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Properties

Label 585.2.dp.a.37.4
Level $585$
Weight $2$
Character 585.37
Analytic conductor $4.671$
Analytic rank $0$
Dimension $20$
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] = [585,2,Mod(28,585)] 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("585.28"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.dp (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,6,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 37.4
Root \(-1.02262i\) of defining polynomial
Character \(\chi\) \(=\) 585.37
Dual form 585.2.dp.a.253.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.885613 - 0.511309i) q^{2} +(-0.477126 + 0.826407i) q^{4} +(1.45744 + 1.69584i) q^{5} +(-0.481787 + 0.834479i) q^{7} +3.02107i q^{8} +(2.15782 + 0.756660i) q^{10} +(-1.60661 - 0.430490i) q^{11} +(-3.11175 - 1.82127i) q^{13} +0.985368i q^{14} +(0.590448 + 1.02269i) q^{16} +(1.87656 + 7.00342i) q^{17} +(0.707496 + 2.64041i) q^{19} +(-2.09684 + 0.395304i) q^{20} +(-1.64295 + 0.440226i) q^{22} +(0.997344 - 3.72214i) q^{23} +(-0.751762 + 4.94316i) q^{25} +(-3.68704 - 0.0218799i) q^{26} +(-0.459747 - 0.796304i) q^{28} +(0.253107 - 0.146132i) q^{29} +(-0.125649 - 0.125649i) q^{31} +(-4.18683 - 2.41727i) q^{32} +(5.24282 + 5.24282i) q^{34} +(-2.11732 + 0.399166i) q^{35} +(2.04061 + 3.53443i) q^{37} +(1.97663 + 1.97663i) q^{38} +(-5.12326 + 4.40302i) q^{40} +(1.79277 - 6.69071i) q^{41} +(7.67707 - 2.05706i) q^{43} +(1.12232 - 1.12232i) q^{44} +(-1.01990 - 3.80633i) q^{46} +7.84582 q^{47} +(3.03576 + 5.25810i) q^{49} +(1.86171 + 4.76211i) q^{50} +(2.98981 - 1.70259i) q^{52} +(1.99855 - 1.99855i) q^{53} +(-1.61149 - 3.35197i) q^{55} +(-2.52102 - 1.45551i) q^{56} +(0.149437 - 0.258832i) q^{58} +(4.87924 - 1.30739i) q^{59} +(-1.04169 + 1.80425i) q^{61} +(-0.175522 - 0.0470311i) q^{62} -7.30568 q^{64} +(-1.44658 - 7.93142i) q^{65} +(6.32050 - 3.64915i) q^{67} +(-6.68304 - 1.79071i) q^{68} +(-1.67103 + 1.43611i) q^{70} +(-12.6082 + 3.37837i) q^{71} -3.22747i q^{73} +(3.61437 + 2.08676i) q^{74} +(-2.51962 - 0.675130i) q^{76} +(1.13328 - 1.13328i) q^{77} -13.5845i q^{79} +(-0.873774 + 2.49180i) q^{80} +(-1.83332 - 6.84204i) q^{82} +8.56854 q^{83} +(-9.14173 + 13.3894i) q^{85} +(5.74712 - 5.74712i) q^{86} +(1.30054 - 4.85368i) q^{88} +(0.134207 - 0.500868i) q^{89} +(3.01901 - 1.71922i) q^{91} +(2.60014 + 2.60014i) q^{92} +(6.94836 - 4.01164i) q^{94} +(-3.44659 + 5.04803i) q^{95} +(-6.50662 - 3.75660i) q^{97} +(5.37702 + 3.10442i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{2} + 