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Properties

Label 128.2.k.a.117.14
Level $128$
Weight $2$
Character 128.117
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 117.14
Character \(\chi\) \(=\) 128.117
Dual form 128.2.k.a.93.14

$q$-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+(1.29114 - 0.577034i) q^{2} +(-3.03479 + 0.298901i) q^{3} +(1.33406 - 1.49006i) q^{4} +(1.49551 - 2.79790i) q^{5} +(-3.74585 + 2.13710i) q^{6} +(0.0999849 - 0.502658i) q^{7} +(0.862640 - 2.69367i) q^{8} +(6.17825 - 1.22893i) q^{9} +(0.316420 - 4.47543i) q^{10} +(0.255564 - 0.209736i) q^{11} +(-3.60322 + 4.92077i) q^{12} +(-5.32590 + 2.84675i) q^{13} +(-0.160957 - 0.706694i) q^{14} +(-3.70226 + 8.93805i) q^{15} +(-0.440555 - 3.97566i) q^{16} +(2.10427 + 5.08015i) q^{17} +(7.26783 - 5.15178i) q^{18} +(4.72373 + 1.43293i) q^{19} +(-2.17394 - 5.96097i) q^{20} +(-0.153188 + 1.55535i) q^{21} +(0.208943 - 0.418267i) q^{22} +(2.84331 + 1.89984i) q^{23} +(-1.81279 + 8.43256i) q^{24} +(-2.81385 - 4.21123i) q^{25} +(-5.23379 + 6.74878i) q^{26} +(-9.62788 + 2.92059i) q^{27} +(-0.615604 - 0.819561i) q^{28} +(0.778524 - 0.948634i) q^{29} +(0.377441 + 13.6766i) q^{30} +(-1.67733 - 1.67733i) q^{31} +(-2.86291 - 4.87891i) q^{32} +(-0.712894 + 0.712894i) q^{33} +(5.64832 + 5.34493i) q^{34} +(-1.25686 - 1.03148i) q^{35} +(6.41100 - 10.8454i) q^{36} +(-0.903482 - 2.97838i) q^{37} +(6.92583 - 0.875651i) q^{38} +(15.3121 - 10.2312i) q^{39} +(-6.24653 - 6.44199i) q^{40} +(-3.47135 + 5.19524i) q^{41} +(0.699702 + 2.09656i) q^{42} +(-0.477885 - 0.0470676i) q^{43} +(0.0284193 - 0.660607i) q^{44} +(5.80121 - 19.1240i) q^{45} +(4.76737 + 0.812262i) q^{46} +(0.997026 - 0.412982i) q^{47} +(2.52532 + 11.9336i) q^{48} +(6.22449 + 2.57827i) q^{49} +(-6.06309 - 3.81357i) q^{50} +(-7.90447 - 14.7882i) q^{51} +(-2.86325 + 11.7337i) q^{52} +(4.57222 + 5.57127i) q^{53} +(-10.7456 + 9.32649i) q^{54} +(-0.204622 - 1.02871i) q^{55} +(-1.26774 - 0.702939i) q^{56} +(-14.7638 - 2.93671i) q^{57} +(0.457785 - 1.67405i) q^{58} +(-3.56594 - 1.90604i) q^{59} +(8.37918 + 17.4405i) q^{60} +(-0.319004 - 3.23891i) q^{61} +(-3.13355 - 1.19779i) q^{62} -3.22842i q^{63} +(-6.51171 - 4.64733i) q^{64} +19.1587i q^{65} +(-0.509078 + 1.33181i) q^{66} +(0.677096 + 6.87467i) q^{67} +(10.3769 + 3.64175i) q^{68} +(-9.19672 - 4.91575i) q^{69} +(-2.21797 - 0.606526i) q^{70} +(1.28015 + 0.254638i) q^{71} +(2.01927 - 17.7023i) q^{72} +(-1.26969 - 6.38314i) q^{73} +(-2.88515 - 3.32415i) q^{74} +(9.79819 + 11.9391i) q^{75} +(8.43690 - 5.12702i) q^{76} +(-0.0798730 - 