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Properties

Label 285.2.bh.a
Level $285$
Weight $2$
Character orbit 285.bh
Analytic conductor $2.276$
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] = [285,2,Mod(13,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 27, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.bh (of order \(36\), 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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(20\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 12 q^{6} - 12 q^{7} - 36 q^{16} + 120 q^{20} - 48 q^{21} + 24 q^{22} + 24 q^{23} - 24 q^{25} - 24 q^{26} - 72 q^{28} - 12 q^{32} - 12 q^{33} - 12 q^{36} - 132 q^{38} - 132 q^{40} - 72 q^{41} - 108 q^{43} + 24 q^{47} + 24 q^{51} + 36 q^{53} - 24 q^{57} - 144 q^{58} + 48 q^{60} + 24 q^{61} - 168 q^{62} - 144 q^{67} - 48 q^{68} - 48 q^{70} - 72 q^{72} - 36 q^{73} - 48 q^{76} - 216 q^{77} + 120 q^{78} + 24 q^{80} + 144 q^{82} - 12 q^{83} + 72 q^{85} + 96 q^{86} - 48 q^{87} + 360 q^{88} + 36 q^{90} + 72 q^{91} - 48 q^{92} + 24 q^{95} + 48 q^{96} + 96 q^{97} + 192 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.80249 − 0.245186i 0.906308 − 0.422618i 5.82423 + 1.02697i 0.962684 − 2.01823i −2.64354 + 0.962170i −0.900815 − 3.36189i −10.6359 − 2.84987i 0.642788 − 0.766044i −3.19276 + 5.42003i
13.2 −2.52754 − 0.221131i −0.906308 + 0.422618i 4.36997 + 0.770543i 2.18153 + 0.490834i 2.38419 − 0.867773i 1.13875 + 4.24987i −5.97340 − 1.60057i 0.642788 − 0.766044i −5.40538 − 1.72301i
13.3 −2.41484 − 0.211271i −0.906308 + 0.422618i 3.81721 + 0.673077i −1.58105 + 1.58123i 2.27788 − 0.829079i −0.888135 − 3.31457i −4.39281 − 1.17705i 0.642788 − 0.766044i 4.15205 − 3.48438i
13.4 −1.98980 − 0.174085i 0.906308 − 0.422618i 1.95939 + 0.345493i −2.19664 + 0.418081i −1.87694 + 0.683152i 0.172868 + 0.645153i 0.0200355 + 0.00536850i 0.642788 − 0.766044i 4.44365 − 0.449497i
13.5 −1.59093 − 0.139189i 0.906308 − 0.422618i 0.542082 + 0.0955836i 0.597651 − 2.15472i −1.50070 + 0.546210i 0.889881 + 3.32108i 2.23608 + 0.599155i 0.642788 − 0.766044i −1.25074 + 3.34483i
13.6 −1.40901 − 0.123273i −0.906308 + 0.422618i 0.000501844 0 8.84886e-5i −0.576957 − 2.16035i 1.32910 − 0.483751i 0.0816294 + 0.304645i 2.73170 + 0.731958i 0.642788 − 0.766044i 0.546627 + 3.11508i
13.7 −0.738279 − 0.0645910i −0.906308 + 0.422618i −1.42873 − 0.251924i −2.22921 − 0.175056i 0.696405 − 0.253471i 0.187174 + 0.698543i 2.47022 + 0.661894i 0.642788 − 0.766044i 1.63447 + 0.273226i
13.8 −0.693370 − 0.0606620i 0.906308 − 0.422618i −1.49253 − 0.263174i −1.33943 + 1.79051i −0.654043 + 0.238052i 0.0498847 + 0.186172i 2.36352 + 0.633303i 0.642788 − 0.766044i 1.03734 − 1.16023i
13.9 −0.470334 − 0.0411489i 0.906308 − 0.422618i −1.75009 − 0.308589i 2.17146 + 0.533633i −0.443658 + 0.161478i −1.01663 − 3.79411i 1.72252 + 0.461547i 0.642788 − 0.766044i −0.999353 − 0.340339i
