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Properties

Label 128.2.k.a.101.15
Level $128$
Weight $2$
Character 128.101
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 101.15
Character \(\chi\) \(=\) 128.101
Dual form 128.2.k.a.109.15

$q$-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+(1.34826 + 0.426852i) q^{2} +(0.205628 - 0.109910i) q^{3} +(1.63559 + 1.15101i) q^{4} +(-1.29703 - 1.58044i) q^{5} +(0.324155 - 0.0604147i) q^{6} +(1.60057 - 1.06947i) q^{7} +(1.71389 + 2.25002i) q^{8} +(-1.63651 + 2.44921i) q^{9} +(-1.07412 - 2.68448i) q^{10} +(-2.28487 + 0.693108i) q^{11} +(0.462832 + 0.0569118i) q^{12} +(-1.01011 - 0.828973i) q^{13} +(2.61449 - 0.758710i) q^{14} +(-0.440412 - 0.182425i) q^{15} +(1.35034 + 3.76518i) q^{16} +(-6.19134 + 2.56454i) q^{17} +(-3.25188 + 2.60361i) q^{18} +(-0.717174 - 7.28159i) q^{19} +(-0.302312 - 4.07785i) q^{20} +(0.211577 - 0.395832i) q^{21} +(-3.37645 - 0.0408150i) q^{22} +(6.82091 - 1.35676i) q^{23} +(0.599724 + 0.274293i) q^{24} +(0.159961 - 0.804177i) q^{25} +(-1.00803 - 1.54884i) q^{26} +(-0.135879 + 1.37960i) q^{27} +(3.84886 + 0.0930649i) q^{28} +(-1.11935 + 3.69001i) q^{29} +(-0.515921 - 0.433947i) q^{30} +(-0.0726984 - 0.0726984i) q^{31} +(0.213422 + 5.65283i) q^{32} +(-0.393654 + 0.393654i) q^{33} +(-9.44220 + 0.814866i) q^{34} +(-3.76622 - 1.14247i) q^{35} +(-5.49573 + 2.12227i) q^{36} +(2.87067 + 0.282737i) q^{37} +(2.14123 - 10.1236i) q^{38} +(-0.298819 - 0.0594388i) q^{39} +(1.33305 - 5.62704i) q^{40} +(0.658149 + 3.30874i) q^{41} +(0.454222 - 0.443372i) q^{42} +(3.06034 + 1.63579i) q^{43} +(-4.53490 - 1.49627i) q^{44} +(5.99342 - 0.590300i) q^{45} +(9.77547 + 1.08226i) q^{46} +(1.75405 + 4.23465i) q^{47} +(0.691500 + 0.625811i) q^{48} +(-1.26071 + 3.04363i) q^{49} +(0.558933 - 1.01596i) q^{50} +(-0.991243 + 1.20783i) q^{51} +(-0.697965 - 2.51851i) q^{52} +(-2.91014 - 9.59343i) q^{53} +(-0.772087 + 1.80206i) q^{54} +(4.05896 + 2.71211i) q^{55} +(5.14953 + 1.76837i) q^{56} +(-0.947794 - 1.41847i) q^{57} +(-3.08426 + 4.49728i) q^{58} +(4.93475 - 4.04984i) q^{59} +(-0.510362 - 0.805293i) q^{60} +(-0.733515 - 1.37231i) q^{61} +(-0.0669847 - 0.129048i) q^{62} +5.67033i q^{63} +(-2.12518 + 7.71256i) q^{64} +2.67161i q^{65} +(-0.698778 + 0.362714i) q^{66} +(2.54422 + 4.75990i) q^{67} +(-13.0783 - 2.93178i) q^{68} +(1.25345 - 1.02868i) q^{69} +(-4.59017 - 3.14796i) q^{70} +(1.60349 + 2.39980i) q^{71} +(-8.31556 + 0.515492i) q^{72} +(11.8626 + 7.92631i) q^{73} +(3.74972 + 1.60656i) q^{74} +(-0.0554951 - 0.182943i) q^{75} +(7.20821 - 12.7352i) q^{76} +(-2.91585 + 