Properties

Label 495.2.r.c
Level $495$
Weight $2$
Character orbit 495.r
Analytic conductor $3.953$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(164,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.164");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.r (of order \(6\), degree \(2\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(64\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 68 q^{4} - 12 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q + 68 q^{4} - 12 q^{5} - 24 q^{9} - 12 q^{11} - 12 q^{14} + 18 q^{15} - 36 q^{16} + 12 q^{20} - 4 q^{25} + 4 q^{34} + 24 q^{36} - 18 q^{45} - 76 q^{49} - 4 q^{55} - 36 q^{56} - 168 q^{59} + 78 q^{60} - 144 q^{64} - 24 q^{66} + 36 q^{69} - 44 q^{70} - 36 q^{75} + 72 q^{81} - 84 q^{86} + 64 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1 −2.35894 1.36194i −0.491830 1.66075i 2.70974 + 4.69341i 1.99447 + 1.01099i −1.10164 + 4.58746i −0.885979 + 1.53456i 9.31422i −2.51621 + 1.63362i −3.32794 5.10120i
164.2 −2.35894 1.36194i 0.491830 + 1.66075i 2.70974 + 4.69341i 0.121696 2.23275i 1.10164 4.58746i −0.885979 + 1.53456i 9.31422i −2.51621 + 1.63362i −3.32794 + 5.10120i
164.3 −2.22820 1.28645i −1.41739 + 0.995490i 2.30993 + 4.00091i −2.12513 + 0.695571i 4.43889 0.394744i −0.0241406 + 0.0418127i 6.74064i 1.01800 2.82200i 5.63004 + 1.18401i
164.4 −2.22820 1.28645i 1.41739 0.995490i 2.30993 + 4.00091i −1.66495 + 1.49263i −4.43889 + 0.394744i −0.0241406 + 0.0418127i 6.74064i 1.01800 2.82200i 5.63004 1.18401i
164.5 −2.16245 1.24849i −1.60498 + 0.651194i 2.11746 + 3.66754i 2.12949 + 0.682097i 4.28369 + 0.595622i 2.33858 4.05054i 5.58053i 2.15189 2.09030i −3.75333 4.13365i
164.6 −2.16245 1.24849i 1.60498 0.651194i 2.11746 + 3.66754i 0.474033 2.18524i −4.28369 0.595622i 2.33858 4.05054i 5.58053i 2.15189 2.09030i −3.75333 + 4.13365i
164.7 −2.03042 1.17226i −0.0364388 + 1.73167i 1.74841 + 3.02833i 0.752578 + 2.10562i 2.10396 3.47330i −1.67901 + 2.90812i 3.50932i −2.99734 0.126200i 0.940291 5.15751i
164.8 −2.03042 1.17226i 0.0364388 1.73167i 1.74841 + 3.02833i −1.44723 1.70456i −2.10396 + 3.47330i −1.67901 + 2.90812i 3.50932i −2.99734 0.126200i 0.940291 + 5.15751i
164.9 −1.84053 1.06263i −1.00779 1.40867i 1.25838 + 2.17957i −1.05343 + 1.97238i 0.357974 + 3.66362i 1.55655 2.69602i 1.09824i −0.968712 + 2.83930i 4.03479 2.51082i
164.10 −1.84053 1.06263i 1.00779 + 1.40867i 1.25838 + 2.17957i −2.23485 0.0738892i −0.357974 3.66362i 1.55655 2.69602i 1.09824i −0.968712 + 2.83930i 4.03479 + 2.51082i
164.11 −1.70741 0.985775i −1.59259 0.680924i 0.943504 + 1.63420i −1.79646 1.33143i 2.04797 + 2.73255i 0.0538350 0.0932449i 0.222770i 2.07268 + 2.16887i 1.75481 + 4.04422i
