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Properties

Label 945.2.dm.a
Level $945$
Weight $2$
Character orbit 945.dm
Analytic conductor $7.546$
Analytic rank $0$
Dimension $1296$
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(92,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([26, 9, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.92");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.dm (of order \(36\), degree \(12\), 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: \(1296\)
Relative dimension: \(108\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1296 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1296 q + 12 q^{11} + 84 q^{18} - 96 q^{20} + 24 q^{23} + 36 q^{25} + 48 q^{27} + 60 q^{32} - 12 q^{33} - 72 q^{38} - 48 q^{41} + 60 q^{42} - 60 q^{45} - 12 q^{47} - 120 q^{48} - 336 q^{51} + 60 q^{60} + 72 q^{65} - 192 q^{66} - 72 q^{67} - 144 q^{72} + 168 q^{75} - 132 q^{78} - 84 q^{81} - 120 q^{83} + 144 q^{86} - 120 q^{87} - 228 q^{90} - 372 q^{92} - 240 q^{93} - 120 q^{95} + 192 q^{96} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1 −2.79785 + 0.244780i 1.18078 + 1.26719i 5.79845 − 1.02242i −1.84848 + 1.25822i −3.61383 − 3.25637i −0.819152 + 0.573576i −10.5473 + 2.82613i −0.211525 + 2.99253i 4.86380 − 3.97278i
92.2 −2.75984 + 0.241455i −1.69261 − 0.367514i 5.58880 − 0.985455i 0.561643 + 2.16438i 4.76007 + 0.605590i 0.819152 − 0.573576i −9.83427 + 2.63508i 2.72987 + 1.24412i −2.07264 − 5.83774i
92.3 −2.68377 + 0.234799i −1.59717 + 0.670105i 5.17787 − 0.912998i −1.55567 − 1.60620i 4.12910 − 2.17342i −0.819152 + 0.573576i −8.47739 + 2.27151i 2.10192 − 2.14055i 4.55220 + 3.94540i
92.4 −2.65774 + 0.232522i −0.293566 − 1.70699i 5.03991 − 0.888673i 2.08613 + 0.805024i 1.17714 + 4.46848i −0.819152 + 0.573576i −8.03418 + 2.15275i −2.82764 + 1.00223i −5.73158 − 1.65447i
92.5 −2.58177 + 0.225876i 0.257468 + 1.71281i 4.64490 − 0.819021i −1.03263 − 1.98335i −1.05161 − 4.36392i 0.819152 − 0.573576i −6.80041 + 1.82216i −2.86742 + 0.881988i 3.11400 + 4.88731i
92.6 −2.56052 + 0.224016i 1.67663 − 0.434622i 4.53644 − 0.799898i −0.345620 + 2.20920i −4.19569 + 1.48845i 0.819152 − 0.573576i −6.47102 + 1.73390i 2.62221 − 1.45740i 0.390070 − 5.73411i
92.7 −2.54019 + 0.222238i 0.638457 − 1.61008i 4.43354 − 0.781753i −2.23557 + 0.0469603i −1.26398 + 4.23181i 0.819152 − 0.573576i −6.16229 + 1.65118i −2.18475 − 2.05594i 5.66834 − 0.616117i
92.8 −2.46712 + 0.215845i −0.953054 − 1.44627i 4.07046 − 0.717732i −1.82802 + 1.28777i 2.66346 + 3.36240i −0.819152 + 0.573576i −5.10308 + 1.36737i −1.18338 + 2.75674i 4.23198 − 3.57164i