3.55297i) q^{77} +(-0.377513 - 0.207690i) q^{78} +(3.13533 - 7.56937i) q^{79} +(4.19920 - 7.01768i) q^{80} +(-3.25805 - 7.86562i) q^{81} +(-0.524989 + 4.74196i) q^{82} +(-10.8406 + 1.06770i) q^{83} +(0.801662 - 0.403893i) q^{84} +(12.0834 + 6.45874i) q^{85} +(3.42789 + 3.51178i) q^{86} +(0.175400 + 0.881798i) q^{87} +(-5.47552 - 3.95309i) q^{88} +(12.0007 + 2.38709i) q^{89} +(8.33264 + 1.76243i) q^{90} +(-2.50331 - 0.246555i) q^{91} +(12.7179 + 5.63184i) q^{92} +(-0.0229392 - 0.00695852i) q^{93} +(0.557339 + 6.45812i) q^{94} +(-10.5779 + 10.5779i) q^{95} +(0.665190 + 1.13892i) q^{96} +(-2.03422 - 2.03422i) q^{97} +(-2.99895 + 3.56546i) q^{98} +(2.04164 - 6.73040i) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{9}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.34826 + 0.426852i 0.953362 + 0.301830i
\(3\) 0.205628 − 0.109910i 0.118719 − 0.0634568i −0.410968 − 0.911650i \(-0.634809\pi\)
0.529687 + 0.848193i \(0.322309\pi\)
\(4\) 1.63559 + 1.15101i 0.817797 + 0.575507i
\(5\) −1.29703 − 1.58044i −0.580050 − 0.706793i 0.397475 − 0.917613i \(-0.369887\pi\)
−0.977525 + 0.210820i \(0.932387\pi\)
\(6\) 0.324155 − 0.0604147i 0.132336 − 0.0246642i
\(7\) 1.60057 − 1.06947i 0.604960 − 0.404221i −0.215023 − 0.976609i \(-0.568983\pi\)
0.819983 + 0.572388i \(0.193983\pi\)
\(8\) 1.71389 + 2.25002i 0.605951 + 0.795502i
\(9\) −1.63651 + 2.44921i −0.545503 + 0.816403i
\(10\) −1.07412 − 2.68448i −0.339666 − 0.848906i
\(11\) −2.28487 + 0.693108i −0.688914 + 0.208980i −0.615284 − 0.788306i \(-0.710959\pi\)
−0.0736306 + 0.997286i \(0.523459\pi\)
\(12\) 0.462832 + 0.0569118i 0.133608 + 0.0164290i
\(13\) −1.01011 − 0.828973i −0.280153 − 0.229916i 0.483814 − 0.875171i \(-0.339251\pi\)
−0.763967 + 0.645255i \(0.776751\pi\)
\(14\) 2.61449 − 0.758710i 0.698752 − 0.202774i
\(15\) −0.440412 − 0.182425i −0.113714 − 0.0471019i
\(16\) 1.35034 + 3.76518i 0.337584 + 0.941295i
\(17\) −6.19134 + 2.56454i −1.50162 + 0.621991i −0.973808 − 0.227370i \(-0.926987\pi\)
−0.527812 + 0.849361i \(0.676987\pi\)
\(18\) −3.25188 + 2.60361i −0.766476 + 0.613678i
\(19\) −0.717174 − 7.28159i −0.164531 − 1.67051i −0.623320 − 0.781967i \(-0.714217\pi\)
0.458789 − 0.888545i \(-0.348283\pi\)
\(20\) −0.302312 − 4.07785i −0.0675989 − 0.911836i
\(21\) 0.211577 − 0.395832i 0.0461698 − 0.0863777i
\(22\) −3.37645 − 0.0408150i −0.719861 − 0.00870179i
\(23\) 6.82091 − 1.35676i 1.42226 − 0.282905i 0.576773 − 0.816905i \(-0.304312\pi\)
0.845485 + 0.534000i \(0.179312\pi\)
\(24\) 0.599724 + 0.274293i 0.122418 + 0.0559898i
\(25\) 0.159961 − 0.804177i 0.0319922 − 0.160835i
\(26\) −1.00803 − 1.54884i −0.197692 − 0.303752i
