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Properties

Label 304.2.x.c
Level $304$
Weight $2$
Character orbit 304.x
Analytic conductor $2.427$
Analytic rank $0$
Dimension $144$
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] = [304,2,Mod(27,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 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("304.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.x (of order \(12\), degree \(4\), 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: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(36\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7} + 24 q^{10} - 28 q^{11} + 6 q^{13} - 6 q^{14} - 14 q^{16} - 4 q^{17} - 10 q^{19} + 4 q^{20} - 12 q^{21} + 6 q^{22} - 28 q^{23} - 2 q^{24} + 76 q^{26} - 30 q^{28} - 6 q^{29} - 64 q^{30} - 36 q^{32} - 36 q^{33} + 48 q^{34} - 40 q^{35} + 6 q^{36} - 66 q^{38} + 24 q^{39} + 24 q^{40} + 6 q^{42} - 26 q^{43} - 16 q^{44} - 8 q^{45} - 30 q^{48} + 112 q^{49} - 18 q^{51} + 66 q^{52} - 30 q^{53} + 16 q^{54} + 16 q^{55} + 12 q^{58} - 24 q^{59} - 72 q^{60} - 26 q^{61} - 46 q^{62} + 52 q^{64} + 38 q^{66} - 12 q^{67} + 24 q^{68} - 120 q^{70} - 24 q^{71} + 6 q^{72} - 10 q^{74} - 38 q^{76} - 56 q^{77} + 90 q^{78} - 66 q^{80} + 48 q^{81} - 22 q^{82} - 92 q^{83} + 2 q^{85} - 30 q^{86} + 120 q^{87} - 138 q^{90} - 84 q^{91} + 2 q^{92} + 192 q^{96} + 60 q^{97} - 42 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.41390 − 0.0299689i 0.390480 + 1.45729i 1.99820 + 0.0847457i −0.158482 + 0.0424650i −0.508424 − 2.07216i 3.43393 −2.82271 − 0.179705i 0.626857 − 0.361916i 0.225349 − 0.0552916i
27.2 −1.38830 + 0.269509i −0.839224 − 3.13203i 1.85473 − 0.748317i −0.918322 + 0.246064i 2.00920 + 4.12200i −3.18154 −2.37323 + 1.53875i −6.50721 + 3.75694i 1.20859 − 0.589105i
27.3 −1.38208 + 0.299745i −0.279639 − 1.04363i 1.82031 − 0.828544i 3.93514 − 1.05442i 0.699307 + 1.35856i 0.668682 −2.26746 + 1.69074i 1.58711 − 0.916321i −5.12263 + 2.63683i
27.4 −1.33528 + 0.465850i 0.140953 + 0.526044i 1.56597 − 1.24408i −3.13996 + 0.841349i −0.433270 − 0.636755i −2.76421 −1.51146 + 2.39071i 2.34122 − 1.35171i 3.80079 − 2.58619i
27.5 −1.31048 − 0.531639i −0.258915 − 0.966283i 1.43472 + 1.39341i −2.89224 + 0.774973i −0.174411 + 1.40394i 0.236175 −1.13938 − 2.58878i 1.73141 − 0.999631i 4.20223 + 0.522041i
27.6 −1.30491 + 0.545165i 0.844029 + 3.14996i 1.40559 − 1.42278i 1.94681 − 0.521647i −2.81863 − 3.65029i −3.12011 −1.05852 + 2.62289i −6.61179 + 3.81732i −2.25604 + 1.74204i
27.7 −1.24333 − 0.673887i −0.422568 − 1.57705i 1.09175 + 1.67573i 1.00145 − 0.268336i −0.537358 + 2.24556i −1.29270 −0.228158 − 2.81921i 0.289564 − 0.167180i −1.42596 − 0.341229i
27.8 −1.18640 + 0.769715i −0.552843 − 2.06324i 0.815079 − 1.82638i −0.617154 + 0.165366i 2.24399 + 2.02229i 4.97623 0.438780 + 2.79419i −1.35323 + 0.781290i 0.604905 − 0.671222i
