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Properties

Label 861.2.bs.a
Level $861$
Weight $2$
Character orbit 861.bs
Analytic conductor $6.875$
Analytic rank $0$
Dimension $448$
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] = [861,2,Mod(178,861)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(861, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([0, 4, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("861.178");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 861.bs (of order \(24\), 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: \(6.87511961403\)
Analytic rank: \(0\)
Dimension: \(448\)
Relative dimension: \(56\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 448 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q - 8 q^{14} + 224 q^{16} - 48 q^{19} + 80 q^{22} + 72 q^{24} + 48 q^{26} - 64 q^{29} + 16 q^{30} + 40 q^{32} - 80 q^{35} - 48 q^{37} + 72 q^{38} - 64 q^{42} + 64 q^{43} - 48 q^{44} + 24 q^{46} - 48 q^{47} + 80 q^{49} - 128 q^{50} - 72 q^{52} + 16 q^{53} - 112 q^{56} - 8 q^{58} + 64 q^{65} - 32 q^{67} + 144 q^{68} + 40 q^{70} - 32 q^{71} + 72 q^{73} - 56 q^{74} - 96 q^{75} + 192 q^{77} - 96 q^{78} + 8 q^{79} - 72 q^{80} - 144 q^{82} - 32 q^{85} + 88 q^{88} - 96 q^{89} + 48 q^{91} - 216 q^{94} - 64 q^{95} - 120 q^{96} + 240 q^{98} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
178.1 −0.712061 + 2.65745i −0.130526 + 0.991445i −4.82294 − 2.78453i 0.341909 − 1.27602i −2.54177 − 1.05284i 1.81852 − 1.92171i 6.94319 − 6.94319i −0.965926 − 0.258819i 3.14750 + 1.81721i
178.2 −0.687550 + 2.56597i 0.130526 − 0.991445i −4.37943 − 2.52846i 0.137071 − 0.511556i 2.45427 + 1.01659i −2.26597 + 1.36580i 5.74219 − 5.74219i −0.965926 − 0.258819i 1.21839 + 0.703440i
178.3 −0.681572 + 2.54366i −0.130526 + 0.991445i −4.27362 − 2.46737i −0.566931 + 2.11582i −2.43294 − 1.00775i −2.31333 − 1.28394i 5.46476 − 5.46476i −0.965926 − 0.258819i −4.99551 − 2.88416i
178.4 −0.642248 + 2.39690i 0.130526 − 0.991445i −3.60060 − 2.07881i 0.439983 − 1.64204i 2.29257 + 0.949612i 2.16571 + 1.51977i 3.78588 − 3.78588i −0.965926 − 0.258819i 3.65323 + 2.10919i
178.5 −0.598655 + 2.23421i −0.130526 + 0.991445i −2.90126 − 1.67504i 0.336798 − 1.25695i −2.13696 − 0.885156i −0.764530 + 2.53288i 2.20813 − 2.20813i −0.965926 − 0.258819i 2.60666 + 1.50496i
178.6 −0.597119 + 2.22848i 0.130526 − 0.991445i −2.87751 − 1.66133i 0.946503 − 3.53240i 2.13147 + 0.882885i −0.529582 − 2.59221i 2.15773 − 2.15773i −0.965926 − 0.258819i 7.30669 + 4.21852i
178.7 −0.567930 + 2.11955i −0.130526 + 0.991445i −2.43788 − 1.40751i 0.531045 − 1.98189i −2.02728 − 0.839728i 2.10207 + 1.60664i 1.26459 − 1.26459i −0.965926 − 0.258819i 3.89910 + 2.25115i
178.8 −0.560938 + 2.09345i −0.130526 + 0.991445i −2.33583 − 1.34859i 0.663499 − 2.47621i −2.00232 − 0.829389i −2.45951 − 0.975090i 1.06844 − 1.06844i −0.965926 − 0.258819i 4.81164 + 2.77800i
