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Properties

Label 345.3.p.c
Level $345$
Weight $3$
Character orbit 345.p
Analytic conductor $9.401$
Analytic rank $0$
Dimension $880$
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] = [345,3,Mod(29,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 18]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 345.p (of order \(22\), degree \(10\), 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: \(9.40056912043\)
Analytic rank: \(0\)
Dimension: \(880\)
Relative dimension: \(88\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 880 q - 220 q^{4} - 22 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 880 q - 220 q^{4} - 22 q^{6} - 18 q^{9} + 18 q^{10} + 60 q^{15} - 316 q^{16} - 52 q^{19} - 110 q^{21} + 112 q^{24} + 66 q^{25} - 357 q^{30} - 68 q^{31} - 808 q^{34} - 302 q^{36} + 98 q^{39} + 1324 q^{40} - 56 q^{45} - 44 q^{46} + 372 q^{49} + 242 q^{51} - 28 q^{54} - 172 q^{55} - 2877 q^{60} - 32 q^{61} + 1420 q^{64} + 160 q^{66} + 88 q^{69} + 316 q^{70} - 80 q^{75} + 1772 q^{76} + 848 q^{79} + 2034 q^{81} + 438 q^{84} + 120 q^{85} + 126 q^{90} - 312 q^{91} - 356 q^{94} - 5088 q^{96} + 278 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.62797 + 3.56477i −2.12813 + 2.11449i −7.43781 − 8.58369i −1.37010 + 4.80862i −4.07311 − 11.0286i −8.23374 − 1.18383i 27.6668 − 8.12369i 0.0578866 − 8.99981i −14.9111 − 12.7124i
29.2 −1.62797 + 3.56477i −0.647120 − 2.92937i −7.43781 − 8.58369i −4.94323 − 0.751288i 11.4960 + 2.46211i 8.23374 + 1.18383i 27.6668 − 8.12369i −8.16247 + 3.79131i 10.7256 − 16.3984i
29.3 −1.58272 + 3.46566i 2.01565 + 2.22197i −6.88639 − 7.94731i 4.97631 + 0.486177i −10.8908 + 3.46880i 2.48926 + 0.357901i 23.8194 − 6.99400i −0.874334 + 8.95743i −9.56100 + 16.4767i
29.4 −1.58272 + 3.46566i 2.89696 − 0.779502i −6.88639 − 7.94731i 1.62499 − 4.72857i −1.88357 + 11.2736i −2.48926 − 0.357901i 23.8194 − 6.99400i 7.78475 − 4.51637i 13.8157 + 13.1157i
29.5 −1.47570 + 3.23132i −0.553132 + 2.94857i −5.64434 − 6.51391i −1.83695 − 4.65033i −8.71152 − 6.13854i −0.117452 − 0.0168870i 15.7441 − 4.62289i −8.38809 − 3.26189i 17.7375 + 0.926683i
29.6 −1.47570 + 3.23132i 1.12879 − 2.77954i −5.64434 − 6.51391i 3.46699 + 3.60277i 7.31584 + 7.74925i 0.117452 + 0.0168870i 15.7441 − 4.62289i −6.45166 − 6.27504i −16.7580 + 5.88638i
29.7 −1.39767 + 3.06047i −2.97841 + 0.359276i −4.79356 − 5.53206i −2.35063 − 4.41300i 3.06328 − 9.61748i 0.756576 + 0.108779i 10.7176 − 3.14697i 8.74184 − 2.14014i 16.7913 − 1.02612i
29.8 −1.39767 + 3.06047i −2.31136 − 1.91249i −4.79356 − 5.53206i 3.03772 + 3.97143i 9.08364 − 4.40081i −0.756576 − 0.108779i 10.7176 − 3.14697i 1.68475 + 8.84091i −16.4002 + 3.74609i
29.9 −1.38800 + 3.03931i 1.45914 + 2.62124i −4.69138 − 5.41415i −3.91331 + 3.11223i −9.99205 + 0.796461i 6.73017 + 0.967652i 10.1433 − 2.97834i −4.74185 + 7.64950i −4.02733 − 16.2135i
