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Properties

Label 888.2.bh.a.565.2
Level $888$
Weight $2$
Character 888.565
Analytic conductor $7.091$
Analytic rank $0$
Dimension $152$
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(565,888)] 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("888.565"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.bh (of order \(6\), degree \(2\), 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: \(152\)
Relative dimension: \(76\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 565.2
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.2

$q$-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+(-1.41300 - 0.0585472i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(1.99314 + 0.165455i) q^{4} +(1.69671 - 0.979598i) q^{5} +(1.25297 - 0.655797i) q^{6} +(1.99537 + 3.45609i) q^{7} +(-2.80663 - 0.350480i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-2.45481 + 1.28483i) q^{10} +0.0532273i q^{11} +(-1.80884 + 0.853284i) q^{12} +(-5.73084 + 3.30870i) q^{13} +(-2.61712 - 5.00028i) q^{14} +(-0.979598 + 1.69671i) q^{15} +(3.94525 + 0.659549i) q^{16} +(-3.42288 + 5.92860i) q^{17} +(-0.757204 + 1.19442i) q^{18} +(1.82670 - 1.05464i) q^{19} +(3.54387 - 1.67175i) q^{20} +(-3.45609 - 1.99537i) q^{21} +(0.00311631 - 0.0752103i) q^{22} -3.38506 q^{23} +(2.60585 - 1.09979i) q^{24} +(-0.580777 + 1.00594i) q^{25} +(8.29140 - 4.33968i) q^{26} +1.00000i q^{27} +(3.40524 + 7.21863i) q^{28} -7.59693i q^{29} +(1.48351 - 2.34010i) q^{30} -2.20359 q^{31} +(-5.53603 - 1.16293i) q^{32} +(-0.0266137 - 0.0460962i) q^{33} +(5.18363 - 8.17671i) q^{34} +(6.77115 + 3.90933i) q^{35} +(1.13986 - 1.64339i) q^{36} +(0.919653 + 6.01284i) q^{37} +(-2.64287 + 1.38326i) q^{38} +(3.30870 - 5.73084i) q^{39} +(-5.10537 + 2.15470i) q^{40} +(-2.81624 - 4.87787i) q^{41} +(4.76663 + 3.02181i) q^{42} -3.09445i q^{43} +(-0.00880670 + 0.106090i) q^{44} -1.95920i q^{45} +(4.78310 + 0.198186i) q^{46} +9.24930 q^{47} +(-3.74646 + 1.40144i) q^{48} +(-4.46303 + 7.73020i) q^{49} +(0.879533 - 1.38738i) q^{50} -6.84575i q^{51} +(-11.9698 + 5.64653i) q^{52} +(-4.12825 - 2.38345i) q^{53} +(0.0585472 - 1.41300i) q^{54} +(0.0521414 + 0.0903115i) q^{55} +(-4.38898 - 10.3993i) q^{56} +(-1.05464 + 1.82670i) q^{57} +(-0.444779 + 10.7345i) q^{58} +(5.60039 + 3.23339i) q^{59} +(-2.23321 + 3.21971i) q^{60} +(-10.5999 + 6.11985i) q^{61} +(3.11367 + 0.129014i) q^{62} +3.99075 q^{63} +(7.75433 + 1.96734i) q^{64} +(-6.48239 + 11.2278i) q^{65} +(0.0349063 + 0.0666922i) q^{66} +(-3.81538 + 2.20281i) q^{67} +(-7.80320 + 11.2502i) q^{68} +(2.93155 - 1.69253i) q^{69} +(-9.33877 - 5.92031i) q^{70} +(3.00191 + 5.19946i) q^{71} +(-1.70684 + 2.25537i) q^{72} -5.34491 q^{73} +(-0.947436 - 8.54999i) q^{74} -1.16155i q^{75} +(3.81536 - 1.79982i) q^{76} +(-0.183958 + 0.106208i) q^{77} +(-5.01072 + 7.90397i) q^{78} +(4.88448 + 8.46017i) q^{79} +(7.34005 - 2.74569i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(3.69376 + 7.05731i) q^{82} +(-8.44226 - 4.87414i) q^{83} +(-6.55834 - 4.54889i) q^{84} +13.4122i q^{85} +(-0.181172 + 4.37246i) q^{86} +(3.79847 + 6.57914i) q^{87} +(0.0186551 - 0.149389i) q^{88} +(-3.61025 + 6.25314i) q^{89} +(-0.114705 + 2.76835i) q^{90} +(-22.8703 - 13.2042i) q^{91} +(-6.74692 - 0.560074i) q^{92} +(1.90836 - 1.10179i) q^{93} +(-13.0693 - 0.541521i) q^{94} +(2.06625 - 3.57885i) q^{95} +(5.37580 - 1.76089i) q^{96} +6.72084 q^{97} +(6.75885 - 10.6615i) q^{98} +(0.0460962 + 0.0266137i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 8 q^{8} + 76 q^{9} - 4 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 6 q^{22} - 16 q^{23} + 80 q^{25} + 6 q^{28} + 8 q^{30} + 8 q^{32} + 2 q^{34} + 4 q^{36} + 76 q^{38}+ \cdots + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41300 − 0.0585472i −0.999143 − 0.0413991i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 1.99314 + 0.165455i 0.996572 + 0.0827273i
\(5\) 1.69671 − 0.979598i 0.758793 − 0.438089i −0.0700692 − 0.997542i \(-0.522322\pi\)
0.828862 + 0.559453i \(0.188989\pi\)
\(6\) 1.25297 − 0.655797i 0.511522 − 0.267728i
\(7\) 1.99537 + 3.45609i 0.754180 + 1.30628i 0.945781 + 0.324806i \(0.105299\pi\)
−0.191600 + 0.981473i \(0.561368\pi\)
\(8\) −2.80663 − 0.350480i −0.992293 − 0.123914i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −2.45481 + 1.28483i −0.776279 + 0.406300i
\(11\) 0.0532273i 0.0160486i 0.999968 + 0.00802432i \(0.00255425\pi\)
−0.999968 + 0.00802432i \(0.997446\pi\)
\(12\) −1.80884 + 0.853284i −0.522167 + 0.246322i
\(13\) −5.73084 + 3.30870i −1.58945 + 0.917669i −0.596052 + 0.802946i \(0.703265\pi\)
−0.993398 + 0.114723i \(0.963402\pi\)
\(14\) −2.61712 − 5.00028i −0.699455 − 1.33638i
\(15\) −0.979598 + 1.69671i −0.252931 + 0.438089i
\(16\) 3.94525 + 0.659549i 0.986312 + 0.164887i
\(17\) −3.42288 + 5.92860i −0.830170 + 1.43790i 0.0677339 + 0.997703i \(0.478423\pi\)
−0.897903 + 0.440192i \(0.854910\pi\)
\(18\) −0.757204 + 1.19442i −0.178475 + 0.281528i
\(19\) 1.82670 − 1.05464i 0.419073 − 0.241952i −0.275608 − 0.961270i \(-0.588879\pi\)
0.694681 + 0.719318i \(0.255546\pi\)
\(20\) 3.54387 − 1.67175i 0.792434 − 0.373815i
\(21\) −3.45609 − 1.99537i −0.754180 − 0.435426i
