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Properties

Label 888.2.r.e.43.14
Level $888$
Weight $2$
Character 888.43
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
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(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.14
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.14

$q$-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+(-0.476177 - 1.33164i) q^{2} -1.00000i q^{3} +(-1.54651 + 1.26819i) q^{4} +(0.0505532 - 0.0505532i) q^{5} +(-1.33164 + 0.476177i) q^{6} +3.49253i q^{7} +(2.42518 + 1.45551i) q^{8} -1.00000 q^{9} +(-0.0913908 - 0.0432462i) q^{10} -5.51911i q^{11} +(1.26819 + 1.54651i) q^{12} +(-0.670670 + 0.670670i) q^{13} +(4.65078 - 1.66307i) q^{14} +(-0.0505532 - 0.0505532i) q^{15} +(0.783387 - 3.92254i) q^{16} +(-0.942300 - 0.942300i) q^{17} +(0.476177 + 1.33164i) q^{18} +(-4.56982 - 4.56982i) q^{19} +(-0.0140700 + 0.142292i) q^{20} +3.49253 q^{21} +(-7.34944 + 2.62807i) q^{22} +(-0.553542 + 0.553542i) q^{23} +(1.45551 - 2.42518i) q^{24} +4.99489i q^{25} +(1.21245 + 0.573731i) q^{26} +1.00000i q^{27} +(-4.42920 - 5.40124i) q^{28} +(-5.14542 - 5.14542i) q^{29} +(-0.0432462 + 0.0913908i) q^{30} +(-5.21554 - 5.21554i) q^{31} +(-5.59642 + 0.824638i) q^{32} -5.51911 q^{33} +(-0.806099 + 1.70350i) q^{34} +(0.176559 + 0.176559i) q^{35} +(1.54651 - 1.26819i) q^{36} +(-5.01880 - 3.43680i) q^{37} +(-3.90930 + 8.26139i) q^{38} +(0.670670 + 0.670670i) q^{39} +(0.196181 - 0.0490202i) q^{40} -9.29748i q^{41} +(-1.66307 - 4.65078i) q^{42} +(2.52987 + 2.52987i) q^{43} +(6.99928 + 8.53536i) q^{44} +(-0.0505532 + 0.0505532i) q^{45} +(1.00070 + 0.473532i) q^{46} +4.06057i q^{47} +(-3.92254 - 0.783387i) q^{48} -5.19779 q^{49} +(6.65137 - 2.37845i) q^{50} +(-0.942300 + 0.942300i) q^{51} +(0.186661 - 1.88774i) q^{52} +4.72864i q^{53} +(1.33164 - 0.476177i) q^{54} +(-0.279009 - 0.279009i) q^{55} +(-5.08340 + 8.47003i) q^{56} +(-4.56982 + 4.56982i) q^{57} +(-4.40170 + 9.30196i) q^{58} +(2.77976 + 2.77976i) q^{59} +(0.142292 + 0.0140700i) q^{60} +(0.0255129 + 0.0255129i) q^{61} +(-4.46168 + 9.42872i) q^{62} -3.49253i q^{63} +(3.76301 + 7.05973i) q^{64} +0.0678091i q^{65} +(2.62807 + 7.34944i) q^{66} -11.5810i q^{67} +(2.65229 + 0.262261i) q^{68} +(0.553542 + 0.553542i) q^{69} +(0.151039 - 0.319185i) q^{70} -7.25122i q^{71} +(-2.42518 - 1.45551i) q^{72} +14.6743i q^{73} +(-2.18673 + 8.31975i) q^{74} +4.99489 q^{75} +(12.8627 + 1.27187i) q^{76} +19.2757 q^{77} +(0.573731 - 1.21245i) q^{78} +(8.02289 - 8.02289i) q^{79} +(-0.158694 - 0.237900i) q^{80} +1.00000 q^{81} +(-12.3809 + 4.42725i) q^{82} -10.7224 q^{83} +(-5.40124 + 4.42920i) q^{84} -0.0952726 q^{85} +(2.16420 - 4.57353i) q^{86} +(-5.14542 + 5.14542i) q^{87} +(8.03309 - 13.3848i) q^{88} +(-5.51025 + 5.51025i) q^{89} +(0.0913908 + 0.0432462i) q^{90} +(-2.34234 - 2.34234i) q^{91} +(0.154062 - 1.55805i) q^{92} +(-5.21554 + 5.21554i) q^{93} +(5.40720 - 1.93355i) q^{94} -0.462038 q^{95} +(0.824638 + 5.59642i) q^{96} +(-1.38805 - 1.38805i) q^{97} +(2.47507 + 6.92157i) q^{98} +5.51911i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.476177 − 1.33164i −0.336708 − 0.941609i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.54651 + 1.26819i −0.773255 + 0.634095i
