1 inner
945.2.bv.e 160 315.bs even 12 1 inner
945.2.cj.e 160 7.d odd 6 1
945.2.cj.e 160 9.c even 3 1
945.2.cj.e 160 35.k even 12 1
945.2.cj.e 160 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{80} + 2 T_{2}^{79} + 2 T_{2}^{78} - 4 T_{2}^{77} + 289 T_{2}^{76} + 568 T_{2}^{75} + 566 T_{2}^{74} + \cdots + 17161 \) Copy content Toggle raw display
\( T_{11}^{80} - 8 T_{11}^{79} + 258 T_{11}^{78} - 1752 T_{11}^{77} + 34377 T_{11}^{76} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
\( T_{13}^{160} - 6473 T_{13}^{156} + 1314 T_{13}^{155} - 58698 T_{13}^{153} + 23984127 T_{13}^{152} + \cdots + 58\!\cdots\!81 \) Copy content Toggle raw display