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Properties

Label 360.2.q.d.121.3
Level $360$
Weight $2$
Character 360.121
Analytic conductor $2.875$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [360,2,Mod(121,360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("360.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.3
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 360.121
Dual form 360.2.q.d.241.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.62241 - 0.606458i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(2.62241 + 4.54214i) q^{7} +(2.26442 - 1.96784i) q^{9} +(-1.33641 - 2.31473i) q^{11} +(-1.90841 + 3.30545i) q^{13} +(-0.285997 + 1.70828i) q^{15} +3.52884 q^{17} -4.67282 q^{19} +(7.00924 + 5.77883i) q^{21} +(2.47842 - 4.29275i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.48040 - 4.56592i) q^{27} +(0.928007 + 1.60735i) q^{29} +(4.33641 - 7.51089i) q^{31} +(-3.57199 - 2.94497i) q^{33} -5.24482 q^{35} -2.67282 q^{37} +(-1.09159 + 6.52016i) q^{39} +(1.83641 - 3.18076i) q^{41} +(-1.76442 - 3.05606i) q^{43} +(0.571993 + 2.94497i) q^{45} +(-4.63164 - 8.02224i) q^{47} +(-10.2541 + 17.7605i) q^{49} +(5.72522 - 2.14009i) q^{51} -2.85601 q^{53} +2.67282 q^{55} +(-7.58123 + 2.83387i) q^{57} +(-2.10083 + 3.63875i) q^{59} +(3.98040 + 6.89425i) q^{61} +(14.8765 + 5.12483i) q^{63} +(-1.90841 - 3.30545i) q^{65} +(0.429983 - 0.744753i) q^{67} +(1.41764 - 8.46766i) q^{69} -15.1625 q^{71} +6.28797 q^{73} +(-1.33641 - 1.10182i) q^{75} +(7.00924 - 12.1404i) q^{77} +(-2.81681 - 4.87886i) q^{79} +(1.25518 - 8.91204i) q^{81} +(1.94958 + 3.37678i) q^{83} +(-1.76442 + 3.05606i) q^{85} +(2.48040 + 2.04499i) q^{87} -11.0000 q^{89} -20.0185 q^{91} +(2.48040 - 14.8156i) q^{93} +(2.33641 - 4.04678i) q^{95} +(1.91764 + 3.32145i) q^{97} +(-7.58123 - 2.61168i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} - 3 q^{5} + 5 q^{7} + 5 q^{9} + 2 q^{11} - q^{15} + 4 q^{17} - 8 q^{19} + 12 q^{21} + 7 q^{23} - 3 q^{25} + 2 q^{27} + 7 q^{29} + 16 q^{31} - 20 q^{33} - 10 q^{35} + 4 q^{37} - 18 q^{39} + q^{41}+ \cdots - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.62241 − 0.606458i 0.936698 − 0.350138i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.62241 + 4.54214i 0.991177 + 1.71677i 0.610374 + 0.792113i \(0.291019\pi\)
0.380803 + 0.924656i \(0.375648\pi\)
\(8\) 0 0
\(9\) 2.26442 − 1.96784i 0.754806 − 0.655948i
\(10\) 0 0
\(11\) −1.33641 − 2.31473i −0.402943 − 0.697918i 0.591136 − 0.806572i \(-0.298679\pi\)
−0.994080 + 0.108653i \(0.965346\pi\)
\(12\) 0 0
\(13\) −1.90841 + 3.30545i −0.529296 + 0.916768i 0.470120 + 0.882603i \(0.344211\pi\)
−0.999416 + 0.0341656i \(0.989123\pi\)
\(14\) 0 0
\(15\) −0.285997 + 1.70828i −0.0738440 + 0.441075i
\(16\) 0 0
\(17\) 3.52884 0.855869 0.427934 − 0.903810i \(-0.359241\pi\)
0.427934 + 0.903810i \(0.359241\pi\)
\(18\) 0 0
\(19\) −4.67282 −1.07202 −0.536010 − 0.844212i \(-0.680069\pi\)
−0.536010 + 0.844212i \(0.680069\pi\)
\(20\) 0 0
\(21\) 7.00924 + 5.77883i 1.52954 + 1.26105i
\(22\) 0 0
\(23\) 2.47842 − 4.29275i 0.516787 − 0.895101i −0.483023 − 0.875608i \(-0.660461\pi\)
0.999810 − 0.0194933i \(-0.00620531\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 2.48040 − 4.56592i 0.477353 − 0.878712i
\(28\) 0 0
\(29\) 0.928007 + 1.60735i 0.172327 + 0.298478i 0.939233 − 0.343281i \(-0.111538\pi\)
−0.766906 + 0.641759i \(0.778205\pi\)
\(30\) 0 0
\(31\) 4.33641 − 7.51089i 0.778843 − 1.34899i −0.153767 − 0.988107i \(-0.549140\pi\)
0.932609 − 0.360888i \(-0.117526\pi\)
\(32\) 0 0
\(33\) −3.57199 − 2.94497i −0.621804 − 0.512653i
\(34\) 0 0
\(35\) −5.24482 −0.886536