0.149432i) q^{77} +(13.8662 - 22.0455i) q^{78} +(-12.4806 - 5.16965i) q^{79} +(-11.7824 - 4.71301i) q^{80} +(10.8863 - 4.50924i) q^{81} +(-1.48415 + 8.71084i) q^{82} +(-3.39463 + 11.1906i) q^{83} +(2.11320 + 2.30319i) q^{84} +(17.3607 + 1.70988i) q^{85} +(-0.644174 + 0.214985i) q^{86} +(-2.07911 + 3.11161i) q^{87} +(-0.344500 - 0.869332i) q^{88} +(0.951634 - 0.635861i) q^{89} +(-3.54507 - 28.0392i) q^{90} +(0.898434 + 2.96174i) q^{91} +(6.62403 - 1.70220i) q^{92} +(5.59171 + 4.58900i) q^{93} +(1.04899 - 1.10853i) q^{94} +(11.0736 - 11.0736i) q^{95} +(10.1466 + 13.9507i) q^{96} +(-13.2359 - 13.2359i) q^{97} +(9.52441 - 0.262851i) q^{98} +(1.32119 - 1.60987i) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{21}{32}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29114 − 0.577034i 0.912971 − 0.408025i
\(3\) −3.03479 + 0.298901i −1.75214 + 0.172570i −0.922568 − 0.385834i \(-0.873914\pi\)
−0.829569 + 0.558405i \(0.811414\pi\)
\(4\) 1.33406 − 1.49006i 0.667031 − 0.745030i
\(5\) 1.49551 − 2.79790i 0.668812 − 1.25126i −0.287183 − 0.957876i \(-0.592719\pi\)
0.955995 − 0.293384i \(-0.0947813\pi\)
\(6\) −3.74585 + 2.13710i −1.52924 + 0.872467i
\(7\) 0.0999849 − 0.502658i 0.0377907 − 0.189987i −0.957279 − 0.289167i \(-0.906622\pi\)
0.995070 + 0.0991797i \(0.0316219\pi\)
\(8\) 0.862640 − 2.69367i 0.304989 − 0.952356i
\(9\) 6.17825 − 1.22893i 2.05942 − 0.409644i
\(10\) 0.316420 − 4.47543i 0.100061 − 1.41526i
\(11\) 0.255564 − 0.209736i 0.0770555 − 0.0632378i −0.595077 − 0.803669i \(-0.702878\pi\)
0.672133 + 0.740431i \(0.265378\pi\)
\(12\) −3.60322 + 4.92077i −1.04016 + 1.42050i
\(13\) −5.32590 + 2.84675i −1.47714 + 0.789548i −0.996191 − 0.0872008i \(-0.972208\pi\)
−0.480949 + 0.876749i \(0.659708\pi\)
\(14\) −0.160957 − 0.706694i −0.0430175 − 0.188872i
\(15\) −3.70226 + 8.93805i −0.955920 + 2.30780i
\(16\) −0.440555 − 3.97566i −0.110139 − 0.993916i
\(17\) 2.10427 + 5.08015i 0.510360 + 1.23212i 0.943675 + 0.330875i \(0.107344\pi\)
−0.433315 + 0.901243i \(0.642656\pi\)
\(18\) 7.26783 − 5.15178i 1.71304 − 1.21429i
\(19\) 4.72373 + 1.43293i 1.08370 + 0.328736i 0.781115 − 0.624387i \(-0.214651\pi\)
0.302583 + 0.953123i \(0.402151\pi\)
\(20\) −2.17394 − 5.96097i −0.486107 − 1.33291i
\(21\) −0.153188 + 1.55535i −0.0334284 + 0.339405i
\(22\) 0.208943 − 0.418267i 0.0445468 − 0.0891749i
\(23\) 2.84331 + 1.89984i 0.592871 + 0.396144i 0.815498 − 0.578760i \(-0.196463\pi\)
−0.222627 + 0.974904i \(0.571463\pi\)
\(24\) −1.81279 + 8.43256i −0.370034 + 1.72129i
\(25\) −2.81385 − 4.21123i −0.562770 − 0.842245i
\(26\) −5.23379 + 6.74878i −1.02643 + 1.32354i
