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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 43.7
Character \(\chi\) \(=\) 888.43
Dual form 888.2.r.e.475.7

$q$-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+(-1.25296 - 0.655808i) q^{2} -1.00000i q^{3} +(1.13983 + 1.64341i) q^{4} +(-1.22832 + 1.22832i) q^{5} +(-0.655808 + 1.25296i) q^{6} -2.71379i q^{7} +(-0.350405 - 2.80664i) q^{8} -1.00000 q^{9} +(2.34458 - 0.733495i) q^{10} +0.762598i q^{11} +(1.64341 - 1.13983i) q^{12} +(4.46535 - 4.46535i) q^{13} +(-1.77973 + 3.40028i) q^{14} +(1.22832 + 1.22832i) q^{15} +(-1.40157 + 3.74641i) q^{16} +(4.46463 + 4.46463i) q^{17} +(1.25296 + 0.655808i) q^{18} +(-1.62049 - 1.62049i) q^{19} +(-3.41870 - 0.618551i) q^{20} -2.71379 q^{21} +(0.500118 - 0.955507i) q^{22} +(-0.894308 + 0.894308i) q^{23} +(-2.80664 + 0.350405i) q^{24} +1.98247i q^{25} +(-8.52334 + 2.66650i) q^{26} +1.00000i q^{27} +(4.45986 - 3.09326i) q^{28} +(-0.601979 - 0.601979i) q^{29} +(-0.733495 - 2.34458i) q^{30} +(-6.04007 - 6.04007i) q^{31} +(4.21304 - 3.77495i) q^{32} +0.762598 q^{33} +(-2.66607 - 8.52196i) q^{34} +(3.33339 + 3.33339i) q^{35} +(-1.13983 - 1.64341i) q^{36} +(-0.195721 - 6.07961i) q^{37} +(0.967685 + 3.09315i) q^{38} +(-4.46535 - 4.46535i) q^{39} +(3.87785 + 3.01703i) q^{40} -3.38722i q^{41} +(3.40028 + 1.77973i) q^{42} +(-3.90961 - 3.90961i) q^{43} +(-1.25326 + 0.869233i) q^{44} +(1.22832 - 1.22832i) q^{45} +(1.70703 - 0.534040i) q^{46} +0.697764i q^{47} +(3.74641 + 1.40157i) q^{48} -0.364654 q^{49} +(1.30012 - 2.48396i) q^{50} +(4.46463 - 4.46463i) q^{51} +(12.4281 + 2.24864i) q^{52} -12.6987i q^{53} +(0.655808 - 1.25296i) q^{54} +(-0.936713 - 0.936713i) q^{55} +(-7.61662 + 0.950926i) q^{56} +(-1.62049 + 1.62049i) q^{57} +(0.359475 + 1.14904i) q^{58} +(-3.41073 - 3.41073i) q^{59} +(-0.618551 + 3.41870i) q^{60} +(3.27678 + 3.27678i) q^{61} +(3.60685 + 11.5291i) q^{62} +2.71379i q^{63} +(-7.75443 + 1.96692i) q^{64} +10.9697i q^{65} +(-0.955507 - 0.500118i) q^{66} -11.7656i q^{67} +(-2.24828 + 12.4261i) q^{68} +(0.894308 + 0.894308i) q^{69} +(-1.99055 - 6.36269i) q^{70} -7.52995i q^{71} +(0.350405 + 2.80664i) q^{72} -9.01634i q^{73} +(-3.74183 + 7.74588i) q^{74} +1.98247 q^{75} +(0.816042 - 4.51022i) q^{76} +2.06953 q^{77} +(2.66650 + 8.52334i) q^{78} +(-1.64275 + 1.64275i) q^{79} +(-2.88021 - 6.32336i) q^{80} +1.00000 q^{81} +(-2.22137 + 4.24407i) q^{82} -14.8303 q^{83} +(-3.09326 - 4.45986i) q^{84} -10.9680 q^{85} +(2.33464 + 7.46255i) q^{86} +(-0.601979 + 0.601979i) q^{87} +(2.14034 - 0.267219i) q^{88} +(4.83482 - 4.83482i) q^{89} +(-2.34458 + 0.733495i) q^{90} +(-12.1180 - 12.1180i) q^{91} +(-2.48907 - 0.450352i) q^{92} +(-6.04007 + 6.04007i) q^{93} +(0.457600 - 0.874272i) q^{94} +3.98096 q^{95} +(-3.77495 - 4.21304i) q^{96} +(6.89861 + 6.89861i) q^{97} +(0.456898 + 0.239143i) q^{98} -0.762598i 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.25296 − 0.655808i −0.885978 − 0.463727i
\(3\) − 1.00000i − 0.577350i
\(4\) 1.13983 + 1.64341i 0.569915 + 0.821703i
\(5\) −1.22832 + 1.22832i −0.549320 + 0.549320i −0.926244 − 0.376924i \(-0.876982\pi\)
