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Properties

Label 869.2.r.a
Level $869$
Weight $2$
Character orbit 869.r
Analytic conductor $6.939$
Analytic rank $0$
Dimension $624$
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] = [869,2,Mod(24,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.r (of order \(30\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(78\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 5 q^{2} - 9 q^{3} - 77 q^{4} - 3 q^{5} - 15 q^{6} - 15 q^{7} - 40 q^{8} - 83 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 624 q - 5 q^{2} - 9 q^{3} - 77 q^{4} - 3 q^{5} - 15 q^{6} - 15 q^{7} - 40 q^{8} - 83 q^{9} - 10 q^{11} - 10 q^{13} + 93 q^{16} + 30 q^{18} - 5 q^{19} - 28 q^{20} - 48 q^{22} - 2 q^{23} + 15 q^{24} + 55 q^{25} - 4 q^{26} - 30 q^{28} - 15 q^{29} + 60 q^{30} - 7 q^{31} - 60 q^{34} - 15 q^{35} + 28 q^{36} + 9 q^{37} - 100 q^{38} - 60 q^{39} + 45 q^{40} - 33 q^{42} + q^{44} - 62 q^{45} - 180 q^{46} - 21 q^{47} + 63 q^{48} + 97 q^{49} + 20 q^{50} + 75 q^{51} - 30 q^{52} - 38 q^{55} - 18 q^{56} - 42 q^{59} + 24 q^{60} - 20 q^{62} + 30 q^{63} + 256 q^{64} + 9 q^{66} - 68 q^{67} - 15 q^{68} + 66 q^{70} - 85 q^{72} + 95 q^{73} - 90 q^{74} - 147 q^{75} + 120 q^{77} + 115 q^{79} + 10 q^{80} + 63 q^{81} + 111 q^{82} - 40 q^{83} + 70 q^{84} - 15 q^{85} - 3 q^{86} + 63 q^{88} - 76 q^{89} - 40 q^{90} + 38 q^{92} + 45 q^{95} - 166 q^{97} - 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −2.06676 + 1.86092i 1.14133 − 0.119958i 0.599421 − 5.70311i −0.674937 + 0.749593i −2.13562 + 2.37184i 0.199686 − 1.89988i 6.10479 + 8.40253i −1.64621 + 0.349912i − 2.80523i
24.2 −1.95705 + 1.76214i −1.35137 + 0.142034i 0.515869 − 4.90816i 1.10158 − 1.22343i 2.39442 − 2.65927i 0.176070 − 1.67519i 4.54345 + 6.25352i −1.12842 + 0.239853i 4.33545i
24.3 −1.91396 + 1.72334i 0.924991 − 0.0972205i 0.484295 − 4.60776i −0.385664 + 0.428323i −1.60285 + 1.78015i −0.124509 + 1.18462i 3.98613 + 5.48644i −2.08829 + 0.443879i − 1.48442i
24.4 −1.90397 + 1.71434i 0.845263 − 0.0888407i 0.477073 − 4.53905i 2.29998 − 2.55439i −1.45705 + 1.61822i −0.305352 + 2.90523i 3.86127 + 5.31459i −2.22787 + 0.473548i 8.80643i
24.5 −1.89025 + 1.70199i −2.14256 + 0.225192i 0.467225 − 4.44535i −0.998035 + 1.10843i 3.66671 − 4.07229i −0.260085 + 2.47454i 3.69261 + 5.08244i 1.60541 − 0.341241i − 3.79386i
24.6 −1.82302 + 1.64145i 2.88512 − 0.303238i 0.419969 − 3.99574i −2.32664 + 2.58399i −4.76187 + 5.28859i −0.112610 + 1.07141i 2.90940 + 4.00445i 5.29751 − 1.12602i − 8.52972i
24.7 −1.80693 + 1.62697i −1.91247 + 0.201009i 0.408917 − 3.89059i −1.13347 + 1.25884i 3.12866 − 3.47473i 0.440300 − 4.18918i 2.73262 + 3.76112i 0.682693 − 0.145111i − 4.11876i