6 q^{4} - 2 q^{7} - 2 q^{10} + 16 q^{11} - 4 q^{13} - 2 q^{16} - 4 q^{17} - 20 q^{19} + 16 q^{22} + 10 q^{23} + 18 q^{25} + 24 q^{26} + 18 q^{28} - 48 q^{32} + 2 q^{34} - 40 q^{35} - 4 q^{37}+ \cdots + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{7}{12}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.885613 − 0.511309i 0.626223 − 0.361550i −0.153065 − 0.988216i \(-0.548914\pi\)
0.779288 + 0.626666i \(0.215581\pi\)
\(3\) 0 0
\(4\) −0.477126 + 0.826407i −0.238563 + 0.413204i
\(5\) 1.45744 + 1.69584i 0.651785 + 0.758404i
\(6\) 0 0
\(7\) −0.481787 + 0.834479i −0.182098 + 0.315404i −0.942595 − 0.333938i \(-0.891622\pi\)
0.760497 + 0.649342i \(0.224956\pi\)
\(8\) 3.02107i 1.06811i
\(9\) 0 0
\(10\) 2.15782 + 0.756660i 0.682364 + 0.239277i
\(11\) −1.60661 − 0.430490i −0.484411 − 0.129797i 0.00834492 − 0.999965i \(-0.497344\pi\)
−0.492756 + 0.870168i \(0.664010\pi\)
\(12\) 0 0
\(13\) −3.11175 − 1.82127i −0.863043 − 0.505130i
\(14\) 0.985368i 0.263351i
\(15\) 0 0
\(16\) 0.590448 + 1.02269i 0.147612 + 0.255671i
\(17\) 1.87656 + 7.00342i 0.455133 + 1.69858i 0.687697 + 0.725998i \(0.258622\pi\)
−0.232564 + 0.972581i \(0.574711\pi\)
\(18\) 0 0
\(19\) 0.707496 + 2.64041i 0.162311 + 0.605752i 0.998368 + 0.0571095i \(0.0181884\pi\)
−0.836057 + 0.548642i \(0.815145\pi\)
\(20\) −2.09684 + 0.395304i −0.468867 + 0.0883927i
\(21\) 0 0
\(22\) −1.64295 + 0.440226i −0.350277 + 0.0938565i
\(23\) 0.997344 − 3.72214i 0.207961 − 0.776120i −0.780566 − 0.625073i \(-0.785069\pi\)
0.988527 − 0.151046i \(-0.0482643\pi\)
\(24\) 0 0
\(25\) −0.751762 + 4.94316i −0.150352 + 0.988632i
\(26\) −3.68704 − 0.0218799i −0.723087 − 0.00429099i
\(27\) 0 0
\(28\) −0.459747 − 0.796304i −0.0868839 − 0.150487i
\(29\) 0.253107 − 0.146132i 0.0470008 − 0.0271360i −0.476315 − 0.879274i \(-0.658028\pi\)
0.523316 + 0.852139i \(0.324695\pi\)
\(30\) 0 0
\(31\) −0.125649 − 0.125649i −0.0225673 − 0.0225673i 0.695733 − 0.718300i \(-0.255080\pi\)
−0.718300 + 0.695733i \(0.755080\pi\)
\(32\) −4.18683 − 2.41727i −0.740134 − 0.427317i
\(33\) 0 0
\(34\) 5.24282 + 5.24282i 0.899136 + 0.899136i
\(35\) −2.11732 + 0.399166i −0.357892 + 0.0674713i
\(36\) 0 0
\(37\) 2.04061 + 3.53443i 0.335474 + 0.581057i 0.983576 − 0.180496i \(-0.0577703\pi\)
−0.648102 + 0.761553i \(0.724437\pi\)
\(38\) 1.97663 + 1.97663i 0.320652 + 0.320652i
\(39\) 0 0
\(40\) −5.12326 + 4.40302i −0.810059 + 0.696178i
\(41\) 1.79277 − 6.69071i 0.279984 − 1.04491i −0.672444 − 0.740148i \(-0.734755\pi\)
0.952427 − 0.304765i \(-0.0985780\pi\)
