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Properties

Label 784.2.x.o
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
Analytic rank $0$
Dimension $48$
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] = [784,2,Mod(165,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), not 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{2} - 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{8} + 2 q^{10} - 4 q^{11} - 2 q^{12} + 24 q^{13} - 40 q^{15} + 16 q^{16} - 8 q^{17} + 18 q^{18} + 4 q^{19} + 16 q^{20} - 18 q^{24} + 10 q^{26} + 24 q^{27} + 24 q^{29} - 4 q^{30} - 28 q^{31} + 16 q^{32} - 16 q^{33} + 44 q^{34} - 72 q^{36} - 24 q^{37} - 20 q^{38} - 26 q^{40} - 40 q^{43} + 6 q^{44} + 28 q^{45} - 14 q^{46} + 20 q^{47} - 56 q^{48} + 56 q^{50} + 24 q^{51} + 16 q^{52} - 16 q^{53} - 64 q^{54} - 6 q^{58} + 20 q^{59} + 46 q^{60} - 8 q^{61} - 24 q^{62} + 80 q^{64} + 8 q^{65} + 20 q^{66} + 48 q^{67} + 40 q^{69} - 32 q^{72} - 8 q^{74} + 4 q^{75} + 36 q^{76} + 116 q^{78} - 36 q^{79} + 28 q^{80} - 2 q^{82} + 8 q^{83} - 20 q^{86} - 42 q^{88} + 20 q^{90} + 76 q^{92} + 8 q^{93} + 72 q^{94} - 4 q^{95} + 120 q^{96} + 48 q^{97} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1 −1.41419 + 0.00789795i −0.814813 − 3.04092i 1.99988 − 0.0223384i −0.501787 + 1.87270i 1.17632 + 4.29401i 0 −2.82803 + 0.0473857i −5.98523 + 3.45557i 0.694833 − 2.65231i
165.2 −1.32577 − 0.492271i 0.221819 + 0.827840i 1.51534 + 1.30528i 1.10095 − 4.10878i 0.113440 − 1.20672i 0 −1.36644 − 2.47646i 1.96196 − 1.13274i −3.48224 + 4.90534i
165.3 −1.08254 − 0.909999i 0.672759 + 2.51077i 0.343804 + 1.97023i −0.780276 + 2.91203i 1.55651 − 3.33023i 0 1.42072 − 2.44572i −3.25328 + 1.87828i 3.49463 − 2.44235i
165.4 −0.743894 + 1.20276i 0.222846 + 0.831674i −0.893243 − 1.78945i −0.543265 + 2.02749i −1.16607 − 0.350647i 0 2.81674 + 0.256804i 1.95606 − 1.12933i −2.03445 − 2.16165i
165.5 −0.504093 − 1.32132i −0.589510 − 2.20008i −1.49178 + 1.33214i 0.622192 − 2.32205i −2.60985 + 1.88798i 0 2.51218 + 1.29960i −1.89476 + 1.09394i −3.38182 + 0.348414i
165.6 −0.478757 + 1.33071i −0.659252 − 2.46036i −1.54158 − 1.27417i 0.502490 − 1.87532i 3.58965 + 0.300642i 0 2.43360 − 1.44138i −3.02070 + 1.74400i 2.25494 + 1.56649i
165.7 0.0582062 − 1.41302i 0.523249 + 1.95279i −1.99322 − 0.164492i 0.256983 − 0.959072i 2.78978 − 0.625694i 0 −0.348448 + 2.80688i −0.941530 + 0.543593i −1.34023 − 0.418944i
165.8 0.438190 + 1.34461i −0.0145698 − 0.0543752i −1.61598 + 1.17839i 0.337028 − 1.25781i 0.0667293 − 0.0434174i 0 −2.29259 − 1.65651i 2.59533 − 1.49842i 1.83895 − 0.0979855i
165.9 0.559929 − 1.29865i −0.224854 − 0.839165i −1.37296 − 1.45430i −0.847576 + 3.16320i −1.21568 − 0.177868i 0 −2.65738 + 0.968682i 1.94444 − 1.12262i 3.63329 + 2.87187i
165.10 1.01114 − 0.988731i −0.430984 − 1.60845i 0.0448227 − 1.99950i 0.522232 − 1.94900i −2.02612 − 1.20025i 0 −1.93164 − 2.06610i 0.196696 − 0.113563i −1.39898 − 2.48706i
165.11 1.07234 + 0.922005i 0.781062 + 2.91496i 0.299815 + 1.97740i −0.199758 + 0.745506i −1.85005 + 3.84597i 0 −1.50167 + 2.39687i −5.28887 + 3.05353i −0.901568 + 0.615256i
165.12 1.40944 − 0.116052i 0.312249 + 1.16533i 1.97306 − 0.327139i 0.262843 − 0.980942i 0.575337 + 1.60623i 0 2.74296 − 0.690063i 1.33758 − 0.772254i 0.256621 − 1.41309i