0.376924 + 0.926244i \(0.376982\pi\)
\(6\) −0.655808 + 1.25296i −0.267733 + 0.511520i
\(7\) − 2.71379i − 1.02572i −0.858473 − 0.512858i \(-0.828587\pi\)
0.858473 − 0.512858i \(-0.171413\pi\)
\(8\) −0.350405 − 2.80664i −0.123887 − 0.992296i
\(9\) −1.00000 −0.333333
\(10\) 2.34458 − 0.733495i 0.741420 − 0.231951i
\(11\) 0.762598i 0.229932i 0.993369 + 0.114966i \(0.0366759\pi\)
−0.993369 + 0.114966i \(0.963324\pi\)
\(12\) 1.64341 − 1.13983i 0.474411 − 0.329041i
\(13\) 4.46535 − 4.46535i 1.23847 − 1.23847i 0.277838 − 0.960628i \(-0.410382\pi\)
0.960628 − 0.277838i \(-0.0896180\pi\)
\(14\) −1.77973 + 3.40028i −0.475652 + 0.908762i
\(15\) 1.22832 + 1.22832i 0.317150 + 0.317150i
\(16\) −1.40157 + 3.74641i −0.350393 + 0.936603i
\(17\) 4.46463 + 4.46463i 1.08283 + 1.08283i 0.996244 + 0.0865882i \(0.0275964\pi\)
0.0865882 + 0.996244i \(0.472404\pi\)
\(18\) 1.25296 + 0.655808i 0.295326 + 0.154576i
\(19\) −1.62049 − 1.62049i −0.371767 − 0.371767i 0.496354 − 0.868120i \(-0.334672\pi\)
−0.868120 + 0.496354i \(0.834672\pi\)
\(20\) −3.41870 − 0.618551i −0.764444 − 0.138312i
\(21\) −2.71379 −0.592197
\(22\) 0.500118 − 0.955507i 0.106626 − 0.203715i
\(23\) −0.894308 + 0.894308i −0.186476 + 0.186476i −0.794171 − 0.607695i \(-0.792094\pi\)
0.607695 + 0.794171i \(0.292094\pi\)
\(24\) −2.80664 + 0.350405i −0.572903 + 0.0715262i
\(25\) 1.98247i 0.396495i
\(26\) −8.52334 + 2.66650i −1.67156 + 0.522945i
\(27\) 1.00000i 0.192450i
\(28\) 4.45986 − 3.09326i 0.842834 − 0.584571i
\(29\) −0.601979 − 0.601979i −0.111785 − 0.111785i 0.649002 − 0.760787i \(-0.275187\pi\)
−0.760787 + 0.649002i \(0.775187\pi\)
\(30\) −0.733495 − 2.34458i −0.133917 − 0.428059i
\(31\) −6.04007 − 6.04007i −1.08483 − 1.08483i −0.996052 − 0.0887761i \(-0.971704\pi\)
−0.0887761 − 0.996052i \(-0.528296\pi\)
\(32\) 4.21304 − 3.77495i 0.744768 − 0.667323i
\(33\) 0.762598 0.132751
\(34\) −2.66607 − 8.52196i −0.457228 − 1.46150i
\(35\) 3.33339 + 3.33339i 0.563447 + 0.563447i
\(36\) −1.13983 − 1.64341i −0.189972 − 0.273901i
\(37\) −0.195721 − 6.07961i −0.0321763 − 0.999482i
\(38\) 0.967685 + 3.09315i 0.156979 + 0.501776i
\(39\) −4.46535 − 4.46535i −0.715029 − 0.715029i
\(40\) 3.87785 + 3.01703i 0.613142 + 0.477035i
\(41\) − 3.38722i − 0.528996i −0.964386 − 0.264498i \(-0.914794\pi\)
0.964386 − 0.264498i \(-0.0852062\pi\)
\(42\) 3.40028 + 1.77973i 0.524674 + 0.274618i
\(43\) −3.90961 − 3.90961i −0.596210 − 0.596210i 0.343092 − 0.939302i \(-0.388526\pi\)
−0.939302 + 0.343092i \(0.888526\pi\)
\(44\) −1.25326 + 0.869233i −0.188936 + 0.131042i
\(45\) 1.22832 − 1.22832i 0.183107 − 0.183107i
\(46\) 1.70703 − 0.534040i 0.251688 − 0.0787399i
\(47\) 0.697764i 0.101779i 0.998704 + 0.0508897i \(0.0162057\pi\)
−0.998704 + 0.0508897i \(0.983794\pi\)
\(48\) 3.74641 + 1.40157i 0.540748 + 0.202299i
\(49\) −0.364654 −0.0520934
\(50\) 1.30012 − 2.48396i 0.183865 − 0.351286i
\(51\) 4.46463 − 4.46463i 0.625174 − 0.625174i
\(52\) 12.4281 + 2.24864i 1.72347 + 0.311831i
\(53\) − 12.6987i − 1.74429i −0.489244 − 0.872147i \(-0.662727\pi\)
0.489244 − 0.872147i \(-0.337273\pi\)
