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Properties

Label 128.2.k.a.109.12
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 109.12
Character \(\chi\) \(=\) 128.109
Dual form 128.2.k.a.101.12

$q$-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+(0.873141 - 1.11249i) q^{2} +(-1.98130 - 1.05903i) q^{3} +(-0.475248 - 1.94271i) q^{4} +(0.218926 - 0.266762i) q^{5} +(-2.90810 + 1.27949i) q^{6} +(-0.368096 - 0.245954i) q^{7} +(-2.57620 - 1.16756i) q^{8} +(1.13730 + 1.70208i) q^{9} +(-0.105615 - 0.476472i) q^{10} +(-0.341220 - 0.103508i) q^{11} +(-1.11578 + 4.35240i) q^{12} +(2.88278 - 2.36584i) q^{13} +(-0.595020 + 0.194749i) q^{14} +(-0.716264 + 0.296686i) q^{15} +(-3.54828 + 1.84654i) q^{16} +(6.12855 + 2.53853i) q^{17} +(2.88656 + 0.220934i) q^{18} +(0.500227 - 5.07889i) q^{19} +(-0.622286 - 0.298532i) q^{20} +(0.468836 + 0.877130i) q^{21} +(-0.413085 + 0.289225i) q^{22} +(4.80482 + 0.955738i) q^{23} +(3.86775 + 5.04154i) q^{24} +(0.952218 + 4.78712i) q^{25} +(-0.114884 - 5.27277i) q^{26} +(0.209836 + 2.13050i) q^{27} +(-0.302881 + 0.831994i) q^{28} +(-1.12675 - 3.71441i) q^{29} +(-0.295341 + 1.05588i) q^{30} +(-1.23628 + 1.23628i) q^{31} +(-1.04390 + 5.55970i) q^{32} +(0.566441 + 0.566441i) q^{33} +(8.17517 - 4.60143i) q^{34} +(-0.146197 + 0.0443482i) q^{35} +(2.76616 - 3.01835i) q^{36} +(-10.0376 + 0.988614i) q^{37} +(-5.21342 - 4.99108i) q^{38} +(-8.21714 + 1.63449i) q^{39} +(-0.875456 + 0.431623i) q^{40} +(0.432861 - 2.17614i) q^{41} +(1.38516 + 0.244286i) q^{42} +(4.05046 - 2.16502i) q^{43} +(-0.0389223 + 0.712086i) q^{44} +(0.703033 + 0.0692427i) q^{45} +(5.25853 - 4.51080i) q^{46} +(-1.49454 + 3.60814i) q^{47} +(8.98573 + 0.0991688i) q^{48} +(-2.60378 - 6.28609i) q^{49} +(6.15703 + 3.12051i) q^{50} +(-9.45412 - 11.5199i) q^{51} +(-5.96619 - 4.47607i) q^{52} +(-2.98330 + 9.83462i) q^{53} +(2.55337 + 1.62679i) q^{54} +(-0.102314 + 0.0683639i) q^{55} +(0.661123 + 1.06340i) q^{56} +(-6.36977 + 9.53304i) q^{57} +(-5.11604 - 1.98971i) q^{58} +(9.60613 + 7.88354i) q^{59} +(0.916780 + 1.25050i) q^{60} +(0.312866 - 0.585330i) q^{61} +(0.295896 + 2.45479i) q^{62} -0.906251i q^{63} +(5.27362 + 6.01573i) q^{64} -1.28696i q^{65} +(1.12474 - 0.135574i) q^{66} +(-1.69431 + 3.16984i) q^{67} +(2.01906 - 13.1125i) q^{68} +(-8.50763 - 6.98203i) q^{69} +(-0.0783135 + 0.201364i) q^{70} +(-7.84873 + 11.7465i) q^{71} +(-0.942620 - 5.71276i) q^{72} +(8.50764 - 5.68462i) q^{73} +(-7.66439 + 12.0298i) q^{74} +(3.18306 - 10.4931i) q^{75} +(-10.1046 + 1.44193i) q^{76} +(0.100144 + 0.122025i) q^{77} +(-5.35638 + 10.5686i) q^{78} +(-5.13428 - 12.3952i) q^{79} +(-0.284222 + 1.35080i) q^{80} +(4.19065 - 10.1171i) q^{81} +(-2.04298 - 2.38163i) q^{82} +(7.66719 + 0.755152i) q^{83} +(1.48120 - 1.32767i) q^{84} +(2.01888 - 1.07911i) q^{85} +(1.12808 - 6.39645i) q^{86} +(-1.70122 + 8.55262i) q^{87} +(0.758200 + 0.665052i) q^{88} +(1.30486 - 0.259552i) q^{89} +(0.690879 - 0.721656i) q^{90} +(-1.64303 + 0.161824i) q^{91} +(-0.426755 - 9.78861i) q^{92} +(3.75869 - 1.14019i) q^{93} +(2.70906 + 4.81307i) q^{94} +(-1.24534 - 1.24534i) q^{95} +(7.95614 - 9.90991i) q^{96} +(1.24876 - 1.24876i) q^{97} +(-9.26665 - 2.59197i) q^{98} +(-0.211889 - 0.698504i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.873141 − 1.11249i 0.617404 − 0.786646i
\(3\) −1.98130 − 1.05903i −1.14390 − 0.611429i −0.213129 − 0.977024i \(-0.568365\pi\)
−0.930774 + 0.365595i \(0.880865\pi\)
\(4\) −0.475248 − 1.94271i −0.237624 − 0.971357i
\(5\) 0.218926 − 0.266762i 0.0979065 − 0.119299i −0.721752 − 0.692151i \(-0.756663\pi\)
0.819659 + 0.572852i \(0.194163\pi\)
\(6\) −2.90810 + 1.27949i −1.18723 + 0.522348i
\(7\) −0.368096 − 0.245954i −0.139127 − 0.0929618i 0.484059 − 0.875035i \(-0.339162\pi\)
−0.623186 + 0.782074i \(0.714162\pi\)
\(8\) −2.57620 − 1.16756i −0.910824 − 0.412794i
\(9\) 1.13730 + 1.70208i 0.379098 + 0.567361i
\(10\) −0.105615 − 0.476472i −0.0333985 − 0.150674i
\(11\) −0.341220 − 0.103508i −0.102882 − 0.0312088i 0.238424 − 0.971161i \(-0.423369\pi\)
−0.341306 + 0.939952i \(0.610869\pi\)
\(12\) −1.11578 + 4.35240i −0.322097 + 1.25643i
\(13\) 2.88278 − 2.36584i 0.799541 − 0.656166i −0.143200 − 0.989694i \(-0.545739\pi\)
0.942740 + 0.333528i \(0.108239\pi\)
\(14\) −0.595020 + 0.194749i −0.159026 + 0.0520488i
\(15\) −0.716264 + 0.296686i −0.184939 + 0.0766041i
\(16\) −3.54828 + 1.84654i −0.887070 + 0.461636i
\(17\) 6.12855 + 2.53853i 1.48639 + 0.615684i 0.970528 − 0.240987i \(-0.0774711\pi\)
0.515864 + 0.856671i \(0.327471\pi\)
\(18\) 2.88656 + 0.220934i 0.680369 + 0.0520748i
\(19\) 0.500227 − 5.07889i 0.114760 − 1.16518i −0.749456 − 0.662055i \(-0.769685\pi\)
0.864215 − 0.503122i \(-0.167815\pi\)
\(20\) −0.622286 − 0.298532i −0.139147 − 0.0667538i
\(21\) 0.468836 + 0.877130i 0.102308 + 0.191406i
\(22\) −0.413085 + 0.289225i −0.0880700 + 0.0616631i
\(23\) 4.80482 + 0.955738i 1.00187 + 0.199285i 0.668662 − 0.743567i \(-0.266867\pi\)
0.333212 + 0.942852i \(0.391867\pi\)
\(24\) 3.86775 + 5.04154i 0.789500 + 1.02910i
\(25\) 0.952218 + 4.78712i 0.190444 + 0.957425i
\(26\) −0.114884 − 5.27277i −0.0225306 − 1.03408i
\(27\) 0.209836 + 2.13050i 0.0403829 + 0.410015i