27.9 −1.15370 − 0.817916i 0.769874 + 2.87321i 0.662026 + 1.88725i −3.08015 + 0.825325i 1.46184 − 3.94450i −1.81654 0.779839 − 2.71880i −5.06454 + 2.92402i 4.22861 + 1.56714i
27.10 −0.989298 − 1.01059i 0.593484 + 2.21491i −0.0425791 + 1.99955i 2.31851 − 0.621242i 1.65123 − 2.79098i 2.51727 2.06284 − 1.93512i −1.95554 + 1.12903i −2.92151 − 1.72846i
27.11 −0.813818 + 1.15659i −0.0528597 − 0.197275i −0.675402 − 1.88251i 1.96441 − 0.526363i 0.271185 + 0.0994090i −3.88727 2.72694 + 0.750854i 2.56195 − 1.47914i −0.989888 + 2.70038i
27.12 −0.795954 + 1.16896i 0.483350 + 1.80389i −0.732913 − 1.86087i 1.45300 − 0.389331i −2.49339 − 0.870796i 3.13160 2.75864 + 0.624425i −0.422301 + 0.243815i −0.701413 + 2.00838i
27.13 −0.755680 − 1.19539i −0.387947 − 1.44784i −0.857896 + 1.80666i 2.03482 − 0.545228i −1.43756 + 1.55785i 2.43502 2.80795 − 0.339739i 0.652342 − 0.376630i −2.18943 − 2.02038i
27.14 −0.542428 − 1.30605i −0.844169 − 3.15048i −1.41154 + 1.41688i −3.53813 + 0.948038i −3.65679 + 2.81144i 2.14351 2.61618 + 1.07500i −6.61483 + 3.81907i 3.15737 + 4.10674i
27.15 −0.537664 − 1.30802i 0.183465 + 0.684700i −1.42184 + 1.40655i −0.293648 + 0.0786828i 0.796959 − 0.608114i −4.76129 2.60427 + 1.10354i 2.16292 − 1.24876i 0.260803 + 0.341793i
27.16 −0.500262 + 1.32278i 0.600884 + 2.24253i −1.49948 − 1.32347i −2.58651 + 0.693054i −3.26696 − 0.327017i −0.348676 2.50079 − 1.32139i −2.06980 + 1.19500i 0.377178 − 3.76809i
27.17 −0.312826 + 1.37918i −0.276668 − 1.03254i −1.80428 − 0.862887i −4.03218 + 1.08042i 1.51061 − 0.0585701i 2.77902 1.75450 − 2.21849i 1.60849 − 0.928659i −0.228723 − 5.89909i
27.18 −0.302641 + 1.38145i −0.662726 − 2.47333i −1.81682 − 0.836167i −0.338025 + 0.0905735i 3.61735 − 0.166994i −1.90034 1.70497 − 2.25679i −3.08006 + 1.77827i −0.0228227 − 0.494376i
27.19 0.103117 + 1.41045i −0.0465071 − 0.173567i −1.97873 + 0.290882i 2.65024 − 0.710129i 0.240011 − 0.0834935i 1.52112 −0.614315 − 2.76091i 2.57011 − 1.48386i 1.27488 + 3.66480i
27.20 0.122861 − 1.40887i −0.703198 − 2.62437i −1.96981 − 0.346189i 3.33319 − 0.893126i −3.78378 + 0.668280i −4.18010 −0.729746 + 2.73267i −3.79476 + 2.19091i −0.848777 − 4.80575i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
19.d odd 6 1 inner
304.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.x.c ✓ 144
16.f odd 4 1 inner 304.2.x.c ✓ 144
19.d odd 6 1 inner 304.2.x.c ✓ 144
304.x even 12 1 inner 304.2.x.c ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.x.c ✓ 144 1.a even 1 1 trivial
304.2.x.c ✓ 144 16.f odd 4 1 inner
304.2.x.c ✓ 144 19.d odd 6 1 inner
304.2.x.c ✓ 144 304.x even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} + 12 T_{3}^{143} + 72 T_{3}^{142} + 288 T_{3}^{141} + 366 T_{3}^{140} + \cdots + 89\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display