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Properties

Label 215.2.p.a
Level $215$
Weight $2$
Character orbit 215.p
Analytic conductor $1.717$
Analytic rank $0$
Dimension $120$
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(4,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.p (of order \(14\), 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: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 8 q^{4} - 3 q^{5} - 28 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 8 q^{4} - 3 q^{5} - 28 q^{6} + 16 q^{9} - q^{10} - 14 q^{11} + 12 q^{14} - q^{15} - 68 q^{16} - 2 q^{19} - 23 q^{20} + 44 q^{21} - 54 q^{24} + 3 q^{25} - 22 q^{26} - 34 q^{29} - 96 q^{30} - 18 q^{31} - 26 q^{34} + 18 q^{35} + 4 q^{36} + 18 q^{39} + 43 q^{40} + 6 q^{41} + 164 q^{44} + 54 q^{45} - 22 q^{46} - 76 q^{49} + 38 q^{50} - 162 q^{51} - 68 q^{54} + 55 q^{55} + 106 q^{56} - 38 q^{59} - 101 q^{60} + 12 q^{61} - 116 q^{64} - 57 q^{65} - 100 q^{66} + 32 q^{69} - 8 q^{70} - 22 q^{71} + 72 q^{74} + 68 q^{75} + 106 q^{76} + 4 q^{79} + 58 q^{80} - 60 q^{81} + 28 q^{84} + 24 q^{85} - 60 q^{86} - 22 q^{89} - 62 q^{90} + 100 q^{91} + 134 q^{94} - 91 q^{95} + 72 q^{96} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.11627 − 1.68767i −1.16738 + 0.930951i 1.18533 + 5.19326i −2.00130 − 0.997390i 4.04162 2.34493i 3.90715 − 8.11327i −0.171467 + 0.751244i 2.55203 + 5.48828i
4.2 −2.06728 − 1.64860i 2.52234 − 2.01150i 1.11072 + 4.86638i 1.03407 + 1.98260i −8.53055 1.64558i 3.43205 − 7.12672i 1.64851 − 7.22259i 1.13079 − 5.80336i
4.3 −1.66301 − 1.32621i 0.0825926 − 0.0658654i 0.561743 + 2.46116i −1.28535 + 1.82972i −0.224704 − 3.24607i 0.484015 − 1.00507i −0.665080 + 2.91390i 4.56415 − 1.33821i
4.4 −1.55765 − 1.24218i 1.30079 − 1.03734i 0.438207 + 1.91991i −0.146161 − 2.23129i −3.31474 − 1.96804i −0.0265503 + 0.0551322i −0.0515959 + 0.226056i −2.54400 + 3.65712i
4.5 −1.18481 − 0.944855i −2.23204 + 1.78000i 0.0659841 + 0.289095i −0.0293562 − 2.23588i 4.32639 − 0.513568i −1.12007 + 2.32584i 1.14607 − 5.02128i −2.07780 + 2.67683i
4.6 −0.944674 − 0.753352i 0.157780 − 0.125825i −0.120173 − 0.526511i 0.0154535 + 2.23601i −0.243842 3.77369i −1.33163 + 2.76517i −0.658500 + 2.88508i 1.66991 − 2.12395i
4.7 −0.834905 − 0.665814i −0.918999 + 0.732877i −0.191285 − 0.838073i 2.18714 − 0.465203i 1.25524 − 1.48083i −1.32497 + 2.75133i −0.360113 + 1.57776i −2.13579 − 1.06783i
4.8 −0.532153 − 0.424377i 1.77638 − 1.41661i −0.341952 − 1.49819i 2.00134 + 0.997316i −1.54648 − 3.22213i −1.04447 + 2.16887i 0.481158 − 2.10809i −0.641780 − 1.38005i
4.9 −0.462141 − 0.368545i −0.766098 + 0.610943i −0.367293 − 1.60922i −2.15006 − 0.614198i 0.579206 1.49705i −0.936265 + 1.94418i −0.453908 + 1.98870i 0.767271 + 1.07624i
4.10 −0.142756 − 0.113844i 2.05187 − 1.63631i −0.437623 − 1.91735i 0.825421 − 2.07814i −0.479201 4.59604i −0.314253 + 0.652554i 0.865088 − 3.79020i −0.354418 + 0.202698i
4.11 0.142756 + 0.113844i −2.05187 + 1.63631i −0.437623 − 1.91735i −1.11012 + 1.94104i −0.479201 − 4.59604i 0.314253 − 0.652554i 0.865088 − 3.79020i −0.379452 + 0.150715i
4.12 0.462141 + 0.368545i 0.766098 − 0.610943i −0.367293 − 1.60922i −1.82074 − 1.29804i 0.579206 − 1.49705i 0.936265 − 1.94418i −0.453908 + 1.98870i −0.363053 − 1.27090i
4.13 0.532153 + 0.424377i −1.77638 + 1.41661i −0.341952 − 1.49819i 2.02755 + 0.942894i −1.54648 3.22213i 1.04447 − 2.16887i 0.481158 − 2.10809i 0.678822 + 1.36221i
4.14 0.834905 + 0.665814i 0.918999 − 0.732877i −0.191285 − 0.838073i 0.999950 + 2.00003i 1.25524 1.48083i 1.32497 − 2.75133i −0.360113 + 1.57776i −0.496783 + 2.33561i
4.15 0.944674 + 0.753352i −0.157780 + 0.125825i −0.120173 − 0.526511i 1.75782 − 1.38205i −0.243842 − 3.77369i 1.33163 − 2.76517i −0.658500 + 2.88508i 2.70174 + 0.0186722i
4.16 1.18481 + 0.944855i 2.23204 − 1.78000i 0.0659841 + 0.289095i −1.76638 + 1.37109i 4.32639 0.513568i 1.12007 − 2.32584i 1.14607 − 5.02128i −3.38831 − 0.0444872i
4.17 1.55765 + 1.24218i −1.30079 + 1.03734i 0.438207 + 1.91991i −1.83562 + 1.27691i −3.31474 1.96804i 0.0265503 − 0.0551322i −0.0515959 + 0.226056i −4.44541 − 0.291199i
4.18 1.66301 + 1.32621i −0.0825926 + 0.0658654i 0.561743 + 2.46116i 0.629130 − 2.14574i −0.224704 3.24607i −0.484015 + 1.00507i −0.665080 + 2.91390i 3.89195 − 2.73404i
4.19 2.06728 + 1.64860i −2.52234 + 2.01150i 1.11072 + 4.86638i 2.19479 − 0.427657i −8.53055 − 1.64558i −3.43205 + 7.12672i 1.64851 − 7.22259i 5.24228 + 2.73425i
4.20 2.11627 + 1.68767i 1.16738 − 0.930951i 1.18533 + 5.19326i −2.02758 − 0.942819i 4.04162 − 2.34493i −3.90715 + 8.11327i −0.171467 + 0.751244i −2.69974 − 5.41714i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
43.e even 7 1 inner
215.p even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.p.a ✓ 120
5.b even 2 1 inner 215.2.p.a ✓ 120
43.e even 7 1 inner 215.2.p.a ✓ 120
215.p even 14 1 inner 215.2.p.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.p.a ✓ 120 1.a even 1 1 trivial
215.2.p.a ✓ 120 5.b even 2 1 inner
215.2.p.a ✓ 120 43.e even 7 1 inner
215.2.p.a ✓ 120 215.p even 14 1 inner

Hecke kernels

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