\(28\) −0.302881 + 0.831994i −0.0572391 + 0.157232i
\(29\) −1.12675 − 3.71441i −0.209233 − 0.689749i −0.997251 − 0.0740967i \(-0.976393\pi\)
0.788018 − 0.615652i \(-0.211107\pi\)
\(30\) −0.295341 + 1.05588i −0.0539216 + 0.192777i
\(31\) −1.23628 + 1.23628i −0.222042 + 0.222042i −0.809358 − 0.587316i \(-0.800185\pi\)
0.587316 + 0.809358i \(0.300185\pi\)
\(32\) −1.04390 + 5.55970i −0.184537 + 0.982826i
\(33\) 0.566441 + 0.566441i 0.0986048 + 0.0986048i
\(34\) 8.17517 − 4.60143i 1.40203 − 0.789139i
\(35\) −0.146197 + 0.0443482i −0.0247117 + 0.00749622i
\(36\) 2.76616 − 3.01835i 0.461027 − 0.503059i
\(37\) −10.0376 + 0.988614i −1.65017 + 0.162527i −0.879888 − 0.475182i \(-0.842382\pi\)
−0.770277 + 0.637709i \(0.779882\pi\)
\(38\) −5.21342 − 4.99108i −0.845728 − 0.809660i
\(39\) −8.21714 + 1.63449i −1.31580 + 0.261728i
\(40\) −0.875456 + 0.431623i −0.138422 + 0.0682456i
\(41\) 0.432861 − 2.17614i 0.0676016 − 0.339856i −0.932152 − 0.362068i \(-0.882071\pi\)
0.999753 + 0.0222118i \(0.00707081\pi\)
\(42\) 1.38516 + 0.244286i 0.213734 + 0.0376941i
\(43\) 4.05046 − 2.16502i 0.617690 − 0.330162i −0.132702 − 0.991156i \(-0.542365\pi\)
0.750391 + 0.660994i \(0.229865\pi\)
\(44\) −0.0389223 + 0.712086i −0.00586776 + 0.107351i
\(45\) 0.703033 + 0.0692427i 0.104802 + 0.0103221i
\(46\) 5.25853 − 4.51080i 0.775328 − 0.665081i
\(47\) −1.49454 + 3.60814i −0.218001 + 0.526302i −0.994611 − 0.103681i \(-0.966938\pi\)
0.776609 + 0.629982i \(0.216938\pi\)
\(48\) 8.98573 + 0.0991688i 1.29698 + 0.0143138i
\(49\) −2.60378 − 6.28609i −0.371969 − 0.898013i
\(50\) 6.15703 + 3.12051i 0.870735 + 0.441306i
\(51\) −9.45412 − 11.5199i −1.32384 − 1.61311i
\(52\) −5.96619 − 4.47607i −0.827362 − 0.620719i
\(53\) −2.98330 + 9.83462i −0.409788 + 1.35089i 0.472977 + 0.881075i \(0.343179\pi\)
−0.882765 + 0.469815i \(0.844321\pi\)
\(54\) 2.55337 + 1.62679i 0.347469 + 0.221378i
\(55\) −0.102314 + 0.0683639i −0.0137960 + 0.00921819i
\(56\) 0.661123 + 1.06340i 0.0883463 + 0.142103i
\(57\) −6.36977 + 9.53304i −0.843697 + 1.26268i
\(58\) −5.11604 − 1.98971i −0.671770 − 0.261262i
\(59\) 9.60613 + 7.88354i 1.25061 + 1.02635i 0.998275 + 0.0587103i \(0.0186988\pi\)
0.252336 + 0.967640i \(0.418801\pi\)
\(60\) 0.916780 + 1.25050i 0.118356 + 0.161439i
\(61\) 0.312866 − 0.585330i 0.0400583 − 0.0749439i −0.861096 − 0.508443i \(-0.830221\pi\)
0.901154 + 0.433499i \(0.142721\pi\)
\(62\) 0.295896 + 2.45479i 0.0375788 + 0.311758i
\(63\) − 0.906251i − 0.114177i
\(64\) 5.27362 + 6.01573i 0.659202 + 0.751966i
\(65\) − 1.28696i − 0.159628i
