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Properties

Label 867.2.q.a
Level $867$
Weight $2$
Character orbit 867.q
Analytic conductor $6.923$
Analytic rank $0$
Dimension $3200$
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] = [867,2,Mod(19,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(136))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.q (of order \(136\), degree \(64\), 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: \(6.92302985525\)
Analytic rank: \(0\)
Dimension: \(3200\)
Relative dimension: \(50\) over \(\Q(\zeta_{136})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{136}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3200 q + 8 q^{5} + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3200 q + 8 q^{5} + 8 q^{6} - 8 q^{11} + 16 q^{14} + 176 q^{16} + 8 q^{17} + 8 q^{19} - 16 q^{20} + 8 q^{22} - 8 q^{23} - 8 q^{24} + 16 q^{25} - 16 q^{26} + 8 q^{28} - 8 q^{31} - 16 q^{33} + 8 q^{34} - 8 q^{36} + 8 q^{37} - 16 q^{39} + 8 q^{40} + 24 q^{41} - 8 q^{42} + 8 q^{43} + 8 q^{45} + 8 q^{49} + 48 q^{52} + 32 q^{53} - 128 q^{54} + 16 q^{56} + 16 q^{57} - 24 q^{58} - 16 q^{59} + 8 q^{60} - 16 q^{61} - 16 q^{62} - 24 q^{65} + 16 q^{66} - 32 q^{67} + 32 q^{69} - 40 q^{70} + 16 q^{71} - 48 q^{73} - 64 q^{74} - 16 q^{75} - 24 q^{76} - 1360 q^{77} - 8 q^{78} + 16 q^{80} + 8 q^{82} - 32 q^{83} - 48 q^{84} - 176 q^{85} + 64 q^{86} - 24 q^{87} + 8 q^{88} + 16 q^{91} - 512 q^{92} + 32 q^{93} + 40 q^{95} + 16 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.13341 + 1.77156i 0.986955 + 0.160996i 1.04551 − 5.59300i −1.16035 + 0.919103i −2.39080 + 1.40498i 2.89412 − 0.200884i 4.97908 + 8.93916i 0.948161 + 0.317791i 0.847263 − 4.01647i
19.2 −2.05349 + 1.70519i 0.986955 + 0.160996i 0.941623 − 5.03724i 0.928132 − 0.735163i −2.30123 + 1.35235i −4.72496 + 0.327964i 4.05819 + 7.28585i 0.948161 + 0.317791i −0.652312 + 3.09229i
19.3 −1.97663 + 1.64137i −0.986955 − 0.160996i 0.845463 − 4.52283i 2.32055 − 1.83808i 2.21510 − 1.30173i 1.99873 − 0.138734i 3.25204 + 5.83853i 0.948161 + 0.317791i −1.56989 + 7.44210i
19.4 −1.93688 + 1.60836i −0.986955 − 0.160996i 0.797164 − 4.26445i −1.12585 + 0.891775i 2.17055 − 1.27555i 5.22358 − 0.362574i 2.86462 + 5.14298i 0.948161 + 0.317791i 0.746340 − 3.53804i
19.5 −1.83188 + 1.52117i 0.986955 + 0.160996i 0.674315 − 3.60727i −2.91609 + 2.30980i −2.05289 + 1.20640i −0.102436 + 0.00711018i 1.93468 + 3.47342i 0.948161 + 0.317791i 1.82832 − 8.66716i
19.6 −1.82352 + 1.51423i −0.986955 − 0.160996i 0.664832 − 3.55654i 1.40059 − 1.10939i 2.04352 − 1.20090i −2.38021 + 0.165213i 1.86634 + 3.35072i 0.948161 + 0.317791i −0.874129 + 4.14382i
19.7 −1.75953 + 1.46110i −0.986955 − 0.160996i 0.593651 − 3.17575i −2.75152 + 2.17945i 1.97181 − 1.15876i −0.899402 + 0.0624284i 1.36972 + 2.45912i 0.948161 + 0.317791i 1.65701 − 7.85506i
