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Properties

Label 322.3.p.a
Level $322$
Weight $3$
Character orbit 322.p
Analytic conductor $8.774$
Analytic rank $0$
Dimension $640$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,3,Mod(11,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([44, 27]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.p (of order \(66\), degree \(20\), 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: \(8.77386451240\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 640 q + 64 q^{4} + 16 q^{6} + 112 q^{9} + 128 q^{16} + 128 q^{18} + 88 q^{20} - 132 q^{21} - 42 q^{23} - 16 q^{24} - 264 q^{25} + 80 q^{26} - 540 q^{27} + 264 q^{28} - 120 q^{29} - 352 q^{30} - 12 q^{31}+ \cdots + 2244 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.25700 − 0.648030i −3.89257 − 3.06115i 1.16011 + 1.62915i −5.21143 + 5.46559i 2.90925 + 6.37038i −6.68556 + 2.07445i −0.402527 − 2.79964i 3.65962 + 15.0852i 10.0926 − 3.49310i
11.2 −1.25700 − 0.648030i −3.85387 − 3.03072i 1.16011 + 1.62915i 5.16879 − 5.42087i 2.88033 + 6.30705i 4.35506 + 5.48028i −0.402527 − 2.79964i 3.54525 + 14.6137i −10.0101 + 3.46452i
11.3 −1.25700 − 0.648030i −3.73796 − 2.93957i 1.16011 + 1.62915i 3.97965 − 4.17374i 2.79370 + 6.11735i −4.89473 − 5.00416i −0.402527 − 2.79964i 3.20948 + 13.2297i −7.70714 + 2.66747i
11.4 −1.25700 − 0.648030i −3.02934 − 2.38230i 1.16011 + 1.62915i −1.89902 + 1.99164i 2.26409 + 4.95767i 6.80071 + 1.65840i −0.402527 − 2.79964i 1.37973 + 5.68733i 3.67772 − 1.27287i
11.5 −1.25700 − 0.648030i −2.00962 − 1.58038i 1.16011 + 1.62915i 1.20550 − 1.26430i 1.50196 + 3.28884i −6.90349 + 1.15834i −0.402527 − 2.79964i −0.580867 − 2.39437i −2.33462 + 0.808021i
11.6 −1.25700 − 0.648030i −1.41063 − 1.10934i 1.16011 + 1.62915i −4.46910 + 4.68706i 1.05429 + 2.30857i 1.67113 − 6.79760i −0.402527 − 2.79964i −1.36257 − 5.61657i 8.65504 − 2.99554i
11.7 −1.25700 − 0.648030i −0.755926 − 0.594467i 1.16011 + 1.62915i 3.32345 − 3.48553i 0.564969 + 1.23711i 1.11071 − 6.91132i −0.402527 − 2.79964i −1.90380 − 7.84756i −6.43631 + 2.22763i
11.8 −1.25700 − 0.648030i 0.0985099 + 0.0774690i 1.16011 + 1.62915i −4.68358 + 4.91199i −0.0736249 − 0.161216i 1.08599 + 6.91525i −0.402527 − 2.79964i −2.11813 − 8.73104i 9.07039 − 3.13929i
11.9 −1.25700 − 0.648030i 0.383835 + 0.301851i 1.16011 + 1.62915i −1.43428 + 1.50423i −0.286873 − 0.628164i 5.66810 + 4.10764i −0.402527 − 2.79964i −2.06562 − 8.51458i 2.77768 − 0.961364i
11.10 −1.25700 − 0.648030i 1.22290 + 0.961696i 1.16011 + 1.62915i 5.14971 − 5.40086i −0.913976 − 2.00133i 6.81593 − 1.59470i −0.402527 − 2.79964i −1.55122 − 6.39420i −9.97313 + 3.45173i
11.11 −1.25700 − 0.648030i 1.24577 + 0.979686i 1.16011 + 1.62915i 2.88072 − 3.02121i −0.931073 − 2.03877i −3.79398 + 5.88266i −0.402527 − 2.79964i −1.52967 − 6.30537i −5.57891 + 1.93088i
11.12 −1.25700 − 0.648030i 2.57257 + 2.02309i 1.16011 + 1.62915i −2.00945 + 2.10745i −1.92271 − 4.21014i −3.84021 − 5.85259i −0.402527 − 2.79964i 0.403390 + 1.66279i 3.89157 − 1.34689i
11.13 −1.25700 − 0.648030i 2.74918 + 2.16198i 1.16011 + 1.62915i −0.940848 + 0.986733i −2.05470 − 4.49917i −6.99974 − 0.0597914i −0.402527 − 2.79964i 0.762015 + 3.14107i 1.82208 − 0.630629i
11.14 −1.25700 − 0.648030i 3.56094 + 2.80035i 1.16011 + 1.62915i 0.809396 − 0.848871i −2.66140 − 5.82765i 4.21811 − 5.58637i −0.402527 − 2.79964i 2.71647 + 11.1975i −1.56751 + 0.542520i
11.15 −1.25700 − 0.648030i 4.36265 + 3.43082i 1.16011 + 1.62915i 5.58209 − 5.85433i −3.26058 − 7.13968i −0.862587 + 6.94665i −0.402527 − 2.79964i 5.14030 + 21.1886i −10.8105 + 3.74155i
11.16 −1.25700 − 0.648030i 4.36393 + 3.43183i 1.16011 + 1.62915i −6.08549 + 6.38228i −3.26154 − 7.14178i 2.75429 + 6.43536i −0.402527 − 2.79964i 5.14458 + 21.2063i 11.7854 − 4.07897i
11.17 1.25700 + 0.648030i −4.41158 − 3.46931i 1.16011 + 1.62915i −0.984942 + 1.03298i −3.29716 − 7.21977i −5.92223 − 3.73192i 0.402527 + 2.79964i 5.30414 + 21.8640i −1.90748 + 0.660184i
11.18 1.25700 + 0.648030i −3.80389 − 2.99141i 1.16011 + 1.62915i 3.89919 − 4.08935i −2.84298 − 6.22525i 4.30373 + 5.52068i 0.402527 + 2.79964i 3.39920 + 14.0117i 7.55132 − 2.61354i
11.19 1.25700 + 0.648030i −3.42677 − 2.69484i 1.16011 + 1.62915i −2.69686 + 2.82839i −2.56112 − 5.60808i 6.97589 − 0.580476i 0.402527 + 2.79964i 2.35876 + 9.72294i −5.22285 + 1.80765i
11.20 1.25700 + 0.648030i −2.29767 − 1.80691i 1.16011 + 1.62915i 6.45892 − 6.77392i −1.71725 − 3.76025i −6.66189 − 2.14924i 0.402527 + 2.79964i −0.107458 − 0.442949i 12.5086 − 4.32926i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
23.d odd 22 1 inner
161.p odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.3.p.a ✓ 640
7.c even 3 1 inner 322.3.p.a ✓ 640
23.d odd 22 1 inner 322.3.p.a ✓ 640
161.p odd 66 1 inner 322.3.p.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.p.a ✓ 640 1.a even 1 1 trivial
322.3.p.a ✓ 640 7.c even 3 1 inner
322.3.p.a ✓ 640 23.d odd 22 1 inner
322.3.p.a ✓ 640 161.p odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).