\(66\) 1.12474 − 0.135574i 0.138446 − 0.0166880i
\(67\) −1.69431 + 3.16984i −0.206993 + 0.387257i −0.963899 − 0.266266i \(-0.914210\pi\)
0.756906 + 0.653523i \(0.226710\pi\)
\(68\) 2.01906 − 13.1125i 0.244846 − 1.59012i
\(69\) −8.50763 − 6.98203i −1.02420 − 0.840538i
\(70\) −0.0783135 + 0.201364i −0.00936025 + 0.0240676i
\(71\) −7.84873 + 11.7465i −0.931473 + 1.39405i −0.0124161 + 0.999923i \(0.503952\pi\)
−0.919057 + 0.394125i \(0.871048\pi\)
\(72\) −0.942620 − 5.71276i −0.111089 − 0.673256i
\(73\) 8.50764 − 5.68462i 0.995744 − 0.665335i 0.0529105 − 0.998599i \(-0.483150\pi\)
0.942833 + 0.333264i \(0.108150\pi\)
\(74\) −7.66439 + 12.0298i −0.890968 + 1.39844i
\(75\) 3.18306 − 10.4931i 0.367548 − 1.21164i
\(76\) −10.1046 + 1.44193i −1.15907 + 0.165401i
\(77\) 0.100144 + 0.122025i 0.0114124 + 0.0139061i
\(78\) −5.35638 + 10.5686i −0.606491 + 1.19666i
\(79\) −5.13428 − 12.3952i −0.577651 − 1.39457i −0.894915 − 0.446236i \(-0.852764\pi\)
0.317264 − 0.948337i \(-0.397236\pi\)
\(80\) −0.284222 + 1.35080i −0.0317770 + 0.151024i
\(81\) 4.19065 − 10.1171i 0.465628 − 1.12413i
\(82\) −2.04298 − 2.38163i −0.225609 − 0.263007i
\(83\) 7.66719 + 0.755152i 0.841584 + 0.0828887i 0.509616 − 0.860402i \(-0.329788\pi\)
0.331968 + 0.943291i \(0.392288\pi\)
\(84\) 1.48120 − 1.32767i 0.161612 − 0.144861i
\(85\) 2.01888 − 1.07911i 0.218978 − 0.117046i
\(86\) 1.12808 − 6.39645i 0.121644 − 0.689747i
\(87\) −1.70122 + 8.55262i −0.182390 + 0.916937i
\(88\) 0.758200 + 0.665052i 0.0808244 + 0.0708948i
\(89\) 1.30486 − 0.259552i 0.138314 − 0.0275124i −0.125448 − 0.992100i \(-0.540037\pi\)
0.263762 + 0.964588i \(0.415037\pi\)
\(90\) 0.690879 − 0.721656i 0.0728250 − 0.0760692i
\(91\) −1.64303 + 0.161824i −0.172236 + 0.0169638i
\(92\) −0.426755 − 9.78861i −0.0444922 − 1.02053i
\(93\) 3.75869 − 1.14019i 0.389758 − 0.118232i
\(94\) 2.70906 + 4.81307i 0.279418 + 0.496431i
\(95\) −1.24534 − 1.24534i −0.127769 − 0.127769i
\(96\) 7.95614 − 9.90991i 0.812020 − 1.01143i
\(97\) 1.24876 − 1.24876i 0.126793 − 0.126793i −0.640863 − 0.767655i \(-0.721423\pi\)
0.767655 + 0.640863i \(0.221423\pi\)
\(98\) −9.26665 − 2.59197i −0.936073 − 0.261829i
\(99\) −0.211889 − 0.698504i −0.0212956 − 0.0702023i
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.109.12 yes 240
4.3 odd 2 512.2.k.a.273.12 240
128.27 odd 32 512.2.k.a.497.12 240
128.101 even 32 inner 128.2.k.a.101.12 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.12 ✓ 240 128.101 even 32 inner
128.2.k.a.109.12 yes 240 1.1 even 1 trivial
512.2.k.a.273.12 240 4.3 odd 2
512.2.k.a.497.12 240 128.27 odd 32