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Properties

Label 585.2.bu.a.361.1
Level $585$
Weight $2$
Character 585.361
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
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(316,585)] 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("585.316"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bu (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 361.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 585.361
Dual form 585.2.bu.a.316.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.633975 - 0.366025i) q^{2} +(-0.732051 + 1.26795i) q^{4} -1.00000i q^{5} +(-3.86603 - 2.23205i) q^{7} +2.53590i q^{8} +(-0.366025 - 0.633975i) q^{10} +(3.00000 - 1.73205i) q^{11} +(0.866025 - 3.50000i) q^{13} -3.26795 q^{14} +(-0.535898 - 0.928203i) q^{16} +(3.36603 - 5.83013i) q^{17} +(-4.73205 - 2.73205i) q^{19} +(1.26795 + 0.732051i) q^{20} +(1.26795 - 2.19615i) q^{22} +(0.267949 + 0.464102i) q^{23} -1.00000 q^{25} +(-0.732051 - 2.53590i) q^{26} +(5.66025 - 3.26795i) q^{28} +(-1.36603 - 2.36603i) q^{29} +3.19615i q^{31} +(-5.07180 - 2.92820i) q^{32} -4.92820i q^{34} +(-2.23205 + 3.86603i) q^{35} +(3.46410 - 2.00000i) q^{37} -4.00000 q^{38} +2.53590 q^{40} +(-4.56218 + 2.63397i) q^{41} +(-0.133975 + 0.232051i) q^{43} +5.07180i q^{44} +(0.339746 + 0.196152i) q^{46} +0.196152i q^{47} +(6.46410 + 11.1962i) q^{49} +(-0.633975 + 0.366025i) q^{50} +(3.80385 + 3.66025i) q^{52} +6.92820 q^{53} +(-1.73205 - 3.00000i) q^{55} +(5.66025 - 9.80385i) q^{56} +(-1.73205 - 1.00000i) q^{58} +(-6.29423 - 3.63397i) q^{59} +(-2.23205 + 3.86603i) q^{61} +(1.16987 + 2.02628i) q^{62} -2.14359 q^{64} +(-3.50000 - 0.866025i) q^{65} +(-10.7942 + 6.23205i) q^{67} +(4.92820 + 8.53590i) q^{68} +3.26795i q^{70} +(11.0263 + 6.36603i) q^{71} -15.3923i q^{73} +(1.46410 - 2.53590i) q^{74} +(6.92820 - 4.00000i) q^{76} -15.4641 q^{77} +1.92820 q^{79} +(-0.928203 + 0.535898i) q^{80} +(-1.92820 + 3.33975i) q^{82} -2.53590i q^{83} +(-5.83013 - 3.36603i) q^{85} +0.196152i q^{86} +(4.39230 + 7.60770i) q^{88} +(1.09808 - 0.633975i) q^{89} +(-11.1603 + 11.5981i) q^{91} -0.784610 q^{92} +(0.0717968 + 0.124356i) q^{94} +(-2.73205 + 4.73205i) q^{95} +(-14.2583 - 8.23205i) q^{97} +(8.19615 + 4.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 4 q^{4} - 12 q^{7} + 2 q^{10} + 12 q^{11} - 20 q^{14} - 16 q^{16} + 10 q^{17} - 12 q^{19} + 12 q^{20} + 12 q^{22} + 8 q^{23} - 4 q^{25} + 4 q^{26} - 12 q^{28} - 2 q^{29} - 48 q^{32} - 2 q^{35}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.633975 − 0.366025i 0.448288 − 0.258819i −0.258819 − 0.965926i \(-0.583333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) −0.732051 + 1.26795i −0.366025 + 0.633975i