\(22\) 0.00311631 − 0.0752103i 0.000664400 − 0.0160349i
\(23\) −3.38506 −0.705834 −0.352917 − 0.935655i \(-0.614810\pi\)
−0.352917 + 0.935655i \(0.614810\pi\)
\(24\) 2.60585 − 1.09979i 0.531917 − 0.224494i
\(25\) −0.580777 + 1.00594i −0.116155 + 0.201187i
\(26\) 8.29140 − 4.33968i 1.62608 − 0.851080i
\(27\) 1.00000i 0.192450i
\(28\) 3.40524 + 7.21863i 0.643530 + 1.36419i
\(29\) − 7.59693i − 1.41071i −0.708852 − 0.705357i \(-0.750787\pi\)
0.708852 − 0.705357i \(-0.249213\pi\)
\(30\) 1.48351 − 2.34010i 0.270851 − 0.427243i
\(31\) −2.20359 −0.395776 −0.197888 − 0.980225i \(-0.563408\pi\)
−0.197888 + 0.980225i \(0.563408\pi\)
\(32\) −5.53603 − 1.16293i −0.978641 − 0.205578i
\(33\) −0.0266137 − 0.0460962i −0.00463285 − 0.00802432i
\(34\) 5.18363 − 8.17671i 0.888985 − 1.40229i
\(35\) 6.77115 + 3.90933i 1.14453 + 0.660797i
\(36\) 1.13986 − 1.64339i 0.189977 − 0.273898i
\(37\) 0.919653 + 6.01284i 0.151190 + 0.988505i
\(38\) −2.64287 + 1.38326i −0.428730 + 0.224395i
\(39\) 3.30870 − 5.73084i 0.529816 − 0.917669i
\(40\) −5.10537 + 2.15470i −0.807230 + 0.340688i
\(41\) −2.81624 − 4.87787i −0.439822 − 0.761795i 0.557853 − 0.829940i \(-0.311625\pi\)
−0.997675 + 0.0681450i \(0.978292\pi\)
\(42\) 4.76663 + 3.02181i 0.735508 + 0.466275i
\(43\) − 3.09445i − 0.471900i −0.971765 − 0.235950i \(-0.924180\pi\)
0.971765 − 0.235950i \(-0.0758201\pi\)
\(44\) −0.00880670 + 0.106090i −0.00132766 + 0.0159936i
\(45\) − 1.95920i − 0.292060i
\(46\) 4.78310 + 0.198186i 0.705229 + 0.0292209i
\(47\) 9.24930 1.34915 0.674575 − 0.738207i \(-0.264327\pi\)
0.674575 + 0.738207i \(0.264327\pi\)
\(48\) −3.74646 + 1.40144i −0.540755 + 0.202280i
\(49\) −4.46303 + 7.73020i −0.637576 + 1.10431i
\(50\) 0.879533 − 1.38738i 0.124385 − 0.196206i
\(51\) − 6.84575i − 0.958597i
\(52\) −11.9698 + 5.64653i −1.65992 + 0.783033i
\(53\) −4.12825 − 2.38345i −0.567059 − 0.327392i 0.188915 − 0.981993i \(-0.439503\pi\)
−0.755974 + 0.654602i \(0.772836\pi\)
\(54\) 0.0585472 − 1.41300i 0.00796726 − 0.192285i
\(55\) 0.0521414 + 0.0903115i 0.00703074 + 0.0121776i
\(56\) −4.38898 − 10.3993i −0.586502 − 1.38966i
\(57\) −1.05464 + 1.82670i −0.139691 + 0.241952i
\(58\) −0.444779 + 10.7345i −0.0584023 + 1.40951i
\(59\) 5.60039 + 3.23339i 0.729109 + 0.420951i 0.818096 − 0.575081i \(-0.195030\pi\)
−0.0889870 + 0.996033i \(0.528363\pi\)
\(60\) −2.23321 + 3.21971i −0.288306 + 0.415663i
\(61\) −10.5999 + 6.11985i −1.35718 + 0.783566i −0.989242 − 0.146286i \(-0.953268\pi\)
−0.367934 + 0.929852i \(0.619935\pi\)
\(62\) 3.11367 + 0.129014i 0.395437 + 0.0163848i
\(63\) 3.99075 0.502787
\(64\) 7.75433 + 1.96734i 0.969291 + 0.245917i
