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Properties

Label 215.2.u.a
Level $215$
Weight $2$
Character orbit 215.u
Analytic conductor $1.717$
Analytic rank $0$
Dimension $240$
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] = [215,2,Mod(9,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.u (of order \(42\), 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: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(20\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 16 q^{4} - 15 q^{5} - 14 q^{6} - 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 16 q^{4} - 15 q^{5} - 14 q^{6} - 58 q^{9} - 11 q^{10} - 16 q^{11} - 72 q^{14} - 23 q^{15} - 28 q^{16} - 34 q^{19} + 11 q^{20} - 74 q^{21} - 6 q^{24} - 27 q^{25} - 26 q^{26} - 14 q^{29} + 114 q^{30} - 6 q^{31} - 46 q^{34} + 9 q^{35} - 10 q^{36} + 60 q^{39} + 83 q^{40} - 12 q^{41} - 212 q^{44} + 6 q^{45} + 22 q^{46} + 52 q^{49} - 38 q^{50} + 24 q^{51} - 112 q^{54} - 70 q^{55} - 190 q^{56} + 44 q^{59} + 113 q^{60} - 36 q^{61} + 56 q^{64} - 6 q^{65} + 106 q^{66} - 50 q^{69} + 17 q^{70} - 8 q^{71} + 78 q^{74} - 131 q^{75} - 106 q^{76} + 14 q^{79} - 70 q^{80} + 102 q^{81} + 278 q^{84} - 84 q^{85} + 36 q^{86} + 46 q^{89} + 23 q^{90} - 100 q^{91} + 148 q^{94} + 49 q^{95} - 90 q^{96} + 216 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.13317 + 1.70115i 0.416231 − 2.76151i 1.21148 − 5.30784i 2.06552 − 0.856509i 3.80985 + 6.59886i −1.75568 − 1.01364i 4.07748 + 8.46698i −4.58598 − 1.41459i −2.94907 + 5.34085i
9.2 −1.89883 + 1.51427i −0.0257783 + 0.171028i 0.867512 − 3.80082i −1.73551 − 1.40997i −0.210033 − 0.363788i −0.172040 − 0.0993276i 2.00065 + 4.15440i 2.83813 + 0.875448i 5.43051 + 0.0492737i
9.3 −1.66488 + 1.32770i −0.179208 + 1.18897i 0.564006 − 2.47107i 2.11126 + 0.736610i −1.28023 − 2.21742i 2.98013 + 1.72058i 0.493957 + 1.02571i 1.48519 + 0.458121i −4.49299 + 1.57675i
9.4 −1.52188 + 1.21366i −0.475748 + 3.15638i 0.398114 − 1.74425i 0.498524 − 2.17979i −3.10675 − 5.38104i −1.93267 − 1.11583i −0.178117 − 0.369863i −6.86968 − 2.11902i 1.88683 + 3.92242i
9.5 −1.36812 + 1.09104i 0.307562 − 2.04054i 0.236340 − 1.03547i −0.232198 + 2.22398i 1.80552 + 3.12725i 1.25799 + 0.726298i −0.712096 − 1.47868i −1.20248 − 0.370917i −2.10877 − 3.29600i
9.6 −0.912410 + 0.727623i 0.431561 − 2.86322i −0.141985 + 0.622076i −1.23251 − 1.86572i 1.68958 + 2.92644i 0.820534 + 0.473735i −1.33579 − 2.77379i −5.14506 − 1.58704i 2.48210 + 0.805496i
9.7 −0.827799 + 0.660148i −0.315162 + 2.09096i −0.195586 + 0.856917i −2.03930 + 0.917204i −1.11945 − 1.93895i 4.30182 + 2.48365i −1.32257 − 2.74635i −1.40607 − 0.433716i 1.08264 − 2.10550i
9.8 −0.818009 + 0.652341i 0.0250402 − 0.166131i −0.201451 + 0.882614i −2.22810 + 0.188603i 0.0878908 + 0.152231i −3.37636 − 1.94934i −1.31890 − 2.73872i 2.83975 + 0.875946i 1.69957 − 1.60776i
