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Properties

Label 888.2.bh.a.565.13
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 565.13
Character \(\chi\) \(=\) 888.565
Dual form 888.2.bh.a.877.13

$q$-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+(-1.23251 + 0.693489i) q^{2} +(-0.866025 + 0.500000i) q^{3} +(1.03815 - 1.70946i) q^{4} +(0.867653 - 0.500940i) q^{5} +(0.720638 - 1.21683i) q^{6} +(0.764837 + 1.32474i) q^{7} +(-0.0940330 + 2.82686i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.721993 + 1.21912i) q^{10} -2.32794i q^{11} +(-0.0443321 + 1.99951i) q^{12} +(4.45316 - 2.57104i) q^{13} +(-1.86136 - 1.10234i) q^{14} +(-0.500940 + 0.867653i) q^{15} +(-1.84450 - 3.54934i) q^{16} +(0.268903 - 0.465753i) q^{17} +(-0.0156747 + 1.41413i) q^{18} +(-0.199723 + 0.115310i) q^{19} +(0.0444155 - 2.00327i) q^{20} +(-1.32474 - 0.764837i) q^{21} +(1.61440 + 2.86920i) q^{22} -5.25512 q^{23} +(-1.33200 - 2.49515i) q^{24} +(-1.99812 + 3.46084i) q^{25} +(-3.70557 + 6.25704i) q^{26} +1.00000i q^{27} +(3.05860 + 0.0678137i) q^{28} -6.98430i q^{29} +(0.0157042 - 1.41679i) q^{30} +4.17278 q^{31} +(4.73479 + 3.09545i) q^{32} +(1.16397 + 2.01606i) q^{33} +(-0.00842996 + 0.760525i) q^{34} +(1.32723 + 0.766275i) q^{35} +(-0.961362 - 1.75379i) q^{36} +(4.15426 - 4.44321i) q^{37} +(0.166194 - 0.280627i) q^{38} +(-2.57104 + 4.45316i) q^{39} +(1.33450 + 2.49984i) q^{40} +(-2.07374 - 3.59182i) q^{41} +(2.16315 + 0.0239772i) q^{42} -5.68123i q^{43} +(-3.97952 - 2.41675i) q^{44} -1.00188i q^{45} +(6.47697 - 3.64437i) q^{46} -5.24610 q^{47} +(3.37206 + 2.15157i) q^{48} +(2.33005 - 4.03576i) q^{49} +(0.0626399 - 5.65118i) q^{50} +0.537805i q^{51} +(0.227959 - 10.2816i) q^{52} +(9.14588 + 5.28038i) q^{53} +(-0.693489 - 1.23251i) q^{54} +(-1.16616 - 2.01985i) q^{55} +(-3.81677 + 2.03752i) q^{56} +(0.115310 - 0.199723i) q^{57} +(4.84353 + 8.60820i) q^{58} +(0.0202239 + 0.0116763i) q^{59} +(0.963169 + 1.75709i) q^{60} +(-5.57946 + 3.22130i) q^{61} +(-5.14298 + 2.89377i) q^{62} +1.52967 q^{63} +(-7.98232 - 0.531637i) q^{64} +(2.57587 - 4.46154i) q^{65} +(-2.83272 - 1.67760i) q^{66} +(12.5263 - 7.23207i) q^{67} +(-0.517025 - 0.943198i) q^{68} +(4.55107 - 2.62756i) q^{69} +(-2.16722 - 0.0240223i) q^{70} +(0.370844 + 0.642321i) q^{71} +(2.40112 + 1.49487i) q^{72} +13.9787 q^{73} +(-2.03884 + 8.35722i) q^{74} -3.99624i q^{75} +(-0.0102239 + 0.461128i) q^{76} +(3.08391 - 1.78050i) q^{77} +(0.0806006 - 7.27154i) q^{78} +(6.74970 + 11.6908i) q^{79} +(-3.37839 - 2.15561i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(5.04679 + 2.98883i) q^{82} +(9.50385 + 5.48705i) q^{83} +(-2.68273 + 1.47057i) q^{84} -0.538816i q^{85} +(3.93987 + 7.00215i) q^{86} +(3.49215 + 6.04858i) q^{87} +(6.58077 + 0.218903i) q^{88} +(0.245607 - 0.425404i) q^{89} +(0.694792 + 1.23482i) q^{90} +(6.81189 + 3.93285i) q^{91} +(-5.45559 + 8.98341i) q^{92} +(-3.61373 + 2.08639i) q^{93} +(6.46585 - 3.63811i) q^{94} +(-0.115527 + 0.200099i) q^{95} +(-5.64817 - 0.313341i) q^{96} +15.2338 q^{97} +(-0.0730458 + 6.58996i) q^{98} +(-2.01606 - 1.16397i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.23251 + 0.693489i −0.871514 + 0.490371i
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 1.03815 − 1.70946i 0.519074 − 0.854730i
