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Properties

Label 935.2.bq.a
Level $935$
Weight $2$
Character orbit 935.bq
Analytic conductor $7.466$
Analytic rank $0$
Dimension $720$
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] = [935,2,Mod(12,935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(935, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([4, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("935.12");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 935 = 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 935.bq (of order \(16\), 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: \(7.46601258899\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(90\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 720 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 720 q - 48 q^{10} + 32 q^{14} - 48 q^{15} - 64 q^{19} - 32 q^{20} - 32 q^{26} + 96 q^{27} - 128 q^{30} - 128 q^{36} - 64 q^{39} - 80 q^{41} - 96 q^{42} - 32 q^{46} - 208 q^{48} - 64 q^{50} + 48 q^{53} + 128 q^{54} - 80 q^{57} + 32 q^{59} + 192 q^{60} - 96 q^{62} - 192 q^{63} - 64 q^{64} - 96 q^{68} - 128 q^{70} + 32 q^{71} + 240 q^{72} - 96 q^{73} + 80 q^{74} - 112 q^{75} + 128 q^{76} - 80 q^{78} + 64 q^{79} + 128 q^{82} + 48 q^{83} + 96 q^{85} + 176 q^{87} + 96 q^{90} + 80 q^{92} - 48 q^{93} + 32 q^{94} - 160 q^{95} + 128 q^{96} - 96 q^{97} + 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −1.07573 + 2.59703i −0.530146 + 2.66522i −4.17318 − 4.17318i −0.760998 + 2.10259i −6.35138 − 4.24386i −2.69087 − 1.79798i 10.1330 − 4.19724i −4.05072 − 1.67786i −4.64187 − 4.23815i
12.2 −1.04511 + 2.52311i 0.428229 − 2.15285i −3.85961 − 3.85961i −2.13050 + 0.678961i 4.98433 + 3.33042i 0.483407 + 0.323002i 8.72569 − 3.61430i −1.67975 − 0.695774i 0.513500 − 6.08505i
12.3 −1.03815 + 2.50632i 0.147949 − 0.743789i −3.78967 − 3.78967i 1.44521 − 1.70627i 1.71058 + 1.14297i 0.697290 + 0.465914i 8.41972 − 3.48756i 2.24031 + 0.927965i 2.77612 + 5.39353i
12.4 −1.00220 + 2.41953i 0.0176918 − 0.0889426i −3.43550 − 3.43550i −1.56184 − 1.60021i 0.197468 + 0.131944i −1.98356 − 1.32537i 6.91630 − 2.86483i 2.76404 + 1.14490i 5.43703 − 2.17518i
12.5 −0.999044 + 2.41191i −0.428683 + 2.15514i −3.40499 − 3.40499i 2.19530 + 0.425027i −4.76971 − 3.18702i 2.42110 + 1.61773i 6.79044 − 2.81269i −1.68920 − 0.699691i −3.21833 + 4.87024i
12.6 −0.970764 + 2.34363i 0.296832 − 1.49227i −3.13601 − 3.13601i 0.690353 + 2.12683i 3.20919 + 2.14431i −3.61610 − 2.41620i 5.70673 − 2.36380i 0.632868 + 0.262143i −5.65468 − 0.446719i
12.7 −0.937964 + 2.26445i 0.184581 − 0.927953i −2.83372 − 2.83372i 1.51133 + 1.64799i 1.92817 + 1.28836i 1.84685 + 1.23403i 4.54585 − 1.88295i 1.94461 + 0.805485i −5.14936 + 1.87658i
12.8 −0.932766 + 2.25190i −0.351103 + 1.76512i −2.78677 − 2.78677i 0.772720 − 2.09831i −3.64736 − 2.43709i −2.96488 − 1.98107i 4.37114 − 1.81059i −0.220722 − 0.0914259i 4.00441 + 3.69732i
