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Properties

Label 720.2.q.j.241.1
Level $720$
Weight $2$
Character 720.241
Analytic conductor $5.749$
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] = [720,2,Mod(241,720)] 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("720.241"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,-3,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
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: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 241.1
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 720.241
Dual form 720.2.q.j.481.1

$q$-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 \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 −0.380803 + 0.924656i \(0.624352\pi\)
−0.610374 + 0.792113i \(0.708981\pi\)
\(8\) 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 \(-0.701321\pi\)
0.994080 + 0.108653i \(0.0346538\pi\)
\(12\) 0 0
\(13\) −1.90841 − 3.30545i −0.529296 − 0.916768i −0.999416 − 0.0341656i \(-0.989123\pi\)
0.470120 − 0.882603i \(-0.344211\pi\)
\(14\) 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 \(-0.319931\pi\)
0.536010 + 0.844212i \(0.319931\pi\)
\(20\) 0 0
\(21\) 7.00924 − 5.77883i 1.52954 − 1.26105i
\(22\) 0 0
\(23\) −2.47842 − 4.29275i −0.516787 − 0.895101i −0.999810 − 0.0194933i \(-0.993795\pi\)
0.483023 − 0.875608i \(-0.339539\pi\)
\(24\) 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 −0.766906 − 0.641759i \(-0.778205\pi\)
0.939233 + 0.343281i \(0.111538\pi\)
\(30\) 0 0
\(31\) −4.33641 − 7.51089i −0.778843 − 1.34899i −0.932609 − 0.360888i \(-0.882474\pi\)
0.153767 − 0.988107i \(-0.450860\pi\)
\(32\) 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 0.973044 − 0.230620i \(-0.0740754\pi\)
−0.686245 + 0.727371i \(0.740742\pi\)
\(42\) 0 0
\(43\) 1.76442 − 3.05606i 0.269071 − 0.466045i −0.699551 − 0.714583i \(-0.746617\pi\)
0.968622 + 0.248538i \(0.0799499\pi\)
\(44\) 0 0
\(45\) 0.571993 − 2.94497i 0.0852677 − 0.439010i
\(46\) 0 0
\(47\) 4.63164 − 8.02224i 0.675595 − 1.17016i −0.300700 − 0.953719i \(-0.597220\pi\)
0.976295 − 0.216446i \(-0.0694464\pi\)
\(48\) 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 \(-0.0784837\pi\)
−0.696252 + 0.717797i \(0.745150\pi\)
\(60\) 0 0
\(61\) 3.98040 − 6.89425i 0.509638 − 0.882719i −0.490300 − 0.871554i \(-0.663113\pi\)
0.999938 − 0.0111647i \(-0.00355392\pi\)
\(62\) 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 \(-0.183395\pi\)
−0.891095 + 0.453817i \(0.850062\pi\)
\(68\) 0 0
\(69\) 1.41764 + 8.46766i 0.170664 + 1.01939i
\(70\) 0 0
\(71\) 15.1625 1.79945 0.899726 − 0.436454i \(-0.143766\pi\)
0.899726 + 0.436454i \(0.143766\pi\)
\(72\) 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 \(-0.730686\pi\)
0.979843 + 0.199770i \(0.0640194\pi\)
\(80\) 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 \(-0.901981\pi\)
0.738966 + 0.673742i \(0.235314\pi\)
\(84\) 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 −0.752097 − 0.659052i \(-0.770958\pi\)
0.946804 + 0.321810i \(0.104291\pi\)
\(98\) 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 720.2.q.j.241.1 6
3.2 odd 2 2160.2.q.j.721.1 6
4.3 odd 2 360.2.q.d.241.3 yes 6
9.2 odd 6 6480.2.a.bu.1.3 3
9.4 even 3 inner 720.2.q.j.481.1 6
9.5 odd 6 2160.2.q.j.1441.1 6
9.7 even 3 6480.2.a.bx.1.3 3
12.11 even 2 1080.2.q.d.721.3 6
36.7 odd 6 3240.2.a.r.1.1 3
36.11 even 6 3240.2.a.q.1.1 3
36.23 even 6 1080.2.q.d.361.3 6
36.31 odd 6 360.2.q.d.121.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.d.121.3 ✓ 6 36.31 odd 6
360.2.q.d.241.3 yes 6 4.3 odd 2
720.2.q.j.241.1 6 1.1 even 1 trivial
720.2.q.j.481.1 6 9.4 even 3 inner
1080.2.q.d.361.3 6 36.23 even 6
1080.2.q.d.721.3 6 12.11 even 2
2160.2.q.j.721.1 6 3.2 odd 2
2160.2.q.j.1441.1 6 9.5 odd 6
3240.2.a.q.1.1 3 36.11 even 6
3240.2.a.r.1.1 3 36.7 odd 6
6480.2.a.bu.1.3 3 9.2 odd 6
6480.2.a.bx.1.3 3 9.7 even 3