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) −3.86603 − 2.23205i −1.46122 − 0.843636i −0.462152 − 0.886801i \(-0.652923\pi\)
−0.999068 + 0.0431647i \(0.986256\pi\)
\(8\) 2.53590i 0.896575i
\(9\) 0 0
\(10\) −0.366025 − 0.633975i −0.115747 − 0.200480i
\(11\) 3.00000 − 1.73205i 0.904534 − 0.522233i 0.0258656 − 0.999665i \(-0.491766\pi\)
0.878668 + 0.477432i \(0.158432\pi\)
\(12\) 0 0
\(13\) 0.866025 − 3.50000i 0.240192 − 0.970725i
\(14\) −3.26795 −0.873396
\(15\) 0 0
\(16\) −0.535898 − 0.928203i −0.133975 − 0.232051i
\(17\) 3.36603 − 5.83013i 0.816381 − 1.41401i −0.0919509 − 0.995764i \(-0.529310\pi\)
0.908332 − 0.418250i \(-0.137356\pi\)
\(18\) 0 0
\(19\) −4.73205 − 2.73205i −1.08561 − 0.626775i −0.153203 − 0.988195i \(-0.548959\pi\)
−0.932403 + 0.361419i \(0.882292\pi\)
\(20\) 1.26795 + 0.732051i 0.283522 + 0.163692i
\(21\) 0 0
\(22\) 1.26795 − 2.19615i 0.270328 − 0.468221i
\(23\) 0.267949 + 0.464102i 0.0558713 + 0.0967719i 0.892608 − 0.450833i \(-0.148873\pi\)
−0.836737 + 0.547605i \(0.815540\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −0.732051 − 2.53590i −0.143567 − 0.497331i
\(27\) 0 0
\(28\) 5.66025 − 3.26795i 1.06969 − 0.617584i
\(29\) −1.36603 − 2.36603i −0.253665 − 0.439360i 0.710867 − 0.703326i \(-0.248303\pi\)
−0.964532 + 0.263966i \(0.914969\pi\)
\(30\) 0 0
\(31\) 3.19615i 0.574046i 0.957924 + 0.287023i \(0.0926656\pi\)
−0.957924 + 0.287023i \(0.907334\pi\)
\(32\) −5.07180 − 2.92820i −0.896575 − 0.517638i
\(33\) 0 0
\(34\) − 4.92820i − 0.845180i
\(35\) −2.23205 + 3.86603i −0.377285 + 0.653478i
\(36\) 0 0
\(37\) 3.46410 − 2.00000i 0.569495 − 0.328798i −0.187453 − 0.982274i \(-0.560023\pi\)
0.756948 + 0.653476i \(0.226690\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 2.53590 0.400961
\(41\) −4.56218 + 2.63397i −0.712492 + 0.411358i −0.811983 − 0.583681i \(-0.801612\pi\)
0.0994908 + 0.995038i \(0.468279\pi\)
\(42\) 0 0
\(43\) −0.133975 + 0.232051i −0.0204309 + 0.0353874i −0.876060 − 0.482202i \(-0.839837\pi\)
0.855629 + 0.517589i \(0.173170\pi\)
\(44\) 5.07180i 0.764602i
\(45\) 0 0
\(46\) 0.339746 + 0.196152i 0.0500928 + 0.0289211i
\(47\) 0.196152i 0.0286118i 0.999898 + 0.0143059i \(0.00455386\pi\)
−0.999898 + 0.0143059i \(0.995446\pi\)
\(48\) 0 0
\(49\) 6.46410 + 11.1962i 0.923443 + 1.59945i
\(50\) −0.633975 + 0.366025i −0.0896575 + 0.0517638i
\(51\) 0 0
\(52\) 3.80385 + 3.66025i 0.527499 + 0.507586i
\(53\) 6.92820 0.951662 0.475831 − 0.879537i \(-0.342147\pi\)
0.475831 + 0.879537i \(0.342147\pi\)
\(54\) 0 0
\(55\) −1.73205 − 3.00000i −0.233550 − 0.404520i
\(56\) 5.66025 − 9.80385i 0.756383 − 1.31009i
\(57\) 0 0
\(58\) −1.73205 − 1.00000i −0.227429 − 0.131306i
\(59\) −6.29423 − 3.63397i −0.819439 − 0.473103i 0.0307841 − 0.999526i \(-0.490200\pi\)
−0.850223 + 0.526423i \(0.823533\pi\)
\(60\) 0 0
