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Properties

Label 128.2.k.a.5.11
Level $128$
Weight $2$
Character 128.5
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 5.11
Character \(\chi\) \(=\) 128.5
Dual form 128.2.k.a.77.11

$q$-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+(0.674225 - 1.24315i) q^{2} +(0.865940 + 0.262680i) q^{3} +(-1.09084 - 1.67632i) q^{4} +(1.24951 + 0.123066i) q^{5} +(0.910389 - 0.899388i) q^{6} +(0.832309 + 1.24564i) q^{7} +(-2.81939 + 0.225861i) q^{8} +(-1.81356 - 1.21178i) q^{9} +(0.995438 - 1.47035i) q^{10} +(-0.865009 + 1.61832i) q^{11} +(-0.504267 - 1.73814i) q^{12} +(-0.0973539 - 0.988451i) q^{13} +(2.10968 - 0.194844i) q^{14} +(1.04967 + 0.434788i) q^{15} +(-1.62013 + 3.65721i) q^{16} +(0.0244568 - 0.0101304i) q^{17} +(-2.72917 + 1.43751i) q^{18} +(-0.837843 + 1.02091i) q^{19} +(-1.15672 - 2.22882i) q^{20} +(0.393525 + 1.29728i) q^{21} +(1.42860 + 2.16645i) q^{22} +(1.65899 + 8.34031i) q^{23} +(-2.50076 - 0.545017i) q^{24} +(-3.35780 - 0.667909i) q^{25} +(-1.29443 - 0.545413i) q^{26} +(-2.97432 - 3.62422i) q^{27} +(1.18018 - 2.75401i) q^{28} +(4.72421 - 2.52514i) q^{29} +(1.24822 - 1.01175i) q^{30} +(0.299761 + 0.299761i) q^{31} +(3.45413 + 4.47984i) q^{32} +(-1.17415 + 1.17415i) q^{33} +(0.00389586 - 0.0372337i) q^{34} +(0.886680 + 1.65886i) q^{35} +(-0.0530320 + 4.36197i) q^{36} +(5.59401 - 4.59089i) q^{37} +(0.704255 + 1.72989i) q^{38} +(0.175344 - 0.881512i) q^{39} +(-3.55065 - 0.0647561i) q^{40} +(-8.72362 + 1.73524i) q^{41} +(1.87804 + 0.385447i) q^{42} +(-1.65222 + 0.501196i) q^{43} +(3.65641 - 0.315294i) q^{44} +(-2.11692 - 1.73731i) q^{45} +(11.4868 + 3.56087i) q^{46} +(1.47462 + 3.56006i) q^{47} +(-2.36361 + 2.74135i) q^{48} +(1.81991 - 4.39365i) q^{49} +(-3.09422 + 3.72393i) q^{50} +(0.0238392 - 0.00234796i) q^{51} +(-1.55077 + 1.24144i) q^{52} +(-3.53843 - 1.89133i) q^{53} +(-6.51080 + 1.25399i) q^{54} +(-1.27999 + 1.91565i) q^{55} +(-2.62795 - 3.32396i) q^{56} +(-0.993695 + 0.663966i) q^{57} +(0.0460485 - 7.57541i) q^{58} +(1.13686 - 11.5428i) q^{59} +(-0.416180 - 2.23388i) q^{60} +(2.71284 - 8.94304i) q^{61} +(0.574754 - 0.170541i) q^{62} -3.26761i q^{63} +(7.89797 - 1.27358i) q^{64} -1.24706i q^{65} +(0.668001 + 2.25128i) q^{66} +(2.33581 - 7.70012i) q^{67} +(-0.0436603 - 0.0299470i) q^{68} +(-0.754247 + 7.65799i) q^{69} +(2.66004 + 0.0161695i) q^{70} +(-9.86712 + 6.59300i) q^{71} +(5.38683 + 3.00688i) q^{72} +(3.02267 - 4.52375i) q^{73} +(-1.93554 - 10.0495i) q^{74} +(-2.73221 - 1.46040i) q^{75} +(2.62534 + 0.290840i) q^{76} +(-2.73579 + 