\(5\) 0.867653 − 0.500940i 0.388026 − 0.224027i −0.293278 − 0.956027i \(-0.594746\pi\)
0.681305 + 0.732000i \(0.261413\pi\)
\(6\) 0.720638 − 1.21683i 0.294199 − 0.496770i
\(7\) 0.764837 + 1.32474i 0.289081 + 0.500704i 0.973591 − 0.228300i \(-0.0733169\pi\)
−0.684509 + 0.729004i \(0.739984\pi\)
\(8\) −0.0940330 + 2.82686i −0.0332457 + 0.999447i
\(9\) 0.500000 − 0.866025i 0.166667 − 0.288675i
\(10\) −0.721993 + 1.21912i −0.228314 + 0.385520i
\(11\) − 2.32794i − 0.701901i −0.936394 − 0.350950i \(-0.885859\pi\)
0.936394 − 0.350950i \(-0.114141\pi\)
\(12\) −0.0443321 + 1.99951i −0.0127976 + 0.577208i
\(13\) 4.45316 − 2.57104i 1.23509 − 0.713077i 0.267000 − 0.963697i \(-0.413968\pi\)
0.968086 + 0.250620i \(0.0806343\pi\)
\(14\) −1.86136 − 1.10234i −0.497469 − 0.294613i
\(15\) −0.500940 + 0.867653i −0.129342 + 0.224027i
\(16\) −1.84450 − 3.54934i −0.461125 − 0.887335i
\(17\) 0.268903 − 0.465753i 0.0652185 − 0.112962i −0.831572 − 0.555416i \(-0.812559\pi\)
0.896791 + 0.442455i \(0.145892\pi\)
\(18\) −0.0156747 + 1.41413i −0.00369457 + 0.333313i
\(19\) −0.199723 + 0.115310i −0.0458197 + 0.0264540i −0.522735 − 0.852495i \(-0.675088\pi\)
0.476915 + 0.878949i \(0.341755\pi\)
\(20\) 0.0444155 − 2.00327i 0.00993160 − 0.447944i
\(21\) −1.32474 − 0.764837i −0.289081 − 0.166901i
\(22\) 1.61440 + 2.86920i 0.344192 + 0.611717i
\(23\) −5.25512 −1.09577 −0.547884 − 0.836554i \(-0.684567\pi\)
−0.547884 + 0.836554i \(0.684567\pi\)
\(24\) −1.33200 − 2.49515i −0.271893 − 0.509321i
\(25\) −1.99812 + 3.46084i −0.399624 + 0.692168i
\(26\) −3.70557 + 6.25704i −0.726723 + 1.22711i
\(27\) 1.00000i 0.192450i
\(28\) 3.05860 + 0.0678137i 0.578021 + 0.0128156i
\(29\) − 6.98430i − 1.29695i −0.761235 − 0.648476i \(-0.775407\pi\)
0.761235 − 0.648476i \(-0.224593\pi\)
\(30\) 0.0157042 − 1.41679i 0.00286718 − 0.258668i
\(31\) 4.17278 0.749453 0.374727 − 0.927135i \(-0.377737\pi\)
0.374727 + 0.927135i \(0.377737\pi\)
\(32\) 4.73479 + 3.09545i 0.837000 + 0.547203i
\(33\) 1.16397 + 2.01606i 0.202621 + 0.350950i
\(34\) −0.00842996 + 0.760525i −0.00144573 + 0.130429i
\(35\) 1.32723 + 0.766275i 0.224342 + 0.129524i
\(36\) −0.961362 − 1.75379i −0.160227 − 0.292299i
\(37\) 4.15426 − 4.44321i 0.682956 − 0.730459i
\(38\) 0.166194 − 0.280627i 0.0269602 − 0.0455237i
\(39\) −2.57104 + 4.45316i −0.411695 + 0.713077i
\(40\) 1.33450 + 2.49984i 0.211003 + 0.395260i
\(41\) −2.07374 − 3.59182i −0.323864 − 0.560949i 0.657418 − 0.753526i \(-0.271649\pi\)
−0.981282 + 0.192577i \(0.938315\pi\)
\(42\) 2.16315 + 0.0239772i 0.333782 + 0.00369977i
\(43\) − 5.68123i − 0.866379i −0.901303 − 0.433190i \(-0.857388\pi\)
0.901303 − 0.433190i \(-0.142612\pi\)
\(44\) −3.97952 − 2.41675i −0.599936 − 0.364338i
\(45\) − 1.00188i − 0.149351i
\(46\) 6.47697 − 3.64437i 0.954977 − 0.537332i
\(47\) −5.24610 −0.765222 −0.382611 − 0.923910i \(-0.624975\pi\)
−0.382611 + 0.923910i \(0.624975\pi\)
\(48\) 3.37206 + 2.15157i 0.486714 + 0.310552i
\(49\) 2.33005 − 4.03576i 0.332864 − 0.576537i
\(50\) 0.0626399 − 5.65118i 0.00885863 − 0.799198i
\(51\) 0.537805i 0.0753078i
\(52\) 0.227959 − 10.2816i 0.0316122 − 1.42580i
\(53\) 9.14588 + 5.28038i 1.25628 + 0.725316i 0.972350 − 0.233528i \(-0.0750272\pi\)
0.283933 + 0.958844i \(0.408361\pi\)
