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Properties

Label 920.2.x.c
Level $920$
Weight $2$
Character orbit 920.x
Analytic conductor $7.346$
Analytic rank $0$
Dimension $264$
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] = [920,2,Mod(413,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.413");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.x (of order \(4\), 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: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(132\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 4 q^{2} - 16 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 4 q^{2} - 16 q^{6} - 4 q^{8} - 16 q^{12} + 64 q^{16} + 16 q^{18} - 4 q^{23} - 8 q^{25} - 8 q^{26} + 16 q^{31} + 16 q^{32} - 104 q^{36} - 16 q^{41} - 16 q^{46} + 8 q^{47} - 12 q^{48} + 12 q^{50} - 96 q^{52} - 96 q^{55} + 20 q^{58} - 132 q^{62} + 20 q^{70} - 16 q^{71} + 16 q^{72} - 8 q^{73} + 92 q^{78} - 120 q^{81} + 84 q^{82} - 32 q^{87} - 20 q^{92} + 144 q^{95} + 80 q^{96} - 52 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
413.1 −1.41152 + 0.0872300i 1.39583 − 1.39583i 1.98478 − 0.246254i −2.22438 − 0.228324i −1.84849 + 2.09201i −0.328173 − 0.328173i −2.78008 + 0.520725i − 0.896700i 3.15968 + 0.128251i
413.2 −1.41152 + 0.0872300i 1.39583 − 1.39583i 1.98478 − 0.246254i 2.22438 + 0.228324i −1.84849 + 2.09201i 0.328173 + 0.328173i −2.78008 + 0.520725i − 0.896700i −3.15968 − 0.128251i
413.3 −1.40870 − 0.124723i 0.569673 − 0.569673i 1.96889 + 0.351396i −1.11428 − 1.93865i −0.873551 + 0.731448i −2.21503 − 2.21503i −2.72975 − 0.740579i 2.35095i 1.32789 + 2.86997i
413.4 −1.40870 − 0.124723i 0.569673 − 0.569673i 1.96889 + 0.351396i 1.11428 + 1.93865i −0.873551 + 0.731448i 2.21503 + 2.21503i −2.72975 − 0.740579i 2.35095i −1.32789 − 2.86997i
413.5 −1.40742 − 0.138403i −1.22169 + 1.22169i 1.96169 + 0.389583i −0.126470 − 2.23249i 1.88853 − 1.55036i −0.715528 − 0.715528i −2.70701 − 0.819812i 0.0149250i −0.130985 + 3.15956i
413.6 −1.40742 − 0.138403i −1.22169 + 1.22169i 1.96169 + 0.389583i 0.126470 + 2.23249i 1.88853 − 1.55036i 0.715528 + 0.715528i −2.70701 − 0.819812i 0.0149250i 0.130985 − 3.15956i
413.7 −1.40687 + 0.143912i −0.790180 + 0.790180i 1.95858 − 0.404932i −2.23004 + 0.164085i 0.997965 − 1.22540i 2.29410 + 2.29410i −2.69719 + 0.851551i 1.75123i 3.11377 − 0.551776i
413.8 −1.40687 + 0.143912i −0.790180 + 0.790180i 1.95858 − 0.404932i 2.23004 − 0.164085i 0.997965 − 1.22540i −2.29410 − 2.29410i −2.69719 + 0.851551i 1.75123i −3.11377 + 0.551776i
413.9 −1.38153 − 0.302280i 2.33354 − 2.33354i 1.81725 + 0.835218i −0.491368 + 2.18141i −3.92925 + 2.51848i −0.767580 − 0.767580i −2.25812 − 1.70320i − 7.89085i 1.33824 − 2.86516i
413.10 −1.38153 − 0.302280i 2.33354 − 2.33354i 1.81725 + 0.835218i 0.491368 − 2.18141i −3.92925 + 2.51848i 0.767580 + 0.767580i −2.25812 − 1.70320i − 7.89085i −1.33824 + 2.86516i
