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Properties

Label 588.8.i.a.373.1
Level $588$
Weight $8$
Character 588.373
Analytic conductor $183.682$
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] = [588,8,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-27,0,-378] 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: \(183.682394985\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.8.i.a.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-13.5000 - 23.3827i) q^{3} +(-189.000 + 327.358i) q^{5} +(-364.500 + 631.333i) q^{9} +(1242.00 + 2151.21i) q^{11} -14870.0 q^{13} +10206.0 q^{15} +(-11151.0 - 19314.1i) q^{17} +(-8150.00 + 14116.2i) q^{19} +(57564.0 - 99703.8i) q^{23} +(-32379.5 - 56082.9i) q^{25} +19683.0 q^{27} +157086. q^{29} +(-8228.00 - 14251.3i) q^{31} +(33534.0 - 58082.6i) q^{33} +(74633.0 - 129268. i) q^{37} +(200745. + 347701. i) q^{39} +241110. q^{41} -443188. q^{43} +(-137781. - 238644. i) q^{45} +(461376. - 799127. i) q^{47} +(-301077. + 521481. i) q^{51} +(348813. + 604162. i) q^{53} -938952. q^{55} +440100. q^{57} +(435078. + 753577. i) q^{59} +(1.03353e6 - 1.79013e6i) q^{61} +(2.81043e6 - 4.86781e6i) q^{65} +(840374. + 1.45557e6i) q^{67} -3.10846e6 q^{69} -1.07028e6 q^{71} +(-1.20167e6 - 2.08135e6i) q^{73} +(-874246. + 1.51424e6i) q^{75} +(-1.15076e6 + 1.99317e6i) q^{79} +(-265720. - 460241. i) q^{81} -4.70869e6 q^{83} +8.43016e6 q^{85} +(-2.12066e6 - 3.67309e6i) q^{87} +(2.07184e6 - 3.58854e6i) q^{89} +(-222156. + 384785. i) q^{93} +(-3.08070e6 - 5.33593e6i) q^{95} +1.62297e6 q^{97} -1.81084e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27 q^{3} - 378 q^{5} - 729 q^{9} + 2484 q^{11} - 29740 q^{13} + 20412 q^{15} - 22302 q^{17} - 16300 q^{19} + 115128 q^{23} - 64759 q^{25} + 39366 q^{27} + 314172 q^{29} - 16456 q^{31} + 67068 q^{33}+ \cdots - 3621672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −13.5000 − 23.3827i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) −189.000 + 327.358i −0.676187 + 1.17119i 0.299933 + 0.953960i \(0.403036\pi\)
−0.976120 + 0.217230i \(0.930298\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −364.500 + 631.333i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1242.00 + 2151.21i 0.281350 + 0.487313i 0.971718 − 0.236146i \(-0.0758844\pi\)
−0.690367 + 0.723459i \(0.742551\pi\)
\(12\) 0 0
\(13\) −14870.0 −1.87719 −0.938597 − 0.345015i \(-0.887874\pi\)
−0.938597 + 0.345015i \(0.887874\pi\)
\(14\) 0 0
\(15\) 10206.0 0.780793
\(16\) 0 0
\(17\) −11151.0 − 19314.1i −0.550481 − 0.953462i −0.998240 − 0.0593071i \(-0.981111\pi\)
0.447758 − 0.894155i \(-0.352222\pi\)
\(18\) 0 0
\(19\) −8150.00 + 14116.2i −0.272596 + 0.472151i −0.969526 − 0.244989i \(-0.921216\pi\)
0.696930 + 0.717140i \(0.254549\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 57564.0 − 99703.8i 0.986515 − 1.70869i 0.351512 − 0.936183i \(-0.385668\pi\)
0.635002 − 0.772510i \(-0.280999\pi\)
\(24\) 0 0
\(25\) −32379.5 − 56082.9i −0.414458 − 0.717862i
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 157086. 1.19604 0.598018 − 0.801482i \(-0.295955\pi\)
0.598018 + 0.801482i \(0.295955\pi\)
\(30\) 0 0
\(31\) −8228.00 − 14251.3i −0.0496053 − 0.0859190i 0.840157 − 0.542344i \(-0.182463\pi\)
−0.889762 + 0.456425i \(0.849130\pi\)
\(32\) 0 0
\(33\) 33534.0 − 58082.6i 0.162438 − 0.281350i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 74633.0 − 129268.i 0.242228 − 0.419552i −0.719120 − 0.694885i \(-0.755455\pi\)
0.961349 + 0.275334i \(0.0887884\pi\)
\(38\) 0 0
\(39\) 200745. + 347701.i 0.541899 + 0.938597i
\(40\) 0 0
\(41\) 241110. 0.546351 0.273175 − 0.961964i \(-0.411926\pi\)
0.273175 + 0.961964i \(0.411926\pi\)
