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Properties

Label 588.3.d.a.97.1
Level $588$
Weight $3$
Character 588.97
Analytic conductor $16.022$
Analytic rank $0$
Dimension $2$
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] = [588,3,Mod(97,588)] 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("588.97"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.97
Dual form 588.3.d.a.97.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.73205i q^{3} +1.73205i q^{5} -3.00000 q^{9} -3.00000 q^{11} +6.92820i q^{13} +3.00000 q^{15} +17.3205i q^{17} +10.3923i q^{19} +36.0000 q^{23} +22.0000 q^{25} +5.19615i q^{27} -51.0000 q^{29} +12.1244i q^{31} +5.19615i q^{33} +22.0000 q^{37} +12.0000 q^{39} +24.2487i q^{41} +10.0000 q^{43} -5.19615i q^{45} +90.0666i q^{47} +30.0000 q^{51} +51.0000 q^{53} -5.19615i q^{55} +18.0000 q^{57} -74.4782i q^{59} +69.2820i q^{61} -12.0000 q^{65} +68.0000 q^{67} -62.3538i q^{69} +20.7846i q^{73} -38.1051i q^{75} +125.000 q^{79} +9.00000 q^{81} +154.153i q^{83} -30.0000 q^{85} +88.3346i q^{87} +72.7461i q^{89} +21.0000 q^{93} -18.0000 q^{95} -147.224i q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} - 6 q^{11} + 6 q^{15} + 72 q^{23} + 44 q^{25} - 102 q^{29} + 44 q^{37} + 24 q^{39} + 20 q^{43} + 60 q^{51} + 102 q^{53} + 36 q^{57} - 24 q^{65} + 136 q^{67} + 250 q^{79} + 18 q^{81} - 60 q^{85}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 1.73205i 0.346410i 0.984886 + 0.173205i \(0.0554123\pi\)
−0.984886 + 0.173205i \(0.944588\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.272727 −0.136364 − 0.990659i \(-0.543542\pi\)
−0.136364 + 0.990659i \(0.543542\pi\)
\(12\) 0 0
\(13\) 6.92820i 0.532939i 0.963843 + 0.266469i \(0.0858571\pi\)
−0.963843 + 0.266469i \(0.914143\pi\)
\(14\) 0 0
\(15\) 3.00000 0.200000
\(16\) 0 0
\(17\) 17.3205i 1.01885i 0.860514 + 0.509427i \(0.170143\pi\)
−0.860514 + 0.509427i \(0.829857\pi\)
\(18\) 0 0
\(19\) 10.3923i 0.546963i 0.961877 + 0.273482i \(0.0881753\pi\)
−0.961877 + 0.273482i \(0.911825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 36.0000 1.56522 0.782609 − 0.622514i \(-0.213889\pi\)
0.782609 + 0.622514i \(0.213889\pi\)
\(24\) 0 0
\(25\) 22.0000 0.880000
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −51.0000 −1.75862 −0.879310 − 0.476249i \(-0.841996\pi\)
−0.879310 + 0.476249i \(0.841996\pi\)
\(30\) 0 0
\(31\) 12.1244i 0.391108i 0.980693 + 0.195554i \(0.0626505\pi\)
−0.980693 + 0.195554i \(0.937349\pi\)
\(32\) 0 0
\(33\) 5.19615i 0.157459i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 22.0000 0.594595 0.297297 − 0.954785i \(-0.403915\pi\)
0.297297 + 0.954785i \(0.403915\pi\)
\(38\) 0 0
\(39\) 12.0000 0.307692
\(40\) 0 0
\(41\) 24.2487i 0.591432i 0.955276 + 0.295716i \(0.0955582\pi\)
−0.955276 + 0.295716i \(0.904442\pi\)
\(42\) 0 0
\(43\) 10.0000 0.232558 0.116279 − 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(44\) 0 0
\(45\) − 5.19615i − 0.115470i
\(46\) 0 0
\(47\) 90.0666i 1.91631i 0.286247 + 0.958156i \(0.407592\pi\)
−0.286247 + 0.958156i \(0.592408\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 30.0000 0.588235
\(52\) 0 0