13.10 0.124871 + 0.0109248i −0.906308 + 0.422618i −1.95414 − 0.344568i 2.18886 + 0.457041i −0.117788 + 0.0428715i 0.208667 + 0.778755i −0.482404 − 0.129260i 0.642788 − 0.766044i 0.268332 + 0.0809839i
13.11 0.408739 + 0.0357601i 0.906308 − 0.422618i −1.80383 − 0.318063i −1.61592 − 1.54558i 0.385557 − 0.140331i −0.297640 − 1.11081i −1.51856 − 0.406897i 0.642788 − 0.766044i −0.605218 − 0.689525i
13.12 0.431412 + 0.0377436i −0.906308 + 0.422618i −1.78492 − 0.314730i −0.270782 + 2.21961i −0.406943 + 0.148115i −0.836073 − 3.12027i −1.59476 − 0.427316i 0.642788 − 0.766044i −0.200595 + 0.947346i
13.13 0.806152 + 0.0705291i 0.906308 − 0.422618i −1.32471 − 0.233582i 1.45963 + 1.69396i 0.760428 − 0.276773i 1.21299 + 4.52694i −2.61476 − 0.700622i 0.642788 − 0.766044i 1.05721 + 1.46853i
13.14 1.18722 + 0.103868i −0.906308 + 0.422618i −0.570922 − 0.100669i −0.0855107 − 2.23443i −1.11988 + 0.407603i −1.02110 − 3.81081i −2.96964 − 0.795712i 0.642788 − 0.766044i 0.130566 − 2.66164i
13.15 1.19597 + 0.104634i −0.906308 + 0.422618i −0.550219 − 0.0970185i −2.22258 + 0.245198i −1.12814 + 0.410608i 1.20231 + 4.48710i −2.96716 − 0.795047i 0.642788 − 0.766044i −2.68380 + 0.0606919i
13.16 1.77165 + 0.154999i 0.906308 − 0.422618i 1.14509 + 0.201911i 1.49531 − 1.66255i 1.67116 − 0.608253i 0.833960 + 3.11238i −1.43823 − 0.385372i 0.642788 − 0.766044i 2.90685 − 2.71367i
13.17 1.99687 + 0.174703i 0.906308 − 0.422618i 1.98735 + 0.350423i 0.846671 + 2.06958i 1.88361 − 0.685578i −1.05176 − 3.92524i 0.0348574 + 0.00934001i 0.642788 − 0.766044i 1.32913 + 4.28059i
13.18 2.18751 + 0.191383i −0.906308 + 0.422618i 2.77898 + 0.490009i 0.980413 − 2.00967i −2.06344 + 0.751032i 0.833171 + 3.10944i 1.74318 + 0.467084i 0.642788 − 0.766044i 2.52929 − 4.20856i
13.19 2.21755 + 0.194011i 0.906308 − 0.422618i 2.91027 + 0.513160i −1.42534 − 1.72291i 2.09178 − 0.761344i −0.192174 − 0.717202i 2.05378 + 0.550308i 0.642788 − 0.766044i −2.82650 − 4.09717i
13.20 2.30867 + 0.201982i −0.906308 + 0.422618i 3.31953 + 0.585323i 0.311908 + 2.21421i −2.17772 + 0.792627i 0.125097 + 0.466867i 3.06843 + 0.822183i 0.642788 − 0.766044i 0.272861 + 5.17487i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.f odd 18 1 inner
95.r even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.bh.a ✓ 240
3.b odd 2 1 855.2.dl.b 240
5.c odd 4 1 inner 285.2.bh.a ✓ 240
15.e even 4 1 855.2.dl.b 240
19.f odd 18 1 inner 285.2.bh.a ✓ 240
57.j even 18 1 855.2.dl.b 240
95.r even 36 1 inner 285.2.bh.a ✓ 240
285.bj odd 36 1 855.2.dl.b 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.bh.a ✓ 240 1.a even 1 1 trivial
285.2.bh.a ✓ 240 5.c odd 4 1 inner
285.2.bh.a ✓ 240 19.f odd 18 1 inner
285.2.bh.a ✓ 240 95.r even 36 1 inner
855.2.dl.b 240 3.b odd 2 1
855.2.dl.b 240 15.e even 4 1
855.2.dl.b 240 57.j even 18 1
855.2.dl.b 240 285.bj odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(285, [\chi])\).