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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(80,567)] 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("567.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 404.4
Character \(\chi\) \(=\) 567.404
Dual form 567.2.p.e.80.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41253 - 0.815523i) q^{2} +(0.330156 + 0.571847i) q^{4} +(-1.00291 + 1.73708i) q^{5} +(-2.29841 - 1.31047i) q^{7} +2.18509i q^{8} +(2.83327 - 1.63579i) q^{10} +(-1.13523 + 0.655423i) q^{11} -3.06236i q^{13} +(2.17784 + 3.72548i) q^{14} +(2.44231 - 4.23020i) q^{16} +(-0.683325 - 1.18355i) q^{17} +(4.92262 + 2.84207i) q^{19} -1.32446 q^{20} +2.13805 q^{22} +(5.70312 + 3.29270i) q^{23} +(0.488359 + 0.845863i) q^{25} +(-2.49742 + 4.32566i) q^{26} +(-0.00944366 - 1.74700i) q^{28} -6.25128i q^{29} +(6.20195 - 3.58070i) q^{31} +(-3.11496 + 1.79842i) q^{32} +2.22907i q^{34} +(4.58148 - 2.67824i) q^{35} +(-2.17992 + 3.77573i) q^{37} +(-4.63555 - 8.02902i) q^{38} +(-3.79569 - 2.19144i) q^{40} +5.59934 q^{41} +3.13805 q^{43} +(-0.749603 - 0.432784i) q^{44} +(-5.37054 - 9.30205i) q^{46} +(-3.86407 + 6.69277i) q^{47} +(3.56533 + 6.02399i) q^{49} -1.59307i q^{50} +(1.75120 - 1.01106i) q^{52} +(8.54669 - 4.93443i) q^{53} -2.62931i q^{55} +(2.86350 - 5.02223i) q^{56} +(-5.09806 + 8.83011i) q^{58} +(-5.78441 - 10.0189i) q^{59} +(12.5188 + 7.22776i) q^{61} -11.6806 q^{62} -3.90260 q^{64} +(5.31957 + 3.07126i) q^{65} +(-5.21234 - 9.02803i) q^{67} +(0.451208 - 0.781515i) q^{68} +(-8.65564 + 0.0467894i) q^{70} -9.13734i q^{71} +(1.16332 - 0.671641i) q^{73} +(6.15840 - 3.55555i) q^{74} +3.75331i q^{76} +(3.46812 - 0.0187475i) q^{77} +(6.78595 - 11.7536i) q^{79} +(4.89881 + 8.48498i) q^{80} +(-7.90922 - 4.56639i) q^{82} +0.413616 q^{83} +2.74124 q^{85} +(-4.43258 - 2.55915i) q^{86} +(-1.43216 - 2.48057i) q^{88} +(-2.39824 + 4.15388i) q^{89} +(-4.01313 + 7.03854i) q^{91} +4.34842i q^{92} +(10.9162 - 6.30248i) q^{94} +(-9.87384 + 5.70067i) q^{95} +18.0448i q^{97} +(-0.123433 - 11.4167i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} - 8 q^{7} - 28 q^{16} + 24 q^{22} - 16 q^{25} - 16 q^{28} - 48 q^{31} - 4 q^{37} + 56 q^{43} + 12 q^{46} - 4 q^{49} + 48 q^{52} + 36 q^{58} + 12 q^{61} - 80 q^{64} - 20 q^{67} + 120 q^{70}+ \cdots + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41253 − 0.815523i −0.998808 − 0.576662i −0.0909125 − 0.995859i \(-0.528978\pi\)
−0.907895 + 0.419197i \(0.862312\pi\)
\(3\) 0 0
\(4\) 0.330156 + 0.571847i 0.165078 + 0.285924i
\(5\) −1.00291 + 1.73708i −0.448513 + 0.776848i −0.998290 − 0.0584643i \(-0.981380\pi\)
0.549776 + 0.835312i \(0.314713\pi\)
\(6\) 0 0
\(7\) −2.29841 − 1.31047i −0.868716 − 0.495311i