\(36\) 0 0
\(37\) −2.67282 −0.439410 −0.219705 − 0.975566i \(-0.570509\pi\)
−0.219705 + 0.975566i \(0.570509\pi\)
\(38\) 0 0
\(39\) −1.09159 + 6.52016i −0.174795 + 1.04406i
\(40\) 0 0
\(41\) 1.83641 − 3.18076i 0.286799 − 0.496751i −0.686245 − 0.727371i \(-0.740742\pi\)
0.973044 + 0.230620i \(0.0740754\pi\)
\(42\) 0 0
\(43\) −1.76442 − 3.05606i −0.269071 − 0.466045i 0.699551 − 0.714583i \(-0.253383\pi\)
−0.968622 + 0.248538i \(0.920050\pi\)
\(44\) 0 0
\(45\) 0.571993 + 2.94497i 0.0852677 + 0.439010i
\(46\) 0 0
\(47\) −4.63164 − 8.02224i −0.675595 − 1.17016i −0.976295 − 0.216446i \(-0.930554\pi\)
0.300700 − 0.953719i \(-0.402780\pi\)
\(48\) 0 0
\(49\) −10.2541 + 17.7605i −1.46486 + 2.53722i
\(50\) 0 0
\(51\) 5.72522 − 2.14009i 0.801691 − 0.299673i
\(52\) 0 0
\(53\) −2.85601 −0.392304 −0.196152 − 0.980574i \(-0.562845\pi\)
−0.196152 + 0.980574i \(0.562845\pi\)
\(54\) 0 0
\(55\) 2.67282 0.360403
\(56\) 0 0
\(57\) −7.58123 + 2.83387i −1.00416 + 0.375355i
\(58\) 0 0
\(59\) −2.10083 + 3.63875i −0.273505 + 0.473724i −0.969757 − 0.244073i \(-0.921516\pi\)
0.696252 + 0.717797i \(0.254850\pi\)
\(60\) 0 0
\(61\) 3.98040 + 6.89425i 0.509638 + 0.882719i 0.999938 + 0.0111647i \(0.00355392\pi\)
−0.490300 + 0.871554i \(0.663113\pi\)
\(62\) 0 0
\(63\) 14.8765 + 5.12483i 1.87426 + 0.645668i
\(64\) 0 0
\(65\) −1.90841 − 3.30545i −0.236709 − 0.409991i
\(66\) 0 0
\(67\) 0.429983 − 0.744753i 0.0525308 − 0.0909860i −0.838564 − 0.544803i \(-0.816605\pi\)
0.891095 + 0.453817i \(0.149938\pi\)
\(68\) 0 0
\(69\) 1.41764 − 8.46766i 0.170664 − 1.01939i
\(70\) 0 0
\(71\) −15.1625 −1.79945 −0.899726 − 0.436454i \(-0.856234\pi\)
−0.899726 + 0.436454i \(0.856234\pi\)
\(72\) 0 0
\(73\) 6.28797 0.735952 0.367976 − 0.929835i \(-0.380051\pi\)
0.367976 + 0.929835i \(0.380051\pi\)
\(74\) 0 0
\(75\) −1.33641 − 1.10182i −0.154316 − 0.127227i
\(76\) 0 0
\(77\) 7.00924 − 12.1404i 0.798777 − 1.38352i
\(78\) 0 0
\(79\) −2.81681 − 4.87886i −0.316916 − 0.548914i 0.662927 − 0.748684i \(-0.269314\pi\)
−0.979843 + 0.199770i \(0.935981\pi\)
\(80\) 0 0
\(81\) 1.25518 − 8.91204i 0.139465 − 0.990227i
\(82\) 0 0
\(83\) 1.94958 + 3.37678i 0.213995 + 0.370650i 0.952961 − 0.303093i \(-0.0980191\pi\)
−0.738966 + 0.673742i \(0.764686\pi\)
\(84\) 0 0
\(85\) −1.76442 + 3.05606i −0.191378 + 0.331477i
\(86\) 0 0
\(87\) 2.48040 + 2.04499i 0.265927 + 0.219246i
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 − 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) −20.0185 −2.09851
\(92\) 0 0
\(93\) 2.48040 − 14.8156i 0.257205 − 1.53630i
\(94\) 0 0
\(95\) 2.33641 − 4.04678i 0.239711 − 0.415191i
\(96\) 0 0
\(97\) 1.91764 + 3.32145i 0.194707 + 0.337242i 0.946804 − 0.321810i \(-0.104291\pi\)
−0.752097 + 0.659052i \(0.770958\pi\)
\(98\) 0 0
\(99\) −7.58123 − 2.61168i −0.761942 − 0.262483i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 360.2.q.d.121.3 ✓ 6
3.2 odd 2 1080.2.q.d.361.3 6
4.3 odd 2 720.2.q.j.481.1 6
9.2 odd 6 1080.2.q.d.721.3 6
9.4 even 3 3240.2.a.r.1.1 3
9.5 odd 6 3240.2.a.q.1.1 3
9.7 even 3 inner 360.2.q.d.241.3 yes 6
12.11 even 2 2160.2.q.j.1441.1 6
36.7 odd 6 720.2.q.j.241.1 6
36.11 even 6 2160.2.q.j.721.1 6
36.23 even 6 6480.2.a.bu.1.3 3
36.31 odd 6 6480.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.d.121.3 ✓ 6 1.1 even 1 trivial
360.2.q.d.241.3 yes 6 9.7 even 3 inner
720.2.q.j.241.1 6 36.7 odd 6
720.2.q.j.481.1 6 4.3 odd 2
1080.2.q.d.361.3 6 3.2 odd 2
1080.2.q.d.721.3 6 9.2 odd 6
2160.2.q.j.721.1 6 36.11 even 6
2160.2.q.j.1441.1 6 12.11 even 2
3240.2.a.q.1.1 3 9.5 odd 6
3240.2.a.r.1.1 3 9.4 even 3
6480.2.a.bu.1.3 3 36.23 even 6
6480.2.a.bx.1.3 3 36.31 odd 6