\(5\) 0.0505532 − 0.0505532i 0.0226081 − 0.0226081i −0.695712 − 0.718320i \(-0.744911\pi\)
0.718320 + 0.695712i \(0.244911\pi\)
\(6\) −1.33164 + 0.476177i −0.543638 + 0.194399i
\(7\) 3.49253i 1.32005i 0.751242 + 0.660027i \(0.229455\pi\)
−0.751242 + 0.660027i \(0.770545\pi\)
\(8\) 2.42518 + 1.45551i 0.857431 + 0.514599i
\(9\) −1.00000 −0.333333
\(10\) −0.0913908 − 0.0432462i −0.0289003 − 0.0136756i
\(11\) − 5.51911i − 1.66407i −0.554720 − 0.832037i \(-0.687175\pi\)
0.554720 − 0.832037i \(-0.312825\pi\)
\(12\) 1.26819 + 1.54651i 0.366095 + 0.446439i
\(13\) −0.670670 + 0.670670i −0.186011 + 0.186011i −0.793969 − 0.607958i \(-0.791989\pi\)
0.607958 + 0.793969i \(0.291989\pi\)
\(14\) 4.65078 − 1.66307i 1.24297 − 0.444473i
\(15\) −0.0505532 − 0.0505532i −0.0130528 − 0.0130528i
\(16\) 0.783387 − 3.92254i 0.195847 − 0.980635i
\(17\) −0.942300 − 0.942300i −0.228541 − 0.228541i 0.583542 − 0.812083i \(-0.301666\pi\)
−0.812083 + 0.583542i \(0.801666\pi\)
\(18\) 0.476177 + 1.33164i 0.112236 + 0.313870i
\(19\) −4.56982 − 4.56982i −1.04839 − 1.04839i −0.998768 − 0.0496212i \(-0.984199\pi\)
−0.0496212 − 0.998768i \(-0.515801\pi\)
\(20\) −0.0140700 + 0.142292i −0.00314614 + 0.0318175i
\(21\) 3.49253 0.762133
\(22\) −7.34944 + 2.62807i −1.56691 + 0.560307i
\(23\) −0.553542 + 0.553542i −0.115421 + 0.115421i −0.762459 − 0.647037i \(-0.776008\pi\)
0.647037 + 0.762459i \(0.276008\pi\)
\(24\) 1.45551 − 2.42518i 0.297104 − 0.495038i
\(25\) 4.99489i 0.998978i
\(26\) 1.21245 + 0.573731i 0.237780 + 0.112518i
\(27\) 1.00000i 0.192450i
\(28\) −4.42920 − 5.40124i −0.837040 − 1.02074i
\(29\) −5.14542 − 5.14542i −0.955481 − 0.955481i 0.0435697 − 0.999050i \(-0.486127\pi\)
−0.999050 + 0.0435697i \(0.986127\pi\)
\(30\) −0.0432462 + 0.0913908i −0.00789564 + 0.0166856i
\(31\) −5.21554 − 5.21554i −0.936739 − 0.936739i 0.0613761 − 0.998115i \(-0.480451\pi\)
−0.998115 + 0.0613761i \(0.980451\pi\)
\(32\) −5.59642 + 0.824638i −0.989318 + 0.145777i
\(33\) −5.51911 −0.960753
\(34\) −0.806099 + 1.70350i −0.138245 + 0.292148i
\(35\) 0.176559 + 0.176559i 0.0298439 + 0.0298439i
\(36\) 1.54651 − 1.26819i 0.257752 − 0.211365i
\(37\) −5.01880 − 3.43680i −0.825086 − 0.565007i
\(38\) −3.90930 + 8.26139i −0.634171 + 1.34017i
\(39\) 0.670670 + 0.670670i 0.107393 + 0.107393i
\(40\) 0.196181 − 0.0490202i 0.0310190 − 0.00775078i
\(41\) − 9.29748i − 1.45202i −0.687683 − 0.726011i \(-0.741372\pi\)
0.687683 − 0.726011i \(-0.258628\pi\)
\(42\) −1.66307 − 4.65078i −0.256617 − 0.717632i
\(43\) 2.52987 + 2.52987i 0.385801 + 0.385801i 0.873187 − 0.487386i \(-0.162049\pi\)
−0.487386 + 0.873187i \(0.662049\pi\)
\(44\) 6.99928 + 8.53536i 1.05518 + 1.28675i
\(45\) −0.0505532 + 0.0505532i −0.00753603 + 0.00753603i
\(46\) 1.00070 + 0.473532i 0.147545 + 0.0698185i
\(47\) 4.06057i 0.592295i 0.955142 + 0.296147i \(0.0957019\pi\)
−0.955142 + 0.296147i \(0.904298\pi\)
\(48\) −3.92254 − 0.783387i −0.566170 − 0.113072i
\(49\) −5.19779 −0.742542
\(50\) 6.65137 − 2.37845i 0.940646 − 0.336364i
\(51\) −0.942300 + 0.942300i −0.131948 + 0.131948i
\(52\) 0.186661 − 1.88774i 0.0258852 − 0.261782i
\(53\) 4.72864i 0.649528i 0.945795 + 0.324764i \(0.105285\pi\)
