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Properties

Label 888.2.o.a.517.8
Level $888$
Weight $2$
Character 888.517
Analytic conductor $7.091$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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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] = [888,2,Mod(517,888)] 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("888.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 517.8
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.7

$q$-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+(-1.38047 + 0.307088i) q^{2} +1.00000i q^{3} +(1.81139 - 0.847851i) q^{4} +3.72444 q^{5} +(-0.307088 - 1.38047i) q^{6} -1.69680 q^{7} +(-2.24021 + 1.72669i) q^{8} -1.00000 q^{9} +(-5.14148 + 1.14373i) q^{10} -4.68385i q^{11} +(0.847851 + 1.81139i) q^{12} +2.13131 q^{13} +(2.34239 - 0.521068i) q^{14} +3.72444i q^{15} +(2.56230 - 3.07159i) q^{16} +2.33582i q^{17} +(1.38047 - 0.307088i) q^{18} +4.49539 q^{19} +(6.74643 - 3.15777i) q^{20} -1.69680i q^{21} +(1.43835 + 6.46591i) q^{22} +4.43424i q^{23} +(-1.72669 - 2.24021i) q^{24} +8.87145 q^{25} +(-2.94221 + 0.654500i) q^{26} -1.00000i q^{27} +(-3.07358 + 1.43864i) q^{28} +2.87264 q^{29} +(-1.14373 - 5.14148i) q^{30} -5.38804i q^{31} +(-2.59392 + 5.02708i) q^{32} +4.68385 q^{33} +(-0.717304 - 3.22454i) q^{34} -6.31964 q^{35} +(-1.81139 + 0.847851i) q^{36} +(1.40098 - 5.91923i) q^{37} +(-6.20576 + 1.38048i) q^{38} +2.13131i q^{39} +(-8.34352 + 6.43095i) q^{40} +5.38413 q^{41} +(0.521068 + 2.34239i) q^{42} +4.73817 q^{43} +(-3.97121 - 8.48429i) q^{44} -3.72444 q^{45} +(-1.36170 - 6.12133i) q^{46} -4.50697 q^{47} +(3.07159 + 2.56230i) q^{48} -4.12086 q^{49} +(-12.2468 + 2.72432i) q^{50} -2.33582 q^{51} +(3.86064 - 1.80704i) q^{52} +3.54067i q^{53} +(0.307088 + 1.38047i) q^{54} -17.4447i q^{55} +(3.80120 - 2.92986i) q^{56} +4.49539i q^{57} +(-3.96559 + 0.882154i) q^{58} -12.2870 q^{59} +(3.15777 + 6.74643i) q^{60} +13.5409 q^{61} +(1.65460 + 7.43803i) q^{62} +1.69680 q^{63} +(2.03708 - 7.73630i) q^{64} +7.93794 q^{65} +(-6.46591 + 1.43835i) q^{66} +2.62610i q^{67} +(1.98043 + 4.23110i) q^{68} -4.43424 q^{69} +(8.72408 - 1.94069i) q^{70} +5.83268 q^{71} +(2.24021 - 1.72669i) q^{72} +4.35697 q^{73} +(-0.116280 + 8.60154i) q^{74} +8.87145i q^{75} +(8.14293 - 3.81143i) q^{76} +7.94757i q^{77} +(-0.654500 - 2.94221i) q^{78} -5.01900i q^{79} +(9.54312 - 11.4399i) q^{80} +1.00000 q^{81} +(-7.43262 + 1.65340i) q^{82} +14.1557i q^{83} +(-1.43864 - 3.07358i) q^{84} +8.69964i q^{85} +(-6.54090 + 1.45504i) q^{86} +2.87264i q^{87} +(8.08756 + 10.4928i) q^{88} +16.4080i q^{89} +(5.14148 - 1.14373i) q^{90} -3.61642 q^{91} +(3.75957 + 8.03215i) q^{92} +5.38804 q^{93} +(6.22174 - 1.38404i) q^{94} +16.7428 q^{95} +(-5.02708 - 2.59392i) q^{96} -0.171488i q^{97} +(5.68872 - 1.26547i) q^{98} +4.68385i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38047 + 0.307088i −0.976140 + 0.217144i
\(3\) 1.00000i 0.577350i
