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Properties

Label 128.2.k.a.109.3
Level $128$
Weight $2$
Character 128.109
Analytic conductor $1.022$
Analytic rank $0$
Dimension $240$
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] = [128,2,Mod(5,128)] 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("128.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(128, base_ring=CyclotomicField(32)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 128.k (of order \(32\), degree \(16\), 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: \(1.02208514587\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(15\) over \(\Q(\zeta_{32})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label 109.3
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.3

$q$-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+(-1.29537 + 0.567464i) q^{2} +(-1.12781 - 0.602827i) q^{3} +(1.35597 - 1.47015i) q^{4} +(-1.29871 + 1.58248i) q^{5} +(1.80302 + 0.140892i) q^{6} +(-1.93875 - 1.29543i) q^{7} +(-0.922221 + 2.67386i) q^{8} +(-0.758155 - 1.13466i) q^{9} +(0.784306 - 2.78687i) q^{10} +(-4.58482 - 1.39079i) q^{11} +(-2.41552 + 0.840640i) q^{12} +(-1.77365 + 1.45559i) q^{13} +(3.24651 + 0.577892i) q^{14} +(2.41866 - 1.00184i) q^{15} +(-0.322701 - 3.98696i) q^{16} +(-0.698095 - 0.289160i) q^{17} +(1.62597 + 1.03958i) q^{18} +(0.355280 - 3.60722i) q^{19} +(0.565481 + 4.05509i) q^{20} +(1.40562 + 2.62973i) q^{21} +(6.72827 - 0.800135i) q^{22} +(-0.824744 - 0.164052i) q^{23} +(2.65196 - 2.45966i) q^{24} +(0.157851 + 0.793571i) q^{25} +(1.47153 - 2.89201i) q^{26} +(0.547088 + 5.55468i) q^{27} +(-4.53337 + 1.09370i) q^{28} +(2.64076 + 8.70543i) q^{29} +(-2.56455 + 2.67026i) q^{30} +(4.31573 - 4.31573i) q^{31} +(2.68048 + 4.98147i) q^{32} +(4.33241 + 4.33241i) q^{33} +(1.06838 - 0.0215743i) q^{34} +(4.56786 - 1.38565i) q^{35} +(-2.69616 - 0.423958i) q^{36} +(-3.24347 + 0.319454i) q^{37} +(1.58675 + 4.87429i) q^{38} +(2.87781 - 0.572431i) q^{39} +(-3.03363 - 4.93195i) q^{40} +(2.34453 - 11.7868i) q^{41} +(-3.31308 - 2.60884i) q^{42} +(-7.17325 + 3.83418i) q^{43} +(-8.26155 + 4.85452i) q^{44} +(2.78020 + 0.273825i) q^{45} +(1.16144 - 0.255505i) q^{46} +(-1.13781 + 2.74691i) q^{47} +(-2.03950 + 4.69107i) q^{48} +(-0.598174 - 1.44412i) q^{49} +(-0.654799 - 0.938393i) q^{50} +(0.613005 + 0.746949i) q^{51} +(-0.265062 + 4.58127i) q^{52} +(2.85064 - 9.39731i) q^{53} +(-3.86076 - 6.88491i) q^{54} +(8.15524 - 5.44916i) q^{55} +(5.25175 - 3.98927i) q^{56} +(-2.57522 + 3.85409i) q^{57} +(-8.36079 - 9.77822i) q^{58} +(-3.31394 - 2.71968i) q^{59} +(1.80676 - 4.91426i) q^{60} +(0.0672870 - 0.125885i) q^{61} +(-3.14144 + 8.03948i) q^{62} +3.18196i q^{63} +(-6.29902 - 4.93177i) q^{64} -4.69715i