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Properties

Label 945.2.bs.a
Level $945$
Weight $2$
Character orbit 945.bs
Analytic conductor $7.546$
Analytic rank $0$
Dimension $276$
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] = [945,2,Mod(16,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([4, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.bs (of order \(9\), degree \(6\), 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: \(276\)
Relative dimension: \(46\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 276 q - 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 276 q - 6 q^{3} + 6 q^{9} + 6 q^{11} + 42 q^{12} + 3 q^{13} + 36 q^{14} - 24 q^{17} + 42 q^{22} + 9 q^{23} + 60 q^{24} - 24 q^{27} - 12 q^{31} - 78 q^{33} - 60 q^{34} + 18 q^{36} - 96 q^{37} - 60 q^{38} - 30 q^{39} + 48 q^{41} - 63 q^{42} - 15 q^{43} + 6 q^{45} - 120 q^{46} + 27 q^{47} - 66 q^{48} + 18 q^{49} - 48 q^{51} + 102 q^{52} - 75 q^{54} - 33 q^{56} - 12 q^{57} - 30 q^{59} - 12 q^{61} - 81 q^{62} - 162 q^{63} - 54 q^{64} - 3 q^{65} + 54 q^{66} - 18 q^{67} - 57 q^{68} + 90 q^{69} + 9 q^{70} + 12 q^{71} - 147 q^{72} + 36 q^{73} - 78 q^{74} - 108 q^{76} - 45 q^{77} + 102 q^{78} - 9 q^{79} + 102 q^{80} + 6 q^{81} - 42 q^{83} + 21 q^{84} + 18 q^{85} + 3 q^{86} - 30 q^{87} + 84 q^{88} + 30 q^{90} + 18 q^{91} - 24 q^{92} - 12 q^{93} + 156 q^{94} + 168 q^{96} + 45 q^{97} - 24 q^{98} - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.58274 + 0.940040i −0.0787941 − 1.73026i 4.25477 − 3.57018i −0.766044 + 0.642788i 1.83002 + 4.39473i 2.23295 + 1.41914i −4.88436 + 8.45996i −2.98758 + 0.272668i 1.37425 − 2.38026i
16.2 −2.44799 + 0.890994i −1.61831 − 0.617321i 3.66667 − 3.07670i −0.766044 + 0.642788i 4.51162 + 0.0692910i −2.57880 + 0.591434i −3.62955 + 6.28656i 2.23783 + 1.99803i 1.30255 − 2.25608i
16.3 −2.34197 + 0.852406i −1.62273 + 0.605600i 3.22612 − 2.70704i −0.766044 + 0.642788i 3.28416 − 2.80152i 0.871871 − 2.49797i −2.75571 + 4.77302i 2.26650 − 1.96545i 1.24613 − 2.15837i
16.4 −2.31654 + 0.843153i 0.0834639 + 1.73004i 3.12338 − 2.62083i −0.766044 + 0.642788i −1.65204 − 3.93734i 2.59677 + 0.506729i −2.56048 + 4.43488i −2.98607 + 0.288791i 1.23261 − 2.13494i
16.5 −2.29988 + 0.837089i 1.55284 + 0.767252i 3.05665 − 2.56483i −0.766044 + 0.642788i −4.21362 − 0.464721i −0.0558907 + 2.64516i −2.43546 + 4.21834i 1.82265 + 2.38285i 1.22374 − 2.11958i
16.6 −2.16738 + 0.788862i 1.43346 + 0.972214i 2.54314 − 2.13395i −0.766044 + 0.642788i −3.87379 − 0.976357i −2.17250 − 1.51005i −1.52209 + 2.63634i 1.10960 + 2.78725i 1.15324 − 1.99747i
16.7 −1.94782 + 0.708949i 0.611221 − 1.62062i 1.75931 − 1.47624i −0.766044 + 0.642788i −0.0416114 + 3.59000i −2.62818 − 0.304383i −0.307418 + 0.532464i −2.25282 − 1.98111i 1.03641 − 1.79512i
16.8 −1.93152 + 0.703015i −0.375415 + 1.69088i 1.70444 − 1.43020i −0.766044 + 0.642788i −0.463591 − 3.52988i −2.32557 + 1.26163i −0.231230 + 0.400503i −2.71813 − 1.26956i 1.02774 − 1.78010i