373.1 −1.40462 + 0.164410i 0.839165 + 0.224854i 1.94594 − 0.461868i 3.16320 − 0.847576i −1.21568 − 0.177868i 0 −2.65738 + 0.968682i −1.94444 − 1.12262i −4.30375 + 1.71059i
373.2 −1.36184 − 0.381311i 1.60845 + 0.430984i 1.70920 + 1.03857i −1.94900 + 0.522232i −2.02612 − 1.20025i 0 −1.93164 − 2.06610i −0.196696 − 0.113563i 2.85335 + 0.0319777i
373.3 −1.25281 + 0.656100i −1.95279 − 0.523249i 1.13907 − 1.64394i −0.959072 + 0.256983i 2.78978 − 0.625694i 0 −0.348448 + 2.80688i 0.941530 + 0.543593i 1.03293 − 0.951197i
373.4 −0.892252 + 1.09722i 2.20008 + 0.589510i −0.407774 − 1.95799i −2.32205 + 0.622192i −2.60985 + 1.88798i 0 2.51218 + 1.29960i 1.89476 + 1.09394i 1.38918 − 3.10295i
373.5 −0.805226 − 1.16259i −1.16533 − 0.312249i −0.703221 + 1.87229i −0.980942 + 0.262843i 0.575337 + 1.60623i 0 2.74296 − 0.690063i −1.33758 − 0.772254i 1.09546 + 0.928784i
373.6 −0.246810 + 1.39251i −2.51077 − 0.672759i −1.87817 − 0.687371i 2.91203 − 0.780276i 1.55651 − 3.33023i 0 1.42072 − 2.44572i 3.25328 + 1.87828i 0.367824 + 4.24761i
373.7 0.236566 + 1.39429i −0.827840 − 0.221819i −1.88807 + 0.659683i −4.10878 + 1.10095i 0.113440 − 1.20672i 0 −1.36644 − 2.47646i −1.96196 − 1.13274i −2.50703 − 5.46838i
373.8 0.262311 − 1.38967i −2.91496 − 0.781062i −1.86239 − 0.729053i 0.745506 − 0.199758i −1.85005 + 3.84597i 0 −1.50167 + 2.39687i 5.28887 + 3.05353i −0.0820438 − 1.08841i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.o 48
7.b odd 2 1 112.2.w.c ✓ 48
7.c even 3 1 784.2.m.k 24
7.c even 3 1 inner 784.2.x.o 48
7.d odd 6 1 112.2.w.c ✓ 48
7.d odd 6 1 784.2.m.j 24
16.e even 4 1 inner 784.2.x.o 48
28.d even 2 1 448.2.ba.c 48
28.f even 6 1 448.2.ba.c 48
56.e even 2 1 896.2.ba.e 48
56.h odd 2 1 896.2.ba.f 48
56.j odd 6 1 896.2.ba.f 48
56.m even 6 1 896.2.ba.e 48
112.j even 4 1 448.2.ba.c 48
112.j even 4 1 896.2.ba.e 48
112.l odd 4 1 112.2.w.c ✓ 48
112.l odd 4 1 896.2.ba.f 48
112.v even 12 1 448.2.ba.c 48
112.v even 12 1 896.2.ba.e 48
112.w even 12 1 784.2.m.k 24
112.w even 12 1 inner 784.2.x.o 48
112.x odd 12 1 112.2.w.c ✓ 48
112.x odd 12 1 784.2.m.j 24
112.x odd 12 1 896.2.ba.f 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.c ✓ 48 7.b odd 2 1
112.2.w.c ✓ 48 7.d odd 6 1
112.2.w.c ✓ 48 112.l odd 4 1
112.2.w.c ✓ 48 112.x odd 12 1
448.2.ba.c 48 28.d even 2 1
448.2.ba.c 48 28.f even 6 1
448.2.ba.c 48 112.j even 4 1
448.2.ba.c 48 112.v even 12 1
784.2.m.j 24 7.d odd 6 1
784.2.m.j 24 112.x odd 12 1
784.2.m.k 24 7.c even 3 1
784.2.m.k 24 112.w even 12 1
784.2.x.o 48 1.a even 1 1 trivial
784.2.x.o 48 7.c even 3 1 inner
784.2.x.o 48 16.e even 4 1 inner
784.2.x.o 48 112.w even 12 1 inner
896.2.ba.e 48 56.e even 2 1
896.2.ba.e 48 56.m even 6 1
896.2.ba.e 48 112.j even 4 1
896.2.ba.e 48 112.v even 12 1
896.2.ba.f 48 56.h odd 2 1
896.2.ba.f 48 56.j odd 6 1
896.2.ba.f 48 112.l odd 4 1
896.2.ba.f 48 112.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{48} - 8 T_{3}^{45} - 162 T_{3}^{44} + 24 T_{3}^{43} + 32 T_{3}^{42} + 1116 T_{3}^{41} + \cdots + 194481 \) Copy content Toggle raw display
\( T_{5}^{48} + 4 T_{5}^{47} + 8 T_{5}^{46} + 40 T_{5}^{45} - 186 T_{5}^{44} - 1252 T_{5}^{43} + \cdots + 7076456545921 \) Copy content Toggle raw display