q^{65} +(-8.07056 - 3.15358i) q^{66} +(3.12347 - 5.84360i) q^{67} +(-1.37171 + 0.634214i) q^{68} +(0.831260 + 0.682197i) q^{69} +(-5.13077 + 4.38702i) q^{70} +(-6.30859 + 9.44148i) q^{71} +(3.73310 - 0.980791i) q^{72} +(-4.87067 + 3.25448i) q^{73} +(4.02022 - 2.25437i) q^{74} +(0.300360 - 0.990154i) q^{75} +(-4.82141 - 5.41359i) q^{76} +(7.08715 + 8.63572i) q^{77} +(-3.40299 + 2.37456i) q^{78} +(4.87747 + 11.7752i) q^{79} +(6.72838 + 4.66723i) q^{80} +(1.16482 - 2.81213i) q^{81} +(3.65153 + 16.5986i) q^{82} +(-13.3221 - 1.31211i) q^{83} +(5.77209 + 1.49936i) q^{84} +(1.36421 - 0.729186i) q^{85} +(7.11625 - 9.03725i) q^{86} +(2.26959 - 11.4100i) q^{87} +(7.94700 - 10.9765i) q^{88} +(-13.5456 + 2.69438i) q^{89} +(-3.75677 + 1.22296i) q^{90} +(5.32428 - 0.524395i) q^{91} +(-1.35951 + 0.990050i) q^{92} +(-7.46896 + 2.26568i) q^{93} +(-0.0848921 - 4.20393i) q^{94} +(5.24694 + 5.24694i) q^{95} +(-0.0201028 - 7.23402i) q^{96} +(4.07614 - 4.07614i) q^{97} +(1.59434 + 1.53123i) q^{98} +(1.89793 + 6.25664i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9} - 16 q^{10} - 16 q^{11} - 16 q^{12} - 16 q^{13} - 16 q^{14} - 16 q^{15} - 16 q^{16} - 16 q^{17} - 16 q^{18}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{23}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29537 + 0.567464i −0.915965 + 0.401258i
\(3\) −1.12781 − 0.602827i −0.651142 − 0.348043i 0.112529 − 0.993648i \(-0.464105\pi\)
−0.763671 + 0.645606i \(0.776605\pi\)
\(4\) 1.35597 − 1.47015i 0.677984 − 0.735077i
\(5\) −1.29871 + 1.58248i −0.580800 + 0.707706i −0.977665 − 0.210171i \(-0.932598\pi\)
0.396865 + 0.917877i \(0.370098\pi\)
\(6\) 1.80302 + 0.140892i 0.736078 + 0.0575190i
\(7\) −1.93875 − 1.29543i −0.732779 − 0.489627i 0.132334 − 0.991205i \(-0.457753\pi\)
−0.865112 + 0.501578i \(0.832753\pi\)
\(8\) −0.922221 + 2.67386i −0.326054 + 0.945351i
\(9\) −0.758155 − 1.13466i −0.252718 − 0.378220i
\(10\) 0.784306 − 2.78687i 0.248019 − 0.881285i
\(11\) −4.58482 − 1.39079i −1.38238 − 0.419339i −0.490430 − 0.871481i \(-0.663160\pi\)
−0.891946 + 0.452141i \(0.850660\pi\)
\(12\) −2.41552 + 0.840640i −0.697302 + 0.242672i
\(13\) −1.77365 + 1.45559i −0.491921 + 0.403709i −0.847383 − 0.530982i \(-0.821823\pi\)
0.355463 + 0.934690i \(0.384323\pi\)
\(14\) 3.24651 + 0.577892i 0.867666 + 0.154448i
\(15\) 2.41866 − 1.00184i 0.624495 − 0.258674i
\(16\) −0.322701 − 3.98696i −0.0806752 − 0.996740i
\(17\) −0.698095 − 0.289160i −0.169313 − 0.0701317i 0.296417 − 0.955059i \(-0.404208\pi\)
−0.465730 + 0.884927i \(0.654208\pi\)
\(18\) 1.62597 + 1.03958i 0.383245 + 0.245031i
\(19\) 0.355280 − 3.60722i 0.0815068 − 0.827553i −0.864333 − 0.502920i \(-0.832259\pi\)
0.945840 − 0.324633i \(-0.105241\pi\)
\(20\) 0.565481 + 4.05509i 0.126445 + 0.906746i