16.9 −1.86185 + 0.677656i −1.53145 − 0.809118i 1.47516 − 1.23781i −0.766044 + 0.642788i 3.39962 + 0.468659i 1.74691 + 1.98704i 0.0736208 − 0.127515i 1.69065 + 2.47824i 0.990667 − 1.71589i
16.10 −1.74210 + 0.634073i 0.802306 − 1.53503i 1.10078 − 0.923662i −0.766044 + 0.642788i −0.424379 + 3.18289i 1.83925 − 1.90188i 0.521908 − 0.903972i −1.71261 − 2.46312i 0.926952 − 1.60553i
16.11 −1.45058 + 0.527968i −0.944667 + 1.45176i 0.293345 − 0.246146i −0.766044 + 0.642788i 0.603833 − 2.60465i −0.379194 − 2.61844i 1.24811 − 2.16179i −1.21521 − 2.74286i 0.771838 − 1.33686i
16.12 −1.41939 + 0.516615i 1.62485 − 0.599885i 0.215684 − 0.180980i −0.766044 + 0.642788i −1.99638 + 1.69089i −2.51365 + 0.825563i 1.29784 − 2.24792i 2.28028 − 1.94945i 0.755241 − 1.30812i
16.13 −1.32073 + 0.480707i 1.02004 + 1.39983i −0.0188340 + 0.0158036i −0.766044 + 0.642788i −2.02010 − 1.35846i 1.88959 − 1.85187i 1.42277 − 2.46431i −0.919049 + 2.85576i 0.702747 − 1.21719i
16.14 −1.29715 + 0.472123i −1.14935 − 1.29576i −0.0723995 + 0.0607504i −0.766044 + 0.642788i 2.10264 + 1.13815i −1.71504 − 2.01461i 1.44563 − 2.50390i −0.357967 + 2.97857i 0.690197 − 1.19546i
16.15 −0.850428 + 0.309531i −1.73203 − 0.00753957i −0.904670 + 0.759108i −0.766044 + 0.642788i 1.47530 − 0.529706i −2.58814 − 0.549126i 1.43940 − 2.49311i 2.99989 + 0.0261176i 0.452503 − 0.783759i
16.16 −0.843609 + 0.307049i 0.836967 + 1.51641i −0.914691 + 0.767517i −0.766044 + 0.642788i −1.17168 − 1.02226i 2.12222 + 1.57993i 1.43373 − 2.48329i −1.59897 + 2.53836i 0.448875 − 0.777474i
16.17 −0.837955 + 0.304991i 0.562930 − 1.63802i −0.922940 + 0.774439i −0.766044 + 0.642788i 0.0278709 + 1.54427i 0.264620 + 2.63248i 1.42892 − 2.47496i −2.36622 − 1.84418i 0.445866 − 0.772263i
16.18 −0.766438 + 0.278961i 1.68449 + 0.403121i −1.02248 + 0.857963i −0.766044 + 0.642788i −1.40351 + 0.160938i −0.999201 − 2.44982i 1.35996 − 2.35551i 2.67499 + 1.35810i 0.407813 − 0.706353i
16.19 −0.736718 + 0.268143i 1.69896 − 0.336949i −1.06124 + 0.890484i −0.766044 + 0.642788i −1.16130 + 0.703801i 1.87165 + 1.87001i 1.32705 − 2.29852i 2.77293 − 1.14493i 0.391999 − 0.678963i
16.20 −0.616113 + 0.224247i −0.875941 − 1.49423i −1.20278 + 1.00925i −0.766044 + 0.642788i 0.874755 + 0.724188i 2.63649 + 0.221132i 1.17038 − 2.02716i −1.46546 + 2.61772i 0.327827 − 0.567812i
See next 80 embeddings (of 276 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.bs.a ✓ 276
7.c even 3 1 945.2.bu.a yes 276
27.e even 9 1 945.2.bu.a yes 276
189.u even 9 1 inner 945.2.bs.a ✓ 276
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.bs.a ✓ 276 1.a even 1 1 trivial
945.2.bs.a ✓ 276 189.u even 9 1 inner
945.2.bu.a yes 276 7.c even 3 1
945.2.bu.a yes 276 27.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{276} + 1855 T_{2}^{270} + 33 T_{2}^{269} + 108 T_{2}^{268} - 456 T_{2}^{267} + \cdots + 92\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display