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Properties

Label 845.2.bh.a
Level $845$
Weight $2$
Character orbit 845.bh
Analytic conductor $6.747$
Analytic rank $0$
Dimension $2112$
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] = [845,2,Mod(4,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([39, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.bh (of order \(78\), degree \(24\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(2112\)
Relative dimension: \(88\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2112 q + 34 q^{4} - 26 q^{5} - 46 q^{6} - 128 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2112 q + 34 q^{4} - 26 q^{5} - 46 q^{6} - 128 q^{9} - 26 q^{10} - 52 q^{11} - 64 q^{14} + 32 q^{15} + 26 q^{16} - 66 q^{19} - 50 q^{20} - 52 q^{21} - 64 q^{24} - 30 q^{25} - 34 q^{26} - 48 q^{29} - 234 q^{30} - 104 q^{31} - 52 q^{34} - 28 q^{35} - 132 q^{36} + 32 q^{39} - 123 q^{40} - 52 q^{41} - 104 q^{44} + 27 q^{45} - 146 q^{46} + 98 q^{49} - 14 q^{50} + 62 q^{51} - 82 q^{54} + 110 q^{55} - 40 q^{56} - 200 q^{59} + 78 q^{60} - 72 q^{61} - 124 q^{64} - 2 q^{65} + 106 q^{66} - 118 q^{69} - 64 q^{71} + 164 q^{74} + 2 q^{75} - 240 q^{76} - 100 q^{79} - 57 q^{80} + 28 q^{81} - 540 q^{84} - 224 q^{85} - 208 q^{86} - 126 q^{89} - 62 q^{90} - 40 q^{91} - 8 q^{94} + 5 q^{95} - 52 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.110652 + 2.74579i 0.214913 + 0.0622503i −5.53362 − 0.446720i 2.23535 − 0.0566076i −0.194707 + 0.583218i 0.663575 − 0.282723i 1.17643 − 9.68876i −2.49326 − 1.57664i −0.0919127 + 6.14407i
4.2 −0.107931 + 2.67828i −1.85275 − 0.536655i −5.16805 − 0.417208i −1.66192 − 1.49601i 1.63728 − 4.90426i −3.61541 + 1.54038i 1.02901 − 8.47464i 0.609105 + 0.385175i 4.18610 − 4.28962i
4.3 −0.107646 + 2.67120i 0.429126 + 0.124298i −5.13020 − 0.414152i −0.0321225 − 2.23584i −0.378218 + 1.13290i 1.33631 − 0.569348i 1.01405 − 8.35144i −2.36687 − 1.49672i 5.97582 + 0.154872i
4.4 −0.105665 + 2.62204i −1.65324 − 0.478867i −4.87042 − 0.393181i −2.14953 + 0.616041i 1.43030 − 4.28426i 2.55334 − 1.08788i 0.912950 − 7.51882i −0.0316805 − 0.0200336i −1.38815 − 5.70126i
4.5 −0.104440 + 2.59166i −2.54622 − 0.737523i −4.71228 − 0.380415i 1.93731 + 1.11662i 2.17734 − 6.52192i 0.431901 − 0.184016i 0.852770 − 7.02319i 3.40375 + 2.15240i −3.09624 + 4.90422i
4.6 −0.103277 + 2.56279i 0.518410 + 0.150159i −4.56370 − 0.368420i −0.220673 + 2.22515i −0.438366 + 1.31307i −3.77927 + 1.61020i 0.797186 − 6.56541i −2.28937 − 1.44771i −5.67980 − 0.795345i
4.7 −0.101078 + 2.50821i 3.01151 + 0.872294i −4.28741 − 0.346115i 0.00790235 − 2.23605i −2.49230 + 7.46534i 3.45945 − 1.47393i 0.696338 − 5.73486i 5.77272 + 3.65044i 5.60770 + 0.245836i
4.8 −0.0971559 + 2.41090i 1.15056 + 0.333263i −3.80948 − 0.307533i −0.387552 + 2.20223i −0.915247 + 2.74150i 1.68410 − 0.717527i 0.529869 − 4.36387i −1.32285 − 0.836521i −5.27169 − 1.14831i