\(4\) 1.81139 − 0.847851i 0.905697 − 0.423926i
\(5\) 3.72444 1.66562 0.832810 − 0.553559i \(-0.186731\pi\)
0.832810 + 0.553559i \(0.186731\pi\)
\(6\) −0.307088 − 1.38047i −0.125368 − 0.563574i
\(7\) −1.69680 −0.641332 −0.320666 − 0.947192i \(-0.603907\pi\)
−0.320666 + 0.947192i \(0.603907\pi\)
\(8\) −2.24021 + 1.72669i −0.792034 + 0.610477i
\(9\) −1.00000 −0.333333
\(10\) −5.14148 + 1.14373i −1.62588 + 0.361679i
\(11\) − 4.68385i − 1.41223i −0.708095 − 0.706117i \(-0.750445\pi\)
0.708095 − 0.706117i \(-0.249555\pi\)
\(12\) 0.847851 + 1.81139i 0.244754 + 0.522904i
\(13\) 2.13131 0.591119 0.295560 − 0.955324i \(-0.404494\pi\)
0.295560 + 0.955324i \(0.404494\pi\)
\(14\) 2.34239 − 0.521068i 0.626029 − 0.139261i
\(15\) 3.72444i 0.961646i
\(16\) 2.56230 − 3.07159i 0.640574 − 0.767896i
\(17\) 2.33582i 0.566521i 0.959043 + 0.283260i \(0.0914160\pi\)
−0.959043 + 0.283260i \(0.908584\pi\)
\(18\) 1.38047 − 0.307088i 0.325380 − 0.0723813i
\(19\) 4.49539 1.03131 0.515657 − 0.856795i \(-0.327548\pi\)
0.515657 + 0.856795i \(0.327548\pi\)
\(20\) 6.74643 − 3.15777i 1.50855 − 0.706099i
\(21\) − 1.69680i − 0.370273i
\(22\) 1.43835 + 6.46591i 0.306658 + 1.37854i
\(23\) 4.43424i 0.924602i 0.886723 + 0.462301i \(0.152976\pi\)
−0.886723 + 0.462301i \(0.847024\pi\)
\(24\) −1.72669 − 2.24021i −0.352459 − 0.457281i
\(25\) 8.87145 1.77429
\(26\) −2.94221 + 0.654500i −0.577015 + 0.128358i
\(27\) − 1.00000i − 0.192450i
\(28\) −3.07358 + 1.43864i −0.580852 + 0.271877i
\(29\) 2.87264 0.533436 0.266718 − 0.963775i \(-0.414061\pi\)
0.266718 + 0.963775i \(0.414061\pi\)
\(30\) −1.14373 − 5.14148i −0.208816 − 0.938701i
\(31\) − 5.38804i − 0.967721i −0.875145 − 0.483861i \(-0.839234\pi\)
0.875145 − 0.483861i \(-0.160766\pi\)
\(32\) −2.59392 + 5.02708i −0.458545 + 0.888671i
\(33\) 4.68385 0.815353
\(34\) −0.717304 − 3.22454i −0.123017 − 0.553003i
\(35\) −6.31964 −1.06821
\(36\) −1.81139 + 0.847851i −0.301899 + 0.141309i
\(37\) 1.40098 − 5.91923i 0.230319 − 0.973115i
\(38\) −6.20576 + 1.38048i −1.00671 + 0.223944i
\(39\) 2.13131i 0.341283i
\(40\) −8.34352 + 6.43095i −1.31923 + 1.01682i
\(41\) 5.38413 0.840859 0.420430 − 0.907325i \(-0.361879\pi\)
0.420430 + 0.907325i \(0.361879\pi\)
\(42\) 0.521068 + 2.34239i 0.0804026 + 0.361438i
\(43\) 4.73817 0.722565 0.361282 − 0.932456i \(-0.382339\pi\)
0.361282 + 0.932456i \(0.382339\pi\)
\(44\) −3.97121 − 8.48429i −0.598682 − 1.27906i
\(45\) −3.72444 −0.555207
\(46\) −1.36170 − 6.12133i −0.200772 − 0.902541i
\(47\) −4.50697 −0.657409 −0.328705 − 0.944433i \(-0.606612\pi\)
−0.328705 + 0.944433i \(0.606612\pi\)
\(48\) 3.07159 + 2.56230i 0.443345 + 0.369836i
\(49\) −4.12086 −0.588694
\(50\) −12.2468 + 2.72432i −1.73195 + 0.385276i
\(51\) −2.33582 −0.327081
\(52\) 3.86064 − 1.80704i 0.535375 − 0.250591i
\(53\) 3.54067i 0.486349i 0.969983 + 0.243174i \(0.0781888\pi\)
−0.969983 + 0.243174i \(0.921811\pi\)
\(54\) 0.307088 + 1.38047i 0.0417894 + 0.187858i
\(55\) − 17.4447i − 2.35224i
\(56\) 3.80120 − 2.92986i 0.507956 − 0.391518i