\(53\) 51.0000 0.962264 0.481132 − 0.876648i \(-0.340226\pi\)
0.481132 + 0.876648i \(0.340226\pi\)
\(54\) 0 0
\(55\) − 5.19615i − 0.0944755i
\(56\) 0 0
\(57\) 18.0000 0.315789
\(58\) 0 0
\(59\) − 74.4782i − 1.26234i −0.775644 − 0.631171i \(-0.782575\pi\)
0.775644 − 0.631171i \(-0.217425\pi\)
\(60\) 0 0
\(61\) 69.2820i 1.13577i 0.823108 + 0.567886i \(0.192238\pi\)
−0.823108 + 0.567886i \(0.807762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −0.184615
\(66\) 0 0
\(67\) 68.0000 1.01493 0.507463 − 0.861674i \(-0.330584\pi\)
0.507463 + 0.861674i \(0.330584\pi\)
\(68\) 0 0
\(69\) − 62.3538i − 0.903679i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 20.7846i 0.284721i 0.989815 + 0.142360i \(0.0454692\pi\)
−0.989815 + 0.142360i \(0.954531\pi\)
\(74\) 0 0
\(75\) − 38.1051i − 0.508068i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 125.000 1.58228 0.791139 − 0.611636i \(-0.209488\pi\)
0.791139 + 0.611636i \(0.209488\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 154.153i 1.85726i 0.371008 + 0.928630i \(0.379012\pi\)
−0.371008 + 0.928630i \(0.620988\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.352941
\(86\) 0 0
\(87\) 88.3346i 1.01534i
\(88\) 0 0
\(89\) 72.7461i 0.817372i 0.912675 + 0.408686i \(0.134013\pi\)
−0.912675 + 0.408686i \(0.865987\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 21.0000 0.225806
\(94\) 0 0
\(95\) −18.0000 −0.189474
\(96\) 0 0
\(97\) − 147.224i − 1.51778i −0.651221 − 0.758888i \(-0.725743\pi\)
0.651221 − 0.758888i \(-0.274257\pi\)
\(98\) 0 0
\(99\) 9.00000 0.0909091
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 588.3.d.a.97.1 2
3.2 odd 2 1764.3.d.c.685.1 2
4.3 odd 2 2352.3.f.c.97.2 2
7.2 even 3 84.3.m.a.73.1 yes 2
7.3 odd 6 84.3.m.a.61.1 ✓ 2
7.4 even 3 588.3.m.a.313.1 2
7.5 odd 6 588.3.m.a.325.1 2
7.6 odd 2 inner 588.3.d.a.97.2 2
21.2 odd 6 252.3.z.b.73.1 2
21.5 even 6 1764.3.z.e.325.1 2
21.11 odd 6 1764.3.z.e.901.1 2
21.17 even 6 252.3.z.b.145.1 2
21.20 even 2 1764.3.d.c.685.2 2
28.3 even 6 336.3.bh.b.145.1 2
28.23 odd 6 336.3.bh.b.241.1 2
28.27 even 2 2352.3.f.c.97.1 2
35.2 odd 12 2100.3.be.c.1249.2 4
35.3 even 12 2100.3.be.c.649.2 4
35.9 even 6 2100.3.bd.b.1501.1 2
35.17 even 12 2100.3.be.c.649.1 4
35.23 odd 12 2100.3.be.c.1249.1 4
35.24 odd 6 2100.3.bd.b.901.1 2
84.23 even 6 1008.3.cg.b.577.1 2
84.59 odd 6 1008.3.cg.b.145.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.m.a.61.1 ✓ 2 7.3 odd 6
84.3.m.a.73.1 yes 2 7.2 even 3
252.3.z.b.73.1 2 21.2 odd 6
252.3.z.b.145.1 2 21.17 even 6
336.3.bh.b.145.1 2 28.3 even 6
336.3.bh.b.241.1 2 28.23 odd 6
588.3.d.a.97.1 2 1.1 even 1 trivial
588.3.d.a.97.2 2 7.6 odd 2 inner
588.3.m.a.313.1 2 7.4 even 3
588.3.m.a.325.1 2 7.5 odd 6
1008.3.cg.b.145.1 2 84.59 odd 6
1008.3.cg.b.577.1 2 84.23 even 6
1764.3.d.c.685.1 2 3.2 odd 2
1764.3.d.c.685.2 2 21.20 even 2
1764.3.z.e.325.1 2 21.5 even 6
1764.3.z.e.901.1 2 21.11 odd 6
2100.3.bd.b.901.1 2 35.24 odd 6
2100.3.bd.b.1501.1 2 35.9 even 6
2100.3.be.c.649.1 4 35.17 even 12
2100.3.be.c.649.2 4 35.3 even 12
2100.3.be.c.1249.1 4 35.23 odd 12
2100.3.be.c.1249.2 4 35.2 odd 12
2352.3.f.c.97.1 2 28.27 even 2
2352.3.f.c.97.2 2 4.3 odd 2