19.8 −1.67928 + 1.39446i 0.986955 + 0.160996i 0.507978 − 2.71744i 0.369099 − 0.292359i −1.88188 + 1.10591i −1.12685 + 0.0782157i 0.812029 + 1.45787i 0.948161 + 0.317791i −0.212139 + 1.00565i
19.9 −1.60813 + 1.33538i 0.986955 + 0.160996i 0.435360 − 2.32897i −0.475304 + 0.376483i −1.80214 + 1.05905i −0.00792955 0.000550398i 0.375646 + 0.674413i 0.948161 + 0.317791i 0.261605 − 1.24014i
19.10 −1.51593 + 1.25881i 0.986955 + 0.160996i 0.345938 − 1.85060i 2.35803 − 1.86777i −1.69882 + 0.998334i 3.55530 − 0.246777i −0.112510 − 0.201993i 0.948161 + 0.317791i −1.22344 + 5.79972i
19.11 −1.35920 + 1.12867i −0.986955 − 0.160996i 0.206042 − 1.10223i 0.115713 − 0.0916548i 1.52318 − 0.895118i −4.16760 + 0.289277i −0.755398 − 1.35620i 0.948161 + 0.317791i −0.0538293 + 0.255179i
19.12 −1.19519 + 0.992472i −0.986955 − 0.160996i 0.0759760 − 0.406436i 3.11896 − 2.47049i 1.33938 − 0.787105i 1.40278 − 0.0973684i −1.19935 − 2.15324i 0.948161 + 0.317791i −1.27585 + 6.04818i
19.13 −1.15298 + 0.957424i 0.986955 + 0.160996i 0.0452078 − 0.241841i −2.07986 + 1.64743i −1.29208 + 0.759309i 0.882360 − 0.0612455i −1.27910 − 2.29643i 0.948161 + 0.317791i 0.820747 − 3.89076i
19.14 −1.14136 + 0.947774i −0.986955 − 0.160996i 0.0369304 − 0.197560i 0.984002 − 0.779417i 1.27906 − 0.751656i 2.46469 − 0.171077i −1.29873 − 2.33167i 0.948161 + 0.317791i −0.384391 + 1.82221i
19.15 −1.05410 + 0.875316i 0.986955 + 0.160996i −0.0225443 + 0.120601i 1.56352 − 1.23845i −1.18127 + 0.694191i −3.09003 + 0.214482i −1.41524 − 2.54085i 0.948161 + 0.317791i −0.564079 + 2.67403i
19.16 −1.04036 + 0.863906i −0.986955 − 0.160996i −0.0314789 + 0.168397i −2.46397 + 1.95168i 1.16588 − 0.685142i 2.35947 − 0.163773i −1.42879 − 2.56517i 0.948161 + 0.317791i 0.877351 − 4.15910i
19.17 −0.906781 + 0.752981i 0.986955 + 0.160996i −0.112228 + 0.600367i −2.99698 + 2.37388i −1.01618 + 0.597171i −3.89828 + 0.270583i −1.49738 − 2.68831i 0.948161 + 0.317791i 0.930122 − 4.40926i
19.18 −0.731924 + 0.607782i 0.986955 + 0.160996i −0.201185 + 1.07625i −0.0901988 + 0.0714454i −0.820226 + 0.482017i 1.06851 − 0.0741663i −1.43276 − 2.57229i 0.948161 + 0.317791i 0.0225954 − 0.107114i
19.19 −0.711831 + 0.591097i −0.986955 − 0.160996i −0.210191 + 1.12442i −3.12554 + 2.47571i 0.797709 − 0.468784i −2.97681 + 0.206623i −1.41549 − 2.54129i 0.948161 + 0.317791i 0.761474 − 3.60978i
19.20 −0.645824 + 0.536285i 0.986955 + 0.160996i −0.238013 + 1.27325i 1.85817 − 1.47183i −0.723739 + 0.425314i 3.92229 − 0.272250i −1.34608 − 2.41668i 0.948161 + 0.317791i −0.410726 + 1.94705i
See next 80 embeddings (of 3200 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
289.i even 136 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.q.a ✓ 3200
289.i even 136 1 inner 867.2.q.a ✓ 3200
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.q.a ✓ 3200 1.a even 1 1 trivial
867.2.q.a ✓ 3200 289.i even 136 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(867, [\chi])\).