\(42\) 0 0
\(43\) 7.67707 − 2.05706i 1.17074 − 0.313699i 0.379494 − 0.925194i \(-0.376098\pi\)
0.791248 + 0.611495i \(0.209432\pi\)
\(44\) 1.12232 − 1.12232i 0.169195 − 0.169195i
\(45\) 0 0
\(46\) −1.01990 − 3.80633i −0.150376 − 0.561212i
\(47\) 7.84582 1.14443 0.572215 − 0.820103i \(-0.306084\pi\)
0.572215 + 0.820103i \(0.306084\pi\)
\(48\) 0 0
\(49\) 3.03576 + 5.25810i 0.433680 + 0.751156i
\(50\) 1.86171 + 4.76211i 0.263286 + 0.673464i
\(51\) 0 0
\(52\) 2.98981 − 1.70259i 0.414612 − 0.236107i
\(53\) 1.99855 − 1.99855i 0.274522 − 0.274522i −0.556395 − 0.830918i \(-0.687816\pi\)
0.830918 + 0.556395i \(0.187816\pi\)
\(54\) 0 0
\(55\) −1.61149 − 3.35197i −0.217293 − 0.451979i
\(56\) −2.52102 − 1.45551i −0.336886 − 0.194501i
\(57\) 0 0
\(58\) 0.149437 − 0.258832i 0.0196220 − 0.0339863i
\(59\) 4.87924 − 1.30739i 0.635223 − 0.170207i 0.0731843 − 0.997318i \(-0.476684\pi\)
0.562039 + 0.827111i \(0.310017\pi\)
\(60\) 0 0
\(61\) −1.04169 + 1.80425i −0.133374 + 0.231011i −0.924975 − 0.380027i \(-0.875915\pi\)
0.791601 + 0.611038i \(0.209248\pi\)
\(62\) −0.175522 − 0.0470311i −0.0222914 − 0.00597296i
\(63\) 0 0
\(64\) −7.30568 −0.913209
\(65\) −1.44658 − 7.93142i −0.179426 − 0.983771i
\(66\) 0 0
\(67\) 6.32050 − 3.64915i 0.772173 − 0.445814i −0.0614765 − 0.998109i \(-0.519581\pi\)
0.833649 + 0.552294i \(0.186248\pi\)
\(68\) −6.68304 − 1.79071i −0.810437 − 0.217156i
\(69\) 0 0
\(70\) −1.67103 + 1.43611i −0.199726 + 0.171648i
\(71\) −12.6082 + 3.37837i −1.49632 + 0.400939i −0.911867 − 0.410486i \(-0.865359\pi\)
−0.584457 + 0.811425i \(0.698692\pi\)
\(72\) 0 0
\(73\) − 3.22747i − 0.377746i −0.982001 − 0.188873i \(-0.939517\pi\)
0.982001 − 0.188873i \(-0.0604835\pi\)
\(74\) 3.61437 + 2.08676i 0.420163 + 0.242581i
\(75\) 0 0
\(76\) −2.51962 − 0.675130i −0.289020 − 0.0774427i
\(77\) 1.13328 − 1.13328i 0.129149 − 0.129149i
\(78\) 0 0
\(79\) − 13.5845i − 1.52838i −0.644992 − 0.764190i \(-0.723139\pi\)
0.644992 − 0.764190i \(-0.276861\pi\)
\(80\) −0.873774 + 2.49180i −0.0976909 + 0.278592i
\(81\) 0 0
\(82\) −1.83332 − 6.84204i −0.202456 − 0.755577i
\(83\) 8.56854 0.940519 0.470260 − 0.882528i \(-0.344160\pi\)
0.470260 + 0.882528i \(0.344160\pi\)
\(84\) 0 0
\(85\) −9.14173 + 13.3894i −0.991560 + 1.45228i
\(86\) 5.74712 − 5.74712i 0.619728 − 0.619728i
\(87\) 0 0
\(88\) 1.30054 − 4.85368i 0.138638 − 0.517404i
\(89\) 0.134207 − 0.500868i 0.0142259 − 0.0530919i −0.958448 − 0.285267i \(-0.907918\pi\)
0.972674 + 0.232175i \(0.0745843\pi\)
\(90\) 0 0
\(91\) 3.01901 − 1.71922i 0.316479 − 0.180223i