\(54\) 0.655808 − 1.25296i 0.0892442 − 0.170507i
\(55\) −0.936713 − 0.936713i −0.126306 − 0.126306i
\(56\) −7.61662 + 0.950926i −1.01781 + 0.127073i
\(57\) −1.62049 + 1.62049i −0.214640 + 0.214640i
\(58\) 0.359475 + 1.14904i 0.0472013 + 0.150876i
\(59\) −3.41073 − 3.41073i −0.444039 − 0.444039i 0.449328 − 0.893367i \(-0.351663\pi\)
−0.893367 + 0.449328i \(0.851663\pi\)
\(60\) −0.618551 + 3.41870i −0.0798546 + 0.441352i
\(61\) 3.27678 + 3.27678i 0.419549 + 0.419549i 0.885048 − 0.465499i \(-0.154125\pi\)
−0.465499 + 0.885048i \(0.654125\pi\)
\(62\) 3.60685 + 11.5291i 0.458070 + 1.46420i
\(63\) 2.71379i 0.341905i
\(64\) −7.75443 + 1.96692i −0.969304 + 0.245865i
\(65\) 10.9697i 1.36063i
\(66\) −0.955507 − 0.500118i −0.117615 − 0.0615603i
\(67\) − 11.7656i − 1.43740i −0.695319 − 0.718701i \(-0.744737\pi\)
0.695319 − 0.718701i \(-0.255263\pi\)
\(68\) −2.24828 + 12.4261i −0.272644 + 1.50689i
\(69\) 0.894308 + 0.894308i 0.107662 + 0.107662i
\(70\) −1.99055 − 6.36269i −0.237916 − 0.760487i
\(71\) − 7.52995i − 0.893641i −0.894624 − 0.446821i \(-0.852556\pi\)
0.894624 − 0.446821i \(-0.147444\pi\)
\(72\) 0.350405 + 2.80664i 0.0412957 + 0.330765i
\(73\) − 9.01634i − 1.05528i −0.849467 − 0.527642i \(-0.823076\pi\)
0.849467 − 0.527642i \(-0.176924\pi\)
\(74\) −3.74183 + 7.74588i −0.434979 + 0.900441i
\(75\) 1.98247 0.228916
\(76\) 0.816042 − 4.51022i 0.0936064 − 0.517358i
\(77\) 2.06953 0.235845
\(78\) 2.66650 + 8.52334i 0.301922 + 0.965078i
\(79\) −1.64275 + 1.64275i −0.184824 + 0.184824i −0.793454 − 0.608630i \(-0.791719\pi\)
0.608630 + 0.793454i \(0.291719\pi\)
\(80\) −2.88021 − 6.32336i −0.322017 − 0.706973i
\(81\) 1.00000 0.111111
\(82\) −2.22137 + 4.24407i −0.245309 + 0.468679i
\(83\) −14.8303 −1.62783 −0.813917 − 0.580981i \(-0.802669\pi\)
−0.813917 + 0.580981i \(0.802669\pi\)
\(84\) −3.09326 − 4.45986i −0.337502 − 0.486611i
\(85\) −10.9680 −1.18964
\(86\) 2.33464 + 7.46255i 0.251751 + 0.804707i
\(87\) −0.601979 + 0.601979i −0.0645390 + 0.0645390i
\(88\) 2.14034 − 0.267219i 0.228161 − 0.0284856i
\(89\) 4.83482 − 4.83482i 0.512490 − 0.512490i −0.402799 − 0.915289i \(-0.631963\pi\)
0.915289 + 0.402799i \(0.131963\pi\)
\(90\) −2.34458 + 0.733495i −0.247140 + 0.0773172i
\(91\) −12.1180 − 12.1180i −1.27031 − 1.27031i
\(92\) −2.48907 − 0.450352i −0.259504 − 0.0469525i
\(93\) −6.04007 + 6.04007i −0.626326 + 0.626326i
\(94\) 0.457600 − 0.874272i 0.0471978 − 0.0901743i
\(95\) 3.98096 0.408438
\(96\) −3.77495 − 4.21304i −0.385279 − 0.429992i
\(97\) 6.89861 + 6.89861i 0.700448 + 0.700448i 0.964507 − 0.264059i \(-0.0850613\pi\)
−0.264059 + 0.964507i \(0.585061\pi\)
\(98\) 0.456898 + 0.239143i 0.0461536 + 0.0241571i
\(99\) − 0.762598i − 0.0766440i
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.43.7 ✓ 72
8.3 odd 2 inner 888.2.r.e.43.24 yes 72
37.31 odd 4 inner 888.2.r.e.475.24 yes 72
296.179 even 4 inner 888.2.r.e.475.7 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.r.e.43.7 ✓ 72 1.1 even 1 trivial
888.2.r.e.43.24 yes 72 8.3 odd 2 inner
888.2.r.e.475.7 yes 72 296.179 even 4 inner
888.2.r.e.475.24 yes 72 37.31 odd 4 inner