\(8\) 2.18509i 0.772547i
\(9\) 0 0
\(10\) 2.83327 − 1.63579i 0.895957 − 0.517281i
\(11\) −1.13523 + 0.655423i −0.342283 + 0.197617i −0.661281 − 0.750138i \(-0.729987\pi\)
0.318998 + 0.947755i \(0.396654\pi\)
\(12\) 0 0
\(13\) − 3.06236i − 0.849345i −0.905347 − 0.424673i \(-0.860389\pi\)
0.905347 − 0.424673i \(-0.139611\pi\)
\(14\) 2.17784 + 3.72548i 0.582053 + 0.995676i
\(15\) 0 0
\(16\) 2.44231 − 4.23020i 0.610577 − 1.05755i
\(17\) −0.683325 − 1.18355i −0.165731 − 0.287054i 0.771184 − 0.636612i \(-0.219665\pi\)
−0.936914 + 0.349559i \(0.886332\pi\)
\(18\) 0 0
\(19\) 4.92262 + 2.84207i 1.12933 + 0.652016i 0.943765 − 0.330616i \(-0.107257\pi\)
0.185560 + 0.982633i \(0.440590\pi\)
\(20\) −1.32446 −0.296159
\(21\) 0 0
\(22\) 2.13805 0.455834
\(23\) 5.70312 + 3.29270i 1.18918 + 0.686574i 0.958121 − 0.286365i \(-0.0924469\pi\)
0.231061 + 0.972939i \(0.425780\pi\)
\(24\) 0 0
\(25\) 0.488359 + 0.845863i 0.0976718 + 0.169173i
\(26\) −2.49742 + 4.32566i −0.489785 + 0.848332i
\(27\) 0 0
\(28\) −0.00944366 − 1.74700i −0.00178468 − 0.330151i
\(29\) − 6.25128i − 1.16083i −0.814320 − 0.580417i \(-0.802890\pi\)
0.814320 − 0.580417i \(-0.197110\pi\)
\(30\) 0 0
\(31\) 6.20195 − 3.58070i 1.11390 − 0.643112i 0.174065 − 0.984734i \(-0.444310\pi\)
0.939837 + 0.341622i \(0.110976\pi\)
\(32\) −3.11496 + 1.79842i −0.550652 + 0.317919i
\(33\) 0 0
\(34\) 2.22907i 0.382282i
\(35\) 4.58148 − 2.67824i 0.774412 − 0.452706i
\(36\) 0 0
\(37\) −2.17992 + 3.77573i −0.358377 + 0.620727i −0.987690 − 0.156425i \(-0.950003\pi\)
0.629313 + 0.777152i \(0.283336\pi\)
\(38\) −4.63555 − 8.02902i −0.751986 − 1.30248i
\(39\) 0 0
\(40\) −3.79569 − 2.19144i −0.600151 − 0.346497i
\(41\) 5.59934 0.874470 0.437235 − 0.899347i \(-0.355958\pi\)
0.437235 + 0.899347i \(0.355958\pi\)
\(42\) 0 0
\(43\) 3.13805 0.478548 0.239274 − 0.970952i \(-0.423091\pi\)
0.239274 + 0.970952i \(0.423091\pi\)
\(44\) −0.749603 − 0.432784i −0.113007 − 0.0652446i
\(45\) 0 0
\(46\) −5.37054 − 9.30205i −0.791843 − 1.37151i
\(47\) −3.86407 + 6.69277i −0.563633 + 0.976241i 0.433543 + 0.901133i \(0.357263\pi\)
−0.997175 + 0.0751077i \(0.976070\pi\)
\(48\) 0 0
\(49\) 3.56533 + 6.02399i 0.509333 + 0.860569i
\(50\) − 1.59307i − 0.225294i
\(51\) 0 0
\(52\) 1.75120 − 1.01106i 0.242848 − 0.140208i
\(53\) 8.54669 − 4.93443i 1.17398 − 0.677796i 0.219364 − 0.975643i \(-0.429602\pi\)
0.954614 + 0.297847i \(0.0962684\pi\)
\(54\) 0 0
\(55\) − 2.62931i − 0.354536i
\(56\) 2.86350 − 5.02223i 0.382651 − 0.671123i
\(57\) 0 0
\(58\) −5.09806 + 8.83011i −0.669409 + 1.15945i
\(59\) −5.78441 − 10.0189i −0.753066 − 1.30435i −0.946330 − 0.323202i \(-0.895241\pi\)
0.193264 − 0.981147i \(-0.438093\pi\)
\(60\) 0 0