0.269452i) q^{77} +(-0.977631 - 0.812316i) q^{78} +(4.78445 - 11.5507i) q^{79} +(-2.47444 + 4.37033i) q^{80} +(0.880494 + 2.12570i) q^{81} +(-3.72452 + 12.0147i) q^{82} +(-2.11424 - 1.73511i) q^{83} +(1.74539 - 2.07480i) q^{84} +(0.0318057 - 0.00964816i) q^{85} +(-0.490907 + 2.39188i) q^{86} +(4.75419 - 0.945667i) q^{87} +(2.07329 - 4.75805i) q^{88} +(-3.59212 + 18.0588i) q^{89} +(-3.58702 + 1.46031i) q^{90} +(1.15022 - 0.943964i) q^{91} +(12.1714 - 11.8790i) q^{92} +(0.180834 + 0.338316i) q^{93} +(5.41991 + 0.567100i) q^{94} +(-1.17253 + 1.17253i) q^{95} +(1.81431 + 4.78661i) q^{96} +(1.86334 + 1.86334i) q^{97} +(-4.23493 - 5.22472i) q^{98} +(3.52979 - 1.88671i) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.674225 − 1.24315i 0.476749 − 0.879040i
\(3\) 0.865940 + 0.262680i 0.499951 + 0.151658i 0.530181 − 0.847885i \(-0.322124\pi\)
−0.0302301 + 0.999543i \(0.509624\pi\)
\(4\) −1.09084 − 1.67632i −0.545421 − 0.838162i
\(5\) 1.24951 + 0.123066i 0.558797 + 0.0550367i 0.373473 − 0.927641i \(-0.378167\pi\)
0.185323 + 0.982678i \(0.440667\pi\)
\(6\) 0.910389 − 0.899388i 0.371665 − 0.367174i
\(7\) 0.832309 + 1.24564i 0.314583 + 0.470807i 0.954742 − 0.297436i \(-0.0961314\pi\)
−0.640159 + 0.768243i \(0.721131\pi\)
\(8\) −2.81939 + 0.225861i −0.996807 + 0.0798539i
\(9\) −1.81356 − 1.21178i −0.604519 − 0.403927i
\(10\) 0.995438 − 1.47035i 0.314785 − 0.464966i
\(11\) −0.865009 + 1.61832i −0.260810 + 0.487941i −0.977909 − 0.209030i \(-0.932969\pi\)
0.717099 + 0.696971i \(0.245469\pi\)
\(12\) −0.504267 − 1.73814i −0.145569 − 0.501758i
\(13\) −0.0973539 − 0.988451i −0.0270011 − 0.274147i −0.999200 − 0.0399906i \(-0.987267\pi\)
0.972199 − 0.234156i \(-0.0752328\pi\)
\(14\) 2.10968 − 0.194844i 0.563835 − 0.0520744i
\(15\) 1.04967 + 0.434788i 0.271024 + 0.112262i
\(16\) −1.62013 + 3.65721i −0.405032 + 0.914303i
\(17\) 0.0244568 − 0.0101304i 0.00593166 − 0.00245697i −0.379716 − 0.925103i \(-0.623978\pi\)
0.385647 + 0.922646i \(0.373978\pi\)
\(18\) −2.72917 + 1.43751i −0.643271 + 0.338825i
\(19\) −0.837843 + 1.02091i −0.192214 + 0.234214i −0.860196 − 0.509964i \(-0.829659\pi\)
0.667981 + 0.744178i \(0.267159\pi\)
\(20\) −1.15672 − 2.22882i −0.258650 − 0.498380i
\(21\) 0.393525 + 1.29728i 0.0858742 + 0.283089i
\(22\) 1.42860 + 2.16645i 0.304579 + 0.461888i
\(23\) 1.65899 + 8.34031i 0.345924 + 1.73907i 0.626662 + 0.779291i \(0.284420\pi\)
−0.280739 + 0.959784i \(0.590580\pi\)
\(24\) −2.50076 − 0.545017i −0.510465 − 0.111251i
\(25\) −3.35780 − 0.667909i −0.671561 − 0.133582i
\(26\) −1.29443 − 0.545413i −0.253859 − 0.106964i