\(65\) −6.48239 + 11.2278i −0.804042 + 1.39264i
\(66\) 0.0349063 + 0.0666922i 0.00429667 + 0.00820924i
\(67\) −3.81538 + 2.20281i −0.466122 + 0.269116i −0.714615 − 0.699518i \(-0.753398\pi\)
0.248493 + 0.968634i \(0.420065\pi\)
\(68\) −7.80320 + 11.2502i −0.946277 + 1.36429i
\(69\) 2.93155 − 1.69253i 0.352917 − 0.203757i
\(70\) −9.33877 − 5.92031i −1.11620 − 0.707613i
\(71\) 3.00191 + 5.19946i 0.356261 + 0.617062i 0.987333 − 0.158662i \(-0.0507180\pi\)
−0.631072 + 0.775724i \(0.717385\pi\)
\(72\) −1.70684 + 2.25537i −0.201153 + 0.265798i
\(73\) −5.34491 −0.625574 −0.312787 − 0.949823i \(-0.601263\pi\)
−0.312787 + 0.949823i \(0.601263\pi\)
\(74\) −0.947436 − 8.54999i −0.110137 − 0.993916i
\(75\) − 1.16155i − 0.134125i
\(76\) 3.81536 − 1.79982i 0.437652 − 0.206454i
\(77\) −0.183958 + 0.106208i −0.0209640 + 0.0121036i
\(78\) −5.01072 + 7.90397i −0.567353 + 0.894948i
\(79\) 4.88448 + 8.46017i 0.549547 + 0.951844i 0.998305 + 0.0581905i \(0.0185331\pi\)
−0.448758 + 0.893653i \(0.648134\pi\)
\(80\) 7.34005 − 2.74569i 0.820642 − 0.306978i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 3.69376 + 7.05731i 0.407908 + 0.779350i
\(83\) −8.44226 − 4.87414i −0.926658 − 0.535006i −0.0409052 − 0.999163i \(-0.513024\pi\)
−0.885753 + 0.464157i \(0.846357\pi\)
\(84\) −6.55834 − 4.54889i −0.715574 − 0.496325i
\(85\) 13.4122i 1.45475i
\(86\) −0.181172 + 4.37246i −0.0195362 + 0.471495i
\(87\) 3.79847 + 6.57914i 0.407238 + 0.705357i
\(88\) 0.0186551 − 0.149389i 0.00198864 − 0.0159250i
\(89\) −3.61025 + 6.25314i −0.382686 + 0.662831i −0.991445 − 0.130524i \(-0.958334\pi\)
0.608759 + 0.793355i \(0.291667\pi\)
\(90\) −0.114705 + 2.76835i −0.0120910 + 0.291809i
\(91\) −22.8703 − 13.2042i −2.39746 − 1.38418i
\(92\) −6.74692 − 0.560074i −0.703415 − 0.0583917i
\(93\) 1.90836 − 1.10179i 0.197888 − 0.114251i
\(94\) −13.0693 − 0.541521i −1.34799 − 0.0558536i
\(95\) 2.06625 − 3.57885i 0.211993 − 0.367183i
\(96\) 5.37580 − 1.76089i 0.548666 − 0.179720i
\(97\) 6.72084 0.682398 0.341199 − 0.939991i \(-0.389167\pi\)
0.341199 + 0.939991i \(0.389167\pi\)
\(98\) 6.75885 − 10.6615i 0.682747 − 1.07697i
\(99\) 0.0460962 + 0.0266137i 0.00463285 + 0.00267477i
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.bh.a.565.2 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.52 yes 152
37.26 even 3 inner 888.2.bh.a.877.52 yes 152
296.285 even 6 inner 888.2.bh.a.877.2 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.2 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.52 yes 152 8.5 even 2 inner
888.2.bh.a.877.2 yes 152 296.285 even 6 inner
888.2.bh.a.877.52 yes 152 37.26 even 3 inner