\(27\) −9.62788 + 2.92059i −1.85289 + 0.562067i
\(28\) −0.615604 − 0.819561i −0.116338 − 0.154882i
\(29\) 0.778524 − 0.948634i 0.144568 − 0.176157i −0.695692 − 0.718340i \(-0.744902\pi\)
0.840260 + 0.542183i \(0.182402\pi\)
\(30\) 0.377441 + 13.6766i 0.0689111 + 2.49699i
\(31\) −1.67733 − 1.67733i −0.301258 − 0.301258i 0.540248 − 0.841506i \(-0.318330\pi\)
−0.841506 + 0.540248i \(0.818330\pi\)
\(32\) −2.86291 − 4.87891i −0.506096 − 0.862477i
\(33\) −0.712894 + 0.712894i −0.124099 + 0.124099i
\(34\) 5.64832 + 5.34493i 0.968678 + 0.916648i
\(35\) −1.25686 − 1.03148i −0.212448 − 0.174352i
\(36\) 6.41100 − 10.8454i 1.06850 − 1.80757i
\(37\) −0.903482 − 2.97838i −0.148532 − 0.489643i 0.850827 − 0.525445i \(-0.176101\pi\)
−0.999359 + 0.0358025i \(0.988601\pi\)
\(38\) 6.92583 − 0.875651i 1.12352 − 0.142049i
\(39\) 15.3121 − 10.2312i 2.45190 − 1.63831i
\(40\) −6.24653 − 6.44199i −0.987664 − 1.01857i
\(41\) −3.47135 + 5.19524i −0.542133 + 0.811360i −0.996853 − 0.0792750i \(-0.974739\pi\)
0.454720 + 0.890635i \(0.349739\pi\)
\(42\) 0.699702 + 2.09656i 0.107966 + 0.323506i
\(43\) −0.477885 − 0.0470676i −0.0728768 − 0.00717773i 0.0615130 − 0.998106i \(-0.480407\pi\)
−0.134390 + 0.990929i \(0.542907\pi\)
\(44\) 0.0284193 − 0.660607i 0.00428437 − 0.0995903i
\(45\) 5.80121 − 19.1240i 0.864793 − 2.85084i
\(46\) 4.76737 + 0.812262i 0.702911 + 0.119762i
\(47\) 0.997026 − 0.412982i 0.145431 − 0.0602395i −0.308781 − 0.951133i \(-0.599921\pi\)
0.454212 + 0.890894i \(0.349921\pi\)
\(48\) 2.52532 + 11.9336i 0.364499 + 1.72247i
\(49\) 6.22449 + 2.57827i 0.889213 + 0.368324i
\(50\) −6.06309 − 3.81357i −0.857450 − 0.539321i
\(51\) −7.90447 − 14.7882i −1.10685 − 2.07077i
\(52\) −2.86325 + 11.7337i −0.397062 + 1.62717i
\(53\) 4.57222 + 5.57127i 0.628043 + 0.765273i 0.985687 − 0.168585i \(-0.0539198\pi\)
−0.357644 + 0.933858i \(0.616420\pi\)
\(54\) −10.7456 + 9.32649i −1.46229 + 1.26917i
\(55\) −0.204622 − 1.02871i −0.0275913 − 0.138711i
\(56\) −1.26774 − 0.702939i −0.169409 − 0.0939342i
\(57\) −14.7638 − 2.93671i −1.95552 − 0.388977i
\(58\) 0.457785 − 1.67405i 0.0601102 − 0.219814i
\(59\) −3.56594 − 1.90604i −0.464246 − 0.248145i 0.222653 − 0.974898i \(-0.428528\pi\)
−0.686899 + 0.726753i \(0.741028\pi\)
\(60\) 8.37918 + 17.4405i 1.08175 + 2.25156i
\(61\) −0.319004 − 3.23891i −0.0408443 − 0.414699i −0.993766 − 0.111484i \(-0.964440\pi\)
0.952922 − 0.303216i \(-0.0980603\pi\)
\(62\) −3.13355 − 1.19779i −0.397961 − 0.152119i
\(63\) − 3.22842i − 0.406743i
\(64\) −6.51171 − 4.64733i −0.813963 − 0.580916i
\(65\) 19.1587i 2.37634i