\(61\) 12.5188 + 7.22776i 1.60287 + 0.925419i 0.990909 + 0.134534i \(0.0429537\pi\)
0.611964 + 0.790885i \(0.290380\pi\)
\(62\) −11.6806 −1.48343
\(63\) 0 0
\(64\) −3.90260 −0.487825
\(65\) 5.31957 + 3.07126i 0.659812 + 0.380942i
\(66\) 0 0
\(67\) −5.21234 − 9.02803i −0.636788 − 1.10295i −0.986133 − 0.165955i \(-0.946929\pi\)
0.349345 − 0.936994i \(-0.386404\pi\)
\(68\) 0.451208 − 0.781515i 0.0547170 − 0.0947726i
\(69\) 0 0
\(70\) −8.65564 + 0.0467894i −1.03455 + 0.00559240i
\(71\) − 9.13734i − 1.08440i −0.840249 − 0.542201i \(-0.817591\pi\)
0.840249 − 0.542201i \(-0.182409\pi\)
\(72\) 0 0
\(73\) 1.16332 − 0.671641i 0.136156 − 0.0786097i −0.430375 − 0.902650i \(-0.641619\pi\)
0.566531 + 0.824041i \(0.308285\pi\)
\(74\) 6.15840 − 3.55555i 0.715899 − 0.413324i
\(75\) 0 0
\(76\) 3.75331i 0.430535i
\(77\) 3.46812 − 0.0187475i 0.395229 − 0.00213647i
\(78\) 0 0
\(79\) 6.78595 − 11.7536i 0.763479 − 1.32238i −0.177569 − 0.984108i \(-0.556823\pi\)
0.941047 − 0.338275i \(-0.109844\pi\)
\(80\) 4.89881 + 8.48498i 0.547703 + 0.948650i
\(81\) 0 0
\(82\) −7.90922 − 4.56639i −0.873427 − 0.504274i
\(83\) 0.413616 0.0454002 0.0227001 − 0.999742i \(-0.492774\pi\)
0.0227001 + 0.999742i \(0.492774\pi\)
\(84\) 0 0
\(85\) 2.74124 0.297329
\(86\) −4.43258 − 2.55915i −0.477978 − 0.275960i
\(87\) 0 0
\(88\) −1.43216 − 2.48057i −0.152669 − 0.264430i
\(89\) −2.39824 + 4.15388i −0.254213 + 0.440310i −0.964681 − 0.263419i \(-0.915150\pi\)
0.710468 + 0.703729i \(0.248483\pi\)
\(90\) 0 0
\(91\) −4.01313 + 7.03854i −0.420690 + 0.737839i
\(92\) 4.34842i 0.453354i
\(93\) 0 0
\(94\) 10.9162 − 6.30248i 1.12592 − 0.650051i
\(95\) −9.87384 + 5.70067i −1.01303 + 0.584876i
\(96\) 0 0
\(97\) 18.0448i 1.83217i 0.400982 + 0.916086i \(0.368669\pi\)
−0.400982 + 0.916086i \(0.631331\pi\)
\(98\) −0.123433 − 11.4167i −0.0124686 − 1.15326i
\(99\) 0 0
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 567.2.p.e.404.4 yes 32
3.2 odd 2 inner 567.2.p.e.404.13 yes 32
7.3 odd 6 inner 567.2.p.e.80.13 yes 32
9.2 odd 6 567.2.i.g.215.4 32
9.4 even 3 567.2.s.g.26.13 32
9.5 odd 6 567.2.s.g.26.4 32
9.7 even 3 567.2.i.g.215.13 32
21.17 even 6 inner 567.2.p.e.80.4 ✓ 32
63.31 odd 6 567.2.i.g.269.13 32
63.38 even 6 567.2.s.g.458.13 32
63.52 odd 6 567.2.s.g.458.4 32
63.59 even 6 567.2.i.g.269.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.4 32 9.2 odd 6
567.2.i.g.215.13 32 9.7 even 3
567.2.i.g.269.4 32 63.59 even 6
567.2.i.g.269.13 32 63.31 odd 6
567.2.p.e.80.4 ✓ 32 21.17 even 6 inner
567.2.p.e.80.13 yes 32 7.3 odd 6 inner
567.2.p.e.404.4 yes 32 1.1 even 1 trivial
567.2.p.e.404.13 yes 32 3.2 odd 2 inner
567.2.s.g.26.4 32 9.5 odd 6
567.2.s.g.26.13 32 9.4 even 3
567.2.s.g.458.4 32 63.52 odd 6
567.2.s.g.458.13 32 63.38 even 6