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Properties

Label 322.3.n.a
Level $322$
Weight $3$
Character orbit 322.n
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(3,322)]
 
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([11, 48]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 322.n (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.

\(\operatorname{Tr}(f)(q) = \) \( 640 q + 64 q^{4} + 12 q^{5} - 12 q^{7} - 80 q^{9} - 12 q^{11} + 64 q^{14} - 32 q^{15} + 128 q^{16} - 128 q^{18} - 72 q^{19} + 84 q^{21} - 32 q^{22} - 74 q^{23} - 48 q^{24} + 16 q^{25} - 48 q^{26} - 216 q^{28}+ \cdots + 4324 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.11165 + 0.874209i −4.74563 + 3.37935i 0.471518 − 1.94362i 1.84102 − 9.55212i 2.32121 − 7.90531i −6.00323 + 3.60017i 1.17497 + 2.57283i 8.15740 − 23.5693i 6.30398 + 12.2280i
3.2 −1.11165 + 0.874209i −4.15068 + 2.95569i 0.471518 − 1.94362i −0.473192 + 2.45515i 2.03021 − 6.91425i 6.83869 + 1.49408i 1.17497 + 2.57283i 5.54847 − 16.0313i −1.62029 − 3.14293i
3.3 −1.11165 + 0.874209i −3.78233 + 2.69339i 0.471518 − 1.94362i −1.41059 + 7.31886i 1.85004 − 6.30064i −2.81835 − 6.40756i 1.17497 + 2.57283i 4.10810 − 11.8696i −4.83013 − 9.36914i
3.4 −1.11165 + 0.874209i −2.80080 + 1.99444i 0.471518 − 1.94362i −0.737564 + 3.82685i 1.36994 − 4.66560i −4.16628 + 5.62513i 1.17497 + 2.57283i 0.923072 − 2.66704i −2.52555 − 4.89889i
3.5 −1.11165 + 0.874209i −2.44387 + 1.74027i 0.471518 − 1.94362i 0.735244 − 3.81481i 1.19536 − 4.07102i 1.44533 − 6.84916i 1.17497 + 2.57283i 0.000339994 0 0.000982349i 2.51761 + 4.88347i
3.6 −1.11165 + 0.874209i −1.89275 + 1.34782i 0.471518 − 1.94362i 0.837424 − 4.34497i 0.925792 − 3.15296i 6.26522 + 3.12203i 1.17497 + 2.57283i −1.17773 + 3.40283i 2.86749 + 5.56216i
3.7 −1.11165 + 0.874209i −1.37861 + 0.981707i 0.471518 − 1.94362i 0.380300 − 1.97318i 0.674316 − 2.29651i −6.71241 − 1.98583i 1.17497 + 2.57283i −2.00678 + 5.79822i 1.30221 + 2.52594i
3.8 −1.11165 + 0.874209i 0.504676 − 0.359378i 0.471518 − 1.94362i −1.00974 + 5.23903i −0.246850 + 0.840694i −4.18462 + 5.61151i 1.17497 + 2.57283i −2.81807 + 8.14227i −3.45753 − 6.70667i
3.9 −1.11165 + 0.874209i 0.506631 − 0.360771i 0.471518 − 1.94362i −0.655707 + 3.40213i −0.247806 + 0.843951i 4.62081 − 5.25815i 1.17497 + 2.57283i −2.81709 + 8.13945i −2.24526 − 4.35519i
3.10 −1.11165 + 0.874209i 1.30880 − 0.931990i 0.471518 − 1.94362i 0.541366 − 2.80887i −0.640166 + 2.18021i −0.762236 + 6.95838i 1.17497 + 2.57283i −2.09927 + 6.06543i 1.85374 + 3.59574i
3.11 −1.11165 + 0.874209i 2.04236 − 1.45436i 0.471518 − 1.94362i 1.52399 − 7.90719i −0.998972 + 3.40219i 6.56953 − 2.41687i 1.17497 + 2.57283i −0.887529 + 2.56435i 5.21840 + 10.1223i
3.12 −1.11165 + 0.874209i 2.28306 − 1.62576i 0.471518 − 1.94362i 1.59535 − 8.27748i −1.11670 + 3.80313i −6.01615 − 3.57854i 1.17497 + 2.57283i −0.374354 + 1.08162i 5.46278 + 10.5963i
3.13 −1.11165 + 0.874209i 2.48903 − 1.77243i 0.471518 − 1.94362i −1.06810 + 5.54185i −1.21745 + 4.14625i −4.56730 − 5.30470i 1.17497 + 2.57283i 0.110154 − 0.318270i −3.65738 − 7.09432i
3.14 −1.11165 + 0.874209i 3.23801 − 2.30578i 0.471518 − 1.94362i −1.00442 + 5.21145i −1.58379 + 5.39391i 5.49532 + 4.33606i 1.17497 + 2.57283i 2.22450 − 6.42726i −3.43933 − 6.67136i
3.15 −1.11165 + 0.874209i 3.86295 − 2.75080i 0.471518 − 1.94362i −1.13909 + 5.91015i −1.88947 + 6.43494i −6.16630 − 3.31311i 1.17497 + 2.57283i 4.41191 − 12.7474i −3.90044 − 7.56580i
3.16 −1.11165 + 0.874209i 4.39124 − 3.12699i 0.471518 − 1.94362i 0.316061 − 1.63988i −2.14787 + 7.31497i 5.65268 − 4.12882i 1.17497 + 2.57283i 6.56133 − 18.9577i 1.08225 + 2.09927i
3.17 1.11165 − 0.874209i −4.41945 + 3.14707i 0.471518 − 1.94362i 0.0422989 − 0.219467i −2.16166 + 7.36195i −4.20941 − 5.59293i −1.17497 − 2.57283i 6.68382 − 19.3116i −0.144839 − 0.280948i
3.18 1.11165 − 0.874209i −3.99264 + 2.84315i 0.471518 − 1.94362i −1.71132 + 8.87916i −1.95290 + 6.65098i 5.85687 + 3.83367i −1.17497 − 2.57283i 4.91410 − 14.1984i 5.85986 + 11.3665i
3.19 1.11165 − 0.874209i −3.70995 + 2.64184i 0.471518 − 1.94362i 0.762734 − 3.95744i −1.81463 + 6.18007i −1.67925 + 6.79560i −1.17497 − 2.57283i 3.84078 − 11.0972i −2.61174 − 5.06607i
3.20 1.11165 − 0.874209i −2.23862 + 1.59412i 0.471518 − 1.94362i −0.717669 + 3.72362i −1.09497 + 3.72912i −0.530757 − 6.97985i −1.17497 − 2.57283i −0.473387 + 1.36776i 2.45743 + 4.76674i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.c even 11 1 inner
161.n 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.n.a ✓ 640
7.d odd 6 1 inner 322.3.n.a ✓ 640
23.c even 11 1 inner 322.3.n.a ✓ 640
161.n odd 66 1 inner 322.3.n.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.3.n.a ✓ 640 1.a even 1 1 trivial
322.3.n.a ✓ 640 7.d odd 6 1 inner
322.3.n.a ✓ 640 23.c even 11 1 inner
322.3.n.a ✓ 640 161.n odd 66 1 inner

Hecke kernels

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