29.10 −1.38800 + 3.03931i 2.64465 − 1.41626i −4.69138 − 5.41415i −4.45663 + 2.26681i 0.633663 + 10.0037i −6.73017 − 0.967652i 10.1433 − 2.97834i 4.98840 − 7.49105i −0.703690 − 16.6914i
29.11 −1.38051 + 3.02289i −2.01001 + 2.22708i −4.61264 − 5.32327i 4.99874 + 0.112099i −3.95738 − 9.15054i 13.5215 + 1.94410i 9.70512 − 2.84968i −0.919736 − 8.95288i −7.23968 + 14.9559i
29.12 −1.38051 + 3.02289i −0.486878 − 2.96023i −4.61264 − 5.32327i 1.97458 − 4.59358i 9.62060 + 2.61484i −13.5215 − 1.94410i 9.70512 − 2.84968i −8.52590 + 2.88254i 11.1600 + 12.3104i
29.13 −1.23440 + 2.70296i 2.76471 + 1.16463i −3.16281 − 3.65008i 1.69156 + 4.70517i −6.56072 + 6.03529i −8.19195 − 1.17783i 2.36570 − 0.694633i 6.28727 + 6.43974i −14.8060 − 1.23584i
29.14 −1.23440 + 2.70296i 2.95547 + 0.514966i −3.16281 − 3.65008i −3.57727 − 3.49330i −5.04017 + 7.35285i 8.19195 + 1.17783i 2.36570 − 0.694633i 8.46962 + 3.04394i 13.8580 − 5.35710i
29.15 −1.21348 + 2.65714i 0.618224 + 2.93561i −2.96844 − 3.42577i 2.63495 − 4.24936i −8.55053 − 1.91958i −6.48852 − 0.932909i 1.49372 − 0.438595i −8.23560 + 3.62973i 8.09369 + 12.1579i
29.16 −1.21348 + 2.65714i 2.10719 − 2.13535i −2.96844 − 3.42577i 4.95995 − 0.631592i 3.11691 + 8.19032i 6.48852 + 0.932909i 1.49372 − 0.438595i −0.119476 − 8.99921i −4.34055 + 13.9457i
29.17 −1.21067 + 2.65100i −2.99930 − 0.0646667i −2.94264 − 3.39598i −2.39049 + 4.39153i 3.80260 − 7.87286i 9.54783 + 1.37277i 1.38006 − 0.405223i 8.99164 + 0.387910i −8.74786 − 11.6539i
29.18 −1.21067 + 2.65100i −2.55814 − 1.56714i −2.94264 − 3.39598i −4.98772 + 0.350156i 7.25156 − 4.88432i −9.54783 − 1.37277i 1.38006 − 0.405223i 4.08812 + 8.01794i 5.11023 − 13.6464i
29.19 −1.12285 + 2.45871i −2.54695 + 1.58526i −2.16500 − 2.49854i 4.95407 − 0.676193i −1.03785 − 8.04222i −8.80328 − 1.26572i −1.79975 + 0.528456i 3.97388 − 8.07516i −3.90013 + 12.9399i
29.20 −1.12285 + 2.45871i −1.28557 − 2.71059i −2.16500 − 2.49854i 2.67308 − 4.22548i 8.10806 − 0.117242i 8.80328 + 1.26572i −1.79975 + 0.528456i −5.69462 + 6.96931i 7.38773 + 11.3169i
See next 80 embeddings (of 880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner
115.j even 22 1 inner
345.p odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.3.p.c ✓ 880
3.b odd 2 1 inner 345.3.p.c ✓ 880
5.b even 2 1 inner 345.3.p.c ✓ 880
15.d odd 2 1 inner 345.3.p.c ✓ 880
23.c even 11 1 inner 345.3.p.c ✓ 880
69.h odd 22 1 inner 345.3.p.c ✓ 880
115.j even 22 1 inner 345.3.p.c ✓ 880
345.p odd 22 1 inner 345.3.p.c ✓ 880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.3.p.c ✓ 880 1.a even 1 1 trivial
345.3.p.c ✓ 880 3.b odd 2 1 inner
345.3.p.c ✓ 880 5.b even 2 1 inner
345.3.p.c ✓ 880 15.d odd 2 1 inner
345.3.p.c ✓ 880 23.c even 11 1 inner
345.3.p.c ✓ 880 69.h odd 22 1 inner
345.3.p.c ✓ 880 115.j even 22 1 inner
345.3.p.c ✓ 880 345.p odd 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{440} + 143 T_{2}^{438} + 10880 T_{2}^{436} + 583434 T_{2}^{434} + 24719696 T_{2}^{432} + \cdots + 26\!\cdots\!21 \) acting on \(S_{3}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display