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Properties

Label 888.2.f.b.445.13
Level $888$
Weight $2$
Character 888.445
Analytic conductor $7.091$
Analytic rank $0$
Dimension $44$
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] = [888,2,Mod(445,888)] 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("888.445"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 445.13
Character \(\chi\) \(=\) 888.445
Dual form 888.2.f.b.445.14

$q$-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+(-0.784871 - 1.17643i) q^{2} +1.00000i q^{3} +(-0.767956 + 1.84668i) q^{4} +2.04572i q^{5} +(1.17643 - 0.784871i) q^{6} +2.83247 q^{7} +(2.77523 - 0.545965i) q^{8} -1.00000 q^{9} +(2.40664 - 1.60563i) q^{10} +0.148585i q^{11} +(-1.84668 - 0.767956i) q^{12} -5.95266i q^{13} +(-2.22312 - 3.33219i) q^{14} -2.04572 q^{15} +(-2.82049 - 2.83635i) q^{16} +5.00506 q^{17} +(0.784871 + 1.17643i) q^{18} -1.84992i q^{19} +(-3.77780 - 1.57102i) q^{20} +2.83247i q^{21} +(0.174799 - 0.116620i) q^{22} +9.18677 q^{23} +(0.545965 + 2.77523i) q^{24} +0.815031 q^{25} +(-7.00287 + 4.67207i) q^{26} -1.00000i q^{27} +(-2.17521 + 5.23068i) q^{28} +7.39851i q^{29} +(1.60563 + 2.40664i) q^{30} -7.74902 q^{31} +(-1.12303 + 5.54426i) q^{32} -0.148585 q^{33} +(-3.92832 - 5.88808i) q^{34} +5.79444i q^{35} +(0.767956 - 1.84668i) q^{36} +1.00000i q^{37} +(-2.17630 + 1.45195i) q^{38} +5.95266 q^{39} +(1.11689 + 5.67735i) q^{40} -0.418741 q^{41} +(3.33219 - 2.22312i) q^{42} +12.4090i q^{43} +(-0.274389 - 0.114107i) q^{44} -2.04572i q^{45} +(-7.21043 - 10.8076i) q^{46} +8.07530 q^{47} +(2.83635 - 2.82049i) q^{48} +1.02288 q^{49} +(-0.639694 - 0.958824i) q^{50} +5.00506i q^{51} +(10.9927 + 4.57138i) q^{52} +4.21989i q^{53} +(-1.17643 + 0.784871i) q^{54} -0.303963 q^{55} +(7.86076 - 1.54643i) q^{56} +1.84992 q^{57} +(8.70380 - 5.80687i) q^{58} -5.45357i q^{59} +(1.57102 - 3.77780i) q^{60} +9.80288i q^{61} +(6.08198 + 9.11615i) q^{62} -2.83247 q^{63} +(7.40384 - 3.03036i) q^{64} +12.1775 q^{65} +(0.116620 + 0.174799i) q^{66} -15.3029i q^{67} +(-3.84366 + 9.24276i) q^{68} +9.18677i q^{69} +(6.81673 - 4.54788i) q^{70} +6.84982 q^{71} +(-2.77523 + 0.545965i) q^{72} -3.32175 q^{73} +(1.17643 - 0.784871i) q^{74} +0.815031i q^{75} +(3.41623 + 1.42066i) q^{76} +0.420861i q^{77} +(-4.67207 - 7.00287i) q^{78} -10.8437 q^{79} +(5.80237 - 5.76993i) q^{80} +1.00000 q^{81} +(0.328658 + 0.492618i) q^{82} -1.66981i q^{83} +(-5.23068 - 2.17521i) q^{84} +10.2389i q^{85} +(14.5983 - 9.73947i) q^{86} -7.39851 q^{87} +(0.0811220 + 0.412357i) q^{88} -12.5465 q^{89} +(-2.40664 + 1.60563i) q^{90} -16.8607i q^{91} +(-7.05504 + 16.9651i) q^{92} -7.74902i q^{93} +(-6.33807 - 9.49999i) q^{94} +3.78443 q^{95} +(-5.54426 - 1.12303i) q^{96} -7.92559 q^{97} +(-0.802827 - 1.20334i) q^{98} -0.148585i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 2 q^{4} + 2 q^{6} - 6 q^{8} - 44 q^{9} - 4 q^{10} - 4 q^{12} - 2 q^{14} - 4 q^{15} + 10 q^{16} + 20 q^{17} + 8 q^{20} + 4 q^{24} - 76 q^{25} + 8 q^{26} - 10 q^{28} - 4 q^{30} - 4 q^{31} + 10 q^{32}+ \cdots + 106 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.784871 − 1.17643i −0.554987 − 0.831859i