\(27\) −2.97432 − 3.62422i −0.572408 − 0.697481i
\(28\) 1.18018 − 2.75401i 0.223032 − 0.520460i
\(29\) 4.72421 − 2.52514i 0.877264 − 0.468907i 0.0296741 − 0.999560i \(-0.490553\pi\)
0.847589 + 0.530652i \(0.178053\pi\)
\(30\) 1.24822 − 1.01175i 0.227893 − 0.184720i
\(31\) 0.299761 + 0.299761i 0.0538386 + 0.0538386i 0.733514 − 0.679675i \(-0.237879\pi\)
−0.679675 + 0.733514i \(0.737879\pi\)
\(32\) 3.45413 + 4.47984i 0.610610 + 0.791932i
\(33\) −1.17415 + 1.17415i −0.204393 + 0.204393i
\(34\) 0.00389586 − 0.0372337i 0.000668134 − 0.00638552i
\(35\) 0.886680 + 1.65886i 0.149876 + 0.280399i
\(36\) −0.0530320 + 4.36197i −0.00883867 + 0.726995i
\(37\) 5.59401 − 4.59089i 0.919650 − 0.754737i −0.0501236 − 0.998743i \(-0.515962\pi\)
0.969774 + 0.244006i \(0.0784615\pi\)
\(38\) 0.704255 + 1.72989i 0.114245 + 0.280625i
\(39\) 0.175344 − 0.881512i 0.0280775 − 0.141155i
\(40\) −3.55065 − 0.0647561i −0.561407 − 0.0102388i
\(41\) −8.72362 + 1.73524i −1.36240 + 0.270998i −0.821558 − 0.570125i \(-0.806895\pi\)
−0.540843 + 0.841124i \(0.681895\pi\)
\(42\) 1.87804 + 0.385447i 0.289787 + 0.0594757i
\(43\) −1.65222 + 0.501196i −0.251962 + 0.0764317i −0.413739 − 0.910396i \(-0.635777\pi\)
0.161777 + 0.986827i \(0.448277\pi\)
\(44\) 3.65641 − 0.315294i 0.551225 − 0.0475323i
\(45\) −2.11692 − 1.73731i −0.315572 − 0.258984i
\(46\) 11.4868 + 3.56087i 1.69363 + 0.525022i
\(47\) 1.47462 + 3.56006i 0.215096 + 0.519288i 0.994192 − 0.107616i \(-0.0343218\pi\)
−0.779096 + 0.626904i \(0.784322\pi\)
\(48\) −2.36361 + 2.74135i −0.341158 + 0.395680i
\(49\) 1.81991 − 4.39365i 0.259987 − 0.627664i
\(50\) −3.09422 + 3.72393i −0.437589 + 0.526643i
\(51\) 0.0238392 − 0.00234796i 0.00333816 − 0.000328780i
\(52\) −1.55077 + 1.24144i −0.215053 + 0.172157i
\(53\) −3.53843 − 1.89133i −0.486041 − 0.259794i 0.210126 − 0.977674i \(-0.432613\pi\)
−0.696167 + 0.717880i \(0.745113\pi\)
\(54\) −6.51080 + 1.25399i −0.886008 + 0.170646i
\(55\) −1.27999 + 1.91565i −0.172594 + 0.258306i
\(56\) −2.62795 − 3.32396i −0.351174 − 0.444183i
\(57\) −0.993695 + 0.663966i −0.131618 + 0.0879444i
\(58\) 0.0460485 − 7.57541i 0.00604646 − 0.994700i
\(59\) 1.13686 − 11.5428i 0.148007 − 1.50274i −0.578558 − 0.815641i \(-0.696384\pi\)
0.726565 − 0.687098i \(-0.241116\pi\)
\(60\) −0.416180 − 2.23388i −0.0537286 − 0.288392i
\(61\) 2.71284 − 8.94304i 0.347344 − 1.14504i −0.593111 − 0.805121i \(-0.702100\pi\)
0.940455 − 0.339919i \(-0.110400\pi\)
\(62\) 0.574754 − 0.170541i 0.0729938 − 0.0216588i
\(63\) − 3.26761i − 0.411680i
\(64\) 7.89797 − 1.27358i 0.987247 − 0.159198i
\(65\) − 1.24706i − 0.154678i