−0.945795 + 0.324764i \(0.894715\pi\)
\(54\) 1.33164 − 0.476177i 0.181213 − 0.0647995i
\(55\) −0.279009 − 0.279009i −0.0376215 − 0.0376215i
\(56\) −5.08340 + 8.47003i −0.679298 + 1.13186i
\(57\) −4.56982 + 4.56982i −0.605288 + 0.605288i
\(58\) −4.40170 + 9.30196i −0.577971 + 1.22141i
\(59\) 2.77976 + 2.77976i 0.361894 + 0.361894i 0.864510 − 0.502616i \(-0.167629\pi\)
−0.502616 + 0.864510i \(0.667629\pi\)
\(60\) 0.142292 + 0.0140700i 0.0183698 + 0.00181642i
\(61\) 0.0255129 + 0.0255129i 0.00326660 + 0.00326660i 0.708738 − 0.705472i \(-0.249265\pi\)
−0.705472 + 0.708738i \(0.749265\pi\)
\(62\) −4.46168 + 9.42872i −0.566634 + 1.19745i
\(63\) − 3.49253i − 0.440018i
\(64\) 3.76301 + 7.05973i 0.470376 + 0.882466i
\(65\) 0.0678091i 0.00841068i
\(66\) 2.62807 + 7.34944i 0.323494 + 0.904654i
\(67\) − 11.5810i − 1.41484i −0.706794 − 0.707419i \(-0.749859\pi\)
0.706794 − 0.707419i \(-0.250141\pi\)
\(68\) 2.65229 + 0.262261i 0.321638 + 0.0318038i
\(69\) 0.553542 + 0.553542i 0.0666386 + 0.0666386i
\(70\) 0.151039 − 0.319185i 0.0180526 − 0.0381500i
\(71\) − 7.25122i − 0.860562i −0.902695 − 0.430281i \(-0.858415\pi\)
0.902695 − 0.430281i \(-0.141585\pi\)
\(72\) −2.42518 − 1.45551i −0.285810 − 0.171533i
\(73\) 14.6743i 1.71750i 0.512397 + 0.858749i \(0.328758\pi\)
−0.512397 + 0.858749i \(0.671242\pi\)
\(74\) −2.18673 + 8.31975i −0.254202 + 0.967151i
\(75\) 4.99489 0.576760
\(76\) 12.8627 + 1.27187i 1.47545 + 0.145894i
\(77\) 19.2757 2.19667
\(78\) 0.573731 − 1.21245i 0.0649622 − 0.137283i
\(79\) 8.02289 − 8.02289i 0.902646 − 0.902646i −0.0930187 − 0.995664i \(-0.529652\pi\)
0.995664 + 0.0930187i \(0.0296516\pi\)
\(80\) −0.158694 − 0.237900i −0.0177425 − 0.0265980i
\(81\) 1.00000 0.111111
\(82\) −12.3809 + 4.42725i −1.36724 + 0.488908i
\(83\) −10.7224 −1.17694 −0.588470 − 0.808519i \(-0.700270\pi\)
−0.588470 + 0.808519i \(0.700270\pi\)
\(84\) −5.40124 + 4.42920i −0.589324 + 0.483265i
\(85\) −0.0952726 −0.0103338
\(86\) 2.16420 − 4.57353i 0.233371 − 0.493176i
\(87\) −5.14542 + 5.14542i −0.551647 + 0.551647i
\(88\) 8.03309 − 13.3848i 0.856330 − 1.42683i
\(89\) −5.51025 + 5.51025i −0.584086 + 0.584086i −0.936023 − 0.351938i \(-0.885523\pi\)
0.351938 + 0.936023i \(0.385523\pi\)
\(90\) 0.0913908 + 0.0432462i 0.00963343 + 0.00455855i
\(91\) −2.34234 − 2.34234i −0.245544 − 0.245544i
\(92\) 0.154062 − 1.55805i 0.0160620 − 0.162438i
\(93\) −5.21554 + 5.21554i −0.540826 + 0.540826i
\(94\) 5.40720 − 1.93355i 0.557710 − 0.199431i
\(95\) −0.462038 −0.0474041
\(96\) 0.824638 + 5.59642i 0.0841643 + 0.571183i
\(97\) −1.38805 − 1.38805i −0.140935 − 0.140935i 0.633119 − 0.774054i \(-0.281774\pi\)
−0.774054 + 0.633119i \(0.781774\pi\)
\(98\) 2.47507 + 6.92157i 0.250020 + 0.699184i
\(99\) 5.51911i 0.554691i
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.r.e.43.14 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.32 yes 72
37.31 odd 4 inner 888.2.r.e.475.32 yes 72
296.179 even 4 inner 888.2.r.e.475.14 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.14 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.32 yes 72 8.3 odd 2 inner
888.2.r.e.475.14 yes 72 296.179 even 4 inner
888.2.r.e.475.32 yes 72 37.31 odd 4 inner