\(3\) 1.00000i 0.577350i
\(4\) −0.767956 + 1.84668i −0.383978 + 0.923342i
\(5\) 2.04572i 0.914874i 0.889242 + 0.457437i \(0.151232\pi\)
−0.889242 + 0.457437i \(0.848768\pi\)
\(6\) 1.17643 − 0.784871i 0.480274 − 0.320422i
\(7\) 2.83247 1.07057 0.535286 − 0.844671i \(-0.320204\pi\)
0.535286 + 0.844671i \(0.320204\pi\)
\(8\) 2.77523 − 0.545965i 0.981193 − 0.193028i
\(9\) −1.00000 −0.333333
\(10\) 2.40664 − 1.60563i 0.761046 − 0.507743i
\(11\) 0.148585i 0.0448000i 0.999749 + 0.0224000i \(0.00713073\pi\)
−0.999749 + 0.0224000i \(0.992869\pi\)
\(12\) −1.84668 − 0.767956i −0.533092 − 0.221690i
\(13\) − 5.95266i − 1.65097i −0.564423 − 0.825486i \(-0.690901\pi\)
0.564423 − 0.825486i \(-0.309099\pi\)
\(14\) −2.22312 − 3.33219i −0.594154 − 0.890565i
\(15\) −2.04572 −0.528203
\(16\) −2.82049 − 2.83635i −0.705122 − 0.709086i
\(17\) 5.00506 1.21390 0.606952 − 0.794738i \(-0.292392\pi\)
0.606952 + 0.794738i \(0.292392\pi\)
\(18\) 0.784871 + 1.17643i 0.184996 + 0.277286i
\(19\) − 1.84992i − 0.424402i −0.977226 − 0.212201i \(-0.931937\pi\)
0.977226 − 0.212201i \(-0.0680631\pi\)
\(20\) −3.77780 − 1.57102i −0.844741 − 0.351291i
\(21\) 2.83247i 0.618095i
\(22\) 0.174799 − 0.116620i 0.0372672 − 0.0248634i
\(23\) 9.18677 1.91557 0.957787 − 0.287479i \(-0.0928171\pi\)
0.957787 + 0.287479i \(0.0928171\pi\)
\(24\) 0.545965 + 2.77523i 0.111445 + 0.566492i
\(25\) 0.815031 0.163006
\(26\) −7.00287 + 4.67207i −1.37337 + 0.916268i
\(27\) − 1.00000i − 0.192450i
\(28\) −2.17521 + 5.23068i −0.411076 + 0.988505i
\(29\) 7.39851i 1.37387i 0.726719 + 0.686935i \(0.241044\pi\)
−0.726719 + 0.686935i \(0.758956\pi\)
\(30\) 1.60563 + 2.40664i 0.293146 + 0.439390i
\(31\) −7.74902 −1.39177 −0.695883 − 0.718156i \(-0.744987\pi\)
−0.695883 + 0.718156i \(0.744987\pi\)
\(32\) −1.12303 + 5.54426i −0.198526 + 0.980096i
\(33\) −0.148585 −0.0258653
\(34\) −3.92832 − 5.88808i −0.673702 − 1.00980i
\(35\) 5.79444i 0.979439i
\(36\) 0.767956 − 1.84668i 0.127993 − 0.307781i
\(37\) 1.00000i 0.164399i
\(38\) −2.17630 + 1.45195i −0.353042 + 0.235538i
\(39\) 5.95266 0.953189
\(40\) 1.11689 + 5.67735i 0.176596 + 0.897668i
\(41\) −0.418741 −0.0653964 −0.0326982 − 0.999465i \(-0.510410\pi\)
−0.0326982 + 0.999465i \(0.510410\pi\)
\(42\) 3.33219 − 2.22312i 0.514168 − 0.343035i
\(43\) 12.4090i 1.89236i 0.323646 + 0.946178i \(0.395091\pi\)
−0.323646 + 0.946178i \(0.604909\pi\)
\(44\) −0.274389 − 0.114107i −0.0413657 − 0.0172022i
\(45\) − 2.04572i − 0.304958i
\(46\) −7.21043 − 10.8076i −1.06312 − 1.59349i
\(47\) 8.07530 1.17790 0.588952 − 0.808168i \(-0.299541\pi\)
0.588952 + 0.808168i \(0.299541\pi\)
\(48\) 2.83635 − 2.82049i 0.409391 − 0.407102i
\(49\) 1.02288 0.146126
\(50\) −0.639694 − 0.958824i −0.0904664 − 0.135598i
\(51\) 5.00506i 0.700848i