164.12 −1.70741 0.985775i 1.59259 + 0.680924i 0.943504 + 1.63420i 0.254824 + 2.22150i −2.04797 2.73255i 0.0538350 0.0932449i 0.222770i 2.07268 + 2.16887i 1.75481 4.04422i
164.13 −1.62560 0.938540i −0.155415 + 1.72506i 0.761715 + 1.31933i 2.19111 0.446145i 1.87168 2.65840i 1.01834 1.76382i 0.894560i −2.95169 0.536200i −3.98059 1.33119i
164.14 −1.62560 0.938540i 0.155415 1.72506i 0.761715 + 1.31933i 1.48193 1.67448i −1.87168 + 2.65840i 1.01834 1.76382i 0.894560i −2.95169 0.536200i −3.98059 + 1.33119i
164.15 −1.58655 0.915995i −1.69477 0.357410i 0.678093 + 1.17449i 0.985404 + 2.00723i 2.36146 + 2.11945i −1.65812 + 2.87195i 1.17946i 2.74452 + 1.21146i 0.275221 4.08720i
164.16 −1.58655 0.915995i 1.69477 + 0.357410i 0.678093 + 1.17449i −1.24561 1.85700i −2.36146 2.11945i −1.65812 + 2.87195i 1.17946i 2.74452 + 1.21146i 0.275221 + 4.08720i
164.17 −1.29115 0.745448i −1.09243 + 1.34410i 0.111386 + 0.192926i 0.565909 2.16327i 2.41245 0.921083i −0.573487 + 0.993309i 2.64966i −0.613184 2.93667i −2.34328 + 2.37126i
164.18 −1.29115 0.745448i 1.09243 1.34410i 0.111386 + 0.192926i 2.15640 + 0.591544i −2.41245 + 0.921083i −0.573487 + 0.993309i 2.64966i −0.613184 2.93667i −2.34328 2.37126i
164.19 −1.12460 0.649288i −1.13838 + 1.30540i −0.156849 0.271671i −2.22900 + 0.177667i 2.12781 0.728919i −1.61390 + 2.79536i 3.00452i −0.408162 2.97210i 2.62209 + 1.24746i
164.20 −1.12460 0.649288i 1.13838 1.30540i −0.156849 0.271671i −1.26836 + 1.84154i −2.12781 + 0.728919i −1.61390 + 2.79536i 3.00452i −0.408162 2.97210i 2.62209 1.24746i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
11.b odd 2 1 inner
45.h odd 6 1 inner
55.d odd 2 1 inner
99.g even 6 1 inner
495.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.r.c 128
5.b even 2 1 inner 495.2.r.c 128
9.d odd 6 1 inner 495.2.r.c 128
11.b odd 2 1 inner 495.2.r.c 128
45.h odd 6 1 inner 495.2.r.c 128
55.d odd 2 1 inner 495.2.r.c 128
99.g even 6 1 inner 495.2.r.c 128
495.r even 6 1 inner 495.2.r.c 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.r.c 128 1.a even 1 1 trivial
495.2.r.c 128 5.b even 2 1 inner
495.2.r.c 128 9.d odd 6 1 inner
495.2.r.c 128 11.b odd 2 1 inner
495.2.r.c 128 45.h odd 6 1 inner
495.2.r.c 128 55.d odd 2 1 inner
495.2.r.c 128 99.g even 6 1 inner
495.2.r.c 128 495.r even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{64} - 49 T_{2}^{62} + 1320 T_{2}^{60} - 24557 T_{2}^{58} + 348458 T_{2}^{56} - 3963045 T_{2}^{54} + 37266104 T_{2}^{52} - 295506029 T_{2}^{50} + 2003338167 T_{2}^{48} - 11720212940 T_{2}^{46} + \cdots + 227889216 \) Copy content Toggle raw display
\( T_{23}^{64} + 303 T_{23}^{62} + 53520 T_{23}^{60} + 6281307 T_{23}^{58} + 546857901 T_{23}^{56} + 36402572421 T_{23}^{54} + 1912157107605 T_{23}^{52} + 79918113532788 T_{23}^{50} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display