\(21\) 1.40562 + 2.62973i 0.306732 + 0.573855i
\(22\) 6.72827 − 0.800135i 1.43447 − 0.170589i
\(23\) −0.824744 − 0.164052i −0.171971 − 0.0342071i 0.108354 − 0.994112i \(-0.465442\pi\)
−0.280325 + 0.959905i \(0.590442\pi\)
\(24\) 2.65196 − 2.45966i 0.541330 − 0.502077i
\(25\) 0.157851 + 0.793571i 0.0315702 + 0.158714i
\(26\) 1.47153 − 2.89201i 0.288591 − 0.567170i
\(27\) 0.547088 + 5.55468i 0.105287 + 1.06900i
\(28\) −4.53337 + 1.09370i −0.856726 + 0.206689i
\(29\) 2.64076 + 8.70543i 0.490378 + 1.61656i 0.757458 + 0.652884i \(0.226441\pi\)
−0.267080 + 0.963674i \(0.586059\pi\)
\(30\) −2.56455 + 2.67026i −0.468220 + 0.487520i
\(31\) 4.31573 − 4.31573i 0.775127 − 0.775127i −0.203871 − 0.978998i \(-0.565352\pi\)
0.978998 + 0.203871i \(0.0653522\pi\)
\(32\) 2.68048 + 4.98147i 0.473846 + 0.880608i
\(33\) 4.33241 + 4.33241i 0.754175 + 0.754175i
\(34\) 1.06838 − 0.0215743i 0.183226 − 0.00369996i
\(35\) 4.56786 − 1.38565i 0.772110 − 0.234217i
\(36\) −2.69616 − 0.423958i −0.449359 − 0.0706596i
\(37\) −3.24347 + 0.319454i −0.533224 + 0.0525179i −0.361046 − 0.932548i \(-0.617580\pi\)
−0.172178 + 0.985066i \(0.555080\pi\)
\(38\) 1.58675 + 4.87429i 0.257405 + 0.790714i
\(39\) 2.87781 − 0.572431i 0.460818 − 0.0916624i
\(40\) −3.03363 − 4.93195i −0.479659 − 0.779810i
\(41\) 2.34453 − 11.7868i 0.366154 − 1.84078i −0.155775 − 0.987793i \(-0.549787\pi\)
0.521929 − 0.852989i \(-0.325213\pi\)
\(42\) −3.31308 − 2.60884i −0.511219 − 0.402552i
\(43\) −7.17325 + 3.83418i −1.09391 + 0.584707i −0.916745 − 0.399474i \(-0.869193\pi\)
−0.177166 + 0.984181i \(0.556693\pi\)
\(44\) −8.26155 + 4.85452i −1.24548 + 0.731847i
\(45\) 2.78020 + 0.273825i 0.414447 + 0.0408195i
\(46\) 1.16144 − 0.255505i 0.171245 − 0.0376722i
\(47\) −1.13781 + 2.74691i −0.165966 + 0.400678i −0.984880 − 0.173239i \(-0.944577\pi\)
0.818913 + 0.573917i \(0.194577\pi\)
\(48\) −2.03950 + 4.69107i −0.294377 + 0.677098i
\(49\) −0.598174 − 1.44412i −0.0854535 − 0.206303i
\(50\) −0.654799 − 0.938393i −0.0926025 − 0.132709i
\(51\) 0.613005 + 0.746949i 0.0858379 + 0.104594i
\(52\) −0.265062 + 4.58127i −0.0367575 + 0.635308i
\(53\) 2.85064 − 9.39731i 0.391566 − 1.29082i −0.510745 − 0.859732i \(-0.670630\pi\)
0.902310 − 0.431087i \(-0.141870\pi\)
\(54\) −3.86076 − 6.88491i −0.525383 − 0.936917i
\(55\) 8.15524 − 5.44916i 1.09965 − 0.734764i
\(56\) 5.25175 − 3.98927i 0.701795 − 0.533088i
\(57\) −2.57522 + 3.85409i −0.341096 + 0.510486i
\(58\) −8.36079 − 9.77822i −1.09783 − 1.28394i
\(59\) −3.31394 − 2.71968i −0.431439 − 0.354073i 0.393471 − 0.919337i \(-0.371274\pi\)
−0.824910 + 0.565264i \(0.808774\pi\)
\(60\) 1.80676 − 4.91426i 0.233252 − 0.634429i
\(61\) 0.0672870 − 0.125885i 0.00861521 − 0.0161179i −0.877577 − 0.479436i \(-0.840841\pi\)
