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Properties

Label 585.2.h.e.64.1
Level $585$
Weight $2$
Character 585.64
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
CM discriminant -195
Inner twists $8$

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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(64,585)] 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("585.64"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 5x^{2} - 4x + 69 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 64.1
Root \(-1.30278 + 2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 585.64
Dual form 585.2.h.e.64.3

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -2.23607i q^{5} -3.60555 q^{7} +2.23607i q^{11} +3.60555 q^{13} +4.00000 q^{16} +8.06226i q^{17} +4.47214i q^{20} +8.06226i q^{23} -5.00000 q^{25} +7.21110 q^{28} +8.06226i q^{35} -3.60555 q^{37} -11.1803i q^{41} -4.47214i q^{44} +6.00000 q^{49} -7.21110 q^{52} +8.06226i q^{53} +5.00000 q^{55} +8.94427i q^{59} -7.00000 q^{61} -8.00000 q^{64} -8.06226i q^{65} -14.4222 q^{67} -16.1245i q^{68} +15.6525i q^{71} +7.21110 q^{73} -8.06226i q^{77} +11.0000 q^{79} -8.94427i q^{80} +18.0278 q^{85} +2.23607i q^{89} -13.0000 q^{91} -16.1245i q^{92} -3.60555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 24 q^{49} + 20 q^{55} - 28 q^{61} - 32 q^{64} + 44 q^{79} - 52 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) 0 0
\(7\) −3.60555 −1.36277 −0.681385 − 0.731925i \(-0.738622\pi\)
−0.681385 + 0.731925i \(0.738622\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.23607i 0.674200i 0.941469 + 0.337100i \(0.109446\pi\)
−0.941469 + 0.337100i \(0.890554\pi\)
\(12\) 0 0
\(13\) 3.60555 1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 8.06226i 1.95538i 0.210042 + 0.977692i \(0.432640\pi\)
−0.210042 + 0.977692i \(0.567360\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.47214i 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) 8.06226i 1.68110i 0.541736 + 0.840548i \(0.317767\pi\)
−0.541736 + 0.840548i \(0.682233\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 7.21110 1.36277
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.06226i 1.36277i
\(36\) 0 0
\(37\) −3.60555 −0.592749 −0.296374 − 0.955072i \(-0.595778\pi\)
−0.296374 + 0.955072i \(0.595778\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 11.1803i − 1.74608i −0.487652 − 0.873038i \(-0.662147\pi\)
0.487652 − 0.873038i \(-0.337853\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) − 4.47214i − 0.674200i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −7.21110 −1.00000
\(53\) 8.06226i 1.10744i 0.832704 + 0.553718i \(0.186791\pi\)
−0.832704 + 0.553718i \(0.813209\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.94427i 1.16445i 0.813029 + 0.582223i \(0.197817\pi\)
−0.813029 + 0.582223i \(0.802183\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) − 8.06226i − 1.00000i
\(66\) 0 0
\(67\) −14.4222 −1.76195 −0.880976 − 0.473160i \(-0.843113\pi\)
−0.880976 + 0.473160i \(0.843113\pi\)
\(68\) − 16.1245i − 1.95538i
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6525i 1.85761i 0.370572 + 0.928804i \(0.379162\pi\)
−0.370572 + 0.928804i \(0.620838\pi\)
\(72\) 0 0
\(73\) 7.21110 0.843996 0.421998 − 0.906597i \(-0.361329\pi\)
0.421998 + 0.906597i \(0.361329\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.06226i − 0.918780i
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0278 1.95538
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.23607i 0.237023i 0.992953 + 0.118511i \(0.0378122\pi\)
−0.992953 + 0.118511i \(0.962188\pi\)
\(90\) 0 0
\(91\) −13.0000 −1.36277
\(92\) − 16.1245i − 1.68110i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.60555 −0.366088 −0.183044 − 0.983105i \(-0.558595\pi\)
−0.183044 + 0.983105i \(0.558595\pi\)
\(98\) 0 0
\(99\) 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.h.e.64.1 ✓ 4
3.2 odd 2 inner 585.2.h.e.64.3 yes 4
5.4 even 2 inner 585.2.h.e.64.2 yes 4
13.12 even 2 inner 585.2.h.e.64.4 yes 4
15.14 odd 2 inner 585.2.h.e.64.4 yes 4
39.38 odd 2 inner 585.2.h.e.64.2 yes 4
65.64 even 2 inner 585.2.h.e.64.3 yes 4
195.194 odd 2 CM 585.2.h.e.64.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.h.e.64.1 ✓ 4 1.1 even 1 trivial
585.2.h.e.64.1 ✓ 4 195.194 odd 2 CM
585.2.h.e.64.2 yes 4 5.4 even 2 inner
585.2.h.e.64.2 yes 4 39.38 odd 2 inner
585.2.h.e.64.3 yes 4 3.2 odd 2 inner
585.2.h.e.64.3 yes 4 65.64 even 2 inner
585.2.h.e.64.4 yes 4 13.12 even 2 inner
585.2.h.e.64.4 yes 4 15.14 odd 2 inner