\(42\) 0 0
\(43\) −443188. −0.850058 −0.425029 − 0.905180i \(-0.639736\pi\)
−0.425029 + 0.905180i \(0.639736\pi\)
\(44\) 0 0
\(45\) −137781. − 238644.i −0.225396 − 0.390397i
\(46\) 0 0
\(47\) 461376. − 799127.i 0.648205 − 1.12272i −0.335346 − 0.942095i \(-0.608853\pi\)
0.983551 − 0.180629i \(-0.0578134\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −301077. + 521481.i −0.317821 + 0.550481i
\(52\) 0 0
\(53\) 348813. + 604162.i 0.321830 + 0.557427i 0.980866 − 0.194685i \(-0.0623686\pi\)
−0.659035 + 0.752112i \(0.729035\pi\)
\(54\) 0 0
\(55\) −938952. −0.760981
\(56\) 0 0
\(57\) 440100. 0.314767
\(58\) 0 0
\(59\) 435078. + 753577.i 0.275794 + 0.477690i 0.970335 − 0.241764i \(-0.0777258\pi\)
−0.694541 + 0.719453i \(0.744393\pi\)
\(60\) 0 0
\(61\) 1.03353e6 − 1.79013e6i 0.583001 − 1.00979i −0.412121 − 0.911129i \(-0.635212\pi\)
0.995121 − 0.0986576i \(-0.0314549\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.81043e6 − 4.86781e6i 1.26933 − 2.19855i
\(66\) 0 0
\(67\) 840374. + 1.45557e6i 0.341359 + 0.591250i 0.984685 − 0.174341i \(-0.0557796\pi\)
−0.643327 + 0.765592i \(0.722446\pi\)
\(68\) 0 0
\(69\) −3.10846e6 −1.13913
\(70\) 0 0
\(71\) −1.07028e6 −0.354890 −0.177445 − 0.984131i \(-0.556783\pi\)
−0.177445 + 0.984131i \(0.556783\pi\)
\(72\) 0 0
\(73\) −1.20167e6 − 2.08135e6i −0.361538 − 0.626202i 0.626676 − 0.779280i \(-0.284415\pi\)
−0.988214 + 0.153077i \(0.951082\pi\)
\(74\) 0 0
\(75\) −874246. + 1.51424e6i −0.239287 + 0.414458i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.15076e6 + 1.99317e6i −0.262596 + 0.454830i −0.966931 − 0.255038i \(-0.917912\pi\)
0.704335 + 0.709868i \(0.251245\pi\)
\(80\) 0 0
\(81\) −265720. − 460241.i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −4.70869e6 −0.903914 −0.451957 − 0.892040i \(-0.649274\pi\)
−0.451957 + 0.892040i \(0.649274\pi\)
\(84\) 0 0
\(85\) 8.43016e6 1.48891
\(86\) 0 0
\(87\) −2.12066e6 − 3.67309e6i −0.345266 − 0.598018i
\(88\) 0 0
\(89\) 2.07184e6 − 3.58854e6i 0.311525 − 0.539576i −0.667168 − 0.744907i \(-0.732494\pi\)
0.978693 + 0.205331i \(0.0658271\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −222156. + 384785.i −0.0286397 + 0.0496053i
\(94\) 0 0
\(95\) −3.08070e6 − 5.33593e6i −0.368652 − 0.638524i
\(96\) 0 0
\(97\) 1.62297e6 0.180555 0.0902777 − 0.995917i \(-0.471225\pi\)
0.0902777 + 0.995917i \(0.471225\pi\)
\(98\) 0 0
\(99\) −1.81084e6 −0.187567
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 588.8.i.a.373.1 2
7.2 even 3 588.8.a.d.1.1 1
7.3 odd 6 588.8.i.h.361.1 2
7.4 even 3 inner 588.8.i.a.361.1 2
7.5 odd 6 12.8.a.a.1.1 ✓ 1
7.6 odd 2 588.8.i.h.373.1 2
21.5 even 6 36.8.a.c.1.1 1
28.19 even 6 48.8.a.e.1.1 1
35.12 even 12 300.8.d.c.49.2 2
35.19 odd 6 300.8.a.g.1.1 1
35.33 even 12 300.8.d.c.49.1 2
56.5 odd 6 192.8.a.o.1.1 1
56.19 even 6 192.8.a.g.1.1 1
63.5 even 6 324.8.e.a.217.1 2
63.40 odd 6 324.8.e.f.217.1 2
63.47 even 6 324.8.e.a.109.1 2
63.61 odd 6 324.8.e.f.109.1 2
84.47 odd 6 144.8.a.j.1.1 1
168.5 even 6 576.8.a.d.1.1 1
168.131 odd 6 576.8.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.8.a.a.1.1 ✓ 1 7.5 odd 6
36.8.a.c.1.1 1 21.5 even 6
48.8.a.e.1.1 1 28.19 even 6
144.8.a.j.1.1 1 84.47 odd 6
192.8.a.g.1.1 1 56.19 even 6
192.8.a.o.1.1 1 56.5 odd 6
300.8.a.g.1.1 1 35.19 odd 6
300.8.d.c.49.1 2 35.33 even 12
300.8.d.c.49.2 2 35.12 even 12
324.8.e.a.109.1 2 63.47 even 6
324.8.e.a.217.1 2 63.5 even 6
324.8.e.f.109.1 2 63.61 odd 6
324.8.e.f.217.1 2 63.40 odd 6
576.8.a.d.1.1 1 168.5 even 6
576.8.a.e.1.1 1 168.131 odd 6
588.8.a.d.1.1 1 7.2 even 3
588.8.i.a.361.1 2 7.4 even 3 inner
588.8.i.a.373.1 2 1.1 even 1 trivial
588.8.i.h.361.1 2 7.3 odd 6
588.8.i.h.373.1 2 7.6 odd 2