413.11 −1.34566 + 0.434979i −2.00419 + 2.00419i 1.62159 − 1.17067i −1.28359 + 1.83095i 1.82517 − 3.56874i 1.08315 + 1.08315i −1.67289 + 2.28067i − 5.03357i 0.930846 − 3.02217i
413.12 −1.34566 + 0.434979i −2.00419 + 2.00419i 1.62159 − 1.17067i 1.28359 − 1.83095i 1.82517 − 3.56874i −1.08315 − 1.08315i −1.67289 + 2.28067i − 5.03357i −0.930846 + 3.02217i
413.13 −1.33616 − 0.463335i −2.22481 + 2.22481i 1.57064 + 1.23818i −2.22212 + 0.249389i 4.00353 − 1.94187i −2.54743 − 2.54743i −1.52494 − 2.38214i − 6.89955i 3.08465 + 0.696361i
413.14 −1.33616 − 0.463335i −2.22481 + 2.22481i 1.57064 + 1.23818i 2.22212 − 0.249389i 4.00353 − 1.94187i 2.54743 + 2.54743i −1.52494 − 2.38214i − 6.89955i −3.08465 − 0.696361i
413.15 −1.33328 + 0.471562i −0.296434 + 0.296434i 1.55526 − 1.25745i −0.251163 − 2.22192i 0.255442 − 0.535016i 3.54670 + 3.54670i −1.48063 + 2.40993i 2.82425i 1.38264 + 2.84399i
413.16 −1.33328 + 0.471562i −0.296434 + 0.296434i 1.55526 − 1.25745i 0.251163 + 2.22192i 0.255442 − 0.535016i −3.54670 − 3.54670i −1.48063 + 2.40993i 2.82425i −1.38264 − 2.84399i
413.17 −1.30237 + 0.551202i 1.10544 − 1.10544i 1.39235 − 1.43574i −1.04628 + 1.97619i −0.830375 + 2.04902i −0.867616 − 0.867616i −1.02198 + 2.63734i 0.556000i 0.273363 − 3.15044i
413.18 −1.30237 + 0.551202i 1.10544 − 1.10544i 1.39235 − 1.43574i 1.04628 − 1.97619i −0.830375 + 2.04902i 0.867616 + 0.867616i −1.02198 + 2.63734i 0.556000i −0.273363 + 3.15044i
413.19 −1.27320 − 0.615599i −1.39301 + 1.39301i 1.24208 + 1.56756i −1.79988 − 1.32682i 2.63112 − 0.916046i 1.14201 + 1.14201i −0.616423 − 2.76044i − 0.880959i 1.47482 + 2.79731i
413.20 −1.27320 − 0.615599i −1.39301 + 1.39301i 1.24208 + 1.56756i 1.79988 + 1.32682i 2.63112 − 0.916046i −1.14201 − 1.14201i −0.616423 − 2.76044i − 0.880959i −1.47482 − 2.79731i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 413.132
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
23.b odd 2 1 inner
40.i odd 4 1 inner
115.e even 4 1 inner
184.e odd 2 1 inner
920.x even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 920.2.x.c ✓ 264
5.c odd 4 1 inner 920.2.x.c ✓ 264
8.b even 2 1 inner 920.2.x.c ✓ 264
23.b odd 2 1 inner 920.2.x.c ✓ 264
40.i odd 4 1 inner 920.2.x.c ✓ 264
115.e even 4 1 inner 920.2.x.c ✓ 264
184.e odd 2 1 inner 920.2.x.c ✓ 264
920.x even 4 1 inner 920.2.x.c ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.x.c ✓ 264 1.a even 1 1 trivial
920.2.x.c ✓ 264 5.c odd 4 1 inner
920.2.x.c ✓ 264 8.b even 2 1 inner
920.2.x.c ✓ 264 23.b odd 2 1 inner
920.2.x.c ✓ 264 40.i odd 4 1 inner
920.2.x.c ✓ 264 115.e even 4 1 inner
920.2.x.c ✓ 264 184.e odd 2 1 inner
920.2.x.c ✓ 264 920.x even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(920, [\chi])\):

\( T_{3}^{132} + 960 T_{3}^{128} + 425250 T_{3}^{124} + 115495592 T_{3}^{120} + 21558614287 T_{3}^{116} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
\( T_{11}^{66} - 364 T_{11}^{64} + 62516 T_{11}^{62} - 6745432 T_{11}^{60} + 513616327 T_{11}^{58} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display