\(66\) 0.668001 + 2.25128i 0.0822252 + 0.277113i
\(67\) 2.33581 − 7.70012i 0.285364 − 0.940720i −0.690682 − 0.723158i \(-0.742690\pi\)
0.976047 − 0.217561i \(-0.0698103\pi\)
\(68\) −0.0436603 − 0.0299470i −0.00529459 − 0.00363161i
\(69\) −0.754247 + 7.65799i −0.0908006 + 0.921914i
\(70\) 2.66004 + 0.0161695i 0.317935 + 0.00193262i
\(71\) −9.86712 + 6.59300i −1.17101 + 0.782445i −0.979972 − 0.199137i \(-0.936186\pi\)
−0.191040 + 0.981582i \(0.561186\pi\)
\(72\) 5.38683 + 3.00688i 0.634844 + 0.354364i
\(73\) 3.02267 − 4.52375i 0.353777 − 0.529465i −0.611311 − 0.791390i \(-0.709358\pi\)
0.965088 + 0.261926i \(0.0843575\pi\)
\(74\) −1.93554 − 10.0495i −0.225002 − 1.16823i
\(75\) −2.73221 − 1.46040i −0.315489 − 0.168632i
\(76\) 2.62534 + 0.290840i 0.301147 + 0.0333616i
\(77\) −2.73579 + 0.269452i −0.311772 + 0.0307069i
\(78\) −0.977631 − 0.812316i −0.110695 − 0.0919767i
\(79\) 4.78445 − 11.5507i 0.538293 − 1.29955i −0.387621 − 0.921819i \(-0.626703\pi\)
0.925914 − 0.377735i \(-0.123297\pi\)
\(80\) −2.47444 + 4.37033i −0.276650 + 0.488618i
\(81\) 0.880494 + 2.12570i 0.0978327 + 0.236189i
\(82\) −3.72452 + 12.0147i −0.411305 + 1.32680i
\(83\) −2.11424 − 1.73511i −0.232068 − 0.190453i 0.511171 − 0.859479i \(-0.329212\pi\)
−0.743240 + 0.669025i \(0.766712\pi\)
\(84\) 1.74539 − 2.07480i 0.190437 − 0.226380i
\(85\) 0.0318057 − 0.00964816i 0.00344981 − 0.00104649i
\(86\) −0.490907 + 2.39188i −0.0529359 + 0.257923i
\(87\) 4.75419 − 0.945667i 0.509702 − 0.101386i
\(88\) 2.07329 − 4.75805i 0.221013 − 0.507210i
\(89\) −3.59212 + 18.0588i −0.380763 + 1.91423i 0.0237989 + 0.999717i \(0.492424\pi\)
−0.404562 + 0.914510i \(0.632576\pi\)
\(90\) −3.58702 + 1.46031i −0.378106 + 0.153931i
\(91\) 1.15022 − 0.943964i 0.120576 − 0.0989543i
\(92\) 12.1714 − 11.8790i 1.26895 − 1.23847i
\(93\) 0.180834 + 0.338316i 0.0187516 + 0.0350818i
\(94\) 5.41991 + 0.567100i 0.559021 + 0.0584919i
\(95\) −1.17253 + 1.17253i −0.120299 + 0.120299i
\(96\) 1.81431 + 4.78661i 0.185172 + 0.488531i
\(97\) 1.86334 + 1.86334i 0.189193 + 0.189193i 0.795347 − 0.606154i \(-0.207288\pi\)
−0.606154 + 0.795347i \(0.707288\pi\)
\(98\) −4.23493 − 5.22472i −0.427793 − 0.527777i
\(99\) 3.52979 − 1.88671i 0.354757 − 0.189622i
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.5.11 ✓ 240
4.3 odd 2 512.2.k.a.113.6 240
128.51 odd 32 512.2.k.a.145.6 240
128.77 even 32 inner 128.2.k.a.77.11 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.5.11 ✓ 240 1.1 even 1 trivial
128.2.k.a.77.11 yes 240 128.77 even 32 inner
512.2.k.a.113.6 240 4.3 odd 2
512.2.k.a.145.6 240 128.51 odd 32