9.9 −0.347628 + 0.277224i −0.0699697 + 0.464218i −0.401050 + 1.75711i 1.20248 − 1.88522i −0.104369 − 0.180773i 0.824132 + 0.475813i −0.733536 − 1.52320i 2.65612 + 0.819303i 0.104612 + 0.988710i
9.10 −0.176340 + 0.140626i −0.317938 + 2.10938i −0.433722 + 1.90026i 1.55802 + 1.60392i −0.240570 − 0.416679i −1.79382 − 1.03566i −0.386467 − 0.802507i −1.48169 − 0.457042i −0.500296 − 0.0637365i
9.11 0.176340 − 0.140626i 0.317938 − 2.10938i −0.433722 + 1.90026i 2.06226 + 0.864343i −0.240570 − 0.416679i 1.79382 + 1.03566i 0.386467 + 0.802507i −1.48169 − 0.457042i 0.485208 − 0.137590i
9.12 0.347628 − 0.277224i 0.0699697 − 0.464218i −0.401050 + 1.75711i −1.31558 + 1.80810i −0.104369 − 0.180773i −0.824132 − 0.475813i 0.733536 + 1.52320i 2.65612 + 0.819303i 0.0439155 + 0.993259i
9.13 0.818009 − 0.652341i −0.0250402 + 0.166131i −0.201451 + 0.882614i −0.638451 − 2.14298i 0.0878908 + 0.152231i 3.37636 + 1.94934i 1.31890 + 2.73872i 2.83975 + 0.875946i −1.92021 − 1.33649i
9.14 0.827799 − 0.660148i 0.315162 − 2.09096i −0.195586 + 0.856917i 0.108762 − 2.23342i −1.11945 − 1.93895i −4.30182 − 2.48365i 1.32257 + 2.74635i −1.40607 − 0.433716i −1.38435 − 1.92062i
9.15 0.912410 − 0.727623i −0.431561 + 2.86322i −0.141985 + 0.622076i −2.18704 − 0.465691i 1.68958 + 2.92644i −0.820534 − 0.473735i 1.33579 + 2.77379i −5.14506 − 1.58704i −2.33432 + 1.16644i
9.16 1.36812 − 1.09104i −0.307562 + 2.04054i 0.236340 − 1.03547i 1.98541 − 1.02866i 1.80552 + 3.12725i −1.25799 − 0.726298i 0.712096 + 1.47868i −1.20248 − 0.370917i 1.59397 − 3.57348i
9.17 1.52188 − 1.21366i 0.475748 − 3.15638i 0.398114 − 1.74425i −1.84698 + 1.26043i −3.10675 − 5.38104i 1.93267 + 1.11583i 0.178117 + 0.369863i −6.86968 − 2.11902i −1.28115 + 4.15983i
9.18 1.66488 − 1.32770i 0.179208 − 1.18897i 0.564006 − 2.47107i 1.45702 + 1.69620i −1.28023 − 2.21742i −2.98013 − 1.72058i −0.493957 − 1.02571i 1.48519 + 0.458121i 4.67781 + 0.889489i
9.19 1.89883 − 1.51427i 0.0257783 − 0.171028i 0.867512 − 3.80082i −1.94656 − 1.10042i −0.210033 − 0.363788i 0.172040 + 0.0993276i −2.00065 − 4.15440i 2.83813 + 0.875448i −5.36251 + 0.858098i
9.20 2.13317 − 1.70115i −0.416231 + 2.76151i 1.21148 − 5.30784i −0.0426808 + 2.23566i 3.80985 + 6.59886i 1.75568 + 1.01364i −4.07748 − 8.46698i −4.58598 − 1.41459i 3.71215 + 4.84166i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.g even 21 1 inner
215.u even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.u.a ✓ 240
5.b even 2 1 inner 215.2.u.a ✓ 240
43.g even 21 1 inner 215.2.u.a ✓ 240
215.u even 42 1 inner 215.2.u.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.u.a ✓ 240 1.a even 1 1 trivial
215.2.u.a ✓ 240 5.b even 2 1 inner
215.2.u.a ✓ 240 43.g even 21 1 inner
215.2.u.a ✓ 240 215.u even 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(215, [\chi])\).