\(27\) −0.135879 + 1.37960i −0.0261500 + 0.265505i
\(28\) 3.84886 + 0.0930649i 0.727366 + 0.0175876i
\(29\) −1.11935 + 3.69001i −0.207858 + 0.685217i 0.789577 + 0.613651i \(0.210300\pi\)
−0.997436 + 0.0715665i \(0.977200\pi\)
\(30\) −0.515921 − 0.433947i −0.0941938 − 0.0792274i
\(31\) −0.0726984 − 0.0726984i −0.0130570 − 0.0130570i 0.700548 − 0.713605i \(-0.252939\pi\)
−0.713605 + 0.700548i \(0.752939\pi\)
\(32\) 0.213422 + 5.65283i 0.0377280 + 0.999288i
\(33\) −0.393654 + 0.393654i −0.0685263 + 0.0685263i
\(34\) −9.44220 + 0.814866i −1.61932 + 0.139748i
\(35\) −3.76622 − 1.14247i −0.636607 − 0.193113i
\(36\) −5.49573 + 2.12227i −0.915956 + 0.353711i
\(37\) 2.87067 + 0.282737i 0.471936 + 0.0464816i 0.331190 − 0.943564i \(-0.392550\pi\)
0.140746 + 0.990046i \(0.455050\pi\)
\(38\) 2.14123 − 10.1236i 0.347353 − 1.64226i
\(39\) −0.298819 − 0.0594388i −0.0478493 − 0.00951783i
\(40\) 1.33305 − 5.62704i 0.210773 − 0.889712i
\(41\) 0.658149 + 3.30874i 0.102786 + 0.516738i 0.997535 + 0.0701698i \(0.0223541\pi\)
−0.894749 + 0.446569i \(0.852646\pi\)
\(42\) 0.454222 − 0.443372i 0.0700880 − 0.0684137i
\(43\) 3.06034 + 1.63579i 0.466698 + 0.249455i 0.687945 − 0.725762i \(-0.258513\pi\)
−0.221247 + 0.975218i \(0.571013\pi\)
\(44\) −4.53490 − 1.49627i −0.683662 − 0.225572i
\(45\) 5.99342 − 0.590300i 0.893446 − 0.0879968i
\(46\) 9.77547 + 1.08226i 1.44131 + 0.159570i
\(47\) 1.75405 + 4.23465i 0.255854 + 0.617687i 0.998656 − 0.0518243i \(-0.0165036\pi\)
−0.742802 + 0.669511i \(0.766504\pi\)
\(48\) 0.691500 + 0.625811i 0.0998094 + 0.0903280i
\(49\) −1.26071 + 3.04363i −0.180102 + 0.434805i
\(50\) 0.558933 − 1.01596i 0.0790451 − 0.143678i
\(51\) −0.991243 + 1.20783i −0.138802 + 0.169130i
\(52\) −0.697965 − 2.51851i −0.0967904 − 0.349255i
\(53\) −2.91014 − 9.59343i −0.399738 − 1.31776i −0.893852 − 0.448362i \(-0.852007\pi\)
0.494114 − 0.869397i \(-0.335493\pi\)
\(54\) −0.772087 + 1.80206i −0.105068 + 0.245229i
\(55\) 4.05896 + 2.71211i 0.547310 + 0.365701i
\(56\) 5.14953 + 1.76837i 0.688135 + 0.236309i
\(57\) −0.947794 − 1.41847i −0.125538 − 0.187882i
\(58\) −3.08426 + 4.49728i −0.404984 + 0.590522i
\(59\) 4.93475 − 4.04984i 0.642450 − 0.527245i −0.255800 − 0.966730i \(-0.582339\pi\)
0.898250 + 0.439485i \(0.144839\pi\)
\(60\) −0.510362 − 0.805293i −0.0658875 − 0.103963i
\(61\) −0.733515 − 1.37231i −0.0939169 − 0.175706i 0.830576 − 0.556905i \(-0.188011\pi\)
−0.924493 + 0.381199i \(0.875511\pi\)
\(62\) −0.0669847 − 0.129048i −0.00850706 − 0.0163891i
\(63\) 5.67033i 0.714394i
\(64\) −2.12518 + 7.71256i −0.265647 + 0.964070i
\(65\) 2.67161i 0.331373i