92.9 −2.41062 + 0.210902i −1.45929 + 0.932996i 3.79701 − 0.669514i 2.14041 − 0.647036i 3.32102 − 2.55687i −0.819152 + 0.573576i −4.33719 + 1.16215i 1.25904 − 2.72302i −5.02325 + 2.01118i
92.10 −2.39215 + 0.209286i −1.66108 − 0.490729i 3.70895 − 0.653987i 1.18031 − 1.89918i 4.07625 + 0.826255i 0.819152 − 0.573576i −4.09655 + 1.09767i 2.51837 + 1.63028i −2.42601 + 4.79013i
92.11 −2.38712 + 0.208846i 0.502209 + 1.65764i 3.68510 − 0.649783i 1.95025 − 1.09385i −1.54502 − 3.85211i −0.819152 + 0.573576i −4.03189 + 1.08034i −2.49557 + 1.66497i −4.42704 + 3.01846i
92.12 −2.33713 + 0.204472i −0.894380 + 1.48327i 3.45075 − 0.608460i −2.01728 + 0.964662i 1.78699 − 3.64947i 0.819152 − 0.573576i −3.40820 + 0.913225i −1.40017 − 2.65321i 4.51740 − 2.66702i
92.13 −2.31868 + 0.202858i 1.03070 + 1.39200i 3.36552 − 0.593431i 1.79168 + 1.33786i −2.67224 − 3.01852i 0.819152 − 0.573576i −3.18672 + 0.853879i −0.875324 + 2.86946i −4.42573 − 2.73862i
92.14 −2.27239 + 0.198808i −0.242239 − 1.71503i 3.15461 − 0.556243i 2.18502 − 0.475048i 0.891423 + 3.84905i 0.819152 − 0.573576i −2.65122 + 0.710393i −2.88264 + 0.830894i −4.87078 + 1.51389i
92.15 −2.27205 + 0.198779i 1.55291 + 0.767126i 3.15309 − 0.555974i −0.542220 − 2.16933i −3.68077 − 1.43427i 0.819152 − 0.573576i −2.64743 + 0.709376i 1.82303 + 2.38255i 1.66317 + 4.82105i
92.16 −2.21166 + 0.193495i 1.72923 − 0.0987658i 2.88437 − 0.508593i −2.01647 − 0.966358i −3.80536 + 0.553034i −0.819152 + 0.573576i −1.99192 + 0.533734i 2.98049 − 0.341578i 4.64673 + 1.74708i
92.17 −2.01811 + 0.176561i 0.429292 − 1.67801i 2.07196 − 0.365343i −0.821888 − 2.07954i −0.570085 + 3.46219i −0.819152 + 0.573576i −0.203358 + 0.0544895i −2.63142 − 1.44071i 2.02582 + 4.05162i
92.18 −1.99235 + 0.174308i −1.69755 + 0.343958i 1.96946 − 0.347269i 0.832753 + 2.07522i 3.32217 − 0.981182i −0.819152 + 0.573576i 0.000306795 0 8.22054e-5i 2.76339 − 1.16777i −2.02086 − 3.98940i
92.19 −1.98615 + 0.173766i 1.58408 + 0.700487i 1.94499 − 0.342954i 1.14823 − 1.91874i −3.26795 − 1.11601i −0.819152 + 0.573576i 0.0481591 − 0.0129042i 2.01864 + 2.21926i −1.94715 + 4.01043i
92.20 −1.97230 + 0.172554i 1.54471 − 0.783504i 1.89058 − 0.333360i 2.21866 + 0.278486i −2.91143 + 1.81185i 0.819152 − 0.573576i 0.153478 − 0.0411243i 1.77224 − 2.42057i −4.42392 − 0.166420i
See next 80 embeddings (of 1296 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.dm.a ✓ 1296
5.c odd 4 1 inner 945.2.dm.a ✓ 1296
27.f odd 18 1 inner 945.2.dm.a ✓ 1296
135.q even 36 1 inner 945.2.dm.a ✓ 1296
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.dm.a ✓ 1296 1.a even 1 1 trivial
945.2.dm.a ✓ 1296 5.c odd 4 1 inner
945.2.dm.a ✓ 1296 27.f odd 18 1 inner
945.2.dm.a ✓ 1296 135.q even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).