\(52\) 10.9927 + 4.57138i 1.52441 + 0.633937i
\(53\) 4.21989i 0.579647i 0.957080 + 0.289823i \(0.0935966\pi\)
−0.957080 + 0.289823i \(0.906403\pi\)
\(54\) −1.17643 + 0.784871i −0.160091 + 0.106807i
\(55\) −0.303963 −0.0409863
\(56\) 7.86076 − 1.54643i 1.05044 − 0.206650i
\(57\) 1.84992 0.245029
\(58\) 8.70380 − 5.80687i 1.14287 − 0.762480i
\(59\) − 5.45357i − 0.709994i −0.934867 − 0.354997i \(-0.884482\pi\)
0.934867 − 0.354997i \(-0.115518\pi\)
\(60\) 1.57102 − 3.77780i 0.202818 − 0.487712i
\(61\) 9.80288i 1.25513i 0.778564 + 0.627565i \(0.215948\pi\)
−0.778564 + 0.627565i \(0.784052\pi\)
\(62\) 6.08198 + 9.11615i 0.772412 + 1.15775i
\(63\) −2.83247 −0.356858
\(64\) 7.40384 − 3.03036i 0.925481 − 0.378795i
\(65\) 12.1775 1.51043
\(66\) 0.116620 + 0.174799i 0.0143549 + 0.0215163i
\(67\) − 15.3029i − 1.86954i −0.355250 − 0.934771i \(-0.615604\pi\)
0.355250 − 0.934771i \(-0.384396\pi\)
\(68\) −3.84366 + 9.24276i −0.466113 + 1.12085i
\(69\) 9.18677i 1.10596i
\(70\) 6.81673 − 4.54788i 0.814755 − 0.543576i
\(71\) 6.84982 0.812924 0.406462 − 0.913668i \(-0.366762\pi\)
0.406462 + 0.913668i \(0.366762\pi\)
\(72\) −2.77523 + 0.545965i −0.327064 + 0.0643426i
\(73\) −3.32175 −0.388782 −0.194391 − 0.980924i \(-0.562273\pi\)
−0.194391 + 0.980924i \(0.562273\pi\)
\(74\) 1.17643 − 0.784871i 0.136757 − 0.0912394i
\(75\) 0.815031i 0.0941117i
\(76\) 3.41623 + 1.42066i 0.391868 + 0.162961i
\(77\) 0.420861i 0.0479616i
\(78\) −4.67207 − 7.00287i −0.529008 − 0.792918i
\(79\) −10.8437 −1.22001 −0.610005 − 0.792398i \(-0.708833\pi\)
−0.610005 + 0.792398i \(0.708833\pi\)
\(80\) 5.80237 − 5.76993i 0.648724 − 0.645097i
\(81\) 1.00000 0.111111
\(82\) 0.328658 + 0.492618i 0.0362942 + 0.0544006i
\(83\) − 1.66981i − 0.183286i −0.995792 − 0.0916429i \(-0.970788\pi\)
0.995792 − 0.0916429i \(-0.0292118\pi\)
\(84\) −5.23068 − 2.17521i −0.570714 − 0.237335i
\(85\) 10.2389i 1.11057i
\(86\) 14.5983 − 9.73947i 1.57417 − 1.05023i
\(87\) −7.39851 −0.793204
\(88\) 0.0811220 + 0.412357i 0.00864764 + 0.0439574i
\(89\) −12.5465 −1.32993 −0.664964 − 0.746875i \(-0.731553\pi\)
−0.664964 + 0.746875i \(0.731553\pi\)
\(90\) −2.40664 + 1.60563i −0.253682 + 0.169248i
\(91\) − 16.8607i − 1.76748i
\(92\) −7.05504 + 16.9651i −0.735538 + 1.76873i
\(93\) − 7.74902i − 0.803536i
\(94\) −6.33807 − 9.49999i −0.653722 − 0.979850i
\(95\) 3.78443 0.388274
\(96\) −5.54426 − 1.12303i −0.565858 − 0.114619i
\(97\) −7.92559 −0.804722 −0.402361 − 0.915481i \(-0.631810\pi\)
−0.402361 + 0.915481i \(0.631810\pi\)
\(98\) −0.802827 − 1.20334i −0.0810978 − 0.121556i
\(99\) − 0.148585i − 0.0149333i
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.f.b.445.13 ✓ 44
4.3 odd 2 3552.2.f.b.1777.15 44
8.3 odd 2 3552.2.f.b.1777.30 44
8.5 even 2 inner 888.2.f.b.445.14 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.f.b.445.13 ✓ 44 1.1 even 1 trivial
888.2.f.b.445.14 yes 44 8.5 even 2 inner
3552.2.f.b.1777.15 44 4.3 odd 2
3552.2.f.b.1777.30 44 8.3 odd 2