\(66\) −0.698778 + 0.362714i −0.0860136 + 0.0446470i
\(67\) 2.54422 + 4.75990i 0.310826 + 0.581515i 0.988027 − 0.154284i \(-0.0493070\pi\)
−0.677200 + 0.735799i \(0.736807\pi\)
\(68\) −13.0783 − 2.93178i −1.58598 − 0.355530i
\(69\) 1.25345 − 1.02868i 0.150897 − 0.123838i
\(70\) −4.59017 − 3.14796i −0.548630 − 0.376254i
\(71\) 1.60349 + 2.39980i 0.190300 + 0.284804i 0.914334 − 0.404960i \(-0.132714\pi\)
−0.724035 + 0.689764i \(0.757714\pi\)
\(72\) −8.31556 + 0.515492i −0.979998 + 0.0607513i
\(73\) 11.8626 + 7.92631i 1.38841 + 0.927704i 0.999981 + 0.00622507i \(0.00198151\pi\)
0.388427 + 0.921479i \(0.373018\pi\)
\(74\) 3.74972 + 1.60656i 0.435896 + 0.186758i
\(75\) −0.0554951 − 0.182943i −0.00640802 − 0.0211244i
\(76\) 7.20821 − 12.7352i 0.826838 − 1.46083i
\(77\) −2.91585 + 3.55297i −0.332291 + 0.404898i
\(78\) −0.377513 − 0.207690i −0.0427450 − 0.0235163i
\(79\) 3.13533 − 7.56937i 0.352753 − 0.851620i −0.643526 − 0.765425i \(-0.722529\pi\)
0.996278 − 0.0861956i \(-0.0274710\pi\)
\(80\) 4.19920 − 7.01768i 0.469485 − 0.784600i
\(81\) −3.25805 − 7.86562i −0.362005 − 0.873958i
\(82\) −0.524989 + 4.74196i −0.0579754 + 0.523662i
\(83\) −10.8406 + 1.06770i −1.18991 + 0.117195i −0.673525 − 0.739164i \(-0.735221\pi\)
−0.516380 + 0.856360i \(0.672721\pi\)
\(84\) 0.801662 − 0.403893i 0.0874685 − 0.0440684i
\(85\) 12.0834 + 6.45874i 1.31063 + 0.700548i
\(86\) 3.42789 + 3.51178i 0.369639 + 0.378685i
\(87\) 0.175400 + 0.881798i 0.0188049 + 0.0945386i
\(88\) −5.47552 − 3.95309i −0.583692 − 0.421401i
\(89\) 12.0007 + 2.38709i 1.27207 + 0.253031i 0.784543 − 0.620075i \(-0.212898\pi\)
0.487531 + 0.873106i \(0.337898\pi\)
\(90\) 8.33264 + 1.76243i 0.878337 + 0.185776i
\(91\) −2.50331 − 0.246555i −0.262418 − 0.0258459i
\(92\) 12.7179 + 5.63184i 1.32593 + 0.587160i
\(93\) −0.0229392 − 0.00695852i −0.00237868 − 0.000721564i
\(94\) 0.557339 + 6.45812i 0.0574851 + 0.666104i
\(95\) −10.5779 + 10.5779i −1.08527 + 1.08527i
\(96\) 0.665190 + 1.13892i 0.0678907 + 0.116241i
\(97\) −2.03422 − 2.03422i −0.206544 − 0.206544i 0.596253 − 0.802797i \(-0.296656\pi\)
−0.802797 + 0.596253i \(0.796656\pi\)
\(98\) −2.99895 + 3.56546i −0.302939 + 0.360166i
\(99\) 2.04164 − 6.73040i 0.205193 − 0.676431i
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.101.15 ✓ 240
4.3 odd 2 512.2.k.a.497.7 240
128.19 odd 32 512.2.k.a.273.7 240
128.109 even 32 inner 128.2.k.a.109.15 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.101.15 ✓ 240 1.1 even 1 trivial
128.2.k.a.109.15 yes 240 128.109 even 32 inner
512.2.k.a.273.7 240 128.19 odd 32
512.2.k.a.497.7 240 4.3 odd 2