\(92\) 2.60014 + 2.60014i 0.271084 + 0.271084i
\(93\) 0 0
\(94\) 6.94836 − 4.01164i 0.716669 − 0.413769i
\(95\) −3.44659 + 5.04803i −0.353613 + 0.517917i
\(96\) 0 0
\(97\) −6.50662 − 3.75660i −0.660648 − 0.381425i 0.131876 − 0.991266i \(-0.457900\pi\)
−0.792524 + 0.609841i \(0.791233\pi\)
\(98\) 5.37702 + 3.10442i 0.543161 + 0.313594i
\(99\) 0 0
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 585.2.dp.a.37.4 20
3.2 odd 2 65.2.t.a.37.2 yes 20
5.3 odd 4 585.2.cf.a.388.2 20
13.6 odd 12 585.2.cf.a.487.2 20
15.2 even 4 325.2.s.b.193.2 20
15.8 even 4 65.2.o.a.63.4 yes 20
15.14 odd 2 325.2.x.b.232.4 20
39.2 even 12 845.2.k.e.577.3 20
39.5 even 4 845.2.o.e.587.4 20
39.8 even 4 845.2.o.f.587.2 20
39.11 even 12 845.2.k.d.577.8 20
39.17 odd 6 845.2.t.e.657.2 20
39.20 even 12 845.2.o.g.357.2 20
39.23 odd 6 845.2.f.d.437.8 20
39.29 odd 6 845.2.f.e.437.3 20
39.32 even 12 65.2.o.a.32.4 ✓ 20
39.35 odd 6 845.2.t.f.657.4 20
39.38 odd 2 845.2.t.g.427.4 20
65.58 even 12 inner 585.2.dp.a.253.4 20
195.8 odd 4 845.2.t.e.418.2 20
195.23 even 12 845.2.k.d.268.8 20
195.32 odd 12 325.2.x.b.318.4 20
195.38 even 4 845.2.o.g.258.2 20
195.68 even 12 845.2.k.e.268.3 20
195.83 odd 4 845.2.t.f.418.4 20
195.98 odd 12 845.2.t.g.188.4 20
195.113 even 12 845.2.o.e.488.4 20
195.128 odd 12 845.2.f.d.408.3 20
195.149 even 12 325.2.s.b.32.2 20
195.158 odd 12 845.2.f.e.408.8 20
195.173 even 12 845.2.o.f.488.2 20
195.188 odd 12 65.2.t.a.58.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.o.a.32.4 ✓ 20 39.32 even 12
65.2.o.a.63.4 yes 20 15.8 even 4
65.2.t.a.37.2 yes 20 3.2 odd 2
65.2.t.a.58.2 yes 20 195.188 odd 12
325.2.s.b.32.2 20 195.149 even 12
325.2.s.b.193.2 20 15.2 even 4
325.2.x.b.232.4 20 15.14 odd 2
325.2.x.b.318.4 20 195.32 odd 12
585.2.cf.a.388.2 20 5.3 odd 4
585.2.cf.a.487.2 20 13.6 odd 12
585.2.dp.a.37.4 20 1.1 even 1 trivial
585.2.dp.a.253.4 20 65.58 even 12 inner
845.2.f.d.408.3 20 195.128 odd 12
845.2.f.d.437.8 20 39.23 odd 6
845.2.f.e.408.8 20 195.158 odd 12
845.2.f.e.437.3 20 39.29 odd 6
845.2.k.d.268.8 20 195.23 even 12
845.2.k.d.577.8 20 39.11 even 12
845.2.k.e.268.3 20 195.68 even 12
845.2.k.e.577.3 20 39.2 even 12
845.2.o.e.488.4 20 195.113 even 12
845.2.o.e.587.4 20 39.5 even 4
845.2.o.f.488.2 20 195.173 even 12
845.2.o.f.587.2 20 39.8 even 4
845.2.o.g.258.2 20 195.38 even 4
845.2.o.g.357.2 20 39.20 even 12
845.2.t.e.418.2 20 195.8 odd 4
845.2.t.e.657.2 20 39.17 odd 6
845.2.t.f.418.4 20 195.83 odd 4
845.2.t.f.657.4 20 39.35 odd 6
845.2.t.g.188.4 20 195.98 odd 12
845.2.t.g.427.4 20 39.38 odd 2