\(61\) −2.23205 + 3.86603i −0.285785 + 0.494994i −0.972799 − 0.231650i \(-0.925588\pi\)
0.687014 + 0.726644i \(0.258921\pi\)
\(62\) 1.16987 + 2.02628i 0.148574 + 0.257338i
\(63\) 0 0
\(64\) −2.14359 −0.267949
\(65\) −3.50000 − 0.866025i −0.434122 − 0.107417i
\(66\) 0 0
\(67\) −10.7942 + 6.23205i −1.31872 + 0.761366i −0.983524 − 0.180780i \(-0.942138\pi\)
−0.335201 + 0.942146i \(0.608804\pi\)
\(68\) 4.92820 + 8.53590i 0.597632 + 1.03513i
\(69\) 0 0
\(70\) 3.26795i 0.390595i
\(71\) 11.0263 + 6.36603i 1.30858 + 0.755508i 0.981859 − 0.189613i \(-0.0607234\pi\)
0.326720 + 0.945121i \(0.394057\pi\)
\(72\) 0 0
\(73\) − 15.3923i − 1.80153i −0.434304 − 0.900767i \(-0.643006\pi\)
0.434304 − 0.900767i \(-0.356994\pi\)
\(74\) 1.46410 − 2.53590i 0.170198 − 0.294792i
\(75\) 0 0
\(76\) 6.92820 − 4.00000i 0.794719 − 0.458831i
\(77\) −15.4641 −1.76230
\(78\) 0 0
\(79\) 1.92820 0.216940 0.108470 − 0.994100i \(-0.465405\pi\)
0.108470 + 0.994100i \(0.465405\pi\)
\(80\) −0.928203 + 0.535898i −0.103776 + 0.0599153i
\(81\) 0 0
\(82\) −1.92820 + 3.33975i −0.212934 + 0.368813i
\(83\) − 2.53590i − 0.278351i −0.990268 − 0.139176i \(-0.955555\pi\)
0.990268 − 0.139176i \(-0.0444452\pi\)
\(84\) 0 0
\(85\) −5.83013 − 3.36603i −0.632366 − 0.365097i
\(86\) 0.196152i 0.0211517i
\(87\) 0 0
\(88\) 4.39230 + 7.60770i 0.468221 + 0.810983i
\(89\) 1.09808 − 0.633975i 0.116396 − 0.0672012i −0.440672 − 0.897668i \(-0.645260\pi\)
0.557068 + 0.830467i \(0.311926\pi\)
\(90\) 0 0
\(91\) −11.1603 + 11.5981i −1.16991 + 1.21581i
\(92\) −0.784610 −0.0818012
\(93\) 0 0
\(94\) 0.0717968 + 0.124356i 0.00740527 + 0.0128263i
\(95\) −2.73205 + 4.73205i −0.280302 + 0.485498i
\(96\) 0 0
\(97\) −14.2583 − 8.23205i −1.44771 − 0.835838i −0.449369 − 0.893346i \(-0.648351\pi\)
−0.998345 + 0.0575081i \(0.981685\pi\)
\(98\) 8.19615 + 4.73205i 0.827936 + 0.478009i
\(99\) 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.bu.a.361.1 4
3.2 odd 2 195.2.bb.a.166.2 yes 4
13.2 odd 12 7605.2.a.y.1.2 2
13.4 even 6 inner 585.2.bu.a.316.1 4
13.11 odd 12 7605.2.a.bk.1.1 2
15.2 even 4 975.2.w.f.49.1 4
15.8 even 4 975.2.w.a.49.2 4
15.14 odd 2 975.2.bc.h.751.1 4
39.2 even 12 2535.2.a.s.1.1 2
39.11 even 12 2535.2.a.n.1.2 2
39.17 odd 6 195.2.bb.a.121.2 ✓ 4
195.17 even 12 975.2.w.a.199.2 4
195.134 odd 6 975.2.bc.h.901.1 4
195.173 even 12 975.2.w.f.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 ✓ 4 39.17 odd 6
195.2.bb.a.166.2 yes 4 3.2 odd 2
585.2.bu.a.316.1 4 13.4 even 6 inner
585.2.bu.a.361.1 4 1.1 even 1 trivial
975.2.w.a.49.2 4 15.8 even 4
975.2.w.a.199.2 4 195.17 even 12
975.2.w.f.49.1 4 15.2 even 4
975.2.w.f.199.1 4 195.173 even 12
975.2.bc.h.751.1 4 15.14 odd 2
975.2.bc.h.901.1 4 195.134 odd 6
2535.2.a.n.1.2 2 39.11 even 12
2535.2.a.s.1.1 2 39.2 even 12
7605.2.a.y.1.2 2 13.2 odd 12
7605.2.a.bk.1.1 2 13.11 odd 12