24.8 −1.78660 + 1.60866i −3.41535 + 0.358968i 0.395087 − 3.75901i −0.590224 + 0.655510i 5.52440 − 6.13547i −0.0778598 + 0.740787i 2.51490 + 3.46147i 8.60134 − 1.82827i − 2.12060i
24.9 −1.66581 + 1.49990i 0.736119 − 0.0773692i 0.316161 − 3.00807i −2.30483 + 2.55977i −1.11019 + 1.23299i −0.435390 + 4.14246i 1.35003 + 1.85816i −2.39856 + 0.509829i − 7.72113i
24.10 −1.64838 + 1.48420i −2.95327 + 0.310402i 0.305223 − 2.90400i 2.64521 − 2.93780i 4.40740 − 4.89492i 0.104472 − 0.993983i 1.19947 + 1.65093i 5.69104 − 1.20967i 8.76862i
24.11 −1.64705 + 1.48301i 2.63854 − 0.277321i 0.304395 − 2.89613i 1.91114 − 2.12253i −3.93453 + 4.36974i −0.133960 + 1.27454i 1.18819 + 1.63540i 3.95053 − 0.839711i 6.33014i
24.12 −1.61597 + 1.45503i 0.0221858 − 0.00233182i 0.285204 − 2.71354i 2.28973 − 2.54300i −0.0324588 + 0.0360492i 0.372611 − 3.54516i 0.931104 + 1.28156i −2.93396 + 0.623632i 7.44105i
24.13 −1.58685 + 1.42881i 2.79503 − 0.293769i 0.267550 − 2.54557i 0.169374 − 0.188109i −4.01556 + 4.45973i 0.397723 − 3.78409i 0.702345 + 0.966695i 4.79145 − 1.01845i 0.540505i
24.14 −1.57608 + 1.41911i −1.79377 + 0.188533i 0.261102 − 2.48422i −2.51947 + 2.79815i 2.55957 − 2.84270i 0.387749 − 3.68918i 0.620682 + 0.854295i 0.247615 − 0.0526321i − 7.98551i
24.15 −1.56924 + 1.41295i 1.71332 − 0.180077i 0.257028 − 2.44546i −0.967290 + 1.07428i −2.43416 + 2.70341i 0.309117 − 2.94105i 0.569615 + 0.784008i −0.0314187 + 0.00667826i − 3.05254i
24.16 −1.47628 + 1.32925i −0.536471 + 0.0563854i 0.203443 − 1.93563i −0.341377 + 0.379137i 0.717030 − 0.796343i −0.363891 + 3.46219i −0.0627148 − 0.0863196i −2.64982 + 0.563237i − 1.01348i
24.17 −1.30986 + 1.17940i −1.48942 + 0.156544i 0.115685 − 1.10067i −0.664184 + 0.737651i 1.76630 − 1.96168i −0.116486 + 1.10829i −0.925448 − 1.27377i −0.740577 + 0.157414i − 1.74956i
24.18 −1.25324 + 1.12842i −0.978902 + 0.102887i 0.0882147 − 0.839307i 1.70059 − 1.88869i 1.11070 − 1.23355i −0.309604 + 2.94568i −1.14594 − 1.57725i −1.98678 + 0.422303i 4.28595i
24.19 −1.24659 + 1.12243i 1.81382 − 0.190640i 0.0850702 − 0.809389i 1.13052 − 1.25557i −2.04710 + 2.27354i −0.234696 + 2.23298i −1.16953 − 1.60971i 0.319140 − 0.0678353i 2.83412i
24.20 −1.24002 + 1.11652i −0.289102 + 0.0303858i 0.0819799 − 0.779987i −2.22060 + 2.46623i 0.324567 − 0.360468i 0.147852 − 1.40672i −1.19236 − 1.64114i −2.85179 + 0.606166i − 5.53754i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
79.d odd 6 1 inner
869.r even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.r.a ✓ 624
11.d odd 10 1 inner 869.2.r.a ✓ 624
79.d odd 6 1 inner 869.2.r.a ✓ 624
869.r even 30 1 inner 869.2.r.a ✓ 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.r.a ✓ 624 1.a even 1 1 trivial
869.2.r.a ✓ 624 11.d odd 10 1 inner
869.2.r.a ✓ 624 79.d odd 6 1 inner
869.2.r.a ✓ 624 869.r even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(869, [\chi])\).