0.886192 + 0.463318i \(0.153341\pi\)
\(62\) −3.14144 + 8.03948i −0.398963 + 1.02102i
\(63\) 3.18196i 0.400889i
\(64\) −6.29902 − 4.93177i −0.787377 − 0.616472i
\(65\) − 4.69715i − 0.582609i
\(66\) −8.07056 − 3.15358i −0.993417 − 0.388179i
\(67\) 3.12347 − 5.84360i 0.381593 − 0.713910i −0.615658 − 0.788014i \(-0.711110\pi\)
0.997251 + 0.0741040i \(0.0236097\pi\)
\(68\) −1.37171 + 0.634214i −0.166344 + 0.0769098i
\(69\) 0.831260 + 0.682197i 0.100072 + 0.0821269i
\(70\) −5.13077 + 4.38702i −0.613244 + 0.524350i
\(71\) −6.30859 + 9.44148i −0.748692 + 1.12050i 0.240034 + 0.970764i \(0.422841\pi\)
−0.988727 + 0.149733i \(0.952159\pi\)
\(72\) 3.73310 − 0.980791i 0.439950 − 0.115587i
\(73\) −4.87067 + 3.25448i −0.570069 + 0.380908i −0.806940 − 0.590633i \(-0.798878\pi\)
0.236871 + 0.971541i \(0.423878\pi\)
\(74\) 4.02022 − 2.25437i 0.467341 − 0.262065i
\(75\) 0.300360 − 0.990154i 0.0346826 − 0.114333i
\(76\) −4.82141 − 5.41359i −0.553054 − 0.620981i
\(77\) 7.08715 + 8.63572i 0.807656 + 0.984132i
\(78\) −3.40299 + 2.37456i −0.385313 + 0.268866i
\(79\) 4.87747 + 11.7752i 0.548758 + 1.32482i 0.918403 + 0.395645i \(0.129479\pi\)
−0.369646 + 0.929173i \(0.620521\pi\)
\(80\) 6.72838 + 4.66723i 0.752256 + 0.521812i
\(81\) 1.16482 − 2.81213i 0.129425 − 0.312459i
\(82\) 3.65153 + 16.5986i 0.403244 + 1.83301i
\(83\) −13.3221 − 1.31211i −1.46229 − 0.144023i −0.664649 − 0.747155i \(-0.731419\pi\)
−0.797641 + 0.603132i \(0.793919\pi\)
\(84\) 5.77209 + 1.49936i 0.629786 + 0.163593i
\(85\) 1.36421 − 0.729186i 0.147970 − 0.0790914i
\(86\) 7.11625 − 9.03725i 0.767365 − 0.974511i
\(87\) 2.26959 − 11.4100i 0.243326 − 1.22328i
\(88\) 7.94700 − 10.9765i 0.847153 − 1.17010i
\(89\) −13.5456 + 2.69438i −1.43583 + 0.285604i −0.850834 − 0.525435i \(-0.823903\pi\)
−0.584993 + 0.811038i \(0.698903\pi\)
\(90\) −3.75677 + 1.22296i −0.395998 + 0.128911i
\(91\) 5.32428 − 0.524395i 0.558136 − 0.0549716i
\(92\) −1.35951 + 0.990050i −0.141738 + 0.103220i
\(93\) −7.46896 + 2.26568i −0.774495 + 0.234940i
\(94\) −0.0848921 − 4.20393i −0.00875595 − 0.433603i
\(95\) 5.24694 + 5.24694i 0.538325 + 0.538325i
\(96\) −0.0201028 − 7.23402i −0.00205173 − 0.738319i
\(97\) 4.07614 − 4.07614i 0.413870 − 0.413870i −0.469214 − 0.883084i \(-0.655463\pi\)
0.883084 + 0.469214i \(0.155463\pi\)
\(98\) 1.59434 + 1.53123i 0.161053 + 0.154677i
\(99\) 1.89793 + 6.25664i 0.190749 + 0.628816i
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 128.2.k.a.109.3 yes 240
4.3 odd 2 512.2.k.a.273.10 240
128.27 odd 32 512.2.k.a.497.10 240
128.101 even 32 inner 128.2.k.a.101.3 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.3 ✓ 240 128.101 even 32 inner
128.2.k.a.109.3 yes 240 1.1 even 1 trivial
512.2.k.a.273.10 240 4.3 odd 2
512.2.k.a.497.10 240 128.27 odd 32