12.9 −0.876389 + 2.11579i 0.334197 − 1.68012i −2.29429 − 2.29429i 0.444877 − 2.19137i 3.26189 + 2.17953i 3.30102 + 2.20567i 2.63336 − 1.09077i 0.0605233 + 0.0250696i 4.24658 + 2.86175i
12.10 −0.864160 + 2.08627i −0.295632 + 1.48624i −2.19152 − 2.19152i −0.0429392 + 2.23566i −2.84523 − 1.90112i 1.41147 + 0.943112i 2.29340 − 0.949957i 0.650118 + 0.269288i −4.62707 − 2.02155i
12.11 −0.822398 + 1.98544i −0.588094 + 2.95655i −1.85144 − 1.85144i −1.76054 − 1.37858i −5.38641 − 3.59909i −0.806256 − 0.538723i 1.22766 − 0.508513i −5.62368 − 2.32941i 4.18497 − 2.36172i
12.12 −0.819011 + 1.97727i 0.509318 − 2.56051i −1.82459 − 1.82459i −1.17297 − 1.90372i 4.64568 + 3.10415i −3.27711 − 2.18970i 1.14754 − 0.475325i −3.52518 − 1.46018i 4.72483 − 0.760112i
12.13 −0.818824 + 1.97682i 0.513774 − 2.58292i −1.82312 − 1.82312i −1.48209 + 1.67434i 4.68526 + 3.13059i 0.437070 + 0.292041i 1.14315 − 0.473509i −3.63585 − 1.50602i −2.09628 − 4.30081i
12.14 −0.790122 + 1.90752i −0.113078 + 0.568483i −1.60014 − 1.60014i −0.910957 + 2.04210i −0.995048 − 0.664870i 0.592905 + 0.396166i 0.501551 − 0.207749i 2.46125 + 1.01948i −3.17558 − 3.35118i
12.15 −0.788563 + 1.90376i −0.601134 + 3.02210i −1.58825 − 1.58825i 1.57800 − 1.58427i −5.27933 − 3.52753i 1.98760 + 1.32807i 0.468572 − 0.194089i −6.00011 − 2.48533i 1.77171 + 4.25343i
12.16 −0.755422 + 1.82375i 0.413773 − 2.08018i −1.34119 − 1.34119i 1.82751 − 1.28849i 3.48115 + 2.32603i −1.47987 − 0.988820i −0.188349 + 0.0780168i −1.38429 − 0.573392i 0.969333 + 4.30628i
12.17 −0.722931 + 1.74531i 0.155513 − 0.781815i −1.10926 − 1.10926i −2.22413 − 0.230768i 1.25208 + 0.836615i 3.78941 + 2.53200i −0.752696 + 0.311777i 2.18459 + 0.904886i 2.01065 − 3.71496i
12.18 −0.707980 + 1.70921i 0.524289 − 2.63578i −1.00596 − 1.00596i 2.23435 − 0.0877080i 4.13392 + 2.76220i −1.82614 − 1.22019i −0.986817 + 0.408753i −3.90080 − 1.61576i −1.43196 + 3.88107i
12.19 −0.704869 + 1.70170i 0.0543971 − 0.273473i −0.984741 − 0.984741i −2.17092 + 0.535828i 0.427027 + 0.285330i −3.35230 − 2.23994i −1.03356 + 0.428113i 2.69981 + 1.11830i 0.618392 − 4.07195i
12.20 −0.698611 + 1.68660i −0.284373 + 1.42964i −0.942332 − 0.942332i 0.988212 − 2.00585i −2.21256 − 1.47838i −0.224396 − 0.149937i −1.12553 + 0.466211i 0.808633 + 0.334947i 2.69268 + 3.06802i
See next 80 embeddings (of 720 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.r even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 935.2.bq.a yes 720
5.c odd 4 1 935.2.bl.a ✓ 720
17.e odd 16 1 935.2.bl.a ✓ 720
85.r even 16 1 inner 935.2.bq.a yes 720
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.2.bl.a ✓ 720 5.c odd 4 1
935.2.bl.a ✓ 720 17.e odd 16 1
935.2.bq.a yes 720 1.a even 1 1 trivial
935.2.bq.a yes 720 85.r even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(935, [\chi])\).