4.9 −0.0964198 + 2.39263i 3.03915 + 0.880300i −3.72188 − 0.300461i 1.39780 + 1.74533i −2.39927 + 7.18668i 0.334376 − 0.142464i 0.500487 − 4.12188i 5.92592 + 3.74733i −4.31070 + 3.17613i
4.10 −0.0942849 + 2.33966i 2.02843 + 0.587542i −3.47158 − 0.280255i −1.92947 − 1.13011i −1.56590 + 4.69043i −3.03057 + 1.29121i 0.418532 − 3.44692i 1.23376 + 0.780181i 2.82600 − 4.40774i
4.11 −0.0904999 + 2.24573i 2.08806 + 0.604813i −3.04161 − 0.245544i 2.21900 − 0.275755i −1.54722 + 4.63448i −2.25175 + 0.959382i 0.284866 − 2.34608i 1.45861 + 0.922371i 0.418453 + 5.00824i
4.12 −0.0886387 + 2.19955i −2.04076 − 0.591114i −2.83663 − 0.228997i 1.43991 − 1.71074i 1.48107 − 4.43635i −4.41359 + 1.88046i 0.224442 − 1.84844i 1.27972 + 0.809247i 3.63523 + 3.31880i
4.13 −0.0861099 + 2.13680i −0.549995 − 0.159308i −2.56497 − 0.207066i 1.48410 − 1.67256i 0.387769 − 1.16151i 0.481068 − 0.204964i 0.147783 − 1.21711i −2.25845 − 1.42816i 3.44612 + 3.31524i
4.14 −0.0851929 + 2.11404i 1.18326 + 0.342735i −2.46839 − 0.199269i −1.22364 − 1.87155i −0.825360 + 2.47225i −1.25739 + 0.535725i 0.121500 − 1.00064i −1.25294 − 0.792311i 4.06077 − 2.42739i
4.15 −0.0831141 + 2.06246i −2.88792 − 0.836495i −2.25330 − 0.181905i −0.880534 + 2.05540i 1.96526 − 5.88667i 0.658788 − 0.280683i 0.0648458 − 0.534054i 5.10476 + 3.22805i −4.16598 − 1.98689i
4.16 −0.0815877 + 2.02458i −0.807417 − 0.233871i −2.09875 − 0.169428i −1.64821 + 1.51109i 0.539366 − 1.61560i −2.46351 + 1.04961i 0.0257852 − 0.212360i −1.93834 − 1.22573i −2.92485 − 3.46023i
4.17 −0.0803434 + 1.99370i −0.517986 − 0.150036i −1.97487 − 0.159428i 1.74802 + 1.39443i 0.340745 − 1.02065i 4.40912 − 1.87855i −0.00449856 + 0.0370490i −2.28977 − 1.44796i −2.92052 + 3.37299i
4.18 −0.0787148 + 1.95329i −1.91449 − 0.554540i −1.81562 − 0.146572i −0.276787 − 2.21887i 1.23387 − 3.69591i 4.66925 − 1.98938i −0.0420542 + 0.346347i 0.822201 + 0.519928i 4.35588 − 0.365987i
4.19 −0.0715639 + 1.77584i −2.19402 − 0.635506i −1.15497 − 0.0932387i 1.73306 + 1.41297i 1.28557 − 3.85075i −1.54519 + 0.658344i −0.180224 + 1.48428i 1.87429 + 1.18523i −2.63324 + 2.97652i
4.20 −0.0697152 + 1.72996i −3.28984 − 0.952913i −0.994404 − 0.0802766i −1.84540 − 1.26274i 1.87786 − 5.62487i −0.819255 + 0.349052i −0.209186 + 1.72280i 7.37941 + 4.66646i 2.31315 − 3.10444i
See next 80 embeddings (of 2112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
169.k even 78 1 inner
845.bh even 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.bh.a ✓ 2112
5.b even 2 1 inner 845.2.bh.a ✓ 2112
169.k even 78 1 inner 845.2.bh.a ✓ 2112
845.bh even 78 1 inner 845.2.bh.a ✓ 2112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.bh.a ✓ 2112 1.a even 1 1 trivial
845.2.bh.a ✓ 2112 5.b even 2 1 inner
845.2.bh.a ✓ 2112 169.k even 78 1 inner
845.2.bh.a ✓ 2112 845.bh even 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(845, [\chi])\).