\(66\) −0.509078 + 1.33181i −0.0626632 + 0.163934i
\(67\) 0.677096 + 6.87467i 0.0827204 + 0.839874i 0.943605 + 0.331074i \(0.107411\pi\)
−0.860884 + 0.508800i \(0.830089\pi\)
\(68\) 10.3769 + 3.64175i 1.25839 + 0.441628i
\(69\) −9.19672 − 4.91575i −1.10715 − 0.591787i
\(70\) −2.21797 − 0.606526i −0.265099 − 0.0724938i
\(71\) 1.28015 + 0.254638i 0.151926 + 0.0302200i 0.270467 − 0.962729i \(-0.412822\pi\)
−0.118541 + 0.992949i \(0.537822\pi\)
\(72\) 2.01927 − 17.7023i 0.237974 − 2.08624i
\(73\) −1.26969 − 6.38314i −0.148605 − 0.747090i −0.981168 − 0.193158i \(-0.938127\pi\)
0.832562 − 0.553932i \(-0.186873\pi\)
\(74\) −2.88515 − 3.32415i −0.335391 − 0.386425i
\(75\) 9.79819 + 11.9391i 1.13140 + 1.37861i
\(76\) 8.43690 − 5.12702i 0.967779 − 0.588110i
\(77\) −0.0798730 − 0.149432i −0.00910237 − 0.0170293i
\(78\) 13.8662 − 22.0455i 1.57004 − 2.49616i
\(79\) −12.4806 − 5.16965i −1.40418 − 0.581631i −0.453348 − 0.891334i \(-0.649770\pi\)
−0.950833 + 0.309703i \(0.899770\pi\)
\(80\) −11.7824 − 4.71301i −1.31731 − 0.526931i
\(81\) 10.8863 − 4.50924i 1.20959 − 0.501027i
\(82\) −1.48415 + 8.71084i −0.163897 + 0.961952i
\(83\) −3.39463 + 11.1906i −0.372609 + 1.22833i 0.547547 + 0.836775i \(0.315562\pi\)
−0.920156 + 0.391552i \(0.871938\pi\)
\(84\) 2.11320 + 2.30319i 0.230569 + 0.251299i
\(85\) 17.3607 + 1.70988i 1.88303 + 0.185463i
\(86\) −0.644174 + 0.214985i −0.0694630 + 0.0231825i
\(87\) −2.07911 + 3.11161i −0.222904 + 0.333599i
\(88\) −0.344500 − 0.869332i −0.0367238 − 0.0926711i
\(89\) 0.951634 − 0.635861i 0.100873 − 0.0674012i −0.504111 − 0.863639i \(-0.668180\pi\)
0.604984 + 0.796238i \(0.293180\pi\)
\(90\) −3.54507 − 28.0392i −0.373684 − 2.95559i
\(91\) 0.898434 + 2.96174i 0.0941815 + 0.310475i
\(92\) 6.62403 − 1.70220i 0.690603 − 0.177466i
\(93\) 5.59171 + 4.58900i 0.579834 + 0.475857i
\(94\) 1.04899 − 1.10853i 0.108195 − 0.114336i
\(95\) 11.0736 − 11.0736i 1.13612 − 1.13612i
\(96\) 10.1466 + 13.9507i 1.03559 + 1.42384i
\(97\) −13.2359 − 13.2359i −1.34390 − 1.34390i −0.892139 − 0.451761i \(-0.850796\pi\)
−0.451761 − 0.892139i \(-0.649204\pi\)
\(98\) 9.52441 − 0.262851i 0.962111 − 0.0265520i
\(99\) 1.32119 − 1.60987i 0.132785 − 0.161798i
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.117.14 yes 240
4.3 odd 2 512.2.k.a.305.14 240
128.35 odd 32 512.2.k.a.465.14 240
128.93 even 32 inner 128.2.k.a.93.14 ✓ 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.k.a.93.14 ✓ 240 128.93 even 32 inner
128.2.k.a.117.14 yes 240 1.1 even 1 trivial
512.2.k.a.305.14 240 4.3 odd 2
512.2.k.a.465.14 240 128.35 odd 32