178.9 −0.556093 + 2.07537i 0.130526 − 0.991445i −2.26586 − 1.30819i −0.877498 + 3.27487i 1.98503 + 0.822225i 0.786197 − 2.52624i 0.936459 − 0.936459i −0.965926 − 0.258819i −6.30858 − 3.64226i
178.10 −0.518286 + 1.93427i 0.130526 − 0.991445i −1.74072 − 1.00501i −0.667452 + 2.49097i 1.85007 + 0.766324i −0.997677 + 2.45044i 0.0141820 − 0.0141820i −0.965926 − 0.258819i −4.47227 − 2.58206i
178.11 −0.515492 + 1.92384i 0.130526 − 0.991445i −1.70339 − 0.983454i −0.114871 + 0.428704i 1.84010 + 0.762194i 1.46716 − 2.20169i −0.0466046 + 0.0466046i −0.965926 − 0.258819i −0.765545 − 0.441988i
178.12 −0.510665 + 1.90583i −0.130526 + 0.991445i −1.63935 − 0.946477i −0.887445 + 3.31199i −1.82287 − 0.755056i 2.64527 − 0.0504690i −0.149346 + 0.149346i −0.965926 − 0.258819i −5.85890 − 3.38263i
178.13 −0.438214 + 1.63544i 0.130526 − 0.991445i −0.750577 − 0.433346i −0.114450 + 0.427132i 1.56425 + 0.647933i 1.92949 + 1.81027i −1.35682 + 1.35682i −0.965926 − 0.258819i −0.648395 − 0.374351i
178.14 −0.422090 + 1.57526i −0.130526 + 0.991445i −0.571238 − 0.329805i 0.622799 − 2.32432i −1.50669 − 0.624092i 0.218380 − 2.63672i −1.54570 + 1.54570i −0.965926 − 0.258819i 3.39853 + 1.96214i
178.15 −0.421333 + 1.57243i −0.130526 + 0.991445i −0.562978 − 0.325036i −0.488621 + 1.82356i −1.50399 − 0.622972i −1.99550 + 1.73723i −1.55391 + 1.55391i −0.965926 − 0.258819i −2.66155 − 1.53665i
178.16 −0.363617 + 1.35704i −0.130526 + 0.991445i 0.0227185 + 0.0131165i −0.740473 + 2.76348i −1.29797 − 0.537635i −1.96578 − 1.77080i −2.01290 + 2.01290i −0.965926 − 0.258819i −3.48090 − 2.00970i
178.17 −0.347242 + 1.29592i 0.130526 − 0.991445i 0.173210 + 0.100003i 1.01362 − 3.78290i 1.23951 + 0.513423i −0.189624 + 2.63895i −2.08711 + 2.08711i −0.965926 − 0.258819i 4.55038 + 2.62716i
178.18 −0.291469 + 1.08778i 0.130526 − 0.991445i 0.633744 + 0.365892i 0.764215 − 2.85209i 1.04043 + 0.430959i −2.32988 + 1.25366i −2.17534 + 2.17534i −0.965926 − 0.258819i 2.87970 + 1.66259i
178.19 −0.290665 + 1.08478i −0.130526 + 0.991445i 0.639799 + 0.369388i 0.501991 − 1.87345i −1.03756 − 0.429770i 2.20989 + 1.45478i −2.17489 + 2.17489i −0.965926 − 0.258819i 1.88637 + 1.08909i
178.20 −0.281123 + 1.04916i 0.130526 − 0.991445i 0.710334 + 0.410111i −0.189317 + 0.706541i 1.00350 + 0.415661i −1.82419 − 1.91634i −2.16605 + 2.16605i −0.965926 − 0.258819i −0.688057 − 0.397250i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 178.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
41.e odd 8 1 inner
287.w even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 861.2.bs.a ✓ 448
7.d odd 6 1 inner 861.2.bs.a ✓ 448
41.e odd 8 1 inner 861.2.bs.a ✓ 448
287.w even 24 1 inner 861.2.bs.a ✓ 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
861.2.bs.a ✓ 448 1.a even 1 1 trivial
861.2.bs.a ✓ 448 7.d odd 6 1 inner
861.2.bs.a ✓ 448 41.e odd 8 1 inner
861.2.bs.a ✓ 448 287.w even 24 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(861, [\chi])\).