\(57\) 4.49539i 0.595430i
\(58\) −3.96559 + 0.882154i −0.520708 + 0.115832i
\(59\) −12.2870 −1.59964 −0.799818 − 0.600243i \(-0.795070\pi\)
−0.799818 + 0.600243i \(0.795070\pi\)
\(60\) 3.15777 + 6.74643i 0.407667 + 0.870960i
\(61\) 13.5409 1.73373 0.866865 − 0.498544i \(-0.166132\pi\)
0.866865 + 0.498544i \(0.166132\pi\)
\(62\) 1.65460 + 7.43803i 0.210135 + 0.944631i
\(63\) 1.69680 0.213777
\(64\) 2.03708 − 7.73630i 0.254635 − 0.967037i
\(65\) 7.93794 0.984580
\(66\) −6.46591 + 1.43835i −0.795899 + 0.177049i
\(67\) 2.62610i 0.320829i 0.987050 + 0.160415i \(0.0512831\pi\)
−0.987050 + 0.160415i \(0.948717\pi\)
\(68\) 1.98043 + 4.23110i 0.240163 + 0.513096i
\(69\) −4.43424 −0.533819
\(70\) 8.72408 − 1.94069i 1.04273 − 0.231956i
\(71\) 5.83268 0.692212 0.346106 − 0.938195i \(-0.387504\pi\)
0.346106 + 0.938195i \(0.387504\pi\)
\(72\) 2.24021 − 1.72669i 0.264011 − 0.203492i
\(73\) 4.35697 0.509945 0.254973 − 0.966948i \(-0.417934\pi\)
0.254973 + 0.966948i \(0.417934\pi\)
\(74\) −0.116280 + 8.60154i −0.0135173 + 0.999909i
\(75\) 8.87145i 1.02439i
\(76\) 8.14293 − 3.81143i 0.934058 − 0.437201i
\(77\) 7.94757i 0.905710i
\(78\) −0.654500 − 2.94221i −0.0741075 − 0.333140i
\(79\) − 5.01900i − 0.564682i −0.959314 − 0.282341i \(-0.908889\pi\)
0.959314 − 0.282341i \(-0.0911109\pi\)
\(80\) 9.54312 − 11.4399i 1.06695 − 1.27902i
\(81\) 1.00000 0.111111
\(82\) −7.43262 + 1.65340i −0.820796 + 0.182588i
\(83\) 14.1557i 1.55379i 0.629631 + 0.776894i \(0.283206\pi\)
−0.629631 + 0.776894i \(0.716794\pi\)
\(84\) −1.43864 − 3.07358i −0.156968 − 0.335355i
\(85\) 8.69964i 0.943608i
\(86\) −6.54090 + 1.45504i −0.705324 + 0.156901i
\(87\) 2.87264i 0.307979i
\(88\) 8.08756 + 10.4928i 0.862137 + 1.11854i
\(89\) 16.4080i 1.73925i 0.493717 + 0.869623i \(0.335638\pi\)
−0.493717 + 0.869623i \(0.664362\pi\)
\(90\) 5.14148 − 1.14373i 0.541959 − 0.120560i
\(91\) −3.61642 −0.379104
\(92\) 3.75957 + 8.03215i 0.391963 + 0.837409i
\(93\) 5.38804 0.558714
\(94\) 6.22174 − 1.38404i 0.641723 − 0.142752i
\(95\) 16.7428 1.71778
\(96\) −5.02708 − 2.59392i −0.513074 − 0.264741i
\(97\) − 0.171488i − 0.0174119i −0.999962 − 0.00870597i \(-0.997229\pi\)
0.999962 − 0.00870597i \(-0.00277123\pi\)
\(98\) 5.68872 − 1.26547i 0.574647 − 0.127831i
\(99\) 4.68385i 0.470745i
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 888.2.o.a.517.8 yes 76
4.3 odd 2 3552.2.o.a.2737.73 76
8.3 odd 2 3552.2.o.a.2737.56 76
8.5 even 2 inner 888.2.o.a.517.70 yes 76
37.36 even 2 inner 888.2.o.a.517.69 yes 76
148.147 odd 2 3552.2.o.a.2737.55 76
296.147 odd 2 3552.2.o.a.2737.74 76
296.221 even 2 inner 888.2.o.a.517.7 ✓ 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.7 ✓ 76 296.221 even 2 inner
888.2.o.a.517.8 yes 76 1.1 even 1 trivial
888.2.o.a.517.69 yes 76 37.36 even 2 inner
888.2.o.a.517.70 yes 76 8.5 even 2 inner
3552.2.o.a.2737.55 76 148.147 odd 2
3552.2.o.a.2737.56 76 8.3 odd 2
3552.2.o.a.2737.73 76 4.3 odd 2
3552.2.o.a.2737.74 76 296.147 odd 2