\(54\) −0.693489 − 1.23251i −0.0943718 − 0.167723i
\(55\) −1.16616 − 2.01985i −0.157245 − 0.272356i
\(56\) −3.81677 + 2.03752i −0.510038 + 0.272275i
\(57\) 0.115310 − 0.199723i 0.0152732 − 0.0264540i
\(58\) 4.84353 + 8.60820i 0.635987 + 1.13031i
\(59\) 0.0202239 + 0.0116763i 0.00263293 + 0.00152012i 0.501316 − 0.865264i \(-0.332849\pi\)
−0.498683 + 0.866784i \(0.666183\pi\)
\(60\) 0.963169 + 1.75709i 0.124345 + 0.226839i
\(61\) −5.57946 + 3.22130i −0.714376 + 0.412445i −0.812679 − 0.582711i \(-0.801992\pi\)
0.0983030 + 0.995157i \(0.468659\pi\)
\(62\) −5.14298 + 2.89377i −0.653159 + 0.367510i
\(63\) 1.52967 0.192721
\(64\) −7.98232 − 0.531637i −0.997789 − 0.0664546i
\(65\) 2.57587 − 4.46154i 0.319497 − 0.553385i
\(66\) −2.83272 − 1.67760i −0.348683 − 0.206499i
\(67\) 12.5263 − 7.23207i 1.53033 − 0.883538i 0.530986 − 0.847380i \(-0.321822\pi\)
0.999346 − 0.0361574i \(-0.0115118\pi\)
\(68\) −0.517025 − 0.943198i −0.0626985 − 0.114380i
\(69\) 4.55107 − 2.62756i 0.547884 − 0.316321i
\(70\) −2.16722 − 0.0240223i −0.259032 − 0.00287122i
\(71\) 0.370844 + 0.642321i 0.0440111 + 0.0762295i 0.887192 − 0.461401i \(-0.152653\pi\)
−0.843181 + 0.537630i \(0.819320\pi\)
\(72\) 2.40112 + 1.49487i 0.282975 + 0.176172i
\(73\) 13.9787 1.63608 0.818041 − 0.575160i \(-0.195060\pi\)
0.818041 + 0.575160i \(0.195060\pi\)
\(74\) −2.03884 + 8.35722i −0.237010 + 0.971507i
\(75\) − 3.99624i − 0.461446i
\(76\) −0.0102239 + 0.461128i −0.00117276 + 0.0528950i
\(77\) 3.08391 − 1.78050i 0.351444 − 0.202906i
\(78\) 0.0806006 − 7.27154i 0.00912622 − 0.823340i
\(79\) 6.74970 + 11.6908i 0.759401 + 1.31532i 0.943157 + 0.332349i \(0.107841\pi\)
−0.183756 + 0.982972i \(0.558826\pi\)
\(80\) −3.37839 − 2.15561i −0.377716 − 0.241005i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 5.04679 + 2.98883i 0.557325 + 0.330061i
\(83\) 9.50385 + 5.48705i 1.04318 + 0.602282i 0.920733 − 0.390192i \(-0.127592\pi\)
0.122450 + 0.992475i \(0.460925\pi\)
\(84\) −2.68273 + 1.47057i −0.292710 + 0.160452i
\(85\) − 0.538816i − 0.0584428i
\(86\) 3.93987 + 7.00215i 0.424847 + 0.755061i
\(87\) 3.49215 + 6.04858i 0.374398 + 0.648476i
\(88\) 6.58077 + 0.218903i 0.701513 + 0.0233352i
\(89\) 0.245607 − 0.425404i 0.0260343 − 0.0450928i −0.852715 − 0.522377i \(-0.825045\pi\)
0.878749 + 0.477284i \(0.158379\pi\)
\(90\) 0.694792 + 1.23482i 0.0732375 + 0.130162i
\(91\) 6.81189 + 3.93285i 0.714080 + 0.412275i
\(92\) −5.45559 + 8.98341i −0.568784 + 0.936585i
\(93\) −3.61373 + 2.08639i −0.374727 + 0.216348i
\(94\) 6.46585 − 3.63811i 0.666902 − 0.375242i
\(95\) −0.115527 + 0.200099i −0.0118528 + 0.0205297i
\(96\) −5.64817 − 0.313341i −0.576464 − 0.0319802i
\(97\) 15.2338 1.54676 0.773378 − 0.633945i \(-0.218566\pi\)
0.773378 + 0.633945i \(0.218566\pi\)
\(98\) −0.0730458 + 6.58996i −0.00737874 + 0.665687i
\(99\) −2.01606 − 1.16397i −0.202621 − 0.116983i
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.565.13 ✓ 152
8.5 even 2 inner 888.2.bh.a.565.39 yes 152
37.26 even 3 inner 888.2.bh.a.877.39 yes 152
296.285 even 6 inner 888.2.bh.a.877.13 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.bh.a.565.13 ✓ 152 1.1 even 1 trivial
888.2.bh.a.565.39 yes 152 8.5 even 2 inner
888.2.bh.a.877.13 yes 152 296.285 even 6 inner
888.2.bh.a.877.39 yes 152 37.26 even 3 inner